Electrical Power and Energy Systems 30 (2008) 486–490

Contents lists available at ScienceDirect

Electrical Power and Energy Systems

journal homepage: www.elsevier .com/locate / i jepes

Optimum cost of generation for maximum loadability limit of power systemusing hybrid particle swarm optimization

A. Shunmugalatha a,*, S. Mary Raja Slochanal b,1

a Electrical and Electronics Engineering Department, K.L.N College of Engineering, Pottapalayam 630 611, Tamil Nadu, Indiab Electrical and Electronics Engineering Department, Thiagarajar College of Engineering, Madurai 625 015, Tamil Nadu, India

a r t i c l e i n f o

Article history:Received 10 October 2007Received in revised form 9 April 2008Accepted 14 April 2008

Keywords:Hybrid particle swarm optimization,Maximum loadability limitVoltage stability

0142-0615/$ - see front matter � 2008 Elsevier Ltd. Adoi:10.1016/j.ijepes.2008.04.001

* Corresponding author. Tel.: +91 452 2698971; faxE-mail addresses: surya_subramanian@yahoo.

smrseee@tce.edu (S.M.R. Slochanal).1 Tel.: +91 452 2482240.

a b s t r a c t

To estimate voltage stability, maximum loadability limit (MLL) is one approach. MLL is the marginbetween the operating point of the system and the maximum loading point. The optimum cost of gener-ation for MLL of power system can be formulated as an optimization problem, which consists of two stepsnamely computing MLL and the optimum cost of generation for MLL. This paper utilizes the hybrid par-ticle swarm optimization, which incorporates the breeding and subpopulation process in genetic algo-rithm into particle swarm optimization. The implementations of HPSO to test systems show that itconverges to better solution much faster.

� 2008 Elsevier Ltd. All rights reserved.

1. Introduction

The problem of voltage stability is one of the main concerns inthe operation of power system. Maximum loadability limit is themargin between the operating point of the system and the maxi-mum loading point. The maximum loadability limit problem hasbeen formulated as a non-linear optimization problem with a mix-ture of discrete and continuous variables. The main objective ofeconomic load dispatch is to minimize the fuel cost while satisfy-ing the load demand. Various mathematical techniques to solvemaximum loadability limit can be categorized as (i) continuationpower flow method (CPF); (ii) successive quadratic programming(SQP); (iii) interior point method (IP); (iv) repetitive power flowsolution.

If the system is already near the maximum loading point, thecontinuation power flow technique [1] may face some convergenceproblems. The SQP [2] approach uses the second order derivativesto improve the convergence rate. These methods become too slowas the number of control variables becomes very large. Interiorpoint methods [2] are computationally efficient. However, if thestep size is not chosen properly, the sub-linear problem may havea solution that is infeasible in the original non-linear domain.Another technique is the repetitive power flow solution [3] byincreasing the load on the system in some direction in steps andsolving the load flow at each step until the load flow solution

ll rights reserved.

: +91 452 2698280.com (A. Shunmugalatha),

diverges. The divergence of the load flow [4] does not representthe maximum loading point. In general, conventional optimizationmethods are not able to locate global optimum, can only lead to alocal optimum and sometimes result in divergence.

Recently, evolutionary techniques have been developed to solveMLL problem. The premature convergence of genetic algorithm(GA) [5] degrades its performance and reduces its search capabil-ity, which leads to a higher probability towards obtaining a localoptimum. The particle swarm optimization (PSO) technique [6]can generate high quality solutions within short calculation timeand have more global searching ability at the beginning of therun and a local search near the end of the run. Chau [7] has appliedPSO technique for Algal bloom prediction, training algorithm forANN in stage prediction of Shing Mun river and rainfall–runoff cor-relation. A hybrid particle swarm optimization (HPSO) [8] addsbreeding and subpopulation process of GA to PSO. It can jump fromthe current searching point into the effective area directly by thebreeding and subpopulation process. Therefore, HPSO has the po-tential to reach a better optimum solution than the standardPSO. The MLL of power system and the optimum cost of generation[9] for modified IEEE 30 bus, IEEE 57 bus and IEEE 118 bus systemsare evaluated using HPSO. The results indicate that the HPSO iscapable of undertaking a global search with a faster convergencerate and the feature of robust computation.

2. Mathematical formulation

The first objective is to maximize the active power load appliedto the transmission network. The second objective is to find the

Nomenclature

hij voltage angle difference between buses i and j (rad)Bij transfer susceptance between bus i and j (p.u.)Gij transfer conductance between bus i and j (p.u.)f total real power demand of the systemNo set of number of total buses excluding slack busNB set of number of total busesND set of number of load busesNG set of number of generator busesNi set of number of buses adjacent to bus i, including

bus iNT set of number of transformer branchesNlim

V set of number of buses violating voltage limitsPDi real power demand at bus i (p.u.)PDi,0 base case real power demand at bus i (p.u.)Ki load increment factor at bus i in %PL,k real power flow of line k

k number of transmission linesPGi real power generation at bus i (p.u.)PS real power at slack bus (p.u)QDi reactive power demand at bus i (p.u.)QGi reactive power generation at bus i (p.u.)Tk tap position of transformer kVi voltage magnitude of bus i (p.u.)VPQ voltage vectors of PQ buses (p.u.)VPV voltage vectors of PV buses (p.u.)VS voltage magnitude of slack bus. (p.u.)Pd max maximum loadability limit (p.u.)Fcj (Pj) fuel cost function of unit jPj min minimum real power output of unit jPj max maximum real power output of unit jPloss real power loss of the system (p.u.)Qloss reactive power loss of the system (p.u.)

A. Shunmugalatha, S.M.R. Slochanal / Electrical Power and Energy Systems 30 (2008) 486–490 487

optimum cost of generation for the maximum loadability limit.Objective I:

Maximize f ð1Þ

where f is the loading factor, which represents the increase in thesystem load from base case without violating the voltage limit.The load at all the load buses are increased in steps from base caseto maximum loading point until the load flow solution diverges,whose load model is given as below:

PDi ¼ PDi;0ð1þ KiÞ ð2Þ

The maximization of the above function is subjected to number ofconstraints,

0 ¼ PGi � PDi � ViPj2Ni

VjðGij cos hij þ Bij sin hijÞi 2 No

0 ¼ QGi � QDi � ViPj2Ni

VjðGij sin hij þ Bij cos hijÞi 2 Npq

9>=>;

ð3Þ

Vmini 6 Vi 6 Vmax

i i 2 NB

Tmink 6 Tk 6 Tmax

k k 2 NT

PL;k 6 PmaxL;k for k ¼ 1;2; . . . ; L

9>=>;

ð4Þ

In the maximum loadability problem, the generator bus voltages(Vpv and Vs) and the tap position of transformer (T) are control vari-ables that are self-constrained. Voltages of PQ buses (Vpq) and realpower flow of line are constrained by adding them as penalty termto the objective function.

The above problem is generalized as

F1 ¼ f þXi2N

kviðVi � V limi Þ

2 þXk2L

kL;kðPL;k � PlimL;k Þ

2 ð5Þ

where kvi and kL,k are the penalty factors. V limi and Plim

L;k are defined as

V limi ¼ Vmax

i for Vi > Vmaxi

V limi ¼ Vmin

i for Vi < Vmini

PlimL;k ¼ Pmax

L;k for PL;k > PmaxL;k

9>>=>>;

ð6Þ

Objective II:

Minimize F2 ¼Xn

j¼1

FcjðPjÞ ð7Þ

where Fcj(Pj) is the fuel cost function of unit j and Pj is the real powergenerated by the unit j, subject to power balance constraints,

Pd max ¼Xn

j¼1

Pj � PL ð8Þ

where Pd max is the maximum loadability limit and PL is the trans-mission loss. The generator capacity constraint is given by

Pj min 6 Pj 6 Pj max for j ¼ 1;2;3; . . . ;n ð9Þ

where Pj min and Pj max are the minimum and maximum real poweroutput of generating unit j. The fuel cost function of the generatingunit j is given by

FcjðPjÞ ¼ aiP2j þ biPj þ ci ð10Þ

where ai, bi and ci are the fuel cost coefficients of generating unit j.

3. Hybrid particle swarm optimization

3.1. PSO algorithm

In PSO, each single solution is called as ‘‘particle” in the searchspace. All particles have fitness values and have velocities, whichdirect the flying of the particles towards the optimal solution byupdating its velocity and positions in each iterations.

3.2. Hybrid PSO

In PSO, if a particle’s current position coincides with the globalbest position and if their previous velocities are very close to zero,then all the particles will stop moving, which may lead to a prema-ture convergence of the algorithm known as stagnation. To avoidthis problem, HPSO incorporates the breeding and subpopulationprocess used in GA into PSO, which allows the search to escapefrom local optima and to search in different zones of the searchspace. The breeding and subpopulation process is employed bythe following equations:

child1 ¼ x1 � parent1 þ ð1� x1Þ � parent2 ð11Þchild2 ¼ x1 � parent2 þ ð1� x1Þ � parent1 ð12Þ

where x1 is a random value between 0 and 1.

childðV1Þ ¼parentðV1Þ þ parentðV2Þ

jV1 þ V2jjV1j ð13Þ

childðV2Þ ¼parentðV1Þ þ parentðV2Þ

jV1 þ V2jjV2j ð14Þ

Parent 1 is the best particle 1.Parent 2 is the best particle 2.

V1 is the velocity of parent 1.V2 is the velocity of parent 2.

Table 2Comparison of MLL for modified IEEE 30 bus system

Algorithm Pd max

(p.u.)Percentage rise of MLL(%)

Convergence Iteration number/time (s)

PSO 2.601001 5.7489 37/61.3615HPSO 2.603455 5.8487 24/39.8020

Table 5Optimum cost of generation for MLL of modified IEEE 30 bus system

Generators Pg [Pd max (MW) kiteration method]

Pg [Pd max

(MW) GA]Pg [Pd max (MW)HPSO]

1 56.4131 67.2532 57.692 71.615 63.0303 72.846913 26.052 24.7312 27.377522 60.3432 52.0968 53.191523 25.1305 14.2815 24.604527 25.1305 38.9052 24.6351Total generation

cost ($/h)868.8696 862.2010 851.00

Computation time(s)

1.03 0.875 0.203

Table 3Comparison of MLL for IEEE 57 bus system

Algorithm Pd max

(p.u.)Percentage rise ofMLL (%)

Convergence Iteration number/time (s)

PSO 14.039 2.035 39/67.2340HPSO 14.062 2.2021 29/58.1046

Table 4Comparison of MLL for IEEE 118 bus system

Algorithm Pd max

(p.u.)Percentage rise ofMLL (%)

Convergence iteration number/time (s)

PSO 56.443 2.3519 41/69.9055HPSO 56.445 2.3555 30/51.1500

488 A. Shunmugalatha, S.M.R. Slochanal / Electrical Power and Energy Systems 30 (2008) 486–490

3.2.1. Pseudocode for HPSO algorithm

begincreate and initializewhile (stop condition is false)beginevaluationupdate velocity and positionmovebreeding and subpopulationendend

The HPSO, like the original PSO algorithm was originally pro-posed for continuous problems and extended to discrete optimiza-tion problems as well.

4. Algorithm for optimum cost of generation for MLL usingHPSO

The objective function is to maximize the load and to minimizethe cost of generation using HPSO. Load is assumed as the particleto be optimized. Either maximum number of iterations or the max-imum value of load without violating the voltage constraints is setas a stopping criterion.

Following is the HPSO algorithm for optimum cost of generationfor MLL of power system:

Step 1: Input parameters of system and algorithm and specifythe lower and upper boundaries of each variable.

Step 2: Generate n number of particles, i.e., the parameters to beoptimized.

Step 3: Evaluate the fitness value of each particle based on theNewton–Raphson power flow analysis.

Step 4: Update the time counter t = t + 1.Step 5: Execute PSO operator on the particles.Step 6: Update the velocity of each particle in PSO.Step 7: Perform the breeding and subpopulation process to

replace the worst particles, i.e., particles with low fitnessvalue according to the Eqs. (11) and (12). Breeding is per-formed between two best parents.

Step 8: If one of the stopping criteria is satisfied, then go to Step9. Otherwise go to Step 4.

Step 9: Output the particle with the maximum fitness value inthe last generation.

Step 10: For the maximum loadability limit obtained in Step 9,obtain the optimum cost of generation with the objectivefunction (Eq. (7)) subjected to the constraints (Eqs. (8)and (9)) using HPSO.

Table 1Parameter values for PSO and HPSO

Parameter IEEE 30 bus IEEE 57 bu

GA PSO HPSO GA

No. of variables 24 24 24 50Population size 50 50 50 50No. of iterations 100 100 100 100C1 – 2 2 –C2 – 2 2 –W – 0.3–0.95 0.3–0.95 –Crossover probability 0.9 – 0.05 0.9Mutation probability 0.003 – 0.1 0.003

5. Numerical results

From the literature [1,2,6], it is understood that HPSO to MLLoutperforms the techniques namely CPF, IP and PSO. To verifythe effectiveness and efficiency of the HPSO method, it has beenapplied to modified IEEE 30 bus, IEEE 57 bus and IEEE 118 bus sys-tems [10] using MATLAB 7.0 programming language and the pro-gram is run on a Pentium-IV, 2.6 GHZ processor to evaluateoptimum cost of generation for MLL of Power System. The initial

s IEEE 118 bus

PSO HPSO GA PSO HPSO

50 50 64 64 6450 50 50 50 50

100 100 100 100 1002 2 – 2 22 2 – 2 20.3–0.95 0.3–0.95 – 0.3–0.95 0.3–0.95– 0.05 0.9 – 0.05– 0.1 0.003 – 0.1

Table 6Optimum cost of generation for MLL of IEEE 57 bus system

Generators Pg [Pd max (MW) kiteration method]

Pg [Pd max

(MW) GA]Pg[Pd max

(MW) HPSO]

1 150.7722 226.2989 151.89392 114.2741 58.26 100.00003 46.7873 64.0469 41.58456 114.2741 77.2239 92.26568 526.3578 547.8495 550.00009 114.2741 85.7283 100.000012 362.6013 349.4819 370.4560Total generation cost ($/h) 48543.84 48377.00 47626.00Computation time (s) 1.562 1.532 0.219

Table 7Optimum cost of generation for MLL of IEEE 118 bus system

Generators Pg [Pd max (MW) k iterationmethod]

Pg [Pd max

(MW) GA]Pg [Pd max (MW)HPSO]

1 41.5593 25.8065 47.2954 41.5593 98.7292 61.80306 41.5593 92.1799 37.42188 41.5593 9.4819 70.772810 468.7022 409.1398 342.871912 88.5326 59.3157 80.737615 41.5593 57.6735 80.360518 41.5593 99.8045 58.012719 41.5593 79.8631 50.309924 41.5593 52.4927 46.808825 229.1433 280.8992 113.652626 327.0500 264.6686 301.643527 41.5593 91.0068 20.118731 7.2909 4.7067 14.113532 41.5593 58.1623 64.252834 41.5593 2.2483 31.286236 41.5593 53.9589 87.184240 41.5593 30.0098 70.128642 41.5593 42.0332 57.429746 19.7896 35.3627 17.332949 212.4781 249.0244 164.026554 49.9949 46.8739 92.431855 41.5593 33.9198 66.976756 41.5593 24.5357 47.819659 161.4418 223.3431 178.739161 166.6495 56.4223 132.181862 41.5593 28.5435 52.427265 407.2505 222.7019 391.809366 408.2913 431.8827 492.000069 537.8622 595.8323 569.723070 41.5593 15.1515 47.401472 41.5593 19.6481 51.494473 41.5593 90.8113 10.000074 41.5593 10.6549 43.276276 41.5593 68.9150 74.675777 41.5593 9.6774 28.611480 496.8229 372.2581 363.935385 41.5593 5.5718 35.739287 4.1662 16.7742 9.641689 632.2252 521.0929 465.916990 41.5593 92.4731 50.955991 41.5593 27.5660 51.070292 41.5593 43.6950 41.305899 41.5593 11.7302 11.661100 262.4732 244.6452 100.4175103 41.6624 19.8436 26.8338104 41.5593 83.1867 71.1620105 41.5593 69.5992 70.6862107 41.5593 18.8661 42.4079110 41.5593 9.7752 27.2280111 37.4961 10.9013 54.0164112 41.5593 99.8045 27.5900113 41.5593 8.1134 0.0000116 41.5593 11.3392 96.6016Total cost ($/h) 197462.11 186530.00 186470.00Computation

time (s)2.187 1.95 1.094

A. Shunmugalatha, S.M.R. Slochanal / Electrical Power and Energy Systems 30 (2008) 486–490 489

load, total generations and power losses for modified IEEE 30 bussystem are given as follows: Pload = 1.8920 p.u.; Qload = 1.0720 p.u.;P

PG ¼ 1:9164 p:u:;P

QG ¼ 1:0041 p:u:; Ploss = 0.0244 p.u; Qloss =0.0899 p.u.

Maximum loadability limit for modified IEEE 30-bus system byrepetitive power flow solution is 2.4596 p.u.

The initial load, total generations and power losses for IEEE 57bus system are given as follows:

Pload ¼ 12:5 p:u:; Q load ¼ 3:364 p:u:;X

PG ¼ 12:7866 p:u:;XQG ¼ 3:2108 p:u:; Ploss ¼ 0:27864 p:u:; Q loss ¼ 1:2167 p:u:

Maximum loadability limit for IEEE 57-bus system by repetitivepower flow solution is 13.759 p.u.

The initial load, total generations and power losses for IEEE 118bus system are given as follows:

Pload ¼ 42:42 p:u:; Q load ¼ 14:38 p:u:;X

PG ¼ 43:7486 p:u:;XQG ¼ 7:9568 p:u:; Ploss ¼ 1:32863 p:u:; Q loss ¼ 7:8379 p:u:

Maximum loadability limit for IEEE 118-bus system by repetitivepower flow solution is 55.146 p.u.

The adopted parameters in the algorithms are given in Table 1.The role of inertia weight W results in a trade off between globaland local exploration capabilities. C1 and C2 are the learning factorsand they play a major role in obtaining a global optimum solution.The crossover probability is always high enough to have goodconvergence characteristics. The mutation probability should betoo low for good optimal solution. HPSO moves the lowly evalu-ated agents to the current effective area directly using the selection

0 10 20 30 40 50 60 70 80 90 1002.46

2.48

2.5

2.52

2.54

2.56

2.58

2.6

2.62

ITERATIONS

Max

imum

Loa

dabi

lity

Lim

it (p

.u)

PSO

HPSO

Fig. 1. Convergence characteristics of MLL for modified IEEE 30 bus system.

Fig. 2. Convergence characteristics of MLL for IEEE 57 bus system.

0 10 20 30 40 50 60 70 80 90 100

5500

5550

5600

5650

ITERATIONS

Max

imum

Loa

dabi

lity

Lim

it (M

W)

P S O

HP S O

Fig. 3. Convergence characteristics of MLL for IEEE 118 bus system.

490 A. Shunmugalatha, S.M.R. Slochanal / Electrical Power and Energy Systems 30 (2008) 486–490

method and concentrated search in the effective area is realized.For comparison purposes, MLL and optimum cost of generationof modified IEEE 30 bus, IEEE 57 bus and IEEE118 bus systemsare tabulated in Tables 2–7, respectively. From Tables 2–4, we ob-serve that MLL by HPSO is greater than PSO with less convergencetime. The convergence characteristics for PSO and HPSO for modi-fied IEEE 30 bus, IEEE 57 bus and IEEE 118 bus systems are given inFigs. 1–3, respectively. HPSO converges nearly 1.5416 times,1.1571 times and 1.366676 times faster than PSO for modified IEEE30 bus system, IEEE 57 bus system and IEEE 118 bus system,respectively. From Tables 5–7, it is observed that the optimum costof generation by HPSO is less than PSO with less computation time.The computation time by HPSO method is 5.0739 times, 7.1324times and 1.999 times faster than k iteration method and 4.3103times, 6.9954 times and 1.7824 times faster than GA for modifiedIEEE 30 bus system, IEEE 57 bus system and IEEE 118 bus system,respectively. On the whole, it is clear that HPSO converges at betteroptimum solution with less execution time.

6. Conclusion

The algorithm named hybrid particle swarm optimizationincorporating breeding and subpopulation process of genetic algo-

rithm into particle swarm optimization is applied to estimate theoptimum cost of generation. From the results obtained, it is con-cluded that this algorithm is an efficient way of reducing the com-putational effort required to compute the maximum loadabilitylimit and also reduces the cost of generation for the same. It isobvious from the simulation study that this approach is simple,easy to implement and converges at a faster rate and can be usedto other optimization problems as well. Maximum loadability limitdoes not differ much when considering both the voltage and lineflow limit constraints. It is expected that this HPSO model can alsobe applied in the other areas of power system stability studies likereactive power allocation, voltage control, etc.

Acknowledgements

The authors are grateful to the Principal and the Managementof Thiagarajar College of Engineering, Madurai and K.L.N Collegeof Engineering, Pottapalayam for providing all facilities for theresearch work.

References

[1] Ajjarapu V, Christy C. The continuation power flow: a tool for steady statevoltage stability analysis. IEEE Trans Power Syst 1992;7(1):416–23.

[2] Irisarri GD, Wang X, Tong J, Mokhtari S. Maximum loadability of powersystems using interior point nonlinear optimization method. IEEE Trans PowerSyst 1997;12(1):162–72.

[3] Chiang HD, Jean-Jumeau R. A more efficient formulation for computation of themaximum loading points in electric power systems. IEEE Trans Power Syst1995;10(2):635–46.

[4] sato H. Computation of power system loadability limits. IEEE Trans Power Syst2002;4(2):1707–11.

[5] Abido MA. Optimal power flow using particle swarm optimization. ElectricPower Energy Syst 2002;24:563–71.

[6] Settles Matthew. An Introduction to Particle SwarmOptimization. Moscow: University of Idaho; 2005. November, pp. 1–8.

[7] Chau KW. A split-step particle swarm optimization algorithm in river stageforecasting. J Hydrol 2007;346(3–4):131–5.

[8] Esmin AAA, Lambert-Torres G, Zambroni de souza AC. A hybrid particle swarmoptimization applied to loss power minimization. IEEE Trans Power Syst2005;20(2):859–66.

[9] Gaing Zwe-Lee. Particle swam optimization to solving the economic dispatchconsidering the generator constraints. IEEE Trans Power Syst 2003;18(3):1187–95.

[10] Zimmerman R, Gan D. MATPOWER: A Matlab Power System SimulationPackage, Ithaca. NY: Cornell University Press; 1997.