*2.1. (Structural) Flexibility Analysis*

Flexibility analysis of HENs has been subject to research since the early 1980s. In 1982, Marselle et al. [36] first introduced the concept of resilient HENs with respect to a certain disturbance range in the inlet conditions. In 1985, Saboo et al. [37] introduced a resilience index to quantify the resilience of HENs. In the same year, Swaney and Grossmann [38] extended the concept of the resilience index to a flexibility index, which is applicable not only to HENs but also to chemical processes in general. Both the resilience and the flexibility index indicate the maximum disturbance range in which inlet conditions may vary while at the same time achieve feasible operation. This maximum disturbance range can be interpreted as a hyperrectangle in the space of the varying inlet conditions. In this context, both indices are defined as the ratio between the largest scaled hyperrectangle within the feasible region and the hyperrectangle defined by an expected disturbance range. Therefore, feasibility is achieved if the respective index is larger or equal to 1. For two varying inlet conditions, this can be visualized as in Figure 1. The largest scaled rectangle within the feasible region can mathematically be expressed by the following set of equations (δ corresponds to flexibility/resilience index):

$$\text{CT}\_{1,\text{N}} - \delta \star \Delta \text{T}\_{1}^- \le \text{T}\_{1} \le \text{T}\_{1,\text{N}} + \delta \star \Delta \text{T}\_{1}^+,\tag{1}$$

$$\text{tr}\,\mathbf{T}\_{2,\text{N}} - \delta \star \Delta \mathbf{T}\_{2}^{-} \le \mathbf{T}\_{2} \le \mathbf{T}\_{2,\text{N}} + \delta \star \Delta \mathbf{T}\_{2}^{+}.\tag{2}$$

**Figure 1.** (Hyper-)rectangle with respect to expected variations and maximum scaled (hyper-)rectangle inscribed within the feasible region in the space of the varying inlet temperatures of a heat exchanger network [38].

In Figure 1, the largest scaled rectangle within the feasible region is smaller than the rectangle according to the expected variations. Thus, feasibility with respect to the expected variations is not achieved (flexibility or resilience index is smaller than 1). In both index formulations, the physical performance of the HEN or the chemical process is described by the following set of constraints:

$$\mathbf{h}\_{\mathbf{l}}(\mathbf{d}, \mathbf{x}, \mathbf{z}, \Theta) = 0; \mathbf{i} \in \mathcal{I}, \tag{3}$$

$$\lg\_{\clubsuit}(\mathbf{d}, \mathbf{x}, \mathbf{z}, \Theta) \le 0; \mathbf{j} \in \mathbf{J}, \tag{4}$$

where d is the vector of design variables, x corresponds to the state variables, z is used for the control variables, and the varying inlet conditions or uncertain parameters are depicted by Θ (see [37,38]). In HENs, typical design variables are the area values of HEXs while typical uncertain parameters are the inlet temperatures or heat capacity flow rates of process and/or utility streams. State and control variables correspond to the internal network temperatures and duties of HEX. Furthermore, the operational equality constraints of HENs are heat and mass balances while operational inequality

constraints ensure a minimum temperature difference as well as that heat transfer is only possible from hot to cold streams. Additionally, design constraints of HEXs (in form of heat transport equations) may also be considered as operational inequality constraints.

Feasibility is achieved when all constraints i ∈ I and j ∈ J are satisfied at the point of operation. With both formulations, it is possible to describe the resilience or flexibility for convex problems in which the solution lies at a vertex point of the largest scaled (hyper-)rectangle within the feasible region (see Figure 1). In 1987, Grossmann and Floudas [39] developed an active set approach to guarantee a global solution of the flexibility index problem also for some non-convex problems. More recently, Li et al. [40] suggested a framework to calculate the flexibility index by means of an alternating direction matrix embedded in a simulated annealing algorithm. Furthermore, Zhao and Chen [41] proposed to explicitly calculate the shape of the uncertainty space via cylindrical algebraic decomposition and quantifier elimination.

In the context of flexibility analysis of HENs, the term structural flexibility is used to define the set of constraints that are included in the flexibility analysis. Often, it is distinguished between structural constraints (e.g., heat and mass balances) and design constraints (e.g., heat transport equations of HEX). Marselle et al. [36] distinguished, for example, between a resilient network structure and a resilient network itself. Marselle et al. [36] further suggest that a network structure is resilient if it remains feasible for the specified disturbance range independent of the HEX areas (i.e., HEX areas are not specified). A resilient network structure is, thus, the premise for a resilient network, which remains feasible for the specified disturbance range for specified HEX areas [36]. This definition of structural resilience was later used by Li et al. [42] to describe the structural flexibility of HENs. In accordance with the literature, our work distinguishes between the structural feasibility of a design and the (general) feasibility of a design. Both can be assessed by solving the flexibility index problem. In the case of structural feasibility assessment, design constraints are discarded, and only structural constraints are considered, i.e., the heat transferred by HEXs is not limited by design characteristics. In the case of (general) feasibility assessment, design constraints are included, i.e., the heat transferred by HEXs is limited by design characteristics (i.e., the installed surface area and the overall heat transfer coefficient of exchangers).
