Resilience test constraints

The solution 8C of max-min problem (8) defines a critical point for feasible operation—it is the point where uncertainty range 0 is closest to feasible region R if (d) 0 (Fig. 3a), or it is the point where maximum constraint violations occur if (d) > 0 (Fig. 3b). In qualitative terms, the critical points in the resilience test are the worst case conditions for feasible operation. 

The nonlinear resilience tests developed by Saboo et al. (1987a,b) are each for a rather specific case. A more general resilience analysis technique based on the active constraint strategy of Grossmann and Floudas (1985,1987) is also presented. The active constraint strategy can be used to test the resilience of a HEN with minimum or more units, with or without stream splits or bypasses, and with temperature and/or flow rate uncertainties (Floudas and Grossmann, 1987b). 

The resilience test correctly identifies that the HEN is not resilient in the specified uncertainty range. In order to apply the resilience test, the values °f vk.max and gjj max shown in Table V are calculated. The values of v max are all nonpositive thus the load constraints are satisfied throughout the... 

The resilience test incorrectly identifies the network as not being resilient. The values of vk max and gy max for this uncertainty range are listed in Table VI. Since all the vk max are nonpositive, the load constraints are satisfied throughout the uncertainty range. To test the stream split constraints, LP (29) is applied ... 

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) ... 

A HEN is resilient in a specified uncertainty range 0 if and only if X(d) 0. If (d) > 0, then at least one of the feasibility constraints fm is violated somewhere in the uncertainty range. Geometrically, the resilience test determines whether uncertainty range 0 lies entirely within feasible region R. 

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. 

Now the active constraint strategy for performing the resilience test can be summarized (Grossmann and Floudas, 1987) as follows. 

Note that in Table VIII, u3i violates the stream split constraints (0 s u31 < 1) at some of the solutions of NLP (34 ) for example, m3, < 0 at the solution of NLP (3418). This means that potential set of active constraints MA(18) = (/5, /7) is not active at the solution of the resilience test problem, since u31 violates the nonnegativity constraint [which was not included in NLP (3418)]. However, u31 does satisfy the constraint m3i = 1 (f7 0), since this constraint was included in NLP (3418). 

Resilience Test with Active Constraint Strategy for Example 11... 

Develop techniques to test the resilience of class 2 HENs with stream splits and/or bypasses, temperature and/or flow rate uncertainties, and temperature-dependent heat capacities and phase change. It may be possible to extend the active constraint strategy to class 2 problems. This would allow resilience testing of class 2 problems with stream splits and/or bypasses and temperature and/or flow rate uncertainties. However, the uncertainty range would still have to be divided into pinch regions (as in Saboo, 1984). 

The operability test (resilience test in Section III,A,2, but without the energy recovery constraint) is applied at the stage of matches to determine whether the predicted stream matches can lead to an operable HEN. The general form of this operability test is... 

Example 8 (from Saboo et al., 1987b). Resilience of the HEN in Fig. 12 is tested in Example 9. In this example, the constraints on the stream split fractions are formulated. 

The minimum unit HEN with stream splits is resilient if < 0. This test is sufficient, but not necessary, for HEN resilience. It is necessary only if the same critical corner point maximizes all of the and vk constraint functions simultaneously. 

Example 9. Resilience of the minimum-unit stream-splitting HEN shown in Fig. 12 is to be tested in the uncertainty range 415 K < rf 515 K. The stream split constraints were derived in Example 8 from the ATm constraints on the hot ends of the exchangers. 

Example 11. The active constraint strategy is to be used to test the resilience of the same stream splitting HEN as in Example 9 (Fig. 12) in the uncertainty range 0 = TII415 < Tf < 515 K. ... 

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. 

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. 

Several powerful HEN resilience analysis algorithms have been reviewed in this chapter, including the active constraint strategy (Grossmann and Floudas, 1985, 1987) which can test the resilience of the most general HEN. However, many common industrial HENs still cannot be analyzed with present techniques. For example, no technique has been... 

Develop techniques to test the resilience of HENs with uncertain heat transfer coefficients (e.g., heat transfer coefficients as a function of flow rate, but with uncertain function parameters). It is possible to extend the active constraint strategy to heat transfer coefficients with bounded uncertainties (not as a function of flow rate), but then the active constraint strategy may not have a single local optimum solution. 

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