**5. Conclusions**

In this work, a novel framework for retrofitting multi-period HENs was developed and presented. The proposed framework divides the retrofit design process into five sub-steps, which allows for the combining of the beneficial designer interaction of graphical approaches (e.g., Pinch based) at an early stage in the design process with the efficiency of mathematical programming to derive flexible and cost-efficient retrofit measures. In this context, inefficient trial and error procedures are avoided. Additionally, by means of splitting the design process in five different sub-steps, the complexity of the sub-problems is decreased compared to the overall problem.

The proposed framework utilizes the designer interaction of graphical approaches to derive different design proposals, which can be very beneficial for large and complex HENs usually found in industrial heat recovery systems. Well-proven, single-period retrofit design methods, such as Bridge analysis, but also "experience-based" retrofit design proposals may be employed. As the single-period character of graphical retrofitting methodologies does not ensure the flexibility and cost-efficiency of the generated proposals for the entire operating period, different mathematical evaluation strategies are incorporated in the proposed framework. By means of structural feasibility assessment based on the calculation of the flexibility index, structurally infeasible design proposals can be identified and discarded from further analysis. Additionally, critical point analysis was suggested to identify those operating points within the uncertainty span that determine necessary overdesign of the HEXs to ensure feasibility of the HEN retrofit proposals for the entire operating range. In order to identify the most cost-efficient proposal, a multi-period MI(N)LP optimization problem was formulated, which considerers the identified critical points (for feasibility) as well as representative operating points (for cost-efficiency). A final feasibility check was suggested to ensure that the complete set of critical points has been identified.

The determination of critical points based on [44] was automated and the achieved results were discussed. Compared to the available literature examples, the HEN examples included in this study are more complex, which caused difficulties when determining critical points. Modifications of the existing methodologies (see Appendix A) were suggested in order to identify the complete set of critical points for the HEN retrofit example presented in this study. It is worth mentioning that by means of the above-mentioned final feasibility check, an incomplete set of critical points is recognized. Additionally, by means of the obtained results of the final feasibility check, the missing critical point(s) can be identified to be included in the multi-period MI(N)LP design problem.

The framework presented in this paper yields opportunities to increase heat recovery in applications operating at multiple periods, e.g., industrial applications. As decarbonization of industry is essential to limit the increase of the global average temperature well below 2 ◦C, systematic approaches to successfully retrofit industrial heat recovery systems are in demand more than ever. In comparison to existing approaches for retrofitting multi-period HENs, the presented framework splits the design process in five different sub-steps, which decreases the complexity of the sub-problems. The decreased complexity can be the decisive factor for a successful retrofit project when large and complex industrial applications are addressed. The novel contributions of this paper are summarized in the following list:


**Supplementary Materials:** The following are available online at http://www.mdpi.com/1996-1073/13/6/1472/s1: Directory with source code of the automated determination of critical points including the following files: Main\_KKT.py Main\_2\_level.py, CalcA.py, ExtendedSetCovAlgorithm.py, Lower\_Control\_Design.py, SetCovAlgorithm.py, Upper\_Uncertainty.py. Additionally, model instances are provided to test the automated KKT-formulation and two-level formulation on each of the 5 different design proposals (in the reduced superstructure) discussed in Section 4 (see e.g., Figure 3 and Table 6).

**Author Contributions:** Conceptualization, C.L. and E.S; methodology, C.L. and E.S; software, C.L.; validation, C.L.; formal analysis, C.L.; investigation, C.L.; resources, S.H.; data curation, C.L.; writing—original draft preparation, C.L.; writing—review and editing, E.S. and S.H; visualization, C.L.; supervision, E.S.; project administration, S.H.; funding acquisition, E.S. and S.H. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research was funded by the Swedish Energy Agency, grant number P42326, and the Södra Foundation for Research, Development and Education.

**Acknowledgments:** We would like to thank Gulnara Shavalieva, Holger Wiertzema, Paraskevi Karka, Sofie Marton and Stavros Papadokonstantakis for interesting discussions and useful suggestions. Furthermore, we would like to acknowledge three anonymous reviewers for their valuable comments.

**Conflicts of Interest:** The authors declare no conflict of interest. The funders had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript, or in the decision to publish the results.
