Formulation of Loss minimization Problem Using Genetic Algorithm and Line-Flow-based Equations

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1 Formuation of Loss minimization Probem Using Genetic Agorithm and Line-Fow-based Equations Sharanya Jaganathan, Student Member, IEEE, Arun Sekar, Senior Member, IEEE, and Wenzhong Gao, Senior member, IEEE Abstract Optimization probems in the steady state anaysis of power systems aim at minimizing or maximizing some objective function. Traditiona methods use mathematica programming techniques to obtain the optimum soution. Artificia Inteigence (AI) methods have been shown to exhibit greater fexibiity in soving the optimization probem. Genetic Agorithm (GA) technique is used in the paper with a new formuation of system equations based on ine fow variabes. MATLAB GA toobox is appied to sove the equations. An exampe system demonstrates the effectiveness of the formuation. Future extensions are indicated. Index Terms Bus incidence matrix, Genetic Agorithm (GA), Matab GA toobox, Line fow based equation, Loss minimization, Power system, Optimization. I. NOMENCLATURE The nomencature used in this paper are isted and defined as foows: i and j are bus numbers is ine or branch number p, are active and reactive power fows at the receiving q end of ine P gi, Qgi, PLi, QLi are active and reactive generator power and oad power at bus i n, are active and reactive power osses in ine k m S ( R X ) V j V i is the square of the votage magnitude at bus i R, are the resistance and reactance of the ine X Z R X A is the bus incidence matrix A. Sekar is with the Department of Eectrica & Computer Engineering, Tennessee Tech University, Cookevie, TN 3855 (e-mai: arunsekar@tntech.edu). S. Jaganathan and W. Gao are with Department of Eectrica & Computer Engineering, Center for Energy Systems Research, Tennessee Tech University, Cookevie, TN 3855 (emai: sjaganath@tntech.edu, wgao@tntech.edu). C is the oop incidence matrix R and X are diagona matrices of ine resistance and reactance p and q are ine fow vectors II. INTRODUCTION ptimization probems in the steady state anaysis of Opower systems aim at minimizing or maximizing some objective function []. The most common objectives are to obtain the minimum vaue of oss and/or production cost. Various mathematica programming techniques such as inear programming, gradient search, mixed integer programming etc., have been investigated in the soution of such probems []. Deveopment of Artificia Inteigence (AI) techniques augmented by the impact of powerfu computer technoogica toos is finding increasing appications in power system optimization during the ast decade [3]. The various advantages offered by the non-agorithmic AI techniques are a great attraction from practica operationa viewpoint. This paper presents the resuts of the appication of Genetic Agorithm (GA) methodoogy to oss minimization. Optima power fow (OPF) formuation forms the basis for many techniques reported in the iterature. The important objective of OPF is to achieve the maximum utiization of the resources subject to constraints that arise due to eectrica, operationa, environmenta, economic, and other reasons. Some of the common objective functions of the OPF formuation are oss minimization, tota fue cost of therma pants, emission minimization and reactive power depoyment. OPF agorithms reported in the iterature foow the traditiona use of bus votage/phase ange variabe formuation foowed by the iterative soution techniques such as Gauss-Seide, Newton-Raphson, and Fast decouped variations [4]. The bus votage/phase ange based formuations pose some difficuties in the incusion of ine fow imits. Initia estimates for the phase ange are difficut to assume with fu certainty [5], [6]. Line-fow based formuation (LBF) of the steady state power fow equations is shown to be advantageous in directy handing the ine fows and bus votage magnitudes in power fow anaysis [4].LBF formuation forms the basis for the deveopment of a GA soution technique for oss minimization in this paper. Line fow constraints are directy incuded and since no bus phase anges are necessary, initiaization is easy to achieve. The resuts obtained using the GA toobox of Authorized icensed use imited to: Universidad Pabo de Oavide. Downoaded on March 3, at 9:8:43 EDT from IEEE Xpore. Restrictions appy.

2 Matab on the we-tested -ine, 6-bus power system from Wood and Woenberg provide evidence of the potentia of this approach [7]. Some genera principes of the GA technique are first described in the next section. LFB equations are presented in the next section using the six bus exampe. Loss minimization probem soution using GA is described in the foowing sections and resuts are attached in the ater sections. Concusion section describes the advantages and future direction of the proposed technique. III. GENETIC ALGORITHM GA derives from the principes of genetics and evoution. Genetic agorithms are heuristic agorithms based on the mechanism of natura seection [8]. It aows a popuation composed of many individuas to evove under specified seection rues to a state that maximizes the fitness [9]. It introduces the principe of evoution and genetics to search for possibe soution to a given probem. The main principe of evoution used in GA is surviva of the fittest meaning the best soution survives whie the bad ones are discarded. Unike the other methods GA has the abiity to hande any type of objective functions, variabes and constraints. GA, one of the evoutionary methods is very comprehensive. The method offers not one idea soution but a set of appicabe near optima soutions at reasonabe amount of computation time []. The advantages of GA can be summarized as:. Optimizes with continuous or discrete variabe.. Simutaneousy searches from a wide samping of the cost surface. 3. Deas with a arge number of variabes. 4. Provides a ist of optimum variabes, not just a singe soution. 5. Optimizes variabes with extremey compex cost surfaces. 6. May encode the variabes so that the optimization is done with the encoded variabes and 7. Works with numericay generated data, experimenta data, or anaytica functions. A. GA Soution Procedure The genetic agorithm is a method for soving both constrained and unconstrained optimization probems that is based on natura seection, the process that drives bioogica evoution. The genetic agorithm repeatedy modifies a popuation of individua soutions. Some of the commony used terminoogies in GA are fitness function- which we want to minimize and popuation- an array of individuas. The agorithm starts by creating random initia popuation. If the initia popuation is specified, it creates a sequence of new popuation. To create new popuation it scores each member of the current popuation by computing the fitness vaue. It then eects members, caed parents, based on their fitness. The individuas in the current popuation that have ower fitness are chosen as eite. These eite individuas are passed to the next popuation to form chidren. Chidren are produced either by making random changes to a singe parent ike mutation or by combining pair of parents i.e. crossover. The agorithm then checks for the constraint vioation. The agorithm stops when the fitness vaue converges or unti a pre-specified number of generations have been reached. A generaized procedure for GA is summarized beow.. The first step is to define the objective function and variabes.. In the second step the GA parameters are seected and the popuation is initiaized. 3. After initiaization the eite and parents are seected based on the fitness vaue. 4. Once the parents are seected either mutation or crossover is performed form offspring. 5. The offspring is then inserted into the new popuation. 6. The ast step is to check if the optimum soution has been achieved. IV. PROBLEM FORMULATION The ist of variabes in the LFB formuation consists of the square of bus votage magnitudes, and rea and reactive power fows at the receiving end of each ine. The set of bus power and reactive power baance equations together with ine votage equations form the static power equations. These are augmented by oop equations of ine phase ange drops. In the oss minimization probem the tota generation is the objective function to be minimized, since minimum generation for a given oad means minimum osses. Line fow, generation and bus votage imits are provided as feasibe ranges in the GA toobox. A. LFB Equations The three sets of LFB equations used in this paper are ine votage equations, power baance equations and oop equations. Fu detais of LFB equations are avaiabe in [4]. Reevant parts are given here using the exampe system for iustration. Line mode used in this exampe are simpe impedances. There are no tap changing transformers. The six bus exampe system with the assumed directions of power fow is shown in Fig.. In this paper the LFB equations have been written for the six bus exampe system. Line Votage Equations: For ine the votage equation is: p q V V ( pr q X ) Z () V Authorized icensed use imited to: Universidad Pabo de Oavide. Downoaded on March 3, at 9:8:43 EDT from IEEE Xpore. Restrictions appy.

3 3 m is a vector of m. Loop Equations: For oop formed by ines, and 5 equation is used: X p R q X p R q X p R q (7) For a the oops the foowing matrix equation is: C [(X p)-(r q)] (8) C matrix is given in appendix. B. GA Formuation The fowchart for this process is as expained beow. Fig.. Six Bus Exampe System. For a the ines the foowing matrix equation is used: D V - (R p X q) k () D is a variation of incidence matrix given in appendix. V is a vector of bus votages X, R, p and q are given in the nomencature k is a vector of k Active Power Baance Equations: For bus the active power baance equation is: p q p q p3 q 3 p p p 3 P g PL R R R3 V V4 V5 (3) For a the buses the foowing matrix equation is used: A p - (P g - P L ) E n (4) P g, P L are the generator active power vector and oad active power vector. n is a vector of n E is a variation of incidence matrix given in appendix. A matrix is given in appendix Reactive Power Baance Equations: For bus the reactive power baance equation is: p q p q p3 q3 q q3 Q g QL X X X3 V V4 V5 q For a the buses the foowing matrix equation is used: A q - (Q g - Q L ) - E m (6) Q g, Q L are the generator reactive power vector and oad reactive power vector. (5) Fig.. Genetic Agorithm Fowchart. The fitness function can be can be either minimizing the oss or minimizing the tota generation cost. Here the overa fitness function or objective function is to minimize the ine oss which is equivaent to the minization of tota injection. n i.e. J ( P gi P Li ) i For this probem the equaity constraint equations are the LFB equations given by (), (4) and (8) V. RESULTS The buit-in MATLAB genetic agorithm toobox was used to sove the optima power fow probem. The objective and constraint equations were written in Matab m-fie. GA parameters ike popuation size, initia range, upper and ower Authorized icensed use imited to: Universidad Pabo de Oavide. Downoaded on March 3, at 9:8:43 EDT from IEEE Xpore. Restrictions appy.

4 4 imit, seection criteria, crossover function, mutation function, stopping criteria and output function were set before running the program. The ranges for the variabes were set in the GA toobox whie for reference the imits are specified in the appendix. The program was run and the resuts are presented in form of bar graphs. The convergence pattern is shown in Fig. 3. The fitness vaue settes down to.7457 after 86 generations. The generation schedue for the minimum oss scenario is shown in Fig. 4. Fig. 5. shows the resuts of bus votages and ine fows are given in Fig. 6. Fig. 6. Bar graph for Active and Reactive Power Fow. Fig. 3.Fitness vaue after 86 generations. VI. CONCLUSION Appication of Genetic Agorithm technique to the soution of oss minimization probem using the ine fow based formuation of power fow equations shows good promise in the determination of optimum operating scenarios. The main advantages of the new formuation are the direct impementation of ine fow imits and the ease of making the initia estimates. Another feature of interest is the absence of defining a swing bus and fixing generation and votage of generator busses. There is some ambiguity in setting ine fow imits and a vaue of about 5% of the steady state stabiity imit for rea power is used in the exampe. Since ine fow orientations pay a roe in the baance equations, oop fows at generators can be avoided by defining unidirectiona range to the power fow in the ines going out of a generator bus. Future work wi investigate the effect of tap changing transformers and cost minimization. i. Six Bus Data Base MVA VII. APPENDIX TABLE I LINE DATA Fig. 4. Bar graph for Generator Active and Reactive power. From bus To bus R(pu) X (pu) Fig. 5. Bar graph for Bus Votages. TABLE II POWER FLOW LIMITS Authorized icensed use imited to: Universidad Pabo de Oavide. Downoaded on March 3, at 9:8:43 EDT from IEEE Xpore. Restrictions appy.

5 5 p imit q imit TABLE III BUS VOLTAGE LIMITS Bus number Bus type Poad (pu MW) Qoad (pu MVAR) Gen... Gen... 3 Gen... 4 Load Load Load.7.7 TABLE IV GENERATOR POWER AND BUS VOLTAGE LIMITS Pg imit Qg imit V imit ii. Agorithm Parameter Setting for GA toobox: The popuation size is 34 Crossover: Scattered Mutation: Adaptive feasibe Eite count: Initia Popuation: [ ] iii. The various incidence matrices are given for the exampe. A C D E VIII. REFERENCES [] S.-Y. Lin, Y.-C. Ho, and C.-H. Lin, An ordina optimization theorybased agorithm for soving the optima power fow probem with discrete contro variabes, IEEE Trans. Power Syst., vo. 9, no., pp , Aug. 3 [] J. A. Momoh, M. E. E-Hawary, and R. Adapa, A review of seected optima power fow iterature to 993, Part: Noninear and quadratic programming approaches, IEEE Trans. Power Syst., vo. 4, no., pp. 96-4, Feb [3] Jesús Riqueme Santos, Aicia Troncoso Lora, Antonio Gómez Expósito, Finding improved oca minima of power system optimization probems by interior point methods, IEEE Trans. Power Syst., vo. 8, no., pp , Feb. 3. [4] P. Yan. And A. Sekar, Steady-state anaysis of power system having mutipe facts devices using ine-fow-based equations, IEE Proc- Gener. Transm. Distrib., vo. 5, no., pp. 3-39, Jan. 5. [5] Mirko Todorovski and Dragosav Rajičić, A power fow method suitabe for soving OPF probems using genetic agorithm, in Proc. IEEE Region 8 EUROCON, vo., 3, pp. 5-9 Authorized icensed use imited to: Universidad Pabo de Oavide. Downoaded on March 3, at 9:8:43 EDT from IEEE Xpore. Restrictions appy.

6 6 [6] Mirko Todorovski and Dragosav Rajičić, An initiaization procedure in soving optima power fow by genetic agorithm, in IEEE Trans. Power Syst.,, vo., no., pp , May 6. [7] Aen J. Wood, and Bruce F. Woenberg, Power Generation, Operation and Contro. New York: John Wiey and Sons, 996 [8] Jason Yuryevich and Kit Po Wong, Evoutionary programming based power fow agorithm, in IEEE Trans. Power Syst., vo. 4, no. 4, pp. 45-5, Nov [9] N. Mwakabuta and A. Sekar, Study of the appication of evoutionary agorithms for the soution of capacitor depoyment probem in distribution systems in Proc. The 4th Southern Symposium on System Theory, USA. pp.78-8, March 8. [] A.G.Bakairtzis, P.N.Bikas, C.E.Zoumas, and V.Petridis, Optima power fow by enhanced genetic agorithm, IEEE Trans. Power Syst., vo. 7, no., pp. 9-36, May. Authorized icensed use imited to: Universidad Pabo de Oavide. Downoaded on March 3, at 9:8:43 EDT from IEEE Xpore. Restrictions appy.