Optimal Operation of Large Power System by GA Method

Transcription

1 Journal of Emerging Trends in Engineering and Applied Sciences (JETEAS) (1): 1-7 Scholarlink Research Institute Journals, 01 (ISSN: ) jeteas.scholarlinkresearch.org Journal of Emerging Trends in Engineering and Applied Sciences (JETEAS) (1):1-7(ISSN: ) Optimal Operation of Large Power System by GA Method 1 Sangita Das Biswas and Anupama Debbarma 1 Department of Electrical Engineering, Tripura University, Tripura, India.7990 Ex-P.G. Scholar, Department of Electrical Engineering, Tripura University, Tripura, India Corresponding Author: Sangita Das Biswas Abstract The optimal operation of a power system, involves many considerations like economy of operation, system security, emission at certain fossil-fuel plants etc. All this considerations cannot be taken into account at the same time and as a result compromise has to be made for the optimal operation of the power system. In our work we shall consider the economy of operation, called the economic load dispatch. In an Interconnected power system it is necessary to find power generation scheduling of each power plant to minimize the operating cost. This means the power generations are allowed to vary within a certain limit to meet a particular load demand with minimum fuel cost. The main objectives of economic load dispatch are to minimize generation cost, while meeting the system load and the transmission loss of the system. The aim of this work is to solve the economic load dispatch problem with Genetic Algorithm keeping the power generation of each plant within security limit and meeting the demand of the power system. A MATLAB program will be developed to solve the economic load dispatch using Genetic Algorithm Keywords: crossover, economic load dispatch, fitness function, genetic algorithm, and optimal power flow INTRODUCTION The energy requirements are rising very rapidly with the development of civilization. To meet this energy requirement the sizes of electric power systems have also increased rapidly. As a result numbers of power plants are interconnected to supply the system load. With the development of integrated power systems it has become necessary to operate the plant units most economically. Economic load dispatch is used to determine optimal output of generators at lowest possible generation cost, while the system is operating within its security limit. Traditionally, Classical methods were used to solve the economic load dispatch but more recently due to growing size of power sectors, the traditional concepts are superimposed by advanced optimization techniques. Genetic Algorithm (GA) is one of the advanced optimization method to solve the optimal generation scheduling. GA, invented by Holland in the early 1970s is a stochastic global search method that mimics the metaphor of natural biological evolution. GA can solve the optimization problem keeping the outputs of generators within the security limit and also meeting the power demand of the system. The basic elements of Genetic Algorithms are reproduction, crossover and mutation which will be discussed later. The first step in GA is to code the control variable into binary strings and then the genetic operators are applied to generate better strings. So GA works with 1 the coding of the parameters and not the parameters themselves. Other methods usually deal with functions and their control variables directly. In GA the main objective is to find string with maximum fitness and if we want better fitness than the present solution, it can be achieved by simply increasing the string bit length. OPTIMAL SYSTEM OPERATION The optimal system operation, in general, involved the consideration of economy operation, system security, emissions at certain fossil-fuel plants, and optimal releases of water at hydro generation, etc. All these consideration may make for conflicting requirements and usually a compromise has to be made for optimal system operation. The main aim in the economic dispatch problem is to minimize the total cost of generating real power at various stations while satisfying the loads and the losses in transmission links. Generator Operating Cost The major component of generator operating cost is the fuel input/hr, while maintenance contributes only to a small extent. The fuel cost is meaningful in case of thermal and nuclear stations, but for hydro stations, where the energy storage is apparently free, the operating cost as such as not meaningful. For dispatching purposes, the cost is usually approximated by one or more quadratic segments.

2 So, the fuel cost curve is modified as a quadratic in the active power generation. Rs/hr The fuel cost curve may have a number of discontinuities. The discontinuities occur when the output power has to be extended by using additional boilers, steam condensers, or other equipment. Discontinuities also appear if the cost represents the operation of an entire power station, so that cost has discontinuities on paralleling of generators. Within the continuity range the incremental fuel cost may be expressed by a number of short line segments or piece-wise linearization. Economic Dispatch Problem on a Bus Bar Assuming that it is known a priori which generators are to be run to meet a particular load demand on the station. Suppose there is a station with NG generators committed and the active power load P D is given, the real power generation P gi for each generator has to be allocated so as to minimize the total cost. The optimization problem can therefore be stated as Minimize (1.1) Subject to (i) the energy balance equation (ii) and the inequality constraint P D NG is the decision variable. (1.) (1.) is real power demand. is the number of generation plants. is the lower limit of real power generation is the upper power limit of real power generation. is the operating fuel cost of the ith plant and is given by the quadratic equation Rs/hr (1.) The above constrained optimization problem is converted into an unconstraint optimization problem. Lagrange multiplier method is used in which a function is minimized (or maximized) with side conditions in the form of equality constraints. Using the method, an augmented function is defined as (1.5) A necessary condition for a function F (P gi ), subject to energy balance constraint to have a relative minimum at point P gi is the partial derivative of the Lagrange function defined by L = L(P gi, λ) with respect to each of its arguments must be zero. So, the necessary conditions for the optimization problem are And From equation (1.6), (i = 1,,, NG) (1.8) Incremental fuel cost of the ith generators (Rs/M Wh) Optimal loading of generators corresponds to the equal incremental cost point of all the generators equation (1.8), called the coordination equations numbering NG are solved simultaneously with the load demand to yield a solution for Lagrange multiplier λ and the optimal generation of NG generators. Considering the cost function given by equation (1.), the incremental cost can be defined as (1.9) Substituting the incremental cost into equation (1.8), this equation becomes (1.10) Rearranging equation (1.10) to get Substituting the value of (1.11) in equation (1.7), we get (1.1) Thus, λ can be calculated using equation (1.1) and can be calculated using equation (1.11) Now consider the effect of the generator limits given by the inequality constraint of equation (1.). If a particular generator loading Pgi reaches the limits P gi min or P gi max, its loading is held fixed at this value and the balance load is shared between the remaining generators on an equal incremental cost basis. Optimal Generation Scheduling The economic dispatch problem is defined as to minimize the total operating cost of a power system while meeting the total load plus transmission losses within the generator limits. Generally, the problem is formulated as follows:- Minimize (1.1), P gi is the real power generation and will act as decision variable. NG is the number of generation buses. a i, b i and c i are the cost co-efficients.

3 In minimizing the cost, the equality constraint (Energy balance) and inequality constraint (Energy limits) should be satisfied. Equality Constraint The equality constraints of the optimal power flow reflect the physics of the power system as well as desired voltage set points throughout the system. The physics of the power system are enforced through the power flow equations (1.1), is the load demand. is the transmission power loss. NG is the number of generation buses. is the real power generation. (1.15) is the reactive power generation and will act as decision variable. ND is no. of demand buses. It is also common for power system operations to have voltage set points for each generation. An equality constraint for each generation is added. V gi - V gi set point =0 (1.16) One of the most important, simple but approximate method of expressing transmission loss is quadratic in the injected bus real powers. The general form of the loss formula using B-Coefficients is MW (1.17), and are the real power injections at the ith and jth buses, respectively. are the loss coefficients which are constant under certain assumed condition NG is the number of generation buses. The B-coefficients are found through the Z- bus calculation technique. Inequality Conatraint The inequality constraints of the OPF reflect the limits on physical devices in the power system as well as the limits created to ensure system security.the generation capacity of each generators have maximum and minimum output powers and reactive powers and it can be expressed as:- min max P gi P gi P gi Q gi min Q gi Q gi max V gi min V gi V gi max Penalty Factor Optimization problem is converted into an unconstrained optimization problem. Lagrange multiplier method is used in which function is minimized (or maximized) with side conditions in the form of equality constraints. Using Lagrange multipliers, an augmented function is defined as (1.18) λ is the Lagrange multiplier. Necessary conditions for the optimization problem are Rearranging the above equation,, Is the incremental cost of ith (i=1,,., N G) generator (Rs/MWh) is the incremental transmission losses (1.19) Equation (1.19) is known as the exact coordination equation, and (1.0) Equation (1.19), the so-called coordination equation, numbering NG is solved simultaneously with equation (1.0) to yield a solution for Lagrange multiplier λ and the optimal generation of NG generators. By differentiating the transmission loss equation, which is:- MW With respect to loss can be obtained as,, the incremental transmission are loss coefficients. (1.1) By differentiating cost function equation (1.1); with respect to, the incremental cost be obtained as Equation (1.19) can be rewritten as Or (i=1,, NG) (1.) (1.) is called penalty factor of i th plant To obtain the solution, substitute equation (1.1) and (1.) into equation (1.19) (i=

4 Rearranging the above equation to get P gi, we have The value of P gi can be obtained (1.) If the initial values of P gi (i=1,, NG) and λ are known, the above equation can be solved iteratively until equation (1.0) is satisfied by modifying λ. This technique is known as successive approximation. GENETIC ALGORITHM IN OPTIMAL POWER FLOW Description of Genetic Algorithm Genetic Algorithm is a stochastic optimization method which starts from multiple points to obtain a solution. It combines an artificial, i.e. the Darwinian Theory of Survival of the fittest with genetic operation, abstract from nature to form a robust mechanism that is very effective at finding optimal solution to complex-real world problems. They operate on string structure which is a combination of binary digits representing a coding of the control parameters for a given problem. Many such strings are considered simultaneously, with the most fit of these structures receiving exponentially increasing opportunities to pass on genetically important material to successive generation of string structures. A more striking difference between Genetic Algorithms and most of the traditional optimization methods is that GA uses a population of points at one time in contrast to the single point approach by traditional optimization methods. This means that GA processes a number of designs at the same time. Working of Genetic Algorithm The basic operators of GA are reproduction, crossover and mutation. The first step is the coding of the control variables as string in binary numbers. In reproduction, the individuals are selected based on their fitness values relative to those of the population. In the crossover operation, two individual strings are selected at random from the mating pool and a crossover site is selected at random along the string length. The binary digits are interchanged between the two strings at the crossover site. In mutation, an occasional random alteration of binary digit is done. From the above description, a simple Genetic Algorithm can be as follows: 1. Generate randomly a population of binary string.. Calculate the fitness for each string in the population.. Create offspring strings through reproduction, crossover and mutation operations.. Evaluate the new strings and calculate the fitness for each string (chromosome). 5. If the search goal is achieved, or an allowable generation is attained, return the best chromosome as the solution; otherwise go to step three. Encoding & Decoding Decoding a binary string into an unsigned integer can play very important roles in genetic algorithm implementation. The inequality power limit constraint is performed in such a way that the individual string is normalized over the unit s operation region. The inequality constraints are handled in the manner, which efficiently reduces the searching space & thus enhances the performance of the system. Binary coded strings having 1s & 0s are used. The equivalent decimal integer of binary string λ is obtained as y j =Σ l i=1 i-1 b j i (j=1,,..l) (.1) b j i is the ith binary digit of the jth string. l is the length of the string L is the number of strings or population size. The continuous variable λ can be obtained to represent a point in the search space according to a fixed mapping rule. i.e. λ j =λ min + (λ max - λ min ) y j /( l - 1) (j= 1,,,L) (.) λ min is the minimum value of variable, λ λ max is the maximum value of variable, λ y j is the binary coded value of the string l is the length of the string L is the number of strings or population size. The number of binary digits needed to represent a continuous variation in accuracy of Δλ can be computed from the relation l log ((λ max - λ min )/ Δλ +1 ) Reproduction The first GA operator is the reproduction. The reproduction genetic algorithm operator selects good strings in a population and forms a mating pool to produce offspring. So, sometimes the operator is also called as selection operator. The commonly used reproduction operator is the proportionate reproduction operator where a string is selected for the mating pool with a probability proportional to its fitness. The various methods of selecting chromosomes for parents to crossover are: 1. Roulette-wheel selection. Boltzmann selection. Tournament selection. Rank selection 5. Steady-state selection Among these Roulette-wheel selection is mostly used.

5 Fitness Function & Parent Selection Implementation of a problem in a genetic algorithm is realized within the fitness function. Since the proposed approach uses the equal incremental cost criterion as its basis, the constraints equation can be rewritten as ε j = P D + P L j Σ NG i=1 P i j (.) Then the converging rule is when ε decreases to within a specific tolerance. In order to emphasize the best chromosomes & speed up convergence of the iteration procedure fitness is normalized into range between 0 &1. The fitness function adopted is f j =1 (1+ α ε j /P D ) (j= 1,, L) (.) α is the scaling constant. When the fitness of each chromosome is calculated, the Roulette wheel selection technique is used to select the best parents according to their fitness. Roulette Wheel Selection The commonly used reproduction operator is the proportionate reproduction operator where a string is selected for the mating pool with a probability proportional to its fitness. Therefore, the probability for selecting the ith string is P i =f i / Σ L j=1 f j (.5) L is the population size, fi is the fitness of the ith population One way to implement this selection scheme is to imagine a Roulette Wheel. The Roulette Wheel is spun L times, each time the pointer of the Roulette Wheel selects the string. As the circumference of the wheel is marked according to a string s fitness, the Roulette Wheel mechanism is expected to make fi/fav copies of the ith string in the mating pool. The average fitness of the population is obtained as F av =( Σ L i=1 f i )x 1/ L (.6) Using the fitness value fi of all strings, the probability of selecting a string Pi can be calculated. Crossover After the reproduction phase is over, the population is enriched with better individuals. Reproduction makes clones of good strings, but does create new ones. Crossover operator is applied to the mating pool with a hope that it would create a better string. Crossover is a recombination operator, which proceeds in three steps. First, the reproduction operator selects at random a pair of two individual strings for mating, then a cross-site is selected at random along the length of the string and then the position values are swapped between two strings following the cross-site. Parent 1: x 1 = { } Parent : x = { } Figure: - Crossover site selected at Mutation After crossover, the strings are subjected to mutation. Mutation of a bit involves flipping it, changing 0 to 1 and vice versa with a small mutation probability P m. The bit-wise mutation is performed bit-by-bit by flipping a coin with a probability P m. A number between 0 and1 is chosen at random. If the random number is smaller than P m then the outcome of coin flipping is true, otherwise the outcome is false. If at any bit, the outcome is true then the bit is altered, otherwise the bit is kept unchanged. The bit of the strings is independently muted, that is, the mutation of a bit does not affect the probability of mutation of other bits. Child A : New Child A: Figure: - New child produced after mutation In the above figure, the mutation is performed at the rd string and the bit 1 is changed to 0 and a new chromosome or offspring is produced. Mutation is used to prevent premature stopping of the algorithm. Economic load dispatch using Genetic Algorithm Economic load dispatch solution can be achieved with the help of Genetic Algorithm. The detailed solution methodology includes: the encoding and decoding techniques, constrained generation output calculation, the fitness function, parent selection and parameter selection. Encoding and Decoding of Incremental Cost Binary code strings having 1s and 0s are generated randomly for the incremental cost λ. Then the equivalent decimal integer of λ is obtained by using the equation.1. Calculation of Generation and Transmission loss After knowing the incremental cost the power generations can be obtained by using the equation NG k=1 A j ikp j k = C i (i=1,, NG).1 (j=1, L) A j ii=(a i + λ j B ii ) A j ik = λb ik C i = λ - b i The above equation can be represented in matrix form and after putting the values of A and C, P i.e. power generations can be calculated by Gauss Elimination method. After the calculation of generations of all the generators the transmission loss can be calculated using the equation Fitness Function and Parent Selection After the calculation of the transmissios loss P L fitness is calculated using equation no.. f j =1 (1+ α ε j /P D ) (j= 1,, L ) where ε j = P D + P L j Σ NG i=1 P i j Algorithm for GA base Economic Dispatch The step by step procedure for solving GA based Economic Dispatch:- 5

6 Journal of Emerging Trends in Engineering and Applied Sciences (JETEAS) (1):1-7(ISSN: ) Read data: cost coefficients, loss coefficients, size of population, string length, maximum and minimum value of λ, power demand, number of generators, probability of crossover, probability of mutation etc. Randomly generate the binary strings to generate the population of λ. Set the generation counter, itr=0 Increment the generation counter, itr=itr+1 and set the population counter, pop=0. Increment the population counter, pop=pop+1. Decode the binary string of λ to their decimal values using the equation.1 and.. Using the Gauss Elimination method, calculate the power generation from equation.1. Calculate the transmission loss using the equation Calculate the fitness from equation.. If (pop<size of population) GOTO step 5 and repeat the procedure. Select the parents for mating from the population by the Roulette Wheel selection. Perform the crossover for the selected parent strings. Perform the mutation. If (itr<maximum iteration) then GOTO step and repeat. Find the generation with maximum fitness. If any power generation limit is violated then GOTO step and repeat. Stop. Figure1: 6 Bus System Table I: Fuel Cost Coefficients Generator no a b C Table II: Operating generator Real Power limits P(min) (MW) RESULTS AND DISCUSSION In this work an economic load dispatch is performed in 6 bus system with six generators. The six generators are connected to bus1, bus, bus, bus, bus5 and bus6 respectively. The generator s operating costs are in $/h. The generators have maximum and minimum generation limits and the optimal generation scheduling is performed within the generation limits. The total load demand of the system is 16 MW and it is supplied by the six generators. The transmission loss coefficients are shown in Table-1 and transmission loss is calculated for all generators. Genetic Algorithm is used to search the incremental cost of the generators and genetic operators are used to find better solution. The string length is taken 16bit and the size of the population is 0. After producing several generations by GA the best solution is found in 17th iteration where the generators supply MW and the transmission loss is 15.78MW. P(max) (MW) Table III: Mvar Limits Bus No. 5 6 Min. Mvar Capacity Max. Mvar Capacity Table IV: Genetic Algorithm Parameters Serial. No 1 G A Parameters Probability of Crossover Probability of Mutation Population Size Length of Population Value The optimal generation scheduling is done with the help of a MATLAB program and the result is obtained in the 17th iteration after running the program. The result is given in table V. 6

7 Table V: Optimal Generation Scheduling Using Genetic Algorithm Gen. no Power Generated (MW) The six generators of the system have generated MW and the transmission loss is MW. The fitness of the system is and the incremental cost is 1.65 $/MWh. CONCLUSION The economic load dispatch using Genetic Algorithm is done with the help of MATLAB program. Before running the program some inputs are required which are to be given by the user. In this program the inputs are: 1. The size of population,. The length of the string,. The minimum value of incremental cost,. The maximum value of incremental cost, 5. The loss coefficient, 6. The cost coefficient, 7. Number of generators in the system, 8. The crossover probability and 9. The mutation probability. The loss coefficient, the cost coefficients, the maximum & minimum value of incremental cost and the number of generators are given for the power system for which we are doing the load dispatch. The rest inputs are given according to user s choice. The incremental cost is searched with the help of string of 0 bit length. It is already said that with increase of the string length, the accuracy will also increase. So, in our future work we can test longer string length and work on larger power system with more numbers of generators and buses. Like the optimal scheduling of active power done in this project scheduling of reactive power can also be done using Genetic Algorithm. So it can also be a part of future work. As it is already said that in this work we are considering only the economy of the system called economic load dispatch, other considerations for optimal operation of power system can also be included in future work. D. Lucknan, T. R. Blackbunn, Modified Algorithm of Load Flow Simulation for Loss Minimization in power system. D. P. Kothari and I. J. Nagrath, Modern Power System Analysis, Tata McGraw-Hill Publishing Company Limited, New Delhi. D. P. Kotheri, J. S. Dhillon, Power system Optimization, Printice-Hall of India Private Limited, New Delhi. David E. Goldberg, Genetic Algorithm in Search, Optimization and Machine Learning, Pearson Education. G. A. Vijayalakshmi Pai, S. Rajasekaran, Neural Networks, Fuzzy Logic, and Genetic Algorithm, Printice-Hall of India Private Limited, New Delhi. Hadi Saadat, Power System Analysis, Tata McGraw- Hill Publishing Company Limited, New Delhi. K. S. Pandya and S. K. Joshi, A survey of optimal power flow methods, Journal of Theoretical and Applied Information Technology. K.Vaisakh, L.R.Srinivas, Differential Evolution Approach for Optimal Power Flow Solution, Journal of Theoretical and Applied Information Technology. L. Abdelhakem-Koridak, M. Rahi, M. Younes, 007. Optimal power flow based on Hybrid Genetic Algorithm, Journal of Information Science and Engineering. : L. Slimani, M. Belkacemi, T. Bouktir, 00. A Genetic Algorithm for solving the Optimal Power flow Problem, Leonardo Journal of Sciences.: M.S. Sukhija, T. K. Nagsarkar, Power System Analysis, Oxford University Press. REFERENCES A. Allali, A. Chaker, K. Hachemi, M. Laouer, 008. New Approach of Optimal Power Flow with Genetic Algorithms, Acta Electrotechnica et Informatica Vol. 8, No., 5. A. Chokrobarti, S. Halder, Power system Analysis operation and Control, Printice-Hall of India Private Limited, New Delhi. B. H.Kim, C. Chase, R. Baldick, Y. Luo, A First Distributed Implementation of Optimal Power Flow, IEEE Transactions on Power Systems, vol. 1, No.. 7