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What is necessary to make modeling, control, and state estimation of high-temperature fuel cells trustworthy and to simultaneously optimize their energy efficiency?

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Andreas Rauh at Carl von Ossietzky University of Oldenburg
Andreas Rauh
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Abstract

High-temperature fuel cells, such as solid oxide fuel cells (SOFCs), are characterized by strong nonlinearities with respect to their thermal and electrochemical behavior if they are operated in a non-stationary regime. However, complex partial differential equation models taking into account these phenomena, are not only excessively complex for the implementation of control procedures but they also have the disadvantage that a large number of parameters can hardly be identified experimentally with the required accuracy. Therefore, this presentation gives an overview of interval-based and stochastic techniques that allow for a reliable modeling of SOFCs, for the real-time capable implementation of controllers that prevent the violation of thermal state constraints, and for an online identification of the electric power characteristic. With the help of this identification, strategies for maximizing the degree of fuel utilization can be implemented for time-varying desired electric power profiles so that operating points within the region of Ohmic polarization are guaranteed to be obtained. The presented methods are visualized by simulations in combination with experiments obtained from a laboratory SOFC test rig.
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What is necessary to make modeling, control,
and state estimation of high-temperature fuel
cells trustworthy and to simultaneously opti-
mize their energy efficiency?
S´
eminaire du D´
epartement DO – LAAS, Toulouse
Andreas Rauh, March 14, 2022
Carl von Ossietzky Universit¨
at Oldenburg, Germany
Contents
– Reliable modeling of high-temperature fuel cells (thermal behavior)
– Interval methods for parameter identification
– Interval observer design: Estimate worst-case bounds of operating conditions
– Robust nonlinear control: Prevent the violation of state constraints with certainty
– Reliable data- and measurement-driven identification of the electrochemical behavior
– Interval-based predictive control
slide 2/41
March 14, 2022
Modeling, control, and state estimation of high-temperature fuel cells — S´
eminaire du D´
epartement DO – LAAS, Toulouse
A. Rauh — Carl von Ossietzky Universit¨
at Oldenburg, Department of Computing Science (Distributed Control in Interconnected Systems)
Control-Oriented Modeling of SOFC Systems (1)
Configuration of an SOFC test rig
– Supply of fuel gas (hydrogen and/or mix-
ture of methane, carbon monoxide, water
vapor)
– Supply of air
– Independent preheaters for fuel gas and
air
– Stack module containing fuel cells in elec-
tric series connection
– Electric load as disturbance
slide 3/41
March 14, 2022
Modeling, control, and state estimation of high-temperature fuel cells — S´
eminaire du D´
epartement DO – LAAS, Toulouse
A. Rauh — Carl von Ossietzky Universit¨
at Oldenburg, Department of Computing Science (Distributed Control in Interconnected Systems)
Control-Oriented Modeling of SOFC Systems (2)
Mathematical representation of the piecewise homogeneous temperature
distribution =⇒spatial finite-volume semi-discretization
˙
ϑI(t) = 1
cImI
˙
QI
HT(t)+X
G∈{AG,CG}
˙
QI
G,I−
j
(t)+˙
QI
R(t)+˙
QI
EL(t)
–HT: Heat transfer (heat conduction and con-
vection)
–G: Enthalpy flows of supplied gases
–R: Exothermic reaction enthalpy
–EL: Ohmic losses
˙
Q
HT,
j
˙
Q
HT,
j
+
˙
Q
HT,
i
˙
Q
HT,
k
+
˙
Q
HT,
k
˙
Q
HT,
i
+
˙
Q
R
,
˙
Q
EL
I
i
=
I
=
I
+
G
˙
Q
G,
j
∑
slide 4/41
March 14, 2022
Modeling, control, and state estimation of high-temperature fuel cells — S´
eminaire du D´
epartement DO – LAAS, Toulouse
A. Rauh — Carl von Ossietzky Universit¨
at Oldenburg, Department of Computing Science (Distributed Control in Interconnected Systems)
Control-Oriented Modeling of SOFC Systems (2)
Mathematical representation of the piecewise homogeneous temperature
distribution =⇒spatial finite-volume semi-discretization
˙
ϑI(t) = 1
cImI
˙
QI
HT(t)+X
G∈{AG,CG}
˙
QI
G,I−
j
(t)+˙
QI
R(t)+˙
QI
EL(t)
–HT: Heat transfer (heat conduction and con-
vection)
–G: Enthalpy flows of supplied gases
–R: Exothermic reaction enthalpy
–EL: Ohmic losses
˙
Q
HT,
j
˙
Q
HT,
j
+
˙
Q
HT,
i
˙
Q
HT,
k
+
˙
Q
HT,
k
˙
Q
HT,
i
+
˙
Q
R
,
˙
Q
EL
I
i
=
I
=
I
+
G
˙
Q
G,
j
∑
slide 4/41
March 14, 2022
Modeling, control, and state estimation of high-temperature fuel cells — S´
eminaire du D´
epartement DO – LAAS, Toulouse
A. Rauh — Carl von Ossietzky Universit¨
at Oldenburg, Department of Computing Science (Distributed Control in Interconnected Systems)
Control-Oriented Modeling of SOFC Systems (2)
Mathematical representation of the piecewise homogeneous temperature
distribution =⇒spatial finite-volume semi-discretization
˙
ϑI(t) = 1
cImI
˙
QI
HT(t)+X
G∈{AG,CG}
˙
QI
G,I−
j
(t)+˙
QI
R(t)+˙
QI
EL(t)
–HT: Heat transfer (heat conduction and con-
vection)
–G: Enthalpy flows of supplied gases
–R: Exothermic reaction enthalpy
–EL: Ohmic losses
˙
Q
HT,
j
˙
Q
HT,
j
+
˙
Q
HT,
i
˙
Q
HT,
k
+
˙
Q
HT,
k
˙
Q
HT,
i
+
I
i
=
I
=
I
+
G
˙
Q
G,
j
∑
˙
Q
R
,
˙
Q
EL
slide 4/41
March 14, 2022
Modeling, control, and state estimation of high-temperature fuel cells — S´
eminaire du D´
epartement DO – LAAS, Toulouse
A. Rauh — Carl von Ossietzky Universit¨
at Oldenburg, Department of Computing Science (Distributed Control in Interconnected Systems)
Control-Oriented Modeling of SOFC Systems (2)
Mathematical representation of the piecewise homogeneous temperature
distribution =⇒spatial finite-volume semi-discretization
˙
ϑI(t) = 1
cImI
˙
QI
HT(t)+X
G∈{AG,CG}
˙
QI
G,I−
j
(t)+˙
QI
R(t)+˙
QI
EL(t)
–HT: Heat transfer (heat conduction and con-
vection)
–G: Enthalpy flows of supplied gases
–R: Exothermic reaction enthalpy
–EL: Ohmic losses
˙
Q
HT,
j
˙
Q
HT,
j
+
˙
Q
HT,
i
˙
Q
HT,
k
+
˙
Q
HT,
k
˙
Q
HT,
i
+
I
i
=
I
=
I
+
G
˙
Q
G,
j
∑
˙
Q
R
,
˙
Q
EL
slide 4/41
March 14, 2022
Modeling, control, and state estimation of high-temperature fuel cells — S´
eminaire du D´
epartement DO – LAAS, Toulouse
A. Rauh — Carl von Ossietzky Universit¨
at Oldenburg, Department of Computing Science (Distributed Control in Interconnected Systems)
Parameter Identification for Dynamic System Models (1)
Branch-and-bound procedure: Simulation over complete measurement horizon
time t
t0
t
1
t
2
t3
...
measured data ym(tk)
[
y
m
](
t
0
)
[
y
m
](
t
1
)
[
y
m
](
t
3
)
[
y
m
](
t
2
)
y
m
(
t
k
)∈[
y
m
](
t
k
)
– Measured data are available at discrete points of time
– Worst-case bounds for measurement tolerances
– Necessity for information about uncertain initial states and bounds on uncertain para-
meters
slide 5/41
March 14, 2022
Modeling, control, and state estimation of high-temperature fuel cells — S´
eminaire du D´
epartement DO – LAAS, Toulouse
A. Rauh — Carl von Ossietzky Universit¨
at Oldenburg, Department of Computing Science (Distributed Control in Interconnected Systems)
Parameter Identification for Dynamic System Models (2)
Branch-and-bound procedure: Simulation over complete measurement horizon
time t
t0
t
1
t
2
t3
...
measured data ym(tk)
simulated output enclosure
[
y
m
](
t
0
)
[
y
m
](
t
1
)
[
y
m
](
t
3
)
[
y
m
](
t
2
)
– Prerequisite: Correctness of model structure
– Initial state/ parameter intervals are subdivided for candidates, for which no decision
about admissibility can be made
– Intersection of directly measured and simulated state intervals possible
slide 6/41
March 14, 2022
Modeling, control, and state estimation of high-temperature fuel cells — S´
eminaire du D´
epartement DO – LAAS, Toulouse
A. Rauh — Carl von Ossietzky Universit¨
at Oldenburg, Department of Computing Science (Distributed Control in Interconnected Systems)
Parameter Identification for Dynamic System Models (3)
Branch-and-bound procedure: Simulation over complete measurement horizon
time t
t0
t
1
t
2
t3
...
measured data ym(tk)
[
y
m
](
t
0
)
[
y
m
](
t
1
)
[
y
m
](
t
3
)
[
y
m
](
t
2
)
– Search for guaranteed admissible initial state/ parameter intervals
– Subdivision until undecided region is sufficiently small
– Needs to be fulfilled for each available sensor if dim(ym)>1
– Parallelize on the GPU (collaboration with Ekaterina Auer, Univ. of Appl. Sc. Wismar)
slide 7/41
March 14, 2022
Modeling, control, and state estimation of high-temperature fuel cells — S´
eminaire du D´
epartement DO – LAAS, Toulouse
A. Rauh — Carl von Ossietzky Universit¨
at Oldenburg, Department of Computing Science (Distributed Control in Interconnected Systems)
Parameter Identification for Dynamic System Models (3)
Branch-and-bound procedure: Simulation over complete measurement horizon
time t
t0
t
1
t
2
t3
...
measured data ym(tk)
[
y
m
](
t
0
)
[
y
m
](
t
1
)
[
y
m
](
t
3
)
[
y
m
](
t
2
)
– Search for guaranteed admissible initial state/ parameter intervals
– Subdivision until undecided region is sufficiently small
– Needs to be fulfilled for each available sensor if dim(ym)>1
– Parallelize on the GPU (collaboration with Ekaterina Auer, Univ. of Appl. Sc. Wismar)
slide 7/41
March 14, 2022
Modeling, control, and state estimation of high-temperature fuel cells — S´
eminaire du D´
epartement DO – LAAS, Toulouse
A. Rauh — Carl von Ossietzky Universit¨
at Oldenburg, Department of Computing Science (Distributed Control in Interconnected Systems)
Definition of Point-Valued Bounding Systems
Bounding systems: x∈[v;w]
A·v+Bu =˙
v≤˙
x≤˙
w=A·w+Bu
with the element-wise bounding matrices
A≤A(x,p)≤Aand 0≤B≤B(x,p)≤Bas well as Bu ≥0
Definition of the system and input matrices (L=N= 1,M= 3)
A(x,p) = "a11 a12 0
a21 a22 a23
0a32 a33#and B(x,p) = "b11 b12 b13 b14
b21 0 0 b24
b31 0 0 b34#
with the state vector
x=ϑ(1,1,1) ϑ(1,2,1) ϑ(1,3,1)T
slide 8/41
March 14, 2022
Modeling, control, and state estimation of high-temperature fuel cells — S´
eminaire du D´
epartement DO – LAAS, Toulouse
A. Rauh — Carl von Ossietzky Universit¨
at Oldenburg, Department of Computing Science (Distributed Control in Interconnected Systems)
Definition of Point-Valued Bounding Systems
Bounding systems: x∈[v;w]
A·v+Bu =˙
v≤˙
x≤˙
w=A·w+Bu
with the element-wise bounding matrices
A≤A(x,p)≤Aand 0≤B≤B(x,p)≤Bas well as Bu ≥0
Definition of the system and input matrices (L=N= 1,M= 3)
A(x,p) = "a11 a12 0
a21 a22 a23
0a32 a33#and B(x,p) = "b11 b12 b13 b14
b21 0 0 b24
b31 0 0 b34#
with the input vector
u=ϑAϑAG,in ϑCG,in 1
3IT
slide 8/41
March 14, 2022
Modeling, control, and state estimation of high-temperature fuel cells — S´
eminaire du D´
epartement DO – LAAS, Toulouse
A. Rauh — Carl von Ossietzky Universit¨
at Oldenburg, Department of Computing Science (Distributed Control in Interconnected Systems)
Definition of Point-Valued Bounding Systems
Bounding systems: x∈[v;w]
A·v+Bu =˙
v≤˙
x≤˙
w=A·w+Bu
with the element-wise bounding matrices
A≤A(x,p)≤Aand 0≤B≤B(x,p)≤Bas well as Bu ≥0
Example for the sign pattern of the system matrix A(x,p)for L=N= 1,M= 3
A(x,p) = "−+ 0
+−+
0 + −#
slide 8/41
March 14, 2022
Modeling, control, and state estimation of high-temperature fuel cells — S´
eminaire du D´
epartement DO – LAAS, Toulouse
A. Rauh — Carl von Ossietzky Universit¨
at Oldenburg, Department of Computing Science (Distributed Control in Interconnected Systems)
Verified Interval-Based Parameter Identification
Use of Verified (Global Optimization) Procedures
– General Branch-and-Bound procedure excluding all unphysical parameter domains:
Comparison of simulated system outputs with interval-bounded measurements
– Extension by exploiting stability and cooperativity properties as well as constraints
A. Rauh, T. D¨
otschel, E. Auer, H. Aschemann: Interval Methods for Control-Oriented Modeling of the Thermal Behavior of High-Temperature Fuel Cell
Stacks. Proceedings of the 16th IFACSymposium on System Identification, Br ussels,Belgium, pp. 446-451. (2012)
DOI:10.3182/20120711-3-BE-2027.00374.
A. Rauh, L. Senkel, J. Kersten, H. Aschemann: Reliable control of high-temperature fuel cell systems using interval-based sliding mode techniques. IMA
Journal of Mathematical Control and Information, Vol. 33, No. 2, pp.457–484. (2016) DOI:10.1093/imamci/dnu051.
A. Rauh, J. Kersten, H. Aschemann: An Interval Observer Approach for the Online Temperature Estimation in Solid Oxide Fuel Cell Stacks. Proceedings of
the 17th European Control Conference, Limassol, Cyprus, pp. 1596–1601. (2018) DOI:10.23919/ECC.2018.8550158.
S. Ifqir, A. Rauh, J. Kersten,D. Ichalal, N. Ait-Oufroukh, S. Mammar: Interval Observer-Based Controller Design for Systems with State Constraints:
Application to Solid Oxide Fuel Cells Stacks. Proceedings of the 24th International Conference on Methods and Models in Automation and Robotics,
pp. 372–377, Mi
edzyzdroje, Poland. (2019) DOI:10.1109/MMAR.2019.8864718.
E. Auer, A. Rauh, J. Kersten:Experiments-based parameter identification on the GPU for cooperative systems. Journal of Computational and Applied
Mathematics. Vol. 371. Paper-id:112657. (2019) DOI:10.1016/j.cam.2019.112657.
slide 9/41
March 14, 2022
Modeling, control, and state estimation of high-temperature fuel cells — S´
eminaire du D´
epartement DO – LAAS, Toulouse
A. Rauh — Carl von Ossietzky Universit¨
at Oldenburg, Department of Computing Science (Distributed Control in Interconnected Systems)
Constraints for Parameter Identification of the SOFC
State-independent parameters (3parameters)
– Heat conductivity λ > 0
– Heat transfer α > 0
– Ohmic cell resistance R > 0
Possibility to split up the identification into heating and reaction phases
Stability-based subdivision of the heat conductivity intervals [λ]so that the bounding
systems become stable prior to the simulation-based identification
slide 10/41
March 14, 2022
Modeling, control, and state estimation of high-temperature fuel cells — S´
eminaire du D´
epartement DO – LAAS, Toulouse
A. Rauh — Carl von Ossietzky Universit¨
at Oldenburg, Department of Computing Science (Distributed Control in Interconnected Systems)
Constraints for Parameter Identification of the SOFC
State-dependent characteristics (15 parameters) with
θ∈ {ϑ(1,1,1), ϑ(1,2,1) , ϑ(1,3,1)}
– Heat capacities of all gases cχ(θ) = γ0,χ +γ1,χθ+γ2,χθ2>0
– Monotonicity of the heat capacities of all gases ∂cχ(θ)
∂θ =γ1,χ + 2γ2,χ θ > 0
– Reaction enthalphy HRχ(θ) = γHR,0+γHR,1θ+γHR,2θ2>0
– Monotonicity of the reaction enthalphy ∂ HRχ(θ)
∂θ =γHR,1+ 2γHR,2θ > 0
Implementation of a parameter pre-identification, e.g.
γHR,0+γHR,1θ+γHR,2θ2>0
γHR,1+ 2γHR,2θ > 0if γHR,2>0
γHR,1+ 2γHR,2θ > 0if γHR,2<0
slide 11/41
March 14, 2022
Modeling, control, and state estimation of high-temperature fuel cells — S´
eminaire du D´
epartement DO – LAAS, Toulouse
A. Rauh — Carl von Ossietzky Universit¨
at Oldenburg, Department of Computing Science (Distributed Control in Interconnected Systems)
Structural Analysis of the Bounding Systems
Cooperativity
Verified by the sign pattern of the system matrix =⇒Metzler matrix for all physically
reasonable parameterizations
Open-loop stability analysis using the Gershgorin circle theorem
ℜ{λi} ≤ aii +
nx=3
X
j=1,j̸=i
|aij |=
1
cImI·−α(2lNlM+lLlN+ 2lLlM)
−CAG,(1,1,1)(ϑ(1,1,1) , t)
−CCG,(1,1,1)(ϑ(1,1,1) , t)<0for i= 1
−2α
cImI·(lNlM+lLlM)<0for i= 2
−α
cImI·(2lNlM+lLlN+ 2lLlM)<0for i= 3
slide 12/41
March 14, 2022
Modeling, control, and state estimation of high-temperature fuel cells — S´
eminaire du D´
epartement DO – LAAS, Toulouse
A. Rauh — Carl von Ossietzky Universit¨
at Oldenburg, Department of Computing Science (Distributed Control in Interconnected Systems)
Efficient Representation of Measured Data and Inputs
Time-dependent Bernstein polynomials with interval remainder
Reduction of memory requirements (full experiment with >300,000 sampling instants)
slide 13/41
March 14, 2022
Modeling, control, and state estimation of high-temperature fuel cells — S´
eminaire du D´
epartement DO – LAAS, Toulouse
A. Rauh — Carl von Ossietzky Universit¨
at Oldenburg, Department of Computing Science (Distributed Control in Interconnected Systems)
Cooperative Bounds for Point-Valued Parameters
Floating-point parameter identification =⇒Full verification as ongoing work
slide 14/41
March 14, 2022
Modeling, control, and state estimation of high-temperature fuel cells — S´
eminaire du D´
epartement DO – LAAS, Toulouse
A. Rauh — Carl von Ossietzky Universit¨
at Oldenburg, Department of Computing Science (Distributed Control in Interconnected Systems)
Interval Observer Design (1)
Bounding systems: x∈[v;w]and ˆ
x∈[ˆ
v;ˆ
w] =⇒x∈[ˆ
v;ˆ
w]
AOˆ
v+Bu +Hym≤˙
ˆ
x≤AOˆ
w+Bu +Hym
with the observer system matrices
AO=A−HC and AO=A−HC
Uncertain measurements
[ym] := ym;ym=ym+ [−∆ym; ∆ym]
with a linear output model
y=Cx
slide 15/41
March 14, 2022
Modeling, control, and state estimation of high-temperature fuel cells — S´
eminaire du D´
epartement DO – LAAS, Toulouse
A. Rauh — Carl von Ossietzky Universit¨
at Oldenburg, Department of Computing Science (Distributed Control in Interconnected Systems)
Interval Observer Design (1)
Bounding systems: x∈[v;w]and ˆ
x∈[ˆ
v;ˆ
w] =⇒x∈[ˆ
v;ˆ
w]
AOˆ
v+Bu +Hym≤˙
ˆ
x≤AOˆ
w+Bu +Hym
with the observer system matrices
AO=A−HC and AO=A−HC
Guaranteed stabilizing, cooperativity preserving parameterization
H=κCTwith κ > 0
leading to matrices H=HC =κCTCwhich are purely diagonal with the elements 0and
κ, if Ccontains a single element 1per matrix row (direct temperature measurements in the
finite-volume model)
slide 15/41
March 14, 2022
Modeling, control, and state estimation of high-temperature fuel cells — S´
eminaire du D´
epartement DO – LAAS, Toulouse
A. Rauh — Carl von Ossietzky Universit¨
at Oldenburg, Department of Computing Science (Distributed Control in Interconnected Systems)
Interval Observer Design (2)
Bounding systems: x∈[v;w]and ˆ
x∈[ˆ
v;ˆ
w] =⇒x∈[ˆ
v;ˆ
w]
AOˆ
v+Bu +Hym≤˙
ˆ
x≤AOˆ
w+Bu +Hym
with the observer system matrices
AO=A−HC and AO=A−HC
Essential properties of this observer parameterization
– Metzler structure of the observer system matrix is strictly preserved =⇒Cooperative
dynamics of the state observer
– Stability proof
– Linear matrix inequalities for further optimizations (e.g., H∞synthesis for a combined
parameterization of a robust linear control and observer structure (Sara Ifqir))
slide 16/41
March 14, 2022
Modeling, control, and state estimation of high-temperature fuel cells — S´
eminaire du D´
epartement DO – LAAS, Toulouse
A. Rauh — Carl von Ossietzky Universit¨
at Oldenburg, Department of Computing Science (Distributed Control in Interconnected Systems)
Estimation of all Segment Temperatures for C=0 0 1
tin 103s
ˆ
vi(t) in K
0 2 4 6 8 10 1612 14
300
1100
900
500
700
ym(t)
ˆv1(t)
ˆ
v2(t)
ˆv3(t)
Lower temperature bounds ˆvi
tin 103s
ˆ
wi(t) in K
0 2 4 6 8 10 1612 14
300
1100
900
500
700
ym(t)
ˆ
w1(t)
ˆw2(t)
ˆw3(t)
Upper temperature bounds ˆwi
slide 17/41
March 14, 2022
Modeling, control, and state estimation of high-temperature fuel cells — S´
eminaire du D´
epartement DO – LAAS, Toulouse
A. Rauh — Carl von Ossietzky Universit¨
at Oldenburg, Department of Computing Science (Distributed Control in Interconnected Systems)
Interval-Based Sliding Mode Control (1)
Definition of tracking error signals for the FV element with the maximum
temperature
– Specification of a sufficiently smooth desired output trajectory yd=ξ1,d
– Introduction of the error vector
˜
ξ=h(ξ1−ξ1,d)ξ(1)
1−ξ(1)
1,d. . . ξ(δ−1)
1−ξ(δ−1)
1,diT
∈Rδ
– Desired operating points are located on the sliding surface
s:= s˜
ξ(t)=˜
ξ(δ−1)
1(t) +
δ−2
X
r=0
αr·˜
ξ(r)
1(t) = 0
–α0,...,αδ−2are coefficients of a Hurwitz polynomial of order δ−1
slide 18/41
March 14, 2022
Modeling, control, and state estimation of high-temperature fuel cells — S´
eminaire du D´
epartement DO – LAAS, Toulouse
A. Rauh — Carl von Ossietzky Universit¨
at Oldenburg, Department of Computing Science (Distributed Control in Interconnected Systems)
Interval-Based Sliding Mode Control (1)
Definition of tracking error signals for the FV element with the maximum
temperature
– Specification of a sufficiently smooth desired output trajectory yd=ξ1,d
– Introduction of the error vector
˜
ξ=h(ξ1−ξ1,d)ξ(1)
1−ξ(1)
1,d. . . ξ(δ−1)
1−ξ(δ−1)
1,diT
∈Rδ
– Desired operating points are located on the sliding surface
s:= s˜
ξ(t)=˜
ξ(δ−1)
1(t) +
δ−2
X
r=0
αr·˜
ξ(r)
1(t) = 0
– Guaranteed stabilizing control: Lyapunov function candidate
V=1
2s2>0with ˙
V=s·˙s < 0for s̸= 0
slide 18/41
March 14, 2022
Modeling, control, and state estimation of high-temperature fuel cells — S´
eminaire du D´
epartement DO – LAAS, Toulouse
A. Rauh — Carl von Ossietzky Universit¨
at Oldenburg, Department of Computing Science (Distributed Control in Interconnected Systems)
Interval-Based Sliding Mode Control (2)
Guaranteed stabilization despite uncertainty: Interval formulation of a
variable-structure control law, x∈[x]
[vCG,d]:=
−˜a(x,[p],[d]) + ξ(δ)
1,d−
δ−2
P
r=0
αr·˜
ξ(r+1)
1−˜η·sign{s}
˜
b(x,[p])
with a suitably chosen parameter ˜η > 0and 0̸∈ ˜
b(x,[p])
Guaranteed stabilizing control: Extraction of suitable point values
– Optimal input allocation (gas mass flow and temperature)
– Consideration of actuator dynamics for reduction of chattering
slide 19/41
March 14, 2022
Modeling, control, and state estimation of high-temperature fuel cells — S´
eminaire du D´
epartement DO – LAAS, Toulouse
A. Rauh — Carl von Ossietzky Universit¨
at Oldenburg, Department of Computing Science (Distributed Control in Interconnected Systems)
Interval-Based Sliding Mode Control (3)
Handling of one-sided state constraint: Extended Lyapunov function
˜
V=V+ρV·X
i∈{I}
ln ¯
θmax
¯
θmax −ϑi>0with V=1
2s2, ρV>0
Constraint ϑI
!
≤θmax is expressed by the strict barrier ϑI<¯
θmax
Corresponding time derivative and control law, x∈[x]
˙
˜
V=˙
V+ρV·X
i∈{I} ˙
ϑi
¯
θmax −ϑi!
<0
[˜vCG,d] := [vCG,d]−s
s2+ ˜ϵ·ρV
˜
b(x,[p]) ·X
i∈{I} ˙
ϑi
¯
θmax −ϑi
Combination with an online gain tuning approach with guaranteed stability properties
slide 20/41
March 14, 2022
Modeling, control, and state estimation of high-temperature fuel cells — S´
eminaire du D´
epartement DO – LAAS, Toulouse
A. Rauh — Carl von Ossietzky Universit¨
at Oldenburg, Department of Computing Science (Distributed Control in Interconnected Systems)
Simulation Results: Stack Temperatures
tin 103s
ϑ(1,j,1) in K
4 8 1612
900
700
500
300
0
ξ1,d(t > t∗)
Offline parameterization
tin 103s
ϑ(1,j,1) in K
4 8 1612
900
700
500
300
0
ξ1,d(t>t∗)
Online parameterization
slide 21/41
March 14, 2022
Modeling, control, and state estimation of high-temperature fuel cells — S´
eminaire du D´
epartement DO – LAAS, Toulouse
A. Rauh — Carl von Ossietzky Universit¨
at Oldenburg, Department of Computing Science (Distributed Control in Interconnected Systems)
Simulation Results: Cathode Gas Preheater Inputs
tin 103s
ϑCG,din 102K
tin 103s
˙
mCG in 10−3kg/s
1680
5
4
3
2
1
08 160
3
5
7
9
11
Offline parameterization
tin 103s
ϑCG,din 102K
tin 103s
˙
mCG in 10−3kg/s
1680
5
4
3
2
1
08 160
3
5
7
9
11
Online parameterization
slide 22/41
March 14, 2022
Modeling, control, and state estimation of high-temperature fuel cells — S´
eminaire du D´
epartement DO – LAAS, Toulouse
A. Rauh — Carl von Ossietzky Universit¨
at Oldenburg, Department of Computing Science (Distributed Control in Interconnected Systems)
Quasi-Linear, Input-Affine Neural Network Structure
A(x,q)·x
b(x,q)·ϑCG,in
d(x,q)
.
.
.
.
.
..
.
.
P
.
.
.
P
•
.
.
.
•
P
.
.
.
P
P
.
.
.
P
•
.
.
.
•
P
.
.
.
P
P
.
.
.
P
Bias Bias
ϑCG,in(tk)
q1(tk)
qm(tk)
x1(tk)
xn(tk)
H1
HL2
˙x1(tk)
˙xn(tk)
˙xR,1(tk)
˙xR,n(tk)
states
A. Rauh, J. Kersten, W.Frenkel, N. Kruse, T. Schmidt: Physically
motivated structuring and optimization of neural networks for
multi-physics modelling of solid oxide fuel cells. Mathematical and
Computer Modelling of Dynamical Systems. Vol. 27, pp.586–614,
2021.
A. Rauh: Kalman Filter-Based Real-Time Implementable
Optimization of the Fuel Efficiency of Solid Oxide Fuel Cells. Clean
Technologies.Vol. 3, pp. 206–226. (2021)
DOI:10.3390/cleantechnol3010012.
slide 23/41
March 14, 2022
Modeling, control, and state estimation of high-temperature fuel cells — S´
eminaire du D´
epartement DO – LAAS, Toulouse
A. Rauh — Carl von Ossietzky Universit¨
at Oldenburg, Department of Computing Science (Distributed Control in Interconnected Systems)
Equivalent Circuit Modeling: Electric Power Characteristics
Istack
IRp,1 ICp,1
Rp,1 Up,1
UN,1
Cp,1 UCell,1
Rs,1
Ustack
UCell
Cp
Up
Rp
ICp
Rs
US,1
US
IRp
UN
– Lumped parameter model for the electric stack
voltage
– Drawback: Large number of parameters to be
identified
– Design of optimal identification experiments
possible:
W. Frenkel,A. Rauh, J. Kersten, H. Aschemann: Optimization Techniques for
the Design of Identification Procedures for the Electro-Chemical Dynamics of
High-TemperatureFuel Cells, Proc. of MMAR 2019, Mie¸dzyzdroje, Poland.
– However, possible temporal resolutions are too
fine and the numbers of state variables too
large to optimize the fuel efficiency
slide 24/41
March 14, 2022
Modeling, control, and state estimation of high-temperature fuel cells — S´
eminaire du D´
epartement DO – LAAS, Toulouse
A. Rauh — Carl von Ossietzky Universit¨
at Oldenburg, Department of Computing Science (Distributed Control in Interconnected Systems)
Static Neural Network Modeling: Electric Power Characteristics
Static, data-driven nonlinear function approximation
Rauh et al. Interval Neural Network Models for High-Temperature Fuel Cells
.
.
..
.
.P
Bias Bias
ϑm,(1,1,1),k
ϑAG,m,k
ϑm,(1,3,1),k
Ik
˙mCG,k
ϑCG,m,k
˙mN2,k
˙mH2,k
σ(·)
σ(·)
Uk
Input
layer
Hidden
layer
Ouput
layer
•
Ik
PEL,k
Figure 7. Static neural network model for the electric power characteristic of the SOFC stack with a
multiplicative output layer to compute PEL,k from the voltage Uk.
•
the measurable stack temperatures
ϑm,(1,1,1),k
and
ϑm,(1,3,1),k
close to the gas inlet and outlet manifolds,
389
•the electric current Ik,390
•the inlet temperature ϑCG,m,k and ϑAG,m,k at the stack’s cathode and anode sides, and391
•the nitrogen and hydrogen mass flows ˙mN2,k and ˙mH2,k at the anode.392
Moreover, we consider the influence of the mass flow
˙mCG,k
of preheated air at the cathode as a further
393
NN input to achieve larger flexibility with respect to temporally varying operating conditions. In Fig. 8,
394
plots for selected measured input and output data are shown that are used throughout the remainder of this
395
section. At the test rig, these inputs
qk
are sampled with a frequency of
10 Hz
. Prior to the NN identification,
396
these values are averaged to obtain the reduced sampling frequency of
1 s
. Our goal is to predict, based on
397
these data, the SOFC stack voltage
Uk
as the system output from which the instantaneous electric power
398
PEL,k
is determined by multiplying with the measured electric current
Ik
(cf. Fig. 7). Interval enclosures
399
for the electric power can be used to forecast not only the uncertainty in the predicted system output but
400
also for implementing set-based generalizations of the maximum power point tracking from Rauh (2021)
401
and to forecast the uncertainty of fuel efficiency factors for specific operating points.402
In this section, we first show results for traditional, point-valued NN models for the electric power
403
obtained with the help of static (Sec. 4.1) and dynamic (Sec. 4.2) networks. After that, we show how to
404
employ the procedure described in Sec. 3 to parameterize interval extensions of both types of models
405
reliably in Secs. 4.3, 4.4. For that purpose, we exploit the fact shown in Rauh et al. (2021) and Rauh
406
(2021) that using seven neurons in the hidden layer is sufficient for representing the system behavior in an
407
accurate way if these neurons are parameterized by the sigmoid function (7).408
4.1 Point-Valued Static Neural Network Model409
As the fundamental, static electric power model, the NN representation in Fig. 7 is employed. This
410
network is trained with the measured data described above by using the MATLA B NN toolbox. We rely on
411
the Bayesian regularization back-propagation algorithm (parallelized on four CPU cores) with a maximum
412
number of 5,000 epochs to solve the training task. During the network training, a worsening of the validation
413
This is a provisional file, not the final typeset article 16
slide 25/41
March 14, 2022
Modeling, control, and state estimation of high-temperature fuel cells — S´
eminaire du D´
epartement DO – LAAS, Toulouse
A. Rauh — Carl von Ossietzky Universit¨
at Oldenburg, Department of Computing Science (Distributed Control in Interconnected Systems)
Dynamic Neural Network Modeling: Electric Power Characteristics
Data-driven NARX models (nonlinear vs. linear output feedback)
Rauh et al. Interval Neural Network Models for High-Temperature Fuel Cells
.
.
.
.
.
.
.
.
.
P
.
.
.
P
bd,1bd,2
Wd,q,1
Wd,y,1
Wd,2
q1,k−M:k
qm,k−M:k
y1,k−M:k−1
yn,k−M:k−1
σd(·)
σd(·)
y1,k
yn,k
Input
layer
Hidden
layer
Ouput
layer
A: NARX model N
d,Awith input and state
nonlinearity.
.
.
.
.
.
.
.
.
.
P
.
.
.
P
bd,1bd,2
Wd,q,1
Wd,y,1
Wd,2
q1,k−M:k
qm,k−M:k
y1,k−M:k−1
yn,k−M:k−1
σd(·)
σd(·)y1,k
yn,k
Input
layer
Hidden
layer
Ouput
layer
B: NARX model N
d,Bwith pure input nonlinearity.
Figure 6. Different options for structuring NARX models for dynamic system representations, where the
activation functions
σd
are set to the hyperbolic tangent function according to (7) for the rest of this paper.
with the sigmoid activation functions (5)–(7), while the previous outputs are fed back in a linear manner.
343
For a compact mathematical representation of the networks344
yk=N
d,Aqk−M:k,yk−M:k−1,Wd,q,1,Wd,y,1,Wd,2,bd,1,bd,2
=Wd,2·gWd,q,1·qk−M:k+Wd,y,1·yk−M:k−1+bd,1+bd,2
(20)
and345
yk=N
d,Bqk−M:k,yk−M:k−1,Wd,q,1,Wd,y,1,Wd,2,bd,1,bd,2
=Wd,2·gWd,q,1·qk−M:k+bd,1+Wd,y,1·yk−M:k−1+bd,2
(21)
with M≥1as the number of previous sampling points, we introduce the stacked vectors346
qk−M:k=qT
k−M. . . qT
kT∈Rm·(M+1) and yk−M:k−1=yT
k−M. . . yT
k−1T∈Rn·M
(22)
for input and output variables, respectively.347
As explained in Sec. 2.4, we suggest to extend both types of networks (
ι∈ {A,B}
in Fig. 6) using an
348
additive connection that is implemented by means of a static NN with interval parameters such that the
349
enclosure property350
ym,k ∈N
d,ι qk−M:k,yk−M:k−1,Wd,q,1,Wd,y,1,Wd,2,bd,1,bd,2
+ [N]q′
k,[W1],[W2],[b1],[b2], ι ∈ {A,B}(23)
This is a provisional file, not the final typeset article 14
slide 26/41
March 14, 2022
Modeling, control, and state estimation of high-temperature fuel cells — S´
eminaire du D´
epartement DO – LAAS, Toulouse
A. Rauh — Carl von Ossietzky Universit¨
at Oldenburg, Department of Computing Science (Distributed Control in Interconnected Systems)
Interval Generalization of Static NN and NARX Models
Interval neural network models
Rauh et al. Interval Neural Network Models for High-Temperature Fuel Cells
.
.
..
.
.
P
.
.
.
P
b1b2
W1W2
q1,k
qm,k
σ(·)
σ(·)
y1,k
yn,k
Input
layer
Hidden
layer
Ouput
layer
Figure 4. Interval parameterization of a feedforward neural network model.
B1 Compute interval correction bounds for the parameter ˇ
b2according to292
∆b2; ∆b2=ym,k −
kmax
G
k=1
Nqk,ˇ
W1,ˇ
W2,ˇ
b1,ˇ
b2(12)
so that all measured samples are included in the interval-valued NN outputs. In (12), the
293
symbol
F
denotes the tightest axis-aligned interval enclosure around all arguments of this
294
operator.295
B2 Determine the intermediate minimum cost function value296
J∗=
kmax
X
k=1
1T·wNqk,ˇ
W1,ˇ
W2,ˇ
b1,ˇ
b2+∆b2; ∆b2 (13)
with
w([x])
being the element-wise extension of the interval diameter definition from (1) and
297
the vector298
1=1. . . 1T∈Rn.(14)
B3 Minimize the cost function299
J=
kmax
X
k=1
1T·w(N(qk,[W1],[W2],[b1],[b2])) + J∗·Pk(15)
by searching for optimal parameterizations of the additive interval bounds
∆bi; ∆bi
and
300
∆Wi; ∆Wiin (10). Here, Pkis a sufficiently large penalty term301
Pk= 100 ·
δ+
k
2
2+
δ−
k
2
2+
∆y+
k
2
2+
∆y−
k
2
2(16)
with the squared Euclidean norms
∥x∥2
2
preventing minima that are inadmissible due to a
302
violation of the network’s output intervals by individual measurements. These points are
303
Frontiers 11
– Weighting matrices and offset vectors are replaced by interval variables to include
measured identification data within the computed output bounds
slide 27/41
March 14, 2022
Modeling, control, and state estimation of high-temperature fuel cells — S´
eminaire du D´
epartement DO – LAAS, Toulouse
A. Rauh — Carl von Ossietzky Universit¨
at Oldenburg, Department of Computing Science (Distributed Control in Interconnected Systems)
Algorithm for the Identification of Interval NARX Models (1)
– Definition of the point-valued NARX model
yk=Nd(qk−M:k,yk−M:k−1,Wd,q,1,Wd,y,1,Wd,2,bd,1,bd,2)
=Wd,2·g(Wd,q,1·qk−M:k+Wd,y,1·yk−M:k−1+bd,1) + bd,2
with the augmented input vector
qk−M:k=qT
k−M. . . qT
kT∈Rm·(M+1)
and the delayed output feedback
yk−M:k−1=yT
k−M. . . yT
k−1T∈Rn·M
– Find an additive, interval-valued error correction network
[N]q′
k,[W1],[W2],[b1],[b2]= [W2]·g[W1]·q′
k+ [b1]+ [b2],
where q′
k=qT
k−M:kyT
k−M:k−1T
slide 28/41
March 14, 2022
Modeling, control, and state estimation of high-temperature fuel cells — S´
eminaire du D´
epartement DO – LAAS, Toulouse
A. Rauh — Carl von Ossietzky Universit¨
at Oldenburg, Department of Computing Science (Distributed Control in Interconnected Systems)
Algorithm for the Identification of Interval NARX Models (2)
Parameterization of [N] (q′
k,[W1],[W2],[b1],[b2])
– Point-valued minimization of the cost function
JN=X
k∈T
(eN,k −ek)2, eN,k =Nq′
k,ˇ
W1,ˇ
W2,ˇ
b1,ˇ
b2
– Inflation of ˇ
b2according to [b2] = ˇ
b2+∆b2; ∆b2with
∆b2; ∆b2=ek−
kmax
G
k=1
Nq′
k,ˇ
W1,ˇ
W2,ˇ
b1,ˇ
b2
to include all measured data
slide 29/41
March 14, 2022
Modeling, control, and state estimation of high-temperature fuel cells — S´
eminaire du D´
epartement DO – LAAS, Toulouse
A. Rauh — Carl von Ossietzky Universit¨
at Oldenburg, Department of Computing Science (Distributed Control in Interconnected Systems)
Algorithm for the Identification of Interval NARX Models (3)
Parameterization of [N] (q′
k,[W1],[W2],[b1],[b2])
– Compute the intermediate cost function
J∗=
kmax
X
k=1
wNq′
k,ˇ
W1,ˇ
W2,ˇ
b1,[b2]
– Minimize
J=
kmax
X
k=1 wNq′
k,[W1],[W2],[b1],[b2]+J∗·Pk
where a maximum percentage of outliers can be tolerated by a suitable choice of Pk
slide 30/41
March 14, 2022
Modeling, control, and state estimation of high-temperature fuel cells — S´
eminaire du D´
epartement DO – LAAS, Toulouse
A. Rauh — Carl von Ossietzky Universit¨
at Oldenburg, Department of Computing Science (Distributed Control in Interconnected Systems)
Algorithm for the Identification of Interval NARX Models (4)
Typical results of the interval NARX modeling
Outliers inadmissible 2% outliers admissible
slide 31/41
March 14, 2022
Modeling, control, and state estimation of high-temperature fuel cells — S´
eminaire du D´
epartement DO – LAAS, Toulouse
A. Rauh — Carl von Ossietzky Universit¨
at Oldenburg, Department of Computing Science (Distributed Control in Interconnected Systems)
Kalman Filter Based Online Identification (1)
Polynomial representation of the electric power characteristic
PEL,k ≈I[m1−1]
k⊗˙
m[m2−1]
H2,k ·xk
with the vectors
I[m1−1]
k=I0
kI1
kI2
k. . . Im1−1
k
and
˙
m[m2−1]
H2,k =˙m0
H2,k ˙m1
H2,k ˙m2
H2,k . . . ˙mm2−1
H2,k
of monomials of the stack current and hydrogen mass flow
slide 32/41
March 14, 2022
Modeling, control, and state estimation of high-temperature fuel cells — S´
eminaire du D´
epartement DO – LAAS, Toulouse
A. Rauh — Carl von Ossietzky Universit¨
at Oldenburg, Department of Computing Science (Distributed Control in Interconnected Systems)
Kalman Filter Based Online Identification (2)
Polynomial representation of the electric power characteristic
PEL,k ≈I[m1−1]
k⊗˙
m[m2−1]
H2,k ·xk
Kalman filter synthesis
– Online estimation of the vector xkof (slowly varying) coefficients
– Identification of the domain of Ohmic polarization by
∂PEL,k
∂Ik
≈0 1 . . . (m1−1) ·Im1−2
k⊗˙
m[m2−1]
H2,k ·µe
x,k >0
– Expected value of the estimated parameters in the innovation stage: µe
x,k
– Covariance information can be used as an indicator for the identification quality
slide 33/41
March 14, 2022
Modeling, control, and state estimation of high-temperature fuel cells — S´
eminaire du D´
epartement DO – LAAS, Toulouse
A. Rauh — Carl von Ossietzky Universit¨
at Oldenburg, Department of Computing Science (Distributed Control in Interconnected Systems)
Kalman Filter Based Online Identification (3)
Typical results: Online identification of the electric power characteristic
Comparison using
measured data Approx. for t= 25200 s Approx. for t= 41400 s
Note
Possible violations of the domain of Ohmic polarization can be detected by figuring out
whether the operating point exceeds the SOFC’s maximum power point
slide 34/41
March 14, 2022
Modeling, control, and state estimation of high-temperature fuel cells — S´
eminaire du D´
epartement DO – LAAS, Toulouse
A. Rauh — Carl von Ossietzky Universit¨
at Oldenburg, Department of Computing Science (Distributed Control in Interconnected Systems)
Online Optimization of the Fuel Efficiency
Fundamental cost function =⇒Goal: Tracking the desired power PEL,d,k in an
energy-efficient way
Jk= (PEL,k −PEL,d,k)2+γ1˙m2
H2,k +γ2ln Imax,k
Imax,k −Ik
Feasibility constraints (Faraday’s law)
– Maximum stack current
Ik< Imax,k := z·F·(MH2·Nc)−1·˙mH2,k
– Minimum hydrogen mass flow (alternative formulation)
˙mH2,k >˙mH2,min,k =MH2·Nc·(z·F)−1·Ik
slide 35/41
March 14, 2022
Modeling, control, and state estimation of high-temperature fuel cells — S´
eminaire du D´
epartement DO – LAAS, Toulouse
A. Rauh — Carl von Ossietzky Universit¨
at Oldenburg, Department of Computing Science (Distributed Control in Interconnected Systems)
Kalman Filter Based Control Optimization
Gradient-based update of the vector-valued control
uk+1 =uk−α· ∇Jk·Jkwith uk=Ik˙mH2,k T
Control law can be extended by a slow PI output feedback
Estimation of the required gradient vector
∇Jk=
∂Jk
∂Ik
∂Jk
∂˙mH2,k
=
2·(PEL,k −PEL,d,k)·∂ˆ
PEL,k
∂Ik+γ2·1
Imax−Ik
2·(PEL,k −PEL,d,k)·∂ˆ
PEL,k
∂˙mH2,k + 2γ1˙mH2,k
Measured power PEL,k Desired power PEL,d,k Estimated power ˆ
PEL,k
slide 36/41
March 14, 2022
Modeling, control, and state estimation of high-temperature fuel cells — S´
eminaire du D´
epartement DO – LAAS, Toulouse
A. Rauh — Carl von Ossietzky Universit¨
at Oldenburg, Department of Computing Science (Distributed Control in Interconnected Systems)
Sensitivity-Based Control Optimization
Extended cost function
Jk=
k+Nc
X
κ=k
(PEL,κ −PEL,d)2+γ1˙m2
H2,κ +γ2ln Imax,κ
Imax,κ −Iκ
+γ3∥(∆uκ−∆uκ−1)∥2
2+γ4·∆P2
EL,κ +γ5˙mH2,κ +γ6Iκ
with the prediction horizon Nc, the updated control vector u(tκ) = u(tk−1) + ∆uκ, and
the increment vector ∆u=∆uT
k. . . ∆uT
k+NcT
Interval-based control update rule
∆u=−α·sup ∂[Jk]
∂∆u+
·[Jk]!
– Derivative computation by algorithmic differentiation
– Underlying evaluation of the interval-based NARX model
slide 37/41
March 14, 2022
Modeling, control, and state estimation of high-temperature fuel cells — S´
eminaire du D´
epartement DO – LAAS, Toulouse
A. Rauh — Carl von Ossietzky Universit¨
at Oldenburg, Department of Computing Science (Distributed Control in Interconnected Systems)
Optimization Results (1)
Optimal reference tracking: Kalman filter based optimization
Optimized fuel cell power Power tracking error Current & fuel mass flow
Improvement of the energy efficiency
– Open-loop efficiency (measured data for t≤1000 s): η∈[15.2 ; 16.8] %
– Kalman filter based optimization: η∈[19.4 ; 21.8] %
slide 38/41
March 14, 2022
Modeling, control, and state estimation of high-temperature fuel cells — S´
eminaire du D´
epartement DO – LAAS, Toulouse
A. Rauh — Carl von Ossietzky Universit¨
at Oldenburg, Department of Computing Science (Distributed Control in Interconnected Systems)
Optimization Results (2)
Optimal reference tracking: Sensitvity-based predictive control
Optimized fuel cell power Power tracking error Current & fuel mass flow
Improvement of the energy efficiency
– Open-loop efficiency (measured data for t≤1000 s): η∈[15.2 ; 16.8] %
– Sensitivity-based predictive control: η∈[68.5 ; 76.5] %
slide 39/41
March 14, 2022
Modeling, control, and state estimation of high-temperature fuel cells — S´
eminaire du D´
epartement DO – LAAS, Toulouse
A. Rauh — Carl von Ossietzky Universit¨
at Oldenburg, Department of Computing Science (Distributed Control in Interconnected Systems)
Further related publications
A. Rauh, L. Senkel, H. Aschemann: Interval-Based Sliding Mode Control Design for Solid Oxide Fuel Cells With State and Actuator Constraints. IEEE
Transactionson Industr ial Electronics.Vol. 62. No. 8, pp. 5208–5217, 2015.
A. Rauh, L. Senkel, H. Aschemann: Reliable Sliding Mode Approaches for the Temperature Control of Solid Oxide Fuel Cells with Input and Input Rate
Constraints. IFAC-PapersOnLine. Vol. 48, No. 11, pp. 390–395, 2015.
S. Ifqir, A. Rauh, J. Kersten,D. Ichalal, N. Ait-Oufroukh, S. Mammar: Interval Observer-Based Controller Design for Systems with State Constraints:
Application to Solid Oxide Fuel Cells Stacks, Proc. of the 24th Intl. Conference on Methods and Models in Automation and Robotics (MMAR), pp. 372–377,
Miedzyzdroje, Poland, 2019. DOI: 10.1109/MMAR.2019.8864718.
A. Rauh, W. Frenkel,J. Kersten: Kalman Filter-Based Online Identification of the Electric Power Characteristic of Solid Oxide Fuel Cells Aiming at
Maximum Power PointTracking. Algorithms. Vol 13, No. 3, paper 58, 2020.
N. Cont, W. Frenkel,J. Kersten, A. Rauh, H. Aschemann: Interval-Based Modeling of High-Temperature Fuel Cells for a Real-Time Control Implementation
Under State Constraints. IFAC-PapersOnLine,Vol. 53, No. 2, pp. 12542–12547, 2020.
A. Rauh: Kalman Filter-Based Real-Time Implementable Optimization of the Fuel Efficiency of Solid Oxide Fuel Cells, Clean Technologies, vol. 3,
pp. 206–226, 2021. DOI: 10.3390/cleantechnol3010012.
slide 40/41
March 14, 2022
Modeling, control, and state estimation of high-temperature fuel cells — S´
eminaire du D´
epartement DO – LAAS, Toulouse
A. Rauh — Carl von Ossietzky Universit¨
at Oldenburg, Department of Computing Science (Distributed Control in Interconnected Systems)
Further related publications
A. Rauh, L. Jaulin: A Computationally Inexpensive Algorithm for Determining Outer and Inner Enclosures of Nonlinear Mappings of Ellipsoidal Domains,
International Journal of Applied Mathematics and Computer Science, vol. 31, no. 3, pp. 399–415, 2021.
A. Rauh, A. Bourgois, L. Jaulin, J. Kersten: Ellipsoidal Enclosure Techniques for a Verified Simulation of Initial Value Problems for Ordinary Differential
Equations, Proc. of Intl. Conference on Control, Automation and Diagnosis (ICCAD), pp. 1–6, Grenoble,France, 2021. DOI:
10.1109/ICCAD52417.2021.9638755.
A. Rauh, E. Auer: Comparison of Stochastic and Interval-Based Modeling Approaches for the Online optimization of the Fuel Efficiency of SOFC Systems,
Proc. of 9th Intl. Conference on Systems and Control (ICSC), pp. 536-541, Caen, France,2021.
A. Rauh, E. Auer: Interval Extension of Neural Network Models for the Electrochemical Behavior of High-Temperature Fuel Cells, Frontiers in Control
Engineering, 2022. Accepted for publication. DOI: 10.3389/fcteg.2022.785123.
Thank you very much for your attention!
www.interval-methods.de
slide 41/41
March 14, 2022
Modeling, control, and state estimation of high-temperature fuel cells — S´
eminaire du D´
epartement DO – LAAS, Toulouse
A. Rauh — Carl von Ossietzky Universit¨
at Oldenburg, Department of Computing Science (Distributed Control in Interconnected Systems)
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