Equation-based optimization

Simulation Sciences, Inc. Documentation for ROMEO (Rigorous On-line Modeling with Equation-based Optimization. Brea, CA (1999). 

Instead of using repeated solution of a suitable eigenvalue equation to optimize the orbitals, as in conventional forms of SCF theory, we have found it more convenient to optimize by a gradient method based on direct evaluation of the ener functional (4), ortho normalization being restored after every parameter variation. Although many iterations are required, the energy evaluation is extremely rapid, the process is very stable, and any constraints on the parameters (e.g. due to spatial symmetry or choice of some type of localization) are very easily imposed. It is also a simple matter to optimize with respect to non-linear parameters such as orbital exponents. 

Optimization Using Equation-Based Process Simulators.525... 

Finally, we should mention that in addition to solving an optimization problem with the aid of a process simulator, you frequently need to find the sensitivity of the variables and functions at the optimal solution to changes in fixed parameters, such as thermodynamic, transport and kinetic coefficients, and changes in variables such as feed rates, and in costs and prices used in the objective function. Fiacco in 1976 showed how to develop the sensitivity relations based on the Kuhn-Tucker conditions (refer to Chapter 8). For optimization using equation-based simulators, the sensitivity coefficients such as (dhi/dxi) and (dxi/dxj) can be obtained directly from the equations in the process model. For optimization based on modular process simulators, refer to Section 15.3. In general, sensitivity analysis relies on linearization of functions, and the sensitivity coefficients may not be valid for large changes in parameters or variables from the optimal solution. 

OPTIMIZATION USING EQUATION-BASED PROCESS SIMULATORS... 

In this section we consider general process simulator codes rather than specialized codes that apply only to one plant. To fnesh equation-based process simulators with optimization codes, a number of special features not mentioned in Chapter 8 must be implemented. 

EXAMPLE 15.2 PROCESS OPTIMIZATION VIA GRG (EQUATION-BASED SOFTWARE)... 

Comparison of the results of equation-based and simultaneous modular-based optimization for two connected distillation columns... 

Effective computer codes for the optimization of plants using process simulators require accurate values for first-order partial derivatives. In equation-based codes, getting analytical derivatives is straightforward, but may be complicated and subject to error. Analytic differentiation ameliorates error but yields results that may involve excessive computation time. Finite-difference substitutes for analytical derivatives are simple for the user to implement, but also can involve excessive computation time. 

Wolbert et al. in 1991 proposed a method of obtaining accurate analytical first-order partial derivatives for use in modular-based optimization. Wolbert (1994) showed how to implement the method. They represented a module by a set of algebraic equations comprising the mass balances, energy balance, and phase relations ... 

SLM Simple flux equation based on film theory considering aqueous boundary layer resistance and membrane diffusion Optimal condition determined with respect to pH of the aqueous phase [57,58]... 

In order for a process to be controllable by machine, it must represented by a mathematical model. Ideally, each element of a dynamic process, for example, a reflux drum or an individual tray of a fractionator, is represented by differential equations based on material and energy balances, transfer rates, stage efficiencies, phase equilibrium relations, etc., as well as the parameters of sensing devices, control valves, and control instruments. The process as a whole then is equivalent to a system of ordinary and partial differential equations involving certain independent and dependent variables. When the values of the independent variables are specified or measured, corresponding values of the others are found by computation, and the information is transmitted to the control instruments. For example, if the temperature, composition, and flow rate of the feed to a fractionator are perturbed, the computer will determine the other flows and the heat balance required to maintain constant overhead purity. Economic factors also can be incorporated in process models then the computer can be made to optimize the operation continually. 

Nielsen, J. E. and Vriend, G. (2001) Optimizing the hydrogen-bond network in Poisson-Boltzmann equation-based pKa calculations. Proteins 43,403-412. 

Otto and Wegscheider [562, 563] applied the window diagram method for the simultaneous optimization of the (binary methanol-water) mobile phase composition, the ionic strength and the pH for the separation of ionic solutes in RPLC. They fitted the experimental data to a semi-empirical 13-parameter equation based on eqn.(3.45) for the composition effect, eqn.(3.71) for the effect of the ionic strength and eqn.(3.70) for the... 

Equations 33-35 are the basic thermoeconomic governing equations for the Second Law based optimization. It should be noted that although column entropy productions due to heat transfer are neglected, the analysis nevertheless includes the fact that the column "buys" thermal available-energy from the reboiler in the thermoeconomic governing equations. 

Optimization Procedure. Given a set of design variables (a working design), capital cost equations as functions of the design variables (9), the unit costs of utilities, and Equations 33-35, the Second-Law based optimization may be performed. 

If you formulate a material balance problem that results in nonlinear equations, you should review Sec. L.2. There you wiU find an outline of techniques to solve sets of nonlinear equations, and also recommendations as to computer codes that are state of the art. In the pocket in the back of the text is a simple Fortran code to solve sets of nonlinear equations based on Newton s method and another one based on an optimization technique. You can use them to solve one or more nonlinear equations, and they will be effective for most of the problems in this book. 

Certain Schrodinger equation based methods, such as coupled cluster theory, are not based on a variational principle. They fall outside schemes that use the energy expectation value as a optimization function for simulated annealing, although these methods could be implemented within a simulated annealing molecular dynamics scheme with alternative optimization function. 

The two basic flowsheet software architectures are sequential modular and equation-based. In sequential modular, we write each unit model so that it calculates output(s), given feed(s), and unit parameters. This is the most commonly used flowsheeting architecture at present, and examples include Aspen+ plus Hysys (AspenTech), ChemCAD, and PROll (SimSci). In equation-based (or open-system) architectures, all equations are written describing material and energy balances as algebraic equations in the form/(x) = 0. This is the preferred architecture for new simulators and optimization, and examples include Speedup (AspenTech) and gPROMS (PSE pic). Each is discussed in turn. 

The most important part of process optimization is linking the process flowsheeting tool to the optimization algorithm. With an equation-based architecture, the unit equations (material and energy balances, operating constraints, and specifications) are constraints in a general nonlinear programming formulation. The main problems are... 

Numerical Methods and Data Structure. Both EQ3NR and EQ6 make extensive use of a combined method, using a "continued fraction" based "optimizer" algorithm, followed by the Newton-Raphson method, to make equilibrium calculations. The method uses a set of master or "basis" species to reduce the number of iteration variables. Mass action equations for the non-basis species are substituted into mass balance equations, each of which corresponds to a basis species. 

Perturbation-based optimization methods were introduced, both for inhomogeneous and homogeneous systems, by the Russian school (52) and by Lewins (5J). Methods based on material density perturbations have been developed and applied (53-55, 57, 61, 69, 70) to several problems described by the inhomogeneous Boltzmann equation and related to radiation shields (5i, 55), fission reactors (54, 57), and fusion reactors (61, 69, 116). Similar methods based on boundary displacement perturbations have also been developed and applied (91, 117) for the optimization of radiation shield problems. Perturbation-based optimization methods associated with the homogeneous Boltzmann equations have been developed and applied to several fission reactor problems (56, 92, 96 99). 

Unlike most of the other methods used for optimizing nuclear systems, perturbation-based optimization methods are not restricted to simplified models and to low-order approximations of the Boltzmann equation. They are highly versatile, and for the following reasons ... 

Based on the theoretical explosion reaction equation, the optimal ratio (mass fraction) for the former should be 85.2 hydrazine nitrate/7.2 hydrazine/7.6 ammonia, whereas the optimal ratio for the latter should be 49.05 hydrazine nitrate/ 34.17 perchloric acid hydrazine/12.40 hydrazine/4.38 ammonia. 

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