Modelling Demand Uncertainties in Generation-Transmission Expansion planning: A case study of the Nigerian Power System

Degree project in

Modelling Demand Uncertainties in Generation-Transmission Expansion

planning

A case study of the Nigerian Power System

Omobobola Omolola Faleye

Stockholm, Sweden 2012 Electric Power Systems

Second Level,

Modelling Demand Uncertainties in Generation-Transmission Expansion planning: A case study of the Nigerian Power System

___________________________________________________________________________

Master Thesis Project

Omobobola Omolola Faleye

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

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.

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 generation- transmission 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.

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

Appendix II ... 2

Gams code for deterministic Model ... 2

Appendix III ... 24

Gams code for Regret analysis ... 24

Appendix IV ... 49

GAMS code for Stochastic Model ... 49

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,

... 25

Table 11 Deterministic Model: Transmission line construction results,

... 25

Table 12 Deterministic model: power plant construction result,

... 27

Table 13 Deterministic Model: transmission line construction result,

... 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, V

≥ ... 36 0 Table 21 Stochastic Model: transmission line construction results, V

≥ ... 36 0 Table 22 Stochastic Model: Power plant construction result V

≥ ... 38 1 Table 23 Stochastic Model: Power line construction V

≥ ... 39 1 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

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

Nomenclature

The symbols and notations used in the GAMS model are listed here for easy reference Sets

N - All network nodes I - All power plants L - All transmission lines S - Random scenarios Subsets

NP - Candidate new power plants EP - Existing power plants NL - Candidate new lines Decision Variables

V - Integer variable of power plant units to be built.

V - Integer variable of transmission line to be built.

G - Power generation in each power plant i.

G

- Power generation in each power plant i for every scenario.

Variables

P - Power flow on each line l.

P - Power flow on each line l in each scenario.

D - Demand at each node n.

D

- Load served at each node n in each scenario Parameters

G

- Installed capacity in plant i

P

- Maximum power flow on line l

D

- Maximum demand at each node

Length

- Length of each line l

α - Cost /km of transmission line construction

β

- Variable generation cost in power plant i

γ

- Cost/MW of construction of power plant i

λ - Penalty for un-served node

.

PTDF

- Power transfer distribution factor matrix of lines based on active power injection

at network nodes

,

GCM

- Generation connection matrix

ELV -Vector of existing lines

EGV -Vector of existing generation plant

p

- Probability of scenario s

1. Introduction

1.1 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].

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.

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.

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

Land cost 6

Studies 9

Miscellaneous 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

i i l l l oi i i

i l i

c v + c v L + c v G

∑ ∑ ∑ ^(2.1)

Where

c is the cost of building a unit of power plant i

c is the cost/km of building power line l

c is the operation cost/MWh of power produced in plant i

, i l

v v is the construction integer variable of plant i and line l respectively.

, ,

, 1

i l

i l

v v o then no construction v v then construct

 =

 ≥



L, is the length of power line l and G _i is the total generation in plant i.

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.

i i i

G ≤ G ≤ G (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:

l l l

F ≤ F ≤ F (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:

k j

sin

l kj

l

F V V

X θ

≈ (2.5)

Where

,

k j

kj

V V are bus voltages X is the line reactance

is the angular difference between buses k and j θ

Taken into consideration assumptions 2 and 3, equation (2.5) becomes

1

l l kj

l l

F B

where B X θ

= ⋅

= (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 F l the vector of power flows then equation (2.6) can be written as

T

=

A θ XF (2.7)

1 T

 

= 



F

X A θ (2.8)

If F denotes the vector of all injected power at each node, then applying Kirchhoff’s

“current” law

=

F AF (2.9)

-1 T

 

=   =

F AX A θ Bθ (2.10)

By combining equation (2.8) with equation(2.10), the phase angles can be eliminated resulting in

1 T -1





=  

F

X A B F (2.11)

l

=

F TF (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 :

n n n

F = G − D (2.13)

Then equation (2.12) can then be substituted in equation (2.13)

,

( )

l l n n n l

F ≤ T G − D ≤ F (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 u g and u l

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 s t

x ∈ χ

(3.1)

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 .

0

T

s t

x

≤

≥ C x

Ax b

(3.2)

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

[ ( , )]

x

Q

A

ω

ω

+ Ε

≤

C x x

x b (3.3)

Where

{ }

( , ) min

, 0

Q x ω = q y W y ≤ − h T x y ≥

Ε 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 = E Q x

[ ( , )] ω , then a deterministic equivalent of the above problem can be written as:

min ( )

.

0

T

s t x

+ Ω

≤

≥

C x x

Ax b (3.4)

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( ω 1 ,…, ω s) with probabilities (p 1 ,…p s ). This algorithm is known as scenario wise decomposition.

min ( )

,

T T

s s s s

s

s

p s S

+

≤ ∈

∑ ^{C x q y}

Ax b

(3.5)

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 km ² . 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.

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

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

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.

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.

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

47 GBARIAN G.S 330

48 EGBEMA G:S 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.

Table 6 Transmission Branch numbering for this study.

From Bus Number

From Bus Name

To Bus Number

To Bus Name

GAMS numbering

Length, km

1 B.KEBBI 2 KAINJI G.S L1 310

1 B.KEBBI 3 JEBBA G.S L2 470

2 KAINJI G.S 4 JEBBA T.S

L3

81

2 KAINJI G.S 4 JEBBA T.S

2 KAINJI G.S 14 KADUNA L4 383,05

3 JEBBA G.S 4 JEBBA T.S

L5

8

3 JEBBA G.S 4 JEBBA T.S

3 JEBBA G.S 6 AYEDE L6 263

4 JEBBA T.S 5 OSOGBO L7 157

4 JEBBA T.S 5 OSOGBO

4 JEBBA T.S 13 GANMO L8 *92,39

4 JEBBA T.S 15 SHIRORO G.S L9 244

4 JEBBA T.S 15 SHIRORO G.S

5 OSOGBO 6 AYEDE L10 104,82

5 OSOGBO 9 IKEJA WEST L11 296

5 OSOGBO 13 GANMO L12 97

5 OSOGBO 20 BENIN T.S L13 251

6 AYEDE 7 OLORUNSOGO L14 *78,97

7 OLORUNSOGO 9 IKEJA WEST L15 70

8 SAKETE 9 IKEJA WEST L16 70

8 SAKETE 9 IKEJA WEST

9 IKEJA WEST 10 AKANGBA T.S

L17

18 18

9 IKEJA WEST 10 AKANGBA T.S

9 IKEJA WEST 11 EGBIN G.S

L18

62

9 IKEJA WEST 11 EGBIN G.S

9 IKEJA WEST 11 EGBIN G.S

9 IKEJA WEST 17

OMOTOSHO

G.S L19 *173,95

10 AKANGBA T.S 12 AJA T.S L120 20

11 EGBIN G.S 12 AJA T.S

L21

14

11 EGBIN G.S 12 AJA T.S

11 EGBIN G.S 20 BENIN T.S L22 296,87

11 EGBIN G.S 38 AES

L23

1

11 EGBIN G.S 38 AES

13 GANMO 16 KATAMPE T.S L24 401

14 KADUNA 15 SHIRORO G.S

L25

96

14 KADUNA 15 SHIRORO G.S

14 KADUNA 18 JOS T.S L26 197

14 KADUNA 19 KANO T.S L27 230

15 SHIRORO G.S 16 KATAMPE T.S

L28

144

15 SHIRORO G.S 16 KATAMPE T.S

16 KATAMPE T.S 18 JOS T.S L29 211,44

17

OMOTOSHO

G.S 20 BENIN T.S L30 *140,86

18 JOS T.S 26 GOMBE T.S L31 265

19 KANO T.S 26 GOMBE T.S L32 415

20 BENIN T.S 21 DELTA G.S L33 107

20 BENIN T.S 22 SAPELEG.S

L34 50

20 BENIN T.S 22 SAPELEG.S

20 BENIN T.S 22 SAPELEG.S

20 BENIN T.S 24 AJAOKUTAT.S L35 195

20 BENIN T.S 24 AJAOKUTAT.S

20 BENIN T.S 25 GEREGU G.S L36 216,9

20 BENIN T.S 29 ONITSHA T.S L37 137

20 BENIN T.S 41 IHOVBOR L38 10

21 DELTA G.S 23 ALADJA T.S L39 30

22 SAPELEG.S 23 ALADJA T.S L40 63

24 AJAOKUTAT.S 25 GEREGU G.S

L41

6,3

24 AJAOKUTAT.S 25 GEREGU G.S

24 AJAOKUTAT.S 27 ASCO G.S L42 6,3

24 AJAOKUTAT.S 27 ASCO G.S

25 GEREGU G.S 27 ASCO G.S L43 6,3

26 GOMBE T.S 35 YOLA T.S L44 225

26 GOMBE T.S 36 MAIDUGURI T. L45 310

27 ASCO G.S 28

NEW HAVEN

T. L46 172,5

28 NEW HAVEN T. 29 ONITSHA T.S L47 96

28

NEW HAVEN

T. 45 OBITE G.S L48 172

29 ONITSHA T.S 30 OKPAI P.S

L49

56

29 ONITSHA T.S 30 OKPAI P.S

29 ONITSHA T.S 31 ALAOJI T.S L50 154

31 ALAOJI T.S 32 AFAM

L51

25 25

31 ALAOJI T.S 32 AFAM

31 ALAOJI T.S 33 EKET T.S L52 70

31 ALAOJI T.S 42 ALAOJI G.S L53 5

31 ALAOJI T.S 44 CALABAR L54 145

32 AFAM 37 PH MAIN T.S

L55

37,8

32 AFAM 37 PH MAIN T.S

33 EKET T.S 34 IBOM G.S L56 45

33 EKET T.S 43 EKET G.S L57 5

37 PH MAIN T.S 39 T. AMADI G.S L58 10

37 PH MAIN T.S 39 OMOKU G.S L59 100

37 PH MAIN T.S 45 OBITE G.S L60 70

37 PH MAIN T.S 47 GBARIAN G.S L61 99,3

37 PH MAIN T.S 48 EGBEMA L62 90

40 OMOKU G.S 46 OMOKU T.S L63 5

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.

Table 7 Load/demand nodal distribution

Current % total load 2012 2015 2020

N1(B. Kebbi) 124.40 3.45 454.21 631.07 1 093.80

N4(jebba t.s) 7.47 0.21 27.27 37.89 65.68

N5(osogbo) 129.77 3.60 473.82 658.31 1 141.02

N6(ayede) 190.43 5.28 695.30 966.03 1 674.38

N8(sakete) 140.00 3.89 511.17 710.20 1 230.97

N9(ikeja west) 230.78 6.40 842.62 1 170.72 2 029.16

N10(akangba) 247.62 6.87 904.11 1 256.15 2 177.23

N12(egbin) 200.00 5.55 730.24 1 014.58 1 758.53

N13(Ganmo) 42.83 1.19 156.38 217.27 376.59

N14(kaduna) 203.71 5.65 743.79 1 033.40 1 791.15

N15(shiroro) 73.39 2.04 267.96 372.30 645.29

N16(katampe) 280.00 7.77 1 022.34 1 420.41 2 461.94

N18(jos) 82.59 2.29 301.55 418.97 726.18

N19(kano) 292.66 8.12 1 068.56 1 484.63 2 573.25

N20(Benin) 173.08 4.80 631.95 878.02 1 521.83

N24(ajaokuta) 68.16 1.89 248.87 345.77 599.31

N26(Gombe) 74.81 2.08 273.15 379.50 657.78

N28(N.heaven) 113.05 3.14 412.77 573.49 994.01

N29(onitsha) 130.51 3.62 476.52 662.06 1 147.53

N31(alaojI) 219.79 6.10 802.50 1 114.97 1 932.53

N33(eket) 50.50 1.40 184.39 256.18 444.03

N35(yola) 26.29 0.73 95.99 481.57 834.68

N36(maiduguri 14.70 0.41 53.67 133.37 231.16

N37(PH main) 94.93 2.63 346.61 74.57 129.25

Auxilliary 192.00 5.33 701.03 973.99 1 688.19

Total, MW 3603.47 13 157.00 18 280.00 31 684.00

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. The full GAMS codes are provided in the appendix.

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.

Objective function

The objective function is to minimize the sum of generation investment cost , the transmission investment cost and operating costs

Min

( ) ( ) 8760

i i i i l l l l i i

i l i

G V EGV length V ELV G

γ ⋅ ⋅ − + α ⋅ ⋅ − + ⋅ β ⋅

∑ ∑ ∑ ^(5.1)

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

i n

i n

G = D

∑ ∑

(5.2) Generation constraint

0.15 ⋅ ⋅ ≤ G V

G

≤ 0.85 ⋅ ⋅ G V

, i ∈ I (5.3)

Transmission constraint

( )

0.85 ( ) 0.85

l l l i n l l

P V PTDF GCM G D P V

− ⋅ ⋅ ≤ ⋅ ⋅ − ≤ ⋅ ⋅ (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.

• 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 :

, , ,

i j i j opt j

r = a − a (5.5)

Where a is the value of a particular attribute for a particular plan and

a

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

( ) ( ) 8760

( )

s

i i i i l l l l s i i

i l s i

s

s n n

s n

G V EGV length V ELV p G

p D D

γ α β

λ

⋅ ⋅ − + ⋅ ⋅ − + ⋅ ⋅

+ ⋅ −

∑ ∑ ∑ ∑

∑ ∑

(5.6) Constraints

The constraints are similar to the ones for the deterministic model.

Energy balance constraint

s s

i n

i n

G = D

∑ ∑ ^{, s ∈ S (5.7)}

Generation constraint

0.15 ⋅ ⋅ ≤ G V

G

≤ 0.85 ⋅ ⋅ G V

, i ∈ I s , ∈ S (5.8) Transmission constraint

(

)

l l l i n l l

P V PTDF GCM G D P V

− ⋅ ≤ ⋅ ⋅ − ≤ ⋅ ^, s ∈ S (5.9)

Additional constraints

s

,

n n

D ≤ D s ∈ S (5.10)

0.8 ,

s

n n

D ≥ D ∀ ∈ 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.

i

0 V ≥

2. Expansion plan with compulsory construction of new power plants i.e. V

≥ 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.

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

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, V

≥ . Results on power plant construction are given in table 10 0

Table 10 Deterministic model: Power plant construction result,

Power plant No of units G(i),MW

Kainji 1 646,00

Jebba 1 491,64

Olorunshogo 1 603,50

Egbin 1 1122,00

Shiroro 1 510,00

Omotosho 1 284,750

Delta 1 765,000

Sapele 1 612,000

Geregu 1 331,313

Asco 1 16,500

Okpai 1 408,000

Afam 1 948,600

Ibom 1 131,750

AES 1 256,700

Transamadi 1 127,500

omoku 1 85,00

Ihovbor 3 1150,050

Alaoji 3 1285,200

Eket 0 0

Calabar 3 1216,00

Obite 3 1017,153

Gbarian 3 573,75

Egbema 2 574,600

TOTAL 13157,006

Results on power line construction are presented in table 11

Table 11 Deterministic Model: Transmission line construction results,

Branch

Curren

t V

l1 1 2

l2 0 2

l3 2 2

l4 0 2

l5 2 2

l6 0 1

l7 2 2

l8 1 1

l9 2 3

l10 1 1

l11 1 1

l12 1 2

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 3

l22 1 2

l23 2 4

l24 0 2

l25 2 2

l26 1 1

l27 1 2

l28 2 2

l29 0 1

l30 1 2

l31 1 1

l32 0 1

l33 1 1

l34 3 3

l35 2 2

l36 0 1

l37 1 4

l38 0 2

l39 1 1

l40 1 1

l41 2 2

l42 2 8

l43 0 1

l44 1 1

l45 1 1

l46 0 2

l47 1 1

l48 0 3

l49 2 2

l50 1 4

l51 2 2

l52 1 1

l53 0 3