Modelling and Control of Inverter Sources within a Low ...

Modelling and Control of InverterSources within a Low Voltage Distributed

Generation System

Michaela Nicole GriffithsB.Eng (Comp)

A thesis submitted for the degree of

Doctor of PhilosophyDiscipline of Electrical Engineering

School of Electrical Engineeringand Computer Science

The University of NewcastleCallaghan, N.S.W. 2308

Australia

November 2012

This thesis contains no material which has been accepted for the award of any other degreeor diploma in any university or other tertiary institution and, to the best of my knowledgeand belief, contains no material previously published or written by another person, exceptwhere due reference has been made in the text. I give consent to this copy of my thesis,when deposited in the University Library, being made available for loan and photocopyingsubject to the provisions of the Copyright Act 1968.

Acknowledgements

This research was jointly funded by Australia’s Commonwealth Scientific and IndustrialResearch Organisation (CSIRO) Energy Centre, Newcastle; and the University of New-castle, Australia.

It would not have been possible without the support of my husband and technical support,Ian, my family, and my supervisor, Dr Colin Coates.

Contents

Abstract xi

1 Distributed Generation and Microgrids 1

1.1 Project Context and Description . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Distributed Generation - An Historical Perspective . . . . . . . . . . . . . . 2

1.3 The Microgrid Concept - Technical Challenges . . . . . . . . . . . . . . . . 6

1.4 Microgrid Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.5 Control Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

1.6 Other Research . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

1.7 Thesis Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2 Microgrid Transient Modelling 17

2.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.2 Source Structure and Models . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.2.1 Inverter Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.2.1.1 D-Q Transform . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.2.1.2 The Per-Unit System . . . . . . . . . . . . . . . . . . . . . 24

2.2.2 Output Filter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

2.2.3 Coupling Inductance . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

2.2.4 Measurement Block . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

2.3 Modelling of Other Circuit Elements . . . . . . . . . . . . . . . . . . . . . . 28

2.3.1 Load Resistors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

2.3.2 Inverter Coupling Inductances . . . . . . . . . . . . . . . . . . . . . 29

2.3.3 Non-linear Loads . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

Contents v

2.4 SimPowerSystems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

2.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

3 Inverter Control 35

3.1 Principles of Droop Control . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

3.2 Controller Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

3.2.1 Voltage Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

3.2.1.1 Voltage Limits . . . . . . . . . . . . . . . . . . . . . . . . . 40

3.2.2 Power Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

3.2.3 Power Sharing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

3.2.3.1 Power Source Block . . . . . . . . . . . . . . . . . . . . . . 44

3.3 Problems with Droop Control . . . . . . . . . . . . . . . . . . . . . . . . . . 44

3.3.1 Simulation Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

3.3.2 Different Voltage Setpoints . . . . . . . . . . . . . . . . . . . . . . . 46

3.3.3 Reactive Power Sharing . . . . . . . . . . . . . . . . . . . . . . . . . 48

3.3.4 Harmonics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

3.3.5 Resistive Line Impedances . . . . . . . . . . . . . . . . . . . . . . . . 51

3.3.6 Unbalanced Grid Voltages . . . . . . . . . . . . . . . . . . . . . . . . 52

3.3.7 Unbalanced Loads . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

3.4 Variations on Droop Control . . . . . . . . . . . . . . . . . . . . . . . . . . 57

3.4.1 Reactive Power Sharing . . . . . . . . . . . . . . . . . . . . . . . . . 58

3.4.1.1 Q-E Droop . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

3.4.1.2 Virtual Output Impedance . . . . . . . . . . . . . . . . . . 62

3.4.2 Frequency Restoration . . . . . . . . . . . . . . . . . . . . . . . . . . 64

3.4.3 Harmonic Current Sharing . . . . . . . . . . . . . . . . . . . . . . . 67

3.4.4 Resistive Line Impedance . . . . . . . . . . . . . . . . . . . . . . . . 67

3.5 Evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

Contents vi

4 Unbalanced Loads 70

4.1 Unbalance Compensation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

4.1.1 Symmetrical Components Theory . . . . . . . . . . . . . . . . . . . . 73

4.1.2 Sequence Representation of Grid . . . . . . . . . . . . . . . . . . . . 75

4.2 Unbalance Controller Design . . . . . . . . . . . . . . . . . . . . . . . . . . 81

4.3 Simulation Details . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

4.3.1 Performance of Unbalance Control . . . . . . . . . . . . . . . . . . . 86

4.4 Power Sharing and voltage regulation . . . . . . . . . . . . . . . . . . . . . 88

4.4.1 Controller Performance . . . . . . . . . . . . . . . . . . . . . . . . . 88

4.4.2 Frequency Issues with the Sequence Analyzer . . . . . . . . . . . . . 90

4.4.3 Variable Frequency Sequence Analyzer . . . . . . . . . . . . . . . . . 93

4.4.4 Performance of Control with Variable Frequency Sequence Analyzer 96

4.5 Sharing of Negative Sequence Generation . . . . . . . . . . . . . . . . . . . 98

4.6 Grid Connection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

5 Proposed Controller Design 101

5.1 Controller Components . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

5.1.1 Power-frequency Droop Control . . . . . . . . . . . . . . . . . . . . . 102

5.1.2 Voltage Regulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

5.1.3 Unbalance Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

5.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

5.2.1 Reference Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

5.2.2 Different Power Setpoints . . . . . . . . . . . . . . . . . . . . . . . . 112

5.2.3 Large Load Change . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

5.2.4 Different Sized Sources . . . . . . . . . . . . . . . . . . . . . . . . . . 114

5.2.5 Different Voltage Setpoints . . . . . . . . . . . . . . . . . . . . . . . 115

5.2.6 Different Coupling Inductances . . . . . . . . . . . . . . . . . . . . . 117

Contents vii

5.2.7 Reactive Load . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

5.2.8 Unbalanced Load . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

5.2.9 Unbalanced Reactive Load . . . . . . . . . . . . . . . . . . . . . . . 121

5.2.10 Unbalanced Load and Different Coupling Inductances . . . . . . . . 124

5.3 Controller Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128

5.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128

6 Power Sources and Storage 130

6.1 Power Source Block . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

6.1.1 Transfer Function Models . . . . . . . . . . . . . . . . . . . . . . . . 132

6.1.1.1 Ideal DC Source (Infinite Capacity Battery) . . . . . . . . 132

6.1.1.2 Microturbines . . . . . . . . . . . . . . . . . . . . . . . . . 132

6.1.1.3 Fuel Cells . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134

6.1.1.4 Renewable Power Sources . . . . . . . . . . . . . . . . . . . 135

6.1.2 Behaviour of Power Source Blocks . . . . . . . . . . . . . . . . . . . 135

6.2 Modifications to Source Models . . . . . . . . . . . . . . . . . . . . . . . . . 138

6.2.1 Forcing Drop in Voltage when Power is not Available . . . . . . . . 139

6.2.2 Shifting the Droop Characteristic . . . . . . . . . . . . . . . . . . . 141

6.3 Directly Connected Storage . . . . . . . . . . . . . . . . . . . . . . . . . . . 145

6.3.1 Controller Design and Performance . . . . . . . . . . . . . . . . . . . 146

6.3.2 The Parameter k . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149

6.3.3 Controller Limitations . . . . . . . . . . . . . . . . . . . . . . . . . . 150

7 Summary and Further Work 151

7.1 Modelling Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151

7.1.1 Further Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151

7.2 Pf-droop . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152

Contents viii

7.3 Controller Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152

7.3.1 Further Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152

7.4 Grid Connected Storage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153

7.4.1 Further Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153

A MATLAB/Simulink Diagrams 154

A.1 Simulation using Developed Model Library . . . . . . . . . . . . . . . . . . 154

A.1.1 Top Level Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . 155

A.1.2 Source Block . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156

A.1.2.1 Calculations Block . . . . . . . . . . . . . . . . . . . . . . . 156

A.1.2.2 Power-frequency Droop . . . . . . . . . . . . . . . . . . . . 156

A.1.2.3 Voltage Control . . . . . . . . . . . . . . . . . . . . . . . . 157

A.1.2.4 Inverter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158

A.1.3 Coupling Inductance . . . . . . . . . . . . . . . . . . . . . . . . . . . 158

A.1.4 Voltage Magnitude Calculation . . . . . . . . . . . . . . . . . . . . . 158

A.1.5 Frequency Calculation . . . . . . . . . . . . . . . . . . . . . . . . . . 159

A.1.6 Reactive Load . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160

A.2 Harmonic Simulation using Developed Model Library . . . . . . . . . . . . . 161


A.2.2 Source Block . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162

A.2.2.1 Calculations Block . . . . . . . . . . . . . . . . . . . . . . . 162

A.2.2.2 Inverter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163

A.2.3 Coupling Inductance . . . . . . . . . . . . . . . . . . . . . . . . . . . 164

A.2.4 Rectifier Load . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164

A.3 SimPowerSystems Unbalanced Load Simulation . . . . . . . . . . . . . . . . 165


A.3.2 Source Block . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166

Contents ix

A.3.2.1 Voltage Magnitude Calculation . . . . . . . . . . . . . . . 166


A.3.2.3 Ideal Inverter . . . . . . . . . . . . . . . . . . . . . . . . . . 167

A.3.3 Signal to Voltage Conversion . . . . . . . . . . . . . . . . . . . . . . 168

A.3.4 Load Block . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168

A.4 Power Control Comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . 170


A.4.2 Source Block . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171

A.4.2.1 Power-Frequency Droop Control Block . . . . . . . . . . . 171

A.5 Simulation with Unbalance Control . . . . . . . . . . . . . . . . . . . . . . . 172


A.5.2 Source Block . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173

A.5.2.1 Power-Frequency Droop . . . . . . . . . . . . . . . . . . . . 173


A.5.2.3 Unbalance Control . . . . . . . . . . . . . . . . . . . . . . . 174

A.5.2.4 Ideal Inverter . . . . . . . . . . . . . . . . . . . . . . . . . . 175

A.5.3 Frequency Calculation Block . . . . . . . . . . . . . . . . . . . . . . 175

A.5.4 Symmetrical Components Calculation Block . . . . . . . . . . . . . . 176

A.6 New Voltage Source . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177


A.6.1.1 Power-Frequency Droop . . . . . . . . . . . . . . . . . . . . 178


A.6.1.3 Unbalance Control . . . . . . . . . . . . . . . . . . . . . . . 179

A.6.1.4 Load Block . . . . . . . . . . . . . . . . . . . . . . . . . . . 180

A.7 Simulation to Investigate Power Sources using Developed Model Library . . 181


Contents x

A.7.2 Source Block (Micro-turbine and Ideal) . . . . . . . . . . . . . . . . 182

A.7.2.1 Power-frequency Droop . . . . . . . . . . . . . . . . . . . . 182

A.7.2.2 Microturbine and Ideal Source Block . . . . . . . . . . . . 183

A.7.2.3 Inverter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183

A.7.3 AC Storage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183

A.7.3.1 Power-frequency Droop for AC Storage Block . . . . . . . . 183

A.7.3.2 DC Source . . . . . . . . . . . . . . . . . . . . . . . . . . . 184

A.7.4 Load Block . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185

Bibliography 186

Abstract

The Microgrid has been proposed as a way of combining distributed generation which can

have a positive impact on power quality. To solve many of the challenges a microgrid

presents, transient modelling is important. This work develops a library for creating a

transient model of a microgrid based on transfer function representations of microgrid

components, and drawing from techniques to implement simple models of inverters. The

developed library allows for the fast, accurate simulation of multiple parallel sources con-

nected to a single load through coupling inductances.

Typically, in a microgrid, a controller with a power-frequency droop is used in the sources.

This allows for power sharing when the microgrid is in island mode. In this work, the model

library was used to develop simulations which illustrated the known problems with the

basic droop control implementation and some of the solutions proposed in the literature.

One area which had not been addressed in detail in relation to droop control, was the high

likelihood of unbalanced loads in a stand-alone system. In this work, a modification to the

droop controller was developed which allowed for the reduction of voltage imbalance in the

presence of unbalanced loads. The controller calculates the amount of negative sequence

voltage at the load, and compensates for this at the source. This unbalance controller was

then integrated into a new controller developed to address many of the weaknesses of the

standard droop control configuration.

Due to the lack of inertia in the power sources, the typical droop controller setup requires

storage on the DC side of the inverter (typically a battery) in order to allow for step changes

in load demand. In this work, a controller was developed for an AC grid connected storage

device (such as a flywheel) which was effective in reducing the need for individual storage

for each source. The advantage to this approach is that the grid connected storage device

can be chosen to be more environmentally friendly than the batteries which are typically

used for the DC storage.

Chapter 1

Introduction to Distributed

Generation and Microgrids

This thesis presents a method for modelling inverter sources within a low voltage dis-

tributed generation system. A control strategy is developed that addresses the major

challenges unique to low voltage distributed generation systems. These include the high

likelihood of unbalanced loads, small impedances between sources and the need for power

sharing, voltage regulation and storage in island operation.

1.1 Project Context and Description

This research has been performed in conjunction with Australia’s Commonwealth Scientific

and Industrial Research Organisation (CSIRO) Energy Centre, Newcastle. The Energy

Centre has a small low voltage distributed generation system which provides power for the

Energy Centre buildings [1]. It consists of microturbines (120kW), photovoltaics (90kW)

and wind turbines (60kW) and can operate both connected to the main power grid, or in

stand-alone mode.

The motivation for installing this generation was to facilitate research into microgrids. Due

to the large number of inverters contained in the PV arrays, a particular area of interest

is the interactions between the controllers of these inverters. In order to investigate this,

models were developed which allowed simulation of a simple microgrid containing several

sources, in a realistic timeframe. These models were then used to examine the system

behaviour when using droop control techniques. This particular type of control was chosen

1. Distributed Generation and Microgrids 2

as it is the most common technique proposed for use in microgrids, due to its ability to

share power amongst different sized sources, and regulate voltage levels, without the need

for communications between devices [2–6]. This investigation led to the development of an

improved droop controller which incorporates a technique for minimising voltage imbalance

at the load. This research also investigated the feasibility of reducing the dependence on

batteries in microgrid sources by directly connecting centralised storage to the microgrid.

A controller for a centralised storage device was designed to achieve this aim.

The following sections provide an overview of distributed generation and microgrids, with

more specific information on modelling and inverter control techniques.

1.2 Distributed Generation - An Historical Perspective

Distributed Generation (DG) is defined as “electric power generation within the distribu-

tion network or on the customer side of the network” [7]. It is generally in the form of

small sources such as wind turbines, solar cells, fuel cells and microturbines or gas tur-

bines, with generation typically in the kW to MW range. This contrasts with traditional

power generation that usually occurs at centralised power plants which produce power in

the hundreds of MW to GW range. This research focusses on low voltage distributed gen-

eration (LVDG) where the sources connect to the low voltage (< 1000Vrms) distribution

network.

DG is becoming increasingly important as demand for power grows, and solutions are

sought that are less harmful to the environment, and produce better quality power. Typ-

ically DG has been deployed as stand alone units connected directly to the grid (possibly

through power electronics), or groups of sources (such as a wind park) combined to pro-

duce higher voltages. The focus has been on providing power in remote areas, or increasing

the overall capacity of the power grid.

DG potentially has several advantages over standard power generation [8]. These include:


• The possibility of combined heat and power (CHP) applications, increasing overall

efficiency

• Reduction in transmission losses and lower capacity requirements in lines, trans-

formers and other transmission system components due to the reduction in power

being supplied from centralised generators

• Low cost and quicker implementation meaning major distribution network upgrades

can be delayed

• Peak power shaving

• Environmental benefits if renewable, or low emissions technologies are used

• Provision of power to remote areas

Conversely, DG can lead to problems which degrade the quality of the power supply

[9]. DG can interfere with protection mechanisms [10], decrease system stability [11] and

increase harmonic levels [12]. DG has typically been seen as a “problem” to be solved, and

researchers have calculated maximum penetration levels before grid instability was likely

to occur [11–13]. In an attempt to overcome these issues, researchers in the US [14, 15]

have proposed grouping different microsources together with local loads as a unit which

can operate in stand alone or grid-connected mode. This concept has been termed a

‘microgrid’.

Figure 1.1 shows a single line diagram of a simple microgrid system. A microgrid is a

collection of microsources and loads grouped together in one location. It interfaces to the

main power grid at one point (commonly referred to as a Point of Common Coupling)

through a switch and appears to the power grid as a single entity. Generally it operates in

grid-connected mode, but if the main power grid is unavailable (due to a power outage),

the microgrid disconnects from the main power grid and operates on its own to maintain

uninterrupted power to the loads (providing there is sufficient local generation, or that

non-critical loads can be shed).


SensitiveLoad

SensitiveLoad Source

SensitiveLoad

Switch(to IsolateSensitiveLoads)

Source

Main Power Grid

Non-Senstive

Load

Non-Sensitive

Load

Point ofCommonCoupling

Figure 1.1: Simple microgrid

The potential advantages of the microgrid are [14,15]:

• Microgrids offer a solution to provide power to remote areas.

• The scale of the generation units means that the customer can own his own power

generation, which can offer financial savings if the technology can be operated more

cheaply than the market price of electricity.

• The microgrid can be set up with sufficient capacity that it can offer Uninterruptible

Power Supply (UPS) type functionality.

• The power electronics that are used to interface microsources to the microgrid are

such that they can be controlled to offer superior power quality in terms of limiting

voltage sags, phase imbalances and harmonics, for customers with sensitive loads.

The most common proposal is to use droop characteristics to allow independent

control of each of the microsources.

• The proximity of generation to loads means that combined heat and power applica-

tions are possible, increasing the overall efficiency of the generators.

The microgrid is not an entirely new concept. Stand-alone low voltage distributed genera-

tion has been around for decades, particularly in remote areas where extensions to existing


power grids were not considered feasible. This was most commonly in the form of wind

turbines, which have been around since the late 1800s [16]. In the US, wind turbines were

used to provide power on many remote rural properties until the rural electrification of the

1930s. More recently, systems have been built which combine several types of generation

with some form of storage [17, 18]. What differentiates the microgrid from these systems

is that it is designed to work in conjunction with the main power grid most of the time, to

seamlessly disconnect in the case of a fault, and to provide high quality power to sensitive

loads.

The microgrid concept has generated significant interest and several research groups have

formed projects to further develop the concept. International groups working in this area

include the Consortium for Electric Reliability Technology Solutions (CERTS) [19], and

the IRED (“Integration of Renewable Energy Sources and Distributed Generation into the

European Electricity Grid”) cluster [20].

CERTS is a group of co-operating universities, commercial companies and research labo-

ratories in the US sponsored by the US Department of Energy and the California Energy

Commission. CERTS has performed considerable work outlining the need for microgrids,

and their proposal for the basic structure of the microgrid [21, 22]. Some of the areas

they have looked at include control strategies [3], protection [23] and customer adoption

models [24]. CERTS has placed a lot of importance on implementing a working microgrid

and have made significant progress towards this goal, including a working demonstration

system operated by American Electric Power [25]. Understandably this means a lot of

their energy has focussed on single solutions rather than investigating a wide range of

possibilities. Detailed descriptions are available for many aspects of the project, and for

this reason, the CERTS project is frequently referred to in this work.

The IRED cluster is a series of projects funded by the European Commission and co-

ordinated by the Institut fuer Solare Energieversorgungstechnik e.V (ISET). Of interest

to this research is the now completed ‘Microgrids’ [26] and ‘More Microgrids’ [27] projects.


Like the CERTS project, the initial Microgrids project was primarily focussed on achieving

a prototype microgrid which can be demonstrated in a laboratory. To this end, research

was performed in a wide range of areas including modelling and simulation [28], con-

trol [2,29,30], protection [31,32], communications requirements, laboratory testing [33,34]

and regulatory, commercial, economic and environmental issues [35]. Further to this, the

More Microgrids project expands on this research and investigates alternative solutions

(e.g DC microgrids, centralised vs. decentralised control). Unlike the CERTS project,

these projects focussed more on research of various alternatives than on building a spe-

cific solution. Rather than focus on a single large scale implementation of a microgrid,

the projects include several pilot installations, both in the laboratory, and in real world

distribution networks.

1.3 The Microgrid Concept - Technical Challenges

Researchers quickly discovered that the nature of the microgrid is such that its behaviour

is not like that of the main power grid [36]:

• Different types of power sources are likely to be used [37]. These sources are gener-

ally small distributed generation sources which connect to the grid through power

electronics such as inverters. As such they have little or no rotating inertia, meaning

that they cannot quickly adjust their power output to provide for load changes. For

stand-alone microgrid operation this means that some form of storage is required

to provide the transient power during load changes. To compound this problem,

renewable sources such as wind turbines or photovoltaic arrays are not dispatchable,

meaning they cannot be relied upon to produce power on demand.

• The power electronics interfaces require different control strategies than in a non-

microgrid environment - in stand-alone mode, the inverter controllers will be required

to regulate the voltage and to share power amongst the sources. The smaller sources

also affect protection strategies, because generally the fault currents are not high


enough for over current protection, so alternatives are needed [38].

• The nature of the loads present may differ [39, 40]. Unbalanced loads are far more

likely, since the usual approach of apportioning loads between the phases such that

the loads are virtually the same on each phase, relies on the large number of loads

in a typical power grid.

• Significant harmonics are a possibility, as many office loads, such as computer equip-

ment, lighting and motors, produce high levels of harmonics. The converters used

to interface distributed generation with the grid also produce harmonics.

• Transmission lines are different in a microgrid [2,41]. This is due to both the shorter

distances involved, and different physical cabling used in low voltage distribution

networks. Small line impedances between sources can allow sizeable reactive currents

to circulate between the sources, and other interactions are possible between the line

impedances and the reactances in the inverters. The line impedances also tend to

be more resistive than reactive.

• If the microgrid is to operate in stand-alone mode, then there needs to be systems

for disconnecting from the main power grid, and resynchronising when a fault event

is over [42]. The safety of line maintenance staff needs to be considered by making

sure lines are not active when they shouldn’t be.

• There are challenges in the relationship between the owner of the microgrid and the

power distributor, both regulatory, and financial.

These differences mean that the behaviour of a microgrid is not fully understood. There

are a number of challenges to be met before widespread adoption of the concept is possible.

Microgrid research can be broken up into the following areas:

• Regulatory constraints

• Market strategies


• Protection

• Technologies, including power sources and storage

• Transient and steady state models

• Control algorithms

• Load management (also known as demand side management)

• Systems for disconnecting and reconnecting to main power grid

This research will focus on issues surrounding the modelling and control of microgrids.

1.4 Microgrid Models

In order to evaluate the performance of microgrids both transient and steady state models

are required. Transient models are more applicable when considering inverter controllers,

however steady state modelling will be briefly discussed for completeness.

The challenges for developing steady state models of microgrids include sources which

interface via power electronic devices, load unbalance and the combination of single phase

and 3-phase generation and loads [43]. Meliopoulos et al. [44] propose a model which

addresses these concerns using multiphase power flow analysis. A distributed slack bus

model is proposed to model distributed generation in the distribution system [45].

Transient modelling is important when it comes to examining problems like the production

of harmonics, voltage sags and spikes and other short lived phenomenon. The major focus

of transient modelling work has been on the various components of a microgrid. Wind

turbines, both with doubly-fed induction generators [46] and self-excited induction gener-

ators [47] have been modelled in several papers. Photovoltaics have also been modelled

significantly. The most common method are the single diode [48, 49] or double diode [50]

equivalent circuit models. Models can be found for inverters [22], rectifiers, microturbines,


fuel cells [51], loads and transmission lines. There have also been quite a few efforts to

model partial systems (see [52] for a model of a system with diesel, microturbine and

fuel cell). The problem with the majority of these models, however, is that they are too

detailed to allow simulation of reasonably sized systems.

CERTS has proposed several simplifications which go some way to addressing this problem.

In [22], a technique is outlined for modelling a simplified inverter. The model is designed

assuming balanced 3 phase operation and no inverter switching dynamics. The document

also outlines a technique for creating simplified models of microturbines and fuel cells.

These maximum power models are based on the idea that the sources can be assumed to

behave like fixed DC sources except during changes to power demands i.e. during load

transients. At these times the models differ depending on the specific source. A further

simplification is to assume sufficient battery capacity on the DC side of the inverter so

that the source can be modelled as an ideal DC power source.

Noticeably lacking in current models is a way of modelling the main grid. One of the pos-

sible benefits of a microgrid is that it provides a way of connecting distributed generation

to the main power grid without degrading the power quality. Therefore the effects of the

microgrid on the main power grid need to be modelled in some way. Generally the main

power grid is assumed to be a stiff voltage, with fixed magnitude and frequency. This

model does not allow for any observation of possible problems that the microgrid may

cause to the main power grid, particularly in the case of a weak grid.

There are software and libraries available that can be used for transient modelling such as

the MATLAB/Simulink SimPowerSystems library and RPM-SIM which were investigated

as possible modelling platforms.

RPM-SIM [53] was created by The National Renewable Energy Laboratory and is a library

of components in a program called VisSim. This library includes the following components:

Point of Common Coupling, Diesel Generator, Rotary Converter with Battery Bank, Wind

Turbine, Village Load, Dump Load, PV Array, Transmission Line Impedance and Power


Factor Correction measures.

MATLAB/Simulink was chosen, as it is a platform which is commonly available and

familiar to a lot of people, along with possessing the required flexibility. Some SimPow-

erSystems components were incorporated into the model as these simplified some aspects

of the modelling.

1.5 Control Algorithms

A key area in microgrid research is the development of new control techniques. Since

most of the technologies being considered for use in microgrids connect through power

electronics, these techniques are generally aimed at inverter control.

Control algorithms for parallel inverters can be categorised into two main types - those

that use a communications medium (either wired or wireless) to carry control signals to

each of the inverters, and those that operate without external control signals, using only

information that is locally available [54]. The majority of systems which utilise communi-

cations have some kind of central controller, which introduces a single point of failure in

the system. Also this kind of control is only as reliable as the system carrying the signals.

If the system is sufficiently distributed geographically, the delay in communications could

become significant. In some cases [55], the electrical power system itself is used to carry

control signals. This removes an extra point of failure from the system, however, power

lines with inverter based sources are not an ideal communication environment.

To avoid the problems with communications and the single point of failure in a system,

control algorithms which only require local information and allow sources to act indepen-

dently are therefore desirable. In some cases it is beneficial to employ the two types of

control in conjunction with each other, provided the individual units can still function in-

dependently in the event of a communications failure. In this case a centralised controller

would provide slow transients like power and voltage setpoints, and the fast transients


(the inverter voltage magnitude and phase) would be determined independently by the

source. In this case the purpose of the external control would be to provide the cheapest,

most fuel efficient, or most environmentally friendly power, depending on the aims of the

producer. Algorithms have been proposed to achieve some of these aims [56].

The main approach to achieving independent, local control is to use variations of the droop

control method. Droop control is based on the idea that when the output impedance is

mainly inductive, reactive power and voltage magnitude are strongly coupled, as are real

power and voltage angle [3]. The standard droop relationships are based on controlling real

and reactive power and employ power vs frequency (P-f) droop (Equation 1.1) and reactive

power vs inverter voltage (Q-V) droop (Equation 1.2), where ωi(t) is the frequency output

by the ith inverter at time t, w∗ is the nominal or base system frequency (generally 50Hz or

60Hz), Pi(t) is the power output of the source i at time t, and mi is the droop co-efficient,

or the gradient of the power-frequency droop characteristic, for source i. Vi(t) is the desired

RMS inverter voltage magnitude at time t, V ∗ is the inverter voltage magnitude setpoint

(or the desired RMS voltage at the load), ni is the reactive power droop co-efficient, and

Qi(t) is the reactive power output by the source at time t.

ωi(t) = ω∗ −miPi(t)) (1.1)

Vi(t) = V ∗ − niQi(t) (1.2)

In a microgrid environment it is necessary to regulate the system voltage and so a voltage

regulator (e.g. Equation 1.4) needs to be employed, wherekps+ki

s is a PI controller, Ei(t)

is the measured RMS load voltage magnitude at time t, and E∗ is the desired RMS

load voltage setpoint. Equation 1.3 shows the controller using the power-frequency droop

relationship, where δvi(t) is the desired inverter voltage phasor angle at time t. The


addition of the P ∗i term provides for a power setpoint in grid-connected mode.

δvi(t) =

ˆ(ω∗ −mi(Pi(t)− P ∗i )) (1.3)

Vi(t) =kps+ ki

s(E∗ − Ei(t)) (1.4)

CERTS has chosen to use an inverter controller with voltage and frequency droop (P-V

droop control) with the addition of a Q-E droop (reactive power vs grid voltage magnitude)

to the voltage controller in order to improve reactive power sharing. This is essentially a

combination of Equation 1.2 and Equation 1.4. The EU Microgrids project also chose a

droop controller for the inverter control, however in a slightly different configuration to

the CERTS controller. In this case the inverter output voltage was regulated to control

the flows of reactive power and prevent reactive currents from exceeding inverter ratings.

Voltage control is achieved by suitable layout of the low voltage grid, and by chokes. Power

sharing was achieved with a power frequency droop, similar to the CERTS controller.

Droop control has the advantage that it is compatible with standard synchronous generator

control. Problems with droop control include steady state error in the frequency output

by the inverter, slow transient response, the possibility of circulating reactive currents,

amplitude variations, poor unbalanced harmonic current sharing, high dependency on

output impedance and lack of compensation for unbalanced loads or grid voltages [57].

Several controllers have been proposed to alleviate these problems. To solve the prob-

lem of the steady state frequency error, Lasseter and Piagi [3] have added a frequency

restoration loop that restores frequency once power sharing has been achieved. The fre-

quency restoration must operate much slower than the power sharing loop (typically 10s

of seconds), so the frequency spends significant amounts of times deviated from the base

frequency. In order to address the problem of sharing harmonic power between different

sized sources, Lee and Cheng [58] have introduced a droop between harmonic conductance

and harmonic VAr consumption (G-H droop).


Another approach to droop control is to use a control loop to alter the output impedance.

A parallel combination of resistor and inductor allows good sharing of linear and non-

linear loads. Guerrero et al. have developed a variation on this idea which they claim

provides good output impedance response, good power sharing, fast transient response and

stable output voltage frequency and amplitude [4]. Their design consists of a transient

frequency droop loop for controlling active power and an adaptive output impedance

loop for controlling reactive power. This output impedance loop also gives improved

performance when ‘hot swapping’ components and provides non-linear load sharing.

One area that has received little attention is dealing with load unbalance while using a

droop controller. CERTS’ approach to unbalance [21] is to filter the voltage and current

signals so that the second harmonic oscillations caused by the negative sequence com-

ponents (i.e. the unbalance) do not affect the inverter controller. If the grid unbalance

becomes too large then the microgrid simply disconnects itself. The possibility of injecting

negative sequence currents to address imbalance is mentioned, but is discarded as requiring

additional complexity in the controller. While the former solution is effective for dealing

with unbalanced grid voltages, it does nothing to address unbalanced voltages which oc-

cur due to load imbalance within the microgrid itself. Even small amounts of unbalance

can decrease the lifespan of motors due to large negative sequence currents, and create

a need for oversized components in non-linear loads such as static converters due to the

low frequency harmonics. Several solutions for improving unbalance have been proposed,

most of which are based around measuring the negative sequence components generated

by the unbalance. Solutions include static VAR generators [59], active power filters [60–62]

and measures which use current and voltage source inverters [63]. Controllers have been

proposed to improve the performance of 3 phase PWM converters in the presence of un-

balanced input voltages [64, 65]. A combination of series/shunt inverters is proposed [66]

to correct for unbalance at the point of connection to the power grid in a microgrid. A

lot of the focus is on inverters which compensate for unbalanced supplies. There are some

solutions which address unbalanced loads/line voltages, however, until recently [67] these


had not been applied to droop controlled inverters, and so lacked the voltage regulation

and power sharing functionality required in a microgrid. Cheng et al. [67] have proposed

using a Q−-G droop (the reactive power produced by the negative sequence current vs

conductance) in conjunction with standard droop control to achieve these three aims.

In the case where energy storage devices (e.g. flywheels, battery banks, supercapacitors)

are connected directly to the common AC bus in the microgrid, control techniques are

needed for these devices. This is a separate problem to controlling power sources as the

storage device needs to supply power quickly during load transients, but should supply no

power during steady state. Jayawarna et al. [68] propose a controller for a flywheel which

allows the flywheel to supply sufficient fault currents when the microgrid is in islanded

mode. This controller is effective, but other techniques are required to apply to storage

devices generally, and for the simpler case where fault currents are not supplied by the

energy storage.

1.6 Other Research

There are several other areas of microgrid research which are outside the scope of this

work, but are mentioned here in brief.

One of these research areas is the connection with the main power grid. In [69] the

transients which occur when the microgrid switches to stand-alone mode are examined,

along with the ability of properly controlled power electronics to minimise the effect of

these transients. Other issues include when to disconnect the microgrid from the main

power grid and how to resynchronise the microgrid voltages with the power grid when the

fault has been removed. CERTS proposes exploiting the frequency difference between the

main grid and the microgrid, which is introduced by the power-frequency droop control,

in order to minimise the voltage across the switch as the connection is re-established [21].

Several authors have discussed the applicability of various types of technologies to the


microgrid concept. In [70] the benefits of using a mini gas turbine with high speed axial

flux generator in a microgrid are examined and the gas turbine is compared to a micro-

turbine. Mini gas turbines are suitable for combined heat and power (CHP) applications

and can provide electricity much cheaper than a microturbine and at a higher efficiency.

In [71] fuel cells are discussed including their impact on protection schemes. Fuel cells

are also suitable for CHP. A disadvantage of fuel cells however, is that changes in power

output cannot be achieved instantaneously so they generally are used in conjunction with

some form of storage. In [72] the suitability of wind and PV for stand-alone applications

are examined. These renewable sources offer the advantage that no fuel is needed, and

minimal environmental damage is caused. The disadvantage is that renewable resources

cannot be controlled and substantial storage, or backup generation is required for reli-

able generation. Also renewable sources can only be situated where the environment is

suitable i.e. locations with sufficient wind, sun etc. The storage requirements of these

various technologies can be met in various ways, including various types of batteries [73],

supercapacitors [74] and AC storage such as flywheels [75,76].

Load control is important in a microgrid in order to minimise storage requirements. The

idea is that non-essential loads can be interrupted in the case of a fault in the main

power grid where the microgrid is forced to operate in stand-alone mode. The loads may

then stay disconnected until the connection to the main power grid is restored, or may

be reconnected as generating capability comes online in the microgrid. What defines a

non-essential load may differ depending on the situation. Standard refrigeration, heating

and cooling can be switched off for minutes at a time with no noticeable effect, an office

environment using laptops and with sufficient natural lighting might consider standard

office lighting and power non-essential. The main focus on load control and load shedding

has been on increasing the stability of the main power grid by shedding loads based on

grid frequency [77] or voltage levels [78]. These measures are designed to prevent the grid

from shutting down under high load conditions, and should the grid shut down, to increase

effectiveness of cold starts. There is, however, some research which focusses on minimising


storage in stand-alone systems [79].

Protection in microgrids is another issue which needs investigation. Standard protection

strategies generally utilise overcurrent protection to detect and isolate a fault. This relies

on the fault current being substantially higher than normal current levels, which is unlikely

when dealing with the smaller sources proposed in a microgrid. The reason for this is that

protection measures in the power electronics generally limit the output current to about

twice the rated output current. In [80] Al-Nasseri et al. outline a fault detection scheme,

based on converting signals to the d-q reference frame, which can overcome this problem

and also detect whether a fault lies inside or outside a pre-defined zone. Other options for

protection in a microgrid include differential protection and zero sequence detection.

1.7 Thesis Overview

Chapter 2 outlines the techniques used to develop a model library in MATLAB/Simulink to

address the problem of simulating multiple parallel inverters in a reasonable timeframe. In

Chapter 3 this model is used to examine the droop controller in detail, including significant

weaknesses and possible solutions. Chapter 4 addresses the absence of solutions for dealing

with unbalanced loads in distributed generation by developing a controller which works

alongside a droop controller to minimise voltage imbalance in the presence of unbalanced

loads. In Chapter 5, the unbalance controller is incorporated into a new controller design

which draws ideas from various existing techniques to address many of the weaknesses with

the basic droop controller. This new controller is evaluated by simulation and performs

well under a wide range of conditions. Chapter 6 examines the need for storage in stand-

alone systems due to the low inertia of typical distributed generation systems. Techniques

are outlined for modelling these limitations, and a controller developed for a grid-connected

storage device which reduces the requirement for storage on the DC bus of each source.

Chapter 7 provides a summary of this work, and examines areas where further work is

required.

Chapter 2

Microgrid Transient Modelling

2.1 Overview

This chapter examines the typical structure of a microgrid and the characteristics of

the sources found within a microgrid. It then describes the method in which MAT-

LAB/Simulink has been utilised to create a simple microgrid model. This model is for the

purposes of testing control strategies for inverters, which are typically used to connect the

sources to the microgrid. An important part of this work has been to create an inverter

source model simple enough that it can be used to simulate a large system with several

inverters. To this end, the techniques outlined in [81], [82] and [5] have been used. Another

aspect of this work has been the unique approach of modelling coupling/line impedances

and loads as transfer functions relating voltages and currents in the d-q reference frame.

The benefit of the d-q reference frame is that it allows sinusoidal voltages and currents to

be represented as DC quantities, providing a further increase in simulation speeds.

SensitiveLoad

SensitiveLoad Source

SensitiveLoad

Switch(to IsolateSensitiveLoads)

Source

Main Power Grid

Non-Senstive

Load

Non-Sensitive

Load

Point ofCommonCoupling

Figure 2.1: Typical microgrid system

2. Microgrid Transient Modelling 18

Source/Inverter 1

Source/Inverter 3 L

oad

Source/Inverter 2

Source/Inverter 4

Source/Inverter 5

Figure 2.2: Multi-inverter system with resistive load.

Figure 2.1 shows a generic representation of a microgrid. The important elements are

the connection to the main power grid, a static switch, the sources, and the loads. The

microgrid connects to the main power grid through a single point, called the Point of

Common Coupling (PCC). Sensitive and non-sensitive loads are placed on separate buses,

and the switch allows the sensitive loads to be disconnected from the main power grid in

the event of a power quality disruption. The non-sensitive loads are left to ride through

the disturbance. The sources are typically small (¡500kW) and connect to the microgrid

through power electronics. Examples include wind turbines, photovoltaics and micro or

gas turbines. The loads are most likely those of a typical commercial or residential building

e.g. lighting, air-conditioning and refrigeration, although it could include industrial plant.

The main purpose of this modelling is to examine the interactions between the inverter

controllers, particularly the power-frequency droop control. For this reason, the behaviour

of the system in stand-alone mode is of particular interest, therefore the modelling focuses

on the sources and the loads. To simplify the models, the loads are considered to be one

lumped load, and the sources connect at a single point (through coupling impedances).

The line impedances are ignored, as the sources and loads are in close proximity. The small

distances involved mean that the cable impedances are small compared to the coupling

impedances.

Figure 2.2 shows a single line diagram of a multi-inverter system which represents a sim-

plified microgrid in stand-alone mode. Multiple sources are connected in parallel via


DC Source

+OutputFilter

DC Storage Inverter Coupling Inductance

V ∠δV E∠δE

Figure 2.3: Typical Microsource

Voltage Controller

Power-frequency

Droop Control

Inverter

δv

V

E vdq0Calculationsidq0

vdq0

E∗

P ∗

P

edq0

Figure 2.4: Model of Voltage Source

inductances to feed a load. This system is the general configuration used for testing

purposes and is used for most of the simulations described in Chapter 3. Different load

configurations are available as well as various source configurations, including an ideal volt-

age source which can used be to simulate a stiff grid connection. The following sections

describe the theory behind the various components of the model.

2.2 Source Structure and Models

The representation of a typical Source is shown in Figure 2.3. It is defined as including

the Power Source (e.g. Fuel Cell, Microturbine), possible storage on the DC bus (battery

or capacitors), the inverter (and associated control), an output filter, and a coupling

inductance to decouple the inverter from the grid. The Power Source is represented here

as a DC Source, and it is assumed that if the source is AC, then AC/DC conversion is

encompassed by this block.

A functional block diagram of the simulation model of a Source is shown in Figure 2.4.

This block diagram corresponds to the Simulink implementation shown in Figure A.2

(Appendix A). The Source model developed in Simulink is a single block with Current

and Voltage Measurements as inputs and the outputs are the inverter voltages and the


source real and reactive power measurements (for display only). This model represents the

physical components of the Source shown in Figure 2.3, excluding the coupling inductance

which is modelled in a separate block. Voltages and currents are fed to a Calculations

block, which calculates the feedback values required by the controllers. These controllers

determine the desired inverter output voltage magnitude and phase. These values are

then used by the inverter to produce the output voltages for the source. The simplifying

assumption being made here is that the inverter is able to produce the exact voltage

requested by the control. This ignores the inverter switching harmonics. This assumption

is made as the switching frequency (generally in the kHz range) is much greater than the

line frequencies (50, 60Hz) under consideration, meaning the harmonic content is mostly

removed by the inverter output filter. The inverter output voltages then feed into the

coupling inductance (not shown). The Power Source and DC Storage are modelled as

part of the Power-frequency Droop Control block. These are examined in more detail in

Chapter 6. For most of this work, the assumption is made that the DC source is ideal, aside

from a limit on the maximum amount of power available from the source. This assumption

relies on the presence of sufficient storage on the DC bus to provide the transient power

demand in the event of a load change.

2.2.1 Inverter Model

An inverter is used to convert DC voltages and currents into AC by rapidly switching

power electronic devices. Since most distributed generation sources produce either DC

voltages, or AC voltages at a frequency incompatible with the grid, inverters are key

to interfacing DG to the grid. The inverter switching electronics can be controlled to

give the desired voltage waveform in terms of magnitude and phase, allowing control of

load voltage and power. In this work, only voltage source inverters are being considered.

Current source inverters are not considered here as they are not as well suited to the

application. A current source inverter cannot act independently of other sources, and

thus cannot perform the desired voltage regulation in stand alone mode.


Time domain voltageCalculation abc - dq transform

V

!v

va

vb

vc

vd

vq

v0

Figure 2.5: Ideal Inverter

It is possible to simulate an inverter down to the component level [81] in electronic sim-

ulators such as SABRE or SPICE. However, this generally results in a model which is

relatively slow to simulate. Since this work is concerned with simulating several inverters

in parallel, and is mainly interested in the interactions between the inverter controllers, a

simplified model is used instead. In a physical inverter system, measurements are used to

calculate feedback values (e.g. voltage magnitude) required by the control system. Using

these calculated values, the control system generates desired values of voltage magnitude

and phase which the inverter switching algorithm uses to create a voltage waveform. The

proposed model ignores the switching process in the inverter, assuming it to be ideal in

the respect that the inverter can generate the exact voltages requested by the controllers.

These voltages are assumed to be balanced (i.e. of the same magnitude and 1200 out

of phase). This assumption means that a different model is necessary if the inverter is

required to generate unbalanced voltages. The inverter output is limited by the amount

of power available from the power source and in that respect the inverter differs from an

ideal voltage source. This model is based on the CERTS model [81], however the flux

quantities have been replaced with voltages. The CERTS model is a simplified version

of a full inverter model based on a voltage flux control strategy and so the use of flux

quantities was carried through to their simplified model. Voltage quantities are used here

instead, because these are more easily recognised.

The ideal inverter block (labelled as Inverter in Figure 2.4) takes as inputs the voltage

magnitude and angle requested by the control, and outputs voltages in the d-q reference

frame (see Section 2.2.1.1 for an explanation of the d-q transform). It theoretically consists

of two blocks (see Figure 2.5), one which calculates the time domain voltages from voltage

magnitude and phase, and system frequency (as given in Equations 2.1, 2.2 and 2.3) and


the other is a simple abc-dq transformation in the synchronously rotating reference frame.

Note that V and δv are the outputs from the voltage and power control blocks (refer to

Chapter 3 for more details). In practice the calculations have been simplified to convert

directly from the V , δv inputs to the vdq0 outputs (this is described further in Section

2.2.1.1). This approach loses some flexibility e.g. the inverter output frequency cannot be

easily altered, however it requires less calculations.

va(t) = V cos(ωt+ δv) (2.1)

vb(t) = V cos(ωt+ δv −2

3π) (2.2)

vc(t) = V cos(ωt+ δv +2

3π) (2.3)

2.2.1.1 D-Q Transform

The simulation described in this section has been modelled in a synchronously rotating d-q

reference frame. The d-q transformation maps 3-phase quantities (fabc) onto two perpen-

dicular vectors, d (direct) and q (quadrature), which are rotating at a fixed speed ω (the

choice of which depends on the application), and a zero sequence vector which represents

the in phase component of the 3-phase quantities. In the case of the synchronously rotat-

ing reference frame this speed is ω = 50Hz or 314rads−1 - the standard mains frequency

in Australia. The reason for choosing the synchronously rotating reference frame is that

it simplifies calculations involving 3-phase quantities, allowing an ideal 3 phase voltage

to be represented as a DC quantity. The reader should be aware that the stationary d-q

reference frame (ω = 0) is often chosen instead.

The general equations for mapping a 3-phase quantity fabc onto the d-q axis (fdq0) are

given here (Equation 2.4) where T is the transform matrix from abc to dq0 (Equation 2.5),

θ = ωt and ω = 314rads−1 as explained above.

fdq0 = Tfabc (2.4)


a

c

b

d

q

Figure 2.6: 3 phase and d-q vectors

T =2

3

cosθ cos(θ − 2π

3 ) cos(θ + 2π3 )

−sinθ −sin(θ − 2π3 ) −sin(θ + 2π

3 )

12

12

12

(2.5)

Note that the direction of the d and q vectors is somewhat arbitrary, but the transform

equations will differ for a different choice of vectors. The abc and dq vectors corresponding

to these equations when θ = 0, can be seen in Figure 2.6.

The synchronously rotating d-q reference frame has several advantages over the standard

abc reference frame. In simulation, less frequent calculations are required because the d-q

quantities change more slowly than the 3-phase voltages they represent; the transform

separates the abc quantities into two independent vectors simplifying the design of the

model; and since the vectors change much more slowly, changes in frequency or magnitude

are much more easily observed. One disadvantage of using d-q transforms is that when

it comes to implementing the controller in hardware, extra calculations are required to

perform the transform. Another issue, when using d-q transforms in conjunction with

droop control, is that the system frequency may alter from ω = 50Hz, which causes the

d-q quantities to slowly oscillate. This can be fixed by allowing the transform frequency ω

to change so that it matches the current system frequency, however, the voltage magnitude

and frequency calculations are not affected, so this was not considered necessary for this


work. Further information on the d-q reference frame can be found in many books on

power systems or electric machinery including [83].

Equations 2.6, 2.7 and 2.8 show the conversion from the time domain inverter voltages to

the d-q synchronously rotating reference frame, where θ = ωt.

vd =2

3(vacos(θ) + vbcos(θ −

2π

3) + vccos(θ +

2π

3) (2.6)

vq = −2

3(vasin(θ) + vbsin(θ − 2π

3) + vcsin(θ +

2π

3) (2.7)

v0 =1

3(va + vb + vc) (2.8)

The frequency ω in the time domain voltage equations (2.1, 2.2 and 2.3) is fixed at 50Hz

(since δv is controlled to change the frequency of the voltage waveforms as desired), and

the d-q axes also rotate at 50Hz, therefore the time domain voltage equations (2.1, 2.2 and

2.3) can be substituted into the dq voltage equations (2.6, 2.7 and 2.8), which simplifies

to Equations 2.9, 2.10 and 2.11. This is the form of the inverter equations implemented

in the simulation (Appendix A.1.2.4).

vd = V cos(δv) (2.9)

vq = V sin(δv) (2.10)

v0 = 0 (2.11)

2.2.1.2 The Per-Unit System

The per-unit system converts measurements from an absolute value (e.g. 5kW) to a

relative value which represents the fraction of the maximum value (e.g. 0.5pu). Per-unit

values are calculated by dividing the actual value by a base value which is generally the

rating of the source.

In this model, the per-unit system is used for the power signal required by the controller,


Figure 2.7: Inverter Output Filter

and also for the display of real and reactive power values. The advantage of using the per-

unit system in this case is that it makes the controller scalable, i.e. the same controller can

be used for sources with different power ratings. It also gives more meaningful information

when viewing power outputs than the actual power in kW would. Generally when using

the per-unit system, all values in a system would be converted to per-unit values, including

currents, voltages and loads. While the per-unit system is useful for the real and reactive

power measurements, no advantage is gained from converting the rest of the system to

per-unit values and it was decided to leave the currents, voltages and loads as their actual

values.

Equation 2.12 gives the conversion from power in kW to per-unit power. To allow the

controller to be scalable, Pbase must be chosen as the rated power of the individual source.

If the sources have different power ratings then this value will differ for each source.

Ppu =PactualPbase

(2.12)

2.2.2 Output Filter

The switching electronics in an inverter generate harmonics at the switching frequency

(in fact, the output voltage from the switches is a PWM square wave). For this reason,

a filter is generally included as part of an inverter for the purposes of removing the high

frequency noise and producing sinusoidal voltages at the inverter output. Figure 2.7 shows


the typical configuration for an inverter output filter. The inductance and capacitance

values are chosen so as to give a suitable cutoff frequency for the filter, one which filters

out the switching harmonics, but has minimal effect on the output sine wave. For a 50Hz

system with a 10kHz switching frequency this would typically be around 500Hz.

Due to the use of a simplified model for the inverter, the output filter is omitted from the

model. This ignores any phase shift caused by the filter, however, it is assumed that this

will be negligible compared to the phase shift due to the coupling inductance.

2.2.3 Coupling Inductance

The coupling inductance connects the output of the inverter filter to the microgrid. It

decouples the source from the microgrid allowing the source to independently set the

voltage magnitude and phase and thus allows the operation of the voltage regulation and

power sharing control.

The size of the coupling inductance needs to be chosen based on the power output of

the source. If δP , the voltage angle across the inductor, is assumed to be small, and the

magnitudes of the inverter voltage magnitude, V , and the common bus voltage magnitude,

E, are close together, then the power transfer relationship across the coupling inductance

can be approximated as Equation 2.13 [3].

P =3

2

V E

ωLsin(δP ) (2.13)

The maximum size of the inductor is based on the requirement that the P , δP relationship

remains close to linear. This means that the maximum power output (in Watts) should

be obtainable given the maximum δP allowable. The minimum size of the inductor is

governed by the accuracy with which the inverter can synthesize the voltage angle. This

is not a hard limit, rather, the performance degrades once the inductance becomes too

small.


If we assume V and E are approximately 230V and δPmax = 300 and ω = 50 ∗ 2π radians

then the maximum inductance can be calculated as:

Lmax =3

2× 2302 × 1

2

50× 2π × Pmax=

2302

400πPmax(2.14)

For a 1kW source Lmax 1kW = 126mH. This means that a good choice of L is something

close to, but not larger than, 126mH, e.g. 100mH. For larger sources, the maximum

inductance can be obtained by dividing Lmax 1kW by the maximum power output in kW.

This is a simplified analysis, with restrictions on Q, V , I etc. ignored, but since a precise

value is not required, this method will suffice for the purposes of this research.

2.2.4 Measurement Block

The detail for the control blocks will be outlined in Chapter 3, however there are some

aspects of the controllers which are common and these will be dealt with here.

The general structure of the controller consists of a Calculations block, which feeds values

into separate power and voltage controllers (Figure 2.4). The output of these controllers is

fed to the inverter which then generates the required voltages. The required calculations

are real and reactive power (Equations 2.15 and 2.16), and the load voltage magnitude

and phase angle (Equations 2.17 and 2.18). Technically, the latter two terms are the mag-

nitude and phase of the load voltage space vector, and the terms are used interchangeably

throughout this thesis. The form of the d-q transform used in this work is not power


invariant resulting in the 32 term in the power calculations.

P =3

2(vdid + vqiq) (2.15)

Q =3

2(vqid − vdiq) (2.16)

E =√e2d + e2

q (2.17)

δe = tan−1(eqed

) (2.18)

Note that the range of the inverse tan function is −π2 < δe <

π2 , however the actual angle

δe is not restricted at all. To accommodate this, some code has been written to allow the

angle δe to continue rotating beyond these limits. This avoids problems with calculations

of frequency, since the frequency is calculated as the rate of change of the angle δe. In a

real system, the frequency is not calculated this way and it is desirable for the range of δe

to be limited since this prevents the register containing the value of δe from overflowing.

Also, as mentioned previously, the power control is performed using the per unit system,

so the power needs to be divided by the base power of the source before being fed into the

power control block. i.e. Ppu = PactualPbase

.

2.3 Modelling of Other Circuit Elements

2.3.1 Load Resistors

Resistors are governed by the equation V = IR, therefore, assuming that the load is

balanced i.e. Ra = Rb = Rc = R, a resistive load can be modelled as a gain R, with

current as the input and voltage as the output. In the case where unbalanced loads are

required, the model is a little more complicated. Equation 2.19 gives the dq0 resistance

for a general 3 phase load Rabc, where T is the transform matrix from abc to dq0 defined

earlier in Equation 2.5.

Rdq0 = TRabcT−1 (2.19)


To convert from currents to voltages the equation V = IR is again used. Equations 2.20,

2.21 and 2.22 give the resulting dq0 voltages. Note that Rdq0 is shortened to R for clarity.

The subscripts on R indicate the position in the matrix.

vd = R11id +R12iq +R13i0 (2.20)

vq = R21id +R22iq +R23i0 (2.21)

v0 = R31id +R32iq +R33i0 (2.22)

2.3.2 Inverter Coupling Inductances

Inductances are governed by the relationship that vL = L δiδt . They are slightly more com-

plicated to model in the d-q reference frame as they are frequency dependent components

and thus the rotating reference frame needs to be taken into account.

It can be shown [83] that in a rotating reference frame the equations relating voltages to

flux linkages are 2.23, 2.24 and 2.25, where s represents a derivative, ω is the frequency of

the rotating reference frame, and λd, λq, λ0 are the flux linkages.

vd = −ωλq + sλd (2.23)

vq = ωλd + sλq (2.24)

v0 = sλ0 (2.25)

Now λdq0s = TL(T )−1idq0 and for a diagonal matrix (i.e. balanced 3 phase inductance)

TL(T )−1 = L =

L 0 0

0 L 0

0 0 L

therefore substituting these two relationships into Equations 2.23, 2.24 and 2.25, and


adding an internal series resistance Rs, the equations become 2.26, 2.27 and 2.28.

vd = −ωLiq + sLid +Rsid (2.26)

vq = ωLid + sLiq +Rsiq (2.27)

v0 = sLi0 +Rsi0 (2.28)

If these equations are rewritten as Equation 2.29 then the voltage across the inductor can

be determined in terms of the current flowing through it. Since, in the case of the inverter

coupling inductances, it is desired to calculate the current flowing through the inductor,

the equation is rearranged to give Equation 2.30, noting that the resulting negative has

been incorporated into the matrix. This equation forms the basis for the simulation

implementation of the coupling inductance shown in Figure A.6 (Appendix A). VT is the

total voltage across the inductor Vin − Vout, VR = Rsidq0 is the voltage drop across the

inductor’s equivalent series resistance, and ω is the frequency of the rotating reference

frame in radians e.g. 100π radians (50Hz).

vdq0 = Ls(sidq0 + ωidq0

0 −1 0

1 0 0

0 0 0

) +Rsidq0 (2.29)

idq0 =1

s∗ (idq0 ∗ ω∗

0 1 0

−1 0 0

0 0 0

+vT − vRLs

) (2.30)

2.3.3 Non-linear Loads

Non-linear loads (e.g. rectifiers) are viewed as loads that generate harmonic currents. It

is proposed to model these as producing additional harmonic components in the current

and voltage vectors. Rather than simply including the harmonics superimposed on the

fundamental voltage and current terms, the vectors have been expanded to include a 5th


harmonic terms e.g. Equation 2.31. Note that the term v05 is not strictly necessary, as

the zero sequence terms are independent of frequency, (see Equation 2.4), however the

inclusion of this term allows for the re-use of simulation component blocks for the 5th

harmonic calculations.

V =

vd1

vq1

v01

vd5

vq5

v05

(2.31)

The d-q reference frame for the 5th harmonic terms rotates at 250Hz, or the 5th harmonic

frequency. To calculate voltage magnitude, angle, and real and reactive power the har-

monics are then converted back to a 50Hz reference frame and then the standard d-q

equations are used. The transform between the two reference frames can be performed

by converting the 5th harmonic terms back to the abc reference frame, and then into the

50Hz rotating reference frame. If these two steps are simplified the resulting equations are

2.32 and 2.33 where θ = ∆ωt and ∆ω is the frequency difference between the two rotating

reference frames, in this case 200Hz or 400π radians. Note that these equations do not

include the original fundamental quantities.

vd = vd5cosθ + vq5sinθ (2.32)

vq = vd5sinθ − vq5cosθ (2.33)

The final 50Hz dq quantities (2.34, 2.35) are the sum of the original 50Hz quantities (vd1

and vq1) and the converted 5th harmonics (2.32, 2.33). Since the zero sequence quantities

are independent of the speed of rotation of the d-q reference frame (the f0 term in Equation

2.4 only involves the original abc vectors and no terms in ω or θ) the final zero sequence


voltage is just the sum of the two zero sequence voltages (Equation 2.36).

vd = vd1 + vd5cosθ + vq5sinθ (2.34)

vq = vq1 + vd5sinθ − vq5cosθ (2.35)

v0 = v01 + v05 (2.36)

Another approach would be to alter the calculations so that they could directly deal with

the fundamental and harmonic terms, however it is unclear if there is any benefit to this,

and so it has not been implemented.

Clearly this work can be extended to include further harmonics by introducing further

terms in the current and voltage vectors, and applying equations 2.34, 2.35 and 2.36 to

the different harmonic terms. The terms vd5 and vq5 will need to be replaced by the

relevant harmonic terms (i.e. vdn and vqn for nth harmonics) and the frequency of ω will

need to be changed to the harmonic frequency - 50Hz (i.e. (50n−50)Hz for nth harmonics).

A typical load is demonstrated in Equation 2.37. Fundamental components of the load

voltage are generated in proportion to the fundamental components of the load current.

Harmonic components of the load voltage are importantly generated in proportion to both

the harmonic and fundamental components of the current. This is typical of non-linear

loads (e.g. rectifiers) where harmonic currents are generated in response to the applied

fundamental voltages.

V =

10 0 0 0

0 10 0 0

1 0 3 0

0 1 0 3

id1

iq1

id5

iq5

(2.37)

Separating the harmonics in this manner has two major advantages: in most sections of

the model 5th harmonics are represented as DC quantities, simplifying the calculations;

and it makes it much easier to observe the behaviour of the harmonics when they are


considered separately.

2.4 SimPowerSystems

Due to mathematical constraints it proved too difficult to incorporate transformers into

the model described in this chapter. For this reason, the unbalanced load simulations

have been implemented using MATLAB/Simulink’s SimPowerSystems library. This con-

tains model blocks for the line inductances, transformers and loads. SimPowerSystems is

significantly slower than the library developed in this work. The developed library has

consequently been used in preference to SimPowerSystems in some situations.

SimPowerSystems is a Simulink library which allows simulation of electrical systems. Since

the connections between the SimPowerSystems electrical components are representing elec-

trical wiring rather than the simple signal connections of standard Simulink, SimPower-

Systems uses different connections to standard Simulink. This means it is necessary to

interface the Voltage Source blocks (which have been created in the standard Simulink

format) with the SimPowerSystems connectors. The simplest way to achieve this is to use

the Controlled Voltage Source, which generates an output voltage based on a standard

Simulink signal input.

The circuit is configured with three single phase Controlled Voltage Sources connected

in a Y configuration (the library doesn’t contain a 3 phase Controlled Voltage Source).

These voltages feed into the coupling inductance (with internal series resistance) which

is represented using a 3 Phase Series RLC Branch block configured as an inductor and

resistor. This is then connected to a ∆ − Y n configured Three-Phase Transformer (Two

Windings) block, which feeds a star connected load (three single-phase Series RLC Branch

blocks connected in star). The neutral of the load is connected to the neutral of the

transformer Y . The source neutral is floating. Simulation details can be found in Appendix

A.3.


2.5 Summary

The model outlined above has been designed to analyse the behaviour of multiple inverters

running in parallel. It can be adjusted to account for any number of different sources, and

the load profile can be adjusted in various ways including reactive, unbalanced, and non-

linear loads (in fact any load that can be described with a transfer function relating voltage

to current). The magnitude or type of the load could also be configured to change during

the simulation. Multiple loads can be included, provided they are modelled as a single

lumped element. Line impedances can be included by modifying the coupling inductances.

Chapter 3

Inverter Control

Chapter 1 gave an overview of the various control techniques proposed for use with parallel

inverters. This Chapter examines in detail the principles of the droop controller, and

its limitations. The performance of a specific implementation of the droop controller

is evaluated by simulation. Improvements to the standard droop control techniques are

considered.

3.1 Principles of Droop Control

Consider a single grid connected inverter as shown in Figure 3.1. The goal of the inverter

control is to regulate the power supplied to the grid. Depending on the application,

the controller may also regulate the grid voltage E or the reactive power Q. Using a

droop controller, voltage regulation or reactive power control is achieved by controlling

the inverter voltage magnitude V , and power dispatch is performed by controlling the

inverter voltage angle δv. This control is possible because of the strong coupling between

real power P and the power angle δP (where δP = δv − δe), and reactive power Q and

inverter voltage magnitude V . This means that a small change in δP causes a significant

change in the real power P , and only a small change in the reactive power Q. Conversely a

small change in the inverter voltage magnitude V , causes a significant change the reactive

source inverter gridV !!V E!!E

L

I

Figure 3.1: Simple inverter system

3. Inverter Control 36

power, but only a small change to the real power. This can be shown by substituting

appropriate values into Equations 3.1 and 3.2. These equations are the real and reactive

power transfer from the inverter assuming (V −E) and δP are small. Since reactive power

can be used to control the system voltage E, the inverter voltage V can also be used to

regulate E.

P3φ =3

2

V E

ωLsin(δP ) (3.1)

Q3φ =3

2

V

ωL(V − EcosδP ) (3.2)

Now consider a stand-alone system with multiple inverters (Figure 3.2). The aim of the

power control changes from dispatching a pre-determined amount of power, to sharing the

load between the sources. Load sharing is achieved by adjusting each source’s frequency in

response to load power. As real output power is increased the system frequency is reduced.

The power-frequency droop characteristic is based on Equation 3.3, where ωi(t) is the

frequency output by the ith inverter at time t, w∗ is the nominal or base system frequency

(generally 50 or 60Hz), Pi(t) is the 3 phase power output of the source i at time t, and mi

is the droop co-efficient, or the gradient of the power-frequency droop characteristic, for

source i. Note that this equation represents the steady-state characteristic. The dynamic

behaviour may deviate from this.

ωi(t) = ω∗ −miPi(t) (3.3)

Another form of the droop characteristic is also common. This second form (Equation

3.4) introduces the term P ∗, called the power setpoint, which allows an operating point to

be specified for the source when the system is operating at nominal frequency. Essentially

this allows the user to specify a power output level for the source when the grid is con-

nected. For this reason, this second form would generally be chosen for a grid-connected


Source/Inverter 1

Source/Inverter 3 L

oad

Source/Inverter 2

Source/Inverter 4

Source/Inverter 5

Figure 3.2: Stand-alone multi-inverter system

environment. Note that in this case mi would be negative.

ωi(t) = ω∗ −mi(P∗i − Pi(t)) (3.4)

The basic principle of power-frequency droop control is similar to the power frequency

relationship in conventional rotating machines. Large generators are driven by a speed

regulated turbine. As power demand increases, the system temporarily slows as the turbine

control acts to increase its power output in response. The inertia in the turbine ensures

that the speed change is small. There is a corresponding reduction in system frequency.

With power-frequency droop control the frequency also droops when the power demand

increases, however rather than being a result of the interactions between the mechanical

and electrical systems in the generator, this droop is deliberately induced in the inverter

output voltage frequency by the controller. With a typical droop algorithm, this altered

frequency is maintained until the load level changes again.

Figure 3.3 shows the power-frequency droop characteristics for a two source system with

grid connection. When the grid is connected, the two sources operate at power levels

P ∗1 and P ∗2 . It is assumed some additional load is being supplied locally from the grid.

If the grid connection is broken (due to a fault or power outage etc.), the sources now


P

!min

!

P !1 P !

2

Pmax1 Pmax2

!!

!new

Pnew1 Pnew2

Figure 3.3: Power-frequency droop characteristic

have to supply the missing power. The droop relationship built into the inverter control

causes the frequency to droop until the two sources reach an equilibrium at a different

frequency ωnew. In the case where the sources were supplying the grid with power when

the connection was broken (i.e. the generated power is greater than the power demand on

the microgrid), the sources are required to reduce their power output and the frequency

will increase in order to find the equilibrium point.

3.2 Controller Implementation

The following sections describe the controller implementation which has been used to

examine the performance of a droop controller. This implementation draws on the work

performed by CERTS [21,22,84]. The controller consists of voltage regulation and power

sharing components. Voltage regulation functionality was chosen over reactive power

droop because, in a stand-alone environment (or temporary island mode), it is desirable

to be able to regulate the system voltage.

3.2.1 Voltage Control

A possible configuration for the voltage controller is shown in Figure 3.4 [21]. The user

defined setpoint for the grid voltage magnitude (E∗) is compared with the actual (calcu-


+- PI

Vmax

E!Eerr

Vmin

VE

++

Figure 3.4: Voltage Control

lated according to Equation (3.5)) grid voltage magnitude (E) to generate a voltage error

term. This is fed into a PI controller ( s+10s ) which is used to determine a value for V , the

desired magnitude of the inverter terminal voltage.

E =√e2d + e2

q + e20 (3.5)

In Chapter 1, it is mentioned that CERTS uses a Q-E droop as part of their voltage

controller. This droop has been included in the section on variations on droop control

later in this chapter.

Chandorkar et al. [85] add a ‘command feedforward’ term to the output of the PI control

in order to speed up the response of the controller. The feedforward term is chosen as

the value of V required to achieve E∗ with an unloaded filter and is given in Equation

3.6. Lf and Cf are the filter inductance and capacitance. The implementation of the

inverter in this system means that the filter is not explicitly defined so instead of using

Equation 3.6, the feedforward term was set to E∗ (making the assumption this is fixed),

since this is relatively close to the likely value of V . The reason for including this modifi-

cation was to speed up the acquisition of simulation results. It mainly affects the voltage

controller’s startup behaviour, making it much quicker to reach the desired value and does

not otherwise alter the behaviour of the controller significantly.

E∗(1− ω2LfCf

ω) (3.6)


3.2.1.1 Voltage Limits

The amount of reactive power injected into the system is related to the difference between

the voltages V and E. This means that the value of V is limited by the amount of reactive

power that can be injected by the inverter (assuming E ∼= E∗). The limits Vmax and Vmin

on the PI controller represent this physical limitation.

The following mathematics shows how Lasseter and Piagi [22] have derived values for Vmax

and Vmin. This is the method used for the simulations performed in this Chapter. Equation

3.7 gives the reactive power transfer across an inductor with impedance X. The maximum

amount of reactive power that can be generated by a given source (Equation 3.8), can be

found by substituting appropriate values into Equation 3.7. This maximum occurs when

the real power of the source is zero, giving cos δP = 1. The maximum allowed voltage

(Equation 3.9) can be calculated by rearranging Equation 3.8. Iinverter = (Vmax−E∗

X ) is

the inverter current when the inverter voltage is Vmax. An additional constraint is placed

on Vmax that it cannot exceed the voltage rating of the inverter.

Q1φ =V 2 − V EcosδP

X(3.7)

Qmax1φ =V 2max − VmaxE∗

X(3.8)

Vmax =E∗ ±

√E∗2 + 4Qmax1φX

2(3.9)

Similarly the minimum voltage (Equation 3.10) can be found from Qmin , the minimum

reactive power available.

Vmin =E∗ ±

√E∗2 + 4Qmin1φX

2(3.10)

An example calculation is provided here for clarity. In the case where E∗ = 230V, Qmax =

10000VAr, Qmin = −1000VAr, L = 1mH, the reactance of the inductor is calculated


according to Equation 3.11. Values can then be substituted into Equations 3.9 and 3.10

as given in Equations 3.12 and 3.13.

|XL| = ωL = 2π ∗ 50 ∗ 0.001 = 0.3Ω (3.11)

Vmax =230±

√2302 + 4 ∗ 10000 ∗ 0.3

2= 242,−12 (3.12)

Vmin =230±

√2302 + 4 ∗ (−10000) ∗ 0.3

2= 216, 14 (3.13)

Since Vmax = −12 and Vmin = 14 are clearly not sensible values, the solutions are Vmax =

242 and Vmin = 216. Note that this calculation is for single phase. Q values will needed

to be adjusted to per phase values for 3 phase calculations.

This method of calculating voltage limits makes some assumptions. The first is that

the measured grid voltage E is always closely matched to the setpoint E∗, which may

not always be the case (Australian standards allow voltages to deviate by +10, -6% AS

60038). Secondly it assumes that the inverter can always supply reactive power Q = Qmax

regardless of the real power being supplied. The first problem can be solved by making

Vmax and Vmin change dynamically depending on the measured value E rather than being

statically calculated using the setpoint E∗. This can be approximated by calculating the

maximum allowed deviation of V from E∗ (e.g. Vmax and Vmin are 12V and 14V away

from E∗ in the example above) and instead allowing V to deviate that amount from the

measured E. This would result in Equations 3.14 and 3.15.

Vmax adj = (Vmax − E∗) + E (3.14)

Vmin adj = (E∗ − Vmin) + E (3.15)

The second problem could be solved by intelligent selection of Qmax and Qmin (for example

set them based on the current ratings on the inverter).


+-

0

!Pmax

!V

P !

P

+-

!EPowerSource

PI

0

Pmax

Figure 3.5: Power dispatch

3.2.2 Power Control

If power sharing is not required for a particular source, then power control can be achieved

as shown in Figure 3.5 [22]. This controller is not used in the testing, but is described

here for completeness. As with the voltage controller, an error term is calculated from

the power setpoint and the actual power (as calculated by Equation (3.16) or (3.17)). An

additional constraint is that the power setpoint P ∗ is limited to being between 0 and Pmax

(shown on the diagram as the Power Source block). Note that the Power Source block

only models the limit in the power available, not the dynamics of the source. Power Source

blocks are covered in more detail in Chapter 6. The power error is fed into a PI controller

which calculates the required value of δP . The PI controller has limits placed on the angle

δPmax. This is generally between 00 and 300 to satisfy the requirement that the P (δP )

function be close to linear.

P = vaia + vbib + vcic (3.16)

P =3

2(vdid + vqiq) (3.17)

3.2.3 Power Sharing

When the microgrid is disconnected from the main power grid for whatever reason, the

power sources in the microgrid need to share the load between them. Power-frequency

droop control can be used to achieve this. Figure 3.6 shows an implementation of a power-


m

P

!V

Pmax

0

PowerSource

+- +

-1s

!!

P!

Figure 3.6: Power frequency droop control

frequency droop controller according to Equation 3.4 [21]. In order to provide effective

control in the presence of noise, the measurements need to be filtered before the real power

(P ) is fed into the controller.

CERTS propose two different methods for calculating the droop co-efficient m. In earlier

work [22, 84] they use the form in Equation 3.18. In later work [86], m is calculated

according to Equation 3.19. In Equation 3.18 ωmin is the user-defined minimum value

the frequency is allowed to droop to e.g. in a 50Hz system this might be 49.5Hz. In

Equation 3.19 ∆ω = ωmax−ωmin, e.g. if the frequency is allowed to vary between 49.5Hz

and 50.5Hz then ∆ω = 1Hz. These values will be governed by the frequency requirements

of the system. Pmax,i is the rated maximum power for the source in per unit.

mi =ω∗ − ωminP ∗i − Pmax,i

(3.18)

mi =∆ω

Pmax,i(3.19)

The earlier form (Equation 3.18) ensures that both sources reach Pmax at the same fre-

quency regardless of the values of Pmax,i and P ∗i (this can be observed by substituting

Equation 3.18 into the droop characteristic Equation 3.4), but if P ∗ is dynamic this means

that m must change when P ∗ changes. It also makes it necessary to ensure that m does

not become too large as the value of P ∗ approaches Pmax. The later form (Equation 3.19)

is independent of the power setpoint, and is fixed. It also avoids the possibility of m


becoming too large. Since Pmax is in per unit form, it is generally the same for all sources

(although it would be possible to define Pmax < 1 for a particular source, if desired),

meaning m is also the same for all sources. This form of the equation does not ensure that

all sources reach Pmax at the same frequency if Pmax is not the same for all sources.

For this work, P ∗ is chosen to be fixed and the same value for each of the sources, so the

form chosen isn’t important.

3.2.3.1 Power Source Block

Part of the power controller described in Figure 3.6 is the Power Source block. The

detail for the Power Source blocks is in Chapter 6, but a brief summary is provided here.

The basic idea of the Power Source block is to provide limits for P ∗ such that P ∗ doesn’t

exceed the maximum power output of the source. These so called Maximum Power Output

models simplify the dynamics of the power source into a transfer function which describes

the change in the maximum power available from the source over time. Various source

models are discussed, including an ideal source, microturbine, fuel cell and renewable

sources.

3.3 Problems with Droop Control

Chapter 1 briefly mentions several known problems with droop controllers. The following

gives more details of these problems.

1. Power-frequency droop control gives rise to an unavoidable steady-state frequency

error. The value of the droop co-efficient m can be adjusted to decrease this er-

ror, however this leads to slower responses from the controller, and the error cannot

be eliminated altogether. In most cases this is not a significant issue, and a fre-

quency difference between the microgrid and the main power grid can aid in resyn-

chronisation with the main grid once the problem that caused the disconnection is


corrected [22].

2. Reactive currents can circulate between inverters due to small errors in voltage

setpoints, differences in the line voltages measured at different inverters, or different

output inductances. This problem is exacerbated by the close proximity of sources

to each other. This means that the line impedances between sources can be quite

small. These circulating currents can consequently be quite large, causing inverter

ratings to be exceeded, possibly damaging equipment.

3. The droop controller does not share the generation of reactive power in the presence

of different output inductances. If reactive power is not shared, this could lead to

inverter ratings being exceeded in the source taking the greater share of the reactive

load.

4. Droop control is not designed to perform well with non-linear loads, and makes no

attempt to minimise or share harmonic currents. This is important, because in some

situations loads can be highly non-linear and contain a high proportion of harmonic

currents. If these are not shared, then the ratings of the systems would need to be

reduced to make allowances for this. The harmonic current drawn by the inverter

can cause resonance and undesirable interactions in the microgrid, which can have

a serious effect on the power quality [87].

5. The relationship between P and δP , and Q and V indicated in Equations 3.1 and

3.2 assumes that the overall impedance between the inverter and the load is mainly

inductive. In a low voltage system this is not necessarily the case, and in some

circumstances the line impedance is mainly resistive [6]. In the case of pure resistive

impedance, the real power would be linked with the voltage magnitude, and the

reactive power with the voltage angle. This is completely opposite to the assumption

which is made in droop control and could render the controller ineffective.

6. The standard droop controller does not attempt to alleviate voltage imbalance caused

by unbalanced loads. This imbalance has been shown to decrease the lifespan of


Source/Inverter 1

Source/Inverter 3 L

oad

Source/Inverter 2

Source/Inverter 4

Source/Inverter 5

Figure 3.7: Multi-inverter system with 3-phase load.

polyphase induction motors due to negative sequence currents [88], and can lead

to low frequency harmonics in non-linear loads which meaning switches and other

electronics need to have higher ratings to compensate. Unbalanced systems can be

less stable, lead to higher losses and problems with overheating [89].

3.3.1 Simulation Overview

The following simulations were performed in order to investigate the performance of a

typical droop controller, and to illustrate some of the problems with these systems. The

scenario used for the simulations is shown in Figure 3.7. It consists of five 6kW sources,

in parallel, connected by coupling inductances (1mH + 50mΩ) to a single 3-phase load.

For most of the tests this was a balanced resistive load of 10Ω (5.3kW) per phase. Unless

otherwise noted E∗ = 230V and P ∗ = 0.5pu for each of the sources. The power-frequency

droop coefficient is m = 1Hz/pu. The Simulink files for these simulations have been

printed in Appendix A.1.

3.3.2 Different Voltage Setpoints

To reproduce the problem of reactive currents circulating between inverters, the voltage

setpoint (E∗) of Source 1 was altered to be E∗ = 229.9V. This is an “error” of 0.1V

from the desired E∗ = 230V. Note that in reality this situation can only occur if the

voltage setpoint E∗ can be altered. This is not the case with the controller described here,


Figure 3.8: Effects of different voltage setpoints

however, the test is equivalent to having a fixed offset error in the measured value of |E|

for one of the sources, so the problem could still apply to a controller where E∗ is fixed.

Figure 3.8 shows the Load Voltage Magnitude and Frequency, and the Real and Reactive

Power output of the altered source and one of the unaltered sources. From these results it

can be seen that this small difference in voltage setpoints is enough to start the affected

inverter absorbing reactive power produced by the other inverters. Because of the absence

of line impedances in the model, the inverter in question is unable to bring the load voltage

down to match the desired setpoint. The amount of reactive power continues to increase

(in the negative direction) until the inverter output voltage magnitude (V ) reaches the

minimum allowed value as defined by the calculations in Section 3.2.1. This behaviour is

consistent with what would happen in practice if two inverters were connected at nearby

points on the grid.

Adjusting the setpoint error to be smaller causes similar behaviour over a longer timescale.

The rate of increase (or decrease) of the reactive power is smaller, but after a period of


time, the reactive power for at least one of the sources will still reach its maximum (or

minimum) value as dictated by the voltage limits.

3.3.3 Reactive Power Sharing

The aim of this simulation was to test the reactive power sharing in the presence of

coupling inductances with different impedances. The coupling inductance of Source 2

is changed to 0.9mH + 50mΩ, with the coupling inductances for the other four sources

remaining at 1mH + 50mΩ. The load used was 22.8kW + 15.5kV Ar (7.6kW + 5.2kV Ar

per phase) configured as a 15Ω resistor in parallel with an inductor (20mH) . The load

inductor contained a series resistance (5Ω).

Figure 3.9 shows the Voltage Magnitude and Frequency for the load, and the Real and

Reactive Power outputs for Sources 1 and 2, where Source 1 has the standard 1mH+50mΩ

coupling inductance, and Source 2 has the altered (0.9mH+50mΩ) inductance. The results

show that the reactive power output for Source 1 is lower than that for Source 2. This

confirms that differences in the coupling inductances do affect the reactive power sharing

amongst the sources. Coupling a source to the load through a larger inductor means

that source will provide less of the reactive load, while a source with a smaller coupling

inductance will provide more of the reactive load.

3.3.4 Harmonics

This test aims to examine the performance of the controller when the source is supplying

non-linear loads, such as a rectifier. The Simulink files for these simulations have been

printed in Appendix A.2. The non-linear load is modelled as a resistor (10Ω / 5.3kW per

phase) plus an injection of 5th harmonic current which is 23% of the magnitude of the

fundamental current. This value was chosen by simulating the performance of an actual

rectifier load and looking at the current harmonic components.

Figure 3.10 shows the baseline performance when an ideal source is supplying the load.


Figure 3.9: Reactive power sharing in the presence of different coupling induc-

tances

The results show that a non-linear load causes oscillations in the voltage waveforms.

Figure 3.11 shows the performance when the droop controller supplies the same load.

The results show that the harmonics cause no obvious disruptions to the performance of

the controller, with the Load Voltage Magnitude still regulated to the desired 230V. The

results are similar in both cases, indicating that the droop controller performs at least as

well as an ideal source with a non-linear load.

Figure 3.12 shows the results if the power setpoint P ∗ is changed to 0.25 for Source 1.

The results show that the power output for Source 1 is higher than the power output for

Source 2, which is the desired result in this case. This simulation has been included to

illustrate that the Power-Frequency droop control is still able to function in the presence

of non-linear loads.


Figure 3.10: Ideal source supplying non-linear load

Figure 3.11: Microgrid performance with non-linear load


Figure 3.12: Microgrid performance with non-linear load and different power

setpoints. P ∗ = 0.25 for Source 1, P ∗ = 0.5 for Sources 2-5

3.3.5 Resistive Line Impedances

In order to examine the effect of highly resistive or highly inductive impedances on voltage

regulation and power sharing on the droop control algorithm being tested here, simulations

were performed using the system shown in Figure 3.17 (excluding the transformers) with

different coupling impedance configurations. The Simulink files for these simulations are

identical to those in Appendix A.3, with the exception of the removal of the transformers.

In this case the sources are 3kW and the load is 30Ω (1.8kW) per phase. It was simpler

to use SimPowerSystems in this case, because the output impedance in the model library

would need to be reconfigured if an inductance L = 0 was required. To show the effects

on the power sharing functionality, the Source 2 coupling impedances are 10% higher than

the Source 1 coupling impedances.

Figure 3.13 shows that in the case where the coupling impedance is purely resistive (b)

(Source 1: R = 1.2Ω, Source 2: R = 1.32Ω), the system is unstable, and the power sharing


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.6

0.65

0.7

0.75

0.8

0.85

0.9

0.95

1

time(s)

Pow

er (

pu)

(a)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.6

0.7

0.8

0.9

1

1.1

time(s)

Pow

er (

pu)

(b)

Figure 3.13: Power output for two sources with (a) purely inductive output

impedance and (b) purely resistive output impedance

functionality does not work. In the case where the coupling impedance is purely inductive

(a) (Source 1: L = 40mH, Source 2: L = 44mH), the power sharing is still functional.

Figure 3.14 shows that the droop controller can perform with a mainly inductive cou-

pling impedance (a), or a mainly resistive coupling impedance (b). In (a) the inductive

impedance is 10x the resistive impedance ( Source 1: L = 40mH, R = 1.2Ω, Source 2:

L = 44mH, R = 1.32Ω), and vice versa for (b) (Source 1: L = 0.4mH, R = 1.2Ω, Source

2: L = 0.44mH, R = 1.32Ω).

These simulations indicate that the presence of a coupling impedance, provided that it is

not a pure inductance or a pure resistance, allows the power-frequency droop control to

operate correctly.

3.3.6 Unbalanced Grid Voltages

To test the effect of unbalanced grid voltages on the inverter control, one of the droop

sources was replaced by an ideal voltage source. The ideal voltage source is connected to

the load through the same coupling inductance as the droop sources. This represents a

relatively weak grid connection. An imbalance was introduced into the grid voltage by


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.6

0.65

0.7

0.75

0.8

0.85

0.9

0.95

1

time(s)

Pow

er (

pu)

(a)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.6

0.65

0.7

0.75

0.8

0.85

0.9

0.95

1

time(s)P

ower

(pu

)

(b)

Figure 3.14: Power output for two sources with (a) mainly inductive output

impedance and (b) mainly resistive output impedance

setting Vb = 220V in the Ideal Voltage Source. This represents a 3% imbalance.

Figure 3.15 shows that in the presence of unbalanced grid voltages, oscillations can be

observed in the voltage and power measurements. This indicates the presence of negative

sequence components, meaning that the load voltages are unbalanced. Additionally, there

is an increase in the amount of reactive power generated (or absorbed) by the inverters.

The generation of reactive power is due to the inverter control attempting to maintain the

voltage magnitude at the load. A stiffer grid connection would result in increased reactive

power generation. This increase in reactive power means an increase in the currents at

the source, which could be a problem if the inverter current ratings are exceeded.

Figure 3.16 shows the behaviour with balanced grid voltages as a reference.

3.3.7 Unbalanced Loads

To test the performance of the droop controller under unbalanced load conditions the

system shown in Figure 3.17 was simulated. To properly see the effects of unbalanced

loads on the system, delta-wye transformers were added to provide a neutral point. For


Figure 3.15: Unbalanced Grid Voltages

Figure 3.16: Balanced Grid Voltages


Source/Inverter

Load

Source/Inverter

L

L! Y

! Y

Figure 3.17: Overall system diagram for unbalanced load test

this reason the SimPowerSystems library was used instead of the model library. Due to

the slower performance of SimPowerSystems, only two sources are used in this simulation.

The details of using SimPowerSystems are described in Chapter 2. The Simulink files for

these simulations have been printed in Appendix A.3.

Inverters with source rating 50kW are connected through coupling inductances (L = 2mH)

to Delta-Wye transformers with a neutral connection on the Wye. The transformer blocks

are left with their default settings, apart from changing the nominal power to 50kW,

the nominal frequency to 50Hz, and the phase-phase voltages to 398V. The output of

these transformers then feeds an unbalanced 3-phase resistive load of Ra = 4Ω (13.2kW),

Rb = 2Ω(26.5kW), Rc = 4Ω (13.2kW).

Figure 3.18 shows the phase-phase voltages at the load, and Figure 3.19 shows the load

voltage magnitude. Note that these both represent measurements of the same quantities.

In Figure 3.18 it can be observed that the phase-phase voltages have different peak values

which indicates that the load voltages are unbalanced. In Figure 3.19 this unbalance shows

as oscillations at twice the system frequency (100Hz) indicating the presence of a negative

sequence component in the load voltages. The amount of unbalance is approximate 3.5%

which exceeds the NER limit in Australia for less than 10kV of between 2% and 3% [90]

Figures 3.20 and 3.21 show the baseline performance of the system without power-frequency

droop control and voltage regulation. The increased size of the oscillations in the voltage


Figure 3.18: Phase-Phase Load voltages in the presence of unbalanced loads.

Figure 3.19: Load voltage magnitude response in the presence of unbalanced

loads.


Figure 3.20: Phase-Phase Load voltages in the presence of unbalanced loads

with controllers off.

magnitude without the control indicate that the performance is slightly improved by the

control. This is most likely due to the voltage control acting to maintain the voltage at

the correct value.

3.4 Variations on Droop Control

Several variations on the droop control algorithm have been proposed to improve the per-

formance of the controller. Details of some of these are described in this section, as well as

the results of testing performed on selected algorithms. Methods include a reactive power

vs grid voltage (Q-E) droop to reduce circulating reactive currents, techniques to eliminate

steady-state frequency error, techniques to improve performance under resistive impedance

conditions and harmonic current sharing (including virtual output impedances).


Figure 3.21: Load voltage magnitude response in the presence of unbalanced

loads with controllers off.

+- PI

Vmag

Vmax

Vmin

Eerror

E!

E

+-Q mQ

Figure 3.22: Voltage control with Q-E droop

3.4.1 Reactive Power Sharing

3.4.1.1 Q-E Droop

Standard power-frequency droop control has a limitation in that slight errors in voltage

setpoints or measurements can cause reactive currents to circulate between the inverters.

If the currents are large enough this could cause the inverter ratings to be exceeded.

To alleviate this, CERTS proposes using a Q-E droop in conjunction with the power-

frequency droop control (Lasseter and Piagi, [21]). A similar idea was initially proposed

by Chandorkar et al. [85]. This droop is also effective in improving reactive power sharing

for reactive loads.

The droop is a modification to the voltage controller and functions by adjusting the voltage


setpoint E∗ in proportion to the amount of reactive power. This simple measure is able

to significantly reduce the amount of reactive current between the inverters. The droop

relationship is given in Equation 3.20 where Eadj is the adjusted voltage setpoint fed to

the voltage controller, E∗ is the actual voltage setpoint desired in the system, mQ is the

Q-E droop co-efficient and Q is the reactive power output of the source (in per unit for

this implementation). The overall voltage controller is shown in Figure 3.22. As with all

droops, Q-E droop contains a trade-off: more effective suppression of circulating currents

requires an increase in the Q-E droop co-efficient mQ which leads to less accurate voltage

regulation. If the net amount of reactive power in the system is negligible (i.e. most of

the reactive power is flowing between the sources), then the effect of this trade-off will be

negligible, since more effective suppression of circulating currents will reduce the value of

Q.

Eadj = E∗ −mQQ (3.20)

To illustrate the trade-offs inherent in Q-E droop, consider two sources with voltage Equa-

tions 3.21 and 3.22.

E1adj = E∗1 −mQQ1 (3.21)

E2adj = E∗2 −mQQ2 (3.22)

Since E∗ is the desired voltage level, the terms mQQ1 and mQQ2 represent the voltage

error due to the Q-E droop.

The system is designed so that E∗1 = E∗2 . Substituting in Equations 3.21 and 3.22 gives

Equation 3.23. Rearranging this gives Equation 3.24 , soE2adj−E1adj

mQis the Q error term,

or the error in the reactive power sharing. These error terms show that a larger mQ will


Figure 3.23: Effect of Q-E droop on circulating reactive currents, E∗ = 229.9V

give a larger voltage magnitude error, but a smaller reactive power sharing error.

E1adj +mQQ1 = E2adj +mQQ2 (3.23)

Q1 −Q2 =E2adj − E1adj

mQ(3.24)

Figure 3.23 shows the results if Q-E droop (with mQ = 1) is implemented in the system

tested in Section 3.3.2, where an error has been introduced into the voltage setpoint of one

of the sources. The solid lines show the results with the Q-E droop applied and the dashed

lines show the results from Figure 3.8 without the droop. The comparison shows that the

amount of reactive power being absorbed is significantly reduced from the previous case

and no longer reaches the limit provided by the voltage control. Even increasing the error

so that the voltage setpoint is E∗ = 229V (1V lower than that used by the other inverters),

the controller was still able to achieve good reactive power sharing (Figure 3.24). Two

things can be observed from the graphs - the steady state voltage magnitude is slightly

low, and the real power supplied is higher.


Figure 3.24: Effect of Q-E droop on circulating reactive currents, E∗ = 229V

Figure 3.25 shows a comparison of the results when the system in Section 3.3.3 (Reactive

Power Sharing) is simulated with and without the Q-E droop. The solid lines show the

results with Q-E droop and the dashed lines are the results from Section 3.3.3 without

Q-E droop. Looking at the reactive power for the two sources shown, it is clear that

the adjusted controller makes a significant improvement in the reactive power sharing.

Looking at the graph for the voltage magnitude shows that this is to the detriment of the

steady state voltage magnitude which is E ' 229.7V instead of the desired E = 230V .

This steady state error is not dependent on the difference in coupling inductor size, but

on the size of the inductive load i.e. the amount of reactive power being absorbed. A

larger inductive load (i.e. smaller inductance value) gives a lower steady state voltage.

The error is quite small, and well within the voltage standards.


Figure 3.25: Effect of Q-E droop on reactive load sharing

3.4.1.2 Virtual Output Impedance

Several authors have proposed emulating the output impedance rather than having a phys-

ical coupling inductor. The most obvious advantage to this is the reduction in components,

which reduces the cost, size and weight of the inverter system. Other possible advantages

include the ability to vary the output impedance online allowing reactive power sharing,

a reduction in losses when resistive output impedance is desired and improved harmonic

current sharing.

De Brabandere et al. [6] emulate the output impedance by simultaneously controlling

voltage and current using a Linear Quadratic Regulator in conjunction with a Kalman Es-

timator. A representation of the system is shown in Figure 3.26. The major components

are the plant (or real system) model (R), the virtual system model (V ), the input distur-

bance model (W ), the Kalman Estimator (Ew), and the control (K). A more detailed

diagram and explanation can be found in [6]. Their results show that the controller was


Plant

-K

Ew

++

-+

W

R V

V

-+

usrc

isrc

ucap

igrid

d

Figure 3.26: Output impedance emulation using linear quadratic regulator

able to maintain sinusoidal voltages in the presence of highly non-linear loads.

Guerrero et al. [91] again use a fast control loop to emulate an output impedance. This

loop droops the inverter output voltage proportionally to the derivative of the fundamental

current, and to the current harmonics up to the 11th (Equation 3.25). In this equation,

LD is the virtual output inductance and Rh is the resistive co-efficient for each of the

harmonic current terms.

vref = v∗ref − sLDi01 −11∑

h=3,odd

Rhi0h (3.25)

The authors also propose a soft-start operation for connecting inverters seamlessly to a

common bus. It involves adjusting the value of LD such that it is high initially, to prevent

an initial current spike due to any PLL error, and gradually decreases to the desired

steady-state value. The adjusted value L∗D is given in Equation 3.26 or 3.27.

L∗D = LD final + (LD initial − LD final)e−tτST (3.26)

L∗D = LD final + (LD initial − LD final)(τ

τs+ 1) (3.27)

The reactive power is regulated by controlling the output impedance according to equation

3.28.

LD = L∗D + kLQ (3.28)


The results in [91] show that this method is effective in sharing reactive power and pre-

venting circulating reactive currents.

The disadvantage of these virtual impedance methods is that they are quite complex and

require more computing power due to the fast calculations involved.

3.4.2 Frequency Restoration

It is desirable that the steady state frequency in a system be as close to ω∗ as possible.

To achieve this, Lasseter and Piagi [21] propose modifying Equation (3.3) to introduce a

parameter P ∗c.i resulting in Equation 3.29.

ωi(t) = ω∗ −mi(P∗c,i − Pi(t)) (3.29)

P ∗c,i is gradually adjusted so that at steady state its value is such that ωi(t) = ω∗. This

frequency restoration loop works over a longer time scale than the power sharing loop in

order to allow the power sharing to reach steady state before the frequency is restored.

Figure 3.27 shows the implementation of the power-frequency droop controller with fre-

quency restoration loop. The constants k′ and k′′ are chosen to ensure that the rate of

change of frequency of each of the sources remains the same in order to maintain the same

power angle between the sources. Changing the power angle will change the power output

which is undesirable once a steady state has been reached. In [84] it is shown that this

requirement leads to the conclusion that mk′′ and k′

m must be the same for all sources.

Another option to improve steady-state frequency performance is proposed by Sao and

Lehn [92]. The authors suggest adding a gain on the integrator which calculates the

inverter angle (Equation 3.30).

δ1 − δcom = Kp1

ˆ(ω1 − ωcom)dt (3.30)


m

0

PowerSource

+- +

-+

++-1

s

0

PI!!k!!

s

+-

k!

+ -

1s

!P !V

!E

!6

+

P

P !

Pmax

Figure 3.27: Power frequency droop control with frequency restoration

Their results show that tuning this gain can provide a faster response without increasing

the steady-state frequency error. This method does not eliminate the error altogether, but

it provides a significant improvement and is much simpler than the frequency restoration

loop. One disadvantage of this is that it requires accurate calculation of ωcom, the fre-

quency of the common bus voltage. All the previous power sharing controllers have only

required the power output of the inverter.

A third approach to minimising steady state frequency error (Guerrero et al. [93]) is to

use P (the real power with the DC component excluded), instead of P , as the input to

the controller (Equation 3.31). This is achieved by high pass filtering the power signal.

The addition of the derivative term is to improve the transient response. This approach

shows good results, and is less complex to implement than other methods.

ω = ω∗ −mP P −mdδP

δt(3.31)

Figure 3.28 shows a comparison between the performance of the droop algorithm in Equa-

tion 3.4 and the droop proposed here in Equation 3.31. The simulations were performed

with identical systems, except for the power controller. Two sources are connected through

coupling impedances (L1 = 20mH, R1 = 0.2Ω, L2 = 12mH, R2 = 0.12Ω). Source 1 is 5kW

and Source 2 is 10kW. The coupling impedances connect to a balanced 3-phase resistive


0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20.6

0.62

0.64

0.66

0.68

0.7

0.72

0.74

0.76

0.78

0.8

(a) Power output

0 1 2 3 4 5 6 7 8 9 1049.9

49.92

49.94

49.96

49.98

50

50.02

50.04

50.06

50.08

50.1

(b) Frequency response

Figure 3.28: Comparison of power-frequency droop responses

load of R = 15Ω (3.5kW) per phase. Note that the ratios between the source power rating,

and the coupling impedance are not the same for the two sources. This is to illustrate the

power sharing functionality. The Simulink files for these simulations have been printed in

Appendix A.4.

The dotted lines show the responses of two sources using the algorithm in Equation 3.4

and the solid lines show the responses of two sources using the algorithm in Equation

3.31. The frequency response has been graphed over a longer timescale in order to show

the effects of the frequency restoration. The results show that this new power-frequency

droop algorithm has a faster response, without the steady-state frequency error. There is

some steady state error in the power sharing due to the frequency restoration. Increasing or

decreasing the time taken for frequency restoration, correspondingly decreases or increases

this error. In Figure 3.28 (b) it can be seen that to achieve this faster response, the new

algorithm requires a higher frequency deviation initially. If the frequency restoration

component (i.e. the high pass filter) were removed, the new algorithm could obtain a

faster response with the same frequency deviation. In both cases, however, the frequency

deviation required is quite small. Note that the steady state frequency error achieved by


Equation 3.4 is dependent on the value of P ∗. In cases where the power required by the

load is close to P ∗, the steady state frequency error will be minimal.

3.4.3 Harmonic Current Sharing

Lee and Cheng [94] suggest a G-H droop, (harmonic conductance vs harmonic var con-

sumption) in order to ensure harmonic current sharing between the sources without com-

munications (Equation 3.32). The harmonic conductance is the conductance ( 1R) seen by

the harmonic voltages and currents. The harmonic var consumption is that component of

the reactive power due to harmonic voltages and currents. This droop is designed to work

in conjunction with standard P − f and Q− V (or Q− E) droops.

G(t) = G∗ − bi(H∗i −Hi(t)) (3.32)

Results show that this method can effectively share harmonic current and reduce total

harmonic distortion (THD) while maintaining real and reactive power sharing.

3.4.4 Resistive Line Impedance

Standard droop control relies on the impedance between the inverter and the load being

mainly inductive. In a low voltage network, this is not necessarily the case. It should be

noted that the use of inductive coupling impedances, as is proposed in this thesis, will tend

to ensure this, however not all droop control algorithms require a coupling impedance.

De Brabandere et al. ( [6]) propose a transform matrix (Equation 3.33) which uses the

ratio between the resistance and reactance of the line impedance. This matrix is used to

transform the quantities in the droop equations to allow the control to work with any line

impedance. The authors have chosen to control the real and reactive currents instead of

the real and reactive power, although it should be noted that the transform matrix can

equally be applied to real and reactive power.


The droop equations are 3.34 and 3.35, where I ′a is the modified active current, I ′r is

the modified reactive current, I ′a,0 and I ′r,0 are the modified setpoints for the active and

reactive currents, and ka and kr are the droop co-efficients for the active and reactive

currents. I ′a

I ′r

=

XZ −R

Z

RZ

XZ

Ia

Ir

(3.33)

ωi(t)− ω∗ = −ka(I ′a − I ′a,0) (3.34)

V − E∗ = −kr(I ′r − I ′r,0) (3.35)

T =

XZ −R

Z

RZ

XZ

is the transform matrix. According to the authors, the benefit of

controlling the currents rather than the voltages is that current can be limited in the

event of a short circuit. Additionally, power sharing and voltage regulation is improved

in the presence of non-zero line resistance, and the controller is able to mitigate voltage

harmonics.

Note that, as shown in Section 3.3.5, the inclusion of a coupling impedance means that

correction for resistive line impedances is unnecessary.

3.5 Evaluation

The literature, and the tests performed here, show that there are several limitations to

basic droop control, including circulating reactive currents, steady-state frequency and

voltage magnitude errors, lack of harmonic current sharing, poor performance under un-

balanced load, and poor short circuit performance. Solutions, or at least improvements,

have been found for many of these areas, with the notable exception of techniques for cor-

recting voltage unbalance in the presence of unbalanced loads. In addition to this, most

of these techniques address only one or two of the several problems, and a combination of


techniques could result in an improved controller.

Chapter 4

Unbalanced Loads

Unbalanced loads are a problem as they can lead to unbalanced phase voltages. This is of

particular interest in Low Voltage Distributed Generation systems because the likelihood

of unbalanced loads is increased. Chapter 3 showed the possible effects of unbalanced loads

on a system employing droop control. None of the proposed implementations of droop

control outlined in Chapter 3 are designed to improve performance under unbalanced

conditions. This Chapter details a method of limiting voltage imbalance at the load.

4.1 Unbalance Compensation

In the literature, the most common solution to unbalanced voltages in any system is to

measure and compensate for the negative sequence components caused by the voltage

imbalance [95–97].

These solutions typically have been designed for grid connected applications [95, 96], or

single source stand-alone systems [97] and therefore lack the power sharing and voltage

regulation functionality necessary in microgrids.

Recently, some authors have addressed the problem of correcting for unbalanced loads in

microgrids. Li et al. [98] have proposed a power quality compensator which minimises the

impact of unbalanced utility grid voltages on sensitive loads in microgrids. The power

quality compensator uses two inverters. A shunt inverter ensures balanced voltages, and

a series inverter balances the line currents. The basic principle of the control is to can-

cel the negative sequence components of the voltages coming from the grid by injecting

negative sequence components of the opposite sign. The disadvantage of this system is

4. Unbalanced Loads 71

that it cannot be implemented only by modifying the source inverter as it requires the two

inverters.

Another approach is to use instantaneous p-q theory to compensate for unbalanced volt-

ages [99]. This theory allows for the calculation of instantaneous values of real and reactive

power. It is similar in concept to using sequence components, in that voltages and currents

are measured and then transformed into other values (in this case real and reactive power)

which are then used by the controller. Watanabe et al. outline how instantaneous p-q

theory can be used for harmonic, reactive power or unbalance compensation, and voltage

regulation. This solution utilises a shunt inverter, so again it cannot be implemented only

by using the source inverter.

Cheng et al. [67] have proposed using a Q−-G droop (the reactive power produced by the

negative sequence current vs conductance) in conjunction with standard droop control.

This is similar to the approach developed here and first reported in [100], except that their

controller takes the negative sequence reactive power as the input rather than the negative

sequence voltage, requiring several extra calculations.

Hojo et al. [101] have proposed another similar approach, instead using d-q transforms

rather than symmetrical components.

In this work, the principle of negative sequence control is adapted to operate in conjunction

with a standard droop controller in a Voltage Source Inverter. The aim is to compensate

for the unbalanced voltages within the microgrid caused by unbalanced loads, while still

providing voltage regulation and power sharing functionality. Using this approach, un-

balanced voltages which occur as a result of unbalanced loads can be minimised without

requiring additional hardware. It is designed to improve power quality in stand-alone

operation rather than compensate for unbalanced utility grid voltages.


a1

c1

b1

a2

b2

c2

a0 b0 c0

Positive Sequence

Negative Sequence

Zero Sequence

a

b

c

Sequences combine to give 3-phase phasors

Direction of Rotation

Figure 4.1: Symmetrical Components


4.1.1 Symmetrical Components Theory

The theory of symmetrical components states that any 3 phase voltage can be represented

as the sum of positive, negative and zero sequence components. As shown in Figure 4.1,

the positive sequence components are balanced 3 phase voltages which are rotating in the

positive sequence (i.e. normal) direction, the negative sequence components are balanced

3 phase voltages which rotate in the negative sequence direction (i.e. the reverse direction)

and zero sequence components are the in phase components (the same as in the dq reference

frame). Note that the magnitude and phase angles of the positive, negative and zero

sequences can be any value, and may differ from each other.

A perfectly balanced 3 phase voltage source will yield only positive sequence components,

however, the introduction of unbalance in the voltages will yield positive and negative

sequence components, along with a zero sequence components if there is a suitable circuit

connection (i.e. earth or neutral connection). Conversely, negative sequence components

are an indicator of unbalance and can be used in the control of unbalance.

Equation 4.1 is the transformation from 3 phase phasor quantities to symmetrical compo-

nents for an arbitrary function Fabc. The reverse transformation is Equation 4.2.

F0

F1

F2

=1

3

1 1 1

1 a a2

1 a2 a

Fa

Fb

Fc

(4.1)

Fa

Fb

Fc

=

1 1 1

1 a2 a

1 a a2

F0

F1

F2

(4.2)

The operator a represents a rotation by 1200 (a = 1∠1200). For the purposes of calculation

a is represented in complex form, a = −0.5 + j√

32 or ej

2π3 . The sub-script 1 represents the


Source/Inverter

Load

L! Y

Figure 4.2: Single line diagram of 3-phase single inverter system

positive sequence, the sub-script 2 represents the negative sequence and the sub-script 0

represents the zero sequence.

If the matrix A is defined as in Equation 4.3 then the above transformations (Equations 4.1

and 4.2) can be simplified as F012 = A−1Fabc and Fabc = AF012.

A =

1 1 1

1 a2 a

1 a a2

(4.3)

All these equations are based on phasor quantities, meaning it is necessary to calculate

the magnitude and phase of the current or voltage you wish to transform. In a dynamic

system, it is sometimes helpful to have a transform which uses the instantaneous quantities.

The instantaneous symmetrical components transform is given in Equation 4.4 where S90

represents a time delay shift equivalent to 900 or a quarter of a period.

f0

f1

f2

=1

3

1 1 1

1 −12 +

√3

2 S90 −12 −

√3

2 S90

1 −12 −

√3

2 S90 −12 +

√3

2 S90

fa

fb

fc

(4.4)


Va V !a

V !b

V !cVc

Vb

N

ZL

ZL

ZLIa

Ib

Ic

Figure 4.3: Three phase coupling inductance

4.1.2 Sequence Representation of Grid

Symmetrical components are not only used to transform voltages and currents. By trans-

forming each of the elements in a system (e.g. impedances, transformers), they can be used

to create sequence representations of networks. This approach is typically used for fault

analyses, as it allows a balanced system to be represented by three independent sequence

circuits, simplifying the calculations. In an unbalanced system, the sequence networks are

no longer independent, but the technique is useful to show the system as it appears to the

sequence voltages and currents.

Consider the single line diagram of a three-phase, single inverter system shown in Fig-

ure 4.2. The impedances can be transformed into symmetrical components form using the

relationship in Equation 4.5.

Z012 = A−1ZabcA (4.5)

Coupling Inductance The coupling inductance consists of a series connected impedance

of ZL in each phase as shown in Figure 4.3. Calculating the voltage/current relation-

ship in the circuit results in Equation 4.6, where ZLabc is the impedance matrix given

in Equation 4.7. Applying the transform in Equation 4.5 gives the symmetrical compo-

nents impedance matrix ZL012 given in Equation 4.8. This means that for the coupling

inductance, the impedance transform results in a series impedance of ZL for each of the


sequences.

Vabc − V ′abc =

Va − V ′aVb − V ′bVc − V ′c

= ZLabc

Ia

Ib

Ic

(4.6)

ZLabc =

ZL 0 0

0 ZL 0

0 0 ZL

(4.7)

ZL012 =

ZL 0 0

0 ZL 0

0 0 ZL

(4.8)

Delta-Wye Transformer Transformers can be represented by their equivalent circuit

as shown in Figure 4.4 (single line diagram) or Figure 4.5 (three phase diagram). RP and

LP are the primary side impedance, R′S and L′S are the secondary impedance referred to

the primary side, and Rm and Lm are the magnetising impedance (also called the exci-

tation branch). In this case the turns ratio of the transformerN1N2

is 1, so R′S and L′S are

the same as RS and LS . The primary and secondary impedances model the effects of

leakage inductance and winding resistance,and the magnetising impedance models mag-

netising inductance and core losses due to magnetic hysteresis and eddy currents. The

ideal transformer with ratio 1 : ej30 represents the phase shift which occurs in a delta-wye

transformer.

The simplest way to convert this equivalent circuit model into symmetrical components

is to consider the primary, secondary and shunt magnetising impedances individually.

By inspection it can be seen that the primary and secondary referred impedances are of


N

Rm Lm

Rp Lp Rs' Ls'

1:ej30

Vap Vas

N

Figure 4.4: Single line diagram of transformer equivalent circuit model

N

Rm Lm

1:ej30Rp Lp Rs' Ls'

Rp Lp R2' Ls'

Rp Lp R2' Ls'

1:ej30

1:ej30

Rm Lm

Rm Lm

a

b

c

a'

b'

c'

Figure 4.5: Three phase diagram of transformer equivalent circuit model


the same form as the coupling impedance and the sequence impedances are ZP012 for the

primary (Equation 4.9) and ZS012 for the secondary (Equation 4.10), where ZP = RP +LP

and ZS = R′S + L′S .

ZP012 =

ZP 0 0

0 ZP 0

0 0 ZP

(4.9)

ZS012 =

ZS 0 0

0 ZS 0

0 0 ZS

(4.10)

To transform the magnetising resistance the phase-neutral voltages VaN , VbN and VcN

are considered. This results in Equation 4.11. It can be observed that the magnetising

impedance matrix Zmabc (Equation 4.12) is of the same form as the previous impedance

matrices. Transforming this matrix results in Equation 4.13 which means there is an

impedance of Zm between the positive sequence and the neutral, and negative sequence

and the neutral. There is no path for zero sequence currents in the primary of a delta-wye

transformer, so the zero sequence is open circuited at the transformer.


Va

Vc

Vb

N

Ia

Ib

Ic

ZLoada

ZLoadb

ZLoadc

Figure 4.6: Three phase diagram of unbalanced Y connected load

Vabc − VN =

Va − VNVb − VNVc − VN

= Zmabc

Ia

Ib

Ic

(4.11)

Zmabc =

Zm 0 0

0 Zm 0

0 0 Zm

(4.12)

Zm012 =

Zm 0 0

0 Zm 0

0 0 0

(4.13)

Unbalanced Star Connected Load Figure 4.6 shows an unbalanced three phase Y

connected load. Equation 4.14 shows the voltage current relationship in the circuit, where

ZLoad−abc is defined in Equation 4.15. Applying the impedance transform equation (Equa-

tion 4.5) to the load impedance (Equation 4.15) gives ZLoad−102 as in Equation 4.16. It

can be observed that cross-coupling occurs between the zero, positive and negative se-

quence networks when an unbalanced load is connected in a three phase system. It is for

this reason that the sequence networks cannot be considered individually when there is

unbalance present.


Vabc − VN =

Va − VNVb − VNVc − VN

= ZLoad−abc

Ia

Ib

Ic

(4.14)

ZLoad−abc =

Za 0 0

0 Zb 0

0 0 Zc

(4.15)

ZLoad−012 = A−1ZLoad−abcA

=

Za + Zb + Zc Za + a2Zb + aZc Za + aZb + a2Zc

Za + aZb + a2Zc Za + Zb + Zc Za + a2Zb + aZc

Za + a2Zb + aZc Za + aZb + a2Zc Za + Zb + Zc

(4.16)

Using these results, three equivalent circuits can be drawn for the symmetrical components.

The positive and negative sequence circuits are shown in Figure 4.7. The delta connection

on the transformer means there is no path for zero sequence currents from the source

to the load, so the zero sequence network is ignored. Since this work is concerned with

eliminating negative sequence voltage at the load (VLoad2), then it is the negative sequence

branch which is of interest. To cancel the negative sequence voltages at the load, negative

sequence voltages are injected at the source, represented by Vs2. The control problem,

therefore, is to choose the magnitude and phase of Vs2 such that VLoad2 is minimised. In

designing the controller it is necessary to account for the losses and phase change across

the coupling impedance and the transformer. Using this sequence network it would be

possible to calculate the impedance between the negative sequence voltage injection point

and the load, but it is more useful to design a controller which doesn’t assume accurate

knowledge of various network parameters.


N

1:ej30Rp Lp Rs' Ls'

Rp Lp Rs' Ls' 1:e-j30

Rm Lm

Rm Lm

Vs1

Vs2

ZLoad

TransformerVoltage Source

VLoad1ZL

ZL VLoad2

Figure 4.7: Sequence Network

4.2 Unbalance Controller Design

The aim of an unbalance compensator is to cancel the negative sequence components of

the relevant voltage (in this case VLoad2−meas, the load voltage). This can be achieved by

Equation 4.17.

VLoad2 injected = − |VLoad2 meas|∠VLoad2 meas (4.17)

Since the voltage will be injected at the inverter (Vs2), not at the load, the controller

needs to account for the losses across the coupling inductance and the transformer. This

is achieved by using a PI controller to determine the negative sequence magnitude to be

injected. This is shown in Figure 4.8. It is not necessary to compensate for the phase shift

across the inductor, provided it can be guaranteed that the magnitude of the phase shift

is less than 900. The following analysis shows why this is the case.

Assume that ∆VLoad2 inj∠∆δLoad2 inj is the incremental change in the injected negative

sequence voltage measured at the load and VLoad2 cur∠δV Load2 cur is the measured negative

sequence voltage at the load before the latest incremental change is applied (note that in

practice it is not possible to separate out these components). The aim is for |VLoad2 next|

(Equation 4.18) to be less than the original |VLoad2 cur|, resulting in Equation 4.19, which

can be expanded out to Equation 4.20. Squaring both sides of Equation 4.20 and then


simplifying, results in the Equation 4.21.

|VLoad2 next| = |VLoad2 cur∠δV Load2 cur + ∆VLoad2 inj∠∆δLoad2 inj | (4.18)

|VLoad2 cur| > |VLoad2 cur∠δV Load2 cur + ∆VLoad2 inj∠∆δV Load2 inj | (4.19)

|VLoad2 cur| >√

(VLoad2 cur cos δV Load2 cur + ∆VLoad2 inj cos ∆δV Load2 inj)2 . . .

. . .+(VLoad2 cur sin δV Load2 cur + ∆VLoad2 inj sin ∆δV Load2 inj)2 (4.20)

cos(δV Load2 cur −∆δV Load2 inj) <−∆VLoad2 inj

2VLoad2 cur(4.21)

Since ∆VLoad2 inj is the magnitude of the incrementally injected voltage, it can be as-

sumed to be very small, therefore the term−∆VLoad2 inj2VLoad2 cur

approaches zero, resulting in

Equation 4.22.

cos(δV Load2 cur −∆δV Load2 inj) < 0 (4.22)

Solving this equation gives the result that 900 < δV Load2 cur−∆δV Load2 inj < 270. Taking

into account the negative in Equation 4.17, which means that the phase angle of the

injected voltage at the inverter is δV s2 inj = δV Load2 cur − 1800, this gives the result that

the phase shift across the coupling inductance (δV s2 inj −∆δV Load2 inj) is between ±900

(Equations 4.23 and 4.24). Since the coupling inductance has been designed to ensure that

the phase shift across the inductor is small (this is a requirement of the droop control, see

Section 2.2.3), this condition will be met.

90 < δV s2 inj + 180−∆δV Load2 inj < 270 (4.23)

− 90 < δV s2 inj −∆δV Load2 inj < 90 (4.24)

In this simulation the sequence components are measured from line-line quantities. Line-

line quantities are√

3 times the magnitude of line-neutral quantities, and they are also

shifted by −300 for the negative sequence (∠V2Ph−g = ∠V2Ph−Ph + 300). Additionally,


+

|V2 req|PIControlDeadzone

|Vload2 meas|

+600

∠Vload2 meas

∠V2 req

Figure 4.8: Magnitude and phase angle calculations for unbalance control

if the load voltage is measured on the transformer secondary then the transformer phase

shift needs to be accounted for (∠VLoadP = ∠VLoadS + 300). These adjustments result in

Equation 4.25. This equation will have to be adjusted if the phase-neutral voltages are

measured instead, or if the voltages are measured on the primary side of the transformer.

The controller is shown in Figure 4.8. The deadzone block is to account for inaccuracies

in the measurement of the sequence components at small negative sequence voltage levels.

This is further explained in Section 4.3. The scaling, and the negative, are applied in the

Inverter block, so are not included in the figure.

V2 req = −|VLoad2 meas|√3

∠(VLoad2 meas + 600) (4.25)

Equation 4.26 is the positive sequence voltage for the source. Note that this controller

only attempts to compensate for unbalance, and is not performing any power sharing, or

voltage regulation (meaning in this case |V1 req| is a fixed number, not the output of the

voltage control block). Equation 4.27 gives the zero sequence voltage for the source, since

the controller is not attempting to control the zero sequence.


V1 req = |V1 req|∠0 (4.26)

V0 req = 0 (4.27)

Once the required symmetrical component phasors have been calculated, Equation 4.28

is used to transform these into the phasor quantities for the inverter. Equations 4.29 -

4.31 calculate voltages va, vb and vc which are the time domain voltages output by the

inverter. Note that the voltage controller provides V1 as an RMS value so this needs to

be multiplied by√

2 to obtain the peak value.

Va

Vb

Vc

=

1 1 1

1 a2 a

1 a a2

V0

V1

V2

(4.28)

va = |Va| sin(ωt+ ∠Va) (4.29)

vb = |Vb| sin(ωt+ ∠Vb) (4.30)

vc = |Vc| sin(ωt+ ∠Vc) (4.31)

4.3 Simulation Details

In order to demonstrate the performance of the unbalance controller, simulations were

performed using SimPowerSystems. Figure 4.9 shows an outline of the system. Two 50kW

Sources are connected through coupling inductances (2mH+0.1Ω per phase) to a ∆−Y n

configured transformer, which feeds a star connected resistive load Ra = 4Ω (13.2kW),

Rb = 2Ω (6.6kW), Rc = 4Ω (13.2kW). The load has a neutral which is connected to

the neutral of the secondary of the transformer. Diagrams of the Simulink files for these

simulations can be found in Appendix A.5.


Source/Inverter

Load

Source/Inverter

L

L∆ Y

∆ YVLoadP VLoadS

Figure 4.9: Overall system diagram for unbalanced load test

This is a fairly simplistic model, but it is standard practice to perform initial testing of

inverter controllers with a simple system consisting of one or two sources [85,92,102].

The SimPowerSystems library includes a Sequence Analyzer block, which is used to calcu-

late the phasor magnitude and angle of the sequence components required for the unbal-

ance control. The block uses the phasor version of the symmetrical components transform

(Equation 4.1), meaning it calculates the phasor quantities from the time domain voltages

before performing the transform. This gives rise to one of the limitations of a sequence

analyzer - it initially takes 1 cycle (0.02s at 50Hz) before a sensible value is available at

the output of the block. To minimise the effects of this, the controllers are configured not

to operate for this time.

Another limitation of the sequence analyzer block is that it sets the phase for the negative

and zero sequence to zero if their magnitudes are less than 0.2% of the positive sequence

magnitude. Presumably this is to account for the fact that the phase angle calculations

become less accurate at smaller magnitudes, in which case a practical sequence analyzer

would also need to include this feature. With a positive sequence magnitude of V1 = 325V,

this means the sequence analyzer phase is inaccurate if V2 < 0.65V. To ensure that this

does not cause problems for the controller for small amounts of imbalance, ‘Deadzone’

blocks have been added to turn the controller input off (i.e the output holds its last value)


(a) (b)

Figure 4.10: Load voltage response in the presence of unbalanced loads. (a)

Load voltage space vector and (b) Phase-phase voltages.

(a) (b)

Figure 4.11: Load voltage magnitude with unbalance compensation on: (a) time

range 0-2s and (b) zoomed in view showing 1.6-2s.

if the negative sequence is smaller than a threshold, currently 1V. To avoid a step in

the ‘error’ term (the output of the Deadzone block) the magnitude of the error term is

Verr = VLoad2 meas − 1.

4.3.1 Performance of Unbalance Control

In order to give a baseline performance for comparison, Figure 4.10 shows the load voltage

magnitude in a system where unbalance correction is not applied. There is no voltage reg-

ulation or power sharing - a fixed voltage of 230V is applied by each of the sources. Figure

4.10 (a) shows 100Hz oscillations in the amplitude of the load voltage space vector. This


(a) (b)

Figure 4.12: Source/line currents (a) from 0-0.2s (before unbalance compensa-

tion) and (b) from 1.8-2s (after unbalance compensation)

indicates the presence of unbalanced voltages as shown by the different peak magnitudes

of the phase-phase voltages in Figure 4.10 (b). The unbalanced load creates unbalanced

phase currents, which leads to unbalanced voltage drops across the coupling inductances.

This causes the unbalanced voltages at the load.

Figure 4.11 shows the effect if the unbalance correction is switched on. Voltage regula-

tion and power sharing remain inactive. As mentioned previously, the sequence analyzer

requires one period before sensible measurements are available, and for this reason the

controllers do not operate until T=0.02s. The results show that oscillations in the volt-

ages are substantially reduced compared to the case with no unbalance correction. The

reduction in the voltage imbalance is slower than is desirable, as faster operation of the

control causes it to become unstable. It should be possible to improve this by improving

the sequence analyzer and the filtering algorithms, but this is outside the scope of this

thesis. Due to the lack of voltage regulation the load voltage is lower than the desired

230V.

Figure 4.12 shows the three phase currents at the source at two different time periods: the

first before the unbalance correction is applied, and the second after it is applied. Note

that there are slight differences between the currents in (a) and (b), but this is not obvious

in the figures shown. These results have been included to show that the unbalance control


Sequence Analyzer

Voltage Controller

Power-frequency

Droop Control

Inverter

Negative SequenceCalculation

!V1req

|V1req|

V2req

|V1meas|

!V1meas

Vabcinverter

V2meas

Calculations

Vabcload

Iabcinverter

Pmeas

Vabcinverter

Figure 4.13: Overall control

does not cause large currents to be generated in any of the phases.

4.4 Power Sharing and voltage regulation

Incorporating power sharing and voltage regulation into the unbalance controller involves

changing the angle of the positive sequence from zero to δv (the value output by the power-

frequency droop controller), and the magnitude of the positive sequence to the value output

by the voltage controller (|V1 req|). The result is Equations 4.32 and 4.33. Figure 4.13

shows the overall controller with unbalanced control as well as voltage regulation and

power sharing.

V1 req = |V1 req|∠δv (4.32)


∠(VLoad2 meas + 600) (4.33)

4.4.1 Controller Performance

For this case, the tests are performed with voltage regulation and power sharing switched

on. Figure 4.14 graphs the voltage magnitude of a system with the unbalance correction

switched off, again to show the baseline performance. Figure 4.15 shows the simulated

Load Voltage Magnitude with the unbalance controller switched on. The results show that


Figure 4.14: Load voltage magnitude with voltage regulation and power sharing

(a) (b)

Figure 4.15: Load voltage magnitude with unbalance compensation, voltage

regulation and power sharing (a) Time range 0-2s and (b) zoomed in view showing

1.8-2s.


with the voltage regulation and power sharing added in, the unbalance control is no longer

stable. This is due to the use of a fixed frequency sequence analyzer and is explained in

the following section.

4.4.2 Frequency Issues with the Sequence Analyzer

The problem with combining unbalanced control with power-frequency droop control is

that the sequence analyzer block used to measure the sequence components assumes a

fixed voltage frequency. As power-frequency droop control alters the system frequency,

errors are introduced into the sequence analyzer measurements. The following example

illustrates this problem.

Consider the effect on the sequence components if 3 balanced voltages have their frequency

altered: va(t)

vb(t)

vc(t)

=

V cos(ωt+ δv)

V cos(ωt+ δv − 23π)

V cos(ωt+ δv + 23π)

(4.34)

To do this analysis, it is easier to use the time domain transform (Equation 4.4 repeated

here for convenience):

f0

f1

f2

=1

3

1 1 1

1 −12 +

√3

2 S90 −12 −

√3

2 S90

1 −12 −

√3

2 S90 −12 +

√3

2 S90

fa

fb

fc

(4.35)

The problem lies with the S90 term, which represents a time delay equivalent to a rotation

through 900. For these calculations, S90 is implemented as a phase rotation, however in

a simulation where the voltages are not necessarily pure sinusoids, this would be imple-

mented as a time delayed signal. Assuming a voltage frequency of 50Hz, S90 is assumed

to be 0.005s (T4 s). If the voltage under consideration is actually 51Hz, for example, the


phase rotation S90 then becomes (π2 × 51×2π50×2π ) radians or to generalise (π2 ×

ωactualωfixed

) radians.

Applying the instantaneous symmetrical components transform (Equation 4.35) to Equa-

tion 4.34 gives Equation 4.36 where θ = ωt + δv, and φ = π2 ×

factual−ffixedffixed

, which is

the difference between the phase rotation which occurs in Told4 s at frequencies factual and

ffixed. v0

v1

v2

=

0

12V cos(θ) + 1

2V cos(θ + φ)

12V cos(θ)− 1

2V cos(θ + φ)

(4.36)

Since the original voltages were balanced, they should contain only positive sequence

components. These calculations show, however, that there is apparent negative sequence

(v2 6= 0), which clearly will affect the accuracy of the control. The positive sequence

has also been altered, the expected value being V cos θ. Note that the error is depen-

dent on φ, which is dependent on the difference between the actual frequency and the

expected frequency, so small frequency deviations will not have a significant effect on the

performance.

Extending this analysis to include voltages with actual negative sequence (as opposed to

the apparent negative sequence in the illustration above), shows that the problem worsens.

Equation 4.37 shows a set of unbalanced 3-phase voltages. Assume that applying the sym-

metrical components transform (Equation 4.1) to Equation 4.37 results in Equation 4.38.

To simplify the expressions, any phase offset in the sequence voltages is ignored. These

voltages are transformed into abc voltages in terms of V1, V2, θ1 and θ2 using Equation


4.2, and then the above analysis is repeated to give Equation 4.39.

va

vb

vc

=

Va cos(ωt+ δv + θa)

Vb cos(ωt+ δv + θb)

Vc cos(ωt+ δv + θc)

(4.37)

v0

v1

v2

=

0

V1 cos(θ1)

V2 cos(θ2)

(4.38)

v0

v1

v2

=

0

12V1 cos(θ1) + 1

2V2 cos(θ2) + 12V1 cos(θ1 + φ)− 1

2V2 cos(θ2 + φ)

12V1 cos(θ1) + 1

2V2 cos(θ2)− 12V1 cos(θ1 + φ) + 1

2V2 cos(θ2 + φ)

(4.39)

It is not clear just from looking at these equations what the calculated sequence compo-

nent waveforms would be. A simulation was performed to illustrate the effect. Figure 4.16

shows the magnitude response of the three sequence components when the voltages from

Equation 4.39 with frequency 51Hz are measured by the SimPowerSystems Sequence An-

alyzer operating at 50Hz. V1 = 230Vrms (325Vpeak ), V2 = 75Vrms (106Vpeak), θ1 = ωt

and θ2 = ωt, where ω = 51 ∗ 2π. The oscillation in the negative sequence magnitude

predominantly caused by the 12V1 cos(θ1) − 1

2V1 cos(θ1 + φ) term can clearly be seen.

The positive sequence magnitude also oscillates, but in this instance the oscillations are

small compared to those in the negative sequence. Even if the oscillations are filtered

out, the resulting magnitudes are incorrect since the expected V2 cos(θ2) is replaced by

12V2 cos(θ2) + 1

2V2 cos(θ2 + φ). This error is, however, fairly small, and if φ is small the

effect is insignificant. The effects on the phase measurements is more of a problem, as can

be seen in Figure 4.17. The figure graphs the negative sequence phase angle as calculated

by the sequence analyzer. The actual negative sequence phase angle was a constant 220

for the test. The difference between the actual voltage frequency and the frequency spec-

ified in the sequence analyzer will cause the phase angle measurement from the Sequence

Analyzer to rotate at a rate equal to |factual − ffixed|. Since the unbalance control relies


Figure 4.16: Frequency effects on sequence analyzer magnitude (From top:

positive, negative and zero sequence)

Figure 4.17: Frequency effects on sequence analyzer phase (negative sequence

only)

on having knowledge of the negative sequence phase angle, this error from the Sequence

Analyzer will eventually cause the controller to become unstable and explains the results

shown in Figure 4.15.

4.4.3 Variable Frequency Sequence Analyzer

The solution to this issue is to build a sequence analyzer block which calculates the se-

quence components based on the measured frequency of the input signals, instead of

assuming a fixed frequency.


K1 PLL LPF K2va !f

Figure 4.18: Frequency Calculation

Figure 4.18 shows the method used to calculate the voltage frequency. Gain K1 = 1230∗

√6

normalises the magnitude of va to 1. This normalised voltage signal is fed into a phase

locked loop (PLL) block. The resulting frequency is filtered and then converted to radians

by gain K2 = 2π. It turns out that the sequence analyzer is very sensitive to changes in

frequency and this sensitivity can lead to instabilities in the unbalance control. To prevent

this, the frequency signal is quantised and sampled. This frequency is then used by the

modified sequence analyzer.

The operation of the sequence analyzer is fairly simple. First, the phasor quantities for each

of the three-phase voltages are calculated using Equations 4.40 and 4.41 (the equations

show the a phase calculation only). Equation 4.1 is then applied to the resulting phasor

quantities. The modification simply uses the calculated voltage frequency to determine

the phasor quantities, instead of using a fixed frequency value.

<Va =

ˆva(t)× 2f × sin(2πft)−

ˆva(t− τ)× 2f × sin(2πf(t− τ)) (4.40)

=Va =

ˆva(t)× 2f × cos(2πft)−

ˆva(t− τ)× 2f × cos(2πf(t− τ)) (4.41)

Figures 4.19 and 4.20 show the improvement over Figures 4.16 and 4.17 if the test is

repeated with the modified sequence analyzer block. This time there are no oscillations in

the negative sequence magnitude and the phase remains constant at close to the expected

220.

A slight modification is required to the unbalance control for it to work with the new

sequence analyzer. In the original control, the frequency change was only being applied to


Figure 4.19: Frequency effects on improved sequence analyzer magnitude (From

top: positive, negative and zero sequence)

Figure 4.20: Frequency effects on improved sequence analyzer phase (negative

sequence only)


Figure 4.21: Baseline performance: Load voltage magnitude with variable fre-

quency sequence analyzer. Power-frequency droop control and voltage regulation

are on, unbalance control is off.

the positive sequence. Since the measurements from the sequence analyzer are now being

calculated using this new frequency, the negative sequence needs to be adjusted to take

this into account. Equation 4.42 shows the final control equation.


∠(VLoad2 meas + 600 + δv) (4.42)

4.4.4 Performance of Control with Variable Frequency Sequence Ana-

lyzer

Figures 4.22 and 4.23 show the response of the controller with the modified sequence

analyzer, with Figure 4.21 provided to show the baseline performance of the system when

the unbalance control is off. The controller response is no longer unstable. As previously

mentioned, an issue with the modified sequence analyzer is that it is very sensitive to

changes in the frequency measurement. This means that the frequency measurement

needs to be heavily filtered, which means that a longer time is required before the sequence

analyzer outputs are correct. For this reason, the controllers which rely on data from the

sequence analyzer (voltage regulation and unbalance control) are not switched on until

t = 0.5s.


(a) (b)

Figure 4.22: Load voltage magnitude with variable frequency sequence ana-

lyzer (a) time range 0-2 seconds and (b) zoomed in view showing 1.8-2s. Power-

frequency droop control, voltage regulation and unbalance control are on.

(a) (b)

Figure 4.23: Source/line currents with variable frequency sequence analyzer (a)

from 0-0.2s (before unbalance compensation) and (b) from 1.8-2s (after unbalance

compensation) . Power-frequency droop control, voltage regulation and unbalance

control are on.


Figure 4.24: Magnitude of generated negative sequence voltages

4.5 Sharing of Negative Sequence Generation

Figure 4.24 shows the negative sequence magnitude requested by the controller of two

sources which are supplying different power levels (P ∗ = 0.4 for Source 1 and P ∗ = 0.8 for

Source 2). The negative sequence control is turned on at t = 1.5s. The load is a resistive

load Ra = 4Ω (13.2kW), Rb = 2Ω (6.6kW), Rc = 4Ω (13.2kW). The two plots are identical

so only one line can be seen on the graph. These results show that the two sources are

generating the same amount of negative sequence despite their different power outputs.

It could be argued that it may be desirable for different sized sources to output different

amounts of negative sequence, however the size of the negative sequence correction is

generally small enough that it has very little effect on the real and reactive power output

of the sources (as shown in Figure 4.25), so it is not necessary to implement some form of

sharing mechanism.

4.6 Grid Connection

The unbalance control was tested with a stiff grid and a weak grid. The stiff grid was

simulated by connecting an ideal (unlimited power) three phase voltage source directly to

the load. The weak grid was tested by connecting the three phase voltage source to the


(a) (b)

Figure 4.25: Power output of source with unbalanced load. Unbalance control

turns on at t=1.5s (a) Time Scale 1-3s (b) zoomed in showing 1.4-1.6s

(a) (b)

Figure 4.26: Unbalance correction performance with stiff grid (a) load voltage

magnitude (b) negative sequence voltage output for sources


(a) (b)

Figure 4.27: Unbalance correction performance with weak grid (a) load voltage

magnitude (b) negative sequence current from grid

load through an inductive impedance (to simulate line impedance).

For the stiff grid simulation (Figure 4.26), the grid supplies the negative sequence currents

for the load, and balances the load voltages. The unbalance control is effectively inactive.

This is expected, since the load voltages are forced to be the same as the grid voltages in

this case.

For the weak grid simulation, the grid is unable to completely balance the load voltages,

which means that the unbalance control is active, and will reduce the negative sequence

currents supplied by the grid. Figure 4.27 shows the load voltage magnitude (a) and the

negative sequence current magnitude from the grid (b). The results show that while the

grid initially provides the negative sequence currents for the load (indicated by only small

magnitude oscillations in the load voltage magnitude), when the controller starts up at

t = 2s, the negative sequence in the grid currents begins to decrease. This means the

control is able to minimise the effect of the unbalanced load on the grid.

Chapter 5

Proposed Controller Design

In Chapter 3 the historical development of droop controller design and its variants were

examined. In Chapter 4, the controller was further developed to improve performance un-

der unbalanced load conditions. In this Chapter, a complete controller is outlined which

combines the developed unbalance controller with modified power-frequency droop and

voltage control. This complete controller performs well under a wider range of condi-

tions than previous designs. The performance of this new controller will be evaluated by

simulation.

5.1 Controller Components

There are 4 main aspects to the controller

• Power-frequency droop control with fast response, low transient frequency deviation

and zero steady-state frequency error

• Simple voltage magnitude regulation

• Reactive power sharing even under conditions with different output or line impedances,

and

• Unbalance correction to minimise voltage imbalance in the presence of unbalanced

loads

The controller is split into three main components which are outlined below: Power-

frequency droop control, Voltage Regulation (including reactive power sharing functional-

ity) and Unbalance Control. Figure 5.1 shows the overall controller.

5. Proposed Controller Design 102

Voltage Controller

Power-frequency

Droop ControlInverter

δv

Vmag

|V1 meas|

vabc

Calculationsiabc

vabc

E∗

P ∗

P

Q

Unbalance Control

|V2 meas|

∠V2 meas

V2 reqSequenceAnalyzer

Figure 5.1: Overall outline of controller

5.1.1 Power-frequency Droop Control

The purpose of this component of the controller is to share the real power generated by

different sources, without requiring explicit communication between sources.

The design for this component is based on the work performed by Guerrero et al. [91].

This algorithm (Equation 3.31, repeated here in Equation 5.1) was chosen because it has a

fast response due to the use of the differential term, no steady state frequency error and is

simple to implement, only requiring a bandpass filter and a proportional differential (PD)

controller. A comparison between this algorithm, and the standard droop controller was

performed in Section 3.4.2.

ω = ω∗ −mP P −mdδP

δt(5.1)

The controller has been modified (Equation 5.2) to give the user some control over which

sources generate more power by including the power setpoint P ∗ (the term Perr = P−P ∗).

The dynamic nature of this control (i.e. the use of P , the real power with the DC

component excluded, instead of simply the real power P ) means that a certain power

output cannot be guaranteed at a given system frequency, but the inclusion of the power

setpoint does allow some control over the relative amounts of power generated by different


+-

δV

P ∗

PPD

ControlBandpassFilter

1

s

ω

Figure 5.2: Power frequency droop control

sources. P ∗ should possibly be viewed more as a control parameter rather than a setpoint.

If precise control is required, it would be possible to add an extra control loop which adjusts

the value of P ∗ to give the desired power output, however this has not been implemented

here.

ω = ω∗ −mP˜Perr −md

δ ˜Perrδt

(5.2)

Sources of various sizes are accounted for by using the per unit system to measure power.

Use of the per unit system means that the measured power signal received by the controller

indicates the proportion of maximum power that the source is currently generating, rather

than the actual power generated.

Figure 5.2 shows a diagram of the power-frequency droop control. The controller takes

the power error term P −P ∗ and passes it through a bandpass filter (F (s) = 10∗π∗s(s+π)(s+40π)),

which is effectively a combination of a lowpass filter (cutoff 20Hz) and a highpass filter

(cutoff 0.5Hz) with a gain of 0.25. The purpose of the lowpass filter is to reduce the oscilla-

tions caused by unbalanced voltages (as well as any other high frequency noise in the power

signal). This is particularly important since the unbalance controller will instruct the in-

verter to deliberately generate unbalanced voltages at the source in order to remove them

at the load. The choice of cutoff point requires a balance between eliminating undesired

high frequency signals, and ensuring the response of the controller is not too slow. The

highpass filter eliminates the DC components of the power signal ensuring that the con-

troller only acts to change the frequency while the power output of the source is changing,

thus restoring the frequency to the nominal value once power sharing is achieved. Again

a balance is required in the choice of cutoff frequency, this time between the effectiveness


of the power sharing, and the speed of frequency restoration. A Proportional Differential

(PD) controller (F (s) = −20− 5× 10−3s, chosen by experimentation) is used to generate

a desired frequency which is passed through an integrator to achieve the desired positive

sequence angle δv.

5.1.2 Voltage Regulation

The Voltage Regulation aims to regulate the voltage level at the load, and also to ensure

the sharing of reactive power generation.

The simple voltage controller developed by Lasseter and Piagi [21] based on PI control

of the error between the measured voltage and the desired setpoint has been shown to

be effective in regulating the voltage and is used here with the addition of the command

feedforward term as outlined in Section 3.2.1.

Guerrero et al. [91] are able to obtain good reactive power sharing by using an adaptive

virtual output impedance. The Q-E droop outlined by Lasseter and Piagi [21] is far simpler

and still effective at reactive power sharing. For this reason the latter approach has been

chosen instead. Equation 5.3 and Figure 5.3 show the details of the complete controller.

V =kps+ ki

s(E∗ −mQQ− E) + E∗ (5.3)

The lowpass filtering of the reactive power feedback (cutoff 2Hz) is to minimise the distur-

bance caused by oscillations due to unbalanced voltages, as well as eliminate any other high

frequency noise in the signal. The specific cutoff frequency was chosen by experimentation

to give a balance between the speed of the reactive power sharing, and the elimination of

unwanted high frequency signals. The Q-E droop gradient is mQ = 10V/V Ar(pu). This

value was chosen to be as small as possible (in order to minimise the steady state error

in the voltage magnitude at the load) while still performing the reactive power sharing

adequately. This value of mQ means for every 1pu of reactive power, the voltage will droop


PIVmag

Vmax

Vmin

E∗

E

+-Q

LowpassFilter

mQ

LowpassFilter

+-

+

Figure 5.3: Voltage control with Q-E droop

by 10V RMS. This means at full reactive power load, the error in voltage magnitude at

the load will be 10V (i.e. this is the maximum error). At this maximum the voltage is

still within the allowed tolerances for voltage levels (+10%, −6% AS 60038), and most of

the time the system would be operating at much lower reactive power levels. If stricter

tolerances were required, it should be possible to modify the controller to restore the volt-

age to nominal, similar to the frequency restoration in Section 5.1.1, however this has not

been implemented.

The PI controller was tuned to give a fast stable response. The load voltage magnitude

E is lowpass filtered (cutoff 10Hz) to minimise the effects of any noise in the voltage

measurements used by the sequence analyzer to determine the value of E. The cutoff for

the filter is a compromise between eliminating unwanted noise, and minimising the phase

delay caused by the filter.

5.1.3 Unbalance Control

The Unbalance Control aims to reduce the level of unbalance in the load voltage by

injecting negative sequence voltages at the source.

The unbalance control is implemented as detailed in Chapter 4 and is shown here in

Figure 5.4. The negative sequence is measured at the load and this measurement is used

to determine the negative sequence voltage to be injected by the inverter. Note that the

phase angle correction here is 300, not 600, due to the absence of the transformer, which


+

|V2 req|PIControlDeadzone

|Vload2 meas|

+300

∠Vload2 meas

∠V2 req

LowpassFilter

Figure 5.4: Magnitude and angle calculations for unbalance control

is explained in the following section.

The use of a power-frequency droop control algorithm that includes frequency restoration

means that there is no steady state frequency error. During transients there is some devi-

ation, but this is small and only for short periods of time. This means that it is possible

to use the original, unmodified fixed frequency sequence analyzer from the SimPowerSys-

tems library. It was explained previously that the modified variable frequency sequence

analyzer is very sensitive to changes in frequency, and so better results are obtained with

a fixed frequency sequence analyzer.

5.2 Results

This section outlines the testing performed on the proposed controller. The simulated

system contains two sources, connected to a 3-phase load through a coupling inductance

as shown in Figure 5.5. For the baseline system, the sources are 50kW, the load is a three-

phase balanced load of 39.7kW (4Ω per phase) and the coupling inductance is 20mΩ +

2mH. At t = 2s an extra load is switched in to show the performance of the controllers

during a load change. The overall load for t >= 2s is 48.1kW (3.3Ω per phase), which

represents a 20% load increase. The power setpoint for each of the sources is P ∗ = 0.8,

and the voltage setpoint is E∗ = 230V. The descriptions of the individual tests detail

which values have been altered from the baseline system.


Source/Inverter

Load

Source/Inverter

L

L

Figure 5.5: System used for testing new controller

0 0.5 1 1.5 2215

220

225

230

235Load Voltage Magnitude

Vol

tage

(V

)

Time (s)

(a)

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 249.5

49.6

49.7

49.8

49.9

50

50.1

50.2

50.3

50.4

50.5Load Voltage Frequency

Time (s)

Fre

quen

cy (

Hz)

(b)

Figure 5.6: (a)Load Voltage magnitude and (b)Load Voltage frequency for un-

balanced load (large unbalance) with transformers included

The choice of coupling inductance is based on Section 2.2.3 which indicates that the

maximum coupling inductance for a 50kW source is approximately Lmax = 2.5mH. The

actual value was chosen to be slightly less at L = 2mH. The series resistance is chosen

to be a typical value for an inductor in this power rating. The resistance must be small

enough so that power loss is not significant. Smaller values also result in less voltage drop

(or rise) during a load change, decreasing the disturbance caused to the controllers.

To show the performance of this controller, it is desirable to run the simulations for longer

periods of time than in previous tests. For this reason, the transformers (which require

more time steps to be run, and thus use more memory and take longer to simulate) have

been omitted from these tests. Figures 5.6, 5.7, 5.8 and 5.9 shows the results of a short


0 0.5 1 1.5 20.3

0.35

0.4

0.45

0.5

0.55

0.6

0.65

Real Power

Time (s)

P (

pu)

(a)

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2−0.1

−0.05

0

0.05

0.1

0.15

0.2

0.25

0.3Reactive Power

Time (s)

Q (

pu)

(b)

Figure 5.7: (a)Real Power output and (b)Reactive Power output of Sources for

unbalanced load (large unbalance) with transformers included.

0.4 0.45 0.5 0.55 0.6215

220

225

230


Vol

tage

(V

)

Time (s)

(a)

0.4 0.45 0.5 0.55 0.60.3

0.35

0.4

0.45

0.5

0.55

0.6

0.65

Real Power

Time (s)

P (

pu)

(b)

Figure 5.8: (a)Load Voltage magnitude and (b)Real Power output for unbalanced

load (large unbalance) with transformers included, zoomed in to show detail


0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

5

10

15

20

25Negative Sequence Voltage

Time (s)

Vol

tage

(V

)

Figure 5.9: Negative sequence voltage magnitude at the load for unbalanced

load (large unbalance) with transformers included

simulation with the transformers included, to illustrate that the controller does still work

if they are included. This simulation is based on the large imbalance simulation in Section

5.2.8 later in this Chapter. The rapid reduction in size of the voltage magnitude oscillations

in Figure 5.6 when the controller is switched on at t = 0.5s can clearly be seen. It is not

clear, however, that the oscillations continue to decrease in size. A graph of the magnitude

of the negative sequence voltages at the load (Figure 5.9) has been included to illustrate

this more clearly.

The first five tests illustrate the performance of the controller under balanced load con-

ditions, and the last four examine the behaviour of the unbalance control under different

conditions. Diagrams of the Simulink files for these simulations can be found in Appendix

A.6.

5.2.1 Reference Case

The first simulation is the reference case, the purpose of which is to provide a baseline for

the rest of the tests. The configuration is as described in the previous section, with a bal-

anced 3-phase load. Each inverter is connected through an identical coupling inductance.


0 1 2 3 4 5 6 7 8 9 100.3

0.32

0.34

0.36

0.38

0.4

0.42

0.44

0.46

0.48

0.5Real Power

Time (s)

P (

pu)

(a)

0 1 2 3 4 5 6 7 8 9 100

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1Reactive Power

Time (s)

Q (

pu)

(b)

Figure 5.10: (a) Real Power output and (b) Reactive Power output of Sources

for reference case. Power outputs for both sources are identical.

0 1 2 3 4 5 6 7 8 9 10226

227

228

229

230

231


Vol

tage

(V

)

Time (s)

(a)

0 1 2 3 4 5 6 7 8 9 1049.5

49.6

49.7

49.8

49.9

50

50.1

50.2

50.3

50.4


Time (s)

Fre

quen

cy (

Hz)

(b)

Figure 5.11: (a) Load Voltage magnitude and (b) Load Voltage frequency for

reference case


0 1 2 3 4 5 6 7 8 9 100.1

0.2

0.3

0.4

0.5

0.6

Real Power

Time (s)

P (

pu)

(a)

0 1 2 3 4 5 6 7 8 9 100

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1Reactive Power

Time (s)

Q (

pu)

(b)


with different power setpoints.

Figure 5.10 shows the real and reactive power responses for both sources in the reference

case. The responses are identical, so the two plots (one for each source), which are overlaid

directly on top of each other, appear as a single line. As expected, we see a change in

power output as a result of the load change at t = 2s.

Figure 5.11 shows the voltage magnitude and frequency of the three phase voltage at the

load. The results show that the voltage is regulated to approximately 230V. A slight drop

in voltage occurs during the load change, but the controller quickly corrects for this. The

small amount of steady state error is caused by the Q-E droop. The frequency response

shows that, after a load change, the frequency initially deviates from 50Hz (in this case by a

maximum of 0.2Hz), and then is gradually restored to nominal. Since the DC components

of the signal are filtered out, the input to the PD control which governs the frequency

response corresponds to the change in the term P − P ∗. At t = 0 this term changes from

0 to some negative value (P ∗ = 0.8 which is larger than P in this configuration), and

since the PD control parameters are negative (see Section 5.1.1) the result is an increase

in frequency. Conversely, at t = 2s, P (and correspondingly P − P ∗) increases, resulting

in a decrease in frequency.


0 1 2 3 4 5 6 7 8 9 10226

227

228

229

230

231


Vol

tage

(V

)

Time (s)

(a)

0 1 2 3 4 5 6 7 8 9 1049.5

49.6

49.7

49.8

49.9

50

50.1

50.2

50.3

50.4


Time (s)

Fre

quen

cy (

Hz)

(b)


sources with different power setpoints

5.2.2 Different Power Setpoints

This simulation shows the effect of changing the power setpoint. P ∗ = 0.8 for Source 1

and P ∗ = 0.4 for Source 2. All other parameters are identical to the reference case.

Figure 5.12 (a) shows the power output of the two sources. As expected, the power outputs

for the two sources are no longer the same. As explained in Section 5.1.1, the dynamic

nature of the control does not give precise control over the power output, but it does allow

some control over the relative power outputs of different sources. It can be seen from

Figures 5.12 and 5.13 that the system otherwise functions as expected.

5.2.3 Large Load Change

This simulation looks at the effects of a larger load change on the system. In this case the

load changes to 59.5kW at t = 2s (2.6Ω/phase), representing a 50% increase in the load

power consumption. All other parameters are identical to the reference case.

Figures 5.14 and 5.15 graph the response of the system to a large load change. The

results show that the larger change in load causes a larger drop in load voltage than

in the reference case. During the transient, the voltage level remains within acceptable

tolerances, and the voltage drop is quickly corrected by the voltage controller.


0 1 2 3 4 5 6 7 8 9 100.3

0.35

0.4

0.45

0.5

0.55

0.6

0.65

Real Power

Time (s)

P (

pu)

(a)

0 1 2 3 4 5 6 7 8 9 100

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1Reactive Power

Time (s)Q

(pu

)

(b)


for reference case with large load change. Power outputs for both sources are

identical.

0 1 2 3 4 5 6 7 8 9 10220

222

224

226

228

230

232

234

236

238


Vol

tage

(V

)

Time (s)

(a)

0 1 2 3 4 5 6 7 8 9 1049.5

49.6

49.7

49.8

49.9

50

50.1

50.2

50.3

50.4


Time (s)

Fre

quen

cy (

Hz)

(b)


reference case with large load change.


5.2.4 Different Sized Sources

The aim of this test is to show that the power-frequency droop control shares power

between sources with different power ratings. For this test Source 1 is 50kW and Source

2 is 100kW. Section 2.2.3 demonstrated that the size of coupling inductance should vary

depending on source size. A doubling in power rating means the coupling inductance

should be halved, so the chosen inductance should be 10mΩ + 1mH. Since an identical

ratio between the coupling inductance and the source size will automatically cause the

real power to be shared evenly without need for any control, the value of the coupling

inductance was chosen to be 11mΩ + 1.1mH to allow the performance of the controller

to be shown. It is likely, in practice, that this situation would frequently occur, since

inductor ratings are only guaranteed to be accurate to within a certain percentage of the

rated value. The load is a balanced three-phase load of 39.7kW (4Ω per phase), which is

increased to 48.1kW (3.3Ω per phase) at t = 2s.

The aim of the power sharing is for each source to share power proportional to its size, so

in this case it is expected that the 100kW source will supply twice the power of the 50kW

source. Since the power measurements are displayed in per unit, successful power sharing

is indicated by the same per unit power output for each source.

Figure 5.16 shows the real and reactive power responses for both sources. Source 1 is

shown as a dotted line, and Source 2 is a solid line. The results show that the sources

are generating similar amounts of power in per unit terms (i.e. the same proportion of

their maximum power output), although there is a noticeable steady state error in the real

power sharing. This error is a result of the controller tuning. It is possible to decrease the

error, which would result in either a larger deviation in the system frequency (changing mP

or md), or a longer time before the frequency returned to nominal (changing the bandpass

filter cutoff frequencies).

Figure 5.17 shows the load voltage magnitude and frequency response of the three phase

voltage at the load. As expected, the voltage is regulated to approximately 230V, with a


0 1 2 3 4 5 6 7 8 9 100.2

0.22

0.24

0.26

0.28

0.3

0.32

0.34

0.36

0.38

0.4Real Power

Time (s)

P (

pu)

(a)

0 1 2 3 4 5 6 7 8 9 100

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0.045

0.05Reactive Power

Time (s)

Q (

pu)

(b)

Figure 5.16: (a) Real Power output and (b) Reactive Power output for different

sized sources

0 1 2 3 4 5 6 7 8 9 10226

227

228

229

230

231


Vol

tage

(V

)

Time (s)

(a)

0 1 2 3 4 5 6 7 8 9 1049.5

49.6

49.7

49.8

49.9

50

50.1

50.2

50.3

50.4


Time (s)

Fre

quen

cy (

Hz)

(b)

Figure 5.17: (a) Load voltage magnitude and (b) Load voltage frequency for

different sized sources

small amount of steady state error caused by the Q-E droop (the mQQ term) in the voltage

controller (Eadj = E∗ − mQQ). The frequency vs. time plot shows that the frequency

deviation is slightly smaller in this case. This is because the smaller per unit load change

(due to the larger source) leads to a smaller output from the PD controller that governs

the output voltage frequency of the source (see Section 5.1.1).

5.2.5 Different Voltage Setpoints

The aim of this test is to show how the system behaves if there is an error in the setpoint

used for the voltage regulation. To perform this test, the voltage setpoint for Source 2 is


0 1 2 3 4 5 6 7 8 9 100.3

0.32

0.34

0.36

0.38

0.4

0.42

0.44

0.46

0.48

0.5Real Power

Time (s)

P (

pu)

(a)

0 1 2 3 4 5 6 7 8 9 100

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1Reactive Power

Time (s)

Q (

pu)

(b)


voltage setpoints

changed from the original E∗ = 230V to E∗ = 229.9V . All other settings remain identical

to the reference case.

Figure 5.18 graphs the real and reactive power output for both sources. It was shown

in Section 3.3.2 that without effective control, an error in the voltage setpoint will cause

one source to produce increasingly more reactive power, and the other source will absorb

increasingly more reactive power. The results here show that the reactive power output for

Source 1 begins increasing, and the reactive power output for Source 2 begins decreasing,

but the control is able to level this off, preventing large reactive currents from circulating

between the sources in steady state. It is possible to decrease the steady state difference

in the reactive power output by increasing the Q-E droop gradient mQ but this would

increase the steady state voltage error which becomes significant at higher reactive power

levels. A more detailed explanation for this was presented in Section 3.4.1.1.

Figure 5.19 graphs the load voltage magnitude and frequency. These results show that

there is a small steady state error in the voltage regulation, which again is due to the Q-E

droop. This error remains quite small and well within acceptable tolerances for voltage

levels. Again we see the frequency reacting to the load change and gradually being restored

to nominal.


0 1 2 3 4 5 6 7 8 9 10226

227

228

229

230

231


Vol

tage

(V

)

Time (s)

(a)

0 1 2 3 4 5 6 7 8 9 1049.5

49.6

49.7

49.8

49.9

50

50.1

50.2

50.3

50.4


Time (s)

Fre

quen

cy (

Hz)

(b)


different voltage setpoints

0 1 2 3 4 5 6 7 8 9 100.3

0.32

0.34

0.36

0.38

0.4

0.42

0.44

0.46

0.48

0.5Real Power

Time (s)

P (

pu)

(a)

0 1 2 3 4 5 6 7 8 9 100

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1Reactive Power

Time (s)

Q (

pu)

(b)


coupling inductances

5.2.6 Different Coupling Inductances

The aim of this test is to show the performance of the real and reactive power sharing in

the presence of different coupling impedances between the source and the load in the case

of a balanced load. To perform the simulation, the model is changed from the reference

case so that the coupling impedance for Source 2 is 22mΩ and 2.2mH. This is an increase

of 10% over the original impedance value. All other settings are unchanged from the

reference case.

Figure 5.20 graphs the real and reactive power output for both sources. The solid line


0 1 2 3 4 5 6 7 8 9 10226

227

228

229

230

231


Vol

tage

(V

)

Time (s)

(a)

0 1 2 3 4 5 6 7 8 9 1049.5

49.6

49.7

49.8

49.9

50

50.1

50.2

50.3

50.4


Time (s)

Fre

quen

cy (

Hz)

(b)


different coupling inductances

shows Source 1, and the dotted line is Source 2. These results show that the controller

is able to achieve real and reactive power sharing in the presence of different coupling

inductances. Figure 5.21 graphs the voltage magnitude and frequency at the load. These

results shows that there is a small steady state error in the voltage regulation, which again

is due to the reactive power sharing Q-E droop. The frequency response is similar to in

previous simulations.

To show the impact of a large difference in coupling inductance, the coupling impedance

for Source 1 is changed to 10mΩ and 1mH and the impedance for Source 2 is restored to

the reference case values (20mΩ and 2mH).

Figures 5.22 and 5.23 show the results in this case. It can be seen that the reactive power

sharing still functions as expected, but the steady state error in the power sharing is more

significant. This is because the different impedances, in the absence of power sharing,

would cause the inverters to supply different amounts of real power, dependent on the size

of impedance. The power sharing therefore has a larger error to correct, resulting in a

larger steady state error.


0 1 2 3 4 5 6 7 8 9 100.2

0.25

0.3

0.35

0.4

0.45

0.5

0.55

Real Power

Time (s)

P (

pu)

(a)

0 1 2 3 4 5 6 7 8 9 100

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1Reactive Power

Time (s)

Q (

pu)

(b)

Figure 5.22: (a) Real Power output and (b) Reactive Power output for large

difference in coupling inductances

0 1 2 3 4 5 6 7 8 9 10226

227

228

229

230

231


Vol

tage

(V

)

Time (s)

(a)

0 1 2 3 4 5 6 7 8 9 1049.5

49.6

49.7

49.8

49.9

50

50.1

50.2

50.3

50.4


Time (s)

Fre

quen

cy (

Hz)

(b)


large difference in coupling inductances

0 1 2 3 4 5 6 7 8 9 100.3

0.35

0.4

0.45

0.5

0.55

Real Power

Time (s)

P (

pu)

(a)

0 1 2 3 4 5 6 7 8 9 100.2

0.22

0.24

0.26

0.28

0.3

0.32

0.34

0.36

0.38

0.4Reactive Power

Time (s)

Q (

pu)

(b)

Figure 5.24: (a) Real Power output and (b) Reactive Power output for reactive

load


0 1 2 3 4 5 6 7 8 9 10210

212

214

216

218

220

222

224

226

228


Vol

tage

(V

)

Time (s)

(a)

0 1 2 3 4 5 6 7 8 9 1049.5

49.6

49.7

49.8

49.9

50

50.1

50.2

50.3

50.4


Time (s)

Fre

quen

cy (

Hz)

(b)


reactive load

5.2.7 Reactive Load

The aim of this test is to show how the system behaves when there is a balanced reactive

load. The load (25.2kV Ar + 39.9kW ) is an inductor (with a small series resistance) of

0.1Ω + 20mH per phase, in parallel with a resistor of 4Ω per phase. At t = 5s the 4Ω

resistor is changed to 3.3Ω, adjusting the total load to (25.2kV Ar + 47.8kW ). As in the

previous test, the coupling inductance for Source 2 was was 22mΩ and 2.2mH. All other

values are the same as the reference case.

Figure 5.24 graphs the real and reactive power output for both sources. The results show

that the real power output for both sources is closely matched, with the power-frequency

droop control causing a slight reduction in the difference in real power output between the

two sources. Figure 5.24 (b) shows that the Q-E droop is able to significantly reduce the

difference between the reactive power output of the two sources. This illustrates that the

Q-E droop is able to function correctly when the load is reactive.

Figure 5.25 graphs the load voltage magnitude and frequency response. As expected,

these results show that there is a small steady state error in the voltage regulation, which

again is due to the Q-E droop. This error remains quite small and well within acceptable

tolerances. The frequency response is again similar to previous simulations.


5.2.8 Unbalanced Load

The aim of this test is to show how the system behaves when there is an unbalanced load.

Two sets of results are obtained - one for a small amount of imbalance, and one for a large

imbalance. In both cases the load is purely resistive, and Rb is altered from the original.

For the small unbalance, initially Rb = 3.5Ω (giving a total load of 41.4kW, which is

increased to 49.7kW at t = 2s) and for the large imbalance initially Rb = 2Ω (giving a

total load of 52.9kW, which is increased to 63.5kW at t = 2s).

Figures 5.26 and 5.29 graph the real and reactive power output for both sources for the

small and large unbalance cases. Note that the values shown are the unfiltered 3-phase

power calculations. The oscillations in the power that can be seen in these plots are caused

by the Unbalance Control, which generates negative sequence voltages at the source in

order to decrease the negative sequence voltages at the load.

Figures 5.27 and 5.30 graphs the load voltage magnitude and frequency of both sources for

the small and large unbalance cases. In both cases the decrease in the magnitude of the

oscillations in the load voltage magnitude indicates that the amount of unbalance in the

load voltages is decreasing, which is the desired response of the controller. The frequency

response is similar to the balanced load cases, indicating that the unbalanced load is not

significantly interfering with the operation of the power-frequency droop.

Figures 5.28 and 5.31 show the voltage magnitude and real power output for the time

period just after the load change in more detail. This has been included here to show the

detail that is not clear from the normal view. The reactive power response is similar and

is therefore not shown.

5.2.9 Unbalanced Reactive Load

The aim of this test is to show how the system behaves when there is an unbalanced

reactive load. The test is similar to Section 5.2.7 with the phase b inductor changed to


0 1 2 3 4 5 6 7 8 9 100.3

0.35

0.4

0.45

0.5

0.55

Real Power

Time (s)

P (

pu)

(a)

0 1 2 3 4 5 6 7 8 9 100

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1Reactive Power

Time (s)

Q (

pu)

(b)

Figure 5.26: (a) Real Power output and (b) Reactive Power output for simulation

with unbalanced load (small imbalance)

0 1 2 3 4 5 6 7 8 9 10225

226

227

228

229

230

231

232

233

234


Vol

tage

(V

)

Time (s)

(a)

0 1 2 3 4 5 6 7 8 9 1049.5

49.6

49.7

49.8

49.9

50

50.1

50.2

50.3

50.4


Time (s)

Fre

quen

cy (

Hz)

(b)


unbalanced load (small imbalance)

2 2.05 2.1 2.15 2.2 2.25 2.3 2.35 2.4228

228.5

229

229.5

230

230.5


Vol

tage

(V

)

Time (s)

(a)

2 2.05 2.1 2.15 2.2 2.25 2.3 2.35 2.40.45

0.46

0.47

0.48

0.49

0.5

0.51

0.52

0.53

0.54

0.55Real Power

Time (s)

P (

pu)

(b)

Figure 5.28: (a) Load voltage magnitude and (b) Real Power output for unbal-

anced load (small imbalance). Graphs are zoomed in to show detail.


0 1 2 3 4 5 6 7 8 9 100.2

0.3

0.4

0.5

0.6

0.7

0.8Real Power

Time (s)

P (

pu)

(a)

0 1 2 3 4 5 6 7 8 9 10−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

0.2

0.25

0.3Reactive Power

Time (s)

Q (

pu)

(b)

Figure 5.29: (a) Real Power output and (b) Reactive Power output for unbal-

anced load (large imbalance)

0 1 2 3 4 5 6 7 8 9 10220

222

224

226

228

230

232

234

236

238


Vol

tage

(V

)

Time (s)

(a)

0 1 2 3 4 5 6 7 8 9 1049.5

49.6

49.7

49.8

49.9

50

50.1

50.2

50.3

50.4


Time (s)

Fre

quen

cy (

Hz)

(b)


anced load (large unbalance)

2 2.05 2.1 2.15 2.2 2.25 2.3 2.35 2.4225

226

227

228

229

230

231

232

233

234


Vol

tage

(V

)

Time (s)

(a)

2 2.05 2.1 2.15 2.2 2.25 2.3 2.35 2.40.4

0.45

0.5

0.55

0.6

0.65

0.7

0.75

0.8Real Power

Time (s)

P (

pu)

(b)


anced load (large unbalance). Graphs are zoomed in to show detail.


0 1 2 3 4 5 6 7 8 9 100.2

0.3

0.4

0.5

0.6

0.7

0.8Real Power

Time (s)

P (

pu)

(a)

0 1 2 3 4 5 6 7 8 9 100

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5Reactive Power

Time (s)

Q (

pu)

(b)

Figure 5.32: (a) Real Power output and (b) Reactive Power output for simulation

with unbalanced reactive load

0.2Ω+40mH, and the resistor to 2Ω, giving a total load of 21kV Ar+40kW . At t = 5s the

resistive load in each phase is increased by 20% (giving a new load of 21kV Ar + 48kW ).

Figure 5.32 and 5.33 show the results for an unbalanced reactive load. The graphs show

that the controllers perform as expected. The voltage magnitude response shows the

oscillations reducing in size, and the steady-state error in the voltage magnitude is due to

the Q-E droop.

Figure 5.34 shows the voltage magnitude and real power output for the time period just

after the load change in more detail. This has been included here to show the detail that

is not clear from the normal view. The reactive power response is similar and is therefore

not shown.

5.2.10 Unbalanced Load and Different Coupling Inductances

The aim of this test is to show that the negative sequence control performs in the presence

of different coupling inductances between the sources and the load. The test is a combina-

tion between the Unbalanced Load test, and the Different Output Inductances test, so for

this test both the load and the coupling inductances are altered from the reference case.

The load is configured with a large unbalance (Rb = 2Ω, total load 52.9kW, increasing to


0 1 2 3 4 5 6 7 8 9 10210

215

220

225

230

235


Vol

tage

(V

)

Time (s)

(a)

0 1 2 3 4 5 6 7 8 9 1049.5

49.6

49.7

49.8

49.9

50

50.1

50.2

50.3

50.4


Time (s)

Fre

quen

cy (

Hz)

(b)


unbalanced reactive load

5 5.05 5.1 5.15 5.2 5.25 5.3 5.35 5.4220

221

222

223

224

225

226

227

228

229


Vol

tage

(V

)

Time (s)

(a)

5 5.05 5.1 5.15 5.2 5.25 5.3 5.35 5.40.2

0.3

0.4

0.5

0.6

0.7

0.8Real Power

Time (s)

P (

pu)

(b)

Figure 5.34: (a) Load voltage magnitude and (b)Real Power output for unbal-

anced reactive load. Graphs are zoomed in to show detail.


0 1 2 3 4 5 6 7 8 9 100.2

0.3

0.4

0.5

0.6

0.7

0.8Real Power Source 1

Time (s)

P (

pu)

(a)

0 1 2 3 4 5 6 7 8 9 100.2

0.3

0.4

0.5

0.6

0.7


Time (s)

P (

pu)

(b)

Figure 5.35: Real Power output for (a) Source 1 and (b) Source 2 for unbalanced

load with different coupling inductances

63.5kW at t = 2s), and the configuration for the coupling inductances is identical to the

Different Output inductances test (coupling impedance for Source 2 is 22mΩ and 2.2mH).

Figure 5.35 graphs the real power output for both sources. Figure 5.37 graphs the reactive

power for both sources. For this simulation the sources have been graphed separately

because the different coupling inductances mean that the sources have slightly different

power outputs, and the oscillations make it difficult to see this when both plots are on the

same axes. From the graphs, it can be seen that the different output inductances causes

the sources to have slightly different magnitude oscillations, but the average value of the

power output is the same for both sources. The oscillations in the power are again caused

by the negative sequence control, which generates negative sequence voltages at the source

in order to decrease the negative sequence voltages at the load. A more detailed view is

shown in Figure 5.36.

Figure 5.38 shows the load voltage magnitude and frequency. A more detailed view of

the load voltage magnitude is shown in Figure 5.39. The decrease in the magnitude of

the oscillations over time indicates that the unbalance at the load is decreasing, thus the

unbalance control still functions correctly with different coupling inductances.


2 2.05 2.1 2.15 2.2 2.25 2.3 2.35 2.40.2

0.3

0.4

0.5

0.6

0.7


Time (s)

P (

pu)

(a)

2 2.05 2.1 2.15 2.2 2.25 2.3 2.35 2.40.2

0.3

0.4

0.5

0.6

0.7


Time (s)

P (

pu)

(b)

Figure 5.36: Real Power output for (a) Source 1 and (b) Source 2 for unbalanced

load with different coupling inductances. Graphs are zoomed in to show detail.

0 1 2 3 4 5 6 7 8 9 10−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

0.2

0.25

0.3Reactive Power Source 1

Time (s)

Q (

pu)

(a)

0 1 2 3 4 5 6 7 8 9 10−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

0.2

0.25

0.3Reactive Power Source 2

Time (s)

Q (

pu)

(b)

Figure 5.37: Reactive Power output for (a) Source 1 and (b) Source 2 for

unbalanced load with different coupling inductances

0 1 2 3 4 5 6 7 8 9 10220

222

224

226

228

230

232

234

236

238


Vol

tage

(V

)

Time (s)

(a)

0 1 2 3 4 5 6 7 8 9 1049.5

49.6

49.7

49.8

49.9

50

50.1

50.2

50.3

50.4


Time (s)

Fre

quen

cy (

Hz)

(b)


unbalanced load with different coupling inductances


2 2.05 2.1 2.15 2.2 2.25 2.3 2.35 2.4220

222

224

226

228

230

232

234

236

238


Vol

tage

(V

)

Time (s)

Figure 5.39: Load voltage magnitude for unbalanced load with different coupling

inductances. Graph is zoomed in to show detail.

5.3 Controller Parameters

This section investigates the effects on the controller response of varying the controller

parameters in the unbalance control.

Figure 5.40 shows the effects of varying the gains on the unbalance control. The graphs

measure the time taken for the load voltage unbalance to decrease below 2% unbalance.

This % unbalance is obtained by v2v1

where v2 is the negative sequence voltage at the load,

and v1 is the positive sequence voltage. Figure 5.40 (a) shows the effects of modifying the

proportional gain kp. Figure 5.40 (b) shows the effects of modifying the integral gain ki.

The results show that when kp is increased the speed of response is increased. When ki is

increased the speed of response only changes slightly.

5.4 Conclusion

A new controller has been developed by combining techniques from various authors with

a new technique for unbalanced loads. The results from this Chapter show that the con-

troller performs well in various conditions, including unbalanced loads, different coupling

inductances, frequency errors and voltage setpoint errors. It should be noted that the

ability of the controller to remove unbalance at the load is limited by the voltage and

current ratings of the inverter. This is because, in the process of injecting negative se-


0 2 4 6 8 100

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2T

ime

(s)

kp

(a)

0 2 4 6 8 100

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Tim

e (s

)

ki

(b)

Figure 5.40: Response time for load voltage unbalance to decrease below 2%

(a) Proportional gain kp (b) Integral Gain ki

quence voltages, the voltage and current in some of the phases may increase. For the

system configuration and load scenarios presented in these simulations, the effect is trivial

(increases of up to a few percent). In other configurations, this may become a signifi-

cant consideration, necessitating either an increase in the inverter ratings, or toleration of

higher amounts of unbalance.

Chapter 6

Power Sources and Storage

Different power sources (e.g. microturbines, fuel cells and batteries) exhibit different

responses to changing load demands. A limitation of the previous work in this thesis,

and in much of the body of work concerned with power sharing algorithms, is that the

dynamics of these power sources are ignored.

For the purposes of this research, most of these dynamics can be assumed to have little

effect on the common AC bus due to the high frequency switching of the inverter. One

effect that does need to be allowed for, however, is the limit on the amount of power that

can be supplied by a power source. Most of the power sources of interest in a microgrid

have little or no “inertia” and are thus incapable of instantaneously supplying increases

in power.

Generally this problem is solved by placing storage (i.e. a battery) on the DC bus of

each voltage source (a voltage source consists of a power source, DC storage and power

electronics), however this can be an expensive and inefficient solution. A better solution

is to place storage devices on the AC bus and minimise the DC storage required for

each voltage source. In order for these AC storage devices to operate correctly alongside

droop controlled sources, a control algorithm is required which does not interfere with the

power-frequency droops.

In this Chapter, transfer function models of these power limits are examined, and issues

with these models are discussed. Several methods are then outlined which try to prevent

voltage sources from exceeding these power limits in steady state. Finally, these models

are used in the development of a controller for a system based storage device connected to

6. Power Sources and Storage 131

m

P

!V

Pmax

0

PowerSource

+- +

-1s

!!

P!

Figure 6.1: Power-frequency droop controller including Power Source block

the common AC bus (referred to as grid connected storage), aimed at minimising the DC

storage required by the individual voltage sources as well as allowing the power sharing

to operate as required.

The simulations in this Chapter are developed using the model library in Chapter 2 in

preference to SimPowerSystems since the model library is able to simulate faster and

varying the load resistance is easier. The controller design in this Chapter is based on the

controller outlined in Chapter 3. It could equally have been applied to the controller from

Chapter 5, however this work was completed prior to the design of the new controller.

6.1 Power Source Block

In order to model the power limits of various different power sources, CERTS [22] proposed

introducing a Power Source or Prime Mover Rate block (shown in Figure 3.5 repeated

here in Figure 6.1), which calculates the maximum power output available from the power

source at the current time. This is implemented as a limit which is imposed on the power

setpoint P ∗, such that P ∗ cannot be larger than the power available from the Power Source

at that time.

These source blocks would have different behaviour depending on the source that is being

modelled. It is not the aim of this work to model the exact mechanical and or chemical

processes involved in each source. Instead, this research will draw on published work that

models the typical transient power output of common sources [22].


DC Source

+OutputFilter

DC Storage Inverter Coupling Inductance

V ∠δV E∠δE

Figure 6.2: Voltage Source

6.1.1 Transfer Function Models

In [22] the authors propose that the maximum power output for the various sources can

be modelled using transfer functions. Transfer function models for various sources are

described here.

6.1.1.1 Ideal DC Source (Infinite Capacity Battery)

The simplest possible model of a power source is an ideal DC source. This source provides

the required DC voltage with a certain maximum power. The source can provide any

amount of power at any given time, provided that it is less than the maximum power.

This is equivalent to having sufficient battery capacity on the DC side of the inverter to

supply any power gap between the power supplied by the source, and the power output by

the inverter (see Figure 6.2). The model ignores the possibility of the storage becoming

exhausted. In the simulation an ideal DC source is implemented as fixed limits on P ∗ of

0− Pmax .

6.1.1.2 Microturbines

A microturbine is a form of gas-turbine. Air and fuel are both supplied to a combustion

chamber where the fuel burns, creating high pressure gases which are then used to power

the turbine. The power output is controlled by adjusting a valve to increase or decrease

the amount of fuel entering the combustion chamber. Since it takes time for the flow of

air into the combustion chamber to increase, a microturbine cannot instantly supply an


Figure 6.3: Micro-turbine response to step change in Power request (sourced

from [22])

0 10 20 30 40 50 600

0.1

0.2

0.3

0.4

0.5

0.6

time(s)

Pow

er (

W)

Power RequestPower Output

Figure 6.4: Approximation of Micro-turbine response to step change in Power

request

increase in power. In [22] the authors found that initially, when the fuel intake is increased,

there is a drop in power output, then a gradual increase until the power output reaches

the desired value. The plot of their results is reproduced here in Figure 6.3. The response

of a microturbine to a step change in the fuel valve pressure can be found in [22] and the

response to a load change can be found in [103] and [104]. Clearly the load change response

is dependent on the control algorithm used to determine the desired fuel valve pressure,

but generally there will be some delay between the load change, and the microturbine

reaching the required power output, unless there is some form of storage.

In [22] (CERTS), this behaviour is approximated by Equation 6.1, with the function

limited to between 0 and Pmax. The response is shown in Figure 6.4. Testing outlined


P

t

P0

P0 + 20%

Preq

t1

10s of seconds

Figure 6.5: Fuel cell response to step change in power request

in [22] shows the validity of this model. The initial decrease in power output due to a step

change in fuel input does not appear to translate into a decrease in power output due to

a load change. Other results in literature ( [103], [104]) agree with this result. In [22] it

is suggested that the time constant τ could be as much as 10s. The microturbine transfer

function model has been used in the simulations later in this section.

1

1 + τs(6.1)

6.1.1.3 Fuel Cells

A fuel cell uses a chemical reaction to produce electricity, similar to a battery, except that,

unlike a battery, a fuel cell requires a continuous supply of fuel. By regulating the amount

of reactant (fuel) supplied to the fuel cell, the power output of the cell can be controlled.

When an increase in power output is requested, the amount of reactant in the fuel cell is

increased. According to [22] a fuel cell can provide a 20% change in power output in a

few seconds, but after that it can require up to several minutes to establish equilibrium

before it can provide another step change in power output. An illustration of the response

of a fuel cell to a step change in the power request can be found in [22]. Figure 6.5 shows

an approximation of this behaviour. At t = t1 the requested power level is changed from

P0 to Preq. Immediately, at t = t1, the power output begins to increase until it reaches

P0 + 20%. It remains at this level until it reaches chemical equilibrium (10s of seconds or


minutes depending on the fuel cell) and then increases until it reaches Preq.

6.1.1.4 Renewable Power Sources

Renewable power sources generally rely on energy which is directly (as in photovoltaics),

or indirectly (wind, hydro etc.) taken from the sun. This means that the power available

from these sources cannot be controlled, and their behaviour is unpredictable. No transfer

function approximation has previously been proposed for renewable sources. Since the

model is dealing with short term phenomena, renewable power sources could be approxi-

mated by a step change, in order to examine the worst case scenario. If a more accurate

representation is required then a possibility is to choose a suitable random distribution as

the power output of a renewable source.

6.1.2 Behaviour of Power Source Blocks

These transfer function models have a limitation; they are only effective in grid-connected

mode. In grid-connected mode, the controller is designed so that each source will output

power according to the setpoint P ∗. If a limit is placed on this setpoint, then this limits

the steady state power output of the source (provided the grid frequency is at its nominal

value). In stand-alone mode, placing a limit on P ∗ does not place any limit on the actual

power output by the inverter, since the frequency is allowed to vary from nominal. When

the microgrid switches to stand-alone mode, the power controller is designed to increase

or decrease the power output in order to match the load. This design means that the

inverter power output can exceed P ∗, and will, every time the microgrid disconnects from

the main power grid after it has been importing power from the main grid. This means

that some other measure for limiting the inverter power output is required for stand-alone

mode. It is also important for the power reference for the power source (i.e. microturbine,

fuel cell) to be adjusted to match the requirements for supplying the load, but that is

outside the scope of this work. Note that the inverter would be designed with protection


!min

!

P ! P !+

Pmax! Pmax+

!! P !"

Pmax

P

Figure 6.6: Droop characteristics for different P ∗ values

mechanisms to disconnect it if the power output became too high.

If the droop gradient m is maintained constant, then changing P ∗ just moves the power-

frequency droop characteristic to the left or right (see Figure 6.6). This alters the theo-

retical value of Pmax (as can be seen from Pmax+ and Pmax−), but it does not prevent the

power from exceeding P ∗, which is the desired maximum power output in this case. Since

there is nothing in the existing design to prevent the voltage frequency from exceeding the

limits ωmax and ωmin, then the value of Pmax is meaningless in any case.

Simulations were performed to illustrate these problems. Diagrams of the Simulink files

for these simulations can be found in Appendix A.7. Two 6kW inverters are connected

in parallel through coupling inductances (0.01H + 0.5Ω) to a balanced 3-phase load of

3.5kW (15Ω) per phase. The power setpoint P ∗ for both inverters was 0.5pu. Figure 6.7

shows the inverter power output of one of the inverters. The graph clearly shows that

the inverter power output can exceed P ∗. If the load is increased to 4.4kW (12Ω) per

phase (Figure 6.8(a)), the power output of the inverter now exceeds 1pu, theoretically the

maximum power of the source. If the model was accurate, some voltage drop would be

expected, however Figure 6.8 (b) shows that the voltage remains regulated to 230V.

Figure 6.9 compares the power output of two sources, one using the Ideal DC Source

model with Pmax = 1pu and the other using the Microturbine model. The load is again


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

time(s)

Pow

er (

pu)

Figure 6.7: Inverter power output for P ∗ = 0.5 and load per phase of 15Ω

showing P exceeding P ∗

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.2

0.4

0.6

0.8

1

1.2

1.4

time(s)

Pow

er (

pu)

(a)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1200

205

210

215

220

225

230

235

240

245

250

time(s)

Load

Vol

tage

(V

)

(b)

Figure 6.8: (a) Power output and (b) Load voltage for P ∗ = 0.5 and load per

phase of 12Ω, showing P exceeding Pmax

0 2 4 6 8 10 12 14 16 18 200

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

time(s)

Pow

er (

pu)

Microturbine Source (6kW) Power OutputMicroturbine Maximum Power Available (Source 1)Ideal Source (12kW) Power Output

Figure 6.9: Power output with microturbine source, showing poor match between

microturbine power available and inverter power output


a balanced 3-phase load of 3.5kW (15Ω) per phase. The dotted line at the bottom of the

figure shows the maximum power available from the microturbine slowly increasing with

time. The solid line in the middle is the inverter power output from the Microturbine

Source block, and the dashed line at the top shows the power output from the other

inverter using the Ideal DC Source block. The figure shows that the power source models

do have some effect on the power output of the inverters (note the difference in the power

outputs of the two inverters), however the difference between the power available from the

microturbine, and the power output of the inverter is considerable. Note that the change

in power output is only in the way it is shared between the sources. As the model stands,

the power required by the load will be supplied by the sources, regardless of how much

power is ‘available’, which is clearly unrealistic.

The purpose of these source models is to show the effects of different power sources on

the microgrid. Since, as these simulations clearly show, these models do not restrict the

power used by the inverter, then another method is needed for modelling the power limits

imposed by the sources. These results also highlight a problem with the control scheme; if

there were some kind of storage (i.e. battery or capacitor), on the DC bus of the inverter,

allowing for the provision of more than 1pu power, the controller would quickly exhaust

this storage.

6.2 Modifications to Source Models

As illustrated in the previous section, inclusion of power source blocks in the Power-

frequency droop controller will not limit the power output to Pmax. This creates two

problems. Firstly, their use in the controller design will allow the inverter to drain the DC

storage and secondly, they cannot be used to create realistic models, which makes them

unsuitable for the task of designing an AC connected storage device. Two methods are

outlined below which address these problems.


+- PI

Vmag

Vmax

Vmin

Eerror

E!

E

+-nQ

Q

+-

+-

P

Pmax PI

Vmag!adj

Figure 6.10: Voltage Control with voltage drop

6.2.1 Forcing Drop in Voltage when Power is not Available

In a physical system, insufficient power availability would cause the voltage to drop, loads

may be shed, and if the deficit were severe enough protection mechanisms would be trig-

gered and possibly the system would go into shutdown. In the simulation, there is currently

no mechanism to induce this behaviour. Ignoring the operation of protection mechanisms,

since this work is not attempting to model faults, this problem could be avoided by altering

the voltage magnitude at the inverter output until the correct amount of power is being

output (Equation 6.2). This technique would also avoid draining the batteries excessively.

Note that this technique will only work in the case of resistive (V = IR) loads, or when

there is another source with more power available that can pick up the slack. In the case

of a constant power load, the load will keep drawing the power it needs until the voltage

collapses on the DC bus, and the power source shuts down.

Vmag adj = Vmag −1

s(P − Pmax) (6.2)

The term 1s (P − Pmax) is a measure of the energy (or work) which would be required to

come from storage. A negative value indicates that the storage is being recharged. If this

energy is plotted, it can be used to give an approximation of how much storage is required,

although a more accurate method is recommended for actual design decisions. To improve

the dynamic performance of the corrections, a PI controller is employed instead of the


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.2

0.4

0.6

0.8

1

time(s)

Pow

er (

pu)

(a)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1200

205

210

215

220

225

230

235

240

245

250

time(s)

Load

Vol

tage

(V

)

(b)

Figure 6.11: (a) Inverter power output and (b) Load voltage magnitude for

P ∗ = 0.5 and load per phase of 12Ω with forced voltage drop

integral term. The adjusted voltage controller is shown in Figure 6.10 and Equation 6.3.

Vmag adj = Vmag −kps+ ki

s(P − Pmax) (6.3)

Figure 6.11 shows the altered response of the inverter power output (a) and the load

voltage (b). While the inverter power is still able to increase slightly beyond Pmax (1pu),

the steady state response is significantly more realistic than in Figure 6.8.

Figure 6.12 shows the results if the test in Figure 6.9 is repeated using the voltage droop

algorithm. The power output of Inverter 1 is now much closer to the maximum power

available from the microturbine source.

This form of correction has some use in simulating scenarios where there is not enough

power in the system and so a drop in voltage occurs. In reality, the system should be

designed to avoid this scenario, either by having enough generation power available at all

times, sufficient energy storage, or by having intelligent load shedding capabilities.

There are limitations to this method of trying to track maximum power output. It is

possible to end up with an error in the load voltage magnitude even when the overall

power output of all the sources is enough to supply the load (see Figure 6.12 (b)). This


0 2 4 6 8 10 12 14 16 18 200

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

time(s)

Pow

er (

pu)


(a)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1150

160

170

180

190

200

210

220

230

240

250

time(s)

Load

Vol

tage

(V

)

(b)

Figure 6.12: (a) Power output and (b) Load voltage magnitude with forced

voltage drop and microturbine source

is because, if the output impedance is mainly reactive, voltage levels only have minimal

effect on output power, so a source would have to reduce its output voltage substantially

in order to significantly reduce its power output.

6.2.2 Shifting the Droop Characteristic

The microgrid design proposed in [21, 22] relies on having some kind of energy storage

(e.g. a battery bank) on the DC bus between the power source and the inverter. In order

to prevent the inverter supplying more power in steady state than the source provides and

thus depleting the storage, the authors have implemented a measure to limit the power

supplied by the inverter [21]. Similarly this measure prevents the inverter from injecting a

negative amount of power in the steady state (i.e. drawing power) so as not to overcharge

the storage.

The idea is to alter the droop characteristic of the inverter such that when the maximum

power level is reached, the droop characteristic will be translated down on the frequency

axis. If, on the other hand, the minimum power level is reached, the droop characteristic

will be translated up the frequency axis. As a result of this, the system equilibrium

frequency is altered, causing the other sources to increase or decrease their power output


!2

!

S1

!1

Pmax

PP2newP2 P1

S2 S1max

Figure 6.13: Droop characteristics with frequency shift

as required.

Figure 6.13 illustrates the case where a load increase causes the droop characteristic to be

translated down the frequency axis. The solid line marked S1 shows the original droop

characteristic for Source 1. At frequency ω1 the power output for Source 1 is P1, which

exceeds Pmax. When the frequency shifting is implemented, the droop characteristic for

Source 1 becomes S1max, and the new power output is Pmax. The system frequency

changes to ω2, and the power output for Source 2 (S2) becomes P2new. Note that, since

Source 2 has not exceeded its maximum power output, the droop characteristic for the

source remains unchanged. Equation 6.4 shows this relationship mathematically.

ωi(t) = ω∗ −mi(P∗i − Pi(t)) + ∆ωmax + ∆ωmin (6.4)

∆ωmax =Ki

s(Pmax − P ) (6.5)

∆ωmin =Ki

s(Pmin − P ) (6.6)

The term ∆ωmax (Equation 6.5) relates to the difference between the actual power and

the maximum power term and is limited so that it cannot become positive. Conversely

∆ωmin (Equation 6.6) relates to the difference between the actual power and the minimum

power term (usually zero) and is limited so that it cannot become negative. During normal

steady-state operation of the controller (i.e. when the power output of the source remains

within the limits), these two terms will be zero, and the droop characteristic will be


0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

0.2

0.4

0.6

0.8

1

1.2

1.4

time(s)

Pow

er (

pu)

Figure 6.14: Inverter power output for P ∗ = 0.5 and load per phase of 12Ω with

shifted droop characteristic and insufficient power available

unaffected. If the power output exceeds the limits, and then returns to within the limits,

∆ωmax (or ∆ωmin) will remain non-zero for a short amount of time, due to the integrator

terms.

This measure attempts to alter the relative power output of different sources, so that

sources with more power available take more of the load. It is only useful if sufficient

power is available to supply the load. If it is applied to the case shown in Figure 6.8,

with two identical sources and insufficient power to supply the load, this measure alters

the droop characteristics of both sources, but has no effect on their power outputs. This

is illustrated in Figure 6.14. Comparing this to Figure 6.8 shows the results are identical

and the shift of the droop characteristic has no effect on the power output of the source.

Figure 6.15 shows the results of a simulation where the sources do have sufficient power to

supply the load. One of the sources is replaced with a microturbine, and the power rating

of the other source is increased to allow it to power the entire load. The results show that

shifting the droop characteristic allows the inverter to track the power available from the

source much more closely. There is still some difference between the power output of the

microturbine and the inverter, but significantly less than in Figure 6.9.


0 2 4 6 8 10 12 14 16 18 200

0.2

0.4

0.6

0.8

1

1.2

time(s)

Pow

er (

pu)


Figure 6.15: Inverter power output for microturbine source and ideal source with

shifted droop characteristic. P ∗ = 0.8 and load per phase is 12Ω. Match between

microturbine power available and inverter power output is improved with droop

characteristic shifting.

The area between the solid line (inverter power output) and the dotted line (power available

from microturbine) will give the energy in Joules required to provide the missing power.

Since W = 1/2CV 2i − 1/2CV 2

f , where W is the work (or energy), this can be used to

calculate the capacitance required on the DC bus to provide the missing power depending

on the energy required. An example is provided here.

The minimum DC bus voltage required to sustain 230V at the inverter output is approxi-

mately 325V. If the voltage is regulated to 400V, then the final voltage can be allowed to

fall to 90% of the initial voltage without affecting the inverter output voltage, while still

leaving some room for switching, and other, losses. The area under the graph is approxi-

mately 913J. Rearranging the work equation gives Equation 6.7 which shows that, in this

particular scenario, a 60mF capacitor would be required to store enough energy to supply

the power gap between the power provided by the inverter, and the power available from

the microturbine.

C =2 ∗ 913

0.19 ∗ 4002∼= 60mF (6.7)


0 2 4 6 8 10 12 14 16 18 200

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

time(s)

Pow

er (p

u)

Inverter 1 Power OutputMicroturbine Maximum Power Available (Source 1)Inverter 2 Power Output


with two microturbine sources. Significant storage is required to supply power gap

between microturbine power available and inverter power output.

6.3 Directly Connected Storage

As explained previously, some form of storage is required in a microgrid due to the lack

of inertia in many of the distributed generation sources [21]. CERTS suggests that this

storage takes the form of batteries on the DC side of the inverter (as shown in Figure 6.2).

This approach gives a high degree of flexibility, allowing the sources to be considered as

having an instantaneous response to load changes if the battery is sized appropriately.

There are, however, various cost and environmental issues with forcing each of the sources

to have a dedicated battery, and it would be preferable if this could be avoided.

As an illustration of the kind of storage that might be required, Figure 6.16 shows a

simulation with two 6kW microturbine sources, as modelled in 6.1.1.2. The load is 20Ω

(2.6kW) per phase. It can be seen that the power being output by the inverter exceeds

the power available from the microturbines for a significant amount of time ( 17s in this

case). Energy output from the storage peaks at around 160kJ before the microturbine has

enough power to supply the load. Obviously this value is strongly dependent on the power

transients in the microturbine, but as an indication, this particular scenario would require

a capacitance of 10.5F (see Equation 6.8). If the storage were able to recharge between


load changes, then this does not equate to a particularly high capacity requirement (for

a 400V battery this equates to a capacity of EV ∗3600Ah ' 100mAh), but the fast rate at

which the battery is required to discharge might be more of an issue as batteries will

typically have peak Ampere ratings.

C =2 ∗ 160e3

0.19 ∗ 4002∼= 10.5F (6.8)

Another approach to storage is to have some form of storage directly connected to the

microgrid. Storage devices which have been considered for use in microgrid style systems

include flywheels and supercapacitors. Both of these technologies are well suited to this

application as they have a higher power density than batteries, meaning they can discharge

their energy faster than batteries. This extra storage, in conjunction with the use of the

droop shifting method on sources such as microturbines and fuel cells, should reduce the

amount of storage needed for each inverter and should allow the battery on the DC bus

to be replaced by a capacitor, or at least a smaller battery. The DC storage cannot be

removed entirely as some storage is necessary to allow the inverter to have some flexibility

in the control.

6.3.1 Controller Design and Performance

For this work, the chosen form of storage is assumed to be connected to the grid through

an inverter and coupling inductance, and to have no limits on how fast it can provide

power (a valid assumption due to the high power density of available solutions). The

aim of this grid connected storage is to provide (or absorb) power during transients, but

during steady state the power output should be zero. This means that a controller is

required which reacts quickly to load changes and then gradually reduces its power output

as the other sources increase their output. Standard droop control is not sufficient in this

case, because it will generally leave the storage device with a non-zero power output in

steady state. However, since the controller is being designed to work alongside sources


0 2 4 6 8 10 12 14 16 18 20−0.2

0

0.2

0.4

0.6

0.8

time(s)

Pow

er (

pu)



with droop controllers, it is desirable for the storage device to have some form of modified

droop control. Without this there would be nothing to indicate to the other sources that

their power output needed to change. A simple solution is to use the standard droop,

with the addition of an integrator term −ksPi(t) to ensure a steady-state power output of

zero. For the grid-connected storage device, the value of mi (the power-frequency droop

co-efficient) does not have to be chosen to ensure a particular steady-state power output

(since this will be zero), so mi is chosen to be small to ensure a fast response to load

changes.

ωi(t) = ω∗ −mi(P∗i − Pi(t)−

k

sPi(t)) (6.9)

Figure 6.17 shows the results if the test shown in Figure 6.15, with one microturbine, and

one ideal source, is repeated with the inclusion of a grid-connected storage device. As is

the case in the previous simulation, the droop characteristic shifting method is used on

the microturbine to minimise the power gap between the microturbine power output, and

the inverter power output for the source. The results show that this technique reduces the

amount of time that the inverter power output is above the microturbine available power.

In this case, the maximum energy required is 30.5J, so, using the same assumptions as

previously, the required capacitance is calculated by Equation 6.10 to be 2mF. This is a


0 10 20 30 40 50 60 70 80 90 1000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

time(s)

Pow

er (

pu)

Microturbine Source 1 and 2 (6kW) Power OutputMicroturbine Maximum Power Available (Source 1)Grid Connected Storage Power Output

(a)

0 10 20 30 40 50 60 70 80 90 1000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

time(s)

Pow

er (

pu)

Microturbine Source 1 and 2 (6kW) Power OutputMicroturbine Maximum Power Available (Source 1)Grid Connected Storage Power Output

(b)

Figure 6.18: Performance of system with grid connected storage (a) standalone

mode (b) with grid connection. At t = 50s load changes from 20Ω to 15Ω

significant reduction on the 60mF capacitance required without the grid connected storage,

which shows that the control algorithm proposed here could be used on directly connected

storage devices to reduce the size of the storage components required for each of the

individual micro-sources.

C =2 ∗ 30.5

0.19 ∗ 4002∼= 2mF (6.10)

To show the behaviour of the controller during a load change, a simulation was performed

of a 2 microturbine system with a change of load from 2.6kW (20Ω) per phase to 3.5kW

(15Ω) per phase at t = 50s. Figure 6.18 (a) shows the power output of Source 1 and Source

2 as well as the power available from the microturbine associated with Source 1, and the

power output of the grid connected storage. The microturbine sources are identical, so

the plots for Sources 1 and 2 overlap on the graph. Power mismatch occurs mainly at

startup (t = 0), and the load change (t = 50s), where spikes occur on the graph. The

results show that the system performs well for a change in load, with the power output

from the sources matching the available power fairly closely. In this case the maximum

energy used is about 64J during the load change transient.

A simulation was run to show the performance of the same system when a stiff grid is

connected. Since the grid is able to supply any power gap between the load power and the


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

time(s)

Pow

er (

pu)

Microturbine Source (6kW) Power OutputMicroturbine Maximum Power Available (Source 1)Grid Connected Storage Power Output

(a) k=0.5

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

time(s)

Pow

er (

pu)

Microturbine Source (6kW) Power OutputMicroturbine Maximum Power Available (Source 1)Grid Connected Storage Power Output

(b) k=1.5

Figure 6.19: Power output for different values of k

microturbine power output, the ideal behaviour for the grid connected storage would be to

supply no power in this case. Figure 6.18 (b) shows the power output of the two sources;

the microturbine for Source 1; and the grid connected storage device. The behaviour

in this case is not ideal, as the graph shows that the grid connected storage is initially

injecting power, and takes some time to reach zero power output. The reason for this

is that the power setpoint for the grid connected storage is non-zero. This improves the

transient response with no grid connected, but means it takes longer for the power output

to reach zero when the grid is connected. The effect due to a load change is minimal,

indicating the problem may only be significant during start-up transients.

6.3.2 The Parameter k

The performance of the system is fairly sensitive to the parameter k in the controller.

This needs to be chosen to approximately match the speed at which the microturbine (or

relevant micro-source) can change power output. If k is too small then the grid connected

storage will continue supplying power beyond when it is required, exhausting the storage.

If k is too large, the grid connected storage will stop supplying power too soon, and the

storage on the dc bus of the power sources will be discharged. Figure 6.19 illustrates the

effects if k is too small (a), or too large (b).


6.3.3 Controller Limitations

The controller presented in this chapter is based on simplified source models. It is designed

to show how a grid connected storage device can work alongside sources interfaced with

inverters using droop control to minimise storage on the DC bus of the source, while

allowing correct operation of the source inverter controllers. The controller assumes that

the storage can provide any amount of power, provided it has not been exhausted, and

does not take into account the nature of the storage device, or any limitations on the

storage. A more complete controller would need to take into account mechanical stresses

on devices such as flywheels, charging cycles on batteries, and provide intelligent control

to ensure that storage did not become exhausted, or overcharged.

Chapter 7

Summary and Further Work

7.1 Modelling Techniques

Historically, power system modelling focuses on steady state or slow dynamic behaviours.

In distributed generation systems, modelling needs to take into account faster dynamic

behaviours. Modelling techniques developed for this are typically processor intensive and

slow to simulate. A modelling approach has been developed that:

• treats the inverter as an ideal controllable source

• approximates the inverter power source as a transfer function

• considers power system components in a d-q reference frame allowing for more effi-

cient simulation of parallel inverters.

This model allows for the simulation of different load types, and dynamic load behaviours.

The modelling approach has also been integrated with components available in the MAT-

LAB SimPowerSystems library. This allowed the incorporation of standard power system

components (transformers, circuit breakers) into the developed models. It should be noted

that SimPowerSystems models are in phase variables which does lead to slower simulations.

7.1.1 Further Work

While the new modelling system shows potential as a fast method for simulating dis-

tributed generations systems, there is still some further development required before it is

suitable for widespread use. In particular, the components that are currently utilised from

7. Summary and Further Work 152

the SimPowerSystems library could be modelled using the d-q modelling approach.

A fast model could be used to investigate the effect of many inverters in parallel, as in

PV systems. This would include investigating the assumption that inverter switching and

other source dynamics can be ignored, and developing a source model which reflects the

behaviour of PV sources.

7.2 Pf-droop

Power-frequency droop control is presented as the solution to power sharing between

sources in a distributed generation system without requiring fast communications, how-

ever in the literature, several problems have been found with the standard droop control

algorithm. The model library was used to perform simulations which verified these prob-

lems. The proposed solutions in the literature were outlined and key ones evaluated. It

was found that the area in which research is lacking is solutions to limit voltage unbalance

at the load.

7.3 Controller Design

A controller was designed to limit voltage unbalance at the load by injecting negative

sequence at the source. Initially the controller was designed to work with the CERTS

droop controller, then the unbalance controller was incorporated into a new controller

which takes ideas from several authors in order to create something to work in a wide

range of conditions. The previous tests were revisited to ensure that new controller is able

to solve the problems present in standard droop control.

7.3.1 Further Work

The component design for the system is fairly simplistic. There is potential for performance

improvements through more sophisticated control techniques in the unbalance controller,

7. Summary and Further Work 153

better filters throughout the system, and an improved algorithm for the sequence analyzer.

Also, harmonic current sharing is not addressed in the new controller.

In this work the controller performance has been evaluated by simulation. The next step

is experimental validation of the controller design.

There is potential for applying the technique used for unbalance control to the problem of

improving zero sequence behaviour. The complexity of this problem is considerable and

has not been investigated as part of this work.

7.4 Grid Connected Storage

An issue with the kinds of sources usually used in distributed generation is that they have

little or no inertia and cannot instantly supply changes in load. The typical solution to

this is to supply storage (i.e. a battery) on the DC bus of the power source. Another

possibility is to provide a grid-connected AC storage device (such as a flywheel) with the

aim of reducing, or eliminating, the DC storage required for addressing the gap between

the power required by the load, and the power supplied by the source. With this aim, a

simple controller based on power-frequency droop control was designed for a grid-connected

storage device. As a part of this work, the literature on simple Power Source models was

examined. Simulation shows that the controller is able to reduce the amount of storage

required on the DC bus, although not eliminate it.

7.4.1 Further Work

The simple controller is able to reduce DC storage requirements significantly, but it is likely

that with fine tuning, or a different control scheme, better results can be achieved. Also

of interest is the effect of non-dispatchable sources such as PV and wind. To investigate

this, Power Source models are required for these.

Appendix A

MATLAB/Simulink Diagrams

This Appendix contains screen captures of the Simulink files used throughout this thesis.

A.1 Simulation using Developed Model Library

This Section contains the Simulink diagrams for the simulations performed in Chapter 3

based on the model library outlined in Chapter 2. The model represents five inverters in

parallel each connected to a single load through coupling inductances.

A. MATLAB/Simulink Diagrams 155

A.1.1 Top Level Simulation

Figure A.1 shows the top-level of the Simulink model.

Figure A.1: Simulation of five parallel sources: Top-level simulation


A.1.2 Source Block

Figure A.2 contains the detail of the blocks labelled Source 1 - Source 5 in Figure A.1.

Figure A.2: Simulation of five parallel sources: Detail of Source Blocks

A.1.2.1 Calculations Block

Figure A.3 shows the detail of the Calculations block from Figure A.2.

E mag, the output of the Load Voltage Magnitude block is calculated using the following

embedded MATLAB code (note that the /2 converts to RMS):

E mag = sqrt((vd^2+vq^2)/2);

P and Q, the output of the Total Real and Reactive Power block are calculated:

P=3/2 * (vd*id + vq*iq);

Q=-3/2 * (vd*iq - vq*id);

A.1.2.2 Power-frequency Droop

Figure A.4 contains the detail of the Power-frequency Droop Control block in Figure A.2.

The Lowpass Filter block is a Simulink Transfer Fcn block which implements a first order


Figure A.3: Simulation of five parallel sources: Detail of Calculations Block

lowpass filter LPF = 2πfs+2πf , f = 10Hz.

Figure A.4: Simulation of five parallel sources: Detail of Power-frequency Droop

Control Block

A.1.2.3 Voltage Control

Figure A.5 shows the detail of the Voltage Control with Q droop block in Figure A.2. To

disable the reactive power droop, mQ is set to zero.

Figure A.5: Simulation of five parallel sources: Detail of Voltage Control Block


A.1.2.4 Inverter

The Ideal Inverter block in Figure A.2 contains the following Embedded MATLAB code:

function [vd,vq] = fcn(V,delta v)

vd = sqrt(2)*V*cos(delta v);

vq = sqrt(2)*V*sin(delta v);

A.1.3 Coupling Inductance

Figure A.6 shows the detail of the Coupling Inductance blocks in A.1. The Matrix Multiply

block multiplies by

0 1

−1 0

.

Figure A.6: Simulation of five parallel sources: Detail of Coupling Inductance

Block

A.1.4 Voltage Magnitude Calculation

The details of Voltage Magnitude Calculation block in A.1 are identical to the Load Voltage

Magnitude block in Section A.1.2.1.


Figure A.7: Simulation of five parallel sources: Detail of Frequency Calculation

Block

A.1.5 Frequency Calculation

Figure A.7 shows the detail of the Frequency Calculation block in A.1. The delta e Cal-

culation block contains the following Embedded MATLAB code:

function delta e = fcn(vd,vq)

persistent sum1;

persistent last;

%Initialse sum1

ifisempty(sum1)

sum1=0;

end

%Initialise last

if isempty(last)

last=0;

end

%Avoid divide by zero

if vd==0

if vq==0

delta e=0;

else

delta e=pi/2+sum1;

end

else %vd is non-zero

temp= atan(vq/vd)+sum1;

%Discontinuity in atan function would lead to errors in frequency measurement

%Correct for discontinuity

if (last - temp>2)

sum1=sum1+pi;


temp=temp+pi;

end

if (last - temp<-2)

sum1=sum1-pi;

temp=temp-pi;

end

last=temp;

delta e=temp;

end

A.1.6 Reactive Load

Figure A.8 shows the detail of the load for Section 3.3.3. The Series Inductance block is

the same as the Coupling Inductance block shown in Figure A.6. This block replaces the

Multiplier block labelled I to V and the Step Change block labelled Load Resistance in

Figure A.1.

Figure A.8: Simulation of five parallel sources: Detail of Reactive Load Block


A.2 Harmonic Simulation using Developed Model Library

This Section contains the Simulink diagrams for the simulations performed in Chapter 3

based on the model library outlined in Chapter 2. The model represents five inverters in

parallel each connected to a single non-linear load through coupling inductances.



Figure A.9: Harmonic simulation with five parallel sources: Top-level simulation


A.2.2 Source Block

Figure A.10 contains the detail of the blocks labelled Source 1 - Source 5 in Figure A.9.

The Voltage Control with Q droop block and the Power-frequency Droop Control blocks

are identical to Section A.1.

Figure A.10: Harmonic simulation with five parallel sources: Detail of Source

Blocks

A.2.2.1 Calculations Block

Figure A.11 shows the detail of the Calculations block from Figure A.10. The details of

the Frequency Conversion blocks are shown in Figure A.12. These blocks convert the 5th

harmonic d-q quantities into fundamental d-q quantities, in order to perform calculations.

The DQ Frequency Conversion block contains the following embedded MATLAB code:

vd1 = (vd3*cos(theta)-vq3*sin(theta));

vq1 = (vd3*sin(theta)+vq3*cos(theta));

E mag, the output of the Load Voltage Magnitude block is calculated using the following

embedded MATLAB code:

E mag = sqrt((vd^2+vq^2)/2);

P and Q, the output of the Total Real and Reactive Power block are calculated:

P=3/2 * (vd*id + vq*iq);

Q=-3/2 * (vd*iq - vq*id);


Figure A.11: Harmonic simulation with five parallel sources: Detail of Calcula-

tions Block

Figure A.12: Harmonic simulation with five parallel sources: Detail of Frequency

Conversion Block

A.2.2.2 Inverter


function [vd1,vq1,vd3,vq3] = fcn(V,delta v)

vd1 = sqrt(2)*V*cos(delta v);

vq1 = sqrt(2)*V*sin(delta v);

vd3 = 0;

vq3 = 0;


A.2.3 Coupling Inductance

Figure A.13 shows the detail of the Coupling Inductance blocks in A.9.

Figure A.13: Harmonic simulation with five parallel sources: Detail of Coupling

Inductance Block

A.2.4 Rectifier Load

Figure A.14 shows the detail of the Rectifier Load block in A.9. The Matrix Multiply

block is a gain block configured with a Gain of [10 0 0 0;0 10 0 0;1 0 3 0;0 1 0 3] and

Multiplication configured as Matrix(K*u) (u vector).

Figure A.14: Harmonic simulation with five parallel sources: Detail of Rectifier

Load Block


A.3 SimPowerSystems Unbalanced Load Simulation

This Section contains the Simulink diagrams for the simulations performed in Section 3.3.7

based on the model library outlined in Section 2.4. The model represents two inverters in

parallel each connected to a single load through coupling inductances.



Figure A.15: Unbalanced load testing: Top Level Simulation


A.3.2 Source Block

Figure A.16 contains the detail of the blocks labelled Source 1 and Source 2 in Figure

A.15. The Power-Frequency Droop control block is identical to Section A.1.

Figure A.16: Unbalanced load testing: Detail of Source Blocks

A.3.2.1 Voltage Magnitude Calculation

Figure A.17 shows the detail of the Voltage Magnitude Calculation block in Figure A.16.

In order to calculate the Voltage Magnitude, the time domain voltage (e) is converted to

dq voltages and then the voltage magnitude calculation from A.1.2.1 can be used.

Figure A.17: Unbalanced load testing: Detail of Voltage Magnitude Calculation

Block

Figure A.18 shows the detail of the dq to Voltage Magnitude Calculation block.


Figure A.18: Unbalanced load testing: Detail of abc-dq Calculation Block

The abc-dq Calculation block contains the following embedded MATLAB code:

function [vd,vq] = fcn(a,b,c,theta)

vd = 2/3*(a*cos(theta) + b*cos(theta -2*pi/3) + c*cos(theta + 2*pi/3));

vq = -2/3*(a*sin(theta) + b*sin(theta- 2*pi/3) + c*sin(theta + 2*pi/3));

The dq to Voltage Magnitude Calculation block contains the following embedded MAT-

LAB code (note that the /6 converts line-line peak values into RMS line - neutral):

function V mag = fcn(vd,vq)

V mag = sqrt((vd^2+vq^2)/6);


Figure A.19 shows the detail of the Voltage Control block in Figure A.16.

Figure A.19: Unbalanced load testing: Detail of Voltage Control Block

A.3.2.3 Ideal Inverter



function [a,b,c] = fcn(V,delta v,t)

omega=50*2*pi;

a=V*sqrt(2)*sin(omega*t + delta v);

b=V*sqrt(2)*sin(omega*t + delta v -2*pi/3);

c=V*sqrt(2)*sin(omega*t + delta v + 2*pi/3);

A.3.3 Signal to Voltage Conversion

Since SimPowerSystems uses different connections to represent electrical wires, the stan-

dard Simulink signals output by the Source block have to be converted into voltages. This

conversion is performed by the Signal to Voltage Conversion block using a SimPowerSys-

tems Controlled Voltage Source component (see Figure A.20).

Figure A.20: Unbalanced load testing: Signal to Voltage Conversion Block

A.3.4 Load Block

Figure A.21 shows the detail of the Load Block in Figure A.15. Resistors Ra, Rb and Rc

are Series RLC branch components from the SimPowerSystems library.


Figure A.21: Unbalanced load testing: Load Block


A.4 Power Control Comparison

This Section contains the Simulink diagrams for the simulations performed in Section 3.4.2

based on the model library outlined in Section 2.4. The model represents two inverters in

parallel each connected to a single load through coupling inductances. The two simulations

differ only in the Power-Frequency Droop Contol block


Figure A.22 shows the top-level of the Simulink model. The Signal to Voltage Conversion

blocks, Load block, and Frequency Calculation block are identical to Section A.3. Note

that, in this case, the load is balanced.

Figure A.22: Power control comparison testing: Top Level Simulation


A.4.2 Source Block


A.22.

Figure A.23: Power control comparison testing: Detail of Source Blocks

A.4.2.1 Power-Frequency Droop Control Block

Figures A.24 and A.25 contains the detail of the Power-frequency Droop Control block in

Figure A.23 for the two different sets of results.

Figure A.24: Power control comparison testing: Detail of Power-frequency

Droop Control block

Figure A.25: Power control comparison testing: Detail of revised Power-

frequency Droop Control block


A.5 Simulation with Unbalance Control

This Section contains the Simulink diagrams for the simulations performed in Chapter 4.

The model represents two inverters in parallel each connected to a single load through

coupling inductances.



block, Voltage Magnitude Calculation block and Load block are identical to Section A.3.

Figure A.26: Unbalance control testing: Top-level Simulation


A.5.2 Source Block


A.26.

Figure A.27: Unbalance control testing: Detail of Source Blocks

A.5.2.1 Power-Frequency Droop

Figure A.28 contains the detail of the Power-frequency Droop Control block in Figure

A.27. The Lowpass Filter block is a Simulink Transfer Fcn block which implements a first

order lowpass filter LPF = 2πfs+2πf with f = 1Hz.

Figure A.28: Unbalance control testing: Detail of Power-Frequency Droop Con-

trol Block




Figure A.29: Unbalance control testing: Detail of Voltage Control Block

A.5.2.3 Unbalance Control

Figure A.30 shows the detail of the Unbalance Control block in Figure A.27.

Figure A.30: Unbalance control testing: Detail of Unbalance Control Block

The purpose of the Step block is to introduce a delay before the controller begins operating.

The need for this delay is explained in Section 4.3. At t = 0, the output of the block is 0,

and at t = 0.05s the output is 1.


The Deadzone block contains the following Embedded MATLAB code:

function out = fcn(in)

if abs(in)<1

out=0;

else

out=in-1;

end

A.5.2.4 Ideal Inverter

The Ideal Inverter block contains the following Embedded MATLAB code:

function [va,vb,vc] = fcn(pos mag,pos angle,neg mag,neg angle,t)

a=-0.5+(sqrt(3)/2)*1j;

Pos=pos mag*sqrt(2)*(cos(pos angle)+sin(pos angle)*1j);

Neg=-neg mag/sqrt(3)*(cos(neg angle*pi/180+pos angle)

+sin(neg angle*pi/180+pos angle)*1j);

V=[1,1,1;1,a^2,a;1,a,a^2]*[0;Pos;Neg];

omega=50*2*pi;

va=abs(V(1))*sin(omega*t + angle(V(1)));

vb=abs(V(2))*sin(omega*t + angle(V(2)));

vc=abs(V(3))*sin(omega*t + angle(V(3)));

A.5.3 Frequency Calculation Block

Figure A.31 shows the detail of the Frequency Calculation block in Figure A.26.

Figure A.31: Unbalance control testing: Detail of Frequency Calculation Block


A.5.4 Symmetrical Components Calculation Block

Figure A.32 shows the detail of the Symmetrical Components Calculation block in Figure

A.26. The block is based heavily on the built-in SimPowerSystems 3-phase Sequence

Analyzer block, modified to allow the voltage frequency to vary.

Figure A.32: Unbalance control testing: Detail of Symmetrical Components

Calculation Block


A.6 New Voltage Source

This Section contains the Simulink models for the simulations performed in Chapter 5.

The model represents two inverters in parallel each connected to a single load through

coupling inductances.



block and Voltage Magnitude Calculation block are identical to section A.3. The Ideal

Inverter in the Source block and the Frequency Calculation Block are identical to Section

A.5. The top level Source block is shown here in Figure A.34.

Figure A.33: New Controller: Top Level Simulation


Figure A.34: New Controller: Detail of Source Blocks

A.6.1.1 Power-Frequency Droop


A.34. The Bandpass Filter block is a Simulink Transfer Fcn block which implements a

first order bandpass filter BPF = 2πfs(s+2πfL)(s+2πfH) .

Figure A.35: New Controller: Detail of Power-Frequency Droop Control Block




Figure A.36: New Controller: Detail of Voltage Control Block

A.6.1.3 Unbalance Control

Figure A.37 shows the detail of the Unbalance Control block in Figure A.34. Note that

the Gain block is not necessary (as it could be incorporated in the PI controller) and was

included for tuning purposes.

Figure A.37: New Controller: Detail of Unbalance Control Block


A.6.1.4 Load Block

Figure A.38 shows the Load block in Figure A.33. At t = 5s the Step block changes output

from 0 to 1 closing the Breakers. This switches in resistors Ra2, Rb2 and Rc2.

Figure A.38: New Controller: Detail of Load Block


A.7 Simulation to Investigate Power Sources using Devel-

oped Model Library

This Section contains the Simulink diagrams for the simulations performed in Chapter

6. The model represents a Micro-turbine, and Ideal Source and a grid-connected storage

device in parallel each connected to a single load through coupling inductances. Different

combinations of sources are used for different simulations, but the information here is

sufficient to reconstruct each of the different scenarios.


Figure A.39 shows the top-level of the Simulink model. The Coupling Inductance blocks

shown allow for the inclusion of line impedance, however, as this is not being used, func-

tionally the Coupling Inductance blocks are identical to those in Section A.1 (see Figure

A.6). The Voltage Magnitude Calculation block uses the embedded MATLAB code in

Section A.1.2.1.

Figure A.39: Investigation of Power Sources: Top-level Simulation


A.7.2 Source Block (Micro-turbine and Ideal)

Figure A.40 contains the detail of the blocks labelled Micro-turbine Source and Ideal

Source in Figure A.39. The Calculations block, the Voltage Control blocks are identical

to Section A.1.

Figure A.40: Investigation of Power Sources: Detail of Source Blocks

A.7.2.1 Power-frequency Droop


A.40.

Figure A.41: Investigation of Power Sources: Detail of Power-frequency droop

control Block


A.7.2.2 Microturbine and Ideal Source Block

Figure A.42 shows the detail of the Microturbine block shown in Figure A.40. In the

Transfer Fcn block the time constant Tau=10s. For the Ideal Source, the Microturbine

block is replaced with a Constant block with a value of 1.

Figure A.42: Investigation of Power Sources: Detail of Micro-turbine Block

A.7.2.3 Inverter


function [vd,vq] = fcn(V,delta v)

vd = sqrt(2)*V*cos(delta v);

vq = sqrt(2)*V*sin(delta v);

vq = 0;

A.7.3 AC Storage

Figure A.43 contains the detail of the block labelled AC Storage in Figure A.26. The

Voltage Control block is the same in the AC Storage block as in the Micro-turbine and

Ideal Source blocks and the detail can be found in Section A.1.

A.7.3.1 Power-frequency Droop for AC Storage Block

Figure A.44 shows the detail of the Power-frequency droop control block in Figure A.43.


Figure A.43: Investigation of Power Sources: Detail of AC Storage Source Blocks

Figure A.44: Investigation of Power Sources: Detail of Power-frequency droop

control Block modified for grid-connected storage device

A.7.3.2 DC Source

Figure A.45 shows the detail of the DC Source block in Figure A.43. The DC Source block

provides a simplistic calculation of the charge remaining in the grid-connected storage

device.

Figure A.45: Investigation of Power Sources: Detail of DC Source Block


A.7.4 Load Block

Figure A.46 shows the detail of the Load block in Figure A.39. The Embedded MATLAB

function block contains the following code:

function [vd,vq,v0] = fcn(a,b,c,theta,id,iq,i0)

%If load is balanced, perform simple calculation

if ((a==b) && (b==c))

vd=a*id;

vq=a*iq;

v0=a*i0;

%Otherwise perform full calculation

else

T=2/3.*[ cos(theta), cos(theta - 2*pi/3),cos(theta + 2 * pi / 3);

-sin(theta), -sin(theta - 2 *pi/3),-sin(theta+ 2 * pi / 3);

1/2,1/2,1/2];

T inv=[cos(theta),-sin(theta),1;

cos(theta- 2 *pi/3),-sin(theta- 2 *pi/3),1;

cos(theta+ 2 * pi / 3),-sin(theta+ 2 * pi/ 3),1];

R=[a,0,0;0,b,0;0,0,c];

Rdq=T*R*T inv;

vd=Rdq(1,1)*id + Rdq(1,2)*iq + Rdq(1,3)*i0;

vq=Rdq(2,1)*id + Rdq(2,2)*iq + Rdq(2,3)*i0;

v0=Rdq(3,1)*id + Rdq(3,2)*iq + Rdq(3,3)*i0;

end

Figure A.46: Investigation of Power Sources: Detail of DC Source Block

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Related Publications

Griffiths, M.; Coates, C.; , “Modelling and Performance of Low Voltage Distributed Gen-

eration Systems,” Power Engineering Conference, 2006. AUPEC 2006. Australasian Uni-

versities, vol., no., pp., 10-13 Dec. 2006

Griffiths, M.; Coates, C.; , “Behaviour of microgrids in the presence of unbalanced loads,”

Power Engineering Conference, 2007. AUPEC 2007. Australasian Universities , vol., no.,

pp.1-5, 9-12 Dec. 2007

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