Multi-Objective Particle Swarm Optimization for Optimal Reservoir Operation M. Janga Reddy and D. Nagesh Kumar Department of Civil Engineering, Indian Institute of Science, Bangalore-560 012, India
[email protected],
[email protected]
Abstract. In multipurpose reservoir operation, with multiple objectives, developing an efficient operating policy is a complex problem, where it is almost implausible to satisfy all the commitments involved for different purposes. In this context, it is very important to generate various alternatives in order to perceive the effect of a particular decision on the performance of other goals. To provide better understanding of various alternatives, it is highly desirable to optimize reservoir operation problem using multi-objective evolutionary algorithm. This paper aims to develop a robust operation policy by using an evolutionary algorithm based Multi-Objective Particle Swarm Optimization (MOPSO) technique. The solutions of multi-objective PSOs yield a trade-off curve, identifying a population of points that define optimal solutions to the problem. The proposed approach is applied to Bhadra reservoir system in India. The model results show that the alternatives developed by MOPSO are very much useful and can yield better operating policies.
Key Words : reservoir operation, multi objective optimization, Particle Swarm Optimization, hydropower, irrigation
1 Introduction Real life systems, whether they are large or small scale, are characterized by multiple objectives and goals, which often conflict and compete with one another. The system modeling in water resources is not an exception to this, which deals with problems, which are mostly multi-objective in nature. Optimization of multipurpose reservoir system is to solve multi-objective problems. Suppose, for a reservoir system having hydropower and flood control as key purposes, the two major objectives can be, (i) maximization of the hydropower generation from that reservoir; (ii) minimization of flood risk or flood damage. Obviously these two objectives are in conflict and competition with each other. The higher the level of the reservoir, the more the hydropower generation possible because of the high water head, yet less water storage
will be available for flood control purposes and vice-versa. Clearly, one can identify, within the active storage capacity of that reservoir, a Pareto optimum region where the enhancement of the first objective can be achieved only at the expense or degradation of the second, namely flood control. Also the units of these two objectives are non-commensurable. The first objective, which maximizes the hydroelectric power, is generally measured in units of energy and not necessarily in monetary units, where the second objective can be measured in terms of acres of land, livestock, or human lives saved. If the objectives are non-commensurate the classic methods of optimization cannot easily be applied. Of the several approaches developed to deal with multiple objectives, tradeoff methodologies have shown promise as effective means for considering non-commensurate objectives that are to be subjectively compared in operation determinations [1]. Most of the researchers on reservoir operation problems have tried conventional techniques to generate the tradeoffs between multiple objectives ([1], [2], [3], [4], [5], [6], [7], [8], [9]). The conventional optimization methods such as Dynamic Programming (DP), Linear Programming (LP), and Non-Linear Programming (NLP) are not suitable to solve multi-objective optimization problems, because these methods use a point-by point approach, and the outcome of these classical optimization methods is a single optimal solution. Recently it is found that Evolutionary Algorithms (EAs) are attractive alternatives in solving multi-objective optimization problems [10]. These techniques have some advantages over the classical optimization techniques. They use objective function information directly, and do not require its derivatives or any other auxiliary information. EAs use a population of solutions in each iteration instead of a single solution, so they are called as population-based approaches and offer a set of alternatives in a single run. EAs use randomized initialization and stochastic algorithm in their operation. Hence EAs can locate the search at any place in the search space, and overcome the problems of local optima. Thus EAs are more appropriate to solve multi objective optimization problems. Among the EAs, until recently another population-based optimization technique Particle Swarm Optimization (PSO) has been applied only to single objective optimization tasks. However the high speed of convergence of PSO algorithm attracted the researchers to develop multi objective optimization algorithms using PSO [11]. With this in view, the present study explores the applicability of multiple-objective PSO, for reservoir operation problems. The proposed approach for solving a multi objective decision problem in reservoir operation has a great potential for application, due to its attractive feature of generation of multiple solutions in a single run. It makes it easy for the decision maker to choose the desired alternative, according to individual preferences. For illustration, the proposed approach is applied to a realistic reservoir operation problem, namely Bhadra reservoir project, involving irrigation releases, power generation and water quality requirements as objectives. In the following sections first a brief description of PSO is presented; then the working of MOPSO is explained. The next section gives the details of case study and model formulation. Model application and other details were presented in subsequent section. Finally results are discussed, followed by conclusions.
2 Particle Swarm Optimization In recent years, population-based optimization methods such as Evolutionary Algorithms have become increasingly popular for solving single and multiobjective optimization problems. EA’s success is due to its generic ability to handle large and complex real-world problems. Among EAs there are number of algorithms that can be used, viz., Evolutionary Programming (EP), Evolutionary Strategies (ES), Genetic Algorithms (GAs) etc. More recently Particle Swarm Optimization (PSO) is proposed, which has proved to be an efficient optimization method for single objective optimization ([11], [12]). Recently, Janga Reddy and Nagesh Kumar [13] have applied PSO technique to solve a multi-reservoir operation model and reported that PSO is performing quite well in achieving the near global optimal solutions. To give an idea about PSO, a brief description of PSO and its working is presented below. 2.1 PSO Algorithm PSO technique is inspired by studies of social behavior of insects and animals [11]. The social behavior is modeled in a PSO to guide a population of particles (so-called swarm) moving towards the most promising area of the search space. In PSO, each particle represents a candidate solution, If the search space is D-dimensional, the ith individual (particle), of the population (swarm), can be represented by a Ddimensional vector, Xi = (xi1, xi2, …, xiD)T. The velocity (position change) of this particle, can be represented by another D-dimensional vector Vi = (vi1, vi2,.., viD)T. The best previously visited position of the ith particle is denoted as Pi = (pi1, pi2, …, piD)T. Defining g as the index of the best particle in the swarm (i.e. the gth particle is the best), and superscripts denoting the iteration number, the swarm is manipulated according to the following two equations:
vidn 1
xidn 1
n [ w vidn c1 r1n ( pidn xidn ) c 2 r2n ( p gd xidn )]
xidn vidn 1
(1) (2)
where d = 1,2,...,D; i =1,2,…,N; N is the size of the swarm population; χ is a constriction factor which controls and constricts the velocity’s magnitude; w is the inertia weight, which is often used as a parameter to control exploration and exploitation in the search space; c1 and c2 are positive constant parameters called acceleration coefficients; r1 and r2 are random numbers, uniformly distributed in [0,1]; and n is iteration number. Utilizing the fact of PSO efficiency of quick convergence in obtaining efficient solutions for single objective optimization, and non-dominated set principle, MOPSO algorithms are proposed. Since MOPSOs maintain a population of solutions, this allows exploration of different parts of the Pareto front simultaneously. Thus PSO seems particularly suitable for multiobjective optimization due to its efficiency in better quality solutions and requiring less computational time [11]. Recently researchers reported that MOPSO is fast, more reliable, and often converging to the true Pareto-optimal front with a good solution spread and coverage within just a few
steps ([14], [15]). The proposed algorithm uses non-dominated set principle and elitism in its application [16]. In the following section a brief description of MOPSO algorithm is presented. 2.2 Multi Objective Particle Swarm Optimization Algorithm The proposed MOPSO methodology uses a historical archive that records some of the non-dominated solutions previously found [17] and crowded comparison operator with niche count [16] to propagate the non-dominated front in further generations. This strategy consequently helps to better exploration of Pareto front and keeps diversity in the population. Also for better exploration of non-dominated solutions a special mechanism is incorporated into the algorithm, i.e., few number of the overcrowded particles are replaced with isolated particles after some random perturbation created by mutation operator. In this way, the performance of the algorithm is improved by effectively exploring the Pareto front in a wider search space. The proposed MOPSO can be summarized in the following steps. 1.
Initialize the population and store the population in a list PSOList: a) The current position of the i-th particle, Xi and its current velocity Vi, are initialized with random real numbers within the specified decision variable range; The personal best position Pi, is set to Xi ; b) Evaluate each particle in the population; iteration counter t := 0. 2. t := t + 1. 3. Identify particles that give non-dominated solutions in the population and store them in a list nonDomPSOList. 4. Calculate - niche count/crowding distance value, for each particle. Resort the nonDomPSOList according to niche count/crowding distance values. 5. Repeat the loop (step through PSOList): a) Select randomly a global best Pg for the i-th particle from a specified top part (e.g. top 5%) of the sorted nonDomPSOList. b) Calculate the new velocity Vi, based on the equation (1), and the new Xi by equation (2). c) Go to step a) if i < numParticles. 6. Set nonDomPSOListTemp to nonDomPSOList and Empty nonDomPSOList. 7. Evaluate each particle in the population. Perform special mutation operation on specified number of overcrowded particles. 8. Identify particles that give non-dominated solutions in current iteration and add them to nonDomPSOListTemp. 9. Find the non-dominated solutions in nonDomPSOListTemp and store them in nonDomPSOList. The size of nonDomPSOList is restricted to the desired set of non-dominated solutions. 10. Empty the nonDomPSOListTemp. 11. If t < maxIterations, go to step 2; else go to step 12. 12. Print the non-dominated solution set obtained so far.
This Algorithm is coded in user friendly mathematical software package Matlab 6.5 and the model is implemented in PC based 512 MB RAM computer.
3 Case Study To demonstrate the efficacy of the proposed approach, Bhadra reservoir system, in India is taken up as a case study for developing optimal reservoir operation policy. The Bhadra dam is located at latitude 13°42′ N and longitude 75°38′20″ E. Bhadra project is a multipurpose project providing facilities for irrigation and hydropower generation and meeting downstream water quality requirements. The reservoir provides water for irrigation of 6,367 ha and 87,512 ha under left and right bank irrigation canals respectively. The irrigated area, spread over the districts of Chitradurga, Shimoga, Chikmagalur, and Bellary in Karnataka state, comprises predominantly of red loamy soil, except in some portions of the right bank canal area, which consists of black cotton soil. Major crops grown in the command area are paddy, sugarcane, permanent garden, and semidry crops. Also in this project there are three sets of turbines, one set each on the left bank canal and the right bank canal and the other set at the river bed level of the dam, for generating hydropower. Salient features of the reservoir are given in Table 1. Table 1. Salient features of Bhadra reservoir system
Description Gross storage capacity
Quantity 2,025 Mm3
Live storage capacity
1,784 Mm3
Dead storage capacity
241 Mm3
Average Annual inflow
2,845 Mm3
Left bank canal capacity
10 m3/s
Right bank canal capacity
71 m3/s
Left bank turbine capacity (PH1)
2,000 kW
Right bank turbine capacity (PH2)
13,200 kW
Riverbed turbine capacity (PH3)
24,000 kW
3.1 Model Formulation The objectives of the model are minimization of irrigation deficits and maximization of hydropower generation. These two are mutually conflicting objectives, since one tries for minimization of the irrigation deficits, requiring more water to be released to satisfy irrigation demands and the other tries to maximize hydropower production, requiring higher level of storage in the reservoir to produce more power. These two competing objectives of the system are expressed as follows:
Minimize sum of squared deficits for irrigation annually, Minimize SQDV
D 12
t 1
1,t
R1,t D2 ,t R2, t 2
12
2
(3)
t 1
where SQDV is the sum of squared deviations of irrigation demands and releases. D1,t and D2,t are the irrigation demands for the left bank canal and right bank canal command areas respectively in period t in Mm3; R1,t and R2,t are the releases into the left and right bank canals respectively in period t in Mm3. Maximize annual energy production, Maximize P
12
p( R t 1
1, t
H 1,t R2, t H 2,t R3, t H 3,t )
(4)
where P is the total energy produced in M kWh; p is power production coefficient; R1,t, R2,t and R3,t are the releases to left bank canal, right bank canal and riverbed turbines respectively in period t in Mm3. H1,t , H2,t , H3,t are the net heads available to left bank, right bank and bed turbines respectively in meters during period t. The optimization is subject to the following constraints: Storage continuity: t=1, 2,…, 12 (5) S t 1 S t I t ( R1,t R2,t R3,t Et Ot ) where St = Active reservoir storage at the beginning of period t in Mm3; It = inflow to the reservoir during period t in Mm3; Et = the evaporation losses during period t in Mm3 (a nonlinear function of initial and final storages of period t); Ot = overflow from the reservoir in period t in Mm3; Storage limits: Smin ≤ St ≤ Smax t=1, 2,…, 12 (6) where Smin and Smax are the minimum and maximum active storages of the reservoir. Maximum power production limits: p R1,t H1,t ≤ E1,max t=1, 2,…, 12 (7) t=1, 2,…, 12 (8) p Rr,t H2,t ≤ E2,max p R3,t H3,t ≤ E3,max t=1, 2,…, 12 (9) where, E1,max , E2,max , and E3,max are the maximum amounts of power in M kWh, that can be produced (turbine capacity) by the left, right and bed level turbines respectively. Canal capacity limits: R1,t ≤ C1,max t=1, 2,…, 12 (10) R2,t ≤ C2,max t=1, 2,…, 12 (11) where, C1,max and C2,max are the maximum canal carrying capacities of the left and right bank canals respectively. Irrigation demands: t=1, 2,…, 12 (12) D1min, t R1,t D1max, t D2min, t R2,t D2max, t t=1, 2,…, 12 (13) where, D1min, t and D1max, t are minimum and maximum irrigation demands for left bank canal respectively; D2min, t and D2max, t are minimum and maximum irrigation demands for right bank canal respectively in time period t.
Water Quality Requirements: R3,t ≥ MDTt t=1, 2,…, 12 (14) where, MDTt= minimum release to meet downstream water quality requirement in Mm3. The formulated final model consists of minimizing irrigation deficits (equation (3)) and maximizing the hydropower (equation (4)) subject to satisfying the constraints from equations (5) to (14). In this study, it is assumed that a minimum release of 9 Mm3/month to downstream is to be made in each month to meet the water quality requirements.
4 Results and Discussion The proposed MOPSO approach is applied to the above formulated model. To apply the model, the following parameters are decided to be used after through sensitivity analysis. The initial population of the MOPSO was set to 200; c1 and c2 were set to 1.0 and 0.5 respectively; w was gradually decreased from 1.0 to 0.4; and χ set to 0.9; the number of non-dominated solutions to be found is set to 200. Then MOPSO is run for 200 iteration steps. To limit any abnormal fluctuation in velocity values, the maximum change in velocity in a time step is set to the upper and lower bounds of the decision variable range. To handle the constraints of the problem, the natural self-adaptation mechanism of the EAs is useful. To perform that, the following criteria are used to select the best individuals from one generation, viz., (i) between two feasible solutions, the one with the higher fitness value is preferred; (ii) if one solution is feasible and the other one is infeasible, the feasible one is preferred; (iii) if both solutions are infeasible, the one with the lowest sum of constraint violation is preferred [5]. The average inflow into the reservoir over a period of 69 years (from 1930-31 to 1998-99) is given as inflow data to the developed model. The model is run for 10 trials; it is observed that, the generated Pareto front is more or less same in all the generations. For demonstration purpose, this section presents a sample result of the model output. Fig. 1 shows, the trade-off between the two objectives of the model, i.e., irrigation and hydropower production. It can be seen that a set of well distributed Pareto-optimal solutions are available (Fig. 1). There are number of alternatives that can be chosen at various satisfaction levels of both the objectives. Depending on the circumstances prevailing under the reservoir system and by analyzing the tradeoff between the two objectives, the reservoir operator can make the appropriate decision. For example, if the operator chooses the optimal point shown in Fig. 1, the model readily gives the corresponding release decisions and storage policy. Fig. 2 shows the corresponding releases to be made into left bank canal, right bank canal and bed turbine. Fig. 3 shows the corresponding initial storages required for ensuring such releases.
225
optimal point
210
f2
195 180 165 150 0
15000
30000
45000
60000
75000
90000
f1 Fig. 1. Pareto optimal front for multiple objectives shows the trade-off between
irrigation and hydropower for average inflows into the reservoir. (f1 is squared irrigation deficit, (Mm3)2; f2 is hydropower generated in M kWh).
200 R1
R2
R3
6
7
Release (Mm3)
150
100
50
0 1
2
3
4
5
8
9
10
11
12
Month
Fig. 2. Release policy obtained for selected optimal point, showing releases for left bank canal (R1), right bank canal (R2) and river bed (R3).
Storage (Mm 3)
2000
1600
1200
800
400 1
2
3
4
5
6
7
8
9
10
11
12
13
Month
Fig. 3. Rule curve for selected optimal point showing the initial storages to be maintained in the reservoir.
5 Conclusions To improve the effectiveness in developing efficient operating policies for multiobjective reservoir operation problem, a multi-objective evolutionary algorithm, namely Multi-objective Particle Swarm Optimization (MOPSO) is proposed. Utilizing the strengths of PSO, quick convergence and yielding efficient solutions for single objective optimization, the multi-objective PSO algorithm is evolved. The developed MOPSO uses the non-dominated sorting concept, and parameter free-niching scheme to promote solution diversity. The proposed approach is applied to Bhadra project. The multiple objectives involve minimization of irrigation deficit and maximization of hydropower while meeting downstream water quality requirements and are modeled with MOPSO. A variety of alternatives are generated for average inflows into the reservoir. Once the alternatives are generated, depending on the preference of the decision maker, he can select a suitable policy. Thus this study demonstrates the usefulness of MOPSO technique to obtain robust and efficient operating policy for real life problems.
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