動態控制理論與遺傳演算法應用於地下水之管理與污染整治

Abstract

本研究於地下水之水量管理與污染整治模式中，同時考量整數型態之鑿井固定成本與時變抽水量所引致之操作成本。傳統演算法如線性規劃或非線性規劃對於目標函數中包含整數型態之變數如設井之位置與數量並不容易考量，而離散型態之演算法如整數規劃或離散動態規劃，雖可求解整數型態變數問題，然若求解之問題中包含時變之變數，如水井隨時間改變抽水量，則有變數過多與產生大量計算量之問題，因此雖然系統開發總成本理應包括固定成本與操作成本，以往並未有任何地下水管理規劃模式能同時考量之。本研究乃藉由遺傳演算法（GA）與限制型微分動態規劃（CDDP），將整數型態與時變非線性之變數同時整合於一個模式（GCDDP）中。GA染色體之二進位編碼方式可非常容易的考量設井之位置與數量，然採用GA求解時變抽水量時，則亦有計算量大之問題，故CDDP乃用於求解每一條染色體所對應之時變最佳抽水量與操作成本。拘限含水層與非拘限含水層分別用於驗證本優選模式可應用於地下水之水量管理與污染整治系統中。由數值之優選結果顯示，鑿井固定成本對最佳方案的決定有顯著的影響，當固定成本愈高時，所需之設井數就愈少，因此可節省許多總成本支出。另一方面，含水層地質參數的變化亦影響抽水井的位置與數目，此一優選結果對於採用傳統梯度型演算法並不容易考量。由於本模式能真正考量系統總成本，因此可作為提昇地下水資源管理效率的有力工具，並可將優選結果提供決策者作決策時之參考。

Obtaining optimal solutions for groundwater management and remediation problems, while simultaneously considering both fixed costs and time-varying pumping rates, is a challenging task. Application of conventional optimization algorithms such as linear and nonlinear programming is difficult due to the discontinuity of the fixed cost function in the objective function and the combinatorial nature of assigning discrete well locations. Use of conventional discrete algorithms such as integer programming or discrete dynamic programming is hampered by the large computational burden caused by varying pumping rates over time. A novel procedure that integrates a genetic algorithm (GA) and constrained differential dynamic programming (CDDP), calculates optimal solutions for a groundwater planning problem while simultaneously considering fixed costs and time-varying pumping rates. GA can easily incorporate the fixed costs associated with the installation of wells. However, using GA to solve for time-varying policies would dramatically increase the computational resources required. Therefore, the CDDP is used to handle the sub-problems associated with time-varying operating costs. Numerical experiment for confined and unconfined aquifer that incorporates fixed and time-varying operating costs is presented to demonstrate the effectiveness of the proposed algorithm. Simulation results indicate that the fixed costs can significantly influence the number and locations of wells and a notable total cost saving can be realized by applying the novel algorithm.