prr.hec.gov.pkprr.hec.gov.pk/jspui/bitstream/123456789/1800/1/910S.pdfv ABSTRACT Unit Commitment is...

UNIT COMMITMENT USING HYBRID

APPROACHES

BY

Engr. Aftab Ahmad 2K1-UET/PhD-EE-04

Supervisor

Prof. Dr. Azzam ul Asar

DEPARTMENT OF ELECTRICAL ENGINEERING UNIVERSITY OF ENGINEERING AND TECHNOLOGY

TAXILA (PAKISTAN)

JUNE 2010

DECLARATION

The substance of this thesis is original work of the author and due reference and

acknowledgement has been made, where necessary, to the work of others. No part of the thesis

has already accepted for any degree, and it is not being currently submitted in candidature of any

degree.

Engr. Aftab Ahmad 01-UET/Ph D-EE-04

Thesis Scholar

DEDICATED TO

My father, mother

Wife, Children

and

Dr. Aftab Ahmad

v

ABSTRACT

Unit Commitment is an important and vital optimization task in a power control centre.

After load forecasting it is the second step in the planning process. It consists of two linked

optimization problems. It comprises unit on/off scheduling problem and the economic dispatch

sub-problem. The on/off scheduling problem is a 0-1 combinatorial problem with equality and

inequality constraints, while the economic dispatch sub-problem is a nonlinear constrained

optimization problem.

Unit commitment is a nonlinear, large scale, combinatorial, constrained optimization

problem. The complete unit commitment optimization problem is to minimize the total

production cost (TPC) of utility in such a way that the constraints such as load demand, spinning

reserve, minimum and maximum power limits of units, minimum up (MUT) and minimum down

times (MDT) are satisfied. Therefore, based on the forecasted load demand, preparing proper

on/off schedule of generators can result in cost saving for utility. It is much more difficult

problem to solve due to its high dimensionality.

The present work is based on the scheduling of thermal units. The generation of initial

feasible UC schedules is much important, for the UCP. When initial feasible schedules

(generation > load + spinning reserve) are generated randomly, it is difficult to get feasible

schedules for the whole daily forecasted load curve.

In this work initial schedule is generated by considering the peak, off peak load of the

forecasted load curve, must run and must out units based on a new priority list method. The

proposed method is very efficient and fast in generating initial unit commitment schedules. In

this work the MUT and MDT constraints are checked and repaired by using bit change operator.

The trial solutions were generated by taking upper four units in the priority list at each hour to

avoid entrapment in local minimum.

vi

In the unit commitment problem, the economic load dispatch (ELD) sub-problem is an

intensive part and its calculations consume a large amount of time. Convex economic dispatch

using load to efficient unit and incremental cost criterion methods have been solved. In this

work, the ELD calculation for non convex problem has been solved using genetic algorithm

(GA) based on real power search method.

In the present work, three hybrid approaches have been developed for the convex and

non-convex cost functions and applied first time to solve the unit commitment problem. To

implement these algorithms a flexible and extensible computational framework has been

developed to run in visual C++ environment. The proposed algorithms are (i) “hybrid of dynamic

programming, particle swarm optimization and artificial neural network algorithms (DP-PSO-

ANN) for convex cost function, (ii) “Neuro-Genetic hybrid approach for non-convex cost

function” and (iii) “hybrid of full load average production cost and maximum power output, for

convex and non convex cost functions”. For comparison the neural network trained with back

propagation learning rule has also been developed. The proposed models have been tested on

IEEE 3 and 10 units standard test systems. The significant improvement in the total production

cost shows the promise of these hybrid models.

National utility system, National Transmission and despatch Company (NTDC) has been

reviewed with reference to its operation problems. Four test systems consisting of 12, 15, 25 and

34 units of NTDC system have been tested.

vii

ACKNOWLEDGEMENTS

My greatest thanks to the ALMIGHTY ALLAH The Most Gracious, the Most

Bountiful, the Omni potent and the Omni present, the MASTER OF THE WORLD, who gave

me the strength and spirit to fulfill the requirements of this thesis. All the respect to our HOLY

PROPHET (May Peace Be Upon Him), who after a lot of hardships and difficulties, made me

able to recognize our ALLAH and to distinguish virtue and evil.

I wish to place on record my deep sense of admiration to Dr. Aftab Ahmad (Late),

Director, NTDC, PAKISTAN for his advice, encouragement and help in my work. May

Almighty ALLAH rest the departed soul in eternal peace!

I would like to express my profound gratitude, most sincere appreciation and special

thanks to my supervisor Prof. Dr. Azzam ul Asar, for his moral support, valuable suggestions,

guidance and encouragement that made me able to complete this research work.

I would like to thank Dr. Habibullah Jamal, Professor, Department of electrical

engineering, Vice Chancellor (Jan. 2001- April 2009) University of Engineering and

Technology, Taxila, PAKISTAN for providing the financial assistance for conducting this

research. Thanks to Prof. Dr. Ahmad Khalil Khan, Dean Faculty of Electrical and Electronics

Engineering, Prof. Dr. Mumtaz Ahmad Kamal, Dean Faculty of Civil and Environmental

Engineering, Prof. Dr. Abdur Razzaq Ghumman, Prof. Dr. Shahab Kushnood, Dean Faculty of

Mechanical and Aeronautical Engineering and Prof. Dr. Adeel Akram, Dean Faculty of

Telecommunication and Information Engineering for encouragement, cooperation and support.

Special thanks to the members of my PhD research monitoring committee for guidance and

suggestions regarding the research work.

Thanks are also to my friends and colleagues especially to Prof. Dr. Tahir Nadeem Malik,

Chairman Department of Electrical Engineering for his helpful discussions and valuable

suggestions throughout this research work.

viii

The assistance in the work by Muhammad Ibrahim, Mian Muhammad Usman, Muhammad

Suleman, Adeel Mukhtar, and Syed Azhar Ali Zaidi.

Finally, I would like to give my special thanks to my wife and children for their patience

and support.

ix

CONTENTS

ABSTRACT

ACKNOWLEDGEMENT

LIST OF FIGURES

LIST OF TABLES

CHAPTER 1

Introduction 1.1 General........................................................................................................................ 1

1.2 Problem statement....................................................................................................... 2

1.3 Objectives ................................................................................................................... 3

1.4 Scope of the work ....................................................................................................... 4

1.5 Thesis Organization .................................................................................................... 5

CHAPTER 2

Unit Commitment Problem --- A Brief Literature Survey 2.1 Introduction................................................................................................................. 7

2.2 Power System Operational Planning .......................................................................... 7

2.3 Unit commitment --- Literature Survey ...................................................................... 9

2.4 Single Classical/Deterministic Approaches................................................................ 9

2.4.1. Priority List ................................................................................................................. 9

2.4.2. Dynamic Programming............................................................................................. 10

2.4.3 Branch and Bound..................................................................................................... 11

2.4.4 Integer and Mixed Integer programming.................................................................. 11

2.4.5 Lagrange Relaxation Method.................................................................................... 12

2.4.6 Straight Forward Method.......................................................................................... 14

2.4.7 Secant Method .......................................................................................................... 15

2.5 Non classical approaches .......................................................................................... 15

2.5.1 Tabu search ............................................................................................................... 15

x

2.5.2 Simulated Annealing (SA)........................................................................................ 15

2.5.3 Expert System ........................................................................................................... 16

2.5.4 Artificial Neural Network (ANN)............................................................................. 17

2.5.5 Evolutionary Programming (EP) .............................................................................. 18

2.5.6 Genetic Algorithm (GA) ........................................................................................... 18

2.5.7 Fuzzy logic................................................................................................................ 20

2.5.8 Particle Swarm Optimization (PSO)......................................................................... 20

2.5.9 Ant Colony Optimization (ACO).............................................................................. 21

2.5.10 Greedy Randomized Adaptive Search Procedure (GRASP) .................................... 21

2.6 Hybrid approaches .................................................................................................... 21

2.7 Unit Commitment --- Issues and Bottlenecks........................................................... 26

2.8 Discussion................................................................................................................. 26

CHAPTER 3

Optimization Tools for Unit Commitment

3.1 Introduction............................................................................................................... 28

3.2 Particle swarm optimization (P.S.O) ........................................................................ 28

3.3 Artificial Neural Networks (ANN) ........................................................................... 30

3.3.1 Feedforward Neural Network ................................................................................... 31

3.4 Dynamic Programming (DP) or Recursive Optimization......................................... 32

3.4.1 Forward Dynamic Programming Approach.............................................................. 34

3.4.2 Mathematical Formulation of the Dynamic Programming for unit commitment

problem ..................................................................................................................... 34

3.5 Genetic Algorithm (GA) ........................................................................................... 36

CHAPTER 4

Unit Commitment --- Problem Formulation and Single Solution Approaches

4.1 Introduction............................................................................................................... 39

4.2 Characteristics of Power Generation Units............................................................... 39

xi

4.2.1 Unit’s Input-Output characteristic (Heat or Cost) .................................................... 40

4.2.2 Non convex fuel cost characteristic due to valve point effect .................................. 41

4.2.3 Incremental heat or cost characteristic...................................................................... 41

4.2.4 Unit Heat rate (HR) characteristic ............................................................................ 42

4.3 Unit Commitment Problem (UCP) ........................................................................... 43

4.3.1 Objective Function.................................................................................................... 44

4.3.1.1 Fuel Cost ................................................................................................................... 44

4.3.1.2 Start up cost............................................................................................................... 44

4.3.1.3 Shut down cost.......................................................................................................... 45

4.4 Constraints ................................................................................................................ 45

4.4.1 System constraints or coupling constraints............................................................... 45

4.4.2 Unit constraints or local constraints.......................................................................... 46

4.5 Unit Commitment mathematical formulation as an optimization problem .............. 47

4.6 Generation of initial feasible unit commitment schedules........................................ 50

4.6.1 Initial unit commitment scheduling by using priority List method and focusing on

peak and off-peak loads of the daily load curve. ...................................................... 50

4.6.2 Generation of trial solutions / neighbors................................................................... 51

4.7 Minimum up and minimum down Time Constraint Handling ................................. 51

4.8 Minimum up and down time constraint repairing by using bit change operator ...... 52

4.9 Algorithm for the construction of initial unit commitment schedule and M.U.T and

M.D.T constraint handling........................................................................................ 52

4.10 Unit commitment schedule and determination of number of units to be operated... 53

4.11 Economic dispatch Problem (Allocation of Generation).......................................... 54

4.11.1 Economic load dispatch (ELD) calculations............................................................. 54

4.11.1.1 Equal Incremental Cost Criterion ............................................................................. 54

4.11.1.2 Loading to most efficient load .................................................................................. 57

4.11.1.3 Using Genetic Algorithm (GA) (Real Power – Search) .......................................... 58

4.12 Economic Dispatch versus Unit Commitment......................................................... 59

4.13 Conventional/Classical Single Approaches for convex fuel cost function............... 60

4.13.1 Single Approach – I --- Complete Enumeration ...................................................... 61

4.13.2 Single Approach – II --- Conventional Priority List ................................................ 61

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4.13.3 Single Approach – III --- Proposed Single Approach.............................................. 62

4.14 Case studies --- Convex cost function ..................................................................... 62

4.14.1 Numerical Results of test system – I........................................................................ 62

4.14.2 Numerical results for test system-III --- 10 unit system .......................................... 70

CHAPTER 5

Proposed New Hybrid Models for Unit Commitment problem based on

convex and Non-Convex Cost Functions

5.1 Introduction............................................................................................................... 80

5.2 Hybrid Model – I: A hybrid of particle swarm optimization (PSO), artificial neural

network (ANN) and dynamic programming (DP). ................................................... 81

5.2.1 Generation of test and training data .......................................................................... 83

5.2.2 Artificial Neural Network using SI learning Rule .................................................... 83

5.2.3 Input and Output of the PSO-ANN Model ............................................................... 83

5.2.4 Scaling of the input and output data ......................................................................... 84

5.2.5 Training process........................................................................................................ 84

5.2.6 Parameters Settings................................................................................................... 85

5.3 Case Studies---Convex cost function........................................................................ 85

5.3.1 Numerical results of test system-I: Three unit system: Hybrid Model-I .................. 85

5.3.2 Numerical Results of Test system-II: Three unit system: Hybrid Model-I .............. 95

5.3.3 Numerical Results of Test system-III: Ten unit system: Hybrid Model-I:............... 97

5.4 Hybrid Model-II: Neuro-Genetic Hybrid Approach............................................... 107

5.5 Case Studies --- non convex cost function.............................................................. 109

5.5.1 Numerical Results of Test Systems–IV and V: 3 units systems: Hybrid Model-II 109

5.6 Hybrid Model –III: Scaleable deterministic hybrid approach. ............................... 115

5.7 Case Studies ---Convex fuel cost function.............................................................. 117

5.7.1 Numerical results of test system –I: three unit system: Hybrid Model-III ............. 117

5.7.2 Numerical Results of Test System-II: Three units system: Hybrid Model-III:

(Convex Fuel Cost Curve) ...................................................................................... 123

5.7.3 Numerical Results of Test System-III: Ten units system: Hybrid Model-III: (Convex

fuel cost function) ................................................................................................... 128

xiii

5.8 Case Studies: Hybrid model-III ---Non-Convex fuel cost function........................ 139

5.8.1 Numerical results of test systems – IV and V: 3 units systems: Hybrid Model- III:

(Non-convex fuel cost Curve)................................................................................. 140

CHAPTER 6

Unit Commitment of National Transmission & Despatch Company Limited

(NTDC)

6.1 Introduction............................................................................................................. 143

6.2 WAPDA --- Brief Overview................................................................................... 143

6.3 National Transmission and Despatch Company ..................................................... 144

6.4 Operational Constraints in NTDC System.............................................................. 144

6.4.1 Hydro-Electric Generation Constraints................................................................... 144

6.4.2 Thermal Generation Constraints ............................................................................. 146

6.4.3 Transmission line Constraints................................................................................. 146

6.4.4 Seasonal Variations in Power Demand................................................................... 146

6.4.5 Spinning Reserve Constraint……………………...……………………………….146

6.4.6 Minimum up and down time constraints………………………………………….146

6.4.7 Start up cost consideration…………….…………..………………………………146

6.4.8 Maintenance cost…………………………………………………………………147

6.4.9 Fuel constraint……………………………………………………………………147

6.4.10 Ramping rates…………………………………………………………………….147

6.4.11 Unit deration………………………………………...……………………………147

6.5 Test systems for NTDC system………….......……...……………………………147

6.6 Case Studies ............................................................................................................ 147

6.7 Numerical results .................................................................................................... 148

CHAPTER 7

Conclusions and Suggestions.................................................................................................. 152

References............................................................................................................................... 156

xiv

APPENDIX A

A.1 Three unit standard test systems ............................................................................. 167

A.1.1 Test System-I --- Three Unit Test System --- Convex Fuel Cost Curve ................ 167

A.1.2 Test System-II --- Three Unit Test System --- Convex Fuel Cost Curve ............... 168

A.2 Ten Unit Standard Test system............................................................................... 169

A.2.1 Test System-III --- Convex Fuel Cost Curve.......................................................... 169

A.3 Three Unit Standard Test Systems --- Non-Convex Fuel Cost Curve………….…170

A.3.1 Test Sysem-IV --- Non- Convex Fuel Cost Curve………………………………..170

A.3.2 Test System-V --- Non-Convex Fuel Cost Curve................................................... 170

A.4 Pakistani Utility NTDC Systems ............................................................................ 171

A.4.1 12 Unit NTDC System............................................................................................ 171




APPENDIX B List of Abbreviations .............................................................................................................. 172

APPENDIX C

Notation …………………………………………………………………………………….174

APPENDIX D

Derived Publications............................................................................................................... 175

xv

List of Figures

Figure 3.1 Multilayer feedforward Neural Network Architecture......................................... 31

Figure 3.2 Supervised learning .............................................................................................. 32

Figure 4.1 Boiler-Turbine-Generators Unit ........................................................................... 40

Figure 4.2 Input-output Curve of a Steam Turbine Generator................................................ 41

Figure 4.3 Input-output curve of a multi valve steam turbine generator with four steam

admission valves ................................................................................................... 41

Figure 4.4 Incremental Heat Rate or Cost characteristic ........................................................ 42

Figure 4.5 Incremental Heat Rate Characteristics of a steam turbine with four valves. ........ 42

Figure 4.6 Heat rate and incremental heat rate curves for convex cost function ................... 43

Figure 4.7 Unit Commitment................................................................................................. 44

Figure 4.8 Representation of Unit Commitment Problem (UCP) .......................................... 49

Figure 4.9 Generation of initial solution by priority list method a graphical representation . 51

Figure 4.10 Generation of new schedule by taking upper 4 units ........................................... 51

Figure 4.11 Repairing of minimum up time ............................................................................ 52

Figure 4.12 Repairing of minimum down time ....................................................................... 52

Figure 4.13 Forecasted load curve ........................................................................................... 53

Figure 4.14 Representation of initial unit commitment schedule ............................................ 53

Figure 4.15 Pseudo Code for Genetic Algorithm Real Power-Search Method........................ 59

Figure 5.1 Flowchart for DP-PSO-ANN Hybrid Model-I…………………………………..82

Figure 5.2 Load patterns for training ..................................................................................... 83

Figure 5.3 Steps for SI learning ANN.................................................................................... 84

Figure 5.4 MSE graph for a load of 500 MW........................................................................ 94



Figure 5.7 Absolute Percentage Error graph for Operating Fuel Cost .................................. 94

Figure 5.8 Absolute Percentage Error (APE) for a load of 900MW (DP-BP-ANN Model):

Ten unit system: Test system-III......................................................................... 106

Figure 5.9 Flowchart for GA-PSO-ANN, Neuro-Genetic Hybrid Model-II........................ 108

Figure 5.10 Flow chart for Hybrid Model–III (PMAX-FLAPC) ........................................... 116

xvi

List of Tables

Table 4.1 Priority order based on single approach-II and III: test system-I ............................. 62

Table 4.2 Comparison of the operating fuel cost ($) for 3 Unit Systems :( Test System I) ..... 63

Table 4.3 Summary of Unit Commitment Schedules for 3 Unit Systems (Test System I)…...63

Table 4.4 Number of Units in Operation for 3 Unit Systems (Test System I).......................... 65

Table 4.5 Comparison of the Three Single approaches with Genetic Algorithm and Hopfield

Neural Network Methods for 3 unit systems: Test system-I .................................... 66

Table 4.6 Unit Commitment Schedule and Power Sharing (MW) of the best solution obtained

from Single Approach-I (Enumeration): Test system-I ............................................ 67

Table 4.7 Unit Commitment Schedule and Power Sharing (MW) of the solution obtained from

the Single Approach-II (Conventional Priority List): Test system-I ........................ 68


the Proposed Single approach-III (PMAX): Test system-I....................................... 69

Table 4.9 Priority order based on single approach-II and proposed single approach III:......... 70

Test system-III --- ten unit system............................................................................ 70

Table 4.10 Comparison of the operating fuel cost ($) for 10 unit systems considering 10%

spinning reserve and minimum up/down time constraints without transition cost:

Test System III......................................................................................................... 71


spinning reserve and minimum up/down time constraints with transition cost Test

System III……………………………….…………………………………….……72

Table 4.12 Comparison of the results of the proposed single approach-III with Genetic

Algorithm and conventional priority list.................................................................. 72

Table 4.13 Summary of Unit Commitment schedules for 10 unit systems: .............................. 73

Single approach-I, II and III: Test System III.......................................................... 73


from Single approach-I (Enumeration) considering MUT and MDT constraints with

10 % spinning reserve: Test system-III ................................................................... 74

xvii

Table 4.15 Unit Commitment Schedule, fuel cost, start up cost and total production cost of the

best solution obtained from Single approach-I (Enumeration) considering MUT and

MDT constraints with 10 % spinning reserve: Test system-III ............................... 75


from Single approach-II (FLAPC) considering MUT and MDT constraints with 10

% spinning reserve: Test system-III ........................................................................ 76

Table 4.17 Unit Commitment Schedule, fuel cost, startup cost and total production cost of the

best solution obtained from Single approach-II (FLAPC) considering MUT and



from Proposed Single approach-III (PMAX) considering MUT and MDT

constraints with 10 % spinning reserve: Test system-III ......................................... 78

Table 4.19 Unit Commitment Schedule fuel cost, startup cost and total production cost of the

best solution obtained from Single approach-III (PMAX) considering MUT and


Table 5.1 Comparison of proposed Hybrid Model-I with Genetic Algorithm, Conventional

Priority List and Hopfield Neural Network methods for 3 unit systems: Test

System-I: .................................................................................................................. 88

Table 5.2 Best Results by the proposed Hybrid Model-I (DP-PSO-ANN) among the sixty

samples: Test system-I............................................................................................. 89


from the proposed Hybrid Model-I (DP-PSO-ANN): Test system-I........................ 90

Table 5.4 Best output results obtained by the Hybrid Model-I (DP-BP-ANN) amongst sixty

samples: Test system-I.............................................................................................. 91

Table 5.5 Comparison of the outputs (MW) obtained by the proposed SI – ANN learning and

BP - ANN learning for 3 unit system: Test System-I ............................................... 92


by using Hybrid Model-I (DP-BP-ANN): Test system-I.......................................... 93

Table 5.7 Comparison of the best Operating fuel cost ($) obtained amongst the sixty samples

for 3 unit system by using Proposed Hybrid Model-I: Test system-II ..................... 95

xviii

Table 5.8 Comparison of the outputs (MW) obtained by the proposed SI –ANN learning

(Hybrid Model-I, DP-PSO-ANN) and BP-ANN learning (Hybrid Model-I (DP-BP-

ANN) for 3 unit systems: Test System-II ................................................................. 96

Table 5.9 Targets for the 10 unit system: Test system-III ........................................................ 99

Table 5.10 Best results obtained by the proposed Hybrid Model-I (DP-PSO-ANN) amongst the

sixty samples: Test system-III ............................................................................... 100


from proposed Hybrid Model-I (DP-PSO-ANN): Test system-III........................ 101

Table 5.12 Load demand and Total Production Cost (TPC) obtained by proposed Hybrid Model-

I (DP-PSO-ANN): Test system-III: ....................................................................... 102

Table 5.13 Comparison of the best results of the Hybrid Model-I (DP- PSO-ANN) with other

approaches available in the literature: Test system-III .......................................... 103

Table 5.14 Comparison between proposed Hybrid model-I and other approaches for daily

saving and Percentage saving in fuel cost: ten unit system ................................... 104

Table 5.15 Comparison of the results of the proposed Hybrid Model-I with Genetic Algorithm,

dynamic programming, simulated annealing and Lagrange relaxation method:... 105


by using (DP-BP-ANN), considering MUT and MDT constraints with 10 %

spinning reserve: Test system-III........................................................................... 105

Table 5.17 Load demand, operating fuel cost, startup cost and Total Production Cost obtained

by Using (DP-BP-ANN), considering M.U.T and M.D.T Constraints with 10 %

spinning reserve: Test system-III:.......................................................................... 106

Table 5.18 Comparison of the best Operating fuel cost ($) obtained amongst the sixty samples

for 3 unit system by using Proposed Hybrid Model-II: (Non-convex).................. 110


from the Hybrid Model-II: (GA-PSO-ANN): Test system-IV .............................. 111


the Hybrid Model-II (GA-BP-ANN): Test system-IV........................................... 112


the Hybrid Model-II: (GA-PSO-ANN): Test system-V ........................................ 113

xix


the Hybrid Model-II: (GA-BP-ANN) Test system-V............................................ 114

Table 5.23 Proposed Priority order based on Hybrid Model-III (PMAX-FLAPC)................. 115

Table 5.24 Comparison of proposed Hybrid Model-III with Genetic Algorithm, Conventional

Priority List and Hopfield Neural Network methods for 3 unit systems:Test

System-I: ................................................................................................................ 119

Table 5.25 Comparison of the Summary of unit commitment schedules of the proposed Hybrid

Models I and III for 3 Unit Systems: Test System -I ............................................. 120

Table 5.26 Comparison of Number of Units in Operation for 3 unit systems for hybrid models-I

and III with three single approaches: Test System I .............................................. 121


from the Proposed Hybrid Model-III: (PMAX-FLAPC2, ED based on average load

assigned method): Test system-I............................................................................ 122


from the proposed Hybrid Model-III (PMAX-FLAPC3) (ED based on lambda

Iteration method): Test system-I............................................................................ 123

Table 5.29 Comparison of the operating fuel cost ($) for proposed Hybrid Models -I and III for

3 unit system: Test System –II (Convex Fuel Cost Curve) .................................. 124

Table 5.30 Summary of Unit Commitment Schedules for 3 unit systems: Test System II: .... 125

Table 5.31 Summary of number of Units in Operation for 3 unit system: Test system II: ...... 126

Table 5.32 Unit commitment schedule and Power Sharing (MW) of the best solution obtained

by Hybrid Model-III (PMAX-FLAPC with ED based on average load) for 3 unit

systems: Test System-II ......................................................................................... 127

Table 5.33 Unit commitment schedule and Power Sharing (MW) of the best solution obtained

by Hybrid Model-III (PMAX-FLAPC with ED) for 3 unit systems Test System-

II…......................................................................................................................... 128

Table 5.34 Comparison of the best results of the Hybrid Model-III with other approaches

available in the literature: Test system-III ............................................................. 131

Table 5.35 Daily Saving and Percentage saving in fuel cost compared with other approaches132

xx


from proposed Hybrid Model-III, with MUT and MDT constraints with 10 %

spinning reserve: (ED by average load to economic unit): Test system-III: ......... 133

Table 5.37 Load demand , fuel cost , start up cost and Total Production Cost (TPC) obtained


spinning reserve. (ED by average load to economic unit): Test system-III........... 134



spinning reserve. (ED by lambda iteration method): Test system-III.................... 135

Table 5.39 Load demand , fuel cost , start up cost and Total Production Cost (TPC) of the best

solution obtained from proposed Hybrid Model-III, with MUT and MDT

constraints with 10 % spinning reserve: Test system-III (ED by Lambda iteration

method) .................................................................................................................. 136

Table 5.40 Summary of Unit Commitment schedules for 10 unit systems with 10% spinning

reserve and considering minimum up time and down time constraints:Test System

III............................................................................................................................ 137

Table 5.41 Comparison of Transition Cost for 10 unit system without considering s. r and

MUT and MDT: Test System III (10 unit system) ................................................ 138

Table 5.42 Start up cost comparison………………………………………………………….138

Table 5.43 Comparison of operating fuel cost ($) for 10 unit systems considering 10% spinning

reserve and considering minimum up/down time constraints without transition cost:

Test System III....................................................................................................... 138


spinning reserve, considering minimum up/down time constraints and transition

cost: Test System III .............................................................................................. 139

Table 5.45 Comparison of the Operating fuel cost ($) for 3 unit system obtained by using

Proposed Hybrid Model-III compared with Hybrid model -II: (Non-convex)... 140

Table 5.46 Unit Commitment Schedule and Power Sharing (MW) of the best solution

obtained from the Hybrid Model-III (PMAX-FLAPC3): (non-convex):Test

system-IV:........................................................................................................... 141

xxi

Table 5.47 Unit Commitment Schedule and Power Sharing (MW) of the best solution

obtained from the Hybrid Model-III (non-convex UC123, ED 321): Test system-

V.......................................................................................................................... 142

Table 6.1 Comparison of the results for NTDC 12, 15, 25 and 34 unit systems .................... 148

Table 6.2 Summary of UC schedule and operating fuel cost for 12 unit NTDC systems ...... 148




1

CHAPTER 1

Introduction

1.1 General

Optimization is the process of maximizing or minimizing a desired objective function

while satisfying the equality and inequality constraints. The majority of the engineering

problems involve constrained minimization. Optimization has vital and dominant role in the field

of engineering.

The Unit Commitment (UC) is an important research challenge and very significant

optimization task in the daily operational planning of modern power systems, because the

improvement of unit commitment schedules results in the reduction of total production cost

(economic benefits). It is a non-linear, high dimensional, mixed-integer combinatorial

optimization problem with both binary/integer (unit status variable) and continuous (unit output

power) variables. The number of combinations grows exponentially for a large scale system.

Therefore UC is one of the most difficult optimization problems in power systems to solve.

The Unit Commitment Problem (UCP) can be considered as two linked optimization

problems: the Unit Scheduling Problem (Allocation of Generators) and the Economic dispatch

problem (Allocation of Generation). The unit schedule problem is the on/off or 0/1 combinatorial

optimization problem. A feasible unit schedule must satisfy the forecasted load demand, system

spinning reserve requirements, and the constraints on the start up and shut down times during

each planning period. The economic dispatch sub problem is the constrained non-linear

optimization problem. This thesis investigates the application of hybrid models based on

Artificial Intelligence (AI) and conventional techniques.

2

1.2 Problem statement

The unit commitment problem (UCP) in power system is defined as determining the

start-up and shut-down schedules of units to meet the forecast load demand and spinning reserve

over a scheduling period so that the total production cost is minimized while satisfying various

system and unit constraints.

The main issues in the UCP are complexity (high dimensions) of search space, generation

of initial feasible schedules, generation of trial solutions, minimum up and down time and

spinning reserve constraint handling, calculations of non linear economic dispatch sub problem,

handling of non convexity in economic dispatch sub problem due to valve point effects. A major

source of infeasibility is the generation of infeasible schedules, which have to be discarded

immediately. Another major source of infeasibility in feasible solutions is the violation of

nonlinear minimum up/down-time constraints, which has to be checked and repaired. By

considering the minimum up and down time constraints the operating fuel cost rises and an

alternate solution needs to found. The unit commitment schedule is based on forecasted load and

spinning reserve requirements. The spinning reserve constraint also changes the on/off schedule

of units; more units are to be operated to satisfy this constraint. These constraints introduce

problems and complicate the unit commitment problem. The units that have the minimum cost

are kept on line. Some of the more expensive units are kept as standby or peaking. Thus the

actual unit commitment problem has high dimensionality, non linear in nature requiring optimal,

robust and fast solution methodology.

It is the standard practice to represent the characteristic of a unit by a quadratic function

which is convex in nature. Presently large units with multi-valve steam turbines exhibit a large

variation in this characteristic, as a result non convexity appears. Non-convexity can not be

handled by conventional single approaches. Derivative based approaches like lambda iteration

method fails to solve the non-convex economic dispatch sub problem. Various mathematical

programming based optimization techniques have been used to solve unit commitment problem.

Most of these are calculus-based optimization techniques that are based on first and second order

differentiations of objective function and its constraint equations as the search direction. They

require input-output characteristics to be of monotonically increasing nature and thus can not

3

solve non-convex ED problem. AI techniques have the ability to take into account the

nonlinearities and discontinuities commonly present in the power systems.

Many classical approaches so far have been developed and successfully implemented.

Some of the most commonly used approaches are Enumeration, priority list, dynamic

programming, Integer and Mixed integer programming, Lagrange relaxation, Benders

Decomposition and Branch and Bound. Enumeration (Brute force) scheme can be used for global

optimal solution. Priority list (PL) methods are fast but they are highly heuristic and produce

near optimal schedules. The dynamic programming method based on a priority list is flexible,

but the computational time depends upon the dimensions of the problem. Integer programming

(IP) and Mixed-integer programming (MIP) require considerable computational efforts when

dealing with large number of units. In LR method it is difficult to obtain initial feasible solutions.

The deterministic approaches are fast and simple. The main problem with these techniques is

numerical convergence and quality of solution.

The non classical single approaches like Tabu Search (TS), Genetic Algorithms (GA),

simulated annealing (SA), Particle Swarm Optimization (PSO), greedy random adaptive search

procedure (GRASP) and Evolutionary Programming (EP) etc. attract the researchers, because of

their ability to search not only for local optimal but also for global optimal. These approaches

provide near optimal solutions, but also suffer from the curse of dimensionality. For large system

of units they consume a lot of computational time to reach near global minimum and quality of

solution is also affected. Thus there is an incentive to explore hybrid methods to get around

above mentioned problems. These methods combine the strength of one approach with weakness

of other approach.

1.3 Objectives

The primary objectives of this thesis are outlined below:

1. To develop flexible and extensible computational framework as general environment for

implementing the various algorithms for unit commitment solution.

2. To develop efficient PSO based Artificial Neural Network (ANN) hybrid models for

convex and non convex UCP.

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3. To compare the results of the proposed hybrid models with GA and other approaches

available in the literature.

4. To develop and formulate the scaleable unit commitment hybrid model, with reference to

the following:

• Easy minimum up and down time constraint handling

• Consideration of convex and non convex cost function

• Quick economic dispatch Calculations

• Low transition cost

• Generating quality solution (low total production cost)

Its (Scaleable unit commitment hybrid model) implementation in computational framework

and finally testing on IEEE standard test systems

5. To review the operational problems of the PAKISTANI utility system National

Transmission and Despatch Company “NTDC” and to develop its test systems for unit

commitment studies.

1.4 Scope of the work

The contributions made in this thesis are:

1. Implementation of two single approaches, Brute Force technique, and conventional

priority list methods to generate results for comparison and development of a new

proposed single approach.

2. Development of generation of quick initial feasible unit commitment schedules based on

forecasted load curve, which satisfy the spinning reserve requirements of the system and

also satisfy the Minimum up Time (M.U.T) and Minimum down Time (M.D.T)

constraints. The generation of initial feasible solution is much important, for the Unit

Commitment problem. When initial feasible schedules are generated randomly, it is

difficult to get feasible schedules for 24 hours loads considering minimum up and

minimum down time constraints. Initial unit commitment scheduling is generated by

using a new hybrid priority list method by focusing on peak and off-peak loads of the

daily load curve.

3. Development of a methodology to generate trial/neighbor solutions in order to achieve

global minimum which also reduce the dimensionality of the unit commitment problem.

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5. Development and testing of three new hybrid algorithms for solving the UCP for convex

and non convex fuel cost functions. The proposed models are listed as follows:

• Hybrid Model–I: PSO-ANN Hybrid Approach.

• Hybrid Model–II: Neuro-Genetic Hybrid Approach.

• Hybrid Model–III: Scaleable Unit Commitment Hybrid Approach. A hybrid of

maximum power (PMAX) output of units and full load average production cost

(FLAPC).

6. Development of Swarm Intelligence (SI) and back propagation (BP) learning based

feedforward neural network for hybrid models-I and II.

7. Formulation of unit commitment using Dynamic Programming (DP) approach for the

generation of test and training data for ANN for convex cost function, for hybrid model-I.

8. Formulation of unit commitment using Genetic Algorithm (GA) approach for the

generation of test and training data for ANN for non-convex cost function

9. For the national utility system “NTDC” the tasks achieved includes:

• Operational constraints of NTDC systems

• Four test systems with convex cost characteristics curves close to the original

machines in the NTDC system have been prepared for unit commitment

problem. The test systems consist of 12, 15, 25 and 34 thermal units.

• Unit commitment study and testing of the four test systems of NTDC.

1.5 Thesis Organization

This thesis consists of seven chapters and four appendices organized as follows:

Chapter 2 presents the literature survey on unit commitment problem based on classical,

non-classical and hybrid approaches with the view to develop some observations and potential

avenues for further investigations.

Chapter 3 discusses the optimization tools used in this thesis. These include the Particle

Swarm Optimization, artificial neural network, dynamic programming and genetic algorithm.

Chapter 4 deals with the mathematical modeling of unit commitment problem and

discussion on three single solution approaches, efficient generation of initial feasible and trial

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solutions, minimum up and down time constraint handling and economic dispatch calculations.

For validation of the three single approaches the algorithms has been implemented on 3 and 10

unit standard test systems.

Chapter 5 presents the proposed new three hybrid models for convex and non convex fuel

cost function.

• Hybrid Model–I: combines the Dynamic Programming (DP), with feed forward neural

network using Swarm Intelligence (SI) learning rule and has been tested on two 3 unit

and one 10 unit standard test systems for convex fuel cost functions. For comparison the

neural network trained with back propagation learning (BP) rule has also been developed.

• Hybrid Model–II: combines the Genetic Algorithm (GA), with feed forward neural

network using Swarm Intelligence (SI) learning rule and has been tested on two 3 unit

standard test systems for non-convex fuel cost functions. For comparison the neural

network trained with back propagation (BP) learning rule has also been developed.

• Hybrid Model–III: is the integration of maximum power output of units and full load

average production cost. The model has been tested on four 3 unit standard test systems

for convex and non-convex fuel cost functions, one 10 unit standard system, and four 12,

15, 25 and 34 unit NTDC systems.

Chapter 6 gives the discussion on operational problems on PAKISTANI utility system

National Transmission and Despatch Company (NTDC) along with its unit commitment studies.

Chapter 7 discusses the conclusions and suggestions for future research.

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CHAPTER 2

Unit Commitment Problem --- A Brief Literature Survey

2.1 Introduction

The Unit Commitment Problem (UCP) is a large scale, non-linear, 0-1 combinatorial

optimization problem.

This chapter presents an overview and literature survey on UCP. Final section includes

the directions on which the new approaches evolved with time, discussion and potential avenues

for further investigations including hybrid approaches.

2.2 Power System Operational Planning

The objectives of the power system operational planning involves the best utilization of

available energy resources subjected to various constraints and to transfer electrical energy from

generating stations to the consumers with maximum safety of personal/equipment, continuity,

and quality at minimum cost.

The operational planning involves many steps such as short term load forecasting, unit

commitment, economic dispatch, hydrothermal coordination, control of active/reactive power

generation, voltage, and frequency as well as interchanges among the interconnected systems in

power pools etc.

In the early days the power system consisted of isolated stations and their individual

loads. But at present the power systems are highly interconnected in which several generating

stations run in parallel and feed a high voltage network which then supplies a set of consuming

centers. Such system has the advantages of running the number of stations with greater reliability

and economy, but at the same time the complexity in the operational and control procedures has

increased. The power industry therefore requires the services of the group of men who are

specially trained to look after the operation of the system. These men are known as the system

engineers and are responsible for the operation, control and operational planning of the system.

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Unit Commitment involves the hour-to-hour ordering of the units on/off in the system to

match the anticipated load and to allow a safety margin. Having solved the unit commitment

problem and having ensured through security analysis that present system is in a secure state

then the efforts are made to adjust the loading on the individual generators to achieve minimum

production cost on minute-to-minute basis. This loading of generators subjected to minimum

operation cost is in essence the economic dispatch.

Load forecasting gives an accurate picture of the expected demand over the following

few hours. In an anticipation of the variations in demand and for reasons of economic operation

of the system the unit commitment activity is carried out.

The solution methods being used to solve the UCP can be divided into three categories as:

• Single classical/Deterministic approaches: A variety of classical/deterministic single

techniques in this context have been reported such as: Priority List (PL), Dynamic

Programming (DP), Exhaustive Enumeration (Brute Force Technique), Branch and

Bound (B&B), Integer /Mixed integer programming (IP/MIP), Lagrangian Relaxation

(LR), Straight Forward (SF) and Secant Methods.

• Single non classical approaches: The popular single non classical approaches which got

attention in recent years are such as: Tabu Search (TS), simulated annealing (SA), Expert

System (ES), Artificial Neural Networks (ANN), Evolutionary Programming (EP),

Genetic algorithms (GA), Fuzzy Logic (FL), Particle Swarm Optimization (PSO), Ant

Colony Optimization (A.C.O), and Greedy Randomized Adaptive Search Procedure

(GRASP).

• Hybrid techniques based on classical and non-classical approaches: More recently

hybrid techniques combining two or more of the above mentioned optimization

techniques were proposed to solve UCP such as: Particle Swarm Optimization Based

Simulated Annealing, Enhanced Lagrange Relaxation, Augmented Lagrange Relaxation,

Fuzzy Adaptive Particle Swarm Optimization, Hybrid Particle Swarm Optimization,

Lagrange Relaxation Parallel Particle Swarm Optimization, Lagrange Relaxation Parallel

Relative Particle Swarm Optimization, Unit Characteristics Classification-Genetic

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Algorithm, Tabu Search based Hybrid Particle Swarm Optimization, Annealing Genetic

Algorithm, Ant Colony Simulated Annealing, Dynamic Programming- Lagrange

Relaxation, Lagrange Relaxation- Genetic Algorithm, Lagrange Relaxation-Particle

Swarm Optimization, Enhanced Merit Order- Augmented Lagrange Hopfield Network,

Priority List based Evolutionary Algorithm, Memetic Algorithm seeded with Lagrange

Relaxation, Dynamic Programming based Hopfield Neural Network etc.

2.3 Unit commitment --- Literature Survey

Unit commitment is the problem to determine the optimal subset of units to be used

during the next 24 to 168 hours [1]. This section presents a survey of the research work based on

techniques both using conventional as well as Artificial Intelligence (AI) approaches.

Traditionally the UC problem is to minimize the total production costs (TPC), (operating

fuel cost, start-up and shut-down costs) and is referred as the cost-based-unit-commitment

(CBUC) problem [2–3]. A 0.5 percent saving of the operating fuel cost gives savings of millions

of dollars per year for large utilities [4]. A number of methodologies to solve the UCP exist and

are under investigation [5-14].

The next section gives the review of several classical approaches which have been reported

in the literature

2.4 Single Classical/Deterministic Approaches

Classical methods give good results. They are heuristic and have dimensionality problem.

2.4.1. Priority List

In 2003, T. Senjyu, et al. [15] introduced extended priority list (EPL) method. The

approach consists of two steps. The initial UC schedules are produced by priority list method and

then modified using the problem specific heuristics to fulfill unit and system constraints. Some

heuristics are also applied. The Economic Dispatch is performed only on the feasible schedules.

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In 2006, T. Senjyu, et al. [16] proposed Stochastic Priority List (SPL) method. Some

initial feasible UC schedules are generated by Priority List method and priority based stochastic

window system. Some heuristics are used to reduce search space and computational time.

2.4.2. Dynamic Programming

In 1966, P. G. Lowery, [17] proposed DP in solving UCP. The main concern was to

determine the feasibility of using Dynamic Programming to solve the UCP. Results of the study

show that simple, straightforward constraints are adequate to produce a usable optimum

operating policy. The computer time to produce a solution is small.

In 1981, C. K. Pang, et al. [18] presented a study of three different DP algorithms. The

Dynamic Programming-Sequential Combinations (DPSC) and Dynamic Programming-Truncated

Combinations (DP-TC) and Dynamic Programming-Sequential/Truncated Combinations (DP-

STC), is a combination of the DP-SC and DP-TC methods. Four methods were used to establish

the savings and computer resource requirements.

In 1986, S. D. Bond, et al. [19] presented a dynamic programming which is capable of solving

the generation scheduling problem. The solutions are guaranteed to be optimal and are obtained

by using a state definition which includes the length of time a unit has been on or off. This

information is required to assess the effect of present commitment decisions on future flexibility.

When similar combinations of with the lowest accumulated cost is pursued further. The reduced

search effort lowers run times by an order of magnitude compared with a mixed integer-linear

programming approach relying mainly on constraint violations as a truncation mechanism.

Storage requirements are reduced even more significantly. The algorithm has been tested

successfully on data for a small and medium sized thermal power system. In addition to the usual

upper and lower limits on unit outputs, emergency reserve and MUT and MDT down time

constraints were incorporated.

In 1987, W. L. Snyder, et al. [20] proposed an approach to save computational time. This

algorithm incorporates a number of special features and effectively deals with the control of

problem size. To achieve the computational time saving, individual units were assigned status

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restriction in any given hour. This approach features the classification of units into groups so as

to minimize the number of unit combinations. Programming techniques are described which

maximize efficiency. This approach has been proved on a medium size utility for which sample

results were presented.

In 1988, W. L. Hobbs, et al. [21] developed an enhanced DP approach. This approach

saves predecessor options. The approach was implemented in an on-line energy management

system. A merit order list is formed which excludes all unavailable, fixed output, peaking, and

must run units. Subsequent combinations of units are formed by decommitting one unit at a time.

The method creates several states from each unique combination and links each state to one of

the possible paths to that combination.

In 1991, C. C. Su, et al. [22] developed a technique using fuzzy DP. The errors in the

forecasted load are considered and membership functions are derived for the load demand, the

total cost, and the spinning reserve using fuzzy set notations. With these membership functions at

hand, a recursive algorithm for fuzzy dynamic programming is presented. The developed

algorithm is used to solve the unit UC of Taiwan power. The proposed fuzzy dynamic

programming approach requires more computer time than the DP approach.

In 1991, Z. Ouyang, et al. [23], presented a heuristic improvement of the truncated

window DP and used a variable window size according to forecast load demand increments. The

corresponding experimental results show a considerable saving in the computation time.

2.4.3 Branch and Bound

In 1983, A. I. Cohen, et al. [24] proposed a new approach based on branch-and-bound

techniques. The method incorporates start-up costs, load demand, spinning reserve, MUT and

MDT constraints.

2.4.4 Integer and Mixed Integer programming

In 1978, T. S. Dillon, et al. [25] proposed an extended and modified version of applying

branch and bound technique for Integer Programming and treats the commitments of both hydro

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and thermal systems. The method is computationally practical for realistic system. The present

method constitutes a basis for the development of unit commitment programs using integer

programming for practical use in electric utilities.

In 2000, S. Takriti, et al. [26] presented a technique for refining the schedules obtained

by Lagrangian method. Given the schedules generated by Lagrangian iterations, and improved

schedule was found by solving the mixed integer program. The model was an integer program

with non linear constraints and solved for optimal solution using branch and bound technique.

The method gives a significant improvement in terms of quality of the solution for large number

of units.

In 2005, Li Tao, et al. [27] formulated the price-based unit commitment problem based

on the mixed integer programming method. The proposed PBUC solution is for a generating

company having cascaded-hydro, thermal, pump storage and combined-cycle, units. The results

are compared with LR method. The major obstacles are more computation time and memory

requirement to solve large UC problems.

In 2007, B. Venkatesh, et al. [28] demonstrated advantages of using the fuzzy

optimization model and presents fuzzy linear optimization formulation of UC using a mixed

integer linear programming (MILP) routine. In this formulation, start up cost is modeled using

linear variables. The fuzzy formulation provides modeling flexibility, relaxation in constraint

enforcement and allows the method to seek a practical solution. The use of MILP technique

makes the proposed solution method rigorous and fast. The method is tested on a 24 h, 104-

generator system demonstrating its speed and robustness gained by using the LP technique. A

five-generator system is additionally used to create a see-through example demonstrating

advantages of using the fuzzy optimization model.

2.4.5 Lagrange Relaxation Method

In 1983, A. Merlin, et al. [29] proposed a new implementation in solving UCP by

Lagrangian relaxation method. Numerous developments were envisaged, to make the algorithm

flexible such as simultaneous management of pumping units, probabilistic determination of the

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spinning reserve. This decomposition method used is flexible and Lagrange multiplier provides a

new solution to the conventional problem.

In 1987, R. Nieva, et al. [30] proposed an approach to solve very large and complex

UCP. The proposed approach gives an estimate of suboptimality that indicates the closeness of

the solution near to the optimum. In contrast with the technique of Lagrangian Relaxation, this

approach makes no attempt of maximizing the dual function.

In 1988, F. Zhuang, et al. [31] presented an LR method for large scale problem. The

algorithm in divided into three phases. First the Lagrangian dual of the unit commitment is

maximized with standard subgradient techniques, second a reserve-feasible dual solution is find,

and finally ED is performed. On 100 units to be scheduled over 168 hours, gives a reliable

performance and low execution times. Both spinning and time-limited reserve constraints are

treated.

In 1989, S. Virmani et al. [32] presented a paper in which they provide an understanding

of the practical aspects of the Lagrangian Relaxation methodology for solving the thermal UCP.

In 1995, R. Baldick [33] formulated UCP in generalized form and solved using LR

method. The algorithm, presented, approximately solves the dual optimization problem. The

algorithm was slower in solving the special cases of the generalized UCP than algorithms

demonstrated by other authors. The approach has been tested for ten units for a time period of 24

hours.

In 1995, W.L. Peterson, et al. [34] proposed a Lagrange Relaxation to incorporate unit

minimum capacity and unit ramp rate constrains. The proposed method is used in finding a

feasible UC schedule considering a new approach for ramping constraints. The algorithm

incorporates other practical features such as boiler fire-up characteristics and non-linear ramp up

sequences.

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In 2000, A. G. Bakirtzis, et al. [35] demonstrated the difference between the lambda

values of the economic dispatch and the UCP based on economic interpretation of the

Lagrangian Relaxation solution framework. During the LR solution of the UCP two sets of

lambdas are used. Although both set of lambdas represent marginal cost of electricity. The first

one, is assigned as a Lagrange multiplier (Lambda) to the UC power balance equations and

second one, is the Lagrange multiplier of the power balance equation in the economic dispatch

problem.

In 2004, W. P. Ongsakul, et al. [36] proposed an enhanced adaptive Lagrangian

relaxation (ELR). Enhanced LR approach consists of heuristic search and adaptive LR. ALR is

enhanced by introducing new 0-1 decisions. After the ALR the best feasible schedule is obtained.

The heuristic search is used to fine tune the schedule. The total system production costs are less

for the large scale system. The computational time is much less compared with others

approaches.

In 2005, D. Murtaza, et al. [37] presented an algorithm for the unit commitment schedule

using the Lagrange relaxation method by taking into account the transmission losses. For better

convergence and faster calculation, a two stage Lagrange relaxation was provided. First,

conventional Lagrange relaxation was applied in order to determine the unit commitment

schedule neglecting transmission loss. The results are then input to the proposed method, and the

unit commitment schedule including transmission losses was produced.

2.4.6 Straight Forward Method

In 2007, S. H. Hosseini, et al. [38] presented, a novel fast straightforward method (SF).

This new approach decomposes the UCP into three sub-problems. The quadratic cost functions

of units are linearized and hourly optimum solution of UC is obtained considering all constraints

except the MUT and MDT constraints and then the MUT/MDT constraints are introduced by

modifying the schedule obtained in the first step through a proposed novel optimization

processing. Finally, by using a new de-commitment algorithm the extra spinning reserve is

minimized.

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2.4.7 Secant Method

In 2008, K. Chandram, et al. [39] proposed an application of Secant method and

Improved Pre-Prepared Power Demand (IPPD) table (based on the units having low minimum

incremental cost) for solving the UCP. The problem is divided into two sub problems, the unit

on/off scheduling and ED sub problem. Initially, IPPD table obtains the unit 0-1 status

information and then the optimal solution is achieved by Secant method. For solving large scale

problems the convergence is in less iteration.

2.5 Non classical approaches

The growing interest is the application of non classical approaches like Artificial

Intelligence (AI) and Swarm Intelligence (SI) in solving the UCP. AI methods like Neural

Networks, Simulated Annealing, Genetic Algorithm, expert system, evolutionary programming,

and fuzzy logic are used to solve the UCP. The SI techniques like PSO and ACO also gained

prominence for solving UCP. In the following section a survey of the AI and SI methods for

UCP are presented.

2.5.1 Tabu search

In 1998, A. H. Mantawy, et al. [40] presented an approach based on the Tabu Search

method. Initial feasible UC schedules are generated randomly using new proposed rules. TSA is

used to solve the combinatorial optimization sub problem while the quadratic programming is

used to solve the EDP subproblem. Numerical results show an improvement in the quality of

solution compared with other approaches.

2.5.2 Simulated Annealing (SA)

In 1990, F. Zhuang, et al. [41] presented a general optimization method, known as

simulated annealing, and is applied to generation unit commitment. SA was used to generate

feasible solutions randomly and moves among these solutions using a strategy leading to a global

minimum with high probabilities. The method assumes no specific problem structures and is

highly flexible in handling unit commitment constraints. Numerical results on test systems of up

to 100 units were reported.

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In 1998, A. H. Mantawy, et al, [42] presented a Simulated Annealing Algorithm (SAA)

and proposed new rules for randomly generating initial feasible UC schedules. SAA is used to

solve the combinatorial optimization sub problem while the quadratic programming is used to

solve the EDP subproblem. Numerical results show an improvement in the total production cost

compared with other approaches.

In 1998, S. Y. W Wong, [43] developed an enhanced SA-approach for solving the UCP

by adopting mechanisms to ensure that the candidate solutions produced are feasible and satisfy

all the constraints. During the solution process, the solutions are generated in the neighbor of the

current one and the extent of perturbation of the solutions decreased with decreasing

temperature.

In 2006, D. N. Simopoulos, et al. [44] developed a new enhanced SA combined with a

dynamic ED method. SA is used for generator scheduling. The dynamic ED method is used to

incorporate the ramp rate constraints in the UCP. New rules for the tuning of the control

parameters of the SA algorithm are also presented.

In 2006, A. Y. Saber, et al. [45] presented fuzzy UCP using the absolutely stochastic

simulated annealing method.

2.5.3 Expert System

In 1988, S. Mokhtari, et al. [46] presented in setting up an expert system which combines

the knowledge of the unit commitment programmer and an experienced operator. In scheduling

units an expert system based on consultant has been formulated. This expert system will lead an

inexperienced operator to a better unit schedule. The basic expert system used 56 rules for the

experiments. The authors estimate that 300 rules will be required to satisfy all operational

requirements.

In 1991, S. K. Tong, et al. [47] demonstrated PL based heuristic to form initial UC

schedules based on the given forecasted load. A new expert system approach was used to handle

short term UC problem. In the proposed approach one of the previous schedules as the staring

17

point is used to find the new schedule that will satisfy the present load requirements. A rule

based approach is applied to implement PL scheme for modifying the previous schedule so that a

sub-optimal feasible schedule can be obtained quickly.

In 1993, S. Li, et al. [48] presented a graphics package and a new heuristic method for

unit commitment. The principles of this method can be expanded to consider more complicated

cases with additional constraints. The units are divided into three categories the base, medium

and peak. The computational time is less than two seconds.

2.5.4 Artificial Neural Network (ANN)

In 1992, Sasaki, et al. [49] explored the feasibility of using the Hopfield neural network

to unit commitment in which a large number of inequality constraints are handled by the

dedicated neural network instead of including them in the energy function. Once the states of

generators are determined, their outputs are adjusted according to the priority order in fuel cost

per unit output.

In 1999, T. Yalcinoz, et al. [50] presented an improved Hopfield neural networks method.

A new mapping process was used and a computational method for obtaining the weights and

biases using a slack variable technique for handling inequality constraints. Transmission

capacity, transmission losses, start-up and shutdown costs and MUT/MDT constraints have been

taken into account. The HFNN approach has been tested on a 3 unit and a 10 unit systems.

In 2000, M. H. Wong, et al. [51] used GA to evolve the weight and the interconnection of

the neural network to solve the UC problem. The back-propagation was used to train the weights.

Three selection methods Roulette Wheel, Tournament and Ranking was used as well as two

options for Weight and Connections are combined for running the GA. Roulette Wheel has the

best performance.

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2.5.5 Evolutionary Programming (EP)

In 1999, K. A. Juste, et al. [52] proposed an algorithm that uses the EP technique in

which populations of initial population is generated randomly and then the solutions are evolved

through selection, competition, and random changes.

2.5.6 Genetic Algorithm (GA)

In 1993, D. Dasgupta et al. [53] presented a genetic approach for determining the priority

order in the commitment of thermal units in power generation. The paper examined the

feasibility of using genetic algorithms and reports some simulation results in near optimal

commitment of thermal units. The genetic-based UC system evaluates the priority of the units

dynamically considering the system parameters, operating constraints and load profile at each

time period in the scheduling horizon.

In 1995, X. Ma, et al. [54] developed a forced mutation operator and the efficiency of the

GA was improved significantly using this operator. Two different coding schemes were devised

and tested. It was observed that the two-point crossover operation is considerably more efficient

than the single-point crossover commonly used in GAs In addition, the effects of GA’s control

variables on convergence were extensively studied. The approach was tested on a 10 unit system.

Test results clearly reveal the robustness and promise of the proposed approach.

In 1996, S. O. Orero, et al. [55] presented an enhanced genetic algorithm incorporating

sequential decomposition logic for faster search mechanism Unit commitment constraints

including ramp rates are considered. The method relies on the selection and grading of the

penalty functions to allow the fitness function to differentiate between good and bad solutions.

The method guarantees the production of solutions that do not violate system or unit constraints,

so long as there are enough generators available in the selection pool to meet the required load

demand. The algorithm has been tested on 26 generators.

In 1996, S. A. Kazarlis, et al. [56] presented Genetic Algorithm by using Varying Quality

Function technique and adding problem specific operators. The coding was implemented in a

binary form. With the technique of varying quality function, the GA finally manages to locate

19

the exact global solution. A nonlinear transformation was used for fitness scaling. New operators

swap-mutation and swap-window hill-climb was implemented. The algorithm is applied to 100

units.

In 2002, T. Senjyu, et al. [57] presented new genetic operator based on unit characteristic

classification and intelligent technique for generating initial populations. The initial population is

generated base on load curve. To handle MUT/MDT constraints new mutation operators were

introduced. New cross over operator, shift operator, and intelligent mutation operators were

proposed. Units are classified in several groups depending upon their MUT/MDT constraints.

For every violated constraint, a penalty term is added to the total cost.

In 2003, E. Gil, et al. [58] proposed a new method for hydrothermal systems. The

proposed GA, using new specialized operators, has demonstrated excellent performance in

dealing with this kind of problem, obtaining near-optimal solutions in reasonable times.

In 2004, G. Loannis, et al. [59] presented a new solution based on an integer-coded

genetic algorithm (ICGA), in which the chromosome size is reduction compared to the binary

coding. The non linear MUT and MDT constraints are directly coded in the chromosome. The

use of penalty functions is avoided because they distort the search space. The ICGA is robust and

execution time is less than other approaches.

In 2006, C. Dang, et al. [60] proposed a floating-point genetic algorithm (FPGA). In

which a floating-point chromosome representation is used based on the forecasted load curve. To

handle MUT and MDT constraints encoding and decoding schemes are used. The fitness

function, constraints, population size, selection, crossover and mutation probabilities are

characterized in detail. The FPGA is also applicable for non-convex cost function.

In 2006, L. Sun, et al. [61] introduced a matrix real-coded genetic algorithm (MRCGA).

A real number matrix representation of chromosome is used that can solve the UC problem

through genetic operations. The search performance is improved through a window mutation.

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The proposed new mechanism of chromosome repair guarantees that the UC schedule satisfies

unit and system constraints.

2.5.7 Fuzzy logic

In 1997, S. Saneifard, et al. [62] formulated the fuzzy logic to the UCP. A comparison of

results presented in the paper indicates that the use of fuzzy logic provides outcomes comparable

to those of conventional dynamic programming. It is claimed that this approach gives

economical cost of operation.

In 2004, S. C. Pandian, et al. [63] presented a fuzzy logic approach that is very useful to

consider the uncertainty in the forecasted load curve, derating and line losses. Numerical results

are compared w.r.t the operating cost and computation time obtained by using fuzzy dynamic

programming and other conventional methods like dynamic programming, Lagrangian relaxation

methods. For validation of the approach in respect of total production cost and computational

time, case studies on 10, 26 and 34 units have been performed.

2.5.8 Particle Swarm Optimization (PSO)

In 2003, Z. L. Gaing, [64] proposed binary particle swarm optimization (BPSO). The

BPSO is used to solve the combinatorial unit on/off scheduling problem for operating fuel and

transition costs. The ED subproblem is solved using the lambda iteration method for obtaining

the total production cost.

In 2006, B. Zhao, et al. [65] presented an improved particle swarm optimization

algorithm (IPSO) for UC which utilizes more particles information to control the process of

mutation operation. For proper selection of parameters some new rules are also proposed. The

proposed method combines LR technique to 0-1 variable.

In 2007, T. Y. Lee, et al. [66] presented a new approach for UCP named the iteration

particle swarm optimization (IPSO). The proposed method improves the quality of solution in

terms of total production cost and also improves the computation efficiency. A standard 48 unit

system has been tested for validation.

21

In 2009, X. Yuan, et al. [67] proposed a new improved binary PSO (IBPSO). The

standard PSO is improved the using the priority list and heuristic search to improve the MUT and

MDT constraints. The 10-100 units have been tested to validate the proposed approach.

Numerical performance shows that the proposed approach is superior in terms of low total

production cost and short computational time compared with other published results.

2.5.9 Ant Colony Optimization (ACO)

In 2003, T. Sum-im, et al. [68] proposed, ant colony search algorithm (ACSA), which is

inspired by the observation of the behaviors of real ant colonies, and is a new cooperative agent’s

approach based on parallel search. In the proposed approach, a set of cooperating agents called

“ants” cooperates to find good unit schedules the ED sub-problem is solved by the λ-iteration

method.

In 2008, A.Y. Saber, et al. [69] proposed memory-bounded ant colony optimization

(MACO). The proposed approach is applicable for large number of units and solves the

computer memory limit requirements. A heuristic is also incorporated to enhance local search.

2.5.10 Greedy Randomized Adaptive Search Procedure (GRASP)

In 2003, A Viana, et al. [70] presented, an adaptive algorithmic framework based on

another meta-heuristic principle (GRASP – Greedy Randomized Adaptive Search Procedure).

The philosophy applied is slightly different from standard meta-heuristics, the decisions taken by

the method, when building a solution, are somehow adapted according to decisions previously

taken. This dynamic learning-process often leads to very good solutions.

2.6 Hybrid approaches

Hybrid approaches are also used to solve many difficult engineering problems. The aim

of the hybrid methods is to improve the performance of single approaches. The objective of

hybrid of two or more methods is to speed up the convergence and to get better quality of

solution compared with single approaches. A brief review of different hybrid approaches which

have been reported in the literature is presented in this section.

22

In 1992, Z. Ouyang et al. [71] utilized neural networks to generate a pre-schedule

according to the forecasted load curve. The proposed approach significantly reduced the

computational time. Case studies are performed on 26 unit system. A 35 training pattern are used

in the study. The training of each load takes approximately eight to ten minutes.

In 1992, Z. Ouyang, et al. [72] proposed a multi-stage Neural Network-expert system

approach. Through inference the feasible UC schedule is obtained. A load pattern matching

scheme is performed at the pre-processor stage. The trained network performs adjustments in

the schedule to achieve the optimal solution at the post processor stages.

In 1995, D. P. Kothari, et al [73] described a hybrid expert system dynamic programming

approach. The output scheduling of the usual dynamic programming is enhanced by

supplementing it with the rule based expert system. The proposed system limits the number of

constraints and also checks the possible constraint violations in the generated schedule. The

expert system communicates with the operator in a friendly manner and hence the various

program parameters can be adjusted to have an optima1, operationally acceptable schedule.

In 1997, H. Shyh-Jier, et al. [74] proposed genetic algorithm based neural network and

dynamic programming approach for UCP. At the initial stage a set of feasible UC schedules are

generated by genetic-enhanced neural networks. In the second stage these schedules are

optimized by using the DP approach. The computational efficiency is more compared with other

methods.

In 2000, M. H. Wong, et al. [75] presented a technique in which genetic algorithm is

evolved to intelligently decide the initial weights and the connections in the ANN. This approach

prevents the stagnation during training. The approach converges into global minimum for a given

range of space. The evolving neural network has lower training error compared to neural

network with random initial weights.

In 2000, R. Nayak, et al. [76] proposed a hybrid of feed forward neural network and the

simulated annealing. The ANN is used to solve the unit scheduling sub problem and the SA is

23

used to solve the ED sub problem. A set of inputs based on the forecasted load curve and

corresponding UC schedules as outputs satisfying the system and unit constraints are used to

train the network. A reduction in computational time is achieved by this approach.

In 2000, C. P. Cheng, et al. [77] presented an application of Genetic Algorithms and

Lagrangian Relaxation (LRGA) method. The proposed approach incorporates GA into LR

method to improve the performance of LR and to update the Lagrangian multipliers. The method

is easy to implement, better in convergence.

In 2002, C. P. Cheng, et al. [78] proposed the application of the annealing–genetic (AG)

algorithm. The AG is a hybrid of GA into the SA to improve the performance of the SA

approach. The method improves the computational time of the Simulated Annealing and the

quality of solution of Genetic Algorithm and gives near optimal solution of a large scale system.

In 2002, J. Valenzuela, et al. [79] presented memetic algorithm, a hybrid of GA, and LR

is efficient and effective for solving large UC problems. The implementations of standard GA or

MA are not competitive compared with the traditional methods of DP and LR. However, an MA

incorporated with LR proves to be superior to other approaches on large scale problems.

In 2003, T. O. Ting, et al. [80] proposed a Hybrid Particle Swarm Optimization (HPSO).

Problem formulation, representation and the numerical results for a 10 unit are presented. Results

shown are acceptable at this early stage.

In 2003, C. C. A. Rajan, et al. [81] presented a neural based tabu search (NBTS) method.

The algorithm is based on the short term memory procedure of the tabu search method. Systems

consisting of 10, 26, and 34 units have been tested and the results are compared with other

approaches. The results in terms of total production cost and computational time are better than

single approaches like DP and LR.

24

In 2004, D. Srinivasan, at al. [82] proposed an efficient algorithm for aiding unit

commitment decisions. To solve the UCP an evolutionary algorithm with problem specific

heuristic and genetic operators has been employed

In 2004, L. Shi, et al. [83] developed and demonstrated a novel ant colony optimization

algorithm with random perturbation behavior (RPACO). The approach is based on the

combination of colony optimization and stochastic mechanism is developed for the solution of

optimal UC with probabilistic spinning reserve.

In 2004, H. H. Balci, et al. [84] presented a hybrid of PSO and LR. UCP is divided into

sub problems and each sub problem is solved using DP. PSO is used to update the Lagrangian

multipliers. The comparison of results shows that the proposed approach uses less computational

time and gives good quality solutions.

In 2005, T.A.A Victoire, [85] introduced an application of hybrid-PSO and sequential-

quadratic programming technique (SQP) guiding the tabu search (TS). The unit scheduling

problem is solved using an improved random-perturbation scheme. A simple procedure for

generating initial feasible UC schedules is proposed for the TS method. The nonlinear ED

subproblem is solved using the hybrid PSO-SQP technique.

In 2005, S. Chusanapiputt, et al. [86] presented Parallel Relative Particle Swarm

Optimization (PRPSO) and LR for a large-scale system. To reduce the dimensionality problem

and to improve the UC schedules the neighborhood solutions are divided into sub-

neighborhoods.

In 2005, P. Sriyanyong, et al. [87] proposed PSO based LR method for optimal setting of

Lagrange multipliers. In the proposed work, the PSO was used to adjust the lagrange multipliers

in order to improve the performance of lagrange relaxation method.

In 2005, T. Aruldoss, et al. [88] presented a solution model using fuzzy logic. Hybrid of

Simulated annealing, particle swarm optimization and sequential quadratic programming

25

technique (hybrid SA-PSO-SQP) is used to schedule the generating units based on the fuzzy

logic decisions.

In 2006, T. O. Ting, et al. [89] introduced a hybrid particle swarm optimization (HPSO)

which is a combination of binary and real coded particle swarm optimization (BPSO and

RCPSO). The term “hybrid particle swarm optimization” was first mentioned by S. Naka, et al.

where by the term hybrid meant the combination of PSO and GA. The BPSO is used to solve

unit scheduling problem and RPSO is used to solve the ED subproblem.

In 2006, V.N. Dieu et al. [90] proposed an enhanced merit order (EMO) and augmented

Lagrange Hopfield network (ALHN) for solving hydrothermal scheduling (HTS) problem with

pumped-storage units. The proposed approach is based on merit order approach enhanced by

heuristic search based algorithms. The ALHN is a continuous Hopfield network and its energy

function is based on augmented Lagrangian function. EMO is efficient in unit scheduling,

whereas ALHN can properly handle generation ramp rate limits, and time coupling constraints.

In 2007, A. Y. Saber, et al. [91] presented a twofold simulated annealing (twofold-SA)

method. A hybrid of SA and fuzzy logic is used to obtain SA probabilities from fuzzy

membership function. The initial feasible UC schedules are generated by a priority list method

and are modified by de-composed SA using a bit flipping operator. Results indicate a low total

production cost and low execution time compared with other approaches.

In 2007, S. Nasser, et al. [92] presented hybrid particle swarm optimization based

simulated annealing (PSO-B-SA) approach. The unit scheduling sub problem is solved by using

binary PSO and ED sub problem is solved by using real valued PSO. Numerical results

demonstrated show that the PSO-B-SA approach can perform well compared with the other

solutions.

In 2007, A. Y. Saber, et al. [93] proposed a fuzzy adaptive Particle Swarm Optimization

(FAPSO) for UCP. FAPSO precisely tracks a changing schedule. Based on the diversity of

26

fitness the fuzzy adaptive criterion is used for the PSO inertia weight. Using fuzzy IF/THEN

rules the weights are dynamically adjusted.

In 2007, S. S. Kumara, et al. [94] developed DP based direct Hopfield computation method. The

proposed approach solves the UCP in two steps. The generator scheduling problem is solved

using DP and generation scheduling problem is solved using Hopfield neural network.

2.7 Unit Commitment --- Issues and Bottlenecks

The issues and bottlenecks in the UCP may be listed as:

1. High dimensionality

2. Handling of cost base and profit base unit commitment

3. Handling of non convex fuel cost function

4. Generation of infeasible solutions

5. Handling of constraints such as:

i. Minimum up and down time

ii. Transmission

iii. Emission

iv. Security

2.8 Discussion

The global optimal solution of the UCP can be obtained by Brute Force (complete

enumeration) technique, which is not applicable for a power system having large number of units

due to its long computational time. Priority list (PL) methods are highly heuristic but very fast

and give UC solutions with high total production cost. The DP methods are based on

enumeration and PL, but suffer from the curse of dimensionality. Integer programming (IP) and

Mixed-integer programming (MIP) require considerable computational efforts when dealing with

large-scale problems. The main problem with Lagrangian relaxation (LR) method is the

difficulty in obtaining the feasible UC schedules.

The non classical approaches such as Evolutionary Computation, Genetic Algorithm, and

Particle Swarm Optimization etc. attract much attention, because they are able to solve convex

and non-convex fuel cost functions, have the ability to search for near global and can deal easily

with non linear constraints. In case of large-scale problem these single approaches consume long

27

computational time. The main difficulty is their sensitivity on the choice of parameters. Hence,

there is an incentive to explore hybrid algorithms. From the selected above mentioned literature

review, it is observed that the hybrid techniques reduce the search space are more efficient and

have better quality of solutions for small and large scale problems, gives solution in an

acceptable computation time and can accommodate more constraints. Thus enhancing existing

classical and non classical optimization approaches and exploring new single and hybrid

approached to solve unit commitment problem has great importance. Among the hybrid

approaches the Swarm Intelligence techniques are new to apply to the UC problem. PSO is new,

flexible and efficient tool for UCP. The potential avenues for further exploration may be listed

as:

1. How to generate initial feasible schedules considering spinning reserve requirements.

2. How to satisfy MUT and MDT constraints.

3. Exploration of new operators for MUT and MDT constraint handling.

4. To reduce the high dimensionality of the UCP.

5. Hybrid methodology is the useful tool for efficient solution by exploiting the strength of

single classical approaches, and non classical approaches.

6. Exploration of fast and efficient method for utility system.

7. Exploration of PSO based approaches.

8. Hybrid models based on the integration of classical and non classical approaches for

enhancing the computational efficiency and handling of non-convex cost function for the

UC problem.

28

CHAPTER 3

Optimization Tools for Unit Commitment

3.1 Introduction

Optimization is the process of making some thing better. Up till now several heuristic

tools have evolved that facilitates solving optimization problems that were previously difficult or

impossible to solve. These tools include particle swarm optimization, tabu search, artificial

neural networks, genetic algorithm, simulated annealing, etc.

This chapter presents the general overview of the optimization techniques on particle swarm

optimization (PSO), artificial neural networks (ANN), dynamic programming (DP) and genetic

algorithm (GA), with the view to their use in the subsequent chapters.

3.2 Particle swarm optimization (P.S.O)

Particle swarm optimization (PSO) first presented by Dr. Kennedy and Dr. Eberhart

[95,96] in 1995 is one of the evolutionary computational techniques based on the social

behaviours of bird flocking and fish schooling. This is a population based stochastic global

optimization technique. PSO has a population with random search solution. Each potential

solution is represented as a particle in population called swarm. Since its introduction it has

attracted lot of attentions from the researchers around the world. PSO models problem as a set of

n particles each representing a dimension of solution space. These particles move in solution

space in search of optimal solution. The particles follow three principles as described by

Kennedy [96] including evaluating, comparing: and Imitating. PSO been used to optimize real

and discrete functions which otherwise are difficult to solve. It can be easily implemented and

high quality solution with stable convergence.

The velocity and position update equations are given by

( ) ( ) ( ) ( )[ ] ( ) ( )[ ]txtytrctxtytrctwvtv ijjijijijij −+−+=+ ˆ)()(1 22,11 (3.1)

( ) ( ) ( )11 ++=+ tvtxtx ijijij i = 1, 2, 3 …s j= 1, 2, 3…n (3.2)

29

Where, xi,j : is the current position of the particle at iteration j, w : inertia weight, vi,j : is the

current velocity of the particle at iteration j, vi(t+1) : is the updated velocity of the particle, yi : is

the personal best position of the particle (every particle tries to adjust its velocity according to

best positions ever visited that is stored in its memory), and y^i is an instance of xi that is visited

by the particle and yielded best positions currently found. c1, c2 are positive numbers and

represent cognitive and social components respectively, c1, c2 controls the movement of the

particle, r1, r2 are the uniform distribution numbers in the range [0, 1], n be the dimension of the

optimization problem, t the current instant (iteration) and s is the swarm size.

The original formula developed by Kennedy and Eberhart was improved by Shi and

Eberhart [97, 98] with the introduction of an inertia parameter w to prevent premature

convergence and provides balance between local and global exploration. Although

experimentation with the inertia weight is still in progress, it appears that a good general

approach is to decrease the inertia weight linearly from 0.9 to 0.4 over 1,000 iterations. The

equation for w is as given below:

w = wmax – (wmax - wmin) x iter

itermax (3.3)

Where

wmax is the initial weight = 0.9

wmin is the final weight = 0.4

itermax is the maximum number of iterations

iter is the current number of iteration

Maurine Clerc [99] introduced a constriction factor χ that improves the ability of PSO and

developed the following update rule.

The complete PSO formula is:

( ) ( ) ( ) ( )( ) ( ) ( )( )( )txtytrctxtytrctvtv ijijijijijij −+−+=+ ˆ)()(1 2211χ (3.4)

Where

ϕϕϕχ

42

22 −−−

= (3.5)

30

4,21 >+= ϕϕ cc (3.6)

Bergh showed that a dangerous condition arises when a particle position approaches closer to

the global best position i.e. yyx ii ˆ== , the particle now will only depends on the term ( )twvij .

Bergh modified the equation of standard PSO by the guaranteed convergence PSO (GCPSO).

The new equation used is represented by the global best particle as follows:

( ) ( ) ( ) ( ) ( ) ( )( )trttwvtytxtv jjjj 21ˆ1 ,,, −+++−=+ ρτττ (3.7)

Where τ is the index of global best particle so that yy ˆ=τ . The term ( )1, +− tx jτ reset the

particle position so that the particle only depends on jy . The term ( ) ( )( )trt 21−ρ generates

random sample around jy for searching better value of jy . The position update equation for the

global best particle is given by

( ) ( ) ( ) ( ) ( )( )trttwvtytx jjj 21ˆ1 ,, −++=+ ρττ (3.8)

The value of ( )tρ is changed after each iteration using the following rule

( )( )( )

( )⎪⎩

⎪⎨

⎧>>

=+otherwiseif5.0if2

1t

fNtsNt

t cF

cS

ρρ

ρρ (3.9)

Where NS number of successes denotes the number of improvements in the y and NF denotes

the number of failures to improve the y . A single failure is given by ( ) ( )1ˆˆ −= tyty cs and cf are

upper threshold values. The following two rules are required for implementation: ( ) ( ) ( ) 011 =+⇒>+ tNtNtN FFS ( ) ( ) ( ) 011 =+⇒>+ tNtNtN SFF

The optimal choice for the values of cs and cf depends upon the objective function. In high

dimension search spaces it is difficult to obtain better values using random search in only a few

iterations, so it is recommended to set sc = 15, fc = 5.

3.3 Artificial Neural Networks (ANN)

Artificial Neural Networks (ANN) in general is massively interconnected network of a large

number of processing elements called neurons in an architecture, inspired by the brain. ANN

exhibits characteristics such as mapping or pattern recognition, generalization, and fault

tolerance. Neural network learn from examples and various learning laws exist of which

31

supervised and unsupervised are popular. For a particular application the neural network is

defined by its architecture and learning rule.

3.3.1 Feedforward Neural Network

Back propagation is a systematic method of training multilayer feedforward artificial

neural networks. It has been built on high mathematical foundation and has very good

application potential. However, it has limitations. BP algorithm works well on small set of data

and simple networks with few neurons. But when the problem under consideration has several

parameters and many hidden layers, it becomes very difficult for backward propagation to

minimize error. The ANN models using back propagation algorithm for training does not ensure

convergence and hangs in local optima and requires much longer training time. The neural

network can be trained based on swarm intelligence learning rule [100].

A neural network consists mainly of three layers: input layer, hidden layer and output

layer. These layers are arranged in some way to have a multilayer feed-forward structure. A

general network model consists of simple processing elements called neurons with adjustable

parameters called weights. These neurons are arranged in a distinct layered topology and perform

a biased weighted sum of their inputs and pass this activation level through a transfer relation

(sigmoid function) to produce their output. Thus, the parameters of data flow from the input

neurons, forwards through many hidden neurons, eventually reaching the output neurons. All of

the layers are fully interconnected with each other by weights as shown in Fig. 3.1.

Figure 3.1 Multilayer feedforward Neural Network Architecture

Neural network learn by examples without using any programmed rules. The ability to learn

through training is the most important feature of ANN. A supervised learning is given in Fig. 3.2.

32

A neural network uses a learning function to modify the variable connection weights at the

inputs of each processing element according to some neural based algorithm. Multiple layers of

neurons with nonlinear activation functions allow the network to learn nonlinear and linear

relationships between the input and output of the network. The training process requires a set of

examples of proper network behavior. The back propagation searches on the error surface by

means of gradient decent technique in order to minimize error. The performance function of the

neural network is normally chosen to be the mean squared error for each pattern on the training

set,

MSE = 1/ P ∑=

P

i 1(tPi – opi) 2 (3.10)

Where tpi is the target at ith pattern, opi is a network’s output at ith pattern and P is number of

neural network pattern.

Figure.3.2 supervised learning

3.4 Dynamic Programming (DP) or Recursive Optimization

Dynamic programming is a mathematical technique that is applicable to many types of

problem. Dynamic programming is a recursive optimization approach to solving sequential

decision problems. By recursive optimization procedure, we mean one that optimizes on a

step-by-step basis using information from the preceding steps. In short, we “optimize as we go.”

The dynamic programming approach involves the optimization of multistage decision processes.

That is, it basically divides a given problem into stages or sub-problems and then solves the sub-

problems sequentially until the invited problem is finally solved. In some mathematical

programming algorithms, optimization was also achieved on a step-by-step basis, but it was

33

iterative rather than recursive; that is, each step represented a unique solution that was non-

optimal. In dynamic programming, a single step is sequentially related to preceding steps and is

not itself a solution to the problem. A single step contains information that identifies a segment

of the optimal solution.

Because of these features, dynamic programming is most often applied to problems

requiring a sequence of interrelated decisions. Many time-dependent (dynamic) processes are

characterized by sequential decision problems that need to be solved; hence the term dynamic

programming (DP). Other applications particularly well-suited to dynamic programming

involve interrelationships rather than time dependencies, although time dependencies are a

common basis for expressing interrelationships among variables. A more appropriate term for

dynamic programming, therefore, might be recursive optimization.

The fundamental approach of dynamic programming involves (1) the breaking down of a

multistage problem into its subparts, steps, or single stages, a process called decomposition; (2)

making decisions one at a time, or recursively, at each stage, according to a specific optimization

objective; and (3) combining the results at each stage to solve the entire problem, a process

called composition. The act of composition results in a set of sequential decision rules called a

policy. For example, dynamic programming would optimize an n-decision variable problem by

decomposing it into a series of n stages (each decision variable a stage), assigning an optimal

value to each variable, and combining the results from each stage to generate the overall solution

to the problem.

At each stage, the decision rule is determined by evaluating a criterion or objective

function, called the recursive equation or functional equation (functional because it is a

function yielding a single real number).

3.4.1 Forward Dynamic Programming Approach

The dynamic programming algorithm can be run back ward in time starting from the final

hour to be studied back to the initial hour. Conversely, the algorithm can be run forward in time

from the initial hour to the final hour. Dynamic programming sub-divide the 24-h day into

34

discrete intervals [101]. The unit commitment procedure then searches for the most economic

feasible combination of generating units to serve the forecast load and spinning reserve

requirement of the system at each interval of the load cycle. In unit commitment problem the DP

is based on enumeration scheme and priority list methods.

For example, if we have four units, N = 4, then 15 possible combinations for each interval

are

Combinations or states (xi) Units x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 x14 x15

1 1 1 1 1 0 0 1 0 1 1 0 1 0 0 0 2 1 1 1 0 1 1 0 0 1 0 1 0 1 0 0 3 1 1 0 1 1 0 0 1 0 1 1 0 0 1 0 4 1 0 1 1 1 1 1 1 0 0 0 0 0 0 1

Where xi denotes combinations or states i of the four units. Of course, not all combinations

are feasible because of the constraints imposed by the load level and other practical operating

requirements of the system.

3.4.2 Mathematical Formulation of the Dynamic Programming for unit commitment

problem

Let us set

xi (h) = combination xi of interval h (3.11)

and then

xj (h + 1) represents combination xj of interval (h + 1) (3.12)

The production cost incurred in supplying power over any interval of the daily load cycle

depends on which combination of units is on-line during that interval. For a given combination

xi, the minimum production cost Pi equals the sum of the economic dispatch costs of the

individual units. Accordingly, we designate

Pi (h) = minimum production cost of combination xi (h) (3.13)

and then

Pj (h+ 1) is the minimum production cost of combination xj (h + 1) (3.14)

35

Besides production cost, in the unit commitment problem the transition cost, is also

considered which is the cost associated with changing from one combination of power-producing

units to another combination. Thus, transition cost associated with changing from one

combination of operating units to another will be denoted by

Tjj (h) = cost of transition from combination xi (h) to combination xj (h + 1) between

intervals h and (h + 1) (3.15)

Accordingly, the problem of minimizing costs at one stage is tied to the combinations of

units chosen for all the other stages, thus the unit commitment is a multistage or dynamic cost-

minimization problem.

The multistage decision process of the UCP can be dimensionally reduced by practical

constraints of system operations and by a search procedure based on the following observations:

• The daily schedule has h discrete time intervals or stages, the durations of which are not

necessarily equal. Stage 1 precedes stage 2, and so on to the final stage H.

• A decision must be made for each stage h regarding which particular combination of

units to operate during that stage. This is the stage h sub problem.

• To solve for the H decisions, h sub problems are solved sequentially in such a way that

the combined best decisions for the h sub problems yield the best overall solution for the

original problem.

This strategy greatly reduces the amount of computation to solve the original unit

commitment problem.

The cost fij (h) associated with any stage h has two components given by

fij (h) = Pi (h) + Tij (h) (3.16)

Which are the combined transition and production cost incurred by combination xi during

interval h plus the transition cost to combination xj of the next interval? The cost fij(h) is tied by

Tij (h) to the decision of the next hour (h + 1). The best policy or set of decisions (in the sense of

minimum cost) over the first (h - 1) hours of the daily load cycle; this is equivalent to assuming

36

that we know how to choose the best combination of units for each of the first (h — 1) intervals.

If we agree that combination xi* is the best combination for stage (h - 1), then by searching

among all the feasible combinations xi of the final stage h, we can find

Fi* (h – 1) = )}({

minhx j

{Pi*(h – 1) + Ti*j(h – 1) + Fi(h)} (3.17)

Where Fi* (h — 1) is the minimum cumulative cost of the final two stages starting with

combination xi* (h - 1) and ending with combination xj ((h); the cumulative cost Fj(h) of stage h

equals the production cost Pj (h) since there is no further transition cost involved.

Similarly, starting with the combination xi* (h — 2) at interval (h - 2), the minimum

cumulative cost of the final three stages of the study period is given by

Fi* (h – 2) = )}1({

min−hx j

{Pi* (h - 2) + Ti* (h – 2) + Fj (h – 1} (3.18)

Where the search is now made among the feasible combinations xj of stage (h — 1),

continuing the above logic, we find the recursive formula

Fi* (h) = )}1({

min+hx j

{Pi* (h) + Ti*j (h) + Fj (h + 1)} (3.19)

for the minimum cumulative cost at stage h, where h ranges from 1 to H.

At each stage in the dynamic programming solution of the unit commitment problem the

economic dispatch outputs of the available generating units must be calculated before the

evaluation of the production costs Pi*(h).

3.5 Genetic Algorithm (GA)

Genetic algorithms are optimization and search algorithms based on the mechanics of

natural selection, genetics and evolution. A GA is a random search procedure, which is based on

the survival of the fittest theory. Genetic Algorithms were first proposed by John Holland and

more recently reviewed and enhanced by one of his student David Gold berg [102] and many

37

others. Their basic principle is the maintenance of a population of solutions to a problem in the

form of encoded individuals that evolve in time.

GAs manipulates strings of binary digits (‘1’ and ‘0’) called chromosomes, which represents

multiple points in the search space. Each bit in a string is called an allele. They carry out

simulated evolution on populations of chromosomes. Genetic algorithms, using simple

manipulations of chromosomes such as simple encoding and reproduction mechanisms, can

display complicated behavior and solve some extremely difficult problems without knowledge of

the decoded world.

The mechanics of a simple genetic algorithm is based on three operators’ reproduction,

crossover and mutation.

a. Reproduction: In this process individual strings are copied according to their fitness

value. This process is conducted by spinning a simulated biased roulette

wheel, which is called roulette wheel parent selection.

b. Crossover: The main operator working on the parents is crossover, which happens for

a selected pair with a crossover probability. At first, two strings from the

reproduced population are mated at random and a crossover point is

selected randomly. At the crossover point, the partial strings are

interchanged between the selected pair to produce two new strings.

c. Mutation: Although first two operators produce many new strings, they do not

introduce any new information in to the population at the bit level. As a

source of new bits, mutation is introduced and is applied with a low

probability. It inverts a randomly chosen bit on a string.

These three operators are applied repeatedly until the offspring’s take over the entire

population. When new solution of strings is produced, they are considered as a new generation

and they totally replace the ‘parents’ in order for the evolution to proceed. It is necessary to

produce many generations for the population converging to the near optimum or an optimum

solution, the number increasing according to the problem complexity.

38

This approach consists of the following steps:

Step1: Read system data and select GA parameters (population size, probability of cross

over, probability of mutation).

Step 2: Randomly generate a population of chromosomes, each consisting of bits ‘0’ and

‘1’. Which is a two dimensional array (H x N) Where H = Load duration in hours,

N = Number of units.

Step 3: Obtain a population of P strings of feasible solutions or chromosomes of

population that could satisfy the load demand and spinning reserve.

Step 4: Get the row value and check for the feasibility of the solution corresponding to

the satisfaction of the equality constraint (Generation ≥ Load + spinning reserve)

for each hour of the load duration ‘H’.

Step 5: Decode chromosomes and perform economic dispatch prior to genetic operation.

Step 6: Evaluate the fitness function (objective function).

Step 7: If terminating criteria satisfied GO TO Step 10.

Step 8: else

{Perform Genetic Operators on the population of the strings i.e. Selection,

Crossover and Mutation. Make a new population}

Step 9: GO TO Step 4.

Step 10: Exit

Step 11: Print the best array, the economic loading and the operating fuel cost for each

load.

39

CHAPTER 4

Unit Commitment --- Problem Formulation and Single

Solution Approaches

4.1 Introduction

The Unit Commitment (UC) is an important research challenge and vital optimization

task in the daily operational planning of modern power systems due to its combinatorial nature.

Because the total load of the power system varies throughout the day and reaches a different

peak value from one day to another, the electric utility has to decide in advance which generators

to start up and when to connect them to the network and the sequence in which the operating

units should be shut down and for how long. The computational procedure for making such

decisions is called unit commitment, and a unit when scheduled for connection to the system is

said to be committed. In this work the commitment of fossil-fuel units has been considered which

have different production costs because of their dissimilar efficiencies, designs, and fuel types.

Unit commitment plans for the best set of units to be available to supply the predict forecast load

of the system over a future time period.

In general, the UC problem may be formulated as a non-linear, large scale, mixed-integer

combinatorial optimization problem with both binary (unit status variable) and continuous (unit

output power) variables. This chapter presents the characteristics of power generation unit, unit

commitment problem formulation, modeling aspects of single approaches to solve UCP for

convex and non convex fuel cost function. The remaining discussions in this chapter focus on

algorithm development and their implementation, and case studies.

4.2 Characteristics of Power Generation Units

In analyzing the problems associated with the operation of power system, there are many

possible parameters of interest. Fundamental is the basic cost data and set of input-output

40

characteristics of generation units. Different types of fuel are used in thermal power plants.

Depending on the types of turbine such as single value or multi value, the characteristic differs.

Although the operating cost of these units consists of both fuel and maintenance costs,

only the fuel cost varies directly with the units and also with the level of generation. The fuel

cost is incurred during the running (either at no-load or at any load), start-up and sometimes

shutdown conditions of the steam units.

4.2.1 Unit’s Input-Output characteristic (Heat or Cost)

Unit (Boiler, turbine and generator) input-output curve establishes the relationship

between energy input to the driving system and the net energy output from the generator. A

typical boiler-turbine-generator unit is represented in Figure 4.1.

Figure 4.1 Boiler -Turbine -Generators Unit

In this characteristic the gross input (Rs. / h or tons of coal/h or millions of cubic feet of

gas/h or any other unit) being measured in millions of B.T.U. per hour (MBTU/h) is plotted

against the output in MW of the unit. The input is taken along y-axis. The output is normally the

net electrical output of the unit and is taken along x-axis. Z-axis represent the time axis, on

which usually one hour is taken to convert the output power P in MW to energy in MWh in order

to evaluate the per unit cost of input i.e., Plant is loaded at P (MW) for one hour, then input is

measured in Rs. / h or MBTU/h. (Z-axis can be omitted as each point loading pertains one hour).

B T

G

A/P

Boiler fuel input Steam Turbine

Generator

(Gross)

(Net)

Auxiliary Power supply

41

The input-output characteristics of a steam unit in idealized form are represented in Figure

4.2.

Figure 4.2 Input-output Curve of a Steam Turbine Generator

For a single value turbine the governing is done by throttling of steam and for such units,

the input-output curve is substantially a straight line within its operating range.

4.2.2 Non convex fuel cost characteristic due to valve point effect

Non convex characteristic results due to valve point effect, multiple fuels and prohibited

operating zones. The valve point effects produce a ripple, which is highly non-smooth and

discontinuous as represented in Figure 4.3.

Figure 4.3 Input-output curve of a multi valve steam turbine generator with four steam

admission valves

A= primary valve B= secondary valve C= Tertiary valve D=Quaternary valve

E= Quinary valve

4.2.3 Incremental heat or cost characteristic

The incremental heat rate characteristic is the derivative of the input–output characteristic

(∆H/∆P or ∆F/∆P). This characteristic is widely used in economic dispatching of units. It is

converted to an incremental fuel cost characteristic by multiplying the incremental heat rate. The

42

incremental heat rate characteristics for single and multi value units are represented in Figures

4.3 and 4.4 respectively. The incremental heat rate characteristic of multi valve steam turbine is

discontinuous type.

Figure 4.4 Incremental Heat Rate or Cost characteristic.

Figure 4.5 Incremental Heat Rate Characteristics of a steam turbine with four valves.

4.2.4 Unit Heat rate (HR) characteristic

The heat rate curve is obtained from the unit input-output curve by dividing the input by the

corresponding output (H/P) at any loading condition versus the megawatt output of the unit. This

characteristic is plotted between H/P versus P. While incremental heat rate is given by the ratio

of the change in input (∆H) to the corresponding change in output (∆P) at any operating point.

Heat rate (HR) = MW in put

MBTU/hin input Out

(4.1)

The units of Hear rate are KWhBtuorMWhorMBtuMW

hRs /,,//.

and

43

Incremental heat rate = ∆Input = ∆H or ∆F ∆Output ∆P ∆P (4.2)

This is an important characteristic and defines the average heat rate per KWH of output. The

incremental efficiency, which is the ratio of the change in output to the corresponding change in

input at any loading condition, is clearly the reciprocal of the incremental heat rate. Thus lower is

the incremental heat rate, higher is the incremental efficiency. Since H/P = 1/η, therefore this

characteristic if the reciprocal of the usual efficiency characteristic developed for a machine.

Both these quantities have the same unit which is B.T.U. per KWh. The heat rate and

incremental heat rate curves are represented in Figure 4.6.

Figure 4.6 Heat rate and incremental heat rate curves for convex cost function

The heat rate and incremental heat rate can be converted into fuel cost function and incremental

fuel cost by multiplying them with the cost of the fuel (Dollars per million of B.T.U.).

Fi(Pi) = Hi(Pi) x fuel cost $/h (4.3)

Incremental cost = λdPdF

i

T = $/MWh (4.4)

4.3 Unit Commitment Problem (UCP)

UCP was defined as preparing on/off schedule of generating units in order to minimize the

total production cost of utility and constraints such as system power balance, system spinning

reserve, and unit’s minimum up and down times are satisfied. Figure 4.7 represents the

configuration that represent UC problem with on/off switches. The unit commitment problem is

discussed as follows:

44

Figure 4.7 Unit Commitment

4.3.1 Objective Function

The principal objective is to prepare on/off schedule of the generating units in every sub-

period (typically 1h) of the given planning period (typically 1 day or 1 week) in order to serve

the load demand and spinning reserve at minimum total production cost (fuel cost, start up cost,

shut down cost), while meeting all unit, and system constraints. The following costs are

considered.

4.3.1.1 Fuel Cost

The quadratic approximation is the most widely used by the researchers, which is

basically a convex shaped function. The curve shown in Fig. 4.1 is the operating fuel cost

equation for unit i and is mathematically represented as:

Fi(Pih) = ∑=

N

i 1

[ ai + bi Pih + ci Pih

2 ] (4.5)

(Units without valve point effects)

To take the effects of valve points as shown in Fig. 4.6 a sinusoidal function is added to the

convex cost function and represented as:

Fi(Pih) = ∑=

N

i 1[ ai + bi Pih

+ ci Pih2 + | ei sin fi ( Pih min - Pih )|] (4.6)

(Units with valve point effects)

4.3.1.2 Start up cost

Start up cost is warmth-dependent. Mathematically it is represented as a step function:

45

STih = h-costi: Ti down ≤Xi off (h) ≤ Ti down + c-s-houri $/h (4.7)

c-costi: Xi off (h) > Ti down + c-s-houri $/h (4.8)

4.3.1.3 Shut down cost

The typical value of the shut down cost is zero in the standard systems. This cost is

considered as a fixed cost.

SDih = KPih $/h (4.9)

Where K is the incremental shut-down cost

4.4 Constraints

The UCP is subjected to many constraints that include:

• The total power generated must meet the load demand.

• There must be enough spinning reserve to cover any shortfalls in generation.

• The loading of each unit must be within its minimum and maximum allowable rating

(limits).

• The minimum up and down times of each unit must be observed.

• Unit availability constraint is either unit is available / not available, out aged/Must out,

Must run, and Fixed Output Power (F.O.P).

• Unit initial status +/- either already up or already down.

The constraints which are taken into consideration in this work may be classified into two

groups: system constraints and unit constraints.

4.4.1 System constraints or coupling constraints

Constraints that concern all the units of the system are called system or coupling constraints.

These constraints have two categories: the system power balance and system spinning reserve

constraints.

(i) System Power balance or load constraint:

The system power balance constraint is the most important constraint in the UCP. The

generated power from all committed units must be equal to the load demand. This is formulated

in the balance equation as:

46

∑=

N

i 1Uih(Pih) = Dh h = 1,2,…….H (4.10)

(ii) System Spinning reserve requirements

In this work spinning reserve is computed as an amount which is a percentage of the

forecasted load demand and is represented as:

∑=

N

i 1

Uih(Pimax) ≥ Dh + SRh h = 1,2,…H (4.11)

4.4.2 Unit constraints or local constraints

Constraints that concern individual units are called unit constraints or local constraints are

described as follow:

(i) Units minimum and maximum generation limits

The generation limits represent the minimum loading limit below which it is not economical

to load the unit, and the maximum loading limit above which the unit should not be loaded. Each

unit has generation range, which is represented as:

Uih Pimin ≤ Pih ≤ Pimax Uih (4.12)

for i = 1,2,…..N, h = 1,2,….,H

(ii) Minimum up and down time limits

Once the unit is running, it should not be turned off immediately. Once the unit is de-committed, there is a minimum time before it can be recommitted. These constraints can be represented as:

Xi on (h) ≥ Tiup

Xi off (h) ≥ Tidown (4.13)

For i = 1, 2, …., N. h = 1, 2, … , H.

(iii) Unit availability constraints.

The availability constraint specifies the unit to be in one of the following different

situations: Available/ not available, out aged/Must out, Must run, Fixed Output (F.O.P).

Must run

47

Some units are given a must run status during certain times of the year for reasons of voltage

support on the transmission network or for such purposes as supply of steam for uses outside the

steam plant itself, and to increase the reliability or stability of the system.

Must out units:

Units which are on forced outages and maintenance are unavailable for commitment and these

are the must out units.

Units on fixed generation: (F.O.P)

These are the units which have been prescheduled and have their generation specified for certain

time period. A unit of fixed generation is automatically a must run unit for the designated time

period.

(iv) Unit initial status

The initial status value, if it is positive indicates the number of hours the unit is already up,

and if it is negative indicates the number of hours the unit has been already down. The status of

the unit +/- before the first hour in the schedule is an important factor to determine whether its

new state violates the MUT/MDT constraints. The initial status also affects the start up cost

calculations.

(v) Unit derating constraint

During the life time of a unit its performance could be change due to aging and cause

derating of the unit. Therefore, the unit minimum and maximum limits are changed.

4.5 Unit Commitment mathematical formulation as an optimization problem

The objective function of the unit commitment problem is to minimize the total

production cost and is mathematically formulated as:

Objective function

Minimize TPC = ∑=

H

h 1∑=

N

i 1[ Fi (Pih) + STih + SDih ] $/h (4.14)

Subject to:

• The system constraints (4.10, 4.11) and unit constraints (4.12, 4.13).

The UCP can be considered as two linked optimization problems: the Unit Scheduling Problem

(Allocation of Generators) and the Economic dispatch problem (Allocation of Generation) and

is represented in Figure 4.8. The unit schedule problem is the on/off or 0/1 combinatorial

optimization problem. A feasible unit schedule must satisfy the forecasted load demand, system

48

spinning reserve requirements, and the constraints on the start up and shut down times during

each planning period. The economic dispatch problem is the constrained non-linear optimization

problem. The economic load dispatch is to allocate the generation requirement among the

available units so that the total cost of energy supplied to meet the load demand within

recognized constraints is minimized on minute to minute basis.

49

Figure 4.8 Representation of Unit Commitment Problem (UCP)

50

4.6 Generation of initial feasible unit commitment schedules

The generation of initial feasible solution is much important, for the Unit Commitment

problem. When initial feasible schedules (generation > load + spinning reserve) are generated

randomly, it is difficult to get feasible schedules for 24 hours loads considering MUT and MDT

constraints It takes a very long time. These randomly generated solutions are also far from the

optimal solution. The convergence is slow and likely to get trapped in the local minimum during

the exploration of the solution space.

4.6.1 Initial unit commitment scheduling by using priority List method and focusing on

peak and off-peak loads of the daily load curve.

In this work initial solution is generated using the priority list method. The priority list

method is very fast and efficient method but the solution obtained cannot fulfill all the

constraints particularly the MUT and MDT constraints. Graphical representation is given in

Figure 4.9. Generally more generators are started up at around the peak load, and few units are

started up at light loads based on full load average production cost. To satisfy minimum up time

constraint the units are set continuously ON.

The full load average production cost (FLAPC) is the Heat Rate (HR) multiplied by the fuel

cost Fi(Pi).

Mathematically it is represented as:

(FLAPC) = HRi x Fi(Pi) = Fi(Pimax)/ (Pimax) (4.15)

(MBTU / MWh x $/ MBTU) = ($/MWh)

= ( ai + bi Pimax + ci Pimax

2 ) / (Pimax) (4.16)

Start up of the base units

In power system some units have “must run” status. These are base units having large output

power. Two units located at the bottom of the priority list as base load units, the units have “ON”

fixed status.

51

Unit10 ▓

Unit9 ▓ ▓

Unit8 ▓ ▓ ▓ ▓ ▓

Unit7 ▓ ▓ ▓ ▓ ▓ ▓ ▓ ▓

Unit6 ▓ ▓ ▓ ▓ ▓ ▓ ▓ ▓

Unit3 ▓ ▓ ▓ ▓ ▓ ▓ ▓ ▓ ▓ ▓ ▓ ▓ ▓ ▓ ▓

Unit4 ▓ ▓ ▓ ▓ ▓ ▓ ▓ ▓ ▓ ▓ ▓ ▓ ▓ ▓ ▓ ▓ ▓ ▓ ▓

Unit5 ▓ ▓ ▓ ▓ ▓ ▓ ▓ ▓ ▓ ▓ ▓ ▓ ▓ ▓ ▓ ▓ ▓ ▓ ▓ ▓ ▓

Unit2 ▓ ▓ ▓ ▓ ▓ ▓ ▓ ▓ ▓ ▓ ▓ ▓ ▓ ▓ ▓ ▓ ▓ ▓ ▓ ▓ ▓ ▓ ▓ ▓

Unit1 ▓ ▓ ▓ ▓ ▓ ▓ ▓ ▓ ▓ ▓ ▓ ▓ ▓ ▓ ▓ ▓ ▓ ▓ ▓ ▓ ▓ ▓ ▓ ▓

Priority List ↑

1 2 3 4 5 6 7 8 9 10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

Figure 4.9 Generation of initial solution by priority list method a graphical representation

4.6.2 Generation of trial solutions / neighbors

At first step in solving the combinatorial optimization problems is to have good

neighbors/trial solutions from an existing solution. More solutions were obtained taking upper

four units in the priority list at every time interval as shown in Figure 4.10. Introduction of these

feasible solutions makes the search closer to the optimum, leading to a faster convergence and

better results.

Unit10 ▓ ▓ ▓ ▓ ▓ ▓ ▓ ▓ Unit9 ▓ ▓ ▓ ▓ ▓ ▓ ▓ ▓ ▓ ▓ ▓ ▓ ▓ ▓ ▓ Unit8 ▓ ▓ ▓ ▓ ▓ ▓ ▓ ▓ ▓ ▓ ▓ ▓ ▓ ▓ ▓ ▓ ▓ ▓ ▓ Unit7 ▓ ▓ ▓ ▓ ▓ ▓ ▓ ▓ ▓ ▓ ▓ ▓ ▓ ▓ ▓ ▓ ▓ ▓ ▓ ▓ ▓ � Unit6 ▓ ▓ ▓ ▓ ▓ ▓ ▓ ▓ ▓ ▓ ▓ ▓ ▓ ▓ ▓ ▓ ▓ ▓ ▓ ▓ ▓ ▓ ▓ ▓ Unit3 ▓ ▓ ▓ ▓ ▓ ▓ ▓ ▓ ▓ ▓ ▓ ▓ ▓ ▓ ▓ ▓ ▓ ▓ ▓ ▓ ▓ ▓ ▓ ▓ Unit4 ▓ ▓ ▓ ▓ ▓ ▓ ▓ ▓ ▓ ▓ ▓ ▓ ▓ ▓ ▓ ▓ ▓ ▓ ▓ ▓ ▓ ▓ ▓ ▓ Unit5 ▓ ▓ ▓ ▓ ▓ ▓ ▓ ▓ ▓ ▓ ▓ ▓ ▓ ▓ ▓ ▓ ▓ ▓ ▓ ▓ ▓ ▓ ▓ ▓ Unit2 ▓ ▓ ▓ ▓ ▓ ▓ ▓ ▓ ▓ ▓ ▓ ▓ ▓ ▓ ▓ ▓ ▓ ▓ ▓ ▓ ▓ ▓ ▓ ▓

Unit1 ▓ ▓ ▓ ▓ ▓ ▓ ▓ ▓ ▓ ▓ ▓ ▓ ▓ ▓ ▓ ▓ ▓ ▓ ▓ ▓ ▓ ▓ ▓ ▓

Priority List ↑

1 2 3 4 5 6 7 8 9 10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

▓ either on/off sate ▓ On state Off state Figure 4.10 Generation of new schedule by taking upper 4 units

4.7 Minimum up and minimum down Time Constraint Handling

During the optimization process of unit commitment schedules the MUT and MDT

constraints may be violated. They will be checked and repaired if violation occurs.

52

4.8 Minimum up and down time constraint repairing by using bit change operator

A bit change operator is used to modify the bit positions. This operator overcomes the

problems of the minimum up/down time constraint violation. The operator looks at the past

states, the future states and the present state itself of all units to decide whether or not the unit’s

status for the present hour should be flipped or not. The units with small minimum up and down

times have more changes in their status, while the units with large minimum up and down times

will require less change. A simple way to achieve this is by the categorization of units as base

load, sub-base load, peaking units, must run, and can run. An example of repairing the minimum

up and down time is represented in Figures 4.11 and 4.12 respectively.

Units/Hr t-1 t t+1 i 0 0 1 1 0 0 0 j 0 0 1 1 1 0 0

Figure 4.11 Repairing of minimum up time

Units/Hr

t-1 t t+1 i 1 1 0 0 1 1 1 j 1 1 1 1 1 1 1

Figure 4.12 Repairing of minimum down time

4.9 Algorithm for the construction of initial unit commitment schedule and M.U.T and

M.D.T constraint handling

Step 1: Generate a feasible unit commitment schedule satisfying load demand and spinning

reserve using priority list method. Which is a matrix (H x N).

Step 2: Get the row values of the matrix and calculate the total generation in each scheduling

hour.

Step 3: Check that the generated power is greater than load demand plus spinning reserve?

Step 4: Get the column values of the matrix and calculate units start up and shut down times.

(Xi on h, Xi off h)

Step 5: Xi on h ≥ Tiup and Xi off h ≥ Ti

down if NO repair minimum up and down time violations

using bit change operator and modify start up cost else go to step 6.

53

Step 6: Get the row values of the matrix and calculate power output of each unit, operating fuel

cost of each unit, total operating fuel cost of each row, start up cost and total production

cost of each row for each hour.

Step 7: The final UC solution is one having the lowest total production cost.

4.10 Unit commitment schedule and determination of number of units to be operated

Consider a system having a forecasted load as shown in Fig. 4.9 Assuming that 10 units

are available to carry the load.

.

Figure 4.13 Forecasted load curve

An initial unit commitment schedule may be as represented in Figure 4.14.

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 units Priority 10 Unit10 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 9 Unit9 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 8 Unit8 0 0 0 0 0 0 0 0 0 1 1 1 1 0 0 0 0 0 0 1 0 0 0 0 7 Unit7 0 0 0 0 0 0 0 0 1 1 1 1 1 0 0 0 0 0 1 1 0 0 0 0 6 Unit6 0 0 0 0 0 0 0 0 1 1 1 1 1 0 0 0 0 0 1 1 0 0 0 0 5 Unit3 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 4 Unit4 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 3 Unit5 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 2 Unit2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 Unit1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

1 on state 0 off state Figure 4.14 Representation of initial unit commitment schedule

Withdrawal of the unit contributes to the saving of its running cost during the reduced load

period of the system, but either start up cost or shut down cost is incurred when the unit is

restored to service. If these costs are less than the spinning cost, the withdrawal is economically

justified. In fact, other schedules can also be considered, but before the shut down of any unit,

the cost of its continued operation should be weighed against the shut down or start up cost. The

54

problem is thus based on the evaluation of the total production cost in supplying the forecasted

load with a specified spinning reserve with various schedules of commitment. 4.11 Economic dispatch Problem (EDP, Allocation of Generation)

The economic dispatch problem can be classified as convex and non convex. Most of the

researches used convex economic dispatch the non-convex economic dispatch may result due to

valve point effect.

4.11.1 Economic load dispatch (ELD) calculations

In the unit commitment problem, the economic dispatch calculations consume a large

amount of calculation time. In this work, the ELD calculation is performed only for feasible

solutions by gradient method, merit order method, load assigned by operational engineer and

genetic algorithm.

4.11.1.1 Equal Incremental Cost Criterion

Let there be N thermal generating units connected to a bus-bar supplying a load demand of

Pload. It is required to load the units so that the cost of operating fuel in minimum. This problem

may be solved by using “Equal Incremental Cost Criterion”. Working philosophy of their

criterion is as: “When the incremental costs of all the machines are equal, and then cost of

generation would be minimum subject to equality constraints”.

The economic dispatch problem mathematically may be defined as:

Minimize: ( )∑=

=N

1iiiT PFF (4.17)

Subject to: equality constraint ∑=

−=N

1iiD PPΦ (4.18)

And inequality constraint ( )

( ) ⎪⎭

⎪⎬⎫

≤−

≤−

0PP0PP

imin.i

max.ii (4.19)

for i =1,2,…,N.

Where

FT = F1 + F2 + ……FN is the total fuel input to the system

Fi = Fuel input to ith unit

55

Pi = the real power generation of ith unit

However, in this analysis method of Lagrange multiplier will be used. The working

philosophy of this method is that constrained problem can be converted into an unconstrained

problem by forming the Lagrange, or augmented function. Optimum is obtained by using

necessary conditions.

⎟⎠

⎞⎜⎝

⎛−+=

+=

∑=

N

1iiDT

T

PPλ.F

λ.ΦFL (4.20)

The necessary conditions for constrained local minima of L are the following:

0PL

i

=∂∂ (4.21)

0λL=

∂∂

(4.22)

Condition-I

First condition gives

( ) 010λ.PF

PL

i

T

i

=−+∂∂

=∂∂

or

λPF

0λPF

i

T

i

T =∂∂

⇒=−∂∂

Q N21T FFFF −−−−−−−++=

Then

λdPdF

PF

i

T

i

T ==∂∂

And therefore the condition for optimum dispatch is

56

λdpdF

i

T = (4.23)

or

λP2cb iii =+ (4.24)

Where 2iiiiiT PcPbaF ++=

Condition-II

Second condition results in

∑=

=−=∂∂ N

1iiD 0PP

λL

or

∑=

=N

1iDi PP (4.25)

Condition for economic operation

“For most economical operation, all plants must operate at equal incremental production cost

while satisfying the equality constraint given by equation (4.25).”

i

ii 2c

bλP

−= (4.26)

The relations given by equations (4.26) are known as the co-ordination equations. They are

function of λ. An analytical solution for λ is given by substituting the value of Pi in equation

(4.25), i.e.

D

N

1i i

i P2c

bλ=

−∑=

(4.27)

∑

∑

=

=

+= N

1i i

N

1i i

iD

2c1

2cbP

λ (4.28)

Optimal schedule of generation is obtained by substituting the value of λ from equation (4.28)

into equation (4.26).

57

Iterative Method of economic dispatch

The equation (4.27) is a function of λ and can be expressed as: ( ) DPλf = (4.29)

Expanding by Taylor’s series about an operating point λ (k) and neglecting higher order terms

results in

( )( ) ( ) ( )D

kk

k P∆λdλλdfλf =⎟⎠⎞

⎜⎝⎛+ (4.30)

( ) ( )( )

( ) ( )k

kDk

dλλdfλfP∆λ

⎟⎠⎞

⎜⎝⎛

−=

( )( )

( ) ( )k

kk

dλλdf

∆P∆λ

⎟⎠⎞

⎜⎝⎛

=

( )( )

( )k

kk

dλdPi∆P∆λ

⎟⎠⎞

⎜⎝⎛

= (4.31)

or

( )k∆λ( )

∑=

i

k

2c1

∆P (4.32)

and therefore ( ) ( ) ( )kk1k ∆λλλ +=+ (4.33)

Where

( ) ( )∑=

−=N

1i

kiD

k PP∆P (4.34)

The process is continued until ∆P (k) is less than a pre-specified accuracy.

4.11.1.2 Loading to most efficient load

Although the criterion of equal incremental production costs will result in the optimum

economic scheduling of generation. The above method is still in use by utilities. In this method

units are loaded in ascending order of their heat rates, to their most efficient loads, based on the

58

forecasted load at that hour. The most economical unit is loaded first and the remaining load is

shifted on the next unit in the priority list. It the remaining load is less the minimum limit of that

unit than the unit is loaded up to its minimum power output and the remaining loaded is shifted

on the previous unit. This is very quick method and gives near optimal solution in very short

time.

4.11.1.3 Economic Dispatch using Genetic Algorithm (Real Power – Search)

GA works better on non convex fuel cost function. The pseudo code for Real Power-

search method is shown in Figure 4.15.

59

Figure 4.15 Pseudo Code for Genetic Algorithm Real Power-Search Method

4.12 Economic Dispatch versus Unit Commitment

The unit commitment assumes that there are N units available to meet the forecasted load

demand, satisfying spinning reserve and MUT and MDT constraints. The economic dispatch

load the units economically within their limits satisfying system and unit constraints

60

4.13 Conventional/Classical Single Approaches for convex fuel cost function

The single approaches used to solve the UCP are, Complete Enumeration, Merit Order based

on Full Load Average Production Cost, and Merit order based on maximum power output of

each unit.

The algorithm of the classical single approaches is as follows:

Step 1: Read in system data ai, bi, ci, load demand (PD), Pmin and Pmax of each unit.

Step 2: Calculate the full load average production cost (FLAPC) cost of each unit.

Step 3-a: Generate all combinations (2N - 1). Check for the feasible combination

corresponding to the satisfaction of the equality constraint (Generation ≥ Load

demand plus spinning reserve) for each hour of the forecasted load.

b. Generate initial schedule based on Full Load Average Production Cost (FLAPC).

c. Generate initial schedule based on Maximum Power Output of each unit (PMAX).

Step 4: Calculate systems lambda using equation 4.28.

Step 5: Calculate economic loading of machines for each feasible combination (based on

equal incremental cost criteria).

Step 6: Check that the Power output of each unit is within minimum and maximum

generating limits of machines.

Step 7: Is there any violation of minimum or maximum power limit (yes-clamp at

minimum or maximum limit), (No - go to step 9).

Step 8: Recalculate the system lambda and output power of each machine using equations

4.32, 4.33 and 4.34.

Step 9: Satisfying the power balance equation (Generation = Load).

Step 10: Print the unit commitment schedule, power output of each machine, operating fuel

cost of each machine, and total production cost and system lambda for each

feasible schedule.

Step 11: Print the best schedule.

61

4.13.1 Single Approach – I --- Complete Enumeration

This method takes all the combinations (2N – 1) H and than calculated the economic

dispatch of each unit. Where N = number of units, H = number of hours (24). For a total period

of H interval, the maximum number of possible combinations is (2N-1) H.

For example, take a 24-h period (e.g. 24 one hour intervals) and consider with 5, 10, 20

and 40 units. The value of (2N-1) H becomes the following.

N (2N-1)H

5 6.25 x (10)35

10 1.73 x (10)72

20 3.12 x (10)144

40 Too Big

There very large numbers are the upper bounds for the numbers of enumerations

required. The constraints on the units and the loading capacity of the units limit the search space.

Never the less, the real practical barrier in the UCP is the high dimensionality of the possible

search space.

4.13.2 Single Approach – II --- Conventional Priority List

This approach is based on conventional priority list method. A priority order is created

based on the Full Load Average Production Cost (FLAPC). The UC schedule is based on

FLAPC and ED is based on Lambda Iteration Method.

In this method, units are committed to service by observing their heat rate values. Units with

the lowest heat rate are put into operation first. For shutting down the reserve order is followed,

i.e. the units with the highest heat rate is withdrawn first. The load dispatcher takes into account

the hourly forecasted load and spinning reserve requirement, and then schedules the units to

match the load and spinning reserve. The industry, however, still mostly used the simple "merit

order" method.

62

4.13.3 Single Approach – III --- Proposed Single Approach

This approach is also based on priority list method. But the priority list in this case is

based on the Maximum Power (PMAX) limit of each unit. The UC schedule is based on PMAX

and ED is based on Lambda Iteration Method.

4.14 Case studies --- Convex cost function

Following two standard test systems have been selected for the validation of the single

approaches.

a. Test System I --- 3 units with 24 hours load.

b. Test System III --- 10 units system with 24 hours load

Input Data: The description and input data of test systems used for investigation in the case

studies are given in Tables A.1 and A.3 placed in Appendix-A.

Computer Implementation: All the three single approaches have been implemented in C++ on

P-IV Personal Computer.

Output Results: The summary and comparison of results is given in this chapter.

4.14.1 Numerical Results of test system – I

The priority order based on conventional and proposed method is given in Table 4.1.

Table 4.1 Priority order based on single approach-II and III: test system-I

Unit No.

PMAX FLAPC (S/MWH)

Single approach-II Conventional Priority

order (FLAPC)

Single approach-III Proposed Priority order

(PMAX)

1 600 9.7922 2 1

2 400 9.4010 1 2

3 200 11.1888 3 3

The output results of test system – I are shown in the following tables:

Table 4.2 presents the comparison of operating fuel cost ($),

Table 4.3 presents the comparison of Summary of Unit Commitment Schedules for single

approaches,

Table 4.4 presents the Number of Units in Operation for single approaches,

63

Table 4.5 presents the Comparison of single approaches with Genetic Algorithm, and Hopfield

Neural Network methods.

Tables 4.6 to 4.8 presents the best solutions obtained by the single approach-I, II and III.

The Salient features of the conventional approaches in the light of the observations from the

results are as follows:

1. Enumeration method gives good results. But the number of transitions is 10. In test

system III the number of transitions is 4.

2. The operating fuel cost obtained by all the three single approaches remains low compared

to genetic algorithm and Hopfield neural network methods.

3. Single approach-I gives $606.28 saving per day compared with GA and $ 691.28 saving

per day compared with HFNN.

4. Single approach-II gives $381.15 saving per day compared with GA and $ 466.15 saving


5. Single approach-III gives $389.28 saving per day compared with GA and $ 474.28 saving


All the three approaches are simple, fast and give fair amount of reduction in operating cost as

compared to GA and HFNN.

Table 4.2 Comparison of the operating fuel cost ($) for 3 Unit Systems :( Test System I)

200800201000201200201400201600201800202000202200202400202600

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Table 4.3 Summary of Unit Commitment Schedules for 3 Unit Systems (Test System I)

Single approach- I

Single approach- II

Proposed Single approach-III

Load (MW)

Complete Enumeration

Merit Order (FLAPC) with ED

UC(2,1,3)

Merit Order (PMAX) with ED

UC(1,2,3) 1200 1200 1150 1100 1000 900 800 600 550 500 500 500 500 500 600 800 850 900 950 1000 1050 1100 1200 1200

111 111 111 111 110 110 110 100 100 011 011 011 011 011 100 110 110 110 110 110 111 111 111 111

111 111 111 111 110 110 110 110 110 110 110 110 110 110 110 110 110 110 110 110 111 111 111 111

111 111 111 111 110 110 110 100 100 100 100 100 100 100 100 110 110 110 110 110 111 111 111 111

201415.789 202393.346 201632.089 No. of Transitions No. of Transitions No. of Transitions 10 2 4

65

Table 4.4 Number of Units in Operation for 3 Unit Systems (Test System I)

Single approach-I Single approach-II Proposed Single

approach-III


Merit Order (FLAPC) with ED

UC(2,1,3)

Merit Order (PMAX) with ED

UC(1,2,3)

Hr. Load (MW)

No. of units No. of units No. of units

1 1200 3 3 3 2 1200 3 3 3 3 1150 3 3 3 4 1100 3 3 3 5 1000 2 2 2 6 900 2 2 2 7 800 2 2 2 8 600 1 2 1 9 550 1 2 1 10 500 2 2 1 11 500 2 2 1 12 500 2 2 1 13 500 2 2 1 14 500 2 2 1 15 600 1 2 1 16 800 2 2 2 17 850 2 2 2 18 900 2 2 2 19 950 2 2 2 20 1000 2 2 2 21 1050 3 3 3 22 1100 3 3 3 23 1200 3 3 3 24 1200 3 3 3

66

Table 4.5 Comparison of the Three Single approaches with Genetic Algorithm and Hopfield Neural Network Methods for 3 unit systems: Test system-I

Algorithm Daily

Operating Fuel Cost

($)

Amount of Daily Saving

(compared with GA)

% saving in fuel cost compared

with GA

Amount of Daily

Saving (compared

with Hopfield Neural

Network

% saving in fuel

cost compared

with Hopfield Neural

Network(HFNN)

Genetic Algorithm[109]

202021.360 - - - -

Hopfield Neural

Network Method[110]

202106.360

-

-

-

-

Single Approach-I

(Enumeration)

201415.089 606.28 0.300 691.28 0.342

Single Approach-II

(Conventional Priority List)

202393.346 -371.98 -0.184 -0.286.98 -0.142

Proposed Single

Approach-III

201632.089 389.28 0.1926 474.28 0.2346

67

Table 4.6 Unit Commitment Schedule and Power Sharing (MW) of the best solution obtained from Single Approach-I (Enumeration): Test system-I

Power Output of each

unit(MW) Fuel Cost of each unit($/h)

Hour

UC Schedule

P-1 P-2 P-3

Load (MW)

F-1 F-2 F-3

Operating Fuel Cost

($/h)

1 111 600.00 400.00 200.00 1200 5875.320 3760.400 2237.760 11873.480 2 111 600.00 400.00 200.00 1200 5875.320 3760.400 2237.760 11873.480 3 111 600.00 400.00 150.00 1150 5875.320 3760.400 1658.340 11294.060 4 111 600.00 400.00 100.00 1100 5875.320 3760.400 1107.840 10743.560 5 110 600.00 400.00 0 1000 5875.320 3760.400 0 9635.720 6 110 500.00 400.00 0 900 4911.500 3760.400 0 8671.900 7 110 433.18 366.82 0 800 4284.887 3450.577 0 7735.464 8 100 600.00 0.00 0 600 5875.320 0.000 0 5875.320 9 100 550.00 0.00 0 550 5389.505 0.000 0 5389.505 10 011 0 400.00 100.00 500 0 3760.400 1107.840 4868.240 11 011 0 400.00 100.00 500 0 3760.400 1107.840 4868.240 12 011 0 400.00 100.00 500 0 3760.400 1107.840 4868.240 13 011 0 400.00 100.00 500 0 3760.400 1107.840 4868.240 14 011 0 400.00 100.00 500 0 3760.400 1107.840 4868.240 15 100 600.00 0 0 600 5875.320 0.000 0 5875.320 16 110 433.18 366.82 0 800 4284.887 3450.577 0 7735.464 17 110 460.88 389.12 0 850 4542.955 3658.336 0 8201.290 18 110 500.00 400.00 0 900 4911.500 3760.400 0 8671.900 19 110 550.00 400.00 0 950 5389.505 3760.400 0 9149.905 20 110 600.00 400.00 0 1000 5875.320 3760.400 0 9635.720 21 111 600.00 400.00 50.00 1050 5875.320 3760.400 586.260 10221.980 22 111 600.00 400.00 100.00 1100 5875.320 3760.400 1107.840 10743.560 23 111 600.00 400.00 200.00 1200 5875.320 3760.400 2237.760 11873.480 24 111 600.00 400.00 200.00 1200 5875.320 3760.400 2237.760 11873.480

SUM 201415.789

68

Table 4.7 Unit Commitment Schedule and Power Sharing (MW) of the solution obtained from the Single Approach-II (Conventional Priority List): Test system-I



Hour

UC Schedule

P-1 P-2 P-3

Load (MW)

F-1 F-2 F-3

Operating Fuel Cost

($/h)

1 111 600.00 400.00 200.00 1200 5875.320 3760.400 2237.760 11873.480 2 111 600.00 400.00 200.00 1200 5875.320 3760.400 2237.760 11873.480 3 111 600.00 400.00 150.00 1150 5875.320 3760.400 1658.340 11294.060 4 111 600.00 400.00 100.00 1100 5875.320 3760.400 1107.840 10743.560 5 110 600.00 400.00 0 1000 5875.320 3760.400 0 9635.720 6 110 500.00 400.00 0 900 4911.500 3760.400 0 8671.900 7 110 433.18 366.82 0 800 4284.887 3450.577 0 7735.464 8 110 322.39 277.61 0 600 3276.676 2638.749 0 5915.425 9 110 294.69 255.31 0 550 3030.592 2440.639 0 5471.231 10 110 266.99 233.01 0 500 2786.906 2244.458 0 5031.364 11 110 266.99 233.01 0 500 2786.906 2244.458 0 5031.364 12 110 266.99 233.01 0 500 2786.906 2244.458 0 5031.364 13 110 266.99 233.01 0 500 2786.906 2244.458 0 5031.364 14 110 266.99 233.01 0 500 2786.906 2244.458 0 5031.364 15 110 322.39 277.61 0 600 3276.676 2638.749 0 5915.425 16 110 433.18 366.82 0 800 4284.887 3450.577 0 7735.464 17 110 460.88 389.12 0 850 4542.955 3658.336 0 8201.290 18 110 500.00 400.00 0 900 4911.500 3760.400 0 8671.900 19 110 550.00 400.00 0 950 5389.505 3760.400 0 9149.905 20 110 600.00 400.00 0 1000 5875.320 3760.400 0 9635.720 21 111 600.00 400.00 50.00 1050 5875.320 3760.400 586.260 10221.980 22 111 600.00 400.00 100.00 1100 5875.320 3760.400 1107.840 10743.560 23 111 600.00 400.00 200.00 1200 5875.320 3760.400 2237.760 11873.480 24 111 600.00 400.00 200.00 1200 5875.320 3760.400 2237.760 11873.480

SUM 202393.346

69

Table 4.8 Unit Commitment Schedule and Power Sharing (MW) of the solution obtained from the Proposed Single approach-III (PMAX): Test system-I



Hour

UC Schedule

P-1 P-2 P-3

Load (MW)

F-1 F-2 F-3

Operating Fuel Cost

($/h)

1 111 600.00 400.00 200.00 1200 5875.320 3760.400 2237.760 11873.480 2 111 600.00 400.00 200.00 1200 5875.320 3760.400 2237.760 11873.480 3 111 600.00 400.00 150.00 1150 5875.320 3760.400 1658.340 11294.060 4 111 600.00 400.00 100.00 1100 5875.320 3760.400 1107.840 10743.560 5 110 600.00 400.00 0 1000 5875.320 3760.400 0 9635.720 6 110 500.00 400.00 0 900 4911.500 3760.400 0 8671.900 7 110 433.18 366.82 0 800 4284.887 3450.577 0 7735.464 8 100 600.00 0 0 600 5875.320 0 0 5875.320 9 100 550.00 0 0 550 5389.505 0 0 5389.505 10 100 500.00 0 0 500 4911.500 0 0 4911.500 11 100 500.00 0 0 500 4911.500 0 0 4911.500 12 100 500.00 0 0 500 4911.500 0 0 4911.500 13 100 500.00 0 0 500 4911.500 0 0 4911.500 14 100 500.00 0 0 500 4911.500 0 0 4911.500 15 100 600.00 0 0 600 5875.320 0 0 5875.320 16 110 433.18 366.82 0 800 4284.887 3450.577 0 7735.464 17 110 460.88 389.12 0 850 4542.955 3658.336 0 8201.290 18 110 500.00 400.00 0 900 4911.500 3760.400 0 8671.900 19 110 550.00 400.00 0 950 5389.505 3760.400 0 9149.905 20 110 600.00 400.00 0 1000 5875.320 3760.400 0 9635.720 21 111 600.00 400.00 50.00 1050 5875.320 3760.400 586.260 10221.980 22 111 600.00 400.00 100.00 1100 5875.320 3760.400 1107.840 10743.560 23 111 600.00 400.00 200.00 1200 5875.320 3760.400 2237.760 11873.480 24 111 600.00 400.00 200.00 1200 5875.320 3760.400 2237.760 11873.480

SUM 201632.089

70

4.14.2 Numerical results: Test system-III --- (10 unit system)

The case studies are conducted based on the following considerations:

• UC schedules without considering spinning reserve and without considering MUT and

MDT constraints.

• Without s. r. but with considering MUT and MDT constraint

• 10% s. r. without MUT and MDT constraint

• With 10 % s. r. without ED considering MUT and MDT constraint

• With 10 % s. r. with ED considering MUT and MDT constraint without transition cost.

• With 10% spinning reserve, minimum up/down time constraints, transition cost and

Economic Dispatch.

The priority order for ten unit system is given in Table 4.9.

Table 4.9 Priority order based on single approach-II and proposed single approach III: Test system-III --- ten unit system

Unit No.

PMAX FLAPC (S/MWH)

Single approach-II Conventional Priority order

(FLAPC)

Proposed Single approach-III

(PMAX)

Proposed hybrid approach

(PMAX-FLAPC)

1 455 18.6062 1 1 1

2 455 19.5329 2 2 2

3 130 22.2446 4 5 5

4 130 22.0051 3 3 4

5 162 23.1225 5 4 3

6 80 27.4546 6 7 7

7 85 33.4542 7 6 6

8 55 38.1472 8 8 8

9 55 39.4830 9 9 9

10 55 40.0670 10 10 10

The output results of test system–III are shown in the following tables:

Table 4.10 presents the comparison of the results for single approaches based upon above

considerations.

71

Table 4.11 presents the comparison of operating fuel cost ($) considering 10% spinning reserve

and minimum up/down time constraints with transition cost.

Table 4.12 presents the comparison of the results with Genetic Algorithm.

Table 4.13 presents the summary of Unit Commitment schedules for Single approaches-I, II and

III.

Tables 4.14-4.19 presents the Unit Commitment Schedules and Power Sharing (MW) , operating

fuel cost, start up cost and total production cost of the best solutions obtained from Single

approaches-I, II, and III, with MUT and MDT constraints with 10 % spinning reserve.

The Salient features of the conventional single approach-III in the light of the observations

from the results are as follows:

1. Proposed Single approach III give better results than GA. The proposed approach gives

cost saving of $949.39 per day equivalent to 0.167% compared with GA.

2. Proposed single approach III give better results the conventional priority list. The

proposed approach gives a cost saving of $1247.37 per day equivalent to 0.220%

compared with conventional priority list.

Table 4.10 Comparison of the operating fuel cost ($) for 10 unit systems considering 10% spinning reserve and minimum up/down time constraints without transition cost: Test System III

560200560400560600560800561000561200561400561600561800

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Table 4.11 Comparison of the operating fuel cost ($) for 10 unit systems considering 10% spinning reserve and minimum up/down time constraints with transition cost:

Test System III

Single Approaches

Single approach - I

(Enumeration) With ED, 10 % s. r

MUT, MDT

Single approach - II (Conventional

Priority List FLAPC) With ED, 10 % s. r

MUT, MDT

Proposed Single approach - III

(Priority List PMAX) With ED, 10 % s. r.

MUT, MDT

UC Schedule

UC12543768910

UC12435678910 UC12534768910

Operating fuel cost ($)

560744.47 561682.98 560775.6

Transition cost ($)

4090 4440 4100

Total Production

Cost ($)

564834.47

566122.98 564875.61

Table 4.12 Comparison of the results of the proposed single approach-III with Genetic Algorithm and conventional priority list

Approach/Model Total

Production Cost ($)

Amount of Daily

Saving compared with GA

($)

% Cost Saving

compared with GA

Amount of daily saving

compared with CPL

($)

% Cost Saving

compared with CPL

Genetic Algorithm 565825.00 - - - - Single Approach-I (Complete Enumeration)

564834.47 990.53 0.175 - -

Single Approach-II (Conventional Priority list, CPL)

566122.98 -297.98 -0.052 - -

Proposed Single Approach-III

564875.61 949.39 0.167 1247.37 0.220

73

Table 4.13 Summary of Unit Commitment schedules for 10 unit systems: Single approach-I, II and III: Test System III

Single

approach-I Single

approach-II Proposed Single

approach-III Hour Load

(MW) Enumeration with 10 % s.r with ED, MUT

and MDT constraint

12543768910

(FLAPC) with 10% s.r.

with ED, MUT and MDT constraint

12435678910

(PMAX) with 10% s.r. with ED, MUT and

MDT constraint

12534768910 UC Schedule UC Schedule UC Schedule 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

700 750 850 950

1000 1100 1150 1200 1300 1400 1450 1500 1400 1300 1200 1050 1000 1100 1200 1400 1300 1100 900 800

1100000000 1100000000 1100100000 1100100000 1101100000 1111100000 1111100000 1111100000 1111111000 1111111100 1111111110 1111111111 1111111100 1111111000 1111100000 1111100000 1111100000 1111100000 1111100000 1111111100 1111111000 1111111000 1100100000 1100000000

1100000000 1100000000 1101000000 1111000000 1111000000 1111100000 1111100000 1111100000 1111111000 1111111100 1111111110 1111111111 1111111100 1111111000 1111100000 1111100000 1111100000 1111100000 1111100000 1111111100 1111111000 1111111000 1101000000 1100000000

1100000000 1100000000 1100100000 1100100000 1110100000 1111100000 1111100000 1111100000 1111111000 1111111100 1111111110 1111111111 1111111100 1111111000 1111100000 1111100000 1111100000 1111100000 1111100000 1111111100 1111111000 1111111000 1100100000 1100000000


564834.47 566122.98 564875.61

74

Table 4.14 Unit Commitment Schedule and Power Sharing (MW) of the best solution obtained from Single approach-I (Enumeration) considering MUT and MDT constraints with 10

% spinning reserve: Test system-III

Power output of each unit (MW) Hour UC schedule 12543768910

Load (MW)

1 2 3 4 5 6 7 8 9 10 1 1100000000 700 455 245 0 0 0 0 0 0 0 0 2 1100000000 750 455 295 0 0 0 0 0 0 0 0 3 1100100000 850 455 370 0 0 25 0 0 0 0 0 4 1100100000 950 455 455 0 0 40 0 0 0 0 0 5 1101100000 1000 455 390 0 130 25 0 0 0 0 0 6 1111100000 1100 455 360 130 130 25 0 0 0 0 0 7 1111100000 1150 455 410 130 130 25 0 0 0 0 0 8 1111100000 1200 455 455 130 130 30 0 0 0 0 0 9 1111111000 1300 455 455 130 130 85 20 25 0 0 0 10 1111111100 1400 455 455 130 130 162 33 25 10 0 0 11 1111111110 1450 455 455 130 130 162 73 25 10 10 0 12 1111111111 1500 455 455 130 130 162 80 25 43 10 1013 1111111100 1400 455 455 130 130 162 33 25 10 0 0 14 1111111000 1300 455 455 130 130 85 20 25 0 0 0 15 1111100000 1200 455 455 130 130 30 0 0 0 0 0 16 1111100000 1050 455 310 130 130 25 0 0 0 0 0 17 1111100000 1000 455 260 130 130 25 0 0 0 0 0 18 1111100000 1100 455 360 130 130 25 0 0 0 0 0 19 1111100000 1200 455 455 130 130 30 0 0 0 0 0 20 1111111100 1400 455 455 130 130 162 33 25 10 0 0 21 1111111000 1300 455 455 130 130 85 20 25 0 0 0 22 1111111000 1100 455 315 130 130 25 20 25 0 0 0 23 1100100000 900 455 420 0 0 25 0 0 0 0 0 24 1100000000 800 455 345 0 0 0 0 0 0 0 0

75

Table 4.15 Unit Commitment Schedule, fuel cost, start up cost and total production cost of the best solution obtained from Single approach-I (Enumeration) considering MUT and MDT

constraints with 10 % spinning reserve: Test system-III

Hour UC schedule 12543768910

Load (MW)

Fuel cost ($/h)

Startup Cost ($/h)

Total Production

Cost ($)

1 1100000000 700 13683.13 0 13683.13 2 1100000000 750 14554.50 0 14554.50 3 1100100000 850 16809.45 900 17709.45 4 1100100000 950 18597.67 0 18597.67 5 1101100000 1000 20020.02 560 20580.02 6 1111100000 1100 22387.04 1100 23487.04 7 1111100000 1150 23261.98 0 23261.98 8 1111100000 1200 24150.34 0 24150.34 9 1111111000 1300 27251.06 860 28111.06 10 1111111100 1400 30057.55 60 30117.55 11 1111111110 1450 31916.06 60 31976.06 12 1111111111 1500 33890.16 60 33950.16 13 1111111100 1400 30057.55 0 30057.55 14 1111111000 1300 27251.06 0 27251.06 15 1111100000 1200 24150.34 0 24150.34 16 1111100000 1050 21513.66 0 21513.66 17 1111100000 1000 20641.82 0 20641.82 18 1111100000 1100 22387.04 0 22387.04 19 1111100000 1200 24150.34 0 24150.34 20 1111111100 1400 30057.55 490 30547.55 21 1111111000 1300 27251.06 0 27251.06 22 1111111000 1100 23592.97 0 23592.97 23 1100100000 900 17684.69 0 17684.69 24 1100000000 800 15427.42 0 15427.42

SUM 21700 560744.47 4090.00 564834.47

76

Table 4.16 Unit Commitment Schedule and Power Sharing (MW) of the best solution obtained from Single approach-II (FLAPC) considering MUT and MDT constraints with 10 %

spinning reserve: Test system-III


Load (MW)

1 2 3 4 5 6 7 8 9 10 1 1100000000 700 455 245 0 0 0 0 0 0 0 0 2 1100000000 750 455 295 0 0 0 0 0 0 0 0 3 1101000000 850 455 265 0 130 0 0 0 0 0 0 4 1111000000 950 455 235 130 130 0 0 0 0 0 0 5 1111000000 1000 455 285 130 130 0 0 0 0 0 0 6 1111100000 1100 455 360 130 130 25 0 0 0 0 0 7 1111100000 1150 455 410 130 130 25 0 0 0 0 0 8 1111100000 1200 455 455 130 130 30 0 0 0 0 0 9 1111111000 1300 455 455 130 130 85 20 25 0 0 0 10 1111111100 1400 455 455 130 130 162 33 25 10 0 0 11 1111111110 1450 455 455 130 130 162 73 25 10 10 0 12 1111111111 1500 455 455 130 130 162 80 25 43 10 1013 1111111100 1400 455 455 130 130 162 33 25 10 0 0 14 1111111000 1300 455 455 130 130 85 20 25 0 0 0 15 1111100000 1200 455 455 130 130 30 0 0 0 0 0 16 1111100000 1050 455 310 130 130 25 0 0 0 0 0 17 1111100000 1000 455 260 130 130 25 0 0 0 0 0 18 1111100000 1100 455 360 130 130 25 0 0 0 0 0 19 1111100000 1200 455 455 130 130 30 0 0 0 0 0 20 1111111100 1400 455 455 130 130 162 33 25 10 0 0 21 1111111000 1300 455 455 130 130 85 20 25 0 0 0 22 1111111000 1100 455 315 130 130 25 20 25 0 0 0 23 1101000000 900 455 315 0 130 0 0 0 0 0 0 24 1100000000 800 455 345 0 0 0 0 0 0 0 0

77

Table 4.17 Unit Commitment Schedule, fuel cost, startup cost and total production cost of the best solution obtained from Single approach-II (FLAPC) considering MUT and MDT constraints

with 10 % spinning reserve: Test system-III

Hour UC schedule Load (MW)

Fuel cost ($/h)

Startup Cost ($/h)

Total Production

Cost ($)

1 1100000000 700 13683.13 0.00 13683.13 2 1100000000 750 14554.50 0.00 14554.50 3 1101000000 850 16892.15 560.00 17452.15 4 1111000000 950 19261.50 550.00 19811.50 5 1111000000 1000 20132.56 0.00 20132.56 6 1111100000 1100 22387.04 1800.00 24187.04 7 1111100000 1150 23261.98 0.00 23261.98 8 1111100000 1200 24150.34 0.00 24150.34 9 1111111000 1300 27251.06 860.00 28111.06 10 1111111100 1400 30057.55 60.00 30117.55 11 1111111110 1450 31916.06 60.00 31976.06 12 1111111111 1500 33890.16 60.00 33950.16 13 1111111100 1400 30057.55 0.00 30057.55 14 1111111000 1300 27251.06 0.00 27251.06 15 1111100000 1200 24150.34 0.00 24150.34 16 1111100000 1050 21513.66 0.00 21513.66 17 1111100000 1000 20641.82 0.00 20641.82 18 1111100000 1100 22387.04 0.00 22387.04 19 1111100000 1200 24150.34 0.00 24150.34 20 1111111100 1400 30057.55 490.00 30547.55 21 1111111000 1300 27251.06 0.00 27251.06 22 1111111000 1100 23592.97 0.00 23592.97 23 1101000000 900 17764.14 0.00 17764.14 24 1100000000 800 15427.42 0.00 15427.42

SUM 561682.99 4440.00 566122.99

78

Table 4.18 Unit Commitment Schedule and Power Sharing (MW) of the best solution obtained from Proposed Single approach-III (PMAX) considering MUT and MDT constraints



Load (MW)

1 2 3 4 5 6 7 8 9 10 1 1100000000 700 455 245 0 0 0 0 0 0 0 0 2 1100000000 750 455 295 0 0 0 0 0 0 0 0 3 1100100000 850 455 370 0 0 25 0 0 0 0 0 4 1100100000 950 455 455 0 0 40 0 0 0 0 0 5 1110100000 1000 455 390 130 0 25 0 0 0 0 0 6 1111100000 1100 455 360 130 130 25 0 0 0 0 0 7 1111100000 1150 455 410 130 130 25 0 0 0 0 0 8 1111100000 1200 455 455 130 130 30 0 0 0 0 0 9 1111111000 1300 455 455 130 130 85 20 25 0 0 0 10 1111111100 1400 455 455 130 130 162 33 25 10 0 0 11 1111111110 1450 455 455 130 130 162 73 25 10 10 0 12 1111111111 1500 455 455 130 130 162 80 25 43 10 1013 1111111100 1400 455 455 130 130 162 33 25 10 0 0 14 1111111000 1300 455 455 130 130 85 20 25 0 0 0 15 1111100000 1200 455 455 130 130 30 0 0 0 0 0 16 1111100000 1050 455 310 130 130 25 0 0 0 0 0 17 1111100000 1000 455 260 130 130 25 0 0 0 0 0 18 1111100000 1100 455 360 130 130 25 0 0 0 0 0 19 1111100000 1200 455 455 130 130 30 0 0 0 0 0 20 1111111100 1400 455 455 130 130 162 33 25 10 0 0 21 1111111000 1300 455 455 130 130 85 20 25 0 0 0 22 1111111000 1100 455 315 130 130 25 20 25 0 0 0 23 1100100000 900 455 420 0 0 25 0 0 0 0 0 24 1100000000 800 455 345 0 0 0 0 0 0 0 0

79

Table 4.19 Unit Commitment Schedule fuel cost, startup cost and total production cost of the best solution obtained from Single approach-III (PMAX) considering MUT and MDT constraints


Hour UC schedule Load (MW)

Fuel cost ($/h)

Startup Cost ($/h)

Total Production

Cost ($)

1 1100000000 700 13683.13 0 13683.13 2 1100000000 750 14554.50 0 14554.50 3 1100100000 850 16809.45 900 17709.45 4 1100100000 950 18597.67 0 18597.67 5 1110100000 1000 20051.16 550 20601.16 6 1111100000 1100 22387.04 1120 23507.04 7 1111100000 1150 23261.98 0 23261.98 8 1111100000 1200 24150.34 0 24150.34 9 1111111000 1300 27251.06 860 28111.06 10 1111111100 1400 30057.55 60 30117.55 11 1111111110 1450 31916.06 60 31976.06 12 1111111111 1500 33890.16 60 33950.16 13 1111111100 1400 30057.55 0 30057.55 14 1111111000 1300 27251.06 0 27251.06 15 1111100000 1200 24150.34 0 24150.34 16 1111100000 1050 21513.66 0 21513.66 17 1111100000 1000 20641.82 0 20641.82 18 1111100000 1100 22387.04 0 22387.04 19 1111100000 1200 24150.34 0 24150.34 20 1111111100 1400 30057.55 490 30547.55 21 1111111000 1300 27251.06 0 27251.06 22 1111111000 1100 23592.97 0 23592.97 23 1100100000 900 17684.69 0 17684.69 24 1100000000 800 15427.42 0 15427.42

SUM 21700 560775.61 4100 564875.61

80

CHAPTER 5

Proposed New Hybrid Models for Unit Commitment

problem based on convex and Non-Convex Cost Functions 5.1 Introduction

Hybrid models deal with the integration of two or more approaches. Recently, hybrid

techniques combining different classical and non classical optimization techniques have been

proposed to solve UCP. This chapter presents three hybrid models for the solution of convex and

non-convex fuel cost functions.

For the past several years, ANNs methods received a great deal of attention. H. Sasaki, et al.

[49] presented the application of Hopfield neural network to unit commitment. T. Yalcinoz, et al.

[50] presents a new mapping process and a computational method for obtaining the weights

using a slack variable technique for handling inequality constraints. M. H. Wong, et al. [51] used

GA to evolve the weight and the interconnection of the neural network to solve the UC problem.

Ouyang et al. [71] utilizes neural networks to generate a initial feasible schedule according to the

input load curve and then refines the schedule, using a dynamic search. Z. Ouyang, et al. [72]

proposes a multi-stage NN-expert system. H. Shyh-Jier et al. [74] proposes genetic algorithm

based NN and DP to solve the UC problem. H. Sasaki et al. [103] utilizes the Hopfield neural

network in which a large number of inequality constraints are handled by the dedicated neural

network instead of including them in the energy function.

One of the first uses of PSO was for evolving neural network weights and, indirectly, to

evolve the structure. R. C. Eberhart, et al. [104] replaced the standard back propagation learning

algorithm with the swarm intelligence learning rule. Y. D. Valle et al. [105] presented a detailed

overview of PSO including basic concepts, its variants and applications to power system. S.

Mohagheghi et al. [106] investigates the efficiency of swarm intelligence learning rule. The

81

applications of PSO as a training algorithm of ANN have been reported in [107-108]. The

advantage of using Particle Swarm Optimization (PSO) algorithm over other techniques is that it

is computationally inexpensive, easy to implement.

5.2 Hybrid Model – I: A hybrid of particle swarm optimization (PSO), artificial neural

network (ANN) and dynamic programming (DP).

Unit commitment problem has been addressed independently by Dynamic Programming

[20], Artificial Neural Network [49] and Particle Swarm Optimization [64]. In this thesis, a new

approach of PSO based artificial neural network (PSO-ANN) has been proposed to solve UC

problem. The proposed Hybrid Model-I combines the Dynamic Programming (DP) with

Artificial Neural Networks (ANN) using Swarm Intelligence (SI) learning rule. In this model

dynamic programming produces near optimal solution based on training data for neural network

model. The neural network fine tunes the data subject to the target values of power output of

units. The best tuned solution is considered the required solution.

The swarm intelligence learning rule based feed forward network has been used for fine

tuning the near optimal dynamic programming results. The standard back propagation learning

rule neural network has also been used in this hybrid approach for the comparison of results.

Flow chart is represented in Figure 5.1.

82

Figure 5.1 Flowchart for DP-PSO-ANN Hybrid Model-I

The PSO algorithm is applied to the neural network to obtain a set of weights that will minimize

the error function. Weights are progressively updated until the convergence criterion is satisfied.

The UC problem has been decomposed in to discrete load level and formed small ANN models

based on hourly load. Three and ten unit standard test systems have been tested for validation of

the proposed approach. The discussion to follow in next section of this approach is with

reference to context given in chapter 3.

83

5.2.1 Generation of test and training data

In this model the test data and training data are generated by using Dynamic

Programming (DP). Artificial Neural Network (ANN) is trained using Swarm Intelligence (S.I.)

learning rule. For proper training of ANN, a pair of load as input and their corresponding

generation schedules as output are prepared off-line by using DP. Each pair is referred as

input/output database pair. A multilayer feed forward neural network with swarm intelligence

learning rule has been programmed for tuning the power generation of the units.

5.2.2 Artificial Neural Network using SI learning Rule

PSO trains the ANN by changing its weight such that mean square error (MSE) for the

training set of an ANN model is reduced. It is a feedback process which runs until the batch error

falls under an acceptable range or iteration cutoff threshold is reached. In the process of training

ANN, particle position corresponds to the weights in the network. The fitness function

corresponds to the mean square error (MSE) of the network [107].

5.2.3 Input and Output of the PSO-ANN Model

The input to the network is the forecasted load demand as shown in Fig.5.2. For a three

unit system, in the ANN model three inputs (P1, P2, P3) the power generation of the machines and

outputs in the range of P1-P3 has been taken according to the load demand. The networks is

trained with input/ output data pattern with 3 neurons as input, 2 neurons for hidden layers and 3

neurons for output layer. Total we have 24 networks for 24 hour duration each representing an

hourly schedule h. Each network has been trained separately over hourly load data.

0

200

400

600

800

1000

1200

1400

1 3 5 7 9 11 13 15 17 19 21 23

Time (hours)

Load

Dem

and

(MW

)

Figure 5.2 Load patterns for training

84

The target has been taken as the best solution attained so far. The percentage error is calculated

as follows: ((target – network output) / target)) x 100

5.2.4 Scaling of the input and output data

The network input and output data will have different ranges if actual hourly load data is

used so it was normalized to fall within the range [0, 1], to avoid convergence problems during

the training process.

5.2.5 Training process

The training process of an ANN model by PSO approach has the following steps as given

in Figure 5.3.

Initialize swarm with random velocity and position Repeat for each particle i in swarm do if MSE( current network ) < MSE (personal best network) then Personal best network = current network end if if MSE ( current network )< the MSE (global best network) then Global best network = current network end if end for Update velocity and position of each particle according to equations (3.1) and (3.2) Until stop criteria (epoch < 30000 or MSE (global best network)< 1.0e-12) being satisfied

Figure 5.3 Steps for SI learning ANN

The perception training algorithm is a form of supervised learning algorithm where the

weights and biases are updated to reduce errors whenever the network output does not match the

desired values. The load patterns and its corresponding feasible schedules are obtained using

dynamic programming technique in a training set for the ANN. Most of the time is spend on off-

line training of the network. With trained network the on-line operation time is very short. The

training set covers 24 hours load data. We used 24 networks for 3 units system each having

architecture of 3:2:3. The training set contained 20 patterns for each hour. Total number of

training patterns is 480.

The terminating criteria are the maximum number of epochs and the maximum mean square

error.

Swarm intelligence learning rule is used for network training.

85

5.2.6 Parameters Settings

a. The PSO model parameters:

The parameter settings are: Number of particles = 60, scaling constants, c1 = c2 =1.4962,

Constriction factor χ = 0.7298, the range for individual particle position 21≤x≤-21, the range

for individual dimension of particle velocity, is v_max = k*x_max, x_max=21, k=0.1 and is

equal to 2.1≤x≤-2.1, Maximum Epoch = 30000, Maximum Error = 1.0e-12, max_memory =

30, success_counter sc =15, fail_counter fc =5.

b. The BP model parameters:

The parameters selected are: No. of samples = 60, No. of inputs = No. of units, No. of

outputs = No. of committed units according to load demand at each hour, momentum rate α =

0.85, learning rate η = 0.8, No. of hidden layers = 2.

5.3 Case Studies---Convex cost function

Following three standard test systems have been selected for the validation of the proposed

approaches.


b. Test System II --- 3 units with 24 hours load

c. Test System III --- 10 units system with 24 hours load

Input data: The description and input data of test systems used for investigation in the case

studies are given in Table A. 1, Table A. 2 and Table A. 3 placed in Appendix A.

Computer Implementation: The algorithms have been implemented in C++ on P-IV Personal

Computer.


5.3.1 Numerical results of test system-I: Three unit system: Hybrid Model-I

Sixty samples consisting of three inputs for three unit system to the neural network

produce fine tuned sixty outputs. The outputs consist of one to three power generations

depending upon forecasted load.

The best fine tuned results generated from both the ANN models amongst the sixty samples

and corresponding near optimal output generated by DP have been tabulated. The mean square

86

error and absolute percentage error has been calculated with reference to the target generation

schedule. The plot of mean square error for three different cases is shown in Figures 5.4, 5.5 and

5.6. The plot of absolute percentage error is shown in Figure 5.7.

The output results of test system–I are shown in the following tables:

Table (5.1) gives the comparison of proposed Hybrid Model-I with Genetic Algorithm,

Conventional Priority List and Hopfield Neural Network methods.

Table (5.2) presents the best results by the proposed Hybrid Model-I (DP-PSO-ANN) among the

sixty samples.

Table (5.3) presents the Unit Commitment Schedule and Power Sharing (MW) of the best

solution obtained from the proposed Hybrid Model-I amongst sixty samples for SI learning.

Table (5.4) shows the best output results obtained by the Hybrid Model-I (DP-BP-ANN)

amongst sixty samples.

Table (5.5) shows the comparison of the outputs (MW) obtained by the proposed SI –ANN land

BP-ANN learning for 3 unit system.


solution obtained from the proposed Hybrid Model-I amongst sixty samples for BP learning.

The Salient features of the proposed approach in the light of the observations from the results are

as follows:

1. PSO trains the ANN up to the target values as shown in Table 5.2.

2. In Hybrid Model-I (DP-PSO-ANN) the dynamic programming is used to generate

training and test data. S.I. learning rule is used to train the neural network. This model

gives accurate results.

3. This approach hits the target due to the use of S.I. learning. This model is able to explore

more solution space; therefore it takes long training time, but gives high quality solution.

The Absolute Percentage Error (APE) in this method is almost zero.

4. Figures 5.4-5.6 presents the MSE graph of SI-learning algorithm and BP learning

algorithm. SI learning graph decays slowly and finally hits the target. BP learning graph

decays rapidly and then slows down and unable to hit the target.

87

5. Figure 5.7 indicates the APE graph of both the SI and BP learning algorithms. The APE

in SI learning is zero, but in BP the APE is fluctuating.

6. The Absolute Percentage Error in BP trained model is more than PSO trained model.

The BP learning stuck in local minima and unable to reach to the target value. BP-ANN

model gives results near to target quickly due to the use of B.P. learning algorithm.

7. For three unit test system-I the operating fuel cost for committed units in the 24 hours

for PSO-ANN and BP-ANN models are $201632.089 and $201630.923 respectively.

The cost of BP-ANN learning approach is slightly lower than PSO-ANN learning as BP-

learning approach does not reach the target.

8. The operating fuel cost obtained by all both trained models remains low compared to

genetic algorithm, Hopfield neural network and Conventional Priority List methods.

9. SI-learning gives $389.28 saving per day equivalent to 0.1926% compared with GA and

$474.28 saving per day equivalent to 0.2346% compared with HFNN and $761.26

saving per day equivalent to 0.37612% compared with conventional priority list

methods.

10. All the two learning models give fair amount of reduction in operating cost as compared

to GA, HFNN and conventional priority list methods.

11. The PSO trained model gives accurate and good quality results.

12. Hybrid Model-I, incorporates dynamic programming as the backbone.

13. Hybrid Model-I using SI-learning generates better solutions than the other methods,

mainly because of its intrinsic nature of updates of positions and velocities. The second

reason is due to the hourly basis solution. This is somehow similar to the “divide and

conquer” strategy of solving a problem. Owning to this hourly solution, the complexity

of the search is greatly reduced.

14. The result obtained from the simulation is most encouraging in comparison to the best

known solution so far.

88

Table 5.1 Comparison of proposed Hybrid Model-I with Genetic Algorithm, Conventional Priority List and Hopfield Neural Network methods for 3 unit systems: Test system-I:

Algorithm Daily

Operating Cost ($)

Amount of Daily Saving (comparedwith GA)

% Cost saving in fuel cost compared with GA

Amount of Daily Saving (compared with conventional PL)

% Cost saving in fuel cost compared with PL

Amount of Daily Saving (compared with Hopfield Neural Network

% Cost saving in fuel cost compared with Hopfield Neural Network

Genetic Algorithm

202021.360 - - - - - -

Conventional Priority List (FLAPC)

202393.346 - - - - - -

Hopfield Neural Network

202106.36 - - - - - -

Hybrid Model–I (DP-PSO-ANN) SI- Learning

201632.089 389.28 0.1926 761.26 0.37612 474.28 0.2346

Hybrid Model–I (DP-BP-ANN) BP-Learning

201630.923 390.44 0.1932 762.42 0.37670 475.44 0.2352

89

Table 5.2 Best Results by the proposed Hybrid Model-I (DP-PSO-ANN) among the sixty samples: Test system-I

Targets

Outputs obtained by the proposed

DP-PSO-ANN model Hours P-1 P-2 P-3 P-1 P-2 P-3

1 600.00 400.000 200.000 600.000 400.000 200.000 2 600.00 400.000 200.000 600.000 400.000 200.000 3 600.00 400.000 150.000 600.000 400.000 150.000 4 600.00 400.000 100.000 600.000 400.000 100.000 5 600.00 400.000 0 600.000 400.000 0 6 500.00 400.000 0 500.000 400.000 0 7 433.18 366.820 0 433.180 366.820 0 8 600.00 0 0 600.000 0 0 9 550.00 0 0 550.000 0 0 10 500.00 0 0 500.000 0 0 11 500.00 0 0 500.000 0 0 12 500.00 0 0 500.000 0 0 13 500.00 0 0 500.000 0 0 14 500.00 0 0 500.000 0 0 15 600.00 0 0 600.000 0 0 16 433.18 366.820 0 433.180 366.820 0 17 460.88 389.120 0 460.880 389.120 0 18 500.00 400.000 0 500.000 400.000 0 19 550.00 400.000 0 550.000 400.000 0 20 600.00 400.000 0 600.000 400.000 0 21 600.00 400.000 50.000 600.000 400.000 50.000 22 600.00 400.000 100.000 600.000 400.000 100.000 23 600.00 400.000 200.000 600.000 400.000 200.000 24 600.000 400.000 200.000 600.000 400.000 200.000

90

Table 5.3 Unit Commitment Schedule and Power Sharing (MW) of the best solution obtained from the proposed Hybrid Model-I (DP-PSO-ANN): Test system-I

Power output of each


Hour

UC Schedule

P-1 P-2 P-3

Load (MW)

F-1 F-2 F-3

Operating Fuel Cost

($/h)

1 111 600 400 200 1200 5875.320 3760.400 2237.760 11873.480 2 111 600 400 200 1200 5875.320 3760.400 2237.760 11873.480 3 111 600 400 150 1150 5875.320 3760.400 1658.340 11294.060 4 111 600 400 100 1100 5875.320 3760.400 1107.840 10743.560 5 110 600 400 0 1000 5875.320 3760.400 0 9635.720 6 110 500 400 0 900 4911.500 3760.400 0 8671.900 7 110 433.18 366.82 0 800 4284.887 3450.577 0 7735.464 8 100 600 0 0 600 5875.320 0 0 5875.320 9 100 550 0 0 550 5389.505 0 0 5389.505 10 100 500 0 0 500 4911.500 0 0 4911.500 11 100 500 0 0 500 4911.500 0 0 4911.500 12 100 500 0 0 500 4911.500 0 0 4911.500 13 100 500 0 0 500 4911.500 0 0 4911.500 14 100 500 0 0 500 4911.500 0 0 4911.500 15 100 600 0 0 600 5875.320 0 0 5875.320 16 110 433.18 366.82 0 800 4284.887 3450.577 0 7735.464 17 110 460.88 389.12 0 850 4542.955 3658.336 0 8201.290 18 110 500 400 0 900 4911.500 3760.400 0 8671.900 19 110 550 400 0 950 5389.505 3760.400 0 9149.905 20 110 600 400 0 1000 5875.320 3760.400 0 9635.720 21 111 600 400 50 1050 5875.320 3760.400 586.260 10221.980 22 111 600 400 100 1100 5875.320 3760.400 1107.840 10743.560 23 111 600 400 200 1200 5875.320 3760.400 2237.760 11873.480 24 111 600 400 200 1200 5875.320 3760.400 2237.760 11873.480

SUM 20650 201632.089

91

Table 5.4 Best output results obtained by the Hybrid Model-I (DP-BP-ANN) amongst sixty samples: Test system-I

Targets


DP-BP-ANN model Hours P-1 P-2 P-3 P-1 P-2 P-3

1 600.00 400.000 200.000 599.999 399.999 199.999 2 600.00 400.000 200.000 599.999 399.999 199.999 3 600.00 400.000 150.000 599.999 399.999 149.999 4 600.00 400.000 100.000 599.999 399.999 99.999 5 600.00 400.000 0 599.984 400.000 0 6 500.00 400.000 0 499.998 399.950 0 7 433.18 366.820 0 433.192 366.849 0 8 600.00 0 0 599.999 0 0 9 550.00 0 0 549.999 0 0 10 500.00 0 0 499.999 0 0 11 500.00 0 0 499.999 0 0 12 500.00 0 0 499.999 0 0 13 500.00 0 0 499.999 0 0 14 500.00 0 0 499.999 0 0 15 600.00 0 0 599.999 0 0 16 433.18 366.820 0 433.192 366.849 0 17 460.88 389.120 0 460.880 389.118 0 18 500.00 400.000 0 499.998 399.950 0 19 550.00 400.000 0 550.009 399.969 0 20 600.00 400.000 0 599.984 400.000 0 21 600.00 400.000 50.000 599.999 399.999 49.99 22 600.00 400.000 100.000 599.999 399.999 99.999 23 600.00 400.000 200.000 599.999 399.999 199.999 24 600.000 400.000 200.000 599.999 399.999 199.999

92

Table 5.5 Comparison of the outputs (MW) obtained by the proposed SI – ANN learning and BP - ANN learning for 3 unit system: Test System-I

Targets

Output obtained by the proposed SI-ANN learning

rule

Output obtained by proposed BP-ANN learning rule

P-1 (MW)

P-2 (MW)

P-3 (MW)

P-1 (MW)

P-2 (MW)

P-3 (MW)

P-1 (MW)

P-2 (MW)

P-3 (MW)

600.00 400.000 200.000 600.000 400.000 200.000 599.999 399.999 199.999 600.00 400.000 200.000 600.000 400.000 200.000 599.999 399.999 199.999 600.00 400.000 150.000 600.000 400.000 150.000 599.999 399.999 149.999 600.00 400.000 100.000 600.000 400.000 100.000 599.999 399.999 99.999 600.00 400.000 0 600.000 400.000 0 599.984 400.000 0 500.00 400.000 0 500.000 400.000 0 499.998 399.950 0 433.18 366.820 0 433.180 366.820 0 433.192 366.849 0 600.00 0 0 600.000 0 0 599.999 0 0 550.00 0 0 550.000 0 0 549.999 0 0 500.00 0 0 500.000 0 0 499.999 0 0 500.00 0 0 500.000 0 0 499.999 0 0 500.00 0 0 500.000 0 0 499.999 0 0 500.00 0 0 500.000 0 0 499.999 0 0 500.00 0 0 500.000 0 0 499.999 0 0 600.00 0 0 600.000 0 0 599.999 0 0 433.18 366.820 0 433.180 366.820 0 433.192 366.849 0 460.88 389.120 0 460.880 389.120 0 460.880 389.118 0 500.00 400.000 0 500.000 400.000 0 499.998 399.950 0 550.00 400.000 0 550.000 400.000 0 550.009 399.969 0 600.00 400.000 0 600.000 400.000 0 599.984 400.000 0 600.00 400.000 50.000 600.000 400.000 50.000 599.999 399.999 49.99 600.00 400.000 100.000 600.000 400.000 100.000 599.999 399.999 99.999 600.00 400.000 200.000 600.000 400.000 200.000 599.999 399.999 199.999 600.00 400.000 200.000 600.000 400.000 200.000 599.999 399.999 199.999

93

Table 5.6 Unit Commitment Schedule and Power Sharing (MW) of the best solution obtained by using Hybrid Model-I (DP-BP-ANN): Test system-I

Power output of each


Hour

UC Schedule

P-1 P-2 P-3

Load (MW)

F-1 F-2 F-3

Operating Fuel Cost

($/h)

1 111 599.999 399.999 199.999 1200 5875.310 3760.391 2237.748 11873.449 2 111 599.999 399.999 199.999 1200 5875.310 3760.391 2237.748 11873.449 3 111 599.999 399.999 149.999 1150 5875.310 3760.391 1658.329 11294.030 4 111 599.999 399.999 99.999 1100 5875.310 3760.391 1107.829 10743.530 5 110 599.984 400 0 1000 5875.163 3760.400 0 9635.563 6 110 499.998 399.95 0 900 4911.481 3759.930 0 8671.411 7 110 433.192 366.849 0 800 4284.998 3450.846 0 7735.845 8 100 599.999 0 0 600 5875.310 0 0 5875.310 9 100 549.999 0 0 550 5389.495 0 0 5389.495 10 100 499.999 0 0 500 4911.491 0 0 4911.491 11 100 499.999 0 0 500 4911.491 0 0 4911.491 12 100 499.999 0 0 500 4911.491 0 0 4911.491 13 100 499.999 0 0 500 4911.491 0 0 4911.491 14 100 499.999 0 0 500 4911.491 0 0 4911.491 15 100 599.999 0 0 600 5875.310 0 0 5875.310 16 110 433.192 366.849 0 800 4284.998 3450.846 0 7735.845 17 110 460.88 389.118 0 850 4542.955 3658.317 0 8201.272 18 110 499.998 399.95 0 900 4911.481 3759.930 0 8671.411 19 110 550.009 399.969 0 950 5389.592 3760.109 0 9149.700 20 110 599.984 400 0 1000 5875.163 3760.400 0 9635.563 21 111 599.999 399.999 49.99 1050 5875.310 3760.391 586.159 10221.859 22 111 599.999 399.999 99.999 1100 5875.310 3760.391 1107.829 10743.530 23 111 599.999 399.999 199.999 1200 5875.310 3760.391 2237.748 11873.449 24 111 599.999 399.999 199.999 1200 5875.310 3760.391 2237.748 11873.449

SUM 20650 201630.923

94

Case - I

-0.02

0.00

0.02

0.04

0.06

0.08

0.10

1 11 21 31 41 51 61 71 81 91 101

No. of Epochs

MSE

BP SI

Figure. 5.4 MSE graph for a load of 500 MW.

Case - II

-0.06

0.04

0.14

0.24

0.34

0.44

0.54

0.64

0.74

0.84

1 13 25 37 49 61 73 85 97 109

No. of Epochs

MSE

BP SI

Figure. 5.5 MSE graph for a load of 800 MW.

Case - III

-0.02

0.03

0.08

0.13

0.18

0.23

1 11 21 31 41 51 61 71 81 91 101No. of Epochs

MSE

BP SI

Figure. 5.6 MSE graph for a load of 850 MW

-0.0010

-0.0005

0.0000

0.0005

0.0010

0.0015

0.0020

0.0025

1 5 9 13 17 21 25 29 33 37 41 45 49 53 57

No. of Samples

Abs

olut

e Pe

rcen

tage

Err

or

Figure. 5.7 Absolute Percentage Error graph for Operating Fuel Cost

95

5.3.2 Numerical Results of Test system-II: Three unit system: Hybrid Model-I:

The output results are shown in the following tables:

Table (5.7) shows the comparison of the best operating fuel cost obtained amongst the sixty

samples.

Table (5.8) gives the comparison of the outputs (MW) obtained by the proposed SI–ANN

learning and BP-ANN learning.

The Salient features of the proposed approach in the light of the observations from the


1. For three unit test system-II the operating fuel cost for committed units in the 24 hours

for PSO-ANN and BP-ANN models are $1992262.671 and $1992262.310 respectively.

The cost of BP-ANN learning approach is slightly lower than PSO-ANN learning as BP-

learning approach does not reach the target.

2. The PSO trained model gives accurate and good quality results.

Table 5.7 Comparison of the best Operating fuel cost ($) obtained amongst the sixty samples for 3 unit system by using Proposed Hybrid Model-I: Test system-II

(GA-PSO-ANN) (GA-BP-ANN) Operating fuel cost Test system-II $199262.671 $199262.310

96

Table 5.8 Comparison of the outputs (MW) obtained by the proposed SI –ANN learning (Hybrid Model-I, DP-PSO-ANN) and BP-ANN learning (Hybrid Model-I (DP-BP-ANN) for 3

unit systems: Test System-II

Targets

Output obtained by the proposed SI-ANN learning

rule

Output obtained by proposed BP-ANN learning rule

Unit-1 (MW)

Unit-2 (MW)

Unit-3 MW

Unit-1 (MW)

Unit-2 (MW)

Unit-3 MW

Unit-1 (MW)

Unit-2 (MW)

Unit -3 MW

600.00 400.000 200.000 600.000 400.000 200.000 599.999 399.999 199.999 600.00 400.000 200.000 600.000 400.000 200.000 599.999 399.999 199.999 600.00 400.000 150.000 600.000 400.000 150.000 599.999 399.999 149.999 600.00 400.000 100.000 600.000 400.000 100.000 599.999 399.999 99.999 600.00 400.000 0 600.000 400.000 0 599.984 400.000 0 500.00 400.000 0 500.000 400.000 0 499.998 399.950 0 433.18 366.820 0 433.180 366.820 0 433.192 366.849 0 600.00 0 0 600.000 0 0 599.999 0 0 550.00 0 0 550.000 0 0 549.999 0 0 500.00 0 0 500.000 0 0 499.999 0 0 500.00 0 0 500.000 0 0 499.999 0 0 500.00 0 0 500.000 0 0 499.999 0 0 500.00 0 0 500.000 0 0 499.999 0 0 500.00 0 0 500.000 0 0 499.999 0 0 600.00 0 0 600.000 0 0 599.999 0 0 433.18 366.820 0 433.180 366.820 0 433.192 366.849 0 460.88 389.120 0 460.880 389.120 0 460.880 389.118 0 500.00 400.000 0 500.000 400.000 0 499.998 399.950 0 550.00 400.000 0 550.000 400.000 0 550.009 399.969 0 600.00 400.000 0 600.000 400.000 0 599.984 400.000 0 600.00 400.000 50.000 600.000 400.000 50.000 599.999 399.999 49.99 600.00 400.000 100.000 600.000 400.000 100.000 599.999 399.999 99.999 600.00 400.000 200.000 600.000 400.000 200.000 599.999 399.999 199.999 600.00 400.000 200.000 600.000 400.000 200.000 599.999 399.999 199.999

97

5.3.3 Numerical Results of Test system-III: Ten unit system: Hybrid Model-I:

For a ten machine system, in the ANN model ten inputs (P1, P2, P3…P10) the power

generation of the machines and two to ten outputs has been taken Training data with different

samples consisting of various combinations have been generated by dynamic programming. The

networks are trained with 10 neurons as input, 2 neurons for hidden layers and 10 neurons for

output layer for ten unit systems. Total we have 24 networks each representing an hour h. Each

network was trained separately over hourly load data.

The PSO-ANN trainer program is written in C++. A training set covers 24 hours load data.

We use 24 networks for 10 units system each having architecture of 10:2:10. The training set

contained 60 patterns for each hour. Total number of training patterns is 1440 for 10 unit system.

Sixty samples consisting of ten inputs for ten unit system to the neural network produce fine

tuned sixty outputs. The outputs consist of one to ten power generations depending upon

forecasted load.

The best fine tuned results generated from both the A.N.N models amongst the sixty

samples and corresponding near optimal output generated by DP have been tabulated. The plot

of Absolute percentage error against the numbers of samples is shown in Figure 5.8.

The output results of test system – III are shown in the following tables:

Table (5.9) shows the target values for 10 unit system.

Table (5.10) presents the best results obtained by the proposed Hybrid Model-I (DP-PSO-ANN)

amongst the sixty samples.

Table (5.11) presents the unit commitment schedule and power sharing (MW) of the best

solution obtained from proposed Hybrid Model-I (DP-PSO-ANN) considering MUT and MDT

constraints with 10 % spinning reserve.

Table (5.12) shows the load demand and total production cost (TPC) obtained by proposed

Hybrid Model-I (DP-PSO-ANN).

Table (5.13) presents the comparison of the best results of the Hybrid Model-I (DP-PSO-ANN)

with other approaches available in the literature.

Table (5.14) Comparison between proposed Hybrid model-I and other approaches for daily

saving and Percentage saving in fuel cost.

Table (5.15) Comparison of the results of the proposed Hybrid Model-I with Genetic

Algorithm, dynamic programming, simulated annealing and Lagrange relaxation method:

98


solution obtained by using Hybrid Model-I (DP-BP-ANN), considering MUT and MDT

constraints with 10 % spinning reserve:

Table (5.17) gives the hourly load demand, operating fuel cost, start-up cost and Total

Production Cost obtained by using (DP-BP-ANN), considering MUT and MDT constraints with

10 % spinning reserve:

The Salient features of the proposed approach in the light of the observations from the results

are as follows:

1. The total production cost for 10 unit system is $563942.3.

2. When compared with other thirty four approaches available in the literature, the

proposed hybrid models-I gives low total production cost compared as given in Table

5.13.

3. The daily costs saving of the proposed Hybrid model-III compared with other

approaches have a range of $0.02 to $24812.02 per day. Which is equivalent to a saving

of 0.00000355% to 4.399% as given in Table 5.14?

4. The daily cost saving compared with GA, DP, SA and LR is $1877 0.33 per day. Which

is equivalent to saving of 0.333% as given in Table 5.15?

99

Table 5.9 Targets for the 10 unit system: Test system-III

Targets

Hour Load (MW)

P-1 P-2 P-3 P-4 P-5 P-6 P-7 P-8 P-9 P-10

1 700 455 245 0 0 0 0 0 0 0 0 2 750 455 295 0 0 0 0 0 0 0 0 3 850 455 370 0 0 25 0 0 0 0 0 4 950 455 455 0 0 40 0 0 0 0 0 5 1000 455 390 0 130 25 0 0 0 0 0 6 1100 455 360 130 130 25 0 0 0 0 0 7 1150 455 410 130 130 25 0 0 0 0 0 8 1200 455 455 130 130 30 0 0 0 0 0 9 1300 455 455 130 130 85 20 25 0 0 0 10 1400 455 455 130 130 162 33 25 10 0 0 11 1450 455 455 130 130 162 73 25 10 10 0 12 1500 455 455 130 130 162 80 25 43 10 10 13 1400 455 455 130 130 162 33 25 10 0 0 14 1300 455 455 130 130 85 20 25 0 0 0 15 1200 455 455 130 130 30 0 0 0 0 0 16 1050 455 310 130 130 25 0 0 0 0 0 17 1000 455 260 130 130 25 0 0 0 0 0 18 1100 455 360 130 130 25 0 0 0 0 0 19 1200 455 455 130 130 30 0 0 0 0 0 20 1400 455 455 130 130 162 33 25 10 0 0 21 1300 455 455 130 130 85 20 25 0 0 0 22 1100 455 455 0 0 145 20 25 0 0 0 23 900 455 425. 0 0 0 20 0 0 0 0 24 800 455 345 0 0 0 0 0 0 0 0

100

Table 5.10 Best results obtained by the proposed Hybrid Model-I (DP-PSO-ANN) amongst the sixty samples: Test system-III


DP-PSO-ANN model P-1 P-2 P-3 P-4 P-5 P-6 P-7 P-8 P-9 P-10455 245 0 0 0 0 0 0 0 0 455 295 0 0 0 0 0 0 0 0 455 370 0 0 25 0 0 0 0 0 455 455 0 0 40 0 0 0 0 0 455 390 0 130 25 0 0 0 0 0 455 360 130 130 25 0 0 0 0 0 455 410 130 130 25 0 0 0 0 0 455 455 130 130 30 0 0 0 0 0 455 455 130 130 85 20 25 0 0 0 455 455 130 130 162 33 25 10 0 0 455 455 130 130 162 73 25 10 10 0 455 455 129.96 129.92 162.14 80 25.01 42.96 10 10 455 455 130 130 162 33 25 10 0 0 455 455 130 130 85 20 25 0 0 0 455 455 130 130 30 0 0 0 0 0 455 310 130 130 25 0 0 0 0 0 455 260 130 130 25 0 0 0 0 0 455 360 130 130 25 0 0 0 0 0 455 455 130 130 30 0 0 0 0 0 455 455 130 130 162 33 25 10 0 0 455 455 130 130 85 20 25 0 0 0 455 455 0 0 145 20 25 0 0 0 455 425.01 0 0 0 20 0 0 0 0 455 345 0 0 0 0 0 0 0 0

101

Table 5.11 Unit Commitment Schedule and Power Sharing (MW) of the best solution obtained from proposed Hybrid Model-I (DP-PSO-ANN): Test system-III

Power output of each unit (MW)

Hour UC schedule

Load (MW)

1 2 3 4 5 6 7 8 9 10

1 1100000000 700 455 245 0 0 0 0 0 0 0 0 2 1100000000 750 455 295 0 0 0 0 0 0 0 0 3 1100100000 850 455 370 0 0 25 0 0 0 0 0 4 1100100000 950 455 455 0 0 40 0 0 0 0 0 5 1101100000 1000 455 390 0 130 25 0 0 0 0 0 6 1111100000 1100 455 360 130 130 25 0 0 0 0 0 7 1111100000 1150 455 410 130 130 25 0 0 0 0 0 8 1111100000 1200 455 455 130 130 30 0 0 0 0 0 9 1111111000 1300 455 455 130 130 85 20 25 0 0 0 10 1111111100 1400 455 455 130 130 162 33 25 10 0 0 11 1111111110 1450 455 455 130 130 162 73 25 10 10 0 12 1111111111 1500 455 455 129.96 129.92 162.14 80 25.01 42.96 10 1013 1111111100 1400 455 455 130 130 162 33 25 10 0 0 14 1111111000 1300 455 455 130 130 85 20 25 0 0 0 15 1111100000 1200 455 455 130 130 30 0 0 0 0 0 16 1111100000 1050 455 310 130 130 25 0 0 0 0 0 17 1111100000 1000 455 260 130 130 25 0 0 0 0 0 18 1111100000 1100 455 360 130 130 25 0 0 0 0 0 19 1111100000 1200 455 455 130 130 30 0 0 0 0 0 20 1111111100 1400 455 455 130 130 162 33 25 10 0 0 21 1111111000 1300 455 455 130 130 85 20 25 0 0 0 22 1111111000 1100 455 455 0 0 145 20 25 0 0 0 23 1100010000 900 455 425.01 0 0 0 20 0 0 0 0 24 1100000000 800 455 345 0 0 0 0 0 0 0 0

102

Table 5.12 Load demand and Total Production Cost (TPC) obtained by proposed Hybrid Model-I (DP-PSO-ANN): Test system-III:

Hour Load

(MW)

Fuel cost ($.h)

Startup Cost ($/h)

Total Production Cost ($/h)

1 700 13683.13 0 13683.13 2 750 14554.50 0 14554.50 3 850 16809.45 900 17709.45 4 950 18597.67 0 18597.67 5 1000 20020.02 560 20580.02 6 1100 22387.04 1100 23487.04 7 1150 23261.98 0 23261.98 8 1200 24150.34 0 24150.34 9 1300 27251.06 860 28111.06 10 1400 30057.55 60 30117.55 11 1450 31916.06 60 31976.06 12 1500 33890.28 60 33950.28 13 1400 30057.55 0 30057.55 14 1300 27251.06 0 27251.06 15 1200 24150.34 0 24150.34 16 1050 21513.66 0 21513.66 17 1000 20641.82 0 20641.82 18 1100 22387.04 0 22387.04 19 1200 24150.34 0 24150.34 20 1400 30057.55 490 30547.55 21 1300 27251.06 0 27251.06 22 1100 22735.52 0 22735.52 23 900 17645.54 0 17645.54 24 800 15427.42 0 15427.42

Sum 27100 559847.98 4090.00 563937.98

103

Table 5.13 Comparison of the best results of the Hybrid Model-I (DP- PSO-ANN) with other approaches available in the literature: Test system-III

Approach DP

[56] LR

[56] ELR [36] EPL [15] SPL [16] RPACO

Total production cost($)

565825 565825 563977 563977 564950 565302

Approach HPSO [80]

IPSO LR-PPSO GA-UCC [57]

EMOALHN

TPC ($) 563942 563954 563977 563977 563977 Approach FPGA

[60] PSO [65]

MRCGA [61]

GA-LR LR-PSO ALR

TPC ($) 564094 564212 564244 564800 565275 565508 Approach GA

[56] GRASP MA ICGA [59] DPHNN BPSO

TPC ($) 565825 565825 565827 566404 588750 565804 Approach AG EP [52] ESA [44] ASSA [45] AG SA TPC ($) 564005 564551 565828 563938 564005 565825

Approach PLEA [82]

DP-LR EP-LR ACSA PLEA [82]

TPC ($) 563977 564049 564049 564059 563977 Proposed Hybrid Model-I (DP-PSO-ANN) 563937.98

• Abbreviations are given in Appendix-B.

104

Table 5.14 Comparison between proposed Hybrid model-I and other approaches for daily saving and Percentage saving in fuel cost: ten unit system

S.

No.

Approach

Total Production

Cost ($)

Amount of daily saving ($) compared

with proposed hybrid model –I

(DP-PSO-ANN).

% saving in fuel cost compared with proposed

model. (DP-PSO-ANN).

1 PSO-SA 563938 0.02 3.55E-06 2 ASSA 563938 0.02 3.55E-06 3 HPSO 563942 4.02 0.000713 4 IPSO 563954 16.02 0.002841 5 ELR 563977 39.02 0.006919 6 EPL 563977 39.02 0.006919 7 LR-PPSO 563977 39.02 0.006919 8 GA-UCC 563977 39.02 0.006919 9 EMOALHN 563977 39.02 0.006919 10 PLEA 563977 39.02 0.006919 11 AG 564005 67.02 0.011884 12 DP-LR 564049 111.02 0.019687 13 EP-LR 564049 111.02 0.019687 14 ACSA 564059 121.02 0.021460 15 FPGA 564094 156.02 0.027666 16 PSO 564212 274.02 0.04859 17 MRCGA 564244 306.02 0.054265 18 EP 564551 613.02 0.108703 19 GA-LR 564800 862.02 0.152857 20 SPL 564950 1012.02 0.179456 21 LR-PSO-1 565275 1337.02 0.237086 22 RPACO 565302 1364.02 0.241874 23 ALR 565508 1570.02 0.278403 24 BPSO 565804 1866.02 0.330891 25 DP 565825 1887.02 0.334615 26 LR 565825 1887.02 0.334615 27 SA 565825 1887.02 0.334615 28 GA 565825 1887.02 0.334615 29 GRASP 565825 1887.02 0.334615 30 MA 565827 1889.02 0.334969 31 LR-PSO-2 565869 1931.02 0.342417 32 ICGA 566404 2466.02 0.437286 33 BCGA 567367 3429.02 0.608049 34 DPHNN 588750 24812.02 4.399778 35 Proposed DP-PSO-ANN 563937.98

105

Table 5.15 Comparison of the results of the proposed Hybrid Model-I with Genetic Algorithm, dynamic programming, simulated annealing and Lagrange relaxation method:

Approach/Model Total

ProductionCost ($)

Amount of Daily Saving ($)

% Saving

GA,DP,SA and LR 565825.00 - - Hybrid Model-I (SI learning) 563937.98 1887.02 0.333 Hybrid Model-I (BP learning) 563935.58 1889.42 0.333

Table 5.16 Unit Commitment Schedule and Power Sharing (MW) of the best solution obtained by using (DP-BP-ANN), considering MUT and MDT constraints with 10 % spinning

reserve: Test system-III

Power output of each unit (MW) Hour UC schedule

Load (MW)

1 2 3 4 5 6 7 8 9 10 1 1100000000 700 455 245 0 0 0 0 0 0 0 0 2 1100000000 750 455 294.99 0 0 0 0 0 0 0 0 3 1100100000 850 455 370 0 0 24.99 0 0 0 0 0 4 1100100000 950 455 455 0 0 40 0 0 0 0 0 5 1101100000 1000 455 390 0 130 25 0 0 0 0 0 6 1111100000 1100 455 360 130 130 25 0 0 0 0 0 7 1111100000 1150 455 410 130 130 25 0 0 0 0 0 8 1111100000 1200 455 455 130 130 30 0 0 0 0 0 9 1111111000 1300 455 455 130 130 85 19.99 25 0 0 0 10 1111111100 1400 455 455 130 130 162 33 25 10 0 0 11 1111111110 1450 455 455 130 130 162 73 25 10 10 0 12 1111111111 1500 455 455 130 130 162 80 25 42.99 10 1013 1111111100 1400 455 455 130 130 162 33 25 10 0 0 14 1111111000 1300 455 455 130 130 85 19.99 25 0 0 0 15 1111100000 1200 455 455 130 130 30 0 0 0 0 0 16 1111100000 1050 455 310 130 130 25 0 0 0 0 0 17 1111100000 1000 455 260 130 130 25 0 0 0 0 0 18 1111100000 1100 455 360 130 130 25 0 0 0 0 0 19 1111100000 1200 455 455 130 130 30 0 0 0 0 0 20 1111111100 1400 455 455 130 130 162 33 25 10 0 0 21 1111111000 1300 455 455 130 130 85 19.99 25 0 0 0 22 1111111000 1100 455 455 0 0 145 19.99 25 0 0 0 23 1100010000 900 454.99 424.99 0 0 0 19.99 0 0 0 0 24 1100000000 800 455 345.000 0 0 0 0 0 0 0 0

106

Table 5.17 Load demand, operating fuel cost, startup cost and Total Production Cost obtained by Using (DP-BP-ANN), considering M.U.T and M.D.T Constraints with 10 %

spinning reserve: Test system-III:

Hour Load (MW)

Fuel cost ($)

Startup Cost ($)

Total Production

Cost ($) 1 700 13683.13 0 13683.13 2 750 14554.33 0 14554.33 3 850 16809.25 900 17709.25 4 950 18597.67 0 18597.67 5 1000 20020.02 560 20580.02 6 1100 22387.04 1100 23487.04 7 1150 23261.98 0 23261.98 8 1200 24150.34 0 24150.34 9 1300 27250.83 860 28110.83 10 1400 30057.55 60 30117.55 11 1450 31916.06 60 31976.06 12 1500 33889.90 60 33949.90 13 1400 30057.55 0 30057.55 14 1300 27250.83 0 27250.83 15 1200 24150.34 0 24150.34 16 1050 21513.66 0 21513.66 17 1000 20641.82 0 20641.82 18 1100 22387.04 0 22387.04 19 1200 24150.34 0 24150.34 20 1400 30057.55 490 30547.55 21 1300 27250.83 0 27250.83 22 1100 22735.30 0 22735.30 23 900 17644.7968 0 17644.80 24 800 15427.42 0 15427.42

Sum 27100 559845.58 4090.00 563935.58

0.0000.0020.0040.0060.0080.0100.012

1 7 13 19 25 31 37 43 49 55

No. of Samples

AP

E

Series1

Figure 5.8 Absolute Percentage Error (APE) for a load of 900MW (DP-BP-ANN Model):

Ten unit system: Test system-III

107

5.4 Hybrid Model-II: Neuro-Genetic Hybrid Approach

The proposed Hybrid Model-II (Neuro-Genetic approach) combines the Genetic

Algorithm (GA) with Artificial Neural Networks (ANN) using Swarm Intelligence (SI) learning

rule. Unit commitment problem has been addressed independently by Genetic Algorithm [56],

Particle Swarm Optimization and Artificial Neural Network. GA has the ability to search better

for non-convex fuel cost function than convex fuel cost function. Three machine standard test

system has been tested for validation of the proposed model. The discussion to follow is with

reference to context of the discussion in chapter 3.

In this model Genetic Algorithm produces near optimal solution based training data for

neural network model. The neural network fine tunes the data subject to the target values. The

best fine tuned solution is considered the required solution. The target values may be taken by

randomly generated values by GA around the near optimal solutions satisfying both equality and

inequality constraints. The SI-learning rule based feed forward neural network has been used for

fine tuning the near optimal GA results. The standard back propagation learning rule has also

been used in this hybrid approach for the comparison of results. Flow chart in Figure 5.9

highlights the steps in the hybrid methodology

In this model the test data and training data are generated by using Genetic Algorithm. For

proper training of ANN, a pair of load as input and their corresponding generation schedules as

output are prepared off-line by using Genetic Algorithm (GA) and are stored in a data base. Each

pair is referred as input/output database pair.

108

No Yes

Figure 5.9 Flowchart for GA-PSO-ANN, Neuro-Genetic Hybrid Model-II

Start

Read System Data

GA is used to generate training and test data for 24 hour based on forecasted load.

Initialization

ANN training Using SI learning to obtain UC schedules

Testing of test cases and Calculation of fuel cost

End

Check for comparison ANN using BP

Learning

ANN using BP learning to obtain UC Schedules

109

5.5 Case Studies --- non convex cost function Following two standard test systems have been selected for the illustration of the

effectiveness of the proposed approaches.

a. Test System IV --- 3 units system with 24 hours load.

b. Test System V --- 3 units system with 24 hours load.

Input data: The description and input data of test systems used for investigation in the case

studies is given in Table A. 4 and Table A.5 placed in Appendix A.


Computer.


5.5.1 Numerical Results of Test Systems –IV and V: 3 units systems: Hybrid Model-II

Sixty samples consisting of three inputs for three unit system to the neural network produce

fine tuned sixty outputs. The outputs consist of one to three power generations depending upon

forecasted load.

The best fine tuned results generated from both the ANN models amongst the sixty samples

and corresponding near optimal output generated by GA have been tabulated.

The output results of non convex test system-IV and V are shown in the following tables.

Table (5.18) presents the comparison of the best operating fuel cost ($) obtained amongst the

sixty samples for 3 unit system IV and V.


solution obtained from the Hybrid Model-II: (GA-PSO-ANN) for Test system-IV.

Table (5.20) presents the Unit Commitment Schedule and Power Sharing (MW) of the solution

obtained from the Hybrid Model-II (GA-BP-ANN) for Test system-IV.


obtained from the Hybrid Model-II: (GA-PSO-ANN): for Test system-V.

110


obtained from the Hybrid Model-II: (GA-BP-ANN) for Test system-V.

The Salient features of the proposed approach in the light of the observations from the


1. For non-convex cost function of test systems IV and V the hybrid model-II gives better results than hybrid model-III.

Table 5.18 Comparison of the best Operating fuel cost ($) obtained amongst the sixty samples for 3 unit system by using Proposed Hybrid Model-II: (Non-convex)

(GA-PSO-ANN) (GA-BP-ANN) Operating fuel cost Test system-IV $ 207773.614 $ 207772.477 Operating fuel cost Test system-V $ 200665.137 $ 200662.723

111

Table 5.19 Unit Commitment Schedule and Power Sharing (MW) of the best solution obtained from the Hybrid Model-II: (GA-PSO-ANN): Test system-IV



Hour

UC Schedule

P-1 P-2 P-3

Load (MW)

F-1 F-2 F-3

Operating Fuel Cost

($/h)

1 111 600.00 400.00 200.00 1200 6175.105 3767.125 2241.543 12183.773 2 111 600.00 400.00 200.00 1200 6175.105 3767.125 2241.543 12183.773 3 111 600.00 400.00 150.00 1150 6175.105 3767.125 1660.862 11603.092 4 111 600.00 400.00 100.00 1100 6175.105 3767.125 1109.101 11051.331 5 110 600.00 400.00 0 1000 6175.105 3767.125 0 9942.230 6 110 551.64 348.36 0 900 5430.872 3449.030 0 8879.902 7 110 542.05 257.95 0 800 5377.510 2532.705 0 7910.216 8 100 600.00 0 0 600 6175.105 0 0 6175.105 9 100 550.00 0 0 550 5399.592 0 0 5399.592 10 100 500.00 0 0 500 5211.370 0 0 5211.370 11 100 500.00 0 0 500 5211.370 0 0 5211.370 12 100 500.00 0 0 500 5211.370 0 0 5211.370 13 100 500.00 0 0 500 5211.370 0 0 5211.370 14 100 500.00 0 0 500 5211.370 0 0 5211.370 15 100 600.00 0 0 600 6175.105 0 0 6175.105 16 110 542.05 257.95 0 800 5377.510 2532.705 0 7910.216 17 110 459.37 390.63 0 850 4623.302 3742.912 0 8366.214 18 110 551.64 348.36 0 900 5430.872 3449.030 0 8879.902 19 110 550.00 400.00 0 950 5399.592 3767.125 0 9166.717 20 110 600.00 400.00 0 1000 6175.105 3767.125 0 9942.230 21 111 600.00 400.00 50.00 1050 6175.105 3767.125 586.260 10528.490 22 111 600.00 400.00 100.00 1100 6175.105 3767.125 1109.101 11051.331 23 111 600.00 400.00 200.00 1200 6175.105 3767.125 2241.543 12183.773 24 111 600.00 400.00 200.00 1200 6175.105 3767.125 2241.543 12183.773

SUM 207773.614

112

Table 5.20 Unit Commitment Schedule and Power Sharing (MW) of the solution obtained from the Hybrid Model-II (GA-BP-ANN): Test system-IV



Hour

UC Schedule

P-1 P-2 P-3

Load (MW)

F-1 F-2 F-3

Operating Fuel Cost

($/h)

1 111 599.999 399.999 199.999 1200 6175.096 3767.107 2241.522 12183.724 2 111 599.999 399.999 199.999 1200 6175.096 3767.107 2241.522 12183.724 3 111 599.999 399.999 149.999 1150 6175.096 3767.107 1660.841 11603.044 4 111 599.999 399.999 99.999 1100 6175.096 3767.107 1109.081 11051.284 5 110 599.984 400 0 1000 6174.954 3767.125 0 9942.079 6 110 551.64 348.36 0 900 5430.872 3449.030 0 8879.902 7 110 542.05 257.95 0 800 5377.510 2532.705 0 7910.216 8 100 599.999 0 0 600 6175.096 0 0 6175.096 9 100 549.999 0 0 550 5399.573 0 0 5399.573 10 100 499.999 0 0 500 5211.361 0 0 5211.361 11 100 499.999 0 0 500 5211.361 0 0 5211.361 12 100 499.999 0 0 500 5211.361 0 0 5211.361 13 100 499.999 0 0 500 5211.361 0 0 5211.361 14 100 499.999 0 0 500 5211.361 0 0 5211.361 15 100 599.999 0 0 600 6175.096 0 0 6175.096 16 110 542.05 257.95 0 800 5377.510 2532.705 0 7910.216 17 110 459.37 390.63 0 850 4623.302 3742.912 0 8366.214 18 110 551.64 348.36 0 900 5430.872 3449.030 0 8879.902 19 110 550.009 399.969 0 950 5399.764 3766.573 0 9166.337 20 110 599.984 400 0 1000 6174.954 3767.125 0 9942.079 21 111 599.999 399.999 49.99 1050 6175.096 3767.107 586.253 10528.456 22 111 599.999 399.999 99.999 1100 6175.096 3767.107 1109.081 11051.284 23 111 599.999 399.999 199.999 1200 6175.096 3767.107 2241.522 12183.724 24 111 599.999 399.999 199.999 1200 6175.096 3767.107 2241.522 12183.724

SUM 207772.477

113

Table 5.21 Unit Commitment Schedule and Power Sharing (MW) of the solution obtained from the Hybrid Model-II: (GA-PSO-ANN): Test system-V



Hour

UC Schedule

P-1 P-2 P-3

Load (MW)

F-1 F-2 F-3

Operating Fuel Cost

($/h)

1 111 600.00 400.00 200.00 1200 5887.927 3767.125 1868.583 11523.635 2 111 600.00 400.00 200.00 1200 5887.927 3767.125 1868.583 11523.635 3 111 600.00 400.00 150.00 1150 5887.927 3767.125 1384.472 11039.524 4 111 500.36 400.00 199.63 1100 4928.400 3767.125 1861.425 10556.950 5 110 600.00 400.00 0 1000 5887.927 3767.125 0 9655.052 6 110 500.00 400.00 0 900 4921.587 3767.125 0 8688.712 7 110 408.20 391.80 0 800 4138.137 3744.593 0 7882.730 8 100 600.00 0 0 600 5887.927 0 0 5887.927 9 100 550.00 0 0 550 5689.290 0 0 5689.290 10 100 500.00 0 0 500 4921.587 0 0 4921.587 11 100 500.00 0 0 500 4921.587 0 0 4921.587 12 100 500.00 0 0 500 4921.587 0 0 4921.587 13 100 500.00 0 0 500 4921.587 0 0 4921.587 14 100 500.00 0 0 500 4921.587 0 0 4921.587 15 100 600.00 0 0 600 5887.927 0 0 5887.927 16 110 408.20 391.80 0 800 4138.137 3744.593 0 7882.730 17 110 593.47 256.53 0 850 5860.308 2508.824 0 8369.131 18 110 500.00 400.00 0 900 4921.587 3767.125 0 8688.712 19 110 599.48 350.52 0 950 5877.847 3477.965 0 9355.812 20 110 600.00 400.00 0 1000 5887.927 3767.125 0 9655.052 21 111 600.00 400.00 50.00 1050 5887.927 3767.125 488.550 10143.602 22 111 600.00 400.00 100.00 1100 5887.927 3767.125 924.461 10579.513 23 111 600.00 400.00 200.00 1200 5887.927 3767.125 1868.583 11523.635 24 111 600.00 400.00 200.00 1200 5887.927 3767.125 1868.583 11523.635

SUM 200665.137

.

114

Table 5.22 Unit Commitment Schedule and Power Sharing (MW) of the solution obtained from the Hybrid Model-II: (GA-BP-ANN) Test system-V



Hour

UC Schedule

P-1 P-2 P-3

Load (MW)

F-1 F-2 F-3

Operating Fuel Cost

($/h)

1 111 600 400 199.999 1200 5887.927 3767.125 1868.564 11523.615 2 111 600 400 199.999 1200 5887.927 3767.125 1868.564 11523.615 3 111 600 400 149.999 1150 5887.927 3767.125 1384.453 11039.505 4 111 500.36 399.99 199.63 1100 4928.400 3766.947 1861.425 10556.772 5 110 599.99 399.99 0 1000 5887.735 3766.947 0 9654.682 6 110 499.99 399.99 0 900 4921.398 3766.947 0 8688.344 7 110 408.20 391.80 0 800 4138.137 3744.593 0 7882.730 8 100 599.999 0 0 600 5887.908 0 0 5887.908 9 100 549.999 0 0 550 5689.281 0 0 5689.281 10 100 499.999 0 0 500 4921.568 0 0 4921.568 11 100 499.999 0 0 500 4921.568 0 0 4921.568 12 100 499.999 0 0 500 4921.568 0 0 4921.568 13 100 499.999 0 0 500 4921.568 0 0 4921.568 14 100 499.999 0 0 500 4921.568 0 0 4921.568 15 100 599.999 0 0 600 5887.908 0 0 5887.908 16 110 408.20 391.80 0 800 4138.137 3744.593 0 7882.730 17 110 593.47 256.53 0 850 5860.308 2508.824 0 8369.131 18 110 499.99 399.99 0 900 4921.398 3766.947 0 8688.344 19 110 599.48 350.52 0 950 5877.847 3477.965 0 9355.812 20 110 599.99 399.99 0 1000 5887.735 3766.947 0 9654.682 21 111 600.00 400.00 49.99 1050 5887.927 3767.125 488.560 10143.612 22 111 600.00 400.00 99.99 1100 5887.927 3767.125 924.277 10579.329 23 111 600.00 400.00 199.99 1200 5887.927 3767.125 1868.389 11523.441 24 111 600.00 400.00 199.99 1200 5887.927 3767.125 1868.389 11523.441

SUM 200662.723

115

5.6 Hybrid Model –III: Scaleable deterministic hybrid approach

The proposed hybrid model-III combines the Maximum Power Output (PMAX) and Full

Load Average Production Cost (FLAPC) of each unit. This model is applicable for both convex

and non convex fuel cost functions.

The priority list method has been used independently [15, 16] and by incorporating the

priority list in evolutionary programming (PLEA) [82]. The extended priority list (EPL) in [15]

is one of the most significant works. The main drawback in these methods is that the final

solution is non deterministic. The choice of the merit-order method for the hybridization process

was based on the following merits:

• Simple to implement

• Computationally fast

• Most widely used technique by the electricity utilities

The proposed unit commitment algorithm hybrid model-III incorporates the unit

commitment solution by the deterministic priority list scheme for generation of initial schedules.

This hybrid model solves the UC problem for convex and non-convex fuel cost functions. The

UC schedule is prepared according to PMAX and FLAPC. Unit with higher PMAX will be at

higher priority. If the two units have the same PMAX then unit with lower FLAPC will be of

higher priority. Economic Dispatch is based on lambda iteration method and average load

assigned methods. Priority order is given in Table 5.23. Flow chart is represented in Figure 5.10.

Table 5.23 Proposed Priority order based on Hybrid Model-III (PMAX-FLAPC)

Unit No.

PMAX FLAPC (S/MWH)

Proposed Priority order

(PMAX-FLAPC)

Unit No.

PMAX FLAPC (S/MWH)

Proposed Priority order (PMAX-

FLAPC) 1 455 18.6062 1 6 80 27.4546 7

2 455 19.5329 2 7 85 33.4542 6

3 130 22.2446 5 8 55 38.1472 8

4 130 22.0051 4 9 55 39.4830 9

5 162 23.1225 3 10 55 40.0670 10

116

Figure 5.10 Flow chart for Hybrid Model–III (PMAX-FLAPC)

117

5.7 Case Studies ---Convex fuel cost function

The following standard test systems have been selected for the validation of the proposed

approach.


b. Test System-II --- 3 units with 24 hours load

c. Test System III --- 10 units with 24 hours load

Input data: The description and data of test system used for investigation in the case studies is

given in Table A.1, Table A.2 and Table A.3 placed in Appendix A.


Computer.

Output Results: The summary and comparison of results is given in this section.

5.7.1 Numerical results of test system –I: three unit system: Hybrid Model-III

The output results of the test system-I are shown in the following tables:

Table (5.24) presents the Comparison of proposed hybrid model-III with Genetic Algorithm,

Conventional Priority List and Hopfield Neural Network methods.

Table (5.25) presents the comparison of the Summary of unit commitment schedules of the

proposed Hybrid Models I and III for 3 units System.

Table (5.26) gives the comparison of Number of Units in Operation for 3 unit systems for hybrid

models I and III with three single approaches.

Table (5.27) shows the Unit Commitment Schedule and Power Sharing (MW) of the best

solution obtained from the Proposed Hybrid Model-III: (PMAX-FLAPC2, ED based on average

load assigned method).

Table (5.28) shows the Unit Commitment Schedule and Power Sharing (MW) of the best

solution obtained from the proposed Hybrid Model-III (PMAX-FLAPC3) (ED based on lambda

Iteration method).

The Salient features of the conventional approaches and the proposed hybrid model-I in the light

of the observations from the results are as follows:

118

1. Table 5.24 presents the comparison of the proposed hybrid model-III with other methods

available in the literature i.e. Genetic Algorithm (GA), Conventional priority list (PL)

method and Hopfield Neural Network (HFNN).

2. Hybrid model-III gives better results than GA, conventional priority list and HFNN.

3. The daily cost saving of Hybrid Model-III amount to $389.28 compared with Genetic

Algorithm. This is equivalent to a percentage saving of 0.1926 as shown in Table 5.24.

4. The daily cost saving of Hybrid Model-III amount to $474.28 compared with Hopfield

Neural Network. This is equivalent to a percentage saving of 0.2346 as shown in Table

5.24.

5. The daily cost saving of Hybrid Model-III amount to $761.26 compared with

Conventional priority list. This is equivalent to a percentage saving of 0.37612 as shown

in Table 5.24.

6. Hybrid Model-III (PMAX-FLAPC2, average load assigned method) is applicable to both

convex and non-convex fuel cost curves. The operating fuel cost is $201640.215. This

method is very fast in speed in calculating the optimal schedule of the generators. This

method gives fuel cost $8.13 higher than hybrid model-III (ED method) as given in Table

5.24.

7. In the proposed hybrid models-I and III when the load is 500 MW only unit 1 is in

operation thereby giving saving in startup cost.

119

Table 5.24 Comparison of proposed Hybrid Model-III with Genetic Algorithm, Conventional Priority List and Hopfield Neural Network methods for 3 unit systems:

Test system-I:

Algorithm Daily Operating Cost ($)

Amount of Daily Saving (comparedwith GA)

% saving in fuel cost compared with GA

Amount of Daily Saving (compared with conventional PL)

% saving in fuel cost compared with PL

Amount of Daily Saving (compared with Hopfield Neural Network

% saving in fuel cost compared with Hopfield Neural Network

Genetic Algorithm

202021.360 - - - - - -

Conventional Priority List (FLAPC)

202393.346 - - - - - -

Hopfield Neural Network

202106.36 - - - - - -

Hybrid Model–III (PMAX-FLAPC2)

201640.215 381.15 0.1886 753.13 0.37211 466.15 0.2306

Hybrid Model–III (PMAX-FLAPC3)

201632.089 389.28 0.1926 761.26 0.37612 474.28 0.2346

120

Table 5.25 Comparison of the Summary of unit commitment schedules of the proposed Hybrid Models I and III for 3 Unit Systems: Test System –I

Single

approach- I

Single approach –

II

Proposed Single

approach-III

Hybrid Model-

III

Hybrid Model-

III

Hybrid Model-

I

Hybrid Model-I

Load (MW)


Conventional priority list (FLAPC) with ED

UC(2,1,3)

Merit Order

(PMAX) with ED

UC(1,2,3)

Merit Order

(PMAX-FLAPC-

2) UC(1,2,3)ED(2,1,3)

Merit Order

(PMAX-FLAPC-3) with

ED UC(1,2,3

)

DP-PSO-ANN (SI-

learning)

DP-BP-ANN (BP-

learning)

1200 1200 1150 1100 1000 900 800 600 550 500 500 500 500 500 600 800 850 900 950 1000 1050 1100 1200 1200

111 111 111 111 110 110 110 100 100 011 011 011 011 011 100 110 110 110 110 110 111 111 111 111

111 111 111 111 110 110 110 110 110 110 110 110 110 110 110 110 110 110 110 110 111 111 111 111

111 111 111 111 110 110 110 100 100 100 100 100 100 100 100 110 110 110 110 110 111 111 111 111

111 111 111 111 110 110 110 100 100 100 100 100 100 100 100 110 110 110 110 110 111 111 111 111

111 111 111 111 110 110 110 100 100 100 100 100 100 100 100 110 110 110 110 110 111 111 111 111

111 111 111 111 110 110 110 100 100 100 100 100 100 100 100 110 110 110 110 110 111 111 111 111

111 111 111 111 110 110 110 100 100 100 100 100 100 100 100 110 110 110 110 110 111 111 111 111

201415. 789

202393. 346

201632. 089

201640. 215

201632. 089

201632. 089

201630. 923

No. of Transitions

No. of Transitions

No. of Transitions

No. of Transitions

No. of Transitions

No. of Transitions

No. of Transitions

10 2 4 4 4 4 4

1 on state 0 off state

121

Table 5.26 Comparison of Number of Units in Operation for 3 unit systems for hybrid models I and III with three single approaches: Test System I

Single approach-

I

Single approach-

II

Proposed Single

approach-III

Hybrid Model-

I

Hybrid Model-

I

Hybrid Model-

III

Hybrid Model-

III

Complete Enumerat-

ion

(Conventi-onal priority

list )

(PMAX) with ED

DP- PSO-ANN

DP- BP-ANN

(PMAX-FLAPC2)

(PMAX-FLAPC3)

Hr.

Load

units units units Units units units units

1 1200 3 3 3 3 3 3 3 2 1200 3 3 3 3 3 3 3 3 1150 3 3 3 3 3 3 3 4 1100 3 3 3 3 3 3 3 5 1000 2 2 2 2 2 2 2 6 900 2 2 2 2 2 2 2 7 800 2 2 2 2 2 2 2 8 600 1 2 1 1 1 1 1 9 550 1 2 1 1 1 1 1

10 500 2 2 1 1 1 1 1 11 500 2 2 1 1 1 1 1 12 500 2 2 1 1 1 1 1 13 500 2 2 1 1 1 1 1 14 500 2 2 1 1 1 1 1 15 600 1 2 1 1 1 1 1 16 800 2 2 2 2 2 2 2 17 850 2 2 2 2 2 2 2 18 900 2 2 2 2 2 2 2 19 950 2 2 2 2 2 2 2 20 1000 2 2 2 2 2 2 2 21 1050 3 3 3 3 3 3 3 22 1100 3 3 3 3 3 3 3 23 1200 3 3 3 3 3 3 3 24 1200 3 3 3 3 3 3 3

122

Table 5.27 Unit Commitment Schedule and Power Sharing (MW) of the best solution obtained from the Proposed Hybrid Model-III: (PMAX-FLAPC2, ED based on average load

assigned method): Test system-I

Power Output of each unit(MW)

Fuel Cost of each unit($/h)

Hour

UC Schedule

P-1 P-2 P-3

Load (MW)

F-1 F-2 F-3

Operating Fuel Cost

($/h)

1 111 600.00 400.00 200.00 1200 5875.320 3760.400 2237.760 11873.480 2 111 600.00 400.00 200.00 1200 5875.320 3760.400 2237.760 11873.480 3 111 600.00 400.00 150.00 1150 5875.320 3760.400 1658.340 11294.060 4 111 600.00 400.00 100.00 1100 5875.320 3760.400 1107.840 10743.560 5 110 600.00 400.00 0 1000 5875.320 3760.400 0 9635.720 6 110 500.00 400.00 0 900 4911.500 3760.400 0 8671.900 7 110 400.00 400.00 0 800 3978.920 3760.400 0 7739.320 8 100 600.00 0 0 600 5875.320 0 0 5875.320 9 100 550.00 0 0 550 5389.505 0 0 5389.505 10 100 500.00 0 0 500 4911.500 0 0 4911.500 11 100 500.00 0 0 500 4911.500 0 0 4911.500 12 100 500.00 0 0 500 4911.500 0 0 4911.500 13 100 500.00 0 0 500 4911.500 0 0 4911.500 14 100 500.00 0 0 500 4911.500 0 0 4911.500 15 100 600.00 0 0 600 5875.320 0 0 5875.320 16 110 400.00 400.00 0 800 3978.920 3760.400 0 7739.320 17 110 450.00 400.00 0 850 4441.305 3760.400 0 8201.705 18 110 500.00 400.00 0 900 4911.500 3760.400 0 8671.900 19 110 550.00 400.00 0 950 5389.505 3760.400 0 9149.905 20 110 600.00 400.00 0 1000 5875.320 3760.400 0 9635.720 21 111 600.00 400.00 50.00 1050 5875.320 3760.400 586.260 10221.980 22 111 600.00 400.00 100.00 1100 5875.320 3760.400 1107.840 10743.560 23 111 600.00 400.00 200.00 1200 5875.320 3760.400 2237.760 11873.480 24 111 600.00 400.00 200.00 1200 5875.320 3760.400 2237.760 11873.480

SUM 201640.215

123

Table 5.28 Unit Commitment Schedule and Power Sharing (MW) of the best solution obtained from the proposed Hybrid Model-III (PMAX-FLAPC3) (ED based on lambda Iteration

method): Test system-I

Power output of each unit(MW)


Hour

UC Schedule

P-1 P-2 P-3

Load (MW)

F-1 F-2 F-3

Operating Fuel Cost

($/h)

1 111 600.00 400.00 200.00 1200 5875.320 3760.400 2237.760 11873.480 2 111 600.00 400.00 200.00 1200 5875.320 3760.400 2237.760 11873.480 3 111 600.00 400.00 150.00 1150 5875.320 3760.400 1658.340 11294.060 4 111 600.00 400.00 100.00 1100 5875.320 3760.400 1107.840 10743.560 5 110 600.00 400.00 0 1000 5875.320 3760.400 0 9635.720 6 110 500.00 400.00 0 900 4911.500 3760.400 0 8671.900 7 110 433.18 366.82 0 800 4284.887 3450.577 0 7735.464 8 100 600.00 0.00 0 600 5875.320 0.000 0 5875.320 9 100 550.00 0.00 0 550 5389.505 0.000 0 5389.505 10 100 500.00 0.00 0 500 4911.500 0.000 0 4911.500 11 100 500.00 0.00 0 500 4911.500 0.000 0 4911.500 12 100 500.00 0.00 0 500 4911.500 0.000 0 4911.500 13 100 500.00 0.00 0 500 4911.500 0.000 0 4911.500 14 100 500.00 0.00 0 500 4911.500 0.000 0 4911.500 15 100 600.00 0.00 0 600 5875.320 0.000 0 5875.320 16 110 433.18 366.82 0 800 4284.887 3450.577 0 7735.464 17 110 460.88 389.12 0 850 4542.955 3658.336 0 8201.290 18 110 500.00 400.00 0 900 4911.500 3760.400 0 8671.900 19 110 550.00 400.00 0 950 5389.505 3760.400 0 9149.905 20 110 600.00 400.00 0 1000 5875.320 3760.400 0 9635.720 21 111 600.00 400.00 50.00 1050 5875.320 3760.400 586.260 10221.980 22 111 600.00 400.00 100.00 1100 5875.320 3760.400 1107.840 10743.560 23 111 600.00 400.00 200.00 1200 5875.320 3760.400 2237.760 11873.480 24 111 600.00 400.00 200.00 1200 5875.320 3760.400 2237.760 11873.480

SUM 201632.089

5.7.2 Numerical Results of Test System-II: Three units system: Hybrid Model-III:

(Convex Fuel Cost Curve)

In this study the fuel cost of the unit-3 which is coal fired unit is decreased from 1.2 to

1.0, whereas the fuel cost of unit 1 and 2 is same as in Test System-I. By decrease in fuel cost of

unit 3, the unit 3 which is uneconomical unit in study-1 and has the lowest priority now becomes

124

the most economical unit in this study. The priority of units based on FLAPC in this study is

Unit 3, 2, 1. Whereas the priority in test system-I is unit 2, 1, 3.

The output results of the test system-II are shown in the following tables:

Table (5.29) shows the comparison of the operating fuel cost ($) for proposed Hybrid Models -I

and III for 3 unit system.

Table (5.30) gives the summary of Unit Commitment Schedules for 3 unit systems

Table (5.31) presents the summary of number of Units in Operation for 3 unit system.

Table (5.32) presents the Unit commitment schedule and Power Sharing (MW) of the best

solution obtained by Hybrid Model-III (PMAX-FLAPC with ED based on average load) for 3

unit systems.

Table (5.33) presents the Unit commitment schedule and Power Sharing (MW) of the best

solution obtained by Hybrid Model-III (PMAX-FLAPC with ED) for 3 unit systems.

The salient features of the proposed approaches in the light of the observations from the


1. Hybrid model-I using SI-learning hits the target values while the BP-learning give results

very close to the target values.

2. Hybrid model-III gives high quality solution very quickly.

Table 5.29 Comparison of the operating fuel cost ($) for proposed Hybrid Models -I and III

for 3 unit system: Test System –II (Convex Fuel Cost Curve)

Proposed Hybrid Approaches

Hybrid Model-I (SI-Learning)

Hybrid Model-I (BP-Learning)

Hybrid Model-III (ED based on load to Economic unit)

Hybrid Model-III (Lambda Iteration

Method) DP-PSO-ANN DP-BP-ANN (PMAX-FLAPC2)

UC(1,2,3) ED(3,2,1)

(PMAX-FLAPC3) with ED


199262.671 199262.310 199299.8125 199262.671

125

Table 5.30 Summary of Unit Commitment Schedules for 3 unit systems: Test System II:

Hybrid Model-I

Hybrid Model-I

Hybrid Model-III

Hybrid Model-III

Load (MW)

DP-PSO-ANN DP-BP-ANN (PMAX-LAPC2)

(PMAX-FLAPC3)

1200 1200 1150 1100 1000 900 800 600 550 500 500 500 500 500 600 800 850 900 950 1000 1050 1100 1200 1200

111 111 111 111 110 110 110 100 100 100 100 100 100 100 100 110 110 110 110 110 111 111 111 111

111 111 111 111 110 110 110 100 100 100 100 100 100 100 100 110 110 110 110 110 111 111 111 111

111 111 111 111 110 110 110 100 100 100 100 100 100 100 100 110 110 110 110 110 111 111 111 111

111 111 111 111 110 110 110 100 100 100 100 100 100 100 100 110 110 110 110 110 111 111 111 111

199262.671 199262.310 199299.812 199262.671 No. of

Transitions No. of

Transitions No. of

Transitions No. of

Transitions 4 4 4 4

126

Table 5.31 Summary of number of Units in Operation for 3 unit system: Test system II:

Hybrid Model-I

Hybrid Model-I

Hybrid Model-III

Hybrid Model-III

DP-PSO-ANN DP-BP-ANN (PMAX-FLAPC2)

(PMAX-FLAPC3)

Hr Load units units units units 1 1200 3 3 3 3 2 1200 3 3 3 3 3 1150 3 3 3 3 4 1100 3 3 3 3 5 1000 2 2 2 2 6 900 2 2 2 2 7 800 2 2 2 2 8 600 1 1 1 1 9 550 1 1 1 1 10 500 1 1 1 1 11 500 1 1 1 1 12 500 1 1 1 1 13 500 1 1 1 1 14 500 1 1 1 1 15 600 1 1 1 1 16 800 2 2 2 2 17 850 2 2 2 2 18 900 2 2 2 2 19 950 2 2 2 2 20 1000 2 2 2 2 21 1050 3 3 3 3 22 1100 3 3 3 3 23 1200 3 3 3 3 24 1200 3 3 3 3

127

Table 5.32 Unit commitment schedule and Power Sharing (MW) of the best solution obtained by Hybrid Model-III (PMAX-FLAPC with ED based on average load) for 3 unit systems:

Test System-II



Hr.

UC Sch.

P-1 P-2 P-3

Load (MW)

F-1 F-2 F-3

Operating Fuel Cost

($/h)

1 111 600.00 400.00 200.00 1200 5875.320 3760.400 1864.800 11500.520 2 111 600.00 400.00 200.00 1200 5875.320 3760.400 1864.800 11500.520 3 111 550.00 400.00 200.00 1150 5389.505 3760.400 1864.800 11014.705 4 111 500.00 400.00 200.00 1100 4911.500 3760.400 1864.800 10536.700 5 111 600.00 400.00 0.00 1000 5875.320 3760.400 0.000 9635.720 6 111 500.00 400.00 0.00 900 4911.500 3760.400 0.000 8671.900 7 110 400.00 400.00 0.00 800 3978.920 3760.400 0.000 7739.320 8 100 600.00 0.00 0.00 600 5875.320 0.000 0.000 5875.320 9 100 550.00 0.00 0.00 550 5389.505 0.000 0.000 5389.505 10 100 500.00 0.00 0.00 500 4911.500 0.000 0.000 4911.500 11 100 500.00 0.00 0.00 500 4911.500 0.000 0.000 4911.500 12 100 500.00 0.00 0.00 500 4911.500 0.000 0.000 4911.500 13 100 500.00 0.00 0.00 500 4911.500 0.000 0.000 4911.500 14 100 500.00 0.00 0.00 500 4911.500 0.000 0.000 4911.500 15 100 600.00 0.00 0.00 600 5875.320 0.000 0.000 5875.320 16 110 400.00 400.00 0.00 800 3978.920 3760.400 0.000 7739.320 17 110 450.00 400.00 0.00 850 4441.305 3760.400 0.000 8201.705 18 110 500.00 400.00 0.00 900 4911.500 3760.400 0.000 8671.900 19 110 550.00 400.00 0.00 950 5389.505 3760.400 0.000 9149.905 20 110 600.00 400.00 0.00 1000 5875.320 3760.400 0.000 9635.720 21 111 450.00 400.00 200.00 1050 4441.305 3760.400 1864.800 10066.505 22 111 500.00 400.00 200.00 1100 4911.500 3760.400 1864.800 10536.700 23 111 600.00 400.00 200.00 1200 5875.320 3760.400 1864.800 11500.520 24 111 600.00 400.00 200.00 1200 5875.320 3760.400 1864.800 11500.520

SUM 199299.825

128

Table 5.33 Unit commitment schedule and Power Sharing (MW) of the best solution obtained by Hybrid Model-III (PMAX-FLAPC with ED) for 3 unit systems:

Test System-II



Hr.

UC Sch.

P-1 P-2 P-3

Load (MW)

F-1 F-2 F-3

Operating Fuel Cost

($/h)

1 111 600.00 400.00 200.00 1200 5875.320 3760.400 1864.800 11500.520 2 111 600.00 400.00 200.00 1200 5875.320 3760.400 1864.800 11500.520 3 111 570.35 400.00 179.65 1150 5586.289 3760.400 1665.372 11012.061 4 111 532.59 400.00 167.41 1100 5222.177 3760.400 1547.344 10529.921 5 110 600.00 400.00 0 1000 5875.320 3760.400 0 9635.720 6 110 500.00 400.00 0 900 4911.500 3760.400 0 8671.900 7 110 433.18 366.82 0 800 4284.887 3450.577 0 7735.464 8 100 600.00 0 0 600 5875.320 0 0 5875.320 9 100 550.00 0 0 550 5389.505 0 0 5389.505 10 100 500.00 0 0 500 4911.500 0 0 4911.500 11 100 500.00 0 0 500 4911.500 0 0 4911.500 12 100 500.00 0 0 500 4911.500 0 0 4911.500 13 100 500.00 0 0 500 4911.500 0 0 4911.500 14 100 500.00 0 0 500 4911.500 0 0 4911.500 15 100 600.00 0 0 600 5875.320 0 0 5875.320 16 110 433.18 366.82 0 800 4284.887 3450.577 0 7735.464 17 110 460.88 389.12 0 850 4542.955 3658.336 0 8201.290 18 110 500.00 400.00 0 900 4911.500 3760.400 0 8671.900 19 110 550.00 400.00 0 950 5389.505 3760.400 0 9149.905 20 110 600.00 400.00 0 1000 5875.320 3760.400 0 9635.720 21 111 494.83 400.00 155.17 1050 4862.520 3760.400 1430.760 10053.679 22 111 532.59 400.00 167.41 1100 5222.177 3760.400 1547.344 10529.921 23 111 600.00 400.00 200.00 1200 5875.320 3760.400 1864.800 11500.520 24 111 600.00 400.00 200.00 1200 5875.320 3760.400 1864.800 11500.520

SUM 199262.671 5.7.3 Numerical Results of Test System-III: Ten units system: Hybrid Model-III: (Convex

fuel cost function)

Using the priority list in Table 5.23, the units are committed based on their priority with the

highest priority (higher capacity) being on first followed by other units. Units are committed

until the load demand plus the spinning reserve requirements are fulfilled.

129

The output results of test system – III are shown in the following tables:

Table (5.34) shows the comparison of the best results of the Hybrid Model-III with other

approaches available in the literature.

Table (5.35) presents the daily saving and percentage saving in fuel cost compared with other

approaches.

Table (5.36) and Table (5.37) presents the Unit Commitment Schedule, Power Sharing (MW) ,

Load demand and Total Production Cost (TPC) of the best solution obtained from proposed

Hybrid Model-III, with MUT and MDT constraints with 10 % spinning reserve: (ED by load to

economic unit).

Table (5.38) and Table (5.39) shows the detailed results of Unit Commitment Schedule, Power

Sharing (MW), Load demand and Total Production Cost (T.P.C) of the best solution obtained

from proposed Hybrid Model-III, with MUT and MDT constraints with 10 % spinning reserve.

(ED by lambda iteration method)

Table (5.40) presents the summary of Unit Commitment schedules for 10 unit systems with 10%

spinning reserve and considering minimum up time and down time constraints.

Table (5.41) shows the Comparison of Transition Cost for 10 unit system without considering s.r

and MUT and MDT.

Table (5.42) shows the Comparison of operating fuel cost ($) for 10 unit systems considering

10% spinning reserve and considering minimum up/down time constraints without transition

cost.

Table (5.43) presents the Comparison of the operating fuel cost ($) for 10 unit systems

considering 10% spinning reserve, considering minimum up/down time constraints and transition

cost.

Table (5.44) gives the Comparison of the results of the proposed Hybrid Models-I and III with

Genetic Algorithm.

130

The Salient features of the proposed approach in the light of the observations from the results

are as follows:

1. When compared with other 30 approaches available in the literature. The proposed hybrid

models III gives low total production cost compared with other approaches as given in

Table 5.33.

2. Hybrid Model-I explores more search space than Hybrid model-III. At hour 23, Table

5.34 when the load is 900 MW units 6 is committed and unit 5 is off in model-I, thus

gives a optimum unit commitment schedule at hour 23 and reduction in operating fuel

cost. The total production cost in this case is $563937.98.

3. The daily costs saving of the proposed Hybrid model-III compared with other approaches

have a range of $28.00 to $24773.00 per day. Which is equivalent to a percentage saving

of 0.00496 to 04.3925 as given in Table 5.34?

4. Table 5.40 gives the comparison of transition cost. The operating fuel cost in

Enumeration approach is $543479.1, which is less than the proposed Hybrid model-III,

but the start up cost is $8180, which is very high compared with proposed model. The

startup cost of the proposed model is $4900. The total production cost of the proposed

Hybrid model-III is $1604 less than single approach-I. Therefore, the proposed hybrid

model-III fulfills the requirement of the objectives of the UC problem, to minimize the

fuel cost, and transition cost.

5. The total production cost (operating fuel cost + transition cost) of the hybrid models-I and

III is low compared with single approaches.

6. The daily costs saving of the proposed Hybrid model-III compared with Genetic

Algorithm has $ 1847.98 per day. Which is equivalent to a percentage saving of 0.326 as

given in Table 5.43?

7. The unit scheduling problem is basically depends upon the spinning reserve

requirements. With less spinning reserve required less number of units is committed. For

5% or 10% spinning reserve the unit schedule is different. The UC schedule and hence

the total production cost depends upon spinning reserve. In this study 10 % spinning

reserve is assumed.

8. Comparing the results of the proposed hybrid model-III for Test System-III, it is clear

that the hybrid model III, provides a better quality of solution at a faster speed. Hybrid

131

Model -III is simple and more efficient than conventional priority list method. The fast

preparation of UC schedules and fast economic dispatch calculation leads to a

deterministic and efficient method.

Table 5.33 Comparison of the best results of the Hybrid Model-III with other approaches

available in the literature: Test system-III

Approach DP [56]

LR [56]

ELR [36]

EPL [15]

SPL [16]

Total production cost($)

565825 565825 563977 563977 564950

Approach HPSO [80]

RPACO LR-PPSO GA-UCC [57]

EMOALHN

TPC ($) 563942 565302 563977 563977 563977 Approach FPGA

[60] PSO [65]

MRCGA [61]

GA-LR LR-PSO ALR

TPC ($) 564094 564212 564244 564800 565275 565508 Approach GA

[56] GRASP MA ICGA

[59] DPHNN BPSO

TPC ($) 565825 565825 565827 566404 588750 565804 Approach AG EP

[52] ESA [44]

ASSA [45]

AG SA

TPC ($) 564005 564551 565828 563938 564005 565825 Approach PLEA

[82] DP-LR EP-LR ACSA RPACO

TPC ($) 563977 564049 564049 564059 565302 Proposed Hybrid Model-III 56397564875.617.02 Proposed Hybrid Model-III 56397563977.027.02

• Abbreviations are given in Appendix-B.

132

Table 5.35 Daily Saving and Percentage saving in fuel cost compared with other approaches

S. No.

Approach

Total Fuel Cost ($/h)

Amount of daily saving ($/h) compared with proposed hybrid

model –III

% Cost saving in fuel cost compared

with proposed model-III

1 ELR 563977 0 0 2 EPL 563977 0 0 3 LR-PPSO 563977 0 0 4 GA-UCC 563977 0 0 5 EMOALHN 563977 0 0 6 PLEA 563977 0 0 7 AG 564005 28.00 0.00496 8 DP-LR 564049 72.00 0.01276 9 EP-LR 564049 72.00 0.01276 10 ACSA 564059 82.00 0.01453 11 FPGA 564094 117.00 0.0207 12 PSO 564212 235.00 0.0416 13 MRCGA 564244 245.00 0.0434 14 EP 564551 574.00 0.0107 15 GA-LR 564800 823.00 0.1459 16 SPL 564950 973.00 0.1725 17 LR-PSO-1 565275 1298.00 0.2301 18 RPACO 565302 1325.00 0.2349 19 ALR 565508 1531.00 0.2714 20 BPSO 565804 1827.00 0.3239 21 DP 565825 1848.00 0.3276 22 LR 565825 1848.00 0.3276 23 SA 565825 1848.00 0.3276 24 GA 565825 1848.00 0.3276 25 GRASP 565825 1848.00 0.3276 26 MA 565827 1850.00 0.3280 27 LR-PSO-2 565869 1892.00 0.3354 28 ICGA 566404 2427.00 0.4303 29 BCGA 567367 3390.00 0.6010 30 DPHNN 588750 24773.00 4.3925 31 Proposed approach 563977

133

Table 5.36 Unit Commitment Schedule and Power Sharing (MW) of the best solution obtained from proposed Hybrid Model-III, with MUT and MDT constraints with 10 % spinning

reserve: (ED by average load to economic unit): Test system-III:

Power output of each unit (MW) Hour Unit Commitment

schedule 12543768910

Load (MW)

1 2 3 4 5 6 7 8 9 10 1 1100000000 700 455 245 0 0 0 0 0 0 0 02 1100000000 750 455 295 0 0 0 0 0 0 0 03 1100100000 850 455 370 0 0 25 0 0 0 0 04 1100100000 950 455 455 0 0 40 0 0 0 0 05 1101100000 1000 455 455 0 65 25 0 0 0 0 06 1111100000 1100 455 455 35 130 25 0 0 0 0 07 1111100000 1150 455 455 85 130 25 0 0 0 0 08 1111100000 1200 455 455 130 130 30 0 0 0 0 09 1111111000 1300 455 455 130 130 85 20 25 0 0 010 1111111100 1400 455 455 130 130 162 33 25 10 0 011 1111111110 1450 455 455 130 130 162 73 25 10 10 012 1111111111 1500 455 455 130 130 162 80 58 10 10 1013 1111111100 1400 455 455 130 130 162 33 25 10 0 014 1111111000 1300 455 455 130 130 85 20 25 0 0 015 1111100000 1200 455 455 130 130 30 0 0 0 0 016 1111100000 1050 455 455 20 95 25 0 0 0 0 017 1111100000 1000 455 455 20 45 25 0 0 0 0 018 1111100000 1100 455 455 35 130 25 0 0 0 0 019 1111100000 1200 455 455 130 130 30 0 0 0 0 020 1111111100 1400 455 455 130 130 162 33 25 10 0 021 1111111000 1300 455 455 130 130 85 20 25 0 0 022 1111111000 1100 455 455 20 100 25 20 25 0 0 023 1100100000 900 455 420 0 0 25 0 0 0 0 024 1100000000 800 455 345 0 0 0 0 0 0 0 0

134

Table 5.37 Load demand , fuel cost , start up cost and Total Production Cost (TPC) obtained from proposed Hybrid Model-III, with MUT and MDT constraints with 10 % spinning reserve.

(ED by average load to economic unit): Test system-III

Hour Load (MW)

Fuel cost ($/h)

Startup Cost ($/h)

Total Production

Cost ($)

Spinning reserve (MW)

1 700 13683.13 0 13683.13 210 2 750 14554.50 0 14554.50 160 3 850 16809.45 900 17709.45 222 4 950 18597.67 0 18597.67 122 5 1000 20059.70 560 20619.70 202 6 1100 22442.40 1100 23542.40 232 7 1150 23284.40 0 23284.40 182 8 1200 24150.34 0 24150.34 132 9 1300 27251.06 860 28111.06 197 10 1400 30057.55 60 30117.55 152 11 1450 31916.06 60 31976.06 157 12 1500 33945.16 60 34005.16 162 13 1400 30057.55 0 30057.55 152 14 1300 27251.06 0 27251.06 197 15 1200 24150.34 0 24150.34 132 16 1050 21597.63 0 21597.63 282 17 1000 20757.86 0 20757.86 332 18 1100 22442.40 0 22442.40 232 19 1200 24150.34 0 24150.34 132 20 1400 30057.55 490 30547.55 152 21 1300 27251.06 0 27251.06 197 22 1100 23674.23 0 23674.23 137 23 900 17684.69 0 17684.69 172 24 800 15427.42 0 15427.42 110

Sum 21700 561253.53 4090.00 565343.53 4357

135

Table 5.38 Unit Commitment Schedule and Power Sharing (MW) of the best solution obtained from proposed Hybrid Model-III, with MUT and MDT constraints with 10 % spinning

reserve. (ED by lambda iteration method): Test system-III

Power output of each unit (MW) Hr. Unit Commitment

Schedule

Load (MW)

1 2 3 4 5 6 7 8 9 101 1100000000 700 455 245 0 0 0 0 0 0 0 0 2 1100000000 750 455 295 0 0 0 0 0 0 0 0 3 1100100000 850 455 370 0 0 25 0 0 0 0 0 4 1100100000 950 455 455 0 0 40 0 0 0 0 0 5 1101100000 1000 455 390 0 130 25 0 0 0 0 0 6 1111100000 1100 455 360 130 130 25 0 0 0 0 0 7 1111100000 1150 455 410 130 130 25 0 0 0 0 0 8 1111100000 1200 455 455 130 130 30 0 0 0 0 0 9 1111111000 1300 455 455 130 130 85 20 25 0 0 0 10 1111111100 1400 455 455 130 130 162 33 25 10 0 0 11 1111111110 1450 455 455 130 130 162 73 25 10 10 0 12 1111111111 1500 455 455 130 130 162 80 25 43 10 1013 1111111100 1400 455 455 130 130 162 33 25 10 0 0 14 1111111000 1300 455 455 130 130 85 20 25 0 0 0 15 1111100000 1200 455 455 130 130 30 0 0 0 0 0 16 1111100000 1050 455 310 130 130 25 0 0 0 0 0 17 1111100000 1000 455 260 130 130 25 0 0 0 0 0 18 1111100000 1100 455 360 130 130 25 0 0 0 0 0 19 1111100000 1200 455 455 130 130 30 0 0 0 0 0 20 1111111100 1400 455 455 130 130 162 33 25 10 0 0 21 1111111000 1300 455 455 130 130 85 20 25 0 0 0 22 1111111000 1100 455 455 0 0 145 20 25 0 0 0 23 1100100000 900 455 420 0 0 25 0 0 0 0 0 24 1100000000 800 455 345 0 0 0 0 0 0 0 0

136

Table 5.39 Load demand , fuel cost , start up cost and Total Production Cost (TPC) of the best solution obtained from proposed Hybrid Model-III, with MUT and MDT constraints with 10

% spinning reserve: Test system-III (ED by Lambda iteration method)

Hour Load (MW)

Fuel cost ($/h)

Startup Cost ($/h)

Total Production

Cost ($)

Spinning reserve (MW)

1 700 13683.13 0 13683.13 210 2 750 14554.50 0 14554.50 160 3 850 16809.45 900 17709.45 222 4 950 18597.67 0 18597.67 122 5 1000 20020.02 560 20580.02 202 6 1100 22387.04 1100 23487.04 232 7 1150 23261.98 0 23261.98 182 8 1200 24150.34 0 24150.34 132 9 1300 27251.06 860 28111.06 197 10 1400 30057.55 60 30117.55 152 11 1450 31916.06 60 31976.06 157 12 1500 33890.16 60 33950.16 162 13 1400 30057.55 0 30057.55 152 14 1300 27251.06 0 27251.06 197 15 1200 24150.34 0 24150.34 132 16 1050 21513.66 0 21513.66 282 17 1000 20641.82 0 20641.82 332 18 1100 22387.04 0 22387.04 232 19 1200 24150.34 0 24150.34 132 20 1400 30057.55 490 30547.55 152 21 1300 27251.06 0 27251.06 197 22 1100 22735.52 0 22735.52 137 23 900 17684.69 0 17684.69 172 24 800 15427.42 0 15427.42 110

Sum 27100 559887.02 4090.00 563977.02 4357

137

Table 5.40 Summary of Unit Commitment schedules for 10 unit systems with 10% spinning reserve and considering minimum up time and down time constraints:

Test System III

Hybrid Model-I

Hybrid Model-I

Hybrid Model-III

Hybrid Model-III

DP-PSO-ANN SI-Learning

DP-BP-ANN BP-Learning

(PMAX-FLAPC2)

(PMAX-FLAPC3)

1100000000 1100000000 1100100000 1100100000 1101100000 1111100000 1111100000 1111100000 1111111000 1111111100 1111111110 1111111111 1111111100 1111111000 1111100000 1111100000 1111100000 1111100000 1111100000 1111111100 1111111000 1111111000 1100010000 1100000000

1100000000 1100000000 1100100000 1100100000 1101100000 1111100000 1111100000 1111100000 1111111000 1111111100 1111111110 1111111111 1111111100 1111111000 1111100000 1111100000 1111100000 1111100000 1111100000 1111111100 1111111000 1111111000 1100010000 1100000000

1100000000 1100000000 1100100000 1100100000 1101100000 1111100000 1111100000 1111100000 1111111000 1111111100 1111111110 1111111111 1111111100 1111111000 1111100000 1111100000 1111100000 1111100000 1111100000 1111111100 1111111000 1111111000 1100100000 1100000000

1100000000 1100000000 1100100000 1100100000 1101100000 1111100000 1111100000 1111100000 1111111000 1111111100 1111111110 1111111111 1111111100 1111111000 1111100000 1111100000 1111100000 1111100000 1111100000 1111111100 1111111000 1111111000 1100100000 1100000000

559847.98 559845.58 560775.61 559887.01

138

Table 5.41 Comparison of Transition Cost for 10 unit system without considering s. r and MUT and MDT: Test System III (10 unit system)

Operating

fuel cost ($/h)

Start-upcost ($/h)

Total Production

Cost ($)

Difference in T.P.C

($)

% Saving in total production cost(T.P.C)

Single Approach-I (Enumeration)

543479.10

8180.00

551659.10

0

0

Proposed Hybrid Method-III 545154.82

4900.00

550054.82

1604.28

0.29081

Table5.42 Start up cost comparison

0

2000

4000

6000

8000

10000

Enumeration Proposed approach

Series1

Table 5.43 Comparison of operating fuel cost ($) for 10 unit systems considering 10% spinning reserve and considering minimum up/down time constraints without transition cost:

Test System III

Single Approaches

Single approach–I

Single approach-II

Single approach–III

UC Schedule

UC12534768910 UC12435678910 UC12534768910


559887.02 561682.98 560775.6

Hybrid Models

Hybrid Model–I Hybrid Model–I Hybrid Model–III

DP-PSO-ANN

DP-BP-ANN

(PMAX-FLAPC)


559847.98 559845.58 559887.02

139

Table 5.44 Comparison of the operating fuel cost ($) for 10 unit systems considering 10% spinning reserve, considering minimum up/down time constraints and transition cost: Test

System III

Single Approaches

Single approach- I

(Enumeration)

Single approach-II

(Conventional Priority List)

Single approach– III

(Proposed Priority List)

UC Schedule

12543768910 12435678910 12534768910


560744.47 561682.98 560775.6

Transition cost ($) 4090 4440 4100 Total Production Cost ($)

564834.47 566122.98 564875.61

% Cost Saving compared with conventional priority list

-

-

0.22

Hybrid Approaches

Proposed Hybrid Model –I

Proposed Hybrid Model –II

Proposed Hybrid Model–III

DP-PSO-ANN DP-BP-ANN (PMAX-FLAPC3) UC Schedule

12543768910 12543768910 12543768910


559847.98 559845.58 559887.02

Transition Cost ($) 4090 4090 4090 Total Production Cost ($)

563937.98 563935.58 563977.02

% Cost Saving compared with conventional priority list

0.385

0.386

0.379

5.8 Case Studies: Hybrid model-III ---Non-Convex fuel cost function

The following two standard test systems have been selected for the illustration of the

effectiveness of the proposed approach.

a. Test System IV --- 3 units with 24 hours load.

b. Test System-V --- 3 units with 24 hours load

Input data: The description and data of test system used for investigation in the case studies is

given in Table A.4 and Table A.5 placed in Appendix A.

140


Computer.

Output Results: The summary and comparison of results is given in this section.

5.8.1 Numerical results of test systems – IV and V: 3 units systems: Hybrid Model- III: (non-convex fuel cost function)

The output results of test systems IV and V are shown in the following tables:

Table (5.45) gives the Comparison of the Operating fuel cost ($) for 3 unit system obtained by

using Proposed Hybrid Model-III compared with Hybrid model –II.

Tables (5.46) and (5.47) showing the load demand, Unit Commitment Schedule, Power Sharing

(MW) of the committed units and Total Production Cost (TPC) in the 24 hours.

The salient features of the proposed approaches in the light of the observations from the results are as follows:

1. The G.A. approach is suitable for non convex curves. Lambda iteration method fails for

non convex curves. Hybrid Model-III (load to economic unit) is also applicable for non

convex curves, but the results obtained by fine tuning by using Hybrid Model-II (GA-

PSO-ANN and GA-BP-ANN) are encouraging.

2. Hybrid model-II gives better results in terms of solution quality and gives a reduction of

$ 318.49 and $ 478.42 per day in operating fuel cost compared with Hybrid Model-III for

test systems IV and V respectively as given in Table 5.45.

Table 5.45 Comparison of the Operating fuel cost ($) for 3 unit system obtained by using

Proposed Hybrid Model-III compared with Hybrid model -II: (Non-convex)

Hybrid Model-II Hybrid Model-III

Reduction in fuel cost ($/day)

(GA-PSO-ANN)

(GA-BP- ANN)

Test system-IV

Operating fuel cost

$ 207773.614 $ 207772.477 $ 208092.107 $318.49

Test system-V

Operating fuel cost

$ 200665.137 $ 200662.723 $ 201143.551 $478.72

141

Table 5.46 Unit Commitment Schedule and Power Sharing (MW) of the best solution obtained from the Hybrid Model-III (PMAX-FLAPC3): (non-convex):

Test system-IV:

Power Output of each unit(MW)


Hour

UC Schedule

P-1 P-2 P-3

Load (MW)

F-1 F-2 F-3

Operating Fuel Cost

($/h)

1 111 600.00 400.00 200.00 1200 6175.105 3767.125 2241.543 12183.773 2 111 600.00 400.00 200.00 1200 6175.105 3767.125 2241.543 12183.773 3 111 600.00 400.00 150.00 1150 6175.105 3767.125 1660.862 11603.092 4 111 600.00 400.00 100.00 1100 6175.105 3767.125 1109.101 11051.331 5 110 600.00 400.00 0 1000 6175.105 3767.125 0 9942.230 6 110 500.00 400.00 0 900 5211.370 3767.125 0 8978.495 7 110 400.00 400.00 0 800 4278.854 3767.125 0 8045.978 8 100 600.00 0 0 600 6175.105 0 0 6175.105 9 100 550.00 0 0 550 5399.592 0 0 5399.592 10 100 500.00 0 0 500 5211.370 0 0 5211.370 11 100 500.00 0 0 500 5211.370 0 0 5211.370 12 100 500.00 0 0 500 5211.370 0 0 5211.370 13 100 500.00 0 0 500 5211.370 0 0 5211.370 14 100 500.00 0 0 500 5211.370 0 0 5211.370 15 100 600.00 0 0 600 6175.105 0 0 6175.105 16 110 400.00 400.00 0 800 4278.854 3767.125 0 8045.978 17 110 450.00 400.00 0 850 4448.871 3767.125 0 8215.995 18 110 500.00 400.00 0 900 5211.370 3767.125 0 8978.495 19 110 550.00 400.00 0 950 5399.592 3767.125 0 9166.717 20 110 600.00 400.00 0 1000 6175.105 3767.125 0 9942.230 21 111 600.00 400.00 50.00 1050 6175.105 3767.125 586.260 10528.490 22 111 600.00 400.00 100.00 1100 6175.105 3767.125 1109.101 11051.331 23 111 600.00 400.00 200.00 1200 6175.105 3767.125 2241.543 12183.773 24 111 600.00 400.00 200.00 1200 6175.105 3767.125 2241.543 12183.773

SUM 208092.107

142

Table 5.47 Unit Commitment Schedule and Power Sharing (MW) of the best solution obtained from the Hybrid Model-III (non-convex UC123, ED 321): Test system-V



Hour

UC Schedule

P-1 P-2 P-3

Load (MW)

F-1 F-2 F-3

Operating Fuel Cost

($)

1 111 600.00 400.00 200.00 1200 5887.927 3767.125 1869 11523.635 2 111 600.00 400.00 200.00 1200 5887.927 3767.125 1869 11523.635 3 111 550.00 400.00 200.00 1150 5689.290 3767.125 1869 11324.998 4 111 500.00 400.00 200.00 1100 4921.587 3767.125 1869 10557.294 5 110 600.00 400.00 0 1000 5887.927 3767.125 0 9655.052 6 110 500.00 400.00 0 900 4921.587 3767.125 0 8688.712 7 110 400.00 400.00 0 800 3986.486 3767.125 0 7753.610 8 100 600.00 0 0 600 5887.927 0 0 5887.927 9 100 550.00 0 0 550 5689.290 0 0 5689.290 10 100 500.00 0 0 500 4921.587 0 0 4921.587 11 100 500.00 0 0 500 4921.587 0 0 4921.587 12 100 500.00 0 0 500 4921.587 0 0 4921.587 13 100 500.00 0 0 500 4921.587 0 0 4921.587 14 100 500.00 0 0 500 4921.587 0 0 4921.587 15 100 600.00 0 0 600 5887.927 0 0 5887.927 16 110 400.00 400.00 0 800 3986.486 3767.125 0 7753.610 17 110 450.00 400.00 0 850 4741.175 3767.125 0 8508.300 18 110 500.00 400.00 0 900 4921.587 3767.125 0 8688.712 19 110 550.00 400.00 0 950 5689.290 3767.125 0 9456.415 20 110 600.00 400.00 0 1000 5887.927 3767.125 0 9655.052 21 111 450.00 400.00 200.00 1050 4741.175 3767.125 1869 10376.883 22 111 500.00 400.00 200.00 1100 4921.587 3767.125 1869 10557.294 23 111 600.00 400.00 200.00 1200 5887.927 3767.125 1869 11523.635 24 111 600.00 400.00 200.00 1200 5887.927 3767.125 1869 11523.635

SUM 201143.551

143

CHAPTER 6

Unit Commitment of National Transmission & Despatch

Company Limited (NTDC)

6.1 Introduction

This chapter gives discussion of the National Transmission and Despatch Company

(NTDC) network, its operational problem, and the results of proposed hybrid models-II and III

for convex and non convex cost function.

6.2 WAPDA --- Brief Overview

The Pakistan Water and Power Development Authority, (WAPDA) was created in 1958

as a Semi-Autonomous Body with water and power wing. WAPDA worked in vertically

integrated environment up to 2001.

Since October 2007, WAPDA has been bifurcated into two distinct entities i.e. WAPDA

and Pakistan Electric Power Company (PEPCO). WAPDA is responsible for water and

hydropower development whereas PEPCO is vested with the responsibility of thermal power

generation, transmission, distribution and billing.

WAPDA is now fully responsible for the development of Hydel Power and Water Sector

Projects. PEPCO has been fully responsible for the management of all the affairs of nine

Distribution Companies (DISCOs), four Generation Companies (GENCOs) and a National

Transmission and Despatch Company (NTDC. The public sector hydel and thermal generation

are in the control of WAPDA and GENCOs respectively. Independent Power Producers (IPPs)

are in private sector.

144

6.3 National Transmission and Despatch Company

National transmission and despatch company links the power generation units and load

centers dotting the entire country, thus creating one of the largest contiguous grid systems of the

world. At present NTDC is operating and maintaining nine 500 kV and twenty four 220 kV grid

stations along with 10,167 km length of associated transmission lines [111]. NTDC power

system has following mainly five types of power stations connected to the National Grid system:

a. Hydro power stations

b. Steam power stations

c. Gas turbine power stations

d. Combined cycle power plant stations.

e. Nuclear power stations.

The generating units are loaded according to the merit order determined by their cost of

operation and synchronized with system with the rising trend of load curve. However sometimes

it is necessary to take generation at high operational cost subjected to constraints such as less

transmission or transformer capacity. All the functions of 500/220 kV power system and power

houses are monitored by SCADA system through R.T.Us installed at the grid/power stations.

NTDC is responsible to purchase the power from hydel stations in the north, thermal

units in public and private sectors installed mostly in the central southern regions of the country

and to sell power to distribution companies through its large network of transmission lines and

grid stations of 500 kV and 220 kV voltage capacities.

6.4 Operational Constraints in NTDC System

The various constraints in the NTDC system are:

6.4.1 Hydro-Electric Generation Constraints

Pakistan is the one of the most fortunate countries of the world having the lot of water

potentials. Their total estimated capacity is about 30,000 MW, and all are almost in the northern

areas of Pakistan. WAPDA's hydel power stations consists of five major station located at

Tarbela, Mangla, Warsak, Chashma, Ghazi Boratha and nine small hydel power stations. Two

145

hydel power stations are presently commissioned in private sector namely Jagran (AJK) and

Malakand ( NWFP).

Tarbela and Mangla dams are multi-purpose projects with main emphasis on irrigation

under the indus water treaty. Tarbela and Mangla reservoirs are the major hydel power

generation sources of the WAPDA system, representing the only significant capacity for the

seasonal storage of water. There are two principal effects of wide seasonal variations in hydel

generation capability. These are the variations in water releases from the reservoirs and in

hydraulic heads available for power generation. Operation of the NTDC system is in practice

dominated by these variations.

Water management i.e., the use of reservoir storage and planning of water release is

dominated by irrigation needs rather than power requirements. Seasonal water management

schedules are derived on the basis of data of current and historic water levels in the reservoirs.

The seasonal pattern of reservoir and hydro electric plant operation has considerable

implications for WAPDA system operation in general. The system experiences shortages of:

• Generation capacity and energy during late winter to early summer when reservoir levels

are low.

• Capacity during the period later in summer, before the reservoirs are filled.

• Energy during the canal closure period in January to early February when reservoir

releases are severely restricted.

Considerable load shedding takes place on NTDC system during the above periods. The load

distribution pattern of NTDC system is such that 75% of the total load is located to the north of

the Multan. So primary transmission system has to transfer the blocks of power from north to

south when full hydel generation capacity is available and from south to north in winter when

water is in short supply and reliance has to be placed on thermal generation.

146

6.4.2 Thermal Generation Constraints

Seasonal variations in thermal plant capability also have considerable influence on the

system operation. Gas turbines are inherently sensitive to ambient temperature. Hence the power

capabilities of gas turbine components of combined cycle plant are significantly lower in

summer.

6.4.3 Transmission line Constraints

Various transmission constraints have effected the system operation in recent years.

Power flows on NTDC network are from north to south in summer and from south to north in

winter and both the times power transfer is limited by transmission capability subjected to the

constraints such as line or transformer overloading, stability problems and difficulty with control

of voltage or reactive power.

6.4.4 Seasonal Variations in Power Demand

The seasonal variation in electricity demand also effects the system operation. This

variation arises principally from increased air-conditioning loads in summer and heating load in

winter.

6.4.5 Spinning Reserve Constraint

The load demand is more than the power generated. There is a short fall of about

5000MW.

6.4.6 Minimum up and down time constraints

For a 210 MW Thermal Power Station the minimum up and down times is approximately

3.5 and 1.5 hours respectively.

6.4.7 Start up cost considerations

Start up costs depends on the number of hours the unit has been off.

• Up to 8 hours : hot state

• After 8 hours and up to 150 hours : warm state

• After 150 hours : cold state

147

For a thermal power station the cold startup cost may be Rs.1, 53,869.52 and warm

Rs. 1, 17,890.08 approximately.

6.4.8 Maintenance cost

The maintenance cost consists of fixed maintenance cost and variable maintenance cost.

6.4.9 Fuel constraint

Due to limited fuel this constraint presents a most challenging unit commitment problem.

6.4.10 Ramping rates

Ramping rate is the maximum rate of change of the output in MW/min. Steam units have

ramping rate of 1MW/min.

6.4.11 Unit deration

Older units give output less than their installed capacity.

6.5 Test systems for NTDC system

The following four test systems with cost curves close to the original machines in the

system have also been prepared for unit commitment.

1. 12-Unit NTDC Test Circuit




However in the discussion to follow the Hybrid models IV and V for unit commitment will be

investigated.

6.6 Case Studies

The cost curve characteristics and load data for 12 machine system are given in Table A.7

placed in Appendix A. The cost curves of the 15, 25, 34 units are given in Tables A. 8, A. 9 and

A.10 respectively placed in appendix A.

148

6.7 Numerical results

Table 6.1 Comparison of the results for NTDC 12, 15, 25 and 34 unit systems

Test System Hybrid Model-

III(ED by load to

economic unit

Hybrid Model-III (ED by

lambda iteration method)

Difference in cost

12-unit 120700024 120692850.78 7173.22 15-unit 150549504 150548559.52 944.48 25-unit 176237968 176234285.37 3682.63

Operating Fuel Cost

($) 34-unit 208266768 208266739.93 28.07

Table 6.2 Summary of UC schedule and operating fuel cost for 12 unit NTDC systems

S. No.

Load demand (MW)

Unit Commitment

schedule

Operating fuel cost by Hybrid Model-II ($)

(load to economic unit)

Operating fuel cost by Hybrid Model-III ($)

(Lambda iteration method) 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

1600 1610 1620 1630 1625 1650 1705 1710 1705 1680 1675 1700 1750 1800 1850 1900 1950 1975 2000 1980 1970 1800 1750 1700

101011001110 101011001110 101011001110 101011101110 101011101110 101011101110 101011101110 101011101110 101011101110 101011101110 101011101110 101011101110 101011111110 101011111110 101111111110 101111111110 111111111110 111111111110 111111111111 111111111110 111111111110 101011111110 101011111110 101011101110

4649995.0 4701900.5 4754483.0 4524162.0 4503106.5 4609900.0 4864742.5 4890442.0 4864742.5 4743059.5 4720487.0 4839212.5 4845135.0 5074447.5 5108391.5 5337704.0 5625712.0 5780787.5 5939454.0 5817434.5 5745929.0 5074447.5 4845135.0 4839212.5

4649994.13 4701899.88 4754482.21 4524161.51 4503106.29 4609899.74 4864742.03 4890441.19 4864742.03 4743059.13 4720486.56 4839212.02 4845134.73 5074447.01 5108390.86 5337703.14 5625710.97 5780786.05 5932296.44 5817433.40 5745927.70 5074447.01 4845134.73 4839212.02

SUM 120700024 120692850.78

149


S. No.

Load demand (MW)

Unit Commitment

schedule

Operating fuel cost by Hybrid Model-III ($) (load to economic unit)


(Lambda iteration method) 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

2400 2425 2450 2475 2500 2525 2550 2575 2600 2625 2650 2675 2700 2725 2750 2775 2800 2825 2850 2875 2900 2925 2950 2958

000011111111111 000011111111111 000011111111111 000011111111111 000011111111111 000011111111111 001011111111111 001011111111111 001011111111111 001011111111111 001011111111111 001111111111111 001111111111111 001111111111111 001111111111111 001111111111111 001111111111111 011111111111111 011111111111111 011111111111111 111111111111111 111111111111111 111111111111111 111111111111111

5605847.0 5666131.5 5737777.0 5836529.0 5939074.0 6045412.5 5909131.5 5970236.5 6056239.5 6156508.5 6260571.0 6153930.5 6213804.5 6278376.0 6376369.0 6478155.5 6583735.5 6538807.5 6640594.0 6746174.0 6701246.0 6803032.5 6908612.5 6943199.0

5605524.78 5666130.72 5737776.32 5836528.06 5939073.22 6045411.80 5909082.52 5970236.13 6056239.05 6156508.16 6260570.69 6153381.67 6213803.84 6278375.29 6376368.35 6478154.83 6583734.73 6538806.72 6640593.20 6746173.10 6701245.09 6803031.57 6908611.47 6943198.20

SUM 150549504 150548559.52

150


S. No.

Load demand (MW)

Unit Commitment schedule Operating fuel cost by Hybrid Model-III ($)

(Load to economic unit)

Operating fuel cost by Hybrid Model-

III ($) (Lambda iteration

method) 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

2800 2825 2850 2875 2900 2925 2950 2975 3000 3025 3050 3075 3100 3125 3150 3175 3200 3225 3250 3275 3400 3425 3450 3555

000011111111111110000000000001111111111111000000000000111111111111100000000000011111111111110000000000101111111111111000000000010111111111111100000000001011111111111110000000000101111111111111000000000010111111111111100000000001011111111111110000000000111111111111111000000000011111111111111100000000001111111111111110000000000111111111111111000000000011111111111111100000000011111111111111110000000001111111111111111000000000111111111111111100000000111111111111111110000000011111111111111111000000001111111111111111111000000111111111111111111100000111111111111111111110000011111111111111111111111111

6620140.5 6713427.0 6814454.5 6919276.0 6802118.0 6862402.5 6934048.0 7032800.0 7135345.0 7241683.5 7106462.5 7167567.5 7253570.5 7353839.5 7457902.0 7416009.0 7516278.0 7620340.5 7578447.5 7678716.5 8092939.0 8116494.0 8221770.5 8581921.0

6620130.05 6713426.00 6814453.80 6919275.01 6800094.40 6862047.31 6934047.41 7032799.16 7135344.32 7241682.90 7105270.37 7167496.82 7253569.77 7353838.88 7457901.41 7416008.14 7516277.25 7620339.78 7578446.51 7678715.62 8092938.01 8116493.14 8221769.56 8581919.76

SUM 176237968 176234285.37

151


S. No.

Load demand (MW)

Unit Commitment schedule Operating fuel cost by Hybrid Model-III ($)

(load to economic unit)


(Lambda iteration method)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

3500 3490 3480 3475 3470 3480 3750 3650 3700 3680 3690 3695 3695 3696 3698 3700 3715 3700 3690 3575 3775 3760 3785 3700

111111111111111111100000100100000011111111111111111110000010010000001111111111111111111000001001000000111111111111111111100000100100000011111111111111111110000010010000001111111111111111111000001001000000111111111111111111111101100111111111111111111111111110000010011111111111111111111111111010011001111111111111111111111111100001100111111111111111111111111110000110011111111111111111111111111010011001111111111111111111111111101001100111111111111111111111111110100110011111111111111111111111111010011001111111111111111111111111101001100111111111111111111111111111100110011111111111111111111111111010011001111111111111111111111111100001100111111111111111111111111110000010010001111111111111111111111111111011111111111111111111111111111111100111111111111111111111111111111111111111111111111111111111111010011001111111

8354777.5 8311605 8269039 8247984 8227080 8269039 8965442 8609265 8786917 8719249 8762240 8765619 8765619 8769866 8778379 8786917 8833262 8786917 8762240 8582748 9049066 8989955 9086636 8786917

8354776.62 8311603.89 8269038.11 8247982.83 8227079.28 8269038.11 8965440.36 8609263.51 8786914.93 8719247.69 8762238.33 8765616.87 8765616.87 8769864.34 8778377.5 8786914.93 8833259.84 8786914.93 8762238.33 8582746.22 9049064.11 8989953.09 9086634.29 8786914.93

SUM 208266768 208266739.93 Execution time 3370.49 s

152

CHAPTER 7

Conclusions and Suggestions

The present work deals with scheduling of thermal units, which could be the major part

of the hydrothermal power system. The scheduling of thermal generating units is considered as

two linked optimization problems as it consists of the unit scheduling problem and economic

dispatch sub problem.

Unit commitment is the essential and vital step in the daily operational planning of the

power system. The UCP is a combinatorial optimization problem with equality and inequality

constraints and ED is a nonlinear optimization problem.

The applications of artificial intelligence (AI) techniques have the potential in solving the

UCP. These techniques have the ability to handle nonlinearities and discontinuities commonly

found in power systems.

Research has been focused on UCP techniques with respect to dimensionality problem,

difficult constraint handling, spinning reserve considerations, computational time, and quality of

solution. The exact solution of the UCP may be obtained from exhaustive enumeration because it

calculates all the feasible and infeasible combinations of units. Due to the high dimensionality of

the search space it is impractical.

In the present study two new AI-based hybrid algorithms have been developed to solve

the UCP for convex and non convex fuel cost functions. The algorithms are presented in chapter

5. The other hybrid algorithm based on classical approaches is applied first time to solve the

UCP.

153

These algorithms are:

• Hybrid Model – I: A hybrid of Particle Swarm Optimization (PSO), Artificial Neural

Network (ANN) and Dynamic Programming (DP).

• Hybrid Model-II: Neuro-Genetic Hybrid Approach

• Hybrid Model –III: Scaleable Deterministic Hybrid Approach

The proposed Hybrid Model-I combines the Dynamic Programming (DP) with Artificial

Neural Networks (ANN) using Swarm Intelligence (SI) learning rule. In this model dynamic

programming produces near optimal solution based on training data for neural network model.

The neural network fine tunes the data subject to the target values of power output of units. The

best tuned solution is considered the required solution. The swarm intelligence learning rule

based feed forward network has been used for fine tuning the near optimal dynamic

programming results. The standard back propagation learning rule neural network has also been

used in this hybrid approach for the comparison of results. The particle swarm optimization

algorithm is applied to the neural network to obtain a set of weights that will minimize the error

function. Weights are progressively updated until the convergence criterion is satisfied. The UC

problem has been decomposed in to discrete loads on hourly basis. These small models get

trained faster due to simple network structure and perform efficiently due to swarm intelligence

learning rule. Three and ten unit standard test systems have been tested for validation of the

proposed approach. The results of the standard test systems obtained using this approach show

the comprehensive reduction in total production cost thus indicating the promise of the approach.

The proposed algorithm explores wider solution space, and gives better quality solution for

convex cost function.

The proposed Hybrid Model-II (Neuro-Genetic approach) combines the Genetic Algorithm

(GA) with Artificial Neural Networks (ANN) using Swarm Intelligence (SI) learning rule. GA

has the ability to search better for non-convex fuel cost function than convex fuel cost function.

Three machine standard test system has been tested for validation of the proposed model. In this

model Genetic Algorithm works as global optimizer and produces near optimal solution based

training data for neural network model. The neural network fine tunes the data subject to the

target values. The best fine tuned solution is considered the required solution. The target values

may be taken by randomly generated values by GA around the near optimal solutions satisfying

154

both equality and inequality constraints. The SI-learning rule based feed forward neural network

has been used for fine tuning the near optimal GA results. The standard back propagation

learning rule has also been used in this hybrid approach for the comparison of results. In this

model the test data and training data are generated by using Genetic Algorithm (GA). For proper

training of ANN, a pair of load as input and their corresponding generation schedules as output

are prepared off-line by using Genetic Algorithm (GA) and are stored in a data base. Each pair is

referred as input/output database pair. The case studies for three unit test system IV and V shows

that SI learning based ANN produces better results than back propagation learning network. The

total production cost results are also better than conventional approaches.

The proposed hybrid model-III combines the Maximum Power Output (PMAX) and Full

Load Average Production Cost (FLAPC) of each unit. This model is applicable for both convex

and non convex fuel cost functions. The proposed unit commitment algorithm hybrid model-III

incorporates the unit commitment solution by the deterministic priority list scheme for

generation of initial population. This hybrid model solves the UC problem with valve point

effect and without valve point effect. The UC schedule is prepared according to PMAX and

FLAPC. Unit with higher PMAX will be at higher priority. If the two units have the same

PMAX then unit with lower FLAPC will be of higher priority. Economic Dispatch is based on

lambda iteration method and average load assigned methods. Five test systems consisting of

three and ten units have been tested. Three test systems are convex, and two test system are non

convex. Final results are better than other many approaches available in the literature. The issue

in solving the combinatorial optimization problem is to have good feasible neighbour/trial

solutions from an existing feasible solution. The proposed algorithm is fast and efficient in

generating feasible initial and trial solutions and is applicable of solving a large scale power

system in reasonable computational time.

A major step in solving the UCP is the solution of the economic dispatch sub problem. In

this regard, a new GA based real power search algorithm has been implemented and tested.

PAKISTANI utility system and its operational problems have been reviewed with a view to

carry out its unit commitment studies. The data of 12, 15, 25 and 34 unit systems have also been

prepared to carry out unit commitment studies.

155

The demonstration of validity of applying the proposed models to the solution of UCP in

this thesis gives rise to the number of topics for further research in this area. Some of the

recommendations for future work may be summarized as follows:

• Exploration of new AI based hybrid models which give good quality of solution,

explore a wider search space, able in solving a large scale system in acceptable

computational time and is applicable to PAKISTANI utility system.

• Some other costs could also be taken into consideration in the objective function

such as: maintenance cost etc.

• The objective function may be taken as Profit Based Unit Commitment (PBUC)

instead of Cost Based Unit Commitment (CBUC).

• Some other constraints could also be taken into consideration such as: security,

transmission line capacity, fuel and emission.

• The hydro generation system can be included in the proposed algorithms to solve the

hydrothermal scheduling problem.

156

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APPENDIX A

Test Systems --- Description and Data

A.1 Three Unit Test Systems --- Convex Fuel Cost Curve A.1.1 Test System-I --- 3 -Unit System

The data for the 3 unit base system for comparison was taken from [1].

Unit 1: Coal-fired steam unit: Input = output curve:

600P150 00142.02.7510 12

111 ≤≤++=⎟⎠⎞

⎜⎝⎛ PP

hMBtuH


400P100 00194.085.7310 22

222 ≤≤++=⎟⎠⎞

⎜⎝⎛ PP

hMBtuH


MWPPh

MBtuH 200P50 00482.097.778 32

333 ≤≤++=⎟⎠⎞

⎜⎝⎛

Unit 1: Fuel cost = 1.1 $/MBtu Unit 2: Fuel cost = 1.0 $/MBtu Unit 3: Fuel cost = 1.2 $/MBtu

TABLE A.1 Unit Data:

Unit No. P max (MW)

P min (MW)

a ($/hr)

b ($/MWhr)

c ($/MW2 hr)

1 600 150 561 7.92 0.001562 2 400 100 310 7.85 0.001940 3 200 50 93.6 9.564 0.005784

Load Data Hour 1 2 3 4 5 6 7 8 9 10 11 12 Load (MW)

1200 1200 1150 1100 1000 900 800 600 550 500 500 500

Hour 13 14 15 16 17 18 19 20 21 22 23 24 Load (MW)

500 500 600 800 850 900 950 1000 1050 1100 1200 1200

168

A.1.2 Test System-II



600P150 00142.02.7510 12

111 ≤≤++=⎟⎠⎞

⎜⎝⎛ PP

hMBtuH


400P100 00194.085.7310 22

222 ≤≤++=⎟⎠⎞

⎜⎝⎛ PP

hMBtuH


MWPPh

MBtuH 200P50 00482.097.778 32

333 ≤≤++=⎟⎠⎞

⎜⎝⎛

Unit 1: Fuel cost = 1.1 $/MBtu Unit 2: Fuel cost = 1.0 $/MBtu Unit 3: Fuel cost = 1.0 $/MBtu

TABLE A.2

Unit Data: Unit No. P max

(MW) P min

(MW) a

($/hr) b

($/MWhr) c

($/MW2hr) 1 600 100 561 7.92 0.001562 2 400 100 310 7.85 0.001940 3 200 50 78 7.97 0.004820

Load Data: Hour 1 2 3 4 5 6 7 8 9 10 11 12 Load (MW)

1200 1200 1150 1100 1000 900 800 600 550 500 500 500

Hour 13 14 15 16 17 18 19 20 21 22 23 24 Load (MW)

500 500 600 800 850 900 950 1000 1050 1100 1200 1200

169

A.2 Ten Unit Standard Test System

A.2.1 Test System-III


TABLE A.3 Unit Data:

Unit 1 Unit 2 Unit 3 Unit 4 Unit5 P max (MW) 455 455 130 130 162 P min (MW) 150 150 20 20 25 a ($/h) 1000 970 700 680 450 b ($/MWh) 16.19 17.26 16.60 16.50 19.70 c (S/MW2-h) 0.00048 0.00031 0.002 0.002211 0.00398 Min up time (MUT) (h) 8 8 5 5 6 Min down time (MDT) (h) 8 8 5 5 6 Hot start cost ($) 4500 5000 550 560 900 Cold start cost ($) 9000 1000 1100 1120 1800 Cold start hrs (h) 5 5 4 4 4 Initial status (h) 8 8 -5 -5 -6

Unit 6 Unit 7 Unit 8 Unit 9 Unit10 P max (MW) 80 85 55 55 55 P min (MW) 20 25 10 10 10 a ($/h) 370 480 660 665 670 b ($/MWh) 22.26 27.74 25.92 27.27 27.79 c (S/MW2-h) 0.00712 0.0079 0.00413 0.00222 0.00173 Min up time (MUT) (h) 3 3 1 1 1 Min down time (MDT) (h) 3 3 1 1 1 Hot start cost ($) 170 260 30 30 30 Cold start cost($) 340 520 60 60 60 Cold start hrs (h) 2 2 0 0 0 Initial status (h) -3 -3 -1 -1 -1


700 750 850 950 1000 1100 1150 1200 1300 1400 1450 1500

Hour 13 14 15 16 17 18 19 20 21 22 23 24 Load (MW)

1400 1300 1200 1050 1000 1100 1200 1400 1300 1100 900 800

170

A.3 Three Unit Test Systems --- Non-Convex Fuel Cost Curve A.3.1 Test System-IV (Non-Convex Fuel Cost Curve)

TABLE A. 4

Unit Data:

Unit No. P max (MW)

P min (MW)

a $/h

b $/ MWhr

c $/ MW2hr

e f

1 600 150 561 7.92 0.001562 300 0.0315 2 400 100 310 7.85 0.001940 200 0.0420 3 200 50 93.6 9.564 0.005784 150 0.0630


1200 1200 1150 1100 1000 900 800 600 550 500 500 500

Hour 13 14 15 16 17 18 19 20 21 22 23 24 Load (MW)

500 500 600 800 850 900 950 1000 1050 1100 1200 1200

A.3.2 Test System-V (Non-Convex Fuel Cost Curve)

TABLE A.5

Unit Data: Unit No. P max

(MW) P min

(MW) a

$/h b

$/ MWhr c

$/ MW2hr e f

1 600 100 561 7.92 0.001562 300 0.0315 2 400 100 310 7.85 0.001940 200 0.0420 3 200 50 78 7.97 0.004820 150 0.0630


1200 1200 1150 1100 1000 900 800 600 550 500 500 500

Hour 13 14 15 16 17 18 19 20 21 22 23 24 Load (MW)

500 500 600 800 850 900 950 1000 1050 1100 1200 1200

171

A.4 Pakistani Utility NTDC Systems

A.4.1 12 Unit NTDC System




TABLE A.6

12 Unit NTDC system data: Unit No.

P max (MW)

P min (MW)

a ($/hr)

b ($/MWhr)

c ($/MW2hr)

1 210 170 117965.5955 3871.2446 03.3829 2 65 25 38044.3947 3930.3768 35.7800 3 300 230 21835.2917 1300.7907 02.0633 4 100 40 121270.2094 1364.9392 00.5492 5 120 40 27983.7444 1143.7230 10.6067 6 250 125 117845.1040 3073.0320 03.0347 7 110 70 21927.9678 1631.1822 03.2305 8 100 25 88181.4304 964.2456 05.0629 9 210 100 59533.4194 1920.6626 01.6412 10 210 110 65184.4252 1701.8310 01.8056 11 320 240 42799.8444 2212.5550 00.0078 12 65 25 38044.3947 3930.3768 35.7800

Load Data:

Hour 1 2 3 4 5 6 7 8 9 10 11 12 Load

(MW) 1600 1610 1620 1630 1625 1650 1705 1710 1705 1680 1675 1700

Hour 13 14 15 16 17 18 19 20 21 22 23 24 Load

(MW) 1750 1800 1850 1900 1950 1975 2000 1980 1970 1800 1750 1700

172

APPENDIX B

List of Abbreviations

EP Evolutionary Programming

DP Dynamic Programming

LR Lagrange Relaxation

ESA Enhanced Simulated Annealing

MA Memetic Algorithm

GA Genetic Algorithm

SGA Standard Genetic Algorithm

FPGA Floating Point Genetic Algorithm

ICGA Integer Coded Genetic Algorithm

BCGA Binary Coded Genetic Algorithm

PSO Particle Swarm Optimization

BPSO Binary Particle Swarm Optimization

IPSO Improved Particle Swarm Optimization

PSO-B-SA1 Particle Swarm Optimization Based Simulated Annealing

ELR Enhanced Lagrange Relaxation

ALR Augmented Lagrange Relaxation

EPL Extended Priority List

FAPSO Fuzzy Adaptive Particle Swarm Optimization

HPSO Hybrid Particle Swarm Optimization

LR-PPSO Lagrange Relaxation Parallel Particle Swarm Optimization

LR-PRPSO Lagrange Relaxation Parallel Relative Particle Swarm

Optimization

GA-UCC Genetic Algorithm -Unit Characteristics Classification

TS-HPSO-SQP-TS-IRP Tabu Search based Hybrid Particle Swarm Optimization -

Sequential Quadratic Programming- Tabu Search- Improved

Random Perturbation

173

TS-HPSO-SQP-TS-RP Tabu Search based Hybrid Particle Swarm Optimization -

Sequential Quadratic Programming- Tabu Search - Random

Perturbation

AG Annealing Genetic Algorithm

ACSA Ant Colony Simulated Annealing

DP-LR Dynamic Programming- Lagrange Relaxation

LR-GA Lagrange Relaxation- Genetic Algorithm

LR-PSO Lagrange Relaxation-Particle Swarm Optimization

ASSA Absolutely Stochastic Simulated Annealing

EMO-ALHN Enhanced Merit Order- Augmented Lagrange Hopfield Network

SPL Stochastic Priority List

PLEA Priority List based Evolutionary Algorithm

MRCGA Matrix Real Coded Genetic Algorithm

SF Straight Forward

MA-LR Memetic Algorithm seeded with Lagrange Relaxation

DPHNN Dynamic Programming based Hopfield Neural Network

FPGA Floating Point Genetic algorithm

Twofold- SA Twofold Simulated Annealing

RPACO Ant Colony Optimization with Random Perturbation

MA seeded with LR Memetic Algorithm seeded with Lagrange Relaxation

SA-PSO-SQP Simulated Annealing based Hybrid Particle Swarm Optimization

- Sequential Quadratic Programming

LR-EP Hybrid Lagrange Relaxation Evolutionary Programming

MACO memory-bounded ant colony optimization

GRASP Greedy Randomized Adaptive Search Procedure

EALHN Enhanced Augmented Lagragian Hopfield Network

174

APPENDIX C

Notation

The following notation is throughout the thesis:

Pih real power output of unit i at hour h,(MW)

Uih the on/off status of unit i at hour h. Uih = 0 when OFF, Uih = 1 when ON,

Fi (Pih ) fuel cost function or fuel cost rate of unit i, ($/h)

ai, bi, ci positive fuel cost coefficients of unit i measured in $/h, $/MW h and $/MW2 h,

respectively,

ei , fi cost coefficients from the valve point effect of the unit i,

Xi on h duration during which the unit i is continuously ON, (h)

Xi off h duration during which the unit i is continuously OFF, (h)

h-costi hot start cost of unit i,($)

c-costi cold start cost of unit i, ($)

c-s-houri cold start time of unit i, (h)

N the number of units,

H the number of hours, (24 h)

Dh load demand at hour h, (MW)

SRh spinning reserve at hour h, (MW)

Tiup minimum up time (MUT) of unit i, (h)

Tidown minimum down time (MDT) of unit i, (h)

TPC total production cost, ($/h)

STih start up cost of unit i in hour h, ($/h)

SDih shut down cost of unit i in hour h, ($/h)

Pimin minimum generations limit of unit i, (MW)

Pimax maximum generations limit of unit i, (MW)

HR heat rate (BTU/KWH)

FLAPC Full Load Average Production Cost ($/MWH)

ELD Economic load dispatch

175

APPENDIX D Derived Publications [1] Engr. Aftab Ahmad , Engr. Tahir Nadeem Malik, and Dr. Aftab Ahmad,” Unit

Commitment Problem Of Thermal Generation Units For Short Term Operational Planning using simple genetic algorithm” IEEEP New Horizon 2006 vol.52,April to June 2006 pp.22-26.

[2] Prof. Aftab Ahmad , Prof. Tahir Nadeem Malik, and Engr. Shahid-ur-Rehman Farooque

“Reconfiguration of Distribution Feeder to Reduce Active Losses” IEEEP New Horizons Vol. No.56, April to June 2007 pp.3-8.

[3] Prof. Aftab Ahmad , Prof. Tahir Nadeem Malik, and Engr. Shahid-ur-Rehman Farooque

“Hybrid Approach to Reduce Voltage Drop and Loading of a Problematic Feeder” IEEEP New Horizons Vol. No.56, April to June 2007 pp.14-17.

[4] Engr. Aftab Ahmad, Dr. Aftab Ahmad, and Engr. Tahir Nadeem Malik, “Unit

Commitment Problem of Thermal Generation Units for Short Term Operational Planning” IEEEP New Horizons Vol. No.56, April to June 2007 pp.20-23.

[5] Aftab Ahmad and Azzam ul Asar, “A PSO based Artificial Neural Network approach

for short term unit commitment problem” accepted for publication in Mehran University Research Journal of Engineering and Technology on June, 2009.

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