UNIVERSITY OF NAIROBI

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1 UNIVERSITY OF NAIROBI FACULTY OF ENGINEERING DEPARTMENT OF ELECTRICAL AND INFORMATION ENGINEEERING HYDROTHERMAL ECONOMIC DISPATCH USING PARTICLE SWARM OPTIMIZATION (P.S.O) PROJECT INDEX: 055 SUBMITTED BY KENNEDY WANJAHI KIMANI F17/1356/2010 SUPERVISOR: PROF. NICODEMUS ABUNGU ODERO EXAMINER: DR. C WEKESA PROJECT REPORT SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENT FOR THE AWARD OF THE DEGREE OF BACHELOR OF SCIENCE IN ELECTRICAL AND ELECTRONIC ENGINEERING OF THE UNIVERSITY OF NAIROBI 2015 SUBMITTED ON: 23 TH APRIL 2015 i

2 DECLARATION OF ORIGINALITY NAME OF STUDENT REGISTRATION NUMBER COLLEGE FACULTY DEPARTMENT TITLE OF WORK KIMANI KENNEDY WANJAHI F17/1356/2010 ARCHITECTURE AND ENGINEERING ENGINEERING ELECTRICAL AND INFORMATION ENGINEERING HYDROTHERMAL ECONOMIC DISPATCH USING PARTICLE SWARM OPTIMIZATION (P.S.O) 1) I understand what plagiarism is and I am aware of the university policy on this regard. 2) I declare that this final year project is my original work and has not been submitted elsewhere for examination, award degree or publication. Where other people s work or my own work has been used, this has been properly acknowledged and referenced in accordance with the University of Nairobi s requirements. 3) I have not sought or used the service of any professional agencies to produce this work. 4) 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. 5) 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: i

3 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:... ii

4 DEDICATION I dedicate my work to my family, for their continued support and love. iii

5 ACKNOWLEDGEMENTS First I would like to thank God, for His uning grace and love throughout my academic life. Without it, I would not have made it this far. I would also like to ext my sincere gratitude to my supervisor, Prof. Nicodemus Abungu Odero for his priceless motivation, support and guidance. I also ext my appreciation to Mr. Peter Musau for his time, for patiently being there for us, and for his valuable insights into my project, and above all, the encouragement. I appreciate all my lecturers and all non-teaching staff at the Department of Electrical and Electronic Engineering, University of Nairobi, for their contribution towards my degree. I would also like to ext my heartfelt gratitude to my classmates and especially my fris Ng etich Kiplimo Brian and Nyasimi Ronald for their encouragement and contribution to this project. To my classmates, I am thankful for their moral support as I did the project and the conducive environment. Finally I would like to thank my family for their help and understanding throughout my academic life. iv

6 DECLARATION OF ORIGINALITY... i CERTIFICATION... ii DEDICATION... iii ACKNOWLEDGEMENTS... iv LIST OF FIGURES... viii LIST OF TABLES... ix LIST OF ABBREVIATIONS... x ABSTRACT... xi CHAPTER INTRODUCTION Hydrothermal Economic Dispatch What is Economic Dispatch? What is Hydrothermal Economic Dispatch? What is Particle Swarm Optimization (P.S.O)? Survey of Earlier Work Optimization Techniques Conventional Methods Unconstrained optimization approaches Linear programming (L.P) Non-linear programming (N.L.P) Quadratic programming Newton s method Interior points (I.P) methods Mixed integer programming Network flow programming Intelligent Search Methods Neural network (NN) Evolutionary algorithms Genetic algorithm (GA)... 7 v

7 1.4.3 Tabu Search (T.S) Ant colony optimization (A.C.O) Particle Swarm Optimization (P.S.O) Simulated Annealing (S.A) Non-Quantity Approaches Problem Statement Organization of the Report CHAPTER Literature Review Literature Review of Hydrothermal Economic Load Dispatch Classical Economic Dispatch Input Output Characteristics of Generator Units Calculation of input-output parameters of a thermal system Hydrothermal System Economic Dispatch System Constraints Power balance constraints (Demand Constraints) Thermal Generator Constraints Hydraulic Network Constraints Particle Swarm Optimization PSO Terminologies PSO Parameter Selection Swarm Size Velocity components Inertia Weight Constriction Factor CHAPTER Solution of Hydrothermal Economic Dispatch Using P.S.O Formulation of Hydrothermal Dispatch Problem Economic Load Dispatch with Network losses Formulation of P.S.O Pso Algorithm vi

8 3.5 Flow Chart CHAPTER Bus Test Network : Particle Swarm Optimization Method Analysis and Discussion : Case Study: 600mw Case For both Thermal and Hydrothermal System : Convergence Characteristics of the PSO Method for 30 Bus system : Power Losses CHAPTER Conclusion and Recommation for Future Works Conclusion Recommation REFERENCES APPENDICES APPENDIX 1: Program for the Thermal Economic Dispatch APPENDIX 2: Program for the Hydrothermal Economic Dispatch APPENDIX 3: Code for plotting the Convergence Characteristics APPENDIX 3: Main program code for the PSO Toolbox Used vii

9 LIST OF FIGURES Fig 1. 1 The Multilevel Optimal System Operation Fig 1. 2 The Genetic Algorithm Process... 8 Fig 2. 1 : A Thermal Unit Fig 2. 2 Input - output characteristic of a thermal generating unit Fig 2. 3 :A Fundamental Hydrothermal System Fig 4. 1 : 30 Bus IEEE test network with six generating units Fig 4. 2: Thermal optimum generation Fig 4. 3: Hydrothermal optimum generation Fig 4. 4: Generation tr for combined thermal and hydrothermal system Fig 4. 5: Convergence Characteristics for 30 bus hydrothermal generating system. (700 MW) Fig 4. 6: Convergence Characteristics for 30 bus all thermal generating system. (700 MW) Fig 4. 7: Hydrothermal Power Loss Fig 4. 8 : Thermal power loss viii

10 LIST OF TABLES Table 4. 1 : Thermal System Generator Coefficients Table 4. 2 : B-Coefficient Matrix Table 4. 3 : Hydrothermal System Generator Coefficients; 3 Hydro and 3 Thermal Generators Table 4. 4 : B-Coefficient Matrix, Hydrothermal System Table 4. 5 : Optimal Scheduling of Thermal Generators of a 6-unit system by PSO Method (Loss included) Table 4. 6 : Optimal scheduling of a 6-unit system, 3 Hydro and 3 Thermal Generators by PSO Method (Loss included) Table 4. 7: Comparison of optimal power output for both thermal and hydrothermal system (Load=600MW) ix

11 LIST OF ABBREVIATIONS N.L.P I.P N.F.P L.P N.L.P N.F.P L.P G.A NN TS Non-Linear Programming Interior Points Methods Network Flow Programming Linear Programming Non-Linear Programming Network Flow Programming Linear Programming Genetic Algorithm Neural Network Tabu Search x

12 ABSTRACT The objective of this project is to solve the hydrothermal economic load dispatch using PSO. The Energy Management System or (EMS) as we know it today had its origin in the need for electric utility companies to operate their generators as economically as possible. To operate the system as economically as possible, one requires that the characteristics of all generating units be available so that the most efficient units could be dispatched properly along with the less efficient. In addition, there is a requirement that the on/off scheduling of generators units be done in an efficient manner as well. Basically, the optimal scheduling of generation in a hydrothermal system involves the allocation of generation among the hydroelectric and thermal plants so as to minimize the total operation costs of thermal plants while satisfying the various constraints on the hydraulic and power system network. Various methods have been used to solve this economic dispatch problem. These include but not limited to goal programming, neural networks and genetic algorithm. In this paper, a heuristic method called the Particle Swarm Optimization (PSO) has been developed and used to solve a hydrothermal economic dispatch problem, having multi-reservoirs cascaded hydro plants and thermal plants. The developed algorithm has been tested on a 6 unit IEEE 30-bus network and has been found to be robust, and quick, converging at just over a hundred iterations. Hydrothermal power was found to be much cheaper compared to thermal power as cost dropped from Rs/Hr. 36, when using thermal power to just Rs/Hr for a power demand of 700 MW. This signifies a drop of over a thousand percent, and hence the need to use hydropower to supply the base load as it s much cheaper and efficient. xi

13 CHAPTER 1 INTRODUCTION 1.1 Hydrothermal Economic Dispatch What is Economic Dispatch? Economic dispatch is the operation of generation facilities to produce energy at the lowest cost to reliably serve customers while recognizing any operational limits of generation and transmission facilities. Economic dispatch is part of multilevel problems in power generation since different power plants, with different units have different characteristics which give different generating costs at any load at any point in time [2,21]. The main aim of economic load dispatch is to reduce the cost. However, for optimal system operation other important factors may come into play namely, reliability, the system s security, emissions (in the case of fossil-fuel plants), and optimal water discharge (for hydro generation). Ideally this multilevel problem, up to where we have the solution of economic load dispatch can be summarized as shown in Figure 1.1 below What is Hydrothermal Economic Dispatch? Hydrothermal economic dispatch is where we have both hydro and thermal units being used in the generation of power. Planning or scheduling has to be done to determine the amount of hydro and thermal power needed to satisfy a given load [15, 20]. A hydrothermal system is a more complex one as compared to an all thermal system since it is dynamic as the water levels keep changing, whereas the thermal problem is a static one. The main constraints in the hydrothermal system may include ;the time coupling effect of the water flow in an earlier time interval that affects the discharge capability at a later period of time, the time varying system long demand, the cascade nature of the hydraulic network, the varying hourly reservoir inflows, the physical limitations on the reservoir storage and turbine flow rate and loading limits of both the thermal and hydro plants. 1

14 Load Flow Casting (What load do we need to supply?) Number of Power Plants Available (Each power plant has several units) Maintenance (Units on maintenance are not available for power generation) Unit Commitment (Which units should be on and which should be off?) Economic Dispatch (How do we effectively and cheaply produce the energy?) Fig 1. 1 The Multilevel Optimal System Operation What is Particle Swarm Optimization (P.S.O)? Particle swarm optimization (PSO) is a population based stochastic optimization technique developed by Eberhart and Kennedy in 1995 [3], inspired by social behavior of bird flocking or fish schooling. Hence, it can also be described as swarm intelligence algorithm 2

15 inspired by social dynamics and an emergent behavior that arises in socially organized colonies. In this context, the population is called the swarm and the individuals are called the particles. The particles change their positions by flying around in a multidimensional search space until a relatively unchanged position has been encountered or the maximum number of complications has been reached [22]. Eberhart and Jim Kennedy describe a swarm as an apparently disorganized collection of moving individuals that t to cluster together while each individual seems to be moving in a random direction for instance schools of fish or flocks of birds. P.S.O has the following advantages over other optimization methods: P.S.O is easier to implement as there are fewer parameters to adjust In P.S.O, each particle remembers its own previous best as well as the neighborhood s best, hence a more effective memory than Genetic Algorithm (G.A) P.S.O is more efficient in maintaining the diversity of the swarm since all the particles use the information related to the best or most successful particle in order to improve themselves whereas in G.A, worse solutions are discarded and only the good ones are saved 1.2. Survey of Earlier Work Optimization Techniques Optimization is the art of achieving the best possible solution to a problem in which there are a number of competing or conflicting parameters [10] Optimization techniques, both traditional and modern can be classified into 3 major groups: i) Conventional methods ii) Intelligence search methods iii) Non-quantity approaches to address uncertainties in objectives and constraints. 3

16 1.3. Conventional Methods From the book, Conventional Optimization Techniques by Mark S. Hillier and Fredrick S. Hillier [1], before solving an optimization model, it is important to consider the form and mathematical properties of the objective function, constraints and decision variables. For instance, the objective function might be linear or non-linear, differentiable or non-differentiable concave or convex. The decision variables might be continuous or discrete. Those differences impact how the model can be solved and hence are used to classify optimization techniques.under conventional optimization methods we have the following: i) Unconstrained optimization approaches ii) Non-linear programming (N.L.P) iii) Linear programming (L.P) iv) Quadratic programming v) Generalized reduced gradient method vi) Newton method vii) Network flow programming (N.F.P) viii) Mixed inter programming ix) Interior points (I.P) methods The review of each and every method follows in the pages below: Unconstrained optimization approaches The major unconstrained optimization methods used in power systems operations are the gradient method, linear search, Lagrange multiplier, Newton-Raphson optimization, trust region optimization, quasi-newton and conjugate gradient optimization. The term unconstrained means that no restriction is placed on the range of x. It is important to note that unconstrained approaches form the basis of constrained algorithms Linear programming (L.P) This is a widely used method of solving power system operation problems such as optimal power flow, reactive power optimization or even security constrained economic dispatch, as practical operations show that this method generally meets the requirements of engineering precision. Its main advantages are: Reliable Quickly identifies infeasibility Accommodates a large variety of power system operating limits. 4

17 This technique basically linearizes the non-linear power system optimization problem so that the objective function and the constraints of power system operation have linear forms Non-linear programming (N.L.P) Power systems operation problems are non-linear; hence this method can easily solve such problems. NLP based methods have higher accuracy than LP methods Quadratic programming This is a special form of non-linear programming where the objective function of the model is quadratic and the constraints are in linear form. A good example is the generator cost function, which is generally quadratic Newton s method This is also called the second-order method as it requires the computation of the partial second order derivatives of the power flow equations. It is favored for its quadratic convergence properties Interior points (I.P) methods This is generally used to solve linear programming problems as it is faster and perhaps better than the simplex algorithm used in L.P Mixed integer programming This is mostly used to solve O.P.F problems, M.I.P methods mostly used are the recursive mixed integer programming technique using an approximation method and the branch & bound (B&B) method. This method is extremely demanding of computer resources Network flow programming This is a special linear programming method that is faster and has simpler calculations. It is mainly used to solve simplified O.P.F problems such as security constrained, economic dispatch and multi-area systems economic dispatch. 5

18 1.4. Intelligent Search Methods Artificial intelligence is the intelligence shown by machines or software. According to Jacques Ferber, in his book Multi-Agent System, an introduction to Distributed Artificial Intelligence [5], an intelligent agent can be a physical or virtual entity that can act, perceive its environment in a partial way and communicate with others. It is autonomous and has skills to achieve its goals and tencies. It is a Multi-Agent System (MAS) that contains an environment, objects and agents, relations between all the entities, a set of operation that can be performed by the entities and the changes due to these actions. In optimization techniques, intelligence search methods can be classified as follows: i) Neural network (NN) ii) Evolutionary algorithms (EAS) iii) Tabu Search (TS) iv) Particle swarm optimization v) Ant colony vi) Bee Colony Neural network (NN) This method was first to solve linear programming problems thou it has been exted to solve non-linear programming problems in recent times. This method is different from the traditional methods of optimization as it does change the solution of an optimization problem into an equilibrium point (state) of non-linear dynamic system and changes the optimal criterion into energy functions for dynamic systems. This approach is mainly used to solve the classic economic dispatch, multi area systems economic dispatch and the reactive power optimization. Neural networks are modeled on the mechanism of the brain and hence the networks have the ability to learn from examples and their tolerance to noise and damage is considerable Evolutionary algorithms Natural evolution is a population based optimization problem. These algorithms are based on the mechanics of natural selection such as mutation, recombination, reproduction, crossover or even selection and hence no need to differentiate cost functions and constraints. Some of the methods that belong to this category include; evolutionary programming (EP); evolutionary strategy (ES) and genetic algorithm (GA). 6

19 Genetic algorithm (GA) This technique first appeared in the 1950s and early 1960s while biologists were explicitly seeking the model of a natural evolution and hence we can say that this is a family of computation models, inspired by evolution. These algorithms encode a potential solution to a specific problem on a simple chromosome lie data structure. This method utilizes the operators of selection, crossover and mutation. Ideally, it combines survival of the fittest among string structures with a structured; yet random information exchange. In every generation, a new set of artificially developed string is produced using elements of the fittest of the old; an occasional new element is experimented with for enhancement. Then, a starting population is built with random genes values and it evolves through several generations in which selection, crossover and mutation are repeated until a satisfactory value has been found or the maximum number of interactions has been reached. The algorithm then identifies the individuals with optimizing fitness values while those with lower fitness are discarded. One limitation, however is that GA cannot ensure constant optimization response times; therefore limiting its application in real time applications. 7

20 The above G.A process can be summarized as follows: Starting Of a Population with Random Genes Evolving of the Population through Several Generations (selection, mutation) Choosing Values with Optimal Levels of Fitness. Discarding Values with Less Levels of Fitness Fig 1. 2: The Genetic Algorithm Process Tabu Search (T.S) This is an interactive search algorithm characterized by use of flexible memory and mainly used for solving combinatorial optimization problems. It eliminates the local mining and is able to search areas beyond local minima. This method works by the means of an evaluation function; that chooses the highest evaluation situation at each interaction. It then selects the more that produces the most improvement or the least deterioration in the objective function. A tabu list is then employed to store the above information to be used for classification in later interactions. To avoid recycling, a forbidding strategy is used to control and update the tabu list to avoid previously visited paths, hence the need for a flexible memory. 8

21 Ant colony optimization (A.C.O) As the name suggests, this optimization method was inspired by the behavior of real ants. The A.C.O algorithm forms part of the swarm intelligence algorithms with other being particle swarm optimization and swarm robotics. According to Marco Dorigo and Glanni Di Caro in their scholarly paper [6], the ant colony optimization meta-heuristic, a colony of ants is able to succeed in its task to find the shortest path between the next and the food by depositing a chemical substance trail, called pheromone on the ground as they move. This pheromone can then be observed by other ants and this motivates them to follow the path with the highest probability hence, in this technique, simple agents called artificial ants are created and mediated by an artificial pheromone trail. This pheromone trail serves as distributed, numerical information which ants use to probabilistically (by use of probability) construct solutions to the problem. By this way the best solution has the highest levels of pheromone and hence a much higher probability to be chosen Particle Swarm Optimization (P.S.O) This is a swarm intelligence algorithm inspired by social dynamics and an emergent behavior that arises in socially organized colonies. In this context, the population is called the swarm and the individuals are called the particles. The particles change their positions by flying around in a multidimensional search space until a relatively unchanged position has been encountered or the maximum number of complications has been reached. Particle Swarm Optimization is reviewed in detail in Chapter Simulated Annealing (S.A) Annealing is the physical process of heating up a solid and then cooling it down slowly until it crystallizes. At high temperatures, the atoms have higher energies which decrease as the temperature is reduced. At minimum energy, a crystal ball with regular structure is obtained. In this method, if the cooling is done very quickly, then widespread defects and irregularities are seen as opposed to slow cooling. We can then deduce and say that: The state of the solid represents feasible solutions to the optimization problem, and the energies of the state correspond to the values of the objective function. 9

22 The minimum energy state represents the optimal solution to the problem while rapid cooling represents the local optimization Non-Quantity Approaches This is the application of fully set theory. This theory accounts for uncertainties in information and goals related to multiple and usually conflicting objectives in power system optimization. The satisfactory parameters (fully sets) for objectives and constraints represent the degree of closeness to the optimum and the degree of enforcement of constraints respectively Problem Statement The main aim of this project is to understand optimal power system and come up with an effective, robust and reliable Particle Swarm Optimization method (PSO) which will be used to solve the Hydrothermal Economic Load Dispatch Problem. Since the Optimization Technique to be used is the Particle Swarm Algorithm, PSO operation and the operators involved are to be thoroughly understood. This knowledge will be used to write a software program in MATLAB programming software package to solve the Hydrothermal Economic Load Dispatch Problem. The Matlab programme written is then to be tested on standard IEEE buses. Hence, in summary, the objectives can be stated as follows: To obtain an optimal solution to the Hydrothermal Economic Load Dispatch Problem, having satisfied all the constraints. To develop a Particle Swarm Algorithm to be used in obtaining the optimal solution to the Hydrothermal Economic Load Dispatch and test it on standard IEEE buses. 10

23 1.7 Organization of the Report. The project report has been organized into the following chapters: Chapter 2 gives review of the Hydrothermal Economic Load Dispatch and Particle Swarm Optimization method. Here, the PSO concept is explained in detail. Chapter 3 formulates the Hydrothermal Economic load dispatch problem.formulation of PSO and its implementation is also discussed. Chapter 4 documents the simulation results obtained from programming in MATLAB, for one line diagram of IEEE 30 bus test network. Comparison of an all thermal system and a hydrothermal system is also done and the results analyzed. Chapter 5 contains the conclusions and the recommations for further work. 11

24 CHAPTER 2 Literature Review 2.1 Literature Review of Hydrothermal Economic Load Dispatch The Economic Load dispatch problem is also known as load scheduling. The optimal scheduling of generation in a hydrothermal system involves the allocation of generation among the hydroelectric and thermal plants so as to minimize the total operation costs of thermal plants while satisfying the various constraints on the hydraulic and power system network[11,20]. A hydrothermal system is usually more complex than the economic operation of an all thermal system. This is due to: The natural differences in watersheds. The differences in man-made storage and release elements used to control the water flows. The different types of natural man-made constraints imposed on the operation of hydro systems. The coordination of the operation of hydroelectric plants involves the scheduling of water release where this can be grouped into two; Long range scheduling This is for hydro schemes with a capacity for impounding water over several seasons. Typically, this ranges between one day to one week. Short term scheduling- This involves the hour by hour scheduling of all generations on a hydrothermal system to achieve minimum production cost (minimum consumption of fuel) for the given period. The below constraints also exist in a hydrothermal system and they up making it even more complex [15, 17, 20]. The time coupling effect of the hydro sub problem, where the water flow in an earlier time Intervals affects the discharge capability at a later period of time. The time varying system long demand. The cascade nature of the hydraulic network. The varying hourly reservoir inflows. The physical limitations on the reservoir storage and turbine flow rate and loading limits of both thermal and hydro plants. 12

25 Further constraints could be deping on the particular requirements of a given power system, such as the need to satisfy activities including, flood control, irrigation, fishing, or water supply. [11] 2.2 Classical Economic Dispatch This is the determination of the power output of each generating unit under the constraint conditions of the system load demands so as to minimize the operating costs of the power system; where the line security constraints are neglected. The fundamental of the economic dispatch problem (ECD) is the set of input output characteristics [8] Input Output Characteristics of Generator Units. We will briefly analyze the input-output characteristics of thermal units and then combine both hydro and thermal to obtain hydrothermal systems. For thermal units, the input-output characteristic is called the generating unit fuel consumption function or simply operating cost function. The unit of the operating cost function is the BTU (British thermal unit) per hr. or MBtu/hr fuel. {1Btu= w}. In addition to the fuel consumption costs, there are other additional cots like labor, maintenance cost, fuel transportation cost that are represented by a constant C. The thermal unit system generally consists of the boiler, the steam turbine and the generator; as shown below [8]. 13

26 Boiler Generator Q in Q in Turbine Pump Condenser W P Q out Fig 2. 1 : A Thermal Unit. For the: Boiler-input = fuel Output = volume of steam Turbine-generator unit- input = volume of steam Output = electrical power Hence, the input-output characteristics of the whole generating unit system can be obtained by directly combining the above; and it is a convex curve shown below. 14

27 Fig 2. 2 Input - output characteristic of a thermal generating unit.. From the curve above, the power output P G, is limited by: PG max PG PG min (2.1) The maximum value is normally determined by the design capacity. Generally, the input-output characteristic equation of the generating unit is non-linear and is given by: F i (P i ) =ap i 2 +bp i +c (2.2) Where the constant C, is the equivalent to the fuel consumption of the generating unit operation without power output Calculation of input-output parameters of a thermal system. This can be done by using of the following methods: Based on the experiments of the generating unit efficiency. Based on historic records of the generating unit operation. Based on the design data of the generating unit provided. In practical however, we can easily obtain the fuel statistic data and power output static data (FK, PK) and then use the least squares method to get the line of best fit [16]. 15

28 Let (FK, PK) be obtained from the statistical data where k=1, 2 n and the fuel curve will be a quadratic function. To determine the co-efficiency a, b and c, we compute the error for each data pair [FK, PK] FK = (Ap 2 K + BpK + c) FK (2.3) According to the method /principle of least squares: = ( FK=1) 2 = n K (Ap 2 K+BpK + c FK ) 2 (2.4) We then differentiate with respect to a, b, c respectively; j = n K=1 =2P 2 K {Ap 2 K + BpK + c FK} =0 (2.4.1) a j = n K=1= 2PK (Ap 2 K+BpK+c-FK) = 0 (2.4.2) b j = n K=1 2(Ap 2 K+BpK + c FK) = 0 (2.4.3) c Hence from (i-iii) we get: ( n K=1 P 2 K) a + ( n K=1 PK) b + n c = n K=1 FK (2.5) This is the main equation from here you just multiply with PK to obtain the below equations: ( n K=1 P 2 K) a + ( n K=1 PK) b + n c = n K=1 FK ( n K=1 P 3 K) a + n K=1 (P 2 K) b + ( n K=1 PK) c = n K=1 FK PK (2.6.2) ( n K=1 P 4 K) a + ( n K=1 P 3 K) b + ( n K=1 P 2 K) c = n K=1 FK P 2 K (2.6.3) Hence, coefficient a, b, and c can be obtained by solving equations ( ). 16

29 2.3. Hydrothermal System Economic Dispatch Operation of a system having both hydro and thermal plants is far more complex as hydro plants have negligible operating cost, but are required to operate under constraints of water available for hydro generation in a given period of time. Hence, this is a problem of dynamic optimization. This problem of minimizing the operating cost of a hydrothermal system can be viewed as one of minimizing the fuel cost of thermal plants under the constraint of water availability (storage and inflow) for hydro generation over a given period of operation.[12] A fundamental hydrothermal system is shown in the figure below: J (inflow) X PGT PGH Thermal plant Hydro plant PD Fig 2. 3 :A Fundamental Hydrothermal System. q (Discharge) System Constraints Generally there are two types of constraints [17]. i) Equality constraints. ii) Inequality constraints 17

30 This non-linear constrained hydrothermal scheduling optimization problem is subjected to a variety of constraints deping upon practical implications like the varying system load demand or the time coupling effect of hydro subsystem as discussed below[11]: Power balance constraints (Demand Constraints) This constraint is based on the principle of equilibrium between the total active power generation from the hydro and thermal plants and the total system demand plus the system losses in each time interval of scheduling. PH(t) + PT(t) =PD(t) +PL(t) (2.7) Where; PT(t) = thermal generation of the i th unit. PH(t) = hydro generation in the i th unit. PL(t) =transmission losses Thermal Generator Constraints The operating limit of equivalent thermal generator has a lower and upper bound so that it lies in between these bounds. PT(t) MAX > PT(t) > PT(t)MIN (2.8) Hydro Generator Constraints The operating limit of hydro plant must also lie in between its upper and lower bounds. PH(t) MAX > PH(t) > PH(t)MIN (2.9) Hydraulic Network Constraints The hydraulic operational constraints comprise the water balance (Continuity) equations for each hydro unit (System) as well as the bounds on reservoir storage and release targets. These bounds are determined by the physical reservoir and plant limitations as well as the multipurpose requirements of the hydro system. These constraints include: Reservoir Capacity Constraints The Water Discharge Constraints Reservoir conditions Water Continuity Equation Constraint Power Generation Characteristics 18

31 2.5 Particle Swarm Optimization Particle swarm optimization is one of the most recent developments in the category of combinatorial metaheuristic optimizations. This method has been developed under the scope of artificial life where PSO is inspired by the natural phenomenon of fish schooling or bird flocking. PSO is basically based on the fact that in quest of reaching the optimum solution in a multi-dimensional space, a population of particles is created whose present coordinate determines the cost function to be minimized. After each iteration the new velocity and hence the new position of each particle is updated on the basis of a summated influence of each particle s present velocity, distance of the particle from its own best performance, achieved so far during the search process and the distance of the particle from the leading particle, that is the particle which at present is globally the best particle producing till now the best performance i.e. minimum of the cost function achieved so far.[11] The set of neighbor connections between all of the particles forms the swarms topology or sociometry [13,16] and affects the swarm s exploitation and exploration behavior [13]. There have been two basic topologies used in the literature: Ring Topology Star Topology (global neighborhood) PSO Terminologies Particle (X): This is a candidate solution represented by an m-dimensional vector, where m is the number of optimized parameters. Swarm: According to R.C Eberhart and Y. Shi, a swarm is an apparently disorganized population of moving particles that t to cluster together towards a common optimum while each particle seems to be moving in a random direction[18]. Personal best (Pbest): The personal best position associated with i th particle is the best position that the particle has visited yielding the highest fitness value for that particle. Global best (Gbest): This is the best position associated with i th particle that any particle in the swarm has visited yielding the highest fitness value for that particle. This represents the best fitness of all the particles of a swarm at any point of time. 19

32 The optimization process uses a number of particles constituting a swarm that moves around a pre-defined search space looking for the best solution. Each particle is treated as a point in the D- dimensional space in which the particle adjusts its flying according to its own flying experience as well as the flying experience of other neighbouring particles of the swarm. Each particle keeps track of its coordinates in the pre-defined space which are associated with the best solution (fitness) that it has achieved so far. This value is called Pbest. Another best value that is tracked by the PSO is the best value obtained so far by any particle in the whole swarm. This value is called Gbest. The concept consists of changing the velocity of each particle toward its Pbest and the Gbest position at the of every iteration. Each particle tries to modify its current position and velocity according to the distance between its current position and Pbest, and the distance between its current position and Gbest PSO Parameter Selection Swarm Size. Swarm size or population size is the number of particles n in the swarm. A big swarm generates larger parts of the search space to be covered per iteration. A large number of particle may reduce the number of iterations need to obtain a good optimization result. In contrast, huge amounts of particles increase the computational complexity per iteration, and more time consuming Velocity components This is an important parameter in PSO and typically the only one adjusted. It is important since it clamps particles velocities on each dimension and determines fineness with which regions are searched [16]. It is also important to note that: Large values of Vmax could result in particles moving past optimal solutions, Small values could result in insufficient exploration of the search space. This lack of a control mechanism for the velocity resulted in low efficiency whereby PSO located the area of the optimum faster than techniques, but once in the region of the optimum, it could not adjust its velocity step size to continue the search at a finer grain. This has since been rectified by incorporating an inertia weight [19]. 20

33 Inertia Weight This helps in controlling the PSO convergence behavior. It controls the impact of the previous history of velocities on the current one.[16] Suitable selection of inertia weight provides a balance between global and local explorations, thus requiring less iteration on an average to find a sufficiently optimal solution. From experiments conducted by Shi and Eberhart, W decreases linearly from about 0.9 to 0.4 quite often during a run, and hence the following weighing function is used. W=W max - {W max -W min }*Iter (2.10) Where; Iter max Wmax is the initial weight. Wmin is the final weight. Iter max is the maximum iteration number. Iter is the current iteration number. The inertia constant can be either implemented as a fixed value or can be dynamically changing. Essentially, this parameter controls the exploration of the search space, therefore an initially higher value (typically 0.9) allows the particles to move freely in order to find the global optimum neighborhood fast. Once the optimal region is found, the value of the inertia weight can be decreased (usually to 0.4) in order to narrow the search, shifting from an exploratory mode to an exploitative mode. However, one main disadvantage of the inertia method is that once the inertia weight is decreased, the swarm loses its ability to search new areas because it is not able to recover its exploration mode. This led to the use of the constriction factor to alternatively limit velocity Constriction Factor Constriction factor controls the magnitude of the velocities in a way similar to Vmax parameter, resulting in a variant of PSO, different with the one for inertia weight. This method was developed by Clerc and Kennedy. Ideally, in general, the constriction factor improves the convergence of the particle over time by damping the oscillations once the particle is focused on the best point in an optimal region. The main disadvantage, however, of this method is that the particles may follow wider cycles and may not converge when the individual best performance is far from the neighborhood s best performance. 21

34 CHAPTER 3 Solution of Hydrothermal Economic Dispatch Using P.S.O 3.1 Formulation of Hydrothermal Dispatch Problem For a certain period of operation T (which can be one year, one month or one day, deping upon the requirement), it is assumed that: (i) (ii) Storage of hydro reservoir at the reservoir at the beginning and the are specified. Water inflow to reservoir (after accounting for irrigation use) and load demand on the system are known as functions of time with complete certainty (deterministic case). The main aim here is to achieve minimum production cost (or minimum consumption fuel) for the given time period. Let PT; F (PY) be the power output and input-output characteristics of a hydro-electric plant. Then, the hydrothermal system economic dispatch problem can be expressed as: T 0 Min F = F[PT (t) ].. (Objective Function) (3.1) S.t; PH(t) + PT(t) - P (t) = 0 (3.1.1) T 0 W[PH(t) dt] W =0 (3.1.2) Equation 3.1 is the objective function for the hydrothermal system economic discharge. The integration sign, from 0 to T covers the hydro part of the dispatch problem as it enables us to divide the operating period into time stages, either hourly or an interval of one s own choice.this may result in short-term hydrothermal dispatch if the planning task is done for just a day or long term if the planning is done over a longer period of time. 22

35 As the source for hydropower is the natural water resources, the operational cost of hydroelectric plants is insignificant. Thus, the objective of minimizing the operational cost of a hydrothermal system essentially reduces to minimize the fuel cost of thermal plants, which is also covered in Equation 3.1(denoted as F[PT (t) ]) where F denotes the cost of fuel of the thermal plants. T 0 Hence, equation 3.1, Min F = F[PT (t) ], fully covers for both the hydro and thermal part of the hydrothermal economic dispatch problem. 3.2 Economic Load Dispatch with Network losses The active power transmission losses may amount to 20 to 30% of the total load demand, ideally, the exact power flow equations should be used to obtain the active power transmission losses in the system, however, and the electric power system engineer may use OPF for expressing the losses in terms of power generations only. One common practice for including the effect of transmission losses is to express the total transmission loss as a quadratic function of the generator power outputs in one of the following forms according to Daniel S. Kirschen,2004[24]. Simple form: P L N N i1 j1 p B i ij p j (3.2) P L Kron s loss formula: N N p B p N i ij j i1 j1 j1 B0 p B (3.3) j j 00 For hydrothermal economic dispatch, being a relatively new problem, one has to calculate the generator losses using Kron s loss formula.the B coefficients used in this paper were adapted from an International Journal, Improved Particle Swarm Optimization Algorithm for Hydrothermal Generation Scheduling [20] and were obtained using Kron s loss formula. 23

36 A review of the formula is given below: P L Kron s loss formula: N N p B p N i ij j i1 j1 j1 B0 p B (3.3.1) j j 00 Bij are called the loss coefficients, which are assumed to be constant for a base range of loads, and reasonable accuracy is expected when actual operating conditions are close to the base case conditions used to compute the coefficients. The economic dispatch problem is to minimize the overall generation cost, C, which is a function of plant output, constrained by: The generation equals the total load demand plus transmission Losses, Each plant output is within the upper and lower generation limits inequality constraints. Mathematically: N i1 N 2 a i bi pi ci pi F : C C (3.3.2) g h total i i1 N : pi PD P (3.3.3) L i1 : (min) i i(max) pi p p i 1,..., N (3.3.4) The resulting optimization equation becomes: L C p i total p P i(max) D P L N i1 p i N i1 i(max) N i1 ( p i(min) i(max) ( p p ) i i p : i(max) 0 pi pi(min) : i(min) i(min) 0 ) (3.3.5) 24

37 25 The effect of transmission losses introduces a penalty factor that deps on the location of the plant. The minimum cost is obtained when the incremental cost of each plant multiplied by its penalty factor is the same for all plants. The incremental transmission loss is obtained from Kron s loss formula as, N j oi j ij i L B p B p P 1 2 (3.3.6) By setting the fuel cost equal to 1 $/MBTU can be rewritten as: i i i i i p c b dp dc 2 Further substitution equation of equations (3.3.6) and (3.3.7) when generator limits are not constrained yields: N j oi j ij i i i i L i i B p B p c b p P dp dc (3.3.8) Rearranging the above equation as: N i j j i oi j ij i ii i b B p B p B c (3.3.9) And exting equation (3.3.9) for all plants results in the following linear equations (in matrix form), N on o N NN N N N N N b B b B b B p p p B c B B B B c B B B B c (3.3.10) To find the optimal dispatch for in estimated value of λ (1),the simultaneous linear equation given by (25) is solved. In MATLAB, the command P=E \ D is used. (3.3.7)

38 3.3 Formulation of P.S.O Let X and V denote a particle position and its corresponding velocity in a search space, respectively. Hence, the i th particle is represented as: Xi=Xi1, Xi2, Xi3, Xi4 Xid, in the d dimensional space. The best previous known positions of the i th particle recorded are presented as P best1, P best2, P best3, and the best particle among all the particles is represented by G bestd. Hence, the modified velocity and the position of each particle can be calculated by using the current velocity and the distance from P bestd to Gbestd as shown in the formulas below: V n+1 id= {v n id +cr 1 (p n id x n id) +cr 2 n id (p n gd x n id) } (3.4) x n+1 id = x n id + v n+1 id (3.5) Equations (3.4) and (3.5) are the initial version of PSO where no actual mechanism for controlling the velocity of a particle, maximum value Vmax was imposed. This lack of a control mechanism for the velocity resulted in low efficiency for PSO[19] and the problem was addressed by incorporating a weight for the previous velocity of the particle. Thus in the largest versions of PSO, equations (3.6) and (3.7) below are now used. V n+1 id= k {wv n id +c 1 r 1 (p n id x n id) +c 2 r 2 2 id (p n gd x n id) (3.6) x n+1 id = x n id + v n+1 id (3.7) 26

39 Where: w = inertia weight c1, c2 = two positive constants; cognitive and social parameter respectively; K = constriction factor which is used, alternatively to w to limit velocity. Hence, equation (3.6) has three important parts and can be analyzed as follows: The first component is referred to as inertia,.it models the tency of the particle to continue in the same direction it has been traveling. This component can be scaled by a constant as in the modified versions of PSO. The second component is a linear attraction towards the best position ever found by the given particle, whose corresponding fitness value is called the particle s best (Pbest) scaled by a random weight (c 1 r 1 ).This component is referred to as memory, or remembrance. The third component of the velocity update equation is a linear attraction towards the best position found by any particle, whose corresponding fitness value is called global best(g best ) scaled by another random weight (c 2 r 2 ).This component is referred to as cooperation, social knowledge, or shared information. 3.4 PSO Algorithm The step by step procedure of PSO algorithm is given as follows [16]: Initialize a population of particles as; Pi = (Pi1, Pi2, Pi3. Pi N). N is number of generating units. Population is initialized with random values and velocities within the d-dimensional search space. 1) Initialize the maximum allowable velocity magnitude of any particle Vmax. 2) Evaluate the fitness of each particle and assign the particle's position to P-best position and fitness to P-best fitness. Identify the best among the P-best as G-best and store the fitness value of G-best. 3) Change the velocity and position of the particle according to equations (3.6) and (3.7), respectively. 4) For each particle, evaluate the fitness, if all decisions variable are within the search ranges. 27

40 5) Compare the particle s fitness evaluation with its previous P-best. If the current value is better than the previous P-best, then set the P-best value equal to the current value and the P-best location equal to the current location in the d-dimensional search space. 6) Compare the best current fitness evaluation with the population G-best. If the current value is better than the population G-best, then reset the G-best to the current best position and the fitness value to current fitness value. 7) Repeat steps (2-5) until a stopping criterion, such as sufficiently good G-best fitness or a maximum number of iterations/function evaluations is met. 28

41 Next Iteration 3.5 Flow Chart START Initialize PSO parameters Iter=0 Compute fitness function f Iteration iter=iter+1 Measureme nt data Calculate fitness for each N If pbest(i)>fitnesspbest(i-1) Pbest(i)=pbest(i) Y Pbest (i)=pbest(i-1) Compute gbesst If gbest(i)>gbest(i-1) Y Gbest(i)=gbest(i--1) N Gbest(i)=gbest(i) N Update velocity and particle position Check stopping criteria Y Gbest is the optimal END 29

42 CHAPTER Bus Test Network The one line diagram of an IEEE-30 bus system is shown in Fig. 4.I. The generator cost coefficients, data are provided in Table 1. The B-loss coefficients matrix of the system is given in Table 1.2. Fig 4. 1 : 30 Bus IEEE test network with six generating units. Comparison was done between a whole thermal system and a hydrothermal system, both being 30 bus and having six generating units. For the hydrothermal system, the coefficients were obtained from a respected journal [20], the results having been experimentally obtained. 30