1

Control relevant modeling and nonlinear state estimation applied

to SOFC-GT power systems

Rambabu Kandepu04-12-2007

2

Contents

• Motivation• Modeling and control of SOFC-GT

power system• Nonlinear state estimation• Conclusions

3

Motivation

• Increase in energy demand– Population growth– Industrialization

• Dependency on oil and gas• Global warming

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

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

SOFC-GT system

• Tight integration between SOFC and GT• Low complexity models

– Relevant dynamics

Fuel cellstack

Gas turbine

Load

Fuel

Air

7

SOFC-GT system

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

4 2 2

2 2 2

4 2 2 2

3

2 4

CH H O CO HCO H O CO HCH H O CO H

+ ⇔ ++ ⇔ ++ ⇔ +

22 2 2H O H O e− −+ → +

22

1 22O e O− −+ →

9

Modeling - SOFC

• Energy balance (one volume)

, , , ,1 1 1

( ) ( )N N M

s ss P an i an i i ca i ca i i j j

i i j

dTm C P F h h F h h H rdt = = =

= − + − + − − Δ∑ ∑ ∑

• Mass balance (anode and cathode)

, ,1

Mi

i in i out ij jj

dN N N a rdt

• •

=

= − +∑Anode

Cathode

Electrolyte

10

Modeling - SOFC

• Fuel Utilization (FU) = fuel utilized / fuel supplied• Distributed nature of SOFC• All models are developed in gPROMS

12

2 2

2

0 ln2

H O

H O

p pRTE EF p

⎛ ⎞= + ⎜ ⎟⎜ ⎟

⎝ ⎠

• VoltagelossV E V= −

Volume I− Volume II−

Anodeinlet Anodeoutlet

Cathodeinlet Cathodeoutlet

Anodeinlet Anodeoutlet

Cathodeinlet Cathodeoutlet

Fuel

Air

11

SOFC model evaluation

• Evaluated against a detailed model

0 100 200 300 400 500 600 700950

1000

1050

1100

1150

1200

Time (min)

Tem

pera

ture

(K

)

Detailed modelSimple model with one volumeSimple model with two volumes

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

Control structure design

• Three possible options– Air blow-off– Extra fuel source– Air by-pass

Hybrid system

Fuel flow

refFU

TController 2

Controller 1

-

refT

FU

Air blow-off

Load disturbance

• Control structure

14

SOFC-GT control

15

SOFC-GT control

Hybrid System

PI Controller 1

rω

ω

fuelm

P

ω

PI Controller 2

rFUFU

FU

PI Controller 3

rSOFCT

SOFCT

SOFCT

gI

16

SOFC-GT control – double shaft

0 5 10 15 20 25 30

2

4

6

8

time (sec)

fuel

flow

rate

(g/s

)

0 5 10 15 20 25 300

0.05

0.1

time (sec)

air b

low

-off

rate

(kg/

s)

Controlled variables

Manipulated variables

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

State estimation

• Need for state estimation• Nonlinear state estimation

– Extended Kalman Filter (EKF)– Unscented Kalman Filter (UKF)– Comparison– Constraint handling– Results

• Conclusions

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

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

EKF principle

( ); a random vector: , nonlinear function

n

n m

y g x xg= ∈

→

( )How to compute the pdf of , given the Gaussian pdf , of ?xy x P x

( ) ( )( )

( )

where is the Jacobian of ( ) at

EKF

TEKFy x

y g x

P g P g

g g x x

=

= ∇ ∇

∇

22

UKF principle

• UKF principle

( ); a random vector: , nonlinear function

n

n m

y g x xg= ∈

→

( )How to compute the pdf of , given the Gaussian pdf , of ?xy x P x

UKF approximates the pdf. It uses true nonlinear process and observation models.

23

UKF principle

• UKF principle

24

Comparison

• Example

= 58.26

= 2686

25

Comparison

0 5 10

0

10

20

30

40

50

60

70

80

90

100

110

x

y=g(

x)=x

2

EKF

Xmean

YmeanEKF

Ymeantrue

linearization

0 5 10

0

10

20

30

40

50

60

70

80

90

100

110

x

y=g(

x)=x

2

UKF

Xmean

Ymeanukf

Ymeantrue

sigma pointstransformed sigma points

Px=16

Pytrue=2686

PyEKF=2304

Px=16

Pytrue=2686

PyUKF=2816

58.26

26

Algorithms: EKF and UKFNonlinear system

27

Algorithms: EKF and UKFEKF UKF

Prediction step: Calculate Jacobians / sigma points transformation

Prediction step: Calculate mean and covariance

Correction step: Calculate Jacobians/ sigma points transformation

Correction step: Kalman update equations

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

State constraint handling - EKF

xkEKF, C

xkEKF

covariance

1kx−

30

State constraint handling - UKF

UKF, t=k

Transformed sigma points

covariancex-

kUKF

1kx−

31

Constraint handling

32

UKFConstraint handling

• Constraint handling method– Projections at different steps

• Sigma points• Transformed sigma points• Transformed sigma points

through measurement function

– Inequality constraints

33

Constraint handling- example

• Gas phase reversible reaction

0 1 2 3 4 5 6 7 8 9 10-1

0

1

2

3

time (sec)

CA

trueUKFEKF

0 1 2 3 4 5 6 7 8 9 101

2

3

4

time (sec)

CB

trueUKFEKF

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

Comparison (EKF and UKF)

0 5 10 15 20 25 30 35 40 45 50

0

0.5

1

time (sec)

x 1

trueUKFEKF

0 5 10 15 20 25 30 35 40 45 50-1.5

-1

-0.5

0

time (sec)

x 2

trueUKFEKF

Comparison of estimated states of an induction motor: components of stator flux

36

Comparison (EKF and UKF)

• SOFC-GT system– Higher order

nonlinear system (18 states)

– Turbine shaft speed plot

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

38

Thank youfor

yourattention ☺