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
A Novel Thévenin-Based Voltage Droop Control Improving Reactive Power Sharing with Structures to Identify
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
A Novel Thévenin-Based Voltage Droop Control Improving Reactive Power Sharing with Structures to Identify
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.
A Novel Thévenin-Based Voltage Droop Control Improving Reactive Power Sharing with Structures to Identify
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
A Novel Thévenin-Based Voltage Droop Control Improving Reactive Power Sharing with Structures to Identify
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
A Novel Thévenin-Based Voltage Droop Control Improving Reactive Power Sharing with Structures to Identify
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Bibliography ........................................................................................................... 114
A Novel Thévenin-Based Voltage Droop Control Improving Reactive Power Sharing with Structures to Identify
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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
A Novel Thévenin-Based Voltage Droop Control Improving Reactive Power Sharing with Structures to Identify
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
A Novel Thévenin-Based Voltage Droop Control Improving Reactive Power Sharing with Structures to Identify
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
A Novel Thévenin-Based Voltage Droop Control Improving Reactive Power Sharing with Structures to Identify
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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
A Novel Thévenin-Based Voltage Droop Control Improving Reactive Power Sharing with Structures to Identify
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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
A Novel Thévenin-Based Voltage Droop Control Improving Reactive Power Sharing with Structures to Identify
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.
A Novel Thévenin-Based Voltage Droop Control Improving Reactive Power Sharing with Structures to Identify
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
Chapter 1: Introduction 2
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
Chapter 1: Introduction 3
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.
Chapter 1: Introduction 4
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
Chapter 1: Introduction 5
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
Chapter 2: Literature Review 8
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].
Chapter 2: Literature Review 9
“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
Chapter 2: Literature Review 10
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
Chapter 2: Literature Review 11
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
Chapter 2: Literature Review 12
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
Chapter 2: Literature Review 13
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.
Chapter 2: Literature Review 14
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
Chapter 2: Literature Review 15
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
Chapter 2: Literature Review 16
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
Chapter 2: Literature Review 17
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
Chapter 2: Literature Review 18
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.
Chapter 2: Literature Review 19
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
Chapter 2: Literature Review 20
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
Chapter 2: Literature Review 21
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.
Chapter 2: Literature Review 22
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
Chapter 2: Literature Review 23
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
Chapter 2: Literature Review 24
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
Chapter 2: Literature Review 25
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
Chapter 2: Literature Review 26
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.
Chapter 3: Signal Processing Concepts Relevant to Local Identification Problems 28
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.
Chapter 3: Signal Processing Concepts Relevant to Local Identification Problems 29
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