GEP and TEP Nigeria Power System.pdf

May 11, 2018 | Author: joe | Category: Mathematical Optimization, Linear Programming, Electric Power Transmission, Power Station, Electricity
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GEP and TEP : Nigeria Power SystemMaster Thesis Project February, 2012 Supervised by Assistant professor Mohammad R. Hesamzadeh Examined by: Assistant professor Mikael Amelin Electrical Power Division School of electrical Engineering Royal Institute of Technology (KTH) Stockholm, Sweden. To every Nigerian who longs for a day when Nigeria can boast of uninterrupted power supply ii Acknowledgement Firstly, I would like to thank my examiner, Mikael Amelin for accepting the masters’ thesis proposal and seeing a prospect in under going this project. I would also like to thank my supervisor, Mohammad R. Hemsazadeh, who agreed to supervise the project and was very helpful throught the period of carrying out this thesis Project. Special thanks to Engr. Charles K. Nnajide and Engr Deji Ojo of the Power Holding Company of Nigeria who assisted me in data collection. Without your help, the work would have been next to impossible. I cannot fail to thank my colleagues, Mahir Sarfati, Maria A. Noriega, John Laury, Olga Galland for the amiable and friendly ambience of Bobenko room during the execution of this project. It was fun sharing a work space with you all. I cannot but thank these special people to me, Olayinka Irerua, Eric Okhiria, Olajumoke Oke. These years in Sweden and at KTH would have been difficult without your support, encouragement and love. Very special thanks to my parents, Mr and Mrs Faleye, my siblings, Folake and Wuyi and my family for your support throughout my studies. Blood is indeed thicker than water. Finally, I want to thank God for his mercies and grace through out my life. Without You, none of this would have ever happened. iii Abstract The Nigerian power system is one plagued with incessant load shedding due to inadequate generation and transmission capacities. Currently, less than 40% of the population is connected to the national grid and less than 50% of the available installed capacity is actively used in meeting demand. A new wave of energy reforms is on-going in the nation. There are proposed generation and expansion plans. These reforms have only fully taken into consideration present demands and not future energy demands. This means that even with new plants and transmission lines being constructed; there may still be inefficient generation and transmission capacities due to demand increase. This thesis models the uncertain future demands in the integrated generation-transmission planning model. An optimal investment plan is found using the deterministic optimization model of integrated generationtransmission planning. A decision analysis method was initially used to study the introduction of uncertain demand into the deterministic model. Then, a two-stage stochastic model of the generation-transmission planning taking into account uncertainties in energy demand is developed using scenario-wise decomposition method. The demand was modelled as taking discrete values with certain probabilities. These models are mixed-integer linear programming problems. They are implemented in the GAMS platform and solved using the CPLEX solver. A stylized version of the Nigerian power system is developed and tested. A thorough analysis and comparison of results from the models were carried out using the developed version of the Nigerian transmission grid. iv Table of Contents Acknowlegement……………………………………………………………………………………………………...…iii Abstract……………………………………………………………………………………………………………………...iv List of tables……………………………………………………………………………………………………………….vi List of figures…………………………………………………………………………………………………….………vii 1. Introduction......................................................................................................................... 1 1.1 Background ................................................................................................................ 1 1.2 Problem Definition and Objective.............................................................................. 2 1.3 Methodology and tools used ...................................................................................... 2 1.4 Report Overview ........................................................................................................ 3 2. Introduction to Generation-Transmission Expansion planning .......................................... 4 3. Basics of Optimization Theory ........................................................................................... 8 3.1. Deterministic Linear Programming............................................................................ 8 3.2. Stochastic Linear Programming ................................................................................. 8 3.2.1. Two-stage stochastic linear programming with recourse........................................... 8 4. Case study ......................................................................................................................... 10 4.1. Background on the Nigerian Society ....................................................................... 10 4.2. Overview of the Current Nigerian Power System.................................................... 12 4.3. Data and Inputs......................................................................................................... 14 5. Models............................................................................................................................... 20 5.1. Introduction .............................................................................................................. 20 5.2. Deterministic Model................................................................................................. 20 5.3. Scenario-based Decision analysis Model ................................................................. 21 5.4. Stochastic Model ...................................................................................................... 22 5.5. Simulation parameters.............................................................................................. 23 6. Simulation Results and Discussion ................................................................................... 25 6.1. Individual Model Results ......................................................................................... 25 6.1.1. Deterministic Model................................................................................................. 25 6.1.2. Regret analysis ......................................................................................................... 32 6.1.3. Two-stage stochastic model ..................................................................................... 36 6.2. Comparison of Deterministic and Stochastic results................................................ 43 7. Conclusions....................................................................................................................... 46 8. Future Work ...................................................................................................................... 47 References ................................................................................................................................ 48 Appendix I.................................................................................................................................. 1 v List of tables Table 1 Percentage cost component of a HV transmission line ................................................. 4 Table 2 Construction time(years) for different types of power plants ....................................... 5 Table 3 Power Generation capacity of the current Nigerian grid............................................. 12 Table 4 Generation capacity of proposed new plants .............................................................. 13 Table 5 Bus numbering of the network for this study .............................................................. 15 Table 6 Transmission Branch numbering for this study. ......................................................... 16 Table 7 Load/demand nodal distribution .............................................................................. 18 Table 8 Cost parameters used in the GAMS Model................................................................ 24 Table 9 Operation costs as used in the models......................................................................... 24 Table 10 Deterministic model: Power plant construction result, V ≥ 0 ................................... 25 Table 11 Deterministic Model: Transmission line construction results, V ≥ 0 ......................... 25 Table 12 Deterministic model: power plant construction result, V ≥ 1 ..................................... 27 Table 13 Deterministic Model: transmission line construction result, V ≥ 1 ............................ 28 Table 14 Deterministic model: Optimal Investment Costs comparison for compulsory and non compulsory construction of new plants .................................................................................... 30 Table 15 Future demand for the regret analysis ....................................................................... 32 Table 16 power plant construction plan for regret analysis ..................................................... 33 Table 17 Generation in power plant for regret analysis ........................................................... 33 Table 18 Investment cost realization for regret analysis ......................................................... 34 Table 19 Regret in cost and served demand ............................................................................ 34 Table 20 Stochastic Model: power plant construction result, Vi ≥ 0 ...................................... 36 Table 21 Stochastic Model: transmission line construction results, Vi ≥ 0 ............................. 36 Table 22 Stochastic Model: Power plant construction result Vi ≥ 1 .......................................... 38 Table 23 Stochastic Model: Power line construction Vi ≥ 1 ..................................................... 39 Table 24 Cost comparison for the stochastic models ............................................................... 41 Table 25 Investment cost comparison for both deterministic and stochastic model. ............... 44 Table 26 Optimal network topology result ............................................................................. 45 i i i i vi List of Figures Figure 1 Map of Nigeria showing population density.............................................................. 11 Figure 2 Map of Nigeria showing transmission grid layout.................................................... 11 Figure 3 One line diagram of the Nigerian transmission network showing existinglines(black lines) and proposed new branches(pink lines) ......................................................................... 14 Figure 4 Deterministic Model :Power generation in power plants .......................................... 30 Figure 5 Deterministic model: Investment cost distribution ................................................... 31 Figure 6 Deterministic model: Network topology showing new line construction ................ 32 Figure 7 Regret calculated for demand .................................................................................... 35 Figure 8 Regret calculated for investment cost ........................................................................ 35 Figure 9 Stochastic Model :Average power generation in power plants ................................. 41 Figure 10 Stochastic Model: Investment cost distribution ....................................................... 42 Figure 11 Stochastic Model: network topology showing new line construction. .................... 42 Figure 12 Power plant construction variable for new plants for deterministic and stochastic model. ....................................................................................................................................... 43 Figure 13 Power generation in new power plants for deterministic and stochastic model. .... 43 Figure 14 Cost comparisons of deterministic and stochastic model ........................................ 44 vii 1. 1.1 Introduction Background Electricity is very important to the social and economic development of any country. All aspects of the life of the citizenry is affected by power supply, ranging from keeping a clean home to running multinational companies. Without adequate power supply, businesses, homes and the society at large cannot function to their full capacity. Goods and services would cost more than they should if every business owner has to own a private generating unit; running a home will be rigorous if there is no means of storing food due to non functional refrigerating systems; health care provision would be substandard; unemployment would increase due to fewer companies and these may lead to high crime rates; life will be boring if access to entertainment is limited due to inadequate power supply. The electric power system of a country should be built to meet the electricity demand of the citizens. Every household and business office should have access to adequate power supply. The problem of inadequate power supply can be tackled by generation upgrade and/or expansion. This means that more generating units can be added to existing power plants or new power plants can be built at new locations in a nation’s power grid. Additional generation always result in increased power flow on transmission lines in the grid. If the existing transmission network is not capable of transferring this added generation, then an upgrade or expansion of the transmission system is also needed. Any expansion planning involves determining where, when and how many new units must be added to an existing system at lowest cost taking into consideration future demand values [1]. Traditionally, transmission and generation expansion planning have been done independently but in this thesis a simultaneous expansion plan is considered. In general, expansion plans have been formulated to minimise investment cost of new units while meeting technical and social constraints. Since the number of transmission lines and generating units are integer values, an expansion plan is a mixed integer problem. In addition, the planning can either be a one stage, two-stage or a multistage problem [2],[3]. The planning problem is generally an optimisation problem which can be solved using a variety of methods like those described in [4]-[6], [9]. When considering uncertainty in any of the parameters of an optimisation problem, the problem becomes a stochastic optimisation problem. Stochastic optimisation problems are classified into different categories such as those described in [7]-[9]. Based on the type of stochastic problem, there exist different solution algorithms such as convex approximations, stage wise decomposition and scenario wise decomposition to mention a few. The Nigerian power system is currently suffering from inadequate generation and transmission capacity. The demand is much higher than the generation and this has led to constant load shedding and erratic power supply [11]. The installed generation capacity of the Nigerian power system is currently estimated at 8800MW of which 25% is hydro and 75% is gas fired thermal plants [12]. 1 At present, only about 4200 MW is generated on the average compared to the installed capacity. The Nigerian government is working on rural electrification and connection of more consumers to the grid. To meet this growing demand, the government has given permission to individual organizations to build thermal power plants. This means that the system is moving from a vertically integrated electricity market structure to a more bilateral electricity market structure. The existing transmission network which currently consists of mostly 330 KV power lines and a few 132 KV lines are weak with high energy losses close to 44% [14]. The network is also mostly radial. The existing transmission system is not sufficient to transfer the additional power injected to the grid by the new power plants. 1.2 Problem Definition and Objective In this thesis, we consider the proposed generation expansion and suggest a few transmission expansion paths for the Nigerian power grid. The proposed transmission expansion paths were aimed at making the current network which is radial to become more meshed. The expansion plans for both generation and transmission includes construction of new power plants and transmission lines as well as upgrade of existing lines. This thesis work aims to propose an expansion plan. In simple terms we aim to answer the following questions: • Which of the proposed power plants should be built and how much power should it produce? • Which of the proposed new lines should be built and how many units? • Which existing power lines should be upgraded? It most be stated however that this study does not aim to solve all the power adequacy problems of Nigeria or even propose a comprehensive generation/transmission investment planning. The study aims to give suggestions on which power plants (i.e. The location in the grid and their power capacity) and transmission lines the Nigerian government should be planning to build. It also aims to provide a good basis for further in-depth study into the problem of power supply and long term transmission and generation planning in Nigeria. 1.3 Methodology and tools used Traditionally, generation and transmission planning has been done separately but in this study, the generation and transmission expansion plan was modelled simultaneously to consider their feasibility and necessity. A deterministic optimisation plan was modelled. The deterministic model was run for two cases: an expansion plan where construction of all new power plants is compulsory and an expansion plan where construction of all new power plants is not compulsory. For both of these plans, the transmission network is considered for both upgrade of old lines and expansion. In addition, a two-stage stochastic model was developed. The changes expected in demand over the coming years were modelled. The demand was modelled as an uncertain parameter with discrete probabilistic distribution. The expected value of the objective function and the generation and transmission plan for this case was optimized. 2 In the two stage stochastic model, the first stage decision variables are the numbers of power plants units and transmission lines to be built. The optimisation models were written in GAMS platform and solved using a CPLEX solver. Results were analysed in Microsoft Excel. A PSSE model of the system was also developed. PSS/E has been used for a better graphic presentation of the Nigerian grid. 1.4 Report Overview Chapter 2 gives the reader an introduction to the formulation of an expansion planning problem. Chapter 3 introduces optimization theory and specifically stochastic programming. Chapter 4 focuses on the case study of this project i.e. the Nigerian system while Chapter 5 focuses of the models developed. Chapter 6 discusses simulation results while Chapter 7 is a summary of the thesis work, conclusions based on the work carried out. Finally, Chapter 8 is a proposal for future work in this area. 3 2. Introduction to Generation-Transmission Expansion planning Expansion planning can involve transmission or generation expansion planning. In the case of generation expansion, this involves the increase of power generation in an existing power system. Transmission expansion on the other hand involves the expansion of the power grid which usually entails building new transmission lines. These two aspects of power system expansion usually go hand in hand and it’s therefore reasonable to combine the generation and transmission expansion planning. Expansion planning is a very typical example of an optimisation problem. The cost of investment is minimised while meeting physical and social constraints. Decision variables for an expansion plan are the number of units to build, the location of the units and the capacity of the units. The constraints are the so called energy-balance constraints, transmission constraints and generation constraints. An expansion plan can formulated as follows: Minimise the transmission investment cost + generation investment cost+ operation cost Subject to energy balance constraint Generation constraint Transmission constraint The parameters for the expansion plan include data on the existing power system i.e. the generation and transmission capacity, the network topology, load forecast at each node, possible candidate new generation lines and candidate new lines. The different terms of the optimisation plan are explained in details below: Costs Transmission Investment Cost This is the cost of building a new transmission line. It usually includes the cost of towers, conductors , insulators , clamps and also the cost of labour. Reference [13] has details on the percentage of each cost component. It must be noted that this cost may vary based on topography and location and labour cost. Transmission cost is usually expressed in ¤/ km. Table 1 shows a sample of percentage cost component for a high voltage line. Table 1 Percentage cost component of a HV transmission line Component % cost Conductors 30 Towers 24 Foundation 14 Insulators 7 Earth wire 3 4 Land cost Studies Miscellaneous 6 9 8 Generation Investment cost This is the cost of building new power plant. The cost varies based on the type of power plants. Cost is expressed in ¤/MW or ¤/kW. The gestation period i.e. the construction time of different power plants also varies. Coal- fired, gas -fired, nuclear and wind power plants have higher costs but shorter gestation period while the hydro power plants have lesser costs but longer gestation period. Table 2 shows the approximate construction time for different types of power plants. Table 2 Construction time(years) for different types of power plants Power plant type Construction period, years Coal-fired 4 Gas- fired 2-3 Nuclear 5 Wind 1-2 Hydro 8 Operation costs These are the cost of running a power plant. It is mainly the cost of fuel. Generator maintenance costs are also included and for thermal power plants, start-up costs are may also be included. It is obvious that hydro and wind power plants have the lowest operation costs. Mathematically, the objective function can be expressed as follows ∑c v + ∑c v L + ∑c i i i l l l l v Gi oi i (2.1) i Where ci is the cost of building a unit of power plant i cl is the cost/km of building power line l coi is the operation cost/MWh of power produced in plant i vi , vl is the construction integer variable of plant i and line l respectively. vi , vl = o , then no construction  vi , vl ≥ 1 then construct L, is the length of power line l and Gi is the total generation in plant i. 5 Constraints Energy balance constraint This constraint in general form ensures that the sum of generation is equal to the sum of demand and losses in the transmission system. G ∑ D + losses ∑= (2.2) Generation constraint This constraint ensures that the generation in each power plant does not exceed the maximum generation capacity of that power plant. In case of plants where there is a minimum allowable generation, the constraint compels the power plant to generate within this range. G i ≤ Gi ≤ Gi (2.3) Transmission constraint This constraint reflects the physical laws governing electricity transmission. Power flow on each line should not exceed the thermal capacity of the line. It can be written simply as: F l ≤ Fl ≤ Fl (2.4) When considering only active power flow, DC power flow calculation can be used to model the transmission constraint. The following assumptions are made for a DC flow: • All line resistances are negligible in comparison to line reactances i.e. r <<X • Angular differences between two nodes are very small. • A flat voltage profile is assumed. The third assumption can be achieved by using per unit calculations. Active power flow on line between buses k and j taking into account the above assumptions is given by: Fl ≈ VkV j Xl sin θ kj (2.5) Where Vk , V j are bus voltages X is the line reactance θ kj is the angular difference between buses k and j Taken into consideration assumptions 2 and 3, equation (2.5) becomes 6 F= Bl ⋅ θ kj l where Bl = 1 Xl (2.6) If A denotes the branch-node incidence matrix, θ the vector of bus phase angles reduced by removing the slack bus row, X denotes the diagonal matrix of line reactances then equation and Fl the vector of power flows then equation (2.6) can be written as AT θ = XFl (2.7) (2.8) Fl =  X-1AT  θ If F denotes the vector of all injected power at each node, then applying Kirchhoff’s “current” law F = AFl (2.9) (2.10) = F = AX-1AT  θ Bθ By combining equation (2.8) with equation(2.10), the phase angles can be eliminated resulting in Fl =  X −1AT B -1  F Fl = TF (2.11) (2.12) Where T is the matrix of the Power transfer distribution factors (PTDF) showing the effect of nodal injections on the power flow on a line. If the power injection in each node, n , is calculated as : F= Gn − Dn n (2.13) Then equation (2.12) can then be substituted in equation (2.13) Fl ≤ Tl ,n (Gn − Dn ) ≤ Fl (2.14) For more details on transmission constraints and calculation of the PTDF matrix, see [15] and [16]. Note: Other constraints that may be included in the expansion plan formulation include limits on variables such as ug and ul 7 3. Basics of Optimization Theory In optimization theory, a given function often called the objective function is to be minimised or maximised subject to a set of constraints. The controllable variables of the optimisation problem are called optimisation or decision variables. Since it is common that the objective function is a cost function that needs to be minimised, a standard minimization problem is expressed as follows: min f ( x) (3.1) s.t x∈χ The set of constraints can be either a set of functions, a set of variable limits or a combination of both. If the function f(x) and the constraint functions are linear functions, then the optimisation problem is referred to as a linear programming problem. If any of the element of x is an integer variable, the problem becomes an integer linear problem. 3.1. Deterministic Linear Programming A deterministic linear programming problem is one in which all the parameters are known with certainty and all functions describing the problem are linear functions. In general form, it is formulated as follows: min CT x s.t (3.2) Ax ≤ b x≥0 Where x is a vector of variables to be determined by the optimization process, b and c are vectors of known coefficients and A is a matrix of known coefficients. 3.2. Stochastic Linear Programming 3.2.1. Two-stage stochastic linear programming with recourse In stochastic linear programming, one parameter or a set of parameters are uncertain. The variables which have to be determined without full information of the uncertain parameter are called first stage variables. This means that optimisation is carried out on an expectation. It is generally stated as follows: min CT x + Ε ω [Q(x, ω )] x Ax ≤ b x≥0 (3.3) 8 Where Q(x, ω ) min {qT y Wy ≤ h − Tx, y ≥ 0} = Ε ω Denotes the expectation with respect to ω. ω is a random vector. x, represent the first stage decision variables and y , the second stage decision variables. The vectors q, W, h, and T describe the actual mathematical realisation of the random parameter. If we define Ω( x) = Eω [Q( x, ω )] , then a deterministic equivalent of the above problem can be written as: min CT x + Ω(x) (3.4) s.t Ax ≤ b x≥0 The method and algorithm for solving a stochastic linear problem is very much dependent on the expression of the uncertain parameter. References [7] and [8] have details on different algorithms and methods for solving stochastic linear problems. In the case where a discrete probabilistic distribution of the uncertain parameter is known as is assumed to be the case in this study as explained later in chapter 5, then the two-stage stochastic problem can be formulated as equation (3.5) if ω takes a set of values which can be also called scenarios(ω1,…, ωs) with probabilities (p1,…ps). This algorithm is known as scenario wise decomposition. min ∑ p (C x T s s Ax s ≤ b, s ∈ S s + q sT y s ) (3.5) 9 4. Case study This chapter gives a background on the Nigerian power system which is the main focus of this thesis work. All models developed were done for this system. 4.1. Background on the Nigerian Society Geography and Demography Nigeria is a country situated in the western part of Africa. It shares borders with Benin Republic in the west, Cameroon and Chad in the east, and Niger in the north. The coast on the south of Nigeria lies on the gulf of Guinea. Nigeria is the most populous country in Africa with a population of approximately 155 million people according to the last census conducted in July 2011 and has a population growth rate of 2% [20] . The total land area is 923,768 km2. The country is divided into 36 states and the Federal Capital Territory (FCT). There three major tribes i.e Hausa, Igbo and Yoruba and over 250 ethnic groups in Nigeria. The south-western part of the country is dominated by the Yorubas who are widely educated while the eastern part of the country is mostly dominated by the Igbos generally known for there business acumen. The northern part of Nigeria is dominated by the Hausas and Fulanis whose majority live in villages and small towns. A great number of the Hausas are not educated and are predominantly crop and livestock farmers. Nigeria has two major rivers: Niger and Benue. River Niger has two dams located on it, namely Kainji and Jebba and these are used for hydro electric power generation. The two rivers meet and empty into the Niger delta region in the south of Nigeria. This region is known as the oil region of Nigeria. Approximately 70% of the nation’s oil and gas is produced from this region. figure 1 and figure 2 shows the map of Nigeria showing the population density and the transmission grid respectively. There are higher population densities in the south west and south-south than in the east and north. These population densities reflect even in the distribution of transmission lines across the nation. 10 Figure 1 Map of Nigeria showing population density Figure 2 Map of Nigeria showing transmission grid layout Economic Activities Major economic activities include agriculture, petroleum exploration, telecommunication and banking. Nigeria is the 8th largest exporter of oil in the world and crude oil is currently 11 the major export product. Agriculture used to be the major economic sector in the country before the oil boom and even today, a large percentage of the population are farmers. Telecommunication has been growing very fast since year 2000 and continues to grow. Aside from these developed sectors, there are other industries such as textile, leather, food processing and movie which depend on power supply to run in the country. Current power Situation Nigeria has more than enough energy sources to meet the power demand of the people. However, only a small percentage of the populace have constant access to power. According to the Nigerian power sector review of 2010, only about 45 % of the population is connected to the grid [17]. On an average basis, approximately 45% of demand is met. This means that most homes have access to electricity only 60% of the time while some even have power only 30 % of the time. Firms and companies also report outages and it is very common for homes and firms to have their own generation units [17]. Aside from the economic implications of these current power problems, there are also environmental and health issues associated. 4.2. Overview of the Current Nigerian Power System. Generation There are 16 existing power plants in the system. Table 3 shows the installed capacity of the power plants and the current actual generation. Table 3 Power Generation capacity of the current Nigerian grid Power plant Installed capacity (MW) Average availability (MW) Hydro power plants Kainji 760 412,55 Jebba 578.4 431,83 Shiroro 600 390,21 Thermal power plants Egbin 1320 819,55 Sapele 720 125,17 Delta II-IV 900 342,95 AfamII,IV,V, VI 1166 457,2 Geregu 414 208,69 Omotosho 335 118 Olorunshogo I,II 710 324 Okpai 480 441,57 Omoku 150 80,18 Ajaokuta G.S 110 0 Ibom G.S 155 82,89 AES 302 208,20 Trans-Amadi 100 32,63 Total 8800,4 4475,87 12 As shown in table 3 above, the average availability of the power plant is around 50%. This according to information from the PHCN(Power Holding Company of Nigeria) is due to faulty generators, lack of machine maintenance and generally aging generatiors in the old power plants e.g Kainji hydro power plant which was commissioned in 1968. Due to the government’s commitment to improve the power situation of the country, there are a number of projects under way to expand the generation and consequently the grid. There are several government owned and independent power plant projects under way. Table 4 shows the power plants under construction at the moment and expected to be connected to the grid between 2012 and 2020 which were considered in this study [21]. Table 4 Generation capacity of proposed new plants Power plant Installed capacity, MW Calabar 561 Egbema 338 Ihovbor 451 Gbarian 225 Alaoji I 504 Eket 500 Obite 450 Total 3029 It must be noted that in this study, these power plants with their installed capacities were used symbolically. The installed capacities shown in table 2 are used to represent a unit of these power plants. Their location in the network topology was of much importance. Transmission The transmission grid currently consists of 40 nodes, 44 branches and 63 lines. The transmission grid consists mainly of 330KV transmission lines but there are a few 132KV lines. For the purpose of this study, 20 new branches have been proposed to reinforce the system and to effectively evacuate the proposed additional generation. The proposed system has 8 additional nodes for the generation nodes and one suggested transmission node. Figure 3 shows the single line diagram of the Nigerian power system modelled in PSS/E software . A manual on modelling in PSS/E can be found in [22] . The diagram shows the current system with the proposed transmission expansion plan. 13 Figure 3 One line diagram of the Nigerian transmission network showing existinglines(black lines) and proposed new branches(pink lines) 4.3. Data and Inputs Due to the strong role of the case study in this thesis, data on the Nigerian power system was very important to carry out the study. These data were not easily available online or even with the PHCN (Power Holding company of Nigeria) due to confidentiality issues. Access to data was limited but some data were finally accessible. Line data for most of existing transmission lines, generation capacities of existing lines and sample of power flow outputs were some of the data that was finally available throught PHCN. However some of the data were not 100% accurate as changes in the grid in recent years have not been properly documented. In addition to this, information on future expansion plans, demand forecast studies and investment cost were not readily available. Some of the parameters used in the model have been assumed based on research done by reading papers, books and online articles and Web pages. For the purpose of this study, the network nodes have been numbered and named as shown in table 5 below. Buses 41-48 are proposed new buses for the new power plants. Bus 46 however, is a proposed transmission node. 14 Table 5 Bus numbering of the network for this study Bus number Bus name Bus KV 1 B.KEBBI 330 2 KAINJI G.S 330 3 JEBBA G.S 330 4 JEBBA T.S 330 5 OSOGBO 330 6 AYEDE 330 7 OLORUNSOGO 330 8 SAKETE 330 9 IKEJA WEST 330 10 AKANGBA T.S 330 11 EGBIN G.S 330 12 AJA T.S 330 13 GANMO 330 14 KADUNA 330 15 SHIRORO G.S 330 16 KATAMPE T.S 330 17 OMOTOSHO G.S 330 18 JOS T.S 330 19 KANO T.S 330 20 BENIN T.S 330 21 DELTA G.S 330 22 SAPELEG.S 330 23 ALADJA T.S 330 24 AJAOKUTAT.S 330 25 GEREGU G.S 330 26 GOMBE T.S 330 27 ASCO G.S 132 28 NEW HAVEN T. 330 29 ONITSHA T.S 330 30 OKPAI P.S 330 31 ALAOJI T.S 330 32 AFAM 330 33 EKET T.S 132 34 IBOM G.S 132 35 YOLA T.S 330 36 MAIDUGURI T. 132 37 PH MAIN T.S 132 38 AES 132 39 T. AMADI G.S 330* 40 OMOKU G.S 330* 41 IHOVBOR 330 42 ALAOJI G.S 330 43 EKET G.S 132 44 CALABAR 330 45 OBITE G.S 330 46 OMOKU T.S 330 15 47 48 GBARIAN G.S EGBEMA G:S 330 330 * Omoku G.S and Trans-amadi G.S are in reality connected to the distribution grid through 33kv lines. The actual connection point and parameters of the line were not available in the data collected. As a result, they have been connected to PH MAIN through a 330 KV line. This is done so as to simplify the modelling of the PTDF matrix in GAMS. The numbering and notation system for lines used are given in table 6 below. From Bus Number 1 1 2 2 2 3 3 3 4 4 4 4 4 5 5 5 5 6 7 8 8 9 9 9 9 9 9 10 11 11 11 11 11 13 14 14 Table 6 Transmission Branch numbering for this study. From Bus To Bus To Bus GAMS Name Number Name numbering B.KEBBI 2 KAINJI G.S L1 B.KEBBI 3 JEBBA G.S L2 KAINJI G.S 4 JEBBA T.S KAINJI G.S 4 JEBBA T.S L3 KAINJI G.S 14 KADUNA L4 JEBBA G.S 4 JEBBA T.S JEBBA G.S 4 JEBBA T.S L5 JEBBA G.S 6 AYEDE L6 JEBBA T.S 5 OSOGBO L7 JEBBA T.S 5 OSOGBO JEBBA T.S 13 GANMO L8 JEBBA T.S 15 SHIRORO G.S L9 JEBBA T.S 15 SHIRORO G.S OSOGBO 6 AYEDE L10 OSOGBO 9 IKEJA WEST L11 OSOGBO 13 GANMO L12 OSOGBO 20 BENIN T.S L13 AYEDE 7 OLORUNSOGO L14 OLORUNSOGO 9 IKEJA WEST L15 SAKETE 9 IKEJA WEST L16 SAKETE 9 IKEJA WEST IKEJA WEST 10 AKANGBA T.S IKEJA WEST 10 AKANGBA T.S L17 IKEJA WEST 11 EGBIN G.S IKEJA WEST 11 EGBIN G.S IKEJA WEST 11 EGBIN G.S L18 OMOTOSHO IKEJA WEST 17 G.S L19 AKANGBA T.S 12 AJA T.S L120 EGBIN G.S 12 AJA T.S EGBIN G.S 12 AJA T.S L21 EGBIN G.S 20 BENIN T.S L22 EGBIN G.S 38 AES EGBIN G.S 38 AES L23 GANMO 16 KATAMPE T.S L24 KADUNA 15 SHIRORO G.S KADUNA 15 SHIRORO G.S L25 Length, km 310 470 81 383,05 8 263 157 *92,39 244 104,82 296 97 251 *78,97 70 70 18 18 62 *173,95 20 14 296,87 1 401 96 16 14 14 15 15 16 17 18 19 20 20 20 20 20 20 20 20 20 21 22 24 24 24 24 25 26 26 27 28 28 29 29 29 31 31 31 31 31 32 32 33 33 37 37 37 37 37 40 KADUNA KADUNA SHIRORO G.S SHIRORO G.S KATAMPE T.S OMOTOSHO G.S JOS T.S KANO T.S BENIN T.S BENIN T.S BENIN T.S BENIN T.S BENIN T.S BENIN T.S BENIN T.S BENIN T.S BENIN T.S DELTA G.S SAPELEG.S AJAOKUTAT.S AJAOKUTAT.S AJAOKUTAT.S AJAOKUTAT.S GEREGU G.S GOMBE T.S GOMBE T.S 18 19 16 16 18 JOS T.S KANO T.S KATAMPE T.S KATAMPE T.S JOS T.S 20 26 26 21 22 22 22 24 24 25 29 41 23 23 25 25 27 27 27 35 36 ASCO G.S NEW HAVEN T. NEW HAVEN T. ONITSHA T.S ONITSHA T.S ONITSHA T.S ALAOJI T.S ALAOJI T.S ALAOJI T.S ALAOJI T.S ALAOJI T.S AFAM AFAM EKET T.S EKET T.S PH MAIN T.S PH MAIN T.S PH MAIN T.S PH MAIN T.S PH MAIN T.S OMOKU G.S L26 L27 197 230 144 L28 L29 211,44 L30 L31 L32 L33 *140,86 265 415 107 L34 50 L35 195 L36 L37 L38 L39 L40 216,9 137 10 30 63 6,3 28 29 BENIN T.S GOMBE T.S GOMBE T.S DELTA G.S SAPELEG.S SAPELEG.S SAPELEG.S AJAOKUTAT.S AJAOKUTAT.S GEREGU G.S ONITSHA T.S IHOVBOR ALADJA T.S ALADJA T.S GEREGU G.S GEREGU G.S ASCO G.S ASCO G.S ASCO G.S YOLA T.S MAIDUGURI T. NEW HAVEN T. ONITSHA T.S 45 30 30 31 32 32 33 42 44 37 37 34 43 39 39 45 47 48 46 OBITE G.S OKPAI P.S OKPAI P.S ALAOJI T.S AFAM AFAM EKET T.S ALAOJI G.S CALABAR PH MAIN T.S PH MAIN T.S IBOM G.S EKET G.S T. AMADI G.S OMOKU G.S OBITE G.S GBARIAN G.S EGBEMA OMOKU T.S L41 L42 6,3 L43 L44 L45 6,3 225 310 L46 L47 172,5 96 L48 172 56 L49 L50 L51 L52 L53 L54 L55 L56 L57 L58 L59 L60 L61 L62 L63 154 25 25 70 5 145 37,8 45 5 10 100 70 99,3 90 5 17 45 OBITE G.S 46 OMOKU T.S L64 75 In this table, all proposed branches are in bold letters while those in italics represent the ones that have been modified. The distances marked with ‘*’ were not supplied data but were estimated using Google maps and online distance calculator. The mean travel/road distances were calculated between nearby towns and depending on the road network of the area, 5 or more km were added. E.g. line L8 between Jebba and Ganmo was calculated using the distance between the town of Jebba and Ilorin. This same method was used to calculated approximate distances for all proposed new lines which are denoted by bold letters in the table. Load The word load is used to represent the present power consumption in the system and demand is used to mean the actual power need and future power consumption of the country. At present, only 30%, which is approximately 3600 MW, of the actual electric load is supplied by the system. The projected demand for the country in 2011 is approximately 12000MW. For the purpose of this study, a few approximations and assumptions have been made as to what the demand is. Based on the current load distribution, future load distributions have been calculated. These load distributions have then been applied to the network topology. Table 1 in appendix I shows a typical load allocation table used by the TSO for a day. Since this load allocation is regional, there was a need to adapt it to the network such that regions are associated with nodes. Table 7 shows the load nodal distribution for the demand forecast for years 2012, 2015 and 2030. Thisforecast were gotten from power point presentation on demand forcast that was done by the Nigerian society of engineers for PHCN. These values have been used as an assumption and not necessarily as a known fact. They provided a good look into the future. In table Table 7, the coloumn ‘current’ represent the current load which is been served in the system. The values were adapted from the load allocation of a typical day that was provided by the TSO. This load flow capture is presented in appendix 1. The column ‘2012’ represents a projection into the future that was done a few years back. Since in this year 2012, the demand is not yet been served in the system, It has been used to represent a future scenario. Therefore simulations were not run for values shown in the column ‘2020’ as they represent a future that is quite furtherin reality than 2020. N1(B. Kebbi) N4(jebba t.s) N5(osogbo) N6(ayede) N8(sakete) N9(ikeja west) N10(akangba) N11(aja) Table 7 Load/demand nodal distribution Current % total load 2012 2015 124.40 3.45 454.21 631.07 7.47 0.21 27.27 37.89 129.77 3.60 473.82 658.31 190.43 5.28 695.30 966.03 140.00 3.89 511.17 710.20 230.78 6.40 842.62 1 170.72 247.62 6.87 904.11 1 256.15 200.00 5.55 730.24 1 014.58 2020 1 093.80 65.68 1 141.02 1 674.38 1 230.97 2 029.16 2 177.23 1 758.53 18 N12(egbin) N13(Ganmo) N14(kaduna) N15(shiroro) N16(katampe) N18(jos) N19(kano) N20(Benin) N24(ajaokuta) N26(Gombe) N28(N.heaven) N29(onitsha) N31(alaojI) N33(eket) N35(yola) N36(maiduguri N37(PH main) Auxilliary Total, MW 200.00 42.83 203.71 73.39 280.00 82.59 292.66 173.08 68.16 74.81 113.05 130.51 219.79 50.50 26.29 14.70 94.93 192.00 3603.47 5.55 1.19 5.65 2.04 7.77 2.29 8.12 4.80 1.89 2.08 3.14 3.62 6.10 1.40 0.73 0.41 2.63 5.33 730.24 156.38 743.79 267.96 1 022.34 301.55 1 068.56 631.95 248.87 273.15 412.77 476.52 802.50 184.39 95.99 53.67 346.61 701.03 13 157.00 1 014.58 217.27 1 033.40 372.30 1 420.41 418.97 1 484.63 878.02 345.77 379.50 573.49 662.06 1 114.97 256.18 481.57 133.37 74.57 973.99 18 280.00 1 758.53 376.59 1 791.15 645.29 2 461.94 726.18 2 573.25 1 521.83 599.31 657.78 994.01 1 147.53 1 932.53 444.03 834.68 231.16 129.25 1 688.19 31 684.00 19 5. Models This chapter describes the different optimization models that were written in GAMS. These models represent the different cases of the power system studied. The different assumptions and simplifications made for different cases are explained and mathematical formulas are given. 5.1. Introduction We assume the existing and proposed plants as built. This assumption is also applied to transmission lines. This was done to eliminate having to build dynamic matrices in GAMS which can be a complicated process. Proposed new lines and power plants were assumed to have zero thermal and generation capacities respectively at the initial stage of planning. This is achieved by building system topology vectors for both transmission lines and generators. Zeroes in the vector represent non existing units while integers represent the number of existing units. All network nodes were assumed to be demand nodes. A generation connection matrix, GCM was built to represent the power plant distribution in the network topology. Here again, ‘0’ mean no power plant connection to that particular node while ‘1’ mean that a power plant is connected to that node. Assumptions and simplifications We modelled only the active power generation and consumption. DC load flow is used to model the transmission constraint. A PTDF (Power transmission distribution factor) matrix was constructed to model the transmission topology of the network. An explanation on how this matrix is built is discussed in chapter 3 of this report and more explanation can me found in [16]. For the purpose of the study, we assume a vertically integrated electricity market even though some of the existing and proposed power plants are owned by independent producers. This assumption is valid since the Nigerian electricity market is still operated as such. We also assume that the investment costs are fixed and that a power plant can be built in a year. This assumption of course is not the case in reality but the focus on this study is not the cost of investment but the object of investment i.e. lines and generating plants. For a more detailed investment plan, interest rates and time lines will have to be introduced into the model. 5.2. Deterministic Model The model used is the standard optimization model which optimizes an objective function subject to a set of constraints. 20 Objective function The objective function is to minimize the sum of generation investment cost , the transmission investment cost and operating costs Min (5.1) ∑ γ i ⋅ Gi ⋅ (Vi − EGVi ) + ∑ αl ⋅ lengthl ⋅ (Vl − ELVl ) + 8760 ⋅ ∑ βi ⋅ Gi i l i Constraints The constraints are standard transmission planning constraints. Additional existing network topology constraints were added such as maximum permitted number of existing lines between nodes and maximum number of generating units that can be built at each generation node. Energy balance constraint ∑G = ∑ D i i n n (5.2) Generation constraint 0.15 ⋅ Gi ⋅ Vi ≤ Gi ≤ 0.85 ⋅ Gi ⋅ Vi , i ∈ I (5.3) Transmission constraint − P l ⋅ Vl ⋅ 0.85 ≤ PTDFl ⋅ ( (GCM ⋅ Gi ) − Dn ) ≤ 0.85 ⋅ Pl ⋅ Vl (5.4) The multipliers 0.15 and 0.85 in the generation constraints were added to model the average availability of the generating plants. As was shown in Table 3, currently most power plants are available 50% of the time. The multiplier 0.85 in the transmission constraint was also added model average availability since the thermal capacity does not always reflect transmission capacity of a line. 5.3. Scenario-based Decision analysis Model This model was developed to tackle the introduction of uncertain parameter in the deterministic model. The uncertain parameter i.e demand is expressed with finite values for different scenarios. These scenarios represent demand forecast for the future. The main difference between this model and the model described in section 5.4 is that for each scenario, an individual optimal expansion plan is obtained. Decision analysis based on the regret method described in [18] is used in choosing which of the optimal plans is less regrettable for all futures. The idea is to choose a plan that minimizes regret or maximises benefit. Some definitions as used in [17] are repeated here to allow for easy understanding of this method. • Attributes: are measures of goodness of a plan. In our model, investment cost and served load is used • Plan: is a set of specified option which in this case are the optimal plans for each future demand. 21 • Risk: is the hazard to which one is exposed because of uncertainty. • Regret: is a measure of risk. It is the difference between the value of an attribute for a particular plan and the value of that attribute for the optimal plan. For the analysis, the mean for all future demand and investment cost was calculated. Regret for each plan using the formula : r= ai , j − aopt , j i, j (5.5) Where ai , j is the value of a particular attribute for a particular plan and aopt , j is the value of that attribute for the optimal plan. 5.4. Stochastic Model The two stage stochastic model described in section 2.2 is applied. Here the demand is the uncertain parameter .A new set, s, is introduced for the scenarios and a uniform probability distribution of the demand is used i.e. probability of scenario s is the inverse of the number of scenarios. This approach to the probability distribution was adopted because there is no information on the randomness of the demand forecast available. Even probability distribution is an assumption. Where real information is available, more accurate probabilities can still be put into the model. The parameters are the same as in the deterministic model. However an additional parameter of the scenario probability is added. Objective function A penalty cost, λ, is introduced for unserved load. This is done to ensure that the optimization result is not a global optimum i.e. one that satisfies the highest demand. The objective function is to minimize the sum of generation investment cost, the transmission investment cost, expected operating costs and expected penalty cost of unserved load. Min ∑γ i i ⋅ Gi ⋅ (Vi − EGVi ) + ∑ α l ⋅ lengthl ⋅ (Vl − ELVl ) + 8760 ⋅ ∑ ps ∑ βi ⋅ Gi l s s i + ∑ ps ∑ λ ⋅ ( D n − Dns ) s n (5.6) Constraints The constraints are similar to the ones for the deterministic model. Energy balance constraint ∑ Gis = ∑ Dns , s ∈ S i (5.7) n 22 Generation constraint 0.15 ⋅ Gi ⋅ Vi ≤ G si ≤ 0.85 ⋅ Gi ⋅ Vi , i ∈ I , s ∈ S (5.8) Transmission constraint − P l ⋅ Vl ≤ PTDFl ⋅ ( GCM ⋅ Gi s − Dn s ) ≤ Pl ⋅Vl , s ∈ S (5.9) Additional constraints Dns ≤ Dn , s ∈ S (5.10) Dns ≥ 0.8 Dn , ∀s ∈ S (5.11) The last constraint was added to compel the model to meet at least 80 % of the demand in each scenario there by ensuring that the program does not chose to pay penalties instead of constructing new plants and lines. 5.5. Simulation parameters The deterministic model was run for a future demand forecast of approximately 13200 MW. This total demand was projected to nodal demands based on the present distribution of load in the system( See table 7)This value was used as the mean for the stochastic model which was run for 200 scenarios uniformly distributed, U (0.4, 1.2). The value for the random number distribution was chosen to ensure that the range of the random number is suitable and they fall within reasonable boundaries. The maximum number of generating units was set at 3 for new generating units and the existing power plants were not considered for upgrade. The value 3 was chosen to repreesent a more practical model since a unit here is figuratively a power plant. It was the opinion that in real life, more than 3 power plants may not be connected to one node in the grid network. The deterministic and stochastic models were run for two cases namely: 1. Expansion plan with no compulsory construction of new power plants i.e. Vi ≥ 0 2. Expansion plan with compulsory construction of new power plants i.e. Vi ≥ 1 The investment cost parameters were set to values shown in Table 8. These values are very close to real costs of these units. However λ is set at twice the generation investment cost so as to ensure the program does not choose to pay a penalty for more demand than necessary. Furthermore, operation cost for existing and new power plants were calculated based on current operation cost as reported in [12]. Operation cost for new power plants were calculated as average costs for existing thermal power plants. Table 9 shows the operation cost parameter used in the models. 23 Table 8 Cost parameters used in the GAMS Model Parameter Values α, $million/km 1 γ, $million/ MW 2.5 λ, $million/ MW 5 Table 9 Operation costs as used in the models Power plant Operation cost, $/MWh Kainji 1,4 Jebba 1,4 Olorunshogo 2,135625 Egbin 2,265 Shiroro 1,4 Omotosho 2,280625 Delta 2,35375 Sapele 2,840625 Geregu 3,0725 Asco 2,41 Okpai 2,41 Afam 1,65 Ibom 2,41 AES 2,64375 Transamadi 2,41 omoku 2,41 24 6. Simulation Results and Discussion 6.1. Individual Model Results 6.1.1. Deterministic Model Expansion with no compulsory construction of new plants For this model, Vi ≥ 0 . Results on power plant construction are given in table 10 Table 10 Deterministic model: Power plant construction result, Power plant Kainji Jebba Olorunshogo Egbin Shiroro Omotosho Delta Sapele Geregu Asco Okpai Afam Ibom AES Transamadi omoku Ihovbor Alaoji Eket Calabar Obite Gbarian Egbema TOTAL No of units 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 3 3 0 3 3 3 2 Vi ≥ 0 G(i),MW 646,00 491,64 603,50 1122,00 510,00 284,750 765,000 612,000 331,313 16,500 408,000 948,600 131,750 256,700 127,500 85,00 1150,050 1285,200 0 1216,00 1017,153 573,75 574,600 13157,006 Results on power line construction are presented in table 11 Table 11 Deterministic Model: Transmission line construction results, Branch l1 l2 l3 l4 l5 Curren t 1 0 2 0 2 Vi ≥ 0 Vl 2 2 2 2 2 25 l6 l7 l8 l9 l10 l11 l12 l13 l14 l15 l16 l17 l18 l19 l20 l21 l22 l23 l24 l25 l26 l27 l28 l29 l30 l31 l32 l33 l34 l35 l36 l37 l38 l39 l40 l41 l42 l43 l44 l45 l46 l47 l48 l49 l50 l51 l52 l53 0 2 1 2 1 1 1 1 1 1 1 2 3 1 0 2 1 2 0 2 1 1 2 0 1 1 0 1 3 2 0 1 0 1 1 2 2 0 1 1 0 1 0 2 1 2 1 0 1 2 1 3 1 1 2 3 1 1 1 2 3 2 1 3 2 4 2 2 1 2 2 1 2 1 1 1 3 2 1 4 2 1 1 2 8 1 1 1 2 1 3 2 4 2 1 3 26 l54 l55 l56 l57 l58 l59 l60 l61 l62 l63 L64 0 2 1 0 1 1 0 0 0 0 0 2 3 1 0 2 1 1 1 1 1 1 For this case, only the construction of power plant Eket is not proposed by the program and all other proposed new plants are needed to meet the demand. It is also obvious that plants like Ihovbor, Alaoji, Calabr and Obite are very much needed as they will be producing over 1000MW each. The transmission line construction as show in Table 11 shows that all proposed transmission line except for the one connecting Eket to the grid need to be built. It can also be seen that some of the existing lines such as L13, L23 , L42, L50 need reinforcement as they are connected to major transmission nodes and power plants which will be producing more power e,g L50 connecting Onitsha to Alaoji T.S. Expansion with compulsory construction of new plants Result of power plant construction is shown in table 12. Table 12 Deterministic model: power plant construction result, Power plant Kainji Jebba Olorunshogo Egbin Shiroro Omotosho Delta Sapele Geregu Asco Okpai Afam Ibom AES Transamdi omoku Ihovbor Alaoji Eket Calabar Obite Vi 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 3 3 1 2 3 Vi ≥ 1 G(i), MW 646,000 491,640 603,500 1122,000 510,000 284,750 765,000 612,000 273,566 93,500 408,000 948,600 131,750 256,700 85,000 127,500 1150,050 1285,200 425,000 608,000 1147,50 27 Gbarian Egbema TOTAL 3 3 573,750 608,00 13157,006 Results of power line construction are presented in table 13 below. Table 13 Deterministic Model: transmission line construction result, Branch Curent Vl, l1 1 2 l2 l3 0 2 2 2 l4 l5 0 2 2 2 l6 l7 0 2 1 2 l8 1 1 l9 l10 2 1 3 1 l11 1 1 l12 1 2 l13 l14 1 1 3 1 l15 1 1 l16 1 1 l17 2 1 l18 3 3 l19 1 2 l20 0 1 l21 2 3 l22 1 2 l23 2 4 l24 l25 0 2 2 2 l26 1 1 l27 l28 1 2 2 2 l29 0 1 l30 l31 1 1 2 1 l32 l33 0 1 1 1 l34 3 3 l35 2 2 l36 0 1 l37 1 4 l38 0 2 Vi ≥ 1 28 l39 1 1 l40 1 1 l41 2 2 l42 2 9 l43 l44 0 1 1 1 l45 1 1 l46 l47 0 1 2 1 l48 0 3 l49 2 2 l50 l51 1 2 4 2 l52 1 6 l53 0 3 l54 0 1 l55 2 2 l56 l57 1 0 1 1 l58 l59 1 1 2 1 l60 0 1 l61 0 1 l62 0 1 l63 l64 0 0 1 1 The results are very similar to the former case with only a few changes. Since construction of all power plants was made mandatory, Eket is now to be built and therefore lesser units of Calabar is needed. The number of power lines to be constructed on some branches also change although these changes are not significant. (See table 26 ). Figure 4 shows the graphical presentation of table 10 and table 12. . 29 1400 G(i), MW 1200 G(i),MW Vi≥0 G(i), MW Vi≥1 1000 800 600 400 200 Kainji Jebba Olorunshogo Egbin Shiroro Omotosho Delta Sapele Geregu Asco Okpai Afam Ibom AES Transamadi omoku Ihovbor Alaoji Eket Calabar Obite Gbarian Egbema 0 Figure 4 Deterministic Model :Power generation in power plants A comparison of investment cost for the above two cases is shown in table 14. Table 14 Deterministic model: Optimal Investment Costs comparison for compulsory and non compulsory construction of new plants Costs Vi ≥ 0 Vi ≥ 1 Generation Investment, $ Million. 18122,50 18815 Transmission investment, $ Million. 8117,22 8295,72 Operation , $ Million. 258,08 257,747 Penalty , $ Million. 0 0 Total, $ Million. 26497,80 27368,47 From the table above, it can be seen that the investment costs when Vi ≥ 0 is lesser than when Vi ≥ 1 . Since all the demand is met in both cases, it would therefore seem reasonable to follow the investment plan where Vi ≥ 0 . A pie chart of the investment cost distribution is shown in figure 5. Generation cost has the highest percentage of total cost. 30 Generation Investment, Million $ Transmission Investment, Million $ Operation cost, Million $ Figure 5 Deterministic model: Investment cost distribution The network topology showing line construction on new branches for the the case preferable case of non compulsory construction of all new power plant(i.e. Vi ≥ 0 ) is shown in Figure 6 below. The crosses represent how many lines are to be built of each branch. This figure is a graphical representation of table 11. 31 Figure 6 Deterministic model: Network topology showing new line construction 6.1.2. Regret analysis The deterministic model was run for four cases of future demands based on the forecast shown in table 7. These futures represent demand forecast considered as explained earlier. The optimal plan for these futures is shown in table 16-18. Here the limit on the number of power plant units to be constructed was increased. Table 15 Future demand for the regret analysis Future 1 Future 2 Future 3 13156,02 15787,18 17256,02 Demand, MW 16154,41 Average, MW Future 4 18418,42 32 Table 16 power plant construction plan for regret analysis Power plant Future 1 Future 2 Future 3 Future 4 Kainji 1 1 1 1 Jebba 1 1 1 1 Olorunshogo 1 1 1 1 Egbin 1 1 1 1 Shiroro 1 1 1 1 Omotosho 1 1 1 1 Delta 1 1 1 1 Sapele 1 1 1 1 Geregu 1 1 1 1 Asco 1 1 1 1 Okpai 1 1 1 1 Afam 1 1 1 1 Ibom 1 1 1 1 AES 1 1 1 1 Transamadi 1 1 1 1 omoku 1 1 1 1 Ihovbor 3 4 6 6 Alaoji 3 4 6 6 Eket 1 1 1 1 Calabar 2 4 5 3 Obite 3 4 1 6 Gbarian 3 4 4 4 Egbema 3 4 5 5 Table 17 Generation in power plant for regret analysis Power plant Future 1 Future 2 Future 3 Kainji 646 646 646,00 Jebba 491,64 491,64 491,64 Olorunshogo 603,5 603,50 603,50 Egbin 1122 1122 1122,00 Shiroro 510 510 510,00 Omotosho 284,75 284,75 284,75 Delta 765 765 765,00 Sapele 612 612 612,00 Geregu 339,807 193,03 342,40 Asco 93,5 16,50 16,50 Okpai 408 408 408,00 Afam 948,6 948,60 948,60 Ibom 131,75 131,75 131,75 AES 256,7 256,70 256,70 Transamadi 85 85 85,00 omoku 127,5 127,50 127,50 1150,05 1533,40 2300,10 Ihovbor 1285,20 1713,60 2570,40 Alaoji 125,80 162,67 304,60 Eket 608,00 1824 2384,25 Calabar 1147,50 1437,34 300,83 Obite Future 4 646,00 491,64 603,50 1122,00 510,00 284,75 765,00 612,00 347,09 93,50 408,00 948,60 116,77 256,70 85,00 127,50 2300,10 2570,40 425,00 1430,55 2295,00 33 Gbarian Egbema Total Generation, MW 573,75 839,97 765 1149,20 608,00 1436,50 608,00 1371,32 13156,02 15787,18 17256,02 18418,42 Table 18 Investment cost realization for regret analysis Costs Future 1 Future 2 Future 3 Generation Investment,$million 18122,5 26540 30187,5 Transmission Investment, $million 8117,22 11459,3 13615,41 Operation cost, $million 258,08 312,808 344,681 Total cost, $million 26497,8 38312,11 44147,59 Future 4 33007,5 14547,02 369,249 47923,77 Based on these plans, regrets were calculated on investment cost and demand for each future according to equation (5.5). Table 19 shows the values of regrets for each optimal plan. For example, ‘regret 2’ under ‘future 1’ means the regret of going for plan 1 if future 2 happens. Furthermore, regret was calculated on the case with the mean future demand as the optimal plan. This regret was found to be equal to the average regret on all four futures .On the demand attribute, “+“ signify average un-served load if optimal plan for a particular future is adopted and another future occurs in reality, while “-“ signify excess generated power. However , on investment cost attribute, “+” represent average investment saved on a particular plan if other plans future occurs other than the one whose optimal plan was chosen; and “-“ regret signify excess investment . Table 19 Regret in cost and served demand Attribute Demand Regret 1 Regret 2 Regret 3 Regret 4 Future 1 0 2631,16 4100 5262,40 Future 2 -2631,16 0 1468,84 2631,24 Future 3 -4100 -1468,84 0 1162,40 Future 4 -5262,40 -2631,24 -1162,40 0 Regret On Mean, MW 2998,39 367,23 -1101,61 -2264,01 Investment Cost Regret 1 Regret 2 Regret 3 Regret 4 0 11814,31 17649,79 21425,97 -11814,31 0 5835,48 9611,66 -17649,79 -5835,48 0 3776,18 -21426 -9611,66 -3776,18 0 Regret On Mean, $million 12722,52 908,21 -4927,27 -8703,45 As can be seen in the table above, zero regret means that going for the optimal plan of a particular future scenario brings no regret if that future scenario occurs. 34 4000 3000 Demand , MW 2000 1000 demand regret 0 Future 1 Future 2 Future 3 Future 4 -1000 -2000 -3000 Figure 7 Regret calculated for demand 15000 Total investment cost, $million 10000 5000 Investment cost regret 0 Future 1 Future 2 Future 3 Future 4 -5000 -10000 Figure 8 Regret calculated for investment cost From table 19 which is also graphically presented by figure 7 and figure 8, it can be seen that minimizing regret on demand and investment cost will result in choosing optimal plan for future 4 and 1 respectively. However, these two plans are regrettable for all other futures as they are both extreme cases. If the plan is to benefit on both demand and cost i.e. meet as much demand as possible with minimum investment cost then plans for future 2 and 3 will be considered. Of these two, for vertically integrated market system such as the Nigerian power 35 system where it is more important to meet demand than save money, optimal plan for future 3 will be chosen. 6.1.3. Two-stage stochastic model Expansion with no compulsory construction of new plants , Vi ≥ 0 Optimal power plant construction plan is shown in table 20 below. Table 20 Stochastic Model: power plant construction result, POWER PLANT Kainji Jebba Olorunshogo Egbin Shiroro Omotosho Delta Sapele Geregu Asco Okpai Afam Ibom AES Transamadi omoku Ihovbor Alaoji Eket Calabar Obite Gbarian Egbema TOTAL Vi ≥ 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 3 3 0 1 3 3 2 Vi ≥ 0 Gave, 646 491,640 603,500 1084,560 510 260,531 659,625 228,817 119,886 49,080 206,528 948,600 92,817 104,708 59,481 99,304 864,432 713,389 0 269,872 642,835 414,122 393,903 9463,63 Results of power line construction are presented in table 21. Table 21 Stochastic Model: transmission line construction results, Branch Current Vl, l1 1 2 l2 0 1 l3 2 2 l4 l5 0 2 2 2 l6 0 1 Vi ≥ 0 36 l7 2 2 l8 1 1 l9 2 2 l10 1 1 l11 1 1 l12 1 1 l13 1 3 l14 1 1 l15 1 1 l16 1 1 l17 2 2 l18 3 3 l19 1 2 l20 0 1 l21 2 2 l22 1 2 l23 2 4 l24 l25 l26 0 2 1 2 2 1 l27 l28 1 2 2 2 l29 0 1 l30 l31 1 1 2 1 l32 l33 l34 l35 0 1 3 2 1 1 3 2 l36 0 1 l37 l38 l39 l40 l41 1 0 1 1 2 4 2 1 1 2 l42 l43 l44 l45 2 0 1 1 7 1 1 1 l46 l47 l48 0 1 0 2 1 3 l49 2 2 l50 l51 1 2 4 2 l52 1 1 37 l53 0 2 l54 0 1 l55 2 2 l56 l57 1 0 1 0 l58 l59 1 1 1 1 l60 l61 l62 l63 l64 0 0 0 0 0 1 1 1 1 1 From table 20 above, Eket is again not suggested and the transmission plan again shows that all proposed new branches are needed even though the maximum allowable number of lines on each branched is not reached. However in this case, the program chooses to only meet 72% of the demand and instead pay a penalty cost for every MW not served. Table 24 shows the expected investment costs for this case. Expansion with compulsory construction of new plants Power plant construction results for the deterministic model with compulsory construction of all plants are shown in table 22. Table 22 Stochastic Model: Power plant construction result Vi Power plant Kainji Jebba Olorunshogo Egbin Shiroro Omotosho Delta Sapele Geregu Asco Okpai Afam Ibom AES Transamadi omoku Ihovbor Alaoji Eket Calabar Obite Vi 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 3 3 1 1 2 ≥1 Gave 646 491,64 603,50 1108,925 510 264,753 677,554 232,53 120,738 67,977 242,041 948,600 70,776 106,691 61,200 102,694 871,355 698,388 239,637 264,774 357,980 38 Gbarian Egbema TOTAL 3 2 411,299 381,912 9480,964 The result of the transmission expansion for this case is shown in table 23. Table 23 Stochastic Model: Power line construction Vi Branch Current Vl, l1 1 2 l2 l3 0 2 1 2 l4 l5 0 2 2 2 l6 l7 0 2 1 2 l8 1 1 l9 2 2 l10 1 1 l11 1 1 l12 1 1 l13 l14 1 1 3 1 l15 1 1 l16 1 1 l17 2 2 l18 3 3 l19 1 2 l20 l21 0 2 1 2 l22 1 2 l23 2 4 l24 l25 0 2 2 2 l26 1 1 l27 l28 1 2 2 2 l29 0 1 l30 l31 1 1 2 1 l32 l33 0 1 1 1 l34 3 3 l35 2 2 l36 0 1 l37 1 4 ≥1 39 l38 l39 0 1 2 1 l40 1 1 l41 2 2 l42 2 7 l43 l44 0 1 1 1 l45 1 1 l46 l47 l48 0 1 0 2 1 2 l49 2 2 l50 l51 1 2 4 2 l52 1 5 l53 0 2 l54 0 1 l55 l56 2 1 4 1 l57 0 1 l58 l59 1 1 1 1 l60 0 1 l61 0 1 l62 0 1 l63 l64 0 0 1 1 From the results, it can be seen again that when construction of new power plant is made compulsory, the number of units of Calabar needed reduces as Eket is built. However Ihovbor and Alaoji remain at the maximum allowed number of units. This result seems to suggest that power plants such as Ihovbor and Alaoji are highly needed. There is need for power generation at these nodes and consequently power line construction around these nodes such as L37 which needs reinforcement. Figure 9 shows the graphical representation of table 20 and table 22. 40 Figure 9 Stochastic Model :Average power generation in power plants A comparison of expected investment cost for the above two cases is shown in table 24. It can be observed that the cost is lesser when Vi ≥ 0 just as is the case using the deterministic model. However the results show that 20 MW more of demand can be met when Vi ≥ 1 . Since the difference in cost, approximately $300 million is more significant than these 20 MW and a penalty has already being paid for these unserved load, It would again seem realistic to chose to follow the plan where not all the power plants are constructed. Table 24 Cost comparison for the stochastic models Costs Vi ≥ 0 Vi ≥ 1 Generation Investment,$ Million. 15317,50 15442,50 Ttansmission Investment , $ Million. 7106,12 7294,72 Operation Cost, $ Million. 177,26 177,41 Penalty Cost, $ Million. 1483,26 1446,59 Total, $ Million. 24084,15 24361,22 41 Generation Investment, Million $ Transmission Investment, Million $ Operation cost, Million $ Penalty cost, Million $ Figure 10 Stochastic Model: Investment cost distribution The network topology showing line construction on new branches for the preferable case of non compulsory construction of all new power plant(i.e. Vi ≥ 0 )is shown below. Figure 11 Stochastic Model: network topology showing new line construction. 42 6.2. Comparison of Deterministic and Stochastic results In this section, we compare results from the two models studied. Figure 12 shows the power plant construction variable for both deterministic and stochastic model while figure 13 shows the power generation in these plants. It can be seen that result on power plant construction for both models are similar even though the expected investment cost using the Stochastic model is lesser than the deterministic one as shown in table 25. 3,5 Number of generating units 3 2,5 2 Deterministic Stochastic 1,5 1 0,5 0 Ihovbor Alaoji Eket Calabar Obite Gbarian Egbema Figure 12 Power plant construction variable for new plants for deterministic and stochastic model. 1400 Power generation, MW 1200 1000 800 G(i),MW Vi≥0 Gave(i), MW Vi≥0 600 400 200 0 Ihovbor Alaoji Eket Calabar Obite Gbarian Egbema Figure 13 Power generation in new power plants for deterministic and stochastic model. 43 Table 25 Investment cost comparison for both deterministic and stochastic model. COSTS DETERMINISTIC MODEL STOCHASTIC MODEL Vi ≥ 0 Vi ≥ 1 Vi ≥ 0 Vi ≥ 1 Generation Investment, Million $ Transmission Investment, Million $ 18122,50 18815 15317,50 15442,50 8117,22 8295,72 7106,12 7294,72 Operation cost, Million $ 258,08 257,747 177,26 177,41 Penalty cost, Million $ 0 0 1483,26 1446,59 Total cost, Million $ 26497,80 27368,47 24084,15 24361,22 Table 25 shows that using the stochastic model, a reduction in investment cost of up to $2413 Million when Vi ≥ 0 and $3007 Million when Vi ≥ 1 is achieved while meeting approximately 70% of the forecasted demand. If the penalty cost is considered as a cost which can be used to build a new power plant, and if the number of proposed unit are similar as shown in figure 12Figure 12. Then it is obvious that we can meet more demand than the 9500MW proposed by the stochastic model. The advantage of using the stochastic model is the cost minimisation which is the main objective of an optimal expansion plan. This also means that we can prevent over estimation of investment cost. 30000 Costs,(million dollars) 25000 20000 Deterministic 15000 Stochastic 10000 5000 0 Generation Transmission Operation Penalty cost, Total cost, Investment, Investment, cost, Million Million $ Million $ Million $ Million $ $ Figure 14 Cost comparisons of deterministic and stochastic model Table 26 shows an overview of the optimal transmission expansion plan for the Nigerian power system when using both the deterministic and the stochastic model. The stochastic models show lesser number of transmission lines to be built than the deterministic model. This is reflected in the costs as well as shown in table 24. 44 Table 26 Optimal network topology result CURRENT DETERMINISTIC MODEL SYSTEM Branches Lines New branches New lines on new branches Total new lines STOCHASTIC MODEL Vi ≥ 0 Vi ≥ 1 Vi ≥ 0 Vi ≥ 1 44 63 - 63 118 19 64 122 20 63 109 19 64 115 20 - 29 29 26 26 55 59 46 52 45 7. Conclusions In this thesis, we have applied not only deterministic programming method but also considered the option of carrying out a decision analysis based on different options. Stochastic mixed integer linear programming method has also been applied to transmissiongeneration expansion planning of the Nigerian power system. When some parameters of an expansion planning are uncertain, deterministic programming becomes inadequate to decide an optimal plan. Running the deterministic model for a set of scenarios can be an option. However, from the simple simplification of the decision analysis method explained in section 5.3 and 6.1.2 , it can be seen that making a decision on the optimal plan when demand is uncertain can be a rigorous task especially when there are a lot of future scenarios to be considered. As a result the stochastic model offers a better approach to solving uncertainty problems in expansion planning. Stochastic optimisation provides a more realistic approach to planning for the future. The stochastic model yields optimum results which are comparable to the deterministic ones at a lower investment cost. In addition to this, decisions made based on results from a stochastic optimisation are better informed than in a deterministic model since more options have been considered. From the results of both models, it is evident that power plants, Ihovbor; alaoji; Calabar; Obite; and Egbema need to be built. The locations of these plants in the system are very strategic and more power generation is needed at these nodes. The transmission network also needs reinforcement and all suggested branches are needed to transmit the generated power. The only branches which may not be built are those that connect a proposed power plant which is not to be built to the transmission grid. 46 8. Future Work The study carried out in this thesis is a pilot study on modelling uncertainties in expansion planning for the Nigerian power system. The work has also included creating a model of the System that can be improved with accurate data. Other improvement on the models used and ideas on further studies are as follows: • Use more accurate data on transmission line properties and investment costs • Use a more accurate probabilistic distribution of the demand forecast. • Implement a multistage plan with interest rates on investment taken into consideration. • Model the system as a deregulated market system. This might involve huge modifications to the model as a deregulated system is quite different from a vertically integrated market system. • Apply primal and dual linear decision rules approximation to the stochastic model. 47 References [1] G. Latorre, R. D. Cruz, J. M. Areiza and A. 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[19] GAMS user guide . www.gams.com. [20] http://www.state.gov/r/pa/ei/bgn/2836.htm. [21] D.J Obadote “Energy Crisis in Nigeria: technical issues and solutions” Power Sector Prayer conference June 2009. [22] PSSE manual. https://www.ptius.com/pti/software/psse/university/index.htm 49 Appendix I Typical Load regional distribution for a day as provided bytheTSO
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