The document discusses control and state estimation techniques applied to solid oxide fuel cell-gas turbine (SOFC-GT) power systems. It motivates modeling and control of SOFC-GT systems to meet increasing energy demands efficiently with low emissions. Nonlinear state estimation using the Extended Kalman Filter (EKF) and Unscented Kalman Filter (UKF) is examined. The UKF is found to provide faster state estimation convergence and better robustness to model errors compared to the EKF for nonlinear systems like the SOFC-GT.
Ponencia Jornada técnica “Proyectos europeos en eficiencia energética en edif...
Thesis Presentation
1. 1
Control relevant modeling and
nonlinear state estimation applied
to SOFC-GT power systems
Rambabu Kandepu
04-12-2007
2. 2
Contents
• Motivation
• Modeling and control of SOFC-GT
power system
• Nonlinear state estimation
• Conclusions
3. 3
Motivation
• Increase in energy demand
– Population growth
– Industrialization
• Dependency on oil and gas
• Global warming
4. 4
Motivation
• Solution to energy demand increase
– Efficient of energy conversion
– Technology with low emissions
– Using renewable energy sources
• Distributed generation
– Avoid transmission and distribution losses
– Wind turbines, biomass, small scale hydro, fuel cells etc
5. 5
Fuel cells
• Electrochemical device
• Advantages
– High efficiency
– Low emissions
– No moving parts
• Different types
– Electrolyte
– Temperature
• SOFC
– Solid components
– High operating temperature
– More fuel flexibility
– Internal reforming
6. 6
SOFC-GT system
Fuel Fuel cell
stack
Load
Gas
Air turbine
• Tight integration between SOFC and GT
• Low complexity models
– Relevant dynamics
8. 8
Modeling - SOFC
• Assumptions
– All variables are uniform
– Thermal inertia of gases is neglected
– Pressure losses are neglected for energy balance
– Ideal gas behavior
• Reactions
CH 4 + H 2O ⇔ CO + 3H 2
1 O + 2e − → O 2 −
2 2 CO + H 2O ⇔ CO2 + H 2
2− −
H2 + O → H 2 O + 2e CH 4 + 2 H 2O ⇔ CO2 + 4 H 2
9. 9
Modeling - SOFC
• Mass balance (anode and cathode)
dN i • • M
= N i ,in − N i ,out + ∑ aij rj
Anode
Electrolyte
dt j =1
Cathode
• Energy balance (one volume)
N N M
dTs
ms C s
P = − P + ∑ Fan ,i (han ,i − hi ) + ∑ Fca ,i (hca ,i − hi ) − ∑ ΔH j rj
dt i =1 i =1 j =1
10. 10
Modeling - SOFC
• Voltage
RT ⎛ pH 2 pO22 ⎞
1
E = E0 + ln ⎜ ⎟ V = E − Vloss
2 F ⎜ pH 2O
⎝
⎟
⎠
• Fuel Utilization (FU) = fuel utilized / fuel supplied
• Distributed nature of SOFC
• All models are developed in gPROMS
Fuel Anode inlet Anode outlet Anode inlet Anode outlet
Volume − I Volume − II
Air Cathode inlet Cathode outlet Cathode inlet Cathode outlet
11. 11
SOFC model evaluation
• Evaluated against a detailed model
1200
Detailed model
Simple model with one volume
1150 Simple model with two volumes
Temperature (K)
1100
1050
1000
950
0 100 200 300 400 500 600 700
Time (min)
12. 12
Control structure design
• Dynamic load operation is necessary
• Manipulated variable (1)
– Fuel flow rate
• Controlled variables (2)
– Fuel utilization (FU)
– SOFC temperature
• Load as a disturbance
• Need for a process redesign
13. 13
Control structure design
• Three possible options
– Air blow-off
– Extra fuel source
– Air by-pass
• Control structure
Load disturbance
FU ref
Fuel FU
Controller 1
flow
Tref Hybrid system T
Controller 2
Air blow-off
-
17. 17
SOFC-GT control
• Model Predictive Control (MPC) to
include constraints
– FU
– Steam to carbon ratio
– SOFC temperature change
• Not all states are measurable
• State estimation is necessary
18. 18
State estimation
• Need for state estimation
• Nonlinear state estimation
– Extended Kalman Filter (EKF)
– Unscented Kalman Filter (UKF)
– Comparison
– Constraint handling
– Results
• Conclusions
19. 19
State estimation
• Important for process control and
performance monitoring
• Uncertainties; Model, measurement and
noise sources
• Represent the model state by an probability
distribution function (pdf)
• State estimation propagates the pdf over time
in some optimal way
• Gaussian pdf
20. 20
Nonlinear state estimation
• Extended Kalman Filter (EKF)
– Most common way to apply KF to a nonlinear system
• High order EKFs
– Computationally not feasible
• Ensemble Kalman Filter (EnKF)
– Mostly for large scale systems (reservoir models)
• Unscented Kalman Filter (UKF)
– Simple and effective
• Moving Horizon Estimation (MHE)
– Computationally demanding
21. 21
EKF principle
y = g ( x); x ∈ n
a random vector
g: n
→ m
, nonlinear function
(
How to compute the pdf of y, given the Gaussian pdf x, Px of x ?)
EKF
y = g ( x)
PyEKF = ( ∇g ) Px ( ∇g )
T
where ( ∇g ) is the Jacobian of g ( x) at x
22. 22
UKF principle
• UKF principle
y = g ( x); x ∈ n
a random vector
g: n
→ m
, nonlinear function
( )
How to compute the pdf of y, given the Gaussian pdf x, Px of x ?
UKF approximates the pdf.
It uses true nonlinear process and observation models.
28. 28
State constraint handling
• No general way in KF theory
– Projecting unconstrained state estimate
onto boundary
• Systematic approach in MHE
– Solving a nonlinear problem at each time
step
• A simple method is introduced in UKF
29. 29
State constraint handling - EKF
xk−1
covariance
xkEKF, C
xkEKF
32. 32
Constraint handling
UKF
• Constraint handling
method
– Projections at different steps
• Sigma points
• Transformed sigma points
• Transformed sigma points
through measurement
function
– Inequality constraints
33. 33
Constraint handling- example
• Gas phase reversible reaction
3
true
UKF
2 EKF
A
1
C
0
-1
0 1 2 3 4 5 6 7 8 9 10
time (sec)
true
4
UKF
EKF
3
B
C
2
1
0 1 2 3 4 5 6 7 8 9 10
time (sec)
34. 34
Comparison (EKF and UKF)
• Nonlinear systems
– Induction motor and Van der Pol Oscillator
– Faster convergence with UKF
• Robustness to model errors
– Van der Pol oscillator
• Better performance with UKF
• Higher order nonlinear system
– SOFC-GT hybrid system (18 states)
35. 35
Comparison (EKF and UKF)
Comparison of estimated states of an induction motor:
components of stator flux
1
true
UKF
EKF
0.5
1
x
0
0 5 10 15 20 25 30 35 40 45 50
time (sec)
0 true
UKF
EKF
-0.5
2
x
-1
-1.5
0 5 10 15 20 25 30 35 40 45 50
time (sec)
36. 36
Comparison (EKF and UKF)
• SOFC-GT system
– Higher order
nonlinear system
(18 states)
– Turbine shaft
speed plot
37. 37
Conclusions – state estimation
• The UKF is a promising option
– Simple and easy to implement
– No need for Jacobians
– Computational load is comparable to EKF
– Improved performance
• Faster convergence
• Robustness to model errors and initial choices
• Simple constraint handling method works