ACEEE International Journal on Electrical and Power Engineering, Vol. 1, No. 2, July 2010         Hybrid Particle Swarm Op...
ACEEE International Journal on Electrical and Power Engineering, Vol. 1, No. 2, July 2010                    II PROBLEM FO...
ACEEE International Journal on Electrical and Power Engineering, Vol. 1, No. 2, July 2010rand ( ) 2 : Random number betwee...
ACEEE International Journal on Electrical and Power Engineering, Vol. 1, No. 2, July 2010                             N PQ...
ACEEE International Journal on Electrical and Power Engineering, Vol. 1, No. 2, July 2010Case B: Multi-objective RPD (RPD ...
ACEEE International Journal on Electrical and Power Engineering, Vol. 1, No. 2, July 2010                                 ...
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Hybrid Particle Swarm Optimization for Multi-objective Reactive Power Optimization with Voltage Stability Enhancement

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This paper presents a new hybrid particle swarm
optimization (HPSO) method for solving multi-objective real
power optimization problem. The objectives of the
optimization problem are to minimize the losses and to
maximize the voltage stability margin. The proposed method
expands the original GA and PSO to tackle the mixed –integer
non- linear optimization problem and achieves the voltage
stability enhancement with continuous and discrete control
variables such as generator terminal voltages, tap position of
transformers and reactive power sources. A comparison is made
with conventional, GA and PSO methods for the real power
losses and this method is found to be effective than other
methods. It is evaluated on the IEEE 30 and 57 bus test system,
and the simulation results show the effectiveness of this
approach for improving voltage stability of the system.

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Hybrid Particle Swarm Optimization for Multi-objective Reactive Power Optimization with Voltage Stability Enhancement

  1. 1. ACEEE International Journal on Electrical and Power Engineering, Vol. 1, No. 2, July 2010 Hybrid Particle Swarm Optimization for Multi-objective Reactive Power Optimization with Voltage Stability Enhancement P.Aruna Jeyanthy1, and Dr.D.Devaraj 2 1 N.I.C.E ,Kumarakoil/EEE Department,Kanyakumari,India Email: arunadarwin@yahoo.com 2 Kalasingam University/EEE Department, Srivillipithur,India Email: deva230@yahoo.comAbstract —This paper presents a new hybrid particle swarm It is a non- linear optimization problem and variousoptimization (HPSO) method for solving multi-objective real mathematical techniques have been adopted to solve thispower optimization problem. The objectives of the optimal reactive power dispatch problem. These include theoptimization problem are to minimize the losses and tomaximize the voltage stability margin. The proposed method gradient method [4, 5], Newton method [6] and linearexpands the original GA and PSO to tackle the mixed –integer programming [7].The gradient and Newton methods suffernon- linear optimization problem and achieves the voltage from the difficulty in handling inequality constraints. To applystability enhancement with continuous and discrete control linear programming, the input- output function is to bevariables such as generator terminal voltages, tap position of expressed as a set of linear functions, which may lead to losstransformers and reactive power sources. A comparison is made of accuracy. Recently, global optimization techniques suchwith conventional, GA and PSO methods for the real power as genetic algorithms have been proposed to solve thelosses and this method is found to be effective than other reactive power optimization problem [8-15]. Genetic algorithmmethods. It is evaluated on the IEEE 30 and 57 bus test system, is a stochastic search technique based on the mechanics ofand the simulation results show the effectiveness of thisapproach for improving voltage stability of the system. natural selection [16].In GA-based RPD problem it starts with the randomly generated population of points, improves theKeywords: Hybrid Particle Swarm Optimization (HPSO), real fitness as generation proceeds through the application ofpower loss, reactive power dispatch (RPD), Voltage stability the three operators-selection, crossover and mutation. Butconstrained reactive power dispatch (VSCRPD). in the recent research some deficiencies are identified in the GA performance. This degradation in efficiency is apparent I. INTRODUCTION in applications with highly epistatic objective functions i.e. Optimal reactive power dispatch problem is one of the where the parameters being optimized are highly correlated.difficult optimization problems in power systems. The sources In addition, the premature convergence of GA degrades itsof the reactive power are the generators, synchronous performance and reduces its search capability. In addition tocondensers, capacitors, static compensators and tap this, these algorithms are found to take more time to reachchanging transformers. The problem that has to be solved in the optimal solution. Particle swarm optimization (PSO) isa reactive power optimization is to determine the optimal one of the stochastic search techniques developed byvalues of generator bus voltage magnitudes, transformer tap Kennedy and Eberhart [17]. This technique can generate highsetting and the output of reactive power sources so as to quality solutions within shorter calculation time and stableminimize the transmission loss. In recent years, the problem convergence characteristics than other stochastic methods.of voltage stability and voltage collapse has become a major But the main problem of PSO is poor local searching abilityconcern in power system planning and operation. To enhance and cannot effectively solve the complex non-linear equationsthe voltage stability, voltage magnitudes alone will not be a needed to be accurate. Several methods to improve thereliable indicator of how far an operating point is from the performance of PSO algorithm have been proposed and somecollapse point [1]. The reactive power support and voltage of them have been applied to the reactive power and voltageproblems are intrinsically related. Hence, this paper formulates control problem in recent years [18-20]. Here a fewthe reactive power dispatch as a multi-objective optimization modifications are made in the original PSO by including theproblem with loss minimization and maximization of static mutation operator from the real coded GA. Thus the proposedvoltage stability margin (SVSM) as the objectives. Voltage algorithm identifies the optimal values of generation busstability evaluation using modal analysis [2] is used as the voltage magnitudes, transformer tap setting and the outputindicator of voltage stability enhancement. The modal of the reactive power sources so as to minimize theanalysis technique provides voltage stability critical areas transmission loss and to improve the voltage stability. Theand gives information about the best corrective/preventive effectiveness of the proposed approach is demonstratedactions for improving system stability margins. It is done by through IEEE-30and IEEE-57 bus system.evaluating the Jacobian matrix, the critical eigen values/vector[3].The least singular value of converged power flow jacobianis used an objective for the voltage stability enhancement. 16© 2010 ACEEEDOI: 01.ijepe.01.02.04
  2. 2. ACEEE International Journal on Electrical and Power Engineering, Vol. 1, No. 2, July 2010 II PROBLEM FORMULATION N o is set of numbers of total buses excluding slack bus Power systems are expected to operate economically N c is the set of numbers of possible reactive power(minimize losses) and technically (good stability).Therefore source installation busesreactive power optimization is formulated as a multi-objectivesearch which includes the technical and economic functions. N t is the set of numbers of transformer branchesA. Economic function: S l is the power flow in branch l the subscripts ‘min’ and “max” in Eq. (2-7) denote the corresponding The economic function is concerned mainly to minimize lower and upper limits respectively.the active power transmission loss and it is stated as, sincereduction in losses reduces the cost. B. Technical function: 2 2Min P = f ( x1 , x2 )  k g k (Vi  V  2ViV j cos  ij ) loss N j (1) The technical function is to minimize the bus voltage E deviation from the ideal voltage and to improve the voltageSubject to stability margin (VSM) and it is stated as PGi  PDi  Vi  V j (Gij cos  ij  Bij sin  ij ) Max (VSM=max (min|eig (jacobi)) (8) i  NB (2) where jacobi is the load flow jacobian matrix , eig (jacobi) returns all the eigen values of the Jacobian matrix,QGi  QDi  Vi  V j (Gij sin ij  Bij cos ij ) k  N min(eig(Jacobi)) is the minimum value of eig (Jacobi) , max PQ (3) ( min ( eig (Jacobi))) is to maximize the minimal eigen value in Vi min  Vi  Vi max i  NB (4) the Jacobian matrix.Tkmin  Tk  Tkmax k  NT III. PARTICLE SWARM OPTIMIZATION (PSO) (5) A. OVERVIEW:Q min  QGi  QGi Gi max i  NG PSO is a population based stochastic optimization (6) technique developed by Kennedy and Eberhart [17]. A population of particles exists in the n-Dimensional searchSl  Slmax l  Nl (7) space. Each particle has a certain amount of knowledge, andwhere f ( x1 , x 2 ) denotes the active power loss function of will move about the search space based on this knowledge. The particle has some inertia attributed to it and so it willthe system. continue to have a component of motion in the direction it isVG is the generator voltage (continuous) moving. It knows where in the search space, it will encounterTk is the transformer tap setting (integer) with the best solution. The particle will then modify its direction such that it has additional components towards itsQc is the shunt capacitor/ inductor (integer) own best position, pbest and towards the overall bestVL is the load bus voltage position, gbest. The particle updates its velocity and position with the following Equations (9) to (11)QG is the generator reactive power Vik1 W*Vik C1 *rand)1 *( pbestSik )C2 *rand)2 *(gbest Sik ) (9) ( (k  (i , j ), i  N B , J  N i , g k is the conductance of branch k. i iij is the voltage angle difference between bus I &j Wmax  W min W  Wmax  * iter (10)PGi is the injected active power at bus i itermaxPDi is the demanded active power at bus iGij is the transfer conductance between bus i and j Bij is the transfer susceptance between bus i and j Vi k 1 : Velocity of particle i at the iteration k  1 QGi is the injected reactive power at bus i Vi k : Velocity of particle i at the iteration k QDi is the demanded reactive power at bus i S ik 1 : Position of particle i at the iteration k  1 N e is the set of numbers of network branches Sik : Position of particle i at the iteration k N PQ is the set of number of PQ buses C1 : Constant weighting factor related to pbest Nb is the set of numbers of total buses C2 : Constant weighting factor related to gbest N i is the set of numbers of buses adjacent to bus i rand ( )1 : Random number between 0 and 1 (including bus i ) 17© 2010 ACEEEDOI: 01.ijepe.01.02.04
  3. 3. ACEEE International Journal on Electrical and Power Engineering, Vol. 1, No. 2, July 2010rand ( ) 2 : Random number between 0 and 1 C. HPSO Algorithm Procedure:pbesti : pbest position of particle i Step 1: Initialization of the parametersgbesti : gbest position of swarm Step 2: Randomly set the velocity and position of all the particles.Usually the constant weighting factor or the acceleration Step 3: Evaluate the fitness of the initialcoefficients C1 , C2  2 , control how far a particle moves in particles by conducting Newton-Raphson power flowa single iteration. The inertia weight’ W’ is used to control analysis results.pbest of e ach particle is set to initialthe convergence behavior of PSO. Suitable selection of the position. The initial best evaluation value among theinertia weight provides a balance between global and local particles is set to gbest.exploration and exploitation of results in lesser number of Step 4: Change the velocity and position of the particleiterations on an average to find a sufficient optimal according to the equations (9) to (11).solution. In the PSO method, there is only one population Step 5: Select the best particles come into mutationin an iteration that moves towards the global optimal point. operation according to (12).This makes PSO computationally faster and the Step 6: If the position of the particle violates the limitconvergence abilities of this method are better than the of variable, set it to the limit value.other evolutionary computation techniques such as GA. Step 7: Compute the fitness of new particles. If the fitness of each individual is better than theB. Proposed Algorithm: previous pbest; the current value is set to The main drawback of the PSO is the premature pbest value. If the best pbest is better thanconvergence. During the searching process, most particles gbest, the value is set to be gbest.contract quickly to a certain specific position. If it is a local Step 8: The algorithm repeats step 4 to step 7 until theoptimum, then it is not easy for the particles to escape from it. convergence criteria is met, usually a sufficiently goodIn addition, the performance of basic PSO is greatly affected fitness or a maximum number of iterations.by the initial population of the particles, if the initial populationis far away from the real optimum solution. A natural evolution IV .HPSO IMPLEMENTATION OF THE OPTIMALof the PSO can be achieved by incorporating methods that REACTIVE POWER DISPATCH PROBLEM:have already been tested in other evolutionary computation When applying HPSO to solve a particular optimizationtechniques. Many researchers have considered incorporating problem, two main issues are taken into consideration namely:selection, mutation and crossover as well as differentialevolution into the PSO algorithm. The main goal is to increase (i) Representation of the decision variables andthe diversity of the population by: preventing the particles (ii) Formation of the fitness functionto move too close to each other and collide, to self-adapt These issues are explained in the subsequent section.parameters such as constriction factor, acceleration constants A. Representation of the decision variablesor inertia weight. As a result, hybrid versions of PSO havebeen created and tested in different applications. In the While solving an optimization problem using HPSO, eachproposed approach, mutation which is followed in genetic individual in the population represents a candidate solution.algorithm is carried out. Mutation is one of the effective In the reactive power dispatch problem, the elements of themeasures to prevent loss of diversity in a population of solution consists of the control variables namely; Generatorsolution, which can cover a greater region of the search bus voltage (Vgi), reactive power generated by the capacitorspace.Hence in this algorithm the addition of mutation into (QCi), and transformer tap settings (tk).Generator bus voltagesPSO will expand its global search space, add variability into are represented as floating point numbers ,whereas thethe population and prevent stagnation of the search in local transformer tap position and reactive power generation ofoptima. capacitor are represented as integers. With thisThe mutation operator works by changing a particle representation the problem will look like the following:position dimension S i  delta (iter , U  S i ) : rb  1using: mutate( S i )  S  delta (iter , S  L ) : rb  0  i i B. Formation of the fitness function (12) Where iter is the current iteration number, In the optimal reactive power dispatch problem, the U is the upper limit of variable spac objective is to minimize the total real power loss while L is the lower limit of variable space satisfying the constraints (14) to (20). For each individual, rb is the randomly generated bit the equality constraints are satisfied by running Newton- delta (iter, y) return a value in the range [0: y] Raphson algorithm and the constraints on the state variablesIt provides a balance between adding variability and allowing are taken into consideration by adding penalty function tothe particles to converge. Hence in this method it reduces the objective function. With the inclusion of the penaltythe probability of getting trapped into local optima. function, the new objective function then becomes, 18© 2010 ACEEEDOI: 01.ijepe.01.02.04
  4. 4. ACEEE International Journal on Electrical and Power Engineering, Vol. 1, No. 2, July 2010 N PQ Ng Ni solution was obtained with the following parameter setting: Min F  Ploss  wEig max   VPi   QPgi   LPl (13) Population size : 30 i 1 i 1 l 1where w, KV , K q , K l are the penalty factors for the eigen wmax : 0.9value,load bus voltage limit violation, generater reactive wmin : 0.4power limits violation and line flow limit violation respectively.In the above expressions C1 :2 C2 :2 K V (Vi  Vi max ) 2 if Vi  Vi max  Maximum generations: 50VPi  K V (Vi  Vi min ) 2 if Vi  Vi min (14) Mutate rate : 0.1 0 otherwise  Figure 1 illustrates the relationship between the best fitness values against the number of generations. Figure . 1. Convergence characteristics Generally, PSO searches for a solution with maximum fitness From the figure it can be seen that the proposed algo-function value. Hence, the minimization objective function rithm converges rapidly towards the optimal solution. The optimal values of the control variables along with the mini-given in (17) is transformed into a fitness function ( f ) to be mum loss obtained are given in Table I for IEEE-30 bus sys-maximized as, tem. Corresponding to this control variable setting, it was               f  K / F (17) found that there are no limit violations in any of the state variables. To show the performance of the HPSO in solvingwhere K is a large constant. This is used to amplify (1/F), the this integer nonlinear optimization problem, it is compared tovalue of which is usually small, so that the fitness value of the well known conventional, GA &PSO techniques. But inthe chromosomes will be in a wider range. HPSO the best solution is achieved. This shows HPSO is capable of reaching better solutions and is superior compared V.SIMULATION RESULTS to other methods. This means less execution time and less In order to demonstrate the effectiveness and robustness memory requirements.of the proposed technique, minimization of real power loss TABLE Iunder two conditions, without and with voltage stability RESULTS OF PSO-RPD OPTIMAL CONTROL VARIABLESmargin (VSM) were considered. The validity of the proposedPSO algorithm technique is demonstrated on IEEE- 30andIEEE-57 bus system. The IEEE 30-bus system has 6 generatorbuses, 24 load buses and 41 transmission lines of whichfour branches are (6-9), (6-10) , (4-12) and (28-27) - are withthe tap setting transformers. The IEEE 57-bus system has 7generator buses, 50 load buses and 80 transmission lines ofwhich 17 branches are with tap setting transformers. The realpower settings are taken from [1]. The lower voltagemagnitude limits at all buses are 0.95 p.u. and the upper limitsare 1.1 for all the PV buses, 0.05 p.u. for the PQ buses and thereference bus for IEEE 30-bus system. The PSO –basedoptimal reactive power dispatch algorithm was implementedusing the MATLAB programmed and was executed on aPentium computer.Case A : RPD with loss minimization objective Here the PSO-based algorithm was applied to identify theoptimal control variables of the system .It was run withdifferent control parameter settings and the minimization 19© 2010 ACEEEDOI: 01.ijepe.01.02.04
  5. 5. ACEEE International Journal on Electrical and Power Engineering, Vol. 1, No. 2, July 2010Case B: Multi-objective RPD (RPD including voltage stability from the simulation work, it is concluded that PSO performsconstraint) better results than the conventional methods. In this case, the RPD problem was handled as a multi-objective optimization problem where both power loss and REFERENCESmaximum voltage stability margin of the system wereoptimized simultaneously. The optimal control variable [1] C.A. Canizares, A.C.Z.de Souza and V.H. Quintana,settings in this case are given in the last column of Table I. To “Comparison of performance indices for detection of proximity tomaximize the stability margin the minimum eigen value should voltage collapse,’’ vol. 11. no.3 , pp.1441-1450, Aug 1996.be increased. Here the VSM has increased to 0.2437 from [2] B.Gao ,G.K Morison P.Kundur ,’voltage stability evaluation0.2403, an improvement in the system voltage stability. 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