Hydroelectric Power Generation and Distribution Planning Under Supply Uncertainty

Home , Distributed generation, Electrical grid, Energy engineering, Hydroelectricity, Power engineering, Power transmission

Govind R. Joshi

A Thesis

Submitted to

The Department of Engineering

Colorado State University-Pueblo

In partial fulfillment of requirements for the degree of

Master of Science

Completed on December 16th, 2016

ABSTRACT

Govind Raj Joshi for the degree of Master of Science in Industrial and Systems Engineering presented on December 16th, 2016.

Hydroelectric Power Generation and Distribution Planning Under Supply Uncertainty

Abstract approved: ------

Ebisa D. Wollega, Ph.D.

Hydroelectric power system is a renewable energy type that generates electrical energy from water flow. An integrated hydroelectric power system may consist of water storage dams and run-of-river (ROR) hydroelectric power projects. Storage dams store water and regulate water ﬂow so that power from the storage projects dispatch can follow a pre-planned schedule. Power supply from ROR projects is uncertain because water ﬂow in the river, and hence power production capacity, is largely determined by uncertain weather factors. Hydroelectric generator dispatch problem has been widely studied in the literature; however, very little work is available to address the dispatch and distribution planning of an integrated ROR and storage hydroelectric projects. This thesis combines both ROR projects and storage dam projects and formulate the problem as a stochastic program to minimize the cost of energy generation and distribution under ROR projects supply uncertainty. Input data from the Integrated Nepal Power System are used to solve the problem and run experiments. Numerical comparisons of stochastic solution (SS), expected value (EEV), and wait and see (W&S) solutions are made. These solution approaches give economic dispatch of generators and optimal distribution plan that the power system operators (PSO) can use to coordinate, control, and monitor the power generation and distribution system. The W&S solution approach provided the least cost plan. The EEV solution was worse than the SS. The PSO may invest in advanced technologies to more accurately reveal the uncertainties in the planning process, to operate at the W&S operational cost. However, the tradeoff between using the SS solution and investing in new technologies to operate at W&S solution may require rigorous feasibility study.

Certificate of Acceptance

This thesis presented in partial fulfillment of the requirements for the degree of

Master of Science

has been accepted by the program of Industrial and Systems Engineering,

Colorado State University - Pueblo

APPROVED

------

Dr. Ebisa D. Wollega Date

Assistant Professor of Engineering and Committee Chair

------

Dr. Leonardo Bedoya-Valencia Date

Associate Professor of Engineering and Committee Member

------

Dr. Jane M. Fraser Date

Professor and Director of Engineering and Committee Member

Master’s Candidate Govind R. Joshi

Date of thesis presentation December 16, 2016

iii

ACKNOWLEDGEMENTS

This thesis would not have been possible without the guidance and the help of several individuals who contributed and extended their valuable assistance in the preparation and completion of this research.

First and foremost, I offer my utmost gratitude to my academic advisor and committee chair, Dr. Ebisa Wollega, for his support throughout the development of this thesis with his knowledge and patience. I also would like to thank the Department of Engineering at Colorado State University -Pueblo for giving me the opportunity to work on this thesis and for providing me the required resources to carry out this research.

Last but not the least, I would like to thank my parents as well as my uncle Rajendra Awasthi and aunt Kacie Awasthi for their guidance and support throughout my master degree at Colorado State University - Pueblo.

ABBREVIATIONS

Table I List of abbreviations

Current measurement unit A (Amperes) MWhr/MWh Mega Watt Hour

AC Alternating Current NEA Nepal Electricity Authority

AI Artificial Intelligence OPF Optimal Power Flow

CF Capacity Factor pf power factor

DC Direct Current PF Plant Factor

DGs Distributed Generators PP Power Plant

DL Distribution Line Q Water discharge

Expectation of Expected Water discharge during dry EEV Value Qdry season

Water discharge during wet FY Fiscal Year Qwet season

H Head or Height ROR Run-of -River

HEP Hydroelectric Project SS Stochastic Solution

Integrated Nepal Power INPS System TL Transmission Line

kV Kilo Volts TP Transportation Problem

Active power measurement unit LOLP Loss of Load Probability W (Watts)

MW Mega Watts W&S Wait and See

LIST OF FIGURES

Figure 1.1 Power system network block diagram consisting of hydroelectric power projects...... 5 Figure 2.1 Integrated power system ...... 7 Figure 3.1 Electromechanical governor ...... 34 Figure 3.2 Integrated power system network ...... 37 Figure 4.1 Plant factors of the ROR projects ...... 43 Figure 4.2 Monthly energy generation by the ROR projects...... 44 Figure 4.3 Seasons of a year in the INPS...... 45 Figure 4.4 Decision variable values from EEV approach...... 52 Figure 4.5 Decision variable values from SS approach...... 54 Figure 4.6 Decision variable values from W&S approach...... 56 Figure A.1 Integrated Nepal Power System map………………………………………………...68 Figure A.2 Integrated Nepal Power System single line diagram………………………………...69 Figure A.3 MATLAB simulation diagram of the Integrated Nepal Power System……………..69 Figure B.1 Monthly energy generation by the ROR projects in Nepal………………………….71 Figure B.2 Total monthly energy generation by the ROR projects in Nepal……………………73 Figure B.3 Average monthly plant factor or the ROR projects in Nepal (in Percentage)……….74 Figure C.1 Hourly peak demand in the INPS……………………………………………………75 Figure E.1 MATLAB and PSAT results for bus voltages in the INPS………………………….79 Figure E.2 INPS bus voltage level for each season……………………………………………...80

LIST OF TABLES

Table I List of abbreviations ...... vi Table 3.1 Notation ...... 25 Table 4.1 Existing major hydroelectric power plants and their types (source: NEA [56]) ...... 42 Table 4.2 Nepali to English calendar conversion ...... 44 Table 4.3 Seasons in a year and their probabilities ...... 46 Table 4.4 Existing transmission and distribution lines capacity in the INPS ...... 47 Table 4.5 Existing substations in the INPS and their capacities ...... 49 Table 4.6 Cost parameter computation ...... 50 Table 4.7 Total operation and maintenance cost from the NEA report ...... 51 Table 4.8 Legend ...... 52 Table 4.9 The objective function values from SS, W&S, and EEV approach ...... 59 Table C.1 Current power supply and demand in Nepal………………………………………….75 Table D.1 Cost and capacity parameters computation of the power system components……….76

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TABLE OF CONTENTS

ABSTRACT ...... II

ACKNOWLEDGEMENTS ...... IV

ABBREVIATIONS ...... V

LIST OF FIGURES ...... VI

LIST OF TABLES ...... VII

CHAPTER 1. INTRODUCTION ...... 1

1.1. BRIEF OVERVIEW ...... 1

1.2. RESEARCH MOTIVATION ...... 2

1.3. OBJECTIVE OF THE STUDY ...... 4

1.4. ORGANIZATION OF THE THESIS ...... 5

CHAPTER 2. LITERATURE REVIEW ...... 6

2.1. HYDROELECTRIC POWER SYSTEMS ...... 6

2.2. HYDROELECTRIC POWER SYSTEM COMPONENTS ...... 8 2.2.1. Generation Station ...... 8 2.2.2. Transmission Line ...... 9 2.2.3. Distribution Substation ...... 9 2.2.4. Distribution Line ...... 10

2.3. HYDROELECTRIC POWER ENERGY SOURCES ...... 10 2.3.1. Storage Projects ...... 10 2.3.2. Run-of-River Projects ...... 11 2.3.3. Pumped Storage ...... 11

2.4. ELECTRICITY AS A COMMODITY ...... 11

2.5. ELECTRICITY MARKETS ...... 12

2.6. MATHEMATICAL OPTIMIZATION ...... 13

2.7. HYDROELECTRIC POWER PLANNING APPROACHES ...... 15 2.7.1. Hydroelectric Power Planning as a Transportation Problem ...... 15 2.7.2. Deterministic Hydroelectric Power Planning ...... 16 2.7.3. Stochastic Hydroelectric Power Planning ...... 17 viii

2.7.4. Large Scale Hydroelectric Power Planning ...... 20

CHAPTER 3. PROBLEM FORMULATION ...... 23

3.1. PROBLEM DEFINITION ...... 23

3.2. DEFINITION OF TERMS ...... 25

3.3. MATHEMATICAL MODEL...... 27 3.3.1. Objective Function ...... 27 3.3.2. Constraints ...... 28

3.4. COST PARAMETER DEFINITION ...... 29 3.4.1. Generation Cost ...... 31 3.4.2. Transmission Line, Distribution Line, and Inter Substation Line Cost ...... 31 3.4.3. Substation Cost ...... 32

3.5. RESOURCE CAPACITY DEFINITION ...... 33 3.5.1. Generation Station Capacity ...... 33 3.5.2. Transmission Line, Distribution Line, and Inter Substation Line Capacity ...... 34 3.5.3. Transmission Substation Capacity ...... 35 3.5.4. Distribution Substation or Demand Centers Capacity ...... 35 3.5.5. Load Flow Analysis ...... 36

CHAPTER 4. NUMERICAL EXPERIMENTS ...... 40

4.1. MODEL INPUT DATA ...... 42 4.1.1. Resource Capacity Parameters ...... 42 4.1.2. Cost Parameters ...... 49

4.2. ANALYSIS OF NUMERICAL RESULTS ...... 51 4.2.1. Operation and Maintenance Cost ...... 51 4.2.2. Expected Value Solution (EEV) ...... 51 4.2.3. Stochastic Solution (SS) ...... 54 4.2.4. Wait & See Solution (W&S)...... 56 4.2.5. Objective Function Values ...... 58

CHAPTER 5. CONCLUSIONS AND RECOMMENDATIONS ...... 60

5.1. CONCLUSIONS ...... 60

5.2. FUTURE WORK ...... 63

BIBLIOGRAPHY ...... 64

APPENDIX ...... 68

Appendix A: The Integrated Nepal Power System ...... 68 Appendix B: Monthly Power Generation of the Projects ...... 70 Appendix C: The INPS Hourly Load...... 75 Appendix D: Cost and Capacity Parameters Computation ...... 76 Appendix E: Load Flow Analysis Results ...... 79

CHAPTER 1. INTRODUCTION

1.1. Brief Overview

Hydroelectric power plants consist of turbine-generator sets to produce electrical energy from the potential and kinetic energy of water flow. Water is tapped from rivers and instantly supplied to turbine-generator sets, or the water is stored in a dam first and then its flow is regulated through the turbine-generator sets to generate electricity. The run-of-river (ROR) hydroelectric projects have diversion channels to tap water from rivers while storage projects use dam to store and supply water to turbines. The ROR projects are cheaper to install compared to storage projects of the same capacity, which would make the per unit energy generation cost of ROR projects cheaper than the per unit generation cost of the storage projects. However, the variation of the ROR projects power output due to fluctuation in the river discharge makes the ROR projects less reliable.

Furthermore, an integrated power system network is complicated. It usually consists of generation stations, transmission lines, power substations, distribution lines, and demand locations and these components are grouped as the power system components. The operational characteristic of each component of the system varies based on the component capacity, location, and weather conditions among others. For example, the power loss in transmission line is a function of the conductor diameter and the length of the line. Longer transmission lines have more power loss than shorter ones and similarly a thicker power conductor has less power loss than a thinner conductor. Another important aspect of the integrated power system is the power system operator (PSO) or electricity utility. The PSO or utility coordinates, controls, and monitors the power system and it has the authority and responsibility of supplying electricity to its consumers optimally. As the number of power system components increases the PSO needs to deal with a more complex problem of optimal electrical power distribution. A comprehensive mathematical model which addresses the operational complexities of the integrated power system is necessary for economic dispatch of the hydroelectric generators and distribution planning.

The purpose of this thesis is to develop a mathematical model and propose solution approaches for an integrated hydroelectric power system that the PSO can use for optimal

1 generation and distribution planning. Hydroelectric generators dispatch problem and electricity distribution planning problem has been widely studied in the literature that have been reviewed to work on this research. However, very little work is available to address the dispatch and distribution planning for a combined ROR and storage hydroelectric projects. A detailed mathematical model and solution approach is provided in this thesis paper.

Uncertainties in the generators dispatch are considered to formulate the problem as stochastic program. The distribution planning is treated as a two-stage transportation problem to minimize the cost of energy generation, transmission, and distribution under constraints such as energy demand requirement, transmission and distribution line capacity, substation capacity, and supply uncertainty from the ROR hydroelectric power plants. The problem is solved and the comparisons of stochastic solution (SS), expectation of expected value (EEV), and wait and see (W&S) solutions are made. The Integrated Nepal Power System (INPS), consisting of ROR projects and storage projects, is used as a case study to run experiments.

The results from these solution approaches provide an economical dispatch of the ROR and the storage projects, and optimal distribution plan to meet the energy demand. Among the experimental results of the three methods, the W&S solution approach provided the least cost plan. The EEV solution was worse than the SS. The PSO may invest in advanced technology to accurately reveal the uncertainties and to operate at the W&S operational cost. However, the tradeoff between using the SS solution and investing in new technologies to operate at W&S solution may require rigorous feasibility study.

1.2. Research Motivation

Electricity is used for various purposes including lighting, heating and cooling, transportation, manufacturing, to run data centers in service industries. Electricity has been traded similar to other commodities in the deregulated energy market. The main challenge of considering electricity as a commodity is its reliability. Disruption of energy supply would not affect simple household lighting as much as it would the lighting in hospitals and manufacturing industries. Some of the major reasons behind unreliable power supply would be variability in generation station due to the change in input power, unexpected disturbances in the system (like major faults), and unexpected increment in the energy demand. For a power system consisting of renewable

2 energy sources like ROR projects, the variability in input power would be due to the change in water discharge availability in different weather patterns.

Electricity is an essential service, a vital input required for almost every business and personal activity. As businesses grow in a country, the country’s overall economy continues to grow. Serious reliability problems could have overall economic impacts and result in bankruptcies, job losses, and even loss of life. The electricity trade can occur through wholesale transactions (bids and offers): which use supply and demand principles to set the price or through the long term trades which are similar to power purchase agreements between the sellers and the buyers [1, 2]. The decision maker in the power system market is not only responsible for the reliable energy supply, but also at possible minimum cost. The cost of energy generation at power plants is different for different types of power plants (PP). This cost depends on the location of the power plant, its size, fuel cost, and maintenance cost. For example, the hydroelectric power plants need to be installed closer to the riverside, which is most of the time far away from the major consumption areas such as commercial buildings, industries, and residential. The fuel (water) is free of cost for HEP but due to the requirement of building water reservoir and access roads to the power plant the initial investment cost becomes expensive. Larger initial investment makes higher per unit energy cost during its operation to the energy users.

The variable energy generators like ROR hydro, solar, and wind have changes in the supply of primary fuel (water flow, solar insolation, or wind velocity) resulting in fluctuations in the plant’s output every time. In other words, the magnitude and timing of variable generation output is less predictable than the conventional generation. Since the storage technologies, like compressed air or battery storage system, has not been well advanced to store energy in bulk in the customer’s house, the integrated power system operator must ensure that the supply matches load instantaneously. If such variability and uncertainty of generators are not properly addressed before integrating with the existing power system, serious reliability issues may arise and cause the whole system shutdown due to unbalanced energy demand and generation. One of the options to balance energy generation and load, for power system consisting of multiple variable energy generators (VEG) would be by integrating the VEG with non-variable energy generators. This non-variable energy generator supplies electricity when the VEG is not available, and also pick up the load when there is deficit in energy generation from the VEG. Examples of non-variable energy generators include storage dam hydro, natural gas or coal, and nuclear power plants. 3

Integrating both the ROR generators and the storage dam hydroelectric power generators will provide a reliable energy. This requires optimal planning to balance the fluctuating output power from ROR power plants by storage type hydroelectric power plants in real time. The optimal planning of the available energy generators (both ROR and storage) benefits the system operator to balance the system load.

1.3. Objective of the Study

The main purpose of this research is to formulate and solve a comprehensive mathematical model that can be used to optimize hydroelectric energy generation and distribution. As described above, the hydroelectric power generation stations are far away from the consumption locations. The transmission and distribution lines, and substations are required to deliver the power to the consumers located in regions far away from each other.

Furthermore, the nature of demand is different from voltage level perspective; low voltage level to high voltage level (i.e. from residential consumers to industrial consumers). Due to the difference in voltage level of a transmission or distribution line the current flow through the line varies. A high voltage lines have smaller current flower than a low voltage line of same capacity. Hence, the power loss in each line and at each power system component differ from one path to another path. A generation station of having the lowest per unit cost and a transmission/distribution line of lowest power loss cost would always be the cheapest option to supply power if those components can supply all the required power. Once the capacity of a power system component is fully utilized another option needs to be explored to supply the energy demand. Hence, the goal of this research is to explore different combinations of the power system components to supply the load optimally. For example, in the following network Figure 1.1 the electrical energy demand at demand center 푘 = 1 can be supplied by a various combination of generators, transmission lines, substations, and distribution lines. In an integrated power network, it is not possible to find exactly which generator is delivering its power to which of the demand; however, it is expected that by implementing the methodology presented in this thesis the power system operator can dispatch each of its generators at their optimal capacity to minimize the total energy supply cost.

Figure 1.1 Power system network block diagram consisting of hydroelectric power projects

1.4. Organization of the Thesis

The remaining sections of this research are organized as follows: in the second chapter, literature study on basic concepts that are needed to formulate the research problem, the previous work related to optimization of electrical energy planning in general, and the hydroelectric power generation and distribution planning in specific are presented. The third chapter presents the formulation of the research problem as a stochastic mathematical model. Description of the solution approaches with numerical examples using real data are discussed in chapter four. Conclusions and future recommendations are highlighted in chapter five.

CHAPTER 2. LITERATURE REVIEW

2.1. Hydroelectric Power Systems

Electrical energy is a form of energy which is produced due to the flow of electric charge particles present in the electrically conducting materials. The moving electric charge particles are called electrons and a flow of electrons is called electric current and measured in Amperes (A). The potential energy stored in charged particles is converted into kinetic energy due to the force of electric field. The capacity of an electric field to do work on an electric charge is called electric potential or voltage and measured in volts (V). An electric circuit connects an electrical energy source and an electric load. The rate at which electrical energy is transferred by an electric circuit is call electric power and it is measured in watts (W). Generally, amount of electrical energy supplied to an electric load is measured in kilowatt hour (kWh) and it is calculated by taking the time into consideration during which the electrical power was supplied [3-5].

Electric potential or voltage is induced in a closed circuit because of moving electromagnetic field. The electric potential can be fluctuating in nature (like alternating current or AC) or it can be a constant value (like direct current or DC). A battery is an example of DC voltage source. AC voltage is generated in the power plants like hydroelectric power stations. In a hydroelectric power plant, the potential energy of water stored in a water reservoir is converted into kinetic energy by the gravitational flow of water. Then the kinetic energy of water is applied to a mechanical component called turbine which runs an electromechanical generator and the electrical energy is generated by the generator. The amount of electrical power (P) in Watts produced from the water of flow rate (Q) cubic meter per second, and from head height (H) meters is given by the following equation [6]:

P = Q ∗ ρ ∗ g ∗ H ∗ η 2-1

Where, ρ = water density in kg/m3

푚2 g = acceleration due to gravity 푠

η = efficiency of all electromechanical components

Several hydroelectric power plants are connected together to form an integrated power system or a grid. An integrated power system is defined as a complex network which assembles all the equipment and circuits for electrical power generating, transmitting, transforming, and distributing the electrical energy. In power systems, electrical energy is generated at the electrical power generation stations located at the different regions of the country and multiple transmission networks are used to transfer power to the demand centers scattered around the region far away from the generation stations. The integrated power system, also known as a power grid, is considered a huge source of energy (infinite bus). It is because of the fact that multiple large sized generators are connected together to supply variable energy demand in the network. The following figure shows an integrated electrical power system network.

Figure 2.1 Integrated power system

In the above figure, residential, commercial, and industrial electrical loads are supplied from renewable as well as non-renewable energy sources. The run-of-river hydroelectric power plants and other renewables are shown to represent the variable energy generators. Constant power generators are represented by a storage hydroelectric power and other non-renewables. The power generators and power users are connected together by transmission lines, substations, and

7 distribution lines. This whole power system network is coordinated, monitored, and controlled by the PSO or the government or a public entity in the country.

In addition to difference in per unit energy generation cost of power plants, the operational characteristic is also different from one type of power source to another. Here the operational characteristics of a power plant means how fast the plant can respond to load change, this characteristic of a generator is called ramp rate. For example, hydroelectric projects (both ROR and storage) have high ramp rate means that they can respond to the load change faster than other types of power plants. This unique ability of HEP helps the PSO to dispatch hydroelectric generators quickly to provide peaking power and hence frequency control in the grid. Unbalanced generation and demand leads to grid frequency change which could lead to the damage of power system equipment’s and even brown or blackouts [7].

Furthermore, hydropower is one of the least expensive and environmentally clean energy options. As stated in above, hydropower stations with storage reservoirs have high ramp rate: power generation can be scheduled in less than an hour, and even frequent startups and shutdowns can be executed without a significant damaging effect on the infrastructure service life. These qualities make it an excellent complement to other renewable energy sources such as wind. Water reservoirs are suitable to be used as energy storage facilities, “batteries”, to store water during high wind periods, and release this water to produce electricity when it is needed [8]. Hence, the optimal scheduling of HEP could optimize the use of electrical power from other variable energy sources.

2.2. Hydroelectric Power System Components

2.2.1. Generation Station

Electricity is most often generated at a power station by electromechanical generators driven by hydraulic turbines in hydroelectric power plants, wind turbines in wind power plants, and steam or gas turbines connected with the heat engines fueled by chemical combustion or nuclear fission in thermal and nuclear power plants respectively. The hydroelectric power plants are site specific because the electrical power generation from such plants depends on water flow available and the hydraulic head (or height) from turbine to the reservoir. A generation station which has relatively high flow but low head uses a different type

8 of turbine than another generation station with lower discharge but high head. An electromechanical device called governor is used to regulate water flow through the turbine. This governor is integrated with the control systems at generation stations so that output of generators can be controlled based on load change. In this thesis, the phrases “generation station” and “power plant” represent the same component of a power system: power station, where electricity is generated. The generated power is supplied to the transmission lines.

2.2.2. Transmission Line

The transmission network is another important component in an electrical power system. The electrical power transmission lines consist of power carrying conductors to transmit power from the generation stations to distribution substations. The transmission lines may be overhead so that the conductors are mechanically supported with towers or they may be underground where the conductors are buried underground with appropriate insulation. The power generated from generation plant is first fed to the generation substation consisting of step up transformers and other switching as well as protection components. The generation substation steps up the voltage level of the generated power so that the power can be transmitted at a higher voltage in order to minimize power loss in the lines.

An interconnection of multiple transmission lines is called the transmission network. The combined transmission and distribution network is known as the "power grid" in North America, or just "the grid" in rest of the world.

2.2.3. Distribution Substation

The distribution substations are at the end of the transmission lines. These substations are required to step down the voltage level of the transmitted power. Distribution substations consist of the components like step down transformers and switching as well as protection devices. The step down transformers step down the voltage level according to the needs of consumers. For example, industrial customers use power at higher voltage to run high powered machines than the residential customers who use power for lighting purpose. Some common distribution voltage levels being used are: 69 kV, 66 kV, 36 kV, 33 kV, 12.47 kV, 11 kV, to 220 V (or 110 V). The

9 distribution line power conductors are connected at the output terminal of a distribution substation to carry electrical power to the end users.

2.2.4. Distribution Line

The distribution lines consist of power carrying conductors to distribute the electrical power among the consumers located at different location in a demand area. Since the electrical appliances being used in regular life use the power at lower voltage, the distribution lines carry electrical power at lower voltage level. Due to lower operating voltage, the amount of power loss in distribution line becomes higher than the power lines operation at higher voltage (see section 3.4.2).

2.3. Hydroelectric Power Energy Sources

A brief introduction to type of hydroelectric power plant (HEP) schemes is presented as follows [9]:

2.3.1. Storage Projects

A storage project also called impoundment facility is typically a large hydropower system that uses a dam to store river or rain water in a reservoir. Water released from the reservoir flows through a turbine, spinning the turbine, which in turn activates a generator to produce electricity. The water may be released either to meet changing electricity needs or to maintain a constant reservoir level. The storage projects are expensive to build because of larger dam requirement to store water and other civil structures. The dam of a storage HEP has been a huge environmental issue in recent years. It is because as the river flow is blocked by a dam, other living species in upstream as well as downstream get affected due to overflow of water or no-flow respectively. Hence, to discourage building of large water reservoirs, the governments charge some additional taxes in the investment cost of storage HEP which makes such projects more expensive. Once the project is built and started generating electrical power, the operational cost is very minimal as compared to similar size of other sources of energy. It is because the fuel required to generate electricity, which is water, is freely available.

2.3.2. Run-of-River Projects

Run-of-river (ROR) or diversion projects generate electricity proportional to the river’s flow. A diversion channel is used to divert the river water through the turbine. Since the diversion channel does not have or lead to a water storing facility before the turbine-generator set of the ROR project, such power plant cannot regulate water flow and hence the power production is not constant. ROR projects are less reliable power plants because of variable power generation. To construct a ROR project, a natural drop in elevation is required. Utilization of natural drops in elevation make the cost of a ROR project lower than the cost of a storage project of the same capacity.

2.3.3. Pumped Storage

This type of hydropower works like a battery, storing the electricity generated by other power sources like solar, wind, and nuclear for later use. It stores energy by pumping water uphill to a reservoir at higher elevation from a second reservoir at a lower elevation. When the demand for electricity is low, a pumped storage facility stores energy by pumping water from a lower reservoir to an upper reservoir. During periods of high electrical demand, the water is released back to the lower reservoir and turns a turbine, generating electricity.

2.4. Electricity as a Commodity

Considering the fact that there are producers, sellers, and buyers of electrical energy, electricity is a commodity that can be traded like any other products used in our daily life. Furthermore, electrical energy is an undifferentiated good that can be traded in quantity because it is easily measured. In economic terms, electricity (both power and energy) is a commodity capable of being bought, sold, and traded. Therefore, microeconomic theory suggests that consumers of electricity, like consumers of all other commodities, will increase their demand up to the point where the marginal benefit they derive from the electricity is equal to the price they have to pay. For example, a manufacturer will not produce widgets if the cost of the electrical energy required to produce these widgets makes their sale unprofitable. The owner of a fashion boutique will increase the lighting level only up to the point where the additional cost translates into additional

11 profits by attracting more customers. Finally, at home during a cold winter evening, there comes a point where most people will put on some extra clothes rather than turning up the thermostat and face a very large electricity bill.

However, due to some distinct features of electricity, it may not be totally right to compare with other commodities. These distinct features include: electricity is a real-time product, cannot be stored in bulk amount because of expensive cost and therefore must be used when it is produced; it cannot be separated from its means of transportation—transmission and distribution lines, which are owned primarily by utility companies; and it faces an inelastic consumer demand curve. This inelastic behavior of the electricity has been observed both in industrial and domestic uses. Furthermore, unlike other products, it is not possible, under normal operating conditions to have customers queue for it. The manufacturing industries who use electricity for their production process will not cut off the supply just because of a small change in electricity prices. Similarly, the residential users also will not carryout cost-benefit due to the electricity price change while turning on the lights at home or offices [10-13]. The commodities within an electricity market generally consist of two types: power and energy. Power is the metered net electrical transfer rate at any given moment and is measured in megawatts (푀푊). Energy is electricity that flows through a metered point for a given period and is measured in megawatt hours (푀푊ℎ푟).

2.5. Electricity Markets

Literature shows that power system market around the world is either controlled by a single supplier or multiple suppliers. In other words, either a monopolistic energy market or a deregulated energy market. Such market policies can be seen in all generation, transmission, and distribution section of power system.

Monopoly:

A monopoly exists when a specific person or company is the only supplier of a particular commodity. The single supplier in markets means that there is no economic competition to produce the goods or service. No competition in a market leads to a lack of viable substitute goods and the possibility of a high monopoly price well above the firm's marginal cost. In a monopoly electricity market, a single utility manages all three aspects of the power system. The supplier may be a public

12 utility that is responsible for the supply of electricity to homes, offices, shops, factories, farms, and mines or it may be the private organizations subject to monopoly regulation or public authorities owned by local, state or national governments.

Deregulated:

Deregulation is the process of removing or reducing government regulations typically in the economic sphere. When the government reduces regulations over private competitors to generate, transmit, and distribute electrical power, there will be more than a single company responsible for energy supply. Hence, deregulation in electricity market will increase the competition (in generation, transmission, and distribution) and also consumers will have multiple options to buy electricity from. In monopoly situation, the prices are defined by the government. In a deregulated market an electricity stock exchange market is made, the electricity price is established on the base of offer and demand.

2.6. Mathematical Optimization

Mathematical optimization is an analytical tool that is utilized to make decisions. In mathematical terms, an optimization problem is the problem of finding the best solution from among the set of all feasible solutions. The mathematical model can be defined as static vs dynamic models, linear vs nonlinear, integer vs non integer, and deterministic vs stochastic. Generally, an optimization model consists of decision variables, objective function, and constraints.

Decision variables

The decision variables are the components of the system which we change to improve a system performance. In a manufacturing setting, the variables may be the amount of each resource consumed or the time spent on each activity, whereas in data fitting, the variables would be the parameters of the model. In the electrical power system problem, the amount of energy generation per unit time can be considered as a decision variable.

Objective function

The objective function measures the performance of the system that we want to minimize or maximize. For example, in production planning we may want to maximize the profits or

13 minimize the cost of production, whereas in fitting experimental data to a model, we may want to minimize the total deviation of the observed data from the predicted data. Similarly, in power systems minimization of power loss or minimization of cost of energy supply would be an objective function.

Constraints

The constraints are the functions that describe the relationships among the variables and that define the allowable range of decision variable changes. The constraints can be of the types greater than, less than, equal to, or combination of equal to with others, depending on the system. For example, the amount of a resource consumed should be less than or equal to the available amount. Similarly, in a reliable power system network amount of electrical energy supply should be greater than or equal to the amount of energy required at any time.

Optimization techniques have been widely applied in the power systems. Some of these optimization techniques consider the variability in power systems like generation variation and demand variation while other optimization methods consider certainty in future conditions for a small period of time.

The distributed generations (DGs) are also the important aspects of a modern power system (or ‘smart grid’) like generators, transmission/distribution lines, and substations. A smart grid is an electrical grid which includes a variety of energy efficient generating sources (renewables and non-renewables), digital processing and communications to the power grid, smart meters, smart appliances, and business processes that are necessary to capitalize on the investments in smart technology. The concept of integrating small generating units (also called the distributed generations or DGs) in the power system has been a field of study in the last few decades. The presence of DGs introduce new problems on distribution system planning. The major problem is the uncertainties in power generation which make the optimization problem more complex to be solved. On the other hand, a DG helps the main generating station in supplying the growing power demand. Properly planned and operated DGs have many benefits like economic savings, decrement of power losses, greater reliability, and higher power quality. Optimal location and capacity of DGs is an important aspect to gain maximum benefits from them in an existing grid. Different optimization techniques have been implemented in this field by the researchers to find optimal size and site of DGs in the existing network.

The following sections summarize some of the exciting hydroelectric power planning works in the field of power system optimization. This literature study is grouped into multiple sections depending on whether the past research considers variability in power system or not.

2.7. Hydroelectric Power Planning Approaches

2.7.1. Hydroelectric Power Planning as a Transportation Problem

Studies have been carried out on the optimization and scheduling of electrical energy supply by different researchers. The supply of energy occurs in two stages: from generation stations to transmission substations and from transmission substations to demand points. The problem occurs because of the limited capacity of the transmission network and substations, power loss in transmission lines, and changing demand of consumers over the time period. The problem of optimizing the supply of electrical energy in two stages is called the two-stage transportation problem.

The transportation problem (TP) deals with shipping commodities from different sources to destinations. The objective function in the TP is to determine the shipping schedule that minimizes the total shipping cost while satisfying supply and demand limits. The TP finds application in industry, planning, communication network, scheduling, transportation and, allotment etc. [14] [15]. H. Y. Yamin [39] presented a literature review on different kind of optimization models used by various researchers and classified into few main categories to help the development of new techniques for generation scheduling. For example,

Deterministic techniques: Priority list, Integer or mixed-integer programming, Dynamic and linear programming, Branch-and-bound method, Lagrangian relaxation, Security constrained unit commitment, Decomposition techniques. Development of the modern computational tools have helped the researchers to use new artificial intelligence on solving complicated mathematical models for the optimization problems.

Meta-heuristic techniques: Expert system, Artificial neural networks, Fuzzy logic approach, Genetic algorithm, Evolutionary programming, Simulated annealing algorithm, Tabu search, Hybrid techniques.

Hydrothermal coordination: In a hydrothermal system, short-term hydro scheduling is done as part of hydrothermal coordination.

2.7.2. Deterministic Hydroelectric Power Planning

In deterministic optimization, it is assumed that the data for the given problem is accurate. The deterministic approach provides definite and unique conclusions. This approach includes priority list (PL), integer/mixed-integer programming method, dynamic programming (DP), branch-and-bound method, and Lagrangian Relaxation (LR) [16].

Bensalem et al. presented a deterministic optimal management strategy of hydroelectric power plants. The main objective was to maximize the reservoir’s water content which means maximizing the potential energy stored in the reservoir at the end of a “planning horizon” (1 week). The major constraint of this model was to satisfy electrical energy demand during that horizon period. The storage capacity of reservoir, total inflow and outflow of water discharge, and capacity of other components besides the reservoir were also the constraints taken into consideration. The problem was solved by using the discrete maximum principle described by many other authors [17, 18]. The authors concluded that the amount of water stored in all reservoirs at the end of the ‘planning horizon’ is less than water stored at the end of sub-periods of the ‘planning horizon’ [19]. The contribution of the research is appealing to the power system operator because of increase in water storage; however, the authors did not talk about how many iterations were performed and how long it took to solve the problem. It wouldn’t make sense if the solution time was longer than the time period of ‘sub-periods’. Furthermore, water content in reservoir changes due to the natural flow which varies with time, so in the solution presented by the authors there is a possibility of decreasing the natural flow itself during long planning horizon which could result in lesser water storage from first strategy than from second strategy.

Luo et al. developed a mathematic model for small hydropower sustainability. Generally, other models related to optimization of small hydropower by other researchers consider maximal power output as the objective of optimal scheduling, but this model takes into consideration sustainability of small hydropower and adverse effects on environment as well. Its objective is water control and minimal reservoir spill, and it regards the down stream’s minimum flow requirement as the constraint for the controlled reservoir releases for hydropower generation. The

16 case simulation presented in their paper proves that sound economic benefits could be achieved by using the model because water is main source of the electrical energy in hydropower [20].

Nick et al. in the field of optimization, their objective function of the optimization problem accounts for the minimization of different aims: (i) energy cost from the external grid, (ii) penalty deviations from the day ahead schedule of the energy import/export from/to external grid, (iii) cost of changing the transformer tap-changer position and (iv) total network and Battery Energy Storage Systems (BESSs) losses [21].

Subramanian et al. investigated the real-time power scheduling to deferrable loads such as electric vehicles and thermostatically controlled loads. They explored three scheduling algorithms for renewable generation: earliest deadline first (EDF), late laxity first (LLF), and receding horizon control (RHC). In the RHC algorithm, the objective function is to minimize the cost of grid capacity and grid energy with some of the constraints like generation capacity limit and energy requirement. They showed with the help of simulations that the coordinated scheduling, via any of these three policies, would decrease the required reserve energy to meet load requirements while only EDF and RHC reduce the reserve capacity requirement [22].

The authors Y. G. Hegazy et.al applied the Big Bang Big Crunch optimization algorithm on balanced/ unbalanced distribution networks for optimal placement and sizing of distributed generators. W. El-Khattam proposed a combined methodology of using an optimization model and the distribution system planner’s experience to achieve optimal sizing and location of distributed generation to meet the growing demand. Similarly, L. Xia et.al used the improved adaptive genetic algorithm to optimize the siting and sizing of DG. The objectives of the studies include power loss minimization, maintaining the voltage level, minimizing the operation and maintenance cost of DGs, and optimal capacity of the power system components to meet the demand. The constraints of the studies are: power balance, distribution network thermal capacity, substation capacity, voltage limits etc. [23-25].

2.7.3. Stochastic Hydroelectric Power Planning

The stochastic optimization problems are also called optimization under uncertainty. The uncertainty is due to unknown values of data which may be due to measurement error, error in forecasting the data about future. For example, in the modern power system the renewable energy 17 power plants have been included which adds a lot of unpredictability in the generation as well as in demand. There is a need of modeling the growing uncertainties due to the wind and PV power penetration into the grid. Such modeling would make a smooth operation of power systems. The stochastic problems consider many types of uncertainties in the field of demand and supply. Addition of renewable energy sources into the power system increases the supply and demand uncertainties and hence, stochastic optimization model is necessary to solve such problems to get an optimal solution [26].

Unlike the unique result from deterministic model, the results obtained for stochastic loads may not be exact. In stochastic problems, the uncertainty is incorporated within the model. The stochastic formulations are based on some probability distributions and can be solved by using any of the deterministic algorithms after changing the original stochastic constraints into determinate constraints [16, 27].

The authors B. Saravanan et al. used a Dynamic Programming (DP) algorithm to solve the unit commitment problem in a stochastic generation and demand environment. The objective of this model was to minimize the total cost of energy generation. When the demand and generation uncertainties are included in the unit commitment problem, the conventional deterministic approaches might not give a good result as compare to the stochastic models. Therefore authors formulated the stochastic unit commitment problem in a deterministic framework and solved the model using the DP algorithm [16].

A coordinated deterministic and stochastic framework for maintenance scheduling of generators, using improved binary particle swarm optimization (IBPSO) was presented by K. Suresh et.al [28]. The main objective of the proposed maintenance scheduling model was to reduce the generator failures and to extend the generator lifespan, which would increase the system reliability. The deterministic leveled reserve rate method was used to minimize the deviation in annual supply reserve ratio for a power system. The stochastic method considers the loss of load probability (LOLP) and uses the Weibull distribution to analyze the impacts of aging failures of the generators in order to calculate the unavailability of power system.

A stochastic model can be single stage, two stage or multistage depending upon how many times we can correct the decision. Single stage stochastic optimization is the study of optimization problems with a random objective function or constraints where a decision is implemented with

18 no subsequent recourse. In other words, in the single stage stochastic optimization problems, a decision made at time 0 and the result is observed later (at time 1, say) but no further decisions can be made. The two-stage stochastic optimization problems, if the decision made at time 0, can be corrected at a later time (time 1, say) by some corrective decision variable. In multistage stochastic optimization problems, the corrective decisions may be taken at several time 1, 2,…T [29].

Optimal Power Flow (OPF) problem in electrical power system may have single objective or multi objectives. The necessity of solution approaches in the field of optimal power flow is to calculate the optimal amount of power supply from the generator buses to load buses in an integrated power system. Since the power companies have been moving in to a more competitive environment, OPF has been used as a tool to define the level of inter-utility power exchange.

P. Smita et al. provided an approach to solve the single objective Optimal Power Flow (OPF) problem considering the objective function of minimizing the reactive loss in an electrical network, while satisfying a set of system operating constraints including constraints dedicated by the electrical network. The Particle Swarm Optimization, a stochastic optimization technique was used to solve this problem [14].

A group of researchers, T. Sicong et al. explained in a paper [30] about the necessity of integration of the micro grid economic load dispatch problem and network configuration problem in order to benefit the whole network. In this paper, an integrated solution that takes care of both micro grid load dispatch and network reconfiguration was proposed. The authors considered the stochastic nature of wind, PV and energy demand in the mathematical model. The proposed scheme used the power flow technique to minimize the total operating cost of a distribution network with multiple micro grids. In addition to minimizing the operating cost, the scheme was also able to alter the network structure in order to handle faults occurred on the distribution feeders.

In a similar way, another application of stochastic modeling technique in electrical power system is presented by J. A. Momoh et al. [31]. The changing behavior of energy demand and variability in energy production from the renewable energy sources change the voltage and reactive power in the network. The voltages beyond limit may damage costly sub-station devices and equipment at consumer end as well. The renewable energy sources like Photovoltaic and Wind power cannot generate the reactive power (measured in VAR) hence, such renewable energy sources pose more disturbance in the system because of the variable reactive power demand. This

19 paper develops a stochastic optimization for Voltage/Var for Optimal Power Flow (OPF) by considering the load variation. The variability in electrical load was modeled using a probability distribution function generated from demand data. The objective of this model was to minimize the cost of energy supply by satisfying the constraints like demand, voltage limit in transmission lines, VAR limit of energy sources, and thermal limit of transmission lines.

The electromechanical components used in the hydroelectric power plants are affected by frequent start-up/shut-down. Higher the frequency of shut-down and start-up shorter would be the service life of a component. This reduction in service life of a generator can be modeled as increase in operational cost of the component. Hence, smaller the frequency of shut-down and start-up better the service life of the electromechanical components. On the other hand, depending upon the variations in tailrace elevation, penstock head losses, and turbine-generator efficiencies some units could be started-up/shut-down and hence total hydro power efficiency could be improved. Arce et al. developed a dynamic programming model to optimize the number of hydroelectric generating units in operation at each hour of the day. This model highlights the tradeoff between start-up/shut-down of the generating units and hydro power efficiency [32].

2.7.4. Large Scale Hydroelectric Power Planning

The large scale optimization models use some sort of modern computational techniques like the Artificial Intelligence (AI). Development of the modern computational tools have helped the researchers on solving such complicated mathematical models.

K. Antony et al. proposed a genetic algorithm (GA) approach to solve two stage transportation problem so that the total cost of transporting goods would be minimum in a supply- chain. A two-stage transportation problem has a transportation network with some associated costs, from plants to customers through distribution centers (DCs). These DCs have some constraints of capacity and also they have some operation and maintenance cost [33]. To implement a similar approach in the power system problem a lot of modifications in the constraints are needed because of the nature of flow of electric power. The electric power cannot be stored in warehouse like other goods.

L. Shuangquan et al. applied a decomposition approach combined with linear approximation method to the short-term hydropower optimal scheduling problem of multi 20 reservoir system. The constraints taken into account include the maximum and minimum water level of reservoir storage and reservoir release, water traveling time, power ramp, minimum startup/shutdown time, maximum startup/shutdown frequency, prohibitive operational regions, and so on. The model developed in this paper was capable of handling the complex objectives (allocating water among power plants to maximize the profit and pumping water to pump-storage plants) and constraints of multi reservoir system and taking pump-storage plant into account [34].

An exciting approach to find an optimal way of connecting a new transmission line into the existing power grid was explored by authors G. Vinasco et. al. The authors described the possible interconnections and investments options involved in connecting a 2400MW hydroelectric power plant (HidroItuango, Columbia) with the grid in order to strengthen the Colombian national transmission system. A Mixed Binary Linear Programming (MBLP) model was used to solve the transmission expansion problem of the Colombian electrical system. The model explores the connection options, accounting for uncertainty in the installed capacity with various generation and demand scenarios. The model result was a unique Transmission Expansion Plan valid in all generation-demand scenarios and grid contingencies [35].

Some optimization approaches based on artificial neural networks (ANNs) have been developed to optimize energy generation by the hydroelectric power plants. R. H. Liang et al. proposed an optimal scheduling strategy of hydroelectric generations. The purpose of hydroelectric generation scheduling was to figure out the optimal amounts of generated powers for the hydro units. The objective is similar to the hydro-thermal scheduling problem however the proposed ANN approach is much faster than conventional dynamic programming approach [36]. Similalry, R. Naresh and et. al proposed a technique for optimizing the operation of interconnected hydropower plants by using ANN. The model gave an optimal water release plan to meet the objectives: maximize the annual power generation and satisfy the irrigation demand. The objectives were defined based upon the constraints like load balancing between generation and demand, water storage limit in reservoir, water spillage from reservoir etc. [37].

Some of the researchers explored new approaches for the scheduling of hydro-thermal units. The optimal hydro-thermal schedule provides a solution with minimum thermal production cost while maximizing energy output from the hydro power plants and maximizing power system reliability. The objective function subjects to the constraints like electric power balance, water

21 flows, hydro discharge limits and reservoir limits. H. Shyh-Jier proposed an Ant Colony System (ACS) based optimization approach for the enhancement of hydro-thermal generation scheduling [38]. P. Doganis et al. [39] presented a convex model called mixed integer quadratic programming (MIQP) model and a paper by J. Wu and et al. [40] describes a hybrid approach to meet the objective of hydro thermal units. The same objective of minizing the total cost of energy generation from hydro-thermal combination is descbribed by J. Zábojník et al. They presented a comprehensive mixed-integer linear programming (MILP) model of transmission network and power plants. This comprehensive model formulation includes thermal, pumped-storage and conventional hydro plants as well as renewable resources [41] [42].

As summarized above, several researchers working in the field of operation research and power system have explored deterministic, meta-heuristic as well as hybrid approaches to find the optimal schedules for supplying electrical power to the load, maximizing power system reliability, and minimizing cost both in monopoly energy market as well in deregulated energy market. Maximization of energy output from the generation stations, maximization of fuel supply to the generators, minimization of generation failures, minimization of greenhouse gas emissions from the generators, and optimal dispatch scheduling of a set of generating units are some of the objectives of the optimization models that have been developed to improve the performance of generation stations in a power system. Each method has its own pros and cons based on model complexity, computational time, and output results.

Furthermore, some of these models consider the variability in power systems like generation variation and demand variation while some of the model consider certainty in future conditions for a small period of time. These articles present methodologies for optimization only in the generator side or in demand side. However, this thesis presents a stochastic optimization methodology that consider all the components of the integrated power system (generator, transmission and distribution line, and substation). The power generation, transmission, and distribution problem of an integrated power system and its components are combined together to form a two-stage transportation problem. Power generated by the storage makes the first stage of the problem while the power generated by ROR projects makes the second stage of the two-stage transportation problem.

CHAPTER 3. PROBLEM FORMULATION

3.1. Problem Definition

The formulation of this research problem is based on the tenet of two-stage stochastic transportation problem. The dam generation and the ROR generation projects are considered as energy supply sources. The generation stations supply energy to the demand locations via transmission lines and sub-stations.

The transmission system carries electric power efficiently, and in large amounts, from ge nerating stations to consumption areas. Depending upon the location of power generation stations from the users, the length of transmission lines is different. The diameter of power conductor used to transmit the power depends on the voltage level of the transmission network. Diameter of the conductor and length of the line affect the electrical power loss in the network which would affect operational cost of the lines. In addition to the design parameters, the operational and maintenance cost of the transmission lines varies from one line to another based on how many times in a year the network is subjected to disturbances like electrical faults (short-circuits) and lightning. Similarly, power loss, operation, and maintenance cost at every substation and distribution line is different from another set of the similar components. A combination of a generator, a transmission line, a substation, and a distribution line to supply a demand may not be the same for another combination of the different generators, transmission and distribution lines, and substations due to the cost difference.

The electric power utility companies aim to minimize costs while providing a reliable and affordable power to their customers. A solution from a comprehensive mathematical model which would incorporate the variability in generation, transmission/distribution networks (failures due to faults), and their various cost could give the power system operator (PSO) or the utility an idea of optimal power generation and distribution. Incorporating all kind of variabilities in a power system would make the model complex and almost impossible to solve however, the major uncertainties like changing weather pattern and system faults (short-circuits) would give an easily solvable and practically applicable mathematical model.

A stochastic mathematical model is developed for an integrated power system consisting of ROR and storage HEP. The mathematical model consists of two stages: first stage is to represent the storage type projects and the second stage represents the ROR projects. This model provides a solution for the amount of power generation, transmission and distribution over one-year period by the ROR and storage power plants. The study took place over a one-year period. Power generated by the storage projects is more certain than the ROR hence the probability of power generation by the storage plants during the study period is considered as 1. In contrast, in case of the ROR plants one-year period is divided into 푡 number of seasons based on the past year’s monthly power generation pattern from the ROR projects. Each season has a certain probability of happening over the study period. For a given season, the power plant can only generate up to a certain percentage of their installed capacity during that season. The past history data on power generation by the ROR projects during each season of a year can be used to find the probability of happening of the season and the maximum capacity of a power plant during that season. Similarly, the capacity of transmission lines, substations, and distribution lines are used to as a constraint in the mathematical model. These capacity parameters are known to the PSO or the utility at the time of detailed engineering design of the power system. In addition to the capacity parameters, the cost parameters: per unit energy generation cost, per unit energy transmission/distribution cost, and the per unit energy loss cost at the substations are modeled in the formulation. A step by step methodology on computing the cost parameters is described in this thesis. This mathematical model can be applied to any kind of energy source (like wind, PV, or non-renewables) if the cost information, capacity parameters, and past data on energy generation are available.

The following table describes all notation that are being used in this thesis.

3.2. Definition of Terms

Table 3.1 Notation

Notation

퐶푔푖 Cost of energy generation at storage power plant 푖 in dollar per 푀푊ℎ푟 = Fuel cost in dollar per 푀푊ℎ푟 + Operation and Maintenance cost in dollar per 푀푊ℎ푟.

퐶푡푙푖푗 Cost of energy transmission in line 푖푗 in dollar per 푀푊ℎ푟 = Power loss cost in dollar per 푀푊ℎ푟 + Operation/Maintenance cost in dollar per 푀푊ℎ푟

퐶푠푗 Cost of energy loss and operation/maintenance cost at substation 푗 in dollar per 푀푊ℎ푟

퐶푧푗푝 Cost of energy transmission in line 푖푗 in dollar per 푀푊ℎ푟 = Power loss cost in dollar per 푀푊ℎ푟 + Operation/Maintenance cost in dollar per 푀푊ℎ푟.

퐶푑푗푘 Cost of energy transmission in line 푗푘 in dollar per 푀푊ℎ푟 = Energy loss cost in dollar per 푀푊ℎ푟 + Operation/Maintenance cost in dollar per 푀푊ℎ푟.

푛1 Total number of storage Power plants

푛2 Total number of ROR plants

푚 Total number of substations (transmission substations)

푙 Total number of demand centers (distribution substations)

푖 Generation stations 푖 = 1,2,3 … . 푛

푗 표푟 푝 Substations 푗 표푟 푝 = 1,2,3...... 푚

푘 Demand centers 푘 = 1,2,3 … . . 푙

푡 Number of seasons in a year 1, 2, 3,4,5,6

푠 Number of scenarios per season or sample size

Decision variables

푥푖푡 Energy generated by storage power plant 푖 during season 푡 in 푀푊ℎ푟

푦푖푡푠 Energy generated by ROR Power plant 푖 during season 푡 and scenario 푠 in 푀푊ℎ푟

푥푖푗푡 Energy transmitted from storage plant 푖 to substation 푗 in 푀푊ℎ푟 during season t in 푀푊ℎ푟

푦푖푗푡푠 Energy transmitted from ROR power plant 푖 to substation 푗 in 푀푊ℎ푟 during season 푡 and scenario 푠 in 푀푊ℎ푟

푥푗푝푡 Energy from storage projects transmitted from substation 푗 to substation 푝 in 푀푊ℎ푟 during season t in 푀푊ℎ푟

푦푗푝푡푠 Energy from ROR projects transmitted from substation 푗 to substation 푝 in 푀푊ℎ푟 during season t and scenario 푠 in 푀푊ℎ푟

푑푗푘푡푠 Energy distributed from substation 푗 to demand center 푘 during season 푡 and scenario 푠 in 푀푊ℎ푟.

푓푖푡푠 Fraction of total installed capacity during season 푡 and scenario 푠 also known as the capacity factor. Capacity fraction for each season follows the uniform distribution (see in chapter 4).

Capacity factor of a power plant is the ratio of its actual energy output over a period of time to the energy that it would have generated during the same time period if it was operated at its installed capacity. It is given as:

퐶푎푝푎푐푖푡푦 퐹푎푐푡표푟 (%) 푇표푡푎푙 푎푚표푢푛푡 표푓 푒푛푒푟푔푦 푡ℎ푒 푝푙푎푛푡 푝푟표푑푢푐푒푑 푑푢푟푖푛푔 푎 푝푒푟푖표푑 표푓 푡푖푚푒 = 푋100 퐴푚표푢푛푡 표푓 푒푛푒푟푔푦 푡ℎ푒 푝푙푎푛푡 푤표푢푙푑 ℎ푎푣푒 푝푟표푑푢푐푒푑 푎푡 푓푢푙푙 푐푎푝푎푐푖푡푦

푝푡푠 Probability of occurring season 푡 and scenario 푠

푍푗푝 표푟 푍푝푗 Power carrying capacity of inter substation transmission line 푗푝 or 푝푗 in 푀푊

퐷퐿푗푘 Power carrying capacity of a distribution line 푗푘 in 푀푊

푇퐿푖푗 Power carrying capacity of a transmission line 푖푗 in 푀푊

푆푖 Installed capacity of a power plant 푖 in 푀푊

푆푆푗 Installed capacity of a substation 푗 in 푀푉퐴 표푟 푀푊

퐷푘 Power demand at demand center 푘 in 푀푊

3.3. Mathematical Model

3.3.1. Objective Function

The objective function of the problem is to minimize the total energy supply cost over one- year period. The cost of energy supply consists of energy generation cost, power transmission cost, power loss cost at the substations, and the power distribution cost. The per unit energy cost ($/푀푊ℎ푟) is assumed to be constant for all seasons of the year. For each season, the product of per unit energy generation cost (in $/푀푊ℎ푟) of a generator and energy generation by that generator (in 푀푊ℎ푟) gives the power generation cost of a generator. The generation cost for all the generators is computed in the same way for all seasons and added together. Then, the sum is multiplied with the probability of the season to get the actual energy generation cost during that season. The same process is applied to get the energy transmission cost, power loss cost at substations, and distribution cost. The final sum of all the costs gives the total energy supply cost in a year. Mathematically, the objective function is given as below.

푇 푆 푛1 푛2 푀푖푛푖푚푖푧푒 푍 = Time ∗ [ ∑푡=1 ∑푠=1 푝푡푠 [∑푖=1 퐶푔푖(푥푖푡) + ∑푖=1 퐶푔푖(푦푖푡푠) +

푛1 푚 푇 푆 푛2 푚 ∑푖=1 ∑푗=1 퐶푡푙푖푗(푥푖푗푡)] + ∑푡=1 ∑푠=1 푝푡푠 [∑푖=1 ∑푗=1 퐶푡푙푖푗 ∗ 푚 푚−1 푦푖푗푡푠 + ∑푗=1 ∑푝=1 퐶푧푖푗 ∗ (푥푗푝푡 + 푦푗푝푡푠) ] + 푇 푆 푚 푙 ∑푡=1 ∑푠=1 푝푡푠 [∑푖=1 ∑푗=1 퐶푠푗 ∗ (푥푖푗 + 푦푖푗푡푠) + 푚 푙 ∑푗=1 ∑푘=1 퐶푑푗푘(푑푖푗푡푠)] ] 3-1

Where,

푛1 ∑푖=1 퐶푔푖(푥푖푡) – total generation cost of all storage projects

푛2 ∑푖=1 퐶푔푖(푦푖푡푠) – total generation cost of all ROR projects

푛1 푚 ∑푖=1 ∑푗=1 퐶푡푙푖푗(푥푖푗푡) – total cost of power transmission from storage projects to distribution substations

푛2 푚 ∑푖=1 ∑푗=1 퐶푡푙푖푗 ∗ 푦푖푗푡푠 - total cost of power transmission from storage projects to distribution substations

푚 푚−1 ∑푗=1 ∑푝=1 퐶푧푖푗 ∗ (푥푗푝 + 푦푗푝푡푠) – total cost of power transmission between inter- substation lines

푚 푙 ∑푖=1 ∑푗=1 퐶푠푗 ∗ (푥푖푗 + 푦푖푗푡푠) – total power loss cost at the substation components like transformers and switching devices

푚 푙 ∑푗=1 ∑푘=1 퐶푑푗푘(푑푖푗푡푠) – total power distribution from substations to the demand centers

Time − Time of a year in hours = 8760 hours

3.3.2. Constraints

I. Generation capacity: The amount of power (energy) produced by each power plant

푖 during season 푡 and scenario 푠 (푥푖푡 표푟 푦푖푡푠) must be less than or equal to its

designed full capacity (푆푖) during that season.

푥푖푡 ≤ 푆푖 푦푖푡푠 ≤ 푓푖푡푠 ∗ 푆푖 ∀ 푖, 푡, 푠 3-2

Also, power (energy) produced by a plant 푖 must be equal to total energy

transmitted from a plant 푖 to substation 푗. i.e. zero energy storage at the power plant at

any season.

푚 푚 ∑푗=1(푥푖푗푡 − 푥푖푡) = 0 ∑푗=1 (푦푖푗푡푠 − 푦푖푡푠) = 0 ∀ 푖, 푡, 푠 3-3

II. Transmission line capacity (from generation stations to substations) and

Distribution line (from substations to demand center) capacity: The amount of

power (energy) transmitted through a transmission line and distribution line must

be less than the power carrying capacity of the line.

푥푖푗푡 ≤ 푇퐿푖푗 푦푖푗푡푠 ≤ 푇퐿푖푗 푥푗푝푡 + 푦푗푝푡푠 ≤ 푍푗푝 푑푗푘푡푠 ≤ 퐷퐿푗푘 3-4

III. Substation capacity: Total power input at substation 푗 must be less than or equal to

the capacity of that substation.

푛1 푛2 푚−1 ∑푖=1 푥푖푗푡 + ∑푖=1 푦푖푗푡푠 + ∑푝=1 (푥푝푗푡 + 푦푝푗푡푠) ≤ 푆푆푗 ∀푗, 푡, 푠 3-5

IV. The power that enters a substation should be equal to the power that leaves that

substation. Assume that there is no power loss at substation.

푛1 푛2 푚−1 푚−1 푙 ∑푖=1 푥푖푗푡 + ∑푖=1 푦푖푗푡푠 + ∑푝=1 (푥푗푝푡 + 푦푗푝푡푠) − ∑푝=1 (푥푝푗푡 + 푦푝푗푡푠) − ∑푘=1 푑푗푘푡푠 = 0 ∀푗, 푡, 푠 3-6

V. Demand Constraint: All demand must be met.

푚 ∑푗=1 푑푗푘푡푠 ≥ 퐷푘 ∀푘, 푡, 푠 3-7

VI. Power or energy cannot be negative.

푥푖푡, 푦푖푡푠, 푥푖푗푡, 푦푖푗푡푠, 푥푗푝푡 , 푦푗푝푡푠, 푑푗푘푡푠 ≥ 0 ∀푖, 푗, 푘, 푡, 푠 3-8

3.4. Cost Parameter Definition

Generally, the cost of a power plant consists of four parts – associated costs, induced costs, external costs, and the opportunity cost of water. Associated costs are costs that are associated with the need to produce power, such as costs of engineering structures and equipment as well as their operation and maintenance costs. Induced costs are costs that are needed to mitigate the adverse impact produced by the project on nature, people or existing ground conditions, such as costs of environmental mitigation measures. Furthermore, there are external costs also that may be needed for the smooth construction and operation of the project though they may also serve other purposes, such as new infrastructure development required for accessibility to the project site. In addition, the opportunity cost of using the natural resource – water - in a hydropower project should be added to the total cost to reach the full cost or the opportunity cost can be replaced by the fuel cost in case of power plants like thermal/nuclear generators [43]. To be more general, these costs are grouped into two categories: Fixed cost and Variable cost.

Fixed cost:

Fixed cost is a one-time cost that is spent during construction and installation and does not depend on intended loading variation to be served after operation. Fixed cost consists of

29 construction, installation, equipment, land, permits, site developing and preparation, taxes, insurance, labor, and testing costs. The capital cost of building a generator station varies from region to region, largely as a function of labor costs and "regulatory costs," which include things like obtaining siting permits, environmental approvals, and so on. It is important to realize that building a generation station takes an enormous amount of time. In a state such as Texas (where building power plants is relatively easy), the time-to-build can be as short as two years. In California, where bringing new energy infrastructure to fruition is much more difficult (due to higher regulatory costs), the time-to-build can exceed ten years. Because of these policy differences there exists an uncertainty in the final cost of erecting power plants.

Variable cost:

The cost which varies with time is variable cost. For example, the operation maintenance cost of a power plant varies year after year. It is because older plants require more maintenance. The operational costs and the cost of new components to replace older ones increase due to economic factors like inflation. Operating costs for power plants include fuel, labor and maintenance costs. Also, the operating cost depends on how much electricity that the plant produces over a period of time. The operating cost required to produce each MWh of electric energy is referred to as the "marginal cost." Fuel costs dominate the total cost of operation for fossil-fired power plants. For renewables, fuel is generally free (perhaps with the exception of biomass power plants in some scenarios); and the fuel costs for nuclear power plants are actually very low. For these types of power plants, labor and maintenance costs dominate total operating costs [44].

The mathematical formula for the costs used in the optimization model are formulated as given in the following section. An experimental study is presented in chapter 4 by using the mathematical model developed in this chapter. The experiments are carried out by using data from the Integrated Power System Nepal. The Nepal Electricity Authority (NEA) is the only electricity utility in the country hence all the data needed for the experimental study were obtained from the utilities website. Following section also describes how each of the parameters can be obtained to get the per unit energy cost of the power system components.

3.4.1. Generation Cost

The general formula used in this research for computing the cost of energy generation is,

퐺푒푛푒푟푎푡푖표푛 푐표푠푡 푖푛 $ 푝푒푟 푀푊ℎ푟 (퐶푔) = (푒 + 푔 + ℎ)/푓 3-9

Where

(1 + 푐)푛 ∗ 푑 퐴푛푛푢푎푙 퐼푛푣푒푠푡푚푒푛푡 푅푒푡푢푟푛 $ (푒) = ((1 + 푐)푛 − 1)

푛 = 푇표푡푎푙 푦푒푎푟푠 표푓 표푝푒푟푎푡푖표푛 (푛)

푎 = 퐼푛푠푡푎푙푙푒푑 푐푎푝푎푐푖푡푦 푀푊 (푎) 푓푟표푚 푁퐸퐴 푟푒푝표푟푡

푏 = 퐼푛푣푒푠푡푚푒푛푡 푝푒푟 푘푖푙표푤푎푡푡 $/푘푊 (푏) 푓푟표푚 푁퐸퐴 푟푒푝표푟푡

푐 = 퐸푡푒푟푛푎푙 푅푎푡푒 표푓 푅푒푡푢푟푛 % (푐 ) 푓푟표푚 푁퐸퐴 푟푒푝표푟푡

푑 = 푇표푡푎푙 푖푛푣푒푠푡푚푒푛푡 푖푛 푝푟표푗푒푐푡 (푑) = 푎 ∗ 푏

푓 = 퐴푛푛푢푎푙 푀푊ℎ푟 퐺푒푛푒푟푎푡푖표푛 (푀푊ℎ푟) (푓) 푓푟표푚 푁퐸퐴 푟푒푝표푟푡

푔 = 퐸푒푛푒푟푔푦 푡푎푥푒푠 (표푟 퐸푛푒푟푔푦 푅표푦푎푙푡푦)푖푛 $ (푔)푓푟표푚 푁퐸퐴 푟푒푝표푟푡

ℎ = 푂푝푒푟푎푡푖표푛 푎푛푑 푀푎푖푛푡푒푛푎푐푒 퐶표푠푡 푖푛 $ (ℎ) 푓푟표푚 푁퐸퐴 푟푒푝표푟푡

3.4.2. Transmission Line, Distribution Line, and Inter Substation Line Cost

Power loss in transmission line depends upon the amount of current flow through the line and electric resistance of the line.

Power loss cost is calculated using following equation;

2 퐴푐푡푖푣푒 푝표푤푒푟 푙표푠푠 푖푛 푝. 푢. (푃푙표푠푠) = 퐼푖푗 ∗ 푅푖푗 3-10

푊ℎ푒푟푒,

퐼푖푗 = 퐶푢푟푟푒푛푡 푓푙표푤 푡ℎ푟표푢푔ℎ 푙푖푛푒 푖푗 푖푛 푝푒푟 푢푛푖푡 푤ℎ푖푐ℎ 푖푠 푐표푚푝푢푡푒푑 푓푟표푚

푙표푎푑 푓푙표푤 푎푛푎푙푦푠푖푠 (푠푒푒 푠푒푐푡푖표푛 3.5.5)

푅푖푗 = 퐷퐶 푟푒푠푖푠푡푎푛푐푒 표푓 푙푖푛푒 푖푗 푖푛 푝푒푟 푢푛푖푡

Only dc resistance is considered in this thesis because it is assumed that electrical loads are active loads and hence the power factor of the power flowing through line is approximately unity. DC resistance of a transmission line conductor is calculated as:

푅푡 = 푅표(1+ ∝ (푡 − 푡표))

푊ℎ푒푟푒,

푅푡 = 퐷퐶 푟푒푠푖푠푡푎푛푐푒 푎푡 푡푒푚푝푒푟푎푡푢푟푒 푡 ℃

푅표 = 퐷퐶 푟푒푠푖푠푡푎푛푐푒 푎푡 푖푛푖푡푎푙 푡푒푚푝푒푟푎푡푢푟푒 푡표℃ ∝ = 푇푒푚푝푒푟푎푡푢푟푒 푐표푒푓푓푖푐푖푒푛푡 표푓 푡ℎ푒 푐표푛푑푢푐푡표푟

Thus the cost associated with power loss in line is calculated as:

푃푙표표푠 푖푛 푀푊ℎ푟 = 푃푙표푠푠푖푗 ∗ 퐵푎푠푒 푀푉퐴 ∗ 푇푖푚푒 푖푛 ℎ표푢푟푠

Average power purchase rate = $70.30 per 푀푊ℎ푟 (assuming 1$=100 Nepalese rupees (NRs.))

푎 = 퐴푛푛푢푎푙 (푇 푎푛푑 퐷) 푑푒푝푟푒푐푖푎푡푖표푛 푐표푠푡 푖푛 $ (푎)

푏 = 퐴푛푛푢푎푙 표푝푒푟푎푡푖표푛 푚푎푖푛푡푒푛푎푛푐푒 푐표푠푡 푖푛 $ (푏)

푐 = 푃표푤푒푟 푙표푠푠 푐표푠푡 푖푛 $ (푐) = 푃푙표푠푠 푖푛 푀푊ℎ푟 ∗ 퐴푣푒푟푎푔푒 푝표푤푒푟 푝푢푟푐ℎ푎푠푒 푟푎푡푒

푑 = 퐻표푢푟푠 푖푛 푎 푦푒푎푟 (푑) = 8760 hours

푒 = 푇퐿 푐푎푝푎푐푖푡푦 푖푛 푀푊 (푒)

푓 = 푇표푡푎푙 푐표푠푡 표푓 푎 푇&퐷 푙푖푛푒 ($/푀푊ℎ푟) (푓) = (푎 + 푏 + 푐)/(푑 ∗ 푒) 3-11

3.4.3. Substation Cost

푎 = 퐴푛푛푢푎푙 푠푢푏푠푡푎푡푖표푛 푑푒푝푟푒푐푖푎푡푖표푛 푐표푠푡 푖푛 $

푏 = 퐴푛푛푢푎푙 표푝푒푟푎푡푖표푛 푎푛푑 푚푎푖푛푡푒푛푎푛푐푒 푐표푠푡 푖푛 $

푐 = 푆푢푏푠푡푎푡푖표푛 푐푎푝푎푐푖푡푦 푖푛 푀푊

푑 = 퐻표푢푟푠 푖푛 푎 푦푒푎푟 = 8760 ℎ표푢푟푠

퐶표푠푡 푎푡 푠푢푏푠푡푎푡푖표푛 ($/푀푊ℎ푟) = (푎 + 푏)/(푐 ∗ 푑) 3-12

3.5. Resource Capacity Definition

3.5.1. Generation Station Capacity

The electrical power (푃 or 푆푖 in the objective function) in Watts produced from the water of flow rate (Q) cubic meter per second, and from head height (H) meters is given by the equation 2-1. A hydropower project is very site specific. A particular site for a hydropower project will produce a specific amount of power and energy that no other site can duplicate. Here are some lists of parameters that are useful while designing the hydroelectric project (HEP):

Hydraulic Head (H): From above equation 2-1, power produced from hydro energy is dependent on hydraulic head (H) which varies according to location of power plants. For a constant discharge, as the values of H increases the electrical power output will also increase. However, increasing H will increase the length of penstock connecting the headrace and tailrace. Hence, a comparative study on amount of power increased versus the cost of penstock will give the economic value of H.

Design discharge (Q): Discharge available in most of the rivers varies monthly due to change in rainfall pattern and climate change. Generally, HEPs are designed on the basis of the normal available flow for the river throughout a year. As an example, if a power plant is designed on the assumption that Q amount of flow will be available 60% of a year then Q60 is the design discharge of that HEP and it means the power plant will operate at full capacity only during 60% of the year. Hence, depending upon the project site, design discharge varies like Q45 or Q60 or Q100. The storage hydroelectric projects are designed to have almost constant discharge (Q) throughout a year so design discharge of storage HEP is Q100.

In power stations, an electromechanical machine is installed next to the turbine that regulates water flow (or steam flow in case of a steam turbine) through the turbine blades. This electromechanical device is called the governor. In other words, the generation limit is determined by controlling the governor. The following figure shows schematic diagram of a governor.

Figure 3.1 Electromechanical governor

The main purpose of governor is to regulate water flow through the turbine based on load change. If the electrical load on generator increases, speed of the turbine decreases which is sensed by the governor and it opens the water valve to increase water flow through the turbine blades resulting in increased speed. Similarly, a decrease in electrical load increases turbine speed and the governor closes water valve to reduce the water flow and hence reduce the increased speed back to original value (a reference value). One example of increased load could be, when a ROR doesn’t supply power to the grid, then the governor in a storage power plant increases water flow to turbine and hence balances the energy generation and demand.

Capacity of hydroelectric power plants is computed by using equation 2-1. Value of Q and H are obtained from a field visit to the project site.

3.5.2. Transmission Line, Distribution Line, and Inter Substation Line Capacity

Transmission line and distribution line capacity (TL and DL) is calculated as follows:

푉퐿 2 푙 푁푐∗150 푃 = (( ) − ) ∗ 푐표푠(∅) ∗ 3-13 5.5 6.6 1000

푉퐿2 푆퐼퐿 = 3-14 푍표

푊ℎ푒푟푒, 푆퐼퐿 = 푆푢푟푔푒 퐼푚푝푒푑푎푛푐푒 퐿표푎푑푖푛푔 표푓 푙푖푛푒 푃 = 푃표푤푒푟 푡푟푎푛푠푚푖푡푡푒푑 푡ℎ푟표푢푔ℎ 푙푖푛푒 푖푛 푀푊 푙 = 퐿푒푛푔푡ℎ 표푓 푡푟푎푛푠푚푖푠푠푖표푛 푙푖푛푒 푖푛 푘푚

푉퐿 = 퐿푖푛푒 푣표푙푡푎푔푒 푖푛 푘푉 푁푐 = 푁푢푚푏푒푟 표푓 푐푖푟푐푢푖푡푠

푐표푠(∅) = 푃표푤푒푟 푓푎푐푡표푟 표푓 푡푟푎푛푠푚푖푠푠푖표푛 푙푖푛푒

퐿 푍 = 퐶ℎ푎푟푎푐푡푒푟푖푠푡푖푐 푖푚푝푒푑푎푛푐푒 표푓 푡푟푎푛푠푚푖푠푠푖표푛 푙푖푛푒 = √ 표 퐶

퐿 = 퐼푛푑푢푐푡푎푛푐푒 표푓 푙푖푛푒

퐶 = 퐶푎푝푎푐푖푡푎푛푐푒 표푓 푙푖푛푒

푉퐿, 푁푐, 푙 are obtained from the NEA’s annual report. The power factor is assumed to be 0.95 because most of the loads to be supplied are active loads (pf is unity).

From the equation above, power transmitted through line decreases with length. To find out the actual power that can be transmitted with stability, the SIL is multiplied by MF (multiplying factor). MF is obtained from the capability curve. The capability curve is a curve plotted between MF and transmission line length in kilometers. Most of the power transmission lines in Nepal are short transmission lines of about 80km to 100km hence in this study MF of 2.5 is used. This value would be even larger for shorter transmission lines.

3.5.3. Transmission Substation Capacity

Substation capacities are given in NEA’s annual report. It depends upon the voltage level of incoming and outgoing transmission lines and power carrying capacity of the lines.

3.5.4. Distribution Substation or Demand Centers Capacity

In this thesis, peak demand at each demand center has been considered to solve the optimization model. Peak demand in the INPS recorded by the NEA was 1200MW in the year 2014. The peak system demand is divided among different demand points by considering the load centers capacity as follows;

퐷푘 ∗ 푁푎푡푖표푛푎푙 푝푒푎푘 푑푒푚푎푛푑 푃푘 = 3-15 푆푢푚 표푓 푎푙푙 푙표푎푑 푐푒푛푡푒푟 푐푎푝푎푐푖푡푖푒푠

푊ℎ푒푟푒,

푃푘 = 푃푒푎푘 푑푒푚푎푛푑 푎푡 푑푒푚푎푛푑 푐푒푛푡푒푟 푘

퐷푘 = 퐷푖푠푡푟푖푏푢푡푖표푛 푠푢푏푠푡푎푡푖표푛 푐푎푝푎푐푖푡푦 푔푖푣푒푛 푓푟표푚 푡ℎ푒 푁퐸퐴’푠 푟푒푝표푟푡

푆푢푚 표푓 푎푙푙 푙표푎푑 푐푒푛푡푒푟 푐푎푝푎푐푖푡푖푒푠 > 푁푎푡푖표푛푎푙 푝푒푎푘 푑푒푚푎푛푑

In a three-phase power system, active and reactive power flow from the generating station to the load through the power system components. The system operator has to make sure to balance total load and generation in the system at every time possible. The optimization model presented in this thesis considers only active power because the reactive power can be generated locally at the load center. Some capacitor banks or synchronous machines are directly connected in parallel with an electrical load that supply the reactive power required by the load locally.

3.5.5. Load Flow Analysis

In addition to supplying energy at minimum cost, the system operator needs to check whether the network parameters (voltages and power flow in the transmission lines) are within limit every time when generation or load changes. For example, it might be cheaper to supply power at minimum cost but the power system voltage or frequency would be out of national grid limits. The optimal results from the optimization model at off limit voltage and frequency would not be a practical result. To address this issue, a load flow analysis is performed using Power System Analysis Toolbox (PSAT) and Matrix Laboratory (MATLAB) tools and results are given in the appendix section of this thesis. The flow of active and reactive power is called power flow or load flow. Power flow is a systematic mathematical approach for determination of various bus voltages, phase angles, active and reactive power flows through different branches, generators and loads under steady state condition. These parameters are necessary for planning, operation, economic scheduling and exchange of power between utilities. The principal information of power flow analysis is to find the magnitude and phase angle of voltage at each bus and the real and reactive power flowing in each transmission lines [45-48].

Some important terminologies used in the load flow analysis are explained with the help of an integrated power system network given in Figure 3.2.

Figure 3.2 Integrated power system network

The basic load flow equations are given below:

1 (푃푖+푗푄푖)∗ 푉푖 = [{ } − ∑푁 푌 ∗ 푉 ] 3-16 푌푖푗 푉푖 푗=1 푖푗 푗 푖≠푗

∗ 푁 푃푖 − 푄푖 = 푉푖 ∑푗=1 푌푖푗 ∗ 푉푗 3-17

Where,

1 1 푌푖푗 = 퐴푑푚푖푡푡푎푛푐푒 푚푎푡푟푖푥 (푝. 푢. ) = = 푍푖푗 (푅푖푗 + 푗푋푖푗)

푅푖푗 = 푅푒푠푖푠푡푎푛푐푒 표푓 푙푖푛푒 푖푗 (푝. 푢. )

푋푖푗 = 푅푒푎푐푡푎푛푐푒 표푓 푙푖푛푒 푖푗 (푝. 푢. )

푃푖 = 퐴푐푡푖푣푒 푝표푤푒푟 푖푛푗푒푐푡푒푑 푎푡 푏푢푠 (푝. 푢. )

= 퐺푒푛푒푟푎푡푒푑 푎푐푡푖푣푒 푝표푤푒푟 푎푡 푏푢푠 푖 (푃푔푖 푖푛 푝. 푢. )– 퐿표푎푑 푎푐푡푖푣푒 푝표푤푒푟 푎푡 푏푢푠 푖 (푃푙푖 푖푛 푝. 푢)

푄푖 = 푅푒푎푐푡푖푣푒 푝표푤푒푟 푖푛푗푒푐푡푒푑 푎푡 푏푢푠 (푝. 푢. )

= 퐺푒푛푒푟푎푡푒푑 푟푒푎푐푡푖푣푒 푝표푤푒푟 푎푡 푏푢푠 푖 (푄푔푖 푖푛 푝. 푢. )– 퐿표푎푑 푟푒푎푐푡푖푣푒 푝표푤푒푟 푎푡 푏푢푠 푖(푄푙푖 푖푛 푝. 푢. )

푉푖 = 푃ℎ푎푠표푟 푣표푙푡푎푔푒 푎푡 푏푢푠 푖 (푝. 푢. ) = |푉푖| < 훿푖

|푉푖| = 푀푎푔푛푖푡푢푑푒 표푓 푣표푙푎푡푔푒 푎푡 푏푢푠 푖 (푝. 푢. )

훿푖 = 푃ℎ푎푠표푟 푎푛푔푙푒 표푓 푣표푙푡푎푔푒 푎푡 푏푢푠 푖 푖푛 푑푒푔푟푒푒

The terminologies used in load flow analysis are described below: 37

Bus:

A bus is a node at which one or many lines, one or many loads and generators are connected. In a power system each node or bus is associated with 4 quantities, such as magnitude of voltage, phage angle of voltage, active or true power and reactive power. In load flow problems two out of these 4 quantities are specified and remaining 2 are required to be determined by solving the load flow equations. Depending on the quantities that have been specified, the buses are classified into 3 categories:

Load Buses: In these buses no generators are connected and hence the generated real power

푃푔푖 and reactive power 푄푔푖 are taken as zero. The objective of the load flow is to find the bus voltage magnitude |푉푖| and its angle 훿푖.

Voltage Controlled Buses: These are the buses where generators are connected. Therefore, the power generation in such buses is controlled through a prime mover or turbine with governor while the terminal voltage is controlled through the generator excitation. Keeping the input power constant through turbine-governor control and keeping the bus voltage constant using automatic voltage regulator, we can specify constant 푃푔푖 and | 푉푖 | for these buses.

Slack or Swing Bus: This is a generator bus which is considered as a reference bus for all the other buses. Since it is the angle difference between two voltage sources that dictates the real and reactive power flow between them, the particular angle of the slack bus is not important. However, it sets the reference against which angles of all the other bus voltages are measured. For this reason, the angle of this bus is usually chosen as 0°. Furthermore, it is assumed that the magnitude of the voltage of this bus is known and assumed as 1 per unit (p.u.).

Power flow analysis is an important tool involving numerical analysis applied to a power system. In this analysis, iterative techniques are used due to absence of a known analytical method to solve the problem. Load flow analysis is used to ensure that electrical power transfer from generators to consumers through the grid system is stable. Conventional techniques for solving the load flow problem are iterative, using the Newton-Raphson or the Gauss-Seidel methods.

Once the bus voltages are computed using load flow analysis, power loss in line 푖푗 is calculated as:

2 퐴푐푡푖푣푒 푝표푤푒푟 푙표푠푠 푖푛 푝. 푢. (푃푙표푠푠푖푗) = 푅푒(퐼푖푗 ) ∗ 푅푖푗 3-18

Where,

(푉푖 – 푉푗 ) 퐼푖푗 (푝. 푢. ) = 푍푖푗

Thus the cost associated with power loss in line is calculated as:

Ploss in line ij in MWhr = Ploss in line ij in p. u.∗ Base MVA ∗ Time of operation in hours

In this thesis, the time of operation is 1 year (365days) period. Assuming loss of load factor (LOLF) of 1 (i.e. load is supplied all the time), total hours in 365 days are:

Time of operation in hours (Time) = 365 ∗ 24 = 8760 hrs

LOLF: It is the ratio of actual energy supplied to a load over a given period of time, to potential energy that the load could consume without any disturbances over the same period of time.

Actual Energy supplied to a load over a given period of time LOLF = 3-19 Total load capacity∗Total time over the same period

In a deregulated pool market, the solution of an optimization problem has to serve a double task. First, the solution aims to minimize the power supply costs. At the same time the solution needs to meet the load demand as much as possible. Loss of load is the loss in revenue for the utility as the customers will not pay if they do not get the energy supply. The optimization model calculates the optimal way of meeting the demand by exploring various combinations of generators, transmission lines, substations, and distribution lines.

CHAPTER 4. NUMERICAL EXPERIMENTS

To address the issue of variability in power generation, a random set of data is generated in each season. Each set of the random data has random scenarios (number of scenarios depends on the sample size chosen by the decision maker), these scenarios represent the plant factor of the

ROR power plants (given by 푓푖푡푠 in the mathematical model). It is assumed that the data set follows a uniform distribution during each season. Also, it should be noted that the higher the sample size for the random scenarios is the more accurate the result will be, with the computation time increases proportionally. The variation in electrical power generation during a season does not guarantee the load-generation balance for that time. The ROR power plants which have

Qwet amount of discharge during the wet season may get a water discharge of about one third of

Qwet in the dry season. Hence, from equation 2-1, electrical power generation goes down to one third of designed capacity during dry season (Qdry) . The stochastic model is expected to address the uncertainty of energy generation. To address the problem of excess demand (due to the decreased generation) the deficit power needs to be supplied from some external sources. These external sources could be the private energy producers within the country (also called independent power producers) or power purchased from neighboring countries through cross boarder transmission lines. To discourage the importing energy from external source when it is not needed, a penalty function is used in the mathematical model. The penalty can be interpreted as the cost due to importing the unbalanced energy from the external sources. The stochastic model uses an algorithm called sample average approximation [49].

The computational tool used to solve the model is Gurobi optimizer version 6.0.4 along with Python software version 2.7 as a programming language. The Gurobi optimizer is a commercial optimization solver for linear programming, quadratic programming and other types of mathematical model.

The mathematical model is solved by using data of an existing power system. The integrated power system chosen for the experimental analysis is Integrated Nepal Power System (INPS) owned and operated by the government of Nepal. Nepal Electricity Authority, a government entity, is the only utility in the country which owns and operates major power plants and all the transmission and distribution network. Some necessary data were collected from the

40 annual reports given in NEA’s website and some data were collected from the Government of Nepal- Ministry of Energy’s website. The major power plants in the INPS are hydroelectric plants and some diesel plants as well as solar farms. The reason behind the majority of hydroelectric projects is that Nepal has more than 6,000 rivers (including rivulets and tributaries) due to glaciers in the Himalayas, regular monsoon rain, and local topography. These rivers have high water discharge rates that increases the potential of hydropower generation in the country. The country has 83,290 MW megawatts (MW) of theoretical hydroelectric potential, about 42,140 MW of which is technically and economically viable [50]. Also, it is estimated that about 50 MW of electricity can be generated by installing micro-hydro plants [51].

The river discharge varies with the amount of rain fall, snow fall, and snow melting in the region of study over a time period. Hydropower is the only resource available to generate electricity in almost every part of the country, both for large export projects and small village mini grid projects. Currently, Nepal has both ROR and storage type hydroelectric plants to supply the current demand. Since the total installed capacity of ROR projects is more than the storage projects, total power generation from the existing power plants varies monthly which is mainly due to the seasonal fluctuation of river discharges.

Hydrological seasons in Nepal can be categorized in three different groups: (a) dry pre- monsoon season (March–May) with almost no rain; (b) rainy monsoon season (June–September); and (c) post-monsoon season (October–February) with little rain [52]. The average precipitation in Nepal in mm per year is 1500 mm (59”) [53]. Four months of rainy monsoon season provide almost 80% rainfall but rest of eight months are almost dry months in the Nepalese rainfall pattern [54, 55]. As a result, river discharge also shows large seasonal fluctuations closely following the annual precipitation cycles. The rivers are over flooded during four months of monsoon and power plants produce full capacity of electrical power, while during the rest of eight months the generated power goes down to one third of the total capacity [56].

Despite having a large potential of hydropower the country hasn’t been able to harvest the full potential energy rather the government has been importing power from neighboring country like India. As the INPS has both variable energy generators (ROR) as well non-variable energy generators (storage and power purchased from India), it is the best example to use data from and hence show the practical significance of this thesis. A descriptive map of the INPS showing all the

41 existing power plants, transmission and distribution lines, substations, and demand centers is given in Appendix A: The Integrated Nepal Power System. The following sections have the data about major power plant capacity, transmission and distribution lines capacity, substation capacity, and hourly average value of peak demand in the system. The input data are estimated as follows.

4.1. Model Input Data

4.1.1. Resource Capacity Parameters

A list of the major existing power plants in the INPS and their installed capacity is given in the following table.

Table 4.1 Existing major hydroelectric power plants and their types (source: NEA [56])

Installed SN Generation stations name capacity Type in MW

1 Kaligandaki-A (KGA) 144 ROR

2 Lower Marsyandhi (LMRS) and Middle Marsyandhi (MMRS) 139 ROR

3 Kulekhani-I (KL-I) 60 Storage

4 Kulekhani-II (KL-II) 30 Storage

5 Modi (MODI) 24.8 ROR

6 Trishuli (TRSLI) and Chilime (CHLME) 82 ROR

7 Devighat (DVGHT) 14.1 ROR

8 Sunkoshi (SNKSHI) and Indrawati (INDRWTI) 17.55 ROR

9 Khimti (KHMTI) 60 ROR

10 Bhotekoshi (BTKSHI) 45 ROR

11 Dhalkebar (DALKEBAR) 400 Storage

Total 1016 MW

As stated earlier, there is a variability in power generation from the ROR projects. Total energy generation per month is plotted from fiscal year 2010/11 to fiscal year 2014-15 (see Appendix B: Monthly Power Generation of the Projects). The plant factor of each power plant is plotted in Figure 4.1.

Figure 4.1 Plant factors of the ROR projects

The Figure 4.1 that maximum power generation occurs during the months of May, June, July, and August and minimum power generation is during the months of November, December, January, and February. The actual month of minimum or maximum generation from ROR project is not same for all fiscal years but time period is approximately same for each year. A sample graph is given in Figure 4.2.

FY 2013-2014 Monthly Power Generation of the ROR Projects in MWhr Kaligandaki‘A’ Mid-Marsyangdi Marsyandi 120,000.00 Trishuli Gandak Modi 100,000.00 Devighat Sunkoshi 80,000.00

60,000.00

40,000.00

20,000.00

0.00

Figure 4.2 Monthly energy generation by the ROR projects

Table 4.2 Nepali to English calendar conversion

Nepali month Equivalent English month

Shrawan July-15 to Aug-15

Bhadra Aug-15 to Sept-15

Ashwin Sept-15 to Oct-15

Kartik Oct-15 to Nov-15

Mangsir Nov-15 to Dec-15

Poush Dec-15 to Jan-15

Magh Jan-15 to Feb-15

Falgun Feb-15 to Mar-15

Chaitra Mar-15 to Apr-15

Baishakh Apr-15 to May-15

Jestha May-15 to June-15

Ashad June-15 to July-15

In order to get the data as required by the model, the continuously variable energy generation is grouped into six intervals. Each interval presents a season during which the power plants generate power at certain percentage of its installed capacity (also called the plant factor or capacity factor). It should be clear to the reader that the word ‘season’ does not represent the conventional seasons like winter, summer, or spring. Twelve months are grouped into six seasons and their plant factors are plotted as shown in the Figure 4.3. Here in this case study, it is assumed that there are random 5 scenarios of power generation by a power plant during each season. In other words, the value of 푠 (sample size) in the objective function defined in equation 3-1 is 5. The sample size of 5 is chosen because higher sample size would result longer computation time and also 5 random scenarios for each season give acceptable output results (given in section 4.2).

Figure 4.3 Seasons of a year in the INPS

The Table 4.3 is an example probability table which shows the name of seasons (1 to 6) and their probabilities of happening in a year. Probability of a season is different for every ROR project (see Appendix B: Monthly Power Generation of the Projects).

Table 4.3 Seasons in a year and their probabilities

Percentage Seasons Number of Chance of Maximum possible generation (% of (s) months in occurrence generation during designed capacity) season s (n/12) season ‘s’ (in % of

(n) designed capacity) (fs)

From 100 to 90 1 2 1/6 1

From 90 to 80 2 4 1/3 0.9

From 80 to 70 3 1 1/12 0.8

From 70 to 60 4 2 1/6 0.7

From 60 to 50 5 2 1/6 0.6

From 50 to 40 6 1 1/6 0.5

Each generation station (both ROR and storage) is connected to single or multiple substations via transmission lines. Then the power is delivered to the demand centers via distribution lines from the substations. The capacity of exiting transmission and distribution lines is given in Table 4.4.

Table 4.4 Existing transmission and distribution lines capacity in the INPS

TL line capacity Type Capacity Substation (kV), length (km), Demand PP# Power Plant (Storage/R (MW) connected to single or double center OR) circuit

6 hrs peaking Lekhnath 132kV, 65.5 single Pokhara 1 Kaligandaki'A' 144 ROR (LKHNTH) circuit (PKHRA)

6 hrs peaking Butwal 132kV, 39.1 double Butwal 2 Kaligandaki'A' 144 ROR (BTWL) circuit (BTWL)

Middle 5 hrs Bharatpur 132kV, 38.2 single Bharatpur 3 Marsyangdi 70 peaking (BRTPR) circuit (BRTPR)

Balaju 132kV, 83 single Kathmandu 4 Marsyandi 69 ROR (BLJU) circuit (KTM)

132kV, 25 single Bharatpur 5 Marsyandi 69 ROR Bharatpur circuit (BRTPR)

Hetauda-1 132kV, 7.9 single Hetauda-1 6 Kulekhani I 60 Storage (HTDA-1) circuit (HTDA-1)

Hetauda-2 132kV, 7.9 single Hetauda-2 7 Kulekhani II 32 Storage (HTDA-2) circuit (HTDA-2)

Suichatar-1 132kv, 33.4 single 8 Kulekhani II 32 Storage (SUCH-1) circuit Kathmandu

Suichatar-2 66kV, 27.36 double 9 Trishuli 24 ROR (SUCH-2) ciruit Kathmandu

Pokhara 132kv, 30 single 10 ModiKhola 14.8 ROR (PKHRA) circuit Pokhara

Chabel (N- 132kV, 37 single 11 Devighat 15 ROR CHBL) circuit Kathmandu

132kV, 27 single 12 Devighat 15 ROR Balaju circuit Kathmandu

66kV, 50 single 13 Sunkoshi 10.05 ROR Kathmandu circuit Kathmandu

33kV, 35 single 14 Ilam(PuwaKhola) 6.2 ROR Illam circuit Illam

33kV, 20 single 15 Panauti 2.4 ROR Kathmandu circuit Kathmandu

16 Seti 1.5 ROR Pokhara 11kV Pokhara

17 Fewa 1 ROR Pokhara 11kV Pokhara

18 Sundarijal 0.64 ROR Kathmandu 11kV Kathmandu

11kV, 10 single 19 Pharping 0.5 ROR Kathmandu circuit Kathmandu

33kV, 14 single 20 Chatara 3.2 ROR Sunsari circuit Sunsari

132kV, 20 single 21 Gandak 15 ROR Bharatpur circuit Bharatpur

132kV, 18 single 22 Gandak 15 ROR Bharatpur ciruit Bharatpur

The Table 4.5 lists the existing substations (both transmission and distribution substations) and their capacities in Mega Volt Ampere (MVA) as well as average hourly demand at each demand center. The MVA is a unit to measure capacity of transformers.

Table 4.5 Existing substations in the INPS and their capacities

Substation Substations Demand points Average Demand MW Capacity MVA

LKHNTH and PKHRA 57.5 BRTPR 25

BTWL 142.6 SHVPR 41

BRTPR 30 BARDGHT 51.5

SUCH-1 113.4 CHAPUR 36

HTDA-1 40 HTDA-8 20

HTDA-2 20 TEKU 90

SUCH-2 36 PATN 36

N-CHBL 45 BAN 36

BLJU 90 BALJU 90

LMOSAGU 15 N-CHBL 45

BKTPR 91.5 LNCHR 45

DHLKE 400 LKHNTH / PKHRA 43

Total 1081 Total 558.5

4.1.2. Cost Parameters

The section 3.4 describes how the cost parameters for each power system component are calculated in this thesis. In addition to the formula given in section 3.4, the following table summarizes cost parameters used in different literature. The numerical values of costs computed using above formula are approximately same as in the literature.

Table 4.6 Cost parameter computation

INPS Description Literature study data Comments and Sources data

http://www.eia.gov/electrici ty/annual/html/epa_08_04.h • Cost of energy from of tml ROR generator in $ per $7-14 per MWhr in the US, MWhr (Cg) http://bze.org.au/ average of $ 22.62 per MWhr in $ 6 - 113 France, $40 per MWhr in Australia per MWhr and from NEA website

$70.5 per • Cost of energy from http://www.eia.gov/electrici MWhr storage hydroelectric ty/annual/html/epa_08_04.h (assuming power plant in $ per tml $7-14 in the US, average of $ 22.62 $1 = 100 MWhr (Cg) in France, $40 in Australia NRs.) and from NEA website

• Cost of power loss in http://ec.europa.eu/enlarge transmission lines from ment/archives/seerecon/infr $3- 11 per MWhr in European $1-9 per storage and ROR astructure/sectors/energy/do countries MWhr generators to substations cuments/benchmarking/tarif

in $ per MWhr (Ctl) fs_study.pdf

Power loss is very low as compared • Cost of power loss at to loss in transmission lines and $ 2-5 per Load flow analysis of INPS substation $ per MWhr also depreciation rate of the MWhr network (Cs) equipment is small

Since the length of • Cost of power loss in distribution line smaller distribution lines in $ per $3- 11 per MWhr in European $1-3 per than transmission line, loss MWhr (Cdl) countries MWhr is low

Finally, all the data presented above were fit into the mathematical model and following results were obtained.

4.2. Analysis of Numerical Results

4.2.1. Operation and Maintenance Cost

Table 4.7 Total operation and maintenance cost from the NEA report

Types of Operating costs FY 2014 FY 2013

Generation Expenses $ 16,792,000.0 $ 16,043,100.0

Power Purchase $ 163,883,100.0 $ 135,724,600.0

Royalty $ 9,300,000.0 $ 8,904,900.0

Transmission Expenses $ 4,500,000.0 $ 4,167,400.0

Distribution Expenses $ 48,555,700.0 $ 40,879,700.0

Depreciation Expenses $ 32,564,800.0 $ 32,286,800.0

Total Operating Expenses $ 275,595,600.0 $ 238,006,500.0

Average $ 256,801,050.0

The solution approaches performed in this research are expectation of expected value solution (EEV), stochastic solution (SS), and wait and see solution. The analysis of all the three methods is presented as follows.

4.2.2. Expected Value Solution (EEV)

Expectation of expected value (EEV) model is constructed by replacing the expected value (EV) with its expectation. The EEV solution is performed in two stages. First, the optimization problem is solved by using the average of random scenario data (i.e. power generated by ROR projects during each season scenario) which is called the expected value (EV). This stage of solution approach provides power generated by both the ROR and storage projects. Since we use the average of the random scenarios, such an EV model is thus a deterministic optimization problem because the uncertainty is dealt with before it is introduced to the optimization problem. Then in second stage, by fixing the values of first stage independent variables (i.e. power generated

51 from storage projects) from the EV problem, the expectation of EV can be obtained by allowing the optimization problem to choose the values for the random variables (i.e. power generated by ROR projects).

Power generated by each power plant in MW (Y-axis) during each season of a year (X- axis) are given in the following graphs:

Table 4.8 Legend

Legend Kaligandaki A (ROR) LMrs &MMrs (ROR) MODI (ROR) TRSLI & CHLME (ROR) DVGHT (ROR) SNKSHI & INDRWTI (ROR) KHMTI (ROR) BTKSHI (ROR) Kulekhani – I (storage) Kulekhani – II (storage) From India (Dhalkebar- storage)

Season-1 Season-2 600.00 600.00

500.00 500.00

400.00 400.00

300.00 300.00

200.00 200.00

100.00 100.00

0.00 0.00 Season Season 1 2

Season-3 Season-4 600.00 600.00 500.00 500.00 400.00 400.00 300.00 300.00 200.00 200.00 100.00 100.00 0.00 0.00 Season Season 3 4

Season-5 Season-6 600.00 600.00 500.00 500.00 400.00 400.00 300.00 300.00 200.00 200.00 100.00 100.00 0.00 0.00 Season Season 5 6

Figure 4.4 Decision variable values from EEV approach

As an example, in the above graph for season-1, the power generated by a storage project (let us say Kulekhani-I) is the average of 5 random values of power generation by the plant. Thus, the power output from the storage project Kulekhani-I is constant for all scenarios in season-1. This solution makes the first stage of EV solution. Then, the second stage solution of the EV model gives the power needs to be generated by a ROR project (let us say Kaligandaki-A) to meet the total demand. These values are different for each of the 5 scenarios in season-1. The same analysis is true for other projects as well. The graphs for season-2, 3, 4, 5, and 6 can be interpreted in the same way. As we take the average of random scenarios in EEV solution, the chances of getting inaccurate power generation is higher than using the random scenarios themselves. This is the reason why the total operational cost from EEV solution is higher than SS solution explained in section 4.2.3.

Total cost per year obtained from the mathematical model using EV approach = $ 113,772,257.8

4.2.3. Stochastic Solution (SS)

In real world problems, the decision variables (power generation, transmission and distribution here in this case) are uncertain. Their uncertainty can be modeled by a probability distribution. The solution approach of specifying the uncertainty by using the probability distributions is called stochastic solution. As the uncertainty requires approximation, the result of the stochastic solution has some inaccuracies. This research approximates the uncertainties of future weather conditions (i.e. power output from ROR plants) with a scenario-based approach. Some random scenarios are generated for each season and such scenarios determines the maximum capacity of the plant during that season. The scenario size is same as for the EV solution approach.

The graph for season -1 in Figure 4.5 represents the output energy generation in MW (Y- axis) during first scenario of season 1 (X-axis) and so on.

Season-1 Season-2 600 600

500 500 400 400 300 300 200 200 100 100 0 0 Season Season 1 2

Season-3 Season-4 600 600

500 500

400 400

300 300 200 200 100 100 0 0 Season 4 Season 3

Season-5 Season-6 600 600

500 500

400 400

300 300

200 200

100 100

0 0 Season 5 Season 6

Figure 4.5 Decision variable values from SS approach

The approach shows the total power output from all the generators (both ROR and storage) from season-1 to season-6. Each scenario of a season is a random number with some probability of happening. This random number represents the power generated by ROR projects. When the ROR projects have lower power generation (like in season-6) due to weather change the storage projects supply the rest of the energy demand. As an example, as the water flow rate decreases from season-1 to season-6 (given in Figure 4.1) the power purchased from India (a storage project) increases accordingly shown in Figure 4.5.

Total cost per year obtained from the mathematical model using SS approach = $ 113,759,512.2 which slightly less than the total cost from EV solution. This makes sense because the second stage of EV solution approach is same as the SS solution but the first stage considers the average of the random numbers which is not as accurate as considering the random numbers themselves.

4.2.4. Wait & See Solution (W&S)

The example output from the wait & see algorithm is plotted in the following graphs. The graph for season-1-1 in Figure 4.6 represents the output energy generation in MW (Y-axis) during first scenario of season 1 (X-axis) and so on. The wait-and-see solution approach gives the value of the objective function (i.e. cost of energy supply in this research) if complete information about the decision variable was known prior to the decision is made. Thus, it can be inferred that the objective function value is obtained by running each scenario independently as if each scenario was certain and then final objective function value (cost) would be the average of all the individual costs. The cost of energy generation obtained from the W&S solution is specific to each scenario which is always less than the cost obtained if the random scenarios are used like in SS approach.

Total cost per year obtained from the mathematical model using W&S approach = $ 112,679,077.76 which cheapest among all three solutions explained above.

Season-1-1 600 Season-1-2 600 500 500 400 400 300 300 200 200

100 100

0 0 Season 1 Season 1

Season-1-3 Season-1-4 600 600

500 500

400 400

300 300

200 200

100 100

0 0 Season 1 Season 1

600 Season-6-1 600 Season-6-2

500 500

400 400

300 300

200 200

100 100

0 0 Season 6 Season 6

Season-6-3 Season-6-4 600 600

500 500

400 400

300 300

200 200

100 100

0 0 Season 6 Season 6

Season-6-5 Season-1-5 600 600

500 500

400 400 300 300 200 200 100 100 0 0 Season 1 Season 6

Figure 4.6 Decision variable values from W&S approach

Let us take an example of the graph for season-6-5 in Figure 4.6 to explain the results of the W&S approach. Scenario 5 of the season 6 is a random number which represents the power generated by the ROR projects. For a moment, the power generated by the ROR project during season-6 is certain which equals to the random scenario-5 of season-6. The PSO now have perfect information about how much power generation will be obtained from ROR projects in the future. Then, the storage projects are dispatched accordingly to meet the total demand. Similar dispatch strategy can be applied for other scenarios of the season-6. Finally, the average of the 5 solutions (objective function from 5 scenarios) is computed to get the cost of energy supply during season- 6 from W&S algorithm. Same analysis is true for season-1, 2, 3, 4, and 5 as well.

4.2.5. Objective Function Values

The objective function values from the stochastic analysis (SA) and wait & wee solution are given in following table.

Table 4.9 The objective function values from SS, W&S, and EEV approach

Stochastic Wait & See $ EEV $ Per VSS in $ Per Season Solution $ Per EVPI in $ Per Year Per Year Year Year Year

1 (0.9-1.0) 105,180,286.0 102,505,429.6 105,184,108.6 3,822.6 2,674,856.4

2(0.8-0.9) 106,382,788.9 104,025,860.2 106,382,788.9 - 2,356,928.7

3(0.7-0.8) 112,823,116.0 111,970,603.8 112,873,031.7 49,915.8 852,512.1

4(0.6-0.7) 115,744,467.2 115,337,384.1 115,764,199.6 19,732.4 407,083.1

5(0.5-0.6) 121,375,733.3 121,356,994.1 121,378,736.0 3,002.7 18,739.2

6(0.4-0.5) 121,050,682.2 120,878,194.7 121,050,682.2 - 172,487.5

Average 113,759,512.2 112,679,077.7 113,772,257.8 12,745.5 1,080,434.5

Total annual cost from Table 4.7 is equal to $ 256,801,050.0. This cost represents the cost of energy supply when considering the total INPS load. Here in this thesis only some major portion of the total is considered which is only about 45 % of the total national peak load.

Total annual savings = $ 256,801,050.0 * 0.45 - $ 113,759,512.2 = $ 1,800,960.3

Expected Value of Perfect Information (EVPI) - is the absolute value of the difference between SS and W&S i.e.

EVPI = SS-W&S = $ 1,080,434.5

The significance of EVPI is that it gives information about how much the decision maker should be willing to pay for the perfect information about the future.

 Value of stochastic solution (VSS)- is the difference in the objective function between the solutions of EEV and SS i.e.

VSS = EV – SS = $ 12,745.5

The significance of VSS is that it gives the possible gain by considering all possible random scenarios.

CHAPTER 5. CONCLUSIONS AND RECOMMENDATIONS

5.1. Conclusions

In this thesis, a stochastic mathematical model was developed for an integrated power system that consists of both the run-of-river and storage projects. The ROR projects are cheaper to install and operate than the storage hydroelectric project of the same capacity. Also, the ROR projects are environmentally friendly compared to the storage projects, because ROR projects have simple water diversion channel. However, due to the fluctuating energy output from the ROR power plants they pose a reliability threat to the integrated power system. A mathematical model that could predict the energy output from the ROR by using the probability of future weather conditions would help the power system operator to schedule all power plants in an optimal way. The optimal solution would ensure load and generation balance at all times and also the total cost of energy supply would be minimum.

The Integrated Nepal Power System consisting of ROR and storage type hydroelectric power plants was taken as a case study power system in this research. Power produced from the ROR depends on the available water discharge in the river which varies continuously with weather conditions. On the other hand, the storage projects are designed to produce power at their installed capacity independent of the temporary weather change. The operational cost of a power plant is different from another power plant because of the power plant’s location, its size, and maintenance cost. As an example, a power plant located in the remote area of a country would require larger investment due to required civil structures like access roads. This investment cost affects the per unit energy cost of the power plant during its operation. The generated power from the generators is supplied to demand centers at distant location via power transmission and distribution lines and substations. The power loss cost in the lines and substations and their operational and maintenance cost is not the same for all the components. Power loss in the transmission and distribution lines depends on the length of the line and diameter of the power carrying conductor. A combination of a generator, a transmission and distribution line, and a substation to supply a demand could be cheaper than another combination of the different generators, transmission and distribution lines, and substations due to the cost difference. In addition to the cost parameters, the power system components have some constraints like maximum capacity of the component which restricts the

60 further operation of that component even though it would be cheaper to meet the energy demand. As the number of these power system components increases the possible combination also increases, which makes the problem of optimal energy supply more complex. The mathematical model presented in research considered all the possible combinations and solves the model to give a most economical way of suppling the load by taking into account the system constraints.

The mathematical model developed is called a two-stage stochastic model which consists of two stages: first stage is to represent the storage type projects and the second stage represents the ROR projects. This model provided a solution for the amount of power generation, transmission and distribution over one-year period by the ROR and storage power plants. Power generated by the storage projects is more certain than the ROR, hence, the probability of power generation by the storage plants during the study period was considered as unity. In contrast, in the case of ROR plants one-year period was divided into six seasons based on the past year’s power generation pattern from the ROR projects. Each season had a certain probability of happening over the study period and also the power plant could only generate up to a certain percentage of their installed capacity during that season. The historic data on power generation by the ROR projects was used to find the probability of occurrence of the season and the maximum capacity of a power plant during that season.

The problem was solved and comparisons of the stochastic solution (SS), expected value (EEV), and wait and see (W&S) was made. The results from these solution approaches provided an economical dispatch of the ROR and storage projects and optimal distribution plan to meet the energy demand. The PSO should interpret the results from these solution techniques as follows:

The months in a year when the power plants can generate power from 90% to 100% of their installed capacity were grouped into the season-1. Similarly, the months in a year when the power plants can generate power from 80% to 90% of their installed capacity were grouped into the season-2 and so on. The results of the decision variables showed that the power generated by a ROR plant during season-1 was more than the power generated by the same ROR plant in other seasons. This means the storage power plants should be operating at minimum capacity and store some water during season-1 so that the stored water could be used to run the plants at increased capacity during other seasons. If the domestic generation was not sufficient to meet the energy demand, then power should be purchased from out of state (like from India in case of INPS).

For all the experimental runs, the W&S solution approach provided the least cost plan. The EEV solution was worse than the SS. The PSO may invest in advanced technology to more accurately reveal the uncertainties to operate at the W&S operational cost. However, the tradeoff between using the SS solution and investing in new technologies to operate at W&S solution may require rigorous feasibility study.

Some of the benefits for the energy users, power producers, and the government from the proposed methodology are:

1. Reliable power supply for the customers.

2. Optimal use of ROR projects which would reduce the cost and also reduce the environmental effects caused by storage power plants. This would benefit the power producers and also the government.

3. Optimal use of available domestic power plants.

4. In addition to the optimal energy generation and distribution planning for an existing system, the government can use the given mathematical model in energy planning. This can be achieved by using a forecasted energy demand (in place of current demand) for a certain planning horizon, the solution of the proposed mathematical model would give an idea of what should be the strategy of energy generation and distribution in coming future. If the solution produces an infeasible solution due to greater forecasted demand than total available generation then the decision maker can virtually place a new power plant with some estimated operational cost to see what would be the best way to supply the future need of energy.

In addition to the above benefits, this research adds a contribution to the state-of-art by presenting the integrated power system network as a two-stage transportation problem and minimizing the total cost of energy supply to the customers. As given in the literature review section, various research articles can be found in the field of deterministic, stochastic as well as large scale optimization that present different objective functions and methodologies of solving problems. Some major objective functions include the optimal scheduling hydro-thermal unit, optimization of water content in the reservoir for a storage hydroelectric project, minimization of carbon emission from power plants, scheduling of electrical loads like electric vehicles and

62 thermostatically controlled loads, and optimal sizing and siting of DGs. These articles present methodologies for optimization only in the generator side or in demand side. However, this thesis presents a stochastic optimization methodology that considers all the components of the integrated power system (i.e. generators, transmission and distribution lines, and substations). Power produced by the generators and transmitted to the distribution substation makes the first stage of the problem while the power distributed from the substations to the demand centers makes the second stage of the two-stage transportation problem.

5.2. Future Work

Here, in this research, the energy demand is assumed to be constant over an hour period (60 minutes), and the optimal energy supply planning is obtained for one-hour period. This assumption is because of the fact that the energy demand can change in every fraction of seconds, and finding an optimal solution with the variable demands, and giving commands to the generators according to the solution would not be practical. However, if the analysis period is reduced to may be 15 minutes or 30 minutes from 60 minutes that would make better approximation with the real load change and more accurate results of optimal energy supply planning. Hence, it is a recommendation for the future researchers in the field of optimization to add a variable energy demand along with variable energy generation and use time period less than one-hour. This research considers the flow of active power only in the network because the reactive power required by the loads can be supplied locally from reactive power generators. However, the power system consists of components like transmission lines and transformers which require some reactive power during their operation. The required reactive power needs to supplied by the generators in the power system. Hence, an optimal reactive power flow methodology needs to developed. The future researches in this field are recommended to use complex power (active power and reactive power) in place of active power presented in this thesis.

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APPENDIX

Appendix A: The Integrated Nepal Power System

Figure A.1 Integrated Nepal Power System map

Figure A.2 Integrated Nepal Power System single line diagram

Figure A.3 MATLAB simulation diagram of the Integrated Nepal Power System

Appendix B: Monthly Power Generation of the Projects

Kaligandaki‘A’ Mid-Marsyangdi 120,000.00 60,000.00

100,000.00 50,000.00

80,000.00 40,000.00

60,000.00 30,000.00

40,000.00 20,000.00

20,000.00 10,000.00

0.00 0.00 1 2 3 4 5 6 7 8 9 10 11 12 1 2 3 4 5 6 7 8 9 10 11 12

Marsyandi Trishuli 60,000.00 12,000.00

50,000.00 10,000.00

40,000.00 8,000.00

30,000.00 6,000.00

20,000.00 4,000.00

10,000.00 2,000.00

0.00 0.00 1 2 3 4 5 6 7 8 9 10 11 12 1 2 3 4 5 6 7 8 9 10 11 12

KulekhaniI KulekhaniII 25,000.00 12,000.00

10,000.00 20,000.00

8,000.00 15,000.00 6,000.00 10,000.00 4,000.00

5,000.00 2,000.00

0.00 0.00 1 2 3 4 5 6 7 8 9 10 11 12 1 2 3 4 5 6 7 8 9 10 11 12

Modi Gandak 6,000.00 3,000.00

5,000.00 2,500.00

4,000.00 2,000.00

3,000.00 1,500.00

2,000.00 1,000.00

1,000.00 500.00

0.00 0.00 1 2 3 4 5 6 7 8 9 10 11 12 1 2 3 4 5 6 7 8 9 10 11 12

Figure B.1 Monthly energy generation by the ROR projects in Nepal

FY 2010-2011 Monthly Power Generation of ROR Projects in MWhr

Kaligandaki'A' Mid-Marsyangdi Marsyandi Trishuli Gandak Modi Devighat Sunkoshi 250,000.00

200,000.00

150,000.00

100,000.00

50,000.00

0.00

FY 2011-2012 Monthly Power Generation of ROR Projects in MWhr

Kaligandaki'A' Mid-Marsyangdi Marsyandi Trishuli Gandak Modi Devighat Sunkoshi 250,000.00

200,000.00

150,000.00

100,000.00

50,000.00

0.00

FY 2013-2014 Monthly Power Generation of ROR Projects in MWhr Kaligandaki‘A’ Mid-Marsyangdi Marsyandi Trishuli Gandak Modi Devighat Sunkoshi

120,000.00

100,000.00

80,000.00

60,000.00

40,000.00

20,000.00

0.00

FY 2014-2015 Monthly Power Generation of ROR Porjects in MWhr

Kaligandaki'A' MiddleMarsyangdi Marsyandi Kulekhani II Trishuli ModiKhola Devighat Sunkoshi 250000

200000

150000

100000

50000

Figure B.2 Total monthly energy generation by the ROR projects in Nepal

Average Montly Plant Factor of ROR Projects in % 120.00%

100.00% Kaligandaki A

80.00% Mid-Marsyangdi and Marsyangdi

60.00% Trishuli PF in % PF Modi 40.00%

Devighat 20.00% Sunkoshi 0.00%

Figure B.3 Average monthly plant factor or the ROR projects in Nepal (in Percentage)

Appendix C: The INPS Hourly Load

Total INPS Load in MW 1400

1200

1000

800

600

Load in MW in Load 400

200

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 Hours of the day

Demand in MW Average Demand (629 MW)

Figure C.1 Hourly peak demand in the INPS

Table C.1 Current power supply and demand in Nepal

Demand type Demand Generation type Generation in MW

Peak demand 1291.8 MW Hydroelectricity (NEA 473.394 grid connected)

Unmet demand More than 410 MW Hydroelectricity (NEA 4.536 isolated)

Total number of consumers in 2,868,012 Hydro Electricity (IPP 255.647 fiscal year 2014/15 grid connected)

Annual average consumer’s 1.31 MWh Thermal (NEA) 53.41 energy demand

Total annual energy demand of 311979.955 MWh Solar (NEA) 0.1 an average consumer

Total 787.087

Appendix D: Cost and Capacity Parameters Computation

Table D.1 Cost and capacity parameters computation of the power system components

Symbols Literature study data INPS data Comments and Sources

• Number of storage dams (n1) 4

• Number of ROR hydroelectric generators (n2) 8

• Number of substations (m) 14

• Number of demand and export regions (l) 12

Few megawatts (1MW) – Considering • Design capacity of 22, 500MW (Three Gorges 30-200MW DHLKEBAR sub-station hydroelectric dam (Xi) Dam in China) as a generation station

http://www.eia.gov/electri city/annual/html/epa_08_ $7-14 per MW in the US, 04.html • Cost of hydroelectric dam average of $ 22.62 per MW http://bze.org.au/ in France, $40 per MW in $ 13 - 113 per Australia MWhr and NEA website

• ROR generator design Few kilowatts (0.1 MW) to capacity (Yi) 22,5000MW 14-144 MW google.com

• Cost of ROR generator during construction 100 - 160 $ 6 - 113 per dollars per MW MWhr

http://www.eia.gov/electri city/annual/html/epa_08_ • Cost of energy from $70.5 per 04.html hydroelectric power plant per $30-100 in the US, average MWhr MWhr http://bze.org.au/ of $ 22.62 in France, $40 in (assuming $1 Australia = 100 NRs. and NEA website

33kV – 750 kV and • Capacity of transmission line depending upon the length from storage and ROR of power transmission 33kV - 220 kV Considering Nepal-India generator to substation (TLij) capacity changes as (100 MW to cross border transmission described in equation above 400MW) line

http://ec.europa.eu/enlarg • Cost of transmission line ement/archives/seerecon/i $3- 11 per MWhr in $1-9 per from storage and ROR nfrastructure/sectors/ener European countries MWhr generator to substation gy/documents/benchmark

ing/tariffs_study.pdf

It can be any depending on connecting transmission • Capacity of substation (Sj) lines capacities, load center DHKLEBAR is treated as size 15-143 MW a generation station

Power loss is very low as compared to loss in • Cost of substation transmission lines and also depreciation rate of the $ 2-5 per Load flow analysis of equipment is small MWhr INPS network

33kV – 750 kV and depending upon the length • Capacity of transmission line of power transmission between substations (Spj) capacity changes as described in equation above 100MW lines

• Cost of transmission line $ 3- 11 per MWhr in $1-9 per between substations European countries MWhr

33kV – 750 kV and depending upon the length • Capacity of distribution line of power transmission (Ljk) capacity changes as described in equation above 100MW lines

Since the length of distribution line smaller • Cost of distribution line $3- 11 per MWhr in $1-3 per than transmission line, European countries MWhr loss is low

• Regional demand distribution mean (dkω) 20-90 MWhr

• ROR generation sample uniformly distributed between 60 - 100 percent of the ROR generator design capacity.

Appendix E: Load Flow Analysis Results

Figure E.1 MATLAB and PSAT results for bus voltages in the INPS

In the above figure, all the bus voltages are between 90% to 100% of the rated value. The range of 0.9 to 1.0 per unit is within the national grid requirement in Nepal. The following figure shows same analysis for all the 6 seasons of a year.

Figure E.2 INPS bus voltage level for each season.