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Framework for Identifying Optimal Allocations

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A spreadsheet modeling framework has been developed to identify optimal allocations under a variety of watershed conditions. This framework has been applied to a small watershed representing conditions in rural Idaho. The goal of this framework is to identify management practices and technical controls (i.e., decisions) that:

This process is called optimization, and the set of management decisions that satisfy the allocation objective, subject to constraints and requirements is called the optimal solution. As an example, optimization can be used to find manure management practices and wastewater treatment technology levels that will minimize the cost (i.e., objective) of satisfying average and 90% upper-bound fecal coliform criteria (i.e., requirements) subject to nutrient needs for crop production (i.e., constraints).

If pollutant requirements include upper-bound, in addition to to average criteria then the framework must also be capable of estimating concentration distributions, not just single concentration values . This framework is therefore stochastic and incorporates Monte Carlo simulations to account for sources of uncertainty (e.g., rainfall and stream flow) that affect predicted concentrations over time.

Figure 1: Flow chart explaining the relationship between load models, fate models, and cost models in the iterative optimization process.

Optimization consists of an iterative search for an optimal solution, and each iteration begins by selecting values for decision variables representing management decisions (figure 1). Cost functions, load models, and hydrologic models are then used to estimate costs and concentration distributions. If water quality criteria or constraints are not satisfied, or if the objective is not being met, then the framework selects new values for decision variables. Iterations continue until criteria, constraints and objectives are met.

It is relatively easy to incorporate cost and load functions into spreadsheets, but integrating dynamic hydrologic models requires more complex linkages and extensive simulation times. To eliminate this problem, direct links to hydrologic models are replaced with stochastic impact coefficients.

Impact coefficients describe how in-stream concentrations at a point in the water body change as source loads change. The coefficients are defined as probability density functions to capture coefficient variability associated with weather and other sources of uncertainty.

βti = Impact coefficient for source i, during month t
βti ~ Probability density function
Concentrationt = Σi(βti*Loadti)

In the consolidated optimization process (see figure below), a sample of impact coefficients are randomly drawn from the impact coefficient pdfs, using risk simulations (e.g., Monte Carlo analysis). Source loads are multiplied by the sample of impact coefficients to generate populations of concentrations, and mean and upperbound concentrations from this population can then be compared to criteria.

Figure 2: Flow chart explaining the relationship between the optimizer, the monte carlo simulator, loading models, and cost models in the consolidated modeling framework.

Direct links to hydrologic models have been replaced by a concentration "forecasting" process that relies on impact coefficients derived from calibrated hydrologic modeling output. Computational time is saved because there is no need to repeat hydrologic modeling for each iteration. The framework itself is developed within a spreadsheet format to take advantage of the ubiquitous and flexible nature of spreadsheets. Optimization and rish analysis tasks are accomplished with the help of spreadsheet addins.

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