This document is posted to help you gain knowledge. Please leave a comment to let me know what you think about it! Share it to your friends and learn new things together.
Transcript
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
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.
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
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
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:
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
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
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.
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
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.
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)
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.
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.
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 ($)
Fig. 11. Convergence characteristics obtained by RCCRO for case 2 of test system 2.
K. Bhattacharjee et al. / Applied Soft Computing 24 (2014) 962–976 973
Table 7Hourly hydro discharge and hydro power generation obtained by RCCRO for test system 3.
Hour Hydro discharges (×105 m3) Hydro power generation (×102 MW)
974 K. Bhattacharjee et al. / Applied Soft Computing 24 (2014) 962–976
0 5 10 15 20 250.6
0.8
1
1.2
1.4
1.6
1.8 x 106
Time (Hr.)
Res
ervo
ir St
orag
e V
olum
e (m
3)
plant1plant2plant3plant4
Fig. 12. Hydro reservoir storage volume for test system 3.
0.5
1
1.5
2
2.5
3 x 105
Wat
er D
isch
arge
(m3)
plant1plant2plant3plant4
pti
4
arRabDo
1.64
1.645
1.65
1.655
1.66
1.665
1.67
1.675
1.68 x 105
Min
imum
Cos
t ($)
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
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 ($)
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.
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.
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.
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
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.
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.
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.