*2.2. Multi-Period Design Problem and Critical Point Analysis*

In a multi-period design problem, independently of the chosen approach, sufficiently many sets of operating points need to be considered to guarantee the flexibility and cost efficiency of the achieved solution. However, with an increasing number of sets of operating points, the complexity of the problem increases. Especially for large-scale industrial applications, the computational capacity can be a limiting factor. To overcome these difficulties, different works in the literature deal with the reduction of the sets of operating points, e.g., by sensitivity analyses [43] or by the identification of critical operating points [44] of a given design proposal. The main idea of these approaches is to identify those variations (within a given uncertainty span) of the uncertain parameters, which require the largest equipment size and are, thus, critical for the design and operation of the HEN. This implies that the HEN itself must be structurally feasible and the feasibility may only be limited by design characteristics, i.e., HEX surface area and heat transfer coefficients (see Section 2.1). The identified critical operating points can be utilized to find the necessary overdesign of HEXs in a fixed HEN structure. By including the identified critical points in the design problem, feasibility for the entire uncertainty span can be achieved. Additionally, representative operating points/scenarios may be identified, e.g., by the screening of historical data, and included in the design problem to achieve a cost-efficient solution (cost optimal for the set of critical points and representative operating points/scenarios). In conclusion, the design problem subject to critical and representative operating points/scenarios can be expressed as the following multi-period MI(N)LP optimization problem:

$$\begin{array}{l}\underset{\begin{subarray}{c}\mathbf{x},\mathbf{d},\mathbf{d}\end{subarray}}{\min}\text{TAC}=\sum\_{\mathbf{s}\in\mathcal{S}}\Big[\mathbf{w}\_{\mathbf{s}}\ast\mathbf{c}\_{\mathrm{operating},\mathbf{s}}\Big]+\mathbf{c}\_{\mathrm{investment}}\ast\mathbf{C}\mathrm{CF}\\\text{s.t.}\\\\\mathbf{h}\_{\mathrm{i}}(\mathbf{x},\mathbf{z},\mathbf{d},\boldsymbol{\theta}\_{\mathrm{op}})=0;\mathbf{i}\in\mathcal{I}\\\mathbf{g}\_{\mathrm{j}}(\mathbf{x},\mathbf{z},\mathbf{d},\boldsymbol{\theta}\_{\mathrm{op}})\leq 0;\mathbf{j}\in\mathcal{J}\\\mathbf{g}\_{\mathrm{d}}(\mathbf{x},\mathbf{z},\boldsymbol{\theta}\_{\mathrm{op}})-\mathbf{d}\leq 0;\mathbf{d}\in\mathrm{DV}\\\mathbf{d}\geq 0\\\mathbf{x},\mathbf{z},\mathbf{d},\boldsymbol{\theta}\_{\mathrm{op}}\in\mathcal{R},\ \boldsymbol{\theta}\_{\mathrm{L}}\leq\boldsymbol{\theta}\_{\mathrm{op}}\leq\boldsymbol{\theta}\_{\mathrm{U}}\end{array} \tag{5}$$

In (5), TAC represents the total annualized cost of the HEN design, which is described by the annual operating cost coperating in €/year (utility cost) of a number of defined representative operating points/scenarios s ∈ S, which are weighted with their normalized duration factor ws (time period represented by operating point divided by entire time period), and the investment cost cinvestment in € (e.g., HEX investment cost), which are annualized with a given capital recovery factor CRF. The set of equality constraints is depicted by hi with i ∈ I (heat and mass balances) and the set of inequality constraints is represented by gj with j ∈ J (temperature and other operational restrictions). The set of design constraints gd depends on the set of the non-negative design variables d with d ∈ DV. In hi, gj , and gd, x is the vector of the state variables, z corresponds to the control variables, and the varying inlet conditions or uncertain parameters are depicted by θ. Consequently, in order to find the optimal value for TAC (for the respective operating points), the degrees of freedom of the optimization problem are the non-negative design variables d with d ∈ DV and the control variables (of the HEN) z. The set of constraints must be satisfied at all representative operating points/scenarios s ∈ S and at all identified critical operating points c ∈ CP, i.e., for certain, previously identified, fixed combinations of values for the uncertain parameters. In (5), the set of constraints must be satisfied at each operating point present in the union of the two sets S and CP: op ∈ (S ∪ CP). This way, a flexible and cost-efficient design is achieved.

To identify those variations (within a given uncertainty span) that require the largest equipment size, sensitivity analyses can be used if monotonic correlations between uncertain parameters and design variables exist. One example is the correlation between the uncertain heat transfer coefficient and the surface area of a HEX—the maximum area of the HEX is obtained at the smallest value of the heat transfer coefficient within the uncertainty span [45]. However, it can be assumed that not many parameters follow these clear correlations and critical operating points can be determined following the procedure introduced by Pintariˇc and Kravanja in [44], which is based on the following idea:

"The main idea is to identify points with the largest values of design variables at optimum objective function. This may be achieved by maximizing design variables one by one, while allowing uncertain parameters to obtain any value between the specified bounds, and simultaneously minimizing cost function." [44] (p. 1607)

This implies a max-min problem for each design variable d ∈ DV, i.e., maximizing the design variable of interest di while minimizing the cost function C(x, z, d, θ), which is stated in problem (6). Compared to the cost function in (5) (TAC), in the cost function of (6) (C(x, z, d, θ)), the operating cost of the representative operating points is not included:

$$\begin{array}{ll}\underset{\theta}{\text{max}}\,\mathrm{d}\_{\mathrm{i}} = \underset{\mathbf{x},\mathbf{z},\mathbf{d}}{\min}\,\mathrm{C}(\mathbf{x},\mathbf{z},\mathbf{d},\theta) = \mathbf{c}\_{\text{operating}} + \underset{\mathbf{c}\_{\text{investment}}}{\text{cCRF}}\,\mathrm{CRF} \\\\\text{s.t.}\\\mathrm{h}\_{\mathrm{i}}(\mathbf{x},\mathbf{z},\mathbf{d},\theta) = 0; \mathbf{i} \in \mathcal{I} \\\\\mathrm{g}\_{\mathrm{j}}(\mathbf{x},\mathbf{z},\mathbf{d},\theta) \leq 0; \mathbf{j} \in \mathcal{J} \\\\\mathrm{d} = \mathrm{g}\_{\mathrm{d}}(\mathbf{x},\mathbf{z},\theta);\mathbf{d} \in \mathrm{DV} \\\\\mathbf{x},\mathbf{z},\mathbf{d},\theta \in \mathbb{R}, \ \mathrm{d} \geq 0, \ \theta\_{\mathrm{L}} \leq \theta \leq \theta\_{\mathrm{U}}.\end{array} \tag{6}$$

To solve the max-min problem, Pintariˇc and Kravanja [44] developed different formulations:


By means of these formulations, possible candidates for critical points are identified. In a second step, a set covering algorithm is applied to the identified candidates to merge them into a final set of critical points. Both the two-level and the approximate one-level formulation approximate the solution of the KKT-formulation. Moreover, the approximate one-level formulation is dependent on a heuristically chosen Big-M parameter and solutions can be very sensitive to this parameter. Additionally, if the system is convex, all critical points are vertices of the uncertain space. These critical vertices can be identified by solving problem (5), excluding the representative operating points, sequentially at all vertices. Those vertices at which the maximum values for each design variable are obtained are critical [44].
