a novel thÉvenin-based oltage droop control ... raghami...a novel thévenin-based voltage droop...

A NOVEL THÉVENIN-BASED VOLTAGE

DROOP CONTROL IMPROVING REACTIVE

POWER SHARING WITH STRUCTURES TO

IDENTIFY THÉVENIN PARAMETERS

Alireza Raghami

BS.c. and MS.c. in Electrical Engineering

A Thesis Submitted in fulfilment of the requirements for the degree of

Doctor of Philosophy

Science and Engineering Faculty

School of Electrical Engineering and Computer Science

Queensland University of Technology

Queensland, Australia

2019

A Novel Thévenin-Based Voltage Droop Control Improving Reactive Power Sharing with Structures to Identify

Thévenin Parameters i

Keywords

Correlation

Customers’ inverters

Distribution system’s identification

Identification through power lines

Reactive power sharing

Real-time identification

Residential load

Thévenin parameters identification

Voltage droop control


Thévenin Parameters ii

Abstract

The prevalent radial configuration of distribution systems and changes in

customer load have made serious voltage magnitude issues concerning the operation

of these systems. The voltage issues relating to urban low voltage systems with a

relatively reactive impedance can be handled by reactive power compensation.

Handling these voltage issues is an increasingly challenging problem. Insufficient

reactive power sources, lack of flexibility in the existing sources and their control are

some of the major problems.

However, a growing number of photovoltaic/battery inverter systems with

reactive power capability creates an opportunity to accurately meet the reactive power

compensation needs. On the one hand, some utilities are providing local compensation

at inverters installation points and many researchers are investigating distributed droop

based voltage control strategies. On the other hand, the cost-effectiveness of the local

compensation may concern individual customers about getting involved for reactive

power support. One of the major concerns is the relative amount of support provided

by each customer.

In fact, when inverters are coordinated using the conventional droop control

strategies, their reactive power contributions could be adversely affected by their

positions. This positioning is the relative distance of each inverter from the power

transformer connecting the feeder to the higher voltage system. This drawback is

investigated in this thesis. Critical elements that can impede an even provision of

reactive power support are diagnosed. Elements of the Thévenin equivalent circuit

model are used to systematically develop a novel droop control strategy improving the

reactive power sharing. Identification of Thévenin equivalent circuit parameters is then

required.

However, the real-time Thévenin parameters identification through power lines

is not a straightforward task. Challenges are detailed regarding the status of loads and

inverters and some innovative solutions are provided. Different scenarios are studied

from the uncompensated conventional systems with ideally unchanged loads to the

voltage compensated systems with continual demand variations. Some statistical and


Thévenin Parameters iii

non-statistical approaches are proposed to enable inverters for this challenging

identification task.

Results show that voltage magnitude is regulated through an even distribution of

reactive power compensation among customers’ inverters using the proposed

Thévenin based droop control. Each droop controller is regularly adjusted via the real-

time Thévenin identification in a fundamentally local process. The statistical and the

non-statistical-based identification approaches are assessed based on locally

measurable metrics of performance.


Thévenin Parameters iv

Table of Contents

Keywords .................................................................................................................................. i

Abstract .................................................................................................................................... ii

Table of Contents .................................................................................................................... iv

List of Figures ........................................................................................................................ vii

List of Tables ............................................................................................................................ x

Statement of Original Authorship ........................................................................................... xi

Acknowledgements ................................................................................................................ xii

Publications Arising from the Thesis .................................................................................... xiii

Chapter 1: Introduction ...................................................................................... 1

1.1 Background .................................................................................................................... 1

1.2 Aims and Objectives of the Thesis ................................................................................. 2

1.3 Significance of the Research .......................................................................................... 2

1.4 Key Contributions of this Research ............................................................................... 3

1.5 Thesis Outline ................................................................................................................ 4

Chapter 2: Literature Review ............................................................................. 7

2.1 Phasor Analysis and Thévenin Equivalent Circuit ......................................................... 7

2.2 Norton Equivalent Circuit ............................................................................................ 11

2.3 Significance of Thévenin Equivalent Circuit in Power System Studies ...................... 12

2.4 Typical Power Elements Connected to an Inverter ...................................................... 13

2.5 A Typical Hierarchical Control for an Inverter ............................................................ 13 2.5.1 Tertiary Control ................................................................................................. 14 2.5.2 Secondary Control ............................................................................................. 15 2.5.3 Synchronisation of Inverters .............................................................................. 16 2.5.4 Primary Control and Basics of Droop Control .................................................. 17

2.6 Advantages, Limitations and Variations of the Conventional Droop .......................... 21

2.7 Adaptiveness of Droop ................................................................................................. 22

2.8 Summary and Implications .......................................................................................... 24

Chapter 3: Signal Processing Concepts Relevant to Local Identification Problems .......................................................................................................... 27

3.1 Overview ...................................................................................................................... 27

3.2 Signal Processing Basics Required for Understanding of a Local Identification ........ 27 3.2.1 Continuous and Discrete Random Process ........................................................ 27 3.2.2 Deterministic and Nondeterministic Random Process....................................... 29 3.2.3 Expected Value and Stationarity ........................................................................ 29 3.2.4 Time Average and Ergodicity ............................................................................ 30 3.2.5 Statistical Concepts for Discrete-Time Processes .............................................. 31 3.2.6 Orthogonality ..................................................................................................... 32 3.2.7 Signal to Noise Ratio ......................................................................................... 33


Thévenin Parameters v

3.2.8 Crest Factor ........................................................................................................33 3.2.9 Continuous Uniform and Normal Distributions .................................................34 3.2.10 White Noise ........................................................................................................35 3.2.11 Random Walk .....................................................................................................36

3.3 Mathematical Framework and Possible Solutions to a Multiple Access Problem .......36 3.3.1 Ordinary Least Squares Estimation and Pseudo-Inverse Estimator ...................37 3.3.2 Properly Posed and Well-Conditioned Problems ...............................................38 3.3.3 FDMA ................................................................................................................40 3.3.4 TDMA ................................................................................................................40 3.3.5 CDMA ................................................................................................................41

3.4 Orthogonal Functions and Their Properties ..................................................................42 3.4.1 Walsh Functions .................................................................................................43 3.4.2 Application of Walsh Functions .........................................................................46 3.4.3 An Example of Walsh Based CDMA Multiple Access Problem .......................48

3.5 Nature of Variation of Loads in Distribution Systems .................................................49

3.6 Applications of the Signal Processing Concepts for Identification Problems in Power Engineering .............................................................................................................................52

3.7 Summary and Implications ...........................................................................................55

Chapter 4: A Novel Thévenin-Based Voltage Droop Control Improving Reactive Power Sharing with Structures to Learn Thévenin Parameters.......... 57

4.1 Overview ......................................................................................................................57

4.2 Proposed Adaptive Droop Control Strategy .................................................................58

4.3 Proposed Load-Based Thévenin Identification Strategies ............................................61 4.3.1 Thévenin Parameters Identification in Ideally No Noise Condition ..................64 4.3.2 Thévenin Parameters Identification in Conventional Distribution Systems .......65

4.4 Proposed Inverter-Based Thévenin Identification Strategies .......................................70 4.4.1 Walsh-Based Identification Strategies in a Highly Reactive System .................70 4.4.2 Practicality Challenges of Simultaneous Walsh-Based Identification ...............74 4.4.3 Identification Strategies in an Uncompensated Distribution System .................75 4.4.4 Identification Strategies in Droop Compensated Systems .................................77

4.5 Synopsis of the Proposals .............................................................................................80

Chapter 5: Results and Discussion ................................................................... 82

5.1 Voltage Control via the Conventional Droop Strategy .................................................82

5.2 Correlation ....................................................................................................................84

5.3 SNR ..............................................................................................................................87

5.4 The Proposed Voltage Droop Control with the Identification Structures .....................89 5.4.1 Significance of Lower Correlation .....................................................................92 5.4.2 Significance of Higher SNR ...............................................................................95 5.4.3 Significance of Lower Numerical Sensitivity ....................................................96 5.4.4 Significance of More Controlled Droop Dynamics ............................................97

5.5 Crest Factor .................................................................................................................101

5.6 Summary .....................................................................................................................106

Chapter 6: Conclusions and Recommendation ............................................. 109

6.1 Conclusions ................................................................................................................109

6.2 Recommendations for Future Research ......................................................................113


Thévenin Parameters vi

Bibliography ........................................................................................................... 114


Thévenin Parameters vii

List of Figures

Figure 2.1. Circuits’ energy storing elements with the graphical voltage-current

relationship in the complex phasor and the time domain planes ................... 8

Figure 2.2. Schematic of the first method of Thévenin impedance derivation .......... 10

Figure 2.3. Norton equivalent circuit of the Thévenin equivalent in Figure 2.2 ....... 11

Figure 2.4. n-bus radial feeder connected to the voltage robust main grid ............... 11

Figure 2.5. One configuration of power elements connected to a PV module .......... 13

Figure 2.6. A typical hierarchical control strategy implemented to coordinate

the inverters of a microgrid (a) tertiary control and secondary control

(b) secondary control and primary control................................................... 14

Figure 2.7. A current source grid supporting inverter controlled by the primary

level .............................................................................................................. 18

Figure 2.8. (a) Schematic of a voltage droop controlled current source grid

supporting inverter (b) simplified presentation of the schematic ................ 19

Figure 2.9. (a) Configuration of a typical n bus modern radial feeder with

customers having loads and PV/battery inverter systems (b) Circuit

model of the ith droop-controlled inverter connected to the system’s

Thévenin equivalent ..................................................................................... 19

Figure 2.10. Voltage reactive current droop characteristic........................................ 20

Figure 2.11. (a) Norton descriptor and (b) Thévenin descriptor of an inverter

managed by the droop control characterised in (2.13) ................................. 21

Figure 3.1. A continuous random process ................................................................. 28

Figure 3.2. A discrete-time random process (or a continuous random sequence)

formed by sampling the waveforms of Figure 3.1 ....................................... 29

Figure 3.3. PDFs of zero-mean normal distribution (the bell-shape solid curve)

versus zero-mean uniform distribution of the same variance (the

dotted rectangle)........................................................................................... 35

Figure 3.4. Auto-correlation of a white noise process ............................................... 36

Figure 3.5. Dimensions of two widely applied multiple access techniques .............. 41

Figure 3.6. CDMA dimensions .................................................................................. 41

Figure 3.7. Walsh function ensemble of length eight ................................................ 44

Figure 3.8. Superimposition of a Walsh ensemble and sine-cosine function ............ 46

Figure 3.9. Extraction of Walsh codes employed by two CDMA terminals ............. 48

Figure 3.10. The sent messages with the reconstructed messages ............................ 49

Figure 4.1. (a) Vector diagram of the circuit in Figure 2.9 (b) in a common

reference frame (b) Magnified components of the voltage change ............. 59


Thévenin Parameters viii

Figure 4.2. Effective equivalent circuit model of the system connected to a

droop controlled inverter .............................................................................. 59

Figure 4.3. Newmarket pilot smart grid geographical map ....................................... 61

Figure 4.4. Cross-correlation of demand changes of the neighbours’ loads (a)

Five randomly selected customers (b) Twenty five randomly selected

customers ..................................................................................................... 63

Figure 4.5. Circuit model of the network as seen from ith customer viewpoint

connected to an unchanged system .............................................................. 64

Figure 4.6. Circuit model of the network as seen from ith customer viewpoint

in a conventional distribution system ........................................................... 65

Figure 4.7. Circuit model of the network as seen from the viewpoint of ith

customer’s passive inverter .......................................................................... 70

Figure 4.8. Flowchart of the proposed Walsh-based Thévenin parameters

identification structure from the viewpoint of ith passive inverter .............. 73

Figure 4.9. Interference limitation in CDMA exemplified for Walsh-based

identification of inverters of different probing power ................................. 74

Figure 4.10. (a) Thévenin equivalent of the network from the perspective of ith

active inverter, (b) Thévenin equivalent of the network obtainable by

probing of ith inverter .................................................................................. 78

Figure 4.11. The control strategy flowchart from the perspective of ith inverter ...... 80

Figure 5.1. Single-line diagram of IEEE 33-bus system ........................................... 82

Figure 5.2. Contoured reactive power contribution (p.u.×100) of the inverters

coordinated by the conventional voltage droop strategy ............................. 84

Figure 5.3. Magnitude change correlation between the inverter at bus 28 and

some of the neighbour inverters: (a) droop, (b) Walsh-based probing

(c) normal distribution-based probing (d) uniform distribution-based

probing ......................................................................................................... 86

Figure 5.4. Voltage compensation via the novel droop with Walsh based-

identification (a) Inverters’ total current (b) Bus voltage (c) Identified

Thévenin source magnitude (d) Identified system’s Thévenin

reactance ....................................................................................................... 91

Figure 5.5. Length-32 Walsh-based probing (a) Identified Thévenin source

magnitude (b) Identified system’s Thévenin reactance ............................... 93

Figure 5.6. Neglecting a part of Walsh codes in Walsh-based probing (a)

Identified Thévenin source magnitude (b) Identified system’s

Thévenin reactance ...................................................................................... 94

Figure 5.7. Neglecting the processing gain and synthesising single

measurement for each observation (a) Identified Thévenin source

magnitude (b) Identified system’s Thévenin reactance ............................... 95

Figure 5.8. Neglecting the numerical properties in this low-level probing

identification problem (a) Identified Thévenin source magnitude (b)

Identified system’s Thévenin reactance ....................................................... 97


Thévenin Parameters ix

Figure 5.9. System operated using the novel Thévenin droop without LPF (a)

Inverters’ total current (b) Bus voltage ........................................................ 98

Figure 5.10. Voltage compensation via the novel droop with normal

distribution based-identification (a) Inverters’ total current (b) Bus

voltage (c) Identified Thévenin source magnitude (d) Identified

system’s Thévenin reactance ....................................................................... 99

Figure 5.11. Voltage compensation via the novel droop with uniform

distribution based-identification (a) Inverters’ total current (b) Bus

voltage (c) Identified Thévenin source magnitude (d) Identified

system’s Thévenin reactance ..................................................................... 100

Figure 5.12. Voltage perturbation crest factor from sequential probing to

simultaneous probing ................................................................................. 104

Figure 5.13. Zoomed voltage perturbation at buses 15, 31 and 33 (a) Walsh

probing (b) Normal probing (c) Uniform probing ..................................... 105


Thévenin Parameters x

List of Tables

Table 2.1 Summary of some of the shortcomings of droop-based control

strategies and the solutions .......................................................................... 22

Table 4.1 Different Thévenin parameters identification scenarios based on the

status of customers’ equipment and identification principle ....................... 64

Table 5.1 Electrical parameters of the adopted 12.66 kV cable ................................. 83

Table 5.2 Walsh sequency allocation ......................................................................... 85

Table 5.3 Parameters of zero-mean probing with the same energy density ............... 87

Table 5.4 SNRs of single measurement-single observation ....................................... 88

Table 5.5 SNRs of five samples-single observation .................................................. 89

Table 5.6 Settings of the novel control strategy ......................................................... 90

Table 5.7 Voltage perturbation crest factor in sequential probing ........................... 102

Table 5.8 Voltage perturbation crest factor from sequential probing to

simultaneous probing ................................................................................. 103

Table 5.9 Current crest factor ................................................................................... 106


Thévenin Parameters xi

Statement of Original Authorship

The work contained in this thesis has not been previously submitted to meet

requirements for an award at this or any other higher education institution. To the best

of my knowledge and belief, the thesis contains no material previously published or

written by another person except where due reference is made.

Signature:

Date: June 2019

QUT Verified Signature


Thévenin Parameters xii

Acknowledgements

I wish to express my deepest gratitude to my principal supervisor, Professor

Gerard Ledwich, for guiding me through this research and also teaching me valuable

life lessons. I also extend my appreciation to my associate supervisor, Dr. Yateendra

Mishra, for his support and advice during my PhD.

I convey my special thanks to the QUT power engineering discipline leader

Associate Professor Geoff Walker for his unwavering support. I also would like to

thank the discipline coordinator, Dr. Adriana Bondarova, QUT EECS school librarian

liaison, Mr. Graham Dawson, QUT Research Student Centre staff members, Ms.

Janelle Fenner and Ms. Judy Liu, EECS staff members, Mrs. Joanne Kelly, Ms. Joanne

Reaves and Ms. Ellainne Steele. I wish to thank all the academics and non-academic

staff from our discipline and outside of the discipline at QUT who have assisted me in

undertaking this research in different ways and creating a supportive environment

during undertaking this PhD.

I would like to sincerely thank QUT for providing scholarships to undertake this

PhD. Also, I would gratefully acknowledge Energex, South East Queensland power

distribution company for providing a set of data from their Newmarket pilot smart grid

used in this research study.

Last but not least, I cannot thank enough my sister, Farnaz, my mom, Parveen

and my father, Yahya for their unconditional support and self-less love without them

accomplishing this PhD study was impossible.


Thévenin Parameters xiii

Publications Arising From the Thesis

Journal papers

1. A. Raghami, G. Ledwich and Y. Mishra "Improved reactive power sharing among customers’ inverters using online Thévenin estimates” Accepted for

publication at IEEE Transactions on Power Systems, DOI:

10.1109/TPWRS.2019.2918312 (Received great feedback from the reviewers,

2017 Journal Impact Factor 5.255, H index 205, Review citation median 26)

2. A. Raghami, et al. "Designing power-frequency probing for simultaneous identification of Thévenin parameters of residential distribution systems” (In

preparation for submission to IEEE Transactions on Power Systems)

Conference papers

3. A. Raghami, G. Ledwich, and Y. Mishra, "Improved reactive power sharing among photovoltaic inverters using Tévenin's impedance based approach"

presented at IEEE Power & Energy Society General Meeting, 2017, pp. 1-5

(Awarded, power engineering flagship conference).

4. A. Raghami, et al. " Simultaneous Demand-Based Identification of the Power

System’s Thévenin Equivalent” (Ready for submission to IEEE Power &

Energy Society General Meeting)

Chapter 1: Introduction 1

Chapter 1: Introduction

This chapter lays out the context of this research and its significance in relation

to the analysis of the state-of-the-art electrical distribution systems. It brings the great

potential of customer’s inverters for voltage compensation into the spotlight. Initially,

a brief background to this research is given in which the research context is framed. It

is followed by putting forward the research purposes. After that, the importance of the

research is highlighted. The final section gives an outline of the remainder of the thesis.

1.1 Background

A regulated voltage magnitude has always been an important requirement in the

power quality context [1]. Increasing on-site renewable generation and proliferation of

sensitive loads are nascent driving forces that have heightened the concern over the

voltage magnitude regulation [1]. On the one hand, renewable power intermittency on

a daily cycle causes continual voltage magnitude fluctuation. On the other hand, more

strict voltage regulation is required by a growing number of more sensitive loads (e.g.

households’ air-conditioning systems and personal computers). These conflicting

forces have recently caused momentary outages in some utilities [2].

Voltage dip/sag typically happens in relatively long radial distribution branches

during peak hours. The edge of the systems experiences the largest voltage variations.

It has conventionally been the utilities’ responsibility to mitigate any violation of

voltage standards otherwise malfunction of customers’ devices would be likely [3].

Traditionally, systems have been reinforced in overhaul phases by replacing the

existing cables with ones having a higher ampacity or by manual adjustment of the

transformers’ taps [1]. Installation of switched capacitors has also been practised

which could adequately meet the voltage regulation needs for slow variation of

demand. However, the latter falls short handling today’s highly dynamic systems.

Dynamic transformer tapping has alternatively been employed. Strategies have been

investigated to set on-load tap changer according to instantaneous network

information. However not only these strategies lead to hunting effect but also they are

possibly too complex and communication intensive to be reliable for distribution

systems’ application [4, 5]. DSTATCOMs have also been recently installed at some


low and medium voltage distribution systems at a considerable overhead cost of

installation. DSTATCOMs can address a range of power quality issues including

voltage magnitude regulation. However not only are they expensive devices but also

their controllers are highly complex [6, 7].

A growing penetration of rooftop solar and battery inverter systems can be an

asset to the system when controlled properly [8]. Fast response and high accuracy of

these inverters are the key factors to differentiate between assets that help versus assets

that hurt the system. Since switching losses of state-of-the-art inverters are limited,

reactive power can be provided almost energy source free by the inverters. In addition

to that, a slight increase in inverter size gives a substantial reactive power capability.

All the preceding advantages have led to worldwide attention to the potential of

reactive power support from customers’ inverter interfaced equipment [9-11].

1.2 Aims and Objectives of the Thesis

The following items are elaborated as the key objectives of this research:

1. Development of a novel droop control strategy to compensate voltage while

improving reactive power sharing among inverters

2. Development of online strategies to robustly identify the elements of the

system’s Thévenin equivalent from a customer’s inverter perspective using

power lines at power frequency

1.3 Significance of the Research

Today’s electric power-hungry world is continually asking for higher power

quality. Increasing number of customers’ inverters is not a power quality problem per

se. When these inverters are managed appropriately, their advantages far outweigh the

challenges they pose to the system. In other words, inverters can facilitate building

grids with an improved power quality [2, 11, 12].

Energy saving and the associated bill reduction benefits of photovoltaic/battery

inverter systems have already been exploited widely [13]. Cutting the carbon emission

and reducing the reliance on the main grid support also encourage some customers to

adopt inverter-based renewable-centric local generations [14]. These inverters have

been mostly operated according to a unity power factor strategy. This conservative

strategy has worsened the voltage problems at some locations [9]. Although customers


are satisfied in this fashion as long as they don’t see any malfunction in their home

appliances, utilities have to deal with the pressing voltage regulation need to avoid the

likely malfunction. On the one hand, operation at unity power factor is required by

some utilities and on the other hand, the same utilities have to regulate the voltage

magnitude at an overhead cost of grids’ upgrade in overhaul phases.

Power system experts are unanimous about the advantages of local voltage

compensation [15]. These broad advantages come under the umbrella of higher

efficiency and improved performance of the system assets since the local solutions

avert involvement of voltage compensators of the neighbouring distribution systems

and upper grids [16]. Inspired by the preceding benefits and realised the potentials of

the customers’ inverters, some utility companies have initialised voltage control at

inverters installation points [9, 17].

There are also strong arguments in favour of coordination of local assets by more

distributed control strategies [18]. This includes droop based strategies that offer more

straightforward cost-effective solutions by avoiding communication means [2].

However when the inverters on a radial feeder are coordinated by a conventional

droop strategy, a heavier burden of the voltage compensation is imposed on the

inverters installed down the feeder. This drawback of the conventional droop control

strategies is systematically investigated in this research using Thévenin theorem.

1.4 Key Contributions of this Research

Cost-effective straightforwardness of decentralised control strategies has been

the prime factor driving their wide applications [2]. Despite the significant advantages,

the conventional droop strategies are open to criticism because of lack of adaptiveness

to the system in which they are employed.

When the shunt inverters on a radial feeder are coordinated by the non-adaptive

conventional droop strategies, the inverters are loaded unevenly. This uneven loading

is an unsatisfactory compromise that has not been investigated yet according to our

latest literature review.

Adaptiveness needs a greater extent of knowledge. Thus, we have investigated

methods to locally gain knowledge of the system from an inverter’s perspective. We

draw on the circuit theory fundamentals to locally broaden the inverter’s perspective.


Maintaining the straightforwardness of decentralised control strategies is our

fundamental condition. The Thévenin theorem as the most significant circuit theorem

is employed in this direction to provide as much knowledge of the system as possible.

Then, Thévenin parameters identification at the terminals of any customer becomes

the next challenge of this work. The big picture of this work can be dissected in two

key contributions as follows.

1. The novel adaptive droop control strategy

Geometry of phasors is investigated to accurately determine the critical factors

impeding an even distribution of the compensation effort in a droop controlled multi-

inverter system. Elements of the Thévenin equivalent circuit model are incorporated

in the development of a novel droop control strategy.

2. On-line identification of the Thévenin elements at the supply frequency

Following the arguments and the intuitions provided in the early chapters,

Thévenin elements integrated into the novel droop strategy need to be identified.

Structures for Thévenin parameters identification are proposed. There is a significant

degree of difficulty when the local identification is undertaken in a real system with

interference caused by loads and other inverters. To this end, challenges regarding the

processing of the measured signals are discussed, and effective solutions are provided.

The outcome of this research is a novel droop-based voltage compensation

strategy improving power-sharing among customers’ inverters.

1.5 Thesis Outline

Following the research overview provided in the first chapter, the remainder of

this thesis is organised as:

A literature review is presented in Chapter 2. A preamble phasor analysis,

methods for equivalent circuit parameters derivation and some applications of

equivalent circuit knowledge in power system studies are briefly reviewed. Then a

literature survey on control of modern inverter-based systems is reported. While

advantages of the droop control strategies are highlighted, we shed light on the

downsides of conventional droop strategies and the corresponding alternatives

suggested to overcome the downsides. Inspired by many alternatives that focused on


enhancing the adaptiveness of the droop strategies, we draw on Thévenin theorem to

provide any customer with the greatest possible knowledge of the system.

Thus the Thévenin parameters need to be identified. As conventional distribution

systems have no means of inter-inverter communication, a local identification problem

through power lines is desired. Relevant signal processing concepts relevant to this

challenging identification are reviewed and appropriate metrics of performance are

delineated in Chapter 3. This is along with a review of the previous research that have

applied these concepts in similar context and the necessary implications are taken into

account for Chapter 4.

Our contributions are presented in Chapter 4. A Thévenin-based droop control

strategy is proposed through detailed phasor analysis. Some reconciliation with our

observations of a real modern residential distribution system is put forward.

Identification of Thévenin parameters as a complex problem is progressively

addressed in Chapter 4 based on the topics discussed in Chapter 3. Challenges of each

step are theoretically elaborated. Dependency of the desired signals and the

interference are determined through temporal analysis of the measurements.

The methodologies designed in Chapter 4 are tested in Chapter 5. The tests’

results are presented along with interpretations. Points of note drawn from this research

as well as some directions for future work are given in Chapter 6.

Chapter 2: Literature Review 7

Chapter 2: Literature Review

This chapter can be partitioned into two parts. The importance of a phasor

context is briefly reviewed for sinusoidal steady-state analysis of distribution systems.

Derivation of equivalent circuit parameters is delineated in this context. It is followed

by setting a synopsis of some of the previous applications of Thévenin theorem in

power system studies.

The second part of the chapter starts from section 2.4. The inverter’s position

among the electrical apparatus of a typical customer’s PV/battery system is pinpointed.

A typical hierarchical control structure is further described for the inverter. The

primary level of this hierarchical control is deeply investigated. In particular, droop-

based power control is delineated as the research focus. Some limitations of the

conventional droop control are given with the corresponding solutions. Introducing

adaptiveness to the conventional droop has been of paramount importance. Examples

of adaptive droop-based control strategies are cited. Finally, a summary of the

reviewed literature with regard to our research question and implications for the

methodology chapter, i.e., Chapter 4 is highlighted in Section 2.8.

2.1 Phasor Analysis and Thévenin Equivalent Circuit

Linear operations (e.g. summation, subtraction, differentiation, and integration)

on sinusoidal functions result in more sinusoidal functions [19]. Time-variant

equations made of sinusoidal functions, i.e., excitations and the corresponding

responses, can be represented as algebraic equations of complex quantities

synthesising the phasor notation and the concept of impedance. Phasors in these

algebraic equations are generally different in magnitude and angle. Steady-state power

system studies rely on the analysis of these equations where integration in time is

replaced by division by 𝑗𝜔, and differentiation in time is replaced by multiplication by

𝑗𝜔 [19, 20].

The purely dissipative nature of resistors causes their voltage and current phasors

to be in the same phase angle. Whereas nondissipative nature of ideal energy storage

elements causes their voltage to be perpendicular to their current phasors [19]. These

energy storage elements, i.e., capacitors and inductors are presented as reactances as


shown in Figure 2.1. Voltage-current relationships of these reactances are presented as

follows

𝑣 = 𝐿𝑑𝑖

𝑑𝑡�⃗� = 𝑗𝜔 ∙ 𝐿 ∙ 𝐼 �⃗� = 𝑗𝑋 ∙ 𝐼 𝐼 = 𝐼𝑚∠(𝜃) �⃗� = 𝑉𝑚∠(𝜃 +

𝜋

2) (2.1)

𝑖 = 𝐶𝑑𝑣

𝑑𝑡𝐼 = 𝑗𝜔 ∙ 𝐶 ∙ �⃗� 𝐼 = 𝑗𝑋 ∙ �⃗� �⃗� = 𝑉𝑚∠(𝜃) 𝐼 = 𝐼𝑚∠(𝜃 +

𝜋

2)(2.2)

Figure 2.1. Circuits’ energy storing elements with the graphical voltage-current relationship in the

complex phasor and the time domain planes

Voltage and current phasors are aligned unless phase differences are induced by

energy storage elements.

This research is based on sinusoidal steady-state analysis. All circuit relations

and theorems that apply to resistive circuits under DC conditions apply for sinusoidal

steady-state analysis in the frequency domain to circuits consisting of resistance,

inductance, and capacitance, with voltages and currents represented as phasors and

impedances of circuit elements replacing resistance [19]. When power system analysis

is conducted using a digital computer, writing nodal equations based on the current

sources and admittances of the circuit is highly desirable [16]. Once, a reference bus

is selected for the circuit and the circuit admittances and their bus connections are

given as the computer input data, the admittance matrix can be formed. This matrix

together with the input currents vector are employed to determine the bus voltage

vector solving simultaneous linear equations using standard computer programs [16].


“Brute force” and mechanistic methods are undesirable in electrical circuit

analysis, due to an important guiding principle expressing “always seek the simplest

solution” thereby saving time and effort. Creativity and drawing on particular insights

into circuit behaviour play a significant role in reducing a circuit to the simpler

equivalence. In this direction, circuit theorems are the cornerstones to develop creative

ways [19, 20].

The Thévenin theorem is described as the circuit analysis most fundamental

theorem [19]. According to this theorem, the voltage and current characteristic at any

specified pair of terminals of a circuit can be expressed with a two-element circuit.

These elements are an ideal voltage source, namely Thévenin voltage, in series with a

source impedance, namely Thévenin impedance. Thévenin circuit is the simplest

possible equivalent circuit as it consists of just an ideal source and an impedance [20].

The Thévenin theorem applies to linear time-invariant circuits; thus the Thévenin

impedance and the Thévenin voltage need updating following circuit changes over

time.

If a comprehensive knowledge of the circuit was available, nodal equations of

the system could be summarized as

𝑉 = 𝑍𝑏𝑢𝑠 ∙ 𝐼 (2.3)

where 𝑉 and 𝐼 respectively denote the vector of the bus voltages and the vector of the

current sources. 𝑍𝑏𝑢𝑠 is a symmetric matrix called the bus impedance matrix [21]. The

diagonal elements of 𝑍𝑏𝑢𝑠, i.e., 𝑍11, 𝑍22, … , 𝑍𝑁𝑁 are the self-impedances. The ith nodal

equation has a general form as indicated in (2.4)

𝑉𝑖⃗⃗ = 𝑍𝑖𝑖⃗⃗ ⃗⃗ ∙ 𝐼𝑖⃗⃗ + 𝑓(𝐼1⃗⃗ , … , 𝐼�⃗⃗� , … , 𝐼𝑁⃗⃗ ⃗) (𝑓𝑜𝑟 𝑖 ≠ 𝑗) (2.4)

Referring to the Thévenin impedance definition, 𝑍𝑖𝑖⃗⃗ ⃗⃗ denotes the Thévenin

impedance seen from ith bus. Nonetheless, since the comprehensive knowledge of the

system is mostly unavailable, Thévenin source and Thévenin impedance are generally

determined as follows.

The Thévenin source is simply the voltage at the specified terminals when these

terminals are open circuited. There are four different methods to determine the

Thévenin impedance [19]. The first method is conceptually shown in Figure 2.2 from

the perspective of the ith bus in a sinusoidal steady-state, i.e., kth instant. The open


circuit voltage phasor is denoted by the argument and the angle of the Thévenin voltage

estimate, i.e., |𝑉𝑖,𝐾𝑇ℎ𝑣⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ̂⃗| ∠arg (𝑉𝑖,𝐾

𝑇ℎ𝑣⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ̂⃗). From the distribution system analysis perspective, it is

worth noting that dynamically varying demand of the loads and supply of the inverters

might need updating of the Thévenin equivalent model. Updates at subsequent time

steps are indexed by the subscript K. Active and reactive components of the Thévenin

impedance estimate are respectively denoted by 𝑅𝑖,𝐾𝑇ℎ�̂� and 𝑋𝑖,𝐾

𝑇ℎ�̂�. According to the first

Thévenin derivation method, the terminals of interest are short-circuited to measure

the current, dividing the open circuit voltage by the short circuit current is calculated

as the Thévenin impedance [20].

Figure 2.2. Schematic of the first method of Thévenin impedance derivation

In second method, independent sources are essentially set to zero. Thévenin

impedance is then derived by applying circuit theorems (e.g. delta-star

transformation).

Alternatively, the Thévenin impedance is determined by applying a parametric

test source (e.g. a voltage source) to the terminals of interest and the current is found

as a function of the test source or vice versa. The Thévenin impedance is the

proportional coefficient relating the voltage to the current, and the Thévenin voltage is

seen as the constant offset term of the relationship [19].

The fourth method is the most practical approach to determine the Thévenin

impedance. While a test voltage or current is applied to the terminals of interest in this

method, independent sources are considered effective at the stage of Thévenin

impedance determination. Change of the voltage made by the test current is divided

by the test current to calculate Thévenin impedance [19]. The fourth approach is also

known as Tellegen’s Theorem method [22].

It is noteworthy that elements of the Thévenin equivalent circuit are affected by

the dependent sources. Their impact can be considered using superposition. Any

dependent source is characterised by a law relating its current to the voltage. When

this law can be translated straightforwardly to an equivalence of an independent source


and an impedance, the dependent source is replaced by the equivalence. Otherwise, it

is treated as an independent source. An unknown can be assigned to the dependent

source voltage and another unknown as its current. Following the analysis of the

circuit, a new relationship is provided relating the two unknowns. The new relationship

with the characteristic relationship are then used to calculate all the circuit parameters

including the determination of Thévenin equivalent parameters seen from the terminals

of interest [19].

2.2 Norton Equivalent Circuit

When a Thévenin voltage source, in conjunction with a Thévenin impedance, is

transformed to an equivalent current source, a Norton equivalent circuit is obtained. It

follows from the source transformation theorem [20]. The Norton equivalent of the

circuit of Figure 2.2 is shown in Figure 2.3.

Figure 2.3. Norton equivalent circuit of the Thévenin equivalent in Figure 2.2

It may be noted that a circuit can have Thévenin equivalent but not a Norton

equivalent and conversely [19]. The Thévenin equivalent circuit of an ideal voltage

source is the source itself with zero Thévenin impedance. This makes the Norton

current infinity, so the Norton equivalent circuit does not exist. The preceding case is

exemplified for a typical n-bus radial distribution feeder connected to a robust main

grid at Bus_1 with unity voltage shown in Figure 2.4. This modelling of the upper

network is used in this thesis.

Figure 2.4. n-bus radial feeder connected to the voltage robust main grid


2.3 Significance of Thévenin Equivalent Circuit in Power System Studies

Today’s active distribution systems prompt a need for continual monitoring of

the systems’ operation condition. Continual equivalent network derivation is in

alignment with the need to the current broader observability of the system for decision-

making processes [23, 24]. This is where attempts for online Thévenin learning fit in.

Thévenin theorem has already been utilised to important wide-ranging topics of

power system studies [9, 25]. Thévenin impedance extraction has been proved as a

worthwhile knowledge for all phases of microgrid study [26, 27]. The significance of

equivalent impedance application has been reflected for network upgrades. Ancillary

services and power flow analysis have also been pointed out as two use cases of the

equivalent network derivation [28]. The Thévenin equivalent circuit of a system has

been derived for cyber security analysis [29]. Thévenin equivalent potential was first

explored for voltage stability analysis in [30]. The preceding exploration was limited

to the equivalent seen from the generator in a radial structure. Two Thévenin-based

alternative approaches were provided for steady-state and transient voltage stability of

a single transmission line connecting a generation side to a load centre side [31]. The

difference in alternatives lays in the fact that to either model load side linearly or model

a Thévenin equivalent for either side of the PMU [31]. The presented highly simplified

models in [31] were appropriate for online stability analysis. Sequential load variation

has been applied to work out the Thévenin based voltage stability margin following

the system’s contingencies [32]. FACTS and HVDC impact on voltage stability was

also studied using Thévenin equivalent [33]. Locally extracted Thévenin equivalent of

the system was a significant milestone that was brought Thévenin-based voltage

stability analysis to attention in [34]. The authors gave significant insights into possible

undervoltage protection enhancement via Thévenin knowledge of the system, and

concisely raised some practical challenges in this pathway [34]. An alternative simple

and computationally efficient voltage stability index was given in [35] based on real-

time Thévenin equivalent identification of the system. The Thévenin equivalent has

been used to determine the voltage disturbance of an unbalanced 3-phase 3-wire and a

3-phase 4-wire network [36]. A novel Thévenin-based distributed control strategy has

been developed to coordinate battery systems of a multi-agent multi-zone distribution

system [37]. Some challenges concerning synchronisation of inverters have been met

by conducting a Thévenin based stability analysis for paralleled inverters [38]. Power


transfer capability of a PV plant for exchange with the main system has been improved

by the development of an adaptive reactive power Thévenin based droop control [39].

The permissible extent of wind penetration is evaluated undertaking a Thévenin based

stability analysis [40]. Maximum voltage stability margin and maximum loadability

have been increased utilising Thévenin knowledge. Unsymmetrical fault location has

also been determined leveraging this knowledge [41]. Leveraging Thévenin

impedance knowledge to locate unsymmetrical fault has been elaborated [42].

2.4 Typical Power Elements Connected to an Inverter

Increasing uptake of inverters by customers has led to the introduction of control

strategies that actively organise clusters of inverters. These clusters are commonly

known as microgrids and they are part of the distribution systems. Conceptually,

microgrids can be operated in parallel or disconnected from the main grid as an isolated

island.

One configuration of a typical PV system is shown in Figure 2.5. While only

power elements have been shown in Figure 2.5. The control system managing this

interconnected system can be partitioned into input side and grid side. DC bus of the

power converters is the boundary between the two sides [43].

Figure 2.5. One configuration of power elements connected to a PV module

There is a broad field of research on control aspects of any power element shown

in Figure 2.5. The inverter’s control is highlighted in this research.

2.5 A Typical Hierarchical Control for an Inverter

Large synchronous generators of the conventional power systems have been

operated using a hierarchical control structure which typically consists of three levels

[44]. Analogously primary, secondary and tertiary control levels can be described for

management of inverters [45]. A typical hierarchical control structure is shown in

Figure 2.6 for a microgrid including two inverters.


Grid-feeding, grid-forming and grid-supporting are classes of inverters

depending on their functions in an AC microgrid [46]. When real or reactive power

delivery of PV/battery inverter systems are controlled to regulate the frequency and/or

magnitude of the grid voltage, they are considered as grid-supporting inverters. Two

different types of grid-supporting inverters are identified as noted in Figure 2.6 (b).

When they can be independently operated in an islanded microgrid, they would be

modelled as voltage source, i.e., the one at the bottom panel of Figure 2.6(b). Whereas

grid-supporting inverters whose operation is limited to the voltage regulated systems

are modelled as a current source, i.e., the one at the top panel of Figure 2.6(b) [46].

Figure 2.6. A typical hierarchical control strategy implemented to coordinate the inverters of a

microgrid (a) tertiary control and secondary control (b) secondary control and primary control

2.5.1 Tertiary Control

The microgrid shown in Figure 2.6 is connected to the main grid through a tie-

switch. The microgrid exchanged power with the main grid in parallel operation mode

is usually controlled by tertiary control. This level is also known as grid level where


functions can be found implemented by distribution network operator (DNO) and

market operator (MO) [47, 48]. This level of control also facilitates synchronisation of

an islanded microgrid to smoothly reconnect with the main grid [45]. Magnitude and

frequency of the voltage of the microgrid side of the tie-switch are controlled for this

purpose. A typical proportional integral controller of the tertiary control can be

expressed as

𝜔𝑀𝐺∗ = 𝑘𝑝𝑃 ∙ (𝑃𝐺

∗ − 𝑃𝐺) + 𝑘𝑖𝑃 ∙ ∫(𝑃𝐺∗ − 𝑃𝐺)𝑑𝑡 (2.5)

𝑉𝑀𝐺∗ = 𝑘𝑝𝑄 ∙ (𝑄𝐺

∗ − 𝑄𝐺) + 𝑘𝑖𝑄 ∙ ∫(𝑄𝐺∗ − 𝑄𝐺)𝑑𝑡 (2.6)

2.5.2 Secondary Control

When frequency and magnitude droop based control strategies are utilized as the

primary controller in an islanded microgrid, frequency and magnitude of the voltage

deviate following any change in demand or supply. Deviations within an allowable

limit can be compensated using a secondary control level. These limits are dictated by

grid code standards (e.g. ±6% for magnitude required by the Australian standards).

Secondary control basically adjusts the reference points for the primary control of all

inverters [45].

In this direction, measured voltage frequency and magnitude, i.e., 𝜔𝑀𝐺 and 𝑉𝑀𝐺

are compared with the references, i.e., 𝜔𝑀𝐺∗ and 𝑉𝑀𝐺

∗ ; all inverters are updated with the

processed errors, i.e., 𝛿𝜔 and 𝛿𝑉 to restore frequency and magnitude to the rated

values. Typical secondary control function for frequency and magnitude are presented

as follow

𝛿𝜔 = 𝑘𝑝𝜔 ∙ (𝜔𝑀𝐺∗ − 𝜔𝑀𝐺) + 𝑘𝑖𝜔 ∙ ∫(𝜔𝑀𝐺

∗ − 𝜔𝑀𝐺) ∙ 𝑑𝑡 (2.7)

𝛿𝑉 = 𝑘𝑝𝑉 ∙ (𝑉𝑀𝐺∗ − 𝑉𝑀𝐺) + 𝑘𝑖𝑉 ∙ ∫(𝑉𝑀𝐺

∗ − 𝑉𝑀𝐺) ∙ 𝑑𝑡 (2.8)

where 𝛿𝜔 and 𝛿𝑉 are frequency and magnitude of the voltage as the secondary

controller’s outputs respectively, 𝑘𝑝𝜔 and 𝑘𝑝𝑉 are the corresponding proportional

gains with 𝑘𝑖𝜔 and 𝑘𝑖𝑉 are the corresponding integral coefficients of the controller.

In the reconnection process of an islanded microgrid to the main grid, the grid

side frequency and magnitude of the voltage of the tie-switch are the references for the

secondary controller. Any phase difference between the isolated microgrid and the

main grid is corrected by a synchronisation control loop which can be a conventional


phase locked loop (PLL) [49]. In this process ∆𝜔𝑠 would be the correction term added

to the (2.7) and sent out to all inverters as follow

𝛿𝜔 = 𝑘𝑝𝜔 ∙ (𝜔𝑀𝐺∗ − 𝜔𝑀𝐺) + 𝑘𝑖𝜔 ∙ ∫(𝜔𝑀𝐺

∗ − 𝜔𝑀𝐺) ∙ 𝑑𝑡 + ∆𝜔𝑠 (2.9)

Following the synchronisation, there would be zero exchanged power between

the paralleled microgrid and the main grid.

Secondary control level is also known as the management level where microgid

central controller (MGCC) is the crucial element. MGCC is the DNO’s and MO‘s main

interface with the microgrid. MGCC can handle considerations like market prices for

electricity and even other commodities like gas. MGCC can also be taken responsible

for optimisation of local production. When there are deferrable loads in the microgird,

they are typically under the MGCC control.

Implementation of tertiary and secondary control levels needs a central system

using communication infrastructures [45].

2.5.3 Synchronisation of Inverters

Overall performance of the coordinated inverters is influenced by the precision

of the estimation of the voltage parameters. Voltage magnitude, frequency, and phase

angle need to be accurately estimated using a synchronisation algorithm to enable

precise control of the active and the reactive power of each inverter module. Moreover,

as it was mentioned in the previous section, any maneuver between the parallel and

the isolated operation modes requires the grid condition monitoring [46].

Synchronisation system of grid-forming and voltage source grid-supporting

inverters should work in the parallel and the isolated operation modes of the microgrid.

In the isolated mode, the synchronisation system oscillates at an unchanged frequency,

i.e., 𝜔𝑓𝑓. In the transition of operation modes, phase angle and frequency of the

isolated microgrid’s voltage are slowly varied to resynchronise with the main grid’s

voltage. A stable and secure manoeuvre is required as all grid-feeding inverters are

under the influence of the reconnection frequency and phase-angle transients [45].

Synchronous reference frame phase-locked loop (SRF-PLL) has been

extensively used in nearly balanced three-phase systems. It is shown as a constituent

part of the primary control in Figure 2.7. Park transformation is employed to obtain

signals in dq reference from the abc reference frame. The 𝑣𝑞 component is driven to


zero through a feedback control loop and the angular position of the dq reference frame

is regulated. Phase estimation dynamics is normally improved by feed forwarding 𝜔𝑓𝑓

[46].

The considerations above should also be received under grids with unbalanced

and distorted voltage conditions. Frequency-locked loop (FLL) can alternatively be

used as the synchronisation system. Compared to PLL systems, FLL systems are

generally less affected by likely phase-angle jumps during transient abnormal grid

conditions [50, 51].

2.5.4 Primary Control and Basics of Droop Control

Active modern distribution grids host customers with loads and PV/battery

inverters. Coordination of power-sharing among these inverters is the crucial role of

primary control level [45]. In this context, proliferation of uninterruptable paralleled

systems (UPS) has taken place before the advent of PV/battery inverters. UPS active

and reactive power control strategies have already been examined for inverters’

coordination. Centralized, master-slave, average-load sharing and circular-chain

control architectures are some of the common UPS control categories [52]. However,

UPS inverters have often been located close to each other equipped with

communication channels. The need for technically complex and costly communication

infrastructure impedes the application of the typical UPS control strategies to spatially

dispersed customers’ inverters without any means of communication at first place [46].

Privacy of individual customers might hinder the application of the UPS control

strategies to PV/battery inverters’ coordination. Alternatively, these inverters can be

controlled via decentralised strategies independent of communication means. Droop

based strategies with an enduring legacy from the straightforward operation principles

of large synchronous generators have been widely applied for decentralised

coordination of the inverters. In fact, the microgrid concept of the Consortium for

Electrical Reliability Technology Solutions (CERTS) strongly discourages

communication-based control strategies for power-sharing purposes [53, 54]. Droop-

based designs have particularly become a prominently robust control strategy since

they are immune to likely disruptions to communication systems [55].

The current regulation loop is considered as the inner part of the primary control.

Droop control as the outer part of the primary control level provides references for the


current loop [56]. A typical control structure for a current-source-based grid-

supporting inverter is delineated in Figure 2.7.

Figure 2.7. A current source grid supporting inverter controlled by the primary level

This control structure has been implemented in dq reference frame. There have

been abc to dq transformations at different parts of the structure. The transformations’

outcome has been DC signals rotating synchronously with the frequency of the grid

voltage. These transformations have specifically required the voltage phase angle

information. This information has been provided by a phase locked loop block as

shown in the top part of Figure 2.7. Proportional-integral (PI) controllers used in the

structure have had a typical transfer function given by

𝐺𝑃𝐼(𝑠) = 𝐾𝑝 +𝐾𝑖

𝑠 (2.10)

where 𝐾𝑝 and 𝐾𝑖 respectively denote the integral gain and the proportional gain. From

circuit perspective, a current-source-based grid-supporting inverter regulating its

injection according to the bus voltage can be modelled as a voltage-controlled

dependent current source. Droop control is further inspected as the focus of this

research. In particular, voltage magnitude droop control is examined. The focus is

highlighted in Figure 2.7 and Figure 2.8.


Figure 2.8. (a) Schematic of a voltage droop controlled current source grid supporting inverter (b)

simplified presentation of the schematic

A radial feeder consisting of n customers all having loads and PV/battery

inverter systems is shown in Figure 2.9(a). The equivalent circuit model of the whole

feeder from the ith inverter’s perspective is depicted in Figure 2.9(b). According to

this model, a voltage controlled current source has been connected to the system’s

Thévenin equivalent.

Figure 2.9. (a) Configuration of a typical n bus modern radial feeder with customers having loads and

PV/battery inverter systems (b) Circuit model of the ith droop-controlled inverter connected to the

system’s Thévenin equivalent

Time steps in the inverter’s action have been indexed by k for any inverter

compensating voltage magnitude. Taking the dynamic nature of the compensators and

varying loads into account, parameters of the system’s Thévenin equivalent model are

subject to change. However, there is a priori assumption that these parameters are

unchanged for a short time span that they are being identified [57]. Updating the

estimates of system’s Thévenin parameters has been indexed by K. Thévenin

resistance, reactance and voltage have been denoted by 𝑅𝑖,𝐾𝑇ℎ�̂�, 𝑋𝑖,𝐾

𝑇ℎ�̂�and 𝑉𝑖,𝐾𝑇ℎ𝑣⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ̂⃗ in which

subscript i shows the inverter bus number in a multi-inverter network. The preceding


notations have been consistently used in this thesis. The exchanged power of an

inverter with the rest of the gird is presented as follow

𝑃𝑖,𝑘 =|𝑉𝑖,𝑘⃗⃗ ⃗⃗ ⃗⃗ ⃗|

𝑅𝑖,𝐾𝑇ℎ�̂�

2+𝑋𝑖,𝐾

𝑇ℎ�̂�2 ∙ [𝑅𝑖,𝐾

𝑇ℎ�̂� ∙ (|𝑉𝑖,𝑘⃗⃗⃗⃗⃗⃗ | − |𝑉𝑖,𝐾𝑇ℎ𝑣⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ̂⃗| ∙ cos (arg(𝑉𝑖,𝑘⃗⃗⃗⃗⃗⃗ ) − arg (𝑉𝑖,𝐾

𝑇ℎ𝑣⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ̂⃗))) + 𝑋𝑖,𝐾𝑇ℎ�̂� ∙ |𝑉𝑖,𝐾

𝑇ℎ𝑣⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ̂⃗| ∙

sin (arg (𝑉𝑖,𝑘⃗⃗⃗⃗⃗⃗ ) − arg (𝑉𝑖,𝐾𝑇ℎ𝑣⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ̂⃗))] (2.11)

𝑄𝑖,𝑘 =|𝑉𝑖,𝑘⃗⃗ ⃗⃗ ⃗⃗ ⃗|

𝑅𝑖,𝐾𝑇ℎ�̂�

2+𝑋𝑖,𝐾

𝑇ℎ�̂�2 ∙ [−𝑅𝑖,𝐾

𝑇ℎ�̂� ∙ |𝑉𝑖,𝐾𝑇ℎ𝑣⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ̂⃗| ∙ sin (arg(𝑉𝑖,𝑘⃗⃗⃗⃗⃗⃗ ) − arg (𝑉𝑖,𝐾

𝑇ℎ𝑣⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ̂⃗)) + 𝑋𝑖,𝐾𝑇ℎ�̂� ∙ (|𝑉𝑖,𝑘⃗⃗⃗⃗⃗⃗ | − |𝑉𝑖,𝐾

𝑇ℎ𝑣⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ̂⃗| ∙

cos (arg(𝑉𝑖,𝑘⃗⃗⃗⃗⃗⃗ ) − arg (𝑉𝑖,𝐾𝑇ℎ𝑣⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ̂⃗)))] (2.12)

where active power and reactive power delivered by ith inverter to the grid have been

respectively denoted by 𝑃𝑖,𝑘 and 𝑄𝑖,𝑘 [46]. The magnitude of the ith inverter’s terminal

voltage has been denoted by |𝑉𝑖,𝑘⃗⃗ ⃗⃗⃗⃗ |. 𝑅𝑖,𝐾𝑇ℎ�̂�

and 𝑋𝑖,𝐾𝑇ℎ�̂� respectively denote the active and

the reactive component of the Thévenin equivalent impedance estimate of the rest of

the system. 𝑉𝑖,𝐾𝑇ℎ𝑣⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ̂⃗ shows the estimated phasor of the Thévenin voltage of the equivalent

of the rest of the system. When the system’s impedance is relatively reactive,

mathematical manipulation of (2.12) results in the conventional voltage reactive

current droop control strategy as follows

𝐼𝑖,𝑘𝐶⃗⃗⃗⃗ ⃗ = 𝑚 ∙ (1 − |𝑉𝑖,𝑘⃗⃗⃗⃗⃗⃗ |)∠(arg(𝑉𝑖,𝑘⃗⃗⃗⃗⃗⃗ ) − 𝜋 2⁄ ) (2.13)

where phasor of the inverter’s compensating reactive current has been denoted by 𝐼𝑖,𝑘𝐶⃗⃗⃗⃗ ⃗

[9]. 𝑚 shows the droop gain. The phase angle of the pure reactive compensation is

ninety degree offset with regard to the terminal voltage phasor. This lagging offset has

been due to the load convention adopted in this research. The conventional droop

characteristic of an inverter is depicted in Figure 2.10. Inverters’ reactive current is

reduced to zero at the rated voltage.

Figure 2.10. Voltage reactive current droop characteristic


As synthesis of dependent sources in Thévenin circuit derivation was earlier

described in section 2.1, the dependent source modelled by (2.13) can also be

represented with an independent source connected to an impedance as depicted in

Figure 2.11. This presentation is known as Thévenin descriptor [9]. The Norton

descriptor is also deliverable using the voltage source transformation to current source.

Figure 2.11. (a) Norton descriptor and (b) Thévenin descriptor of an inverter managed by the droop

control characterised in (2.13)

2.6 Advantages, Limitations and Variations of the Conventional Droop

Droop-based strategies are closely tied with the concept of decentralisation in

the control systems. Minimum dependency of these strategies on inter-module

communications results in outstanding flexibility, excellent reliability and last but not

least, economic benefits [54]. These all together have made droop-based strategies

almost an obvious choice for the operation of large power systems over the decades

and have created a huge interest for their applications in modern distribution systems.

Despite the obvious advantages, droop control strategies have some limitations

too [47]. Some of these limitations with the proposed alternate strategies are

summarized in Table 2.1.


Table 2.1

Summary of some of the shortcomings of droop-based control strategies and the solutions

Limitation of the conventional droop Alternate strategy

Trade-off between load sharing and

voltage regulation

Dynamic droop gain

Restoration control

High gain angle droop with supplementary loop

Slow and oscillating dynamic response

Adaptive derivative droop to damp transients

Angle droop

Droop based on coupling filter parameters

Droop based on H infinite control theory

Adverse influence of system impedance

between inverters

Voltage drooped as a function of mixed active and

reactive powers output

Additional loop with the grid Thévenin impedance

and voltage estimation

Interfacing a virtual inductor

Poor harmonic sharing

Virtual impedance

Cooperative harmonic filtering strategy

Additional loop reducing the bandwidth

2.7 Adaptiveness of Droop

System’s characteristics strongly impact the accuracy of droop-based reactive

power sharing [55]. There has been a considerable effort to develop adaptive droop

strategies. Adaptiveness has served different purposes.

Oscillating dynamic response of power-sharing among paralleled inverters has

been improved [58]. While a robust steady-state power-sharing has been ensured by

the static droop gain, transient droop gains have been dynamically set to damp the

oscillatory modes.

Reactive power sharing has become less dependent on line impedances and

active power control using a voltage droop as a non-linear function of inverters’ active

and reactive power [59].

There is an intrinsic trade-off between the accuracy of reactive power sharing

and the voltage regulation in the conventional droop control. Reference of each module

has been adaptively controlled to provide a proper current sharing in a single bus multi-

inverter configuration. This adaptive reference modulation could also limit the

variation of operating voltage [60].

A combination of reactive power control and adaptive droop-based active power

curtailment has been proposed for PV inverters with loss minimisation and voltage

regulation as the objectives [61]. The objectives have been adaptively prioritised based


on the recommended range of operating point’s voltage. When the voltage has been in

the operating range, loss minimisation has had priority however when the voltage has

violated the rated operating range; the voltage regulation has been prioritised. When

the reactive power supply has been exhausted in controlling over-voltages, the active

power has been curtailed evenly according to a droop law with a gain adjusted

according to the voltage sensitivity of the PV bus [61].

Droop coefficients have been adaptively tuned to improve reactive power

sharing using a communication system. Thus sharing could match the inverters’

relative ratings despite differences in the output impedances. A floating term has been

basically added to the conventionally fixed droop gain [62]. This floating term has

been tuned according to the inverter’s active to reactive power ratio as well as the

mismatch between the connecting impedance of different inverter modules to the

common microgrid bus [62].

Droop gain has been adaptively adjusted. A controller consisting of an estimator

and an adaptive droop has been proposed with the objective of tight active and reactive

power regulation decoupled from grid parameters. The estimated parameters have

been equivalent impedance and voltage of the grid that the inverter has been connected

to [63]. The proposed controller has been developed based on an offline static

estimation where the sensitivity of the proposal has only been analysed to different

system impedance in separate simulation scenarios. In other words, there has been no

dynamical change in the impedance to really test the proposed estimation robustness

[63]. The emphasis of this proposal has been on a single inverter case and has neglected

the estimation challenges in a multi-inverter microgrid that in turn has led to significant

limitation to the practicality of the proposal. A second order general integral frequency

locked loop has been simplistically proposed as the solution to challenges of the

Thévenin circuit’s parameters estimation with a justification revolving around the

capability of integration of the voltage frequency [63]. Even when the connection and

the disconnection of a single inverter to the grid is studied, one needs to discuss the

variation in the grid side as well.

Regulation of average voltage in a microgrid has been addressed, and reactive

power has been shared proportionally using an adaptive consensus droop based control

strategy [64]. This strategy has had two modules for each inverter to process the

information gathered locally and also the data sent by the neighbours. Inverters have


reached to a consensus about overall voltage deviation, and they have consequently

lowered/elevated their droop characteristic for voltage compensation. Proportional

reactive power supply of each inverter has also been set according to the rating by

droop gain adjustment.

Voltage magnitude has been regulated using a combination of PV inverters’

reactive power and battery inverters’ active power [4]. A variable droop gain based on

the voltage sensitivity analysis has been applied for droop controlled battery inverters

to realise even investment for battery storage capacity by all customers and has

minimised the total capacity installation. However, this adaptiveness has been

introduced as either set and forget process or communication dependent for regular

update.

A smooth operating mode transition has been realised for an inverter-based

microgrid from the grid-connected operation mode to the isolated mode. In this

direction, an adaptive droop-based control has been proposed to coordinate

charge/discharge of the batteries with the generation of the other inverters to facilitate

likely multi-transition. This droop characteristic’s reference has been shifted up and

down accordingly to control the isolated microgrid’s frequency and to share the loads

[65].

2.8 Summary and Implications

Since the infancy of large power systems, a higher observability of network

operations has always been of the systems’ operators’ interest. The Thévenin theorem

as the most significant electrical circuit theorem has been extensively employed in

studies conducted at higher voltage levels to hone the adaptiveness of the system

operation drawing on a higher observability.

Penetration of distributed generations for the past two decades has made a

significant transformation in the status of distribution systems from the formerly

passive to the currently active. The proliferation of inverter-interfaced customers calls

for coordination of these distributed generators. This coordination can be undertaken

in a hierarchical scheme similar to what has been being applied to large generators of

the conventional power systems. In this realm, we refine the research foci to the

primary control of inverters. Then it should be noted that the power controller module

of the primary control is investigated in particular. Power controllers can be


categorised based on the extent of their dependency on communication infrastructure.

Considering the fact that inverters are being introduced to the existing conventional

distribution systems that often do not have any means of inter-inverter communication,

we focus on decentralised power sharing strategies. To this end, scalable and modular

droop-based control strategies have already attracted the attention of the decision

makers of the new distribution systems. Droop control applications in modern

distribution systems are developed by imitating the self-regulation capability of

synchronous generators.

Despite the advantages, there are some drawbacks to droop-based control

strategies that researchers have tried to address. In this direction, introducing

adaptiveness to the conventional droop strategy has been widely applied for different

objectives. Analogous to the high voltage systems, pursuit of adaptiveness is

commonly tied up with an improved observability in distribution systems. A higher

observability of the new age active distribution systems is advantageous not only to

the actors at low voltage level but also to the higher voltage systems’ operators.

However, most of the attempts at the introduction of adaptiveness have hinged

on using communication means to some extent. The employment of the

communication means is in contrast with the local nature of the droop-based control

strategies as it incurs an increasing amount of cost and complexity.

Saving cost and reducing complexity justifies leveraging circuit theorems for

innovative methods in distribution systems’ analysis. Since Thévenin theorem gives

the equivalent model of the whole system from any pair of terminals, the theorem’s

potentials are exploited in this research to provide communication-free adaptiveness

without compromising the local nature of droop-based control strategies.

Previous attempt to acquire Thévenin parameters has been mostly limited to

single point probing, single point measurement [36, 57, 66]. This kind of approach

does not align with the coordination need of a multi-inverter system.

In sum, a study of the literature has not revealed an in-depth investigation of

Thévenin-based observability for local control of real distribution systems with

dynamic loads and dynamic compensators. Thévenin parameters identification is a

highly challenging task in this dynamic noisy system compared to the hypothetically


static noise-free system. The background signal processing concepts for a typical local

identification through power lines is detailed in the next chapter.

Chapter 3: Signal Processing Concepts Relevant to Local Identification Problems 27

Chapter 3: Signal Processing Concepts Relevant to Local Identification

Problems

3.1 Overview

In this research, Thévenin parameters of the distribution system are identified

locally through power lines at power frequency. The relevant background signal

processing concepts and the mathematical topics relevant to this scene are discussed

in this chapter.

3.2 Signal Processing Basics Required for Understanding of a Local Identification

Theoretical circuit principles of Thévenin parameters derivation were presented

in section 2.1. Working with time waveforms is needed in many real-world science

and engineering practices including local identification problems. Desired waveforms

frequently appear as random time signals. In this direction, random waveforms need

to be described in a probabilistic sense [67].

3.2.1 Continuous and Discrete Random Process

Enlarging the random variable concept across time gives rise to the concept of

random process. Since the possible outcome of an experiment, i.e., s dictates the value

of a random variable X, the random process becomes a function of both s and t. The

random process can be denoted as X(s,t) representing a family or ensemble of time

functions where s and t are variables. Each member of the ensemble as a specific

waveform of a random process is called a sample function commonly represented as

x(t) [67, 68].

When t can have any value from a continuum and X is continuous too, X(s,t) is

considered a continuous random process. A few sample functions of a continuous

random process are illustrated in Figure 3.1.


Figure 3.1. A continuous random process

When X is continuous, but t has only discrete values, the random process is

considered as a continuous random sequence. A continuous random sequence is often

referred to as a discrete-time (DT) random process as it is defined at only discrete

(sample) times. Sample functions of a DT random process are frequently called DT

random signal [67, 68]. A DT random process is technically a set of random variables

denoted by {𝑥𝑖(𝑙 ∙ 𝑇𝑠) ∶ 𝑖 = 1,2, … , 𝑙 = 1,2, … } given for sample times with 𝑇𝑠 known

as the sampling interval. 1 𝑇𝑠⁄ is called the sampling rate stated as samples per second.

This type of random processes are frequently encountered in real-world local

identification problems since data loggers have limited sampling rates. In practice, it

is often sufficient to refer to a DT random process as 𝑋(𝑙 ∙ 𝑇𝑠). When the constant 𝑇𝑠

is already known, 𝑋[𝑙] is adopted as a brief notation, where l is the time index. DT

signal phasors are denoted by 𝑋𝑖,𝑙⃗⃗ ⃗⃗ ⃗ in this thesis [69]. A few members of an ensemble

of a discrete-time random process formed by sampling the waveform of Figure 3.1 are

depicted in Figure 3.2.


Figure 3.2. A discrete-time random process (or a continuous random sequence) formed by sampling

the waveforms of Figure 3.1

3.2.2 Deterministic

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