*4.3. Solution of the Multi-Period Design Problem and Final Feasibility Check*

In the final step, the multi-period design problem was solved. In order to calculate the TAC of the different design proposals, representative operating points and their respective normalized duration factors were defined. For the illustrative example, 11 representative operating points were assumed. Furthermore, it was assumed that each operating point had the same normalized duration factor. The different representative operating points and their normalized duration factors are given in Table A4 in Appendix B. As can be seen in Table A4, extreme values (which can be calculated with the values given in Table 3) were not considered as representative operating points since these values represent extreme situations, which are usually not representative for longer operating periods.

To reduce the computational complexity, the multi-period design problem was solved individually for each retrofit proposal. The problems were solved using the global solver BARON. Table 6 shows the TAC for the proposals in the reduced superstructure, the annual net savings (difference between the annual operating cost of the initial HEN and the TAC of the proposals), and the flexibility index. The annual operating cost of the initial HEN was used as a benchmark for the TAC of the different retrofit proposals and was calculated considering the cost data presented in Section 4.2. The annual operating cost of the initial network is 314,600 €/y. Additionally, the derived design values of the process-to-process HEXs of the different retrofit design proposals are shown in Tables A6–A10 in Appendix B. The results in Table 6 indicate that proposal MER is the most cost efficient.

**Table 6.** Total annualized cost, annual net savings, and flexibility index of the different retrofit design proposals in the reduced superstructure.


The flexibility index was calculated using the active set approach developed by Floudas and Grossman [39] and the results were verified by searching for the minimum direction matrix using simulated annealing as it is reported in [40]. In contrast to [40], the subproblem to find the maximum value for δ (maximum feasible variation of the uncertain parameters) in a given direction was obtained by utilizing BARON. The parameters that were used for the simulated annealing algorithm are shown in Table A11 in Appendix B.

Introducing different representative operating points to the problem increases the problem size and complexity, which may cause longer CPU times to guarantee the globality of the obtained solution. The solutions presented above were obtained on an Intel i7-6600 2.6 GHz processor with 16.0 GB RAM in 1800 s. It is worth mentioning, that globality (within default EpsA range of BARON: 1e-6) could be achieved in the case of proposals 2, 3, and 4. However, it should be mentioned that the optimality gap for proposal 1 was less than 500 €/y while the largest remaining optimality gap was 37,000 €/y in case of proposal MER. Although the remaining optimality gap of the proposal MER is considerable in comparison to the possible savings, the found local solution was considered satisfactory as it indicates that proposal MER is the most cost efficient.
