DSpace at VNU: Cuckoo search algorithm for non-convex economic dispatch

Published in IET Generation, Transmission & DistributionReceived on 8th March 2012 Revised on 14th January 2013 Accepted on 10th February 2013 doi: 10.1049/iet-gtd.2012.0142 ISSN 1751-86

Published in IET Generation, Transmission & Distribution

Received on 8th March 2012

Revised on 14th January 2013

Accepted on 10th February 2013

doi: 10.1049/iet-gtd.2012.0142

ISSN 1751-8687 Cuckoo search algorithm for non-convex economic dispatch

1

Department of Power Systems, Ho Chi Minh City University of Technology, Ho Chi Minh City, Vietnam

2 Institute of Electrical Power Systems and High Voltage Engineering, Technische Universität Dresden, 01069 Dresden, Germany

3 Energy Field of Study, School of Environment, Resources and Development, Asian Institute of Technology,

Pathumthani 12120, Thailand

E-mail: ongsakul@ait.asia

Abstract: This study proposes a cuckoo search algorithm (CSA) for solving non-convex economic dispatch (ED) considering generator and system characteristics including valve-point effects, multiple fuels, prohibited zones, spinning reserve and power loss CSA is a new meta-heuristic optimisation method inspired from the obligate brood parasitism of some cuckoo species by laying their eggs in the nests of other host birds of other species When the host birds discover an alien egg in their nest, they can either throw it away or simply abandon their nest and build a new one elsewhere The CSA idealised such breeding behaviour in combination with Lévy flights behaviour of some birds and fruit flies for applying to various constrained optimisation problems The effectiveness of the proposed method has been tested on different non-convex ED problems Test results have indicated that the proposed method can obtain less expensive solutions than many other methods reported in the literature Accordingly, the proposed CSA is a promising method for solving the practical nonconvex ED problems

Nomenclature

ai, bi, ci fuel cost coefficients of unit i

aij, bij, cij fuel cost coefficients for fuel type j of unit i

ei, fi fuel cost coefficients of unit i reflecting

valve-point effects

eij, fij fuel cost coefficients for fuel type j of unit i

reflecting valve-point effects

Bij, B0i,

B00

B-matrix coefficients for transmission power

loss

N total number of generating units

ni number of prohibited zones of unit i

Pi power output of unit i

Pi, max maximum power output of unit i

Pi, min minimum power output of unit i

Pij, min minimum power output for fuel j of unit i

Puik upper bound for prohibited zone k of unit i

Plik lower bound for prohibited zone k of

generator i

PD total system load demand

PL total transmission loss

Si spinning reserve from unit i

Si, max maximum spinning reserve contribution of

unit i

SR total system spinning reserve requirement

1 Introduction

Economic dispatch (ED) is to optimally allocate the real power output among the online thermal units so that their total production cost is minimised while satisfying the unit and system operating constraints [1,2] Conventionally, the objective function of the ED was approximated by a single quadratic function for mathematical convenience [3] Nevertheless, the input–output characteristics of thermal generating units are essentially more complicated because

of the effects of valve point effects [4], multiple fuels (MFs) [5] or prohibited zones [6] Therefore the practical

ED problem can be formulated as non-convex objective function subject to non-linear constraints, which is difficult

to be solved by the classical mathematical programming techniques

Several conventional methods have been applied for solving ED problems such as gradient search, Newton’s method, dynamic programming (DP) [3], hierarchical approach based on the numerical method (HNUM) [5], decomposition method [6] and Maclaurin series-based Lagrangian (MSL) method [7] Among these methods, only MSL method can directly deal with the non-convex ED problem with non-differentiable objective by using the Maclaurin expansion of non-convex terms in the objective function Although this method can quickly find a solution

for the problem, the obtained result is still far from optimum,

especially for the large-scale systems In general, the

conventional methods are not effective for non-convex ED

problems Recently, many methods based on artificial

intelligence have been developed for solving ED problems

such as Hopfield neural network (HNN) [8], genetic

algorithm (GA) [9–12], evolutionary programming (EP)

[13], Taguchi method (TM) [14], biogeography-based

optimisation (BBO) [15], and particle swarm optimisation

(PSO) [16–26] Among of them, the HNN method based on

the minimisation of its energy function can be only applied

to the convex optimisation problems with differentiable

objective function This method can be implemented on

large-scale problems but it suffers many drawbacks such as

local optimum solution and long computation time The

others are the meta-heuristic search methods which can

overcome the drawbacks of the HNN method because of

their ability to find near optimal solution for non-convex

optimisation problems However, for the large-scale and

non-smooth problems with multiple minima, these methods

may suffer low solution quality and long computational

time In addition, hybrid methods have been also developed

for dealing with the non-convex ED problems such as

combining of chaotic differential evolution and quadratic

programming (DEC-SQP) [27], simulated annealing like

particle swarm optimisation (SA-PSO) [28], combination of

differential evolution and BBO (DE/BBO) [29], and hybrid

Hopfield neural network quadratic programming based

technique (HNN-QP) [30] These hybrid methods utilise the

advantages of each element method to enhance their search

ability for the complex problems However, the hybrid

methods contain many controllable parameters which may

not be properly selected

In this paper, a cuckoo search algorithm (CSA) is proposed

for solving non-convex ED problems considering generator

and system characteristics including valve point loading

effects (VPE), MF options, prohibited operating zones,

spinning reserve and power loss CSA is a new

meta-heuristic optimisation method inspired from the

obligate brood parasitism of some cuckoo species by laying

their eggs in the nests of other host birds of other species

When the host birds discover an alien egg in their nest, they

can either throw it away or simply abandon their nest and

build a new one elsewhere The CSA idealised such

breeding behaviour in combination with Lévy flights

behaviour of some birds and fruit flies for applying to

various constrained optimisation problems The proposed

CSA is tested on several large-scale and non-convex

systems and the obtained results are compared to those

from many other methods in the literature

2 Problem formulation

2.1 ED problem with valve point effects

The ED with VPE is a non-smooth and non-convex problem

with multiple minima considering ripples in the heat-rate

curves of boilers The model of VPE has been proposed in

[9] by adding a sinusoidal function to the quadratic fuel

cost function The objective of the problem is written as

Min F=
N

i=1

Fi Pi

(1)

where the fuel cost function of unit i is represented by [13]

Fi Pi

= ai+ biPi+ ciPi2 + ei× sin fi× Pi, min− Pi

subject to Real power balance: The total real power output of generating units satisfies total real load demand plus power loss

N i=1

Pi= PD+ PL (3)

where the power loss PL can be approximated by Kron’s formula [3]

PL=
N

i=1

N j=1

PiBijPj+
N

i=1

B0iPi+ B00 (4)

Generator capacity limits: The real power output of generating units should be within their upper and lower operating limits as

The cost curve function of units with valve point effects is depicted in Fig.1

2.2 ED problem with multiple fuels

In practical power system operation conditions, many thermal generating units being supplied with MF sources such as coal, natural gas and oil require that their fuel cost functions may be segmented as piecewise quadratic cost functions for different fuel types Therefore, in the ED problem with MFs, the piecewise quadratic function is used to represent the MFs that are available for each generating unit [5] The fuel cost function of unit i is represented by [8]

Fi Pi

=

ai1+ bi1Pi+ ci1P2i, fuel 1, Pi, min≤ Pi≤ Pi1

ai2+ bi2Pi+ ci2P2i, fuel 2, Pi1, Pi≤ Pi2

aij+ bijPi+ cijP2i, fuel j, Pij−1 , Pi≤ Pi, max

⎧

⎪

⎨

⎪

⎩

(6)

Fig 1 Fuel cost curve of units with valve-point effects

For generator i with j fuel options in (6), its cost curve is

divided into j discrete segments between lower limit Pi, min

and upper limit Pimax, in which each fuel type is

represented by a quadratic function with lower power

output limit Pij-1 and upper power output limit Pij The cost

curve function of units with MFs is depicted in Fig.2

The objective of the ED problem with MF is to minimise

the total cost (1) where the fuel cost function for each

generator is given in (6) subject to the real power balance

constraint (3) and generator capacity limits (5)

2.3 ED problem with VPE and MFs

In practical power system operation conditions, thermal

generating units can be supplied with MF sources and their

boilers also have valve points for controlling their power

outputs Therefore for more accurate in determination of the

solution for the practical ED problem the fuel cost function

of units should consider both VPE and MF [12] The fuel

cost function of generating unit i is represented by

Fi Pi

=

Fi1 Pi , fuel 1, Pi, min≤ Pi≤ Pi1

Fi2 Pi , fuel 2, Pi1, Pi≤ Pi2

Fij Pi , fuel j, Pij−1, Pi≤ Pi, max

⎧

⎪

where the fuel cost function for fuel type j of unit i is

determined by

Fij Pi

= aij+ bijPi+ cijP2i

+ e ij× sin( fij× (Pij, min− Pi)) (8)

subject to the real power balance constraint (3) and generator

capacity limits (5)

2.4 ED problem with prohibited operating zones

Thermal generating units may have prohibited operating

zones (POZ) because of physical limitations on components

of units Consequently, the whole operating region of a

generating unit with POZ will be broken into several

isolated feasible sub-regions [16] The fuel cost function for

each unit in the ED problem with POZ can be a quadratic

function (2) or a quadratic function with VPE (8) The

equality and inequality constraints for this problem include

the real power balance constraint (3), generator capacity limits (5) for units having no POZ, and

POZ: For units with POZ, their feasible operating points should be located at one of the sub-regions as follows

Pi[

Pi, min≤ Pi≤ Pl

i1

Puik−1≤ Pi≤ Pl

ik; k= 2, , ni

Puin

i ≤ Pi≤ Pi, max

⎧

⎪

Equation (9) indicates that if unit i has ni POZ, it will have (ni+ 1) feasible disjoint operating regions which will form a non-convex set The cost curve function of units with prohibited zones is depicted in Fig.3

Spinning reserve constraint: The spinning reserve constraint for all units is defined as

N i=1

where the operating margin of each unit Siis determined by

Si= min Pi, max− Pi, Si, max

; ∀i  V (11)

Si= 0; ∀i [ V (12) whereΩ is the set of units with POZ

Equation (10) shows that the spinning reserve contribution

of all units should satisfy a required threshold and the contributed spinning reserve in the system is only from the units without prohibited zones as in (11) This is because the ability to regulate system load of units with prohibited zones are strictly limited by their prohibited zones Therefore the required spinning reserve is mainly contributed from the units without prohibited zones

3 CSA for ED problems

3.1 CSA Cuckoo search is a new meta-heuristic algorithm inspired from the nature for solving optimisation problems developed by Yang and Deb in 2009 [31] The basic idea

of this algorithm is based on the obligate brood parasitic behaviour of some cuckoo species in combination with the Lévy flight behaviour of some birds and fruit flies There are three idealised rules for the new CSA described as follows [32]

Fig 2 Fuel cost curve of units with MFs Fig 3 Fuel cost curve of units with prohibited zones

† Each cuckoo lays one egg (a design solution) at a time and

dumps its egg in a randomly chosen nest among the fixed

number of available host nests;

† The best nests with a high quality of egg (better solution)

will be carried over to the next generation;

† A host bird can discover an alien egg in its nest with a

probability of pa∈ [0, 1] In this case, it can simply either

throw the egg away or abandon the nest and find a new

location to build a completely new one

Based on these rules, a general mathematical model for the

CSA is summarised in [31,32]

3.2 Calculation of power output for slack unit

To guarantee that the equality constraint (3) is always

satisfied, a slack generating unit is arbitrarily selected and

therefore its power output will be dependent on the power

output of remaining N-1 generating units in the system The

method for calculation of power output for the slack unit is

as follows [33] Suppose that the power output of the N− 1

generating units are known, the power output of the slack

unit is calculated by

Ps= PD+ PL−
N

i=1 i=s

where s is an arbitrary unit selected among the N units

The power loss in (4) is rewritten by considering Psas an

unknown variable

PL= BssP2s + 2

N i=1

i =s

BsiPi+ B0s

⎛

⎜

⎞

⎟

⎠Ps

+
N

i =1 i=s

N

j =1 j=s

PiBijPj+
N

i =1 i=s

B0iPi+ B00

(14)

By substituting (14) into (13), a quadratic equation is obtained

as follows

A× P2

s+ B × Ps+ C = 0 (15) where the coefficients A, B and C are given by

B= 2
N

i=1 i=s

BsiPi+ B0s− 1 (17)

C=
N

i=1

i=s

N

j=1

j=s

PiBijPj+
N

i=1 i=s

B0iPi+ B00+ PD −
N

i=1 i=s

Pi (18)

The power output of the slack generator is the positive root of

(15) between the two ones obtained as below

Ps=−B +



B2− 4 × A × C

√

where B2− 4 × A × C ≥ 0

(19)

3.3 Handling of prohibited operating zones violation

When a unit operates in one of its POZ, a repairing strategy is used to force the unit either to move towards the lower bound

or upper bound of that zone For making a decision based on the operating point of a unit located in one of its prohibited zones, the middle point of each prohibited zone is firstly determined as follows

Pmik = Pl

ik+ Pu ik

This middle point divides a prohibited zone in two sub-zones, the left and right prohibited sub-zones with respect to the point Therefore the operating point Piof unit i violating its prohibited zone k will be adjusted by

Pnewi = Plik if Pi≤ Pm

ik

Puik if Pi Pm

ik



(21)

The modification of unit’s limits in (21) for the cases with prohibited zones violation is depicted in Fig.4

3.4 Implementation of CSA The proposed CSA is a population-based method similar to other meta-heuristic methods The structure of the CSA includes two main operations including a direct search based on Lévy flights and a random search based on the probability for a host bird to discover an alien egg in its nest With the combined two operations, the proposed CSA becomes a more powerful search method than other meta-heuristic search methods for complex and large-scale optimisation problems Therefore, the proposed CSA is very effective in solving non-convex and large-scale ED problems

In the proposed CSA method, each nest represents a solution and a population of nests is used for finding the best solution for the problem similar to many other meta-heuristic search methods The main steps for the proposed CSA are described as follows:

Initialisation: A population of Nphost nests is represented

by X = [X1, X2,…, XNp]T, where each nest Xd= [Pd1,…, Pds-1,

Pds + 1,…, PdN] (d = 1,…, Np) represents power output of units except the slack unit is initialised by

Xdi= Pi, min+ rand∗

1 Pi, max− Pi, min

(22)

Fig 4 Adjustment of unit ’s prohibited zone violation

where rand1 is a uniformly distributed random number in

[0,1] for each population of the host nests

This initial solution is further checked for POZ violation

If the violation is found, the repairing strategy in Section

3.3 is used to move the operating point to a feasible region

Based on the initial population of nests, thefitness function

to be minimised corresponding to each nest for the

considered problem is calculated

FTd =
N

i =1

Fi Xdi + Ks× Pds− Plim

s

+ Kr× max 0, SR−

N i=1

Sdi

where Ks and Kr are penalty factors for the slack unit and

spinning reserve constraint, respectively; Pds is power

output of the slack unit calculated from Section 3.2

corresponding to nest d in the population; and Sdi is the

spinning reserve of unit i corresponding to nest d in the

population calculated from (11) and (12)

The limits for the slack unit in (23) are determined based on

its calculated power output as follows

Pslim=

Ps, max if Pds Ps, max

Ps, min

Pds

if Pds , Ps, min

otherwise

⎧

⎨

where Ps, max and Ps, min are the maximum and minimum

power outputs of the slack unit, respectively

The initial population of the host nests is set to best value of

each nest Xbestd(d = 1,…, Nd) and the nest corresponding to

the bestfitness function in (23) is set to the best nest Gbest

among all nests in the population

Generation of new solution via Lévy flights: The new

solution is calculated based on the previous best nests via

Lévyflights In the proposed method, the optimal path for

the Lévyflights is calculated by Mantegna’s algorithm [34]

The new solution by each nest is calculated as follows

Xdnew = X bestd+a × rand2× DXnew

where α > 0 is the updated step size; rand2 is a normally

distributed stochastic number; and the increased value

ΔXdnewis determined by

DXnew

d = n ×sx(b)

sy(b)× X best d− Gbest (26)

n = randx

randy



where randxand randyare two normally distributed stochastic

variables with standard deviationσx(β) and σy(β) given by

sx(b) = G(1 +b) × sin (pb)/2

G (1 + b)/2×b × 2((b−1)/2)

(28)

whereβ is the distribution factor (0.3 ≤ β ≤ 1.99) and Γ(.) is

the gamma distribution function

For the newly obtained solution, its lower and upper limits should be satisfied according to the unit’s limits

Xdinew=

Pi, max if Xdinew Pi, max

Pi, min

Xdi

if Xdinew, Pi, min

otherwise

⎧

⎨

In addition, the newly adjusted solution should be further checked for POZ violation and the repairing strategy in Section 3.3 is used to move the solution out of the prohibited zones if any violation is found

The fitness function (23) will be re-evaluated for the new solution to determine the newly best value of each nest Xbestdand the best nest of all nests Gbest by comparing the stored fitness values in Section 3.4.1 and the newly calculated ones

Alien egg discovery and randomisation: The action of discovery of an alien egg in a nest of a host bird with the probability of paalso creates a new solution for the problem similar to the Lévyflights The new solution because of this action is calculated as follows

Xddis= X bestd+ K × DXdis

where K is the updated coefficient determined based on the probability of a host bird to discover an alien egg in its nest

K= 1 if rand3, pa

0 otherwise



(32) and the increased valueΔXddisis determined by

DXdis

d = rand4× randp1X bestd

− randp2X bestd

(33) where rand3and rand4are the distributed random numbers in [0,1] and randp1(Xbestd) and randp2(Xbestd) are the random perturbation for positions of nests in Xbestd

Similar to the solution obtained via Lévy flights, this new solution is also redefined as in (30) if the upper or lower limit is violated and Section 3.3 if any prohibited zones are violated The newly best value for each nest Xbestd and the best value of all nests Gbest are also determined based on comparing the calculated fitness function in (23) from this new solution and the stored one

in Section 3.4.2

Stopping criteria: The proposed algorithm is terminated when the predefined maximum number of iterations is reached

The flowchart of the proposed CSA method for solving non-convex ED problem is given in Fig 5

4 Numerical results

The proposed CSA is coded in Matlab platform and run 100 independent trials for each test case on a 2.1 GHz PC with 2

GB of RAM

4.1 Selection of parameters

In the proposed CSA method, four main parameters that have to be predetermined are the number of nests Np, maximum number of iterations Nmax, distribution factor β,

and the probability of an alien egg to be discovered in host

nests pa

Among these parameters, the number of nests can be easily

fixed Since CSA is a powerful search method, it only needs a

small number of nests for dealing with different systems

By experiment, the number of nests is fixed at 10 for all

test systems On the other hand, the maximum number of

iterations for the CSA can be also easily fixed depending

on the complexity and scale of the considered problems

The maximum number of iterations for the CSA ranges

from 300 for small systems up to 10 000 for large-scale

systems The value of distribution factor β can be fixed in

the range [0.3, 1.99] as in the Mantegna’s algorithm

However, different values of β have not much impact on

thefinal solution Therefore the value of β is fixed at 1.5 as

in [31] for all test systems in this paper The value of the

probability for an alien egg to be discovered can be chosen

in the range [0, 1] However, different values of pa may

lead to different optimal solutions for large-scale systems

To select the optimal probability, its value is varied from

0.1 to 0.9 with the step size of 0.1 for large-scale problems

with complicated objective function and fixed at a certain

value in the range [0.1, 0.5] for small-scale problems

4.2 Systems with VPE

The test system consists of 40 units with VPE supplying to a

load demand of 10 500 MW [13] System power loss in this

case is neglected The maximum number of iterations for the CSA isfixed at 8000 The results obtained by the proposed CSA including the minimum total cost, average total cost, maximum total cost, standard deviation, and average computational time with different values of pafrom 0.1 to 0.9 with the increase step of 0.1 are given in Table 1 For this system, the best of minimum total costs is $/h

121 412.5355 obtained at the probability of 0.5 whereas the best of average total costs, maximum total costs and standard deviations are respectively 121 494.7789, $/h

121 657.9782 and $/h 47.8170 obtained at the probability

of 0.2 As observed from Table 1 for this case, a good value of pa lies in the range [0.2, 0.5] The smaller and larger values of pawill not lead to optimal solution The average computational time for the proposed method to obtain the optimal solution of this system is around 3 s The optimal solution for the system is given in the Appendix

The best total cost and average computational time obtained by the proposed CSA for this system is compared

to those from many other methods such as MSL [7], improved fast evolutionary programming (IFEP) [13],

TM [14], modified PSO (MPSO) [17], combining of chaotic differential evolution and quadratic programming (DEC-SQP) [27], new PSO with local random search (NPSO-LRS) [22], self-organising hierarchical PSO (SOH_PSO) [20], PSO with recombination and dynamic linkage discovery (PSO-RDL) [23], quantum-inspired PSO (QPSO) [21], SA-PSO [28], BBO [15], improved coordinated aggregation-based PSO (ICA-PSO) [24, 25], DE/BBO [29], new adaptive PSO (NAPSO) [26] and CCPSO [18,19] as shown in Table 2 The minimum total cost obtained by the CSA is less than that from the other methods Moreover, the CSA method can obtain better solution in a faster computing manner than many other methods except MSL and BBO methods The computational times for the MSL, IFEP, NPSO-LRS, BBO and DE-BBO, ICA-PSO, and NAPSO methods are from a Pentium IV 1.5-GHz with 512-MB RAM PC, Pentium II 350-MHz with 128-MB RAM PC, Pentium IV 1.5-GHz with 128-MB RAM processor, Pentium IV 2.3-GHz PC with 512-MB RAM, Pentium IV 1.4-GHz PC and Pentium

IV 3-GHz PC with 2-GB RAM, respectively There is no computational time or computer processor reported for the other methods

4.3 Systems with multiple fuels The test systems here [8] comprise 10, 30, 60 and 100 units The basic 10-unit system supplies to a load demand of 2700

MW neglecting power loss For obtaining the large-scale

Fig 5 Flowchart of the proposed CSA method for solving

non-convex ED problem

Table 1 Results for 40-unit system with valve-point loading effects

p a Min total cost, $/h

Avg total cost, $/h

Max total cost, $/h

Std.

dev., $/h

Avg CPU, s 0.1 121 419.1726 121 567.6383 121 961.0546 111.2286 2.98 0.2 121 412.7163 121 494.7789 121 657.9782 47.8170 3.00 0.3 121 412.6466 121 502.9722 121 811.3775 79.9659 3.01 0.4 121 415.0031 121 513.5459 122 095.5398 111.4726 3.02 0.5 121 412.5355 121 520.4106 121 810.2538 81.5705 3.03 0.6 121 420.8949 121 579.6568 122 076.7172 131.5436 3.02 0.7 121 421.1218 121 645.0421 122 264.3211 166.3095 3.01 0.8 121 420.8949 121 862.9429 122 898.0568 306.0020 3.01 0.9 121 435.6459 122 105.6920 123 330.6021 429.1208 3.03

systems with 30, 60 and 100 units, the basic 10-unit system is

duplicated with the load demand proportionally adjusted to

the system size The maximum numbers of iterations for the

CSA for these systems are set to 300, 1000, 2000 and

3000, respectively The value of pa is fixed at 0.25 for all

systems The results obtained by the CSA are given in

Table3

Table4shows a comparison from the average total costs and

computational times obtained by the CSA for the systems to

those from conventional GA (CGA) and IGA_AMUM in

[10] Test result has indicated that the proposed CSA can

obtain less total costs and faster computational times than

both CGA and IGA-AMUM methods for all systems Note

the computational times for both CGA and IGA AMUM

methods were from a PIII-700 PC

4.4 Systems with prohibited operating zones

The test systems from [11] include 15, 30, 60 and 90 units

The basic 15-unit system with four units having prohibited

zones supplies to a load demand of 2650 MW neglecting power loss with a required spinning reserve of 200 MW The large-scale systems including 30, 60 and 90 units are formed by duplicating the basic 15-unit system with the corresponding load demand proportionally adjusted to the system size The maximum number of iterations for the CSA is set to 400, 1000, 1500 and 1900 for the systems, respectively The probability pa is fixed at 0.25 for all systems The results obtained by the CSA for the systems are given in Table5

A comparison of the average total costs and computational times from the CSA and other methods for these systems is shown in Table 6 The proposed CSA can obtain better solution quality than CGA and improved GA with multiplier updating method (IGAMUM) in [11] in terms of total cost and computational time for the large-scale systems The computational times for CGA and IGAMUM were from a PIII-700 PC

4.5 Systems with valve point effects and multiple fuels

The test systems in [12] consist of 10, 20, 40, 80 and 160 units The basic 10-unit system supplies to a load demand

of 2700 MW neglecting power loss The large-scale systems are created by duplicating the basic 10-unit system with the load demand proportionally adjusted to the system size The maximum number of iterations for the CSA for these systems is set to 500, 1000, 2000, 4000 and 6000, respectively The value of pa isfixed at 0.1 for all systems Table 7 shows the results obtained by the CSA for these systems

Table 3 Results for systems with MFs

min total cost, $/h 623.8092 1871.4275 3742.8559 6238.1144

avg total cost, $/h 623.8092 1871.4603 3743.2089 6240.4449

max total cost, $/h 623.8097 1872.9701 3753.3834 6250.5385

std Deviation, $/h 0.0001 0.2158 1.1403 3.3561

CPU time, s 0.679 2.517 5.765 10.268

Table 4 Comparison of average total costs and CPU times for

systems with MFs

Method No of units Total cost, $ CPU time, s

Table 5 Results for systems with POZ

No of units

min cost,

$/h

32 544.9704 65 084.9949 130 170.3949 195 258.7847 avg cost,

$/h

32 545.0068 65 085.1878 130 171.5986 195 264.3818 max cost,

$/h

32 546.6734 65 089.8697 130 174.0722 195 271.7057 Std dev.,

$/h

0.2386 0.6779 0.7531 2.4857 CPU

time, s

Table 6 Comparison of average total costs and CPU times for systems with POZ

Method No of units Total cost, $ CPU time, s

60 131 992.310 563.81

90 198 831.690 940.93

60 130 180.030 162.58

90 195 274.060 255.45

Table 2 Comparison of best total cost and average CPU time

for 40-unit system with valve-point loading effects

ICA-PSO [ 24 , 25 ] 121 422.1000 139.92

CCPSO [ 18 , 19 ] 121 412.5362 19.3

In Table8, the average total costs and computational times

obtained by the CSA are compared to those from CGA_MU

and IGA_MU in [12] In all cases, the CSA achieves better

solution quality than both CGA_MU and IGA_MU

methods, especially for the large-scale systems Note the

CGA_MU and IGA_MU methods were implemented on a

PIII-700 PC

4.6 Systems with valve point effects and

prohibited operating zones

The test systems for this problem consist of 40 and 140 units

The data for the 40-unit system with VPE are given in [13]

where units 10–14 have POZ as given in [26] (POZ 2) The

load demand for this system is 10 500 MW The complete

data of the 140-unit system can be found in [18], in which

12 units have the fuel cost function with VPE and four other units have POZ The total load demand of this system

is 49 342 MW Power loss is neglected in both systems The maximum number of iterations for the CSA for both systems is set to 10 000 The results obtained by the CSA for these systems with different values of pa are given in Table 9 The optimal solution for the 40-unit system is given in Appendix

A comparison of best total cost and average computational time for the 40 and 140-unit systems is shown in Table10 Obviously, the CSA can obtain less total cost than PSO, fuzzy adaptive PSO (FAPSO), and NAPSO in [26] for the 40-unit system and conventional PSO with the constraint treatment strategy (CTPSO), PSO with chaotic sequences (CSPSO), PSO with crossover operation (COPSO) and PSO with both chaotic sequences and crossover operation (CCPSO) in [18] for the 140-unit system Moreover, the computational time from the CSA is also faster than the other methods for the two systems except NAPSO for the 40-unit system Note that the PSO methods in [18] were implemented on a Pentium IV 2.0-GHz PC

In all test cases, the proposed CSA method dominates many other methods in the literature in finding better optimal solutions with faster computational times for the non-convex ED problems In the CSA method, two main features contributing to its search process are the Lévy flights and the behaviour of a host to discover an alien egg

in its nest, in which the Lévy flights are used to guide the search direction while the behaviour of alien egg discovery

is used to search the optimal solution The two features are combined together to constitute a powerful search ability for the CSA method Between the two features, the behaviour of a host bird to discover an alien egg in its nest

is the most effective one since it plays a key role to find an

Table 7 Results for systems with VPE and MF

Table 8 Comparison of average total costs and CPU times for

systems with VPE and MF

Method No of units Total cost, $ CPU time, s

160 10 143.7263 621.30

160 10 042.4742 174.62

Table 9 Results for 40-unit and 140-unit systems with VPE and POZ

No of units p a Min total cost, $/h Avg total cost, $/h Max total cost, $/h Std dev., $/h Avg CPU, s

optimal solution and can be independently used or combined

with other methods

5 Conclusion

In this paper, the CSA method has been efficiently

implemented for solving non-convex ED with practical

nonlinear characteristics of generators The proposed CSA is

a powerful search method with few controllable parameters

The obtained results from the several test systems have

indicated that the proposed CSA method has a much better

performance than the other optimisation methods reported in

the literature Therefore, the proposed CSA method is a

promising method for online non-convex ED

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7 Appendix

The optimal solutions for the 40-unit system from different problems are given in Table11

Table 10 Comparison of best total costs and average CPU

times for 40-unit and 140-unit systems with VPE and POZ

No of units Method Total cost, $/h CPU time, s

FAPSO [ 26 ] 122 261.3706 19.6

NAPSO [ 26 ] 121 491.0662 12.7

CSA 121 487.7727 14.71

CSPSO [ 18 ] 1 657 962.73 99

COPSO [ 18 ] 1 657 962.73 150

CCPSO [ 18 ] 1 657 962.73 150

CSA 1 655 746.14 38.90

Table 11 Optimal solutions of 40-unit system for different

problems

Unit With

VPE

With VPE and POZ

Unit With

VPE

With VPE and POZ

1 110.7998 110.7998 21 523.2794 523.2794

2 110.7998 110.7999 22 523.2794 523.2794

3 97.3999 97.3998 23 523.2794 523.2794

4 179.7331 179.7331 24 523.2794 523.2793

5 87.7999 87.7998 25 523.2794 523.2794

6 140.0000 140.0000 26 523.2794 523.2794

7 259.5997 259.5996 27 10.0000 10.0000

8 284.5997 284.5995 28 10.0000 10.0000

9 284.5997 284.5997 29 10.0000 10.0000

10 130.0000 130.0000 30 87.7999 87.7998

11 94.0000 168.7982 31 190.0000 190.0000

12 94.0000 168.0414 32 190.0000 190.0000

13 214.7598 125.0000 33 190.0000 190.0000

14 394.2794 400.0000 34 164.7998 164.7997

15 394.2794 394.2790 35 194.3978 164.7998

16 394.2794 394.2792 36 200.0000 164.7998

17 489.2794 489.2794 37 110.0000 110.0000

18 489.2794 489.2793 38 110.0000 110.0000

19 511.2794 511.2793 39 110.0000 109.9988

20 511.2794 511.2793 40 511.2794 511.2793