Solution of the Optimisation Problem

In addition to the above-mentioned problem, numerical difficulties may arise. The system (model equations) describing the multicomponent off-cut recycle operation needs to be reinitialised at the end of each main-cut and off-cut to accommodate the next off-cut to the reboiler. To optimise these initial conditions (new mixed reboiler charge and its composition) it is essential to obtain the objective function gradients with respect to these initial conditions. 

In this chapter, a decomposition strategy considered by Mujtaba (1989) and Mujtaba and Macchietto (1992) is presented in which the whole multiperiod optimisation problem is divided into a series of independent dynamic optimisation problems. This is presented in the next section. 

Decomposition of the Optimisation Problem Formulation for Multicomponent Mixtures 

Before discussing how the general optimisation problem mentioned in the previous section is decomposed into a series of independent dynamic optimisation problem and before presenting the mathematical formulation for such problem the following discussions are worthwhile. 

Usually separation specifications in multicomponent mixtures are imposed on a particular (key) component of the cut, e.g. on component 1 in main-cut 1, on component 2 in main-cut 2 and likewise. Of course, it is possible to set intuitively some of the heavier component compositions to zero during lighter cuts and some of the lighter component compositions to zero during heavier cuts. But it is really difficult to specify independently the compositions of the preceding and few successive component compositions in a particular cut. 

---

A set of variables that satisfy the items 2 and 3 precisely will provide feasible solution of the optimisation problem. 

Tran and Mujtaba (1997), Mujtaba et al. (1997) and Mujtaba (1999) have used an extension of the Type IV- CMH model described in Chapter 4 and in Mujtaba and Macchietto (1998) in which few extra equations related to the solvent feed plate are added. The model accounts for detailed mass and energy balances with rigorous thermophysical properties calculations and results to a system of Differential and Algebraic Equations (DAEs). For the solution of the optimisation problem the method outlined in Chapter 5 is used which uses CVP techniques. Mujtaba (1999) used both reflux ratio and solvent feed rate (in semi-continuous feeding mode) as the optimisation variables. Piecewise constant values of these variables over the time intervals concerned are assumed. Both the values of these variables and the interval switching times (including the final time) are optimised in all the SDO problems mentioned in the previous section. 

At any time t, the true estimation of the state variables requires instantaneous values of the unknown mismatches eft). To find the optimal control policies in terms of any decision variables (say z) of a dynamic process using the model will require accurate estimation of ex(t) for each iteration on z during repetitive solution of the optimisation problem (see Chapter 5). Although estimation of process-model mismatches for a fixed operating condition (i.e. for one set of z variables) can be obtained easily, the prediction of mismatches over a wide range of the operating conditions can be very difficult. 

Case 1 of Table 9.3 is the base case. It shows the optimisation results using the cost parameters presented in Table 9.2. The maximum profit and optimal batch time obtained by optimisation shows very good agreement to those shown in Figure 9.8. The maximum profit shown in Figure 9.8 is between 3.99-4.13 ( /hr) with an optimum batch time between 12-14 hr. Each of the optimisation problems (i.e. solution of P2 with Equation 9.6) presented in Table 9.3 requires approximately 3- 4 iterations and about 3- 4 cpu sec using a SPARC-1 Workstation (Mujtaba and Macchietto, 1997). 

Due to the complexity of the optimisation problem, a solution in a closed form is not feasible and for this reason simulation methodologies are usually used. The respective tools utilise logistic models of the power components and are based on... 

To simulate a real-time operation, a set of case studies (Table 3) were proposed, where changes in the process behaviour were introduced by changing the model parameters. The objective was to verify if the adaptation procedure would be able to change the base metamodels in order to allow acceptable solutions to the optimisation problem. The selected model parameters were the feed composition (I and II), the global heat coefficient of the atmospheric column pumparoimd (UppA - III and IV) and the global heat coefficient of the condenser of column N753 (Ucond - V). 

The algorithmic treatment depends on the architecture of the flowsheeting system. In Equation-Oriented mode, the approach consists of solving all the equations describing the problem simultaneously. In Sequential-Modular approach the mathematical solution must take into account the convergence of units and tear streams, as well as of all design specifications. Supplementary equations must be added, so that the general formulation of the optimisation problem (3.10) becomes ... 

Proof In order to find the solution to the optimisation problem given by Eq. (3.3), it is necessary to determine the points at which the derivative of the objective function are zero and then solve for the desired unknown values. 

The performance criteria of a batch distillation column can be measured in terms of maximum profit, maximum product or minimum time (Mujtaba, 1999). In distillation, whether batch, continuous or extractive, purity of the main products must be specified as it is driven by the customer demand and product prices. The amount of product and the operation time can be dictated by economics (maximum profit) or one of them can be fixed and the other is obtained (minimum time with fixed amount of product or maximum distillate with fixed operation time). The calculation of each of these will require formulation and solution of optimisation problems. A brief description of these optimisation problems is presented below. Further details will be provided in Chapter 5. 

Reklaitis et al. (1983), Edgar and Himmelblau (1988) have discussed several solution methods for solving linear and nonlinear optimisation problems. Here, some of the optimisation techniques used in batch distillation will be discussed. 

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. 

The above-mentioned strategy requires the solution of just 8 inner loop problems for calculating the gradients with respect to the 4 decision variables. Note, additional efficiency can be achieved by using the corresponding optimal reflux ratio profiles from the previous pass as the initial estimate of the optimisation variables for each inner loop problem. This will significantly reduce the number of iterations required for each inner loop problem, and in particular for gradient evaluation. 

In this chapter first, the optimisation method of Al-Tuwaim and Luyben (1991) for single separation duty is presented. Then the optimisation problem formulation and solution considered by Mujtaba and Macchietto (1996) is explained. Finally, the optimisation problem formulations considered by Logsdon et al. (1990) and Bonny et al. (1996) are presented. 

Computationally, the solution of the dynamic optimisation problem is time consuming and expensive. Mujtaba and Macchietto (1997) reported that the number of "Function" and "Gradient Evaluations" for each maximum conversion problem is between 7-9. A fresh solution would require approximately 600 cpu sec in a SPARC-1 Workstation. However, subsequent solutions for different but close values of tf could take advantage of the good initialisation values available from the previous solutions. 

Using the above profit function, the solution of problem P2 will automatically determine the optimum batch time (tf), conversion (C), reflux ratio (r) and the amount of product (Di). However, as the cost parameters (CDh CB0, etc.) can change from time to time, it will require a new solution of the dynamic optimisation problem P2 (as outlined in Mujtaba and Macchietto, 1993, 1996), to give the optimal amount of product, optimal batch time and optimal reflux ratio. And this is computationally expensive. To overcome this problem Mujtaba and Macchietto (1997) calculated the profit of the operation using the results of the maximum conversion problem (PI) which were obtained independent of the cost parameters. 

The dynamic optimisation problem P2 now results in a single variable algebraic optimisation problem. The only variable to be optimised is the batch time t. The solution of the problem does no longer require full integration of the model equations. This method will solve the maximum profit problem very cheaply under frequently changing market prices of (CD/, CB0, C ) and will thus determine new optimum batch time for the plant. The optimal values of C, Dh r, QR, etc. can now be determined using the functions represented by Equations 9.2-9.5. 

Mujtaba (1999) considered the conventional configuration of BED processes for the separation of binary close boiling and azeotropic mixtures. Dynamic optimisation technique was used for quantitative assessment of the effectiveness of BED processes. Two distinct solvent feeding modes were considered and their implications on the optimisation problem formulation, solution and on the performance of BED processes were discussed. A general Multiperiod Dynamic Optimisation (MDO) problem formulation was presented to obtain optimal separation of all the components in the feed mixture and the recovery of solvent while maximising the overall profitability of the operation. 

For cases 2-4, the optimal values of R in both intervals were less than Rmax meaning the column was never flooded during the operation. This is also evident from the reboiler holdup values at the end of each interval. Also for cases 2-4 the constraint given by Equation 10.5 was not active. Mujtaba (1999) noted that cases 2-4 were re-run with only one time interval for the distillate Task. The constraint given by Equation 10.5 was imposed on the optimisation problem. For cases 2-3 the maximum productivity obtained was about 0.55 and for case 4 no solution was obtained. This shows the importance of having time-sequenced operation. 

Dynamic sets of process-model mismatches data is generated for a wide range of the optimisation variables (z). These data are then used to train the neural network. The trained network predicts the process-model mismatches for any set of values of z at discrete-time intervals. During the solution of the dynamic optimisation problem, the model has to be integrated many times, each time using a different set of z. The estimated process-model mismatch profiles at discrete-time intervals are then added to the simple dynamic model during the optimisation process. To achieve this, the discrete process-model mismatches are converted to continuous function of time using linear interpolation technique so that they can easily be added to the model (to make the hybrid model) within the optimisation routine. One of the important features of the framework is that it allows the use of discrete process data in a continuous model to predict discrete and/or continuous mismatch profiles. 

Solution of optimisation problems using rigorous mathematical methods have received considerable attention in the past (Chapter 5). It is worth mentioning here that these techniques require the repetitive solution of the model equations (to evaluate the objective function and the constraints and their gradients with respect to the optimisation variables) and therefore computationally can be very expensive. 

---