(246r) Optimal Design of Process Systems Under Uncertainty

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

²Kazan
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]

                         
                                                 (1)

        
                      (2)

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