Incorporation of Non-Convex Objective Functions Describing Real Costs Into a Groundwater Remediation and Optimization Algorithm

Optimization of groundwater remediation is a rapidly developing technology with the potential to significantly improve the cost-effectiveness of groundwater remediation design. Although a wide variety of optimization techniques are bring tested, most have assumed surrogates for cost, such as total pumping, and have assumed steady-state pumping. Both these assumptions have been shown to lead to sub-optimal designs. This work evaluates the performance of a dynamic control application for time-varying pump-and-treat remediation using a realistic cost function for granular activated carbon treatment (GAC).

Granular activated carbon technology is one of the most widely used techniques to remove organic contaminants from groundwater. Due to the partitioning of the contaminant from the aqueous to sorbed phase, the performance and costs of GAC depend on both the volume and concentrations of the treated water. Air stripping, another common removal technique for volatile organics, is based on air-water partitioning, and thus is also sensitive to both the volume and concentration of the influent. Total pumping rates alone are poor surrogates for the costs of these partitioning-based treatment technologies. Furthermore, the cost functions for these treatment technologies are non-linear functions with significant economies-of-scale. Together the economies-of-scale and the nonlinear response of concentration with pumping make the realistic cost function non-convex.

In this work, a quasi-Newton differential dynamic programming algorithm for groundwater remediation was modified to include a realistic GAC cost function. Formulations were run with and without fixed costs of treatment.· Various numerical techniques, including shifting the Hessian matrix (Yakowitz, 1984) and iterative approximations of the non-convex objective, were used to control the non-convexity of the problem.

Numerical experiments demonstrated that the dynamic control algorithm had . difficulty handling a realistic cost function, even when using a shift in the Hessian matrix. Remediation designs wer(! successfully found using iterative approximations to the non-convex function, but the approximation process does not guarantee overall optimality. This work suggests that improvements in the control theory algorithm are needed for it to be a practical technique for these realistic objective functions. Non-gradient based optimization techniques may be more robust for these dynamic nonconvex problems.