An Indirect Simulation-Optimization Model for Determining Optimal TMDL Allocation under Uncertainty

An indirect simulation-optimization model framework with enhanced computational efficiency and risk-based decision-making capability was developed to determine optimal total maximum daily load (TMDL) allocation under uncertainty. To convert the traditional direct simulation-optimization model into our indirect equivalent model framework, we proposed a two-step strategy: (1) application of interval regression equations derived by a Bayesian recursive regression tree (BRRT v2) algorithm, which approximates the original hydrodynamic and water-quality simulation models and accurately quantifies the inherent nonlinear relationship between nutrient load reductions and the credible interval of algal biomass with a given confidence interval; and (2) incorporation of the calibrated interval regression equations into an uncertain optimization framework, which is further converted to our indirect equivalent framework by the enhanced-interval linear programming (EILP) method and provides approximate-optimal solutions at various risk levels. The proposed strategy was applied to the Swift Creek Reservoir’s nutrient TMDL allocation (Chesterfield County, VA) to identify the minimum nutrient load allocations required from eight sub-watersheds to ensure compliance with user-specified chlorophyll criteria. Our results indicated that the BRRT-EILP model could identify critical sub-watersheds faster than the traditional one and requires lower reduction of nutrient loadings compared to traditional stochastic simulation and trial-and-error (TAE) approaches. This suggests that our proposed framework performs better in optimal TMDL development compared to the traditional simulation-optimization models and provides extreme and non-extreme tradeoff analysis under uncertainty for risk-based decision making.