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Dynamic Modeling of Reliability Constraints in Solid Oxide Fuel Cells and Implications for Advanced Control AIChE Fall 2010 Annual Meeting Benjamin Spivey Advisor: Dr. Thomas F. Edgar University of Texas at Austin November 10, 2010 *GE, 2007

Dynamic Modeling of Reliability Constraints in Solid Oxide ... · Dynamic Modeling of Reliability Constraints in Solid Oxide Fuel Cells and Implications for Advanced Control AIChE

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Page 1: Dynamic Modeling of Reliability Constraints in Solid Oxide ... · Dynamic Modeling of Reliability Constraints in Solid Oxide Fuel Cells and Implications for Advanced Control AIChE

Dynamic Modeling of Reliability Constraints in Solid Oxide Fuel

Cells and Implications for Advanced Control

AIChE Fall 2010 Annual Meeting

Benjamin Spivey

Advisor: Dr. Thomas F. Edgar

University of Texas at Austin

November 10, 2010

*GE, 2007

Page 2: Dynamic Modeling of Reliability Constraints in Solid Oxide ... · Dynamic Modeling of Reliability Constraints in Solid Oxide Fuel Cells and Implications for Advanced Control AIChE

The University of Texas at Austin 2Benjamin James Spivey

Modeling and Controls Objective:

Maintain solid oxide fuel cell (SOFC) performance and operational integrity subject to load-following, efficiency maximization, and disturbances using advanced process control.

Agenda:

• Motivation and Overview of Tubular SOFC System

• Distributed-Parameter SOFC Modeling

• SS and Dynamic Simulations of Fuel Cell Operation

• Conclusion

Overview

Page 3: Dynamic Modeling of Reliability Constraints in Solid Oxide ... · Dynamic Modeling of Reliability Constraints in Solid Oxide Fuel Cells and Implications for Advanced Control AIChE

The University of Texas at Austin 3Benjamin James Spivey

Benefits

• High efficiencies at full and partial load: 40-50% (LHV) for SOFC for 200 kW, 60-70% for GT-SOFC, 90%+ for cogeneration.

• Fuel flexibility:– Hydrogen, natural gas, propane

– Alcohols, biomass, coal gas

• Suitability for cogeneration with high exhaust temperatures

• Low noise and emission levels

Operational Challenges

• Micro-cracking, catalyst poisoning, and air & fuel starvation decrease the lifetime and increase cost of electricity.

• Majority of real plants have used SISO PID and PLC control –operations experience with advanced control is limited.

Research Motivation

Page 4: Dynamic Modeling of Reliability Constraints in Solid Oxide ... · Dynamic Modeling of Reliability Constraints in Solid Oxide Fuel Cells and Implications for Advanced Control AIChE

The University of Texas at Austin 4Benjamin James Spivey

• The SOFC is often recommended to be coupled with a gas-turbine to utilize waste heat, maximize efficiencies, and supplement power production.

• Model of manipulated variables assumes an external variable speed compressor and recuperators.

Primary

Fuel

Secondary

Fuel from HX

Downstream

CHP BOP

Recuperator

Compressor

TurbineCombustor

Air Supply Tube

Tubular

SOFC

Internal

Reformer

Ejector

Inverter

Grid

Ambient

Air Cool AirHeated Air

Fuel

Spent Fuel

Electrical Current

HX

Generator

Pre-reformer

ACDC

~

SOFC system within a GT-SOFC Plant

SOFC System Overview

Page 5: Dynamic Modeling of Reliability Constraints in Solid Oxide ... · Dynamic Modeling of Reliability Constraints in Solid Oxide Fuel Cells and Implications for Advanced Control AIChE

Distributed-Parameter SOFC Modeling

Page 6: Dynamic Modeling of Reliability Constraints in Solid Oxide ... · Dynamic Modeling of Reliability Constraints in Solid Oxide Fuel Cells and Implications for Advanced Control AIChE

The University of Texas at Austin 6Benjamin James Spivey

Tubular SOFC Modeling

Type: high-temperature tubular SOFC

Structure: cathode-supported

Fuel: prereformation and direct

internal reformation of methane

Balance of Plant in Model: ejector and

prereformer

Pressurized to 3 bar

Based on a Siemens-Westinghouse

plant at National Fuel Cell Research

Center.

*Image from Singhal, S.C., 2006.

Tubular Solid Oxide Fuel Cell Assembly

Parameters and inlet conditions are well known in literature versus other SOFC designs

Page 7: Dynamic Modeling of Reliability Constraints in Solid Oxide ... · Dynamic Modeling of Reliability Constraints in Solid Oxide Fuel Cells and Implications for Advanced Control AIChE

The University of Texas at Austin 7Benjamin James Spivey

Dynamic Modeling Challenges

• Distributed parameter approach produces a large number of states:

220 states for 10 finite volumes in the axial direction.

• Dynamic system of differential and algebraic equations to be solved

simultaneously (without algebraic loops).

• Algebraic equations are in an implicit form.

• Nonlinearities introduced by reaction and electrochemical terms.

• Multiple time scales varying from milliseconds to hours.

Page 8: Dynamic Modeling of Reliability Constraints in Solid Oxide ... · Dynamic Modeling of Reliability Constraints in Solid Oxide Fuel Cells and Implications for Advanced Control AIChE

The University of Texas at Austin 8Benjamin James Spivey

Anode

Electrolyte

Cathode

Air Tube

rInner Air Chamber

Outer Air Chamber

Fuel Chamber

• Total States per Element = 22 : Tga, Tsa, Tse, Tsc, Tgc2, Tsat, Tgc1, Nga1-Nga7, Ngc21-Ngc27,

I (current)

• Total States per SS Model = 878

• Total States per Dynamic Model = 220

∆xGas Element States:

Tg, Ng1, …Ng7

Solid Element States:

TsH2, CH4, H2O, etc.

N2, O2

2224

224

42

3

2

1

HCOOHCH

HCOOHCH

r

r

+→+

+→+

x

SOFC Cross-Section in Radial (r) and Axial (x) Directions

Quasi-2D SOFC Model Discretization

Page 9: Dynamic Modeling of Reliability Constraints in Solid Oxide ... · Dynamic Modeling of Reliability Constraints in Solid Oxide Fuel Cells and Implications for Advanced Control AIChE

The University of Texas at Austin 9Benjamin James Spivey

( )

( )

( ) rxnesopps

s

i

sgiis

p

rxnr

i

gsiip

i jjiiinout

QTTAF

dx

dTkTTAh

dt

dTVc

QTTAhdx

dTcm

rvnn

i

i

,

4,

4

,0

+−+

+−=

+−+=

+=

∑ ∑

σε

ρ

&

&&

effr

cl

jc

al

ja

c

jcja

a

r

iR

ii

F

RT

ii

F

RT

ii

F

RT

ii

F

RT

F

GE

=

−+

−=

+

=

∆−=

η

η

ααη

,

,

,

,

0

,

0

,

0

1ln4

1ln2

ln4

ln2

2Electrochemical Model

VolK

XXXXkr

AreapRT

EAr

ps

COH

OHCO

CHsa

a

−=

−=

22

2

4

2

1 exp

Anode Steam Reformation ModelSpecies and Energy Balances

rcacell EV ηηη −−−=

+=

OH

OHavg

p

pp

F

RTEE

2

22

2/1

0 ln2

( )

0)(

),(0

),(..

,min

=

=Ω∈

xh

uxg

uxfxts

uxJx

&

The complete SOFC model is solved simultaneously via constrained NLP using the APMonitor Modeling Language.

2224

224

42

3

2

1

HCOOHCH

HCOOHCH

r

r

+→+

+→+

Key SOFC Model Equations

Page 10: Dynamic Modeling of Reliability Constraints in Solid Oxide ... · Dynamic Modeling of Reliability Constraints in Solid Oxide Fuel Cells and Implications for Advanced Control AIChE

The University of Texas at Austin 10Benjamin James Spivey

MAP for Electrolyte Temperature = 3.85%The mean absolute percentage (MAP) error is used to compare the models.

The tubular SOFC steady-state model is validated based upon experimental data and model data from standard practice (Campanari, 2004; Seume, 2009).

Model Results Campanari Model

∑ −=

t t

tt

A

PA

n

1MAP

SOFC Steady-State Model Validation

Page 11: Dynamic Modeling of Reliability Constraints in Solid Oxide ... · Dynamic Modeling of Reliability Constraints in Solid Oxide Fuel Cells and Implications for Advanced Control AIChE

The University of Texas at Austin 11Benjamin James Spivey

Comparison of the concentration profiles also indicates that the steady-

state model matches well versus the standard models used for tubular,

high-temperature SOFC modeling.

Model Results Campanari Model

SOFC Steady-State Model Validation

Page 12: Dynamic Modeling of Reliability Constraints in Solid Oxide ... · Dynamic Modeling of Reliability Constraints in Solid Oxide Fuel Cells and Implications for Advanced Control AIChE

The University of Texas at Austin 12Benjamin James Spivey

Radiation Analysis for Plant B : Air channel radiation is significant

Final Steady-State Model = Validated Campanari Model + Air Channel Radiation + Model Validation

Without Radiation With Radiation

Radiation Effects:

•Increased peak temperature

•Inlet air and solid PEN is closer in temperature

• Higher T gradients at outlet

• Molar flow exhibits no noticeable change.

SOFC Radiation Sensitivity

Page 13: Dynamic Modeling of Reliability Constraints in Solid Oxide ... · Dynamic Modeling of Reliability Constraints in Solid Oxide Fuel Cells and Implications for Advanced Control AIChE

The University of Texas at Austin 13Benjamin James Spivey

SOFC Electrical Characterization

SS Electrical Characterization for Plant A: 120 kW, 1.05 bar

• LHV efficiencies are 45% and 38% for Plants A and B respectively - typical for 100-300 kW SOFC.

•Nominal efficiency is based upon provided inputs, not plant modeling.

• Fixed fuel flow rate condition

0.0

0.2

0.4

0.6

0.8

1.0

1.2

0 500 1000 1500 2000 2500 3000

Current Density [A/m^2]

Vo

ltage

[V

]

0

20

40

60

80

100

120

140

160

Po

we

r [W

]

Voltage Power

0%

20%

40%

60%

80%

100%

0 1000 2000 3000

Current Density [A/m^2]

Eff

icie

ncy

[%

]

0%

20%

40%

60%

80%

100%

Fuel U

tiliza

tion [

%]

Efficiency

Fuel Utilization

in,4CH4CHin,COCOin,O2HO2H NLHVNLHVNLHV

VI

+⋅+⋅

⋅=η

Page 14: Dynamic Modeling of Reliability Constraints in Solid Oxide ... · Dynamic Modeling of Reliability Constraints in Solid Oxide Fuel Cells and Implications for Advanced Control AIChE

SS and Dynamic Simulations of Fuel Cell Operation

Page 15: Dynamic Modeling of Reliability Constraints in Solid Oxide ... · Dynamic Modeling of Reliability Constraints in Solid Oxide Fuel Cells and Implications for Advanced Control AIChE

The University of Texas at Austin 15Benjamin James Spivey

Prereformer:

Quasi-Steady-State

ODE Model

8 States

Matlab

SOFC :

Dynamic

DAE Model

220 States

APMonitor

Ejector:

Quasi-Steady-State

Algebraic Model

9 States

Matlab

PT PC

FC

TT TC

PT PC

TT

FC

Air

FT

TC

SOFC Exhaust

Variable Speed

Compressor

Fuel

FC

FC

Fuel Tanks

MV MV

MV MV

MV

MV

MV

MV

..

..

TT

TT

TT

..

.

CV

CV

Cell

Temperatures

Power &

Efficiency

MV Voltage

CV

Air/Fuel Utilization

Steam-Carbon Ratio

Manipulated and controlled variables are chosen based upon the SOFC+BOP system

SOFC and Balance of Plant (BOP) System

Page 16: Dynamic Modeling of Reliability Constraints in Solid Oxide ... · Dynamic Modeling of Reliability Constraints in Solid Oxide Fuel Cells and Implications for Advanced Control AIChE

The University of Texas at Austin 16Benjamin James Spivey

Relevance of Controlled Variables

Objective or Risk Controlled Variable

DC Power Delivery Power (W)

Efficiency (%)

Thermal Stress Minimization Minimum Stack Temperature (K)

Radial Thermal Gradient (K/m)

Avoid Catalyst Poisoning Steam-to-Carbon Ratio

Avoid Air and Fuel Starvation Air and Fuel Utilization (%)

Recent studies report that the minimum stack temperature and radial thermal gradient are responsible for the highest and second-highest thermal stress levels

(Seume, 2009).

Page 17: Dynamic Modeling of Reliability Constraints in Solid Oxide ... · Dynamic Modeling of Reliability Constraints in Solid Oxide Fuel Cells and Implications for Advanced Control AIChE

The University of Texas at Austin 17Benjamin James Spivey

-2,500

-2,000

-1,500

-1,000

-500

0

500

1,000

0.15 0.3 0.45 0.6 0.75 0.9 1.05 1.2 1.35 1.5

dT

dx

(K/m

)

Length (m)

Radial

Axial

Radial versus Axial Temperature Gradient

The radial gradient is negative near the fuel inlet placing the anode in tension. The radial gradient is several times the axial gradient.

Simulation results agree with prior studies indicating that radial thermal gradients are most significant.

Page 18: Dynamic Modeling of Reliability Constraints in Solid Oxide ... · Dynamic Modeling of Reliability Constraints in Solid Oxide Fuel Cells and Implications for Advanced Control AIChE

The University of Texas at Austin 18Benjamin James Spivey

Tstack, min (K)

Radial ∆T/L, max

(K/m)

Power (W)

CVs

Air Mass Flow

(kg/s)

System Pressure

(bar)Voltage (V)

Air Inlet

Temperature (K)

Fuel Inlet Pressure

(bar)

Fuel Inlet

Temperature (K)MVs

All scales are consistent for a given CV to compare slopes.

Variable Steady-State Gains

• Most MVs affect the minimum stack temperature.

• Air temperature and voltage affect the radial gradient significantly.

• Fuel pressure and temperature have most affect on power in this operating region.

Manipulated variables (MVs) are adjusted 10-20% of nominal on both sides of the nominal

value. Nominal is close to Plant B conditions.

800

1150

0 5

800

1150

0.55 0.7

800

1150

500 1500

800

1150

5 10

800

1150

200 500

800

1150

0.3 0.9

1000

3500

0.3 0.9

1000

3500

0 5

1000

3500

0.55 0.7

1000

3500

500 1500

1000

3500

5 10

1000

3500

200 500

140

165

0.3 0.9

140

165

0 5140

165

0.55 0.7140

165

500 1500

140

165

5 10

140

165

200 500

Page 19: Dynamic Modeling of Reliability Constraints in Solid Oxide ... · Dynamic Modeling of Reliability Constraints in Solid Oxide Fuel Cells and Implications for Advanced Control AIChE

The University of Texas at Austin 19Benjamin James Spivey

Air Utilization (%)

Fuel Utilization

(%)

Steam-Carbon

Ratio

CVs

Air Mass Flow

(kg/s)

System Pressure

(bar)Voltage (V)

Air Inlet

Temperature (K)

Fuel Inlet Pressure

(bar)

Fuel Inlet

Temperature (K)MVs

Variable Steady-State Gains

• Air mass flow is a key MV for managing air utilization.

• Fuel temperature and system pressure manage fuel utilization.

• Fuel pressure and temperature and system pressure significantly affect

the steam-to-carbon ratio.

15

40

0.3 0.9

15

40

0 5

15

40

0.55 0.7

15

40

500 1500

15

40

5 10

15

40

200 500

40

100

0.3 0.9

40

100

0 5

40

100

0.55 0.7

40

100

500 1500

40

100

5 10

40

100

200 500

0

8

0.55 0.70

8

0.3 0.9

0

8

0 5

0

8

500 1500

0

8

5 10

0

8

200 500

Page 20: Dynamic Modeling of Reliability Constraints in Solid Oxide ... · Dynamic Modeling of Reliability Constraints in Solid Oxide Fuel Cells and Implications for Advanced Control AIChE

The University of Texas at Austin 20Benjamin James Spivey

Simulation Time Discretization: Power Response to Voltage Step

• Decreasing time steps below 1 s yields little change in dynamic response.

• The QSS gas transport assumption is only valid to 1s time steps.

153.2

153.6

154

154.4

154.8

155.2

155.6

0 200 400 600 800

Po

we

r (W

)

Time (s)

1 s

0.5 s

60 s

Transport Time Delays

• Three transport time delays are added to the QSS gas transport models to improve dynamic accuracy.

• Delays are important for sub-60 s response.

• Delays are updated by mass flows

153.2

153.6

154

154.4

154.8

155.2

155.6

0 20 40 60 80

Po

we

r (W

)

Time (s)

All Time Delays

No Time Delays

Dynamic Simulation Design

Page 21: Dynamic Modeling of Reliability Constraints in Solid Oxide ... · Dynamic Modeling of Reliability Constraints in Solid Oxide Fuel Cells and Implications for Advanced Control AIChE

The University of Texas at Austin 21Benjamin James Spivey

110

120

130

140

150

160

170

180

0 500 1000 1500 2000

Po

we

r (W

)

Time (s)

110

120

130

140

150

160

170

180

0 500 1000 1500 2000

Po

we

r (W

)

Time (s)

110

120

130

140

150

160

170

180

0 500 1000 1500 2000

Pow

er (W

)

Time (s)

Fuel Pressure

Fuel Temperature

Voltage

MV:

• Staircase tests are run over 5 minutes in response to MV steps over 1 s.

• Increasing gains further from the nominal illustrate non-linearity.

• Decreasing the voltage to 0.55 V reduces power output – crossed peak voltage and shows nonlinear V-P relationship.

5.0

6.0

7.0

8.0

9.0

10.0

11.0

0 1000 2000

Pfu

el (

ba

r)

Time (s)

Staircase Dynamic Simulations: Power Plots

Example of staircase input:

Page 22: Dynamic Modeling of Reliability Constraints in Solid Oxide ... · Dynamic Modeling of Reliability Constraints in Solid Oxide Fuel Cells and Implications for Advanced Control AIChE

The University of Texas at Austin 22Benjamin James Spivey

110

610

1110

1610

2110

2610

3110

3610

0 500 1000 1500 2000

dT

rad

,ma

x (K

/m)

Time (s)

110

610

1110

1610

2110

2610

3110

3610

0 500 1000 1500 2000

dTr

ad,m

ax

(K/m

)

Time (s)

110

610

1110

1610

2110

2610

3110

3610

0 500 1000 1500 2000

dT

rad

,ma

x (

K/m

)

Time (s)

Fuel Pressure

Fuel Temperature

Air Mass Flow

MV:

• Numerator dynamics response is real and due to quickly changing anode chamber gas conditions.

• Discontinuous curves can result between step changes when location of max gradient changes.

• The maximum radial thermal gradient does not change quickly due to varying cathode-side conditions.

Staircase Dynamic Simulations: Max Radial dTdr Plots

Page 23: Dynamic Modeling of Reliability Constraints in Solid Oxide ... · Dynamic Modeling of Reliability Constraints in Solid Oxide Fuel Cells and Implications for Advanced Control AIChE

The University of Texas at Austin 23Benjamin James Spivey

Electrolyte Temperature and Radial Gradients

Power Increase

Dynamic Evolution of the Temperature Profile

Open-Loop Fuel Pressure Step Increase

-Lower Min Cell Temp.

-Higher Max dTdr

Increased Thermal Stresses

5.0

6.0

7.0

8.0

9.0

10.0

11.0

0 200 400 600

Pfu

el (

ba

r)

Time (s)

110

130

150

170

190

0 200 400 600

Po

we

r (W

)

Time (s)

Minimum temperature and radial gradients undergo unique dynamics

Page 24: Dynamic Modeling of Reliability Constraints in Solid Oxide ... · Dynamic Modeling of Reliability Constraints in Solid Oxide Fuel Cells and Implications for Advanced Control AIChE

The University of Texas at Austin 24Benjamin James Spivey

Conclusions

• Modeling:

Distributed parameter modeling for SOFC is critical to accurately

capture overall performance as well as local gradients and minimum

temperatures – key reliability criteria.

• Simulation

- Radial thermal gradients may increase quickly in response to

changing input conditions. Control algorithms should account for

dynamics of minimum cell temperature and maximum radial

gradients.

- Most properties have an initial rise time < 1 min in response to 1 s

input steps despite final settling times of hours. Control input

intervals should be several times less than the rise time.

Page 25: Dynamic Modeling of Reliability Constraints in Solid Oxide ... · Dynamic Modeling of Reliability Constraints in Solid Oxide Fuel Cells and Implications for Advanced Control AIChE

The University of Texas at Austin 25Benjamin James Spivey

Acknowledgments

• Dr. Thomas Edgar

• Dr. John Hedengren

• Dr. Jeremy Meyers

• Dr. Dongmei Chen

• APMonitor Modeling Language

• Labmates