The study of optimizing power system performation by using optimal FACTS devices.pdf

Power Flow Control devices such as Flexible AC Transmission Systems FACTS provide the technical solutions to address the new operating challenges being presented today.. Especially, Thyr

学位申请人姓名： DUONG THANH LONG

The Study of Optimizing Power System Performance by Using Optimal

FACTS Devices

By

DUONG THANH LONG

B.E (University of Technical Education Hochiminh City, Vietnam) 2003

M.E (University of Technical Education Hochiminh City, Vietnam) 2005

A dissertation submitted in partial satisfaction of the

Requirements for the degree of Doctor of Engineering

In Electrical Engineering

In the Graduate School

Of Hunan University

Supervisor

Professor YAO JIAN GANG

April 2014

湖 南 大 学 学位论文原创性声明

本人郑重声明：所呈交的论文是本人在导师的指导下独立进行研究所取得的研究成果。除了文中特别加以标注引用的内容外，本论文不包含任何其他个人或集体已经发表或撰写的成果作品。对本文的研究做出重要贡献的个人和集体，均已在文中以明确方式标明。本人完全意识到本声明的法律后果由本人承担。

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摘要

本文研究基于这样一个事实，就是由于输电系统的扩展无法与日益增长的输电服务相协调而造成了目前电力系统的过载严重问题日益凸显。之所以造成这样的问题是由于环境和经济的因素导致建造新的传输线路是困难的。因此，通过研究提高当前电力系统的输电能力来满足日益增长的电力需求以确保电力系统的安全运行是一个亟待解决的问题。

潮流控制设备比如柔性交流输电设备(FACTS)对时下电网新的运行方式挑战提供了解决方案。特别的，晶闸管控制的串联补偿设备（TCSC）是一种被选来研究的很有效的FACTS。TCSC通过改变输电线路参数来控制网络潮流。TCSC对网络的影响可以看成是在相关的输电线中嵌入一个可控的电抗。串联电容补偿通过抵消感抗部分减少传输线的等效串联阻抗。因而提高了功率传输能力。这样在不用重新安排发电计划和改变网络拓扑结构的情况下，通过控制潮流可极大提高系统的性能。而且这样也不会超越导线的热容量的限制同时提高了电力系统的稳定边界。为了通过采用这些设备而获得最大的效益，需要一个有效的控制方式。在研究调查柔性交流输电设备 (FACTS) 的大量应用情况后，可知针对不同目的的有效控制依赖于控制装置的安装位置。因此，运行人员面临这样一个问题就是在什么位置安装FACTS设备能达到期望的目标。目前已有大量研究在于通过FACTS设备的最优配置来改善现有电力系统性能，但对如何缩减最优安装位置的搜索范围显得无能为力。基于此，本文重点研究了如何使用和选择柔性交流输电(TCSC)设备的最优安装位置来优化电力系统性能。提出了一种用于确定晶闸管控制的串联补偿设备（TCSC）最优位置的最小切割算法（Min-cut algorithm）。一旦TCSC设备的安装位置确定了，为其寻求最佳参数的最优化问题就可以得到相应的解决。该方法可以寻找出系统中最需要安装设备的配置点，从而帮助调度人员以一种更加安全和高效的方式进行调度。采用本文方法大大减少了待选位置，从而大大减少了为提高电力系统性能而需要运行人员实际调查才能确定TCSC配置点的工作量。

本文的研究重点是通过TCSC设备的最优配置点的选择及其最优参数的确定来优化电力系统性能。本文尤其能够通过解决电力系统阻塞来提高电力系统的传输能力和考虑了电力系统静态稳定与暂态稳定约束下的电力系统性能。本文所提到的电力系统性能通过对解决安全约束最优潮流问题，全面提高电力系统传输能力，阻塞管理和最大负荷承载能力等四个问题来进行衡量评价。仿真结果表明在TCSC设备优化配置后，大大提高了电力系统性能

cut algorithm），最优化，电力系统，电力市场

关键词：柔性交流输电（FACTS），可控串联补偿器（TCSC），最小切割算法（Min-Abstract

This research stems from the fact that today’s power systems are heavy stressed due to expansion in the transmission network have not kept pace with the increasing demand for transmission services because building of new lines is difficult for environmental as well as political reasons Hence, enhancement the transfer capability of existing power networks to satisfy the increased power demand and ensure its secure operation has become an important issue for the Independent System Operator in the new electricity markets

Power Flow Control devices such as Flexible AC Transmission Systems (FACTS) provide the technical solutions to address the new operating challenges being presented today Especially, Thyristor-Controlled Series Compensator (TCSC), which is one of the most effective FACTS devices, can control the power flows in the network by changing transmission lines parameters The effect of TCSC on the network can be seen as a controllable reactance inserted in the related transmission line Series capacitive compensation works by reducing the effective series impedance of the transmission line by canceling part of the inductive reactance Hence the power transferred is increased In this way, the system performance can be considerably improved by controlling the power flows without generation rescheduling or topological changes Furthermore, the thermal limits are not violated and the stability margin is increased In order to obtain benefits from these devices, an appropriate control is necessary Studies that investigate the deployment of FACTS are indicated that the effectiveness of the controls for different purposes mainly depends on the location of control device Therefore, Operators are facing the problem of where FACTS should be installed in order to achieve require goal? Many studies have been proposed for improving existing power networks via optimal location of FACTS devices, these however studies are not able to limit search space. From the viewpoint, this thesis focuses on look at the problem of how to place and use the TCSC devices optimally to enhance of power system performance A Min-Cut algorithm is proposed in this thesis to determine proper location of TCSC Once the locations are determined, an optimization problem of finding the best settings for the installed TCSC is formulated and solved The proposed method can identify the weakest location of the system and therefore helps the System Operators to operate the system in a more secure and sufficient way With this method search space and the number of branches which need to be investigated to determine the position of TCSC for optimizing of power system performance will be significantly decreased

The focus of this work is to optimize power system performance by the optimal

placement of TCSC devices and their settings Particularly, it is intended to improve the power system transfer capability by resolving congestions, and the performance of the power system considering steady state operating condition as well as the system subjected to small disturbances The power system performance in this thesis is evaluated via solution problem concern in Security Constrained Optimal Power Flow, Enhancing Total Transfer Capability of Power System, Transmission Congestion Management and Maximum Loadability Simulation results throughout this research show a significant improvement of the power system performance after the TCSC devices is optimized

Keywords: FACTS, TCSC, Min Cut Algorithm, Optimization, Power System, Electricity

Market.

Table of Contents

位论文原创性声明和学位论文版权使用授权书 I

摘要 II Abstract IV Table of Contents VI List of figures X List of tables XII List of symbols and acronyms XV

CHAPTER 1 INTRODUCTION 1

1.1 Problem statement 1

1.2 Aims of the research 2

1.3 Contributions 2

1.4 Literature review 3

1.4.1 Current methods for solving the FACTS allocation problem 3

1.4.1.1 Classical optimization methods 3

1.4.1.2 Methods based on technical criteria 4

- Sensitivity analysis 4

- Modal analysis 5

1.4.1.3 Evolutionary computation techniques 6

1.5 Contents of the thesis 7

CHAPTER 2 TRANSMISSION CONGESTION, FACTS DEVICES AND MIN CUT ALGORITHM 9

2.1 Introduction 9

2.2Transmission Congestion 10

2.3 An Overview of FACTS 11

2.4 Thyristor-Controlled Series Compensator (TCSC) 13

2.4.1 Definition 13

2.4.2 Structure and operation 13

2.4.3 Static modeling of TCSC 15

2.5 Optimal Power Flow in Electricity Market 16

2.6 Min – Cut Algorithm 17

2.6.1 Max-Flow/Min-Cut-Theorem 17

2.6.2 Modeling power network using Min cut algorithm 18

CHAPTER 3 SECURED OPTIMAL POWER FLOW UNDER NORMAL AND

NETWORK CONTIGENCIES VIA OPTIMAL LOCATION OF TCSC 23

3.1 Introduction 23

3.2 Problem formulation 25

3.2.1 Objective function 26

3.3 Case study and discussions 26

3.3.1 Six bus system 27

3.3.1.1 OPF under normal operation 27

3.3.1.2 OPF under network contingencies 30

3.3.2 IEEE 14-bus test system 32

3.3.2.1 OPF under normal operation 32

3.3.2.2 OPF under network contingencies 33

3.3.3 IEEE 30-bus test system 35

3.3.4 IEEE 118-bus test system 40

3.4 Conclusion 41

CHAPTER 4 ENHANCING TOTAL TRANSFER CAPABILITY VIA OPTIMAL LOCATION OF TCSC 42

4.1 Introduction 42

4.2 Problem formulations using RPF with and without TCSC 43

4.3 Case study and discussions 45

4.3.1 IEEE 14-bus test system 45

4.3.2 IEEE 30-bus test system 46

4.4 Conclusion 48

CHAPTER 5 OPTIMAL LOCATION OF TCSC FOR TRANSMISSION CONGESTION MANAGEMENT 49

5.1 Introduction 49

5.2 Problem formulation 51

5.2.1 Objective function 51

5.2.2 TCSC cost function 51

5.2.3 Benefit index 52

5.3 Case study and discussions 53

5.3.1 The 5-bus system 53

5.3.2 IEEE 14-bus test system 55

5.3.3 IEEE 30-bus test system 57

5.4 Conclusion 58

CHAPTER 6 APPLICATION OF MIN CUT ALGORITHM FOR OPTIMAL

LOCATION OF TCSC DEVICES CONSIDERING SYSTEM LOADABILITY AND

COST OF INSTALLATION 60

6.1 Introduction 60

6.2 Problem formulation 62

6.3 Optimal setting of TCSC 63

6.4 Cost function 65

6.5 Case study and discussions 65

6.5.1 IEEE-6 bus system 66

6.5.2 IEEE-30 bus system 68

6.5.3 IEEE-118 bus system 70

6.6 Conclusion 71

CHAPTER 7 IMPROVING THE TRANSIENT STABILITY - CONSTRAINED OPTIMAL POWER FLOW WITH TCSC 72

7.1 Introduction 72

7.2 TSCOPF Problem formulation 73

7.2.1 Objective function 74

7.2.2 Formulation of TSCOPF 76

7.3 Case study and discussions 76

7.3.1 WECC Nine-Bus, Three-Machine System 76

7.3.2 IEEE 30-bus system 79

7.4 Conclusion 82

CHAPTER 8 SUMMARY, CONCLUSION AND FUTURE WORK 83

8.1 Thesis Summary 83

8.2 Conclusion 84

8.3 Future Work 85

References 86

Appendix A: Chapter 3 97

Appendix B: Chapter 5 102

Appendix C: List of publications 103

Appendix D: 中文摘要 104

Acknowledgment 113

List of figures

Figure 2.1: Basic structure of the TCSC 13

Figure 2.2: Operating range of the TCSC 14

Figure 2.3: Operation modes of the TCSC 14

Figure 2.4: Practical structure of a TCSC 15

Figure 2.5: Model of transmission line with TCSC 15

Figure 2.6: Modeling of a network with some cuts 18

Figure 2.7: Example power system with generators of 8 at 1, 24 at 2 and 12 at 3 and loads of 20 and 24 19

Figure 2.8: Power network shown as a directed flow graph with virtual nodes s and t Edges are labeled with (flow/capacity) 20

Figure 2.9: The units of flow along s-2-5-t 20

Figure 2.10: The units of flow along s-3-5-t 20

Figure 2.11: The units of flow along s-2-4-t 20

Figure 2.12: The units of flow along s-1-4-t 21

Figure 2.13: Some possible cuts 21

Figure 2.14: Flow chart of min cut algorithm 21

Figure 3.1: Flow chart for secured optimal power flow under normal and network contingencies 28

Figure 4.1: IEEE 14-bus system 45

Figure 4.2: Flow chart for power transfer capability with TCSC and without TCSC 47

Figure 5.1: Flow chart for determination optimal location of TCSC in congestion management 52

Figure 6.1: Model of transmission line with TCSC 62

Figure 6.2: The 4-bus system 63

Figure 6.3: Flow chart for achieving MSL and optimal installation cost of TCSC 66

Figure 7.1: One-line diagram of the IEEE 9 bus 77

Figure 7.2: Relative rotor angles for fault at bus 7 for IEEE 9 bus (Base case) 77

Figure 7.3: Relative rotor angles for fault at bus 7 for IEEE 9 bus (TSCOPF) 78

Figure 7.4: Relative rotor angles for fault at bus 7 for IEEE 9 bus (TSCOPF with TCSC) 78

Figure 7.5: IEEE 30-bus system 79

Figure 7.6: Relative rotor angles for fault at bus 2 for IEEE 30 bus (case 1) 80

Figure 7.7: Relative rotor angles for fault at bus 2 for IEEE 30 bus (case 2) 80

Figure 7.8: Relative rotor angles for fault at bus 2 for IEEE 30 bus (case 3) 81

Figure 7.9: Relative rotor angles for fault at bus 2 for IEEE 30 bus (case 4) 81

Figure B.1: The 5-bus system 102

List of Tables

Table 3.1: Optimal generation profile for six bus system 27

Table 3.2: Optimal power flow profile (in p.u) for six bus system 27

Table 3.3: The minimum cut of six bus system 29

Table 3.4: Optimal generation profile for six bus system (Case-3) 30

Table 3.5: Total cost of active power generation ($/h) computed for different locations of the TCSC 30

Table 3.6: OPF profile (in p.u) for six bus system under 1–5, 2-3 and 4-5 line outage 31

Table 3.7: OPF solution with TCSC in line 1–4 under 1–5, 2-3, 4-5 line outage 31

Table 3.8: OPF solution with TCSC in line 2–6 under 1–5, 2-3, 4-5 line outage 31

Table 3.9: OPF profile for six bus system for line outage cases with TCSC in line 1-4 31

Table 3.10: Optimal generation profile for IEEE -14 bus system 32

Table 3.11: Optimal power flow profile (in p.u) for IEEE -14 bus system 32

Table 3.12: The minimum cut of IEEE – 14 bus system 33

Table 3.13: OPF profile (in p.u) for IEEE -14 bus system under 2–3, 2-4, 2-5 line outage 33

Table 3.14:OPF profile for IEEE-14 bus system for line outage cases with TCSC in line1-5 34 Table 3.15: OPF solution with TCSC in line 1–5 under 2–3, 2-4, 2-5 line outage 35

Table 3.16: Optimal generation profile for IEEE -30 bus system 35

Table 3.17: Optimal power flow profile (in p.u) for IEEE -30 bus system 36

Table 3.18: The minimum cut of IEEE -30 bus system 37

Table 3.19: OPF profile (in p.u) for IEEE -30 bus system under line outage cases 38

Table 3.20: OPF profile (in p.u) for IEEE -30 bus system under line outage cases with TCSC in line 8-28 and line 10-22 39

Table 3.21: OPF solution with TCSC in line 8-28 and line 10-22 under line outage cases 40

Table 3.22: OPF solution for IEEE 118-bus system 40

Table 4.1: TTC from bus to bus without TCSC and with TCSC placed in best location in the minimum cut for IEEE 14-bus 45

Table 4.2: Test results of bilateral transaction from bus 2(area1) to bus 21(area3) of the IEEE

30 bus test system 46

Table 4.3: Test results of multilateral transaction from area 3 to area 2 of the IEEE 30 bus test system 46

Table 5.1: Re-dispatch with TCSC in line 2-5 objective function minimum generation 53

Table 5.2: Line loadings from load flow with initial scheduled and re-dispatch with TCSC in line 2-5 53

Table 5.3: The minimum of the 5-bus system 54

Table 5.4: Benefit index computed for different branches in the minimum cut of the TCSC 54 Table 5.5: Benefit index computed for different locations of the TCSC 55

Table 5.6: Line loadings from load flow without TCSC and with TCSC in line 1-5 55

Table 5.7: Re-dispatch with TCSC in line 1-5 objective function minimum generation 56

Table 5.8: The minimum cut of IEEE 14-bus system 56

Table 5.9: Benefit index computed for different locations of the TCSC for IEEE 14-bus 56

Table 5.10: Re-dispatch with TCSC in line 8-28 and 10-22 objective function minimum generation for IEEE-30 bus system 57

Table 5.11: Line loadings from load flow with initial scheduled and re-dispatch with two TCSC in line 8-28 and 10-22 for IEEE-30 bus system 58

Table 5.12: The minimum cut of IEEE 30-bus system 58

Table 6.1: The minimum cut of IEEE 6-bus system 66

Table 6.2: The real and reactive power flow in the line i–j at λ= 0.15(MSL=115%) before and after installing TCSC devices, optimal setting and optimal cost of installation of TCSC devices (IC) and MSL in IEEE 6 bus system 67

Table 6.3: The loops which contain branch overloaded and branch in the minimum cut for IEEE 6-bus system 67

Table 6.4: Power flow of base case at λ= 0, at λ= 0.15 with FACTS devices 67

Table 6.5: MSL and the optimal installation cost (IC) of FACTS devices obtained by proposed method and [101] algorithm in IEEE-6 bus system 68

Table 6.6: The real and reactive power flow in the line i–j at λ= 0.36(MSL=136%) before and after installing TCSC devices, optimal setting and optimal cost of installation of TCSC

devices (IC) and MSL in IEEE 30 bus system 69

Table 6.7: The minimum cut of IEEE 30-bus system 70

Table 6.8: MSL and the optimal installation cost (IC) of FACTS devices obtained by proposed method and [101] algorithm in IEEE 30-bus system 70

Table 6.9: MSL and the optimal installation cost (IC) of FACTS devices obtained by proposed method and [101] algorithm in IEEE 118-bus system 70

Table 7.1: Optimal solutions for Nine-bus, Three-Machine System 77

Table 7.2: Optimization results for IEEE 30-bus system 79

Table A.1: Cost coefficients 97

Table A.2: Transmission line data for IEEE 118-bus system 97

Table B.1: 5-bus system line data 100

Table B.2: Generator data 100

List of Symbols and Acronyms

a, b and c Cost coefficients for the generator

Ci(Pgi) The bid curve of ith generator

Pgi, Qgi The active and reactive power generation at bus-i

Pdi, Qdi The active and reactive power demand at bus i

Vi The voltage magnitude at bus i

Vi,min The minimum voltage limits

Vi,max The maximum voltage limits

Pgi,min The minimum limits of real power generation

Pgi,max The maximum limits of real power generation

Nb The total number of buses

Ng The total number of generation buses

NTCSC The set of TCSC indices

Sl The apparent power flow in transmission line connecting nodes i and j

Sl,max Maximum limit in transmission line connecting nodes i and j

α The capital recovery factor (CRF);

r The interest rate;

n The capital recovery plan

BI The benefit index for the investment in FACTS devices

K Branch which TCSC is located

Loopi Loop ith contains branch overloaded and branch k in the minimum cut

∆S (∆I) Power flow on branch overloaded needs to be reduced

Sk(Ik) Power flow on branch k before installing TCSC

loopi

.

Z Impedance of loop i

Xij The line reactance between bus-i and bus-j

XTCSC Size of series capacitive compensation of TCSC

X2 Size of series capacitive compensation which can lead to overload of

branches

X3 Size of series capacitive compensation which does not lead to

overcompensation state

δi Rotor angle of i-th generator

ωi Rotor speed of i-th generator

Di Damping constant of i-th generator

Pmi Mechanical input power of i-th generator

Pei Electrical output power of i-th generator

ω0 Synchronous speed

FACTS Flexible AC Transmission Systems

TCSC Thyristor-controlled Series Compensator

UPFC Unified Power Flow Controller

SVC Static VAR Compensator

TCPST Thyristor-Controlled Phase Shifting Transformer

STATCOM STATic synchronous COMpensator

IPFC Interline Power Flow Controller

SSSC Static Synchronous Series Compensator

TCVR Thyristor-Controlled Voltage Regulator

RPF Repeated Power Flow

TTC Total Transfer Capability

CPF Continuation Power Flow

RPF Repetitive Power Flow

MILP Mixed Integer Linear Programming

MINLP Mixed Integer Non-Linear Programming

OPF Optimal Power Flow

SCOPF Security Sonstrained Optimal Power Flow

TSCOPF Transient Stability Constrained Optimal Power Flow

NERC North American Electric Reliability Council

ISO Independent System Operator

CHAPTER 1 INTRODUCTION

The topic of this thesis stems from the fact that today’s power systems are exposed to

an increasing stress and measures have to be taken to ensure system security and economic benefit This first chapter describes brief literature review, scope and contributions of the thesis In addition, the outline of the thesis is also given in this chapter

Due to the rapid technological progress, the consumption of electric energy increases continuously Meanwhile, the transmission systems are not extended to the same extent because building of new lines is difficult for environmental and political reasons Hence, the systems are driven closer to their limits resulting in congestions and critical situations endangering the system security From this fact, the study of enhancement the transfer capability of existing power networks to satisfy the increased power demand and ensure its secure operation has become an important issue for the Independent System Operator in the new electricity markets In order to solve this problem without building more transmission lines, installation of FACTS devices can be a better alternative

Flexible AC Transmission Systems (FACTS) devices [1-2] provide technical solutions

to address the new operating challenges being presented today Devices, such as a TCSC, SVC and UPFC can be connected in series or shunt or a combination of the two to achieve numerous control functions, including voltage regulation, power flow control, and system damping In this way, the system performance can be considerably improved by controlling the power flows without generation rescheduling or topological changes Furthermore, the thermal limits are not violated, losses are minimized, and the stability margin is increased

The potential benefits of FACTS equipment are nowadays widely recognized by the power systems engineering community, however, the current challenge is to obtain the benefits from these devices Studies that investigate the deployment of FACTS must address the following questions [3]

• Which type of FACTS devices should be used?

• How many should be used?

• What is their best allocation?

• What should be their parameter settings?

• What is their installation cost?

In which, it is indicated that the effectiveness of the controls for different purposes

mainly depends on the location of control device [4] Therefore, Operators are facing the problem of where FACTS should be installed in order to achieve require goal? Many studies have been proposed for improvement existing power networks via optimal location of FACTS devices, these however studies are not able to limit search space. From the viewpoint, this thesis focuses on look at the problem of how to place and use the FACTS devices optimally to enhance power system performance A Min-cut algorithm is applied in this thesis to determine proper location of FACTS Once the locations are determined, an optimization problem of finding the best settings for the installed FACTS is formulated and solved The proposed method can identify the weakest location of the system and therefore helps the System Operators to operate the system in a more secure and sufficient way Using this method that reduces search space, the number of branches which need to be investigated to determine the position of FACTS for optimizing of power system performance will be significantly decreased

1.2 Aims of the Research

In this thesis, Thyristor-Controlled Series Compensator (TCSC), which is one of the most effective FACTS devices, is selected to study The main aim of this research is to optimize power system performance by the optimal placement of TCSC devices and their settings Particularly, it is intended to improve the power system transfer capability by resolving congestions, and the performance of the power system considering steady state operating condition as well as the system subjected to small disturbances The scope of this work is covers solution problem concern in Security Constrained Optimal Power Flow, Enhancing Total Transfer Capability of Power System, Transmission Congestion Management and Maximum Loadability

1.3 Contributions

The main contributions of this thesis are

• A Min cut algorithm is applied in power system to determine proper location of TCSC

• An Optimal Power Flow problem is formulated to determine the optimal settings for TCSC with the objective to obtain economic benefit and steady-state security in power system operation

• A benefit index is used to decide proper location of TCSC in transmission congestion management

• A formulation for determining the best setting of TCSC devices is carried out to obtain maximum system loadability and optimal cost of installation of this devices

• An Optimal Power Flow problem is formulated in order to determine the optimal settings of TCSC in power system with the aim to ensure that the system would be transiently stable following the fault disturbances

Models and applications of FACTS devices in power system operation have been often analyzed in recent years [1-3] For example, in [5] Chung et al suggest a load-equivalent model for TCSC, TCPST and UPFC for use in optimal active power flow and Varma in [6] provides an overview of the different FACTS controller configurations and their operating principles FACTS devices applications to Economic Dispatch (ED) are presented in [7], while applications in Optimal Power Flow (OPF) researches are investigated in [8] Under restructured operation of power systems, it became even more important to enhance total Transmission Capacity (TTC) of transmission networks and avoid congestion Since FACTS devices can affect line flows, they became main candidates in attempts to resolve these difficulties

1.4.1 Current methods for solving the FACTS allocation problem

There are several methods for allocating FACTS devices in operation and

control power systems to achieve these objectives But mostly, these works are commonly focused on the following methods and techniques [9]: classical optimization methods (section 1.1.3.1), methods based on technical criteria (section 1.1.3.2), and evolutionary computation

techniques (section 1.3.3.3)

1.4.1.1 Classical optimization methods

Classical optimization theory has been applied in the literature to the FACTS allocation problem in the form of Mixed Integer Linear Programming (MILP) and Mixed Integer Non-Linear Programming (MINLP)

In the MILP formulation, the approach is based on DC power flow that allows the power system to be represented in a linear manner The performance of the system is analyzed

in steady state conditions considering maximum loadability of the system and total transfer capability (TTC) [10-11] This DC simplification is also used by Aygen and Abur in [12] where they use DC OPF to find the optimal placement of TCPST devices by a two-step optimization procedure to obtain the candidate branches for TCPS installments for each contingency In [13] Kazemi and Sharifi find optimal location of TCPST devices to maximize the social welfare using DC load flow and quadratic programming The concluding remarks

of the MILP approach indicate that the optimization process is performed in an efficient manner However DC power flow is not suitable for performing transient analysis, therefore

AC models should be considered and then the problem becomes non linear For the formulation based on MINLP, the optimal allocation of FACTS devices is determined using,

as main criteria, the power price in deregulated markets [14], optimal economic dispatch and transmission losses [15], and security enhancement [16-17] Particularly in the case of security enhancement, the complexity of the problem is exacerbated since several states are defined to describe the operation of the power system (normal, collapse, corrective, and preventive) These states occur as a consequence of stochastic events (failures, topological changes), therefore there are certain probabilities associated with each one of them Furthermore, the feasibility of the problem must be guaranteed by considering load shedding

as a last resort solution to avoid voltage collapse In all the studies reported in literature, the algorithm used to solve the optimization problem is the Bender’s decomposition that consists

of separating the main problem into multiple sub-problems that are simpler to solve However,

in the case where security enhancement is considered, the complexity of the problem requires the optimization process to be aided by GA [16-17] The main conclusions about the MINLP formulation indicate that the size and the non-convexity of the problem, which depend on the system parameters, are critical issues that may cause a convergence problem in the algorithm

1.4.1.2 Methods based on technical criteria

Another group of methods that the literature have presented to solve the allocation of FACTS devices correspond to those based on pure technical criteria, in particular, sensitivity analysis for steady state performance and modal analysis for transient and dynamic conditions

of the power system

- Sensitivity analysis

Sensitivity analysis is a widely used terminology to describe the analysis based on the evaluation of the rate of change of one group of variables in a system with respect to another group There are many different ways to perform the analysis depending on the selected variables and methodologies used to calculate the sensitivities On the one hand, from the perspective of the classical optimization theory, the sensitivities can be calculated using the Lagrange multipliers [18], which provide the rate of change of the quantity being optimized as

a function of the specific constraint variable On the other hand, technically speaking, different rates of change may be of interest depending on the application, therefore many performance indexes can be defined such as the real power performance index [19-20] For example, this index considers the derivatives of the power flow equations with respect to the

steady state representation of the FACTS devices Singh and David in [21] consider the TCSC and TCPST costs along with the production costs in the objective function and solve the optimization problem by a sensitivity-based approach where each transmission line is first rated by using an optimal dispatch without considering the line flow limit and FACTS devices, then the Sensitivity factors are calculated for FACTS devices placed in every line one at a time and finally the optimal dispatch problem is solved to select the optimal location and parameter settings Another such index is the single contingency sensitivity index [22] This index considers the percentage of overload in the system’s branches for different contingencies and assigns a probability of occurrence to each of them These authors

in [23] present a two step method where the optimal location of the TCSC and TCPST is ascertained first and in the next step the settings of their control parameters is optimized The approach is based on the sensitivity of three objectives: loss on a transmission line in which a device is installed, the total system real power loss and the real power flow performance index

For transient stability, a sensitivity analysis based on critical clearing time (CCT) is proposed for optimal allocation of a TCSC in a 10 bus system [24] The FACTS devices are located in those lines where the CCT has its maximum improvement All the indices defined above, with the exception of the Lagrange multipliers, constitute a methodology for evaluating the impact of FACTS devices in the system However, to find the best location for each device an exhaustive evaluation of all possible locations is required Therefore, the basis

of these methodologies is deficient in the formulation and implementation of an appropriate search process to avoid the exhaustive search and corresponding computational burden In addition, these methodologies evaluate the sensitivity indices independently for each FACTS device In this way it is not possible to evaluate the combined effect of several of these devices installed in the system

- Modal analysis

Modal analysis is the technical method most commonly used to allocate FACTS devices when the dynamic and transient conditions for the power system are considered Since modal analysis includes the calculation of the eigenvalues and eigenvectors, the method

is not suitable for large power systems The Extended Phillips-Heffron method, proposed in [25], is able to handle larger power systems by reducing the order of the matrix to a number

no larger the number of the machines in the system However this method is validated using a very small power system of 5 buses and 3 machines Another method proposed in literature defines a controllability index as a measurement of the damping of inter-area modes of oscillation [26] This controllability index is based on the relative participation of the

parameters of each FACTS device to the critical mode Additionally, in this study a steady state criterion, based on sensitivity indices, is applied to also find the optimal locations of the FACTS devices The result of this last search process is different than the case where the controllability indices are used In general, from the perspective of finding the optimal location of FACTS devices, the modal analysis method offers technical feasibility but no guarantee about the optimality of the solution Moreover, there is a discrepancy when both steady state and transient analysis are considered that makes the method not suitable for multi-objective optimization

1.4.1.3 Evolutionary computation techniques

A third group of methods to address the problem of optimally allocate FACTS devices corresponds to evolutionary computation techniques such as GA [27-32], PSO [33-35], SA [36-37], TS [38-39], and EP [40-41] In this case, the main objective is to find the optimal types, number, sizes, and locations for the FACTS devices in the system To achieve this goal, several criteria are considered such as maximum loadability, minimum cost installation, transmission loss minimization, improvement of security margin, and maximization of TTC Some studies also include N-1 contingency analysis [42] and the power generation and dispatch problem in deregulated markets [43] The investigation mostly limited to steady state conditions; just two cases consider dynamic analysis of the power system In one of these cases ([44]), the transient analysis is not used to allocate the FACTS device but to determine optimal control settings for power system stabilizers (PSS) The other study ([45]) uses small signal analysis in order to determine the optimal location and types of FACTS devices The objective function is formulated based on three measurements: overshoot coefficient, damping ratio, and a penalty term for those unstable eigenvalues aligned in the right hand side plane Several evolutionary computation techniques and hybrid versions can be used to address power system problems but the results are highly dependent on the nature of the problem and the implementation of the algorithm In general the evolutionary computation techniques perform well in solving mixed integer non linear problems However the scalability of these methods as well as their applications to dynamic and transient analysis requires further investigation

Chapter 2 focuses on the overview of FACTS devices and Min-Cut algorithm sector

The modeling power network using Min cut algorithm and procedure for determining the minimum cut of example power system is described in detail in this chapter The

concept of bottleneck of power system and proper location of TCSC is also discussed

In addition, some important issues about the transmission sector in the deregulated environment and the impact of transmission congestion and the required features of a suitable approach for resolving transmission congestion are also introduced

Chapter 3 concentrates on security constraints in optimal power flow under normal

and network contingencies via optimal location of TCSC In this chapter, proper location of TCSC is determined base on the minimum cut of power system An Optimal Power Flow problem is formulated to determine the optimal settings for TCSC with the objective to obtain economic benefit and ensure steady-state security

in power system operation.The results on Six-bus, IEEE 14-, IEEE 30- and IEEE 118- bus show that the effectiveness of the proposed method

The problem of total transfer capacity (TTC) enhancement of transmission system is

described in chapter 4 A procedure for evaluating power transfer capability with

TCSC and without TCSC is developed in this chapter The simulation results demonstrate the impact of one TCSC unit (including its optimal location and size) on the TTC The TTC with TCSC has been significantly improved compare with case without TCSC

An index called the benefit index, which is used to decide proper location of TCSC in

transmission congestion management, is presented in chapter 5 This index is formed

from the difference between the minimum generation cost with and without TCSC It

is calculated for each location of TCSC and used to evaluate the suitability of a given branch for placing a TCSC The branch that gives a maximum benefit index is the main proper location of TCSC

Chapter 6 introduces a formulation for determining the best setting of TCSC devices

with the aim to obtain maximum system loadability and optimal cost of installation of these devices Kirchhoff's law of current and criteria for optimal location of TCSC are presented in this chapter From simulation results show that, the proposed method is capable of finding the location, quantity and size of TCSC in such effective way for enhancing system loadability and minimum installation cost of TCSC devices

Chapter 7 presents the improvement of the transient stability constraints in optimal

power flow (TSCOPF) under small disturbances using TCSC Simulations are carried out in Matlab environment to analyze the effects of TCSC on transient stability constraints in OPF The simulation results on WECC Nine-Bus, Three-Machine

System and IEEE 30-bus system have proved the effectiveness of using TCSC in TSCOPF problem

In Chapter 8, the main conclusion of the thesis is presented, and proposals for future

work in this area of research, are made

CHAPTER 2 TRANSMISSION CONGESTION, FACTS

DEVICES AND MIN CUT ALGORITHM

This chapter focuses on the overview of FACTS devices and Min-Cut algorithm sector The modeling power network using Min cut algorithm and procedure for determining the minimum cut of example power system is described in detail in this chapter The concept of bottleneck of power system and proper location of TCSC is also discussed In addition, some important issues about the transmission sector in the deregulated environment and the impact

of transmission congestion and the required features of a suitable approach for resolving transmission congestion are also introduced

2.1 Introduction

The function of transmission system was to connect the utility’s generators to the utility’s customers and to operate the system reliably The transmission systems were interconnected by different utilities to increase reliability, share reserves and take advantage

of economic exchanges Although, transmission plays a key role in making the electricity

market work but investments in the transmission network have not kept pace with the

increasing demand for transmission services This has led to heavy stress on the power networks Hence, enhancement the transfer capability of existing power networks to satisfy the increased power demand and ensure its secure operation has become an important issue for the Independent System Operator in the new electricity markets

Electricity, unlike many other commodities, can’t be stored easily and its delivery is constrained by some physical transmission limits that have to be satisfied all the time to keep the operating security of the power system With transmission limits, the deregulation of the power industry is more difficult These limits are the main causes for transmission congestion and can be listed as [3, 46]:

• Thermal limits - Colliding electrons in the AC power line cause electrical resistance and resistance interferes with current in a wire, producing heat As a wire heats up, it softens Since power lines are heavy, their weight makes them sag as heat builds Beyond a certain temperature the overloaded line will be permanently damaged It is caused not only by real power flow but also by reactive power flow

• Voltage magnitude limits - Voltage constraints define operating bounds that can limit the amount of power flowing on transmission lines Voltage constraints inevitably require attention to both the real and reactive power loads and transfers in the AC transmission system Consumption of reactive power tends to make the voltage sag

Often this must be corrected by injecting reactive power locally because reactive power is not easily transmitted over long distances

• Stability limits on power lines - Power flows through AC power lines because the voltage at the generator end reaches its maximum slightly ahead of the voltage at the load end The amount by which the generation voltage is ahead is called the “phase angle.” Beyond 90 degrees, power flow decreases become completely unstable This is the line’s physical stability limit Angle stability can be classified into two categories: small-signal stability, which is the ability of the system to maintain synchronism under small disturbance; transient stability, which is the ability to maintain synchronism when subjected to a severe transient disturbance

• Voltage stability limits - Voltage stability is the ability of a power system to maintain steady acceptable voltages at all buses in the system under normal condition or after being subjected to a disturbance The main factor causing voltage instability is the inability of the power system to meet the demand for reactive power The heart of the problem is usually the voltage drop that occurs when active power and reactive power flow through inductive reactance associated with transmission grid

• Contingency constraints - Transmission system operators leave some unused capacity

on power lines in case an unexpected event (a contingency) occurs somewhere on the system If, for example, a large power line drops out of service, the power flows will shift to other lines at the speed of light The power system operators’ job is to ensure that none of those power lines overloads Contingency constraints are fundamental element of economy-security control Contingency analysis identifies potential emergencies through extensive “what if?” simulations on the power system network A more conservative estimation of transmission capability will be obtained after considering the post-contingency constraints

In the research of this thesis, thermal, voltage magnitude limits and contingency constraints are considered in the optimal power flow problems

2.2 Transmission Congestion

Transmission congestion can be defined as the condition where desired transmission line-flows exceed reliability limits Congestion is a term that has come to power systems from economics in conjunction with deregulation, although congestion was present on power systems before deregulation Before deregulation, the transmission system was designed so that when the generation was dispatched economically there would be no limit violations

Hence, just solving economic dispatch was usually sufficient However, with the deregulation

of the electric utility industry, the transmission system is becoming increasingly stress due to

need to satisfy increased power demand and ensure its secure operation In the markets, security is measured through “system congestion” levels Congestion leads to inefficient use

of the system, increasing total generation costs and effecting direct on market transactions and electricity prices (prices in some areas will increase and in others decrease) Congestion therefore distorts the market

Congestion relief through transmission enhancements is desirable if it is cost-effective There are usually several alternatives to relieve congestion and the goal should be to devise systems of incentives that produce cost-effective means to reduce such congestion where it is economical to do so From the viewpoint of planning, effective relief methods can include installation and/or operation of large or small-scale generation in the congested area for energy production, for voltage support, to enhance stability, or to reduce flows on specific lines Transmission-based solutions can include construction of new lines or facilities, upgrading of lines or facilities, installation of voltage support (capacitors, inductors, voltage regulating transformers, static condensers, or static VAr compensators), or installation of flow-control devices (phase angle regulators or FACTS devices), and power system stabilizers

at generating stations The technologies allow more power to be delivered over a line or to operate the system more reliably Load management approaches (including bidding interruptible load in response to different market clearing prices) can also provide congestion relief under certain circumstances The incentives (and moreover, disincentives) for a particular type of relief depend on various economic, technical, informational, and regulatory elements

2.3 An Overview of FACTS

In order to increase the amount of power to be transmitted one is faced with the option

of either upgrading the existing transmission lines or building new ones The upgrade of transmission lines by upgrading the conductor may not be effective if loop flows already exist This action may be self defeating [47] The building of new transmission lines almost is impossible task due to environmental, right-of-way and cost constraints The foregoing difficulties can be partly overcome by the use of power electronic devices collectively referred to as Flexible AC Transmission System controllers (FACTS)

FACTS are a new technology developed in recent three decades, and it has been widely put in practice in the world FACTS are defined by the IEEE as a power electronic-based system and other static equipment that has the ability to enhance controllability,

increase power transfer capability These devices are not an alternative to constructing new transmission networks or upgrading transmission links but make it possible to use existing transmission network up to or close to their thermal limits FACTS could be connected either

in series or in shunt with the power system or even in a combined pattern to provide compensation for the power system Due to such features of FACTS, using it to improve

power system performance becomes more and more popular

There are different ways to class the FACTS devices [3, 27] According to the technology of the used semi- conductor they can be classified as:

• Thyristor-based FACTS Controllers

• IGBT-based FACTS Controllers

The first group employs reactive impedances or a tap-changing transformer thyristor switches as controlled-elements in circuit arrangements which are similar to breakers witched capacitors and reactors and conventional (mechanical) tap-changing transformers, but have much faster response and are operated by sophisticated controls while the second group uses self-commutated static converters as controlled voltage sources [2] The STATic synchronous COMpensator (STATCOM), the Static Synchronous Series Compensator (SSSC), the Unified Power Flow Controller (UPFC) and the Interline Power Flow Controller (IPFC) are in this category

According to the type of compensation the FACTS controllers may be classed in one

of three categories:

• Series controllers such as TCSC (Thyristor Controlled Series Capacitor), TCPST (Thyristor-Controlled Phase Shifting Transformer) and SSSC (Static Synchronous Series Compensator)

• Shunt controllers such as SVC (Static Var Compensator) and STATCOM (Static Synchronous Compensator)

• Combined series-shunt controllers such as TCVR (Thyristor-Controlled Voltage Regulator) and UPFC (Unified Power Flow Controller)

The choice of the appropriate device is important since it depends on the goals to be reached TCSC and TCPST affect the line reactance and the angle of the voltage and can control the active power flow in the transmission line The SVC is used to absorb or inject reactive power while the TCVR is picked up to act on the difference between the voltage magnitudes at the sending and receiving ends of the line The UPFC can independently control real and reactive power by being integrated into a generalized power controller combining the functions of TCSC, TCPST and SVC Different approaches and algorithms,

which are stated in the literature review, are used to investigate the best location, parameter settings, the number and the installation cost of the FACTS devices The necessity to investigate these issues stems from the fact that this is a useful but expensive technology, so its application has to be well planned and fully justified on technical and economic terms

In this thesis, FACTS devices, specifically the Thyristor-Controlled Series Capacitor (TCSC) is used with the aim to enhance power system performance in electricity market

2.4.1 Definition

The IEEE defines the TCSC as an impedance compensator which is applied in series

on an AC transmission system to provide smooth control of series reactance

2.4.2 Structure and operation

The basic element of a TCSC is shown in Figure 2.1 It is composed of a fixed capacitor and a Thyristor-Controlled Reactor The effective reactance of the TCSC is equal to the parallel connection of the reactance of the capacitor and the reactance of the TCR resulting in

C TCR

C TCR

TCSC

X ) ( X

X ).

( X ) ( X

XTCR ,



−

=C

1

v C (t)

C L

i s (t)

i L (t)

Figure 2.1: Basic structure of the TCSC

The firing angle α of the thyristors is defined as the firing delay with respect to the zero-crossing of the line current [47]

The resonance angle αres is the firing angle for which it holds that

and the TCSC is in resonance The operation close to this angle is prohibited Therefore, the operating range of the TCSC is limited to

0 ≤ α ≤ αres - ∆α, αres + ∆α ≤ α ≤ π/2 (2.4)

which is illustrated in Figure 2.2 For the effective reactance, this yields a capacitive range

reaching from Xmin to Xblock and an inductive range from Xbypass to Xmax leaving a deadband

between Xblock to Xbypass

Defined by the firing angle, four different modes of operation are distinguished

(schematically illustrated in Figure 2.3)

Figure 2.2: Operating range of the TCSC

C

L

C L

C L

C L

(c) (d)

Figure 2.3: Operation modes of the TCSC: (a) bypass mode, (b) inductive boost mode, (c) capacitive boost

mode and (d) blocking mode

Bypass mode (α=0): The thyristor valve is triggered continuously The basic circuit behaves

like a parallel connection of the series capacitor and the inductor

Inductive boost mode (0<α<αres): For α between zero and the resonance angle the equivalent

reactance is positive corresponding to an inductance

Capacitive boost mode (αres<α<π/2): If the firing angle is larger than the resonance angle, the equivalent reactance is negative resulting in capacitive behavior

Blocking mode (α=π/2): The thyristor is not triggered and therefore kept in non conducting

state Simply the fixed capacitor contributes to the effective reactance

C L

C L

C L

Figure 2.4: Practical structure of a TCSC

In practice, a TCSC consists of several basic circuits and also fixed capacitors in series (Figure 2.4) Thus, the equivalent reactance of these circuits is added such that the total effective reactance of the TCSC can be continuously varied between a minimal and maximal value without deadband

2.4.3 Static modeling of TCSC

The effect of TCSC on the network can be seen as a controllable reactance inserted in the related transmission line [48] Series capacitive compensation works by reducing the effective series impedance of the transmission line by canceling part of the inductive reactance Hence the power transferred is increased In this case study, TCSC only operates as

a capacitor The model of the network with TCSC is shown in Figure 2.5 TCSC can be considered as a static reactance –jXTCSC under steady state

Figure 2.5: Model of transmission line with TCSC

TCSC is integrated in the OPF problem by modifying the line data The maximum compensation by TCSC is limited to 70% of the reactance of the un-compensated line where TCSC is located A new line reactance (Xnew) is given as follows

Where L = XTCSC/ Xij is the degree of series compensation and Xij is the line reactance

between bus-i and bus-j

The power flow equations of the line with a new reactance can be derived as follows

) sin Bij cos

Gij ( V V Gij V Pij i j ij ij 2

) cos Bij sin

Gij ( V V Bij V

Gij(VVGijV

Gij ( V V Bij V

ij

ij ij

X R

R G

+

New 2

ij

New ij

X R

X B

+

Optimal power flow (OPF) was first discussed by Carpentier in 1962 It describes

OPF as a large non linear mathematical programming Any power network problem in

a steady state involves the adjustment of some controllable quantities to achieve a

desired operating condition and can be formulated as an optimal power flow problem

OPF has been playing a very important role in power system operation and planning and has

perhaps been the most significant technique for obtaining minimum cost generation patterns

in a power system with existing transmission and operational constraints Today OPF has

been extended and has a variety of applications in competitive electricity market

Transmission system with its natural monopoly characteristic is the major source of technical

complication in a competitive electricity market and therefore has brought about many new

potential applications and technical challenges to the OPF It maximizes the social welfare in

spot market clearing and pricing, minimizes generation cost or maximizes consumer net

benefit in transmission pricing, minimizes the cost of congestion management in congestion

management, maximizes the TTC in ATC evaluation, minimizes the cost of ancillary services

in ancillary services procurement, and maximizes the revenue of transmission rights auction

in transmission rights allocation

The progress in numerical optimization techniques and computer technology has

brought development to the OPF techniques too These techniques may be classified as

Gradient methods, Quadratic Programming, Newton-based methods, Linear Programming,

Interior Point methods, Heuristic optimization methods and Nonlinear Programming

2.6 Min - Cut Algorithm

Enhancing of power system performance by using FACTS device such as TCSC to improve the usage of existing capacities of power system is a major function of any Independent System Operator (ISO) Searching space and number of FACTS devices need to

be installed can be strongly reduced if bottleneck of power system is determined The bottleneck is location that demonstrates maximum possible power flow from source(s) to sink(s) When the system load is increased, the bottleneck is the first location where congestion occurs Therefore, in order to eliminate/alleviate congestion, the transfer capability

at the bottleneck should be examined

Furthermore, the distribution of power flow is independent from capacity loading of line but it is rely on impedance This leads to the result that the bottleneck can be overloaded though the capacity loading of bottleneck is higher than the power demand Therefore, the placement of TCSC on the branch bottleneck to modify the line impedance is a method which rapidly rebalances the power by redirecting the power flow across this branch to eliminate/alleviate overload In order words, preventing congestion using TCSC means redistributes power flow to increase the use of available capacity of the existing lines

Using the Min-cut algorithm to find the minimum cut has been introduced by Ford and Fulkerson (1956) [49] In this thesis, the Min-cut algorithm will be used to determine the minimum cut of power system The basic idea of the algorithm is to find the cut that has the minimum cut value over all possible cuts in the network That is the cut which contains bottleneck branches with sum of capacity through its smallest In other words, the power system can satisfy sufficient the power to the loads, but due to the limit of the minimum cut,

so maximum possible power flow from source(s) to sink(s) equals the minimum cut value for all the cuts in the network Therefore, if the minimum cut is identified, the branch that has the ability to contribute to adjust impedance will be recognized and only that branch is able to install TCSC to help the congested branch Hence, searching space will be reduced from n branch to m branch (m is the branches that minimum cut passes through)

2.6.1 Max-Flow/Min-Cut-Theorem

There are several methods to find minimum cut for any network having a single origin node and single destination node One of the usual approaches to solve this problem is to use its close relationship to the maximum flow problem The famous Max-Flow/Min-Cut-Theorem by Ford and Fulkerson (1956) showed the duality of the maximum flow and the so-called minimum s-t-cut There, s and t are two vertices that are the source and the sink in the flow problem and have to be separated by the cut, that is, they have to lie in different parts of

the partition as figure 2.6 Maximum amount can flow between node i and j is called capacity

of arc Cij

• Max-Flow is the maximum possible flow from origin to destination equals the

minimum cut values for all cuts in the network

• Minimum Cut

A cut is any set of directed links containing at least one link in every path from origin node to destination node This means if the links in the cut are removed the flow from the origin to destination is completely cut off The cut value is the sum of the flow capacities in the origin to node direction over all the links The minimum cut problem

is to find the cut across the network that has the minimum cut value over all possible cuts

Figure 2.6: Modeling of a network with some cuts

2.6.2 Modeling Power Network Using Min Cut algorithm

The power network is modeled as a directed graph G(N,A) where power flow is represented as flow in the graph [50] The set of nodes, N, corresponds to the buses of the power network The power line between buses n i , n j ∈ N is represented by an arc a ij ∈ A Each arc is assigned u ij, denoting the maximum allowable power flow through that line The min-cut algorithm is added two nodes, the virtual source and the virtual sink, representing the combination of the generators and loads, respectively Each line out of the virtual source has a maximum flow that matches the generation of the connected node, and each line into the virtual sink represents the load demanded by the connected node The modeling of an example power system depicted in Figure 2.7 is shown in Figure 2.8

G G G

5 4

Figure 2.7: Example power system with generators of 8 at 1, 24 at 2 and 12 at 3 and loads of 20, and 24

The algorithm works by successively assigning flow f(aij) to arcs along a directed path from s

to t until no more flow can be added [49]

- The steps in the method are summarized as follow:

1 Find any path from the origin node to the destination node If there are no more such path, exit

2 Detemine f, the maximum flow along this path, which will be equal to the smallest flow capacity on any arc in the path ( the bottleneck arc)

3 Subtract f from the remaining flow capacity according to the direction from the origin node

to the destination node for each arc in the path

4 Go to Step 1

- The algorithm will be used to determine the minimum cut of power system (Figure 2.8)

• The arcs along the path s - 2 – 5 - t are labeled using 12 units of flow The bottleneck here is the arc 2 – 5 as shown in Figure 2.9

• The arcs along the path s - 3 – 5 - t are labeled using 10 units of flow The bottleneck here is the arc 3 – 5 Note that with the simultaneous flow on path s – 2 – 5 – t, the total flow on arc 5 – t is now 22 units of flow as Figure 2.10

• The arcs along the path s - 2 – 4 - t are labeled using 8 units of flow The bottleneck on this path is arc 2 – 4 as Figure 2.11

• The arcs along the path s - 1 – 4 - t are labeled using 6 units of flow The bottleneck on this path is arc 1 – 4 as Figure 2.12

3

5 4

2 1

0/12

0/10 0/6

0/24 0/8

0/2

0/12 0/8

0/2

0/24 0/20

2 1

0/12

0/10 0/6

12/24 0/8

0/2

12/12 0/8

0/2

12/24 0/20

0/6

t

Figure 2.8: Power network shown as a directed flow graph with Figure 2.9: The units of flow along s-2-5-t

virtual nodes s and t Edges are labeled with (flow/capacity)

10/10

s

3

5 4

2 1

10/12

0/6

12/24 0/8

0/2

12/12 0/8

0/2

22/24 0/20

2 1

10/12

0/6

20/24 0/8

0/2

12/12 8/8

0/2

22/24 8/20