fuel cell reformer control

Fuel Cell Reformer Control

Karel SchnebeleMay 5, 2006

Presentation Outline

IntroductionDevelopment of the state space modelModeling the systemSISO controlMultivariable controlRGA analysis and pairingDisturbance rejectionDirectional sensitivity

Introduction

Purpose: create final projectModel

Steam reformer for residential fuel cell plantFrom Jahn and Schroer, 2005

Development of State Space Model:Model Component Relationships

Single lines depict heat transfer (solid is conduction, dashed is radiation), double lines depict the burner gas flow, and triple lines depict the reformategas flow.

Development of State Space Model:Dynamic Equations

( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( ) ( )

( ) ( ) ( )

( ) ( )

( ) ( ) ( )

( ) ( )ERCHCHpEROHOHp

RFGFFpERRERBBRR

R

AEACHECHCHp

OHEOHOHpiOHERFFpERREE

E

GBBGRBBRWBBBpBFFBB

B

GFFFpFGWGGWGBBGG

G

AWBBpAWWAWGGWW

W

TTncTTncnhnh

TTncTTkTTkdt

dTC

TTkTTnc

TTncnrTTncTTkdt

dTC

TTkTTkTTncTTkdt

dTC

TTnckTTkTTkdt

dTC

TTncTTkTTkdt

dTC

−⋅−−⋅−Δ⋅Δ−Δ⋅Δ−

−⋅+−−−=

−−−⋅−

−⋅−⋅−−⋅+−=

−−−−−⋅−−=

−⋅⋅+−−−=

−⋅−−−−=

⋅⋅⋅⋅

⋅

⋅

⋅⋅⋅

⋅

⋅

⋅

4422

444

2222

,,1100

,44

E,

,,,

44,

,

,

Development of State Space Model:Changing nCH4i

0 100 200 300 400 500 600 700 800 900400

500

600

700

800

900

1000

1100

1200

1300

time (sec)

Tem

pera

ture

(K

)

Tw =151.937Tg =368.1042Tb =487.445Te =993.785Tr =754.7771

Tw

Tg

TbTe

Tr

0 100 200 300 400 500 600 700 800 900400

600

800

1000

1200

1400

1600

time (sec)

Tem

pera

ture

(K

)

Tw =172.8318Tg =428.0062Tb =577.0143Te =1208.8189Tr =898.7824

Tw

Tg

TbTe

Tr

initial methane flow rate=10 SLPMsteam to carbon ratio=3.5excess air ratio=5

initial methane flow=15 SLPMsteam to carbon ratio=3.5excess air ratio=5

Development of State Space Model:Changing Steam to Carbon Ratio

0 100 200 300 400 500 600 700 800 900400

500

600

700

800

900

1000

1100

1200

1300

time (sec)

Tem

pera

ture

(K

)

Tw =146.5796Tg =352.8994Tb =464.8156Te =916.397Tr =709.2319

Tw

Tg

TbTe

Tr

0 100 200 300 400 500 600 700 800 900400

500

600

700

800

900

1000

1100

1200

1300

1400

time (sec)

Tem

pera

ture

(K

)

Tw =158.9264Tg =388.0229Tb =517.3146Te =1069.7957Tr =807.7388

Tw

Tg

TbTe

Tr

initial methane flow rate=10 SLPMsteam to carbon ratio=3excess air ratio=5

initial methane flow rate=10 SLPMsteam to carbon ratio=4excess air ratio=5

Development of State Space Model:Changing Excess Air Ratio

0 100 200 300 400 500 600 700 800 900400

500

600

700

800

900

1000

1100

1200

1300

1400

time (sec)

Tem

pera

ture

(K

)

Tw =213.1884Tg =487.5032Tb =666.8927Te =1124.999Tr =907.7055

Tw

Tg

TbTe

Tr

0 100 200 300 400 500 600 700 800 900400

500

600

700

800

900

1000

1100

1200

1300

time (sec)

Tem

pera

ture

(K

)

Tw =151.937Tg =368.1042Tb =487.445Te =993.785Tr =754.7771

Tw

Tg

TbTe

Tr

initial methane flow rate=10 SLPMsteam to carbon ratio=3.5excess air ratio=4

initial methane flow rate=10 SLPMsteam to carbon ratio=3.5excess air ratio=5

Development of State Space Model:Specified values

Reformer temp = 700 deg CSteam to carbon ~ 3.5Methane flow rate ~ 10 SLPM

Development of State Space Model:Output Temperatures

0 100 200 300 400 500 600 700 800400

500

600

700

800

900

1000

1100

1200

time (sec)

Tem

pera

ture

(K

)

Tw =145.5671Tg =350.0323Tb =460.524Te =897.0203Tr =700.003

Tw

Tg

TbTe

Tr

Methane flow rate = 9.5 SLPMSteam to carbon=3.0076Excess air ratio=5

Development of State Space Model:States, Inputs, and Outputs

States =

⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢

⎣

⎡

−−−−−

=

⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢

⎣

⎡

RsR

EsE

BsB

GsG

WsW

TTTTTTTTTT

xxxxx

5

4

3

2

1

Inputs = ⎥⎦

⎤⎢⎣

⎡=⎥

⎦

⎤⎢⎣

⎡nv

ratioairexcessuu )(

2

1 λOutputs = ⎥

⎦

⎤⎢⎣

⎡=⎥

⎦

⎤⎢⎣

⎡=⎥

⎦

⎤⎢⎣

⎡

R

B

TT

yy

gg

2

1

2

1

Development of State Space Model:Matrices

''''''

DuCxyBuAxx

+=+=&

j

iij x

fA∂∂

=j

iij u

fB∂∂

=

j

iij x

gC∂∂

=j

iij u

gD∂∂

=

Development of State Space Model:Matrices

⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢

⎣

⎡

−×−

−−−

×−

=

−

−

007322.0004675.0104285.100004436.000472.0000008232.00058625.0020455.0034462.0

000018443.0004911.0002115.000010098.7001593.0

4

4

A

⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢

⎣

⎡

−−−−−−

−−

=

0119.80075294.06894.5705429.0

8766.20320898.20762.15614687.0888.2502424.0

B ⎥⎦

⎤⎢⎣

⎡=

1000000100

C ⎥⎦

⎤⎢⎣

⎡=

0000

D

Development of State Space Model:Final Subsystem

2

Tr

1

Tb

Uniform RandomNumber3

Uniform RandomNumber2

x' = Ax+Bu y = Cx+Du

State-Space1

Product

-C-

Constant4

-C-

Constant3

-C-

Constant2

5

Constant1

-C-

Constant

2

nv

1

lambda

Development of State Space Model:Subsystem in Large System

Tr

reformer tempmethane flow rate

excess air ratio

Tb

burner templambda

nv

Tb

Tr

Subsystem3

Model Development

0 1000 2000 3000 4000 5000440

445

450

455

460

465

Time (sec)

Tb

(deg

C)

0 1000 2000 3000 4000 5000675

680

685

690

695

700

705

Time (sec)

Tr

(deg

C)

0 1000 2000 3000 4000 5000

4.8

5

5.2

5.4

5.6

Time (sec)

Exc

ess

Air

Rat

io

0 1000 2000 3000 4000 50005.5

6

6.5

7

7.5

8

Time (sec)

Met

hane

Flo

w R

ate

to B

urne

r (S

LPM

)

Temperature responses to excess air ratio change of 0.5

Lead-lag First order

0 1000 2000 3000 4000 5000440

445

450

455

460

465

Time (sec)

Tb

(deg

C)

0 1000 2000 3000 4000 5000675

680

685

690

695

700

705

Time (sec)

Tr

(deg

C)

0 1000 2000 3000 4000 50004

4.5

5

5.5

6

Time (sec)

Exc

ess

Air

Rat

io

0 1000 2000 3000 4000 50006.5

6.6

6.7

6.8

6.9

7

7.1

7.2

7.3

Time (sec)

Met

hane

Flo

w R

ate

to B

urne

r (S

LPM

)

Temperature responses to a methane flow rate change of 0.5658 SLPM

Lead-lag First order

Model Parameters

1+=

skpg

pp τ ⎟

⎟⎠

⎞⎜⎜⎝

⎛

++

=11

sskpg

p

np τ

τ

uykp

ΔΔ

=

First order equation: Lead-lag equation:

uykp

ΔΔ

=

occurs change of 63.2% when time=pτ pn ττ , find toiterate

( )( )

( )( )

( ) ( )⎥⎦

⎤⎢⎣

⎡=⎥

⎦

⎤⎢⎣

⎡

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

+°−

+°−

++

°−+

°−

21

21

1sec71069.35

1sec730506.45

1sec4001sec52039.19

sec4001sec500548.28

yy

uu

sC

sC

ssC

ssC

Process Transfer Functions

Process vs Model

0 1000 2000 3000 4000440

445

450

455

460

465

Time (sec)

Tb

(deg

C)

0 1000 2000 3000 4000675

680

685

690

695

700

705

Time (sec)

Tr

(deg

C)

0 1000 2000 3000 40004.5

5

5.5

6

Time (sec)

Exc

ess

Air

Rat

io

0 1000 2000 3000 40005.5

6

6.5

7

7.5

8

Time (sec)

Met

hane

Flo

w R

ate

to B

urne

r (S

LPM

)

Model and process responses to setpoint change in excess air ratio

Process vs Model

0 1000 2000 3000 4000440

445

450

455

460

465

Time (sec)

Tb

(deg

C)

0 1000 2000 3000 4000675

680

685

690

695

700

705

Time (sec)

Tr (d

eg C

)

0 1000 2000 3000 40004

4.5

5

5.5

6

Time (sec)

Exc

ess

Air

Rat

io

0 1000 2000 3000 40006.5

6.6

6.7

6.8

6.9

7

7.1

7.2

7.3

Time (sec)

Met

hane

Flo

w R

ate

to B

urne

r (S

LPM

)

Model and process responses to setpoint change in methane flow rate

SISO Controller Development:IMC-based PID control strategy

⎟⎟⎠

⎞⎜⎜⎝

⎛+=

skcg

Ic τ

11

PI controller w/ disturbance rejection for first order transfer functions

PI controller w/ filter term for lead-lag transfer functions

λλτ

kpkc p −=

2

p

pI τ

λλττ

22 −=

⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝

⎛+=

1111ss

kcgFI

c ττ

λτkp

kc p=

pI ττ = nF ττ =

Simulink Diagram: SISO control

step reformer temp

step burner temp

r

setpoint

nv

methane flow rate

1

500s+1

filter

lambda

excess air ratio

lambda

nv

Tb

Tr

Subsystem1

Tr

Reformer Temperature

PID

5

Tb

Burner Temperatue

Burner temperature controlled by the excess air ratio

Burner Temperature Control

0 10 20 30 40460

470

480

490

500

510

Time (min)

Tb

(deg

C)

0 10 20 30 40 50 60690

700

710

720

730

740

750

760

770

Time (min)

Tr (d

eg C

)

0 10 20 30 403.6

3.8

4

4.2

4.4

4.6

4.8

5

5.2

Time (min)

Exc

ess

Air

Rat

io

0 10 20 30 405.5

6

6.5

7

7.5

8

Time (min)

Met

hane

Flo

w R

ate

to B

urne

r (S

LPM

)

lambda = 25

lambda = 75

lambda = 100lambda = 150

setpoint

0 10 20 30 40460

470

480

490

500

510

Time (min)Tb

(deg

C)

0 20 40 60 80680

700

720

740

760

780

Time (min)

Tr (d

eg C

)

0 20 40 60 80 1004

4.5

5

5.5

6

Time (min)

Exc

ess

Air

Rat

io

0 20 40 60 80 1004.5

5

5.5

6

6.5

7

Time (min)

Met

hane

Flo

w R

ate

to B

urne

r (S

LPM

)

lambda = 25

lambda =75

lambda =100lambda = 150

setpoint

Control by excess air ratio

Control by methane flow rate

Reformer Temperature Control

0 10 20 30 40 50450

500

550

600

Time (min)

Tb

(deg

C)

0 10 20 30 40 50680

700

720

740

760

780

Time (min)

Tr (d

eg C

)

0 10 20 30 40 500

1

2

3

4

5

6

Time (min)

Exc

ess

Air

Rat

io

0 10 20 30 40 505.5

6

6.5

7

7.5

8

Time (min)

Met

hane

Flo

w R

ate

to B

urne

r (S

LPM

)

lambda = 400

lambda = 500

lambda = 600lambda = 700

setpoint

0 10 20 30 40 50440

460

480

500

520

540

560

580

Time (min)Tb

(deg

C)

0 10 20 30 40 50680

700

720

740

760

780

Time (min)

Tr (d

eg C

)

0 10 20 30 40 504

4.5

5

5.5

6

Time (min)

Exc

ess

Air

Rat

io

0 10 20 30 40 501

2

3

4

5

6

7

Time (min)

Met

hane

Flo

w R

ate

to B

urne

r (S

LPM

)

lambda = 400

lambda = 500

lambda = 600lambda = 700

setpoint

Control by excess air ratio

Control by methane flow rate

SISO Controllersy1-u1 y1-u2

y2-u1 y2-u2

⎟⎠⎞

⎜⎝⎛

+⎟⎠⎞

⎜⎝⎛ +−=

15001

400111401.011 ss

gc ⎟⎠⎞

⎜⎝⎛

+⎟⎠⎞

⎜⎝⎛ +−=

15201

400112063.012 ss

gc

⎟⎠⎞

⎜⎝⎛ +−=

sgc 85.706

110315.021 ⎟⎠⎞

⎜⎝⎛ +−=

sgc 96.692

110383.022

Multivariable Control: RGA Analysis

⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

−−−

−−

−=Λ

21122211

2211

21122211

1221

21122211

2112

21122211

2211

kkkkkk

kkkkkk

kkkkkk

kkkkkk

⎥⎦

⎤⎢⎣

⎡−−−−

=69.35506.4539.19548.28

K ⎥⎦

⎤⎢⎣

⎡−

−=Λ

463.7463.6463.6463.7

Process Gain Matrix Relative Gain Array

Do not pair on negative relative gain y1-u1 and y2-u2 pairings

Simulink Diagram: y1-u1, y2-u2 pairing

Tr

reformer temperature

Trset

reformer setpoint

nv

methane flow rate

1

500s+1

filter

lambda

excess air ratioTb

burner temperature

Tbset

burner setpoint

lambda

nv

Tb

Tr

Subsystem6Step6

Step13

PID

PID

6.5856

5

Multivariable Control:Setpoint changes in both temperatures

0 50 100 150 200 250 300440

460

480

500

520

540

Time (min)

Tb

(deg

C)

0 50 100 150 200 250 300680

700

720

740

760

780

Time (min)

Tr

(deg

C)

0 50 100 150 200 250 3004.6

4.8

5

5.2

5.4

5.6

Time (min)

Exc

ess

Air

Rat

io

0 50 100 150 200 250 3003

4

5

6

7

Time (min)

Met

hane

Flo

w R

ate

to B

urne

r (S

LPM

)

Multivariable Control:Setpoint changes in only one temperature

0 50 100 150 200 250 300 350455

460

465

470

475

480

485

Time (min)

Tb

(deg

C)

0 100 200 300 400695

700

705

710

715

720

Time (min)

Tr

(deg

C)

0 100 200 300 4000

1

2

3

4

5

6

Time (min)

Exc

ess

Air

Rat

io

0 100 200 300 4006

7

8

9

10

11

12

13

Time (min)

Met

hane

Flo

w R

ate

to B

urne

r (S

LPM

)

0 50 100 150 200 250 300455

460

465

470

475

480

485

Time (min)T

b (d

eg C

)0 100 200 300 400

690

700

710

720

730

740

Time (min)

Tr

(deg

C)

0 100 200 300 4004

5

6

7

8

9

10

Time (min)

Exc

ess

Air

Rat

io

0 100 200 300 4000

1

2

3

4

5

6

7

Time (min)

Met

hane

Flo

w R

ate

to B

urne

r (S

LPM

)

Setpoint change in burner temperature

Setpoint change in reformer temperature

Disturbance Rejection:First-order controller differences

⎟⎠⎞

⎜⎝⎛ +−=

sgc 710

110332.0

⎟⎟⎠

⎞⎜⎜⎝

⎛+=

skcg

Ic τ

11

PI controller w/ disturbance rejection for first order transfer functions

λλτ

kpkc p −=

2

p

pI τ

λλττ

22 −=

⎟⎟⎠

⎞⎜⎜⎝

⎛+=

skcg

Ic τ

11

PI controller w/o disturbance rejection for first order transfer functions

λτkp

kc p=

pI ττ =

⎟⎠⎞

⎜⎝⎛ +−=

sgc 96.692

110383.0

Simulink Diagram:System with catalyst sintering disturbance

Tr

reformer temp

Trset

reformer setpoint

nv

methane flow rate

lambda

excess air ratio

Tb

burner temp

Tbset

burner setpoint

1

500s+1

Transfer Fcn7

lambda

nv

Sintering (%)

Tb

Tr

Subsystem9

Step16

Step15

Step14

PID

PID

6.5856

5

Disturbance Rejection:100% Sintering

0 50 100 150 200440

460

480

500

520

540

Time (min)

Tb

(deg

C)

0 100 200 300 400680

700

720

740

760

780

800

Time (min)

Tr

(deg

C)

0 100 200 300 4001

2

3

4

5

6

Time (min)

Exc

ess

Air

Rat

io

0 100 200 300 4003

4

5

6

7

8

9

10

Time (min)

Met

hane

Flo

w R

ate

to B

urne

r (S

LPM

)

w/ dist rejection

w/o dist rejection

Directional Sensitivity:Scaling the ranges

6.584613.17126.58460u 2 (methane)

51050u 1 (excess air)

249.997950700.003450.006y 2 (reformer)

200660.524460.524260.524y 1 (burner)

½ RangeMax ValueNominal ValueMin Value

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

=

⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

=

997.24910

02001

)2(1

0

0)1(

1

21

21

yrange

yrangeSo

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

=

⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

=

5846.610

051

)2(1

0

0)1(

1

21

21

urange

urangeSI

Scaled Output Matrix Scaled Input Matrix

Directional Sensitivity:Scaled gain matrix

1−∗ ××= IO SGSG

⎥⎦

⎤⎢⎣

⎡−−−−

=9402.09101.06385.07137.0

*G

Directional Sensitivity:SVD analysis

TVUG Σ=∗

T

⎥⎦

⎤⎢⎣

⎡−⎥

⎦

⎤⎢⎣

⎡⎥⎦

⎤⎢⎣

⎡−

−−=⎥

⎦

⎤⎢⎣

⎡−−−−

7133.07009.07009.07133.0

0555.0006206.1

5903.08072.08072.05903.0

9402.09101.06385.07137.0

strongest output direction

weakest output direction

strongest input direction

weakest input direction

Directional Sensitivity:Scaling back to the process

∗− ×= ySy O1

⎥⎦

⎤⎢⎣

⎡−−

=⎥⎦

⎤⎢⎣

⎡7976.20106.118

2

1

yy

Strong Direction

⎥⎦

⎤⎢⎣

⎡−=⎥

⎦

⎤⎢⎣

⎡5732.147

44.161

2

1

yy

Weak Direction

-1 -0.5 0 0.5 1-1

-0.5

0

0.5

1

u1

u2

-1 0 1

-1.5

-1

-0.5

0

0.5

1

1.5

y1

y2

Input to Output Mapping

Directional Sensitivity:Changes in the strong direction

0 50 100 150 200 250 300440

445

450

455

460

465

Time (min)

Tb

(deg

C)

0 50 100 150 200 250 300675

680

685

690

695

700

705

Time (min)

Tr

(deg

C)

0 50 100 150 200 250 300

4.9

5

5.1

5.2

5.3

Time (min)

Exc

ess

Air

Rat

io

0 50 100 150 200 250 3006.4

6.6

6.8

7

7.2

7.4

Time (min)

Met

hane

Flo

w R

ate

to B

urne

r (S

LPM

)

Setpoint changes were only 10% of the total

Directional Sensitivity:Changes in the weak direction

0 50 100 150 200 250 300440

445

450

455

460

465

470

Time (min)

Tb

(deg

C)

0 100 200 300 400680

690

700

710

720

Time (min)

Tr

(deg

C)

0 100 200 300 4004

6

8

10

12

Time (min)

Exc

ess

Air

Rat

io

0 100 200 300 400-2

0

2

4

6

8

Time (min)

Met

hane

Flo

w R

ate

to B

urne

r (S

LPM

)

Setpoint changes were only 10% of total

Negative flow rate

Conclusion

2

Tr

1

Tb

Uniform RandomNumber3

Uniform RandomNumber2

x' = Ax+Bu y = Cx+Du

State-Space1

Product

-C-

Constant4

-C-

Constant3

-C-

Constant2

5

Constant1

-C-

Constant

2

nv

1

lambda