Information Technology Reference
In-Depth Information
⎧
⎨
if
ψ
(2)
(
T
heatex
,T
evap
)
<
0
0
c
p,vap
(
ξ
in
)
· T
in
− c
p,liq
(
ξ
in
)
· T
heatex
+
h
V
(
ξ
in
)
F
in,vap
if
ψ
(2)
(
T
heatex
,T
evap
)
≥
0
Q
=
(3)
⎩
T
evap
if
ψ
(2)
(
T
heatex
,T
evap
)
<
0
T
heatex
=
(4)
if
ψ
(2)
(
T
heatex
,T
evap
)
≥
0
T
evap
+
Q/kA
with the temperature of the incoming steam
T
in
, the temperature inside the heatex-
changer
T
heatex
, the mass flow rate of the steam
F
in,vap
, the composition of the steam
ξ
in
, the specific evaporation enthalpy
h
V
, the heat capacities of liquid and vapor, the
heat transfer coefficient
k
and the heating area
A
.
F
in,vap
is calculated using an approx-
imation of the Bernoulli equation [17].
At the zero-crossing points of
ψ
(1)
and
ψ
(2)
, the state variables immediately before
the switch
x
−
have to be mapped onto the state variables immediately after the switch
x
+
using the so-called transition functions
x
+
=
T
(
x
−
)
. For instance, for the vapor
mass fraction
ξ
B
the transition function from the non-evaporating modes (
m
=1
,
2
)to
the evaporating modes (
m
=3
,
4
) reads
w
B
P
B
(
T
)
w
A
P
A
(
T
)+
w
B
P
B
(
T
)+
w
C
P
C
(
T
)
ξ
B
=
ξ
B
+
(5)
with the temperature
T
+
=
T
−
=
T
and the liquid mass fractions
w
+
=
w
−
=
w
.
3.2
A Smooth Evaporator Model
State trajectories are in general non-smooth or even discontinuous at the transition
points. If such a model is included into an optimization problem, these points are severe
obstacles for gradient-based optimization algorithms. In order to make the optimiza-
tion of hybrid systems accessible to NLP solvers, the complementarity condition of
the original problem is relaxed, i.e., the strict complementarity conditions are fulfilled
only approximately. In our smoothing approach, we replace the if-else-statement usu-
ally used to implement Eq. (1) and (2) by the smoothing function
−
1
ψ
(
x
)
τ
ϕ
(
x
)=
1+exp
−
(6)
where
τ>
0
is the small smoothing parameter. The model equations are combined in
one single set of equations according to
f
(
x, x, p
)=
ϕ
(
x
)
f
(1)
(
x,x,p
)
+(1
− ϕ
(
x
))
f
(2)
(
x,x,p
)
.
(7)
Eq. 7 is expected to reproduce the switching behavior of the hybrid model in the limit
τ →
0
.
Search WWH ::
Custom Search