國立中正大學電機工程研究所
碩士論文
太陽能換流器之控制策略與諧波抑制研究
Research on Control Strategy and Harmonic
Suppression of PV inverter
葉曜魁
指導教授：張文恭
博士
中華民國一百零六年七月
太陽能換流器之控制策略與諧波抑制研究
Research on Control Strategy and Harmonic
Suppression of PV inverter
研 究 生：葉曜魁
指導教授：張文恭
Student：Y. K. Yeh
博士
Advisor：G. W. Chang, Ph.D.
國立中正大學工學院
電機工程研究所
碩士論文
A Thesis
Submitted to
Institute of Electrical Engineering
College of Engineering
National Chung Cheng University
in Partial Fulfillment of the Requirements
for the Degree of
Master
in Electrical Engineering
July 2017
Chiayi, Taiwan, Republic of China
ACKNOWLEDGMENTS
I would like express my sincere appreciation to my advisor, Dr. Gary W. Chang,
for his instruction, guidance, encouragement, and assistance during this research work.
His knowledge, experiences, advices and creative thinking have always been a great
help to my research work. This work would not be completed without his support and
encouragement.
Additionally, I want to give my special thanks to the thesis committee, Dr. ChinChung Wu, and Dr. Chi-Jui Wu for their valuable discussions, insight and comments
on this work.
I also highly appreciate my colleagues in the Power Quality Research Laboratory
(PQRL): Dr. Yi-Ying Chen, Dr. Heng-Jiu Lu, Li-Yuan Hsu, Nguyễn Công Chính,
Yueh-Lin Han, Che-Hsien Lin, Hsin-Lin Li, Chun-Chieh Wang, Chien-Hao Lai,
Shang-Yi Chen, Cheng-Wei Lin, Chi-Yang Huang, Yung-Han Hung, Tai-Chieh Yeh,
Yu-Lu Lin, Kun-Min Lin, Hung-Pin Chan, Pham Dinh Thai, Phạm Duy Phước and
Nguyen Tung Kha who stimulate me to make further efforts in the studies. The kindly
friendships will be deeply kept in my mind.
Lastly, I am very grateful to my parents, my family, and all my friends for their
supports, encouragement, and love.
Yao-Kuei Yeh
2017-08
I
太陽能換流器之控制策略與諧波抑制研究
研究生 : 葉曜魁
指導教授：張文恭 博士
國立中正大學電機工程研究所
中文摘要
本論文主要以三相太陽能併網發電系統的控制策略為研究對象，分析其操作
原理並建立數學模型，針對太陽能電池模組的最大功率點跟蹤方法、併網發電系
統的控制策略以及併網換流器得濾波器設計原理等問題進行各方面的探討。
一開始本論文根據太陽能電池的電氣特性以及太陽能電池模組在不同的光
照強度和環境溫度下的輸出特性，使用實際的太陽能模組的參數資料作驗證。也
分別對目前幾種經典的最大功率跟蹤方法的動作原理作介紹。
接著分析了三相太陽能併網換流器的工作原理和拓墣架構，以並網換流器的
濾波器為切入點，建立三相太陽能併網換流器的狀態空間數學模型，以利於第四
章研究不同的三相太陽能併網換流器控制方法。在控制方法上，分別針對 PI 和
PR 控制器的原理，透過應用至換流器中的控制架構與轉移函數對其作分析，並
根據 PR 控制器的特性介紹的新型的准 PR 控制器。准 PR 控制器也能應用在抑制
電網諧波，然而傳統架構上的准 PR 控制器諧波補償效果並不佳，因此探討了一
種改良式的准 PR 控制器諧波補償架構。
最後在 Matlab/Simulink 中建立了三相太陽能併網發電系統，並透過模擬驗
證前述 PI 和 PR 控制器的特性，以及其併網效果。也驗證了改良式的諧波補償，
比起傳統的補償架構有更良好的效果。
關鍵詞：三相太陽能併網系統、換流器、諧波補償、PR 控制器、准 PR 控制器
II
Research on Control Strategy and Harmonic
Suppression of PV inverter
Student: Y. K. Yeh
Advisor: G. W. Chang, Ph. D.
Institute of Electrical Engineering
Collage of Engineering
National Chung Cheng University
Abstract
In this thesis, the control strategy of three-phase grid-connected PV power
generation system is the main research object, and establish the mathematical model to
analyze its principle. The maximum power point tracking method of solar cell module,
control strategy of grid-connected power generation system and the designing principle
of grid-connected inverter filter are also discussed.
At the beginning of this thesis, according to the electrical characteristics of solar
cells, using the actual solar module parameters to verify the output characteristics of
PV array in different irradiation and ambient temperature. And also introduces the
operating principle of several classic maximum power tracking method.
Then, the working principle and topology of the three-phase grid-connected PV
inverter is analyzed to establish the state space mathematical model. In the control
method, through the application of the inverter control structure and transfer function
to analyze the principle of PI and PR controller. According to the characteristics of the
PR controller, a new type of quasi-PR controller is introduced. The quasi-PR controller
can also be used to suppress the harmonics of the power grid. However, the harmonic
III
compensation effect of the PR controller on the normal structure is not well. Therefore,
an improved quasi-PR controller harmonic compensation structure is discussed.
Finally, a three-phase grid-connected PV generation system is established in
Matlab/ Simulink, and the characteristics of the PI and PR controllers are verified by
the simulation. Also verified the improved harmonic compensation with a better effect
on the harmonics compression which compared to the normal one.
Keywords: Three-phase grid-connected PV generation system, inverter, harmonic
compensation, PR controller, quasi-PR controller
IV
TABLE OF CONTENTS
ACKNOWLEDGMENTS ................................................................................. I
中文摘要 ............................................................................................................ II
ABSTRACT ..................................................................................................... III
TABLE OF CONTENTS ................................................................................. V
LIST OF FIGURES .......................................................................................... 1
LIST OF TABLES ............................................................................................. 4
I.
INTRODUCTION ..................................................................................... 5
1.1
1.2
OVERVIEW ...................................................................................................... 5
DEVELOPMENT AND CURRENT SITUATION OF SOLAR POWER GENERATION .... 6
1.2.1
Statistics of International Solar Energy ................................................. 7
1.2.2
Development and Current Situation of Domestic Solar Energy ............ 9
1.3
INTRODUCTION OF GRID-CONNECTED PV SYSTEM ....................................... 11
1.3.1
Structure of Grid-connected PV System .............................................. 11
1.3.2
Maximum Power Point Tracking ......................................................... 13
1.3.3
Grid-connected Control Technology.................................................... 14
1.4
ORGANIZATION OF THESIS ............................................................................ 15
II.
PV CELL MODEL AND MPPT METHODS ................................... 17
2.1
OVERVIEW .................................................................................................... 17
2.2
MATHEMATICAL MODEL AND CHARACTERISTICS OF PV CELL...................... 17
2.3
MAXIMUM POWER POINT TRACKING ............................................................ 21
2.3.1
Constant Voltage Tracking Method ...................................................... 22
2.3.2
Perturbation and Observation Method ................................................. 23
2.3.3
Incremental Conductance Method ....................................................... 25
III. THEORETICAL ANALYSIS OF THREE-PHASE
PHOTOVOLTAIC GRID-CONNECTED SYSTEM .................................. 27
3.1
TOPOLOGY AND MODEL OF PV GRID-CONNECTED INVERTER ....................... 27
3.1.1
State-Space Model of PV Inverter with L Filter .................................. 27
3.1.2
State-Space Model of PV Inverter with LC Filter ............................... 30
3.1.3
State-Space Model of PV Inverter with LCL Filter ............................. 32
3.2
PERFORMANCE ANALYSIS OF LCL FILTER .................................................... 34
3.3
PARAMETER DESIGN OF LCL FILTER IN THREE-PHASE GRID-CONNECTED
INVERTER .................................................................................................................. 36
V
IV.
CONTROL STRATEGY OF PHOTOVOLTAIC INVERTER ....... 38
4.1
GRID-CONNECTED INVERTER CONTROL METHOD AND STRATEGIES ............. 38
4.1.1
Design of Grid-Connected Inverter Control System ........................... 38
4.1.2
State-Space Model of PV Inverter with L Filter .................................. 39
4.2
GRID CONTROL TECHNOLOGY OF INVERTER BASED ON PI CONTROLLER ..... 42
4.3
GRID CONTROL TECHNOLOGY OF INVERTER BASED ON PR CONTROLLER .... 44
4.4
GRID CONTROL TECHNOLOGY OF INVERTER BASED ON QUASI-PR
CONTROLLER ............................................................................................................ 48
4.4.1
Principle and Analysis of Quasi-PR Controller ................................... 48
4.4.2
Parameter Design of Quasi-PR Controller........................................... 50
4.5
IMPROVED QUASI-PR HARMONIC COMPENSATION DESIGN ............................ 53
4.5.1
Disadvantage of Normal Harmonic Compensation ............................. 53
4.5.2
Improved Quasi-PR Harmonic Compensation .................................... 55
V.
CASE STUDY .......................................................................................... 58
5.1
THREE-PHASE GRID-CONNECTED PV SYSTEM MODEL ................................ 58
5.2
SIMULATION UNDER THE IDEAL POWER GRID ............................................... 61
5.2.1
Simulation with PI Controller .............................................................. 61
5.2.2
Simulation with PR Controller ............................................................ 62
5.3
SIMULATION UNDER THE NONIDEAL POWER GRID ........................................ 65
5.3.1
Simulation in grid-connected PV System Model ................................. 65
5.3.2
Overview of INER Microgrid Model .................................................. 67
5.3.3
Simulation Result in INER Microgrid Model ...................................... 70
VI.
6.1
6.2
CONCLUSION AND FUTURE WORK ........................................... 73
CONCLUSION................................................................................................. 73
FUTURE WORK.............................................................................................. 74
REFERENCES ................................................................................................ 75
VI
LIST OF FIGURES
Fig. 1.1 Global annual new installed power plant capacities 2000 till 2015 ................. 6
Fig. 1.2 Solar PV Global Capacity, by Country/Region, 2005-2015 [4] ....................... 7
Fig. 1.3 Germany renewable energy generation ratio from 2005 to 2015 [7] ............... 9
Fig. 1.4 Accumulated capacity of domestic solar power generation [10] .................... 10
Fig. 1.5 Off-grid PV system ......................................................................................... 11
Fig. 1.6 Grid-connected PV system ............................................................................. 11
Fig. 1.7 Structure of single-stage grid-connected PV System ..................................... 12
Fig. 1.8 Structure of two-stage grid-connected PV System ......................................... 12
Fig. 1.9 Structure diagram of grid-connected PV generation system .......................... 15
Fig. 2.1 PV cell modeled as diode circuit .................................................................... 18
Fig. 2.2 PV cell output characteristics curve ............................................................... 19
Fig. 2.3 PV cell output characteristics curve with various irradiation ......................... 19
Fig. 2.4 PV cell output characteristics curve with various temperature ...................... 20
Fig. 2.5 PV cell output characteristics curve with 25 °C and various temperature ..... 21
Fig. 2.6 Flow chart of the constant voltage tracking method....................................... 22
Fig. 2.7 The operation principle of perturbation and observation method .................. 24
Fig. 2.8 Flow chart of the perturbation and observation method ................................. 24
Fig. 2.9 Relationship in the incremental conductance method .................................... 26
Fig. 2.10 Flow chart of the incremental conductance method ..................................... 26
Fig. 3.1 Topology of three-phase grid-connected PV power system with L filter ....... 27
Fig. 3.2 The relationship of coordinate transformations between abc, αβ and dq
reference frames ........................................................................................................... 29
Fig. 3.3 L-type grid-connected inverter model in the dq coordinate system ............... 30
Fig. 3.4 Topology of three-phase grid-connected PV power system with LC filter .... 31
Fig. 3.5 Topology of three-phase grid-connected PV power system with LCL filter .. 32
Fig. 3.6 L-type grid-connected inverter model in the dq coordinate system ............... 34
Fig. 3.7 Three types of filter for inverter ..................................................................... 34
Fig. 3.8 Bode diagram of L-filter and LCL-filter ........................................................ 35
Fig. 4.1 Control system model of voltage control mode.............................................. 38
Fig. 4.2 Control system model of current control mode .............................................. 39
Fig. 4.3 Voltage and Current Dual-Loop Control Structure Diagram in Three-Phase
System .......................................................................................................................... 40
Fig. 4.4 Structure of single-loop control using the grid current................................... 41
Fig. 4.5 Structure of dual-loop control using the grid current and capacitor C2 current
...................................................................................................................................... 41
Fig. 4.6 Bode diagram of current single-loop and dual-loop control........................... 42
1
Fig. 4.7 System step response with PI controller and without PI controller ................ 43
Fig. 4.8 Bode diagram of comparison between PI and PR controller.......................... 45
Fig. 4.9 Closed-loop disturbance transfer function of the PI and the PR controller .... 46
Fig. 4.10 Current dual-loop control block diagram with PI controller ........................ 47
Fig. 4.11 Current dual-loop control block diagram with PR controller ....................... 47
Fig. 4.12 Bode diagram of current dual-loop control with PI controller ..................... 48
Fig. 4.13 Bode diagram of current dual-loop control with PR controller .................... 48
Fig. 4.14 Bode diagram of PR controller and quasi-PR controller .............................. 49
Fig. 4.15 Closed-loop disturbance transfer function of PI and quasi-PR controller .... 50
Fig. 4.16 Bode diagram of the quasi-PR controller with different ωc values .............. 51
Fig. 4.17 Bode diagram of the quasi-PR controller with different KR values.............. 52
Fig. 4.18 Bode diagram of the quasi-PR controller with different Kp values .............. 52
Fig. 4.19 Normal quasi-PR harmonic compensation structure diagram [32] .............. 54
Fig. 4.20 Bode diagram of current dual-loop control with normal quasi-PR harmonic
compensation ............................................................................................................... 55
Fig. 4.21 Improved quasi-PR harmonic compensation structure diagram [32] ........... 56
Fig. 4.22 Bode diagram of current dual-loop control with improved quasi-PR harmonic
compensation ............................................................................................................... 57
Fig. 5.1 Two-stage grid-connected PV System model ................................................. 58
Fig. 5.2 Control strategy block .................................................................................... 60
Fig. 5.3 Current i1a of inverter side with PI controller ................................................. 61
Fig. 5.4 Current i2a of grid-connected side with PI controller ..................................... 61
Fig. 5.5 Grid-connected current and the reference current with PI controller ............. 62
Fig. 5.6 Dc voltage of the inverter side and the reference voltage with PI controller . 62
Fig. 5.7 Phase current and phase voltage of the grid with PI controller ...................... 62
Fig. 5.8 Current i1a of inverter side with PR controller................................................ 63
Fig. 5.9 Current i2a of grid-connected side with PR controller .................................... 63
Fig. 5.10 FFT analysis frequency spectrum ................................................................. 63
Fig. 5.11 Grid-connected current and the reference current with PR controller .......... 64
Fig. 5.12 Dc voltage of the inverter side and the reference voltage with PR controller
...................................................................................................................................... 64
Fig. 5.13 Phase current and phase voltage of the grid with PR controller ................... 65
Fig. 5.14 Grid-connected current i2a without harmonic compensation ........................ 65
Fig. 5.15 Grid-connected current i2a with normal harmonic compensation................. 66
Fig. 5.16 Grid-connected current i2a with improved harmonic compensation............. 66
Fig. 5.17 FFT analysis of the grid-connected current i2a ............................................. 66
Fig. 5.18 The structure of the zone 1 to zone 3 microgrid systems ............................. 68
Fig. 5.19 The architecture of the zone 1 area ............................................................... 68
2
Fig. 5.20 Matlab/Simulink model of the zone 1 area ................................................... 69
Fig. 5.21 Grid-connected current at PCC without harmonic compensation ................ 70
Fig. 5.22 FFT analysis of the grid-connected current at PCC without harmonic
compensation ............................................................................................................... 70
Fig. 5.23 Grid-connected current at PCC with harmonic compensation ..................... 71
Fig. 5.24 FFT analysis of the grid-connected current at PCC with harmonic
compensation ............................................................................................................... 71
3
LIST OF TABLES
Table 1.1 ISF scenario analysis – solar photovoltaic cumulative capacities [1] ............ 7
Table 1.2 advantages and disadvantages of single-stage and two-stage PV System ... 13
Table 4.1 Gain and phase angle of each harmonic in Fig.4.20 .................................... 55
Table 5.1 Parameters of PV array ................................................................................ 59
Table 5.2 Parameters of the system .............................................................................. 60
Table 5.3 Summary FFT analysis of the grid-connected current i2a ............................ 67
Table 5.4 System parameters for simulation in the zone 1 area ................................... 69
Table 5.5 Summary FFT analysis of the grid-connected current at PCC .................... 72
Table 5.6 Current distortion limits for systems rated 120 V through 69 kV [35] ........ 72
4
I. INTRODUCTION
1.1
Overview
With the rapidly development of economic, population growth, and the increasing
demand for the energy, the fossil fuel reserves become deplete and cause energy
shortages. In addition, the widespread use of fossil fuel has increasingly deteriorated
the quality of environment. Nowadays, these problems have become the world major
problem that each country needs to face them. In Taiwan, energy supply is highly
dependent on imports and with the variation of international energy price and the
increasing pressure of greenhouse gas reduction, our energy development challenges
are more severe than other countries.
Today, our world is looking for renewable energy and expects it can change the
present energy structure and achieve the sustainable development of human society.
Fig. 1.1 shows that the capacity of renewable energy construction in the world has
increased year by year [1]. In view of this, our government has expanded all kinds of
renewable energy projects, including “Thousand Wind Turbines Project” and “Million
Rooftop PVs Project”. From the energy supply and many other factors to be considered,
solar energy is the major green energy for sustainable development and is one of the
most important energy in the 21st century.
At present, the use of solar energy can be divided into photoelectric conversion
and photo-thermal conversion and the photoelectric conversion of solar power has the
following obvious advantages:
1.
No pollution: No noise when generating electricity, nor will it generate
greenhouse gases or contaminated waste.
2.
Renewable: For the Earth, solar energy is abundant.
3.
Adaptability: The electrical energy generated from photovoltaic can be
5
connected to utility grid for transmission. Due to the solar energy can
generate the largest energy at noon, that the time same with the largest
demand of electric power, grid-connected photovoltaic system can reduce the
pressure on the power grid withstanding the peak period and reduce the
possibility of collapse.
4.
Distributed generation system: It can improve the safety and reliability of the
entire power system.
5.
Flexibility: Power generation system can be modularized as needed and the
capacity can be easily expansion.
6.
Independence: Not affected by the energy crisis and the stability of fuel
market.
7.
Building-integrated photovoltaics (BIPV): Use solar photovoltaic materials
to replace traditional building materials, because the consideration in the
build design stage, the green building has the best ratio of power generation
and cost [2].
Fig. 1.1 Global annual new installed power plant capacities 2000 till 2015
1.2
Development and Current Situation of Solar Power Generation
6
1.2.1 Statistics of International Solar Energy
In order to achieve the goal of the average increasing of the Earth temperature is
limited to 2 ℃ (2DS) at the end of this century that proposed by Energy Technology
Perspectives (ETP), each country efforts to promote the development of renewable
energy and its technology to make the related products can be mass production [3]. In
terms of solar power, the solar PV capacity of the whole world had reached to 227 GW
in the year 2015 as shown in Fig. 1.2 and Germany, China, the United States, Japan and
Italy are the top 5 countries of the solar PV capacity [4]. There are total 22 countries
can meet the national demand for electricity more than 1% by using solar PV power,
among them, Italy can reach 7.8%, Greece can reach 6.5% and Germany can reach
6.4% [5]. According to the scenario analysis of Institute for Sustainable Futures (ISF)
in University of Technology Sydney (UTS), the estimation of the solar photovoltaic
cumulative capacities in the world by 2050 is shown in Table 1.1 [1].
Fig. 1.2 Solar PV Global Capacity, by Country/Region, 2005-2015 [4]
Table 1.1 ISF scenario analysis – solar photovoltaic cumulative capacities [1]
7
The United States in 2016 has proposed indefinite extension of renewable energy
investment tax relief to encourage the development of renewable energy, especially
wind, solar photovoltaic and geothermal, and hope that the renewable energy power
generation can double than in 2014 before 2040. The United States Congress passed
the amendment bill of the tax credit eligibility on new installations of wind and solar
power generation in December 2015 [6].
Germany's solar power output began to significantly increase in 2010 to 2012, and
the main reason is the guidance of government policy. The German government at that
time began to set the goal of solar power, this measure not only increase the proportion
of solar power generation and help the transition from fossil fuel to renewable energy.
According to the German Association of Energy and Water Industries (Bundesverband
der Energie- und Wasserwirtschaftl; BDEW), total solar power generation in 2015 is
8
38.5TWh that equivalent to 7.5% in the use of German electricity and 38% in the
renewable energy, is shown in Fig. 1.3 [7]. In statistics, Germany has total solar energy
capacity of 40 GW by the end of 2015, and the German government is expected to have
about 200GW of solar capacity by 2050.
Fig. 1.3 Germany renewable energy generation ratio from 2005 to 2015 [7]
China has become the world's major energy producers and consumers. Under the
requirements of carbon reduction and the safety energy, the Chinese government has
gradually adjusted the diversification of energy supply, including the expansion of
natural gas, nuclear power, hydroelectric power, and other renewable energy
applications, and gradually reduce the proportion of coal use to work to reduce serious
air pollution problem. China sets the goal of non-fossil fuel energy consumption
accounted for 15% and 20% in 2030 ，to relieve dependence on coal [8]. In the
development of solar energy, China is currently rich in solar resources such as Qinghai,
Xinjiang, Gansu and Inner Mongolia that has many idle land resources to construct
large scale solar power generation. It is estimated that by 2020, the centralized and
distributed solar power generation capacity in China will reach 100GW.
1.2.2 Development and Current Situation of Domestic Solar Energy
9
In 2012 the government of Taiwan began to implement the“Million Rooftop PVs
Project”, through promoting the public rental roof to enhance people's awareness and
willingness to install. Due to the costs decline in solar power and the wholesale price
per unit is between $ 4.6 and $ 6, Bureau of Energy (BOE) announced the original goal
to install 6,200MW solar power in "Million Rooftop PVs Project" in 2030 has ahead of
schedule in 2025 reached. In addition, the goal of the annual solar photovoltaic power
generation equipment to promote is set to 500MW. In recent years, the accumulated
capacity of domestic solar power generation is shown in Fig. 1.4 [10]. To the end of
May 2017, the total installed capacity of the solar photovoltaic power generation system
is 1121WM.
Fig. 1.4 Accumulated capacity of domestic solar power generation [10]
The government had carried out “Greenhouse Gas Reduction and Management
Act” on July 1, 2015 to achieve a long term carbon reduction target till 2050 and push
“Two-year Solar Power Promotion Plan(July 2016～June 2018” to accomplish the
building of solar photovoltaic system capacity to 1,520 MW. It is hoped that the total
solar photovoltaic capacity can reach 20 GW until 2025 through the “Forward-looking
Infrastructure Plan” [11].
10
1.3
Introduction of Grid-connected PV System
At present, solar photovoltaic power system is divided into two categories, namely
off-grid PV system and grid-connected PV system and their basic structures are shown
in Fig. 1.5 and Fig. 1.6, respectively. It is seen that all the power generated from offgrid PV system is used for local consumption, and the power of the grid-connected PV
system can be transmitted in utility grid, in addition to the local consumption.
PV Array
DC / DC
Converter
Charge
Controller
Battery
Bank
DC / AC
Inverter
AC Load
Center
DC Load
Center
Fig. 1.5 Off-grid PV system
DC / DC
Converter
PV Array
DC / AC
Inverter
Utility
Grid
DC Load
Center
AC Load
Center
Fig. 1.6 Grid-connected PV system
1.3.1 Structure of Grid-connected PV System
In general, grid-connected PV system is composed of PV array, inverter and
controller. Among them, grid-connected inverter is a very important part in gridconnected PV system which connected with the utility grid. Grid-connected PV system
generally can be divided into single-stage and two-stage grid-connected PV System.
Single-stage grid-connected PV System is composed of PV array, DC/AC inverter,
controller, static switch and local load, and is shown as Fig. 1.7. Its operational principle
is that direct current generated by PV array is converted to alternating current through
inverter and feed to the grid.
11
DC / AC
Inverter
PV Array
Static
Switch
Current
Detection
Voltage and
Current
Detection
Utility
Grid
Local AC
Load Center
Voltage and
Frequency
Detection
Control System
Fig. 1.7 Structure of single-stage grid-connected PV System
Two-stage grid-connected PV System is composed of PV array, DC/DC converter,
DC/AC inverter, energy storage system, controller, static switch and local load, and is
shown as Fig. 1.8. Its operational principle is that the direct current generated by PV
array is converted to another voltage level (in general, use boost converter) and be
converted to alternating current through inverter to feed into the grid.
The first stage converts the direct current generated by the PV array through DC /
DC converter, after that stores the electric to the energy storage unit or provides it to
inverter in the next stage, furthermore, the maximum power tracking (MPPT) function
for PV cell array needed to be achieve in this stage. The second stage is to convert the
current into alternating current for local load and excess energy will fed into the grid.
PV Array
Voltage and
Current
Detection
DC / DC
Converter
MPPT
Control
DC / AC
Inverter
Charge
Controller
Static
Switch
Current
Detection
Utility
Grid
Local AC
Load Center
Voltage and
Frequency
Detection
Energy
Storage
System
Control System
Fig. 1.8 Structure of two-stage grid-connected PV System
12
The advantages and disadvantages of single-stage and two-stage grid-connected PV
System is shown in Table 1.2.
Table 1.2 advantages and disadvantages of single-stage and two-stage PV System
Single-stage
1.
2.
System
Two-stage
topology
is 1.
Can
be
independently
relatively simple.
controlled that the controller
The entire grid-connected
design is simpler.
system without intermediate 2.
In the DC / DC part is easy
energy storage links that
to achieve high-frequency
saving investment costs.
isolation.
It is more efficient that has 3.
With the energy storage
only one energy conversion.
system, it can achieve grid
Advantages
3.
and standalone two modes of
operation to switch.
1.
Disadvantages
Control
system
in
the 1.
System topology is more
inverter need to achieve
complex than the single-
MPPT and grid-connected
stage
function at the same time,
increasing
that the controller require
storage device will increase
more precise control.
the entire system investment
system,
the
while
energy
costs.
1.3.2 Maximum Power Point Tracking
At different irradiation and temperatures, the maximum output power point of the
PV cell is not the same. In order to obtain the best efficiency, it is necessary to take
measures to automatically track the changes in the climatic conditions and that why the
13
maximum power tracking technology is proposed for this problem. The most common
methods for maximum power tracking are constant voltage tracking method,
incremental conductance method and perturbation and observation method. Constant
voltage tracking method typically use 76% of the open circuit voltage as the maximum
power point voltage [12]. Perturbation and observation method is by regularly
increasing and reducing the output voltage of photovoltaic cells to adjust the PV system
output voltage to the maximum power point. When the irradiation changes little, this
method is easy to track the maximum power point, but it will causing a certain power
loss with the output voltage oscillates near the maximum power point. Moreover, the
perturbation and observation method may fail when the irradiation and ambient
temperature change drastically [13, 14]. Incremental conductance method is widely
used in the photovoltaic power generation system [15, 16]. This method is change the
control signal by comparing the conductance increment and the transient conductance
value of the PV array to track the maximum power point and this algorithm needs to be
determined by measuring the output voltage and current variation of the PV array.
Incremental conductance method is accurate and fast response, but the requirements of
the accuracy of the sensor are relatively high that the entire system cost more.
1.3.3 Grid-connected Control Technology
In grid-connected PV system, the inverter acts as an important interface device for
grid-connected PV system systems and utility grid and its control performance become
the direct factor for the current quality to feeds in grid that may causes the resonance
phenomena at the point of common coupling (PCC) [17, 18].
Grid-connected PV generation system is shown as Fig. 1.9. The main control
object is the grid-connected current [19]. The resonance problem is usually reflected in
the current loop in control system and the current control technology for the inverter is
essential which has faster dynamic response and accuracy in steady state. Commonly
14
used control methods are PI control, repeat control, proportional resonance control, and
dead-beat control.
Grid-connected
Inverter
PCC
ig
Filter
Vdc
Ug
Utility
Grid
PV
Array
ig
Vdc
Control
System
Ug
Fig. 1.9 Structure diagram of grid-connected PV generation system
1.4
Organization of Thesis
Chapter I. Introduction
This chapter introduces the motivation and outline of this thesis.
Chapter II. PV Cell Model and MPPT Methods
This chapter introduces the mathematical model and the most common methods
for maximum power tracking. Through the mathematical equation to build the PV cell
model in MATLAB/Simulink and compare the characteristics curve of PV cell with the
real data from the datasheet of Mitsubishi Electric photovoltaic module PVAE125MF5N.
Chapter III. Model Analysis of Grid-connected PV System
In this chapter, the topology grid-connected PV system is analyzed in
15
mathematical and also analyzed the performance of three type passive filters,
furthermore, will determine the parameter of the filter.
Chapter IV. Control Strategy of Grid-connected PV inverter
This chapter introduce several types of control strategy used in the PV inverter. By
using mathematical and simulating in MATLAB/Simulink to analyze the performance
of each control method.
Chapter V. Simulation Results
The theories above are used to build the model of grid-connected PV system, and
analyze the performance of harmonic suppression between certain control strategies
and the passive filter efficiency.
Chapter VI. Conclusions and Future Works
Conclusions and future works of this thesis are provided in this chapter.
16
II. PV cell model and MPPT methods
2.1
Overview
The output of PV array has non-linear characteristic and its output voltage and
current are effect by the irradiation, ambient temperature and load conditions. At any
irradiation and ambient temperature, PV array can work at different output voltages,
but only at certain output voltage that PV array output power can reach the maximum
value. When the PV array operating point reach the highest point of P -V characteristics
curve, called the maximum power point (MPP). Therefore, in the PV generation system,
an important way to improve the overall efficiency of the system is to timely adjust the
PV array operating point which makes it work near the maximum power point. This
process is called the maximum power point tracking.
This chapter describes the operation principle of PV cell, which includes analyzing
the constant voltage tracking method, perturbation and observation method, and
incremental conductance method.
2.2
Mathematical Model and Characteristics of PV Cell
PV cell is a device made of a semiconductor material and use the photoelectric
effect to change the solar irradiation to the surface of the battery into DC power. The
photoelectric effect refers to the object by the impact of irradiation, the internal
distribution state of the charges change in voltage and current. In general, liquid, solid
and gas have this effect, but the solid, especially the conversion of photoelectric effect
in semiconductor has highest efficiency that is most suitable for the production of PV
cells.
The equivalent circuit of a PV cell is as shown in Fig. 2.1 [20]. The current source
Iph represents the cell photocurrent. R sh and R s are the internal shunt and series
resistances of the cell.
17
Ipv
+
Rs
Iph
Io
Vpv
Rsh
-
Fig. 2.1 PV cell modeled as diode circuit
The current-voltage characteristic of the ideal p-n junction diode can be expressed
by the following equation:
𝑞𝑉𝑝𝑣
𝐼𝑑 = 𝐼𝑜 (𝑒 𝐴𝐾𝐵 𝑇 − 1)
(2.1)
Where Io is the diode saturation current, A is the diode ideality factor, q is the Electron
charge, T is the module operating temperature and K B is Boltzmann constant.
From Fig. 2.1 and (2.1), the current output of PV module is
𝐼𝑝𝑣 = 𝐼𝑝ℎ − 𝐼𝑜 [𝑒
𝑞(𝑉𝑝𝑣 +𝑅𝑠 𝐼𝑝𝑣 )
𝐴𝐾𝐵 𝑇
− 1] −
𝑉𝑝𝑣 +𝑅𝑠 𝐼𝑝𝑣
𝑅𝑠ℎ
(2.2)
Usually the value of R sh is very large and that of R s is very small, hence they may
be neglected to simplify the analysis that the (2.2) can be simplify as
𝐼𝑝𝑣 = 𝐼𝑝ℎ − 𝐼𝑜 [𝑒
𝑞(𝑉𝑝𝑣 +𝑅𝑠 𝐼𝑝𝑣 )
𝐴𝐾𝐵 𝑇
− 1]
(2.3)
Under certain condition of irradiation and ambient temperature, Fig. 2.2 shows the
output characteristics curve of PV cell. Fig. 2.2(a) shows the characteristic curve of
output currentIpv versus output voltage Vpv and Fig. 2.2(b) shows the characteristic
curve of output power versus output voltage Vpv . In Fig. 2.2, Isc indicates the short
circuit current of the photovoltaic cell, that is, PV cell can produce the maximum current;
Voc indicates the open circuit current of the photovoltaic cell, that is, PV cell can
produce the maximum voltage; 𝑃𝑚𝑎𝑥 indicates the maximum power that the
photovoltaic cell can produce, which the voltage Vm corresponding to this point is
called the maximum power point voltage and the current Im corresponding to this
18
point is called the maximum power point current.
Isc
Pmax
Im
Vm
Vm
Voc
(a) I-V characteristics
Voc
(b) P-V characteristics
Fig. 2.2 PV cell output characteristics curve
(a) I-V characteristics
(b) P-V characteristics
Fig. 2.3 PV cell output characteristics curve with various irradiation
The operating status of PV cells is influenced by external factors, and the external
factors mainly include irradiation and ambient temperature. Fig. 2.3 shows PV cell
output characteristics curve with various temperature.
The following conclusions can be drawn from the figure：
(1) For PV cells, the short-circuit current and irradiation is approximately direct
proportion.
(2) The intensity of irradiation little effect on the value of the open circuit voltage.
19
(3) Increased irradiation will increase the maximum output power and there is
only one point will make the PV cell reach the maximum output power at the
same irradiation.
(a) I-V characteristics
(b) P-V characteristics
Fig. 2.4 PV cell output characteristics curve with various temperature
Fig. 2.4 shows PV cell output characteristics curve with various temperature. The
following conclusions can be drawn from the figure：
(1) The temperature has little effect on the short circuit current of the PV cell.
(2) The temperature has affected on the open circuit voltage of the PV cell, while
the temperature rises, the open circuit voltage value will drop.
(3) The maximum output power point that the PV cell can reach and the
maximum operating voltage corresponding to that point decreases as the
temperature increases.
The curve of PV cell output characteristics above is simulate in MATLAB/
Simulink. To verify its accuracy, we use the specification sheet data of Mitsubishi
Electric photovoltaic module PV-AE125MF5N [21] to compare the result. Fig. 2.5(a)
shows the characteristic curves of PV-AE125MF5N for various values of irradiance at
temperature 25 °C and Fig. 2.5(b) shows the characteristic curves of PV cell that
simulate in MATLAB/ Simulink with the same parameter in datasheet. The result shows
20
that the curve between them are almost the same.
(a) PV-AE125MF5N
(b) Simulink model
Fig. 2.5 PV cell output characteristics curve with 25 °C and various temperature
2.3
Maximum Power Point Tracking
By the analysis of the previous section, the output power of photovoltaic cell
influenced by irradiation and ambient temperature, is shown as Fig. 2.4. At any
irradiation and ambient temperature, PV array can work at different output voltages,
but only at certain output voltage that PV array output power can reach the maximum
value which is called the maximum power point. Therefore, in the PV generation
system, the instantaneous detection of the output voltage or current of the photovoltaic
cell, calculate and compare the photovoltaic cell output power to predict the position
where the maximum power output is and adjusting its operating point to make it always
around the maximum power point. These process is called maximum power point
tracking. The most common methods for maximum power tracking are constant voltage
tracking method, incremental conductance method and perturbation and observation
method.
21
2.3.1 Constant Voltage Tracking Method
When the temperature of the PV cell is constant, the maximum power point voltage
on the output P-V curve of the PV cell is almost a fixed voltage value that can be seen
by analyzing Fig. 2.3. Therefore, the thought of constant voltage tracking method is to
set the command value of maximum power point voltage of the PV cell in the control
system, and control the output voltage of the solar cell at the set voltage command value
when the system is working. When the external environmental conditions change is not
large, it can approximate that the solar cell always work at the maximum power point.
Flow chart of the constant voltage tracking method is shown in Fig. 2.6.
Start
Read Vpv(n)
Yes
No
Yes
Vpv(n)-Vconst=0
No
Vref(n)=Vref(n-1)-∆V
Vref(n)=Vref(n-1)+∆V
Vpv(n)=Vpv(n-1)
Return
Fig. 2.6 Flow chart of the constant voltage tracking method
Constant voltage tracking method has the advantages as follows:
(1) The control is simple and is easy to realize: The system only needs to read the
output voltage of the PV cell and compare it with the setting value.
(2) The system operating voltage has good stability: Through reading the feedback
of PV cell output voltage, it can set the PI regulator for a simple PI adjustment which
22
can make PV cell output voltage stable.
However the constant voltage tracking method is too simple, there has some
shortcomings cannot be ignored, and thus limit the method can apply only in specific
moments. The disadvantages is as follows:
(1) The maximum power point tracking accuracy is not well. The maximum power
point voltage command value of the PV cell has a great influence on the operating
efficiency of the system, and the output power of the PV system may be zero when the
voltage value setting is improper.
(2) Constant voltage tracking method has poor adaptability and does not have the
ability to track the maximum power point while the external environment conditions
change.
2.3.2 Perturbation and Observation Method
Perturbation and observation method is one of the commonly used methods to
achieve maximum power point tracking control. This method is to constantly disturb
the operating point of the PV system to obtain the direction of the maximum power
point. The operation principle of perturbation and observation method is shown in Fig.
2.7 and the action can be divided into following four parts:
A.
When reference voltage increases and causes output power increase, then
keeping the direction of perturbing and still increasing the reference voltage.
B.
When reference voltage increases but causes output power decrease, then
changing the direction of perturbing and decreasing the reference voltage.
C.
When reference voltage decreases but causes output power increase, then
keeping the direction of perturbing and still decreasing the reference voltage.
D.
When reference voltage decreases and causes output power increase, then
changing the direction of perturbing and increasing the reference voltage.
23
Fig. 2.7 The operation principle of perturbation and observation method
Start
Read Vpv(n), Ipv(n)
Calculate
Ppv(n) = Vpv(n)*Ipv(n)
Ppv(n)>Ppv(n-1)
Yes
Yes
Vpv(n)>Vpv(n-1)
Vref(n)=Vref(n-1)+∆V
No
Yes
Vref(n)=Vref(n-1)-∆V
No
Vpv(n)>Vpv(n-1)
Vref(n)=Vref(n-1)-∆V
No
Vref(n)=Vref(n-1)+∆V
Return
Fig. 2.8 Flow chart of the perturbation and observation method
Flow chart of the perturbation and observation method is shown in Fig. 2.8. This
method can use the disturbance of PV cell output voltage to achieve the PV cell output
power control. Therefore, when the external environmental conditions change,
24
Therefore, when the external environmental conditions change, the system
controlled by the perturbation and observation method can automatically detect the
change of the output power of the PV cell, and then use the appropriate control to ensure
that the operating point of the PV cell can move to maximum power point which can
improve the efficiency of PV cells. The disadvantage of this method is that it can only
operate in the vicinity of the maximum power point, so it will lose part of the power. It
can only reduce the perturbation frequency and choose the proper perturbation quantity
to reduce the power loss.
2.3.3 Incremental Conductance Method
From the P-V characteristics curve of PV cell, it can be seen that there has
dP⁄dV = 0 at the maximum power point. Through simple derivation, it can be
concluded that the following equation holds at maximum power point:
𝑑𝐼
𝐼
=−
𝑑𝑉
𝑉
(2.4)
This equation is used as a basis for judging whether the solar cell is operating at
the maximum power point, then draw the relationship between dI⁄dV and −I⁄V on
the two sides of the maximum power point on I-V characteristic curve that shown on
Fig. 2.9. When the PV cell is not working on the line of 𝑑𝐼 ⁄𝑑𝑉 = −𝐼 ⁄𝑉 , it can be
determined by the value of 𝑑𝐼 ⁄𝑑𝑉 whether the output voltage of the PV cell is in the
interval of [0, Vm] or the interval of [Vm, Voc], and adjusts PV cell can work at the
maximum power point. Flow chart of the incremental conductance method is shown in
Fig. 2.10. This method is based on the physical characteristics of the PV cell, so there
is no power fluctuation in the steady state, the control system has a higher stability.
When the external environment changes, the output voltage can track the maximum
power point smoothly. But the algorithm of incremental conductance method is more
complex, the control system requirements are relatively high. Besides, the initial value
25
of the voltage on the system startup process has a greater impact on the performance, if
the value is set incorrectly, it is likely to cause greater power loss.
Im
Voc
Vm
Fig. 2.9 Relationship in the incremental conductance method
Start
Read Vpv(n), Ipv(n)
Calculate dV pv(n), dIpv(n)
dVpv(n)=Vpv(n)-Vpv(n-1)
dIpv(n)=Ipv(n)-Ipv(n-1)
dVpv(n)=0
Yes
No
Yes
Yes
dIpv(n)/dVpv(n)
= -Ipv(n)/Vpv(n)
dIpv(n) = 0
No
Yes
No
dIpv(n)/dVpv(n)
> -Ipv(n)/Vpv(n)
dIpv(n) > 0
No
Vref(n)=Vref(n-1)+∆V
No
Vref(n)=Vref(n-1)-∆V
Vref(n)=Vref(n-1)-∆V
Vref(n)=Vref(n-1)+∆V
Return
Fig. 2.10 Flow chart of the incremental conductance method
26
III. Theoretical Analysis of Three-Phase Photovoltaic GridConnected System
3.1
Topology and Model of PV Grid-Connected Inverter
In order to satisfy the international standards of IEC 61727:2004, IEEE 929-2000,
IEC 61000-3-2 and other relevant standards, selection of appropriate control strategy
and harmonic filter in three-phase grid-connected inverter are key points. There are
generally three kinds of filters in three-phase grid-connected inverter, namely, L-type,
LC-type and LCL-type. In this chapter, three-phase photovoltaic inverters with three
different grid-connected filters are used to establish the state space mathematical model
and analyzed respectively.
3.1.1 State-Space Model of PV Inverter with L Filter
The structure diagram of the three-phase photovoltaic grid-connected power
system with L filters is shown in Fig. 3.1, where the PV array represents the solar cell
array, C1 is the DC side capacitor, T1~T6 are the 6 IGBT switch tubes in three-phase
inverter, L1 is the L filter, R1 is the internal resistance in inductance L1, Vdc is DC side
output voltage to inverter, idc is DC side output current, Va, Vb and Vc are the output
voltage of inverters, i1a, i1b, i1c are the output current of inverters, Vsa, Vsb and Vsc are
three-phase voltage of power grid.
iPV
idc
T1
T3
T5
ic
i1
C1
Va
Vc
T6
Usa
o
PV
Array
T4
R1
Usb
Vb
Vdc
L1
Usc
T2
n
Fig. 3.1 Topology of three-phase grid-connected PV power system with L filter
27
Before establishing the mathematical model of the inverter, we first define the
switching function Sk(k=a, b, c):
Sk = {
1,
0,
T1 , T3 , T5 closed and T2 , T4 , T6 open.
T1 , T3 , T5 open and T2 , T4 , T6 closed.
(3.1)
In phase a, when Sk=1, that means T1 is closed and T4 is opened, then VaN=Vdc;
when Sk=0, that means T1 is opened and T4 is closed, then VaN=0. Through the relation
above, (3.2) can be derived and k means phase a, b or c.
𝑉𝑘𝑛 = 𝑉𝑑𝑐 ∙ 𝑆𝑘
(3.2)
By Kirchhoff's voltage law, the phase a, loop equation in three-phase inverter is
𝐿1
𝑑𝑖1𝑎
+ 𝑅1 𝑖1𝑎 = 𝑉𝑎𝑛 − 𝑉𝑠𝑎 − 𝑉𝑜𝑛
𝑑𝑡
(3.3)
Similarly, the grid side voltage equation is derive as
𝐿1
𝑑𝑖1𝑘
+ 𝑅1 𝑖1𝑘 = 𝑉𝑘𝑛 − 𝑉𝑠𝑘 − 𝑉𝑜𝑛
𝑑𝑡
(3.4)
By using Kirchhoff’s current law on the DC side capacitor, it can derive
𝐶1
𝑑𝑉𝑑𝑐
= 𝑖𝑃𝑉 − ∑ 𝑖1𝑘 𝑆𝑘
𝑑𝑡
(3.5)
𝑘=𝑎,𝑏,𝑐
In the three-phase balanced system, (3.19) must hold.
∑ 𝑖1𝑘 = ∑ 𝑉𝑠𝑘 = 0
𝑘=𝑎,𝑏,𝑐
(3.6)
𝑘=𝑎,𝑏,𝑐
From (3.2), (3.4) and (3.5), it can derive (3.7).
𝑉𝑜𝑛 =
𝑉𝑑𝑐
∑ 𝑆𝑘
3
(3.7)
𝑘=𝑎,𝑏,𝑐
In order to facilitate the analysis, the Clarke transformation is used to transform
the grid-connected inverter model from abc three-phase coordinate system to αβ twophase coordinate system. Let the α axis be set in the direction of the grid phase-a voltage,
as shown in Fig. 3.2 and can obtain the Clarke transformation matrix 𝑇𝐴𝐵𝐶→𝛼𝛽 and
Park transformation matrix 𝑇𝛼𝛽→𝑑𝑞 .
28
b axis
β axis
q axis
d axis
U
uβ
ud
uq
ωt
uα
α axis
a axis
c axis
Fig. 3.2 The relationship of coordinate transformations between abc, αβ and dq
reference frames
𝑇𝐴𝐵𝐶→𝛼𝛽 =
(3.8)
[0
1
1
−
2
2
√3
√3
− ]
2
2
𝑐𝑜𝑠𝜔𝑡
𝑠𝑖𝑛𝜔𝑡
𝑠𝑖𝑛𝜔𝑡
]
−𝑐𝑜𝑠𝜔𝑡
(3.9)
2 1 −
3
𝑇𝛼𝛽→𝑑𝑞 = [
According to Clark transformation matrix (3.8), we can get the state space
equation of L filter in αβ coordinate system:
𝑑𝑖1𝛼
𝑅1
1
1
= − 𝑖𝛼 + 𝑉𝛼 − 𝑉𝑠𝛼
𝑑𝑡
𝐿1
𝐿1
𝐿1
𝑑𝑖1𝛽
𝑅1
1
1
= − 𝑖𝛽 + 𝑉𝛽 − 𝑉𝑠𝛽
𝐿1
𝐿1
𝐿1
{ 𝑑𝑡
(3.10)
Therefore, according to the Park transformation matrix (3.9), we can get the state space
equation in dq coordinate system:
𝑑𝑖1𝑑
𝑅1
1
1
= − 𝑖1𝑑 + 𝜔𝑖1𝑞 + 𝑉𝑑 − 𝑉𝑠𝑑
𝑑𝑡
𝐿1
𝐿1
𝐿1
𝑑𝑖1𝑞
𝑅1
1
1
= − 𝑖1𝑞 − 𝜔𝑖1𝑑 + 𝑉𝑞 − 𝑉𝑠𝑞
𝐿1
𝐿1
𝐿1
{ 𝑑𝑡
(3.11)
In dq coordinate system, the d-axis and q-axis components of the inverter AC side
29
voltage Vko are Vd and Vq, respectively, is shown as (3.12) and (3.13).
𝑉𝑑 = 𝑆𝑑 𝑉𝑑𝑐
{𝑉 = 𝑆 𝑉
𝑞
𝑉𝑑𝑐 =
(3.12)
𝑞 𝑑𝑐
3
𝑖𝑝𝑣 − 2 (𝑆𝑑 𝑖1𝑑 + 𝑆𝑞 𝑖1𝑞 )
𝑠𝐶1
𝑖𝑑𝑐 = 𝑆𝑎 𝑖1𝑎 + 𝑆𝑏 𝑖1𝑏 + 𝑆𝑐 𝑖1𝑐 =
𝑖𝑃𝑉 − 𝑖𝑑𝑐
𝑠𝐶1
(3.13)
3
(𝑆 𝑖 + 𝑆𝑞 𝑖1𝑞 )
2 𝑑 1𝑑
(3.14)
=
Based on the above mathematical analysis, we can draw the L-type grid-connected
inverter model in the dq coordinate system, as shown in Fig. 3.3.
Fig. 3.3 L-type grid-connected inverter model in the dq coordinate system
3.1.2 State-Space Model of PV Inverter with LC Filter
The structure diagram of the three-phase photovoltaic grid-connected power
system with LC filters is shown in Fig. 3.4. The circuit topology is basically the same
as the three-phase photovoltaic grid-connected power system with L filters, and the AC
side with L1 and C2 are represent the LC-type filter. ica, icb, icc are capacitance currents
through C2; Vca, Vcb, Vcc are capacitance voltages of C2.
30
iPV
idc
T1
T3
T5
ic
i1
C1
Va
L1
R1
i1
Usb
Vb
Vdc
T6
o
Usc
Vc
PV
Array
T4
Usa
iC2
T2
C2
Vca Vcb Vcc
n
Fig. 3.4 Topology of three-phase grid-connected PV power system with LC filter
Similar to the analysis in section 3.1.1, we can obtain the following equations:
𝐶1
𝑑𝑉𝑑𝑐
= 𝑖𝑃𝑉 − ∑ 𝑖1𝑘 𝑆𝑘
𝑑𝑡
(3.15)
𝑑𝑖1𝑘
+ 𝑅1 𝑖1𝑘 = 𝑉𝑘𝑜 − 𝑉𝑠𝑘
𝑑𝑡
(3.16)
𝑑𝑉𝑐𝑘
𝑑𝑡
(3.17)
𝑘=𝑎,𝑏,𝑐
𝐿1
𝑖2𝑘 = 𝑖1𝑘 − 𝑖𝑐𝑘 = 𝑖1𝑘 − 𝐶2
𝑉𝑐𝑘 = 𝑉𝑠𝑘
(3.18)
And in the three-phase balanced system, there has
∑ 𝑉𝑘𝑜 = ∑ 𝑉𝑐𝑘 = ∑ 𝑖1𝑘 = ∑ 𝑖𝑐𝑘 = ∑ 𝑖2𝑘 = 0
𝑘=𝑎,𝑏,𝑐
𝑘=𝑎,𝑏,𝑐
𝑘=𝑎,𝑏,𝑐
𝑘=𝑎,𝑏,𝑐
(3.19)
𝑘=𝑎,𝑏,𝑐
According to Clark transformation and Park transformation, we can get the state
space equation of LC filter in αβ coordinate system and dq coordinate system:
𝑑𝑖1𝛼
𝑅1
1
1
= − 𝑖𝛼 + 𝑉𝛼 − 𝑉𝑠𝛼
𝑑𝑡
𝐿1
𝐿1
𝐿1
𝑑𝑖1𝛽
𝑅1
1
1
= − 𝑖𝛽 + 𝑉𝛽 − 𝑉𝑠𝛽
𝑑𝑡
𝐿1
𝐿1
𝐿1
𝑑𝑉𝑠𝑎
1
= 𝑖𝑐𝛼
𝑑𝑡
𝐶2
𝑑𝑉𝑠𝛽
1
= 𝑖𝑐𝛽
{
𝑑𝑡
𝐶2
31
(3.20)
𝑑𝑖1𝑑
𝑅1
1
1
= − 𝑖1𝑑 + 𝜔𝑖1𝑞 − 𝑉𝑑 − 𝑉𝑠𝑑
𝑑𝑡
𝐿1
𝐿1
𝐿1
𝑑𝑖1𝑞
𝑅1
1
1
= − 𝑖1𝑞 − 𝜔𝑖1𝑑 − 𝑉𝑞 − 𝑉𝑠𝑞
𝑑𝑡
𝐿1
𝐿1
𝐿1
𝑑𝑉𝑠𝑑
1
= 𝑖𝑐𝑑 + 𝜔𝑉𝑠𝑞
𝑑𝑡
𝐶2
𝑑𝑉𝑠𝑞
1
= 𝑖𝑐𝑞 − 𝜔𝑉𝑠𝑑
{
𝑑𝑡
𝐶2
(3.21)
3.1.3 State-Space Model of PV Inverter with LCL Filter
The structure diagram of the three-phase photovoltaic grid-connected power
system with LCL filters is shown in Fig. 3.5. The circuit topology is basically the same
as the system above, and the AC side with L1, L2 and C2 are represent the LCL-type
filter, that i2a, i2b, i2c are inductance currents through L2.
iPV
idc
T1
T3
T5
ic
L1
i1
C1
Va
R1
L2
o
iC2
T2
T6
Usa
Usc
Vc
PV
Array
T4
R2
Usb
Vb
Vdc
i2
C2
Vca Vcb Vcc
n
Fig. 3.5 Topology of three-phase grid-connected PV power system with LCL filter
Same as the analysis in section 3.1.1, we can obtain the following equations:
𝐶1
𝑑𝑉𝑑𝑐
= 𝑖𝑃𝑉 − ∑ 𝑖1𝑘 𝑆𝑘
𝑑𝑡
(3.22)
𝑘=𝑎,𝑏,𝑐
𝑑𝑖1𝑘
+ 𝑅1 𝑖1𝑘 = 𝑉𝑘𝑚 − 𝑉𝑐𝑘
𝑑𝑡
𝑉𝑑𝑐
𝑉𝑚𝑛 = 𝑉𝑜𝑛 =
∑ 𝑆𝑘
3
𝐿1
(3.23)
(3.24)
𝑘=𝑎,𝑏,𝑐
𝑉𝑘𝑛 = 𝑉𝑑𝑐 𝑆𝑘
𝑑𝑉𝑐𝑘
𝑑𝑡
(3.26)
𝑑𝑖2𝑘
+ 𝑅2 𝑖2𝑘
𝑑𝑡
(3.27)
𝑖2𝑘 = 𝑖1𝑘 − 𝑖𝑐𝑘 = 𝑖1𝑘 − 𝐶2
𝑉𝑐𝑘 = 𝑉𝑠𝑘 + 𝐿2
(3.25)
32
And in the three-phase balanced system, (3.28) must hold.
∑ 𝑉𝑘𝑜 = ∑ 𝑉𝑐𝑘 = ∑ 𝑖1𝑘 = ∑ 𝑖𝑐𝑘 = ∑ 𝑖2𝑘 = 0
𝑘=𝑎,𝑏,𝑐
𝑘=𝑎,𝑏,𝑐
𝑘=𝑎,𝑏,𝑐
𝑘=𝑎,𝑏,𝑐
(3.28)
𝑘=𝑎,𝑏,𝑐
According to Clark transformation matrix (3.8), we can get the state space
equation of LCL filter in αβ coordinate system as follow.
𝑑𝑖1𝛼
𝑅1
1
1
= − 𝑖𝛼 + 𝑉𝛼 − 𝑉𝑐𝛼
𝑑𝑡
𝐿1
𝐿1
𝐿1
𝑑𝑖1𝛽
𝑅1
1
1
= − 𝑖𝛽 + 𝑉𝛽 − 𝑉𝑐𝛽
𝐿1
𝐿1
𝐿1
{ 𝑑𝑡
(3.29)
𝑑𝑖2𝛼
𝑅2
1
1
= − 𝑖𝛼 + 𝑉𝑐𝛼 − 𝑉𝑠𝛼
𝑑𝑡
𝐿2
𝐿1
𝐿2
𝑑𝑖2𝛽
𝑅2
1
1
= − 𝑖𝛼 + 𝑉𝑐𝛽 − 𝑉𝑠𝛽
𝐿2
𝐿2
𝐿2
{ 𝑑𝑡
(3.30)
𝑑𝑉𝑐𝛼
1
1
= 𝑖1𝛼 − 𝑖2𝛼
𝑑𝑡
𝐶2
𝐶2
𝑑𝑉𝑐𝛽
1
1
= 𝑖1𝛽 − 𝑖2𝛽
𝐶2
𝐶2
{ 𝑑𝑡
(3.31)
Therefore, according to the Park transformation matrix (3.9), we can get the state space
equation in dq coordinate system:
𝑑𝑖1𝑑
𝑅1
1
1
= − 𝑖𝑑 + 𝜔𝑖1𝑞 + 𝑉𝑑 − 𝑉𝑐𝑑
𝑑𝑡
𝐿1
𝐿1
𝐿1
𝑑𝑖1𝑞
𝑅1
1
1
= − 𝑖𝑞 − 𝜔𝑖1𝑑 + 𝑉𝑞 − 𝑉𝑐𝑞
𝐿1
𝐿1
𝐿1
{ 𝑑𝑡
(3.32)
𝑑𝑖2𝑑
𝑅2
1
1
= − 𝑖𝑑 + 𝜔𝑖2𝑞 + 𝑉𝑐𝑑 − 𝑉𝑠𝑑
𝑑𝑡
𝐿2
𝐿1
𝐿2
𝑑𝑖2𝑞
𝑅2
1
1
= − 𝑖𝑞 − 𝜔𝑖2𝑑 + 𝑉𝑐𝑞 − 𝑉𝑠𝑞
𝐿2
𝐿2
𝐿2
{ 𝑑𝑡
(3.33)
𝑑𝑉𝑐𝑑
1
1
= 𝑖1𝑑 − 𝑖2𝑑 + 𝜔𝑉𝑐𝑞
𝑑𝑡
𝐶2
𝐶2
𝑑𝑉𝑐𝑞
1
1
= 𝑖1𝑞 − 𝑖2𝑞 − 𝜔𝑉𝑐𝑑
𝐶2
𝐶2
{ 𝑑𝑡
(3.34)
Based on the above mathematical analysis, we can draw the LCL-type gridconnected inverter model in the dq coordinate system, as shown in Fig. 3.6.
33
Fig. 3.6 L-type grid-connected inverter model in the dq coordinate system
3.2
Performance Analysis of LCL Filter
There are generally three kinds of filters in three-phase grid-connected inverter,
namely, L-type, LC-type and LCL-type, as shown in Fig. 3.7, and the selection of
different kinds of filters will directly affect the power quality of the grid-connected
current. The filter with a reasonable design can be achieved to mitigate the harmonic
which caused by the inverter to make the grid-connected system operation more stable
and safer. According to the inverter operating in different states, the selection of filter
is not the same. For the inverter operating in off-grid mode, normally, using the LCtype filter [22]; while the inverter work in grid-connected mode, three kinds of filter
may be used under different needs [23, 24]. Here, we analyze LCL-type filter and
compare it with L-type filter, which is the most basic type filter.
L
L
L1
C
(a) L-type filter
L2
C
(b) LC-type filter
(c) LCL-type filter
Fig. 3.7 Three types of filter for inverter
34
For the grid-connected inverter with L-type filter, transfer function of the grid
current Igrid and inverter output voltage Vinv is:
𝐺𝐿 (𝑠) =
𝐼𝑔𝑟𝑖𝑑
1
=
𝑉𝑖𝑛𝑣 𝐿𝑠 + 𝑅
(3.35)
And the grid-connected inverter with LCL-type filter, transfer function of the grid
current Igrid and inverter output voltage Vinv becomes
𝐺𝐿𝐶𝐿 (𝑠) =
1
𝐿1 𝐿2 𝐶𝑠 3 + (𝐿1 𝐶𝑅2 + 𝐿2 𝐶𝑅1 )𝑠 2 + (𝐿1 + 𝐿2 + 𝑅1 𝑅2 𝐶)𝑠 + 𝑅1 + 𝑅2
(3.36)
where R, R1, R2 are internal resistances in L, L1 和 L2, respectively.
In order to illustrate their characteristics, let L1 + L2 = L , and plot the Bode
diagram of L-type and LCL-type filter transfer function, as shown in Fig. 3.8. It can be
seen from the figure that at high frequency, LCL filter attenuation rate is larger than that
of L-type. Therefore, the LCL filter have a better attenuation effect on higher current
harmonics. At low frequency, both of the frequency responses are exactly the same.
Fig. 3.8 Bode diagram of L-filter and LCL-filter
From the analysis above, it can be obtained that at low frequency, LCL-type filter
can be equivalent as the L-type with inductance L = L1 + L2. At high frequency,
comparing LCL-type filter to L-type filter, the effect of mitigating higher harmonics is
35
better. Due to the L-type filter belongs to the first-order system, the structure is simple
and easy to design, but its high-frequency mitigating effect is not ideal, which needs to
set a larger inductance value to get the ideal filtering effect and will limit its range of
application. The LCL-type filter belongs to the third-order system, although the
structure is complex, the high frequency filtering effect is better than L-type. If both are
with the same harmonic attenuation requirement, LCL-type filter equivalent inductance
L =L1 + L2 is smaller than L-type, and then LCL filter impedance is relatively smaller,
which can reduce the filter size and weight and will relatively reduce loss, so it can be
used on higher power equipment.
3.3
Parameter Design of LCL Filter in Three-Phase GridConnected Inverter
The inductance core of the LCL filter occupies the vast majority of the volume and
cost of the filter. Based on this structural feature, the smaller inductance L is selected
as far as achieving the filtering requirements. On this basis, it also needs to ensure that
the resonant frequency of the LCL filter cannot be too small, so as not to affect the
performance of the inverter control strategy [25]. In LCL-type grid-connected filter, L1
(inductor at the inverter side) can transfer the inverter output voltage into the gridconnected current and the value of inductor L1 will affect the power quality and system
dynamic performance. In addition, although L2 (AC grid side inductor) and C can filter
the high frequency components of the grid current, but also need to set L2 and C to
avoid the resonance phenomenon caused by improper settings.
At present, many literatures have introduced the design methods of LCL-type gridconnected filter. The most widely cited literature [26] introduces the restriction
conditions and design steps of LCL filter parameters. Combing and summarizing
literature [26][27][28], the design constraints of filter parameters are show as follows:
(1) The voltage loss generated on LCL filter-inductance should less than 10% of
36
the normal grid voltage.
1
𝐿 +𝐿
(2) The resonant frequency 𝑓𝑟𝑒𝑠 = 2𝜋 √𝐿 1𝐿 𝐶2 should be 10𝑓1 < 𝑓𝑟𝑒𝑠 < 0.5𝑓𝑠𝑤 ,
1 2 2
where f1 is the grid voltage frequency and fsw is the switch frequency.
(3) In order to let the high-frequency harmonics go through the capacitor path.
The switch frequency impedances XC2 and XL2 should be satisfied.
𝑋𝐶2 = (0.1~0.2)𝑋𝐿2
(3.37)
(4) Moreover, generally the reactive power absorbed by the filter-capacitor
should be less than the rated active power of the system.0
𝐶2 ≤
𝜆𝑃
2
3 × 2𝜋𝑓1 𝐸𝑚
(3.38)
where Em is the phase voltage (RMS), and λ is the ratio of the reactive power
absorbed by capacitor C2 . P is the rated power.
37
IV. Control Strategy of Photovoltaic Inverter
4.1
Grid-Connected Inverter Control Method and Strategies
4.1.1 Design of Grid-Connected Inverter Control System
From the output control mode, it can be divided into voltage control and current
control mode. The voltage control mode sees the inverter as a voltage source and the
regulation of the active and reactive power which input to the grid by the inverter is
controlled by the magnitude and phase of the grid voltage. The grid-connected inverter
model with voltage control is shown in Fig. 4.1.
igrid
i0
iR
u0
Zg
ugrid
ZR
Fig. 4.1 Control system model of voltage control mode
In this figure, u0 is the inverter output voltage, ZR is the load of the inverter, Zg is the
grid side impedance, ugrid is the grid voltage, i0 is the inverter output current, iR is the
load current, igird is the grid side current. In this control mode, the inverter is seen as a
voltage source, so the inverter output voltage is the normal sinusoidal wave. The
connection of inverter and the power grid is equivalent to two voltage sources in parallel,
which makes the grid current be susceptible to the voltage and occurs distortion. In
addition, circulating current may occur in the circuit, which makes the inverter output
voltage is not easy to control, so the grid-connected system is usually not to use voltage
control mode.
The current control mode, refers to the inverter as a current source [29] and the
model is shown in Fig. 4.2. When the inverter is connected to the grid, the grid38
connected inverter performs a high impedance characteristic for the grid, so the inverter
output current is usually not affected by the grid voltage disturbance, which improves
the inverter output current waveform that the inverter output can have the high power
quality.
i0
igrid
iR
i0
Zg
ugrid
ZR
Fig. 4.2 Control system model of current control mode
4.1.2 State-Space Model of PV Inverter with L Filter
As can be seen from the analysis of Section 4.1.1, the grid-connected inverter
adopts the current control mode. In the current control mode, the current loop control
strategy is usually used to control the AC side inductor current, which is very good to
follow the grid voltage. In order to control the DC side of three-phase inverter, usually
add a voltage loop, which is the voltage and current dual-loop control strategy. The
photovoltaic grid-connected system in this thesis adopts two-stage structure, as shown
in Fig. 4.3. The first stage is DC/DC link, which mainly achieves the maximum power
point tracking and DC boost of the PV array and the second stage is DC/AC link using
grid-connected inverter. Because the control of photovoltaic array side voltage is in the
first stage, the DC bus voltage control needs to achieve by the voltage and current dualloop control system in second stage.
By mathematical model analysis of LCL filter three-phase photovoltaic inverter in
chapter Ⅲ, with the voltage and current dual-loop control mentioned above, the system
control structure can be shown as Fig. 4.3. In the part of the current loop, grid-connected
39
current for single-loop control is the basic way to use. Although the single-loop control
has a very fast response speed, the anti-disturbance ability is very weak (such as input
voltage disturbance, load fluctuation, etc.), that the gird current is susceptible to grid
voltage and is difficult to ensure the power quality in the nonlinear load conditions. In
order to increase the stability of the system, the capacitor current feedback of the filter
is increased in the current loop control of the grid current to form the dual-loop control
of the current loop.
i1
iPV
VPV
R1
L1
L2
R2
i2
Vdc
DC / DC
Converter
usa
isa
usb
isb
PV
ic
PWM
C2
uc
PWM
PLL
uabc*
MPPT
abc
abc
αβ
dq
uα
ud
Feedforward
+
Vdc* +
−
PI
o
usc
isc
iq*
id*
+
uq*
PI
uq
ud*
uβ
αβ
abc
αβ
iα
iβ
αβ
dq
dq
iq
id
−
PI
−
Fig. 4.3 Voltage and Current Dual-Loop Control Structure Diagram in Three-Phase System
Capacitor current feedback can be contained in the current loop of grid-connected
current i2k, which can timely suppress the grid current disturbance. Due to the (4.1) and
(4.2), the magnitude of grid current i2k is directly related to the capacitor voltage and
the grid voltage, and the use of capacitor current feedback can stabilize the capacitor
voltage uck, which is also good to the stability of the grid current i2k.
𝑖𝑐𝑘 = 𝐶2 (𝑑𝑢𝑐𝑘 /𝑑𝑡)
(4.1)
𝐿2 (𝑑 𝑖𝑐𝑘 ⁄𝑑𝑡) + 𝑅2 𝑖2𝑘 = 𝑢𝑐𝑘 − 𝑢𝑠𝑘
(4.2)
Fig.4.4 shows the structure of the single-loop control using the grid current for the
current loop. Through the figure, the open loop transfer function GT1(s) can be derived:
40
𝐺𝑇1 (𝑠) =
𝐺𝑖 (𝑠)𝐺1 (𝑠)𝐺2 (𝑠)𝐺3 (𝑠)
1 + 𝐺𝑖 (𝑠)𝐺1 (𝑠)𝐺2 (𝑠)𝐺3 (𝑠) + 𝐺1 (𝑠)𝐺2 (𝑠) + 𝐺2 (𝑠)𝐺3 (𝑠)
(4.3)
us
I2*
Gi(s)
+
G1(s)
+
−
−
i1
+
ic
G2(s)
uc +
−
G3(s)
i2
−
Fig. 4.4 Structure of single-loop control using the grid current
And Fig.4.5 shows the dual-loop control structure using grid current i2k and capacitor
C2 current ick, and the open loop transfer function GT2(s) can be derived by this graph,
where 𝐺1 (s) = 1⁄(𝐿1 s + 𝑅1 )，𝐺2 (s) = 1⁄𝐶2 s，，𝐺3 (s) = 1⁄(𝐿2 s + 𝑅2 )，𝐺c (s) =
𝑘𝑐 𝑘𝑃𝑊𝑀 and 𝐺i (s) = 𝑘𝑃 + 𝑘𝐼 ⁄s.
us
I2*
Gi(s)
+
−
Ic*
+
Gc(s)
−
G1(s)
+
−
i1
+
ic
G2(s)
uc +
−
G3(s)
i2
−
Fig. 4.5 Structure of dual-loop control using the grid current and capacitor C2 current
𝐺𝑇2 (𝑠) =
𝐺𝑖 (𝑠)𝐺𝑐 (𝑠)𝐺1 (𝑠)𝐺2 (𝑠)𝐺3 (𝑠)
1 + 𝐺𝑖 (𝑠)𝐺𝑐 (𝑠)𝐺1 (𝑠)𝐺2 (𝑠)𝐺3 (𝑠) + 𝐺1 (𝑠)𝐺2 (𝑠) + 𝐺2 (𝑠)𝐺3 (𝑠) + 𝐺1 (𝑠)𝐺𝑐 (𝑠)
(4.4)
In addition, in (4.3) and (4.4), Gi(s) can be expressed as a different function form
depending on the selection controller. The specific performance and function form of
the controller Gi(s) will be described in detail in the other sections of this chapter.
Respectively, plot the Bode diagram of the two open loop transfer functions, (4.3)
and (4.4), shown as Fig.4.6. From the Bode diagram, it can be seen that the dual-loop
control is more powerful than the single-loop control, and the system damping can be
increased to suppress the resonant peak. According to the previous analysis and the
comparison of two kinds of control using Bode diagram, it is proved that the dual-loop
control using the grid current i2k and the capacitor C2 current ick is more stable than the
single-loop control using grid current.
41
Fig. 4.6 Bode diagram of current single-loop and dual-loop control
4.2
Grid Control Technology of Inverter Based on PI Controller
PI controller has the advantages of simple algorithm, easy implementation and
high reliability, and the current loop of the three-phase grid-connected inverter is
controlled by the PI controller with the grid current control and capacitance current
control. The transfer function of the PI controller is expressed as:
𝐺𝑃𝐼 (𝑠) = 𝐾𝑃 +
𝐾𝐼
𝑠
(4.5)
The control block diagram of the grid-connected inverter system after adding the
PI controller is shown in Fig.4.5. Ignoring the disturbance of the grid voltage, that is,
ug equal to 0, the closed loop transfer function with PI controller can be derived as (4.6).
𝐺𝐾𝑃𝐼 (𝑠) =
𝐺𝑖 (s)𝐺𝑐 (s)𝐺1 (s)𝐺2 (s)𝐺3 (s)
1 + 𝐺𝑖 (s)𝐺𝑐 (s)𝐺1 (s)𝐺2 (s)𝐺3 (s) + 𝐺1 (s)𝐺2 (s) + 𝐺2 (s)𝐺3 (s) + 𝐺1 (s)𝐺𝑐 (s)
(4.6)
If the system does not use the PI controller, the system closed-loop transfer function
can be shown as (4.7).
𝐺𝐾1 (𝑠) =
𝐺𝑖 (s)𝐺𝑐 (s)𝐺1 (s)𝐺2 (s)𝐺3 (s)
1 + 𝐺𝑖 (s)𝐺𝑐 (s)𝐺1 (s)𝐺2 (s)𝐺3 (s) + 𝐺1 (s)𝐺2 (s) + 𝐺2 (s)𝐺3 (s) + 𝐺1 (s)𝐺𝑐 (s)
(4.7)
As shown in Fig.4.7, comparing the step response diagram before and after adding the
PI controller. In the case of grid voltage disturbance, after adding the PI controller, it
42
can be seen that the speed of response of the system has a more substantial increase.
Fig. 4.7 System step response with PI controller and without PI controller
From the control diagram of Fig.4.5, the output current i2 of grid-connected PV
system is
𝑖2 =
𝐺𝑖 (s)𝐺𝑐 (s)𝐺2 (s)
𝐺𝑐 (s)[𝐺2 (s)𝐺𝑖 (s) + 𝐺3 (s)] + [𝐺1 (s) + 𝐺2 (s)][1 + 𝐺3 (s)]
−
𝑖2∗
𝐺𝑖 (s)𝐺𝑐 (s)𝐺2 (s)
𝐺𝑐 (s)[𝐺2 (s)𝐺𝑖 (s) + 𝐺3 (s)] + [𝐺1 (s) + 𝐺2 (s)][1 + 𝐺3 (s)]
(4.8)
𝑢𝑠
According to (4.5), (4.6) and (4.7), the (4.8) can be rewritten as
𝑖2 =
1
1
𝑖2∗ −
𝑢
1
𝐺𝑐 (s)[𝐺2 (s)𝐺𝑖 (s) + 𝐺3 (s)] + [𝐺1 (s) + 𝐺2 (s)][1 + 𝐺3 (s)] 𝑠
1+
𝐺𝐾1 (𝑠)𝐺𝑃𝐼 (𝑠)
𝐺1 (s) + 𝐺2 (s) + 𝐺𝑐 (s)
(4.9)
When the fundamental frequency of the grid is 𝜔0 , that is, s = 𝑗𝜔0 , then the PI
controller amplitude-frequency characteristic is
𝐾𝐼 2
2
√
𝑀𝑃𝐼 (𝜔0 ) = 𝐾𝑝 + (
)
𝑗𝜔0
(4.10)
Since (4.10) must be a finite value, the first term of (4.7) has the following inequality:
0<
1
1+
1
𝐺𝐾1 (𝑠)𝐺𝑃𝐼 (𝑠)
<1
(4.11)
From (4.9) and (4.11), it can be seen that the output current i2 of the grid-connected
inverter is less than the reference current i*2, which indicates that there is steady state
43
error in the system.
In summary, the use of PI controller to control the grid-connected inverter system
has the advantages of simple algorithm and easy to implement, and can better improve
the performance of grid-connected inverter system. In spite of this, there is still a
steady-state error in the control process, and a small anti-disturbance ability to the grid
voltage disturbance, which causes the PI controller having a highly dependency of
feedforward compensate.
4.3
Grid Control Technology of Inverter Based on PR Controller
PI control has the advantages of simple algorithm, easy to implement and high
reliability, so it is widely used in industrial control. However, the conventional PI
control is difficult to eliminate the steady-state error for the sinusoidal reference current
in the stationary coordinate system. In order to solve the problem generally use ABCdq coordinate transformation, in the synchronous rotation coordinate system, the
sinusoidal reference signal becomes DC reference signal, and PI control can be used to
zero steady-state error. However, from the mathematical model analysis of the threephase photovoltaic inverter with LCL-type filter in Chapter Ⅲ and the control
structure diagram drawn in Fig.4.3, it can be seen that there is a strong coupling between
d-axis and q-axis during the ABC-dq coordinate transformation and requiring complex
feedforward decoupling calculation, which increases the complexity of the system. In
order to solve this problem, using the PR(proportion and resonant) controller instead of
the PI controller, which can track the sinusoidal reference current in the stationary
coordinate system to achieve zero steady-state error [30][31]. PR controller has an
infinite gain at the fundamental frequency and a small gain at the non-fundamental
frequency, so the system can achieve zero steady-state error at the fundamental
frequency, and eliminates the complex calculation of the ABC-dq coordinate
transformation process to simplify the design of the controller.
44
The transfer function of the PR controller is expressed as:
𝐺𝑃𝑅 (𝑠) = 𝐾𝑃 +
𝐾𝑅 𝑠
𝑠 2 + 𝜔0 2
(4.12)
Where 𝜔0 denotes to the fundamental frequency. Fig.4.8 shows the comparison of
amplitude-frequency characteristic between PI and PR controller. Under the same
proportion coefficient KP and the integral coefficient KI is equal to resonant coefficient
KR, the gain of the PI controller is greater than the PR controller before the fundamental
frequency of 60 Hz, and the gain of the PI controller and the PR controller is almost
equal after the fundamental frequency of 60 Hz. Taking into account the frequency of
controlled system is 60Hz and 60Hz above, the design of the PR controller parameters
can be used PI controller parameters instead.
Fig. 4.8 Bode diagram of comparison between PI and PR controller
According to (4.6), (4.7) and (4.12), (4.8) can be rewritten as:
𝑖2 =
1
1+
1
𝐺𝐾1 (𝑠)𝐺𝑃𝑅 (𝑠)
𝑖2∗ −
1
𝑢
𝐺𝑐 (s)[𝐺2 (s)𝐺𝑃𝑅 (s) + 𝐺3 (s)] + [𝐺1 (s) + 𝐺2 (s)][1 + 𝐺3 (s)] 𝑠
𝐺1 (s) + 𝐺2 (s) + 𝐺𝑐 (s)
(4.13)
When the fundamental frequency of the grid is 𝜔0 , that is, s = 𝑗𝜔0 , then the PR
controller amplitude-frequency characteristic is
2
𝐾𝑅 𝜔0
)
−𝜔02 + 𝜔02
𝑀𝑃𝑅 (𝜔0 ) = √𝐾𝑝2 + (
(4.14)
From (4.14), it can be seen that 𝑀𝑃𝑅 (𝜔0 ) approaches infinity, and the first term of
(4.13) has
45
1
1+
1
𝐺𝐾1 (𝑠)𝐺𝑃𝑅 (𝑠)
≅1
(4.15)
Similarly, the second term of (4.13) approaches 0, where I2=I*2, so the PR control can
achieve zero steady-state error and has the ability of anti-disturbance.
If only the grid voltage disturbance is taken into account, the closed-loop
disturbance transfer function 𝜑𝑃𝐼 (𝑠) based on the PI controller can be obtained from
(4.8).
𝜑𝑃𝐼 (𝑠) =
𝑖2
𝐺1 (s) + 𝐺2 (s) + 𝐺𝑐 (s)
=−
𝑢𝑠
𝐺𝑐 (s)[𝐺2 (s)𝐺𝑃𝐼 (s) + 𝐺3 (s)] + [𝐺1 (s) + 𝐺2 (s)][1 + 𝐺3 (s)]
(4.16)
Similarly, the closed-loop disturbance transfer function 𝜑𝑃𝑅 (𝑠) based on the PR
controller is
𝜑𝑃𝑅 (𝑠) =
𝑖2
𝐺1 (s) + 𝐺2 (s) + 𝐺𝑐 (s)
=−
𝑢𝑠
𝐺𝑐 (s)[𝐺2 (s)𝐺𝑃𝑅 (s) + 𝐺3 (s)] + [𝐺1 (s) + 𝐺2 (s)][1 + 𝐺3 (s)]
(4.17)
The closed-loop disturbance transfer function of the PI controller and the PR controller
is presented in the Bode diagram as shown in Fig.4.9. It can be seen that the attenuation
effect on the fundamental frequency disturbance signal of PR controller is remarkable,
and its effect is far more than the PI controller. So the anti-disturbance ability of PR
controller superior than the PI controller.
Fig. 4.9 Closed-loop disturbance transfer function of the PI and the PR controller
46
According to the above analysis, the PR controller is adopted into the current dualloop control to realize the zero steady-state error tracking of the sinusoidal reference
current. The control block diagram is shown in Fig.4.11. In addition, according to
Fig.4.5, the control block diagram with PI controller is shown in Fig.4.10.
us
I2*
GPI(s)
+
Ic*
+
−
Gc(s)
G1(s)
+
i1
+
−
−
ic
G2(s)
−
uc +
G3(s)
i2
−
Fig. 4.10 Current dual-loop control block diagram with PI controller
us
I2*
GPR(s)
+
−
Ic*
+
Gc(s)
−
G1(s)
+
−
i1
+
ic
G2(s)
uc +
−
G3(s)
i2
−
Fig. 4.11 Current dual-loop control block diagram with PR controller
Therefore, according to the control block diagram shown in Fig.4.11, the closed-loop
transfer function can be shown as:
𝐺𝐾𝑃𝑅 (𝑠) =
𝐺𝑃𝑅 (s)𝐺𝑐 (s)𝐺1 (s)𝐺2 (s)𝐺3 (s)
1 + 𝐺𝑃𝑅 (s)𝐺𝑐 (s)𝐺1 (s)𝐺2 (s)𝐺3 (s) + 𝐺1 (s)𝐺2 (s) + 𝐺2 (s)𝐺3 (s) + 𝐺1 (s)𝐺𝑐 (s)
(4.18)
And the closed-loop transfer function of dual-loop current control based with PI
controller is shown in (4.6). Plot the Bode diagram under the current dual-loop control
strategy with PI and PR controller, as shown in Fig.4.12 and Fig.4.13. It can be seen
from the comparison that the dual-loop control strategy with PI controller has a gain of
-0.00432dB at the fundamental frequency that there is a steady state error, and the phase
difference is -7.2 °. With the dual-loop control strategy with PR controller, the gain at
the fundamental frequency is -0.0000119dB, the steady-state error is almost zero, and
the phase difference is only -0.000345 °. It is shown that with PR controller, the zero
steady error can be realized and the phase at the fundamental frequency can be
compensated well.
47
Fig. 4.12 Bode diagram of current dual-loop control with PI controller
Fig. 4.13 Bode diagram of current dual-loop control with PR controller
4.4
Grid Control Technology of Inverter Based on Quasi-PR
Controller
4.4.1 Principle and Analysis of Quasi-PR Controller
Through the analysis in section 4.3, setting a tracking frequency point, the of the
gain of the controller under that frequency is infinite, so that the PR controller can do
48
the zero steady state error tracking to the AC current, while the other frequency point
of the signal is not tracked. However, since the PR control requires a very precise
determination of the frequency point, it can be seen from the Bode diagram of Fig.4.8
that the gain is small at non-fundamental frequency. If the frequency of the grid is
shifted, this will cause the controller not able to track reference current. In order to
overcome the shortcomings of the PR controller with low bandwidth, this thesis adopts
quasi-PR controller, which can keep the advantages of high gain control and increase
the bandwidth and reduce the influence of the frequency shifting.
The transfer function of quasi-PR controller is expressed as:
𝐺𝑄𝑃𝑅 (𝑠) = 𝐾𝑃 +
2𝐾𝑅 𝜔𝑐 𝑠
2
𝑠 + 2𝜔𝑐 𝑠 + 𝜔0 2
(4.19)
Where 𝜔𝑐 denotes to the cutoff frequency. It can be seen from Figure 4.13 that the
quasi-PR controller overcomes the shortcoming of the amplitude gain and the
bandwidth of the PR controller in the vicinity of the fundamental frequency, and the
amplitude gain and bandwidth of the quasi-PR controller can be changed by changing
the parameter of (4.17).
Fig. 4.14 Bode diagram of PR controller and quasi-PR controller
49
In order to discuss the attenuation effect of quasi-PR control on grid voltage
disturbance, the closed-loop system disturbance transfer function 𝜑𝑄𝑃𝑅 (𝑠) is
expressed as:
𝜑𝑄𝑃𝑅 (𝑠) =
𝑖2
𝐺1 (s) + 𝐺2 (s) + 𝐺𝑐 (s)
=−
𝑢𝑠
𝐺𝑐 (s)[𝐺2 (s)𝐺𝑄𝑃𝑅 (s) + 𝐺3 (s)] + [𝐺1 (s) + 𝐺2 (s)][1 + 𝐺3 (s)]
(4.20)
The closed-loop disturbance transfer function of the PI controller and the quasi-PR
controller is presented in the Bode diagram as shown in Fig.4.15. At the fundamental
frequency, the attenuation of the control system to the grid is 89.1dB, and its attenuation
effect is still larger than that of the PI controller. Therefore, the QPR controller is
superior to the PI controller in anti-disturbance ability.
Fig. 4.15 Closed-loop disturbance transfer function of PI and quasi-PR controller
4.4.2 Parameter Design of Quasi-PR Controller
From the transfer function of the quasi-PR controller, we can see that the
parameters that affect its performance are 𝐾𝑃 ,𝐾𝑅 and 𝜔𝑐 . To analyze their respective
effects on the PR controller, two of the parameters are set and the other changes, and
through the Bode diagram to observe the characteristics.
(1) Fixed 𝐾𝑃 and 𝐾𝑅 , changing 𝜔𝑐 :
50
It can be seen from Fig.4.16 that when 𝜔𝑐 becomes larger, the bandwidth of
the system becomes larger and the system resonant gain is constant. That
means 𝜔𝑐 affects the system bandwidth.
(2) Fixed 𝐾𝑃 and 𝜔𝑐 , changing 𝐾𝑅 :
It can be seen from Fig.4.17 that when 𝐾𝑅 becomes larger, both of the
bandwidth and resonant gain of the system becomes larger. That means 𝐾𝑅
affects the system bandwidth and resonant gain.
(3) Fixed 𝐾𝑅 and 𝜔𝑐 , changing 𝐾𝑃 :
It can be seen from Fig.4.18 that when 𝐾𝑃 becomes larger, the system
resonant gain becomes larger and the bandwidth of the system is constant.
That means 𝐾𝑃 affects the system resonant gain.
Fig. 4.16 Bode diagram of the quasi-PR controller with different ωc values
51
Fig. 4.17 Bode diagram of the quasi-PR controller with different KR values
Fig. 4.18 Bode diagram of the quasi-PR controller with different Kp values
Usually, according to the (4.19), we can make 𝜔𝑐 = 𝜉𝜔0 , where 𝜉 is the correlation
coefficient between the cutoff frequency and the resonant frequency, which choses 𝜉 =
0.01 here. (4.19) can be written as
52
𝐺𝑄𝑃𝑅 (𝑠) = 𝐾𝑃 +
2𝐾𝑅 𝜉𝜔0 𝑠
2
𝑠 + 2𝜉𝜔0 𝑠 + 𝜔0 2
(4.21)
Through the factors above and comparing the multiple group of parameters, considering
the peak value of the resonance point and the system gain and bandwidth, the
parameters selected here are𝐾𝑃 = 20, 𝐾𝑅 = 400 and 𝜉 = 0.01.
4.5
Improved quasi-PR harmonic compensation design
Quasi-PR controller designed in previous section can do the zero steady state error
tracking, however in reality, the power grid is not ideal which may contain certain
current harmonics, and in severe cases there may cause imbalance in three-phase power
grid. So it is needed to suppress the harmonic current, to avoid the impact to the threephase current into the grid. The current international standards for power quality include:
IEEE Standard 519-2014 standard, IEC 61000-3 series of standards and IEC 61400021 standard.
4.5.1 Disadvantage of Normal Harmonic Compensation
From the (4.12) and the analysis in previous sections, we can see that the PR
controller can do the zero steady state error tracking at the setting resonant frequency,
so that the gain at the setting frequency tends to infinity, while the gain of other
frequency is very small. If the harmonic frequency is set to the resonant frequency, the
system has a higher gain at this harmonic frequency, which eliminates the harmonics
so that the PR controller can be used as a compensation for a particular harmonics.
According to the (4.12), its transfer function for a particular harmonics compensation
is shown as (4.22), where h is the harmonic order.
2𝐾ℎ 𝑠
𝐺𝑃𝑅𝑆 (𝑠) = ∑ 2
𝑠 + (ℎ𝜔0 )2
(4.22)
ℎ
Since the quasi-PR controller is similar to the PR controller, the transfer function
𝐺𝑄𝑃𝑅 (𝑠) of the quasi-PR harmonic compensation can be derived from equations (4.12)
and (4.22).
53
𝐺𝑄𝑃𝑅 (𝑠) = 𝐻1 (𝑠) + 𝐻2 (𝑠)
2𝜉𝜔0 𝑠
2
𝑠 + 2𝜉𝜔0 𝑠 + 𝜔0 2
(4.24)
2𝜉ℎ 𝜔ℎ 𝑠
2
𝑠 + 2𝜉ℎ 𝜔ℎ 𝑠 + 𝜔ℎ 2
(4.25)
𝐻1 (𝑠) = 𝐾𝑃 + 𝐾𝑅
𝐻2 (𝑠) = ∑ 𝐾𝑅ℎ
ℎ
(4.23)
Where 𝐻1 (𝑠) is s the fundamental frequency controller and 𝐻2 (𝑠) is the harmonic
compensation controller. From the analysis in previous chapter, PR controller can be
directly control in the αβ coordinate system, similarly, quasi-PR controller is the same.
According to the control structure diagram of Fig.4.3, we can draw the normal quasi-PR
harmonic compensation structure diagram, as shown in Fig.4.19.
iα,β* +
H1(s)
−
iα,β
H2(s)
+
+
uα,β*
+
−
uα,β
Fig. 4.19 Normal quasi-PR harmonic compensation structure diagram [32]
The quasi-PR controller with harmonic compensation is added to the system and
the (4.23) is substituted into the control block diagram of Fig.4.10. Fig.4.20 shows the
Bode diagram of current dual-loop control with normal quasi-PR harmonic
compensation.
According to Fig.4.20, the gain and phase angle of each harmonic order of the
closed-loop transfer function can be shown as Table 4.1. As can be seen from Fig.4.20
and Table 4.1, the gain at each harmonic is close to 0 dB. If the grid exists harmonics,
such as the 5th order and 7th order harmonics, then the reference current calculated by
the basis of the (4.6) will also exist 5th order and 7th order harmonics. In here, the normal
harmonic compensation in the vicinity of the harmonic frequency can completely track
the input current harmonics, once the reference current exists harmonics, the grid
current i2 must also exist harmonics, which does not match with our target to reduce the
harmonics go into the grid. It can be seen that there is harmonics in the reference current
54
when the harmonics exist in the grid current and the system cannot suppress the current
harmonics. Therefore, this thesis studies an improved harmonic compensation method,
which can suppress the current harmonic in the case of harmonics in the grid [32][33].
Fig. 4.20 Bode diagram of current dual-loop control with normal quasi-PR harmonic
compensation
Table 4.1 Gain and phase angle of each harmonic in Fig.4.20
Harmonic order
Magnitude(dB)
Phase(°)
Fundamental
-0.000158
-0.0056
3rd
0.000398
-0.0119
5th
0.00482
-0.0283
7th
0.00189
-0.0331
4.5.2 Improved Quasi-PR Harmonic Compensation
The current control diagram of the improved harmonic compensation method is as
follows:
55
iα,β* +
H3(s)
−
+
+
uα,β*
+
−
H4(s)
iα,β
uα,β
Fig. 4.21 Improved quasi-PR harmonic compensation structure diagram [32]
Where 𝐻3 (𝑠) is s the fundamental frequency controller and 𝐻4 (𝑠) is the harmonic
compensation controller. The transfer function 𝐺𝑄𝑃𝑅_𝑛𝑒𝑤 (𝑠) of the improved quasi-PR
harmonic compensation is
𝐺𝑄𝑃𝑅_𝑛𝑒𝑤 (𝑠) = 𝐻3 (𝑠) + 𝐻4 (𝑠)
(4.24)
2𝜉𝜔0 𝑠
2
𝑠 + 2𝜉𝜔0 𝑠 + 𝜔0 2
(4.25)
𝐻3 (𝑠) = 𝐾𝑅
𝐻4 (𝑠) = 𝐾𝑃 + ∑ 𝐾𝑅ℎ
ℎ
2𝜉ℎ 𝜔ℎ 𝑠
2
𝑠 + 2𝜉ℎ 𝜔ℎ 𝑠 + 𝜔ℎ 2
(4.26)
It can be seen from Fig.4.19 and Fig.4.21 that in the normal harmonic compensation,
∗
the error signal 𝑖𝛼,𝛽
− 𝑖𝛼,𝛽 pass through the fundamental frequency controller and
harmonic compensation controller; while in the improved harmonic compensation
∗
method, the error signal 𝑖𝛼,𝛽
− 𝑖𝛼,𝛽 only pass through the fundamental frequency
controller, and do the harmonic compensation for the feedback signal 𝑖𝛼,𝛽 .
Improved quasi-PR harmonic compensation is added to the system to obtain the
Bode diagram of its closed-loop transfer function, as shown in Fig.4.22. The most
striking difference between Fig.4.22 and Fig.4.20 is that, in Fig.4.22, there is 0 dB of
gain near the fundamental frequency and at the other frequencies is negative, especially
in the setting harmonics compensation, 3rd , 5th and 7th order, gain are all less than 40dB. So the improved quasi-PR harmonic compensation control system at the
fundamental frequency can basically track the reference current, and suppress the
harmonics in other frequency. Therefore, this improved harmonic compensation
method can effectively suppress the harmonic, reduce the THD of the grid current and
56
improve the power quality.
Fig. 4.22 Bode diagram of current dual-loop control with improved quasi-PR
harmonic compensation
In this section, for the harmonics that may exist in the grid current, the normal
method cannot effectively suppress the harmonic, and an improved harmonic
compensation method is designed. By comparing Bode diagram of the closed-loop
transfer function, it is concluded that the harmonic suppression effect of the improved
method is more significant than the normal harmonic compensation method.
57
V. Case Study
Through the modeling and analysis of the chapters above, this chapter is modeled
and simulated by Matlab/ Simulink. First, each model of the whole system is introduced.
Here, each type of control strategy which introduced in chapter 4 is simulated under the
case of ideal power grid and nonideal power grid.
5.1
Three-Phase Grid-Connected PV System Model
Model of three-phase grid-connected photovoltaic system is shown in Fig.5.1.
According to the analysis of previous chapters, the system here is two-stage gridconnected PV System. The former stage is to achieve the maximum power point
tracking function, and the second stage of DC / AC inverter is to achieve the stability
of DC voltage and the grid-connected control strategy.
Fig. 5.1 Two-stage grid-connected PV System model
The PV Array model is created using the equation in Chapter 2, and the PV array
parameters used here are shown in Table 5.1. The MPPT function is implemented in the
first stage, and the perturbation and observation method of maximum power tracking is
used here which is based on the concept of the flow chart of perturbation and
observation method in Fig. 2.7. The inverter is full bridge topology with six IGBT, and
the inverter output go through the LCL filter and the transformer, which’s voltage level
is 11.4kV, and then go into the ideal power grid. The green part of the figure is the focus
of this thesis, namely PI, PR and quasi-PR control technology, the internal structure is
58
shown in Fig.5.2. The Control strategy block is included the Coordinate Transformation
function, Voltage loop and the controller blocks, which the current loop is inside of
them. As described in the analysis of chapter 3 of the mathematical model, the use of
PI controller needs to implement in dq coordinate and has to do the feedforward
decoupling so that the decoupling function is inside the contains the PI controller model.
Finally, the parameters used for the system simulation are shown in Table 5.2.
Table 5.1 Parameters of PV array
Array data
Parallel strings
66
Series-connected per string
5
Module data
Maximum Power (W)
30.43
Open circuit voltage Voc (V)
20.31
Short circuit current Isc (A)
1.88
Voltage at maximum power Vmp (V)
17.29
Current at maximum power Imp (A)
1.76
Temperature coefficient of Voc (%/℃)
-0.2727
Temperature coefficient of Isc (%/℃)
0.06175
59
Fig. 5.2 Control strategy block
Table 5.2 Parameters of the system
Grid-connected inverter
DC voltage Vdc (V)
500
Grid phase-to-phase voltage Vrms (kV)
11.4
Grid frequency (Hz)
60
Output power (kW)
10
LCL filter
Inductance L1 (mH)
4.58
Internal resistance R1 in inductance L1 (mΩ)
1.885
Inductance L2 (mH)
0.92
Internal resistance R2 in inductance L1 (mΩ)
1.885
Capacitance C2 (μF)
5
Controller
Voltage loop KP
7
Voltage loop KI
800
Current loop KP
0.3
Current loop KI
20
Current loop KR
20
correlation coefficient 𝜉
0.01
60
5.2
Simulation under the Ideal Power Grid
5.2.1 Simulation with PI Controller
While the inverter is operating in the maximum output power, the current
waveform of inverter side current i1a is shown as Fig. 5.3 and the grid side current i2a is
shown as Fig. 5.4. Fig. 5.5 and Fig. 5.6 shows the dc voltage of the inverter side can
well track the reference signal, in the voltage loop. But in current loop, grid-connected
current has steady-state error with the reference current, just like the analysis in chapter
3. Fig.5.7 shows the phase current and phase voltage that connected to the grid.
Fig. 5.3 Current i1a of inverter side with PI controller
Fig. 5.4 Current i2a of grid-connected side with PI controller
61
Fig. 5.5 Grid-connected current and the reference current with PI controller
Fig. 5.6 Dc voltage of the inverter side and the reference voltage with PI controller
Fig. 5.7 Phase current and phase voltage of the grid with PI controller
Through the simulation and the figures above, it can see that the PI controller can
well control the grid-connected inverter with feedforward decoupling in the dq
coordinate system.
5.2.2 Simulation with PR Controller
While the inverter is operating in the maximum output power, the current
62
waveform of inverter side current i1a is shown as Fig.5.8 and the grid side current i2a is
shown as Fig.5.9. Through the FFT analysis, Fig.5.5 shows the frequency spectrum of
current i1a and current i2a. The THDi value of Current i2a is only 1.02% which is smaller
than the Current i1a and is also comply with the standard. That means the parameter
design of LCL filter in chapter 3 is well and effective.
Fig. 5.8 Current i1a of inverter side with PR controller
Fig. 5.9 Current i2a of grid-connected side with PR controller
(a) Current i1a of inverter side
(b) Current i2a of grid-connected side
Fig. 5.10 FFT analysis frequency spectrum
63
Fig.5.11 and Fig.5.12 shows the grid-connected current and the dc voltage of the
inverter side can well track the reference signal, respectively, in the voltage loop and
current loop. Fig.5.13 shows the phase current and phase voltage that connected to the
grid. From Fig.5.11, it can see that the grid-connected current 𝑖𝛼 almost overlaps on
reference current 𝑖𝛼∗ . In summary, this section verify the PI and PR control analysis of
the chapter 4 of this thesis.
Fig. 5.11 Grid-connected current and the reference current with PR controller
Fig. 5.12 Dc voltage of the inverter side and the reference voltage with PR controller
64
Fig. 5.13 Phase current and phase voltage of the grid with PR controller
5.3
Simulation under the Nonideal Power Grid
5.3.1 Simulation in grid-connected PV System Model
The power grid is not always ideal in the reality and it may contain certain
harmonic. Through the two harmonic compensation methods discussed in Chapter 4,
normal harmonic compensation and improved harmonic compensation, simulations are
performed. In order to compensation usefulness clearly, we set up a nonideal power
grid which contains 3rd and 5th order harmonics and the simulation shows as follows.
Fig. 5.14 - Fig. 5.16 show the grid connected current i2a without harmonic
compensation, with normal compensation, and with the improved harmonic
compensation, respectively. The FFT analysis of each method is shown in Fig. 5.17 and
results are summarized in Table 5.3.
Fig. 5.14 Grid-connected current i2a without harmonic compensation
65
Fig. 5.15 Grid-connected current i2a with normal harmonic compensation
Fig. 5.16 Grid-connected current i2a with improved harmonic compensation
(a) Without harmonic compensation
(b) Normal harmonic compensation
(c) Improved harmonic compensation
Fig. 5.17 FFT analysis of the grid-connected current i2a
66
Table 5.3 Summary FFT analysis of the grid-connected current i2a
Vsa
i2a
Without harmonic compensation
4.24%
7.90%
Normal harmonic compensation
4.24%
5.05%
Improved harmonic compensation
4.24%
0.76%
By observing the above results, the total harmonic voltage distortion is 4.24%, due
to the harmonic distortion in the grid voltage. Since the voltage distortion of the grid
will cause the reference current 𝑖𝛼∗ and 𝑖𝛽∗ with harmonics and if harmonic
compensation is not performed, the grid current of the system will also appear the same
order of current distortion. Compared with the normal harmonic compensation, the
improved harmonic compensation shows superior mitigation of harmonic distortions.
5.3.2 Overview of INER Microgrid Model
In order to see the performance of the harmonic compensation, we use the
microgrid of Institute of Nuclear Energy Research (INER) to simulate the control
strategies [34]. The microgrid of INER is the first autonomous microgrid in Taiwan
with 3-phase 4-wire configuration and the one-line diagram is shown in Fig. 5.18. The
static switch is used to interconnect between Zone 1, Zone 2 and Zone 3.
The distributed energy resources in the microgrid are photovoltaics, wind turbines
and micro-turbines. To show the performance of the control strategy in photovoltaic
system, we consider only connecting Zone 1 to the utility grid. The structure of Zone 1
is shown in Fig. 5.19 and the Matlab/Simulink model is shown in Fig. 5.20. System
parameters for simulation in the Zone 1 area are shown in Table 5.4.
67
12×5 kW 150 kW 65 kW
HCPV WT
µT
2×5 kW 25 kW 2×2 kW
HCPV WT
WT
inverter
inverter
380 V
380 V
400 kVA
380 V
25 m
69 kV TPC
GCB
SCC 1714.2 MVA
10 MVA
X/R 8.02
7.26%
Cable
2.5 km
B0
500 kVA
3.85%
6×50 kVAr
380 V
25 m
25 m
to Zone 2
L
to Zone 2 and Zone 3L
Zone 2
30 kW 30 kW
11.4 kV INER power system
380 V microgrid
NO
ACB
NC
B1
ss
150 kVA
NC
NO
B2
L
Zone 3
30 kW
380 V
U
NC
NFB
NC
4%
4%
150 kVA
150 kVA
L
DMSC
: Digital power meter, protective relay, and magnetic contactor
: No-fuse breaker (NFB)
NO : Normal open
NC : Normal close
B3
B4
100 kVA
208V
L2
µ T 480V L1
30 kW 30 kW
65 kW
B5
380 V
inverter 12×3.6 kVA
HCPV
21×1.5 kW
Zone 1
Fig. 5.18 The structure of the zone 1 to zone 3 microgrid systems
PCC
DG1
STS
LC Filter
Grid-forming
Load 1
Load 2
DG 2
LC Filter
Grid-following
DG 3
LC Filter
Grid-following
Fig. 5.19 The architecture of the zone 1 area
68
Grid
Fig. 5.20 Matlab/Simulink model of the zone 1 area
Table 5.4 System parameters for simulation in the zone 1 area
Zone 1
Battery
Solar
Microturbine
Controller
Parameters
Values
Inverter switching frequency
10 kHz
Inverter filter inductance
Inverter filter capacitance
2 mH
68.5  F
DC-link Voltage
700 V
Maximum active power
60 kW
Maximum reactive power
100 kVAr
Inverter switching frequency
10 kHz
Inverter filter inductance
Inverter filter capacitance
2 mH
68.5  F
DC-link Voltage
700 V
Maximum active power
51.5 kW
Inverter switching frequency
10 kHz
Inverter filter inductance
Inverter filter capacitance
2 mH
68.5  F
DC-link Voltage
700 V
Maximum active power
31.5 kW
Kp_PQ
5
Ki_PQ
500
Kp_VF
6
Ki_VF
30
69
5.3.3 Simulation Result in INER Microgrid Model
In the study, the nonideal power grid contains the 3rd and 5th order harmonic
voltages. Without harmonic compensation, the grid-connected current at PCC is shown
in Fig 5.21 and three-phase FFT analysis is shown in Fig 5.22. After using the control
method with harmonic compensation, the grid-connected current at PCC is shown in
Fig 5.23 and three-phase FFT analysis is shown in Fig 5.24.
Fig. 5.21 Grid-connected current at PCC without harmonic compensation
(a) Ia
(b) Ib
(c) Ic
Fig. 5.22 FFT analysis of the grid-connected current at PCC without harmonic
compensation
70
Fig. 5.23 Grid-connected current at PCC with harmonic compensation
(a) Ia
(b) Ib
(c) Ic
Fig. 5.24 FFT analysis of the grid-connected current at PCC with harmonic
compensation
The FFT analyses in Fig. 5.22 and Fig. 5.24 are summarized in Table 5.5. The results
show that the control strategy with harmonic compensation can prevent the reference
current containing harmonics while the voltage distortion occurs at the grid side.
Through the current distortion limits in IEEE Standard 519-2014, the harmonic
distortion caused by a single consumer should be limited to an acceptable level at any
point in the system, which is shown in Table 5.6 [35]. It is obviously seen that current
distortion in all three phases exceed the limits without harmonic compensation and the
71
current distortion in all three phases can be reduced to much smaller than the limits with
harmonic compensation.
Table 5.5 Summary FFT analysis of the grid-connected current at PCC
THDIa
THDIb
THDIc
Without harmonic compensation
4.48%
4.52%
4.54%
With harmonic compensation
1.30%
1.27%
1.21%
Current distortion limit
5.0%
5.0%
5.0%
Table 5.6 Current distortion limits for systems rated 120 V through 69 kV [35]
Maximum harmonic current distortion in percent of IL
Individual harmonic order (odd harmonics)
11≦h<17 17≦h<23
23≦h<35
35≦h≦50
THD
1.5
0.6
0.3
5.0
3.5
2.5
1.0
0.5
8.0
10.0
4.5
4.0
1.5
0.7
12.0
100 < 1000
12.0
5.5
5.0
2.0
1.0
15.0
>1000
15.0
7.0
6.0
2.5
1.4
20.0
ISC/IL
3≦h<11
< 20
4.0
2.0
20 < 50
7.0
50 < 100
72
VI. Conclusion and Future Work
6.1
Conclusion
In this thesis, three-phase grid-connected PV system model has been built in
Matlab/ Simulink. The maximum power point tracking function is implemented in the
first stage of the system, and the perturbation and observation method of maximum
power tracking is used in here. The second stage of DC / AC inverter is to achieve the
stability of DC voltage and the grid-connected control strategy by using voltage loop
and current loop, respectively. In order to analyze and implement each control strategy,
the mathematical model of the grid-connected inverter with several types of filters has
been built in chapter 3.
Through the mathematical analysis, the performance of three kinds of controllers
has been discussed in chapter 4. PI control has the advantages of simple algorithm, easy
to implement and high reliability, so it is widely used in industrial control. However, it
is difficult to eliminate the steady-state error for the sinusoidal reference current in the
stationary coordinate system. Compare with the PI controller, PR controller can do the
zero steady state error tracking, but it requires a very precise determination of the
frequency point. The quasi-PR controller has been proposed, which can keep the
advantages of high gain control and increase the bandwidth and reduce the influence of
the frequency shifting.
At the end of this thesis, three-phase grid-connected PV system model with several
control strategies has been built. Under the ideal power grid, the simulation result shows
that the PI controller cannot do the zero steady-state error, same as the theoretical
analysis in previous chapter. In the control strategy of the grid-connected inverter, PR
controller shows the well performance in both theoretical analysis. Through the FFT
analysis, THDi of the current between inverter side and grid-connected side can be well
73
reduced, that means the parameter design of inverter LCL filter in chapter 3 is effective.
To compare the normal harmonic compensation and improved harmonic compensation
of the quasi-PR controller, the system is simulated under the nonideal power grid. The
result shows that the improved harmonic compensation of the quasi-PR controller can
reduce the current harmonics while controlling the grid-connected inverter.
6.2
Future Work
The future works are recommended as follows.
1.
There are still other control methods for the controller, such as fuzzy theory,
particle swarm optimization, and artificial neural network. In order to get the better
result in simulation, these methods are needed to research and model.
2.
To analyze the effective and the interaction of the passive filter between the power
grid and the distributed generation system, the inverter side can be seen as a current
source with the impedance corresponding to each harmonic order. The
corresponding impedance to each harmonic order is an important index to design
a well perform pass filter and needs to be find out
74
REFERENCES
[1] Teske, S., et al. "Renewables Global Futures Report: Great Debates Towards 100%
Renewable Energy," Global Status Report RENEWABLE ENERGY POLICY
NETWORK FOR THE 21st CENTURY (REN21), 2017.
[2] James, T., A. Goodrich, M. Woodhouse, R. Margolis, and S. Ong, “BuildingIntegrated Photovoltaics (BIPV) in the residential sector: an analysis of installed
rooftop system prices,” National Renewable Energy Laboratory. NREL/TP-6A2053103, 2011.
[3] M. Hoeven., et al. “Tracking Clean Energy Progress 2015,” IEA, 2015.
[4] PVPS, IEA. "Photovoltaic Power Systems Programme: Annual Report 2006," IEA
PVPS, 2006.
[5] Adib, Rana, et al. "Renewables 2016 Global Status Report," Global Status Report
RENEWABLE ENERGY POLICY NETWORK FOR THE 21st CENTURY (REN21),
2016.
[6] [Online], SHORT-TERM ENERGY OUTLOOK, January, 2016. Available:
https://www.eia.gov/outlooks/steo/report/
[7] T. Schwencke, C, Bantle, “BDEW-Strompreisanalyse,” BDEW, GERMANY,
MAY, 2016.
[8] [Online], Facts Global Energy (FGE), January 2015 Available:
https://www.fgenergy.com/services/product-types/research-and-analysis/chinagas-service/china-gas-monthly.aspx
[9] 2016 年能源產業技術白皮書，經濟部，中華民國，2016.
75
[10] [Online], Taipei Power Company, 2017, Available:
http://www.taipower.com.tw/content/new_info/new_info-b36-2.aspx
[11] 前瞻基礎建設計畫（核定本），行政院，中華民國，2017
[12] J.H.R.Enslin, M.S.Wolf, D.B.Snyman, W.Swiegers, “Integrated Photovoltaic
Maximum Power Point Tracking Converter,” IEEE Transactions on Industrial
Electronics, Vol.44, No.6, December 1997
[13] P. Huynh and B. H. Cho, “Design and analysis of microprocessor controlled peak
power tracking system,” in Proc. 27th IECEC, 1992,
[14] O. Wasynczuk, “Dynamic behavior of a class of photovoltaic power systems,”
IEEE Trans. Power App. Syst., vol. PAS-102, Sept. 1983
[15] C. R. Sullivan and M. J. Powers, “A high-efficiency maximum power point tracker
for photovolatic array in a solar-powered race vehicle,” in Proc. IEEE PESC, 1993
[16] Wu T F, Chang C H, Chen Y K. “A fuzzy-logic-controlled single-stage converter
for PV-powered lighting system application,” IEEE Trans. On Industrial
Electronics. 2000
[17] Mekhilef, S., and N. A. Rahim. "Implementation of grid-connected photovoltaic
system with power factor control and islanding detection." Power Electronics
Specialists Conference, 2004. PESC 04. 2004 IEEE 35th Annual. Vol. 2. IEEE,
2004.
[18] Heskes, P. J. M., and J. L. Duarte. "Harmonic reduction as ancillary service by
inverters for distributed energy resources (DER) in electricity distribution
networks," Proceedings of the 19th International Conference on Electricity
Distribution (CIRED 2007). Vol. 21. 2007
76
[19] Nishida K, Ahmed T, Nakaoka M. “A novel finite-time settling control algorithm
designed for grid connected three-phase inverter with an LCL type filter[J],”
IEEE Trans. on Industry Applications, 2014, 50(3): 2005-2020.
[20] Pandiarajan, Natarajan, and Ranganath Muthu. "Mathematical modeling of
photovoltaic module with Simulink," Electrical Energy Systems (ICEES), 2011 1st
International Conference on. IEEE, 2011.
[21] [Online], Mitsubishi Electric photovoltaic module AE series, 2017, Available:
http://www.mitsubishi-electric.co.nz/materials/solar/brochures/PV-AE125MF5N.pdf
[22] Kim, Kwang-Seob, Byung-Ki Kwon, and Chang-Ho Choi. "A novel control
algorithm of a three-phase PWM inverter with output LC filter," Power
Electronics and Applications, 2007 European Conference on. IEEE, 2007.
[23] Serpa, Leonardo Augusto, et al. "A modified direct power control strategy
allowing the connection of three-phase inverters to the grid through LCL
filters," IEEE Transactions on Industry Applications 43.5 (2007): 1388-1400.
[24] Mariethoz, S., A. G. Beccuti, and M. Morari. "Analysis and optimal current control
of a voltage source inverter connected to the grid through an LCL filter," Power
Electronics Specialists Conference, 2008. PESC 2008. IEEE. IEEE, 2008.
[25] Twining, Erika, and Donald Grahame Holmes. "Grid current regulation of a threephase voltage source inverter with an LCL input filter," IEEE Transactions on
Power Electronics 18.3 (2003): 888-895.
[26] Liserre, Marco, Frede Blaabjerg, and Steffan Hansen. "Design and control of an
LCL-filter-based three-phase active rectifier," IEEE Transactions on industry
applications 41.5 (2005): 1281-1291.
77
[27] Teodorescu, Remus, et al. "A new control structure for grid-connected LCL PV
inverters
with
zero
steady-state
error
and
selective
harmonic
compensation," Applied Power Electronics Conference and Exposition, 2004.
APEC'04. Nineteenth Annual IEEE. Vol. 1. IEEE, 2004.
[28] Liu, Fei, et al. "Design and research on parameter of LCL filter in three-phase gridconnected inverter," Power Electronics and Motion Control Conference, 2009.
IPEMC'09. IEEE 6th International. IEEE, 2009.
[29] Kjæ r, Søren Bæ khøj. “Design and control of an inverter for photovoltaic
applications,” Institute of Energy Technology, Aalborg University, 2005.
[30] Zmood, Daniel Nahum, Donald Grahame Holmes, and Gerwich H. Bode.
"Frequency-domain analysis of three-phase linear current regulators," IEEE
Transactions on Industry Applications 37.2 (2001): 601-610.
[31] Yuan, Xiaoming, et al. "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
transactions on industry applications 38.2 (2002): 523-532.
[32] Castilla, Miguel, et al. "Reduction of current harmonic distortion in three-phase
grid-connected photovoltaic inverters via resonant current control," IEEE
transactions on industrial electronics 60.4 (2013): 1464-1472.
[33] Blaabjerg, Frede, et al. "Overview of control and grid synchronization for
distributed power generation systems," IEEE Transactions on industrial
electronics 53.5 (2006): 1398-1409.
[34] C. Y. Yu, “Modeling, Control Strategies, and Real-time Simulation for an AC
78
Microgrid,” Master dissertation, National Chung Cheng University, Taiwan, 2015.
[35] IEEE Recommended Practice and Requirements for Harmonic Control in Electric
Power Systems, IEEE Std 519-2014,
79