cuckoo search algorithm for economic dispatch

lable at ScienceDirect

Energy xxx (2013) 1e10

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Energy

journal homepage: www.elsevier .com/locate/energy

Cuckoo search algorithm for economic dispatch

M. Basu*, A. Chowdhury 1

Department of Power Engineering, Jadavpur University, Kolkata 700098, India

a r t i c l e i n f o

Article history:Received 2 January 2013Received in revised form18 June 2013Accepted 2 July 2013Available online xxx

Keywords:Cuckoo search algorithmMicro gridEconomic dispatchMultiple fuelValve-point loadingProhibited operating zone

* Corresponding author. Fax: þ91 33 23357254.E-mail addresses:[email protected] (M. B

(A. Chowdhury).1 Fax: þ91 33 23357254.

0360-5442/$ e see front matter � 2013 Elsevier Ltd.http://dx.doi.org/10.1016/j.energy.2013.07.011

Please cite this article in press as: Basu M,10.1016/j.energy.2013.07.011

a b s t r a c t

This paper presents cuckoo search algorithm for solving both convex and nonconvex ED (economicdispatch) problems of fossil fuel fired generators considering transmission losses, multiple fuels, valve-point loading and prohibited operating zones. This paper also presents cuckoo search algorithm for microgrid power dispatch problem. Cuckoo search algorithm is a new meta-heuristic algorithm. It is a nature-based searching technique which is inspired from the obligate brood parasitism of some cuckoo speciesby laying their eggs in the nests of other host birds of other species. The effectiveness of the proposedalgorithm has been verified on different test systems. Compared with the other existing techniques,considering the quality of the solution obtained, the proposed algorithm seems to be a promisingalternative approach for solving the ED problems in practical power system and micro grid powerdispatch problem.

� 2013 Elsevier Ltd. All rights reserved.

1. Introduction

ED (Economic dispatch) is one of the most important optimi-zation problems in power system operation and planning. ED al-locates the load demand among the committed generators mosteconomically while satisfying the physical and operational con-straints. Since the cost of power generation in fossil fuel fired plantsis exorbitant, an optimum dispatch saves a considerable amount ofmoney. Classical methods such as lambda iteration, base pointparticipation factor, gradient method, Newton’s method andLagrange multiplier method can solve economic dispatch problemunder the assumption that the incremental cost curves of thegenerating units are monotonically increasing piecewise-linearfunctions. However, in reality, large steam turbines have a num-ber of steam admission valves which contribute non convexity inthe fuel cost function of the generating units. Classical calculus-based techniques fail to address these types of problems satisfac-torily and lead to sub optimal solutions producing huge revenueloss over time. DP (Dynamic programming) can solve ELD problemwith inherently nonlinear and discontinuous cost curves. But itsuffers from the curse of dimensionality or local optimality.

In this respect, stochastic search algorithms such as SA (simulatedannealing) [2], GA (genetic algorithm) [3,4], EP (evolutionary

asu), [email protected]

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Chowdhury A, Cuckoo search

programming) [5], ANN (artificial neural networks) [6], ACO (antcolony optimization) [7], PSO (particle swarm optimization) [8], AIS(artificial immune system) [9], DE (differential evolution) [10], BFA(bacterial foraging algorithm) [11], BBO (biogeography-based opti-mization) [12], etc., have been applied successfully to solve complexED problem without any restriction in the shape of the cost curves.

Recently, different hybridization andmodification of EP, GA, PSO,DE methods like PSO-SQP (particle swarm optimization-sequentialquadratic programming) [13], IFEP (improved fast evolutionaryprogramming) [14], IGA (improved genetic algorithm) [15], DEC-SQP(chaotic differential evolution and quadratic programming) [16],NPSO-LRS (new particle swarm optimization-local random search)[17], SOH-PSO (self organizing hierarchical- particle swarm opti-mization) [18], ICA-PSO (improved coordinated aggregation-basedparticle swarmoptimization) [19], hybrid differential evolutionwithbiogeography-based optimization [20] etc., have been proposed forsolving EDproblem in search of better quality solution.However, thehybrid methods contain many controllable parameters which maynot be properly selected.

Renewable energy can reduce power losses, replace large-scalegenerators and prevent system black out if they are placed andoperating properly in the power systems [21e24]. A micro gridrefers to a small-scale power system which consists of load, gen-erators, energy storages and a control system. However, when amain utility grid is facing with serious problems, the power can besupplied to the customers constantly in the islanding operation.

Many papers have reported studies on micro grid optimaloperation. Krishnamurthy proposed the plug-and-play approach to

algorithm for economic dispatch, Energy (2013), http://dx.doi.org/

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Delta:1_surname

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M. Basu, A. Chowdhury / Energy xxx (2013) 1e102

make it possible to conveniently expand the micro grid and scale itto meet the customer requirements, discussing the modeling andcontrol issues related to such gen-sets for their operation in a dis-tribution system that contains multiple distributed generations(DGs) including inverter-based sources [25].

CSA (Cuckoo search algorithm) is a new metaheuristic optimi-zation algorithm developed by Yang and Deb in 2009 [26]. Thisalgorithm is based on the obligate brood parasitic behavior of somecuckoo species in combination with the Le’vy flight behavior ofsome birds and fruit flies. CSA has been successfully applied inmulti-objective scheduling problem [30], reliability optimizationproblems [31] and DG allocation in distribution network [32].

In this paper, cuckoo search algorithm (CSA) is proposed forsolving both convex and non-convex ED problem and micro gridpower dispatch problem.

The convex ED problem assumes quadratic cost function alongwith system power demand and operational limit constraints. Thepractical non-convex ED problem, in addition, considers generatornonlinearities such as valve point loading effect, prohibited oper-ating zones, andmulti-fuel options. Here, four types of ED problemshave been considered. These are A) EDQCTL (ED with quadratic costfunction and transmission loss): 20 Generator system B) EDQC-POZTL (ED with quadratic cost function, prohibited operating zonesand transmission loss): 6 Generator system C) EDVPL (ED withvalve-point loading effect and without transmission loss): 40Generator system D) EDVPLMF (ED with valve-point loading effectand multi-fuel options): 10 Generator system.

The performance of the proposed cuckoo search algorithm interms of solution quality has been compared 1) with BBO [12],NPSO-LRS [17] SOH_PSO [18], and PSO [8] for a six-generator systemincluding prohibited operating zone and transmission loss; 2) withBBO [12], IGA [15], NPSO [17] and NPSO_LRS [17] for a ten-generatorsystemwith valve-point loading andmulti-fuel options; 3)withBBO[12], Lambda Iteration [6] andHopfieldmodel [6] for a 20-generatorsystem with quadratic cost function and transmission loss; and 4)with BBO [12], NPSO-LRS [17] and SOH_PSO [18] for a 40-generatorsystem including valve-point loading.

A simple micro grid test problem has also been used todemonstrate the performance of the proposed CSA. Test resultsobtained from CSA have been compared with those obtained fromdifferential evolution (DE) and particle swarm optimization (PSO).

2. Problem formulation

Problem formulation consists of two sections. Section 1 con-siders ED problem and Section 2 considers micro grid powerdispatch problem.

2.1. Economic dispatch problem

The ED may be formulated as a nonlinear constrained optimi-zation problem. Four different types of ED problems have consid-ered here.

2.1.1. EDQCTL (Economic dispatch with quadratic cost function andtransmission loss)

The objective function Ft, total cost of N committed generators,of ED problem may be written as

Ft ¼XNi¼1

FiðPiÞ ¼XNi¼1

ai þ biPi þ ciP2i (1)

where Fi(Pi) is the cost function of ith generator, and is usuallyexpressed as a quadratic polynomial; ai, bi and ci are the cost

Please cite this article in press as: Basu M, Chowdhury A, Cuckoo search10.1016/j.energy.2013.07.011

coefficients of ith generator; N is the number of committed gen-erators; Pi is the power output of ith generator. The ED problemminimizes Ft subject to the following constraints.

2.1.1.1. Real power balance constraint.

XNi¼1

Pi � PD � PL ¼ 0 (2)

The transmission loss PL may be expressed by using B-co-efficients as

PL ¼XNi¼1

XNj¼1

PiBijPj þXNi¼1

B0iPi þ B00 (3)

2.1.1.2. Real power generation capacity constraints. The powergenerated by each generator should be within its lower limitPmini and upper limit Pmax

i , so that

Pmini � Pi � Pmax

i i˛N (4)

2.1.2. Economic dispatch with quadratic cost function, prohibitedoperating zones and transmission loss (EDQCPOZTL)

The objective function Ft of this type of ED problem is same asmentioned in EDQCTL. Here, the objective function is to be mini-mized subject to the constraints (2), (4) and constraint arises fromprohibited operating zone.

2.1.2.1. Prohibited operating zone. The prohibited operating zonesare the range of power output of a generator where the operationcauses undue vibration of the turbine shaft bearing caused byopening or closing of the steam valve. This undue vibration mightcause damage to the shaft and bearings. Normally operation isavoided in such regions. The feasible operating zones of unit can bedescribed as follows:

Pmini � Pi � Pli;1

Pui;j�1 � Pi � Pli;j j ¼ 2;3;.;niPui;ni

� Pi � Pmaxi

(5)

where j represents the number of prohibited operating zones of ithe generator. Pui;j�1 is the upper limit of (j�1)th prohibited oper-ating zone of i the generator. Pli;j is the lower limit of jth prohibitedoperating zone of i the generator. Total number of prohibitedoperating zone of i the generator is ni.

2.1.3. EDVPL (Economic dispatch with valve-point loading effectand without transmission loss)

The generator cost function is obtained from data points takenduring “heat run” tests,when input andoutput data aremeasured asthe unit is slowly varied through its operating region. Wire drawingeffects, occurring as each steamadmissionvalve in a turbine starts toopen, produce a rippling effect on the unit curve. Tomodel the effectof valve-points, a recurring rectified sinusoid contribution is addedto the quadratic function [3]. The fuel cost function consideringvalve-point loadings of the generator is given as

Ft ¼XNi¼1

FiðPiÞ

¼XNi¼1

ai þ biPi þ ciP2i þ

��di � sinnei �

�Pmini � Pi

�o�� (6)


M. Basu, A. Chowdhury / Energy xxx (2013) 1e10 3

where di and ei are cost coefficients of ith generator due to valve-point effect. The objective of EDVPL is to minimize Ft subject tothe constraints given in (2) and (4). Here transmission loss (PL) isnot considered.

2.1.4. Economic dispatch with valve-point loading effect and multi-fuel options (EDVPLMF)

Since generators are practically supplied withmulti-fuel sources[15], each generator should be represented with several piecewisequadratic functions superimposed sine terms reflecting the effect offuel type changes and the generator must identify the mosteconomical fuel to burn. The fuel cost function of the ith generatorwith NF fuel types considering valve-point loading is expressed as

FiðPiÞ ¼ aij þ bijPi þ cijP2i þ

��dij � sinneij �

�Pminij � Pi

�o�� (7)

if Pminij � Pi � Pmax

ij for fuel type j and j ¼ 1,2,.,NF

The objective function Ft is given by

Ft ¼XNi¼1

FiðPiÞ (8)

The objective function Ft is to be minimized subject to theconstraints given in (2) and (4). Here transmission loss (PL) is notconsidered.

2.1.5. Determination of generation level of slack generatorN committed generators deliver their power output subject to

the power balance constraint (2) and the respective capacity con-straints (4). Assuming the power loading of first (N�1) generatorsare known, the power level of the Nth generator (i.e. the slackgenerator) is given by

PN ¼ PD þ PL �XN�1

i¼1

Pi (9)

The transmission loss PL is a function of all generator outputsincluding the slack generator and it is given by

PL ¼XN�1

i¼1

XN�1

j¼1

PiBijPj þ 2PN

XN�1

i¼1

BNiPi

!þ BNNP

2N þ

XN�1

i¼1

B0iPi

þ B0NPN þ B00(10)

Expanding and rearranging, equation (9) becomes

BNNP2N þ

2XN�1

i¼1

BNiPi þ B0N � 1

!PN

þ0@PD þ

XNi¼1

XNj¼1

PiBijPj þXN�1

i¼1

B0iPi �XN�1

i¼1

Pi þ B00

1A ¼ 0

(11)

The loading of the slack generator (i.e. Nth) can then be found bysolving equation (11) using standard algebraic method.

2.2. Micro grid power dispatch problem

This paper also considers an operation of a micro grid withthree types of generation: diesel generators, fuel-cell plants, andwind-turbine generators. The objective function in a dieselgenerator consists of the fuel cost function similar to the costfunctions used for the conventional fossil-fuel generating plants.


The wind-turbine generator is modeled by the characteristics ofvariable output because the generation in a wind-turbine gener-ator is determined by the strength of the wind. The operating costin fuel-cell system takes the fuel costs and includes the efficiencyfor fuel to generate electric power. The constraints include powerbalance equations and the power generation capacity limits withinthe grid [25].

2.2.1. Distributed generation modeling in micro grid2.2.1.1. Diesel generator. The objective function in a diesel gener-ator consists of the fuel cost function similar to the conventionalcost functions used for the fossil-fuel generating plants.

FDiesel ¼XTt¼1

XNd

i¼1

�adi þ bdiPDiesel;it þ gdiP

2Diesel;it

�(12)

where FDiesel is the diesel generation cost; PDiesel,it is the dieselgeneration output in KW of unit i at time t; adi, bdi, gdi are the co-efficients of i generator fuel cost; Nd is the number of dieselgenerator; t and T are the time index and scheduling period.

2.2.1.2. Wind-turbine generator. The wind-turbine generator ismodeled by the characteristics of variable output because thegeneration in a wind-turbine generator is determined by thestrength of the wind.

Fwind ¼XTt¼1

XNw

i¼1

bwiPwind;it (13)

Pwind;it ¼ Pwind_rated;i ��Vit � Vcutin;i

��Vrated;i � Vcutin;i

� KW;

Vcutin;i � Vit � Vrated;i; t˛T

Pwind;it ¼ Pwind_rated;i KW Vrated;i � Vit � Vcutout;i; t˛T

Pwind;it ¼ 0 Vit < Vcutin;i and Vit > Vcutout;i; t˛T

(14)

where Vit, Vcutin,i, Vcutout,i and Vrated,i are the tth time wind velocity,cut-in velocity, cut-out velocity and rated velocity in m/sec of unit irespectively; Pwind,it and Pwind,rated,i are the wind power generationof unit i at time t and rated wind power output of unit i respec-tively; bwi maintenance and operating cost in $/KW; Nw is thenumber of wind-turbine generators; Fwind is the wind generationcost.

2.2.1.3. Fuel-cell plant. The operating cost in fuel-cell system takesthe fuel costs and includes the efficiency for fuel to generate electricpower. When fuel is transformed into power, the cost functionconsiders the efficiency of fuel cell. Fuel-cell is the most efficientsystem among all fossil-fuel energy sources.

FFC ¼XTt¼1

bnatural

XNFC

i¼1

PFC;ithFC;i

!(15)

where FFC is the fuel-cell generation cost; bnatural is the natural gascost in $/Kg; PFC,it is the fuel-cell generation of the ith unit at time t;hFC,i is the fuel-cell efficiency of unit i; NFC is the number of fuel-cellplants.

2.2.2. Objective functionTotal cost function consists of cost functions of all distributed

generations and is used as the objective function:



Ftotal ¼ FDiesel þ Fwind þ FFC þ FBuy (16)

where, the cost of the buying power from a transmission grid, FBuy,can be calculated as follows:

FBuy ¼XTt¼1

�KePs;Buy;t

�(17)

where, Ke is the purchasing cost per electric power and Ps,Buy,t is thebuying power from the network at time t.

2.2.3. Constraints2.2.3.1. Power balance equations.

XNd

i¼1

PDiesel;it þXNw

i¼1

Pwind;it þXNFC

i¼1

PFC;it þ Ps;Buy;t

¼ PLoad;t þ Ploss;t ; t˛T (18)

The load balance equation consists of the load, power losses,distributed generation in a micro grid and the buying power fromnetwork at each time interval over the scheduling horizon.

2.2.3.2. Power generation capacity limits. The power generated byeach unit should be within its lower limit and upper limit, so that

ðiÞPminDiesel;i � PDiesel;it � Pmax

Diesel;i; i˛Nd; t˛T (19)

ðiiÞPminwind;i � Pwind;it � Pmax

wind;i; i˛Nw t˛T (20)

ðiiiÞ PminFC;i � PFC;it � Pmax

FC;i ; i˛NFC; t˛T (21)

3. Cuckoo search algorithm

Cuckoo search is a meta-heuristic algorithm developed by YangandDeb in 2009 [21]. The basic idea of this algorithm is based on theobligate brood parasitic behavior of some cuckoo species in com-bination with the Levy flight behavior of some birds and fruit flies.

Cuckoos are fascinating birds, not only because of the beautifulsounds they can make, but also because of their aggressive repro-duction strategy. Some species such as the ani and guira cuckoos laytheir eggs in communal nests, though theymay remove others’ eggsto increase the hatching probability of their own eggs. Quite anumber of species engage the obligate brood parasitism by layingtheir eggs in the nests of other host birds. Some host birds canengage direct conflict with the intruding cuckoos. If a host birddiscovers the eggs are not its own, it will either throw these alieneggs away or simply abandon its nest and build a new nest else-where. Some cuckoo species such as the new world brood-parasiticTapera have evolved in such away that female parasitic cuckoos areoften very specialized in themimicry in color andpattern of the eggsof a few chosen host species. This reduces the probability of theireggs being abandoned and thus increases their reproductively.

In nature, animals search for food in a random or quasi-randommanner. In general, the foraging path of an animal is effectively arandom walk because the next move is based on the current loca-tion/state and the transition probability to the next location. Whichdirection it chooses depends implicitly on a probability which canbe modeled mathematically. A recent study shows that fruit flies orDrosophila melanogaster explore their landscape using a series ofstraight flight paths punctuated by a sudden 90� turn, leading to aLe’vy-flight-style intermittent scale-free search pattern.


Cuckoo search is based on three idealized rules [27]:

a. Each cuckoo lays one egg (a design solution) at a time, anddumps its egg in a randomly chosen nest among the fixednumber of available host nests;

b. The best nests with high quality of egg (better solution) will becarried over to the next generation;

c. The number of available hosts nests is fixed, and the egg laid bya cuckoo is discovered by the host bird with a probability ofpa˛[0,1]. In this case, it can simply either throw the egg away orabandon the nest and find a new location to build a completelynew one.

Based on these rules, a general mathematical model for thecuckoo search algorithm (CSA) is summarized in Refs. [26,27,29](Fig. 1).

4. Implementation of cuckoo search algorithm for EDproblem

The proposed Cuckoo Search Algorithm (CSA) is a populationbased method similar to other meta-heuristic methods. Thestructure of CSA includes two main operations including a directsearch based on Levy flights and a random search based on theprobability for a host bird to discover an alien egg in its nest.With the combination of two operations, the proposed CSA be-comes a more powerful search method than other meta-heuristicsearch methods for complex and large-scale optimization prob-lems. Therefore, the proposed CSA is very effective for solvingnon-convex and large-scale ED problems.

In the proposed CSA method, each nest represents a solutionand a population of nest is used for finding the best solution of theproblem. Themain steps of the proposed CSAmethod are describedbelow:

Step1: initialization

A population of NP host nests is represented byX ¼ [X1,X2,.,XNP]T where each nest Xi ¼ [Pi1,Pi2,.,Pij,.,PiN] repre-sents power output of units. The power output of unit is initializedby:

Pij ¼ Pminj þ rand1*ðPmax

j � Pminj Þ

where rand1 is a uniformly distributed random number be-tween 0 and 1.

Step2: generation of new solution via Levy flights

The new solution is calculated based on the previous best nestsvia Levy flights. In the proposed method, the optimal path for theLevy flights is calculated by Mantegna’s algorithm [28]. The newsolution for each nest is calculated as follows:

Xnewi ¼ Xbesti þ a� rand2 � DXnew

iwhere a > 0 is the updated step size and rand2 is a normally

distributed stochiastic number. DXnewi is calculated as follows:

DXnewi ¼ v� sxðbÞ=syðbÞ � ðXbesti � GbestÞ

v ¼ randx=��randy��1=b

where randx and randy are two normally distributed stochasticvariables with standard deviation sx(b) and sy(b) given by

sxðbÞ ¼hGð1þ bÞ � sin

�Pb2

��G�1þb2

�� b� 2ðb�1

2 Þi1=b

syðbÞ ¼ 1where b is the distribution factor 0.3 � b � 1.99 and G(.) is the

gamma distribution function.


Fig. 1. Flow chart of Cuckoo Search Algorithm


The newly obtained solution should be satisfied according to itslower and upper limits.

Step3: alien egg discovery and randomization

The action of discovery of an alien egg in a nest of a host birdwith the probability of pa also creates a new solution for the


problem similar to the Levy flights. The new solution due to thisaction can be found out in the following way.

Xdisi ¼ Xbesti þ K � DXdis

iwhere K is the updated coefficient determined based on the

probability of a host bird to discover an alien egg in its nest:

K ¼�1 if rand3 < pa0 otherwise


Table 1Power Output for 6-Generator System (PD ¼ 1263 MW).

Unit poweroutput (MW)

CSA BBO[12]

SOH-PSO[18]

NPSO-LRS[17]

PSO[8]

P1 447.4768 447.3997 438.21 446.96 447.50P2 173.2234 173.2392 172.58 173.3944 173.32P3 263.3787 263.3163 257.42 262.3436 263.47P4 138.9524 138.0006 141.09 139.5120 139.06P5 165.4120 165.4104 179.37 164.7089 165.48P6 87.0024 87.07979 86.88 89.0162 87.13Total power

output (MW)1275.447 1275.446 1275.55 1275.94 1276.01

Ploss (MW) 12.447 12.446 12.55 12.936 12.958Total cost ($/h) 15443.08 15443.096 15446.02 15450 15450

0 50 100 150 200 250 3001.5442

1.5444

1.5446

1.5448

1.545

1.5452

1.5454

1.5456

1.5458x 10

4

iteration

cost

($/h

our)

Fig. 2. Cost convergence characteristic of 6-generator system.

0 50 100 150 200 250 300 350 400 450 500590

600

610

620

630

640

650

660

iteration

cost

($/h

our)



Unit poweroutput (MW)

CSA BBO [12] Hopfieldmodel [6]

Lambdaiteration [6]

P1 512.8467 513.0892 512.7804 512.7805P2 168.8534 173.3533 169.1035 169.1033P3 126.8549 126.9231 126.8897 126.8898P4 102.8784 103.3292 102.8656 102.8657P5 113.6863 113.7741 113.6836 113.6386P6 73.5482 73.06694 73.5709 73.5710P7 115.4766 114.9843 115.2876 115.2878P8 116.4497 116.4238 116.3994 116.3994P9 100.7505 100.6948 100.4063 100.4062P10 106.1438 99.99979 106.0267 106.0267P11 150.2221 148.977 150.2395 150.2394P12 292.7736 294.0207 292.7647 292.7648P13 118.9029 119.5754 119.1155 119.1154P14 30.8736 30.54786 30.8342 30.8340P15 115.7864 116.4546 115.8056 115.8057P16 36.2102 36.22787 36.2545 36.2545P17 66.8828 66.85943 66.8590 66.8590P18 87.8848 88.54701 87.9720 87.9720P19 100.7805 100.9802 100.8033 100.8033P20 54.1771 54.2725 54.3050 54.3050Total Power

ouput (MW)2555.80 2592.1011 2591.9670 2591.9670

Ploss (MW) 55.80 92.1011 91.5670 91.9670Total cost ($/h) 62456.63 62456.7926 62456.63 62456.63


The increased value DXdisi is determined by

DXdisi ¼ rand3 � ½randp1ðXbestiÞ � randp2ðXbestiÞ�

where rand3 is the distributed random number in [0,1].randp1(Xbesti) and randp2(Xbesti) are the random perturbation forpositions of nests in Xbesti.

Step4: stopping criteria

The above algorithm is stopped when the number of iterationsreaches the predefined value.


Unit power output (MW) CSA BBO [12] NPSO-LRS [17] NPSO [17] IGA [15]

Fuel Fuel FFuel FFuel Fuel

P1 236.4387 2 212.96 2 223.33 2 220.657 2 219.126 2P2 230.0000 1 209.43 1 212.19 1 211.785 1 211.164 1P3 417.3113 2 332.02 3 276.21 1 280.402 1 280.657 1P4 135.9952 1 238.34 3 239.41 3 238.601 3 238.477 3P5 328.6017 1 269.25 1 274.64 1 277.562 1 276.417 1P6 197.6450 1 237.64 3 239.79 3 239.120 3 240.467 3P7 257.0953 1 280.61 1 285.53 1 292.139 1 287.739 1P8 228.2969 3 238.47 3 240.63 3 239.153 3 240.761 3P9 411.4391 3 414.85 3 429.26 3 426.114 3 429.337 3P10 257.1768 1 266.38 1 278.95 1 274.463 1 275.851 1Total cost ($/h) 598.0243 605.6387 624.1273 624.1624 624.5178

Please cite this article in press as: Basu M, Chowdhury A, Cuckoo search algorithm for economic dispatch, Energy (2013), http://dx.doi.org/10.1016/j.energy.2013.07.011

0 50 100 150 200 250 300 350 400 450 5006.245

6.25

6.255

6.26x 10

4

iteration

cost

($/h

our)


0 50 100 150 200 250 300 350 400 450 5001.21

1.22

1.23

1.24

1.25

1.26

1.27

1.28

1.29

1.3

1.31x 10

5

iteration

cost

($/h

our)


0 5 10 15 20 250

5

10W

ind

Spe

ed (m

/sec

)

Hour0 5 10 15 20 25

0

1000

2000

Load

Dem

and

(KW

)

Fig. 6. Wind speed and load demand for a day.


5. Simulation results

The proposed cuckoo search algorithm has been applied to solveED problems and micro grid power dispatch problem for verifyingits feasibility. The software has been written in MATLAB 7 on a PC(Pentium e IV, 80 GB, 3.0 GHZ). Simulation results consist of twosections. The first section considers simulation results of ED prob-lem and second section considers the simulation results of microgrid dispatch problem.

5.1. Simulation results of ED problem

The proposed cuckoo search algorithm has been applied to solveED problems in four different test systems.

Test System 1: A six generator systemwith prohibited operatingzone is considered here. The generator data and B-coefficients havebeen taken from Ref. [8]. The load demand is 1263MW. For this testsystem, the population size (NP), maximum number of iterationsand the value of probability pa have been selected 50, 300 and 0.7


Output(MW) CSA BBO [12] NPSO-LRS [17] SOH-PSO [18] Output (MW) CSA BBO [12] NPSO-LRS [17] SOH-PSO [18]

P1 112.0518 111.0465 113.9761 110.80 P21 523.3012 523.417 523.2916 523.28P2 111.4948 111.5915 113.9986 110.80 P22 523.2928 523.2795 523.2853 523.28P3 97.5626 97.6077 97.4141 97.40 P23 523.2892 523.3793 523.2797 523.28P4 179.8000 179.7095 179.7327 179.73 P24 523.4340 523.3225 523.2994 523.28P5 88.9834 88.3060 89.6511 87.80 P25 523.2839 523.3661 523.2865 523.28P6 140.0000 139.9992 105.4044 140.00 P26 523.2810 523.4362 523.2936 523.28P7 299.9993 259.6313 259.7502 259.60 P27 10.0000 10.05316 10.0000 10.00P8 284.9506 284.7366 288.4534 284.60 P28 10.0009 10.01135 10.0001 10.00P9 284.9653 284.7801 284.6460 284.60 P29 10.0014 10.00302 10.0000 10.00P10 130.0006 130.2484 204.8120 130.00 P30 92.0666 88.47754 89.0139 97.00P11 94.0000 168.8461 168.8311 94.00 P31 190.0000 189.9983 190.0000 190.00P12 94.0000 168.8239 94.00 94.00 P32 190.0000 189.9881 190.0000 190.00P13 214.7621 214.7038 214.7663 304.52 P33 190.0000 189.9663 190.0000 190.00P14 304.5194 304.5894 394.2852 304.52 P34 199.9998 164.8054 199.9998 185.20P15 394.2799 394.2761 304.5187 394.28 P35 199.9999 165.1267 165.1397 164.80P16 394.2793 394.2409 394.2811 394.28 P36 200.0000 165.7695 172.0275 200.00P17 489.2802 489.2919 489.2807 489.28 P37 110.0000 109.9059 110.0000 110.00P18 489.2776 489.4188 489.2832 489.28 P38 110.0000 109.9971 110.0000 110.00P19 511.2797 511.2997 511.2845 511.28 P39 110.0000 109.9695 93.0962 110.00P20 511.2799 511.3073 511.3049 511.27 P40 511.2824 511.2794 511.2996 511.28Total cost ($/h) 121425.61 121426.95 121664.43 121501.14


Table 5Generation capacity, cost coefficients, efficiency of fuel-Cell, cut-in velocity, cut-out velocity and rated velocity of wind generator.

Generation type a ($/h) b ($/KWh) g ($/(KW)2h) Pmin (KW) Pmax (KW) h (%) Vcutin (m/sec) Vcutout (m/sec) Vrated (m/sec)

Diesel 1 0.4333 0.2333 0.0074 0 400Diesel 2 0.2731 0.1453 0.0042 0 800F_Cell 1 0 0.05 0 0 150 90F_Cell 2 0 0.05 0 0 100 90F_Cell 3 0 0.07 0 0 100 85Wind 1 0 0.022 0 0 300 5 15 10Wind 2 0 0.032 0 0 300 5 15 10

Table 6Micro grid generation (KW) schedule and cost ($) using CSA.

Hour PDiesel,1 PDiesel,2 Pwind,1 Pwind,2 PFC,1 PFC,2 PFC,3 Cost

1 0 298.2899 171.2722 130.3794 53.6585 0 0 33824.102 0 139.3229 300.0000 43.3855 67.5462 0 0.14533 199.6980 176.0579 0 84.0951 4.4504 100.0000 80.69864 16.2097 171.5906 300.0000 200.1996 0 0 05 0 356.1842 0 300.0000 150.0000 36.6158 06 400.0000 451.9106 0 0147.8926 100.0000 18.19687 306.5509 515.3107 0 257.1533 81.1944 64.1908 100.00008 400.0000 422.2449 300.0000 20.9551 150.0000 100.0000 09 400.0000 707.5096 0 0125.1895 98.4089 96.492110 0 643.1133 277.5463 300.0000 72.5404 100.0000 011 206.9417 151.3608 300.0000 300.0000 150.0000 30.1269 99.970612 0 323.2937 288.1575 300.0000 25.4824 62.5794 84.086913 0 682.0000 0 0150.0000 100.0000 100.000014 292.5128 298.4082 300.0000 99.2525 7.4265 0 015 228.9831 450.0469 4.6599 299.9101 0 0 100.000016 0 436.6821 300.0000 40.2296 149.5381 6.8379 98.712317 349.4064 218.5936 0300.0000 150.0000 100.0000 018 156.4700 469.5300 300.0000 300.0000 150.0000 0 019 305.1299 609.8976 300.0000 278.3494 0 100.0000 75.023120 400.0000 645.6064 300.0000 0144.1661 100.0000 61.427521 400.0000 337.7474 300.0000 300.0000 138.5387 100.0000 57.713922 321.9382 374.7723 300.0000 300.0000 65.2895 0 100.000023 286.4981 466.4268 300.0000 65.0652 23.7098 99.9001 100.000024 0 466.4000 300.0000 300.0000 0 0 0

Table 7Micro grid generation (KW) schedule and cost ($) using DE.


1 66.6480 25.9270 192.0000 192.0000 150.0000 23.6733 3.3517 33930.942 5.5009 66.8991 114.0000 114.0000 150.0000 0 100.00003 85.3674 162.7418 36.0000 36.0000 148.3530 76.5378 100.00004 115.1099 50.9179 165.0000 165.0000 0 91.9722 100.00005 64.7310 76.4247 252.0000 252.0000 0 100.0000 97.64446 272.6012 511.0701 0 0 148.3989 100.0000 85.92987 284.9755 578.6040 60.0000 60.0000 150.0000 92.1588 98.66188 215.7252 474.0957 186.0000 186.0000 148.6036 82.7754 100.00009 400.0000 723.8947 0 0 147.9034 79.7891 76.012810 338.6740 368.5260 168.0000 168.0000 150.0000 100.0000 100.000011 163.6135 408.0517 210.0000 210.0000 150.0000 96.7347 012 215.9637 228.8870 180.0000 180.0000 150.0000 28.7493 100.000013 290.2655 391.7345 0 0150.0000 100.0000 100.000014 264.2401 428.2957 48.0000 48.0000 150.0000 0 59.064215 76.5865 495.0286 108.0000 108.0000 150.0000 45.9849 100.000016 143.9892 178.2737 225.0000 225.0000 109.2153 53.4447 97.077217 211.6718 383.0754 90.0000 90.0000 150.0000 100.0000 93.252818 400.0000 298.6280 192.0000 192.0000 150.0000 92.7435 50.628519 324.9751 607.4806 198.0000 198.0000 141.7662 100.0000 98.178120 361.7894 699.4106 120.0000 120.0000 150.0000 100.0000 100.000021 400.0000 764.0000 60.0000 60.0000 150.0000 100.0000 100.000022 177.7931 694.2069 120.0000 120.0000 150.0000 100.0000 100.000023 189.6284 467.6628 228.0000 228.0000 150.0000 0 78.308824 161.9239 348.8853 120.0000 120.0000 141.8495 98.8400 74.9013



Table 8Micro grid generation (KW) schedule and cost ($) using PSO.


1 66.1616 106.4451 192.0000 192.0000 69.2539 27.3742 0.3652 38189.312 92.5537 32.4934 114.0000 114.0000 89.2949 54.6474 53.41063 241.5360 182.3276 36.0000 36.0000 76.7130 33.2812 39.14224 42.2118 51.4620 165.0000 165.0000 147.3141 26.3460 90.66605 0 108.7443 252.0000 252.0000 150.0000 28.8063 51.24946 324.6861 549.0500 0 0 07.7500 100.0000 36.51397 393.6437 494.6299 60.0000 60.0000 150.0000 89.4320 76.69438 186.1055 498.3208 186.0000 186.0000 148.6369 100.0000 88.13689 367.6265 715.3484 0 0 150.0000 100.0000 94.625110 268.1017 567.9695 168.0000 168.0000 38.6890 82.4398 100.000011 205.5394 378.4093 210.0000 210.0000 75.1091 84.6016 74.740712 187.3128 336.0428 180.0000 180.0000 110.5747 3.2478 86.421913 319.8232 426.0081 0 0 120.8976 100.0000 65.271014 197.6576 519.6057 48.0000 48.0000 82.7991 82.6094 18.928115 259.1079 448.6219 108.0000 108.0000 61.7093 39.9444 58.216516 186.9911 157.8641 225.0000 225.0000 134.2793 25.4353 77.430117 300.3640 347.1187 90.0000 90.0000 150.0000 79.2817 61.235618 198.6275 507.1232 192.0000 192.0000 149.1925 65.3591 71.697819 290.8133 694.6384 198.0000 198.0000 126.5239 100.0000 60.424420 393.7356 722.7663 120.0000 120.0000 97.7331 100.0000 96.964921 389.1652 790.1448 60.0000 60.0000 147.2911 95.9937 91.405322 297.9510 714.2537 120.0000 120.0000 128.6081 81.1873 023 130.9227 588.0110 228.0000 228.0000 58.9867 27.6753 80.004324 131.7268 432.4161 120.0000 120.0000 128.9226 33.3345 100.0000

0 10 20 30 40 50 60 70 80 90 1003.3

3.4

3.5

3.6

3.7

3.8

3.9

4

4.1

4.2

4.3x 10

4

Cost

($)

Generation

CSADEPSO

Fig. 7. Cost convergence characteristic of micro grid power dispatch.


respectively. Results obtained from proposed CSA, BBO [12], SOH-PSO [18], NPSO-LRS [17] and PSO [8] have been presented inTable 1. It is found that the proposed CSA based approach providesthe lowest minimum cost among all the methods. The costconvergence characteristic of six generator system obtained fromCSA is shown in Fig. 2.

Test System 2: This system consists of ten generators with valve-point loading and multi-fuel sources. The generator data has beenadopted from Ref. [15]. The load demand is 2700MW. Transmissionloss has not been considered here. For this test system, the popu-lation size (NP), maximum number of iterations and the value ofprobability pa have been selected 50, 500 and 0.7 respectively.Results obtained from proposed CSA, BBO [12], NPSO-LRS [17],NPSO [17] and IGA [15] have been summarized in Table 2. It is seenfrom Table 2 that the proposed CSA based approach provides thelowest minimum cost among all methods. The cost convergencecharacteristic of this test system obtained from CSA is shown inFig. 3.

Test System 3: A twenty generator system with quadratic costfunction is considered here. The generator data and B-coefficientshave been taken from Ref. [6]. The load demand is 2500 MW. Forthis test system, the population size (NP), maximum number ofiterations and the value of probability pa have been selected 50, 500and 0.7 respectively. Results obtained from proposed CSA, BBO [12],Hopfield Model [6], and Lambda Iteration [6] have been shown inTable 3. It is found that the proposed CSA based approach providesthe lowest minimum cost among all the methods. The costconvergence characteristic of twenty generator system obtainedfrom CSA is shown in Fig. 4.

Test System 4: This system consists of forty generators withvalve-point loading. The generator data has been adopted from Ref.[14]. The load demand is 10,500 MW. Transmission loss has notbeen considered here. For this test system, the population size (NP),maximum number of iterations and the value of probability pa havebeen selected 50, 500 and 0.7 respectively. Results obtained fromproposed CSA, BBO [12], NPSO-LRS [17], and SOH-PSO [18] havebeen depicted in Table 4. It is seen from Table 4 that the proposedCSA based approach provides the lowest minimum cost among allmethods. The cost convergence characteristic of this test systemobtained from CSA is shown in Fig. 5.


5.2. Simulation results of micro grid power dispatch problem

The proposed cuckoo search algorithm has been applied to solvemicro grid power dispatch problem in a single test system. Thewind speed and the load demand for a day are shown in Fig. 6.Table 5 shows generation capacity and cost coefficients of all typesunits, efficiency of fuel-Cell, cut-in velocity, cut-out velocity andrated velocity of wind generator.

The problem is solved by using CSA. For this test system, thepopulation size (NP), maximum number of iterations and the valueof probability pa have been selected 50, 100 and 0.7 respectively.Micro grid generation schedule and cost obtained from CSA havebeen shown in Table 6.

To validate the proposed CSA based approach, the same testsystem is solved by using differential evolution (DE) and particleswarm optimization (PSO). In case of DE, the population size (NP),scaling factor (F), crossover rate (CR)and the maximum iterationnumber (Nmax) have been selected as 100, 0.75, 1.0 and 100



respectively for the test system under consideration. Parametersare taken as NP ¼ 50, wmax ¼ 0.2, wmin ¼ 0.05, c1 ¼ 0.35, c2 ¼ 0.35and Nmax ¼ 100 for this test system in case of PSO. Table 7 andTable 8 summarize the micro grid generation schedule and costobtained from DE and PSO respectively. It is seen that the proposedCSA based approach provides the lower minimum cost than DEand PSO. Fig. 7 shows the cost convergence obtained from CSA, DEand PSO.

6. Conclusion

In this paper, CSA has been successfully implemented to solveboth convex and nonconvex ED problems and micro grid powerdispatch problem. It is seen that the proposed CSA has the ability toconverge to a better quality solution.

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