-
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
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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
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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
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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.
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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
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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
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A Novel Thévenin-Based Voltage Droop Control Improving Reactive
Power Sharing with Structures to Identify
Thévenin Parameters vi
Bibliography
...........................................................................................................
114
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A Novel Thévenin-Based Voltage Droop Control Improving Reactive
Power Sharing with Structures to Identify
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
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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
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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
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A Novel Thévenin-Based Voltage Droop Control Improving Reactive
Power Sharing with Structures to Identify
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
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A Novel Thévenin-Based Voltage Droop Control Improving Reactive
Power Sharing with Structures to Identify
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
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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.
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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)
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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
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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.
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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
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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
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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