MPEC Problem Formulations in Chemical Engineering Applications

6 Case StudiesFrom the results of Sections 3 and 4, we find that the penalty formulationperforms well, particularly with an active set solver such as CONOPT. With thisexamination of NLP reformulations for MPCC as well as a systematic strategyfor developing complementarity formulations, we now consider two large-scalecase studies that illustrate both of these concepts. The first considers discretedecisions in dynamic systems with the reformulation of the signum function,while the second considers the disappearance of phases in optimization modelsfor distillation.6.1 Dynamic Optimization with Direct TranscriptionThe goal of this case study is to demonstrate the NLP formulations in Section 3on an example with arbitrarily many complementarity conditions. The exampleproblem is derived from discretization of a differential inclusion (ẋ ∈ sgn(x)+2),where the derivative can be defined by a complementarity system [29]. While wedo not consider limiting properties of the discretization, our numerical resultswill focus on how the solution time varies as a function of the problem size(controlled by the discretization strategy).The differential inclusion can be reformulated as the following optimal controlproblem:∫ tendmin (x end − 5/3) 2 + x 2 · dt (47a)t 0s.t. ẋ = u + 2 (47b)x = s + − s −(47c)0 ≤ 1 − u ⊥ s + ≥ 0 (47d)0 ≤ u + 1 ⊥ s − ≥ 0 (47e)Applying the implicit Euler’s method with a fixed step size, the discretizedproblem becomes:NFE∑min (x end − 5/3) 2 + h · x 2 ii=1(48a)s.t. ẋ i = u i + 2 i = 1, . . .,NFE (48b)x i = x i−1 + h · ẋ i i = 1, . . .,NFE (48c)x i = s + i− s − ii = 1, . . .,NFE (48d)0 ≤ 1 − u i ⊥ s + i ≥ 0 i = 1, . . .,NFE (48e)0 ≤ u i + 1 ⊥ s − i ≥ 0 i = 1, . . .,NFE (48f)The resulting discretized problem can be scaled up to be arbitrarily large byincreasing the number of finite elements NFE, with a discrete decision in each20