A hybrid of real coded genetic algorithm and artificial fish swarm algorithm for short-term optimal hydrothermal scheduling

Electrical Power and Energy Systems 62 (2014) 617–629

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Electrical Power and Energy Systems journal homepage: www.elsevier.com/locate/ijepes

A hybrid of real coded genetic algorithm and artificial fish swarm algorithm for short-term optimal hydrothermal scheduling Na Fang a,b, Jianzhong Zhou a,⇑, Rui Zhang a, Yi Liu a, Yongchuan Zhang a a b

School of Hydropower and Information Engineering, Huazhong University of Science and Technology, Wuhan 430074, PR China School of Electrical and Electronic Engineering, Hubei University of Technology, Wuhan 430068, PR China

a r t i c l e

i n f o

Article history: Received 2 July 2013 Received in revised form 21 April 2014 Accepted 10 May 2014

Keywords: Short-term hydrothermal scheduling Real coded genetic algorithm Artificial fish swarm algorithm Constraints handling

a b s t r a c t The short-term hydrothermal scheduling (SHS) is a complicated nonlinear optimization problem with a series of hydraulic and electric system constraints. This paper presents a hybrid algorithm for solving SHS problem by combining real coded genetic algorithm and artificial fish swarm algorithm (RCGA–AFSA), which takes advantage of their complementary ability of global and local search for optimal solution. Real coded genetic algorithm (RCGA) is applied as global search, which can explore more promising solution spaces and give a good direction to the global optimal region. Artificial fish swarm algorithm (AFSA) is used as local search to obtain the final optimal solution for improving the exploitation capability of algorithm. The water transport delay between connected reservoirs is taken into account in this paper. Moreover, new coarse and fine adjustment methods without any penalty factors and extra parameters are proposed to deal with all equality and inequality constraints. To verify the feasibility and effectiveness of RCGA–AFSA, the proposed method is tested on two hydrothermal systems. Compared with other methods reported in the literature, the simulation results obtained by hybrid RCGA–AFSA are superior in fuel cost and computation time. Ó 2014 Elsevier Ltd. All rights reserved.

Introduction Short-term hydrothermal scheduling (SHS) is one of the most important issues in the economic operation of power system. The objective of SHS is to determine the optimal amount of water discharges of hydro plants and power generations of thermal plants over a scheduling horizon so as to minimize the total fuel cost of thermal plants while satisfying various hydraulic and electric system constraints. Among these constraints, equality constraints include system load balance, water dynamic balance, and initial and terminal reservoir storage volumes. Inequality constraints are hydrothermal generation limits, ramp rate limits of thermal units, reservoir storage volumes limits, water discharge rate limits and prohibited discharge zones of hydro units. Furthermore, the valve-point effects of thermal units intensify non-linearity and non-convexity of SHS problem [1,2]. Therefore, SHS is a large-scale, dynamic, non-linear, non-convex and complicated constrained optimization problem. Various methods have been proposed to solve SHS problem in the past several decades. The major methods include dynamic programming (DP) [3,4], linear programming (LP) [5], network flow ⇑ Corresponding author. Tel.: +86 02 78 7543 127. E-mail address: [email protected] (J. Zhou). http://dx.doi.org/10.1016/j.ijepes.2014.05.017 0142-0615/Ó 2014 Elsevier Ltd. All rights reserved.

programming (NFP) [6,7], mixed-integer programming (MIP) [8,9], non-linear programming (NLP) [10] and Lagrangian relaxation (LR) [11]. These methods are not able to give optimal solution due to their drawbacks. DP can overcome the difficulty of nonlinearity and non-convexity of SHS problem. However, it suffers from the curse of dimensionality which leads to long computation time and large memory storage. LP is applicable only to problems with linear objective function and constraints. SHS is solved by linear approximation, which would lead to errors of scheduling result. Although NFP is more efficient than LP in terms of computation time and space resources, the network flow model of SHS is often simplified to a linear or piecewise linear one. For MIP, poor calculating efficiency is widely recognized especially when applied to large-scale optimization problem such as SHS problem. NLP can accurately express the characteristics of SHS problem, while it also has some weaknesses of slow convergence, large memory requirement and inability to deal with constraints. Compared with other methods, LR is more flexible for handling different constraints. However, LR leads to the oscillation of solutions, and the convergence and accuracy of LR depend on the Lagrange multipliers updating methods. In recent years, heuristic methods such as artificial neural network (ANN) [12,13], genetic algorithm (GA) [1], simulated annealing (SA) [14], tabu search (TS) [15], ant colony optimization (ACO)

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[16], particle swarm optimization (PSO) [17–22], differential evolution (DE) [23], quantum-inspired evolutionary algorithm (QEA) [24,25] and artificial bee colony (ABC) [26,27] have been employed for solving SHS problem. Moreover, some combinatorial methods such as fuzzy satisfying method based on evolutionary programming technique [28] and hybrid differential evolution and sequential quadratic programming (DE–SQP) algorithms [29] have been successfully applied to this problem. These methods are able to provide good solution and deal with complicated nonlinear constraints more simply and effectively. However, the above mentioned methods require a large amount of computation time especially for large-scale SHS problems. Besides, they are inclined to trap into the local optimum in the later evolution period and sensitive to initial points. In order to overcome the drawbacks of heuristic optimization methods, a hybrid method that combines real coded genetic algorithm (RCGA) with artificial fish swarm algorithm (AFSA) is proposed to accelerate convergence and enhance the performance of searching global optimal solution. RCGA is one of the most popular stochastic search algorithms. It is very suitable for solving continuous optimization problems because of its real-number representation. RCGA is able to promote the calculation efficiency and improve the hill-climbing ability of binary coded genetic algorithm (BCGA). It has been widely and successfully applied to many optimization problems [30–32]. However, the basic RCGA also has disadvantages such as premature convergence and poor local search ability, because it cannot exploit local information of individual in the population. AFSA is a population-based intelligent algorithm, which was inspired by the various social behaviors of fish. Each fish search its own local optimum and pass on information in its selforganized system and finally obtain the global optimum [33]. AFSA can enhance the searching ability and avoid being trapped into local optimum. It has been proved effective in many engineering problems [33–35]. The combination of RCGA and AFSA make full use of their advantages. RCGA is capable of exploring new and more promising solution spaces and give a good direction to the global optimal region; AFSA is able to fine tune the solution to reach the global optimal solution. Thus, this hybrid algorithm has good global exploration capability of RCGA, as well as the local exploitation capability of AFSA. It can obtain better solution with faster convergence speed. What’s more, the constraints handling methods, which require neither penalty factors nor any extra parameters, are provided to deal with complicated constraints of SHS problem. Hybrid RCGA–AFSA is applied to two test systems for solving SHS problem. The simulation results demonstrate the feasibility and effectiveness of the proposed method. The rest of paper is organized as follow. Section ‘Problem formulation’ provides the mathematical formulation of SHS problem. The proposed algorithm is introduced in Section ‘A hybrid algorithm combining RCGA with AFSA’. Section ‘Application of the proposed algorithm to solve SHS problem’ describes the application of hybrid RCGA–AFSA to solve SHS problem. Simulation results are presented and analyzed in Section ‘Simulation results’. In the last section, conclusions and future research are given.

Objective function The fuel cost function of thermal plant considering valve-point effects is expressed as the sum of a quadratic and a sinusoidal function. The superimposed sine component represents the rippling effects produced by the opening of each steam admission valve in a turbine [36]. For a given hydrothermal system, the total fuel cost function can be described as follows

min F ¼

Ns T X X fi ðPs ði; tÞÞ t¼1 i¼1

N s h T X  X 2 asi þ bsi Ps ði; tÞ þ csi ðPs ði; tÞÞ ¼

   i   min þdsi sin esi Ps ðiÞ  Ps ði; tÞ  where fi(Ps(i, t)) is fuel cost of the ith thermal plant at time interval t, Ps(i, t) is power generation of the ith thermal plant at time interval t, asi, bsi and csi are cost coefficients of the ith thermal plant, dsi and esi are valve-point effects coefficients of the ith thermal unit, Ps(i)min, Ps(i)max are the lower and upper generation limits of the ith thermal plant, Ns is the number of thermal plants, i is thermal plant index, T is the number of total intervals over a scheduling horizon, t is time interval index. Constraints (1) System power balance Nh Ns X X P s ði; tÞ þ Phðj;tÞ ¼ PDðtÞ þ PLðtÞ i¼1

Since the operation cost of hydropower is almost negligible, the SHS problem is aimed to minimize the thermal cost when making full use of hydro resources as much as possible. Typically, the total scheduling period is 1 day and each scheduling time interval is 1 h. The objective function and associated constraints of SHS problem are formulated as follows.

ð2Þ

j¼1

where Ph(j, t) is power generation of the jth hydro plant at time interval t, Nh and j are the number of hydro plants and hydro plant index, PD(t) is load demand at time interval t and PL(t) is transmission loss at the corresponding time. The power generated from a hydro plant is related to the reservoir characteristics as well as the water discharge rate. In general, hydropower generation is a function of net head and turbine discharge. The model can be written in terms of reservoir volume instead of the reservoir net head, and a frequently used expression [37] is 2

2

Ph ðj; tÞ ¼ C 1j ðV h ðj; tÞÞ þ C 2j ðQ h ðj; tÞÞ þ C 3j V h ðj; tÞQ h ðj; tÞ þ C 4j V h ðj; tÞ þ C 5j Q h ðj; tÞ þ C 6j

ð3Þ

where C1j, C2j, C3j, C4j, C5j, and C6j are generation coefficients of the jth hydro plant, Vh(j, t) is storage volume of the jth reservoir at the end of time interval t, Qh(j, t) is water discharge rate of the jth reservoir at time interval t. (2) Generation limits

P s ðiÞ

min

6 Ps ði; tÞ 6 Ps ðiÞ

min

6 Ph ðj; tÞ 6 Ph ðjÞ

P h ðjÞ

min

Problem formulation

ð1Þ

t¼1 i¼1

max

max

ð4Þ ð5Þ

max

where Ph(j) and Ph(j) are the minimum and maximum generation of the jth hydro plant, respectively. (3) Ramp rate limits

P s ði; tÞ  P s ði; t  1Þ 6 URs ðiÞ

ð6Þ

P s ði; t  1Þ  Ps ði; tÞ 6 DRs ðiÞ

ð7Þ

where URs(i) and DRs(i) are the ramp-up and ramp-down rate limit of the ith thermal unit, respectively.

N. Fang et al. / Electrical Power and Energy Systems 62 (2014) 617–629

(4) Water discharge rate limits 8

min LB;1 > 6 Q h ðj; tÞ 6 Q h ðjÞ > < Q h ðjÞ UB;m1 LB;m Q h ðjÞ 6 Q h ðj; tÞ 6 Q h ðjÞ > > : UB;n max Q h ðjÞ j 6 Q h ðj; tÞ 6 Q h ðjÞ

m ¼ 2; . . . ; nj

ð8Þ

where Qh(j)min and Qh(j)max are the minimum and maximum water discharge rate of the jth hydro plant, nj is the number of prohibited discharge zones of the jth hydro unit, Qh(j)LB,m and Qh(j)UB,m are the lower and upper limits of the mth prohibited discharge zones of the jth hydro unit, respectively. (5) Reservoir storage volumes limits

V h ðjÞ

min

max

6 V h ðj; tÞ 6 V h ðjÞ

min

ð9Þ

max

where Vh(j) and Vh(j) are the minimum and maximum storage volume of the jth reservoir, respectively. (6) Water dynamic balance

V h ðj; tÞ ¼ V h ðj; t  1Þ þ Ih ðj; tÞ  Q h ðj; tÞ  Sh ðj; tÞ þ

Ruj X ½Q h ðk; t  skj Þ þ Sh ðk; t  skj Þ

ð10Þ

k¼1

where Ih(j, t) is the natural inflow rate of the jth reservoir at time interval t, Sh(j, t) is the spillage of the jth reservoir at time interval t, Ruj is the number of upstream hydropower plants directly above the jth hydro plant, skj is water transport delay from reservoir k to j. (7) Initial and terminal reservoir storage volumes

V h ðj; tÞjt¼0 ¼ V h ðjÞ

begin

ð11Þ

end

ð12Þ

V h ðj; tÞjt¼T ¼ V h ðjÞ begin

where Vh(j) is the initial storage volume of the jth reservoir at the beginning of scheduling horizon, Vh(j)end is the final storage volume of the jth reservoir at the end of scheduling horizon.

paper, RCGA and AFSA are combined to solve SHS problem. This hybrid algorithm employs RCGA to obtain sub-optimal solutions, and then uses AFSA to fine tune the sub-optimal solutions to reach the global optimum. So, the proposed RCGA–AFSA has not only the strong global search ability of RCGA, but the fast local search ability of AFSA. Real coded genetic algorithm Real coded genetic algorithm (RCGA) is expected to be global optimizer to encourage global exploration of the optimal solution. Elitist strategy reserves the best individuals and ensures the convergence of algorithm. Crossover operator combines the features of two or more individuals and creates potentially better offspring. Mutation operator randomly changes some genes of individual to increase the diversity of population. The evolution process of RCGA is discussed in detail in this section. Elitist strategy Elitist strategy is to retain the best parent individuals directly to the next generation, which can balance the disruptive nature of crossover and mutation and improve the chance of finding global optimal solution. In reproduction process, new individuals are created through crossover and mutation. Crossover Crossover is to make chromosomes cross mutually. The simulated binary crossover (SBX) [45,46] uses a probability distribution around two parents to create two children individuals. The proce(1) (2) (1) dure of generating two children Y(1) = (y(1) = 1 , y2 , . . . , yn ) and Y (2) (1) (2) (1) (1) (y(2) , y , . . . , y ) from the parent individuals X = (x , 1 2 n 1 x2 , (2) (2) . . . , xn(1)) and X(2) = (x(2) 1 , x2 , . . . , xn ) are described as follows. First, a uniform random number u between 0 and 1 is created. Then, a parameter c is generated which follows the polynomial probability distribution.

(

A hybrid algorithm combining RCGA with AFSA Genetic algorithm (GA) is a stochastic search algorithm based on the principle of natural selection and genetics. In GA, a population of potential solutions, termed as chromosomes or individuals, is evolved over successive generations using a set of genetic operators called selection, crossover and mutation [38]. The traditional GA is also called binary coded genetic algorithm (BCGA) because the chromosomes are represented as binary strings. Although BCGA has a robust evolution capability, it suffers from loss of precision and premature convergence especially for high-dimensional or high-precision problem. For real valued optimization problems, floating point representations are superior to binary representations because they are more consistent, more precise and lead to faster convergence [39]. The same conclusion was also reached by other researchers [40–42]. Hence, real coded genetic algorithm (RCGA) is applied to solve SHS problem. In practical engineering applications, problems are complex and search spaces are continuous not discrete, RCGA is more suitable and convenient than BCGA. Kumar and Naresh [1] indicated that RCGA has outperformed BCGA in terms of solution quality and execution time for the continuous and non-convex hydrothermal scheduling problem. As with most heuristics stochastic search algorithms, it is very difficult for the pure RCGA to effectively explore the solution space. Instead of providing the global optimal solution, RCGA only may offer a sub-optimal solution because of the rapid decline of population diversity during search process. A combination of a global search optimization method with a local search optimization method can usually improve the performance of algorithm [43]. Some successful applications of combination methods in optimization problems have been reported in the literatures [29,44]. In this

619

c¼

ðauÞ1=ðgcþ1 Þ  1 1=ðgcþ1 Þ 2au

if u 6 a1 otherwise

ð13Þ

where a ¼ 2  bðgc þ1Þ and b is calculated as follows

b¼1þ

2 ð2Þ xi



ð1Þ xi

min

h   i ð1Þ ð2Þ xi  xli ; xui  xi

ð14Þ

The parameter gc is the distribution index with nonnegative value. A small value of gc allows the creation of children solutions far away from parents and a large value restricts only near parent solutions to be created as children solutions [36]. xli and xui are the lower and upper limits of the ith decision variable, respectively. Here, it is assumed that xi(1) 6 xi(2). The children individuals are given by the equation ð1Þ

¼ 0:5

 i h   ð2Þ ð1Þ ð2Þ ð1Þ   cxi  xi  xi þ x i

ð15Þ

ð2Þ

¼ 0:5

i  h   ð2Þ ð1Þ ð2Þ ð1Þ  xi þ x i þ cxi  xi 

ð16Þ

yi yi

Mutation Mutation is to randomly change some genes of individual. Michalewicz’s Non-Uniform Mutation (NUM) is one of the widely used mutation operators in RCGA [39,47]. One parent individual is represented as X = (x1, x2, . . . , xn) that is taken from the population. When a vector xi of the parent X is selected to be mutated, 0 its new vector xi after mutation is

x0i ¼



xi þ Dðt; xui  xi Þ ðs ¼ 0Þ xi  Dðt; xi  xli Þ ðs ¼ 1Þ

ð17Þ

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N. Fang et al. / Electrical Power and Energy Systems 62 (2014) 617–629

where s is a Boolean value, t is the current iteration number. The function D(t, y) is given as follows

  b Dðt; yÞ ¼ y 1  r ð1t=tmax Þ

ð18Þ

where r is a uniformly distributed random number in [0, 1], tmax is the maximum iteration number. Artificial fish swarm algorithm Artificial fish swarm algorithm (AFSA) is used as a local optimizer to improve the exploitation capability in the search space. In nature, the fish can find the more nutritious area by individual search or following after other fish, the area with much more fish is generally most nutritious. AFSA is an efficient, random and parallel search method, which was initially introduced by Li [48] in 2002. This method uses the local search of fish individuals to reach the global optimal solution by simulating the fish behaviors such as preying, swarming, and following. The position of each artificial fish is a potential solution, the current position of artificial fish i can be represented as Xi = (xi1, xi2, . . . , xin). The food consistence of artificial fish i in the current position can be indicated as Yi = f(Xi), where Yi is the objective 0 function. The new position Xi can be formulated as:

X 0i ¼ X i þ RandðÞ  step 

Xj  Xi kX j  X i k

ð19Þ

where Rand() is a random number in the interval [0, 1], step is the moving step length, Xj is the position within vision scope. Here, Xj is defined by different behaviors as below. (1) Following behavior: The artificial fish may trail the neighboring partner, who has reached the most food consistence position, in its vision scope to find more food. Xj is the position of partner with the most food consistence in visual distance. (2) Swarming behavior: The artificial fish may move to the center of assembled group, which is a kind of living habits in order to ensure the existence of colony and avoid dangers. Here, Xj is the center of gathering area. (3) Preying behavior: The artificial fish perceives the consistence of food to determine the movement by vision or sense. When artificial fish discovers an area with more foods, it will go directly toward the area and Xj is given as

X j ¼ X i þ ð2RandðÞ  1Þ  v isual

ð20Þ

where visual represents the visual distance. Combination of RCGA and AFSA This hybrid algorithm makes full use of their advantages of RCGA and AFSA. First, RCGA is applied to SHS problem as global search. When the termination criteria of RCGA is reached, AFSA uses the final elitist population from RCGA as initial population and performs local search by simulating and performing artificial fish behaviors to obtain the final optimal solution. The proposed RCGA–AFSA can be summarized as follows, and the flowchart is illustrated in Fig. 1. Step 1: Initialize population in the feasible region, set the iteration number g = 0. Step 2: Check terminal condition of RCGA: If g = gmax_RCGA, then go to Step8, where gmax_RCGA is the maximum iteration number of RCGA; otherwise go to Step 3.

Step 3: Calculate the fitness of each individual in the current population and sort all individuals according to their fitness value. Step 4: Elitist strategy: Copy the best Pe individuals from the current population to the next generation, where Pe is the number of elite individuals. The rest of the new population is created by crossover and mutation. Step 5: Crossover operation: Randomly select Pc from the rest population (NP - Pe) and the opponents from the elitist population respectively. Recombine each pair parents using SBX operator and select the better offspring in next generation. Step 6: Mutation operation: Randomly select Pm parents from the rest population (NP - Pe), apply NUM operator and generate the equal number of offspring. Step 7: Set g = g + 1, and go to Step 2. Step 8: Set parameters of AFSA, select elitist population of RCGA as the initial population of AFSA, update the bulletin board with the best individual in the elitist population and set g = 0. Step 9: Check terminal condition of AFSA: If g = gmax_AFSA, then terminate the procedure, and output the result of bulletin board, where gmax_AFSA is the maximum iteration number of AFSA; Otherwise go to Step 10. Step 10: Select behavior: Each artificial fish try to find better food consistence position by simulating swarming behavior and following behavior respectively. If success, choose the better one to perform; otherwise, perform preying behavior. Step 11: Update the bulletin board: Compare the food consistence between each artificial fish and bulletin board. If the food consistence of artificial fish is superior to that of the bulletin board, then update the bulletin board. Step 12: Set g = g + 1, and go to Step 9. Application of the proposed algorithm to solve SHS problem Initialization For SHS problem, the initial population of NP individuals is randomly generated and covers the entire search space uniformly. Each individual consists of a set of decision variables which represent the water discharge rate of each hydro plant and power generated by each thermal plant over a scheduling horizon. Thus an individual X can be expressed as: 2 3 Q h ð1;1Þ Q h ð2; 1Þ  Q h ðNh ; 1Þ P s ð1;1Þ P s ð2;1Þ  P s ðNs ; 1Þ 6 Q h ð1;2Þ Q h ð2; 2Þ  Q h ðNh ; 2Þ P s ð1;2Þ P s ð2;2Þ  P s ðNs ; 2Þ 7 6 7 X¼6 7 .. .. .. .. .. .. 4 5 . .  . . .  . Q h ð1;TÞ Q h ð2;TÞ  Q h ðNh ;TÞ P s ð1;1Þ P s ð2;1Þ  P s ðN s ;TÞ ð21Þ

Each element of an individual is randomly generated within the feasible range as follows: min

Q h ðj; tÞ ¼ Q h ðjÞ

min

Ps ði; tÞ ¼ Ps ðiÞ

  max min þ Uð0; 1Þ  Q h ðjÞ  Q h ðjÞ

  max min þ Uð0; 1Þ  Ps ðiÞ  P s ðiÞ

ð22Þ

ð23Þ

where U(0, 1) is a uniform distributed random number in [0, 1]. Generally, the newly generated individuals do not satisfy all the constraints and need to be modified by the constraints handling methods which will be discussed in details in Section ‘Constraints handling’.

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N. Fang et al. / Electrical Power and Energy Systems 62 (2014) 617–629

Fig. 1. The flowchart of the proposed RCGA–AFSA.

Considering the feasible boundaries, the handling method for reservoir storage volumes limits (9) is as follows:

Constraints handling SHS problem has a series of equality and inequality constraints, and how to handle these constraints is a key step for solving this problem effectively. At present, penalty function methods have been the most popular approach for handling constraints because of their simplicity and ease of implementation. However, penalty function methods have several drawbacks. Among those drawbacks, the main one is to find appropriate penalty parameters that can guide the search towards the constrained optimum. In contrast to the penalty function methods, the new constraints handling methods do not require any penalty factors and extra parameters and are more suitable for solving problem with complicated constraints. Constraints handling for hydro plants Water discharge rate limits (8) and reservoir storage volumes limits (9) are inequality constraints. For water discharge rate limits (8), the handling method considering prohibited discharge zones is carried out as follows:

Q h ðj; tÞ ¼

8 min > Q h ðjÞ > > > max > Q h ðjÞ > > > > LB;m > > < Q h ðjÞ > > UB;m > > > > Q h ðjÞ > > > > > : Q h ðj; tÞ

min

if Q h ðj; tÞ < Q h ðjÞ max if Q h ðj; tÞ > Q h ðjÞ LB;m

if Q h ðjÞ

ðQ h ðj; tÞ  Q h ðjÞ LB;m

if Q h ðjÞ

UB;m

and

UB;m

 Q h ðj; tÞÞ

UB;m

and

UB;m

 Q h ðj; tÞÞ

< Q h ðj; tÞ < Q h ðjÞ LB;m

Þ < ðQ h ðjÞ

< Q h ðj; tÞ < Q h ðjÞ

ðQ h ðj; tÞ  Q h ðjÞ other

LB;m

Þ > ðQ h ðjÞ

ð24Þ

8 min > < V h ðjÞ V h ðj; tÞ ¼ V h ðj; tÞ > : max V h ðjÞ

if V h ðj; tÞ < V h ðjÞ if V h ðjÞ

min

min

6 V h ðj; tÞ 6 V h ðjÞ

if V h ðj; tÞ > V h ðjÞ

max

ð25Þ

max

Water dynamic balance (10), initial reservoir storage volume (11) and terminal reservoir storage volume (12) are equality constraints. Compared with inequality constraints, equality constraints are more difficult to be handled. The water spillages are neglected for simplicity, the details of implementation are shown with pseudo codes in Fig. 2. DVh(j) is the storage volume violation of the jth hydro plant, DQh(j, t) is the adjustable water discharge range of the jth hydro plant at time interval t.eVcoarse and eVfine are the precision of storage volume violation for coarse and fine adjustment. Iterationcoarse and Iterationfine are the maximum number of iterations for coarse and fine adjustment respectively, and k is the scheduling interval index with the maximum adjustable discharge. Constraints handling for thermal plants For thermal generation limits (4) and ramp rate limits (6, 7), the handling method can be expressed as follows:

8 min > < Ps ði; tÞ Ps ði; tÞ ¼ Ps ði; tÞ > : max Ps ði; tÞ

min

if Ps ði; tÞ < Ps ði; tÞ if Ps ði; tÞ

min

6 Ps ði; tÞ 6 P s ði; tÞ max

if Ps ði; tÞ > Ps ði; tÞ

max

ð26Þ

622

N. Fang et al. / Electrical Power and Energy Systems 62 (2014) 617–629

Fig. 2. Pseudo codes of equality constraints handling for hydro plants.

where Ps(i,t)min and Ps(i,t)max are the minimum and maximum power generation of the ith thermal plant at time interval t. The expressions for Ps(i,t)min and Ps(i,t)max are given by min

n o min ¼ max Ps ðiÞ ; Ps ði; t  1Þ  DRs ðiÞ

max

 max ¼ min Ps ðiÞ ; Ps ði; t  1Þ þ URs ðiÞ

Ps ði; tÞ Ps ði; tÞ

Fig. 3. Pseudo codes of equality constraints handling for thermal plants.

ð27Þ ð28Þ

To keep system power balance (2), the power generations of thermal plants are modified by coarse and fine adjustment. Meanwhile, the water discharge rate, reservoir storage volume and hydro generation are not changed, which do not influence the previous constraints handling for hydro plants. The pseudo codes of handling constraints (2) are described in Fig. 3. DP(t) is the power generation violation at time interval t, ePcoarse and ePfine are the precision of generation violation for coarse and fine adjustment respectively, and r is the thermal plant index with the maximum adjustable generation. Application of RCGA–AFSA to SHS problem Application of the proposed RCGA–AFSA to solve SHS problem can be described in the following steps: Step 1: Initialize the individuals of the population according to (22) and (23), and set the iteration number g = 0. Considering that the new generated individuals may violate the constraints, constraints handling methods, as described in Section ‘Constraints handling’, should be implemented to make each individual satisfy all constraints. Step 2: Evaluate all individuals based on the objective function (1) and perform elitist strategy.

Step 3: Perform crossover and mutation operation as introduced in Section ‘Real coded genetic algorithm’, and modify the offspring in the new population by means of constraints handing methods. Step 4: If g = gmax_RCGA, then go to Step 5; otherwise, update the iteration number as g = g + 1 and go back to Step 2. Step 5: The elitist population obtained by RCGA is taken as an initial population of AFSA, initialize the bulletin board with the best individual in the elitist population and set g = 0. Step 6: Select and perform one of the artificial fish behaviors as given in Section ‘Artificial fish swarm algorithm’, ensure that each individual is within its limits, and update the bulletin board. Step 7: If g = gmax_AFSA, then output the result on the bulletin board as the final optimal solution of SHS problem; otherwise, increase the iteration number by g = g + 1 and go back to Step 6.

Simulation results To verify the feasibility and effectiveness of RCGA–AFSA, two hydrothermal test systems are introduced. The proposed method is coded in Visual C++ 6.0 and executed on a PC (Pentium-IV, 2.0 GHz, 2 GB RAM). The scheduling period is 1 day with 24 intervals of 1 h each. The hybrid RCGA–AFSA and RCGA are used to solve SHS problem. The parameters of each algorithm are given in the Table 1.

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N. Fang et al. / Electrical Power and Energy Systems 62 (2014) 617–629 Table 1 Parameters setting for RCGA and RCGA–AFSA. Method

NP

Pe

Pc

Pm

step

visual

eVcoarse

eVfine

ePcoarse

ePcoarse

Iterationcoarse

Iterationfine

RCGA RCGA–AFSA

100 100

30 30

30 30

40 40

– 0.5

– 1.0

1 1

0.01 0.01

2 2

0.01 0.01

20 20

10 10

Table 2 Comparison of simulation results obtained by different methods for case 1 of test system 1. Method

Minimum cost ($)

Average cost ($)

Maximum cost ($)

Average CPU time (s)

IFEP [49] EPSO [50] IPSO [19] MAPSO [2] MDE [51] TLBO [52] RCGA RCGA–AFSA

930129.82 922904 922553.49 922421 922555.44 922373.39 923966.285 922339.625

930290.13 923527 – 922544 – 922462.24 924108.731 922346.323

930881.92 924808 – 923508 – 922873.81 924232.072 922362.532

1033.20 – 38.46 – 45 – 17 11

Table 3 Hourly water discharge (104 m3) obtained by RCGA–AFSA for case 1 of test system 1. Hour

Qh(1, t)

Qh(2, t)

Qh(3, t)

Qh(4, t)

Hour

Qh(1, t)

Qh(2, t)

Qh(3, t)

Qh(4, t)

1 2 3 4 5 6 7 8 9 10 11 12

9.9662 10.3606 9.0423 8.4943 8.2006 8.1249 8.2087 8.4758 8.6517 8.7412 8.6206 8.6424

7.3727 6.1291 6.0066 6.0073 6.0507 6.2520 6.7027 7.2551 7.8284 8.0758 8.1391 8.3962

29.9997 29.9998 29.9998 29.9997 18.1716 18.1311 16.9315 15.9157 15.0679 15.3308 15.6147 15.9508

13.0000 13.0000 13.0000 13.0000 13.0000 13.0000 13.0000 13.0000 13.0001 13.0000 13.0000 13.2629

13 14 15 16 17 18 19 20 21 22 23 24

8.5524 8.4315 8.2629 8.0517 7.9846 7.7348 7.6985 7.6144 7.5732 7.4881 5.0780 5.0006

8.4599 8.6576 8.7890 8.9658 9.3201 9.6517 10.2591 11.0455 11.7427 9.5570 10.2863 11.0500

16.4085 16.5983 17.0362 17.1101 16.7070 15.4775 14.4136 13.4434 10.0034 10.0041 10.0046 10.0084

14.7559 15.3301 15.6149 15.9504 16.4084 16.6013 17.0361 17.1101 17.9932 19.3579 20.4967 22.1897

Table 4 Hourly hydrothermal generation (MW) obtained by RCGA–AFSA for case1 of test system 1. Hour

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

Hydro generation Ph(1, t)

Ph(2, t)

Ph(3, t)

Ph(4, t)

85.8562 87.1046 80.4008 76.8428 74.3530 73.4972 73.9166 75.6546 77.1403 78.4466 78.8915 79.4414 79.6371 79.8591 79.4510 78.4917 78.2430 76.5936 76.2163 75.3281 74.9289 74.4331 55.4238 55.0255

58.5622 51.3655 52.1669 53.7703 55.0874 56.9095 59.6945 63.2452 66.9648 68.9638 69.7986 71.1112 71.2363 72.5543 73.4088 73.8657 74.4678 74.0145 74.8856 76.2703 77.0353 67.3715 69.1805 70.1467

0.0000 0.0000 0.0000 0.0000 25.3219 25.6624 30.1260 33.2346 35.4262 35.2196 35.1194 35.4194 36.0319 36.8396 36.7504 37.2578 39.3532 43.6533 46.7361 49.2696 50.5555 52.7488 54.5855 56.0758

200.0937 187.7553 173.7332 156.7916 178.7417 198.9579 217.4401 234.1883 238.9401 243.4947 246.8781 251.7822 266.4279 271.5927 274.0777 276.9413 280.7377 282.2949 285.7270 286.2993 291.6205 296.8066 296.8544 294.6941

Thermal generation

Total generation

1025.4879 1063.7747 1053.6990 1002.5953 956.4960 1054.9730 1268.8228 1593.6772 1821.5286 1893.8752 1799.3125 1872.2459 1776.6669 1739.1543 1666.3121 1603.4436 1657.1982 1663.4437 1756.4351 1792.8326 1745.8597 1628.6399 1373.9558 1114.0579

1370 1390 1360 1290 1290 1410 1650 2000 2240 2320 2230 2310 2230 2200 2130 2070 2130 2140 2240 2280 2240 2120 1850 1590

624

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Test system 1 This test system consists of four cascaded hydro plants and an equivalent thermal plant. The detail data of the test system can be found in [37]. The additional data of valve-point effects and prohibited discharge zones are given in [49]. In test system 1, the decision variables are water discharge rate of each hydro plant at each time interval. Transmission losses and ramp rate limits are neglected. The fuel cost function of the equivalent thermal plant with valve-point effects is expressed as follows: 2

fi ðPs ði; tÞÞ ¼ 5000 þ 19:2P s ði; tÞ þ 0:002ðPs ði; tÞÞ      min þ 700 sin 0:085ðPs ðiÞ  Ps ði; tÞÞ  Fig. 4. Hourly reservoir storage volumes obtained by RCGA–AFSA for case 1 of test system 1.

ð29Þ

The lower and upper generation limits of thermal plant are 500 MW and 2500 MW. The maximum iteration number of RCGA

Table 5 Comparison of simulation results obtained by different methods for case 2 of test system 1. Method

Minimum cost ($)

Average cost ($)

Maximum cost ($)

Average CPU time (s)

IFEP [49] IPSO [19] MDE [51] RCGA RCGA–AFSA

933949.25 925978.84 925960.56 930565.242 927899.872

938508.87 – – 930966.356 927963.764

942593.02 – – 931427.212 928025.343

1450.90 31.11 27 20 13

Table 6 Hourly water discharge (104 m3) obtained by RCGA–AFSA for case 2 of test system 1. Hour

Qh(1, t)

Qh(2, t)

Qh(3, t)

Qh(4, t)

Hour

Qh(1, t)

Qh(2, t)

Qh(3, t)

Qh(4, t)

1 2 3 4 5 6 7 8 9 10 11 12

9.6633 9.3225 9.4662 10.2688 7.4793 6.8055 7.4192 6.6542 9.8827 7.3710 9.7296 10.3012

9.1602 8.6851 9.0846 9.2172 8.6639 8.1280 6.3974 6.1676 8.1969 8.0194 8.7835 9.6643

27.6261 27.4647 21.5761 27.2872 28.1040 27.9187 20.6637 19.0597 16.7477 14.0518 16.9690 17.4264

13.0503 13.0803 13.0271 13.0286 13.0212 13.0190 13.0212 13.0613 13.0161 13.0247 18.0288 18.1215

13 14 15 16 17 18 19 20 21 22 23 24

9.6691 7.7200 9.4922 7.6351 7.3613 7.2272 7.0613 7.2337 7.0363 6.7757 6.6490 6.7757

8.6055 8.6468 8.8074 9.2509 6.4497 8.1029 8.0739 8.9263 8.9726 9.0056 8.1496 8.8410

16.4398 15.7821 14.8772 14.3231 14.0937 13.4325 11.0920 10.2719 11.1699 10.4191 10.8205 12.2625

17.8113 14.8818 18.0000 15.8196 15.8932 15.8985 14.7051 14.4526 14.0329 18.0118 18.0003 18.0002

Table 7 Hourly hydrothermal generation (MW) obtained by RCGA–AFSA for case 2 of test system 1. Hour

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

Hydro generation Ph(1, t)

Ph(2, t)

Ph(3, t)

Ph(4, t)

84.5211 82.6956 82.9163 85.3751 69.8114 65.2812 69.6744 65.0686 84.3620 71.1895 85.7231 88.2911 85.7545 75.4479 86.5195 75.6589 74.0065 73.1513 71.9089 72.9915 71.5242 69.7255 69.0397 70.3873

67.6965 64.7248 66.8032 67.3481 63.9822 60.2387 49.1146 48.0562 60.7896 60.3975 64.8317 68.2306 62.3683 62.8291 63.8240 65.2912 49.0940 58.0360 57.1375 61.1566 61.4166 61.5830 56.8411 60.0233

0.0000 0.0000 19.7438 0.0000 0.0000 0.0000 15.5382 22.7214 31.4348 37.6592 30.7676 29.2846 34.8238 39.1941 43.9393 46.2251 48.5472 50.7014 52.2549 52.3659 54.7304 55.1847 56.8321 58.7778

200.4838 188.2886 173.7587 156.7685 175.6979 193.2001 203.0049 218.6806 233.1577 246.6136 292.2162 293.7852 290.5328 265.8986 290.0693 274.6336 275.7561 275.6947 265.3344 262.8817 258.9617 287.7355 280.6158 272.4013

Thermal generation

Total generation

1017.2987 1054.2911 1016.7780 980.5082 980.5085 1091.2799 1312.6679 1645.4732 1830.2559 1904.1401 1756.4615 1830.4085 1756.5207 1756.6302 1645.6479 1608.1911 1682.5962 1682.4167 1793.3643 1830.6044 1793.3671 1645.7713 1386.6713 1128.4103

1370 1390 1360 1290 1290 1410 1650 2000 2240 2320 2230 2310 2230 2200 2130 2070 2130 2140 2240 2280 2240 2120 1850 1590

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tion of AFSA, which can speeds up convergence and enhances search ability. In addition, the minimum, average and maximum cost obtained by RCGA–AFSA are very close to each other, which indicates the proposed method has good robustness for solving SHS problem. The optimal hourly water discharge and hydrothermal generation obtained by RCGA–AFSA are given in Tables 3 and 4 respectively. The optimal reservoir storage volumes of four hydro plants are depicted in Fig. 4. The scheduling results satisfy all kinds of constraints. It demonstrates that the proposed constraints handling methods are useful to deal with constraints of SHS problem.

Fig. 5. Hourly reservoir storage volumes obtained by RCGA–AFSA for case 2 of test system 1.

gmax_RCGA = 600, the maximum iteration number of AFSA gmax_AFSA = 400. All programs were run 30 times from different random initial population respectively. Case 1: Valve-point effects and prohibited discharge zones are not considered In this case, the fuel cost of thermal plant is a quadratic function and the prohibited discharge zones of hydro units are not considered. The minimum, average and maximum fuel cost and average CPU time among 30 runs are shown in Table 2. The minimum fuel cost obtained by the proposed RCGA–AFSA is 922339.625 $, and the average CPU time is 11 s. Compared with the simulation results reported in the literatures [2,19,49–52], RCGA–AFSA outperforms other methods in terms of total fuel cost and computation time. Moreover, it is clear from Table 2 that hybrid RCGA–AFSA realizes the combination of global exploration of RCGA and local exploita-

Case 2: Valve-point effects and prohibited discharge zones are considered In this case, the valve-point effects of thermal unit and prohibited discharge zones of hydro units are considered. The simulation results and average CPU time obtained by different methods are given in Table 5. The minimum fuel cost obtained by RCGA–AFSA is 927899.872 $, and the average execution time is 13 s. Although the proposed method cannot provide the best scheduling result among all methods, the average execution time is the shortest of all. The optimal hourly water discharge obtained by RCGA–AFSA is shown in Table 6, and the optimal hydrothermal generation is given in Table 7. Fig. 5 illustrates the optimal reservoir storage volume of each hydro plant for this case. From Table 6 and 7, we can see that the water discharge and hydrothermal generation are all within the limits. It is obvious that the constraints handling method are effective for SHS problem with valve-point effects and prohibited discharge zones. Test system 2 The test system 2 consists of four cascaded hydro plants and three equivalent thermal plants. The data of this test system are

Table 8 Comparison of simulation results obtained by different methods for case 1 of test system 2. Method

Minimum cost ($)

Average cost ($)

Maximum cost ($)

Average CPU time (s)

Fuzzy EP[28] PSO[18] IQPSO[54] MDE[51] MHDE[53] CSA[55] TLBO[52] QOTLBO[56] RCGA RCGA–AFSA

45063.004 44740 42359.00 42611.14 41856.50 42440.574 42385.88 42187.49 42886.352 40913.828

– – – – – – 42407.23 42193.46 43032.334 41235.726

– – – – – – 42441.36 42202.75 43261.912 41362.575

– 232.73 – 125 31 109.12 – 21.6 30 21

Table 9 Hourly water discharge (104 m3) obtained by RCGA–AFSA for case 1 of test system 2. Hour

Qh(1, t)

Qh(2, t)

Qh(3, t)

Qh(4, t)

Hour

Qh(1, t)

Qh(2, t)

Qh(3, t)

Qh(4, t)

1 2 3 4 5 6 7 8 9 10 11 12

9.7239 10.8554 10.1991 9.1474 10.2073 8.9588 7.4334 9.3055 6.4336 7.9313 8.6288 7.6128

7.7539 8.9633 9.8437 7.0580 7.7048 7.3324 6.4131 6.9693 6.7557 7.6494 9.0166 6.8893

30.0000 18.7567 18.7529 22.3745 29.9985 17.9826 18.4908 16.5125 18.5854 17.8077 16.4681 17.4360

7.5233 7.7615 7.7842 7.5293 8.2910 8.3954 8.4044 11.3184 19.8426 17.9394 18.4474 16.5578

13 14 15 16 17 18 19 20 21 22 23 24

8.4539 8.8459 8.3960 6.0866 5.1731 5.0383 11.0183 9.8099 5.8190 6.9716 7.8968 5.0533

7.4869 8.0432 12.0744 8.5325 6.6898 6.0731 11.3419 12.9391 8.1189 11.7934 9.4353 7.1219

15.4219 14.6210 17.8377 20.9749 14.0292 13.7439 13.0728 12.1238 12.1300 12.7140 12.3459 15.8179

18.5924 17.8497 16.5068 17.4235 15.6585 14.3215 19.8266 19.9866 16.4290 18.9641 18.7610 17.6765

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Table 10 Hourly hydrothermal generation (MW) obtained by RCGA–AFSA for case1 of test system 2. Hour

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

Hydro generation

Thermal generation

Ph(1, t)

Ph(2, t)

Ph(3, t)

Ph(4, t)

Ps(1, t)

Ps(2, t)

Ps(3, t)

Total generation P P Ph(t) + Ps(t)

84.7948 89.0080 85.4566 79.6435 82.7934 76.1656 67.6076 78.0022 61.8750 73.0252 78.3071 72.7861 78.8181 82.0049 79.9760 63.9802 56.4258 55.3598 95.3064 88.6910 61.7551 71.1545 77.9854 55.5196

60.6815 66.9481 70.8374 56.4062 60.6463 58.1365 51.8407 55.5766 54.9136 61.4399 69.2756 57.2262 61.3738 65.3273 82.4354 65.9978 54.8365 50.5397 76.6035 78.3687 58.3861 73.3156 62.3983 50.1198

0.0000 39.1151 37.2166 18.3897 0.0000 38.2717 37.1198 44.2015 36.4949 38.8810 42.1151 39.0056 45.9910 49.0034 41.6368 27.2191 50.4233 52.5626 53.3625 53.5392 55.2238 57.6511 58.9486 56.8433

147.3285 145.0537 139.0883 128.1595 159.2424 170.1124 179.1875 222.8103 304.8904 292.3649 295.9822 281.8322 296.9220 291.6686 281.3397 288.4975 274.0590 262.2675 302.7906 304.7551 277.6002 290.6155 283.3103 270.2258

102.7138 175.0000 102.6675 102.7089 102.6542 102.6789 175.0000 175.0000 102.7122 174.9946 174.9992 175.0000 102.6452 102.6810 175.0000 175.0000 175.0000 175.0000 102.6303 174.9999 102.6533 102.6883 102.6837 102.6619

124.9539 124.9194 124.8899 124.9177 124.9203 125.0285 209.7814 294.6402 209.7576 209.7790 209.8076 294.6658 294.7030 209.8065 209.7770 209.7764 209.7355 294.6880 209.7907 209.7279 124.8895 124.8041 124.9183 124.9182

229.5275 139.9558 139.8438 139.7746 139.7434 229.6064 229.4629 139.7691 319.3565 229.5155 229.5130 229.4841 229.5468 229.5083 139.8353 229.5290 229.5200 229.5825 229.5160 139.9182 229.4920 139.7708 139.7553 139.7114

750 780 700 650 670 800 950 1010 1090 1080 1100 1150 1110 1030 1010 1060 1050 1120 1070 1050 910 860 850 800

the same as in [53]. Besides, the prohibited discharge zones of four hydro units and ramp rate limits of three thermal units are given in [23]. To evaluate the performance of RCGA–AFSA, the proposed method is applied to solve SHS problem based on three different cases. Moreover, the complexity of these cases is increased step by step. The parameters for test system 2 are set as follows: gmax_RCGA = 1000, gmax_AFSA = 500.

Fig. 6. Hourly reservoir storage volumes obtained by RCGA–AFSA for case 1 of test system 2.

Case 1: Valve-point effects is considered In this case, the valve-point effects of thermal units are considered. The comparison of minimum, average and maximum cost and average CPU execution time obtained by RCGA–AFSA and other methods [18,28,51–56] are summarized in Table 8. The minimum cost obtained by RCGA–AFSA is 40913.828 $, and the average CPU execution time is 21 s. It is quite evident that hybrid RCGA–AFSA can obtain better solution with shorter time. This

Table 11 Comparison of simulation results obtained by different methods for case 2 of test system 2. Method

Minimum cost ($)

Average cost ($)

Maximum cost ($)

Average CPU time (s)

MHDE [53] HDE[53] RCGA RCGA–AFSA

42679.87 43656.62 43465.244 41707.964

– – 43643.365 41832.355

– – 43717.273 41894.628

40 68 32 25

Table 12 Hourly water discharge (104 m3) obtained by RCGA–AFSA for case 2 of test system 2. Hour

Qh(1, t)

Qh(2, t)

Qh(3, t)

Qh(4, t)

Hour

Qh(1, t)

Qh(2, t)

Qh(3, t)

Qh(4, t)

1 2 3 4 5 6 7 8 9 10 11 12

8.9693 8.9644 11.1111 8.1305 10.9076 8.8940 8.9676 7.1885 8.5909 8.0883 8.8453 6.6499

6.9692 9.4586 7.8474 6.9794 7.9820 7.8995 9.2246 6.6314 7.0744 7.7201 7.8375 7.2981

21.7041 19.8123 18.8988 20.9185 29.9826 19.3579 19.5217 19.6397 19.5201 18.5536 16.5771 17.4685

7.6068 7.5679 8.5341 7.5366 9.0292 10.0564 8.6584 7.9695 12.5807 18.3717 19.4057 19.6044

13 14 15 16 17 18 19 20 21 22 23 24

8.9575 8.6044 8.8088 6.1444 7.2704 5.6259 6.4406 10.3870 6.5400 7.8023 7.8940 5.2175

7.9917 9.6811 10.8708 7.6717 7.4695 6.6342 7.5327 11.5933 10.8123 11.4569 10.4338 6.9298

14.7400 15.7487 17.3716 16.3243 14.2035 15.9084 14.5984 12.6819 12.9865 13.0172 14.0437 13.7889

19.4859 19.1071 15.9896 18.0193 14.8340 15.7178 18.1272 19.3179 15.6227 19.4667 19.3777 18.3444

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N. Fang et al. / Electrical Power and Energy Systems 62 (2014) 617–629 Table 13 Hourly hydrothermal generation and transmission loss (MW) obtained by RCGA–AFSA for case 2 of test system 2. Hour

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

Hydro generation

Thermal generation

Ph(1,t)

Ph(2,t)

Ph(3,t)

Ph(4,t)

Ps(1,t)

Ps(2,t)

Ps(3,t)

Total generation P P Ph(t) + Ps(t)

81.1471 81.1319 89.7707 74.9252 86.4247 76.7199 76.6992 66.8794 75.9795 74.1175 79.5774 66.3028 81.9174 80.8170 82.5822 64.5111 73.5268 60.5425 67.4862 91.9565 67.8073 77.1754 77.9942 57.0445

56.2185 69.6920 61.5384 57.2959 63.5758 62.5431 67.8992 52.7649 56.2414 61.0664 62.4779 59.5490 63.8256 72.5992 76.8182 60.5580 59.0254 53.2371 58.7007 76.3077 71.9752 72.7100 66.8244 48.9030

32.4567 37.4598 39.0536 28.7583 0.0000 33.7720 33.6135 32.7836 32.6379 36.1614 42.4503 39.7217 48.1994 46.9945 43.5338 47.1235 52.6059 50.9895 53.4260 54.9021 55.3877 56.9775 57.7417 58.8764

148.2860 142.7980 147.2225 127.5546 158.2298 179.6119 173.2731 175.7627 243.8403 295.1116 302.0498 303.3210 302.6215 299.6634 277.0740 292.3134 266.3825 274.2108 292.2794 297.1668 268.6281 292.8689 287.1962 274.6434

175.0000 103.7096 103.3729 102.8115 103.1260 102.8506 175.0000 175.0000 175.0000 102.8104 102.7144 175.0000 102.8003 102.6660 102.6775 175.0000 175.0000 175.0000 175.0000 102.6803 102.6713 102.7362 102.5452 102.6628

126.3960 125.3854 125.3507 124.9587 125.0175 125.2294 209.8358 294.7882 294.7585 209.8323 209.8134 294.7513 209.8452 209.9299 209.8998 294.7300 209.8210 294.5477 209.6834 209.7140 124.9332 124.9180 125.0030 124.9622

140.7079 230.3382 139.9995 139.8531 139.9920 229.7482 229.5343 229.8384 229.4751 319.2500 319.3140 229.5150 319.1828 229.5816 229.5276 139.8538 229.6663 229.3845 229.5726 229.5619 229.4537 139.7263 139.7306 139.6893

760.2122 790.5149 706.3082 656.1573 676.3657 810.4751 965.8552 1027.8173 1107.9327 1098.3496 1118.3972 1168.1608 1128.3923 1042.2516 1022.1131 1074.0898 1066.0278 1137.9120 1086.1482 1062.2893 920.8566 867.1122 857.0352 806.7815

Loss PL(t) 10.2122 10.5149 6.3082 6.1573 6.3657 10.4751 15.8552 17.8173 17.9327 18.3496 18.3972 18.1608 18.3923 12.2516 12.1131 14.0898 16.0278 17.9120 16.1482 12.2893 10.8566 7.1122 7.0352 6.7815

comparison also verifies that the combination of RCGA and AFSA improves the global search ability and convergence speed. To validate the effectiveness of the constraints handling methods, the optimal hourly water discharge and hydrothermal generation obtained by RCGA–AFSA are provided in Tables 9 and 10. The optimal reservoir storage volumes of hydro plants are shown in Fig. 6. From Tables 9 and 10 and Fig. 6, it is obvious that the scheduling results obtained by RCGA–AFSA satisfy all hydraulic and electric system constraints. Thus, it can be easily concluded that the proposed constraints handling methods are able to deal with various constraints of SHS problem with valve-point effects.

Fig. 7. Hourly reservoir storage volumes obtained by RCGA–AFSA for case 2 of test system 2.

Case 2: Valve-point effects and transmission losses are considered In this case, the valve-point effects and transmission losses are considered. The transmission losses are computed using the loss coefficients and can be formulated as follows:

Table 14 Comparison of simulation results obtained by different methods for case 3 of test system 2. Method

Minimum cost ($)

Average cost ($)

Maximum cost ($)

Average CPU time (s)

IDE[23] RCGA RCGA–AFSA

43790.33 43474.261 41818.422

43800.51 43622.126 41906.384

43812.01 43738.632 41962.575

782.23 32 26

Table 15 Hourly water discharge (104 m3) obtained by RCGA–AFSA for case 3 of test system 2. Hour

Qh(1, t)

Qh(2, t)

Qh(3, t)

Qh(4, t)

Hour

Qh(1, t)

Qh(2, t)

Qh(3, t)

Qh(4, t)

1 2 3 4 5 6 7 8 9 10 11 12

8.0000 11.5832 10.9372 10.1427 8.0000 8.0000 9.5950 7.4351 10.7826 7.7883 10.2249 7.3362

8.1647 10.0675 8.4477 8.7425 6.9758 8.2710 6.6144 6.9775 6.9873 9.0910 9.9407 8.6513

30.0000 30.0000 18.0260 30.0000 21.2837 18.4990 19.6252 17.9576 14.6829 18.0934 16.2640 15.9018

9.0294 9.2738 9.1490 9.0825 9.1984 9.4451 9.3250 12.2469 20.0000 18.1320 19.6525 18.0507

13 14 15 16 17 18 19 20 21 22 23 24

6.2990 10.0774 8.0000 9.1032 6.1357 6.4246 5.6806 6.1986 6.8474 7.7748 7.2166 5.4168

6.6766 9.3049 11.5811 6.7542 6.6508 6.9586 6.9636 9.1255 9.6656 11.1366 11.5572 6.6938

16.1753 14.2125 14.5823 15.1521 16.8472 15.0966 14.0364 12.3420 11.9094 12.0407 12.6378 11.9132

14.9666 18.4330 15.9931 15.7664 18.0794 15.0533 18.2178 13.5980 15.5866 19.4833 19.4353 18.3797

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Table 16 Hourly hydrothermal generation and transmission loss (MW) obtained by RCGA–AFSA for case 3 of test system 2. Hour

Hydro generation

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

Ph(1,t)

Ph(2,t)

Ph(3,t)

Ph(4,t)

Ps(1,t)

Ps(2,t)

Ps(3,t)

Total generation P P Ph(t)+ Ps(t)

75.7032 91.9909 88.5239 84.0340 72.2059 71.8289 79.6197 68.3002 84.8646 71.4169 84.8736 69.9416 63.6751 87.6692 76.9654 83.8514 64.0617 66.6327 60.6345 64.9670 70.1278 76.9016 73.2070 58.8649

62.8608 71.4429 63.5170 65.3096 55.4011 62.5242 51.8596 54.2662 54.9560 67.2204 70.8194 63.8348 52.6911 67.9500 76.1726 52.1889 51.7214 53.1394 53.1946 65.0120 67.2489 72.1452 71.0670 47.3774

0.0000 0.0000 35.0202 0.0000 17.3950 32.6214 27.8607 34.2988 44.4995 34.2845 41.0081 42.2282 44.2339 49.7379 50.3393 50.6212 47.8568 53.6098 54.7405 55.6130 56.0435 56.7699 57.8717 58.5587

163.8721 159.5886 150.1304 139.2568 163.0614 185.7051 191.9164 239.8968 305.3456 293.5947 303.6012 292.9005 267.7483 294.9645 276.6197 274.8262 291.0329 266.1214 287.5234 250.7002 270.0200 294.0230 287.9446 274.8687

103.2355 106.2261 103.1500 102.7379 103.0196 103.1444 175.0000 102.6437 174.9241 102.9481 174.9548 175.0000 175.0000 102.4477 102.5727 175.0000 175.0000 174.7955 102.8409 102.8914 102.7048 102.6645 102.4588 102.2570

125.2936 131.2682 125.3511 124.9926 125.1765 125.1656 210.1819 294.4616 211.4684 294.7555 299.5179 294.7581 294.6090 209.8585 209.8446 209.8105 294.5492 294.1803 210.0285 209.6659 125.2041 124.8921 124.9362 124.9455

229.5651 230.3384 140.6367 139.9633 139.9979 229.4811 229.4472 229.5295 230.3214 229.5339 139.6113 229.4312 230.0005 229.5906 229.5515 229.8053 139.8361 229.3693 319.2235 319.1810 229.5183 139.7198 139.5394 139.9036

760.5303 790.8550 706.3292 656.2941 676.2574 810.4706 965.8854 1023.3968 1106.3795 1093.7539 1114.3863 1168.0945 1127.9578 1042.2185 1022.0658 1076.1034 1064.0581 1137.8484 1088.1857 1068.0305 920.8672 867.1162 857.0248 806.7759

NX s þN h NX s þN h

Thermal generation

NX s þN h

Loss PL(t) 10.5303 10.8550 6.3292 6.2941 6.2574 10.4706 15.8854 13.3968 16.3795 13.7539 14.3863 18.0945 17.9578 12.2185 12.0658 16.1034 14.0581 17.8484 18.1857 18.0305 10.8672 7.1162 7.0248 6.7759

ð30Þ

AFSA satisfy all constraints of SHS problem considering valve-point effects and transmission losses.

where B, B0 and B00 are transmission loss coefficients and are given in [53]. The proposed RCGA–AFSA and RCGA are applied to solve SHS problem separately for 30 times. The minimum, average and maximum fuel cost and average CPU time obtained by RCGA–AFSA are compared with those of other methods, as shown in Table 11. The minimum fuel cost is 41707.964 $ and average CPU time is 25 s. It is clearly seen that the proposed method can obtain better quality solution with higher efficiency. Meanwhile, from the results given in Table 11, it is quite evident that hybrid RCGA–AFSA can avoid premature convergence and has better convergence property compared to the pure RCGA. The optimal hourly water discharge of each hydro plant is listed in Table 12, and the optimal hydrothermal generation and transmission losses are shown in Tables 13. Fig. 7 depicts the optimal reservoir storage volumes of hydro plants for this case. The optimal scheduling results obtained by RCGA–

Case 3: Valve-point effects, transmission losses, prohibited discharge zones and ramp rate limits are considered In this case, the valve-point effects and ramp rate limits of thermal units, prohibited discharge zones of hydro units and transmission losses of hydrothermal system are taken into account. For the convenience of comparisons, the scheduling results of RCGA–AFSA and other methods are summarized in Table 14. The minimum cost and average CPU time obtained by RCGA–AFSA are 41818.422 $ and 26 s respectively. It is proved again that RCGA– AFSA is superior to other methods in terms of solution quality and execution time. Table 15 shows the optimal water discharge obtained by RCGA–AFSA, and Table 16 gives the optimal hydrothermal generation and transmission losses. The optimal reservoir storage volume of each hydro plant is illustrated in Fig. 8. The water discharge, reservoir storage volume and hydrothermal generation satisfy equality and inequality constraints of SHS problem. Therefore, the feasibility and effectiveness of the proposed constraints handling methods are successfully verified again.

PL ðtÞ ¼

i¼1

j¼1

Pði; tÞBij Pðj; tÞ þ

B0i Pði; tÞ þ B00

i¼1

Conclusions

Fig. 8. Hourly reservoir storage volumes obtained by RCGA–AFSA for case 3 of test system 2.

In this paper, a new hybrid algorithm combining real coded genetic algorithm (RCGA) and artificial fish swarm algorithm (AFSA) is proposed to solve short-term hydrothermal scheduling (SHS) problem. RCGA–AFSA adopts global search advantage of RCGA and local search advantage of AFSA, which speeds up convergence and enhances search ability. Moreover, the effective constraints handling methods are utilized to deal with complicated hydraulic and electric system constraints. To evaluate the performance of the proposed method, RCGA–AFSA has been successfully applied to two hydrothermal systems. Simulation results show that hybrid RCGA–AFSA can obtain better scheduling results in terms of total fuel cost and average CPU execution time. It is superior to other optimization methods reported in the literature. Hence, the proposed RCGA–AFSA is an effective and promising method for solving SHS problem and can be extended to largescale hydrothermal scheduling problem.

N. Fang et al. / Electrical Power and Energy Systems 62 (2014) 617–629

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