Flowsheets optimization

Schmid, C. and L. T. Biegler. A Simultaneous Approach for Flowsheet Optimization with Existing Modeling Procedures. Trans Inst Chem Eng 12A May (1994b). 

Wolbert, D. X. Joulia B. Koehret and L. T. Biegler. Flowsheet Optimization and Optimal Sensitivity Analysis Using Analytical Derivatives. Comput Chem Eng 18 1083-1095... 

Kisala, T. P. R. A. Trevino-Lozano J. F. Boston H. I. Britt et al. Sequential Modular and Simultaneous Modular Strategies for Process Flowsheet Optimization. Comput Chem Eng 11 567-579 (1987). 

Finally, a great advantage to SQP is that it does not require convergence of the equality constraints, h(x) = 0, at intermediate points. Consequently, the process model (or at least the part directly incorporated into the optimization problem) can be solved simultaneously with the optimization problem. In the next section we discuss the application of the SQP algorithm to flowsheet optimization. Here, if the number of variables in the optimization problem is small, application is straightforward. On the other hand, when the number of variables, n, becomes large (n > 100, say), special-purpose extensions to SQP are required. These are discussed in the remainder of this section. 

Nevertheless, in process design problems where model evaluations are frequently expensive, application of the SQP algorithm alone has provided great impetus toward solving optimization problems efficiently. Flowsheet optimization, one class of problems in this category, is discussed in the next section. 

Fig, 2. Ammonia process case study. Reprinted with permission from Comp. Chem. Eng., 11, 143-152, Y.-D. Lang and L. T. Biegler, A Unified Algorithm for flowsheet optimization, Copyright 1987, Pergamon Press PLC. 

As shown in this example, flowsheet optimization, which previously required several hundred simulation time equivalents (Gaines and Gaddy, 1971), now requires about 10. Thus, many complex design problems, which can be treated through formulation and solution of an optimization problem, can now be handled routinely. In the remainder of this section, we briefly summarize three important problems of this type ... 

Flowsheet optimization is also regarded as a key task in the structural optimization of a flowsheet. As a described in the introduction, structural optimization for process design can be formulated as a mixed integer nonlinear program (MINLP). This then allows for addition or replacement of existing units, and consideration of a number of design options simultaneously. In these formulations individual units are turned on and off over the course of the optimization, as suggested by the MINLP master problem. 

Before leaving this section we consider a slightly different optimization problem that may also be expensive to solve. In flowsheet optimization, the process simulator is based almost entirely on equilibrium concepts. Separation units are described by equilibrium stage models, and reactors are frequently represented by fixed conversion or equilibrium models. More complex reactor models usually need to be developed and added to the simulator by the engineer. Here the modular nature of the simulator requires the reactor model to be solved for every flowsheet pass, a potentially expensive calculation. For simulation, if the reactor is relatively insensitive to the flowsheet, a simpler model can often be substituted. For process optimization, a simpler, insensitive model will necessarily lead to suboptimal (or even infeasible) results. The reactor and flowsheet models must therefore be considered simultaneously in the optimization. 

Fig. 3. Reactor-based flowsheet tom for simultaneous optimization. Reprinted with permission from Chem. Eng. Res. Des. 66, 396, S. Vasantharajan and L. T. Biegler, Simultaneous Solution of Reactor Models within Flowsheet Optimization, 1988.

Note that this formulation illustrates an interesting trade-off for the optimization problem. In the modular mode the optimization problem remains fairly small and function evaluations (e.g., the reactor model) are expensive. With the simultaneous formulation, the model becomes a set of equations whose right-hand sides are much cheaper to evaluate, but the size of the optimization problem increases. Nevertheless, Vasantharajan and Biegler (1988b) showed that, even without SQP decomposition, the simultaneous approach for the reactor was 38% cheaper for the entire flowsheet optimization than the modular approach. Moreover, the number of function evaluations for the reactor model decreased by over an order of magnitude. 

The application of simultaneous optimization to reactor-based flowsheets leads us to consider the more general problem of differentiable/algebraic optimization problems. Again, the optimization problem needs to be reconsidered and reformulated to allow the application of efficient nonlinear programming algorithms. As with flowsheet optimization, older conventional approaches require the repeated execution of the differential/algebraic equation (DAE) model. Instead, we briefly describe these conventional methods and then consider the application and advantages of a simultaneous approach. Here, similar benefits are realized with these problems as with flowsheet optimization. 

We first considered applications of this approach within process engineering. Steady-state flowsheeting or simulation tools are the workhorse for most process design studies the application of simultaneous optimization strategies has allowed optimization of these designs to be performed within an order of magnitude of the effort required for the simulation problem. An application of this strategy to an ammonia synthesis process was presented. Currently, flowsheet optimization is widely available commercially and has also been installed on the FLOWTRAN simulator for academic use. 

C.L. Chen, Ph.D. thesis, A class of successive quadratic programming methods for flowsheet optimization, University of London, 1988. 

At the process level, efficient flowsheet optimization strategies based on lumped parameter models are now widely used in practice (Biegler et al., 1997). At this scale, the PEFC is embedded within a power plant flowsheet model, as shown in Figure 3. The process comprises... 

Kocis, G. R., A Mixed-Integer Nonlinear Programming Approach to Structural Flowsheet Optimization. Ph.D. thesis, Carnegie Mellon University, Pittsburgh, 1988. 

The optimization model in the sequential case had 342 equations and 362 variables for the reactor flowsheet optimization (96 CPU seconds on the VAX 6320) and 200 equations and 161 variables for the energy integration (170 CPU seconds). The simultaneous optimization model (542 equations, 523 variables) was solved in 1455 CPU seconds and was initialized with the solution to the sequential model. Table I presents a comparison between the results for sequential and simultaneous modes to synthesis. A target production rate of 40,000 Ib/h is assumed for the desired product B. 

---