Dynamic optimal control for groundwater remediation with flexible management periods

Abstract

A successive approximation linear quadratic regulator (SALQR) method with management periods is combined with a finite element groundwater flow and transport simulation model to determine optimal time-varying groundwater pump-and-treat reclamation policies. Management periods are groups of simulation time steps during which the pumping policy remains constant. In an example problem, management periods reduced the total computational demand, as measured by the CPU time, by as much as 85% compared to the time needed for the SALQR solution without management periods. Conversely, the optimal costs increased as the number of times that the control can change is reduced. With two simulation periods per management period, the optimal cost increased by less than !% compared to the optimal cost with no management periods, yet the computational work was reduced by a third. The optimal policies, including the number and locations of wells, changed significantly with the number of management periods. Complexity analysis revealed that the SALQR algorithm with management periods can significantly reduce the computational requirements for nonsteady optimization of groundwater reclamation and other management applications. INTRODUCTION The design of pump-and-treat groundwater remediation projects is typically done by trial-and-error simulation of feasible combinations of pump locations and pumping rates. Given the countless number of feasible reclamation designs for a given site, the optimal management solution might never be identified by trial-and-error simulation. Mathematical optimization techniques combined with simulation can efficiently search through the possible design options and thus can be a useful tool for planning cost-effective pumpand-treat reclamation. Most mathematical optimization models for groundwater reclamation have determined static (or steady state) policies in which the pumping does not change over time. Given that groundwater remediation may occur over a period of decades, steady state policies would not be expected to be as cost-effective as dynamic policies in which the pumping policies are allowed to change as the contaminant plume moves. Control theory algorithms were developed specifically for dynamic systems and can readily handle timevarying pumping policies. In the few previous applications of control theory techniques to groundwater remediation [Andricevic and Kitanidis, 1990; Lee and Kitanidis, 1991; L.-C. Chang, C. A. Shoemaker, and P. L.-F. Liu, Application of a constrained optimal control algorithm to groundwater remediation, submitted to Water Resources Research, 1991 (hereinafter Chang et al., submitted, 1991)], the time steps of the optimization algorithms were identical to the time steps of the numerical models describing contaminant transport. The focus of this paper is the development of an algorithm for time-varying optimization of groundwater remediation in which the time steps in the optimization model (called "management periods") may be longer than the time periods in the numeri...

Integration of Optimal Dynamic Control and Neural Network for Groundwater Quality Management

“…Besides, optimal operating cost in ANN-CDDP and ISOQUAD-CDDP is illustrated in Table 3. Proposed model (ANN-CDDP) accuracy can be quantified when compared with ISOQUAD-CDDP (Chang et al 1992;Culver and Shoemaker 1992). The cost results demonstrate that relative error is 2.16% or less.…”

“…Applying an optimization technique such as GA and simulated annealing (SA) to solve time-varying policies would dramatically increase required computational resources (Mckinney and Lin 1994;Wang and Zheng 1998;Rao et al 2003). Accommodating these situations, an optimal dynamic groundwater remediation design is required to use constrained differential dynamic programming (CDDP) (Jones et al 1987;Chang et al 1992;Culver and Shoemaker 1992). Furthermore, the CDDP significantly reduces working dimensionality of the algorithm over that of mathematical programming algorithms, by taking advantage of the dynamic groundwater supply or water quality optimization problems through stage-wise decomposition .…”

A Primal Method for the Solution of the Groundwater Quality Management Problem

“…This rate of growth is lower if optimal control techniques are used (in this case the decision variables become the control variables of the problem), but the computer time in this case is proportional to n 3 , if n is the number of nodes in the simulation model. For further information in this regard see Culver and Shoemaker (1992). In the method introduced for the first time in this paper the exponent m is significantly reduced.…”

Optimal remediation of unconfined aquifers: Numerical applications and derivative calculations

“…Each management period is d simulation time steps long. The problem of computing the least-cost set of pump-andtreat pumping rates for a given set of wells over all management periods may be solved using the following formulation [Culver and Shoemaker, 1992 …”

“…In addition, we describe derivatives for management periods, which may be used both for time-varying optimization, such as the policies given by Chang et al [1992], and for time-invariant policies, such as those considered by Gorelick et al [1984]. Management periods are sequential groups of simulation model time steps over which pumping rates are held steady [Culver and Shoemaker, 1992;Rizzo and Dougherty, 1996], allowing implementation of pumping strategies which call for fewer adjustments to pumping rates than would be required if rates were adjusted at every simulation time step.…”