Resilience index uncertainties

Example 5 (adapted from Grossmann and Floudas, 1987). The resilience index is to be calculated for the HEN shown in Fig. 4. The following series of LPs is formulated to determine the maximum load uncertainty allowed in a positive direction on each stream ... 

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 resilience index, another measure of the largest uncertainties for which the HEN remains feasible, is the size of the largest total uncertainty load (polytope region) which the HEN can tolerate. 

The resilience index can be determined by maximizing the load uncertainty /, which the network can tolerate in either a positive or negative direction in each stream supply temperature. Then the resilience index is... 

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 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+). 

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. 

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