MPEC Problem Formulations in Chemical Engineering Applications

Simplifying the complementarity conditions to eliminate the variable y inthe first two equations leads to:z = f(x) + s b , f(x) − z a = s a − s b (33)0 ≤ s b ⊥ s a ≥ 0• The min operator y = min (f(x), y b ) can be treated in a similar way bydefining the problem:z = f(x) + (f(x) − z b )yy = arg{minŷ(f(x) − z b )ŷ s.t. 0 ≤ ŷ ≤ 1} (34)Applying the KKT conditions and simplifying leads to the following complementaritysystem:5.2 Flow Reversalz = f(x) + s a , z b − f(x) = s b − s a (35)0 ≤ s b ⊥ s a ≥ 0Flow reversal can occur in many places in chemical processes including fuelheaders and pipeline distribution networks. This is problematic for modelingsince physical properties and other equations implicitly assume the sign of theflowrate to be positive. These situations may be modeled with the absolute valueoperator, using complementarities as in (31). The magnitude of the flowrate canthen be used in the sign sensitive equations.On the other hand, it may not be possible to implement these changes insome modeling environments, such as commercial process optimization softwarewith directional streams and integrated thermo-physical models. In this case,each stream with the potential for flow reversal can be modeled as two parallelcounter-current streams. The stream flowrates are then complemented, suchthat either one stream or the other has zero flow. Using mixers and splitters asappropriate, the two streams can then be integrated into any process flowsheet[9]. While this method may obscure the original representation of the system,it remains equivalent.5.3 Relief Valves, Check Valves, Compressor Kick-BackRelief valves, check valves and compressor kick-back operation are all relatedphenomenon that can be modeled using complementarities in a manner similarto the functions used in Section 5.1.• Check valves prevent flows in reverse directions that may arise from changesin differential pressure. Assuming that the directional flow is a monotonicfunction of the differential pressure, we model flow through the check valveas:flow = max{0, f(∆p)}and rewrite the max operator as the complementarity system given in (33).16