# ابزار مدیریت آبهای زیرزمینی برای حل مشکل به حداقل رساندن هزینه پمپاژ برای حوضه تاهتالی(ازمیر ترکیه) با استفاده از الگوریتم بهینه سازی ترکیبی حل کننده جستجوی هارمونی

کد مقاله | سال انتشار | تعداد صفحات مقاله انگلیسی |
---|---|---|

6597 | 2013 | 14 صفحه PDF |

**Publisher :** Elsevier - Science Direct (الزویر - ساینس دایرکت)

**Journal :** Journal of Hydrology, Volume 478, 25 January 2013, Pages 63–76

#### چکیده انگلیسی

This study proposes a linked simulation–optimization model to solve the groundwater pumping cost minimization problem for existing and new wells to satisfy any given water demand. The proposed model integrates MODFLOW-2000 with HS-Solver which is a recently proposed global–local hybrid optimization algorithm that integrates heuristic harmony search (HS) algorithm with the spreadsheet Solver add-in. Using the proposed model, a pumping cost minimization problem is solved for different number of wells by considering the pumping rates as well as the locations of additional new wells as the decision variables. Some physical and managerial constraints are defined for this problem. These constraints that need to be satisfied in the optimization process are set up using the penalty function approach. The performance of the proposed model is evaluated on the groundwater flow model of the Tahtalı watershed (Izmir-Turkey), an urban watershed which is a key component of Izmir’s water supply system. Also, a sensitivity analysis is performed to evaluate the model results for different sets of HS solution parameters. Results indicate that the proposed simulation–optimization model is found to be efficient in identifying the optimal numbers, locations, and pumping rates of the pumping wells for satisfying the given constraints. Results also show that the model is not only capable of obtaining just any mathematically plausible solution but a realistic one that can be confirmed by repetitive runs of the model.

#### مقدمه انگلیسی

Managing groundwater systems is a challenging task since the quality and quantity of groundwater resources are continuously deteriorating mostly due to anthropogenic factors. Linked simulation–optimization models are essential tools in the development of management strategies for groundwater systems. The main idea of these tools is to evaluate modeling results and select the best management strategy using optimization models with respect to prescribed physical or managerial constraints (Singh and Datta, 2006). Many studies in the literature deal with the solution of groundwater management problems through simulation–optimization models. These studies differ among themselves in the way governing PDEs are solved in the simulation stage and the types of algorithms used in the optimization stage (Ahlfeld et al., 2005). A detailed literature survey of groundwater management models and an overview of the applications can be found in Gorelick, 1983 and Willis and Yeh, 1987, and Das and Datta (2001). Although the groundwater management studies have some differences in terms of the simulation models, the main differences stem from the variety of the optimization models. Initial studies about linked simulation–optimization based groundwater management were usually performed using gradient-based optimization methods. These methods include linear programming (LP) (Willis, 1983, Hallaji and Yazıcıgil, 1996 and Mantoglou, 2003), nonlinear programming (NLP) (Gorelick et al., 1984 and Finney et al., 1992), and dynamic programming (DP) (Jones et al., 1987 and Culver and Shoemaker, 1992). Although these methods are efficient in finding optimum solutions with reasonable computational times, their accuracy is mostly tied to the initial solutions since groundwater management problems usually have non-convex solution spaces and hence obtaining the global optimum is not guaranteed at all times (McKinney and Lin, 1994 and Ayvaz, 2009). Therefore, heuristic optimization algorithms are generally preferred for the solution of groundwater management problems through simulation–optimization models. Many heuristic optimization algorithms mimic some natural phenomena. These include natural selection and evolution in genetic algorithm (GA) (Holland, 1975 and Goldberg, 1989), physical annealing process in simulated annealing (SA) (Kirkpatrick et al., 1983), social behaviors of birds or fishes in particle swarm optimization (PSO) (Kennedy and Eberhart, 1995), finding shortest paths between nest and a food source in ant colony optimization (ACO) (Dorigo et al., 1996), and musical improvisation process in harmony search (HS) (Geem et al., 2001). Many studies that apply heuristic optimization methods to the solution of groundwater management problems are published in the literature. One of the first applications was done by McKinney and Lin (1994) in which three separate management problems were solved including maximization of total pumping and minimization of pumping cost to satisfy the given water demand, and minimization of the remediation cost. McKinney and Lin (1994) performed a detailed analysis on solving these three problems and showed the superiority of GA by solving the same problems with LP and NLP. In a later study, Wang and Zheng (1998) compared the performance of GA and SA by solving the same management problems of McKinney and Lin (1994). Their results showed that both GA and SA give nearly identical or better solutions than those obtained by several traditional solution methods. Wu et al. (1999) developed a solution approach called GA based SA penalty function approach (GASAPF) to solve groundwater management problems. Their results showed that GASAPF model can be effective in solving these problems. Zhu et al. (2006) developed a groundwater management model which uses the shuffled complex evolution (SCE) algorithm in the optimization model. They applied their SCE based solution model to the solution of the management problems of the Yangtze Delta and compared their identification results with GA. Their results indicated that SCE based optimization model provides better results than GA for the same flow conditions. Ayvaz (2009) developed a combined simulation–optimization model where MODFLOW (Harbaugh, 2000) was used as the simulation model and HS as the optimization model. Ayvaz (2009) solved the same problem of McKinney and Lin (1994) with this model and obtained identical or better results than those obtained by GA. Sedki and Ouazar (2011) compared the performance of two common swarm intelligence techniques namely PSO and ACO for the solution of several groundwater management problems. Their results indicated that while the convergence speed and solution quality of PSO is better than ACO for the problems with small number of decision variables, ACO provides better results for the solutions with high number of decision variables. It should be noted that all these examples considered fixed locations of pumping wells such that the coordinates of wells remain constant during the search process. Although fixed well based solution approaches are successful in determining the optimum pumping schedule for existing wells, they are not suitable to find potential locations of new wells to satisfy the given groundwater demand. Therefore, pumping well locations as additional decision variables must be considered together with associated pumping rates. In the current literature, studies that consider variable well locations to solve groundwater management problems through heuristic approaches are limited. Huang and Mayer (1997) developed a GA based solution model that considers the well locations as discrete decision variables for solving remediation system design problems. Park and Aral (2004) developed a multi-objective optimization model which is based on the progressive GA (PGA). They determined pumping rates and corresponding well locations in coastal aquifers. Ayvaz and Karahan (2008) developed a GA based solution approach to determine the number, locations and pumping rates of illegally drilled pumping wells. Similarly, Lin and Yeh (2008) developed a SA based solution model to determine the pumping source information including source locations, pumping rates, and pumping periods. Gaur et al. (2011a) developed a groundwater management model to determine the locations and pumping rates of potential wells by integrating the analytical elements method (AEM) to a PSO based optimization model. Similarly, Gaur et al. (2011b) compared the performance of two management models which were developed by combining AEM with PSO (AEM-PSO) and finite difference solution of governing PDEs with PSO (FDM-PSO). It should be noted that although heuristic optimization approaches were successively employed to solve several management problems, they may require long computation times to precisely find the optimum solution and satisfy the given physical and managerial constraints (Ayvaz et al., 2009). Recently, hybrid optimization algorithms were successfully used for the solution of complex optimization problems with a non-convex solution space. These algorithms integrate heuristic and gradient-based algorithms such that the main objective is to utilize the global exploring capability of heuristic and strong fine-tuning property of gradient-based algorithms. In this integration, the global search process starts with multiple starting points and explores the entire search space, and then, gradient-based search methods find the optimum solution by assigning the results of global search as their initial values (Ayvaz et al., 2009). Although hybrid optimization algorithms can be effective in finding the global optimum precisely, programming of them is usually difficult since most of the gradient-based algorithms require some advanced mathematical calculations such as partial derivatives, Jacobian/Hessian matrices, and inversions (Ayvaz et al., 2009). Therefore, developing a robust hybrid algorithm can be a challenging task. Nowadays, spreadsheet programs have become an essential tool for performing engineering calculations due to their popularity and availability. Most commercial spreadsheet packages contain a “Solver” add-in to solve optimization problems (Frontline Systems, 2012). The main advantage of “Solver” is that it does not only effectively solve many linear, nonlinear, and mixed-integer type optimization problems, but also successfully satisfies the given constraints. In addition, it is easy to use since it does not require much knowledge about programming gradient-based optimization algorithms. Recently, a new hybrid optimization algorithm, HS-Solver, is proposed by (Ayvaz et al., 2009). HS-Solver consists of the integration of heuristic HS optimization algorithm with the spreadsheet Solver add-in. In this integration, a set of multiple solutions are generated using HS based on its computational structure, and then, the generated solutions are improved by the Solver. This type of solution sequence makes finding the global optimum solution easier than both global and local searches by HS and Solver, respectively. Ayvaz and Elçi (2013) recently applied HS-Solver for the solution of two groundwater management applications. Their first application deals with the solution of groundwater pumping maximization problem on a hypothetical aquifer model with fixed well locations. They applied HS-Solver to the solution of the related problem and compared their identification results with those obtained by different solution approaches (McKinney and Lin, 1994, Wang and Zheng, 1998, Wu et al., 1999 and Ayvaz, 2009). In their second application, they applied HS-Solver to the solution of pumping maximization problem for the groundwater flow model of the Tahtalı watershed (Izmir-Turkey). They considered 17 pumping wells with fixed locations and maximized their pumping rates using both HS and HS-Solver. The results of both of their applications indicate that HS-Solver does not only provide better results than other deterministic and heuristic based solution approaches, but also solves the pumping maximization problems with much lower iteration numbers. The main objective of the present study is to propose a new groundwater management model for existing and new wells that solves the pumping cost minimization problem by satisfying any given water demand. In the proposed model, the groundwater flow process is simulated using MODFLOW-2000 (Harbaugh, 2000), a modular three-dimensional finite difference groundwater flow model. This MODFLOW based simulation model is then integrated with a HS-Solver based optimization model. Since the solution is based on a hybrid approach, the optimization process starts with randomly generated solutions by HS, and then, the identified solutions are subjected to local search by Solver. The main objective of the HS-Solver based management model is to solve the pumping cost minimization problem for different number of wells by considering the pumping rates and the locations of additional new wells as decision variables. During the solution, some physical and managerial constraints are considered by the model such that the amount of pumped groundwater must not be lower than the given water demand, hydraulic head values at proposed well locations must not be lower than the bottom elevation at that point, locations of the proposed wells must not be placed into any inactive grid cells within the finite difference model domain, depth of the proposed wells must not exceed a permissible limit, and saltwater intrusion along the coast line must not occur due to excessive pumping. All of these constraints are included in the optimization model using the penalty function approach. The proposed management model is evaluated on a groundwater flow model of the Tahtalı watershed (Izmir-Turkey) which is a key component of Izmir’s water supply system. Also, a sensitivity analysis is conducted to determine the effects of different HS solution parameters on the solution accuracy. Identified results indicate that the proposed model is not just efficient in identifying optimal numbers, locations, and pumping rates of the wells, but also satisfies all the constraints to maintain minimum cost. The presentation of this study is organized as follows: first, basic concepts of simulation and optimization models are described; second, brief information about the study area and previously developed groundwater flow model are presented; third, the application of the proposed management model to the solution of pumping cost minimization problem of the Tahtalı watershed is described; and finally, results of the performed sensitivity analysis for different sets of HS solution parameter sets are presented.

#### نتیجه گیری انگلیسی

In this study, a linked simulation–optimization model is proposed to solve the groundwater pumping cost minimization problem. In the proposed model, the simulation part uses MODFLOW-2000 to simulate the groundwater flow process. This part is then linked to an optimization model where the global–local hybrid HS-Solver algorithm is used. The proposed simulation–optimization model is demonstrated on the problem of pumping cost minimization for the groundwater flow model of the Tahtalı watershed (Izmir-Turkey). The main problem here is to find alternative well field locations that can replace in the future the Halkapınar well field in the city center. With this purpose, the problem is solved for different number of wells by taking the locations and pumping rates of the new wells as the decision variables. Some physical and managerial constraints have been integrated to the solution using the penalty function approach. Finally, a sensitivity analysis is performed to evaluate the variability of model results for different HS solution parameters sets. The following conclusions can be drawn from the results of this study. Based on the identification results, it can be concluded that the use of the Halkapınar well field alone is not suitable to satisfy the given water demand due to the potential saltwater intrusion problem. This issue can be clearly seen from the results in Table 2. The identified rates for Halkapınar well field in all solutions are lower than 20,000 m3/day. This means that higher pumping rates for this well field leads to obtain penalized objective function values since the problem of saltwater intrusion is controlled by Eq. (16). Therefore, alternative well locations are sought by the optimization model. The main objective of HS-Solver is to determine the new potential well field locations with a minimum cost of pumping. The results shown in Table 2 indicate that the minimum pumping cost is obtained for the solution with four pumping wells. Since the objective of the proposed model is to seek new well locations, exact satisfaction of the given water demand is a crucial step to obtain a solution with minimum cost. Although heuristic optimization algorithms with penalty functions are successful on satisfying the given constraints, they may require high iteration numbers to exactly satisfy them. This situation can be observed in Table 1 where the amount of total water supply is greater than the given water demand. Although these values are very close to the given demand, their exact satisfaction is required to ensure minimum pumping cost. Therefore, all the solutions of HS are subjected to local search by Solver since it has an ability of satisfying the constraint equations without requiring more iterations. After solving the problem with Solver, the water demand constraint is satisfied exactly for all solutions. This result shows HS-Solver’s strong capability of finding the optimum solutions. As can be seen from the identified well locations in Fig. 7, all the new pumping wells are placed around the main streams. This outcome can be explained by the high water potential of the grid cells around the streams. Since more water potential of a grid cell corresponds to lower drawdown, the obtained solutions are associated with lower pumping cost. Furthermore, since the results of the HS based optimization model depend on the values of its related solution parameters, a sensitivity analysis is performed to determine the influence of this parameters on solution accuracy. Analysis results indicate that the response of the proposed model for different parameter sets is practically acceptable, and therefore, the proposed simulation–optimization model is as effective way of solving the pumping cost minimization problem discussed here.