Active constraint strategy

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). 

Active Constraint Strategy for Nonlinear Resilience Analysis... 

In addition, the active constraint strategy can save significant computational time in linear resilience analysis (86% in one example Grossmann and Floudas, 1987) by eliminating the need to check HEN feasibility at every corner point. Thus, this strategy makes it practical to analyze the resilience of HENs with a large number of streams. 

Grossmann and Floudas (1985, 1987) present both nonlinear and specialized linear forms of the active constraint strategy. Only the nonlinear form is discussed in this chapter. 

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 difficulty with problem (34) is that the set of active constraint equations MA can change for different values of uncertain variables 8. The key feature of the active constraint strategy is its ability to identify potential sets of active constraints. 

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

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. ... 

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

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. 

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). 

Step 4. Apply the operability test at the stage of matches (with the active constraint strategy). The form of this operability test is described in the next section, (a) If the set of matches cannot lead to an operable HEN, then add the critical point for operability as another period of operation and return to step 2. (b) If the set of matches can lead to a HEN operable in the specified uncertainty range, then go 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... 

Grossmann, I. E., and Floudas, C. A., Active constraint strategy for flexibility analysis in chemical processes. Comp. Chem. Eng. 11, 675 (1987). 

There is a second reason why Feasible Direction Methods should not be referred to as methods that use the active constraints strategy the direction d is feasible only for linear constraints. 

There is a third reason why Feasible Direction Methods should not be called methods that use the active constraints strategy it is possible to ejqjloit the direction di also including the bound for the variables (beyond the bounds already existing) to limit the search region. In this case, the direction d is not used to perform a onedimensional search, since a Trust region method or Reduced-step method is used. 

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