Flexibility indices

In this section, general mathematical formulations and graphic interpretations are presented for several resilience analysis problems (1) feasibility test, (2) resilience (flexibility) test, (3) flexibility index, and (4) resilience index. 

Fig. 5. Flexibility index F defines largest scaled hyperrectangle 0(F) for which HEN is resilient.

The flexibility index can be defined by considering how much larger (or smaller) the expected uncertainty range 0 must be scaled so that it exactly fits inside feasible region R (Fig. 5). The family of scaled hyperrectangles 0( ) can be parameterized by scale factor s as... 

A HEN is resilient in an expected uncertainty range 0(1) if and only if Fs 1. This is the same information which the resilience (flexibility) test gives. But the flexibility index tells us even more. For instance if F = 0.5, we know that the HEN is not resilient in the specified uncertainty range in addition, we know that the HEN can tolerate uncertainties only half as large as those expected. 

Whether a HEN is resilient in a specified uncertainty range is independent of the choice of nominal values 0N of the uncertain variables in that range. However, the actual value of the flexibility index F does depend on the choice of 0N. 

The problem of calculating the flexibility index can be mathematically formulated as... 

When the critical point must be a vertex of the hyperrectangle 0(F), the simplest approach to calculating the flexibility index F is to maximize s in each vertex direction 61 (Fig. 5) by the following (N)LP (Swaney and Grossmann, 1985a) ... 

Example 4 (from Grossmann and Floudas, 1987). The flexibility index of the HEN structure in Fig. 4 is to be calculated with respect to an expected uncertainty range of + 10 K in all stream supply temperatures. 

Solution of the 16 LPs yields the results in Table II. The flexibility index is 0.5 thus the HEN can only tolerate uncertainties of 5 K in each stream supply temperature instead of the 10 K expected. In particular, there are four critical vertices (s in = 0.5). At all four of these critical vertices, HEN flexibility is limited by ATm violations in exchanger 2. 

The RI can be scaled in terms of maximum expected uncertainties like the flexibility index. Then the RI would be an upper bound on the flexibility index, since the RI only considers uncertainties in each stream individually, while the flexibility index considers uncertainties in all streams simultaneously. However, for HENs it seems reasonable to assume that the uncertainty in each stream s supply temperature is roughly inversely proportional to that stream s heat capacity flow rate (i.e., that each stream is subject to roughly the same load uncertainty ). Thus the RI is calculated in terms of load uncertainty 81 or 8t, where... 

In order to compare the resilience index with the flexibility index, suppose that the RI is scaled in terms of temperature rather than load. Then the temperature RI is 6.67 K (Table III), limited by positive uncertainty in the supply temperature of stream 4. (Note that the limiting uncertainty direction changes when the RI is rescaled from load to temperature.) This means that the HEN can tolerate uncertainty of 6.67 K in any individual stream supply temperature in either a positive or negative direction. Because of linearity and convexity, it also means that the HEN can tolerate a total temperature uncertainty S,jTf - 7fN of 6.67 K, no matter how the uncertainty is distributed among the streams. Note that this does not mean that the HEN can tolerate uncertainties of 6.67 K in all... 

The active constraint strategy has been developed for both the resilience (flexibility) test and the flexibility index (Grossmann and Floudas, 1985, 1987). However, only the active constraint strategy for the resilience test will be discussed here. Recall that the resilience test is based upon a resilience measure x(d) ... 

The basic idea of the active constraint strategy is to use the Kuhn-Tucker conditions to identify the potential sets of active constraints at the solution of NLP (4) for feasibility measure ip. Then resilience test problem (6) [or flexibility index problem (11)] is decomposed into a series of NLPs with a different set of constraints (a different potential set of active constraints) used in each NLP. 

Floudas and Grossmann (1987b) have shown that for HENs with any number of units, with or without stream splits or bypasses, and with uncertain supply temperatures and flow rates but with constant heat capacities, the active constraint strategy decomposes the resilience test (or flexibility index) problem into NLPs which have a single local optimum. Thus the resilience test (or flexibility index) also has a single local optimum solution. 

The preceding theorem describes an operability test for class 2 HENs. Similarly, by omitting the energy recovery constraint from flexibility index problem (15) or resilience index problem (19), an operability index could be defined for class 2 HENs. 

The flexibility index determines the largest uncertainty range (scaled hyperrectangle) for which the HEN is resilient. The flexibility index is scaled in terms of an expected uncertainty range (0N - F Afl <0<0N + F A0+). 

The flexibility index can be calculated by determining the largest scaled uncertainty sk which the network can tolerate in the direction of each of the k corner points of the uncertainty range. Then the flexibility index is... 

Different algorithms are required if the HEN resilience problem is nonlinear. Special algorithms were presented for testing the resilience of minimum unit HENs with piecewise constant heat capacities, stream splits, or simultaneous flow rate and temperature uncertainties. A more general algorithm, the active constraint strategy, was also presented which can test the resilience or calculate the flexibility index of a HEN with minimum or more units, stream splits and/or bypasses, and temperature and/or flow rate uncertainties, but with constant heat capacities. 

In the synthesis of resilient HENs, use of the class 1 flexibility index target offers two important features (Colberg et al., 1988) ... 

Step 6. Apply the active constraint strategy to the flexibility index (F) at the stage of structure (without the energy recovery constraint). The form of this flexibility index problem is described in a later section, (a) If F a 1, then the HEN is operable in the specified uncertainty range. Stop, (b) If F< 1, then add the critical point for operability as another period of operation and return to step 5. 

---