Big Bang-Big Crunch Algorithm: Application to

Proceeding of the 12th International Conference on QiR (Quality in Research) Bali, Indonesia, 4-7 July 2011 ISSN 114-1284

Big Bang-Big Crunch Algorithm: Application to Reservoir Operation problems M.H. Afshara, I. Motaeib a

Department of Civil Engineering, Iran University of Science And Technology, Enviro-Hydroinformatic Center of Excellence ,Tehran, Iran E-mail : [email protected] b

Department of Civil Engineering, Iran University of Science And Technology, Tehran, Iran E-mail : [email protected]

ABSTRACT In this paper, one of the newest algorithms, namely the Big Bang-Big Crunch (BB-BC) algorithm is used to solve the problem of reservoir operation problem. BB-BC relies on one of the theories of the evolution of universe; namely, the Big Bang and Big Crunch theory. The purpose of this study is evaluating the performance of algorithm in solving the reservoir operation problem. For this, the problem of water supply and hydropower operation of Dez Reservoir in Iran over 5 and 20 years operation periods is taken as case study and the results are presented and compared with those obtained by Max-Min Ant System (MMAS) algorithm. the result confirms the ability of the BB-BC algorithm to optimally solve the problem of reservoir operation. Keywords Big Bang-Big Crunch algorithm, reservoir operation, optimization

1. INTRODUCTION In water resource management literature, optimal reservoir operation problem is not a new issue. Various optimization techniques have been used to solve this problem. These techniques include mathematical approaches such as Linear Programming (LP), Nonlinear Programming (NLP), Dynamic Programming (DP) and Heuristic algorithms such as Genetic algorithms (GA), Particle swarm optimization (PSO), Ant Colony Optimization (ACO) and Honey Bee Mating Optimization (HMBO) algorithms. Amongst mathematical methods, LP is well known as the simple optimization technique because it is so easy to understand and does not need any initial solution. Some application of LP to reservoir operation can be found in Yeh [1]. After LP, DP is the most useful optimization method that is widely used in water resources planning and management. Dynamic Programming disintegrates the primary problem into sub problems, which are solved over each period [2]. Some applications of dynamic programming to reservoir operation problems can be found in [3], [4] and [5]. Nonlinear programming (NLP) has not become as popular as LP and DP in the literature of water resource management problems. A reason for this is that NLP approaches are slow, iterative and require large amount of time and computational costs [6]. Nonetheless, Hicks et al. [7] and Haimes [8] applied NLP technique to reservoir operation problems. Despite the widespread application of mathematical methods in water resource management, these techniques fail to find the optimal solution when the objective functions are complex and the number of decision variables is too large. To overcome this limitation, recently meta-heuristic algorithms such as GA, PSO and ACO, are used to solve the problem of reservoir operation. A comprehensive survey on the application of evolutionary algorithms (EAs), with particular focus on GA, to water resource management problems can be found in [9]. Kumar and Reddy [10] used PSO for multiobjective optimization operation of the Bhadra reservoir system in India. They [11] also applied ACO algorithm to optimize the operation of multi-purpose reservoir system located in India. Afshar and Moeini [15] used a version of ACO algorithm called the Max-Min Ant System (MMAS) Algorithm for the solution of simple and hydropower reservoir operation problems. Montalvo et al [12] applied one of the variants of Particle Swarm Optimization (PSO) to two case studies: the Hanoi water distribution network and the New York City water supply tunnel system. Haddad et al [13] applied the Honey Bee Mating Optimization (HBMO) algorithm to single reservoir operation problem. In this paper, the Big Bang-Big Crunch (BB-BC) algorithm is used to solve the problem of simple and hydropower optimal operation of a single reservoir. Then the Dez reservoir in Iran is taken as a case study and the results are presented and compared with those obtained by Max-Min Ant System (MMAS) algorithm. The results show that the BB-BC method is much more efficient and effective than those obtained by MMAS for both of the problems considered in this article.

Page 2294


2. BIG BANG-BIG CRUNCH (BB-BC) ALGORITHM Big Bang-Big Crunch is one of the newest algorithm that relies on the famous theories about the evolution of universe; namely, the Big Bang and Big Crunch theory [14]. This algorithm is composed of two phases: a Big Bang phase, and a Big Crunch phase. In the Big Bang phase, the randomly generated candidates are uniformly distributed over search space and then in the Big Crunch phase, these candidates are concentrated to a point by using a convergence operator namely Center of Mass. Using center of mass, the new position of each candidates are calculated. This process is repeated until the convergence is achieved. In original version of the algorithm, center of mass is calculated as follows:

where is the jth component of center of mass, is the jth component of ith candidate, is fitness of the ith candidate, and N is the number of candidates. The algorithm updates the new position of each solution by using of the center of mass. Updating rule can be written as:

where

is the new value of the jth component from ith candidate

distribution, is the constant value and component for variable x, respectively.

is the iteration index.

and

is a random number with a standard normal are the maximum and minimum value of th

Other forms of the center of mass can also be used. Here, the best solution of each iteration is used as the center of mass.

3. RESERVOIR OPERATION The objective function of simple operation, can be defined by the minimization of release deviation from a pre-defined pattern of demands over an operation period. This can be written mathematically as:

Subject to continuity equations at each period

and minimum and maximum allowable values of release and storage volume at each period

where

is water demand in period in million cubic meters (MCM), is release from the reservoir in period t (MCM), is the maximum demand over the whole operation period (MCM), NT is number of periods. is storage at the start of period t (MCM), is water inflow to the reservoir in period t (MCM), and are the minimum and maximum allowable storage in reservoir (MCM), respectively and and are the minimum and maximum allowable release from reservoir (MCM), respectively. For hydropower operation of a single reservoir, the objective is to find a set of release volumes so that the total power generated is maximized. This objective can be defined in different ways. Here the following objective function is used:

Page 2295


where is power generated in Mega Watt (MW) by the hydro-electric plant in period t, is the total capacity of hydro-electric plant (MW). The constraints of this problem are the same as those of the simple operation problem.

The power generated can be stated as follows:

with

and

Where is the gravity acceleration equal to 9.81 m2/s, is the efficiency of the hydroelectric plant, is release from reservoir at period (m3/s), is the plant factor, is the effective head of the hydroelectric plant (m), is the elevation of water in reservoir at period (m), is the tail water level of the hydroelectric plant (m), , , and are four constant which can be obtained by fitting equation (10) to the existing data.

4. CASE STUDY Here, the water supply and hydropower operation of Dez reservoir in southern Iran is taken as the case study to evaluate the performance of the BB-BC algorithm. The total storage volume of Dez reservoir is equal to 2510 MCM; average annual inflow is 5303 MCM over 5 years and 5900 MCM over 40 years. The maximum and minimum allowable storage volumes are taken to be 3340 MCM and 830 MCM respectively, while the maximum and minimum monthly water releases are taken as 1000 MCM and 0 MCM, respectively. The coefficients of the volume–elevation curve defined by Equation 10 are a=249.83364, b= 0.058720, c= -1.37 3×10–5 and d=1.526 3×10–9. The hydroelectric plant consists of 8 units. the capacity of each unit is 80.8 MW working 10 hours per day leading to a plant factor of 0.417. The total capacity of the plant is 650 MW and its efficiency is considered to be %90 that is, 0.9. The tail water level (TWL) from sea level is 172 m. These problems are solved here using BB-BC algorithm for monthly operation over 60 and 240 monthly periods with 400,000 function evaluation, penalty coefficient of 10000 and equal to 7. Table 1 shows the maximum, minimum and average results obtained by BB-BC and MMAS [15] algorithms over 10 runs. Figures 1-4 show the convergence curves of simple and hydropower operation over 60 and 240 monthly periods, respectively. It can be seen in Table 1 that the minimum solutions obtained using BB-BC for water supply operation over 60 and 240 months are 0.739 and 5.261, respectively and for hydropower operation over 60 and 240 months are 7.461 and 22.362, respectively. These results can be compared with the global solutions obtained by Lingo software that are 0.7316 and 4.7684 for water supply operation, respectively 7.372 and 20.622 for hydropower operation over 60 and 240 months, respectively. As it can be seen BB-BC results compared with those obtained by MMAS algorithm are much closer to global solutions especially for longer period operation i.e. 240 months. So that the objective costs for simple and hydropower using MMAS are 10.313 and 35.300 whereas using BB-BC this costs are 5.261 and 22.362, respectively. This clearly shows that BB-BC is very successful to solve the large-scale reservoir operation problems.

Page 2296

Proceeding of the 12th International Conference on QiR (Quality in Research) Bali, Indonesia, 4-7 July 2011 ISSN 114-1284 Table 1: maximum, minimum and average results obtained by BB-BC and MMAS for 10 runs

Method

Model Simple

BB-BC hydropower Simple MMAS hydropower

Month 60 240 60 240 60 240 60 240

Minimum 0.739 5.261 7.461 22.362 0.785 10.313 7.913 35.300

Maximum 0.745 5.427 7.557 22.646 0.814 13.326 8.063 39.980

Average 0.742 5.381 7.488 22.458 0.799 12.013 8.001 37.568

Figure 1: Convergence curves for the simple operation over 60 month period

Figure 2 : Convergence curves for the simple operation over 240 month period

Page 2297


Figure 3: Convergence curves for the hydropower operation over 60 month period

Figure 4: Convergence curves for the hydropower operation over 240 month period

5. CONCLUSION Big Bang Big Crunch algorithm is a new optimizer technique that was not applied to water resource management problems yet and Simplicity and high speed of convergence can be justify its use in this field of engineering. This article had two main goals: 1-use of this algorithm in water resources management, and more importantly 2- study efficiency of algorithm with solving a real engineering problem and compare the results with those previously obtained with the Max-Min algorithm. The real engineering problem was simple and hydropower operation of Dez reservoir in Iran over 60 and 240 operation periods. The results showed that the proposed methods have the ability to solve large-scale reservoir-operation problems.

Page 2298


REFERENCES [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15]

W .Yeh, “Reservoir management and operations models: a state-of-the-art review,” Water Resources Research, Vol. 21, No. 12, Pages 1797-1818, December, 1985 W. J. Labadie, “Optimal operation of multireservoir systems: state-of-the-art review,” Journal of Water Resources Planning and Management, Vol. 130, No. 2, March 1, 2004 G.K. Young, “Finding reservoir operating rules,”. Journal of the Hydraulic Division of ASCE, 93(HY6), 297-321. 1967 W. Hall, „„Optimal state dynamic programming for multireservoir hydroelectric systems,‟‟ Technical Rep., Dept. of Civil Engineering, Colorado State Univ., Ft. Collins, Colo. 1970 N. Buras, „„Scientific allocation of water resources; water resources development and utilization-a rational approach,‟‟ American Elsevier, New York. S. P. Simonovic, “Reservoir systems analysis: closing gap between theory and practice,” Journal Of Water Resources Planning And Management, Vol. 118, No. 3, May/June, 1992 R. H. Hicks, C. R. Gagnor, S. L. Jacoby, and J. S. Kowalik, “Large scale non-linear optimization of energy capability for the pacific northwest hydroelectric system,” Proc. Institute of Electrical and Electronics Engineers. 1974. Y. Y. Haimes, “Hierarchical analysis of water resources systems,” McGraw Hill, New York, N.Y. 1977 J. Nicklow, et al., “State of the art for genetic algorithms and beyond in water resources planning and management,” Journal of Water Resources Planning And Management ASCE / July/August 2010 M. J. Reddy, D. N. Kumar, „„Multi-Objective particle swarm optimization for generating optimal trade-offs in reservoir operation,‟‟. Hydrological Processes, 2007, 21, 2897–2909. M. J. Reddy, D. N. Kumar, „„Ant Colony Optimization for Multi-Purpose Reservoir Operation,‟‟. Water Resources Management (2006) 20: 879–898 I. Montalvoa, J. Izquierdoa, R. Pereza, M. M. Tungb, “Particle Swarm Optimization applied to the design of water supply systems,” Computers and Mathematics with Applications 56 (2008) 76 776 O. B. Haddad, A. Afshar, M. A. Marino, “Honey-Bees Mating Optimization (HBMO) algorithm: a new heuristic approach for water resources optimization,” Water Resources Management (2006) 20: 661–680 O. K. Erol, I. Eksin, “New optimization method: Big Bang-Big Crunch,” Advances in Engineering Software 37 106–111, 2006 R. Moeini, M. H. Afshar, “Application of an Ant Colony Optimization algorithm for optimal operation of reservoirs: a comparative study of three proposed formulations,” Civil Engineering Vol. 16, No. 4, pp. 273-285; Sharif University of Technology, August 2009

Page 2299