SHORT TERM HYDROTHERMAL COORDINATION USING AN …prr.hec.gov.pk/jspui/bitstream/123456789/8282/1/PhD... · 2018-07-23 · SHORT TERM HYDROTHERMAL COORDINATION USING AN EVOLUTIONARY

SHORT TERM HYDROTHERMAL COORDINATION

USING AN EVOLUTIONARY APPROACH

Engr. Shaikh Saaqib Haroon

2011-UET/PhD-EE-40

Supervisor

Prof. Dr. Tahir Nadeem Malik

Department of Electrical Engineering

University of Engineering and Technology

Taxila (Pakistan)

April 2017

Dedicated To

My Ami (Mother) and Abu (Father)

who taught me how to write

“A, B, C, ….” and “1, 2, 3, ….”

and finally now I am able to write my thesis

for

Doctor of Philosophy

Short-term Hydrothermal Coordination using an Evolutionary Approach PhD Thesis Engr. Sh. Saaqib Haroon, U.E.T. Taxila, Pakistan, 2017 ii

TABLE OF CONTENTS

LIST OF TABLES ......................................................................................................................................... vii

LIST OF FIGURES .......................................................................................................................................... xi

LIST OF ABBREVIATIONS AND SYMBOLS ..................................................................................... xii

ABSTRACT .................................................................................................................................................... xiv

ACKNOWLEDGEMENTS ......................................................................................................................... xvi

CHAPTER NO. 1 INTRODUCTION ..................................................................................... 1

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

1.2 Problem Statement .......................................................................................................................... 2

1.3 Objectives ............................................................................................................................................. 4

1.4 Scope of Work .................................................................................................................................... 5

1.5 Thesis Organization ........................................................................................................................ 7

CHAPTER NO. 2 HYDROTHERMAL COORDINATION --- A COMPREHENSIVE

REVIEW ...................................................................................................................... 8

2.1 Basic Mathematical Modelling .................................................................................................. 8

2.1.1 Objective Function................................................................................................................. 8

2.1.1.1 Convex objective function ........................................................................................ 8

2.1.1.2 Non-convex objective function .............................................................................. 9

2.1.2 Constraints ................................................................................................................................ 9

2.1.2.1 Power balance constraint ......................................................................................... 9

2.1.2.2 Water dynamic balance constraint ................................................................... 10

2.1.2.3 Generation capacity constraint ........................................................................... 10

2.1.2.4 Discharge rates limit & prohibited discharge zones constraints ........ 11

2.1.2.5 Reservoir volume storage constraint ............................................................... 11

Short-term Hydrothermal Coordination using an Evolutionary Approach PhD Thesis Engr. Sh. Saaqib Haroon, U.E.T. Taxila, Pakistan, 2017 iii

2.1.2.6 Reservoir end conditions constraint ................................................................ 11

2.1.2.7 Ramp rate limit constraint .................................................................................... 11

2.2 Literature Review ......................................................................................................................... 12

2.2.1 Classical Derivative Based Methods .......................................................................... 13

2.2.2 Deterministic Methods ..................................................................................................... 15

2.2.2.1 Lagrange relaxation (LR) & benders decomposition (BD) .................... 15

2.2.2.2 Dynamic programming (DP) ................................................................................ 19

2.2.2.3 Mixed integer linear programming (MILP) ................................................... 22

2.2.3 Artificial Intelligence Based Methods ....................................................................... 22

2.2.3.1 Neural networks (NN) ............................................................................................. 22

2.3.3.2 Fuzzy logic (FL) .......................................................................................................... 22

2.2.4 Evolutionary/Heuristic and Hybrid Methods ....................................................... 23

2.2.4.1 Genetic algorithm (GA) ........................................................................................... 23

2.2.4.2 Particle swarm optimization (PSO) .................................................................. 27

2.2.4.3 Differential evolution (DE) .................................................................................... 32

2.2.4.4 Gravitational search algorithm (GSA) .............................................................. 35

2.2.4.5 Bacterial foraging algorithm (BFA)................................................................... 36

2.2.4.6 Simulated annealing (SA) & tabu search (TS) .............................................. 37

2.2.4.7 Others .............................................................................................................................. 38

2.3 Discussion ......................................................................................................................................... 40

2.4 Challenges & Bottlenecks ............................................................................................................... 42

CHAPTER NO. 3 WATER CYCLE ALGORITHM --- ESSENTIAL BACKGROUND

FOR HYDROTHERMAL COORDINATION ........................................................................ 43

3.1 Introduction ..................................................................................................................................... 43

3.2 Steps of WCA .................................................................................................................................... 43

3.2.1 Initialization .......................................................................................................................... 43

Short-term Hydrothermal Coordination using an Evolutionary Approach PhD Thesis Engr. Sh. Saaqib Haroon, U.E.T. Taxila, Pakistan, 2017 iv

3.2.2 Movement of streams to the rivers or sea .............................................................. 44

3.2.3 Evaporation & raining process ..................................................................................... 45

3.3 Significance of WCA ...................................................................................................................... 47

3.4 Chaos Theory ................................................................................................................................... 48

3.4.1 Chaotic Sequences .............................................................................................................. 48

3.4.1.1 Logistic Map ................................................................................................................. 49

3.4.1.2 Tent Map ........................................................................................................................ 49

3.4.1.3 Sinusoidal Iterative Map ........................................................................................ 49

3.4.1.4 Lozi Iterative Map ...................................................................................................... 50

3.4.1.5 Gauss Iterative Map .................................................................................................. 50

3.4.2 Application of Chaos Theory in Evolutionary Algorithms .............................. 50

CHAPTER NO. 4 HYDROTHERMAL COORDINATION MODELLING USING

WATER CYCLE ALGORITHM AND PROPOSED HYBRID CHAOTIC WATER CYCLE

ALGORITHM .................................................................................................................. 52

4.1 Initialization of Solution Structure ....................................................................................... 52

4.2 Constraint Handling ..................................................................................................................... 52

4.2.1 Constraint Handling Mechanism for Inequality Constraints ......................... 53

4.2.2 Constraint Handling Mechanism for Equality Constraints ............................. 53

4.2.2.1 Water dynamic balance constraint handling mechanism ...................... 53

4.2.2.2 Active power balance constraint handling mechanism .......................... 54

4.3 Modelling of SHTCP as per WCA ............................................................................................ 54

4.4 Steps of Standard WCA for SHTCP ........................................................................................ 55

4.5 Proposed Hybrid Chaotic Water Cycle Algorithm ......................................................... 56

4.5.1 Initialization .......................................................................................................................... 56

4.5.2 WCA with Chaotic Evaporation Process .................................................................. 56

4.5.3 Chaotic Local Search .......................................................................................................... 57

Short-term Hydrothermal Coordination using an Evolutionary Approach PhD Thesis Engr. Sh. Saaqib Haroon, U.E.T. Taxila, Pakistan, 2017 v

CHAPTER NO. 5 IMPLEMENTATION & CASE STUDIES ............................................ 59

5.1 Development of a Computational Framework ................................................................ 59

5.2 Strategy for Implementation ................................................................................................... 59

5.3 Test Systems Investigated ......................................................................................................... 60

5.3.1 Fixed Head Hydroelectric Units ................................................................................... 60

5.3.1.1 Test System 1 ............................................................................................................... 60

5.3.1.2 Test System 2 ............................................................................................................... 64

5.3.2 Multi-Chain Hydroelectric Units .................................................................................. 67

5.3.2.1 Test System 3 ............................................................................................................... 68

5.3.2.2 Test System 4 ............................................................................................................... 83

5.3.2.3 Test System 5 ............................................................................................................... 96

5.3.2.4 Test System 6 ............................................................................................................ 105

5.3.2.5 Test System 7 ............................................................................................................ 111

5.3.2.6 Test System 8 ............................................................................................................ 116

5.3.2.7 Test System 9 ............................................................................................................ 116

CHAPTER NO. 6 MULTI-OBJECTIVE HYDROTHERMAL COORDINATION USING

PROPOSED HCWCA ............................................................................................................ 119

6.1 Multi-Objective Hydrothermal Coordination Problem ............................................ 119

6.2 Literature Review --- MOSHTCP ......................................................................................... 119

6.3 Mathematical Formulation Of MOSHTCP ....................................................................... 126

6.3.1 Economic Cost Coordination (ECC) ......................................................................... 127

6.3.2 Economic Environmental Coordination (EEC) .................................................. 127

6.3.3 Economic Cost & Environmental Coordination ................................................ 127

6.4 Test System Investigated ........................................................................................................ 128

6.4.1 Economic Cost Coordination ...................................................................................... 129

6.4.2 Economic Environmental Coordination ............................................................... 129

Short-term Hydrothermal Coordination using an Evolutionary Approach PhD Thesis Engr. Sh. Saaqib Haroon, U.E.T. Taxila, Pakistan, 2017 vi

6.4.3 Economic Cost & Environmental Coordination ................................................ 130

CHAPTER NO. 7 HYDROTHERMAL COORDINATION OF UTILITY SYSTEM USING

PROPOSED HYBRID CHAOTIC WATER CYCLE ALGORITHM ................................. 138

7.1 Utility Systems ............................................................................................................................. 138

7.2 Indian Utility System ................................................................................................................ 138

CHAPTER NO. 8 CONCLUSION & SUGGESTIONS ..................................................... 141

Future Work ......................................................................................................................................... 143

Practical Applications ...................................................................................................................... 143

DERIVED PUBLICATIONS ................................................................................................. 144

REFERENCES ......................................................................................................................... 145

Short-term Hydrothermal Coordination using an Evolutionary Approach PhD Thesis Engr. Sh. Saaqib Haroon, U.E.T. Taxila, Pakistan, 2017 vii

LIST OF TABLES

Table 5.1 Test Systems Investigated .......................................................................................... 61

Table 5.2 Test System 1 --- Complete Data ............................................................................. 62

Table 5.3 Test System 1 --- Evolution model ......................................................................... 63

Table 5.4 Test System 1 --- Optimal Hydroelectric and Thermal Power

Generations ...................................................................................................................... 63

Table 5.5 Test System 1 --- Comparison of Results ............................................................ 64

Table 5.6 Test System 2 --- Complete Data ............................................................................. 65



Generations ...................................................................................................................... 66

Table 5.9 Test System 2 --- Comparison of Results ............................................................ 67

Table 5.10 Test System 3 to 9 --- Complete Data of Hydroelectric Units .................... 70

Table 5.11 Test System 3 --- Complete Data of Thermal Unit and Hourly Load

Demand .............................................................................................................................. 71


Table 5.13 Test System 3: Case-I --- Optimal Hydroelectric Discharges ..................... 72

Table 5.14 Test System 3: Case-I --- Optimal Hydroelectric & Thermal Powers .... 73

Table 5.15 Test System 3: Case-I --- Comparison of Results ............................................. 74

Table 5.16 Test System 3: Case-II --- Optimal Hydroelectric Discharges ................... 74

Table 5.17 Test System 3: Case-II --- Optimal Hydroelectric & Thermal Powers ... 75

Table 5.18 Test System 3: Case-II --- Comparison of Results ........................................... 76

Table 5.19 Test System 3: Case-III --- Optimal Hydroelectric Discharges.................. 76

Table 5.20 Test System 3: Case-III --- Optimal Hydroelectric & Thermal Powers . 77

Table 5.21 Test System 3: Case-III --- Comparison of Results ......................................... 78

Table 5.22 Test System 3: Case-IV --- Optimal Hydroelectric Discharges .................. 79

Short-term Hydrothermal Coordination using an Evolutionary Approach PhD Thesis Engr. Sh. Saaqib Haroon, U.E.T. Taxila, Pakistan, 2017 viii

Table 5.23 Test System 3: Case-IV --- Optimal Hydroelectric & Thermal Powers . 80

Table 5.24 Test System 3: Case-IV --- Comparison of Results .......................................... 83

Table 5.25 Test System 4 --- Complete Data of Thermal Units and Hourly Load

Demand .............................................................................................................................. 85



Table 5.28 Test System 4: Case-I --- Optimal Hydroelectric & Thermal Powers .... 87

Table 5.29 Test System 4: Case-I --- Comparison of Results ............................................. 88

Table 5.30 Test System 4: Case-II --- Optimal Hydroelectric Discharges ................... 88

Table 5.31 Test System 4: Case-II --- Optimal Hydroelectric & Thermal Powers ... 89

Table 5.32 Test System 4: Case-II --- Comparison of Results ........................................... 90

Table 5.33 Test System 4: Case-III --- Optimal Hydroelectric Discharges.................. 91

Table 5.34 Test System 4: Case-III --- Optimal Hydroelectric & Thermal Powers . 92

Table 5.35 Test System 4: Case-III --- Comparison of Results ......................................... 93

Table 5.36 Test System 4: Case-IV --- Optimal Hydroelectric Discharges .................. 94

Table 5.37 Test System 4: Case-IV --- Optimal Hydroelectric & Thermal Powers . 95

Table 5.38 Test System 4: Case-IV --- Comparison of Results .......................................... 96


Demand .............................................................................................................................. 98



Table 5.42 Test System 5: Case-I --- Optimal Hydroelectric Powers ......................... 100

Table 5.43 Test System 5: Case-I --- Optimal Thermal Powers .................................... 101

Table 5.44 Test System 5: Case-I --- Comparison of Results .......................................... 101

Table 5.45 Test System 5: Case-II --- Optimal Hydroelectric Discharges ................ 102

Table 5.46 Test System 5: Case-II --- Optimal Hydroelectric Powers ....................... 103

Short-term Hydrothermal Coordination using an Evolutionary Approach PhD Thesis Engr. Sh. Saaqib Haroon, U.E.T. Taxila, Pakistan, 2017 ix

Table 5.47 Test System 5: Case-II --- Optimal Thermal Powers .................................. 104

Table 5.48 Test System 5: Case-II --- Comparison of Results ........................................ 104


Demand ........................................................................................................................... 105

Table 5.50 Test System 6 --- Evolution model ...................................................................... 106

Table 5.51 Test System 6 --- Optimal Hydroelectric Discharges ................................. 107

Table 5.52 Test System 6 --- Optimal Hydroelectric Powers ......................................... 108

Table 5.53 Test System 6 --- Optimal Thermal Powers .................................................... 109

Table 5.54 Test System 6 --- Comparison of Results ......................................................... 109


Demand ........................................................................................................................... 112


Table 5.57 Test System 7 --- Optimal Hydroelectric Discharges ................................. 113

Table 5.58 Test System 7 --- Optimal Hydroelectric Powers ......................................... 114

Table 5.59 Test System 7 --- Optimal Thermal Powers .................................................... 115






Table 6.1 Test System 10 --- Emission Data of Thermal Units ................................... 129

Table 6.2 Test System 10 --- Evolution model ................................................................... 129

Table 6.3 Test System 10: ECC--- Optimal Hydroelectric Discharges ..................... 130

Table 6.4 Test System 10: ECC --- Optimal Hydroelectric & Thermal Powers ... 131

Table 6.5 Test System 10: EEC--- Optimal Hydroelectric Discharges ..................... 132

Table 6.6 Test System 10: EEC --- Optimal Hydroelectric & Thermal Powers ... 133

Short-term Hydrothermal Coordination using an Evolutionary Approach PhD Thesis Engr. Sh. Saaqib Haroon, U.E.T. Taxila, Pakistan, 2017 x

Table 6.7 Test System 10: ECEC--- Optimal Hydroelectric Discharges .................. 134

Table 6.8 Test System 10: ECEC --- Optimal Hydroelectric & Thermal Powers 135

Table 6.10 Test System 10 --- Comparison of Results ...................................................... 136

Table 7.1 Indian Utility System --- Complete Data of Hydroelectric Units, Thermal

Units and Hourly Load Demand .......................................................................... 139

Table 7.2 Indian Utility System --- Evolution model ....................................................... 140

Table 7.3 Indian Utility System --- Comparison of Results .......................................... 140

Short-term Hydrothermal Coordination using an Evolutionary Approach PhD Thesis Engr. Sh. Saaqib Haroon, U.E.T. Taxila, Pakistan, 2017 xi

LIST OF FIGURES

Fig. 2.1 No. of paper published on SHTCP w.r.t. years ....................................................... 12

Fig. 2.2 No. of papers published on the use of potential tools ....................................... 13

Fig. 3.1 Graphical view of a stream flowing towards a river .......................................... 45

Fig. 4.1 Flowchart of Proposed HCWCA for SHTCP ............................................................ 58

Fig. 5.1 Network Configuration of Test System 1 ................................................................ 62

Fig. 5.2 Network Configuration of Test System 2 ................................................................ 64

Fig. 5.3 Configuration of Multi-chain Hydroelectric Units for Test System 3-10 . 67

Fig. 5.4 Test System 3: Case-I --- Convergence Characteristics ..................................... 81

Fig. 5.5 Test System 3: Case-II --- Convergence Characteristics ................................... 81

Fig. 5.6 Test System 3: Case-III --- Convergence Characteristics ................................. 82

Fig. 5.7 Test System 3: Case-IV --- Convergence Characteristics .................................. 82

Fig. 5.8 Test System 4: Case-I --- Convergence Characteristics ..................................... 93

Fig. 5.9 Test System 4: Case-II --- Convergence Characteristics ................................... 96

Fig. 5.10 Test System 4: Case-III --- Convergence Characteristics ................................ 97

Fig. 5.11 Test System 5 --- Convergence Characteristics ................................................ 110





Fig. 6.2 Test System 10: ECC --- Convergence Characteristics ................................... 137

Fig. 6.3 Test System 10: EEC --- Convergence Characteristics ................................... 137

Fig. 7.1 Network Configuration of Indian Utility System .............................................. 138

Short-term Hydrothermal Coordination using an Evolutionary Approach PhD Thesis Engr. Sh. Saaqib Haroon, U.E.T. Taxila, Pakistan, 2017 xii

LIST OF ABBREVIATIONS AND SYMBOLS

ABC Artificial Bee Colony

AI Artificial Intelligence

AL Augmented Lagrangian

BD Bender’s Decomposition

BFA Bacterial Foraging Algorithm

CA Cultural Algorithm

CSA Clonal Selection Algorithm

DE Differential Evolution

DP Dynamic Programming

EA Evolutionary Algorithm

ED Economic Dispatch

EP Evolutionary Programming

EPS Electrical Power System

ERWCA Evaporation Rate based Water Cycle Algorithm

FL Fuzzy Logic

GA Genetic Algorithm

GSA Gravitational Search Algorithm

HCWCA Hybrid Chaotic Water Cycle Algorithm

HTC Hydrothermal Coordination

IP Interior Point

LR Lagrange Relaxation

MFM Multi Fuel Mix

MILP Mixed Integer Linear Programming

MOSHTCP Multi-Objective Short Term Hydrothermal Coordination Problem

Short-term Hydrothermal Coordination using an Evolutionary Approach PhD Thesis Engr. Sh. Saaqib Haroon, U.E.T. Taxila, Pakistan, 2017 xiii

MVPE Multi Valve Point Effect

NN Neural Networks

IP Interior Point

NR Newton Raphson

OPF Optimal Power Flow

PDZ Prohibited Discharge Zones

POZ Prohibited Operating Zones

PPO Predator Prey Optimization

PSO Particle Swarm Optimization

PSOP Power System Operational Planning

RR Ramp Rate

SA Simulated Annealing

SHTCP Short Term Hydrothermal Coordination Problem

TLBO Teaching Learning Based Optimization

TS Tabu Search

UC Unit Commitment

WCA Water Cycle Algorithm

Short-term Hydrothermal Coordination using an Evolutionary Approach PhD Thesis Engr. Sh. Saaqib Haroon, U.E.T. Taxila, Pakistan, 2017 xiv

ABSTRACT

Short-term hydrothermal coordination problem (SHTCP) is one of the vital

step in power system operational planning. It controls both the economics and the

environmental aspects of electrical energy as SHTCP decides the optimum share of

thermal energy in the absence/unavailability of hydroelectric energy.

In actuality SHTCP is highly non-linear, non-convex, dynamic and constrained

optimization problem and generally, SHTCP is formulated as an approximate

problem by neglecting the prohibited discharge zones and coupling time required

between the cascade reservoirs of hydroelectric units, non-convexity in thermal unit

fuel cost curves, prohibited operating zones and ramp rates.

Evolutionary algorithms are potential solution methodologies for such non-

convex problems. Many evolutionary algorithms have been applied in the literature

for the solution of SHTCP but many complex constraints discussed above are mostly

neglected. In addition to this, the larger test systems are also usually neglected while

solving SHTCP. The research is still focusing in finding out a robust and strong

technique which is able to solve complete SHTCP with all constraints as well as the

larger test systems of SHTCP.

Water cycle algorithm (WCA) is a new meta-heuristic algorithm inspired from

the natural hydrologic cycle and has certain inherent strengths over other

evolutionary algorithms. It basically works on the principle of raining which cause

the formation of streams that flow downhill towards the river and eventually into the

sea which is the optimal solution. This algorithm has not yet been investigated for

SHTCP.

In the proposed research, SHTCP has been modelled in WCA environment

along with the constraints like cascade nature of hydroelectric units, varying

reservoir inflows, limits on the reservoir storage and discharge capacity, water

transport delay, the prohibited discharge zones of hydroelectric units, multiple valve

point effects in thermal fuel cost curves, ramp rates and prohibited operating zones

of thermal units, the varying load demand, and the limitations on the generation of

hydroelectric and thermal units. This algorithm has been tested on both fixed head

and multi-chain cascade variable head hydroelectric standard test systems. In multi-

Short-term Hydrothermal Coordination using an Evolutionary Approach PhD Thesis Engr. Sh. Saaqib Haroon, U.E.T. Taxila, Pakistan, 2017 xv

chain cascade hydroelectric test system, four hydroelectric units are configured with

different number of thermal units as per standard test systems. Different standard

case studies along with some larger proposed test systems and utility system

available in literature have been investigated in detail.

To further improve the performance of WCA in context of getting trapped in

local solutions, a hybrid chaotic water cycle algorithm (HCWCA) is proposed for the

solution of SHTCP. The conventional WCA has been hybridized with the logistic

mapping of the chaotic paradigm to improve its performance to avoid premature

convergence. The proposed algorithm when tested on all the above mentioned test

systems and case studies showed improved results from WCA which already has

outperformed other recent methods in literature.

The conventional SHTCP is now being actively investigated as due to the

environmental concerns. Further, the standard multi-objective SHTCP (MOSHTCP)

test system has been successfully modelled in the environment of WCA and HCWCA.

In all cases, the obtained results are better than recent available results indicating

the promise of approach.

The proposed work is a valuable addition in the bank of SHTCP solution

methodologies. It has the strength of incorporating complex constraints and

capability of solving complex search space with superior results as compared to

other recent techniques available in the literature.

Short-term Hydrothermal Coordination using an Evolutionary Approach PhD Thesis Engr. Sh. Saaqib Haroon, U.E.T. Taxila, Pakistan, 2017 xvi

ACKNOWLEDGEMENTS

To begin with the name of ALMIGHTY ALLAH, who gave me the strength and

spirit to fulfill the mandatory requirements of this dissertation.

I am highly indebted to my supervisor, Prof. Dr. Tahir Nadeem Malik because

without his guidance, his strictness (in some cases) and his trust; this work could not

have been completed. His valuable knowledge and vast experience of the subject and

the area removed the difficulties at all the critical junctures.

I would also like to thank to the members of my Research Monitoring Committee

for their guidance and valuable inputs.

I would like to thank the authorities of University of Engineering & Technology,

Taxila for providing the financial means and resources for conducting this research.

My friends and colleagues, Dr. Salman, Dr. Sarmad, Dr. Intisar and Dr. Azhar

deserve acknowledgement and a debt of gratitude for their time and energy in

reviewing various section of the manuscripts and the published articles.

Finally, I would like to give my special gratitude to my parents, wife and children

for their patience and support.

Chapter 1 Introduction

Short-term Hydrothermal Coordination using an Evolutionary Approach PhD Thesis Engr. Sh. Saaqib Haroon, U.E.T. Taxila, Pakistan, 2017 1

CHAPTER NO. 1 INTRODUCTION

1.1 GENERAL

Electrical power system is one of the vital and critical infrastructure

because it enables all other infrastructures. The electrical power may be regarded

as the heart of all other systems of life; which when works let al.l other organs or

systems to work, and its failure causes failure of all other systems of daily life.

With the passage of time and development, an optimized supply of electrical

energy which should be clean and has limited fuel emissions must be made

available to the consumer with proper reliability and stability.

The energy consumption per capita per annum along with the indicators

for emissions and pollutants are also being considered while deciding about the

development of any country. The power system planners who plan for the next

twenty to thirty years also have to consider the environmental aspects in addition

to the economic aspects while designing new power plants. The plants based on

fossil fuels generate the noxious fuel emissions while generating electrical energy

thereby causing global warming. In addition to the increase in the global warming,

they are also responsible of diminishing the natural oil resources which have a

number of other uses. Therefore, to save these resources and to preserve the

environment, alternate sources of generating electrical energy having

zero/negligible emissions must be exploited and planned. Among all the

renewable energy sources, hydroelectric energy is the most abundant, the densest

and the most tested source of generation of electrical energy. But due to its

unavailability at certain times, it has to be used in conjunction with thermal units.

The power system operational planning (PSOP) deals: short term load

forecasting, unit commitment, hydrothermal coordination, economic dispatch,

voltage control and frequency control. HTCP is one of the vital step in PSOP not

only coordinating the hydroelectric and thermal machines but also address the

environmental aspects. It is considered as a highly non-linear, non-convex,

dynamic and combinatorial optimization problem involving numerous



constraints. Keeping in view the complexity of the problem, evolutionary

algorithms are the potential solution methodologies for the solution of HTCP. In

this thesis efforts are made to investigate the SHTCP.

1.2 PROBLEM STATEMENT

The optimum utilization of both the hydroelectric and thermal energy is the

key objective of solving HTCP. Depending upon the time consideration from an hour

or a day to a week or months or even years, the HTCP can be classified as short-term

HTCP (SHTCP), mid-term HTCP and long-term HTCP. The SHTCP is one of the active

area of research in the field of PSOP where an optimum coordinated operation of both

hydroelectric and thermal units for a given time interval depending upon the water

availability for the operation of hydroelectric units is calculated. The well-timed

apportionment of hydroelectric units is an essential task because of the availability

and utilization of water during different time intervals. The SHTCP controls both the

economics and environmental aspects of electrical power system as it decides the

optimum share of thermal power in the absence/unavailability of hydroelectric

power. Thermal power is only responsible for the economics and environmental

effects of an electrical power system. As the operating cost of hydroelectric units is

insignificant, the objective function of SHTCP reduces only to minimize the fuel cost

of thermal units; subject to a variety of constraints like cascade nature of

hydroelectric units, varying reservoir inflows, limits of the reservoir storage, limits

on the discharge capacity, water transport delay, the prohibited discharge zones

(PDZ) of hydroelectric units, multiple valve point effects on thermal fuel cost curves,

ramp rates (RR) and prohibited operating zones (POZ) of thermal units, the varying

load demand, and the limitations on the generation of the hydroelectric and thermal

units. The association of these large number of constraints make the SHTCP a highly

complex, non-linear, dynamic and non-convex combinatorial optimization problem.

Based on head variations, the SHTCP can be classified as, fixed head and

variable head SHTCP. The fixed head SHTCP is the type in which the head of the

reservoir remain fixed, meaning the reservoir is too large or the time interval taken

into account for SHTCP is too short i.e. an hour, such that the water used in this



interval does not affect the head of the reservoir. In variable head SHTCP the head of

the reservoir is not fixed rather it varies due to the inflows and water discharges in

all time intervals.

The objective function of SHTCP has been assumed to be a convex function for

many years by ignoring the real and practical effects of multiple valve points and

prohibited operating zones in the fuel cost curves of the thermal units. This

assumption was basically the requirement for the application of mathematical

approaches. This assumption leads to inaccurate results because in actuality, fuel

cost curves of thermal units are highly non-convex due to the involvement of

different practical effects as mentioned above. On the other hand, if these constraints

are taken into account, the conventional deterministic methods fail to apply. The

failure of the conventional deterministic methods opened a gateway for the

development and application of evolutionary algorithms on the non-convex and non-

linear optimization problems like SHTCP. Many EA like genetic algorithms, simulated

annealing, particle swarm optimization, differential evolution etc. have been

successfully applied to investigate this SHTCP. Even while applying EA, some

important constraints like transmission losses, PDZ of hydroelectric units, RR and

POZ of thermal units are usually not taken into account while solving SHTCP.

Due to the concerns regarding zero emission act in many of the countries, the

noxious fuel emissions have to be minimized in addition to the fuel costs in SHTCP.

Therefore, the two contradictory objective functions i.e. minimizing the fuel cost and

the fuel emissions have to be solved simultaneously in MOSHTCP resulting in more

complex problem as compared to single objective SHTCP.

The EA give solution to non-linear, non-convex and highly complex single

objective and multi-objective problems due to independence from restriction of

differentiability & continuity, random nature and better exploration and exploitation

capabilities. The scope of application of EA is wide and are being applied in different

areas of electrical power system such as fuzzy based optimization algorithm for

optimizing the level of energy [1], multi-objective distribution system configuration

optimization [2], load pattern grouping [3], fault diagnosis of power system [4], the

modelling of the operation strategies of hydroelectric resources in presence of wind



farms [5], voltage control optimization with distributed generation [6] and non-

convex economic dispatch of thermal units [7].

Water Cycle Algorithm (WCA) proposed by Hadi Eskandar et al. [8] is a

relatively recent meta-heuristic algorithm. The working philosophy of this algorithm

has been derived from water cycle process of nature in which rainfall causing the

formation of streams, flowing downhill towards the rivers and eventually merging

into the sea. The main features of WCA for optimization may be listed as:

i. superior global searching capability because of evaporation and raining

process

ii. quick convergence to the global optimum,

iii. streams or rivers are not static points rather they move and there is an

unintentional move towards the optimum solution,

iv. population based search,

WCA has not yet been investigated for SHTCP, so in this research work, WCA has

been selected for the solution of SHTCP and MOSHTCP. In EA there are the chances

of premature convergence and getting struck in local optima. There are ways to come

out of these problems. Chaos phenomenon may be the one of the options. In this

work, chaos paradigm has been hybridized with standard WCA to propose a hybrid

chaotic water cycle algorithm (HCWCA). Both single objective SHTCP and MOSHTCP

have been modelled in standard WCA and HCWCA environment and tested on

standard test systems available in the literature.

1.3 OBJECTIVES

The objectives of the proposed research work are listed as:

i. Modelling of SHTCP by integrating the following sub-problems:

a) Hydro sub-problem

b) Unit commitment sub-problem

c) Economic dispatch problem



ii. To investigate the performance of standard WCA for the solution of SHTCP

and MOSHTCP.

iii. To develop a hybrid model based on WCA for the optimal solution of SHTCP

and MOSHTCP which is called as HCWCA.

iv. To include all possible constraints of practical SHTCP.

v. To implement the algorithms in C++ to work in personal computer

environment and testing on standard test systems for validation and

comparison.

1.4 SCOPE OF WORK

The contributions made in this research work are outlined as:

I. Design of SHTCP

Non-linear, non-convex, dynamic constrained optimization problem

a. Single objective function

Convex and non-convex cost functions due to valve point effect

b. Multi-objective function

Non-convex fuel cost function and emission function

c. Constraints

power balance constraint with and without transmission loss

generation limit constraint of hydroelectric and thermal units

water continuity constraint

discharge limit and reservoir limit constraints

initial and final reservoir storage constraints

water transport delay constraint

prohibited discharge zones of hydroelectric units

ramp rate constraint of thermal units

prohibited operating zones of thermal units



II. Modelling of SHTCP & MOSTCHP using WCA

Mapping of designed SHTCP & MOSHTCP in WCA environment

III. Proposed HCWCA

Mapping of designed SHTCP & MOSHTCP in proposed HCWCA

environment

IV. Selection of Test Systems

Standard test systems from the literature have been

1. Fixed Head SHTCP consisting of 1 hydroelectric and 1 thermal unit

2. Fixed Head SHTCP consisting of 1 hydroelectric and 3 thermal units

3. Multi-chain Variable Head SHTCP consisting of:

a. 4 hydroelectric and 1 thermal unit with following case studies:

I. Convex Cost Function

II. Convex Cost Function with PDZ

III. Non-Convex Cost Function

IV. Non-Convex Cost Function with PDZ

b. 4 hydroelectric and 3 thermal units with following case studies:

I. Non-Convex Cost Function

II. Non-Convex Cost Function with Transmission Losses

III. Non-Convex Cost Function with Losses, PDZ and RR

IV. Non-Convex Cost Function with POZ

c. 4 hydroelectric and 6 thermal units with following case studies:

I. Non-Convex Cost Function

II. Non-Convex Cost Function with POZ

d. 4 hydroelectric and 10 thermal units with non-convex cost

function

e. 4 hydroelectric and 10 thermal units for a mixed binary

hydrothermal problem

f. For large scale test studies, two larger test systems have been

designed to investigate the performance of WCA. The larger test

systems consist of 20 thermal and 40 thermal units.

g. For MOSHTCP standard test system of 4 hydroelectric and 3

thermal units.

h. Practical utility system available in the literature



V. Computer Implementation

The standard WCA and HCWCA have been implemented in C++

environment on a personal computer and a computational framework

has been developed for the solution of SHTCP and MOSHTCP.

VI. Testing & Simulation

All the above mentioned standard test systems and their case studies

have been tested using the standard WCA and HCWCA.

VII. Validation of Results

The results obtained have been validated by comparing them with the

results available in the literature obtained using other EA.

1.5 THESIS ORGANIZATION

Chapter 2 discusses the detailed mathematical modelling of SHTCP along with all

its constraints. A detailed literature review of the work done on SHTCP especially

using the evolutionary computation methods is presented in detail.

Chapter 3 presents the basics and the working philosophy of standard WCA and

discusses the essential background required for the implementation of standard

WCA for SHTCP and MOSHTCP.

Chapter 4 explains the detailed working of standard WCA and the proposed

HCWCA for the solution of SHTCP. The structure of the solution for SHTCP using these

methods and the pragmatic set of rules to handle the equality and inequality

constraints along with the flow chart of the proposed methodology to solve SHTCP.

Chapter 5 contains the simulation results and discussion. The results of all case

studies are presented in tabular form with discussions.

Chapter 6 presents the MOSHTCP along with all case studies and discussion.

Chapter 7 presents the SHTCP of utility system available in the literature.

Lastly, Chapter 8 gives conclusions and suggestions for future work.

Chapter 2 Hydrothermal Coordination --- A Comprehensive Review


CHAPTER NO. 2 HYDROTHERMAL COORDINATION

--- A COMPREHENSIVE REVIEW

2.1 BASIC MATHEMATICAL MODELLING

The optimal solution of SHTCP involves the optimization of an objective function

which is highly complex and non-convex and is subjected to a variety of non-linear

hydraulic and thermal constraints. The objective of SHTCP is to curtail the total fuel

cost of thermal units by using the optimal amount of water from the hydroelectric

sources as per their release and volume constraints and satisfying the thermal

constraints as well. Mathematically, the SHTCP can be formalized as:

2.1.1 Objective Function

The mathematical representation of the objective function of a hydrothermal

coordination problem is written as:

𝑚𝑖𝑛𝑖𝑚𝑖𝑧𝑒 𝐹 = ∑∑𝑓𝑖(𝑃𝑠𝑖𝑡)

𝑁𝑠

𝑖=1

𝑇

𝑡=1

(2.1)

where 𝐹 is the total fuel cost, 𝑃𝑠𝑖𝑡 is the power generation of 𝑖𝑡ℎ thermal generating

unit at time 𝑡, 𝑓𝑖 is the fuel cost of 𝑖𝑡ℎ thermal unit, 𝑁𝑠 is the total number of thermal

units and 𝑇 is the total number of time intervals for the scheduled period.

The objective function of both convex and non-convex nature will be handled in this

research work.

2.1.1.1 Convex objective function

Conventionally, the fuel cost function of thermal units can be represented as a

quadratic function as follows:

𝑓𝑖(𝑃𝑠𝑖𝑡) = 𝑎𝑖 + 𝑏𝑖𝑃𝑠𝑖𝑡 + 𝑐𝑖𝑃𝑠𝑖𝑡2 (2.2)



where 𝑎𝑖, 𝑏𝑖, 𝑐𝑖 are the fuel cost coefficients of 𝑖𝑡ℎ thermal unit.

2.1.1.2 Non-convex objective function

For the precise and real-world modeling of problem, the above mentioned fuel cost

function needs to be reviewed. The real-world characteristics involve valve point

effect and the objective function is re-written as:

𝑓𝑖(𝑃𝑠𝑖,𝑡) = 𝑎𝑖 + 𝑏𝑖𝑃𝑠𝑖𝑡 + 𝑐𝑖𝑃𝑠𝑖𝑡2 + |𝑑𝑖 × sin {𝑒𝑖 (𝑃𝑠𝑖

𝑚𝑖𝑛 − 𝑃𝑠𝑖𝑡)}| (2.3)

where 𝑑𝑖, 𝑒𝑖 are the fuel cost coefficients of 𝑖𝑡ℎ thermal unit showing valve point

effect.

2.1.2 Constraints

The solution of SHTCP involves many hydroelectric and thermal constraints

described as follows:

2.1.2.1 Power balance constraint

The total hydroelectric and thermal generations at each time interval 𝑡 should meet

the forecasted load demand and the transmission line losses.

∑𝑃ℎ𝑗𝑡 + ∑𝑃𝑠𝑖𝑡 =

𝑁𝑠

𝑖=1

𝑁ℎ

𝑗=1

𝑃𝐷𝑡 + 𝑃𝐿𝑡 (2.4)

where 𝑁ℎ is total number of hydroelectric units, 𝑃ℎ𝑗𝑡 is generated power of 𝑗𝑡ℎ

hydroelectric unit at interval 𝑡, 𝑃𝐷𝑡 is power demand at interval 𝑡.

The power loss will be calculated as:

𝑃𝐿𝑡 = ∑ ∑ 𝑃𝑖𝑡𝐵𝑖𝑗𝑃𝑗𝑡

𝑁𝑠+𝑁ℎ

𝑗=1

𝑁𝑠+𝑁ℎ

𝑖=1

+ ∑ 𝐵0𝑖𝑃𝑖𝑡

𝑁𝑠+𝑁ℎ

𝑖=1

+ 𝐵00 (2.5)

As per the power balance equation both the hydroelectric and thermal powers share

the total load demand. The power output of the thermal units increases with the



decrease in the power from hydroelectric units and therefore, the fuel costs of the

thermal units increases.

The hydroelectric power generation is the function of reservoir storage volume and

water discharge rate and it is expressed as:

𝑃ℎ𝑗𝑡 = 𝐶1𝑗𝑉ℎ𝑗𝑡2 + 𝐶2𝑗𝑄ℎ𝑗𝑡

2 + 𝐶3𝑗𝑉ℎ𝑗𝑡𝑄ℎ𝑗𝑡 + 𝐶4𝑗𝑉ℎ𝑗𝑡 + 𝐶5𝑗𝑄ℎ𝑗𝑡 + 𝐶6𝑗 (2.6)

where 𝐶1𝑗, 𝐶2𝑗 , 𝐶3𝑗, 𝐶4𝑗, 𝐶5𝑗 , 𝐶6𝑗 are the generation coefficients of 𝑗𝑡ℎ hydroelectric

unit, 𝑉ℎ𝑗𝑡 is the reservoir storage volume of 𝑗𝑡ℎ plant at time 𝑡 and 𝑄ℎ𝑗𝑡 is the water

release of 𝑗𝑡ℎ plant at time 𝑡.

2.1.2.2 Water dynamic balance constraint

𝑉ℎ𝑗𝑡 = 𝑉ℎ𝑗,𝑡−1 + 𝐼ℎ𝑗𝑡 − 𝑄ℎ𝑗𝑡 − 𝑆ℎ𝑗𝑡 + ∑(𝑄ℎ𝑛,𝑡−𝜏𝑛𝑗+ 𝑆ℎ𝑛,𝑡−𝜏𝑛𝑗

)

𝑅𝑢𝑗

𝑛=1

(2.7)

where 𝐼ℎ𝑗𝑡 is the natural inflow of 𝑗𝑡ℎ hydroelectric unit respectively at time 𝑡, 𝑆ℎ𝑗𝑡 is

the spillage discharge rate of 𝑗𝑡ℎ hydroelectric unit respectively at time 𝑡, 𝑅𝑢𝑗 is the

number of upstream hydroelectric generating units immediately above the 𝑗𝑡ℎ

reservoir and 𝜏𝑛𝑗 is the water transport time delay from reservoir 𝑛 to reservoir 𝑗.

2.1.2.3 Generation capacity constraint

𝑃𝑠𝑖𝑚𝑖𝑛 < 𝑃𝑠𝑖𝑡 < 𝑃𝑠𝑖

𝑚𝑎𝑥 (2.8)

𝑃ℎ𝑗𝑚𝑖𝑛 < 𝑃ℎ𝑗𝑡 < 𝑃ℎ𝑗

𝑚𝑎𝑥 (2.9)

Where 𝑃𝑠𝑖𝑚𝑖𝑛, 𝑃𝑠𝑖

𝑚𝑎𝑥 are the minimum & maximum generating capacity of 𝑖𝑡ℎ thermal

unit and 𝑃ℎ𝑗𝑚𝑖𝑛, 𝑃𝑠𝑗

𝑚𝑎𝑥 are the minimum & maximum generation capacity of 𝑗𝑡ℎ

hydroelectric unit.



2.1.2.4 Discharge rates limit & prohibited discharge zones constraints

𝑄ℎ𝑗𝑚𝑖𝑛 < 𝑄ℎ𝑗𝑡 < 𝑄ℎ𝑗

𝑙𝑏,1

𝑄ℎ𝑗𝑙𝑏,𝑛−1 < 𝑄ℎ𝑗𝑡 < 𝑄ℎ𝑗

𝑢𝑏,1 𝑛 = 2,3, … .𝑁𝐷𝑗

𝑄ℎ𝑗𝑢𝑏,𝑛 < 𝑄ℎ𝑗𝑡 < 𝑄ℎ𝑗

𝑚𝑎𝑥

(2.10)

where 𝑄ℎ𝑗𝑚𝑖𝑛, 𝑄ℎ𝑗

𝑚𝑎𝑥 are the minimum & maximum discharge limits of 𝑗𝑡ℎ reservoir and

𝑄ℎ𝑗𝑙𝑏 , 𝑄ℎ𝑗

𝑢𝑏 are the lower & upper limits of prohibited discharge zones of 𝑗𝑡ℎ reservoir.

2.1.2.5 Reservoir volume storage constraint

𝑉ℎ𝑗𝑚𝑖𝑛 < 𝑉ℎ𝑗 < 𝑉ℎ𝑗

𝑚𝑎𝑥 (2.11)

where 𝑉ℎ𝑗𝑚𝑖𝑛, 𝑉ℎ𝑗

𝑚𝑎𝑥 are the minimum & maximum reservoir storage limits of 𝑗𝑡ℎ

reservoir.

2.1.2.6 Reservoir end conditions constraint

𝑉𝑗0 = 𝑉𝑗

𝐼𝑛𝑖 , 𝑉𝑗𝑇 = 𝑉𝑗

𝐸𝑛𝑑; 𝑗 = 1,2, ……𝑁ℎ (2.12)

where 𝑉𝑗𝐼𝑛𝑖, 𝑉𝑗

𝐸𝑛𝑑 are the initial & final reservoir volume storage restrictions of 𝑗𝑡ℎ

plant.

2.1.2.7 Ramp rate limit constraint

The upper and lower ramp rate limits of thermal units limit the difference of power

generated by the 𝑖𝑡ℎ thermal unit in a certain interval of time than the previous

interval. Mathematically, it is written as:

𝑃𝑠𝑖𝑡 − 𝑃𝑠𝑖,(𝑡−1) ≤ 𝑈𝑅𝑖 , 𝑃𝑠𝑖,(𝑡−1) − 𝑃𝑠𝑖𝑡 ≤ 𝐿𝑅𝑖 (2.13)

where 𝑈𝑅𝑖, 𝐿𝑅𝑖 are the upper & lower ramp rate limits of 𝑖𝑡ℎ thermal unit



2.2 LITERATURE REVIEW

The problems of power system optimal operation and planning have been

investigated by the researchers for last many decades using various optimization

methods. The earliest methods were the base load, the incremental and the best point

loading methods. I. A. Farhat and M. E. El-Hawary in [9] presented a complete

overview of majority of the optimization methods applied to solve SHTCP. In addition

to many research papers, some PhD works have also been done on the application of

many methods/techniques on SHTCP [10, 11]. Many mathematical programming

methods, iterative procedures, artificial intelligence tools and the evolutionary meta-

heuristics have been used to solve this SHTCP. With the development of new

evolutionary methods, the additional details of the problems are taken into account.

Initially, only the thermal units were considered and the problem of economic load

dispatch have been solved. And now for a long time the hydroelectric topology and

its constraints are also considered. On the basis of the types of the optimization

methods, they can be divided into three broad categories as follows. Fig. 2.1 shows a

graph of no. of paper published on SHTCP during last 10 years and Fig. 2.2 shows the

no. of paper published on the use of the potential tools for the solution of SHTCP.

Fig. 2.1 No. of paper published on SHTCP w.r.t. years

0

10

20

30

40

50

60

70

2016 2015 2014 2013 2012 2011 2010 2009 2008 2007 2006 2005

No

. of

Pap

ers

Years



Fig. 2.2 No. of papers published on the use of potential tools

2.2.1 Classical Derivative Based Methods

The classical derivative based methods include the gradient method, Gauss-Seidel

and Newton-Raphson (NR) method, Simplex method and Interior-Point (IP) method.

These methods are basically simple and easy to program as they only make steps

using the jacobian and/or hessian operators. These methods can solve only the

simple, small scale, differentiable, continuous and convex objective functions and can

solve for local optimum with a considerable strength. These methods were very less

applied to SHTCP.

A derivative based NR method was applied to solve the SHTCP in [12]. But here only

a simple and convex fuel cost characteristics of thermal units have been considered.

In 1984, Karmarkar [13] introduced the IP method and it is been used widely since

then. This paper claimed that the proposed IP method is faster for large scale

optimization problems than simplex methods.

The IP method combined with the Gauss-Newton method [14] was used by M. Kleina

et al. in 2012, to solve the Brazilian interconnected system. The method had a good

computational time and better results.

In 1997 a study was carried out which compared the different codes of IP applied to

the medium term HTCP [15]. The pros and cons of commercial and researched codes

were compiled. A Spanish Hydrothermal System was used for testing out all of these

codes.

01020304050607080

No

. of

Pap

ers

Different Potential Tools



In 1998 a decoupled method based on Lagrangian Relaxation (LR) was proposed.

Hydrothermal Optimal Power Flow (HTOPF) was decomposed into OPF sub-

problems of thermal and hydroelectric units [16]. It was suitable for the large scale

problems and greatly reduced the memory requirements. The calculations were fast

and quick in achieving convergence.

The clipping off method was merged with IP method in 1999 [17]. The main

advantage of the clipping off method was the setting of control variables to their

lower and upper bounds. The number of required iterations and trials were reduced

as compared to the standard IP solution. The results were same nonetheless. It was

the first time that IP was used in combination with clipping off method for SHTCP.

In 2000, R. F. Loyola et al. carried out a comparison of direct and indirect methods of

solving SHTCP in terms of computation [18]. The paper’s contribution was based on

to have the Quasi-Optimal solution in reasonable time. The direct method was used

in combination with the indirect method approach to achieve that. The direct method

Primal-Dual IP was used to relax the binary variables of thermal unit’s status and the

indirect method LR was used for decomposing the primal problem into thermal and

hydro sub-problems. Cutting plane method was used for the maximization of dual

function and the hydro and thermal units were solved using DP. The results showed

that the solution provided by both approaches was practically equal. However, LR

provided the solution faster.

In 2000, H. Wei et al. [19] used IP method for the HTOPF. The main difference

between the HTOPF and OPF is that the first is a dynamic nature optimization. The

algorithm was tested on the six systems; the largest one contained 1047 buses with

72 time intervals. The study concluded that the algorithm is very fast as well

compared to other existing techniques and has the ability to handle the large scale

problem like HTOPF. A high accuracy was achieved with half the CPU time.

In 2001, the genetic algorithm (GA) in combination with IP was presented by J. L. M.

Ramos et al. in [20]. GA was used for the ON/OFF status of binary variables of thermal

units and the solution of hydraulically coupled hydroelectric and thermal units was

obtained by IP method. The constraints like, maximum up down ramps of thermal

units and temporal constraints of cascaded reservoir were also considered.



This assumption of simplifying the fuel cost characteristics by neglecting the non-

convexity in the fuel cost curves of thermal units due to the multiple valve point effect

(MVPE) and POZ leads to inaccurate results causing huge loss of revenue.

These classical derivative based methods have a major drawback of trapping into

local optima as they never reach to the global optima for non-convex search space.

Due to the inability of these methods to find the accurate results for complicated,

large scale, non-convex and non-differentiable systems, the use of these methods

have almost been limited to theory.

2.2.2 Deterministic Methods

The deterministic methods offer a variety of different optimization methods based

on some deterministic and mathematical background. These methods usually work

either by decomposing the problem by relaxation or by making sub-problems based

on the principle of optimality. The main deterministic methods which have been

investigated by the researchers for the solution of SHTCP are discussed in this section

and their merits and demerits are presented here.

2.2.2.1 Lagrange relaxation (LR) & benders decomposition (BD)

In the field of mathematical optimization, LR is a relaxing method which

approximates a difficult problem of constrained optimization by a simpler problem.

By using LR violation of inequality constraints is penalized, posing a cost on

violations. The problem of maximizing the lagrangian function of the dual variables

is the lagrangian dual problem.

Although, a variety of techniques which are advance and computationally efficient

than LR exist these days but still we need LR to satisfy the constraints. LR is an

excellent and best suited method to satisfy the constraints of the problem present

days. The co-evolutionary techniques are used to find optimum values in SHTCP. But

they use the solution provided by the LR to optimize the objective function. It is

present in these days and other advance and nature inspired techniques in particular

are used to overcome the difficulties faced by the system.

Techniques like linear and nonlinear programming are among mostly used

techniques to solve SHTCPs. The BD’s objective function and LR are the most



renounced techniques. T. Forrest and others introduced the LR method while solving

the dual problem. The study showed that we can acquire very promising results for

the large scale systems by using LR method [21]. Formulating the sub-problems of

the original problem was found to be the most effective approach.

The sub-problems we get by using LR method are either piecewise linear or linear

functions. The solutions to these problems tend to oscillate between optima. In 1994,

H. Yan et al. used an Augmented Lagrangian (AL) decomposition and coordination

technique [22]. Adding the quadratic penalty term to Lagrangian resulted in the

oscillations and smoother dual function. AL greatly reduced the oscillations,

increased the convergence speed and it is also computationally efficient but it tends

to damage the lower bound property.

To deal with the inherent oscillation of LR, G. Xiaohong et al. [23] introduced a

nonlinear approximation method in 1995. There is a huge difference between the

solutions of individual sub-problems and the solution of primal problem due to these

oscillations. Nonlinear functions are utilized to solve these sub-problems e.g.

quadratic function. By using this method the singularity is avoided as well.

In 1998, the optimal distance method, based on Kuhn Tucker optimality principles

was used by S. Ruzic and R. Rajakovic to update the multipliers [24]. All the

constraints were satisfied and a near optimal solution is obtained as the result of the

minimization of the optimal distance function. This method gave better results in

term of convergence and accuracy when compared to sub-gradient method.

In 1998, an improved LR method is introduced by M. S. Salam and others in [25]. LR

method was used to satisfy the reserve requirements and system demand. The

problem was decomposed into sub-problems. For thermal sub-problem, the dynamic

programming without discretizing generation level was used. Many constraints e.g.

power balance, spinning reserve, ramp rate, capacity limits, minimum up/down time,

hydro constraints, transmission losses and non-linear cost functions were

considered. This new method performed better than the standard LR method.

N. J. Redondo and A. J. Conejo presented a novel, computationally efficient, and non-

oscillating procedure in 1999 [26]. Dual problem was solved using LR method



despite of the original problem. The duality gap was as low as 0.3%. The paper

focused on the updating of Lagrangian multipliers.

In 1999, a new variation of LR method was used in [27] to solve the power system

scheduling problem with head-dependent and cascaded reservoirs. The constraints

like hydro river catchments, discrete operating states, discontinuous operating

regions and hydraulic coupling of cascaded reservoirs were considered.

The comparative study of LR and IP was done in terms of their performance by R.

Fuentes-Loyola and others in 2000, for SHTCP [18]. The results were same but the

LR showed very quick convergence.

In 2000, LR technique was used for the scheduling of large scale hydrothermal

problem by J. Ngundam et al. in [28]. The problem included random load demand,

variation of water head, non-linear cost function of thermal, nonlinear function of

hydroelectric output and regulation of reservoirs in cascaded case with limited

spillage capacity. The real system considerations like fluctuation of power

interchange cost make the model very flexible.

S. Al-Agtash used LR with the AL while considering transmission constraints in 2001

[29]. Transmission constraints were not considered in SHTCP as the complexity of

the problem is increased.

In 2003, A. Borghetti et al. [30] solved the hydrothermal unit commitment problem

using Lagrangian heuristics by exploiting the results obtained from dual problem.

This was achieved using warm starting method and primal bundle method that

improved both quality of the solution and convergence time. The main advantage is

the disaggregated methods are employed to exploit the available primal information.

Hydroelectric unit commitment is the key feature of SHTCP. A realistic approach for

hydroelectric unit commitment was presented in 2006 by E. C. Finardi and E. L. da

Silva [31]. The temporal and spatial coupling relaxation was presented. The Bundle

method was used to update the lagrangian multipliers. Linear programming,

sequential quadratic programming, bundle method, and mixed-integer linear

programming are used in combination with LR to solve the optimization problem.

In 2007, LR in combination with variable splitting (LRVS) was used for hydro and

thermal variable duplication as well as spillage variables and turbine outflow [32].



LRVS can’t find the feasible solution; therefore AL was used to tackle this issue. The

problem considered was large scale with nonlinear cost function and binary

variables for ON/OFF states. Forbidden operating zones were also considered. Some

of the variables were duplicated to achieve decomposition.

In 2009, the co-evolutionary methods were also used with the LR method by L. Ruey-

Hsun et al. [33]. This method had two steps; Lagrangian function was formed using

primal solution by LR method. The algorithm employed the two GAs (1st and 2nd

populations) at the same time, for the evolution of lagrangian multipliers. The control

variables were updated by the fitness function minimization using 1st population

while maximization was used for the adjustments of multipliers by 2nd population.

Lagrangian multipliers and control variables were updated simultaneously and the

results showed that proposed method found the optimal solution effectively.

The LR was presented in combination with artificial variables technique by F. Y.

Takigawa et al. in 2010 [34]. An introduction of new possible constraint variables

was made. If hydro production is modeled by nonlinear programming the two phase

approach proves to be very efficient. LR relaxes the constraints but hydro sub

problems are still coupled in space and time.

The convergence of LR is not satisfactory because of the inherent oscillations in dual

solution. The problem is non-convex due to the presence of the network constraints

and integer variables. The violated constraints cannot be eliminated iteratively. LR

was used in combination with piecewise linear approximation of penalty to improve

dual solution and avoid oscillations as well. Lagrangian was made decomposable

using block descent coordination technique by C. Liu at el. in 2010 [35].

LR in combination with the AL was used in 2012 by R. N. Rodrigues et al. [36]. As LR

faces difficulties in finding a near feasible solution for non-convex, nonlinear, and

complex optimization problem, the AL was used with LR. The LR method became

very efficient by using decomposition technique.

The authors in [37] presented a novel approach based on BD to take care of

hydrothermal UC problem with AC power flow and security constraints in 2013. The

proposed strategy disintegrates the problem into an expert problem and two

arrangements of sub-problems. The expert problem applies number programming



strategy to solve UC, while the sub-problems apply nonlinear programming

arrangement technique to solve ED for each time period. The technique was

investigated on the 9-bus and IEEE 118-bus test systems. The results obtained show

the worth of the proposed approach.

2.2.2.2 Dynamic programming (DP)

The Risk-Constrained Stochastic DP Approach (RCSDP) was discussed in [38] for the

operation planning of hydrothermal system in 1985. The technique was applied on

the hydro-dominated Brazilian generating system which is characterized by large

reservoirs and acceptable results were obtained.

In 1989 another effort was made by J.S. Yang and N. Chen [39] to decrease large

storage memory requirements, long computation time and production cost. The

techniques used were multi-pass DP combined with successive approximation. In

solving the SHTCP, minimum production cost is achieved by optimizing the hour-by-

hour scheduling of all generators available on a system.

I. Erkmen and B. Karataş [40] also used the same techniques of multi-pass DP with

successive approximation in 1994, for solving SHTCP. Initial feasible solution wasn’t

required and also the technique was able to detect the infeasible problems

systematically. Case study was done on the system of Turkish Electricity Authority.

It was concluded that the approach offers a number of advantages over other

techniques.

In 1995, T. Jianxin and P. B. Luh [41] employed the new techniques of extended

differential DP and mixed coordination for solving SHTCP, by decomposing the main

problem into a hydro sub problem and thermal sub-problem by relaxing the supply

demand constraints. Analytical methods were used to solve the thermal sub-problem

but a set of smaller sub problems were made and solved in parallel in order to solve

the hydro sub-problem. The advantages were prevention of dimensionality problem

and accurate estimation of the impact of natural inflow change on total production

cost. The test results indicated that the proposed algorithm was fast and numerically

stable.

To overcome these troubles a new technique of DP two-stage algorithm was used in

[42] in 1998. The algorithm restricted all allowable states in reservoirs to the stages



before and after the supposed stage. Discretization of control and state variables

wasn’t required, so dimensionality issue is solved. The algorithm required less

computing time and reduced storage space also as compared to DP with successive

approximations method.

In 2004, a comparative study was done by S. M. Salam between truncated DP

methods and LR in [43]. The commitment states were obtained by solving thermal

units and thermal sub-problems using truncated DP and LR respectively. Load

demand, the capacity limits, spinning reserve, minimum up and down time, hydro

constraints and ramp rate were also considered in formulation of the problem. Non-

linear cost function was used and accurate transmission losses were also

incorporated. The two methods were compared for operating cost and speed of

execution by testing on a practical utility system. The truncated version of DP reduces

computational time requirement with loss of accuracy.

Another comparison study was performed in 2004, between Primal and Dual

Stochastic DP (DSDP) [44] with the intension of removing the curse of dimensionality

problems. The comparison was done by taking only one hydroelectric unit of

Brazilian Hydroelectric system and simulating the historical inflows records. Lag-one

parametric auto-regression was used to model the stochastic variable of the system.

For dual approach, a parametric auto-regressive model of superior model is also

considered. DSDP also avoided the discretization of the state space in solving the

recursive equation of the DP. The approach used the piecewise linear functions to

estimate the expected cost-to-go function of SDP at each stage. These approximate

functions were achieved from the dual solution of the problem at each stage.

A study was performed on SDP by T. Siqueira et al. for long term HTCP to identify the

effect of different stream flows on the stochastic DP [45]. By considering

progressively complex stream flow models, the benefits of growing sophistication of

stream flow modeling on the stochastic DP’s performance were identified. The first

one was simplest model taking the inflows by their average values; inflows of the

second model were taken as independent probability distribution functions; and the

third model adopted a Markov chain based on a lag-one periodical auto-regressive

model. Brazilian inflow records were used for simulation purposes and it was

concluded that both the stochastic and deterministic approaches provided similar



performance which means the stochastic models considered didn’t provide much

improvement in DP for long term hydrothermal coordination. The results showed

that multi reservoir systems can be easily dealt with deterministic approaches

without any modeling manipulation.

In 2006, a research study was performed for reducing the computation time while

considering different requirements of spinning reserve in [46] and the techniques

used were the combination of Multi-Pass DP and Hybrid EP. The paper dealt with two

types of spinning reserve requirements, frequency relating reserve requirements

(FRRR) and the instantaneous reserve requirements (IRR). Multi pass DP has been

used, which is fast, requires less storage memory but it took solutions from three

discrete values. For making the performance better Evolutionary programming was

then combined with MPDP to form EMPDP to for obtaining optimal volume of

reservoir and double filtration algorithm was used to obtain UC and ED of thermal

units.

In 2011, a comprehensive case study was then done [47] on stopping criteria and

sampling strategies for stochastic dual DP. The problem was formulated to achieve

an optimal policy, over multi-annual planning horizon under water energy resources

uncertainty, for thermal and hydroelectric units. The problem was modeled as a

multistage stochastic program and an algorithm was developed for it. The paper

applied two alternative sampling methods named randomized Quasi-Monte Carlo

and Latin hypercube sampling for the sampling and generation of scenario trees in

SDDP algorithm. And an alternative criterion for formulating the stopping criteria for

the optimization algorithm in was also discussed. The authors tested these ideas for

three year planning on a problem which was associated with the whole Brazilian

power system.

K. S. Gjerden et al. in [48] in 2015 solved the problem of HTCP using SDDP. The

examination demonstrates that hydropower planning issue can be taken care of by

connecting the SDDP methodology to expansive framework sizes through proper

optimization. However, this can be very time-consuming as compared to other

representations based on other principles.



2.2.2.3 Mixed integer linear programming (MILP)

In 2001, G. W. Chang et al. [49] published a paper which described experiences for

solving SHTCP with MILP based approaches. The SHTCP was solved with a state-of-

the-art package which included a MILP solver and an algebraic modeling language.

While solving the SHTCP with MILP approaches, piecewise linear approximation can

be used to incorporate the nonlinearities and the constraints can be easily added to

the problem. The mathematical models were described for both units and plant

based hydro scheduling, where both pumped storage unit and conventional with I/O

characteristics, minimum up/down time limits, unit startup/shutdown, and other

hydraulic constraints were also modeled in detail. Algebraic modeling language,

AMPL is used to formulate the problem with flexibility.

2.2.3 Artificial Intelligence Based Methods

2.2.3.1 Neural networks (NN)

In 1999, R. Naresh and J. Sharma [50] solved the SHTCP using a two-phase neural

system. AL energy function was transformed to get the solution of set of differential

equations. The cascaded hydroelectric units and their dynamics were also

considered. The proposed methodology takes into account the simultaneous

relationship between all the decision variables of this problem. However careful

consideration is required for setting of network parameters. In terms of achieving

optimality the proposed technique performed better as compared to Lagrangian

method.

M. Basu used Hopfield neural systems in 2003 for optimal forecasting of permanent

head hydrothermal system [51]. The power dispatches from both hydroelectric units

and thermal units were considered. The results were found to be far better than

Newton’s technique.

2.3.3.2 Fuzzy logic (FL)

In 1998, FL was utilized for process policy of large hydrothermal power systems [52].

Optimal operation rules were given for the attached process of hydroelectric power

plants via fuzzy logic. A connection was established between the state of every pool

and the aggregated condition of the whole hydroelectric system on an optimal



operation context. These rules were then used by a simulator and the consequences

were compared with the consequences of others fitting methods. The Brazilian South

East System was used as test system. The proposed technique showed the great

advantage of being totally regular and presented a best performance in the

imitations.

In 2009, [53] an adaptive Neuro-Fuzzy Inference System (NFIS) was presented as a

simpler and less difficult alternative, in parallel with a deterministic optimization

model, for the optimization of SHTCP. The presented approach was compared to

other policies like SDP, using inflow records of a large Brazilian hydroelectric power

unit. The results were similar for NFIS and SDP.

The authors in [54] used Takagi-Sugeno Fuzzy inference system to develop an

energetic operation policy. This policy was compared with adopted process policy in

Brazilian system. The proposed policy was found to be much better in achieving the

optimum generation cost.

In 2012, fuzzy based PSO system was proposed by A. L. Rabelo, et al., [55] for

hydrothermal operational planning. An imprecise system was planned for each

system. The relationship function to represents fuzzy system was adjusted using PSO.

Similarly in [56], a genetic based fuzzy system was proposed for hydrothermal

operational planning. The rules of fuzzy genetic system were used for reservoir

operation. Simulation results showed the effectiveness of the proposed scheme.

2.2.4 Evolutionary/Heuristic and Hybrid Methods

2.2.4.1 Genetic algorithm (GA)

In 1998 Carneiro et al. [57] adopted GA based approach as a substitute of classical

methods for the operational planning optimization of hydrothermal power system.

Earlier works done on the same problem suffer from over simplification,

convergence, and approximation of the problem. The results showed that this

approach as one of the best substitute of classical techniques.

In 1998 Orero and M. R. Irving [58] suggested to solve the SHTCP using GA

framework considering various constraints. The problem considered was a multi-

reservoir cascaded hydro system with complex relationships of various variables.



The problem was successfully solved using GA due to its capability of handling such

a complex problem with simplicity and robustness.

In 2000 Xiangping, M., Z. Huaguang, et al. [59] addressed the main disadvantages of

the GA that are premature convergence and slow speed. A new algorithm named Fast

Synthetic Genetic Algorithm (FSGA) was proposed to overcome these drawbacks.

The new algorithm had fast speed of convergence, higher precision and it required

small population size. A hybrid algorithm was also developed by combining back

propagation (BP) with FSGA and applied for solving SHTCP. It solved the problem of

long training time of BP. The proposed algorithm was tested using three units and six

bus test system. The result showed reduced run time.

In [60] SHTCP was solved using GA and simulated annealing and two hybrid

techniques. All inequality and equality constraints were taken into account. The

thermal generators were not been considered as single unit. The results showed that

proposed algorithms can be more reliable and efficient.

GA based model was proposed for SHTCP in 2003 [61]. The proposed algorithm

divided the problem into two sub problems which are ED and UC. For optimizing the

hydro energy required during a specific period Future cost curve was worn. The

performance was improved by a new technique used for representing the candidate

solution and applying a set of expert operators. Results obtained were compared

with LR, mimetic algorithm, and GA and proved to be competitive with the previous

results.

In 2003, a GA model was proposed to handle the sub problems of SHTCP, UC and ED

simultaneously [61]. Period of a week was considered as a scheduling horizon and

hourly generation schedules were obtained for hydro and thermal units. The amount

of hydro energy to be used during the week was optimized by using the future cost

curves of hydro generation obtained from long and mid-term models. Candidate

solutions of GA are represented by a new technique, and the behavior of the

algorithm was improved by incorporating a set of expert operators.

In 2004, Zoumas et al. [62] used Enhanced Genetic Algorithm (EGA) with priority list

method to solve SHTCP. SHTCP was divided into two sub problems. Priority list

method was used for solving the thermal part and hydro part was modeled as



nonlinear, diverse integer optimization problem and was solved by EGA. Hydro as

well as thermal constraints were taken into consideration. GA performance was

improved by adapting problem specific genetic operators. The main advantage of

EGA is the flexibility in modeling.

In 2006, Leite et al. [63] used a hybrid GA for optimizing the operation of Brazilian

hydrothermal power networks. A comparison was also presented between GA and

gradient method. All inequality and equality constraints were taken into account. The

proposed hybrid algorithm used two new genetic operators, gradient direct mutation

and gradient mutation. Results proved to be quite promising for hybrid technique.

In 2007, Kumar, S. and R. Naresh [64] solved SHTCP with non-convex cost using real

coded GA (RGA) technique. Travel time between cascaded reservoirs and valve point

effect in addition to all equality and inequality constraints were also considered. A

comparison between binary coded GA (BGA) and RGA was also done and it showed

RGA performs better than BGA in its simplicity, efficiency, ease of implementation,

small population size and effective constraints handling without using penalty

parameters.

In 2009 M. Kumar et al. [65] solved SHTCP using GA and decomposition based OPF.

The problem was divided into two sub problem, the thermal sub problem was solved

by lambda iteration method including line losses and discharge proportional to

demand method (DPDM) was used to solve hydro sub problem. GA based optimal

power flow was used to control line losses. Results of DPDM were compared with

Average Inflow method (AIFM) and it was found that DPDM is simple, efficient and

reliable.

In 2010 Sasikala, J. and M. Ramaswamy [66] introduced a novel optimal gamma based

technique using GA to improve the accuracy, robustness and computational speed in

hydrothermal coordination problem. The results showed that the proposed

technique is fast and accurate with has small population size.

In 2011, Kumar, V. S. and M. Mohan [67] solved SHTCP using GA. As usual problem

was divided into two sub problem; thermal and hydro sub problems. Lambda

iterative technique was used to solve the thermal problem and GA was adopted for

solving hydro problem. Line discharge limitations and line losses were also



considered for GA based OPF. GA based OPF was applied whenever line discharge

limitations were violated. Fast decoupled load flow (FDLF) was used to compute line

losses. The proposed technique reduced complexity, computational speed and

provided near optimal global solution.

In 2011 [68] SHTCP was solved using real coded genetic algorithm (RGA). The

scheduling problem of hydrothermal system has probabilistic nature in many aspects

e.g. load demand and inflow in the reservoir are probabilistic. These uncertain

parameters were treated as random variables. All inequality and equality constraints

were considered. Exterior penalty method was used for operation limits violations.

In 2013, M. M. Salama et al. [69] solved the short term fixed head HTCP with line

losses, using a GA in combination with constriction factor based PSO technique. The

proposed technique was tested on a hydrothermal test system of one hydro and three

thermal units. Many hydraulic and thermal constraints like maximum and minimum

limits of hydro and thermal units, active power balance constraint, discharge rate

limit and water availability limit were considered. The results of the proposed

technique were compared with the results of GA and it was concluded that the

proposed technique provided the same solution as obtained by GA but with less

computation time.

In 2013 M. M. Salama et al. [70] used the same technique for solving the SHTCP having

non liner fuel cost objective functions. The proposed methods were analyzed using

hydrothermal test system comprising of four hydro power units and three thermal

units. Many constraints were taken into account like equality and inequality

constraints, reservoir storage volume limits water flow rate limits, water dynamic

balance limits, and reservoir volume constraints were considered. The simulation

results proved that the PSO method is far better than GA in terms of precision and

computational time.

In 2014 [71] a hybrid technique combining artificial fish swarm algorithm (AFSA)

with real coded genetic algorithm (RCGA) was suggested to improve the performance

and accelerate convergence. RCGA is exceptionally suitable for taking care of nonstop

streamlining issues due to its genuine number representation. AFSA has good

searching ability and avoids being trapped into local optimum. RCGA can give a



decent course to the worldwide ideal area; AFSA can calibrate the answer for span

the worldwide ideal arrangement. It can acquire better arrangement with quicker

joining speed.

2.2.4.2 Particle swarm optimization (PSO)

In 2007, S. Titus and A. E. Jeyakumar [72] used an improved PSO technique to solve

the hydrothermal coordination problem with POZ. The problem was formulated as

non-convex due to these zones. PSO resulted in much better solution. The constraints

like reservoir volume, power balance, ramp limits and water discharge were also

considered. Craziness function was used to improve the algorithm. A cheaper and

better quality solution was obtained using this approach.

In 2008 Lee, T. [73] solved the problem of hydroelectric scheduling using multiple

pass iteration PSO. Wind turbine generators were also considered in this work.

Solution quality was improved using a new index called iteration best. The idea of

multi pass DP was used for improving and modifying the computation efficiency. The

technique started with a same time period and a penetrating space.

In 2008 Mandal K. K et al. in [74] used PSO to solve SHTCP considering valve point

effect on the objective function and practical constraints. A test framework

comprising of many thermal and hydroelectric units with no pumped-storage units

was employed. The results showed PSO algorithm to be superior as compared to

evolutionary programming and simulated annealing (SA) method.

In 2008, C. Samudi et al. [75] used PSO to solve SHTCP. This work analyzed different

particle selections and finally the reservoir volume was considered as a particle. The

proposed scheme performed better in comparison to other techniques. Success rate

of finding global optimum was found to be 100% among the 300 tests.

In 2008 J. Wu and coworkers introduced a hybrid technique of Particle Swarm

Optimization with Chance constrained programming [76]. The Hybrid PSO (HPSO)

surely converged to the global optimum result. HPSO was combined with Monte

Carlo simulations to solve the model. A cascaded hydropower plant was used to test

the technique which comprised of three reservoirs and three power houses. The

hybrid approach resulted in a better solution while meeting all the constraints with

a specified probability.



In 2008 X. Yuan et al. [77] introduced an enhanced PSO for optimal scheduling of

hydro system. In this technique the particle remembered its worst position. For

handling the constraints effectively viability based selection technique, chaotic

sequences and random selection was employed. The technique was tested

for the daily best production plan and the results were superior in comparison to

other methods.

In 2009 P. K. Hota et al. [78] solved the SHTCP using a new method named improved

particle swarm optimization. The vibrant search space minimizing technique was

used for speeding up the optimization and handling the inequality constraints. The

test system was multi-reservoir cascaded system having restricted operating zones.

Valve point loading effect of thermal unit was also considered. The results were

compared with nonlinear programming, dynamic programming, differential

advancement techniques and transformative programming.

In 2009, S. Liu and J. Wang presented another improved PSO approach [79]. The work

employed a self-adaptive inertia weight technique. Nonlinear constraints were

handled using a penalty function. The results showed better performance and good

results.

In 2009 P.-H. Chen presented the algorithms with in an excellent optimized cost [80].

The work described encoding/decoding methods. A test system comprising of three

cascaded hydroelectric units with 22 thermal units was used and convergence of the

solution was robust.

In 2010 a new modified PSO was used for solving daily SHTCP by Amjady N and H. R

Soleymanpour [81]. The technique was tested on different systems and comparison

with GA was also done. It was concluded that this approach can solve the SHTCP with

high computational efficiency.

In 2010 Thakur S. and C. Boonchay [82] employed a Self-organizing Hierarchical PSO

in combination with Time Varying Acceleration Coefficients (SPSO-TVAC). This

approach reduced the thermal operating cost and met all the thermal and hydraulic

generation limitations. Scheduling was done for multiple periods and non-convex

fuel cost of thermal units was also considered. The technique was tested on different



systems and its comparison with Inertia Weight Approach Particle Swarm

Optimization (IWAPSO) was also done. The results in all cases were found better.

In 2010 a fuzzy adaptive PSO was introduced by W. Chang et al. [83] to solve the

problems of premature convergence. Fuzzy laws were incorporated in the inertia

weight approach. The proposed technique was tested on a hydrothermal system with

four hydroelectric units and one thermal unit. The results were compared with

simple PSO and GA. It was concluded that the proposed technique is better in terms

of solution quality and computational efficiency.

In 2010 Singh S. and N. Narang [84] solved SHTCP using PSO. The technique avoided

local minima and reached to the global optimum very quickly. The technique reached

a best solution while satisfying all the constraints.

A novel fuzzy adaptive PSO (FAPSO) was presented in 2010 by W. Chang [85] for

finding the optimal scheduling of hydrothermal power system. For solving the

problems of local optima and premature convergence of the standard PSO, the fuzzy

adaptive criterion was applied for inertia weight. The inertia weight was changed

using the fuzzy rules to adapt to nonlinear optimization process, in each iteration

process. The PSO technique is simple, robust efficient and it is implemented easily.

To overcome the disadvantages of PSO the proposed technique adjusted the inertia

weight with respect to its environment. The FAPSO proved to be fast and it had

powerful search capabilities of generating better results.

In 2011, self-organizing hierarchical PSO was proposed by K.K. Mandal and N.

Chakraborty [86] for cascaded hydrothermal system. Time changing acceleration

coefficients were imposed for avoiding premature convergence. The technique was

tested on a multi chain cascaded hydrothermal system having non-linear generated

power, water discharge rates and total head. The technique performed better in

optimizing fuel cost and emission output.

In 2011, the SHTCP was solved by S. Padmini et al. using PSO [87, 88]. The technique

was tested on a system of each one hydro and thermal unit. The comparison with

earlier works was also done and it was concluded that the method had excellent

convergence characteristics and the results were better and effective in terms of

computation time and fuel cost.



In 2012 Y. Wang et al. [89] introduced an improved self-adaptive PSO. Premature

convergence was avoided by changing the evolution path of every particle. Different

constraints were tackled by using a new scheme. The technique was tested on a

system of four hydro and thermal units. The results showed that the proposed

technique was accurate and robust as compared to the other methods.

J. Zhang et al. introduced a Small Population-Based PSO (SPPSO) approach for the

first time in 2012 [90]. The paper used a new mutation operation for improving the

diversity of the population and differential evolution (DE) for rushing process. The

convergence speed was increased but the optimal result had no significant

improvements after many iterations. The crowding variety of the swarm was kept

above a desired level by using a relocation operation. The complex equality

constraints were taken care of by a special patch up procedure. The technique was

tested on three hydrothermal systems and found to be effective. The results were

compared with different evolutionary techniques. It was concluded that the SPPSO

gives a best solution with less effort.

In 2012 V. Hinojosa and C. Leyton [91] proposed a Mixed-Binary Evolutionary

Particle Swarm Optimizer (MB-EPSO) to solve SHTCP. Not only the results were

improved but noteworthy change was also found because of the utilization of

possible arrangements. The results were compared with the ones already reported

in the literature such as GA, PSO and DP.

In 2013, M. M. Salama et al. [92] proposed the PSO technique with choking variable

to handle the problems of multi chain hydro planning with non-linear function of fuel

cost. The proposed technique was tested on a system having three thermal and four

hydroelectric units. Many constraints like, minimum and maximum limits of thermal

and hydroelectric units, discharge rate limits, initial and final volume limitation and

water dynamic constraints were also considered. The results were compared with

different techniques like, evolutionary programming (EP) and simulated annealing

(SA) to show the feasibility of the proposed technique. The proposed algorithm

achieved minimum fuel cost with less computational time.

In 2014 K. Dasgupta et al. [93] determined the optimal hourly schedule of

hydrothermal system using PSO with inertia weight and constriction factor



approach. The problem formulation involved cascaded multi-reservoirs

hydroelectric units and valve point loading effect of thermal units. The authors

considered the cost characteristics of individual thermal units instead of an

equivalent thermal unit. The result showed the proposed technique to be very

encouraging and innovative compared to other techniques.

Another approach named predator prey based optimization (PPO) technique was

applied to solve SHTCP by N. Narang et al. in 2014 [94]. PPO belongs to the family of

swarm intelligence and is a good option for solving non-linear and large scale

optimization problems. In PPO idea of PSO is combined with the concept of predator

effect which avoids premature convergence and maintains diversity in the swarm.

Equality constraints are handled by variable elimination method which eliminates

the variables explicitly. The eliminated variables are then used by penalty approach

to restrict the slack units in their limits. Power balance equality constraint for each

interval, is handled by slack thermal unit and slack hydroelectric units are used to

handle water balance constraint. The proposed technique was applied on fixed as

well as variable-head hydrothermal power systems. The results were compared with

other existing techniques and it was shown that the proposed technique performs

better.

In 2015 Vinay Kumar Jadoun et al. [95] used Dynamically Controlled PSO technique

to solve the SHTCP. The technique efficiently dealt with many hydro constraints like

discharge rate limits, reservoir storage capacity limits, initial and final reservoir

storage quantity limits, water balance constraint etc. for a given time period.

Moreover the performance of the swarm was modified for exploiting the search

space and better investigation. The proposed technique was tested on a standard test

system.

In 2015, A. Rasoulzadeh-Akhijahani et al. [96] solved the SHTCP using a Modified

Dynamic Neighborhood Learning based PSO (MDNLPSO). The particles were

assembled in various neighborhoods and each particle learns only from its particular

neighborhood. The information among the particles was exchanged by changing the

neighborhood of a particle at refreshing periods with a refreshing operation. This

causes the improvement in both exploitation and exploration of basic PSO. The



proposed approach was tried on three different cascaded multi-reservoir

hydrothermal systems and the results were compared with fresh techniques.

2.2.4.3 Differential evolution (DE)

In 2006 L. Lakshminarasimman and S. Subramanian [97] solved SHTCP using the

modified DE (MDE) algorithm. Differential evolution is an improved version of a GA

and is fast, very simple and robust technique. Loss coefficients were used to

incorporate transmission losses. The study was extended to solve the combined

economic emission dispatch. The modifications in basic DE algorithm were done as

it had difficulties while handling the equality constraints, especially the reservoir

end-volume constraints. The proposed technique didn’t require any use of penalty

functions and reached the optimum solution with less computational effort.

In 2008 X. Yuan et al. [98] proposed a chaotic hybrid DE algorithm for solving SHTCP.

Chaos theory was applied for obtaining self-adaptive parameter settings in DE. For

handling the constraints effectively heuristic rules were embedded into DE which

guided the process towards the feasible region of the search space. The SHTCP is

often programmed as a linear or piecewise linear one. The values of DE parameters

were determined by chaotic sequences. To guide the search toward the optimum,

three simple comparison mechanisms were devised on the basis of feasibility and

heuristic rules. No penalty function or any extra parameters were needed to

effectively handle the all the constraints.

In 2008, K. Mandal and N. Chakraborty [99] used DE to solve SHTCP. A cascaded

multi-reservoir hydrothermal system with non-linear discharge rate, net head and

power generation was considered. The water transport delay was also taken into

account. Many inequality and equality constraints of thermal as well as hydroelectric

units with valve point loading effect were also included in problem formulation. The

proposed method was tested on two test systems. The results were compared with

other evolutionary algorithms like GA and PSO. It was concluded that DE can produce

better results in terms of computation time and fuel cost.

In 2008, L. Lakshminarasimman and S. Subramanian [100] used a modified hybrid

differential evolution (MHDE) algorithm for solving SHTCP of cascaded reservoirs.

The DE was modified for handling the equality constraints especially power balance



and reservoir end volume constraints. The MHDE was combined with a new equality

constraint handling method which required less effort to provide better solutions.

The proposed algorithm was tested on different case studies considering prohibited

discharge zones, valve point effects, transmission losses and multiple thermal units.

The technique employed two additional operations, migration operation which

upgraded the exploration and acceleration operation that improved the fitness. The

acceleration phase accelerated the convergence while the migration phase helped in

escaping the local optima. This work emphasized the satisfaction of equality

constraints as they have major effect on the cost of the overall schedule.

In 2009, EP was used to optimize hydrothermal power system in [101]. Cauchy

mutation was inserted in basic EP for modifying it to generate better results while

solving SHTCP. Generally both DE and EP take exceptionally high time to solve

complex problems like SHTCP. To handle these problems DE is used as hybrid

approach in combination with other Deterministic and Heuristic approaches to solve

SHTCPs e.g. in 2010 in [102] Y. Lu et al. used Chaotic Local Search with DE (CLS) to

cope with problems like poor convergence and high computational time. The results

were comparable to both DE and PSO. In [103] in 2011 DE has been assisted with

Sequential Quadratic Programming (SQP) to cover these problems.

In 2010, K. K. Mandal et al. presented a comparison of different variants of DE in

[104]. Different strategies of mutation phase of DE have been investigated for the

same standard system and the best option was recommended.

In 2013, V. Sharma and R. Naresh [105] solved the SHTCP using DE based algorithm

while considering valve point loading effect for fixed head hydro-thermal problem.

The technique was compared with real variable genetic algorithm (RVGA). The

control parameters of RVGA and DE were optimized for the case study under

consideration. The algorithm was tested on a system comprising of one hydro and

three thermal units with valve point loading effects.

In 2013 K. Mandal and N. Chakraborty [106] conducted the study of control

parameters of DE for optimizing the scheduling of hydrothermal systems having

cascaded reservoir. A cascaded multi reservoir hydrothermal system having non-

linear power generation, discharge rates and net head was considered. The optimum



setting of the control parameters is dependent upon the problem, computation time

requirement and accuracy. DE has three control parameters i.e. population size,

mutation factor and crossover rate and all of them were investigated for the effects

of variation. The results showed that wrong parameter selection can cause

premature convergence and even stagnation. Recommendations were then given for

the range of control parameters of DE.

In 2014, M. Basu [107] introduced an Improved DE (IDE) to solve SHTCP with multi-

reservoir cascaded hydroelectric units. Prohibited operating zones, valve point

loading effect and ramp rate limits for thermal units were also considered. The

technique was tested on three hydrothermal test systems and the results were

compared with other population based evolutionary techniques. It was found that

the proposed approach gives best solutions.

In 2015, Jingrui Zhang et al. [108] introduced a Modified Chaotic DE (MCDE)

algorithm for solving short term optimal hydrothermal problem. The constraints of

the problem were handled by introducing a novel selection operator and a repairing

procedure in MCDE technique. The repairing procedure was also used to avoid

penalty factor approach and to preserve the feasibility of generated solutions. The

usage of introduced selection parameter made unclear distinction between infeasible

and feasible solution in start of the algorithm but later on this distinction was made

clear. To avoid trapping in local optima and to enhance the diversity of solutions, an

adaptive regeneration operation was also introduced by authors. And the searching

process was accelerated by a chaotic local search technique. The proposed technique

was tested on three well known hydrothermal test systems and the results were

compared with other existing techniques. The authors concluded that MCDE

produces competitive and efficient solutions.

In 2015, Jingrui Zhang et al. [109] solved the SHTCP using an Improved DE approach.

To avoid local optima a regeneration operation was also incorporated in this

approach. Furthermore, constraints handling was done by a novel mechanism to

avoid using the penalty factor approach and increase the effectiveness of the

algorithm. The proposed technique was tested on a hydrothermal system to confirm

its effectiveness. The results were compared to other population-based heuristic

approaches.



In 2015, an Improved Chaotic Hybrid DE (ICHDE) algorithm [110] was introduced

for finding the optimal solution to SHTCP considering many practical constraints.

Chaos theory was applied to get a self-adjusted parameter setting for differential

evolution (DE) and a chaotic hybridized local search technique was embedded to

cope with premature convergence effectively. Self-adjusted crossover parameter

setting was obtained using a logic map based chaotic operator. Then local search was

done by a chaotic hybridized local search (CHLS) mechanism to avoid stopping at

local optima.

2.2.4.4 Gravitational search algorithm (GSA)

In 2015, N. Gouthamkumar et al. [111] solved the problem of SHTCP using an

Oppositional Learning Based GSA. GSA is a stochastic search algorithm which works

on the principles of gravitational law. To improve the joining rate of GSA, the

technique utilized opposite numbers in the evolution process of GSA. Finally, the

proposed technique was tested on two systems, one with four hydro and thermal

units and the other one with three thermal and nine cascaded hydroelectric units.

The results showed that the proposed approach performed better than other

techniques, with reduction in fuel cost, better convergence characteristics and

computation time.

In 2015, N. Gouthamkumar et al. [112] solved the SHTCP using Disruption Based GSA

(DGSA) while considering limits of hydroelectric and thermal units and valve point

effect of thermal units. To enhance the performance of the algorithm a disruption

operator based on astrophysics was embedded in GSA, which also increased the

exploration as well as exploitation abilities. Power balance and end storage volume

constraints were handled by introducing an effective strategy. The proposed

algorithm was then tested on two test systems. The first one comprised of four hydro

and four thermal units and the other one had three thermal and four hydroelectric

units. The result comparison analysis was done which concluded that DGSA performs

better than GSA in terms of convergence accuracy and better solutions.

In 2016, N. Gouthamkumar et al. [113] solved the SHTCP using disruption based

oppositional GSA (DOGSA). The opposition based learning concept was embedded in

a GSA to improve the quality of solution and a disruption operator was integrated for



accelerating the convergence of solutions. The proposed algorithm was tested on two

test systems. The first one comprised of four hydro and four thermal units and the

other one had three thermal and four hydroelectric units with and without valve

point loading effect of thermal units.

2.2.4.5 Bacterial foraging algorithm (BFA)

In 2009, I. Farhat and M. El-Hawary [114] employed BFA to solve the SHTCP. And

results showed satisfactory results for Short term Variable head hydrothermal

problem and multi-objective optimization of SHTCP.

In 2010, I.A. Farhat and M. E. El-Hawary [115] solved fixed head short term

hydrothermal coordination using BFA algorithm. Fixed head problem is supposed to

have very large reservoir, so that it has virtually no effect on water head. Although

authors approximated fixed head but still due to the consideration of real time

constraints the system is complex. BFA proved to be excellent for solving this

complex problem but there were some problems like poor convergence and trapping

to local minima. This problem was fixed by modifying the chemo-taxis step of BFA

and this version of BFA was termed as modified Bacterial Foraging Algorithm, MBFA.

The accuracy of algorithm was tested on two fixed head test systems. First system

contained two fixed head hydroelectric units and one thermal unit while second test

system contained three thermal and one fixed head hydroelectric unit.

In 2011 enhanced bacterial BFA (EBFA) was introduced by I.A. Farhat and M. E. El-

Hawary [116] to solve dynamic, multi constrained and highly nonlinear variable head

SHTCP. Real time constraints were also considered for both thermal and hydro

power plants. Only linear fuel costs were used for thermal machines. Raw BFA

technique required long computational time and it showed poor convergence criteria

while solving SHTCP. The EBFA algorithm was tested on two test systems. First one

contained two hydro and two thermal machines and other test system comprised of

four hydro and five thermal machines. The technique showed promising results.

In 2011, I.A. Farhat and M. E. El-Hawary [117] solved a hydrothermal system

containing three thermal and four hydroelectric units. The modeled test system was

short term cascaded multi reservoir which is highly nonlinear, complicated and

dynamic. Real-time constraints were also taken into consideration for both thermal



and hydro power plants. The technique managed to show promising results for the

SHTCP.

2.2.4.6 Simulated annealing (SA) & tabu search (TS)

SA is an evolutionary algorithm based technique. It is among some of the very

popular and well established techniques. It works on the principles of metallurgical

annealing process. But if SA is not able to achieve global optimum, it takes

exceptionally high time. Cooling rate is an important decision for the perfect state to

be achieved. SA has found to be excellent for power system operation problems like

SHTCP [118, 119].

In 1994 in [120, 121] SA was proposed by K. Wong and Y. Wong to solve the SHTCP

while satisfying the constraints like Active power balance constraint, reservoir

volume limits, water discharge limits. A relaxation method was also embedded in the

algorithm for checking the limits. A new SHTCP formulation was developed which

considers the relative operation limits of the hydroelectric unit and the equivalent

thermal units. Then simulated annealing based algorithm was established by

combining this formulation with the SA technique. In another work by the same

authors a coarse-grained parallel simulated annealing algorithm (PSA) was

developed for SHTCP. The parallel algorithm proved to be much faster than the

sequential algorithm.

In 2013, N. C. Nayak and C. C. A. Rajan presented an EP technique embedded with TS

[122] for solving UC problem of hydrothermal coordination. UC schedule was coded

as a string of symbols in this method. Initial population is generated at random from

a pre-defined set of solutions. Random recommitment is then done with respect to

the unit’s minimum down times. TS maintain the short term memory of recent

solutions that helps it to avoid trapping in local minima. The memory structure helps

TS in preventing certain movements that can make the solution weak. But the Tabu

status is overruled if some specific conditions are satisfied which are expressed in

the form of aspiration level (AL). TS becomes more flexible by using AL because it

directs the search towards attractive solutions. The proposed technique is compared

with conventional methods like DP and LR in terms of the fuel cost and computation

time.



In 2011, V. Ferreira and G. Silva solved SHTCP using SA [123]. A detailed comparison

between GA and SA was done for solving a complex medium term SHTCP. Results

shown by both the techniques were found to be equally suitable.

2.2.4.7 Others

In 2012 H. Baradaran Tavakoli et al. [124] solved SHTCP, via honey-bee mating

optimization (HBMO) algorithm. The proposed algorithm was tested on a

hydrothermal system and the results showed that the technique can give far higher

convergence speed and minimum cost as compared to other optimization techniques

such as PSO and GA.

In 2012, a novel effective Differential Real-coded Quantum inspired Evolutionary

Algorithm (DRQEA) was used by Y. Wang et al. [125] to solve SHTCP. The proposed

technique was tested on two hydrothermal systems and the acquired results were

compared with different techniques, and it was concluded that DRQEA can perform

better than other reported techniques in terms of speed and solution quality.

In 2012, Y. Wang et al. [126] solved the SHTCP using a colonel real-coded quantum-

inspired evolutionary algorithm (CRQEA) with Cauchy transformation. The proposed

technique was tested using three test systems and the results were compared with

other reported techniques. It was concluded that CRQEA can perform better than

other algorithms.

In 1996, P. C. Yang et al. [127] presented a novel evolutionary programming (EP)

based algorithm in order to solve the SHTCP. Non-linear generation models with non-

linear curves and prohibited operating zones of hydroelectric units as well as thermal

units were used. This paper proved that there is still potential to find a more optimal

solution by using the proposed EP-based algorithm.

In 2003, fast evolutionary programming (FEP) techniques was presented by N. Sinha

et al. [128] for solving SHTCP. EP based algorithms with gaussian and other fast

mutation techniques were developed. Conventional methods required the hydro and

thermal models to be represented as a polynomial or piecewise linear approximation

of monotonically increasing nature. More realistic generation models are

represented by non-linear cost curves with prohibited areas. The study obtained

impressive results by using LR technique to generate near optimal solutions.



In 2011, Clonal Section Algorithm (CSA) was proposed by R. Swain et al. [129] for

solving the SHTCP. CSA is a new technique in the family of evolutionary computation.

It proves to be fast, simple and a robust when solving complex hydrothermal

coordination problems. The hydrothermal coordination is a non-linear optimization

problem with a set of operational and physical constraints. The constraints like water

transport delay, the cascading nature of hydro-plants, power balance constraints,

reservoir storage limits, water discharge limits and hydraulic continuity constraint

were fully taken into account. The simulation results showed that proposed

algorithm is capable of finding the optimized solutions with less computational effort.

In 2006, X. Yuan and Y. Yuan [130] introduced a Cultural Algorithm (CA) for solving

the daily scheduling of hydrothermal power systems. Water transport delay time was

also considered between connected reservoirs and complicated hydraulic coupling

constraints were also incorporated. The proposed cultural algorithm was verified

using a test system, and comparing the results with both the GA and Lagrange

method. It was concluded that the proposed algorithm avoids premature

convergence and gives a better quality solution with quick convergence speed.

In 2013, P. K. Roy [131] proposed a novel Teaching Learning Based Optimization

(TLBO) for solving SHTCP considering valve point loading effects and water release

limitations of hydroelectric units. TLBO has two essential stages. In first phase,

teaching methodology is used to improve the knowledge of learners and in second

phase learners interact with each other for increasing their knowledge. The proposed

algorithm was robust in nature as it didn’t require any specific parameters. The

proposed technique was tested on three unique cases to be specific, quadratic cost

without prohibited discharge zones; quadratic cost with prohibited discharge zones

and valve point loading effect with prohibited discharge zones. The comparison with

other techniques showed that the proposed technique performs better.

In 2013, X. Liao et al. [132] presented an Adaptive Chaotic Artificial Bee Colony

Algorithm (ACABC). For escaping the local optima, chaotic local search and Control

parameter setting were introduced. As original ABC technique didn’t consider

constrained problems, a new constraint handling method presented to solve

constrained SHTCP. The proposed technique performed better in terms of

convergence and computational speed when compared to other methods.



In 2014, T. T. Nguyen et al. [133] suggested cuckoo search algorithm (CSA) for solving

fixed head SHTCP while considering line losses and valve point effect of thermal

units. The technique was tested and compared with GA and PSO. The results proved

that the CSA technique is far better than both GA and PSO strategies for all test

systems. The CSA technique required less number of control parameters.

In 2015, T. T. Nguyen et al. [134] solved the SHTCP using a Modified Cuckoo Search

Algorithm (MCSA). The search capacity of CSA was upgraded to form MCSA

technique. The proposed technique was tried on various test systems with complex

constraints and the results are compared with other techniques. The comparison

showed that MCSA performs much better that CSA.

In 2015, T. T. Nguyen et al. [135] introduced One Rank Cuckoo Search Algorithm

(ORCSA) for solving SHTCP with reservoir limitation. The proposed technique had

two major steps, lévy distribution and cauchy distribution. The results were

compared with different algorithms and it was shown that the proposed technique

performs better.

In 2016, Dubey et al. [136] proposed another nature inspired Ant Lion Optimization

(ALO) for solving SHTCP with wind integration. The ALO models the unique six step

natural hunting activity of ant lions. The random walk mechanism and roulette wheel

operation increases the exploration capability of algorithm. The shrinking of trap

boundaries and elitism increase exploitation efficiency. The algorithm when tested

on four standard test systems produced better results.

2.3 DISCUSSION

On the basis of the detailed literature review given above, the non-linearity, non-

convexity and dynamic nature of both the objective function and constraints of

SHTCP is crystal cleared. Optimization methods are classified based on the type of

search space and the objective function along with equality and inequality

constraints. In general, the objective function and/or constraints contain

nonlinearity giving rise to non-linear problem. The optimal scheduling of

hydrothermal power system is basically a complex programming problem involving

nonlinear objective function and a mixture of linear and nonlinear constraints. The



objective of hydrothermal coordination is to generate the power economically,

whereas satisfying various constraints. The equality constraints include generation-

demand balance and balance of available volume-water discharge. The inequality

constraints include generation limit on thermal and hydroelectric unit, limit on water

discharge rate and reservoir storage limit.

The SHTCP is broadly classified into convex and non-convex optimization problem.

In convex hydrothermal coordination problem the input–output characteristics are

often assumed piecewise linear and monotonically increasing. In this case, the

optimization algorithms that are based on mathematical programming can be

applied.

In past, majority of the researchers have used the convex objective function to solve

SHTCP and the factors like POZ, VPE, MFI causing non-convexity and the

transmission loss have not been incorporated by the most of the researchers.

But in actuality, SHTCP is a highly non-linear, non-convex and a complex optimization

problem. Such problems require fast, accurate and robust solution methodology as

they cannot be handled effectively by mathematical programming based

optimization methods. Generally, evolutionary methods are used as tool for the

solution of complex optimization problems because of their strength to overcome the

shortcomings of the traditional optimization methods. Evolutionary methods have a

number of exclusive advantages like robust and reliable performance, global search

capability, little information requirement, ease of implementation, parallelism, no

requirement of differentiable and continuous objective function.

Besides from the traditional approaches discussed in this chapter, several

evolutionary algorithms have also been applied to obtain the solution for SHTCP.

These approaches have proved to be more efficient and have got more attraction due

to their robustness and their capability to provide a reasonable solution. However

these methods have also a drawback of premature convergence [4] and some of these

techniques also requires a huge computational time especially for large scale SHTCP.

Therefore, the researchers are emphasizing on these tools. These tools have been

applied alone or with some modifications / improvements in their basic parameters

and they have been hybridized with any other conventional / non-conventional tool.



Chaos phenomenon has also been proposed with certain evolutionary algorithms to

cater the major drawback of evolutionary algorithms like premature convergence

and local optimal stucking.

Further, many practical constraints like PDZ of hydroelectric units, POZ and RR of

thermal units have also not been considered by many of the researchers.

2.4 CHALLENGES & BOTTLENECKS

Based on the above discussion, the following challenges and bottlenecks may be

enlisted while solving SHTCP.

i. A robust and strong algorithm/method for the solution of SHTCP is needed

that takes into account all the complex constraints

ii. Non-convexity in the fuel cost characteristics of thermal units

iii. Inflow of the hydroelectric units are not fixed rather it varies hourly

iv. Reservoir storage and water discharge limitations

v. Coupling time required for water from one reservoir to next downward

reservoir

vi. Consideration of PDZs of hydroelectric units (rarely taken into account)

vii. Consideration of RRs of thermal units (rarely taken into account)

viii. Consideration of POZs of thermal units (not taken into account)

All the above mentioned are the challenges for the solution of SHTCP and almost all

have been successfully solved and investigated in this thesis.

Chapter 3 Water Cycle Algorithm --- Essential Background for Hydrothermal Coordination


CHAPTER NO. 3 WATER CYCLE ALGORITHM ---

ESSENTIAL BACKGROUND FOR HYDROTHERMAL

COORDINATION

3.1 INTRODUCTION

Water cycle algorithm (WCA) imitates the formation of streams from rain

and then their flow towards rivers and then flow of these rivers towards the sea.

It is basically derived from the water cycle process of the nature. It starts with the

assumption of rain or precipitation so that a population of streams is randomly

generated.

3.2 STEPS OF WCA

3.2.1 Initialization

An initial population of design variables i.e. the population of streams is

generated randomly. The individual with the best fitness value i.e. the best stream

is chosen as sea. Then the individuals with the next fitness values are designated

as rivers, while all other streams flow to the rivers and sea [137]. The total

population of stream as mentioned in [138] is:

𝑇𝑜𝑡𝑎𝑙 𝑃𝑜𝑝𝑢𝑙𝑎𝑡𝑖𝑜𝑛 =

[

𝑆𝑒𝑎

𝑅𝑖𝑣𝑒𝑟1



⋮

𝑆𝑡𝑟𝑒𝑎𝑚𝑁𝑠𝑟+1



⋮

𝑆𝑡𝑟𝑒𝑎𝑚𝑁𝑝𝑜𝑝 ]

=

[

𝑥11 𝑥2

1 ⋯ 𝑥𝑁𝑣𝑎𝑟1

𝑥12 𝑥2

2 ⋯ 𝑥𝑁𝑣𝑎𝑟2

⋮ ⋮ ⋱ ⋮

𝑥1𝑁𝑝𝑜𝑝 𝑥2

𝑁𝑝𝑜𝑝 ⋯ 𝑥𝑁𝑣𝑎𝑟𝑁𝑝𝑜𝑝

]

(3.1)



The initial population is randomly generated in the viable range so as to

cloak the whole search region homogeneously as:

𝑥𝑖𝑗= 𝑥𝑖

𝑗,𝑚𝑖𝑛+ 𝑟𝑎𝑛𝑑 × (𝑥𝑖

𝑗,𝑚𝑎𝑥− 𝑥𝑖

𝑗,𝑚𝑖𝑛) (3.2)

In fact, 𝑁𝑠𝑟 (a predefined parameter) is the total of number of rivers and a

single sea as given in Eq. (3.3). The remaining number of streams 𝑁𝑠𝑡𝑟𝑒𝑎𝑚which

flow towards the rivers or may directly flow towards the sea is calculated using

Eq. (3.4) as follows:

𝑁𝑠𝑟 = 𝑁𝑜. 𝑜𝑓 𝑅𝑖𝑣𝑒𝑟𝑠 + 1 (𝑆𝑒𝑎) (3.3)

𝑁𝑠𝑡𝑟𝑒𝑎𝑚 = 𝑁𝑝𝑜𝑝 − 𝑁𝑠𝑟 (3.4)

The amount of water entering a specific river or sea depends on the

intensity of flow (fitness value). The no. of streams entering the sea and the no. of

streams entering the river are calculated using the Eq. (3.5).

𝑁𝑆𝑛 = 𝑟𝑜𝑢𝑛𝑑 {|𝐶𝑜𝑠𝑡𝑛

∑ 𝐶𝑜𝑠𝑡𝑖𝑁𝑠𝑟

𝑖=1

| × 𝑁𝑠𝑡𝑟𝑒𝑎𝑚 } , 𝑛 = 1, 2, 3, …… , 𝑁𝑠𝑟 (3.5)

3.2.2 Movement of streams to the rivers or sea

As per the hydrologic cycle, streams are formed from the raindrops and they

join each other to form new rivers. All rivers and sea end up in the sea which has

the best fitness value [137]. Out of 𝑁𝑝𝑜𝑝 streams, one stream is designated as sea

and other 𝑁𝑠𝑟 − 1 streams are designated as rivers.

Fig. 3.1 shows the graphical view of a stream flowing towards a specific river.

The connection lines are also shown. The distance 𝑋 between the stream and the

river is randomly updated as:

𝑋 ∈ (0, 𝐶 × 𝑑), 𝐶 > 1 (3.6)



d

x

Old position of stream

New position of stream

Fig. 3.1 Graphical view of a stream flowing towards a river

The updated positions of streams, rivers and sea are given using the

following equations:

𝑋𝑠𝑡𝑟𝑒𝑎𝑚𝑖+1 = 𝑋𝑠𝑡𝑟𝑒𝑎𝑚

𝑖 + 𝑟𝑎𝑛𝑑 × 𝐶 × (𝑋𝑅𝑖𝑣𝑒𝑟𝑖 − 𝑋𝑠𝑡𝑟𝑒𝑎𝑚

𝑖 ) (3.7)

𝑋𝑠𝑡𝑟𝑒𝑎𝑚𝑖+1 = 𝑋𝑠𝑡𝑟𝑒𝑎𝑚

𝑖 + 𝑟𝑎𝑛𝑑 × 𝐶 × (𝑋𝑠𝑒𝑎𝑖 − 𝑋𝑠𝑡𝑟𝑒𝑎𝑚

𝑖 ) (3.8)

𝑋𝑅𝑖𝑣𝑒𝑟𝑖+1 = 𝑋𝑅𝑖𝑣𝑒𝑟

𝑖 + 𝑟𝑎𝑛𝑑 × 𝐶 × (𝑋𝑠𝑒𝑎𝑖 − 𝑋𝑅𝑖𝑣𝑒𝑟

𝑖 ) (3.9)

Eq. (3.7) depicts streams flowing towards the corresponding river and Eq.

(3.8) depicts streams flowing directly towards the sea. If the fitness of the streams

comes out to be better than its connecting rivers then the streams and river is

swapped with each other. The same is done for the river and sea.

3.2.3 Evaporation & raining process

The exploitation phase of WCA helps it to avoid premature convergence. This

exploitation is done through evaporation process. The evaporation process causes

sea water to vaporize as the streams or rivers flow to sea. This results in rainfall to

form new streams. It is therefore checked if the rivers or streams have approached



the sea to make the evaporation process occur [137]. The following condition is used

to check this evaporation condition:

𝐸𝐶1: ‖𝑋𝑠𝑒𝑎𝑖 − 𝑋𝑅𝑖𝑣𝑒𝑟

𝑖 ‖ < 𝑑𝑚𝑎𝑥 𝑜𝑟 𝑟𝑎𝑛𝑑 < 0.1, 𝑖 = 1, 2, 3, …… . . , 𝑁𝑠𝑟 − 1

if the above condition 𝐸𝐶1 becomes true then start the raining process as per Eq.

(3.10), where 𝑑𝑚𝑎𝑥 is a small number (close to zero).

𝑋𝑠𝑡𝑟𝑒𝑎𝑚𝑛𝑒𝑤 = 𝑀𝑖𝑛𝐿𝑖𝑚 + 𝑟𝑎𝑛𝑑 × (𝑀𝑎𝑥𝐿𝑖𝑚 − 𝑀𝑖𝑛𝐿𝑖𝑚) (3.10)

The same condition of evaporation is checked for those streams which flow

directly to the sea. The condition for evaporation for the streams flowing directly

towards the sea is

𝐸𝐶2: ‖𝑋𝑠𝑒𝑎𝑖 − 𝑋𝑆𝑡𝑟𝑒𝑎𝑚

𝑖 ‖ < 𝑑𝑚𝑎𝑥 , 𝑖 = 1, 2, 3, …… . . , 𝑁𝑆1

If the above condition 𝐸𝐶2 becomes true, then start the raining process as per

Eq. (3.11)

𝑋𝑠𝑡𝑟𝑒𝑎𝑚𝑛𝑒𝑤 = 𝑋𝑠𝑒𝑎 + √𝜇 × 𝑟𝑎𝑛𝑑𝑛 (1, 𝑁) (3.11)

where 𝜇 depicts the area being searched around the sea. The smaller value for 𝜇 leads

the algorithm to search in smaller region near the sea. A suitable value for 𝜇 is set to

0.1. A better of exploration in the vicinity of sea is achieved through ERWCA by

boosting up the exploitation.

The value of 𝑑𝑚𝑎𝑥 comes from Eq. (3.12) and is decreasing adaptively. If a

higher value of 𝑑𝑚𝑎𝑥 is selected it avoids extra searches and its smaller value

intensify the search closer to the sea.



𝑑𝑚𝑎𝑥𝑖+1 = 𝑑𝑚𝑎𝑥

𝑖 −𝑑𝑚𝑎𝑥

𝑖

𝑀𝑎𝑥 𝐼𝑡𝑒𝑟𝑎𝑡𝑖𝑜𝑛⁄ (3.12)

This raining process is similar to mutation phase in GA.

The merging of streams into river or sea or the merging of river into sea is

dependent on the intensity of flow of streams or rivers. Those streams and rivers

which have low flow and are not able to reach the sea will definitely evaporate after

some time. The evaporation process in ERWCA is altered slightly by adding the

concept of evaporation rate [137]. Therefore the evaporation rate (휀) is defined as:

휀 = {∑ 𝑁𝑆𝑛

𝑁𝑠𝑟𝑛=2

𝑁𝑠𝑟 − 1} × 𝑟𝑎𝑛𝑑 (3.13)

The above Eq. (3.13) clearly depicts a lower value of 휀 for the solutions having

better fitness values and a relatively higher value of 휀 for the solutions having poor

fitness values. Meaning, that the rivers having more number of streams have lower

probability to evaporate compared to those having lesser number of streams.

Therefore, one more evaporation condition for those rivers having fewer streams has

to be satisfied to perform the raining process again. These conditions are:

𝐸𝐶3: exp (− 𝐼𝑡𝑒𝑟𝑎𝑡𝑖𝑜𝑛 𝑁𝑜

𝑀𝑎𝑥 𝐼𝑡𝑒𝑟𝑎𝑡𝑖𝑜𝑛) < 𝑟𝑎𝑛𝑑 & 𝑁𝑆𝑖 < 휀

If the above conditions 𝐸𝐶3 are satisfied, then the raining process is started

again using Eq. (3.10). If the evaporation condition is satisfied for any river, then that

specific river along with its streams will be removed and new streams and a river will

be created but in a different position.

3.3 SIGNIFICANCE OF WCA

WCA is a new meta-heuristic algorithm which has outperformed many other EAs

wherever it has been applied in the literature. In WCA, rivers which are selected as

the best points except the sea act as “guidance/direction points” to direct other



population members towards better positions. This also helps algorithm not to jump

into inappropriate regions like in many other EAs where the next step is random and

no guidance is there. Furthermore, streams are not fixed points and their movement

downhill towards the rivers and eventually into the sea is an unintentional move

towards the optimum solution i.e. the sea. The evaporation and raining process acts

as the mutation operator in many EAs like GA and DE. The evaporation and raining

process helps escape the algorithm from getting trapped in local solutions. Many

other EAs do not possess such mechanism.

All these advantages have encouraged the researchers to apply WCA to constrained

optimization problems with single and multi-objective problems. Many

mathematical benchmark problem as well as practical research problems of different

areas have been solved using WCA and in all cases, WCA has proven to be better in

bringing good quality results. The successful application of WCA has certainly

encouraged the author to investigate this method on a highly complex and non-

convex problem of SHTCP and MOSHTCP.

3.4 CHAOS THEORY

Chaos theory is a field of study in mathematics; however it has applications in several

disciplines, including engineering, metrology, physics and social sciences. Chaos

theory studies the behavior of dynamical systems that are highly sensitive to initial

conditions----an effect which is referred as butterfly effect. Small difference in initial

conditions yields widely diversified outcomes for such dynamical systems, rendering

long term prediction models. This happen even though these systems are

deterministic, meaning that their future behavior is fully determined by their initial

conditions, with no randomness involved. This behavior is known as deterministic

chaos [139]. It is a mathematical fact that the chaotic systems are highly dependent

on the involved parameters and the conditions on which the objective function has

been formulated. Hence, the chaotic systems are random and unpredictable.

3.4.1 Chaotic Sequences



There are several variations of chaotic time series sequence strategies that can be

applicable in EAs which are capable to enhance the algorithm exploitation capability

in search space and improve its convergence characteristics. Some of strategies for

generation of chaotic sequences are described below [140]:

3.4.1.1 Logistic Map

One of the simplest dynamic systems indicating chaotic behavior is the iterative

named logistic map, whose mathematical representation as follows:

𝑥𝑡+1 = 𝑎 × 𝑥𝑡(1 − 𝑥𝑡) (3.14)

The following values for the parameter 𝑎 = 4 have been generally used for

simulation purposes.

3.4.1.2 Tent Map

The operator also resembles the logistic iterative map and assumes the following

form:

𝑥𝑡+1 = 𝑌(𝑥𝑡) (3.15)

With

(𝑥𝑡) = {

𝑥𝑡0.7⁄ 𝑖𝑓 𝑥 < 0.7

1

0.3⁄ (𝑥𝑡 × (1 − 𝑥𝑡)), 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒 (3.16)

The initial condition of 𝑥 has been taken in the range of [0, 1].

3.4.1.3 Sinusoidal Iterative Map

This is also used to generate the chaotic sequences and it is mathematically

represented as:

𝑥𝑡+1 = 𝑎 × 𝑥𝑡2 × 𝑠𝑖𝑛 (𝜋𝑥𝑡) (3.17)

In what follows, it is treated with 𝑎 = 2.3 and 𝑥 = 0.7 and it is simplified by using the

following relation:



𝑥𝑡+1 = 𝑠𝑖𝑛 (𝜋𝑥𝑡) (3.18)

3.4.1.4 Lozi Iterative Map

Lozi’s piecewise linear model is a simplified version of Henon’s attractor. The

mathematical representation of this map is given as:

(𝑥𝑡+1, 𝑦𝑡+1) = 𝐻(𝑥𝑡, 𝑦𝑡) (3.19)

With

𝐻(𝑥𝑡, 𝑦𝑡) = (1 + 𝑦𝑡 − 𝑎|𝑥𝑡|, 𝑏𝑥𝑡) (3.20)

Lozi suggested the following values for the parameters 𝑎 = 1.7 and 𝑏 = 0.5.

3.4.1.5 Gauss Iterative Map

This transformation is similar to a quadratic one and it is widely used in literature

for testing purposes because it allows a comprehensive analysis of its chaotic

qualitative and quantitative features. The mathematical representation is as follows:

𝑥𝑡+1 = 𝐺(𝑥𝑡) (3.21)

With

𝐺(𝑥𝑡) = {

0 𝑖𝑓 𝑥 = 0

(1 𝑥⁄ )𝑚𝑜𝑑 1, 𝑥 ∈ (0,1) (3.22)

3.4.2 Application of Chaos Theory in Evolutionary Algorithms

All of the evolutionary algorithms work on the principle of exploration and

exploitation. But due to their nature, majority of these EAs suffer with the problem

of premature convergence or getting trapped in local optima. To improve their

performance by improving the exploitation and the exploration phases of the EAs

different solutions have been proposed. e.g. wavelet transfer was combined with

ANN to improve the ability of algorithm [141], real coding technique has been used

with GA to improve the search ability [142], parallel computation has been suggested

to improve PSO [143], some internal modifications in the algorithms themselves [2,



3]. But recent development of the hybridization of chaos sequences with the

evolutionary algorithms has proven its strength in bringing the optimum results by

avoiding the premature convergence and local optima trapping. In [102, 108, 110,

144] DE has been hybridized with chaotic sequences to improve its search capability

and in [132] chaos paradigm is combined to improve ABC.

The sequences generated from chaotic systems can substitute random numbers in all

phases of EAs where it is necessary to make a random based choice. Each time a

random number is needed by the classical EA it is generated by iterating one step the

chaotic iterative operator that has been started from the random initial condition at

the first generation of the EA. In particular, the use of chaotic sequences affects the

EA in following phases [145].

During the creation of initial population, the chaotic sequences can be used to

generate the individuals.

During the mutation or crossover, the chaotic sequences can be used for the

choice of points inside the chromosomes or far the generation of bit masks

and to decide whether or not to apply the desired operation.

During the selection operation, the chaotic sequences can be used for the

probabilistic choice of individuals according to the roulette wheel method.

Therefore chaotic sequences influence the behavior of all operators especially the

mutation and crossover, not because new strategies has been developed, but because

all the existing standard operators work following the outcomes of a chaotic

sequences instead of a random number generator.

Chapter 4 Hydrothermal Coordination Modelling using WCA and Proposed HCWCA



MODELLING USING WATER CYCLE ALGORITHM AND

PROPOSED HYBRID CHAOTIC WATER CYCLE

ALGORITHM

4.1 INITIALIZATION OF SOLUTION STRUCTURE

The structure of solution for SHTCP consists of two control variables; the discharge

of water for hydroelectric units and the power generation by thermal units. Both the

variables are initialized within their prescribed limits as:

𝑄ℎ𝑗𝑡 = 𝑄ℎ𝑗𝑚𝑖𝑛 + 𝑟𝑎𝑛𝑑 × (𝑄ℎ𝑗

𝑚𝑎𝑥 − 𝑄ℎ𝑗𝑚𝑖𝑛) (4.1)

𝑃𝑠𝑖𝑡 = 𝑃𝑠𝑖𝑚𝑖𝑛 + 𝑟𝑎𝑛𝑑 × (𝑃𝑠𝑖

𝑚𝑎𝑥 − 𝑃𝑠𝑖𝑚𝑖𝑛) (4.2)

where, 𝑟𝑎𝑛𝑑 is the random number generated in [0,1]. A candidate population of

streams will be initialized as:

𝑋𝑘 =

(

𝑄ℎ11 𝑄ℎ2

1 𝑄ℎ𝑗1 𝑄ℎ𝑁ℎ

1 ; 𝑃𝑠11 𝑃𝑠2

1 𝑃𝑠𝑖1 𝑃𝑠𝑁𝑠

1

𝑄ℎ12 𝑄ℎ2

2 𝑄ℎ𝑗2 𝑄ℎ𝑁ℎ

2 ; 𝑃𝑠12 𝑃𝑠2

2 𝑃𝑠𝑖2 𝑃𝑠𝑁𝑠

2

𝑄ℎ1𝑡 𝑄ℎ2

𝑡 𝑄ℎ𝑗𝑡 𝑄ℎ𝑁ℎ

𝑡 ; 𝑃𝑠1𝑡 𝑃𝑠2

𝑡 𝑃𝑠𝑖𝑡 𝑃𝑠𝑁𝑠

𝑡

𝑄ℎ1𝑇 𝑄ℎ2

𝑇 𝑄ℎ𝑗𝑇 𝑄ℎ𝑁ℎ

𝑇 ; 𝑃𝑠1𝑇 𝑃𝑠2

𝑇 𝑃𝑠𝑖𝑇 𝑃𝑠𝑁𝑠

𝑇)

(4.3)

where 𝑋𝑘 is the 𝑘𝑡ℎ stream or candidate solution.

4.2 CONSTRAINT HANDLING

The complexity of SHTCP increases drastically due to the involvement of many

equality and inequality constraints. Therefore, the satisfaction of all these

constraints is very important and tedious task in this problem. In this work,

empirical set of rules have been settled to satisfy these constraints.



4.2.1 Constraint Handling Mechanism for Inequality Constraints

As a result of raining process, new streams are created, which might violate the

minimum and maximum limits. If any stream candidate violates the limits, then the

following equation are used to bring them within their limits.

𝑃𝑠𝑖𝑡 = {𝑃𝑠𝑖

𝑚𝑖𝑛 𝑖𝑓 𝑃𝑠𝑖𝑡 < 𝑃𝑠𝑖𝑚𝑖𝑛

𝑃𝑠𝑖

𝑚𝑎𝑥 𝑖𝑓 𝑃𝑠𝑖𝑡 > 𝑃𝑠𝑖𝑚𝑎𝑥

, 𝑄ℎ𝑗𝑡 = {𝑄ℎ𝑗

𝑚𝑖𝑛 𝑖𝑓 𝑄ℎ𝑗𝑡 < 𝑄ℎ𝑗𝑚𝑖𝑛

𝑄ℎ𝑗

𝑚𝑎𝑥 𝑖𝑓 𝑄ℎ𝑗𝑡 > 𝑄ℎ𝑗𝑚𝑎𝑥

(4.4)

4.2.2 Constraint Handling Mechanism for Equality Constraints

The equality constraints are tougher to be handled in SHTCP as compared to

inequality constraints. The dynamic water balance constraint and power balance

constraint are required to be handled after the initialization and every time

whenever the raining process starts. The conventional method of penalty factor is

not suitable as the constraints are higher in number. Therefore another method to

balance these constraints is devised as follows:

4.2.2.1 Water dynamic balance constraint handling mechanism

To meet exactly the restrictions on the initial and final reservoir the water

discharge rate of the 𝑗𝑡ℎ hydroelectric unit 𝑄ℎ𝑗𝑑 in the dependent interval 𝑑 is then

calculated by:

𝑄ℎ𝑗𝑑 = 𝑉ℎ𝑗0 − 𝑉ℎ𝑗𝑇 − ∑𝑄ℎ𝑗𝑡 − ∑ ∑(𝑄ℎ𝑚,𝑡−𝜏𝑚𝑗)

𝑅𝑢𝑗

𝑚=1

+ ∑𝐼ℎ𝑗𝑡

𝑇

𝑡=1

𝑇

𝑡=1

𝑇

𝑡=1𝑡≠𝑑

(4.5)

If the water release element violates the constraint, then it is attuned according to

Eq. (4.5) and another random interval is selected. The practice reiterates until the

computed element fulfills the constraint.



4.2.2.2 Active power balance constraint handling mechanism

With the fulfillment of the water dynamic balance constraint, reservoir storage

volumes and resultant hydroelectric generations are computed but active power

balance constraint still remains unsatisfied. To fulfill the power balance constraint

exactly the dependent thermal unit d from the thermal units is randomly selected and

dependent thermal generation 𝑃𝑠,𝑑𝑡 is calculated using the following equation:

𝑃𝑠𝑑𝑡 = 𝑃𝐷𝑡 + 𝐵𝑑𝑑𝑃𝑠𝑑𝑡2 + ∑ ∑ 𝑃𝑖𝑡𝐵𝑖𝑗𝑃𝑗𝑡

𝑁𝑠+𝑁ℎ

𝑗=1,𝑗≠𝑑

𝑁𝑠+𝑁ℎ

𝑖=1,𝑖≠𝑑

+ ∑ 𝐵0𝑖𝑃𝑖𝑡

𝑁𝑠+𝑁ℎ

𝑖=1,𝑖≠𝑑

+ ∑ 𝑃𝑗𝑡(𝐵𝑗𝑑 + 𝐵𝑑𝑗)𝑃𝑑𝑡

𝑁𝑠+𝑁ℎ

𝑗=1,𝑗≠𝑑

+ 𝐵0𝑑𝑃𝑑𝑡 + 𝐵00 − ∑ 𝑃𝑠𝑖𝑡

𝑁𝑠

𝑖=1,𝑖≠𝑑

− ∑𝑃ℎ𝑗𝑡

𝑁ℎ

𝑗=1

(4.6)

The above step is iterated if the dependent thermal power generation doesn’t fulfill

the inequality constraint. It is ensured that the dependent thermal unit is not

repeatedly selected while selecting a new random thermal unit.

4.3 MODELLING OF SHTCP AS PER WCA

The complete SHTCP has been modelled as per the environment of WCA. The

modelling process for SHTCP can be described as:

1. The first step is to generate random water discharges of hydroelectric units

and power outputs of thermal units. This initial population is regarded as

streams.

2. The hydroelectric discharges and the reservoir limitations are used to check

the satisfaction of water continuity equation.

3. Upon satisfaction of water dynamic balance, the hydroelectric power outputs

are calculated and then the randomly generated thermal powers are used to

satisfy the power balance equation.

4. Only those population members will be considered which satisfy all the

constraints.



5. Then, the fitness of all the populations are calculated and designated as sea,

rivers and streams.

6. After this, the steps described in the flow chart will be followed.

4.4 STEPS OF STANDARD WCA FOR SHTCP

The detailed steps of the standard WCA for SHTCP are as under:

Step1: Generate initial population of streams i.e. water discharges of hydroelectric

units and power output of thermal units, while satisfying all the constraints

and evaluate the fitness function.

Step2: Designate the Sea, Rivers and Streams.

Step3: Determine the intensity of flow for rivers and sea using Eq. (3.5).

Step4: Streams flow to the rivers and sea using Eq. (3.7) & (3.8).

Step5: If the stream has lower fitness value than the corresponding river and sea

then Go to Step 6 else Go to Step 7.

Step6: Exchange positions of rivers/sea with streams.

Step7: Rivers flow to the sea using Eq. (3.9).

Step8: If the River has lower fitness value than the corresponding sea then Go to

Step 9 else Go to Step 10.

Step9: Exchange positions of sea with river.

Step10: Calculate Evaporation Rate using Eq. (3.13)

Step11: If the evaporation condition Cond3 has been satisfied then Go to Step 12 else

Go to Step 13.

Step12: Calculate new positions of rivers and streams using Eq. (3.10)

Step13: If the evaporation condition Cond1 & Cond2 have been satisfied then Go to

Step 14 else Go to Step 15.

Step14: Calculate new positions of rivers and streams using Eq. (3.10) and Eq. (3.11).

Step15: Reduce 𝑑𝑚𝑎𝑥 using Eq. (3.12).



Step16: Check for the stopping condition else Go to Step 2.

4.5 PROPOSED HYBRID CHAOTIC WATER CYCLE ALGORITHM

4.5.1 Initialization

Both the variables are initialized within their prescribed limits as:

𝑄ℎ𝑗𝑡 = 𝑄ℎ𝑗𝑚𝑖𝑛 + 𝑐ℎ𝑜𝑥 × (𝑄ℎ𝑗

𝑚𝑎𝑥 − 𝑄ℎ𝑗𝑚𝑖𝑛) (4.7)

𝑃𝑠𝑖𝑡 = 𝑃𝑠𝑖𝑚𝑖𝑛 + 𝑐ℎ𝑜𝑥 × (𝑃𝑠𝑖

𝑚𝑎𝑥 − 𝑃𝑠𝑖𝑚𝑖𝑛) (4.8)

4.5.2 WCA with Chaotic Evaporation Process

The chaotic sequences when applied to EAs, enhance their exploitation capability. In

WCA, among other initial user controlled parameters, the value of 𝑑𝑚𝑎𝑥 is very vital..

The logistic iterative map has been proposed in WCA to self-adjust 𝑑𝑚𝑎𝑥 in chaotic

evaporation between (1E-17, 1E-1). The evaporation condition EC1 will be modified

as:

𝐸𝐶1 (new): ‖𝑋𝑠𝑒𝑎

𝑖 − 𝑋𝑅𝑖𝑣𝑒𝑟𝑖 ‖ < 𝑐ℎ𝑜𝑥𝑡 𝑜𝑟 𝑐ℎ𝑜𝑥𝑡 < 0.1,

𝑖 = 1, 2, 3, …… . . , 𝑁𝑠𝑟 − 1

If the above condition EC1 (new) becomes true, then perform chaotic raining

process as per Eq. (3.23)

𝑋𝑠𝑡𝑟𝑒𝑎𝑚𝑛𝑒𝑤 = 𝑀𝑖𝑛𝐿𝑖𝑚 + 𝑐ℎ𝑜𝑥𝑡 × (𝑀𝑎𝑥𝐿𝑖𝑚 − 𝑀𝑖𝑛𝐿𝑖𝑚) (4.9)

And EC2 will be modified as:

𝐸𝐶2 (new): ‖𝑋𝑠𝑒𝑎𝑖 − 𝑋𝑆𝑡𝑟𝑒𝑎𝑚

𝑖 ‖ < 𝑐ℎ𝑜𝑥𝑡 , 𝑖 = 1, 2, 3, …… . . , 𝑁𝑆1

If the above condition EC2 (new) becomes true, then perform chaotic raining

process as per Eq. (3.24)



𝑋𝑠𝑡𝑟𝑒𝑎𝑚𝑛𝑒𝑤 = 𝑋𝑠𝑒𝑎 + √𝜇 × 𝑐ℎ𝑜𝑥𝑡 (4.10)

𝜇 (𝑡) = { 𝜇𝑖 + (𝜇𝑓 − 𝜇𝑖) × (𝑡 𝑡𝑚⁄ )} (4.11)

This 𝑐ℎ𝑜𝑥𝑡 is calculated as per logistic map of chaos paradigm as follows:

𝑐ℎ𝑜𝑥𝑡+1 = 4 × 𝑐ℎ𝑜𝑥𝑡 × (1 − 𝑐ℎ𝑜𝑥𝑡)

such that 𝑐ℎ𝑜𝑥𝑡 ≠ 0, 0.25, 0.5, 0.75 𝑜𝑟 1 (4.12)

4.5.3 Chaotic Local Search

Like other evolutionary computation methods, the performance of WCA is not

satisfactory in local search. On the other hand, chaotic search techniques perform

quite well for local search. Therefore, a chaotic local search technique has been

proposed and implemented using the standard logistic mapping. The detailed

procedure for HCWCA for SHTCP will be as:

i. Find the best solution named Sea and calculate its fitness

ii. Set iteration count to 0, and generate the initial chaotic vectors as per Eq.

(4.12)

iii. Calculate the chaotic variables for the next iteration using Eq. (4.12)

iv. Convert the chaotic variable 𝑐ℎ𝑜𝑥𝑡 into the control variables as per Eq.

(4.7) & (4.8)

v. Calculate the fitness value of control variable and compare it with Sea to

find new Sea

vi. Increment iteration count and go to Step (b) and repeat until max

iteration is reached. Otherwise terminate chaotic local search.

The flowchart of the proposed HCWCA has been shown in Fig. 4.1.



Start

Generate initial population of streams i.e. water discharges and thermal powers

Fulfill the constraints of Thermal Plants and Hydel Network

Evaluate the fitness function

Designate the Sea, Rivers and Streams

Determine the intensity of flow for rivers and sea using Eq. (3.5)

Streams flow to the rivers and sea using Eq. (3.7) and (3.8)

Exchange positions of rivers/sea with streams

Rivers flow to the sea using Eq. (3.9)

Exchange positions of sea with river

A

A

Apply chaotic local search on sea

Apply chaotic sequence to search for dmax

If the stopping criteria has been met?

Stop

B

B

No

Yes

No

Yes

Yes

No

If the River has lower fitness value than the corresponding sea ?

Read the Data of Hydroelectric & Thermal Units from Input Files

If the Stream has lower fitness value than the corresponding

river and sea ?

Calculate new positions of rivers and streams using Eq. (4.9) and (4.10)

If the evaporation condition EC1 (new) &

EC2 (new)have been satisfied?

No

Yes

Fig. 4.1 Flowchart of Proposed HCWCA for SHTCP

Chapter 5 Implementation & Case Studies


CHAPTER NO. 5 IMPLEMENTATION & CASE

STUDIES

5.1 DEVELOPMENT OF A COMPUTATIONAL FRAMEWORK

A computational framework for the implementation of WCA and HCWCA has

been developed in the environment of Visual C++ to test the different standard

hydrothermal test system and a practical utility system. This framework has been

modelled on a system Intel Dual Core with 4 GB RAM and the environment used was

Visual C++ 6.0. The key features of this framework are:

declaration and initialization of variables,

definition of evolution model and control parameters,

selection of convex or non-convex test system,

selection of feasible parameters of WCA or HCWCA,

reading of input data of test system from text files,

displaying and printing of output results in text files

5.2 STRATEGY FOR IMPLEMENTATION

The strategy for the implementation of WCA and HCWCA depends upon the

quality and characteristics of the test system under investigation. In case of SHTCP,

it highly depends upon the model and configuration of hydroelectric units used. The

standard inputs and outputs will be:

I. Standard Inputs

i. Thermal fuel cost curve coefficients and their generation capacities

ii. Hydroelectric units generation coefficients and their capacities

iii. Hydroelectric units discharge and volume capacities with initial and

end conditions of reservoirs

iv. Hourly inflows of all reservoirs

v. Hourly load demand

II. Standard Outputs



i. Optimal discharges of hydroelectric units

ii. Optimal hydroelectric and thermal generation schedule

iii. Optimal total fuel cost and fuel emission (for MOSHTCP)

5.3 TEST SYSTEMS INVESTIGATED

WCA and the proposed HCWCA both have been implemented on different

standard test systems available in the literature with fixed head hydroelectric units

and multi-chain variable head hydroelectric units with different no. of thermal units

as well as some practical utility systems. All the test systems investigated in this

thesis are enlisted in Table 5.1.

5.3.1 Fixed Head Hydroelectric Units

Glimn-Kirchmayer mathematical model for the fixed head hydroelectric

configuration has been used in this research. As per the Glimn-Kirchmayer model,

the water discharge is a function of power output and the effective head. For the

reservoirs with comparatively larger capacity, the effective head is assumed to be

constant or fixed over the under consideration interval of time. The water discharge

𝑄𝑗𝑡 from the 𝑗𝑡ℎ reservoir at time 𝑡 is written as:

𝑄𝑗𝑡 = 𝑥𝑗𝑃2 + 𝑦𝑗𝑃 + 𝑧𝑗 (5.1)

where 𝑥𝑗 , 𝑦𝑗 and 𝑧𝑗 are the discharge coefficients of 𝑗𝑡ℎ hydroelectric unit. Four

different types of fixed head hydrothermal coordination standard test systems have

been investigated using the WCA and proposed HCWCA.

5.3.1.1 Test System 1

This test system consists of one hydroelectric unit and one thermal unit. The

configuration of the hydroelectric units with thermal unit is shown in Fig. 5.1. The

power generation coefficients, power generation limits and total amount of water

available of hydroelectric unit, fuel cost coefficients, generation limits of thermal unit

and hourly load demand are shown in Table 5.2.



Table 5.1 Test Systems Investigated

Test Systems

No

. of

Hy

dro

-

ele

ctri

c U

nit

s

No

. of

Th

erm

al

Un

its

Case Studies Investigated

Fixed Head Test System 1 1 1 Convex Cost Function

Test System 2 1 3 Convex Cost Function

Multi-chain

Variable

Head

Test System 3 4 1

I) Convex Cost Function

II) Convex Cost Function with

Prohibited Discharge Zones

III) Non-Convex Cost Function

IV) Non-Convex Cost Function with


Test System 4 4 3

I) Non-Convex Cost Function

II) Non-Convex Cost Function with

Transmission Losses

III) Non-Convex Cost Function with

Prohibited Discharge Zones and

Ramp Rates

IV) Non-Convex Cost Function with

Prohibited Operating Zones

Test System 5 4 6

I) Non-Convex Cost Function

II) Non-Convex Cost Function with


Test System 6 4 10 Non-Convex Cost Function

Test System 7 4 10 Mixed Binary Problem

Test System 8 4 20 Non-Convex Problem

Test System 9 4 40 Non-Convex Problem

Multi-

objective

(discussed in

Chapter 6)

Test System 10 4 3

I) Economic Cost Coordination

II) Environmental Economic

Coordination

III) Economic Environmental &

Cost Coordination

Practical

Utility

(discussed in

Chapter 7)

Indian Utility

System 11 12 Convex Cost Function



The evolution model i.e. the parameter setting for Test System 1 is shown in Table

5.3.

Fig. 5.1 Network Configuration of Test System 1

Table 5.2 Test System 1 --- Complete Data

Hydroelectric Unit Data

I. Generation Coefficients and Generation Limits

𝒙𝒋 𝒚𝒋 𝒛𝒋 𝑷𝒉𝒎𝒊𝒏 𝑷𝒉

𝒎𝒂𝒙

0.000219427 0.00025709 1.74233 100 300

II. Available Amount of Water

𝑊 = 72 𝑚3

Thermal Unit Data


𝒂𝒊 𝒃𝒊 𝒄𝒊 𝑷𝒔𝒎𝒊𝒏 𝑷𝒔

𝒎𝒂𝒙

373.3 9.606 0.001991 150 500

Hourly Load Demand

Hour Load (MW) Hour Load (MW) Hour Load (MW)

1 455 9 665 17 721

2 425 10 675 18 740

3 415 11 695 19 700

4 407 12 705 20 678

5 400 13 580 21 585

6 420 14 605 22 540

7 487 15 616 23 540

8 604 16 653 24 503



Table 5.3 Test System 1 --- Evolution model

Hydroelectric

Units

Thermal

Units 𝑵𝒑𝒐𝒑 𝑵𝒔𝒓

𝒅𝒎𝒂𝒙 Max.

Generations

No. of

Runs WCA HCWCA

1 1 50 4 0.01 Adaptive 300 30

The results obtained for the optimal water discharges, hydroelectric power and

thermal power have been presented in Table 5.4. Table 5.5 gives the comparison of

the obtained results using HCWCA and WCA with other methods available in the

literature.

Table 5.4 Test System 1 --- Optimal Hydroelectric and Thermal Power Generations

Hour

Water

Discharge

(m3/hr.)

Hydroelectric

Power (MW)

Thermal

Power (MW)

Total Power

Generated

(MW)

𝑸𝒋𝒕 𝑷𝒉𝒕 𝑷𝒔𝒕 𝑷𝑫

1 2.6016 192.1160 262.8840 455

2 2.8272 216.5742 208.4258 425

3 2.9560 229.3956 185.6044 415

4 3.2499 256.3248 150.6752 407

5 3.0260 236.0841 163.9159 400

6 2.7773 211.3995 208.6005 420

7 2.4451 173.1968 313.8032 487

8 2.7880 212.5194 391.4806 604

9 3.3610 265.8055 399.1945 665

10 3.1610 248.4794 426.5206 675

11 3.0271 236.1904 458.8096 695

12 3.2288 254.4824 450.5176 705

13 2.6858 201.5804 378.4196 580

14 2.9062 224.5207 380.4793 605

15 3.6200 286.7300 329.2700 616

16 2.8521 219.1127 433.8873 653

17 3.4078 269.7030 451.2970 721

18 3.7962 300.1413 439.8587 740

19 3.0674 239.9476 460.0524 700

20 3.1954 251.5438 426.4562 678

21 2.3163 155.9771 429.0229 585

22 3.1142 244.2539 295.7461 540

23 3.3169 262.0798 277.9202 540

24 2.7524 208.7704 294.2296 503

Total Generation Cost = 93,988.87 $



Table 5.5 Test System 1 --- Comparison of Results

Methods

Total

Generation

Cost ($)

Computational

Time (s)

%age

Improvement

in Generation

Cost

HCWCA 93,988.87 3.4 -

WCA 94,510.78 3.0 0.55%

Classical Iterative Method [146] 96,024.39 --- 2.17%


This test system consists of one hydroelectric unit and three thermal units. The

configuration of the hydroelectric unit with thermal units is shown in Fig. 5.2. The

power generation coefficients, power generation limits and total amount of water

available of hydroelectric unit, fuel cost coefficients, generation limits of thermal

units and hourly load demand are shown in Table 5.6.

Fig. 5.2 Network Configuration of Test System 2


5.7.



Table 5.6 Test System 2 --- Complete Data

Hydroelectric Unit Data


𝒙𝒋 𝒚𝒋 𝒛𝒋 𝑷𝒉𝒎𝒊𝒏 𝑷𝒉

𝒎𝒂𝒙

0.06 20 140 10 100

II. Available Amount of Water

𝑊 = 25,000 𝑚3

Thermal Unit Data


𝒂𝒊 𝒃𝒊 𝒄𝒊 𝑷𝒔𝒎𝒊𝒏 𝑷𝒔

𝒎𝒂𝒙

100 0.1 0.01 50 200

120 0.1 0.02 40 170

150 0.2 0.01 30 215

Hourly Load Demand


1 175 9 440 17 425

2 190 10 475 18 400

3 220 11 525 19 375

4 280 12 550 20 340

5 320 13 565 21 300

6 360 14 540 22 250

7 390 15 500 23 200

8 410 16 450 24 180




Hydroelectric

Units

Thermal



Generations

No. of

Runs WCA HCWCA

1 3 50 4 0.01 Adaptive 300 30

The results obtained for the optimal water discharges, hydroelectric power and

thermal powers have been presented in Table 5.8. Table 5.9 gives the comparison of

the obtained results using HCWCA and WCA with other methods available in the

literature.


Generations

Hour

Water

Discharge

(m3/hr.)

Hydroelectric

Power (MW) Thermal Power (MW)

𝑸𝒋𝒕 𝑷𝒉𝒕 𝑷𝒔𝟏𝒕 𝑷𝒔𝟐𝒕 𝑷𝒔𝟑𝒕

1 346.0000 10.0000 95.0000 40.0000 30.0000

2 346.0000 10.0000 109.4407 40.0000 30.5593

3 346.0000 10.0000 69.6408 40.0000 100.3592

4 705.1649 26.1991 98.9349 40.0000 114.8660

5 1157.6317 44.8477 140.3877 40.0000 94.7646

6 1464.7368 56.6195 138.8280 66.5406 98.0119

7 1131.0197 43.7966 130.1429 49.3314 166.7292

8 1054.5318 40.7459 173.5646 46.7934 148.8961

9 1412.9953 54.6800 153.7129 71.2458 160.3613

10 1006.9406 38.8249 189.3810 65.7874 181.0067

11 1873.0289 71.3703 200.0000 80.7650 172.8647

12 1571.3639 60.5641 200.0000 90.8793 198.5565

13 977.9208 37.6447 200.0000 112.3853 214.9700

14 1228.0159 47.6027 200.0000 77.8540 214.5433

15 2099.9828 79.1873 189.0060 85.8069 145.9998

16 1440.0994 55.6981 200.0000 53.1948 141.1071

17 2051.5036 77.5385 146.9428 45.3971 155.1215

18 912.3088 34.9508 167.3522 72.1043 125.5927

19 1124.0981 43.5223 176.6838 54.3925 100.4014

20 1019.8583 39.3481 139.5684 40.2189 120.8645

21 404.1697 12.7229 119.1277 40.0000 128.1494

22 634.6289 23.1269 145.6827 40.0002 41.1902

23 346.0000 10.0000 120.0000 40.0000 30.0000

24 346.0000 10.0000 97.2458 40.0000 32.7542





Methods

Total

Generation

Cost ($)

Computational

Time (s)

%age

Improvement

in Generation

Cost

HCWCA 21,893.94 6.2 -

WCA 22,246.43 6.1 1.61%

MBFA [147] 24,267.41 9.96 10.84%

Classical Iterative Method [146] 24,276.00 --- 10.88%

5.3.2 Multi-Chain Hydroelectric Units

In multi-chain hydroelectric units, there are four hydroelectric units attached in a

configuration as shown in Fig. 5.3. The discharges of reservoir no. 1 and reservoir no.

2 are added to reservoir no. 3 with the specific time delays, and then the discharge of

reservoir no. 3 is added to reservoir no. 4 with another specified time delay. This is

the IEEE standard test system for the multi-chain hydroelectric units also known as

cascade hydroelectric units. Various IEEE standard test systems have been

formulated by configuring different number of thermal units with the above

mentioned multi-chain hydroelectric units.

Fig. 5.3 Configuration of Multi-chain Hydroelectric Units for Test System 3-10




This test system consists of above mentioned four multi-chain hydroelectric units

and numerous thermal units represented by an equivalent thermal unit. Four

different case studies have been performed by selecting four different characteristics

and constraints of the hydroelectric and/or thermal unit. The power generation

coefficients, power generation limits, water discharge limits, prohibited discharge

zones, reservoir storage limits, initial and end conditions of reservoirs and hourly

inflows of hydroelectric units have been shown in Table 5.10 and fuel cost

coefficients and generation limits of thermal unit and hourly load demand are shown

in Table 5.11.


5.12.

The four different case studies investigated are as described below:

I. Test System 3: Case I --- Quadratic Fuel Cost Function

In this case, the valve point effect of thermal unit is neglected and the fuel cost

function of thermal unit is assumed to be a quadratic function only. The prohibited

discharge zones of hydroelectric units have also not been taken into account. The

results obtained by the proposed HCWCA and WCA for the optimal water discharges

have been presented in Table 5.13 and for the optimal hydroelectric powers and

thermal power have been presented in Table 5.14. Table 5.15 gives the comparison

of the obtained results using HCWCA and WCA with other methods available in the

literature. Fig. 5.4 shows the convergence characteristics of this system. It can be seen

that the convergence is very fast in the first 100-150 iterations and then become

slower for the next 50-100 iteration up to 200 and finally it is almost constant till 500

iterations. The algorithm was tested for more than 500 generations but no significant

improvement was found. The improvement in the cost is up to 0.9% as per the

comparison table. This improvement means a total savings of approx. 8253$ per day

which accumulates to more than 3 million $ per annum. This much improvement is a

very significant improvement.



II. Test System 3: Case II --- Quadratic Fuel Cost Function with


In this case, the valve point effect of thermal unit is neglected and the fuel cost

function of thermal unit is assumed to be a quadratic function only. The prohibited

discharge zones of hydroelectric units have been taken into account. The results

obtained by the proposed HCWCA and WCA for the optimal water discharges have

been presented in Table 5.16 and for the optimal hydroelectric powers and thermal

power have been presented in Table 5.17. Table 5.18 gives the comparison of the

obtained results using HCWCA and WCA with other methods available in the

literature. Fig. 5.5 shows the convergence characteristics of this system. It can be seen

that the convergence is very fast in the first 200 iterations and then become slower

up to 400 and finally it is almost constant till 500 iterations. The algorithm was tested

for more than 500 generations but no significant improvement was found. The

improvement in the cost is up to 0.85% as per the comparison table. This

improvement means a total savings of approx. 7767$ per day which accumulates to

more than 2.8 million $ per annum. This much improvement is a very significant

improvement.

III. Test System 3: Case III --- Non-Convex Fuel Cost Function

In this case, the valve point effect of thermal unit is considered and the fuel cost

function of thermal unit has been assumed to be a non-convex function. The

prohibited discharge zones of hydroelectric units have not been taken into account.

The results obtained by the proposed HCWCA and WCA for the optimal water

discharges have been presented in Table 5.19 and for the optimal hydroelectric

powers and thermal power have been presented in Table 5.20. Table 5.21 gives the

comparison of the obtained results using HCWCA and WCA with other methods

available in the literature. Fig. 5.6 shows the convergence characteristics of this

system. It can be seen that due to the inclusion of valve point effect, convergence has

become slower. It is fast in the first 200-300 iterations and then it is almost constant

till 500 iterations. The algorithm was tested for more than 500 generations but no

significant improvement was found. The improvement in the cost is up to 0.46% as

per the comparison table. This improvement means a total savings of approx. 4203$



per day which accumulates to more than 1.5 million $ per annum. This much

improvement is a very significant improvement.

Table 5.10 Test System 3 to 9 --- Complete Data of Hydroelectric Units


Unit 𝑪𝟏𝒋 𝑪𝟐𝒋 𝑪𝟑𝒋 𝑪𝟒𝒋 𝑪𝟓𝒋 𝑪𝟔𝒋 𝑷𝒉𝒋𝒎𝒊𝒏 𝑷𝒉𝒋

𝒎𝒂𝒙

1 -0.0042 -0.42 0.030 0.90 10.0 -50 0 500

2 -0.0040 -0.30 0.015 1.14 9.5 -70 0 500

3 -0.0016 -0.30 0.014 0.55 5.5 -40 0 500

4 -0.0030 -0.31 0.027 1.44 14.0 -90 0 500

II. Discharge and Volume Limits, Initial and End Conditions

Unit 𝑽𝒉𝒋𝒎𝒊𝒏 𝑽𝒉𝒋

𝒎𝒂𝒙 𝑽𝒉𝒋𝑰𝒏𝒊 𝑽𝒉𝒋

𝑬𝒏𝒅

Test System

3 & 5

Test System

4, 6-9

𝑸𝒉𝒋𝒎𝒊𝒏 𝑸𝒉𝒋

𝒎𝒂𝒙 𝑸𝒉𝒋𝒎𝒊𝒏 𝑸𝒉𝒋

𝒎𝒂𝒙

1 80 150 100 120 5 15 5 15

2 60 120 80 70 6 15 6 15

3 100 240 170 170 10 30 10 30

4 70 160 120 140 13 25 6 20

III. Transport Delays and Prohibited Discharge Zones

Unit

Upstream Hydroelectric units & Time

Delay Prohibited Discharge Zones

𝑹𝒖 𝒕𝒅 𝑸𝒉𝒋𝒍 𝑸𝒉𝒋

𝒉

1 0 2 8 9

2 0 3 7 8

3 2 4 22 27

4 1 0 16 18

IV. Hourly Inflows

Hour Reservoir

Hour Reservoir

1 2 3 4 1 2 3 4

1 10 8 8.1 2.8 13 11 8 4 0

2 9 8 8.2 2.4 14 12 9 3 0

3 8 9 4 1.6 15 11 9 3 0

4 7 9 2 0 16 10 8 2 0

5 6 8 3 0 17 9 7 2 0

6 7 7 4 0 18 8 6 2 0

7 8 6 3 0 19 7 7 1 0

8 9 7 2 0 20 6 8 1 0

9 10 8 1 0 21 7 9 2 0

10 11 9 1 0 22 8 9 2 0

11 12 9 1 0 23 9 8 1 0

12 10 8 2 0 24 10 8 0 0



Table 5.11 Test System 3 --- Complete Data of Thermal Unit and Hourly Load

Demand


Unit 𝒂𝒊 𝒃𝒊 𝒄𝒊 Case I & II Case III & IV

𝑷𝒔𝒎𝒊𝒏 𝑷𝒔

𝒎𝒂𝒙 𝒅𝒊 𝒆𝒊 𝒅𝒊 𝒆𝒊

1 5000 19.2 0.002 0 0 700 0.085 500 2500

Hourly Load Demand


1 1370 9 2240 17 2130

2 1390 10 2320 18 2140

3 1360 11 2230 19 2240

4 1290 12 2310 20 2280

5 1290 13 2230 21 2240

6 1410 14 2200 22 2120

7 1650 15 2130 23 1850

8 2000 16 2070 24 1590


Hydroelectric

Units

Thermal



Generations

No. of

Runs WCA HCWCA

4 1 50 4 0.01 Adaptive 500 30



Table 5.13 Test System 3: Case-I --- Optimal Hydroelectric Discharges

Hour

Optimal Hydroelectric Discharges (× 𝟏𝟎𝟒𝒎𝟑)

𝑸𝒉𝟏 𝑸𝒉𝟐 𝑸𝒉𝟑 𝑸𝒉𝟒

1 7.255 6.000 30.000 13.000

2 7.292 6.000 30.000 13.000

3 7.503 6.000 30.000 13.000

4 7.838 6.000 30.000 13.000

5 8.204 6.004 30.000 13.000

6 8.483 6.760 30.000 13.000

7 8.873 7.841 30.000 13.000

8 9.654 8.338 30.000 13.000

9 9.608 8.485 29.999 13.000

10 9.474 8.662 10.505 13.000

11 9.035 8.569 10.860 13.000

12 9.096 8.991 11.406 13.108

13 8.821 9.056 11.491 14.265

14 8.898 8.820 11.479 14.986

15 8.682 9.449 11.230 15.349

16 8.190 9.603 11.130 15.849

17 8.269 9.990 10.405 16.927

18 8.219 10.520 10.007 17.389

19 8.022 11.252 10.000 18.503

20 7.891 12.061 10.000 19.374

21 7.826 12.810 10.000 20.216

22 7.866 6.314 10.001 21.225

23 5.000 6.894 10.000 21.987

24 5.000 7.580 10.000 23.133



Table 5.14 Test System 3: Case-I --- Optimal Hydroelectric & Thermal Powers

Hour

Hydroelectric Generation (MW)

Thermal

Generation

(MW)

Total

Generation

(MW) 𝑷𝒉𝟏 𝑷𝒉𝟐 𝑷𝒉𝟑 𝑷𝒉𝟒 𝑷𝒔

1 70.939 50.164 0.000 200.094 1048.803 1370

2 71.621 51.298 0.000 187.756 1079.326 1390

3 73.201 52.935 0.000 173.733 1060.131 1360

4 75.237 54.500 0.000 156.792 1003.471 1290

5 76.955 55.534 0.000 178.742 978.769 1290

6 78.186 60.982 0.000 198.958 1071.873 1410

7 80.137 67.039 0.000 217.441 1285.383 1650

8 83.966 69.292 0.000 234.189 1612.553 2000

9 83.882 69.875 0.000 249.204 1837.039 2240

10 83.741 71.070 7.404 262.483 1895.302 2320

11 82.363 70.783 11.911 274.030 1790.913 2230

12 82.966 72.576 16.360 285.038 1853.060 2310

13 82.022 72.320 21.691 306.086 1747.881 2230

14 83.265 71.137 26.259 311.806 1707.534 2200

15 82.512 74.227 30.772 313.242 1629.247 2130

16 79.774 74.059 34.592 315.787 1565.787 2070

17 80.421 74.110 37.954 322.670 1614.846 2130

18 80.057 73.572 41.096 322.782 1622.494 2140

19 78.567 73.611 43.999 326.584 1717.239 2240

20 77.323 73.497 46.754 326.349 1756.077 2280

21 76.724 72.794 49.574 323.224 1717.684 2240

22 77.021 43.811 52.169 318.097 1628.902 2120

23 54.705 48.388 54.319 309.022 1383.566 1850

24 55.022 52.933 56.060 298.211 1127.774 1590




Table 5.15 Test System 3: Case-I --- Comparison of Results

Methods

Total

Generation

Cost ($)

Computational

Time (s)

%age

Improvement

in Generation

Cost

HCWCA 917,130.51 8.9 -

WCA 917,893.94 8.6 0.08%

MHDE [100] 921,893.94 8.0 0.52%

SPPSO [90] 922,336.31 16.3 0.57%

DE [100] 923,574.31 50.0 0.70%

LWPSO [148] 925,383.80 82.9 0.90%

Table 5.16 Test System 3: Case-II --- Optimal Hydroelectric Discharges

Hour



1 7.5165 6.1134 29.9986 13.0000

2 7.7846 6.0162 29.9820 13.0202

3 7.6307 6.0022 29.9358 13.0034

4 8.0000 6.3558 29.9719 13.0093

5 7.6954 6.6693 30.0000 13.0017

6 7.8866 6.7764 29.9854 13.0001

7 9.8420 8.4305 29.9616 13.0010

8 9.7272 6.9875 29.9398 13.0096

9 10.3733 8.5103 29.9803 13.0028

10 9.7982 8.6472 11.1070 13.0028

11 7.9969 6.9949 11.0656 13.1767

12 9.2314 10.1182 10.5171 13.5200

13 7.7602 8.3776 11.1314 13.8774

14 10.2591 9.4506 11.5676 15.5084

15 7.9999 9.7024 10.7243 14.8564

16 7.9521 9.6022 10.9505 15.2035

17 7.8185 10.3837 10.8015 16.0000

18 7.8628 11.1942 10.4848 18.0403

19 7.8606 10.8488 10.0430 19.2816

20 7.9294 12.4772 10.0788 19.3393

21 7.8161 11.6784 10.0075 20.7753

22 7.9608 6.9730 10.0338 20.5470

23 5.2771 6.7237 10.0707 22.0022

24 5.0206 6.9661 10.0004 22.8484



Table 5.17 Test System 3: Case-II --- Optimal Hydroelectric & Thermal Powers

Hour


Thermal

Generation

(MW)

Total

Generation


1 72.6688 50.9033 0.0000 200.0937 1046.3342 1370

2 74.7763 51.3397 0.0000 187.9048 1075.9792 1390

3 73.8493 52.8791 0.0000 173.7320 1059.5396 1360

4 76.0009 56.7895 0.0000 156.8214 1000.3882 1290

5 73.5594 59.6539 0.0000 178.7115 978.0752 1290

6 74.5454 60.5000 0.0000 198.8976 1076.0570 1410

7 84.7734 69.5963 0.0000 217.3266 1278.3037 1650

8 83.9628 60.6504 0.0000 234.1364 1621.2504 2000

9 86.7216 69.7759 0.0000 249.0994 1834.4031 2240

10 84.6038 70.7511 8.5099 262.3848 1893.7503 2320

11 75.8932 61.6753 12.7086 275.7458 1803.9771 2230

12 83.3496 78.3314 18.1543 289.2793 1840.8855 2310

13 75.3769 68.9347 22.9124 301.4427 1761.3333 2230

14 89.9858 74.5446 26.5133 317.0961 1691.8603 2200

15 78.0138 75.4074 31.2632 308.2217 1637.0939 2130

16 78.0918 73.9509 35.4085 309.2721 1573.2768 2070

17 77.3932 75.5661 38.7689 314.3384 1623.9334 2130

18 77.7203 75.5143 41.9052 328.3263 1616.5338 2140

19 77.5552 71.4076 44.2466 331.4631 1715.3276 2240

20 77.6592 73.8469 47.0211 324.9315 1756.5413 2280

21 76.7340 69.1410 49.8580 325.1437 1719.1234 2240

22 77.7128 47.5757 52.3712 313.9715 1628.3687 2120

23 57.2375 46.8643 54.6260 308.6877 1382.5845 1850

24 55.2133 49.1345 56.0607 297.2091 1132.3823 1590




Table 5.18 Test System 3: Case-II --- Comparison of Results

Methods

Total

Generation

Cost ($)

Computational

Time (s)

%age

Improvement

in Generation

Cost

HCWCA 917,428.69 9.0 -

WCA 917,950.57 8.8 0.06%

TLBO [131] 923,041.91 --- 0.61%

IPSO [149] 923,443.17 --- 0.66%

ORCCRO [150] 925,195.87 8.15 0.85%

Table 5.19 Test System 3: Case-III --- Optimal Hydroelectric Discharges

Hour



1 5.6909 6.9647 30.0000 13.0008

2 10.5254 7.2412 30.0000 13.0049

3 6.6316 7.6754 29.7284 13.4622

4 11.8099 7.1622 29.7797 13.1386

5 8.0479 6.5274 29.7540 13.0854

6 6.4460 6.4445 29.9833 13.0812

7 5.0640 8.3536 27.9158 13.0558

8 5.3796 7.7304 29.5301 13.2181

9 9.2083 7.4883 12.1379 13.0469

10 8.0031 8.4249 15.3460 13.0374

11 11.7236 6.7827 15.4388 13.0097

12 10.1120 8.6853 10.5743 14.6777

13 10.6028 11.7414 10.7812 15.9422

14 13.0607 8.8793 11.2076 13.0623

15 6.1722 9.4767 10.1227 16.0869

16 8.7893 10.6892 11.3651 14.8284

17 5.3255 12.3610 10.9393 15.9317

18 9.9695 9.3155 11.8963 15.0994

19 6.1937 9.2540 12.5024 17.3706

20 8.8231 8.2210 11.6466 22.9375

21 8.6439 8.6733 10.3835 17.0798

22 5.8826 6.0035 11.3072 22.5925

23 5.2979 6.8378 10.1176 15.2509

24 7.5963 11.0667 10.0399 22.4486



Table 5.20 Test System 3: Case-III --- Optimal Hydroelectric & Thermal Powers

Hour


Thermal

Generation

(MW)

Total

Generation


1 59.2954 56.1920 0.0000 200.0998 1054.4127 1370

2 89.3143 58.4294 0.0000 187.7907 1054.4656 1390

3 66.7429 61.9330 0.0000 176.9101 1054.4141 1360

4 92.6755 59.7586 0.0000 157.0717 980.4942 1290

5 74.5732 56.3396 0.0000 178.5980 980.4892 1290

6 63.7818 56.0687 0.0000 198.7940 1091.3555 1410

7 53.2253 66.8586 0.0000 216.8148 1313.1013 1650

8 56.6732 62.7292 0.0000 234.8571 1645.7405 2000

9 83.7288 61.5266 16.0576 248.1469 1830.5401 2240

10 77.3166 67.4773 9.2519 261.4943 1904.4599 2320

11 95.8667 58.5012 10.4614 271.5906 1793.5802 2230

12 89.2533 69.8413 21.4948 298.8705 1830.5401 2310

13 91.6085 81.7223 27.9382 309.0707 1719.6603 2230

14 99.3725 68.7577 31.8564 280.3530 1719.6604 2200

15 64.6499 71.5901 36.4879 311.5315 1645.7405 2130

16 83.4681 75.5202 42.8138 296.3772 1571.8207 2070

17 57.8732 78.0401 44.7222 303.6241 1645.7405 2130

18 90.4020 63.2008 47.7450 292.9117 1645.7405 2140

19 65.2910 61.2721 49.1331 307.6836 1756.6203 2240

20 83.6058 55.6506 52.3132 331.8101 1756.6203 2280

21 82.1675 58.3552 52.8452 290.0117 1756.6203 2240

22 62.4147 43.9862 55.7494 312.1092 1645.7405 2120

23 57.6366 50.3678 55.5776 262.4369 1423.9811 1850

24 76.5940 70.2121 56.1346 295.7147 1091.3445 1590




Table 5.21 Test System 3: Case-III --- Comparison of Results

Methods

Total

Generation

Cost ($)

Computational

Time (s)

%age

Improvement

in Generation

Cost

HCWCA 921,775.23 9.2 -

WCA 921,893.16 9.1 0.01%

RCGA-AFSA [71] 922,339.63 11.0 0.06%

RCGA [71] 923,966.28 17.0 0.24%

IPSO [149] 925,978.84 31.11 0.46%

DE [100] 929,755.94 45.0 0.87%

IV. Test System 3: Case IV --- Non-Convex Fuel Cost Function with




prohibited discharge zones of hydroelectric units have also been taken into account.






system. It can be seen that the convergence was very slow till 300 iterations and it is

almost constant till 500 iterations. The algorithm was tested for more than 500

generations but no significant improvement was found. The improvement in the cost

is up to 0.45% as per the comparison table. This improvement means a total savings

of approx. 4118$ per day which accumulates to more than 1.5 million $ per annum.

This much improvement is a very significant improvement.



Table 5.22 Test System 3: Case-IV --- Optimal Hydroelectric Discharges

Hour



1 5.8142 6.8037 29.8371 13.0001

2 10.8908 6.9534 29.3767 13.0536

3 6.1568 8.7367 29.8518 13.1648

4 12.2892 6.9937 29.1735 13.1214

5 7.6950 6.9453 29.9621 13.0606

6 7.1054 6.0016 29.4462 13.0022

7 9.5523 10.2411 21.9962 13.0465

8 7.3380 9.2134 16.3681 13.1673

9 7.8685 6.0263 29.5177 13.0911

10 6.2435 6.9254 16.0785 15.1633

11 7.9997 6.4898 21.9963 13.0784

12 7.7275 6.2374 14.2962 15.9236

13 9.6708 6.6387 11.6826 15.6700

14 10.1709 13.9791 10.6334 16.4018

15 7.9207 6.3898 12.0182 13.2520

16 7.0167 6.8485 11.8707 14.4702

17 11.0563 12.1864 12.2669 15.4483

18 7.9173 12.1553 10.1027 15.0618

19 7.6388 11.8766 10.2934 19.1263

20 7.2887 12.1817 10.2563 15.5592

21 7.4445 13.3912 10.1045 20.2998

22 10.6862 6.6087 10.2191 22.2318

23 5.4257 6.0753 10.0334 22.0956

24 6.0824 6.1009 11.6257 22.3346



Table 5.23 Test System 3: Case-IV --- Optimal Hydroelectric & Thermal Powers

Hour


Thermal

Generation

(MW)

Total

Generation


1 60.2943 55.2269 0.0000 200.0946 1054.3842 1370

2 90.6298 56.8315 0.0000 188.1521 1054.3866 1390

3 63.0588 67.7402 0.0000 174.8170 1054.3840 1360

4 93.8304 58.4072 0.0000 157.2958 980.4666 1290

5 72.3035 58.6829 0.0000 178.5505 980.4631 1290

6 68.3232 52.7132 0.0000 197.6205 1091.3431 1410

7 82.2105 75.2134 0.0000 216.4335 1276.1426 1650

8 69.9487 68.7883 18.9100 233.5709 1608.7821 2000

9 74.0743 50.4002 0.0000 248.0240 1867.5014 2240

10 63.7250 57.7561 14.0486 280.0090 1904.4613 2320

11 77.2603 56.2115 0.0000 265.9868 1830.5413 2230

12 75.9415 55.3895 16.7135 294.4536 1867.5018 2310

13 87.9487 58.9008 24.1901 302.3394 1756.6211 2230

14 90.9317 89.8759 27.5615 308.9262 1682.7047 2200

15 78.3953 55.9257 30.4432 282.5348 1682.7010 2130

16 72.2209 59.6830 33.6161 295.6973 1608.7828 2070

17 95.9546 83.7104 38.3203 303.2325 1608.7822 2130

18 78.5283 79.4187 39.8570 296.4553 1645.7407 2140

19 76.4885 74.9474 42.9879 325.9152 1719.6610 2240

20 73.7711 72.7944 46.3502 293.5033 1793.5810 2280

21 74.8370 72.6722 49.3922 323.4373 1719.6614 2240

22 93.0622 44.1640 52.1726 321.8149 1608.7863 2120

23 58.6663 41.9328 54.0506 308.3280 1387.0224 1850

24 64.7026 43.3981 58.3234 295.2705 1128.3054 1590




Fig. 5.4 Test System 3: Case-I --- Convergence Characteristics

Fig. 5.5 Test System 3: Case-II --- Convergence Characteristics

915000

920000

925000

930000

935000

940000

0 100 200 300 400 500

Gen

erat

ion

Co

st (

$)

No. of Iterations

HCWCA WCA

915000

920000

925000

930000

935000

940000

0 100 200 300 400 500

Gen

erat

ion

Co

st (

$)

No. of Iterations

HCWCA WCA



Fig. 5.6 Test System 3: Case-III --- Convergence Characteristics

Fig. 5.7 Test System 3: Case-IV --- Convergence Characteristics

920000

922000

924000

926000

928000

930000

932000

934000

936000

938000

940000

0 100 200 300 400 500

Gen

erat

ion

Co

st (

$)

No. of Iterations

HCWCA WCA

920000

922000

924000

926000

928000

930000

932000

934000

936000

938000

940000

0 100 200 300 400 500

Gen

erat

ion

Co

st (

$)

No. of Iterations

HCWCA WCA



Table 5.24 Test System 3: Case-IV --- Comparison of Results

Methods

Total

Generation

Cost ($)

Computational

Time (s)

%age

Improvement

in Generation

Cost

HCWCA 921,860.93 9.3 -

WCA 922,884.40 9.2 0.11%

TLBO [131] 924,550.78 --- 0.29%

MHDE [100] 925,547.31 9.0 0.40%

IPSO [149] 925,978.84 31.1 0.45%


This test system consists of above mentioned four multi-chain hydroelectric units

and three different thermal units. Three different case studies have been performed

by selecting four different characteristics and constraints of the hydroelectric and/or

thermal unit. The power generation coefficients, power generation limits, water

discharge limits, prohibited discharge zones, reservoir storage limits, initial and end

conditions of reservoirs and hourly inflows of hydroelectric units have been shown

in Table 5.10 and fuel cost coefficients and generation limits of thermal units and

hourly load demand are shown in Table 5.25.


5.26.

The four different case studies investigated are as described below:

I. Test System 4: Case I --- Non-Convex Fuel Cost Function without

Transmission Losses



transmission losses of the network have been ignored. The results obtained by the

proposed HCWCA and WCA for the optimal water discharges have been presented in

Table 5.27 and for the optimal hydroelectric powers and thermal power have been



presented in Table 5.28. Table 5.29 gives the comparison of the obtained results

using HCWCA and WCA with other methods available in the literature. Fig. 5.8 shows

the convergence characteristics of this system. It can be seen that the convergence

was very fast till 100 iterations and then it is almost constant till 500 iterations. The

algorithm was tested for more than 500 generations but no significant improvement

was found. The improvement in the cost is up to 3.58% as per the comparison table.

This improvement means a total savings of approx. 1448$ per day which accumulates

to more than 0.5 million $ per annum.

II. Test System 4: Case II --- Non-Convex Fuel Cost Function with

Transmission Losses

In this case, the valve point effect of thermal unit is assumed to be a non-convex

function and the transmission losses of the network have also been taken into

account. The results obtained by the proposed HCWCA and WCA for the optimal

water discharges have been presented in Table 5.30 and for the optimal hydroelectric




system. It can be seen that the convergence was very slow and solution has

converged in about 400 iterations and after that it is almost constant till 500




which accumulates to more than 0.5 million $ per annum.




Demand


Unit 𝒂𝒊 𝒃𝒊 𝒄𝒊 𝒅𝒊 𝒆𝒊 𝑷𝒔𝒎𝒊𝒏 𝑷𝒔

𝒎𝒂𝒙

1 100 2.45 0.0012 160 0.038 20 175

2 120 2.32 0.0010 180 0.037 40 300

3 150 2.10 0.0015 200 0.035 50 500

Hourly Load Demand


1 750 9 1090 17 1050

2 780 10 1080 18 1120

3 700 11 1100 19 1070

4 650 12 1150 20 1050

5 670 13 1110 21 910

6 800 14 1030 22 860

7 950 15 1010 23 850

8 1010 16 1060 24 800


Hydroelectric

Units

Thermal



Generations

No. of

Runs WCA HCWCA

4 3 100 8 0.01 Adaptive 500 30




Hour



1 6.7790 6.3390 29.9908 10.2694

2 10.5048 7.1457 18.5215 7.5155

3 7.8079 6.4916 17.2487 10.2078

4 9.6028 7.0442 29.9840 8.3623

5 9.0029 8.4422 29.9395 7.7044

6 12.7935 10.3449 29.8984 9.2882

7 7.1998 7.2971 25.7488 17.3183

8 6.1078 6.7490 11.3730 12.1055

9 10.4688 7.0873 29.1210 10.8142

10 11.4440 6.1354 10.2672 13.7222

11 9.0851 9.4272 14.2501 13.7919

12 11.4567 9.7743 17.1953 10.3613

13 9.3836 7.7905 12.6787 15.2936

14 9.6063 6.2277 15.8748 15.7624

15 5.1285 6.0345 13.2339 16.6692

16 5.7569 8.9492 10.6716 19.1865

17 7.1771 9.0028 10.3481 17.5926

18 5.1783 6.2384 14.4410 19.9514

19 7.5631 12.1381 10.0002 19.5502

20 5.9672 13.1586 12.8639 19.2402

21 5.0737 7.8650 12.6839 17.2698

22 7.8568 8.2244 12.3318 19.0327

23 5.1480 12.9351 12.2014 19.4409

24 6.7790 6.3390 29.9908 10.2694



Table 5.28 Test System 4: Case-I --- Optimal Hydroelectric & Thermal Powers

Hour

Hydroelectric Generations (MW) Thermal Generations

(MW)

Total

Gener

ation

(MW) 𝑷𝒉𝟏 𝑷𝒉𝟐 𝑷𝒉𝟑 𝑷𝒉𝟒 𝑷𝒔𝟏 𝑷𝒔𝟐 𝑷𝒔𝟑

1 67.63 52.35 0.00 176.34 103.31 209.93 140.44 750

2 88.85 58.24 40.09 139.57 103.28 209.82 140.15 780

3 74.44 55.32 41.96 160.83 102.65 124.96 139.84 700

4 83.86 60.12 0.00 131.51 20.01 125.06 229.44 650

5 79.72 68.48 0.00 147.12 20.15 124.89 229.65 670

6 90.98 76.24 0.00 174.88 102.82 125.36 229.73 800

7 67.11 58.85 0.00 247.43 137.09 209.95 229.58 950

8 60.18 55.42 40.03 222.40 102.80 209.96 319.20 1010

9 85.55 58.18 0.00 224.61 102.76 209.82 409.07 1090

10 88.87 53.32 37.13 268.76 102.76 209.91 319.25 1080

11 80.12 73.21 36.87 278.15 102.61 209.82 319.23 1100

12 89.57 73.88 30.99 239.09 102.52 294.79 319.17 1150

13 81.71 63.11 42.53 303.40 175.00 124.86 319.39 1110

14 83.68 54.40 40.93 304.32 102.63 124.85 319.19 1030

15 54.52 54.58 48.11 310.97 102.43 209.82 229.58 1010

16 60.65 72.56 50.22 329.82 102.56 125.05 319.15 1060

17 72.13 71.70 50.62 313.59 102.66 209.72 229.58 1050

18 56.20 54.35 50.49 327.53 102.53 209.71 319.18 1120

19 75.31 82.19 51.66 319.23 102.53 209.67 229.42 1070

20 63.01 81.73 54.22 309.21 102.68 209.63 229.51 1050

21 55.38 59.51 55.01 288.63 102.22 209.67 139.59 910

22 77.71 62.11 56.72 296.55 102.45 124.83 139.64 860

23 56.35 78.87 58.08 289.17 102.80 124.87 139.86 850

24 85.34 70.57 58.66 284.40 25.80 45.66 229.57 800





Methods

Total

Generation

Cost ($)

Computational

Time (s)

%age

Improvement

in Generation

Cost

HCWCA 40,408.29 19.6 -

WCA 40,878.06 19.2 1.16%

RCGA-AFSA [71] 40,913.82 21.0 1.25%

ACABC [132] 41,074.42 16.0 1.65%

MHDE [100] 41,856.50 31.0 3.58%


Hour



1 9.1602 6.1652 21.0710 6.1177

2 7.9710 7.6371 18.2182 8.2176

3 6.3674 7.1777 29.9984 6.3458

4 9.6776 9.0357 29.7288 7.6336

5 11.6287 7.2665 29.0309 8.8523

6 7.2960 7.6390 12.1250 9.5574

7 8.5301 6.0739 16.0064 6.6658

8 5.5044 6.2244 28.8517 7.8700

9 5.7445 6.5063 16.8290 10.3586

10 5.5371 7.2840 13.9238 8.5507

11 7.6378 7.3359 13.7359 17.2223

12 6.2258 6.3117 27.9721 19.6849

13 8.3313 8.0353 13.5440 17.0720

14 9.1914 6.7421 15.4957 18.0912

15 5.7815 7.2231 10.9692 18.0019

16 9.4729 8.7901 12.9051 19.6359

17 8.5725 8.5942 10.6803 18.8511

18 6.3726 10.7765 10.3947 18.2640

19 10.5522 10.684 11.1650 19.8893

20 9.2024 11.6353 11.7047 18.7456

21 6.0337 9.2803 10.0171 16.4457

22 7.0439 12.7648 12.0972 19.7836

23 9.2040 7.9576 11.9198 19.6587

24 13.9612 14.8594 12.1691 19.6343



Table 5.31 Test System 4: Case-II --- Optimal Hydroelectric & Thermal Powers

Hour


(MW)

Total

Generat

ion


1 82.12 51.24 35.93 130.50 102.22 123.71 229.68 755.39

2 75.48 61.15 44.40 151.48 102.19 209.77 141.43 785.91

3 64.58 59.32 0.00 123.57 103.22 124.56 230.06 705.30

4 84.76 70.26 0.00 131.83 101.69 125.44 140.79 654.76

5 90.32 60.30 0.00 158.75 102.22 124.04 139.93 675.54

6 68.96 62.27 43.59 175.43 103.84 123.56 229.62 807.26

7 76.42 51.83 41.39 159.63 102.42 208.89 318.20 958.78

8 56.33 53.32 0.00 191.67 102.13 209.54 408.34 1021.34

9 59.29 56.11 34.21 236.13 102.71 207.58 408.69 1104.73

10 58.58 62.23 40.07 214.48 102.76 293.79 319.35 1091.26

11 75.65 63.47 40.01 309.38 100.36 207.99 321.34 1118.18

12 65.50 57.41 0.00 335.63 101.33 292.44 317.26 1169.58

13 81.43 68.71 36.72 314.91 102.38 294.34 229.11 1127.60

14 87.24 61.52 33.61 320.19 100.78 124.46 319.51 1047.32

15 62.36 65.63 40.86 316.09 100.10 124.18 319.07 1028.27

16 89.68 74.82 42.93 335.01 102.13 293.51 139.80 1077.89

17 84.11 72.88 44.21 325.03 102.83 208.67 229.38 1067.12

18 67.54 81.21 46.56 318.34 102.83 291.40 229.51 1137.38

19 95.27 78.56 49.29 322.15 102.79 209.08 230.40 1087.54

20 87.46 80.06 50.77 309.10 103.29 205.46 229.85 1065.99

21 64.48 69.43 52.38 286.31 102.16 207.64 141.68 924.08

22 72.84 81.06 56.58 301.04 102.45 120.88 139.69 874.53

23 87.72 59.77 57.82 291.31 20.00 207.60 139.43 863.64

24 105.53 80.73 58.73 282.39 20.64 124.50 140.67 813.18





Methods

Total

Generation

Cost ($)

Computational

Time (s)

%age

Improvement

in Generation

Cost

HCWCA 41,223.41 19.8

WCA 41,969.59 19.2 1.81%

RCGA_AFSA [71] 41,707.96 25.0 1.18%

SPPSO [90] 42,470.23 32.7 3.02%

MHDE [100] 42,679.87 40.0 3.53%

RCGA [71] 43,465.24 32.0 5.44%

DE [100] 45,773.99 110.0 11.04%

III. Test System 4: Case III --- Non-Convex Fuel Cost Function with

Transmission Losses, Ramp Rates and Prohibited Discharge Zones


function and the transmission losses of the network have also been taken into

account. Further to make the system more complex, the ramp rates of thermal units

and prohibited discharge zones of hydroelectric units have also been considered. The

results obtained by the proposed HCWCA and WCA for the optimal water discharges

have been presented in Table 5.33 and for the optimal hydroelectric powers and

thermal power have been presented in Table 5.34. Table 5.35 gives the comparison

of the obtained results using HCWCA and WCA with other methods available in the

literature. Fig. 5.10 shows the convergence characteristics of this system. It can be

seen that the convergence was very slow and solution has converged in about 400

iterations and after that it is almost constant till 500 iterations. The algorithm was

tested for more than 500 generations but no significant improvement was found. The

improvement in the cost is up to 3.39% as per the comparison table. This

improvement means a total savings of approx. 1435$ per day which accumulates to

more than 0.5 million $ per annum.



IV. Test System 4: Case IV --- Non-Convex Fuel Cost Function with



function. Further to make the system more complex, the prohibited operating zones

of thermal units have also been considered. This inclusion of POZ in SHTCP has not

been investigated before. The results obtained by the proposed HCWCA and WCA for

the optimal water discharges have been presented in Table 5.36 and for the optimal

hydroelectric powers and thermal power have been presented in Table 5.37. Table

5.38 shows the obtained results using HCWCA and WCA.

Table 5.33 Test System 4: Case-III --- Optimal Hydroelectric Discharges

Hour



1 6.8028 8.8220 29.8579 9.5061

2 7.9817 6.0000 18.3888 9.4091

3 7.6056 6.4529 29.9950 6.2401

4 6.3271 8.7068 19.6457 7.2625

5 6.3433 6.7556 16.3698 7.5456

6 6.2246 6.7905 18.1134 11.1770

7 13.7545 8.3807 17.8447 11.3264

8 7.8303 6.4930 18.9666 12.5233

9 6.1944 7.0000 19.5876 11.7858

10 6.3435 6.0130 21.0360 12.3215

11 6.7484 8.4866 18.8007 10.6298

12 7.1038 13.8202 14.4040 19.8261

13 13.0290 10.5977 11.2743 15.4765

14 7.9493 8.0726 12.9170 19.7211

15 7.1739 11.6904 15.8454 15.0174

16 12.9077 8.2603 14.3692 19.9571

17 5.7551 6.7305 11.6048 14.5923

18 12.9236 10.6338 13.5133 15.6159

19 5.9438 9.2348 12.6438 14.7649

20 5.1068 6.1311 11.3818 15.1212

21 10.6439 8.1471 17.1788 14.2019

22 7.3857 13.1528 15.2584 19.9869

23 9.4414 9.4859 13.5829 19.7546

24 7.4800 6.1416 21.0184 19.5965



Table 5.34 Test System 4: Case-III --- Optimal Hydroelectric & Thermal Powers

Hour


(MW)

Total

Generat

ion


1 67.80 66.12 0.00 168.79 104.18 209.19 140.55 756.63

2 76.19 49.69 40.67 160.44 102.51 125.95 230.78 786.24

3 73.82 54.32 0.00 117.79 102.10 124.02 233.19 705.24

4 64.64 68.49 27.35 123.02 101.68 127.01 142.30 654.49

5 64.70 57.25 39.41 148.95 102.50 123.11 139.07 674.99

6 63.92 57.60 33.28 197.51 102.47 123.64 229.56 807.97

7 97.33 66.10 34.35 216.73 102.78 124.02 319.42 960.72

8 74.37 54.53 28.19 235.66 101.88 207.18 319.92 1021.72

9 63.52 58.50 26.30 231.79 21.88 291.31 409.92 1103.22

10 65.58 53.34 17.10 242.33 101.12 205.47 409.98 1094.91

11 69.57 69.62 25.66 228.74 102.88 292.94 323.97 1113.38

12 72.68 87.77 39.13 311.36 38.47 210.19 409.07 1168.67

13 102.13 75.09 43.29 282.01 92.74 209.34 320.36 1124.96

14 79.06 62.80 45.04 315.91 99.61 124.08 320.83 1047.33

15 73.80 78.26 45.92 281.92 102.35 120.66 322.14 1025.06

16 103.28 62.03 50.42 315.71 101.26 125.43 319.14 1077.27

17 62.07 52.76 52.86 270.70 98.08 208.66 318.77 1063.90

18 102.85 70.86 56.19 277.71 102.25 295.02 229.16 1134.03

19 63.52 62.80 56.84 270.96 102.40 209.17 319.66 1085.34

20 56.06 45.54 57.89 273.59 101.71 208.64 320.70 1064.11

21 94.09 58.84 54.15 262.80 100.77 123.47 227.64 921.76

22 74.80 76.77 57.93 301.09 100.65 121.30 141.69 874.24

23 88.02 61.93 59.97 292.17 100.71 123.04 137.54 863.37

24 75.75 43.68 40.35 282.18 102.89 124.06 143.43 812.34




Table 5.35 Test System 4: Case-III --- Comparison of Results

Methods

Total

Generation

Cost ($)

Computational

Time (s)

%age

Improvement

in Generation

Cost

HCWCA 42,355.35 19.8 -

WCA 43,115.71 19.5 1.80%

IDE [107] 43,790.33 782.23 3.39%

Fig. 5.8 Test System 4: Case-I --- Convergence Characteristics

40000

41000

42000

43000

44000

45000

46000

47000

48000

0 100 200 300 400 500

Gen

erat

ion

Co

st (

$)

No. of Iterations

HCWCA WCA



Table 5.36 Test System 4: Case-IV --- Optimal Hydroelectric Discharges

Hour



1 5.0103 6.0000 29.5252 13.3981

2 7.1888 8.2885 26.7761 13.3184

3 11.9472 6.0000 29.9838 13.1660

4 9.9895 9.0541 11.3526 13.0836

5 6.4686 8.2378 30.0000 14.0148

6 8.6070 9.7071 27.8333 13.0000

7 10.9664 11.4117 14.9482 13.3027

8 7.8103 10.0824 15.0476 13.3313

9 8.2406 6.0000 24.0899 13.3846

10 5.0056 7.7401 16.0297 18.1216

11 11.6702 8.8029 11.6835 13.4537

12 8.5749 7.6543 10.0000 13.4767

13 8.5120 6.1852 21.8140 13.4030

14 9.1899 6.6983 28.4031 13.4794

15 10.3274 9.3520 10.8574 14.5786

16 5.3511 9.5653 11.3490 13.0000

17 6.0807 8.4938 19.6320 13.0000

18 9.6845 6.0000 11.1128 13.0000

19 5.0000 11.7568 10.0000 13.0000

20 6.9227 9.9680 11.8180 21.5425

21 13.5748 7.9183 10.6608 16.6495

22 6.6275 7.6433 17.0494 24.5054

23 6.6172 10.2660 10.0000 21.8985

24 5.6330 9.1744 10.0000 15.9477



Table 5.37 Test System 4: Case-IV --- Optimal Hydroelectric & Thermal Powers

Hour


(MW)

Total

Generat

ion


1 53.54 50.16 0.00 203.13 20.00 300.00 123.17 750.00

2 71.43 64.73 0.00 189.60 20.00 300.00 134.23 780.00

3 94.52 51.69 0.00 173.98 20.00 300.00 59.80 700.00

4 85.90 70.76 43.41 156.15 20.00 223.78 50.00 650.00

5 64.36 66.06 0.00 183.84 20.00 285.74 50.00 670.00

6 78.05 72.29 0.00 192.40 20.00 294.79 142.47 800.00

7 87.93 76.10 35.91 213.87 20.00 300.00 216.20 950.00

8 72.67 68.38 37.33 211.97 20.00 300.00 299.64 1010.00

9 75.90 46.09 0.00 229.80 32.11 295.94 410.16 1090.00

10 53.28 58.23 35.79 277.67 20.00 298.31 336.72 1080.00

11 94.12 64.40 44.22 241.24 24.77 300.00 331.26 1100.00

12 80.21 58.06 44.16 242.91 20.00 300.00 404.66 1150.00

13 80.48 49.61 16.66 251.67 20.00 300.00 391.57 1110.00

14 85.14 54.49 0.00 254.60 21.28 294.79 319.71 1030.00

15 91.16 69.93 45.53 262.64 21.17 298.51 221.07 1010.00

16 58.00 70.00 47.68 244.76 20.00 300.00 319.56 1060.00

17 64.61 63.39 30.60 252.14 21.67 300.00 317.58 1050.00

18 89.33 47.64 49.03 263.91 20.00 300.00 350.09 1120.00

19 54.99 74.98 49.65 262.36 20.00 300.00 308.02 1070.00

20 71.38 66.28 53.19 321.10 20.00 292.31 225.74 1050.00

21 102.03 56.37 52.82 291.18 20.59 300.00 87.02 910.00

22 68.40 55.65 49.15 321.80 20.00 295.00 50.00 860.00

23 68.65 67.75 55.36 298.36 20.00 289.88 50.00 850.00

24 60.80 61.74 56.06 257.51 20.00 293.89 50.00 800.00




Table 5.38 Test System 4: Case-IV --- Comparison of Results

Methods

Total

Generation

Cost ($)

Computational

Time (s)

%age

Improvement in

Generation Cost

HCWCA 44,041.12 20.5 -

WCA 44,911.93 20.3 1.98%


This is a comparatively larger system than the previous systems and consists of

above mentioned four multi-chain hydroelectric units and six different thermal units.

The power generation coefficients, power generation limits, water discharge limits,

prohibited discharge zones, reservoir storage limits, initial and end conditions of

reservoirs and hourly inflows of hydroelectric units have been shown in Table 5.10

and fuel cost coefficients and generation limits of thermal units and hourly load

demand are shown in Table 5.39.


5.40.

Fig. 5.9 Test System 4: Case-II --- Convergence Characteristics

41000

42000

43000

44000

45000

46000

47000

48000

49000

50000

0 100 200 300 400 500

Gen

erat

ion

Co

st (

$)

No. of Iterations

HCWCA WCA



Fig. 5.10 Test System 4: Case-III --- Convergence Characteristics

I. Test System 5: Case-I: Non-Convex Fuel Cost Function without


In this case, the fuel cost has been considered as non-convex but the prohibited

operating zones of thermal units have not been considered. The results obtained

by the proposed HCWCA and WCA for the optimal water discharges have been

presented in Table 5.41 and for the optimal hydroelectric powers and thermal

powers have been presented in Table 5.42 and Table 5.43 respectively. Table 5.44

gives the comparison of the obtained results using HCWCA and WCA with other

methods available in the literature. Fig. 5.11 shows the convergence characteristics

of this system. It can be seen that the convergence was relatively faster and solution

converged in about 200 iterations and after that it is almost constant till 500




which accumulates to an approx. 0.8 million $ per annum.

II. Test System 5: Case-II: Non-Convex Fuel Cost Function with Prohibited

Operating Zones

In this case, the fuel cost has been considered as non-convex and the prohibited

operating zones of thermal units have also considered. The results obtained by

41700

42700

43700

44700

45700

46700

47700

48700

49700

0 100 200 300 400 500

Gen

erat

ion

Co

st (

$)

No. of Iterations

HCWCA WCA



the proposed HCWCA and WCA for the optimal water discharges have been

presented in Table 5.45 and for the optimal hydroelectric powers and thermal

powers have been presented in Table 5.46 and Table 5.47 respectively. Table 5.48

shows the obtained results using HCWCA and WCA.


Demand



𝒎𝒂𝒙

1 150 1.89 0.0050 300 0.035 40 415

2 115 2.00 0.0055 200 0.042 35 350

3 40 3.50 0.0060 200 0.042 35 425

4 122 3.15 0.0050 150 0.063 35 410

5 125 3.05 0.0050 150 0.063 50 450

6 120 2.75 0.0070 150 0.063 75 550

Hourly Load Demand


1 1270 9 1640 17 1330

2 1290 10 1520 18 1540

3 1260 11 1330 19 1340

4 1190 12 1310 20 1280

5 1190 13 1430 21 1540

6 1310 14 1500 22 1120

7 1450 15 1130 23 1450

8 1800 16 1270 24 1590


Hydroelectric

Units

Thermal



Generations

No. of

Runs WCA HCWCA

4 6 100 8 0.01 Adaptive 500 30




Hour



1 5.0646 6.7934 29.8550 13.0563

2 8.4799 7.1062 26.5394 13.0163

3 12.4767 6.0002 27.9914 13.0002

4 6.6646 6.9833 29.1730 13.0316

5 7.1852 6.6761 29.7237 13.2507

6 10.5738 7.8460 29.8220 13.1593

7 9.1716 10.6486 28.7560 13.0232

8 7.0984 6.7698 29.9334 14.5365

9 6.8659 7.4005 24.2984 14.2371

10 10.8260 8.9881 11.2361 13.0639

11 7.6358 6.0868 10.1016 13.1848

12 7.4984 10.7647 10.6408 13.0725

13 8.8089 7.1589 11.2334 15.3215

14 7.9604 9.1370 10.1175 13.8271

15 9.1869 9.4994 17.2396 13.8010

16 10.8822 10.6043 12.0102 13.8993

17 8.3578 8.6544 10.5154 14.8993

18 9.6289 9.4340 10.6594 17.0427

19 9.2927 9.9231 10.1255 15.2547

20 6.2369 7.8050 10.0554 21.7698

21 7.8239 11.1323 10.7923 18.9041

22 6.3412 7.4741 10.5117 16.8374

23 5.8436 11.1643 10.0811 24.8188

24 5.0957 7.9496 10.3605 20.8193



Table 5.42 Test System 5: Case-I --- Optimal Hydroelectric Powers

Hour

Hydroelectric Generations (MW)

𝑷𝒉𝟏 𝑷𝒉𝟐 𝑷𝒉𝟑 𝑷𝒉𝟒

1 54.0102 55.1648 0.0000 200.5300

2 79.6275 57.7434 0.0000 187.8086

3 95.2375 51.9064 0.0000 173.6432

4 66.3193 59.7927 0.0000 156.9029

5 69.7600 58.4440 0.0000 180.2084

6 87.1518 65.5446 0.0000 195.6069

7 80.2646 77.4312 0.0000 211.1655

8 68.3442 56.1582 0.0000 239.9824

9 67.5683 60.6053 0.0000 252.4324

10 89.7398 69.8843 19.4293 255.3340

11 74.1072 53.3965 21.7990 268.2659

12 73.7560 78.5603 26.5223 278.0119

13 82.6634 59.6355 30.8302 307.4650

14 78.1745 71.1905 33.2958 290.1709

15 86.1753 72.7640 25.4400 287.6958

16 94.4837 76.3340 37.8595 286.7587

17 81.0773 65.6585 41.2889 294.7573

18 88.2110 67.5148 45.3104 309.5978

19 85.8382 67.8680 47.6906 294.8291

20 64.9680 56.8051 50.0156 335.6845

21 76.5872 71.6435 53.2872 310.2571

22 65.9362 54.4093 54.5525 289.1874

23 62.2200 70.5534 55.0829 316.4503

24 55.9156 55.1096 56.6987 288.5998



Table 5.43 Test System 5: Case-I --- Optimal Thermal Powers

Hour

Thermal Generations (MW)

𝑷𝒔𝟏 𝑷𝒔𝟐 𝑷𝒔𝟑 𝑷𝒔𝟒 𝑷𝒔𝟓 𝑷𝒔𝟔

1 309.320 259.640 35.000 127.622 153.712 75.000

2 312.721 258.667 35.008 128.778 154.647 75.000

3 394.128 254.193 35.000 35.001 145.891 75.000

4 317.638 261.783 35.001 123.582 93.982 75.000

5 311.328 275.620 35.000 35.000 149.640 75.000

6 317.969 349.516 35.000 35.000 149.212 75.000

7 399.114 259.643 35.000 109.367 203.015 75.000

8 308.694 342.330 173.515 223.878 312.099 75.000

9 407.596 341.074 35.000 99.353 301.371 75.000

10 401.948 335.796 35.000 78.351 159.518 75.000

11 316.472 259.587 35.000 131.970 94.402 75.000

12 308.995 259.991 35.000 35.000 139.163 75.000

13 306.884 259.518 35.000 132.718 140.286 75.000

14 411.235 186.357 35.002 182.372 137.202 75.001

15 125.651 337.274 35.000 35.000 50.000 75.000

16 209.953 276.756 35.000 35.000 142.855 75.000

17 305.555 346.663 35.000 35.000 50.000 75.000

18 395.513 189.508 35.001 133.365 200.979 75.000

19 307.337 341.438 35.000 35.000 50.000 75.000

20 222.333 259.512 35.000 35.004 145.677 75.001

21 400.616 343.087 35.000 35.001 139.521 75.000

22 118.624 342.284 35.000 35.004 50.002 75.000

23 215.502 340.215 35.000 35.000 244.977 75.000

24 407.106 335.269 35.000 72.265 209.037 75.000



Methods

Total

Generation

Cost ($)

Computational

Time (s)

%age

Improvement

in Generation

Cost

HCWCA 104,137.48 37.2

WCA 104,772.63 36.4 0.61%

SOHPSO-TVAC [86] 104,232.38 76.25 0.09%

PSO [86] 106,322.23 --- 2.10%




Hour



1 8.8440 6.1148 30.0000 13.0769

2 12.7982 8.0604 29.9999 13.1304

3 5.1381 8.7486 29.9997 13.0000

4 8.1162 10.1375 29.5312 13.0001

5 12.3805 6.6539 29.8205 13.0000

6 5.3767 7.3875 15.5109 13.0937

7 7.2262 8.2970 29.9409 13.0000

8 7.6244 8.0935 13.9304 13.1714

9 11.5150 7.1736 26.6129 14.6941

10 8.1462 6.9537 10.0708 14.7560

11 8.0423 6.5093 16.6204 13.7986

12 10.8294 8.3756 11.9308 14.9153

13 11.3815 6.5434 11.2546 17.8600

14 10.2319 8.1338 13.7353 14.8890

15 8.3764 8.1448 13.9713 13.6729

16 5.8065 7.6192 15.6431 13.0322

17 9.4743 12.8964 11.5381 14.0045

18 6.0892 8.5991 17.6207 15.6219

19 5.4182 12.0281 11.6892 20.5898

20 5.7526 10.0837 10.4664 13.4261

21 9.2310 8.8194 10.0925 20.5743

22 5.3218 7.0129 10.1043 17.1474

23 5.8101 7.8582 10.3034 19.5795

24 6.0693 11.7556 10.4066 23.6531



Table 5.46 Test System 5: Case-II --- Optimal Hydroelectric Powers

Hour



1 80.4917 50.9122 0.0000 200.6893

2 94.3809 63.4748 0.0000 188.6248

3 53.7539 67.5420 0.0000 173.4712

4 75.5685 73.8494 0.0000 156.5140

5 91.2084 54.8638 0.0000 178.4851

6 54.3701 59.4182 30.2909 199.4435

7 68.5582 63.4758 0.0000 217.1282

8 71.6388 61.6121 31.5423 234.9726

9 89.8641 56.5125 0.0000 263.1907

10 75.3504 56.3243 31.6266 264.4300

11 75.9115 54.7914 24.9139 268.6227

12 89.8928 66.3112 35.8304 278.7695

13 91.7852 55.6330 39.0064 311.3373

14 87.8831 66.2671 40.2567 281.6410

15 78.8686 66.8218 43.3529 271.7558

16 61.1179 63.8436 41.6441 264.1910

17 86.1407 84.4933 47.8955 272.2704

18 63.6874 64.5120 38.2459 286.2102

19 58.1944 76.3284 49.3691 316.9083

20 61.1272 67.1280 51.0172 261.5283

21 85.2141 61.0582 51.9035 309.9218

22 57.4208 51.8558 53.9235 287.6019

23 62.0337 57.1433 56.0193 296.4442

24 64.5915 72.7632 56.7749 299.9166



Table 5.47 Test System 5: Case-II --- Optimal Thermal Powers

Hour


𝑷𝒔𝟏 𝑷𝒔𝟐 𝑷𝒔𝟑 𝑷𝒔𝟒 𝑷𝒔𝟓 𝑷𝒔𝟔

1 397.0637 345.8420 35.0000 35.0000 50.0011 75.0000

2 318.3104 188.3483 35.0000 230.0000 96.8609 75.0000

3 310.2986 185.7822 35.0000 183.0433 176.1088 75.0000

4 310.0001 339.8175 35.0000 35.0000 89.2505 75.0000

5 310.0474 260.7534 35.0000 35.0000 149.6419 75.0000

6 389.3271 336.2751 36.6791 35.0022 92.3517 76.8420

7 396.2499 349.5879 35.0000 35.0000 210.0000 75.0000

8 398.5626 277.7456 107.7929 129.2749 411.8581 75.0000

9 399.0908 336.7960 35.0221 35.7110 346.9697 76.8430

10 395.2731 340.2599 36.7356 35.0001 210.0000 75.0000

11 373.8559 334.2199 35.0000 36.3694 51.3153 75.0000

12 310.0000 334.1961 35.0000 35.0000 50.0000 75.0000

13 399.1289 335.6580 35.2131 35.0135 52.2246 75.0000

14 398.5792 261.6092 35.2032 121.1573 132.4032 75.0000

15 140.0000 334.2009 35.0000 35.0000 50.0000 75.0000

16 310.0000 334.2034 35.0000 35.0000 50.0000 75.0000

17 310.0000 334.2001 35.0000 35.0000 50.0000 75.0000

18 398.3463 185.5342 35.0000 183.4640 210.0000 75.0000

19 310.0000 334.1998 35.0000 35.0000 50.0000 75.0000

20 310.0000 334.1992 35.0000 35.0000 50.0000 75.0000

21 310.0015 343.5114 35.0000 35.0000 233.3896 75.0000

22 140.0000 334.1980 35.0000 35.0000 50.0000 75.0000

23 395.0372 347.8367 35.0000 35.0000 90.4857 75.0000

24 397.8016 343.1523 35.0000 35.0000 210.0000 75.0000



Methods

Total

Generation

Cost ($)

Computational

Time (s)

%age

Improvement in

Generation Cost

HCWCA 105,031.25 41.1

WCA 109,061.43 40.3 3.84%




This is an even larger test system than all the previous systems and is consisted of

above mentioned four multi-chain hydroelectric units and ten different thermal

units. The power generation coefficients, power generation limits, water discharge

limits, prohibited discharge zones, reservoir storage limits, initial and end conditions

of reservoirs and hourly inflows of hydroelectric units have been shown in Table 5.10

and fuel cost coefficients and generation limits of thermal units and hourly load



5.50.


Demand



𝒎𝒂𝒙

1 150 1.89 0.0050 300 0.035 50 455

2 115 2.00 0.0055 200 0.042 50 450

3 40 3.50 0.0060 200 0.042 20 130

4 122 3.15 0.0050 150 0.063 20 130

5 125 3.05 0.0050 150 0.063 25 470

6 120 2.75 0.0070 150 0.063 40 460

7 70 3.45 0.0070 200 0.053 45 465

8 70 3.45 0.070 150 0.063 35 300

9 130 2.45 0.0050 180 0.043 25 160

10 130 2.45 0.0050 100 0.062 25 180

Hourly Load Demand


1 1750 9 2090 17 2050

2 1780 10 2080 18 2120

3 1700 11 2100 19 2070

4 1650 12 2150 20 2050

5 1670 13 2110 21 1910

6 1800 14 2030 22 1860

7 1950 15 2010 23 1850

8 2010 16 2060 24 1800




Hydroelectric

Units

Thermal



Generations

No. of

Runs WCA HCWCA

4 10 150 12 0.01 Adaptive 500 30



powers and thermal powers have been presented in Table 5.52 and Table 5.53

respectively. Table 5.54 gives the comparison of the obtained results using

HCWCA and WCA with other methods available in the literature. Fig. 5.12 shows

the convergence characteristics of this system. It can be seen that the convergence

was relatively faster and solution converged in about 200-300 iterations and after

that it is almost constant till 500 iterations. The algorithm was tested for more than

500 generations but no significant improvement was found. The improvement in the

cost is up to 11.99% as per the comparison table.



Table 5.51 Test System 6 --- Optimal Hydroelectric Discharges

Hour



1 11.8225 7.3331 28.4824 6.8302

2 5.1381 6.6465 29.2865 8.4334

3 11.6367 9.4206 29.7962 11.3030

4 6.7947 6.3629 21.6090 7.2910

5 6.0665 6.2293 24.2271 7.9197

6 7.3870 6.0000 14.5882 12.8064

7 5.5084 11.3072 29.0013 9.9354

8 7.3355 6.5687 18.8073 7.8995

9 10.6651 6.5971 21.0732 9.7132

10 5.2990 6.1321 16.7815 11.4270

11 5.6569 6.5157 20.6203 15.9604

12 11.6119 6.1187 13.2652 20.0000

13 10.3459 6.3163 17.6205 20.0000

14 11.8619 8.8771 11.0971 9.2367

15 5.7985 9.8573 13.3234 20.0000

16 6.6657 8.9381 14.0416 20.0000

17 8.1874 9.2286 11.9413 17.9882

18 6.2542 13.3279 11.9742 20.0000

19 10.3551 13.6548 11.0013 20.0000

20 8.9497 11.4697 10.7510 20.0000

21 9.5858 11.3291 10.0000 19.7324

22 5.3169 6.1853 13.4313 19.6822

23 11.0384 11.5838 15.6129 19.9721

24 5.7181 6.0000 10.4415 19.9575



Table 5.52 Test System 6 --- Optimal Hydroelectric Powers

Hour



1 92.2189 58.3370 0.0000 139.1975

2 54.1266 54.6598 0.0000 153.1030

3 91.7396 70.7919 0.0000 173.4223

4 66.5654 53.9878 11.8803 121.8130

5 61.0458 54.0330 0.0000 150.5006

6 70.5788 52.9381 40.0377 219.8839

7 57.0090 79.3583 0.0000 207.1084

8 71.3858 54.1983 19.2996 191.4012

9 89.4875 55.1996 2.8170 224.7310

10 56.4754 53.5591 24.3136 247.8532

11 60.4570 57.5780 5.7219 304.7224

12 96.6427 55.7473 31.5517 336.4577

13 91.5755 58.0014 19.4261 337.3359

14 97.7651 74.2731 36.2888 231.5662

15 62.1152 78.9651 37.7695 343.7912

16 69.6335 73.6407 39.3845 338.4956

17 80.9994 73.9626 42.8252 323.1212

18 66.3984 85.7686 45.1243 330.7610

19 93.4354 81.8237 47.1293 324.8711

20 85.1965 72.5602 48.6231 319.3890

21 88.3072 70.2885 51.6744 310.2506

22 57.8203 45.0333 56.2687 302.2341

23 95.4342 70.6304 55.7058 294.4841

24 61.5533 42.7000 56.8315 284.1710



Table 5.53 Test System 6 --- Optimal Thermal Powers

Hour


𝑷𝒔𝟏 𝑷𝒔𝟐 𝑷𝒔𝟑 𝑷𝒔𝟒 𝑷𝒔𝟓 𝑷𝒔𝟔 𝑷𝒔𝟕 𝑷𝒔𝟖 𝑷𝒔𝟗 𝑷𝒔𝟏𝟎

1 399.44 351.83 123.06 27.74 87.44 44.23 45.00 46.33 156.32 178.85

2 246.08 411.93 63.01 104.52 179.82 40.11 45.00 92.61 155.43 179.59

3 231.63 437.67 22.77 56.12 123.89 40.00 45.00 68.61 158.36 180.00

4 233.78 426.06 23.09 91.88 162.49 40.00 45.00 35.00 158.45 180.00

5 319.42 427.89 31.30 80.41 83.24 40.00 45.00 37.16 160.00 180.00

6 241.20 431.78 20.00 89.68 180.23 40.00 45.00 35.00 157.91 175.77

7 412.13 348.29 94.08 62.80 73.06 147.50 45.00 83.67 160.00 180.00

8 318.28 429.79 84.32 82.78 282.09 67.80 45.00 35.35 151.42 176.89

9 328.97 342.66 95.21 74.84 365.35 40.00 45.00 85.74 160.00 180.00

10 411.20 349.55 95.75 78.96 71.15 86.61 45.00 237.11 159.78 162.69

11 425.44 418.86 76.60 60.30 95.64 79.47 45.61 129.60 160.00 180.00

12 325.06 351.83 81.78 99.66 201.17 46.22 45.59 138.49 159.79 180.00

13 406.00 432.14 20.00 69.82 227.82 40.00 45.00 35.00 147.88 180.00

14 365.75 413.78 87.76 115.94 157.68 40.00 45.00 35.00 158.36 170.84

15 391.34 350.77 21.87 111.14 170.83 40.00 45.00 35.00 155.04 166.37

16 296.71 404.52 82.00 69.63 224.72 40.00 45.00 37.30 160.00 178.96

17 318.56 429.14 65.25 118.05 124.19 40.80 45.00 54.02 154.08 180.00

18 325.81 408.58 99.38 119.32 64.81 85.99 45.00 103.90 159.15 180.00

19 336.66 282.59 26.54 94.73 323.73 40.00 45.00 84.42 155.15 133.93

20 410.27 279.76 92.22 57.67 224.30 40.00 45.00 35.00 160.00 180.00

21 142.57 353.92 22.44 43.04 367.51 40.00 45.00 35.00 160.00 180.00

22 410.39 275.12 20.00 62.39 170.75 40.00 45.00 35.00 160.00 180.00

23 249.44 429.49 20.00 57.28 120.85 40.00 45.00 35.00 156.69 180.00

24 329.81 368.77 21.96 116.74 58.44 40.00 45.00 35.00 159.04 180.00



Methods

Total

Generation

Cost ($)

Computational

Time (s)

%age

Improvement

in Generation

Cost

HCWCA 175,010.84 27.3 -

WCA 179,005.39 27.0 2.28%

DE [99] 196,000.01 --- 11.99%



Fig. 5.11 Test System 5 --- Convergence Characteristics


102000

104000

106000

108000

110000

112000

114000

0 100 200 300 400 500

Gen

erat

ion

Co

st (

$)

No. of Iterations

HCWCA WCA

174000

176000

178000

180000

182000

184000

186000

188000

190000

0 100 200 300 400 500

Gen

erat

ion

Co

st (

$)

No. of Iterations

HCWCA WCA




This test system consists of four multi-chain hydroelectric units and ten different

thermal units. In this test system, the SHTCP is formulated by taking into account two

decision variables simultaneously i.e. water discharge as continuous and thermal

states as binary. The cooling and banking constraints of thermal units have been

taken into account. The power generation coefficients, power generation limits,

water discharge limits, prohibited discharge zones, reservoir storage limits, initial

and end conditions of reservoirs and hourly inflows of hydroelectric units have been

shown in Table 5.10 and fuel cost coefficients, generation limits, minimum up & down

times, start-up costs, start-up hours and initial states of thermal units and hourly load



5.56.



powers and thermal powers have been presented in Table 5.58 and Table 5.59

respectively. Table 5.60 gives the comparison of the obtained results using

HCWCA and WCA with other methods available in the literature.




Demand


Unit 𝒂𝒊 𝒃𝒊 𝒄𝒊 𝑷𝒔𝒎𝒊𝒏 𝑷𝒔

𝒎𝒂𝒙

1 1000 16.19 0.00048 150 592

2 970 17.26 0.00031 150 592

3 700 16.60 0.00200 20 169

4 680 16.50 0.00211 20 169

5 450 19.70 0.00398 25 211

6 370 22.26 0.00712 20 104

7 480 27.74 0.00079 20 114

8 660 25.92 0.00413 10 72

9 665 27.27 0.00222 10 72

10 670 27.79 0.00173 10 72

II. Minimum Up & Down Times, Start Up Costs, Start Up Hours and Initial Status

Unit MUT MDT CSC HSC CSH IS

1 8 8 4500 9000 5 8

2 8 8 5000 1000 5 8

3 5 5 550 1100 4 -5

4 5 5 560 1120 4 -5

5 6 6 900 1800 4 -6

6 3 3 170 340 2 -3

7 3 3 260 520 2 -3

8 1 1 30 60 0 -1

9 1 1 30 60 0 -1

10 1 1 30 60 0 -1

Hourly Load Demand


1 1370 9 2240 17 2130

2 1390 10 2320 18 2140

3 1360 11 2230 19 2240

4 1290 12 2310 20 2280

5 1290 13 2230 21 2240

6 1410 14 2200 22 2120

7 1650 15 2130 23 1850

8 2000 16 2070 24 1590




Hydroelectric

Units

Thermal



Generations

No. of

Runs WCA HCWCA

4 10 150 12 0.01 Adaptive 1000 30

Table 5.57 Test System 7 --- Optimal Hydroelectric Discharges

Hour



1 6.6788 6.0091 29.9911 13.0015

2 7.3759 6.0000 29.9985 13.0004

3 7.6136 6.0931 29.9996 13.0002

4 7.9615 6.0653 29.9986 13.0041

5 8.6959 6.1416 29.9998 13.0003

6 9.0485 6.9499 29.9945 13.0006

7 8.4549 6.1855 29.9979 13.0064

8 9.5916 6.6075 29.8361 13.0131

9 9.9282 8.0642 12.5722 13.0062

10 10.7023 11.1818 12.0851 16.8618

11 8.8994 8.6236 12.4175 13.0016

12 9.2438 9.9435 13.0249 14.1036

13 9.7698 9.7419 13.1565 15.8654

14 9.0850 9.0168 13.4803 13.6055

15 7.9746 8.8839 13.2200 13.4545

16 8.1855 9.0359 12.4484 14.8539

17 8.0069 9.1750 12.6223 16.2629

18 7.6637 11.1029 11.7792 16.7003

19 7.3504 10.4307 10.1353 17.8121

20 7.6868 12.0976 10.0468 19.4444

21 7.5356 12.2416 10.0000 20.4546

22 7.5059 7.3897 10.0133 21.5198

23 5.0033 6.6788 10.0284 20.6498

24 5.0382 8.3399 10.0035 21.9819



Table 5.58 Test System 7 --- Optimal Hydroelectric Powers

Hour



1 66.9084 50.2237 0.0000 200.1051

2 72.3253 51.2909 0.0000 187.7563

3 74.0487 53.5475 0.0000 173.7323

4 76.1035 54.8861 0.0000 156.8164

5 79.8121 56.3751 0.0000 178.7254

6 81.1180 62.0168 0.0000 198.9438

7 77.6270 56.6227 0.0000 217.4751

8 83.5490 59.7895 0.0000 234.2815

9 85.1770 69.1922 11.7397 249.2295

10 88.6181 83.8374 14.9988 295.9238

11 81.0808 71.4993 17.4520 271.5425

12 83.1947 77.3497 20.8222 293.3396

13 86.2754 75.3340 26.2376 309.2945

14 83.5481 71.5767 29.3778 285.0481

15 77.5274 70.9233 34.0803 282.7590

16 79.2896 71.1385 38.2979 296.4329

17 78.3144 70.5675 40.7202 307.9204

18 76.0410 76.1913 43.6541 309.5192

19 73.7602 71.0103 45.3123 315.3364

20 75.8170 74.4174 47.4709 321.5350

21 74.6632 72.3860 50.1300 320.7651

22 74.5512 51.0974 52.4477 316.8620

23 54.7390 47.5103 54.5348 301.1529

24 55.3788 57.3195 56.0666 293.8450



Table 5.59 Test System 7 --- Optimal Thermal Powers

Hour


𝑷𝒔𝟏 𝑷𝒔𝟐 𝑷𝒔𝟑 𝑷𝒔𝟒 𝑷𝒔𝟓 𝑷𝒔𝟔 𝑷𝒔𝟕 𝑷𝒔𝟖 𝑷𝒔𝟗 𝑷𝒔𝟏𝟎

1 592.00 460.76 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00

2 592.00 486.63 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00

3 592.00 466.67 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00

4 592.00 410.19 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00

5 592.00 383.09 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00

6 592.00 475.92 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00

7 592.00 592.00 0.00 114.28 0.00 0.00 0.00 0.00 0.00 0.00

8 592.00 592.00 169.00 169.00 100.38 0.00 0.00 0.00 0.00 0.00

9 592.00 592.00 169.00 169.00 211.00 91.66 0.00 0.00 0.00 0.00

10 592.00 592.00 169.00 169.00 211.00 103.62 0.00 0.00 0.00 0.00

11 592.00 592.00 169.00 169.00 211.00 55.43 0.00 0.00 0.00 0.00

12 592.00 592.00 169.00 169.00 21s1.00 102.29 0.00 0.00 0.00 0.00

13 592.00 592.00 169.00 169.00 210.86 0.00 0.00 0.00 0.00 0.00

14 592.00 592.00 169.00 169.00 208.45 0.00 0.00 0.00 0.00 0.00

15 592.00 592.00 169.00 169.00 142.71 0.00 0.00 0.00 0.00 0.00

16 592.00 592.00 169.00 169.00 62.84 0.00 0.00 0.00 0.00 0.00

17 592.00 592.00 169.00 169.00 110.48 0.00 0.00 0.00 0.00 0.00

18 592.00 592.00 169.00 169.00 112.59 0.00 0.00 0.00 0.00 0.00

19 592.00 592.00 169.00 169.00 192.58 20.00 0.00 0.00 0.00 0.00

20 592.00 592.00 169.00 169.00 211.00 27.76 0.00 0.00 0.00 0.00

21 592.00 592.00 169.00 169.00 180.06 20.00 0.00 0.00 0.00 0.00

22 592.00 592.00 169.00 169.00 103.04 0.00 0.00 0.00 0.00 0.00

23 592.00 592.00 39.06 169.00 0.00 0.00 0.00 0.00 0.00 0.00

24 592.00 535.39 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00



Methods

Total

Generation

Cost ($)

Computational

Time (s)

%age

Improvement

in Generation

Cost

HCWCA 701,024.03 35.2 -

WCA 702,498.98 36.1 0.21%

MB-EPSO [91] 705,329.29 --- 0.61%




This test system has been formulated by replicating the thermal units as in Test

System 6 and thus the total system consists of same four multi-chain hydroelectric

units and twenty different thermal units. The data of the system is same as of Test

System 6 with the difference that thermal units have been increased to twenty by

replicating those ten thermal units and the load demand has also been doubled for

every time interval. The evolution model i.e. the parameter setting for Test System 8

is shown in Table 5.61.

Table 5.62 gives the comparison of the obtained results using HCWCA and WCA.


Hydroelectric

Units

Thermal



Generations

No. of

Runs WCA HCWCA

4 20 200 16 0.01 Adaptive 1000 30


Methods

Total

Generation

Cost ($)

Computational

Time (s)

%age

Improvement

in Generation

Cost

HCWCA 412,540.58 29.1 -

WCA 417,024.80 28.7 1.09%


This test system has been formulated by replicating the thermal units as in Test

System 8 and thus the total system consists of same four multi-chain hydroelectric

units and forty different thermal units. The data of the system is same as of Test

System 8 with the difference that thermal units have been increased to forty by

replicating those twenty thermal units and the load demand has also been doubled

for every time interval. The evolution model i.e. the parameter setting for Test System

9 is shown in Table 5.63.

Table 5.64 gives the comparison of the obtained results using HCWCA and WCA.




Hydroelectric

Units

Thermal



Generations

No. of

Runs WCA HCWCA

4 40 200 16 0.01 Adaptive 1000 30


Methods

Total

Generation

Cost ($)

Computational

Time (s)

%age

Improvement

in Generation

Cost

HCWCA 882,433.90 29.8 -

WCA 888,500.62 29.3 6.87%


700000

705000

710000

715000

720000

0 200 400 600 800 1000

Gen

erat

ion

Co

st (

$)

No. of Iterations

HCWCA WCA





410000

415000

420000

425000

430000

435000

0 200 400 600 800 1000

Gen

erat

ion

Co

st (

$)

No. of Iterations

HCWCA WCA

880000

885000

890000

895000

900000

905000

910000

915000

920000

0 200 400 600 800 1000

Gen

erat

ion

Co

st (

$)

No. of Iterations

HCWCA WCA

Chapter 6 Multi-Objective Hydrothermal Coordination using Proposed HCWCA


CHAPTER NO. 6 MULTI-OBJECTIVE

HYDROTHERMAL COORDINATION USING PROPOSED

HCWCA

6.1 MULTI-OBJECTIVE HYDROTHERMAL COORDINATION PROBLEM

In many of the developed countries the concerns regarding environment are

increasing a lot and the authorities are trying their best to make the harmful

emissions zero. The harmful emissions released by the burning of fossil fuels is the

root cause of the increase in temperature of the atmosphere known as global

warming effect. To control these harmful emissions, the power producing companies

are also directed to keep their emissions within certain permissible limits. Hence, the

power producing companies have to face a multi-objective problem of optimization

of fuel emissions in addition to that of optimization of generation cost. This multi-

objective optimization problem solved with SHTCP leads to MOSHTCP.

6.2 LITERATURE REVIEW --- MOSHTCP

In 2001, J. Dhillon et al. [151], introduced fuzzy decision making for multi-objective

long-term hydrothermal forecasting. Uncertainties like nitrogen emission, power

production cost data, power demand and water inflows were also considered. The

problem formulated was a tri-objective in which fuel cost, load demand and nitrogen

emission are minimized. A specific technique was used to convert stochastic models

into deterministic models. Decomposition approach was used to reduce the

complexity of the problem.

In 2002 J. Dhillon et al. [152], solved the fixed head SHTCP using fuzzy logic while

considering five objectives namely cost, NOx, SO2 & CO2 emission and variance of

generation mismatch. It was hard to find a tradeoff between these five contradicting

objectives. FL was used to choose the weighting patterns which in turn locate a

compromise between the objectives.



In 2004, M. Basu [153] used an evolutionary programming based interactive fuzzy

fulfilling technique for MOSHTCP while considering fuel cost and emission as

objectives. The author first converted this multi objective problem into a mini-max

problem so that EP can be used to solve the problem. A cascaded multi-reservoir

hydroelectric system with water discharge rate limitations, net head and water

transport delay was considered. The technique performed better in attaining

optimality. A similar EP based technique; fuzzy satisfying technique was also utilized

by the author to solve MOSHTCP [93]. The results were compared with other

techniques.

In 2006 [154], fixed head SHTCP was solved using FL approach. Three objective

functions namely fuel cost, CO2 emissions and SO2 emissions were considered. The

tradeoff among these objectives was found using weighting method. After the

compromised solution was found fuzzy decision approach was used for locating the

minima.

In 2007 M. Basu [155], PSO based interactive fuzzy satisfying method for MOSHTCP

of fixed head thermal units and hydroelectric units with non-smooth fuel rates and

discharge level functions is investigated. The multi-objective problem is changed into

a mini-max problem, which is then handled by the PSO method. The results obtained

from the proposed technique are evaluated to those establish by interactive fuzzy

satisfying technique based on evolutionary programming method.

In 2009 Ozyon et al. [156] solved environmental economic power dispatch problem

of a hydrothermal power system using GA based technique. Reduction of NOx and

total thermal cost were considered as objectives of the study. The multi-objective

problem was converted into a single objective optimization problem by using

Weighted Sum Method (WSM). The GA was used for solving the single objective ED

optimization problem.

In 2011, M. Basu [157] solved the problem of ED of fixed head hydrothermal system

using Non-dominated sorting GA-II. The problem was formulated as multi-objective,

constrained and non-linear. All the constraints were considered and the real coded

GA was used. The reduction of NOx, SOx and fuel cost were taken as objectives. The



results from proposed technique were compared with previous techniques like

multi-objective DE and some others, and found better.

In 2010, Akbari Foroud [158] used the efficient PSO technique for solving a multi-

objective optimization problem of short term economic emission hydrothermal

scheduling. The problem was converted into a single objective problem by the

weighted sum method. Some decision variables were used to tune the weighing

factors until a desired solution was obtained. The star topology and random topology

were combined in order to guide the particles in searching. This topology resulted in

overall better search capability and convergence. The proposed technique was tested

on a hydrothermal system with four cascaded hydro and three thermal units and the

results were compared with some other techniques. The robustness and

effectiveness of the proposed technique was verified by these results.

In 2010, Lu S. et al. [159] solved the multi-objective problem of short term combined

economic emission hydrothermal scheduling using an improved Quantum behaved

PSO with Differential Mutation (QPSO-DM). The pollutant emission and fuel cost

were taken as objectives considering many constraints. Differential Mutation was

combined with Quantum based PSO to form this novel technique. In differential

mutation, the population was diversified by combining the simple arithmetic

operations with the classical evolution operator mutation. The technique was applied

on various test systems and the results were compared with other techniques. The

proposed method converged quickly and gave better quality solutions.

In 2010, C. Sun and S. Lu [160] improved the previously presented QPSO-DM

technique. Heuristic strategies were employed and the many constraints were

considered. Different tests on hydrothermal systems with economic emission

dispatch were carried out and the test results were compared with other techniques

to prove the improvement of the proposed technique.

In 2011 K. Mandal and N. Chakraborty [161] afterwards solved the combined

economic emission dispatch problem using an efficient PSO based algorithm. The

problem considered had cascaded reservoirs and the fuel cost and pollutant emission

were taken as objectives. The proposed technique was tested on a hydrothermal



system having four cascaded hydro and three thermal units and the results were

compared with other EP methods.

2012 K. K. Mandal et al. afterwards developed an efficient PSO based algorithm [162].

The algorithm was applied to cascaded reservoirs for combined economic emission

scheduling. For problem formulation the cost and emission were considered. The

algorithm was evaluated on a system with four cascaded hydro and three thermal

units, and was also compared with other evolutionary programming methods.

In 2012, J. Sasikala and M. Ramaswamy [163] solved the problem of hydrothermal

economic emission dispatch by using a PSO based technique. The proposed

technique had less number of variables and improved results were shown on three

test systems.

In 2009, the MOSHTCP was solved by K. Mandal and N. Chakraborty [164] using a DE

based algorithm while considering many inequality and equality constraints. The

authors also considered the water transport delay between connected reservoirs.

Fuel cost and pollutant emission were taken as objectives of the problem. The

MOSHTCP was converted into a single objective problem using a penalty factor

approach. The performance of the proposed algorithm was evaluated on a sample

test systems with cascaded multi-reservoir hydroelectric units and three thermal

units. The results were compared with other techniques like fuzzy satisfying

evolutionary programming.

In 2010, H. Qin et al. [165] used the Multi-Objective DE with Adaptive Cauchy

Mutation (MODE-ACM) for solving MOSHTCP. The fuel cost and pollutant emission

were taken as objectives of the problem. The valve point loading effect and water

transport delay were also considered in problem formulation. An adaptive cauchy

mutation was used for preventing the premature convergence. The results of the

proposed technique were compared with several previous techniques and it was

found that the results were superior with small computation time.

In 2011, S. Lu and C. Sun [166] proposed a Quadratic Approximation based DE with

Valuable Trade-off approach (QADEVT) for solving the MOSHTCP. This study

employed heuristic rules for handling the water dynamic balance constraints and

active power balance constraints were handled by heuristic strategies based on



priority list. For satisfying the reservoir storage volume constraints, a feasibility-

based selection technique was introduced. Premature convergence can also be

avoided using these techniques.

In 2013, H. Zhang et al. [167] solved the SHTCP by Multi-Objective DE with three

Chaotic Sequences (CS-MODE). The technique employed elitist archive mechanism

for keeping the non-dominated individuals to improve the convergence of DE. The

equality and inequality constraints were handled by a heuristic two-step constraint-

handling technique. Furthermore, premature convergence was prevented by

integrating three chaotic mappings into DE. The compromised solution was then to

be selected from the non-dominated set. The simulation results of the proposed

technique also revealed the effectiveness and feasibility of proposed CS-MODE.

In 2013, H. Zhang et al. [168] used a Culture Belief based Multi-Objective Hybrid

Differential Evolution (CB-MOHDE) for solving SHTCP. The key knowledge source of

culture algorithm (CA) is cultural belief, which provides the computational model for

hybrid DE algorithm. Premature convergence was prevented using an adaptive

chaotic factor which was integrated into mutation mechanism. The coupled complex

constraints were handled using an iteration based constraint handling technique.

The results were compared with other techniques and it was proved that CB-MOHDE

can be a attractive alternative for solving SHTCP.

In 2013 H. Zhang et al. [169] developed a Simulated Annealing based Multi-Objective

Cultural DE (SAMOCDE) in order to solve MOSHTCP. Many constraints like line losses

and water transport delay were considered. The non-convex fuel cost and pollutant

emission were taken as objectives. The SA technique was combined with multi-

objective differential evolution. Premature convergence was avoided by proper

controlling of the population space evolution. The comparison of the proposed

technique was done to other alternatives, and the proposed algorithm performed

better which confirms that SA-MOCDE can be a robust and effective alternative to

solve MOSHTCP.

In 2015 Arnel Glotic et al. [170] solve the problem of short term combined economic

emission dispatch of hydrothermal system by using a surrogate differential evolution

technique. The effectiveness of the proposed technique was shown by studying it for



different problems like optimum load scheduling and combined economic emission

dispatch. The test system comprised of four hydro and three thermal units.

In 2014, H. Tian et al. [171] solved the MOSHTCP using a Non-dominated Sorting GSA

with Chaotic Mutation (NSGSA-CM). The proposed technique was used to optimize

the pollutant emission and fuel cost by introducing the concept of crowding distance

and non-dominated sorting. The authors introduced the concept of population social

information and particle memory character in velocity update process to improve the

performance. And premature convergence was avoided by using a chaotic mutation.

Furthermore, elitism strategy was also embedded in update process to select better

solutions in offspring and parent populations based on their crowding distance and

non-domination rank. The constraints were dealt without penalty factor approach.

The performance of the proposed technique was compared with other methods and

it was concluded that the proposed technique is efficient and feasible for solving

MOSHTCP.

In 2015, Chunlong Li et al. [172] proposed an Improved Multi-Objective GSA

(IMOGSA) in order to solve the short term economic environmental hydrothermal

coordination problem. The proposed technique was tested on two unique test

systems to confirm its usefulness. The results were compared with other algorithms

and the proposed technique was found to give better results.

In 2016, N. Gouthamkumar et al. [173] MOSHTCP using non-dominated sorting

disruption based GSA. The authors included the valve point effect and transmission

losses in their case study. The fixed head and variable head MOSHTCP has been

successfully solved and tested on different standard test systems.

In 2011, I.A. Farhat and M. E. El-Hawary [174] solved the complex and dynamic

problem of STHTC with environmental impacts by using Improved Bacterial

Foraging Algorithm (IBFA). The fuel cost as well as fuel emissions were to be

minimized. So this multi-objective and many constraints made this problem very

unique. The authors considered linear fuel costs of thermal units with real-time

constraints. The proposed technique was tested on a hydrothermal test system

having two hydro and two thermal machines. According to authors IBFA was found

to be promising for solving the bi-objective SHTCP.



In 2011 I. A. Farhat and M. E. El-Hawary [175] solved the dynamic multi-constrained

non-linear multi-objective SHTCP using an Improved Bacterial Foraging Algorithm

(IBFA). The authors introduced critical improvements in basic BFA like changes in

chemo-taxis step. Fuel cost and pollutant emission were to be minimized at the same

time. The authors adapted weighting factors to have an acceptable tradeoff among

both objective functions. The algorithm was tested on a hydrothermal system

containing two hydro and two thermal machines.

In 2012, N. Narang et al. [176] proposed an integrated PPO to solve the fixed head

MOSHTCP. The constraints were handled without using the penalty factor approach.

The proposed algorithm was then tested on three hydrothermal test systems

considering transmission losses and valve-point loading effect for thermal units. The

authors claimed that the proposed technique yields better quality solution while

satisfying all constraints.

In 2014, short term scheduling of hydrothermal energy system was done by J. Zhou

et al. [177] using a Multi-Objective Artificial Bee Colony (MOABC) algorithm. The

problem was formulated as non-linear short-term hydrothermal coordination

problem with combined economic emission dispatch with a group of difficult

constraints. The worker selection of ABC algorithm was improved to adapt the multi-

objective problem. Furthermore, the local search ability of the algorithm is improved

using a progressive optimality algorithm. The performance of the proposed

algorithm was verified on three different hydrothermal test systems and the results

were compared with other possible algorithms. The results showed that the

proposed technique can perform better with less fuel cost and environment

pollution.

In 2015, Abdollah Ahmadi et al. [178] solved the MOSHTCP using lexicographic

optimization and Normal Boundary Intersection (NBI). The authors tested the

proposed model on a hydrothermal test system of three thermal and four cascaded

hydroelectric units. The proposed technique was also applied on IEEE 118 bus test

system. The results showed that the proposed technique is competent in solving

MOSHTCP when compared to newly employed techniques.



In 2014, N. Narang et al. [179] solved the MOSHTCP, by using the weighting

technique. In the proposed technique sufficient weights were to be decided so that

no critical data was lost. Ordinary factual measures were used for figuring out the

weight examples. A hybrid search technique was also proposed which includes PPO

and PPS to get the optimum solution. Basically, the pattern search techniques were

employed to begin the searching from the nearby best arrangements acquired by PPO

strategy.

In 2014, [150] SHTCP was solved using Quasi-Opposition based Learning (QOBL)

combined with Real Coded Chemical Reaction Optimization (RCCRO) to form

Oppositional Real Coded Chemical Reaction Optimization (ORCCRO). OBL uses

converse numbers instead of asymmetrical numbers in population initialization to

develop the population rapidly. It was seen that the ORCCRO gave better results while

considering different constraints which proved that the proposed technique is

competent for handling SHTCP.

In 2013, A. Immanuel Selvakumar [180] solved the MOSHTCP using Civilized Swarm

Optimization (CSO) which is the hybrid of PSO and Society Civilization Algorithm

(SCA). CSO was formed by embedding the communication strategy of CSA in food

searching strategy of PSO. The problem was formulated by considering economic and

emission as objectives. Pareto-optimal front are found using a new ideal guide.

Cascaded multi-reservoir hydroelectric units with nonlinear characteristics and

thermal units with nonlinear cost curves were considered to analyze the algorithm.

Other constraints like water availability, water transport delay, power loss, storage

conformity and operating limits were also fully accounted in this work. The algorithm

was tested on two hydrothermal test systems and the results were compared with

other techniques. The authors concluded that the proposed technique outperformed

all the previous approaches.

6.3 MATHEMATICAL FORMULATION OF MOSHTCP

The mathematical problem formulation of MOSHTCP is studies under three different

case studies, (i) Economic Cost Coordination (ii) Economic Environmental

Coordination (iii) Economic Cost & Environmental Coordination.



6.3.1 Economic Cost Coordination (ECC)

For a given hydrothermal energy system, the objective of pure economic cost

coordination (ECC) problem is the minimization of total fuel cost of thermal units. All

the mathematical equations can be seen in section 2.1.

6.3.2 Economic Environmental Coordination (EEC)

The objective of economic environmental coordination (EEC) problem is to minimize

the amount of harmful emissions from thermal units due to burning of fossil fuels

used for generation of electricity. The emissions released by thermal unit can be

formulated as summation of an exponential function with a quadratic one [181]. The

EEC problem is written mathematically as:

𝑚𝑖𝑛𝑖𝑚𝑖𝑧𝑒 𝐸 = ∑∑𝑒𝑖𝑡(𝑃𝑠𝑖𝑡)

𝑁𝑠

𝑖=1

𝑇

𝑡=1

(6.1)

where, 𝑒𝑖𝑡(𝑃𝑠𝑖𝑡) is the fuel emissions caused by the 𝑖𝑡ℎ thermal unit and it is defined

as:

𝑒𝑖𝑡(𝑃𝑠𝑖𝑡) = 𝛼𝑠𝑖 + 𝛽𝑠𝑖𝑃𝑠𝑖𝑡 + 𝛾𝑠𝑖𝑃𝑠𝑖𝑡2 + 𝜂𝑠𝑖exp (𝛿𝑠𝑖𝑃𝑠𝑖𝑡) (6.2)

where 𝛼𝑖, 𝛽𝑖, 𝛾𝑖, 𝜂𝑖 , & 𝛿𝑖 are the emission coefficients of 𝑖𝑡ℎ thermal unit.

6.3.3 Economic Cost & Environmental Coordination

The combined economic cost & environmental coordination problem (ECEC) seeks a

trade-off relation between generation cost and fuel emissions. Emission coordination

discussed in section 6.2.2 above is incorporated in the conventional SHTCP by the

addition of fuel emission minimization objective function of Eq. 6.1 in conventional

ECC. This becomes a multi-objective ECEC problem, converted into a single one by

adapting a cost penalty approach as follows [182]:

𝑀𝑖𝑛 𝑇𝐶 = ∑∑[𝑓𝑖𝑡(𝑃𝑠𝑖𝑡) + 𝐶𝑃𝐹𝑡 × 𝑒𝑖𝑡(𝑃𝑠𝑖𝑡) ]

𝑁𝑠

𝑖=1

𝑇

𝑡=1

(6.3)

where, 𝐶𝑃𝐹𝑡 is the cost penalty factor at time interval 𝑡.



The trade-off relation between generation cost and fuel emissions is developed as:

𝑀𝑖𝑛 𝑇𝐶 = ∑∑[𝐾1 × 𝑓𝑖𝑡(𝑃𝑠𝑖𝑡) + 𝐾2 × 𝐶𝑃𝐹𝑡 × 𝑒𝑖𝑡(𝑃𝑠𝑖𝑡) ]

𝑁𝑠

𝑖=1

𝑇

𝑡=1

(6.4)

where 𝐾1 , 𝐾2 are the weight factors.

The procedure of finding the cost penalty factors is given below:

i. Compute the average generation cost & average fuel emissions of each

generating unit at its maximum rated power.

ii. Obtain the ratio ℎ𝑠𝑖 by dividing the computed average generation cost with

the average emissions according to following equation as:

ℎ𝑠𝑖($

𝑙𝑏) =

𝐹(𝑃𝑠𝑖𝑚𝑎𝑥)/𝑃𝑠𝑖

𝑚𝑎𝑥

𝐸(𝑃𝑠𝑖𝑚𝑎𝑥)/𝑃𝑠𝑖

𝑚𝑎𝑥 (6.5)

iii. Re-arrange the computed values of ℎ𝑠𝑖 in an ascending order.

iv. Starting from the smallest ℎ𝑠𝑖 add full load capacity of each generating unit

one at a time until ∑𝑃𝑠𝑖𝑚𝑎𝑥 ≥ 𝑃𝐷𝑡 is achieved.

v. At this phase, ℎ𝑠𝑖 related with last unit in this process is the cost penalty factor

𝐶𝑃𝐹𝑡 for a given power demand at time 𝑡.

From above procedure it is obvious that the value of cost penalty factor 𝐶𝑃𝐹𝑡 is

dependent on the total power demand during each time interval 𝑡 and it varies

according to power demand.

6.4 TEST SYSTEM INVESTIGATED

The test system investigated in section 5.3.2.2 i.e. Test System 4 is the standard test

system for the MOSHTCP. The test system consists of a multi-chain of four

hydroelectric units and three thermal units. The power generation coefficients,

power generation limits, water discharge limits, prohibited discharge zones,

reservoir storage limits, initial and end conditions of reservoirs and hourly inflows

of hydroelectric units are the same as described in Table 5.10 and fuel cost

coefficients and generation limits of thermal units and hourly load demand are also



same as in Table 5.25. The data for the fuel emissions of thermal units is shown in

Table 6.1 and the evolution model i.e. the parameter setting for this Test System is

shown in Table 6.2.

Table 6.1 Test System 10 --- Emission Data of Thermal Units

I. Fuel Emission Coefficients

Unit 𝜶𝒊 𝜷𝒊 𝜸𝒊 𝜂𝒊 𝜹𝒊

1 60 -1.355 0.0105 0.4968 0.0192

2 45 -0.600 0.0080 0.4860 0.01694

3 30 -0.555 0.0120 0.5035 0.01478


Hydroelectric

Units

Thermal



Generations

No. of

Runs WCA HCWCA

4 3 100 8 0.01 Adaptive 500 30

The three different cases for this Test System 10 are discussed as below:

6.4.1 Economic Cost Coordination

In this case only the minimization of generation cost objective is considered.

Therefore, the value of weight factors will be 𝐾1 = 1, 𝐾2 = 0. For satisfaction of

active power balance constraint, the priority list of thermal units is same over the

whole scheduling horizon in this case. Table 6.3 shows the optimal discharges of

hydroelectric units. Table 6.4 shows the hourly optimal hydroelectric and thermal

power generation schedules obtained from the proposed HCWCA method.

6.4.2 Economic Environmental Coordination

In this case the objective is to only minimize the harmful emissions of thermal units.

The value of weight factors will be 𝐾1 = 0, 𝐾2 = 1/𝐶𝑃𝑓𝑡. In this case the priority

sequence of thermal plants is also same for whole scheduled period for the

satisfaction of active power balance constraint. Table 6.5 shows the optimal

discharges of hydroelectric units. Table 6.6 shows the hourly optimal hydroelectric

and thermal power generation schedules obtained from the proposed HCWCA

method.



6.4.3 Economic Cost & Environmental Coordination

In this case an amalgamated objective function with an attempt to optimize both

generation cost and harmful emissions is engaged. The value of weight factors for

this case is 𝐾1 = 1,𝐾2 = 1. The optimal hydroelectric discharges and optimal hourly

generation schedules of hydroelectric and thermal units for this case study are

presented in Table 6.7 and Table 6.8 respectively.

The generation cost and emissions for above three studies are collectively

summarized in Table 6.9. It is clearly seen from results that generation cost and fuel

emissions are contradictory to one another. In ECC problem the minimum generation

cost is achieved but the amount of emissions in this case is higher than EEC and ECEC

while in EEC the minimum emissions are obtained but the generation cost is higher

than ECC and ECEC. However, the ECEC yields an optimized better solution with a

reasonably reduced generation cost and reduced harmful emissions.

Table 6.3 Test System 10: ECC--- Optimal Hydroelectric Discharges

Hour



1 6.1046 8.1792 29.6402 9.1269

2 5.4318 6.1949 29.9045 6.3594

3 5.9362 6.7230 30.0000 6.2664

4 8.9485 7.4538 29.9646 8.3694

5 8.4847 6.5939 13.2366 6.0991

6 11.0949 7.2175 28.3448 8.8944

7 12.7148 10.2435 26.2268 11.3994

8 6.5577 7.6731 14.6733 9.3565

9 7.6597 10.6396 15.6895 14.0828

10 8.1283 11.7518 11.7939 11.0585

11 7.4684 7.3311 14.3385 14.7082

12 6.6907 11.6322 14.7312 18.3573

13 7.6446 6.4461 14.7778 15.4409

14 5.4657 7.5652 10.8462 17.4978

15 9.8851 9.1026 12.8334 19.8362

16 8.6102 7.6774 17.0467 19.3362

17 6.4993 8.0413 13.1449 19.9229

18 5.7685 7.7920 10.6561 19.4905

19 10.2888 8.9853 13.4212 19.9049

20 8.1666 8.5522 11.1023 19.5773



21 6.3243 6.4746 10.1603 16.5704

22 10.9906 7.1757 15.4951 18.2782

23 6.6369 9.2974 10.8587 19.8235

24 13.4993 13.2565 10.5476 19.4153

Table 6.4 Test System 10: ECC --- Optimal Hydroelectric & Thermal Powers

Hour


(MW)

Total

Gener

ation


1 62.59 62.94 0.00 164.89 109.05 211.02 139.51 750

2 57.65 51.33 0.00 127.21 20.00 296.29 227.52 780

3 62.26 56.26 0.00 121.49 105.78 214.04 140.17 700

4 82.95 61.90 0.00 138.48 20.00 296.67 50.00 650

5 79.56 57.03 35.39 133.41 20.00 294.61 50.00 670

6 90.92 61.06 0.00 187.45 24.95 295.32 140.30 800

7 93.69 75.27 0.00 232.02 106.27 213.12 229.63 950

8 64.86 61.09 28.01 221.59 104.05 210.71 319.70 1010

9 73.24 74.87 28.12 275.82 107.08 212.07 318.79 1090

10 77.05 77.38 36.51 252.91 20.00 297.24 318.91 1080

11 73.77 56.68 34.82 301.41 20.00 296.98 316.34 1100

12 68.67 75.55 36.72 333.06 21.55 295.41 319.04 1150

13 76.25 49.61 40.14 306.83 20.00 298.79 318.39 1110

14 59.39 57.73 44.69 322.01 20.00 297.23 228.95 1030

15 91.41 66.36 47.97 335.89 106.71 213.34 148.32 1010

16 84.04 58.55 39.64 328.60 106.87 212.98 229.33 1060

17 68.56 60.04 48.83 328.25 20.00 297.88 226.42 1050

18 62.24 57.41 50.82 317.66 20.00 297.09 314.78 1120

19 93.98 62.71 51.95 313.67 20.47 297.97 229.25 1070

20 81.00 60.03 52.61 309.15 105.20 212.66 229.35 1050

21 67.00 49.04 53.65 284.20 20.00 298.49 137.62 910

22 96.51 54.79 54.44 289.66 20.00 294.60 50.00 860

23 69.55 66.03 56.28 292.67 20.00 295.47 50.00 850

24 104.57 77.34 57.00 281.15 20.00 209.94 50.00 800


Total Fuel Emissions = 26,060.02 lbs.



Table 6.5 Test System 10: EEC--- Optimal Hydroelectric Discharges

Hour



1 8.2892 7.1573 30.0000 6.1427

2 6.1844 7.5938 29.9964 6.1544

3 9.5081 6.1081 29.9820 6.0450

4 5.9062 6.2439 29.9629 6.0717

5 7.8051 6.7318 29.9703 6.0741

6 8.9871 6.2913 29.9922 8.0334

7 11.8804 8.0337 29.7577 11.5543

8 9.7033 8.1483 13.4735 13.3032

9 8.2713 10.7655 14.8042 14.6578

10 8.3964 6.8008 11.1870 15.8722

11 8.9473 7.4121 15.9622 17.6727

12 9.3741 11.3317 12.8390 19.9830

13 11.8918 7.9828 11.4521 16.7531

14 8.3741 7.7955 12.6205 17.0243

15 7.4581 7.5702 14.2190 17.1806

16 10.5005 7.5293 12.7237 18.1009

17 5.8814 11.5377 10.9849 19.6596

18 9.6708 10.9204 11.3354 19.8057

19 5.4946 11.2345 10.6427 19.9033

20 7.9418 9.5109 10.6822 19.4710

21 6.4736 10.1611 11.3976 20.0000

22 6.2181 9.7637 10.5932 19.9931

23 6.6165 7.0792 11.7861 19.9881

24 5.2260 8.2962 15.9536 19.9438



Table 6.6 Test System 10: EEC --- Optimal Hydroelectric & Thermal Powers

Hour


(MW)

Total

Gener

ation


1 77.42 57.32 0.00 130.81 169.68 204.32 110.45 750

2 63.36 60.31 0.00 127.52 175.00 263.16 90.64 780

3 84.64 52.14 0.00 121.89 170.63 187.06 83.63 700

4 61.03 54.62 0.00 116.35 174.96 167.26 75.78 650

5 74.54 58.68 0.00 138.34 154.95 177.66 65.83 670

6 81.01 56.00 0.00 182.18 174.98 240.01 65.81 800

7 91.63 66.26 0.00 239.06 175.00 293.36 84.70 950

8 83.02 66.28 19.12 270.42 175.00 286.34 109.81 1010

9 75.98 77.92 18.71 294.91 175.00 299.97 147.51 1090

10 77.56 57.47 27.32 315.89 174.89 299.76 127.10 1080

11 81.68 62.35 20.39 340.02 174.98 296.16 124.43 1100

12 84.18 80.55 30.88 355.19 174.98 285.74 138.47 1150

13 94.35 63.96 35.31 326.56 174.96 300.00 114.85 1110

14 79.25 63.54 37.82 325.49 173.35 275.98 74.56 1030

15 74.02 62.96 40.92 326.13 175.00 232.53 98.44 1010

16 91.39 62.97 44.52 330.53 175.00 275.36 80.24 1060

17 62.25 80.43 46.50 336.37 174.97 275.08 74.41 1050

18 87.78 74.68 49.32 331.50 175.00 299.87 101.86 1120

19 58.90 72.93 49.83 327.26 174.90 292.86 93.33 1070

20 77.43 64.34 52.73 318.36 174.99 281.09 81.07 1050

21 66.85 66.53 55.00 313.25 142.07 187.27 79.03 910

22 65.07 64.15 56.15 304.53 160.77 140.44 68.89 860

23 68.59 50.05 58.39 294.63 157.97 156.37 64.00 850

24 57.12 57.08 56.62 284.10 164.10 130.98 50.00 800





Table 6.7 Test System 10: ECEC--- Optimal Hydroelectric Discharges

Hour



1 5.4881 6.1855 29.2250 6.0945

2 9.6407 6.0428 29.6236 6.2112

3 7.2166 7.8006 29.9063 7.0524

4 5.5513 6.1135 29.5655 6.0509

5 10.1502 8.9212 29.9292 7.3606

6 7.4044 6.4938 29.5976 11.4772

7 8.1696 7.3634 29.8869 8.3197

8 8.2896 8.3656 15.7577 11.5971

9 11.6955 8.2415 28.4685 17.4346

10 9.3504 6.1397 11.0267 16.4344

11 8.0399 7.5928 11.9844 15.9789

12 10.0015 8.0939 12.2512 19.9012

13 9.1895 9.2308 10.9264 14.5536

14 7.9284 6.3998 10.2971 19.3240

15 6.1006 7.2593 11.3488 18.0898

16 8.2267 10.1202 10.5459 19.7542

17 11.1537 8.9114 10.3155 17.7255

18 7.5194 8.6889 10.2112 17.3149

19 8.7415 13.0228 11.3725 19.9739

20 7.2189 14.0380 10.0207 19.8521

21 6.7188 6.4683 11.2368 19.0049

22 7.6700 12.1316 10.5670 19.8998

23 8.0226 10.8243 11.0427 19.8673

24 5.5121 7.5502 10.1521 19.7878



Table 6.8 Test System 10: ECEC --- Optimal Hydroelectric & Thermal Powers

Hour


(MW)

Total

Gener

ation


1 57.62 51.37 0.00 130.21 175.00 211.36 124.44 750

2 85.58 51.47 0.00 128.26 174.99 289.69 50.00 780

3 71.14 63.78 0.00 133.86 175.00 206.01 50.21 700

4 58.45 54.20 0.00 115.09 175.00 197.26 50.00 650

5 87.38 71.38 0.00 152.66 101.79 125.52 131.26 670

6 71.66 56.66 0.00 217.30 102.80 211.64 139.94 800

7 76.59 61.69 0.00 196.64 175.00 300.00 140.09 950

8 77.53 66.98 13.42 249.67 175.00 295.97 131.44 1010

9 92.75 66.12 0.00 317.22 175.00 300.00 138.91 1090

10 83.51 54.11 17.62 317.49 175.00 300.00 132.27 1080

11 77.03 64.61 21.81 321.36 175.00 300.00 140.20 1100

12 88.02 67.62 25.30 353.23 175.00 300.00 140.83 1150

13 84.36 73.32 29.29 311.10 175.00 300.00 136.92 1110

14 77.61 57.36 33.57 352.43 175.00 221.83 112.20 1030

15 64.57 64.07 37.51 338.32 175.00 209.65 120.88 1010

16 80.77 79.07 40.49 346.54 174.91 288.23 50.00 1060

17 96.09 71.78 41.79 325.61 175.00 216.93 122.80 1050

18 75.63 68.99 44.09 317.04 175.00 299.83 139.41 1120

19 83.46 83.38 48.32 330.27 175.00 210.16 139.41 1070

20 73.00 81.50 49.13 321.19 174.83 214.41 135.94 1050

21 69.25 49.50 52.51 307.51 174.65 206.59 50.00 910

22 76.33 75.82 54.42 303.63 174.70 125.10 50.00 860

23 78.92 68.91 57.03 294.49 175.00 125.65 50.00 850

24 59.72 52.75 56.34 283.24 175.00 122.94 50.00 800






Methods

ECC EEC ECEC

Generation

Cost ($)

Fuel

Emissions

(lbs.)

Generation

Cost ($)

Fuel

Emissions

(lbs.)

Generation

Cost ($)

Fuel

Emissions

(lbs.)

HCWCA 40,906.20 26,060.02 47,114.98 16,342.68 42,470.99 16,390.69

WCA 41,376.87 27,737.85 46,977.52 16,518.41 42,705.68 16,477.48

SOHPSO-

TVAC [86] 41,983 24,482 44,432 16,803 43,045 17,003

IQPSO [160] 42,359 31,298 45,271 17,767 44,259 18,229

DE [164] 43,500 21,092 51,449 18,257 44,914 19,615

PSO [161] 42,474 28,132 48,263 16,928 43,280 17,899



Fig. 6.2 Test System 10: ECC --- Convergence Characteristics

Fig. 6.3 Test System 10: EEC --- Convergence Characteristics

40000

41000

42000

43000

44000

45000

46000

47000

48000

0 100 200 300 400 500

Gen

erat

ion

Co

st (

$)

No. of Iterations

HCWCA WCA

16000

16500

17000

17500

18000

18500

19000

19500

20000

0 100 200 300 400 500

Gen

erat

ion

Co

st (

$)

No. of Iterations

HCWCA WCA

Chapter 7 Hydrothermal Coordination of Utility System using Proposed Hybrid Chaotic WCA



OF UTILITY SYSTEM USING PROPOSED HYBRID

CHAOTIC WATER CYCLE ALGORITHM

7.1 UTILITY SYSTEMS

There are many practical utility systems available in the literature which are being

used by the researchers to validate their research in the real time systems. In this

work, two utility systems have been investigated using the proposed HCWCA.

7.2 INDIAN UTILITY SYSTEM

Indian Utility System consist of 66 buses, 93 transmission lines, 11 hydroelectric

units and 12 thermal units. Network configuration of this test system is shown in Fig.

7.1. The complete data of hydroelectric units and of the thermal units is shown in

Table 7.1 and the evolution model used for investigating the utility system is shown

in Table 7.2.

Fig. 7.1 Network Configuration of Indian Utility System



Table 7.1 Indian Utility System --- Complete Data of Hydroelectric Units, Thermal

Units and Hourly Load Demand

I. Hydroelectric Units Data

Unit 𝑽𝒉𝒋𝒎𝒊𝒏 𝑽𝒉𝒋

𝒎𝒂𝒙 𝑽𝒉𝒋𝑰𝒏𝒊 𝑽𝒉𝒋

𝑬𝒏𝒅 𝑸𝒉𝒋𝒎𝒊𝒏 𝑸𝒉𝒋

𝒎𝒂𝒙 𝑷𝒉𝒋𝒎𝒊𝒏 𝑷𝒉𝒋

𝒎𝒂𝒙 𝑯𝒐/𝑮 𝑰𝒉

1 0 12000 10072 9912 0 17 0 40 2.35 0

2 0 21000 19946 19662 0 56.6 0 160 2.83 1.40

3 0 21000 19956 19670 0 28.6 0 175 6.05 0

4 0 21000 19956 19650 0 52.4 0 180 3.43 6.76

5 0 10000 8652 8657 0 68.3 0 48 0.70 0.22

6 0 30000 27530 27414 0 56.7 0 100 1.77 1.42

7 0 15000 13835 13694 0 9.6 0 70 7.26 0

8 0 8000 6333 6135 0 49.5 0 140 2.83 2.59

9 0 13000 11873 11668 0 21 0 60 3.10 1.70

10 0 13500 12888 12077 0 76.9 0 65 0.78 4.70

11 0 4500 3980 3901 0 16.9 0 60 3.53 0.076

II. Thermal Units Data

Unit 𝒂𝒊 𝒃𝒊 𝒄𝒊 𝑷𝒔𝒎𝒊𝒏 𝑷𝒔

𝒎𝒂𝒙

1 0 40 1.6 20 210

2 0 40 1.6 20 210

3 0 40 1.6 20 210

4 0 40 1.6 20 210

5 0 50 1.6 20 210

6 0 50 1.6 20 210

7 0 50 1.6 20 210

8 0 52 1.2 10 60

9 0 52 1.2 10 60

10 0 56 1.6 10 110

11 0 56 1.6 10 110

12 0 56 1.6 10 110

III. Hourly Load Demand


1 980 9 1520 17 1250

2 1025 10 1610 18 1340

3 1115 11 1655 19 1430

4 1205 12 1700 20 1610

5 1250 13 1610 21 1520

6 1340 14 1520 22 1340

7 1385 15 1430 23 1160

8 1430 16 1295 24 1070



Table 7.2 Indian Utility System --- Evolution model

Hydroelectric

Units

Thermal



Generations

No. of

Runs WCA HCWCA

11 12 150 12 0.01 Adaptive 500 30

The optimal generation cost obtained from the proposed HCWCA for this utility

system is 4,107,653.01 which is less as compared to other methods.

Table 7.3 Indian Utility System --- Comparison of Results

Methods Total Generation Cost

($)

HCWCA 4,107,653

DA-GA 4,138,118

GA 4,341,229

Chapter 8 Conclusion & Suggestions


CHAPTER NO. 8 CONCLUSION & SUGGESTIONS

This chapter discusses the general aspects of the work proposed and described in the

thesis. The relevant chapters have discussed the detailed aspects of the work done.

The research work presented is a complete computer oriented and simulation based

work and the key motivation was to develop a complete framework to solve the short

term hydrothermal coordination problem, model this problem in the environment of

WCA standard version and as per the proposed hybrid model of WCA, named as

HCWCA.

SHTCP is a vital step in power system operational planning, which is carried out both

off-line and on-line. The practical and real SHTCP is a non-linear and non-convex in

nature. Trend is to solve it as a convex function by neglecting the effects of valve point

loadings, multiple fuel mix and the prohibited operating zones. This assumption

infact results in an inaccurate schedule leading to huge loss of revenue.

Evolutionary computation algorithms have come up with a solution of solving these

non-convex and non-linear functions. These EAs have proven them successful in

finding the optimum solutions without any requirements of simplicity of the

objective function. Many EAs have been proposed in the literature since their origin

and have been applied in different fields of optimization. Their applications have

been equally useful in the field of power system operational planning e.g. economic

dispatch, hydrothermal coordination or unit commitment etc.

WCA proposed in 2012 as a new meta-heuristic and evolutionary computation

algorithm working on the basic principle of hydrologic cycle. Like all other EAs it also

starts working by generating random population initially, then the evolutionary

operations of evaporation and raining like mutation and crossover and finally

selecting the best optimum solution.

Very limited work has been done on the application of WCA and also it had not been

investigated for its application on any area of power system operation before. WCA

proved it to be successful in different applications which motivated the author to



investigate the performance of standard WCA on SHTCP and further to propose some

modification or hybrid model of WCA for the investigation of the same SHTCP.

Like all other EAs, WCA also suffers from the problems of premature convergence

and trapping in local optima. To avoid both the problem, a hybrid method based on

chaos theory has been proposed in this thesis. Chaos phenomenon has already been

hybridized with many EAs to improve their performance.

In this proposed work, logistic mapping of the chaos paradigm has been hybridized

with standard WCA to develop a HCWCA to investigate the problem of SHTCP both

as a single objective as well as a multi-objective problem. The standard test systems

have been successfully investigated using the proposed HCWCA and the standard

WCA in this thesis. Moreover, many larger test systems have been proposed by

replicating the previous test systems.

The proposed work also includes the inclusion of many practical constraints which

have been very rarely taken into account by the researchers. These include the

prohibited discharge zones of the hydroelectric units as well as the ramp rates of the

thermal units.

SHTCP has never been solved by considering the prohibited operating zones of

thermal units. This thesis also successfully investigates the performance of proposed

HCWCA and standard WCA for SHTCP with the consideration of POZ of thermal units.

Further, SHTCP has also been investigated as a multi-objective SHTCP known as

MOSHTCP and both the proposed HCWCA and the standard WCA have successfully

investigated the MOSHTCP.

The Indian utility system available in the literature has also been successfully

investigated using both the proposed HCWCA and standard WCA.

In all the cases, the results of best generation cost, along with the execution time for

minimum 30 runs have been compared with the results of recently available

methods/algorithms in the literature. In almost all the cases, the proposed HCWCA

outperforms the standard WCA as well as the recently available methods/algorithms

of literature.



Hence, it can be concluded that the proposed HCWCA as well as the standard WCA

can be worthwhile methodologies for the solution of SHTCP and MOSHTCP even with

the inclusion of all practical constraints like transmission losses, prohibited

discharge zones of hydroelectric units, prohibited operating zones and ramp rates of

thermal units. It can further be concluded that both these methods are successful for

the solution of larger and practical utility systems.

Future Work

Some recommendations regarding future research work may be summarized as

below:

performance enhancements of standard WCA by developing new operators

and the control of WCA parameters,

exploration of hybrid models of standard WCA with other meta-heuristics,

exploration of other hybrid models of WCA with other chaotic paradigms,

evaluating the performance of all the above for the other areas of power

system operation like load forecasting, economic dispatch and unit

commitment.

Practical Applications

The practical applications of the proposed works may be underlined as:

enhancement of the visual environment for the provision of increased

flexibility, interactivity and convenience,

graphical user interface

It is hoped that in future this highly non-linear, non-convex SHTCP and MOSHTCP

will be extensively used for the continuous, secure and most economical supply of

electrical energy with increased efficiency.

Derived Publications____________________________________________________________________________________________


DERIVED PUBLICATIONS

J1. Shaikh Saaqib Haroon and Tahir Nadeem Malik, “Short-term hydrothermal

coordination using water cycle algorithm with evaporation rate”, International

Transactions on Electrical Energy Systems formerly European Transactions on

Electrical Power, (2017), DOI 10.1002/etep.2349, (JCR 2015 I.F = 1.084)

J2. Shaikh Saaqib Haroon and Tahir Nadeem Malik, “Evaporation rate based water

cycle algorithm for the environmental economic scheduling of hydrothermal

energy systems”, AIP Journal of Renewable & Sustainable Energy, 8, 044501-15,

2016, http://dx.doi.org/10.1063/1.4958995, (JCR 2015 I.F = 0.961)

J3. Shaikh Saaqib Haroon and Tahir Nadeem Malik, “Evaporation rate based water

cycle algorithm for short-term hydrothermal scheduling”, Arabian Journal of

Science & Engineering, DOI 10.1007/s13369-016-2262-8, (JCR 2015 I.F = 0.728)

http://dx.doi.org/10.1063/1.4958995

References_______________________________________________________________________________________________________


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