Energy Conversion and Management 49 (2008) 3627–3633

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Energy Conversion and Management

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

Hydrothermal scheduling using chaotic hybrid differential evolution

Xiaohui Yuan a,*, Bo Cao b, Bo Yang b, Yanbin Yuan c

a School of Hydropower and Information Engineering, Huazhong University of Science and Technology, 430074 Wuhan, Chinab Central China Grid Company Limited, 430077 Wuhan, Chinac School of Resource and Environmental Engineering, Wuhan University of Technology, 430070 Wuhan, China

a r t i c l e i n f o a b s t r a c t

Article history:Received 21 March 2007Received in revised form 15 December 2007Accepted 21 July 2008Available online 31 August 2008

Keywords:Differential evolutionChaosHydrothermal systemGeneration scheduling

0196-8904/$ - see front matter � 2008 Elsevier Ltd. Adoi:10.1016/j.enconman.2008.07.008

* Corresponding author. Tel.: +86 2762994000; faxE-mail address: yxh71@163.com (X. Yuan).

This paper proposes a chaotic hybrid differential evolution algorithm to solve short-term hydrothermalsystem generation scheduling problem. In the proposed method, chaos theory is applied to obtain self-adaptive parameter settings in differential evolution (DE). In order to handle constraints effectively, fea-sibility-based selection comparison techniques and heuristic rules embedded into DE are devised toguide the process toward the feasible region of the search space. A test hydrothermal system is usedto verify the feasibility and effectiveness of the proposed method. Results from the proposed methodare compared with those obtained by augmented Lagrange and two-phase neural network methods interms of solution quality. It is shown that the proposed method is able to obtain higher quality solutions.

� 2008 Elsevier Ltd. All rights reserved.

1. Introduction

The short-term hydrothermal generation scheduling problem(SHGSB) is one of the most important and challenging optimizationproblems in the economic operation of power systems. In a hydro-thermal power system, water resources available for electrical gen-eration are represented by the inflows to the hydro plants and thewater stored in their reservoirs. Thus, the available resources ateach stage of the operation-planning horizon depend on the previ-ous use of the water, which establishes a dynamic relationshipamong the operation decisions made along the whole horizon.The purpose of short-term daily hydrothermal scheduling is to findthe optimal amount of the water release for the hydro and thermalgeneration in the system to meet the load demands over a sched-uling horizon of one day. As the source for hydropower is the nat-ural water resources with almost zero operation cost, the objectiveof short-term optimal hydrothermal scheduling problem essen-tially reduces to minimize the fuel cost of thermal plants over allthe scheduling period of time while satisfying various constraints.The practical constraints to be satisfied include generator-loadpower balance equations, total water discharge constraint as theequality constraints and reservoir storage limits and the operationlimits of the hydro as well as thermal generators as the inequalityconstraints. Thus, the short-term hydrothermal generation sched-uling problem becomes a typical large-scale non-convex nonlinearconstrained optimization problem.

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The importance of hydrothermal generation scheduling is wellrecognized. Therefore, many methods have been developed tosolve the SHGSB problem in the past decades. The major methodsinclude dynamic programming (DP) [1,2], linear programming (LP)[3], network flow and mixed-integer linear programming (MILP)[4,5], progressive optimality algorithm (POA) [6] and Lagrangianrelaxation (LR) [7,8]. Although DP can handle the non-convexitiesand the nonlinear characteristics in SHGSB problem, it suffers fromthe well-known curse of dimensionality. As compared to DP, thePOA can significantly reduce the dimensionality of the SHGSB opti-mization problem. Unfortunately, the ‘‘trapping” phenomenon canseverely hinder solution optimality for SHGSB problem when usingPOA approach. The network flow model of SHGSB is often pro-grammed as a linear or piecewise linear one. Linear programmingtypically considers that power generation is linearly dependent onwater discharge, thus ignoring the head change effect, leading to asolution schedule with less power generation. Also, the discretiza-tion of the nonlinear dependence between power generation,water discharge and head, used in MILP to model head variations,augment the computational burden required to solve SHGSB prob-lem. The implementation of LR is complicated and its efficiencyheavily depends on the size of the duality gap. Furthermore, solu-tion quality of LR depends on the method to update Lagrangemultipliers.

Aside from the above methods, meta-heuristic approaches suchas artificial neural networks [9], evolutionary algorithm [10–12],cultural algorithm [13], Tabu search [14], simulated annealing[15], ant colony [16] and particle swarm optimization (PSO) [17]have been applied to solve SHGSB problem during the last decade.These meta-heuristic methods optimization methods have

3628 X. Yuan et al. / Energy Conversion and Management 49 (2008) 3627–3633

received more interest because of their ability to provide a reason-able solution (suboptimal near globally optimal), flexibility in deal-ing with various nonlinear constraints and simplicity inimplementation. However, these meta-heuristic methods requirea considerable amount of computational time to find the globalminimum especially for a large-scale SHGSB problem. Also, thesemethods have drawbacks such as premature phenomena and trap-ping into local optimum.

Although extensively investigated, the SHGSB problem still at-tracts the attention of researchers because of the stronger needfor lower cost operating schedules. Thus, improving current opti-mization techniques and exploring new method to solve theSHGSB problem has great significance so as to efficiently utilizewater resources, which can be regarded as a renewable source ofenergy. In recent years, a new optimization method known as dif-ferential evolution (DE) has gradually become more popular andhas been applied to solve optimization problems in power system[18,19]. However, DE’s parameters usually are constant throughoutthe entire search process and it is difficult to properly set controlparameters. At the same time, canonical version of DE lacks amechanism to handle constraints effectively for optimization prob-lem. Therefore, this paper proposes an improved DE approach tosolve SHGSB problem, which is focuses on self-adaptive parametersetting and handling constraints when it finds optimal solutionduring evolutionary search process. It is enhanced by simple feasi-bility-based selection comparison techniques and heuristic searchstrategies to handling constraints effectively, especially for thewater dynamic balance equation and reservoir end-volume equal-ity constraints in SHGSB problem. At the same time, the applicationof chaotic sequences based on logistic map instead of random se-quences is a powerful strategy to diversify the DE population andimprove the DE’s performance to find optimal solution. Finally,the proposed DE method is applied to solve SHGSB of a hydrother-mal system, which consists of four interconnected cascade hydroplants and a thermal plant. Simulated results demonstrate the fea-sibility and effectiveness of the proposed method for solvingSHGSB problem compared with both of augmented Lagrange andtwo-phase neural network methods reported in literature.

This paper is organized as follows. Section 2 provides the math-ematical formulation of SHGSB. Section 3 briefly describes thestandard differential evolution. Section 4 proposes an improveddifferential evolution algorithm for solving SHGSB problem. Sec-tion 5 presents the numerical example. Section 6 outlines theconclusions.

2. Problem formulations

The short-term hydrothermal system generation schedulingproblem is aimed to minimize the total thermal plant cost whilemaking use of the availability of hydro resource as much aspossible. The objective function and associated constraints of theproblem are formulated as follows.

min F ¼XT

t¼1

XNs

i¼1

fiðPtsiÞ ¼

XT

t¼1

XNs

i¼1

ai þ biPtsi þ ciðP

tsiÞ

2n o

ð1Þ

Subject to the following constraints

� System load balance

XNh

i¼1

Pthi þ

XNs

j¼1

Ptsj ¼ Pt

D t ¼ 1; 2; � � � ; T ð2Þ

� Thermal plant power limits

Pminsi 6 Pt

si 6 Pmaxsi i ¼ 1; 2; � � �Ns; t ¼ 1; 2; � � � ; T ð3Þ

� Hydro plant power limits

Pminhi 6 Pt

hi 6 Pmaxhi i ¼ 1; 2; � � � ; Nh; t ¼ 1; 2 . . . T ð4Þ

� Hydro plant discharge limits

Qmini 6 Q t

i 6 Q maxi i ¼ 1; 2; � � � ; Nh; t ¼ 1; 2; � � � ; T ð5Þ

� Reservoir storage volumes limits

Vmini 6 Vt

i 6 Vmaxi i ¼ 1; 2; � � � ; Nh; t ¼ 1; 2; � � � ; T ð6Þ

� Initial and terminal reservoir storage volumes

V0i ¼ VB

i ; VTi ¼ VE

i i ¼ 1;2; � � �Nh ð7Þ

� Water dynamic balance equation with travel time

Vti ¼ Vt�1

i þM � Iti � Q t

i � Sti þXNu

m¼1

½Q t�sm;im þ S

t�sm;im �

( )

i ¼ 1;2; � � � ;Nh; t ¼ 1; 2; � � � ; T ð8Þ

where fiðPtsiÞ is fuel cost of ith thermal plant at time interval t; Pt

si ispower generation of thermal plant i at time interval t, ai,bi,ci isthermal generation coefficients of ith plant, Pmin

si is minimumpower generation of thermal plant i; Pmax

si is maximum power gen-eration of thermal plant i, Ns is number of thermal plants, Nh isnumber of hydro plants, T is total time horizon, t is time index,Pt

hi is power generation of hydro plant i at time interval t; Pminhi is

minimum power generation of hydro plant i; Pmaxhi is maximum

power generation of hydro plant i;Vti is water volume of reservoir

i at the end of time interval t;Vmini is minimum water volume of

reservoir i;Vmaxi is maximum water volume of reservoir i;Q t

i iswater discharge of hydro plant i at time interval t;Q min

i is minimumwater discharge of hydro plant i;Qmax

i is maximum water dischargeof hydro plant i;VB

i is initial storage volume of reservoir i at thebegin of dispatching horizon, VE

i is final storage volume of reservoiri at the end of dispatching horizon, St

i is water spillage of hydroplant i at time interval t; It

i is natural inflow into reservoir i at timeinterval t, it Nu is number of upstream hydropower plants directlyabove ith hydro plant, sm,i is water transport delay time from res-ervoir m to i, M is conversion factor of water discharge into storedwater.

3. Differential evolution

Differential evolution (DE) [20], invented by Price and Storn in1995, is a simple yet powerful heuristic method for solving nonlin-ear, non-differentiable and multi-modal optimization problem.This technique combines simple arithmetic operators with theclassical events of crossover, mutation and selection to evolve fromrandomly generated initial population to final individual solution.The key idea behind DE is a scheme for generating trial parametervectors. Mutation and crossover are used to generate new vectors(trial vectors), and selection then determines which of the vectorswill survive the next generation.

A set of D optimization parameters is called an individual,which is represented by a D-dimensional parameter vector. A pop-ulation consists of NP parameter vectors Xi,G, (i = 1,2, � � �,NP for eachgeneration G). According to Storn and Price, DE’s basic strategy canbe described as follows.

3.1. Mutation

For each target vector Xi,G(i = 1,2, � � �,NP), a mutant vector Vi,G+1

is generated according to

Vi;Gþ1 ¼ Xr1;G þ F � ðXr2;G � Xr3;GÞ; r1–r2–r3–i ð9Þ

X. Yuan et al. / Energy Conversion and Management 49 (2008) 3627–3633 3629

with randomly chosen integer indexes r1,r2, r3 2 {1,2, � � �,NP}. Notethat indexes have to be different from each other and from the run-ning index. F is called mutation factor between [0,1] which controlsthe amplification of the differential variation (Xr2,G-Xr3,G).

3.2. Crossover

In order to increase the diversity of the perturbed parametervectors, crossover is introduced. The target vector is mixed withthe mutated vector, using the following scheme, to yield the trialvector Ui,G+1 = (u1i,G+1,u2i,G+1, . . . uDi,G+1), that is

uji;Gþ1 ¼vji;Gþ1 if randðjÞ 6 CR or j ¼ rnbðiÞxji;G otherwise

�j ¼ 1; 2; � � � ; D

ð10Þ

where rand(j) is the jth evaluation of a uniform random numbergenerator between [0,1]. CR is the crossover constant between[0,1] which has to be determined by the user. rnb(i) is a randomlychosen index from 1,2, � � �, D which ensures that Ui,G+1 gets at leastone parameter from Vi,G+1. Otherwise, no new parent vector wouldbe produced and the population would not alter.

3.3. Selection

To decide whether or not it should become a member of thenext generation G + 1, the trial vector Ui,G+1 is compared to the tar-get vector Xi,G using the greedy criterion. Assume that the objectivefunction is to be minimized, according to the following rule:

Xi;Gþ1 ¼Ui;Gþ1 if f ðUi;Gþ1Þ 6 f ðXi;GÞXi;G otherwise

�ð11Þ

That is, if vector Ui,G+1 yields a better evaluation function value thanXi,G, then Xi,G+1 is set to Ui,G+1; otherwise, the old value Xi,G is retained.As a result, all the individuals of the next generation are as good as orbetter than their counterparts in the current generation.

4. Improved differential evolution for solution SHGSB

Chaotic sequences display an unpredictable long-term behaviordue to their sensitiveness to initial conditions. This feature is usefulto track the chaotic variable as it travels ergodically over the spaceof interest, so it is can be applied in DE. In order to obtain high-quality solution for SHGSB, an improved DE combination chaoticsequences for parameters setting with selection comparison tech-nique based on individual feasibility and heuristic rules for con-straints handling is proposed in this paper.

4.1. Self-adaptive parameters setting for DE with chaos theory

Recently, some applications of chaotic sequences in evolution-ary algorithm(EA) have been investigated by the literature. Numer-ous examples and statistical results show that some chaoticsequences applied to EA are able to increase the algorithm-exploi-tation capability in the search space and enhance its convergence[21]. The DE’s parameters CR and F that need to be adjusted bythe user are generally the key factors affecting the DE’s conver-gence. Choosing suitable parameter valus are difficult for DE,which is usually a problem-dependent task. The trial-and-errormethod adopted frequently for tuning the parameters in DE re-quires multiple optimization runs. However, the parameter CRand F cannot ensure the optimization’s ergodicity completely inthe search phase because they are often constant factors in tradi-tional DE. Therefore, this paper adopts chaotic sequences to self-adaptive adjust parameters CR and F during the evolutionaryprocess. The utilization of chaotic sequences in DE can be useful

to escape more easily from local minima than with the standardDE and improve its global convergence. Application of chaotic se-quences to obtain DE’s parameters CR and F has two advantages:First, user does not need to guess the good values for F and CR. Sec-ond, the rules for self-adapting adjusted parameters F and CR arequite simple.

One of the simplest dynamic systems evidencing chaotic behav-ior is the iterator named the logistic map, whose equation is thefollowing:

yðtÞ ¼ l � yðt � 1Þ � ½1� yðt � 1Þ� ð12Þ

where l is a control parameter, 0 6 l 6 4.Eq. (12) is deterministic, displaying chaotic dynamics when

l = 4 and y(0) R {0,0.25,0.5,0.75,1}. y(t) is distributed in the range(0,1) provided the initial y(0) 2 (0,1).

The parameters value of F and CR in DE are modified by the for-mula (12) through the following expresses:

FðGÞ ¼ l � FðG� 1Þ � ½1� FðG� 1Þ�CRðGÞ ¼ l � CRðG� 1Þ � ½1� CRðG� 1Þ�

ð13Þ

where G is the current iteration generation; F(0) and CR(0) are ran-domly number between 0 and 1, respectively.

4.2. Initialization individuals

In the initialization process, a set of individuals is created atrandom. The structure of an individual for SHGSB problem is com-posed of a set of discharge decision variables for each hydro plantin over the scheduling horizon. Each individual’s contains realnumbers randomly generated, representing the water dischargein each hydro plant at every dispatch horizon t as a vectorX = [Q1,1, . . . ,Q1,T,Q2,1, . . . ,Q2,T, . . . ,QN,1, . . . , QN,T] with length T � N.These values are within the hydro plant discharge bounds (5). Notethat it is very important to create a population of individuals satis-fying the equality constraints (7) and (8). To satisfy the constraints(7) on the final reservoir storage volume and hydro plant water dy-namic balance Eq. (8), a dependent hydro discharge Qi,l is randomlyselected. The initial population excluding the dependent hydro dis-charge is expressed as

X0 ¼ ½Q 1;1; . . . ;Q 1;l�1;Q1;lþ1; . . . ;Q1;T ;Q 2;1; . . . ;Q 2;l�1;

Q 2;lþ1; . . . ;Q 2;T ; . . . ;Q N;1; . . . ;QN;l�1;Q N;lþ1; . . . ;Q N;T �

The dependent hydro discharge Qli is computed from (8) as

Q i;l ¼ VBi � VE

i þXT

t¼1

Iti �XT

t¼1t–l

Q i;t �XT

t¼1

Sti

þXNu

m¼1

XT

t¼1

Q m;t�sm;iþ S

t�sm;im

h i; i ¼ 1;2; . . . ;N

ð14Þ

This process is repeated until the dependent hydro discharge Qi,l

does not violate its bound constraints (5). Using these hydro dis-charges, the volumes at different intervals are determined. Accord-ing to hydro plant generation characteristics, hydro plantgeneration power can be obtained using its hydro discharges andstorage volumes. From the calculated hydro generation power, thethermal generation power is calculated using (2). The objective func-tion of the hydro scheduling problem can be calculated using (1).Then, constraint violations can be evaluated using current valuesof discharge, storage and thermal powers over the scheduling period.

4.3. Mutation operation considering equality constraints

For each individual in population space, applying the muta-tion operator of DE generates offspring individual. The resulting

3630 X. Yuan et al. / Energy Conversion and Management 49 (2008) 3627–3633

values of offspring individuals are not always guaranteed to sat-isfy the equality constraints of terminal reservoir storage vol-umes (7) and hydro plant water dynamic balance Eq. (8). Toresolve the equality constraints (7) and (8) without interveningthe dynamic process inherent in the DE algorithm, we proposethe following heuristic search strategies for all offspringindividuals.

Step 1: Let the present iteration be k.Step 2: Choose a hydro discharge element l of offspring

individual at random as the dependent hydro discharge.Let l = 1.

Step 3: The dependent hydro discharge is computed using (14). Ifthe computed hydro discharge does not violate the con-straints (5) then go to step 6; otherwise go to step 4.

Step 4: The dependent discharge is fixed either to its maximum orminimum limit, then a new random hydro discharge ele-ment is chosen and l = l + 1.

Step 5: If l< = T, then go to step 3; otherwise go to step 6.Step 6: The modification process is terminated.

The new set of Np individuals thus obtained by the modificationprocess will satisfy the final storage volume constraints (7) and hy-dro plant water dynamic balance Eq. (8).

4.4. Selection operation based on handling inequality constraints

According to hydro discharges of each individual by the muta-tion modification process, reservoirs storage volumes over thescheduling period are calculated using (8). Each hydro plantpowers over the scheduling period are determined by dischargeand storage with hydro plant generation characteristics. Fromthe hydro generation power, the thermal generation power iscalculated using (2), and then the objective function value ofthe hydro scheduling problem is calculated using (1). The indi-vidual may not guarantee to satisfy the hydro generation powerconstraints (4) and reservoir storage volumes constraints (6). Thestrategy for handling such inequality constraint is usually the useof penalty function methods. Despite the popularity of penaltyfunctions, they have several drawbacks among which the mainone is that they require a careful fine tuning of the penalty fac-tors that accurately estimates the degree of penalization to beapplied as to approach efficiently the feasible region. In orderto keep the advantages of the penalty function approach andovercome drawback of choice penalty factors, this paper applyan effective constraint handling method for DE, which does notrequire to set any additional parameters in comparison withthe original DE.

Motivated by [22], the following three simple feasibility-basedselection comparison rules are adopted in this paper:

(1) Any feasible solution is preferred to any infeasible solution.(2) Between two feasible solutions, the one having better objec-

tive function value is preferred.(3) Between two infeasible solutions, the one having smaller

constraint violation is preferred.

Based on the above criteria, objective function and constraintviolation information are considered separately. Consequently,penalty factors are not used at all. Different from [22] where anadditional fitness function was designed to evaluate solutions;there is no need to design the additional fitness function in this pa-per by incorporating the rules into DE.

The short-term optimal generation scheduling of hydrothermalsystems in Section 2 can be converted into the following con-strained optimization problem:

min f ðQÞs:t:

gjðQÞ 6 0Q min 6 Q 6 Q max

� ð15Þ

where Q = [Q1,Q2, � � � ,Qn]T is a vector of n discharge decision vari-ables of the optimization problem; n = T � Nh;j = 1,2, � � � ,(4 � T � Nh +2 � T � Ns).

If any element of an individual violates inequality constraint (4)or (6), the sum of constraint violations are calculated. Based on theabove constraint handling mechanism, the individuals of the nextgeneration can be obtained by replace the individual with the off-spring if the offspring is better, which use the objective functionvalue of individual and corresponding total constraint violations.The constraint violation value of an infeasible solution is calculatedas follows:

violðQÞ ¼XM

j¼1

½maxðgjðQÞ;0Þ�

M ¼ 1;2; . . . ; ð4 � T � Nh þ 2 � T � NsÞð16Þ

Suppose that Qi,G represents the ith individual at iteration genera-tion G and Ui,G+1 represents the newly generated the ith offspringat iteration generation G + 1. In the classical DE, Qi,G+1 = Ui,G+1 onlyif f(Ui,G+1) < f(Qi,G). Whiles in our DE, the feasibility-based rule is em-ployed. That is, Qi,G will be replaced by Ui,G+1 at any of the followingscenarios; Otherwise old vector Qi,G is preserved.

(1) Qi,G is infeasible, but Ui,G+1 is feasible.(2) Both Qi,G and Ui,G+1 are feasible, but f(Ui,G+1) < f(Qi,G).(3) Both Qi,G and Ui,G+1 are infeasible, but viol(Ui,G+1) < viol(Qi,G).

4.5. Stopping criteria

Check the termination condition. If the predefined maximumiteration number is reached, then the DE is terminated and obtainthe optimal results. Otherwise, it is repeated to carry out until ter-mination condition is satisfied.

5. Numerical example

In order to verify the feasibility and effectiveness of the pro-posed method, a test system taken from Ref. [9] is used. The systemconsists of an equivalent thermal plant and four cascaded hydroplants along a river. The scheduling period is one day with 1-h timeinterval. The test system configuration is shown in Fig. 1. Thishydraulic test network models most of the complexities encoun-tered in practical hydro networks.

The details data used for the present test network are given inTables 1–3. Load demand data for 24 h is given in Table 1, whileTable 2 gives hydro plant power generation coefficients. Boundson reservoir storage volume, water discharge rates and boundaryconditions on reservoir storage volume are given in Table 3. InTable 3, the units of storage are 103 m3, while units of water dis-charge rate are 103 m3/h. The vector [10,000, 8000, 1000, 0]Tm3/hgives reservoir hourly inflows. The water transportation delay timeconsidered are s1,3 = 1 h; s2,3 = 2 h; s3,4 = 2 h. The composite ther-mal plant fuel cost coefficients a,b and c taken are 1000.0, 10.0and 0.5, respectively.

With the data given, the proposed method coded by MicrosoftVisual C++ 6.0 language on a Pentium-4 2.0GHz-based processorwith 512MB of RAM PC computer is applied to solve the short-termoptimal generation scheduling of this hydrothermal system. Theparameters used by our experiment are the following: populationsize takes 80, initial mutation factor F(0) takes 0.4, crossover factor

Fig. 1. Hydraulic system test network.

Table 1Load demand

Time 1 2 3 4 5 6 7 8Load 190 170 170 190 190 210 230 250Time 9 10 11 12 13 14 15 16Load 270 310 350 310 350 350 310 290Time 17 18 19 20 21 22 23 24Load 270 250 230 210 210 210 190 190

Table 2Hydro plant power generation coefficients ðPt

hi ¼ C1i � ðVti Þ

2 þ C2i � ðQti Þ

2 þ C3i � Vti �

Q ti þ C4i � Vt

i þ C5i � Q ti þ C6iÞ

Plant i C1i C2i C3i C4i C5i C6i

1 �0.001 �0.1 0.01 0.40 4.0 �302 �0.001 �0.1 0.01 0.38 3.5 �303 �0.001 �0.1 0.01 0.30 3.0 �304 �0.001 �0.1 0.01 0.38 3.8 �30

Table 3Characteristics of hydro plant

Plant i Vmini Vmax

i VBi VE

i Qmini Qmax

i

1 80 150 100 120 5 152 60 120 80 70 6 153 100 240 170 170 10 304 70 160 120 120 13 25

Fig. 2. Hourly hydro plant discharge.

Fig. 3. Hourly hydro plant storage.

X. Yuan et al. / Energy Conversion and Management 49 (2008) 3627–3633 3631

CR(0) takes 0.9 and the maximal evolutionary iterative number is2000. Under those chosen parameters, we run our method 20 timesfrom different initial populations in succession and select the bestresult as the final optimization solution. The total thermal plant

Table 4Hourly hydrothermal power generation scheduling (Unit: MW)

Hour 1 2 3 4 5 6

Ph1 23.12 24.32 25.50 26.60 27.63 33.30Ph2 16.51 17.04 17.58 18.12 18.65 19.16Ph3 45.04 40.84 38.16 36.48 34.52 32.94Ph4 47.50 43.66 42.97 45.32 47.17 48.81Ps 57.82 44.14 45.78 63.48 62.01 75.81

Hour 13 14 15 16 17 18

Ph1 57.43 55.86 52.01 46.88 44.18 40.05Ph2 33.78 32.68 28.88 27.59 24.53 23.00Ph3 35.38 37.26 39.14 41.05 42.30 42.88Ph4 65.15 67.26 66.77 67.29 68.08 68.41Ps 158.26 156.94 123.2 107.19 90.90 75.67

cost in all interval time is 154,338$. The hourly each hydro plantpower generation are showed in Table 4. The hourly each reservoirrelease and storage trajectories are showed in Figs. 2 and 3,respectively.

To validate the results obtained with the proposed method, thesame problem was solved using the augmented Lagrangian andtwo-phase neural network methods. Table 5 gives the total ther-mal plant cost obtained with the three-solution techniques. FromTable 5, it is clear that the final total thermal plant cost obtainedwith the proposed method is better than those of the two-phaseneural network and augmented Lagrangian methods.

7 8 9 10 11 12

42.05 47.54 51.89 58.49 60.01 55.2919.73 22.39 25.20 31.54 35.08 29.1631.61 30.67 30.16 30.52 31.59 32.9950.24 51.48 52.63 53.73 59.88 58.6786.37 97.93 110.12 135.72 163.45 133.9

19 20 21 22 23 24

35.44 29.06 32.30 30.95 30.24 26.6119.79 18.15 18.91 21.17 12.46 13.0342.90 41.26 40.90 39.06 36.72 39.3368.69 69.51 69.72 69.73 68.01 65.5563.18 52.03 48.16 49.08 42.58 45.48

Table 5Comparison of thermal plant cost with other methods

Method Thermal plant total cost ($)

Augmented Lagrange method 154739.0Two-phase neural network 154808.5Proposed method 154338.1

Fig. 4. Thermal plant cost with generation numbers.

Fig. 5. Standard deviation with generation numbers.

Fig. 6. Distribution of thermal plant cost of each trail.

3632 X. Yuan et al. / Energy Conversion and Management 49 (2008) 3627–3633

In the meantime, we examine the variation for the best totalthermal plant cost and population’s standard deviation during evo-lutionary process, which show the convergence property of theproposed method. Figs. 4 and 5 show the variation of the best totalthermal plant cost and population’s standard deviation with gener-ation numbers during evolutionary process respectively. In twoFigures, there are a sharp declines for the best total thermal plantcost and corresponding standard deviation at the beginning evolu-tionary stages, while it declines slowly during later stages and fi-nally the best total thermal plant cost and population standarddeviation stabilize at constant values. From Fig. 5, it can be seenthat population standard deviation is much smaller in the end.As it can be seen, the proposed method has rapid convergencespeed and better thermal plant cost, thus verifying that the pro-posed method has better quality of solution and convergenceproperty.

In order to check solution quality, we inspected the variation intotal thermal plant cost and its standard deviation from 20 trialsusing the proposed method. The best, average and worst total ther-mal plant costs are 154,338, 154,366 and 154,410, respectively andthe corresponding standard deviation is 20. Fig. 6 shows the distri-bution of the best thermal plant cost of each trail, which generatesvariation in a very small range with trial numbers, thus verifyingthat the proposed method can obtain better quality solution.

6. Conclusions

In this paper, application of chaotic sequences based on logisticmap to determine the values of parameters F and CR in DE andthree simple comparison mechanisms based on feasibility andheuristic rules to guide the search toward the optimum are devisedto effectively handling constraints which does not require a pen-alty function or any extra parameters to bias the search towardsthe feasible region of a problem. Thus, an improved chaotic hybridDE algorithm is proposed to solve short-term hydrothermal gener-ation scheduling problem. In the optimization model, not onlycomplicated spatial-temporal coupling among reservoirs can bedealt with conveniently, but also nonlinear non-convex relation-ships for power generation characteristics and the water transportdelay time in hydrothermal system are all taken into account. Fi-nally the proposed method is applied to solve short-term optimalscheduling of hydrothermal system with four cascaded plantsalong a river. Simulation results show that the proposed methodcan obtain better quality solution with higher precision, so it pro-vides an effective method to solve the short-term optimal genera-tion scheduling of hydrothermal system.

Acknowledgements

The authors gratefully acknowledge the financial supports fromNational Natural Science Foundation of China under Grant Nos.50779020, 40572166 and 50539140.

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