*Appendix A.2. A Karush–Kuhn–Tucker (KKT-)Formulation*

The main idea of the KKT-formulation is to apply the KKT conditions to the minimization problem of problem (6) (min x,z,d C(x, z, d, θ)) and then solve the remaining single-level optimization problem (max di <sup>θ</sup> ) for each design variable di ∈ DV individually. Similar to the result of the two-level formulation, the result of the KKT-formulation is the operating point in the uncertainty span at which the maximum value for di is obtained while the cost function C(x, z, d, θ) is minimized (i.e., a potential critical point) [44]. If one is able to solve the KKT-formulation globally, the solution represents the global solution of problem (6). In contrast to the two-level formulation, the solution process of the KKT-formulation does not involve an iterative procedure. However, a global NLP solver is necessary to solve the resulting problem, which may cause difficulties for large industrial applications. Additionally, as global solvers usually do not provide marginal values, an (local) NLP solver needs to be applied afterwards to obtain marginal values, which are necessary to identify the influencing uncertain parameters (compare [44]).

In order to obtain marginal values, in this study, the KKT-formulation was first solved using BARON and the found solution was used as an initialization for IPOPT in GAMS. However, the calculation of BARON was interrupted after 300 s and the best (local) obtained solution was used for the initialization of IPOPT. In all tested cases, IPOPT returned the same value as BARON and marginal values were provided. The KKT-formulation was solved for the five different design proposals. For design proposals 1, 2, 3, and 4, the sets of critical points which are shown in Table A5 in Appendix B (i.e., the complete sets of critical points) were obtained by solving the KKT-formulation and applying the set covering algorithm as it was reported by Pintariˇc and Kravanja in [44]. However, the situation was different for the design proposal MER. The set of critical points that was obtained by the KKT-formulation and the set covering algorithm for proposal MER together with the corresponding TAC and flexibility index are shown in Table A3.


**Table A3.** Results of the Karush–Kuhn–Tucker (KKT-)formulation of design proposal MER.

Comparing the TAC and the flexibility index shown in Table A3 to the results for the proposal MER shown in Table 6 indicates that the set of critical points (shown in Table A3) is incomplete. A first solution attempt was to investigate the influence of the uncertain parameters that have "0" elements in the set of critical points (e.g., FcpH2 in the first identified critical point of the set shown in Table A3). These "0" elements result from the calculation of the marginal values of the uncertain parameters, meaning that the corresponding uncertain parameter was discovered to have no influence on at least one of the design variables when solving the KKT formulation (compare [44]). When solving the design problem, the uncertain parameters corresponding to these "0" elements are defined as variables for the respective critical point, i.e., these uncertain parameters can obtain any value within the uncertainty span at the solution of the design problem. As the identified sets were incomplete, it seemed that the influence of one or several uncertain parameters was not correctly detected. This was investigated by extending the set covering algorithm. In addition to the three steps reported in the appendix of [44], a fourth step was added. In this step, each "0" element present in the set of critical points, obtained after the first three steps, was replaced with all possible combinations with respect to the other points in the set and the set covering formulation (AP6 in [44]) is solved.

*Example:* If a set consists of the following three points [1,2,1,1], [1,1,3,1] and [2,0,0,1], the third point is replaced by all possible combinations with respect to elements of the first two points. For each of the two "0" elements of the third point (second and third element), the nonzero values in the other two points are considered to replace the 0: (1,2) for the second element and (1,3) for the third element. Eventually, four additional candidates to replace the third point are found: [2,1,1,1], [2,1,3,1], [2,2,1,1], and [2,2,3,1]. After applying the previously mentioned set covering formulation, one candidate point remains: [2,2,1,1]. This candidate point replaces the third point of the initial set.

With this additional step in the set covering algorithm, the set of critical points shown in Table A3 was transformed to the set of critical points that is shown in Table A5 in Appendix B. Eventually, with the additional step in the set covering algorithm, the complete set of critical points was found, and the feasible design for the proposal MER was achieved (see Table 6).

Obviously, the KKT-formulation is easier to solve for less complex HENs as the different HEX units are less dependent on each other. This is reflected in the achieved results as the extended set covering algorithm was only necessary to identify the complete set of critical points for proposal MER. In all other cases, the extra step in the extended set covering algorithm was not necessary. It was further investigated if increasing the maximum calculation time of BARON would result in different solutions for proposal MER. However, even an increase to 1000 s did not have any impact on the results. Actually, for several design variables, the best (local) solution was found before BARON started iterating, i.e., during the local search before the iteration scheme of BARON is executed. This implies that different starting points could have an impact. However, the influence of different starting points was not studied.

The achieved results indicate that more complex HEN structures can be handled by means of the KKT-formulation compared to the two-level formulation. Future work should focus on rather complex HEN structures and investigate if modifications of the KKT-formulation are necessary to ensure that, in combination with the extended set covering algorithm, the complete set of critical points can be identified.
