Optimization of multi-reservoir management using Harmony Search Algorithm (HSA). I. P. Kougias* • N. P. Theodossiou Division of Hydraulics and Environmental Engineering, Department of Civil Engineering, Aristotle University of Thessaloniki, 541 24 Thessaloniki, Greece *Corresponding author: E-mail:
[email protected], Tel +30 2310 995856
Abstract The potential of Harmony Search Algorithm (HSA) which is an optimization technique presented a few years ago, has encouraged scientists from various fields to solve complex, large scale problems. Because of its successful implementations, it is considered as a robust optimization tool and publications regarding its applications show a remarkable increase. In the present paper, along with a brief presentation of HSA, an application on the optimal management of a multi-reservoir system is presented. This particular application concerns the optimal operation of a four-reservoir system on a yearly basis. The objective is to maximize the annual benefits gained from the four reservoirs (hydropower generation and irrigation) through the optimal management of the system. The main difference between the problem we deal with and typical Dam-Scheduling problems, is that decision variables can take continuous values. This minor difference causes a huge impact and leads to a significant expansion of the search-space. As a result, convergence to the optimum solution with the use of stochastic methods becomes a challenging target. This problem was first introduced by V.T. Chow and G. Cortes-Rivera who proposed a solution using Dynamic Programming. In this paper, for the first time, a solution is derived with the application of a stochastic Algorithm. A program was created in order to optimize this particular problem using MATLAB. The successful convergence to optimum management is presented along with the main characteristics and results. The comparison between the proposed solution and the one achieved with Dynamic Programming reveals some interesting information regarding the efficiency of HSA. Keywords:. Dam Scheduling ;• Metaheuristic algorithm ;• Harmony Search ; • Hydro informatics ; • Continuous variables
1.
INTRODUCTION
The operation of a multiple dam system is as a rule an intricate process. Usually, it is needed to fulfill different kind of constraints and select values which will optimize the various objectives. These objectives have, normally, a competitive nature. Thus, finding a good balance is not always an easy task, since the improvement of one parameter of the system may result to the deterioration of others. In the first attempts to optimize multi-reservoir systems, researchers used mathematical optimization techniques. In the late 1960’s, linear programming and dynamic programming were used in order to find the best schedule to different dam systems.
However, a couple of decades later, stochastic methods drew the attention of those involved in optimization techniques. Genetic Algorithms (GA) and other Evolutionary techniques had already been used with success in various scientific projects. They practically replaced the previously used methods, because they had shown particular advantages. These advantages include the fact that they require less initial information and have a simple structure. Stochastic methods are also suitable for computer implementation. For this reason, and although in some occasions traditional mathematical techniques were giving excellent results, stochastic techniques prevailed. Nowadays, one can find many successful implementations of stochastic methods in Dam Scheduling. GA is the main tool used along with other algorithms (Simulated Annealing, Tabu Search). Recently, Zong Woo Geem presented a new metaheuristic algorithm [4], Harmony Search Algorithm (HSA). HSA is based on an analogy between music and optimization. In the same way musicians try to improve their music, HSA seeks for certain values that optimize the objective function and at the same time satisfy the problem’s constraints. Furthermore, in the same way musicians improve rehearsal after rehearsal, HSA improves iteration after iteration. This Algorithm was initially used in order to optimize water distribution networks [6]. Since then, a sustained and increasing interest in HSA applications resulted in its successful use in a vast variety of implementations [7]. Z.W. Geem presented an optimal scheduling of a multiple-dam system using HSA in 2007 [5]. His implementation converged to the optimum solution, providing five different, global-optimum solutions. During 2011 the authors presented the complete optimization of a multiple dam system, providing all possible alternatives [9]. For the first time ever, a complete solution of this problem, which was initially presented by R.B. Larsson in 1968 [10], was achieved. 2. APPLICATION OF HSA ON A MULTI-RESERVOIR SYSTEM 2.1. Description of the problem In the present paper the optimization of the dam system of Fig.1 is presented. This problem was introduced in 1974 by Ven Te Chow and Gonzalo Cortes-Rivera [2]. It was designed as a demanding application to be optimized with Discrete Differential Dynamic Programming (DDDP). Its main difference, compared to conventional problems, is that inflows and constraints are not restricted to integer values. As a result, the solution space is considerably wider and convergence to optimum management becomes a challenging task. In the present paper this particular problem is optimized for the first time with the use of a stochastic algorithm (HSA). In this system benefits derive from the use of water towards irrigation and hydropower generation. The aim is to determine the optimum operation of the four-reservoir system over a period of 12 time steps (months). FIGURE 1. The multi-reservoir system
2.2. Problem’s Constraints The Inflow parameters IN1 and IN2 representing the amount of water flowing in Reservoirs 1 and 2 in each of the 12 time steps are given in Table 1. The initial and final states of the reservoirs are identical and are required to be: S1 (0) S 2 (0) S3 (0) S1 (12) S 2 (12) S3 (12) 6 (1) (2) S 4 (0) S 4 (12) 8 water units The releases from the four dams through turbines are as follows (values in water units): (3) 0.005 R1 4.0 (4) 0.005 R2 4.5 (5) 0.005 R3 4.5 (6) 0.005 R4 8.0 It can be easily understood that this minor change in the constraints is causing a huge impact in the size of the search space. In the “classic problem” R1 would range between 0≤ R1 ≤4, taking one of the 5 possible integer values. However, this change allows R1 to take thousands different possible values (decimal precision of third digit). If we consider that this adjustment applies also to R2, R3, R4, it can be safely said that the search space is becoming incomparably wider. This adjustment has an even stronger effect in stochastic methods. These techniques detect good solutions by scanning the search space. As a result any significant increase of the space’s size will raise difficulties. Moreover, a vast search space usually includes various local optima. This leads to an increased possibility of non-convergence to the problems’ global-optimum. Table 2 shows the maximum permissible storage in the 4 reservoirs during the time period of the 12 time steps. The minimum permissible storage is equal to 1. This minimum storage is common for all 4 reservoirs and it is constant during the function of the system As a result, the equations that express the water stored in each reservoir throughout the 12 timesteps are: (7) S1 t 1 S1 t R1 t IN1 (t ) (8) S 2 t 1 S 2 t R2 t IN 2 (t ) S3 t 1 S3 t R3 t R2 t (9) S 4 t 1 S 4 t R4 t R1 t R3 t (10) The objective function considers the use of water for power generation and irrigation. Reservoirs 1, 2 and 3 generate electricity through turbines, while water from Reservoir 4 is used for hydropower generation and irrigation. The objective function to be maximized is: maximize:
4
11
i 1
t 0
b (t ) R (t ) i
i
(11)
Table 3 presents the benefit coefficients (bi, i= 1-4) included in Equation 11.
3. HARMONY SEARCH ALGORITHM MODEL 3.1. The basic concept of the Harmony Search Algorithm Many optimization techniques imitate a natural or artificial procedure. Harmony Search Algorithm was inspired from music and imitates the way musicians perform. During a performance, a musician will obviously play the basic melody (theme) of each song. All members of the music band know their part in this theme and can play it from memory. The second option musicians have, is to play a variation of this theme. In this way the theme is altered and enriched with new melodies. A third option is to start an improvisation, creating new musical material. 3.2. Mechanisms of HSA Harmony Search Algorithm uses a place in the memory in order to store good solutions, found during the optimization process. This place is denominated as Harmony Memory (HM). The optimization of a problem begins with the filling of Harmony Memory with random values which correspond to random solutions. After that the Algorithm starts producing new feasible solutions based on 3 basic mechanisms. Each mechanism imitates one of the 3 musicians’ options mentioned above. These 3 mechanisms are: i.
HSA chooses a variable’s value from Harmony Memory in order to create a new harmony. This process is defined as Memory Consideration and it ensures that good harmonies (values that give good results) will be considered through the solution. Harmony Memory Considering Rate (HMCR) is defined as the probability that the New Harmony will include a value from the historic values stored in the HM.
ii.
Every component of the New Harmony, chosen from HM, is likely to be pitch-adjusted. This procedure is called Pitch Adjustment and it is the local-search mechanism. Pitch Adjusting Rate (PAR), indicates the probability that algorithm will choose neighboring values for the PAR% of the components chosen from HM: x inew = x i Random·b w xi is the existing pitch stored in HM, Random is a random number between 0 and 1, and bw is the bandwidth of the adjustment
iii.
The third choice is to select a totally random value from the possible value range. Randomization occurs instead of Memory Consideration, with probability equal to (100HMCR)% and increases the diversity of the solutions. Randomization can drive the algorithm to explore the whole range and attain the global optimality.
HSA creates a new harmony-solution in every repetition, following the operators mentioned above. If the new harmony is better than the worst harmony stored in HM, it replaces it. This repetitive process continues until the termination criterion is satisfied.
4. RESULTS The multiple reservoir system was optimized with HSA using Matlab. A program was designed and developed, providing some interested results. As it is already mentioned this was the first successful optimization of this particular problem with a stochastic technique. This is due to the continuous nature of the decision variables. This particularity makes this problem really demanding for stochastic techniques, because the solution search space is vast. Figure 2 shows a typical run of the program demonstrating the objective’s functions gradual evolution. Chow and Cortes Rivera reported that the linear programming solution converged to a maximum benefit equal to 308.2665 [2]. Soon, Murray and Yakowitz presented a solution using Differential Dynamic Programming (DDP) which converged to 308.234. The same authors also used Discrete Differential Dynamic Programming which resulted to a convergence to 307.98 [11]. HSA converged to a maximum of 307.218. This solution represents the 99.66% of the best known solution. Thus, it is safe to say that HSA achieved very satisfying results. This convergence was the best result of few runs. An extra advantage was that there was no need for calibration of HSA’s parameters throughout the solution. The values of the parameters proposed in the HSA literature, produced the best results. However, the number of iterations needed was considerably higher than the iterations in simpler problems. This resulted to a slow convergence to the optimum solution. Convergence to the global-optimal solution in the integer problem appeared only after ≈30,000 iterations [9]. On the contrary, in the present application, the program converged after several hundred thousand iterations. 1,000,000 iterations takes approximately 2 minutes in a computer with a U7300-1.3Ghz Intel CPU. Considering that the total search space comprises from ≈10169 possible solutions, Harmony Search Algorithm converged to an optimum management only after searching 2*10-161% of the total space. These results show that HSA is a robust optimization tool, even for complicated continuous problems. Table 4 shows the optimum water release for each dam, as it resulted from HSA. In Table 5 the Water storage of the reservoirs is presented. These storages correspond to the optimum solution and along with the releases of Table 4, show that all constraints are satisfied by the optimum solution.
5. CONCLUSIONS Harmony Search Algorithm is a recently developed optimization technique. However, many applications have already been implemented, covering a variety of scientific fields. In the present paper, HSA successfully optimized a complex four-reservoir problem. This particular problem was solved for the first time with the use of a stochastic technique. Solution was implemented creating a program in Matlab. This program showed a remarkable performance and converged to a solution that results to a very high benefit, equal to more than 99.6% of the highest benefit ever achieved. This solution shows that Harmony Search Algorithm is a powerful tool, which can be successfully applied not only to typical optimization problems, but also to problems that include continuous variables and involve a vast solutions search-space. FIGURE 2. Gradual evolution of the objective function REFERENCES Ayvaz M.T. (2009) “Application of Harmony Search algorithm to the solution of groundwater management models”, Advances in Water Resources, Vol.32, pp. 916-924. 2. Chow V.T and Cortes Rivera (1974), “Applications of DDDP in water resources planning, Research Report 78, University of Illinois, Water Resources Center, Urbana. 3. Diao R., Shen Q. (2010), “Deterministic Parameter Control in Harmony Search”, Computational Intelligence UK Workshop (UKCI), September 2010. 4. Geem Z.W., Kim J.H., Loganathan G.V. (2001) “A New Heuristic Optimization Algorithm: Harmony Search” Simulation, Vol.76:2, pp 60-68. 5. Geem Z.W. (2007), “Optimal scheduling of Multiple Dam System Using Harmony Search Algorithm”, F. Sandoval et al (Eds.): IWANN 2007, LNCS 4507, pp. 316-323 6. Geem Z.W., Kim J.H., Loganathan G.V. (2002) “Harmony search optimization: Application to pipe network design” Intern. journal of modeling & simulation, Vol.22, No.2, pp 125-133 7. Kougias I. and Theodossiou N. (2010), A new music-inspired harmony based optimization algorithm. Theory and applications, International Conference on Protection and Restoration of the Environment X. 8. Kougias I. and Theodossiou N. (2010), A new music-inspired harmony based optimization algorithm. Application in water resources management problems, International Conference: Protection and Restoration of the Environment X. 9. Kougias I. and Theodossiou N. (2011), Application of the Harmony Search optimization algorithm for the solution of the multiple dam scheduling, (to be published – under review). 10. Larson R.E. (1968) “State increment dynamic programming” Elsevier Science, New York 11. Murray D.M., Yakowitz S. (1979) “Constrained differential dynamic programming and its application to multireservoir control” Water Resources Research, 15(5), 1017-1027 12. Wardlaw R. and Sharif M. (1999) “Evaluation of Genetic Algorithms for optimal reservoir system operation, Journal of Water Resources Planning & Management, ASCE 125(1), 2533 1.
