Optimisation solver

More generally, the process optimisation problem is solved using an optimisation solver, which interacts with the process simulator to minimise the objective function. The optimisation solver can be based on any one of many optimisation techniques. One group of possible optimisation techniques are the gradient-search methods. These methods rely on analytic or semi-analytic expressions for the objective function and objective function... 

As an integral component of Microsoft Office, the spreadsheet program Excel is installed on many personal computers. Thus, a widespread basic expertise can be assumed. Although initially designed for business calculations and graphics, Excel is also extremely useful for scientific purposes. Its matrix capabilities, as well as the optimisation add-in solver, are not widely known but can often be applied in order to quickly resolve quite complex multivariate problems. We have used Excel 2002 but any other version will do equally well. 

Depending on the spectra of the components, wrong results cannot be identified by impossible, negative spectra, (also see Chapter 4.4.3, Optimisation in Excel, the Solver, Figure 4-62 and Figure 4-63)... 

Note also that the Newton-Gauss algorithm for function optimisation is the standard option in Excel s solver. 

Equilibria and 3.3.4 Solmng Non-Linear Equations. The Solver includes optimisation as one of the options. Its main application, within this chapter on data analysis, is data fitting, based on the minimisation of sum of squares. 

In this approach, the process variables are partitioned into dependent variables and independent variables (optimisation variables). For each choice of the optimisation variables (sometimes referred to as decision variables in the literature) the simulator (model solver) is used to converge the process model equations (described by a set of ODEs or DAEs). Therefore, the method includes two levels. The first level performs the simulation to converge all the equality constraints and to satisfy the inequality constraints and the second level performs the optimisation. The resulting optimisation problem is thus an unconstrained nonlinear optimisation problem or a constrained optimisation problem with simple bounds for the associated optimisation variables plus any interior or terminal point constraints (e.g. the amount and purity of the product at the end of a cut). Figure 5.2 describes the solution strategy using the feasible path approach. 

A model reduction-based optimisation framework for large-scale simulators using iterative solvers... 

The first term of Eq. 7 vanishes if feasible points (i.e. steady states) are computed at each iteration [6]. Clearly the calculation of the basis Z from Eq. 5 is expensive, as the large system Jacobians need to be constructed and inverted. Furthermore, in the case of input/output simulators Jacobians are not explicitly available to the optimisation procedure, or even to the solver itself as is the case of solvers using iterative linear algebra (e.g. Newton-Krylov solvers). In such cases the Jacobian can only be numerically approximated, with great computational expense in terms of CPU and memory requirements. For this purpose here we compute only reduced-order Jacobians... 

A Model Reduction-Based Optimisation Framework for Large-Scale Simulators Using Iterative Solvers... 

In the present model the pre-exponential factors and activation energies are optimised by non-linear regression using the Solver module on Microsoft Excel. The amounts of coke associated with CO2 evolution at Pi and P2 are also optimised, while a mass balance determines the coke quantity associated with CO at P5 and CO2 at P3. The oxygen partial... 

The new chemical industry will have two main types of companies. The molecule providers will focus on delivering commodity and fine chemicals at the lowest cost. These will require all the traditional skills associated with chemical engineering with a particular emphasis on supply chain and logistics optimisation, plant efficiency and reliability. The problem solvers will provide customised effects. Whether these are speciality additives or personalised drug systems, the customer is more concerned with the efficacy of the product than with specification of the composition of the product. 

Equation 4 was discretised by a 5-point central difference formula and thereafter first-order differential equations 1, 2, 4 and 6 were solved by a backward difference method. Apparent reaction rate was solved by summing the average rates of each discretisation piece of equation 4. The reactor model was integrated in a FLOWBAT flowsheet simulator [12], which included a databank of thermodynamic properties as well as VLE calculation procedures and mathematical solvers. The parameter estimation was performed by minimising the sum of squares for errors in the mole fractions of naphthalene, tetralin and the sum of decalins. Octalins were excluded from the estimation because their content was low (<0.15 mol-%). Optimisation was done by the method of Levenberg-Marquard. 

Since an FE solver can calculate all modal parameters for all modes of interest at once, the computational cost to calculate one objective function is equal to the computational cost to calculate all objective functions. Generic non-linear optimisers use only one of these objective functions. Other results however can be useful for other optimisations, for instance to select a suitable start vector. In most cases some optima - especially these located on a vertex - can even be found without performing additional FE analyses. Especially for larger FE models, storing all EE analysis results can cut the computational cost significantly. 

Also, when using a standard FE solver, the computational cost to evaluate all three objective functions for all modes is equal to the computational cost to evaluate one objective function for one mode. Because of the very low computational cost of a single evaluation of the response surface approximation function, the computational cost of a global optimisation on the response surface is feasible for most practical applications. These properties can be used to develop a very efficient optimisation algorithm for the full fuzzy FE algorithm (Munck et al, 2006). 

The objective of the optimisation is to minimise the batch time. Feed and reflux ratio profiles are considered as decision variables within the optimisation problem. Constraints to be taken into account are product specifications (acid value, carboxyl number, viscosity, water content), feed amount and limitation of the feed flow rate. The model DAE s are discretised and the resulting algebraic system is optimised with a NLP algorithm (e.g. a SQP solver). The objective function and the constraints can be defined as VBA macros and then be computed by CHEMCAD. Present work is concerned with the implementation of the optimisation algorithm. 

Different solution approaches can be used to converge problems when a good initial point is not available. These include Homotopy or Continuation techniques (e.g., Allgower et al., 1990), Interval solvers (e.g., Neumaier, 1990) and Global Optimisation (e.g., Maranas and Floudas, 1995). Instead of using a different solution method, we propose to find a better initial point by using the same equations that have to be solved as guidance. 

Type of Optimisation what type of optimisation is desired maximisation (Max), minimisation (Min), or force the solver to obtain a particular value (Value of). For regression, the minimisation option should be used. 

In the proposed framework, the objective functions are formulated in Excel for the modelled process in HYSYS. The multi-objective optimisation technique, e-constraint, is formulated with the Premium Solver Platform (by Frontline Systems), which is an upgrade of the standard Excel solver, that uses the standard non-linear GRG (Generalized Reduced Gradient) method. However, any other optimisation method, such as that mentioned earlier, can be easily formulated in Excel and evaluated accordingly. 

As the MATLAB software packages with Optimisation Toolbox provides both effective ordinary differential equation (ODE) solvers as well as powerful optimization algorithms, the dynamic simulations reported in this paper are carried out by using the MATLAB Optimisation Toolbox (8). 

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