Optimal design of process systems under uncertainty
G. M. Ostrovsky1, N. N. Ziyatdinov2, T. V. Lapteva2
1Karpov Institute of Physical Chemistry, Vorontsovo Pole, 10, Moscow, 103064, Russia
Email: email@example.com, Phone: 8-499-4792991
2Kazan State Technological University, Karl Marx str., 68, Kazan, 420111, Russia
Email: firstname.lastname@example.org ; email@example.com
We will consider the problem of the optimal design of a process systems and chemical process (CP) for the case when inexact mathematical models are used. The inexactness of mathematical models arises because of the original uncertainty of chemical, physical, and economic data which are used during the CP design. The problem of the CP optimal design under uncertainty can be formulated as follows: it is necessary to create an optimal CP that would guarantee the satisfaction (exactly or with some probability) of all design specifications in the case when inexact mathematical models are used and internal and external factors change during the CP operation stage.
Usually, the following two formulations of this problem are used
1) The formulation of the two-stage optimization problem (TSOP) takes into account possibility of the control variables change at the operation stage. Here we suppose that at each time instant during the operation stage (a) values of all or some of the uncertain parameters can be either measured or calculated using the experimental data and (b) during the operation stage the control variables are adjusted depending on a chemical process state. This formulation can be used if it is possible to accurately estimate all or some of the uncertain parameters at the operation stage of CP.
2) The formulation of the one-stage optimization problem (OSOP) supposes that the control variables are constant at the operation stage. We will consider the two-stage optimization problem (TSOP) with joint chance constraints has the following form 
where is a goal function, is the expected value of the goal function , d , z are vectors of design and control variables, respectively, q is a vector of uncertain parameters, is the probability of joint satisfaction of all the constraints, i.e. it is the probability measure of the region
where , is the probability density function. We will assume that the uncertain parameters are normally distributed, random variables. The main issue in solving optimization problems under uncertainty is the computation of multiple integrals for calculation of the expected value of the objective function and probabilities of the satisfaction of the constraints. The known methods of nonlinear programming (e.g., SQP) require the calculation of multiple integrals at each iteration. This operation is very intensive computationally even when the dimensionality of the uncertain parameters vector is low.
We have developed a new approach for solving the TSOP. This approach is based on the following two operations. The first operation is the approximate transformation of the chance constraints into deterministic constraints (based on the approximation of the region by a union of multidimensial rectangles). The second operation is based on the approximation of the goal function by a piece wise linear function. This permits to exclude operations of multiple integration when solving the OSOP. An example is given that illustrates efficiency of this approach. Removal of the operations of multiple integration permits to decrease significantly computational time of solving the OSOP.
 Ostrovsky GM, Ziyatdinov NN, Lapteva TV, Zaitsev IV. Two-stage optimization problem with chance constraints. Chem Eng Sci. 2011; 66: 3815–3828
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