Nonlinear Programming Models

The best way to approach the retrofit synthesis of the heat-exchanger network is to model all five tasks simultaneously. A mixed-integer nonlinear programming model is usually formulated to accomplish this goal. 

A. R. Ciric and C. A. Floudas. A mixed-integer nonlinear programming model for retrofitting heat exchanger networks. I EC Res., 29 239,1990. 

The FCC simulator program was converted to subroutine form a few years ago and incorporated into a nonlinear programming model representing a complex of process units in the Toledo refinery. It is this subroutine version which has been linked with the LP preprocessor. 

The hydrocracker simulator was also converted to subroutine form for inclusion in the nonlinear programming model of the Toledo process complex. The subroutine was considerably simplified, however, to save computer time and memory. The major differences are (1) the fractionation section is represented by correlations instead of by a multi-stage separation model, (2) high pressure flash calculations use fixed equilibrium K-values instead of re-evaluating them as a function of composition, and (3) the beds in each reactor are treated as one isothermal bed, eliminating the need for heat balance equations. 

The reformer simulator was converted to subroutine form for inclusion in nonlinear programming models of two refinery complexes. To save computer time and memory, the subroutine uses a linearized version of the original kinetic model, with 28 components and 33 reactions. Instead of numerical integration, the linearized model is solved analytically at constant temperature, pressure, and total mols using special subroutines to find the eigenvalues and eigenvectors of the reaction rate constant matrix. 

Alhajri, I., Elkamel, A., Albahri, T., and Douglas, P. A nonlinear programming model for refinery planning and optimisation with rigorous process models and product quality specifications. International Journal of Oil, Gas and Coed Technology, l(3) 283-307,... 

Due to major computational difficulties, nonlinear configuration models have not been frequently encountered in the supply chain configuration literature (see Wu and O Grady (2004) for a brief discussion of nonlinear programming models in supply chain configuration). The main nonlinear factors relevant to supply chain configuration, such as inventory and transportation costs, are usually represented using piece-wise linear functions (e.g., Tsiakis et al. 2001). 

Waste Heat Integration Between Processes III Mixed Integer Nonlinear Programming Model... 

One important class of nonlinear programming techniques is called quadratic programming (QP), where the objective function is quadratic and the constraints are hnear. While the solution is iterative, it can be obtained qmckly as in linear programming. This is the basis for the newest type of constrained multivariable control algorithms called model predic tive control. The dominant method used in the refining industiy utilizes the solution of a QP and is called dynamic matrix con-... 

The specific problem characteristics are modeled most appropriately by a combination of concepts from various general modeling frameworks leading to a mixed-integer nonlinear programming (MINLP) model. 

Both the mixing process and the approximation of the product profiles establish nonconvex nonlinearities. The inclusion of these nonlinearities in the model leads to a relatively precise determination of the product profiles but do not affect the feasibility of the production schedules. A linear representation of both equations will decrease the precision of the objective but it will also eliminate the nonlinearities yielding a mixed-integer linear programming model which is expected to be less expensive to solve. 

The nonlinear programming problem based on objective function (/), model equations (b)-(g), and inequality constraints (was solved using the generalized reduced gradient method presented in Chapter 8. See Setalvad and coworkers (1989) for details on the parameter values used in the optimization calculations, the results of which are presented here. 

Although, as explained in Chapter 9, many optimization problems can be naturally formulated as mixed-integer programming problems, in this chapter we will consider only steady-state nonlinear programming problems in which the variables are continuous. In some cases it may be feasible to use binary variables (on-off) to include or exclude specific stream flows, alternative flowsheet topography, or different parameters. In the economic evaluation of processes, in design, or in control, usually only a few (5-50) variables are decision, or independent, variables amid a multitude of dependent variables (hundreds or thousands). The number of dependent variables in principle (but not necessarily in practice) is equivalent to the number of independent equality constraints plus the active inequality constraints in a process. The number of independent (decision) variables comprises the remaining set of variables whose values are unknown. Introduction into the model of a specification of the value of a variable, such as T = 400°C, is equivalent to the solution of an independent equation and reduces the total number of variables whose values are unknown by one. 

If the model includes nonlinear constraints, the problem can be solved using nonlinear programming (Chapter 8). 

In this sense, the application of Q-R factorizations constitutes an efficient alternative for solving bilinear data reconciliation. Successive linearizations and nonlinear programming are required for more complex models. These techniques are more reliable and accurate for most problems, and thus require more computation time. 

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. 

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