*3.4. Multi-Period Design Problem for Reduced Superstructure*

With all critical points obtained, the multi-period design problem for the reduced superstructure and the representative operating points can be formulated. This problem is very similar to problem (5), with the exception that each design proposal p ∈ P may include a not generalizable number of individual equality constraints i ∈ Ip, inequality constraints j ∈ Jp, and design constraints/variables d ∈ DVp as well as an individual set of critical points c ∈ CPp, which results in an individual set of operating points opp ∈ S ∪ CPp at which the set of constraints of the respective design proposal must be satisfied. Therefore, problem (5) is reformulated:

$$\begin{aligned} \text{minim} & \mathbf{TAC} = \sum\_{\mathbf{p} \in \mathbf{P}} \mathbf{y}\_{\mathbf{p}} \cdot \left\{ \sum\_{\mathbf{s} \in \mathbf{S}} \left[ \mathbf{w}\_{\mathbf{s}} \ast \mathbf{c}\_{\mathbf{o}\operatorname{arg\,min}, \mathbf{s}, \mathbf{p}} \right] + \mathbf{c}\_{\mathbf{in}\operatorname{vest\,m\,p}, \mathbf{p}} \cdot \mathbf{CRF} \right\} \\ \text{s.t.} & \mathbf{y}\_{\mathbf{p}} \ast \mathbf{h}\_{\mathbf{i}} \{ \mathbf{x}, \mathbf{z}, \mathbf{d}, \mathbf{o}\_{\operatorname{op}, p} \} = 0; \mathbf{i} \in \mathbf{I}\_{\mathbf{P}} \\ & \mathbf{y}\_{\mathbf{p}} \ast \mathbf{g}\_{\mathbf{i}} (\mathbf{x}, \mathbf{z}, \mathbf{d}, \boldsymbol{\theta}\_{\operatorname{op}, p}) \leq 0; \mathbf{j} \in \mathbf{I}\_{\mathbf{P}} \\ & \mathbf{y}\_{\mathbf{p}} \ast \left[ \mathbf{g}\_{\mathbf{d}} (\mathbf{x}, \mathbf{z}, \boldsymbol{\theta}\_{\operatorname{op}, p}) - \left( \mathbf{d}\_{\mathbf{x}\operatorname{sisting}} + \mathbf{d} \right) \right] \leq 0; \mathbf{d} \in \mathbf{D} \mathbf{V}\_{\mathbf{P}} \\ & \mathbf{z}, \mathbf{z}, \mathbf{d}, \boldsymbol{\theta}\_{\operatorname{op}, p} \in \mathbf{R}, \ \ \ \ \ \mathbf{d}\_{\mathbf{L}} \leq \boldsymbol{\theta}\_{\operatorname{op}, p} \leq \boldsymbol{\theta}\_{\mathbf{U}} \\ & \sum\_{\mathbf{p} \in \mathbf{P}} \mathbf{y}\_{\mathbf{p}} = 1 \\ & \mathbf{y}\_{\mathbf{p}} \in \{ \mathbf{0}, 1\}; \mathbf{p} \in$$

In problem (7), the set of individual design proposals of the reduced superstructure is represented by p ∈ P. Each design proposal is expressed via a binary variable yp and one and only one of these binary variables is forced to be 1, which will be the design proposal, which performs most cost efficiently at the representative operating points (and critical operating points). Instead of solving problem (7) simultaneously, it may also be solved proposal-wise, setting one yp to 1 and eventually comparing the TAC of all proposals to identify the most cost-efficient proposal of the reduced superstructure. In this context, it should be noted that all costs that can be associated with the respective retrofit design proposal must be included in the respective cost functions coperating,p and cinvestment,p. This is especially important for retrofitting projects of industrial HENs, which are often very interconnected, and which may cause increased cost in other operational units.
