Control Techniques for LCL-Type Grid-Connected Inverters

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CPSS Power Electronics Series
Control Techniques
for LCL-Type Grid-
Connected Inverters 
Xinbo Ruan · Xuehua Wang
Donghua Pan · Dongsheng Yang
Weiwei Li · Chenlei Bao
CPSS Power Electronics Series
Series editors
Wei Chen, Fuzhou University, Fuzhou, Fujian, China
Yongzheng Chen, Liaoning University of Technology, Jinzhou, Liaoning, China
Xiangning He, Zhejiang University, Hangzhou, Zhejiang, China
Yongdong Li, Tsinghua University, Beijing, China
Jingjun Liu, Xi’an Jiaotong University, Xi’an, Shaanxi, China
An Luo, Hunan University, Changsha, Hunan, China
Xikui Ma, Xi’an Jiaotong University, Xi’an, Shaanxi, China
Xinbo Ruan, Nanjing University of Aeronautics and Astronautics, Nanjing,
Jiangsu, China
Kuang Shen, Zhejiang University, Hangzhou, Zhejiang, China
Dianguo Xu, Harbin Institute of Technology, Harbin, Heilongjiang, China
Jianping Xu, Xinan Jiaotong University, Chengdu, Sichuan, China
Mark Dehong Xu, Zhejiang University, Hangzhou, Zhejiang, China
Xiaoming Zha, Wuhan University, Wuhan, Hubei, China
Bo Zhang, South China University of Technology, Guangzhou, Guangdong, China
Lei Zhang, China Power Supply Society, Tianjin, China
Xin Zhang, Hefei University of Technology, Hefei, Anhui, China
Zhengming Zhao, Tsinghua University, Beijing, China
Qionglin Zheng, Beijing Jiaotong University, Beijing, China
Luowei Zhou, Chongqing University, Chongqing, China
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Xinbo Ruan • Xuehua Wang
Donghua Pan • Dongsheng Yang
Weiwei Li • Chenlei Bao
Control Techniques
for LCL-Type
Grid-Connected Inverters
123
Xinbo Ruan
College of Automation Engineering
Nanjing University of Aeronautics
and Astronautics
Nanjing, Jiangsu
China
Xuehua Wang
Huazhong University of Science
and Technology
Wuhan, Hubei
China
Donghua Pan
Huazhong University of Science
and Technology
Wuhan, Hubei
China
Dongsheng Yang
Nanjing University of Aeronautics
and Astronautics
Nanjing, Jiangsu
China
Weiwei Li
Huazhong University of Science
and Technology
Wuhan, Hubei
China
Chenlei Bao
Huazhong University of Science
and Technology
Wuhan, Hubei
China
ISSN 2520-8853 ISSN 2520-8861 (electronic)
CPSS Power Electronics Series
ISBN 978-981-10-4276-8 ISBN 978-981-10-4277-5 (eBook)
DOI 10.1007/978-981-10-4277-5
Jointly published with Science Press, Beijing, China
ISBN: 978-7-03-043810-2 Science Press, Beijing
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Preface
Renewable energy-based distributed power generation systems (RE-DPGS) repre-
sent promising solutions to mitigate energy crisis and environmental pollution. The
LCL-type grid-connected inverter, being a conversion interface between the
renewable energy power generation units and the power grid, has been widely used
to convert dc power to high-quality ac power and feed it into the grid, and it plays
an important role in maintaining safe, stable, and high-quality operation of
RE-DPGS.
This book aims to present the control techniques for the LCL-type
grid-connected inverter to improve the system stability, control performance, and
suppression of grid current harmonics. The detailed theoretical analysis with design
examples and experimental validations are included.
This book contains twelve chapters.
Chapter 1 gives a brief review of the key techniques for the LCL-type
grid-connected inverter, including the design and magnetic integration of the LCL
filter, design of the controller parameters, the control delay effects in digital control
and the methods of reducing the control delays, suppression of the grid current
distortion caused by the grid voltage harmonics, and the grid impedance effects on
the system stability and the methods to improve the system stability.
Chapter 2 introduces the modulation strategies for the single-phase and
three-phase inverters, and presents the design methods of LCL filters for both
single-phase and three-phase inverters.
Chapter 3 presents magnetic integration methods for LCL filters, aiming to
reduce volume and weight.
In Chap. 4, the resonance hazard of LCL filters is analyzed, and six basic
passive-damping solutions are discussed in terms of their effects on the charac-
teristics of LCL filters. It is pointed out that adding a resistor in parallel with the
filter capacitor can effectively damp the resonance peak and does not affect
the frequency response of the LCL filter, but it results in high power loss. The
v
active-damping solutions, equivalent to a virtual resistor in parallel with the filter
capacitor, are derived, and the capacitor-current-feedback active-damping is found
superior for its simple implementation and effectiveness.
Chapter 5 presents a step-by-step parameter design method for the LCL-type
grid- connected inverter with capacitor-current-feedback active-damping, including
the capacitor current feedback coefficient and current regulator parameters.
In Chaps. 6 and 7, methods based on full feedforward of the grid voltage are
proposed for single-phase and three-phase grid-connected inverters with
capacitor-current-feedback active-damping. The feedforward function consists of a
proportional, a derivative, and a second-derivative component. The proposed full
feedforward scheme does not only reduce the steady-state error of the grid current
effectively, but also suppressesthe grid current distortion arising from the har-
monics in the grid voltage.
In Chap. 8, the mechanism of the control delay in digital control systems is
discussed, and the influence of the digital control delay on the system stability and
control performance are analyzed in detail. Then, the range of the LCL filter res-
onance frequency that would lead to instability is identified and hence should
be avoided. Then, the system stability evaluation method is presented by checking
the phase margin and the gain margin at one-sixth of sampling frequency (fs/6) and
the resonance frequency of the LCL filter.
In Chap. 9, a real-time sampling method is presented to reduce the computa-
tional delay, and it is not restricted by the modulation scheme and can be applied to
the single-phase and three-phase grid-connected inverters. Furthermore, a real-time
computational method with dual sampling modes is given to completely eliminate
the computation delay, and it is suitable for the single-phase grid-connected inverter
since it is based on the unipolar SPWM. With the two computation delay reduction
methods, the steady-state and dynamic performances of the LCL-type grid-
connected inverter can be improved, and high robustness against the
grid-impedance variation is obtained.
In Chaps. 10 and 11, the virtual series–parallel impedance shaping method and
weighted-feedforward scheme of grid voltages are proposed, respectively. The
purpose is to improve the harmonic rejection capability and the stability robustness
of the LCL-type grid-connected inverter when connected into a weak grid.
In Chap. 12, the complex-vector-filter method (CVFM) is adopted to derive
various prefilters in the synchronous reference frame phase-locked loops
(SRF-PLLs), and some insights into the relationships among different prefilters are
drawn. A brief comparison is presented to highlight the features of each prefilter.
Moreover, a generalized second-order complex-vector filter (GSO-CVF) with faster
dynamic response and a third-order complex-vector filter (TO-CVF) with higher
harmonic attenuation are proposed with the help of the CVFM, which are useful to
improve the dynamic performance and the harmonic attenuation ability of the PLL
for the grid-connected inverter.
vi Preface
This book is essential and valuable reference for the graduate students and
academics majoring in power electronics and renewable energy generation system
and the engineers being engaged in developing grid-connected inverters for pho-
tovoltaic system and wind turbine generation system. Senior undergraduate students
majoring in electrical engineering and automation engineering would also find this
book useful.
Nanjing, China Xinbo Ruan
Wuhan, China Xuehua Wang
Wuhan, China Donghua Pan
Nanjing, China Dongsheng Yang
Wuhan, China Weiwei Li
Wuhan, China Chenlei Bao
Preface vii
The original version of the book was revised:
Bibliography has been removed from Backmatter.
ix
Acknowledgements
This research monograph summarizes the research work on the control techniques
for LCL-type grid-connected inverters since the key project of National Natural
Science Foundation of China, titled “Research on Energy Conversion, Control, and
Grid-Connection Operation of Renewable Energy Based Distributed Power
Generation Systems”, was funded in 2008.
We wish to thank the members of the key project of National Natural Science
Foundation of China: Prof. Chengxiong Mao, Prof. Buhan Zhang, Prof. Yi Luo,
Prof. Kai Zhang, Prof. Xudong Zou, and Prof. Yu Zhang from Huazhong
University of Science and Technology (HUST), Wuhan, China, and Prof. Weiyang
Wu, Prof. Chunjiang Zhang, Prof. Xiaofeng Sun, and Prof. Xiaoqiang Guo from
Yanshan University, Qinhuangdao, China, for their outstanding contribution to
this key project. We also wish to express my sincere appreciation and gratitude
to Prof. Yuan Pan, Prof. Shijie Cheng, Prof. Xianzhong Duan, Prof. Jian Chen,
Prof. Yong Kang, Prof. KexunYu, Prof. Shanxu Duan, Prof. Hua Lin, Ms. Taomin
Zou, and Ms. Yi Li in the School of Electrical and Electronic Engineering, HUST,
for their great support during the application and research of this key project.
We are grateful to Prof. Lijian Ding, Director of the Fifth Engineering Section,
Engineering and Materials Department, National Natural Science Foundation of
China, and Prof. Weiming Ma from Naval University of Engineering, Wuhan,
China, for their great support and kind encouragement.
We also wish to thank Prof. Chengshan Wang from Tianjin University, Tianjin,
China, and Prof. An Luo from Hunan University, Changsha, China, for inviting me
to participate in the project of National Basic Research Program of China (973
Program), titled “Research on the Fundamentals of Distributed Power Generation
and Supply Systems”.
Special thanks are due to Prof. Chi. K. Tse from Hong Kong Polytechnic
University for his suggestions in the writing of this book, which have led to
improvements in clarity and readability.
The work in this book was supported by the National Natural Science
Foundation of China under Award 50837003, the National Basic Research Program
xi
of China (973 Program) under Award 2009CB219706, and Jiangsu Province 333
Program for Excellent Talents under Award BRA2012141. I would like to express
my sincere thanks to these supports.
It has been a great pleasure to work with the colleagues of Springer, Science
Press, China, and China Power Supply Society (CPSS). The support and help from
Mr. Wayne Hu (the project editor) are greatly appreciated.
January 2017
xii Acknowledgements
Contents
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Energy Situation and Environmental Issues . . . . . . . . . . . . . . . . 1
1.2 Renewable Energy-Based Distributed Power Generation
System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.3 Key Issues of LCL-Type Grid-Connected Inverters . . . . . . . . . . 4
1.3.1 Design and Magnetic Integration of LCL Filter . . . . . . . 6
1.3.2 Resonance Damping Methods of LCL Filter . . . . . . . . . 7
1.3.3 Controller Design of Grid-Connected Inverters . . . . . . . 8
1.3.4 Effects of Control Delay and the Compensation
Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.3.5 Suppression of Grid Current Distortion Caused
by Grid Voltage Harmonics. . . . . . . . . . . . . . . . . . . . . . 16
1.3.6 Grid-Impedance Effects on System Stability
and the Improvement Methods . . . . . . . . . . . . . . . . . . . 22
1.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2 Design of LCL Filter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
2.1 PWM for Single-Phase Full-Bridge Grid-Connected Inverter . . 32
2.1.1 Bipolar SPWM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
2.1.2 Unipolar SPWM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
2.2 PWM for Three-Phase Grid-Connected Inverter . . . . . . . . . . . . . 37
2.2.1 SPWM. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
2.2.2 Harmonic Injection SPWM Control. . . . . . . . . . . . . . . . 41
2.3 LCL Filter Design. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
2.3.1 Design of the Inverter-Side Inductor . . . . . . . . . . . . . . . 47
2.3.2 Filter Capacitor Design . . . . . . . . . . . . . . . . . . . . . . . . . 55
2.3.3 Grid-Side Inductor Design. . . . . . . . . . . . . . . . . . . . . . . 55
2.4 Design Examples for LCL Filter . . . . . . . . . . . . . . . . . . . . . . . . . 56
2.4.1 Single-Phase LCL Filter. . . . . . . . . . . . . . . . . . . . . . . . . 57
xiii
2.4.2 Three-Phase LCL Filter . . . . . . . . . . . . . . . . . . . . . . . . . 58
2.5 Summary . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . 60
References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
3 Magnetic Integration of LCL Filters . . . . . . . . . . . . . . . . . . . . . . . . . 63
3.1 Magnetic Integration of LCL Filters . . . . . . . . . . . . . . . . . . . . . . 64
3.1.1 Magnetic Integration of Single-Phase LCL Filter . . . . . . 64
3.1.2 Magnetic Integration of Three-Phase LCL Filter . . . . . . 66
3.2 Coupling Effect on Attenuating Ability of LCL Filter. . . . . . . . . 67
3.2.1 Magnetic Circuit of Integrated Inductors . . . . . . . . . . . . 67
3.2.2 Characteristics of LCL Filter with Coupled
Inductors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
3.3 Design Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
3.3.1 Magnetics Design for Single-Phase LCL Filter . . . . . . . 71
3.3.2 Magnetics Design for Three-Phase LCL Filter . . . . . . . . 73
3.4 Experimental Verification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
3.4.1 Experimental Results for Single-Phase LCL Filter . . . . . 74
3.4.2 Experimental Results for Three-Phase LCL Filter . . . . . 76
3.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
4 Resonance Damping Methods of LCL Filter . . . . . . . . . . . . . . . . . . . 79
4.1 Resonance Hazard of LCL Filter. . . . . . . . . . . . . . . . . . . . . . . . . 80
4.2 Passive-Damping Solutions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
4.2.1 Basic Passive Damping . . . . . . . . . . . . . . . . . . . . . . . . . 81
4.2.2 Improved Passive Damping . . . . . . . . . . . . . . . . . . . . . . 85
4.3 Active-Damping Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
4.3.1 State-Variable-Feedback Active Damping . . . . . . . . . . . 88
4.3.2 Notch-Filter-Based Active Damping . . . . . . . . . . . . . . . 90
4.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
5 Controller Design for LCL-Type Grid-Connected Inverter
with Capacitor-Current-Feedback Active-Damping . . . . . . . . . . . . . 95
5.1 Modeling LCL-Type Grid-Connected Inverter . . . . . . . . . . . . . . 96
5.2 Frequency Responses of Capacitor-Current-Feedback
Active-Damping and PI Regulator . . . . . . . . . . . . . . . . . . . . . . . 99
5.3 Constraints for Controller Parameters . . . . . . . . . . . . . . . . . . . . . 101
5.3.1 Requirement of Steady-State Error . . . . . . . . . . . . . . . . 101
5.3.2 Controller Parameters Constrained by Steady-State
Error and Stability Margin. . . . . . . . . . . . . . . . . . . . . . . 103
5.3.3 Pulse-Width Modulation (PWM) Constraint . . . . . . . . . 104
5.4 Design Procedure for Capacitor-Current-Feedback
Coefficient and PI Regulator Parameters . . . . . . . . . . . . . . . . . . . 105
5.5 Extension of the Proposed Design Method . . . . . . . . . . . . . . . . . 107
xiv Contents
5.5.1 Controller Design Based on PI Regulator
with Grid Voltage Feedforward Scheme . . . . . . . . . . . . 107
5.5.2 Controller Design Based on PR Regulator. . . . . . . . . . . 108
5.6 Design Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
5.6.1 Design Results with PI Regulator . . . . . . . . . . . . . . . . . 111
5.6.2 Design Results with PR Regulator. . . . . . . . . . . . . . . . . 112
5.7 Experimental Verification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
5.8 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
6 Full-Feedforward of Grid Voltage for Single-Phase LCL-Type
Grid-Connected Inverter. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
6.2 Effects of the Grid Voltage on the Grid Current . . . . . . . . . . . . . 122
6.3 Full-Feedforward Scheme for Single-Phase LCL-Type
Grid-Connected Inverter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
6.3.1 Derivation of Full-Feedforward Function of Grid
Voltage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
6.3.2 Discussion of the Three Feedforward Components . . . . 128
6.3.3 Discussion of Full-Feedforward Scheme with Main
Circuit Parameters Variations . . . . . . . . . . . . . . . . . . . . 130
6.4 Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
6.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
7 Full-Feedforward Scheme of Grid Voltages for Three-Phase
LCL-Type Grid-Connected Inverters . . . . . . . . . . . . . . . . . . . . . . . . . 139
7.1 Modeling the Three-Phase LCL-Type Grid-Connected
Inverter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
7.1.1 Model in the Stationary a–b Frame. . . . . . . . . . . . . . . . 140
7.1.2 Model in the Synchronous d–q Frame. . . . . . . . . . . . . . 143
7.2 Derivation of the Full-Feedforward Scheme
of Grid Voltages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
7.2.1 Full-Feedforward Scheme in the Stationary a–b
Frame . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
7.2.2 Full-Feedforward Scheme in the Synchronous d–q
Frame . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
7.2.3 Full-Feedforward Scheme in the Hybrid Frame . . . . . . . 147
7.3 Discussion of the Full-Feedforward Functions . . . . . . . . . . . . . . 150
7.3.1 Discussion of the Effect of Three Components
in the Full-Feedforward Function . . . . . . . . . . . . . . . . . 151
7.3.2 Harmonic Attenuation Affected by LCL Filter
Parameter Mismatches . . . . . . . . . . . . . . . . . . . . . . . . . . 154
Contents xv
7.3.3 Comparison Between the Feedforward Functions
for the L-Type and the LCL-Type Three-Phase
Grid-Connected Inverter . . . . . . . . . . . . . . . . . . . . . . . . 154
7.4 Experimental Verification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
7.4.1 Description of the Prototype . . . . . . . . . . . . . . . . . . . . . 155
7.4.2 Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 156
7.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162
References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163
8 Design Considerations of Digitally Controlled LCL-Type
Grid-Connected Inverter with Capacitor-Current-Feedback
Active-Damping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165
8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165
8.2 Control Delay in Digital Control System . . . . . . . . . . . . . . . . . . 167
8.3 Effect of Control Delay on Loop Gain and
Capacitor-Current-Feedback Active-Damping . . . . . . . . . . . . . . . 168
8.3.1 Equivalent Impedance of Capacitor-Current-Feedback
Active-Damping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168
8.3.2 Discrete-Time Expression of the Loop Gain . . . . . . . . . 172
8.3.3 RHP Poles of the System Loop Gain . . . . . . . . . . . . . . 174
8.4 Stability Constraint Conditions for Digitally Controlled
System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176
8.4.1 Nyquist Stability Criterion . . . . . . . . . . . . . .. . . . . . . . . 176
8.4.2 System Stability Constraint Conditions . . . . . . . . . . . . . 177
8.5 Design Considerations of the Controller Parameters of
Digitally Controlled LCL-Type Grid-Connected Inverter . . . . . . 179
8.5.1 Forbidden Region of the LCL Filter Resonance
Frequency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179
8.5.2 Constraints of the Controller Parameters . . . . . . . . . . . . 180
8.5.3 Design of LCL Filter, PR Regulator and
Capacitor-Current-Feedback Coefficient. . . . . . . . . . . . . 182
8.6 Design of Current Regulator for Digitally Controlled
LCL-Type Grid-Connected Inverter Without Damping . . . . . . . . 183
8.6.1 Stability Necessary Constraint for Digitally
Controlled LCL-Type Grid-Connected Inverter
Without Damping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184
8.6.2 Design of Grid Current Regulator and Analysis
of System Performance . . . . . . . . . . . . . . . . . . . . . . . . . 184
8.7 Design Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186
8.7.1 Design Example with Capacitor-Current-Feedback
Active-Damping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187
8.7.2 Design Example Without Damping . . . . . . . . . . . . . . . . 190
8.8 Experimental Verification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191
8.8.1 Experimental Validation for the Case
with Capacitor-Current-Feedback Active-Damping . . . . 191
xvi Contents
8.8.2 Experimental Validation Without Damping . . . . . . . . . . 193
8.9 Comparison of System Performance with Three Control
Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194
8.10 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195
References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196
9 Reduction of Computation Delay for Improving Stability and
Control Performance of LCL-Type Grid-Connected Inverters. . . . .. . . . 197
9.1 Effects of Computation and PWM Delays . . . . . . . . . . . . . . . . . 198
9.1.1 Modeling the Digitally Controlled LCL-Type
Grid-Connected Inverter . . . . . . . . . . . . . . . . . . . . . . . . 198
9.1.2 Improvement of Damping Performance
with Reduced Computation Delay . . . . . . . . . . . . . . . . . 202
9.1.3 Improvement of Control Performance
with Reduced Computation Delay . . . . . . . . . . . . . . . . . 205
9.2 Real-Time Sampling Method . . . . . . . . . . . . . . . . . . . . . . . . . . . 208
9.2.1 Sampling-Induced Aliasing of the Capacitor Current . . . 208
9.2.2 Design Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210
9.2.3 Experimental Verification . . . . . . . . . . . . . . . . . . . . . . . 212
9.3 Real-Time Computation Method with Dual Sampling
Modes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215
9.3.1 Derivation of the Real-Time Computation Method . . . . 215
9.3.2 Design Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218
9.3.3 Experimental Verification . . . . . . . . . . . . . . . . . . . . . . . 221
9.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224
References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225
10 Impedance Shaping of LCL-Type Grid-Connected Inverter
to Improve Its Adaptability to Weak Grid . . . . . . . . . . . . . . . . . . . . 227
10.1 Derivation of Impedance-Based Stability Criterion
for Grid-Connected Inverter . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228
10.2 Output Impedance Model of Grid-Connected Inverter . . . . . . . . 229
10.3 Relationship Between Output Impedance and Control
Performances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232
10.4 Output Impedance Shaping Method . . . . . . . . . . . . . . . . . . . . . . 233
10.4.1 Parallel Impedance Shaping Method . . . . . . . . . . . . . . . 234
10.4.2 Series–Parallel Impedance Shaping Method. . . . . . . . . . 236
10.4.3 Discussion of the Series–Parallel Impedance
Shaping Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239
10.5 Experimental Verification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241
10.5.1 Prototype Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241
10.5.2 Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 244
10.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 248
References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 248
Contents xvii
11 Weighted-Feedforward Scheme of Grid Voltages
for the Three-Phase LCL-Type Grid-Connected Inverters
Under Weak Grid Condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249
11.1 Impedance-Based Stability Criterion . . . . . . . . . . . . . . . . . . . . . . 250
11.2 Stability Analysis Under Weak Grid Condition . . . . . . . . . . . . . 251
11.2.1 Derivation of Output Impedance of
Grid-Connected Inverter . . . . . . . . . . . . . . . . . . . . . . . . 251
11.2.2 Stability of Grid-Connected Inverter Under Weak
Grid Condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254
11.3 Characteristics of the Inverter Output Impedance . . . . . . . . . . . . 255
11.3.1 Characteristics of the Inverter Output Impedance
Without Feedforward Scheme . . . . . . . . . . . . . . . . . . . . 256
11.3.2 Inverter Output Impedance Affected by the
Full-Feedforward Scheme . . . . . . . . . . . . . . . . . . . . . . . 257
11.4 Weighted-Feedforward Scheme of Grid Voltages . . . . . . . . . . . . 259
11.4.1 The Proposed Weighted-Feedforward Scheme
of Grid Voltages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259
11.4.2 Realization of the Weighted-Feedforward Scheme
of Grid Voltages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261
11.4.3 Tuning of the Weighted Coefficients . . . . . . . . . . . . . . . 262
11.5 Experimental Verification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265
11.5.1 Stability Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 266
11.5.2 Harmonic Suppression Test . . . . . . . . . . . . . . . . . . . . . . 266
11.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269
References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269
12 Prefilter-Based Synchronous Reference Frame Phase-Locked
Loop Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271
12.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271
12.2 Operation Principle of SRF-PLL. . . . . . . . . . . . . . . . . . . . . . . . . 272
12.3 Prefilter-Based SRF-PLL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274
12.3.1 Complex-Vector-Filter Method (CVFM) . . . . . . . . . . . . 275
12.3.2 Derivation of the Prefilters with the CVFM. . . . . . . . . . 277
12.4 Generalized Second-Order Complex-Vector Filter . . . . . . . . . . . 285
12.5 Third-Order Complex-Vector Filter. . . . . . . . . . . . . . . . . . . . . . . 287
12.6 Simulation and Experimental Verification. . . . . . . . . . . . . . . . . . 289
12.6.1 Simulation Results. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289
12.6.2 Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 291
12.6.3 Brief Comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 294
12.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295
References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . 299
xviii Contents
About the Authors
Xinbo Ruan was born in Hubei Province, China, in 1970. He received the B.S. and
Ph.D. degrees in electrical engineering from Nanjing University of Aeronautics and
Astronautics (NUAA), Nanjing, China, in 1991 and 1996, respectively.
In 1996, he joined the Faculty of Electrical Engineering Teaching and Research
Division, NUAA, where he became a professor in the College of Automation
Engineering in 2002 and has been engaged in teaching and research in the field of
power electronics. From August to October 2007, he was a research fellow in the
Department of Electronic and Information Engineering, Hong Kong Polytechnic
University, Hong Kong, China. From March 2008 to August 2011, he was also with
the School of Electrical and Electronic Engineering, Huazhong University of
Science and Technology, China. He is a guest professor at Beijing Jiaotong
University, Beijing, China, Hefei University of Technology, Hefei, China, and
Wuhan University, Wuhan, China. He is the author or co-author of seven books and
more than 300 technical papers published in journals and conferences. His main
research interests include soft-switching dc–dc converters, soft-switching inverters,
power factor correction converters, modeling the converters, power electronics
system integration, and renewable energy generation system.
Dr. Ruan was a recipient of the Delta Scholarship by the Delta Environment and
Education Fund in 2003 and was a recipient of the Special Appointed Professor
of the Chang Jiang Scholars Program by the Ministry of Education, China, in 2007.
From 2005 to 2013, he served as vice president of the China Power Supply Society.
From 2014 to 2016, he served as vice chair of the Technical Committee on
Renewable Energy Systems within the IEEE Industrial Electronics Society.
Currently, He is an associate editor for the IEEE Transactions on Industrial
Electronics, IEEE Transactions on Power Electronics, IEEE Transactions on
Circuits and System II, and the IEEE Journal of Emerging and Selected Topics on
Power Electronics. He was elevated to IEEE fellow in 2015.
Xuehua Wang was born in Hubei Province, China, in 1978. He received the B.S.
degree in electrical engineering from Nanjing University of Technology, Nanjing,
China, in 2001, and the M.S. and Ph.D. degrees in electrical engineering from
xix
Nanjing University of Aeronautics and Astronautics, Nanjing, China, in 2004 and
2008, respectively.
From October 2008 to March 2011, he was a postdoctoral fellow at Huazhong
University of Science and Technology (HUST), Wuhan, China. Since April 2011,
he joined the School of Electrical and Electronic Engineering, HUST, and he is
currently an associate professor. His main research interests include multilevel
inverter and renewable energy generation system.
Donghua Pan was born in Hubei Province, China, in 1987. He received the B.S.
and Ph.D. degrees in electrical and electronic engineering from Huazhong
University of Science and Technology, Wuhan, China, in 2010 and 2015, respec-
tively.
He is currently a research engineer with Suzhou Inovance Technology Co., Ltd.,
Suzhou, China. His research interests include magnetic integration technique and
renewable energy generation system.
Dongsheng Yang was born in Jiangsu, China, in 1984. He received the B.S., M.S.,
and Ph.D. degrees, all in electrical engineering from Nanjing University of
Aeronautics and Astronautics, Nanjing, China, in 2008, 2011, and 2016, respec-
tively.
He is currently a postdoctoral fellow at Aalborg University, Denmark. His main
research interests include grid-connected inverter control and renewable energy
generation systems.
Weiwei Li was born in Henan Province, China, in 1987. He received the B.S. and
Ph.D. degrees in electrical engineering from Huazhong University of Science and
Technology, Wuhan, China, in 2009 and 2014, respectively.
He is currently a research assistant in SEPRI of China Southern Power Grid Co.,
Ltd, Guangzhou, China. His research interests include HVDC power transmission,
dc distribution, and renewable energy generation systems.
Chenlei Bao was born in Zhejiang Province, China, in 1987. He received the B.S.
degree in electrical engineering and automation from Harbin Institute of
Technology, Harbin, China, in 2010, and the M.S. degree in electrical engineering
from Huazhong University of Science and Technology, Wuhan, China, in 2013.
In April 2013, he joined the Shanghai Marine Equipment Research Institute,
Shanghai, China. His current research interests include digital control technique and
renewable energy generation system.
xx About the Authors
Abbreviations
ANF Adaptive notch filter
ASM Averaged switch model
CVF Complex vector filter
CVFM Complex-vector-filter method
DPGS Distributed power generation system
DSC Delayed signal cancellation
DSP Digital signal processor
E-PLL Enhanced phase-locked loop
FNC Fundamental negative-sequence components
FPC Fundamental positive-sequence components
GSO-CVF Generalized second-order complex-vector filter
LF Loop filter
LPF Low-pass filter
MAF Moving average filter
NF Notch filter
PCC Point of common coupling
PD Phase detector
PF Power factor
PI Proportional integral
PLL Phase-locked loop
PO Percentage overshoot
PR Proportional resonant
PSF Positive-sequence filter
PU Per unit
PWM Pulse-width modulation
Q-PLL Quadrature phase-locked loop
RE-DPGS Renewable energy-based distributed power generation system
RHP Right half plane
RMS Root-mean-square
R/P Reserves to production
xxi
SGT Sliding Goertzel transform
SO Symmetrical optimum
SO-CVF Second-order complex-vector filter
SOF Second-order scalar filter
SOGI Second-order generalized integrator
SPWM Sinusoidal pulse-width modulation
SRF-PLL Synchronous reference frame PLL
THD Total harmonic distortion
TO Technical optimum
TO-CVF Third-order complex-vector filter
VCO Voltage-controlled oscillator
VSI Voltage source inverter
ZC-PLL Zero-crossing PLL
ZOH Zero-order hold
xxii Abbreviations
Chapter 1
Introduction
Abstract After 200 years of continuous extraction and recent massive consump-
tion, fossil fuels have rapidly become depleted. At the same time, the process of
consuming fossil energy has produced a large amount of waste, which has seriously
polluted the environment, jeopardizing the long-term sustainability of development
of our society. The renewable energy-based distributed power generation system
(RE-DPGS) has been attracting a great deal of attention due to its sustainable and
environmental-friendly features, and its use represents an effective approach to
dealing with future energy shortage and environmental pollution. As the energy
conversion interface between the renewable energy power generation units and the
grid, the grid-connected inverter plays an important role for the safe, stable, and
high-quality operation of RE-DPGS. The worldwide energy situation is first
reviewed in this chapter, and then, the typical configurations and the advantages of
the RE-DPGS are introduced. The key control technologies of the LCL-type
grid-connected inverter are also systematically elaborated including: (1) design and
magnetic integration of LCL filter, (2) resonance damping methods, (3) design of
controller parameters, (4) control delay effects and the compensation methods,
(5) suppressing grid current distortion caused by grid-voltage harmonics, and
(6) grid-impedance effects on system stability and the improvement methods.
Keywords Renewable energy � Distributed power generation � Grid-connected
inverter � LCL filter � Phase-locked loop (PLL)
1.1 Energy Situation and Environmental Issues
Fossil energy is the cornerstone of modern civilization. After 200 years of con-
tinuous extraction and recent massive consumption, fossil fuels have rapidly
become depleted. At the same time, the process of consuming fossil energy has
produced a large amount of waste, which has seriously polluted the environment,
jeopardizing the long-term sustainability of developmentof our society. Table 1.1
shows the consumption shares and reserves-to-production (R/P) ratios of various
© Springer Nature Singapore Pte Ltd. and Science Press 2018
X. Ruan et al., Control Techniques for LCL-Type Grid-Connected Inverters,
CPSS Power Electronics Series, DOI 10.1007/978-981-10-4277-5_1
1
primary energy sources in 2015 [1]. The R/P ratio, expressed in time, refers to the
ratio of the energy reserves to the energy production in the same year, reflecting the
remaining amount of energy source or the sustainability of the particular form of
energy supply. As shown in Table 1.1, the fossil energy, including oil, coal, and
natural gas, was still the dominant source of energy, accounting for a total of 85.9%
of the global primary energy consumption. However, of these three kinds of fossil
energy, only coal has an R/P ratio exceeding 100 years, and the others’ are less than
60 years.
To cope with problems associated with environmental pollution and the rapid
depletion of fossil fuels, tremendous efforts have been made to improve the effi-
ciency of energy utilization, reduce the energy consumptions, and lower the amount
of carbon emissions. Meanwhile, new clean energy and renewable energy have
been developed and adopted rapidly for the purpose of sustaining the energy
supply. As listed in Table 1.1, the hydroelectricity and renewable energy accounted
for 6.8% and 2.8% of the global energy consumption, respectively, in 2015. Being
the most promising forms of renewable energy, the use of wind energy and solar
energy has increased exponentially, and they will continue to play an important role
in the future energy markets.
1.2 Renewable Energy-Based Distributed Power
Generation System
Renewable energy sources, including wind and solar energy, are available over
wide geographical areas, and the utilization of renewable energy sources has caused
a significantly lower level of pollution to the environment. As a result, extensive
support policies and financial incentives have been implemented to promote the
deployment and commercialization of renewable energy in many countries [2].
The renewable energy-based distributed power generation system (RE-DPGS)
has recently become a significant development direction toward achieving a
large-scale utilization of renewable energy. The RE-DPGS is usually located in the
proximity of the load center and can be operated flexibly in either standalone mode
or grid-connected mode. The RE-DPGS has many advantages, including:
1. Environmental friendliness. The generation of renewable energy causes less
environmental pollution and produces zero carbon emission.
Table 1.1 Consumption shares and R/P ratios of various primary energy sources in 2015
Energy Oil Natural
gas
Coal Hydroelectricity Nuclear
energy
Renewable
energy
Shares (%) 32.9 23.8 29.2 6.8 4.4 2.8
R/P ratios
(year)
50.7 52.8 114 – – –
2 1 Introduction
2. Enhanced energy security. The utilization of renewable energy helps to alleviate
the energy shortage problem and reduce the dependence on energy import.
3. Low power loss. The RE-DPGS is usually located close to the load center, and
electricity is generated near where it is used. This eliminates the power loss due
to long-distance transmission.
4. High reliability. When a power grid fault happens, the RE-DPGS can be
operated as an uninterrupted power supply for the local loads, and it can help the
grid to restore from faults.
5. Cost-effectiveness. Compared with a large-scale centralized generation station, a
single RE-DPGS has relatively small capacity. Thus, the cost of installation and
construction is significantly reduced.
The electrical power generated by RE-DPGS accounted for 6.7% of the global
power generation in 2015, with a growth of 15.2% over 2014, contributing 97% of
the growth in the global power generation in 2015 [1]. In fact, the renewable energy
sources are already playing an important role in some countries. Denmark leads,
with 66% of power coming from renewables, followed by Portugal with 30%.
Among the larger EU economies, the renewables share is 27% in Germany, 24% in
Spain, and 23% in both Italy and the UK.
Figure 1.1 shows the typical configurations of RE-DPGS integrating the wind
and solar energy, where the energy storage devices, i.e., flywheel, battery, and
supercapacitor, are used to absorb the random power fluctuation of the renewable
energy generators [3–6]. Figure 1.1a shows the RE-DPGS with a dc bus, where all
the renewable energy generators and energy storage devices are connected to the dc
bus through dc–dc converters or ac–dc converters. Then, a dc–ac inverter (i.e., the
grid-connected inverter) converts the dc-bus voltage to an ac voltage and transfers
power to the utility grid through a step-up transformer [7, 8]. The dc-bus config-
uration has been widely used in small-scale DPGS for convenience of control and
the use of interface of renewable energy to the system. The RE-DPGS with ac bus is
shown in Fig. 1.2b, where all the renewable energy generators and energy storage
devices are connected to the ac bus. Then, the ac bus is connected to the utility grid
through a step-up transformer [9, 10]. Since each of the renewable energy gener-
ators and energy storage devices is interfaced with a grid-connected inverter, the
capacity of the grid-connected inverter is reduced and its reliability can be
improved.
As shown in Fig. 1.1, power electronic converters are indispensable in the
RE-DPGS. As the power conversion interface between the renewable energy
sources and the utility grid, the grid-connected inverters are used to convert the dc
power to the high-quality ac power and feed it into the grid, and they play an
important role in the RE-DPGS for achieving safe, stable, and high-quality
operation.
1.2 Renewable Energy-Based Distributed Power Generation System 3
1.3 Key Issues of LCL-Type Grid-Connected Inverters
Grid-connected inverters can be either single-phase ones or three-phase ones.
Single-phase inverters are mainly used in small-volume resident power generation
system, while three-phase inverters are widely employed in large-scale distributed
power station involving renewable energy. In the grid-connected inverters, a filter is
needed to attenuate the switching harmonics generated from pulse-width modula-
tion (PWM). Usually, an L filter and an LCL filter are the two alternatives, as shown
in Fig. 1.2a, b, respectively. The L filter is formed by a single inductor L, and the
LCL filter is composed of two inductors L1 and L2 and a capacitor C. Compared
ac-dc
Load
10 kV ac bus
G
dc-dc
ac-dcM
Wind 
Turbine
Solar 
Array
Flywheel
dc bus
dc-ac Grid
dc-dcBattery
Super
Capacitor
dc-dc
ac-dc-ac
Load
10 kV ac bus
G
dc-ac
ac-dc-acM
Wind 
Turbine
Solar
Array
Flywheel
ac bus
Grid
dc-acBattery
Super
Capacitor
dc-ac
(a) RE-DPGS with dc bus
(b) RE-DPGS with ac bus
Fig. 1.1 Typical configurations of RE-DPGS
4 1 Introduction
with the L filter, the LCL filter has an additional capacitor branch which can bypass
high-frequency current harmonics, thus allowing the use of smaller inductors to
meet the harmonic limits [11–13]. However, the LCL filter suffers from resonance
problem. At the resonance frequency fr, there is a high resonance peak, while a
sharp phase step down of −180° occurs, as shown in Fig. 1.2c. If this resonance
peak is not properly damped, it would lead to grid current oscillation or even system
instability [14, 15]. Due to this resonance hazard, the control of LCL-type
grid-connected inverter has been attracting much more interests and efforts.
The quality of the injected power into the grid and the system stability are the
two important aspects of the LCL-type grid-connected inverter. Specifically, the key
issues are summarized as follows.
1. Design of LCL filter. The LCL filter parameters need to be properly chosen to
limit the grid current harmonics. Furthermore, in order to reduce the volume of
magnetic components, the two inductors of an LCL filtercan be integrated into
one.
2. Damping LCL filter resonance. Resonance of the LCL filter will cause system
instability. In order to ensure system stability, damping is required, and the
controller parameters should be properly designed.
L
vgVin
iL
A
B
+
–
L1
vgVin
i2
A
B
+
–
L2
C
i1 iC
(a) L )b(filter LCL filter
0
M
ag
ni
tu
de
(d
B
)
270
180
90
Ph
as
e
(°
)
Frequency (Hz)
fr
L filter
LCL filter
(c) Frequency responses of the two kinds of filters
Fig. 1.2 Configurations and frequency responses of the L filter and LCL filter
1.3 Key Issues of LCL-Type Grid-Connected Inverters 5
H
Realce
H
Realce
H
Realce
3. Stability problem caused by the digital control delays. If digital control is
employed in the grid-connected inverter, there will be computation and PWM
delays. These control delays will change the characteristics of the resonance
damping and degrade the control performance of thegrid current loop. Thus,
proper control strategies should be adopted to alleviate the effects of control
delays.
4. Impacts of grid voltage harmonics. The local nonlinear loads, e.g., arc welding
machine, electric rail transport, and saturated transformer, always generate
harmonic currents. The harmonic currents flow through the line impedance,
causing distortion of the grid voltage at the point of common coupling (PCC)
[16]. The grid-voltage distortion not only decreases the injected power quality,
but also degrades the tracking performance of the phase-locked loop. Thus,
efforts should be made to reduce the impacts of grid-voltage harmonics by
properly controlling the grid-connected inverters.
5. Effects of grid impedance on system stability. Generally, the grid at the PCC can
be represented by an ideal voltage source in series with grid impedance. The
grid impedance has effects on the system stability of the grid-connected inverter.
To address these issues, extensive work has been conducted in the past decades, and
they are briefly reviewed in the following.
1.3.1 Design and Magnetic Integration of LCL Filter
The LCL filter aims to reduce the switching harmonics at the grid side. When
designing the LCL filter, the following three constraints must be taken into account.
1. Individual harmonic and total harmonic distortion (THD) of the grid current.
Table 1.2 shows the current harmonic limits in IEEE std. 929-2000 [17] and
IEEE std. 1547-2003 [18]. The LCL filter parameters need to be designed to
meet these limitations.
2. Current ripple at the inverter side. To reduce the core loss of the inverter-side
inductor and the conduction loss of power switches, the current ripple at the
inverter side should be limited.
3. Reactive power introduced by the filter capacitor. Limiting the reactive power of
the filter capacitor is helpful to reduce the current stress of power switches.
Table 1.2 Current harmonic limits in percent of rated current
Harmonic order
h (odd
harmonics)*
h < 11 11 � h < 17 17 � h < 23 23 � h < 35 35 � h THD
Percent (%) 4.0 2.0 1.5 0.6 0.3 5.0
*Even harmonics are limited to 25% of the odd harmonic limits above
6 1 Introduction
H
Realce
H
Realce
H
Realce
H
Realce
H
Realce
H
Realce
H
Realce
H
Realce
H
Realce
Based on the above constraints, the design procedure for the LCL filter will be
presented in Chap. 2.
An LCL filter has two individual inductors. In order to reduce the volume of
magnetic components, these two inductors can be integrated into one. Magnetic
integration techniques have been widely used in switching-mode power supplies,
especially in dc–dc converters. According to the presence of coupling between the
integrated magnetic components, the magnetic integration techniques can be clas-
sified into two types, namely decoupled magnetic integration and coupled magnetic
integration [19, 20].
With decoupled magnetic integration, the fluxes generated by the windings of
different magnetic components are independent. Thus, the integrated magnetic
components keep the same characteristics as the discrete ones. The fundamental
principle of decoupled magnetic integration is introduced in Ref. [19]. By utilizing
an ungapped magnetic leg as the common flux path and arranging the windings
properly, the fluxes generated by different windings are largely canceled out in the
common leg. As a result, the cross-sectional area of the common leg can be reduced
due to the low flux, and the size of the magnetic core can be reduced. Based on this
principle, for example, integration can be achieved for the two inductors of an
interleaved quasi-square-wave dc–dc converter [21], the two transformers of an
asymmetrical half-bridge converter [22], as well as the inductor and the transformer
for an LLC resonant converter [23].
With coupled magnetic integration, the fluxes generated by the windings of
different magnetic components are coupled to certain degree, and the characteristics
of integrated magnetic components are thus different from the discrete ones. In
some particular applications, coupled magnetic integration can improve the
steady-state or dynamic performances of the converters. For example, by selecting a
proper method of coupling, the inductor current ripple can be reduced in interleaved
dc–dc converters [24–26], and even zero current ripple can be achieved in the Cuk
converter [27] and multioutput buck-derived dc–dc converters [28].
The decoupled magnetic integration of the two inductors in the LCL filter will be
presented in Chap. 3 of this book for the purpose of reducing the overall size of the
LCL filter while maintaining the same harmonic attenuation ability.
1.3.2 Resonance Damping Methods of LCL Filter
Basically, methods for resonance damping of LCL filter can be classified into two
types, namely passive damping and active damping. Passive-damping methods are
very simple since only a resistor is required to be inserted into the LCL filter.
Among which, connecting a resistor in parallel with the filter capacitor shows the
best damping performance, and the magnitude-frequency characteristics of LCL
filter remain unchanged at the low- and high-frequency ranges, but the power loss
in the damping resistor is relatively large, leading to reduced efficiency [29].
Comparatively, connecting a resistor in series with the filter capacitor has been
1.3 Key Issues of LCL-Type Grid-Connected Inverters 7
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widely used since the power loss in the damping resistor is lower, but the
high-frequencyharmonic attenuation ability of the LCL filter is weakened. In order
to retain the high-frequency harmonic attenuation, the filter capacitor can be split
into two, and the damping resistor is connected in series with one of the two
capacitors. Furthermore, an inductor can be connected in parallel with the damping
resistor to provide the flowing path for the fundamental current of the filter
capacitor, thus reducing the power loss in the damping resistor [30].
In order to avoid power loss in the damping resistor, the concept of virtual
resistor has been proposed to replace the passive one. The virtual resistor is realized
by specific control algorithms, which are referred to as active-damping methods
[31–33]. Through equivalent transformation of the control block diagram, it has
been proven that proportional feedback of the capacitor current is equivalent to a
virtual resistor connected in parallel with the filter capacitor [33]. Besides the use of
a virtual resistor, there are other active-damping methods, which are implemented
with pole-zero placement based on state-space model [34, 35], predictive control
[36, 37], and h-infinity control [38–41], etc.
In Chap. 4, a comparative study of various passive- and active-damping methods
will be given. The capacitor-current-feedback active damping is chosen in this book
due to its effectiveness and simple implementation.
1.3.3 Controller Design of Grid-Connected Inverters
1.3.3.1 Classification of Control Schemes
Besides damping the resonance peak of the LCL filter, appropriate choice of thecontroller parameters is also important to ensure the stable operation of the
grid-connected inverter. The control schemes for the grid-connected inverter can be
classified into voltage-controlled schemes and current-controlled schemes.
Voltage-controlled schemes are usually referred to amplitude-phase control.
Based on the LCL filter model, the amplitude and phase of the inverter bridge
output voltage can be calculated according to the grid voltage and the command of
inverter output power. By regulating the inverter bridge output voltage, the grid
current can be indirectly controlled, thereby the inverter output power can be
controlled [42–44]. The control structure is simple, and no current sensor is needed.
However, the voltage-controlled schemes are based on the steady-state sinusoidal
model and the grid current is under open-loop control. As a result, the dynamics
response of the system is poor, and the ability of suppressing the harmonics and
unbalanced components in the grid current caused by the grid-voltage distortion is
also poor.
Current-controlled schemes can be classified into direct current control and
indirect current control. In the direct current control, the grid current is fed back and
directly regulated with a closed loop [34, 45]. Thus, fast dynamic response and
good disturbance rejection ability of the grid current can be achieved. In the indirect
8 1 Introduction
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current control, however, it is the inverter-side inductor current that is fed back and
regulated [31, 46]. Since the inverter-side inductor current is the sum of the grid
current and the filter capacitor current, the grid current is indirectly controlled. The
indirect current control can be regarded as the direct current control plus partial
capacitor-current-feedback active-damping [47]. In this book, the direct current
control with capacitor-current-feedback active-damping is studied, as shown in
Fig. 1.3, where the phase of the grid current reference is obtained through the
phase-locked loop (PLL) so as to synchronize with the grid voltage. The amplitude
of the grid current reference is determined by the outer voltage loop. Since the
bandwidth of the voltage loop is far narrower than that of the grid current loop, it is
reasonable to consider the voltage loop as being decoupled from the grid current
loop [48].
Three-phase three-wire grid-connected inverters are widely used in high-power
system. The closed-loop system can be designed in the stationary a–b frame [49], as
shown in Fig. 1.4a, or in the synchronous d–q frame, as shown in Fig. 1.4b. The
advantage of the former one is that the three-phase grid-connected inverter can be
equivalently transformed into two independent single-phase grid-connected
inverters, resulting in a simple control algorithm. The advantage of the latter one
is that zero steady-state error of the grid current can be achieved with a simple
proportional-integral (PI) regulator.
1.3.3.2 Closed-Loop Design Targets
In the control systems of the LCL-type grid-connected inverter shown in Figs. 1.3
and 1.4, the capacitor-current-feedback coefficient and the grid current regulator
vg
cos
Vin
L1 L2
C
iC
++ ––
vC
PLL
Control System
vM i2*
Hv
Gi(s) *I
i1 i2+
–
vinv
Sinusoidal PWM
Hi2Hi1
Fig. 1.3 Single-phase LCL-type grid-connected inverter with capacitor-current-feedback
active-damping
1.3 Key Issues of LCL-Type Grid-Connected Inverters 9
should be tuned to meet the performance and stability requirements. The key design
targets are as follows: (1) small steady-state error of the grid current; (2) fast
dynamic response and low overshoot; and (3) low THD of the grid current [50, 51].
iCa iCb iCc i2a i2b i2c vgc vgb vga
PWM
Modulator
Vin
N'
L1
L1
L1
L2
L2
L2
C C C
abc/ abc/abc//abc
N
i2
i2
Gi(s)
Gi(s)
iC
iC
sin
cos
I*
i2
i2+
–
+
–
+
–
+
–
(a) Stationary - frame
iCa iCb iCc i2a i2b i2c vgc vgb vga
PWM
Modulator
Vin
N'
L1
L1
L1
L2
L2
L2
C C C
abc/dq abc/dqabc/dqdq/abc
N
i2d
i2q
Gi(s)
Gi(s)
iCd
iCq
+
–
+
–
I2d*
I2q*
+
–
+
–
(b) Synchronous d-q frame
Fig. 1.4 Control structure of the three-phase LCL-type grid-connected inverter
10 1 Introduction
These design targets are related to the crossover frequency, phase margin, gain
margin, and the loop gain in the low-frequency range [52].
Recently, much work has been devoted to the closed-loop design of the LCL-
type grid-connected inverter. In Refs. [45, 53], the root locus method and pole-zero
placement are adopted to design the closed-loop parameters. In Ref. [46], the LCL
filter is initially approximated to an L filter, and the parameters of the grid current
regulator are adjusted using the symmetrical optimum (SO) method to achieve the
maximum phase margin, and finally, the capacitor-current-feedback coefficient is
computed using the root locus method. These parameters design methods aim to
find optimized closed-loop parameters with iteration or simulation. The technical
optimum (TO) method is widely used in designing controller parameters especially
for second-order systems, and its design target is to set the damping ratio of the
closed-loop system to 0.707. However, if the TO method is applied to high-order
systems such as LCL-type grid-connected inverters, the designed controller
parameters will lead to poor dynamic response and large steady-state error [32].
1.3.3.3 Grid Current Regulator
Usually, a PI regulator or proportional-resonant (PR) regulator is used as the grid
current regulator Gi(s), as shown in Figs. 1.3 and 1.4. The PI regulator has a simple
structure and allows easy implementation, while the PR regulator can provide a
sufficiently high gain at the fundamental frequency or selected harmonic frequen-
cies, so as to eliminate the steady-state error of the grid current or suppress the grid
current distortion caused by the specific grid voltage harmonics [11, 54].
The transfer function of the PI regulator is expressed as
Gi sð Þ ¼ Kp þ Kis ð1:1Þ
where Kp is the proportional coefficient and Ki is the integral coefficient. By
increasing Kp, a high crossover frequency can be obtained. By increasing Ki, a high
loop gain at the low frequencies can be achieved. The steady-state error of the grid
current can thus be reduced, and the harmonics and unbalance of the grid current
can be better suppressed. However, in order to ensure system stability, the selection
of Kp and Ki is subject to upper limits, implying that the harmonics and unbalance
of the grid current cannot be fully eliminated.
As for the single-phase grid-connected inverter, the PI regulator cannot achieve
zero steady-state error of the grid current [51], whereas the PR regulator can
overcome this problem [55]. The transfer function of the PR regulator is expressed
as
Gi sð Þ ¼ Kp þ Krss2 þx2o
ð1:2Þ
1.3 Key Issues of LCL-Type Grid-Connected Inverters 11
where Kp is the proportional coefficient, Kr is the resonant coefficient, and xo = 2pfo
is the angular fundamental frequency. From (1.2), it can be observed that the gain of
the PR regulator is infinite at xo, so the steady-state error of the grid current can be
eliminated.
However, the grid frequency fluctuates when the load varies or when a grid fault
occurs. If the grid frequency deviates from the preset xo, the gain of the PR
regulator will decrease rapidly. As a result, the steady-state error of the grid current
will increase. To achieve a high gain within a grid frequency range around xo, two
solutions can be employed. One solution is to use an adaptive PR regulator whose
resonance frequency is adjusted to actual grid frequency [56, 57]. The actual grid
frequency can be measured using a PLL or other methods. The other solution is to
use a PR regulator which has a high gain within a grid frequency range around xo
[54]. Such a PR regulator is expressed as
Gi sð Þ ¼ Kp þ 2Krxiss2 þ 2xisþx2o
ð1:3Þ
where xi is the bandwidth of the resonant part when concerning −3 dB cutoff
frequency, which means the gain of theresonant part is Kr=
ffiffiffi
2
p
at xo ± xi.
Similarly, for the three-phase grid-connected inverter, adopting the PI regulator
in the stationary a–b frame cannot eliminate the steady-state error of the grid
current [50]. However, the PR regulator can accomplish it [56, 57]. Since the
positive-sequence and negative-sequence fundamental frequencies of the grid
voltage or grid current are the same in the stationary a–b frame, the PR regulator
can eliminate the steady-state error of both the positive-sequence and
negative-sequence fundamental wave components of the grid current.
The three-phase grid-connected inverter can be controlled in the synchronous d–
q frame, as shown in Fig. 1.4b. Note that the fundamental components of the
voltage and current are transformed to dc components in the synchronous d–
q frame. The PI regulator can thus eliminate the steady-state error of the grid
current. In fact, the PI regulator in the d–q synchronous frame is equivalent to the
PR regulator in the stationary a–b frame [58].
In fact, the procedure for finding the controller parameters is consistent for both
the single-phase grid-connected inverter and three-phasegrid-connected inverter
regardless of the representation in the stationary a–b frame or the synchronous d–
q frame. Except for the steady-state error, the phase margin and gain margin are
determined by both the grid current regulator and the capacitor-current-feedback
active-damping. Thus, the controller parameters should be carefully designed.
Taking the single-phase LCL-type grid-connected inverter shown in Fig. 1.3 as an
example, a step-by-step controller parameters design method will be discussed in
Chap. 5.
12 1 Introduction
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1.3.4 Effects of Control Delay and the Compensation
Methods
1.3.4.1 Control Delay Effects
Figure 1.5 shows the structure of a digitally controlled LCL-type grid-connected
inverter. In contrary to Fig. 1.3, the grid voltage vg, grid current i2, and capacitor
current iC are sampled and converted into digital signals by an A/D converter, and
the control algorithm is implemented with a digital signal processor (DSP).
The digitally controlled system contains computation and PWM delays. The
computation delay is one sampling period in the commonly used synchronous
sampling scheme, and it is modeled as z−1 in the z-domain and e�sTs in the s-
domain, where Ts is the sampling period. The PWM delay is caused by the
zero-order hold effect, which can be approximated as Tse�0:5sTs [59]. Therefore, it is
a delay of half sampling period. Hence, the total control delay is one and a half
sampling periods. Since this control delay is included in the
capacitor-current-feedback active-damping, it will certainly affect the damping
performance, thereby affecting the features of the loop gain, which will be discussed
in Chap. 8.
It is shown in Fig. 1.2c that in an analog-controlled LCL-type grid-connected
inverter, the phase plot of the loop gain crosses −180° at the resonance frequency fr.
Thus, the resonance peak must be damped below 0 dB to stabilize the system [51].
While in the digitally controlled system, the −180° crossover might take place at fr
or one-sixth of the sampling frequency (fs/6). Specifically, if fr < fs/6, the phase plot
still crosses −180° at fr, implying that the resonance damping is mandatory [60]. If
fr > fs/6, the phase lag resulted from the control delay makes the phase plot cross
−180° at fs/6 in advance. Thus, as long as the magnitude at fs/6 is below 0 dB, the
vg
cos
Vin
L1 L2
C
iC
++ ––
vC
vM i2* *I
i1 i2+
–
vinv
Gi(z)
Hi1 Hi2 PLL
DSP Controller
Sinusoidal PWM
Fig. 1.5 Structure of a digitally controlled LCL-type grid-connected inverter
1.3 Key Issues of LCL-Type Grid-Connected Inverters 13
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system can be stable even without any damping [31]. If fr = fs/6, the system can be
hardly stable even with damping.
In practice, the real grid contains inductive grid impedance, which makes the
resonance frequency lower. Moreover, the grid impedance might vary over a wide
range depending on the grid configuration, which leads to a wide range of variation
of the resonance frequency [14]. If the LCL filter with a resonance frequency higher
than fs/6 is installed, potential instability will be triggered when the grid impedance
makes the resonance frequency be reduced and pass through fs/6. Therefore, it is
necessary to alleviate the control delay effect to ensure the LCL-type grid-connected
inverter is robust against the grid-impedance variation.
1.3.4.2 Control Delay Compensation Methods
In order to compensate the control delay of one and a half sampling periods, an
ideal approach is to introduce a leading element with one and a half sample periods,
i.e., e1:5sTs , to completely cancel out e�1:5sTs . For ease of implementation, e1:5sTs is
approximated by a first-order Taylor expansion, yielding e1:5sTs � 1þ 1:5sTs.
Noting that 1þ 1:5sTs contains a derivative part, which will lead to an infinite
amplification of high-frequency noises, a lead compensator is usually adopted as an
alternative in practice, which is expressed as [61]
Glead sð Þ ¼ 1þ 1:5sTs1þ 1:5asTs ð1:4Þ
where a < 1. The phase lead introduced by Glead(s) can be regulated by tuning a.
Figure 1.6 shows the Bode diagrams of Glead(s) with two different values of a
and Ts = 50 ls. As shown in Fig. 1.6, a phase lead is achieved, but the gain at
higher frequencies is amplified at the same time. Hence, high-frequency noises will
be amplified to a certain extent. Moreover, a smaller a leads to a better compen-
sation of the phase, but a higher amplification of high-frequency noises arises. So,
the possible phase lead is limited in practice.
To achieve a more satisfactory compensation, the state observer can be used for
predicting the values one sampling period ahead [62]. Figure 1.7 shows the block
diagram of the state observer in discrete domain, where the system state-space
equations are expressed as
x kþ 1ð Þ ¼ Gx kð ÞþHu kð Þ
y kð Þ ¼ Cx kð Þ ð1:5Þ
where x(k) is the state-variable vector, u(k) is the input-variable vector, y(k) is the
output-variable vector, and G, H, and C are the state-space matrices. The observer
equations are
14 1 Introduction
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x̂ kþ 1ð Þ ¼ Gx̂ kð ÞþHu kð ÞþL y kð Þ � ŷ kð Þð Þ
ŷ kð Þ ¼ Cx̂ kð Þ ð1:6Þ
where L is the observer gain matrix, and the variables with hat (^) denote the observed
variables. From (1.6), it can be seen that based on the input and output at time step k,
i.e., u(k) and y(k), the state variable at time step k + 1, i.e., x̂ kþ 1ð Þ, can be estimated.
This means that the estimated values are one sampling period ahead. Hence, if the
observed variable x̂ kþ 1ð Þ is used for feedback control instead of the actual vari-
able x(k), the one-sample computation delay will be completely compensated.
Since the state observer is built based on the system state-space model, its
precision of estimation is dependent on the accuracy of the model. In practice, the
0
|G
le
ad
(s
)|
(d
B
)
∠
G
le
ad
(s
) (
°)
102 103
Frequency (Hz)
104 105
10
8
12
0
4
=1/3
=2/3
20
30
Fig. 1.6 Bode diagrams of
Glead(s) with different a
u(k) y(k)
–
++
+
+
x(k+1)ˆ
L
z 1 x(k)ˆ y(k)ˆ
x(k+1)=Gx(k)+Hu(k)
y(k)=Cx(k)
G
H C
State observer
Fig. 1.7 Block diagram of
the state observer in discrete
domain
1.3 Key Issues of LCL-Type Grid-Connected Inverters 15
inaccuracy of the model caused by the variation of circuit parameters can lead to the
prediction error, which will degrade the control performance or even result in
system instability [63].
Instead of using a delay compensation, a direct reduction of the computation
delay is preferred. For the purpose of improving system stability and control per-
formance of LCL-type grid-connected inverter, methods of reducing or even
eliminating the computation delay will be given in Chap. 9.
1.3.5 Suppression of Grid Current Distortion Caused
by Grid Voltage Harmonics
Asmentioned above, the actual grid voltage contains abundant background har-
monics, which will lead to the grid current distortion. Besides, the three-phase grid
voltages at PCC may be unbalanced during grid faults, and this will cause unbal-
ance of the three-phase grid currents. It is desirable to suppress the harmonics and
unbalanced components in the grid current since they may increase the power loss,
reduce the utilization rate and life span of the electric motors and transformers in the
power system, and reduce the reliability and accuracy of the relay protection and
measurement devices in the utility grid.
In order to guarantee safe, stable, and high-quality operation of power system
when integrating RE-DPGS, various standards for grid-connected inverters have
been established [17, 18, 64–66] to give the mandatory limitations of the grid
current harmonics and the amount of unbalanced components. This poses great
challenges to the control of the grid-connected inverters. According to Figs. 1.3 and
1.4, it can be known that the grid voltage imposes the impacts on the grid current by
two ways. One way is through the PCC, which directly generates the fundamental
positive-sequence component, unbalanced components, and the harmonic compo-
nents in the grid current. The other way is through the PLL, which introduces an
error in the grid current reference and thus generates the unbalanced components
and harmonic components in the grid current. In the following, the three-phase
grid-connected inverter is taken as the example to review the state of the art in the
suppression of grid current distortion.
1.3.5.1 Suppression of Grid Current Distortion and Unbalance
Caused by Grid Voltage
1. Control in Stationary Frame
In order to suppress the grid current harmonics caused by the grid voltagedistortion,
a multiresonant regulator can be used [14, 67], which is expressed as
16 1 Introduction
Gi sð Þ ¼ Kp þ Kr0ss2 þx2o
þ Kr1s
s2 þx21
þ � � � þ Krns
s2 þx2n
ð1:7Þ
where x1, x2, …, xn are the frequencies of the selected harmonics to be sup-
pressed, and Kr1, Kr2, …, Krn are the corresponding resonant gains.
Compared with (1.2), multiple resonant components are introduced in (1.7), of
which the resonance frequencies are set at the harmonic frequencies so that an
infinite loop gain at these frequencies can be obtained and the selected harmonics
can be eliminated. However, when the harmonic frequency is higher than the loop
gain crossover frequency, negative phase shift induced by the resonant components
will reduce the phase margin and even cause system instability. To solve the
problem, a phase-lead compensation has been introduced for improving the system
stability [56]. Therefore, the multiresonant regulator can be used to suppress the
current harmonics above the loop gain crossover frequency.
2. Control in Synchronous Frame
In the positive-sequence synchronous frame, the fundamental negative-sequence
components are transformed into ac components at twice the fundamental fre-
quency, which cannot be eliminated by a PI regulator. To improve the rejection
ability of the fundamental negative-sequence component, an integration regulator is
introduced in the negative-sequence synchronous frame, as shown in Fig. 1.8,
where dq+1 and dq−1 denote the positive- and negative-sequence synchronous
frames, respectively. With the control method, zero steady-state error can be
achieved for both the fundamental positive-sequence and negative-sequence com-
ponents of the grid current. In fact, the regulator shown in Fig. 1.8 is equivalent to a
PR regulator in the stationary frame [56].
To further improve the harmonic rejection ability of the grid-connected inverter,
integration compensators can be also introduced in the harmonic synchronous
frames [56], so that the loop gain at the selected harmonic frequencies can be
increased, leading to higher attenuation of the grid current harmonics. This method
is the so-called multisynchronous frame control.
[eαβ]
[iαβ]
αβ
dq+1
Ki
s
αβ
dq+1
Ki
s
−1
ωt
Kp
+
–
[iαβ]* [vM_αβ]
αβ
dq−1αβ
dq−1
Fig. 1.8 Control structure of the PI regulator in the positive- and negative-sequence synchronous
frames
1.3 Key Issues of LCL-Type Grid-Connected Inverters 17
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When the 6k + 1 positive-sequence and 6k − 1 negative-sequence components
are dominant in grid voltage harmonics (as is usually the case for the utility grid
installed with high-power diode-based or thyristor-based rectifiers), a PI-R regulator
in the positive-sequence synchronous frame has been proposed [57], which can be
expressed as
Gi sð Þ ¼ Kp þ Kis þ
Xn
k¼1
Krks
s2 þ 6kxoð Þ2
: ð1:8Þ
In the positive-sequence synchronous frame, both the 6k + 1 positive-sequence
and the 6k − 1 negative-sequence harmonics are transformed into the 6k harmonics.
Therefore, these two dominant harmonics can be suppressed by only one resonant
compensator placed at the 6kth harmonic frequency, which simplifies the controller
structure.
3. Repetitive Control
Using the multiresonant regulator in the stationary frame or the multisynchronous
frame control can eliminate the harmonics and the unbalance of the grid current.
However, the controller would be too complex when more harmonics are required
to be suppressed. Based on the internal model theory, a repetitive controller has
been proposed which can eliminate numbers of harmonics at the same time [68].
The control block diagram of the repetitive control is shown in Fig. 1.9, where r, e,
d, and y are the reference, error signal, disturbance, and the output of the system,
respectively. The repetitive controller, shown as the dashed block in Fig. 1.9,
contains a repetitive signal generator Q(z)z–N, a delay component z–N, and a com-
pensator C(z). Meanwhile, P(z) represents the transfer function of the controlled
object. Benefiting from the accumulative control, the repetitive controller acquires
high gains at the fundamental and harmonic frequencies, so that the grid current
harmonics and the unbalance can be effectively suppressed [40, 69–71]. The
repetitive control has the shortcoming of poor transient performance [40] and it can
be improved by combining the instantaneous feedback control [72].
4. Feedforward Control of Grid Voltage
All the aforementioned control methods increase the gains of the grid current loop
to suppress the grid current harmonics and unbalance caused by the grid voltage. In
fact, grid-voltage feedforward control method is an alternative way to cancel the
influence of the grid voltage, as shown in Fig. 1.10, where Gi is the current reg-
ulator, Hi2 is sensor gain of the grid current, GiM is the transfer function from the
Q(z)z N
z N C(z)r
Repetitive Controller
+
–
P(z)+
–
e
d
y
Fig. 1.9 Control structure of
the repetitive control
18 1 Introduction
modulation wave vM to the grid current i2, and Yo is the output admittance of the
grid-connected inverter. The grid voltage vg is incorporated in the modulation wave
vM through feedforward function Gff, which can be derived from GiM and Yo to
completely eliminate the influence of vg on the grid current. Compared with the
aforementioned methods, the grid-voltage feedforward control method is very
simple, and it does not change the grid current loop gain, thus ensuring good
dynamic performance.
The grid-voltage feedforward control for the L-type grid-connected inverter has
been extensively studied and the feedforward function is 1/KPWM, where KPWM is
the transfer function from modulation wave vM to the inverter bridge output voltage
[73–75]. The grid-voltage feedforward control can effectively eliminate the
steady-state error, harmonics, and unbalance of the grid current caused by the grid
voltage even when a PI regulator is employed in the stationary frame, and it has
been widely used in the practical applications [50, 76].
As for the LCL-type grid-connected inverter, the positive feedback of capacitor
voltage can eliminate the influence of the grid voltage on the inverter-sideinductor
current [77]. However, the grid current will still be distorted by the harmonics and
the unbalance of the grid voltage. In [16], a positive feedback of capacitor current is
introduced to reduce the influence of the grid voltage on the grid current. Since the
positive feedback function of the capacitor current is derived based on the
low-frequency approximation, the proposed method is only effective to reduce the
harmonic and unbalanced components of the grid current at low frequencies. In
Chaps. 6 and 7, the full feedforward control of the grid voltage for single-phase and
three-phase grid-connected inverters will be discussed to further improve the
quality of the grid current.
1.3.5.2 Suppression of the Grid Current Reference Error
The grid voltages do not only distort the grid current directly, but also cause
significant deviation of the current reference through the PLL, resulting in harmonic
and unbalanced components of the grid current. Therefore, extensive efforts have
been made to improve the performance of the PLL under distorted and unbalanced
grid voltages.
i2vM
vg
Gi
Hi2
i2* +
–
+ + + –GiM
Gff Yo
Fig. 1.10 Control diagram of
the grid-voltage feedforward
control
1.3 Key Issues of LCL-Type Grid-Connected Inverters 19
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H
Nota
Há controversias, principalmente em rede fraca
1. PLL in Synchronous Reference Frame
The synchronous reference frame PLL (SRF-PLL) is the widely used PLL for
three-phase inverters [78–81], as shown in Fig. 1.11a, where h′, x′, and vd are the
extracted phase angle, angular frequency, and amplitude of the grid voltage.
Three-phase grid voltages can be expressed as
vga ¼ Vm sin h
vgb ¼ Vm sin h� 2p=3ð Þ
vgc ¼ Vm sin hþ 2p=3ð Þ
8<
: ð1:9Þ
where Vm and h are the amplitude and the phase of the grid voltage, respectively.
In Fig. 1.11a, the three-phase grid voltages are transformed into the synchronous
d–q frame. Applying Park transformation, the d-axis and q-axis components of the
grid voltages could be written as
vd ¼ Vm cos h� h0ð Þ
vq ¼ Vm sin h� h0ð Þ
�
: ð1:10Þ
dq
PI s
1v
v vd
'
'
vq
abc
vga
vgb
vgc
(a) SRF-PLL
PI s
1 ''vqVm
+
(b) Linearized model of SRF-PLL
dq
PI s
1vg
vg vd
vq
abc
vga
vgb
vgc
Extended
Loop Filter
'
'
(c) Extended-loop-filter based SRF-PLL
dq
PI s
1vg
vg vd
vq
abc
vga
vgb
vgc
v
vPrefilter
'
'
(d) Prefilter based SRF-PLL
Fig. 1.11 SRF-PLL and improved SRF-PLL
20 1 Introduction
when h is very close to h′, (1.10) could be simplified as
vd � Vm
vq � Vm h� h0ð Þ
�
: ð1:11Þ
Therefore, vd is the amplitude information extracted by SRF-PLL, and vq reflects
the phase difference between h and h′. Also, vq is fed into the PI regulator for
closed-loop control. The output of the PI regulator is x′, which is the extracted
angular frequency of the grid voltage. Integrating x′ gives the phase angle h′, which
is used for calculation of the Park transformation. According to (1.11) and
Fig. 1.11a, the linearized model of SRF-PLL can be derived, as shown in
Fig. 1.11b. As shown, the output h′ tracks the reference h through the closed-loop
control, so the grid currents can be synchronized with the grid voltages. However,
h′ extracted by SRF-PLL will contain harmonic and unbalanced components under
the distorted and unbalanced grid conditions.
By lowering the crossover frequency, SRF-PLL can reduce the influence of the
grid voltage harmonics on h′, so that the error of the grid current reference could be
suppressed [79]. However, when the grid voltages contain low-frequency har-
monics and the negative-sequence components, it is difficult to maintain satisfac-
tory steady and dynamic performances at the same time. Therefore, many improved
PLLs have been proposed, which can be categorized into two types [82], one is
extended-loop-filter-based grid synchronization system, as shown in Fig. 1.11c; the
other is the prefilter-based grid synchronization system, as shown in Fig. 1.11d.
2. Extended-Loop-Filter-Based SRF-PLL
As shown in Fig. 1.11c, an extended-loop filter is introduced into the closed loop of
SRF-PLL to eliminate the harmonic components or the fundamental
negative-sequence components (twice the fundamental frequency) in vq, so that
SRF-PLL could extract h′ accurately. This is the so-called extended-loop-filter-
based SRF-PLL, where the extended-loop filter usually takes the form of the
low-pass filter (LPF) [83], adaptive notch filter (ANF) [84], second-order lead
compensator [85], sliding Goertzel transform (SGT) [86], and moving average filter
(MAF) [87]. The extended-loop-filter-based SRF-PLL can quickly and accurately
extract the phase angle and frequency of the fundamental positive-sequence component
of the grid voltages even under largely unbalanced and distorted grid conditions.
3. Prefilter-Based SRF-PLL
As the penetration of RE-DPGS becomes high, the related grid codes, regarding the
power quality, safe running, fault ride-through and so on, are becoming more
stringent [18, 64]. Therefore, not only the phase angle and frequency, but also the
amplitudes of fundamental positive- and negative-sequence components of the grid
voltages are required to be measured in order to ensure the RE-DPGS guarantees
the dynamic grid voltage support and power-oscillation elimination under grid fault
conditions [88–90]. However, the extended-loop-filter-based SRF-PLL has limited
ability of extracting the positive-sequence components. Under largely unbalanced
1.3 Key Issues of LCL-Type Grid-Connected Inverters 21
and distorted grid conditions, even if h′ is identical with the phase of fundamental
positive-components of the grid voltages, vd will be still affected by the harmonic
and unbalanced components, for the reason that the amplitude information vd is not
processed by the extended-loop filter. Thus, it is necessary to filter vd again.
In order to extract the frequency, amplitudes, and phase of the grid voltages
quickly and accurately, the grid voltages should be filtered before they are delivered
to SRF-PLL, as shown in Fig. 1.11d. This kind of method is called prefilter-based
SRF-PLL. Recently, prefilter-based SRF-PLL has been studied extensively, and the
representative prefilters include positive-sequence filter (PSF) based on generalized
integrator [91], nonlinear adaptive filter for the enhanced phase-locked loop (E-
PLL) [92] and the quadrature phase-locked loop (Q-PLL) [93], adaptive filter based
on the second-order generalized integrator (SOGI) [81, 94, 95], decoupled double-
prefilter for SRF-PLL [96], complex coefficient prefilter [97], and delayed signal
cancellation (DSC)-based prefilter [98–103]. The analysis and comparison of the
aforementioned prefilters will be discussed in Chap. 12 in details.
1.3.6 Grid-Impedance Effects on System Stability
and the Improvement Methods
Under the stiff grid condition with small grid impedance, the grid-voltage-induced
harmonic distortion and the unbalance in the grid current can be effectively sup-
pressed by employing the multiresonant regulator, feedforward of the grid voltage,
repetitive control technique, or advanced PLL. As for the weak grid conditions,
however, the grid impedance is relatively large, which causes dynamic interactions
between the power grid and grid-connected inverter. Therefore, the stability
problems of the LCL-type grid-connected inverter may be aroused if the same
techniques are employed to suppress the harmonic distortion and unbalance in the
grid current [14, 104–107].
As pointed out in [14], using the multiresonant regulators in stationary frame, the
LCL-type grid-connected inverter can be operated stably under the stiff grid con-
dition. However, under weak grid condition, it may become unstable due to the
reduction of crossover frequency and the negative phase shift caused by the grid
impedance. In Ref. [104], it also shows that the single-phase LCL-type
grid-connected inverter may be unstable when the multiresonant regulator, feed-
forward of the grid voltage, and repetitive control technique are employedunder the
weak grid conditions. Therefore, the grid impedance must be taken into account
when designing the controller parameters under the weak grid condition.
In Refs. [14, 104], when analyzing the system stability of the grid-connected
inverter under weak grid conditions, the grid-connected inverter and the grid are
taken as a whole, and then, the effects of the grid impedance on crossover fre-
quency, phase-frequency response and locations of poles and zeros of the grid
current loop gain are discussed for determining the system stability, the harmonic
rejection ability, and the transient performances.
22 1 Introduction
H
Realce
H
Realce
H
Realce
Reference [105] extends the impedance-based stability criterion for the dc dis-
tributed power system into the ac grid-connection system. It has been pointed out
that in order to ensure the stability of the grid-connected inverter under weak grid
conditions, the following two conditions should be satisfied: (1) The
current-controlled grid-connected inverter is stable when operating under an ideal
grid with the assumption of Zg(s) = 0; and (2) the impedance ratio Zg(s)/
Zo(s) satisfies the Nyquist criterion. Here, Zg(s) and Zo(s) denote the grid impe-
danceand output impedance of the grid-connected inverter, respectively. The
impedance-based stability criterion avoids the need to remodel each inverter and
repeat its loop stability analysis when the grid impedance changes, or when more
inverters are connected to the same grid. Therefore, it is suitable for the stability
analysis of the complicated RE-DPGS operated under weak grid conditions.
Based on the above-mentioned impedance-based stability criterion, the system
stability of the single-phase and three-phase grid-connected inverters under weak
grid conditions will be discussed in Chaps. 10 and 11, respectively, and the control
strategies will be presented to improve the system stability while improve the
quality of the injected grid currents.
1.4 Summary
The worldwide energy situation is first reviewed in this chapter. The renewable
energy-based distributed power generation system (RE-DPGS) has been attracting a
great deal of attention due to its sustainable and environmental-friendly features,
and its use represents an effective approach to dealing with future energy shortage
and environmental pollution. As the energy conversion interface between the
renewable energy power generation units and the grid, the grid-connected inverter
plays an important role for the safe, stable, and high-quality operation of RE-DPGS.
The typical configurations and the advantages of the RE-DPGS are introduced in
this chapter. Moreover, the key control technologies of the LCL-type
grid-connected inverter are also systematically elaborated including: (1) design and
magnetic integration of LCL filter, (2) resonance damping methods, (3) design of
controller parameters, (4) control delay effects and the compensation methods,
(5) suppressing grid current distortion caused by grid voltage harmonics, and
(6) grid impedance effects on system stability and the improvement methods.
References
1. British Petroleum Company.: BP statistical review of world energy (2016)
2. REN21.: Renewables 2016: global status report (2016)
3. Lopes, J., Moreira, C., Madureira, A.: Defining control strategies for micro grids islanded
operation. IEEE Trans. Power Syst. 21(2), 916–924 (2006)
1.3 Key Issues of LCL-Type Grid-Connected Inverters 23
H
Realce
H
Realce
4. Mitra, J., Vallem, M.: Determination of storage required to meet reliability guarantees on
island-capable micro-grids with intermittent sources. IEEE Trans. Power Syst. 27(4), 2360–
2367 (2012)
5. Lamont, A.: Assessing the economic value and optimal structure of large-scale electricity
storage. IEEE Trans. Power Syst. 28(2), 911–921 (2013)
6. Ghofrani, M., Arabali, A., Amoli, M., Fadali, M.S.: Energy storage application for
performance enhancement of wind integration. IEEE Trans. Power Syst. 28(4), 4803–4811
(2013)
7. Eid, A.: Control of hybrid energy systems micro-grid. In: Proceeding of the IEEE
International Conference on Smart Energy Grid Engineering, pp. 1−6 (2013)
8. Majumder, R.: A hybrid microgrid with dc connection at back-to-back converters. IEEE
Trans. Smart Grid 5(1), 251–259 (2014)
9. Li, Y., Vilathgamuwa, D., Loh, P.: Design, analysis, and real-time testing of a controller for
multi-bus micro-grid system. IEEE Trans. Power Electron. 19(5), 1195–1204 (2004)
10. Katiraei, F., Iravani, M., Lehn, P.: Micro-grid autonomous operation during and subsequent
to islanding process. IEEE Trans. Power Deliv. 20(1), 248–257 (2005)
11. Shen, G., Zhu, X., Zhang, J., Xu, D.: A new feedback method for PR current control of LCL-
filter-based grid-connected inverter. IEEE Trans. Ind. Electron. 57(6), 2033–2041 (2010)
12. Mariéthoz, S., Morari, M.: Explicit model-predictive control of a PWM inverter with an LCL
filter. IEEE Trans. Ind. Electron. 56(2), 389–399 (2009)
13. Liserre, M., Blaabjerg, F., Hansen, S.: Design and control of an LCL-filter-based three-phase
active rectifier. IEEE Trans. Ind. Appl. 41(5), 1281–1291 (2005)
14. Liserre, M., Teodorescu, R., Blaabjerg, F.: Stability of photovoltaic and wind turbine
grid-connected inverters for a large set of grid impedance values. IEEE Trans. Power
Electron. 21(1), 263–272 (2006)
15. Liserre, M., Dell’Aquila, A., Blaabjerg, F.: Stability improvements of an LCL-filter based
three–phase active rectifier. In: Proceeding of the IEEE Power Electronics Specialists
Conference, pp. 1195–1201 (2002)
16. Abeyasekera, T., Johnson, C.M., Atkinson, D.J., Armstrong, M.: Suppression of line voltage
related distortion in current controlled grid connected inverters. IEEE Trans. Power Electron.
20(6), 1393–1401 (2005)
17. IEEE Recommended Practice for Utility Interface of Photovoltaic (PV) Systems, IEEE Std.
929 (2000)
18. IEEE Standard for Interconnecting Distributed Resources with Electric Power Systems,
IEEE Std. 1547 (2003)
19. Zumel, P., Garcia, O., Cobos, J., Uceda, J.: Magnetic integration for interleaved converters.
In: Proceeding of the IEEE Applied Power Electronics Conference and Exposition,
pp. 1143–1149 (2003)
20. Chen, Q., Ruan, X., Yan, Y.: The application of the magnetic-integration techniques in
switching power supply. Trans. China Electrotech. Soc. 19(3), 1–8 (2004) (in Chinese)
21. Chen, W., Lee, F., Zhou, X., Xu, P.: Integrated planar inductor scheme for multi-module
interleaved quasi-square-wave (QSW) dc/dc converter. In: Proceeding of the IEEE Power
Electronics Specialists Conference, pp. 759–762 (1999)
22. Vanwyk, J.D., Lee, F.C., Liang, Z., Chen, R., Wang, S., Lu, B.: Integrating active, passive
and EMI-filter functions in power electronics systems: a case study of some technologies.
IEEE Trans. Power Electron. 20(3), 523–536 (2005)
23. Yang, B., Chen, R., Lee, F.C.: Integrated magnetic for LLC resonant converter. In:
Proceeding of the IEEE Applied Power Electronics Conference and Exposition, pp. 346–351
(2002)
24. Wong, P., Xu, P., Yang, B., Lee, F.C.: Performance improvements of interleaving VRMs
with coupling inductors. IEEE Trans. Power Electron. 16(4), 499–507 (2001)
25. Zhu, G., McDonald, B., Wang, K.: Modeling and analysis of coupled inductors in power
converters. IEEE Trans. Power Electron. 26(5), 1355–1363 (2011)
24 1 Introduction
26. Yang, F., Ruan, X., Yang, Y., Ye, Z.: Interleaved critical current mode boost PFC converter
with coupled inductor. IEEE Trans. Power Electron. 26(9), 2404–2413 (2011)
27. Cuk, S.: Coupled inductor and integrated magnetics techniques in power electronics. In:
Proceeding of the Telecommunications Energy Conference, pp. 269–275 (1983)
28. Maksimović, D., Erickson, R.W., Griesbach, C.: Modeling of cross-regulation in converters
containing coupled inductors. IEEE Trans. Power Electron. 15(4), 607–615 (2000)
29. Peña-Alzola, R., Liserre, M., Blaabjerg, F., Sebastián, R., Dannehl, J., Fuchs, F.W.: Analysis
of the passive damping losses in LCL-filter based gridconverters. IEEE Trans. Power
Electron. 28(6), 2642–2646 (2013)
30. Wang, T., Ye, Z., Sinha, G., Yuan, X.: Output filter design for a grid-interconnected
three-phase inverter. In: Proceeding of the Power Electronics Specialist Conference,
pp. 779–784 (2003)
31. Dannehl, J., Liserre, M., Fuchs, F.W.: Filter-based active damping of voltage source
converters with LCL-filter. IEEE Trans. Ind. Electron. 58(8), 3623–3633 (2011)
32. Blasko, V., Kaura, V.: A novel control to actively damp resonance in input LC filter of a
three-phase voltage source converter. IEEE Trans. Ind. Appl. 33(2), 542–550 (1997)
33. Dahono, P., Bahar, Y., Sato, Y., Kataoka, T.: Damping of transient oscillations on the output
LC filter of PWM inverters by using a virtual resistor. In: Proceeding of the IEEE
International Conference on Power Electronics and Drive Systems, pp. 403–407 (2001)
34. Dannehl, J., Fuchs, F.W., Thøgersen, P.: PI state space current control of grid-connected
PWM converters with LCL filters. IEEE Trans. Power Electron. 25(9), 2320–2330 (2010)
35. Gabe, I., Montagner, V., Pinheiro, H.: Design and implementation of a robust current
controller for VSI connected to the grid through an LCL filter. IEEE Trans. Power Electron.
24(6), 1444–1452 (2009)
36. Espí, J., Castelló, J., García-Gil, R., Figueres, E.: An adaptive robust predictive current
control for three-phase grid-connected inverters. IEEE Trans. Ind. Electron. 58(8), 3537–
3546 (2011)
37. Ahmed, K.H., Massoud, A.M., Finney, S.J., Williams, B.W.: A modified stationary
reference frame-based predictive current control with zero steady-state error for LCL coupled
inverter-based distributed generation systems. IEEE Trans. Ind. Electron. 58(4), 1359–1370
(2011)
38. Doyle, J., Glover, K., Khargonekar, P.: State-space solutions to standard H2 and H∞ control
problems. IEEE Trans. Automat. Control 34(8), 831–847 (1989)
39. Willmann, G., Coutinho, D.F., Pereira, L.F.A., Libano, F.B.: Multiple-loop h-infinity control
design for uninterruptible power supplies. IEEE Trans. Ind. Electron. 54(3), 1591–1602
(2007)
40. Hornik, T., Zhong, Q.: A current-control strategy for voltage-source inverters in microgrids
based on H∞ and repetitive control. IEEE Trans. Power Electron. 26(3), 943–952 (2011)
41. Yang, S., Lei, Q., Peng, F., Qian, Z.: A robust control scheme for grid-connected
voltage-source inverters. IEEE Trans. Ind. Electron. 58(1), 202–212 (2011)
42. Wu, R., Dewan, S., Slemon, G.: Analysis of an AC-DC voltage source converter using
PWM with phase and amplitude control. IEEE Trans. Ind. Appl. 27(2), 355–364 (1991)
43. Mastromauro, R.A., Liserre, M., Kerekes, T., Dell’Aquila, A.: A single-phase
voltage-controlled grid-connected photovoltaic system with power quality conditioner
functionality. IEEE Trans. Ind. Electron. 56(11), 4436–4444 (2009)
44. Ko, S.H., Lee, S.R., Dehbonei, H., Nayar, C.V.: Application of voltage- and
current-controlled voltage source inverters for distributed generation systems. IEEE Trans.
Energy Convers. 21(3), 782–792 (2006)
45. Twining, E., Holmes, D.G.: Grid current regulation of a three-phase voltage source inverter
with an LCL input filter. IEEE Trans. Power Electron. 18(3), 888–895 (2003)
46. Dannehl, J., Fuchs, F.W., Hansen, S., Thogersen, P.B.: Investigation of active damping
approaches for PI-based current control of grid-connected pulse width modulation converters
with LCL filters. IEEE Trans. Ind. Appl. 46(2), 1509–1517 (2010)
References 25
47. Tang, Y., Loh, P., Wang, P.: Exploring inherent damping characteristic of LCL-filters for
three-phase grid-connected voltage source inverters. IEEE Trans. Power Electron. 27(3),
1433–1443 (2012)
48. Figueres, E., Garcera, G., Sandia, J., Gonzalez-Espin, F., Rubio, J.C.: Sensitivity study of the
dynamics of three-phase photovoltaic inverters with an LCL grid filter. IEEE Trans. Ind.
Electron. 56(3), 706–717 (2009)
49. Blaabjerg, F., Teodorescu, R., Liserre, M., Timbus, A.V.: Overview of control and grid
synchronization for distributed power generation systems. IEEE Trans. Ind. Electron. 53(5),
1398–1409 (2006)
50. Holmes, D.G., Lipo, T.A., McGrath, B., Kong, W.Y.: Optimized design of stationary frame
three phase ac current regulator. IEEE Trans. Power Electron. 24(11), 2417–2426 (2009)
51. Bao, C., Ruan, X., Li, W., Wang, X., Pan, D., Weng, K.: Step-by-step controller design for
LCL-type grid-connected inverter with capacitor-current-feedback active-damping. IEEE
Trans. Power Electron. 29(3), 1239–1253 (2014)
52. Erickson, W., Maksimovic, D.: Fundamentals of Power Electronics, 2nd edn. Kluwer,
Boston, MA (2001)
53. Liu, F., Zhou, Y., Duan, S., Yin, J., Liu, B.: Parameter design of a two-current-loop
controller used in a grid-connected inverter system with LCL filter. IEEE Trans. Ind.
Electron. 56(11), 4483–4491 (2009)
54. Zmood, D., Holmes, D.G.: Stationary frame current regulation of PWM inverters with zero
steady-state error. IEEE Trans. Power Electron. 18(3), 814–822 (2003)
55. Pan, D., Ruan, X., Bao, C., Li, W., Wang, X.: Capacitor-current-feedback active damping
with reduced computation delay for improving robustness of LCL-Type grid-connected
inverter. IEEE Trans. Power Electron. 29(7), 3414–3427 (2014)
56. Mattavelli, P.: A closed-loop selective harmonic compensation for active filters. IEEE Trans.
Ind. Appl. 37(1), 81–89 (2001)
57. Liserre, M., Teodorescu, R., Blaabjerg, F.: Multiple harmonics control for three-phase grid
converter systems with the use of PI-RES current controller in a rotating frame. IEEE Trans.
Power Electron. 21(3), 836–841 (2006)
58. Teodorescu, R., Liserre, M., Rodríguez, P.: Grid Converters for Photovoltaic and Wind
Power Systems. IEEE & Wiley, West Sussex, UK (2011)
59. Zhang, X., Spencer, W.J., Guerrero, J.M.: Small-signal modeling of digitally controlled
grid-connected inverters with LCL filters. IEEE Trans. Ind. Electron. 60(9), 3752–3765
(2013)
60. Pan, D., Ruan, X., Bao, C., Li, W., Wang, X.: Optimized controller design for LCL-type
grid-connected inverter to achieve high robustness against grid-impedance variation. IEEE
Trans. Ind. Electron. 62(3), 1537–1547 (2015)
61. Jalili, K., Bernet, S.: Design of LCL filters of active-front-end two-level voltage-source
converters. IEEE Trans. Ind. Electron. 56(5), 1674–1689 (2009)
62. Lee, K., Jahns, T.M., Lipo, T.A., Blasko, V.: New control method including state observer
of voltage unbalance for grid voltage-source converters. IEEE Trans. Ind. Electron. 57(6),
2054–2065 (2010)
63. Mastromauro, R.A., Liserre, M., Aquila, A.D.: Study of the effects of inductor nonlinear
behavior on the performance of current controllers for single-phase PV grid converters. IEEE
Trans. Ind. Electron. 55(5), 2043–2052 (2008)
64. Generating Plants Connected the Medium-Voltage Network. BDEW Technical Guideline
(2008)
65. Technical Rule for Distributed Resources Connected to Power Grid, Q/GDW 480 (2010) (in
Chinese)
66. Technical Rule for Photovoltaic Power Station Connected to Power Grid, Q/GDW 617
(2011) (in Chinese)
67. Timbus, A.V., Ciobotaru, M., Teodorescu, R., Blaabjerg, F.: Adaptive resonant controller
for grid-connected converters in distributed power generation systems. In: Proceeding of the
IEEE Applied Power Electronics Conference and Exposition, pp. 1601–1606 (2006)
26 1 Introduction
68. Hara, S., Yamamoto, Y., Omata, T., Nakano, M.: Repetitive control system: a new type
servo system for periodic exogenous signals. IEEE Trans. Automat. Control 33(7), 659–668
(1988)
69. Tzou, Y., Ou, R., Jung, S., Chang, M.: High-performance programmable ac power source
with low harmonic distortion using DSP-based repetitive control technique. IEEE Trans.
Power Electron. 12(4), 715–725 (1997)
70. Zhang, K., Kang, Y., Xiong, J., Chen, J.: Direct repetitive control of SPWM inverter for UPS
purpose. IEEE Trans. Power Electron. 18(3), 784–792 (2003)
71. Mattavelli, P., Marafão, F.: Repetitive-based control for selective harmonic compensation in
active power filters. IEEE Trans. Ind. Electron. 51(5),1018–1024 (2004)
72. Zhou, L., Jian, X., Zhang, K., Shi, P.: A high precision multiple loop control strategy for
three phase PWM inverters. In: Proceeding of the Annual Conference on IEEE Industrial
Electronics, pp. 1781–1786 (2006)
73. Timbus, A.V., Liserre, M., Teodorescu, R., Rodriguez, P., Blaabjerg, F.: Evaluation of
current controllers for distributed power generation systems. IEEE Trans. Power Electron. 24
(3), 654–664 (2009)
74. Kim, J. Sul, S.: New control scheme for ac-dc-ac converter without dc link electrolytic
capacitor. In: Proceeding of the IEEE Power Electronics Specialists Conference, pp. 300–
306 (1993)
75. Zeng, Q., Chang, L.: An advanced SVPWM-based predictive current controller for
three-phase inverters in distributed generation systems. IEEE Trans. Ind. Electron. 55(3),
1235–1246 (2008)
76. Wang, X., Ruan, X., Liu, S., Tse, C.K.: Full feed-forward of grid voltage for grid-connected
inverter with LCL filter to suppress current distortion due to grid voltage harmonics. IEEE
Trans. Power Electron. 25(12), 3119–3127 (2010)
77. Park, S., Chen, C., Lai, J., Moon, S.: Admittance compensation in current loop control for a
grid-tie LCL fuel cell inverter. IEEE Trans. Power Electron. 23(4), 1716–1723 (2008)
78. Magueed, F., Svensson, J.: Control of VSC connected to the grid through LCL-filter to
achieve balanced currents. In: Proceeding of the IEEE Industry Applications Conference,
pp. 572–578 (2005)
79. Kaura, V., Blasko, V.: Operation of a phase locked loop system under distorted utility
conditions. IEEE Trans. Ind. Appl. 33(1), 58–63 (1997)
80. Chung, S.: A phase tracking system for three phase utility interface inverters. IEEE Trans.
Power Electron. 15(3), 431–438 (2000)
81. Rodríguez, P., Luna, A., Muñoz-Aguilar, R.: A stationary reference frame grid synchro-
nization system for three-phase grid-connected power converters under adverse grid
conditions. IEEE Trans. Power Electron. 27(1), 99–112 (2012)
82. Hoffmann, N., Lohde, R., Fischer, M., Asiminoaei, L., Thogersen, P.B.: A review on
fundamental grid-voltage detection methods under highly distorted conditions in distributed
power-generation networks. In: Proceeding of the IEEE Energy Conversion Congress and
Exposition, pp. 3045–3052 (2011)
83. Liccardo, F., Marino, P., Raimondo, G.: Robust and fast three-phase PLL tracking system.
IEEE Trans. Ind. Electron. 58(1), 221–231 (2011)
84. Eren, S., Karimi-Ghartemani, M., Bakhshai, A.: Enhancing the three-phase synchronous
reference frame PLL to remove unbalance and harmonic errors. In: Proceeding of the Annual
Conference of IEEE Industrial Electronics, pp. 437–441 (2009)
85. Freijedo, F., Yepes, A., López, Ó., Vidal, A., Doval-Gandoy, J.: Three-phase PLLs with fast
postfault retracking and steady-state rejection of voltage unbalance and harmonics by means
of lead compensation. IEEE Trans. Power Electron. 26(1), 85–97 (2011)
86. Carugati, I., Maestri, S., Donato, P., Carrica, D., Benedetti, M.: Variable sampling period
filter PLL for distorted three-phase systems. IEEE Trans. Power Electron. 27(1), 321–330
(2012)
References 27
87. Freijedo, F., Doval-Gandoy, J., López, Ó., Acha, E.: Tuning of phase-locked loops for
power converters under distorted utility conditions. IEEE Trans. Ind. Appl. 45(6), 2039–
2047 (2009)
88. Yuan, X., Merk, W., Stemmler, H., Allmeling, J.: Stationary-frame generalized integrators
for current control of active power filters with zero steady-state error for current harmonics
of concern under unbalanced and distorted operating conditions. IEEE Trans. Ind. Electron.
38(2), 523–532 (2002)
89. Rodríguez, P., Timbus, A.V., Teodorescu, R., Liserre, M., Blaabjerg, F.: Flexible active
power control of distributed power generation systems during grid faults. IEEE Trans. Ind.
Electron. 54(5), 2583–2592 (2007)
90. Castilla, M., Miret, J., Sosa, J., Matas, J., de Vicuña, L.G.: Grid-faults control scheme for
three-phase photovoltaic inverters with adjustable power quality characteristics. IEEE Trans.
Power Electron. 25(12), 2930–2940 (2010)
91. Rodríguez, P., Timbus, A.V., Teodorescu, R., Liserre, M., Blaabjerg, F.: Reactive power
control for improving wind turbine system behavior under grid faults. IEEE Trans. Power
Electron. 24(7), 1798–1801 (2010)
92. Karimi-Ghartemani, M., Iravani, M.: A method for synchronization of power electronic
converters in polluted and variable-frequency environments. IEEE Trans. Power Syst. 19(3),
1263–1270 (2004)
93. Karimi-Ghartemani, M., Karimi, H., Iravani, M.: A magnitude/phase-locked loop system
based on estimation of frequency and in-phase/quadrature-phase amplitudes. IEEE Trans.
Ind. Electron. 51(2), 511–517 (2004)
94. Rodríguez, P., Teodorescu, R., Candela, I., Timbus, A.V., Liserre, M., Blaabjerg, F.: New
positive-sequence voltage detector for grid synchronization of power converters under faulty
grid conditions. In: Proceeding of the IEEE Power Electronics Specialists Conference, pp. 1–
7 (2006)
95. Rodríguez, P., Karimi, H., Luna, A., Candela, I., Mujal, R.: Multiresonant frequency-locked
loop for grid synchronization of power converters under distorted grid conditions. IEEE
Trans. Ind. Electron. 58(1), 127–138 (2011)
96. Rodríguez, P., Pou, J., Bergas, J., Candela, J., Burgos, R.P.I., Boroyevich, D.: Decoupled
double synchronous reference frame PLL for power converters control. IEEE Trans. Power
Electron. 22(2), 584–592 (2007)
97. Guo, X., Wu, W., Chen, Z.: Multiple-complex coefficient-filter-based phase-locked loop and
synchronization technique for three-phase grid-interfaced converters in distributed utility
networks. IEEE Trans. Ind. Electron. 58(4), 1194–1204 (2011)
98. Svensson, J., Bongiorno, M., Sannino, A.: Practical implementation of delayed signal
cancellation method for phase-sequence separation. IEEE Trans. Power Deliv. 22(1), 18–26
(2007)
99. Souza, H., Bradaschia, F., Neves, F., Cavalcanti, M.C., Azevedo, G.M.S., de Arruda, J.P.: A
method for extracting the fundamental-frequency positive-sequence voltage vector based on
simple mathematical transformations. IEEE Trans. Ind. Electron. 56(5), 1539–1547 (2009)
100. Neves, F., Cavalcanti, M., Souza, H., Bradaschia, F., Bueno, E.J., Rizo, M.: A generalized
delayed signal cancellation method for detecting fundamental-frequency positive-sequence
three-phase signals. IEEE Trans. Power Deliv. 25(3), 1816–1825 (2010)
101. Neves, F., Souza, H., Cavalcanti, M., Bradaschia, F., Bueno, E.J.: Digital filters for fast
harmonic sequence component separation of unbalanced and distorted three-phase signals.
IEEE Trans. Ind. Electron. 59(10), 3847–3859 (2012)
102. Wang, Y., Li, Y.: Grid synchronization PLL based on cascaded delayed signal cancellation.
IEEE Trans. Power Electron. 26(7), 1987–1997 (2011)
103. Wang, Y., Li, Y.: Analysis and digital implementation of cascaded delayed-signal-
cancellation PLL. IEEE Trans. Power Electron. 26(4), 1067–1080 (2011)
104. Xu, J., Xie, S., Tang, T.: Evaluations of current control in weak grid case for grid-connected
LCL-filtered inverter. IET Power Electron. 6(2), 227–234 (2013)
28 1 Introduction
105. Sun, J.: Impedance-based stability criterion for grid-connected inverters. IEEE Trans. Power
Electron. 26(11), 3075–3078 (2011)
106. Céspedes, M., Sun, J.: Impedance modeling and analysis of grid-connected voltage-source
converters. IEEE Trans. Power Electron. 29(3), 1254–1261 (2014)
107. Yang, D., Ruan, X., Wu, H.: Impedance shaping of the grid-connected inverter with LCL
filter to improve its adaptability to the weak grid condition. IEEE Trans. Power Electron. 29
(11), 5795–5805 (2014)
References 29
Chapter 2
Design of LCL Filter
Abstract As the interface between renewable energy power generation system and
the power grid, the grid-connected inverter is used to convert the dc power to the
high-quality ac power and feed it into the power grid. In the grid-connected
inverter, a filter is needed as the interface between the inverter and the power grid.
Compared with the L filter, the LCL filter is consideredto be a preferred choice for
its cost-effective attenuation of switching frequency harmonics in the injected grid
currents. To achieve high-quality grid current, the LCL filter should be properly
designed. In this chapter, the widely used pulse-width modulation (PWM) schemes
are introduced, including the bipolar sinusoidal pulse-width modulation (SPWM),
unipolar SPWM and harmonic injection SPWM. The spectrums of the output PWM
voltage with different SPWM are studied and compared. A design procedure for
LCL filter based on the restriction standards of injected grid current is presented and
verified by simulations.
Keywords Grid-connected inverter � LCL filter � Pulse-width modulation
(PWM) � Total harmonics distortion (THD)
As the interface between renewable energy power generation system and the power
grid, the grid-connected inverter is used to convert the dc power to the high-quality
ac power and feed it into the power grid. In the grid-connected inverter, a filter is
needed as the interface between the inverter and the power grid. Compared with the
L filter, the LCL filter is considered to be a preferred choice for its cost-effective
attenuation of switching frequency harmonics in the injected grid currents. To
achieve high-quality grid current, the LCL filter should be properly designed. In this
chapter, the widely used pulse-width modulation (PWM) schemes are introduced,
including the bipolar sinusoidal pulse-width modulation (SPWM), unipolar SPWM
and harmonic injection SPWM. The spectrums of the output PWM voltage with
different SPWM are studied and compared. A design procedure for LCL filter based
on the restriction standards of injected grid current is presented and verified by
simulations.
© Springer Nature Singapore Pte Ltd. and Science Press 2018
X. Ruan et al., Control Techniques for LCL-Type Grid-Connected Inverters,
CPSS Power Electronics Series, DOI 10.1007/978-981-10-4277-5_2
31
2.1 PWM for Single-Phase Full-Bridge Grid-Connected
Inverter
Figure 2.1 shows the topology of a single-phase full-bridge LCL-type grid-
connected inverter, where switches Q1–Q4 compose the two bridge legs, and
inductors L1, L2 and capacitor C compose the LCL filter. Note that the two switches
in the same bridge leg are switched in a complementary manner.
Generally, the bipolar SPWM and unipolar SPWM are usually used for
single-phase full-bridge inverter. For convenience of illustration, the dc input
voltage Vin is split into two ones equally, and the midpoint O is defined as the base
potential.
2.1.1 Bipolar SPWM
Figure 2.2 shows the key waveforms of the bipolar SPWM for single-phase LCL-
type grid-connected inverter, where, vM is the sinusoidal modulation signal with the
amplitude of VM, and vtri is the triangular carrier with the amplitude of Vtri. When
vM > vtri, Q1 and Q4 turn on, Q2 and Q3 turn off, resulting in vAO = Vin/2 and
vBO = −Vin/2; When vM < vtri, Q1 and Q4 turn off, Q2, Q3 turn on, resulting in
vAO = −Vin/2 and vBO = Vin/2. The inverter bridg eoutput voltage vinv is the dif-
ference between vAO and vBO, i.e., vinv = vAO − vBO. As shown in Fig. 2.2, vinv has
only two voltage levels, namely −Vin and +Vin. So, this PWM scheme is often
called as bipolar SPWM.
In the following, xo and xsw denote the angular frequencies of the modulation
signal vM and triangular carrier vtri, respectively, the initial phase of the modulation
signal vM is set to 0, and Mr denotes the ratio of VM and Vtri, i.e.,
Mr ¼ VM=Vtri ð2:1Þ
According to the Fourier transform theory, the time-varying signals vAO and vBO
shown in Fig. 2.2 can be expressed as [1]
A
B
L2
vg
Q3
Q4
Q1
Q2
L1
Cvinv
Vin /2
Vin /2
O
Fig. 2.1 Single-phaseLCL-
type grid-connected inverter
32 2 Design of LCL Filter
H
Realce
vAO tð Þ ¼ �vBO tð Þ
¼ MrVin
2
sinxotþ 2Vinp
X1
m¼1;3;...
X�1
n¼0;�2;�4;...
Jn mMrp=2ð Þ
m
sin
mp
2
cos mxswtþ nxotð Þ
þ 2Vin
p
X1
m¼2;4;...
X�1
n¼�1;�3;...
Jn mMrp=2ð Þ
m
cos
mp
2
sin mxswtþ nxotð Þ
ð2:2Þ
where, Jn(x) is the Bessel function of the first kind [2], expressed as
Jn xð Þ ¼
X1
k¼0
�1ð Þk
k! kþ nð Þ!
x
2
� �2kþ n
ð2:3Þ
According to (2.2), the Fourier series expansion of the inverter bridge output
voltage vinv with bipolar SPWM can be obtained, which is
Vtri
−Vtri
0
Vin /2
t
−Vin /2
0 t
vM vtri
t
Q1
vAO
0
1
Vin
−Vin
0 t
vinv
Vin /2
−Vin /2
0 t
vBO
t
Q4 0
1
Fig. 2.2 Bipolar SPWM for single-phaseLCL-type grid-connected inverter
2.1 PWM for Single-Phase Full-Bridge Grid-Connected Inverter 33
vinv tð Þ ¼ vAO tð Þ � vBO tð Þ
¼ MrVin sinxotþ 4Vinp
X1
m¼1;3;5;...
X�1
n¼0;�2;�4;...
Jn mMrp=2ð Þ
m
sin
mp
2
cos mxswtþ nxotð Þ
þ 4Vin
p
X1
m¼2;4;6;...
X�1
n¼�1;�3;...
Jn mMrp=2ð Þ
m
cos
mp
2
sin mxswtþ nxotð Þ
ð2:4Þ
where, the first term is the fundamental component, the second term is the sideband
harmonics around odd multiples of the carrier frequency, and the third term is the
sideband harmonics around even multiples of the carrier frequency. In the second
and third terms, m is the carrier index variable, and n is the baseband index variable.
m and n determine the harmonics distribution. When m is odd, |sin(mp/2)| = 1;
When m is even, |cos(mp/2)| = 1. With a given Vin, the amplitudes of the harmonics
in vinv are determined by |Jn(mMrp/2)/m|. Moreover, the harmonics in vinv distribute
only at the frequencies where m + n is odd.
According to (2.3), an example of |Jn(mMrp/2)/m| with Mr = 0.9 and m = 1, 2
and 3 is depicted with the dots, as shown in Fig. 2.3, where the three dashed lines
are plotted with Gamma Function C(k + n + 1), where the variable n uses a real
number. As observed, the dot with the maximum value locates at the center fre-
quency xsw, where m = 1, n = 0; the farther the sideband harmonic departs from
the center frequency, the smaller its amplitude is. In contrast to the harmonics
around the center frequency xsw, the amplitudes of the harmonics around twice and
above the carrier frequency are much smaller. Thus, the dominant harmonics in vinv
are at around xsw, which needs to be attenuated by the LCL filter.
In conclusion, the spectrum of the inverter bridge output voltage, vinv, generated
by the bipolar SPWM can be described as
9 7 5 3 1 1 3 5 7 9
0
0.2
0.4
0.6
0.8
1
n
(
)2
n
r
J
m
M m
π
m=3
m=2
m=1
Fig. 2.3 Characteristic curves of Bessel function
34 2 Design of LCL Filter
H
Realce
H
Realce
H
Realce
H
Realce
(1) The harmonics in vinv distribute only at frequencies where m + n is odd. When
m is odd, the harmonics distribute not only at m times of the carrier frequency,
but also at the sideband frequency when n is even; When m is even, the
harmonics only distribute at the sideband frequency when n is odd;
(2) The dominant harmonics in vinv are at around the carrier frequency (e.g., n = 0,
±2, ±4, …). The design of the LCL filter is determined by attenuating these
dominant harmonics.
2.1.2 Unipolar SPWM
As mentioned above, with the bipolar SPWM, the voltage levels of vinv could only
be −Vin and +Vin. In fact, when Q1 and Q3 or Q2 and Q4 turn on simultaneously, vinv
will be 0. The unipolar SPWM is such a kind of the modulation scheme that could
make vinv be not only +Vin and −Vin, but also 0.
Figure 2.4 shows the key waveforms of the unipolar SPWM for single-phase
LCL-type grid-connected inverter, where vM is the sinusoidal modulation signal,
and vtri and −vtri are the two sets of triangular carrier. Comparison of vM and vtri
leads to the control signals for Q1 and Q2, and comparison of vM and −vtri leads to
the control signals for Q3 and Q4. In detail, when vM > vtri, Q1 turns on and Q2 turns
off, thus vAO = Vin/2; When vM < vtri, Q1 turns off and Q2 turns on, thus
vAO = −Vin/2. Likewise, when vM > −vtri, Q4 turns on and Q3 turns off, thus
vBO = −Vin/2; When vM < − vtri, Q4 turns off and Q3 turns on, thus vBO = Vin/2.
Since vinv = vAO − vBO, the voltage levels of vinv could be +Vin, −Vin, and 0. In the
positive period of vM, the voltage levels of vinv could only be +Vin and 0; while in
the negative periodof vM, the voltage levels of vinv could only be −Vin and 0.
Therefore, this modulation scheme is calledunipolar SPWM. Furthermore, the
ripple frequency of vinv is twice the carrier frequency.
Since the control signal for Q1 is obtained by comparing vM and vtri, the Fourier
series expansion of vAO is the same as (2.2). The control signal for Q4 is obtained by
comparing vM and −vtri, and −vtri lags vtri with a phase of p, the Fourier series
expansion of vBO can be obtained by replacing xswt in (2.2) with xswt − p. Thus,
vBO is expressed as
vBO tð Þ ¼ �MrVin2 sinxot
� 2Vin
p
X1
m¼1;3;5;...
X�1
n¼0;�2;�4;...
Jn mMrp=2ð Þ
m
sin
mp
2
cos m xswt � pð Þþ nxotð Þ
� 2Vin
p
X1
m¼2;4;6;...
X�1
n¼�1;�3;...
Jn mMrp=2ð Þ
m
cos
mp
2
sin m xswt � pð Þþ nxotð Þ
ð2:5Þ
2.1 PWM for Single-Phase Full-Bridge Grid-Connected Inverter 35
H
Realce
Equation (2.5) can be further simplified as
vBO tð Þ ¼ �MrVin2 sinxotþ
2Vin
p
X1
m¼1;3;5;...
X�1
n¼0;�2;�4;...
Jn mMrp=2ð Þ
m
sin
mp
2
cos mxswtþ nxotð Þ
� 2Vin
p
X1
m¼2;4;6;...
X�1
n¼�1;�3;...
Jn mMrp=2ð Þ
m
cos
mp
2
sin mxswtþ nxotð Þ
ð2:6Þ
According to (2.2) and (2.6), the Fourier series expansion of vinv with the
unipolar SPWM is expressed as
vinv tð Þ ¼ vAO tð Þ � vBO tð Þ
¼ MrVin sinxotþ 4Vinp
X1
m¼2;4;6;...
X�1
n¼�1;�3;...
Jn mMrp=2ð Þ
m
cos
mp
2
sin mxswtþ nxotð Þ
ð2:7Þ
Vtri
−Vtri
0
Vin /2
t
−Vin /2
0 t
vM vtri
t
Q1
vAO
0
1
Vin
−Vin
0 t
vinv
Vin /2
−Vin /2
0 t
vBO
t
Q4
0
1
−vtri
Fig. 2.4 Unipolar SPWM for single-phase LCL-type grid-connected inverter
36 2 Design of LCL Filter
According to (2.7), the harmonic spectrum of vinv with the unipolar SPWM can
be described as
(1) The harmonics in vinv distribute only at the sideband frequencies where m is
even and n is odd.
(2) The dominant harmonics in vinv are at around twice the carrier frequency, which
is the major consideration of filter design.
Comparing (2.4) and (2.7), it shows that the frequencies of the harmonics in vinv
with the unipolar SPWM are twice that of those with the bipolar SPWM. This is
because the ripple frequencies of vinv with the bipolar and unipolar SPWMs are one
and two times of the carrier frequency, respectively, which can be found from
Figs. 2.2 and 2.4.
2.2 PWM for Three-Phase Grid-Connected Inverter
Figure 2.5a shows the topology of a three-phasegrid-connected inverter, where
switches Q1–Q6 compose the three-phase legs, and three sets of inductors L1, L2,
and capacitor C compose the three-phase LCL filter. Note that the three-phase
capacitors in LCL filter can be either delta- or star-connection. The capacitance
needed in delta-connection is one-third of that in star-connection, and the capacitor
Vin/2 a
b
c
L2
L2
L2
vgc vgb vga
N'
Q3
Q6
Q1
Q4
Q5
Q2
L1
L1
L1
N
Vin/2
O
C C C
vCa
vCb
vCc
(a) Main circuit
L2
L2
L2
vgc vgb vga
N'
L1
L1
L1
N
vao vbo vco
O
ia1
ib1
ic1
ia2
ib2
ic2
C C C
vCa
vCb
vCc
iCa iCb iCc
(b) Equivalent circuit
Fig. 2.5 Three-phase LCL-type grid-connected inverter
2.1 PWM for Single-Phase Full-Bridge Grid-Connected Inverter 37
H
Realce
H
Realce
current and voltage stresses in delta-connection are 1=
ffiffiffi
3
p
and
ffiffiffi
3
p
times of that in
star-connection, respectively. In this book, star-connection is adopted. Similarly, Vin
is split into two ones equally for convenience of illustration, and the midpoint O is
defined as the base potential.
Figure 2.5b shows the equivalent circuit of the three-phase grid-connected
inverter, where vao, vbo, and vco are the three inverter bridge output voltages with
respect to midpoint O; i1x (x = a, b, c) is the inverter-side inductor current; vCx and
iCx are the filter capacitor voltage and current, respectively; i2x is the grid-side
inductor current. From Fig. 2.5b, vao, vbo and vco can be expressed as
vao ¼ jxL1 � i1a þ vCa þ vNO
vbo ¼ jxL1 � i1b þ vCb þ vNO
vco ¼ jxL1 � i1c þ vCc þ vNO
8><
>: ð2:8Þ
where vNO is the voltage across points N and O.
The three-phase filter capacitor voltages can be expressed as
vCa ¼ iCa= jxCð Þ
vCb ¼ iCb= jxCð Þ
vCc ¼ iCc= jxCð Þ
8><
>: ð2:9Þ
For three-phase three-wire system, i1a + i1b + i1c = 0, iCa + iCb + iCc = 0.
According to (2.8), the zero sequence component vNO is derived as
vNO ¼ vao þ vbo þ vcoð Þ=3 ð2:10Þ
Similarly, vNN 0 , the voltage across points N and N′, can be obtained, expressed as
vNN 0 ¼ vga þ vgb þ vgc
� �
=3 ð2:11Þ
With PWM control, vao + vbo + vco 6¼ 0. So, according to (2.10), vNO is not
equal to zero, which means that the potentials of N and O are not equal. When the
three-phase grid voltages are balance, i.e., vga + vgb + vgc = 0, the potentials of
N and N′ are equal according to (2.11).
2.2.1 SPWM
Figure 2.6 shows the key waveforms of SPWM for three-phase grid-connected
inverter, where vtri is the triangular carrier, and vMa, vMb, and vMc are the three-
phase sinusoidal modulation signals, expressed as
38 2 Design of LCL Filter
H
Realce
H
Realce
vMa ¼ VM � sinxot
vMb ¼ VM � sin xot � 2p=3ð Þ
vMc ¼ VM � sin xotþ 2p=3ð Þ
8><
>: ð2:12Þ
where VM is the amplitude of the modulation signals, xo is the angular frequency of
the modulation signals, which is equal to the grid angular frequency.
Obviously, the control signals for Q1 and Q4 are determined by comparing vMa
and vtri, the control signals for Q3 and Q6 are determined by comparing vMb and vtri,
and the control signals for Q5 and Q2 are determined by comparing vMc and vtri.
Thus, the voltages of the midpoints of three-phase legs with respect to O, vao, vbo,
and vco, are obtained. vNO can be determined according to (2.10). The output phase
voltage vaN is equal to vao − vNo, and the output line voltage vab is equal to
vao − vbo.
Vtri
−Vtri
0
Vin /2
t
−Vin /2
0
vMa vtri
vao
Vin
−Vin
0
vab
Vin /2
−Vin /2
0 t
t
vbo
vMb vMc
Vin /2
−Vin /2
0 t
t
t
t
vco
Vin /2
−Vin /2
0
vNO
vaN
−2Vin /3
−Vin/3
Vin /3
2Vin /3
0
Fig. 2.6 SPWM for three-phase LCL-type grid-connected inverter
2.2 PWM for Three-Phase Grid-Connected Inverter 39
According to the modulation scheme, the expression of vao is the same as (2.2).
Since vMb lags vMa with a phase of 2p/3 and vMc leads vMa with a phase of 2p/3, by
replacing xot in (2.2) with xot − 2p/3 and xot + 2p/3, respectively, the expres-
sions of vbo and vco can be obtained as
vbo tð Þ ¼ MrVin2 sin xot �
2p
3
� �
þ 2Vin
p
X1
m¼1;3;...
X�1
n¼0;�2;...
Jn mMrp=2ð Þ
m
sin
mp
2
cos mxswtþ n xot � 2p3
� �� �
þ 2Vin
p
X1
m¼2;4;...
X�1
n¼�1;�3;...
Jn mMrp=2ð Þ
m
cos
mp
2
sin mxswtþ n xot � 2p3
� �� �
ð2:13Þ
vco tð Þ ¼ MrVin2 sin xotþ
2p
3
� �
þ 2Vin
p
X1
m¼1;3;...
X�1
n¼0;�2;...
Jn mMrp=2ð Þ
m
sin
mp
2
cos mxswtþ n xotþ 2p3
� �� �
þ 2Vin
p
X1
m¼2;4;...
X�1
n¼�1;�3;...
Jn mMrp=2ð Þ
m
cos
mp
2
sin mxswtþ n xotþ 2p3
� �� �
ð2:14Þ
Substituting (2.2), (2.13), and (2.14) into (2.10) yields
vNO tð Þ ¼ 2Vin3p
X1
m¼1;3;...
X�1
n¼0;�2;...
Jn mMrp=2ð Þ
m
1þ 2 cos 2np
3
� �
sin
mp
2
cos mxswtþ nxotð Þ
þ 2Vin
3p
X1
m¼2;4;...
X�1
n¼�1;�3;...
Jn mMrp=2ð Þ
m
1þ 2 cos 2np
3
� �
cos
mp
2
sin mxswtþ nxotð Þ
ð2:15Þ
According to (2.2) and (2.15), the output phase voltage vaN is obtained, which is
vaN tð Þ ¼ vao tð Þ � vNO tð Þ
¼ MrVin
2
sinxotþ 2Vinp
X1
m¼1;3;5;...
X�1
n¼0;�2;�4;...
4
3
Jn mMrp=2ð Þ
m
sin
mp
2
sin2
np
3
cos mxswtþ nxotð Þ
þ 2Vin
p
X1
m¼2;4;6;...
X�1
n¼�1;�3;...
4
3
Jn mMrp=2ð Þ
m
cos
mp
2
sin2
np
3
sin mxswtþ nxotð Þ
ð2:16Þ
40 2 Design of LCL Filter
As seen in (2.16), for the harmonics in vaN at around odd times (m = 1, 3, 5, …)
of carrier frequency, when n = 6k (k is an integer), sin2(np/3) = 0; when
n = 6k ± 2, sin2(np/3) = 3/4. Similarly, for the harmonics in vaN at around even
times (m = 2, 4, 6, …) of carrier frequency, when n = 3(2k − 1), sin2(np/3) = 0;
when n = 6k ± 1, sin2(np/3) = 3/4.
So, the harmonics spectrum of the output phase voltages of three-phase inverter
controlled by SPWM can be described as
(1) The harmonics in the output phase voltages vxN (x = a, b, c)only distribute at
frequencies where m + n is odd. When m is odd, the harmonics only distribute
at the sideband frequencies where n = 6k ± 2 (k is an integer); when m is even,
the harmonics only distribute at the sideband frequencies where n = 6k ± 1.
(2) The harmonics in the output phase voltages vxN at around the carrier frequency
(n = ±2, ±4,…) are the dominant harmonics, which is the major consideration
of filter design.
According to (2.2) and (2.13), the output line voltage vab can be obtained,
expressed as
vab tð Þ ¼ vao tð Þ � vbo tð Þ
¼
ffiffiffi
3
p
MrVin
2
sin xotþ p6
� �
þ 2Vin
p
X1
m¼1;3;5;...
X�1
n¼0;�2;�4;...
Jn mMrp=2ð Þ
m
sin
mp
2
2 sin
np
3
cos mxswtþ nxotþ p2 �
np
3
� �
þ 2Vin
p
X1
m¼2;4;6;...
X�1
n¼�1;�3;...
Jn mMrp=2ð Þ
m
cos
mp
2
2 sin
np
3
sin mxswtþ nxotþ p2 �
np
3
� �
ð2:17Þ
By comparing (2.16) and (2.17), it can be observed that: (1) at the fundamental
frequency, the amplitude of line voltage is
ffiffiffi
3
p
times of that of the phase voltage,
and the line voltage leads to the phase voltage with a phase of p/6; (2) The har-
monics of the output phase and line voltages vaN and vab distribute at the same
sideband frequencies, and the amplitudes of harmonics in line voltages are also
ffiffiffi
3
p
times of that of the harmonics in phase voltages, and it leads to the harmonics in the
corresponding phase voltages with a phase of p/2 − np/3.
2.2.2 Harmonic Injection SPWM Control
According to (2.17), when 0 � Mr � 1, the maximum amplitude of output line
voltage vab is only
ffiffiffi
3
p
Vin=2, i.e., 0.866Vin. It means that the dc voltage utilization of
the three-phase inverter controlled by SPWM is only 0.866. However, according to
(2.3) and (2.7), the dc voltage utilization of a single-phase full-bridge inverter is 1.
2.2 PWM for Three-Phase Grid-Connected Inverter 41
To make the dc voltage utilization of three-phase inverter attain 1, a third har-
monic component vz as shown in Fig. 2.7 is injected to the three-phase sinusoidal
modulation signals. It can be observed that the peak of vMa and the valley of vz
appear at the same time. As a result, the peak of the modulation signal vMaz, which
is the sum of vMa and vz, distributes not at but on both sides of the peak of vMa.
When the amplitude of vMaz is equal to that of vtri, the real amplitude of vMa will be
larger than that of vtri. Define the modulation ratio of three-phase inverter is still the
ratio of the amplitudes of vMa and vtri, then according to (2.1), the modulation ratio
larger than 1 will be obtained.
Further study shows that when the amplitude of the injected third harmonic
component vz is one-sixth of that of modulation sinusoidal signal vMa [1], i.e.,
Vtri
−Vtri
0
Vin /2
t
−Vin /2
0 t
vMa vtri
vao
Vin
−Vin
0
t
vab
Vin /2
−Vin /2
0 t
vbo
vMb
z
vMcz
Vin /2
−Vin /2
0 t
vco
t
Vin /2
−Vin /2
0
vNO
t
vaN
−2Vin /3
−Vin /3
Vin /3
2Vin /3
0
vMaz
vz
Fig. 2.7 Third harmonic injection SPWM for three-phase LCL-type grid-connected inverter
42 2 Design of LCL Filter
H
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H
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vz ¼ VM6 � sin 3xot ð2:18Þ
the dc voltage utilization of the three-phase inverter attains 1. A brief proof is
presented as follows.
According to (2.12) and (2.18), the modulation signal vMaz is as follows:
vMaz ¼ VM � sinxotþ VM6 � sin 3xot ¼
3VM
2
� sinxot � 2VM3 sin
3 xot ð2:19Þ
According to (2.19), it can be derived that the peak of vMaz locates at xot = p/3
or 2p/3. If the amplitude of vMaz is set to equal to that of vtri, VM/Vtri can reach 1.15,
which indicates that the modulation ratio of the third harmonic injection SPWM can
reach 1.15. When Mr = 1.15, according to (2.17), the amplitude of line voltage can
attain Vin, which is the same as that of the single-phase full-bridge inverter. In other
words, the dc voltage utilization attains 1.
From the Fourier transform theory, the expansions of vao and vbo in Fig. 2.7 can
be obtained, which are
vao tð Þ ¼ MrVin2 sinxotþ
MrVin
12
sin 3xotþ
X1
m¼1;2;3;...
X�1
n¼0;�1;�2;...
Amn cos mxswtþ nxotð Þ
ð2:20Þ
vbo tð Þ ¼ MrVin2 sin xot �
2p
3
� �
þ MrVin
12
sin 3xot
þ
X1
m¼1;2;3;...
X�1
n¼0;�1;�2;...
Amn cos mxswtþ n xot � 2p3
� �� � ð2:21Þ
where Amn is the amplitude of harmonics, expressed as [1]
Amn ¼ 2Vinmp
J0 mMrp=12ð ÞJk mMrp=2ð Þ sin mþ kð Þp=2½ �jk¼ nj j
þ J0 mMrp=2ð ÞJh mMrp=12ð Þ sin mþ hð Þp=2½ �j3h¼ nj j
þP Jk mMrp=2ð ÞJh mMrp=12ð Þ sin mþ kþ hð Þp=2½ �jkþ 3h¼ nj j
þP Jk mMrp=2ð ÞJh mMrp=12ð Þ sin mþ kþ hð Þp=2½ �jk�3h¼ nj j
þP Jk mMrp=2ð ÞJh mMrp=12ð Þ sin mþ kþ hð Þp=2½ �j3h�k¼ nj j
2
666666664
3
777777775
ð2:22Þ
2.2 PWM for Three-Phase Grid-Connected Inverter 43
H
Realce
H
Realce
Same as the derivation of output phase voltage vaN with SPWM in Sect. 2.2.1,
the expression of vaN with the third harmonic injection SPWM can be derived,
expressed as
vaN tð Þ ¼ MrVin2 sinxotþ
X1
m¼1;2;3;...
X�1
n¼0;�1;�2;...
4
3
sin2
np
3
� Amn cos mxswtþ nxotð Þ
ð2:23Þ
By comparing (2.16) and (2.23), the harmonics spectrum of the output phase
voltages of three-phase inverter with the third harmonic injection SPWM can be
concluded as follows:
(1) The harmonics in the output phase voltages vxN (x = a, b, c) only distribute at
the frequencies where m + n is odd. When m is odd, the harmonics only
distribute at even sideband frequencies where n = 6k ± 2 (k is an integer);
when m is even, the harmonics only distribute at odd sideband frequencies
where n = 6k ± 1.
(2) The harmonics in vxN at around the carrier frequency (n = ±2, ±4, …) are the
dominant harmonics, which is the major consideration of filter design.
According to (2.20) and (2.21), the output line voltage vab can be obtained as
vab tð Þ ¼ vao tð Þ � vbo tð Þ
¼
ffiffiffi
3
p
MrVin
2
sin xotþ p6
� �
þ
X1
m¼1;2;3;...
X�1
n¼0;�1;�2;...
2 sin
np
3
� Amn cos mxswtþ nxotþ p2 �
np
3
� �
ð2:24Þ
Besides (2.18), the harmonic vz injected to the modulation sinusoidal signal can
be generated from the envelope magnitude of vMa, vMb, and vMc [1], which means
that the maximum magnitude of |vMa|, |vMb| and |vMc| is selected, as shown in
Fig. 2.8. In detail, within xot 2 [0, p/6) [ [5p/6, 7p/6) [ [11p/6, 2p), |vMa| is the
largest one, so vz is extracted from vMa; Likewise, within xot 2 [p/6, p/2) [ [7p/6,
3p/2), |vMb| is the largest one, then vz is extracted from vMc; Within xot 2 [p/2, 5p/
6) [ [3p/2, 11p/6), |vMc| is the largest one, and vz is extracted from vMb. Due to
vMa + vMb + vMc = 0, vz can be expressed as follows
vz ¼ �k max vMa; vMb; vMcf gþmin vMa; vMb; vMcf gð Þ ð2:25Þ
It also can be proved that the peak of the modulation signal vMaz locates at
xot = p/3 or 2p/3. When k in (2.25) equals to 0.5, the dc voltage utilization can also
44 2 Design of LCL Filter
H
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attain 1. The result of harmonic injection SPWM shown in Fig. 2.8 is equivalent to
the space vector modulation (SVM) [1]. Since the zero sequence component
extracts directly from the modulation sinusoidal signals, the realization of the
three-phase modulation signals shown in (2.25) is simple and widely used.
The amplitude Amn of output harmonics voltage controlled by the harmonic
injection SPWM shown in Fig. 2.8 is expressed as [2]
Vtri
−Vtri
0
Vin /2
t
−Vin /2
0 t
vMa vtri
vao
Vin
−Vin
0 t
vab
Vin /2
−Vin/2
0 t
vbo
vMbz
z
vMcz
Vin /2
−Vin /2
0 t
vco
t
Vin /2
−Vin /2
0
vNO
t
vaN
−2Vin /3
−Vin /3
Vin /3
2Vin /3
0
vMaz
vz
Fig. 2.8 Harmonic injection SPWM for three-phase LCL-type grid-connected inverter that is
equivalent to SVM
2.2 PWM for Three-Phase Grid-Connected Inverter 45
Amn ¼ 4Vinmp2
p
6
sin
mþ nð Þp
2
Jn
3mMrp
4
� �
þ 2 cos np
6
Jn
ffiffiffi
3
p
mMrp
4
� �	 
þ 1
n
sin
mp
2
cos
np
2
sin
np
6
J0
3mMrp
4
� �
� J0
ffiffiffi
3
p
mMrp
4
� �	 
����
n 6¼0
þ
X1
k¼1
k 6¼�n
1
nþ k sin
mþ kð Þp
2
cos
nþ kð Þp
2
sin
nþ kð Þp
6
:
Jk
3mMrp
4
� �
þ 2 cos 2nþ 3kð Þp
6
Jk
ffiffiffi
3
p
mMrp
4
� �	 
8>>>><
>>>>:
9>>>>=
>>>>;
þ P1
k¼1
k 6¼n
1
n� k sin
mþ kð Þp
2
cos
n� kð Þp
2
sin
n� kð Þp
6
:
Jk
3mMrp4
� �
þ 2 cos 2n� 3kð Þp
6
Jk
ffiffiffi
3
p
mMrp
4
� �	 
8>>>><
>>>>:
9>>>>=
>>>>;
2
66666666666666666666666664
3
77777777777777777777777775
ð2:26Þ
2.3 LCL Filter Design
The PWM output voltage of the grid-connected inverters contains abundant of
switching harmonic components, which results in the harmonic current injecting
into the grid. Therefore, a filter is required to interface between the inverter bridge
and the power grid. The LCL filter is usually employed since it has better ability of
suppressing high frequency harmonics than the L filter. This section will focus on
the design of the LCL filter.
The single-phase full-bridge inverter, as shown in Fig. 2.1, could be simplified to
the equivalent circuit as shown in Fig. 2.9a. Likewise, when the three-phase grid
voltages are balanced, the voltage potentials of node N and N′ are identical. As a
result, the three-phase circuit, as shown in Fig. 2.5b, can be decoupled and each
phase could be simplified to the equivalent circuit as shown in Fig. 2.9b, where
x = a, b, c. As seen, the structures of the equivalent circuits of the single-phase and
three-phase LCL filters are the same, so the design procedures of them are almost
uniform, except that the harmonic spectrum of the imposed PWM voltages are
different. In the following, the grid voltage vg is assumed a pure sinusoidal waveform.
C vC
L2
i2iC
L1
i1
vg
+
–
vinv
+
–
+
–
C vCx
L2
i2xiCx
L1
i1x
vgx
+
–
vxN
+
–
+
–
(a) Single-phase (b) Three-phase
Fig. 2.9 Equivalent circuits of single-phase and three-phase LCL-type grid-connected inverters
46 2 Design of LCL Filter
H
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H
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2.3.1 Design of the Inverter-Side Inductor
From Figs. 2.1 and 2.5a, it can be observed that the current flowing through the
filter inductor L1 and the switches are the same. The larger the inductor current
ripple is, the larger the inductor losses and higher current stress of the switches are.
As a result, the conduction and switching losses will increase. Thus, the inductor
current ripple should be limited.
2.3.1.1 Single-Phase Full Bridge Grid-Connected Inverter
1. Bipolar SPWM
Figure 2.10a gives the key waveforms of the single-phase full-bridge inverter
with bipolar SPWM, where i1_f is the fundamental component in the inverter-side
inductor current, and Tsw is the carrier period.
When vM > vtri, switches Q1 and Q4 turn on simultaneously, and the bridge
output voltage vinv = Vin. The voltage applied on inductor L1 is
L1
di1
dt
¼ Vin � vC ð2:27Þ
where vc is the filter capacitor voltage. Within one carrier period, vC can be regarded
to be constant, and Vin > vC. So, the inductor current i1 increases linearly, and the
increment is
T+
Vtri
−Vtri
0 t
vM
vtri
Tsw
S
t
t1t'1 t2t'2
0
Vin
−Vin
vinv
t3
t
i1_f
i1
0
Vtri
−Vtri
0
t
vM
vtri
Tsw/2
T+
−vtri
S
vinv
t1t'1 t2t'2 t3
t
0
Vin
−Vin
t
i1_ f
i1
0
(a) Bipolar SPWM (b) Unipolar SPWM
Fig. 2.10 Key waveforms of single-phase full bridge inverter
2.3 LCL Filter Design 47
H
Realce
H
Realce
Di1 þð Þ ¼ Vin � vCL1 � T þð Þ ð2:28Þ
where T(+) = t12 is the time interval when Q1 and Q4 conduct simultaneously.
When vM < vtri, Q2 and Q3 turn on simultaneously, and vinv = −Vin. The voltage
across inductor L1 is
L1
di1
dt
¼ �Vin � vC ð2:29Þ
Similarly, il decreases linearly, and the decrement is
Di1 �ð Þ ¼ Vin þ vCL1 � T �ð Þ ð2:30Þ
where T(−) = t23 is the time interval when Q2 and Q3 conduct simultaneously.
The equation for solving the intersection points of vM and vtri is transcendental,
so regular sampling SPWM is usually used to calculate T+. In detail, a horizontal
line is drawn across point S, as shown in Fig. 2.10, and it would intersect the
triangle carrier at t01 and t
0
2. Considering the fundamental frequency is much lower
than the carrier frequency, it is reasonable to have T(+) = t12 � t012. Then, T(+) can
be calculated, which is
T þð Þ ¼ vM þVtri2Vtri Tsw ¼
1
2
Tsw Mr sinxotþ 1ð Þ ð2:31Þ
Likewise, T(−) can be expressed as
T �ð Þ ¼ Tsw � T þð Þ ¼ 12 Tsw 1�Mr sinxotð Þ ð2:32Þ
Generally, the fundamental component in the voltage across inductors L1 and L2
are small, so the filter capacitor voltage vC can be approximated to the grid voltage
vg and it equals to the fundamental component of the bridge output voltage vinv, i.e.,
vC � vg ¼ MrVin sinxot ð2:33Þ
Substituting (2.32) and (2.33) into (2.28) and (2.30), respectively, Di1(+) and Di1
(−) can be derived as
Di1 þð Þ ¼ Di1 �ð Þ ¼ VinTsw2L1 1�M
2
r sin
2 xot
� � ð2:34Þ
As seen in (2.34), either the maximum increment or decrement of the current of
inductor L1 (denoted as Di1_max) within a carrier period appears at sinxot = 0, i.e.,
Di1_max = VinTsw/(2L1). Defining the ripple coefficient as kc_L1 = Di1_max/I1, where
48 2 Design of LCL Filter
H
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H
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H
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H
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H
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I1 is the rated RMS value of the fundamental component of i1, the minimum
inductance of L1 can be obtained as
L1 min ¼ VinTsw2kc L1I1 ð2:35Þ
In practice, kc_L1 is set to be 20–30% [2].
The maximum value of L1 could be determined from the fundamental voltage of
L1, which is defined as vL1_f. The smaller vL1_f is, the lower the dc-link voltage is
required. Defining the ratio of RMS values of vL1_f and vC as kv_L1, the maximum
value of L1 can be obtained, which is
L1 max ¼ kv L1VCxoI1 �
kv L1Vg
xoI1
ð2:36Þ
where Vg is the RMS value of the grid voltage, and kv_L1 is usually set to be about
5%.
2. Unipolar SPWM
Figure 2.10b gives the key waveforms of the single-phase full-bridge inverter
with unipolar SPWM. When vM > vtri and vM > − vtri, switches Q1 and Q4 turn on
simultaneously, and vinv = Vin. As a result, i1 increases linearly. From Fig. 2.10b,
the ratio of T(+) and Tsw/2 can be obtained, which is
T þð Þ
Tsw=2
¼ vM
Vtri
¼ Mr sinxot ð2:37Þ
Substituting (2.33) and (2.37) into (2.28), the increment Di1(+) can be derived as
Di1 þð Þ ¼
VinTsw
2L1
1�Mr sinxotð ÞMr sinxot ð2:38Þ
Similarly, the decrement Di1(−) when both Q2 and Q3 turn on can be calculated,
which is the same as (2.38).
As seen in (2.38), the maximum increment and decrement of i1 appear when
sinxot = 1/(2Mr), and Di1_max = VinTsw/(8L1). Then, the minimum of L1 with
unipolar SPWM is
L1 ¼ VinTsw8kc L1I1 ð2:39Þ
By Comparing of (2.35) and (2.39), it can be seen that the required L1 with
unipolar SPWM is only one-fourth of that with bipolar SPWM when that the
permitted maximum increment (or decrement) of inductor current are identical. The
reasons are: (1) the equivalent carrier frequency with unipolar SPWM is twice that
with bipolar SPWM; (2) the bridge output voltage vinv switches between Vin and
2.3 LCL Filter Design 49
H
Realce
H
Realce
H
Realce
H
Realce
H
Realce
H
Realce
H
Realce
−Vin when bipolar SPWM is used, while it is switched between Vin and 0, or 0 and
−Vin when unipolar SPWM is used.
2.3.1.2 Three-Phase Grid-Connected Inverter
Similar to the single-phase grid-connected inverter, the inverter-side inductor L1 of
the three-phase grid-connected inverter is also determined by the maximum current
ripple. The fundamental voltage of L1 is also ignored here, and the filter capacitor
voltage vCx is approximated to the fundamental voltage of the inverter bridge output
voltage vxN, i.e., vCa � (MrVin/2)sinxot. However, differed from the single-phase
full-bridge inverter, the three-phase inverter bridge output voltage vxN can output five
levels, i.e., 0, ±Vin/3, and ±2Vin/3. As a result, the current ripple of i1x (x = a, b, c) is
more complex. In the following, a detailed analysis about the current ripple of i1x will
be presented. Since the voltages and currents are periodic, only the key waveforms in
a quarter of one cycle, i.e., xot 2 [0, p/2] is given, as shown in Fig. 2.11.
i1a
t
t
t0 t1 t2 t3 t4 t5
0
Vtri
−Vtri
0
t6
vtri vMc
vMa
vMb
t
vaN
Vin /3
Vin /3
0
i1a
t
t
t0 t1 t2 t3 t4 t5 t6
vaN
0
Vtri
−Vtri
0
t
2Vin /3
Vin /3
0
vtri vMa
vMc
vMb
(a) ωo t ∈ [0, /6] (b) ωo t ∈ ( /6, φ]
i1a
t
t
t0 t1 t2 t3 t4 t5 t6
vaN
0
Vtri−Vtri
0
t
2Vin /3
Vin /3
0
vtri vMa
vMc
vMb
(c) ωo t ∈ (φ, /2]
−
Fig. 2.11 Inverter-side inductor current of three-phase inverter
50 2 Design of LCL Filter
From Figs. 2.6, 2.7 and 2.8, it can be observed that no matter SPWM or har-
monic injection SPWM is used, the three-phase filter capacitor voltages satisfy the
relation vMc > vMa > vMb within xot 2 [0, p/6]. Moreover, vMa increases mono-
tonously and reaches its maximum value at xot = p/6. Since vCx is proportional to
vMx in the linear modulation region, vCc > vCa > vCb is also true, and vCa increases
monotonously and reaches its maximum value at xot = p/6. Thus, the maximum
value of vCa equals to (MrVin/2)sin(p/6) = MrVin/4. When SPWM or harmonic
injection SPWM is used, the maximum values of vCa are Vin/4 and 1.15Vin/4,
respectively. Obviously, vCa < Vin/3 is always true within xot 2 [0, p/6]. When
xot 2 [0, p/6], i1a can be divided into six sections in one carrier period, i.e., [t0, t6],
as shown in Fig. 2.11a, and three cases can be found in the six sections.
Case 1: when t 2 [t0, t1) [ [t2, t3), vaN = Vin/3. Since vCa < Vin/3, i1a increases
linearly;
Case 2: when t 2 [t1, t2) [ [t4, t5), vaN = 0. Since vCa > 0, i1a decreases linearly;
Case 3: when t 2 [t3, t4) [ [t5, t6), vaN = −Vin/3. Since vCa < Vin/3, i1a decreases
linearly.
When xot 2 [p/6, p/2], vCa > vCc > vCb is true, and vCa increases monotonously
and reaches its maximum value at xot = p/2. The maximum value of vCa isMrVin/2.
When SPWM or harmonic injection SPWM is used, the maximum values of vCa are
Vin/2 and 1.15Vin/2, respectively. Obviously, vCa < 2Vin/3 is always true within
xot 2 (p/6, p/2]. Similarly, when xot 2 (p/6, p/2], i1a can also be divided into six
sections in one carrier period, i.e., [t0, t6], as shown in Fig. 2.11b, c, and three cases
can also be found in the six sections.
Case 1: when t 2 [t0, t1) [ [t4, t5), vaN = 2Vin/3. Since vCa < 2Vin/3, i1a increases
linearly;
Case 2: when t 2 [t1, t2) [ [t3, t4), vaN = Vin/3. If vCa < Vin/3, i1a increases lin-
early, as shown in Fig. 2.11b. If vCa > Vin/3, i1a decreases linearly, as shown in
Fig. 2.11c;
Case 3: when t 2 [t2, t3) [ [t5, t6), vaN = 0. Since vCa > 0, i1a decreases linearly.
Defining xot when vCa = Vin/3 as /, yields
MrVin
2
sin/ ¼ Vin
3
ð2:40Þ
Then, / can be calculated as
/ ¼ arcsin 2
3Mr
� �
ð2:41Þ
According to (2.41), it can be obtained that only when Mr � 2/3, vCa will be
possible to be larger than Vin/3, thus the case shown in Fig. 2.11c appears; and
when Mr < 2/3, vCa will be never larger than Vin/3, thus the case shown in
Fig. 2.11c does not appear.
2.3 LCL Filter Design 51
As seen from Fig. 2.11a, i1a continues decreasing within [t3, t6]. As seen from
Fig. 2.11b, i1a continues increasing within [t0, t2] or [t3, t5], and decreases within
[t2, t3] or [t5, t6]. As seen from Fig. 2.11c, i1a increases within [t0, t1] or [t4, t5] and
continues decreasing within [t1, t4] or [t5, t6]. As mentioned above, the maximum
increment and decrement of the inverter-side inductor current is identical. In the
following, only the decrements of i1a within [t3, t6] shown in Fig. 2.11a, within [t2,
t3] or [t5, t6] shown in Fig. 2.11b, and within [t1, t4] or [t5, t6] shown in Fig. 2.11c,
will be derived. Based on these decrements, the lower limit of the inverter-side
inductor can be obtained.
According to Fig. 2.11a, the decrement of i1a within [t3, t6] can be expressed as
Di1a 1ð Þ ¼ �Vin=3� vCaL1 t34 þ
0� vCa
L1
t45 þ �Vin=3� vCaL1 t56
����
����
¼ Vin
3L1
t36 � t45ð Þþ vCaL1 t36
����
���� ð2:42Þ
According to Fig. 2.11b, the decrements of i1a within [t2, t3] and [t5, t6] can be,
respectively, expressed as
Di1a 2ð Þ ¼ vCaL1 t23
����
���� ð2:43Þ
Di1a 3ð Þ ¼
vCa
L1
t56
����
���� ð2:44Þ
According to Fig. 2.11c, the decrement of i1a within [t1, t4] can be expressed as
Di1a 4ð Þ ¼ Vin=3� vCaL1 t12 þ
0� vCa
L1
t23 þ Vin=3� vCaL1 t34
����
����
¼ Vin
3L1
t14 � t23ð Þ � vCaL1 t14
����
���� ð2:45Þ
And the expression of the decrement of i1a within [t5, t6] is the same as (2.44).
If the SPWM is used, the following relations can be obtained from Fig. 2.11.
t36 ¼ Tsw � Vtri � vMað Þ=2Vtri
t45 ¼ Tsw � Vtri � vMcð Þ=2Vtri
(
ð2:46Þ
t23 ¼ Tsw � Vtri þ vMbð Þ=2Vtri
t56 ¼ Tsw � Vtri � vMað Þ=2Vtri
(
ð2:47Þ
t14 ¼ Tsw � Vtri þ vMcð Þ=2Vtri
t23 ¼ Tsw � Vtri þ vMbð Þ=2Vtri
(
ð2:48Þ
52 2 Design of LCL Filter
If the harmonic injection SPWM is used, vMa, vMb, and vMc in (2.46)–(2.48)
should be replaced by vMaz, vMbz, and vMcz, respectively.
When the SPWM is used, vMa, vMb, and vMc given in (2.12) andMr = VM/Vtri are
substituted into (2.46), t36 and t45 can be calculated. Then, by substituting t36, t45,
and vCa � (MrVin/2)sinxot into (2.42), Di1a(1) will be obtained. On the base of
MrVinTsw/(2L1), the normalized Di1a(1) is finally expressed as
Di1 SPWM xotð Þ,
Di1a 1ð Þ
MrVinTsw= 2L1ð Þ ¼
1
6
sinxotþ 13 sin xotþ
2p
3
� �
�Mr
2
sin2 xot
����
����
ð2:49Þ
Similarly, according to (2.12), (2.43)–(2.45), (2.47) and (2.48), the normalized
Di1a(2), Di1a(3), and Di1a(4) can be derived as
Di2 SPWM xotð Þ,
Di1a 2ð Þ
MrVinTsw= 2L1ð Þ ¼ sinxot
1
2
þ Mr
2
sin xot � 2p3
� �	 
����
���� ð2:50Þ
Di3 SPWM xotð Þ,
Di1a 3ð Þ
MrVinTsw= 2L1ð Þ ¼ sinxot
1
2
�Mr
2
sinxot
� �����
���� ð2:51Þ
Di4 SPWM xotð Þ,
Di1a 4ð Þ
MrVinTsw= 2L1ð Þ
¼ 2
3
sin xotþ 2p3
� �
� 1
6
sinxot �Mr2 sinxot sin xotþ
2p
3
� �����
����
ð2:52Þ
Same as the above calculation procedure for the SPWM, when the harmonic
injection SPWM is used, the normalized Di1a(1), Di1a(2), Di1a(3), and Di1a(4) can be
derived, expressed as
Di1 HI-SPWM xotð Þ,
Di1a 1ð Þ
MrVinTsw= 2L1ð Þ ¼
1
6
sinxotþ 13 sin xotþ
2p
3
� �
� 3Mr
4
sin2 xot
����
����
ð2:53Þ
Di2 HI-SPWM xotð Þ,
Di1a 2ð Þ
MrVinTsw= 2L1ð Þ
¼ sinxot 12 þ
Mr
2
sin xot � 2p3
� �
þ Mr
4
sin xotþ 2p3
� �	 
����
����
ð2:54Þ
2.3 LCL Filter Design 53
Di3 HI-SPWM xotð Þ,
Di1a 3ð Þ
MrVinTsw= 2L1ð Þ
¼ sinxot 12�
Mr
2
sinxot �Mr4 sin xotþ
2p
3
� �	 
����
����
ð2:55Þ
Di4 HI-SPWM xotð Þ,
Di1a 4ð Þ
MrVinTsw= 2L1ð Þ
¼ 2
3
sin xotþ 2p3
� �
� 1
6
sinxot � 3Mr4 sinxot sin xotþ
2p
3
� �����
����
ð2:56Þ
Note that the harmonic injection SPWM is equivalent to SVM, the discussion of
the inverter-side inductor current ripple with SVM is not repeated here.
Since sinxot = −sin(xot − 2p/3) − sin(xot + 2p/3), by substituting it into
(2.55), it is easy to find that Δi2_HI-PWM = Δi3_HI-PWM.
According to (2.49)–(2.52), the curves of Δi1_SPWM(xot), Δi2_SPWM(xot),
Δi3_SPWM(xot), and Δi4_SPWM (xot) are depicted, as shown in Fig. 2.12a.
According to (2.53)–(2.56), the curves of Δi1_HI-SPWM(xot), Δi2_HI-SPWM(xot),
Δi3_HI-SPWM (xot), and Δi4_HI-SPWM(xot) are depicted, as shown in Fig. 2.12b.
From Fig. 2.12, the maximum value of the inverter-side inductor current ripple can
be obtained. Thus, when the current ripple coefficients kc_L1 are given, the lower
limits of L1 can be determined. In addition, the maximum value of L1 can also be
calculated from (2.36). According to the lower and upper limits of L1, the value of
L1 can be properly selected.
0
0.2
0.4
0 /6 /3 /2
o t
0.6
Mr = 0.6
Mr = 1
φ
Δi1_SPWM( ot)
Δi2_SPWM( ot)
Δi3_SPWM( ot)
Δi4_SPWM( ot)
Mr = 0.6
Mr = 1
0
0.2
0.4
0.6
0 /6 /3 /2
o t
Δi1_HI-SPWM( ot)
Δi3_HI-SPWM( ot)
Δi4_HI-SPWM( ot)
φ
(a) SPWM (b) Harmonic injection SPWM
Fig. 2.12 Curves of inverter-side inductor current ripple
54 2 Design of LCL Filter
2.3.2 Filter Capacitor Design
The filter capacitor will lead to reactive power. The larger the capacitance is, the
higher the reactive power is introduced, and also the larger the current flows
through inductor L1 and the power switches [3]. Thus, the conduction loss of the
switches will increase. Defining kC as the ratio of the reactive power introduced by
the filter capacitor to the rated output active power of the grid-connected inverter,
the maximum value of filter capacitor could be expressed as
C ¼ kC � PoxoV2g
ð2:57Þ
where Po is the rated output active power of single-phase full-bridge inverter or therated output active power of one phase for three-phase full-bridge inverter. In
practice, kC is usually recommended to be about 5% [4].
2.3.3 Grid-Side Inductor Design
According to Fig. 2.9a, the transfer function of the grid current i2 to the inverter
bridgeoutput voltage vinv can be obtained, which is
GLCL sð Þ, i2 sð Þvinv sð Þ ¼
1
L1L2Cs3 þ L1 þ L2ð Þs ¼
1
L1 þ L2ð Þs �
x2r
s2 þx2r
ð2:58Þ
where xr is the resonance angular frequency, which is
xr ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
L1 þ L2
L1L2C
r
ð2:59Þ
The expression of GLCL(s) for three-phase grid-connected inverter is the same as
(2.58).
After the inverter-side inductor and the filter capacitor are determined, the
grid-side inductor L2 could be designed according to the harmonic restriction
standards such as IEEE Std. 929-2000 and IEEE Std. 1547-2003 [5, 6]. Table 2.1
lists the current harmonic restriction, including the limits on individual harmonics
and the limit on the total harmonics distortion (THD) of the injected grid current. If
the specifications of the grid-connected inverter is given, the spectrum of vinv can be
calculated from (2.4) or (2.7), from which the angular frequency xh and amplitude
|vinv(jxh)| of the dominant harmonics can be obtained. Substituting the obtained xh
and |vinv(jxh)| into (2.59), yields
2.3 LCL Filter Design 55
i2 jxhð Þj j
vinv jxhð Þj j ¼
1
L1L2C jxhð Þ3 þ jxh L1 þ L2ð Þ
�� �� ð2:60Þ
According to the spectrum of the inverter bridge output voltage vinv, the angular
frequency xh and harmonic order h of the dominant harmonics can be determined.
Then, according to (2.60), Table 2.1, and the expected harmonics proportion kh, the
minimum value of L2 can be obtained, which is
L2 ¼ 1L1Cx2h � 1
� L1 þ Vinv jxhð Þj jxhkhI2
� �
ð2:61Þ
where Vinv(jxh) and I2 are the RMS value of the inverter bridge output voltage and
the rated injected grid current, respectively. If three-phase grid-connected inverter is
used, Vinv(jxh) in (2.60) and (2.61) is replaced by VaN(jxh).
After L1, C and L2 are determined, the simulation or experimental validations is
conducted to check whether the individual harmonics and the THD of the grid
current satisfy the restriction shown in Table 2.1 or not.
2.4 Design Examples for LCL Filter
To validate the above design methods, two prototypes are designed, where
single-phase full-bridge grid-connected inverter is controlled by the unipolar
SPWM, and three-phase grid-connected inverter is controlled by the harmonic
injection SPWM. The specifications of the single-phase full-bridge grid-connected
inverter are as follows: the dc input voltage is 360 V, the rated power is 6 kW, the
carrier frequency is 10 kHz, and the grid voltage is 220 V/50 Hz. The specifica-
tions of the three-phase grid-connected inverter are as follows: the dc input voltage
is 700 V, the rated power is 20 kW, the carrier frequency is 10 kHz, and the grid
voltage is 380 V/50 Hz.
Table 2.1 Maximum harmonics limits of grid current
Harmonic order h (odd
harmonic)a
h < 11 11 � h < 17 17 � h < 23 23 � h < 35 35 � h THD
Proportion to the rated
grid-connected current (%)
4.0 2.0 1.5 0.6 0.3 5.0
aThe allowable maximum limits of even harmonics is 25% of those of odd harmonics in the table
56 2 Design of LCL Filter
2.4.1 Single-Phase LCL Filter
Setting the inductor current ripple coefficient kc_L1 to 30%, and substituting the
corresponding parameters into (2.39), the minimum value of L1 is calculated as
550 lH. Defining the ratio of the RMS value of the fundamental voltage of L1 to
that of the capacitor voltage as kv_L1, and assuming kv_L1 = 5%, the maximum
value of L1 is calculated from (2.36), which is 1.28 mH. Finally, L1 = 600 lH is
chosen.
Setting kC = 3% and substituting Po = 6 kW, Vg = 220 V, and fo = 50 Hz into
(2.58), yields C < 12 lF. Here, C = 10 lF is chosen.
Assuming that the output power factor (PF) of the grid-connected inverter equals
to 1, the fundamental RMS value of iL1 could be calculated, i.e.,
I1 ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffi
I2C þ I22
p
¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
xoC � Vg
� �2 þ I22
q
= 27.28 (A). According to the dc input
voltage and the magnitude of grid voltage, the modulation ratio can be obtained,
which is Mr = 311/360 = 0.86. By substituting Mr = 0.86 into (2.7), the spectrum
of the bridge output voltage vinv can be depicted, as shown in Fig. 2.13. As seen,
the dominant harmonics locate at fh = 19.95 kHz and 20.05 kHz, and the corre-
sponding |Vinv(j2pfh)|/Vin = 28%. As long as these dominant harmonics in i2 are
attenuated to satisfy the aforementioned standards, the other harmonics in i2 will
naturally satisfy the standards. Since the orders of the dominant harmonics are
higher than 33, the required current harmonic proportion kh should be less than
0.3%. Setting kh = 0.2%, and substituting I1 = 27.28 A, |Vinv(j2pfh)|/Vin = 28%,
Vin = 360 V, L1 = 600 lH, C = 10 lF, xh = 2p 	 19,950, and kh = 0.2% into
(2.61) leads to L2 = 164 lH. Finally, L2 = 150 lH is selected. The final
single-phaseLCL filter parameters are listed in Table 2.2.
Figure 2.14 shows the simulation results. In Fig. 2.14a, the waveforms from top
to bottom are the inverter-side inductor current i1, the grid current i2, the capacitor
current iC and its fundamental component, respectively. In Fig. 2.14b, the wave-
forms from top to bottom are the inverter bridge output voltage vinv, the spectrums
10
30
20
0
100
v in
v/
V i
n
(%
)
f (kHz)
0.05
19.95
20.05
19.85
20.15
20.25
19.75
39.95
40.05
39.75
40. 25
39 . 85
40. 15
40 .35
39.65
Fig. 2.13 Calculated spectrum of vinv in single-phase LCL-type grid-connected inverter with
unipolar SPWM
2.4 Design Examples for LCL Filter 57
of vinv and i2, respectively. The maximum current ripple of i1 is 7.73 A, and the
RMS value of i1 is 27.28 A. As a result, Di1_max/I1 = 28%. As seen from
Fig. 2.14b, the maximum harmonic magnitude of vinv is 100 V, and it appears at
19.95 kHz. The magnitude of the harmonic is 27.8% of Vin, which is agreement
with the calculated results shown in Fig. 2.13. Through the LCL filter, the mag-
nitude of the current harmonic in i2 at 19.95 kHz is suppressed below 0.06 A,
which is 0.15% of the rated injected grid current; and the THD of i2 is 0.8%.
Clearly, both the single harmonic and THD satisfy the restriction standards, which
validate the effectiveness of the design procedure for single-phase LCL filter.
2.4.2 Three-Phase LCL Filter
According to (2.23), the modulation ratio can be obtained as
Mr ¼ 220
ffiffiffi
2
p
=350 ¼ 0:888. As observed from Fig. 2.12b, the maximum current
ripple of the inverter-side inductor appears at xot = 0. Setting the inductor current
ripple coefficient kc_L1 = 30%, and according to Eq. (2.49), the minimum value of
Table 2.2 Parameters of single-phase LCL-type full-bridge grid-connected inverter
Parameter Symbol Value Parameter Symbol Value
Input voltage Vin 360 V Switching frequency fsw 10 kHz
Grid voltage Vg 220 V Inverter-side inductor L1 600 lH
Output power Po 6 kW Filter capacitor C 10 lF
Fundamental frequency fo 50 Hz Grid-side inductor L2 150 lH
0
0
0
i1:[30 A/div]
i2:[30 A/div]
Time:[5 ms/div]
∆i1_max=7.73 A
I1m=40.77 A
I2m=38.83 A
I2=27.42 A
ICf =0.70 A, ICm=5.13 AiC:[5 A/div]
0
vinv:[400 V/div]
Freq.:[10 kHz/div]
Time:[5 ms/div]
50Hz
vinv:[50 V/div]
i2:[2.5 mA/div]
(a) i1, i2 and vg (b) spectrum of vinv and i2
Fig. 2.14 Simulation results of single-phase full-bridge grid-connected inverter
58 2 Design of LCL Filter
L1 is calculated as 988 lH. Assuming kv_L1 = 5%, the maximum value of L1 is
calculated from (2.36), which is 1.16 mH. So, L1 = 1 mH is selected.
Setting kC = 5% and substituting Po = 20/3 kW, Vg = 220 V, and fo = 50 Hz
into (2.58) yields C < 22 lF. Here, C = 20 lF is selected.
Assuming PF = 1, the fundamental RMS value of iL1 could be calculated as
I1 = 30.31 A. Substituting Mr = 0.888 into (2.26), the spectrum of the output phase
voltagevaN is depicted, as shown in Fig. 2.15. As seen, the dominant harmonics
locate at fh = 9.9 kHz and 10.1 kHz, where |VaN(j2pfh)|/Vin = 17.6%. Likewise, as
long as these dominant harmonics in i2 are attenuated to satisfy the aforementioned
standards, the other harmonics in i2 will naturally satisfy the standards. Since the
orders of these dominant harmonics are higher than 33, so the required kh should be
less than 0.3%. Here, setting kh = 0.15%, and substituting I1 = 30.31 A,
|Vinv(j2pfh)|/Vin = 17.6%, Vin = 360 V, L1 = 1 mH, C = 20 lF, xh = 2p 	 9900
and kh = 0.15% into (2.61), produces L2 = 301 lH. Finally, L2 = 300 lH is
selected. The final three-phase LCL filter parameters are listed in Table 2.3.
Figure 2.16 shows the simulation results with the prototype parameters of
Table 2.3. In Fig. 2.16a, the waveforms from top to bottom are the inverter-side
inductor current i1a, the injected grid current i2a, the capacitor current iCa and its
fundamental component, respectively. In Fig. 2.16b, the waveforms from top to
bottom are the output phase voltage vaN, the spectrums of vaN and i2a, respectively.
The maximum current ripple of i1a is 9.5 A, and the RMS value of i1 is 30.31 A. As
a result, Di1_max/I1 = 31.4%. As seen from Fig. 2.16b, the maximum harmonic
magnitude in vaN appears at 9.9 kHz and it is about 60 V, which is 17.1% of Vin/2
and in agreement with the calculated results shown in Fig. 2.15. Through the LCL
10
30
20
0
100
v a
N
/V
in
(%
)
f (kHz)
0.05
9.9
10.1
9.8
10.2
19.95
20 .05
19. 75
20. 25
Fig. 2.15 Calculated
spectrum of vaN when
harmonic injection SPWM is
used
Table 2.3 Parameters for three-phase LCL-type full-bridge grid-connected inverter
Parameter Symbol Value Parameter Symbol Value
Input voltage Vin 700 V Switching frequency fsw 10 kHz
Grid voltage Vgab 380 V Inverter-side inductor L1 1 mH
Output power Po 20 kW Filter capacitor C 20 lF
Fundamental wave frequency fo 50 Hz Grid-side inductor L2 300 lH
2.4 Design Examples for LCL Filter 59
filter, the current harmonic magnitude of i2 at 9.9 kHz is suppressed below 0.05 A,
which accounts for 0.13% of the rated injected grid current. The THD of i2 is 0.8%.
Both the single harmonics and THD satisfy the restriction standards, which validate
the design procedure for three-phase LCL filter.
2.5 Summary
In this chapter, the design procedure of LCL filter is presented. The Fourier series
expansions of the inverter bridge output voltage of single- and three-phase LCL-
type grid-connected inverter with different PWM schemes are derived for the
purpose of determining the dominant harmonics which needs to be suppressed. The
harmonic spectrum shows that for single-phase inverter, the dominant harmonics
with the bipolar SPWM distribute around the carrier frequency, whereas those with
the unipolar SPWM distribute around twice the carrier frequency. For the
three-phase inverter, the dominant harmonics with both the SPWM and the har-
monic injection SPWM distribute around the carrier frequency. Considering the
permitted current ripple of the inverter-side inductor, the allowable reactive power
introduced by the filter capacitor, and the maximum harmonic limit of the grid
current, the filter parameters can be determined. The design procedure for the LCL
filter is given as follows:
(1) By limiting the maximum inductor current ripple in one cycle and the funda-
mental voltage on the inductors, the lower and upper limits of the inverter-side
inductor is obtained, from which, a proper inverter-side inductor can be
selected.
0
0
0
∆ia1_max=9.5 A
ia1:[30 A/div]
Ia2=30.3 Aia2:[30 A/div]
Time:[5 ms/div]
iCa:[5 A/div] ICaf =1.38 A, ICam=6.63 A
0
vaN:[400 V/div]
Freq.:[10 kHz/div]
Time:[5 ms/div]
50Hz
vaN:[50 V/div]
i2a:[2.5 mA/div]
(a) i1a, i2a and vag (b) spectrum of vaN and i2a
Fig. 2.16 Simulation results of three-phase grid-connected inverter
60 2 Design of LCL Filter
(2) According to the maximum reactive power introduced by the filter capacitor,
the upper limit of the filter capacitor can be obtained.
(3) By limiting the single harmonic of the grid current in accord with the restriction
standards, the minimum value of the grid-side inductor can be determined, from
which, the proper grid-side inductor can be selected.
The LCL filter design procedure is verified by simulations.
References
1. Holmes, D.G., Lipo, T.A.: Pulse Width Modulation for Power Converters: Principles and
Practice. IEEE Press & Wiley, New York, NY (2003)
2. Holmes, D.G.: A general analytical method for determining the theoretical harmonic
components of carrier based PWM strategies. In: Proceeding of Annual Conference of IEEE
Industry Applications Society, pp. 1207–1214 (1998)
3. Jalili, K., Bernet, S.: Design of LCL filters of active-front-end two-level voltage-source
converters. IEEE Trans. Ind. Electron. 56(5), 1674–1689 (2009)
4. Liserre, M., Blaabjerg, F., Hansen, S.: Design and control of an LCL-filter-based three-phase
active rectifier. IEEE Trans. Ind. Appl. 41(5), 1674–1689 (2005)
5. IEEE Std. 929: IEEE Recommended Practice for Utility Interface of Photovoltaic (PV) Systems
(2000)
6. IEEE Std. 1547: IEEE Standard for Interconnecting Distributed Resources with Electric Power
Systems (2003)
2.5 Summary 61
Chapter 3
Magnetic Integration of LCL Filters
Abstract An LCL filter has two individual inductors. In order to reduce the volume
of magnetic components, magnetic integration of these two inductors is introduced
in this chapter. First, the integration method of the two inductors of an LCL filter is
proposed, and the magnetic circuit model of integrated inductors is built. Then,
based on this model, the coupling caused by the nonzero reluctance of the common
core is analyzed, and the coupling effect on the ability of attenuating high-frequency
harmonics of LCL filter is evaluated. According to the harmonic limits of the grid
current, the maximum allowable coupling coefficient is derived, which provides the
guidelines for selecting cross-sectional area and magnetic material of the common
core. Finally, with the help of Ansoft Maxwell software, design examples of
integrated magnetics for both single-phase and three-phase LCL filters are pre-
sented, and experiments are performed to verify the proposed method.
Keywords Grid-connected inverter � LCL filter � Coupling coefficient � Magnetic
integration � Magnetic circuit
Chapter 2 presents the design procedure for LCL filter. An LCL filter has two
individual inductors. In order to reduce the volume of magnetic components,
magnetic integration of these two inductors [1] is introduced in this chapter. First,
the integration method of the two inductors of an LCL filter is proposed, and the
magnetic circuit model of integrated inductors is built. Then, based on this model,
the coupling caused by the nonzero reluctance of the common core is analyzed, and
the coupling effect on the ability of attenuating high-frequency harmonics of LCL
filter is evaluated. According to the harmonic limits of the grid current, the maxi-
mum allowable coupling coefficient is derived, which provides the guidelines for
selecting cross-sectional area and magnetic material of the common core. Finally,
with the help of Ansoft Maxwell software, design examples of integrated magnetics
for both single-phase and three-phase LCL filters are presented, and experiments are
performed to verify the proposed method.
© Springer Nature Singapore Pte Ltd. and Science Press 2018
X. Ruan et al., Control Techniques for LCL-Type Grid-Connected Inverters,
CPSS Power Electronics Series, DOI 10.1007/978-981-10-4277-5_3
63
3.1 Magnetic Integration of LCL Filters
3.1.1 Magnetic Integration of Single-Phase LCL Filter
Figure 3.1 shows the topology of a single-phase LCL-type grid-connected inverter,
where L1 is the inverter-side inductor, C is the filter capacitor, L2 is the grid-side
inductor, i1 is the inverter-side inductor current, iC is the capacitor current, and i2 is
the grid current. As illustrated in Chap.2, the LCL filter is designed with the
constraints that the current ripple of i1, ΔI1m, is 20–30% (peak-to-peak) of the rated
fundamental current I2 [2], and the fundamental RMS of iC, ICf, is less than 5% of I2
[3]. Under these constraints, the designed LCL filter parameters are L1 = 360 lH,
C = 10 lF, and L2 = 90 lH, and the simulation waveforms of i1, i2, and iC under
rated load are shown in Fig. 3.2, where I1m, I2m, and ICm are the maximum values of
i1, i2, and iC, respectively. As seen, ΔI1m is about 31% of I2, and ICf is about 2.6% of
I2. I1m and I2m are close to each other, and they are far larger than ICm.
An intuitive choice for inductor design is to use an individual magnetic core for
each inductor of the LCL filter, as shown in Fig. 3.3a, where EI cores are used. Due
to the symmetry of the magnetic circuit, fluxes in the I-type cores of L1 and L2 can
be obtained as
/I1 ¼
L1i1
2N1
; /I2 ¼
L2i2
2N2
ð3:1Þ
where N1 and N2 are the winding turns of L1 and L2, respectively.
If the EI cores for each inductor are with the same width and thickness, L1 and L2
can be integrated with the core structure as shown in Fig. 3.3b, where the E-type
cores and air gaps of L1 and L2 remain unchanged, and the I-type core serving as a
common flux path is arranged between the E-type cores. According to the flux
flows shown in the figure, the fluxes generated by the windings of L1 and L2 go
through the common path in the opposite directions. Thus, if the discrete inductors
are designed to meet L1/N1 = L2/N2, the flux in the common I-type core can be
obtained as
S1
S2
S3
S4
Vin vg
i1
C
iC
L1 L2
i2
vC
+
vinv
Fig. 3.1 Single-phase LCL-
type grid-connected inverter
64 3 Magnetic Integration of LCL Filters
/c ¼ /I1 � /I2 ¼
L1iC
2N1
: ð3:2Þ
It indicates that /c is generated by iC. As discussed above, ICm is far smaller than
I1m and I2m, thereby /cm will be far smaller than /I1m and /I2m (/cm, /I1m, and /I2m
are the maximum values of /c, /I1, and /I2, respectively). According to the sim-
ulation result in Fig. 3.2, we can get
/cm
/I1m þ/I2m
¼ ICm
I1m þ I2m ¼ 6:44%: ð3:3Þ
0
0
0
I1m=8.52 A
I1m=40.77 A
I2m=38.83 A
I2=27.42 A
ICf =0.70 A, ICm=5.13 A
Time:[5 ms/div]
i1:[30 A/div]
i2:[30 A/div]
iC:[5 A/div]
Fig. 3.2 Simulation
waveforms in single-phase
LCL-type grid-connected
inverter
(b) Integrated inductors(a) Discrete inductors
Fig. 3.3 Core structures of
the two inductors for
single-phase LCL filter
3.1 Magnetic Integration of LCL Filters 65
Therefore, letting the E-type cores and the common I-type core operate in the
same maximum flux density, the required cross-sectional area of the common I-type
core is only 6.44% of the sum of the cross-sectional areas of the I-type cores for L1
and L2. As a result, the core volume of integrated inductors can be dramatically
reduced.
In addition, since I1m � I2m, then according to (3.1), /I1m � /I2m under the
condition L1/N1 = L2/N2. That means if the same maximum flux density is chosen,
the two E-type cores for the integrated inductors could have the same
cross-sectional area.
3.1.2 Magnetic Integration of Three-Phase LCL Filter
Figure 3.4 shows the topology of a three-phase LCL-type grid-connected inverter,
where i1a, i1b, and i1c are the inverter-side inductor currents, iCa, iCb, and iCc are the
capacitor currents, and i2a, i2b, and i2c are the grid currents. With the three-wire
connection, three-phase EI cores can be used for both the three inverter-side
inductors and grid-side inductors [4–6]. In this way, the proposed magnetic inte-
gration scheme can be extended to the three-phase LCL filter. The corresponding
core structure of integrated inductors is shown in Fig. 3.5, where /1a, /1b, and /1c
are the fluxes in the three legs of L1, and /2a, /2b, and /2c are the fluxes in the three
legs of L2. Their expressions are given as
/1a ¼
L1i1a
N1
; /1b ¼
L1i1b
N1
; /1c ¼
L1i1c
N1
/2a ¼
L2i2a
N2
; /2b ¼
L2i2b
N2
; /2c ¼
L2i2c
N2
:
ð3:4Þ
Similarly, if the condition L1/N1 = L2/N2 is met, the fluxes in the common I-type
core can be obtained as
a
N'
C
N
i1b
i1c
i2a
i2b
i2c
Vin
iCa iCb iCc
S1 S5S3
S4 S6 S2
b
c
i1a L1
L1
L1
vCa
vCb
vCc
C C
L2
L2
L2
vga
vgb
vgc
Fig. 3.4 Three-phase LCL-type grid-connected inverter
66 3 Magnetic Integration of LCL Filters
/c1 ¼ /1a � /2a ¼
L1iCa
N1
; /c2 ¼ /1c � /2c ¼
L1iCc
N1
: ð3:5Þ
From (3.5), it can be seen that the fluxes in the common I-type core are generated
by the capacitor currents, which show the same features as the single-phase inte-
grated inductors.
3.2 Coupling Effect on Attenuating Ability of LCL Filter
In the previous analysis, the reluctance of the common I-type core is ignored, and
thus, the integrated inductors are considered to be decoupled. However, in practice,
due to the nonzero reluctance of the common I-type core, the coupling between the
integrated inductors can hardly be avoided. The coupling effect on the LCL filter is
analyzed in this section.
3.2.1 Magnetic Circuit of Integrated Inductors
Taking the single-phase LCL filter as the example, the magnetic circuit of the
integrated inductors is shown in Fig. 3.6a where Rc1 and Rc2 are the reluctances of
the outer legs and the center leg for L1, Rc3 and Rc4 are the reluctances of the outer
legs and the center leg for L2, Rg1 and Rg2 are the reluctances of the center-leg air
gaps for L1 and L2, and Rcc is the reluctance of a half of the common I-type core.
Due to the symmetry of the magnetic circuit, Fig. 3.6a can be simplified into
Fig. 3.6b, from which the coupling coefficients of L1 to L2 and L2 to L1 can be
obtained as
Fig. 3.5 Core structure of the
integrated inductors for
three-phase LCL filter
3.1 Magnetic Integration of LCL Filters 67
k12 ¼ RccRcc þRc3 þ 2Rc4 þ 4Rg2
k21 ¼ RccRcc þRc1 þ 2Rc2 þ 4Rg1 :
ð3:6Þ
Note that Rc1–Rc4 are far smaller than Rg1 and Rg2, so (3.6) can be approximated as
k12 � Rcc4Rg2 ; k21 �
Rcc
4Rg1
: ð3:7Þ
Thus, the coupling coefficient between L1 and L2 is
k ¼
ffiffiffiffiffiffiffiffiffiffiffiffi
k12k21
p
� Rcc
4
ffiffiffiffiffiffiffiffiffiffiffiffiffi
Rg1Rg2
p : ð3:8Þ
As seen, k is mainly determined by the reluctances of the air gaps and the
common I-type core. These reluctances are expressed as
Rg1 ¼ d1l0Aee
; Rg2 ¼ d2l0Aee
; Rcc ¼ lc2l0lrAec
ð3:9Þ
where d1 and d2 are the air gaps of the center legs for L1 and L2, respectively; lc is
the width of the EI core, l0 is the absolute permeability of free space, lr is the
relative permeability of the common I-type core, and Aee and Aec are the
cross-sectional areas of the center legs and the common I-type core, respectively.
Substituting (3.9) into (3.8) yields
k ¼ 1
8lr
Aee
Aec
lcffiffiffiffiffiffiffiffiffi
d1d2
p : ð3:10Þ
2Rg1
Rc1
Rg1
Rc2 Rc1
2Rg1
Rcc Rcc
2Rg2 Rg2 2Rg2
Rc3 Rc4 Rc3
N1i1
N2i2
+_
+_
Rg1 Rg1
Rc2
Rg2 Rg2
Rc4
N1i1
N2i2
+_
+_
Rc312
Rc112
Rcc12
(a) (b)Fig. 3.6 Magnetic circuit of
the integrated inductors
68 3 Magnetic Integration of LCL Filters
Note that Aee and lc are specified for a selected EI core, and d1 and d2 are
determined by the values of L1 and L2, respectively; thus, the coupling coefficient
can be specified after lr and Aec are confirmed.
3.2.2 Characteristics of LCL Filter with Coupled Inductors
Considering the coupling between L1 and L2, the equivalent circuit of the LCL filter
with coupled inductors is shown in Fig. 3.7a, where the inverter bridge output
voltage vinv is represented by a voltage source, and M is the mutual inductance,
expressed as M ¼ k ffiffiffiffiffiffiffiffiffiffiL1L2p . Figure 3.7a can be simplified into Fig. 3.7b, which is
further transformed into Fig. 3.7c using Y-D transformation. As seen in Fig. 3.7c,
the LCL filter with coupled inductors is equivalent to a parallel connection of an
L filter and an LCL filter, where the L filter is L3d, and the LCL filter is composed of
L1d, C, and L2d. L1d–L3d are expressed as
L1d ¼ L1L2 �M
2
L2 þM ; L2d ¼
L1L2 �M2
L1 þM ; L3d ¼ �
L1L2 �M2
M
: ð3:11Þ
As seen in Fig. 3.7c, the grid current i2is the summation of i21 and i22, where i21
is supplied by the L filter branch, and i22 is supplied by the LCL filter branch. The
transfer functions from vAB to i21, i22, and i2 can be derived as
Gi21 sð Þ ¼ i21 sð Þvinv sð Þ ¼
1
sL3d
Gi22 sð Þ ¼ i22 sð Þvinv sð Þ ¼
1
s3L1dL2dCþ s L1d þ L2dð Þ
Gi2 sð Þ ¼ i2 sð Þvinv sð Þ ¼
1
sL3d
þ 1
s3L1dL2dCþ s L1d þ L2dð Þ
ð3:12Þ
With the parameters L1 = 360 lH, C = 10 lF, L2 = 90 lH, and k = 0.01, the
magnitude plots of Gi21(s), Gi22(s), and Gi2(s) are shown in Fig. 3.8. As seen, the
magnitude plot of Gi21(s) is a straight line with slope of −20 dB/dec; the magnitude
plot of Gi22(s) has a resonance peak, and it falls with slope of −60 dB/dec above the
L1
vgvinv
L2
i1 i2
M
iC
C
L1+M
vg
L2+M
iC
i2
M
i1
vinv
C
vinv vg
i1 i2
L1d L2d
L3d i21
i22iC
C
(a) (b) (c)
Fig. 3.7 Equivalent circuit of the LCL filter with coupled inductors
3.2 Coupling Effect on Attenuating Ability of LCL Filter 69
resonance frequency, which indicates high harmonics attenuation. The frequency
where the magnitude plots of Gi21(s) and Gi22(s) intersect is called the intersection
frequency, and it is denoted by fint. Solving |Gi21(s)| = |Gi22(s)|, fint can be calculated
as
fint ¼ 12p
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
L1d þ L2d � L3d
L1dL2dC
r
¼ 1
2p
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
L1L2 þ 2M L1 þ L2ð Þþ 3M2
L1L2 �M2ð ÞMC
s
: ð3:13Þ
The magnitude plot of Gi2(s) is the combination of those of Gi21(s) and Gi22(s).
As seen in Fig. 3.8, Gi22(s) dominates at the frequencies lower than fint, and
Gi21(s) dominates at the frequencies higher than fint. Compared with the LCL filter,
the LCL filter with coupled inductors has a poorer harmonic attenuation at the
frequencies higher than fint. Hence, to ensure the effective attenuation of the
switching harmonics, fint must be higher than the frequency where the dominant
switching harmonics lie in. According to (3.13), the curve of fint as the function of
k is depicted in Fig. 3.9. As seen, fint increases with the decrease in k. Thus, to make
the integrated LCL filter applicable, the coupling coefficient must be limited.
20 dB/dec
60 dB/dec
102 103 104 105 106
160
120
80
40
0
40
Frequency (Hz)
M
ag
ni
tu
de
 (d
B
)
fint
Gi21
Gi22
Gi2
Fig. 3.8 Magnitude plots of
Gi21(s), Gi22(s), and Gi2(s)
0.01 0.02 0.03 0.04 0.050
10
30
50
70
90
k
f in
t
(k
H
z)
0.017
Fig. 3.9 Curve of fint as the
function of k
70 3 Magnetic Integration of LCL Filters
3.3 Design Examples
3.3.1 Magnetics Design for Single-Phase LCL Filter
Table 3.1 gives the parameters of a 6-kW single-phase LCL-type grid-connected
inverter, where the unipolar sinusoidal pulse-width modulation (SPWM) is adopted.
Magnetic cores are selected with the well-known area product method [7].
Referring to the product catalog of NCD EE ferrite cores [8], two pairs of
EE70/33/32 are used for L1; 32-turn windings are designed and fabricated by
copper foils with width of 40 mm and thickness of 0.2 mm, and the air gap
d1 = 2.4 mm. Moving the air gaps in the EE core to the end of the window, the
equivalent EI core is obtained, as shown in Fig. 3.10. The core sizes are listed as
A1–H1 in Table 3.2.
As discussed in Sect. 3.1.1, the EI core for L2 is in the same dimensions as
the one for L1 except for the window height. N2 = 8 turns can be derived from
L1/N1 = L2/N2, and the air gap d2 = 0.6 mm. To ensure the same window utiliza-
tion and current density, the windings of L2 are fabricated by copper foils with
width of 10 mm and thickness of 0.8 mm. Considering the isolation requirements, a
margin of 1 mm should be reserved at both ends of the windings [9]. Thus, a
window height of 12 mm is necessary for L2, i.e., F2 = 12 mm. Consequently, the
overall core sizes for L2 are listed as A2 – H2 in Table 3.2. According to Table 3.2,
the core volumes for L1 and L2 can be calculated as
Table 3.1 Parameters of single-phase prototype
Parameter Symbol Value Parameter Symbol Value
Input voltage Vin 360 V Inverter-side inductor L1 360 lH
Grid voltage (RMS) Vg 220 V Filter capacitor C 10 lF
Output power Po 6 kW Grid-side inductor L2 90 lH
Fundamental frequency fo 50 Hz Switching frequency fsw 15 kHz
E
D
E
F
B
A
HG
Fig. 3.10 EI-type magnetic
core
3.3 Design Examples 71
Ve1 ¼ 2A1 B1 þH1ð ÞG1 � 4E1F1G1 ¼ 2:24� 105 mm3
Ve2 ¼ 2A2 B2 þH2ð ÞG2 � 4E2F2G2 ¼ 1:34� 105 mm3:
ð3:14Þ
Using the core structure shown in Fig. 3.3b for the integration of L1 and L2,
while the two parts of the E-type cores remain unchanged, the key issue lies in the
design of the common I-type core. As seen in (3.10), a larger Aec or a higher lr is
expected for a smaller coupling coefficient k. Since the common I-type core keeps
the same width and thickness as those of the E-type cores, its cross-sectional area
Aec is determined by the height Hc. And lr is related to the magnetic material that
used. Thus, to limit the coupling coefficient, the height and magnetic material of the
common I-type core need to be selected with caution.
With the limit of maximum flux density, according to (3.3), the minimum height
of the common I-type core can be obtained as
Hcmin ¼ 6:44% H1 þH2ð Þ � 2 mm: ð3:15Þ
Here, the widely used soft ferrite and silicon steel are investigated. For NCD
ferrite core, lr = 1725 [8], and for the silicon steel, lr = 5660 [10]. Based on the
Ansoft Maxwell 3D model shown in Fig. 3.11a, a more detailed investigation of the
relationship between k and Hc is carried out by simulation. The simulation result is
shown in Fig. 3.11b, where Hc � 2 mm is constrained by the maximum flux
density, and Hc � 22 mm is constrained to ensure that Hc will not exceed the
summation of H1 and H2. From Fig. 3.11b, Hc can be determined according to the
requirement of k.
For the single-phase grid-connected inverter adopting the unipolar SPWM, the
dominant switching harmonics are placed around twice the switching frequency
[11], i.e., 30 kHz. As previously mentioned, fint > 30 kHz is required. To achieve
that, as shown in Fig. 3.9, k < 0.017 has to be satisfied. Recalling Fig. 3.11b, if
NCD ferrite core is used for the common I-type core, k < 0.017 cannot be achieved
even if Hc = 22 mm; if the silicon steel is used for the common I-type core,
k < 0.017 can be achieved if Hc > 8 mm. Therefore, the silicon steel is preferred in
practical application. By making a tradeoff between the core volume and the
coupling coefficient, Hc = 11 mm is chosen since a further increase in Hc only
results in a little decrease in k. Thus, the reduced core volume is
Table 3.2 Parameters of
single-phase prototype
Symbol Value (mm) Symbol Value (mm)
A1 70.5 A2 70.5
B1 55.1 B2 23.3
D1 22 D2 22
E1 13 E2 13
F1 43.8 F2 12
G1 31.6 G2 31.6
H1 11.3 H2 11.3
72 3 Magnetic Integration of LCL Filters
DVe ¼ 2A1 H1 þH2 � Hcð ÞG1 ¼ 5:17� 104 mm3: ð3:16Þ
Compared with the total core volume of the discrete inductors, the reduced core
volume in percentage terms is
DVe% ¼ DVeVe1 þVe2 � 100% ¼ 14:4%: ð3:17Þ
3.3.2 Magnetics Design for Three-Phase LCL Filter
Table 3.3 gives the parameters of a 20-kW three-phase LCL-type grid-connected
inverter, and the space vector modulation is adopted. The three-phase silicon steel
cores are used. Referring to the electronic transformer handbook [10], two pairs of
BSD 25 � 25 � 80 are selected and then cut into two parts with the ratio of 3:1 in
the window height. These two parts are served as three-phase E-type cores for L1
and L2, respectively (see Fig. 3.12b in Sect. 3.4). For L1, N1 = 50 turns, and the
windings are fabricated by copper foils with width of 60 mm and thickness of
0.15 mm, and for L2, N2 = 15 turns, and the windings are fabricated by copper foils
with width of 18 mm and thickness of 0.5 mm. The core structure shown in
Fig. 3.5 is used for the integration of L1 and L2. With the same design procedurementioned above, the common I-type core is fabricated by the silicon steel with a
height of 25 mm. Consequently, the reduced core volume can be calculated as
17.5%.
2 6 10 14 18 22
0.04
0.08
0.12
0.16
0
k
Hc (mm)
4 8 12 16 20
0.017
Soft Ferrite
Silicon Steel
(a) 3-D model (b) Simulation results 
Fig. 3.11 Ansoft Maxwell 3D model and simulation results
3.3 Design Examples 73
3.4 Experimental Verification
Both 6-kW single-phase and 20-kW three-phase prototypes are built and tested in
the laboratory.
3.4.1 Experimental Results for Single-Phase LCL Filter
In the single-phase system, referring to Table 3.2, one pair of EE70/54/32 can be
used for the E-type core of L1. However, the E-type core required for L2 is irregular,
and for simplicity, it is replaced by one pair of EE70/33/32. As for the common
I-type cores, both the soft ferrite and silicon steel are evaluated, and Hc = 11 mm is
chosen in both cases. Figure 3.12a shows the photograph of the integrated
inductors.
According to IEEE std.1547-2003 [12], the harmonics higher than 35th in the
grid current are limited to 0.3% of its rated value. For the 6-kW single-phase
prototype, the rated current is 38.6 A, and thus, the harmonic limit is 116 mA.
Figure 3.13 shows the experimental results with discrete inductors. As seen, the key
switching harmonics in i1 are placed around multiples of twice the switching fre-
quency. Because of the high attenuating ability of the LCL filter, only a little
switching harmonics are injected into the grid. The dominant switching harmonics
in i2 are placed around 30 kHz with maximum amplitude of about 52 mA.
Table 3.13 Parameters of three-phase prototype
Parameter Symbol Value Parameter Symbol Value
Input voltage Vin 700 V Inverter-side inductor L1 1 mH
Grid voltage (RMS) Vg 220 V Filter capacitor C 20 lF
Output power Po 20 kW Grid-side inductor L2 300 lH
Fundamental frequency fo 50 Hz Switching frequency fsw 10 kHz
L1
L2
L1 L1 L1
L2
L2
L2
(a) Single phase (b) Three phase
Fig. 3.12 Photographs of the
integrated inductors
74 3 Magnetic Integration of LCL Filters
The experimental results with integrated inductors are shown in Figs. 3.14 and
3.15. If the soft ferrite is used for the common I-type core, the measured coupling
coefficient is k = 0.045, which is larger than 0.017, and thus, the attenuating ability
of the LCL filter around 30 kHz is weakened. As seen in Fig. 3.14b, the maximum
amplitude of the dominant switching harmonics is about 100 mA, which is nearly
twice the one for discrete inductors. Fortunately, if the silicon steel is used for the
common I-type core, the measured coupling coefficient is k = 0.012, which is lower
than 0.017, and thus, the high attenuating ability of the LCL filter around 30 kHz is
remained. As seen in Fig. 3.15b, the maximum amplitude of the dominant
switching harmonics is about 60 mA, which is close to the one for discrete
inductors.
Time: [5 ms/div]
i1:[30 A/div]
i2:[30 A/div]
i2:[50 mA/div]
Harmonic limit: 116mA
i1:[50 mA/div]
30kHz0 60kHz 90kHz 120kHz
(a) Experimental waveform (b) Harmonic spectra
Fig. 3.13 Experimental results with discrete inductors in single-phase prototype
Time: [5 ms/div]
i1:[30 A/div]
i2:[30 A/div]
Harmonic limit: 116mA
i2:[100 mA/div]
i1:[100 mA/div]
30kHz0 60kHz 90kHz 120kHz
(a) Experimental waveform (b) Harmonic spectra
Fig. 3.14 Experimental results with integrated inductors in single-phase prototype (soft ferrite
used for the common I-type core)
3.4 Experimental Verification 75
3.4.2 Experimental Results for Three-Phase LCL Filter
In the three-phase system, the three-phase integrated inductors are implemented
with the design procedure depicted in Sect. 3.3.2, the photograph is shown in
Fig. 3.12b, and the measured coupling coefficient between L1 and L2 is k = 0.02.
The harmonic limit for the 20-kW three-phase prototype is calculated as 128 mA.
Figure 3.16 shows the experimental results with discrete inductors. As seen, the key
switching harmonics in i1a are placed around multiples of the switching frequency.
And the maximum amplitude of the dominant switching harmonics in i2a is about
92 mA. Figure 3.17 shows the experimental results with integrated inductors, and
the maximum amplitude of the dominant switching harmonics in i2a is about
100 mA, which is close to the one for discrete inductors. Experimental results from
both the single-phase and three-phase prototypes confirm the theoretical
expectations.
(a) Experimental waveform (b) Harmonic spectra
Time: [5 ms/div]
i1:[30 A/div]
i2:[30 A/div]
i2:[50 mA/div]
Harmonic limit: 116mA
i1:[50 mA/div]
30kHz0 60kHz 90kHz 120kHz
Fig. 3.15 Experimental results with integrated inductors in single-phase prototype (silicon steel
used for the common I-type core)
(a) Experimental waveform (b) Harmonic spectra
i1c:[30 A/div] i1b:[30 A/div]
i2a:[30 A/div] i2c:[30 A/div] i2b:[30 A/div]
Time:[5 ms/div]
i1a:[30 A/div]
0 20kHz 40kHz 60kHz 80kHz
i1a:[100 mA/div]
i2a:[100 mA/div]
Harmonic limit: 128mA
Fig. 3.16 Experimental results with discrete inductors in three-phase prototype
76 3 Magnetic Integration of LCL Filters
3.5 Summary
This Chapter proposes the magnetic integration of the LCL filter in both
single-phase and three-phase grid-connected inverters. By sharing an ungapped
core and arranging the windings properly, the fundamental fluxes generated by the
two inductors of the LCL filter cancel out mostly in the common core. The coupling
caused by the nonzero reluctance of the common core is considered, and the
coupling effect on the attenuating ability of the LCL filter is analyzed. It turns out
that the LCL filter with coupled inductors is equivalent to a parallel connection of an
L filter and an LCL filter. In order to meet the harmonic limits, the cross-sectional
area and magnetic material of the common core are properly selected, ensuring the
coupling coefficient of the integrated inductors be limited to a satisfactory range.
With the proposed magnetic integration scheme, core volume is reduced by 14.4%
for a 6-kW single-phase prototype and 17.5% for a 20-kW three-phase prototype,
respectively. Experimental results from both single-phase and three-phase proto-
types confirm the theoretical expectations.
References
1. Pan, D., Ruan, X., Bao, C., Li, W., Wang, X.: Magnetic integration of the LCL filter in
grid-connected inverters. IEEE Trans. Power Electron. 29(4), 1573–1578 (2014)
2. Wang, T.C., Ye, Z., Sinha, G., and Yuan, X.: Output filter design for a grid-interconnected
three-phase inverter. In: Proceeding IEEE Power Electronics Specialists Conference,
pp. 779–784 (2003)
3. Liserre, M., Blaabjerg, F., Hansen, S.: Design and control of an LCL-filter-based three-phase
active rectifier. IEEE Trans. Ind. Appl. 41(5), 1281–1291 (2005)
4. Wei, L., Lukaszewski, R.A.: Optimization of the main inductor in a LCL filter for three phase
active rectifier. In: Proceeding Annual Conference of IEEE Industry Applications Society,
pp. 1816–1822 (2007)
5. Bueno, E.J., Cóbreces, S., Rodríguez, F.J., Hernández, Á., Espinosa, F.: Design of a
back-to-back NPC converter interface for wind turbines with squirrel-cage induction
generator. IEEE Trans. Energy Convers. 23(3), 932–945 (2008)
(a) Experimental waveform (b) Harmonic spectra
Time:[5 ms/div]
i1a:[30 A/div] i1c:[30 A/div] i1b:[30 A/div]
i2a:[30 A/div] i2c:[30 A/div] i2b:[30 A/div]
20kHz0 40kHz 60kHz 80kHz
Harmonic limit: 128mA
i1a:[100 mA/div]
i2a:[100 mA/div]
Fig. 3.17 Experimental results with integrated inductors in three-phase prototype
3.5 Summary 77
6. Wei, L., Patel, Y., Murthy, C.: Evaluation of LCL filter inductor and active front end rectifier
losses under different PWM method. In: Proceeding of the IEEE Energy Conversion Congress
and Exposition, pp. 3019–3026 (2013)
7. Zhao, X.: Utility Power Supply Technology Handbook of Magnetic Components. Liaoning
Science and Technology Publishing House, Shenyang (2002). (in Chinese)
8. EE Ferrite Cores.: Nanjing NewConda Magnetic Industrial Co. Ltd. (2013) [Online].
Available: http://ncd.com.cn/category/eecores-2599-e179/1
9. Dixon, L.H.: Magnetics Design for Switching Power Supplies. Texas Instruments.
(2011) [Online]. Available: http://focus.ti.com/docs/training/catalog/events/event.jhtml?sku=
SEM401014
10. Wang, Q.: Electronic Transformer Handbook. Liaoning Science and Technology Publishing
House, Shenyang (2007). (in Chinese)
11. Holmes, D.G., Lipo, T.A.: Pulse Width Modulation for Power Converters: Principles and
Practice. IEEE Press & Wiley, New York (2003)
12. IEEE Standard for Interconnecting Distributed Resources with Electric Power Systems.: IEEE
Std. 1547-2003 (2003)
78 3 Magnetic Integration of LCL Filters
http://ncd.com.cn/category/eecores-2599-e179/1
http://focus.ti.com/docs/training/catalog/events/event.jhtml?sku=SEM401014
http://focus.ti.com/docs/training/catalog/events/event.jhtml?sku=SEM401014
Chapter 4
Resonance Damping Methods
of LCL Filter
Abstract The control challenges of LCL-type grid-connected inverter arise from
the resonance problem. At the resonance frequency, the LCL filter resonance causes
a sharp phase step down of −180° with a high resonance peak. This resonance peak
would easily lead to system instability and should be damped. In this chapter, the
resonance hazard resulted by the LCL filter is reviewed first, and then, the existing
passive- and active-damping solutions are described systematically to reveal the
relationship among them. Among the six basic passive-damping solutions, adding a
resistor in parallel with capacitor shows the best damping performance, but it results
in a high power loss. In order to avoid the power loss in the damping resistor, the
active-damping solutions equivalent to a resistor in parallel with capacitor are
derived, and the capacitor-current-feedback active damping is superior for its
simple implementation and effectiveness. This chapter provides the basis for the
study of the control techniques of LCL-type grid-connected inverter in the fol-
lowing chapters.
Keywords Grid-connected inverter � LCL filter � Resonance � Passive damping �
Active damping
Chapters 2 and 3 have presented the design and magnetic integration of LCL filters.
In the following chapters, the control techniques for the LCL-type grid-connected
inverter will be discussed. The control challenges of LCL-type grid-connected
inverter arise from the resonance problem. At the resonance frequency, the LCL
filter resonance causes a sharp phase step down of −180° with a high resonance
peak. This resonance peak would easily lead to system instability and should be
damped. In this chapter, the resonance hazard resulted by the LCL filter is reviewed
first, and then, the existing passive- and active-damping solutions are described
systematically to reveal the relationship among them.
© Springer Nature Singapore Pte Ltd. and Science Press 2018
X. Ruan et al., Control Techniques for LCL-Type Grid-Connected Inverters,
CPSS Power Electronics Series, DOI 10.1007/978-981-10-4277-5_4
79
4.1 Resonance Hazard of LCL Filter
Figure 4.1a shows the main circuit of a single-phase LCL-type grid-connected
inverter, where L1 is the inverter-side inductor, C is the filter capacitor, and L2 is the
grid-side inductor. By representing the inverter bridge output voltage vinv with a
voltage source, Fig. 4.1a can be simplified into Fig. 4.1b, from which the transfer
function from vinv to the grid current i2 can be derived as
GLCL sð Þ ¼ i2 sð Þvinv sð Þ ¼
1
s3L1L2Cþ s L1 þ L2ð Þ ¼
1
sL1L2C
� 1
s2 þx2r
ð4:1Þ
where xr is the resonance angular frequency of the LCL filter, expressed as
xr ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
L1 þ L2
L1L2C
r
ð4:2Þ
and the resonance frequency is fr ¼ xr=ð2pÞ.
According to (4.1), the Bode diagram of GLCL(s) is shown with the solid line, as
shown in Fig. 4.2. As seen, at the resonance frequency fr, the LCL filter resonance
causes a sharp phase step down of −180° with a high resonance peak. From a
control perspective, this −180° crossing is a negative crossing, and it will create a
pair of closed-loop right-half plane poles [1], leading to system instability.
Therefore, in order to stabilize the system, the resonance peak must be damped
below 0 dB so that the negative crossing can be avoided. To achieve the resonance
damping, a first-order term related to s needs to be incorporated into the resonant
term s2 þx2r of (4.1), yields
GLCL�d sð Þ ¼ 1sL1L2C �
1
s2 þ 2nxrsþx2r
ð4:3Þ
where n is the damping ratio. According to (4.3), the Bode diagram of GLCL-d(s) is
depicted with the dashed line, as shown in Fig. 4.2. It can be seen that by
S1
S2
S3
S4
Vin vg
i1
C
iC
L1 L2
i2
vC
+
vinv
vC
i1
L1 L2
vinv
+ +
vg
iC
C
i2
(a) Main circuit (b) Simplified circuit
Fig. 4.1 Single-phase LCL-type grid-connected inverter
80 4 Resonance Damping Methods of LCL Filter
introducing the damping term, the resonance peak of LCL filter is effectively
suppressed, while the magnitude-frequency characteristics at the low- and
high-frequency ranges remain unchanged. This is helpful to providing high
low-frequency gains and strong high-frequency harmonic attenuating ability, and is
exactly desirable as expected.
4.2 Passive-Damping Solutions
4.2.1 Basic Passive Damping
As discussed above, the resonance hazard of LCL filter calls for damping solutions
to stabilize the system. A direct way to damp the LCL filter resonance is to insert a
resistor into the filter network, which is called the passive damping. According to
the location of the resistor, there are six basic passive-damping solutions, as shown
in Fig. 4.3. A detailed analysis of these solutions is presented in the following.
As shown in Fig. 4.3a, resistor RL11 is introduced to be in series with L1, and the
transfer function from vinv to i2 can be derived as
GLCL�1 sð Þ ¼ i2 sð Þvinv sð Þ ¼
1
s3L1L2Cþ s2L2CRL11 þ s L1 þ L2ð ÞþRL11 : ð4:4Þ
When resistor RL21 is introduced to be in series with L2, as shown in Fig. 4.3b,
the transfer function from vinv to i2 can be derived as
M
ag
ni
tu
de
 (d
B
)
50
0
50
100
150
Ph
as
e 
(°
)
Frequency (Hz)
10 102 103 104 105
0
90
180
270
360
fr
w/ damping
w/o damping
Fig. 4.2 Frequency response
of the LCL filter
4.1 Resonance Hazard of LCL Filter 81
GLCL�2 sð Þ ¼ i2 sð Þvinv sð Þ ¼
1
s3L1L2Cþ s2L1CRL21 þ s L1 þ L2ð ÞþRL21 : ð4:5Þ
If resistor RL12 is located in parallel with L1, as shown in Fig. 4.3c, the transfer
function from vinv to i2 can be derived as
GLCL�3 sð Þ ¼ i2 sð Þvinv sð Þ ¼
sL1=RL12 þ 1
s3L1L2Cþ s2L1L2=RL12 þ s L1 þ L2ð Þ : ð4:6Þ
The resistor, denoted as RL22, can also be added to be in parallel with L2, as
shown in Fig. 4.3d, and the transfer function from vinv to i2 can be derived as
GLCL�4 sð Þ ¼ i2 sð Þvinv sð Þ ¼
sL2=RL22 þ 1
s3L1L2Cþ s2L1L2=RL22 þ s L1 þ L2ð Þ : ð4:7Þ
When resistor RC1 is placed in series with C, as shown in Fig. 4.3e, the transfer
function from vinv to i2 can be derived as
vC
i1
L1 L2
vinv
+ +
vg
iC
C
i2
RL11 vC
i1
L1 L2
vinv
+ +
vg
iC
C
i2
RL21
(a) Resistor in series with L1 (b) Resistor in series with L2
vC
i1
L1 L2
vinv
+ +
vg
iC
C
i2
RL12
vC
i1
L1 L2
vinv
+ +
vg
iC
C
i2
RL22
(c) Resistor in parallel with L1 (d) Resistor in parallel with L2
vC
i1
L1 L2
vinv
+ +
vg
iC
C
i2
RC1
vC
i1
L1 L2
vinv
+ +
vg
iC i2
RC2C
(e) Resistor in series with C (f) Resistor in parallel with C
Fig. 4.3 Six basic passive-damping solutions
82 4 Resonance Damping Methods of LCL Filter
GLCL�5 sð Þ ¼ i2 sð Þvinv sð Þ ¼
sCRC1 þ 1
s3L1L2Cþ s2 L1 þ L2ð ÞCRC1 þ s L1 þ L2ð Þ : ð4:8Þ
Also, incorporating the resistor, denoted as RC2, to be in parallel with C, as
shown in Fig. 4.3f, can effectively damp the resonance peak, and the transfer
function from vinv to i2 can be derived as
GLCL�6 sð Þ ¼ i2 sð Þvinv sð Þ ¼
1
s3L1L2Cþ s2L1L2=RC2 þ s L1 þ L2ð Þ : ð4:9Þ
Comparing (4.4) and (4.5) with (4.1), it can be seen that when the resistor is
added in series with L1 andL2, respectively, the transfer functions from vinv to i2 are
similar, in which, a damping term (the second-order term related to s) and a con-
stant term are added to the denominator of GLCL(s). Comparing (4.6), (4.7), and
(4.8) with (4.1), it can be seen that when the resistor is added in parallel with L1 and
L2, respectively, or the resistor is added in series with C, the transfer functions from
vinv to i2 are similar, in which, a zero is added besides introducing a damping term.
When the resistor is introduced to be in parallel with C, the transfer function from
vinv to i2, shown in (4.9) is similar to (4.3), which is the desired form with only a
damping term being added.
According to (4.4)–(4.9), the frequency responses of the six basic
passive-damping solutions are depicted, as shown in Fig. 4.4. From which, it can be
seen that:
(1) Resistor in series with inductors will reduce the low-frequency gains of LCL
filter, as shown in Fig. 4.4a, b. This is because that at the low-frequency range,
the inductor reactance is relatively small, and a series resistor distinctly
increases the impedance of inductor branch, making the gains lower. The larger
the series resistor is, the more the low-frequency gains are reduced. While at the
high-frequency range, the inductor reactance is far larger than the value of
series resistor, the series resistor can be ignored, and thus it has no effect on the
high-frequency gains of LCL filter.
(2) Resistor in parallel with inductors will weaken the high-frequency harmonic
attenuating ability of LCL filter, as shown in Fig. 4.4c, d. This is because that at
the high frequencies, the inductor reactance is relatively large, and a parallel
resistor distinctly reduces the impedance of inductor branch, lowering the
harmonic attenuating ability. The smaller the parallel resistor is, the poorer the
high-frequency harmonic attenuating ability becomes. While at the low fre-
quencies, the inductor reactance is far smaller than the value of the parallel
resistor, the parallel resistor can be ignored, and thus it has no effect on the
low-frequency gains of LCL filter.
(3) Resistor in series with capacitor will weaken the high-frequency harmonic
attenuating ability of LCL filter, as shown in Fig. 4.4e. This is because that at
the high-frequency range, the capacitor reactance is relatively small, and a
series resistor distinctly increases the impedance of capacitor branch, lowering
4.2 Passive-Damping Solutions 83
M
ag
ni
tu
de
 (d
B
)
50
0
50
100
150
Ph
as
e 
(°
)
Frequency (Hz)
10 102 103 104 105
0
90
180
270
360
RL11=1 Ω
RL11=0 Ω
RL11=10 Ω
M
ag
ni
tu
de
 (d
B
)
50
0
−50
−100
−150
Ph
as
e 
(°
)
Frequency (Hz)
10 102 103 104 105
0
−90
−180
−270
−−
−
−
−
−
−
−
360
RL21=1 Ω
RL21=0 Ω
RL21=10 Ω
Resistor in series with L1 Resistor in series with L2
M
ag
ni
tu
de
 (d
B
)
50
0
−50
−100
−150
Ph
as
e 
(°
)
Frequency (Hz)
10 102 103 104 105
0
−90
−180
−270
−360
RL12=10 Ω
RL12= ∞ Ω
RL12=1 Ω
M
ag
ni
tu
de
 (d
B
)
50
0
−50
−100
−150
Ph
as
e 
(°
)
Frequency (Hz)
10 102 103 104 105
0
−90
−180
−270
−360
RL22=10 Ω
RL22= ∞ Ω
RL22=1 Ω
Resistor in parallel with L1 Resistor in parallel with L2
M
ag
ni
tu
de
 (d
B
)
50
0
−50
−100
−150
Ph
as
e 
(°
)
Frequency (Hz)
10 102 103 104 105
0
−90
−180
−270
−360
RC1=1 Ω
RC1=0 Ω
RC1=10 Ω
M
ag
ni
tu
de
 (d
B
)
50
0
−50
−100
−150
Ph
as
e 
(°
)
Frequency (Hz)
10 102 103 104 105
0
−90
−180
−270
−360
RC2=10 Ω
RC2= ∞ Ω
RC2=1 Ω
Resistor in series with C Resistor in parallel with C
(a) (b)
(c) (d)
(e) (f)
Fig. 4.4 Frequency responses of the six basic passive-damping solutions
84 4 Resonance Damping Methods of LCL Filter
the harmonic attenuating ability. The larger the series resistor is, the poorer the
high-frequency harmonic attenuating ability becomes. While at the
low-frequency range, the capacitor reactance is far larger than the value of
series resistor, the series resistor can be ignored, and thus it has no effect on the
low-frequency gains of LCL filter.
(4) Resistor in parallel with capacitor will not affect the magnitude-frequency
characteristics of LCL filter at the low- and high-frequency ranges, as shown in
Fig. 4.4f. This is because that at the low-frequency range, the reactance of L2 is
far smaller than the value of parallel resistor, the parallel resistor can be ignored;
while at the high-frequency range, the capacitor reactance is far smaller than the
value of parallel resistor, the parallel resistor can also be ignored.
From the above analysis, it can be known that introducing a resistor in parallel
with the filter capacitor C shows the best damping performance among the six basic
passive-damping solutions. However, since the voltage drop on L2 is relatively
small, the capacitor voltage is much close to the grid voltage, and it is directly
applied on the parallel resistor, resulting in a high power loss. Thus, the
passive-damping solution using a resistor in parallel with the capacitor is not
applicable in practice. Comparatively, the damping solution using a resistor in
series with the capacitor has been widely used for its lower loss [2, 3].
4.2.2 Improved Passive Damping
Based on the passive-damping solution of adding a resistor in series with the
capacitor, several improved solutions has been proposed in [4–7] to further reduce
the power loss in the damping resistor. Figure 4.5 shows four representative
improved passive-damping solutions, which will be analyzed in the following.
(1) Adding a Bypass Inductor
As seen in Fig. 4.5a, an inductor Ld is connected in parallel with the damping
resistor RC1. At the fundamental frequency, the reactance of Ld is far smaller
than the value of RC1, thus the fundamental current in C is almost bypassed by
Ld, leading to reduced power loss in RC1. From Fig. 4.5a, the transfer function
from vinv to i2 can be derived as
GLCL�5a sð Þ ¼ i2 sð Þvinv sð Þ
¼ s
2LdCRC1 þ sLd þRC1
s4L1L2LdCþ s3 L1L2 þ L1 þ L2ð ÞLd½ �CRC1 þ s2 L1 þ L2ð ÞLd þ s L1 þ L2ð ÞRC1
ð4:10Þ
4.2 Passive-Damping Solutions 85
(2) Adding a Bypass Inductor and Capacitor
Based on Fig. 4.5a, a capacitor Cd is further connected in parallel with the
damping resistor RC1, as shown in Fig. 4.5b. At the high-frequency range, the
reactance of Cd is far smaller than the value of RC1, thus the high-frequency
harmonic current in C is almost bypassed by Cd, and the high-frequency loss of
RC1 is reduced. Moreover, Cd also reduces the high-frequency impedance of
capacitor branch, which makes the LCL filter still have a high harmonic
attenuating ability after damping. From Fig. 4.5b, the transfer function from
vinv to i2 can be derived as
GLCL�5b sð Þ ¼ i2 sð Þvinv sð Þ
¼ s
2Ld CþCdð ÞRC1 þ sLd þRC1
s5L1L2LdCCdRC1 þ s4L1L2LdC
þ s3 L1L2Cþ L1 þ L2ð ÞLd CþCdð Þ½ �RC1
þ s2 L1 þ L2ð ÞLd þ s L1 þ L2ð ÞRC1
2
664
3
775
ð4:11Þ
(3) Splitting the Capacitor
Besides adding bypass components to the damping resistor, the capacitor C can
be split into two ones, and resistor RC1 is in series with one of the capacitors,
C1, as shown in Fig. 4.5c. Essentially, this method is equivalent to adding a
bypass capacitor to the resistor RC1. From Fig. 4.5c, the transfer function from
vinv to i2 can be derived as
i1+ +i2
vinv
C
RC1Ld
vg
L1 L2
i1+ +i2
vinv
C
RC1Ld
vg
Cd
L1 L2
(b) Adding a bypass inductor and capacitor 
i1+ +i2
vinv
C1
RC1
vg
L1 L2
C2
i1+ +i2
vinv
C1
RC1
vg
L1 L2
C2
Ld
(c) Splitting the capacitor 
(a) Adding a bypass inductor 
(d) Splitting the capacitor and adding a bypass inductor
Fig. 4.5 Four improved passive-damping solutions
86 4 Resonance Damping Methods of LCL Filter
GLCL�5c sð Þ ¼ i2 sð Þvinv sð Þ
¼ sC1RC1 þ 1
s4L1L2C1C2RC1 þ s3L1L2 C1 þC2ð Þþ s2 L1 þ L2ð ÞC1RC1 þ s L1 þ L2ð Þ
ð4:12Þ
(4) Splitting the Capacitor and Adding a Bypass Inductor
Similarly, based on Fig. 4.5c, an inductor Ld is further connected in parallel
with the damping resistor RC1, as shown in Fig. 4.5d. In this way, thepower
loss in RC1 at the fundamental frequency can be reduced. Actually, this method
is equivalent to the method shown in Fig. 4.5b. From Fig. 4.5d, the transfer
function from vinv to i2 can be derived as
GLCL�5d sð Þ ¼ i2 sð Þvinv sð Þ ¼
s2LdC1RC1 þ sLd þRC1
s5L1L2LdC1C2RC1 þ s4L1L2Ld C1 þC2ð Þ
þ s3 L1L2 C1 þC2ð Þþ L1 þ L2ð ÞLdC1½ �RC1
þ s2 L1 þ L2ð ÞLd þ s L1 þ L2ð ÞRC1
2
64
3
75
ð4:13Þ
According to (4.10)–(4.13), the frequency responses of the four improved
passive-damping solutions are depicted in Fig. 4.6. Compared with the basic
M
ag
ni
tu
de
 (d
B
)
50
0
50
100
150
Ph
as
e 
(°
)
Frequency (Hz)
10 102 103 104 105
0
90
180
270
360
GLCL-5(s)
GLCL(s)
GLCL-5a(s)
GLCL-5b(s)
M
ag
ni
tu
de
 (d
B
)
50
0
50
100
150
Ph
as
e 
(°
)
Frequency (Hz)
10 102 103 104 105
0
90
180
270
360
GLCL-5(s)
GLCL(s)
GLCL-5c(s)
GLCL-5d(s)
(a) Adding bypass components (b) Splitting the capacitor
Fig. 4.6 Frequency responses of the four improved passive-damping solutions
4.2 Passive-Damping Solutions 87
passive-damping solution using a resistor in series with the capacitor, it can be seen
that:
(1) Adding a bypass inductor will not affect the magnitude-frequency character-
istics of the LCL filter at the high-frequency range. This is because that at the
high-frequency range, the reactance of bypass inductor is far larger than the
value of series resistor, the series resistor still plays a dominant role.
(2) Adding a bypass capacitor or splitting the capacitor will improve the
high-frequency harmonic attenuating ability of LCL filter. This is because that
at the high-frequency range, the bypass capacitor or split capacitor offers a
low-impedance branch which absorbs most of the high-frequency harmonic
currents.
From the above analysis, it can be known that among the four improved
passive-damping solutions, by adding a bypass inductor and capacitor or splitting
the capacitor and adding a bypass inductor, both a lower power loss and a high
harmonic attenuating ability can be achieved, but the circuit complexity and the
system volume and cost are also increased.
4.3 Active-Damping Solutions
As illustrated in Sect. 4.2, passive damping is able to suppress the LCL filter
resonance, but it results in power loss, and it might reduce the low-frequency gains
or the high-frequency harmonic attenuating ability of LCL filter. To overcome these
drawbacks, proper control algorithms can be adopted to compensate the frequency
response of LCL filter to achieve the desired damping performance. This method is
called active damping. Generally speaking, active-damping solutions can be clas-
sified into two kinds: one is the state-variable-feedback active damping; the other is
the notch-filter-based active damping.
4.3.1 State-Variable-Feedback Active Damping
The state-variable-feedback active damping is the method that uses the feedback of
proper state variable to mimic a virtual resistor in place of the physical one. As
reported in Sect. 4.2.1, the resistor in parallel with capacitor shows the best
damping performance, thus the active-damping solutions equivalent to a resistor in
parallel with capacitor are derived as follows.
According to Fig. 4.3f, the control block diagram of LCL-type grid-connected
inverter using a resistor in parallel with the capacitor is obtained, as shown in
Fig. 4.7. In which, vr(s) is the modulation signal and KPWM = Vin/Vtri is the transfer
function from vr(s) to the inverter bridge output voltage vinv(s), where, Vin and Vtri
are the input voltage and the amplitude of the triangular carrier, respectively.
88 4 Resonance Damping Methods of LCL Filter
Referring to Fig. 4.7, by moving the feedback node of the capacitor voltage
vC(s) to the input of KPWM, and adjusting its feedback function, an equivalent
control block diagram is obtained, as shown in Fig. 4.8a. From which, it can be
seen that derivative feedback of the capacitor voltage is equivalent to a resistor in
parallel with capacitor.
–+ + +
– –
i2(s)
iC(s)
vC(s)
vr(s) KPWM
vinv(s)
sL1
1
sC
1
sL2
1+ –
vg(s)
1
RC2
Fig. 4.7 Control block diagram of LCL-type grid-connected inverter using a resistor in parallel
with capacitor
+ ++ i2(s)
iC(s)
vC(s)
vr(s) KPWM
vinv(s)
sL1
1
sC
1
sL2
1+
vg(s)
KPWMRC2
sL1
+ ++ i2(s)
iC(s)
vC(s)
vr(s) KPWM
vinv(s)
sL1
1
sC
1
sL2
1+
vg(s)
KPWMRC2
s2L1L2
v2(s)
KPWMRC2
sL1
+ ++ i2(s)
iC(s)
vC(s)
vr(s) KPWM
vinv(s)
sL1
1
sC
1
sL2
1+
vg(s)
KPWMCRC2
L1
(a) Derivative feedback of the capacitor voltage
(b) Second-derivative feedback of the grid current
(c) Proportional feedback of the capacitor current
Fig. 4.8 Equivalent forms of the damping solution using resistor in parallel with capacitor
4.3 Active-Damping Solutions 89
Considering that the capacitor voltage vC(s) is the summation of the grid voltage
vg(s) and the voltage on L2, v2(s), the feedback of vC(s) can be decomposed into the
feedbacks of vg(s) and v2(s). Then, by replacing the feedback variable v2(s) with the
grid current i2(s), and adjusting its feedback function, an equivalent control block
diagram is obtained, as shown in Fig. 4.8b. As seen, the derivative feedback of the
grid voltage plus the second-derivative feedback of the grid current is also equiv-
alent to a resistor in parallel with capacitor. It is worth noting that vg(s) is a
disturbance signal, and it makes no contribution to the damping of LCL filter
resonance. Therefore, from the viewpoint of damping the resonance, the derivative
feedback of the grid voltage can be omitted (see the dashed line in Fig. 4.8b), and
only the second-derivative feedback of the grid current is enough.
Based on Fig. 4.8a, if we replace the feedback variable vC(s) with the capacitor
current iC(s) and adjust its feedback function, an equivalent control block diagram
can be obtained as shown in Fig. 4.8c. As seen, the feedback function of the
capacitor current is a constant L1/(KPWMCRC2). Therefore, proportional feedback of
the capacitor current is equivalent to a resistor in parallel with capacitor as well.
The above analysis shows that either proportional feedback of the capacitor
current [8–11] or derivative feedback of the capacitor voltage [12, 13], or even
second-derivative feedback of the grid current [14, 15] can achieve the same
damping performance as a resistor in parallel with the capacitor. In practice,
derivative will lead to the amplification of high-frequency noise. Moreover, an ideal
derivator can hardly be implemented, and the discretization error introduced by a
digital derivator will degrade the performance of active damping. Comparatively,
proportional feedback of the capacitor current has been widely used for its simple
implementation and effectiveness. For brevity of illustration, hereinafter the
active-damping solution using proportional feedback of the capacitor current is
simply called the capacitor-current-feedback active damping.
Similarly, for the other passive-damping solutions depicted in Sect. 4.2, their
equivalent active-damping representations can also be derived through equivalent
transformation of the control block diagram, and they are not discussed here.
4.3.2 Notch-Filter-Based Active Damping
As depicted in Sect. 4.1, introducing a damping term makes the transfer function of
LCL filter, GLCL(s), become GLCL-d(s), which can be realized by either the
passive-damping solution using a resistor in parallel with the capacitor or the
capacitor-current-feedback active damping. The alternative method of introducing
the damping term is to multiply GLCL(s) by Gtrap(s) directly, and
Gtrap(s) = GLCL-d(s)/GLCL(s). According to (4.1) and (4.3), Gtrap(s) is derived as
90 4 Resonance Damping Methods of LCL Filter
Gtrap sð Þ ¼ GLCL�d sð ÞGLCL sð Þ ¼
s2 þx2r
s2 þ 2nxrsþx2r
ð4:14Þ
Obviously, Gtrap(s) is the transfer function of a notch filter. To realize the
multiplication of GLCL(s) and Gtrap(s) from the control perspective, Gtrap(s) can be
embedded into the control loop in cascade, as shown in Fig. 4.9.This method is
called the notch-filter-based active damping [16–18].
Figure 4.10a gives the Bode diagram of Gtrap(s). At the LCL filter resonance
frequency fr, an anti-resonance peak is provided by Gtrap(s), which cancels out the
resonance peak of LCL filter. While at the low- and high-frequency ranges, the
gains of Gtrap(s) are 0 dB, thus it will not affect the magnitude-frequency charac-
teristics of LCL filter at these frequency ranges. This means that the
notch-filter-based active damping can also achieve the desired damping perfor-
mance, as shown in Fig. 4.10b.
As seen in (4.14), the LCL-filter resonance frequency must be known exactly for
the purpose of implementing the notch-filter-based active damping. However, in
practice, due to the core saturation or aging of the filter components, the LCL filter
–+ +
–
i2(s)
iC(s)vr(s) KPWM
vinv(s)
Gtrap(s) sL1
1
sC
1
vC(s)
sL2
1+ –
vg(s)
Fig. 4.9 Notch-filter-based active damping
M
ag
ni
tu
de
 (d
B
)
50
0
50
100
150
Ph
as
e 
(°
)
Frequency (Hz)
10 102 103 104 105
180
90
0
90
180
fr
M
ag
ni
tu
de
 (d
B
)
50
0
50
100
150
Ph
as
e 
(°
)
Frequency (Hz)
10 102 103 104 105
0
90
180
270
360
fr
w/o damping
w/ notch-filter- 
based active damping
(a) Bode diagram of the notch filter
active damping
(b) LCL filter with notch-filter-based
Fig. 4.10 Frequency response of the notch-filter-based active damping
4.3 Active-Damping Solutions 91
parameters will vary and derivate from the designed ones. As a consequence, the
resonance frequencies of the LCL filter and the notch filter will not match exactly,
and the performance of notch-filter-based active damping becomes poorer or even
ineffective. To address this issue, the LCL filter resonance frequency can be
detected online [19, 20], and the resonant frequency of the notch filter is adjusted to
be adaptive to the resonance frequency variation. But, this will raise the hardware
cost and control complexity.
Taking all these practical issues into account, it can be concluded that the
capacitor-current-feedback active damping is more valuable in practical application.
For this reason, the capacitor-current-feedback active damping is adopted in fol-
lowing chapters of this book.
4.4 Summary
In this chapter, the resonance hazard of LCL filter is analyzed, and six basic
passive-damping solutions are discussed in term of their effects on the character-
istics of LCL filter. The analysis reveals that adding a resistor in parallel with
capacitor shows the best damping performance, but it results in a high power loss;
while adding a resistor in series with capacitor is the most valuable
passive-damping solution due to its low power loss. On the basis of a resistor in
series with capacitor, four improved passive-damping solutions are introduced to
further reduce the power loss of the damping resistor. Meanwhile, the
active-damping solutions equivalent to a resistor in parallel with capacitor are
derived, which can be classified into two kinds: one is the state-variable-feedback
active damping, including proportional feedback of the capacitor current, derivative
feedback of the capacitor voltage, and second-derivative feedback of the grid
current; the other is the notch-filter-based active damping. Among the
active-damping solutions, the capacitor-current-feedback active damping is superior
for its simple implementation and effectiveness. This chapter provides the basis for
the study of the control techniques of LCL-type grid-connected inverter in the
following chapters.
References
1. Goodwin, G.C., Graebe, S.F., Salgado, M.E.: Control System Design. Prentice Hall, Upper
Saddle River, NJ (2000)
2. Liserre, M., Dell’Aquila, A., Blaabjerg, F.: Stability improvements of an LCL-filter based
three-phase active rectifier. In: Proceeding IEEE Power Electronics Specialists Conference,
1195–1201 (2002)
3. Liserre, M., Blaabjerg, F., Hansen, S.: Design and control of an LCL-filter-based three-phase
active rectifier. IEEE Trans. Ind. Appl. 41(5), 1281–1291 (2005)
92 4 Resonance Damping Methods of LCL Filter
4. Wang, T.C., Ye, Z., Sinha, G., Yuan, X.: Output filter design for a grid-interconnected
three-phase inverter. In: Proceeding of the IEEE Power Electronics Specialists Conference,
779–784 (2003)
5. Rockhill, A.A., Liserre, M., Teodorescu, R., Rodriguez, P.: Grid filter design for a
multi-megawatt medium-voltage voltage source inverter. IEEE Trans. Ind. Electron. 58(4),
1205–1217 (2011)
6. Alzola, R.P., Liserre, M., Blaabjerg, F., Sebastián, R., Dannehl, J., Fuchs, F.W.: Analysis of
the passive damping losses in LCL-filter-based grid converters. IEEE Trans. Power Electron.
28(6), 2642–2646 (2013)
7. Mühlethaler, J., Schweizer, M., Blattmann, R., Kolar, J.W., Ecklebe, A.: Optimal design of
LCL harmonic filters for three-phase PFC rectifiers. IEEE Trans. Power Electron. 28(7),
3114–3125 (2013)
8. Tang, Y., Loh, P.C., Wang, P., Choo, F.H., Gao, F., Blaabjerg, F.: Generalized design of high
performance shunt active power filter with output LCL filter. IEEE Trans. Ind. Electron. 59(3),
1443–1452 (2012)
9. He, J., Li, Y.W.: Generalized closed-loop control schemes with embedded virtual impedances
for voltage source converters with LC or LCL filters. IEEE Trans. Power Electron. 27(4),
1850–1861 (2012)
10. Jia, Y., Zhao, J., Fu, X.: Direct grid current control of LCL-filtered grid-connected inverter
mitigating grid voltage disturbance. IEEE Trans. Power Electron. 29(3), 1532–1541 (2014)
11. Zou, Z., Wang, Z., Cheng, M.: Modeling, analysis, and design of multifunction
grid-interfaced inverters with output LCL filter. IEEE Trans. Power Electron. 29(7), 3830–
3839 (2014)
12. Dannehl, J., Fuchs, F.W., Hansen, S., Thøgersen, P.B.: Investigation of active damping
approaches for PI-based current control of grid-connected pulse width modulation converters
with LCL filters. IEEE Trans. Ind. Appl. 46(4), 1509–1517 (2010)
13. Xiao, H., Qu, X., Xie, S., Xu, J.: Synthesis of active damping for grid-connected inverters
with an LCL filter. In: Proceeding of the IEEE Energy Conversion Congress and Exposition,
550–556 (2012)
14. Hanif, M., Khadkikar, V., Xiao, W., Kirtley, J.L.: Two degrees of freedom active damping
technique for LCL filter-based grid connected PV systems. IEEE Trans. Ind. Electron. 61(6),
2795–2803 (2014)
15. Xu, J., Xie, S., Tang, T.: Active damping-based control for grid-connected LCL-filtered
inverter with injected grid current feedback only. IEEE Trans. Ind. Electron. 61(9), 4746–
4758 (2014)
16. Liserre, M., Teodorescu, R., Blaabjerg, F.: Stability of photovoltaic and wind turbine
grid-connected inverters for a large set of grid impedance values. IEEE Trans. Power
Electron. 21(1), 263–272 (2006)
17. Dannehl, J., Liserre, M., Fuchs, F.W.: Filter-based active damping of voltage source
converters with LCL filter. IEEE Trans. Ind. Electron. 58(8), 3623–3633 (2011)
18. Zhang, S., Jiang, S., Lu, X., Ge, B., Peng, F.Z.: Resonance issues and damping techniques for
grid-connected inverters with long transmission cable. IEEE Trans. Power Electron. 29(1),
110–120 (2014)
19. Liserre, M., Blaabjerg, F., Teodorescu, R.: Grid impedance estimation via excitation of LCL-
filter resonance. IEEE Trans. Ind. Appl. 43(5), 1401–1407 (2007)
20. Zhou, X., Fan, J., Huang, A.Q.: High-frequency resonance mitigation for plug-in hybrid
electric vehicles’ integration with a wide range of grid conditions. IEEE Trans. Power
Electron. 27(11), 4459–4471 (2012)
References 93
Chapter 5
Controller Design for LCL-Type
Grid-Connected Inverter
with Capacitor-Current-Feedback
Active-Damping
Abstract For the LCL-type grid-connected inverter, the capacitor-current-feedback
active-damping is equivalent to a resistor in parallel with the filter capacitor to damp
the LCL filter resonance. This active-damping method has no power loss and has
been widely used. Based on the capacitor-current-feedback active-damping and the
proportional-integral (PI) regulator as the grid current regulator, this chapterpro-
poses a step-by-step controller design method for the LCL-type grid-connected
inverter. By carefully examining the steady-state error, phase margin, and gain
margin, a satisfactory region of the capacitor-current-feedback coefficient and PI
regulator parameters for meeting the system specifications is obtained. With this
satisfactory region, it is very convenient to choose the controller parameters and
optimize the system performance. Besides, the proposed design method is extended
to the situations where PI regulator with grid voltage feedforward scheme or
proportional-resonant (PR) regulator is adopted. Finally, design examples of
capacitor-current-feedback coefficient and current regulator parameters are pre-
sented for a single-phase LCL-type grid-connected inverter, and experiments are
performed to verify the proposed design method.
Keywords Grid-connected inverter � LCL filter � Active damping � Controller
design � PI regulator � PR regulator
Chapter 4 has discussed the damping solutions to LCL filter resonance. Among the
six basic passive-damping solutions, adding a resistor in parallel with the filter
capacitor can effectively suppress the resonance peak without affecting the
magnitude-frequency characteristics at the low- and high-frequency ranges, but it
results in a high power loss. Capacitor-current-feedback active-damping is equiv-
alent to a resistor in parallel with the filter capacitor, and it has no power loss and
has been widely used. Based on the capacitor-current-feedback active-damping and
the proportional-integral (PI) regulator as the grid current regulator, this chapter
proposes a step-by-step controller design method for the LCL-type grid-connected
inverter. By carefully examining the steady-state error, phase margin, and gain
margin, a satisfactory region of the capacitor-current-feedback coefficient and PI
© Springer Nature Singapore Pte Ltd. and Science Press 2018
X. Ruan et al., Control Techniques for LCL-Type Grid-Connected Inverters,
CPSS Power Electronics Series, DOI 10.1007/978-981-10-4277-5_5
95
regulator parameters for meeting the system specifications is obtained [1, 2]. With
this satisfactory region, it is very convenient to choose the controller parameters and
optimize the system performance. Besides, the proposed design method is extended
to the situations where PI regulator with grid voltage feedforward scheme or pro-
portional-resonant (PR) regulator is adopted. Finally, design examples of
capacitor-current-feedback coefficient and current regulator parameters are pre-
sented for a single-phase LCL-type grid-connected inverter, and experiments are
performed to verify the proposed design method.
5.1 Modeling LCL-Type Grid-Connected Inverter
Figure 5.1 shows the configuration of the single-phase LCL-type grid-connected
inverter, where switches Q1 to Q4 compose the single-phase inverter bridge, and the
inverter-side inductor L1, the filter capacitor C, and the grid-side inductor L2
compose the LCL filter. The primary objective of the grid-connected inverter is to
control the grid current i2, so that it can be synchronized with the grid voltage vg,
and its amplitude can be regulated as required. Generally, the phase angle of vg is
obtained through a phase-locked loop (PLL), and the current amplitude reference is
generated by the outer voltage loop [3]. Since the dynamics of the voltage loop is
much slower than that of the grid current loop, the grid current loop can be eval-
uated separately, and the current amplitude reference is directly given as I* here. Hv
and Hi2 are the sensor gains of vg and i2, respectively. The sensed grid current is
compared to the current reference i�2, and the current error signal is sent to current
regulator Gi(s). The capacitor current iC is fed back to damp the LCL filter reso-
nance actively, and Hi1 is the feedback coefficient. Subtracting the
capacitor-current-feedback signal vic from the current regulator output vr, the
modulation reference vM is yielded.
vg
L1 L2
C
iC
vC
i1 i2+
–
cosθ
++ ––
PLL
vM i2*
Hv
Gi(s) *I
Hi2Hi1
vinvVin
Q1
Q2
Q3
Q4
Sinusoidal PWM
vr
vic
Control system
Fig. 5.1 Topology and
control scheme of LCL-type
grid-connected inverter
96 5 Controller Design for LCL-Type Grid …
Referring to Fig. 5.1, the mathematical model of LCL-type grid-connected
inverter can be obtained, as shown in Fig. 5.2, in which KPWM = Vin/Vtri is the
transfer function from vM to the inverter bridge output voltage vinv, with Vin and Vtri
as the input voltage and the amplitude of the triangular carrier, respectively. ZL1(s),
ZC(s), and ZL2(s) are the impedances of L1, C, and L2, expressed as
ZL1 sð Þ ¼ sL1; ZC sð Þ ¼ 1sC ; ZL2 sð Þ ¼ sL2 ð5:1Þ
Based on Fig. 5.2, a series of equivalent transformations of the control block
diagrams is shown in Fig. 5.3, where the dashed lines represent the original status,
and the solid lines represent the destination status. First, replacing the feedback of
capacitor voltage vC(s) with capacitor current iC(s), and relocating its feedback node
to the output of Gi(s), an equivalent block diagram is obtained, as shown in
Fig. 5.3a. Second, by combining the two feedback functions of iC(s), and moving
the feedback node of i2(s) from the output of 1/ZL1(s) to the output of Gi(s), the
equivalent block diagram is obtained, as shown in Fig. 5.3b. Third, moving the
feedback node of i2(s) from the output of Gi(s) to the output of ZC(s), and sim-
plifying the forward path from Gi(s) to ZC(s), results in the equivalent block dia-
gram shown in Fig. 5.3c, where
Gx1 sð Þ ¼ KPWMGi sð ÞZC sð ÞZL1 sð Þþ ZC sð ÞþHi1KPWM ð5:2aÞ
Hx1 sð Þ ¼ ZL1 sð ÞZC sð ÞZL1 sð Þþ ZC sð ÞþHi1KPWM ð5:2bÞ
Furthermore, Fig. 5.3c can be simplified to Fig. 5.3d, where
Gx2 sð Þ ¼ ZL1 sð Þþ ZC sð ÞþHi1KPWMZL1 sð ÞZL2 sð Þþ ZL1 sð Þþ ZL2 sð Þð ÞZC sð ÞþHi1KPWMZL2 sð Þ ð5:3Þ
KPWM +
–
Hi1
+
–
+ –
vg(s)
i2(s)ZC(s)
Hi2
vC(s)
+
–Gi(s)
+
–
1
ZL1(s)
1
ZL2(s)
i2(s)*
iC(s)
Fig. 5.2 Mathematical model of LCL-type grid-connected inverter with capacitor-current-
feedback active-damping
5.1 Modeling LCL-Type Grid-Connected Inverter 97
From Fig. 5.3d, and considering (5.1), the loop gain can be obtained as
TA sð Þ ¼ Gx1 sð ÞGx2 sð ÞHi2 ¼ Hi2KPWMGi sð Þs3L1L2Cþ s2L2CHi1KPWM þ s L1 þ L2ð Þ ð5:4Þ
and the grid current i2(s) is expressed as
i2 sð Þ ¼ 1Hi2
TA sð Þ
1þ TA sð Þ i
�
2 sð Þ �
Gx2 sð Þ
1þ TA sð Þ vg sð Þ, i21 sð Þþ i22 sð Þ ð5:5Þ
KPWM +
–
Hi1
+
–
+ –ZC(s)
Hi2
+
–Gi(s)
+
–
1
ZL1(s)
1
ZL2(s)
–
× ×
i2(s)*
ZC(s)
KPWM
vg(s)
i2(s)
(a)
KPWM + + –
+ –ZC(s)
Hi2
+
–Gi(s)
+
–
1
ZL1(s)
1
ZL2(s)–
×
i2(s)*
vg(s)
i2(s)
ZL1(s)
KPWM
Hi1+
ZC(s)
KPWM
(b)
+ –
Hi2
1
ZL2(s)
Gx1(s)+ –
Hx1(s)
–
i2(s)* i2(s)
vg(s)
+ –
Hi2
Gx1(s)+ – Gx2(s)
i2(s)* i2(s)
vg(s)
(c) (d)
Fig. 5.3 Equivalent transformations of the mathematical model of LCL-type grid-connected
inverter
98 5 Controller Design for LCL-Type Grid …
where
i21 sð Þ ¼ 1Hi2
TA sð Þ
1þ TA sð Þ i
�
2 sð Þ ð5:6aÞ
i22 sð Þ ¼ � Gx2 sð Þ1þ TA sð Þ vg sð Þ ð5:6bÞ
From (5.5), it is clear to see that i2(s) consists of two components i21(s) and
i22(s), where i21(s) is related to the reference tracking, and i22(s) is related to the
disturbance caused by the grid voltage.
5.2 Frequency Responses of Capacitor-Current-Feedback
Active-Damping and PI Regulator
According to (5.4), the Bode diagram of uncompensated loop gain (Gi(s) = 1) is
depicted, as shown in Fig. 5.4, where fo is the fundamental frequency, fc is the
crossover frequency of the loop gain, and fr is the LCL filter resonance frequency.
As shown in the figure, introducing the feedback of capacitor current can effectively
damp the resonance peak, and it only affects the magnitude plot of the loop gain
nearby fr. However, this damping solution has significant impact on the phase plot,
and the phase is decreased from −90° at the frequencies lower than fr. A larger Hi1
leads to a better resonance damping but a larger negative phase shift.
Since the phase plot of the loopgain crosses over −180° at fr, the crossover
frequency fc is needed to be lower than fr to preserve an adequate phase margin.
Fig. 5.4 Bode diagram of the
uncompensated loop gain
5.1 Modeling LCL-Type Grid-Connected Inverter 99
When calculating the magnitude of the loop gain at fc and the frequencies lower
than fc, the capacitor branch can be regarded as open circuit since the reactance of
the filter capacitor is far larger than that of the grid-side inductor; thus, the LCL
filter can be approximated as a pure inductor with the inductance of L1 + L2. From
(5.4), the approximated |TA(s)| can be obtained as
TA sð Þj j � Hi2KPWMGi sð Þs L1 þ L2ð Þ
����
���� ð5:7Þ
PI or PR regulator is usually adopted as the current regulator, and their Bode
diagrams are shown in Fig. 5.5. Here, the PI regulator is discussed as an instance,
and it is expressed as
Gi sð Þ ¼ Kp þ Kis ð5:8Þ
where Kp is the proportional gain, and Ki is the integral gain. The corner frequency
of PI regulator is fL = Ki/(2pKp). As shown in Fig. 5.5, at the frequencies around fL,
the slope of the magnitude plot changes from −20 dB/dec to 0 dB/dec, and the
phase escalates from −90° up to 0°. To alleviate the decrease of phase margin
resulted from PI regulator, fL is suggested to be sufficiently lower than fc. Thus, the
magnitude of Gi(s) can be approximated to Kp at fc and the frequencies higher than
fc. Note that the loop gain has unit magnitude at fc, i.e., |TA(j2pfc)| = 1, and sub-
stituting |Gi(j2pfc)| � Kp into (5.7) yields
Kp � 2pfc L1 þ L2ð ÞHi2KPWM ð5:9Þ
Fig. 5.5 Bode diagrams of PI
and PR regulators
100 5 Controller Design for LCL-Type Grid …
5.3 Constraints for Controller Parameters
5.3.1 Requirement of Steady-State Error
The steady-state error of the grid current is an important performance index in the
grid-connected inverter. As depicted in (5.5), the grid current i2 consists of i21 and
i22. Generally, the magnitude of the loop gain is sufficiently large at fo and then
1 + TA(j2pfo) � TA(j2pfo). Thus, according to (5.6a), i21 � i�2/Hi2, which means i21
is in phase with i�2. As discussed above, the capacitor branch can be regarded as
open circuit at fc and the frequencies lower than fc. Therefore, at the fundamental
frequency fo, (5.3) and (5.4) can be approximated as
Gx2 j2pfoð Þ � 1j2pfo L1 þ L2ð Þ ð5:10aÞ
TA j2pfoð Þ � Hi2KPWMGi j2pfoð Þj2pfo L1 þ L2ð Þ ð5:10bÞ
Substituting (5.10a, 5.10b) into (5.6b) yields
i22 � � vgHi2KPWMGi j2pfoð Þ ð5:11Þ
For PI regulator, there is Gi(j2pfo) � Ki/(j2pfo), so i22 � −j2pfovg/(Hi2KPWMKi),
which means that i22 is 90º lagging to vg.
Figure 5.6a shows the phasor diagram of i2, i21, i22, and vg, where h is the phase
angle that i�2 leads to vg and it is set according to the power factor (PF) requirement
of the system. As no active power is absorbed from the grid, there is h 2 [−90°,
90°]. As shown in the figure, the steady-state error of i2 includes the amplitude error
EA and the phase error d, and EA is expressed as
vg
i2
i22
0
δ θ
i2*
i21
−90 −45 0 45 90
0
I 22
θ (°)
I22_δPI I22_EAPI
(a) Phasor diagram of i2, i21, i22, and vg (b) Curves of I22_EAPI and I22_δPI as θ varies
Fig. 5.6 Steady-state error of the grid current with PI regulator
5.3 Constraints for Controller Parameters 101
EA ¼ Hi2I2 � I
�
2
I�2
����
���� ¼ Hi2I�2
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
I221 þ I222 � 2I21I22 sin h
q
� 1
����
���� ð5:12Þ
where I�2 , I2, I21, and I22 are the rms values of i
�
2, i2, i21, and i22, respectively.
Equation (5.12) can be rewritten as
�EA ¼ Hi2I�2
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
I221 þ I222 � 2I21I22 sin h
q
� 1 ð5:13Þ
Substituting I21 � I�2 /Hi2 into (5.13), the four roots of I22 can be solved as
I22 rt1 ¼ I
�
2
Hi2
sin hþ
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
sin2 h� 2EA þE2A
q� �
I22 rt2 ¼ I
�
2
Hi2
sin h�
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
sin2 h� 2EA þE2A
q� �
I22 rt3 ¼ I
�
2
Hi2
sin hþ
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
sin2 hþ 2EA þE2A
q� �
I22 rt4 ¼ I
�
2
Hi2
sin h�
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
sin2 hþ 2EA þE2A
q� �
ð5:14Þ
Apparently, I22_rt4 < 0, and it is an invalid root. The upper boundary of I22 con-
strained by EA is denoted by I22_EAPI, and it is determined by the smallest one of
I22_rt1 * I22_rt3. If h 2 [−90°, −h1] where h1 ¼ arcsin
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2EA � E2A
p
, both I22_rt1 and
I22_rt2 are negative and invalid. If h 2 (−h1, h1), sin2 h� 2EA þE2A < 0, and I22_rt1
and I22_rt2 are inexistent. Thus, for h 2 [−90°, h1), I22_EAPI = I22_rt3. While for
h 2 [h1, 90°], I22_rt1 * I22_rt3 are all valid and I22_rt2 is the smallest, so
I22_EAPI = I22_rt2. In summary, I22_EAPI is expressed as
I22 EAPI ¼
I�2
Hi2
sin hþ
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
sin2 hþ 2EA þE2A
q� �
; h 2 �90�; h1½ Þ
I�2
Hi2
sin h�
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
sin2 h� 2EA þE2A
q� �
; h 2 h1; 90�½ �
8>><
>>: ð5:15Þ
Applying sine law to Fig. 5.6a yields
sin d ¼ I22
I21
sin 90� þ h� dð Þj j ð5:16Þ
From Fig. 5.6a and (5.16), it is obvious that d = 0° when h = ±90°, and when
h 6¼ ±90°, the upper boundary of I22 constrained by d, which is denoted by I22_dPI,
is expressed as
102 5 Controller Design for LCL-Type Grid …
I22 dPI � I
�
2
Hi2
sin d
cos h� dð Þ
����
���� ð5:17Þ
According to (5.15) and (5.17), the curves of I22_EAPI and I22_dPI as the functions
of h are depicted in Fig. 5.6b, from which it can be seen that I22_EAPI is minimum
when h � ±90° and I22_dPI is minimum when h � 0°.
Considering (5.10b) and (5.11), I22 can be approximated as
I22 � VgHi2KPWM Gi j2pfoð Þj j �
Vg
2pfo L1 þ L2ð Þ TA j2pfoð Þj j ð5:18Þ
According to (5.18), the magnitude of the loop gain at the fundamental fre-
quency fo, which is denoted by Tfo, can be expressed as
Tfo ¼ 20 lg TA j2pfoð Þj j � 20 lg Vg2pfo L1 þ L2ð ÞI22 ð5:19Þ
where the unit of Tfo is dB. (5.19) indicates that I22 is related to Tfo; thus, the
requirement of steady-state error can be further converted into the requirement of
Tfo. In order to satisfy the requirements of EA and d at the same time, I22 in (5.19)
should be set as the smaller one between I22_EAPI and I22_dPI.
5.3.2 Controller Parameters Constrained by Steady-State
Error and Stability Margin
Substituting (5.8) into (5.7), the expression of Tfo with PI regulator is given as
Tfo ¼ 20 lg TA j2pfoð Þj j ¼ 20 lg
Hi2KPWM Kp þ Kij2pfo
� �
j2pfo L1 þ L2ð Þ
������
������ ð5:20Þ
Substituting (5.9) into (5.20) and manipulating, the Ki constrained by Tfo is
obtained as
Ki Tfo ¼ 4p
2fo L1 þ L2ð Þ
Hi2KPWM
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
10
Tfo
20 fo
� �2
�f 2c
r
ð5:21Þ
According to (5.4), the phase margin PM can be expressed as
PM ¼ 180� þ\ Hi2KPWMGi sð Þ
s3L1L2Cþ s2L2CHi1KPWM þ s L1 þ L2ð Þ
����
s¼j2pfc
ð5:22Þ
5.3 Constraints for Controller Parameters 103
Substituting (5.8) into (5.22) and manipulating yields
PM ¼ arctan 2pL1 f
2
r � f 2c
� 	
Hi1KPWMfc
� arctan Ki
2pfcKp
ð5:23Þ
Applying tangent on both sides of (5.23) and manipulating, the Ki constrained by
PM is obtained as
Ki PM ¼ 2pfcKp
2pL1 f 2r � f 2c
� 	� Hi1KPWMfc tan PM
2pL1 f 2r � f 2c
� 	
tan PMþHi1KPWMfc
ð5:24Þ
If the selected Ki meets the constraints of Tfo and PM at the same time, then
Ki_Tfo = Ki_PM. Substituting (5.9) and (5.21) into (5.24), the Hi1 constrained by Tfo
and PM is obtained as
Hi1 Tfo PM ¼
2pL1 f 2r � f 2c
� 	
f 2c �fo
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
10
Tfo
20 fo
� �2
�f 2c
r
tan PM
 !
KPWMfc f 2c tan PMþ fo
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
10
Tfo
20 fo
� �2
�f 2c
r ! ð5:25Þ
Since the phase plot of the loop gain crosses over −180° at fr, the gain margin
GM can be expressed as
GM ¼ �20 lg TA j2pfrð Þj j ð5:26Þ
where the unit of GM is dB. It is worth noting that the magnitude of the loop gain
TA(s) given in (5.7) is not accurate at fr. Therefore, substituting the loop gain
without approximation, i.e., (5.4), and (5.9) into (5.26), the Hi1 constrained by GM
is obtained as
Hi1 GM ¼ 10GM20 � 2pfcL1KPWM ð5:27Þ
5.3.3 Pulse-Width Modulation (PWM) Constraint
Figure 5.7 gives the schematic diagram of a modulation reference compared to the
triangular carrier in the PWM inverter, and fsw is the switching frequency. In the
LCL-type grid-connected inverter, the switching ripple on the inverter side is almost
bypassed by the filter capacitor, letting the fundamental sinusoidal current to be
injected into the grid. Hence, the current regulator output vr is nearly constant
during a switching period. As reported in Sect. 5.1, the modulation reference vM is
104 5 Controller Design for LCL-Type Grid …
the difference between vr and the capacitor current feedback signal vic. Therefore,
the rate of change of vM is dependent on that of vic, which has a maximum value of
Hi1Vin/L1 (i.e., multiply the maximum rate of change of the inverter-side inductor
current by the capacitor-current-feedback coefficient). From Fig. 5.7, it can be seen
that the rate of change of the triangular carrier is 4Vtrifsw. In order to avoid the
multiple switching transitions, the maximum rate of change of the modulation
reference should be smaller than that of the triangular carrier [4–6], i.e.,
Hi1Vin
L1
\4Vtrifsw ð5:28Þ
According to (5.28), the Hi1 constrained by PWM can be obtained as
Hi1 PWM ¼ 4fswL1VtriVin ¼
4fswL1
KPWM
ð5:29Þ
5.4 Design Procedure for Capacitor-Current-Feedback
Coefficient and PI Regulator Parameters
Based on the above analysis, a design procedure for capacitor-current-feedback
coefficient and PI regulator parameters is given as follows.
Step 1: Specify the requirements of Tfo, PM, and GM. Specifically, Tfo is deter-
mined by the requirement of the steady-state error, and PM and GM are determined
by the requirements of the dynamic response and robustness of the system. As
shown in Fig. 5.6a, the steady-state error is more notable under light-load condi-
tion, and thus, Tfo needs to be specified by the most severe situation presented in the
standards, e.g., PF must be greater than 0.85 under 10% of the rated load condition
[7] or PF must be greater than 0.98 under half-load condition [8]. Besides, PM is set
in the range (30º, 60º) for good dynamic response, and GM > 3 dB is preserved to
ensure the system robustness.
t
1/fsw
Vtri
−Vtri
0
vic
vr vM
Carrier
Fig. 5.7 Schematic diagram
of a modulation reference
compared to the triangular
carrier
5.3 Constraints for Controller Parameters 105
Step 2: Based on the specific Tfo, PM, and GM, draw the curves of Hi1_Tfo_PM,
Hi1_GM, and Hi1_PWM as the functions of fc according to (5.25), (5.27), and (5.29),
respectively, and then, get the satisfactory region of fc and Hi1.
Figure 5.8 shows the satisfactory region of fc and Hi1. The area upon the dashed
line meets the requirement of GM, and the area under the solid line meets the
requirements of Tfo and PM. Thus, the shaded area between these two lines includes
all the possible fc and Hi1 satisfying the aforementioned specifications. From
Fig. 5.8, it can be seen that:
(1) With the increase of fc, the lower boundary of Hi1 constrained by GM increases.
This is because that as fc approaching fr, the resonance peak should be damped
lower to achieve the same GM, and thus, a larger Hi1 is needed.
(2) With the increase of fc, the upper boundary of Hi1 ascends first and then
descends. This is because that when fc is relatively low and is close to the
corner frequency of PI regulator, the negative phase shift caused by PI regulator
is significant at fc, and thus, a smaller Hi1 has to be chosen to preserve the
desired phase margin. With the increase of fc, the impact of the negative phase
shift caused by PI regulator becomes less, so the upper boundary of Hi1 rises
first. But when fc keeps increasing and approaches fr, the negative phase shift
caused by the capacitor-current-feedback active-damping becomes larger and
plays the dominant role, so the upper boundary of Hi1 falls then.
It is worth noting that if the requirements of Tfo, PM, and GM specified in Step 1
are too strict, the satisfactory region might be very small or even not exist. If so,
return to Step 1 and modify the specifications and then renew Step 2.
Step 3: Select a proper fc from the satisfactory region of fc and Hi1, and then,
calculate Kp from (5.9). A higher fc is expected to improve the dynamic perfor-
mance and low-frequency gains. Nevertheless, in order to suppress the
high-frequency switching noise, fc is usually limited to 1/10 of the switching fre-
quency [9].
Step 4: Select a proper Hi1 according to the requirements of PM and GM. The lower
boundary of Hi1 is Hi1_GM, and the upper boundary of Hi1 is the smaller one
between Hi1_PWM and Hi1_Tfo_PM. For a specific fc, increasing Hi1 will decrease PM
0
H
i1
fc
GM constraint
PM and Tfo
constraint
Hi1_PWM
Fig. 5.8 Satisfactory region
of fc and Hi1 constrained by
Tfo, PM, GM, and PWM
106 5 Controller Design for LCL-Type Grid …
but not affect Tfo. Therefore, while retaining enough GM, a smaller Hi1 is preferred
to improve the dynamic performance.
Step 5: After fc and Hi1 have been determined, select a proper Ki according to the
requirements of Tfo and PM. The upper and lower boundaries of Ki are Ki_PM and
Ki_Tfo, respectively. A larger Ki leads to a higher Tfo but a smaller PM. Therefore, Ki
needs to be chosen by making a trade-off between Tfo and PM.
Step 6: Check the compensated loop gain to ensure all the specifications are well
satisfied.
Moreover, it should be noted that the satisfactory region is an effective tool not
only to choose but also to optimize the controller parameters. While meeting the
basic specifications depicted above, the controller parameters can be further opti-
mized as follows.
(1) For a specific fc, a larger Ki can be chosen for a higher Tfo;
(2) A larger Hi1 can be chosen for a larger GM; and
(3) A smaller Ki and Hi1 can be chosen for a larger PM.
5.5 Extension of the Proposed Design Method
In practical applications, in order to reduce the steady-state error of the grid current,
PI regulator with the grid voltage feedforward scheme (the grid voltage feedforward
scheme will be discussed in Chaps. 6 and 7 of this book) or PR regulator is usually
adopted. The controller design method proposed in Sect. 5.4 is extended to these
cases in this section.
5.5.1 Controller Design Based on PI Regulator with Grid
Voltage Feedforward Scheme
PI regulator is widely used for its simplicity and effectiveness, but it cannot achieve
zero steady-state error of the grid current for a single-phase grid-connected inverter.
To overcome this drawback, a grid voltage feedforward scheme is proposed in [10].
With this scheme, the disturbance component i22 caused by the grid voltage vg can
be eliminated from the grid current. Thus, as shown in Fig. 5.6, only the amplitude
error EA is left to be considered. From (5.6a), EA is expressed as
EA ¼ I
�
2 � Hi2I21
I�2
����
���� ¼ 1� TA j2pfoð Þ1þ TA j2pfoð Þ
����
����
����
���� ¼ 1þ TA j2pfoð Þj j � TA j2pfoð Þj j1þ TA j2pfoð Þj j
����
����
ð5:30Þ
5.4 Design Procedure for Capacitor-Current-Feedback … 107
Since |TA(j2pfo)| 	 1, then |1 + TA(j2pfo)| � 1 + |TA(j2pfo)|, so (5.30) can be
approximated as
EA � 11þ TA j2pfoð Þj j �
1
TA j2pfoð Þj j ¼ 10
�Tfo20 ð5:31Þ
Therefore, the requirement of Tfo in Step 1 can be specified as Tfo � 20lg(1/EA).Since the grid voltage feedforward scheme has no effect on the loop gain, the
controller design method proposed in Sect. 5.4 can be extended to PI regulator plus
the grid voltage feedforward scheme without any other modification, and it is not
repeated here.
5.5.2 Controller Design Based on PR Regulator
Compared with PI regulator, PR regulator can provide far larger gain at the fun-
damental frequency and thus can greatly reduce the steady-state error [11, 12]. In
order to preserve certain adaptability to the grid frequency, a practical alternative of
PR regulator is adopted as
Gi sð Þ ¼ Kp þ 2Krxiss2 þ 2xisþx2o
ð5:32Þ
where Kp is the proportional gain, Kr is the resonant gain, xo = 2pfo is the fun-
damental angular frequency, and xi is the bandwidth of the resonant part con-
cerning −3 dB cutoff frequency, which means the gain of the resonant part is
Kr
 ffiffiffi
2
p
at xo ± xi. For small-scale photovoltaic power stations, the grid-connected
inverter is required to work normally when the grid frequency fluctuates between
49.5 Hz and 50.2 Hz [8], and thus, the maximum frequency fluctuation is
Df = 0.5 Hz. In order to attain enough gain in the entire working frequency range,
xi = 2pDf = p rad/s is set.
The Bode diagram of PR regulator is depicted with the dashed line, as shown in
Fig. 5.5. As seen, PR regulator can provide a large gain at fo, but it also introduces
negative phase shift at the frequencies higher than fo, especially at the frequencies
close to fo. To alleviate the decrease of phase margin caused by this negative phase
shift, the crossover frequency fc is suggested to be far higher than fo. Thus, similar
to PI regulator, PR regulator can also be approximated to Kp in magnitude at fc and
the frequencies higher than fc. Hence, (5.9), (5.27), and (5.29) still work, that is to
say, Kp can be expressed as the function of fc given by (5.9), and Hi1 is constrained
by the requirements of GM and PWM given by (5.27) and (5.29), respectively.
Different from PI regulator, PR regulator given in (5.32) is expressed as
Gi(j2pfo) = Kp + Kr at the fundamental frequency. Substituting it into (5.11) yields
i22 � −vg/[Hi2KPWM(Kp + Kr)], which means i22 and vg are opposite in phase.
108 5 Controller Design for LCL-Type Grid …
Figure 5.9a shows the phasor diagram of i2, i21, i22, and vg, from which it can be
seen that d = 0° when h = 0°. The upper boundaries of I22 constrained by EA and d
are denoted by I22_EAPR and I22_dPR, respectively, and they can be derived from
Fig. 5.9a, i.e.,
I22 EAPR ¼
I�2
Hi2
cos hþ
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
cos2 hþ 2EA þE2A
p� �
; h 2 �90�;�h2½ Þ [ h2; 90�ð �
I�2
Hi2
cos h�
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
cos2 h� 2EA þE2A
p� �
; h 2 �h2; h2½ �
8<
:
ð5:33aÞ
I22 dPR � I
�
2
Hi2
sin d
sin hþ dð Þ
����
���� ð5:33bÞ
where h2 ¼ arcsin
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2EA � E2A
p
:
According to (5.33a, 5.33b), the curves of I22_EAPR and I22_dPR as the functions
of h are depicted, as shown in Fig. 5.9b. As seen, I22_EAPR is minimum when
h � 0° and I22_dPR is minimum when h � ±90°. Substituting the smaller one
between I22_EAPR and I22_dPR into (5.19), the desired Tfo for meeting the require-
ment of steady-state error is obtained. Further, considering that |TA[j2p
(fo ± Df)]| = 0.707Tfo, 3 dB needs to be added to (5.19) to ensure the requirement
of steady-state error is met when the grid frequency fluctuates between fo ± Df.
Substituting (5.32) into (5.7), the expression of Tfo with PR regulator is given as
Tfo ¼ 20 lg TA j2pfoð Þj j ¼ 20 lg
Hi2KPWM Kp þKr
� 	
2pfo L1 þ L2ð Þ ð5:34Þ
Substituting (5.9) into (5.34) and manipulating, the Kr constrained by Tfo is
obtained as
Kr Tfo ¼ 10
Tfo
20 fo � fc
� � 2p L1 þ L2ð Þ
Hi2KPWM
ð5:35Þ
vg
i2
i22
θ
0
δ
i2*
i21
−90 −45 0 45 90
0
I 22
θ (°)
I22_δPRI22_EAPR
(a) Phasor diagram of i2, i21, i22, and vg (b) Curves of I22_EAPR and I22_δPR as θ varies
Fig. 5.9 Steady-state error of the grid current with PR regulator
5.5 Extension of the Proposed Design Method 109
At the crossover frequency fc, PR regulator can be approximated as
Gi(s) � Kp + 2Krxi/s. Substituting it into (5.4), the phase margin PM is derived as
PM ¼ arctan 2pL1 f
2
r � f 2c
� 	
Hi1KPWMfc
� arctan Krxi
pfcKp
ð5:36Þ
Applying tangent on both sides of (5.36) and manipulating, the Kr constrained
by PM is obtained as
Kr PM ¼ pfcKpxi
2pL1 f 2r � f 2c
� 	� Hi1KPWMfc tan PM
Hi1KPWMfc þ 2pL1 f 2r � f 2c
� 	
tan PM
ð5:37Þ
If the selected Kr meets the constraints of Tfo and PM at the same time, then
Kr_Tfo = Kr_PM. Substituting (5.9) and (5.35) into (5.37), the Hi1 constrained by Tfo
and PM is obtained as
H0i1 Tfo PM ¼
2pL1 f 2r � f 2c
� 	
KPWMfc
pf 2c � 10
Tfo
20 fo � fc
� �
xi tan PM
10
Tfo
20 fo � fc
� �
xi þ pf 2c tan PM
ð5:38Þ
According to H0i1 Tfo PM, Hi1_GM, and Hi1_PWM, the satisfactory region of fc and
Hi1 for meeting the requirements of Tfo, PM, and GM can be obtained. Thus, the
controller design method proposed in Sect. 5.4 can also be extended to PR
regulator.
5.6 Design Examples
Based on the system parameters of a single-phase LCL-type grid-connected inverter
given in Table 5.1, design examples of the controller parameters are presented in
this section for PI and PR regulators, respectively.
Table 5.1 Parameters of single-phase prototype
Parameter Symbol Value Parameter Symbol Value
Input voltage Vin 360 V Inverter-side inductor L1 600 lH
Grid voltage (rms) Vg 220 V Filter capacitor C 10 lF
Output power Po 6 kW Grid-side inductor L2 150 lH
Fundamental frequency fo 50 Hz Switching frequency fsw 10 kHz
Amplitude of the
triangular carrier
Vtri 3.05 V Grid current feedback
coefficient
Hi2 0.15
110 5 Controller Design for LCL-Type Grid …
5.6.1 Design Results with PI Regulator
According to the design procedure given in Sect. 5.4, the requirements of Tfo, PM,
and GM are specified at first, which are as follows: (1) Tfo > 52 dB to ensure that
PF is greater than 0.98 under half-load condition [8], which corresponds to
PF > 0.994 and EA 
 0.5% under full-load condition; (2) PM > 45° to preserve a
good dynamic performance; and (3) GM > 3 dB to ensure the system robustness.
Based on these specifications, the satisfactory region of fc and Hi1 is obtained
according to (5.25), (5.27), and (5.29), shown as the shaded area in Fig. 5.10, from
which a group of controller parameters is properly selected as follows.
In order to perform a fast dynamic response, the crossover frequency fc is
suggested to be as high as possible. Since the grid-connected inverter employs the
unipolar sinusoidal PWM, its equivalent switching frequency is 20 kHz, and thus, fc
is set at 2 kHz here. Substituting it into (5.9) yields Kp = 0.45. After fc is selected,
the possible interval of Hi1 can be determined. As shown in Fig. 5.10, the lower
boundary of Hi1 is Hi1_GM, and the upper boundary of Hi1 is Hi1_Tfo_PM.
Substituting fc = 2 kHz into (5.25) and (5.27), respectively, the possible range of
Hi1 is calculated as [0.09, 0.165]. Here, Hi1 = 0.1 is chosen to get a larger phase
margin. At last, substituting fc = 2 kHz into (5.21) yields Ki_Tfo = 1657, and sub-
stituting fc = 2 kHz, Kp = 0.45, and Hi1 = 0.1 into (5.24) yields Ki_PM = 2626, and
thus, the possible interval of Ki is [1657, 2626]. By trading off between the
steady-state error and phase margin, Ki = 2200 is chosen.
With the controller parameters designed above, the Bode diagram of compen-
sated loop gain is depicted in Fig. 5.11, where fc = 2.05 kHz, Tfo = 54.4 dB,
PM = 48.1°, and GM = 4.29 dB can be identified. It is obvious that all the spec-
ifications are satisfied as expected.
Figure 5.12 shows the Bode diagrams of compensated loop gain considering the
variations in the LCL filter parameters. The real grid contains the inductive grid
impedance, which contributes to the grid-side inductor, and canbe regarded as a
part of L2. It is found that even if L1 and C vary in ±20% and L2 varies in −30% to
+100% (considering the grid impedance), the crossover frequency is still higher
0
0.05
0.10
0.15
0.20
H
i1
0.5 1.0
fc (kHz)
1.5 2.0 2.5
GM=3dB 
constraint
0.25
Hi1_PWM
 PM=45°,
Tfo=52dB
constraint
3.0
Fig. 5.10 Satisfactory region
of fc and Hi1 constrained by
Tfo, PM, GM, and PWM with
PI regulator
5.6 Design Examples 111
than 1.77 kHz, the phase margin is larger than 36°, and the gain margin is larger
than 4 dB. All of these results verify a strongly robust system. As shown in
Fig. 5.12a, the variation in L1 has little effect on the loop gain. The variation in
C mainly affects the phase margin, as shown in Fig. 5.12b. This is because that with
the increase of C, the resonance frequency fr decreases, and thus, the impact of the
capacitor-current-feedback active-damping on the phase margin becomes more
significant. The variation in L2 mainly affects the crossover frequency and phase
margin, as shown in Fig. 5.12c. This is because that with the increase of L2, both fc
and fr decrease, and thus, the impact of the negative phase shift caused by PI
regulator and the capacitor-current-feedback active-damping on the phase margin
becomes more significant. Therefore, to deal with the wide-range variations of filter
parameters, a relatively smaller Ki and Hi1 can be selected to improve the phase
margin.
5.6.2 Design Results with PR Regulator
When PR regulator is adopted, the requirements of Tfo, PM, and GM are given as
follows: (1) Tfo > 75 dB to ensure that the amplitude error of the grid current is less
than 1% when the grid frequency fluctuates in ±0.5 Hz; (2) PM > 45° to preserve a
good dynamic performance; and (3) GM > 3 dB to ensure the system robustness.
Based on these specifications, the satisfactory region of fc and Hi1 is obtained
according to (5.27), (5.29), and (5.38), shown as the shaded area in Fig. 5.13.
Fig. 5.11 Bode diagram of
compensated loop gain with
PI regulator
112 5 Controller Design for LCL-Type Grid …
Fig. 5.12 Bode diagrams of compensated loop gain considering the variations in the LCL filter
parameters
5.6 Design Examples 113
Similar to the design procedure in Sect. 5.6.1, fc = 2 kHz is still set here, which
leads to Kp = 0.45 as well. As shown in Fig. 5.13, for fc = 2 kHz, the lower
boundary of Hi1 is Hi1_GM, and the upper boundary of Hi1 is Hi1_PWM. Substituting
fc = 2 kHz into (5.27) and (5.29), respectively, the possible interval of Hi1 is cal-
culated as [0.09, 0.2]. Here, Hi1 = 0.1 is chosen. At last, substituting fc = 2 kHz
into (5.35) yields Kr_Tfo = 74, and substituting fc = 2 kHz, Kp = 0.45, and
Hi1 = 0.1 into (5.37) yields Kr_PM = 418, and thus, the possible interval of Kr is
[74, 418], and here, Kr = 350 is chosen.
Fig. 5.12 (continued)
0
0.05
0.10
0.15
0.20
0.25
0.5 1.0 3.0
fc (kHz)
1.5 2.0 2.5
H
i1
0.30
0.35
Hi1_PWM
 PM=45°,
Tfo=75dB
constraint
GM=3dB 
constraint
Fig. 5.13 Satisfactory region
of fc and Hi1 constrained by
Tfo, PM, GM, and PWM with
PR regulator
114 5 Controller Design for LCL-Type Grid …
With the controller parameters designed above, the Bode diagram of compen-
sated loop gain is depicted in Fig. 5.14, where fc = 2.05 kHz, Tfo = 88.4 dB,
PM = 48.1°, and GM = 4.29 dB can be identified. It is obvious that all the spec-
ifications are satisfied as expected.
5.7 Experimental Verification
In order to verify the theoretical analysis and the effectiveness of the proposed
controller design method, a 6-kW prototype is built in the laboratory according to
the parameters listed in Table 5.1. Figure 5.15 shows the photograph of the
prototype.
Figure 5.16 shows the experimental results with PI regulator designed in
Sect. 5.6.1. The experimental waveform at full load is given in Fig. 5.16a, where
the measured power factor is 0.995, phase error is 3.7°, and fundamental rms value
of i2 is 27.13 A (since the reference is 27.27 A, the amplitude error is 0.5%). All of
these results are in agreement with the design target in Sect. 5.6.1. Figure 5.16b
shows the experimental result when the grid current reference steps between half
load and full load. According to (5.4), the theoretical percentage overshoot and
settling time of i2 are calculated as 45% and 1.5 ms using MATLAB. In practice,
the measured percentage overshoot i.e., r/Istep in Fig. 5.16(b) and settling time are
about 34% and 1.5 ms, respectively. Due to the effect of the parasitic parameters,
the measured percentage overshoot is a little smaller than the theoretical value.
Fig. 5.14 Bode diagram of
compensated loop gain with
PR regulator
5.6 Design Examples 115
Time: [5 ms/div]
vg:[100 V/div]
i2:[20 A/div]
PF = 0.995
(a) Steady-state experimental results under full load condition
vg:[100 V/div] i2:[20 A/div]
σ
Istep
Time: [20 ms/div]
(b) Experimental results when the grid current reference 
steps between half load and full load
Fig. 5.16 Experimental
results with PI regulator
(Kp = 0.45, Ki = 2200,
Hi1 = 0.1)
Fig. 5.15 Photograph of the prototype
116 5 Controller Design for LCL-Type Grid …
Figure 5.17 shows the experimental results with PR regulator designed in
Sect. 5.6.2. The measured power factor is 0.999, fundamental rms value of i2 is
27.1 A (the amplitude error is 0.6%), percentage overshoot is about 35%, and
settling time is about 1.5 ms.
The experimental results in Figs. 5.16 and 5.17 show that with the proposed
controller design method, the LCL filter resonance is damped effectively, and sat-
isfactory steady-state and transient performances are obtained at the same time.
Taking PI regulator for instance, Fig. 5.18 shows the plots of the measured
fundamental rms value, power factor, and percentage overshoot of i2 when Hi1
varies. As Hi1 increases from 0.1 to 0.2, the measured percentage overshoot
increases from 34% to 50%, while the fundamental rms value and power factor of i2
remain 27.13 A and 0.995, respectively. Figure 5.19 shows the experimental results
when Hi1 is reduced intentionally (Kp = 0.45, Ki = 2200, Hi1 = 0.016), where
significant oscillation arises in the grid current. From Figs. 5.18 and 5.19, it can be
seen that increasing Hi1 has no improvement in the steady-state error, but it reduces
the phase margin and thus increases the percentage overshoot, while a too small Hi1
will result in current oscillation or even system instability. The experimental results
confirm the analysis of Hi1 in Sect. 5.3.
Time: [5 ms/div]
vg:[100 V/div]
i2:[20 A/div]
PF=0.999
(a) Steady-state experimental results under full load condition
Time: [20 ms/div]
vg:[100 V/div] i2:[20 A/div]
σ
Istep
(b) Experimental results when the grid current reference 
steps between half and full load
Fig. 5.17 Experimental
results with PR regulator
(Kp = 0.45, Kr = 350,
Hi1 = 0.1)
5.7 Experimental Verification 117
Figure 5.20 shows the plots of the measured fundamental rms value, power
factor, and percentage overshoot of i2 when Ki varies. As Ki increases from 600 to
2600, the measured fundamental rms value of i2 increases from 25.69 A to 27.13 A,
power factor increases from 0.935 to 0.996, and the percentage overshoot increases
from 11% to 37%. Figure 5.21 shows the experimental results when Ki is increased
intentionally (Kp = 0.45, Ki = 7400, Hi1 = 0.1), where significant oscillation arises
0.1 0.12 0.14 0.16 0.18 0.2
0.991
0.992
0.993
0.994
0.995
0.996
0.997
0.998
0.999
1.000
30
32
34
36
38
40
42
44
46
48
50
0.990
Po
w
er
 fa
ct
or
Pe
rc
en
ta
ge
 o
ve
rs
ho
ot
 (%
)
Hi1
26.5
26.6
26.7
26.8
26.9
27.0
27.1
27.2
27.3
27.4
26.4
Fu
nd
am
en
ta
l r
m
s v
al
ue
 (A
)
PO
PF
rms value
Fig. 5.18 Fundamental rms
value, power factor, and
percentage overshoot as Hi1
varies
Time: [2 ms/div]
vg:[100 V/div]
i2:[20 A/div]
Fig. 5.19 Experimental
results with a small Hi1
(Kp = 0.45, Ki = 2200,
Hi1 = 0.016)
600 1000 1400 1800 2200 2600
0.93
0.94
0.95
0.96
0.97
0.98
0.99
1.00
10
15
20
25
30
35
40
45
Ki
Po
w
er
 fa
ctor
Pe
rc
en
ta
ge
 o
ve
rs
ho
ot
 (%
)
25.5
25.8
26.1
26.4
26.7
27.3
27.6
27.0
Fu
nd
am
en
ta
l r
m
s v
al
ue
 (A
)
PO
PF
rms value
Fig. 5.20 Fundamental rms
value, power factor, and
percentage overshoot as Ki
varies
118 5 Controller Design for LCL-Type Grid …
in the grid current. From Figs. 5.20 and 5.21, it can be seen that increasing Ki has
significant improvement in the steady-state error, but it reduces the phase margin
and thus increases the percentage overshoot as well, and a too large Ki will result in
current oscillation or even system instability. The experimental results confirm the
analysis of Ki in Sect. 5.3.
5.8 Summary
In this chapter, the mathematical model of LCL-type grid-connected inverter is
built, and the frequency responses of capacitor-current-feedback active-damping
and current regulators are investigated. The analysis reveals that
(1) capacitor-current-feedback active-damping can effectively suppress the LCL
filter resonance, but it decreases the system phase below the resonance frequency,
and (2) PI and PR regulators determine the crossover frequency and the
low-frequency gains of the system, but they also introduce negative phase shift.
Due to the interaction between the capacitor-current-feedback active-damping and
the current regulator, the negative phase shifts caused by each other are added
together, which would easily lead to system instability. Based on the steady-state
error, phase margin, and gain margin, this chapter proposes a step-by-step controller
design method to determine and optimize the controller parameters. The proposed
method is raised based on PI regulator and extended to PI regulator with grid
voltage feedforward scheme and PR regulator, respectively. Finally, design
examples are presented for a single-phase LCL-type grid-connected inverter, and
experiments are performed on a 6-kW prototype. Experimental results show that
with the proposed controller design method, the LCL filter resonance is damped
effectively, and satisfactory steady-state and transient performances are obtained at
the same time.
Time: [2 ms/div]
vg:[100 V/div]
ig:[20 A/div]
Fig. 5.21 Experimental
result with a large Ki
(Kp = 0.45, Ki = 7400,
Hi1 = 0.1)
5.7 Experimental Verification 119
References
1. Bao, C.: Design of current regulator and capacitor-current-feedback active damping for LCL-
type grid-connected inverter (in Chinese). M.S. thesis. Huazhong University of Science and
Technology, Wuhan, China (2013)
2. Bao, C., Ruan, X., Wang, X., Li, W., Pan, D., Weng, K.: Step-by-step controller design for
LCL-type grid-connected inverter with capacitor-current-feedback active-damping. IEEE
Trans. Power Electron. 29(3), 1239–1253 (2014)
3. Blaabjerg, F., Teodorescu, R., Liserre, M., Timbus, A.V.: Overview of control and grid
synchronization for distributed power generation systems. IEEE Trans. Ind. Electron. 53(5),
1398–1409 (2006)
4. Zargari, N.R., Joós, G.: Performance investigation of a current-controlled voltage- regulated
PWM rectifier in rotating and stationary frames. IEEE Trans. Ind. Electron. 42(4), 396–401
(1995)
5. Kazmierkowski, M.P., Malesani, L.: Current control techniques for three-phase
voltage-source PWM converters: a survey. IEEE Trans. Ind. Electron. 45(5), 691–703 (1998)
6. Martinz, F.O., Miranda, R.D., Komatsu, W., Matakas, L.: Gain limits for current loop
controllers of single and three-phase PWM converters. In: Proceeding of the IEEE
International Power Electronics Conference, 201–208 (2010)
7. IEEE Recommended Practice for Utility Interface of Photovoltaic (PV) Systems, IEEE Std.
929. (2000)
8. Technical Rule for Photovoltaic Power Station Connected to Power Grid, Q/GDW 617
(2011) (in Chinese)
9. Erickson, R.W., Maksimović, D.: Fundamentals of Power Electronics, 2nd edn. Kluwer,
Boston, MA (2001)
10. Wang, X., Ruan, X., Liu, S., Tse, C.K.: Full feed-forward of grid voltage for grid-connected
inverter with LCL filter to suppress current distortion due to grid voltage harmonics. IEEE
Trans. Power Electron. 25(12), 3119–3127 (2010)
11. Zmood, D.N., Holmes, D.G.: Stationary frame current regulation of PWM inverters with zero
steady-state error. IEEE Trans. Power Electron. 18(3), 814–822 (2003)
12. Holmes, D.G., Lipo, T.A., McGrath, B.P., Kong, W.Y.: Optimized design of stationary frame
three phase AC current regulators. IEEE Trans. Power Electron. 24(11), 2417–2426 (2009)
120 5 Controller Design for LCL-Type Grid …
Chapter 6
Full-Feedforward of Grid Voltage
for Single-Phase LCL-Type
Grid-Connected Inverter
Abstract The grid-connected inverter plays an important role in injecting
high-quality power into the power grid. The injected grid current is affected by the
grid voltage at the point of common coupling (PCC). This chapter studies the
feedforward scheme of the grid voltage for single-phase LCL-type grid-connected
inverter. First, the mathematical model for the LCL-type grid-connected inverter
with capacitor-current-feedback active-damping is presented, and then it is sim-
plified through a series of equivalent transformations. After that, a full-feedforward
of the grid voltage is proposed to eliminate the effect of the grid voltage on the
steady-state error and harmonics in the injected grid current. The feedforward
function consists of three parts, namely proportional, derivative, and
second-derivative components. A comprehensive investigation shows that if the
grid voltage contains only the third harmonic, the proportional feedforward com-
ponent is adequate to suppress the harmonic distortion in the grid current caused by
the grid voltage; when the grid voltage contains harmonic distortion up to the
thirteenth harmonic, the proportional and derivative components are required; and
when the grid voltage contains harmonic distortion higher than the thirteenth har-
monic, the second-derivative component must be incorporated, i.e., the
full-feedforward scheme is necessary.
Keywords Grid-connected inverter � LCL filter � Damping resonance � Total
harmonics distortion (THD) � Feedforward � Single-phase
6.1 Introduction
As the interface between the distributed power generation system (DPGS) and
power grid, grid-connected inverter plays an important role in injecting high-quality
power into the power grid. As illustrated in Chap. 5, the injected grid current is
affected by the grid voltage at the point of common coupling (PCC). Generally, lots
of nonlinear equipments such as arc welder, saturable transformer, and electric rail
vehicles are connected to the PCC and produce harmonic current. The produced
© Springer Nature Singapore Pte Ltd. and Science Press 2018
X. Ruan et al., Control Techniques for LCL-Type Grid-Connected Inverters,
CPSS Power Electronics Series, DOI 10.1007/978-981-10-4277-5_6
121
harmonic current flows through the grid impedance and introduces background
harmonics to the grid voltage at PCC. The background harmonics of the grid
voltage will cause the injected grid current of the grid-connected inverter distorted.
Besides, the fundamental component of the grid voltage will also lead to the
steady-state error of the grid current [1, 2]. In order to ensure the grid current to
meet the standards, the effect of the grid voltage on the grid current should be
mitigated, which can be achieved by two ways. One is to use multiple
proportional-resonant (PR) regulator [2, 3] or repetitive regulator [4, 5] as the grid
current controller, which achieve infinite loop gains at the fundamental and har-
monic frequencies. The other way is to use the feedforward schemes of the grid
voltage [6–8]. Through feedforward of the grid voltage, both the steady-state error
and the distortion of the grid current can be mitigated even if a simple regulator
such as proportional-integral (PI) regulator is used. Furthermore, a fast dynamic
response of the inverter can be achieved.
This chapter studies the feedforward scheme of the grid voltage for single-phase
LCL-type grid-connected inverter. First, the mathematicalmodel for the LCL-type
grid-connected inverter with capacitor-current-feedback active-damping is pre-
sented, and then it is simplified through a series of equivalent transformations. After
that, a full-feedforward of the grid voltage is proposed to eliminate the effect of the
grid voltage on the steady-state error and harmonics in the injected grid current. The
feedforward function consists of three parts, namely proportional, derivative and
second-derivative components. A comprehensive investigation shows that if the
grid voltage contains only the third harmonic, the proportional feedforward com-
ponent is adequate to suppress the harmonic distortion in the grid current caused by
the grid voltage; when the grid voltage contains harmonic distortion up to the
thirteenth harmonic, the proportional and derivative components are required; and
when the grid voltage contains harmonic distortion higher than the thirteenth har-
monic, the second-derivative component must be incorporated, i.e., the full-feed-
forward scheme is necessary. Since the full-feedforward function is related to the
transfer function of the PWM modulator, the inverter-side inductor and the filter
capacitor, the impact of the variations of these parameters on the mitigation of the
harmonics in the grid current is investigated. Finally, in order to verify the effec-
tiveness of the proposed full-feedforward scheme of the grid voltage, a 6-kW
single-phase LCL-type grid-connected inverter is built and tested. The experimental
results show the proposed full-feedforward scheme can not only effectively reduce
the steady-state error of the grid current, but also sufficiently suppress the grid
current distortion arising from the background harmonics in the grid voltage.
6.2 Effects of the Grid Voltage on the Grid Current
Figure 6.1 shows the configuration of a single-phase LCL-type grid-connected
inverter, where the LCL filter is composed of L1, C, and L2. The primary objective
of the grid-connected inverter is to control the grid current i2 to synchronize with
122 6 Full-FeedForward of Grid Voltage for Single-Phase …
the grid voltage vg, and its amplitude can be regulated as required. i�2 is the grid
current reference, which includes the amplitude I* and the phase angle h. h is
usually obtained by a phase-locked loop (PLL), and I* is generated by an outer
voltage loop. Since the bandwidth of the voltage loop is much slower than that of
the grid current loop, it is reasonable to ignore the voltage loop and set I* directly
while designing the grid current regulator Gi(s). In this figure, Hi1, Hi2, and Hv
represent the feedback coefficients of the capacitor current, grid current, and grid
voltage, respectively. Here, the capacitor-current-feedback active-damping is used
to damp the resonance of the LCL filter.
According to Fig. 6.1, the mathematical model of the LCL-type grid-connected
inverter can be derived as shown in Fig. 6.2a, where KPWM = Vin/Vtri is the transfer
function from the modulation signal vM to the inverter bridge output voltage vinv,
with Vin and Vtri as the input voltage and the amplitude of the triangular carrier,
respectively; ZL1(s), ZC(s), and ZL2(s) represent the reactance of L1, C, and L2,
respectively, expressed as
ZL1 sð Þ ¼ sL1; ZC sð Þ ¼ 1sC ; ZL2 sð Þ ¼ sL2 ð6:1Þ
In Chap. 5, through a series of equivalent transformations of the control block
diagram, the block diagram shown in Fig. 6.2a can be equivalently simplified to
that shown in Fig. 5.3d. For convenience of illustration, it is given here again, as
shown in Fig. 6.2b, where
Gx1 sð Þ ¼ KPWMGi sð ÞZC sð ÞZL1 sð Þþ ZC sð ÞþHi1KPWM ð6:2Þ
Gx2 sð Þ ¼ ZL1 sð Þþ ZC sð ÞþHi1KPWMZL1 sð ÞZL2 sð Þþ ZL1 sð Þþ ZL2 sð Þð ÞZC sð ÞþHi1KPWMZL2 sð Þ ð6:3Þ
vg
cosθ
Vin
L1 L2
C
iC
++ ––
vC
PLL
Control System
vM i2*
Hv
Gi(s) *I
i1 i2+
–
vinv
Sinusoidal PWM
Hi2Hi1
Fig. 6.1 Topology and
control diagram of LCL-type
grid-connected inverter
6.2 Effects of the Grid Voltage on the Grid Current 123
From Fig. 6.2b, the loop gain TA(s) and the grid current i2(s) can be obtained as
TA sð Þ ¼ Gx1 sð ÞGx2 sð ÞHi2
¼ Hi2KPWMGi sð ÞZC sð Þ
ZL1 sð ÞZL2 sð Þþ ZL1 sð Þþ ZL2 sð Þð ÞZC sð ÞþHi1KPWMZL2 sð Þ
ð6:4Þ
i2ðsÞ ¼ TAðsÞ1þ TAðsÞ
1
Hi2
i�2ðsÞ �
Gx2ðsÞ
1þ TAðsÞ vgðsÞ, i21 sð Þþ i22 sð Þ ð6:5Þ
where
i21 sð Þ ¼ 1Hi2
TA sð Þ
1þ TA sð Þ i
�
2 sð Þ ð6:6aÞ
i22 sð Þ ¼ � Gx2 sð Þ1þ TA sð Þ vg sð Þ ð6:6bÞ
As seen from (6.5), the grid current i2 is composed of two parts. One is the static
tracking component i21, and the other is the disturbance component i22 resulting
from the grid voltage.
It can be observed from (6.6) that if the loop gain TA is large enough in mag-
nitude, both the static tracking error and the variation component i22 will be sub-
stantially reduced. However, TA cannot be designed to be too large, otherwise the
Fig. 6.2 Model of single-phase LCL-type grid-connected inverter with capacitor-current-feedback
active-damping
124 6 Full-FeedForward of Grid Voltage for Single-Phase …
system may become unstable. Basically, when the magnitude of TA at the funda-
mental frequency is larger than 10, the static tracking error can be effectively
reduced, but the disturbance component i22 may be still large.
Figure 6.3 shows the experimental results under half-load and full-load condi-
tions tested from the prototype. The parameters of the prototype are listed in
Table 6.1. Here, PI regulator is used as the grid current loop. As seen from Fig. 6.3,
due to the disturbance component i22 resulting from the grid voltage vg, the grid
current i2 lags to vg. Besides, the distortion in i2 is evident, which is resulted by i22.
Since i22 is independent from the grid current reference i�2, it will keep the same
when i�2 decreases. Meanwhile, the static tracking component i21 will decrease when
i�2 decreases. Therefore, the distortion of i2 becomes more serious at light load than
at heavy load.
Fig. 6.3 Experimental waveforms of single-phase LCL-type grid-connected inverter
Table 6.1 Parameters of single-phase prototype
Parameter Symbol Value Parameter Symbol Value
Input voltage Vin 360 V Filter capacitor C 10 lF
Grid voltage
(RMS)
Vg 220 V Grid-side inductor L2 150 lH
Output power Po 6 kW Carrier amplitude Vtri 3 V
Fundamental
frequency
fo 50 Hz Capacitor-current-feedback
coefficient
Hi1 0.075
Switching
frequency
fsw 10 kHz Grid current feedback
coefficient
Hi2 0.15
Inverter-side
inductor
L1 600 lH Grid voltage feedback
coefficient
Hv 0.017
6.2 Effects of the Grid Voltage on the Grid Current 125
6.3 Full-Feedforward Scheme for Single-Phase LCL-Type
Grid-Connected Inverter
6.3.1 Derivation of Full-Feedforward Function
of Grid Voltage
In (6.5), the function −Gx2/(1 + TA(s)) can be regarded as the admittance between i2
and vg. If an additional path from the grid voltage vg to the grid current i2 with the
transfer function of Gx2 is introduced, as shown in Fig. 6.4, the effect of vg on the
grid current will be eliminated.
By moving the feedforward node from the output of Gx2(s) to the output of
Gx1(s) and modifying the feedforward function as appropriate, the equivalent block
diagram can be obtained, as shown in Fig. 6.5a. Figure 6.5a can be further
equivalently transformed into Fig. 6.5b. As seen, the feedforward of vg with the
function of 1/Gx1(s) will eliminate the effect of vg on the grid current i2.
According to Fig. 6.5b, the block diagram shown in Fig. 6.2a can be
re-configured, as shown in Fig. 6.6a. Note that the numerator of Gx1(s) shown in
(6.2) contains the current regulator function Gi(s). Thus, Fig. 6.6a can be equiva-
lently transformed into Fig. 6.6b and the feedforward component contributes to the
modulation signal.
Substituting (6.2) into Fig. 6.6b, the feedforward function can be expressed as
Gff sð Þ, GiðsÞGx1ðsÞ ¼
1
KPWM
1þ ZL1ðsÞ
ZCðsÞ þ
Hi1KPWM
ZCðsÞ
� �
: ð6:7Þ
+ –
vg(s)
i2(s)Gx1(s)+ –
Gx2(s)
Gx2(s)
++i2 (s)*
Hi2
Fig. 6.4 Block diagram of full-feedforward scheme
+ – i2(s)Gx1(s)+ –
Gx2(s)
vg(s)
+ +i2 (s)*Hi2
+ – i2(s)Gx2(s)
vg(s)
Gx1(s)+ –
+
1
Gx1(s)
i2 (s)*
Hi2(s)
(a) (b)
Fig. 6.5 Derivation of full-feedforward scheme of grid voltage
126 6 Full-FeedForward of Grid Voltage for Single-Phase …
Substituting the expressions of ZL1(s) and ZC(s) given in (6.1) into (6.7) yields
Gff sð Þ ¼ 1KPWM þCHi1 � sþ
L1C
KPWM
� s2: ð6:8Þ
KPWM +
–
Hi1
+
–
+ –
vg(s)
i2(s)
ZC(s)
Hi2
+
–
Gi(s)+ –
1
ZL1(s)
1
ZL2(s)
+
1
Gx1(s)
i2 (s)*
(a)
KPWM +
–
Hi1
+
–
+ –
vg(s)
i2(s)
ZC(s)
Hi2
+
–
Gi(s)+ –
1
ZL1(s)
1
ZL2(s)
+
Gi(s)
Gx1(s)
i2 (s)*
(b)
KPWM +
–
Hi1
+
–
+ –
vg(s)
i2(s)
ZC(s)
Hi2
+
–
Gi(s)+ –
1
ZL1(s)
1
ZL2(s)
+
1
KPWM
L1C·s2
KPWM
CHi1·s
++i2 (s)*
(c)
Fig. 6.6 Block diagrams of feedforward scheme of grid voltage and equivalent representations
6.3 Full-Feedforward Scheme for Single-Phase LCL-Type … 127
Putting the three feedforward components indicated in (6.8) into the feedforward
function in Fig. 6.6b, the equivalent transformation is obtained, as shown in
Fig. 6.6c. It can be seen that the feedforward function of the grid voltage includes
three components, i.e., the proportional, derivative, and second-derivative compo-
nents. If only the proportional component is used, as attempted previously in [6],
the result will not be very satisfactory. To differ from the proportional feedforward
of the grid voltage, the derived feedforward function as shown in (6.8) is defined as
the full-feedforward scheme in this chapter.
6.3.2 Discussion of the Three Feedforward Components
As seen in (6.8), the full-feedforward function of the grid voltage is composed of
the proportional, derivative, and second-derivative components. When the input
voltage Vin and the amplitude of the triangle carrier Vtri are determined, the pro-
portional component is a constant, whereas the magnitudes of the derivative and
second-derivative components increase as the harmonic frequency of the grid
voltage increases. Therefore, when the grid voltage contains different harmonic
distortion, the weight of the three components of the full-feedforward function will
be different.
Substituting s = j2pf into (6.8) yields
Gff j2pfð Þ ¼ 1KPWM þ j2pf � CHi1 � 2pfð Þ
2 L1C
KPWM
,Gff p j2pfð ÞþGff d j2pfð ÞþGff dd j2pfð Þ
ð6:9Þ
where Gff_p(j2pf), Gff_d(j2pf), and Gff_dd(j2pf) are the proportional, derivative, and
second-derivative components, respectively, expressed as
Gff p j2pfð Þ ¼ 1KPWM
Gff d j2pfð Þ ¼ j2pf � CHi1
Gff dd j2pfð Þ ¼ � 2pfð Þ2 L1CKPWM
ð6:10Þ
To investigate the weight of the three feedforward components, comparison is
made among the full-feedforward scheme and two simplified feedforward schemes,
i.e., the proportional feedforward scheme and the proportional and derivative
feedforward scheme. The difference between the full-feedforward scheme and the
proportional feedforward scheme is defined as E1(j2pf); the difference between the
full-feedforward scheme and the proportional and derivative feedforward scheme is
defined as E2(j2pf). So, E1(j2pf) and E2(j2pf) are expressed as
128 6 Full-FeedForward of Grid Voltage for Single-Phase …
E1 j2pfð Þ ¼ Gff d j2pfð ÞþGff dd j2pfð Þ ¼ j2pf � CHi1 � 1KPWM 2pfð Þ
2L1C ð6:11Þ
E2 j2pfð Þ ¼ Gff dd j2pfð Þ ¼ � 1KPWM 2pfð Þ
2L1C ð6:12Þ
Setting the full-feedforward function Gff(j2pf) as the reference, the per-unit
values of E1(j2pf) and E2(j2pf) can be expressed as
E1 p:u:ð Þ fð Þ, E1 j2pfð Þj j
Gff j2pfð Þ
�� �� ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2pfð Þ2L1C
h i2
þ 2pf � CHi1KPWMð Þ2
r
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1� 2pfð Þ2L1C
h i2
þ 2pf � CHi1KPWMð Þ2
r ð6:13Þ
E2 p:u:ð Þ fð Þ, E2 j2pfð Þj j
Gff j2pfð Þ
�� �� ¼ 2pf � CHi1KPWMffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1� 2pfð Þ2L1C
h i2
þ 2pf � CHi1KPWMð Þ2
r ð6:14Þ
Substituting the corresponding parameters in Table 6.1 into (6.13) and (6.14),
the curves of E1(j2pf) and E2(j2pf) can be depicted, as shown in Fig. 6.7. As seen,
E1(j2pf) and E2(j2pf) increase as the harmonic frequency increases. It means that
the harmonic suppression ability of the two simplified feedforward schemes is
reduced. If E1(p.u.)(f) < 0.1, the harmonic suppression ability of the full-feedforward
scheme can be approximated to that of the proportional feedforward scheme.
Setting E1(p.u.)(f) = 0.1 yields fP1 � 181 Hz, which means the proportional feed-
forward scheme is adequate if vg contains only the third harmonic (here, the fun-
damental frequency is 50 Hz). Likewise, if E2(p.u.)(f) < 0.1, the harmonic
suppression ability of the full-feedforward scheme can be approximated to that of
the proportional and derivative feedforward scheme. Setting E2(p.u.)(f) = 0.1 yields
0.1
No Feedforward
Full-Feedforward
Proportional and 
Derivative 
Feedforward (E2(p.u.))
Proportional 
Feedforward
(E1(p.u.))
−0.1
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
0 0.5 1.0 1.5 2.5 3.0
f (kHz)
2.0
P1 P2
P3
fP1 fP2
P4
Fig. 6.7 Curves of E1(p.u.)
and E2(p.u.)
6.3 Full-Feedforward Scheme for Single-Phase LCL-Type … 129
fP2 � 641 Hz. It means that the second-derivative feedforward function can be
omitted if vg contains harmonic distortion up to the thirteenth harmonic. If vg
contains harmonic distortion higher than the thirteenth harmonic, the full-feedfor-
ward scheme is necessary to ensure a satisfying harmonic suppression.
As seen in Fig. 6.7, when the harmonic distortion is higher than the thirtieth
harmonic (i.e., 1.5 kHz), E1(p.u.)(f) > 1 occurs. Compared with no feedforward
scheme, the proportional feedforward scheme will amplify the grid current har-
monics higher than 1.5 kHz. Likewise, when the harmonic distortion is higher than
the fiftieth harmonic (i.e., 2.5 kHz), E2(p.u.)(f) > 1 happens. Compared with no
feedforward scheme, the proportional and derivative feedforward scheme will also
amplify the grid current harmonics higher than 2.5 kHz.
6.3.3 Discussion of Full-Feedforward Scheme with Main
Circuit Parameters Variations
As discussed in Sect. 6.2, the effect of the grid voltage on the grid current can
theoretically be eliminated if the proportional coefficient, derivative, and second-
derivative components are accurate. However, as seen in (6.8), the three feedfor-
ward coefficients are related to KPWM, the capacitor-current-feedback coefficient
Hi1, the filter capacitor C, and the inverter-side filter inductor L1, where KPWM is
determined by the input voltage Vin and the amplitude of the triangle carrier Vtri.
Therefore, if the input voltage Vin fluctuates, or the values of inductor L1 and filter
capacitor C vary, the harmonic suppression ability of the proposed full-feedforward
scheme will be affected. Considering that the capacitor current is sensed by a
high-accuracy current hall, the variation of Hi1 is very little and can be ignored.
Therefore, the following analysis will focus on the feasibility of the full-
feedforward scheme with the variations of Vin, L1, and C.
Supposing the actual input voltage, inverter-side inductor and the filter capacitor
are V′in, L′1, and C′, respectively, the required full-feedforward function which can
completely eliminate the effect of grid voltage is
G0ff j2pfð Þ ¼
1
K 0PWM
þ j2pf � C0Hi1 � 2pfð Þ2 L
0
1C
0
K 0PWM
ð6:15Þ
Setting the full-feedforward function Gff(j2pf) with the designed parameters as
the reference, the per-unit values of the amplitude difference between G′ff(j2pf) and
Gff(j2pf) can be expressed as
130 6 Full-FeedForward of Grid Voltage for Single-Phase …
E3 p:u:ð Þ fð Þ,
Gff j2pfðÞ � G0ff j2pfð Þ
��� ���
Gff j2pfð Þ
�� ��
¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1� KPWM=K 0PWM
� �� 2pfð Þ2 L1C � L01C0KPWM=K 0PWM� �
h i2
þ 2pfHi1KPWM � C � C0ð Þ½ �2
r
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1� 2pfð Þ2L1C
h i2
þ 2pf � CHi1KPWMð Þ2
r
ð6:16Þ
As seen from Table 6.1, the rated input voltage is Vin = 360 V, and the designed
inverter-side inductor and filter capacitor are L1 = 600 µH and C = 10 µF,
respectively. Supposing that the fluctuation of Vin is between 360 V and 400 V, the
variation of L1 is between 500 µH and 700 µH, and the variation of C is between
8 µF and 12 µF. After comparing all kinds of the parameter variations, it can be
observed that the worst deterioration of harmonic suppression ability of the full-
feedforward scheme occurs in three cases: (1) L′1 = 500 µH with the rated Vin and
designed C; (2) C′ = 8 µF with the rated Vin and designed L1; and (3) V′in = 400 V
with the designed L1 and C. By substituting the corresponding parameters of the
three cases into (6.16), the curves of E3(p.u.) can be depicted, as shown in Fig. 6.8.
As seen, when the background harmonics in the grid voltage are below 1.5 kHz, the
full-feedforward scheme of the three cases can suppress the grid currents harmonics
down to 20% of that with no feedforward scheme. Even if the worst case while
C′ = 8 µF happens, the full-feedforward scheme can still suppress the harmonics
down to 35% of that with no feedforward scheme. Thus, it can be concluded that
the full-feedforward scheme is less affected when L1, C, and Vin have a relatively
large variations.
0 0.5 1.0 1.5 2.5 3.0
f (kHz)
1.0
0.4
0.2
0.0
0.6
2.0
−0.1
0.8
1.1
No Feedforward
Vin=400V
L1=500µH
C'=8µF
Full-Feedforward
'
'
E 3
(p
.u
.)
Fig. 6.8 Curves of E3(p.u.)
with main circuit parameter
variation
6.3 Full-Feedforward Scheme for Single-Phase LCL-Type … 131
6.4 Experimental Results
A 6-kW prototype of single-phase LCL-type grid-connected inverter was con-
structed for verification of the full-feedforward scheme and for comparing the
effectiveness of the three constituent feedforward functions. The parameters of the
prototype are listed in Table 6.1. Experimental results of four cases are compared.
Case I is no feedforward of vg. Case II is proportional feedforward of vg, i.e., only
1/KPWM is used as the feedforward function. Case III is proportional and derivative
feedforward of vg. Case IV is full-feedforward of vg, i.e., the proportional,
derivative, and second-derivative feedforward of vg are all used. To clearly check
the harmonic suppression ability of the four cases, a programmable AC source
(Chroma 6590) is used to simulate the grid voltage.
Figure 6.9 shows the experimental results for Case I and Case II under full-load
condition. Here, the grid voltage vg is sinusoidal. It can be seen that the waveforms
of i2 are sinusoidal in both Cases I and II. However, a phase difference of about 3.7°
exists between i2 and vg in Case I, which is caused by the fundamental component
of vg according to the analysis in Sect. 6.1.
Figure 6.10 shows the experimental results for Case I and Case II at full-load
condition. Here, the third harmonic has been injected into vg, and the magnitude and
phase of the injected harmonic is 10% and 0°, respectively, with respect to the
fundamental component. It can be seen that the waveforms of i2 are distorted in
Case I, and a phase difference of about 3.7° exists between i2 and vg. For Case II, as
shown in Fig. 6.10b, i2 is perfectly sinusoidal, and the phase lag has been elimi-
nated. The THDs of the waveforms of i2 shown in Fig. 6.10a, b are 3.21% and
1.2%, respectively. The results show that when the distortion contains only the third
harmonic, the proportional feedforward scheme (Case II) is effective in suppressing
the distortion.
Figure 6.11 shows the experimental results for Cases II and III under full-load
condition. Here, the injected harmonics into vg include the third, fifth, seventh,
ninth, eleventh, and thirteenth harmonics, and the magnitudes of the injected
Δ=3.7o
vg: [100V/div] iLf2: [20A/div]
Time: [5ms/div] Time: [5ms/div]
vg: [100V/div] iLf2: [20A/div]
(a) Case I (b) Case II
Fig. 6.9 Experimental waveforms with idea grid voltage
132 6 Full-FeedForward of Grid Voltage for Single-Phase …
harmonics with respect to the fundamental component of vg are 10%, 5%, 3%, 3%,
2% and 2%, respectively, and the corresponding phase angles are 0°, 90°, 0°, 0°, 0°
and 0°. As seen, for Cases II and III, the phase lag between i2 and vg is eliminated.
The measured THDs of the waveforms of i2 shown in Fig. 6.11a, b are 2.61% and
1.42%, respectively. The results show that when the harmonic distortion in the grid
voltage is up to the thirteenth harmonic, the proportional and derivative feedforward
scheme (Case III) is effective in suppressing the distortion, and the proportional
feedforward scheme (Case II) is inadequate.
Figure 6.12 shows the experimental results for four cases under full-load con-
dition. Here, the thirty-third harmonic, with magnitude and phase of 1% and 0° with
respect to the fundamental, has been injected into vg. As seen from Fig. 6.12a, when
the feedforward of vg is not used, the distortion of i2 is evident. Compared with
Fig. 6.12a, the distortion of i2 shown in Fig. 6.12b is deteriorated with proportional
feedforward of vg, which coincides with the analysis in Sect. 6.2. As seen from
Fig. 6.12c, when the proportional and derivative feedforward of vg is incorporated,
Time: [5ms/div]
vg: [100V/div]
i2: [20A/div]
Time: [5ms/div]
vg: [100V/div]
i2: [20A/div]
(a) Case I (b) Case II
Fig. 6.10 Experimental waveforms when the grid voltage contains only the third harmonic
Time: [5ms/div]
vg: [100V/div]
i2: [20A/div]
Time: [5ms/div]
vg: [100V/div]
i2: [20A/div]
(a) Case II (b) Case III
Fig. 6.11 Experimental waveforms when the grid voltage contains harmonic distortion up to the
thirteenth harmonic
6.4 Experimental Results 133
the distortion of i2 is greatly reduced. As seen from Fig. 6.12d, when the
full-feedforward of vg is adopted, the distortion of i2 is the smallest. The results
show that when the grid voltage contains higher harmonics, the full-feedforward
scheme is necessary for eliminating the distortion in the grid current.
Furthermore, a test to verify the effectiveness of the proposed scheme under
possible voltage dip conditions is conducted. Figure 6.13 shows the experimental
results for the four cases when a 40 V voltage dip occurs at the trough and crest of
the voltage waveform of vg. The THDs of i2 for the four cases are 4.61%, 5.42%,
3.26%, and 2.24%, respectively. The results show that the full-feedforward scheme
can effectively suppress the current distortion even vg experiences a voltage dip.
The transient response of the grid-connected inverter under the proposed full-
feedforward scheme has been studied, and the results are shown in Fig. 6.14a
corresponding to step change of i�2. Note that the grid voltage is taken from the
active power grid. Here, i�2 is stepped up from half load to full load, and vice versa.
The load changes are intentionally set to occur at the peak of i2, which is the worst
case. Results show that i2 is still kept in phase with vg, with small oscillatory
transient observed immediately after the step change of i�2. Also, Fig. 6.14b shows
thetransient response corresponding to step change of vg with the full-feedforward
Time: [5ms/div]
vg: [100V/div]
i2: [20A/div]
Time: [5ms/div]
vg: [100V/div] i2: [20A/div]
(a) Case I (b) Case II
Time: [5ms/div]
vg: [100V/div]
i2: [20A/div]
Time: [5ms/div]
vg: [100V/div]
i2: [20A/div]
(c) Case III (d) Case IV
Fig. 6.12 Experimental waveforms when the grid voltage contains the thirty-third harmonic
134 6 Full-FeedForward of Grid Voltage for Single-Phase …
scheme. Here, vg is stepped down from 220 V to 180 V, and vice versa. The vg
changes are again purposely set to occur at the peak of vg, which is the worst case.
Results show that the amplitude of i2 is kept unchanged, with small oscillatory
transient immediately following the step change of vg.
Time: [5ms/div]
vg: [100V/div]
i2: [20A/div]
Time: [5ms/div]
vg: [100V/div]
i2: [20A/div]
(a) Case I (b) Case II
Time: [5ms/div]
vg: [100V/div]
i2: [20A/div]
Time: [5ms/div]
vg: [100V/div]
i2: [20A/div]
(c) Case III (d) Case IV
Fig. 6.13 Experimental waveforms for the four control strategies under voltage dip conditions
Time: [10ms/div]
i2: [20A/div]
vg: [100V/div]
Time: [50ms/div]
vg: [200V/div]
i2: [50A/div]
(a) step change in *2i (b) step change in vg
Fig. 6.14 Measured transient response under step changes in i�2 and vg
6.4 Experimental Results 135
To verify the adaptability of the proposed full-feedforward scheme to the vari-
ation of Vin, L1, and C, mismatches are intentionally introduced to the three
parameters, and the THDs of the grid current i2 are tested, as shown in Table 6.2.
As seen, the tested THDs changes very little, which indicates a good adaptability of
the proposed full-feedforward scheme. The results verify the analysis in Sect. 6.2.
6.5 Summary
This chapter studies the effect of the grid voltage on the grid current for the
single-phase LCL-type grid-connected inverter. It shows that the fundamental
component of the grid voltage affects the steady-state error, and the harmonic
components cause the grid current distorted. The traditional proportional feedfor-
ward of the grid voltage can suppress the current distortion but the result is not
satisfactory especially when the grid voltage contains high harmonic distortion.
This chapter proposes a full-feedforward of grid voltage scheme to suppress the
grid current distortion arising from the harmonics in the grid voltage. It is composed
by the proportional, derivative, and second-derivative components. Four cases,
namely no feedforward, the proportional feedforward of the grid voltage, the pro-
portional and derivative feedforward of the grid voltage, and the full-feedforward of
the grid voltage, are compared. The results show that if the grid voltage contains
only the third harmonic, the proportional feedforward of the grid voltage is ade-
quate for achieving good suppression of the current distortion. If the grid voltage
contains harmonic distortion up to the thirteenth harmonic, the proportional and
derivative feedforward of the grid voltage is adequate. If the grid voltage contains
higher harmonic distortion, the full-feedforward of the grid voltage is necessary.
Furthermore, the adaptability of the proposed full-feedforward scheme to the
variation of the input voltage, inverter-side inductor and filter capacitor is investi-
gated. A 6-kW single-phase LCL-type grid-connected inverter is fabricated and
tested to verify the effectiveness of the proposed full-feedforward scheme. The
experimental results show that the proposed feedforward scheme can not only
significantly reduce the steady-state error of the grid current, but also effectively
suppress the grid current distortion arising from the harmonics in the grid voltage.
Even if mismatch occurs from the input voltage, inverter-side inductor or filter
capacitor, the proposed full-feedforward scheme can still be effective.
Table 6.2 Measured THDs
of grid current i2 with main
circuit parameters variation
Vin (V) L1 (µH) C (µF) THD of i2 (%)
360–400 600 10 1.3–1.48
360 500–700 10 1.45–1.5
360 600 8–12 1.3–1.7
136 6 Full-FeedForward of Grid Voltage for Single-Phase …
References
1. Prodanović, M., Green, T.: High-quality power generation through distributed control of a
power park microgrid. IEEE Trans. Ind. Electron. 53(5), 1471–1482 (2006)
2. Zmood, D.N., Holmes, D.G.: Stationary frame current regulation of PWM inverters with zero
steady-state error. IEEE Trans. Power Electron. 18(3), 814–822 (2003)
3. Liserre, M., Teodorescu, R., Blaabjerg, F.: Stability of photovoltaic and wind turbine
grid-connected inverters for a large set of grid impedance values. IEEE Trans. Power Electron.
21(1), 888–895 (2006)
4. Bojoi, R.I., Limongi, L.R., Roiu, D., Tenconi, A.: Enhanced power quality control strategy for
single-phase inverters in distributed generation systems. IEEE Trans. Power Electron. 26(3),
798–806 (2011)
5. Zhong, Q.C., Hornik, T.: Cascaded current-voltage control to improve the power quality for a
grid-connected inverter with a local load. IEEE Trans. Ind. Electron. 60(4), 1344–1355 (2013)
6. Wang, X., Ruan, X., Liu, S., Tse, C.K.: Full feed-forward of grid voltage for grid-connected
inverter with LCL filter to suppress current distortion due to grid voltage harmonics. IEEE
Trans. Power Electron. 25(12), 3119–3127 (2010)
7. Wang, X.: Research on control strategies for grid-connected inverter with LCL filter.
Postdoctoral research report, Huazhong University of Science and Technology, Wuhan, China
(2011) (in Chinese)
8. Liu, S.: Control strategy for single-phase grid-connected inverter with LCL filter. M.S. thesis,
Huazhong University of Science and Technology, Wuhan, China (2011) (in Chinese)
References 137
Chapter 7
Full-Feedforward Scheme of Grid
Voltages for Three-Phase LCL-Type
Grid-Connected Inverters
Abstract In order to alleviate the effect of the grid voltage on the grid current,
Chap. 6 presented the full-feedforward scheme of grid voltages for the single-phase
LCL-type grid-connected inverters, and the harmonics of the injected grid current
are effectively suppressed. In this chapter, the full-feedforward scheme is extended
to the three-phase LCL-type grid-connected inverter. In this chapter, the mathe-
matical models of the three-phase LCL-type grid-connected inverter in both the
stationary a–b frame and synchronous d–q frame are derived first. Then, based
on the mathematical models, the full-feedforward schemes of the grid voltages
for the stationary a–b frame, synchronous d–q frame, and decoupled synchronous
d–q frame-controlled three-phase LCL-type grid-connected inverter are proposed.
After that, the full-feedforward functions are discussed, and it will be illustrated that
the simplification of the full-feedforward function should be taken with caution and
simplifying the full-feedforward functions to a proportional feedforward function
will give rise to the amplification of the high-frequency injected grid current har-
monics. The effect of LCL filter parameter mismatches between the actual and
theoretical values is also evaluated. Finally, the effectiveness of the proposed
full-feedforward schemes is verified by the experimental results. Meanwhile, the
performance of the proposed full-feedforward schemes under unbalanced grid
voltage condition is intentionally investigated.
Keywords Grid-connected inverter � LCL filter � Damping resonance � Total
harmonics distortion (THD) � Feedforward � Three-phase
As described in Chap. 6, various nonlinear equipments, such as arc wielding
machine and electric rail transportation, are connected into the power grid, and they
produce harmonic currents. These harmonic currents flow through the grid impe-
dance and distort the grid voltage at the point of common coupling (PCC). The
grid-connected inverter is the interface between the distributed power generation
system (DPGS) and the power grid, and it is required to produce high-quality
current to be injected into the power grid [1]. In order to alleviate the effect of the
gridvoltage on the grid current, Chap. 6 presented the full-feedforward scheme of
© Springer Nature Singapore Pte Ltd. and Science Press 2018
X. Ruan et al., Control Techniques for LCL-Type Grid-Connected Inverters,
CPSS Power Electronics Series, DOI 10.1007/978-981-10-4277-5_7
139
grid voltages for the single-phase LCL-type grid-connected inverters, and the har-
monics of the injected grid current are effectively suppressed. In this chapter, the
full-feedforward scheme is extended to the three-phase LCL-type grid-connected
inverter [2].
Basically, the three-phase grid-connected inverter can be controlled in two
control frames, which are the stationary frame and the synchronous rotating frame.
In this chapter, the mathematical models of the three-phase LCL-type
grid-connected inverter in both the stationary a–b frame and synchronous d–
q frame are derived first. Then, based on the mathematical models, the
full-feedforward schemes of the grid voltages for the stationary a–b frame, syn-
chronous d–q frame, and decoupled synchronous d–q frame-controlled three-phase
LCL-type grid-connected inverter are proposed. After that, the full-feedforward
functions are discussed, and it will be illustrated that the simplification of the
full-feedforward function should be taken with caution and simplifying the
full-feedforward functions to a proportional feedforward function will give rise to
the amplification of the high-frequency injected grid current harmonics. The effect
of LCL filter parameter mismatches between the actual and theoretical values is also
evaluated. Finally, the effectiveness of the proposed full-feedforward schemes is
verified by the experimental results. Meanwhile, the performance of the proposed
full-feedforward schemes under unbalanced grid voltage condition is intentionally
investigated.
7.1 Modeling the Three-Phase LCL-Type
Grid-Connected Inverter
Figure 7.1 shows the three-phase LCL-type grid-connected inverter considered in
this chapter. A standard three-phase voltage source inverter (VSI) consisting of Q1–
Q6 is connected to the grid through an LCL filter. L1 is the inverter-side inductor,
C is the filter capacitor, and L2 is the grid-side inductor. Vin is the dc input voltage
and vga, vgb, and vgc are the three-phase grid voltages.
7.1.1 Model in the Stationary a–b Frame
According to Fig. 7.1, the mathematical model in the stationary a–b–c frame of the
three-phase LCL-type grid-connected inverter is described as
vxN abc tð Þ½ � ¼ vCx abc tð Þ½ � þ L1p i1x abc tð Þ½ �
vCx abc tð Þ½ � ¼ vgx abc tð Þ
� �þ vN 0N tð Þ 1 1 1½ �T þ L2p i2x abc tð Þ½ �
i1x abc tð Þ½ � ¼ i2x abc tð Þ½ � þCp vCx abc tð Þ½ �
ð7:1Þ
140 7 Full-Feedforward Scheme of Grid Voltages for Three-Phase …
where [vxN_abc(t)] = [vaN(t), vbN(t), vcN(t)]
T are the midpoint voltages of the three
inverter legs referred to point N, [vCx_abc(t)] = [vCa(t), vCb(t), vCc(t)]
T are the filter
capacitor voltages referred to point N, [vgx_abc(t)] = [vga(t), vgb(t), vgc(t)]
T are the
grid voltages referred to point N′, vN′N(t) is the voltage between points N′ and N,
[i1x_abc(t)] = [i1a(t), i1b(t), i1c(t)]
T are the inverter-side inductor currents,
[i2x_abc(t)] = [i2a(t), i2b(t), i2c(t)]
T are the injected grid currents, and p = d/dt. The
equivalent series resistors of L1, C, and L2 are relatively small and ignored here.
For the three-wire three-phase grid-connected inverter, there is no zero-sequence
injected grid current. Therefore, the system can be controlled in the stationary a–b
frame. The system schematic diagram of the stationary a–b frame-controlled
three-phase grid-connected inverter is shown in Fig. 7.1.
The relationship between the stationary a–b–c frame, a–b frame, and syn-
chronous d–q frame is shown in Fig. 7.2, where xo is the fundamental angular
frequency of the grid. According to Fig. 7.2, the stationary a–b–c to a–b trans-
formation and its inverse transformation used in this chapter are defined by
xab tð Þ
� � ¼ P½ � xabc tð Þ½ �; P½ � ¼ 23
1 �1=2 �1=2
0
ffiffiffi
3
p �
2 � ffiffiffi3p �2
" #
ð7:2Þ
xabc tð Þ½ � ¼ 32 P½ �
T xab tð Þ
� �
;
3
2
P½ �T¼
1 0
�1=2 ffiffiffi3p �2
�1=2 � ffiffiffi3p �2
2
664
3
775 ð7:3Þ
Fig. 7.1 Schematic diagram of the stationary a–b frame-controlled three-phase grid-connected
inverter
7.1 Modeling the Three-Phase LCL-Type Grid-Connected Inverter 141
where [xab(t)] = [xa(t), xb(t)]
T are the stationary a–b frame time-varying quantities,
[xabc(t)] = [xa(t), xb(t), xc(t)]
T are the stationary a–b–c frame time-varying quanti-
ties, and [P] is the transformation matrix.
Applying (7.3) to transform (7.1), the mathematical model of the main circuit in
the stationary a–b frame is obtained as
vinv ab tð Þ
� � ¼ vC ab tð Þ� �þ L1p i1 ab tð Þ� �
vC ab tð Þ
� � ¼ vg ab tð Þ� �þ L2p i2 ab tð Þ� �
i1 ab tð Þ
� � ¼ i2 ab tð Þ� �þCp vC ab tð Þ� �
ð7:4Þ
where [vinv_ab(t)] = [vinv_a(t), vinv_b(t)]
T, [vC_ab(t)] = [vC_a(t), vC_b(t)]
T,
[vg_ab(t)] = [vg_a(t), vg_b(t)]
T, [i1_ab(t)] = [i1_a(t), i1_b(t)]
T, [i2_ab(t)] = [i2_a(t),
i2_b(t)]
T.
Applying the Laplace transformation to (7.4), the mathematical model in s-
domain can be obtained as
vinv ab sð Þ
� � ¼ vC ab sð Þ� �þ L1s i1 ab sð Þ� �
vC ab sð Þ
� � ¼ vg ab sð Þ� �þ L2s i2 ab sð Þ� �
i1 ab sð Þ
� � ¼ i2 ab sð Þ� �þCs vC ab sð Þ� �
ð7:5Þ
According to Fig. 7.1 and (7.5), the block diagram of the stationary a–b
frame-controlled three-phase LCL-type grid-connected inverter is shown in
Fig. 7.3, where the feedback of capacitor currents is used to damp the resonance of
the LCL filter, which is equivalent to a virtual resistor connected in parallel with
each filter capacitor. i�2 ab sð Þ
ih
¼ i�2 aðsÞ; i�2 bðsÞ
h iT
represents the reference of
the injected grid current, Gsi ðsÞ is the injected grid current regulator in the stationary
a–b frame, [vr_ab(s)] = [vr_a(s), vr_b(s)]
T are the output signals of the injected grid
current regulators, [vM_ab(s)] = [vM_a(s), vM_b(s)]
T are the modulating signals,
ZL1(s), ZC(s), and ZL2(s) are the impedances of L1, C, and L2, expressed as
Fig. 7.2 Relationship
between three reference
frames
142 7 Full-Feedforward Scheme of Grid Voltages for Three-Phase …
ZL1 sð Þ ¼ sL1; ZC sð Þ ¼ 1sC ; ZL2 sð Þ ¼ sL2 ð7:6Þ
KPWM is the transfer function from the modulating signals to the three-phase
inverter bridge voltages. Since the three-phase sine-triangle pulse-width modulation
(PWM) is used here and the switching frequency is assumed to be high enough,
KPWM can be expressed as
KPWM ¼ Vin= 2Vtrið Þ ð7:7Þ
where Vtri is the amplitude of the triangle carrier. Hi1 is the feedback coefficient of
the filter capacitor current, and Hi2 is the sensor gain of the injected grid current.
7.1.2 Model in the Synchronous d–q Frame
According to Fig. 7.2, the stationary a–b frame to synchronous d–q frame trans-
formation and its inverse transformation are defined as
xdq tð Þ
� � ¼ C½ � xab tð Þ� �; C½ � ¼ cosxot sinxot� sinxot cosxot
� �
ð7:8Þ
xab tð Þ
� � ¼ C½ ��1 xdq tð Þ� �; C½ ��1¼ cosxot � sinxotsinxot cosxot
� �
ð7:9Þ
where [xdq(t)] = [xd(t), xq(t)]
T are the synchronous d–q frame time-varying
quantities.
M
M
inv
inv
Fig. 7.3 Block diagram of the stationary a–b frame-controlled grid-connected inverter
7.1 Modeling the Three-Phase LCL-Type Grid-Connected Inverter 143
Applying (7.9) to transform (7.4) and manipulating, the mathematical model of
the main circuit in the synchronous d–q frame is obtained as
vinv dq tð Þ
� � ¼ vC dq tð Þ� �þ L1 A tð Þ½ � i1 dq tð Þ� �
vC dq tð Þ
� � ¼ vg dq tð Þ� �þ L2 A tð Þ½ � i2 dq tð Þ� �
i1 dq tð Þ
� � ¼ i2 dq tð Þ� �þC A tð Þ½ � vC dq tð Þ� �
ð7:10Þ
where [vinv_dq(t)] = [vinv_d(t), vinv_q(t)]
T, [vC_dq(t)] = [vC_d(t), vC_q(t)]
T, [vg_dq(t)] =
[vg_d(t), vg_q(t)]
T, [i1_dq(t)] = [i1_d(t), i1_q(t)]
T, [i2_dq(t)] = [i2_d(t), i2_q(t)]
T,
A tð Þ½ � ¼ p �xo
xo p
� �
.
Applying the Laplace transformation to (7.10), the model in s-domain is given as
vinv dq sð Þ
� � ¼ vC dq sð Þ� �þ L1 A sð Þ½ � i1 dq sð Þ� �
vC dq sð Þ
� � ¼ vg dq sðÞ� �þ L2 A sð Þ½ � i2 dq sð Þ� �
i1 dq sð Þ
� � ¼ i2 dq sð Þ� �þC A sð Þ½ � vC dq sð Þ� �
ð7:11Þ
where A sð Þ½ � ¼ s �xo
xo s
� �
.
According to (7.11) and considering the controller in the synchronous d–
q frame, the block diagram of the synchronous d–q frame-controlled three-phase
LCL-type grid-connected inverter is shown in Fig. 7.4, where Gei (s) is the injected
grid current regulator in the synchronous d–q frame. Again, the feedback of ca-
pacitor currents is used here to damp the resonance of the LCL filter. From Fig. 7.4,
it is clear to see that there are three pairs of cross-coupling quantities, which are the
currents of filter inductors L1 and L2, and the filter capacitor voltages.
M inv
invM
Fig. 7.4 Block diagram of the synchronous d–q frame-controlled grid-connected inverter
144 7 Full-Feedforward Scheme of Grid Voltages for Three-Phase …
7.2 Derivation of the Full-Feedforward Scheme of Grid
Voltages
Based on the model given in Sect. 7.1, the full-feedforward schemes of grid volt-
ages for the three-phase LCL-type grid-connected inverter controlled in stationary
a–b frame, synchronous d–q frame, and hybrid frame are derived in this section.
7.2.1 Full-Feedforward Scheme in the Stationary a–b
Frame
It can be seen from Fig. 7.3 that there are no cross-coupling terms between the a-
axis and b-axis, and the model of each axis is the same as the model of the
single-phase inverter given in Fig. 6.2. Therefore, the full-feedforward scheme of
grid voltages for the stationary a–b frame-controlled three-phase LCL-type
grid-connected inverter can be derived similarly as shown in Sect. 6.2. The block
diagram of the full-feedforward scheme of grid voltages for the stationary a–b
frame-controlled grid-connected inverter is shown in Fig. 7.5, and the same as
(6.8), the full-feedforward function of grid voltages in Fig. 7.5 is expressed as
Gff sð Þ ¼ 1KPWM þHi1C � sþ
L1C
KPWM
� s2 ð7:12Þ
M
inv
invM
Fig. 7.5 Block diagram of the full-feedforward scheme for the stationary a–b frame-controlled
grid-connected inverter
7.2 Derivation of the Full-Feedforward Scheme of Grid Voltages 145
7.2.2 Full-Feedforward Scheme in the Synchronous d–q
Frame
The synchronous d–q frame control has the particular advantage of controlling the
active and reactive current directly, which is very convenient for the power flow
control. Therefore, the full-feedforward scheme for the synchronous d–q frame-
controlled three-phase LCL-type grid-connected inverter is derived here.
Similar to the full-feedforward scheme of the stationary a-b frame controlled
given in Fig. 7.5, the feedforward signals in the synchronous d-q frame, which are
referred as vff_d(s) and vff_q(s), can be added into Fig. 7.4, as shown in Fig. 7.6,
where [Gff_dq(s)] is the full-feedforward function for the synchronous d–
q frame-controlled three-phase LCL-type grid-connected inverter.
From Fig. 7.6, it can be obtained that
vinv dq sð Þ
� � ¼ vr dq sð Þ� �� Hi1 i1 dq sð Þ� �� i2 dq sð Þ� �� 	þ vff dq sð Þ� �
 �KPWM
ð7:13Þ
where [vff_dq(s)] = [vff_d(s), vff_q(s)]
T are the feedforward components added to the
modulating signals.
Substituting (7.13) into (7.11) and manipulating, [i2_dq(s)] can be expressed as
Hi1L2C A sð Þ½ �2 þ 1KPWM L2 A sð Þ½ � þ L1 A sð Þ½ � þ L1L2C A sð Þ½ �
3
� 
� �
i2 dq sð Þ
� �
¼ vr dq sð Þ
� �� 1
KPWM
I½ � þ L1C A sð Þ½ �2
� 
þHi1C A sð Þ½ � I½ �
� �
vg dq sð Þ
� �� vff dq sð Þ� �
� �
ð7:14Þ
M inv
M inv
Fig. 7.6 Block diagram of the full-feedforward scheme for the synchronous d–q frame-controlled
grid-connected inverter
146 7 Full-Feedforward Scheme of Grid Voltages for Three-Phase …
where [I] = diag[1] and [vr_dq(s)] = [vr_d(s), vr_q(s)]
T are the output of injected grid
current regulator Gei (s), expressed as
vr dq sð Þ
� � ¼ Gei sð Þ I½ � i�2 dq sð Þh i� i2 dq sð Þ� �� 
 ð7:15Þ
From (7.14) and (7.15), it can be found that [vg_dq(s)] can be eliminated from
[i2_dq(s)] when [vff_dq(s)] is controlled as depicted as
vff dq sð Þ
� � ¼ 1
KPWM
I½ � þ L1C A sð Þ½ �2
� 
þHi1C A sð Þ½ � I½ �
� �
vg dq sð Þ
� �
, Gff dq sð Þ
� �
vg dq sð Þ
� � ð7:16Þ
where [Gff_dq(s)] is expressed as
Gff dqðsÞ
� � ¼ Gff ðsÞ � DðsÞ �EðsÞ
EðsÞ Gff ðsÞ � DðsÞ
� �
ð7:17Þ
where Gff(s) has been given in (7.12), D sð Þ ¼ L1Cx
2
o
KPWM
, EðsÞ ¼ 2sL1xoCKPWM þxoHi1C.
7.2.3 Full-Feedforward Scheme in the Hybrid Frame
Comparing (7.12) and (7.17), it can be observed that the full-feedforward function
[Gff_dq(s)] in the synchronous d–q frame is more complicated than that in the
stationary a–b frame. This is due to the cross-coupling terms in the model shown in
Fig. 7.6. For the synchronous d–q frame-controlled grid-connected inverter, since
the purpose of introducing the full-feedforward of the grid voltages is to suppress
the injected grid currents caused by the grid voltages, and the feedback of the filter
capacitor currents is to damp the resonance of the LCL filter, which make no
contribution to the active and reactive power flow control, it is unnecessary to
implement them in the synchronous d–q frame. Therefore, the full-feedforward of
grid voltages and the feedback of the filter capacitor currents shown in Fig. 7.6 can
be implemented in the stationary a–b frame, while the regulation of the injected
grid currents is still implemented in the synchronous d–q frame which allows the
direct control of the active and reactive power. The block diagram of this scheme is
shown in Fig. 7.7. Hereinafter, the stationary a–b frame implemented
full-feedforward scheme for the synchronous d–q frame-controlled three-phase
grid-connected inverter is called the full-feedforward scheme for hybrid
frame-controlled three-phase grid-connected inverter.
The full-feedforward function Gff(s) in Fig. 7.7 can be directly derived from the
full-feedforward scheme for the synchronous d–q frame-controlled grid-connected
inverter given in Sect. 7.2.2. The s-domain full-feedforward function in the
7.2 Derivation of the Full-Feedforward Scheme of Grid Voltages 147
synchronous d–q frame shown in Fig. 7.6 is given in (7.17), and in the time
domain, the feedforward function can be expressed as
Gff dq tð Þ
� � ¼ h11 tð Þ h12 tð Þ
h21 tð Þ h22 tð Þ
� �
ð7:18Þ
Therefore, the full-feedforward components of the grid voltages in the syn-
chronous d–q frame shown in Fig. 7.6 is given by
vff d tð Þ ¼ h11 tð Þ � vg d tð Þþ h12 tð Þ � vg q tð Þ
vff q tð Þ ¼ h21 tð Þ � vg d tð Þþ h22 tð Þ � vg q tð Þ
�
ð7:19Þ
where * denotes convolution product.
The stationary a–b to synchronous d–q transformation is given in (7.8), hence
the synchronous grid voltages in terms of the stationary grid voltages can be
expressed as
vg d tð Þ ¼ vg a tð Þ cosxotþ vg b tð Þ sinxot
vg q tð Þ ¼ �vg a tð Þ sinxotþ vg b tð Þ cosxot
(
ð7:20Þ
The synchronous d–q to stationary a–b transformation is given in (7.9), hence
the feedforward components added to the modulating signals in the stationary a–b
frame can be expressed as
vff a tð Þ ¼ vff d tð Þ cosxot � vff q tð Þ sinxot
vff b tð Þ ¼ vff d tð Þ sinxotþ vff q tð Þ cosxot
(
ð7:21Þ
Substituting (7.20) into (7.19), gives
M
inv
M inv
Fig. 7.7 Block diagram of the full-feedforward scheme for the hybrid frame-controlled
grid-connected inverter
148 7 Full-Feedforward Scheme of Grid Voltages for Three-Phase …
vff d tð Þ ¼ h11 tð Þ � vg a tð Þ cosxotþ vg b tð Þ sinxot
� �
þ h12 tð Þ � �vg a tð Þ sinxotþ vg b tð Þ cosxot
� �
vff q tð Þ ¼ h21 tð Þ � vg a tð Þ cosxotþ vg b tð Þ sinxot
� �
þ h22 tð Þ � �vg a tð Þ sinxotþ vg b tð Þ cosxot
� �
8>>><
>>>:
ð7:22Þ
Substituting (7.22) into (7.21), it can be obtained that
vff a tð Þ ¼
h11 tð Þ � vg a tð Þ cosxotþ vg b tð Þ sinxot
� �
þ h12 tð Þ � �vg a tð Þ sinxotþ vg b tð Þ cosxot
� �
( )
cosxot
�
h21 tð Þ � vg a tð Þ cosxotþ vg b tð Þ sinxot
� �
þ h22 tð Þ � �vg a tð Þ sinxotþ vg b tð Þ cosxot
� �
( )
sinxot
vff b tð Þ ¼
h11 tð Þ � vg a tð Þ cosxotþ vg b tð Þ sinxot
� �
þ h12 tð Þ � �vg a tð Þ sinxotþ vg b tð Þ cosxot
� �
( )
sinxot
þ
h21 tð Þ � vg a tð Þcosxotþ vg b tð Þ sinxot
� �
þ h22 tð Þ � �vg a tð Þ sinxotþ vg b tð Þ cosxot
� �
( )
cosxot
8>>>>>>>>>>>>>>>><
>>>>>>>>>>>>>>>>:
ð7:23Þ
Equation (7.23) is transformed into the s-domain by taking the Laplace trans-
formation of each term. (7.24) will be used during the transformation.
L h tð Þ � vg tð Þ cos xotð Þ
� �
cos xotð Þ
 � ¼ L vg tð Þ cos xotð Þ� �H sð Þ
 � � ss2 þx2o
¼ 1
2
H sð Þvg sþ jxoð ÞþH sð Þvg s� jxoð Þ
� � � s
s2 þx2o
¼ 1
4
H sþ jxoð Þvgðsþ j2xoÞþH s� jxoð Þvg sð Þ
þHðsþ jxoÞvgðsÞþHðs� jxoÞvg s� j2xoð Þ
" #
L h tð Þ � vg tð Þ sin xotð Þ
� �
sin xotð Þ
 � ¼ 1
4
�H sþ jxoð Þvg sþ j2xoð ÞþH s� jxoð Þvg sð Þ
þH sþ jxoð Þvg sð Þ � H s� jxoð Þvg s� j2xoð Þ
" #
L h tð Þ � vg tð Þ sin xotð Þ
� �
cos xotð Þ
 � ¼ j
4
H sþ jxoð Þvg sþ j2xoð ÞþH s� jxoð Þvg sð Þ
�H sþ jxoð ÞvgðsÞ � H s� jxoð Þvg s� j2xoð Þ
" #
L h tð Þ � vg tð Þ cos xotð Þ
� �
sin xotð Þ
 � ¼ j
4
H sþ jxoð Þvg sþ j2xoð Þ � H s� jxoð ÞvgðsÞ
þH sþ jxoð Þvg sð Þ � H s� jxoð Þvg s� j2xoð Þ
" #
ð7:24Þ
where h(t) can be any one of h11(t), h12(t), h21(t), and h22(t), vg(t) can be vg_a(t) or
vg_b(t). H(s) and vg(s) are the Laplace forms of h(t) and vg(t), respectively.
As shown in (7.17), we have
7.2 Derivation of the Full-Feedforward Scheme of Grid Voltages 149
H11 sð Þ ¼ H22 sð Þ; H12 sð Þ ¼ �H21 sð Þ ð7:25Þ
Hence, the Laplace transformation of (7.23) is simplified into (7.26) using
(7.24).
vff a sð Þ ¼ 12 H11 sþ jxoð ÞþH11 s� jxoð Þ½ � � j H12 s� jxoð Þ � H12 sþ jxoð Þ½ �f gvg a sð Þ
þ 1
2
H12 sþ jxoð ÞþH12 s� jxoð Þ½ � � j H11 sþ jxoð Þ � H11 s� jxoð Þ½ �f gvg b sð Þ
vff b sð Þ ¼ 12 � H12 sþ jxoð ÞþH12 s� jxoð Þ½ � þ j H11 sþ jxoð Þ � H11ðs� jx0Þ½ �f gvg a sð Þ
þ 1
2
H11 sþ jxoð ÞþH11 s� jxoð Þ½ � � j H12 s� jxoð Þ � H12 sþ jxoð Þ½ �f gvg b sð Þ
8>>>>>>>>>><
>>>>>>>>>>:
ð7:26Þ
Substituting the corresponding terms shown in (7.17) into (7.26) gives
vff aðsÞ ¼ 1KPWM þHi1C � sþ L1CKPWM � s2
� 
vg aðsÞ
vff bðsÞ ¼ 1KPWM þHi1C � sþ L1CKPWM � s2
� 
vg bðsÞ
8<
: ð7:27Þ
According to (7.27), the full-feedforward function in the hybrid frame is
Gff ðsÞ ¼ 1KPWM þHi1C � sþ
L1C
KPWM
� s2 ð7:28Þ
Comparing (7.28) and (7.12), it is apparent that the full-feedforward function in
the hybrid frame is the same as the full-feedforward function in the stationary a–b
frame. Similarly, the feedback coefficient of the capacitor current can also be
transformed into the stationary a–b frame.
As mentioned above, the control strategy in the hybrid frame in Fig. 7.7 has the
following advantages:
(1) The active and reactive injected grid currents are controlled directly and
independently;
(2) The full-feedforward function is simple, which has no cross-coupling terms;
(3) Less transformation between different control frames.
7.3 Discussion of the Full-Feedforward Functions
In the previous section, the full-feedforward functions for the stationary a–b frame,
synchronous d–q frame, and hybrid frame-controlled three-phase LCL-type
grid-connected inverter have been derived. In this section, the effect of the three
components in the full-feedforward function, which are proportional, derivative,
150 7 Full-Feedforward Scheme of Grid Voltages for Three-Phase …
and second-derivative components, is discussed. After that, the harmonic attenua-
tion affected by LCL filter parameter mismatches is studied. Finally, a comparison
between the full-feedforward functions for the L-type and LCL-type three-phase
grid-connected inverters is presented.
7.3.1 Discussion of the Effect of Three Components
in the Full-Feedforward Function
The full-feedforward functions of grid voltages for the three-phase LCL-type
grid-connected inverters are composed of the proportional, derivative, and
second-derivative components. The proportional component is frequency inde-
pendent, and the derivative and second-derivative components will be increased as
the harmonic frequency going high. Therefore, as the frequency of the harmonic
frequency varies, the effect of the three components will be different, and it is
possible to simplify the full-feedforward function.
For the convenience of the demonstration, a 20-kW three-phase LCL-type
grid-connected inverter prototype is taken as the example, and the main parameters
are given in Table 7.1. According to (7.12), taking the proportional component as
the base, the proportional, derivative, and second-derivative components are drawn
in p.u., as shown in Fig. 7.8. As seen, in the low-frequency range, the proportional
component is dominant; as the frequency goes high, the derivative and
second-derivative components become large and dominant in the high-frequency
range. Therefore, if the grid voltages are mainly distorted by the low-frequency
harmonics, fifth harmonic for example, the full-feedforward function in (7.12) can
be simplified to the proportional component. If the harmonic order is not higher
than thirteenth, the full-feedforward function can be simplified to the proportional
plus derivative component [3].
To help investigating the harmonic attenuation performance of the feedforward
schemes, a generalized equivalent block diagram for the stationary a–b frame-
controlled three-phase LCL-type grid-connected inverter with the feedforward
scheme is given in Fig. 7.9, where F(s) comes from the feedforward path, and it can
be derived by taking the inverse procedures shown in Figs. 6.5 and 6.6 in Chap. 6.
Table 7.1 Parameters of the
prototype
Parameter Value Parameter Value
Vin 750 V C 15 lF
Vg (phase, rms) 220 V L2 110 lH
Po 20 kW Vtri 4.58 V
fo 50 Hz Hi1 0.12
fsw 15 kHz Hi2 0.14
L1 700 lH Hv 0.017
7.3 Discussion of the Full-Feedforward Functions 151
Taking the a-axis, for example, vg_a(s) is the actual grid voltage at a-axis, while
v′g_a(s) which is used to evaluate the harmonic attenuation performance, is the
equivalent grid voltage at a-axis with feedforward schemes. Observing Fig. 7.9, it
can be obtained that
v0g a sð Þ ¼ vg a sð Þ 1þF sð Þð Þ ð7:29Þ
If the full-feedforward scheme is used, F(s) equals to −1 and v′g_a(s) is zero,
which means that the injected grid currents caused by the grid voltages are elimi-
nated. In this case, the a-axis in Fig. 7.9 is equivalent to Fig. 6.5a in Chap. 6.
When the full-feedforward scheme is simplified to the proportional feedforward
scheme, according to Fig. 7.5, F(s) can be derived as
F sð Þ ¼ �Gff P sð ÞGx1 sð Þ
Gsi sð Þ
¼ �
1
KPWM
1
KPWM
þHi1C � sþ L1CKPWM � s2
ð7:30Þ
where Gff_P(s) is the proportional component in (7.12).
Substituting (7.30) into (7.29), v′g_a(s) can be expressed as
100 101 102 103 104
Frequency (Hz)
0
1
2
3
Proportional
Derivative
Second
derivative
Fig. 7.8 Amplitude of the
three components of the
full-feedforward function
in p.u
Fig. 7.9 Generalized block
diagram of the three-phase
LCL-type grid-connected
inverter with the feedforward
schemes
152 7 Full-Feedforward Scheme of Grid Voltages for Three-Phase …
v0g a sð Þ ¼
Hi1C � sþ L1CKPWM � s2
1
KPWM
þHi1C � sþ L1CKPWM � s2
vg a sð Þ� ð7:31Þ
According to (7.31), the amplitude of v0g aðsÞ in per-unit values with vg_a(s) as
the base is drawn in Fig. 7.10 using the parameters listed in Table 7.1. It can be
observed that with the proportional feedforward scheme, the low-frequency har-
monics are well suppressed, while the equivalent grid voltage is larger than that
without feedforward scheme at the frequency range higher than ft, and it means the
corresponding injected grid current harmonics are amplified. ft can be derived by
equalizing the amplitude of v′g_a(s) and vg_a(s) from (7.31), i.e.,
j2pftHi1Cþ j2pftð Þ
2L1C
KPWM
1
KPWM
þ j2pftHi1Cþ j2pftð Þ
2L1C
KPWM
������
������ ¼ 1 ð7:32Þ
Solving (7.32), leads to
ft ¼ 1
2p
ffiffiffiffiffiffiffiffiffiffiffi
2L1C
p ð7:33Þ
As seen, ft is only related to the parameters of the LCL filter.
Based on the above analysis, it can be known that the high-order harmonics are
amplified by the proportional feedforward scheme, while the full-feedforward
scheme can yield a relative wide-frequency-rangeharmonic suppression.
Fig. 7.10 Equivalent grid
voltage at a-axis in p.u
7.3 Discussion of the Full-Feedforward Functions 153
7.3.2 Harmonic Attenuation Affected by LCL Filter
Parameter Mismatches
In practice, due to the tolerance or aging of the filter components and the parasitic
parameters of the system, the LCL filter parameter mismatches might happen.
Referring to (7.12), (7.17), and (7.28), the full-feedforward functions are related to
L1 and C. Therefore, the harmonic attenuation performance of the full-feedforward
schemes might be weakened by the LCL filter parameter mismatches. The effect of
LCL filter parameter mismatches is also analyzed with Fig. 7.9.
With the full-feedforward scheme, F(s) is depicted as (7.34) when LCL filter
parameter mismatches happen.
F sð Þ ¼ F0 sð Þ ¼ �
1
KPWM
þHi1C � sþ s2L1CKPWM
1
KPWM
þHi1C0 � sþ s
2L01C
0
KPWM
ð7:34Þ
where C′ and L01 are the actual parameters of the filter capacitance and inverter-side
inductance in the prototype, C and L1 are the parameters in the designer’s mind.
Assuming the variations of C′ and L01 are limited to ±10% and ±20%,
respectively. Through the enumeration method, the worst case is found to be
C′ = 0.9C and L01 = 0.8L1. Substituting (7.34) into (7.29) and taking vg_a(s) as the
base, the amplitude of v′g_a(s) under the worst case can be expressed in per-unit
values and drawn in Fig. 7.10 using the parameters listed in Table 7.1. Since larger
equivalent grid voltages bring larger injected grid currents, it can be seen that the
harmonic attenuation performance of the full-feedforward scheme at low-frequency
range is still outstanding even with large LCL filter parameter mismatches, but it is a
little weakened at higher-frequency range. Besides, using full-feedforward scheme,
no injected grid current harmonic amplification is found with LCL filter parameter
mismatches.
7.3.3 Comparison Between the Feedforward Functions
for the L-Type and the LCL-Type Three-Phase
Grid-Connected Inverter
The full-feedforward function for the three-phase LCL-type grid-connected inverter
derived in this chapter consists of three parts, which are the proportional, derivative,
and second-derivative parts. For the three-phase L-type grid-connected inverter,
C does not exist. So, Letting C = 0 in (7.12) yields the disappearance of derivative
and second-derivative parts, and only the proportional part holds. This means that
the proportional feedforward scheme is valid for the three-phase L-type
grid-connected inverter, which has been proposed in [4–7]. The connection and
differences between the three-phase grid-connected inverters with different filters
154 7 Full-Feedforward Scheme of Grid Voltages for Three-Phase …
are listed as follows to help understanding the new features of the full-feedforward
schemes.
(1) Feedforward function for the three-phase L-type grid-connected inverter is the
same as the proportional part of the full-feedforward functions for the LCL-type
inverter. And, there are two additional parts, which are derivative and
second-derivative components, in the full-feedforward functions for the LCL-
type grid-connected inverter.
(2) Since there are derivative components in the full-feedforward function for the
LCL-type grid-connected inverter, when the grid voltage step happens, the
calculated feedforward signal becomes infinite, which is not applicable in
practical circuits. Therefore, compared with the L-type grid-connected inverter,
the improvement of the transient response under the step change of the grid
voltage using full-feedforward scheme for the LCL-type grid-connected inverter
is quite limited.
(3) The feedforward function for the three-phase L-type grid-connected inverter
stays the same no matter the feedforward scheme is implemented in the sta-
tionary a–b frame or the synchronous d–q frame [5, 6]. In contrast, the
full-feedforward functions for the three-phase LCL-type grid-connected inverter
are different when the feedforward schemes are implemented in different
frames.
Therefore, when applying the feedforward scheme for the three-phase LCL-type
grid-connected inverter, the full-feedforward function should be selected according
to the control strategy being used.
7.4 Experimental Verification
7.4.1 Description of the Prototype
To verify the effectiveness of the full-feedforward scheme, a 20-kW prototype is
built and tested in the laboratory. Figure 7.11 gives the photograph of the prototype.
The key parameters of the prototype have been given in Table 7.1. The power
switches use IGBT module CM100DY-24NF and the driving chip is M57962L.
The current sensors are LA-55P, and the voltage sensors are LV-25P. The controller
is implemented in a DSP (TMS320F2812). The sampling frequency (fs = 1/Ts) of
the digital control system is 20 kHz. Synchronization of the injected grid currents to
grid voltages is achieved by a digital PLL. An RC low-pass filter with the time
constant of 0.1 ls is used in the prototype to suppress the noise in the sampling
circuits of the grid voltages. A very little phase shift of the sampled grid voltage is
introduced by this low-pass filter, and it has little effect on the performance of the
full-feedforward scheme. Moreover, the backward difference approximation, which
7.3 Discussion of the Full-Feedforward Functions 155
is defined as s = (1 − z−1)/Ts, is used to discretize the controller. For example, the
full-feedforward function of the grid voltages given in (7.12) can be discretized as
Gff ðzÞ ¼ 1Hv
1
KPWM
þ 1� z
�1ð ÞHi1C
Ts
þ 1� z
�1ð Þ2L1C
T2s KPWM
" #
ð7:35Þ
Therefore, the output of (7.35) only depends on the past and present input, which
means that Gff(z) is a causal function.
7.4.2 Experimental Results
To get an accurate evaluation of the proposed full-feedforward schemes, the grid
voltages are simulated using a programmable AC source (Chroma 6590). The
simulated grid voltages distorted by fifth, seventh, eleventh, thirteenth, and
twenty-third harmonics. The magnitudes of the simulated grid voltage harmonics
with respect to the fundamental component of the grid voltages are 5%, 3%, 2%,
2%, and 1%, respectively, and the corresponding phases are 180°, 0°, 0°, 0°, and 0°.
Both the full-feedforward schemes for the stationary a–b frame and hybrid
frame-controlled three-phase grid-connected inverter are verified in the experiment,
which are defined as strategy I and strategy II, respectively.
Figures 7.12 and 7.13 show the experimental results with strategies I and II
under the simulated distorted grid voltages. The measured total harmonic distortion
(THD) of the injected grid currents shown in Fig. 7.12a–c are 15.6%, 13.6%, and
C C C
L1
Auxiliary
Power
DC Bus 
Capacitor
DSP Board
Filter capacitor and 
sampling board
L2
IGBT & Drive
Fig. 7.11 Photograph of the prototype
156 7 Full-Feedforward Scheme of Grid Voltages for Three-Phase …
4.6%, respectively. The measured THD of the injected grid currents shown in
Fig. 7.13a–c are 16.4%, 13.2%, and 5.1%, respectively. The harmonic spectrum of
the injected grid currents shown in Figs. 7.12 and 7.13 is presented in Fig. 7.14.
From Figs. 7.12, 7.13, and 7.14, it can be observed that the proposed
full-feedforward schemes suppress the injected grid current harmonics caused by
vga
Time:[5 ms/div]
vgcvgb
i2a i2ci2b
(a) No feedforward scheme 
Time:[5 ms/div]
vga vgcvgb
i2a i2ci2b
(b) Proportional feedforward scheme 
Time:[5 ms/div]
vga vgcvgb
i2a i2ci2b
(c) Full-feedforward scheme 
Fig. 7.12 Experimental
results under distorted grid
voltages with strategy I. Grid
voltage: 200 V/div, injected
grid current: 10 A/div
7.4 Experimental Verification 157
the grid voltage distortion effectively. Compared with the full-feedforward schemes,
the proportional feedforward scheme has a relatively poor performance on sup-
pressing the injected grid current harmonics. Furthermore, Fig. 7.14 shows that the
proportional feedforward scheme amplifies the twenty-third order current harmonic.
It is in agreement withthe conclusion that simplifying the full-feedforward function
Time:[5 ms/div]
vga vgcvgb
i2a i2ci2b
(a) No feedforward scheme 
Time:[5 ms/div]
vga vgcvgb
i2a i2ci2b
(b) Proportional feedforward scheme 
Time:[5 ms/div]
vga vgcvgb
i2a i2ci2b
(c) Full-feedforward scheme 
Fig. 7.13 Experimental
results under distorted grid
voltages with strategy II. Grid
voltage: 200 V/div, injected
grid current: 10 A/div
158 7 Full-Feedforward Scheme of Grid Voltages for Three-Phase …
for the three-phase LCL-type grid-connected inverter to a proportional feedforward
function will give rise to the amplification of the high-frequency harmonics as
shown in Fig. 7.10.
The proposed full-feedforward scheme for the three-phase LCL-type
grid-connected inverter is also investigated under unbalanced grid voltage condi-
tion. In the laboratory, the representative phase-to-phase-fault unbalanced grid
voltages and single-phase-fault unbalanced grid voltages transferred through a
Δy transformer [8] are simulated using the programmable AC source. The
positive-sequence grid voltage is 80% of the rated grid voltage and the phase is 0°.
The negative-sequence grid voltage is 20% of the rated grid voltage and the phase is
also 0°. Thus, the three-phase grid voltages are described as
vga tð Þ ¼ 311 sin xotð Þ
vgb tð Þ ¼ 224 sin xotþ 226:1�ð Þ
vgc tð Þ ¼ 224 sin xotþ 133:9�ð Þ
ð7:35Þ
where vga preserves the nominal grid voltage, and the voltages of the other two
phases have reduced magnitude and present a symmetrical phase deviation of 13.9°.
Figure 7.15 gives the experimental results with strategy I under unbalanced grid
voltage condition. The positive-sequence injected grid current reference is 4 A, and
the negative-sequence injected grid current reference is 0 A. The synchronization of
the positive-sequence grid voltage is achieved with the digital PLL. As shown in
Fig. 7.15a, without the feedforward scheme of the grid voltages, the RMS value of
i2a is 4.70 A, and i2a has a large phase shift with respect to vga. The RMS value of
i2b and i2c are 4.01 A and 4.78 A, respectively. The injected grid currents are
obviously unbalanced. With the proposed full-feedforward scheme of the grid
voltages, as shown in Fig. 7.15b, the measured rms value of i2a is 4.06 A, and the
phase shift with respect to vga is eliminated. The RMS value of i2b and i2c are
4.08 A and 4.09 A, respectively. Therefore, by introducing the proposed
No fee
dforwa
rd
Propor
tional f
eedfor
ward
Full fe
edforw
ard Order
of harm
onicP
er
ce
nt
ag
e
of
in
je
ct
ed
gr
id
cu
rr
en
th
ar
m
on
ic
s(
%
)
0
1
2
3
4
5
6
7
8
9
10
5
7
11
13
23
Strateg
y I
Strateg
y II
Fig. 7.14 Harmonic
spectrum of the injected grid
currents under distorted grid
voltages
7.4 Experimental Verification 159
full-feedforward scheme of the grid voltages, the negative-sequence injected grid
current is well regulated under the unbalanced grid voltage condition.
Figure 7.16 gives the experimental results with strategy I at full load (20 kW)
under a real power grid. Figure 7.16a gives the experimental results without
feedforward scheme. It can be observed that there is a little phase shift between the
injected grid currents and grid voltages. Meanwhile, the injected grid currents are
distorted by the grid voltage harmonics, and the measured THD of the injected grid
currents is 1.18%. Figure 7.16b gives the experimental results with the proposed
full-feedforward scheme. Obviously, the phase shift between grid currents and grid
voltages is eliminated, and the injected grid current harmonics are greatly reduced
with the measured THD of 0.97%. Therefore, the proposed full-feedforward
scheme works well under a real power grid.
Figure 7.17a, b show the transient response of the three-phase LCL-type
grid-connected inverter using strategy I when the step change in the grid current
reference and the grid voltages occur, respectively. The references of the injected
grid currents are stepped between half load and full load in Fig. 7.17a. Note that the
waveforms in Fig. 7.17a are taken under a real power grid. It is observed that the
Time:[5 ms/div]
vga vgcvgb
i2a i2ci2b
(a) No feedforward scheme 
Time:[5 ms/div]
vga vgcvgb
i2a i2ci2b
(b) Proportional feedforward scheme 
Fig. 7.15 Experimental
results under distorted grid
voltages with strategy II. Grid
voltage: 200 V/div, injected
grid current: 5 A/div
160 7 Full-Feedforward Scheme of Grid Voltages for Three-Phase …
references are fast tracked in about 2 ms, and at steady state, the injected grid
currents are well regulated. The overshoot of the injected grid currents is large.
Fortunately, in practice, the step change of the reference is not indispensable and it
can approximately be replaced by a ramp change, for example, the reference ramps
to the final value in 1 ms. This approximation would dramatically improve the
transient performance of the system. In Fig. 7.17b, the three-phase grid voltages are
stepped between 220 V and 180 V. The step changes of the grid voltages are
simulated using the programmable AC source. It is observed that the amplitude of
the injected grid currents is kept unchanged at steady state, but the proposed
full-feedforward scheme seems useless during the transient state. This is because
there are derivative and second-derivative parts in the full-feedforward function
shown in (7.12). The step change of the grid voltages will result in infinite feed-
forward signals, and this is only possible in mathematics but not in practice.
Therefore, the improvement of the transient response under the step change of the
grid voltage using full-feedforward scheme is quite limited.
Time:[5 ms/div]
vga vgb
i2a i2b
(a) No feedforward scheme 
Time:[5 ms/div]
vga vgb
i2a i2b
(b) Proportional feedforward scheme 
Fig. 7.16 Experimental
results under real grid with
strategy I. Grid voltage:
100 V/div, injected grid
current: 20 A/div
7.4 Experimental Verification 161
7.5 Summary
To suppress the harmonic and unbalance components in the grid currents injected
from the grid-connected inverter, the full-feedforward scheme of grid voltages in
the stationary a–b frame, synchronous d–q frame, and hybrid frame for the
three-phase LCL-type grid-connected inverter have been proposed and investigated
in this chapter. The full-feedforward function is mainly composed of the propor-
tional, derivative, and second-derivative components. A brief comparison between
the feedforward functions for the L-type and the LCL-type three-phase
grid-connected inverter is presented to emphasize the new features of the pro-
posed full-feedforward schemes. Moreover, it is important to notice that simplifying
the full-feedforward function for the three-phase LCL-type grid-connected inverter
to a proportional feedforward function will give rise to the amplification of the
high-frequency harmonics. With the proposed full-feedforward schemes, the
injected grid current harmonics and unbalance caused by the grid voltage are
greatly reduced. Besides, the harmonic attenuation affected by LCL filter parameter
vga
i2b i2ci2a
Time:[10 ms/div]
(a) Step change in the grid current reference. 
Grid voltage: 100 V/div, injected grid current: 20 A/div. 
Time:[10 ms/div]
vga
i2a i2b i2c
(b) Step change in the grid voltages. 
Fig. 7.17 Transient response
with strategy I. Grid voltage:
200 V/div, injected grid
current: 5 A/div
162 7 Full-Feedforward Scheme of Grid Voltages for Three-Phase …
mismatches is also discussed, and it is found that the harmonic attenuation per-
formance of the full-feedforward scheme is still outstanding even with large LCL
filter parameter mismatches. Finally, a 20-kW prototype has been built to verify the
effectiveness of the proposed full-feedforward scheme. It should be pointed out that
the improvement of the transient response under step change of the grid voltages is
limited in practice due to the limited amplitude of the feedforward signals.
References
1. Prodanović, M., Green, T.C.: High-qualitypower generation through distributed control of a
power park microgrid. IEEE Trans. Ind. Electron. 53(5), 1471–1482 (2006)
2. Li, W., Ruan, X., Pan, D., Wang, X.: Full-feedforward schemes of grid voltages for a
three-phase LCL-type grid-connected inverter. IEEE Trans. Ind. Electron. 60(6), 2237–2250
(2013)
3. Wang, X., Ruan, X., Liu, S., Tse, C.K.: Full feed-forward of grid voltage for grid-connected
inverter with LCL filter to suppress current distortion due to grid voltage harmonics. IEEE
Trans. Power Electron. 25(12), 3119–3127 (2010)
4. Timbus, A.V., Liserre, M., Teodorescu, R., Rodriguez, P., Blaabjerg, F.: Evaluation of current
controllers for distributed power generation systems. IEEE Trans. Power Electron. 24(3), 654–
664 (2009)
5. Holmes, D.G., Lipo, T.A., McGrath, B.P., Kong, W.Y.: Optimized design of stationary frame
three phase ac current regulators. IEEE Trans. Power Electron. 24(11), 2417–2426 (2009)
6. Kim, J.S., Sul, S.K.: New control scheme for ac-dc-ac converter without dc link electrolytic
capacitor. In: Proceeding of the IEEE Power Electronics Specialists Conference, pp. 300–306.
(1993)
7. Zeng, Q., Chang, L.: An advanced SVPWM-based predictive current controller for three-phase
inverters in distributed generation systems. IEEE Trans. Ind. Electron. 55(3), 1235–1246
(2008)
8. Bollen, M.H.J.: Characterization of voltage sags experienced by three-phase adjustable-speed
drives. IEEE Trans. Power Del. 12(4), 1666–1671 (1997)
7.5 Summary 163
Chapter 8
Design Considerations of Digitally
Controlled LCL-Type Grid-Connected
Inverter with Capacitor-
Current-Feedback Active-Damping
Abstract The capacitor-current-feedback active-damping is an effective approach
for damping the resonance peak of the LCL filter. When the LCL-type
grid-connected inverter is digitally controlled, the control delay will be generated.
This will result in different behavior of the capacitor-current-feedback
active-damping from that with analog control. In this chapter, the mechanism of
the control delay in the digital control system is introduced first. Then, a series of
equivalent transformations of the control block diagram considering the control
delay are performed, and it reveals that the capacitor-current-feedback
active-damping is no longer equivalent to a virtual resistor in parallel with the
filter capacitor, but a virtual frequency-dependent impedance. A forbidden region
for choosing the LCL filter resonance frequency is presented in order to guarantee
the system stability. Then, the controller design for digitally controlled LCL-type
grid-connected inverter with capacitor-current-feedback active-damping is studied.
Since the control delay leads to a phase lag and consequently changes the location
of −180°-crossing in the phase curve of the loop gain, the system stability might be
guaranteed even without damping the resonance of LCL filter. For this case, the
necessary condition for system stability is studied, and the controller design method
is presented. Finally, the controller parameters design examples for the grid current
regulator with and without the capacitor-current-feedback active-damping are
given, and the effectiveness of the theoretical analysis is verified by the experi-
mental results.
Keywords Grid-connected inverter � LCL filter � Active damping � Digital control �
Controller design
8.1 Introduction
In the LCL-type grid-connected inverter, the inherent resonance of LCL filter
exhibits a resonance peak and a sharp phase step down of −180° at the resonance
frequency, which might trigger undesired oscillation or even system instability.
© Springer Nature Singapore Pte Ltd. and Science Press 2018
X. Ruan et al., Control Techniques for LCL-Type Grid-Connected Inverters,
CPSS Power Electronics Series, DOI 10.1007/978-981-10-4277-5_8
165
Therefore, the resonance peak should be damped properly to ensure system sta-
bility. Chapter 4 has presented the methods of damping the LCL filter resonance.
Adding a resistor in parallel with the filter capacitor can effectively damp the
resonance without affecting magnitude-frequency characteristics of the LCL filter at
the low- and high-frequency ranges. However, there is considerable power loss in
the damping resistor, degrading the efficiency of the grid-connected inverter. With a
series of equivalent transformation of control block diagram, it is revealed that the
capacitor-current-feedback active-damping is equivalent to a virtual resistor in
parallel with the filter capacitor, and the power loss in the real resistor is avoided.
A step-by-step controller design method for the LCL-type grid-connected inverter
with capacitor-current-feedback active-damping has been presented in Chap. 5,
where PI and PR regulators are adopted as the grid current regulator. Given the
specified grid current steady-state error, stability margin (including phase margin
and gain margin), a satisfactory region for the capacitor-current-feedback coefficient
and the crossover frequency is obtained. With this satisfactory region, it is very
convenient to choose the controller parameters and optimize the system
performance.
Actually, the equivalent transformation of the control block diagram presented in
Chap. 4 is based on the analog control. When the LCL-type grid-connected inverter
is digitally controlled, the control delay, including the computation and pulse-width
modulation (PWM) delays, will be generated. This will result in different behavior
of the capacitor-current-feedback active-damping. In this chapter, the mechanism of
the control delay in the digital control system will be introduced first. Then, a series
of equivalent transformations of the control block diagram considering the control
delay are performed, and it will reveal that the capacitor-current-feedback
active-damping is no longer equivalent to a virtual resistor, but a virtual
frequency-dependent impedance, which is in parallel with the filter capacitor. The
virtual frequency-dependent impedance consists of a virtual reactor and a virtual
resistor, which are connected in parallel. The virtual reactor makes the resonance
frequency of the system loop gain derivate from the resonance frequency of the
LCL filter. The virtual frequency-dependent resistor might be negative at the res-
onance frequency of the loop gain, which implies the loop gain will contain two
open-loop right-half-plane (RHP) poles. This is different from the characteristics of
the analog control system.
After that, a forbidden region for choosing the LCL filter resonance frequency is
presented in order to guarantee the system stability. Then, the controller design
for digitally controlled LCL-type grid-connected inverter with capacitor-
current-feedback active-damping is studied. Similar to that presented in Chap. 5,
in terms of the specified grid current steady-state error, phase margin, and gain
margin, a satisfactory region for the capacitor-current-feedback coefficient and the
crossover frequency is obtained, from which, proper controller parameters can
easily be selected.
Since the control delay leads to a phase lag and consequently changes the
location of −180°-crossing in the phase curve of the loop gain, the system stability
might be guaranteed even without damping the resonance of LCL filter. For this
166 8 Design Considerations of Digitally Controlled LCL-Type …
case, the necessary condition for system stability will be studied, and the controller
design method is presented [1].
Finally, the controller parameters design examples for the grid current regulator
with and without the capacitor-current-feedback active-damping are given, and the
effectiveness of the theoretical analysis is verified by the experimental results from a
6-kW single-phase LCL-type grid-connected inverter prototype.
8.2 Control Delay in Digital Control System
Figure 8.1 shows the main circuit and control diagram of the digitally controlled
LCL-type grid-connected inverter, where Hv and Hi2 represent the sampling coef-
ficients ofthe grid voltage vg and the grid current i2, respectively. The filter ca-
pacitor current iC is fed back with the coefficient Hi1 for damping the resonance of
the LCL filter. The grid current reference i�2 = I
*cosh, where h is the phase of vg,
which is obtained through a phase-locked loop (PLL), and I* is the current
amplitude reference, which is generated by the outer voltage loop. The error
between i�2 and i2 is sent to the grid current regulator Gi(z). The modulation signal
vM is obtained by subtracting the feedback signal of filter capacitor current from the
output of Gi(z). By comparison with vM and the triangular carrier, the control
signals of the power switches in the grid-connected inverter are generated.
Generally, the crossover frequency of the outer voltage loop is far lower than that of
the grid current loop [2, 3], so the grid current loop can be designed independently.
In digital control system, the currents i2 and iC are usually sampled at the peak
and valley of the triangular carrier to avoid the switching noise, as shown in
Fig. 8.2 [4]. The sampled i2 and iC, for example at step k, are sent to the digital
signal processor (DSP) and calculated by the control algorithm to obtain the
modulation signal vM. To avoid repetitive intersections of vM and the carrier signal,
vgVin
L1 L2
i1 i2
C
iC
Hi1 Hi2 Hv
vC
Sine-triangle PWM
+
–
vinv
I*
+ i2+ –– Gi(z)
vM
DSP Controller
*
PLL
cosθ
Fig. 8.1 Digital control
schematic of single-phase
LCL filtered grid-connected
inverter
8.1 Introduction 167
the calculated vM is updated at step k + 1 [5]. Therefore, one sampling period delay
occurs, and this delay is called computation delay [6, 7]. After that, vM keeps
constant in the following sampling period and compares with the triangular carrier.
The zero-order hold (ZOH) is used to model the PWM process, expressed as [8]
Gh sð Þ ¼ 1� e
�sTs
s
� Tse�0:5sTs : ð8:1Þ
As shown in (8.1), the ZOH induces a half sampling period delay, which is
called PWM delay.
In summary, in the digital SPWM scheme, there exists control delay, including
the computation and the PWM delays. The former one is one sampling period delay
and the latter one is half sampling period delay.
8.3 Effect of Control Delay on Loop Gain
and Capacitor-Current-Feedback Active-Damping
8.3.1 Equivalent Impedance
of Capacitor-Current-Feedback Active-Damping
According to Fig. 8.1, the mathematical model in z-domain of the digitally con-
trolled LCL-type grid-connected inverter is given in Fig. 8.3a where z−1 represents
k k+1 k+2 k+3 k+4
iC
i2
vM
Ts
t
t
t
j
l
Actual current
Sampled current
Actual currentSampled current
Carrier
0
0 t
Fig. 8.2 Key waveforms of
signal sampling and digital
PWM
168 8 Design Considerations of Digitally Controlled LCL-Type …
the computation delay; KPWM = Vin/Vtri is the transfer function from the modulation
signal v′M after the ZOH to the inverter bridge output voltage, with Vin and Vtri
being the input voltage and the amplitude of the triangular carrier, respectively;
ZL1(s) = sL1, ZC(s) = 1/(sC), and ZL2(s) = sL2 are the impedances of L1, C, and L2,
respectively.
To intuitively illustrate the effect of the control delay on the
capacitor-current-feedback active-damping, the z-domain model shown in Fig. 8.3a
is transferred to the s-domain one, as shown in Fig. 8.3b, where the frequency
response of the sampling switch is represented by 1/Ts within the Nyquist fre-
quency, i.e., fs/2, [9, 10], z ¼ esTs , and i2*(s) and Gi(s) are the counterparts of
i2
*(z) and Gi(z) in s-domain, respectively. As observed from Fig. 8.3b, 1/Ts is
(a)
(b)
(c)
(d)
invMM
invMM
invMM
inv
Fig. 8.3 Mathematical model of the digitally controlled LCL filtered grid-connected inverter with
capacitor-current-feedback active-damping
8.3 Effect of Control Delay on Loop Gain … 169
included in both the forward path of i2
*(s) and the feedback paths of i2 and iC, so it
can be merged into the input of the transfer function e�sTs , as shown in Fig. 8.3c.
The product of 1/Ts, e�sTs and Gh(s) is e
−1.5sTs; thus, Fig. 8.3c is simplified to
Fig. 8.3d.
By changing the feedback of capacitor current to that of capacitor voltage, and
relocating the feedback node from the output of Gi(s) to that of 1/ZL1(s), Fig. 8.3d is
equivalently transformed into Fig. 8.4a. As observed, the capacitor-current feed-
back can be equivalent to virtual impedance Zeq in paralleled with the filter
capacitor, and the expression of Zeq is
Zeq ¼ ZL1 sð ÞZC sð ÞKPWMHi1e�1:5sTs ¼
L1
CKPWMHi1
e1:5sTs ¼ RAe1:5sTs ð8:2Þ
where RA = L1/(CKPWMHi1), which is the equivalent virtual resistor of the
capacitor-current-feedback active-damping in analog control system, which has
been presented in Chap. 5.
Substituting s = jx into (8.2) yields
Zeq jxð Þ ¼ RA cos 1:5xTsð Þþ jRA sin 1:5xTsð Þ,Req xð Þ==jXeq xð Þ ð8:3Þ
where
Req xð Þ ¼ RA=cos 1:5xTsð Þ ð8:4aÞ
Xeq xð Þ ¼ RA=sin 1:5xTsð Þ: ð8:4bÞ
KPWM
+ ––1.5
–1.5
Hi1
+
–
+ –
vg(s)
i2(s)
ZC(s)
Hi2
+
–
Gi(s)
+
–
1
ZL1(s)
1
ZL2(s)
e sTs
i2(s)*
KPWMHi1e sTs
ZL1(s)ZC(s)
+
–
1/Zeq(s)
(a) Equivalent transformation of the block diagram 
L1 L2
Cvinv vg
+ +
jXeq Req
(b) Equivalent circuit 
Fig. 8.4 Equivalent virtual impedance of the capacitor-current-feedback active-damping.
a Equivalent transformation of the block diagram. b Equivalent circuit
170 8 Design Considerations of Digitally Controlled LCL-Type …
As shown in Eq. (8.3), Zeq can be represented in the form of parallel connection
of a resistor Req and a reactor Xeq, as shown in Fig. 8.4b.
According to (8.4), the curves of Req and Xeq as the function of frequency can be
depicted, as shown in Fig. 8.5. As observed, when Hi1 > 0, Req is positive in the
range (0, fs/6) and negative in the range (fs/6, fs/2); Xeq is inductive in the range (0,
fs/3) and capacitive in the range (fs/3, fs/2). When Hi1 < 0, the frequency charac-
teristics of Req and Xeq are opposite to that when Hi1 > 0.
Comparing Fig. 8.3d with Fig. 5.2 in Chap. 5, it can be found that the difference
is the control delay e�1:5sTs . Therefore, replacing KPWM in (5.4) by KPWMe�1:5sTs ,
the loop gain of the digitally controlled LCL-type grid-connected inverter can be
obtained as
TD sð Þ ¼ Hi2KPWMe
�1:5sTsGi sð Þ
s3L1L2Cþ s2L2CHi1KPWMe�1:5sTs þ s L1 þ L2ð Þ
¼ 1
sL1L2C
� Hi2KPWMe
�1:5sTsGi sð Þ
s2 þ 1CZeq sð Þ sþx2r
ð8:5Þ
where xr ¼ 2p fr ¼
ffiffiffiffiffiffiffiffiffiffiffi
L1 þL2
L1L2C
q
is the resonance angular frequency of the LCL filter.
As shown in (8.5), both the numerator and denominator of TD(s) contain the
control delay e�1:5sTs . The e�1:5sTs in numerator introduces phase lag, and the
e�1:5sTs in denominator affects the location of the loop gain poles.
Figure 8.6 shows the Bode diagram of the uncompensated loop gain when
Hi1 > 0. Since Xeq behaves as a virtual inductor in the range (0, fs/3), the loop gain
resonance frequency fr′ will be higher than the LCL filter resonance frequency fr, as
shown in Fig. 8.6a, b, and Xeq behaves as a virtual capacitor in the range (fs/3, fs/2);
thus, fr′ will be lower than fr, as shown in Fig. 8.6c. According to (8.4b) and
considering RA = L1/(CKPWMHi1), a larger Hi1 will lead to a smaller RA and thus a
smaller |Xeq|. A smaller |Xeq| means that Xeq may behave as a smaller inductance or a
larger capacitance, which will cause a higher fr′ or lower fr′. That is to say,
increasing Hi1 will cause fr′ to deviate far from fr. Since fs/3 is the boundary for Xeq
is inductive and capacitive, no matter how Hi1 increases, fr′ cannot exceed fs/3. This
fs/6 fs/2fs/3
f (Hz)
0
Req
Xeq
RA
RA
Fig. 8.5 Curves of Req and
Xeq as the functions of
frequency
8.3 Effect of Control Delay on Loop Gain … 171
means that when fr < fs/3, fr′ cannot be higher than fs/3 as Hi1 increases; when
fr > fs/3, fr′ cannot be lower than fs/3 as Hi1 increases.
8.3.2 Discrete-Time Expression of the Loop Gain
As mentioned above, Req is negative in the range (fs/6, fs/2), which implies that the
loop gain might have RHP poles. As shownin (8.5), the loop gain TD(s) contains
the nonlinear term e�1:5sTs , it is difficult to directly calculate the poles in TD(s). So,
the control diagram shown in Fig. 8.3a will be transformed into z-domain. Note that
0
0
−360
−180
fs/6fr fs/2
−540
Hi1
Hi1=0 Hi1=Hi1C
|A
T
|(
dB
)
D
A
ng
(T
)(
º)
D
Frequency (Hz)
0
0
−360
−180
fs/6 fr fs/2
−540
|A
T
|(
dB
)
D
A
ng
(T
)(
º)
D
Frequency (Hz)
Hi1
Hi1=0
fs/3
(a) fr < fs/6 (b) fs/6 ≤ fr < fs/3 
0
0
−360
−180
fs/6 fr fs/2
−540
|A
T
|(
dB
)
D
A
ng
(T
)(
º)
D
Frequency (Hz)
fs/3
Hi1
Hi1=0
(c) fs/3 ≤ fr < fs/2 
Fig. 8.6 Bode diagrams of the uncompensated loop gain TD(s)
172 8 Design Considerations of Digitally Controlled LCL-Type …
vg(s) is a disturbance which does not affect the location of the poles, so it is ignored
in the following transformation.
According to Fig. 8.3a, the transfer function from v′M to i2 can be obtained as
i2 sð Þ
v0M zð Þ
¼ Gh sð Þ � KPWM
sL1L2C
� 1
s2 þx2r
¼ 1� e�sTs� � � KPWM
s2L1L2C
� 1
s2 þx2r
ð8:6Þ
While ignoring vg, i2 can be expressed as
i2 sð Þ¼ 1s2L2C iC sð Þ ð8:7Þ
Substituting (8.7) into (8.6) yields
iC sð Þ
v0M zð Þ
¼ 1� e�sTs� � � KPWM
L1
� 1
s2 þx2r
ð8:8Þ
Applying z-transform to (8.6) and (8.8), respectively, yields
i2 zð Þ
v0M zð Þ
¼ Z 1� e�sTs� � � KPWM
s2L1L2C
� 1
s2 þx2r
� �
¼ KPWM
xr L1 þ L2ð Þ
xrTs
z� 1�
z� 1ð Þ sinxrTs
z2 � 2z cosxrTs þ 1
� �
ð8:9Þ
iC zð Þ
v0M zð Þ
¼ Z 1� e�sTs� � � KPWM
L1
� 1
s2 þx2r
� �
¼ z� 1
xrL1
� KPWM sinxrTs
z2 � 2z cosxrTs þ 1 ð8:10Þ
Defining the output of Gi(z) in Fig. 8.3a as vr(z), v′M can be expressed as
v0M zð Þ ¼ z�1 � vr zð Þ � Hi1iC zð Þð Þ ð8:11Þ
Rearranging (8.11) leads to
v0M zð Þ
vr zð Þ ¼
1
zþHi1 � iC zð Þv0M zð Þ
ð8:12Þ
According to Fig. 8.3a, TD(z) can be expressed as
TD zð Þ,Hi2Gi zð Þ i2 zð Þvr zð Þ ¼ Hi2Gi zð Þ
i2 zð Þ
v0M zð Þ
� v
0
M zð Þ
vr zð Þ ð8:13Þ
8.3 Effect of Control Delay on Loop Gain … 173
Substituting (8.12) into (8.13) yields
TD zð Þ ¼ Hi2Gi zð Þ i2 zð Þv0M zð Þ
� 1
zþHi1 � iC zð Þv0M zð Þ
ð8:14Þ
Substituting (8.9) and (8.10) into (8.14) leads to
TD zð Þ ¼ Hi2Gi zð ÞKPWMxr L1 þ L2ð Þ
� xrTs z
2 � 2z cosxrTs þ 1ð Þ � z� 1ð Þ2sinxrTs
z� 1ð Þ z z2 � 2z cosxrTs þ 1ð Þþ z� 1ð Þ Hi1KPWMxrL1 sinxrTs
h i ð8:15Þ
As shown in (8.15), there is no nonlinear term in TD(z). Thus, it is convenient to
obtain the poles in TD(z) in z-domain.
8.3.3 RHP Poles of the System Loop Gain
As shown in (8.15), since Gi(z) does not contain any open-loop unstable pole, and
the pole z = 1 locates on the unit circle which is not an open-loop unstable pole, the
open-loop unstable poles in TD(z) are determined by the following equation, i.e.,
z z2 � 2z cosxrTs þ 1
� �þ z� 1ð ÞHi1KPWM
xrL1
sinxrTs ¼ 0 ð8:16Þ
In order to easily identify the number of the open-loop unstable poles in
TD(z) easily, w-transform is introduced. Substituting z = (1 + w)/(1 − w) into
(8.16) [9] gives
a0w
3 þ a1w2 þ a2wþ a3 ¼ 0 ð8:17Þ
where
a0 ¼ 1þ cosxrTs þ Hi1KPWMxrL1 sinxrTs
a1 ¼ 1þ cosxrTs � 2 Hi1KPWMxrL1 sinxrTs
a2 ¼ 1� cosxrTs þ Hi1KPWMxrL1 sinxrTs
a3 ¼ 1� cosxrTs
8>><
>>: ð8:18Þ
174 8 Design Considerations of Digitally Controlled LCL-Type …
The Routh array for (8.17) is expressed as
w3 : a0 a2
w2 : a1 a3
w1 : b1 0
w0 : a3
ð8:19Þ
where b1 = (a1a2 − a0a3)/a1. In order to ensure the controllability of the system, fr
must be lower than fs/2, so we have xrTs < p [9]. Given Hi1 � 0, it can be
observed from (8.18) that a0, a2, and a3 are always larger than 0.
Based on the Routh criterion, the number of the RHP roots of (8.17) is equal to
the number of the sign changing in the first row of the Routh array in (8.19), i.e.,
(a0, a1, b1, a3)
T. If (8.17) has the RHP roots, a1 < 0 or b1 < 0 must be true.
If b1 < 0 is true, Hi1 must satisfy
Hi1 [
2 cosxrTs � 1ð ÞxrL1
KPWM sinxrTs
,Hi1C ð8:20aÞ
If a1 < 0 is true, Hi1 must satisfy
Hi1 [
1þ cosxrTsð ÞxrL1
2KPWM sinxrTs
,H0i1C ð8:20bÞ
It is obvious that cosxrTs � 1, so we have H′i1C � Hi1C according to (8.20a,
b). If Hi1 > Hi1C, then b1 < 0. Considering a0 > 0 and a3 > 0, no matter a1 > 0 or
a1 < 0, the sign of (a0, a1, b1, a3)
T changes two times. Therefore, two open-loop
unstable poles must be in TD(z).
Substituting Hi1 = Hi1C into (8.16), the two open-loop unstable poles in
TD(z) can be calculated, which are z1;2 ¼ 12 1� j
ffiffiffi
3
p� �
. Mapping z1,2 back to s-
domain produces s1,2 = ± jpfs/3, which means that the resonance peak of the loop
gain actually locates at fs/6, as shown in Fig. 8.6a. In the range (0, fs/3), a larger Hi1
results in a higher fr′. Therefore, when Hi1 > Hi1C, fr′ > fs/6 happens. As mentioned
above, Req is negative in the range (fs/6, fs/2), so Req at fr′ must be negative when
TD(z) has open-loop unstable poles. Please note that the open-loop unstable poles in
TD(z) correspond to the RHP poles in TD(s).
Substituting x = 2pfr and x = 2pfs/6 into (8.15), respectively, yields
TD ejxTs
� ���
x¼2pfr¼ �
Hi2
Hi1x2r L2C
ð8:21aÞ
TD ejxTs
� ���
x¼2pfs=6¼
Hi2L1
L1 þ L2ð Þ sinxrTs
xrTs 1� 2 cosxrTsð Þþ sinxrTs
Hi1C � Hi1 ð8:21bÞ
8.3 Effect of Control Delay on Loop Gain … 175
As shown in (8.21a), when Hi1 � 0, TD at fr is negative, which means that the
phase curve of TD crosses −180° at fr. Defining g(xrTs) = xrTs(1 − 2cosxrTs)
+ sinxrTs, and considering xrTs � p, it can be calculated that the derivative of g
(xrTs), g′(xrTs), is greater than 0, which means that g(xrTs) is a monotone increasing
function, i.e., g(xrTs) � g(0) = 0. So, according to (8.21b), when Hi1 > Hi1C, TD at
fs/6 is negative, which means the phase curve of TD also crosses −180° at fs/6. This
conclusion is in accord with Fig. 8.6. The analysis when Hi1 < 0 is similar to that
when Hi1 � 0, which is not given here.
8.4 Stability Constraint Conditions for Digitally
Controlled System
8.4.1 Nyquist Stability Criterion
As stated in Sect. 8.3.3, when Req is negative at fr′, the loop gain TD(s) contains two
RHP poles. The stability constraint conditions for the controller design are different
from those in Chap. 5. Fortunately, the Nyquist stability criterion is still applicable
for illustrating the stability constraint conditions of the digitally controlled LCL-
type grid-connected inverter. For the convenience of discussion, this criterion is
given here. Figure 8.7a, b shows the Nyquist diagram and the corresponding Bode
diagram [9], respectively. The −180°-crossing is classified as follows:
1. When the amplitude–phase curve of the loop gain in the Nyquist diagram
encircles (−1, j0) counterclockwise once, a positive crossing is recorded. It is
equivalent to that the phase curve crosses −180° 	 (2k + 1) (k is an integer)
from down to up in the Bode diagram when the corresponding amplitude curve
is above 0 dB.
(–1, j0) Re
Im
0
Negative
Positive
0
Positive Negative
–180 (2k+1)
Mag(dB)
Phase(°)
(a) Nyquist diagram (b) Bode diagram
Fig. 8.7 Positive and negative crossing. a Nyquist diagram. b Bode diagram
176 8 Design Considerations of Digitally Controlled LCL-Type …
2. When the amplitude–phase curve encircles (−1, j0) clockwise once, a negative
crossing is recorded. It is equivalent to that the phase curve crosses
−180° 	 (2k + 1) from up to down in the Bode diagram when the corre-
sponding amplitude curve is above 0 dB.
3. When the amplitude–phase curve ends to or starts from the negative real axis
and encircles (−1, j0) counterclockwise, a half positive crossing is recorded. It is
equivalent to that the phase curve ends to or starts from −180° 	 (2k + 1) from
down to up when the corresponding amplitude curve is above 0 dB.
4. When the amplitude–phase curve ends to or starts from the negative real axis
and encircles (−1, j0) clockwise, a half negative crossing is recorded. It is
equivalent to that the phase curve ends to or starts from −180° 	 (2k + 1) from
up to down when the corresponding amplitude curve is above 0 dB.
According to the Nyquist stability criterion, only when C+ − C− = P/2, the
system is stable, where C+ and C− denote the timesof positive and negative
crossing, respectively, and P denotes the number of RHP poles in the loop gain.
8.4.2 System Stability Constraint Conditions
In order to guarantee system stability and good dynamic response, sufficient sta-
bility margins, i.e., gain margin and phase margin, are required for a compensated
system. As stated in Sect. 8.3.3, when Req is negative at fr′, the loop gain TD(s) has
two RHP poles, i.e., P = 2. According to the Nyquist stability criterion, it requires
C+ − C− = 1. Taking Fig. 8.6 as the example, it requires C+ = 1 and C− = 0, which
means the negative crossing must be disabled, and the positive crossing must be
enabled. Accordingly, the resonance peak of the loop gain cannot be damped below
0 dB. Obviously, the stability constraint conditions are different from those for the
analog-controlled inverter in Chap. 5.
For the convenience of illustration, GM1 and GM2 are defined as the gain
margins at fr and fs/6, respectively, and PM is defined as the phase margin at fc (the
first 0 dB-crossing frequency of the amplitude–frequency curve). Then, the stability
constraint conditions can be concluded as:
Case I When fr < fs/6 and Hi1 � Hi1C, as shown in Fig. 8.8a, P = 0, and the
phase curve only crosses −180° at fr from up to down. If GM1 > 0 and
PM > 0, the system will be stable. Note that since no positive crossing
occurs, GM2 is not required.
Case II When fr < fs/6 and Hi1 > Hi1C, as shown in Fig. 8.8b, P = 2, and the
phase curve crosses −180° at fr and fs/6 from up to down and from down
8.4 Stability Constraint Conditions … 177
to up, respectively. If GM1 > 0, GM2 < 0 and PM > 0, the system will
be stable.
Case III When fr � fs/6, as shown in Fig. 8.8c, it can be observed from (8.20a)
that Hi1C < 0. If Hi1 > 0, P = 2, and the phase curve crosses −180° at fs/
6 and fr from up to down and from down to up, respectively. If GM1 < 0,
GM2 > 0, and PM > 0, the system will be stable.
Note that by comparing Case II and Case III, the frequencies of the two
−180°-crossings of TD(s) are exchanged. Accordingly, the requirements of GM1
and GM2 are also exchanged.
PM
0
0
−180
fs/6fr fs/2
−540
|A
T
|(
dB
)
D
A
ng
(T
)
(º
)
D
Frequency (Hz)
fc fr
Hi1=0
GM1
PM
0
0
−180
fs/6fr fs/2
−540
|A
T
|(
dB
)
D
A
ng
(T
)
(º
)
D
Frequency (Hz)
fc fr
Hi1=0
GM2
GM1
(a) fr < fs/6, Hi1 ≤ Hi1C (b) fr < fs/6, Hi1 > Hi1C
PM
Hi1=0
GM2
GM1
0
0
−180
fs/6 fr fs/2
−540
|A
T
|(
dB
)
D
A
ng
(T
)
(º
)
D
Frequency (Hz)
fc fr
(c) fr ≥ fs/6 
Fig. 8.8 Stability constraints in Bode diagrams
178 8 Design Considerations of Digitally Controlled LCL-Type …
8.5 Design Considerations of the Controller Parameters
of Digitally Controlled LCL-Type Grid-Connected
Inverter
8.5.1 Forbidden Region of the LCL Filter Resonance
Frequency
As observed from Fig. 8.8b, c, if fr = fs/6, GM1 = GM2 will happen. At this time,
the requirements of GM1 and GM2 for Case II and Case III can never be satisfied,
and the system can hardly be stable. Since the gain margins are usually recom-
mended to be no less than 2 dB [11], a forbidden region can be obtained, where the
LCL filter resonance frequency fr cannot fall into.
According to Fig. 8.8, the gain margins GM1 and GM2 can be expressed as
GM1 ¼ �20 lg TD j2pfrð Þj j ð8:22aÞ
GM2 ¼ �20 lg TD j2pfs=6ð Þj j ð8:22bÞ
As stated in Sect. 5.2, no matter PI or PR regulator is used, Gi(s) is approximate
to the proportional coefficient Kp within the crossover frequency fc. In practice, fc is
lower than both fr and fs/6, so we have Gi(2pfr) � Gi(2pfs/6) � Kp. Substituting
Gi(2pfr) � Gi(2pfs/6) � Kp, s = j2pfr, and s = j2pfs/6 into (8.5) yields
GM1 ¼ 20 lgHi1 L1 þ L2ð ÞHi2KpL1 ð8:23Þ
GM2 ¼ 20 lg 2pfs=6ð ÞL1L2CHi2KPWMKp � 2pfrð Þ
2 � 2pfs=6ð Þ2 þ 2pfs=6ð ÞKPWMHi1L1
� 	� �
ð8:24Þ
As observed from Fig. 8.8, the magnitude curve of the uncompensated TD
descends with a slope of −20 dB/dec within fc, which means the effect of the filter
capacitor C is little within fc. Substituting C � 0 into (8.5), TD can be approximated
to
TD sð Þ � Hi2KPWMe
�1:5sTsGi sð Þ
s L1 þ L2ð Þ ð8:25Þ
Since |TD(j2pfc)| = 1 and Gi(2pfc) � Kp, Kp can be calculated from (8.25),
expressed as
8.5 Design Considerations of the Controller Parameters … 179
Kp � 2p fc L1 þ L2ð ÞHi2KPWM ð8:26Þ
Substituting (8.26) into (8.23), the Hi1 constrained by GM1 can be obtained as
Hi1 GM1 ¼ 10
GM1
20 � 2pfcL1=KPWM ð8:27Þ
Substituting (8.26) and (8.27) into (8.24) yields
GM2 ¼ 20 lg 10
GM1
20 � fs=6
fr
� 	2
þ fs=6
fc
� 1� fs=6
fr
� 	2" #( )
ð8:28Þ
(8.28) can be rewritten as
10
GM2
20 � k3fr � k2fr
fr
fc
� 10GM120 � kfr þ frfc ¼ 0 ð8:29Þ
where
kfr ¼ frfs=6 ð8:30Þ
It is worth noting that fc is determined by the phase margin PM, and fc is
commonly set to be 0.3fr so as to achieve a sufficient phase margin [12, 13]. When
fr < fs/6, substituting the expected GM1 and GM2 into (8.29), the lower limit of kfr
is obtained; when fr > fs/6, substituting the expected GM1 and GM2 into (8.29), the
upper limit of kfr is obtained. Since the sampling frequency fs is selected, the
forbidden region of LCL filter resonance frequency fr is obtained.
8.5.2 Constraints of the Controller Parameters
According to the design method of LCL filter in Chap. 2 and the forbidden region of
fr, the LCL filter can be determined. Then, considering the stability constraint
conditions presented in Sect. 8.4.2, the design procedure of the grid current regu-
lator and the capacitor-current-feedback coefficient proposed in Chap. 5 can be
applied to the digitally controlled grid-connected inverter. According to the
requirements of steady-state error of the grid current and the stability margins, the
satisfactory region of the grid current regulator or the capacitor-current-feedback
coefficient can be determined, from which the proper parameters can be selected.
Since PR regulator can provide a sufficiently high gain at the fundamental
frequency to reduce the steady-state error, it is used here. The PR regulator is
expressed as
180 8 Design Considerations of Digitally Controlled LCL-Type …
Gi sð Þ ¼ Kp þ 2Krxiss2 þ 2xisþx2o
ð8:31Þ
where Kp is the proportional gain, Kr is the resonance gain; xo is the angular
fundamental frequency, and xi is the bandwidth of the resonant part concerning
−3 dB cutoff frequency to reduce the sensitivity of the regulator to grid frequency
variations at xo [14], which means the gain of the resonant part of PR regulator is
0.707Kr at xo ± xi. For the sake of the sufficiently high gain with the frequency
fluctuation of 0.5 Hz, xi = p rad/s is set.
As stated in Sect. 8.5.1, at the frequencies lower than fc, the expression of the
uncompensated system loop gain TD can be approximated to (8.25). Comparing
(8.25) with (5.7), it can be observed that the approximated TD has one more term
e�1:5sTs than TA, which means the magnitude curves of TD and TA are the same at the
frequencies lower than fc. Therefore, the requirements of the loop gain at the
fundamental frequency, Tfo, constrained by steady-state value EA, and the grid
current regulator Gi(s) constrained by Tfo, are the same in both digitally controlled
and the analog-controlled inverters. So, the Kr constrained by Tfo is the same as
(5.35), which is given here again as (8.32)
Kr Tfo ¼ 10
Tfo
20 fo � fc
 � 2p L1 þ L2ð Þ
Hi2KPWM
ð8:32Þ
By substituting s = 2pfc into (8.31), and considering the crossover frequency fc
is much higher than fo and fi, Gi(j2pfc) � Kp + 2Krfi/fc can be obtained. Note that
fi = xi /(2p). Substituting s = 2pfc and Gi(j2pfc) � Kp + 2Krfi/fc into (8.5), PM can
be derived, expressed as
PM ¼ arctan 2pL1 f
2
r � f 2c
� �þHi1KPWMfc sin 3pfcTsð Þ
Hi1KPWMfc cos 3pfcTsð Þ � 3pfcTs � arctan
Krxi
pfcKp
ð8:33Þ
Applying tangent on both sides of (8.33) and manipulating, the Kr constrained
by PM is obtained as
Kr PM ¼ pf
2
c L1 þ L2ð Þ
KPWMHi2fi
2p f 2r �f 2cð ÞL1
fcKPWMHi1
þ sin 3pfcTsð Þ
� �
� tan 3pfcTs þ PMð Þ cos 3pfcTsð Þ
2p f 2r �f 2cð ÞL1
fcKPWMHi1
þ sin 3pfcTsð Þ
� �
tan 3pfcTs þ PMð Þþ cos3pfcTsð Þ
ð8:34Þ
If the selected Kr meets the constraints of Tfo and PM simultaneously, we have
Kr_Tfo = Kr_PM. According to (8.32) and (8.34), the Hi1 constrained by Tfo and PM
in digital control system can be obtained as
8.5 Design Considerations of the Controller Parameters … 181
Hi1 Tfo PM ¼
2pL1 f 2r �f 2cð Þ
fcKPWM cos 3pfcTsð Þ f
2
c � 2fi 10
Tfo
20 fo � fc
 �
tan 3pfcTs þ PMð Þ
h i
2fi 10
Tfo
20 fo � fc
 �
tan 3pfcTs þ PMð Þ tan 3pfcTsð Þþ 1½ 
þ f 2c tan 3pfcTs þ PMð Þ � tan 3pfcTsð Þ½ 
( ) ð8:35Þ
Substituting (8.26) into (8.24), the Hi1 constrained by GM2 can be obtained as
Hi1 GM2 ¼
2pL1
KPWM
10
GM2
20
fr
fs=6
� 	2
fc þ fs=6ð Þ
2�f 2r
fs=6
" #
ð8:36Þ
When Tfo, PM, GM1, and GM2 are specified, the satisfactory region formed by fc
and Hi1 can be obtained.
8.5.3 Design of LCL Filter, PR Regulator
and Capacitor-Current-Feedback Coefficient
From the above analysis, the design procedure of the grid current regulator and
capacitor-current-feedback coefficient for the digitally controlled grid-connected
inverter can be concluded as follows.
Step 1: Specify the requirements of Tfo, PM, GM1, and GM2. Tfo is determined
by the requirement of the steady-state error of grid current. GM1 and
GM2 are determined by the relationship between fr and fs/6, as well as
the requirement of system robustness: (1) When fr � fs/6, GM1 < 0 dB
and GM2 > 0 dB are required; (2) when fr < fs/6, if Hi1 � Hi1C,
GM1 > 0 dB is required, and if Hi1 > Hi1C, GM1 > 0 dB and
GM2 < 0 dB are required. To guarantee the system dynamic response
and robustness, PM is usually recommended to be within (30°, 60°), and
the GMs are recommended to be no less than 3–6 dB, i.e., |GM1, 2|
� 3–6 dB.
Note that there is no constraint on Hi1_PWM in digital control system,
since the modulation signal vM keeps constant in one sampling period
after being updated.
Step 2: Substituting the specified GM1 and GM2 into (8.29) yields the lower and
upper limits of kfr. Given the sampling frequency, the forbidden region
of the LCL filter resonance frequency fr is obtained. Referring to the
forbidden region, the designed parameters of LCL filter with the design
method presented in Chap. 2 should be carefully modified.
Step 3: According to the specified requirements of Tfo, PM, GM1, and GM2, the
boundaries of Hi1_GM1, Hi1_Tfo_PM, and Hi1_GM2 as the functions of fc can
182 8 Design Considerations of Digitally Controlled LCL-Type …
be determined according to (8.27), (8.35), and (8.36), respectively.
According to these boundaries, the satisfactory region of Hi1 and fc can
be obtained.
Step 4: Select the proper fc from the satisfactory region. In practice, a higher fc is
recommended so as to attain the better dynamic response and a high gain
in the low-frequency range. Then, Kp can be calculated according to
(8.26).
Step 5: After fc is determined, the proper Hi1 can be selected according to the
boundaries of Hi1_GM1, Hi1_Tfo_PM, and Hi1_GM2. When fr � fs/6, the
lower limit of Hi1 is Hi1_GM2, and the upper limit is the minimum value
of Hi1_GM1 and Hi1_Tfo_PM; when fr < fs/6, the lower limit of Hi1 is
Hi1_GM1, while the upper limit is the minimum value of Hi1_GM2 and
Hi1_Tfo_PM. To improve the dynamic response, a smaller Hi1 is
recommended.
Step 6: After fc and Hi1 are determined, the lower and upper limits of Kr can be
determined from (8.32) and (8.34). The larger Kr is, the larger Tfo is,
whereas the smaller PM is. Therefore, when the required Tfo and PM are
met, trade-off is needed when selecting an appropriate Kr to achieve the
expected performance.
Step 7: Check the compensated loop gain to ensure all the specifications are well
satisfied.
Note that if the requirements of Tfo, PM, GM1, and GM2 in Step 1 are too strict,
the satisfactory region may be very small or null. If so, return to Step 1 and modify
the requirements of Tfo, PM, GM1, and GM2, and then renew Step 2.
8.6 Design of Current Regulator for Digitally Controlled
LCL-Type Grid-Connected Inverter Without Damping
As observed from Fig. 8.8c, when fr > fs/6, the phase curve of the uncompensated
loop gain TD crosses −180° from up to down only one time and it occurs at fs/6. If
the proportional gain of the grid current regulator Gi(s), Kp, is tuned to make the
loop gain TD at fs/6 below 0 dB, the negative crossing is disabled. As a result, the
system stability may be guaranteed by properly designing the grid current regulator
and the resonance damping is not required. In the following, the design of the grid
current regulator for the digitally controlled LCL-type grid-connected inverter
without damping is studied.
8.5 Design Considerations of the Controller Parameters … 183
8.6.1 Stability Necessary Constraint for Digitally Controlled
LCL-Type Grid-Connected Inverter Without Damping
Substituting Hi1 = 0 into (8.5), the loop gain without damping is obtained as
TD nodamp sð Þ ¼ Hi2KPWMe
�1:5sTsGi sð Þ
s3L1L2Cþ s L1 þ L2ð Þ ð8:37Þ
Obviously, it is shown in (8.37) that TD_nodamp contains no RHP poles, i.e.,
P = 0. As shown in Fig. 8.6, it is clear that, when Hi1 = 0, there is only one
−180°-crossing in the phase curve of TD, and it is from up to bottom and occurs at fr
when fr � fs/6 or at fs/6 when fr > fs/6. This means that, for the uncompensated TD,
only one negative crossing is possible, and no positive crossing exists, i.e., C+ = 0.
According to the Nyquist stability criterion, only when C+ − C− = P/2, the
system is stable. Here, P = 0, and C+ = 0. So, in order to guarantee the system
stability, the negative crossing must be disabled, i.e., C− = 0. For the purpose of
disabling the negative crossing, the loop gain should be lower than 0 dB at the
negative crossing frequency. As shown in Fig. 8.6a, when fr � fs/6, the
−180°-crossing occurs at fr. The loop gain at fr is hardly reduced below 0 dB due to
the resonance peak. As shown in Fig. 8.6b, c, when fr > fs/6, the −180°-crossing
occurs at fs/6. The loop gain at fs/6 could be easily reduced below 0 dB by selecting
a small proportional gain Kp when PI or PR regulator is adopted.
From the above analysis, it can be concluded that, for a digitally controlled LCL-
type grid-connected inverter, if fr > fs/6, the system might be stable without
damping the resonance of the LCL filter. Basically, the possibility to guarantee the
system stability when fr > fs/6 is due to the existence of the control delay, which
results in a phase lag. As shown in Fig. 8.6b, c, the phase lag makes the
−180°-crossing occur at fs/6, earlier than fr. And, the loop gain at fs/6 is far smaller
than that at fr, so it is easy to disable the negative crossing of the phase curve.
It is worth noting that when fr < fs/6, if the inverter-side inductor current is
directly controlled, the system stability without damping can also be guaranteed,
which is not discussed here.
8.6.2 Design of Grid Current Regulator and Analysis
of System Performance
Similar to Sect. 8.5, according to the requirements of the steady-state error and the
system stability margins, the design procedure of the grid current regulator without
damping will be presented. In the following, PR regulator is also used as the current
regulator.
184 8 Design Considerations of Digitally Controlled LCL-Type …
8.6.2.1 Constraints of Steady-State Error and Stability Margins
on Grid Current Regulator
Since the closed-loop system without damping is the special case with Hi1 = 0,
(8.26) about the relationship of Kp and fc is still true, and (8.32) about Kr con-
strained by Tfo is also true.
Substituting Hi1 = 0 into (8.34), the Kr, constrained by the phase margin PM
without damping, is obtained as
Kr PM nodamp ¼ pf
2
c L1 þ L2ð Þ
KPWMHi2fi tan 3pfcTs þ PMð Þ ð8:38Þ
Substituting Hi1_GM2 = 0 into (8.36), the crossover frequency fc_GM_nodamp
constrained by the gain margin GM2 can be obtained, expressed as
fc GM nodamp ¼ 10�
GM2
20
fs
6
f 2r � fs=6ð Þ2
f 2r
ð8:39Þ
8.6.2.2 Design Procedure of Grid Current Regulator Parameters
Without Damping
Similar to the design procedure givenin Sect. 8.5.3, the design procedure of the
grid current regulator without damping can be concluded as follows:
Step 1: Specify the requirements of Tfo, PM, and GM2. The detailed require-
ments are the same as given in Sect. 8.5.3. Note that GM1 is not required
here, since the −180°-crossing does not occur at fr when the
capacitor-current-feedback active-damping is not used, as shown in
Fig. 8.8c.
Step 2: According to the specified requirements of Tfo, PM, and GM2 in Step 1,
calculate the boundaries of Kr_Tfo, Kr_PM, fc_GM with respect to fc based
on (8.32), (8.38), and (8.39), respectively. Based on these boundaries,
the satisfactory region of Kr and fc can be determined.
Step 3: Select a proper fc from the satisfactory region. Then, Kp can be calcu-
lated from (8.26).
Step 4: After fc is determined, a proper Kr can be selected according to the
boundaries of Kr_Tfo and Kr_PM.
Step 5: Check the compensated loop gain to ensure all the specifications are well
satisfied.
Here, selecting appropriate fc and Kr in the satisfactory region is the same as that
given in Sect. 8.5.3.
8.6 Design of Current Regulator for Digitally Controlled … 185
8.6.2.3 Analysis of System Performance Without Damping
As illustrated in Chap. 5 and Sect. 8.5.2, the capacitor-current-feedback
active-damping can increase the gain margin, whereas reduce the phase margin
at the frequencies lower than fr. Therefore, compared to the compensated system
with capacitor-current-feedback active-damping, the compensated system without
damping can achieve a larger phase margin when fr > fs/6. Besides, as stated in
Sect. 8.5.1, at the frequencies lower than fc, the loop gain TD can be approximated
to (8.25) and is independent from Hi1. It means that the capacitor-current-feedback
active-damping has little effects on the steady-state error. So, the effect caused by
Hi1 = 0 on the crossover frequency and gain margin is analyzed in the following.
As stated in Sect. 8.5.3, when the capacitor-current-feedback active-damping is
used, if fr � fs/6, the lower limit of Hi1 is Hi1_GM2, and the upper limit is the
minimum value of Hi1_GM1 and Hi1_PM. Clearly, to ensure the expected gain
margins GM1 and GM2, the maximum crossover frequency should be no higher
than the frequency when Hi1_GM1 = Hi1_GM2. For convenience, the maximum
crossover frequency is defined as fc_GM, According to (8.27) and (8.36), fc_GM can
be obtained as
fc GM ¼ f
2
r � fs=6ð Þ2
10
GM2
20 f 2r � fs=6ð Þ210
GM1
20
� fs
6
ð8:40Þ
According to (8.39) and (8.40), the ratio of fc_GM_nodamp and fc_GM is derived as
fc GM nodamp
fc GM
¼ 1� fs=6
fr
� 	2
10
GM1
20 �
GM2
20 ð8:41Þ
Obviously, the ratio of fc_GM_nodamp and fc_GM is less than 1. In other words, with
the same specified gain margins, the maximum crossover frequency without
damping is lower than that with capacitor-current-feedback active-damping. This is
because that the loop gain without damping at fs/6 is not attenuated, the crossover
frequency must be lowered down to reduce the loop gain at fs/6 to reserve the
specified gain margin.
8.7 Design Examples
This section will present the design examples for the LCL-type grid-connected
inverter with and without capacitor-current-feedback active-damping. The main
parameters of the single-phase LCL-type grid-connected inverter are listed in
Table 8.1, where three different LCL filters are intentionally given for the purpose
of verifying the forbidden region of fr. The resonance frequencies of filters I, II, and
III are 2.7 kHz, 3.2 kHz, and 4.6 kHz, respectively. The resonance frequencies of
186 8 Design Considerations of Digitally Controlled LCL-Type …
filters I and II are lower than fs/6 (= 3.33 kHz), and the resonance frequency of filter
III is higher than fs/6.
8.7.1 Design Example with Capacitor-Current-Feedback
Active-Damping
When the capacitor-current-feedback active-damping is employed, Tfo, PM, GM1,
and GM2 are specified as follows:
1. Set Tfo > 73 dB, so as to ensure the steady-state error of the grid current below
1% when the grid frequency variation is ±0.5 Hz.
2) Set PM > 45°, so as to ensure a good dynamic response.
3. When fr < fs/6 and Hi1 � Hi1C, set GM1 = 3 dB; when fr < fs/6 and
Hi1 > Hi1C, set GM1 = 3 dB and GM2 = −3 dB; when fr � fs/6,
GM1 = −3 dB and GM2 = 3 dB. All these requirements are to ensure the sys-
tem robustness.
When fr < fs/6, setting fc to 0.3fr [13], and substituting GM1 = 3 dB and
GM2 = −3 dB into (8.29), three roots of kfr can be obtained, which are −1.08, 0.88,
and 4.9. When fr > fs/6, substituting GM1 = −3 dB and GM2 = 3 dB into (8.29),
also yields three roots of kfr, which are −0.92, 1.25, and 2.04. According to the two
sets of three roots, it can be obtained that the lower limit of kfr is 0.88 and the upper
limit of kfr is 1.25. As a result, the forbidden region of kfr is [0.88, 1.25].
When filter I is used, it can be calculated that kfr = 0.81, which is outside the
forbidden region [0.88, 1.25]. According to (8.27), (8.35), and (8.36), Fig. 8.9a can
be obtained. Referring to Step 5 of the design procedure in Sect. 8.4, the lower and
upper limits of Hi1 are determined by GM1 and PM, respectively. As observed from
Table 8.1 Parameters of prototype
System parameters
Parameter Symbol Value Parameter Symbol Value
Input voltage Vin 360 V Fundamental
frequency
fo 50 Hz
Grid voltage (RMS) Vg 220 V Switching frequency fsw 10 kHz
Output power Po 6 kW Sampling frequency fs 20 kHz
Amplitude of the
triangular carrier
Vtri 3 V Grid
current-feedback
coefficient
Hi2 0.15
LCL filter parameters
Filter Inverter-side
inductor L1 (lH)
Filter
capacitor
C (lF)
Grid-side
inductor L2 (lH)
Resonance
frequency fr (kHz)
I 600 30 150 2.7
II 600 20 150 3.2
III 600 10 150 4.6
8.7 Design Examples 187
Fig. 8.9a, the upper limit of Hi1 is always lower than the lower limit, so the
satisfactory region of fc and Hi1 does not exist. To overcome this problem, the
expected PM is reduced to 36°. Accordingly, the satisfactory region appears, shown
as the shaded area in Fig. 8.9a. Point A is selected, where fc = 1.1 kHz and
Hi1 = 0.05. Substituting fc = 1.1 kHz into (8.26) yields Kp = 0.293. According to
(8.32) and (8.34), Kr_Tfo = 59.2 and Kr_PM = 66.6 can be calculated, respectively.
Here, we choose Kp = 0.29 and Kr = 63. With these designed parameters, the
compensated loop gain TD is depicted, as shown Fig. 8.10a. As shown,
Tfo = 73.6 dB, PM = 36.5°, GM1 = 3.1 dB, and GM2 = −6.1 dB, which meet the
expected requirements.
When filter II is used, it can be calculated that kfr = 0.96, which is in the
forbidden region [0.88, 1.25]. Likewise, according to (8.27), (8.35), and (8.36),
Fig. 8.9b can be obtained. In this case, the lower and upper limits of Hi1 are
determined by GM1 and GM2, respectively. As observed from Fig. 8.9b, the upper
limit of Hi1 is also always lower than the lower limit, so the satisfactory region of fc
and Hi1 does not exist. If both the expected GM1 and GM2 are reduced to 0, i.e.,
0.1
0.08
H
i1
0
0.06
0.04
0.02
fc (Hz)
20001500
GM2
increase
0dB
A
Tfo=73dB
PM=36
−3dB
GM1=3dB
Tfo=73dB
PM=45
B
0.08
H
i1
0
0.06
0.04
0.02
fc (Hz)
20001500
0.1
GM2 increase
Tfo=73dB
PM=45
GM2
=0dB
GM2=−3dB
GM1 increase
GM1=3dB
GM1
=0dB
A
(a) Filter I (b) Filter II 
0.1
0.08
H
i1
0
0.06
0.04
0.02
500 1000 500 1000
500 1000
fc (Hz)
20001500
A
B
GM1 increase
0dB
−3dB
Tfo=73dB
PM=45
GM2=3dB
(c) Filter III 
Fig. 8.9 Satisfactory region constrained by GM1, GM2, PM, and Tfo
188 8 Design Considerations of Digitally Controlled LCL-Type …
GM1 = GM2 = 0, and the satisfactory region appears, shown as the shaded area in
Fig. 8.9b. In the satisfactory region, point A is selected, where fc = 1 kHz and
Hi1 = 0.034. Substituting fc = 1 kHz into (8.26) yields Kp = 0.27. According to
(8.32) and (8.34), Kr_Tfo = 59.2 and Kr_PM = 59.6 can be calculated. Here,
Kp = 0.27 and Kr = 59.3 are chosen. With these design parameters, the compen-
sated loop gain TD is depicted, as shown Fig. 8.10b, from whichit can be measured
that Tfo = 73.7 dB, PM = 45°, GM1 = −0.2 dB, and GM2 = 0.2 dB. Clearly, GM1
and GM2 are too small, which will result in poor dynamic response.
When filter III is used, it can be calculated that kfr = 1.38, which is outside the
forbidden region [0.88, 1.25]. According to (8.27), (8.35), and (8.36), Fig. 8.9c can
be obtained. In this case, the lower and upper limits of Hi1 are determined by GM2
and GM1, respectively. As observed in Fig. 8.9c, the satisfactory region exists.
100
0
−50
−180
−540
0
−360
0101 3 104102
Frequency (Hz)
A
ng
(T
D
)
(º
)
fs/6
50
frfc
|A
T
|(
dB
)
D
fo
Compensated TD(s)
Uncompensated TD(s)
fc: 1.1 kHz; Tfo: 73.6 dB;
GM1: dB; GM2: 6.1 dB;
PM: 36.5º
100
0
−50
−180
−540
0
−360
0101 3 104102
Frequency (Hz)
A
ng
(T
D
)
(º
)
fs/6
50
fr(fc)
|A
T
|(
dB
)
D
fo
Compensated TD(s)
Uncompensated TD(s)
fc: 1.0 kHz; Tfo: 73.7 dB;
GM1: dB; GM2: 0.2 dB;
PM: 45º
(a) Filter I (b) Filter II 
100
0
−50
−180
−540
0
−360
0101 3 104102
Frequency (Hz)
A
ng
(T
D
)
(º
)
fs/6
50
frfc
|A
T
|(
dB
)
D
fo
Compensated TD(s)
Uncompensated TD(s)
fc: 1.3 kHz; Tfo: 73.6 dB;
GM1: dB; GM2: 3.2 dB;
PM: 45º
(c) Filter III 
Fig. 8.10 Bode diagrams of uncompensated and compensated loop gains
8.7 Design Examples 189
From the satisfactory region, point A is selected, where fc = 1.3 kHz and
Hi1 = 0.02. Substituting fc = 1.3 kHz into (8.26) yields Kp = 0.346. According to
(8.32) and (8.34), Kr_Tfo = 59.1 and Kr_PM = 63.1 can be calculated. We choose
Kp = 0.35 and Kr = 63. With these parameters, the compensated loop gain TD is
depicted, as shown Fig. 8.10c, from which it can be measured that Tfo = 73.7 dB,
PM = 45°, GM1 = − 6.4 dB, and GM2 = 3.2 dB. Clearly, all the expected
requirements are achieved.
8.7.2 Design Example Without Damping
As stated before, to guarantee the system stability without damping, the LCL filter
resonance frequency is required to be higher than fs/6. So, filter III is used for the
following design. The specified requirements are Tfo > 73 dB, GM2 � 3 dB, and
PM > 45°. According to (8.32), (8.38), and (8.39), the satisfactory region of Kr and
fc can be obtained, as shown with the shaded area in Fig. 8.11.
To ensure a sufficient gain margin, a higher crossover frequency should be
selected. According to Fig. 8.11, fc = 1.1 kHz, corresponding to the constraint
boundary of GM2 = 3 dB, is selected, and then, Kr = 75 at point A is chosen.
Substituting fc = 1.1 kHz into (8.26), we have Kp = 0.27. Figure 8.12 shows the
Bode diagrams of the uncompensated and compensated loop gains, from which,
fc = 1.1 kHz, PM = 46°, Tfo = 75.2 dB, and GM2 = 3.7 dB can be measured,
which satisfies the specifications. Compared to Fig. 8.10c, the phase margin and the
gain at the fundamental frequency are improved, at the cost of a little reduced
crossover frequency.
75
K
r
0
50
25
500 1000
fc (Hz)
20001500
100
2500
A
Constrained
by PM = 45°
Constrained
by Tfo = 73dB
Constrained
by GM2 = 3dB
Fig. 8.11 Satisfactory region
constrained by GM2, PM, and
Tfo
190 8 Design Considerations of Digitally Controlled LCL-Type …
8.8 Experimental Verification
A 6-kW single-phase LCL-type grid-connected inverter prototype has been fabri-
cated and tested to validate the theoretical analysis and the designed controller
parameters. The specifications of the prototype are listed in Table 8.1, and the
photograph of the prototype has been shown in Fig. 5.15 in Chap 5.
8.8.1 Experimental Validation for the Case
with Capacitor-Current-Feedback Active-Damping
The experimental waveforms of the grid-connected inverter with filters I, II, and III
are shown in Figs. 8.13, 8.14, and 8.15, respectively. Note that the
capacitor-current-feedback active-damping is adopted here. In these figures, the left
figures show the steady-state waveform at full load, and the right ones show the
transient response when the grid current reference step changes between full load
and half load. Table 8.2 shows the measured power factor, RMS value of grid
current, current overshoot, and settling time. It can be seen that the measured power
factors are all larger than 0.995. Since the full-load grid current reference is
27.27 A, the steady-state errors are all less than 1%, which satisfy the design
expectation. As the grid current reference steps from half load to full load, the
current overshoot with filter II is larger than that with the other two filters; the
corresponding settling time is also longer than that with the other two filters. These
results verify that the system dynamic performance is deteriorated when the reso-
nance frequency of LCL filter falls in the forbidden region, which is well in
agreement with the analysis in Sect. 8.5.1.
100
0
−50
−180
−540
0
−360
0101 3 104102
Frequency (Hz)
A
ng
(T
D
_n
od
am
p)
(º)
fs/6
50
frfc
|A
T
|(
dB
)
D
_n
od
am
p
fo
Compensated TD_nodamp(s)
Uncompensated TD_nodamp(s)
fc: 1.1 kHz; Tfo: 75.2 dB;
GM2: dB; PM: 46º.
Fig. 8.12 Bode diagrams of
loop gains without damping
8.8 Experimental Verification 191
PF=0.998Time: [5 ms/div]
vg: [100 V/div] i2: [20 A/div]
%=37%Time: [10 ms/div]
vg: [100 V/div] i2: [20 A/div]
(a) Full-load steady state (b) Dynamic response 
Fig. 8.13 Experimental waveform of the prototype with Filter I. a Full-load steady state.
b Dynamic response
PF=0.998Time: [5 ms/div]
vg: [100 V/div] i2: [20 A/div]
%=78%Time: [10 ms/div]
vg: [100 V/div] i2: [20 A/div]
(a) Full-load steady state (b) Dynamic response 
Fig. 8.14 Experimental waveform of the prototype with Filter II. a Full-load steady state.
b Dynamic response
PF=0.997Time: [5 ms/div]
vg: [100 V/div] i2: [20 A/div]
%=44%Time: [10 ms/div]
vg: [100 V/div] i2: [20 A/div]
(a) Full-load steady state (b) Dynamic response 
Fig. 8.15 Experimental waveform of the prototype with Filter III. a Full-load steady state.
b Dynamic response
192 8 Design Considerations of Digitally Controlled LCL-Type …
Figure 8.16a, b shows the experimental waveforms with filters I and III,
respectively. For filter I, fc = 1.125 kHz and Hi1 = 0.062, which correspond to
point B in Fig. 8.9a. It can be calculated from (8.36) that GM2 = 0 dB. For filter III,
fc = 1.25 kHz and Hi1 = 0.04, which corresponds to point B in Fig. 8.9b. It can be
calculated from (8.27) that GM1 = 0 dB. As shown in Fig. 8.16, large oscillations
occur in the measured grid current. Note that the oscillations do not divergent due to
the parasitic resistors in the LCL filter.
The experimental results shown in Figs. 8.13, 8.14, 8.15, and 8.16 indicate that
the forbidden region of kfr can guide the design of the LCL filter. If the designed LCL
filter resonance frequency falls in the forbidden region, the filter parameter should be
adjusted (e.g., modify the capacitance of the filter capacitor). Moreover, the above
experimental results indicate that the satisfactory region presented in this chapter is a
convenient and intuitive interface to guide the design of the controller parameters,
from which the proper controller parameters can be selected, guaranteeing a low
steady-state error, sufficient stability margins, and good dynamic response.
8.8.2 Experimental Validation Without Damping
Figure 8.17 shows the experimental waveforms of the grid-connected inverter with
filter III and without damping. The steady-state waveforms at full load are shown in
Table 8.2 Prototype parameter of single-phase LCL filtered grid connected inverter
Power
factor
RMS value of grid
current (A)
Overshoot at command
step (%)
Settling time
(ms)
Filter I 0.998 27.07 37 1
Filter
II
0.998 27.18 78 5
Filter
III
0.997 27.09 44 1
Time: [5 ms/div]
vg: [100 V/div] i2: [20 A/div]
Time: [5 ms/div]
vg: [100 V/div] i2: [20 A/div]
(a) Filter I (b) Filter III 
Fig. 8.16 Experimental waveform of prototype in two critical stable cases. a Filter I. b Filter III
8.8 Experimental Verification 193
Fig. 8.17a, the measured power factor is