*4.2. Economic Data and Identification of Critical Points*

In order to identify the critical points of the different retrofit design proposals, design variables were defined. These design variables could be modified in order to achieve feasibility for the entire range of operating conditions. It was assumed that the area of the two existing units could be increased while new units could be designed freely. Furthermore, the structure of the objective function of the overall design problem was specified. All in all, the total annualized cost (TAC) of the design should be minimized. Consequently, different cost functions were specified:


Additionally, representative operating points s ∈ S, their normalized duration factor ws, the capital recovery factor CRF, and the annual operating time toperating were specified, in order to achieve cost efficiency for the entire operating period. The structure of the objective function for the retrofit design proposals is given in Equation (8):

$$\min \text{TAC} = \sum\_{\mathbf{s} \in \mathbf{S}} \left[ \mathbf{w}\_{\mathbf{s}} \left( \mathbf{p}\_{\text{CU}} \mathbf{Q}\_{\text{CU},\mathbf{s}} + \mathbf{p}\_{\text{HU}} \mathbf{Q}\_{\text{HU},\mathbf{s}} \right) \right] \bullet \mathbf{t}\_{\text{operating}} + \left( \text{Inv.cost}\_{\text{Increase}} + \text{Inv.cost}\_{\text{RRW}} \right) \bullet \text{CRF} . \tag{8}$$

As mentioned in Section 2.2, the cost function, which is minimized in problem (6) to identify critical points, does not include the operating cost of the representative operating points. The cost function that was used for determining the critical points is given in Equation (9):

$$\min \mathcal{C} = \left( \mathbf{p}\_{\rm CU} \mathbf{Q}\_{\rm CU} + \mathbf{p}\_{\rm HU} \mathbf{Q}\_{\rm HU} \right) \mathbf{t}\_{\rm operating} + \left( \text{Inv.cost}\_{\rm increase} + \text{Inv.cost}\_{\rm new} \right) \bullet \text{CRF} \tag{9}$$

For simplicity, it was assumed that the increase of the two existing HEX units could be described by the same cost function. The cost for increasing a HEX in € is given in Equation (10) (A in m2):

$$\text{Inv.cost}\_{\text{increase}} = 4000 + 2000 \ast \text{A}\_{\text{increasea}}.\tag{10}$$

For new HEX units, it was also assumed that the same cost function is applicable for all possible new HEX units. The cost of a new HEX in € is given in Equation (11) (A in m2):

$$\text{Inv.cost}\_{\text{new}} = 40,000 + 2000 \ast \text{A}\_{\text{HEX}}.\tag{11}$$

A capital recovery factor of 0.1 (e.g., 15 years lifetime and an interest rate of 7%) and an annual operating time of 8200 h were assumed. Furthermore, for the operating cost, the data in Table 5 were used. It was further assumed that no additional costs are associated with the retrofit proposals.

**Table 5.** Cost data for operational cooling and heating.


For each of the five retrofit design proposals, which together constitute the reduced superstructure, one set of critical points was identified. These sets of critical points are listed in Table A5 in Appendix B. The sets of critical points listed in Table A5 are complete, i.e., the designs based on these sets were proven to be feasible by means of flexibility analysis (see Section 4.3). To avoid numerical difficulties, the logarithmic mean temperature difference in the design constraints of the HEXs was approximated using Paterson's approximation (see, e.g., [47]). As the determination of the critical points was connected to difficulties that were neither reported in the literature nor have been experienced when the results of the available literature examples of critical point analysis of HENs (see [44,45]) were reproduced, a description and a discussion on the determination of the critical points can be found in Appendix A. In this context, modifications on the existing methodologies were suggested to be able to determine critical points for more complex HEN structures but also for retrofit studies of HENs. For more information, see Appendix A.
