**Optimal design of process systems
under uncertainty**

G. M. Ostrovsky^{1}, N. N.
Ziyatdinov^{2}, T. V. Lapteva^{2}

^{1}*Karpov
Institute of Physical Chemistry, Vorontsovo Pole, 10, Moscow, 103064, Russia*

*Email: *ostralex@yandex.ru,*
Phone: 8-499-4792991*

* *

^{2}*Kazan
State Technological University, Karl Marx str., 68, Kazan, 420111, Russia*

*Email:* nnziat@yandex.ru ;
tanlapteva@yandex.ru

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
[1]

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.

** References**

[1] 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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