Expert Systems with Applications 38 (2011) 6556–6564

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Expert Systems with Applications

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

Combined heat and power economic dispatch by mesh adaptive directsearch algorithm

Seyyed Soheil Sadat Hosseini a,⇑, Ali Jafarnejad b, Amir Hossein Behrooz c, Amir Hossein Gandomi d,e

a College of Electrical Engineering, Tafresh University, Tafresh, Iranb Department of Construction Management & Engineering, Islamic Azad University, Science and Research Branch, Tehran, Iranc Faculty of Management and Accounting, Allameh Tabatabai University, Tehran, Irand College of Civil Engineering, Tafresh University, Tafresh, Irane The Highest Prestige Scientific and Professional National Foundation, National Elites Foundation, Tehran, Iran

a r t i c l e i n f o

Keywords:Economic dispatchCombined heat and powerMesh adaptive direct search algorithmOptimization

0957-4174/$ - see front matter � 2010 Elsevier Ltd. Adoi:10.1016/j.eswa.2010.11.083

⇑ Corresponding author. Tel.: +98 912 419 9906.E-mail addresses: s.sadathosseini@gmail.com (S.S. S

gmail.com (A.H. Gandomi).

a b s t r a c t

The optimal utilization of multiple combined heat and power (CHP) systems is a complex problem.Therefore, efficient methods are required to solve it. In this paper, a recent optimization technique,namely mesh adaptive direct search (MADS) is implemented to solve the combined heat and power eco-nomic dispatch (CHPED) problem with bounded feasible operating region. Three test cases taken from theliterature are used to evaluate the exploring ability of MADS. Latin hypercube sampling (LHS), particleswarm optimization (PSO) and design and analysis of computer experiments (DACE) surrogate algo-rithms are used as powerful SEARCH strategies in the MADS algorithm to improve its effectiveness.The numerical results demonstrate that the utilized MADS–LHS, MADS–PSO, MADS–DACE algorithmshave acceptable performance when applied to the CHPED problems. The results obtained using theMADS–DACE algorithm are considerably better than or as well as the best known solutions reported pre-viously in the literature. In addition to the superior performance, MADS–DACE provides significant sav-ings of computational effort.

� 2010 Elsevier Ltd. All rights reserved.

1. Introduction power demands. They can be built in urban areas and utilized as

The conversion of primary fossil fuels to electricity is a rela-tively inefficient process. Even the most modern combined cycleplants can only obtain efficiencies of between 50% and 60% (Vasebi,Fesanghary, & Bathaee, 2007). Most of the energy wasted in thisconversion process is released to the environment as waste heat.Cogeneration or CHP generation is a mature and established tech-nology which has energy efficiency and environmental advantagesover other forms of energy supply. Economic dispatch (ED) must beapplied to obtain the optimal use of CHP units. The main objectiveof economic dispatch problem in a conventional power plant is tofind the optimal solution for the power production such that thetotal demand matches the generation with minimum fuel cost.The mutual dependencies of heat and power generation proposea complication in the integration of cogeneration units into thepower system economic dispatch. The best CHP schemes can ob-tain fuel conversion efficiencies of the order of 90% (Vasebi et al.,2007). Cogeneration systems have extensively been used by theindustry, recently. Some industrial processes have large heatrequirements, either as process steam or piped hot fluid, and large

ll rights reserved.

adat Hosseini), a.h.gandomi@

distributed electrical energy sources.Several researches worked in the field of the CHPED problem.

Non-linear optimization algorithms, such as dual and quadraticprogramming (Rooijers & van Amerongen, 1994), and gradient des-cent approaches, such as Lagrangian relaxation (Guo, Henwood, &van Ooijen, 1996), have been applied to it. However, these algo-rithms cannot deal with discontinuous and/or non-monotonicinput–output models for generator fuel characteristics. Alterna-tives to the conventional mathematical approach: evolutionarycomputation techniques such as genetic algorithm (GA) (Song &Xuan, 1998; Su & Chiang, 2004), evolutionary programming (EP)(Wong & Algie, 2002), multi-objective particle swarm optimization(MPSO) (Wang & Singh, 2008), a hybrid of genetic algorithm withtabu search (GT) (Sudhakaran & Slochanal, 2003), harmony search(HS) (Vasebi et al., 2007), fuzzy decision making (FDM) (Chang &Fu, 1998), improved ant colony search algorithm (ACSA) (Song,Chou, & Stonham, 1999) and self adaptive real-coded genetic algo-rithm (SARGA) (Subbaraj, Rengaraj, & Salivahanan, 2009) have suc-cessfully been applied to the CHPED problem. In the optimizationarea, many interesting results come from the utilization of patternsearch (PS) methods (Torczon, 1997). At this stage, mesh adaptivedirect search (MADS) (Audet & Dennis, 2006) is one of the mostpowerful optimization algorithms that has recently emerged. TheMADS method lends itself to optimization problems with discrete

S.S. Sadat Hosseini et al. / Expert Systems with Applications 38 (2011) 6556–6564 6557

and continuous variables, such as the generator loads of the CHPproblem. MADS does not depend on derivatives of the objectivefunction of the problem to be solved, non-monotonic functionsand accommodating discontinuous. Despite significant advantagesof MADS over other optimization approaches, there have beensome little scientific efforts directed at applying it to academicand practical problems (Audet, Bechard, & Chaouki, 2008; Nicosia& Stracquadanio, 2007).

The main objective of this study is to introduce the MADS algo-rithm to solve the CHPED problem. Three four-unit systems previ-ously presented in the literature have been used as case studies.The Latin hypercube sampling (LHS), particle swarm optimization(PSO) and design and analysis of computer experiments (DACE)surrogate algorithms are used as search strategies in MADS to solveeach of the CHPED problems. The results obtained by the MADS–LHS, MADS–PSO, MADS–DACE methods are further compared withthose generated with other (evolutionary and mathematical pro-gramming) techniques reported in the literature. This paper is or-ganized as follows: Section 2 describes the characteristics of acogeneration unit and formulation of the CHPED problem, Section3 deals with mesh adaptive direct search, Section 4 describesimplementation of MADS to the CHPED problem, and Section 5 dis-cusses the MADS performance on this specific problem.

2. Formulation of the CHPED problem

Combined heat and power generation is a mature and estab-lished technology. It has higher energy efficiency and less greenhouse gas emission compared with the other forms of energy sup-ply (Vasebi et al., 2007). The essential difference between com-bined heat and power units and conventional condensing plant isin the type of the power obtained and the overall efficiency of eachplant. In conventional condensing plants, the energy from the fuelis utilized to produce electrical power only, while in CHP systems,the energy from the fuel is utilized to produce both electrical andthermal power thus increasing its efficiency. The heat productiondepends on power generation and inversely. This introduces com-plexity due to the non-separable heat in the CHP units and natureof electrical power.

The heat-power feasible operation region (FOR) of a combinedcycle cogeneration unit is shown in Fig. 1. The boundary curveMNOPQR enclosed the feasible operation region. The upper andlower bounds of heat and power units are restricted by theirown generation limits. Along the boundary curve BC, the heatcapacity increases as the power generation declines.

Fig. 1. Feasible operation region for a cogeneration unit.

The CHPED problem of a system is to determine the unit heatand power production so that system production cost is minimizedwhile the heat-power demands and other constraints are met. Itcan be mathematically stated as follows:

MinXNp

i¼1

CiðPiÞ þXNb

j¼1

CjðPj;HjÞ þXNh

k¼1

CkðHkÞ ð1Þ

Subject to the equilibrium constraints of heat and electricity pro-duction, and the capacity limits of each unit

XNp

i¼1

Pi þXNb

j¼1

Pj ¼ Pd ð2Þ

XNc

j¼1

Hj þXNh

k¼1

Hk ¼ Hd ð3Þ

Pmini 6 Pi 6 Pmax

i ; i ¼ 1; . . . ;Np ð4Þ

Pminj ðHjÞ 6 Pj 6 Pmax

j ðHjÞ; j ¼ 1; . . . ;Nb ð5Þ

Hminj ðPjÞ 6 Hj 6 Hmax

j ðPjÞ; j ¼ 1; . . . ;Nb ð6Þ

Hmink 6 Hk 6 Hmax

k ; k ¼ 1; . . . ;Nh ð7Þ

where f is the total heat and power production cost, C is the unitproduction cost; P the unit power generation; H the unit heatproduction; Hd and Pd the system heat and power demands, respec-tively; i, j and k the indices of conventional power units, cogenera-tion units and heat-only units; Np, Nb and Nh the numbers of thekinds of units mentioned above; Pmin and Pmax the unit powercapacity limits; Hmin and Hmax the unit heat capacity limits.

The mutual dependencies of heat and power generations fromEqs. (5) and (6) introduce a complication in the integration ofcogeneration units. Hence, the optimization problem of the CHPEDis non-linear and highly constrained in nature.

3. Mesh adaptive direct search algorithm

The PS optimization algorithm is a class of direct search meth-ods. This algorithm is suitable to solve different optimization prob-lems that lie outside the scope of the standard optimizationmethods. In general, PS has the advantage of being very simplein concept, easy to implement and computationally efficient. Auseful review of direct search methods for unconstrained optimi-zation is introduced in Conn, Scheinberg, and Vicente, (2009), inwhich the authors give a modern perspective on the classical fam-ily of derivative-free algorithms, focusing on the development ofdirect search methods.

The MADS algorithm for non-linear optimization extends thegeneralized pattern search (GPS) (Kolda, Lewis, & Torczon, 2004;Lewis & Torczon, 2000) algorithms. A main advantage of MADSover the GPS method for both unconstrained and linearly con-strained optimization is that local exploration of the space of vari-ables is not restricted to a finite number of directions, called polldirections. This is the primary drawback of the GPS algorithms,and the essential motivation in defining MADS was to overcomethis restriction (Audet & Dennis, 2006). The MADS algorithm is aframe-based method. The MADS frames are specifically aimed atensuring an asymptotically dense set of polling directions.

Notation. R, Z, and N, respectively, denote the sets of realnumbers, integers, and non-negative integers.

For a matrix D, the notation d e D indicates that d is a column ofD. The iteration numbers are denoted by the index i.

Fig. 2. Basic MADS algorithm.

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3.1. Features of the MADS algorithm

MADS is an iterative algorithm. The MADS class of algorithmsstarts with an initial point with finite function value. This algo-rithm does not need any derivative information for f. This is impor-tant when rf is unavailable, either because it does not exist, or itcannot be accurately evaluated due to noise in f or other reasons.

Each iteration is divided into two steps. The first one is called theSEARCH step. In this step, any finite set of mesh points can be eval-uated. This allows great flexibility to choose strategies. The SEARCHis expressed to be empty when no trial points are considered. TheSEARCH step adds nothing to the convergence theory, except toprovide counterexamples as in Audet (2004). It is notable thatwell-chosen search strategies can greatly improve algorithm perfor-mance (see Abramson, Audet, & Dennis, 2004; Audet & Orban, 2006;Booker et al., 1999; McKay, Conover, & Beckman, 1979).

The SEARCH step is essential in concentrated on. Lewis andTorczon (1996) identified practice because it is so flexible, but itis a difficulty for the theory for the same reason (Abramson, Audet,& Dennis, 2006). SEARCH can return any point on the underlyingmesh. Of course, it is trying to identify an unfiltered point. Sincethe POLL step is the basis of the convergence analysis, it is the partof the algorithm where most research has been that POLL shouldconsider points on the mesh neighboring the incumbent solutionin a set of directions whose non-negative linear combinations spanthe space. This may seem simple, but it is an essential observation.Coope and Price (2002) extended this concept to the idea of frames,which can be thought of as doing away with the requirement thatthe POLL points be mesh neighbors. Audet and Dennis (2006) pro-posed MADS as a way to implement frames so that the directionsnormalized used in infinitely many POLL steps generate a denseset on the unit sphere at a MADS limit point. This allows strongconvergence (Abramson & Audet, 2006; Audet & Dennis, 2006)and excellent computational results for the MADS algorithms(Audet & Orban, 2006; Marsden, 2004).

3.2. Description of MADS algorithm

The steps in the MADS algorithm are summarized in Fig. 2.These steps are described in the next five subsections.

3.2.1. InitializationIn the MADS algorithm, a starting point x0 must be specified

with finite function value, and an initial mesh size parameterDm

0 2 Rþ. The superscripts m and p stand for mesh and poll,respectively.

The algorithm then defines:

� Dm0 6 Dp

0: the poll parameter size.� D ¼ f�ek; k ¼ 1;2; . . . ;ng: the positive basis needed to generate

polling directions, where ek is the kth coordinate direction, andn is the number of variables.� M0 ¼ fx 2 S0g [ fx0 þ Dm

0 Dz : z 2 N2ng; the mesh, where S0 is theset of points where the objective function f had been evaluatedby the initialization. Then, MADS proceeds to the quest for animproved mesh point.

3.2.2. SEARCH stepThe SEARCH step can be empty. This means that the algorithm

can only be implemented as a sequence of POLL steps. This is anacceptable choice when a local minimizer in the same basin asthe initial guess is adequate. Another reasonable strategy is to trya step in the same direction as a previously successful POLL step.It should be noted that although this seems reasonable, someresearchers have found this approach of limited value at best(Abramson et al., 2006). Three of the SEARCH strategies used in

the present study are briefly described in the followingsubsections.

3.2.2.1. Latin hypercube sampling. An easy and a popular experi-mental design technique to provide a good global search of the fea-sible region is Latin hypercube sampling (LHS). The statisticalmethod of LHS was made to produce a distribution of plausible col-lections of parameter values from a multi-dimensional distribu-tion. For the first time, the technique was described by McKayet al. (1979). In the area of statistical sampling, a square gridincluding sample positions is a Latin square if (and only if) thereis only one sample in each row and each column. A Latin hyper-cube is the generalization of this concept to an arbitrary numberof dimensions, whereby each sample is the only one in each axis-aligned hyperplane including it.

When a function of n variables is sampled, the range of eachvariable is divided into M equally probable intervals. Therefore,M sample points are placed to satisfy the Latin hypercube require-ments; it should be noted that this forces the number of divisions,M, to be equal for each variable. Also note that this samplingscheme does not need more samples for more dimensions (vari-ables); this independence is one of the most important advantagesof this sampling scheme. One of the other advantages is that ran-dom samples can be taken one at a time, remembering which sam-ples were taken so far.

The maximum number of combinations for a Latin hypercube ofM divisions and n variables (i.e., dimensions) can be calculatedwith the following formula:

Qnk¼0ðM � kÞn�1.

For instance, a Latin hypercube of M = 4 divisions with n = 2variables (i.e., a square) has 24 possible combinations. A Latinhypercube of M = 4 divisions with n = 3 variables (i.e., a cube) has576 possible combinations. This is convenient for the MADS algo-rithms, since the feasible region is already equally divided by themesh, and thus the design points must simply be distributedproperly.

On the region [0; 1] £ [0; 1], Fig. 3 illustrates an example of 4random points selected on a two-dimensional Latin hypercubedesign. From the 16 sectors, 4 random points are selected so thatno row or column is repeated.

Fig. 3. Latin hypercube design (2-D, 4-points).

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3.2.2.2. Particle swarm optimization. Particle swarm optimization(PSO) is a population-based, self-adaptive, stochastic optimizationmethod Kennedy and Eberhart (1995). The fundamental idea of thePSO is the mathematical modeling and simulation of the foodsearching activities of a swarm of birds (particles). In the multi-dimensional space where the optimal solution is sought, eachparticle in the swarm is moved toward the optimal point by theaddition of a velocity with its position. The velocity of a particleis influenced by three elements, namely, inertial, cognitive, and so-cial. The inertial element simulates the inertial behavior of the birdto fly in the previous direction. The cognitive element models thememory of the bird about its previous best position, and the socialelement models the memory of the bird about the best positionamong the particles (interaction inside the swarm). The particlesmove toward the multi-dimensional search space until they findthe food (optimal solution).

3.2.2.3. Surrogate-based methods. In problems with expensive func-tion evaluations, it is usually beneficial to substitute a model, orcomputationally cheaper approximation to stand in for the truefunction. This substitution can give an insight into the behaviorof the function without performing excessive costly function eval-uations. The term surrogate is used rather than approximation be-cause there is no tendency for implying that anything is requiredwith respect to how well the surrogates approximate the problemfunctions (Booker et al., 1999). In engineering applications, surro-gates can often be differentiated into two basic types: numericalsolutions of simpler physical models, and interpolation based mod-els of the solutions that are constructed using knowledge of thetrue function values. The former utilizes a simplified model ofthe physics that hopes to capture some aspect of the functionbehavior. The latter utilizes no knowledge of the physics, and isbased only on knowledge of the function values at a set of knowndata points.

A disadvantage may be that some simplified physics models arecostly to evaluate compared to interpolation-based models. Inaddition, a simplified physics model that is not well understoodcould result in a misleading surrogate that slows down the optimi-zation. In the current study, an interpolation-based surrogate isused. It is notable that surrogates are not required to accuratelyrepresent the function. A surrogate may efficiently capture the glo-bal behavior of the function, but be inaccurate for local prediction,

and yet it will serve as a valuable tool for steering the optimizer to-wards a function minimum.

Surrogates are a specifically good match for combining withMADS. One reason for this is the inherent flexibility of MADS, be-cause it can be separated into a SEARCH step, which offers the userflexibility, and a POLL step which provides the basis for proof ofconvergence. Due to this, a surrogate can be incorporated to searchthe design space in a matter well-suited to the problem, and thechoice of surrogate is flexible.

The first step in the optimization process is to choose a set ofinitial data. The main reason of using LHS method is to find a welldistributed set of initial data in the parameter space, thus ensuringthat each input variable has all portions of its range represented inthe chosen data set. Once the initial data set, x1, ... , xm, is chosen,the cost function is evaluated at these points, and an initial surro-gate model is constructed (Marsden, Wang, Dennis, & Moin, 2004).A Kriging surrogate model is utilized to interpolate the data, and topredict the value of the function at a particular location in theparameter space (Marsden, 2004). Kriging is a statistical methodbased on the usage of spatial correlation functions. The Krigingalgorithm is implemented in DACE software package. It is easilyextended to multiple dimensions, making it attractive for optimi-zation problems with several variables. A detailed derivation ofthe Kriging approximation can be found in Marsden et al. (2004),following Lophaven et al. (2002). If the surrogate optimization failsto find an improved mesh point, then the surrogate is updated andmade more accurate with the new point. The use of surrogates usu-ally yields significant improvement in the objective function valueearly on in the iteration process (Abramson, 2007).

Boeing utilizes DACE surrogates (Santner, Williams, & Notz,2003) in their Design Explorer filter implementation (Audet,Booker, Dennis, Frank, & Moore, 2000). They generate data sitesby an orthogonal array, and thereafter fit a DACE model to the data.The SEARCH includes a global Newton SQP method applied to thesurrogate problem to try to generate several good local optimizersfor that problem. Afterwards, they use the expensive ‘‘true’’ prob-lem functions at those points to decide whether the SEARCH hasbeen successful. Whenever new values of the true problem func-tions have been computed, they are utilized to recalibrate the sur-rogates. This surrogate management framework results in verysuccessful methods. Details are given in Audet et al. (2000). AlisonMarsden has solved trailing edge shape design problems utilizingboth types of surrogates in an insightful way. She generates trialpoints using the MATLAB DACE surrogate package (Lophaven,Nielsen, & Sondergaard, 2002) and thereafter uses a less expensiveturbulence model to check whether a trial point is in X. If it is, thenshe runs the more precise simulation. Her SEARCH includes apply-ing an evolutionary algorithm to the DACE surrogates (Marsden,2004). Another interesting application of surrogates is in Marsdenet al. (2004), where a framework to identify good algorithmicparameter values is provided. To demonstrate this framework,MADS was applied to an objective function that measured theCPU time required by a trust-region algorithm (Gould, Orban, &Toint, 2003) to solve a set of difficult problems.

3.2.3. The POLL stepThe POLL step is more rigidly defined than the SEARCH step.

Whenever the SEARCH step fails to generate an improved meshpoint, then the POLL step is invoked before terminating the itera-tion. The POLL step includes a local exploration of the space of opti-mization variables near the current incumbent solution xi (calledthe frame center). The set of trial points considered during thePOLL step is called a frame. If the POLL step fails to produce an im-proved mesh point, Pi is expressed as a minimal frame with mini-mal frame center xi. If both the SEARCH and POLL step is successfulin finding an improved mesh point, the improved mesh point

6560 S.S. Sadat Hosseini et al. / Expert Systems with Applications 38 (2011) 6556–6564

becomes the new current iterate xi+1 and the mesh is either re-tained or coarsened. If none of the steps is successful, then the min-imal frame center is retained as the current iterate (i.e., xi+1 = xi)and the algorithm continues to the parameters update step.

For MADS, the poll size parameter Dpi 2 Rþ for iteration i is pre-

sented. This new parameter determines the magnitude of the dis-tance from the trial points generated by the POLL step to thecurrent incumbent solution xi.

The MADS frame is constructed by the usage of a currentincumbent solution xi and the poll and mesh size parameters Dp

i

and Dmi to obtain a positive spanning set of directions Di. In general,

the MADS set of directions Di is not a subset of D.At iteration i, the MADS frame is defined to be the set

Pi ¼ fxi þ Dmi d : d 2 Dig � Mi ð8Þ

where Di is a positive spanning set such that 0 R Di and for eachd e Di,

� d can be written as a non-negative integer combination of thedirections in D: d = Du for some vector u 2 NnDi that may dependon the iteration number i,� the distance from the frame center xi to a frame point

xi þ Dmi d 2 Pi is bounded above by a constant times the poll size

parameter: Dmi kdk 6 Dp

i maxfkd0k : d0 2 Dg,� limits of the normalized sets Di ¼ d

kdk : d 2 Di

n oare positive

spanning sets.

The set of all poll directions D ¼ [1i¼1Di is said to be asymptoti-cally dense if the closure of D equals Rn.

Graphically, the definition of a hypothetical sequence of threePOLL sets is given by Fig. 4. We can easily check if the above fourproperties are satisfied. It is demonstrated in Audet and Dennis(2006) that the set of normalized directions formed by [1i¼1Di isdense in the unit circle with probability 1.

There are two strategies to evaluate f at the trial POLL points Pi.The first one is to terminate iteration i whenever there is a declinein f. The exhaustive one consists of evaluating the entire POLL set Pi

and thereafter choosing the feasible point leading to the largest de-crease of the objective function value f. In the present paper, thefirst strategy was used since it was practically observed that it usu-ally reduces the overall number of function evaluation. In somecases, incomplete derivative information might be available. Forinstance, in some multi-disciplinary design optimization (MDO)problems, derivatives for some disciplines might be available, butnot for others, and derivatives across multiple disciplines are notavailable. If the full gradient is available, directions can be selectedso that all but one are ascent directions, which can be ignored, thusreducing the required number of function evaluations to one periteration (Abramson et al., 2004). In this case, the MADS algorithmreduces to an approximation of steepest descent. Even if only some

Fig. 4. Examples o

partial derivatives are known, MADS can exploit this informationto decrease the number of function evaluations in each POLL step(Abramson et al., 2004) without sacrificing theoretical convergenceproperties. In related work, Custodio and Vicente (2007) compute asimplex gradient from a subset of formerly evaluated points havingcertain geometrical properties, and they have studied its use as apotential direction of descent in an effort to speed convergence.Because MADS is opportunistic, in that it moves immediately to anew improved mesh point as soon as it is found, the order in whichPOLL points are evaluated can impact performance. One approachin such improvement can be witnessed is what is called dynamicpolling, in which the most recent successful direction is movedto the front of the queue after each successful POLL step. Dynamicpolling was presented useful in Audet and Dennis (2006) on achemical engineering parameter fit problem (Hayes, Bertrand,Audet, & Kolaczkowski, 2003). Custodio and Vicente (2007) havealso seen a reduction in function evaluations by computing a sim-plex gradient and ordering POLL points in accord with how smallan angle the corresponding poll directions make with the negativeof the simplex gradient. However, these strategies (dynamic poll-ing, surrogate and simplex gradient ordering) do not necessarilylead to improved computational times in all cases.

3.2.4. Parameters updateRules for refining and coarsening the mesh are as follows.Given a fixed rational number s > 1 and two integers w� 6 �1

and w+ P 0, the mesh size parameter Dmi is updated according to

the rule

Dmiþ1 ¼ rwi Dm

i for some wi

2f0;1; . . . ;wþg if an improved mesh po int is found

fw�;w� þ 1; . . . ;�1g otherwise

� �

ð9Þ

In MADS, the strategy for updating Dpi must be such that Dm

i 6 Dpi for

all i, and more, it must satisfy

limi2K

Dmi ¼ 0 if and only if lim

i2KDp

i

¼ 0 for any infinite subset of indices K ð10Þ

3.2.5. TerminationSome termination criterion must be specified, such as a mini-

mal value on the mesh size parameter Dmi , a maximal number of

objective function evaluations, or a maximal number of consecu-tive unsuccessful function evaluations. It is shown in Dolan, Lewis,and Torczon (2003) that the mesh size parameter is a measure offirst-order stationarity for GPS in the unconstrained case. As soonas one termination criterion is achieved, the algorithm terminates.Otherwise, it returns to step 2.

f POLL sets Pi.

Table 1Parameter settings for the MADS algorithm.

Parameters Setting

Termination parametersMesh size tolerance 10�6

Maximum number of function evaluations 1000Maximum number of consecutive POLL failures 50

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3.3. Constraints handling

For bound and linearly constrained problems, the only addi-tional requirement over those of unconstrained problems is thatthe set of poll directions must be selected so as to conform tothe geometry of the constraints. In this study, non-linear con-straints are treated using a filter. Filter-based methods attempt

Fig. 5. Feasible operation region for the cogeneration unit 2.

Fig. 6. Feasible operation region for the third unit.

Maximum CPU time (s) 120

Mesh parametersInitial mesh size ðDm

0 Þ 1Mesh refinement factor 0.5Mesh coarsening factor 1

Table 2Optimal results for case study I.

Methods Optimal results

P1 P2 P3

GA [5] 0 159.23 40.77LR [3] 0 160 40ACSA [10] 0.08 150.93 49GT [8] 0 157.92 42.08a

HS [1] 0 160 40SARGA [11] 0 159.99 40.01MADS–LHS 0.0017 159.80 40.2014MADS–PSO 0.0092 157.9392 42.0516MADS–DACE 0 160 40

a Outside the feasible operating region of cogeneration unit 3.

to minimize both the objective function f and a non-negativeaggregate constraint violation function h, where h satisfies the con-dition that h(x) P 0 with h(x) = 0 if and only if x is feasible.

In the context of MADS, a filter is a collection of points, none ofwhich dominates any other in the set with respect to their objec-tive and constraint violation function values. That is to say, forany two points x; y in the filter, the condition f(x) < f(y) andh(x) < h(y), cannot hold. Polling is performed around either the bestfeasible point found thus far, or the least infeasible point in an at-tempt to add new points to the filter-preferentially new poll cen-ters. A more detailed explanation can be found in Audet andDennis (2004).

Equality constraints are difficult to handle because of basicproperties of the MADS algorithm, and also due to numerical con-siderations. Equality constraints are difficult to handle because ofbasic properties of the MADS algorithm, and also due to numericalconsiderations. In order to avoid the problems that equality con-straints present, one variable should be eliminated and these con-straints should be reformulated as inequalities.

4. Case studies

Three examples taken from the optimization literature are usedto show the validity and effectiveness of the MADS algorithm. TheMADS algorithm can handle the cases of multiple heat areas andpower areas. For the demonstration, a single heat area and powerarea system is first considered (case study I). This case study wasoriginally proposed by Guo, Henwood, and van Ooijen (1996).The system tested is comprised of a conventional power unit,two cogeneration units and a heat-only unit. The objective is to ob-tain the minimum overall cost of units subject to constraints onheat and power production and demands. The power capacity lim-its for the conventional power unit are 0.0 and 150 MW. The heatcapacity limits for the heat only unit are 0.0 and 2695.2 MWth. Thefeasible operating regions of the two cogeneration units are shownin Figs. 5 and 6. The system power demand Pd and the heat demandHd are respectively 200 MW and 115 MWth.

Cost ($)

H2 H3 H4

39.94 75.06 0 9267.240 75 0 9257.0748.84 65.79 0.37 9452.226 89a 0 9207.6440 75 0 9257.0739.99 75 0 9257.0742.4042 72.3904 0.2054 9277.13142.4459 72.5522 0.0019 9301.35740 75 0 9257.07

6562 S.S. Sadat Hosseini et al. / Expert Systems with Applications 38 (2011) 6556–6564

The objective function of the CHPED problem is:

C1 ¼ 50P1 ð11ÞC2 ¼ 2650þ 14:5P2 þ 0:0345P2

2 þ 4:2H2 þ 0:03H22 þ 0:031P2H2

ð12ÞC3 ¼ 1250þ 36P3 þ 0:0435P2

3 þ 0:6H3 þ 0:027H23 þ 0:011P3H3

ð13ÞC4 ¼ 23:4H4 ð14Þ

Min f ¼X4

i¼1

Ci ð15Þ

Pd ¼X3

i¼1

Pi ð16Þ

Hd ¼X4

i¼2

Hi ð17Þ

s:t: 0 6 P1 6 150MW ð18Þ0 6 H4 6 2695:2MWth ð19Þ

The CHPED problem formulation of all test cases has been solvedusing MADS incorporated in the NOMADm software (Abramson,

Fig. 7. The convergence nature of the MADS–DACE algorithm for case study I.

Table 3Performance of MADS based on different SEARCH methods for case study I.

Performance SEARCH method

LHS PSO DACE

Best solution 9277.1311 9301.3567 9257.0754Worst solution 10005.3805 9997.6576 9260.4317Mean solution 9547.9151 9551.277 9257.5148Standard deviation 213.3516 251.1569 1.0743CPU time (s) 7.0422 7.5594 2.3078

Table 4Optimal results for case studies II and III.

Cases Demand Demand Optimal results

Pd Hd P1 P2

II 175 110 MADS–LHS 0.0066 134.3325MADS–PSO 0.1381 128.9305MADS–DACE 0 135GA1 [5]GA2 [5]

III 175 110 MADS–LHS 0 181MADS–PSO 0.0144 182.0725MADS–DACE 0 185GA1 [5]GA2 [5]

2007). In order to further test the MADS approach and compare itwith conventional methods, two other tests (case study II and III)are conducted with various power and heat demands. These caseswere presented by Song and Xuan (1998). To verify the performanceof the MADS, the program is ran a 50 times on the examples. Theparameters settings for the MADS algorithm are shown in Table 1.

The poll directions can be set to MADS Positive basis N + 1 and2N, where N is the number of independent variables for the objec-tive function. The MADS Positive basis 2N directions explore morepoints around the current point at each iteration. Therefore, the Po-sitive basis 2N directions are considered as the POLL method in thepresent study to avoid finding a local minimum rather than theglobal minimum. Both of the dynamic polling and the simplex gra-dient strategies are considered to solve the problem. In addition tobetter performance, the simplex gradient strategy led to improvedcomputational times in all cases compared to the dynamic polling.Each of the CHPED problems was solved using the LHS, PSO andDACE SEARCH strategies.

These examples have been previously solved using a variety oftechniques (both evolutionary and traditional mathematical pro-gramming methods). The resulting minimum cost are used to com-pare the performance of MADS with those of other methods. Noconstraints are active for this solution. Table 2 presents the bestsolution of this problem obtained using the MADS algorithm andthose reported by other researchers. It can be seen from Table 2that the result obtained using the MADS–DACE algorithm is thesame as the best known solution reported previously in the litera-ture (Guo et al., 1996; Subbaraj et al., 2009; Vasebi et al., 2007).Although Sudhakaran and Slochanal (2003) reported better solu-tion, this solution is not feasible due to violation of constraints.Also, the results reported by Subbaraj et al. (2009) have someproblem about their considered feasible operating region (SadatHosseini & Gandomi, 2010). For brevity, only the convergence nat-ure of the MADS–DACE algorithm is shown in Fig. 7. It is evidentfrom this figure that MADS–DACE has a considerably fast conver-gence. Moreover, the best, worst and mean solutions, the standarddeviation and the average CPU times obtained the MADS–LHS,MADS–PSO, MADS–DACE algorithms are shown in Table 3. It canbe seen from this table that MADS–DACE provides significant sav-ings of computational effort in addition to its superior performancecompared to the other SEARCH methods. These short computingtimes allow for more cases to be studied with the goal of increasingthe confidence in the final solution.

Optimal results obtained by different methods for case studies IIand III are shown in Table 4. It can be seen from this table that thebest performance is obtained by the MADS–DACE algorithm forboth case studies. For case study III, the MADS–LHS and MADS–PSO algorithms also provide superior performance compared tothe available methods in the literature. The convergence natureof the MADS–DACE algorithm is shown in Fig. 8(a) and (b).

Cost ($)

P3 H2 H3 H4

40.6609 44.371 65.6061 0.0229 8622.074845.9314 29.9893 80.0027 0.0080 8629.415640 35 75 0 8555.9625NA 8622.9NA 8769.6

44 47.3125 77.6875 0 10101.475342.9131 49.0611 75.9298 0.0091 10101.894240 50 75 0 10074.4875NA 10128NA 10132.4

Fig. 8. The convergence nature of the MADS–DACE algorithm for (a) case study II, (b) case study III.

Table 5Performance of MADS based on different SEARCH methods for case studies II and III.

Cases Demand Performance SEARCH method

Pd Hd LHS PSO DACE

II 175 110 Best solution 8622.0748 8629.4156 8555.9625Worst solution 9442.1843 9046.6340 8555.9652Mean solution 8921.7626 8799.8773 8555.9625Standard deviation 252.3315 136.8504 0.0009CPU time (s) 63.6000 56.5656 3.7094

III 225 125 Best solution 10101.4753 10101.8942 10074.4875Worst solution 10833.7313 11393.4252 10074.4883Mean solution 10295.4723 10455.1238 10074.4907Standard deviation 236.1302 478.2026 0.0011CPU time (s) 8.1516 7.0563 5.5281

S.S. Sadat Hosseini et al. / Expert Systems with Applications 38 (2011) 6556–6564 6563

Performance of MADS based on different SEARCH methods forthese case studies is illustrated in Table 5.

5. Conclusion

This paper has introduced a recent optimization method,namely MADS to solve the CHP economic dispatch problem consid-ering the feasible operating region. MADS is a recently developedalgorithm that is supported by a thorough convergence analysis.The MADS method is illustrated using three test cases taken fromthe literature. The LHS, PSO and DACE algorithms are employedas effective search strategies in MADS to solve each of the CHPEDproblems. The performance of the utilized MADS–LHS, MADS–

PSO, MADS–DACE methods are compared to that of other tech-niques reported in the literature. The results clearly demonstratethat the MADS-based methods are practical and valid for CHPEDapplications. The MADS–DACE algorithm performs superior thanor as well as the other recent methods in terms of solution quality,handling constraints and computation time.

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