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of the discrete variables. An embedded NLP is used to find the local optima in the
continuous state space. The complexity study in [10] indicates that for systems with
many decision variables solving the problem becomes computationally expensive. The
second approach applied, e.g., in [11,12] comprises sequential optimization methods.
Here, the optimization layer exclusively contains continuous variables. The hybrid sys-
tem is put into the simulation layer and solved by any simulator which is capable to
treat discontinuities. Again, the necessity of many simulation runs increases the com-
putational cost. Reformulation strategies, which represent the third class of methods,
introduce additional variables and parameters to remove the non-smoothness related to
the complementarity conditions from the problem while retaining the system features.
Reformulation strategies have been studied in [13,14,15]. Most reformulation strate-
gies fall into one of the following two classes: (i) Relaxation methods transform the
complementarities into a set of relaxed equality or inequality constraints, e.g., by the
smoothing discussed in this contribution. A sequence of relaxed problems is solved in
order to approach the solution of the original problem. (ii) Penalization methods intro-
duce a penalization term into the objective function which measures the violation of
the complementarity condition. A comparison of relaxation methods with the heuristic,
simulation-based particle swarm optimization regarding the accuracy of the optimiza-
tion result and the computation time can be found in [16].
3
Model of the Evaporator
The evaporation of volatile components to concentrate non-volatile components within
a mixture is a common technology in process engineering. Usually multi-stage systems
built up from several identical single evaporators are used. A single evaporator model
is considered in this paper following [8].
The system consists of an evaporation tank and a heat exchanger (see Figure 1). The
tank is fed through the valve
V
1
with a mixture of three liquid components A, B, C with
mass fractions
w
A
,w
B
,w
C
, where A is a hydrocarbon of high molar mass and thus has
a very low vapor pressure (implemented as
P
A
=0
in the model) compared to water
(B) and ethanol (C). Inside the tank, the volatile components are evaporated. Hence the
mass fraction of the non-volatile component A in the liquid is increased. This product
will be drained from the tank through the valve
V
2
when the desired concentration of
A is reached. The vapor which consists of B and C with the mass fractions
ξ
B
,ξ
C
determined by the phase equilibrium escapes from the tank through the valve
V
v
1
.In
order to heat the tank, hot steam is supplied to the heat exchanger, where the steam
condensates and leaves the heat exchanger as a liquid.
Depending on the pressure inside the evaporator and the temperature difference be-
tween the heat exchanger and the tank, 4 operating modes can be distinguished: If the
temperature of the heat exchanger is higher than that of the tank, the heat exchanger
operates in the mode 'heating' (H), otherwise 'non-heating' (NH). Inside the tank, the
transition from the mode 'non-evaporating' (NE) to the mode 'evaporating' (E) occurs
as soon as the pressure reaches a certain threshold. Hence, during operation the system
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