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 1œ, the energy consumption may also have to grow about 2œor 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. Other municipalities in the country which were provided with electricity included Kumasi where work on public lighting was commenced. On May 27, 1927, a restricted evening supply arrangement was effected and subsequently, the power station became fully operational on October 1, 1927 (RCEER and ISSER (2005)). The next municipality to be supplied with electricity in 1927 was Winneba where with an initial direct current supply, the service was changed to alternating current (AC) by extending the supply from Swedru.

The total electricity demand before the construction of Akosombo could not be accurately determined due to the dispersed nature of the supply resources and the constraints surrounding the supply of electricity across the country. Most of the towns served had supply for only part of the day. In addition to being inadequate, the supply was also very unreliable. There was therefore very little growth in electricity consumption during the period. Total recorded power demand of about 70 MW with the first switch on of the Akosombo station can be used as a proxy for the level of electricity demand in the country just prior

3 to the construction of Akosombo.

The Hydro Years (1966 - Mid 1980’s) Akosombo Hydroelectric Project

The Akosombo Hydroelectric Project historically, is linked with efforts to develop the huge bauxite reserves of Ghana as part of integrated bauxite to Aluminium industry. In 1913, Sir Albert Kittson, a British- Australian geologist and naturalist was the first to promote the Akosombo hydroelectric project, he was appointed by the British Colonial Office to establish what is known as the Geological Survey Department. In 1915, while Sir Kittson was on a rapid voyage down the Volta River he identified the hydro potential of the Volta River and later outlined a scheme for harnessing the water-power and mineral resources of the then Gold Coast in an official bulletin. Later in the 30’s, Duncan Rose came across Kinston’s proposals in the bulletin and was interested in the idea of a hydroelectric Aluminium scheme, she toke the idea and developed it up to some point.

The VRA was also charged with the responsibility for the construction of the Akosombo dam and a power station near Akosombo and the resettlement of people living in the lands to be inundated as well as the administration of lands to be inundated and lands adjacent thereto. Construction of the Akosombo dam formally commenced in 1962 and the first phase of the Volta River Development project with the installation of four generating units with total capacity of 588 MW each was completed in 1965 and formally commissioned on January 22, 1966. In 1972, two additional generating units were installed at Akosombo bringing the total installed capacity to 912 MW. By 1969, the Volta Lake, created following the completion of the Akosombo dam, had covered an area of about 8,500 km2 and had become the world?s largest manmade lake in surface area. It can hold over 150,000 million m3 of water at its Full Supply Level (FSL)

4 of 278 feet NLD and has a shoreline length of about 7,250 km. The Lake is about 400 km long and covers an approximate area of 3,275 square miles, i.e., 3 percent of Ghana. The drainage area of the Lake comprises a land area of approximately 398,000 km2, of which about 40 percent is within Ghana’s borders. The other portions of the Volta Basin are in Togo, Benin, Mali and la Côte d’Ivoire. The average annual inflow to Lake Volta from this catchment area is about 30.5 MAF (37,600 million m2).

1.1.2 Demand for Electricity during the Volta

Development Period

In 1962, VALCO signed a Master Agreement with the Government of Ghana, which included a Power Supply Agreement for the supply of power from Akosombo to the 200,000-tonne aluminum smelter. VALCO thus served as the anchor customer for VRA, consumes not less than 40. In 1967, the Electricity Corporation of Ghana (ECG), established by the Electricity Corporation of Ghana decree of 1967 (NLCD 125) and Executive Instrument No. 59 dated June 29, 1967 vested all assets and liabilities of the former Electricity Department in ECG PSEC and GRIDCo (2010). Total domestic load (excluding VALCO) supplied from the grid in 1967 was approximately 540 GWh with an associated peak demand of about 100 MW. Domestic consumption in 1967 was therefore less than 20 percent of the installed capacity at the Akosombo Station. Between 1967 and 1976, domestic consumption more than doubled, growing from about 540 GWh to about 1,300 GWh. During this period, supply to VALCO was governed by the VRA- VALCO Power Supply Agreement, VRA commenced supply of electricity to neighbouring Togo and Benin following the construction of a 205-kilometre 161-kV transmission line from Akosombo (Ghana) to Lome (Togo).

5 1.1.3 Thermal Complementation : 1983 on

In 1983, following the drought, VRA as part of its Generation and Transmission planning process undertook a comprehensive expansion study, the Ghana Generation Planning Study (GGPS). The engineering planning study which was completed in 1985 confirmed the need for a thermal plant to provide a reliable complementation to the hydro generating resources at the Kpong and Akosombo power plants. The study concluded that by adding thermal complementation to the all hydro system, the vulnerability of the power system in Ghana would be significantly reduced. This was because in times of insufficient rainfall resulting in low inflows into the Volta Lake, the thermal plants could be used to meet the shortfall in demand resulting from reduced hydro generation. In effect, the thermal generators were to serve as an insurance policy against poor hydrological years to meet the demand for electricity in Ghana. The Takoradi Thermal Plant Feasibility Study was finally completed in 1992 with a recommendation for the construction of a 600-MW plant, with an initial 300-MW combined cycle plant and a 100-MW Combustion Turbine unit to be commissioned by 1995. There were however, delays in financing approvals by the International Development Association, which eventually resulted in the first 330MW tranche of the Takoradi Plant being commissioned in 1999.

1.1.4 Bui dam supplementation

The bulletin of Albert Kitson also captured the Bui dam project in 1925. In the 1960s’ the dam had already been put on the dawning board. Other options studied by the Kaiser Engineers of USA 1971 included the Bui Hydroelectric Plant on the Black Volta, expansion of the Akosombo Plant with the installation of additional units and development of the Pra, Tano and lower White Volta rivers. The Bui thermal hydroelectric plant on the Black Volta at the border of Northern and Brong Ahafo regions is the most recent completed hydroelectricity

6 generation system, producing just 400MW of power at the moment for some areas in the northern sector. The project is a collaboration between the government of Ghana and Sino hydro, a Chinese construction company. In October, 2007, the Ghana government lead by his Excellency John Agykum Kufour, created the Bui Power Authority to oversee the construction and operation of the project. Construction on the main dam began in 2009, its first generator produced power of the grid on May 3, 2013, with completion expected in 2014.

1.2 The need for additional generation

In Ghana today and the world at large, the effect of electrical energy generation and consumption cannot be looked down upon in the sense that regular supply of electrical energy across the country enhances productivity, globalization, economical growth, as well as its adverse effects on climatic change. The need for additional capacity of electricity source to meet the increasing demand of the growing population and industrialization has continued to be at the forefront of economic policy decision making in may developing economies. However, these ideas and principles are yet to be fully considered in Ghana (Essah (2011)). Electricity consumption in Ghana is estimated to be increasing by about 10œper annum due to the demand from the growing population. However, current baseline production sources generate only 66œof the current demand. From this, an estimated 65œis used in the industrial and service sectors while the residential sector accounts for about 47œof total electricity consumed in the country. Though this does not add up (certainly there must be justified reason), this is what has been presented in the Energy Sector Strategy and Development Plan. Current data draws on the fact that electricity generation is primarily obtained from hydropower sources at Akosombo and Kpong Dam located in the Eastern Region of Ghana and another two thermal power plants using light crude oil at Aboadze near Second-Takoradi in the Western Region of Ghana, (Gand 2009). Additional infrastructure has been constructed to boost the capacity, bringing Ghana’s

7 installed capacity to 1960MW (i.e. 2009 figures) .Ghana’s energy strategy and development plan by 2015 (www.ghanaoilwatch.org) predicts baseline production to rise to 80œ

It was planned in 2012, that additional capacity will be met through the establishment of thermal as well as hydro plants such as the Bui Hydroelectric Plant. An attractive candidate for generation expansion is the 300-MW combined cycle thermal power plant to be located at Tema (RCEER and ISSER (2005)). Currently, the Takoradi Thermal Power Station is fuelled with light crude oil, the price for which has appreciated significantly on the world market. In order to secure a sustainable and cost-competitive fuel source, Ghana is involved in the West African Gas Pipeline (WAGP) Project for power generation. The West African Gas Pipeline (WAGP) supply natural gas from Nigeria to meet the energy requirements of Ghana and other West African countries at relatively low cost than light crude oil. Ghana is also involved in the development of the West African Power Pool (WAPP), aimed at establishing a regional market for electricity trades.

1.3 Why the shortage

Will there ever be a time when electricity supply in this country Ghana will be regular, stable and reliable? That should be the day when Ghanaians can thump their chests and say we have arrived. But that day seem very far away from now because Ghana seems to find no lasting solution to the shortage of power which often lead to load shedding. The first power crises occurred in 1984, it was attributed to an unprecedented drought whose impact was felt throughout the West African sub-region. The second and third power crisis, which occurred in 1998 and 2002 respectively, were also attributed to low rainfall in the Volta basin. The fourth crisis following so closely in the heels of the second and third, was subjected to much public debate and political criticisms, most critics

8 concluded that the old reason of low water levels in the Volta Lake could not have accounted for those crisis (Abeeku and Kemawusor (2007)). The crises continued in random fashions which were sometimes attributed to mechanical failure in the various generation centers. Another crisis happened in 2013, this was attributed to the breakage of the West-African-Gas-Pipe-Line, which transports gas from Nigeria to the Tema Thermal Plant. The most recent crises in June 2014, whose load shedding is currently on going has been attributed largely to low generation capacity at the various generation stations. The Bui Power Authority has also announced a 2.22 million kWh/Day reduction in its power generation due to the dwindling water level in the Bui reservoir, (graphic (2014a)). During any of these crises period, the whole country or part suffered load shedding

1.4 Electricity load shedding in Ghana

Due to the above mentioned challenges that have periodically be-fronted the various installed electricity generation centers in the country, the retailing party (ECG) under the circumstance periodically undertake load shedding exercises, part or across the entire country. In some cases, a load shedding exercise time table is drawn for the awareness of consumers, as to when they would and wouldn’t have power. But most at times the exercise is done unannounced and this has detrimental effect on consumers, such as cold store operators, electric welders and homes. These consumers lose huge sums of money in the process (Ghanaweb.com (2014)).

1.5 Statement of problem

Energy crisis over the years has become major economic problem in which many countries are facing. This problem in many countries comes about as a result of political instability coupled with rising demand for energy arising from population growth and industrialization. Other factors including mechanical

9 failure, inadequate funding and insufficient generation capacities primarily cause energy crises in third world countries such as Ghana (Abeeku and Kemawusor (2007)).

The periodic rise and fall of the water level in the Akosombo, Kpong and Dui dams (graphic (2014a)), in addition to unexpected mechanical failure at the various energy generation centers, periodically, cause unexpected shortage of electricity supply. This leaves the Electricity Company of Ghana (ECG) no other option than to undertake load shedding exercise in some parts of the country and sometimes across the entire nation, (graphic (2014b)). This exercise is done mainly in alternating fashion to ensure equity and fair distribution of electrical power without a major concern on revenue maximization. "The Electricity Company of Ghana (ECG) says it has lost almost GH4.2 million due to the on-going load shedding exercise across the country" (Modernghana.com (2013)). As a result, consumers do not get power at the right time and place and this has detrimental effect on productivity and stifling economic growth (CEPA (2007)). This probes my interest to undertake this study which seeks to look at a more scientific approach to the load shedding with keen interest in revenue maximization.

1.6 Objectives

The objectives of this study are: •To model electricity load shedding problem using knapsack. •To improve upon the Knapsack with Game Theory •To apply the models on an existing data and compare the result with a deterministic result by the ECG at the catchment area. •To propose a load shedding approach that satisfies both the retailer and the consumer.

10 CHAPTER 2

REVIEW OF LITERATURE

2.1 Introduction

The problem of finding and optimal solution among a set of feasible solution arises in most operational process, this problem is usually dealt with using combinational optimization approach. One of such approaches is the knapsack problem which has been profoundly used to solve problems of allocating limited resources such as labour, capacity, budget, time etc. as constraint to attaining optimality in a specific objective such as revenue maximization, efficiency and sufficiency,Deniz Dizdar and Gershkov (2011). The knapsack problem like any other operations research method is a linear programming problem with an objective function and a constraint equation. It is a concept of loading a list of items into a bag called knapsack in other to realize the maximum value of the items without exceeding the capacity of the bag. The knapsack approach is usually a maximization problem, but like most linear programming problems the dual of the maximization problem is the solution to the minimization problem and knapsack is of no exception. By this a minimization problem of knapsack load shedding is modeled.

Another of such approach is modeling the real problem into linear programming and subsequently solving it with suitable numerical approach. The Mixed Strategies Game Theory is an (m x n) game theory which bears a strong relation to linear programming Dawka and Amponsah (2007). The game comprise of two players called the row player and column player, with each player selecting multiple strategies base on certain predetermined set of probabilities. This applies

11 to a game which has no saddle point, i.e. the maximin of the row player is not equal to the minimax of the column player. By the nature of the problem at hand (electricity load - shedding), employing the Mixed Strategies Game Theory will also be a better option to modeling a revenue maximization bias problem of load shedding.

2.2 Electric Power in Ghana

The Resource Center for Energy Economics and Regulation (RCEER), institute of statistical, social and economic research, university of Ghana, Legon, in July 2005, published "the Guide to Electric Power in Ghana" to provide comprehensive facts on Ghana’s electric power sector covering the basics, history, regulations and policies affecting electric power generation in Ghana. The guide outlined comprehensive notes on generation from Akosombo, Kpong and Bui hydropower stations, Tema and Takoradi thermal stations, VRA, etc, transmission and distribution by ECG and VRA, consumption dynamics by the general public for the past two decades across the entire country. It was said that in Ghana, electricity consumption has been growing at 10 to 15 percent per annum for the last two decades, thus, projections were made; the average demand growth over the next decade will be about six percent per annum and as a result, consumption of electricity will reach 9,300 GWh by 2010.

The guide also discussed the creation and functions of the two key policy regulatory institutions by acts of Parliament. These were the Public Utilities Regulatory Commission (PURC) established under the Public Utilities Regulatory Commission Act, 1997 (Act 538) and the Energy Commission (EC) established under the Energy Commission Act, 1997 (Act541). The PURC has the responsibility for economic regulation, ensuring fair competition among utilities and monitoring quality of service. The Energy Commission on the other hand is responsible for indicative national planning, licensing of electricity

12 utilities and technical standards, RCEER and ISSER (2005). It further evaluated the future prospects of the industry by highlighting on the inception of advanced technological mode of generation, transmission and distribution electric power, it also discussed at length major issues and challenges facing electric power generation in Ghana, particularly financing and policy regulation. The document also assessed the role played by consumers and their critical contribution towards the maintenance of the sector.

In june 2007, The College of Engineering at the Kwame Nkrumah University of Science and Technology (KNUST) Kumasi, wrote a book from the series of seminars held on the topic "energy crisis in Ghana, drought, technology or policy". The seminars pooled together many stake holders in the energy sector of Ghana and sister departments in the university such as the Volta River Authority (VRA), the electricity company of Ghana (ECG), the Energy Commission, the Public Utility Regulatory Commission (PURC), the KINT and ABANTU for development (NGOs’), the department of mechanical engineering etc. It was acknowledged in the book that as at 2007, it was widely accepted that a fourth power crisis had been experienced in Ghana in recent memory. The first in 1984 was caused by an unprecedented drought whose impacts were felt throughout the West African sub-region. The second and third power crisis, which occurred in 1998 and 2002, were also attributed to low rainfall in the Volta basin. The current and the fourth crisis, following so closely in the heels of the last one, was been subject to much public debate and most critical observers agreed that the old reason of low water levels in the Volta Lake was no longer tenable.

In effect the book outlined many factors that have led to Ghana’s current predicament and argues that, in essence, the current power crisis is due more to the shortage of generation capacity in the country than to a drought in the Volta Lake Basin. The shortage of generation was partly attributed to

13 poor tariffs regulation policies by the stake holders in the industry Abeeku and Kemawusor (2007). Tariffs had not been adjusted for a long period of time as at when this book was written in 2007, to reflect the increased cost of supply with the introduction of thermal generation which had significantly increased the cost of power production. The book also argues that given a favorable policy environment and economic tariffs, the private sector should be able to help address many of the critical issues including the establishment of new power plants using both conventional systems and renewable energy technology. The role played by consumers towards the maintenance and sustenance of the electricity sector was also assessed.

The Power Systems Energy Consulting (PSEC) in collaboration with Ghana Grid Company Limited (GRIDCo) also wrote a report in March 2010, which presented the outlook for Ghana’s bulk power supply system for stakeholders and the general public. It presents an overview of the operating status of the bulk power system and assesses whether it meets minimum reliability standards, a reflection of its ability to reliably supply the anticipated demand in the 2010 operating year. In addition, the report addresses historical and long-term physical or structural limitations that could impair the reliability of the power system going forward and it also estimates the cost to society and the Ghanaian economy of potential reliability failures. PSEC uses industry standard measures of system reliability-reserve margin for generation capacity, and transfer capability for transmission capacity-to assess the overall reliability of Ghana’s power system. The assessment shows that Ghana’s wholesale electricity supply system is not expected to meet minimum reliability standards for the 2010 year.

The reserve margin, a widely used measure of generation adequacy, currently stands at 10.1œ, while this number represents a modest improvement over reserve margins in the past, it is well below the PSEC-estimated minimum reserve

14 margin for reliable operation of 20œ, and the West African Power Pool (WAPP) recommended minimum of 20œ, Likewise, firm transfer capability, a measure of how well the electricity system can move power around to locations that need it, will remain inadequate in many key areas including Accra and Kumasi. A handful of generation and transmission projects scheduled for completion in 2010 will improve reliability if completed and made operational as expected PSEC and GRIDCo (2010). A new generation project, Sunon Asogli (SAPP) in Kpone, will potentially increase the reserve margin to 14.9œ, The above shows that Ghana’s electricity system remains in the balance. On paper, the three to five year generation expansion plan in place will improve the reliability of generation supply to the required levels by 2014.

The book highlighted on the predicament of energy crises is a result of a decade of chronic underinvestment in generation and transmission infrastructure despite robust growth in demand and energy consumption. From 2000 to 2009, the natural (i.e. not curtailed) peak demand and energy growth rates were 44œ, and 100œ, respectively, driven in large by three major factors: economic growth, urbanization, and industrial activity. However, the most significant factor was a tariff structure that was insufficient to meet the true long run marginal cost of generation and transmission, especially considering the rampant currency devaluation and escalating cost of infrastructure of the period. For Ghanaians, the cost to society of insufficient wholesale power supply adequacy and security is massive. PSEC estimates this cost to be between US 320 million to 924 million annually, or 2œ, to 6œ, of GDP, not including a number of indirect costs. For reference Ghana’s GDP growth averaged 5.5œ, over the last 10 years, a rate that is considered quote robust.

In comparison, US$320 million is enough to fund all of GRIDCo’s identified reliability projects from 2010 to 2016 or to add 500-700MW of combined-cycle

15 thermal generation capacity each year (that is the equivalent of adding a new Akosombo facility every 2 years). These figures indicate that the total cost to society of reliable power is truly in excess of the investment needed to attain reliability and avoid or at least dramatically reduce reliability failures. Finally, it is important that all stakeholders in Ghana’s power sector (consumers, regulators and politicians) recognize that the sector is in a new era. The new era will require an emphasis on operational efficiency and excellence in reflection of the increasing presence of thermal generation, which requires nothing less.

Hydropower plants in Ghana have suffered several low level operations within the past two decades. A paper written by y. a. k. Fiabge and Oben (2006) looked at optimum operations of hydropower plants during these low level conditions by employing the Geometric Programming and Search Method of operations research technique. In the research, two hydropower plants were located along the Volta River at Akosombo and Kpong. The Akosombo plant was at the upstream such that its discharge created available head for the Kpong plant which is downstream. The Akosombo plant was designed to operate between the levels of 75.59m minimum and 84.12m maximum while the Kpong plant operates between 14.5m and 17.7m. The mode of unit combinations for optimum power generation was determined in the study. This according to the paper could serve as a guide in operating the two originally installed hydropower plants. The method presented could also function as a power prediction tool during the low level conditions at Akosombo.

2.3 Electricity Load shedding

Electricity load shedding is an inevitable phenomenon which even in the so-call advance countries, shortage of generation arising from many factors sometimes leaves them no other option than to embark on load shedding, Xu and Girgis (2001). Ghana as a third world country faces many economic challenges. As a

16 result the energy sector suffers the most as being the driving force of the economy. Since 2001, load shedding has become part of the routine activates of the major players in the energy sector. The most recent was done by the ECG across the southern part of Ghana, covering areas like Greater Accra, Ashanti, Volta, Western, Eastern and Central regions. Daily graphic, (June 3, 2014) Many papers have been published on electricity load shedding using different mathematical algorithms. Most of these algorithms I have come across are applied at the generation stage of bulk electricity by considering input AC voltage, current, frequency etc whiles, only a few have dealt with it at the distribution stage. Walling and Miller (2002) Considered dispersed generation in optimal load shedding for distribution networks. In 2009, the same group published another paper on a genetic algorithm (GA) based optimal load shedding, that can apply for electrical distribution networks with and without dispersed generators (DG). Also, the proposed method had the ability to consider constant and variable capacity deficiency caused by unscheduled outages in the bulked generation and transmission system of bulked power supply. The genetic algorithm (GA) was employed to search for the optimal load shedding strategy in distribution networks considering (DGs) in two cases of constant and variable modeling of bulked power supply of distribution networks. The objective of the study was to minimize the sum of curtailed load and also system losses within the frame-work of system operational and security constraints. The proposed method was tested on a radial distribution system with 33 load points for more practical applications.

The formulation and computation of an optimal load shedding time algorithm was presented by ?. This was a new concept developed as an enhancement to the application of optimal load shedding for corrective control to support islanded power systems. The methodology combined nonlinear mathematical programming and discretized differential-algebraic power systems equations to estimate the optimal amount of load to be shed as well as the best time to

17 shed it. The methodology represented a significant enhancement to the fields of corrective control and islanding for distributed power systems. When compared to early optimal load shedding approaches, the algorithm captured the most significant missing feature, dynamic trajectories of the system. The optimal load shedding time translates into more efficient and effective use of a shedding scheme as compared to newer methodologies. The optimization formulation is both flexible and robust. It was also observed, that the algorithms could also handle all kinds of scenarios like multiple islands or an optimal load shedding schedule, where more than one shedding stage could be determined. Given the close ties between the dominant machine and the optimal time, it is likely that heavy penetration of very fast devices will reduce the optimal shedding time to a point at which it becomes ineffective. Several simulated scenarios were studied and results presented.

The Navy ships in U.S.A. are supplied with electric energy from on-board electric power systems, to sophisticated systems for weapons, communications, navigation and operation. The reliability and survivability of the shipboard power systems (SPS) are very essential to the mission of a ship, especially under battle conditionsPalaniswamy K. and Misra (1985). Sometimes, due to enemy attack on the ship, part of the shipboard power system may become malfunctioning and power generation becomes unavailable, this leaves the engineers no other option than to embark on emergency load shedding for the survival of the ship and its occupants. A load shedding system disconnects selected loads from a power system to keep the remaining portion of the system operational. It was observed that the current SPS load shedding is normally provided in several stages or levels, which only shed loads based on fixed priority categories. Moreover, majority of these SPS load shedding are done manually. Ding (2006) presented a new real-time dynamic load shedding scheme for Navy SPS. In his proposed load shedding scheme, analytic hierarchy process (AHP) was adopted for dynamically

18 prioritizing loads according to a) changing priorities of loads and b) system critical natures of loads. The AHP is a multi-criteria decision making methodology that produces a ranking among several options. The proposed load shedding scheme, maximizes various system benefits and minimizes load curtailment. An illustration of the developed load shedding scheme (LSS) was presented and the LSS proposed was applied to a test SPS simulated in a Real Time Digital Simulator (RTDS) and the result presented.

2.4 Knapsack

It is the aim of every investor to allocate a given resources to maximize returns, the choice of the best from a list of ventures given their respective value of returns to invest with the available resources can be modelled into a binary (0,1) Knapsack Problem as done in a case study of TRASHY BAGS ACCRA, Gambrah (2013). There are a limited number of recycling industries for plastic waste in Ghana, due to high investment and operation cost. The larger part of plastic waste produced is burnt, sent to the landfill or left on the street. TRASHY BAGS is one of such small scale industries located in ACCRA, the company sorts various plastic waste and make out of them, varieties of useful items such as bags, wallets, puss, porches, etc. A study was conducted to look at more scientific mode of production in the company, by exploring ways of sufficiently and efficiently selecting items to be produced in the waste management industry from a list of items given a fixed capital. A list of items to be produced were valued by their demand and price, and weighted by their unit cost per production, using the classical 0-1 knapsack problem with a single constraint. A data from the case study was fitted onto the model and good returns were realized.

Similar work has been done in the advertising industry in Nigeria, Owoloko (2010), where funds for advert placement were observed to be allocated by try and error through the discretion of persons or departments in charge.

19 This practice mostly yielded bad returns, because a small section of audience was reached, thus investors and advertisers were always at the receiving end of this unfortunate faulted practices. It was therefore important to place adverts in such a way that the combination of adverts would have the largest possible reach to audience, without escalating the amount allocated for that particular combination of adverts placement. The problem was modeled into a single constraint linear (0,1) knapsack problem by considering the medium of advertising as an item, its audience base as the value on the item and the cost of advertising at the media as a resource constraint. An industrial data was fitted onto the model and it was observed that combining 1 print medium, 2 TV, zero billboard and 5 radio advertisement slots for the month under consideration, 125,000,000, number of audience was reached at a total cost of N3,700,000, as compared to 1 TV, 2 print media and 2 billboards advert placement done by a company to a total of N3,900,000 whiles reaching 102,000,000 number of audience.

Moshe Babaioff and Kleinberg (2008), came up with the Knapsack Secretary Problem, which considered situations in which a decision-maker with a fixed budget faces a sequence of options, each with a cost and a value, and must select a subset of them online so as to maximize the total value. Such challenges arise in many practical situation, e.g. jobs scheduling, hiring workers, and bidding in sponsored search auctions. This problem, often called the online knapsack problem, is known to be non approximate. The group therefore, made the enabling assumption that elements arrive in a random order. Hence the problem was thought of as a weighted version of the Classical Secretary Problem, which they called as the Knapsack Secretary Problem. Using the random-order assumption, they designed a constant-competitive algorithm for arbitrary weights and values, as well as a e-competitive algorithm for the special case when all weights are equal (i.e, the multiple-choice secretary problem). In contrast to previous work on on-line knapsack problems, the group did not

20 assume any knowledge regarding the distribution of weights and values beyond the act that the order is random.

As a family of combinational optimization problem, the knapsack approach can take many forms in maximization problems, one of such is the S–Curve returns function which deals effectively with the allocation of a limited resource to a set of activities to maximize returns. Agrail and Geunes (2009) considered the allocation of a limited budget to a set of investments in order to maximize return from the investments. The returns from investments in an activity was effectively modeled using an S-curve, where increasing returns to scale exist at small investment levels, and decreasing returns to scale occur at high investment levels. The group demonstrated that the resulting knapsack problem with S-curve return functions is NP-hard, provide a pseudo-polynomial time algorithm for the integer variable version of the problem, and develop efficient solution methods for special cases of the problem. They also discussed a fully-polynomial-time approximation algorithm for the integer variable version of the problem.

The ant colony optimization algorithms (ACO) and their applications on the multiple knapsack problem (MKP) were introduced by Fidanova (2007). The ACO is a new meta-heuristic method applied in real life industrial problems in which a good solution is required for a short period of time. The ACO achieves good results for problems with restrictive constraints like multiple knapsack problem. It follows closely the heels of the behavior shown by real ants when searching for food Moyson et al (1988). Ants are social insects that live in colonies and whose behavior is aimed more to the survival of the colony as a whole than to that of a single individual component of the colony. Ants communicate information about the shortest distance between their nest and food sources via a chemical substance called pheromone secreted as they move

21 along. Analogously, ACO is based on the indirect communication of a colony of simple agents, called "artificial" ants, mediated by "artificial" pheromone trails, Fidanova et al (2002). The pheromone trails in ACO algorithms serve as distributed numerical information, which ants use to probabilistically construct solutions to the problem to be solved and which ants adapt during the algorithm’s execution to reflect their search experience. The MKP is a hard combinatorial optimization problem with wide application. Problems encounter at different industrial areas can be interpreted as a knapsack problem including financial and other management. The aim of the study was to incorporate different variants of ACO algorithms and their applications on MKP since the MKP is a constraint problem which provides possibilities to use varied heuristic information. The MKP was represented by a graph, and solutions were represented by paths through the graph. Two pheromone models are compared: pheromone on nodes and pheromone on arcs of the graph.

Manual selection approaches in the repairs of cars are usually inadequate and cannot provide the best solution for a company to maximize revenue; hence a tremendous effort has been spent for improving operational productivity through effective and efficient means of selections. This research develops a procedure of using a simple knapsack model to solve the problem of selecting and scheduling cars due for repairs in an auto workshop environment. This is a direct application of a Knapsack problem to an industrial problem of selection and scheduling (Essuman (2012)). This thesis considers the application of the classical 0-1 knapsack problem with a single constraint to the selection of some cars which has to be repaired within a given time. Our objective of selecting these cars is to maximize income from the associated labour charges earned from the repairs of the cars. Our focus is to use a simple scientific approach of Dynamic Programming that can solve the classical 0-1 knapsack problem above.

22 "We study an extension of the Linear Multiple Choice Knapsack (LMCK) Problem that considers two criteria. The problem can be used to find the optimal allocation of an available resource to a group of disjoint sets of activities, while also ensuring that a certain balance on the resource amounts allocated to the activity sets is attained. The first criterion maximizes the profit incurred by the implementation of the considered activities. The second criterion minimizes the maximum difference between the resource amounts allocated to any two sets of activities. We present the mathematical formulation and explore the fundamental properties of the problem. Based on these properties, we develop an efficient algorithm that obtains the entire frontier of non dominated solutions. The algorithm is very efficient compared to generic multiple objective linear programming (MOLP) algorithms. We present theoretical findings which provide insight into the behaviour of the algorithm, and report computational results which demonstrate its efficiency." Kozanidis and Melachrinoudis (2006)

Deniz Dizdar and Gershkov (2011) analyzes maximization of revenue using the dynamic and stochastic knapsack problem where a given resource needed to be allocated to meet a deadline to sequentially arriving agents. Each agent was described by a two-dimensional type that reflects his resource capacity requirement and his willingness to pay per unit of capacity. The group first characterized implementable policies and solved the revenue maximization problem for the special case where there is private information about per-unit values, but capacity needs are observable. After that, two sets of additional conditions on the joint distribution of values and weights under, which the revenue maximizing policy for the case with observable weights is implementable and thus optimal also for the case with two dimensional private information, were derived. In particular, the role of concave continuation revenues for implementation was investigated. The group also constructed a simple policy for which per-unit prices vary with requested weight but not with time, and proved

23 that it is asymptotically revenue maximizing when available capacity and time to the deadline both go to infinity. The work highlighted the importance of nonlinear as opposed to dynamic pricing.

In a thesis written by (Hifi and Sadfi (1997), an exact algorithm for the knapsack sharing problem is proposed. The proposed algorithm seems quite efficient in the sense that it solves quickly some large problem instances. The problem is decomposed into a series of single constraint knapsack problems; and by applying the dynamic programming and another strategy, we solve optimally the orig-inal problem. The performance of the exact algorithm is evaluated on a set of medium and large problem instances (a total of 240 problem instances). This algorithm is parallelizable and this is one of its important feature.

The Multiple Knapsack problem (MKP) is a hard combinatorial optimization problem with large application, which embraces many practical problems from different domains, like cargo loading, cutting stock, bin-packing, financial and other management, etc. It also arise as a subproblem in several more complex problems like vehicle routing problem and the algorithms to solve these problems will benefit from any impr ovement in the field of MKP. (Fidanova (2002))wrote a which seeks to compare different kind of heuristic models, statics and dynamics. The heuristics are used by an Ant Colony Optimization (ACO) algorithm to construct solutions to the MKP.

2.5 Game theory

When in 1950 John Nash was getting his PhD degree on theory of non- cooperative games, no one could foresee that he was more than several decades ahead of his times. One could also not predict that almost half a century later his effort will be celebrated and Nash himself, along with other notable economists, will be given the Nobel Prize. Nowadays, no one undermines

24 the relevance of Game Theory (GT) as science which influences other sciences, starting with mathematics and economy and ending with philosophy and biology, (Tomasz Goluch (2012)). The forefathers of GT are generally agreed to be John von Neumann and Oskar Morgenstern, it was after the publishing of their book "Theory of Games and Economic Behavior" that the potential of GT began to be realized. A player has his own interest in mind when behaving in a particular way. GT tries to explain that behavior and anticipates the best possible solutions. Game theory can be applied to many academic areas, most commonly Economics and Sociological problems but more recently Politics and Evolutionary Theory. It is also becoming popular among Computer Scientists due to its use in AI, (Brook (2007)). used the Mixed Strategy Nash Equilibrium concept to create a system that solves two-player, zero-sum games. His work concentrated on two-player games because, as J. D. Williams says, ’one player games are uninteresting’ and games with more than two players can be affected by decisions separate from the game data such as co-operation. A game where the sum of the payoffs is zero is known as a zero-sum game. Simply put, one person wins what the other loses.

Demand Side Management in smart grid has emerged as a hot topic for optimizing energy consumption. In conventional research works, the energy consumption is optimized from the perspective of either the users or the producers. (Zubair Md and Stojmenovic (2013)) investigated how energy consumption may be optimized by taking into consideration the interaction between both the parties. The group made two new propositions; (1) the energy price model as a function of total energy consumption and (2) the objective function, which optimizes the difference between the value and cost of energy. The power supplier pulls consumers in a round-robin fashion, and provides them with energy price parameter and current consumption summary vector. Each user then optimizes his own schedule and reports it to the supplier, which in

25 turn updates its energy price parameter before pulling the next consumers. This interaction between the producer and its consumers was modelled through a two-step centralized game, from which the Game-Theoretic Energy Schedule (GTES) method was adopted. The objective of studying GTES method was to reduce the Peak to Average power Ratio (PAR) by optimizing the user’s energy schedules. The performance of the GTES approach was evaluated using computer-based simulations. The results from the simulation demonstrate that the proposed algorithm converged in O(n) iterations, and achieved good PAR reduction and energy consumption reduction. These results are desired from the electricity producer’s point of view, and provided it has prow knowledge on the consumers’ energy schedules, the power producer would be able to predict the total energy consumption per hour in order to estimate the required amount of power. Also, every user has his electric cost reduced in an effective manner, and the values of the users’ objective functions (which represent their pay-offs) are increased.

In the theory of games, a player is said to use a mixed strategy whenever he or she chooses to randomize over the set of available actions. Formally, a mixed strategy is a probability distribution that assigns to each available action a likelihood of being selected Saenz R. and X. (2007). Game theory outlines optimal response strategies during mixed-strategy competitions in which available actions are selected probabilistically. An example is a study conducted to examine the neural processes involved in action selection using mixed-strategy competition (Thevarajah (2009)). Two studies were made, the first tested whether the primate superior colliculus (CS) was involved in choosing saccades under strategic conditions. In the second study, the influence of previous actions and rewards on updating Premotor activity in the superior colliculus (CS) in the strategic condition was compared. It was observed that updating Premotor activity with past actions and rewards with more recent events exerted the

26 largest influence. "Together our results highlight the active role played by the brain in choosing strategic actions."

Food allocation in bird broods has been dived into two with different concepts such as the Utopia value and the Prenucleolus. A research conducted by (Sherlin (2012)) looked at the same problem from the perspective of cooperative game theory. The study explored whether or not food distribution data fit into the known solution concepts of cooperative game theory. A first issue to be handled is the fact that in the bird brood data we only see the solutions, while the starting position, the game, is not immediately clear. As such there is the need to reconstruct the game from the solutions given. A second issue is that there are many different solution concepts (e.g. Shapley value, nucleolus, etc) and we want to analyze which of these fits best. Most interesting is to specifically address the properties that lead to these solutions, because these would be most useful in finding a motivation for the specific solution concept found in nature. The study in conclusion emphasized the implementation of the Shapley Value Solution Concept to solve the bird brood food allocation problem and compared its result with already established solution concepts namely the Utopia value and the Prenucleolus. This meaning that the biological problem of blackbird food allocation was translated into a cooperative game approach using different techniques.

The Coalitional Game Theory (CGT) is another methodology which examines the interactions between groups of decision-makers and thus is cooperative in nature; it is a game of n-persons or infinite players Beaumont and Yang. (2005). Scheduling divisible loads in distributed systems is the subject of Divisible Load Theory (DLT) which can be dealt with using CGT. A paper written by C. and Daniel (2007) show that GCT is a natural fit for modelling DLT, because the participants in the scheduling algorithm must cooperate in order to execute a

27 job. The group devised a coalitional scheduling game in which the job owners and the independent organizations that own processors form coalitions in order to maximize their profits. They examine the payoffs to the participants and show that the core of the proposed coalitional scheduling game is non-empty. The “fair sharing” of the payoffs among the participants using the Shapley value was examined and the properties of the proposed coalitional scheduling game considering different distributed systems configurations were finally simulation.

A larger portion of electricity power in the USA is consumed by computers and the number is increasing steadily as more multi-core processors troop into the system. A study by a man Dataquest reported that the world-wide total power dissipation of processors in PCs was 160MW in 1992, and shot up to 9000MW by 2001. Multi-core processors are beginning to revolutionize the landscape of high-performance computing. In a study conducted by Ahmad I. and K. (2008), addressed the problem of power-aware scheduling of tasks onto heterogeneous and homogeneous multi-core processor architectures. The objective of scheduling was to minimize the electric power consumption as well as the makespan of computationally intensive problems. It was observed by the group that the multi-objective optimization problem was not properly utilized by conventional approaches that try to maximize a single objective. They proposed a solution based on game theory and modeled the problem as a cooperate game. "Although we can guarantee the existence of a Bargaining Point in this problem, the classical cooperative game theoretical techniques such as the Nash axiomatic technique cannot be used to identify the Bargaining Point due to low convergence rates and high complexity. Hence, we transform the problem to a maximin problem such that it can generate solutions with fast turnaround time". In conclusion, the proposed algorithm outperformed other already existing algorithms such as greedy and LR heuristics in identifying a smaller makespan and saving the energy.

28 A thesis presented by (Karppien (2011)) aimed at forming a strategic marketing plan for Hotel X, a small privately owned hotel in Helsinki. The theoretical part of this thesis presents tourism and marketing from the hospitality industry’s point of view; what challenges the accommodation providers face when marketing their products and what kind of plans can be formed in order to keep their marketing actions up to date. In the research a qualitative method was used and the data was collected using semi-structured questionnaire. In-depth interviews were conducted with the hotel owner and staff members. The results of the research formed the strategic marketing plan outlining the action plans on how Hotel X should proceed with its marketing.

29 CHAPTER 3

METHODOLOGY

3.1 Introduction

The so-called resource constrained project scheduling problem is a very general scheduling problem which may be used to model many applications in practice e.g. a production process, a school timetable, etc. The objective is to schedule some activities over time such that scarce resource capacities are respected and a certain objective function is optimized. Examples of resources constraints may be machines, people, money or energy, which are only available with limited capacities(Fidanova (2002)), such that some objective function, for example, the project duration, the cost concerning resource and profit etc may be optimized. The problem at hand (electricity load shedding) is of no different form one of the Resource Constrained Project Scheduling Problem. In this problem, a limited amount of electricity (resource constraint) is to be distributed in such a way that profit is maximized. Two mathematical methods are used in a combinational optimization process to solve the problem, this is what I refer to as the Knapsack with Game Theory Load Shedding (K-GTLS)problem. There is a 24 hours scheduling problem which has been divided into two schedules namely Day Schedule and Night Schedule, with 12 hours in each schedule. 1) Day Schedule (DS) (6:00am to 6:00pm) 2) Night Schedule (NS (6:00pm to 6:00am)

30 3.2 Computing the load

The case study, Kumasi Metropolis has been divided into sub locations called feeders. These feeders have different amount of electricity they consume called loads, the total amount of electricity consume by the study area according Mr Raymond at the ECG control station near Ridge Police Station is 250MW at its peak period. The loads of the various feeders constitute the main data used in this study.

The loads are calculated using the formula √ P = 3ILVL cos θ Where P = power or load

IL = line current

VL = line voltage θ = power factor (value = 0.85 in Kumasi)

The line current (IL) and line voltage (VL) values from the various feeders are recorded on hourly bases at the control station of the ECG, from an instrument called mimic board shown below.

3.3 Formulation of the knapsack problem

As discussed earlier in the previous chapter, the knapsack problem comprises of a family of combinational optimization approaches, each depends on the nature of optimization problem to be solved. Examples of some of the types of the knapsack approaches are: The single constraint 0-1 Knapsack Problem which deals with choosing a subset of the n items one after the other such that

31 the corresponding profit sum is maximized. this may be formulated as follows

n X maxmise pixi (3.1) i=1 subject to n X aixi ≤ k (3.2) i=1 where xi(0, 1) i = 1,2,3 ...... n

If we have a bounded amount mi of each item type i, then the Bounded Knapsack Problem arises as:

n X maxmise pixi (3.3) i=1 subject to n X aixi ≤ k (3.4) i=1 where xi(0, 1, ....mi) i = 1,2,3 ...... n

A situation where by an unlimited number of each item type is available, then a generalization of the Bounded Knapsack Problem known as Unbounded Knapsack Problem arise as:

n X maxmise pixi (3.5) i=1 subject to n X aixi ≤ k (3.6) i=1 where xi ≥ 0integer, i = 1,2,3 ...... n

Another generalization of the 0-1 Knapsack problem is to choose exactly one item i from each of k classes Nj, such that the profit sum is maximized. This gives

32 the Multiple-choice Knapsack Problem which is defined as:

m X X maximise pjixji (3.7)

j=1 iNj subject to m X X ajixji ≤ k (3.8)

j=1 iNj X xji ≤ 1 (3.9)

iNj where xi(0, 1) j,i = 1,2,3 ...... mn etc.

The Knapsack model can either be used alone or combined with other mathematical models to solve a problem. Example is the Knapsack with S- Curve returns function(Agrail and Geunes (2009)), which deals with allocating limited resource to a set of investment activities in other to maximize return on the investment. Another one is the Knapsack with the Ant Colony Optimization algorithm,(Fidanova (2007)) etc. By the nature of this electricity load shedding problem, the Single Constraint 0-1 Knapsack Problem which has been referred to as Knapsack Load Shedding (KLS) is adopted.

3.3.1 Mathematical formulation of the KLS model

In the case of the KLS, every schedule (DS and NS) has its own objective function but both linked to a single constraint equation. When there is shortage of generation, the ECG is quickly informed by its suppliers to shed some given amount of load, thus some feeders need to be selected and their power taken off to make up for the shortage. Instead of selecting locations to be distributed with the available power to maximize profit, the KLS rather select locations to be shed off to make up for the shortage to minimize lost. This means that the idea

33 of the dual of a maximization problem is a minimization is employed.

Primal

n X max H = uiyi (3.10) i=1 subject to n X aiyi ≤ k (3.11) i=1 where i = 1,2,...n, a = weight on ith item, k = total weight, v = value on ith item, y = item

Dual

n X min Z = vixi (3.12) i=1 subject to

n X wixi ≥ c (3.13) i=1 where i = 1,2,...n, w = weight on ith item, c = 250-k(load to be shed), v = value on ith item, x = item

The routes and locations of the case study called feeders are ranked base on their peak period of consumption. There are two ranks, rank 1 representing a normal period and rank 2 representing a peak period. The feeders are itemized (x) in the knapsack with a minimum capacity (c) representing the amount of load available to be shed. The items are weighted (w) by the average amount of electricity they consume in 12 hours and summed to at least the maximum capacity of the knapsack (c) and that forms the constraint equation. Each item is also valued (v) by the average cost per unit consumption rate and summed, that forms the objective function (Z) of both the KLS and the GTLS model.

34 3.3.2 Definition of Variables in the KLS model m = number of feeders in a group k = number of groups of feeders in catchment area

th vi = amount of money in cedis loss of shedding the i load

th wi = load of the i feeder C = total amount of load to be shed Z = total amount of money in cedis loss for shedding loads of m number of feeders

3.3.3 Expansion of the KLS model

Objective function: k m X X min Z = vixi, fortheDS (3.14) j=1 i=1

k m X X min Z = vixi, fortheNS (3.15) j=1 i=1 st. k m X X wixi ≥ c (3.16) j=1 i=1 where xi(0, 1) i = 1,2,...k and i = 1,2,...m.

The xi is a binary variable equaling 1 if item i should be included in the knapsack and 0 otherwise.

35 3.4 Formulation of the Game Theory problem

Game Theory deals with taken decisions under uncertainty involving two or more intelligent opponents in which each opponent aspires to optimize his own decision at the expense of the other opponent. Typical examples include lunching advertisement campaigns for competing products, planning war tactics for opposing armies, developing playing tactics in a football match for opponent team etc. In Game Theory, competitors are referred to as players, each player has a number of choices which may be finite or infinite called strategies. The outcome of a game is called payoff (Dawka and Amponsah (2007)). A game with two players selecting pure strategies, where the sum of their individual payoffs is equal to zero is called two-person zero-sum game, when the sum of their payoffs is equal to a constant, it is called two-person constant-sum game, and these games usually have saddle points i.e. minimax = maximin.

A game in which the two players do not select pure strategies, but select multiple strategies guided by certain predetermined probabilities is known as a Mixed Strategies Game Theory (Huang (2011)). Formally, a mixed strategy is a probability distribution that assigns to each available action a likelihood of being selected. Several methods such as the Nash Equilibrium, Prisoner’s Dilemma, Matching Penis, Linear Programming etc, can be used to solve a problem modeled with the Mixed Strategies Game Theory Dataquest. As an optimization model that deals with selection process, it can be fused with other mathematical models to maximize some objective function. A clear example has been presented by (Sherlin (2012)), the study explored whether or not food distribution data fit into the known solution concepts of cooperative game theory. Another example is a study conducted to examine the neural processes involved in action selection using mixed-strategy competition (Thevarajah (2009)). Two studies were made, the first tested whether the primate superior colliculus (CS) was involved in

36 choosing saccades under strategic conditions. In the second study, the influence of previous actions and rewards on updating Premotor activity in the superior colliculus (CS) in the strategic condition was compared.

In this thesis, the load shedding problem is modeled into a Mixed Strategy Game Theory Problem. It is then combined with the KLS model, discussed previously in this chapter. The combination is referred to as Knapsack with Game Theory Load Shedding (K-GTLS)

3.4.1 Definition of variables in the game theory problem m = number of feeders selected by the NS n = number of feeders selected by the DS i = a feeder selected by the NS j = a feeder selected by the DS xi = the optimal strategy of the NS yj = the optimal strategy of the DS v = value of the game equaling the total load to be shed ai,j = load for a feeder available for selection Z = objective variable of NS’s problem W = objective variable of DS’s problem

3.4.2 Mathematical formulation of the Game Theory

problem

To begin with, the two schedules DS and NS mentioned previously are treated as two players in a two-person zero-sum game in a mixed strategies approach. Each player instead of selecting a pure strategy only, may play all his strategies according to a predetermined set of probabilities. Let x1, x2 ,..., xm and y1, y2

,..., ym be the row and column probabilities by which the two players DS and NS respectively select their pure strategies.

37 Then m n X X xi = yi = 1, xi, yj ≥ 0 (3.17) i=1 j=1 for all i and j

th Thus if ai,j represents the consumption capacities of the (i, j) entry of the game matrix, xi and yj appear as in the matrix shown below.

DS

Prob. y1 y2 .. yn

x1 a11 a12 .. a1n

x2 a21 a22 .. a2n NS ......

xm am1 am2 .. amn

The solution of the mixed strategy problem is based also on the minimax criterion.

The only difference is that NS selects xi that maximizes the smallest expected payoff in a column, whereas DS selects yj that minimize the largest expected payoff in a row. The minimax criterion for a mixed strategy case is given as follows. Player NS selects m X xi(xi > 0, xi = 1) (3.18) i=1 that yield

" m m m # X X X max min( ai1xi, ai2xi....., ainxi) (3.19) i=1 i=1 i=1 And player DS selects n X yj(yj > 0, yj = 1) (3.20) j=1 that yield

" n n n # X X X min max( a1jyj, a2jyi....., amjyj) (3.21) j=1 j=1 j=1

38 These values are referred to as the maximin and minimax expected payoffs respectively. As in the case of the pure strategies, the relationship minimax expected payoff ≥ maximin expected payoff holds. Assuming xi and yj are optimal solutions for both players each payoff element aij will be associated with probabilities (xi , yj). Thus the optimal expected value ’v’ of the game from minimax theory is

m n X X v = aijxiyj (3.22) i=1 i=1 NS’s optimal mixed strategies satisfy

" m m m # X X X max min( ai1xi, ai2xi....., ainxi) (3.23) i=1 i=1 i=1

Subject to x1 + x2+,..., +xm = 1 xi ≥ 0, i = 1, 2, ...... m

This problem can be put into a linear programming form as follows. let m m m X X X v = min( ai1xi, ai2xi....., ainxi) (3.24) i=1 i=1 i=1 then the problem becomes maximizeP = v (3.25) subject to m X aijxi ≥ v, j = 1, 2, ...... m (3.26) i=1

m X xi = 1 (3.27) i=1

xi ≥ 0, i = 1, 2, ...... m (3.28)

The linear programming formulation can be simplified by dividing all (n + 1)

39 constraints by ’v’, so long as v > 0, otherwise if v < 0, the direction of the inequality constraints is reversed. Assuming that v > 0, the constraint of the linear programming becomes.

x1 x2 xm a11 v + a21 v + . . . .+ am1 v ≥ 1

x1 x2 xm a12 v + a22 v + . . . .+ am2 v ≥ 1 ......

x1 x2 xm a1n v + a2n v +. . . .+ amn v ≥ 1 and

x1 x2 xm 1 v + v +...... + v = v let

xi Xi = v , i = 1,2,...... ,m

1 Since max P = v, min P = v The linear programming finally becomes

min P = X1 + X2 +...... + Xm subject to

a11X1 + a21X2 +. . . .+ am1Xm ≥ 1

a12X1 + a22X2 +. . . .+ am2Xm ≥ 1 ......

a1nX1 + a2nX2 +. . . .+ amnXm ≥ 1

X1,X2, ...... Xm ≥ 0

Player DS’s problem can also be represented in linear programming by derivation as done for player NS as,

max Q = Y1 + Y2 +...... + Yn

40 subject to

a11Y1 + a21Y2 +....+ a1nYn ≤ 1

a12Y1 + a22Y2 +....+ a2nYn ≤ 1 ......

am1Y1 + am2Y2 +....+ amnYn ≤ 1

Y1,Y2, ...... Yn ≥ 0

1 Yj Where Q = v ,yj = w , j = 1,2,.....,n.

3.4.3 How to compute the strategies

DS’s strategies

1 Yj Q = v , yj = Q , where yj are DS’s strategies

Since the row player’s (NS’s) linear programing problem is the dual of the column player’s (DS’s), the solution of the DS above automatically yield the solution of NS. The matrix from the linear programming is solved using a suitable numerical approach in MATLAB and the results is used to compute the individual strategic probabilities by which the two players DS and NS play the game. The game is played using the minimax theory, loads are selected into column maximums and row minimums. But the total load in each group is far greater than the capacity needed to be shed, this calls for the fusion of the KLS and GTLS into K-GTLS. This is to say that in the K-GTLS model the game theory plays the constraint role selecting the loads and Knapsack optimizes the lost function.

41 3.4.4 Formulation of the knapsack with game theory

problem (K-GTLS)

k m X X min Z = vixi, fortheDS (3.29) j=1 i=1

k m X X min Z = vixi, fortheNS (3.30) j=1 i=1 subject to m X wi ≥ c (3.31) i=1 where xi(0, 1), j = 1,2,...k and i = 1,2,...m.

The xi is a binary variable equaling 1 if item i is selected by the game and 0 otherwise.

42 CHAPTER 4

RESULTS AND CALCULATION

4.1 Computing the knapsack load shedding

The data used in this project and presented in the table below was extracted and restructured from load shedding done in May 2013 by the ECG in Kumasi, a total capacity of 50.00 Mega Watt was shed in the process. The data is fitted onto the two models and the outcomes compared to that of the ECG. For security reasons the values associated with 12hrs Cost per consumption/ are a little bit altered, but to a large extent it is the true reflection of the actual consumption rate values.

Table 4.1: Data extracted and restructured from the May 2013 load shedding in Kumasi Feeder Items Load/MW Ave. load Cost/GH¢ Ranking Rank 2 Rank 1 Day Night Techiman 0.67 1 2 2 0.60 2.16 1.58 N. Asafo 0.53 1 2 G11 1.26 1 2 Edwinasi 1.30 1 2 F11 1.80 1 2 B21 1.80 2 1 8 1.57 5.65 4.12 D41 1.90 2 1 B61 1.64 1 2 OBR 1.35 1 2 E51 1.66 1 2 Yabi 2.21 1 2 C51 2.71 1 2 F41 5 2.90 2.63 9.45 7.04 1 2 Lake road 2.90 2 1 G21 2.41 1 2

43 Feeder Items Load/MW Ave. load Cost/GH¢ Ranking Rank 2 Rank 1 Day Night C31 3.05 1 2 ND/OHL 3.61 2 1 Wahw/GBC 3.44 9.02 2 1 Bare 1 7 3.77 3.43 12.35 2 1 Apire 3.97 1 2 G31 3.20 1 2 Guinness II 3.39 1 2 C17 4.05 1 2 Airport I 4.24 1 2 E21 5 4.47 4.32 15.55 11.35 2 1 E31 4.07 2 1 F51 4.77 1 2 C61 5.07 1 2 E41 5.12 1 2 Airport II 5 5.91 5.41 19.48 14.22 1 2 D51 5.40 2 1 St. Hubert 5.52 1 2 B11 6.05 1 2 D31 6.10 1 2 4 6.29 22.64 16.53 B71 6.29 1 2 E11 6.73 1 2 Bekwae 1 7.53 7.53 27.18 19.84 1 2 M.Nkwanta 1 8.74 8.74 31.46 22.97 1 2 Nsuta Kumawu 1 13.22 13.22 47.59 34.59 1 2

4.1.1 Solving the KLS problem

2 10 15 22 27 X X X X X DS : minZ = vixi + vixi + vixi + vixi + vixi i=1 i=3 i=11 i=16 i=23 32 36 37 38 39 (4.1) X X X X X + vixi + vixi + vixi + vixi + vixi i=28 i=33 i=37 i=38 i=39

44 2 10 15 22 27 X X X X X NS : minZ = vixi + vixi + vixi + vixi + vixi i=1 i=3 i=11 i=16 i=23 32 36 37 38 39 (4.2) X X X X X + vixi + vixi + vixi + vixi + vixi i=28 i=33 i=37 i=38 i=39 s.t.

2 10 15 22 27 X X X X X wixi + wixi + wixi + wixi + wixi i=1 i=3 i=11 i=16 i=23 32 36 37 38 39 (4.3) X X X X X + wixi + wixi + wixi + wixi + wixi ≥ c i=28 i=33 i=37 i=38 i=39 Solution

There are 39! number of basic solution in both the DS and NS problems above. In a nut shell all of them need to be solved in order to obtain the basic feasible solutions, but this will be very tedious manually and require computer simulation to obtain the basic feasible solutions. However few solutions would be considered manually until some basic feasible solutions are obtained.

Considering the following basic solution in the KLS model I. [00,100000111,100001,1000000,10001,10000,1000,1,1,0] Constraint 4(1.57) + 2(2.68) + 3.43+ 2(4.32) + 5.41 + 6.29+7.53+ 8.74 ≥ 50.00 51.65≥ 50.00 feasible solution lost function DS: min Z = 4(4.12) + 2(7.04) + 9.02 + 2(11.35) + 14.22 + 16.53 + 19.84 + 22.97 = GH135.84 NS: min Z = 4(5.65) + 2(9.65) + 12.35 + 2(15.55) +19.48 + 22.64 + 27.18 + 31.46 = GH186.11 Total lost = 186.11 + 135.84 = GH321.95

45 II. [11,01100000,01100,0000110,01000,01001,0100,0,0,1] Constraint 2(0.6) + 2(1.57) + 2(2.68) + 2(3.43) + 4.32 +2(5.41) + 6.29+ 13.22 ≥ 50.00 50.67≥ 50.00 feasible solution lost function DS: min Z =2(1.58)+ 2(4.12) + 2(7.04) + 2(9.02) + 11.35 + 2(14.22) + 16.53 + 34.74 = GH134.62 NS: min Z = 2(2.16)+2(5.65)+ 2(9.65) + 2(12.35) + 15.55 +2(19.48) + 22.64 + 47.59 = GH184.36 Total lost = 184.36 + 134.62 = GH318.98

III.

[00,00011000,00010,0111001,00110,00110,0011,0,0,0] Constraint 2(1.57) + 2.68 + 4(3.43) + 2(4.32) + 2(5.41) + 2(6.29) ≥ 50.00 51.58 ≥ 50.00 feasible solution lost function DS: min Z = 2(5.65) + 9.65 + 4(12.35) + 2(15.55) +19.48 +14.22 + 2(16.56) = GH168.27 NS: min Z = 2(4.12) + 7.04 + 4(9.02) + 2(11.35) + 14.22+19.48 + 2(22.64) = GH153.04 Total lost = 153.04 + 168.27 = GH321.31

IV. [00,11111111,00000,0000000,00000,01111,0000,1,0,0] Constraint 8(1.57) + 3(3.43)+ 4(5.41) + 7.53 ≥ 50.00 51.5 ≥ 50.00 feasible solution lost function

46 DS: min Z = 2(5.65) + 6(4.12) +3(12.35) + 3(14.22) + 19.48 + 19.84 = GH154.57 NS: min Z = 6(5.65) + 2(4.12) +3(9.02) + 3(19.48) + 14.22 + 27.18 = GH167.60 Total lost = 167.60 + 154.57 = GH322.17

V. [11,00011000,00000,1111111,00000,00000,0000,0,1,1] Constraint 2(0.6) + 2(1.57) + 7(3.43) + 8.74 + 13.22 ≥ 50.00 50.31 ≥ 50.00 feasible solution lost function DS: min Z = 2(1.58) + 3(5.65) + 4(12.35) + 3(9.02) + 22.97 + 34.74 = GH154.26 NS: min Z = 2(2.16) + 2(4.12) + 4(9.02) + 3(12.35) + 31.46 + 47.59 = GH164.74 Total lost = 164.74 + 154.26 = GH319.00

VI. [00,01111000,11111,0000000,11111,00000,0000,0,1,0] constraint 4(1.57) + 5(2.68) + 5(4.32) + 8.74 ≥ 50.00 50.02 ≥ 50.00 feasible solution lost function DS: min Z = 2(5.65) + 2(4.12) + 3(7.04) + 2(9.65) + 3(11.53) + 2(15.55) + 22.97 = GH148.6 NS: min Z = 2(5.65) +2(4.12) + 2(7.04)+ 3(9.65) + 2(11.53) + 3(15.55) + 31.46 = GH163.74 Total lost = 163.74 + 148.62 = GH312.36

47 VII. [00,11111111,00000,0000000,00000,00000,1111,0,0,1] constraint 8(1.57) + 4(6.29) + 13.22 ≥ 50.00 50.94 ≥ 50.00 feasible solution lost function DS: min Z = 2(5.65) + 6(4.12) + 4(16.53) + 34.74 = GH136.88 NS: min Z = 6(5.65) + 2(4.12) + 4(22.64) + 47.59 = GH180.29 Total lost = 180.29 + 136.88 = GH317.17

VIII. [01,00111000,01100,0111000,01100,11101,0000,0,0,0] Constraint 0.6 + 3(1.57) + 2(2.68) + 3(3.43) + 2(4.32) + 4(5.41) ≥ 50.00 51.24 ≥ 50.00 feasible solution lost function DS: min Z = 1.58+2(5.65)+4.12+7.04+9.65+3(12.35)+11.35+15.55+19.48 +3(14.22) = GH159.78 NS: min Z = 2.16+5.65+2(4.12)+7.04+9.65+3(9.02)+15.55+11.35+3(19.48) + 14.22 = GH159.36 Total lost = 159.36 + 159.78 = GH319.14

IX.

[10,10000111,10010,1000111,10011,00010,0100,0,0,0] Constraint 0.6 + 4(1.57) + 2(2.68) + 4(3.43) + 3(4.32) + 5.41 + 6.29 ≥ 50.00 51.75 ≥ 50.00 feasible solution lost function

48 DS: min Z = 1.58+4(4.12)+7.04+9.65+12.35+3(9.02)+15.55+2(11.35)+ 19.48+16.53 = GH148.42 NS: min Z = 2.16 +4(5.65)+7.04+9.65+3(12.35)+9.02+2(15.55)+11.35+14.22+22.64 = GH166.83 Total lost = 166.83 + 148.42 = GH315.25

X.

[00,01000000,00001,0000000,00000,00000,1011,1,1,1] Constraint 1.57 + 2.68 + 3(6.29) + 7.53 + 8.74 + 13.22 ≥ 50.00 52.61 ≥ 50.00 feasible solution lost function DS: min Z = 4.12 + 7.04 + 3(16.53) + 19.84 + 22.97 + 34.74 = GH138.30 NS: min Z = 5.65 + 9.65 +3(22.64) + 27.18 + 31.46 + 47.59 = GH189.45 Total lost = 183.80 + 134.18 = GH327.75

4.1.2 Calculation of the KLS in a Load shedding done by

the ECG.

The ECG Kumasi, on 5th may, 2013 used the same data for load shedding as shown in table 5.4. 50.00MW was shed in the process. The load shedding is fitted onto the KLS model and the result presented as follows.

Group A2

[01,10000011,10000,1000011,10001,00100,0100,0,0,1] Constraint 0.6 + 3(1.57) + 2.68 + 3(3.43) + 2(4.32) + 5.41 + 6.29 + 13.22 ≥ 50.00 51.84 ≥ 50.00 feasible solution lost function DS: min Z = 1.58+3(4.12)+7.04+2(9.02)+12.35+2(11.35)+14.22+16.53+34.74

49 = GH142.56 NS: min Z = 2.16+3(5.65)+9.65+2(12.35)+9.02+2(15.55)+19.48+22.64+47.59 = GH183.29 Total lost = 142.56 +183.29 = GH325.56

Group B2

[00,00000100,00111,0011100,01100,01000,0001,0,1,0] Constraint 1.57 + 3(2.68) + 3(3.43) + 2(4.32) + 5.41 + 6.29 + 8.74 ≥ 50.00 48.98 ≥ 50.00 not a feasible solution lost function DS: min Z = 4.14+9.65+2(7.04)+9.45+2(12.35)+15.55+11.35+14.22+16.53+22.97 = GH142.21 NS: min Z = 5.65+2(9.65)+7.04+2(9.45)+12.35+15.55+11.35+19.48+22.64+31.46 = GH162.86 Total lost = 142.64 + 163.72 = GH306.36

Group C2

[10,01111000,01000,0100000,00010,11010,1010,1,0,0] Constraint 0.6 + 4(1.57) + 2.68 + 3.43 + 4.32 + 3(5.41) + 2(6.29) + 7.53 ≥ 50.00 53.65 ≥ 50.00 feasible solution lost function DS: min Z = 1.58+2(5.65)+2(4.12)+7.04+12.35+15.55+2(14.22)+19.48+2(16.53)+19.84= GH156.68 NS: min Z = 2.16+2(5.65)+2(4.12)+9.02+9.45+11.35+2(19.48)+14.22+2(22.64)+27.18 = GH177.18 Total lost = 157.40 + 177.36 = GH333.86

50 4.2 Solving the Game Theory problem

The data structure of the game theory for the two players DS and NS is as shown table 5.6. The column player is the DS and the row player is the NS. The loads for the various feeders are arranged in increasing order in rows and columns.

DS’s linear programming problem

Since the payoff of a game cannot be zero (0), a constant k = 5 is added to the entries of the original matrix before applying the linear programming.

max Q = Y1+Y2+Y3+Y4+Y5+Y6+Y7+Y8+Y9+Y10. Subject to

5.53Y1+5.67Y2+5Y3+5Y4+5Y5+5Y6+5Y7+5Y8+5Y9+5Y10 ≤ 1

5.67Y1+6.26Y2+6.30Y3+6.35Y4+6.64Y5+6.66Y6+6.80Y7+6.80Y8+1.90Y9 +

5Y10 ≤ 1

5Y1+6.30Y2+7.21Y3+7.41Y4+7.71Y5+7.90Y6+7.90Y7+5Y8+5Y9+ 5Y10≤ 1

5Y1+6.35Y2+7.41Y3+8.05Y4+8.20Y5+8.39Y6+8.44Y7+8.61Y8+8.77Y9+

8.97Y10 ≤ 1

5Y1+6.64Y2+7.71Y3+8.20Y4+9.05Y5+9.07Y6+9.25Y7+9.47Y8+9.77Y9+

9.97Y10 ≤ 1

5Y1+6.67Y2+7.90Y3+8.39Y4+9.07Y5+10.07Y6+10.12Y7+10.40Y8+10.24Y9+

10.91Y10 ≤ 1

5Y1+6.80Y2+7.90Y3+8.44Y4+9.25Y5+10.12Y6+11.05Y7+11.10Y8+11.29Y9+

11.73Y10 ≤ 1

5Y1+6.80Y2+5Y3+8.61Y4+9.47Y5+10.40Y6+11.10Y7+12.53Y8+5Y9+5Y10 ≤ 1

5Y1+6.90Y2+5Y3+8.77Y4+9.77Y5+10.24Y6+11.23Y7+5Y8+13.74Y9+5Y10 ≤ 1

5Y1+5Y2+5Y3+8.97Y4+5Y5+10.91Y6+11.73Y7+5Y8+5Y9+18.22Y10 ≤ 1

Y1,Y2 ,Y3 ,Y4, Y5, Y6 ,Y7, Y8, Y9 Y ≥ 0

51 Table 4.2: Matrix of the linear programming above 5.53 5.67 5 5 5 5 5 5 5 5 Y1 1

5.67 6.26 6.30 6.35 6.64 6.66 6.80 6.80 6.90 5 Y2 1

5 6.30 7.21 7.41 7.71 7.90 7.90 5 5 5 Y3 1

5 6.35 7.41 8.05 8.20 8.39 8.44 8.61 8.77 8.97 Y4 1

5 6.64 7.71 8.20 9.05 9.07 9.24 9.47 9.77 5 Y5 = 1

5 6.66 7.90 8.39 9.07 10.07 10.12 10.40 10.24 10.91 Y6 1

5 6.80 7.90 8.44 9.24 10.12 11.05 11.10 11.29 11.73 Y7 1

5 6.80 5 8.61 9.47 10.40 11.10 12.53 5 5 Y8 1

5 6.90 5 8.77 9.77 10.24 11.29 5 13.74 5 Y9 1

5 5 5 8.97 5 10.91 11.73 5 5 18.22 Y10 1

Since DS’s problem is actually the dual of NS’s problem, the numerical solution of the matrix from the above linear programming of player DS automatically yields the solution of the player NS.

4.2.1 Solution to the matrix from the linear programming

The system is of the form Ay = b, where A is the matrix, y is the solution and b is a vector. To obtain the system ’A’ from the linear programming, the inequalities are changed to equations.

The numerical solution of the matrix is computed using MATLAB and the results is as follows.

Y1 = 0.4929 Y6 = 0.0815

Y2 = 0.8961 Y7 = 0.0777

Y3 = 0.0171 Y8 = 0.0170

Y4 = 0.1375 Y9 = 0.0184

Y5 = 0.4171 Y10 = 0.0135

52 4.2.2 Computing the strategies

The value of the game (Q) is given by,

Q = Y1+Y2+Y3+Y4+Y5+Y6+Y7+Y8+Y9+Y10 Q = 0.4929+0.8961+0.0171+0.1375+0.4171+0.0815+0.0777+0.0170+0.0184+0.0135 Q = 2.1688

1 1 Thus the original problem /v/ = Q − k = 2.1688 − 5 = 4.5387 The optimal strategies (probabilities) for DS are obtained as follows

Y1 0.4929 y1 = Q = 2.1688 = 0.2272

Y2 0.8961 y2 = Q = 2.1688 = 0.4132

Y3 0.0171 y3 = Q = 2.1688 = 0.0079

Y4 0.1375 y4 = Q = 2.1688 = 0.0634

Y5 0.4171 y5 = Q = 2.1688 = 0.1923

Y6 0.0815 y6 = Q = 2.1688 = 0.0376

Y7 0.0777 y7 = Q = 2.1688 = 0.0358

Y8 0.0170 y8 = Q = 2.1688 = 0.0078

Y9 0.0184 y9 = Q = 2.1688 = 0.0085

Y10 0.0135 y10 = Q = 2.1688 = 0.0062

The optimal strategies (probabilities) for NS are obtained from the dual solution of the DS’s problem above. This is given by

P = X1+X2+X3+X4+X5+X6+X7+X8+X9+X10 P = 0.0135+0.0184+0.0170+0.0777+0.0815+0.4171+0.1375+0.0171+0.8961+0.4929 P = 2.1688

X1 0.0135 x1 = P = 2.1688 = 0.0062

X2 0.0184 x2 = P = 2.1688 = 0.0085

X3 0.0170 x3 = P = 2.1688 = 0.0078

X4 0.0777 x4 = P = 2.1688 = 0.0358

X5 0.0815 x5 = P = 2.1688 = 0.0376

X6 0.4171 x6 = P = 2.1688 = 0.1923

53 X7 0.1375 x7 = P = 2.1688 = 0.0634

X8 0.0171 x8 = P = 2.1688 = 0.0079

X9 0.8961 x9 = P = 2.1688 = 0.4132

X10 0.4929 x10 = P = 2.1688 = 0.2272

4.2.3 Solving the K-GTLS problem

In a two person-zero-sum game like the game at hand, the aim of the column player (DS) is to maximize the expected payoff (load) whiles that of the row player is to minimize the expected payoff (load) of the game. The game is played with the above probabilities using the minimax theory, loads are selected into column maximums and load minimums. But the total load in each group is far greater than the capacity needed to be shed This is to say that in the K-GTLS model the GTLS plays the constraint role by selecting the loads up to the capacity of the Knapsack to be shed and KLS computes the lost function.

Table 4.3: Load shedding schedules by the K-GTLS model Group Feeders Total load Guinness II, Airport II, Y Apire, C61, M. Nkwanta 51.03 Nsuta Kumawu,F15, B71 C31,C17,HD/OHL,OBR Edwinase,E51,Lake road X 51.73 B21,techiman,N. Asafo Yabi,G31,Airport I,E21 D31,B61,F41,G11,D51 Bekwae,D41,C51,E11,G21, U Wahw/GBC,F11,E31,E41, 51.99 Bare 1,B11,St. Hubert

The selected loads are then applied to the K-GTLS model to determine the lost involve in shedding those loads. Below is the computation of the lost.

Group Y

54 Total load: 51.36 ≥ 50.00 feasible solution lost function DS: min Z = 2(9.02)+11.35+2(14.22)+16.53+22.97+34.59= GH131.92 NS: min Z = 2(12.35)+15.55+2(19.48)+22.64+31.46+47.59 = GH180.90 Total lost = 131.92 + 180.90 = GH312.82

Group X

Total load: 51.74 ≥ 50.00 feasible solution lost function DS: min Z = 2(1.58)+5(4.12)+5.65+9.45+2(7.04)+2(9.02)+12.35+2(11.35)+15.55+19.48+16.53= GH157.38 NS: min Z = 2(2.16)+5(5.64)+4.12+2(9.45)+7.04+9.02+2(12.35)+2(15.55)+11.35+14.22+22.64= GH175.61 Total lost = 157.38 + 175.61 = GH332.99

Group U

Total load: 51.12 ≥ 50.00 feasible solution lost function DS: min Z = 4.12+5.65+2(7.04)+2(12.35)+15.35+2(19.22)+2(16.58)+19.84= GH154.44 NS: min Z = 5.65+4.12+2(9.45)+2(9.02)+11.35+2(14.48)+2(22.64)+27.18= GH160.48 Total lost = 154.44 + 160.48 = GH314.92

55 CHAPTER 5

CONCLUSION

5.1 Introduction

In this chapter, the analysis of the results obtained from the KLS and that of the K-GTLS models in the previous chapter are presented, comparisons are made, conclusions are drawn, load shedding is done and consequently recommendations are made.

5.1.1 Analysis of the result in the KLS

Table 5.5 represents the summary of the result obtained when the data was fitted onto the KLS model in chapter 4. The groups numbering I to X contain feeders selected by the KLS model to be shed, such that a minimum load 50.00MW is obtained in each case. The table also presents the amount of money lost by the company, should power of those feeders be taken off in both the day and the night schedules.

It is obvious from the table, that some groups are better off scheduled in the day than in the night and vice versa. For instance, groups I, II and III as well as VIII, IX and X captured all feeders under study once in every group. It is observed that, in groups I and II, it is more appropriate to shed load in the night than in the day, but the opposite is true for group III. In the same breath, it is more appropriate to shed load in the night for groups IX and X than in the day, whilst it is 50-50 for group VIII.

56 5.1.2 Analysis of the results from the K-GTLS model

It is clear from table 4.3 that, the minimax theory of a pure strategy works perfectly in the mixed strategy theory in that, when the game is control by the NS to play for the row minimums, the group X is obtained. On the other hand when the game is control by the DS to play for the column maximums, the group Y is obtained. The items in these two groups X and Y depict pure maximini and minimaxi values respectively. But the constraint that, some minimum amount of 50MW load must be reached, call for the fusion of the game theory into the knapsack. The items that could not make it for groups X and Y are placed in group Z, to a large extent, they qualify to belong to the Y group.

Table 5.1: Summary of results from the K-GTLS model. Group Amount lost/GH Total/GH DS NS Y 131.92 180.90 312.82 X 157.38 175.61 332.99 U 154.44 160.48 314.92

Table 5.1 above represents the summary of the results obtained by the K-GTLS model. It contains the groups Y, X and U, with the amount lost/GHfor the DS and the NS. It is obvious from the table, that just as in the KLS model, some groups are better off scheduled in the day than others.

5.2 Load shedding

The load shedding is being presented in two ways. These include the 1. The cyclic load shedding, 2. The revenue bias load shedding.

57 5.2.1 The cyclic load shedding

The cyclic mode of load shedding as employed by the ECG, places much emphasis on equity and fair distribution of power to customers. The tables below show a three days cyclic load shedding for three groups. In the table, a zero(0) means OFF and a one(1) means ON for a group.

Table 5.2: A three days cyclic load shedding

Group DS NS 1ST 0 1 Day 1 2ND 1 0 3RD 1 1

Group DS NS 1ST 1 1 Day 2 2ND 0 1 3RD 1 0

Group DS NS 1ST 1 0 Day 3 2ND 1 1 3RD 0 1

There are three(3) days in a cycle and ten (10) cycles in a month. The amount lost per cycle is calculated and the result multiply by ten (10) for the amount lost per month. The result is presented as follows: KLS: GH9621.50 K-GTLS: GH9607.30 ECG: GH9667.70

58 5.2.2 Revenue-bias load shedding

In this proposed mode of load shedding, much emphasis is placed on revenue maximization as well as giving consumer power at the right time and place. There are six (6) days and six (6) nights. The amount lost in this period is calculated and it is multiplied by five (5) to obtain the amount lost in 30 days period;

Considering groups VIII, IX and X in table 5.5. If group VIII is scheduled one day and four nights off, group IX gets two days and one nigh off and group X gets three days and one night off, then the amount lost is computed as follows. Group VIII loses = 159.78 + 4(159.36) = GH797.22 Group IX loses = 2(148.42) + 166.83 = GH463.67 Group X loses = 3(138.30) + 189.45 = GH604.35 Total lost = GH9326.20

In the K-GTLS model, if group Y is scheduled three days and one night off, group X gets one day and one night off and group U two days and four nights off, then the amount lost is computed as follows. Group Y loses = 3(131.92) + 180.90 = GH576.66 Group X loses = 157.38 + 175.61 = GH332.99 Group U loses = 2(154.44) + 4(160.48) = GH950.80 Total lost = GH9302.25

Considering the ECG load shedding, if group A2 is scheduled two days and one night off, group B2 gets three days and four nights off and group C2 one day and one night off, then the amount lost is computed as follows. Group A2 loses = 2(142.56) + 183.29 = GH468.41 Group B2 loses = 3(142.64) + 4(163.72) = GH1082.80 Group C2 loses = 157.40 + 177.36 = GH334.76

59 Total lost = GH9429.85

5.3 Summary of results from the load shedding

Table 5.3: Summary of results from the load shedding Model Amount lost per month/GH Cyclic mode Revenue-bias mode KLS 9621.50 9326.20 K-GTLS 9607.30 9302.25 ECG 9667.70 9429.85

5.4 Conclusion

The following deductions are made from the analysis of the results. 1) It can be seen from table 5.3 that the proposed revenue-bias load shedding: • helps the company to reduce lost than the cyclic mode already employed by the ECG. • serves the consumer energy at the right time and place. 2) Whether revenue bias or cyclic mode, the K-GTLS solves the problem better than the KLS.

3) The fusion of Knapsack and Game Theory is a good approach to solving resource constraint project scheduling problem.

5.5 Recommendation

In a situation where load shedding is done for a large number of feeders, the manual application of the minimax theory in the selection process is very tedious and time consuming, i therefore recommend that a software be developed for the selection process of the K-GTLS.

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64 APPENDIX A

Table 5.4: ECG load shedding groupings in may 2013 Groups Feeder(x) Load/MW C31 3.05 C17 4.05 Techiman 0.67 Nsuta Kumawu 13.22 G11 1.26 F51 4.77 A2 C61 5.07 D31 6.10 G31 3.20 OBR 1.35 E51 1.66 Yabi 2.21 Guinness I 3.39 total F41 2.90 M.Nkwanta 8.74 B11 6.05 Wahw/GBC 3.44 Airport II 5.91 B2 Bare 1 3.77 Apire 3.97 Lake road 2.90 C51 2.71 B61 1.64 Airport I 4.24 E21 4.47 total

65 Groups Feeder(x) Load/MW E31 4.07 Edwinasi 1.30 F11 1.80 N. Asafo 0.53 E41 5.12 ND/OHL 3.61 D51 5.40 St. Hubert 5.52 C2 B21 1.80 B71 6.29 G21 2.41 E11 6.73 Bekwae 7.53 D41 1.90 total

66 APPENDIX B

Table 5.5: Load shedding schedules by the KLS model. G F L A NS DS T G11, B61, OBR, E51, Yabi, G21, C17, F15, I 51.65 186.11 135.84 321.95 Guinness 2, C61, B11, Bekwae, Manso Nkwanta N. Asafo, techiman, F11, Edwinase, C51, F41, Apire, II 50.67 184.36 134.62 318.98 G31,Airport I, C31, E41, D31, St. Hubert,Nsuta Kwaman B21, D41, Lake Road, Bare 1, HD/OHL, Wahw/GBC, E21, E31, III 51.58 153.04 168.27 321.31 Airport II, D51, B71, E11 G11, F11, Edwinase, B21, D41, B61, OBR, E51, E41, IV 51.50 167.60 154.57 322.17 Airport II, D51, St. Hubert, Bekwae Nsuta Kwaman, Manso Nkwanta, C31, HD/OHL, Wahw/GBC, Bare 1, V 50.31 164.74 154.26 319.00 Apire, G31, N. Asafo, Techiman, Guinness 2,B21, D41 Edwinase, F11, B21, D41, Yabi, C51, F41, Lake road, G21, C17 VI 50.02 163.74 148.62 312.36 Airport 1, E21, E31, F51, Manso Nkwanta Nsuta Kumawu, B11, D31, B71, VII E11, G11, Edwinase, F11, B21, 50.94 180.29 136.88 317.17 D41, B61, OBR, E51

67 G F L A NS DS T N. Asafo, F11, B21, D41, C51, F41, HD/OHL, Wahw/GBC, Bare 1 VIII 51.24 159.36 159.78 319.40 Airport 1, E21, C61, E41 Airport II, St. Huber D31, D51, F51, E31, Guinness 2, G31, Apire, C31, IX 51.75 166.83 148.42 315.25 Lake road, Yabi, E51, OBR, B61, Techiman. G21, B11, B71, E11, Bekwae, X 51.04 189.45 138.30 327.63 Manso Nkwanta, Nsuta Kumawu.

G = Group F = Feeder L = Load(MW) A = Amount lost/GH T = Total

68 APPENDIX C

Table 5.6: Data structure of the game theory

DS

Y1 Y2 Y3 Y4 Y5 Y6 Y7 Y8 Y9 Y10

X1 0.53 0.67 0 0 0 0 0 0 0 0

X2 0.67 1.26 1.30 1.35 1.64 1.66 1.80 1.80 1.90 0

X3 0 1.30 2.21 2.41 2.71 2.90 2.90 0 0 0

X4 0 1.35 2.41 3.05 3.20 3.39 3.44 3.61 3.77 3.97

NS X5 0 1.64 2.71 3.20 4.05 4.07 4.25 4.47 4.77 0

X6 0 1.66 2.90 3.39 4.07 5.07 5.12 5.40 5.25 5.91

X7 0 1.80 2.90 3.44 4.25 5.12 6.05 6.10 6.29 6.73

X8 0 1.80 0 3.61 4.47 5.40 6.10 7.53 0 0

X9 0 1.90 0 3.77 4.77 5.52 6.29 0 8.74 0

X10 0 0 0 3.97 0 5.91 6.73 0 0 13.22

69 APPENDIX D

DS’s strategies (probabilities) NS’s strategies(probabilities y1 = 0.2272 x1 = 0.0062 y2 = 0.4132 x2 = 0.0085 y3 = 0.0079 x3 = 0.0078 y4 = 0.0634 x4 = 0.0358 y5 = 0.1923 x5 = 0.0376 y6 = 0.0376 x6 = 0.1923 y7 = 0.0358 x7 = 0.0634 y8 = 0.0078 x8 = 0.0079 y9 = 0.0085 x9 = 0.4132 y10 = 0.0062 x10 = 0.2272

70