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Electric Power Systems Research 80 (2010) 723–732

Contents lists available at ScienceDirect

Electric Power Systems Research

journa l homepage: www.e lsev ier .com/ locate /epsr

aily Hydrothermal Generation Scheduling by a new Modifieddaptive Particle Swarm Optimization technique

ima Amjady ∗, Hassan Rezai Soleymanpourepartment of Electrical Engineering, Semnan University, Semnan, Iran

r t i c l e i n f o

rticle history:eceived 11 June 2009eceived in revised form8 September 2009ccepted 12 November 2009vailable online 23 December 2009

a b s t r a c t

The fundamental requirement of power system hydrothermal scheduling is to determine the optimalamount of generated powers for the hydro and thermal units of the system in the scheduling horizonof 1 day or few days while satisfying the constraints of the hydroelectric system, thermal plants andelectrical power system. Daily Hydrothermal Generation Scheduling (DHGS) is a complicated non-linear,non-convex and non-smooth optimization problem with discontinuous solution space. To deal with thiscomplicated problem, a new Modified Adaptive Particle Swarm Optimization (MAPSO) is proposed in this

eywords:ydroelectric systemhermal plantHGSAPSO

paper. The inertia weight and acceleration coefficients of the PSO are adaptively changed in the MAPSOowning tree topology. We split-up the cognitive behavior of PSO into the best and not-best parts. Theproposed not-best cognitive component, unlike recent methods, retains its dynamic behavior throughoutthe search process. Personal best position exchanging method is proposed to increase activities of par-ticles to explore broad space. New velocity limiter is also proposed in this paper to enhance explorationcapability and convergence behavior of the MAPSO. The proposed MAPSO is tested on six test systems

rece

and compared with some

. Introduction

Daily Hydrothermal Generation Scheduling (DHGS) is anmportant issue in economical operation of power systems. Thehort-term hydrothermal generation scheduling determines opti-al hourly water releases of each reservoir in hydroelectric

lants to generate electrical energy for supporting some part ofhe power demand. In order to satisfy the rest of the poweremand, which is not supported by hydropower generation, DHGSchedules thermal generation units so that the total produc-ion cost is minimized during the scheduling time horizon. The

ain objective is focused on the optimal use of water resourcesor minimizing the production cost of thermal plants consider-ng the practical constraints. In DHGS problem, the constraintsre usually divided into three categories related to hydroelectricystem, thermal plants and electrical power system (satisfyingower demand constraint) [1]. Aside from these constraints, theascading nature of hydrosystems causes dependency between

he performances of hydropower plants. Also, the impact ofteam valve loading on operational cost curve of thermal unitsntensifies non-convexity and non-linearity of the DHGS prob-em. So, DHGS is a complicated non-linear, non-convex and

∗ Corresponding author. Tel.: +98 021 88889096; fax: +98 021 88880098.E-mail address: amjady@tavanir.org.ir (N. Amjady).

378-7796/$ – see front matter © 2009 Elsevier B.V. All rights reserved.oi:10.1016/j.epsr.2009.11.004

nt research works in the area.© 2009 Elsevier B.V. All rights reserved.

non-smooth optimization problem with discontinuous solutionspace.

Several methods, such as dynamic programming (DP) [2], net-work flow [3], decomposition technique [4], mixed integer linearprogramming (MILP) [5], and Lagrangian relaxation (LR) [1,6] havebeen proposed to solve DHGS problem in the recent years. Amongthe existing methods, DP appears to be the most popular despite themajor disadvantage of drastic growth of computational and dimen-sional requirements with increasing system size and planninghorizon [7]. The network flow model of DHGS is often programmedas a linear or piecewise linear one. Linear programming typicallyconsiders that power generation is linearly dependent on water dis-charge, thus ignoring the head change effect, leading to a solutionschedule with less power generation [8]. Handling of the vari-ous constraints increases the number of dual variables and thecomplexity of the optimization task in the decomposition tech-nique [9]. Also, the discretization of the non-linear dependencebetween power generation, water discharge and head, used in theMILP to model head variations, augment the computational burdenrequired to solve DHGS problem. The implementation of LR is com-plicated and its efficiency heavily depends on the size of the duality

gap. Furthermore, solution quality of LR depends on the method toupdate Lagrange multipliers [8]. As a result, conventional methodsrequire models of hydro as well as thermal plants to be representedas piecewise linear or polynomial approximations of monotonicallyincreasing nature. However, such an approximation may lead to a

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24 N. Amjady, H.R. Soleymanpour / Electric

uboptimal solution, resulting in huge loss of revenue over the time.ence, the trend in recent times is to use more realistic models forydro and thermal plants [7].

Recently the artificial intelligence (AI) methods such as geneticlgorithm [10,11], neural network [12], simulated annealing [13],ifferential evolution [14], cultural algorithm [15], evolutionaryrogramming [7], and particle swarm optimization [16–18] haveeen presented for the solution of DHGS problem to overcomehese deficiencies. Moreover, some combinatorial techniques suchs coevolutionary algorithm (CEA) based on the Lagrangian method19] and hybrid simulated annealing-genetic algorithm [20] haveeen also proposed to solve this problem. However, the qualityf solution is a concern for the artificial intelligence based meth-ds as these methods may trap in local minima or even infeasibleolutions for the complex problem of DHGS. Besides, depen-ency of these methods on the correct tuning of their adjustablearameters and initial point is another problem of these meth-ds.

To mitigate the imperfections of the previous AI methods for theolution of DHGS problem, a new Modified Adaptive Particle Swarmptimization (MAPSO) is proposed in this paper. Novel operatorsnd techniques are incorporated in the MAPSO to enhance its explo-ation and exploitation capabilities for the solution of complexon-linear non-convex DHGS problem. Also, the proposed MAPSOas low sensitivity with respect to both its adjustable parametersnd initial point.

The remaining parts of the paper are organized as follows. Inhe second section, the DHGS problem is formulated. The proposed

APSO is introduced in the third section. Application of the MAPSOo solve the DHGS problem is presented in the form of a step by steplgorithm in Section 4. Obtained numerical results are presentednd discussed in Section 5. Section 6 concludes the paper.

. Problem formulation

.1. Objective function

Due to negligibility of hydropower production cost, the mainbjective of the DHGS is to assess optimal water releases ofydroreservoirs and power generation of each thermal unit so thatotal thermal fuel cost (objective function OF) becomes minimized2,21]:

inimize : OF =T∑

t=1

Ns∑i=1

[fi(Psi,t)] (1)

It is assumed that fi(Psi,t) is a quadratic function of Psi,t:

i(Psi,t) = asi + bsiPsi,t + csiPs2i,t (2)

In practice, the fuel cost function has non-differentiable pointsue to valve loading effects. To consider these effects of units, aecurring rectifying sinusoidal term is added to the quadratic costunction as follows [22]:

i(Psi,t)=asi + bsiPsi,t+csiPs2i,t + |dsi × sin(esi × (Psmin

i − Psi,t))| (3)

.2. Constraints

.2.1. Power generation limits

Each power generation unit has generation capacity boundaries:

hminj ≤ Phj,t ≤ Phmax

j (4)

smini ≤ Psi,t ≤ Psmax

i (5)

r Systems Research 80 (2010) 723–732

Phj,t is considered to be a polynomial function of water dischargerate and reservoir storage volume [7,8,21,23]:

Phj,t = Ch1jVh2j,t + Ch2jQh2

j,t + Ch3jVhj,tQhj,t + Ch4jVhj,t + Ch5jQhj,t

+ Ch6j (6)

A more complex function for hydropower generation is pre-sented in [1]. Our proposed solution algorithm (MAPSO) canconsider this hydropower generation function as well, providedthat its data is available.

2.2.2. Prohibited operating zones (POZs)In practice, thermal units can have prohibited operating regions

as discussed in [22]. The power generations for these units shouldbe in the non-forbidden operating zones. So, the power generationfor ith thermal unit considering POZ is modeled as follows:⎧⎨⎩

Psmini

≤ Psi,t ≤ PsLB,1i

PsUB,k−1i

≤ Psi,t ≤ PsLB,ki

k = 2, 3, . . . , NOi

PsUB,ki

≤ Psi,t ≤ Psmaxi

k = NOi

(7)

2.2.3. Ramp rate limitsRamp rate limits of ith thermal generating unit can be described

as:{Psi,t − Psi,t−1 ≤ URi if Psi,t ≥ Psi,t−1Psi,t−1 − Psi,t ≤ DRi if Psi,t−1 ≥ Psi,t

(8)

Combining (5) and (8), the ramp rate constrained operating lim-its of thermal units can be stated as follows:

Psrmaxi,t = min{Psmax

i , Psi,t−1 + URi} (9)

Psrmini,t = max{Psmin

i , Psi,t−1 − DRi} (10)

Psrmini,t ≤ Psi,t ≤ Psrmax

i,t (11)

2.2.4. Water discharge rate limitConsidering prohibited discharge zones (PDZ) of hydro units [7],

the following constraints for Qhj,t should be imposed:⎧⎨⎩

Qhminj

≤ Qhj,t ≤ QhLB,1j

QhUB,m−1j

≤ Qhj,t ≤ QhLB,mj

m = 2, 3, . . . , NDj

QhUB,mj

≤ Qhj,t ≤ Qhmaxj

m = NDj

(12)

It is noted that thermal unit i with NOi POZs and hydro unit j withNDj PDZs will have NOi + 1 and NDj + 1 disjoint operating regions,respectively. These disjoint regions form a non-convex set [22].

2.2.5. Reservoir storage volume limitVhmin

j ≤ Vhj,t ≤ Vhmaxj (13)

2.2.6. Water dynamic balance [21]

Vhj,t = Vhj,t−1+Ihj,t − Qhj,t−Shj,t +UPj∑r=1

(Qhr,t−�r,j+ Shr,t−�r,j

) (14)

2.2.7. Final volume constraint [14]Vhj,T = Vhj,end (15)

2.2.8. Active power balance

Ns∑i=1

Psi,t +Nh∑j=1

Phj,t − PDt − PLt = 0 (16)

PLt can be calculated by the B matrix loss formula [21,22].

N. Amjady, H.R. Soleymanpour / Electric Powe

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Fig. 1. (a) General tree topology and (b) proposed tree topology.

. The proposed MAPSO

PSO was introduced by Kennedy and Eberhart as a moderneuristic optimizer [24]. PSO is a population based search methodhich deals with random particles in search space. The particles, i.e.

rial solutions of the optimization problem, share their informationith each other and run toward best trajectory to find optimum

olution in an iterative process. A velocity vector is defined forach particle and particle position depends on this velocity. In eachteration, the velocity and position of particles are updated:

i,iter+1 = wVi,iter + c1r1(Pbesti,iter − Xi,iter) + c2r2(Gbest

iter − Xi,iter) (17)

i,iter+1 = Xi,iter + Vi,iter+1 (18)

here Vi,iter and Xi,iter represent the velocity vector and the positionector of ith particle at iteration iter, respectively; Pbest

i,iterand Gbest

iterre personal best position of ith particle and global best position ofwarm until iteration iter, respectively; w is inertia weight factorhich controls the global and local exploration capabilities of par-

icles; c1 and c2 are cognitive and social coefficients, respectively;1 and r2 are two random numbers between 0 and 1. To enhancehe efficiency of the PSO, we adjust the inertia weight w to linearlyeduce during the iterations [25]:

= (wmax − wmin) ×(

itermax − iter

itermax

)+ wmin (19)

here itermax is the maximum number of iterations. Also, to effi-iently control the local search and convergence to the globalptimum solution, time-varying acceleration coefficients (TVAC)re introduced in addition to the time-varying inertia weightTVIW) in PSO [26]. A large cognitive component and small socialomponent at the beginning, allows particles to move around theearch space, instead of moving towards the population best pre-aturely. During the latter stage in optimization, a small cognitive

omponent and a large social component allow the particles to con-erge to the global optima [25]. So, the acceleration coefficients aredaptively changed as follows [25,26]:

1 = (cfinal1 − cinitial

1 )(

iter

itermax

)+ cinitial

1 , cfinal1 < cinitial

1 (20)

2 = (cfinal2 − cinitial

2 )(

iter

itermax

)+ cinitial

2 , cfinal2 > cinitial

2 (21)

here the superscripts “initial” and “final” indicate the initial andnal values of the acceleration coefficients, respectively. We con-ider both TVIW as stated in (19) and TVAC as represented in (20)nd (21) in the PSO and call it Adaptive PSO or APSO. Moreover, wesed a specific kind of tree topology for the APSO. In tree topology,ll particles are arranged in a tree and each node of the tree containsxactly one particle [27] as shown in Fig. 1(a). A particle is influ-nced by its own best position so far (Pbest) and by the best position

f the particle that is directly above in the tree (parent). If a particlet a child node has found a solution that is better than the best soar solution of the particle at the parent node, the both particles arexchanged. In this way, this topology offers a dynamic neighbor-ood, which enhances the search ability of the PSO. However, tree

r Systems Research 80 (2010) 723–732 725

topology is based on breadth-first search in which an individualcan move down several levels in the hierarchy but it can move upat most one level in each iteration [27]. Our proposed tree topologyfor the APSO is shown in Fig. 1(b), wherein each level of the treehas only one node or one particle. In each iteration, all particles aresorted from the top of the tree to down, based on their Pbest values.In this topology, each particle can move up and down by any num-ber of levels in each iteration and so the hierarchy is updated in oneiteration. Based on the tree topology, (17) becomes as follows:

Vi,iter+1=wVi,iter + c1r1(Pbesti,iter − Xi,iter) + c2r2(PARi,iter − Xi,iter) (22)

where Gbestiter

is replaced by PARi,iter indicating the best position sofar of the parent (directly above particle) of ith particle at iterationiter.

For the APSO with the tree topology of Fig. 1(b), three newmodifications are introduced to construct a novel heuristic searchtechnique called Modified APSO or MAPSO. These new modifica-tions, enhancing the exploration and convergence capabilities ofthe proposed method, are described in the following, respectively.

3.1. The split-up of the cognitive part into the best and not-bestcomponents

Selvakumar and Thanushkodi [28] proposed a split-up in thecognitive behavior of PSO into Pbest

i,iterand Pworst

i,iterthat changed (17) as

follows:

Vi,iter+1 = wVi,iter + c1br1(Pbesti,iter − Xi,iter) + c1wr2(Xi,iter − Pworst

i,iter )

+ c2r3(Gbestiter − Xi,iter) (23)

where Pbesti,iter

is exactly the same as the cognitive component of the

basic PSO and Pworsti,iter

is personal worst position of ith particle untiliteration iter; r1, r2 and r3 are random numbers between 0 and 1.That is, the particle is made to remember its worst position also.This modification helps to search the solution space more effec-tively compared with the classical PSO and has also been usedin some other research works such as [8]. However, our experi-ence with DHGS problem shows that after a few initial iterations,the swarm particles become better and better and so Pworst

i,iterof ith

particle approximately remains unchanged. In other words, Pworsti,iter

nearly becomes a static limit and loses its dynamic behavior. So,the expected enhancement in the exploration capability of the PSOmay not be obtained. To solve this problem, we propose the newidea of decomposing the cognitive part into the “Best” and “Not-best” components instead of splitting up into the “Best” and “Worst”components as follows:

Vi,iter+1 = wVi,iter + c1br1(Pbesti,iter − Xi,iter) + c1nbr2(Xi,iter − Pnot-best

i,iter )

+ c2r3(PARi,iter − Xi,iter) (24)

{Pbest

i,iter= Xi,iter, if AOF(Xi,iter) < AOF(Pbest

i,iter−1)

Pbesti,iter

= Pbesti,iter−1, if AOF(Xi,iter) ≥ AOF(Pbest

i,iter−1)(25)

{Pnot-best

i,iter= Xi,iter, if AOF(Xi,iter) ≥ AOF(Pbest

i,iter−1)

Pnot-besti,iter

= Pnot-besti,iter−1 , if AOF(Xi,iter) < AOF(Pbest

i,iter−1)(26)

where AOF is the augmented objective function of the optimizationproblem that should be minimized. AOF includes both the objec-

tive function and penalty terms. For the DHGS problem, AOF willbe introduced in the next section. Equation (25) describes Pbest

i,iteraccording to basic PSO. In any iteration that the particle cannotfind a solution better than its best solution until the previous itera-tion Pbest

i,iter−1, the “Not-best” component Pnot-besti,iter

is replaced by Xi,iter

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26 N. Amjady, H.R. Soleymanpour / Electric

s shown in (26). In this way, the limit of Pnot-besti,iter

is continuouslypdated (even in the last iterations) and saves its dynamic behaviorhroughout the search process of the PSO enhancing its explorationapability. It is noted that the both coefficients of the cognitiveart of (24), including c1b and c1nb (related to the “Best” and “Not-est” components, respectively), are adaptively changed along theterations of the MAPSO based on (20).

.2. Personal best position exchanging

When a particle position becomes close to the Gbest of the swarmhere, parent of the particle or PARi,iter), its personal best positionbest will also become close to Gbest after a few iterations and so theffect of both cognitive and social parts will decrease. Therefore,his particle becomes lazy in the swarm degrading the explorationapability of the PSO. This phenomenon is more seen in the lattertages after the initial part of the search. To remedy this problemnd increase the activity of swarm particles, we propose the idea ofersonal best position exchanging. When two particles exchangeheir Pbest, their velocity vectors are suddenly changed based on24), which makes the lazy particle to move abroad the searchpace. So, the lazy particle is renewed, which enhances the explo-ation capability of the proposed MAPSO. We inspired this idea fromhe crossover operator of genetic algorithms (GA) that swaps a partf information of two individuals. In each iteration, a number of par-icles of the swarm are randomly selected and grouped in twofoldlusters, like the crossover operator of GA. Then, Pbest of the twoembers of each cluster is exchanged. While the diversity of the

earch process is enhanced, no information about Pbest of a particles lost in this technique (only it is stored in another particle). It isoted that this technique is different from “Small-World” methodroposed for random leader selection [29]. “Small-World” method

s related to the social behavior and is implementable in some spe-ific PSO topologies. On the other hand, the proposed personal bestosition exchanging technique is related to the cognitive part andan be implemented in any PSO. Indeed, we have never seen thisechnique in the previous research works on PSO.

.3. New velocity limiter

In conventional PSO, an upper limit is placed on the velocity inll dimensions. This upper limit (velocity limiter) prevents parti-les from moving too rapidly from one region in search space tonother. The maximum velocity allowed actually serves as a con-traint that controls the maximum global exploration ability of PSO30]. A fixed velocity limiter for each dimension is usually con-idered as a proportion of the allowable position range. However,electing a proper velocity limiter is difficult, especially for com-lex optimization problems [31]. For the initial stages of the searchrocess, a high maximum velocity allowed is usually required sohat the particles can search different regions of the solution space.n the other hand, a low maximum velocity allowed may be prefer-ble for the final stages of the search process so that the PSO canetter converge. A discussion about this matter can be found in [32]here it has been concluded that the choice of a suitable value for

he velocity limiter can be nontrivial and also very important forhe determination of the overall performance of the algorithm. Tovercome this problem, some modifications have been investigatedo determine proper velocity limiter. For instance, a random veloc-ty limiter was introduced in [33] such that the velocity boundary

lters randomly to prevent the velocity of a particle from stop-ing on a same boundary during the evolution. However, randomelection of maximum velocity allowed may result in inconsistentelocity limiters along the search process of the PSO. In [31], thearticles are divided into several groups and the maximum veloc-

r Systems Research 80 (2010) 723–732

ity of each group is adaptively controlled such that the maximumvelocity of the worst group is changed to approach the maximumvelocity of the best group if the difference of goodness between thebest and worst group is big. However, the exploration capabilityof PSO along the search process is not explicitly considered in thisapproach. A review of different velocity limiters of PSO proposed inthe previous research works in the area can be found in [33]. In thispaper, a new velocity limiter is proposed, which adaptively sets themaximum velocity allowed along the search process of the MAPSOas follows:

R = Rinitial + (Rfinal − Rinitial)(

iter

itermax

), Rinitial > Rfinal (27)

Vj,max = R × (Xj,max − Xj,min) (28)

where R is a dynamic coefficient that adaptively changes along thesearch process; Vj,max is velocity limiter for the jth dimension ofthe particles (jth decision variable) that is |Vj| ≤ Vj,max; Xj,min andXj,max indicate the minimum and maximum allowable position forthe jth dimension. As seen the velocity limiter linearly decreasesfrom a higher value to a lower one along the search process. By thisvelocity limiter the particles can have high velocities in the initialiterations to broadly search the solution space. In the final iterationsthe particles velocity is more limited to avoid jumping particles. Bythe proposed adaptive velocity limiter, the MAPSO can benefit fromboth good exploration capability and convergence behavior.

4. Application of the proposed MAPSO to solve the DHGSproblem

Application of the proposed MAPSO to solve the DHGS problem,formulated in section II, can be summarized as the following stepby step algorithm:

Step 1 – initialization. The decision variables for the DHGS problemare hourly reservoir discharges and thermal generations over theentire scheduling horizon. So, the structure of the position vectorfor the particles of the MAPSO to solve the DHGS problem becomesas follows

X = [Qh1,1, Qh2,1, . . . , QhNh,1, Ps1,1, Ps2,1, . . . , PsNs,1, . . . ,

× Qh1,T , Qh2,T , . . . , QhNh,T , Ps1,T , Ps2,T , . . . , PsNs,T ] (29)

The swarm of the MAPSO has Npar particles with the struc-ture shown in (29). The initial swarm of the MAPSO isgenerated by the reference unit technique [22,34]. Considerthe part [Qh1,t, Qh2,t, . . . , QhNh,t, Ps1,t, Ps2,t, . . . , PsNs,t] of theposition vector X, including reservoir discharges and thermal gen-erations for tth hour (1 ≤ t ≤ T). The initial reservoir dischargesare randomly chosen in the respective ranges such that Qhmin

j≤

Qhj,t ≤ Qhmaxj

(1 ≤ j ≤ Nh). The initial generations of Ns − 1 thermalunits are arbitrary selected in the respective ranges based on (11).The initial generation of one remaining unit, named reference unit,is so determined that the active power balance constraint of (16)is satisfied [22,34]. This initialization process is repeated for allT scheduling hours to generate one particle with the structureshown in (29). In this way, Npar particles of the swarm can be ini-tially generated. Now the velocity vectors of the swarm particlesshould be initialized. The following strategy is used for creatingthe initial velocities:

Vji,0 = Rand

[Xj,min

i− Xj

i,0, Xj,maxi

− Xji,0

](30)

where Xji,0 and Vj

i,0 are the initial position (obtained from the pre-viously mentioned initialization process) and initial velocity for ith

Powe

N. Amjady, H.R. Soleymanpour / Electric

particle in jth dimension, respectively; Xj,mini

and Xj,maxi

indicateminimum and maximum allowable position for the jth dimen-sion of the ith particle, respectively; Rand[a,b] generates a randomnumber in the range of [a,b]. Also, Pnot-best

i,0 = Pbesti,0 = Xi,0 and iter = 0.

Step 2 – Evaluation of the objective function. The fuel cost functionsof the thermal generating units fi(Psi,t) are evaluated according to(3) based on the current values of Psi,t. Then, the objective functionof the DHGS problem, i.e. OF, is computed according to (1)Step 3 – Construction of augmented objective function. The ramprate constrained generation limits of thermal units (except the ref-erence unit), i.e. (11), and the constraint of reservoir dischargesthat is Qhmin

j≤ Qhj,t ≤ Qhmax

j(1 ≤ j ≤ Nh) are already satisfied.

The remaining constraints of the DHGS problem are consideredbased on the penalty factor technique. This technique convertsthe primal constrained problem into an unconstrained problemby penalizing constraint violations. The penalty terms are basedon the deviation from the constraints and they are chosen highenough to make constraint violations prohibitive in the final solu-tion. The following penalty factors are defined to handle theremaining constraints of the DHGS problem:(1) Reference unit constraint penalty

ω1,t =

⎧⎪⎨⎪⎩

(1 − Ps1,t/Psrmin1,t )Fmax if Ps1,t < Psrmin

1,t

(Ps1,t/Psrmax1,t − 1)Fmax if Ps1,t > Psrmax

1,t

0 else

(31)

where ω1,t is in proportion to the reference unit limits violationand zero in case of no violation of these limits at time intervalt. The first thermal unit is considered as the reference unit in(31). Fmax is the penalty factor, which should be chosen highenough. Here, Fmax is calculated as below [22,35]:

Fmax =Ns∑i=1

fi(Psmaxi ) (32)

Total penalty term for this constraint, denoted by PT1, iscalculated as follows:

PT1 =T∑

t=1

ω1,t (33)

(2) POZs constraint penalty

ω2,i,k,t ={

(Psi,t/PsLB,ki

− 1)Fmax if PsLB,ki

< Psi,t ≤ Psave,ki

(1 − Psi,t/PsUB,ki

)Fmax if Psave,ki

< Psi,t ≤ PsUB,ki

0 else

(34)

Psave,ki

= (PsLB,ki

+ PsUB,ki

)

2(35)

where ω2,i,k,t in (34) is penalty term for the violation of kth POZconstraint of thermal unit i at time interval t. Total penalty termfor the POZ constraints PT2 is calculated as follows:

PT2 =T∑

t=1

Ns∑i=1

NOi∑k=1

ω2,i,k,t (36)

(3) PDZs constraint penalty⎧⎨ (Qhj,t/QhLB,mj

− 1)Fmax if QhLB,mj

< Qhj,t ≤ Qhave,mj

UB,m ave,m UB,m

ω3,j,m,t = ⎩ (1 − Qhj,t/Qhj

)Fmax if Qhj

< Qhj,t ≤ Qhj

0 else

(37)

Qhave,mj

=(QhLB,m

j+ QhUB,m

j)

2(38)

r Systems Research 80 (2010) 723–732 727

where ω3,j,m,t in (37) is the penalty term for the violation of mthPDZ constraint of hydro unit j at time interval t. Total penaltyterm for the PDZ constraints PT3 is calculated as follows:

PT3 =T∑

t=1

Nh∑j=1

NDj∑m=1

ω3,j,m,t (39)

(4) Reservoir storage volume limit penalty. By means of (14), reser-voir storage volume Vhj,t at each time interval t (1 ≤ t ≤ T) canbe calculated from initial storage volume Vhj,0. If Vhj,t violatesreservoir storage volume limits, the following penalty termshould be considered:

ω4,j,t =

⎧⎪⎨⎪⎩

(1 − Vhj,t/Vhminj

)Fmax if Vhj,t < Vhminj

(Vhj,t/Vhmaxj

− 1)Fmax if Vhj,t > Vhmaxj

0 else

(40)

Total penalty term ω4,j,t for the reservoir storage volumelimits PT4 is calculated as follows:

PT4 =T∑

t=1

Nh∑j=1

ω4,j,t (41)

(5) Final volume constraint penalty

ω5,j = |Vhj,T − Vhj,end|Vhj,end

Fmax (42)

PT5 =Nh∑j=1

ω5,j (43)

(6) Hydropower generation limits penalty

ω6,j,t =

⎧⎪⎨⎪⎩

(1 − Phj,t/Phminj

)Fmax if Phj,t < Phminj

(Phj,t/Phmaxj

− 1)Fmax if Phj,t > Phmaxj

0 else

(44)

PT6 =T∑

t=1

Nh∑j=1

ω6,j,t (45)

Considering the total thermal fuel cost OF in (1) and the abovepenalty terms PT1 to PT6, the augmented objective function of theDHGS problem, i.e. AOF, for each particle in the swarm is computedas follows:

AOF = OF + PT1 + PT2 + PT3 + PT4 + PT5 + PT6 (46)

The augmented objective function AOF is used for the MAPSO,e.g. in (25) and (26).Step 4 – Sorting particles in the tree topology. The particles of theswarm in the MAPSO are sorted in the tree topology, i.e. Fig. 1(b),based on the AOF of their best positions so far (Pbest

i,iter). For each

particle, the best position so far of its directly above particle isselected as its parent (PARi,iter).Step 5 – Updating velocity and position vectors. The velocities of par-ticles are updated based on (24). The proposed adaptive velocitylimiter, based on (27) and (28), is applied to the particles to preventthem from moving too rapidly. Then, the positions of the particles,as shown in (29), are updated according to (18). In the new position

of each particle, if Qhj,t and Psi,t (except the reference unit) violatetheir allowable ranges [Qhmin

j, Qhmax

j] and [Psrmin

i,t, Psrmax

i,t], respec-

tively, they are limited to their respective ranges. For instance, ifQhj,t becomes greater than Qhmax

j, it is limited to Qhmax

j. Iteration

number is incremented, i.e. iter = iter + 1. Then, TVIW (w), TVAC

728 N. Amjady, H.R. Soleymanpour / Electric Power Systems Research 80 (2010) 723–732

Q

oi(kth

5

pat

Table 2POZ constraints, ramp-up and ramp-down rate limits and initial state of thermalunits for test case 4.

Thermal unit 1 2 3

TT

Fig. 2. Hydrosystem configuration used in the four test cases.

(c1b, c1nb and c2) and the dynamic coefficient of the velocity limiter(R) are updated based on (19), (20), (21) and (27), respectively.Step 6 – Evaluation of the new particle positions and updating the“Best” and “Not-best” components. For the new position of eachparticle, the OF and AOF are calculated as described in the steps2 and 3, respectively. Then, the “Best” and “Not-best” componentsof each particle are updated based on (25) and (26), respectively.The personal best exchanging (PBE) technique is applied to theparticles of the MAPSO.Step 7 – Evaluation of the stopping condition. If iter < itermax, go tostep 4. Otherwise, the MAPSO algorithm terminates. It is noted thatdue to the execution of the PBE technique, the top particle in thetree topology, i.e. Fig. 1(b), does not necessarily have the optimum“Best” component. So, we search among the “Best” components ofthe particles to find the “Best” component owning minimum AOFvalue, which is selected as the final solution of the MAPSO for theDHGS problem.

Note: The constraints of water discharge rate for hydro unitshmin

j≤ Qhj,t ≤ Qhmax

j(1 ≤ j ≤ Nh) and the ramp rate constrained

perating limits of thermal units (except the reference unit) shownn (11) are satisfied in the initialization (step 1) and next iterationsstep 5). The constraint handling techniques of steps 1 and 5 arenown as preservation method and solution repair method, respec-ively [22,35]. The remaining constraints of the DHGS problem areandled in the step 3 by the penalty function method

. Numerical results

The effectiveness of the proposed MAPSO to solve the DHGSroblem is evaluated based on four test cases. All test cases includemulti-chain cascade of four hydroplants, shown in Fig. 2. Water

ransport delays between reservoirs are indicated on the figure.

able 1he characteristics of the four test cases and the constraints of the respective DHGS prob

Test case Characteristics

1 Nh = 4, Ns = 1 quadratic cost function, shown in (2), for thethermal unit

2 Test case 1 + valve loading effects of the thermal unit + PDZof hydro units

3 Nh = 4, Ns = 3 cost functions of thermal units include valveeffects as shown in (3)

4 Test case 3 + PDZ constraints of hydro units + POZ constraithermal units + ramp rate limits of thermal units

POZ (MW) 70–80 160–190 170–180URi (MW/h) 50 90 170DRi (MW/h) 40 70 120Psi,0 (MW) 100 120 230

The scheduling period is one day (24 h). Variable natural inflowrate into each reservoir and variable load demand over schedulingperiod are considered in the four test cases. The characteristics ofeach test case as well as the constraints of the respective DHGSproblem are shown in Table 1. As seen, the complexity of the fourtest cases increases step by step. The data of the test cases 1, 2and 3 can be found in [23,7,36], respectively. The additional dataof test case 4 with respect to test case 3 are shown in Table 2including POZ constraints, ramp-up and ramp-down rate limits andinitial state of thermal units. Moreover, the same PDZ constraintsof test case 2 are also considered in test case 4. The best results ofthe MAPSO and several other methods for the four test cases areshown in Table 3. In order to illustrate the relative contributionof the suggested modifications in improving the performance ofthe proposed MAPSO, the obtained results from the proposed APSO(with the new version of tree topology), MAPSO(1) = APSO + the firstmodification, MAPSO(2) = MAPSO(1) + the second modification, andMAPSO(3) = MAPSO(2) + the third modification for the four test casesare shown in Table 3. The MAPSO(3) is the proposed MAPSO tech-nique. As seen, the proposed MAPSO outperforms all other methodsof Table 3 on all test cases. Also, Table 3 shows that the proposedmodifications enhance the performance of the MAPSO step by step.

Obtained results from the MAPSO for the reservoir discharge ofthe four hydro units (Qhj,t) over the 24 h of the scheduling horizonfor the first test case are shown in Fig. 3. Also, load demand, ther-mal power generation (Ps1,t) and total hydropower generation forthis test case are illustrated in Fig. 4. The obtained values for thedecision variables in Figs. 3 and 4 satisfy all constraints of this testcase, including (4), (5), (13), (15), (16) and Qhmin

j≤ Qhj,t ≤ Qhmax

j

(1 ≤ j ≤ Nh). Similarly, obtained values for the decision variablesin the three other test cases satisfy the respective constraints ofthese cases. In order to also give a graphical view about the conver-gence behavior of the proposed MAPSO, evolution of classical PSO,APSO and MAPSO for the fourth test case (the most complex case)is shown in Fig. 5. In this figure, variation of AOF with the itera-tions is shown. Fig. 5 shows better convergence behavior of theMAPSO with respect to PSO and APSO. Also, variation of AOF–OF(total penalty term) with the iterations for the MAPSO is shown inFig. 6. As seen, from the iteration 3196, AOF–OF becomes zero.

For each test case, the best solution of the proposed MAPSO

among 25 trail runs is shown in Table 3, since the MAPSO beginsfrom a random initial point (step 1 of the step by step algorithm).Similar trial runs have been considered for the other methods intheir respective references and their best solutions are reported inTable 3. In order to further verify the robustness of the proposed

lem.

Constraints of the DHGS problem

single (4), (5), (13), (15), (16) andQhmin

j≤ Qhj,t ≤ Qhmax

j(1 ≤ j ≤ Nh)

constraints Constraints of test case 1 + (12)

loading (4), (5), (13), (15), (16) andQhmin

j≤ Qhj,t ≤ Qhmax

j(1 ≤ j ≤ Nh)

nts of (4), (7), (11), (12), (13), (15) and (16)

N. Amjady, H.R. Soleymanpour / Electric Power Systems Research 80 (2010) 723–732 729

Table 3Obtained results for the four test cases.

Test case 1 Test case 2 Test case 3 Test case 4

Method OF ($) Method OF ($) Method OF ($) Method OF ($)

EGA [8] 934,727.00 NLP [16] 936,709.52 EP [17] 47,306.00 Classical PSO 47,443.40GA [23] 932,734.00 DP [16] 935,617.76 SA [17] 45,466.00 APSO 41,973.19FEP [7] 930,268.00 IFEP [7] 933,949.25 PSO [17] 44,740.00 MAPSO(1) 41,624.65CEP [7] 930,166.00 DE [16] 928,236.94 DE [14] 44,526.10 MAPSO(2) 41,074.12IFEP [7] 930,130.00 HDE [14] 927,895.81 MDE [14] 42,611.14 MAPSO(3) 40,748.38PSO [8] 928,878.00 APSO 925,991.35 HDE [14] 42,337.30BCGA [11] 926,922.71 IPSO [16] 925,978.84 APSO 41,858.27APSO 926,151.54 MAPSO(1) 925,963.72 MHDE [14] 41,856.50RCGA [11] 925,940.03 MDE [14] 925,960.56 MAPSO(1) 41,241.91LWPSO [18] 925,383.80 MHDE [14] 925,547.31 MAPSO(2) 40,875.34MAPSO(1) 924,232.38 MAPSO(2) 925,054.53 MAPSO(3) 40,225.06DE [21] 923,991.08 MAPSO(3) 924,636.88MAPSO(2) 923,074.27EPSO [8] 922,904.00IPSO [16] 922,553.49MAPSO(3) 922,421.66

Table 4The best, average and worst OF values ($).

Method EGA [8] GA [23] FEP [7] CEP [7] IFEP [7] PSO [8] BCGA [11] RCGA [11]

Test case 1 1 1 1 1 1 1 1Best 934,727 932,734 930,268 930,166 930,130 928,878 926,922 925,940Average 936,058 936,969 930,897 930,373 930,290 933,085 927,815 926,120Worst 937,339 939,734 931,397 930,927 930,882 938,012 929,451 926,538

Method LWPSO [18] EPSO [8] MAPSO IFEP [7] MAPSO PSO APSO MAPSO

Test case 1 1 1 2 2 4 4 4933,938,942,

MMFariMm

ltc

Best 925,383 922,904 922,421Average 926,352 923,527 922,544Worst 927,240 924,808 923,508

APSO, the best, average and worst OF values obtained by theAPSO and some other methods of Table 3 are shown in Table 4.

or the other methods of Table 3 not mentioned in Table 4, theverage and worst OF values have not been given in the respectiveeferences. The best OF values in Table 4 are the reported resultsn Table 3. As seen, the best, average and worst OF values of the

APSO are lower than those of all other methods of Table 4 on the

entioned test cases.In order to also evaluate the performance of the MAPSO for

arger DHGS test cases, two test cases 5 and 6 are constructed fromhe test case 4 by repeating the thermal and hydro units of this testase 5 and 10 times, respectively. Also, the load demand of the test

Fig. 3. Reservoir discharge of the four hydro units (Qhj,t) for first test case.

949 924,636 47,443 41,973 40,748508 926,496 49,238 42,521 40,957593 927,431 51,062 42,874 41,695

cases 5 and 6 are 5 and 10 times higher than that of the test case 4.The best, average and worst results of the MAPSO for the test case5 are 206,834, 214,795 and 223,721, respectively and for the testcase 6 are 415,683, 421,581 and 429,758, respectively. As seen, theMAPSO obtained good results for these two test cases as well. Theobtained best, average and worst results for the test cases 5 and 6are about 5 and 10 times higher than those of the test case 4.

The parameters of the MAPSO have been set as followsbased on trial and error (the selected values are the bestones among several runs): wmin = 0.4, wmax = 0.9, cinitial

1b=

1.6, cfinal1b

= 0.7, cinitial1nb

= 0.15, cfinal1nb

= 0.1, cinitial2 = 0.9, cfinal

2 =

Fig. 4. Load demand, thermal power generation (Ps1,t) and total hydropower gen-eration for the first test case.

730 N. Amjady, H.R. Soleymanpour / Electric Power Systems Research 80 (2010) 723–732

1Ni

vtTMrtatatpaniot

rturbso

tions, very small deviations are seen in the results of the proposed

TO

Fig. 5. Evolution of classical PSO, APSO and MAPSO for the fourth test case.

.7, Rinitial = 0.3, Rfinal = 0.1, Npar = 100, itermax = 5000 andPBE = 10, where NPBE indicates number of particles participating

n the personal best position exchanging in each iteration. These

alues of the parameters have been used for all six test cases ofhe paper, although these cases have different conditions and sizes.his matter shows robustness and low sensitivity of the proposedAPSO with respect to its parameters (it is not necessary to sepa-

ately adjust these parameters for each problem). To further verifyhe low sensitivity of the MAPSO, we perturbed its parametersround their original values; however the results of the MAPSO forhe test cases negligibly changed. Sample results of this sensitivitynalysis for the fourth test case are shown in Table 5. In the sensi-ivity analysis, the initial value of each parameter of the MAPSO iserturbed in the down and up directions and its perturbed valuesre shown in the first and fifth columns of Table 5 titled as “Origi-

deviation (%) = solution (with pe

al − �” and “Original + �”, respectively. Here, the perturbation �s 20% of the original value. To also evaluate the cumulative effectf the parameters’ perturbations, we perturbed all parameters ofhe MAPSO simultaneously in the down and up directions, respec-

able 5btained results from the sensitivity analysis for the fourth test case.

Original − � Best Average Worst O

wmin = 0.32 40,798 (0.122%) 41,014 (0.139%) 41,835 (0.337%) w

wmax = 0.72 40,768 (0.049%) 40,975 (0.044%) 41,807 (0.269%) w

cinitial1b

= 1.28 40,771 (0.056%) 40,961 (0.010%) 41,842 (0.353%) c1

cfinal1b

= 0.56 40,749 (0.002%) 40,902 (−0.134%) 41,705 (0.024%) c1

cinitial1nb

= 0.12 40,748 (0.001%) 40,939 (−0.044%) 41,778 (0.199%) c1

cfinal1nb

= 0.08 40,755 (0.017%) 40,944 (−0.031%) 41,813 (0.283%) c1

cinitial2 = 0.72 40,749 (0.002%) 40,959 (0.005%) 41,642 (−0.126%) c2

cfinal2 = 1.36 40,754 (0.014%) 40,991 (0.083%) 41,809 (0.273%) c2

NPBE = 8 40,758 (0.024%) 40,975 (0.044%) 41,801 (0.254%) N

NPar = 80 40,759 (0.027%) 41,030 (0.178%) 41,876 (0.434%) N

itermax = 4000 40,757 (0.022%) 40,966 (0.023%) 41,809 (0.273%) it

Rinitial = 0.24 40,770 (0.054%) 40,962 (0.013%) 41,644 (−0.121%) R

Rfinal = 0.08 40,751 (0.007%) 40,974 (0.042%) 41,805 (0.264%) R

All (down) 40,760 (0.029%) 41,037 (0.195%) 41,763 (0.163%) A

Fig. 6. Variation of AOF–OF with iterations for the MAPSO in the fourth test case.

tively and its results are shown in the last row of Table 5, indicatedby “All”. In each cell of columns 2, 3, 4, 6, 7 and 8, the number in theparentheses indicates deviation from the solution with the originalparameters in terms of percentage, computed as follows:

ed parameter) − solution (with original parameter)lution (with original parameter)

× 100 (47)

For instance, when the initial value of wmin = 0.4 is perturbedby 20% in the down direction to wmin = 0.4 − (0.4 × 20%) = 0.32,the best solution becomes 40,798 having 0.122% deviation withrespect to the initial value of the best solution, i.e. 40,748. InTable 5, negative sign for a deviation means the objective func-tion with the perturbed parameter decreases with respect to itsoriginal value or equivalently the solution with the perturbedparameter is better than the original solution and conversely forthe positive deviation. It is noted that the selected values for theparameters of the MAPSO are not the optimum values; they areonly selected after a few tests. However, Table 5 shows that evenwith considering the cumulative effect of all parameters’ perturba-

MAPSO.The computation times of the proposed MAPSO for the test

cases 1, 2, 3, 4, 5 and 6 are 64 s, 83 s, 97 s, 146 s, 898 s and1827 s, respectively, measured on a simple personal computer Pen-

riginal + � Best Average Worst

min = 0.48 40,772 (0.059%) 41,004 (0.115%) 41,815 (0.288%)

max = 1.08 40,788 (0.098%) 41,034 (0.188%) 41,819 (0.297%)initialb

= 1.92 40,760 (0.029%) 40,958 (0.003%) 41,776 (0.194%)

finalb

= 0.84 40,755 (0.017%) 40,979 (0.054%) 41,659 (−0.084%)

initialnb

= 0.18 40,764 (0.039%) 40,967 (0.024%) 41,773 (0.187%)

finalnb

= 0.12 40,749 (0.002%) 40,988 (0.075%) 41,696 (0.002%)

initial = 1.08 40,754 (0.014%) 40,957 (0.001%) 41,603 (−0.219%)final = 2.04 40,751 (0.007%) 40,963 (0.015%) 41,855 (0.384%)

PBE = 12 40,750 (0.005%) 41,004 (0.117%) 41,804 (0.261%)

Par = 120 40,744 (−0.010%) 40,876 (−0.195%) 41,602 (−0.223%)

ermax = 6000 40,747 (−0.002%) 40,869 (−0.213%) 41,693 (−0.004%)initial = 0.36 40,759 (0.027%) 40,959 (0.004%) 41,742 (0.112%)final = 0.12 40,753 (0.012%) 40,947 (−0.023%) 41,726 (0.075%)

ll (up) 40,758 (0.024%) 41,061 (0.255%) 41,749 (0.130%)

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N. Amjady, H.R. Soleymanpour / Electric

ium P4 2.8 GHz with 1 GB RAM. These computation times showow computation burden of the MAPSO to solve the DHGS prob-em. We did not compare the computation times of the MAPSO

ith those of the other methods, since the computation timesf each method have been measured on a different hardwareet.

. Conclusion

This paper proposes a new MAPSO to solve the DHGS problem,hich is a complicated non-linear, non-convex and non-smooth

ptimization problem with discontinuous solution space. Obtain-ng a good solution is hard for this type of optimization problem.he proposed MAPSO has TVIW, TVAC and a specific kind of treeopology. Also, three new modifications including the Split-up ofhe cognitive part into “Best” and “Not-Best” components, personalest position exchanging and new velocity limiter are incorporated

nto the proposed MAPSO to enhance its exploration capabilitynd convergence behavior. Obtained results from extensive test-ng of the proposed MAPSO on different DHGS test cases confirmhe validity of the developed approach. The research work is underay in order to include unit commitment in the DHGS prob-

em, which converts the DHGS into a Mixed Integer Non-Linearrogramming (MINLP) problem. Also, considering the uncertaintyources of the DHGS problem (such as uncertain load demand andnflow rate of reservoirs), which converts the DHGS into a stochas-ic optimization problem, can be considered as the future researchork.

ppendix A. List of symbols

F total fuel cost ($)scheduling horizon (h)

s, Nh number of thermal generating units and number of hydrogenerating units, respectively

si, bsi, csi, dsi, and esi fuel cost coefficients of ith thermal generat-ing unit with valve loading effects

si,t, fi(Psi,t) power generation (MW) and fuel cost ($/h) of ith ther-mal unit at time interval t

hj,t, Qhj,t hydropower generation (MW) and water discharge rate(m3/h) of reservoir j at time interval t

smini

, Psmaxi

minimum and maximum power generation for iththermal unit, respectively

srmini,t

, Psrmaxi,t

ramp rate constrained minimum and maximumpower generation for ith thermal unit at time interval t,respectively

hminj

, Phmaxj

minimum and maximum power generation for jthhydro unit, respectively

hj,t water storage volume (m3) of reservoir j at time intervalt

hj,end specified storage volume (m3) of reservoir j at the end ofscheduling horizon

h1j–Ch6j hydropower generation coefficients for jth hydro unit

sLB,ki

, PsUB,ki

lower and upper boundaries of kth POZ of ith thermalunit, respectively

Oi, NDj number of POZs of thermal unit i and number of PDZs ofhydro unit j, respectively

Ri, DRi ramp-up and ramp-down rate limits of thermal unit i

(MW/h), respectively

hminj

, Qhmaxj

minimum and maximum water discharge rates ofjth reservoir, respectively

hLB,mj

, QhUB,mj

lower and upper boundaries of mth PDZ of jthhydro unit, respectively

[

[

r Systems Research 80 (2010) 723–732 731

Vhminj

, Vhmaxj

minimum and maximum storage volume of jthreservoir, respectively

Ihj,t, Shj,t natural inflow rate and spillage discharge rate of jth reser-voir at time interval t, respectively

�r,j water transport delay from reservoir r to jUPj total number of upstream units which are immediately

above the jth reservoirPDt, PLt active power demand and total transmission loss at time

interval t, respectively

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