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discontinuous due to the mixed discrete-continuous dynamics. This can, of course, be a
major problem for any optimization task. Several approaches, e.g., mixed-integer pro-
gramming, heuristic methods, relaxation and penalization strategies have been proposed
to tackle this problem. The present work focuses on relaxation strategies because they
are most promising with regard to the computation time. Hitherto, mostly relatively
small systems have been studied using relaxation methods. The present study concerns
a large-scale industrial evaporator with switching behavior as either a hybrid model or
as a relaxed continuous model. Both models are simulated and parameter sensitivities
are calculated over the whole time horizon. For the smooth model, the dependence of
the solution on the reformulation parameter is considered in detail.
This paper is organized as follows. Section 2 starts with a discussion of the
challenges and solution approaches to simulation and optimization of hybrid dynamic
systems. Next, in Section 3, the evaporator model in its hybrid and relaxed form is pre-
sented. In Section 4, the simulation results of the relaxed (and consequently smooth
continuous) model are compared with those of the original hybrid model. Section 5
studies for the evaporator model two fundamental tasks of process engineering, namely
parameter estimation and sensitivity analysis. Section 6 summarizes the results and con-
cludes the paper.
2
Simulation and Optimization of Hybrid Systems
Mathematically, discrete transitions in hybrid dynamic systems are often formulated
in terms of complementarity conditions. In numerical simulation, discrete transitions
are almost always handled through embedded logical statements. At the zero-crossing
points of some switching function, the initial conditions are updated and the appro-
priate set of equations is solved restarting at this point in time [1,2]. Systems with
so-called Filippov solutions that remain for a while at the zero-crossing require addi-
tional analysis. Since they do not pose a particular problem for our approach, we will
not discuss them further here. A profound analysis and numerical simulation results
of hybrid systems can be found in [3,4]. For optimization tasks, the hybrid simulation
can be embedded into a heuristic search algorithm. For instance, an evolutionary algo-
rithm was applied to the start-up of the evaporation system in [5] and particle swarm
optimization to the unit commitment problem [6]. These methods suffer from high com-
putational cost when many function evaluations are needed (i.e., in a high dimensional
search space). Alternatively one can consider the problem as a constrained optimization
problem subject to the dynamic model equations. This leads to a dynamic nonlinear
program (NLP). In the so-called direct method, the DAE system is discretized result-
ing in a large-scale NLP with equality (and possibly inequality) constraints, which can
be solved by means of a NLP solver with a gradient-based search. However this NLP-
based optimization of hybrid systems is an computationally extremely challenging task
due to the non-smoothness of the objective function or constraints which result from
instantaneous mode transitions. As a consequence, NLP regularity cannot be presumed
and NLP solvers may fail [7]. Essentially three different approaches can be used to
overcome this difficulty. Mixed-integer methods have been applied successfully to op-
timal control problems in [8,9], where a graph search algorithm explores the state space
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