(240m) A Study of Differential Evolution and Tabu Search for Benchmark and Phase Stability Problems

           
Phase stability problems are crucial in the computation of phase equilibria and
thus play a significant role in various chemical engineering applications such
as extraction and distillation. The problem involves determining whether a given
phase with certain composition, pressure and temperature is stable or will split
into multiple phases. Phase stability is frequently tested using the well known
Tangent Plane Criterion (Baker et al., 1982). The criterion formulates the
tangent plane distance function (TPDF), defined as the vertical distance between
the molar Gibbs free energy surface and the tangent plane at the given
composition, as the objective function. The problem can be solved using two
approaches namely: solving a system of non-linear equations for stationary
points (Michelsen, 1982) and the direct minimization of TPDF function. The
former approach is a conventional approach where the solution obtained may be
trivial or local, and is mainly dependent on the initial guess. The later
approach employs the global minimization techniques because of the high
non-linearity associated with the objective function. The presence of comparable
minima (i.e., function value at the local minimum will be nearest to that at the
global minimum) in TPDF poses a computationally challenging problem to many of
the global optimization methods. The complexicity in the TPDF is mainly due to
the thermodynamic models that are used to describe the non-ideality in the Gibbs
free energy function.
 

Problem Formulation
           

       
The molar Gibbs energy of a system, g at a given temperature and pressure is the
summation of the product of mole fraction and partial molar Gibbs energy G_i^p
for all components:   

           
g = ∑x_iG_i^p                                  
(i = 1, 2, 3, ... , N)                                            (1)
 

where x_i is the mole
fraction of i^th component and N is the total number of components in
the system. Tangent plane, t at the specified composition x^* = (x₁^*,
x₂^*,?, x_N^*) can be written as:

            t = ∑x_iG_i^p*       
                        (i = 1, 2, 3, ... ,
N)                                            (2)
 

Thus the objective function (TPDF)
can be expressed as
 

            F = g - t = ∑x_i
(G_i^p - G_i^p*)           
(i = 1, 2, 3, ... , N)                                            (3)
 

subject to the equality
constraint
 

             ∑x_i =
1                                     (i = 1, 2, 3, ... ,
N)                                            (4)
 

and the non-negative
conditions:  0 ≤ x_i ≤ 1.

           
With the
recent advances in global optimization methods, several authors addressed this
problem using different methods (Sun and Seider, 1995; McDonald and Floudas,
1995; Hua et al., 1998; and Zhu and Xu, 1999) but not yet examined by the
promising stochastic methods such as Differential Evolution (Storn and Price,
1997; and Babu et al., 2004) and Tabu Search (Chelouah and Siarry, 2000 and Teh
and Rangaiah, 2003).
 

           
DE is simple, robust and requires few control variables. The method is a
population based search and consists of 3 steps namely: mutation, crossover and
selection. The mutation generates a new individual by adding the weighted
difference between two individuals to a third individual in the population.
Crossover is performed mainly to increase the diversity during the search. The
selection step determines whether or not the new individual is allowed into the
next generation using some greedy criterion.
 

           
TS is meta-heuristic method that guides and improves the search in the solution
space. The method escapes from the local minimum by generating solutions that
differs in various ways from those seen in the previous generations. The
algorithm avoids the repeated visits to the same place during the search thus
increasing computational efficiency. After a number of iterations several
promising areas are identified for further in depth search known as
intensification.
 

           
In this work both DE and TS are implemented and tested for benchmark problems
involving 2 to 20 variables and a few to hundreds of local minima. The results
show that the methods are better or comparable to several other methods reported
in the literature. The potential of DE and TS is then examined for the phase
stability problems involving several components with different feed compositions
and thermodynamic models. The results show that both DE and TS are reliable in
solving phase stability problems and are computationally efficient than other
stochastic methods such as genetic algorithms. All these results will be
reported and discussed in the conference presentation.

References

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