Stochastic analysis of process-control models

Most conventional methods of process design ignore the stochastic nature of the process and its param eters ; only deterministic models are employed, and the solutions obtained are at best only approxima tions. To allow for the uncertainty of designs based on such a procedure, gross safety factors must be added to the design variables based on the deterministic models. An alternative approach is to introduce stochastic variability directly into the process model and to use stochastic simulation to estimate the process outputs and design parameters. By using stochastic simulations which incorporate random inputs and ran dom parameters, one can obtain sample estimates of the expected values and expected variances of the output variables. From these and the characteristics of the probability distributions of the output vari ables, one can show when the expected values of the outputs of stochastic process models differ from those of deterministic models in which the random variables are replaced by their expected values and what safety factors are necessary for a satisfactory design. As an example, we have developed a stochastic dynamic mathematical model for a wastewater treatment plant. The output of the plant was simulated for a period of several days under different random input conditions and different probability distributions of parameters. Differences between the deterministic and stochastic responses of the model are described, as is the effect of frequency of fluctuations of the stochastic inputs on responses of the model and on their variances. Of particular interest is the use of the results of the simulations, stochastic and deterministic, to estimate safety (overdesign) factors that should be used in designs based on deterministic models. The validity of the propagation-of-variance formula- i.e., the relation used to compute the sample vari ances of the outputs of a nonlinear stochastic model in the unsteady state from the sample variances of the inputs and parameters - is demonstrated. With the aid of this relation, overdesign factors can be esti mated from the magnitudes of the standard deviations of the outputs. Although stochastic analysis cannot reduce the uncer tainty in design, it can provide a more quantitative assessment of the uncertainty than can deterministic design and so can lead to better designs of process plants.