UNIVERSITY OF NAIROBI FACULTY OF ENGINEEING DEPARTMENT OF ELECTRICAL AND INFORMATION ENGINEERING SECURITY CONSTRAINED ECONOMIC DISPATCH FOR HVDC USING PARTICLE SWARM OPTIMIZATION PROJECT INDEX: 058 SUBMITTED BY CYPRIAN OCHIENG, F17/1430/2011 SUPERVISOR: PROF. N.O. ABUNGU EXAMINER: PROJECT REPORT SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENT FOR THE AWARD OFTHE DEGREEOF BACHELOR OF SCIENCE IN ELECTRICAL AND ELECTRONICENGINEERING OF THE UNIVERSITY OF NAIROBI 2016 SUBMITTED ON:
DECLARATION OF ORIGINALITY NAME OF STUDENT: REGISTRATION NUMBER: COLLEGE: FACULTY/ SCHOOL/ INSTITUTE: DEPARTMENT: COURSE NAME: TITLE OF WORK: Cyprian Ochieng F17/1430/2011 Architecture and Engineering Engineering Electrical and Information Engineering Bachelor of Science in Electrical & Electronic Engineering Security Constrained Economic Dispatch for HVDC Using Particle Swarm Optimization (PSO) I understand what plagiarism is and I am aware of the university policy in this regard. I declare that this final year project report is my original work and has not been submitted elsewhere for examination, award of a degree or publication. Where other people s work or my own work has been used, this has properly been acknowledged and referenced in accordance with the University of Nairobi s requirements. I have not sought or used the services of any professional agencies to produce this work. I have not allowed, and shall not allow anyone to copy my work with the intention of passing it off as his/her own work. I understand that any false claim in respect of this work shall result in disciplinary action, in accordance with University anti-plagiarism policy. Signature:.. Date:.. PAGE 1
CERTIFICATION This report has been submitted to the Department of Electrical and Information Engineering University of Nairobi with my approval as supervisor: Prof. Nicodemus Abungu Odero Date:... PAGE 2
DEDICATION I dedicate this project to the Almighty God and to my family, lecturers and fellow students for their support and encouragement. PAGE 3
ACKNOWLEDGEMENTS I would like to thank God for this life and for His Presence and Guidance throughout my life and studies. I would also like to thank my supervisor, Prof. Nicodemus Abungu for his insight, motivation, support and guidance. I extend my appreciation to Mr. Musau for his valuable insights into my project, criticism and encouragement. I appreciate all my lectures and all staff at the Department of Electrical and Electronic Engineering, and the entire community of the University of Nairobi for their contribution towards my degree. I am thankful to my classmates and friends for their support and availability throughout my studies and most importantly during the period over which I was working on the project. In conclusion I would like to sincerely thank my family for their presence and undying support throughout my studies. PAGE 4
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Contents DECLARATION OF ORIGINALITY... 1 CERTIFICATION... 2 DEDICATION... 3 LIST OF TABLES... 9 LIST OF ABBREVIATIONS... 10 1.1. Definition of terms... 12 1.2. A brief introduction to HVDC system... 14 1.2.1. Components of an HVDC transmission system... 15 1.2.2. HVDC systems control... 16 1.2.3. HVDC configurations... 17 1.2.4. Reasons for HVDC... 18 1.2.5. Comparison between DC and AC systems... 18 1.3. Problem statement... 20 1.4. Objectives... 20 1.5. Research Questions... 21 CHAPTER 2... 22 2.1. Economic dispatch problem neglecting transmission losses... 23 2.1.1. Fuel cost characteristics... 24 2.1.2. Problem formulation... 25 2.2. Economic dispatch problem considering network losses... 27 2.2.1. Transmission line loss equation...29 2.2.2. Losses in HVDC systems...30 2.2.3. HVDC inequality constraints... 31 2.3. Review of solution methods... 32 2.3.1. Particle swarm optimization... 33 2.3.2. Genetic algorithm... 34 2.3.3. Evolutionary Programming... 34 2.3.4. Linear Programming... 35 2.3.5. Lambda iteration...36 CHAPTER 3... 37 SOLUTION OF SECURITY CONSTRAINED ECONOMIC DISPATCH USING PSO... 37 PAGE 6
3.1. PSO Algorithm... 37 3.2. Parameter representation... 42 3.3. PSO Algorithm for SCED...42 CHAPTER 4... Error! Bookmark not defined.45 RESULTS AND ANALYSIS 45 4.1. Case Study: IEEE 30-bus System.45 4.2. Results.46 4.3. Analysis and Discussion.49 CHAPTER 5... 51 CONCLUSION AND RECOMMENDATION..51 5.1 Conclusion.51 5.2 Recommendations..51 REFERENCES..52 APPENDIX.53 PAGE 7
LIST OF FIGURES Fig 1.1: Total cost/distance.....18 Fig 1.2: Typical transmission line structures for approximately 1000MW....19 Fig 2.1: N-thermal units connected to a bus to serve a load......23 Fig2.2: A typical fuel cost characteristics...24 Fig 2.3: N thermal units serving P Load through a transmission network with losses....26 Fig 3.1: PSO Algorithm........36 Fig 4.1: One line diagram of IEEE 30-bus system [1]......43 Fig 4.2: Variation of Real Power Losses with Power Demand for SCED and CED......47 Fig 4.3: Variation of Optimal Cost with Power Demand for SCED and CED.... 48 PAGE 8
LIST OF TABLES Table 4.1: Optimal generation for SCED and CED using PSO for a total demand of 283.4MW...44 Table 4.2: Optimal generation for SCED and CED using PSO for a total demand of 374.3MW...45 Table 4.3: Optimal generation for SCED and CED using PSO for a total demand of 540MW...45 Table 4.4: Comparison of Economic Dispatch using PSO and GA for a total demand of 283.4MW..46 PAGE 9
LIST OF ABBREVIATIONS CED Classical Economic Dispatch ED GA HVAC HVDC IEEE LP MATLAB MW OPF PSO QP SCED HPSO Economic Dispatch Genetic Algorithm High Voltage Alternating Current High Voltage Direct Current Institute of Electrical and Electronic Engineering Linear Programming Matrix Laboratory Megawatts Optimal Power Flow Particle Swarm Optimization Quadratic Programming Security Constrained Economic Dispatch Hybrid Particle Swarm Optimization PAGE 10
ABSTRACT With increasing number of appliances, at house hold, office and institution level, as well as facilities such as transport, communication among others, and machinery that require electrical power, there is need for a reliable and efficient power supply that is at a reasonable cost. With this in mind there is need for generation facilities, while harnessing the abundant renewable energy sources provided by nature, to generate the energy that requires fossil fuel at reduced cost to ensure affordability of the power on the side of the consumer. This is to be done while ensuring that the whole system of generation and supply if operating within their defined safe limits. Considering this growing demand of power, there is need for interconnection of power grids between nations within a region as well as integration of offshore generated energy into the inland grid. With these long distances over which transmission is to be done to achieve this, as well as to harmonize power systems operating at different frequencies, HVDC technology is necessary. Security Constrained Economic Dispatch is as an optimization procedure that attempts to obtain an optimal balance of two conflicting objectives; cost efficiency that aims at serving the demand with minimum cost, and security that requires electricity to be delivered to customers without interruption even when a component of the system fails. SCED problem has been solved with conventional methods as well as intelligent search methods. In this project, Particle Swarm Optimization method is used to solve the problem. The PSO algorithm has been implemented on the IEEE 30-bus network with a load demand of 283.4MW. Six generating units are used to supply the power to meet this demand. PAGE 11
CHAPTER 1 INTRODUCTION 1.1.Definition of terms 1.1.1. Economic Dispatch Describes how the real power output of each controlled generating unit in an area is selected to meet a given load and to minimize the total operating cost in the area. It is the allocation of generation levels to generating units comprising a power system to economically serve the load in entirety while remaining within the operational limits of the generation facilities. 1.1.2. System Security These are measures put in place to keep the system operating when components fail. A measure of this security is the ability of the power system to withstand the effects of contingencies such as generator, transformer or line outage, the effects of which are monitored with specified security limits. An operationally secure system is one with low possibility of system collapse or equipment damage. The following are the three major functions carried out in the energy control center under system security; i. System monitoring; gives up-to-date information on the state of the power system on real time basis with regard to the load and generation change. ii. Contingency analysis; foresees possible system outages before they actually occur. They alert the operators to any potential overloads or serious voltage violation. iii. Corrective action analysis; allows the operator to alter the operation of the power system in such a direction as to avert the occurrence of a serious problem due to a given outage. 1.1.3. Security Constrains These are limits put in a power system to prevent outage of equipment due to overstretching of the performance capabilities of the facilities and therefore ensure continued supply of power to consumers with minimum interruptions. PAGE 12
May be applied as; A temporary constraint to deal with an outage situation when some assets are not available. A permanent constraint when the normal integrated power system capability and expected generation offers and demand may not result in secure operation. 1.1.4. Security Constrained Economic Dispatch This is the operation of generating facilities to produce energy at the lowest cost to reliably serve consumers, recognizing any operational limits of generation and transmission facilities. The varying nature of demand of energy and the variations in costs of different types of generating units, together with the known and unknown conditions on the transmission network determine which generating units to be used to serve the load most reliably.[13] 1.1.5. HVDC system This involves the transmission of power at high voltages with the aim of improving system efficiency and reducing the overall energy cost. PAGE 13
1.2.A brief introduction to HVDC system Power transmission was formerly done in early 1880s using Direct Current (DC). With the introduction of transformers, development of induction motors, availability of synchronous generators and facilities that could convert alternating current to direct current whenever required, alternating current gradually replaced direct current as a method of power transmission. However in 1928 the availability of devices like mercury vapor rectifiers that have the ability to rectify and invert current created possibilities for high voltage direct current transmission. With fast development of converters (rectifiers and inverters) that can work at higher voltages and large currents, dc transmission became a major factor in the planning of power transmission. In the early stages all HVDC schemes used mercury arc valves, invariably single phase in construction in contrast to the low voltage polyphase units used for industrial application. Around 1960, control electrodes were added to silicon diodes, giving silicon-controlledrectifiers (SCRs). Among the early schemes were;[14] The Gotland Scheme in Sweden, commissioned in 1954, capable of transmitting 20MW of power at -100KV and consisting of a single 96km cable with sea return. The Cross Channel link between England and France put in operation in 1961, two single conductor submarine cables 64km at ±100KV with two bridges each rated at 80MW. Sakuma Frequency Changer in 1965 to connect the 50Hz and 60Hz systems of Japan, capable of transmitting 300MW in either direction at 250KV. With the growing application of HVDC transmission, need arises to formulate Economic Dispatch for it. This entails the allocation of generation levels to generating units in a power system employing HVDC lines to economically serve the load in entirety while remaining within the operational limits of the generating and transmitting facilities. The fundamental processes in an HVDC system is the conversion of electrical current from AC to DC (rectification) at the transmitting end, and from DC to AC at the receiving end. There are three ways of achieving conversion;[12] I. Use of Natural Commutated Converters; are most used in HVDC systems today. A thyristor capable of handling 4000A currents and blocking up to 10KV is used. Series connection of these thyristors to form a thyristor valve enables them to block hundreds of KV. The thyristor is operated at net frequency (50Hz or 60Hz) PAGE 14
II. III. and change of DC voltage level is achieved by means of adjusting the control angle of the thyristor. Use of Capacitor Commutated Converters (CCC); this is an improvement of the thyristor-based-commutation. Commutation capacitors are inserted in series between the converter transformers and the thyristor valves. These capacitors improve the commutation failure performance of the converters when connected to weak networks. Use of forced commutated converters; the valves of these converters are made with semiconductors with the ability to turn-on and also turn-off. They are known as Voltage Source Converters (VSCs). Semiconductors normally used to are Gate-Turn-Off Thyristors (GTOs) and Insulated Gate Bipolar Transistors (IGBTs). The VSC commutates with high frequency and its operation is achieved by Pulse Width Modulation (PWM). With PWM it is possible to create any phase angle and/or amplitude by changing the PWM pattern, which can be done almost instantaneously. Thus PWM offers possibility to control both active and reactive power independently, making the PWM VSC a close to ideal component in the transmission network. 1.2.1. Components of an HVDC transmission system The three main elements in an HVDC system are; i. Converter Station Converter stations at each ends are replicas of each other and thus consists of all equipment needed to convert from AC to DC and vice versa. The main components here are; Thyristor valves most common way of arranging thyristor valves is in a twelve-pulse group with three quadruple valves. All communication between the control equipment at earth potential and each thyristor at high potential is done with fiber optics. VSC valves consists of two level or multilevel converter, phase-reactors and AC filters. VSC valves, control equipment and cooling equipment are in enclosures which make transport and installation easy. Transformers adapt the AC voltage level to DC voltage level and contribute to commutation reactance. AC filters and Capacitor banks filters are installed in order to limit the amount of harmonics to the level required by the network. In the conversion process the converter consumes reactive power which is compensated for in part by the filter banks and the rest by capacitor banks. DC filters reduce the disturbances in telecommunication systems created by harmonics due to HVDC converters. PAGE 15
ii. iii. Transmission medium Most frequently overhead line, normally bipolar (that is, two conductors with different polarity) when over land. HVDC cables are used for submarine transmission. Electrodes Are conductors that provide connection to the earth for neutral. They have large surface to minimize current densities and surface voltage gradients. 1.2.2. HVDC systems control Control is done for efficiency and stability of the system, and also for maximum flexibility of power control without compromising on safety of equipment. The parameters mostly controlled are; Direct current from rectifier to inverter I d V dr cos Vdi cos R R R cr L ci V V R R R dr di cr ci L rectifier end voltage inverter end voltage rectifier resistance inverter resistance transmission line resistance Power at the rectifier terminal Pdr Vdr Id Power at the inverter terminal Pdi V di Id Pdr RLI d The means of control is by control of the internal voltages that can be used to control the voltage at any point along the transmission line and the current flow or power. This is done by controlling the firing and extinction angles of the rectifiers and inverters (fast action) or by changing taps on the transformers on the AC side (slow response). Power reversal is done by reversing the polarity of the DC voltages at both ends, current flow remains unchanged (since valves can only conduct in on direction) A control action may be chosen with aim of prevention of large fluctuations in DC voltage/current due to variations in the AC side voltage, maintenance of direct current 2 PAGE 16
voltage near the rated value or to keep power factor at the receiving and transmitting ends as high as possible. Tap changer control is used to keep the converter firing angle ( & ) within the desired range. They are sized to allow for minimum and maximum steady state voltage variation. For current limits, the maximum short circuit current is limited to 1.2 to 1.3 times the normal full load current to avoid thermal damage of equipment. Minimum current limit is set to avoid ripple in the current that may cause it to be discontinuous or intermittent. Minimum firing angle limit is set to prevent reversal of power flow as a result of the inverter station switching to rectification mode in case of a DC fault. For power control, the current order required to transmit a scheduled power is given by; Po Iord, Po is thescheduled power, Vd the controlled voltage V d 1.2.3. HVDC configurations There are three HVDC configurations; monopolar, bipolar and homopolar systems. Monopolar HVDC systems have either ground return or metallic return. A monopolar HVDC system with ground return consists of one or more six-pulse converter units in series or parallel at each end. It can be a cost-effective solution for an HVDC cable transmission and/or the first stage of a bipolar scheme. At each end of the line it requires an electrode line and a ground or sea electrode built for continuous operation. A monopolar HVDC system with metallic return usually consists of one high voltage and one medium voltage conductor. A monopolar configuration is used either in the first stage of a bipolar scheme, avoiding ground currents, or when construction of ground electrode lines and ground electrodes result in an uneconomical solution due to a short distance or high value of earth resistivity. Bipolar HVDC systems consist of two poles, each of which includes one or more twelve-pulse converter units in series or parallel. There are two conductors, one with positive and the other with negative polarity. For power flow in the other PAGE 17
direction, the two conductors reverse their polarities. A bipolar system is a combination of two monopolar schemes with ground return. Back-to-Back HVDC links are special cases of monopolar HVDC interconnections where there is no DC transmission line and both converters are located at the same site. For economic reasons each converter is usually twelvepulse converter unit, and the valves for both converters may be located in one valve hall. The control system, cooling equipment and auxiliary system may be integrated into configurations common to the two converters. Generally for a Back-to-Back HVDC link, the DC voltage rating is low and the thyristor valve current rating is higher in comparison with HVDC interconnections via overhead lines or cables. 1.2.4. Reasons for HVDC Some short comings of AC transmission as well as the need to incorporate the upcoming renewable energy from sources such as solar and wind compel a change and application of DC technology. Some gaps in high voltage AC transmission are;[7] Inductive and capacitive elements of overhead lines and cables put limits to the transmission capacity and the transmission distance for AC transmission links. Depending on the required transmission capacity, the system frequency and the loss evaluation, the achievable transmission distance for an AC cable is in the range of 40 to 100km, mainly limited by charging current. Direct connection between two AC systems with different frequencies is not possible. Direct connection between two AC systems with the same frequency or a new connection within a meshed grid may be impossible because of system instability, too high short-circuit levels or undesirable power flow scenarios 1.2.5. Comparison between DC and AC systems Comparison can be done under technical merits, economic considerations and environmental issues. Technical Merits of HVDC The advantages of a DC link over an AC link are; A DC link allows for power transmission between AC networks with different frequencies or networks which cannot be synchronized for some reasons. PAGE 18
Transmission capacity or the maximum length of a DC line or cable is not limited by inductive and capacitive parameters. The conductor cross-section is also fully utilized because there is no skin effect. A digital control system provides accurate and fast control of the active power flow. Fast modulation of DC transmission power can be used to damp power oscillations in an AC grid and thus improve the system stability. Economic considerations For a given transmission task, feasibility studies are conducted before the final decision on implementation of an HVAC or HVDC system can be made. The figure below shows a typical comparison curve between AC and DC transmission considering AC vs DC station terminal costs, AC vs DC line costs, and AC vs DC capitalized value of losses.[7] Fig 1.1 Total cost/distance[7] The DC curve is not as steep as the AC curve because of considerably lower line costs per kilometer. For long AC lines the cost of intermediate reactive power compensation has to be taken into account. The break even distance is in the range of 500 to 800km depending on factors like country-specific cost elements, interest rates for projects financing, loss evaluation, and cost of right of way, among others. Environmental issues An HVDC transmission system is basically environment friendly because improved energy transmission possibilities contribute to a more efficient utilization of existing PAGE 19
power plants. The land coverage and the associated right of way cost for an HVDC overhead transmission line is not as high as that of an AC line. This reduces the visual impact and saves land compensation for new projects. It is also possible to increase the power transmission capacity for existing right of way. [7] Fig 1.2 Typical transmission line structures for approximately 1000MW[7] There are however some environmental issues which must be considered for the converter stations. These include audible noise, visual impact, electromagnetic compatibility and use of ground or sea return path in monopolar operation, among others. In general it can be said that an HVDC system is highly compatible with any environment and can be integrated into it without the need to compromise on any environmentally important issues of today. 1.3.Problem statement The aim of this paper is to introduce security constraints to the economic dispatch through the analysis of factors affecting the generation of energy at the generating units and the transmission of that energy to the demand centers using High Voltage Direct Current. To solve this problem, adequate knowledge of economic dispatch, system security and HVDC as well as its security aspects and transmission is a fundamental requirement. 1.4.Objectives To obtain an optimal solution to the Security Constrained Economic Dispatch, the following objectives are to be achieved; To formulate the ED problem taking into consideration the constrains in generation of power and its transmission using HVDC PAGE 20
To study the methods of solution of the Security Constrained Economic Dispatch problem, and To come up with an optimal solution for the problem 1.5.Research Questions The process and outcome of the project will attempt to address the following questions; What are the constraint to be considered while undertaking a Security Constrained Economic Dispatch for HVDC? What is the most effective technique to apply in solving this particular SCED problem? PAGE 21
CHAPTER 2 LITERATURE REVIEW Economic Dispatch (ED) entails optimal allocation of the outputs of a large number of participating generators. Security Constrained Economic Dispatch seeks to optimize the process, taking into account all the relevant factors pertaining to selection of the generating units to dispatch so as to deliver a reliable supply of power at the lowest cost possible. The choice as to whether a generating unit should participate in sharing the load at a given interval of time is a problem of unit commitment. The unit commitment problem having been solved, optimal allocation of the available generation units to meet the forecasted load demand for the time interval in question is done. ED process has two stages, also referred to as time periods; PAGE 22
a. Unit commitment stage the operators decide which units to be committed to be online for each hour, usually for the next 24hrs period, based on load forecast taking into account the unit s maximum and minimum output levels, the minimum time a generator in the unit must run once started, the generating costs and the costs of environmental compliance and how quickly the output of the unit can be changed. Also of importance to consider under this stage are the forecasted conditions that can affect transmission grid, that is, the security constrained aspect of commitment analysis, as well as the generation and transmission facilities outages, line capabilities (limits and directions) and weather. If security analysis indicate that optimal ED cannot be done reliably, relatively expensive generators may have to be opted for. b. Unit dispatch stage operators decide in real time the level at which each available resource (as determined from stage (a)) should be operated, given the actual load and grid conditions, such that reliability is maintained and overall production costs are minimized. Optimization techniques are used to determine not only the optimal outputs of the participating generators, but also the optimal settings of various control devices such as the tap settings of Load Tap Changers (LTCs), outputs of VAR compensating devices, desired settings of phase shifters, among others. The desired objectives for optimization problems include minimization of the cost of generation, minimization of the total power loss in the system, minimization of voltage deviations, and maximization of the reliability of the power supplied to customers. While formulating the optimization strategy, one or more of these objectives can be taken into consideration. Determination of the real power output of the generators so that the total cost of generation in the system is minimized is known as Economic Dispatch. The majority of generating systems are of three types; nuclear, hydro and thermal (using fossil fuels such as coal, oil and gas), but due to developments in the technology of renewable energy, and with rising environmental concerns together with rising demand for power, these sources are currently complemented by other sources such as wind, solar and tidal energy. Nuclear plants tend to be operated at constant output power levels. Operating cost of hydro units do not change much with the output. The operating cost of thermal plants however change significantly with the output power level, and therefore are considered in this paper for discussing the ED problem. 2.1.Economic dispatch problem neglecting transmission losses First the ED problem is considered with the transmission losses neglected (like in the case of Back-to-Back HVDC systems where systems operating at different frequencies are joined together and at one location and the transmission distance is essentially zero or where a group of generators are connected to a particular bus-bar like in the case of PAGE 23
individual generating units in a power plant, or when they are physically located very close to each other, and thus the transmission losses can be ignored due to the short distance involved).[10] Consider the Fig2.1 below showing N-thermal units connected to a bus to serve a load. P Load Fig 2.1 N-thermal units connected to a bus to serve a load P Load [9] Input to each unit is expressed in terms of cost rate, say $/h. The total cost rate is the sum of cost rates of each of the individual units. The essential operating constraint is that the sum of the power outputs must be equal to the load (neglecting power losses). 2.1.1. Fuel cost characteristics ED problem is the determination of generation levels with the aim of minimizing the total cost of generation for a defined level of load. For thermal generating units, the cost of fuel per unit power output varies significantly with the power output of the unit. In solving the ED problem, the fuel cost characteristics of the generators are considered while finding their optimal real power outputs. A typical fuel cost characteristics is as shown in Fig2.2 below. PAGE 24
Fig2.2 A typical fuel cost characteristics[9] The labor cost, supply and maintenance are generally fixed. P Min is the output level below which it is uneconomical or technically infeasible to operate the units. PMax on the other hand is the maximum output power limit. In formulating the dispatch problem, fuel costs are usually represented as a quadratic function of output power as shown by the equation below. 2 F( P) ap bp c (2.1) 2.1.2. Problem formulation Total fuel cost for operating N generators is given by; F F ( P) F ( P )... F ( P ) (2.2) T 1 1 2 2 N N N F( P) (2.3) i1 i i With transmission losses neglected, total generation should meet the total load. Therefore, the equality constraint is; N Pi PLoad (2.4) i1 Based on minimum and maximum power limits of the generators, the following inequality constraint is imposed; P P P ; i 1,2,... N (2.5) i( Min) i i( Max ) This is a constrained optimization problem that can be solved by Lagrange multiplier method. The Lagrange method is formulated as; LFT (2.6) N Where P Pi accounts for the equality constraint (2.4), is the Lagrange Load i1 Multiplier. The necessary condition for T F to be minimum is that the derivative of Lagrange function with respect to each independent variable is zero. Thus the necessary conditions for the optimization problem are; PAGE 25
N N L { F ( P) (P P) P P i i i i Load i i1 i1 (2.7) Fi 0 P ; i 1,2,... N (2.8) and i L 0 P i (2.9) Rearranging (2.8), Fi P i ; i 1,2,... N (2.10) Equation (2.10) states that to minimize the fuel cost, the necessary condition is to have all the incremental fuel costs equal. Equation (2.10), along with the equality constraint (2.4) and the inequality constraint (2.5) are the Coordination Equations for Economic Dispatch with network losses neglected. Using equation (2.1), fuel cost characteristics of all the generators are expressed as; F a P b P c ; i 1,2,... N (2.11) 2 i i i i i i Using (2.10), the necessary conditions for the optimal solutions are given by; Fi P i 2a P b ; i 1,2,... N (2.12) i i i Or bi Pi ; i 1,2,... N (2.13) 2a i Substituting P i from (2.13) into (2.4), we have; b N i PLoad (2.14) i1 2ai Or PAGE 26
N i PLoad ( ) i1 2ai [ ] N i1 1 ( ) 2a i b With this, can be calculated from (2.12) and (2.11). P i (2.15) be determined for i 1,2,... N from 2.2.Economic dispatch problem considering network losses This involves economically distributing the load among different plants of a power system. In this case transmission losses are considered as shown in the schematic below depicting such a system. Fig 2.3 N thermal units serving P Load through a transmission network with losses For a unit with low incremental cost, operating cost may be higher if the transmission line losses are very high (for example where there is a large distance between the unit and the load), thus it becomes necessary to take into consideration the transmission line losses when determining Economic Dispatch of units in a power system. The total fuel cost rate is still as expressed by equation (2.2) F F ( P) F ( P )... F ( P ) T 1 1 2 2 N N The power balance (equality) equation now includes the transmission losses. PAGE 27
N P P P 0 (2.16) P Loss Loss Load i i0 is the total transmission loss in the system. The problem is to find Pi s that minimize F T subject to the constraint (2.16). Using the method of Lagrange multipliers, LFT, with given as expressed by equation (2.16). The necessary conditions to minimize F T are as follows; L P i 0 ; i 1,2,... N Or N N { Fi ( Pi ) ( PLoss PLoad Pi )} 0 P (2.17) Or i i1 i1 L Fi P Loss ( 1) 0 ; i 1,2,... N (2.18) P P P i i i This (equation 2.18) is the condition for optimal dispatch. Rearranging (2.18), Fi Pi (2.19) PLoss 1 P i The equation above is often expressed as P fi Fi P i (2.20) With P fi being the Penalty Factor of the plant, and is given by; PAGE 28
P fi 1 (2.21) PLoss 1 P i Here, P P Loss i is the incremental loss for bus i. From Equation (2.20), the minimum cost operation is achieved when the incremental cost (IC) for each unit multiplied by its penalty factor is same for all generating units in the system. Relating to the case of units in the same plant, or generators connected to the same bus, (2.20) implies; F F FN P P... P P P P 1 2 f 1 f 2 fn 1 2 N When units are connected to the same bus, incremental change with transmission loss with change in generation is the same for all the units, thus; P... 1 P 2 P, and therefore; f f fn F1 F2 FN... P P P 1 2 N (2.22) Which is the same as in the case of units connected to a bus. Equation (2.19) and (2.16) are collectively known as coordination equations for Economic Dispatch considering transmission losses. 2.2.1. Transmission line loss equation Transmission line loss equation, known as Kron s loss formula is expressed as; P P BP B P B (2.23) T T Loss O OO Where P is the vector of all generator bus net outputs; B is a square matrix; B O is a vector of same length as P ; B OO is a constant. The B-terms are called Loss-Coefficients or B-Coefficients, and the N by N symmetrical matrix B is simply known as the B-matrix. Equation (2.23) can be written as; PAGE 29
N N N (2.24) P PB P B P B Loss i ij j io i OO i1 j1 i1 Referring to the coordination equation, the equality constraint now becomes; N N N N P [ PB P B P B ] P (2.25) Load i ij j io i OO i i1 j1 i1 i1 The derivative of Lagrange function now becomes; N L F [1 2 Bij Pj BiO ] P P (2.26) i i j 1 The coordination equations are now coupled. Solution of Economic Dispatch problem in this case is a little complex compared to the case with network losses neglected. 2.2.2. Losses in HVDC systems Typically, overall losses in HVDC transmission are 30% to 50% less than HVAC transmission. Although HVDC incurs losses during the conversion process from AC to DC, the line losses in HVDC cable are smaller than HVAC cables, and when used over long distances, lower cable losses compensate for higher conversion losses of HVDC transmission. The power losses produced, per converter station, in VSC HVDC technology are more than the power losses produced per converter station in LCC HVDC technologies. The power losses in VSC per converter station are 4% to 6% of the total power being delivered while that per converter stations in LCC HVDC are 2% to 3%. However VSCs are preferred due to their low levels of harmonics generated hence reduced need to install filters in offshore substations. For a VSC HVDC system, losses can be studied under the following stages; VSC Converter losses; divided into conduction losses and switching losses Transmission losses, including DC cable losses, coupling transformer losses, smoothing reactor losses and losses in AC filters. PAGE 30
2.2.3. HVDC inequality constraints In HVDC the inequality constraints are usually the operation or physical limits. For instance, a transmission line capacity is constrained by its thermal limit, the bus voltages are within their insulation limits and generating units have lower and upper output limits. Such constraints restrict the ED of the generators to range between the maximum and minimum values, and include; The power generator capacity constraint P P P GiMin Gi GiMax The tap ratio of the converter T T T Min Max Ignition angle of the converter Min Max Extinction angle of the converter Min Max DC current I I I dcmin dc dcmax DC voltage V V V dcmin dc dcmax The aim of this project is to minimize the total operating cost of the power system while meeting the total load plus the transmission losses while operating within the generator limits and transmission line limits. The transmission losses were taken to be 40% of the losses obtained by calculation using B coefficients for an HVAC line of equivalent length. Mathematically, the aim is to minimize; N N 2 Fi ( Pi ) aipi bi Pi ci i1 i1, Subject to the following constraints; PAGE 31
The energy balance equation N i0 P P P i Loss Load The generator limits P P P GiMin Gi GiMax The tap ratio of the converter T T T Min Max Ignition angle of the converter Min Max Extinction angle of the converter Min Max DC current IdcMin Idc I dcmax DC voltage VdcMin Vdc VdcMax 2.3.Review of solution methods Two basic approaches are used in the solution; 1. The case of network loss formula, and 2. The case of optimization tools incorporating power flow equations and constraints. PAGE 32
2.3.1. Particle swarm optimization Particle Swarm Optimization was proposed by James Kennedy and Russell C. Eberhart in 1995.[15] It is a technique used to explore the search space of a given problem to find the settings or parameters required to maximize a particular objective. It originates from two concepts; the idea of swarm intelligence based on the observation of swarming habits of certain kinds of animals, and the field of evolutionary computation. Optimization is the mechanism by which the maximum or minimum value of a function or process is obtained. A search space is defined, with elements called candidate solutions of the search space. The number of parameters involved in the optimization problem is the dimensions of the search space. An objective function maps the search space to the function space. For a known function, calculus may be used to easily provide the minima or maxima as desired. In real life however, the objective function is not directly known. Instead, the objective function is a black box to which we apply parameters and receive an output value. The result of this evaluation of a candidate solution becomes the solution s fitness. The final goal of an optimization task is to find the parameters in the search space to maximize or minimize the fitness.[17] The elements of a candidate solution may be subject to certain constraints, in which case the task becomes a constrained optimization task. Each particle is searching for an optimum and is moving, hence has a velocity. Each particle remembers the position at which it had its best so far (its personal best). To improve the effectiveness, particles in the swarm co-operate by exchanging information about what they have discovered in the places they had been to. Thus a particle has neighbors associated with it, it knows the fitness of those in its neighborhood, and uses the position of the one with best fitness to adjust its own velocity. In each time step a particle has to move to a new position by adjusting its velocity. The adjustment is the sum of its current velocity, a weighted random portion in the direction of its personal best, and a weighted random portion in the direction of the neighborhood best. Particles velocity on each dimension are clamped to a maximum velocityv Max. If the sum of accelerations would cause the velocity on that dimension to exceedv Max, a parameter specified by the user. Then the velocity in that direction is limited tov Max. Particle Swarm Optimization is a preferred method of solving ED problems for among others the following reasons; With a population of candidate solutions, a PSO algorithm can maintain useful information about the characteristics of the environment. PAGE 33
PSO, as characterized by its fast convergence behavior, has an inbuilt ability to adapt to a changing environment. Some early works on PSO have shown that PSO is effective for locating and tracking options in both static and dynamic environment. As compared to other optimization techniques, PSO is a simple concept that is easy to implement, cheaper, impervious to failure regardless of user input or unexpected conditions and takes less time to converge. 2.3.2. Genetic algorithm First proposed by Frazer and later by Bremermann and Raed, Genetic Algorithm was popularized by the work of Holland. GA models genetic evolution. Features of individuals are expressed using genotypes. The main driving operators of a GA is selection which models survival of the fittest and recombination through application of a crossover operator that models reproduction.[17] A population of individuals (phenotypes) to an optimization problem is evolved towards better solutions. Each candidate solutions features are mutated. The process starts from a population of individuals and is an iterative process resulting in successive generations. For each generation the fitness of each individual is evaluated, the fitness being the value of the objective function in the optimization problem being solved. The more fit individuals are stochastically selected from the current population and each individual s genome modified to form the next generation. This generation solutions are then used in the next iteration of the algorithm. The algorithm terminates when a satisfactory fitness level has been reached for the population. Although Genetic Algorithm always converges, it does not give assurance that a global optimum will be obtained. It also lacks a constant optimization response time. 2.3.3. Evolutionary Programming Evolutionary Programming (EP) was first used by Lawrence J. Fogel in the US in 1960 to use simulated evaluation as a learning process to generate artificial intelligence. While EP shares the objectives of imitating natural evolutionary processes, with Genetic Algorithm and Genetic Programming, it differs substantially in that EP emphasizes the development of behavioral models and not genetic models. EP considers phenotypic evolution, it iteratively applies two evolutionary operators; variation through application of mutation operators and selection.[17] The evolutionary process, first developed to evolve finite state machines, consists of finding a set of optimal behaviors from a space of observable behaviors. The fitness PAGE 34
function measures the behavioral error of an individual with respect to the environment of that individual. EP utilizes four main components of Evolutionary Algorithm (EA). Initialization; a population of individuals is initialized to uniformly cover the domain of the optimization problem. Mutation; the mutation operator introduces variation in the population to produce new candidate solutions. Each parent produces one or more offspring through application of the mutation operator. Evaluation; a fitness function is used to quantify the behavior error of individuals. While the fitness function provides an absolute fitness measures to indicate how well the individual solves the problem being optimized, survival in EP is usually based on a relative fitness measure. A score is computed to quantify how well an individual compares with a randomly selected group of competing individuals. Individuals that survive to the next generation are selected based on this relative fitness. Selection; the selection operator selects the individuals that survive to the next generation. The setback with EP is its slow convergence in solving some of the multimodal optimization problems. 2.3.4. Linear Programming Linear programming is the most commonly applied form of constrained optimization. The main elements of any constrained optimization problem are; Variables (decision variables); values are unknown at start, usually represent things that can be adjusted or controlled. The goal is to find values of the variables that provide the best value of the objective function. Objective function; is a mathematical expression that combines the variables to express the goal. The requirement is to either maximize or minimize the objective function. Constraints; are mathematical expressions that combine the variables to express limits on the possible solutions. In linear programming all the mathematical expressions for the objective function and the constraints are linear, thus it has an inaccurate evaluation of system losses and a limited ability to find accurate solutions due to its linear approximation of non-linear control parameters as compared to an exact non-linear model of a power system.[17] PAGE 35
2.3.5. Lambda iteration When the minimization is constrained with an equality constraint it can be solved using the method of Lagrange Multiplier. The key idea is to represent a constrained minimization problem as an unconstrained problem.[11] Lambda iteration method requires a unique mapping of from a value of lambda (incremental cost) to each generator s output. For any choice of lambda the generators collectively produce an output. The methods starts with values of lambda below and above the optimal value (corresponding to too little and too much output), and then iteratively brackets the optimum value. Inclusion of losses impact the necessary conditions for an optimal economic dispatch. The analytic calculation of the penalty factor is involving, the problem is that a small change in the generation impacts the flow and hence the losses throughout the entire system. However using a power flow, the function can be approximated by making a small change to the output of individual generators and seeing how the losses change. PAGE 36
CHAPTER 3 SOLUTION OF SECURITY CONSTRAINED ECONOMIC DISPATCH USING PSO 3.1.PSO Algorithm The PSO algorithm works by simultaneously maintaining several candidate solutions in the search space. Initially, the algorithm chooses candidate solutions randomly within the search space composed of all possible solutions. The algorithm uses the objective function to evaluate its candidate solutions and operates upon the resultant fitness values. PAGE 37
Fig 3.1. PSO Algorithm PAGE 38
Each particle maintains its position, composed of the candidate solution and its evaluated fitness, and its velocity. Additionally, it remembers the best fitness value it has achieved thus far during the operation of the algorithm, referred to as the individual best fitness, and the candidate solution that achieved this fitness, referred to as the individual best position or individual best candidate solution. Finally, the PSO algorithm maintains the best fitness value achieved among all particles in the swarm, called the global best fitness, and the candidate solution that achieved this fitness, called the global best position or global best candidate solution. The PSO algorithm consists of just three steps, which are repeated until some stopping condition is met: 1. Evaluate the fitness of each particle 2. Update individual and global best fitness and positions 3. Update velocity and position of each particle The first two steps are fairly trivial. Fitness evaluation is conducted by supplying the candidate solution to the objective function. Individual and global best fitness and positions are updated by comparing the newly evaluated fitness against the previous individual and global best fitness, and replacing the best fitness and positions as necessary. The velocity and position update step is responsible for the optimization ability of the PSO algorithm. The velocity of each particle in the swarm is updated using the following equation: 1, 1 1 1 t 1 Vj t w t Vj t c1r 1 X j pbest t X j t c2r 2 X gbest t X j ] 0 r 1and 0 r 1, x 1 max, x max, 2 The index of the particle is represented by j. Thus, vj () t is the velocity of particle i at time t and xj () t is the position of particle j at time t. The parameters w, c 1 and c 2 0 w1.2,0 c 2, and 0c2 2) are user-supplied coefficients. The values r 1 and ( 1 r 2 ( r1 r2 0 1and 0 1) are random values regenerated for each velocity update. The value X, is the individual best candidate solution for particle i at time t, and X gbest is j pbest the swarm s global best candidate solution at time t. Each of the three terms of the velocity update equation have different roles in the PSO w t V t 1 is the inertia component, responsible for keeping algorithm. The first term j PAGE 39
the particle moving in the same direction it was originally heading. The value of the inertial coefficient w is typically between 0.8 and 1.2, which can either dampen the particle s inertia or accelerate the particle in its original direction. Generally, lower values of the inertial coefficient speed up the convergence of the swarm to optima, and higher values of the inertial coefficient encourage exploration of the entire search space. The second term c r X t X t 11 j, pbest 1 j 1, called the cognitive component, acts as the particle s memory, causing it to tend to return to the regions of the search space in which it has experienced high individual fitness. The cognitive coefficient is usually close to 2, and affects the size of the step the particle takes toward its individual best candidate solution. X gbest The third term c r X t X t 2 2 gbest 1 j 1, called the social component, causes the particle to move to the best region the swarm has found so far. The social coefficient typically close to 2, and represents the size of the step the particle takes toward the global best candidate solution X the swarm has found up until that point. gbest The random values r 1 in the cognitive component and r 2 in the social component cause these components to have a stochastic influence on the velocity update. This stochastic nature causes each particle to move in a semi-random manner heavily influenced in the directions of the individual best solution of the particle and global best solution of the swarm. In order to keep the particles from moving too far beyond the search space, we use a technique called velocity clamping to limit the maximum velocity of each particle. For a search space bounded by the rangex, x, velocity clamping limits the velocity to max the range v, v, where v k * x. max max max max max The value k represents a user-supplied velocity clamping factor, 0.1 k 1.0. In many optimization tasks, such as the ones discussed in the paper, the search space is not centered on 0 and thus the range xmax, xmax, is not an adequate definition of the search space. In such a case where the search space is bounded byx, x, we define v k *( x x ) / 2. max max min Once the velocity for each particle is calculated, each particle s position is updated by applying the new velocity to the particle s previous position: 1 X t X t V t j j j min max c 1 c 2 is PAGE 40
This process is repeated until some stopping condition is met. Some common stopping conditions include: a preset number of iterations of the PSO algorithm, a number of iterations since the last update of the global best candidate solution, or a predefined target fitness value. For this project, the constrained optimization problem is converted as an unconstrained optimization using penalty function method. In fitness penalization of a solution, the fitness function is the sum of the objective function value and the sum of constraint violation.[5] i.e. minimize n F ( P) 1000* P D P i i i l i1 i1 n Power loss is obtained by DC power flow with the following assumptions made; 1. All voltage magnitudes are equal to 1.0 p.u. 2. The resistances of the branches are ignored; i.e., the susceptance of the branch is 1 Bij x ij 3. The angle difference on the two ends of the branch is very small, such that sin ij i j cos 1 ij 4. All ground branches are ignored; i.e., B B i0 j0 0 And therefore the DC power flow model is P1 1 P 2 2 B P n1 n1 The DC power flow is a purely linear equation, so only one iteration calculation is needed to obtain the power flow solution. It is used in calculation of real power flow on transmission lines and transformers. The power flowing on each line using DC power flow is; PAGE 41
i j Pij Bij ( i j ) x ij 3.2.Parameter representation The aim is to minimize the operating cost. The optimization is done by PSO technique. The population of particles P, representing the generators where is the i th unit in the power system, is initialized together with other variables. Each particle is generated randomly within the allowable range specified by the generation limits of the particular i th particle. P P P imin i imax The size of the population, representing the number of generators is initialized along with the initial and final inertia weight, random velocity of the particle, acceleration constant, maximum generation, the number of iterations and Lagrange s multiplier. The fitness of each individual in the population is calculated using the fitness function, which includes the cost function and the penalty function for violation of the equality and inequality constraints. P i C T C ( P) Equality constraints Inequality constraints T i i i1 i1 n Each unit s position and velocity is updated along with the multipliers i for inequality constraints whose value can be 1 or 0 depending on whether the particular constraint has been violated or has not been violated respectively. i for equality and If the number of iterations reaches the maximum, the individual that generates the latest value is the optimal generation power of each unit with the minimum total generation cost. 3.3.PSO Algorithm for SCED PSO is a population based stochastic optimization technique. Each particle in the population represents a candidate solution to the problem. All particles start at randomly initialized positions and fly throughout the search space to find the best possible solution, PAGE 42
while communicating with each other and sharing the best local solutions each of them has achieved. Based on the local and global information obtained, each particle updates its position towards a desired global optimum.[4] The elements of the PSO algorithm are as described below: Particle, X j optimization variables. t each particle is a candidate solution vector containing n X j t is the,,,, th j particle at time t described as; X j t x j 1 t x j 2 t x j n t The particle vector describes the particle s position within the search space. Population, is a set of m particles at time, Pop t t Particle velocity, V j t is the velocity of the dimensional search space represented as; th j particle at the time t in the n-,,,, T Vj t v j 1 t v j 2 t v j n t The velocity of the particle indicates the relative change of the particle within the solution space with respect to its current position vector. For each time increment a particle s velocity demonstrates the time rate of change to the particle s solution vector. Individual best X, j pbest t is the best position achieved by the th j particle so far at time t. Each particle memorizes its best position throughout the entire searching process. X t is the best solution that has been achieved so far among Global best, gbest all the particles. The information of global best is known to all the particles through communication among the particles. The PSO algorithm is implemented in the order below Initialization at the start t = 0, all particles are initialized with a randomly assigned position and velocity value. The i th th dimensional position x ji, of the j particle is initialized with a uniform random value between lower and upper bounds. The i th dimensional velocity of the particle is initialized with a uniform xi ( u) xi ( l) random value between v imax and v imax, with vimax where x iu ( ) and N x il () are the upper and lower bounds of the of the particles position respectively in the i th dimension and N im is the minimum number of steps that change a particles im PAGE 43
position from its lower bound to its upper bound of the dimension, a value chosen by the user. Velocity updating during each iteration cycle the particle velocity is updated according to the following formula. Vj t w t Vj t 1 c1r 1 Xj, pbest t 1 Xj t 1 c2r 2 X gbest t 1 Xj t 1 ] Where wt is the inertia weighting factor, c 1 and th i c 2 are two positive constants, and r 1 and r 2 are uniform random numbers in [0, 1]. Position updates with the velocity updated, each particle changes its position according to the formula; X t X t 1 V t j j j Process termination the process stops when a specified stopping criterion is met, for instance when the number of iterations reach a pre-specified maximum. PAGE 44
CHAPTER 4 RESULTS AND ANALYSIS The proposed Particle Swarm Optimization algorithm was tested on IEEE 30 bus systems and results compared with those obtained from Classical Economic Dispatch neglecting security constraints, as well as with results obtained from Genetic Algorithm. The Network topology, load data, line limits and generator cost data for the systems are taken from [1]. 4.1.Case Study: IEEE 30-Bus System Fig 4.1: One line diagram of IEEE 30-bus system [1] PAGE 45
4.2.Results The optimal total generation, generation for the individual six generating units, the optimal generation costs for each unit, the total generation cost and the system power losses by PSO are given in tables 4.1 and 4.2 for SCED as well as CED for system demand of 540MW and 840MW. Table 4.3 shows the comparison between the results from the proposed PSO method with those from GA for a total demand of 840MW. Table 4.1: Optimal generation for SCED and CED using PSO for a total demand of 283.4MW Generation No. SCED CED PG1 166.003 168.289 PG2 43.8662 50.1362 PG5 20.7474 26.4431 PG8 26.9202 23.1219 PG11 17.5307 10.8661 PG13 16.8103 13.2421 Total generation (MW) 291.878 292.098 Total cost ($/hr) 801.712 803.898 Total loss (MW) 8.47783 8.6982 PAGE 46
Table 4.2: Optimal generation for SCED and CED using PSO for a total demand of 374.3MW Generation No. SCED CED PG1 175.305 193.811 PG2 61.6824 52.011 PG5 30.8379 34.0868 PG8 44.3274 45.164 PG11 40.6352 26.8235 PG13 31.2773 33.2283 Total generation (MW) 384.065 385.125 Total cost ($/hr) 1172.5 1163.01 Total loss (MW) 9.76504 10.8245 Table 4.3: Optimal generation for SCED and CED using PSO for a total demand of 540MW Generation No. SCED CED PG1 109.883 149.128 PG2 80.7397 83.7191 PG5 68.1473 50 PG8 130.108 90.8636 PG11 81.3105 101.135 PG13 82.3566 80 Total generation (MW) 552.545 554.846 Total cost ($/hr) 2268.03 2180.01 Total loss (MW) 12.545 14.8457 PAGE 47
Table 4.4: Comparison of Economic Dispatch using PSO and GA for a total demand of 283.4MW. Generation No. PSO GA PG1 166.003 175.7899 PG2 43.8662 48.2548 PG5 20.7474 22.0974 PG8 26.9202 22.3942 PG11 17.5307 12.3715 PG13 16.8103 11.3669 Total generation (MW) 291.878 292.795 Total cost ($/hr) 801.712 802.3516 Total loss (MW) 8.47783 9.395 PAGE 48
Real Power Losses (MW) 4.3.Analysis and Discussions 16 14 SCED CED 12 10 8 6 4 2 0 0 100 200 300 400 500 600 Demand (MW) Fig 4.2: Variation of Real Power Losses with Power Demand for SCED and CED PAGE 49
Optimal Cost ($/hr) SCED CED 2500 2000 1500 1000 500 0 0 100 200 300 400 500 600 Demand (MW) Fig 4.3: Variation of Optimal Cost with Power Demand for SCED and CED Fig 4.2 is a graph showing the variation of Real Power Losses with Demand for both SCED and CED. The Real Power Losses increase with increasing Total Demand. In comparison, the Real Power Losses are lower for the SCED case than or the CED. A possible explanation for this could be due to the fact that SCED aims at operating within the power flow limits and as a consequence reduces the power loss in the buses. Fig 4.3 shows variation of Optimal Cost with power demand for SCED and CED. The cost of generation is observed to increase with increasing demand, a fact that can be attributed to higher fuel requirement for higher power generation. It can also be observed that the cost of generation under SCED is higher than that under CED. The margin between the two is more pronounced at higher demand. This can be explained from the fact that higher costs occur when the transmission line constraints are violated. At low power demand, the power flow is more likely within the bus limits, or if it goes above the limits it is by a small magnitude. Increasing demand increases the strain on the system resources in an attempt to meet this demand, and with that, increases the costs. PAGE 50
CHAPTER 5 CONCLUSION AND RECOMMENDATIONS 5.1.Conclusion The project utilizes the PSO algorithm to solve the security constrained economic dispatch problem for HVDC. The procedure was tested on the IEEE 30-bus network with six generators. The results of the various parameters from the SCED for HVDC using PSO were compared with results of similar published works obtained using Genetic Algorithm to verify the effectiveness of the proposed PSO algorithm. In this project, the security aspects considered were the generator active power limits and the real power flow limits of the buses. From the comparisons done, the PSO algorithm exhibited the advantages of lower optimal cost, lower total losses and higher probability of convergence to the global optimum. The method is therefore appropriate for network flow analysis. 5.2.Recommendations More research and study is necessary on HVDC systems with regards to security constraints, and to economic dispatch as a whole. Power flow in PSO code was found to be rather slow in execution. The computational time could be reduced by lowering the number of iterations, which on the negative side could increase the chances of settling at a local minimum thus inhibiting achieving of the optimal solution, the global minimum. Use of a Hybrid Particle Swarm Optimization algorithm could enhance the chances of obtaining best results with a lower computational time. Such a method can be considered in future projects. PAGE 51
REFERENCES [1] Jizhong Zhu, Optimization of Power System Operation, New Jersey, Wiley-IEEE Press, 2009. [2] Xi-Fan Wang, Yonghua Song, Malcolm Irving, Modern Power Systems Analysis, New York, Springer, 2008. [3] J. Duncan Glover, Mulukutla S. Sarma, Thomas J. Overbye, Power Systems Analysis and Design: Fifth Edition, Stamford, Cengage Learning, 2012. [4] Caisheng Wang, M. Hashem Nehrir, Le Yi Wang, Feng Lin1and Chris M. Colson (Hybrid Constraint-Handling Mechanism for Particle Swarm Optimization with Applications in Power Systems) [5] Efr en Mezura-Montes and Jorge Isacc Flores-Mendoza (Improved Particle Swarm Optimization in Constrained Numerical Search Spaces) [6] Lizhi Wang, Nan Kong, Security Constrained Economic Dispatch: A Markov Decision Process Approach with Embedded Stochastic Programming [7] Siemens AG, High Voltage Direct Current Transmission Proven Technology for Power Exchange. [8] Jochen Kreusel, High Voltage Direct Current (HVDC) Transmission Workshop Transporting tens of Gigawatt of Green Power to the Market, Potsdam, May 12, 2011 [9] L. Kirchmayer, Economic Operation of Power Systems. New Delhi: Wiley Eastern Limited,rst ed., 1979. [10] A. J. Wood, B. F. Wollenberg, Power Generation Operation and Control. New York: John Wiley & Sons, Inc., second ed., 2006. [11] Rahul Dogra, Nikita Gupta, Harsha Saroa, Economic Load Dispatch Problem and Matlab Programming of Different Methods, International Conference of Advance Research and Innovation (ICARI-2014) [12] Roberto Rudervall, J.P. Charpentier, Raghuveer Sharma, High Voltage Direct Current (HVDC) Transmission Systems Technology Review Paper, Presented at Energy Week 2000, Washington, D.C, USA, March 7-8, 2000 [13] Federal energy commission, USA Security Constrained Economic Dispatch A Report to Congress Regarding Recommendations of the Joint Boards for the Study of Economic Dispatch, July 31, 2006. [14] J. R. Lucas, High Voltage Engineering, 2001 PAGE 52
[15] Vinod Puri, Yogesh K. Chauhan, A Solution to Economic Dispatch Problem Using Augmented lagrangian Particle Swarm Optimization, International Journal of Emerging Technology and Advanced Engineering, Volume 2, Issue 8, August 2012. [16] Moses Peter Musau, Economic Dispatch for HVDC Bipolar System with HVAC and Optimal Power Flow Comparisons using Improved Genetic Algorithm (IGA), International Journal of Engineering Research & Technology, Vol. 4 Issue 08, August- 2015 [17] Andries P. Engelbrecht, Computational Intelligence, An Introduction, Second Edition, West Sussex, John Wiley & Sons, 2007. [18] Dr. L.V.Narasimha Rao, PSO Technique for Solving the Economic Dispatch Problem Considering the Generator Constraints, International Journal of Advanced Research in Electrical, Electronics and Instrumentation Engineering, Vol. 3, Issue 7, July 2014. [19] V.Karthikeyan, S.Senthilkumar and V.J.Vijayalakshmi, A New Approach to the solution of Economic Dispatch using Particle Swarm Optimization with Simulated Annealing, International Journal on Computational Sciences & Applications (IJCSA) Vol.3, No.3, June 2013 PAGE 53
APPENDIX Appendix Table1: Generator data for IEEE 30-bus system [1] Generators No.1 No.2 No.5 No.8 No.11 No.13 Pgimax(pu) 2.00 0.80 0.50 0.35 0.30 0.40 Pgimin(pu) 0.50 0.20 0.15 0.10 0.10 0.12 Qgimax(pu) 2.50 1.00 0.80 0.60 0.50 0.60 Qgimin(pu) -0.20-0.20-0.15-0.15-0.10-0.15 Cost Function ai 0.00375 0.0175 0.0625 0.0083 0.0250 0.0250 bi 2.00000 1.7500 1.0000 3.2500 3.0000 3.0000 ci 0.00000 0.0000 0.0000 0.0000 0.0000 0.0000 Appendix Table 2: Load data for IEEE 30-bus system [1] Bus no. PD(p.u) QD(p.u) Bus no. PD(p.u) QD(p.u) 1 0.000 0.000 16 0.035 0.016 2 0.217 0.127 17 0.090 0.058 3 0.024 0.012 18 0.032 0.009 4 0.076 0.016 19 0.095 0.034 5 0.942 0.190 20 0.022 0.007 6 0.000 0.000 21 0.175 0.112 7 0.228 0.109 22 0.000 0.000 8 0.300 0.300 23 0.032 0.016 9 0.000 0.000 24 0.087 0.067 10 0.058 0.020 25 0.000 0.000 11 0.000 0.000 26 0.035 0.023 12 0.112 0.075 27 0.000 0.000 13 0.000 0.000 28 0.000 0.000 PAGE 54
14 0.062 0.016 29 0.024 0.009 15 0.082 0.025 30 0.106 0.019 Appendix Table 3: Line flow limits data for IEEE 30-bus system [1] PAGE 55