Page 65 - Sustainability in the Process Industry Integration and Optimization

42 Cha p te r T h r ee

possible so that the resulting model remains an adequate
representation of the underlying process. Another pitfall is a
potentially unacceptable increase in combinatorial complexity, which
can result if too many linear segments are used to approximate the
original relationship.
Discretization of Continuous Process Variables
Another approach to avoiding nonlinearity is to define a number of
fixed levels for some variables and then to substitute the original
nonlinear variables in the model with linear combinations of integer
variables and parameters (where the parameters are derived from
the original variables). In this way, the bilinear terms in the mass
balances of contaminants can be reduced to purely linear expressions.
In its consequences and pitfalls, this technique is similar to piecewise
linearization.
Other Techniques
There are other approaches and techniques for coping with nonlinear
models. Two of them are of particular interest to practical process
optimization: successive MILP (SMILP), which is used for model
decomposition and solving; and model reformulation.
Successive MILP can be applied to many process engineering
optimization models—as long as the nonlinearities are not too strong.
One example is the optimization of utility systems, which consist
mainly of a set of steam headers combined with steam turbines, gas
turbines, boilers, and letdowns. Most of the nonlinearities in such
systems are bound to the enthalpy balances of the steam headers, the
steam turbines, and the letdowns. The computational difficulties
imposed by the nonlinearities can be overcome by first fixing the
values of some system properties during optimization (e.g., enthalpies
of steam mains), thereby producing a linear optimization model, and
following this with a rigorous simulation after each optimization
step. The linear optimization steps are repeated, followed again by
simulation, and so on until convergence is achieved; see Figure 3.2.
This procedure converges rapidly when applied to the optimization
of existing utility systems: usually five iterations at most are required
to reach reasonably small error levels.
Model reformulation refers to the symbolic transformation of the
original nonlinear equations into another set of equations that are
equivalent but linear. The resulting set usually contains more
equations than the initial one. One such reformulation technique,
known as the Glover transformation (Floudas 1995), can transform
equations containing the product of a continuous and a binary
variable. The essence of the technique is to replace each term that is a
product of a continuous variable and a binary variable with additional
continuous variables and an additional set of linear inequality
constraints.