fuel cell reformer control
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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
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