**1. Introduction**

After the Paris Agreement 2015, decarbonization of industry and energy-intensive industrial processes has been proposed as one of the key measures to limit the increase of the global average temperature well below 2 ◦C. Within decarbonization of industry, waste heat recovery is dedicated to play a major role.

Systematic approaches to design and synthesize heat recovery systems, especially heat exchanger networks (HENs), have been subject to research since the early 1980s with the introduction of the graphical Pinch analysis method [1]. In addition to the graphical Pinch analysis method, mathematical programming has been well established as a complementary approach to HEN synthesis. Both sequential approaches [2] and simultaneous approaches [3], where targeting and network synthesis are performed separately and simultaneously, respectively, have been developed and reported in the literature. In addition, hybrid methods, which combine graphical methods and mathematical programming [4], have been suggested. Among others, Axelsson et al. [5], Bengtsson et al. [6], as well as Escobar and Trierweiler [7] have applied graphical and mathematical methods to case studies, including industrial applications, and reported results and comparisons.

The above-mentioned methods do not account for the variation and uncertainty in operating data that often occur in industrial applications. One example is the variation of input conditions for HENs, i.e., temperatures and flowrates, but also uncertainty in design characteristics, such as heat transfer coefficients of heat exchangers (HEXs). The consequences of neglecting these variations and uncertainties as well as using average or nominal values for design purposes can be increased utility demand (see, e.g., [8]). Additionally, more severe operability issues may occur, e.g., the controllability of target values may not be guaranteed. In order to face these challenges, different strategies have been proposed and a good overview is provided by Kang and Liu [9] in their recent review paper on the synthesis of flexible HENs. Some literature is mentioned here. Kotjabasakis and Linnhoff [10] developed an approach to mitigate the unwanted response of a HEN to variations based on sensitivity tables and systematic utilization of downstream paths. Hafizan et al. [11] utilized the plus-minus principle, introduced by Linnhoff and Vredeveld [12], to visualize the impact of process modifications on the minimum utility target, to derive heuristics for HEN design synthesis based on graphical methods if the respective HEN is subject to disturbances in inlet temperatures.

In addition to graphical approaches (e.g., Pinch-based), mathematical programming has great potential when dealing with variations and uncertainty in operating data. Floudas and Grossmann [13] introduced a multi-period approach to the HEN synthesis problem, which was automated [14] and developed further in the late 1980s [15]. More recently, the multi-period approach has been developed further by a number of authors. In this context, Aaltola [16] extended the SYNHEAT model of Yee and Grossmann [3] to multiple periods. Additionally, Verheyen and Zhang [17] as well as Short et al. [18] improved the model of Aaltola [16] by considering more specific HEX design data. The multi-period HEN synthesis strategy was further applied by Tveit et al. [19] to industrial applications.

Compared to the synthesis of HENs for greenfield problems, the rearrangement of an existing HEN (retrofitting problem) is a different problem. In contrast to greenfield design situations, a retrofitting problem is often characterized by additional constraints and limitations. Extensive work exists on retrofitting methodologies based on graphical insights (e.g., Lai et al. [20], Kamel et al. [21], Bonhivers et al. [22]), mathematical programming (e.g., Ciric and Floudas [23], Asante and Zhu [24], Athier et al. [25]), and those methodologies that can be characterized as hybrid methodologies combining graphical analysis and mathematical programming (e.g., Smith et al. [26], Jiang et al. [27], Akpomiemie and Smith [28]). A comprehensive overview on different retrofitting methodologies can be found in [29].

The above-mentioned retrofit methodologies for HENs do not account for variations and uncertainties in operating data, e.g., in inlet temperatures or heat capacity flowrates. Usually, steady-state or average values are considered, i.e., the reported methodologies are suited for retrofitting single-period HENs, only. However, the capability to account for variations and uncertainties is essential in certain industry applications. Persson and Berntsson [8] showed that considering annual average values to calculate the steam savings potential of a retrofit heat integration project in a pulp mill can lead to an overestimation of 15% compared to the steam savings potential calculated with monthly average values. Persson and Berntsson further concluded in [30] that the steam savings potential decreases even further if it is calculated with average values of shorter time periods (e.g., daily or 10-min periods). In contrast to the results reported by Persson and Berntsson as well as the excessive amount of methodologies available for retrofitting single-period HENs, little attention has been paid to HEN retrofit methodologies that account for variations and uncertainties in operating data. It is worth mentioning that this imbalance between only a few methodologies for multi-period HEN retrofit in comparison with the reported methodologies for single-period HEN retrofit was observed by authors previously publishing in the field (see, e.g., [31]). In fact, to our knowledge, only two systematic retrofitting methodologies for multi-period HENs are reported in the literature. These two retrofitting methodologies are described in the following paragraph.

Papalexandri et al. [32] developed a multi-period Mixed Integer (Non-)Linear Program (MI(N)LP) model, which is based on multi-period hyperstructures, including the considered retrofit alternatives. Furthermore, Papalexandri et al. [32] proposed an iterative scheme between the developed multi-period MI(N)LP and a flexibility analysis subproblem to identify the retrofit alternative, which is operable

over a priori defined variations and yields the minimum total annualized cost. Another retrofitting methodology was reported by Kang and Liu [31], who introduced a two-step method to achieve retrofit design proposals that can operate cost efficiently at multiple periods. In a first step, the multi-period HEN synthesis model as it is used in greenfield design problems is employed. In the second step, existing exchangers are relocated in order to meet the required area demands identified in step one.

Although the above-mentioned retrofit approaches account for variations and uncertainty in operating data, there are several reasons why these approaches are not easily applied to large and complex industrial applications. Commonly, complexities, such as splitting or recirculation of streams, are present in industrial heat recovery systems, which cause nonlinearities in the mathematical formulation of these systems. Therefore, the resulting multi-period MINLP, including retrofit design proposals incorporated as hyper structures as proposed in [32], becomes complex, which inevitably leads to difficultness in finding feasible solutions. Additionally, the flexibility analysis subproblem for large and complex industrial applications is difficult to solve even with state-of-the-art global MINLP solvers. Moreover, depending on the problem complexity itself, the solution produced by common multi-period HEN synthesis formulations and the current network layout may differ essentially. Consequently, relocating existing exchangers to meet identified area demands as proposed in [31] can easily result in inefficient trial and error procedures.

The above reported methodologies are based on mathematical programming, which coincides with the benefits of mathematical programming when dealing with variations in operating data. However, in comparison to these methodologies, graphical insights-based methods have great potential to be applied to large and complex applications due to their beneficial user interaction. Recently, different strategies have been reported in the literature to utilize approaches based on graphical insights in the design process of multi-period HEN retrofitting case studies. A common strategy to deal with variations in operating conditions from a Pinch analysis perspective is to develop different retrofit proposals for a number of selected sets of operating conditions (e.g., annual, seasonal, and monthly average values) [8]. The different design proposals are then evaluated and may be combined to achieve an, over all considered operating points, operable and energy-efficient retrofit proposal. Another strategy is to develop different retrofit proposals by employing graphical insights-based retrofitting methods (e.g., Bridge analyses [22] or advanced composite curves [33]) for a specific nominal point and analyze the network's response to variations to obtain insights and identify the best performing retrofit design proposal [34]. Additionally, Langner et al. [35] proposed to decouple the design and analysis steps in retrofitting processes. By means of this decoupling, well-proven (single period) retrofit design methodologies can be utilized to generate different design proposals, which thereafter are evaluated and adjusted in a trial and error manner with respect to flexibility and energy efficiency [35]. In conclusion, approaching a retrofit problem subject to variations in operating data from a Pinch analysis perspective relies to a large extent on trial and error procedures as different design proposals must be evaluated and manually combined. Good results can be obtained, but the design process itself is often very inefficient due to the trial and error character of the respective strategies.

As pointed out, systematic methodologies for retrofitting industrial HENs subject to variation in operating data are needed, but the available methodologies fulfill this demand only partly. Therefore, we propose a new framework in this paper to achieve flexible and cost-efficient retrofit measures by combining 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. By means of this combination, inefficient trial and error procedures can be avoided. The proposed framework is based on single-period (e.g., Pinch based) retrofit methodologies, (structural) flexibility analysis, critical point analysis, and multi-period optimization. The proposed framework is outlined in Section 3 of this paper. In the following section, the theoretical background of (structural) flexibility analysis and critical point analysis is provided.
