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Development of Modular Grid Architecture for
Decentralized Generators in Electrical Power Supply
System with Flexible Power Electronics

Alaa Faisal Mohd

A thesis submitted in partial fulfilment of the
requirements of The University of Bolton
for the degree of Doctor of Philosophy

This research programme was carried out
in collaboration with
The University of Applied Sciences
South Westphalia - Soest Campus
Division of Electrical Engineering
Soest, Germany

February 2009

ABSTRACT
Fossil fuels are currently the major source of energy in the world. However, as
the world is considering more economical and environmentally friendly
alternative energy generation systems, the global energy mix is becoming more
complex. Factors forcing these considerations are (a) the increasing demand for
electric power by both developed and developing countries, (b) many
developing countries lacking the resources to build power plants and distribution
networks, (c) some industrialized countries facing insufficient power generation
and (d) greenhouse gas emission and climate change concerns. Renewable
energy sources such as wind turbines, photovoltaic solar systems, solar-thermo
power, biomass power plants, fuel cells, gas micro-turbines, hydropower
turbines, combined heat and power (CHP) micro-turbines and hybrid power
systems will be part of future power generation systems.
Nevertheless, all of these sources require interfacing units to provide the
necessary crossing point to the grid. The core of these interfacing units is
power electronics technologies since they are fundamentally multifunctional and
can provide not only their principle interfacing function but various utility
functions as well.
The focus of this dissertation is on developing different and various robust
control approaches for realistic distributed power system with power electronics
inverters as front-end (by paralleling power electronic inverters). These control
strategies should guarantee real modularity, higher reliability and true
redundancy to qualify to be standardised.
It will be shown that the existing control techniques have their own
characteristics, objectives, limits and appropriate uses. That often makes it
difficult to adapt one control scheme for all applications which most of the
current approaches are trying to introduce “one-size-fits-all solution”. The fact is
that each system’s (customer’s) needs are different and various approaches are
needed to fit their exact specifications.
Based on that, this research study develops a theoretical system concept
including original control concepts, which can assist the current efforts in
designing, building and operating a smart power system that is more flexible,
efficient, reliable and environmentally friendly. This work introduces various
opportunities of control functions for three-phase inverters used to feed various
passive/active grids including different topologies to feed balanced/unbalanced
loads. These are based on standardized system concepts using various control
strategies and no one-size-fits-all solution.
Finally, the developed system concept is verified through simulation models in
MATLAB/SIMULIK to show the feasibility of the new system philosophy and the
effectiveness of the control and management functions.
Laboratory experiments are carried currently by another team in the lab and
initial results are promising.

DEDICATION

Dedicated to my parents:
Faisal Saleem Tellawi and Asma Masoud Al-Shaaer.

ACKNOWLEDGMENTS
This research study was carried out between January 2006 and March 2009 in
the department of Power Systems and Power Economics, Soest division of the
University of Applied Science South Westphalia (Germany), in partial fulfilment
of the requirements of the University of Bolton (UK) for the Doctor of Philosophy
degree. It was financially supported by the Federal Ministry of Education and
Research in Germany.

I would like to express my acknowledgment of the tremendous support provided
to me by my supervisor Prof. Dr.-Ing. Egon Ortjohann from Soest division
throughout the entire research study period. In particular, I would like to
appreciate the long hours he made himself available for me, allowing for a
maximum knowledge transfer.

I would like to acknowledge my supervisor and director of studies Prof. Dr.
Danny Morton from the University of Bolton for his distinguished supervision
and excellent support, which contributed greatly to the successful finalisation of
this research study.

I would like to express my appreciation to Prof. Dr. Henrik Janzen for having
faith in me by recommending me for this position and supporting me in every
possible way.

I would like to express my thanks to Dr. Osama Omari from the Arab American
University-Jenin for his great contribution in improving and revising this work. I
would like as well to express my respect to him for his belief and faith in the
freedom of his country and his people.

I am also very grateful to my colleagues in the Laboratory of Power Systems
and Power Economics. I would like here to particularly acknowledge Mr.
Nedzad Hamsic (Dipl.-Ing.) and Mr. Worpong Sinsukthavorn (M.Sc. Eng.) for

iii

their contributions in developing the simulation models used in this research
study. I would like also to thank Mr. Max Lingemann (Dipl.-Ing., M.Sc.), Mr.
Andreas Schmelter (Dipl.-Ing.) and Mr. Manfred Wilczek (Dipl.-Ing.) for all the
ideas, technical support and discussions. My thanks go also to all
postgraduates, undergraduates and other academics who worked in the lab
through this period.

Finally, I thank every body that has participated in editing and reviewing this
thesis and in revising the presentations prepared during this research study.

DECLARATION
No portion of the work presented in this dissertation has been submitted in
support of another award or qualification either at this institution or elsewhere.

TABLE OF CONTENTS
ABSTRACT

DEDICATION

ACKNOWLEDGMENTS ................................................................................... iii
DECLARATION

LIST OF FIGURES ........................................................................................... ix
LIST OF TABLES

xii

LIST OF ABBREVIATIONS............................................................................ xiii
CHAPTER 1 INTRODUCTION .......................................................................... 1
1.1

A Brief History of Electric Power Systems ........................................... 2

1.2
Electric Power Supply Systems Today ................................................ 4
1.2.1
Conventional Power Supply System............................................. 4
1.2.2
Distributed Generation.................................................................. 6
1.3
Future Power Supply Systems (Smart Grids) ...................................... 8
1.3.1
Drivers Towards Smart Grids ....................................................... 9
1.3.2
Key Challenges for Smart Grids ................................................. 11
1.4

Problem Statement ............................................................................ 12

1.5

Motivation and Justification................................................................ 14

1.6

Research Aims .................................................................................. 15

1.7

Research Contributions ..................................................................... 17

1.8

Organisation of the Thesis ................................................................. 18

CHAPTER 2 STATE-OF-THE-ART ................................................................. 21
Introduction

2.1
Single Inverters .................................................................................. 22
2.1.1 Stand-alone Inverters....................................................................... 22
2.1.2 Grid Connected Inverters ................................................................. 22
2.1.3 Interactive Inverters ......................................................................... 22
2.2
Paralleled Inverters ............................................................................ 23
2.2.1 Master/Slave Control Techniques .................................................... 24
2.2.2 Current/Power Deviation (Sharing) Control Techniques .................. 27
2.2.3 Frequency and Voltage Droop Control Techniques ......................... 30
a.
Adopting Conventional Frequency/Voltage Droop Control ......... 30
b.
Opposite Frequency/Voltage Droop Control ............................... 34
c.
Droop Control in Combination with Other Methods .................... 36
2.3
Discussion ......................................................................................... 39
2.3.1
Master/Slave Control Techniques............................................... 40
2.3.2
Current/Power Deviation (Sharing) Control Techniques ............. 40
2.3.3
Frequency and Voltage Droop Control Techniques .................... 40
2.3.4
Summery ................................................................................... 41
CHAPTER 3 THE PROPOSED SMART GRID PHILOSOPHY
“ARCHITECTURE AND COMPONENTS”.............................. 42

3.1

General Architecture of the Proposed Smart Grid (Feeding Modes) . 42

3.2
Inverter Topologies ............................................................................ 45
3.2.1
Three-phase, Three-leg Voltage Source Inverters...................... 46
3.2.2
Three-phase, Three-leg, Four-wire Voltage Source Inverters .... 46
3.2.3
Three-phase, Four-leg Voltage Source Inverters........................ 47
3.3
Inverter Control .................................................................................. 48
3.3.1
Symmetrical Grid Forming .......................................................... 48
3.3.2
Symmetrical Grid Supporting ...................................................... 49
3.3.3
Symmetrical Grid Parallel ........................................................... 50
3.3.4
Asymmetrical Grid Forming ........................................................ 51
3.3.5
Asymmetrical Grid Supporting .................................................... 52
3.3.6
Asymmetrical Grid Parallel ......................................................... 54
3.4
Space Vector Modulation (SVM) ....................................................... 54
3.4.1
SVM for Three-phase, Three-leg Voltage Source Inverters....... 55
3.4.2
SVM for Three-leg, Four-wire Voltage Source Inverters ............. 61
3.4.3
SVM for Three-phase, Four-leg Voltage Source Inverters.......... 62
3.5

Sequence Decomposition .................................................................. 63

3.6

Discussion ......................................................................................... 66

CHAPTER 4 THE PROPOSED SMART GRID PHILOSOPHY “OPERATION,
CONTROL, AND MANAGEMENT” ........................................ 68
4.1
Multi-inverter Three-wire System Control Philosophy ........................ 71
4.1.1
Supervisory Control and Energy Management Scenario ............ 72
4.1.2
Droop Control Functions Scenario.............................................. 76
4.1.2.1 Analysis of Frequency and Voltage Droop Control Techniques 76
4.1.2.2 Grid Forming Inverter with Droop Control.................................. 81
4.1.2.3 Grid Supporting Inverter with Droop Control ............................. 87
4.1.3
Isochronous Control Functions Scenario .................................... 90
4.1.4
Combined Isochronous/Droop Control Functions Scenario ........ 92
4.1.5
Swing-Inverter/Droop Control Function Scenario ....................... 94
4.2
Multi-inverter Four-wire System Control Philosophy .......................... 96
4.2.1
Supervisory Control and Energy Management Scenario ............ 97
4.2.2
Droop Control Functions Scenario............................................ 100
4.2.2.1 Asymmetrical Grid Forming Inverter with Droop Control ......... 100
4.2.2.2 Asymmetrical Grid Supporting Inverter with Droop Control ..... 101
4.2.3
Isochronous Control Functions Scenario .................................. 104
4.2.4
Combined Isochronous/Droop Control Functions Scenario ...... 107
4.2.5
Swing-Inverter/Droop Control Functions Scenario.................... 110
4.3
Additional Aspects ........................................................................... 112
4.3.1
Role of Energy Storage Systems.............................................. 112
4.3.2
Nonlinearity .............................................................................. 112
4.3.3
Harmonics ................................................................................ 114
4.3.4
Stability ..................................................................................... 115
4.4

Discussion ....................................................................................... 116

CHAPTER 5 THE PROPOSED SMART GRID PHILOSOPHY “VERIFICATION
BY SIMULATION” ................................................................. 118
5.1
Multi-inverter Three-wire System Simulation Models and Results ... 119
5.1.1
Supervisory Control and Energy Management Scenario .......... 120
vii

5.1.2
5.1.3
5.1.4
5.1.5

Droop Control Functions Scenario............................................ 126
Isochronous Control Functions Scenario .................................. 133
Isochronous-droop Control Functions Scenario ....................... 137
Swing-Inverter and Droop Control Functions Scenario............. 141

5.2
Multi-inverter Four-wire System Simulation Models and Results ..... 146
5.2.1
Supervisory Control and Energy Management Scenario .......... 146
5.2.2
Droop Control Functions Scenario............................................ 151
5.2.3
Isochronous Control Functions Scenario .................................. 157
5.2.4
Isochronous-droop Control Functions Scenario ....................... 162
5.2.5
Swing-inverter and Droop Control Functions Scenario ............. 167
5.3

Discussion ....................................................................................... 171

CHAPTER 6 CONCLUSIONS AND FURTHER WORK ................................ 173
6.1

Conclusions ..................................................................................... 173

6.2

Further Work .................................................................................... 175

APPENDIX
177
A.1
SVM for Three-leg, Four-wire Voltage Source Inverters ........... 177
a) Zero-Vector Approach ........................................................................ 177
b) Compensated Vectors Approach ........................................................ 185
A.2
SVM for Three-phase, Four-leg Voltage Source Inverters........ 190
A.3
Inverter control in DQ ............................................................... 196
A.4
Generalised Integrator “The Selective Filter” ............................ 199
LIST OF PUBLICATIONS ............................................................................. 206
REFERENCES

210

viii

LIST OF FIGURES
Fig. 1. 1: Regions of different proportions of population not connected to the electrical
grid. ........................................................................................................................... 6
Fig. 1. 2: Future Network Vision [8]................................................................................. 9
Fig. 1. 3: General grid architecture. ............................................................................... 13
Fig. 1. 4: Desired System Structure. ............................................................................... 17
Fig. 1. 5: The structure of this thesis ............................................................................... 20
Fig. 2. 1: a) Stand-alone inverter, b) Grid connected inverter, Grid interactive inverter.
................................................................................................................................. 23
Fig. 2. 2: Combined voltage and current controlled inverters [57]. ................................ 25
Fig. 2. 3: Proposed Master/Slave configuration in [61]. ................................................. 26
Fig. 2. 4: Proposed distributed control configuration in [67].......................................... 27
Fig. 2. 5: Proposed parallel operation of inverter with current minor loop [68]. ............ 28
Fig. 2. 6: Proposed current sharing control proposed in [69-71]. ................................... 28
Fig. 2. 7: The proposed circular chain control (3C) strategy [72]................................. 29
Fig. 2. 8: Reference voltage and power calculation [81]. ............................................... 30
Fig. 2. 9: Inverter control scheme [82]............................................................................ 31
Fig. 2. 10: Inverter control scheme proposed in [85, 86]. ............................................... 31
Fig. 2. 11: Inverter control scheme proposed in [19, 98-100]......................................... 33
Fig. 2. 12 Regular conventional droop functions (left) and transient droop functions
(right) [95, 101]. ...................................................................................................... 34
Fig. 2. 13 Static droop/boost characteristics for resistive output impedance [101-105]. 35
Fig. 2. 14 Overall scheme for the proposed droop control method [26, 109-111]. ....... 36
Fig. 2. 15 Frequency-dependent droop scheme: (a) the series impedance is created in the
inverter internally; (b) the equivalent inverter circuit at the fundamental frequency;
(c) the equivalent inverter circuit at the harmonic frequency [115, 116]................ 38
Fig. 2. 16 Schematic diagram of implementing the signal injection technique [54]. ..... 39
Fig. 3. 1: System overview of the intermediate DC stage. .............................................. 43
Fig. 3. 2: General control of a system operating in a grid-driven feeding mode (Forming,
Supporting). ............................................................................................................. 45
Fig. 3. 3: General control of a system operating in ECSs-driven feeding mode (parallel).
................................................................................................................................. 45
Fig. 3. 4: A general definition of feeding modes for DER .............................................. 44
Fig. 3. 5: Three leg inverter (balanced output)................................................................ 46
Fig. 3. 6: Three-leg inverter with a neutral point. .......................................................... 47
Fig. 3. 7: Four-leg inverter. ............................................................................................. 48
Fig. 3. 8: Inverter in grid forming mode for balanced loads. .......................................... 49
Fig. 3. 9: P, Q-controlled inverter in grid supporting mode for balanced loads. ............ 50
Fig. 3. 10: Q-controlled inverter in grid parallel mode. .................................................. 50
Fig. 3. 11: Inverter in grid forming mode for unbalanced loads. .................................... 52
Fig. 3. 12: P, Q-controlled Inverter in grid supporting mode for unbalanced loads. ...... 53
Fig. 3. 13: Inverter in grid parallel mode for unbalanced loads. ..................................... 54
Fig. 3. 14: Principle of Space Vector Modulation[135]. ................................................. 55
Fig. 3. 15: Three phase, three leg voltage source inverter. ............................................. 56
Fig. 3. 16: The eight inverter voltage vectors (V0 to V7). .............................................. 56
Fig. 3. 17: Output voltage Space vectors. ....................................................................... 57
Fig. 3. 18: Output voltages in time domain..................................................................... 57
Fig. 3. 19: Space-vector modulation in sector S1. ........................................................... 59
ix

Fig. 3. 20: Symmetric space-vector modulation pulse generation. ................................. 60
Fig. 3. 21 Space vector diagram for five-level diode-clamped inverter.......................... 62
Fig. 3. 22 Sequence decomposition and composition. .................................................... 63
Fig. 3. 23 Getting the d,q-components for phase a [153]................................................ 64
Fig. 3. 24 Sequence decomposition................................................................................. 65
Fig. 3. 25 Sequence composition [153]........................................................................... 66
Fig. A. 6. 26 3D-Space vectors [146]........................................................................... 186
Fig. 4. 1: The control philosophy (example). .................................................................. 69
Fig. 4. 2: Feeding modes at the grid side. ....................................................................... 70
Fig. 4. 3: The proposed scenarios.................................................................................... 71
Fig. 4. 4: Overview of Supervisory control and energy management proposed system
structure. .................................................................................................................. 73
Fig. 4. 5: Supervisory control and energy management scenario. .................................. 74
Fig. 4. 6: Parallel operation of two inverters, inductive impedance. .............................. 77
Fig. 4. 7: Frequency and voltage droop. ......................................................................... 78
Fig. 4. 8: Parallel operation of two inverters. .................................................................. 79
Fig. 4. 9: Phasor diagram. ............................................................................................... 79
Fig. 4. 10: Influence of active and reactive power on voltage and frequency for different
line impedance ratios: (a) R/X=0, (b) R/X=1, (c) R/X= ∞ [111]. ............................ 80
Fig. 4. 11: The classical Grid Forming mode with droop. .............................................. 82
Fig. 4. 12: The proposed grid forming mode with droop. ............................................... 82
Fig. 4. 13: Single phase diagram. .................................................................................... 83
Fig. 4. 14: Voltage-phasor diagrams [125]. .................................................................... 84
Fig. 4. 15: a) Phasor diagram of grid-forming case. b) Phasor diagram of grid-forming
case while minimizing Id. c) Phasor diagram of grid-forming case while
minimizing Iq. .......................................................................................................... 85
Fig. 4. 16: Phasor diagram of an inverter (General view). .............................................. 86
Fig. 4. 17: Grid supporting mode with droop.................................................................. 87
Fig. 4. 18: (a) Frequency vs. active power droop and (b) Voltage vs. reactive power
droop. ...................................................................................................................... 88
Fig. 4. 19: Modular grid using droop-controlled Inverters. ............................................ 89
Fig. 4. 20: (a) Frequency vs. active power isochronous and (b) Voltage vs. reactive
power isochronous. ................................................................................................. 90
Fig. 4. 21: Grid-forming with isochronous control function. .......................................... 90
Fig. 4. 22: Modular grid using grid-forming with isochronous control function............ 92
Fig. 4. 23: Grid-forming with isochronous-droop control function. ............................... 93
Fig. 4. 24: Modular Grid using grid-forming with isochronous-droop control function.94
Fig. 4. 25: Grid-forming as swing inverter. .................................................................... 95
Fig. 4. 26: Modular grid using swing-inverter and droop-controlled Inverters. ............. 96
Fig. 4. 27: Supervisory control and energy management scenario. ................................ 99
Fig. 4. 28: The proposed asymmetrical grid forming mode with droop. ...................... 101
Fig. 4. 29: The proposed asymmetrical grid supporting mode with droop. .................. 102
Fig. 4. 30: Modular grid using droop-controlled inverters. .......................................... 104
Fig. 4. 31: Grid-forming with isochronous control function. ........................................ 105
Fig. 4. 32: Modular grid using Grid-forming with isochronous control function. ........ 107
Fig. 4. 33: Grid-forming with isochronous-droop control function. ............................. 108
Fig. 4. 34: Modular grid using grid-forming with isochronous-droop control function.
............................................................................................................................... 109
Fig. 4. 35: Grid-forming as a swing inverter. ................................................................ 110
Fig. 4. 36: Modular grid using swing-inverter and droop-controlled Inverters. ........... 111
Fig. 4. 37: Output impedance of different sources in function of frequency [169]. ..... 115
x

Fig. 5. 1: Overview of the simulated scinarios. ............................................................ 119
Fig. 5. 2: Topology: supervisory control and energy management modular grid. ........ 121
Fig. 5. 3: The system frequency. ................................................................................... 122
Fig. 5. 4: The active power. .......................................................................................... 123
Fig. 5. 5: The reactive power. ....................................................................................... 124
Fig. 5. 6: The Voltage and current of grid forming at first step .................................... 125
Fig. 5. 7: Voltage and current of grid supporting at second step. ................................. 125
Fig. 5. 8: Voltage and current of grid parallel at first step. ........................................... 126
Fig. 5. 9: Voltage and current at load one during first step........................................... 126
Fig. 5. 10: Topology: Droop modular grid. ................................................................... 127
Fig. 5. 11: The droop factors for the system under study. ............................................ 128
Fig. 5. 12: The time sequence for the system under study. ........................................... 129
Fig. 5. 13: The system frequency. ................................................................................. 129
Fig. 5. 14: The voltage and current response of inverter one to load step at t=3.0
seconds. ................................................................................................................. 130
Fig. 5. 15: The power response of inverter one............................................................. 130
Fig. 5. 16: The current response of inverter 2 to load step at t=3.0. ............................. 131
Fig. 5. 17: The current response of inverter five. .......................................................... 132
Fig. 5. 18: The current response of inverter five to load step at t=6.5. ......................... 132
Fig. 5. 19: The voltage response at load one after the load step at t=6.5. ..................... 133
Fig. 5. 20: Topology: isochronous modular grid. ......................................................... 134
Fig. 5. 21: The system frequency. ................................................................................. 135
Fig. 5. 22: The system total load. .................................................................................. 136
Fig. 5. 23: The current response of inverter one to load step at t=3.0. ......................... 136
Fig. 5. 24: The voltage response of inverter two to load step at t=3.0. ......................... 137
Fig. 5. 25: The voltage and current response of inverter three to load step at t=3.0. .... 137
Fig. 5. 26: Topology: Isochronous-droop control modular grid. .................................. 138
Fig. 5. 27: The system frequency. ................................................................................. 139
Fig. 5. 28: The system total load. .................................................................................. 139
Fig. 5. 29: The current response of load one. ................................................................ 140
Fig. 5. 30: The voltage and current response of inverter three to load step at t=3.0. .... 140
Fig. 5. 31: The current response of inverter one to load step at t=3.0. ......................... 141
Fig. 5. 32: Topology: swing inverter based modular grid. ............................................ 142
Fig. 5. 33: Timing diagram for the modular grid. ......................................................... 143
Fig. 5. 34: The system frequency. ................................................................................. 144
Fig. 5. 35: The system total load. .................................................................................. 144
Fig. 5. 36: The swing inverter total supplied power. .................................................... 145
Fig. 5. 37: Load three voltage and current at t=4 seconds. ........................................... 145
Fig. 5. 38: The total power supplied by inverter two (parallel mode). ......................... 145
Fig. 5. 39: Topology: supervisory control and energy management modular grid (fourwire). ..................................................................................................................... 147
Fig. 5. 40: Timing diagram for the modular grid. ......................................................... 147
Fig. 5. 41: The system frequency. ................................................................................. 148
Fig. 5. 42: The active power.......................................................................................... 148
Fig. 5. 43: The reactive power. ..................................................................................... 149
Fig. 5. 44: The Voltage and current of grid forming at first step .................................. 150
Fig. 5. 45: The Voltage and current of grid supporting at first step .............................. 150
Fig. 5. 46: The voltage and current of load during second step .................................... 151
Fig. 5. 47: The neutral current. ...................................................................................... 151
Fig. 5. 48: Topology: Drooped modular grid (four-wire). ............................................ 152
Fig. 5. 49: The time sequence for the system under study. ........................................... 153
xi

Fig. 5. 50: The system frequency response. .................................................................. 154
Fig. 5. 51: The active power.......................................................................................... 154
Fig. 5. 52: The reactive power. ..................................................................................... 155
Fig. 5. 53: The grid forming voltage response. ............................................................. 155
Fig. 5. 54: The grid forming voltage and current response at the first load step. ......... 156
Fig. 5. 55: The grid supporting voltage and current response at the first load step. ..... 156
Fig. 5. 56: The neutral current response at the first load step. ...................................... 157
Fig. 5. 57: Topology: isochronous modular grid (four-wire). ....................................... 158
Fig. 5. 58:The system frequency response. ................................................................... 159
Fig. 5. 59: The active power.......................................................................................... 159
Fig. 5. 60: The reactive power. ..................................................................................... 160
Fig. 5. 61:The load voltage and current response at the first load step. ........................ 160
Fig. 5. 62:The grid forming (num:2) voltage and current response at the second load
step. ....................................................................................................................... 161
Fig. 5. 63:The neutral current response at the first load step. ....................................... 161
Fig. 5. 64: Topology: Isochronous-droop control modular grid (four wire). ................ 163
Fig. 5. 65:The system frequency response. ................................................................... 164
Fig. 5. 66: The active power.......................................................................................... 164
Fig. 5. 67: The reactive power. ..................................................................................... 165
Fig. 5. 68:The first grid forming voltage amplitude response....................................... 165
Fig. 5. 69:The load voltage and current response at the first load step. ........................ 166
Fig. 5. 70:The current response at the first load step, phase ”a”. .................................. 166
Fig. 5. 71: Topology: swing inverter based modular grid (four wire). ......................... 167
Fig. 5. 72: The system frequency response. .................................................................. 168
Fig. 5. 73: The active power.......................................................................................... 169
Fig. 5. 74: The reactive power. ..................................................................................... 169
Fig. 5. 75:The voltage and current response at load on the first load step. ................... 170
Fig. 5. 76:The neutral current response at the first load step. ....................................... 170

LIST OF TABLES
Table A. 1 Switching states and the corresponding output voltages [147].
Table A. 2 Normalized αβ Components of each switching vector [147].

xii

LIST OF ABBREVIATIONS
BB

Battery Bank

CHP

Combined Heat and Power

DEC

Decentralised Energy Coverter

DG
ECS

Distributed Generation
Energy Conversion Source

Grid Forming

Grid Supporting

Grid Parallel

HPS

Hybrid Power System

IGBT

Insulated Gate Bipolar Transistor

IPS

Isolated Power System

LDU

Load Dispatch Unit

LPF

Low Pass Filter

Management System

PLL

Phase Locked Loop

PV
PWM

Photovoltaic
Pulse Width Modulation

RES

Renewable Energy Source

RMS

Root Mean Square

SHS

Solar Home System

SSD

Symmetrical Sequence Decomposition

SOC

State-Of-Charge

STC

Standard Test Condition

SVM

Space Vector Modulation

THD

Total Harmonic Distortion

UPS

Uninterruptable Power System

VSC

Voltage Source Converter

VSI

Voltage Source Inverter

WEC

Wind Energy Converter

xiii

CHAPTER 1
INTRODUCTION

The electric energy is the backbone of our society. It continues to support the
growth, welfare and progress of the human race since Thomas Edison began
his work on the electric light in 1878.
The electric energy demand of the world is continuously increasing, and the
vast majority of it in most countries is generated by conventional sources of
energy. However, the rapid growth of global climate change along with the fear
of an energy supply shortage and limited fossil fuel is making the global energy
situation tends to become more complex. The increasing demand for electric
power than the offer, along with many developing countries lacking the
resources to build power plants and distribution networks, and the industrialized
countries that face insufficient power generation and greenhouse gas emission
problem forces us to consider a better economical and environmental friendly
alternative. Renewable energy sources (RESs) such as wind turbines, solar
panels and fuel cells could be part of the solution.
All of these sources require interfacing units to provide the necessary interface
to the grid. The core of these interfacing units is power electronics technologies
because they are fundamentally multifunctional and can provide not only their
principle interfacing function but various utility functions as well.
The evolution of the Electric Power System will be briefly discussed in the
following section.
1

1.1

A Brief History of Electric Power Systems

In 1878, Thomas Edison began work on the electric light and by 1879 he
perfected his light. Edison Electric Illuminating Company of New York
inaugurated the Pearl Street Station in 1881. The station had a capacity of four
250-hp boilers supplying steam to six engine-dynamo sets. Edison’s system
used a 110-V dc underground distribution network with copper conductors
insulated with a jute wrapping. In 1882, the first water wheel-driven generator
was installed in Appleton, Wisconsin. The low voltage of the circuits limited the
service area of a central station, and consequently, central stations proliferated
throughout metropolitan areas
Afterwards, due to the introduction of the DC motor by Frank J. Sprague in 1884
and the development of the three wire 220V DC system, loads could be
increased. However, voltage problems were experienced when the transmission
distance and the loads continued to increase.
The invention of the transformer, then known as the “inductorium,” made ac
systems possible. The first practical ac distribution system in the U.S. was
installed by W. Stanley at Great Barrington, Massachusetts, in 1866 for
Westinghouse, which acquired the American rights to the transformer from its
British inventors Gaulard and Gibbs. The ability to transmit power at a high AC
voltage and low current, minimized the voltage drops on the transmission lines;
making AC more attractive than DC.
The Nikola Tesla invention of the two phase induction motor in 1888 helped
replace dc motors and hastened the advance in use of ac systems. This
discovery enhanced the advantages of the poly-phase verse the single–phase
systems, leading to an expansion in the usage of three-phase line in Germany
(1891,179 km at 12 kV) and United States of America (1893, 12 km at 2.3 kV).
In the year 1891 Edison’s steam-driven generators were introduced, a
waterwheel-driven generator was installed in Appleton, Wisconsin. Since then
most electric energy has been generated in steam-powered and water-powered
(hydro) turbine plants. Steam plants are fuelled primary by coal, gas, oil and
uranium.
In the early years, AC systems usually operated at various frequencies starting
from 25, 50, 60 and 133Hz. Today, the only two standard frequencies to be
operated are fixed at 50Hz (in Europe, Russia, Middle East and South America
2

except Brazil) and 60Hz (in United States of America, Canada, Japan, and
Brazil).
Along with increases in load growth, there have been continuing increases in
the size of generating units and in transmission voltages too. Extra higher
voltage (EHV) has become dominant in electric power transmission over great
distances. By 1896, an 11 kV three-phase line was transmitting 10 MW from
Niagara Falls to Buffalo over a distance of 20 miles. Today, transmission
voltages of 230 kV, 287 kV, 345 kV, 500 kV, 735 kV, and 765 kV are
commonplace, with the first 1100 kV line already energized in the early 1990s.
In 1954, the Swedish State Power Board energized the 60-mile, 100 kV dc
submarine cables. Lamm’s Mercury Arc valves at the sending and receiving
ends of the world’s first high-voltage direct current (HVDC) link connecting the
Baltic island of Gotland and the Swedish mainland. Currently, numerous
installations with voltages up to 800-kV dc are in operation around the world.
Many other technologies evolved through the years to support the system such
as [1]:

•

The suspension insulator

•

The high speed relay system, currently capable of detecting short–
circuit currents within one cycle (0.02 s)

•

High speed, extra high voltage (EHV) circuit breakers, capable of
interrupting up to 63 kA three phase short circuit currents within two
cycles (0.04 s)

•

High speed reclosure of EHV line, which enables automatic return to
service within a fraction of a second after a fault has been cleared

•

The EHV surge arrestor, which provides protection against transient
over voltages due to lightning strikes and line switching operations

•

Power line carrier and microwave, as communication mechanisms for
protecting, controlling and metering of transmission lines.

• Energy control centers with supervisory control and data acquisition
(SCADA) and with automatic generation control (AGC).
And as a result we have our current electric power supply system which will be
described in the following section.

1.2

Electric Power Supply Systems Today

The electric power system is a complex system since electricity must be
generated in the exact moment that it is consumed. This means that the prime
directive for power system designers and operators is to balance generation
and load at every instant [2].

1.2.1 Conventional Power Supply System
The electric power system consists today of bulky central generation plants,
transmission network, distribution network and control centres.
•

Generation: power plants (Generators) typically utilize energy sources
such as fossil fuel (gas, oil, and coal), nuclear fuel (uranium), geothermal
energy (hot water, steam), and hydro energy (water falling through a
head) into electricity [3].

•

Transmission: Electric power transmission is the process of transferring
bulk electrical power from distant energy sources (such as hydroelectric
power plants) to consumers. Typically, power transmission is between
the power plant and a substation. Due to the large amount of power

involved, transmission normally takes place at high voltage (110 kV or
above). Electricity is usually transmitted over long distance through
overhead power transmission lines.
• Distribution: Electric power distribution is the portion of the power
delivery infrastructure that takes the electricity from the highly
meshed, high-voltage transmission circuits and delivers it to
customers. Primary distribution lines are “medium-voltage” circuits,
normally thought of as 600 V to 35 kV. At a distribution substation, a
substation transformer takes the incoming transmission-level voltage
and steps it down to several distribution primary circuits, which fan out
from the substation. Close to each end user, a distribution transformer
takes the primary-distribution voltage and steps it down to a lowvoltage secondary circuit (commonly 120/240 V) phase voltage (other
4

utilization voltages are used as well). From the distribution
transformer, the secondary distribution circuits connect to the end
user [4].
• Control Centers (Energy Management): Energy management is the
process of monitoring, coordinating, and controlling the generation,
transmission,

and

distribution

electrical

energy.

Energy

management is performed at control centers, typically called system
control centers, by computer systems called energy management
systems (EMS). Data acquisition and remote control is performed by
computer systems called supervisory control and data acquisition
(SCADA) systems [3].

Today’s traditional power systems are based on large central power plants
transmitting power via high voltage transmission systems, which is then
distributed

medium/low-voltage

local

distribution

systems.

The

transmission and distribution systems are commonly run by national
monopolies (national or regional bodies) under energy authorities’ control.
The overall picture is still one of power flow in one direction from the power
plants, via the transmission and distribution system, to the final customer.
Dispatching of power and network control is typically the responsibility of
regionally centralized facilities. Normally, there is little or no consumer’s
participation and no end-to-end communication. An overview of the
conventional power supply system is illustrated in Fig. 1.1.

Fig. 1. 1: Principal supply strategy in conventional electrical grids.

1.2.2 Distributed Generation
Currently, there is no consensus on how the distributed generation (DG)
should be exactly defined [5]. A very good overview of the different
definitions proposed in the literature is given in [6]. In general, distributed
generation describes electric power generation that is geographically
distributed or spread out across the grid, generally smaller in scale than
traditional power plants and located closer to the load, often on customers’
property [2]. Distributed generation is characterized by some or all of the
following features:
•

Small to medium size, geographically distributed power plants

•

Intermittent input resource, e.g., wind, solar

•

Stand-alone or interface at the distribution or sub-transmission level

•

Utilize site-specific energy sources, e.g., wind turbines require a
sustained wind speed of 20 km/hour. To meet this requirement they
are located on mountain passes or the coast
6

•

Located near the loads

•

Integration of energy storage and control with power generation

Technologies those are involved in Distributed Generation include but are
not limited to: Photovoltaic, Wind energy conversion systems, Mini and micro
hydro, Geothermal plants, Tidal and wave energy conversion, Fuel cell,
Solar-thermal-electric conversion, Biomass, Micro and mini turbines, Energy
storage technologies, including flow and regular batteries, pump-storage
hydro, flywheels and thermal energy storage.
The idea behind DG is not a new concept. In the early days of electricity
generation, DG was the rule, not the exception [7]. However, technological
evolutions and economical reasons developed the current system with its
huge power generation plants, transmission and distribution grids. An
overview of Distributed Generation is illustrated in Fig. 1.2.
Legend
400 – 230 kV High Voltage Level

Turbine

Grid

Transformer
3~

110 kV High Voltage Level
Load
3~
3~

Wind Energy
Converter

...

.
..

...

3~
3~

Grid
Offshore
Wind-Park

Photovoltaic
System

Grid
Wind-Park

3~
3~

Grid
Industry

...

3~
3~

Dieselmotor
10 – 30 kV Medium Voltage Level
Energy Storage
System

3~
3~

Grid
Industry

..
.

...
3~

.
..

Grid
Wind-Park

Battery

Fuel Cell
=

3~
3~

Flywheel

0.4 kV Low Voltage Level

3~
3~

..
.

Grid

House
with RES

House

..
.

3~
=

Frequency
Converter

3~
3~

=
3~

House
with CHP

House
with RES

Inverter

Intelligent Meter

Fig. 1. 2: Principal supply strategy of distributed Generation.

Converter/Inverter
System

Compressed Air
Storage System

Converter

In the last decade, technological innovation, economical reasons and the
environmental policy renew the interest in Distributed Generation. The major
reasons for that are:
• To reduce dependency on conventional power resources
• To reduce emissions and environmental impact
• Market liberalization
• Improve power quality and reliability
• Progress in DG technologies especially RESs
• To reduce transmission costs and losses
• To increase system security by distributing the energy plants instead
of concentrating them in few locations making them easy targets for
attacking

Distributed generation is becoming an increasing important part of the power
infrastructure and the energy mix and is leading the transition to future Smart
Grids. This is as well one of European Commission targets in order to increase
the efficiency, safety and reliability of European electricity transmission and
distribution systems and to remove obstacles to the large-scale integration of
distributed and renewable energy sources.

1.3

Future Power Supply Systems (Smart Grids)

Energy plays a vital role in the development of any nation. The current
electricity infrastructure in most countries consists of bulk centrally located
power plants connected to highly meshed transmission networks. However,
new trend is developing toward distributed energy generation, which means that
energy conversion systems (ECSs) will be situated close to energy consumers
and the few large units will be substituted by many smaller ones. For the
consumer the potential lower cost, higher service reliability, high power quality,
increased energy efficiency, and energy independence are all reasons for the
increasing interest in what is called “Smart Grids”.
Although the “Smart Grid” term was used for a while, there is no agreement on
its definition. It is still a vision, a vision that is achievable and will turn into reality
in near future. One of the best and general definitions of a smart grid is
8

presented in [8]. Smart grid is an intelligent, auto-balancing, self-monitoring
power grid that accepts any source of fuel (coal, sun, wind) and transforms it
into a consumer’s end use (heat, light, warm water) with minimal human
intervention. It is a system that will allow society to optimize the use of RESs
and minimize our collective environmental footprint. It is a grid that has the
ability to sense when a part of its system is overloaded and reroute power to
reduce that overload and prevent a potential outage situation; a grid that
enables real-time communication between the consumer and utility allowing to
optimize a consumer’s energy usage based on environmental and/or price
preferences [8]. The Future network vision (Smart Grid) is shown in Fig. 1.3.

Fig. 1. 3: Future Network Vision [9].

In the following sections we will have a close look at the drivers towards Smart
grid and the key challenges.

1.3.1 Drivers Towards Smart Grids
Many factors are influencing the shape of our future electricity networks
including climate change, aging infrastructure and fossil fuels running out.
According to the International Energy Agency (IEA) Global investments required
in the energy sector for 2003-2030 are an estimated $16 trillion. In Europe
alone, some €500 billion worth of investment will be needed to upgrade the
electricity transmission and distribution infrastructure [9]. The following are the
main drivers towards Smart Grids [9-12]:

•

The Market: Providing benefits to the customers by increasing
competition between companies in the market. Competition has led
many utilities to divest generation assets, agree to mergers and
acquisitions, and diversify their product portfolios. This will give the
customers a wider choice of services and lower electricity prices.

•

Environmental regulations: Another significant driver concerns the
regulation of the environmental, public health, and safety consequences
of electricity production, delivery, and use. The greenhouse gases
contribute to climate change, which is recognised to be one of the
greatest environmental and economic challenges facing humanity. To
meet these environmental policies, rapid deployment of highly effective,
unobtrusive, low-environmental-impact grid technologies is required.

•

Lack of resources: Energy is the main pillar for any modern society.
Countries without adequate reserves of fossil fuels are facing increasing
concerns for primary energy availability. Currently approximately 50%
within EU is imported from politically unstable countries.

•

Security: The need to secure the electric system from threats of
terrorism and extreme weather events are having there effect as well.
Techniques must exist for identifying occurrences, restoring systems
quickly

after

disruptions,

and

providing

services

during

public

emergencies. This is why electricity grids should be redesigned to cope
with the new rule.
•

Aging infrastructure: The aging infrastructure (Europe and USA) of
electricity generation plants, transmission and distribution networks is
increasingly threatening security, reliability and quality of supply. The
most efficient way to solve this is by integrating innovative solutions,
technologies and grid architectures.

•

New generation technologies (Distributed Generation): These forms
of generation have different characteristics from traditional plants. Apart
from large wind farms and large hydropower plants, this type of
10

generation tends to have much smaller electricity outputs than the
traditional type. Some of the newer technologies also exhibit greater
intermittency. However, existing transmission and distribution networks,
were not initially designed to incorporate these kinds of generation
technology in the scale that is required today.
•

Advanced power electronics: Power electronics allow precise and
rapid switching of electrical power. Power electronics are at the heart of
the interface between energy generation and the electrical grid. This
power conversion interface-necessary to integrate direct current or
asynchronous sources with the alternating current grid-is a significant
component of energy systems.

•

Information and communication technologies (ICT): The application
of ICT to automate various functions such as meter reading, billing,
transmission and distribution operations, outage restoration, pricing, and
status reporting. The ability to monitor real-time operations and
implement automated control algorithms in response to changing system
conditions is just beginning to be used in electricity [10]. Distributed
intelligence,

including

“smart”

appliances,

could

drive

the

co-

development of the future architecture.

1.3.2 Key Challenges for Smart Grids
Even though many drivers for smart grids and their benefits are obvious, there
are many challenges and barriers standing in the way and should be cracked
first. These include:
•

Standardisation: Design and development of a modular standardised
architecture of modern power electronic systems for linking distributed
energy converting systems (DECSs) (i.e. PV, wind energy converters,
fuel cells, diesel generators and batteries) to conventional grids and to
isolated grids on the basis of modular power electronic topologies which
fulfil the requirements for integration into the dynamic control system of
the grid [13].

•

Advance communication layer: Development and implementation of a
general communication layer model for simple and quick incorporation of
DECSs in the grid and its superimposed online control system

•

Non-technical challenges: Issues such as pricing, incentives, decision
priorities, risk responsibility and insurance for new technologies
adaptation, interconnection standards, regulatory control and addressing
barriers. This also includes, finding a profitable business model,
attracting resources and developing better public policies [9, 14].

1.4

Problem Statement

Fossil fuels are currently the major source of energy in the world today.
However, as the world is considering more economical and environmentally
friendly alternative energy generation systems, the global energy mix is
becoming more complex. Factors forcing these considerations are (a) the
increasing demand for electric power by both developed and developing
countries, (b) many developing countries lacking the resources to build power
plants and distribution networks, (c) some industrialized countries facing
insufficient power generation and (d) greenhouse gas emission and climate
change concerns. Renewable energy sources such as wind turbines,
photovoltaic solar systems, solar-thermo power, biomass power plants, fuel
cells, gas micro-turbines, hydropower turbines, combined heat and power
(CHP) micro-turbines and hybrid power systems will be part of future power
generation systems [15-23].
This new trend is developing toward DG, which means that energy conversion
systems are situated close to energy consumers and large units are substituted
by smaller ones. For the consumer the potential lower cost, higher service
reliability, high power quality, increased energy efficiency, and energy
independence are all reasons for interest in distributed energy resources (DER).
The use of renewable distributed energy generation and "green power" can also
provide a significant environmental benefit [24-26]. This is also driven by an
increasingly strained transmission and distribution infrastructure as new lines
lag behind demand and to reduce overall system losses in transmission and
distribution. Further, the increased need for reliability and security in electricity

supply, high power quality needed by an increasing number of activities
requiring UPS like systems and to prevent or delay the expansion of central
generation stations by supplying the growing loads locally [27, 28].
However, the exploitation of RESs and more efficient utilisation of energy
sources due to local (distributed generation) result in a large number of sources
at the low and medium voltage Grid. For most micro-turbines, wind plants, fuel
cells and photovoltaic cells electrical power is generated as a direct current
(DC) and converted to an alternating current (AC) by means of inverters [29].
Due to that, the inverter is considered as an essential component at the grid
side of such systems due to the wide range of functions it has to perform. It has
to convert the DC voltage to sinusoidal current for use by the grid in addition to
act as the interface between the ECSs, the local load and the grid. It also has to
handle the variations in the electricity it receives due to varying levels of
generation by the RESs, loads and grid voltages [30]. Inverters influence the
frequency and the voltage of the grid and seem to be the main universal
modular building block of future smart grids mainly at low and medium voltages.
The main problem associated with that is the development of a general, flexible,
integrated, and hierarchical control strategy for DERs to be integrated into the
dynamic grid control and management procedures of electrical power supply
systems (primary control, frequency and power control, voltage and reactive
power control) through flexible power electronics namely inverters.

10 – 30 kV Medium Voltage Level

Grid

3~
3~

Grid
Industry

3~
3~

Grid
WindPark

...

.
..

..
.

3~
=

0.4 kV Low Voltage Level

..
.

Grid

3~
=
=

=
3~

..
.

=
=

Grid

..
.

3~
3~

=
=

Fig. 1. 4: General grid architecture by means of flexible power electronics.

1.5

Motivation and Justification

At present, the integration of decentralised energy conversion systems
(DECSs), such as photovoltaic systems, wind energy converters, etc., into
conventional power systems is based on passive grid utilisation. In most cases,
the decentralised energy conversion systems purely operate as energy sources
and utilise the grid as an infinite energy sink. Active participation of DECSs in
the conventional dynamic grid control almost does not exist, at present, at all.
The problems such a situation cause to grid stability and grid operation
management are clearly evident when considering the large fraction of
decentralised energy systems (solar, wind, fuel cells, etc.) in Europe. With a
growing fraction of decentralised energy conversion systems, it is necessary to
devise an energy supply system design such that the DECSs are actively
integrated in the dynamic grid control strategy and contribute towards supply
stability and supply quality and does not rely on a rigid grid. In the sustainable
development of modern electric supply systems in relation to the targets of
exploiting renewable energy sources, a system standardisation is required such
that the system components and control strategies cover a broad range of
applications in different kinds of grid forms (conventional grids, island grids and
mini-grids) in a way that guarantee their contribution to the grid formulation.
The development of the main grid level control functions for electric power grids
with a large fraction of decentralised (including renewable) energy sources
constitutes the basis for a sustainable design of modern energy supply systems.
Motivated by the above, new system architecture for smart power systems that
can be standardised in terms of power-electronic units and control and
management functions will be launched. The developed system should not
impose any limitations on the types or the sizes of the employed energy
conversion systems. It should also not depend upon a component relevant to a
particular ECS.
The major objective is to design, develop, and validate approaches to speed up
the integration of DECSs, including RESs, in the dynamic grid online-control
strategy and operation procedures through developing control standards for
power electronics inverters to interface them to the grid.

The intention is to achieve appropriate and efficient solutions with a high degree
of system flexibility and modularity with regard to possible extension, and
optimum integration with regard to decentralised electric power.

The developed system architecture should have the following qualifications:

Flexible in terms of the types, sizes, and numbers of energy
conversion systems.

Flexible in terms of the types of electrical loads and their sizes.

Expandable so as to accept the addition or removal of energy
conversion systems.

1.6

Compatible with other conventional electrical systems.

Research Aims

Our present and future power network situation requires extra flexibility in the
integration of distributed generation more than ever. Mainly for the small and
medium energy converting systems including intelligent control and advanced
power electronics conversion systems.
The aim of this research study is to explore forming an electric power supply
system by paralleling power electronic inverters. Even though many
publications are addressing this topic, various issues are still unsolved or not
adequately investigated and standardized. Standardized modular architectures
and techniques for distributed intelligence and smart power systems are still
lagging behind. The main concern here is the inverters and their connection to
the grid. The focus will be in developing a general supply strategy using a
combination of different schemes. The attention is not related to stability issues
in control theory.
A standard system is defined as a system that is being widely recognized or
employed as a model of authority or excellence [31]. This current research
study therefore intends to classify grid components which can be used to
interface ECSs to the grid and devise general control architecture for them to
form smart grids that can be inherited and used in a wide range of applications.
In addition, it will be compatible and work successfully with existing traditional
grid equipment. The proposed architecture will take into account the new

challenges and opportunities in electric networks evolving into decentralized
scheme and will be deployed rapidly and cost-effectively to enable existing grids
to accept new power injections from distributed energy resources and RESs.
And finally, the employed components in the proposed architecture will be as
modular as possible.

The specific aims of this research study may be formulated in the following way:

To develop different and various robust control approaches for a
realistic distributed power system with power electronics inverters as
multi-functional front-end. These control strategies should guarantee
real modularity, higher reliability and true redundancy as well as to
avoid a single point of failure and to qualify it to be standardised. The
proposed control architecture should maintain the three phase
voltages and frequencies in the grid within certain limits and has to
provide power sharing between the units according to their ratings.
See Fig. 1.5.

The designed system should include inverter units of different power
ratings, distributed at varied locations feeding distributed unequal
loads (balanced, unbalanced) taking into account dissimilar line
impedances between them to insure true expandability and
generation placement flexibility. This means that the types, sizes, and
numbers of the inverters, and the sizes and nature of the electrical
loads may all vary without the need to alter the control strategy.

The developed control concept should work with rotating generators
existing in the same grid (diesel generators, small hydro, wind energy
converters…etc). In order to assure flexibility and also compatibility
with the existing structure mostly based on rotating generators.

This research study will develop a standard supply strategy that will fulfil the
mentioned targets.

10 – 30 kV Medium Voltage Level

Grid

3~
3~

Grid
Industry

3~
3~
=

Grid
WindPark

...

..
.

.
..

0.4 kV Low Voltage Level

=
=

..
.

Grid

..
.

Grid

3~
=
=

=
3~

..
.

=
=
3~

=
=

Supervisory Control and
Energy Management

Isochronous Control

Droop Control

Fig. 1. 5: Desired System Structure.

1.7

Research Contributions

Even though, most of the current approaches to build future smart power
systems are trying to introduce one-size-fits-all solution but the fact is that each
system (customer) needs are different and various approaches are needed to fit
their exact specifications. This work will introduce varied opportunities of control
functions for three-phase inverters used to feed passive/active grids including
different topologies to feed balanced/unbalanced loads.
It will be shown how the developed architecture will apply to the three possible
feeding modes, namely grid forming, grid supporting and grid parallel modes in
both symmetrical and asymmetrical manner. In these three modes, the
responsibility of the feeding concerning the frequency and the voltage of the
grid is different. Therefore, it will be shown how the control and management
strategies will change in these cases using different scenarios. These scenarios
will be verified through simulation models.
The success of the proposed control architecture will assist the current efforts in
designing, building and operating a smart power system that is more flexible,
efficient, reliable and environmentally friendly. It will bring new standards that
are expected to result in design, installation and debugging time cost
17

reductions. This will facilitate and accelerate the transition from our current
conventional grid to the future distributed smart grid.

1.8

Organisation of the Thesis

This research study will develop a theoretical system concept, which can assist
the current efforts in designing, building and operating a smart power system
that is more flexible, efficient, reliable and environmentally friendly. This work
will introduce varied opportunities of control functions for three-phase inverters
used to feed passive/active grids including different topologies to feed
balanced/unbalanced loads. These are based on a standardized system
concept using various control strategies and no one-size-fits-all solution.
Chapter two presents the state-of-the-art of power electronic inverters control
used currently in electrical systems. Different system architectures, their modes
of operation, management and control strategies will be analysed. Advantages
and disadvantages will be discussed. This chapter will start by briefly reviewing
the current trends and different modes of operation for single inverters that
exists in the literature. Next, the different techniques to parallel inverters
suggested in the literature will be explained.

Chapter three presents the new system philosophy developed for the smart
grid. This philosophy is taking into account the advantages and disadvantages
of the different approaches reviewed in chapter two.
The chapter is divided into five main parts. In the first part, the general
architecture will be presented and discussed. The second part presents the
main power electronic element of the philosophy, the inverter, showing the
different inverter topologies used in the feeding philosophy. In the third part, the
developed operating principles and control and regulation techniques for these
inverters will be presented. The space vector modulation algorithms built up will
be presented in the fourth part. Next, in part five the sequence decomposition
which is used to develop the advanced control strategy for power electronic
inverters feeding unbalanced loads will be emphasized. Finally, a brief
discussion will be carried out on assessing the new philosophy.

Chapter four will be dedicated to the operation, control, application and
management of the philosophy. The proposed philosophy has two main
categories. The first category is the Multi-inverter Three-wire system and the
second is the Multi-inverter Four-wire system. For each of these categories,
different control function scenarios will be proposed and explored. This will start
by supervisory control and management scenario. Then, droop control scenario
will be introduced. Afterwards, isochronous control function scenario is
explored. Next, a combination of droops and isochronous control scenario will
be proposed. Finally, the basic case of using an inverter as a swing machine
will be investigated.

Chapter five is devoted to the verifications of the new philosophy and the
control

and

management

functions

through

simulation

models

Matlab/Simulink.

Chapter six draws the conclusions of the research study. The advantages and
disadvantages of the new system philosophy are pointed out. Further work to
be done in this field is also recommended.

Chapter One
Introduction

Chapter Two
State of the Art

Chapter Three
Architecture and
Components

Chapter Four
Operation, Control
and Management

Chapter Five
Verification

Chapter Six
Conclusions and
Further Work
Fig. 1. 6: The structure of this thesis.

CHAPTER 2
STATE-OF-THE-ART

Introduction

This chapter presents the state-of-the-art of power electronic inverters control
used currently in electrical systems. Different system architectures, their modes
of operation, management and control strategies will be analysed. Advantages
and disadvantages will be discussed. Though, it is not easy to give a general
view at the state of the art for the research area since it is rapid and going in
different directions. The focus here will be on the main streams in low voltage
grids especially paralleled power electronics inverters.
This chapter will start by briefly reviewing the current trends and different modes
of operation for single inverters that exists in the literature. Next, the different
techniques to parallel inverters suggested in the literature will be checked.
These can be categorized to the following main approaches: master/slave
control techniques, current/power deviation (sharing) control techniques and
frequency and voltage droop control techniques. Finally, based on the reviewed
state of the art, the study presents a discussion comparing the different
approaches reported. In addition, their weaknesses and strengths are explored.

2.1

Single Inverters

Different modes of operation for single inverters exists in the literature, they can
be categorized into the following main approaches:

1. Stand-alone Inverters
2. Grid-connected Inverters
3. Interactive Inverters

2.1.1 Stand-alone Inverters
A stand-alone voltage source inverter, see Fig. 2.1 (a), is the basic building
block of a microgrid. Its primary role is to maintain a regulated voltage and
frequency supply to loads [32]. Stand alone inverters can provide any power or
current up to the rating of the inverter and presupposing that there is enough
energy in the DC-source behind (e.g. battery). It is an inverter that operates only
in stand-alone mode and thus contains no facility to synchronise its output
energy to a local distribution company (Public grid). They are used mainly for
rural and other off-grid electrifications applications.

2.1.2 Grid-connected Inverters
A grid-connected system is connected to an independent grid (typically the
public electricity grid) to which it exports the energy it produces. Grid-connected
inverters are normally ECS (supply) driven, they provide all the power supplied
from a DC/AC source to the grid or mains. Grid-connected inverters are usually
optimised for one specific type of generator, e.g. PV and generally operate at a
higher DC voltage than stand-alone inverters [33]. However, in some cases, the
inverter must also monitor the grid's import capacity in order to prevent
overvoltage [34], see Fig. 2.1 (b).

2.1.3 Interactive Inverters
A grid-interactive (Grid-tie) has two way (bidirectional) grid interconnection to
the utility grid. These systems normally supply local loads and use sophisticated
control equipment so that when ECS produces more power than needed, the
excess power is fed back into the grid. When ECS does not produce enough
power, then power will be imported from the grid [35], see Fig. 2.1 (c).

DC-to-DC
Converter
ECS
No. i

Inverter

Grid

Power Flow

(a) Stand-alone inverter.

DC-to-DC
Converter
ECS
No. i

Inverter

Utility
Grid
Power Flow

(b) Grid connected inverter.

DC-to-DC
Converter
ECS
No. i

Inverter

Utility
Grid

Power Flow
Grid

Fig. 2. 1: a) Stand-alone inverter, b) Grid connected inverter, c) Grid interactive inverter.

Many inverters proposed in the literature can work in more than one mode. The
source is some times connected directly using a DC/AC converter but this will
not influence the general architecture. Different applications of stand-alone, grid
connected and grid interactive inverters especially in combination with RESs
can be found on [36-54]. This will not be taken further into discussion since it is
not the main focus of this study

2.2

Paralleled Inverters

Inverters are often paralleled to construct power systems in order to improve
performance or to achieve a high system rating. Parallel operation of inverters
offers also higher reliability over a single centralized source because in case
one inverter fails the remained (n-1) modules can deliver the needed power to

the load. This is as well driven by the increase of RESs such as photovoltaic
and wind.
There are many techniques to parallel inverters which are already suggested in
the literature, they can be categorized to the following main approaches:

1) Master/Slave Control Techniques
2) Current/Power Deviation (Sharing) Control Techniques
3) Frequency and Voltage Droop Control Techniques
a) Adopting Conventional Frequency/Voltage Droop Control
b) Opposite Frequency/Voltage Droop Control
c) Droop Control in Combination with Other Methods

These will be discussed in the following sections.

2.2.1 Master/Slave Control Techniques
The Master/Slave control method uses a voltage controlled inverter as a master
unit and current controlled inverters as the slave units. The master unit
maintains the output voltage sinusoidal, and generates proper current
commands for the slave units [55-57].
One of the Master/Slave configuration is the scheme suggested in [58, 59] , see
Fig. 2.2, which is a combination of voltage-controlled and current-controlled
PWM inverters for parallel operation of a single-phase uninterruptible power
supply (UPS). The voltage-controlled inverter (master) is developed to keep a
constant sinusoidal wave output voltage. The current-controlled inverter units
are operated as slave controlled to track the distributive current. The inverters
do not need a PLL circuit for synchronization and gives a good load sharing.
However, the system is not redundant since it has a single point of failure.

Vdc

Lf
VCPI
Cf

Vdc

CCPI

Vdc

CCPI

Load

Power
distribution
center

Fig. 2. 2: Combined voltage and current controlled inverters [58].

A comparable scheme is also presented in [60] but it needs even more
interconnection since it is sharing the voltage and current signals. In [61] the
system is redundant by extended monitoring of the status and the operating
conditions of all power electronic equipment. Each block of the UPS system is
monitored by two independent microcomputers that process the same data. The
microcomputers are part of a redundant distributed monitoring system that is
separately interlinked by two serial data buses through which they
communicate. They establish a hierarchy among the participating blocks by
defining one of the healthy inverter blocks as the master.
The scheme proposed in [62], see Fig. 2.3., is based on the Master/Slave
configuration but is using a rotating priority window which provides random
selection of a new master and therefore results in true redundancy and increase
reliability.

Fig. 2. 3: Proposed Master/Slave configuration in [62]

In [63] the system is also redundant since a status line is used to decide about
the master inverter using a logical circuit (flip-flop), if the master is disconnected
one slave becomes automatically the master. The auto-master-slave control
presented in [64] is designed to let the unit with highest output real power act as
a master of real power and derives the reference frequency, the others have to
follow as slaves. The regulation of the reactive power is similar, the highest
output reactive power module acts as master of reactive power and adjusts the
voltage reference amplitude.
In [65, 66] the paper focus on operation of the microgrid when it becomes
isolated under different condition. This was investigated for two main control
strategies, single master operation where a voltage source inverter (VSI) can be
used as voltage reference when the main power supply is lost; all the other
inverters can then be operated in PQ mode. And multi-master operation where
more than one inverter are operated as a VSI, other PQ inverters may also
coexist.
In more recent papers [56, 67, 68] an enhanced approach is introduced, the
master inverter is replaced by a central control block which controls the output
voltages and can influence the output current of the different units, this is
sometimes called central mode control or distributed control. This means that
the voltage magnitude, frequency and power sharing are controlled centrally
(commands are distributed through a low bandwidth communication channels to
the inverters) and other issues such as harmonic suppression are done locally,
see below Fig. 2.4.

Feed
Forward

Fig. 2. 4: Proposed distributed control configuration in [68].

2.2.2 Current/Power Deviation (Sharing) Control Techniques
In this control technique the total load current is measured and divided by the
number of units in the system to obtain the average unit current. The actual
current from each unit is measured and the difference from the average value
is calculated to generate the control signal for the load sharing [55]. In the
approach suggested in [69], see Fig. 2.5, the voltage controller adjusts the small
voltage deviation and keeps the voltage constant. The ∆I signal is detected and
given to the current loop as a correction factor, and the ∆P signal controls the
phase of the reference sine wave. The system has many desirable
characteristics. A very good load sharing can be obtained. Transient response
is very good due to the feed forward control signal [55].

Fig. 2. 5: Proposed parallel operation of inverter with current minor loop [69].

A method of current sharing for paralleled power converters is introduced and
then explored in [70-72], in this approach (see Fig. 2.6.) each converter is
controlled such that its average output current is directly related to its switching
frequency. As a result, the frequency content of the aggregate output ripple
voltage contains information about the individual cell output currents. Each cell
measures the output ripple voltage and uses this information to achieve current
balance with the other cells.

Fig. 2. 6: Proposed current sharing control proposed in [70-72].

In [73] circular chain control (3C) strategy is proposed, see below Fig. 2.7., all
the modules have the same circuit configuration, and each module includes an
inner current loop and an outer voltage loop control. With the 3C strategy, the
modules are in circular chain connection and each module has an inner current

loop control to track the inductor current of its previous module, achieving an
equal current distribution.

Vdc

L
Module1

Vdc

Load

L
Module 2

Vdc

L
Module n

Fig. 2. 7: The proposed circular chain control (3C) strategy [73].

The authors of [74] proposed an inverter current feed-forward compensation
which makes the output impedance resistive rather than inductive in order to get
a precise load sharing. In [75] the paper goes further based on the approach
introduced in [74] and proposes a solution to the noise problem of harmonic
circulating currents due to PWM non-synchronization which is affecting the load
sharing precision. This is done in [76] using a digital control algorithm for
parallel connected three-phase inverters. The digital voltage controller, which
has high-speed current control as a minor loop, provides low voltage distortion
even for nonlinear loads. Output current of each UPS module is controlled to
share the total load current equally and the voltage reference command of each
inverter is controlled to balance the load current. In [77-79] similar approaches
are suggested. In [80] the focus is on developing a solution for the effect of DC
offset between paralleled inverters and its effect on the circulating currents. In
[81] the authors suggest two-line share bus connecting all inverters, one for
current sharing control and the other to adjust the voltage reference.

2.2.3 Frequency and Voltage Droop Control Techniques
Many methods were found in the literature and can be roughly categorized into
the following:

a. Adopting Conventional Frequency/Voltage Droop Control
b. Opposite Frequency/Voltage Droop Control
c. Droop Control in Combination with Other Methods

a. Adopting Conventional Frequency/Voltage Droop Control
In [82] the paper proposes a control technique for operating two or more single
phase inverter modules in parallel with no auxiliary interconnections. In the
proposed parallel inverter system, each module includes an inner current loop
and an outer voltage loop controls, see Fig. 2.8. This technique is similar to the
conventional

frequency/voltage

droop

concept;

uses

frequency

and

fundamental voltage droop to allow all independent inverters to share the load
in proportion to their capacities. However, the paper considered only inductive
lines.

Fig. 2. 8: Reference voltage and power calculation [82].

In [83] scheme for controlling parallel-connected inverters in a stand-alone AC
supply system is presented, see Fig. 2.11. This scheme is suitable for control of
inverters in distributed source environments such as in isolated AC systems,
large and UPS systems, PV systems connected to AC grids. Active and reactive
power sharing between inverters can be achieved by controlling the power
angle (by means of frequency), and the fundamental inverter voltage
magnitude. Simulation results obtained for large units (P=1 MW, Q=500 kvar,
line-to-line voltage is 3.3 kV rms and the DC bus voltage is 10 kV) using Gate
turn-off (GTO) thyristor switches. The control is done in the d-q reference frame;
an inverter flux vector is formed by integrating the voltage space vector. The
choice of the switching vectors is essentially accomplished by hysteresis
30

comparators for the set values and then using a look-up table to choose the
correct inverter output voltage vector. The considerations for developing the
look-up table are dealt with in [84]. However, the inductance connected
between the inverter and the load makes the output impedance high. Therefore,
the voltage regulation as well as the voltage waveform quality is not good under
load change conditions as well as a nonlinear load condition. The authors
explain the same concept but with focusing in control issues of UPS systems in
[85].

Fig. 2. 9: Inverter control scheme [83].

In [86, 87] the inverse droop equations are used to control the inverter. The
inverter is able to work in parallel with a constant-voltage constant-frequency
system, as well as with other inverters or also in stand-alone mode. There is no
communication interface needed. The different power sources can share the
load also under unbalanced conditions. Very good load sharing is achieved by
using an outer control loop with active and reactive power controller, for which
the set point variables are derived out of droops. Furthermore, a relatively big
inductance of 12 mH (C=10 µF) is used in the LC filter and a small decoupling
reactance is used to decouple the inverter from other voltage sources. The
interface inductance make the voltage source converters (VSCs) less sensitive
to disturbances on the load bus [88, 89].

u1 (t ) = U .sin(ωt + ϑ )
2π
+ϑ)
3
4π
u3 (t ) = U . sin(ωt −
+ϑ)
3
u 2 (t ) = U .sin(ωt −

Fig. 2. 10: Inverter control scheme proposed in [86, 87].

In [90] an interesting autonomous load-sharing technique for parallel connected
three-phase voltage source converters is presented. This paper focuses on an
improvement to the conventional frequency droop scheme for real power
sharing and the development of a new reactive power-sharing scheme. The
improved frequency droop scheme computes and sets the phase angle of the
VSC instead of its frequency. It allows the operator to tune the real power
sharing controller to achieve desired system response without compromising
frequency regulation by adding an integral gain into the real power control. The
proposed reactive power sharing scheme introduces integral control of the load
bus voltage, combined with a reference that is drooped against reactive power
output. This causes two VSCs on a common load bus to share the reactive load
exactly in the presence of mismatched interface inductors if the line impedances
are much smaller than the interface reactors (assuming short lines). Moreover,
in the proposed reactive power control, the integrator gain can be varied to
achieve the desired speed of response without affecting voltage regulation.
In [91] the authors are considering that large DC cross currents can flow
between the different inverters which is normally neglected since only the AC
cross current is normally taken into consideration by means of control schemes.
However, this can happen only if we have a considerable DC voltage offset
difference between the inverters which is usually not the case since these errors
are generated by the sensors and they are very small. One more condition to
make this happen is a very small output resistance of each inverter in
comparison to its output impedance. It is to be noted that this droop scheme
can only make inverters have the same DC-offset voltage so as to avoid the DC
cross current, whereas it cannot get rid of the DC-offset voltage. The test was
done using two 1.5 kW single phase units.
In [92-94] the author discusses the application of conventional droops for
voltage source inverters and categorize the system components to form a
modular AC-hybrid power system. Then in [95] by the same author an
investigation of what is called opposite droop (active power/voltage and reactive
power/frequency droop) control is carried out. The focus is on the need of
different droop functions for different types of grids. In [95] it is found that for
high voltage (mainly inductive) grids the regular droop functions can be used
also for distributed generation systems. For low voltage (mainly resistive) grids,
so-called opposite droop functions could be used instead but the regular droop
32

functions are advantageous since it allows connectivity to higher voltage levels
and power sharing also with rotating generators [95-98].
A microgrid control was introduced and implemented in [21, 99-101], the
microgrid has two critical components, the static switch and the micro-source.
The static switch has the ability to autonomously island the microgrid from
disturbances such as faults or power quality events. After islanding, the
reconnection of the microgrid is achieved autonomously after the tripping event
is no longer present. This synchronization is achieved by using the frequency
difference between the islanded microgrid and the utility grid insuring a transient
free operation without having to match frequency and phase angles at the
connection point. Each micro-source can seamlessly balance the power on the
islanded microgrid using a power vs. frequency droop controller. This frequency
droop also insures that the microgrid frequency is different from the grid to
facilitate reconnection to the utility. The introduced micro-source control is
shown in Fig. 2.11.
Ereq

Inverter
Current

Q
Calculation

Low-Pass
Filter

Q versus E
Droop

E0
Load Voltage
Measure

Magnitude
Calculation

Low-Pass
Filter

To Inverter
Gates

V
Voltage Control

Gate Pulse
Generator
v

Inverter or
Line Current

P
Clculation

Low-Pass
Filter

P versus
Frequency Droop

Fig. 2. 11: Inverter control scheme proposed in [21, 99-101].

The authors of [57] present a scheme for controlling parallel connected inverters
using droop sharing method in a standalone ac system. The scheme proposed
a PI regulator to determine the set points for generator angle and flux. The
model of the microgrid power system is simulated in MATLAB/SIMULINK. The
dynamic response of the system is investigated under different impedance load
conditions especially motor loads. Paper [32] analyzes the fault behaviour of
four wire paralleled inverters (in droop mode) based on their control
methodology.
33

b. Opposite Frequency/Voltage Droop Control
In [96, 102] the method selected here is to modify the droop functions of the
source converters so that the regular droop functions are used in the steadystate case and opposite droops are used in transients, see Fig. 2.12. Note that
here ωref = ωn and vref = Vn. The steady-state droop functions are according to:

p s * = K ω (ω ref − ω )

(2.1)

q s = K v (v ref − v q )

(2.2)

where pS* and qS* are the active and reactive power references (index s denotes
source converter, e.g. unit 1). K is the droop gain (slope). For the transient
droop functions according to:

p s * = K v (v ref − v q )

(2.3)

q s * = − K ω (ω ref − ω )

(2.4)

where ωref = ω* and vref = v*

Fig. 2. 12 Conventional droop functions (left) and transient droop functions (right) [96, 102].

In this method the load-sharing is acceptable for the investigated, highly
resistive, network. Still, in the case of line inductance in the same order of
magnitude as the converter output filter inductance there can be a considerable
degradation of power quality in terms of voltage disturbance. The origin of this
degradation is the LC-circuit formed by the line inductance and the converter
AC side capacitors. Furthermore, using this approach it is not possible to

connect with the high level voltage which is using the regular conventional
droop functions. The tested units are 4.5 kVA and 3 kVA.
In [102-106] the authors focus on the transient behaviour of parallel connected
UPS inverters, they claim that damping and oscillatory phenomena of phase
shift difference between the paralleled inverters could cause instabilities, and a
large transient circulating current that can overload and damage the paralleled
inverters. To overcome this they proposed using a method called “droop/boost”
control scheme which adds integral-derivative terms to the droop function. This
can be seen in Fig. 2.13. Stable steady-state frequency and phase and a good
dynamic response are obtained. Further, virtual output impedance is proposed
in order to reduce the line impedance impact and to properly share nonlinear
loads, this is done using a high pass filter, the filter gain and pole values of this
must be carefully chosen. Furthermore, the test results shown are considering a
short resistive line, but the method is not taking into consideration what
happens if the distance between the inverters is considerable, which is normally
the case in distributed generation were an inductive impedance component
appears. Nevertheless, when an inverter is connected suddenly to the common
AC bus, a current peak appears due to the initial phase error [107]. The units
tested are single phase, in [102] six kVA and in [103, 104] is one kVA.
Compatibility problems are expected because of the opposite droop scheme (if
synch generator will be included). The characteristic and the scheme are shown
below:

dP
dt
dQ
ω = ω * −mQ − md
dt
E = E * −nP − nd

Fig. 2. 13 Static droop/boost characteristics for resistive output impedance [102-106].

(2.5)
(2.6)

Where P is active power, Q is reactive power, E is output voltage, ω is angular
frequency and m and n are the droop coefficients for the frequency and
amplitude, respectively. As an addition in [107] a soft-start is included to avoid
the initial current peak as well as a bank of band pass filters in order to share
the significant output-current harmonics. In more recent papers [108, 109] the
authors use the conventional droop equations for a microgrid too.

E = E * −n(Q − Q*)

(2.7)

ω = ω * − m( P − P*)

(2.8)

c. Droop Control in Combination with Other Methods
In [28, 110-112] each inverter supplies a current that is the result of the voltage
difference between a reference AC voltage source and the grid voltage across a
virtual impedance with real and/or imaginary parts. The reference AC voltage
source is synchronized with the grid, with a phase shift, depending on the
difference between nominal and real grid frequency. This method behaviour is
equal to the normal existing droop control methods except that, short-circuit
behaviour is better since it is controlling the active and reactive currents and not
the power. It behaves also better in case of a non-negligible line resistance.

2π

1
s

Fig. 2. 14 Overall scheme for the proposed droop control method [28, 110-112].

In [113, 114] novel fast control loops that adjust the output impedance of the
closed-loop inverters is used in order to ensure resistive behaviour with the
purpose to share the harmonic current content properly. In the measurements

part a notch filter is added to remove the unwanted harmonics, it seems that
without this filter the voltage regulator will not work efficiently. Furthermore, the
control is done in the αβ-coordinates using a discrete controller.
The author of [115] discusses the problem of inverters with very low output
impedance (such as those employing resonant controllers) directly connected in
parallel through a near zero impedance cable. Low total harmonic distortion
(THD) content and good current sharing are simultaneously obtained by
controlling the load angle through an least mean square estimator and by
synthesizing a variable inductance in series with the output impedance of the
inverter, while the harmonic current sharing is achieved by controlling the gain
of the resonant controllers at the selected frequencies.
The proposed droop scheme In [116, 117] is shown below, see Fig. 2.15., in
which the inverter connects with the load via series impedance (Z) like the
conventional approach. However, there exist two differences: (i) the series
impedance is created by the inverter internally, no true impedance is required.
As a result, the inverters connect with the load tightly. (ii) The series impedance
is frequency-dependent; it exhibits a reactive characteristic at the fundamental
frequency and a resistive characteristic at the harmonic frequency.

Vo1∠δ 2

VO ∠0

Vo2∠δ 2

VO,h

Fig. 2. 15 Frequency-dependent droop scheme: (a) the series impedance is created in the
inverter internally; (b) the equivalent inverter circuit at the fundamental frequency; (c) the
equivalent inverter circuit at the harmonic frequency [116, 117].

The authors of [118, 119] introduced fast control loops that adjust the output
impedance of the closed-loop inverters in order to ensure inductive behaviour
with the purpose to share the harmonic current content properly. The paper
presents a small-signal analysis for parallel-connected inverters in stand-alone
AC power systems. The control approaches have an inherent trade-off between
voltage regulation and power sharing [102].
The signal injection technique proposed by [55, 120] is not dependent in the
plant parameters and can share reactive power even if the VSCs have not
perfectly matched output inductors by having each VSC inject a non-60-Hz
signal and use it as a means of sharing a common load with other VSCs on the
network. However, the circuitry required to measure the small real power output
variations due to the injected signal adds to the complexity of the control [90].
Moreover, the controllers use an algorithm which is too complicated to calculate
the current harmonic content, the harmonic current sharing is achieved at the
expense of reducing the stability of the system [104].
38

Fig. 2. 16 Schematic diagram of implementing the signal injection technique [55].

In [121] the proposed control method uses low-bandwidth data communication
signals between each generation system in addition to the locally measurable
feedback signals. The focus is on systems of distributed resources that can
switch from grid connection to island operation without causing problems for
critical loads. This is achieved by combining two control methods: droop control
method and average power control method. In this method, the sharing of real
and reactive powers between each DGS is implemented by two independent
control variables: power angle and inverter output voltage amplitude. However,
adding external communication can be considered as a drawback. Such
communications increase the complexity and reduce the reliability, since the
power balance and the system stability rely on these signals [102]. In [122, 123]
a communication bus is used in addition to the conventional droop, it has to
trigger all inverters to measure their load sharing parameters at the same line
period, this is used to correct the load sharing calculation.

2.3

Discussion

The following sections summarize the advantages and disadvantages of the
different control techniques discussed above.

2.3.1 Master/Slave Control Techniques
The master/slave control configuration has many good characteristics. The
inverters do not need a PLL circuit for synchronisation and give a good load
sharing. The line impedance of the interconnecting lines does not affect the load
sharing and the system is also easily expandable.
There are, however, a few serious disadvantages. One of the major
disadvantages is that most of these systems are not truly redundant, and have
a single point of failure, the master unit. Another disadvantage of this
configuration is that the stability of the system depends upon the number of
slave units in the system [55]. Furthermore, all these master/slave techniques,
need communication and control interconnections, so they are less reliable for a
distributed power supply system.

2.3.2 Current/Power Deviation (Sharing) Control Techniques
The current/power deviation (sharing) control techniques have excellent
features. It has a very good load sharing, transient response and can reduce
circulating currents between the inverters.
There are as well some drawbacks. It is not easily expandable due to the need
for measuring the load current and the need to know the number of inverters in
the system. The needed interconnection makes the system less reliable and not
truly redundant and distributed.

2.3.3 Frequency and Voltage Droop Control Techniques
Droop control methods are based on local measurements of the network state
variables which makes them truly distributed and give them an absolute
redundancy as they do not depend on cables/communication for reliable
operation. It has many desirable features such as expandability, modularity
flexibility and redundancy. Nevertheless, the droop control concept has some
limitations including frequency and amplitude deviations, slow transient
response and possibility of circulating current among inverters due to wire
impedance mismatches between inverter output and load bus and/or
voltage/current sensor measurement error mismatches.

2.3.4 Summary
From the above discussion it can be concluded that each of these control
techniques has its own characteristics, objectives, limits and appropriate uses.
That often makes it difficult to adapt one control scheme for all applications.
However, a deep understanding of these control techniques will help in
enhancing them and though will improve the design and implementation of
future distributed modular grid architectures.
This is why this work will not adapt one control scheme for all applications which
most of the current approaches are trying to introduce. Instead this work will
build modular different approaches that can be customized according to each
system’s (customer’s) needs and exact specifications.
This work introduces various opportunities of control functions for three-phase
inverters used to feed various passive/active grids including different topologies
to feed balanced/unbalanced loads. These are based on standardized system
concepts using various control strategies and no one-size-fits-all solution.

CHAPTER 3
THE PROPOSED SMART GRID PHILOSOPHY
“ARCHITECTURE AND COMPONENTS”

This chapter presents the system philosophy developed for the smart grid. This
philosophy is taking into account the advantages and disadvantages of the
different approaches reviewed in chapter two.
The chapter is divided into five main parts. In the first part, the general
architecture will be presented and discussed. The second part presents the
main power electronic element of the philosophy, the inverter, showing the
different inverter topologies used in the feeding philosophy. In the third part, the
developed operating principles and control and regulation techniques for these
inverters will be presented. The space vector modulation algorithms built up will
be presented in the fourth part. Next, in part five the sequence decomposition
which is used to develop the advanced control strategy for power electronic
inverters feeding unbalanced loads will be emphasized. Finally, a brief
discussion will be carried on assessing the new philosophy.

3.1 General Architecture of the Proposed Smart Grid (Feeding
Modes)
A general philosophy to supply electric energy in isolated power systems
through power electronic inverters is introduced in [124] and is extended here.
The basic system philosophy is illustrated through Fig. 3.1. The power produced
by the ECS is fed through the DC-to-DC converter and after that this DC power
is fed to the grid through the inverter. The inverter produces an AC output of a

specific voltage magnitude and frequency. The intermediate capacitance is
used to decouple the DC current flowing to the input terminal of the grid-inverter
from the DC current flowing from the DC-to-DC converters of the ECS side.
Intermediate DC Stage
DC-to-DC
Converter
ECS
No. i

Inverter
IDC

IINV

Grid

IC
VC

Fig. 3. 1: System overview of the intermediate DC stage.

The mismatches between these two currents result in variations in the voltage
across the intermediate capacitance caused by changes in the capacitor’s
current. This can be expressed using the following equation:

VC =

1
1
I C dt + VC ,0 =
C
C

∫

∫ (I

− I INV )dt + VC, 0

(3.1)

Where the voltage across the intermediate capacitance is VC, the output current
of the DC/DC converter is IDC and the input current to the inverter is IINV.
These voltage variations can be utilised to control the power flow. The size of
the capacitor is determined depending on the maximum possible mismatches
between power production and power consumption. The voltage variations
across the capacitor should be kept within the allowable ranges.
This intermediate DC stage has two important characteristics. First, it provides a
decoupling between the voltages across the terminals of the ECSs from one
side and the grid voltage from the other side. Second, it provides a decoupling
between the frequency of the ECSs (in the case of AC energy conversion
systems) from one side and the grid frequency from the other side.
In this philosophy the power flow from an energy conversion source (ECS) into
the grid may be driven by the grid or by the ECS itself as summarised in Fig.
3.2.

Fig. 3. 2: A general definition of feeding modes for DER

In a grid-driven feeding mode the flow of power from the ECS to the grid is
controlled according to the requirements of the grid while in an ECS-driven
feeding mode, the flow of power is controlled according to the requirements of
the ECS itself. In the second case, ECSs are normally controlled to maximise
their power production despite the requirements of the grid.
The grid-driven feeding mode represents the active integration case while the
ECSs-driven feeding mode represents the passive one. A grid-driven feeding
mode may be realised through two different cases: grid-forming case and gridsupporting case, while an ECS-driven feeding mode may be realised through a
grid-parallel case.
An ECS in a grid-forming case is responsible for establishing the voltage and
the frequency of the grid (state variables) and maintaining them [124]. This is
done by increasing or decreasing its power production in order to keep the
power balance in the electrical system.
An ECS in a grid-supporting case produces predefined amounts of power which
are normally specified by a management unit. Therefore, the power production
in such a case is not a function of the power imbalances in the grid.
Nevertheless, the predefined amounts of power for these units may be
adjusted. The management system may change the reference values according
to the system’s requirements and the units’ own qualifications.
The control strategy of the intermediate DC circuit is derived from the feeding
modes definition. Therefore, in the grid-driven feeding mode the voltage across
the capacitor is kept within the allowable ranges through controlling IDC current
while keeping IINV free to change, see Fig. 3.3.

DC-to-DC
Converter
ECS
No. i

VCON

Icon

Inverter
IDC

IINV

IG
Grid

IC
+
_

PWM

SVM

ϕ
CU

+
PI
Controller

Reference
Values

VREF

Fig. 3. 3: General control of a system operating in a grid-driven feeding mode (Forming,
Supporting).

An ECS in a grid-parallel case is a power production unit that is not controlled
according to the requirements of the electrical system. RES’s such as wind
energy converters and photovoltaic systems may be used to feed their
maximum power into the grid (standard applications in conventional grids). In
such a case, these systems are considered as grid parallel units.
For the ECSs-driven feeding mode control strategy the vice versa applies, IINV is
controlled and IDC is free to change, see Fig. 3.4.

Fig. 3. 4: General control of a system operating in ECSs-driven feeding mode (parallel).

3.2

Inverter Topologies

To articulate the control strategies in relation to power electronic devices a short
introduction of the different used three-phase inverter topologies is given.

3.2.1 Three-phase, Three-leg Voltage Source Inverters
Three single-phase half-bridge inverters can be connected in parallel to form
the three phase inverter configuration, one leg for each phase, see Fig. 3.5. The
gating signals of single-phase inverters should be advanced or delayed by 120
degree with respect to each other in order to obtain three-phase balanced
voltages [125]. In this case it requires that the three currents are a balanced
three-phase set. However, this topology can be used to feed balanced loads
only.

Fig. 3. 5: Three leg inverter (balanced output).

Two configurations able to generate three-phase asymmetrical signals will be
discussed. These are: The three-leg neutral point built by capacitors and the
four-leg inverter with a controlled neutral point by the fourth leg.

3.2.2 Three-phase, Three-leg, Four-wire Voltage Source Inverters
Three-phase inverters with neutral point are an evolution from the single-phase
ones. Three half-bridge single-phase inverters joined together can be seen as a
three-phase neutral point inverter, see Fig. 3.6, where each output feeds one
phase. This topology can be used to feed balanced or unbalanced loads. In
case of unbalanced loads, the sum of the output currents ia, ib, and ic will not be
zero and the neutral current will flow in the connection between the neutral point
and the mid-point of the capacitive divider [124, 126, 127]. To maintain a
symmetrical voltage across the two capacitors an adequate power electronic
and a voltage stage management are needed, this will not be taken further into
discussion.

Fig. 3. 6: Three-leg inverter with a neutral point.

3.2.3 Three-phase, Four-leg Voltage Source Inverters
The general power electronic topology of the four-legged inverter is shown in
Fig. 3.7. The goal of the three-phase four-leg inverter is to supply a desired
sinusoidal output voltage waveform to the load for all load conditions and
transients. By tying the load neutral point to the mid-point of the fourth leg, it can
handle the neutral current caused by an unbalanced load. A balanced output
voltage can be achieved due to the tightly regulated neutral point. The
additional neutral inductor Ln is optional. It can reduce switching frequency
ripple [128]. A four leg inverter can produce sixteen switching states. This
enlarges the space vector modulation to three-dimensional (3-D-SVM), for a
four-leg voltage source inverter the representation of the phase voltage space
vectors is done in the α,β,γ space.
Compared with the four-leg inverter, the three-leg four-wire inverter has a lower
number of semiconductor switches and the control function can be built like
three individual single line inverters. However, the four-leg inverter still has the
advantages of higher utilization of the DC link voltage. This is because the
maximum available peak value of the line-to-neutral output voltage in the threeleg four-wire inverter is equal to half the value of the dc link voltage while the
maximum amplitude of the line-to-line voltage with a four-leg inverter is equal to
the dc bus voltage. Moreover, the high unbalanced current flowing through the
dc link capacitors of the three-leg four-wire inverter requires higher capacitance
[32, 129]. So, the four-leg inverter has small DC link capacitor as no zero
sequence current flow across the DC link capacitor and has an additional
degree of freedom due to the fourth leg [127, 130, 131].

iI
Q

Q
D3

D4
La

Load
VI
0

Q
5

Vab

Vbc

Vca

ib
Load

ic
Load

Cc
N

Fig. 3. 7: Four-leg inverter.

In general, three-leg inverter will use the two-dimensional space vector
modulation (2-D-SVM). On the other hand, the three-leg inverter with neutral
point and the four leg inverter will extend the space vector modulation to threedimensional (3-D-SVM) making the selection of the modulation vectors more
complex. The 3-D-SVM of three-leg with neutral point inverter differ from that of
the four leg inverter. Nevertheless, the control strategies are similar. Both the
control strategies and the SVM algorithms will be discussed in detail in the
following sections.

3.3

Inverter Control

In the following sections, the known control strategies of symmetrical inverters
will be briefly reviewed; Further details can be found in [124]. Afterwards, the
proposed control strategies for the asymmetrical inverters will be introduced,
these were published in papers [131, 132].

3.3.1 Symmetrical Grid Forming
The control strategy of a three-phase inverter in grid forming mode for balanced
load is shown in Fig. 3.8. The inverter in this case determines the voltage and
the frequency of the grid. There is one inner current control loop and a second
voltage control loop. Both loops use only the d-component. The q-component of
the current can not be influenced since the reactive part is depending on the
load condition. Therefore, the q-component is not considered in this case. The
reference angle for the dq-transformation is taken from the reference frequency.

Vdc
LL

vα
iα

iβ

Vejϕ

vβ
ωref
vact
id

Vref

Fig. 3. 8: Inverter in grid forming mode for balanced loads.

3.3.2 Symmetrical Grid Supporting
The grid supporting unit for balanced loads feeds the grid with a specified
amount of power, which might be active, reactive, or a combination of both, see
Fig. 3.9. The control strategy for the grid supporting unit using active and
reactive power has four controllers, two for the current (id and iq), and two for the
power (P and Q). Active power, P, is controlled by the real part of the grid
current “id“, while reactive power, Q, is controlled by the imaginary part ”iq“.
Synchronization is implemented by the generation of the angle for the dq
transformation from the voltage on the grid.
Other control strategies for the grid supporting mode can be implemented
straight forward through controlling the real and the imaginary components of
the grid current or the magnitude of the voltage and the active component of the
power fed into the grid.

Vdc
LN

vα
iβ

iα

Vejϕ

vβ

Qact

Qref

Pact

Pref

Fig. 3. 9: P, Q-controlled inverter in grid supporting mode for balanced loads.

3.3.3 Symmetrical Grid Parallel
In the case of grid-parallel feeding mode, see Fig. 3.10, all of the produced
active power by the ECS is passed to the grid through the inverter. The active
power management is done in this application by the control of the voltage of
the DC stage. The reactive power control is similar to the grid supporting case.

Vdc
LN

vα

iα

iβ

Vejϕ

vβ

Qact
vq

Qref

VDCref

Fig. 3. 10: Q-controlled inverter in grid parallel mode.

3.3.4 Asymmetrical Grid Forming
As a grid forming unit the inverter has to provide both the voltage and the
frequency of the grid. This is done as following: The voltage and the current
sensed values are transformed from the abc-frame to the positive-negative-zero
dq sequence components. The controller block comprises current and voltage
PI controllers for each component. Six controllers are needed for the voltage
and the current components of the load. For the controller only the d-component
of the positive sequence Vp_d_ref is considered. The other reference values are
set to zero since the inverter has to supply symmetrical three phase voltage.
The output reference values from the control unit are transformed to the αβγspace and the SVM block uses them to calculate the pulse pattern for the
switches [132].
Fig. 3.11 shows an inverter in grid forming mode for unbalanced loads. The
control functions can be also described as vectors according to the following
definition:
Vp _ d _ ref 


Vp _ q _ ref 

V
n _ d _ ref

[Vpn0 _ dq _ ref ] = 
Vn _ q _ ref 


V0 _ d _ ref 
V

 0 _ q _ ref 

Vp _ d 


Vp _ q 
V 
n_d

[Vpn0 _ dq ] = 
Vn _ q 


V0 _ d 
V 
 0_q 

Vp _ d _ act 


Vp _ q _ act 
V

n _ d _ act

[Vpn0 _ dq _ act ] = 
Vn _ q _ act 


V0 _ d _ act 
V

 0 _ q _ act 

 I p _ d _ act 


 I p _ q _ act 
I

n _ d _ act

[ I pn0 _ dq _ act ] = 
 I n _ q _ act 


 I 0 _ d _ act 
I

 0 _ q _ act 

(3.2)

(3.3)

Vdc

vα
vβ

vγ

ωref

[Ipn0_dq_act]

[Vpn0_dq_act ]

[Vpn0_dq_ref]

[Vpn0_dq]

Fig. 3. 11: Inverter in grid forming mode for unbalanced loads.

3.3.5 Asymmetrical Grid Supporting
The asymmetrical grid supporting unit has to supply the grid with a specified
amount of power, which might be active, reactive, or a combination of both as
mentioned before. Synchronisation with the grid voltage is done by the voltage
reference angle which has to be generated as in the symmetrical grid
supporting mode. The desired amount of power has to be set by a management
unit in positive, negative and zero sequence components. The power controller
block generates a reference signal for the current controller. The current
controller is delivering a reference voltage signal represented by positive,
negative and zero sequence components. These reference values have to be
transformed (composed) to the αβγ-space vector and the SVM block uses them
to calculate the pulse pattern for the switches [132].
Fig. 3.12 shows a P, Q-controlled Inverter in grid supporting mode for
unbalanced loads, the control functions can be also described as vectors
according to the following definition:

 Pp _ ref 


[ Ppn0 _ ref ] =  Pn _ ref 
P

 0 _ ref 

Qp _ ref 


[Qpn0 _ ref ] = Qn _ ref 
Q

 0 _ ref 

(3.4)

 Pp _ act 


[ Ppn0 _ act ] =  Pn _ act 
P

 0 _ act 

Qp _ act 


[Qpn0 _ act ] = Qn _ act 
Q

 0 _ act 

(3.5)

 I p _ q _ act 


[ I pn0 _ q _ act ] =  I n _ q _ act 
I

 0 _ q _ act 

(3.6)

Vp _ q 


[Vpn0 _ q ] = Vn _ q 
V 
 0_q 

(3.7)

 I p _ d _ act 


[ I pn0 _ d _ act ] =  I n _ d _ act 
I

 0 _ d _ act 

V p _ d 


[V pn0 _ d ] = Vn _ d 
V 
 0_ d 

Other control strategies can be implemented simply through the real and the
imaginary components of the grid current or the magnitude of the voltage and
the active component of the power fed into the grid.
L

Vdc
LN

vα

vβ
vγ

[Ipn0_dq_act]

[Vpn0_dq_act]

Positive sequence
I-Controller

V-Controller

I-Controller

V-Controller

Negative sequence

[Ppn0__ref],
[Qpn0__ref]

[Vpn0_dq]

Zero sequence
I-Controller

V-Controller

Fig. 3. 12: P, Q-controlled Inverter in grid supporting mode for unbalanced loads.

3.3.6 Asymmetrical Grid Parallel
Obviously, in the case of asymmetrical grid-parallel unit, shown in Fig. 3.13, the
values that can be controlled are the flow of the reactive power or reactive
current to the grid. In comparison to the asymmetrical grid supporting
remarkable is the active power control using Vdc and:
 Pn _ ref 
[ Pn 0 _ ref ] = 

 P0 _ ref 

 Pn _ act 
[ Pn 0 _ act ] = 

 P0 _ act 

(3.8)

Vdc

vα
vβ
Vejϕ

vγ

[Ipn0_q_act]

[Qpn0_act]

[Vpn0_q]

[Qpn0_ref ]
[Ipn0_d_act]

Vdc_ref
[Pn0_ref ]

[Vpn0_d]
[Pn0_act ]

Fig. 3. 13: Inverter in grid parallel mode for unbalanced loads.

3.4

Space Vector Modulation (SVM)

After introducing the general architecture of the proposed smart grid (Feeding
Modes), the inverter power electronic topologies and the inverter control, the
space vector algorithms developed will be introduced. Several modulation
strategies, differing in concept and performance, have been developed so far in
order to achieve variety of aims including: wide linear modulation range, less
switching loss, less total harmonic distortion (THD) in the spectrum of switching
waveform, easy implementation and less computation time [133]. With the

development of microprocessors, space-vector modulation (SVM) has become
one of the most important PWM methods for three-phase inverters due to its
ability to reduce the commutation losses and the harmonic contents of the
output voltage, as well as obtaining higher amplitude modulation indexes [134,
135].
Fig. 3.14 shows the basic functioning of SVM in a simplified manner. First, the
calculation of the position of the voltage vector is carried out. The vector is
represented by either Cartesian coordinates (real and imaginary), or by
exponential form (amplitude to phase position). Based on the reference vector
and the DC intermediate voltage, the duty cycle for each switch will be
calculated. Finally, the switching pattern is constructed from the duty cycle.

Calculating
Voltage Vectors
Position

vα vβ

Calculating Duty
Cycles

vDC

Building Switching
Pattern

Grid
Fig. 3. 14: Principle of Space Vector Modulation [136].

In the following sections The Space Vector Algorithms used will be introduced.

3.4.1 SVM for Three-phase, Three-leg Voltage Source Inverters
A two-level three-leg inverter has six states when a voltage is applied to the
load and two states (0 and 7) when the load is connected through the upper or
lower switches resulting in zero volts being applied to the loads. Define eight

ur
voltage vectors V 0 , …, V 7 corresponding to the switch states 0, …, 7
appearing in Table 3.1. Voltage vectors V 1 , …, V 6 have unity lengths while
voltage vectors V 0 and V 7 have zero length.

Fig. 3. 15: Three phase, three leg voltage source inverter.
V5 = [0 0 1]

V4 = [0 1 1]

V7 = [1 1 1]

V6 = [1 0 1]

VDC

V1 = [1 0 0]

V2 = [1 1 0]

V0 = [0 0 0]

V3 = [0 1 0]

VDC

B
C

VDC

B
C

VDC

B
C

Fig. 3. 16: The eight inverter voltage vectors (V0 to V7).

Table 3. 1: Switches combinations and states.

Voltage

Switch

Off Switch

vectors

states

combinations

000

Va/VDC

Vb/VDC

Vc/VDC

S4 S6 S2

100

S1 S6 S2

2/3

-1/3

110

S1 S3 S2

1/3

-2/3

010

S4 S3 S2

-1/3

2/3

-1/3

011

S4 S3 S5

-2/3

1/3

001

S4 S6 S5

-1/3

2/3

101

S1 S6 S5

1/3

-2/3

1/3

111

S1 S3 S5

The result of plotting each of the output voltages in a αβ reference frame is
shown in Fig. 3.17. The figure shows first the three-axes balanced system abc
with 120° phase shift between each two axes. It als o shows the eight voltagevectors, V 0 , …, V 7 , which divide the vector space into six sectors S1, S2, ..., S6.
An orthogonal αβ -axes system is shown as well, which divides the vector
space into four quadrants, Q1 to Q4.
The vectors V 1 , …, V 6 represent the six voltage steps developed by the
inverter with the zero voltages V 0 and V 7 located at the origin. The inverter
switches at each of these states are in a steady state. To develop a sine-wave
at the inverter’s output, a switching pattern that causes the switches to function
between these states is required. This switching pattern may be realised by a
continuously rotating vector that transitions smoothly in the complete space.
This is shown in Fig. 3.17 and Fig. 3.18. To generate the switching signals that
produce the rotating vector, the switching-time intervals of the switches for each
sector should be determined.

Vmax =

1
VDC
3

Fig. 3. 17: Output voltage space vectors.

Fig. 3. 18: Output voltages in time domain.

Here, the maximum length of the target vector is not longer than the radius of
the circle, which is within the hexagon and equals

VDC / 3 . The maximum

radius pointer is also affected through the filter inductance at the output of the
filter. With a higher inductance at the same current a higher voltage arises at
the inductance. This results in a higher inverter voltage, which is represented
through the SVM pointer. Therefore, at the limit of the maximum SVM pointer is
exceeded faster at a higher inductance rather than a smaller inductance.
However, the filter effect is also decreasing if the inductivity is decreased.
Therefore, the choice of inductance in principle is connected to the switching
frequency [136, 137].
The space vector diagram is divided into six sectors, which are limited by three
vectors. In order to generate the reference vector the information about the
sector and the region where the reference vector lies should be obtained, this
can be done by using the equations of the separation lines between the sectors.
There are three major separation lines, l41, l52 and l36. The equations for these
lines are presented as follow:

l 41 : v β = 0

(3.9)

l52 : v β − 3vα = 0

(3.10)

l 36 : v β + 3vα = 0

(3.11)

Then comparisons can be implemented to identify the position of the reference
vector as it is shown in Table 3.2.
Table 3. 2: Sector Identification

Sector

Comparison

Phase

(l41 > 0) & (l52 < 0)

0° ≤ φ < 60°

(l41 > 0) & (l52 > 0) & (l36 > 0)

60° ≤ φ < 120°

(l41 > 0) & (l52 > 0) & (l36 < 0)

120° ≤ φ < 180°

(l41 < 0) & (l52 > 0)

-180° ≤ φ < -120°

(l41 < 0) & (l52 > 0) & (l36 < 0)

-120° ≤ φ < -60°

(l41 < 0) & (l52 > 0) & (l36 > 0)

-60° ≤ φ < 0°

To generate the switching signals that produce the rotating vector, the
switching-time intervals of the switches for each sector should be determined.
Fig. 3.19 shows the case of the rotating vector when it is in the first sector (S1).

2
V

V
b

V
1

Fig. 3. 19: Space-vector modulation in sector S1.

The vector V X can be resolved as:
π

V X sin( − ϕ ) = Va sin ,
3
3

and
V X sin(ϕ ) = Vb sin

(3.12)

π,

(3.13)

where Va is the component of V X in the direction of V 1 , Vb is the component
of V X in the direction of V 2 , and

is the angle of V X from V 1 . From the

above equations the following two equations are achievable:
Va =

2
3

V X sin( − ϕ )
3

(3.14)

and
Vb =

2
3

V X sin(ϕ )

(3.15)

The rotating vector V X can be approximated by applying V 1 for a percentage of
time ta and V 2 for a percentage of time tb over a period T0 which is equal to the
half of the complete cycle duration T.

V X = Va + Vb + (V0 or V7 ) =

(

)

V1 × t a V2 × t b V0 or V7 × t 0
+
+
T0
T0
T0
59

(3.16)

ta, tb, and t0 are given by:

ta =

Va
V1

T0 ,

tb =

Vb
V2

t 0 = T0 − t a − t b .

T0 , and

(3.17)

Substituting for V a from eq. (3.14) gives:



1
t a = V × cosϕ −
sin ϕ  ,
3



(3.18)

and for Vb from eq. (3.15) gives:

tb =

sin ϕ ,

(3.19)

where V is the ratio V X / V1 or V2 for the period T0 in segment S1.
The rotating voltage-vector can have a maximum value of (2VI/3), i.e. the same
value of the voltage-vectors V 1 , …, V 6 , and it can last for a period T0. After
achieving the values of ta and tb, a symmetrical switching pattern for two
consecutive periods of T0 can be constructed. The pattern is shown in Fig. 3.19.
From Fig. 3.20, when phase a is “High” this means that the switch connected to
phase a (refer to Fig. 3.16) is in touch with terminal Q1, and when it is “Low”,
then it is in touch with Q4. The same thing applies for the other two phases.
When they are “High” then they are in touch with the upper contacts and when
they are “Low” then they are in touch with the lower contacts.
V
0

V
1

V
2

V
7

t0/2

V
7

V
1

V
0

Phase a

Phase b

Phase c
t0/2

T0
T

Fig. 3. 20: Symmetric space-vector modulation pulse generation.

The diagram in Fig. 3.19 represents sector S1 only. Phase a is in “Low” position
for t0/2, phase b is in “Low” position for (t0/2 + ta), and phase c is in “Low”
position for (T0 – t0/2). For the rest of the sectors, the same derivation procedure
has to be repeated to find the time intervals for the switches when to be “High”
and when to be “Low”. Table 3.3 shows the “Low” periods for the six sectors.
The “High” periods are the rests of the half cycles T0’s.
The time periods ta and tb are functions of the angle ϕ and the voltage
magnitude V, while t0 is the difference between T0 and the sum of these
periods. Therefore, given ϕ and V it is possible through pulse-width modulation
to create a sinusoidal voltage at the inverter’s output terminals.
Table 3. 3: “Low” periods of the three-phases in the six sectors

Sector

t0/2

t0/2 + ta

T0 - t0/2

t0/2 + ta

t0/2

t0/2 + ta

t0/2

t0/2 + ta

T0 - t0/2

t0/2 + ta

t0/2

t0/2 + ta

Phase

With this SVM, a three-phase sinusoidal function system can be produced. The
introduced control functions build the basics for all discussion related to them.
Details to the introduced principles of the SVM can be found in [124, 138-140].

3.4.2 SVM for Three-leg, Four-wire Voltage Source Inverters
Three-leg four-wire inverters are expected to play an essential role in power
distribution because of their ability to handle the neutral current caused by
unbalanced or non-linear loads. The use of three-dimensional space vector
modulation (3-D-SVM) with four-leg inverters was explored by many authors
using different approaches. On the other hand, the use of 3-D-SVM with threeleg four-wire was discussed briefly in [141-145]. In [146], a discussion about the
3-D-SVM for three-leg four-wire voltage source inverters was made including
theory, implementation and application examples.

In the first approach introduced in [147] and called the zero vector approach, a
zero vector is generated by turning off all power switches to produce zero volts
at the output terminals of the inverter. Here, the switching vectors, separation
planes, the matrices for switching vectors, duty cycles and the switching
sequences are derived. Still, the proposed zero vector approach algorithm has
a drawback of stressing the IGBTs unequally. Therefore, another SVM
algorithm without using a zero-vector was launched in [148]. This algorithm
based on vectors compensation (compensated vectors approach) is more
practical as it is not only stressing the IGBTs equally but less as well.
These two novel algorithms are explained in detail in Appendix A.

3.4.3 SVM for Three-phase, Four-leg Voltage Source Inverters
The purpose of the three-phase four-leg inverters is to achieve a balanced
output voltage waveform over all loading conditions and transients, an
additional neutral inductor Ln can be added to the neutral line where the
switching frequency ripple will be reduced. It is ideal for applications like
industrial automation, military equipment, which require high performance
uninterruptible power supply as well as active power filters. Fig. 3.21 shows the
structure of a four leg inverter.

Va
VDC
Vb
S8

Fig. 3. 21 Space vector diagram for five-level diode-clamped inverter.

The detailed steps of obtaining the needed SVM can bee seen in Appendix A.2
More details can be found in [128, 149, 150].

3.5

Sequence Decomposition

Symmetrical Sequence Decomposition (SSD) is used mainly in power electric
fault analysis. In [151] a controller based on symmetrical components for
handling unbalanced conditions with a multilevel inverter was introduced. In the
present work, sequence decomposition is used in the implementation of the four
leg three-phase-inverter controller. Through SSD it is possible to represent an
asymmetrical three-phase signal as a sum of positive, negative and zero
sequence. Positive and negative components are three-phase symmetrical
signals, while the zero sequence is a single-phase one. The general idea is
shown in Fig. 3.22.

vb
vb _ p

vabc

vabc = {va , vb , vc }

negative

positive

vb _ n

va _ n va _ p

vc _ n
vc

vc _ p

Fig. 3. 22 Sequence decomposition and composition.

In dq0-coordinates, the negative sequence appears as a disturbance in d and q
variables at a frequency of 2ω [152, 153]. This is because the dq reference
frame is rotating in the positive direction at an angular frequency of ω, while the
negative sequence disturbance rotates at an angular frequency of ω in the
opposite direction. The zero sequence appears at a frequency of ω, because
the zero-axis is stationary [154].
An asymmetrical three-phase signal va, vb and vc (the following method may be
applied to currents in exactly the same way) can be decomposed into two
symmetrical-three-phase-waves, the positive and the negative components

{

v p = v a _ p , vb _ p , v c _ p

{

v n = v a _ n , vb _ n , v c _ n

}

(3.20)

(3.21)

and the zero component v0. The asymmetrical signals can be reconfigured by
the sums

v a = v a _ p + v a _ n + v0

(3.22)

vb = vb _ p + vb _ n + v 0

(3.23)

vc = v c _ p + v c _ n + v 0

(3.24)

Here, the sequence decomposition is used to develop an advanced control
strategy for power electronic inverters with unbalanced loads. Therefore, each
voltage of each line is transformed by an ideal filter in vα and vß components
[92, 151]. Then a Park transformation is performed to get the vd and vq
components (see Fig. 3.23). When this transformation process is done, each
line is represented in separate dq-components (see additionally Fig. 3.24). The
signals in the dq-plane can be interpreted as complex values:

V a _ dq = Va _ d + jVa _ q

(3.25)

V b _ dq = Vb _ d + jVb _ q

(3.26)

V c _ dq = Vc _ d + jVc _ q

(3.27)

Park Transformation

va _ α 


va _ β 

va(t)

 cos ϕ
− sin ϕ


sin ϕ  vα  vd 
 = 
cos ϕ  vβ  vq 

va _ d 


va _ q 

ϕ = ω 0t
Fig. 3. 23 Getting the d,q-components for phase a [155].

These unsymmetrical complex dq-values for each line can be now decoupled
by the use of SSD. The relationship between the symmetrical dq-components
corresponding to the three-phase unsymmetrical dq signals is given by equation
(3.58) :

1 a
V p _ dq 

 1
2
V n _ dq  = .1 a
V
 3 1 1
 0 _ dq 

where

a 2  V a _ dq 


a .V b _ dq 
1  V c _ dq 


(3.28)

a=e

2π
3

4π
3

= cos

2π
2π
1
3
+ j sin
=− + j
3
3
2
2

(3.29)

= cos

4π
4π
1
3
+ j sin
=− − j
3
3
2
2

(3.30)

and
a2 = e

The phasors are defined in a complex dq-plane. The complete transformation
process is represented in an illustrated form again, (see Fig. 3.24).

va(t)

1 α α 2  V a 
1 2  
1 α α  V b 
3
1 1 1  V c 



vb(t)

vc(t)

Fig. 3. 24 Sequence decomposition.

The back transformation into α , β and γ components is easier since no
transformation to the complex domain is needed. Summing up v p and v n since
they are both in the αβ -plane, while taking into account that the negative
imaginary component is rotating anticlockwise we get:

vα = v p _ α + vn _ α

(3.31)

vβ = v p _ β − vn _ β

(3.32)

vγ = v0 _ α

(3.33)

From the last equation it can be seen that the zero component v 0 _ α equals the
γ -component and both of them are single-phase signals. The complete

transformation starting from dq plane of each line is shown in (see Fig. 3.25).

Fig. 3. 25 Sequence composition [155].

The composed α , β and γ components can be used for the space vector
modulator (SVM) to produce the pulse pattern for the power electronic switches
[128].

3.6

Discussion

This chapter presented the system components developed for the smart grid. In
the first part, the general feeding architecture was presented and discussed.
The second part presents the main power electronic element of the philosophy,
the inverter, showing the different topologies used.
In the third part, the operating principles and control techniques for these
inverters were presented. This included novel standardized advanced control
concept for four-wire inverters (three-leg four-wire and four-leg) using
symmetrical components based on sequence decomposition to supply
balanced/unbalanced loads. The principle idea is to control the positive,

negative and zero sequence components. Controlling (eliminating) the negative
and zero sequence components helps expanding the inverter based systems by
increasing the distribution network efficiency (consequently leads to less losses
and results in enhancing the power quality). This can be used for shunt active
filters’ applications and also grant the opportunity to supply unbalanced loads
which mean supplying single and three phase loads using the same source.
The space vector modulation algorithms used were presented in the fourth part.
This has introduced different novel algorithms for three-leg, three-leg four-wire
and four-leg inverters. An absolute high light is the novel three dimensional
space vector modulation (3D-SVM) control strategy of three-leg four-wire
inverters able to feed grids with unbalanced loads while reducing the switching
frequency losses. The proposed solution covers an existing gap in this field
since the SVM of three-leg four-wire inverter was discussed briefly in the current
literature according to author's knowledge.
Next, in part five the sequence decomposition which is used to develop the
advanced control strategy for power electronic inverters feeding unbalanced
loads was emphasized.
These different components were published in the following reviewed papers
[131, 132, 147, 148, 155-158]. Successful experimental results for the different
components where accomplished by another team in the lab.
After introducing the main components of the proposed smart grid, a detailed
discussion will be made in the next chapters showing its architect and
application opportunities. This will show that the new philosophy is modular,
expandable, and can accept different types and sizes of power electronic
inverters and loads.

CHAPTER 4
THE PROPOSED SMART GRID PHILOSOPHY
“OPERATION, CONTROL, AND MANAGEMENT”

In the previous chapter, the principles of the proposed smart grid philosophy
and its components have been introduced. In this chapter, the operation,
control, application and management of this philosophy are going to be
presented.
Even though, most of the current approaches to build future smart power
systems are trying to introduce one-size-fits-all solution but the fact is that each
system (customer) needs are different and various approaches are needed to fit
their exact specifications. This chapter will introduce varied opportunities of
control functions for three-phase inverters used to feed passive/active grids
including different topologies to feed balanced/unbalanced loads.
The proposed philosophy will develop different and various robust control
approaches for a realistic distributed power system with power electronics
inverters as front-end, see Fig. 4.1. These control strategies should guarantee
real modularity, higher reliability and avoid a single point of failure to qualify to
be standardised. The proposed control architecture should maintain three
phase voltages and frequency in the grid within certain defined limits and has to
provide power sharing between the units according to their ratings and user
settings.

10 – 30 kV Medium Voltage Level

Grid

3~
3~

Grid
Industry

3~
3~
=

Grid
WindPark

...

..
.

.
..

0.4 kV Low Voltage Level

=
=

..
.

Grid

..
.

Grid

3~
=
=

=
3~

..
.

=
=
3~

=
=

Supervisory Control and
Energy Management

Isochronous Control

Droop Control

Fig. 4. 1: The control philosophy (example).

As discussed in chapter three, the electrical energy produced by ECSs may be
fed into the electrical grid according to one of two possible feeding modes. In
the first mode, the amount of electrical energy fed into the grid is specified
according to the grid requirements. This mode is denoted as a “Grid-driven
feeding mode”. In the second mode, the ECSs specify the amount of energy fed
into the grid. This mode is denoted as an “ECSs-driven feeding mode”.
Fig. 4.2 presents a diagram showing the structure of the control functions
proposed in this research study. These control strategies will be launched in this
chapter.
The system philosophy under discussion is also characterised by an
intermediate DC stage between the energy sources from one side and the
electrical grid from the other side. From the DC-DC converters’ side, it connects
to the ECSs and from the main inverter’s side it connects to the electrical grid,
see chapter three. However, in order to simplify the analysis, the ECSs-side
(the generation sources such as PV and fuel cells) are represented using a DC
voltage source. This will not be taken into consideration here since it was
discussed in detail in a previous dissertation [124].

Fig. 4. 2: Feeding modes at the grid side.

Based on the modes proposed in Fig. 4.2 many scenarios can be obtained. The
key scenarios are taken into account in this research study as shown in Fig. 4.3.
The proposed philosophy has two main categories. The first category is the
Multi-inverter Three-wire system and the second is the Multi-inverter Four-wire
system. For each of these categories different control scenarios will be
proposed and explored.

Fig. 4. 3: The proposed scenarios.

This chapter will start by investigative supervisory control and management
scenario. Then, droop control scenario will be introduced. Afterwards,
isochronous control function scenario will be launched. Next, combination of
droops and isochronous control scenario will be proposed. Finally, the
fundamental case of using an inverter in swing mode will be studied.
This chapter will be dedicated to the operation, control, application and
management of the philosophy. The verification case studies will be presented
in the next chapter.

4.1

Multi-inverter Three-wire System Control Philosophy

Since the inverters are relatively stiff sources, with unique value of open circuit
frequency and voltage (due to components tolerance), large circulating currents
would result if they were simply paralleled without additional control. This can
be done based on information available locally at the inverter (state variables)
for example using droops to make the system less stiff or using data
communication such as in supervisory controlled systems. Recently data
communication between units became easy realized by the rapid advances in
71

the field of communication. However, it is preferred that communication of
information will be used to enhance system performance but must not be critical
for system operation. The following sections will introduce modular approaches
to parallel inverters using different methodologies.

4.1.1 Supervisory Control and Energy Management Scenario
The specific aim of this concept is to develop a standardised control strategy for
a realistic distributed power system with power electronics inverters as frontend. The proposed control architecture will maintain the three phase voltages
and frequencies in the grid precisely and will provide power sharing between
the units according to their ratings, meteorological parameters, economical
dispatch prospective (can include real-time pricing) and user settings. This
allows total energy optimization. The designed system can include inverter units
of different power rating, distributed at various locations feeding distributed
unequal loads taking into account dissimilar line impedances between them to
insure true expandability and generation placement flexibility. This means that
the types, sizes, and numbers of the inverters, and the size and nature of the
electrical loads may all vary without the need to alter the control strategy. The
amount of data exchange can be small if it includes only basic measurements
and set points but will increase proportionally as more functions are added. The
proposed structure is shown in Fig. 4.4. It is worthy to note that the source do
not have to be a single ECS and could be a hybrid power system (HPS).
The supervisory control is responsible for units dispatching, load management,
and power optimization. It can include also many functions like meteorological
forecasting and demand side management as illustrated in [159]. It can also
manage an intelligent switch or a feeder to the main grid or to other mini-grids.
The current and voltage control are done locally at the inverters according to the
definition introduced in chapter three. Moreover, the proposed control can be
implemented not only in distribution system of isolated grid systems, but also in
the interconnected power systems (some times called on-grid micro-grid).

Meteorological Forecasting
User Settings
LG

Measurements

Set points

DC-to-DC
Converter

L1
L2
L3

IINV

IDC
IC

Load

Set points
R

WEC

Grid
Forming

DC-to-DC
Converter
ECS
No. i

Grid
Parallel

DC-to-DC
Converter

Fuel Cell

Grid
Supporting

Battery
Bank

Load

Fig. 4. 4: Overview of supervisory control and energy management proposed system structure.

The control functions of the inverters are shown in Fig. 4.3. As mentioned in
chapter 3, each grid mode has its own character for controlling the inverter. The
grid forming contains inner current control loop and outer voltage control loop.
The reference voltage is given to control the voltage of the system. The angular
speed related to the frequency of the system is also set as constant (2πf). The
control loop produce the voltage of d-axis which will be transformed to αβ
frame, the angle is required for that. These voltages in αβ frame are supplied to
the SVM to calculate the switching sequence and periods. In the next step the
inverter supplies the three-phase currents to the system through the LC filter.
The output currents will be measured to feed the signal to the inner current
loop. The voltages across the capacitor are also measured to feed the outer
control loop.

Measurments

Vdc
LL

v
i

v
L

ref

vact

id
vd

Vref

v
i

Qact

Set points

(Reference values)

Vdc

Qref

Pact
Pref

Vdc

Meteorological
Forecasting
User Setting

Qact

Qref
id

VDC_Ref
vd

Fig. 4. 5: Supervisory control and energy management scenario.

As stated previously, the responsibility of the grid supporting mode is to
maintain the system power balance. The reference power of the grid supporting
inverter is calculated in the supervisory unit based on other inverters in the
system (grid forming and parallel modes) and loads. Moreover, it depends also
on the pre-setting percentages or algorithms used in the supervisory control to
manage the power balance. The reference values of PGS and QGS are
calculated based on that. In the simplest case, the set values can be adjusted
by the percentage value (GSpercent) and the active power load (Pload) and
reactive power load (Qload). As a simple example, the set values of active and
reactive power can be calculated via equations 4.1 and 4.2 respectively:

GS _ ref

(∑
=

QGS _ ref

Where,

∑

n
i =1

Pload i and

n
i =1

(∑
=

∑

n
i =1

)

Pload i − ∑ j =1 PGPj × GSpercent
m

(4.1)

100
n
i =1

)

Qload i − ∑ j =1 QGPj × GSpercent
m

(4.2)

100

Qload i are the summation of the active and reactive

power of load in the system, where, n is the number of loads and i is the
counter.

∑

m
j =1

PGPj and

∑

m
j =1

QGPj are the summation of the active and reactive

power of grid parallel units in the system, where, m is the number of grid parallel
units and j is the counter.
This means that the amount of power needed is deducted from the power of the
grid parallel units since they can not be influenced by the grid, the rest is shared
between the grid forming and supporting according to the percentage GSpercent.
This percentage can be calculated according to an algorithm based on the units’
ratings, meteorological parameters, economical dispatch prospective and user
settings but this will not be taken into discussion over here since its out of the
scope of this study. This was demonstrated in [159].
After the actual active and reactive power of the grid supporting mode is passed
to the outer loop of the controllers, another inner current control loop is used.
The current of d-axis is used to control the active power signal and the current
of q-axis is used to control the reactive power signal.
The grid parallel mode is used to produce maximum amount of active power
and can sometimes supply certain amount of reactive power to the system. In
the voltage control loop, there are two reference inputs, voltage reference and
reactive power reference. There are three inputs measured to calculate the new
reference for Id and Iq controllers. These are first, the DC intermediate stage
which will be passed through the voltage controller to feed into the inner current
loop for Id controller; based on that the new reference of the voltage is
established. The second input, is the three-phase voltage measured from the
line. The three-phase voltage is transformed into dq-frame and the angle of the
voltage can be measured from voltage of q-axis (Vq). The voltage magnitude is
fed to the Iq controller which is compared to the reactive set value to get the
new reference value for Iq controller. Third, the actual output current values
75

measured are used by Id and Iq controllers of the inner control loop. The current
signals are transformed into dq-frame. After the controlled signals passed
through the Id and Iq controllers, both signals are added with the actual values of
the voltage in dq-frame and then transformed into αβ frame to control the
inverter’s output.
It should be also noticed that as a grid parallel unit, if the system frequency is
rising too high the inverter’s output should be reduced or set to zero
(disconnected).

4.1.2 Droop Control Functions Scenario
A grid-driven feeding unit can either be a grid-forming unit, or a grid-supporting
unit. An electrical system should include, at least, one grid-driven feeding unit,
so as to maintain its power balance. If an electrical system has one grid-driven
feeding unit only, then it should be the grid-forming element [124]. If there is
more than one grid-driven feeding point in an electrical system, one of them, at
least, takes the responsibility of forming the grid state-variables and the others
function as grid-supporting elements. Recalling the definitions of grid-forming
and grid-supporting units presented in the introduction of chapter three, a griddriven feeding unit is always active from the control point-of-view.
One possibility that makes the system less stiff and allows load sharing is using
the droop functions depending on the system state variables, the voltage and
frequency.
In the following, the droop control concept will be revised and afterwards
implemented to the inverter control.
4.1.2.1 Analysis of Frequency and Voltage Droop Control Techniques

Droop is a change in speed or frequency, proportional to load. As the load
increases, the speed or frequency decreases. Doing that, the generator will
produce a certain amount of power at each speed (frequency). In the
conventional frequency droop control of generators with speed governors, an
integral relation exists between the angular frequency (ω) and phase angle (δ).
This is imposed by the physical relation between generator’s speed and its rotor
position. No such relation need exist in a voltage source inverter (VSI), though it
is typically convenient to emulate such a relation through control action in order
76

to reach stable operation when the inverters are connected in parallel [90, 119].
Down-scaling the conventional grid control concept into the low voltage grid is a
promising approach. Using this methodology the system architecture is
providing more modularity, redundancy, expandability, maintainability, reliability
and avoids huge communication requirements and costs [82, 96].
Fig. 4.6 shows the conventional equivalent circuit of two inverters connected in
parallel to common load.

V ∠θ

E 2 ∠δ 2

E1 ∠δ1

Fig. 4. 6: Parallel operation of two inverters, inductive impedance.

When the output impedance of the inverters is inductive, which is mostly the
case, the complex power supplied S to the load by inverter one is given by:

S1 = P1 + jQ1 = E1 I1*

(4.3)

where
 E 1 c o s δ 1 + jE 1 s in δ 1 − V 

jX



I 1* = 

 E 1 cos δ 1 + jE 1 sin δ 1 − V 

jX



S 1 = ( E1 cos δ 1 + jE1 sin δ ) 

(4.4)
*

(4.5)

this gives us
E V 
P1 =  1  sin δ 1
 X 

E1 − E1V cos δ 1
X

(4.6)

Q1 =

(4.7)

and for the second inverter,
E V 
P2 =  2  sin δ 2
 X 

Q2 =

E 2 2 − E 2V cos δ1
X

(4.8)
(4.9)

Where X is the output reactance of an inverter, δ is the phase angle between
the output voltage of the inverter and the voltage of the common load, E is the
amplitude of the output voltage of the inverter and V the output voltage at the
load, respectively.
It can be seen that the active power P is dependent on the power angle δ, while
the reactive power Q mostly depends on the output voltage amplitude. Each
inverter should be able to share the load automatically based on its power
rating. As a result, the following droop functions can be expressed as:

ω = ω0 − m ⋅ P

(4.10)

V = V0 − n ⋅ Q

(4.11)

where ω0 and V0 are the output voltage frequency and amplitude at no load,
respectively. m and n (Also refer to as KP and KQ respectively) are the droop
coefficients for the frequency and amplitude, respectively. This can be seen in
the figure below.

ω = 2πf
∆ω

ω ref

V (Volt)
∆V

Vref

kW
∆P

Pmax

∆Q

kvar
Q max

Fig. 4. 7: Frequency and voltage droop curves.

To insure proper load sharing based on the inverters different rating, the droop
coefficients are selected as follows:

m1 ⋅ S1 = m2 ⋅ S2 = ... = mn ⋅ S n

(4.12)

n1 ⋅ S1 = n2 ⋅ S 2 = ... = nn ⋅ S n

(4.13)

where S1, S2…Sn are the apparent power rating of the different inverters.
However, this solution is valid only if the lines are mostly inductive which is not
applicable all the time. To get the general solution, equivalent circuit (see Fig.
4.8) has to be analyzed assuming that the lines impedance is Z = R+jX.

V ∠θ

E1 ∠δ1

E 2 ∠δ 2

Fig. 4. 8: Parallel operation of two inverters.

δ1
θ

Fig. 4. 9: Phasor diagram.

The power supplied by the first inverter is expressed as:

S1 = P1 + jQ1 = E1I1*
 E
I1* = 


cos

+ jE 1 sin δ
R − jX

(4.14)
1

− V 



(4.15)

 E cos δ 1 + jE 1 sin δ 1 − V 
S 1 = (E 1 cos δ 1 + jE 1 sin δ ) 1

R − jX



(
S1 = 

)

 E 2 − E V co s δ + jE V sin δ   R − jX 
1
1
1
1
1 


R2 + X

(4.16)

(4.17)

Based on that follows:

P1 =
Q1 =

[(E
[(E

2
1
2
1

)

]
R]

− E 1V cos δ 1 R + E 1V sin δ 1 X
R2 + X 2

(4.18)

− E 1V cos δ 1 X − E 1V sin δ 1
R2 + X 2

(4.19)

)

Assuming that the lines are pure resistive (the angle is zero) which can be the
case in law voltage lines, then:

P1 =
Q

(E
=

2
1

)

− E 1V cos δ 1
E
≈ 1 .( E 1 − V )
R
R

− E 1 V sin
R

≈

− E 1V
R

.δ

(4.20)
(4.21)

Based on that, it can be seen that active power is related to the inverter output
voltage amplitude while reactive power is related to the inverter frequency. This
is known as the opposite droop method. We can generally say that transmission
lines are inductive for high voltage lines, mixed for medium voltage lines and
more resistive (but not pure resistive) for low voltage lines. The Influence of
active and reactive power on voltage and frequency for different line impedance
ratios can bee seen in Fig. 4.10.

Fig. 4. 10: Influence of active and reactive power on voltage and frequency for different line
impedance ratios: (a) R/X=0, (b) R/X=1, (c) R/X= ∞ [110].

In [97, 98] a comparison between both droop concepts is carried on based on
the voltage control and the active power dispatch which are the major control
issues, see Table 4.1. It concludes that the only advantage of using the inverse
droops is the direct voltage control. But if one would control the voltage this
way, no power dispatch would be possible. Notable, is that the phase angle
should be consider in the control strategy in order to help in eliminating the
effect of the resistive lines since it is affecting reactive power directly.

Table 4. 1 Comparison of Droop Concepts for the Low Voltage Level [97, 98]

Conventional droop

Opposite droop

Compatible with HV-level

Yes

Compatible with generators

Yes

Direct voltage control

Yes

Active power dispatch

Yes

In the following sections, it is attempted to reach a communication less modular
power supply philosophy by adding third control loop (droops). The different
ways of controlling the grid-side inverters introduced in sections 3.3.1 and 3.3.2
of the previous chapter (grid-forming and grid-supporting cases) will be modified
in the next discussion by adding droops as third loop to share load
proportionally.
4.1.2.2 Grid Forming Inverter with Droop Control
The grid forming unit has to form and maintain the frequency and the voltage of
the electrical system. It has also to feed as much current into the grid as
necessary. The basic method (classical) of adding a droop function is
introduced in Fig. 4.11. The active power will be measured continuously at the
grid. This measured power will be reflected into frequency deviation through the
droop factor. This droop factor is dependent on the size of the inverter and the
system approved frequency deviation. This change will be subtracted from the
reference value. By integrating this frequency the rotating angle φ will be
calculated and used afterwards for the transformation. The d string regulates
voltage at the output capacitor through the comparison of the measured voltage
Vact and the reference voltage Vref. The voltage controller regulates voltage
differences by setting the reference current value Id_ref. The output signal is
placed through the αβ-transformation to the space vector modulation, where the
switching sequence and periods are calculated. However, this method does not
take the reactive power into consideration.

Fig. 4. 11: The classical Grid Forming mode with droop.

The new proposed droop grid forming control strategy can be seen in Fig. 4.12
below.

L
Vdc
LL RL
C

Vα

SVM

ωact
PLL

Vß
Iq_act

Droop

∆ω

Pact

ωref
dq

Id_act
-

Vact
-

Droop

∆V

Qact

Vref
Fig. 4. 12: The proposed grid forming mode with droop.

In this approach the current d-component is used for controlling the voltage and
the current q-component is responsible for the frequency control. Here, the
controller is getting in addition to the reference and the actual voltage values,
voltage droop value based on the reactive power change in the system. The

rotating angle for the transformations has to be measured from the grid; it is
used also for the different transformations. This way the dq- transformation can
be synchronized to the grid and allows parallel operation.
If the active power in the grid changes (∆P) the droop function will result in a
similar angular frequency (∆ω). This will be compared with the reference
angular frequency (ωref) and the actual frequency (ωact). In case of positive
active power step (increment) this will build a greater difference which leads to
positive increment at the reference signal of the current controller. In case of
negative active power step (decrement) the current reference signal will be
lower. The actual angular frequency value (ωact) is measured from the network
through phased locked loop (PLL) using the voltage at the output filter
capacitor. The current controller calculates the reference value for Iq based on
the difference. In case of reactive power step (∆Q) the d string will have the
same analogy to the q string when an active power step happens. The droop
function will calculate the voltage change related to this (∆Q). This value will be
compared to the voltage reference and actual values and given to the current
controller. Herewith, an increase in the reactive power will lead to a positive
increment in the reference value and vice versa.
In order to understand the concept behind, it is useful to have a look at the
single phase circuit diagram shown in Fig. 4.13.

Fig. 4. 13: Single phase diagram.

The voltage at the connection point to the grid VG is related to the voltage of the
inverter VINV through the voltage across the inductor VL. The latter is given by:

VL = L

diINV
dt

(4.22)

and the relation between the three voltages is given by:

VINV = VL + VG = L

diINV
+ VG
dt

(4.23)

If the loads of the grid change while VG is constant, this change appears only as
change in the current IG. To maintain VG constant under variation of IG, requires
VINV to be changed accordingly. The voltage-phasor diagram of VG, VL, and VINV
is shown in Fig. 4.14.
The sketch in (a) shows a reference case; If only the magnitude of IG changes
without a change in its angle θ, as demonstrated in (b), the magnitude of VL
changes accordingly, resulting in a change of both the magnitude of VINV and its
angle δ with respect to VG. A change of the angle θ without a change of the
magnitude of IG, as shown in (c), results in a change in the direction of VL
without a change in its magnitude. Consequently, both the magnitude of VINV
and its angle δ are influenced.

δ
θ

δ θ

Fig. 4. 14: Voltage-phasor diagrams [124].

For the circuit above:
V V
P =  G INV
 X


 sin δ


V INV 2 − V INV V cos δ
X

(4.24)
(4.25)

Where X is the output reactance of an inverter, δ is the phase angle between
the output voltage of the inverter and the voltage of the load, VINV is the
amplitude of the output voltage of the inverter and VG the output voltage at the

load, respectively. It can be seen that the active power P is dependent on the
power angle δ, while the reactive power Q mostly depends on the output
voltage amplitude as mentioned before.
While working in the dq frame, the active and reactive power can be influenced
using the voltage and current d,q components of the inverter independently
since the power is given by:

P = U d ⋅ I d +U q ⋅ I q

(4.26)

Q = Uq ⋅ Id −U d ⋅ Iq

(4.27)

Assuming a positive active and reactive power, the inverter’s phasor diagram
can be displayed as in Fig. 4.15. As a reference case in (a), it is assumed that
the dq current components have the same size. The resulting supplied current,
and the dq components, are phase shifted by the angle θ related to the grid
voltage VG. The inductance voltage VL is based on its impedance and the grid
current going through. The load angle δ describes the phase shift of the inverter
voltage to the grid voltage.

Fig. 4. 15: a) Phasor diagram of grid-forming case. b) Phasor diagram of grid-forming case while
minimizing Id. c) Phasor diagram of grid-forming case while minimizing Iq.

As demonstrated in (b) the equivalent current have a great phase shift to the
grid voltage. Through the current phase shift the inductance voltage will be also
displaced (adjusted). The inverter voltage adjusts itself with changed amplitude
and a decreased load angle. By the reduction of the d component the load
angle will be also reduced. Having a look at the power, the decrease of the

load angle entails a decrease of the active power feed. At the same time the
reactive power feed rises.
In (c) the Id is increased and Iq is minimized. That leads to a decrease of the
phase shift of the line current to the mains voltage and that has in consequence
a shift of the inductance voltage. Thus we will get a larger load angle of the
inverter voltage. It is also important to keep in mind that contrary to the
synchronous generators, the inverter can be driven in all four operating
quadrants (depending on the DC source behind). This can be seen in Fig. 4.16.

Re
P<0
Q<0

P>0
Q<0

VG
VINV

VINV

I
δ

δ
I

0° < δ < 90°

90° < δ < 180°

-Im
VL

P<0
Q>0

P>0
Q>0
VG

VINV

-180° < δ < -90°

δ I

-90° < δ < 0°

Fig. 4. 16: Phasor diagram of an inverter (General view)[136].

4.1.2.3 Grid Supporting Inverter with Droop Control
A grid-supporting unit - as mentioned before - is a unit whose power production
under steady-state condition does not depend upon the voltage and the
frequency in the grid. Instead, it produces an amount of power equal to a
reference value, which is specified by another unit like load dispatcher. A gridsupporting unit acquires its frequency from the grid. In the whole grid there is
only one frequency. Therefore, if for some reason, the frequency of the grid
changes, the frequency of the grid-supporting unit follows that change [124].
The proposed grid supporting control strategy can be seen in the figure below.

Fig. 4. 17: Grid supporting mode with droop.

According to the power equations, the frequency is responsible for influencing
active power and the voltage is responsible for the reactive power. Therefore,
frequency droop is added to the d string and voltage droop is added to the q
string. This at the end affects the reference values of the active and reactive
power produced in order to react to the change in the grid state variables
(frequency and voltage).
At the input, the actual frequency will be compared to the reference. In case of a
load step this leads to a frequency difference. From that difference ∆f an
equivalent active power difference can be calculated through the droop function.
Once the grid frequency decreases in comparison to the reference frequency
more active power will be supplied to the grid and vice versa. The active power
87

verses frequency droop can be expressed mathematically through the slope KP ,
where :
∆f
∆P

KP = −

(4.28)

In case the frequency is 50 Hz, the inverter will supply the network with its
reference power (Pref). A change of frequency ∆f leads to an equivalent change
in the active power ∆P. The rate of the change is dependent on the droop factor
KP. Normally, the frequency allowed band is 1-2% [97].

ω = 2πf
∆ω

ω ref

V (Volt)
∆V

Vref

∆P

Pmax

∆Q

kvar
Q max

Fig. 4. 18: (a) Frequency vs. active power droop and (b) Voltage vs. reactive power droop.

At the input of the current controller the actual and the reference active power
will be compared, Id will be calculated from that difference. That will be
forwarded to the voltage controller which is producing the voltage values (α,β)
and pass it to the SVM block.
The q string is comparable to the d string. The droop function is weighing the
voltages against each other and results in an equivalent reactive power
difference ∆Q. This is all passed to the controller. The reactive power verses
voltage droop can be expressed mathematically through the slope KQ, where:

KQ = −

∆V
∆Q

(4.29)

The voltage allowed band is normally 4-5% [97] based on the grid level.
An example is shown in Fig. 4.19. The modular isolated grid is using droopcontrolled inverters (Grid forming, Grid supporting) in addition to grid parallel
and different loads. The network consists of three inverters with altered power
and modes of operation placed at different places and working in different
modes. The first inverter is operating in grid forming (drooped) mode. The grid
forming is normally the most powerful in the grid. It also includes a phased
88

locked loop (PLL) that allows the interaction between the inverter and the grid,
including synchronisation and grid monitoring. The second inverter is in grid
supporting (drooped) mode and will react to the change in the state variables.
The third inverter is working in grid parallel mode and is not controllable by the
grid; it includes no droop and supplies a certain amount of power to the grid
based on the source status and not the grid.

Vdc
C
φ

SVM

Iq_act

PLL

Droop

∆ω

φ
Va

ωist

Pact
L

ωref
Vact
Id_act
-

R
Droop

∆V

Qact

Vref

Vdc

C
Va
SVM

iα

iβ

Vejϕ

Vact

iq
Qact

∆V

∆Q

Vref

Qref

f act
Pact

∆P

∆f
fref

vd
Pref

Vdc

vα

iα

iβ

Vejϕ
φ

vβ

Qact

Qref
id

VDCref

Fig. 4. 19: Modular grid using droop-controlled Inverters.

4.1.3 Isochronous Control Functions Scenario
If the load is frequency/voltage critical then isochronous mode (zero droop) is
the optimal solution. An inverter operating in the isochronous mode will operate
at the same set frequency/voltage during steady state regardless of the load it is
supplying as shown in Fig. 4.20. The isochronous control scheme provides in
comparison to the droop scheme the possibility of precise control of the voltage
and the frequency.

Fig. 4. 20: (a) Frequency vs. active power isochronous and (b) Voltage vs. reactive power
isochronous.

This needs communication in order to measure the grid load and share this
information with all the other inverters in the system. However, the realisation of
such a system needs low-bandwidth communication and is considered practical
especially if the inverters are connected to the same load bus and have no
massive distance between them. This is also needed if sensitive loads exist that
can not accept the voltage and frequency band used in droop schemes.
The proposed grid forming with isochronous control strategy can be seen in the
figure below.

Fig. 4. 21: Grid-forming with isochronous control function.

Here, the total measured load is divided by the total rated power and compared
to the active power supplied by the generator (inverter) divided by its rated
power.

∆Pi ,[%] =

∑P

Grid

i =1
n

∑S
i =1

−

PGen,i
S r ,Gen,i

(4.30)

r ,i

This difference is amplified and added to the summation point of the
actual/reference angular frequency. The difference out of that summation point
is passed to the q current controller. The output of the controller is compared to
the actual current value. The output of that comparison is given to the voltage
controller, this will calculate Vq which is transformed to the αβ frame and used
by the SVM to generate the switching states.
The reactive power is also controlled in the same manner. The total measured
reactive power load is divided by the total rated power and then is compared to
the active power supplied by the generator divided by its rated power.

∆Qi ,[%] =

∑Q

Grid

i =1
n

∑ S r ,i

−

QGen,i
S r ,Gen ,i

(4.31)

i =1

This difference is amplified and added to the summation point of the
actual/reference voltage. The difference out of that summation point is passed
to the d current controller. The output of the controller is compared to the actual
current value. The result of that comparison is given to the voltage controller,
this will calculate Vd which is transformed to the αβ coordinator and used by the
SVM to generate the switching states. The frequency used by the controller is
measured from the grid using PLL and then integrated to get the needed angle.
An example of such a grid can bee seen in Fig. 4.22. The network consists of
three inverters from different power classes working in isochronous mode
(modified grid forming). The inverters will work at the same frequency/voltage in
steady state regardless of the load they are supplying. The reason for the use of
grid forming for the isochronous mode is the basic control structure of inverters.
91

In synchronous machines and grid forming inverters the controller starts from
the power through the frequency (speed) controller to the current (moment)
controller. In the grid supporting case this looks different since the control
sequence is the opposite.

Vdc

Vα
φ

SVM
Vβ

PLL
1

- ∆ω

Vref

∆P%
-

Vact

- ∆V

Ptotal

10-99%
P-Factor

ωref

Power
Calculation

Id_act

Iq_act

ωact

10-99%

Pgen,1

Qtotal

∆Q%
-

Q-Factor

Qgen,1

Vdc

Vα
φ

SVM
Vβ

ωact
Power
Calculation

PLL

Iq_act
-

- ∆ω

φ
ωref
Id_act
-

10-99%

P-Factor

Vact
- ∆V
Vref

Ptotal

Pgen,2

Qtotal

Qgen,2

∆P%

10-99% ∆Q%

L
-

Q-Factor

Vdc

Vα
φ

SVM

ωact
PLL

Power
Calculation

Vβ
Iq_act

Id_act
-

1
- ∆ω

ωref

10-99%

∆P%
-

P-Factor

Vact
-

- ∆V
Vref

10-99%
Q-Factor

Ptotal

Pgen,n

Qtotal

Qgen,n

∆Q%
-

Fig. 4. 22: Modular grid using grid-forming with isochronous control function.

4.1.4 Combined Isochronous/Droop Control Functions Scenario
In this scheme inverter’s active power/frequency is regulated using isochronous
control while the reactive power/voltage is regulated using the droop scheme.

Through that it is possible to minimize the frequency difference and fix it to the
nominal frequency while minimizing the communication as well.
The proposed grid forming with isochronous-droop control strategy can be seen
in Fig. 4.23.

Fig. 4. 23: Grid-forming with isochronous-droop control function.

The total measured active power load is divided by the total rated power and
compared to the active power supplied by the generator divided by its rated
power. This difference is amplified and added to the summation point of the
actual/reference angular frequency. The difference out of that summation point
is passed to the q current controller. The output of the controller is compared to
the actual current value. The output of that comparison is given to the voltage
controller, this will calculate Vq which is transformed to the αβ coordinator and
used by the SVM to generate the switching states.
In case of reactive power step (∆Q), the droop function will calculate the voltage
change related to this (∆Q). This value will be compared to the voltage
reference and actual values and given to the current controller. Herewith, an
increase in the reactive power will lead to a positive increment in the reference
value and vice versa.
A modular grid using grid-forming with isochronous-droop control function is
shown in Fig. 4.24. The grid includes three inverters working in isochronousdroop mode with different power rates. The frequency/active power control is
done using isochronous mode. In contrast to previous case the voltage/reactive
93

power interaction is controlled using droop functions. Through that it is possible
to minimize the frequency difference and fix it to the nominal frequency while
minimizing the communication as well by using the droop function for the
voltage/reactive power.

Vdc

Vα
φ

SVM
Vβ

ωact

PLL
1
- ∆ω

10-99%

Ptotal

∆P%
-

P-Factor

ωref
Id_ac
- t

Power
Calculation

Iq_act
-

Pgen,1

Vact

R
Droop

- ∆V

Qact

∆Q
-

Vref

Qref

Vdc

Vα
φ

SVM
Vβ

ωact
Power
Calculation

PLL

Iq_act
-

- ∆ω

ωref

10-99%

Ptotal

Pgen,2

∆P%
-

P-Factor

Vact
Id_act
-

Droop
- ∆V

Qact

∆Q

Vref

Qref

Vdc

Vα
φ

SVM
Vβ

ωact
PLL

Power
Calculation

Iq_act

1
-

- ∆ω

ωref

10-99%

Ptotal

∆P%
-

P-Factor

Pgen,n

Vact
Id_act
-

Droop
- ∆V

Qact

∆Q
-

Vref

Qref

Fig. 4. 24: Modular grid using grid-forming with isochronous-droop control function.

4.1.5 Swing-Inverter/Droop Control Function Scenario
A further form of grid forming (actually this is the fundamental role) is the swing
inverter (since it is doing the same function carried by a swing generator in an
isolated conventional power system). The swing inverter compares the grid
94

state variables frequency and voltage in the grid and drives them back to their
reference values in the case of deviations. See Fig. 4.25.

L
VDC
RL L L
Vß

C
SVM

ωact

PLL

Vα

Iq_act
-

∆ω

ωref

Vact
Vd

Id_act
-

∆V

Vref

Fig. 4. 25: Grid-forming as swing inverter.

An example of modular grid using swing-inverter and droop-controlled Inverters
is shown in Fig. 4.26. Normally, the swing inverter is the one with the highest
power rating in the system so that the system will accept the largest load
changes within its capacity. The remaining feeding inverters/machines are
switched in droop mode in parallel to the swing inverter. Maximum load for this
type of system is the addition of the output of the swing inverter plus the total
set power of the droop machines. The minimum system load cannot be allowed
to decrease below the output set for the droop inverters/generators. If it does,
the system frequency will change, and the swing inverter can be motorized.
Grid parallel units may co-exist. The following figure shows the grid forming
inverter in swing mode. The controller compares the grid state variables with the
associated references. In case of deviations corresponding current reference
values are given.

Vdc
C

vα

SVM

PLL

vβ

ωact
Iq_act
-

ωref

L
Vact
Id_act
-

R
Vref

Vdc

vα
SVM

iβ

iα

vβ

Power
Calculation

Vejϕ
φ

Vact

Qact

∆Q

V
Vref

φ
Qref

fact
id

Pact

∆P

f
fref

Pref

Vdc

vα

iα

iβ

vβ

Vejϕ
φ

iq
Qact
Qref
id

VDC,ref

Fig. 4. 26: Modular grid using swing-inverter and droop-controlled Inverters.

4.2

Multi-inverter Four-wire System Control Philosophy

One of the desirable characteristics of inverters in three-phase systems is the
ability to feed unbalanced loads with voltage and frequency nominal values.
Four-wire inverters are developed to power unbalanced/nonlinear three-phase
loads. They can also feed three phase and single phase AC loads
simultaneously. Furthermore, four leg inverters can be also used as shunt
96

active power filters to reduce the zero and negative sequence current
components generated by unbalance loads. By compensating these current
components the efficiency of power transmission can be maximized which
means less line losses and better power quality.

4.2.1 Supervisory Control and Energy Management Scenario
In similar manner to the concept proposed in section 4.1.1, the control
architecture will maintain three phase voltages and frequency in the grid
precisely and will provide power sharing between the units according to their
ratings, meteorological parameters, economical dispatch prospective (can
include real-time pricing) and user settings. It has however the advantage of
handling the neutral current which allows supplying unbalanced/non-linear loads
as well as single/three phase loads using the same source.
This scenario allows total energy optimization. The designed system can
include inverter units of different power rating, distributed at various locations
feeding distributed unequal loads taking into account dissimilar line impedances
between them to insure true expandability and generation placement flexibility.
This means that the types, sizes, and numbers of the inverters, and the size
and nature of the electrical loads may all vary without the need to alter the
control strategy. The amount of data exchange is small since it includes only
basic measurements and set points. The supervisory control is responsible for
units’ dispatching, load management, and power optimization. It can include
also many functions like meteorological forecasting and demand side
management. It can also manage an intelligent switch or feeder to the main grid
or to other mini-grids. The current and voltage control are done locally at the
inverters according to the definition introduced in chapter 3. Moreover, the
proposed control can be implemented not only in distribution system of isolated
grid systems, but also in the interconnected power system. Fig. 4.27 shows the
control functions of the inverters. As mentioned in chapter 3, each grid mode
has it own character for controlling the inverters.
As a grid forming unit the inverter has to provide the voltage and the frequency
of the grid. This is done as following. The voltage and the current sensed values
are transformed from the abc-frame to the positive-negative-zero dq sequence
components. The controller block comprises current and voltage PI controllers
for each component. Six controllers are needed for the voltage and the current
97

components of the load. For the controller only the d component of the positive
sequence Vp_d_ref is considered. The other reference values are set to zero
since the inverter has to supply symmetrical three phase voltage. The output
reference values from the control unit are transformed to the αβγ-space and the
SVM block uses them to calculate the pulse pattern for the switches.
The asymmetrical grid supporting unit has to supply the grid with specified
amount of power, which might be active, reactive, or a combination of both.
Synchronisation with the grid voltage is done by the voltage reference angle
which has to be generated as in the symmetrical grid supporting mode. The
desired amount of power can be set by a management unit in positive, negative
and zero sequence components. The power controller block generates
reference signal for the current controller. The current controller is delivering a
reference voltage signal represented by positive, negative and zero sequence
components. These reference values have to be transformed (composed) to the

αβγ-space vector and the SVM block uses them to calculate the pulse pattern
for the switches.
Obviously, in the case of asymmetrical grid-parallel unit, the values that can be
controlled are the flow of the reactive power or reactive current to the grid.
An example of supervisory control and energy management scenario can be
seen in Fig. 4.27. This scenario is working in the following way: the amount of
power needed is deducted from the power of the grid parallel units since they
can not be influenced. The rest is shared between the grid forming and
supporting. This percentage can be calculated according to an algorithm based
on the units’ ratings, meteorological parameters, economical dispatch
prospective and user settings but this will not be taken into discussion over here
since it is out of the scope of this study. It should also be noticed that in grid
parallel, if the system frequency is rising too high the inverter’s output should be
reduced or set to zero (disconnected).

Vdc
LN

vα
vβ

vγ

ωref

[Vpn0_dq_act]

[Ipn0_dq_act]

I-Controller

V-Controller

Positive sequence

Negative sequence
I-Controller

V-Controller

[Vpn0_dq_ref]

[Vpn0_dq]

Zero sequence
I-Controller

V-Controller

Vdc
LN

vα

vβ
vγ

[Vpn0_dq_act]

[Ipn0_dq_act]

Positive sequence
I-Controller

V-Controller

L
R

Negative sequence
I-Controller

V-Controller

[Ppn0__ref],
[Qpn0__ref]

[Vpn0_dq]

Zero sequence
I-Controller

V-Controller

Vdc
LN
C

vα
Power
Calculation

vβ

Vejϕ
vγ

[Ipn0_q_act]

[Qpn0_act]

[Vpn0_q]

[Qpn0_ref ]
[Ipn0_d_act]

Vdc_ref

[Pn0_ref ]

[Vpn0_d]
[Pn0_act ]

Fig. 4. 27: Supervisory control and energy management scenario (four-wire).

4.2.2 Droop Control Functions Scenario
As mentioned previously, when two or more inverters have to work in parallel,
an additional loop is needed to guarantee stability and load sharing. Droop
functions are a common method of doing that, which is adapted from the
conventional power system. Adding this loop into the proposed controller for
four-wire systems will be described in the following sections.

4.2.2.1 Asymmetrical Grid Forming Inverter with Droop Control
As a grid forming unit the inverter has to establish the voltage and the frequency
of the grid. In the approach proposed here, the positive current d-component is
used for controlling the voltage and the current positive q-component is
responsible for the frequency control.
Here, the controller is getting in addition to the reference and the actual voltage
values, a voltage droop value based on the positive reactive power change in
the system, see Fig. 4.28. The rotating angle for the transformations has to be
measured from the grid. It is used also for the different transformations. This
way the dq-transformation can be synchronized to the grid and allows parallel
operation.
If the active power in the grid changes (∆P) the droop function will result in a
similar angular frequency (∆ω). This will be compared with the reference
angular frequency (ωref) and the actual frequency (ωact). In case of a positive
active power step (increment) this will build a greater difference which leads to a
positive increment at the reference signal of the current controller. In case a
negative active power step (decrement) the current reference signal will be
lower. The actual angular frequency value (ωact) is measured from the network
through PLL using the voltage at the output filter capacitor. The current
controller calculates the reference value for positive Iq based on the difference.
In case of reactive power step (∆Q) the d string will have the same analogy to
the q string when an active power step happens. The droop function will
calculate the voltage change related to this (∆Q). This value will be compared to
the voltage reference and actual values and given to the current controller.
Herewith, an increase in the reactive power will lead to a positive increment in
the reference value and vice versa.

100

For the negative and zero sequence the change of the active power is giving a
change in the voltage q component and a change in reactive power is producing
a change in the voltage d component which is affecting the controllers in
analogy way to the positive component.
LL

Vdc

vα
vβ

vγ

[Ipn0_dq_act]

[Vpn0_dq_act]

P ac t
P

[Vpn0_dq]

Q act

Fig. 4. 28: The proposed asymmetrical grid forming mode with droop.

4.2.2.2 Asymmetrical Grid Supporting Inverter with Droop Control
A grid-supporting unit - as mentioned before - is a unit whose power production
under steady-state conditions does not depend upon the voltage and the
frequency in the grid. Instead, it produces an amount of power equal to a
reference value, which is specified by another unit like a load dispatcher. A gridsupporting unit acquires its frequency from the grid. In the whole grid there is
only one frequency. Therefore, if for some reason, the frequency of the grid
changes, the frequency of the grid-supporting unit follows that change [124].
The proposed grid supporting control strategy can be seen in Fig. 4.29.

101

Vdc
LN
PLL

vα

fact

vβ
vγ

[Ipn0_dq_act]

[Vpn0_dq_act]
Positive sequence

I-Controller

fact

V-Controller

fref
Vact
Vref

Negative sequence
I-Controller

V-Controller

I-Controller

V-Controller

[Vpn0_dq]

Zero sequence

Fig. 4. 29: The proposed asymmetrical grid supporting mode with droop.

According to the power equations, the frequency is responsible for influencing
active power and the voltage is responsible for the reactive power. Therefore, a
frequency droop is added to the d string and a voltage droop is added to the q
string. This at the end affects the reference values of the active and reactive
power produced in order to react to the change in the grid state variables
(frequency and voltage).
At the input, the actual frequency will be compared to the reference. In case of a
load step this leads to frequency difference. From that difference ∆f an
equivalent active power difference can be calculated through the droop function.
Once the grid frequency decreases in comparison to the reference frequency
more active power will be supplied to the grid and vice versa. The active power
verses frequency droop can be expressed mathematically through the slope KP ,
where :
KP = −

∆f
∆P

(4.32)

In case the frequency is 50 Hz (nominal frequency), the inverter will supply the
network with its reference power (Pref). A change of frequency ∆f leads to an
102

equivalent change in the active power ∆P. The rate of the change is dependent
on the droop factor KP. Normally, the frequency allowed band is 2%.
At the input of the current controller the actual and the reference active powers
are compared, Id will be calculated from that difference. That will be forwarded
to the voltage controller which is producing the voltage values (α,β) and pass it
to the SVM block.
The q string is comparable to the d string. The droop function is weighting the
voltages against each other and results in an equivalent reactive power
difference ∆Q. This is all passed to the controller. The reactive power verses
voltage droop can be expressed mathematically through the slope KQ, where:

KQ = −

∆V
∆Q

(4.33)

The voltage allowed band is normally 4-5%. An example is shown in Fig. 4.30,
where a modular grid using droop-controlled inverters (Grid forming, Grid
supporting) in addition to grid parallel and different loads is introduced. The
network consists of three inverters from different power classes and modes of
operation placed at different places and working in different modes. Inverter one
is operating in grid forming (drooped) mode. Inverter two is in grid supporting
mode (drooped) and the third inverter is working in grip parallel mode. Inverter
one provides the network with constant voltage and frequency. Inverter two is
based on the network state variables and provides power into the grid. Inverter
three is grid parallel and is not controllable by the grid and includes no droop.

103

Vdc
LN
PLL

vα
vβ

Power
Calculation

vγ

[Vpn0_dq_act]

[Ipn0_dq_act]

Positive sequence
I-Controller

V-Controller

I-Controller

V-Controller

I-Controller

V-Controller

Negative sequence
P

[Vpn0_dq]

Zero sequence
Q

Vdc
LN
PLL

vα

fact

vβ

vγ

[Ipn0_dq_act]

[Vpn0_dq_act]
Positive sequence

I-Controller

fact

V-Controller
fref

Vact
Vref
Negative sequence
I-Controller

V-Controller

I-Controller

V-Controller

[Vpn0_dq]

Zero sequence

Vdc
LN
C

vα
Power
Calculation

vβ
Vejϕ

vγ

[Ipn0_q_act]

[Qpn0_act]

[Vpn0_q]

[Qpn0_ref ]

[Ipn0_d_act]

Vdc_ref
[Pn0_ref ]

[Vpn0_d]
[Pn0_act ]

Fig. 4. 30: Modular grid using droop-controlled inverters.

4.2.3 Isochronous Control Functions Scenario
The isochronous control scheme provides in comparison to the droop scheme
the possibility of precise control of the voltage and the frequency. This needs
communication in order to measure the grid load and share this information with
104

all the other inverters in the system. However, the realisation of such a system
needs a low-bandwidth communication especially if the inverters are connected
to the same load bus and have no huge distance between them. This is also
needed if we have sensitive loads that can not accept the voltage and
frequency band used in droop schemes.
The proposed grid forming with isochronous control strategy can be seen in the
figure below.
LL

Vdc

vα
vβ

vγ

[Ipn0_dq_act]

[Vpn0_dq_act]

P
P

[Vpn0_dq]

Q
Q

Fig. 4. 31: Grid-forming with isochronous control function.

The total measured load is divided by the total rated power and compared to the
active power supplied by the generator divided by its rated power.

∆Pi ,[%] =

∑P
i =1
n

Grid

∑ S r ,i

−

PGen,i
S r ,Gen,i

(4.34)

i =1

This is done for each component separately (+, -, 0). This difference is amplified
and added to the summation point of the actual/reference angular frequency for
the positive component. The difference out of that summation point is passed to
the q current controller. The output of the controller is compared to the actual
current value. The output of that comparison is given to the voltage controller,
this will calculate Vq which is transformed to the αβ coordinator and used by the
SVM to generate the switching states.
105

The reactive power is also controlled in the same manner. The total measured
reactive power load is divided by the total rated power and compared to the
active power supplied by the generator divided by its rated power.

∆Qi ,[%] =

∑Q
i =1
n

Netz

∑ S r ,i

−

QGen ,i
S r ,Gen,i

(4.35)

i =1

This is done for each component separately. This difference is amplified and
added to the summation point of the actual/reference voltage. The difference
out of that summation point is passed to the q current controller. The output of
the controller (Ipd_ref , Ind_ref, I0d_ref) are compared to the actual current value
(Ipd_act, Ind_act, I0d_act). The output of that comparison is given to the voltage
controller. This will calculate Vd which is transformed to the αβ coordinator and
used by the SVM to generate the switching states.
The frequency used by the controller is measured from the grid using PLL and
then integrated to get the needed angle.
An example of a modular grid using grid-forming inverters with isochronous
control function can be seen in Fig. 4.32. The network consists of three
inverters from different power classes working in isochronous mode (modified
grid forming). The inverters will work at the same frequency/voltage in steady
state regardless of the load they are supplying.

106

Vdc
LN
PLL

vα
vβ

vγ

[Ipn0_dq_act]

[Vpn0_dq_act]
Positive sequence

I-Controller

∆

Q total+

V-Controller
-

Qgen,1+

1
Ptotal+
Negative sequence
I-Controller

[Vpn0_dq]

1
1

Zero sequence

1
1

∆

V-Controller

Qtotal-

I-Controller

Pgen,1+

∆

V-Controller

1
1

Qgen,1-

PtotalPgen,1-

Q
Q

Qtotal0
Qgen,10
Ptotal0
Pgen,10

Vdc
LN
PLL

vα
vβ
vγ

[Ipn0_dq_act]

[Vpn0_dq_act]
Positive sequence

I-Controller

∆

Q total+

V-Controller
-

Qgen,2+

1
Ptotal+
-

Negative sequence
I-Controller

∆

V-Controller

[Vpn0_dq]

1
1

I-Controller

∆

V-Controller

P
P

Qtotal-

Zero sequence

Pgen,2+

1
1

Qgen,2PtotalPgen,2-

Qtotal0
Qgen,20
Ptotal0
Pgen,20

Vdc
LN
PLL

vα
vβ
vγ

[Ipn0_dq_act]

[Vpn0_dq_act]
Positive sequence

I-Controller

∆

Q total+

V-Controller
-

Qgen,n+

1
Ptotal+
Negative sequence
I-Controller

V-Controller

1
1

I-Controller

V-Controller

Qtotal-

Zero sequence

Pgen,n+

∆

[Vpn0_dq]

1
1

∆
-

1
1

Qgen,nPtotalPgen,n-

Qtotal0
Qgen,n0
Ptotal0
Pgen,n0

Fig. 4. 32: Modular grid using grid-forming with isochronous control function.

4.2.4 Combined Isochronous/Droop Control Functions Scenario
In this scheme, active power/frequency are regulated using isochronous control
while the reactive power/voltage are regulated using the droop method.
Through that, it is possible to minimize the frequency difference and fix it to the
107

nominal frequency while minimizing the communication as well. The proposed
grid forming with isochronous-droop control strategy can be seen in the figure
below.
L

Vdc
LN
PLL

vα
vβ
vγ

[Ipn0_dq_act]

[Vpn0_dq_act]
Positive sequence

I-Controller

V-Controller

1
Ptotal+
P

Negative sequence

J
I-Controller

Pgen,n+

V-Controller

[Vpn0_dq]
1

Zero sequence
I-Controller

PtotalPgen,n-

Q
Q

V-Controller

1
-

Ptotal0
Pgen,n0

Fig. 4. 33: Grid-forming with isochronous-droop control function.

The total measured active power load is divided by the total rated power and
compared to the active power supplied by the generator divided by its rated
power. This difference is amplified and added to the summation point of the
actual/reference angular frequency. The difference out of that summation point
is passed to the q current controller. The output of the controller is compared to
the actual current value. The output of that comparison is given to the voltage
controller, this will calculate Vq which is transformed to the αβ coordinator and
used by the SVM to generate the switching states.
In case of reactive power step (∆Q), the droop function will calculate the voltage
change related to this (∆Q). This value will be compared to the voltage
reference and actual values and given to the current controller. Herewith, an
increase in the reactive power will lead to positive increment in the reference
value and vice versa.
An example of modular grid using grid-forming with isochronous-droop control
functions can be seen in Fig. 4. 34. The grid includes three inverters working in
isochronous-droop mode with different power rates. The frequency/active power
control is done using isochronous mode. In contrast to previous case the
108

voltage/reactive power interaction is controlled using droop functions. Through
that it is possible to minimize the frequency difference and fix it to the nominal
frequency while minimizing the communication as well by using the droop
function for the voltage/reactive power.
Inverter
=

Vdc

Isochronous-droop Grid Forming 1

Load
RL

~~
~
LN
PLL

vα
vβ

ia
ib
ic

SVM

va
vb
vc

Ip_dq
In_dq
I0_dq

vγ

Vp_dq
Vn_dq
V0_dq

[Ipn0_dq_act]

[Vpn0_dq_act]

Positive sequence

Ip_d

I-Controller

Ip_q

Vp_d

1/K

Q+

∆Vp_d

Vp_d

V-Controller

act

Vp_d_ref

Vp_q

P-Factor

ref

∆P+%

∆

Negative sequence

In_d

I-Controller
Vp_dq
Vn_dq
vα
vß
vγ

In_q -

Vn_d

[Vpn0_dq]

V0_dq

Vn_d

V-Controller

∆Vn_d

Vn_q

1/K

∆P-%

I0_q

V0_d

∆V0_d

V0_q

1/K

SGen,n

V0_q_ref
P-Factor

∆P0%
-

Inverter

Q+
QQ0

Pgen,1-

∆V0_q

V0_d_ref

V0_q

Vdc

Ptotal-

Zero sequence
V-Controller

1
Stotal

P+
PP0

Q-

P-Factor

∆Vn_q

Pgen,1+

Vn_q_ref

I0_d

Ptotal+

1
SGen,n

Vn_d_ref

Vn_q

I-Controller

1
Stotal

Ptotal0

1
SGen,n

Pgen,10

Isochronous-droop Grid Forming 2

~~
~

∑PGrid,i

LN
PLL

vα
vβ

ia
ib
ic

SVM

va
vb
vc

Ip_dq
In_dq
I0_dq

vγ

Vp_dq
Vn_dq
V0_dq

[Ipn0_dq_act]

[Vpn0_dq_act]

Positive sequence

Ip_d

I-Controller

Ip_q

Vp_d

1/K

Q+

∆Vp_d

Vp_d

V-Controller

act

Vp_d_ref

Vp_q

P-Factor

ref

∆

∆P+%
-

Negative sequence
I-Controller
Vp_dq
Vn_dq
vα
vß
vγ

Vn_d

[Vpn0_dq]

V0_dq

In_d

Vn_d

V-Controller

In_q -

Vn_q

1/K

∆Vn_d

Vn_q_ref

P-Factor

∆P-%

∆Vn_q

I0_q

V0_d

V-Controller

1/K

∆V0_d

V0_q

P-Factor

∆P0%
-

Inverter

Ptotal-

1
SGen,n

Pgen,2-

Q+
QQ0

V0_q_ref
∆V0_q

1
Stotal

P+
PP0

V0_d_ref

V0_q

Vdc

Pgen,2+

Q-

Zero sequence

I0_d

Ptotal+

1
SGen,n

Vn_d_ref

Vn_q

I-Controller

1
Stotal

Ptotal0

1
SGen,n

Pgen,20

Isochronous-droop Grid Forming n

~
~
~
LN

vβ

ia
ib
ic

SVM

va
vb
vc

Ip_dq
In_dq
I0_dq

vγ

Load

PLL

vα

[Ipn0_dq_act]

Vp_dq
Vn_dq
V0_dq

[Vpn0_dq_act]

Positive sequence

Ip_d

I-Controller

Ip_q

Vp_d

1/K

Q+

act

∆Vp_d

Vp_d

V-Controller

Vp_d_ref

Vp_q

P-Factor

ref

∆

∆P+%
-

Negative sequence
I-Controller
Vp_dq
Vn_dq
vα
vß
vγ

V0_dq

[Vpn0_dq]

Vn_d

In_d

In_q -

Vn_d

V-Controller

Vn_q

∆Vn_d

1/K

Vn_q_ref

P-Factor

∆Vn_q

V0_d
V0_q

I0_q
-

V-Controller

∆P-%
-

V0_d
V0_q
-

Pgen,n+

1
Stotal

Ptotal-

1
SGen,n

Pgen,n-

P+
PP0

Q+
QQ0

Q-

Zero sequence
I-Controller

Ptotal+

1
SGen,n

Vn_d_ref

Vn_q

I0_d

1
Stotal

∆V0_d

1/K

V0_d_ref
V0_q_ref
P-Factor

∆V0_q

∆P0%
-

1
Stotal

Ptotal0

1
SGen,n

Pgen,n0

Fig. 4. 34: Modular grid using grid-forming with isochronous-droop control function.

109

4.2.5 Swing-Inverter/Droop Control Functions Scenario
A further form of grid forming (actually this is the basic function) is the swing
function. The swing inverter compares the grid state variables frequency and
voltage in the grid and drives them back to their reference values in the case of
deviations. See Fig. 4.35.

Vdc

vα

vβ
vγ

ωref

[Vpn0_dq_act ]

[Ipn0_dq_act]

[Vpn0_dq_ref]

[Vpn0_dq]

Fig. 4. 35: Grid-forming as swing inverter.

An example of a modular grid using swing-inverter and droop-controlled
inverters can be seen in Fig. 4.36. Normally, the Swing inverter is the one with
the

highest

power

rating

the

system.

The

remaining

feeding

inverters/machines are switched in droop mode in parallel to the swing inverter.
Grid parallel units may exist. The following figure shows the grid forming
inverter in swing mode. The controller compares the grid state variables with the
associated references. In the case of deviations corresponding current
reference values are given.

110

Vdc
LN

vα
vβ
vγ

[Ipn0_dq_act]

[Vpn0_dq_act]

Positive sequence
I-Controller

V-Controller

I-Controller

V-Controller

Negative sequence

[Vpn0_dq]
[Vpn0_dq_ref]

Zero sequence
I-Controller

V-Controller

Vdc
LN
PLL

vα

fact

vβ

vγ

[Ipn0_dq_act]

[Vpn0_dq_act]
Positive sequence

I-Controller

fact

V-Controller
fref

Vact
Vref
Negative sequence
I-Controller

V-Controller

I-Controller

V-Controller

[Vpn0_dq]

Zero sequence

Vdc
LN
C

vα
vβ
Vejϕ
vγ

[Ipn0_q_act]

[Qpn0_act]

[Vpn0_q]

[Qpn0_ref ]
[Ipn0_d_act]

Vdc_ref
[Pn0_ref ]

[Vpn0_d]

[Pn0_act ]

Fig. 4. 36: Modular grid using swing-inverter and droop-controlled Inverters.

111

4.3

Additional Aspects

In the following is a brief look at some critical issues that we should not lose
sight of in terms of the new philosophy. Some of these will be covered shortly
including the role of energy storage systems, power system nonlinearity,
harmonics and stability issues.

4.3.1 Role of Energy Storage Systems
Exploitation of renewable energy sources, even when there is a good potential
resource, may be problematic due to their variable and intermittent nature. In
addition, wind fluctuations, lightning strikes, sudden change of a load, or the
occurrence of a line fault can cause sudden momentary dips in system voltage.
Earlier studies have indicated that energy storage can compensate for the
stochastic nature and sudden deficiencies of RESs for short periods without
suffering loss of load events, and without the need to start more generating
plants. Another issue is the integration of RESs into grids at remote points,
where the grid is weak may generate unacceptable voltage variations due to
power fluctuations. Upgrading the power transmission line to mitigate this
problem is often uneconomic. Instead, the inclusion of energy store for power
smoothing and voltage regulation at the remote point of connection would allow
utilization of the power and could offer an economic alternative to upgrading the
transmission lines.
The current status shows that several drivers are emerging and will spur growth
in the demand for energy storage systems. These include: the growth of
stochastic generation from renewables; an increasingly strained transmission
infrastructure as new lines lag behind demand; the emergence of micro-grids as
part of distributed grid architecture; and the increased need for reliability and
security in electricity supply. However, a lot of issues regarding the optimal
active integration (operational, technical and market) of these emerging energy
storage technologies into the electric grid are still not developed and need to be
studied, tested and standardized [160, 161].

4.3.2 Nonlinearity
Many components in a power system such as generators, excitation systems,
governors and loads have non-linear characteristics. These components and

112

their associated controls include saturation and output limitations. The theory of
nonlinear systems can be used to analyze these nonlinearities. However, the
application is restricted to small and simple systems. In the presence of larger
order model complexities such as excitation control, turbine control, dynamic
load and a network with transfer conductance etc., suitable functions are difficult
to obtain. However, the theory of linear system analysis can provide useful
insights into the operating behaviour although the dynamic behaviour of the
system must be assumed linear for such tools to be applicable [162].
Voltage source inverters (VSI) have various non-linearity issues. These are
influenced by several parameters including [163, 164]:

1. Dead time to prevent short circuit
2. Turn-on/off of the power devices
3. The dc-link voltage
4. Snubber circuit and the voltage drops across the switches
5. The slope of the rising and falling edges of the output voltage from the
parasitic capacitance of devices

The main nonlinearity of voltage source inverters is attributed to the necessary
dead time inserted in every pulse-width modulated (PWM) cycle to avoid the
short-through of the dc power supply. During this dead time, the output voltage
is determined according to the direction of the load current. The turn-on/off
delay times for insulated gate bipolar transistor (IGBT) based inverters cannot
be neglected and contributes to the nonlinearity in similar fashion, and can be
treated as part of the dead time. Another important factor is the voltage drop
across the power switches. This voltage drop can be divided into two parts, one
part is constant, which is referred to the threshold value; the other is the
resistance voltage drop, varying according to the load current, which is caused
by the conduct resistance. The conduct resistance, in turn, varies due to
temperature changes [164]. The first three factors, in principle, could be
determined from an accurate measurement or nominal values obtained from
these devices’ respective vendors, the last two may change with the current
amplitude and temperature in a nonlinear fashion [163].

113

An accurate approach of nonlinearity compensation was not taking into account
in this study since it is out of the scope of this work. The main focus of this study
is to feed and stabilise the grid state variables (voltage and frequency). The
study mainly illustrates that the control strategies will in general work under
different grid conditions and combinations.

4.3.3 Harmonics
Harmonics are AC voltages and currents with frequencies that are integer
multiples of the fundamental frequency. On a 50-Hz system, this could include
2nd order harmonics (100 Hz), 3rd order harmonics (150 Hz), 4th order
harmonics (200 Hz), and so on, see Table 5.1. Normally, only odd-order
harmonics (3rd, 5th, 7th, 9th) occur on a 3-phase power system.
Table 5. 1 Harmonics sequencing values [165]

Harmonic order

Fund

2nd

3rd

4th

5th

6th

7th

etc

Phase Sequence

…

We are interested mostly in harmonics 1 through the 25th (50-1500Hz). But
most harmonic problems are due to the 3rd, 5th and 7th. Modelling accuracy is
not good beyond the 25th. Zero sequence currents flow through neutral or
ground paths. Positive and negative sequence currents sum to zero at neutral
and grounding points [166].
Total Harmonic Distortion (THD) is an important index used widely to describe
power quality issues in transmission and distribution systems. It considers the
contribution of every individual harmonic component on the signal [167]. THD is
defined for voltage and current signals, respectively, as follows:

THDv =

THDi =

∑

∞
h= 2

Vh2

(4.36)

∑

∞

2
h=2 h

114

(4.37)

This means that the ratio between rms values of signals including harmonics
and signals considering only the fundamental frequency define the total
harmonics distortion.
In [168] the author shows that modern PWM inverters are excellent generators
even when they supply non-linear loads. This is mainly related to their output
impedance which remains very low up to high frequencies and the output
voltage distortion due to circulating currents, even highly distorted currents, is
negligible, see Fig. 4.37. The PWM inverter is by far the best available
generator as regards its ability to minimize the voltage harmonics distortion. It is
5 to 6 times better than a transformer of the same rating. More details can be
seen in [167-170]

Fig. 4. 37: Output impedance of different sources in function of frequency [168].

4.3.4 Stability
Power system stability is the ability of an electric power system, for a given
initial operation condition, to regain a state of operation equilibrium after being
subjected to a physical disturbance, with most system variables bounded in
such a way that particularly the entire system remain enact. It is a measure of
the inherent ability of the system to recover from extraneous disturbances (such
as faults), as well as planned disturbances (such as switching operations) [29].

115

The designed inverter based grids used in this study showed stable operation.
However, the control approaches should be investigated in detail for stability
issues. This should take into account the effect of inverters output filter, grid
impedance, droop coefficients and the presence of rotating generators. In
addition, the system response and stability should be investigated in the
presence of loads with inertia (motors). The assessment of stability is normally
done using method of Lyapunov or Pole zero map analysis. However, the
attention of this study is to feed and stabilise the grid state variables (voltage
and frequency). The study will mainly point out if the control strategies will work
under different grid conditions and combinations.

4.4

Discussion

Our present and future power network situation requires extra flexibility in the
integration of distributed generation more than ever. Mainly for the small and
medium energy converting systems including intelligent control and advanced
power electronics conversion systems.
This chapter introduced standardized modular architectures and techniques for
distributed intelligence and smart power systems control that can be used to
build an electric power supply system by paralleling power electronic inverters.
It launched different and various robust control approaches for a realistic
distributed power system with power electronics inverters as front-end. These
control strategies guarantee real modularity, high reliability and true
redundancy. The proposed control architectures maintain the three phase
voltages and frequencies in the grid within certain pre-defined limits and provide
power sharing between the units according to their ratings.
The designed systems include inverter units of different power rating, distributed
at various locations feeding distributed unequal loads (balanced, unbalanced)
taking into account dissimilar line impedances between them to ensure true
expandability and generation placement flexibility. The types, sizes, and
numbers of the inverters, and the size and nature of the electrical loads may all
vary without the need to alter the control strategy.
This chapter develops a theoretical system concept including original control
concepts, which can assist the current efforts in designing, building and
operating a smart power system that is more flexible, efficient, reliable and
environmentally friendly. It introduces various opportunities of control functions
116

for three-phase inverters used to feed various passive/active grids including
different topologies to feed balanced/unbalanced loads. These are based on
standardized system concepts using various control strategies and no one-sizefits-all solution.
In the next chapter, the developed system concept is verified through simulation
models in MATLAB/SIMULIK to show the feasibility of the new system
philosophy and the effectiveness of the control and management functions.

117

CHAPTER 5
THE PROPOSED SMART GRID PHILOSOPHY
“VERIFICATION BY SIMULATION”

In the previous chapter, the operation, control and management of the
supplying philosophy was extensively discussed. This chapter is devoted to the
verification of the new philosophy and the control and management functions
through simulation models in Matlab/Simulink.
The proposed philosophy has two main categories as mentioned previously.
The first category is the multi-inverter three-wire system and the second is the
multi-inverter four-wire system. For each of these categories, different control
scenarios have been proposed and explored in chapter four and will be
simulated here.
This chapter will start by the simulation results of the supervisory control and
management scenario. Then, droop control scenario will be simulated.
Afterwards, isochronous control scenario will be tested. Next, a combination of
droops and isochronous control will be verified. Finally, the fundamental case of
using an inverter in swing mode will be studied. This is done for three-wire and
four wire systems respectively. An overview of the simulated scenarios is shown
in Fig. 5.1.
The proposed philosophy will be simulated using full dynamic models of a
realistic model of distributed power system with power electronics inverters. The
simulation shows the operation of inverters in isolated grids. The proposed
control architectures maintain the three phase voltages and frequencies in the
118

grid within certain predefined limits and have to provide power sharing between
the units according to their ratings.

Fig. 5. 1: Overview of the simulated scenarios.

The simulation will show that the designed system can include inverter units of
different power rating, distributed at various locations feeding distributed
unequal loads (balanced, unbalanced) taking into account dissimilar line
impedances between them to insure true expandability and generation
placement flexibility. This means that the types, sizes, and numbers of the
inverters, and the size and nature of the electrical loads may all vary without the
need to alter the control strategy.

5.1 Multi-inverter Three-wire System Simulation Models and
Results
This section will start by showing the simulation results of the supervisory
control and management scenario. Then, droop control scenario will be
simulated. Afterwards, isochronous control scenario will be tested. Next, a
combination of droops and isochronous control scenario will be verified. Finally,

119

the fundamental case of using an inverter in swing mode will be studied. An
overview of the simulated scenarios is shown in Fig. 5.1.

5.1.1 Supervisory Control and Energy Management Scenario
The following simulation case study is carried to validate the proposed inverter
supervisory control approach. The supervisory control is responsible for units
dispatching, load management, and power optimization. However, the current
and voltage control are done locally at the inverters according to the definition
introduced in chapter three. Moreover, the proposed control can be
implemented not only in distribution systems of isolated grid systems, but also
in the interconnected power system (some times called on-grid micro-grid).
In this case study there are three inverters working in grid forming mode, grid
supporting mode and grid parallel mode. They are connected in parallel to
supply two loads including steps as shown in Fig. 5.2. The main technical data
for the network is shown in Table 5.1.
It should be also noticed that when using distribution cables, the impact of the
ohmic coupling especially in low voltage networks for short and medium length
lines is rather high while the shunt capacitance of the line can be ignored. This
makes it acceptable to use the series RL model instead of the π model [29,
171-173]. This applies to all the following simulation cases as well.

120

Q3 = 15.25 kvar

Q2 = 15.25 kvar

P2 = 31.5 kW

Q1 = 30,5 kvar

P1 = 63 kW

P3 = 31.5 kW

Fig. 5. 2: Topology: supervisory control and energy management modular grid.

Table 5. 1 Technical data for the simulated system

Inverter 1

Inverter 2

Inverter 3

Mode

cos φ

0,9

L (mH)

C (µF)

DC Link (V)

700

P (kW)

31.5

Q (kvar)

30.5

15.25

S (kVA)

4.66

121

In the carried case study, the first load step is at t = 1 s and the second load
step is at t = 1.5 s. At t=2 s, the active power of the grid parallel unit is stepped
up from 14 kW to 21 kW. The frequency response of the system is shown in
Fig. 5.3. At t = 1 s, when the load is increased the frequency will drop. On the
other hand at t = 1.5 s, the load is decreased, and then the frequency will rise.
At t = 2 s, the grid parallel gives more power to the system. As response and to
keep the frequency constant, the grid forming and supporting inverters will
supply less power to the system.

Fig. 5. 3: The system frequency.

Fig. 5.4 shows the active power response of the inverters and loads from t= 0.5
s to t = 2.5 s. At the first step (t = 1 s), active power of load one is increased as
shown in Fig. 5.4. Consequently, the active power of grid forming inverter and
grid supporting inverter are increased to balance with the increased load. The
grid supporting inverter takes only 30 percent from the load as pre-set. The
active power of grid parallel inverter supplies to the system is the same. At
second step (t = 1.5 s), the active power of load two is decreased. The active
power of the grid forming and grid supporting inverters are decreased, while the
active power of the grid parallel inverter is still the same. At last step (t = 2 s),
the grid parallel inverter is set to give more active power to the system.
Therefore, the active power of the grid parallel inverter will increase and as
response both active power of grid forming and grid supporting inverters will be
signalled to decrease since the load is kept constant. The exact values are
shown in Table 5.2 and confirm the system power balance.

122

Active Power at Inverters
[ kW ]

45
40
35

Grid Forming

Grid Parallel

20
15
10
5

0.5

Active Power at Loads
[ kW ]

Grid
Supporting

50
45

1.5

2.5

1.5

2.5

Load 1

40
35

Load 2

30
25
20
15
10

0.5

Time [sec]

Fig. 5. 4: The active power.
Table 5. 2 Active power (kW)

Time (sec)

P load 1

P load 2

Σ P load

0 – 1.0

21.7

9.3

1.0 – 1.5

36.8

15.6

1.5 – 2

12.5

2.0 – 2.5

14 k

23.6

9.9

The reactive power behaviour of the inverters and loads are almost the same as
the active power. The difference is that the grid parallel inverter is set only to
give more active power to the system and is not contributing to the reactive
power balance. Therefore, it is not affecting the reactive power of the grid
parallel inverter at last step as shown in Fig. 5.5. The exact values are shown in
Table 5.3.

123

Fig. 5. 5: The reactive power.
Table 5. 3 Reactive power (kvar)

Time (sec)

Q load 1

Q load 2

Σ Q load

0 – 1.0

8.5

17.5

8.6

3.6

5.5

1.0 – 1.5

13.2

22.2

11.9

5.1

5.5

1.5 – 2.0

13.2

5.2

18.4

9.3

3.87

5.5

2.0 – 2.5

13.2

5.2

18.4

9.3

3.87

5.5

Having a look at Fig. 5.6 the response of the grid forming inverter to the load
increase can be seen. The voltage is held constant and the current will
increase. The high dynamic performance of the controller can be seen at t = 1 s
where the voltage is restored rapidly. Another example is the response of the
grid supporting inverter shown in Fig. 5.7, when the load decrease which is the
case at t = 1.5 s. It can be seen that the voltage will stay constant and the
supplied current will decrease.

124

Current at Load 1 [A]

Voltage at Load 1 [V]

Fig. 5. 6: The voltage and current of grid forming at first step.

Fig. 5. 7: The voltage and current of grid supporting at second step.

Since the grid parallel inverter is not dependent on the load and is not actively
dispatch able by the grid. It can be seen in Fig. 5.8 that it does not respond to
the load steps in the grid and instead of that keeps supplying the same amount
of current all the time. Having a look at the load, see Fig. 5.9, it can be seen that
the voltage is kept constant all the time by the system and is restored rapidly in
case of any load step. This shows the controller capabilities to supply high
power quality.
125

Voltage at Load 1 [V]
Current at Load 1 [A]

Fig. 5. 8: The voltage and current of grid parallel at first step.

Fig. 5. 9: The voltage and current at load one during first step.

5.1.2 Droop Control Functions Scenario
In the following case study, it is attempted to verify the communication less
modular power supply philosophy by adding third control loop (droops)
proposed in the last chapter. The different ways of controlling the grid-side

126

inverters introduced in sections 4.1.2 of the previous chapter (drooped gridforming and grid-supporting cases) will be tested in the following. The topology
is shown in Fig. 5.10. The technical specifications of the different inverters can
be seen in Table 5.4. The network consists of five inverters with altered power
and modes of operation placed at different places and working in different
modes. Inverters one and two are operating in grid forming (drooped) mode.
Inverters three and four are in grid supporting (drooped) mode and the fifth

Q3 = 7.2 kvar

P3= 15 kW

Q2 = 9.7 kvar

P2 = 20 kW

Q4 = 20.34 kvar

P4 = 42 kW

Q5 =0 kvar

P5 = 14 kW

Q1 = 29 kvar

P1 = 60 kW

inverter is working in grip parallel mode.

Fig. 5. 10: Topology: Droop modular grid.

The droop factors of the system under study are shown in Fig. 5.11. It is worth
to note that the grid parallel (green line) is always giving the same power at all
frequencies inside the operation range since it does not include a droop factor.

127

V4
IN

INV2

INV3

Fig. 5. 11: The droop factors for the system under study.

Table 5. 4 Technical data for the simulated system

Inverter 1

Inverter 2

Inverter 3

Inverter 4

Inverter 5

Mode

Forming

Supporting

Parallel

cos φ

0,9

L (mH)

C (µF)

100

P (kW)

Q (kvar)

9.7

7.2

20.34

S (kVA)

66.64

22.23

16.64

46.66

∆ω
= 0.1×10−3
∆P

∆ω
= 0 .3 × 1 0 − 3
∆P

∆P
= 11194.2
∆ω

∆P
= 3343.9
∆ω

(2%)

(2%)
∆Q
= 508.5
∆U

Frequency
Droop
Voltage
Droop

(2%)

∆U
= 0.7 ×10−3
∆Q

∆U
= 2.1×10−3
∆Q

∆Q
= 180
∆U

(5%)

The simulation includes a period of 12 seconds. Fig. 5.12 shows the time scale.
Inverters one and two are in grid forming mode and provide the network with
constant voltage and frequency. Inverters three and four are in grid supporting
mode and are based on the network state variables and provide power balance
into the grid. Inverter five is operating in grid parallel mode and is not
controllable by the grid, it includes no droop and supplies active power to the
grid.

128

Fig. 5. 12: The time sequence for the system under study.

In the first three seconds, inverters one to four are active while inverter five and
load three are not included. After three seconds, the transient phase is finished
and the system is reaching steady state. Once three seconds, load three is
switched on which increases the load in the grid. This leads to frequency drop in
the system and to an increase in the power injected to the system, in the time
scale a red arrow is showing that. This load is disconnected from the grid after
6.5 s. At t = 5 s, inverter five is connected in grid parallel mode. The other
inverters sense that (through the state variables) and adapt to the requirements.
The blue arrow shows when that is done. At that time, when inverter five is
switched on, load three is still in the network. This results in an overlap of the
two periods. While load three disconnects from the grid at t = 6.5 s, inverter five
continues active until the ninth second. After that the grid is back to its initial
condition. This period is pointed out on the time scale, as the green phase. The
relation between the load and the frequency in the grid is pointed out in Fig.
5.13.

Fig. 5. 13: The system frequency.

129

With the apparent power of 66 kVA, inverter one which is working in grid
forming mode is the most powerful in the grid and can supply about 30% of load
demand. Having a look at Fig. 5.14 it can be seen that as the reactive power
demand increases at t = 3 s the voltage will drop to compensate for that.
Furthermore, extra current will be supplied to respond to the active power
demand (frequency drop). The power is shown in Fig. 5.15.

Fig. 5. 14: The voltage and current response of inverter one to load step at t = 3.0 s.

Fig. 5. 15: The power response of inverter one.

130

Inverter three is in grid supporting mode; it is rated power is 22 kVA and can
cover 10 percent of the load demand. Once the frequency drops at t = 3 s this
unit will react by injecting more current (power to the system) as can be seen in
Fig. 5.16, the upper graph shows the current amplitude.

Fig. 5. 16: The current response of inverter two to load step at t = 3.0 s.

Inverter five is in grid parallel mode and is supplying a certain amount of current
regardless of the load steps in the grid. The current response can be seen in
Fig. 5.17 and Fig. 5.18 (the upper graphs show the amplitude) where the
system supplies a constant power for a certain time and based on it is source
and not on the grid status. The spike appearing after closing the switch is
because no synchronisation and protection procedure were taking into account
while modelling this case. This can be also solved by adding limiters to the
inverter controller.

131

Fig. 5. 17: The current response of inverter five (GP).

Fig. 5. 18: The current response of inverter five (GP) to load step at t = 6.5 s.

The voltage response at the load can be seen for example through the voltage
response at load one after the load step at t = 3 s where the voltage drops
within the limit due to the droop scheme, see Fig. 5.19.

132

Fig. 5. 19: The voltage response at load one after the load step at t = 6.5 s.

5.1.3 Isochronous Control Functions Scenario
If the load is frequency/voltage critical then isochronous mode (zero droop) is
one of the optimal solutions. An inverter operating in the isochronous mode will
operate at the same set frequency/voltage regardless of the load it is supplying.
The isochronous control scheme provides in comparison to the droop scheme
the possibility of precise control of the voltage and the frequency.
In this case study, the control behaviour of modular isolated grid controlled
inverter in isochronous mode is tested. The topology is shown in Fig. 5.20 and
the technical specification of the different inverters can be seen in Table 5.5.
The network consists of three inverters from different power classes working in
isochronous mode (modified grid forming) as described before in section 4.1.3.
The reason for the use of grid forming for the isochronous mode is the basic
control structure of inverters. In synchronous machines and grid forming
inverters the controller starts from the power through the frequency (speed)
controller to the current (moment) controller. In the grid supporting case this
looks different since the control sequence is the opposite.

133

=
Vdc

=
3~

=
Vdc

Vdc

Q3 = 20 kvar

P3 = 39 kW

Q2 = 14 kvar

P2 = 24 kW

Q1 = 8.5 kvar

P1 = 12 kW
GF

3~
L

L1= 200 m

L2= 200 m

L3= 200 m

LLoad= 1 m

tOn = 3 sec
tOff= 5 sec

S= 17.24 kVA
P = 15 kW
Q= 8.5 kvar

S= 69 kVA
P=60 kW
Q= 34 kvar

Fig. 5. 20: Topology: isochronous modular grid.
Table 5. 5 Technical data for the simulated system

Inverter 1

Inverter 2

Inverter 3

Mode

0,9

0,87

0,85

S (kVA)

P (kW)

13.5

26.1

42.5

Q (kvar)

6.5

14,8

26.3

PInv , n S Inv , n

0.9

0,87

0.85

QInv ,n S Inv ,n

0.43

0.493

0.526

KP,ω

KI,ω

0.25

2.5

134

KP,V

KI,V

0.25

2,5

L (mH)

C (µF)

100

At the beginning of the simulation, the grid demands an apparent power of 69
kVA. At t = 3 s, a load step of 17.24 kVA is included. Later, at t = 5 s, the extra
load is switched off and the grid is back to its original status.
Having a look at the system frequency of the simulated system shown in Fig.
5.21. It can be seen that the frequency is rapidly restored back to 50 Hz (the
nominal frequency) after any load step. This is the advantage over the droop
concept where a frequency gap stays due to the droop response. The speed of
the frequency restoration is related to the control loops parameters and is
adjustable. The system load is shown in the figure after (Fig. 5.22) for
comparison. The swinging response in the beginning of the figure is normal
behaviour of the transient starting phase when switching the inverters since the
voltage is still not stable at the load.

Fig. 5. 21: The system frequency.

135

P [kW]

200
150
72

100

58.7

38.3

S [kVA]

Q [kvar]

100
50
0
200
150
100
50
0

Time [sec]
Fig. 5. 22: The system total load.

At the inverter side and having a look at the inverters outputs shown in the
figures below (Figs. 5.23 to 5.25), it can be recognised that they are supplying
fixed voltage output and respond by changing their current to the different load
steps. At t =3 s once the load will decrease the current supplied by the inverters

Current [A]

will decrease but the voltage will be kept constant by the controllers all the time.

Fig. 5. 23: The current response of inverter one to load step at t = 3.0 s.

136

Voltage [V]
Voltage [V]

Voltage [V]

Current [A]

Fig. 5. 24: The voltage response of inverter two to load step at t = 3.0.

Fig. 5. 25: The voltage and current response of inverter three to load step at t = 3.0.

5.1.4 Isochronous-droop Control Functions Scenario
In this scheme, inverter’s active power/frequency is regulated using isochronous
control while the reactive power/voltage is regulated using the droop scheme.
Through that it is possible to minimize the frequency difference and fix it to the
nominal frequency while minimizing the communication as well.
The grid includes three inverters working in isochronous-droop mode with
different power rates, see Fig. 5.26 and Table 5.6. The frequency/active power

137

control is done using isochronous mode. In contrast to previous simulation the
voltage/reactive power interaction is controlled using droop functions.

Fig. 5. 26: Topology: isochronous-droop control modular grid.
Table 5. 6 Technical data for the simulated system

Inverter 1

Inverter 2

Inverter 3

Mode

0,9

0,87

0,85

S (kVA)

P (kW)

13.5

26.1

42.5

Q (kvar)

6.54

14,8

26.34

PInv ,n S Inv , n

0.9

0,87

0.85

Gain ω

Voltage Droop

∆U
= 3.1 × 10 − 3
∆Q

∆U
= 1.4 ×10−3
∆Q

∆U
= 0.8 × 10 − 3
∆Q

(5%)
2

(5%)

L (mH)

(5%)
2

C (µF)

100

138

The network has a base load of 75 kVA. At t = 3 s, a load step is considered
and the new load demand is 63 kVA. At t = 5 s, this load is switched off and the
grid goes back to the base load.
Having a look at the system frequency of the simulated system shown in Fig.
5.27. It can be seen that the frequency is rapidly restored by the controller back
to 50 Hz (the nominal frequency) after any load step. The load is fixed in steady
state at 50 Hz regardless of the load. The system load is shown in the figure
below for the comparison.

Frequeny [Hz]

52
51
50
49
0

Time [sec]
Fig. 5. 27: The system frequency.

Fig. 5. 28: The system total load.

This consists as well with the amount of current supplied to the load as shown
in Fig. 5.29 (The upper graph shows the amplitude). Once the load drops, less
current is supplied to the load.

139

Fig. 5. 29: The current response at load one.

At the inverter side and looking at the inverter’s output shown in the figure
below (Fig. 5.30), it can be recognised that the voltage will change when a
reactive load step happen. This voltage change is following the droop function

Voltage [V]

Current [A]

used to control the voltage and share reactive power.

Fig. 5. 30: The voltage and current response of inverter three to load step at t = 3.0 s.

Finally, as can be seen in Fig. 5.31 (The upper graph shows the amplitude) as
the load decreases at t = 3 s the inverter will reduce its output current
accordingly.

140

Fig. 5. 31: The current response of inverter one to load step at t = 3.0 s.

5.1.5 Swing-inverter and Droop Control Functions Scenario
As discussed in chapter four a grid forming inverter (actually this is the
fundamental role) can be used as swing inverter. The swing inverter compares
the grid state variables frequency and voltage in the grid and drives them back
to their reference values in the case of deviations. Normally, the swing inverter
is the one with the highest power rating in the system thus the system will
accept the largest load changes within its capacity. The remaining feeding
inverters/machines are switched in droop mode in parallel to the swing inverter.
The investigated simulation model includes a swing inverter (grid forming), two
grid supporting units and a grid parallel unit. The other technical specifications
can be seen in Table 5.7.

141

Vdc

=
Vdc

Vdc

3~
L

L2= 500 m

L1= 50 m

Q4 = 14.79 kvar

GS
=

P4 = 26.1 kW

GS
=

Vdc

Q3 = 22,8 kvar

P3 = 44.5 kW

Q2 =0 kvar

P2 = 14 kW

Q1 = 43.5 kvar

P1 = 90 kW

Swing

L3= 900 m

L4= 30 m

tOn = 7 sec
tOff= 11 sec
L= 200 m

LLoad_3= 300 m
SLoad_1 = 70 kVA
PLoad_1 =52.3 kW
QLoad_1 = 32 kvar

tein = 4 sec
taus= 11 sec

SLoad_3 = 40 kVA
PLoad_3 = 35.6 kW
QLoad_3 = 18.32 kvar

LLoad_2= 500 m

SLoad_2 = 35 kVA
PLoad_2 =31.15 kW
QLoad_2 = 15.96 kvar

Fig. 5. 32: Topology: swing inverter based modular grid.
Table 5. 7 Technical data for the simulated system

Inverter 1

Inverter 2

Inverter 3

Inverter 4

Mode

Swing

0.9

0.89

0.87

P ( kW)

44.5

26.1

Q ( kvar)

43.5

22.8

14.79

S ( kVA)

100

Frequency
Droop

Isochronous (50Hz)
No Droop

Voltage Droop

Isochronous (320V)

∆P
= 3543
∆ω
∆Q
= 570
∆U

∆P
= 2100
∆ω
∆Q
= 370
∆U

142

The grid consists of two supplying nodes, four inverters and loads. While the
first three inverters are supplying at the first node, the fourth inverter is
delivering at the other node. At t = 4 s an extra load is switched on with 40 kVA
apparent power. At t = 7 s inverter two is switched on and finally at t = 9 s the
load will be switched off while all inverters stay in the grid. See Fig. 5.33.
Inverter 2

t [sec]
Starting

Load 3

t [sec]
Starting

Fig. 5. 33: Timing diagram for the modular grid.

The swing inverter (inverter one) is the largest inverter in the grid with an
apparent power of 100 kVA. It states the grid frequency and voltage. In case of
any differences it takes care of compensating it. Inverter two is in grid parallel
mode. It depends basically on the source behind and is not always available to
supply the grid. In this simulation model it will be switched on at t = 7 s and will
deliver 14 kVA. Five seconds later this inverter will be disconnected from the
grid. Inverters three and four are in grid supporting mode. They are dependable
on the grid state variables (voltage and frequency) and are regulated in droop
mode.
Having a look at the system frequency of the simulated case shown in Fig. 5.34,
it can be seen that the frequency is rapidly restored back to 50 Hz (the nominal
frequency) after any load step. The system load is shown in the figure below for
comparison. The system will have the same frequency regardless of the load
since the swing inverter controller will restore it.

143

S [kVA]

Q [kvar]

P [kW]

Fig. 5. 34: The system frequency.

Fig. 5. 35: The system total load.

Having a look at Fig. 5.36 it can be seen that most of the needed power after
the step is compensated by the swing inverter (grid forming). In Fig. 5.37 and
Fig. 5.38 the response of the grid parallel inverter is illustrated which is not
dependent on the system state variables rather on the ECS. The peaks seen in
the graphs are due to the voltage changes due to switching other inverter/loads.

144

Fig. 5. 36: The swing inverter total supplied power.

S [kVA]

Q [kvar]

P[kW]

Fig. 5. 37: Load three voltage and current at t = 4 s.

Fig. 5. 38: The total power supplied by inverter two (parallel mode).

145

5.2 Multi-inverter Four-wire System Simulation Models and
Results
This part will be dedicated to the four- wire inverters used to feed unbalanced
loads with symmetrical voltage and frequency nominal values. This section will
start by showing the simulation results of the supervisory control and
management scenario. Then, droop control scenario will be simulated.
Afterwards, isochronous control scenario will be tested. Next, combination of
droops and isochronous control scenario will be verified. Finally, the
fundamental case of using an inverter in swing mode will be studied. An
overview of the simulated scenarios is shown in Fig. 5.1.

5.2.1 Supervisory Control and Energy Management Scenario
The following simulation case study is carried to validate the proposed inverter
supervisory control approach to supply nonlinear and unbalanced loads. The
supervisory control is responsible for units dispatching, load management, and
power optimization. However, the current and voltage control are done locally at
the inverters according to the definition introduced in chapter three. In this case
study there are three inverters. Grid forming, grid supporting and grid parallel
connected in parallel to supply unbalanced load including load steps as shown
in Fig. 5.39. The main load is a series resistive-inductive load placed at phase
“a” Ra = 8 Ω and L = 5 mH. The resistance at phase “b” has been kept constant
at 7 Ω and the resistance of phase “c” is constant Rc = 9 Ω.

146

Fig. 5. 39: Topology: supervisory control and energy management modular grid (four-wire).

In the carried case study, the first load step is at t = 1 s and the second load
step is at t = 1.5 s. At t = 2 s, the grid parallel unit is switched in to supply active
power to the grid as shown below.
Inverter 3

t [sec]
Load

t [sec]
1

Fig. 5. 40: Timing diagram for the modular grid.

147

The frequency response of the system is shown in Fig. 5.41. At t = 1 s, when
the load is increased, then the frequency will drop. On the other hand at t = 1.5
s the load is decreased, and then the frequency will rise. At t = 2 s, the grid
parallel inverter gives more power to the system. As a response and to keep the
frequency constant, the grid forming and supporting inverters will supply less
power to the system.

Fig. 5. 41: The system frequency.

Fig. 5.42 shows the active power responses of the inverters and loads. At first
step (t = 1 s), the load is increased as shown in Fig. 5.42. The active power of
grid forming and grid supporting inverters are increased to balance with the
increased load. The grid supporting inverter takes only 25 percent from the load
as pre-set. At second step (t = 1.5 s), load is again increased. The active power
of the grid forming and grid supporting inverters will increase again to cover it,
while the active power of the grid parallel is still zero. At last step (t = 2 s), the
grid parallel inverter is set to give active power to the system. As the active
power of the grid parallel inverter increases both active power of grid forming
and grid supporting inverters are decreased since the load is constant. The
power balance can be seen in Table 5.9, the differences are due to the line
losses.

Fig. 5. 42: The active power.

148

Table 5. 8 Active power (kW)

Time (s)

P load

0 – 1.0

1.0 – 1.5

23.5

18.5

1.5 – 2.0

27.8

2.0 – 2.5

27.8

8.5

3.1

The reactive power behaviour of the inverters and loads are almost the same as
the active power. The difference is only at last step. The grid parallel inverter is
set only to give more active power to the system. Therefore, it is not affecting
the reactive power of the grid parallel inverter at last step as shown in Fig. 5.43

Reactive Power [kvar]

and Table 5.9.

Fig. 5. 43: The reactive power.
Table 5. 9 Reactive power (kvar)

Time (s)

Qload

0 – 1.0

2.5

2.2

0.5

1.0 – 1.5

2.8

2.4

0.75

1.5 – 2.0

3.3

2.5

2.0 – 2.5

3.3

2.5

Having a look at Fig. 5.44 the response of the grid forming inverter to the load
increase can be seen. The voltage is constant and symmetrical while the
current will increase and is asymmetrical to compensate for the unbalanced
load. The neutral current flying back is also illustrated (light blue). This is also
the case with the grid supporting inverter shown in Fig. 5.45. The change in the
current can be seen more clearly since a longer period is shown.
149

300
100
-100
-300
40
20
0
-20
-50
1

1.1

1.05

1.15

Time [sec]
Fig. 5. 44: The voltage and current of grid forming inverter at first step.

Fig. 5. 45: The voltage and current of grid supporting inverter at first step.

Having a look at the load during load step at t = 1.5 s it can be seen that the
voltage will stay constant and symmetrical which shows the controller
capabilities. The supply current will increase to fulfil the load demand; it is
asymmetrical since the load is unbalanced. The load neutral current will be
flying back to the grid forming unit as shown in Fig. 5.47. The grid supporting
and grid parallel are not taking part of this.

150

Voltage [V]

300
100
-100
-300

Current [A]

40
20
0
-20
-50
1.4

1.45

1.5

1.55

1.6

Time [sec]

Current [A]

Fig. 5. 46: The voltage and current of load during second step.

Fig. 5. 47: The neutral current.

5.2.2 Droop Control Functions Scenario
In the following case study, it is attempted to verify the communication less
modular power supply philosophy by adding third control loop (droops)
proposed in the last chapter. The different ways of controlling the grid-side
inverters introduced in sections 4.2.2 of the previous chapter (drooped gridforming and grid-supporting cases) will be tested in the following. The topology
is shown in Fig. 5.48. The technical specifications of the different inverters can
be seen in Table 5.10. The network consists of three inverters from different
power classes and modes of operation placed at different places and working in
151

different modes. Inverter one is operating in grid forming (drooped) mode.
Inverter two is in grid supporting mode (drooped) and the third inverter is
working in grid parallel mode.

Fig. 5. 48: Topology: drooped modular grid (four-wire).
Table 5. 10 Technical data for the simulated system

Inverter 1

Inverter 2

Inverter 3

Mode

cos φ

0.75

0.95

L (mH)
C (µF)
P ( kW)

3
10
37.5

3
10
33

2
10
5

Q (kvar)

19.25

5.5

S ( kVA)

Frequency Droop

∆ω
= 0.17 ×10−3
∆P

(2%)

∆P
= 3065.3
∆ω
(2%)

∆U
= 0.6 × 10−4
∆Q

∆Q
= 275
∆U

(5%)

Voltage Droop

152

The simulation includes a period of 5 seconds. Fig. 5.49 shows the time scale.

Fig. 5. 49: The time sequence for the system under study.

Inverter one provides the network with constant voltage and frequency. Inverter
two is based on the network state variables and provides power into the grid.
Inverter three is grid parallel and is not controllable by the grid and includes no
droop.
The main load is resistive at phase “a” and is constant at 10 Ω while a series
resistive-inductive load is placed at phase “b” Rb = 4 Ω and L = 5 mH. The
resistance of phase “c” is constant at Rc = 13 Ω.
In the first two seconds, inverters one and two are active while inverter three
and the load step are not included. After two seconds, the transient phase is
finished and the system is in steady state. Once two seconds, the load step is
switched on which increases the load in the grid. This leads to frequency drop in
the system and to an increase in the power injected to the system. This load is
disconnected from the grid at t = 3 s and it can be seen that the frequency is
restored to its initial value. At t = 4 s, inverter three is connected in grid parallel
mode. The other inverters sense that (through the state variables) and adapt to
the requirements by supplying less power. The relation between the load and
the frequency in the grid is pointed out in Fig. 5.50 and Fig. 5.51. If the load
active power will increase then the frequency will sink and it will not get back to
the nominal value due to the droop effect. If the active load decreases then the
opposite happens. The amount of power delivered by each inverter can be
adjusted using the droop factors in the individual inverters.

153

Fig. 5. 50: The system frequency response.

Fig. 5. 51: The active power.

The relation between the reactive load and the voltage in the grid is pointed out
in Fig. 5.52 and Fig. 5.53. Once the reactive power demand increases the
inverter output voltage will decrease to fulfil the needed reactive power. This
means that the voltage will decrease due to the droop curve. If the reactive load
decreases then the opposite happens. The amount of reactive power delivered
by each inverter can be adjusted using the droop factors in the individual
inverters.

154

Fig. 5. 52: The reactive power.

Fig. 5. 53: The grid forming voltage amplitude response.

With the apparent power of 50 kVA, inverter one which is working in grid
forming mode is the most powerful in the grid and can supply about 60% of load
demand. Having a look at Fig. 5.54 it can be seen that as the reactive power
demand increase at t = 2 s the voltage will drop to compensate for that.
Furthermore, more current will be supplied to response to the active power
demand (frequency drop).

155

Fig. 5. 54: The grid forming voltage and current response at the first load step.

Inverter two is in grid supporting mode; its rated power is 20 kVA and can cover
35% of the load demand. Once the frequency increases at t = 3 s this unit will
react by injecting less current (power to the system) as can be see in Fig. 5.55.
The voltage response is following the grid forming inverter.
Having a look at the neutral currents as shown in Fig. 5.56 it can be seen that
they are handled by the grid forming unit and that they increase as the load
increase.

Fig. 5. 55: The grid supporting voltage and current response at the first load step.

156

Fig. 5. 56: The neutral current response at the first load step.

5.2.3 Isochronous Control Functions Scenario
If the load is frequency/voltage critical then isochronous mode (zero droop) is
the optimal solution. An inverter operating in the isochronous mode will operate
at the same set frequency/voltage regardless of the load it is supplying. The
isochronous control scheme provides in comparison to the droop scheme the
possibility of precise control of the voltage and the frequency.
In this case study, the control behaviour of modular isolated grid controlled in
isochronous mode to supply unbalanced loads is tested. The topology is shown
in Fig. 5.57 and the technical specification of the different inverters can be seen
in Table 5.11. The network consists of three inverters working in isochronous
mode (modified grid forming) as described before in section 4.2.3. The three
inverters in this case study are identical to show the load sharing precisely. At
the beginning of the simulation, the grid demands an apparent power of 33.8
kVA. At t = 2 s, load step of 16.8 kVA is included. Later, at t = 3 s, the extra load
is switched off and the grid is restored to its original status.

157

Fig. 5. 57: Topology: isochronous modular grid (four-wire).
Table 5. 11 Technical data for the simulated system

Inverter 1

Inverter 2

Inverter 3

Mode

0.9

S (kVA)

P (kW)

22.5

Q (kvar)

10.89

PInv , n S Inv , n

0.9

QInv ,n S Inv ,n

0.44

0,44

KP,ω

KI,ω

KP,V

KI,V

0.0001

L (mH)

C (µF)

158

Having a look at the system frequency of the simulated system shown in Fig.
5.58. It can be seen that the frequency is rapidly restored back to 50 Hz (the
nominal frequency) after any load step. This is possible because of the
isochronous load sharing method where all the loads are feedback to the
controller to allow precise load sharing. The system load is shown in the figure
below for comparison. It is also shared equally between the units; this can also
be adjusted through the controller even though normally the load is shared

Frequency [ Hz ]

based on the inverters rated power.

Fig. 5. 58:The system frequency response.

Fig. 5. 59: The active power.

159

Having a look at the reactive power sharing shown in Fig. 5.60, it can be seen
that the three units are sharing it equally and they are doing this as well when

Reactive Power [kvar]

any load step occurs in the grid.

Fig. 5. 60: The reactive power.

When looking at the load, it can be seen that the voltage is kept constant and
symmetrical at any load step while the current is changing to compensate for
that which is intended. The controller strength can be seen at t = 2 s when the
additional load at one phase is switched on. The voltage has a small distortion
but will be restored rapidly and the current will increase.

Fig. 5. 61: The load voltage and current response at the first load step.

160

This can be seen as well from the response of the inverter to load steps as
shown below. It will maintain the voltage constant and vary the supplied current
to the grid. At t = 3 s for example when the load will decrease, the inverter will
supply constant symmetrical voltage while reducing the current at the phase
with less power.

Fig. 5. 62: Second grid forming inverter voltage and current response at the second load step.

Finally, having a look at the neutral current it can be seen that it is shared by the
three units equally.

Fig. 5. 63: The neutral current response at the first load step.

161

5.2.4 Isochronous-droop Control Functions Scenario
In this scheme inverter’s active power/frequency is regulated using isochronous
control while the reactive power/voltage is regulated using the droop scheme.
Through that it is possible to minimize the frequency difference and fix it to the
nominal frequency while minimizing the communication as well.
In this simulation a grid including three inverters working in isochronous-droop
mode with different power rates will be considered. The frequency/active power
control is done using isochronous mode. In contrast to previous simulation the
voltage/reactive power interaction is controlled using droop function.
The network has a base load of 41.8 kVA. At t = 2 s, load step of 23.6 kVA is
considered. The new load demand is 65.4 kVA. At t = 3 s, this load is switched
off and the grid goes back to the base load.
The main load is resistive at phase “a” at 3 Ω while a series resistive-inductive
load is placed at phase “b” Rb = 2 Ω and L = 5 mH. The resistance of phase “c”
is constant at Rc = 4 Ω. The load step is also unbalanced and has a series
resistive-inductive load placed at phase “a” with Ra = 5 Ω and L = 5 mH. At
phase “b” Rb = 4 Ω and L = 5 mH. The resistance of phase “c” is constant at Rc
= 10 Ω.

162

Fig. 5. 64: Topology: Isochronous-droop control modular grid (four wire).
Table 5. 12 Technical data for the simulated system

Inverter 1

Inverter 2

Inverter 3

Mode

0.98

S (kVA)

P (kW)

34.34

24.5

19.6

Q (kvar)

6.78

4.98

3.96

Gain ω

Voltage Droop

∆V
= 1.19 × 10 − 3
∆Q

∆V
= 1.98 ×10−3
∆Q

∆V
= 2.96 × 10 − 3
∆Q

L (mH)

C (µF)

163

Having a look at the system frequency of the simulated system shown in Fig. 5.
65. It can be seen that the frequency is rapidly restored back to 50 Hz (the
nominal frequency) after any load step. The system load is shown in Fig. 5.66
below for comparison. There the load sharing can be seen where the unit with
highest power rating is taking more load than the units with less rating. This is
done using the isochronous control. It can be seen that the steady state
frequency is always 50 Hz at any load level. It can be also seen that the units

Frequency [ Hz ]

are sharing the active power load according to their rating.

Active Power [kW]

Fig. 5. 65: The system frequency response.

Fig. 5. 66: The active power.

164

On the other hand, having a look at the reactive power and voltage response it
can be seen that as the reactive power increase the voltage amplitude will
decrease due to the droop control. This can be seen in Fig. 5.67 and Fig. 5.68.
The load sharing is done using the preset V/Q droop factor. The droop factors
where calculated based on the units rating so that the units with higher power
can take more reactive power.

Fig. 5. 67: The reactive power.

Having a look at the inverters outputs (e.g. inverter one) shown in the figure
below it can be recognised that the voltage amplitude will change when a
reactive load step happens and the voltage is not fixed. This is the effect of the
droop load sharing loop. In this practical case the reactive power demand

Voltage [V]

increase leading to voltage drop.

Fig. 5. 68: The first grid forming voltage amplitude response.

165

Finally, at the load side (asymmetrical load) it can be seen that the voltage will
change within the limits and will stay symmetrical when a load step happen
while the needed current will be supplied by the inverters according to their
rates. See Fig. 5.69 below. The neutral current, see Fig. 5.70 is also shared
between the units according to their rating.

Fig. 5. 69:The load voltage and current response at the first load step.

150

Load
Grid Forming 2

Current [A]

100

Grid Forming 1

50
0
-50

Grid Forming 3

-100
-150
1.85

1.9

1.95

2.05

2.1

2.15

2.2

Time [sec]
Fig. 5. 70:The current response at the first load step, phase ”a”.

166

5.2.5 Swing-inverter and Droop Control Functions Scenario
As was discussed in chapter four a grid forming unit can be used as swing
inverter. The swing inverter compares the grid state variables frequency and
voltage in the grid and drives them back to their reference values in the case of
deviations. Normally, the swing inverter is the one with the highest power rating
in the system thus that system will accept the largest load changes within its
capacity. The remaining feeding inverters/machines are switched in droop mode
parallel to the swing inverter. The investigated simulation model includes a
swing inverter (grid forming), one grid supporting unit and a grid parallel unit.

Fig. 5. 71: Topology: swing inverter based modular grid (four wire).

167

The grid consists of three inverters connected to loads. At t = 2 s an extra load
is switched on with 40 kVA apparent power and at t = 3 s the load will be
switched off. At t = 4 s inverter three is switched on in grid parallel mode while
all inverters stay in the grid. The other technical specifications can be seen in
Table 5.13.
Table 5. 13 Technical data for the simulated system

Inverter 1

Inverter 3

Inverter 2

Mode

Swing

0.9

0.89

P ( kW)

44.5

Q ( kvar)

43.5

22.8

S ( kVA)

100

Frequency Droop

Isochronous (50Hz)

Voltage Droop

Isochronous (325V)

∆P
= 3543
∆ω
∆Q
= 570
∆U

No Droop
No Droop

Having a look at the system frequency of the simulated system shown in Fig.
5.72. It can be seen that the frequency is rapidly restored back to 50 Hz (the
nominal frequency) after any load step. The system load is shown in Fig. 5.73
below for comparison.

Fig. 5. 72: The system frequency response.

168

Having a look at Fig. 5.73 and 5.74 it can be seen that most of the needed
power after the step is compensated by the swing inverter (grid forming). The
grid supporting inverter is also giving its share. The grid parallel inverter is not
dependent on the system variables and is not producing any reactive power.

Fig. 5. 73: The active power.

Fig. 5. 74: The reactive power.

At the load side, see Fig. 5. 75, it can be seen that the voltage is kept constant
and symmetrical when a load step happen while the needed current will be
supplied by the inverters according to their rates.

169

Voltage [V]

300
100
-100

Current [A]

-300
150
100
50
0
-50
-100
-150
1.9

1.95

2.05

2.1

2.15

Time [sec]
Fig. 5. 75: The voltage and current response at load on the first load step.

Finally, the neutral current is compensated using the swing inverter. The grid
supporting and parallel inverters are not contributing into that as shown in Fig.
5. 76.

Fig. 5. 76: The neutral current response at the first load step.

170

5.3

Discussion

The proposed philosophy has two main categories as mentioned previously,
see Fig. 5.1. The first category is the multi-inverter three-wire system and the
second is the multi-inverter four-wire system. For each of these categories,
different control scenarios have been verified by simulation.
These were simulated using full dynamic models of realistic distributed power
system with power electronics inverters. The simulation case studies showed
that the designed systems can include inverter units of different power rating,
distributed at various locations feeding distributed unequal loads (balanced,
unbalanced), taking into account dissimilar line impedances between them to
ensure true expandability and generation placement flexibility. This means that
the types, sizes, and numbers of the inverters, and the size and nature of the
electrical loads may all vary without the need to alter the control strategy.
The simulation shows that the proposed control architecture maintains the three
phase voltages and frequency in the grid within certain predefined limits and
provides power sharing between the units according to their ratings. It has been
shown that the developed models fulfil the requirements of future smart grids.
The supervisory control and energy management scenario maintain the three
phase voltages and frequency in the grid precisely and will provide power
sharing between the units according to their ratings, meteorological parameters,
economical dispatch prospective (can include real-time pricing) and user
settings. This allows total energy optimization. The amount of data exchange
can be small if it includes only basic measurements and set points but will
increase proportionally as more functions are added.
One possibility that makes the system less stiff and allows load sharing is using
the droop functions depending on the system state variables, the voltage and
frequency. The three phase voltages and frequency in the grid will be
maintained within a pre-set allowed band. Using this methodology the system
architecture

providing

modularity,

redundancy,

expandability,

maintainability, reliability and avoids communication requirements and costs.
If the load is frequency/voltage critical then isochronous mode (zero droop) is
the optimal solution. An inverter operating in the isochronous mode will operate
at the same set frequency/voltage during steady state regardless of the load it is
supplying. The isochronous control scheme provides in comparison to the droop

171

scheme the possibility of precise control of the voltage and the frequency. This
needs communication in order to measure the grid load and share this
information with all the other inverters in the system. However, the realisation of
such a system needs low-bandwidth communication and is considered practical
especially if the inverters are connected to the same load bus and have no
massive distance between them.
By

combining

isochronous/droop

control

functions,

inverter’s

active

power/frequency are regulated using isochronous control while the reactive
power/voltage is regulated using the droop scheme. Through that it is possible
to minimize the frequency difference and fix it to the nominal frequency while
minimizing the communication as well.
Finally, Swing-Inverter/Droop scenario; the swing inverter compares the grid
state variables frequency and voltage in the grid and drives them back to their
reference values in the case of deviations. Normally, the swing inverter is the
one with the highest power rating in the system so that the system will accept
the largest load changes within its capacity.
The proposed philosophy using these different control techniques will improve
the design and implementation of future distributed modular grid architectures.

172

CHAPTER 6
CONCLUSIONS AND FURTHER WORK

6.1

Conclusions

The main contributions are summarized below:

This work introduced a variety of standardized modular architectures and
techniques for distributed intelligence and smart power systems control
that can be used to build an electric power supply system by paralleling
power electronic inverters. It launched different and various robust
control approaches based on the feeding mode definition for a realistic
distributed power system with power electronics inverters as front-end.
These control strategies guarantee real modularity, high reliability and
true redundancy. The proposed control architectures maintain the three

173

phase voltages and frequencies in the grid within certain limits and
provide power sharing between the units according to their ratings.
2-

The research led to an original philosophy for supervisory control and
energy management of an Inverter-based modular smart grid for
distributed generation applications. The method developed is based on
the feeding modes definition and supports the active integration of the
inverters (energy converting systems & renewable energy sources). The
main control tasks (voltage/frequency control) are done locally at the
inverters to guarantee modularity and to minimize communication
bandwidth requirements. The supervisory control is used for dispatching
and optimization control. It can also include real time pricing and
meteorological forecasting. The concept was developed and tested for
three-phase, three-wire and four-wire systems.

A droop communication less novel concept of load sharing for inverterbased modular smart grids for distributed generation applications is
introduced. The method can achieve good voltage/frequency regulation
and good load sharing. The method developed is based on the extended
feeding modes definition. All control tasks (voltage/frequency control) are
done locally at the inverters to guarantee absolute modularity without
control and communication interconnection. Furthermore, the total load
can be distributed over the different inverters according to their capacity
by adjusting their droop coefficients while placed at various locations to
ensure flexibility. The concept was developed and tested for three-phase,
three-wire and four-wire systems.

The isochronous mode (zero droop) is the optimal solution if the load is
frequency/voltage critical. An inverter operating in the isochronous mode
will operate at the same set frequency/voltage during steady state
regardless of the load it is supplying. The isochronous control scheme
provides in comparison to the droop scheme the possibility of precise
control of the voltage and the frequency. This needs communication in
order to measure the grid load and share this information with all the
other inverters in the system. However, the realisation of such a system
needs low-bandwidth communication and is considered practical
especially if the inverters are connected to the same load bus and have
no massive distance between them.
174

This work has proposed a novel standardized advanced control concept
for four-wire inverters (three-leg four-wire and four-leg) using symmetrical
components

based

sequence

decomposition

supply

balanced/unbalanced loads. The principle idea is to control the positive,
negative and zero sequence components. Controlling (eliminating) the
negative and zero sequence components helps expanding the inverter
based systems by increasing the distribution network efficiency
(consequently leads to less losses and results in enhancing the power
quality). It also grants the opportunity to supply unbalanced loads which
means supplying single and three phase loads using the same source. It
can be used as well for shunt active filters applications.
6-

This work has introduced a novel three dimensional space vector
modulation (3D-SVM) control strategy of three-leg four-wire inverters
able to feed grids with unbalanced loads while reducing the switching
frequency losses. The proposed solution covers a current gap in the
present literature since the SVM of three-leg four-wire inverter was
discussed briefly in the current literature according author's knowledge.

In this study, the general control functions and the system behaviour have been
investigated. With this investigation it has been shown that the realisation of
smart power systems in general through the new system philosophy is possible
and advantageous.

6.2

Further Work

This work has particularly established new control solutions for the development
of modular grid architecture for decentralized generators in electrical power
supply system with flexible power electronics. This research has established a
foundation and covered the general concepts. However, further investigation is
absolutely necessary, mainly in the following areas:

Experimental tests are needed in order to validate the proposed
concepts under different practical conditions. Fortunately, this is taking
place now by other colleagues in the research group.

To develop very accurate and fast phase locked loop to estimate the
frequency even under highly distorted grid conditions.
175

To assess the stability of inverter-based modular grids especially in
transients, in presence of rotating generators, loads with inertia (motors)
and starting of inverters or machines. Furthermore, the effect of grid
impedance on the controller.

Investigating protection, Fault ride-through and recovery.

To study in detail the effect of the harmonics and non-linearity since
linearised models were used in this research.

Advanced algorithms may be utilised to enhance the management and
supervisory control like accurate prediction of the power production and
consumption, real time pricing and meteorological forecasting.

A quantitative study that identifies the possible cost reduction in the
different application cases due to modularity, flexibility and optimization
would be of great importance.

176

APPENDIX

A.1

SVM for Three-leg, Four-wire Voltage Source Inverters

The following sections present a comprehensive analysis for two novel threedimensional space vector modulation (3-D-SVM) algorithms for three-leg fourwire inverters.
In the first approach introduced in [147] and called the zero vector approach, A
zero vector is generated by turning off all power switches to produce zero volts
at the output terminals of the inverter. Here, the switching vectors, separation
planes, the matrices for switching vectors duty cycles and the switching
sequences are derived. Still, the proposed zero vector approach algorithm has
a drawback of stressing the IGBTs unequally. Therefore, another SVM
algorithm without using zero-vector was launched in [148]. This algorithm based
on vectors compensation (Compensated Vectors Approach) is more practical as
it is not only stressing the IGBTs equally but less as well.

a) Zero-Vector Approach
The proposed SVM algorithm can be achieved through the following steps:

1) Determining the switching combinations and the corresponding vectors.
2) Calculating the voltage drop related to each vector.
3) Identifying the position for each vector in the αβγ-space vector diagram.
4) Identifying the reference vector position.

177

5) Calculating the duty cycles.
6) Building a vector sequence.
7) Computing pulse patterns.

Step-One, Two and Three
Table A.1 presents the nine possible switching vectors and the corresponding
output voltages related to the DC-voltage as reference voltage.
Table A. 1 Switching states and the corresponding output voltages [147].

Output Voltage
Va/VDC Vb/VDC Vc/VDC

Turned On Switches

Vector

S4 S6 S2

-1/2

S1 S6 S2

1/2

-1/2

S1 S3 S2

1/2

-1/2

S4 S3 S2

-1/2

1/2

-1/2

S4 S3 S5

-1/2

1/2

S4 S6 S5

-1/2

1/2

S1 S6 S5

1/2

-1/2

1/2

S1 S3 S5

1/2

S1S 2S3S 4S 5S 6

The switching vectors can be represented in the αβγ-coordinates by using
Clarke’s transformation:

1
Vα 

  2 
V
=
⋅
β
  3 0
Vγ 
1
 
2


1
2
3
2
1
2

−

1 
2  V 
 a
3  
−
. Vb
2  
1  V c 
2 
−

Table A.2 shows the normalized αβγ-values of each switching vector.

(A.1)

The

representation for these switching vectors in αβγ-space is shown in Fig. A.1 (a).
The vectors are distributed on layers according to the value of the γ component
of the switching vectors. Three vectors (v2, v4 and v6) are located on the layer of
Vγ = 1/6VDC. The vectors (v1, v3 and v5) are lying on the layer of Vγ = -1/6VDC,
the zero vector (vZ) is located in the origin. The vectors v7 and v0 are located on
the γ-axis at Vγ = 1/2VDC and Vγ = -1/2VDC , respectively. The projection of the
178

vectors in the αβ-frame is shown in Fig. A.1 (b) which is divided into six prisms.
Fig. A.2 shows the six prisms in the αβγ-space. Each prism is divided into two
tetrahedrons, upper (TH) and lower (TL) tetrahedron. Each tetrahedron is
characterized by three non-zero vectors and the zero vector.
Table A. 2 Normalized αβγ− Components of each switching vector [147].

αβγ-components
Vector

Vγ =

Vα/ VDC Vβ/ VDC Vγ/ VDC

-1/2

2/3

-1/6

1/3

1/ 3

1/6

-1/3

1/ 3

-1/6

-2/3

1/6

-1/3

1/ 3

-1/6

1/3

1/ 3

1/6

1/2

VDC
2

V4
P4

1
Vγ = VDC
6

P11

P2
P1

P3
Vz

Vγ = 0

P4
P5

V3
1
Vγ = − VDC
6
V5

Vz
V7

Vγ = −

VDC
2

Fig. A. 1: Space vectors. (a) 3D diagram. (b) αβ projection [147].

179

Vγ =

VDC
2

Vγ =

VDC
2

V4
1
Vγ = VDC
6

V6
P6

P5
P3

PP23

Vγ = 0

V5
V3

1
Vγ = − VDC
6

1
Vγ = − VDC
6
V1

Prism 1

Prism 2

V
Vγ = − DC
2

Vγ =

VDC
2

Vγ = −
Vγ =

V DC
2

VDC
2

V6
1
Vγ = VDC
6

PP34

P
P54

Vγ = 0

1
Vγ = − VDC
6

1
Vγ = − VDC
6
V3

Prism 3
V
Vγ = − DC
2

VDC
2

Prism 4

V
Vγ = − DC
2

Vγ =

VDC
2

V2
1
Vγ = VDC
6

PP56

1
Vγ = VDC
6

P6
V6

PP16

Vγ = 0

1
Vγ = − VDC
6

1
Vγ = − VDC
6
V5

Prism 5
V
Vγ = − DC
2

P6
P

P3
V4

Vγ = 0

1
Vγ = VDC
6

P4
V4

Vγ =

Vγ = 0

1
Vγ = VDC
6

PP12

Prism 6
Vγ = −

VDC
2

Fig. A. 2: 3D prisms for three-leg four-wire inverter [147].

180

Step Four: Reference Vector Position Identification

There are twelve possibilities for the reference vector position which is a
consequence of having twelve tetrahedrons. If the reference vector lies in one
of these tetrahedrons, it can be identified by using the boundary planes which
are limiting the tetrahedron. Each tetrahedron is limited by three planes. The
boundary planes can be determined by the following linear equations:
1
Vβ = 0
3
E12 : Vα − 3Vβ + 4Vγ = 0
E

E 27 :

(A.2)

(A.3)

3
1
Vα − V β = 0
6
6

(A.4)

E23 : − 2Vα + 2Vγ = 0

(A.5)

3
1
E37 :
Vα + Vβ = 0
6
6

(A.6)

E34 : Vα + 3Vβ + 4Vγ = 0

(A.7)

where Em.,n is the separation plane stretched from the switching vectors m and
n. The identification for the twelve tetrahedrons through the bounding planes for
the upper tetrahedron and for the lower tetrahedron is shown in Table A.3.
Table A. 3 Tetrahedrons Boundaries [147]

Prism

Tetrahedron
Upper

Lower

E71P E12 P E27 P

E71 P E12 N E 27 P

E27 N E23 P E37 P

E 27 N E 23 N E37 P

E37 N E34 P E71P

E37 N E34 N E71 P

E71 N E12 P E27 N

E71N E12 N E27 N

E27 P E23 P E37 N

E27 P E23 N E37 N

E37 P E34 P E71 N

E37 P E34 N E71N

As an example Fig. A.3 shows the upper (TH) and the lower (TL) tetrahedrons in
the first prism and the separation planes. The upper tetrahedron is limited by
the positive region of the planes E71 (E71P), E27 (E27P) and E12 (E12P), while the
lower tetrahedron is limited by the positive region of the planes E71 and E27 and
the negative region of the plane E12 (E12N).

181

Fig. A. 3: Upper and lower tetrahedrons in Prism 1[147].

Step Five: Duty Cycles Calculation
SVM is the approximation of an arbitrary vector in the αβγ-space using the
nearest three vectors (va, vb and vc) and the zero vector the inverter generates
[174]. The nearest three vectors are chosen once the target tetrahedron is
defined, the required on-duration of each of the vectors can be easily computed
using (A.8). These specify that the reference vector is the geometric sum of the
chosen three vectors multiplied by their on-durations (da, db, dc) and that their
on durations must fill the complete cycle:
vα − ref

v ref = v β − ref
 vγ − ref




 = [v a



 da 
v c ]  d b 
 d c 

(A.8)

The duty cycles can be obtained from equation (A.9) as follows:
d a 
 
d b  = Tr
d c 

vα _ ref 


vβ _ ref 


vγ _ ref 

(A.9)

Where Tr is the decomposition matrix obtained by the following equation:
Tr = [v a

182

vc ]

−1

(A.10)

The decomposition matrices for the twelve tetrahedrons are shown in Table A.4.
The rest of the switching period will be equal to the on-duration time for the zero
vector:

d z = 1 − (d a + d b + d c )

(A.11)

Table A. 4 matrices for switching vector duty cycles [147].

Tetrahedron

Prism

Upper


























3
2
0










Lower
0
0

1
2

3
−
2

3
2

−

3
2

0
1
2

3
−
2

0
3
2
1
2

−

 1
 −
 2

 0

 −3
 2










3
−
2

3
2
3
2

−

3
−

−

3
2
3
2
3
2
3

3
−
2

3
2
3
−
2
−

−1

3
2

− 3

1
2

3
2



























1
−
2
3
2

3
2
3
−
2


 1

− 3
 2

 3
 2


0 


0 

2 


 1
−
 2

 0

−3
 2


−2 


0 

0 


 1
 −
 2

 0

 −3
 2


0 


0 

2 












0 


0 

2 











0
3
2
3
2
−

3
2
3

−

3
2

−

3
2
3

−

3
2

3
2
3
2

−

1
2
3
2

−

− 3

−

3
2
3
−
2
3
2
3
2


−2 


0 

0 



− 2

0 

0 


−2 


0 

0 


−2 


0 

0 



−2 


0 

0 


−2 


0 

0 


Step-Six and Seven: Building Vector Sequence and Pulse Pattern
Computation:
In order to reduce the current ripple, switching vectors adjacent to the reference
vector should be selected since the adjacent switching vectors produce nonconflicting voltage pulses (same voltage polarity) [175]. Symmetric modulation
is characterized by using four vectors per modulation sequence.

In this

approach the generated zero-vector will be injected in each modulation
sequence, where the six switches will be turned off at the same time for the time

183

duration tZ. Table A.5 shows the vector sequence for the upper and lower
tetrahedron in each prism.
Table A. 5 Symmetric switching sequence [147]

Prism
Prism 1
Prism 2
Prism 3
Prism 4
Prism 5
Prism 6

Tetrahedron
Upper
Lower
Upper
Lower
Upper
Lower
Upper
Lower
Upper
Lower
Upper
Lower

Sequence
vZ-v1-v2-v7//v7-v2-v1-vZ
vZ-v0-v1-v2//v2-v1-v0-vZ
vZ-v3-v2-v7//v7-v2-v3-vZ
vZ-v0-v3-v2//v2-v3-v0-vZ
vZ-v3-v4-v7//v7-v4-v3-vZ
vZ-v0-v3-v4//v4-v3-v0-vZ
vZ-v5-v4-v7//v7-v4-v5-vZ
vZ-v0-v5-v4//v4-v5-v0-vZ
vZ-v5-v6-v7//v7-v6-v5-vZ
vZ-v0-v5-v6//v6-v5-v0-vZ
vZ-v1-v6-v7//v7-v6-v1-vZ
vZ-v0-v1-v6//v6-v1-v0-vZ

An example for determining the switching sequence for the upper tetrahedron in
the first prism is shown in Fig. A.4. The pulse sequence can be achieved by
comparing the duty cycles with a carrier signal. The pulse sequence for phase
a, b and c are shown in Fig. A.5 for the same mentioned case.

Fig. A. 4: Steps for symmetric modulation for TH in the first Prism [147].

184

Fig. A. 5: Symmetric modulation for the upper tetrahedron in the first prism [147].

b) Compensated Vectors Approach
The proposed SVM algorithm can be achieved through the same steps
mentioned in section a. These steps are described in detail in the following:

Step-One, Two and Three
In similar way to two-level three-phase inverters, there are eight different
switching combinations where the output terminals will be connected to +1/2 or
-1/2 of the input DC voltage. Unlike two-level three-leg inverters none of these
switching combinations is generating a zero voltage at the output terminals,
which complicate the implementation of its SVM. Table I presents the eight
possible switching vectors and the corresponding output voltages related to the
DC-voltage as reference voltage. The switching vectors can be represented in
the αβγ-coordinates using Clarke’s transformation.
Table A.8 shows the normalized αβγ-values of each switching vector.

The

representation for these switching vectors in αβγ-space is shown in Fig. A.6.
The vectors are taking place in layers according to the value of their γ
component. Three vectors (V2, V4 and V6) are located at the layer Vγ = 1/6 VDC.
The vectors (V1, V3 and V5) are lying at the layer Vγ = -1/6 VDC. The vectors V7
and V0 are located at the γ-axis at Vγ = 1/2 VDC and Vγ = -1/2 VDC respectively.
The projection of the vectors in the αβ-frame is shown in Fig. A.6 which is
185

divided into six prisms. Each prism is divided into two tetrahedrons, upper and
lower tetrahedron. Each tetrahedron is characterized by three vectors.

Table A. 6 Switching states, the corresponding output voltages and normalized αβγ-components
of each switching vector

Normalized Output Voltage
Va/VDC
Vb/VDC
Vc/VDC
-1/2
0
-1/2

Switches (On)

Vector

S4 S6 S2

S1 S6 S2
S1 S3 S2
S4 S3 S2

V1
V2
V3

1/2

2/3

-1/6

1/2
-1/2

1/3
-1/3

1/6
-1/6

S4 S3 S5

-1/2

-2/3

1/6

S4 S6 S5

V5
V6

-1/3
1/3

-1/6

S1 S6 S5

-1/2
1/2

S1 S3 S5

1/2

Vγ =

1/6

VDC
2

1
Vγ = VDC
6

Vγ = 0

1
Vγ = − VDC
6

Vγ = −

VDC
2

Fig. A. 6. 26 3D-Space vectors [148].

Step Four: Reference Vector Position Identification
In view of the fact that twelve tetrahedrons exist, there are twelve possibilities
for the reference vector position. Switching states, the corresponding output
voltages and normalized αβγ−components of each switching vector.
We can identify the position of the reference vector using the boundary planes
limiting the tetrahedron. Each tetrahedron is limited by three planes. The
boundary planes can be determined by means of the following linear equations:

186

E 71 :

1
V =0
3 β

E12 : Vα − 3Vβ + 4Vγ = 0
E 27 :

3
1
Vα − V β = 0
6
6

E23 : − 2Vα + 2Vγ = 0
E37 :

3
1
Vα + Vβ = 0
6
6

E 34 : Vα + 3V β + 4Vγ = 0

(A.12)
(A.13)
(A.14)
(A.15)
(A.16)
(A.17)

The zero-vector is compensated by the vectors V0 and V7, both lying against
each other direction on the γ-axis. The reference vector can be expressed as
following:
V ref = d a ⋅ V a + db ⋅ V b + dγ ⋅ V c

(A.18)

where for the upper tetrahedron (A.19) applies and for the lower tetrahedron
(A.20) applies:

V c = V 7 and d γ = d 7 − d 0

(A.19)

V c = V 0 and d γ = d 0 − d 7

(A.20)

187

Vγ =

VDC
2

Vγ =

1
Vγ = VDC
6

VDC
2

Vγ = −

VDC
2

Vγ =

VDC
2

1
Vγ = VDC
6

Vγ = 0

1
Vγ = − VDC
6

Vγ = −

Vγ = 0

1
Vγ = − VDC
6

VDC
2

1
Vγ = − VDC
6

Vγ = −

Vγ =

Vγ = −

VDC
2

1
Vγ = VDC
6

Vγ = 0

1
Vγ = − VDC
6

VDC
2

1
Vγ = VDC
6

Vγ = 0

1
Vγ = − VDC
6

Vγ =

1
Vγ = VDC
6

Vγ = 0

Vγ = −

V DC
2

1
Vγ = − VDC
6

VDC
2

Vγ = −

V DC
2

Fig. A. 7: 3D prisms for three-leg four-wire inverters [148].

Step Five: Duty Cycles Calculation:
Once the target tetrahedron is defined the nearest three vectors are chosen. By
normalizing the standard vectors at the intermediate circuit voltage, the duty
cycles in matrix form can be written as following:

d a 
 
1
−1
d b  = V [ v a v b v c ]
DC
d 
 γ

Vα − ref 


Vβ − ref 


Vγ − ref 

(A.21)

The duty cycles for the vectors V7 and/or V0 can be determined according to the
position of the reference vector. For the upper tetrahedron using (A.22) and for
the lower tetrahedron using (A.23):
d7 =

d0 =

1 − ( d a + d b ) + dγ
2

where

d 0 = d 7 − dγ

(A.22)

where

d 7 = d 0 − dγ

(A.23)

188

Step-Six and Seven: Building Vector Sequence and Pulse Pattern
Computation:
In order to reduce the current ripple, switching vectors adjacent to the reference
vector should be selected since they produce non-conflicting voltage pulses
(same voltage polarity) [133]. An example for determining the switching
sequence for the first prism is shown in Fig. A.8 (a). The pulse sequence can be
achieved by comparing the duty cycles with a carrier signal. The pulse
sequence for phase a, b and c are shown in Fig. A.8 (b) for the same mentioned
case.

Fig. A. 8 :(a) Steps for the modulation in the first prism, (b) Symmetric modulation in the first
prism [148].
Table A. 7 The vector sequence for the upper and lower tetrahedrons in each prism.

Sequence

Prism
1

v0-v1-v2-v7-v7-v2-v1-v0

v0-v3-v2-v7-v7-v2-v3-v0

v0-v3-v4-v7-v7-v4-v3-v0

v0-v5-v4-v7/-v7-v4-v5-v0

v0-v5-v6-v7-v7-v6-v5-v0

v0-v1-v6-v7-v7-v6-v1-v0

189

A.2

SVM for Three-phase, Four-leg Voltage Source Inverters

The purpose of the three-phase four-leg inverters is to achieve a balanced
output voltage waveform over all loading conditions and transients, an
additional neutral inductor Ln can be added to the neutral line where the
switching frequency ripple will be reduced. It is ideal for applications like
industrial Automation, military equipment, which require high performance
uninterruptible power supply. Fig. A.9 shows the structure of a four leg inverter.

Va
VDC
Vb
S8

Fig. A. 9: Space vector diagram for five-level diode-clamped inverter.

There are (24 =16) switching combinations (vectors), there are two zero
switching vector V0 and V15, and fourteen non-zero switching vectors.
The switching combinations are represented by ordered sets (phase-a, phaseb, phase-c, phase-d), when phase-a = ‘1’ denotes that the upper switch (S1) in
phase-a, is turned on, and phase-a = ‘0’ denotes that the bottom switch in
phase A, S4, is turned on. The same notation applies to phase legs B and C and
the fourth neutral leg.
By applying park’s transformation to transform these values for the abccoordinate to the α-β-γ-coordinate and the results are shown in Table A.8.

190

Table A. 8 Output voltage and the α-β-γ-Components related to the DC input voltage.

Output voltage related to the DC
Input
Vector
Va/VDC
Vb/VDC
Vc/VDC
V0
0
0
0
V1
-1
-1
-1
V2
0
0
1
V3
0
1
0
V4
1
0
0
V5
1
1
0
V6
1
0
1
V7
0
1
1
V8
-1
-1
0
V9
-1
0
-1
V10
0
-1
-1
V11
0
0
-1
V12
0
-1
0
V13
-1
0
0
V14
1
1
1
V15
0
0
0

α-β-γ-Components
Vα/ VDC
0
0
− 1/ 3
− 1/ 3

2/3
1/ 3
1/ 3
− 2/3
− 1/ 3
− 1/ 3

2/3
1/ 3
1/ 3
− 2/3
0
0

Vβ/ VDC
0
0

Vγ/ VDC
0
-1

− 1/ 3

1/ 3
1/ 3
1/ 3
2/3
2/3
2/3
− 2/3
− 2/3
− 2/3

1/ 3
0
1/ 3

− 1/ 3

0
− 1/ 3

1/ 3
0
1/ 3

− 1/ 3

0
0
0

− 1/ 3
− 1/ 3
− 1/ 3
1
0

Fig. A.10 shows the space vector diagram for a four leg inverter which is a 3D
diagram, the diagram is divided into seven layers, Two switching vectors are
zero vectors (V0, V15) are located in the origin where Vγ =0. The three vector
1
2
(V2, V3 and V4) are located on the layer of Vγ = V DC , on the layer Vγ = V DC , the
3
3

vectors (V5, V6 and V7) are lying on. Only V14 vector lies on the layer Vγ = VDC .
1
On the layer Vγ = − V DC , the vectors (V11, V12 and V13) are lie on. The vectors
3
2
(V8, V9 and V10) are lying in the layer Vγ = − V DC , and for the layer Vγ = − V DC ,
3

only vector V1 lies on it. Projection of all switching vectors back on the α-β plane
forms a hexagon, similar to that of a conventional three-phase inverter, shown
in Fig. A.10. The projected vectors have a length of

191

2
V DC .
3

Vγ = VDC

2
Vγ = VDC
3

1
Vγ = VDC
3
2
U ZK
3

Vγ = 0

1
Vγ = − VDC
3

2
Vγ = − VDC
3

Vγ = −VDC

Fig. A. 10 Space vector diagram for four-leg inverter.

A three dimensional space vector modulation is applied to the four legs inverter
to estimate the control signals for the power transistors. To estimate the
reference voltage in the 3D space adjacent switching vectors must be identified.
Like the six sectors in the 2D space vector modulation. The 3D space will be
divided into six prisms; they are numbered prism 1 through 6, as shown in Fig.
A.11.

192

Fig. A. 11: Six prisms for four-leg inverter space vector diagram.

The criteria to determine which prism the reference vector is in relies only on
the projections of the reference vector on the α-β plane Vα and Vβ. Therefore,
once the prism where the reference vector locates is found, there are six nonzero switching vectors and two zero switching vectors available to synthesize
the reference vector [151].
Each prism is composed of four tetrahedrons and each tetrahedron is defined
by 3 non-zero switching vectors and two zero-vectors as shown in Fig. A.12.

193

Fig. A. 12: Tetrahedrons 1-4 in the first prism.

With six prisms and each prism contains four tetrahedrons, which means that
there are twenty four possibilities positions for the reference vector that should
be identified, in order to calculate the duty cycles and perform the modulation,
one way to identify the tetrahedrons is by using the separation planes which is
created by connecting the switching vectors together which result the 3D frame
which is shown in Fig. A.13.

Fig. A. 13: Four leg inverter 3D frame.

The equations of these planes are :

2Vγ −Vα − 3Vβ =0

(Green Plane)

(A.24)

2Vγ − Vα + 3Vβ = 0

(Red Plane)

(A.25)

Vγ + Vα = 0

(Blue Plane)

(A.26)

194

Once the target tetrahedron is defined the nearest three vectors are chosen, the
required on-duration of each of the vectors can be easily computed using (4).
These specify that the reference vector is the geometric sum of the chosen
three vectors multiplied by their on-durations (da, db, dc) and that their on
durations must fill the complete cycle.

v ref

vα − ref

= v β − ref
 vγ − ref




 = [v a



 da 
v c ]  d b 
 d c 

(A.27)

The duty cycles can be obtained from equation (3.47)as follows:

d a 
 
d b  = Tr
d c 

vα _ ref

v β _ ref

vγ _ ref







(A.28)

Where Tr is the decomposition matrix obtained by the following equation:

Tr = [v a

vc ]

−1

(A.29)

The rest of the switching period will be equal to the on-duration time for the zero
vector:

d z = 1 − (d a + d b + d c )
More details can be found in [128, 149, 150].

195

(A.30)

A.3

Inverter control in DQ

In the following the basic control equations will be derivate in dq frame including
transformations and their cross decoupling:

Vdc

Fig. A. 14: Inverter basic circuit.

The per phase inverter equation is:
ea − va = Ria + L

dia
dt

(A.31)

In matrix form

ea − va 
ia 
ia 
e − v  = R i  + L. d i 
b
 b b
b
dt  
ec − vc 
ic 
ic 

(A.32)

u a 
ia 
u  =  R + L. d  . i 
 b
 b  
dt   
uc 
ic 

(A.33)

Using αβ transformation we get

1
uα 

  2
u β  = 3  0
u 

 0
1
2


−1
2
3
2
1
2

196

−1 
2 
 u a 
− 3  
. ub
2   
 u 
1   c
2 

(A.34)

1

 3 ( 2ua − ub − uc ) 
uα  

   1
( ub − uc ) 
u β  = 
3
u  

 0
 1 (u + u + u ) 
 3 a b c 

(A.35)

In a balanced system ua+ub+uc=0 and so
1
 1

 3 ( 2ua − [ub + uc ])   3 ( 2ua − [ −ua ])  
ua

uα  
 
 

   1
 =  1 u −u  =  1 u −u 
u
=
u
−
u
( b c)
 β 
  3 ( b c )   3 ( b c )
u   3
 
 

 0
1
0
 1 (u + u + u )  
 

0
(
)
a
b
c
3
 3
 


(A.36)

To transfer to the dq frame we need the following equation:
ud   cos θ
 =
uq   − sin θ

sin θ  uα 
. 
cos θ  uβ 

(A.37)

So we get
ud   uα cos θ + uβ sin θ  
d   iα cos θ + iβ sin θ 
  =  −u sin θ + u cos θ  =  R + L   −i sin θ + i cos θ 
dt   α
β
β
 

u q   α

 iα cos θ + iβ sin θ 
∂
= R
+L

∂t θ =const
 −iα sin θ + iβ cos θ 
 iα cos θ + iβ sin θ 
∂
+L


∂t i=const  −iα sin θ + iβ cos θ 

 iα cos θ + iβ sin θ 
 −i sin θ + i cos θ 
β
 α


(A.38)

(A.39)

Solving these differential equations:

∂
∂t θ =const

∂
∂t i=const

 iα cos θ + iβ sin θ   cos θ
 −i sin θ + i cos θ  = 
β
 α
  − sin θ

d 
i
sin θ   dt β 


cos θ   d 
i
 dt β 

 iα cos θ + iβ sin θ 
∂ ∂θ  iα cos θ + iβ sin θ 
 −i sin θ + i cos θ  = L


∂t ∂t  −iα sin θ + iβ cos θ 
β
 α

197

(A.40)

(A.41)

 iα cos θ + iβ sin θ 
 cos θ
 −i sin θ + i cos θ  = −ω L 
β
 − sin θ
 α

 −i 
= −ω L  q 
 id 
= ωL

∂
∂θ

u d 
 iα cos θ + iβ sin θ 
∂
  = R  −i sin θ + i cos θ  + L
∂t θ =const
β
 α

u q 
 iα cos θ + iβ sin θ 
∂
+L


∂t i=const  −iα sin θ + iβ cos θ 

sin θ  iβ 
 
cos θ  iβ 

 iα cos θ + iβ sin θ 
 −i sin θ + i cos θ 
β
 α


d
Lid + ω Liq
dt
d
uq = Riq + Liq + ω Lid
dt

ud = Rid +

(A.42)

(A.43)

(A.44)
(A.45)

From that we can see that the controllers are coupled but it is easy to decouple
them even if we do not know them as shown in Fig. A.15.

Vd_act
Vd_ref

Id_ref
Id_act
wL
wL
Iq_ref

Iq_act

Vq_ref
Vq_act

Fig. A. 15: Inverter variables decoupling.

198

A.4

Generalised Integrator “The Selective Filter”

A very fast method to measure the fundamental active and reactive power, and
the rms-values of the fundamentals of current and voltage can be realized by
using a structure with the transfer function:

G (s) = K .

s
s + ωn2
2

(A.46)

This structure is called “generalised integrator” or “the selective filter” [100] (Fig.
A.16). The GI has a gain k and is adjusted for a certain frequency ω = 2π f. It is
done using two integrators that are building an oscillator removing any
component that is not at the specified frequency. If the GI is adjusted for the
fundamental frequency f of an incoming signal u and if it is implemented with a
feedback loop (Fig. A.17) it provides a signal y equal to the fundamental of u
and a fundamental signal y which is lagging for 90 degrees [86]. The GI was
first used for similar applications in [176] and is described in [87] and [177] too
[86].
Extracting the α-, β- components can be done using this generalized integrator.
However, the integrator factors must be adjusted according to the grid
frequency.
The transfer function can be expressed as:

⋅

ωn ⋅ s
s 2 + ω n2

(A.47)

G β ( s) = K ⋅

ωn
s + ω n2

(A.48)

Gα ( s ) =

ωn

199

ωn

uα

ωn

uβ

Fig. A. 16: Generalised integrator.

In order to make sure that the integrator outputs will not drift away, a feedback
of the α-component is needed. Ignoring the proportional factor K/ωn, we will get
the structure shown in Fig. A. 17 which result’s in the single phase α-, βcomponents.

ωn

uα

uβ

Fig. A. 17: Generalised integrator with a feedback loop to get α-, β-components.

Transfer Function Derivation:
According to Figure A. 17 follows that the α-component is:
uα = ω n ∫ (ue − uα − u β )dt
u α = ω n (u e − uα − u β )
o

with u β = ω n ∫ uα dt

(

| d / dt

u α = ω n u e − uα − ω n ∫ uα dt

)

oo
o
o

u α = ω n  u e − u α − ω n uα 



u α = ω n u e − ω n u α − ω n2 uα

200

| d / dt

(A.49)
follows

(A.50)
(A.11)

(A.52)

(A.53)

u α + ω n u α + ω n2uα = ω n u e
1

uα +

|
1

u α + uα =

ω n2

(A.54)

(A.55)

(A.56)


s
s2 
s
uα 1 +
+ 2  =
ue .
ω
ω
ω
n
n
n 


(A.57)

2
n

ωn

This leads to:
s2

2
n

uα +

ωn

uα + u α =

ωn

Therefore, for the transfer function the real and imaginary parts can be
expressed as following:
s
Gα ( s ) =

G β ( s) =

uα ( s )
=
u e ( s)

u β (s)
u e ( s)

ωn
1+

ωn

ω n2

ω n uα ( s )
s

⋅

u e ( s)

ωn
s

ωn
sω n
=
ω n2 + sω n + s 2 ω n2 + sω n + s 2
ω n2
⋅

ω n2
sω n
=
.
ω n2 + sω n + s 2 ω n2 + sω n + s 2

(A.58)

(A.59)

These express a transfer function of a second order band pass filter with cut-off
frequency fcut = ωn / 2π = 50Hz, the band width is B = + 25Hz. This way the
measured signals will be filtered (Only the fundamental component will be
measured). The function shown in Fig. A.18 can be used to measure the
voltage as well. The grid voltage RMS value can be calculated from the Realand Imaginary part s as follows:

u = uα2 + u β2

201

(A.60)

Fig. A. 18: Voltage in phase one and its alpha and beta components using the generalised
integrator.

202

203

204

205

LIST OF PUBLICATIONS
1

Egon Ortjohann, Arturo Arias, Danny Morton, Alaa Mohd, Nedzad Hamsic, Osama
Omari:
Grid-Forming Three-phase Inverters for Unbalanced Loads in Hybrid Power
Systems, IEEE 4th World Conference on Photovoltaic Energy Conversion,
Waikoloa, Hawaii, May 2006

Egon Ortjohann, Alaa Mohd, Nedzad Hamsic, Danny Morton, Osama Omari:
Advanced Control Strategy for Three-Phase Grid Inverters with Unbalanced Loads
for PV/Hybrid Power Systems, 21th European PV Solar Energy Conference ,
Dresden, September 2006

N.Hamsic , A.Schmelter , A.Mohd, E.Ortjohann, J.Zimmermann, A.Tuckey,
E.Schultze:
Stabilising the Grid Voltage and Frequency in Isolated Power Systems Using a
Flywheel Energy Storage System, The Great Wall World Renewable Energy Forum
(GWREF), Beijing, China, October 2006

E.Ortjohann, W.Sinsukthavorn, N.Hamsic, A.Schmelter, A.Mohd, D.Morton:
An Innovative Simulation Approach using Dynamic-RMS Model for Hybrid
Isolated Mini-Grids, The Great Wall World Renewable Energy Forum (GWREF),
Beijing, China, October 2006

E.Ortjohann, A.Mohd ,N.Hamsic, Osama Omari, D.Morton:
Control and Representation of Three-Phase Asymmetrical Signals Used by Modular
Inverters to Feed Unbalanced Loads in Hybrid Power Systems, The Great Wall
World Renewable Energy Forum (GWREF), Beijing, China, October 2006

N.Hamsic , A.Schmelter , A.Mohd, E.Ortjohann, J. Zimmermann, A.Tuckey,
E.Schultze:
Increasing Renewable Energy Penetration in Isolated Grids Using a Flywheel
Energy Storage System, the first International Conference on Power Engineering,
Energy and Electrical Drives (POWERENG, IEEE),Portugal, April 2007.

E.Ortjohann, W.Sinsukthavorn, A.Mohd, N.Hamsic, A.Schmelter, D.Morton:
Modeling/Simulation of Power Distribution in Hybrid Power Systems Using
Dynamic-RMS Technique, the first International Conference on Power Engineering,
Energy and Electrical Drives (POWERENG,IEEE),Portugal, April 2007.

E.Ortjohann, A. Mohd, N. Hamsic, A. Al-Daib, M.Lingemann:
Three-Dimensional Space Vector Modulation Algorithm for Three-Leg Four-Wire
Voltage Source Inverters, the first International Conference on Power Engineering,
Energy and Electrical Drives (POWERENG, IEEE), Portugal, April 2007.

Egon Ortjohann, Alaa Mohd, Andreas Schmelter, Nedzad Hamsic, Max
Lingemann: Simulation and Implementation of an Expandable Hybrid Power
System, 2007 IEEE International Symposium on Industrial Electronics, Vigo, Spain,
June 4-7, 2007.

206

Osama Omari, Egon Ortjohann, Alaa Mohd, Danny Morton:
An Optimal Control Strategy for DC Coupled Hybrid Power Systems, 2007 IEEE
International Symposium on Industrial Electronics ,Vigo, Spain, June 4-7, 2007

Osama Omari, Egon Ortjohann, Alaa Mohd, and Danny Morton:
An Online Control Strategy for DC Coupled Hybrid Power Systems, IEEE PES
2007 general meeting, Tampa, Florida, USA, June 24-28, 2007.

E. Ortjohann, A. Mohd, N. Hamsic, M. Lingemann, W. Sinsukthavorn, D. Morton.
A Novel Space Vector Modulation Control Strategy for Three-leg Four-Wire
Voltage Source Inverters, EPE 2007 - 12th European Conference on Power
Electronics and Applications , Aalborg, Denmark , September ,2007.

E. Ortjohann, O. Omari, M. Lingemann, A. Mohd, N. Hamsic, W. Sinsukthavorn1,
D. Morton:
An Online Control Strategy for a Modular DC Coupled Hybrid Power System , EPE
2007 - 12th European Conference on Power Electronics and Applications , Aalborg,
Denmark , September ,2007.

E. Ortjohann, M. Lingemann, A. Mohd, W. Sinsukthavorn, A. Schmelter, N.
Hamsic, D. Morton
Scalable Hybrid Power System for Decentralized Stand-alone AC-Grids, Second
International Renewable Energy Storage Conference (IRES II), Bonn, November,
2007.

E. Ortjohann, W. Sinsukthavorn, A. Mohd, N. Hamsic, A. Schmelter, D. Morton:
Dynamic-RMS Modeling of Distributed Electrical Power Supply Systems, 2008
IEEE PES Transmission and Distribution Conference and Exposition, Chicago,
April, 2008.

E. Ortjohann, M. Lingemann, A. Mohd, W. Sinsukthavorn:
Scalable Hybrid Power System for Decentralized Mini-grids, 4th European PVHybrid and Mini-Grid Conference, Glyfada, Athens, Greece, May, 2008.

Alaa Mohd, E. Ortjohann, Andreas Schmelter, Nedzad Hamsic, Danny Morton:
Challenges in Integrating Distributed Energy Storage Systems into Future Smart
Grid, The IEEE International Symposium on Industrial Electronics, Cambridge, UK,
from June 30 to July 2, 2008.

E.Ortjohann, M. Lingemann, A. Mohd, W. Sinsukthavorn, Andreas Schmelter, N.
Hamsic, D. Morton:
A General Architecture for Modular Smart Inverters, The IEEE International
Symposium on Industrial Electronics, Cambridge, UK, from June 30 to July 2, 2008.

E. Ortjohann, M. Lingemann, O.Omari, A. Schmelter, N. Hmasic, A. Mohd, W.
Sinsukthavorn, D. Morton
Modular Architecture for Decentralized Hybrid Power Systems. 13th International
Conference on Power Electronics and Motion Control (EPE-PEMC 2008), Poznań,
Poland, September 2008.

207

Egon Ortjohann, Worpong Sinsukthavorn, Alaa Mohd, Max Lingemann, Nedzad
Hamsic, Andreas Schmelter, Danny Morton:
General Control Methodology for Interconnected Mini-Grids. 8th WSEAS
International Conference on POWER SYSTEMS (PS 2008). Santander, Cantabria,
Spain, September, 2008

Egon Ortjohann, Worpong Sinsukthavorn, Alaa Mohd, Nedzad Hamsic, Max
Lingemann, Andreas Schmelter, Danny Morton:
Control Methodology of Distributed Generation in Interconnected Grids. IEEE PES
Power Systems Conference & Exhibition (PSCE). Seattle, Washington, March 2009.

Alaa Mohd, Egon Ortjohann, Worpong Sinsukthavorn, Nedzad Hamsic, Max
Lingemann, Andreas Schmelter, Danny Morton:
Supervisory Control and Energy Management of an Inverter-based Modular Smart
Grid. IEEE PES Power Systems Conference & Exhibition (PSCE). Seattle,
Washington, March 2009.

Egon Ortjohann, Alaa Mohd, Nedzad Hamsic, Max Lingemann, Andreas
Schmelter, Worpong Sinsukthavorn, Danny Morton:
Three Dimensional Space Vector Modulation Strategy for Three-Leg Four-Wire
Voltage Source Inverters. IET Power Electronics Research Journal.
(Accepted but subject to revisions)
Alaa Mohd, Egon Ortjohann, Nedzad Hamsic, Max Lingemann, Andreas
Schmelter, Worpong Sinsukthavorn, Danny Morton, Osama Omari:
Review of control techniques for inverters parallel operation. Electric Power
Systems Research Journal.
(Sent but not published yet)
Alaa Mohd, Egon Ortjohann, Nedzad Hamsic, Max Lingemann, Worpong
Sinsukthavorn, Danny Morton:
Isochronous Load Sharing and Control for Inverter-based Distributed Generation.
International Conference on CLEAN ELECTRICAL POWER Renewable Energy
Resources Impact. Capri – Italy, June, 2009.
(Abstract Sent)
E. Ortjohann, W. Sinsukthavorn, A. Mohd, M. Lingemann:
A Hierarchy Control Strategy of Distributed Generation Systems. International
Conference on CLEAN ELECTRICAL POWER Renewable Energy Resources
Impact. Capri – Italy, June, 2009.
(Abstract Sent)
Alaa Mohd, Egon Ortjohann, Nedzad Hamsic, Max Lingemann, Worpong
Sinsukthavorn, Danny Morton:
Inverter-based Distributed Generation Control using Droop/Isochronous Load
Sharing. IFAC Symposium on Power Plants and Power Systems Control. Tampere,
Finland, July 2009.
(Sent but not published yet)
Egon Ortjohann, Worpong Sinsukthavorn, Alaa Mohd, Nedzad Hamsic, Max
Lingemann, Danny Morton:
A Hierarchy Control Strategy and Management of Interconnected Distributed
Generation Systems. IFAC Symposium on Power Plants and Power Systems
Control. Tampere, Finland, July 2009.
(Sent but not published yet)

208

Egon Ortjohann, Alaa Mohd, Nedzad Hamsic, Max Lingemann, Worpong
Sinsukthavorn, Danny Morton:
Design and Experimental Investigation of Space Vector Modulation for Three-leg
Four-Wire Voltage Source Inverters. EPE 2009 - 13th European Conference on
Power Electronics and Applications Barcelona, Spain, September 2009.
(Abstract Sent)

209

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