Grid Inverter

저작자표시-비영리-변경금지 2.0 대한민국 이용자는 아래의 조건을 따르는 경우에 한하여 자유롭게 l

이 저작물을 복제, 배포, 전송, 전시, 공연 및 방송할 수 있습니다.

다음과 같은 조건을 따라야 합니다:

저작자표시. 귀하는 원저작자를 표시하여야 합니다.

비영리. 귀하는 이 저작물을 영리 목적으로 이용할 수 없습니다.

변경금지. 귀하는 이 저작물을 개작, 변형 또는 가공할 수 없습니다.

l l

귀하는, 이 저작물의 재이용이나 배포의 경우, 이 저작물에 적용된 이용허락조건 을 명확하게 나타내어야 합니다. 저작권자로부터 별도의 허가를 받으면 이러한 조건들은 적용되지 않습니다.

저작권법에 따른 이용자의 권리는 위의 내용에 의하여 영향을 받지 않습니다. 이것은 이용허락규약(Legal Code)을 이해하기 쉽게 요약한 것입니다. Disclaimer

A Dissertation for the Degree of Doctor of Philosophy

Power Quality Improvement of Single-Phase Grid-Connected Photovoltaic Inverter

Department of Information and Communications Engineering Graduate School Chungnam National University

By

Trung-Kien Vu

Advisor

Se-Jin Seong

February 2011

Power Quality Improvement of Single-Phase Grid-Connected Photovoltaic Inverter

Advisor

Se-Jin Seong

Submitted to the Graduate School in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy February 2011 Department of Information and Communications Engineering Graduate School

Chungnam National University

By

Trung-Kien Vu

To Approve the Submitted Dissertation for the Degree of Doctor of Philosophy

By Trung-Kien Vu Power Quality Improvement of Single-Phase Grid-Connected Photovoltaic Inverter

February 2011 Committee Chair

차한주 Prof. Dr. Hanju Cha Chungnam National University

Committee

황인호 Prof. Dr. In-ho Hwang Chungbuk Provincial College

Committee

성세진 Prof. Dr. Se-Jin Seong Chungnam National University

Committee

성태경 Prof. Dr. Tae-Kyung Sung Chungnam National University

Committee

김상진 Dr. Sang-Jin Kim President of Junma Engineering Co., Ltd

Graduate School Chungnam National University

To My Family

ABSTRACT* Power Quality Improvement of Single-Phase Grid-Connected Photovoltaic Inverter Trung-Kien Vu Department of Information and Communications Engineering Graduate School, Chungnam National University Daejeon, Korea

(Supervised by Professor Se-Jin Seong)

The economical and environmental impacts of fossil fuels have forced society to investigate sustainable solutions. The interest has focused on the renewable energy sources since the green and clean benefits. Consequently, investments in research and development in the field of power electronics have increased proportionally, especially in high voltage and high power grid-connected systems. * A dissertation submitted to the committee of Graduate School, Chungnam National University in a partial fulfillment of the requirements for the degree of Doctor of Philosophy conferred in February 2011. -i-

The distributed power generation (DG) systems are becoming more common as the need for electric power increases because of taking advantage of using different energy sources such as wind and solar. A few examples are hybrid cars, solar houses or hospitals in remote areas where providing clean, efficient and reliable electric power is critical to the loads. In such systems, the power is distributed from the source side to the load side via power electronic converters in the system. At low and medium power applications, the task is often left to single-phase inverters where they are the only interface between sources connected to dc bus and loads connected to an ac bus. This dissertation investigates the power quality improvements to properly regulate the power flow between renewable source and the utility network. The control method for single-phase inverters used in low and medium power DG systems is based on (and also takes the advantage of) the well-known d-q transformation (which is mostly employed for three-phase

converters’

analysis

and

control

design).

The

transformation requires at least two independent phases for each state variable in the system; thus a second phase must be created. This virtual-phase can be done by DSP implementation, hence there is no need for additional hardware in the system, making it more attractive and cost effective method. The Proportional-Resonant (PR) controller based current control scheme, compared with conventional Proportional-Integral (PI) controller, not only provides a superior transient response but also provides a zero steady-state error as well as a high disturbance rejection and a low output current THD under grid-tie mode operation. The entire controller can be implemented in a DSP digital control board

- ii -

which is becoming more common in power electronics converters within the past decade. Special attention is given to systems which demand a third-order LCL-filter as interface between inverter and grid. This filter configuration is widely employed in high power systems, in which the switching frequency is typically limited by the switching devices. The LCL-filter has the ability to reduce the level of harmonic distortion with less inductance, compared with the first-order L filter. On the other hand, it introduces undesirable characteristics, such as resonance, that must be compensated by the controller. Another issues related to the switching devices such as IGBT, MOSFET and others have very high switching frequency above tens of kilohertz. A blanktime is needed to avoid the conduction overlap of two switching devices in the same leg. This blank-time causes the phase error, output voltage distortions and fundamental voltage drop, which degrade the control performance and hence, may cause the power loss during generation process. Comparative analysis and design procedures of conventional PI and PR controller for current control technique have been presented. Furthermore, an output low-pass LCL-filter design procedure and a dead-time compensation method for a 3kW single-phase grid-connected full-bridge PV VSI have also been introduced in this study with the final results are implemented in a DSP TMS320F2812 based digital controller board.

- iii -

- iv -

Table of Contents

Abstract Table of Contents List of Figures List of Tables Nomenclatures

i v viii xiv xv

Chapter 1 Introduction 1.1 Introduction 1.2 Objectives and Outline 1.3 Single-phase grid-connected PV inverter topologies 1.4 Control methods for single-phase VSI 1.4.1 Single voltage loop control 1.4.2 Multi-loop control 1.4.3 Predictive control 1.4.4 Deadbeat control 1.4.5 Sliding mode control 1.4.3 Conclusion Current Controller for Single-Phase Grid-Connected PV Inverter 2.1 Introduction 2.2 PI controller analysis 2.2.1 PI controller gains 2.2.2 Digital implementation of PI controller

1 1 3 4 6 6 7 7 9 10 10

Chapter 2

-v-

11 11 14 14 20

2.2.3 Non-ideal PI controller 2.3 PR controller analysis 2.3.1 Ideal PR controller 2.3.2 Non-ideal PR controller 2.3.3 PR controller gains 2.3.4 Digital implementation of PR controller 2.4 Single-phase grid-connected PV inverter system 2.4.1 Current control scheme 2.4.2 Digital phase-locked loop (DPLL) 2.5 Simulation results 2.6 Experimental results 2.7 Conclusions Low-Pass LCL Output Filter for Single-Phase Grid-Connected PV Inverter 3.1 Introduction 3.2 Low-pass output LCL-filter analysis and design 3.2.1 Ripple current analysis and filter induction calculation 3.2.2 Current attenuation ratio 3.2.3 Capacitance calculation 3.2.4 Inductance ratio 3.2.5 Damping resistance calculation 3.2.6 LCL-filter design procedure summary 3.3 Comparative analysis of filter topologies for singlephase grid-connected PV VSI 3.3.1 LCL-filter parameters calculation 3.3.2 Comparative analysis of filter topologies 3.4 Simulation results 3.5 Experimental results 3.6 Conclusions

22 22 22 24 25 30 32 32 39 42 45 51

Chapter 3

Chapter 4

Dead-Time Compensator for Single-Phase Grid-Connected PV Inverter

- vi -

52 52 59 62 67 71 72 73 74 75 75 78 98 99 100

101

4.1 Introduction 4.2 Dead-time effect analysis 4.3 Conventional DTCV 4.4 Distorted voltage and current caused by dead-time 4.5 Adaptive harmonics extraction filter 4. 5.1Single-frequency harmonic extraction filter 4.5.2 Combination of SHEF and regular plant control part 4.5.3 Multi single-frequency harmonic extraction filters 4.6 Dead-Time Compensator For Single-phase GridConnected PV Inverter Based on MSHEF 4.7 Inverse plant transfer function analysis 4.8 Simulation results 4.8.1 No dead-time 4.8.2 3ms dead-time without compensation 4.8.3 3ms dead-time with 3rd harmonic compensation 4.9 Experimental results 4.10 Conclusions Chapter 5

Summary

101 103 104 109 112 112 116 117 118 121 126 127 127 127 140 142 143

References

145

Appendix A Simulation tools Appendix B PR controller implementation (PSIM) Appendix C Dead-time compensation implementation (PSIM)

156 158 164

국문 초록

173

Acknowledgements

177

Vita

181

- vii -

List of Figures

1.1

Topology of single-phase PV system (a) two-state (b) single-state

5

1.2

Single-loop control for single-phase PWM inverter

8

1.3

Multi-loop control for single-phase PWM inverter

8

2.1

An equivalent block diagram of a simple closed-loop control system using PI controller

16

2.2

Effect of Kp variation on PI controller (Ki = 1)

16

2.3

Effect of Ki variation on PI controller (Kp = 1)

16

2.4

Frequency response of ideal and non-ideal PI controller (Kp=10, Ki=100, wc=10rad/s)

21

2.5

Frequency response of ideal and non-ideal PR controller (Kp=10, Ki=100, wc=10rad/s, w0 =377rad/s)

21

2.6

Frequency response of non-ideal PR controller when Ki changes (a) Kp=0, wc=1rad/s, wc=377rad/s (b) Kp=1, wc=1rad/s, w0=377rad/s

27

2.7

Frequency response of non-ideal PR controller when wc changes (a) Kp=0, Ki=1, wc=377rad/s (b) Kp=1, Ki=1, w0 =377rad/s

28

- viii -

2.8

Frequency response of non-ideal PR controller when Kp changes, Ki=1, wc=1rad/s, w0=377rad/s

29

2.9

Single-phase grid-connected PV inverter control system (a) Current control circuit structure (b) Equivalent current control block diagram

29

2.10

Single-phase current control structure (a) using PI controller (b) using PR controller

31

2.11

Frequency response of closed-loop current control using PI controller (Kp=10, Ki=100) and PR controller (Kp =15, Ki =200, wc = 15rad/s, w0 = 377rad/s)

36

2.12

Root locus of closed-loop current control system (a) PI controller (b) PR controller

37

2.13

Nyquist diagram of open-loop current control system (a) PI controller (b) PR controller

38

2.14

Digital phase-locked loop block diagram

40

2.15

The simulation model of 3kW single-phase gridconnected PV VSI system using Matlab/Simulink

40

2.16

DPLL operation results (a) grid voltage (b) stationary frame components of grid voltage (c) synchronous frame components of grid voltage (d) stationary frame components of grid current (e) synchronous frame components of grid current (f) grid phase angle

43

2.17

Grid-tie operation results (a) grid voltage (b) grid current using PI controller (c) grid current using PI controller (d) grid phase angle

44

2.18

Frequency spectrum and THD value of grid current (a) using PI controller (b) using PR controller

44

2.19

3kW PV PCS experimental prototype

46

2.20

DPLL implementation

46

2.21

Transient-state operation (a) using PI controller (b)

47

- ix -

using PR controller 2.22

Steady-state operation (a) using PI controller (b) using PR controller

48

2.23

Frequency spectrum and THD value of grid current (a) using PI controller (b) using PR controller

49

3.1

Low-pass filter topologies (a) L-filter (b) LC-filter (c) LCL-filter

54

3.2

Frequency response of L-filter

57

3.3

Frequency response of LC-filter with and without damping

57

3.4

Frequency response of LCL-filter with and without damping

58

3.5

LCL-filter (a) Equivalent circuit diagram (b) LCL-filter modeling

58

3.6

Single-phase grid-connected full-bridge PV inverter using LCL-filter (a) Equivalent circuit (b) Equivalent block diagram of current control

65

3.7

Output voltage and current waveforms of single-phase full-bridge inverter

65

3.8

Equivalent circuit of LCL-filter when considering fundamental and switching harmonic components

66

3.9

Equivalent circuit of LCL-filter when considering only switching harmonic component

66

3.10

LCL-filter design flow chart

76

3.11

Frequency responses of filters in case of inverter-side sensing Frequency responses of filters in case of grid-side sensing Frequency responses of open-loop system using difference filters in case of grid-side sensing Frequency responses of closed-loop system using

82

3.12 3.13 3.14

-x-

82 83 83

3.15 3.16 3.17 3.18 3.19 3.20

difference filters in case of grid-side sensing Root locus of closed-loop system using L-filter in case of grid -side sensing Root locus of closed-loop system using un-damped LCfilter in case of grid-side sensing Root locus of closed-loop system using damped LCfilter in case of grid -side sensing Root locus of closed-loop system using un-damped LCL-filter in case of grid -side sensing Root locus of closed-loop system using damped LCLfilter in case of grid-side sensing The simulation model of 3kW single-phase gridconnected PV VSI system using Matlab/Simulink

84 84 85 85 86 87

3.21

Simulation inverter and grid current waveforms (a) inverter current and (e) its zoomed in; (b) grid current using L-filter and (f ) its zoomed in; (c) grid current using LC-filter and (g) its zoomed in; (d) grid current using LCL-filter and (h) its zoomed in

88

3.22

Frequency spectrum analysis of inverter current

89

3.23

Frequency spectrum analysis of grid current using Lfilter Frequency spectrum analysis of grid current using LCfilter Frequency spectrum analysis of grid current using LCL-filter Grid voltage, inverter voltage and current waveforms

89

3.24 3.25 3.26 3.27 3.28 3.29 3.30

Inverter current and grid current using L-filter waveforms Inverter current and grid current using LC-filter waveforms (a) without damping (b) with damping Inverter current and grid current using LCL-filter waveforms (a) without damping (b) with damping Frequency spectrum of inverter current

- xi -

90 90 91 91 92 93 94

3.31

Frequency spectrum of grid current using L-filter

94

3.32

95

4.4

Frequency spectrum of grid current using LC-filter (a) without damping (b) with damping Frequency spectrum of grid current using LCL-filter (a) without damping (b) with damping Inverter output voltage in case of positive current flow (Ig > 0) (a) positive grid current flow (b) inverter output voltage waveforms Inverter output voltage in case of negative current flow (Ig < 0) (a) positive grid current flow (b) inverter output voltage waveforms Relationship between the grid current’s sign and distorted voltage in PWM inverter Single-frequency harmonic tracking filter

114

4.5

Integration of a SHEF and regular plant control part

114

4.6

Mitigation of harmonic components from single-phase 115 grid current by using MSHEF Proposed dead-time compensator 119

3.33 4.1

4.2

4.3

4.7

96 106

107

108

4.9

Single-phase PV inverter current control scheme with 119 dead-time compensator Injected voltage generation by SHEDC 119

4.10

Bode diagram of plant (L-filter)

124

4.11

Bode diagram of inverse plant (L-filter)

124

4.12

Bode diagram of high-pass filter based equivalent inverse plant Simulation model of 3kW single-phase PV inverter using PSIM Grid current in case of no dead-time (a) grid current waveform (b) grid current’s frequency spectrum Grid current in case of 3ms dead-time with no compensation (a) grid current waveform (b) grid current’s frequency spectrum

124

4.8

4.13 4.14 4.15

- xii -

129 130 131

4.24

Grid current, ideal fundamental component and error between them in case of 3ms dead-time with no compensation Grid current in case of 3ms dead-time with 3rdharmonic compensation (a) grid current waveform (b) grid current’s frequency spectrum Grid current, first harmonic component and error between them in case of 3ms dead-time with 3rdharmonic compensation First and third harmonic components with their corresponding weight functions Grid voltage and grid current in case of 3ms dead-time with no compensation Grid voltage, grid current and its frequency spectrum in case of 3ms dead-time with conventional DTCV compensation Grid voltage, grid current and its frequency spectrum in case of 3ms dead-time with proposed MSHEFDC of 3rd harmonic compensation Grid voltage, grid current, the first harmonic component and the error between grid current and first component in case of 3ms dead-time (a) no compensation (b) 3rd harmonic compensation with proposed MSHEFDC First harmonic component and its weight coefficients

138

4.25

Third harmonic component and its weight coefficients

138

4.26

System’s THD (a) in case of 3ms dead-time without 139 compensation and (b) in case of 3ms dead-time with 3rd harmonic compensation

4.16

4.17

4.18

4.19 4.20 4.21

4.22

4.23

- xiii -

132

133

134

134 135 135

136

137

List of Tables

Table 2.1

Single-phase grid-connected PV system parameters

42

Table 3.1

VSI system parameters

77

Table 3.2

Designed LCL-filter parameters

77

Table 4.1

VSI system parameters for dead-time compensation 128

- xiv -

Nomenclatures

ac

Alternating current

APF

All-pass filter

dc

Direct current

DG

Distributed generation

DLL

Dynamic link library

DPLL

Digital phase-locked loop

DSP

Digital signal processing

DTCT

Dead-time compensation time

DTCV

Dead-time compensation voltage

EMTP

Electro Magnetic Transients Program

ESR

Equivalent series resistance

IGBT

Insulated-gate bipolar transistor

LMS

Least mean square

MOSFET

Metal–oxide–semiconductor field-effect transistor

MSHEF

Multi-single-frequency harmonic extraction filter

MSHEFDC

Multi-single-frequency harmonic extraction based dead-time compensator

PI

Proportional-Integral

- xv -

filter

PLL

Phase-locked loop

PR

Proportional-Resonant

PV PCS

Photovoltaic power conditioning system

PWM

Pulse width modulation

SHEF

Single-frequency harmonic extraction filter

THD

Total harmonic distortion

VSI

Voltage source inverter

- xvi -

Chapter

1

Introduction

1.1 Introduction Distributed power generation has been recently introduced as a new concept for the generation of power and the enhancement of conventionally produced electricity. Global warming issue calls for renewable energy resources in electricity production. Petrol and fuel crisis, as well as environmental issues, are of today’s highest concerns all over the world. In recent years, many places in the world have been experiencing continued shortage of electric power or energy crisis due to their fast increasing demand [1][2]. There are two ways of research and development for solving this problem: a) Improve the efficiency of present power conversion and utilization system. Since various high-power semiconductor devices (IGBT, MOSFET…) were developed and become commercially available. The increased power ratings as well as

-1-

switching speed, ease of control and low power semiconductor devices cost make the new converter topologies possible [1]. Furthermore, since the controller for a power converter requires a high controlling and data-processing speed to achieve high performance, the DSP should be used to control the power conversion. The speed of DSP has been raised dramatically for last 20 years and can operate at a data-processing rate of up to several hundred megahertz now. This can meet the requirements of high-performance control and make the real-time digital control of power converter realizable. b) Develop the efficient renewable energy generation and conversion systems to supplement conventional fossil-fuel based energy supply and eventually replace it. The renewable energy generation and conversion system has many advantages over conventional energy supply, e.g. the ability of regeneration, reusability and less pollution. However, the renewable energy generation and conversion technologies are not completely mature yet. There still exist problems such as low efficiency and high cost. The main sources of renewable energy currently under development include solar, wind, hydro-power and biomass, etc [2]. The solar power system has the potential to become one of the main renewable energy sources due to the commercial availability of semiconductor-based photovoltaic (PV) devices, reduction in the system cost and development of power electronic technologies. The important task is to make solar power generation and conversion system more efficient and more reliable. Photovoltaic or fuel cell power systems, which generate power as dc electricity, require power

-2-

conversion devices from dc to ac in order to provide power to the transmission and distribution network of a utility grid. It has been wellproven that a PV power source must be integrated with other power source, whether used in either a stand-alone or grid-connected mode. The dc to ac power conversion, either in single-phase or three-phase systems, can be considered as the core of the whole system because of an important role in interconnection between dc and ac power part. This power conversion is accomplished in phase-controlled (uncontrollable frequency) converters by means of controllable switches. A phasecontrolled converter requires that the external ac system be a voltage source, typically ac utility line. This condition is necessary because the phase-controlled converter uses the reversal of the ac voltage to drive the commutation process. And the ac frequency in the converter is constrained to be that of the ac source.

1.2 Objectives and Outline The main objective of this dissertation is to evaluate and improve the interconnection quality between dc and ac power of single-phase gridconnected PV system. Firstly, a current control scheme is analyzed and presented for controlling the single-phase PWM inverter. Secondly, to improve the quality of grid current, a single-phase low-pass output LCL-filter design procedure is introduced for reducing the inverter current’s ripple and harmonic components at switching frequency. Finally, for mitigating the current distortion at zero-crossing point caused by dead-time phenomenon, a software-based dead-time

-3-

compensation method for single-phase PWM inverter is proposed for distributed renewable energy conversion system from solar power. Chapter 1 gives a brief introduction and motivations of this work while Chapter 2 covers the current control principle of single-phase grid-connected full-bridge PV inverter system and its operation in both simulation and experiment. Chapter 3 focuses on the low-pass output LCL-filter design procedure and provide a comparative analysis between L-, LC- and LCL-filter. Finally, Chapter 4 presents a deadtime compensation method based on the selective harmonic extraction filter using LMS algorithm with the summary is covered in Chapter 5 of this dissertation.

1.3 Single-Phase Grid-Connected PV Inverter Topologies The power conversion subsystem for a solar power system has two main tasks: a) To control the input terminal conditions to make PV modules operate at a maximum power point (MPP) and to capture the maximum power for the sun [3]. b) To convert the output dc voltage from solar modules to ac voltage and meet the utility requirements [4]. Therefore, most of the solar power conversion systems apply twostage topology [5]. The most common two-stage systems consist of a dc-dc solar module-connected converter to perform maximum power point tracking (MPPT) and a dc-ac PWM inverter as shown in Figure 1.1(a). A drawback of this topology is relatively low efficiency due to the two-stage power conversion. -4-

(a)

(b) Fig. 1.1 Topologies of single-phase PV system (a) two-state (b) single-state

-5-

To improve the conversion efficiency, a single-stage solar power system [4] shown in Figure 1.1 (b) can be employed to improve the conversion efficiency. In this topology, dc voltage generated by solar modules is converted to ac power directly by an inverter thus we can get higher efficiency. However, the control method for this topology is complex since it has to do MPPT, provide desired output and balance power flow between utility and solar module simultaneously [3][4][6].

1.4 Control Methods for Single-Phase VSI First and most important factor used to evaluate the inverter operation in PV system is grid current quality. Grid current should be sinusoidal with a low THD value, usually less than 5%. In addition, the inverter is expected to have good transient response under the softstarting period. Efficiency and cost have also a part in evaluation of different inverters and can impact both topology or control method selected for the inverter [8][9][10]. In literature, many advance control methods were introduced, all aiming to control the instantaneous output voltage of single-phase inverter with superior dynamic response as well as zero steady-state error at the fundamental frequency of the inverter. Several of these methods are introduced in the following sections. 1.4.1 Single voltage loop control This method uses a single feedback loop to regulate the output voltage of the inverter, as shown in Fig. 1.2, where the time-varying dynamic of inverter is ignored. The error between the output voltage -6-

and the sinusoidal reference voltage is compensated to create a sinusoidal duty reference for the modulator. Although this approach is easy to design and implement, it does not provide good regulation of output voltage. Steady state error will also exist due to the finite loop gain of the controller at the fundamental frequency of the inverter [11]. 1.4.2 Multi-loop control By using a multi-loop control strategy where both output voltage and current have a feedback loop, the system can get a better performance. The inner loop is current control loop and has a faster response than the outer voltage control loop. The reference of the current loop is created from error of the outer voltage loop. The inner loop provides fast dynamic response for the inverter as well as damping of the resonant peaking cause by the output filter making it easier to close the outer voltage loop with higher bandwidth [12]. Fig 1.3 shows the block diagram of a multi-loop controller for single-phase inverter. 1.4.3 Predictive control Based on the predictive control concept, which was presented in [13], this method utilizes prediction and optimization to minimize the error in the output voltage. The operation of this method is to predict the behavior of dependent variables (or output signal) with respect to the changes in system’s independent variables (inputs signal). Predictive controllers provide better performance and regulation compared to the single-loop and multi-loop control methods described above. The disadvantage of this method is to require continuous knowledge of load

-7-

Fig. 1.2 Single-loop control for single-phase PWM inverter

Fig. 1.3 Multi-loop control for single-phase PWM inverter

-8-

making it difficult to be used in stand-alone system where knowledge of load is not ensured at all time [14]. 1.4.4 Deadbeat control Deadbeat control techniques have been presented for many years [15]-[17]. Although deadbeat control can theoretically provide the fastest response for a digital implementation, it has some disadvantage such as a very high gain, which makes it extremely sensitive to the noise or dependency on inverter’s parameters. It means that with imperfect estimation or variation of inverter parameters, the controller performance is affected much than the other methods. To clearly understand this method, let consider an analog control system. This analog system can achieve faster dynamic response if its close-loop poles can be moved far away to the left half of the s-plane. In the extreme case, the analog system can respond as fast as possible if the poles are located at-¥. But, unfortunately, the analog systems cannot be realizable since it is impossible to design a continuous-time system with a proper transfer function in which all poles are placed at -¥. However, this is possible in a digital system. The discrete-time system can be obtained from the continuous-time system by applying the Zero-order-hold (ZOH) transformation. It is noted that a pole at s = -¥ in s-plane corresponds to z = 0 in z-plane. Therefore, it is possible for a digital control system to achieve its fastest response if all the close-loop poles are place at the origin of the z-plane. The technique of placing all the poles of a discrete-time system at zero is named as deadbeat control. This method cannot be used for analog control systems. -9-

1.4.5 Sliding mode control Sliding control method is a variable structure control strategy where a reference line is selected so that the system trajectory exhibits desirable behavior when confined to that reference line. The feedback gain is determined to control the system trajectory intersects and stays on the reference line. In principle, this control constrains the system, by using the suitable control strategy, to stay on the sliding surface in which the system will exhibit desirable features. The chattering phenomenon, is a disadvantage of this control, is caused by the presence of parasitic dynamics in series with the plant or the non-idealities of the switches. It causes a small high-frequency amplitude oscillation around the sliding reference line. Normally, these parasitic dynamics are often ignored in control design to help simplifying the process. The sliding mode control provides a great dynamic response for the system as well as low output voltage distortion under nonlinear load operation making it one of the more exotic control methods that can be done for single-phase inverters if carefully designed so that the chattering phenomenon is minimized and eliminated [18]-[21]. 1.4.6 Conclusion Although each of the methods described above can be designed to give desired outcome, this dissertation will be carried out based on single-loop current-control based vector control method, one that is very well known and has been used for many years but most for threephase applications. Chapter 2 includes a detailed design of this control method with all analysis, simulation and experimental results.

- 10 -

Chapter

2

Current Controller for Single-Phase Grid-Connected PV Inverter

2.1 Introduction Over the years, the energy consumption is increased due to more and more appliances that are using electricity. Among the new energy sources, wind and solar energy are receiving higher interest due to their potentials. The wind turbine (WT) and photovoltaic (PV) systems are widely used in countries with high wind potential (e.g. Germany, Denmark, Spain, Korea, Japan, China and other countries). Except a few wind turbine topologies, almost all WT and PV systems are interfacing the utility grid through a pulse-width modulated (PWM) driven Voltage Source Inverter (VSI) [23] whose functionality is to synchronize and transfer the variable produced power over to the grid. Another feature of the adopted dc/ac inverter is that it is usually PWM

- 11 -

at a high switching frequency and is either current- or voltagecontrolled using a selected linear or nonlinear control algorithm. Although current and voltage control schemes are possible and implementable, the current control principle is generally preferred for its excellent dynamic characteristics and its inherent over-current limitation capabilities. The deciding criterion when selecting the appropriate control scheme, which deals with the grid interconnection, usually involves an optimal tradeoff between cost, complexity and waveform quality needed for meeting the power quality standards for distributed generation in low-voltage grids, like IEEE-1547 (in USA) and IEC61727 (in Europe) at a commercially favorable cost. Since the similarities in hardware, initially the control strategies applied to drives applications were also applied in distributed power generation system (DPGS). However, because of the more restrictive standard demands for power quality, various control strategies and controller types have also been investigated [24]–[27]. One of the most common control structures applied to DPGS is based on Voltage Oriented Control (VOC) employing a controller for the dc-link voltage and a controller to regulate the injected current into the utility grid [27]. However, the most controllers with precise reference tracking are either overburdened by complex computational requirements or have high parametric sensitivity (sometimes both). On the other hand, simple linear proportional–integral (PI) controllers are prone to known drawbacks, including the presence of steady-state error in stationary frame and the need to decouple phase dependency in three-phase systems although they are relatively easy to implement [27]. Many approaches have been proposed in the literature

- 12 -

to improve the overall performance of PI controllers such as the addition of a grid voltage feed-forward path, multiple-state feedback and increasing the proportional gain. These variations can expand the PI controller bandwidth but, unfortunately, they also push the systems towards their stability limits. Furthermore, another disadvantage of the modified PI controllers is the possibility of distorting the line current caused by background harmonics introduced along the feed-forward path if the grid voltage is distorted. This distortion can in turn trigger LC resonance especially when a LCL filter is used at the converter AC output for filtering switching current ripple [28][29]. In case of three-phase systems, synchronous frame PI control with voltage feed-forward can be used, but it usually requires multiple frame transformations, and can be difficult to implement using a low-cost fixed-point digital signal processor (DSP). To overcome the computational burden and to achieve virtually similar frequency response characteristics as a synchronous frame PI controller, another controller, proportional-resonant (PR) controller for reference tracking in the stationary frame has been developed in [30][32]. Interestingly, the same control structure can also be used for the precise control of a single-phase converter [31]. In brief, the basic functionality of the PR controller is to introduce an infinite gain at a selected resonant frequency for eliminating steady-state error at that frequency. Removal of the steady-state error in single-phase systems, no need for coupling or voltage feed-forward and easy tuning stand as its main advantages. As already highlighted, the resonant frequency information is necessary in their internal model. This issue may be

- 13 -

regarded as a drawback when implemented in a grid-connection system in case of utility grid frequency variations. With the introduced flexibility of tuning the resonant frequency, the selective low-order harmonics compensation using multiple PR controllers have also been applied for three-phase active power filters[33], for three-phase uninterruptible power supplies (UPS)[34], for single-phase PV inverters[35]. Furthermore, various harmonic reference generators using PR filters have also been proposed for single-phase traction power conditioners [36] and three-phase active power filters [37]. Since the electronic power converters will be increased either as inverters processing DC energy from renewable energy source for grid injection or as rectifiers conditioning grid energy for different load usages, this chapter provides a comprehensive reference for readers on the integration of PR controller to grid-connected inverter for enhancing their tracking performances. To begin, the frequency domain derivation of the ideal and non-ideal PR controllers are reviewed, and the comparative analysis between PR and conventional PI controller are discussed. Generic control block diagrams for illustrating current tracking control are also described before providing readers some practical results as implementation examples.

2.2 PI Controller Analysis 2.2.1 PI controller gains The conventional (ideal) PI controller transfer function is defined as: - 14 -

æ K 1 ö÷ GPI (ideal) = K p + i = K p ç1 + ç T s÷ s i ø è

where Kp, Ki and Ti =

(2.1)

K

p are the proportional gain, integral gain and K i

integral time constant, respectively. Fig. 2.1 depicts a simple control system block diagram using conventional PI controller with one plant, where r is the reference input signal, y is the system output signal, e is the error between the reference and feedback signals, u is the controlled signal, d is the disturbance signal, Gc(s) is the PI controller transfer function shown in (2.1) and Gp(s) is the plant transfer function as shown in (2.2) for a general case.

G p = Ks

1 + a s + a s 2 + ... + a s m - sT m e 1 2 d 2 n 1 + b s + b s + ... + b s 1 2 n

(2.2)

where Ks is the steady-state gain, Td is the plant time delay, ax and by are the numerator and denominator parameters (x = 1,..,m and y = 1,..,n, where m ≤ n), respectively.

By applying the Taylor series, the delay component e

- sT d in (2) can

be expanded as:

( ) ( ) ( )

sT 2 sT 3 sT 4 - sT d d e = 1 - sT + - d + d + ... d 2! 3! 4!

- 15 -

(2.3)

r

e

+

u

Gc(S)

y

Gp(S)

+

Controller

Plant

d

Fig. 2.1 An equivalent block diagram of a simple closed-loop control system using PI controller Bode diagram of ideal PI controller 80 K p=1 K p=5

Gain(dB)

60

K p=10 40

Kp increases

20 0 -4 10

-3

10

-2

10

-1

10

0

1

10

10

2

10

3

4

10

10

0

Phase(deg)

-20 -40 -60

Kp increases

-80 -100 -4 10

-3

10

-2

10

-1

10

0

1

10 10 Frequency (Hz)

2

10

3

4

10

10

Fig. 2.2 Effect of Kp variation on PI controller (Ki =1) Bode diagram of ideal PI controller 100 K i=1

Gain(dB)

80

K i=5

Ki increases

60

K i=10

40 20 0 -4 10

-3

10

-2

10

-1

10

0

10

1

10

2

10

3

10

4

10

0

Phase(deg)

-20 -40

Ki increases

-60 -80 -100 -4 10

-3

10

-2

10

-1

10

0

1

10 10 Frequency (Hz)

2

10

3

10

4

10

Fig. 2.3 Effect of Ki variation on PI controller (Kp = 1) - 16 -

The optimization procedure is considered upon examination of both the input and disturbance transfer functions. The closed-loop transfer function between the output y and the reference input r is (2.4), where the transfer function between the output y and the disturbance d is (2.5):

Gr (s ) =

G (s )G (s ) Y (s ) c p = R(s ) 1 + G (s )G (s ) c p

(2.4)

G (s ) Y (s ) p ( ) = Gd s = D (s ) 1 + G (s )G (s ) c p

(2.5)

It is noted that both the system transfer functions corresponding to the reference and the disturbance have the same characteristic equation. The PI controller gains can be calculated following the symmetric or magnitude optimum criterion, where the latter has a fast and nonoscillatory closed-loop time response for a large class of plants [38][41]. In accordance to the magnitude optimum criterion, the gains of the PI controller may be determined following these criteria: G (s ) =1 r s=0

(2.6)

é d 2 k G (s ) ù r ú lim ê =0 ê 2 w ® 0ê dw k úú ë û s = jw

(2.7)

for k=1,2,3… The closed-loop transfer function between the output and reference input of the system, shown in Fig. 2.1, can be obtained by substituting - 17 -

(2.1) and (2.2) into (2.4) as follows: 1 + N s + N s 2 + N s 3 + ... 1 2 3 Gr (s ) = 2 1 + D s + D s + D s3 + ... 1 2 3

(2.8)

where N1 = a + T - T 1 i d

(2.8.a)

T2 N2 = a + a T - a + T T + d 2 1i 1 i d 2!

(

)

(2.8.b)

T2 T3 N3 = a + a T - a + a T T + a + T d - d 3 2 i 2 1i d 1 i 2! 3!

(

)

(

)

(2.8.c)

T T i =N + i D1 = a + T - T + 1 i d K K 1 K K p s p s

(2.8.d)

T2 T T i = N +b i D2 = a + a T - a + T T + d + b 2 1i 1 i d 1 2 1 2! K K K K p s p s

(2.8.e)

...

(

)

T2 T3 b T b T D3 = a + a T - a + a T T + a + T d - d + 2 i = N + 2 i 3 2 i 2 1i d 1 i 2! 3 K K 3! K K p s p s

(

)

(

)

(2.8.f) ... The condition (2.7) is fulfilled by using the set of equations with the used notations in (2.8) [41]: 2k

2k

i=0

i=0

å (- 1)i + k Di D2k - i = å (- 1)i + k N i N 2k - i

- 18 -

(2.9)

where k is the step order. It depends on each system but should be taken as many as possible. Solving equations (2.9) for k=1 and k=2, the following PI regulator gains are obtained:

(

)

æ 2b - b 2 ö b - a + T - b 1 çè 2 1 ÷ø 1 1 d K = p 2K b -a +T a -b s 1 1 d T = i

(

(

2K K b - a + T p s 1 1 d 1 + 2K K p s

)

)

(2.10)

(2.11)

where T2 a = a -T b -b + a T - a + d 1 d 1 2 1 d 2 2!

(

)

(2.12)

æ T 2 ö÷ T2 T3 ç d d d b = ç - a Td + a + ÷b1 - a1 - Td b2 + b3 + a1 2! + a2Td - a3 + 3! 1 2 2 ! ç ÷ è ø

(

)

(2.13) The effect of each parameter to PI controller operation is analyzed. Assuming one of parameters to be constant, the effect of changes in the other parameter can be easily observed. Fig. 2.2 shows the effect of Kp to the operation of PI controller. It can be concluded that the harmonic impedance depends on Kp value, it increases as Kp is added. A high value of Kp can lead to a low harmonic output caused by the disturbance. Similarly, Fig. 2.3 shows the effect of Ki to the operation of PI controller. The change of Ki has effect on both

- 19 -

magnitude and phase of the regulator. The magnitude and the bandwidth increase when Ki increases. 2.2.2 Digital implementation of PI controller In the time domain, the PI controller is expressed as follows: t

u (t ) = K e(t ) + K ò e(t )dt p i0

(2.14)

where u(t) and e(t) are the output and the error signal input of the PI controller in time domain, Kp and Ki are the PI controller gains, respectively. During a sampling time Ts, equation (2.14) can be rewritten as follows: u (k ) = K e(k ) + K T å e(k ) p i s k

(2.15)

At the previous sampling point: u (k - 1) = K e(k - 1) + K T å e(k ) p i s k

(2.16)

Subtract (2.15) to (2.16), we can get the digital form of the PI controller in DSP as follows:

- 20 -

Bode diagram of PI controller 70 Ideal PI Nonideal PI

Gain(dB)

60 50 40 30 20 -2 10

-1

10

0

10

1

10

2

3

10

10

4

10

5

10

0

Phase(deg)

-20 -40 -60 -80 -100 -2 10

-1

10

0

10

1

2

3

10 10 Frequency (Hz)

10

4

10

5

10

Fig. 2.4 Frequency response of ideal and non-ideal PI controller (Kp=10, Ki=100, wc=10rad/s) Bode diagram of PR controller 70 Ideal PR Nonideal PR

Gain(dB)

60 50 40 30 20 -2 10

-1

10

0

10

1

10

2

10

3

10

4

10

5

10

Phase(deg)

100 50 0 -50 -100 -2 10

-1

10

0

10

1

2

10 10 Frequency (Hz)

3

10

4

10

5

10

Fig. 2.5 Frequency response of ideal and non-ideal PR controller (Kp=10, Ki=100, wc =10rad/s, w0 =377rad/s)

- 21 -

u (k ) = u (k - 1) + K

p

[e(k ) - e(k - 1)] + KiTs å e(k )

(2.17)

k

2.2.3 Non-ideal PI controller In the harmonic synchronous reference frame, the low-pass filtered DC output is the harmonic voltage magnitude. The lower the filter cutoff frequency, the more accurately the harmonic voltage component will be extracted in the steady-state, but the transient response time will increase correspondingly as a consequence. If the simple first-order low-pass filter with cut-off frequency ωc is used in the synchronous reference frame, the transfer function of the ideal PI controller in (2.1) becomes: Kw GPI (nonideal) = K p + i c s +w c

(2.18)

Equation (2.18) can be seen as to be a non-ideal PI controller and be used to obtain the non-ideal PR controller. The bode diagrams of ideal and non-ideal PI controller are depicted in Fig. 2.4.

2.3 PR Controller Analysis 2.3.1 Ideal PR controller In the load current control technique of power electronic converter regulations, the system signals are converted from ac to dc signal. It means that the three-phase signals in the stationary reference frame are transferred into a synchronous reference frame by using Clarke-Park

- 22 -

transformations. These transforms essentially perform frequency shifts on the system signals, which are time-varying quantities, and it is useful to have Laplace-domain presentations of these transformations for controller analysis. In a dc control system, the standard control network is an ideal PI controller with an s-plane transfer function as (2.1). It is also used in a synchronous reference frame current regulation system. In single-phase system, the popularly reference frame transformation cannot be applied directly. Therefore, an alternative approach of transforming the regulator in dc quantities from the synchronous reference frame to an equivalent stationary reference frame is the frequency modulated method. This process can be mathematically expressed as [30][32]:

[ (

)

(

1 G PR (s ) = G ac (s ) = G s + jw + G s - jw PI I 0 0 2 I

)]

(2.19)

where ω0 is the ac frequency. In (2.19), GPI(s) is a low-pass transfer block, this transformation results in a low-pass to band-pass or a frequency shifting transformation to the frequency. By using the first-order low-pass filter or the PI controller in the synchronous reference frame, but centered around frequency ω0, we can get: ù K K Ks 1é i +K + i ú=K + i GPR (s ) = G ac (s ) = ê K + p PI p s - jw ú 2 2 ê p s + jw s + w2 0 0û ë 0

(2.20)

- 23 -

Equation (2.20) can be seen to be an ideal PR controller which achieves infinite gain at the ac frequency ω0. In the other words, the ideal PR controller can be mathematically derived by transforming an ideal PI controller from synchronous reference frame to stationary reference frame. 2.3.2 Non-ideal PR controller Unfortunately, the ideal PR controller acts like a network with an infinite quality factor, which is hard to implement the PR controller in reality because of the following reasons: -

The introduced infinite gain leads to an infinite quality factor which cannot be achieved in either analog or digital system. Furthermore, this gain causes the associated instability problem to system.

-

The PR controller gains are much reduced at other frequencies and they are not introduced adequate to eliminate harmonic influence caused by ac voltage.

To solve the aforementioned problems caused by the ideal PR controller, an approximating ideal (or non-ideal) PR controller using a high-gain low-pass filter is used. By substituting the non-ideal PI controller in (2.18) into (2.19), the non-ideal PR controller can be obtained as:

GPR ( nonideal ) (s ) = K p + Ki

wc s + wc2 s 2 + 2w s + w 2 + w 2 c c 0

- 24 -

(2.21)

Assuming ωc áá ω0, a simpler approximation of non-ideal PR controller as (2.22) can be used instead of (2.21):

G PR (nonideal ) (s ) = K p + K i

w s c 2 s + 2w s + w 2 c 0

(2.22)

The frequency responses of ideal and non-ideal PR controllers are depicted in Fig. 2.5. As shown in Fig. 2.5, the ideal PR controller achieves infinite gain at the ac frequency (in this dissertation, f0=60Hz) to force the steady-state voltage error to zero, and no phase shift and gain at other frequencies. In case of the non-ideal PR controller, we can adjust the gains to obtain the high enough finite gain for eliminating the voltage tracking error. In addition, a wider bandwidth is observed around the resonant frequency, which is minimizes the sensitivity of the controller to slight ac frequency variations. At other harmonic frequencies, the response of the non-ideal PR controller is comparable to those of the ideal one. 2.3.3 PR controller gains It can be seen that in (2.22), there are three parameters affect to the PR controller operation. It means that there are three degrees of freedom in PR controller design. To simplify the analysis, we keep two parameters be constant, and then the effect on the PR controller of the last parameter effect can be easily observed. Firstly, assuming the proportional gain Kp and the cutoff frequency ωc have no change. Fig. 2.6(a) shows that without proportional gain Kp,

- 25 -

the variation of integral gain Ki has effect on the gain of the controller. The gain increases as Ki is added. With proportional gain Kp, the Ki variation has effects on both bandwidth and gain of PR controller, as shown in Fig. 2.6(b). PR controller can achieve a very high gain in a narrow frequency band centered around the resonant frequency. The width of this frequency band depends on the integral gain Ki. A low Ki value leads to a very narrow band and vice versa. Secondly, assuming the proportional gain Kp and integral gain Ki are constant, the cutoff frequency ωc have effect on the bandwidth and small effect on the gain of the PR controller. As shown in Fig. 2.7(a) and (b), the PR controller bandwidth increases as ωc added, but there is only a small change in the controller gain. So that we can say the same PR controller gain can be achieved at the resonant frequency when ωc changes. Using a smaller ωc will make the controller more sensitive to the variation of frequency, hence this leads to a slower transient response and more difficult in implementation in DSP. In practical, the value of ωc can be chosen at 5-15(rad/s) for a good response [36]. Finally, as shown in Fig. 2.8, when Kp is added, the PR controller gain increases. But a high Kp value will decrease the bandwidth of the PR controller. It means that the harmonic impedance increases as Kp added, and then the higher Kp value can lead to a relatively low harmonic component. The value of Kp can be chosen to make sure that the system can achieve high performance in the sinusoidal reference tracking as well as the disturbance rejection. In conclusion, the PR controller gains can be designed following a step-by-step procedure: - First, choose the value of ωc to meet the system bandwidth.

- 26 -

Bode diagram of PR controller when K i changes 50

Gain(dB)

0

-50

Ki increases -100 -1 10

0

10

1

2

10

3

10

10

4

5

10

10

100 Ki = 1

Phase(deg)

50

K i = 10 K i = 100

0

K i = 200

-50 -100 -1 10

0

10

1

10

2

10 Frequency (Hz)

3

10

4

10

5

10

(a) Bode diagram of PR controller when Ki changes 50 Ki = 1

Gain(dB)

40

K i = 10

30 20

K i = 100

Ki increases

K i = 200

10 0 -1 10

0

10

1

10

2

3

10

10

4

10

5

10

Phase(deg)

100 50

Ki increases

0 -50 -100 -1 10

0

10

1

10

2

10 Frequency (Hz)

3

10

4

10

5

10

(b) Fig. 2.6 Frequency response of non-ideal PR controller when Ki changes (a) Kp = 0, wc = 1rad/s, w0 = 377rad/s (b) Kp = 1, wc = 1rad/s, w0 = 377rad/s - 27 -

Bode diagram of PR controller when W c changes

Gain(dB)

0

-50

-100

-150 -1 10

wc increases 0

10

1

2

10

3

10

4

10

5

10

10

100 Wc = 1

Phase(deg)

50 0

W c = 10

wc increases

W c = 100 W c = 200

-50 -100 -1 10

0

10

1

10

2

10 Frequency (Hz)

3

4

10

10

5

10

(a) Bode diagram of PR controller when W c changes 4 Wc = 1

Gain(dB)

3 2

W c = 10 W c = 100

wc increases

W c = 200

1 0 -1 10

0

10

1

10

2

3

10

10

4

10

5

10

Phase(deg)

20

wc increases

10 0 -10 -20 -1 10

0

10

1

10

2

10 Frequency (Hz)

3

10

4

10

5

10

(b) Fig. 2.7 Frequency response of non-ideal PR controller when wc changes (a) Kp = 0, Ki = 1, w0 = 377rad/s (b) Kp = 1, Ki = 1, w0 = 377rad/s - 28 -

Bode diagram of PR controller when Kp changes 50

Gain(dB)

40 30 20

Kp increases

10 0 -1 10

0

10

1

10

2

10

3

10

4

5

10

10

Phase(deg)

20 10

Kp = 1 Kp = 10

0

Kp = 100

-10

Kp = 200

-20 -1 10

0

10

1

10

2

10 Frequency (Hz)

3

10

4

10

5

10

Fig. 2.8 Frequency response of non-ideal PR controller when Kp changes with Ki =1, wc = 1rad/s, w0 = 377rad/s

Fig. 2.9 Single-phase grid-connected PV inverter control system (a) Current control circuit structure (b) Equivalent current control block diagram - 29 -

- Second, choose the integral gain Ki for a suitable controller gain. - Finally, the proportional gain Kp can be selected to make sure that the system can achieve high performances in sinusoidal signal tracking and disturbance rejection. 2.3.4 Digital implementation of PR controller By

applying

the

bilinear

transformation

s=

2 1 - z -1 T 1+ z-1 s

and

substituting into (2.22), the discrete transfer function of the PR controller can be given by (2.23): n + n z-1 + n z- 2 2 GPR ( nonideal ) ( z ) = 0 1 1 1+ d z + d z- 2 1 2

(2.23)

æç 4 + 4T w + w 2T 2 ö÷ K + 4 K T w s c i s c 0 s ø p n =è 0 2 2 4 + 4T w + w T s c 0 s

(2.23a)

æç - 8 + 2w 2T 2 ö÷ K 0 s ø p n =è 1 4 + 4T w + w 2T 2 s c 0 s

(2.23b)

æç 4 - 4T w + w 2T 2 ö÷ K - 4 K T w s c 0 s ø p i s c n =è 2 2 2 4 + 4T w + w T s c 0 s

(2.23c)

- 8 + 2w 2T 2 0 s d = 1 4 + 4T w + w 2T 2 s c 0 s

(2.23d)

4 - 4T w + w 2T 2 s c 0 s d = 2 4 + 4T w + w 2T 2 s c 0 s

(2.23e)

- 30 -

Fig. 2.10 Single-phase current control structure (a) Using PI controller (b) Using PR controller

- 31 -

where Ts is the sampling time, wc is the cut-off frequency (rad/s) and w0 is the grid utility frequency (rad/s) From (2.23), the discrete equation can be obtained as follows: y(k ) = n0u(k ) + n1u(k - 1) + n2u(k - 2) - d1 y(k - 1) - d2 y(k - 2)

(2.24)

where u(k) is the error input and y(k) is the output of the PR controller.

2.4 Single-Phase Grid-Connected PV Inverter System 2.4.1 Current control scheme A single-phase grid-connected PV PCS has built where its circuit block diagram of current control scheme is depicted in Fig. 2.9(a) and its current control block diagram is illustrated in Fig. 2.9(b), respectively. Fig. 2.10 (a) shows the detail single-phase current control structure by using PI controller and Fig. 2.10 (b) shows that of using PR controller, respectively. Since the L-filter achieves low attenuation of the inverter switching components, a capacitor (as a shunt element) is needed to further attenuate the switching frequency components. This capacitor must be selected to produce a low reactance at the switching frequency. But within the control frequency range, this element must present high magnitude impedance. However, a very high capacitance is not recommended since the system may face with inrush current, high current fed on capacitor at the fundamental frequency, resonance phenomenon at the grid side, etc. If a system is connected to the grid

- 32 -

via LC-filter, the resonant frequency varies over time as the inductance value of the grid. The LCL-filter can provide a better decoupling between the filter and the grid impedance. A lower ripple current distortion across the grid-side inductor since the current ripple is reduced by the capacitor. Although LCL-filter can provide a good attenuation ratio even with small L and C values, the LCL-filter design, which will be presented in Chapter 3 in detail, needs to consider various constraints, such as the resonance phenomenon, the current ripple through filter inductor, the total impedance of the filter, the current harmonics attenuation at switching frequency and the reactive power absorbed by capacitor, etc. The relationship between input and output of current control system in Fig. 2.9(b) can be obtained as:

(s ) = H i (s )I *g (s ) - H v (s )V g (s )

(2.25)

I (s ) G (s )G (s )G (s ) g c i f = H (s ) = i * I (s ) 1 + Gc (s )Gi (s )G f (s ) g

(2.26)

I

g

H

(s ) = v

G (s ) (s ) f = V (s ) 1 + G (s )G (s )G (s ) g c i f I

g

(2.27)

where Gc(s) is the controller transfer function. Gc(s) is shown in (2.1) in case of PI controller and is shown in (2.22) in case of PR controller; 1 is the inverter transfer function with time delay Gi (s ) = 1 + 1.5T s s

Rd C f s + 1

consideration; G f (s) = is the Li Lg C f s 3 + Li + L g Rd C f s 2 + Li + Lg s

(

- 33 -

)

(

)

LCL-filter transfer function with damping resistor, where the resonant frequency is calculated as wres

=

L +L i g ). LL C i g f

As shown in (2.25), the output current depends on both the reference current and the grid voltage. In steady state, because the PI controller has a finite gain at the fundamental frequency, the second term of (2.25) cannot be neglected but can be eliminated by using de-coupling technique. However, PR controller introduces an infinite gain, then the first term approaches the inverter current reference and the second term approaches zero. Hence, the PR controller can achieve the zero steadystate error of the controlled inductor current. From (2.26), the closed-loop transfer functions of current control system shown in Fig. 2.9(b) in case of using PI controller can be obtained as (2.28) and in case of using PR controller can be obtained as (2.29), respectively.

H

(s ) = i ( PI )

A s2 + A s + A 2 1 0 5 4 3 B s + B s + B s + B s2 + B s + B 5 4 3 2 1 0

(2.28)

A =K R C p d f 2

(2.28a)

A =K +K R C p i d f 1

(2.28b)

A =K 0 i

(2.28c)

B = 1.5T L L C 5 s i g f

(2.28d)

é ù B = ê1.5T æç L + L ö÷ R + L L úC 4 ë sè i gø d i gû f

(2.28e)

- 34 -

(

)(

B = 1.5Ts + Rd C f Li + L g 3

)

(2.28f)

B = Li + Lg + K p Rd C f 2

(2.28g)

B =K +K R C p i d f 1

(2.28h)

B =K 0 i

(2.28i)

C s3 + C s 2 + C s + C 3 2 1 0 H (s ) = i( PR ) 6 5 4 3 D s + D s + D s + D s + D s2 + D s + D 6 5 4 3 2 1 0

(2.29)

C =K R C p d f 3

(2.29a)

C = K + æç 2 K + K ö÷ R C w 2 p è p iø d f c

(2.29b)

C = K w 2 R C + æç 2 K + K ö÷w p 0 d f è p iø c 1

(2.29c)

C = K w2 0 p 0

(2.29d)

D = 1.5T L L C 6 s i g f

(2.29e)

[ (

)]

D = 1.5T æç L + L ö÷ R C + 1 + 2 1.5T w Li L g C sè i gø d f 5 s c f

[ (

(2.29f)

)]

D = 1.5T æç L + L ö÷ + R C æç L + L ö÷ 1 + 2 1.5T w + sè i gø d fè i gø s c 4 + Li L g C æç 2w + 1.5T w 2 ö÷ fè c s 0ø

[ (

(2.29g)

)]

D = æç L + L ö÷ 1 + 2 1.5T w + æç L + L ö÷æç 2w + 1.5T w 2 ö÷ R C + 3 è i gø s c g øè c s 0ø d f è i + Li L g C w 02 + K p Rd C f f

(2.29h)

D = æç L + L ö÷æç 2w + 1.5T w 2 ö÷ + æç L + L ö÷ R C w 2 + K + 2 è i g øè c s 0ø è i gø d f 0 p + æç 2 K + K ö÷ Rd C f w p iø c è

(2.29i)

- 35 -

Bode diagram of closed-loop current control using PI controller 50

Gain(dB)

0 -50 -100 -150 1 10

2

10

3

10

4

10

5

10

6

10

Phase(deg)

0 Using PI controller Using PR controller

-100

-200

-300 1 10

2

10

3

4

10 10 Frequency (Hz)

5

10

6

10

Fig. 2.11 Frequency response of closed-loop current control using PI controller (Kp=10, Ki=100) and PR controller (Kp =15, Ki =200, wc = 15rad/s, w0 = 377rad/s)

- 36 -

Root locus of system using PI controller 1 0.6p/T 0.8

0.5p/T

0.4p/T 0.10.3p/T 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

0.7p/T

0.6

0.8p/T

0.4

Imaginary Axis

0.9p/T 0.2 0 -0.2

0.2p/T

0.1p/T

p/T p/T

0.9p/T

0.1p/T

-0.4 0.8p/T

-0.6

0.2p/T

0.7p/T

-0.8

0.3p/T 0.6p/T

-1 -1

-0.8

-0.6

-0.4

0.5p/T

-0.2

0

0.4p/T 0.2

0.4

0.6

0.8

1

Real Axis

(a) Root locus of system using PR controller 1 0.6p/T 0.8

0.5p/T

0.4p/T 0.10.3p/T 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

0.7p/T

0.6

0.8p/T

Imaginary Axis

0.4 0.2 0 -0.2

0.9p/T

0.2p/T

0.1p/T

p/T p/T 0.9p/T

0.1p/T

-0.4 -0.6

0.8p/T

0.2p/T 0.7p/T

-0.8

0.3p/T 0.6p/T

-1 -1

-0.8

-0.6

-0.4

0.5p/T

-0.2

0

0.4p/T 0.2

0.4

0.6

0.8

1

Real Axis

(b) Fig. 2.12 Root locus of closed-loop current control system (a) Using PI controller (b) Using PR controller - 37 -

Nyquist diagram of system using PI controller 3 0 dB 2 dB

-2 dB

2 4 dB

Imaginary Axis

1

-4 dB

6 dB 10 dB

-6 dB -10 dB

0

-1

-2

-3 -3

-2

-1

0

1

2

3

Real Axis

(a) Nyquist diagram of system using PR controller 3 0 dB 2 dB

-2 dB

2 4 dB

Imaginary Ax is

1

-4 dB

6 dB 10 dB

-6 dB -10 dB

0

-1

-2

-3 -3

-2

-1

0

1

2

3

Real Axis

(b) Fig. 2.13 Nyquist diagram of open-loop current control system (a) Using PI controller (b) Using PR controller - 38 -

D = æç L + L ö÷w 2 + æç 2 K + K ö÷w + K w 2 Rd C f 1 è i gø 0 è p iø c p 0

D = K w2 0 p 0

(2.29j) (2.29k)

Fig. 2.11 shows the bode diagrams of closed-loop systems using PI and PR controller where the system using PR controller can introduce a higher bandwidth than using PI controller. For stability analysis, assuming the voltage source inverter switching frequency is higher than power system frequency (by this way, the inverter will have negligible impact on the control loop dynamics) and supply voltage is sinusoid. Following these assumptions, the current closed-loop control stability can be analyzed with classical methods (e.g. Nyquist, root locus). Fig. 2.12(a) and (b) show the Nyquist diagrams of the system using PI and PR controller, respectively, where their Nyquist curves do not contain (-1,j0) point. The root locus diagrams of these systems in zdomain are also illustrated in Fig. 2.13(a) and (b), respectively. 2.4.2 Digital phase-locked loop (DPLL) In this paper, a single-phase digital phase-locked loop (DPLL) is implemented by making a virtual orthogonal voltage, which is delayed by 900 from the measured grid voltage and is generated by passing through an all-pass filter (APF) as shown in Fig. 2.14. The phase detection time of DPLL method is faster than conventional zerocrossing detection method. The APF transfer function is given by:

- 39 -

Fig. 2.14 Digital phase-locked loop block diagram

All pass filter I alpha

[Igrid ]

Igrid

From Igrid

I beta

[theta ] vd cmd ibeta

[theta ]

Theta

[Vpeak ]

v peak

vq cmd

From Vgrid

Vgrid

v peak

[Vpeak]

Theta

[theta ]

Single phase PLL

m

[m] PWM Generator

d

q

Vcmd

Signal(s) Pulses A

Current control

[Vgrid ]

theta

ialpha

Command Voltage Generator

[Vdc]

v

g + -

DC Voltage Source

Grid voltage Grid voltage

Igrid

Grid current Grid current

B

+ A

[Vgrid ] Vgrid

A

A

B

B

[Igrid ]

A1

[PWM] -

B

IGBT Inverter

Filter

B1

PWM

PWM voltage

Current & Voltage Measurement [theta ]

Theta

Scope

Fig. 2.15 The simulation model of 3kW single-phase grid-connected PV VSI system using Matlab/Simulink - 40 -

H

PLL

By

(s ) = w - s

(2.30)

w+s

applying

the

bilinear

transformation

s=

2 1 - z -1 T 1+ z-1 s

and

substituting into (2.30), the discrete PLL transfer function can be given by (2.31):

H

(z ) = PLL

( (

) )

wT - 2 + wT + 2 z - 1 s s wT + 2 + wT - 2 z - 1 s s

(2.31)

Hence, the virtual voltage which is generated from the measured grid voltage can be implemented by the following discrete difference equation: y(k ) = au(k ) + u(k - 1) - ay(k - 1) a=

(2.32)

wT - 2 s wT + 2 s

(2.33)

where u(k) is the error input and y(k) is the output of the filter, respectively. In Fig. 2.14, the grid voltage becomes a-component of stationary reference frame while b-component is a virtual voltage through the aforementioned all-pass filter. And then, the stationary reference frame ab-components are converted to the synchronous reference frame dqcomponents, where d-component denotes the difference between utility - 41 -

and estimated phase angles. In addition to the phase estimation, the instantaneous grid voltage peak can be calculated by considering it is the same as q-component.

2.5 Simulation Results The Matlab/Simulink model of the single-phase grid-connected PV inverter system is shown in Fig. 2.15 with the parameters listed in Table 2.1. The performance of the DPLL is depicted in Fig. 2.16, where the grid component becomes α-component and the b-component is generated by making a 900-delayed virtual voltage component. Table 2.1 Single-phase grid-connected PV system parameters Specifications

Value

Maximum rated power Prated

3 (kW)

Dc-link voltage Vdc

400 (V)

Rated grid voltage Vg (rms)

220 (V)

Maximum rated grid current Ig (rms)

13.64 (A)

Grid frequency f0

60 (Hz)

Switching frequency fsw

10 (kHz)

Modulation index ma

0.8

Inverter-side inductance Li

0.481 (mH)

Grid-side inductance Lg

0.348(mH)

Capacitance Cf

8.2 (mF)

Damping resistance Rd

4.95 (W)

Resonant frequency fres

3.9 (kHz)

- 42 -

The grid voltage (V ) g

400 200 (a) 0 -200 -400

(b)

(c)

400 200 0 -200 -400 400 200 0 -200 -400

0

0.005

0.01

0.015 0.02 0.025 0.03 0.035 The stationary frame voltages (V -V )

0.04

0.045

0.05

0

0.005

0.01

0.015 0.02 0.025 0.03 0.035 The synchronous frame voltages (Vd-V q)

0.04

0.045

0.05

0

0.005

0.01

0.015 0.02 0.025 0.03 0.035 The stationary frame currents (I -I )

0.04

0.045

0.05

a b

10

(d)

b

a

0 -10

0

0.005

0.01

0.015 0.02 0.025 0.03 0.035 The synchronous frame currents (Id-Iq)

0.04

0.045

0.05

0

0.005

0.01

0.015

0.035

0.04

0.045

0.05

0.035

0.04

0.045

0.05

10

(e)

0 -10

0.025

0.03

PLL phase angle (q)

10

(f)

0.02

5 0

0

0.005

0.01

0.015

0.02

0.025

0.03

Fig. 2.16 DPLL operation results (a) grid voltage (b) stationary frame components of grid voltage (c) synchronous frame components of grid voltage (d) stationary frame components of grid current (e) synchronous frame components of grid current (f) grid phase angle

- 43 -

The grid voltage 400 200

(a)

0 -200 -400

0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0.045

0.05

0.04

0.045

0.05

The grid current using PI controller 10 5

(b)

0 -5 -10

0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

The grid current using PR controller 10 5

(c)

0 -5 -10

0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0.045

0.05

0.035

0.04

0.045

0.05

PLL phase angle 8 6

(d)

4 2 0

0

0.005

0.01

0.015

0.02

0.025

0.03

Fig. 2.17 Grid-tie operation results (a) grid voltage (b) grid current by using PI controller (c) grid current by using PR controller (d) grid phase angle Fundamental (60Hz) = 6.011 , THD= 1.85% 7

6

6

5

5

4

4

Mag

Mag

Fundamental (60Hz) = 6.138 , THD= 3.71% 7

3

3

2

2

1

1

0

0

0

200

400 600 Frequency (Hz)

800

1000

0

200

400 600 Frequency (Hz)

800

(a) (b) Fig. 2.18 Frequency spectrum and THD value of grid current (a) using PI controller (b) using PR controller

- 44 -

1000

Fig. 2.16 also shows the Park-transformation operation, where the stationary and synchronous reference frame components of grid voltage and current are shown in Fig. 2.16 (b), (c), (d) and (e), respectively. It is noted that all waveforms are in phase and the grid phase angle is locked as shown in Fig. 2.16(f). Fig. 2.17 shows the results of single-phase grid-connected PV inverter current control, where Fig 2.17 (a) shows the grid voltage, Fig. 2.17 (b) and (c) show the grid current by using PI and PR controller, respectively and Fig. 2.17(d) shows the grid phase angle. Fig. 2.18 (a) and (b) show the frequency spectrum of the grid current and its THD value by using the PI and PR controller, respectively. By using PR controller, the THD value is reduced and less than 5% (11.41% in case of using PI controller compared with 4.14% in case of using PR controller)

2.6 Experimental Results The overall system of 3kW single-phase grid-connected PV PCS, as shown in Fig. 2.19, is implemented fully in software adopting a 32-bit fixed-point DSP TMS320F2812. The inverter controller is implemented in software, and the PWM pulses are generated through the internal pulse generator of the DSP. Voltage and current signals are measured by using the 12-bit resolution of internal analog-to-digital converter in the DSP. Also a four-channel 8-bit digital-to-analog converter has been used for debugging. The switching frequency of the inverter is chosen to 10 kHz and the dead-time is 3ms. - 45 -

Fig. 2.19 3kW PV PCS experimental prototype

Grid voltage Real component

Virtual component

Grid phase angle

Fig. 2.20 DPLL implementation

- 46 -

Grid voltage

Grid current

Soft-start reference current

dc-link voltage

(a)

Grid voltage

Grid current

Soft-start reference current

dc-link voltage

(b) Fig. 2.21 Transient-state operation (a) using PI controller (b) using PR controller

- 47 -

Grid voltage

Grid current

Grid phase angle

(a)

Grid voltage

Grid current

Grid phase angle

(b) Fig. 2.22 Steady-state operation (a) using PI controller (b) using PR controller

- 48 -

(a)

(b) Fig. 2.23 Frequency spectrum and THD value of grid current (a) using PI controller (b) using PR controller

- 49 -

The experimental results of the PV PCS are shown in Fig. 2.20 - Fig. 2.23. Fig. 2.20 shows the implementation of the DPLL, where channel “1” shows the grid voltage (250V/div), channel “2” shows the real voltage component in stationary reference frame, channel “3” shows the virtual voltage component generated by APF and the grid phase angle is measured at channel “4”. It is noted that all results are in phase. Fig. 2.21 shows the transient-state performance by using PI and PR controller. As shown in Fig. 2.21 (a), the grid current has fluctuation during transient-state by using PI controller. The PR controller has a better performance, even in transient-state as shown in Fig. 2.21 (b) Fig. 2.22 (a) shows the grid voltage in channel “1” (250V/div), grid current in channel “2” (10A/div) and the grid phase angle in channel “4” (10V/div) by using the PI controller. The experimental results of using PR controller are shown in Fig. 2.22 (b) with the same channel and scale, respectively. It can be seen that the grid current waveform in Fig. 2.22 (b) is nearly perfect sinusoid. It is noted that the experimental results show a good agreement with the simulation results and both waveforms are in phase. The frequency analysis and THD values of grid current using the PI and PR controller are shown in Fig. 2.23 (a) and (b), respectively. It is noted that in this experiment, the PV inverter does not work with the maximum rated power yet, so the THD can be reduced if working in maximum rated power. Compared with all the experimental results mentioned above, it is can be shown that by using PR controller, the current control scheme can achieve the steady-state performance better than the use of conventional PI controller in current-controlled based single-phase

- 50 -

grid-connected PV inverter system. Experimental results that the PR controller can overcome drawbacks of PI controller: inability to track a sinusoidal reference with zero steady-state error and poor disturbance rejection capability.

2.7 Conclusions In this chapter, a comparison between the conventional PI and PR controller has been presented. Firstly, these control schemes have been simulated by Matlab/Simulink and then, have been implemented in a single-phase grid-connected current-controlled based PV PCS with 32bit fixed-point TMS320F2812 DSP. The theoretical analysis has been performed that the PR controller has some advantages compared with the conventional PI controller and it can enable the implemented control inverter system to achieve the high performance. The experimental results of 3KW PV PCS prototype system verified the performance of these current control schemes. Furthermore, these control schemes are suitable for single-phase current-controlled VSI of distributed generation units, as well as threephase systems, and not only photovoltaic but also the other power generation system, such as small wind turbine, fuel-cells, etc...

- 51 -

Chapter

3

Low-Pass Output LCL Filter for Single-Phase Grid-Connected PV Inverter

3.1 Introduction The VSI, with the use of PWM in conjunction with current controller, can generate a sinusoidal current with a low THD below 5%. However, the high switching frequencies of between 2-150 kHz can cause high-order harmonics that can produce loss and disturb sensitive loads on the utility grid. To eliminate the current harmonics around the switching frequency and comply with the standards (e.g. IEEE 1547), the grid-connected inverter for renewable energy source requires an output low-pass filter for interfacing with the grid. Ideally, the filter with low cut-off frequency and high attenuation at the high switching

- 52 -

frequency is better to eliminate switching ripple effectively. A simple L-filter with high inductance value should be used to reduce the current harmonics around the switching frequency. But it is expensive to realize a high value filter inductance, especially in several kilowatts applications, and the dynamic response of system may become poor because of reactive components. Hence, LC-filter and LCL-filter can be used to solve this problem and the optimum results can be obtained in the range of power levels up to hundreds of kilowatts, still using quite small values of inductors and capacitors [45][46]. The LCL-filter type becomes more attractive for grid interfacing VSI since it can render the current harmonics attenuation around the switching frequency by using smaller inductance than L-filter. Moreover, system using LCL-filter does not depend on the grid impedance and has a better output response while comparing with LCfilter. A good criterion to choose LCL filter parameters is to limit the size of reactive elements, which can results in poor power factor [3], and reactive power dropped on LCL-filter because of passive damping. In literature, some issues have been studied: active damping of the filter [46][48], criteria for parameter choices [47][49], control techniques integrated with LCL-filter for VSI [50]–[53]. The design of an LCLfilter to limit switching frequency ripple injection into the grid in the range of 2-150 kHz is often specifically required. However, the comparative analysis between the LCL-filter and other filter types has not been analytically studied to date. The L-, LC- and LCL-filter topologies are shown in Fig. 3.1.

- 53 -

Ri

Li

(a) L Filter Ri

Li Cf

(b)

Rd LC Filter Ri

Lg

Li Cf

(c)

Rd LCL Filter Fig. 3.1 Low-pass filter topologies (a) L-filter (b) LC-filter (c) LCL-filter

- 54 -

Rg

In the literature, the concepts of L-filter, LC-filter and LCL-filter have been introduced and are revised as follows: - Firstly, the L-filter is popular and simple to use. But it requires a high inductance value to have the low ripple attenuation (-20dB/decade) over the whole frequency range. This causes a poor dynamic system response because of the dropped voltage on the inductor. Hence the dynamic system response takes long time to reach steady-state. By using L-filter, the inverter switching frequency must have a high value in order to sufficiently attenuate the harmonics [54]. - Secondly, since the L-filter achieves low attenuation of the inverter switching components, a LC-filter which is modified from Lfilter by adding a shunt element is needed to further attenuate the switching frequency components. A capacitor is selected to be shunt element for a low reactance at the switching frequency and high magnitude impedance within the control frequency range. The LC-filter has an attenuation of -40dB/decade and is suited to configurations where the load impedance across the capacitor is relatively high at and above the switching frequency. To reduce the losses and cost, the capacitance should be high and then, the inductance can be reduced. But a very high capacitance is not recommended since the system may face with inrush current, high reactive current fed on capacitor at the fundamental frequency, possible resonance phenomenon at the grid side, etc. If a system is connected to the grid via LC-filter, the resonant frequency is changed due to grid impedance over the time [55]. The cost and the reactive power consumption of the LC-filter are more than to the L-filter because of the addition of the shunt element. The resonant frequency of LC-filter can be calculated as follows: - 55 -

f res =

1 2p Li C f

(3.1)

- Thirdly, LCL-filter produces better attenuation (-60dB/decade) at the inverter switching frequency when compared with the previous filter topologies. LCL-filter can provide not only a better decoupling between filter and grid impedance but also a good attenuation ratio even with small L and C values. A lower ripple current distortion across the grid-side inductor since the current ripple is reduced by the capacitor. However, the three-order LCL-filter design has to consider various constraints, such as the resonance phenomenon, the current ripple through inductors, the total impedance of the filter, the current harmonics attenuation at switching frequency and the reactive power absorbed by capacitor, etc. With low inductance on the inverter side, it is difficult to comply with IEEE519 standards without an LCL filter. An LCL filter can achieve reduced levels of harmonic distortion with lower switching frequencies and with less overall stored energy. On the other hand the LCL filter may cause both dynamic and steady state input current distortion due to resonance. Key advantages of the LCL-filter are: - Low grid current distortion and reactive power production; - Attenuation of -60dB/decade for frequencies in excess of the resonance frequency; -

Possibility of using a relatively low switching frequency for a

given harmonic attenuation.

- 56 -

Bode diagram of Plant

Gain(dB)

50

0

-50

-100 0 10

1

10

2

10

3

10

4

10

5

6

10

10

-89 Phase(deg)

L -89.5 -90 -90.5 -91 0 10

1

10

2

10

3

10 Frequency (Hz)

4

10

5

10

6

10

Fig. 3.2 Frequency response of L-filter

Bode diagram of Plant

Gain(dB)

50 0

-50

-100 0 10

1

10

2

10

3

10

4

10

5

10

6

10

Phase(deg)

200 LC no damp LC damp

100 0 -100 -200 0 10

1

10

2

10

3

10 Frequency (Hz)

4

10

5

10

6

10

Fig. 3.3 Frequency response of LC-filter with and without damping

- 57 -

Bode diagram of Plant 50

Gain(dB)

0 -50 -100 -150 0 10

1

10

2

10

3

10

10

4

5

10

6

10

Phase(deg)

100 LCL no damp LCL damp

0 -100 -200 -300 0 10

1

10

2

10

3

10 Frequency (Hz)

10

4

5

10

6

10

Fig. 3.4 Frequency response of LCL-filter with and without damping

Fig. 3.5 LCL-filter (a) Equivalent circuit diagram (b) LCL-filter modeling

- 58 -

The resonant frequency of LC-filter can be calculated as follows: f res =

1 2p

Li + Lg Li Lg C f

(3.2)

The frequency responses of L-filter, LC-filter and LCL-filter are shown in Fig. 3.2 ~ Fig. 3.4, respectively. In this chapter, a theoretical analysis and design procedure of output LCL-filter for single-phase grid-connected PV inverter system is presented. Due to the theoretical analysis, a comparative analysis between the designed LCL-filter with L-filter and LC-filter based single-phase grid-connected PV inverter system is carried out The simulation and experimental comparison results are given to validate the theoretical analysis and to show the effectiveness of filter design methodology.

3.2 Low-pass Output LCL-Filter Analysis and Design The LCL-filter equivalent circuit diagram is shown in Fig.3.5 (a) and its equivalent model is shown in Fig.3.5 (b), where Vi and Vg are inverter and grid voltage; Li, Lg, Ri, Rg are the filter inverter-side and grid-side inductor and its equivalent resistors, respectively. A damping resistor Rd for avoiding resonance phenomenon is in series with capacitor Cf. Based on the equivalent model of LCL-filter and adopting the Mason’s rule, the transfer function of LCL filter using the inverter current as feedback can be derived as (3.3). Equation (3.4) is obtained

- 59 -

when Ri and Rg are assumed to be small enough for neglecting.

Rd C f s + 1 ig(s) GLCL(s) = = vi(s) d s3 + d s 2 + d s + d 3 2 1 0

(3.3)

where d 3 = Li L g C f

(3.3.a)

d 2 = Li R g + L g Ri + æç Li + L g ö÷C f è ø

(3.3.b)

d1 = Li + L g + æç Ri Rd + R2 Rd + Ri R g ö÷C f è ø

(3.3.c)

d 3 = Ri + R g

(3.3.d)

Rd C f s + 1 ig(s) GLCL(s) = = vi(s) L L C s3 + L + L R C s 2 + L + L s g d f g i g f i i

(

)

(

)

(3.4)

The resonant frequency of LCL-filter is calculated as (3.2) and the frequency response of LCL-filter based on transfer function (3.4) is illustrated in Fig. 3.4. As shown in (3.4), the LCL-filter has two more zeros and two more poles when compared to a simple L-filter. If the transfer function in (3.4) is discretized and the PI controller is used, these additional zeros and poles can make the system unstable. This problem can be solved by using proper damping. The simple way is connecting a resistor in series with the filter capacitor as shown in Fig. 3.1(c), Fig. 3.5 (a) and the model of LCL-filter of Fig. 3.5 (b). The damping resistor moves the unstable pole more inside the stability area. The task of LCL-filter is to reduce the ripple and the high-order harmonic components on the grid current, but a weak design can make - 60 -

the filter has a lower attenuation ratio than expected. Moreover, a weak design can increase the distortion caused by the oscillation. In the real application, the harmonic components of inverter output current may cause the saturation of filter resonance. Hence, the filter design must consider about the inverter output current ripple and the damping component should be used to avoid the resonance. However, the damping is limited by the filter inductance value, losses and filter performance reduction. To design the LCL-filter, some limits on the parameters values have been studied in [49] and are revised as follows: a) The total inductance should be less than 0.1 (p.u) to limit the ac drop voltage during operation. Otherwise, a higher dc-link voltage will be required and this results in higher switching losses. b) The capacitance is limited by the reactive power factor (less than 5%). ω

c) The resonant frequency should be in range 10ω 0 £ ω res £ sw 2 to avoid resonance problems in the lower and upper parts of the harmonic spectrum, where w0 is the utility frequency (rad/s), wres is the resonant frequency (rad/s) and wsw is the switching frequency (rad/s). d) The damping element losses cannot be so high as to reduce system efficiency. e) Other constraints should be taken into account such as: The filter efficiency at low frequency (first 50 harmonics) and around the switching frequency; the current tracking capability. And some assumptions are given for single-phase LCL-filter design

- 61 -

procedure such as follows: - The switching frequency is much higher than the utility grid frequency. - The utility grid voltage is ideal sinusoid. - The system has unity power factor - The unipolar PWM is adopted To verify the filter effectiveness, the following performance factors are also used: 2 å I (h)

- The THD of the current THD = - The power factor PF =

h I (1)

.

I (1) cos (j ) . I

- The average of the absolute dc voltage error. - The maximum sideband current harmonics around the switching frequency. - The rms value of current harmonic components at switching frequency The following sub-sections will present the whole procedure of LCL-filter design. 3.2.1 Ripple current analysis and filter inductance calculation Fig. 3.6 (a) describes a single-phase grid-connected PV inverter using full-bridge topology. For a simple analysis, the fundamental component of grid current is assumed to be zero. Hence, the fundamental component of the filter inductor voltage is also zero:

- 62 -

V L = Vin - V g = 0

(3.5)

where VL is inductor voltage, Vin is the output inverter voltage and Vg is the grid voltage. By adopting unipolar PWM method, the inverter output voltage Vin has three-step values: Vdc, 0 and –Vdc, where Vdc is the dc-link voltage. Fig. 3.7 shows the output voltage and current waveform of singlephase full-bridge inverter. Because the switching frequency fsw is higher than the grid frequency f0, the time average value of the inverter output voltage Vav can be considered as constant during the switching time Ts. Hence, the filter inductor current of grid-connected singlephase full-bridge inverter during any switching period has a typical waveform as shown in Fig. 3.7 (b). In this case, the peak-to-peak value of the filter inductor current when using unipolar PWM switching method can be obtained as: V -Vav d1 I pp = 2I m = dc T L 2 s

(3.6)

where Ipp and Irpm are the peak-to-peak value and maximum value of filter inductor current ripple, respectively; Vdc is the dc-link voltage; Vav is the average value of inverter output voltage; L is the filter inductance value; d1 is the duty cycle and Ts is the switching time. During the interval of 0 0 g è on S off D ø æç t V + t V ö÷ for I < 0 g è off S on D ø

(4.3)

where VS and VD are the on-switching-period voltage dropped on switching device and diode.

- 105 -

Fig. 4.1 Inverter output voltage in case of positive current flow (Ig > 0) (a) Positive grid current flow (b) Inverter output voltage waveforms

- 106 -

Fig. 4.2 Inverter output voltage in case of negative current flow (Ig < 0) (a) Negative grid current flow (b) Inverter output voltage waveforms

- 107 -

Fig. 4.3 Relationship between the grid current’s sign and distorted voltage in PWM inverter

- 108 -

4.4 Distorted Voltage and Current Caused by Dead-time To simplify analysis, we assumed that the turn-on/turn-off times and the dropped voltages on switching device and diode are small enough to be ignored. The grid current and the distorted voltage caused by deadtime are shown in Fig. 4.3. By applying Fourier series, the single-phase harmonic component of inverter output voltage Vinv(h) in case of squarewave distorted voltage is calculated as follows: 4V ¥ 1 Vinv(h) = dc sin hw0t å p h=1,3,5,7... h

(

)

(4.4)

where Vinv(h) is the harmonic component of inverter output voltage, Vdc is the dc-link voltage, h is harmonic order and w0 is the grid frequency (rad/s). Hence, the distorted voltage in Fig. 4.3 can be obtained as follows:

Vd =

=

¥ 4DV 1 sin hw0t å p h=1,3,5,7... h

(

4DV p

( )

(

)

)

(

)

1 1 é ù êsin w0t + sin 3w0t + sin 5w0t + ...ú 3 5 ë û

(4.5)

The distorted voltages of the stationary reference frame can be obtained as (4.6) and (4.7), where the d-axis voltage is the distorted

d and the q-axis voltage Vd is a virtual orthogonal phase voltage Vds qs voltage, which is delayed 900 from the real one, is generated by using

- 109 -

an all-pass filter.

( )

(

)

(

)

d = Vds

1 1 4DV é ù sin w0t + sin 3w0t + sin 5w0t + ...ú ê p ë 3 5 û

d = Vqs

4DV p

( )

(

)

(

)

1 1 é ù ê- cos w0t + 3 cos 3w0t - 5 cos 5w0t + ...ú ë û

(4.6) (4.7)

The compensation voltages of synchronous reference frame can be obtained by using transformation matrix in (4.8): éV d ù é cos(wt ) sin (wt )ù éV d ù ê de ú = ê ê ds ú êV d ú ë- sin (wt ) cos(wt )úû êV d ú ë qe û ë qs û

(4.8)

Hence,

(

)

(

)

d = 4DV é 8 sin 4w t + 16 sin 8w t + ...ù Vde ú 0 63 0 p êë15 û d = V qe

(

)

(

(4.9)

)

4DV é 2 2 ù cos 8w 0 t + ...ú - 1 + cos 4w 0 t + 15 63 p êë û

(4.10)

Similarly, the relative distorted current can be obtained as follows:

Id =

¥ 4D V 1 sin h w 0t - q å pZ p (h) h=1,3,5,7... h

[(

)]

(4.11)

where Zp(h) is the plant impedance related to the harmonic components and q is related to the power factor.

- 110 -

To simplify the analysis, we assumed that the system has unity power factor, hence q = 00. By applying the Fourier series, the relative distorted current can be expressed as:

Id =

4D V p

é 1 ù 1 1 ê sin w 0t + sin 3w0 t + sin 5w0 t + ...ú 3Z p (3) 5Z p (5) êë Z p (1) úû

( )

(

)

(

)

(4.12) The relative distorted currents in stationary and synchronous reference frames can be obtained as (4.13~ (4.16):

d = I ds

4D V p

é 1 ù 1 1 ê sin w 0 t + sin 3w 0 t + sin 5w 0 t + ...ú Z (1) 3Z p (3) 5Z p (5) úû ëê p

( )

(

)

(

)

(4.13) d = I qs

4 DV p

é ù 1 1 1 êcos w 0 t + cos 3w0 t cos 5w 0 t + ...ú 3Z p (3) 5Z p (5) êë Z p (1) úû

( )

(

)

(

)

(4.14)

(

)

(

)

d = I de

4DV é 8 16 ù sin 4w0 t + sin 8w0 t + ...ú ê 63 p ë15 û

Iqe =

4DV p

(

)

(

)

2 2 é ù ê- 1 + 15 cos 4w0t + 63 cos 8w0t + ...ú ë û

(4.15) (4.16)

In (4.13) and (4.14), the distorted currents in the stationary reference frame contain the 1st, 3rd, 5th and higher harmonic components caused by dead-time. Similarly, the distorted currents of the synchronous reference frame contain the 4th, 8th and higher harmonic components caused by dead-time, as shown in (4.15) and (4.16). The dead-time

- 111 -

effect can be mitigated by the appropriately harmonic component mitigation, either in stationary or in synchronous reference frame.

4.5 Adaptive Harmonics Extraction Filter 4.5.1 Single-frequency harmonic extraction filter To track the unexpected harmonic component from a reference signal, a single-frequency harmonic extraction filter (SHEF), as shown in Fig. 4.4, is used. This adaptive frequency cancellation structure, which is mostly used in digital signal processing for the noise cancellation, was introduced in [71]-[73]. The principle of filter's operation is revised as follows: - The task of filter is to track the unexpected noise with already known frequency (e.g. frequency ω0) from the original signal dk, which contains the desired signal and unexpected noises, during the period t=kTs (where k is the discrete time index and T s is the sampling time). Firstly, the noise is split into two orthogonal (sine and cosine) components x sk and x ck . Then, these orthogonal components are provided to the filter. - Secondly, these two orthogonal components are fed to the LMS algorithm for adaptation process. The outputs of adaptation process are adaptation coefficients (or called weight functions) w sk and w ck . After that, the orthogonal components ( x sk and x ck )

will be multiplied with their corresponding adaptation coefficients ( w sk and w ck ). After multiplying with their corresponding adaptation coefficients,

these

- 112 -

orthogonal components

are

combined to be the adapted noise yk, which has the same magnitude and phase with the noise component to be tracked in the original signal dk. In other words, the noise component yk in the original signal dk can be tracked out. - Finally, the error εk between the original signal dk and the tracked noise yk is fed back to LMS algorithm for adaptation process. In here, LMS algorithm adjusts the adaptive coefficients to force the magnitude and phase of tracked noise yk exactly match with those of unexpected noise component of original signal dk. This error εk can be calculated as (4.17)

(

e k = d k - y k = d k - xks wks + xks wks

)

(4.17)

The LMS algorithm is expressed as follows: wk +1 = wk + 2 me k x k

(4.18)

where m is the adaption gain. The adaptation gain m should be selected to be small for taking enough (five or more) periods of harmonic component for weight functions wk can reach the final values. One of the slow adaptation results is attenuation of high frequency components from input ex to propagate to the output wk of LSM algorithm.

- 113 -

Fig. 4.4 Single-frequency harmonic tracking filter

x c = cos( t)

wc y

LMS algorithm ws

x s = sin( t)

G-1 p Inverse Plant

SHEF i* Regular control

e

PI

Vc

Controller

VSHEF Gp Vd

i

Plant

Fig. 4.5 Integration of a SHEF and regular plant control part

- 114 -

Fig. 4.6 Mitigation of harmonic components from single-phase grid current by using MSHEF

- 115 -

4.5.2 Combination of SHEF and regular plant control part In case of mitigating the high-order harmonic components, some assumptions should be made before integrating the SHEF with regular control part. These assumptions ensure that there is no interaction between the regular control part and SHEF, thus they can be analyzed separately: - The adaptation process of the filter is slow. Therefore, the SHEF does not interfere with dynamics and does not alter the transfer function of the plant and associated control. - The harmonic component to be tracked by the filter has a high frequency highly above the bandwidth of regular control. - The current harmonic magnitude changes slowly to allow SHEF adjusts the weight functions and eliminates the higher harmonic components. A simplified block diagram of SHEF and regular plant control integration in synchronous reference frame is shown in Fig. 4.5. Since the same structure is applied to both d and q reference axis component, the subscript d and q are omitted from variable in the block diagram and analysis. As shown in Fig. 4.5, the primary input d contains the command voltage Vc created by current controller and the harmonic distortion voltage component Vd with known frequency. The current is the error signal and is fed back to the LMS algorithm for adaptation process. To compensate the phase shift caused by the plant, the inverse plant transfer function G -p 1 is integrated to the output of SHEF. The objective is to eliminate undesired harmonic component from plant output which is created by Vd. The elimination is accomplished

- 116 -

when output of the SHEF VSHEF is equal to the harmonic component Vd.

4.5.3 Multi single-frequency harmonic extraction filters Fig. 4.6 shows the harmonics’ mitigations from the grid current by using multi single-frequency harmonic extraction filter (MSHEF). The 3rd and kth (k = 5, 7, 9…) harmonic components are tracked and mitigated from the grid current by using the same filtering structure, where the frequencies of reference signals must be equal to those of harmonic components to be mitigated. The output of SHEFs will be combined and integrated to regular control part. If using the same error input e for LMS algorithm of different filters, the tracked harmonic components may have the different magnitudes when compared with the real harmonic components obtained from grid current. Therefore, the different error signals ek will be used for corresponding frequency to be tracked. As shown in Fig. 4.6, the SHEF_1 is used to track the 1st harmonic component (or the ideal fundamental component) from the grid current. It is clearly to see that the ideal fundamental component acts as the noise to be tracked and the grid current acts as the original signal, which contains the fundamental and harmonic components. Although this step is not necessary, however it helps to reduce the adaptation time of tracking the other harmonic components. Since the ideal fundamental component is tracked out of the grid current by using SHEF_1, the output error of SHEF_1 contains only information of the other harmonic components. Hence, this output error is used for SHEF_k's

adaptation

process,

where

- 117 -

the

kth-order

harmonic

components are tracked for dead-time compensation. The output error between the ideal fundamental component created by the SHEF_1 and the real feedback grid current is also fed-forward to compensate the distorted voltage. To avoid the phase shift caused by the plant, an inverse plant transfer function is added to the output of MSHEF. It is noted that the reference input signals for extracting the fundamental and harmonic components are created from PLL. Furthermore, other harmonic components can be mitigated by applying the same tracking structures, but with different frequencies which to be mitigated. By using the same error input ε for multi-frequencies' mitigations, the total process can have a faster adaptation speed and the noises in other tracking filters are reduced.

4.6 Dead-Time Compensator For Single-phase GridConnected PV Inverter Based on MSHEF Due to the limited bandwidth (£ 1kHz), PI controllers are not able to suppress the high frequency components from current feedback. In case of high harmonic components mitigation from grid current, a dead-time compensator using aforementioned MSHEF (MSHEFDC) is applied, as shown in Fig. 4.7. Fig. 4.7 shows the integration between 1st and 3rd harmonic components' tracking filters and control part by using MSHEF. Since the same structure is applied to both d- and q-axis components, the subscript d and q are omitted from variable in the block diagram and analysis. Based on (4.13) and (4.14), the dominant 3rd harmonic component is chosen to be mitigated.

- 118 -

Fig. 4.7 Proposed dead-time compensator

Fig. 4.8 Single-phase PV inverter current control scheme with dead-time compensator

Fig. 4.9 Injected voltage generation by MSHEFDC

- 119 -

Since the MSHEF can mitigate the selective harmonic components, but it cannot completely compensate the lost voltage caused by deadtime. Hence, in the proposed MSHEFDC, the error between the grid current and the ideal fundamental component created by SHEF_1 is multiplied with the inverse plant transfer function and is feed-forward as another compensating voltage component. This injected voltage can be calculated as follows:

[ (

1 s s s s -1 -1 Ve _ ff = G p e 1 = G p (d - y1 ) = G p d - x1 w1 + x1 w1

)]

(4.19)

where G -p 1 is the estimated inverse plant transfer function. Hence, the injected voltage for dead-time compensation can be obtained as follows:

VMSHEFDC= VSHEF_ 3 + Ve _ ff

(4.20)

By integrating the MSHEFDC with regular current control part, the proposed current controller is shown in Fig. 4.8, where the detail of compensating voltages generation is shown in Fig. 4.9, respectively. Due to the d-q rotating frame theory, in single-phase application, the daxis in stationary reference frame exactly matches with the real signal. Therefore, to simplify the control structure, only d-axis component is considered as shown in Fig. 4.8. The dead-time compensation and harmonic elimination was done in the stationary rather than the synchronous reference frame because of - 120 -

avoiding the coupling problem in synchronous reference frame. In case of need, the other harmonic components can be mitigated by adding the same structure as 3rd harmonic component mitigation. The frequencies of the reference signals must be equal to those of the harmonic components to be eliminated.

4.7 Inverse plant transfer function analysis As descript in [70], the weight functions w sk and w ck change slowly then the variable in the system is essence of the adaptive algorithms. Hence, the slowly changing of w sk and w ck can be assumed to be constant during the sampling time Ts. With this assumption, the plant transfer function in output signal of MSHEF can act as a linear operator of sine and cosine inputs into the combiner. Therefore, G -p 1 can be implemented as a linear combination of sine and cosine functions from reference input. Thus, the dynamic inverse plant model can be avoided direct implementation. To illustrate the integration process, a low-pass output LC-filter is used as a plant. At switching frequency, LC filter acts as an L-filter when considering only the switching harmonic component. Therefore, an equivalent L-filter is used replace LC-filter, which can be considered as an inductance plant. The L-filter transfer function (or plant transfer function):

G p ( s) =

1 R + sL

(4.21)

- 121 -

To simplify analysis, the ESR of inductor is ignored. Hence, the transfer function in s-domain and in frequency-domain can be rewritten as (4.22) and (4.23), respectively:

G p (s ) =

1 sL

G p ( jw ) =

(4.22) 1 jwL

(4.23)

It is noted that the location of the break point for the pole is located at ω → 0, which is all the way to the left of the graph. If inserting 0 in for ω gives us an undefined value (which approaches negative infinity, as the limit). Because there is a single pole at zero, the graph to the right of zero (which is everywhere) has a slope of -20dB/decade and passes through 0dB at:

w =1 L

(4.24)

Adopting L has a value of 4mH, then the slop of the line will pass through 0dB at w = 250(rad/s) or f = 39.78Hz, as shown in Fig. 4.10. Hence, the inverse plant transfer function can be rewritten in sdomain and in frequency-domain can be rewritten as (4.25) and (4.26), respectively: G -p 1 ( s ) = sL

(4.25)

1 Gp ( jw ) = jw L

(4.26)

- 122 -

As shown in Fig. 4.11, the frequency response of the inverse plant is a slope of +20dB/decade, which approaches positive infinity as the limit and passes through 0dB at w = 250(rad/s) or f = 39.78(Hz). Based on the frequency response diagram, the inverse plant may act as a high-pass filter, where its frequency response approaches the positive infinity limitation. This may cause the instability of the whole system in operation. Fortunately, in the dead-time compensation by using selective harmonic component mitigation, only the low-order harmonic components such as 3rd- and 5th-order with known frequencies are considered because higher-order components have little effects to the inverter output voltage distortion. Because of that reason, we should limit the frequency range as 0Hz £ f £ 420Hz, corresponding to the 7th harmonic component (the fundamental frequency f0 = 60Hz), to force the frequency response of inverse plant reaches the steady-state. The transfer function of high-pass filter is defined as follows

G HPF ( s) =

s s+ p

(4.27)

where p is a pole To match the inverse plant transfer function with as shown in (4.27), an additional pole will be added into the inverse plant transfer function. So, the inverse plant transfer function can be rewritten as an equivalent high-pass filter in s-domain and in frequency-domain as (4.28) and (4.29), respectively:

- 123 -

Bode diagram of Plant 50

Gain(dB)

0

-50

-100 0 10

1

10

2

10

3

10

4

10

5

10

6

10

Phase(deg)

-89 -89.5 -90 -90.5 -91 0 10

1

10

2

10

3

10 Frequency (Hz)

4

10

5

10

6

10

Fig. 4.10 Bode diagram of plant (L-filter) Bode diagram of Inverse Plant

Gain(dB)

100 50

0

-50 0 10

1

10

10

2

3

10

10

4

5

10

10

6

Phase(deg)

91 90.5 90 89.5 89 0 10

1

10

10

2

3

10 Frequency (Hz)

10

4

5

10

10

6

Fig. 4.11 Bode diagram of inverse plant (L-filter) Bode diagram of Inverse Plant 40

Gain(dB)

20 0 -20 -40 0 10

1

10

2

10

3

10

4

10

5

10

6

10

100 Phase(deg)

80 60 40 20 0 0 10

1

10

2

10

3

10 Frequency (Hz)

4

10

5

10

6

10

Fig. 4.12 Bode diagram of high-pass filter based equivalent inverse plant

- 124 -

1 Gp ( s) = K

s s+ p

1 Gp ( jw ) = K

(4.28)

jw jw + p

(4.29)

To exactly match with the original inverse plant transfer function’s frequency characteristics as shown in Fig. 4.11, the frequency response of equivalent high-pass filter must pass through 0dB at w = 250 (rad/s) or f = 39.78 (Hz). Hence:

K

jw = jw L jw + p

(4.30)

With w =1/L:

K

1L »1 1 L+ p

Assuming that

(4.31)

1 1.508)

- 160 -

//PR controller iIde_ref = 0; iIqs_ref = 0; iIds_ref = 0; iIqs_ref = 0; iKp_Idqe = (int)(10*1024);

// Q10

iKr_Idqe = (int)(0.001*1024);

// Q10

w0 = (int)(377*1024); //Q20 wcut = (int)(15*1024);// Q10 in0 = (long)(35*1024); //Q10 in1 = (long)(-25*1024); //Q10 in2 = (long)(10.2283*1024); //Q10 id1 = (long)(-1.9956*1024); //Q10 id2 = (long)(0.9970*1024);

//Q10

} if ((samp_T >= 100.0e-6)) // 10kHz sampling { samp_T0

= t;

Theta = wTheta * 326; //Gt = 2048/max_wTheta = 2048 / (2*3.14) = 326 iSin = SIN4TBL[(int)Theta]; //Q15 iCos = COS4TBL[(int)Theta]; iSin1 = iSin;

//Q15

iCos1 = iCos;

//Q15

iVds = iVgrid;

//Q15

//Q6

iVqs = ((-lbeta * iVqs1)>>15) + ((lbeta * iVds)>>15) + (iVds1); iVds1 = iVds; //Q6 iVqs1 = iVqs; //Q6 iVde = (iVds * (long)iCos + iVqs * (long)iSin)>>15;

//Q6

iVqe = (-iVds * (long)iSin + iVqs * (long)iCos)>>15;

//Q6

- 161 -

//PLL iVerr_pll = iVde; lpi_pll

+=

//Q6

(long)iKp_pll

*

(iVerr_pll-iVerr_pll1)

+

(long)iKit_pll * iVerr_pll; //Q16 iVerr_pll1 = iVerr_pll; //Q6 if(lpi_pll > lpi_pll_limit)

lpi_pll = lpi_pll_limit; //Q16

if(lpi_pll < -lpi_pll_limit)

lpi_pll = -lpi_pll_limit; //Q16

wr_est = wr_60 + ((long)lpi_pll>>10);

//Q6

wTheta += ((long)wr_est)/64. * Ts_sec; //Q0 if( wTheta >= TWOPI ) //Q0 { wTheta -= TWOPI; //Q0} wTheta1 = wTheta; if (iIqe_ref > 7168) { iIqe_ref = 7168; } else {iIqe_ref = iIqe_ref+1; } //Reference current from synchronous -> stationary frame iIds_ref=(iIde_ref*(long)iCos + (-iIqe_ref)*(long)iSin)>>15; iIqs_ref

=

(iIde_ref

*

(long)iSin

+

iIqe_ref

*

(long)iCos)>>15;//Q10 iIds = iIgrid; //Q10 iIqs = -((lbeta * iIqs1)>>15) + ((lbeta * iIds)>>15) + (iIds1);//Q10 iIds1 = iIds;

//Q10

iIqs1 = iIqs;

//Q10

iIde = ((iIds * iCos)>>15) + ((iIqs * iSin)>>15);

//Q10

iIqe = ((-iIds * iSin)>>15) + ((iIqs * iCos)>>15); //Q10

- 162 -

//////////////////PR controller in stationary frame/////////////////////////// iIds_err = iIds_ref - iIds;

//Q10

iIds_pr_err += (long)(in0) * iIds_err + (long)(in1)*iIds_err1 + (long)(in2)*iIds_err2

-

(((long)(id1)*iIds_pr_err1)>>10)

-

(((long)(id2) * iIds_pr_err2)>>10); //Q20 iIds_err1 = iIds_err; //Q10 iIds_err2 = iIds_err1; //Q10 iIds_pr_err1 = iIds_pr_err;

//Q20

iIds_pr_err2 = iIds_pr_err1; //Q20 ipr_Ids = (iIds_pr_err)>>14; //Q6 iIqs_err = iIqs_ref - (iIqs);

//Q10

iIqs_pr_err += (long)(in0)*iIqs_err + (long)(in1)*iIqs_err1 + (long)(in2)*iIqs_err2

-

(((long)(id1)*iIqs_pr_err1)>>10)

(((long)(id2)*iIqs_pr_err2)>>10); //Q20 iIqs_err1 = iIqs_err;//Q10 iIqs_err2 = iIqs_err1;//Q10 iIqs_pr_err1 = iIqs_pr_err;//Q20 iIqs_pr_err2 = iIqs_pr_err1;//Q20 ipr_Iqs = (iIqs_pr_err>>14);//Q6 pwms = (long)(ipr_Ids>>6); //////////// Unipolar PWM compensation ///////////////////// pwm_p = pwms; pwm_m = -pwms; } out[0] = pwm_p; out[1] = pwm_m; }

- 163 -

-

Appendix

C

Dead-Time Compensation Implementation (PSIM)

#include #include // Sine and Cosine table //////////////////////// basic init /////////////////////////// static double

Fs= 10000; // sampling frequency

static double

Ts= 100.0;// sampling frequency(us)

static double

DFs

static double

Ts_sec=100e-6; // 1/sampling frequency

= 100e-6; // 1/sampling frequency

static double TWOPI = 2*3.1415926535897932384626433832795; static double ONEPI = 3.1415926535897932384626433832795; static double

samp_T, samp_T0 = 0 ;

static double

samp_T_bet, samp_T0_bet = 0 ;

static int

init=0;

///////////////////////// in //////////////////////////////////

- 164 -

static int

iVgrid=0, iIgrid=0; // grid voltage and current

static int

iVds=0,iVds1=0,iVqs=0,iVqs1=0, iVde=0,iVqe=0;

static int

iIds=0,iIds1=0,iIqs=0,iIqs1=0,iIde=0,iIqe=0;

///////////////////////// pll ///////////////////////////////// static long lbeta=0; static int

iKp_pll=0,iKit_pll=0;

static int

iVerr_pll=0,iVerr_pll1=0;

static long lpi_pll=0,lpi_pll_limit=0; static int

iCos,iSin,iCos1,iSin1,iSint;

///////////////////// PR controller /////////////////////////// static int

iI_ref=0;

static int

ipi_Ide=0,ipi_Iqe=0;

static int

ipi_Vde=0,ipi_Vqe=0;

static int

iKp_Idqe=0,iKit_Idqe=0;

static long lWL=0; static int

iVde_cmd,iVqe_cmd,iVde_cmd1,iVqe_cmd1;

static int

iVds_cmd,iVqs_cmd,iVds_cmd1,iVqs_cmd1;

static int

in0,in1,in2,id1,id2;

static int

w0=0,wcut=0;

static int

iKr_Idqe=0;

static long ipr_Ids=0,iIds_err=0,iIds_err1=0,iIds_err2=0; static long ipr_Iqs=0,iIqs_err=0,iIqs_err1=0,iIqs_err2=0; static long iIds_pr_err=0,iIds_pr_err1=0,iIds_pr_err2=0; static long iIqs_pr_err=0,iIqs_pr_err1=0,iIqs_pr_err2=0; static int

iIde_cmd=0,iIqe_cmd=0;

static int

iIds_cmd=0,iIqs_cmd=0;

static int

iIde_ref=0,iIqe_ref=0,iIds_ref=0,iIqs_ref=0,iWL=0;

- 165 -

//////////////////////// pwm Control ////////////////////////// static double

pwm_dc, pwm, pwm_m,pwm_p;

static double

Vcmd,Vcmd1;

static double

antiu, antid;

static double

fault;

static double

pwms,pwms1;

__declspec(dllexport) void simuser (t, delt, in, out) // Note that all the variables must be defined as "double" double t, delt; double *in, *out; { iIgrid

= (int)(in[2] * 1024); //Q10

iVgrid

= (int)(in[3] * 64);//Q6

samp_T = t -samp_T0 ;//+ 50.0e-6; // + 25.0e-6 ; samp_T_bet = t -samp_T0_bet + 50.0e-6 ; if (init==0) { init

= 1;

lbeta = -31555;

//Q15

iKp_pll = (int)(1 * 1024);//Q10

//28.6

iKit_pll = (int)(0.05 * 1024);//Q10

//1.26998

wr_60 = (int)(TWOPI * 60 * 64);//Q6 wr_3rd = (int)(TWOPI * 180 * 64);//Q6

//3rd harmonic

wr_5th = (int)(TWOPI * 300 * 64);//Q6

//5th harmonic

lpi_pll_limit = (long)(TWOPI * 3 * 65536);//Q16

- 166 -

wTheta =0; wTheta1 = 0; iKp_Idqe = (int)(10 * 1024); //10 iKit_Idqe = (int)(0.2 * 1024);//0.4 lWL = (long)(1.7 * 32768);//Q15 (2mH ->1.508) //////////// dead-time compensation ///////// iepsilon = 0; iepsilon1 = 0; iW_fund_sin = 0;

iW_fund_cos = 0;

iW_fund_sin1 = 0;

iW_fund_cos1 = 0;

iW_3rd_sin = 0;

iW_3rd_cos = 0;

iW_3rd_sin1 = 0;

iW_3rd_cos1 = 0;

Vcmps_ds = 0;

Vcmps_qs = 0;

I_fund_cmps = 0;

I_3rd_cmps = 0;

} if ((samp_T >= 100.0e-6))

// 10kHz sampling

{ samp_T0

=t;

Theta = wTheta * 326; Theta3rd = wTheta3rd * 326; iSin = SIN4TBL[(int)Theta];

//Q15

iCos = COS4TBL[(int)Theta];

//Q15

iSin3rd = SIN4TBL[(int)Theta3rd];

//Q15

iCos3rd = COS4TBL[(int)Theta3rd];

//Q15

iSin1 = iSin;

iCos1 = iCos;

//Q15

iSin3rd1 = iSin3rd;

iCos3rd1 = iCos3rd;

//Q15

- 167 -

//

ALL PASS FILTER for Vq

iVds = iVgrid;

//Q6

iVqs = ((-lbeta * iVqs1)>>15) + ((lbeta * iVds)>>15) + (iVds1); iVds1 = iVds;

//Q6

iVqs1 = iVqs;

//Q6

iVde = ((iVds * iCos)>>15) + ((iVqs * iSin)>>15);

//Q6

iVqe = ((-iVds * iSin)>>15) + ((iVqs * iCos)>>15);

//Q6

iVde = (iVds * (long)iCos + iVqs * (long)iSin)>>15;

//Q6

iVqe = (-iVds * (long)iSin + iVqs * (long)iCos)>>15;

//Q6

iVerr_pll = iVde; lpi_pll

+=

//Q6

(long)iKp_pll

*

(long)iKit_pll * iVerr_pll;

(iVerr_pll-iVerr_pll1) //Q16

iVerr_pll1 = iVerr_pll; //Q6 if(lpi_pll > lpi_pll_limit)

lpi_pll = lpi_pll_limit; //Q16

if(lpi_pll < -lpi_pll_limit)

lpi_pll = -lpi_pll_limit; //Q16

wr_est = wr_60 + ((long)lpi_pll>>10);

//

//1st component wTheta += ((long)wr_est)/64. * Ts_sec; //Q0 if( wTheta >= TWOPI ) //Q0 { wTheta -= TWOPI; //Q0 } wTheta1 = wTheta; //3rd harmonic wTheta3rd += (3.* (long)wr_est)/64. * Ts_sec; //Q0 if( wTheta3rd >= TWOPI ) { wTheta3rd -= TWOPI;

}

wTheta3rd1 = wTheta3rd; // Soft start current control

- 168 -

+

if (iIqe _ref > 7168) { iIqe _ref = 7168; } else { iIqe _ref = iIqe _ref+1; } //

ALL PASS FILTER for Iq

iIds = iIgrid; //Q10 iIqs = -((lbeta * iIqs1)>>15) + ((lbeta * iIds)>>15) + (iIds1); iIds1 = iIds; //Q10 iIqs1 = iIqs;

//Q10

iIde = ((iIds * iCos)>>15) + ((iIqs * iSin)>>15);

//Q10

iIqe = ((-iIds * iSin)>>15) + ((iIqs * iCos)>>15);

//Q10

iIde = (iIds * (long)iCos + iIqs * (long)iSin)>>15; //Q10 iIqe = (-iIds * (long)iSin + iIqs * (long)iCos)>>15; //Q10 ///////////////////////////////////// id ////////////////////////////////////////// iIde_err = 0 - iIde ; //Q10 ipi_Ide

+=

(long)iKp_Idqe

*

(iIde_err-iIde_err1)

+

(long)iKit_Idqe * iIde_err; //Q20 iIde_err1 = iIde_err; //Q10 ipi_Vde = ipi_Ide>>14;//Q6 ///////////////////////////////////// iq ///////////////////////////////////// iIqe_err = iIqe_ref - (iIqe); ipi_Iqe += iKp_Idqe * (iIqe_err-iIqe_err1) + iIqe_err; iIqe_err1 = iIqe_err; ipi_Vqe = ipi_Iqe>>14;//Q6 /////* Dead-time compensation */////// //1ST

- 169 -

iKit_Idqe *

iW_fund_sin =

iW_fund_sin1 + ((int)(3e-2 * 64) *

(long)iepsilon1 * (long)(iSin1>>15)); //Q12 iW_fund_sin1 = iW_fund_sin; iW_fund_cos =

//Q1

iW_fund_cos1 + ((int)(3e-2 * 64) *

(long)iepsilon1 * (long)(iCos1>>15)); //Q12 iW_fund_cos1 = iW_fund_cos; I_fund_cmps

=

//Q12

((iW_fund_sin>>6)

(iW_fund_cos>>6) * (long)iCos)>>15;

*

(long)iSin

+

//Q6

//3RD iW_3rd_sin

=

iW_3rd_sin1

+

((int)(3e-2

*

64)

*

*

64)

*

(long)iepsilon_3rd1 * (long)(iSin3rd1>>15)); //Q12 iW_3rd_sin1 = iW_3rd_sin; //Q12 iW_3rd_cos =

iW_3rd_cos1

+ ((int)(3e-2

(long)iepsilon_3rd1 * (long)(iCos3rd1>>15)); //Q12 iW_3rd_cos1 = iW_3rd_cos; //Q12 I_3rd_cmps = (((iW_3rd_sin) * (long)iSin3rd + (iW_3rd_cos) * (long)iCos3rd)>>15); //Q12 iepsilon_3rd = iepsilon - (long)(I_3rd_cmps>>6); //Q6 iepsilon_3rd1 = iepsilon_3rd; //Q6 //Injected voltage for compensation Icmps = (long)(I_3rd_cmps>>6) ;//Q6 iepsilon = (iIgrid>>4) - (I_fund_cmps); //Q6 iepsilon1 = iepsilon; //Q6 //

HIGH PASS FILTER for Vcmps

lVcmps_in = (long)(Icmps); //Q6 lVcmps_temp = (long)(21.1 * 64) * (lVcmps_in - lVcmps_in1) + (((long)(0.473 * 64)*lVcmps_temp1)>>6);//Q12

- 170 -

iVcmps_out = (long)(lVcmps_temp>>6);//Q6 lVcmps_in1=lVcmps_in;//Q6 lVcmps_temp1=lVcmps_temp;//Q12 //

HIGH PASS FILTER for iepsilon

lepsilon_in = (long)(iepsilon); //Q6 lepsilon_temp

=

(long)(19.4142

*

64)*

(lepsilon_in

-

lepsilon_in1) + (((long)(1 * 64)*lepsilon_temp1)>>6);//Q12 lepsilon_out = lepsilon_temp>>6;//Q6 lepsilon_in1 = lepsilon_in;//Q6 lepsilon_temp1=lepsilon_temp;//Q12 ///////////////////////////////////// cmd ///////////////////////////////////// iVqe_cmd = (ipi_Vqe + (((long)iIde * lWL)>>19) + (iVqe)); iVde_cmd = (ipi_Vde) - (((long)iIqe * lWL)>>19); iVds_cmd =

(iVde_cmd * (long)iCos + (-iVqe_cmd) *

(long)iSin)>>15; iVqs_cmd =

(iVde_cmd * (long)iSin + iVqe_cmd *

(long)iCos)>>15;

//Q6

iVds_cmd1

iVds_cmd

=

-

(long)(lVcmps_out)

(long)(lepsilon_out); pwms1 = (long)(iVds_cmd1>>6); //Q0 //////////// Unipolar PWM compensation ///////////////////// pwm_p = pwms1; pwm_m = -pwms1; } out[0] = pwm_p; out[1] = pwm_m; out[2] = I_fund_cmps/64.;

- 171 -

-

out[3] = iW_fund_sin/4096.; out[4] = iW_fund_cos/4096.; out[5] = I_3rd_cmps/4096.;//iepsilon/64.; out[6] = iW_3rd_sin/4096.;//iW_fund_sin/4096.; out[7] = iW_3rd_cos/4096.;//iW_fund_cos/4096.; out[8] = Icmps/64.;//iW_fund_sin/4096.; out[9] = lepsilon_out/64.;//iW_fund_cos/4096.; out[10] = lVcmps_out/64.; out[11] = iepsilon/64.; out[12] = iepsilon_3rd/64.; }

- 172 -

초록*

단상 태양광 인버터의 전력품질 향상 방안 연구

부우충기엔

충남대학교 대학원 정보통신공학부 통신시스템 전공

(지도교수: 성세진)

최근 들어 경제적이고 환경적인 이유로 화석연료를 대체할 새로운 해결책에 대한 관심이 집중되며, 녹색과 청정의 이미지를 가진 신재생 에너지가 그 대안으로 거론되고 있다. 신재생 에너지원으로부터 고품질, 고효율로 전력을 생산하기 위해서는 계통 연계형 시스템과 같은 전력전자분야에 대한 연구개발 투자가 비약적으로 증대되고 있다. 친환경 전력수요가 증가됨에 따라 풍력과 태양광 같은 친환경 에너지원을 이용한 분산전원시스템이 계통에의 접속이증가하고 있으며, 중대형 * A dissertation submitted to the committee of Graduate School, Chungnam National University in a partial fulfillment of the requirements for the degree of Doctor of Philosophy conferred in February 2011.

- 173 -

시스템은 삼상 계통 연계형 인버터가, 소형에는 단상 계통 연계형 인버터가 사용되고 있다. 본 논문에서는 이와 같은 친환경 신재생 에너지원으로부터 계통으로의 전력 흐름을 조절하며 전력품질을 향상시키는 요소 기술로써 PR (Proportional-Resonant) 전류 제어기법, LCL 필터 설계기법과 온라인 데드타임 보상기법을 연구하였다. 첫번째로, 소형 분산전원시스템에서 사용되는 단상 인버터의 제어 방법은 삼상 인버터의 분석과 제어에 널리 사용되는 d-q 변환 기법을 변형하여 단상용으로 개발하였다. 이 d-q 변환 기법은 시스템의 동작을 위해서는 두 개의 독립적인 90도의 위상차를 가진 물리량이 요구되며, 단상 시스템의 경우 두 번째의 90도 지연된 물리량은 가상으로 만들어진다. 이 가상으로 만들어진 물리량은 추가적인 하드웨어 없이 전역 필터를 사용하여 DSP로 구현하였다. d-q 변환 기법과 연계된 PR (Proportional-Resonant) 제어기는 전통적인 PI (Proportional-Integral) 제어기에 비하여 빠른 응답 특성, 정상상태 에러 제로, 외란 제거, 계통 연계 모드에서 사용될 때 낮은 THD를 발생시킨다. 두 번째로, 인버터와 계통 사이에 사용되는 교류 필터로써, 3차의 LCL (Inductor-Capacitor-Inductor) 필터의 설계, 해석 기법을 연구하였다. LCL 필터는 1차의 L (Inductor) 필터와 비교하면 적은 인덕턴스로 스위칭 주파수 성분의 리플 전류를 더 효과적으로 감쇄시킬 수 있으나, 계통의 왜곡된 전압이나 고조파 전류와 공진을 일으킬 수 있으므로 제어기 설계에 주의가 요구된다. 본 논문에서는 LCL 필터의 설계법을 제안하였으며, 댐핑 저항을 사용하여 공진 현상을 완화하는 방법을 택하였다. 세번 째는 온라인 데드타임 보상 기법이다. 인버터에 사용되는 스위칭 소자인 MOSFET 나 IGBT는 높은 전력품질을 유지하기 위해서 10kHz 정도의 높은 스위칭 주파수로 동작한다. 같은 폴에 직렬로 연결된 두 개의 스위칭 소자의 턴 온/ 오프 시간에 의한 단락을 방지하기 위하여 데드타임을 삽입한다. 이 데드타임은 출력전압 감소, 저차 고조파 전류 발생과 전동기 구동에 사용될 시는 특정 주파수에서 공진을 발생시킬 수 있다. 본 논문에서는 단상 인버터에서 데드 타임에 의해 발생하는 고조파 성분을 해석하였으며, 적응 필터를 사용하여 온라인으로

- 174 -

데드 타임을 보상하는 기법을 개발하였다. 이와 같이 단상 계통 연계형 인버터의 전력 품질 향상을 위한 요소 기술로써 전류 제어기법, 필터 설계기법과 데드 타임 보상기법을 제안, 시뮬레이션, 비교 분석하였으며, 3KW 단상 계통 연계 태양광 인버터를 통한 실험으로 그 성능을 확인하였다.

- 175 -

- 176 -

Acknowledgements This dissertation arose in part out of years of research that has been done since I came to Intelligent Power Conversion Laboratory, Chungnam National University. By that time, I have worked with a great number of people whose contribution in assorted ways to the research and the making of the dissertation deserved special mention. It is a pleasure to convey my gratitude to them all in my humble acknowledgment. In the first place, I would like to record my gratitude to Prof. Se-Jin Seong and Prof. Hanju Cha for their supervisions, advices and guidance from the very early stage of this research as well as giving me extraordinary experiences throughout the work. Above all and the most needed, they provided me unflinching encouragements and supports in various ways. I am indebted to them more than they know. I would like to thank my previous academic advisors, Prof. Phan Xuan-Minh and Prof. Nguyen Doan-Phuoc in Hanoi University of Science and Technology, for their guidance, encouragements and recommendations for my further training at Chungnam National University. I gratefully thank Prof. In-Ho Hwang, Prof. Tae-Kyung Sung and Dr. Sang-Jin Kim for their constructive comments and great suggestions on this dissertation. I am thankful that in the midst of all their activity, they

- 177 -

accepted to be members of the reading committee. I would like to be grateful to many professors at Department of Information and Communication Engineering and Department of Electrical Engineering, Chungnam National University for their encouragement, instruction and stimulation during my doctor course. It is a pleasure to pay tribute also to the sample collaborators. Many thanks go to my seniors, Dr. Jungwan Choi, Mr. Young-Sik Cho, Mr. Byung-Ha Lee and Dr. In-Soo Kim for their encouragements, their helps and sharing moment excitements. I gratefully acknowledge Sanghoey Lee for his advice, supervision, and crucial contribution, which made him a backbone of this research and so to this dissertation. His involvement with his originality has triggered and nourished my intellectual maturity that I will benefit from, for a long time to come. To Soon-Ho Choi and Woo-Jung Kim, I would like to thank for their high quality helps at my beginning time. Many thanks go in particular to Huyn-Joo Lim for her precious times and valuable helps in Korean life. To Seunggoo Lee, Youngju Kang, Wujong Lee, Xia Wei, Seungwoo Chae, Jongkyung Lee and Taesub Kang, it is a pleasure to collaborate with you. I have shared the hardship, daunting difficulties and moment excitement, and happiness with all of you during the working time. Many thanks go to you for giving me such a pleasant time when working together with you since I knew you in Intelligent Power Conversion Laboratory.

- 178 -

Collective and individual acknowledgments are also owed to my colleagues at Department of Electrical Engineering, whose present somehow perpetually refreshed, helpful, and memorable meeting and dry humor about engineer’s life. Where would I be without my family? My parents deserve special mention for their inseparable support and encouragement. My father, Vu Dinh-Quy, in the first place is the person who put the fundament my learning character, showing me the joy of intellectual pursuit ever since I was a child. My mother, Nguyen Thi-Hue, is the one who sincerely raised me with her caring and gently love. Many thanks go to my brother Vu Huy-Giao, his wife Nguyen Thi-Nhung, and his son Vu Thai-Binh, for their helps during years. Also, I gratefully acknowledge my sister Vu Phuong-Lan, her husband Nghiem Quoc-Dat, and her daughter Nghiem Phuong-Thao for their encouragements, supports and helps since the first day in Korea. Words fail me to express my appreciation to my wife Le Thi HongHanh whose dedication, love and persistent confidence in me, has taken the load off my shoulder. I owe her for being unselfishly let her intelligence, passions, and ambitions collide with mine. Therefore, I would also thank my father-in-law Le Ky-Thanh, for letting me take her hand in marriage, for accepting me as a member of the family, and for his thoughtful supports and encouragements during years, warmly. Furthermore, to my wife’s brother Le Song Hiep’s family with their support and encouragement, thank you. To Tran Anh-Dung, it is a pleasure to have you as my best friend. I

- 179 -

have shared the daunting difficulties, moment excitement, and happiness with you during years. Many thanks go to you for giving me such a pleasant time and help when study together with you since I knew you. Hope to see your success in life soon. Thanks go to my friends in Vietnamese student organization in Chungnam National University, who were always willing to help me out. I cannot list all of you here. However, I always appreciate you. Finally, I would like to thank everybody who was important to the successful realization of dissertation, as well as expressing my apology that I could not mention personally one by one. Without them none of this would have been possible.

Trung-Kien Vu January 15th, 2011 Daejon, Korea

- 180 -

VITA Trung-Kien Vu He was born on October 27, 1979, in Hanoi, Vietnam. He obtained his B.S. in Department of Electrical Engineering from Hanoi University of Science and Technology, Hanoi, Vietnam in 2001. After working in company for two years, he returned to academic environment and received his M.S. degrees in Department of Electronics Engineering from Chungnam National University, Daejeon, Korea in 2005. He had finished course-works for his Ph.D. study in Department of Information and Communications Engineering at Chungnam National University in 2007. He is supposed to obtain his doctorate degree on February 25, 2011. His interest areas are power conversion system, control, renewable energy systems and applications of microprocessor to power systems. He has been a student member of Institute of Electrical and Electronics Engineers (IEEE), a member of the Korean Institute of Electrical and Electronics Engineers (KIEE) and the Korean Institute of Power Electronics (KIPE).

January 15th, 2011

- 181 -