ACEEE Int. J. on Control System and Instrumentation, Vol. 02, No. 02, June 2011        Hybrid Particle Swarm Optimization ...
ACEEE Int. J. on Control System and Instrumentation, Vol. 02, No. 02, June 2011                    II PROBLEM FORMULATION ...
ACEEE Int. J. on Control System and Instrumentation, Vol. 02, No. 02, June 2011C1          : Constant weighting factor rel...
ACEEE Int. J. on Control System and Instrumentation, Vol. 02, No. 02, June 2011Raphson algorithm and the constraints on th...
ACEEE Int. J. on Control System and Instrumentation, Vol. 02, No. 02, June 2011Case B: Multi-objective RPD (RPD including ...
ACEEE Int. J. on Control System and Instrumentation, Vol. 02, No. 02, June 2011[10] K.Iba,’ Reactive power optimization by...
Upcoming SlideShare
Loading in …5
×

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

339
-1

Published on

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.

Published in: Technology
0 Comments
0 Likes
Statistics
Notes
  • Be the first to comment

  • Be the first to like this

No Downloads
Views
Total Views
339
On Slideshare
0
From Embeds
0
Number of Embeds
1
Actions
Shares
0
Downloads
14
Comments
0
Likes
0
Embeds 0
No embeds

No notes for slide

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

  1. 1. ACEEE Int. J. on Control System and Instrumentation, Vol. 02, No. 02, June 2011 Hybrid Particle Swarm Optimization for Multi-objective Reactive Power Optimization with Voltage Stability Enhancement 2 P.Aruna Jeyanthy1, and Dr.D.Devaraj 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 is used an objective for the voltage stability enhancement. Itoptimization (HPSO) method for solving multi-objective real is a non- linear optimization problem and various mathematicalpower optimization problem. The objectives of the techniques have been adopted to solve this optimal reactiveoptimization problem are to minimize the losses and to power dispatch problem. These include the gradient methodmaximize the voltage stability margin. The proposed method [4, 5], Newton method [6] and linear programming [7].Theexpands the original GA and PSO to tackle the mixed –integernon- linear optimization problem and achieves the voltage gradient and Newton methods suffer from the difficulty instability enhancement with continuous and discrete control handling inequality constraints. To apply linear programming,variables such as generator terminal voltages, tap position of the input- output function is to be expressed as a set of lineartransformers and reactive power sources. A comparison is made functions, which may lead to loss of accuracy. Recently, globalwith conventional, GA and PSO methods for the real power optimization techniques such as genetic algorithms have beenlosses and this method is found to be effective than other proposed to solve the reactive power optimization problemmethods. It is evaluated on the IEEE 30 and 57 bus test system, [8-15]. Genetic algorithm is a stochastic search technique basedand the simulation results show the effectiveness of this on the mechanics of natural selection [16].In GA-based RPDapproach for improving voltage stability of the system. problem it starts with the randomly generated population ofKeywords: Hybrid Particle Swarm Optimization (HPSO), real points, improves the fitness as generation proceeds throughpower loss, reactive power dispatch (RPD), Voltage stability the application of the three operators-selection, crossoverconstrained reactive power dispatch (VSCRPD). and mutation. But in the recent research some deficiencies are identified in the GA performance. This degradation in I. INTRODUCTION efficiency is apparent in applications with highly epistatic objective functions i.e. where the parameters being optimized Optimal reactive power dispatch problem is one of the are highly correlated. In addition, the premature convergencedifficult optimization problems in power systems. The sources of GA degrades its performance and reduces its searchof the reactive power are the generators, synchronous capability. In addition to this, these algorithms are found tocondensers, capacitors, static compensators and tap take more time to reach the optimal solution. Particle swarmchanging transformers. The problem that has to be solved in optimization (PSO) is one of the stochastic search techniquesa reactive power optimization is to determine the optimal developed by Kennedy and Eberhart [17]. This techniquevalues of generator bus voltage magnitudes, transformer tap can generate high quality solutions within shorter calculationsetting and the output of reactive power sources so as to time and stable convergence characteristics than otherminimize the transmission loss. In recent years, the problem stochastic methods. But the main problem of PSO is poorof voltage stability and voltage collapse has become a major local searching ability and cannot effectively solve theconcern in power system planning and operation. To enhance complex non-linear equations needed to be accurate. Severalthe voltage stability, voltage magnitudes alone will not be a methods to improve the performance of PSO algorithm havereliable indicator of how far an operating point is from the been proposed and some of them have been applied to thecollapse point [1]. The reactive power support and voltage reactive power and voltage control problem in recent yearsproblems are intrinsically related. Hence, this paper formulates [18-20]. Here a few modifications are made in the original PSOthe reactive power dispatch as a multi-objective optimization by including the mutation operator from the real coded GA.problem with loss minimization and maximization of static Thus the proposed algorithm identifies the optimal values ofvoltage stability margin (SVSM) as the objectives. Voltage generation bus voltage magnitudes, transformer tap settingstability evaluation using modal analysis [2] is used as the and the output of the reactive power sources so as to minimizeindicator of voltage stability enhancement. The modal the transmission loss and to improve the voltage stability.analysis technique provides voltage stability critical areas The effectiveness of the proposed approach is demonstratedand gives information about the best corrective/preventive through IEEE-30and IEEE-57 bus system.actions for improving system stability margins. It is done byevaluating the Jacobian matrix, the critical eigen values/vector[3].The least singular value of converged power flow jacobian 12© 2011 ACEEEDOI: 01.IJCSI.02.02.42
  2. 2. ACEEE Int. J. on Control System and Instrumentation, Vol. 02, No. 02, June 2011 II PROBLEM FORMULATION N PQ is the set of number of PQ buses Power systems are expected to operate economically N b is the set of numbers of total buses(minimize losses) and technically (good stability).Thereforereactive power optimization is formulated as a multi-objective N i is the set of numbers of buses adjacent to bus isearch which includes the technical and economic functions. (including bus i )A. Economic function: N o is set of numbers of total buses excluding slack bus The economic function is concerned mainly to minimize N c is the set of numbers of possible reactive powerthe active power transmission loss and it is stated as, since source installation busesreduction in losses reduces the cost. Nt is the set of numbers of transformer branches f ( x1 , x2 )  g (Vi 2  V j2  2ViV j cos  ij ) S l is the power flow in branch l the subscripts ‘min’Min P = loss k N E k (1) and “max” in Eq. (2-7) denote the corresponding lower and upper limits respectively.Subject to B. Technical function: The technical function is to minimize the bus voltage PGi  PDi  Vi  V j (Gij cos  ij  Bij sin  ij ) i  NB deviation from the ideal voltage and to improve the voltage stability margin (VSM) and it is stated as (2) Max (VSM=max (min|eig (jacobi)) (8)QGi  QDi  Vi  V j (Gij sin  ij  Bij cos  ij ) k  N PQ where jacobi is the load flow jacobian matrix , eig (jacobi) returns all the eigen values of the Jacobian matrix, (3) min(eig(Jacobi)) is the minimum value of eig (Jacobi) , maxVi min  Vi  Vi max i  NB (4) ( min ( eig (Jacobi))) is to maximize the minimal eigen value in the Jacobian matrix.Tkmin  Tk  Tkmax k  NT (5) III. PARTICLE SWARM OPTIMIZATION (PSO)Q min  QGi  QGi max A. Overview: Gi i  NG PSO is a population based stochastic optimization (6) technique developed by Kennedy and Eberhart [17]. ASl  Slmax l  Nl (7) population of particles exists in the n-Dimensional search space. Each particle has a certain amount of knowledge, andwhere f ( x1 , x 2 ) denotes the active po wer loss function of will move about the search space based on this knowledge.the system. The particle has some inertia attributed to it and so it willVG is the generator voltage (continuous) continue to have a component of motion in the direction it is 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 itsQc is the shunt capacitor/ inductor (integer) direction such that it has additional components towards its own best position, pbest and towards the overall bestVL is the load bus voltage position, gbest. The particle updates its velocity and positionQG is the generator reactive power with the following Equations (9) to (11)k  (i , j ), i  N B , J  N i , g k is the conductance of branch k. ij is the voltage angle difference between bus I &jPGi is the injected active power at bus iPDi is the demanded active power at bus iGij is the transfer conductance between bus i and j : Velocity of particle i at the iterationBij is the transfer susceptance between bus i and j Vi k : Velocity of particle i at the iteration k QGi is the injected reactive power at bus i S ik 1 : Position of particle i at the iteration k  1 QDi is the demanded reactive power at bus i Sik : Position of particle i at the iteration k N e is the set of numbers of network branches 13© 2011 ACEEEDOI: 01.IJCSI.02.02.42
  3. 3. ACEEE Int. J. on Control System and Instrumentation, Vol. 02, No. 02, June 2011C1 : Constant weighting factor related to pbest It provides a balance between adding variability and allowing the particles to converge. Hence in this method it reducesC2 : Constant weighting factor related to gbest the probability of getting trapped into local optima.rand ( )1 : Random number between 0 and 1 C. HPSO Algorithm Procedure:rand ( ) 2 : Random number between 0 and 1 Step 1: Initialization of the parameterspbest i : pbest position of particle i Step 2: Randomly set the velocity and positiongbest i : gbest position of swarm of all the particles. Step 3: Evaluate the fitness of the initialUsually the constant weighting factor or the acceleration particles by conducting Newton-Raphsoncoefficients C1 , C2  2 , control how far a particle moves in a power flow analysis results. pbest of e achsingle iteration. The inertia weight’ W’ is used to control the particle is set to initial position. The initialconvergence behavior of PSO. Suitable selection of the inertia best evaluation value among the particles isweight provides a balance between global and local 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 solution. according to the equations (9) to (11).In the PSO method, there is only one population in an iteration Step 5: Select the best particles come into mutationthat moves towards the global optimal point. This makes operation according to (12).PSO computationally faster and the convergence abilities of Step 6: If the position of the particle violates the limitthis method are better than the other evolutionary computation of variable, set it to the limit value.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 7optimum, then it is not easy for the particles to escape from it. until the convergence criteria is met,In addition, the performance of basic PSO is greatly affected usually a sufficiently good fitness or aby the initial population of the particles, if the initial population maximum number of iterations.is far away from the real optimum solution. A natural evolutionof the PSO can be achieved by incorporating methods that IV .HPSO IMPLEMENTATION OF THE OPTIMALhave already been tested in other evolutionary computation REACTIVE POWER DISPATCH PROBLEM:techniques. Many researchers have considered incorporatingselection, mutation and crossover as well as differential When applying HPSO to solve a particular optimizationevolution into the PSO algorithm. The main goal is to increase problem, two main issues are taken into consideration namely:the diversity of the population by: preventing the particles (i) Representation of the decision variables andto move too close to each other and collide, to self-adapt (ii) Formation of the fitness functionparameters such as constriction factor, acceleration constants These issues are explained in the subsequent section.or inertia weight. As a result, hybrid versions of PSO have A. Representation of the decision variablesbeen 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. The mutation operator works by changing a particle capacitor are represented as integers. With thisposition dimension using: representation the problem will look like the following: S i  delta (iter , U  S i ) : rb  1 0. 0. ...1. 0. 0. ...1. 3.35 2.10 ...1.50 981 970 017 925 965 000 mutate( S i )   (12)             S i  delta (iter , S i  L ) : rb  0 V1 V2 Vn t1 t2 tn Qc 1 Qc 2 QcnWhere iter is the current iteration number, B. Formation of the fitness function U is the upper limit of variable space L is the lower limit of variable space In the optimal reactive power dispatch problem, the rb is the randomly generated bit objective is to minimize the total real power loss while delta (iter, y) return a value in the range [0: y] satisfying the constraints (14) to (20). For each individual, the equality constraints are satisfied by running Newton- 14© 2011 ACEEEDOI: 01.IJCSI.02.02.42
  4. 4. ACEEE Int. J. on Control System and Instrumentation, Vol. 02, No. 02, June 2011Raphson algorithm and the constraints on the state variables Case A : RPD with loss minimization objectiveare taken into consideration by adding penalty function to Here the PSO-based algorithm was applied to identify thethe objective function. With the inclusion of the penalty optimal control variables of the system .It was run withfunction, the new objective function then becomes, different control parameter settings and the minimization 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 1 wmax : 0.9 wmin : 0.4where w, KV , K q , K l are the penalty factors for the eigen C1 :2value,load bus voltage limit violation, generater reactive C2 :2power limits violation and line flow limit violation respectively Maximum generations: 50.In the above expressions Mutate rate : 0.1 Figure 1 illustrates the relationship between the best fitness K V (Vi  Vi max ) 2 if Vi  Vi max values against the number of generations. VPi  K V (Vi  Vi min ) 2 if Vi  Vi min (14) 0 otherwise   K q (Qi  Qimax ) 2 if Qi  Qimax  QPgi   K q (Qi  Qimin ) 2 if Qi  Qimin (15)  0  otherwise  K ( S  S lmax ) 2 if S l  S lmax Figure . 1. Convergence characteristicsLPl   l l (16) From the figure it can be seen that the proposed algorithm 0 otherwise converges rapidly towards the optimal solution. The optimalGenerally, PSO searches for a solution with maximum fitness values of the control variables along with the minimum lossfunction value. Hence, the minimization objective function obtained are given in Table I for IEEE-30 bus system.given in (17) is transformed into a fitness function ( f ) to be Corresponding to this control variable setting, it was foundmaximized as, that there are no limit violations in any of the state variables. To show the performance of the HPSO in solving this integer               f  K / F (17) nonlinear optimization problem, it is compared to the wellwhere K is a large constant. This is used to amplify (1/F), the known conventional, GA &PSO techniques. But in HPSO thevalue of which is usually small, so that the fitness value of best solution is achieved. This shows HPSO is capable ofthe chromosomes will be in a wider range. reaching better solutions and is superior compared to other methods. This means less execution time and less memory V.SIMULATION RESULTS requirements. In order to demonstrate the effectiveness and robustness T ABLE Iof the proposed technique, minimization of real power loss RESULTS OF PSO-RPD OPTIMAL CONTROL VARIABLESunder two conditions, without and with voltage stabilitymargin (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. 15© 2011 ACEEEDOI: 01.IJCSI.02.02.42
  5. 5. ACEEE Int. J. on Control System and Instrumentation, Vol. 02, No. 02, June 2011Case B: Multi-objective RPD (RPD including voltage CONCLUSIONstability constraint) This paper presents a hybrid particle swarm optimization In this case, the RPD problem was handled as a multi- algorithm approach to obtain the optimum values of theobjective optimization problem where both power loss and reactive power variables including the voltage stabilitymaximum voltage stability margin of the system were constraint. The effectiveness of the proposed method foroptimized simultaneously. The optimal control variable RPD is demonstrated on IEEE-30 and IEEE-57 bus systemsettings in this case are given in the last column of Table I. To with promising results. Simulation results show that the HPSOmaximize the stability margin the minimum eigen value should based reactive power optimization is always better than thosebe increased. Here the VSM has increased to 0.2437 from obtained using conventional, GA and simple PSO methods.0.2403, an improvement in the system voltage stability. For From this multi-objective reactive power dispatch solutionIEEE-57 bus system the minimum power loss obtained is the application of HPSO leads to global search with fast25.6665 MW.The VSM has increased to 0.1568 from 0.1456. convergence rate and a feature of robust computation. HenceTo determine the voltage security of the system, contingency from the simulation work, it is concluded that PSO performsanalysis was conducted using the control variable setting better results than the conventional methods.obtained in case A and case B. The eigen valuescorresponding to the four critical contingencies are given in REFERENCESTable II. From this result it is observed that the eigen valueshas increased appreciably for all contingencies in the second [1] C.A. Canizares, A.C.Z.de Souza and V.H. Quintana, “Comparison of performance indices for detection of proximity tocase. This improvement in voltage stability was achieved voltage collapse,’’ vol. 11. no.3 , pp.1441-1450, Aug 1996.because of the additional objective included in the RPD [2] B.Gao ,G.K Morison P.Kundur ,’voltage stability evaluationproblem in the base case condition. This shows that the using modal analysis ‘ Transactions on Power Systems ,Vol 7, Noproposed algorithm has helped to improve the voltage .4 ,November 1992 [9].stability of the system. To analyze the simulation results it [3] Taciana .V. Menezes, Luiz .C.P.da silva, and Vivaldo F.da Costa,”has been compared with other optimization methods. Table Dynamic VAR sources scheduling for improving voltage stabilityIII summarizes the minimum power loss obtained by these margin,” IEEE Transactions on power systems. vol 18,no.2 ,Maymethods for the IEEE-30 bus system. 2003 [3] O.Alsac, and B. Scott, “Optimal load flow with steady state TABLE II security”, IEEE Transaction. PAS -1973, pp. 745-751. VSM UNDER CONTINGENCY STATE [4] Lee K Y ,Paru Y M , Oritz J L –A united approach to optimal real and reactive power dispatch , IEEE Transactions on power Apparatus and systems 1985: PAS-104 : 1147-1153 [5] A.Monticelli , M .V.F Pereira ,and S. Granville , “Security constrained optimal power flow with post contingency corrective rescheduling” , IEEE Transactions on Power Systems :PWRS-2, No. 1, pp.175-182.,1987. [6] Deeb N, Shahidehpur S.M, Linear reactive power optimization T ABLE III in a large power network using the decomposition approach. IEEE COMPARISON OF OPTIMAL RESULT OBTAINED BY Transactions on power system 1990: 5(2) : 428-435 DIFFERENT METHODS FOR IEEE-30 BUS SYSTEM [7] D. Devaraj, and B. Yeganarayana, “Genetic algorithm based optimal power flow for security enhancement”, IEE proc- Generation. Transmission and. Distribution; 152, 6 November 2005. [8]- Deb, K. (201): Multi – objective optimization using evolutionary algorithms 1st ed. (John Wiley & Sons, Ltd.). [9] Q. H. Wu and J. T. Ma, “Power System Optimal Reactive Power Dispatch Using Evolutionary Programming”, IEEE Trans. on Power Systems, Vol. 10, No. 3, pp. 1243-1249, August 1995. 16© 2011 ACEEEDOI: 01.IJCSI.02.02.42
  6. 6. ACEEE Int. J. on Control System and Instrumentation, Vol. 02, No. 02, June 2011[10] K.Iba,’ Reactive power optimization by genetic algorithm ,” [16] D Goldberg, “Genetic algorithms in search, optimization andIEEE Trans.power syst.vol.9,pp.685-692,May 1992 . machine learning”, Addison-Wesley,1989.[11] D.Devaraj, “Improved genetic algorithm for multi – objective [17] J. Kennedy and R. Eberhart, “Particle swarm optimization”,reactive power dispatch problem,” European Transactions on Proceedings of the IEEE International Conference on Neural Net-Electrical Power, 2007; 17; 569- 581. works, Vol. IV, pp. 1942- 1948,[12] A.J.Urdaneta, J.F.Gomez, E.Sorrentino, L.Flores, and R.Diaz, 1995.‘A hybrid genetic algorithm for optimal reactive power planning [18 ] S.Durairaj, P.S.Kannan and D.Devaraj ,”International journalbased upon successive linear programming,’’ IEEE Trans.power of emerging electric power systems,”The Berkeley Electronic Presssyst, vol.14, pp.1292-1298, Nov. 1999 ,vol 4, issue 1 ,2005,article 1082 pp.1-15.[13] Y.Liu,L.Ma, and Zhang ,” GA/SA/TS Hybrid algorithm for [19] J.G.Vlachogiannis ,K.Y.Lee ,”Contribution of generation toreactive power optimization,’ in Proc. IEEE Power Eng.Soc.Summer transmission system using parallel vector particle swarm optimi-Meeting,vol.1, pp. 245- 249,July 2000. zation”, IEEE Transactions on power systems,20(4),2005,1765-[14] L.L.Lai,J.T.Ma ,’’Application of evolutionary programming 1774.to reactive power planning –Comparison with non linear [20]H.Yoshida,K.Kawata,Y.Fukuyama,”A Particle Swarm optimi-programming approach,’’ IEEE Trans.Power syst.,vol.12,pp.198- zation for reactive power and voltage control considering voltage206,Feb.1997 security assessment,” IEEE Transactions on power[15] Kwang Y. Lee and Frank F.Yang, “Optimal Reactive Power systems,vol.15.no.4,November2000.Planning Using evolutionary Algorithms: A Comparative study forEvolutionary Strategy, Genetic Algorithm and Linear Programming”,IEEE Trans. on Power Systems,Vol. 13, No. 1, pp. 101- 108, February 1998. 17© 2011 ACEEEDOI: 01.IJCSI.02.02. 42

×