Outer Loop Optimisation Problem

A complete solution of the sequence of inner loop problem is required for each function evaluation of the outer loop problem. The gradients of the objective function with respect to the decision variables are obtained by a finite difference technique but in an efficient way, i.e. by utilising the solution of each inner loop problem from the function evaluation stage. This is explained more clearly with 

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. 

The constraints in the outer loop optimisation problem are simple bounds on the decision variables and their gradients can be easily calculated. Mujtaba and Macchietto solved the outer loop optimisation problem using the efficient SQP technique due to Chen (1988). Mujtaba and Macchietto (1993) used 1.0E-3 and 1.0E-2 as the tolerances for the inner and outer loop optimisations respectively. These were quite tight considering the fact that all the optimisation variables and constraints were scaled within the range 0-10. 

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Problem P2-3 is formally similar to P2-1 and P2-2. These are all solved using the same solution procedure. Any of the outer loop variables which have known or fixed values may be dropped from the outer loop optimisation. 

Having design parameters fixed in the outer problem and with a specific choice of D° (discussed in section 7.2) the inner loop optimisation can be partitioned into M independent sequences (one for each mixture) of NTm dynamic optimisation problems. This will result to a total of ND = 2 NTm problems. In each (one for each task) problem the control vector m for each task is optimised. This can be clearly explained with reference to Figure 7.3 which shows separation of M (=2) mixtures (mixture 1 = ternary and mixture 2 = binary) and number of tasks involved in each separation duty (3 tasks for mixture 1 and 2 tasks for mixture 2). Therefore, there are 5 (= ND) independent inner loop optimal control problems. In each task a parameterisation of the time varying control vector into a number of control intervals (typically 1-4) is used, so that a finite number of parameters is obtained to represent the control functions. Mujtaba and Macchietto (1996) used a piecewise constant approximation to the reflux ratio profile, yielding two optimisation parameters (a control level and interval length) for each control interval. For any task i in operation m the inner loop optimisation problem (problem Pl-i) can be stated as ... 

In this separation, there are 4 distillation tasks (NT-4), producing 3 main product states MP= D1, D2, Bf) and 2 off-cut states OP= Rl, R2 from a feed mixture EF= FO. There are a total of 9 possible outer decision variables. Of these, the key component purities of the main-cuts and of the final bottom product are set to the values given by Nad and Spiegel (1987). Additional specification of the recovery of component 1 in Task 2 results in a total of 5 decision variables to be optimised in the outer level optimisation problem. The detailed dynamic model (Type IV-CMH) of Mujtaba and Macchietto (1993) was used here with non-ideal thermodynamics described by the Soave-Redlich-Kwong (SRK) equation of state. Two time intervals for the reflux ratio in Tasks 1 and 3 and 1 interval for Tasks 2 and 4 are used. This gives a total of 12 (6 reflux levels and 6 switching times) inner loop optimisation variables to be optimised. The input data, problem specifications and cost coefficients are given in Table 7.1. 

The optimum number of plates, the optimum values of the decision variables for both outer and inner loop optimisation problems, and optimal amounts and composition of all products are shown in Table 7.2. Typical composition profiles in the product accumulator tank are shown in Figure 7.6. Bold faced mole fractions in Table 2 are the specifications (all satisfied) and underlined mole fractions are decision variables which were optimised. Although the optimum number of plates is almost close to that of the base case, the optimal total operation time is 14% lower than the base case. The profit with the optimal design and operation is 35% higher than that for the base case (calculated using the same cost model). This is obtained... 

Re i, xRi, Re D2) to consider in the outer optimisation problem. At the function evaluation stage the solution (reflux profiles, column profile, duration of task) of each inner loop is stored as A, B, and C, respectively, as shown in Figure 6.3a. 

Mujtaba and Macchietto (1996) formulated a two-level optimisation problem with Dd and D° optimised in an outer loop and U optimised in the inner loop, using an extension of the method of Mujtaba and Macchietto (1993). The outer loop problem can be written as ... 

For each outer loop function and gradient evaluation 4 and 14 inner loop problems were solved respectively (a total of 124 inner loop problems). For the inner loop problems 12-14 iterations for Tasks 1 and 3 and 5-7 iterations for Tasks 2 and 4 were usually required. For this problem size and detail of dynamic and physical properties models the computation time of slightly over 5 hrs (using SPARC-1 Workstation) is acceptable. It is to note that the optimum number of plates and optimum recovery for Task 1 (Table 7.2) are very close to initial number of plates and recovery (Table 7.1). This is merely a coincidence. However, during function evaluation step the optimisation algorithm hit lower and upper bounds of the variables (shown in Table 7.1) a number of times. Note that the choices of variable bounds were done through physical reasoning as explained in detail in Chapter 6 and Mujtaba and Macchietto (1993). 

In this problem, there are 3 outer loop decision variables, N and the recovery of component 1 from each mixture (Re1 D1B0, Re D2,BO)- Two time intervals for reflux ratio were used for each distillation task giving 4 optimisation variables in each inner loop optimisation making a total of 8 inner loop optimisation variables. A series of problems was solved using different allocation time to each mixture, to show that the optimal design and operation are indeed affected by such allocation. A simple dynamic model (Type III) was used based on constant relative volatilities but incorporating detailed plate-to-plate calculations (Mujtaba and Macchietto, 1993 Mujtaba, 1997). The input data are given in Table 7.3. 

The results presented in Table 8.3 are in good agreement. The small differences between the results might be due to the different accuracy set for the optimisation (see Table 8.4). Since the gradients in the two-level formulation were solved by finite difference the inner loop problems PI and P2) were to be solved very tightly (accuracy for the optimiser = 1.0E-4). Whereas, the outer loop problem (P0) of the two-level formulation and the one level problem (P4) were solved using the optimiser accuracy = 1.0E-2. 

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