Page 64 - Sustainability in the Process Industry Integration and Optimization

Pro c ess O p timization 41

simplify network design if the number of candidate operations is
large and the targets can be used as guides in the network synthesis.
Usually the engineer aims either to achieve the targets exactly or to
approach them closely with the final design. If the targeting model is
too idealized, then the estimates produced will serve as loose
performance or cost bounds, not as tight bounds. Yet in many cases
this strategy results in a simple design procedure and a nearly
optimal outcome.
Second, if the targeting model is exact, as in Pinch Analysis for
Heat Integration (Linnhoff and Flower, 1978), or if it at least captures
all key factors at the corresponding design stage, as with Regional
Energy Clustering (Lam, Varbanov, and Klemeš, 2010), then the
targeting procedure also provides a convenient partitioning of the
original design space. This makes it easier to decompose the problem
and to simplify the remaining actions. A good example of partitioning
the design space is the division above/below the Pinch in Heat
Integration (see Chapter 4).
3.10.6 Handling Model Nonlinearity
As discussed in Section 3.5, convex problems are guaranteed to
produce globally optimal results when solved with deterministic
algorithms that employ local search. In contrast, a nonconvex
optimization problem is difficult to solve, and its solution is not
guaranteed to be a global optimum. All linear MP models are convex
(Williams, 1999). With nonlinear models, however, the problem
convexity must be established on a case-by-case basis. Nonlinear
models hinder the computation process of the solvers (e.g., by
requiring that feasible initial solutions be provided), and they often
result in poor numerical convergence. This is why engineers usually
seek ways to obtain linear models in some form. Crucial factors in
this task are preserving the model’s precision and validity.
Trading Off Precision and Linearization
Sometimes it is possible to linearize relationships that are inherently
nonlinear. This can be done, for instance, by replacing a single
nonlinear relationship with two or more linear ones that, together,
approximate the original function over the required range. This
technique is known as piecewise linearization. For example, it can be
applied when the available cost functions (for piping, distillation
columns, or heat exchangers) are too complex. The result of this
approach is a small reduction in the overall model precision and an
increase in the number of integer variables in the model; thus,
combinatorial complexity is increased but computational complexity
(due to nonlinearity) is reduced. The principal advantage is the
resulting linearity of the model, which almost always makes it easier
to solve than the original one. Caution must be exercised in the
process of linearization: the loss of precision must be kept as small as