A NOVEL MULTIOBJECTIVE OPTIMIZATION ALGORITHM, MO HSA. APPLICATION ON A WATER RESOURCES MANAGEMENT PROBLEM I. Kougias 1, L. Katsifarakis 2 and N. Theodossiou 3 Division of Hydraulics and Environmental Engineering Department of Civil Engineering Aristotle University of Thessaloniki 541 24 Thessaloniki, Greece E-mail: ikougias@civil.auth.gr 1, lyska@civil.auth.gr 2, niktheod@civil.auth.gr 3 ABSTRACT Most of the known optimization techniques are inspired by physical phenomena and imitate a natural procedure. In 2001, a new optimization technique, Harmony Search Algorithm (HSA), was presented. This novel metaheuristic is based on the observation that the aim of music creation is the quest of the perfect state of harmony. This quest is relevant to the optimization procedures that seek the optimum solutions. Since then, HSA was used successfully in various engineering optimization problems. In this paper, after a brief explanation of HSA s structure, a Multiobjective HSA is presented. After the first application of HSA in multiobjective optimization in 2010, applications of MO HSA grow in numbers. This application concerns the optimum management of a pumping station which consists of five pumps. The objectives of the optimization problem include the simultaneous maximization of the quantity of the pumped water and the minimization of the pumps wear. At the same time the energy consumption and the electric energy s consumption peak must also remain at a minimum level. The problem was optimized for two different demand curves, which correspond to two different states of the network. The first curve is a typical demand curve with a peak between 10 12am, while the second curve has two peaks, one in the morning and one in the afternoon. The average daily consumption is 55000 m 3 of drinking water. For each state the problem was solved for three different sizes of reservoirs. Varying the size of the reservoir causes a change in the search space and the Pareto front. The optimal solutions that Multiobjective HSA detected, are presented in 3d graphs along with analysis of the results and comments. Keywords ; pump scheduling; Pareto front; multiobjective optimization; Harmony Search Algorithm 79
1. INTRODUCTION Protection and restoration of the environment XI Multiobjective optimization has always been an aim for those involved in optimization techniques. Real problems usually involve the simultaneous optimization of competing objectives. As a rule these problems don t have a single solution which optimizes all objectives of the problem. For this reason, multiobjective optimization leads to a set of alternative solutions. Defining alternative solutions leads to the determination of the Pareto front of the searching space. A candidate solution X is Pareto dominating a solution Y if X is at least as good as Y in all objectives and superior to Y in at least one objective. On the other hand, neither X or Y dominates the other if X is better in some aspects while Y is better in others. The best options, known as Pareto front or Pareto set, are the set of solutions which are non dominated (Luke S., 2009). These solutions are optimal in the sense that an improvement in one objective cannot be achieved without the deterioration of at least one other objective. 1.1. Harmony Search Algorithm (HSA) In the present paper the optimization of a multiobjective problem using Multiobjective Harmony Search Algorithm (MO HSA) is presented. Zong Woo Geem et al. (2001) designed and presented the single objective HSA. HSA was inspired from the music creation process and imitates the procedure that musicians follow during a performance. A musician can play the well-known melody of the performed song, the theme. A second option is to vary this theme and slightly change its melody or rhythm. Finally, the musician can start an improvisation and create totally new melodies. HSA consists of three basic mechanisms that imitate the above three options in order to create new solutions (Harmonies) of the examined problem. These three mechanisms are: Harmony Memory Consideration (occurs with probability HMCR%), Pitch Adjusting Rate (occurs with probability PAR%) and Randomization (occurs with probability 100-HMCR%). Full description of these mechanisms can be found in the literature (Kougias & Theodossiou, 2010). 1.2. Multiobjective Harmony Search Algorithm (MO-HSA) Zong Woo Geem (2010) presented an optimization of a multiobjective problem using HSA. His implementation managed to locate the non dominated set in a bi objective problem concerning time cost trade off. Before that, in a publication by Geem and Hwangbo (2006), HSA was used to optimize a multiobjective problem which included the merging of two objectives of the problem to a single objective function. In other words, they optimized a multiobjective problem by transforming it to a single objective problem. Later on, more scientists presented implementations of Multiobjective HSA. Sivasubramani and Swarup (2011a and 2011b) created a population based, MO - HSA. In order to solve a multiobjective problem the HSA is adjusted. Harmony Memory includes the solutions detected during the repetitive process, the values of the variables of the problem and the corresponding value of the objective function. However, in multi-criterion problems the HM must contain the corresponding values of every objective of the problem. An additional adjustment concerns the Selection procedure of the solutions which are created during the repetitive process. According to Pareto dominance theory, a solution is dominant compared to another only if it shows equal performance for every objective function and better 80
performance for at least one objective of the problem. In MO-HSA the authors have designed a selection procedure which includes routines and subroutines that lead to a Pareto based choice of the solutions. 2. MULTIOBJECTIVE PUMP SCHEDULING A specific model has been chosen to be optimized with the Multiobjective Harmony Search Algorithm. This model is using five pumps and it is based on an operational pumping field in Paraguay (Savic et al., 1997). The model is presented in Figure 1 and it is composed of: A potable water source. A pumping station (5 pumps) used to pump water to an elevated reservoir. The main pipeline used to convey water through the pumps to the reservoir. The elevated reservoir, which supplies water to a community. The combinatorial function of different pumps is assigned with a fixed water discharge and corresponds to fixed electric energy consumption. The characteristics of the different combinations of hourly discharge energy consumption of the pump system is presented in (Savic et al., 1997). Figure 2: Water demand curve 1. 81
The demand in two states of the network presented in Fig. 2 & 3.Their main difference is that in curve 3 two demand peaks occur in 8am and 6 7pm. Figure 3: Water demand curve 2. The average daily demand is 54788 m 3. Solutions that correspond to final discharges equal to the daily demand ± 10% are accepted. In that way, the Pareto front comprises of solutions that cover a range around the daily demand. 2.1. Mathematical formulation of the problem The objectives of the present problem include the minimization of: Electric energy cost All the pumps of the pumping station consume electric energy during their operation. This cost is an important objective of the problem, since it represents the main cost of the operational pumping station. The energy cost per KWh varies during a 24 hour operation and this variation depends on the tariff structure. In the present problem the cost of electric energy is constant between 7 am and 11pm and reduced by 50% during the night (11pm 7 am): High Cost: From 7:00 to 23:00 (C H = 0,05625 / KWh) Low Cost: From 23:00 to 7:00 (C L = 0,02812 / KWh) Electric energy peak consumption Electricity companies have often an additional factor in their charging policy, the peak power consumption which acts as a penalty if the maximum energy demand exceeds a certain limit. Thus, it is important to keep the maximum consumption among all time intervals as low as possible. Pump maintenance cost The cost of maintenance of the pumping system can be an important parameter, sometimes as important as the cost of electricity. Using indirect methods to predict the wear of a pumping system is an advisable method. Such a method suggests that the number of switches corresponds to the maintenance cost, because a change in the state of a pump causes a considerable wear. This idea was presented for the first time by Lansey and Awumah in 1994. Obviously a pump that was working in a preceding time interval and continues to work doesn t constitute a switch. Similarly, a pump that was off in the previous time step and doesn t work in the present interval doesn t constitute a switch, too. Reducing the number of switches in the daily operation of the system has been proved to be beneficial to the pump maintenance cost. 82
3. RESULTS Multiobjective HSA optimized three different scenarios for its demand - curve. In these scenarios the size of the elevated reservoir is altered. For all cases, the results were obtained after several runs of the program. The results were afterwards transferred to excel and 3d diagrams were created for each case. WATER DEMAND CURVE 1 1 st Scenario: Water Deposit Volume = 15400 m 3 Fig. 4a: Peak of electricity demand Fig. 4b: Damage of pumps For each scenario there are 2 diagrams. One contains the solutions that offer maximum flow rate, minimum cost and minimum peak of the electricity demand, while the other shows the solutions that offer maximum flow rate, minimum cost and minimum amount of damage for the pumps. The pumps were considered to be damaged exclusively when switched off and on and for this reason the damage of the pumps was calculated by the number of times a pump was switched on. The diagrams consist of 3 axes. The F1 axis represents the flow rate of each solution and the F2 its cost. The F3 axis represents in a diagrams the peak of the electricity demand, while in the b diagrams the number of times the pumps was switched on. The several dots represent the different solutions. On every diagram, apart from the solutions themselves the projections of them on the F1 F2 level (flow rate cost), can be found. By watching the projections of the solutions one can easily understand how the solutions are spread in the 3d space, without examining the diagrams by different view angles. Consequently, for each case the diagrams are presented once, from a standard view angle. In all cases the total water demand was the same and equal to 54788 m 3. However, every solution offers a different daily water supply. The accepted ones offer a daily supply between 90% 54778 m 3 and 110% 54788 m 3. 83
2 nd Scenario: Water Deposit Volume = 18200 m 3 Fig. 5a: Peak of electricity demand Fig. 5b: Damage of pumps In case a constraint was included and every accepted solution had to offer an exact daily supply of 54788 m 3, the convergence would fail. Furthermore, it is quite realistic to consider that the total daily demand will vary during the days of the year and moreover it may change in the future. 3 rd Scenario: Water Deposit Volume = 21000 m 3 Fig. 6a: Peak of electricity demand Fig. 6b: Damage of pumps 84
WATER DEMAND CURVE 2 1 st Scenario: Water Deposit Volume = 15400 m 2 Fig. 7a: Peak of electricity demand Fig. 7b: Damage of pumps It can be easily seen by the diagrams themselves and the projections of the solutions into F1 F2 level, that as the amount of water offered by a solution increases, the cost increases as well. This was totally expected. 2 nd Scenario: Water Deposit Volume = 18200 m 2 Fig. 8a: Peak of electricity demand Fig. 8b: Damage of pumps What is more interesting is that the projections of the solutions form a stripe the width of which varies. Most of the solutions that offer similar amount of water supply have similar cost and their projections are close to one another. 85
3 rd Scenario: Water Deposit Volume = 21000 m 3 Fig. 9a: Peak of electricity demand Fig. 9b: Damage of pumps However, it is clear that there are also some solutions that are not close to the majority of the rest (for similar water supply). The explanation is that the diagrams include a 3 rd parameter as well. There is a smaller number of solutions that offer smaller peak or less damage to the pumps, even though the cost of them is relevantly higher. These solutions also belong to the optimum ones however. For the greater amounts of water supply this phenomenon becomes more intense, due to the reason that all different combinations of pumps can be used. Consequently, several different electricity peaks and pump damages may occur in these solutions. By looking closely to the diagrams, one can moreover see that, for the minimum amounts of water supply, solutions that include high electricity peak or high pump damage are not numerous (if there are any). This is due to the reason that fewer combinations of pumps can lead to the lower amounts of water supply and the majority of them do not require great amounts of electricity. In addition, one can notice that, for the greater amounts of water supply, there are fewer solutions that lead to very low electricity peak or damage to the pumps, which is totally reasonable. 4. CONCLUSIONS In the present paper, Harmony Search has been applied towards the optimization of a demanding multiobjective problem. MO HSA in the application on a pump scheduling problem has shown a remarkable performance. It optimized three different scenarios for two different water demand curves. The solutions obtained, covered the entire Pareto fronts. In these figures the decision maker can find optimal solutions, members of the Pareto optimal set, and choose applicable solutions to each situation. 86
Applying further multiobjective HSA on water management problems can be an important and a robust tool towards multiobjective optimisation. Besides, in the present paper it is proved that MO HSA provides excellent results in complex multiobjective water management problems. REFERENCES 1. Barán Β., von Lücken C., Sotelo A. (2005). Multi objective pump scheduling optimization using evolutionary strategies. Advances in Engineering Software, vol. 36, pp. 39 47. 2. Deb K. (1999). Multi objective genetic algorithms: Problem difficulties and construction of test problems. Evolutionary Computation, 7(3), pp. 205 230. 3. Fonseca C.M. and Fleming P.J. (1996). On the performance Assesment and Comparison of Stochastic Multiobjective Optimizers. Lecture Notes in Computer Science. Volume 1141/1996, pp. 584 593. 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., Hwangbo H. (2006). Application of Harmony Search to Multi-Objective Optimization for Satellite heat pipe design, In: Proceedings of 2006 US Korea Conference on Science Technology & Entrepreneurship (UKC 2006). 6. Geem Z.W. (2010). Multiobjective Optimization of Time Cost Trade Off Using Harmony Search. Journal of Construction Engineering and Management. Vol. 136, No. 6, pp. 711 716. 7. Kougias I. and Theodossiou N. (2010). A new music-inspired harmony based optimization algorithm. Application in water resources management problems, In: Proceedings of International Conference on Protection and Restoration of the Environment X, Corfu 2010. 8. Kougias I. and Theodossiou N. (2012). Application of the Harmony Search optimization algorithm for the solution of the multiple dam system scheduling. Optimization and Engineering 9. Lansey KE, Awumah K. (1994). Optimal pump operations considering pumpswitches. Water Resources Planning and Management. 120(1). 10. Luke, S (2009), Essentials of Metaheuristics. Available at: http://cs.gmu.edu/~sean/book/metaheuristics/ 11. Mackle G., Savic D., Walters G. (1995). Application of Genetic Algorithms to Pump Scheduling for Water Supply. GALESIA 1995. In: Conference Publication 4/4, 400 405. London: Institute of Electrical Engineers. 12. Savic D., Walters G., Schwab M. (1997). Multi objective Genetic Algorithms for Pump Scheduling in Water Supply. Erasmus, Exeter, UK/Stuttgart, Germany: School of Engineering of Exeter / University of Stuttgart. 13. Sivasubramani S., Swarup K. S. (2011a). Multi-objective harmony search algorithm for optimal power flow problem, Electrical Power and Energy Systems, Vol.33, pp. 745-752. 14. Sivasubramani S., Swarup K. S. (2011b). Environmental / Economic dispatch using multiobjective harmony search algorithm, Electric Power Systems Research, Vol.81, pp. 1778-1785. 15. Theodossiou N., Kougias I., Kakoudakis K., Doikos K. (2011). Harmony Search Algorithm, a novel optimization technique. Parameter calibration and applications on water resources management problems. In: Proceedings of 3 rd International Conference on Environmental Management, Engineering,Planning and Economics (CEMEPE 2011) & SECOTOX Conference, Skiathos. 87