Water quality management under the watershed approach of Total Maximum Daily Load (TMDL) programs requires that water quality standards be maintained throughout the year. The main purpose of this research was to develop a methodology that incorporates inter-temporal variations in stream conditions through statistical distributions of pollution loading variables. This was demonstrated through a cost minimization mixed-integer linear programming (MIP) model that maintains the spatial integrity of the watershed problem. Traditional approaches for addressing variability in stream conditions are unlikely to satisfy the assumptions on which these methodologies are founded or are inadequate in addressing the problem correctly when distributions are not normal. The MIP model solves for the location and the maximum capacity of treatment plants to be built throughout the watershed which will provide the optimal level of treatment throughout the year. The proposed methodology involves estimation of parameters of the distribution of pollution loading variables from simulated data and use of those parameters to re-generate a suitable number of random observations in the optimization process such that the new data preserve the same distribution parameters. The objective of the empirical model was to minimize costs for implementing pH TMDLs for a watershed by determining the level of treatment required to attain water quality standards under stochastic stream conditions. The output of the model was total minimum costs for treatment and selection of the spatial pattern of the least-cost technologies for treatment. To minimize costs, the model utilized a spatial network of streams in the watershed, which provides opportunities for cost-reduction through trading of pollution among sources and/or least-cost treatment. The results were used to estimate the costs attributable to inter-temporal variations and the costs of different settings for the 'margin of safety'. The methodology was tested with water quality data for the Paint Creek watershed in West Virginia. The stochastic model included nine streams in the optimal solution. An estimate of inter-temporal variations in stream conditions was calculated by comparing total costs under the stochastic model and a deterministic version of the stochastic model estimated with mean values of the loading variables. It was observed that the deterministic model underestimates total treatment cost by about 45 percent relative to the 97th percentile stochastic model. Estimates of different margin of safety were calculated by comparing total costs for the 99.9th percentile treatment (instead of an idealistic absolute treatment) with that of the 95th to 99th percentile treatment. The differential costs represent the savings due to the knowledge of the statistical distribution of pollution and an explicit margin of safety. Results indicate that treatment costs are about 7 percent lower when the level of assurance is reduced from 99.9 to 99 percent and 21 percent lower when 95 percent assurance is selected. The application of the methodology, however, is not limited to the estimation of TMDL implementation costs. For example, it could be utilized to estimate costs of anti-degradation policies for water quality management and other watershed management issues.

在总最大日负荷(TMDL)计划的流域方法下的水质管理要求全年保持水质标准。本研究的主要目的是开发一种方法，通过污染负荷变量的统计分布将河流条件的跨时间变化纳入其中。成本最小化混合整数线性规划(MIP)模型证明了这一点，该模型保持了流域问题的空间完整性。当分布不正常时，处理河流条件变化的传统方法不太可能满足这些方法所建立的假设，或者在正确处理问题时是不充分的。MIP模型求解了整个流域内将要建造的处理厂的位置和最大容量，这将提供全年最佳的处理水平。拟议的方法涉及从模拟数据估计污染负荷变量分布的参数，并使用这些参数在优化过程中重新产生适当数量的随机观察结果，以便新数据保持相同的分布参数。该经验模型的目标是通过确定在随机水流条件下达到水质标准所需的处理水平，使实施pH TMDLs的成本最小化。模型的输出是治疗的总最小成本和治疗的最低成本技术的空间格局的选择。为了使成本最小化，该模型利用了流域内河流的空间网络，通过污染源之间的污染交易和/或成本最低的处理提供了降低成本的机会。这些结果被用来估计由于跨时期变化而引起的成本，以及“安全边际”的不同设置的成本。该方法用西弗吉尼亚州油漆溪流域的水质数据进行了测试。该随机模型在最优解中包含9个流。通过比较随机模型下的总成本和负荷变量平均值估计的随机模型的确定性版本，计算了河流条件跨时间变化的估计数。观察到，确定性模型相对于第97百分位随机模型低估了大约45%的总处理费用。通过比较99.9百分位治疗(而不是理想的绝对治疗)和95至99百分位治疗的总成本，计算出了不同的安全边际估计数。不同的费用是由于了解污染的统计分布和明确的安全边际而节省的。结果表明，当医疗保障水平从99.9%降低到99%时，治疗成本大约降低7%，而选择95%时，治疗成本降低21%。然而，该方法的应用并不局限于TMDL执行费用的估计。例如，它可以用来估计水质管理和其他流域管理问题的抗退化政策的费用。