Flexibility index uncertainty range

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. 

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. 

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... 

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. 

To test for operability of the HEN in the specified uncertainty range, the active constraint strategy is applied to the flexibility index at the stage of structure (without the energy recovery constraint). First, constraints (A1)-(A5) and (B1)-(B4) are developed for this network. Since there are nine equations and 12 unknowns, there exist three control variables which have been selected to be zx = w 2, z2 = w 2, z3 = T4 (see Fig. 20). Using the information from the gradients of the feasibility constraints with respect to the control variables, four active sets of constraints are identified. Then, solving an NLP for each active set of constraints, it was found... 

A process design is flexible if it can tolerate uncertainties in parameters and can handle disturbances. A flexibility index is a measure of the amount of uncertainty that can be tolerated with the desired process operation remaining feasible. A schematic of the flexibility index, 5, is shown in Fig. 8. Here, the two degrees of freedom represent uncertain parameters or disturbance variables, which have assumed upper and lower bounds. The feasible operating region lies within the cross-hatched area. The flexibility index is the fraction of the parameter range that still results... 

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