Integrated statistical metric of flexibility for systems with discrete state and continuous parameter uncertainties

This paper addresses the problem of developing a quantitative measure for the flexibility of a design to withstand uncertainties in the continuous parameters and discrete states. The metric is denoted as the expected stochastic flexibility, E(SF). For a given a linear model, a Joint distribution for the parameters and probabilities of failure for the discrete states, the proposed metric predicts the probability of feasible operation of a design. A novel inequality reduction scheme is proposed to aid in performing the integration over the feasible region characterized by inequalities. A bounding scheme is also proposed to avoid the examination of a large number of discrete states when determining the E(SF). An example problem is presented to demonstrate the fact that the proposed measure provides a framework for integrating flexibility and reliability in process design. UNIVERSITY LIBRARIES CARNEGIE MEUMH UNIVERSITY PITTSBURGH, PA 15213-3890 Introduction The operation of chemical plants is subject to uncertainties in the parameters and in the reliability of its design components. The uncertain parameters typically include such items as flowrates, temperatures, and kinetic rate constants, while design uncertainties include the availability of equipment. Note that these uncertainties are of two types: continuous and discrete, respectively. The types of uncertainties are distinguished by the values they take. Continuous uncertainties may take on a range of values while discrete uncertainties take on only specific values. Because the feasible operation of a chemical plant is clearly dependent upon these uncertainties, it is important to be able to quantitatively determine the effect of these uncertainties on plant operation. This has been done for two cases, processes with only continuous parameter uncertaintiesflexibility (Swaney and Grossmann 1985, Saboo and Morari 1984, Pistikopoulos and Mazzuchi 1989) and processes with only discrete state uncertainties-reliability, (Tzafestas 1980, Shooman 1968, Dhillon 1988). Neither of these two problems, however, fully captures the nature of uncertainties in chemical processes. There is obviously a great need to handle both types of uncertainties together, since they show strong interactions in defining the feasible operation of a plant. The goal of this paper is to develop a stochastic metric for feasible operation in systems that can be represented by a linear model, and where we can simultaneously account for both types of uncertainties. The proposed metric represents the probability of feasible operation for given probabilities in the discrete states and for given joint distribution functions of the continuous uncertain parameters. For the latter, a novel inequality reduction scheme is proposed that allows the efficient numerical integration of the joint distribution function over the feasible region for a given state. Since the proposed metric requires the enumeration of a large number of discrete states, an effective bounding scheme is proposed that exploits the structure found in reliability problems. Application of the proposed metric is illustrated with an example problem which shows that flexibility and reliability can be integrated within a common framework. Motivating Example The most common example of a process containing both discrete and continuous parameter uncertainties is a continuous chemical process with redundant equipment. In continuous chemical processes there are typically uncertain continuous parameters such as flowrates, product demands, and thermodynamic constants. There are also uncertainties in the discrete states corresponding to the availability of equipment. For example pumps, compressors, or reactors may experience failure modes and become inoperable. An example of this type of system is shown in Figure 1.