Dhana Raj Markandu: Maximising Generator Connections in Distributed Networks under Stability Constraints

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INSTITUTE FOR ENERGY SYSTEMS
SCHOOL OF ENGINEERING
MSC DISSERTATION THESIS
MSC IN SUSTAINABLE ENERGY SYSTEMS
MAXIMISING GENERATOR CONNECTIONS
IN ELECTRICITY NETWORKS UNDER
STABILITY CONSTRAINTS
DHANA RAJ MARKANDU
S1024415
AUGUST 2011

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ABSTRACT
The twin concerns of climate change and energy security are driving the increased
penetration of renewable generation in distribution networks. Conventional networks are not
designed to handle this change and several technical issues can arise as a result of this.
Transient stability, which is a measure of the dynamic response of a generator to large
disturbances, is one of the issues that requires scrutiny. Tools are needed to model and
analyse its behaviour for networks with increased distributed generation.
This project uses optimal power flow to determine maximum generation capacity under
transient stability constraints for single-generator networks. The Swing Equation and the
Equal Area Criterion were selected as the basis to evaluate stability. A transient stability-
constrained optimal power flow was iteratively developed from basic principles and applied
to a theoretical case study. The project concludes that the model developed was functional
under specific conditions, but additional work is required in order for it to accurately
represent a true distribution network. Finally, suggested paths for future work are described
towards the eventual goal of developing a model capable of analysing generator capacity
allocation for multiple-machine distribution networks under transient stability constraints.

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DECLARATION OF ORIGINALITY
I declare that this thesis is my original work except where stated.
This thesis has never been submitted for any degree or examination to
any other University.
………………….………..……………….
Dhana Raj Markandu
MSc. Sustainable Energy Systems
University of Edinburgh
16th
August 2011

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CONTENTS (I)
1 Introduction 1
2 Objectives & Methodology 3
2.1 Project Objectives 3
2.2 Project Methodology 3
3 Background & Technical Fundamentals 6
3.1 Distributed generation 6
3.2 Generator transient stability 9
3.2.1 Influence of distributed generation on transient stability 10
3.2.2 Methods for evaluating transient stability 10
3.2.3 Classical model of a synchronous generator 11
3.2.4 The Power-angle Curve 14
3.2.5 Equivalent circuit of a faulted power line 15
3.2.6 The Swing Equation 19
3.2.7 Solving the Swing Equation 21
3.2.8 The Equal Area Criterion 22
3.3 Optimal power flow 26
4 Modelling Transient Stability 29
4.1 Modelling the Equal Area Criterion in Microsoft Excel 29
4.1.1 Structure and functionality of the Microsoft Excel model 30
4.1.2 Verification of the Microsoft Excel model against PowerWorld 34
4.1.3 Application of the Microsoft Excel model to determine maximum
generator capacity
37
4.2 Solving the Swing Equation with Matlab 40
4.2.1 Structure and functionality of the Matlab solver 40
4.2.2 Verification of the Matlab solver against PowerWorld 41

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CONTENTS (II)
5 Transient Stability-Constrained Optimal Power Flow
(TSC-OPF)
45
5.1 TSC-OPF Version 1 46
5.1.1 Structure and functionality of TSC-OPF-1 47
5.1.2 Verification of TSC-OPF-1 against the Microsoft Excel model 54
5.1.3 Limitations of TSC-OPF-1 55
5.2.2 Verification of TSC-OPF-2 against the Microsoft Excel model 59
5.3.2 Verification of TSC-OPF-3 against PowerWorld 63
6 Case Study: Transient Stability-Constrained Power
Capacity Allocation
64
6.1 Description of the test network 64
6.2 Results 67
6.3 Discussion 68
6.3.1 Performance of the optimal power flow model 68
6.3.2 Analysis of generation capacity allocation 68
7 Project Status & Future Development 70
7.1 Project closing status 70
7.2 Proposals for future work 70
7.2.1 Improving the optimal power flow formulation 71
7.2.2 Expanding the scope of the component models 71
7.2.3 Analysing the system behaviour at the stability limit 73
8 Conclusions 74

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CONTENTS (III)
Acknowledgements 75
References 76
Appendix A: The Microsoft Excel Model of the Equal
Area Criterion
A1
Appendix B: Visual Basic Application (VBA) Code for
Automatic Maximum Generation
Calculation in the Microsoft Excel Model of
the Equal Area Criterion
B1
Appendix C: Matlab Code for Solving the Non-Linear
Swing Equation
C1
Appendix D: Definition of Transient Stability-
Constrained Optimal Power Flow Version 3
(TSC-OPF-3)
D1
Appendix E: Definition of Transient Stability-
Constrained Optimal Power Flow Version 4
(TSC-OPF-4)
E1

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LIST OF FIGURES
3.1 Conventional power system architecture 8
3.2 Conventional power system architecture with distributed generation 8
3.3 Classical model of a synchronous generator connected to an infinite bus 12
3.4
Power-angle curve for the classical model of the classical model of a synchronous
generator
15
3.5 Non-faulted circuit for transient stability fault modelling 16
3.6 Faulted circuit for transient stability fault modelling 17
3.7 Star representation of a faulted circuit 17
3.8 Delta representation of a faulted circuit 17
3.9 Power-angle curves for prefault, fault and postfault conditions 19
3.10 The Equal Area Criterion 23
4.1 Microsoft Excel model – Network used for the Equal Area Criterion 29
4.2 Microsoft Excel model – Graphical representation of the Equal Area Criterion 33
4.3 PowerWorld model – Circuit used for verification 35
4.4 PowerWorld model – Transient stability simulations for Microsoft Excel verification 36
4.5 Microsoft Excel Goal Seek dialog box 38
4.6
Microsoft Excel model – Stability-constrained maximum generator capacity for
different fault clearing times and inertia constants
39
4.7 PowerWorld model – Transient stability simulations for Matlab verification 43
5.1 Network used for TSC-OPF-1 46
6.1 Test network for case study of transient stability-constrained capacity allocation 65

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LIST OF TABLES
4.1 Structure of the Microsoft Excel model – User defined variable network parameters 31
4.2 Structure of the Microsoft Excel model – User defined fixed network parameters 31
4.3 Structure of the Microsoft Excel model – Calculated network parameters 32
4.4 Structure of the Microsoft Excel model – Calculated complex network parameters 32
4.5 Structure of the Microsoft Excel model – Calculated transient stability parameters 32
4.6 Microsoft Excel model – Consolidated values for verification with PowerWorld 34
4.7 Microsoft Excel model – Verification of steady-state parameters with PowerWorld 35
4.8 Microsoft Excel model – Verification of stability parameters with PowerWorld 37
4.9
Microsoft Excel model with Matlab solver - Consolidated values for verification with
PowerWorld
42
4.10
Microsoft Excel model with Matlab solver - Verification of stability parameters against
PowerWorld
43
5.1 TSC-OPF-1 – Network sets 48
5.2 TSC-OPF-1 – Network optimisation 48
5.3 TSC-OPF-1 – User-defined network parameters 48
5.4 TSC-OPF-1 – Calculated network variables 49
5.5 TSC-OPF-1 – Network constraints 49
5.6 TSC-OPF-1 – Transient stability optimisation 49
5.7 TSC-OPF-1 – Transient stability parameters 50
5.8 TSC-OPF-1 – Calculated transient stability variables 50
5.9 TSC-OPF-1 – Transient stability constraints 51
5.10 TSC-OPF-1 – Input parameters for verification against the Microsoft Excel model 54
5.11 TSC-OPF-1 – Results for verification against the Microsoft Excel model 55
5.12 TSC-OPF-1 – Example of high θs value for network-only optimisation 56
5.13 TSC-OPF-2 – Network sets (added or changed only) 57
5.14 TSC-OPF-2 – User-defined network parameters (added or changed only) 57
5.15 TSC-OPF-2 – Calculated network variables (added or changed only) 58
5.16 TSC-OPF-2 – Network constraints (added or changed only) 58
5.17 TSC-OPF-2 – Results for verification against the Microsoft Excel model 59
5.18 TSC-OPF-2 – Example of constrained θs value for network-only optimisation 60
5.19 TSC-OPF-3 – User-defined network parameters (added or changed only) 61
5.20 TSC-OPF-3 – Calculated network variables (added or changed only) 61
5.21 TSC-OPF-3 – Network constraints (added or changed only) 62
6.1 Line reactances for the test network 65
6.2 Case study results for maximum capacity 67
6.3 Case study sample results for bus phases and voltages 67
B.1 Critical cell references for VBA code B1
D.1 TSC-OPF-3 – Network sets D1
D.2 TSC-OPF-3 – Network optimisation D1
D.3 TSC-OPF-3 – User-defined network parameters D2
D.4 TSC-OPF-3 – Calculated network variables D2
D.5 TSC-OPF-3 – Network constraints D3
D.6 TSC-OPF-3 – Transient stability optimisation D3
D.7 TSC-OPF-3 – Transient stability parameters D4
D.8 TSC-OPF-3 – Calculated transient stability variables D4
D.9 TSC-OPF-3 – Transient stability constraints D5

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LIST OF SYMBOLS
* This list does not include symbols for OPF formulation. Kindly refer to main text where the OPF is defined.
* Symbols with accents in the main text indicate vectors of the magnitude quantities listed here
δ : Generator internal voltage angle (power angle)
δ0 : Steady-state power angle
δc : Clearing angle
δcc : Critical clearing angle
δm : Generator stator synchronous rotating reference axis
δmax : Maximum generator power angle
θr : Infinite bus (receiving bus) voltage angle
θs : Generator terminal (sending bus) voltage angle
ωm : Mechanical angular velocity
ωsm : Synchronous mechanical angular velocity
E’ : Generator internal voltage magnitude
f : System frequency
H : Per-unit inertia constant
I : Generator current magnitude
IL1 : Generator Line 1 current magnitude
IL2 : Generator Line 1 current magnitude
J : Combined moment of inertia of turbine and generator
M : Inertia constant
p : Number of generator poles
Pe : Generator electrical power output
Pe(fault) : Faulted generator electrical power output
Pe(postfault) : Postfault generator electrical power output
Pe(prefault) : Prefault generator electrical power output
Pm : Generator mechanical power input
Pmax : Maximum active power
Q : Reactive power
S : Complex power
SB : System base power
tc : Clearing time
tcc : Critical clearing time
Ta : Accelerating torque
Te : Electromagnetic torque
Tm : Mechanical torque
Vr : Infinite bus (receiving bus) voltage magnitude
Vs : Generator terminal (sending bus) voltage magnitude
Wk : Kinetic energy in a rotation mass
X’d : Generator direct axis transient reactance
XL : Network reactance
XL1 : Network reactance of Line 1
XL2 : Network reactance of Line 2
XL2A : Network reactance of Line 2 before fault location
XL2B : Network reactance of Line 2 after fault location
XAB : Equivalent delta reactance of faulted Line 2
XBC : Equivalent delta reactance of faulted Line 2
XCA : Equivalent delta reactance of faulted Line 2

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GLOSSARY
ACOPF : A non-linear OPF (not abbreviation)
AIMMS : Advanced Integrated Multidimensional Modelling Software
CG : Conventional generation / conventional generator (context-specific)
DCOPF : A linearised OPF (not abbreviation)
DG : Distributed generation / distributed generator (context-specific)
KCL : Kirchhoffs Current Law
KVL : Kirchhoffs Voltage Law
OMIB : One-machine infinite-bus
OPF : Optimal power flow
TSC-OPF : Transient stability-constrained optimal power flow
VBA : Visual Basic Application

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INTRODUCTION
The increasing use of renewable energy sources for power generation is being driven by
the twin concerns of climate change and energy security. Greater capacities of renewable
generation are now being deployed to meet ambitious national targets for the production of
cleaner energy from more diversified sources as well as for the reduction of carbon
emissions. However, these renewable resources tend to be located in remote or non-urban
regions and their integration into power grids occurring typically at the distribution network.
This runs contrary to the conventional design philosophy of power system infrastructure,
which favours large, centralised generation facilities with unidirectional power flow from
source to load through reducing voltage levels.
Incorporating high volumes of distributed generation from renewables at the edges of these
networks introduces several technical challenges that must be addressed, such as bi-
directional power flow, local voltage rise, increasing fault levels and power quality, among
others. A common practise among network operators is to connect distributed generation
facilities on a first-come first-served basis while ensuring that measures are taken, either by
themselves or the project developer, to address these technical issues for the short-term as
part of the project scope.
However, suboptimal allocation of distributed generation sites and connection points may
pose hidden long-term risks such as inadvertently sterilising portions of the network and
limiting future development by prematurely pushing it to its operational limits. Whilst these
issues could generally be overcome with network expansion or reinforcement, such
measures may not always be possible due to planning restrictions, environmental concerns,
public objections as well as reluctance to finance investment in new infrastructure. Given
these potential obstacles, it is often desirable that any new development is carried out in a
manner that maximises the utilisation of existing assets. Therefore, tools and methods are
required to provide network operators with the capability to study the effects of, and
efficiently plan for, the long-term deployment of distributed generation within their regions.

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Studies carried out by the Institute of Energy Systems at the University of Edinburgh have
led to the development of optimal power flow as a technique to model the allocation of
distributed generation capacity. Assessment models for maximising generation capacity
have been produced for constraints such as active network management, loss minimisation,
fault levels, network security and voltage step changes. Continuing work at the Institute in
this field involves expanding this concept by developing models for more network constraints
that can be integrated into the optimal power flow formulation and analysing how they
influence, and can be influenced by, the allocation of distributed generation.
This project seeks to develop generator transient stability as a new optimal power flow
constraint for the allocation of distributed generation capacity. The project aims to describe
the fundamental methods for analysing transient stability, develop and verify mathematical
models of it and eventually use these models to produce a functional prototype of a
transient-stability constrained optimal power flow, which will form the key deliverable of the
project in the time allocated. Within this scope, the project will also propose suitable areas
for future work on the subject that will allow the model to be developed further into a more
accurate representation of the dynamic network that it is intended to simulate.

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II
OBJECTIVES & METHODOLOGY
2.1 Project objectives
The objectives of this project are to:
a) understand the use of optimal power flow (OPF) to assess capacity allocation of
distributed generation.
b) understand the concept of generator transient stability, evaluate some of the available
methods to calculate it and select the most suitable method for use in this project, with a
view to the project duration and deliverables.
c) develop and verify stand-alone mathematical models for transient stability to further
understand its behaviour and cement the theoretical knowledge gained previously.
d) develop and verify a functional transient stability-constrained generation capacity
allocation OPF for a single generator.
e) conduct a theoretical case study using the model developed.
f) propose opportunities for further work in the area.
2.2 Project methodology
The project was initiated with a review of the existing literature and theory relevant to the
subject matter from published papers, textbooks as well as videos of online lectures. From
these sources, the Equal Area Criterion was selected as the basis upon which transient
stability models would be modelled. Although other techniques do exist, as will be described
in Section 3.2.2, it was decided that a realistic goal would be to initially focus on using an
OPF to model the fundamental concepts of stability, with the intention that it would provide
the foundation upon which more advanced models could be built upon in future work.

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The following software were used in this project:
a) Microsoft Excel - version 2007
b) Matlab – version 7.10.0.499 (R2010a)
c) AIMMS [1] – version 3.10 PR – SU3 (Non-commercial educational stand-alone license)
d) PowerWorld Simulator [2] – version 15 with Transient Stability add-on [3][4] (Evaluation
and University Education license)
Microsoft Excel and Matlab are well-known applications commonly used in engineering and
a wide range of other fields.
AIMMS stands for “Advanced Integrated Multidimensional Modelling Software” and is a
modelling package used to develop the optimal power flow for this project. The following
documents were used as references for modelling with AIMMS:
a) AIMMS Tutorial for Beginners [5]
b) AIMMS Users Guide [6]
c) AIMMS Language Reference [7]
d) AIMMS Optimisation Modelling [8]
e) AIMMS Application Examples [9]
PowerWorld is a visual power system simulation and analysis package. The Transient
Stability add-on in PowerWorld was used to verify the calculations and results of the models
developed.
The modelling phase of the project was carried out using the methodology described
below. The concepts mentioned here are given detailed treatment in Section 3.
1. Practical understanding of transient stability was gained by developing an initial
mathematical model of the Equal Area Criterion using Microsoft Excel, which was
selected due to its ease of use and the familiarity of the author with several of its
advanced features that would facilitate development of the model. The model calculated
the fault critical clearing time by utilising the linear formulation of the Swing Equation for
the special case when the fault occurs on the sending bus and the fault power is zero. In
addition, the model also provided a graphical display of the Equal Area Criterion that
dynamically changed based on the input settings.

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2. The Microsoft Excel model was generalised to accommodate a fault occurring anywhere
along the line. Matlab was used as the calculation engine to solve the non-linear
formulation of the Swing Equation to obtain the critical clearing time as it had the
capability to do so more efficiently then Microsoft Excel.
3. A basic DCOPF model for a single generator connected to an infinite bus with transient
stability constraints was implemented in AIMMS. As the author did not have prior
experience using AIMMS, the DCOPF was chosen in order to reduce the configuration
complexity of the initial model. In doing so, more focus could be given to the formulation
of the transient stability constraint within AIMMS and its integration with the OPF.
Furthermore, the transient stability calculations developed previously in Microsoft Excel
did not require reactive power as an input and assumed that network resistance was
negligible. Therefore, it was decided that using the DCOPF would be a suitable starting
point that would not detract from the accuracy of the transient stability calculations.
4. As the authors’ familiarity with AIMMS increased, the basic stability-constrained DCOPF
was developed further to improve its accuracy and extend its applicability to larger
networks. Some characteristics of an ACOPF were introduced to the model in order to
increase its accuracy, creating a hybrid OPF formulation. Additional constraints and
error-checking mechanisms were implemented in AIMMS to make the model more
robust and prevent spurious computation errors.
5. A version of the OPF that could model a single-generator multiple-bus network was used
to conduct a case study of a theoretical network that contained several scenarios for
possible connection of a distributed generation facility. Analysis of the results as well as
critical evaluation of the models’ validity were carried out as part of the study.
6. Work on the model was continued, albeit at a lesser priority, during the process of thesis
writing with the intention of producing a full ACOPF formulation that would be capable of
analysing an actual distribution network. Despite some progress being made, the
ACOPF model that incorporated reactive power was not yet fully functional at the end of
the project. However, the author strongly believes that it is close to completion.

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III
BACKGROUND & TECHNICAL FUNDAMENTALS
The background study carried out for this project can be divided into three main categories,
namely distributed generation, generator transient stability and optimal power flow. An
overview of distributed generation and the issues raised by its increased penetration is first
provided to establish the motivation behind the aims of this project. This is followed by a
review of the effects that increased penetration of distributed generation has on transient
stability and a brief summary of the methods that can be used to evaluate limits of stability.
The technical concepts required to understand and model generator transient stability within
the scope of the project, including derivation of the relevant formulae, are then presented in
detail. Finally, the framework which will be used, capacity-allocation optimal power flow, is
reviewed with reference to the previous work carried out in the area at the University of
Edinburgh.
3.1 Distributed generation
The term “distributed generation”, also known as “embedded generation” or “decentralised
generation”, can have a wide range of definitions, from those that are generic and
descriptive in nature to more specific ones that may include the type of technology used,
size of facility, power or voltage levels and other parameters [10]. These can differ
significantly between countries and technical bodies, as well as the context in which the term
is utilised, with no single standard being prevalent [11]. For the purpose of this project, a
generic definition shall be used where distributed generation (DG) is considered to be the
production of electricity located within the distribution network.
While DGs are now commonly inferred to mean renewable generation, the term itself does
not automatically imply that the source is renewable. Conventional generation, such as
diesel generators and small scale combined heat and power (CHP) plants, can be broadly
classified as DGs as well. However, environmental and energy security concerns have
become key energy policy drivers in many countries, leading to ever-increasing targets for

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the deployment of renewable sources of power generation. The growing penetration of
renewable technologies such as solar photovoltaic (PV), biomass, wind, mini-hydro and
marine which are connected at distribution levels has, therefore, created a greater impetus
to explore their impact upon the electricity network that is currently in place.
The conventional power system architecture has been primarily based upon the concept
of large power plants constructed at strategic locations that permit cost-effective generation
of electricity, usually close to the sources of primary energy or other resources required to
support the process, such as cooling water. The power delivery network was developed to
transport bulk power unidirectionally over great distances from the source to the loads, via a
hierarchical series of reducing voltage levels. This resulted in systems that were designed
and optimised specifically to connect a relatively small number of high-capacity, centralised
power sources to a relatively large number of distributed users, with factors such as loss
minimisation, network redundancy, protection and power quality playing important roles to
define the operational layout of the network [11].
However, sites suitable for renewable generation are typically distributed over wide
geographical regions based on natural resource availability and connections to the network
are usually made at distribution level where these resources are more prevalent. This does
not conform to the philosophy governing the design of conventional networks and must be
accounted for as larger capacities of renewable generation come online in parts of the
network that used to contain only loads. Figures 3.1 and 3.2 illustrate the changes in the
electricity network from the conventional design to the current scenario in areas where DG
capacity is increasing.
The effects of connecting more DG to distribution networks have been the subject of a
large number of studies, with work by Ackerman et. al. [10], Dondi et. al. [11] and Pecas et.
al. [12] being just some of the many examples. The details of these studies are not
presented here, but key issues commonly highlighted include reverse power flows,
increased fault levels, voltage regulation, harmonics, power quality and system stability [13].
These are generally not of significant concern when DG penetration is small as their impact
can be absorbed by the conventional network architecture. However, the risk they pose to
the integrity of the supply network increases with installed capacity.
Compounding the inherent technical issues is the general practise commonly adopted by
network operators for the connection of new DG installations, which is usually done based
on a first-come-first-served basis and may possibly sterilise portions of the network [14].

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This would occur if a poorly-sited DG facility pushes the network close to the brink of its
operating limits, thereby excluding future installations in the area and resulting in the waste
of generating potential from renewable sources.
Figure 3.1: Conventional power system architecture
High
Voltage
Medium
Voltage
Low
Voltage
Load
Distributed
Generation
POWER FLOW
Centralised
Generation
Distributed
Generation
(Bidirectional)
Figure 3.2: Conventional power system architecture with distributed generation
While network reinforcement may help to mitigate the problem, availability of finances to
invest in new equipment and obtaining the required planning permissions are often uncertain
factors and may delay or prevent such projects. It is therefore a more desirable option for

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network operators to leverage upon existing infrastructure in order to achieve the maximum
possible capacity from renewable resources. In order to do so, tools and methods are
needed for network operators to optimally allocate generating capacity by carrying out
holistic, long-term planning. A large number of studies have been carried out with this aim
using various techniques such as optimal power flow [14 - 20], multiobjective impact indices
[21] and genetic algorithms [22].
The limitations of the existing network architecture in a regime of high DG penetration are
becoming increasingly obvious. It is, therefore, critical that the body of knowledge in this
field continues to grow in order to prevent the network from turning into a significant limiting
factor towards the increased deployment of renewable power generation.
3.2 Generator transient stability
The stability of a synchronous generator refers to its ability to maintain synchronism after
being subjected to an external disturbance. Stability analysis can be broadly categorised as
either steady-state stability or transient stability. Steady-state stability is the response of a
generator to small or slow disturbances such as gradual changes to load and generation,
while transient stability involves larger or more abrupt disturbances such as those caused by
system faults, loss of generation or sudden load changes. Only the effects of transient
stability are examined within the scope of this project.
A transient stability event can be summarised as follows. During steady-state operation,
with losses neglected, the mechanical power supplied from the turbine is in equilibrium with
the output electrical power of the generator and the generator rotor rotates synchronously
with the stator magnetic field at a fixed angular offset. The occurrence of a transient fault in
the electrical network alters the properties of the network and reduces the power-producing
capabilities of the generator during the period of the fault. The duration of transient faults
are typically too short to allow the turbine and its upstream processes to respond and an
imbalance occurs where the input power to the generator is greater than the output power.
As a result, the generator rotor gains speed in order to store the excess energy supplied to it
and this increases the angle between the rotor and the stator field. When the fault is
cleared, the rotor acts to discharge this energy causing it to slow down again. This causes
the rotor to swing as it first gains and then discharges energy alternately. If the rotor angle
swings beyond a prescribed limit, then synchronism will be lost.

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3.2.1 Influence of distributed generation on transient stability
As discussed in Section 3.1, the increasing amount of DG in conventional power networks
causes several technical issues, with transient stability being one of properties affected. A
summary of the impact that high DG penetration can have on transient stability has been
carried out by Boemer et. al. [23] based on numerous other studies conducted on the matter.
Their findings indicate that DG can play both a positive and negative influence on the
transient stability properties of a network.
From a positive aspect, they note that high penetration of DG can reduce the overall
loading of large conventional generators (CG) and transmission lines as the power
generated is spread over a wider geographical area and located closer to the loads. As a
result, the imbalance between CGs and loads during any network disturbance is less,
causing smaller rotor swings. Furthermore, as DGs typically have lower power ratings than
CGs, their individual tripping would have less impact on the overall network.
On the other hand, they stress that being smaller machines with lesser inertia, DGs
themselves have lower inherent stability and respond to changes much quicker than larger,
conventional machines. They note observations that networks with high penetration of DG
exhibit increased frequency oscillation after a disturbance with longer settling times. They
also caution that increased replacement of CG with DG would require current methods of
network modelling to be re-evaluated to account for the impact that this change may have.
Within the context of this study, it can be inferred that more effective tools are required to
model the relationship between of DG and transient stability in order to determine the correct
balance in terms of capacity and siting that will accentuate the positive effects and minimise
the negative ones. It is hoped that this project will lay the foundations for developing optimal
power flow methods as a tool for this purpose.
3.2.2 Methods for evaluating transient stability
The concepts of transient stability used in this project are the Swing Equation, also known
as the time-domain approach, and the Equal Area Criterion. The theory behind these are
readily available in many electrical power textbooks and the details outlined in Sections 3.2.3
to 3.2.9 are amalgamated primarily from the writings of Glover and Sarma [24], Weedy and

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Cory [25] and Saadat [26], as well as online video lectures made available by the Indian
National Program of Technology Enhanced Learning [27].
Some other ways of evaluating stability have been reviewed by Suampun and Chiang [28].
These were direct methods such as the transient energy function method and the controlling
unstable equilibrium point method, as well as less commonly used techniques such as the
hyperplane method and the quadratic approximation approach. Each of these were
concluded to embody specific trade-offs in terms of accuracy of results, computational time
and mathematical complexity. Additional methods that have been published include the
Extended Equal Area Criterion [29 - 30], the primal-dual Newton interior point method [31]
and differential evolution algorithms [32].
The Swing Equation and Equal Area Criterion were chosen over the more advanced
methods as they provided the best option to successfully develop a basic, but functional
transient-stability constrained optimal power flow within the duration of this project. With a
view that any selected method would have to be implemented in the AIMMS modelling
environment, which was unfamiliar to the author at the start of the project, it was decided
that using stability concepts that were well established and did not require the application of
advanced or exotic mathematical concepts would keep the project focused on its key
objective. It was envisioned that the models developed in this project would be used as a
foundation for more complex versions in future work that may incorporate some of the other
methods of evaluating transient stability.
Sections 3.2.3 to 3.2.9 that follow present the technical background that was applied to
develop mathematical models for transient stability.
3.2.3 Classical model of a synchronous generator
The classical model of a synchronous generator was used throughout this project as it was
deemed to be sufficient to develop an initial transient stability-constrained capacity dispatch
OPF. Integration of other complex models that would permit more accurate representations
of generator dynamics could be an opportunity for future work in this area.
The classical model represents a generator as a constant internal voltage located behind a
direct axis transient reactance and includes the following assumptions:

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a) The generator operates under balanced three-phase positive sequence conditions.
b) Excitation of the generator is assumed to be constant and input powers do not change
during the period of simulation.
c) Losses, saturation, damping and saliency effects are neglected.
Figure 3.3 depicts the classical generator model connected to an infinite bus through a
purely inductive network. The infinite bus is assumed to have a fixed voltage magnitude and
an angle of 0°, and can absorb all the active power supplied from the generator. The
description is simplified by neglecting generator transformer reactance.
Figure 3.3: Classical model of a synchronous generator connected to an infinite bus
The key parameter required for stability calculations is the generator internal voltage vector,
Ē’. However, this is usually not automatically known in a power system and must be
calculated from other network parameters. The following derivation assumes that Pm, Vs, Vr,
θr, X’d and XL are all known values, and at steady-state, Pm = Pe.
jX’d : Generator direct axis transient reactance
jXL : Network reactance
Pm : Generator mechanical power input
I : Generator current
Synchronous generator Infinite bus
E’ ∠δ
jX’d jXL
Pe
I
Vs
∠θs
Vr
∠θr = 0°
Pm
Generator terminal

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The sending bus voltage angle is:
ߠ௦ = sinିଵ
൬
ܲ௠ܺ௅
ܸ௦ܸ௥
൰ + ߠ௥ (3.1)
The generator current is:
‫ܫ‬̅ = ቤ
ܸ௦
ഥ − ܸ௥
ഥ
݆ܺ௅
ቤ (3.2)
The generator internal voltage can then be calculated by:
‫ܧ‬തᇱ
= ܸ௦
ഥ + ‫ܫ‬̅ܺ′ௗ = ‫ߜ∠ܧ‬ (3.3)
Generator current, I, can also be expressed in terms of the vector Ē’:
‫ܫ‬̅ =
‫ܧ‬ᇱ − ܸ௥
݆ሺܺᇱ
ௗ + ܺ௅ሻ
=
‫ ܧ‬ sin ߜ
ܺᇱ
ௗ + ܺ௅
− ݆ ൬
‫ܧ‬ cos ߜ − ܸ௥
ܺᇱ
ௗ + ܺ௅
൰ (3.4)
The complex power output at the generator terminals is:
ܵ̅ = ܲ௘ + ݆ܳ = ܸത௥‫ܫ‬̅∗
= ܸ௥ ቈ
‫′ܧ‬ sin ߜ
ܺᇱ
ௗ + ܺ௅
+ ݆ ቆ
‫′ܧ‬ cos ߜ − ܸ௥
ܺᇱ
ௗ + ܺ௅
ቇ቉ (3.5)
The active power delivered is:
ܲ௘ =
ܸ௥‫′ܧ‬
ܺᇱ
ௗ + ܺ௅
sin ߜ (3.6)
Maximum active power is delivered when sin δ = 1 at δ = 90°:
ܲ௠௔௫ =
ܸ௥‫′ܧ‬
ܺᇱ
ௗ + ܺ௅
sin 90° =
ܸ௥‫′ܧ‬
ܺᇱ
ௗ + ܺ௅
(3.7)
Equation 3.6 can be written in terms of Pmax as:
ܲ௘ = ܲ௠௔௫ sin ߜ (3.8)

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At steady-state, Pe = Pm, therefore the steady-state operating angle is:
ߜ଴ = sinିଵ
ܲ௠
ܲ௠௔௫
(3.9)
3.2.4 The Power-angle Curve
Equation 3.6 shows that the active power delivered by the generator is dependent upon the
angle between the sending and receiving voltages, known as the power angle, δ.
The relationship between the input and output powers at the generator can be illustrated by
the Power-Angle curve shown in Figure 3.4. Curve Pe represents the electrical output power
from the relationship given by Equation 3.8 while line Pm is the mechanical input power from
the turbine. Pm is in equilibrium with Pe at two operating points, X and Y, where the power
angles are δx and δy respectively. δx is given by Equation 3.9, while δy can be calculated by
ߜ௬ = 180° − sinିଵ
ܲ௠
ܲ௠௔௫
= ߜ௠௔௫ (3.10)
Point X is known as the stable operating point because any small perturbations in the
system will be self-corrected by the generator, as long as the disturbance does not increase
the power angle beyond 90°. If the power angle exceeded δx, Pe would become greater than
Pm. Since there is more power leaving the generator than entering it, the rotor would lose
energy and decelerate, causing the power angle to decrease. The power angle would then
undergo a damped oscillation and eventually restabilise at the equilibrium point X if the
network properties remain the same. Similarly, if the power angle was below δx, Pe would be
less than Pm causing the rotor to accelerate and increasing the power angle back to the
equilibrium point.
Point Y is known as the unstable operating point because any small perturbations around
this region that increase the power angle will not be self-corrected by the generator. If the
power angle exceeded δy, Pe would become less than Pm and the rotor would continue
absorbing power from the turbine. This would cause the power angle to increase further and
reduce Pe even more, resulting in a positive feedback loop that would cause instability.

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Figure 3.4: Power-angle curve for the classical model of a synchronous generator
It is also worth noting that, as Pm is increased, the operating power angle gets closer to 90°
and the margin for stability reduces. It is therefore desirable for the generator to operate in a
region where the power angle is less than 90° and the input power Pm is not close to the
output power capability limit, Pmax.
3.2.5 Equivalent circuit of a faulted power line
Transient faults in this project were modelled as a balanced three-phase-to-ground short-
circuit faults with no impedance that occurs on one branch of a double circuit power line.
The fault is cleared by disconnecting both ends of the faulted line. Modelling other types of
fault conditions could be an opportunity for future work in this area.
Pm : Turbine mechanical power input
δ : Generator power angle
X : Stable operating point
δX : Power angle for stable operating point
Y : unstable operating point
δy : Power angle for unstable operating point

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Figure 3.5 shows a non-faulted network with a double circuit power line that connects a
classical model of a generator to an infinite bus. The active power transferred from the
generator to the infinite bus under prefault conditions is given by Equation 3.6 previously:
ܲ௘ሺ௣௥௘௙௔௨௟௧ሻ =
ܸ௥‫′ܧ‬
ܺᇱ
ௗ + ܺ௅
sin ߜ (3.11)
XL is the parallel line impedance given by:
ܺ௅ =
ܺ௅ଵܺ௅ଶ
ܺ௅ଵ + ܺ௅ଶ
(3.12)
Figure 3.5: Non-faulted circuit for transient stability fault modelling
Figure 3.6 shows the same circuit with a fault occurring at Point F on Line 2. The line
reactances before and after the fault are labelled as XL2A and XL2B respectively, where:
ܺ௅ଶ = ܺ௅ଶ஺ + ܺ௅ଶ஻ (3.13)
Vinf : Infinite bus voltage magnitude
jXL1 : Line 1 reactance
E’ ∠δ
jX’d
jXL1
jXL2
Vr
∠θr = 0°

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Figure 3.6: Faulted circuit for transient stability fault modelling
Figure 3.6 is redrawn as shown in Figure 3.7 to represent it as a star-circuit, with AD, BD
and CD being the branches of the circuit and D as the common node.
Figure 3.7: Star representation of a faulted circuit
Star-delta transformation is applied to convert Figure 3.7 to a delta-circuit as shown in
Figure 3.8, with AB, AC and BC the branches of the circuit.
Figure 3.8: Delta representation of a faulted circuit
E’ ∠δ
jX’d
jXL1
Vr
∠θr = 0°
jXL2A jXL2B
F
Infinite bus
E’ ∠δ
jX’d
jXL1
Vr
∠θr = 0°jXL2A jXL2B
C
A B
D
Infinite bus
E’ ∠δ
jXAB
Vr
∠θr = 0°
jXL2B
C
A B
jXAC jXBC

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The equivalent reactances of the delta-circuit are:
ܺ஺஻ =
ሺܺᇱ
ௗ
ܺ௅ଵሻ + ሺܺᇱ
ௗ
ܺ௅ଶ஺ሻ + ሺܺ௅ଵܺ௅ଶ஺ሻ
ܺ௅ଶ஺
(3.14)
ܺ஻஼ =
ሺܺᇱ
ௗ
ௗ
ܺᇱ
ௗ
(3.15)
ܺ஺஼ =
ሺܺᇱ
ௗ
ௗ
ܺ௅ଵ
(3.16)
The equivalent fault reactance between the generator and the infinite bus is XAB given by
Equation 3.14. Therefore under fault conditions, Equation 3.11 can be modified to calculate
the power delivered by the generator by replacing the reactances with XAB, giving:
ܲ௘ሺ௙௔௨௟௧ሻ =
ܸ௥‫′ܧ‬
ܺ஺஻
sin ߜ (3.17)
When the fault is cleared by disconnecting Line 2, the postfault power delivery is given by:
ܲ௘ሺ௣௢௦௧௙௔௨௟௧ሻ =
ܸ௥‫′ܧ‬
ܺᇱ
ௗ + ܺ௅ଵ
sin ߜ (3.18)
Equations 3.11, 3.17 and 3.18 describe the 3 power-angle curves under prefault, fault and
postfault conditions. Further algebraic manipulation will show that Pe(prefault) > Pe(postfault) >
Pe(fault) (this is not derived here, but can be seen from the example in Section 4.1.1) and the
respective power-angle curves can be plotted as shown in Figure 3.9, which will form the
basis of transient stability analysis using the Equal-Area Criterion detailed in Section 3.2.8.
It is worth noting a special case of the fault power-angle curve when the fault occurs at the
generator, or sending bus. Under such conditions: the line reactance before the fault, XL2A,
is zero, resulting in no output power delivery during the duration of the fault.
ܺ௅ଶ஺ = 0 (3.19)
ܺ஺஻ = ∞ (3.20)
ܲ௘ሺ௙௔௨௟௧ሻ ≈ 0 (3.21)

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Figure 3.9: Power-angle curves for prefault, fault and postfault conditions
3.2.6 The Swing Equation
The Swing Equation describes the relative motion of the generator rotor to the stator
magnetic field and the variation of the power angle with time during a disturbance, or the
amount that the rotor swings. It is the primary relationship used in this project to determine
conditions for transient stability. An abridged derivation of the Swing Equation is
demonstrated below.
Under steady-state conditions and neglecting losses, the input mechanical torque, Tm to a
synchronous generator will be equal to the electromagnetic torque, Te, developed. During a
disturbance, the difference between Tm and Te results in an accelerating torque, Ta.
ܶ௔ = ܶ௠ − ܶ௘ (3.22)
From the laws of rotation, Ta can be described in terms of the combined moment of inertia
of the turbine and generator, J, and the mechanical displacement angle of the rotor with
respect to the synchronously rotating reference axis of the stator, δm.

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ܶ௔ = ‫ܬ‬
݀ଶ
ߜ௠
݀‫ݐ‬ଶ
= ܶ௠ − ܶ௘
(3.23)
It is convenient to express Equation 3.23 in terms of power, P, where:
ܲ = ߱ܶ (3.24)
Hence, multiplying both sides of Equation 3.23 by mechanical angular velocity ωm gives:
‫߱ܬ‬௠
݀ଶ
ߜ௠
݀‫ݐ‬ଶ
= ߱௠ܶ௠ − ߱௠ܶ௘ = ܲ௠ − ܲ௘
(3.25)
The term Jωm is known as the inertia constant and denoted by M, which can also be written
in terms of the kinetic energy in the rotating mass, Wk and the synchronous mechanical
angular velocity ωsm. The definition of M can be summarised as:
‫߱ܬ‬௠ = ‫ܯ‬ =
2ܹ௞
߱௦௠
(3.26)
The mechanical rotor angle displacement, δm, and synchronous mechanical angular
velocity, ωsm, are respectively related to the electrical power angle, δ, and the synchronous
electrical angular velocity, ωs, by the number of generator poles, p. The synchronous
electrical angular velocity, ωs is also more conveniently denoted in terms of frequency, f.
ߜ =
‫݌‬
2
ߜ௠ (3.27)
߱௦ =
‫݌‬
2
߱௦௠ = 2ߨ݂ (3.28)
Equation 3.25 can be rewritten in terms of Equations 3.26, 3.27 and 3.28 as:
ܹ௞
ߨ݂
݀ଶ
ߜ
݀‫ݐ‬ଶ
= ܲ௠ − ܲ௘ (3.29)
Equation 3.29 can be expressed in terms of per-unit values by dividing it with the base
power, SB.

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ܹ௞
ܵ஻ߨ݂
݀ଶ
ߜ
݀‫ݐ‬ଶ
=
ܲ௠
ܵ஻
−
ܲ௘
ܵ஻
= ܲ௠ሺ௣௨ሻ − ܲ௘ሺ௣௨ሻ (3.30)
The per-unit inertia constant, H, can be defined as the ratio of kinetic energy in the rotor to
the machine rating. H is a critical parameter in analysing the dynamics of synchronous
generators and has a value that ranges between 1 to 10 seconds. Smaller machines with
less inertia have a lower value of H and vice versa.
‫ܪ‬ =
ܹ௞
ܵ஻
(3.31)
Substituting Equation 3.31 into Equation 3.30, and omitting the subscript pu for
convenience, results in the final Swing Equation shown in 3.32.
‫ܪ‬
ߨ݂
݀ଶ
ߜ
݀‫ݐ‬ଶ
= ܲ௠ − ܲ௘ (3.32)
3.2.7 Solving the Swing Equation
The electrical power delivered, Pe, from 3.8 can be substituted into Equation 3.32 to give:
‫ܪ‬
ߨ݂
݀ଶ
ߜ
݀‫ݐ‬ଶ
= ܲ௠ − ܲ௠௔௫ ‫ߜ ݊݅ݏ‬ (3.33)
Equation 3.33 is a second-order non-linear differential equation in terms of the power
angle, δ, and cannot be easily solved manually as long as Pe ≠ 0. However, if the transient
line fault occurs on the sending bus as described by Equations 3.19 to 3.21 and Pe ≈ 0, the
Swing Equation becomes
‫ܪ‬
ߨ݂
݀ଶ
ߜ
݀‫ݐ‬ଶ
= ܲ௠ (3.34)
Equation 3.34 is a second-order linear differential equation which can be readily solved
manually and allows the power angle to be expressed in terms of time. This is more useful

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in practice as it allows the stability limits to be described as the time taken to clear a
transient fault.
Rearranging and integrating both sides of Equation 3.34 twice gives:
න ቆන
݀ଶ
ߜ
݀‫ݐ‬ଶ
݀‫ݐ‬ቇ ݀‫ݐ‬ = න ൬න
ߨ݂ܲ௠
‫ܪ‬
݀‫ݐ‬൰ ݀‫ݐ‬ (3.35)
Which solves to:
ߜ =
ߨ݂ܲ௠‫ݐ‬ଶ
2‫ܪ‬
+ ߜ଴
(3.36)
The initial condition term after the first integration is the electrical angular velocity of the
rotor relative to the stator field. At time t = 0 under steady-state conditions, both rotor and
stator field are rotating in synchronism, hence:
ߜሺ0ሻ
݀‫ݐ‬
= 0 (3.37)
The initial condition term after the second integration is the steady-state power angle of the
rotor, δ0, which can be calculated from Equation 3.9.
Rearranging Equation 3.36 to express for time gives:
‫ݐ‬ = ඨ
2‫ܪ‬ሺߜ − ߜ଴ሻ
ߨ݂ܲ௠
(3.38)
3.2.8 The Equal Area Criterion
Solving the non-linear Swing Equation 3.33 allows stability conditions to be determined
from a time-series behaviour of the system. However, for a simplified model containing a
single generator connected to an infinite bus, such as that shown in Section 3.2.3, a concept
called the Equal Area Criterion can be applied to calculate the stability angle limit.

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The Equal Area Criterion uses a graphical interpretation of the energy stored and
dissipated in the generator rotor during a transient fault to evaluate stability and assumes
that synchronism is either retained or lost during the first swing of the rotor. Figures 3.4 and
3.9 described previously are combined to create Figure 3.10, which shall be used to explain
the Equal Area Criterion in further detail.
Under steady-state conditions the input power Pm and output power Pe(prefault) at the
generator are in equilibrium. The generator operates at point U with power angle δ0.
Figure 3.10: The Equal Area Criterion
At the instance the fault occurs, the power delivery capability of the generator is reduced to
curve Pe(fault), as explained previously in Section 3.2.5, and the operating point moves
Pm : Turbine mechanical power input
δ0 : Steady-state power angle
δc : Clearing angle
δmax : Maximum swing angle

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instantaneously to point V, creating an imbalance between Pm and Pe. Assuming that the
input power does not change for the duration of the transient fault, the rotor begins to absorb
the excess energy supplied causing it to accelerate. The rotor angle increases and the
operating point moves along curve Pe(fault).
The fault is cleared by disconnecting the faulted line after a period of time, during which the
rotor has moved to point W at power angle δc, which is called the clearing angle. Upon
clearing of the fault, the power delivery capability changes to curve Pe(postfault) and the
operating point instantaneously moves to point Y. Since Pe is now greater than Pm, the rotor
dissipates energy and decelerates while its momentum continues to cause the power angle
to increase along curve Pe(postfault).
As explained in Section 3.2.4, the power angle should remain less than the unstable
operating point, δmax, in order to prevent the rotor from losing synchronism. Therefore, point
Z represents the maximum limit that the rotor can continue to swing after the fault has been
cleared in order to remain stable. If this condition is satisfied, the rotor will experience a
damped oscillation along Pe(postfault) before eventually reaching a new steady-state point.
The region bounded by points UVWX with area A1 represents the total amount of energy
absorbed by the rotor during the period of the fault. This excess energy must be dissipated
by the rotor before it can return to a new equilibrium point on curve Pe(postfault). The region
bounded by points XYZ with area A2 represents the total amount of energy that can be
dissipated by the rotor after the fault is cleared before it becomes unstable. The condition
for stability is that A1 must be less or equal to A2, as this will allow the rotor to release all the
energy absorbed without violating the limit, δmax.
The specific value of the clearing angle, δc, that causes A1 to be exactly equal to A2 is
called the critical clearing angle, δcc. It represents the maximum amount that the rotor can
swing before the fault must be cleared in order for stability to be retained. Referring to
Section 3.2.7, solving the Swing Equation for δcc will give the maximum time, known as the
critical clearing time tcc, that a fault can be sustained to prevent loss of synchronism. This
readily translates into protection relay settings for transmission or distribution systems.
A mathematical representation of the Equal Area Criterion to determine the critical clearing
angle can be derived as given below.

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From the generic Equation 3.8, the following can be written:
ܲ௘ሺ௣௥௘௙௔௨௟௧ሻ = ܲ௘ሺ௣௥௘௙௔௨௟௧ሻ௠௔௫ sin ߜ (3.39)
ܲ௘ሺ௙௔௨௟௧ሻ = ܲ௘ሺ௙௔௨௟௧ሻ௠௔௫ sin ߜ (3.40)
ܲ௘ሺ௣௢௦௧௙௔௨௟௧ሻ = ܲ௘ሺ௣௢௦௧௙௔௨௟௧ሻ௠௔௫ sin ߜ (3.41)
Area A1 can be calculated as:
‫1ܣ‬ = ܲ௠ሺߜ௖௖ − ߜ଴ሻ − ቌ න ܲ௘ሺ௙௔௨௟௧ሻ௠௔௫ sin ߜ ݀ߜ
ఋ೎
ఋబ
ቍ
= ܲ௠ሺߜ௖௖ − ߜ଴ሻ + ܲ௘ሺ௙௔௨௟௧ሻ௠௔௫ሺcos ߜ௖௖ − cos ߜ଴ሻ (3.42)
Area A2 can be calculated as:
‫2ܣ‬ = ቌ න ܲ௘ሺ௣௢௦௧௙௔௨௟௧ሻ௠௔௫ sin ߜ ݀ߜ
ఋ೘ೌೣ
ఋ೎
ቍ − ܲ௠ሺߜ௠௔௫ − ߜ௖௖ሻ
= ܲ௘ሺ௣௢௦௧௙௔௨௟௧ሻ௠௔௫ሺcos ߜ௖௖ − cos ߜ௠௔௫ ሻ − ܲ௠ሺߜ௠௔௫ − ߜ௖௖ሻ (3.43)
Equating A1 and A2 to satisfy the Equal Area Criterion gives:
ߜ௖௖ = cosିଵ
ቈ
ܲ௠ሺߜ௠௔௫ − ߜ௢ሻ + ܲ௘ሺ௣௢௦௧௙௔௨௟௧ሻ௠௔௫ cos ߜ௠௔௫ − ܲ௘ሺ௙௔௨௟௧ሻ௠௔௫ cos ߜ଴
ܲ௘ሺ௣௢௦௧௙௔௨௟௧ሻ௠௔௫ − ܲ௘ሺ௙௔௨௟௧ሻ௠௔௫
቉ (3.44)
The critical clearing angle, δcc, can then be applied to either Equation 3.33 (if Pe(fault) ≠ 0) or
Equation 3.38 (if Pe(fault) = 0) to obtain the critical clearing time, tcc, which is the key constraint
for the capacity dispatch optimal power flow developed in this project.

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3.3 Optimal power flow
The optimal power flow (OPF) technique has long been used in power systems to solve a
variety of optimisation problems. While its most common use has been for the economic
dispatch of power, applications such as loss minimisation, efficient use of fuel and security
contingencies have also been implemented using the OPF method [33]. At its most basic,
the OPF combines the solving of a networks’ power flow with a linear programming (LP)
problem, such as minimising cost, to arrive at the optimal solution for the LP that also
satisfies the physical and electrical constraints of the network [24].
The Institute of Energy Systems at the University of Edinburgh has developed methods for
using an OPF as a tool to assess the allocation of DG capacity as presented in the works of
Harrison et. al. [14 - 20]. This method has been used to evaluate the impact that various
constraints such as fault-levels [15], network security [16], voltage step changes [17] as well
as active network control methods [18 - 20], have on the allocation of DG capacity.
The formulation of an OPF can generally be stated in terms of the maximisation or
minimisation of an objective function subject to certain constraints. The capacity-allocation
OPF can be defined both descriptively and mathematically as follows [20]:
• The objective function is to maximise the total active DG capacity p of a set of generators
G that are indexed by g.
݉ܽ‫ݔ‬ ෍ ‫݌‬௚
௚ఢீ
(3.45)
The maximisation is done subject to the following constraints:
• Voltage magnitudes V at each bus b (in the set of buses B) are constrained by their
permitted minimum and maximum levels.
ܸ௕_௠௜௡ ≤ ܸ௕ ≤ ܸ௕_௠௔௫ ∀ ܾ ߳ ‫ܤ‬ (3.46)
• Thermal limits constrain the apparent power flow f on each line l (in the set of lines L) to
its maximum level, where p and q are the active and reactive power injections into the
end of line l respectively.

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‫݌‬௟
ଶ
+ ‫ݍ‬௟
ଶ
≤ ݂௟_௠௔௫
ଶ
∀ ݈ ߳ ‫ܮ‬ (3.47)
• Kirchhoffs’ Current Law models the power balance at each bus b (in the set of buses B)
so that the total power p and q generated by all generators g (in the set of buses G)
located at that specific bus is equal to sum of the demands on that bus d p
and d q
and
the power leaving that bus pl
b1b2
and ql
b1b2
on line l (in the set of lines L). b1 and b2
represent the sending and receiving bus of line l respectively.
෍ ‫݌‬௚ = ݀௕
௣
+ ෍ ‫݌‬௟
௕భ௕మ
௟ఢ௅|ఉ೒ୀ௕௚ఢீ|ఉ೒ୀ௕
∀ ܾ ߳ ‫ܤ‬ (3.48)
෍ ‫ݍ‬௚ = ݀௕
௤
+ ෍ ‫ݍ‬௟
௕భ௕మ
௟ఢ௅|ఉ೒ୀ௕௚ఢீ|ఉ೒ୀ௕
∀ ܾ ߳ ‫ܤ‬ (3.49)
• Kirchhoffs’ Voltage Law models the active and reactive power injections p and q into the
end of each line l (in the set of lines L), where b1 and b2 represent the sending and
receiving bus of line l respectively, g is the line conductance, b is the line susceptance
and δ is the voltage phase angle.
‫݌‬௟
௕భ௕మ
= ݃௟ܸ௕భ
ଶ
− ܸ௕భ
ܸ௕మ
ൣ݃௟ cos൫ߜ௕భ
− ߜ௕మ
൯ + ܾ௟ sin൫ߜ௕భ
− ߜ௕మ
൯൧ ∀ ݈ ߳ ‫ܮ‬ (3.50)
‫ݍ‬௟
௕భ௕మ
= −ܾ௟ܸ௕భ
ଶ
− ܸ௕భ
ܸ௕మ
ൣ݃௟ sin൫ߜ௕భ
− ߜ௕మ
൯ + ܾ௟ cos൫ߜ௕భ
− ߜ௕మ
൯൧ ∀ ݈ ߳ ‫ܮ‬ (3.51)
• The reference bus voltage phase angle is defined explicitly as being zero
ߜఉబ
= 0 (3.52)
Equations 3.45 to 3.52 describe the generic formulation for the capacity allocation OPF.
The terms for active power, reactive power, voltage magnitude and voltage phase are the
OPF variables. Additional terms or constraints can be included to model the effects of
constant power factors [15], external connections [18] or other network properties if required.
While some OPFs, such as those for economic dispatch, may also impose constraints for
the active and reactive power capacity limits, this may be optional for a capacity allocation
OPF depending on the context of analysis. Neglecting this constraint will provide the

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theoretical maximum achievable power capacity that can be installed at a given site, but
resource or regulatory limits may impose restrictions on it.
The OPF described by Equations 3.45 to 3.52 is categorised as an ACOPF due to the non-
linear formulation of Kirchhoffs’ Voltage Law (Equations 3.50 and 3.51). The ACOPF can be
simplified by linearising the equations to create a DCOPF, with the assumptions that voltage
level is constant across the network at the nominal voltage V0 and the phase difference is
small such that the terms sin (δb1-δb2) = (δb1-δb2) and cos (δb1-δb2) = 1 [34]. With a further
assumption that line resistance is significantly smaller than reactance and can be neglected,
Equation 3.50 can be simplified to:
‫݌‬௟
௕భ௕మ
= −
ܸ଴
ଶ
‫ݔ‬௟
൫ߜ௕భ
− ߜ௕మ
൯ ∀ ݈ ߳ ‫ܮ‬ (3.53)
The reactive power in Equation 3.51 equates to zero, hence reactive power flows are not
modelled in a DCOPF. This removes Equation 3.49 from the formulation while Equation
3.47 simplifies to:
‫݌‬௟ ≤ ݂௟_௠௔௫ ∀ ݈ ߳ ‫ܮ‬ (3.54)
The assumptions made for DCOPFs are more suited to model transmissions systems, as
they are long-distance, high-voltage and high-capacity [34]. For distribution networks, the
line resistances are comparable to the reactances while voltage level becomes an important
constraint as well, hence the ACOPF is more appropriate for accurate representation of the
network. Nonetheless, a DCOPF is easier to implement and its linearity allows for quicker
convergence, making it an acceptable option for the development of initial prototypes.

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IV
MODELLING TRANSIENT STABILITY
4.1 Modelling the Equal Area Criterion in Microsoft Excel
The concepts outlined in Sections 3.2.3 to 3.2.8 were used to model the Equal Area
Criterion using Microsoft Excel based on the circuit shown in Figure 4.1, which is the One
Machine-Infinite Bus (OMIB) system with a double circuit transmission line described in
Sections 3.2.3 and 3.2.5. The generator is assumed to always supply power at its maximum
capacity. The infinite bus is assumed to have a fixed voltage magnitude and an angle of 0°,
and absorbs all the power supplied by the synchronous generator.
Figure 4.1: Microsoft Excel model - Network used for the Equal Area Criterion
The model simulates a transient fault at the sending bus (point A) that is cleared by
disconnecting the faulted line at both ends (points A and B). Limiting the fault to point A
Pm : Generator power
Vs
∠θs
Generator
terminals
E’ ∠δ
jX’d
jXL1
Vr
∠θr = 0
jXL2A B
Pm

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allows the Excel model to solve the linear case of the Swing Equation (Equation 3.38) in
order to obtain the critical clearing time. The Excel model was not developed to solve the full
non-linear Swing Equation as it can be done more efficiently using Matlab. This is described
further in Section 4.2.
The following sections describe the structure and functionality of the model, the method to
verify the results produced by the model and an example application of the model.
4.1.1 Structure and functionality of the Microsoft Excel model
The structure of the Excel model is shown in Tables 4.1 to 4.5 and a screen-capture of the
actual spreadsheet is provided in Appendix A. The calculations used in the model are cross-
referenced to the equations defined in Section 3 where relevant and the table row numbers
continue sequentially across all five tables. The values displayed in the tables are used in a
sample calculation to verify the accuracy of the model in Section 4.1.2. These values are
not representative of any specific network, but chosen merely to demonstrate the
functionality of the model.
Table 4.1 contains the values that the end user must provide to describe the network based
on parameters that should be readily known and would be typically changed to analyse
different scenarios of stability. These parameters include the sending and receiving bus
voltages, reactances, input power and the generator inertia constant. In addition, Row 9
contains a field that allows the user to toggle the fault on Line 2 and while Row 10 defines
the location of the fault along the line. However, as the Excel model is only capable of
calculating the critical clearing time for the linear version of the Swing Equation when the
fault is at the sending bus, Row 10 should remain at 0% if a result in terms of time is
required. If it is non-zero, then the critical clearing time field in Table 4.5 will display the
word “Matlab”, which is a reference to the solver that will be described further in Section 4.2.
The result in terms of the critical clearing angle, however, can be calculated for any fault
location.
Table 4.2 also contains values that the end user must provide but these will typically remain
unchanged during modelling. They comprise of the system frequency as well as base
values for power and voltage. It should be noted that the calculation of stability parameters
is done purely in per-unit and is independent of the base values defined. The bases are

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included to allow convenient verification of the calculated power and current values with
PowerWorld, which displays them as ohmic values by default.
User Defined Variable Network Parameters Sample Value Units
1 Generator terminal voltage magnitude, Vs 1.000 pu
2 Receiving bus voltage, Vr 1.000 pu
3 Receiving bus voltage angle, θr 0.000 rad
4 Line 1 reactance, XL1 0.400 pu
5 Line 2 reactance, XL2 0.400 pu
6 Generator direct transient reactance, Xd' 0.200 pu
7 Mechanical power input, Pm 1.000 pu
8 Inertia constant, H 3 s
9 Line 2 faulted? Yes -
10 Fault location (0=sending end, 100=receiving end) 0 %
Table 4.1: Structure of Microsoft Excel model – User-defined variable network parameters
User Defined Fixed Network Parameters Sample Value Units
11 Frequency, f 50 Hz
12 Base power 100 MVA
13 Base voltage 138 kV
Table 4.2: Structure of the Microsoft Excel model – User-defined fixed network parameters
The third, fourth and fifth tables contain calculated parameters as detailed in Section 3 and
should not be modified by the user. The network parameters in complex terms are explicitly
defined in Table 4.4 to facilitate calculations in Excel. The model neglects generator losses,
hence the electrical output power of the generator (Table 3, Row 14) is set to be equal to the
mechanical input power from the turbine (Table 1, Row 7).
The end result of the model is to determine the critical clearing time (Table 5, Row 37) in
milliseconds. This represents the maximum time that the fault can be sustained under the
network conditions listed in Table 4.1 before the generator loses stability.

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Equation Calculated Network Parameters Sample Value Units
- 14 Real electrical power output, Pe 1.000 pu
3.12 15 Equivalent line reactance, XL 0.200 pu
3.1 16 Generator terminal voltage angle, θs 0.201 rad
3.2 17 Generator current, I 1.005 pu
3.3 18 Generator internal voltage magnitude, E' 1.040 pu
- 19 Faulted line reactance (sending end), XL2A 0.000 pu
3.14 20 Equivalent delta reactance, XAB 0.000 pu
Table 4.3: Structure of the Microsoft Excel model – Calculated network parameters
Equation Calculated Complex Network Parameters
Sample Value
Real Imaginary Complex
- 21 Receiving bus voltage, Vr 1.000 0.000 1
- 22 Generator terminal voltage, Vs 0.980 0.200 0.98+0.2i
- 23 Generator direct transient reactance, Xd' 0.000 0.200 0.2i
- 24 Line 1 reactance, XL1 0.000 0.400 0.4i
- 25 Line 2 reactance, XL2 0.000 0.400 0.4i
3.12 26 Equivalent line reactance, XL 0.000 0.200 0.2i
3.2 27 Generator current, I 1.000 0.101 1+0.1i
3.3 28 Generator internal voltage, E' 0.960 0.400 0.96+0.4i
Table 4.4: Structure of the Microsoft Excel model – Calculated complex network parameters
Equation Calculated Transient Stability Parameters Sample Value Units
3.11 29 Pre-fault maximum power, Pmax(prefault) 2.599 pu
3.17 30 Fault maximum power, Pmax(fault) 0.000 pu
3.18 31 Post-fault maximum power, Pmax(postfault) 1.733 pu
3.9 32 Steady-state operating angle, δ0 0.395 rad
3.10 33 Stability limit angle, δmax 2.526 rad
3.42 34 Area A1 0.750 -
3.43 35 Area A2 0.750 -
3.44 36 Critical clearing angle, δcc 1.145 rad
3.38 37 Critical clearing time, tcc 169.2 ms
Table 4.5: Structure of the Microsoft Excel model – Calculated transient stability parameters

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The model also includes a graphical representation of the Equal Area Criterion that
displays the three power angles curves as well as key angles δ0, δcc and δmax. This is shown
in Figure 4.2. The curves will automatically change to reflect the input values supplied.
Limitations with Excel do not permit the relevant bounded areas of the diagram to be
emphasised easily, but this can be inferred by comparing Figure 4.2 to Figure 3.10.
Figure 4.2: Microsoft Excel model – Graphical representation of the Equal Area Criterion
The results obtained from the model can be verified with PowerWorld to ascertain its
accuracy. To facilitate verification, an additional table that consolidates all the information
that will be relevant to the PowerWorld simulation is made available for the user. This is
shown in Table 4.6.
0.0
0.5
1.0
1.5
2.0
2.5
3.0
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5
Power,pu
Delta , rad
Microsoft Excel model of the Equal Area Criterion
P (prefault)
P (fault)
P (postfault)
Pm
δδδδ0 δδδδmaxδδδδcc

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PowerWorld network verification parameters Sample Value Units
USER INPUT
Real electrical power output, Pe = Pm 100.00 MW
Generator terminal voltage, Vs 1.000 pu
Infinite bus voltage, Vr 1.000 pu
Infinite bus voltage angle, θr 0.00 deg
Line 1 reactance, XL1 0.400 pu
Generator direct transient reactance, Xd' 0.200 pu
Inertia constant, H 3 s
Fault location (0=sending end, 100=receiving end) 0 %
STAGE 1 VERIFICATION - NETWORK PARAMETERS
Generator terminal voltage angle, θs 11.54 deg
Line 1 current, IL1 210.25 A
STAGE 2 VERIFICATION - GENERATOR TRANSIENT PARAMETERS
Steady-state operating angle, δ0 22.63 deg
Critical clearing angle, δcc 65.58 deg
Stability limit angle, δmax 144.75 deg
Critical clearing time, tcc 169.2 ms
Table 4.6: Microsoft Excel model - Consolidated values for verification with PowerWorld
4.1.2 Verification of the Microsoft Excel model against PowerWorld
Various critical clearing time values calculated by the model under different conditions were
compared with PowerWorld simulations for verification. A verification example for the set of
network parameters and output power shown in Table 4.6 is described below.
Figure 4.3 shows a PowerWorld model that was configured and run for the circuit shown in
Figure 4.1 with the configured network parameters of Table 4.6. The generator terminal was
modelled as a PV bus, with the generator automatic voltage regulator (AVR) switched on in
order to lock the sending bus voltage at 1 pu. The reactive power output of the generator is
not displayed as it is not relevant to the stability model at this stage. The slack generator on
the infinite bus absorbs all the active power output of the modelled generator.

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Figure 4.3: PowerWorld model – Circuit used for verification
The first level of verification was done before the transient stability simulation was carried
out. Only θs, IL1 and IL2 were examined at this stage (shown as boxed values in Figure 4.3),
which are calculated steady-state network values in both the Excel model and PowerWorld.
The results produced by the two models are compared in Table 4.7 and show that the same
values, with negligible decimal differences. This indicated that the Excel model accurately
models the steady-state network.
Parameter Microsoft Excel PowerWorld
Generator terminal voltage angle, θs 11.54° 11.53°
Line 1 current, IL1 210.25 A 210.21 A
Line 2 current, IL2 210.25 A 210.21 A
Table 4.7: Microsoft Excel model – Verification of steady-state parameters with PowerWorld
The next level of verification was done by running transient stability simulations with
PowerWorld and checking the accuracy of the stability parameters δ0, δcc, δmax and tcc. Three
simulations were carried out, where the fault was applied at t = 1s and cleared at different
times in each. The relevant results of each simulation are summarised in Figure 4.4.

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In the first simulation, the fault was cleared instantaneously and the generator remained
stable. This indicated that the configuration was correct. The rotor angle continued to
oscillate between 22° to 46° after the fault was cleared because damping effects were not
modelled. In practise, the oscillations should gradually recede due to damping and the
generator should return to a new stable operating point as described in Section 3.2.8.
In the second simulation, the fault was cleared at the critical clearing time of 169.2 ms
given by the Excel model. While the generator still remained stable, the rotor angle
oscillated between -25° to 144° in a distorted pattern. The distortion of the rotor angle
movement warrants further analysis but this is left as an opportunity for future work as it has
no immediate bearing on the model utilised and lies outside the primary scope of the project.
In the third simulation, the fault was cleared at 169.3 ms, just after the critical clearing time.
The rotor angle continued to increase exponentially after the fault was cleared, indicating
that synchronism was lost and the generator was no longer stable.
Figure 4.4: PowerWorld model – Simulations for Microsoft Excel verification
-50
-25
0
25
50
75
100
125
150
175
200
225
250
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
Rotorangle,degrees
Time, s
Transient stability simulations with Powerworld
Fault duration = 0 ms
Fault duration = 169.2 ms
Fault duration = 169.3 ms
Steady-state
operation
Post-fault
operation
Fault
applied
X

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The “first swing” assumption of the Equal Area Criterion described in Section 3.2.8 is
evident from Figure 4.4, as it can be seen that the 169.3 ms curve tracks the 169.2 ms curve
until the peak of the first swing before it separates at point “X” and increases exponentially.
PowerWorld values for δ0, δcc, δmax and tcc were recorded in Table 4.8 and found to be
similar to the Excel values, with negligible decimal variations in some cases, showing that
the Excel model accurately represented the transient stability response of the generator.
Parameter Microsoft Excel PowerWorld
Steady-state operating angle, δ0 22.63° 22.63°
Critical clearing angle, δcc 65.58° 65.57°
Stability limit angle, δmax 144.75° 144.86°
Critical clearing time, tcc 169.2 ms 169.2 ms
Table 4.8: Microsoft Excel model – Verification of stability parameters with PowerWorld
In this example, the critical clearing time calculated by Excel was exactly the same as that
demonstrated by PowerWorld. However, it is worth noting that the Excel value may not
always be precise due to rounding errors in the calculation. In this situation, the second and
third simulations may need to be repeated several times with minor adjustments in order to
determine to the actual value of the critical clearing empirically.
4.1.3 Application of the Microsoft Excel model to determine maximum
generator capacity
The Microsoft Excel model was used to determine the maximum capacity of a single
generator subjected to transient stability constraints. This was carried out without using an
OPF formulation and neglecting network constraints to test the capabilities of the model.
The ‘Goal Seek’ function of Excel, shown in Figure 4.5, was utilised for this purpose.
The procedure used is listed below:

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1. The ‘Set cell’ field was configured to the cell containing the critical clearing time, tcc
(Table 4.5, Row 37).
2. The ‘To value’ field was set to a user-defined clearing time in milliseconds.
3. The ‘By changing cell’ field was configured to the cell containing the mechanical power
input to the generator, Pm (Table 4.1, Row 7).
Figure 4.5: Microsoft Excel Goal Seek dialog box
When ‘Goal Seek’ was run with the configuration above, the Excel model automatically
iterated through values of generator power in order to obtain a value of critical clearing time
that equates to the user-defined clearing time. The result was the power capacity that
brought the generator to the brink of stability for a specified clearing time.
A simple Visual Basic Application (VBA) program was written in Excel to automatically run
iterations of the ‘Goal Seek’ function to obtain the maximum capacity with different clearing
times. In addition, the inertia constant, H, was also changed to observe the effect that it had
on maximum capacity. The VBA code used for this purpose is provided in Appendix B and
has been configured as a macro within the Excel model to facilitate future users.
Figure 4.6 shows the results obtained. The network parameters used were the same as
listed in Table 4.1, except that different inertia constants were used and mechanical input
power, Pm, was treated as variable that depended upon the user-defined critical clearing
time value, tcc. As this is a generic example to demonstrate the functionality of the model,
the specific values are not relevant and only the trends observed are discussed. It should be
noted that the values of inertia constant used are not representative of any specific
generator or turbine type, but generic values chosen to represent low, medium and high
samples of the range.

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Figure 4.6: Microsoft Excel model - Stability-constrained maximum generator capacity for
different fault clearing times and inertia constants
The results indicate that the maximum stability-constrained capacity of a generator
decreases non-linearly as the clearing time is increased. The longer a sustained fault must
be tolerated, the smaller the output power is permitted to be. Referring to the explanation of
the Equal Area Criterion given in Section 3.2.8, higher input power to the generator would
mean that the rotor would absorb more energy during the period of the fault when there is no
output power delivery (assuming the fault occurs at the sending bus). Dissipating this
energy would require a longer duration, resulting in a larger angular movement for the rotor.
Hence, in order for the respective pre- and post-fault clearing regions illustrated in Figure
3.10 to have the same area without violating the limit imposed by the maximum clearing
angle, the critical clearing angle must reduce, which also reduces the critical clearing time.
Higher inertia constants permit greater generator capacities for the same clearing time.
Machines with larger inertia are generally physically bigger and have slower responses to
transient disturbances. Therefore, it would take a longer period of time for their rotors to
travel to the critical clearing angle and they are able to tolerate a longer clearing time for a
given power output compared to a generator with smaller inertia.
0.00
0.25
0.50
0.75
1.00
1.25
1.50
1.75
0 100 200 300 400 500 600 700 800 900 1000
Power,pu
Time, ms
Stability-constrained maximum generator capacity
(ignoring network constraints)
H = 3
H = 6
H = 9

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The rate of change in capacity is most significant at lower clearing times and plateaus
towards the maximum time. Generators with smaller inertia constants display a faster rate of
change since they are quicker to respond to any transient disturbances.
If the user-defined clearing time in this application of the model is assumed to be a
protection relay setting and only transient stability limits are considered, then the model may
be used to determine the maximum capacity for a given protection relay configuration for a
theoretical single generator system. Conversely, it may also be used to evaluate maximum
relay clearing times for a given generator power output.
The use of the Excel “Goal Seek” function with the Equal Area Criterion in order to
determine maximum generation capacity provided valuable insight into methods of
formulating the optimal power flow model in Section 5, which forms the primary aim of this
project. However, the Excel model on its own has been developed to a stage where it would
serve as a useful tool for learning, teaching or modelling the Equal Area Criterion, which will
benefit any future work carried out to advance the scope of this project.
4.2 Solving the Swing Equation with Matlab
The Microsoft Excel model of the Equal Area Criterion was limited to solving the linear case
of the Swing Equation when the fault occurred at the sending bus and the fault power was
zero (Equation 3.34). By using Matlab to solve the full non-linear version of the Swing
Equation (Equation 3.33), the functionality of the Excel model can be expanded to
encompass a fault at any location on the line between points A and B on Figure 4.1 shown
previously. While it was possible to translate the entire Equal Area Criterion model from
Excel to Matlab, it was decided to retain the main model in Excel as the spreadsheet format
provided more flexibility and was more user-friendly. Hence, Matlab was only used as the
calculation engine to solve the differential equation and calculate the critical clearing time.
4.2.1 Structure and functionality of the Matlab solver
The Matlab program consists of two parts, the first being the main program file swing.m,
while the second is the function file f.m. The function defines the non-linear second-order
differential equation in terms of two first order differential equations, which is required by

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Matlab in order to solve it. The full Matlab code developed is given in Appendix C, while a
description of the codes’ functionality is as follows.
The main program requires input values for Pm, Pmax(fault), δ0, δcc, f and H, which are obtained
manually from the Excel model described previously. In addition, a time range for the
solution is defined, along with the initial values described in Section 3.2.7. The program first
uses the Matlab differential equation solver ode45 to solve the equations defined by function
f.m with the inputs provided and produces a plot of rotor angle against time. A conditional
loop is used to iterate the solver by incrementing the solution time range in fixed steps until
the critical clearing angle is located in the middle of the rotor angle range. This method
provides some flexibility to the program by allowing it to solve up to an optimum time range
depending on the input parameters, hence avoiding a solution time range that is too short
(which would cause a program error and require user intervention to increase the time by
trial-and-error) or too long (which would unnecessarily increase the computation time).
Additionally, accuracy of the solution is improved by locating the critical clearing angle in the
middle of the solved range.
After the solver routine is complete, the Matlab curve-fitting functions polyfit and polyval are
then used to calculate the critical clearing time from the critical clearing angle provided by
the Excel model. Another conditional loop is implemented to improve the accuracy of the
result by increasing the order of the polynomial used until the curve-fitting error calculated by
the polyval function becomes less than 1%. The final result extracted from Matlab is not only
the calculated critical clearing time, but also a minimum and maximum value based on the
curve-fitting error. Verification with PowerWorld is required to determine the exact value,
which should lie within the range given by Matlab.
4.2.2 Verification of the Matlab solver against PowerWorld
Various critical clearing time values calculated by the Matlab solver for different fault
locations were verified against PowerWorld simulations. An example verification exercise for
the set of network parameters shown in Table 4.9 is described below.
The parameters shown were the same as those used in Section 4.1.2, except that the fault
location was set to 50%. This increased the power output during the fault duration, Pmax(fault)

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from 0 pu to 1.04 pu and changed the critical clearing angle values from 65.58°to 110.54°.
All other parameters were unchanged in the model.
PowerWorld network verification parameters Sample Value Units
USER INPUT
Real electrical power output, Pe = Pm 100.00 MW
Generator terminal voltage, Vs 1.000 pu
Infinite bus voltage, Vr 1.000 pu
Infinite bus voltage angle, θr 0.00 deg
Generator direct transient reactance, Xd' 0.200 pu
Inertia constant, H 3 s
Fault location (0=sending end, 100=receiving end) 50 %
STAGE 1 VERIFICATION - NETWORK PARAMETERS
Generator terminal voltage angle, θs 11.54 deg
STAGE 2 VERIFICATION - GENERATOR TRANSIENT PARAMETERS
Steady-state operating angle, δ0 22.63 deg
Critical clearing angle, δcc 110.54 deg
Stability limit angle, δmax 144.75 deg
Critical clearing time, tcc MATLAB ms
Table 4.9: Microsoft Excel model with Matlab solver - Consolidated values for
verification with PowerWorld
The required values were input into Matlab and the solver was run to obtain the critical
clearing time. This result was then used in PowerWorld for verification. The results of the
first stage of verification were same as the previous example shown in Table 4.7 since the
steady-state parameters of the network were unchanged. The results of the second stage of
verification are shown in Table 4.10.
Matlab calculated the critical clearing time to be 411.1 ms. However, running the
PowerWorld simulation with this value resulted in an unstable system. The simulation was

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rerun several times with slightly different time values and the accurate result was found to be
410.4 ms, which was within the minimum range given by Matlab. The rotor angle plots for
both the time values are shown in Figure 4.7.
Parameter Microsoft Excel Matlab PowerWorld
Steady-state operating angle, δ0 22.63° - 22.63°
Critical clearing angle, δcc 110.54° - 110.54°
Stability limit angle, δmax 144.75° - 144.70°
Critical clearing time, tcc - 411.1 ms
410.4 msMinimum critical clearing time, tcc(min) - 407.3 ms
Maximum critical clearing time, tcc(max) - 415.0 ms
Table 4.10: Microsoft Excel model with Matlab solver – Verification of stability parameters
against PowerWorld
Figure 4.7: PowerWorld model - Transient stability simulations for Matlab verification
-50
-25
0
25
50
75
100
125
150
175
200
225
250
275
300
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6
Rotorangle,degrees
Time, s
Transient stability simulations with Powerworld for Matlab verification
tcc = 410.4 ms
tcc = 411.1 msSteady-state
operation
Post-fault
operation
Fault
applied

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In this example, although the critical clearing time value demonstrated by PowerWorld was
not exactly the same as that calculated by Matlab, it remained within the tolerance range
defined by the curve-fitting error. This demonstrated that the Excel model can be used in
conjunction with Matlab to determine the transient stability response of a generator for any
fault location between the sending and receiving buses.
Work on the Matlab solver was not progressed further as it was decided that the OPF
modelling in AIMMS would focus on using only the linear case of the Swing Equation.
Comparing Tables 4.8 and 4.10, it can be seen that the critical clearing time increased as
the fault moved further away from the generator bus. Therefore, limiting the fault condition
modelled by the OPF to the generator bus would simulate a worst-case scenario with the
most stringent clearing time forming the bounding constraint.

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V
TRANSIENT STABILITY-CONSTRAINED
OPTIMAL POWER FLOW (TSC-OPF)
The AIMMS modelling software was used to develop the transient stability-constrained
optimal power flow (TSC-OPF) model. The optimisation concept used is similar to that of the
Microsoft Excel ‘Goal Seek’ function outlined in Section 4.1.3, where the critical clearing time
is defined as a fixed value and generator power is allowed to vary. The computation
algorithm then seeks to determine the maximum generator power that would bring the
system to the brink of stability if the actual fault clearing time was the same as the critical
clearing time. Only the linear case of the Swing Equation will be modelled in the TSC-OPF
as a fault on the sending bus causes represents the worst-case scenario when the critical
clearing time of the transient fault is the shortest.
The approach taken in model development was to start with a basic version of the TSC-
OPF and progressively build upon a working model at each iteration to make it more
complex, with the ultimate goal of eventually having a version that could be applied to any
generic distribution network. As the author did not have prior experience in using AIMMS,
this approach facilitated the configuration and troubleshooting at each stage of the
development and allowed for the gradual improvement of competency in the software and
understanding of the models behaviour. It should be noted that the configurations described
in this section do not represent the only or most optimal method of implementing the TSC-
OPF within AIMMS. The goal of developing prototype models that were functional was given
precedence over programming elegance or computational efficiency.
The AIMMS configuration defines the OPF by the declaration of sets, parameters,
variables, constraints and executables. The declarations used for each TSC-OPF are
provided for each version along with descriptive notes. The mathematical notation of the
OPF is also provided for each version. For ease of reference, the declarations are denoted
both by the name used within AIMMS as well as the mathematical symbol used in the formal
notation. For the final version developed in this project, the complete tabular description and
text representation of the AIMMS code is supplied in Appendix D.

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5.1 TSC-OPF version 1
The first version of the transient stability-constrained optimal power flow (TSC-OPF-1) was
developed to specifically implement the circuit shown in Figure 5.1, which is similar to the
One Machine-Infinite Bus (OMIB) system with a double circuit transmission line described in
Sections 3.2.3 and 3.2.5 and previously modelled with Microsoft Excel in Section 4.1.
The same assumptions used in Section 4.1 also apply to TSC-OPF-1:
a) The generator is assumed to always supply power at its maximum capacity, subject to
the constraints specified.
b) The infinite bus is assumed to have a fixed voltage magnitude of 1 pu and an angle of 0°,
and absorbs all the power supplied by the generator.
c) The fault is simulated at the sending bus (point A) that is cleared by disconnecting the
faulted line at both ends (points A and B), therefore the fault current is zero and the
linear case of the Swing Equation is used to determine the critical clearing time.
Figure 5.1: Network used for TSC-OPF-1
Pm : Generator power
Pm
Vs
∠θs
Generator
terminals
E’ ∠δ
jX’d
jXL12_1
Vr
∠θr = 0
jXL12_2A B
B1 B2

FOR
REFERENCEONLY
47
5.1.1 Structure and functionality of TSC-OPF-1
A pure DCOPF representation of Figure 5.1 was used to implement TSC-OPF-1. The
primary objective with this version of the model was to produce a functional prototype that
successfully integrated the transient stability calculations as an OPF constraint and to verify
the results obtained against the Microsoft Excel model. This goal was eventually achieved
after several iterations of testing and troubleshooting.
The general formulation of TSC-OPF-1 was in accordance with the theory presented in
Section 3. However, there were several key configuration decisions made during the
development process that were instrumental in producing a functional prototype as
described below
a) The network power flow and transient stability calculations were compartmentalised as
separate segments with their own unique variables and constraints. Any common
physical quantities were declared separately in each segment and then explicitly linked
by equality constraints to construct an integrated TSC-OPF. This allowed the network
OPF to be solved with or without the transient stability constraints and facilitated testing
and troubleshooting.
b) The output quantity at each stage of the transient stability calculation was declared as a
free variable which was then constrained by the respective formula. This allowed the
quantities to be bounded by maximum or minimum limits to preserve the physical
representation of the Equal Area Criterion and avoid computational errors. Examples of
these include limiting the post-fault power, Pmax(postfault), to be less than the pre-fault
power, Pmax(prefault), and bounding the clearing angle, δc, between the steady-state
operating angle, δ0, and the maximum stability limit angle, δmax. In addition, some parts
of the calculation were combined into a single formulation in order to reduce the number
of variables.
Tables 5.1 to 5.9 describe the full AIMMS declarations used in the formulation of TSC-OPF-
1, which consisted of 16 parameters, 12 variables and 20 constraints. Tables 5.1 to 5.5 refer
to the network power flow segment while Tables 5.6 to 5.9 refer to the transient stability
segment. Equations 5.1 to 5.22 describe the OPF using mathematical notation. The
formulas used repeated from those presented previously in Section 3.

FOR
REFERENCEONLY
48
Set AIMMS Name AIMMS Index Symbol Description
S1 Buses b B Set of busbars in the network.
S2 Lines l L Set of lines in the network.
S3 DG g G Set of distributed generators in the network.
Table 5.1: TSC-OPF-1 - Network sets
Network
Optimisation
AIMMS
Name
Equation Description
NO1
DG_Network
Power
Equation 5.22 Variable to be maximised.
NO2
NetworkOnly_
Variables
-
Limits the variables used to those listed in Table
5.4 so that only the network OPF is optimised.
Transient stability variables listed in Table 5.8 are
omitted.
NO3
NetworkOnly_
Constraints
-
Limits the constraints used to those listed in Table
5.5 so that only the network OPF is optimised.
Transient stability constraints listed in Table 5.9
are omitted.
NO4
MaxDG_
NetworkOnly
Equation 5.22
Defines the objective function and direction of
optimisation
Table 5.2: TSC-OPF-1 - Network optimisation
Network
Parameter
AIMMS
Name
AIMMS
Index
Symbol Description
NP1 DG_Location - -
The busbar on which the distributed generator is
located.
NP2 Demand b D Active power demand at each busbar.
NP3 Reactance l X Reactance of each line.
NP4 Flow_Limit l fmax
Thermal flow limit of each line. Since reactive
power is not modelled, it is assumed that this
relates to the active power flow limit.
NP5 Capacity g Pmax
Power generating capacity of each distributed
generator.
NP6 Connections b, l c
Two-dimensional matrix that forms the definition
of the network connections. The value ‘1’
indicates the start busbar for a line, while the
value of ‘-1’ indicates the end busbar for a line.
Table 5.3: TSC-OPF-1 - User-defined network parameters

FOR
REFERENCEONLY
49
Network
Variables
AIMMS
Name
AIMMS
Index
Symbol Description
NV1
Power_
Network
g Pm Power output of the distributed generator.
NV2 Phase b θ Voltage phase angle at each busbar.
NV3 Flow l f
Active power flow in each line. Negative values indicate
that the powerflow is in the opposite direction than that
defined by the ‘Connections’ parameter.
Table 5.4: TSC-OPF-1 - Calculated network variables
Network
Constraint
AIMMS
Name
AIMMS
Index
Equation Description
NC1
Phase_
SlackBus
- Equation 5.2
Explicitly defines the voltage phase angle of the
slack/infinite bus to the zero.
NC2 Flow_Min l Equation 5.3
Sets the lower active powerflow limit on the lines
when power flows in the opposite direction than
that defined by the ‘Connections’ parameter
NC3 Flow_Max l Equation 5.4
Sets the upper active power flow limit on the
lines when power flows in the same direction as
that defined by the ‘Connections’ parameter.
NC4
Power_
Capacity
g Equation 5.5
Limits the upper boundary of the power
generated by the distributed generator.
NC5 KVL l Equation 5.6 Defines Kirchhoffs Voltage Law constraint.
NC6 KCL b Equation 5.7
Defines a modified version of Kirchhoffs Current
Law constraint.
Table 5.5: TSC-OPF-1 - Network constraints
Stability
Optimisation
AIMMS Name Equation Description
SO1 DG_StabilityPower Equation 5.1 Variable to be maximised.
SO2 MaxDG_ TSCOPF Equation 5.1
Defines the objective function and direction
of optimisation.
Table 5.6: TSC-OPF-1 - Transient stability optimisation

FOR
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50
Stability
Parameter
Parameter
Type
AIMMS
Name
Symbol Description
SP1
User-
defined
Xd Xd
’
Generator direct-axis transient reactance.
SP2
User-
defined
H H Machine inertia constant.
SP3
User-
defined
F F System frequency.
SP4
User-
defined
t_c tc Desired clearing time limit in milliseconds.
SP5 Fixed X_prefault Xprefault
Prefault reactance between generator and
infinite bus. This is the parallel reactance of
the values in NP3.
SP6 Fixed X_postfault Xpostfault
Postfault reactance between generator and
infinite bus. This is the reactance of L21_1
after the fault is cleared by opening L21_2.
SP7 Fixed Vs_mag Vs Sending bus voltage magnitude fixed at 1 pu.
SP8 Fixed Vr_mag Vr Receiving bus voltage magnitude fixed at 1pu.
SP9 Fixed Vr_ang θr Receiving bus phase angle fixed at 0 radians.
SP10 Fixed Pmax_fault Pmax(fault)
Fault power delivery fixed a 0 pu due to
simulated fault on the sending bus.
Table 5.7: TSC-OPF-1 - Transient stability parameters
Stability
Variables
AIMMS Name Symbol Description
SV1 Power_Stability Pt Power output of the distributed generator.
SV2 Vs_ang θs Voltage phase angle at the sending busbar.
SV3 E_mag E’ Internal voltage magnitude of the generator
SV4 Pmax_prefault Pmax(prefault) Maximum prefault power of the generator
SV5 Pmax__postfault Pmax(postfault) Maximum postfault power of the generator
SV6 delta_0 θ0 Steady-state operating power angle of the generator
SV7 delta_max θmax Maximum allowable power angle of the generator
SV8 delta_cc θcc Critical clearing angle of the generator
SV9 t_cc tcc Critical clearing time of the distributed generator
Table 5.8: TSC-OPF-1 - Calculated transient stability variables

FOR
REFERENCEONLY
51
Stability
Constraint
AIMMS Name Equation Description
SC1
Power_
Equality
Equation 5.8
Explicitly binds the variables Power_Network and
Power_Stability.
SC2
Phase_
GeneratorBus
Equation 5.9 Explicitly binds the variables Phase and Vs_ang.
SC3 Power_Max Equation 5.10
Sets the upper bound of Power_Stability so that it
does not exceed Pmax_postfault.
This constraint is required to satisfy the assumption
made in Section 3.2.3 that the generator output
power does not change during the duration of the
analysis. Without this, then it would be possible for
the prefault output power to exceed the maximum
possible postfault power, which would invalidate the
assumption and result in computational errors.
SC4
Vs_ang_
Formula
Equation 5.11
Calculation for the generator terminal bus voltage
angle, Vs_ang
SC5
E_mag_
Formula
Equation 5.12
Calculation for the internal voltage magnitude of the
generator, E_mag.
SC6
Pmax_
prefault_
Formula
Equation 5.13
Calculation for the maximum prefault output power
delivery of the generator, Pmax_prefault.
SC7
Pmax_
postfault_
Formula
Equation 5.14
Calculation for the maximum postfault output power
delivery of the generator, Pmax_postfault.
SC8
delta_0_
Formula
Equation 5.15
Calculation for the steady-state operating power
angle of the generator, delta_0
SC9
delta_max_
Formula
Equation 5.16
Calculation for the maximum allowable power angle
of the generator, delta_max
SC10
delta_cc_
Formula
Equation 5.17
Calculation for the critical clearing angle of the
generator, delta_cc.
SC11 delta_cc_min Equation 5.18
Sets the lower bound for delta_cc so that it does not
become less than delta_0.
SC12 delta_cc_max Equation 5.19
Sets the upper bound for delta_cc so that it does
not exceed delta_max.
SC13 t_cc_Formula Equation 5.20 Calculation for the critical clearing time, t_cc.
SC14 t_cc_limit Equation 5.21
Sets the upper bound for clearing time, t_c, so that
it does not exceed the critical clearing time, t_cc.
Table 5.9: TSC-OPF-1 - Transient stability constraints
As stated previously, the TSC-OPF can also be described using mathematical notation that
is commonly used to define OPFs. For the full integrated version of TSC-OPF-1, where the
network is solved with transient stability constraints, this can be written as:

Dhana Raj Markandu: Maximising Generator Connections in Distributed Networks under Stability Constraints

Dhana Raj Markandu: Maximising Generator Connections in Distributed Networks under Stability Constraints

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