Rh

Ka

b

c

a

ARRAA

KCHOR

1

hphfpriitbnaw

a

as

h1

Applied Soft Computing 24 (2014) 962–976

Contents lists available at ScienceDirect

Applied Soft Computing

j ourna l ho me page: www.elsev ier .com/ locate /asoc

eal coded chemical reaction based optimization for short-termydrothermal scheduling

untal Bhattacharjeea, Aniruddha Bhattacharyab,∗, Sunita Halder nee Deyc

Dr. B. C. Roy Engineering College, Durgapur, West Bengal 713206, IndiaNational Institute of Technology – Agartala, Tripura 799055, IndiaDepartment of Electrical Engineering, Jadavpur University, Kolkata, West Bengal 700032, India

r t i c l e i n f o

rticle history:eceived 14 April 2013eceived in revised form 18 April 2014ccepted 22 August 2014vailable online 3 September 2014

eywords:hemical reaction

a b s t r a c t

This paper presents a real coded chemical reaction based (RCCRO) algorithm to solve the short-termhydrothermal scheduling (STHS) problem. Hydrothermal system is highly complex and related withevery problem variables in a nonlinear way. The objective of the hydro thermal scheduling is to determinethe optimal hourly schedule of power generation for different hydrothermal power system for certainintervals of time such that cost of power generation is minimum. Chemical reaction optimization mimicsthe interactions of molecules in term of chemical reaction to reach a low energy stable state. A real codedversion of chemical reaction optimization, known as real-coded chemical reaction optimization (RCCRO)

ydrothermal schedulingptimizationeal coded chemical reaction optimization

is considered here. To check the effectiveness of the RCCRO, 3 different test systems are considered andmathematical remodeling of the algorithm is done to make it suitable for solving short-term hydrothermalscheduling problem. Simulation results confirm that the proposed approach outperforms several otherexisting optimization techniques in terms quality of solution obtained and computational efficiency.Results also establish the robustness of the proposed methodology to solve STHS problems.

© 2014 Elsevier B.V. All rights reserved.

. Introduction

The short-term hydrothermal scheduling involves the hour-by-our scheduling to minimize the total operating cost of thermalower plants while satisfying the various constraints related to theydro, thermal plant, and power system network. As the source

or hydropower is the natural water resources, the objective is tolan the usage of available water for hydroelectric generation toeduce the production cost of the thermal plants, while maintain-ng all sets of constraints. Hydraulic and thermal constraints maynclude generation-load power balance, operating capacity limits ofhe hydro and thermal units, water discharge rate, upper and lowerounds on reservoir volumes, water spillage, and hydraulic conti-uity restrictions. Additional constraints such as the need to satisfyctivities including: flood control, irrigation, navigation, fishing,

ater supply, recreation, etc., may also be considered.

The optimal scheduling of hydrothermal power system is usu-lly more complex than all the other thermal system. It is basically

∗ Corresponding author. Tel.: +91 9474188660.E-mail addresses: kunti 10@yahoo.com (K. Bhattacharjee),

ni bhatta2004@rediffmail.com, bhatta.aniruddha@gmail.com (A. Bhattacharya),unitaju@yahoo.com (S. Halder nee Dey).

ttp://dx.doi.org/10.1016/j.asoc.2014.08.048568-4946/© 2014 Elsevier B.V. All rights reserved.

a nonlinear programming problem involving nonlinear objectivefunction and a mixture of linear and nonlinear constraints. Dueto these, classical calculus-based methods like Lagrangian mul-tiplier and gradient search techniques [1] cannot perform verywell for finding the most economical hydrothermal generationschedule under practical constraints. Kirchmayer [2] used coordi-nation equations of variation for short-range scheduling problem.Mixed integer programming [3] and dynamic programming (DP)[4] functional analysis [5–7], network flow and linear programming[8–11], non-linear programming [12,13], mathematical decom-position [14–16], heuristics, expert systems and artificial neuralnetworks [17–20] methods have been widely used to solve suchscheduling problems in different formulations.

In recent years, evolutionary algorithms have been used dueto their flexibility, versatility, and robustness in searching a glob-ally optimal solution. These algorithms are powerful optimizationtechniques corresponding to their natural selection process. Sev-eral evolutionary techniques, such as simulated annealing [21,22],genetic algorithm [23–27], evolutionary programming [28–30] anddifferential evolution [31–34], particle swarm optimization (PSO)

[35–38] have been employed to solve the STHS problem. Improvedversion of PSO [39–43] has also been applied to solve STHS prob-lem to improve the smoothness of the algorithm. Recently, a newoptimization technique called clonal selection algorithm [44] has

Soft C

bsihpd

ctruwtadvpmdoerobRps

bRtcSaows

tcfiT

2

mecpa

2

mumT

M

wo

K. Bhattacharjee et al. / Applied

een used in STHS problem to achieve much better global optimalolution. Recently, another soft computing approach called mixednteger programming [47,48] has been applied in multiobjectiveydro-thermal scheduling problem. In the year 2013, hydrothermalroblems have been scheduled with considering wing power andifferent type of security constraints [49,50].

In recent times, a new optimization technique based on theoncept of chemical reaction, called chemical reaction optimiza-ion (CRO) has been proposed by Lam and Li [45]. In a chemicaleaction, the molecules of initial reactants stay in high-energynstable states and undergo a sequence of collisions either withalls of the container or with other molecules. The reactants pass

hrough some energy barriers, reach in low-energy stable statesnd become the final products. CRO captures this phenomenon ofriving high-energy molecules to stable, low energy states, througharious types of on-wall or intermolecular reactions. CRO has beenroved to be a successful optimization algorithm in discrete opti-ization. Basically, the CRO is designed to work in the discrete

omain optimization problems. In order to make this newly devel-ped technique suitable for continuous optimization domain, Lamt al. [46] have developed a real-coded version of CRO, known aseal-coded CRO (RCCRO). It has been observed that the performancef RCCRO is quite satisfactory when applied to solve continuousenchmark optimization problems. The improved performance ofCCRO to solve different optimization problems has motivated theresent authors to implement this newly developed algorithm toolve short-term hydrothermal scheduling (STHS) problems.

Like other soft computing algorithm, CRO is also a population-ased metaheuristic. In first few stages the operating principle ofCCRO is quite similar to GA but RCCRO uses lesser parameter andherefore lesser computational time required. As the simulationontinues, it tends to keep a single solution in each iteration, likeA. Therefore, RCCRO have the benefit of the advantages of both GAnd SA, and generally it performs the best. Due to the advantagef flexibility of adjusting and combining the elementary reactions,e can adopt the new concept into the short term hydrothermal

cheduling problems.Section 2 of the paper provides a brief mathematical formula-

ion of different types of STHS problems. The concept of real codedhemical reaction is described in Section 3. The parameter settingsor the test system to evaluate the performance of real coded chem-cal reaction and the simulation studies are discussed in Section 4.he conclusion is drawn in Section 5.

. Problem formulation

The optimizing schedule for hydrothermal power systems isodeled as a constrained optimization problem with a nonlin-

ar objective function and a set of linear, nonlinear, and dynamiconstraints. Generating characteristics of hydro as well as thermallants are in non-linear in nature. Hydro plants whose outputs are

nonlinear function of water discharge and net hydraulic head.

.1. Objective function

The problem of short term hydro thermal scheduling aims atinimizing the total generation cost of thermal units while making

se of the available hydro resources in the scheduling horizon asuch as possible, due to the zero incremental cost of hydro plants.

he objective function is expressed as:

Ns∑ T∑

inimized F =

k=1 t=1

fk(Ps(k, t)) (1)

here Ns is the number of thermal plants, T is the total intervalsf the scheduling horizon considered, and Ps(k, t) represents the

omputing 24 (2014) 962–976 963

power generation of the kth thermal plant at time interval t. The fuelcost function with valve point loading effect is usually representedas:

fk(Ps(k, t)) = aks + bks · Ps(k, t) + cks · P2s (k, t)

+ |dks · sin(eks · (Pmins (k) − Ps(k, t)))|

k = 1, 2, . . ., Ns, t = 1, 2, . . ., T (2)

where aks, bks, cks, dks, and eks are the fuel cost coefficients ofthe kth thermal plant and Pmin

s (k) represents the minimum powergeneration of the kth thermal plant.

2.2. Constraints

2.2.1. Continuity equation for hydro reservoirs network

Vh(i, t) = Vh(i, t − 1) + Ih(i, t) − Qh(i, t)

+∑

m∈Ru(i)

Qh(m, t − �m) i = 1, 2, . . ., Nh t = 1, 2, . . ., T

(3)

where Vh(i, t), Ih(i, t), Qh(i, t) are the end storage volume, inflow,discharge of reservoir i at time interval t respectively. Spillage is notconsidered here; Nh is the number of hydro plants; �m is the watertransport delay from reservoir m to its immediate downstream;Ru(i) represents the set of upstream plants directly above hydroplant i.

2.2.2. Physical limitations on reservoir storage volumes anddischargesVmin

h (i) ≤ Vh(i, t) ≤ Vmaxh (i) i = 1, 2, . . ., Nh t = 1, 2, . . ., T (4)

where Vminh

(i) and Vmaxh

(i) are the minimum and maximum storagevolumes of the ith reservoir:

Q minh (i) ≤ Qh(i, t) ≤ Q max

h (i) i = 1, 2, . . ., Nh t = 1, 2, . . ., T (5)

where Q minh

(i) and Q maxh

(i) represent the minimum and maximumwater discharges of the ith reservoir.

2.2.3. Initial and final reservoir storage volumeVh(i, 0) = Vbegin

h(i) i = 1, 2, . . ., Nh (6)

Vh(i, T) = Vendh (i) i = 1, 2, . . ., Nh (7)

2.2.4. Generator capacityPmin

s (k) ≤ Ps(k, t) ≤ Pmaxs (k) k = 1, 2, . . ., Ns t = 1, 2, . . ., T (8)

where Pmins (k) and Pmax

s (k) are the minimum and maximum powergeneration of the kth thermal plant:

Pminh (i) ≤ Ph(i, t) ≤ Pmax

h (i), i = 1, 2, . . ., Nh t = 1, 2, . . ., T (9)

where Ph(i, t) is the power generation of the ith hydro plant at timet; Pmin

h(i) and Pmax

h(i) represent the minimum and maximum power

generation of the ith hydro plant respectively. Ph(i, t) is usuallyassumed to be a function of the water discharge and the storagevolume

Ph(i, t) = c1i · V2h (i, t) + c2i · Q 2

h (i, t) + c3i · Vh(i, t) · Qh(i, t)

+ c4i · Vh(i, t) + c5i · Qh(i, t) + c6i i = 1, 2, . . ., Nh

t = 1, 2, . . ., T (10)

where C1i, C2i, C3i, C4i, C5i and C6i are the constant coefficients.

9 Soft C

2

nttdc

P

P

2∑

wrt

3

rr

raImtpisdaaeri

afpfir

oab

3

3

k(

64 K. Bhattacharjee et al. / Applied

.2.5. Ramp rate limit constraintThe power Pi generated by the ith generator in certain interval

either should exceed that of previous interval by more than a cer-ain amount URi, the up-ramp limit and nor should it be less thanhat of the previous interval by more than some amount DRi, theown-ramp limit of the generator. These give rise to the followingonstraints:

As generation increases

s(k, t) − Ps(k, t − 1) ≤ UR(k) k = 1, 2, . . ., Ns t = 1, 2, . . ., T

(11)

As generation decreases

s(k, t − 1) − Ps(k, t) ≤ DR(k) k = 1, 2, . . ., Ns t = 1, 2, . . ., T

(12)

.2.6. System load balanceNs

k=1

Ps(k, t) +Nh∑i=1

Ph(i, t) = PD(t) + PL(t) t = 1, 2, . . ., T (13)

here PD(t) is the predicted demand at time interval t and PL(t) rep-esents the total transmission losses. In these problem formulationransmission loss is not considered i.e., zero.

. Real-coded chemical reaction optimization (RCCRO)

This section presents an interesting new optimization algo-ithm called chemical reaction optimization (CRO) which has beenecently proposed in [45].

CRO loosely mimics what happens to molecules in a chemicaleaction system. Every chemical reaction tends to release energy,nd thus, products generally have less energy than the reactants.n terms of stability, the lower the energy of the substance, the

ore stable it is. In a chemical reaction, the initial reactants inhe high-energy unstable states undergo a sequence of collisions,ass through some energy barriers, and become the final products

n low-energy stable states. Therefore, products are always moretable than reactants. It is not difficult to discover the correspon-ence between optimization and chemical reaction. Both of themim to seek the global optimum with respect to different objectivesnd the process evolves in a stepwise fashion. With this discov-ry, the chemical-reaction-inspired metaheuristic, called chemicaleaction optimization (CRO) [45] has been developed by Lam et al.n 2010.

However this paper is the extension of CRO. CRO has beenlready proved to be a successful optimization algorithm with dif-erent applications [46], most of which are discrete optimizationroblems. In order to make this optimization technique suitableor continuous optimization problems, Lam et al. presented a mod-fied version of CRO in 2012, which is termed as real-coded chemicaleaction optimization (RCCRO) [46].

In the following sections, major components based on designf the chemical reaction, i.e., molecules and elementary reactionsre described. The basic operational steps of RCCRO are describedelow.

.1. Major components of RCCRO

.1.1. MoleculesThe manipulated agents those are involved in a reaction are

nown as molecules. Three main properties of each molecule are:1) the molecular structure X; (2) current potential energy (PE); (3)

omputing 24 (2014) 962–976

current kinetic energy (KE), etc. The meanings of the attributes inthe profile are given below:

Molecular structure: X actually represents the solution currentlyheld by a molecule. Depending on the problem; X can be in theform of a number, an array, a matrix, or even a graph. In this papermolecular structure has been represented in a matrix form.

Current PE: PE is the value of objective function of the currentmolecular structure X, i.e., PEX = f(X).

Current KE: KE provides the tolerance for the molecule to hold aworse molecular structure with higher PE than the existing one.

3.2. Elementary reactions

In CRO, several types of collisions occur. These collisions occureither between the molecules or between the molecules and thewalls of the container. Depending upon the type of collisions,distinct elementary reactions occur, each of which may have a dif-ferent way of controlling the energies of the involved molecule(s).Four types of elementary reactions normally occur. These are: (1)on-wall ineffective collision; (2) decomposition; (3) intermolecularineffective collision; and (4) synthesis. On wall ineffective collisionand decomposition are unimolecular reactions when the moleculehits a wall of the container. Intermolecular ineffective collision andsynthesis involve more than one molecule. Successful completionof an elementary reaction results in an internal change of a molecule(i.e., updated attributes in the profile). Different types of elementaryreactions are described below:

3.2.1. On wall ineffective collisionWhen a molecule hits a wall and bounces back, a small change

occurs of its molecular structure and PE. As the collision is not sovigorous, the resultant molecular structure is not too different fromthe original one. If X and X′ represent the molecular structure beforeand after the on-wall collision respectively, then this collision triesto transform X to X′, in the close neighborhood of X, that is

X ′ = X + � (14)

where � is a perturbation for the molecule. There are many prob-ability distributions which can be used to produce probabilisticperturbations, e.g., Gaussian, Cauchy, lognormal, exponential, Stu-dent’s T and many others. In this paper, Gaussian distribution hasbeen employed. By the change of molecular structure, PE and KE alsochange from PEX to PEX′ and KEX to KEX′ . This change will happenonly if

PEX + KEX ≥ PEX ′ (15)

If (15) does not hold, the change is not allowed and the moleculeretains its original X, PE and KE. Due to interaction with a wall ofthe container, a certain portion of molecule’s KE will be extractedand stored in the central energy buffer (buffer) when the trans-formation is complete. The size of KE loss depends on a randomnumber a1 ∈ [KELossRate, 1], where KELossRate is a parameter ofCRO. Updated KE and buffer is represented as

KEX ′ = (PEX − PEX ′ + KEX ) × a1 (16)

buffer = buffer + (PEX + KEX − PEX ′ ) × (1 − a1) (17)

3.2.2. Decomposition

In decomposition, one molecule hits the wall and breaks into

two or more molecule e.g., X ′1 and X ′

2. Due to change of molecularstructure, their PE and KE also changes from PEX to PEX ′

1and PEX ′

2,

and KEX to KEX ′1

and KEX ′2. This change is allowed, if the original

Soft C

mr

P

K

w[vadbm

P

K

K

b

w[Xp

3

meombaetcoctTc

P

K

K

wvo

3

pbdft

K. Bhattacharjee et al. / Applied

olecule has sufficient energy (PE and KE) to endow the PE of theesultant ones, that is

EX + KEX ≥ PEX ′1

+ PEX ′2

(18)

Let temp1 = PEX + KEX − PEX ′1

− PEX ′2

Then,

EX ′1

= k × temp1 and KEX ′2

= (1 − k) × temp1 (19)

here k is a random number uniformly generated from the interval0, 1]. (18) holds only when KEX is large enough. Due to the conser-ation of energy, X sometimes may not have enough energy (both PEnd KE) to sustain its transformation into X ′

1 and X ′2. To encourage

ecomposition, a certain portion of energy, stored in the centraluffer (buffer) can be utilized to support the change. In that caseodified condition is

EX + KEX + buffer ≥ PEX ′1

+ PEX ′1

(20)

The new KE for resultant molecules and buffer are

EX ′1

= (temp1 + buffer) × m1 × m2 (21)

EX ′2

= (temp1 + buffer) × m3 × m4 (22)

uffer = buffer + temp1 − KEX

′1

− KEX

′2

(23)

here values of m1, m2, m3 and m4 are taken randomly in between0, 1]. To generate X ′

1 and X ′2, any mechanism which creates X ′

1 and′2 quite different from X, is acceptable. However, in this paper,rocedure mentioned in Section IIIB of [46] is used.

.2.3. Intermolecular ineffective collisionAn intermolecular ineffective collision happens when two

olecules collide with each other and then bounce away. Theffect of energy change of the molecules is similar to that in ann-wall ineffective collision, but this elementary reaction involvesore than one molecule and no KE is drawn to the central energy

uffer. Similar to the on-wall ineffective collision, this collision islso not vigorous, therefore the new molecular structure are gen-rated in the neighborhood of previous molecular structures. Inhis paper, new molecular structures are created using the sameoncept mentioned in on-wall ineffective collision. Suppose, theriginal molecular structures are X1 and X2 are transformed afterollision and two new molecular structures are X ′

1 and X ′2 respec-

ively. The two PE are changed from PEX1 and PEX2 to PEX ′1

and PEX ′2.

he two KE are changed from KEX1 and KEX2 to KEX ′1

and KEX ′2. The

hange to the molecules are acceptable only if

EX1 + PEX2 + KEX1 + KEX2 ≥ PEX ′1

+ PEX ′2

(24)

The new values of KE are calculated as

EX ′1

=(

PEX1 + PEX2 + KEX1 + KEX2 − PEX ′1

− PEX ′2

)× aaa1 (25)

EX ′2

=(

PEX1 + PEX2 + KEX1 + KEX2 − PEX ′1

− PEX ′2

)× (1 − aaa1)

(26)

here aaa1 is a random number uniformly generated in the inter-al [0, 1]. If the condition of (24) fails, the molecules maintain theriginal X1, X2, PEX1 , PEX2 , KEX1 and KEX2 .

.2.4. SynthesisSynthesis is a process when two or more molecules (in present

aper two molecules X1 and X2) collide with each other and com-

ine to form a single molecule X′. The change is vigorous. As inecomposition, any mechanism which combines two molecules toorm a single molecule may be used. In this paper, procedure men-ioned in section IIIB of [46] is used to create X′. The two PE are

omputing 24 (2014) 962–976 965

change from PEX1 and PEX2 to PEX ′ . The two KE are change fromKEX1 and KEX2 to KEX ′ . The modification is acceptable if

PEX1 + PEX2 + KEX1 + KEX2 ≥ PEX ′ (27)

The new value of KE of the resultant molecule is

KEX ′ = PEX1 + PEX2 + KEX1 + KEX2 − PEX ′ (28)

If condition of (27) is not satisfied, X1, X2 and their related PEand KE are preserved. The pseudo codes for all above-mentionedelementary reaction steps are available in [46].

3.3. Sequential steps of RCCRO algorithm

The three stages in CRO: initialization, iteration, and the finalstage are mentioned below:

(1) In initialization stage, choose unknown variables (n) number.Arrange the initial structure for the molecules and the differentparameters i.e., PopSize, KELossRate, MoleColl, buffer, InitialKE, ˛,and ˇ. Also indicate the lower and upper bounds of unknownvariables of the given problem.

(2) Randomly generate each molecule set of the unknown variablesof the problem within their effective lower and upper boundsand the molecule set must satisfying different constraints. Eachmolecule set characterizes a potential solution of the problem.Generate (PopSize × n) molecule set to create Molecular matrix.

(3) Determine PEs of each molecule set, by their correspondingobjective function values. Set their initial KEs to InitialKE.

(4) During iterative process, first check which type of reaction tobe held. Random create an unknown variable number b ∈ [0,1].If b is greater than MoleColl (which is initialized earlier) orthere is only one molecule left, the reaction take place is a uni-molecular reaction, otherwise it is an intermolecular reaction.

(5) In a uni-molecular reaction, choose one molecule from themolecule set randomly and check whether it satisfies thedecomposition criterion: (number of hits − minimum hit num-ber) > ˛. Where ̨ is the tolerance of duration for the moleculewithout obtaining any new local minimum solution.

If decomposition criterion satisfies, perform decompositionsteps; else perform on-wall ineffective collision steps.

For decomposition if (18) or (20) are satisfied, modify KE andbuffer using (19) or (21), (22) and (23) respectively. Similarlyfor on wall ineffective collision if (15) is satisfied then mod-ify KE and buffer using (16) and (17) respectively. For both thecases, modify the PE of each molecule set using their objectivefunction value.

(6) For each intermolecular reaction, select two (or more) moleculesets randomly from the molecular matrix and test the synthe-sis criterion: (KE ≤ ˇ) where, ̌ is the minimum KE a moleculeshould have.

If the condition is satisfied, perform the synthesis steps; oth-erwise, perform different steps of an intermolecular ineffectivecollision.

For synthesis if (27) is satisfied, modify KE using (28). Forintermolecular collision, if (24) is satisfied, modify KE using (25)and (26). PE of each modified molecule set is calculated in thesame way as mentioned in step 5.

(7) If the maximum no. of iterations is reached or specified accuracylevel is achieved, terminate the iterative process, otherwise goto step 4 for continuation.

Detailed procedures of evaluation for RCCRO algorithm throughflow chart have been shown in Fig. 1. Interested readers may refer[46], which contains the detail steps of the CRO Algorithm.

9 Soft C

3h

c

66 K. Bhattacharjee et al. / Applied

.4. Sequential steps of RCCRO algorithm to solve short termydro thermal scheduling

The detailed steps of the RCCRO approach for the STHS probleman be described as follows:

Step 1: For initialization, choose no. of hydro and thermal gen-erator units, number of molecular structure set, PopSize; elitismparameter “p”. Specify maximum and minimum capacity of

S

Selec t m, PopS ize, Pmax, Pmin, KELoss Rate, Mo

Initi alize rando mly molec u

given molec ule set of mole the effecti ve op erating li mi

Calculate PE for each molecu le set

Sort out best “PopS ize” molec ule sets, base

set.

Ite

Selec t b ! [0, 1]

b > MoleCollYes

No

(nu mber of hit s minimum hit

number) > !

Yes

No

Crea te two molec ule

sets using

Decomposition criteria.

Calculate PE , mod ify

KE using (17) or (19),

(20) if (16) or (18) is

sati sfied respecti vely.

Crea te a molec ule s

using on w

ineffective collisio

Calculate PE , mod i

KE if (13) is satisfi

using (14). Mod i

buffer using (15 ).

!

KE

Yes

1 26 5

Fig. 1. Flow chart of R

omputing 24 (2014) 962–976

water volume (Vminh

, Vmaxh

) and water discharge (Q minh

, Q maxh

) foreach hydro generator, power demand for each interval (PD(t)),initial and final reservoir water volume (Vbegin

h, Vend

h). Also ini-

tialize the RCCRO parameters like KELossRate, MoleColl, buffer,InitialKE, ˛, and ˇ, etc. Set maximum number of iterations,

Itermax.Step 2: Initialize each element of a given molecule set of X matrixhaving discharge of water for each hydro plant for T intervalsand output power generation for each thermal power plant for

tart

leColl, buff er, Initi alKE, , and maxIter

lar structure matrix (X). Eac h element of a

cular structure matrix (X) shou ld be wit hin

ts.

d on t he PE s values of molec ule

ration = 1

et

all

n.

fy

ed

fy

Create a new

molec ule set fr om the

two molec ule sets

using Syn thesis

crit eria. Calculate PEusing (26), mod ify

KE if (25) is sati sfied. !

Crea te two new

molec ule sets using

Intermolecular

ineffecti ve colli sion.

Calculate PE, modify

KE if (22) is sati sfied,

using (23) and (24 )!

No

34

Pop = 1

CCRO algorithm.

K. Bhattacharjee et al. / Applied Soft Computing 24 (2014) 962–976 967

1 2 3 4

Is Iteration =maxIter ?

Iteration =iter ation +1

No6

Opti mum molec ular set ou tpu t. Best PE ou tpu t.

5

Sort out best “PopS ize” molec ule sets out of old Molec ule sets and newly generated molec ule sets from

molec ular reaction s, based on t he PE s values of molec ule sets.

Calculate PE for each newly generated molec ule set.

Yes!

Is Pop = PopSize?

Pop = Pop+1 No

Base d on t he PE values i denti fy t he

best molecule set.

Stop

Yes

(Conti

frhu

Q

ihoipc

Fig. 1.

T intervals. As for example, if 4 nos. of hydro units, 3 nos. of ther-mal units are there and scheduling is done for 24 h, then total nos.of elements in each molecule set will be 168 ((4 × 24) + (3 × 24)).Initialization is performed using the following procedure.

For j = 1, 2, . . ., PopSize; initialize discharges of each hydro unitsor first (T − 1) intervals Qh(i, t) t = 1, 2, . . ., (T − 1); i = 1, 2, . . ., Nhandomly within lower and upper discharge limits of individualydro units. The hydro discharge at Tth interval, Qh(i, T) is calculatedsing the following equation

h(i, T) = Vhbegin − Vh

end −T−1∑j=1

Qh(i, j) +T∑

j=1

Ih(i, j)

+Ru∑

m=1

T∑j=1

Qh(m, j − �m) i = 1, 2, . . ., Nh (29)

Knowing hydro discharges, evaluate reservoir volume for eachnterval for each hydro units using (3). Reservoir volume of eachydro unit for each interval should satisfy the inequality constraint

f (4). Find out the power generations of each hydro-unit for eachnterval Ph(i, t) by simple algebraic method of Eq. (10). Power out-ut of each hydro unit for each interval should satisfy the inequalityonstraint of (9). From the calculated generations for all hydro-units

nued ).

of a given interval Ph(i, t), and the given load PD(t) of that interval,compute active power demand for all thermal units for thatparticular interval Pth

D (t) using following equation for t = 1, 2, . . ., T:

PDth(t) = PD −

Nh∑i=1

Ph(i, t) (30)

Initialize power outputs of first (Ns − 1) nos. of thermal unitsrandomly within their minimum and maximum operating limits.Compute power outputs of Nth

s thermal units for each interval usingthe following equation:

Ps(Ns, t) = PDth(t) −

Nh∑i=1

Ph(i, t) −Ns−1∑k=1

Ps(k, t) t = 1, 2, . . ., T (31)

Each molecule set of X matrix should be in the form of

Xj = [Qh(1, 1), Qh(1, 2), . . ., Qh(1, T), Qh(2, 1), Qh(2, 2), . . .,

Qh(2, T), . . ., Qh(Nh, 1), Qh(Nh, 2), . . ., Qh(Nh, T), Ps(1, 1),

P (1, 2), . . ., P (1, T), . . ., P (N , 1), P (N , 2), . . ., P (N , T)]

s s s s s s s s

Evaluated thermal generators output should satisfy the inequal-ity constraint of (8).

968 K. Bhattacharjee et al. / Applied Soft C

Reserv oir 1 Reservoir 2

Reserv oir 3

Reserv oir 4

Ih1Ih2

Ih3

Qh1Qh2

Qh3Ih4

Qh4

sR2o

otass

ate reservoir volume for each interval for each hydro unit using

Fig. 2. Hydraulic system test network.

If any variable for a molecular set do not satisfy any of the con-traints (4), (5), (8), (9); discard the corresponding molecule set.e-initialize the corresponding molecule set randomly using step. Continue the process until the molecule set satisfies the entireperation limit and other constraints of (4), (5), (8) and (9).

Step 3: Calculate the PE value (i.e., fitness function value) for eachmolecule set of the habitat matrix for given initial kinetic energy(KE) InitialKE.Step 4: Based on the PE values identify the elite molecule set.Here, elite term is used to indicate those molecule sets of gen-erator power outputs, which give best fuel cost of thermal powergenerators. Keep top ‘p’ molecule sets unchanged after individualiteration, without making any modification on it.Step 5: Create a random number b ∈ [0,1]. If b is greater thanMoleColl or there is only one molecule left (at the later stage of iter-ative procedure, this condition may hold), perform a unimolecularreaction, else perform an intermolecular reaction on each sets ofmolecular matrix.Step 6: If unimolecular reaction is selected, choose one moleculeset randomly from the whole X matrix and check whether it sat-isfies the decomposition criterion.

If decomposition condition is satisfied, perform decompositionn that particular molecule set. Create two new molecule sets usinghe steps mentioned in section IIIB of [46]. Each newly gener-

ted molecule set is one of the possible solutions of hydro-thermalcheduling problem. Calculate PE i.e., fuel cost of the new moleculeets. If the condition mentioned in (18) or (20) is satisfied, modify

0 5 10 0.7

0.8

0.9

1

1.1

1.2

1.3

1.4

1.5

1.6

1.7x 10

6

Time

Res

ervo

ir St

orag

e V

olum

e (m

3)

Fig. 3. Hourly variation of hydro reservoi

omputing 24 (2014) 962–976

KE of new molecule sets using (19) or (21), (22). Modify buffer using(23).

If decomposition condition is not satisfied, perform on wallineffective collision. Create two new molecule sets using Gaussiandistribution and the procedure mentioned in Section 3.2.1. Calcu-late PE of the modified molecule set. If the condition mentionedin (15) is satisfied then modify KE of new molecule set using (16).Modify buffer using (17).

Step 7: From the condition of step 5, if intermolecular reactionis chosen, select two (or more) molecule sets randomly from themolecular matrix X and test the synthesis criterion (KE ≤ ˇ).

If the condition is satisfied, perform the synthesis steps. Createa new molecule set from the two selected molecule sets followingthe procedure given in Section IIIB of [46]. Calculate PE of the newmolecule set. After new molecule creation, if the condition of (27)is satisfied, modify KE of new molecule set using (28).

If synthesis condition (KE ≤ ˇ) is not satisfied, perform inter-molecular collision. Create two new molecule sets in theneighborhood of selected molecule sets following Gaussian dis-tribution and the procedure mentioned in Section 3.2.1. Calculatefuel cost i.e., PE for the newly generated molecule set. After newmolecule sets creation, if condition presented in (24) is satisfied,modify KE of new molecule sets using (25) and (26).

Step 8: In each iteration any one of the reaction mentioned in steps6 and 7 takes place. It may be possible that either one of the inter-molecular or one of the unimolecular reactions happens that time.After the reaction, molecule sets get modified. For each modifiedmolecule sets, operating limit constraint of (5) is verified for themodified water discharge, Qh(i, t) t = 1, 2, . . ., (T − 1); i = 1, 2, . . .,Nh. If some Qh(i, t) elements of a molecule set violate either upperor lower operating limits, then fix the values of those elements ofthe molecule set at the limit hit by them. The hydro discharge atTth interval Qh(i, T) is calculated using (29). If the value of Qh(i, T)violet their maximum or minimum value, then go to step 5 andreapply step 6 and step 7 again on old value of that molecule setsuntil the value of Qh(i, T) should satisfy the inequality constraints(5). Knowing the value of all feasible hydro discharges, evalu-

(3). Reservoir volume of each hydro unit for each interval shouldsatisfy the inequality constraint of (4). If any values of Reservoirvolume do not satisfy the inequality constraint of (4) then go to

15 20 25(Hr.)

plant1plant2plant3plant4

r storage volume for test system 1.

Soft C

4

ithpC

TO

TS

K. Bhattacharjee et al. / Applied

step 5 and reapply step 6 and step 7 again on old value of thatmolecule sets until the value of all Reservoir volume is satisfied.Calculate the power generations of each hydro-unit for each inter-val Ph(i, t) using Eq. (10). Power output of each hydro unit for eachinterval should satisfy the inequality constraint of (9). If any valuesof Ph(i, t) do not satisfy the inequality constraint then go to step 5and reapply step 6 and step 7 again on old value of that moleculesets until the value of all Ph(i, t) satisfy the inequality constraint(9). From the calculated generations for all hydro-units of a giveninterval Ph(i, t), and the given load PD(t) of that interval, computeactive power demand for all thermal units for that particular inter-val PD

th(t) using Eq. (30) t = 1, 2, . . .,T. Initialize power outputs offirst (Ns − 1) nos. of thermal units randomly within their minimumand maximum operating limits. Compute power outputs of Ns

th

thermal units for each interval using Eq. (31).Step 9: Recalculate the PE of each newly generated feasiblemolecule set i.e., the fuel cost for each thermal power output setof each newly generated molecule set.Step 10: Go to step 5 for next iteration. Stop the process after apredefined number of iterations.

. Numerical results

Three illustrative hydrothermal test systems are considered tonspect and verify the efficiency of the proposed RCCRO approach

o solve short term hydro-thermal scheduling problems. Programsave been written in MATLAB-7 language and executed on aersonal computer with 512-MB RAM and 2.3 GHz Pentium Dualore processor.

able 1utput of hourly water discharge, hydro and thermal power generation of test system 1.

Hour Hydro discharges (×105 m3) Hydr

Q1 Q2 Q3 Q4 H1

1 0.9649 0.9107 2.9854 1.3078 0.8442 0.9640 0.8293 2.9876 1.3128 0.8413 0.8000 0.7000 2.9653 1.3141 0.7504 0.9621 0.9262 2.9072 1.3133 0.8315 0.8000 0.8178 1.7063 1.3053 0.7356 0.7249 0.8000 1.8611 1.3111 0.6867 0.7150 0.6666 1.6466 1.3119 0.6828 0.6895 0.6686 1.5788 1.3094 0.6709 0.7263 0.8000 1.4974 1.3171 0.70410 0.7076 0.8001 1.4110 1.3082 0.70011 0.9000 0.8585 1.4424 1.3242 0.83212 0.9003 0.8470 1.3933 1.3533 0.83413 0.8000 0.8754 1.6152 1.3459 0.78014 0.9000 0.8822 1.6675 1.3340 0.84815 0.8000 0.9372 1.6436 1.3827 0.79016 0.6009 0.8422 1.6803 1.3493 0.64117 0.9000 0.8967 1.7088 1.8000 0.86118 0.7497 0.8491 1.6452 1.6000 0.76019 0.9008 0.8412 1.5212 1.8011 0.85920 0.6459 0.8000 1.3802 1.5780 0.67821 0.7670 0.9430 1.0148 1.8000 0.76922 0.8000 0.6758 1.0224 1.8427 0.79223 0.5915 0.6994 1.0168 1.9302 0.63324 1.1898 1.3330 1.1422 2.3719 0.998

able 2tatistical test comparison result of test system 1 out of 25 trials.

Method Average cost ($) Maximum co

RCCRO 925246.786152 925621.5062Modified DE [31] NAa NA

DE [31] NA NA

IFEP [28,31] 938508.87 942593.02

CEP [28] 938801.47 946795.50

a NA: not available.

omputing 24 (2014) 962–976 969

4.1. Description of hydrothermal test systems

Test system 1: It comprises of four hydro-plants coupledhydraulically and an equivalent thermal plant. The schedule hori-zon is 1 day with 24 intervals of 1 h each. The hydraulic sub-systemis characterized by the following: (a) a multi chain cascade flow net-work, with all of the plants in one stream; (b) river transport delaybetween successive reservoirs; (c) variable head hydro-plants; (d)variable natural inflow rates into each reservoir; (e) prohibitedoperating regions of water discharge rates; (f) variable load demandover scheduling period. The hourly water discharge of differenthydro plants is shown in Fig. 2. The hydrothermal scheduling ofhourly water discharges and hydro generations obtained by RCCROalgorithm are shown in Table 1. Table 1 also presents the output ofthermal generators as obtained by RCCRO algorithm. The minimum,maximum, average system costs obtained using proposed RCCROare much improved than those obtained using modified DE [31],DE [31], IFEP [28,31] and the CEP [28]. These are summarized inTable 2. Table 2 also shows that the simulation time for test systemis 10.21 s, which is much less than the time required by IFEP [28],CEP [31], etc. Fig. 3 depicts the trajectories of cascaded reservoirstorage volumes for the test system 1. The optimal hourly waterdischarge of four hydro-plants obtained by the proposed methodis shown in Fig. 4. The convergence characteristic for the proposedRCCRO algorithm is shown in Fig. 5.

Test system 2: This system consists of four cascaded hydro plants

and three composite thermal plants. The effect of valve point load-ing is considered in case of thermal power plants by superimposinga sinusoidal component on their basic fuel cost characteristic. Thisincreases the complexity of the system.

o power generation (×102 MW) Thermalgeneration (MW)

H2 H3 H4

6 0.6745 0.0000 2.0070 1017.49 0.6282 0.0000 1.8861 1054.43 0.5616 0.0000 1.7446 1054.35 0.6918 0.0000 1.5720 980.505 0.6314 0.3146 1.7834 943.502 0.6148 0.2662 1.9890 1054.42 0.5268 0.3584 2.1711 1276.16 0.5301 0.3831 2.3284 1608.81 0.6126 0.4054 2.3725 1830.69 0.6189 0.4213 2.4141 1904.51 0.6549 0.4187 2.4586 1793.69 0.6455 0.4374 2.5071 1867.51 0.6564 0.4147 2.5130 1793.67 0.6612 0.4160 2.5077 1756.67 0.6876 0.4337 2.5608 1682.77 0.6341 0.4347 2.5318 1645.83 0.6510 0.4330 2.8977 1645.78 0.6087 0.4534 2.7501 1682.74 0.5947 0.4901 2.8896 1756.60 0.5713 0.5189 2.7266 1830.57 0.6449 0.5291 2.8900 1756.68 0.5067 0.5435 2.8996 1645.89 0.5285 0.5542 2.9132 1387.08 0.7752 0.5813 3.0012 1054.4

st ($) Minimum cost ($) Average time (s)

925214.2018 10.21925960.56 NA929755.94 NA933949.25 1450.90934713.18 2790.40

970 K. Bhattacharjee et al. / Applied Soft Computing 24 (2014) 962–976

0 5 10 15 20 250.5

1

1.5

2

2.5

3x 10

5

Time(Hr.)

Wat

er D

isch

arge

(m3)

plant1plant2plant3plant4

Fig. 4. Hourly water discharge of different hydro plants for test system 1.

0 50 100 150 200 250 3009.24

9.26

9.28

9.3

9.32

9.34

9.36

9.38

9.4

9.42

9.44x 10

5

Iterations

Min

imum

Cos

t($)

s obta

tfoifaaqriouhpoi

lprt

algorithm is shown in Table 5. Table 5 also presents the com-plete scheduling of all four hydro and three thermal generators asobtained by RCCRO algorithm for 24 h period. The total minimum,maximum, average system costs obtained by proposed RCCRO out

0.8

1

1.2

1.4

1.6

1.8 x 106

Res

ervo

ir St

orag

e V

olum

e (m

3)

plant1plant2plant3plant4

Fig. 5. Convergence characteristic

Case 1: Here prohibited operating zone and ramp rate limit forhermal power plants are not considered. The detailed input dataor this system are taken from [31]. The hydrothermal schedulingf hourly water discharges obtained by RCCRO algorithm is shownn Table 3. Table 3 also presents the complete scheduling of allour hydro and three thermal generators as obtained by RCCROlgorithm for 24 h period. The total minimum, maximum, aver-ge system costs obtained by proposed RCCRO out of 25 trials areuite close to each other and are summarized in Table 4. Timeequired by the algorithm to converge to the optimum solutions 15.51 s, which is also very less, compared to the complexityf the system. The trajectories of cascaded reservoir storage vol-mes for the test system 2 are presented in Fig. 6. The optimalourly hydro discharge of four hydro-plants obtained by the pro-osed method is shown in Fig. 7. The convergence characteristicf the proposed RCCRO algorithm for this test system is shownn Fig. 8.

Case 2: Here prohibited zone of hydro plants and ramp rate

imit for thermal power plants are considered. Input data for hydrolant and thermal plants for this case study are taken from [28,31],amp rate limit for thermal plants are taken from [51]. The hydro-hermal scheduling of hourly water discharges obtained by RCCRO

ined by RCCRO for test system 1.

0 5 10 15 20 250.6

Time (Hr.)

Fig. 6. Hourly variation of hydro reservoir storage volume for case 1 of test system 2.

K. Bhattacharjee et al. / Applied Soft Computing 24 (2014) 962–976 971

Table 3Hourly hydro plant water discharges, hydro and thermal generation schedules obtained by RCCRO for case 1 of test system 2.

Hour Hydro discharges (×105 m3) Hydro power generation (MW) Thermal generation (MW)

Q1 Q2 Q3 Q4 H1 H2 H3 H4 T1 T2 T3

1 0.8431 0.8007 1.4353 1.1091 78.2298 62.0367 57.2027 184.0167 103.8465 124.9079 139.75982 1.4350 0.9979 2.6783 0.9544 96.1479 71.1958 0.0000 160.1009 102.9799 209.8158 139.75983 0.9266 0.9645 2.1160 0.9219 80.5861 69.1710 27.7288 148.2532 20.0000 124.7414 229.51964 0.7499 0.8876 2.6474 1.0203 70.1603 65.3233 0.0000 146.8023 103.0463 124.9079 139.75985 0.9681 0.9667 2.7342 0.9979 80.8972 68.2388 0.0000 149.6906 101.6539 40.0000 229.51966 0.9330 0.8092 1.7255 0.9132 78.2391 59.1582 42.5991 159.9495 105.6265 124.9080 229.51967 0.6111 0.9408 1.9191 1.7780 59.0150 63.8515 36.6074 238.5163 22.9145 209.8158 319.27948 0.5141 0.6480 1.9150 1.8140 52.3481 47.2017 37.4817 250.9072 97.8179 294.7237 229.51969 0.9172 0.6516 2.2095 1.2871 80.1312 48.4172 19.9283 225.4363 102.0840 294.7237 319.279410 0.9271 1.1973 1.4179 1.5380 81.2777 73.7802 48.5317 249.6777 102.4894 294.7237 229.519611 1.1840 0.9035 1.4456 1.6303 91.7182 61.2600 48.9049 259.9755 109.0462 209.8158 319.279412 0.8393 0.8326 1.6020 1.4886 77.0531 57.2413 46.7477 252.7218 102.2429 294.7139 319.279413 0.7268 0.6542 1.8189 1.3515 70.9025 47.3579 43.9538 248.1200 175.0000 295.1462 229.519614 0.8100 0.6441 2.0756 1.3281 77.4464 48.3586 33.1805 246.6167 100.1545 294.7237 229.519615 0.8084 1.2940 1.5176 1.4967 78.0415 76.5396 52.3120 261.9661 101.8054 209.8158 229.519616 0.5397 0.7128 1.7240 1.7024 58.1737 50.8036 47.6910 277.8661 101.2223 294.7237 229.519617 0.6403 0.6580 1.5133 1.3585 67.0117 47.5504 52.6027 252.3050 101.4351 209.8157 319.279418 0.8288 0.8966 1.6038 1.9511 80.7507 59.5220 52.2553 300.7887 102.4400 294.7237 229.519619 0.6722 0.6933 1.4319 1.5698 69.5541 47.8640 54.8155 272.3013 101.2257 294.7198 229.519620 1.1774 0.8673 1.3737 1.9517 96.7058 57.5494 55.7855 297.9478 102.6761 209.8158 229.519621 0.6595 0.7630 1.2606 1.7666 67.7779 52.6496 57.1497 283.1071 99.7401 209.8158 139.759822 0.7705 0.8509 1.1256 1.9670 75.9619 58.0053 58.0213 292.3427 21.2413 124.9079 229.519623 0.5139 0.7569 1.3889 1.9009 55.9902 52.9244 59.3884 283.3162 20.0000 148.8612 229.519624 0.5038 0.8084 1.8165 1.7873 55.3755 55.8782 51.4103 271.5525 101.1158 124.9079 139.7598

Table 4Comparison of performance for case 1 of test system 2 out of 25 trials.

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

RCCRO 41498.2129 41502.3669 41497.8517 15.51IDE [51] 40708.53 40860.70 40627.92a 627.06TLBO [52] 42407.23 42441.36 42385.88 NACSA [44] NA NA 42440.574 NAIPSO [44] NA NA 44321.236 NAMDE [31] NA NA 42611.142 NADE [31] NA NA 44526.106 NAEP [44] NA NA 45063.004 NA

a Transmission line losses have not been considered here. So total power generations should be equal to load. Sum of all power generation of hydro and thermal reportedin [51] is not equal to their respective demand for each interval. For example, sum of hydro and thermal power generations for 1st hour reported as 756.391 MW but loaddemand of that interval was 750 MW.

Table 5Hourly hydro plant water discharges, hydro and thermal generation schedules obtained by RCCRO for case 2 of test system 2.

Hour Hydro discharges (×105 m3) Hydro power generation (MW) Thermal generation (MW)

Q1 Q2 Q3 Q4 H1 H2 H3 H4 T1 T2 T3

1 0.9556 1.1506 2.8892 1.3353 84.0279 76.5905 0.0000 202.7950 26.2070 126.3650 234.01522 1.0690 0.7000 2.7000 1.0728 88.4932 54.2587 0.0000 168.7400 106.2070 42.3003 319.99893 1.0130 1.2440 1.8666 1.3311 85.3145 78.0580 34.6751 178.3265 53.6683 40.3651 229.59134 0.6042 0.6241 2.2000 1.2005 60.6057 48.7612 19.1823 151.9685 99.3776 130.3651 139.73985 0.7898 1.3459 1.1575 1.1081 73.1152 78.9004 50.7271 165.4674 31.5349 40.3903 229.86566 0.6998 0.6683 2.1008 0.9275 67.1631 48.4324 28.9679 165.8313 103.7367 124.7696 261.09677 0.5255 0.6385 1.5976 1.5335 54.3649 46.1937 47.8205 226.8034 127.1345 126.9201 320.75378 0.9570 0.8465 1.6990 1.0494 83.1827 57.8822 47.1712 193.9987 99.8225 210.4130 317.52999 0.9022 0.8000 2.2000 1.8000 80.7099 55.2746 22.5088 251.1038 156.7423 294.7087 228.951110 0.9864 1.0447 1.8738 0.7701 85.3577 66.4654 37.5819 166.9375 109.7070 294.7100 319.234611 0.8000 0.9293 1.7817 1.0078 76.2461 60.9379 41.1468 200.9853 105.3468 295.3829 319.951412 0.6647 0.8000 1.6055 0.8971 67.5297 54.0420 47.3504 194.3574 173.2149 294.5263 318.985913 0.7447 0.7000 1.0406 1.9887 74.1052 48.7171 53.0630 295.5971 104.3969 299.7672 234.355114 0.5445 0.8737 1.2545 1.6000 58.9503 59.0066 56.0955 271.2349 146.3587 209.7672 228.584515 0.7126 0.9126 1.3636 1.6000 73.2374 60.9311 57.0355 272.9537 103.7157 212.3662 229.759516 0.9000 0.6167 1.3003 1.1511 86.2518 44.5547 57.4948 233.2370 102.9361 214.2940 321.233117 0.6628 0.6137 1.7310 1.6000 69.5433 44.9182 51.9630 271.9629 174.2100 208.5828 228.823318 1.1302 0.6102 1.4082 1.4076 97.8893 44.6095 58.5456 253.5782 144.3994 293.5196 227.461919 0.6080 1.0588 1.6687 1.4990 64.8310 66.9469 54.2783 260.6430 100.3788 293.4466 229.480820 0.7269 0.6290 1.1765 1.2860 74.2953 44.6457 58.9180 240.6364 105.4310 296.3082 229.764621 0.5594 0.8435 1.3910 1.5697 60.5762 58.1960 59.4364 268.3731 106.8262 216.2086 140.378822 0.6362 0.6325 2.2000 1.8000 67.3116 47.0288 35.7274 282.0816 76.7167 211.2344 139.905423 0.9000 0.6667 1.4853 1.8895 86.4904 50.1857 57.9059 285.6345 21.2685 209.6717 138.845524 1.4073 1.2506 1.3817 1.8702 105.7356 75.2213 58.8672 276.9007 20.0004 123.7799 139.5083

972 K. Bhattacharjee et al. / Applied Soft Computing 24 (2014) 962–976

Table 6Comparison of performance for case 2 of test system 2 out of 25 trials.

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

RCCRO 43557.2899 43580.0712 43555.3089 35.85IDE [51] 43800.51 43812.01 43790.33 782.23

0 5 10 15 20 250.5

1

1.5

2

2.5

3 x 105

Time (m3)

Wat

er D

isch

arge

(m3)

plant1plant2plant3plant4

Fs

oTmcsFovt

hmwdtgobitt

F

0 5 10 15 20 250.6

0.8

1

1.2

1.4

1.6

1.8 x 106

Time (Hr.)

Res

ervo

ir St

orag

e V

olum

e (m

3)

plant1plant2plant3plant4

Fig. 9. Hourly variation of hydro reservoir storage volume for case 2 of test system 2.

0 5 10 15 20 250.5

1

1.5

2

2.5

3 x 105

Time (Hr.)

Wat

er D

isch

arge

(m3)

plant1plant2plant3plant4

ig. 7. Hourly variation of water discharge of different plants for case 1 of testystem 2.

f 25 trials are quite close to each other and are summarized inable 6. Time required by the algorithm to converge to the opti-um solution is 35.85 s, which is also very less, compared to the

omplexity of the system. The trajectories of cascaded reservoirtorage volumes for case 2 of the test system 2 are presented inig. 9. The optimal hourly hydro discharge of four hydro-plantsbtained by the proposed method is shown in Fig. 10. The con-ergence characteristic of the proposed RCCRO algorithm for thisest system is shown in Fig. 11.

Test system 3: This system is a more practical representation ofydrothermal systems consisting of four hydro plants and ten ther-al plants. The effect of valve point loading is taken into accountithin the fuel cost characteristics of thermal generators. Theetailed data for this system have been taken from [32]. The hydro-hermal scheduling of hourly water discharges and hydro powerenerations obtained by RCCRO algorithm is shown in Table 7. Forptimal operation, the outputs of 10 thermal generators as obtainedy RCCRO algorithm are presented in Table 8. The minimum, max-

mum, average system costs obtained by proposed RCCRO for thisest system are depicted in Table 9. Time required by the algorithmo converge to the optimum solution for this test system is 22.02 s.

0 50 100 150 200 250 3004.1

4.15

4.2

4.25

4.3

4.35

4.4

4.45

4.5 x 104

Iterations

Min

imum

Cos

t ($)

ig. 8. Convergence characteristics obtained by RCCRO for case 1 of test system 2.

Fig. 10. Hourly variation of water discharge of different plants for case 2 of testsystem 2.

These results are compared with the results obtained using MDE[43], SPSO [43] and SPPSO [43]. Fig. 12 depicts the trajectories ofcascaded reservoir storage volumes for the test system 3. The opti-mal hourly hydro discharge of four hydro-plants obtained by the

0 50 100 150 200 250 3004.35

4.4

4.45

4.5

4.55

4.6

4.65 x 104

Iterations

Min

imum

Cos

t ($)

Fig. 11. Convergence characteristics obtained by RCCRO for case 2 of test system 2.

K. Bhattacharjee et al. / Applied Soft Computing 24 (2014) 962–976 973

Table 7Hourly hydro discharge and hydro power generation obtained by RCCRO for test system 3.

Hour Hydro discharges (×105 m3) Hydro power generation (×102 MW)

Q1 Q2 Q3 Q4 H1 H2 H3 H4

1 0.9098 1.0088 2.2327 0.8461 0.8181 0.7163 0.2879 1.57812 0.6985 0.9213 2.4413 1.0150 0.6906 0.6657 0.0877 1.68783 0.6410 0.6482 2.3755 0.9593 0.6513 0.5188 0.0854 1.54134 0.7055 0.6871 1.8052 0.6088 0.6994 0.5576 0.3881 1.06495 0.8493 0.7157 1.5073 0.8178 0.7872 0.5812 0.4782 1.46776 0.7859 0.6359 1.6473 0.8641 0.7455 0.5316 0.4526 1.67727 0.9955 0.9898 2.7263 1.4184 0.8548 0.7142 0.0000 2.34578 0.5447 0.7313 1.8026 1.3838 0.5692 0.5696 0.3681 2.35759 0.6972 0.7706 2.0332 1.2934 0.6962 0.5957 0.2565 2.293410 0.9595 0.7838 1.9157 1.6716 0.8634 0.6108 0.2989 2.613911 0.7030 0.6405 1.9218 1.4318 0.7130 0.5339 0.2787 2.543612 1.2310 0.8503 1.1218 1.8199 0.9819 0.6623 0.4669 2.852313 0.8625 0.8033 1.3648 1.7011 0.8214 0.6350 0.4788 2.799914 0.8866 0.9164 1.5631 1.9474 0.8425 0.6970 0.4705 2.965415 0.6566 1.0237 1.4900 1.5733 0.6869 0.7417 0.5003 2.727016 0.9737 0.8326 1.2138 1.4473 0.9012 0.6415 0.5363 2.585817 0.9050 0.8641 2.0972 1.5066 0.8619 0.6488 0.3132 2.626118 0.7451 0.7401 1.6569 1.8807 0.7559 0.5672 0.4958 2.878219 0.5610 0.6524 1.7293 1.9851 0.6070 0.5133 0.4805 2.887220 1.0865 0.8755 1.8339 1.9955 0.9502 0.6441 0.4440 2.804321 0.8999 0.9806 1.2177 1.8348 0.8498 0.6921 0.5482 2.739822 0.6237 0.8388 1.2785 1.9524 0.6571 0.6226 0.5645 2.777023 0.7249 0.8445 1.2077 1.5955 0.7387 0.6229 0.5763 2.552424 0.8537 1.4445 1.0401 1.6101 0.8301 0.8000 0.5677 2.5871

Table 8Hourly thermal generation schedules obtained by RCCRO for test system 3.

Hour Thermal generation (MW)

T1 T2 T3 T4 T5 T6 T7 T8 T9 T10

1 230.3025 274.9383 95.3709 123.1488 125.1976 138.1509 104.6664 35.0025 157.5152 125.66732 317.5802 200.3333 95.8884 120.3417 175.0341 139.5721 104.9078 35.4192 159.9432 117.81263 317.9728 274.4509 20.6810 121.7433 175.0589 139.9815 105.4572 35.0452 98.7042 131.23764 227.4650 270.6888 91.0782 119.3222 173.7248 140.0381 45.2006 36.3916 98.2448 176.84615 229.4472 199.7475 95.0257 120.6454 174.3806 139.7994 105.1736 35.0395 99.2728 140.04346 319.7883 201.4342 93.9638 119.4720 225.9310 141.8139 45.3160 35.0069 99.2990 177.28367 319.2841 277.0417 96.4778 120.8139 174.8844 90.2261 105.2600 35.0674 159.9376 179.53258 319.1121 201.9475 95.0505 123.1208 223.2771 242.0368 45.3266 35.1633 159.3107 179.22289 318.8438 271.1458 96.8711 125.7977 230.7164 191.1793 158.5186 35.0203 100.8376 176.897110 321.2010 274.9696 93.6804 119.8193 224.2009 187.9975 45.3738 35.3189 159.8742 178.868311 319.9830 350.0990 95.5005 120.2606 174.3382 151.5100 170.7197 35.0114 97.6855 177.972212 321.8306 275.4961 96.2993 119.2288 220.7197 189.8694 107.3915 35.5371 159.9998 127.289213 311.9251 273.4731 94.5789 125.0533 176.4458 185.0227 163.4696 35.0075 100.4208 171.088614 321.3659 274.2706 95.0383 120.5641 174.6947 190.0774 45.4640 35.2515 98.7438 176.990415 317.6511 276.2721 98.3386 122.6415 172.9123 139.2376 45.0442 35.2127 159.9578 177.153816 408.3044 198.4273 94.0054 119.9417 223.8129 141.1697 102.3134 35.0725 97.9972 172.487217 320.7952 279.4985 95.1986 119.9807 174.9174 140.2111 104.9735 35.1217 159.6787 174.631218 319.1092 273.0885 94.8011 119.8009 177.7027 186.6497 103.1573 36.7375 159.7345 179.507019 319.1587 200.4823 95.6295 123.2740 222.1192 185.2063 102.6235 35.4016 157.7525 179.556020 319.7337 275.7899 99.8742 119.7583 124.4259 189.4039 106.2494 35.7242 159.8588 134.922721 319.5182 200.0157 95.2285 119.6304 174.2652 189.6395 45.0576 35.0029 98.1375 150.506822 318.7104 198.7296 94.3999 119.5077 172.0700 139.5233 45.0507 35.3166 99.3170 175.2598

TC

23 231.3467 275.0957 95.2802 120.3267 124.7636

24 319.1111 200.1282 94.1594 119.6859 124.8806

able 9omparison of performance obtained by RCCRO algorithm for test system 3 out of 25 tria

Method Average cost ($) Maximum cost (

RCCRO 164140.3997 164182.3520

DE [43] NA NA

MDE [43] 179676.35 182172.01

SPSO [43] 190560.31 191844.28

SPPSO [43] 168688.92 170879.30

139.0445 103.9985 35.6827 98.5945 176.833589.4646 45.0395 35.0082 159.6233 134.4133

ls.

$) Minimum cost ($) Average time (s)

164138.6517 22.02170964.15 96.4177338.60 86.5189.350.63 108.1167710.56 24.8

974 K. Bhattacharjee et al. / Applied Soft Computing 24 (2014) 962–976

0 5 10 15 20 250.6

0.8

1

1.2

1.4

1.6

1.8 x 106

Time (Hr.)

Res

ervo

ir St

orag

e V

olum

e (m

3)

plant1plant2plant3plant4

Fig. 12. Hydro reservoir storage volume for test system 3.

0.5

1

1.5

2

2.5

3 x 105

Wat

er D

isch

arge

(m3)

plant1plant2plant3plant4

pti

4

arRabDo

1.64

1.645

1.65

1.655

1.66

1.665

1.67

1.675

1.68 x 105

Min

imum

Cos

t ($)

TE

N

0 5 10 15 20 25Time (Hr.)

Fig. 13. Hourly water discharge of different plants for test system 3.

roposed method is presented in Fig. 13. The convergence charac-eristic for the test system obtained by proposed RCCRO algorithms shown in Fig. 14.

.2. Discussion

Minimum, maximum, average fuel costs obtained by RCCROlgorithm for test systems 1, 2, 3 are presented in Tables 2, 4, 6 and 9espectively (Tables 4 and 6 for case 1 and case 2 of test system 2).esults show that the minimum fuel costs for these test systems

s obtained by RCCRO is quite less compared to those obtainedy different versions of evolutionary programming [28], PSO [43],E [31,43], CSA[44], IDE[51], SPPSO [43], TLBO [52], etc. More-ver, minimum, maximum, average fuel costs obtained by RCCRO

able 10ffect of different parameters on performance of RCCRO (minimum fuel cost obtained for

InitialKE ̌ ̨ MoleColl

0.1 0.2

2000 1000 2000 0.9 164186.35 1641641800 900 1500 0.8 164181.55 1641601600 800 1300 0.75 164178.17 1641551400 700 1000 0.70 164172.40 1641511200 600 800 0.60 164167.50 1641481000 500 600 0.50 164162.26 164144

800 400 400 0.40 164156.93 164142600 300 300 0.30 164150.41 164141400 200 200 0.20 164153.07 164144200 100 100 0.10 164155.44 164149

otes: Bold digit indicates that for those particular settings of RCCRO parameters, result o

0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100Iterations

Fig. 14. Convergence characteristics of algorithms for test system 3.

algorithm out of certain number of trials are quite close to eachother. RCCRO reaches to the minimum solutions 23, 23, 24 timesfor test systems 1, 2, 3 respectively. Therefore, success rate of RCCROis 92%, 92% and 96% respectively for these test systems. This clearlyshows that RCCRO has the ability to reach to the minimum solu-tion consistently. It establishes the improved robustness of thealgorithm.

Results also show that the average simulation time required byRCCRO to converge to minimum solution is quite less compared tothat required by many previously developed techniques. Conver-gence characteristics for test systems 1, 2, 3 obtained by RCCRO, aspresented in Figs. 5, 8, 11 and 14 clearly reflects that RCCRO reachesto the minimum solutions within very few numbers of iterations.These establish the superior computational efficiency of RCCRO.

Therefore, the above results prove the enhanced ability ofRCCRO to solve complex, non linear short term hydro thermalscheduling problem in order to achieve superior quality solutions,in a computationally efficient and robust manner.

4.3. Tuning of RCCRO parameters for short term hydro thermalscheduling problems

It is very essential to get the proper values of different parame-ters like, kinetic energy loss rate (KELossRate), initial kinetic energy(InitialKE) and ̌ to reach optimum solution using RCCRO algorithm.Tuning of other RCCRO parameters like MoleColl, ̨ are also veryimportant. RCCRO algorithm has been run repeatedly for test sys-tem 3 with different combinations of different parameters. Resultsare shown in Table 10. As for example, when InitialKE = 2000; that

time ̌ has been varied from 100 to 1000 in suitable steps. At thesame time for each value of ˇ, ̨ has been varied from 100 to 2000in suitable steps. Similarly for each value of ˛, MoleColl and KELoss-Rate have been varied from 0.1 to 0.9. However, to present all these

test system 3).

KELossRate

0.5 0.6 0.8 0.9

.64 164151.91 164144.85 164140.28 164141.27

.07 164149.58 164142.87 164140.08 164140.84

.28 164145.37 164141.30 164139.90 164140.41

.10 164144.17 164139.91 164139.41 164139.91

.20 164142.15 164139.27 164139.15 164139.68

.98 164141.11 164139.07 164138.98 164139.15

.66 164140.54 164138.88 164138.82 164138.89

.08 164139.09 164138.75 164138.65 164138.71

.00 164139.23 164138.79 164138.72 164138.75

.88 164139.74 164139.01 164138.87 164139.54

btained by RCCRO is optimum. It is only used to highlight the best results.

K. Bhattacharjee et al. / Applied Soft Computing 24 (2014) 962–976 975

Table 11Effect of molecular structure size on test system 3.

Molecular structure size No. of hits to best solution Simulation time (s) Maximum cost ($) Minimum cost ($) Average cost ($)

20 22 16.24 164192.2732 164139.8485 164146.139550 24 22.02 164182.3520 164138.6517 164140.3997

100 15 24.51 164253.5512 164144.9807 164188.4089150 10 28.71 164289.5418 164158.2000 164237.0051

N esult o

rri

nshoocte

oKbi

5

aahorscemrsoRSsarfot

R

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

200 8 30.90

otes: Bold digit indicates that for those particular settings of RCCRO parameters, r

esults in a table, takes lots of space. Therefore, the detail tuningesults are not shown in Table 10. Only a brief summarized results only shown in Table 10.

Too large or small value of molecular structure size mayot be capable to get the optimum value. For each moleculartructure size (PopSize) of 20, 50, 100, 150 and 200, the programas been run for 25 trials. Out of these, molecular structure sizef, 50 achieves best fuel cost of generation for test system 3. Forther molecular structure size, no significant improvement of fuelost has been observed. Moreover, beyond PopSize = 50, simulationime also increases. Best output obtained by RCCRO algorithm forach molecular structure size is presented in Table 11.

Therefore, optimum values of these tuned parameters asbtained from Tables 10 and 11 are PopSize = 50, InitialKE = 600,ELossRate = 0.8, ̌ = 300, MoleColl = 0.3, ̨ = 300. Initial value ofuffer = 0 is not selected using tuning procedure; rather its values assumed based on the value presented in Section IIC of [46].

. Conclusion

In a chemical reaction, molecules start from high-energy statesnd terminate at low-energy states via a sequence of collisionsnd molecular changes. CRO captures this idea to develop a meta-euristic for optimization problems. RCCRO is a real coded versionf original CRO algorithm. In this paper, real coded chemicaleaction optimization (RCCRO) technique is presented to solvehort-term hydrothermal scheduling problem (STHS) to minimizeost of generation for thermal power plants. RCCRO have both goodxploration and exploitation ability, therefore it reaches to opti-al solution within very small number of iterations. Numerical

esults obtained for three test systems and comparative analy-is with previous approaches indicate the superior performancef RCCRO algorithm. Moreover, total simulation time required byCCRO to reach to optimal solution for any test system is quite less.uccessful implementation and superior performance of RCCRO toolve short term hydro thermal scheduling problems has created

new path in the field of power system which may encourage theesearcher to apply this newly developed algorithm to solve dif-erent much complex power system optimization problems likeptimal power flow, loss minimization, optimal placement of dis-ributed generators, FACTS devices, etc.

eferences

[1] A.J. Wood, B.F. Wollenberg, Power Generation, Operation and Control, Wiley,New York, 1984.

[2] L.K. Kirchmayer, Economic Control of Interconnected Systems, Wiley, NewYork, 1959.

[3] O. Nilson, D. Sjelvgren, Mixed integer programming applied to short-term plan-ning of a hydro-thermal system, in: Proc. 1995 IEEE PICA, Salt Lake City, UT, May,1995, pp. 158–163.

[4] L. Engles, R.E. Larson, J. Peschon, K.N. Stanton, Dynamic programming appliedto hydro and thermal generation scheduling, in: IEEE Tutorial Course Text, IEEE,New York, 1976, 76CH1107-2-PWR.

[5] M. Papageorgiou, Optimal multi reservoir network control by the discrete max-imum principle, Water Resour. Res. 21 (2) (1985) 1824–1830.

[6] S.A. Soliman, G.S. Christensen, application of functional analysis to optimisationof variable head multi reservoir power system for long term regulation, WaterResour. Res. 22 (6) (1986) 852–858.

[

164325.5410 164167.6621 164275.0198

btained by RCCRO is optimum. It is only used to highlight the best results.

[7] B.Y. Lee, Y.M. Park, K.Y. Lee, Optimal generation planning for a thermal systemwith pumped storage based on analytical production costing model, IEEE Trans.PWRS 2 (May (2)) (1987) 486–493.

[8] H. Brannud, J.A. Bubenko, D. Sjelvgren, Optimal short term operation planningof a large hydrothermal power system based on a non linear network flowconcept, IEEE Trans. PWRS 1 (November (4)) (1986) 75–82.

[9] F. Wakamori, S. Masui, K. Morita, Layered network model approach to opti-mal daily hydro scheduling, IEEE Trans. PAS 101 (September (9)) (1982)3310–3314.

10] Q. Xia, N. Xiang, S. Wang, B. Zhang, M. Huang, Optimal daily scheduling ofcascaded plants using a new algorithm of non-linear minimum cost networkflow concept, IEEE Trans. PWRS 3 (3) (1988) 929–935.

11] M.R. Piekutowski, T. Litwinowicz, R.J. Frowd, Optimal short term scheduling fora large scale cascaded hydro system, in: Power Industry Computer ApplicationsConference, Phoenix, AZ, 1993, pp. 292–298.

12] T.N. Saha, S.A. Khapade, An application of a direct method for the optimalscheduling of hydrothermal power systems, IEEE Trans. PAS 97 (3) (1978)977–985.

13] H. Habibollahzadeh, J.A. Bubenko, Application of decomposition techniques toshort term operation planning of hydro-thermal power system, IEEE Trans.PWRS 1 (February (1)) (1986) 41–47.

14] C. Wang, S.M. Shahidehpour, Power generation scheduling for multi-areahydrothermal power systems with tie-line constraints, cascaded reservoirs anduncertain data, IEEE Trans. PWRS 8 (3) (1993) 1333–1340.

15] M.V.F. Pereira, L.M.V.G. Pinto, application of decomposition techniques to themid and short term scheduling of hydrothermal systems, IEEE Trans. PAS 102(November (11)) (1983) 3611–3618.

16] S. Soares, C. Lyra, W.O. Tavayes, Optimal generation scheduling of hydrothermalpower system, IEEE Trans. PAS 99 (3) (1980) 1106–1114.

17] R.H. Liang, Y.Y. Hsu, Scheduling of hydroelectric generation units using artificialneural networks, Proc. IEEE Part C 141 (September (5)) (1994) 452–458.

18] G.X. Luo, H. Habibollahzadeh, A. Semylen, Short-term hydrothermal sched-uling, detailed model and solutions”, IEEE Trans. PWRS 1 (October (4)) (1989)1452–1462.

19] S. Soares, T. Ohishi, Hydro dominated short-term hydrothermal scheduling viaa hybrid simulation-optimisation approach: a case study, Proc. IEE Part C 142(November (6)) (1995) 569–575.

20] K.P. Wong, Y.N. Wong, Short term hydrothermal scheduling. Part I: Simulatedannealing approach, Proc. IEE Part C 141 (5) (1994) 497–501.

21] K.P. Wong, Y.W. Wong, Short-term hydrothermal scheduling, Part. I. Simu-lated annealing approach, Proc. Inst. Electron. Eng. Gen. Transm. Distrib. 141(September (5)) (1994) 497–501.

22] K.P. Wong, Y.W. Wong, Short-term hydrothermal scheduling, Part II. Parallelsimulated annealing approach, Proc. Inst. Electron. Eng. Gen. Transm. Distrib.141 (September (5)) (1994) 502–506.

23] S.O. Orero, M.R. Irving, A genetic algorithm modeling framework and solutiontechnique for short term optimal hydrothermal scheduling, IEEE Trans. PowerSyst. 13 (May (2)) (1998) 501–518.

24] E. Gil, J. Bustos, H. Rudnick, Short-term hydrothermal generation schedulingmodel using a genetic algorithm, IEEE Trans. Power Syst. 18 (November (4))(2003) 1256–1264.

25] C.E. Zoumas, A.G. Bakirtzis, J.B. Theocharis, V. Petridis, A genetic algorithm solu-tion approach to the hydrothermal coordination problem, IEEE Trans. PowerSyst. 19 (August (3)) (2004) 1356–1364.

26] J.M. Ramirez, P.E. Ontae, The short-term hydrothermal coordination via geneticalgorithms, Electr. Power Compon. Syst. 34 (1) (2006) 1–19.

27] V.S. Kumar, M.R. Mohan, A genetic algorithm solution to the optimal short-term hydrothermal scheduling, Electr. Power Energy Syst. 33 (6) (2011)827–835.

28] N. Sinha, R. Chakrabarti, P.K. Chattopadhyay, Fast evolutionary programmingtechniques for short-term hydrothermal scheduling, IEEE Trans. Power Syst. 18(February (1)) (2003) 214–219.

29] P.C. Yang, H.T. Yang, C.L. Huang, Scheduling short-term hydrothermal genera-tion using evolutionary programming techniques, Proc. Inst. Electron. Eng. Gen.Transm. Distrib. 143 (July (4)) (1996) 371–376.

30] P. Venkatesh, R. Gnanadass, N.P. Padhy, Comparison and application of evolu-tionary programming techniques to combined economic and emission dispatchwith line flow constraints, IEEE Trans. Power Syst. 18 (May (2)) (2003)688–697.

31] L. Lakshminarasimman, S. Subramanian, Short-term scheduling of hydro-thermal power system with cascaded reservoirs by using modified differentialevolution, Proc. Inst. Electron. Eng. Gen. Transm. Distrib. 153 (November (6))(2006) 693–700.

9 Soft C

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

76 K. Bhattacharjee et al. / Applied

32] K.K. Mandal, N. Chakraborty, Differential evolution technique basedshort-term economic generation scheduling of hydrothermal sys-tems, Elect. Power Syst. Res. 78 (November (11)) (2008) 1972–1979.

33] A.S. Uyar, B. Türkay, A. Keles, A novel differential evolution application to short-term electrical power generation scheduling, Electr. Power Energy Syst. 33 (6)(2011) 1236–1242.

34] Xiaohui Yuan, Bo Cao, Bo Yang, Yanbin Yuan, Hydrothermal scheduling usingchaotic hybrid differential evolution, Energy Convers. Manage. 49 (12) (2008)3627–3633.

35] Z.L. Gaing, Particle swarm optimization to solving the economic dispatchconsidering the generator constraints, IEEE Trans. Power Syst. 18 (August (3))(2003) 1187–1195.

36] J.B. Park, K.S. Lee, J.R. Shin, K.Y. Lee, A particle swarm optimization for economicdispatch with non-smooth cost functions, IEEE Trans. Power Syst. 20 (February(1)) (2005) 34–42.

37] B. Yu, X. Yuan, J. Wang, Short-term hydro-thermal scheduling using particleswarm optimization method, Energy Convers. Manage. 48 (July (7)) (2007)1902–1908.

38] K.K. Mandal, M. Basu, N. Chakraborty, Particle swarm optimization techniquebased short-term hydrothermal scheduling, Appl. Soft Comput. 8 (September(4)) (2008) 1392–1399.

39] P.K. Hota, A.K. Barisal, R. Chakrabarti, An improved PSO technique for short-term optimal hydrothermal scheduling, Elect. Power Syst. Res. 79 (July (7))(2009) 1047–1053.

40] Y. Wang, J. Zhou, C. Zhou, Y. Wang, H. Qin, Y. Lu, An improved self-adaptive

PSO technique for short-term hydrothermal scheduling, Expert Syst. Appl. 39(3) (2012) 2288–2295.

41] X. Fu, A. Li, L. Wang, C. Ji, Short-term scheduling of cascade reservoirs using animmune algorithm-based particle swarm optimization, Comput. Math. Appl.62 (6) (2011) 2463–2471.

[

[

omputing 24 (2014) 962–976

42] A. Mahor, S. Rangnekar, Short term generation scheduling of cascaded hydroelectric system using novel self adaptive inertia weight PSO, Electr. PowerEnergy Syst. 34 (1) (2012) 1–9.

43] Jingrui Zhang, Jian Wang, Chaoyuan Yue, Small population-based particleswarm optimization for short-term hydrothermal scheduling, IEEE Trans.Power Syst. 27 (February (1)) (2012) 142–152.

44] R.K. Swain, A.K. Barisal, P.K. Hota, R. Chakrabarti, Short-term hydrothermalscheduling using clonal selection algorithm, Electr. Power Energy Syst. 33 (3)(2011) 647–656.

45] A.Y.S. Lam, V.O.K. Li, Chemical-reaction-inspired metaheuristic for optimiza-tion, IEEE Trans. Evol. Comput. 14 (3) (2010) 381–399.

46] A.Y.S. Lam, V.O.K. Li, J.J.Q. Yu, Real-coded chemical reaction optimization, IEEETrans. Evol. Comput. 16 (3) (2012) 339–353.

47] A. Ahmadi, J. Aghaei, H.A. Shayanfar, A. Rabiee, Mixed integer programmingof multiobjective hydro-thermal self scheduling, Appl. Soft Comput. 12 (2012)2137–2146.

48] A. Ahmadi, J. Aghaei, H.A. Shayanfar, A. Rabiee, Mixed integer programming ofgeneralized hydro-thermal self-scheduling of generating units, Electr. Eng. 95(2) (2013) 109–125.

49] H. Moghimi, A. Ahmadi, J. Aghaei, M. Najafi, Risk constrained self-schedulingof hydro/wind units for short term electricity markets considering intermit-tency and uncertainty, Renew. Sustain. Energy Rev. 16 (September (7)) (2012)4734–4743.

50] M. Karami, H. Shayanfar, J. Aghaei, A. Ahmadi, Scenario-based security-constrained hydrothermal coordination with volatile wind power generation,Renew. Sustain. Energy Rev. 28 (2013) 726–737.

51] M. Basu, Improved differential evolution for short-term hydrothermal sched-uling, Electr. Power Energy Syst. 58 (2014) 91–100.

52] P.K. Roy, Teaching learning based optimization for short-term hydrothermalscheduling problem considering valve point effect and prohibited dischargeconstraint, Electr. Power Energy Syst. 53 (2013) 10–19.