3. Methodology

The methodology adopted entails the use of a multi-period version of the simplified stage-wise superstructure (SWS) of Yee and Grossmann [19], as presented in Refs. [5, 8]. The SWS is shown in Figure 1. In this superstructure, each hot-process stream and each cold-process stream has the option of splitting within each interval and each period of operation where it exists so as to exchange heat with streams of the opposite kind in intervals 2 and 3. The hotand cold-process streams are then taken to their final temperatures in intervals 1 and 4 through heat exchange with utilities of the opposite kind in each period of operation where the process streams exist. The details of the superstructure can be found in Refs. [5, 7].

#### 3.1. Model equations

The detailed multi-period HENS models used in this chapter are shown in the appendix. For detailed explanations of each of these equations, the reader is referred to the multi-period SWS HENS model of Verheyen and Zhang [5] and Isafiade et al. [7]. The maximum area approach as introduced by Verheyen and Zhang [5] for HENS is also used in this chapter for flexible HENS as shown in Eq. (3)

Figure 1. Multi-period version of SWS model.

$$A\_{i,j,k} \ge \frac{q\_{i,j,k,p}}{(LMTD\_{i,j,k,p})(U\_{i,j})} \tag{3}$$

The maximum area, Ai,j,k, is then included in the objective function shown in Eq. (4). However, it should be known that Eq. (4), which is the objective function used in this study, was introduced by Isafiade and Fraser [6] for multi-period networks having specified process parameter points

$$\begin{split} \min \left\{ \left[ \frac{DOP\_p}{\sum\_{p=1}^{NOP} DOP\_p} \sum\_{i:H} CUC \cdot q\_{i,j,k,p} + \frac{DOP\_p}{\sum\_{p=1}^{NOP} DOP\_p} \sum\_{i:H} CUH \cdot q\_{i,j,k,p} \right] \\ \quad + AF \left[ \frac{\sum\_{\tilde{i}} \sum\_{\tilde{i}} CCF\_{\tilde{i}\tilde{i}} \cdot z\_{i,j,k} + \sum\_{\tilde{i}:H \tilde{j} \in C\epsilon K} \sum\_{i,j,k} A\mathbf{C}\_{i,j,k} \cdot A\frac{dE\_{i,j}}{i,j,k} \right] \forall i \\ \in H, j \in C, k \in K, p \in P \end{split} \tag{4}$$

It is worth mentioning that the terms in the first square bracket of Eq. (4) are the annual operating cost terms. The presentation of these terms by Isafiade and Fraser [6] adequately allocates the contribution of each hot/cold utility to the annual operating cost of the flexible network based on each operating periods of duration. This is unlike the objective functions shown in Eqs. (1) and (2), which are used in most existing methods, and which make an implicit assumption that each operating parameter point within the uncertain range would operate at an equal/average period duration. Whereas this may not always be true because any of the parameter points may dominate at any point in time, hence the network needs to be flexible enough to handle unforeseen period durations.

For the example solved in this chapter, the solver DICOPT, which uses CPLEX for the MILP and CONOPT for the NLP sub-problems, has been used. The solver environment used is GAMS [21]. The machine used operates on Microsoft® Windows 7 Enterprise™ 64 bit, Intel® Core™ i5-3210M processor running at 2.50 GHz with 4 GB of installed memory.

#### 3.2. Solution approach

Ai,j,<sup>k</sup> <sup>≥</sup> qi,j,k,<sup>p</sup>

CUC � qi,j,k,<sup>p</sup> þ

CFij � zi,j,<sup>k</sup> <sup>þ</sup> ∑

iϵH ∑ jϵC ∑ kϵK

It is worth mentioning that the terms in the first square bracket of Eq. (4) are the annual operating cost terms. The presentation of these terms by Isafiade and Fraser [6] adequately allocates the contribution of each hot/cold utility to the annual operating cost of the flexible network based on each operating periods of duration. This is unlike the objective functions shown in Eqs. (1) and (2), which are used in most existing methods, and which make an implicit assumption that each operating parameter point within the uncertain range would operate at an equal/average period duration. Whereas this may not always be true because any of the parameter points may dominate at any point in time, hence the network needs to be

" #9

parameter points

min DOPp ∑NOP <sup>p</sup>¼<sup>1</sup> DOPp

> <sup>þ</sup>AF ∑ iϵH ∑ jϵC ∑ kϵK

flexible enough to handle unforeseen period durations.

∈H, j∈C, k∈K, p∈P

2 4

8 < :

Figure 1. Multi-period version of SWS model.

98 Heat Exchangers– Design, Experiment and Simulation

∑ iϵH

The maximum area, Ai,j,k, is then included in the objective function shown in Eq. (4). However, it should be known that Eq. (4), which is the objective function used in this study, was introduced by Isafiade and Fraser [6] for multi-period networks having specified process

> DOPp ∑NOP <sup>p</sup>¼<sup>1</sup> DOPp

<sup>ð</sup>LMTDi,j,k, <sup>p</sup>ÞðUi,j<sup>Þ</sup> (3)

∑ iϵH

ACi,j,<sup>k</sup> � AAEi,<sup>j</sup> i,j,k

CUH � qi,j,k, <sup>p</sup>

= ; ∀i 3 5

(4)

Based on the foregoing explanations, in this chapter, the following procedure is adopted in generating the candidate initial multi-period network and the subsequent flexibility tests:


in Step 3. The model at this stage is solved with further minor adjustments to exchanger areas, if needed, so as to accommodate as many operating points as possible.
