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Energy 44 (2012) 228e240

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Energy

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

Imperialist competitive algorithm for solving non-convex dynamic economicpower dispatch

Behnam Mohammadi-ivatloo a, Abbas Rabiee b, Alireza Soroudi c,*, Mehdi Ehsan a

aCenter of Excellence in Power System Management and Control, Department of Electrical Engineering, Sharif University of Technology, Tehran, IranbDepartment of Electrical Engineering, Abhar Branch, Islamic Azad University, Abhar, IrancDepartment of Electrical Engineering, Damavand Branch, Islamic Azad University, Damavand, Iran

a r t i c l e i n f o

Article history:Received 8 September 2011Received in revised form11 June 2012Accepted 12 June 2012Available online 15 July 2012

Keywords:Dynamic economic dispatchImperialist competitive algorithmProhibited operation zoneValve-point effectRamp-rate limitsOptimization

* Corresponding author. Tel.: þ98 (0)2166165954;E-mail addresses: ivatloo@ee.sharif.edu (B. Moh

ee.sharif.edu (A. Rabiee), alireza.soroudi@gmail.csharif.edu (M. Ehsan).

0360-5442/$ e see front matter � 2012 Elsevier Ltd.http://dx.doi.org/10.1016/j.energy.2012.06.034

a b s t r a c t

Dynamic economic dispatch (DED) aims to schedule the committed generating units’ output activepower economically over a certain period of time, satisfying operating constraints and load demand ineach interval. Valve-point effect, the ramp rate limits, prohibited operation zones (POZs), andtransmission losses make the DED a complicated, non-linear constrained problem. Hence, in this paper,imperialist competitive algorithm (ICA) is proposed to solve such complicated problem. The feasibility ofthe proposed method is validated on five and ten units test system for a 24 h time interval. The resultsobtained by the ICA are compared with other techniques of the literature. These results substantiate theapplicability of the proposed method for solving the constrained DED with non-smooth cost functions.Besides, to examine the applicability of the proposed ICA on large power systems, a test case with 54units is studied. The results confirm the suitability of the ICA for large-scale DED problem.

� 2012 Elsevier Ltd. All rights reserved.

1. Introduction

A power utility needs to ensure that the electrical power isgeneratedwithminimumcost. Hence, for economic operation of thesystem, the total demand must be appropriately shared among thegenerating units with an objective to minimize the total generationcost of the system. Thus, economic dispatch (ED) is one of theimportant problems of power system operation and control. Tradi-tional ED problem, attempts to minimize the cost of supplyingenergy subject to constraints on static behavior of the generatingunits. It is assumed that the amount of power to be supplied bya given set of committed units is constant for a given interval of time.However, to avoid shortening of the life of their equipment, plantoperators, try to keep thermal gradients inside the turbine withinsafe limits. This mechanical constraint is usually translated intoa limit on the rate of increase of the electrical output. Such ramp-rateconstraints lead to the construction of dynamic economic dispatch(DED) problem, which is an extension of conventional ED problem.

fax: þ98 (0)216602326.ammadi-ivatloo), a_rabiee@om (A. Soroudi), ehsan@

All rights reserved.

DED refers to the problem of determining minimum cost ofdispatch of generators for a given horizon of time, taking intoconsideration the constraints imposed on system operation by thegenerator ramp-rate limitations. To solve DED problem, generatorsare modeled using inputeoutput curves in most of the powersystem operation studies. Traditionally an approximate quadraticfunction used to model the generator inputeoutput curves [1,2].But, the generating units with multi-valve steam turbines exhibita greater variation in the fuel-cost functions; and thus the naturalinputeoutput curve is non-linear and non-smooth due to the effectof multiple steam admission valves (known as valve-points effect)[3,4]. Besides, generating units may have certain prohibited oper-ation zones (POZs) due to limitations of machine components orinstability concerns. Hence, considering the effect of valve-pointsand POZs in generators’ cost function, makes the DED a non-convex optimization problem.

Lots of optimization methods including classical and heuristicalgorithms were applied to solve DED problem. Due to non-convexity of the DED problem, application of classical methodslike Lagrangian relaxation [5] and dynamic programming [6] arerestricted. In recent years, Maclaurin series approximation has beenapplied to model the valve-point effects [7,8] but it has been shownthat this method leads to non-optimal solution.

B. Mohammadi-ivatloo et al. / Energy 44 (2012) 228e240 229

More recent works have been around artificial intelligence (AI)methods, such as artificial neural networks (ANN), simulatedannealing (SA), genetic algorithms (GA), differential evolution (DE),particle swarm optimization (PSO), evolutionary programming(EP), tabu search (TS), and hybrid methods. Optimization methodsbased on AI have shown better performance in solving the DEDproblem with capability of modeling more realistic objectivefunctions and constraints. In [9] hybrid EP and sequential quadraticprogramming (SQP) method has been proposed to solve non-convex DED problem. Chaotic quantum genetic algorithm (CQGA)is used in [10] for solving DED problem considering the effect ofwind generation. DE algorithms have received attention in solvingDED problems [11e17]. Other heuristic search methods have beenapplied to solve DED problems in the past decade. These include GA[1], quantum GA (QGA) [18], artificial immune system method [19],artificial bee colony algorithm (ABC) [20], PSO [15,16,21,22],multiple TS (MTS) algorithm [23], enhanced cross-entropy method[24], and SA algorithm [25]. Hybrid methods such as hybrid artifi-cial immune systems and SQP [26], hybrid EP and SQP method[9,22], hybrid swarm intelligence based harmony search algorithm[4], hybrid seeker optimization algorithm (SOA) and SQP [27],hybrid Hopfield neural network (HNN) and quadratic programming(QP) [28,29], adaptive hybrid DE algorithm [30], hybrid PSO andSQP [31], and artificial immune system (AIS) [32] are found to beeffective in solving complex optimization problems such as DEDproblem. Table 1 summarizes and compares different proposedalgorithms for solution of the DED problem.

In this paper, an imperialist competitive algorithm (ICA) isproposed to solve constrained non-convex DED problems. ICA isrecently proposed by Atashpaz-Gargari and Lucas [38]. This algo-rithm is inspired by the imperialistic competitive. ICA has shown

Table 1Summary of the proposed algorithms for solution of DED problem in literature.

Reference Algorithm POZ Valve

[2] Quadratic programming No No[4] Hybrid harmony search algorithm No Yes[6] Constructive dynamic programming No No[7] Maclaurin series-based Lagrangian method No Yes[9] Hybrid EP and SQP No Yes[10] Chaotic quantum GA No Yes[11] Chaotic sequence based DE No Yes[13] DE No Yes[14] Improved DE No Yes[15] Improved PSO No Yes[16] Modified DE No Yes[17] Chaotic DE No Yes[18] Quantum GA No Yes[19] Artificial immune system No Yes[20] Artificial bee colony No Yes[21] Adaptive PSO No Yes[22] Deterministically guided PSO No Yes[23] Multiple tabu search Yes No[24] Enhanced cross-entropy method No Yes[25] Simulated annealing No Yes[26] Hybrid AIS and SQP No Yes[27] Hybrid SOA and SQP No Yes[28] Hybrid HNN and QP No No[29] Hybrid HNN and QP No No[30] Adaptive hybrid DE No Yes[31] Hybrid PSO and SQP No Yes[32] Artificial immune system No Yes[33] Maclaurin series-based Lagrangian method No Yes[34] Modified hybrid EP and SQP No Yes[35] Hybrid DE No Yes[36] Covariance matrix adapted evolution strategy No Yes[37] Improved chaotic PSO No YesProposed Imperialist competitive algorithm Yes Yes

good performance in solving optimization problems in differentareas such as template matching [39], DG planning [40], optimaldesign of plate-fin heat exchangers [41] and electromagneticproblems [42]. This algorithm also has been successfully applied topower systemproblems like as PSS (power system stabilizer) design[43], linear induction motor design [44], unit commitment [45] andmodel reduction of a detailed transformer model [46]. The chaoticversion of the ICA is presented in [47] for global optimization.

Application of ICA to benchmark and large scale DED test casesshow that ICA is capable to find better results comparingwith otherheuristic algorithms. The rest of the paper is organized as follows:

In Section 2 the mathematical formulation of the DED problemis given, considering POZs, ramp-rate limits, valve-point effects andtransmission losses. Section 3 proposes the ICA and describes itsimplementation on DED problems. Section 4 is devoted to casestudies and numerical results. In this section, four application casesare studied, and the corresponding comparisons with the recentlyapplied methods are presented. Conclusions are finally outlined inSection 5.

2. Dynamic economic dispatch problem formulation

The objective function of DED problem is to minimize thetotal production cost over the operating horizon, which can bewritten as:

minTC ¼XTt¼1

XNi¼1

CitðPitÞ (1)

where Cit (in $/h) is the production cost of unit i at time t, N is thenumber of dispatchable power generation units and Pit (in MW) is

point Transmission loss Test cases and time horizon

No 5-unit (24 h), 10-unit (24-h)Yes 5-unit (24 h), 10-unit (24 h), 30-unit (24 h)No 3-unit (24 h)Yes 5-unit (24 h)No 10-unit (24-h)Yes 6-unit (24-h), 10-unit (24-h)No 10-unit (24 h), 13-unit (24 h)Yes 5-unit (24 h), 10-unit (24-h)Yes 5-unit (24 h), 10-unit (24-h)Yes 10-unit (24 h), 30-unit (24 h)No 10-unit (24 h)No 10-unit (24 h), 30-unit (24 h)Yes 10-unit (24 h)Yes 5-unit (24 h), 10-unit (24-h)Yes 5-unit (24 h), 10-unit (24-h)Yes 5-unit (24 h)Yes 10-unit (24 h), 30-unit (24 h)Yes 6-unit (24 h), 15-unit (24 h)Yes 10-unit (24 h), 30-unit (24 h)Yes 5-unit (24 h)No 10-unit (24 h)Yes 5-unit (24 h), 10-unit (24-h)No 10-unit (12 h)No 10-unit (12 h)Yes 10-unit (24 h)Yes 3-unit (24 h), 10-unit (24 h)Yes 10-unit (24 h)Yes 5-unit (24 h)Yes 10-unit (24 h), 30-unit (24 h)No 10-unit (24 h)Yes 5-unit (24 h), 10-unit (24-h)Yes 10-unit (24 h), 30-unit (24 h)Yes 5-unit (24 h), 10-unit (24-h), 54-unit (24-h)

Fig. 1. The flowchart of the proposed algorithm.

B. Mohammadi-ivatloo et al. / Energy 44 (2012) 228e240230

the power output of ith unit at time t. T is the total number of hoursin the operating horizon. The production cost of a generation unitconsidering valve-point effects is defined as:

CitðPitÞ ¼ aiP2it þ biPit þ ci þ

���eisin�fi�Pmini � Pit

����� (2)

where ai, bi and ci are the fuel cost coefficients of the ith unit, ei andfi are the valve-point coefficients of the ith unit. The units of theabove coefficients are ($/MW2 h), ($/MWh), ($/h), ($/h) and(1/MW), respectively. Pmin

i (in MW) is the minimum capacity limitof unit i. The added sinusoidal term in the production cost functionreflects the effect of valve-points. The DED problem is non-convexand non-differentiable considering valve-point effects [48].

The objective function of the DED problem (1) should be mini-mized subject to the following constraints:

1. Real power balance

Hourly power balance considering network transmission lossesis written as:

XNi¼1

Pit ¼ PDðtÞ þ PlossðtÞ t ¼ 1;2;.; T (3)

where Ploss(t) and PD(t) (both inMW) are total transmission loss andtotal load demand of the system at time t, respectively. System lossis a function of units power production and the topology of thenetwork which can be calculated using the results of load flowproblem [31] or Kron’s loss formula known as B� matrix coeffi-cients [28]. In this work, B� matrix coefficients method is used tocalculate system loss, as follows:

PlossðtÞ ¼XNi¼1

XNj¼1

PitBijPjt þXNi¼1

Bi0Pit þ B00 t ¼ 1;2;.; T (4)

2. Generation limits of units:

Pmini � Pit � Pmax

i i ¼ 1;.;N; t ¼ 1;2;.; T (5)

where Pmaxi (in MW) is the maximum power outputs of ith unit.

3. Rampup and ramp down constraints: The output power changerate of the thermal unit must be in an acceptable range to avoidundue stresses on the boiler and combustion equipments [49].The ramp rate limits of generation units are stated as follows:

Pit � Pit�1 � URi i ¼ 1;.;N; t ¼ 1;2;.; T (6)

Pit�1 � Pit � DRi i ¼ 1;.;N; t ¼ 1;2;.; T (7)

where URi is the ramp up limit of the ith generator (MW/h) and DRiis the ramp down limit of the ith generator (MW/h). Consideringramp rate limits of unit, generator capacity limit Eq. (5) can berewritten as follows:

max�Pmini ; Pit�1 � DRi

�� Pit � min

�Pmaxi ; Pit�1 þ URi

�i ¼ 1;.;N; t ¼ 1;2;.; T

(8)

4. Prohibited operation zones limits (POZs):

Generating units may have certain restricted operation zonedue to limitations of machine components or instabilityconcerns. The allowable operation zones of generation unit canbe defined as:

Pit˛

8><>:

Pmini � Pit � Pli;1

Pui;j�1 � Pit � Pli;j j ¼ 2;3;.;Mi i ¼ 1;.;N t ¼ 1;2;.;TPui;Mi

� Pit � Pmaxi

(9)

where Pli;j and Pui;j are the lower and upper limits of the jth pro-hibited zone of unit i, respectively. Mi is the number of prohibitedoperation zones of unit i.

3. Imperialist competitive algorithm

The ICA was first proposed in [38]. It is inspired by the imperi-alistic competition. It starts with an initial population called colo-nies. The colonies are then categorized into two groups namely,imperialists (best solutions) and colonies (rest of the solutions). Theimperialists try to absorb more colonies to their empire. The colo-nies will change according to the policies of imperialists. Thecolonies may take the place of their imperialist if they becomestronger than it (propose a better solution). The flowchart ofproposed algorithm which is the same as [38] for solving the DEDproblem is depicted in Fig. 1. The imperialist competitive algorithmis very strong in pattern recognition. This aspect is used in this

pap

erto

findtheoptim

algeneratin

gsch

edule

oftherm

alunits

overagiven

period

.Theobjective

function

(OF)is

defined

assu

mmation

oftotalcost

Eq.(1)an

dpen

altiesfor

constrain

tviolation

s.

OFc¼ X

T

t¼1 X

N

i¼1

Cit ðP

itc Þþb1 X

T

t¼1 "X

N

i¼1 ðP

itc Þ�PD ðtÞþ

Ploss ðtÞ #

2

þb2 X

T

t¼1 X

N

i¼1 ��� P

itc �Plimit ��� þ

b3 X

T

t¼1 X

N

i¼1 ��� P

itc �Ppozlim

it

���(10)

where

b1 ,b

2an

db3are

pen

altyparam

etersan

dassu

med

tobe

2000

inthis

pap

er.Pitc

refersto

pow

erprod

uction

ofunitiat

timetin

colonyc.

Plimit

and

Ppozlim

itare

constrain

tviolation

indicator

and

defined

asfollow

s.

Plimit

¼ 8<:max �P

min

i;P

it�1 �

DRi �

ifPit �

max �P

min

i;P

it�1 �

DRi �

min �P

max

i;P

it�1 þ

URi �

ifPit �

min �P

max

i;P

it�1 þ

URi �

Pit

otherw

ise(11)

Ppozlim

it¼ 8>>>>>>>>>>>><>>>>>>>>>>>>:

Pmin

iifPit �

Pmin

i

Pli;1

ifPit �

Pli;1

Pui;j�

1ifPit �

Pui;j�

1

Pli;j

ifPit �

Pli;j

Pui;M

iifPit �

Pui;M

i

Pmax

iifPit �

Pmax

iPit ;

otherw

ise

(12)

Thestep

sof

theproposed

ICA

forminim

izationproblem

sare

described

asfollow

s:

Step1.

Aninitial

setof

colonies

with

thesize

ofNcsh

ould

becreated

.SetIteration

¼1.

Step2.Th

eobjective

function

iscalcu

latedfor

eachcolony

usin

gEq.(2)

andthepow

erof

eachcolony

isset

asfollow

s:

CPc¼Ste

an

IPi ¼StecaimIP

iP

Nim

p

j¼1

Stecr

.3 0.4 0.5 0.6 0.7 0.8 0.9 1

,040,759.762 1,040,857.165 1,040,832.173 1,040,822.788 1,040,759.200 1,040,787.808 1,040,861.282 1,040,834.126,040,801.027 1,040,813.022 1,040,774.911 1,040,822.508 1,040,838.791 1,040,859.513 1,040,819.286 1,040,854.461,040,804.587 1,040,901.610 1,040,866.508 1,040,758.424 1,040,833.076 1,040,775.074 1,040,828.074 1,040,774.610,040,812.858 1,040,759.515 1,040,821.431 1,040,810.875 1,040,845.324 1,040,794.061 1,040,842.433 1,040,818.478,040,787.537 1,040,797.492 1,040,852.562 1,040,838.355 1,040,860.900 1,040,814.770 1,040,771.166 1,040,828.326,040,783.394 1,040,834.461 1,040,825.106 1,040,776.906 1,040,760.407 1,040,841.346 1,040,856.349 1,040,794.907,040,850.795 1,040,858.601 1,040,820.498 1,040,784.803 1,040,856.512 1,040,760.408 1,040,837.067 1,040,859.417,040,835.277 1,040,825.962 1,040,798.368 1,040,779.009 1,040,857.959 1,040,822.541 1,040,760.712 1,040,813.239,040,852.750 1,040,774.224 1,040,803.692 1,040,829.272 1,040,803.259 1,040,777.327 1,040,815.166 1,040,835.049,040,826.316 1,040,786.835 1,040,767.926 1,040,840.978 1,040,817.465 1,040,819.043 1,040,860.415 1,040,787.196,040,776.518 1,040,826.567 1,040,803.601 1,040,817.051 1,040,818.600 1,040,761.290 1,040,853.439 1,040,779.244

B.Moham

madi-ivatloo

etal./

Energy44

(2012)228

e240

1OFc ;

c¼

1:N

c(13)

p3.Th

eNim

pstron

gestcolon

iesare

keptas

theim

perialists

dthepow

erof

eachim

perialist

i.e.IPi ,is

setas

follows:

1OFi ;i¼

1:N

imp

(14)

p4.

Assign

the

colonies

toeach

imperialist

according

tolcu

latedIP

i .Thismean

sthenumber

ofcolonies

owned

byeach

perialist

isprop

ortional

toits

pow

er,i.e.IPi .

IPj � �N

c �Nim

p �

p5.

Thecolon

iesare

moved

toward

their

imperialist

usin

gossover

andmutation

operators.

Table 2Parameter selection of ICA algorithm.

Crossover probability

0 0.1 0.2 0

Mutationprobability

0 1,040,809.203 1,040,793.703 1,040,814.214 10.1 1,040,861.647 1,040,798.899 1,040,805.738 10.2 1,040,892.690 1,040,862.048 1,040,813.833 10.3 1,040,775.147 1,040,814.383 1,040,855.586 10.4 1,040,783.293 1,040,793.448 1,040,855.756 10.5 1,040,809.212 1,040,769.845 1,040,789.243 10.6 1,040,855.085 1,040,834.106 1,040,760.549 10.7 1,040,828.667 1,040,837.343 1,040,804.916 10.8 1,040,764.370 1,040,844.350 1,040,830.527 10.9 1,040,773.374 1,040,791.545 1,040,792.113 11 1,040,793.354 1,040,802.358 1,040,773.989 1

231

Table 3Optimal solution of 5-unit using proposed algorithm.

Hour P1 (MW) P2 (MW) P3 (MW) P4 (MW) P5 (MW) Cost ($)P5

i¼1 PiðMWÞ PD (MW)

1 10 20 30 124.485 229.504 1226.587 413.989 4102 19.078 20 30 140.846 229.52 1418.346 439.444 4353 10 20 30 190.846 229.519 1493.566 480.365 4754 10 20 67.023 209.816 229.52 1662.802 536.359 5305 10 20 95.511 209.816 229.515 1667.456 564.842 5586 13.949 50 112.675 209.816 229.52 1826.62 615.96 6087 10 72.451 112.673 209.816 229.52 1840.605 634.46 6268 12.709 98.54 112.674 209.815 229.52 1797.229 663.258 6549 42.709 102.78 115.353 209.817 229.52 2013.697 700.179 69010 64.03 98.54 112.671 209.799 229.519 1996.68 714.559 70411 75 98.791 117.878 209.816 229.52 2039.988 731.005 72012 75 124.71 112.674 209.816 229.521 2180.027 751.721 74013 64.012 98.54 112.673 209.816 229.52 1996.599 714.561 70414 49.62 98.54 112.673 209.816 229.519 1977.667 700.168 69015 35.892 98.54 112.673 186.5 229.52 2010.648 663.125 65416 10 98.54 112.674 136.5 229.52 1682.8 587.234 58017 10 87.586 112.672 124.905 229.519 1615.305 564.682 55818 10 98.54 112.674 165.218 229.52 1853.472 615.952 60819 12.709 98.54 112.674 209.816 229.52 1797.224 663.259 65420 42.709 119.939 112.674 209.816 229.52 2115.511 714.658 70421 39.353 98.54 112.674 209.816 229.52 1944.597 689.903 68022 10 98.541 110.204 164.619 229.52 1860.868 612.884 60523 10.001 98.54 70.204 124.908 229.52 1643.076 533.173 52724 10 73.366 30.204 124.908 229.519 1455.677 467.997 463Total 43,117.047 14,773.737

5.8

x 104

B. Mohammadi-ivatloo et al. / Energy 44 (2012) 228e240232

Step 6. Exchange the position of a colony and the imperialist if itis stronger (CPc > IPi). If there are several colonies better thanthe imperialist, then the imperialist will be replaced by the bestof them.Step 7. Compute the empire’s power, i.e. EPi for all empires asfollows:

EPi ¼ w1 � IPi þw2 �Xc˛Ei

CPc (15)

where,w1 andw2 are weighting factors which are selected in a waythat the algorithm will not be trapped into a local Minima. For thisreason, the value of w1 is selected as a number about 10e20% andw2 ¼ 1 � w1.

Step 8. Pick the weakest colony and give it to one of the bestempires (select the destination empire probabilistically basedon its power (EPi)).

Table 4Comparison of optimization results for 5-unit test system (case 1).

Method Minimumcost ($)

Averagecost ($)

Maximumcost ($)

Number oftrial runs

SA [25] 47,356 NA NA NAAPSO [21] 44,678 NA NA NAAIS [19] 44,385.43 44,758.8363 45,553.7707 30GA [20] 44,862.42 44,921.76 45,893.95 30PSO [20] 44,253.24 45,657.06 46,402.52 30ABC [20] 44,045.83 44,064.73 44,218.64 30MSL [7] 49,216.81 NA NA NAHS [4] 44,367.23 NA NA 25HHS [4] 43,154.8554 NA NA 25DE [13] 45,372 43,739 44,214 20DE [14] 45,800 NA NA 10Proposed (ICA) 43,117.055 43,144.472 43,209.533 100

NA denotes that the value was not available in the literature.

Step 9. Eliminate the empires that have no colony.Step 10. If more than one empire remained andIteration� Itermax then Iteration¼ Iterationþ 1 and go to Step. 5Step 11. End.

It should be noted that the Nc and Nimp are given constants andare determined by the expert who uses the algorithm. Typically10e20% of Nc would be a good choice for Nimp. The steps of thealgorithm is shown in Fig. 1.

The operating schedule of each country (for all operatingperiods) is binary coded. The generation value in time t of unit iis calculated as follows: suppose that the row t in the columncorresponding to unit i is vec ¼ [string of binary values].

0 50 100 150 200

4.2

4.4

4.6

4.8

5

5.2

5.4

5.6

Iteration

To

tal C

ost ($)

Fig. 2. Convergence characteristics of the ICA algorithm for 5-unit test system.

Table 5Analysis of objective function for different number of trial runs for 5-unit testsystem.

Numberof runs

Minimumcost ($)

Averagecost ($)

Maximumcost ($)

SD ($)

25 43,117.055 43,141.643 43,187.928 19.06150 43,117.055 43,142.606 43,209.533 19.9675 43,117.055 43,143.876 43,209.533 19.834100 43,117.055 43,144.472 43,209.533 19.821

10 20 30 40 50 60 70 80 90 100

4.312

4.314

4.316

4.318

4.32

4.322x 10

4

Run Number

To

tal C

ost ($)

Cost

Minimum

Average

Maximum

Fig. 3. Distribution of the objective function for 100 trial runs for 5-unit test system.

B. Mohammadi-ivatloo et al. / Energy 44 (2012) 228e240 233

Pit ¼ ðPmaxi � Pmin

i Þ�½vec0*½2n�1;n ¼ N : 1��=2N þ Pmini . Where N is

the number of generating units. The binary coding ofeach country (which may become an imperialist or not) canbe helpful in easily using the crossover and mutation operatorsof GA.

Table 6Optimal 24-h schedule of ten-unit test system (case 2).

Hour P1 P2 P3 P4 P5 P6

1 150 135 194.065 60 122.88 122.42 226.624 135 191.461 60 122.867 122.43 303.249 142.266 185.208 60 172.733 142.54 379.874 222.266 196.603 60 172.733 122.55 379.868 222.266 183.675 60 222.6 1606 455.434 302.266 263.674 60 172.601 122.47 379.898 309.534 305.892 110 222.601 122.48 456.497 316.799 297.946 120.418 172.747 1609 456.497 396.799 303.71 132.802 222.6 16010 456.497 460 297.781 182.802 233.328 16011 456.491 460 300.462 232.802 222.598 159.912 456.498 460 318.192 282.802 222.6 16013 456.497 396.8 307.935 238.264 222.6 16014 456.446 396.799 297.407 188.264 172.733 122.415 379.872 393.192 297.301 170.448 122.863 122.416 303.251 313.192 331.753 120.449 73 122.417 226.624 309.533 295.168 113.568 122.755 122.418 303.248 315.523 303.703 120.416 172.751 122.419 379.872 395.523 295.242 120.341 172.671 122.420 456.512 460 340 170.341 222.671 132.521 456.497 389.533 322.67 120.342 222.604 122.422 379.85 309.533 283.231 70.342 172.707 122.423 303.249 229.533 203.235 60 122.867 123.224 226.639 222.267 189.711 60 73 122.4Total

4. Case studies and numerical results

In this section, the proposed ICA is applied to four test systemswith different number of generating units. All the programs aredeveloped using MATLAB 7.1 on a Pentium IV personal computerwith 3.6 GHz speed processor and 2 GB RAM.

4.1. Determination of ICA parameters

The number of colonies is an important factor in determina-tion of optimal solution. If the number of colonies is increasedthen it can better explore the solution space but it wouldaugment the computation burden. In this paper, the number ofcolonies are assumed to be Nc ¼ 100 like as most of the ICApapers [38e40]. The following procedure has been adopted tocalculate optimum value of the mutation and crossover proba-bilities and w1, w2 ¼ 1 � w1. The value of w1 is varied from 0.05 to0.9 the problem is solved for various combinations of mutationand crossover probabilities. There is a unique set of w1 andprobability values the total cost is minimum. The results foroptimal value of w1 for 10-unit test case, which is equal to 0.15are given in Table 2. It should be noted that due to lack of spaceonly the table corresponding to the optimal values are givenhere.

For all cases, the dispatch horizon is selected as one day with24 dispatch periods where each period is assumed to be 1 h. Inthis paper the stopping criteria is defined as reaching to themaximum number of iterations (Itermax ¼ 200 for cases 1e3 and800 iterations for case 4) and when no more than one imperialistexists in the search space. For getting better starting point, firstDED problem is solved without considering valve-point effects,losses, and POZs, which results in a convex quadratic program-ming problem. The initial sets of colonies are generated byrandomly perturbing the results of quadratic programmingproblem.

P7 P8 P9 P10 Cost ($) PD

6 129.594 47 20 55 28,238.754 103657 129.591 47 20 55 29,828.077 111046 129.997 47 20 55 33,347.045 125826 129.997 47 20 55 36,296.715 1406

129.59 47 20 55 37,991.334 148034 129.59 47 20 55 41,387.159 162881 129.594 47 20 55 42,844.529 1702

129.593 47 20 55 44,600.484 1776129.59 47 20.002 55 47,885.318 1924129.59 47 50.002 55 51,887.342 2072

99 129.59 77 52.057 55 53,788.277 2146129.594 85.312 50.002 55 55,605.118 2220129.59 85.312 20.002 55 51,357.359 2072

5 129.59 85.312 20 55 47,818.061 192421 129.59 85.312 20 55 44,649.659 177651 129.592 85.312 20 55 39,816.706 155449 129.59 85.312 20 55 37,983.869 148056 129.59 85.312 20 55 41,294.355 162848 129.59 85.312 20 55 44,374.06 177671 129.592 85.312 20 55 51,862.515 20725 129.591 85.312 20 55 47,915.54 192435 129.59 85.312 20 55 41,280.418 162814 129.59 85.312 20 55 34,952.455 133281 129.591 85.312 20 55 31,462.345 1184

1,018,467.494

Table 8Analysis of objective function for different number of trial runs for 10-unit testsystem without considering losses (case 2).

Numberof runs

Minimumcost ($)

Averagecost ($)

Maximumcost ($)

SD ($)

25 1,018,467.49 1,019,240.406 1,021,284.875 687.5650 1,018,467.49 1,019,264.59 1,021,795.773 713.78675 1,018,467.49 1,019,329.503 1,021,795.773 727.194100 1,018,467.49 1,019,291.358 1,021,795.773 693.487

Table 7Comparison of optimization results for case 2.

Method Minimum cost ($) Average cost ($) Maximum cost ($)

DE [13] 1,019,786.000 NA NAEP-SQP [9] 1,031,746.000 1,035,748.000 NAPSO-SQP [31] 1,027,334.000 1,028,546.000 1,033,986.000DGPSO [22] 1,028,835.000 1,030,183.000 NAMHEP-SQP [34] 1,028,924.000 1,031,179.000 NAIPSO [15] 1,023,807.000 1,026,863.000 NAHDE [35] 1,031,077.000 NA NAIDE [14] 1,026,269.000 NA NAABC [20] 1,021,576.000 1,022,686.000 1,024,316.000MDE [16] 1,031,612.000 1,033,630.000 NACMAES [36] 1,023,740.000 1,026,307.000 1,032,939.000AIS [19] 1,021,980.000 1,023,156.000 1,024,973.000HHS [4] 1,019,091.000 NA NAICPSO [37] 1,019,072.000 1,020,027.000 NAAIS-SQP [26] 1,029,900.000 NA NASOA-SQP [27] 1,021,460.010 NA NACS-DE [11] 1,023,432.000 1,026,475.000 1,027,634.000CDE [17] 1,019,123.000 1,020,870.000 1,023,115.000AHDE [30] 1,020,082.000 1,022,474.000 NAECE [24] 1,022,271.579 1,023,334.930 NAHS [4] 1,046,725.910 NA NADE [11] 1,027,634.000 1,023,432.000 1,026,475.000IDE [14] 1,026,269.000 NA NAProposed (ICA) 1,018,467.490 1,019,291.358 1,021,795.773

NA denotes that the value was not available in the literature.

B. Mohammadi-ivatloo et al. / Energy 44 (2012) 228e240234

4.2. Case 1: five unit system

The first test system is a 5-unit test system. The data for thissystem are adapted from [25] and provided in B. In this test system,transmission losses and ramp rate constraints are considered. Thehourly load profile for this case is presented in last column ofTable 3.

The DED problem of 5-unit system is solved using the proposedalgorithm. The valve-point effects, transmission losses, ramp rateconstraints and generation limits are considered in this system. Theprohibited operating zones are not considered in this test case forthe sake of comparison of results with those reported in literatureusing different methods. Table 3 shows the obtained results for thissystem.

0 50 100 150 200

1

1.05

1.1

1.15

1.2

1.25

1.3

1.35

x 106

Iteration

To

ta

l C

os

t ($

)

Fig. 4. Convergence characteristics of the ICA algorithm for 10-unit test system.

These results are compared with several methods presented inrecent literature in terms of minimum cost, mean cost, andmaximum cost over 100 runs in Table 4. The results of the proposedalgorithm are in bold. Themaximum iteration number is selected tobe 200. The convergence characteristic of the proposed algorithm isdepicted in Fig. 2. By investigating the results presented in Table 4,it can be observed that the obtained results outperform the othercited methods for 5-unit test case. In order to analysis thecomputational efficiency of the proposed algorithm, differentsimulations are done considering number of trial runs and itsdistribution. Table 5 show the minimum, average, maximum andstandard deviation (SD) of objective function for different numbersof trial runs. Fig. 3 shows the distribution of the objective functionfor trial run number of 100.

4.3. Case 2: ten unit system without transmission loss

The second test system is ten-unit test system. In this case,generators capacity limits, ramp rate constraint and valve-pointeffects are considered. The transmission losses are ignored in thiscase for sake of comparison. The data for this system is adaptedfrom [25] and provided in B. The hourly load profile for this case ispresented in last column of Table 6.

Table 6 shows the obtained results for 10-unit system withoutconsidering transmission losses. Theminimum cost, mean cost, andmaximum cost of obtained optimal results are compared withresults of previously developed algorithms such as differentialevolution (DE) [13], hybrid EP and SQP [9], Hybrid PSO-SQP [31],deterministically guided PSO (DGPSO) [22], modified hybrid EP-SQP(MHEP-SQP) [34], improved PSO (IPSO) [15], hybrid DE (HDE) [35],improved DE (IDE) [14], artificial bee colony algorithm (ABC) [20],modified differential evolution (MDE) [16], covariance matrixadapted evolution strategy (CMAES) [36], artificial immune system

10 20 30 40 50 60 70 80 90 100

1.018

1.0185

1.019

1.0195

1.02

1.0205

1.021

1.0215

1.022x 10

6

Run Number

To

tal C

ost ($)

Cost

Minimum

Average

Maximum

Fig. 5. Distribution of the objective function for 100 trial runs for 10-unit test systemwithout losses (case 2).

Table 9Optimal 24-h schedule of ten-unit test system (case 3).

Hour P1 P2 P3 P4 P5 P6 P7 P8 P9 P10 Cost ($) Loss (MW)

1 150 135 206.166 60 122.87 122.499 129.602 47 20 55 28,592.287 12.1372 226.624 135 204.767 60 122.867 122.463 129.592 47 20 55 30,218.122 13.3133 303.249 142.272 186.402 60 172.758 160 129.591 47 20 55 33,728.388 18.2724 379.87 222.267 222.291 60 172.754 122.466 129.613 47 20 55 36,993.096 25.2615 379.873 222.27 211.933 60 222.61 160 129.59 47 20 55 38,788.438 28.2766 456.496 302.27 290.073 67.72 172.718 122.449 129.59 47 20 55 42,039.292 35.3167 379.879 309.534 331.883 117.72 222.718 124.031 129.936 47 20 55 43,737.143 35.7018 456.497 314.872 328.142 130.832 172.732 160 129.591 47 20 55 45,776.26 38.6669 456.497 394.872 297.293 180.832 222.597 160 129.59 55.313 20 55 49,108.729 47.99410 456.5 460 307.325 230.832 222.605 160 129.59 85.313 20 55 53,074.979 55.16511 456.498 460 340 241.936 222.6 160 129.958 115.313 20 55 55,072.649 55.30512 456.497 460 340 264.671 243 160 129.591 120 50 55 57,430.259 58.75913 456.511 396.8 340 250.839 222.73 160 129.949 90 20 55 52,887.564 49.82914 456.499 396.799 297.406 233.434 172.736 122.45 129.591 85.312 20 55 48,916.081 45.22715 379.873 396.557 318.492 183.434 122.87 123.333 129.598 85.312 20 55 45,517.715 38.46916 303.248 316.557 296.785 179.396 73 122.45 129.591 85.312 20 55 40,406.888 27.33917 226.624 309.533 305.063 129.396 122.882 122.649 129.597 85.312 20 55 38,655.592 26.05618 303.249 316.799 333.065 120.416 172.733 122.453 129.591 85.312 20 55 42,178.909 30.61819 379.872 396.799 322.806 130.766 172.733 122.753 129.591 85.312 20 55 45,537.406 39.63220 456.497 460 340 180.766 222.599 160 129.591 101.423 20 55 53,346.47 53.87621 456.497 389.548 340 130.766 222.602 140.445 129.591 85.311 20 55 49,313.097 45.76022 379.873 309.548 304.15 80.884 172.754 122.45 129.591 85.313 20 55 42,008.685 31.56323 303.249 229.548 224.15 60 122.866 122.45 129.591 85.312 20 55 35,495.677 20.16624 226.625 222.267 205.669 60 73 122.626 129.591 85.319 20 55 31,934.698 16.097Total 1,040,758.424 848.797

1.3

1.35

1.4

x 106

B. Mohammadi-ivatloo et al. / Energy 44 (2012) 228e240 235

(AIS) [19], hybrid swarm intelligence based harmony search algo-rithm (HHS) [4], improved chaotic particle swarm optimizationalgorithm (ICPSO) [37], hybrid artificial immune systems andsequential quadratic programming (AIS-SQP) [26], hybrid SOA-SQPalgorithm [27], chaotic sequence based differential evolution algo-rithm (CS-DE) [11], chaotic differential evolution (CDE)method [17],adaptive hybrid differential evolution algorithm (AHDE) [30], andenhanced cross-entropy method (ECE) [24], harmony search (HS)[4], DE [11] and Improved DE [14] in Table 7. Results of the proposedmethod are in bold. Themaximum iteration number and number oftrails are selected to be 200 and 100, respectively. The convergencecharacteristic of the proposed algorithm is depicted in Fig. 4. It canbe observed that the obtained resultswith ICA algorithm is less thanthose of reported in literature. Table 8 show the minimum, average,maximum and standard deviation (SD) of objective function fordifferent numbers of trial runs. Fig. 5 shows the distribution of theobjective function for trial run number of 100.

4.4. Case 3: ten unit system with transmission loss

The data for this case is similar to case 2. In this case, thetransmission losses also considered. The B� matrix coefficients of

Table 10Comparison of optimization results for case 3.

Method Minimum cost ($) Average cost ($) Maximum cost ($)

EP [34] 1,054,685 1,057,323 NAEP-SQP [34] 1,052,668 1,053,771 NAMHEP-SQP [34] 1,050,054 1,052,349 NAGA [20] 1,052,251 1,058,041 1,062,511PSO [20] 1,048,410 1,052,092 1,057,170IPSO [15] 1,046,275 1,048,145 NAECE [24] 1,043,989.154 1,044,470.0849 NAABC [20] 1,043,381 1,044,963 1,046,805AIS [19] 1,045,715 1,047,050 1,048,431Proposed (ICA) 1,040,758.424 1,041,664.622 1,043,173.551

NA denotes that the value was not available in the literature.

this system can be found in [25] which is given in per unit(100 MW base). The proposed algorithm applied to ten-unit testcase with taking into account the transmission losses. The corre-sponding generation dispatch is presented in Table 9. Theminimum cost, mean cost, and maximum cost of obtained optimalresults over 100 runs are compared with the results of evolu-tionary programming (EP) [34], hybrid EP-SQP (EP-SQP) [34],modified hybrid EP-SQP (MHEP-SQP) [34], genetic algorithm (GA)[20], particle swarm optimization (PSO) [20], improved PSO (IPSO)[15], enhanced cross-entropy method (ECE) [24], artificial beecolony algorithm (ABC) [20] and artificial immune system (AIS)[19] in Table 10. Results of the proposed method are in bold.

0 50 100 150 200

1

1.05

1.1

1.15

1.2

1.25

Iteration

To

tal C

ost ($)

Fig. 6. Convergence characteristics of the ICA algorithm for 10-unit test system withloss.

Table 11Analysis of objective function for different number of trial runs for 10-unit testsystem with considering losses (case 3).

Numberof runs

Minimumcost ($)

Averagecost ($)

Maximumcost ($)

SD ($)

25 1,040,758.424 1,041,722.489 1,043,173.551 635.97950 1,040,758.424 1,041,685.962 1,043,173.551 621.07475 1,040,758.424 1,041,668.016 1,043,173.551 612.612100 1,040,758.424 1,041,664.622 1,043,173.551 603.758

10 20 30 40 50 60 70 80 90 100

1.0405

1.041

1.0415

1.042

1.0425

1.043

x 106

Run Number

To

ta

l C

os

t ($

)

Cost

Minimum

Average

Maximum

Fig. 7. Distribution of the objective function for 100 trial runs for 10-unit test systemwith losses (case 3).

0 100 200 300 400 500 600 700 800

1.8

1.9

2

2.1

2.2

2.3

2.4

x 106

Iteration

To

tal C

ost ($)

ICA

GA

PSO

Fig. 8. Convergence characteristics of the ICA algorithm compared with PSO and GAfor case 4.

B. Mohammadi-ivatloo et al. / Energy 44 (2012) 228e240236

The convergence characteristic of the proposed algorithm isdepicted in Fig. 6. The minimum, average, maximum and standarddeviation (SD) of objective function for different numbers of trialruns are shown in Table 11. Distribution of the objective function for100 trial runs are depicted in Fig. 7.

4.5. Case 4: 54 unit system

In this case, a 54-unit test system is employed. The data of thissystem is adopted from [50]. The valve-point effects and POZs areconsidered here. Hence this is a large non-convex test case. Theresults obtained using the ICA are presented in Table A.1 for theload demand which is also given in Table A.1. Beside the ICA, twodifferent algorithms (GA [1] and PSO [51]) are used for optimaldispatch of this system. For GA algorithm, mutation and selectionrates are 0.2 and 0.5, respectively. For PSO algorithm cognitive andsocial parameters are equal to 1 and 2.5, respectively. Themaximum iteration number for PSO and GA are same as ICA. Theobtained results over 25 trial runs are compared in Table 12.Results of the proposed method are in bold. The minimum costobtained using ICA is 1,807,081.174 $/day, whereas for the case ofGA and PSO algorithms theminimum costs are 1,834,373.494 $/day

Table 12Comparison of optimization results for case 4.

Method Minimum cost ($) Average cost ($) Maximum cost ($)

GA 1,834,373.494 1,839,422.714 1,850,775.804PSO 1,832,121.861 1,835,851.611 1,845,937.037Proposed (ICA) 1,807,081.174 1,809,664.219 1,811,388.285

and 1,832,121.861 $/day, respectively. With assumption that thedaily load profile is same as studied day during the entire year, itmeans that using ICA will result in 9,139,850.75 $ annual savingcomparing to PSO and 9,961,696.80 $ annual saving comparing toGA. It should be mentioned that in a practical power system thedaily load profile is changing and DED problem should be solvedfor each day separately and the numbers are provided just forillustration of the economic effect of better solution. It is observedthat the performance of the proposed method is better for largescale test cases too, and the proposed method can be used forscheduling of practical large power systems. The convergencecharacteristics of the ICA algorithm compared with PSO and GA forthis case are given in Fig. 8. The maximum iteration number forthis case is selected to be 800.

5. Conclusion

In this paper, the imperialist competitive algorithm (ICA) hasbeen applied to solve the DED problem of generating unitsconsidering the valve-point effects, prohibited operation zones(POZs), ramp rate limits and transmission losses. The effectivenessof the proposed algorithm has been examined by comprehensivestudies on DED problems of different dimensions and complex-ities. At the first, the ICA is tested on five and ten units test systemfor a 24 h time interval. The results justify the applicability of theproposed method for solving the constrained DED with non-smooth cost functions. Also the proposed algorithm is imple-mented on a 54 units test system and the ICA is compared withtwo well-known heuristic algorithm, i.e. GA and PSO. Numericalexperiments on 4 test systems show that the proposed methodcan obtain lower total generation cost, so it provides a newand efficient approach to solve large-scale constrained DEDproblem.

Appendix A. Hourly optimum dispatch of 54-unit test system

Optimal 24-h schedule of 54-unit test system is provided inTable A.1.

B. Mohammadi-ivatloo et al. / Energy 44 (2012) 228e240 237

Appendix B. Test system’s data

Generating units’ characteristics of 5-unit test system areprovided in Table B.1. The B� matrix coefficients of this test systemare as follows.

Bij ¼

0BBBB@0:000049 0:000014 0:000015 0:000015 0:0000200:000014 0:000045 0:000016 0:000020 0:0000180:000015 0:000016 0:000039 0:000010 0:0000120:000015 0:000020 0:000010 0:000040 0:0000140:000020 0:000018 0:000012 0:000014 0:000035

1CCCCA

Generating units’ characteristics of 10-unit test system areprovided in Table B.2. The B�matrix coefficients of this test systemin per-unit in 100 MW base are as follows.

Bij ¼

0BBBBBBBBBBBBBB@

8:7 0:43 �4:61 0:36 0:32 �0:66 0:96 �1:6 0:8 �0:10:43 8:3 �0:97 0:22 0:75 �0:28 5:04 1:7 0:54 7:2�4:61 �0:97 9 �2 0:63 3 1:7 �4:3 3:1 �20:36 0:22 �2 5:3 0:47 2:62 �1:96 2:1 0:67 1:80:32 0:75 0:63 0:47 8:6 �0:8 0:37 0:72 �0:9 0:69�0:66 �0:28 3 2:62 �0:8 11:8 �4:9 0:3 3 �30:96 5:04 1:7 �1:96 0:37 �4:9 8:24 �0:9 5:9 �0:6�1:6 1:7 �4:3 2:1 0:72 0:3 �0:9 1:2 �0:96 0:560:8 0:54 3:1 0:67 �0:9 3 5:9 �0:96 0:93 �0:3�0:1 7:2 �2 1:8 0:69 �3 �0:6 0:56 �0:3 0:99

1CCCCCCCCCCCCCCA

Table B.1Generating units’ characteristics in 5-unit test system.

Unit ai bi ci ei fi Pmin Pmax UR DR POZs

1 0.008 2 25 100 0.042 10 75 30 30 [25 30], [55 60]2 0.003 1.8 60 140 0.04 20 125 30 30 [45 50], [80 90]3 0.0012 2.1 100 160 0.038 30 175 40 40 [60 70], [125 140]4 0.001 2 120 180 0.037 40 250 50 50 [95 110], [160 180]5 0.0015 1.8 40 200 0.035 50 300 50 50 [85 100], [175 200]

Table B.2Generating units’ characteristics in 10-unit test system.

Unit ai bi ci ei fi Pmin Pmax UR DR

1 0.00043 21.6 958.2 450 0.041 150 470 80 802 0.00063 21.05 1313.6 600 0.036 135 460 80 803 0.00039 20.81 604.97 320 0.028 73 340 80 804 0.0007 23.9 471.6 260 0.052 60 300 50 505 0.00079 21.62 480.29 280 0.063 73 243 50 506 0.00056 17.87 601.75 310 0.048 57 160 50 507 0.00211 16.51 502.7 300 0.086 20 130 30 308 0.0048 23.23 639.4 340 0.082 47 120 30 309 0.10908 19.58 455.6 270 0.098 20 80 30 3010 0.00951 22.54 692.4 380 0.094 55 55 30 30

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Table A.1Optimal 24-h schedule of 54-unit test system (case 4).

Hour 1 2 3 4 5 6 7 8 9 10 11

P1 5 5 5 5 5 5 5 5 5 5 5P2 5 5 5 5 5 5 5 5 5 5 5P3 5 5 5 5 5 5 5 5 5 5 5P4 5 5 5 5 5 5 5 5 5 5 5P5 212.2 249.6 212.2 287 212.2 100 100 212.2 287 287 212.2P6 10 10 10 10 10 10 10 10 10 10 10P7 50 85 91.538 85 85 85 85 91.554 100 92.138 85P8 5 5 5 5 5 5 5 5 5 5 5P9 5 5 5 5 5 5 5 5 5 5 5P10 100 100 100 120 249.6 145 249.6 249.6 120 120 100P11 350 350 350 350 350 350 350 350 350 350 350P12 8 8 8 8 8 8 8 8 8 8 8P13 8 8 8 8 8 8 8 8 8 8 8P14 39.577 56.106 91.538 73.446 41.829 69.015 80.2 91.554 100 92.138 79.262P15 8 8 8 8 8 8 8 8 8 8 8P16 39.577 56.106 91.538 73.446 41.829 69.015 80.2 91.554 100 92.138 79.262P17 8 8 8 8 8 8 8 8 8 8 8P18 8 8 8 8 8 8 8 8 8 8 8P19 39.577 56.106 91.538 73.446 41.829 69.015 80.2 91.554 100 92.138 79.262P20 250 250 250 250 250 250 250 250 250 250 250P21 250 250 250 250 250 250 250 250 250 250 250P22 39.577 56.106 91.538 73.446 41.829 69.015 80.2 91.554 100 92.138 79.262P23 39.577 56.106 91.538 73.446 41.829 69.015 80.2 91.554 100 92.138 79.262P24 200 200 200 200 200 200 200 200 200 200 200P25 200 200 200 200 200 200 200 200 200 200 200P26 39.577 56.106 91.538 73.446 41.829 69.015 80.2 91.554 100 92.138 79.262P27 39.577 56.106 91.538 73.446 41.829 69.015 80.2 91.554 100 92.138 79.262P28 349.333 399.199 399.199 399.199 399.199 399.199 399.199 349.333 399.199 420 399.199P29 273.364 292.819 300 300 276.015 300 300 300 300 300 300P30 36.609 40 70 70 37.237 40 40 70 80 70 40P31 10 10 10 10 10 10 10 10 10 10 10P32 5 5 5 5 5 5 5 5 5 5 5P33 5 5 5 5 5 5 5 5 5 5 5P34 39.577 56.106 91.538 90 50 90 90 91.554 100 92.138 90P35 50 90 91.538 90 50 90 90 91.554 100 92.138 90P36 300 300 300 150 150 150 300 300 300 300 300P37 39.577 56.106 91.538 73.446 41.829 69.015 80.2 91.554 100 92.138 79.262P38 10 10 10 10 10 10 10 10 10 10 10P39 100 100 100 100 100 100 100 100 100 100 100P40 100 100 100 100 100 100 100 100 249.6 174.8 100P41 8 8 8 8 8 8 8 8 8 8 8P42 20 20 20 20 20 20 20 20 26.734 20 20P43 100 100 100 100 100 100 249.6 300 300 300 249.6P44 100 100 100 100 100 249.6 300 300 300 300 300P45 100 100 100 100 174.8 300 300 300 300 300 249.6P46 8 8 8 8 8 8 8 8 8 8 8P47 60 60 91.538 73.446 60 69.015 80.2 91.554 100 92.138 79.262P48 39.577 56.106 91.538 73.446 41.829 69.015 80.2 91.554 100 92.138 79.262P49 8 8 8 8 8 8 8 8 8 8 8P50 25 25 25 25 25 25 25 25 26.734 25 25P51 39.577 56.106 91.538 73.446 41.829 69.015 80.2 91.554 100 92.138 79.262P52 39.577 56.106 91.538 73.446 41.829 69.015 80.2 91.554 100 92.138 79.262P53 39.577 56.106 91.538 73.446 41.829 69.015 80.2 91.554 100 92.138 79.262P54 25 25 25 25 25 25 25 25 26.734 25 25Load (MW) 3900 4300 4800 4500 4100 4600 5200 5500 5800 5600 5100Cost ($/h) 6,1229.265 68,706.276 78,654.861 72,839.349 64,468.602 74,364.815 85,298.934 91,201.079 97,671.104 93,399.176 83,265.125Total cost ($) 1,807,081.174

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12 13 14 15 16 17 18 19 20 21 22 23 24

5 5 5 5 5 5 5 5 5 5 5 5 55 5 5 5 5 5 5 5 5 5 5 5 55 5 5 5 5 5 5 5 5 5 5 5 55 5 5 5 5 5 5 5 5 5 5 5 5174.8 287 137.4 100 100 100 100 100 174.8 287 137.4 249.6 249.610 10 10 10 10 10 10 10 10 10 10 10 1085 65 85 85 85 35 85 56.719 85 85 98.6 65 355 5 5 5 5 5 5 5 5 5 5 5 55 5 5 5 5 5 5 5 5 5 5 5 5249.6 120 249.6 120 270 120 100 120 249.6 249.6 287 174.8 100350 350 350 350 350 350 350 350 350 350 350 350 3508 8 8 8 8 8 8 8 8 8 8 8 88 8 8 8 8 8 8 8 8 8 8 8 868.414 76.523 59.803 51.986 31.915 29.743 58.313 56.719 36.82 71.169 98.6 71.328 43.8068 8 8 8 8 8 8 8 8 8 8 8 868.414 76.523 59.803 51.986 31.915 29.743 58.313 56.719 36.82 71.169 98.6 71.328 43.8068 8 8 8 8 8 8 8 8 8 8 8 88 8 8 8 8 8 8 8 8 8 8 8 868.414 76.523 59.803 51.986 31.915 29.743 58.313 56.719 36.82 71.169 98.6 71.328 43.806250 250 250 250 250 250 250 250 250 250 250 250 250250 250 250 250 250 250 250 250 250 250 250 250 25068.414 76.523 59.803 51.986 31.915 29.743 58.313 56.719 36.82 71.169 98.6 71.328 43.80668.414 76.523 59.803 51.986 31.915 29.743 58.313 56.719 36.82 71.169 98.6 71.328 43.806200 200 200 200 200 200 200 200 200 200 200 200 200200 200 200 200 200 200 200 200 200 200 200 200 20068.414 76.523 59.803 51.986 31.915 29.743 58.313 56.719 36.82 71.169 98.6 71.328 43.80668.414 76.523 59.803 51.986 31.915 29.743 58.313 56.719 36.82 71.169 98.6 71.328 43.806399.199 399.199 399.199 399.199 399.199 399.199 399.199 399.199 399.199 399.199 399.199 349.333 249.6300 300 297.17 287.97 264.346 261.789 295.417 293.54 270.119 300 300 300 278.34240 40 70 40 34.474 33.868 40 40 35.841 70 70 40 37.78810 10 10 10 10 10 10 10 10 10 10 10 105 5 5 5 5 5 5 5 5 5 5 5 55 5 5 5 5 5 5 5 5 5 5 5 568.414 70 90 90 50 29.743 58.313 56.719 90 90 98.6 70 4090 70 90 90 50 29.743 58.313 90 90 90 98.6 70 40300 300 150 150 150 150 150 150 300 150 300 300 15068.414 76.523 59.803 51.986 31.915 29.743 58.313 56.719 36.82 71.169 98.6 71.328 43.80610 10 10 10 10 10 10 10 10 10 10 10 10100 100 100 100 100 100 100 100 100 100 100 100 100100 100 100 100 100 100 100 100 100 100 174.8 100 1008 8 8 8 8 8 8 8 8 8 8 8 820 20 20 20 20 20 20 20 20 20 20 20 20100 100 100 100 100 100 100 249.6 300 300 300 300 300249.6 100 100 100 100 100 100 100 249.6 300 300 300 300100 100 100 100 100 100 100 249.6 300 300 300 300 3008 8 8 8 8 8 8 8 8 8 8 8 868.414 76.523 60 60 60 29.743 60 60 60 71.169 98.6 71.328 4068.414 76.523 59.803 51.986 31.915 29.743 58.313 56.719 36.82 71.169 98.6 71.328 43.8068 8 8 8 8 8 8 8 8 8 8 8 825 25 25 25 25 25 25 25 25 25 25 25 2568.414 76.523 59.803 51.986 31.915 29.743 58.313 56.719 36.82 71.169 98.6 71.328 43.80668.414 76.523 59.803 51.986 31.915 29.743 58.313 56.719 36.82 71.169 98.6 71.328 43.80668.414 76.523 59.803 51.986 31.915 29.743 58.313 56.719 36.82 71.169 98.6 71.328 43.80625 25 25 25 25 25 25 25 25 25 25 25 254700 4600 4300 4000 3800 3500 4000 4300 4700 5200 5700 5100 430075,799.456 74,533.399 68,798.421 63,529.631 59,428.216 54,010.349 63,478.427 68,510.324 75,384.953 85,204.075 95,168.903 83,328.531 68,807.903

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