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7.4 PSA Case Study - Post Combustion CO2 Capture terminated is not optimal, which implies that ROM is not predicting the correct descent direction at the termination point. The reason for this is that the assumption (A3) is not true at the termination point, i.e., the objective function gradient and the constraint Jacobian obtained from ROM do not match the actual ones. Hence, this clearly shows that although we can construct susbstantially accurate ROMs based on the snapshot information of the state variables, such ROMs in general cannot always ensure that the gradient information is also reasonably accurate. Moreover, we infer that to ensure convergence to an optimal point, assumption (A3) is essential and accurate gradients should be incorporated in the ROM- based optimization problem. This can be accomplished with a First-order Correction (FOC), as illustrated in the next section. 7.4.3.3 Optimization with First-order Correction We solve (7.31) with the First-order Correction (7.9) applied for the objective function and the purity constraint. For FOC, we need to evaluate gradients of the objective and the constraints of the original optimization problem with the rigorous model before each trust-region iteration starts. However, this is computed just once and optimization within a trust-region is carried out using ROM. For this case, we evaluate gradients using perturbation. Table B.2 in Appendix B lists all the trust-region iterations with the penalty parameter μ=1000. As in the previous case, we observe that because of the penalty parameter, the al- gorithm focuses on satisfying the CO2 purity constraint for the first few iterations. In fact, within 5 iterations (k=4) feasibility is attained after which we drop the penalty parameter and move the purity constraint into the trust-region subproblem. Because of the exact gradient information, algorithm goes beyond the optimal CO2 recovery of 81.74% obtained in the previ- ous case, up to a recovery of 97.19%. However, the key observation is that the algorithm takes tiny steps to improve CO2 recovery after achieving feasibility, and thus eventually takes 92 iterations to get to the optimum, which is considerably large. After 92nd iteration (k=91), ∆91 gets reduced from 0.021 to 0.005, thus going below ∆min = 0.02, and the algorithm terminates. Chapter 7. Trust-region Framework for ROM-based Optimization 159PDF Image | Design and Operation of Pressure Swing Adsorption Processes
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