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KWAME NKRUMAH UNIVERSITY OF SCIENCE AND TECHNOLOGY, KUMASI OPTIMAL ELECTRICITY LOAD SHEDDING PROBLEM FOR FEEDERS USING KNAPSACK AND GAME THEORY BY JIBRILU ABASS A THESIS SUBMITTED TO THE DEPARTMENT OF MATHEMATICS, KWAME NKRUMAH UNIVERSITY OF SCIENCE AND TECHNOLOGY IN PARTIAL FULFILLMENT OF THE REQUIREMENT FOR THE DEGREE OF MSc INDUSTRIAL MATHEMATICS NOVEMBER, 2015 Declaration I hereby declare that this submission is my own work towards the award of the MSc degree and that, to the best of my knowledge, it contains no material previously published by another person nor material which had been accepted for the award of any other degree of the university, except where due acknowledgment had been made in the text. Jibrilu Abass ....................................... Student Signature Date Certified by: Dr. Peter Amoako Yirenkyi ....................................... Supervisor Signature Date Certified by: Prof. SK Amponsah ....................................... Head of Department Signature Date i Dedication I dedicate this piece of dissertation to my father and mother Mr. and Mis Abass Odoom. Sibling: Iddrisu, Obroni, Mohammed, Sheik and Salim. lastly to my dear wife Aisha Namateng, and all friends who directly or indirectly contributed to making this book a success, My the almithy Allah bless you all and grant you Jannah. ii Abstract A combination of Knapsack and Game Theory to Electricity Load Shedding (K- GTLS)in the Kumasi Metropolis is presented. A minimization problem of single constraint 0-1 Knapsack known as Knapsack Load Shedding (KLS) is modeled to selects items (loads/MW) up to a minimum capacity of the knapsack in other to minimize some objective function (lost). The KLS comprise of two objective functions belonging to a Day schedule (DS) and a Night schedule (NS), with both linked to a single constraint equation. The problem again is modeled into a Mixed Strategy Game Theory in which strategic probabilities of two players in the game are computed by the linear programing approach. The game comprise of two players: the row player NS and the column player DS each competing for loads to be shed. The game is played with the probabilities using the minimax theory and loads are selected. The minimum capacity of load to be shed calls the fusion of the game theory and the knapsack, hence the K-GTLS model. A data from a load shedding already done by the Electricity Company of Ghana, ECG at the catchment area is fitted onto the models, the K-GTLS in a cyclic mode of load shedding obtained a lost of GH¢9607.30 up against GH¢9621.50 by the KLS and GH¢9667.70 by the ECG. In a proposed Revenue bias mode, the K-GTLS obtained a lost of GH¢9302.25 up against GH¢9326.20 by the KLS and GH¢9429.85 by the ECG. In conclusion, the fusion of Knapsack and game theory in a combinational optimization is a good approach to solving resource constraint project scheduling problem. iii Acknowledgments The researcher gives thanks to almighty ALLAH who gave him the knowledge, wisdom and strength to write this book. He owes a debt of gratitude to Dr. Peter Amoako Yirenkyi, his supervisor who contributed immensely to make writing of this book a reality. The researcher also gives special thanks to Mr. Raymond, Mr. Quansa and all the engineers at the ECG control room in Kumasi who assisted and directed in the acquisition of data to make the completion of this book possible. iv CONTENTS DECLARATION ............................... i DEDICATION ................................ ii ABSTRACT ................................. iii ACKNOWLEDGMENT .......................... iv LIST OF TABLES .............................. viii LIST OF FIGURES ............................. ix 1 INTRODUCTION ............................ 1 1.1 Background of the Study . 1 1.1.1 History of electricity in Ghana . 2 1.1.2 Demand for Electricity during the Volta Development Period 5 1.1.3 Thermal Complementation : 1983 on . 6 1.1.4 Bui dam supplementation . 6 1.2 The need for additional generation . 7 1.3 Why the shortage . 8 1.4 Electricity load shedding in Ghana . 9 1.5 Statement of problem . 9 1.6 Objectives . 10 2 REVIEW OF LITERATURE ..................... 11 2.1 Introduction . 11 2.2 Electric Power in Ghana . 12 v 2.3 Electricity Load shedding . 16 2.4 Knapsack . 19 2.5 Game theory . 24 3 METHODOLOGY ............................ 30 3.1 Introduction . 30 3.2 Computing the load . 31 3.3 Formulation of the knapsack problem . 31 3.3.1 Mathematical formulation of the KLS model . 33 3.3.2 Definition of Variables in the KLS model . 35 3.3.3 Expansion of the KLS model . 35 3.4 Formulation of the Game Theory problem . 36 3.4.1 Definition of variables in the game theory problem . 37 3.4.2 Mathematical formulation of the Game Theory problem . 37 3.4.3 How to compute the strategies . 41 3.4.4 Formulation of the knapsack with game theory problem (K-GTLS) .......................... 42 4 RESULTS AND CALCULATION .................. 43 4.1 Computing the knapsack load shedding . 43 4.1.1 Solving the KLS problem . 44 4.1.2 Calculation of the KLS in a Load shedding done by the ECG. 49 4.2 Solving the Game Theory problem . 51 4.2.1 Solution to the matrix from the linear programming . 52 4.2.2 Computing the strategies . 53 4.2.3 Solving the K-GTLS problem . 54 5 CONCLUSION .............................. 56 5.1 Introduction . 56 5.1.1 Analysis of the result in the KLS . 56 5.1.2 Analysis of the results from the K-GTLS model . 57 vi 5.2 Load shedding . 57 5.2.1 The cyclic load shedding . 58 5.2.2 Revenue-bias load shedding . 59 5.3 Summary of results from the load shedding . 60 5.4 Conclusion . 60 5.5 Recommendation . 60 REFERENCES ................................ 64 vii LIST OF TABLES 4.1 Data extracted and restructured from the May 2013 load shedding in Kumasi . 43 4.2 Matrix of the linear programming above . 52 4.3 Load shedding schedules by the K-GTLS model . 54 5.1 Summary of results from the K-GTLS model. 57 5.2 A three days cyclic load shedding . 58 5.3 Summary of results from the load shedding . 60 5.4 ECG load shedding groupings in may 2013 . 65 5.5 Load shedding schedules by the KLS model. 67 5.6 Data structure of the game theory . 69 viii LIST OF FIGURES 1.1 Location of major electricity generation centers in Ghana . 2 ix CHAPTER 1 INTRODUCTION 1.1 Background of the Study Electricity is a very critical factor in the development process of any country. As the economy grows and population increase, people consume more electricity. The country therefore has to produce more electricity to satisfy the increasing demand of population and for industries to increase the output of goods and services which is what economic growth is all about. In Ghana, the economists describe this relationship in terms of elasticity: ’if the economy has to grow by 1, the energy consumption may also have to grow about 2or more’(PSEC and GRIDCo (2010)). Ghana has always boasted and relied on its human resource capacity and capabilities, but no matter the capacity of human resource put out there on the field will not salvage this problem hence, the need for electric power generation infrastructural expansion in the economy of Ghana. Not electricity at any time but it should be available and reliable at the right place and at the right time. Up until very recently. the last 10 - 15 years, all electricity power in Ghana was hydroelectrically supplied by the Akosombo dam in the Volta Region. The dam not only supplied Ghana but produced excess energy which was exported to neighboring countries such as Burkina and Togo, this was the dream the Ghana’s first president Dr. Kwame Nkrumah as an important step in the Pan - African Movement. Ghana is developing rapidly as it is now considered middle-income country by United Nations. With development comes increase in power demand, eventually. Akosombo became inadequate and Ghana began to import electricity. 1 1.1.1 History of electricity in Ghana There are four main periods of electricity production in Ghana. The first period is "Before Akosombo", the use of diesel in generators. The second period is "the 1st Hydro Years", construction of Akosombo, Kpon and VRA supplements. The third period is "Thermal Complementation" the Tema and Aboadzi power plants etc, and the last period is "the 2nd hydro supplementation" completion of the Bui thermal hydro plant. Figure 1.1: Location of major electricity generation centers in Ghana Before Akosombo (1914 to 1966) Before the construction of the Akosombo hydroelectric plant, power generation and electricity supply in Ghana was carried out with a number of isolated diesel generators dispersed across the country as well as standalone electricity supply systems. These were owned by industrial establishments such as mines and factories, municipalities and other institutions such as hospitals, schools, banks etc. The first public electricity supply in Ghana was established at Sekondi in 2 1914. The Gold Coast Railway Administration operated the system which was used mainly to support the operations of the railway system and the ancillary facilities which went with its operations such as offices, workshops etc. In 1928, the supply from the system was extended to Takoradi which was less than 10 km away. This system served the needs of railway operations in the Sekondi and Takoradi cities. In addition to the Railways Administration, the Public Works Department (PWD) also operated public electricity supply systems and commenced limited direct current supply to Accra in 1922. In November, 1924, the PWD started Alternating Current supply to Accra. The first major electricity supply in Koforidua commenced on April 1, 1926 and consisted of three horizontal single cylinder oil-powered engines.
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