Fuel Cell Reformer Control Karel Schnebele May 5, 2006 - - PowerPoint PPT Presentation

Fuel Cell Reformer Control

Karel Schnebele May 5, 2006

Presentation Outline

Introduction Development of the state space model Modeling the system SISO control Multivariable control RGA analysis and pairing Disturbance rejection Directional sensitivity

Introduction

Purpose: create final project Model

Steam reformer for residential fuel cell plant From 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 reformate gas flow.

Development of State Space Model:

Dynamic Equations

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

( )

( ) ( ) ( )

( ) ( )

( )

( )

( ) ( ) ( ) ( )

E R CH CH p E R O H O H p R FG F F p E R RE R B BR R R A EA CH E CH CH p O H E O H O H p i O H E R F F p E R RE E E G B BG R B BR W B B B p B F FB B B G F F F p FG W G GW G B BG G G A W B B p A W WA W G GW W W

T T n c T T n c n h n h T T n c T T k T T k dt dT C T T k T T n c T T n c n r T T n c T T k dt dT C T T k T T k T T n c T T k dt dT C T T n c k T T k T T k dt dT C T T n c T T k T T k dt dT C − ⋅ − − ⋅ − Δ ⋅ Δ − Δ ⋅ Δ − − ⋅ + − − − = − − − ⋅ − − ⋅ − ⋅ − − ⋅ + − = − − − − − ⋅ − − = − ⋅ ⋅ + − − − = − ⋅ − − − − =

⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅

4 4 2 2 4 4 4 2 2 2 2

, , 1 1 , 4 4 E , , , , 4 4 , , ,

Development of State Space Model:

Changing nCH4i

100 200 300 400 500 600 700 800 900 400 500 600 700 800 900 1000 1100 1200 1300 time (sec) Temperature (K) Tw =151.937 Tg =368.1042 Tb =487.445 Te =993.785 Tr =754.7771 Tw Tg Tb Te Tr 100 200 300 400 500 600 700 800 900 400 600 800 1000 1200 1400 1600 time (sec) Temperature (K) Tw =172.8318 Tg =428.0062 Tb =577.0143 Te =1208.8189 Tr =898.7824 Tw Tg Tb Te Tr

initial methane flow rate=10 SLPM steam to carbon ratio=3.5 excess air ratio=5 initial methane flow=15 SLPM steam to carbon ratio=3.5 excess air ratio=5

Development of State Space Model:

Changing Steam to Carbon Ratio

100 200 300 400 500 600 700 800 900 400 500 600 700 800 900 1000 1100 1200 1300 time (sec) Temperature (K) Tw =146.5796 Tg =352.8994 Tb =464.8156 Te =916.397 Tr =709.2319 Tw Tg Tb Te Tr

100 200 300 400 500 600 700 800 900 400 500 600 700 800 900 1000 1100 1200 1300 1400 time (sec) Temperature (K) Tw =158.9264 Tg =388.0229 Tb =517.3146 Te =1069.7957 Tr =807.7388 Tw Tg Tb Te Tr

initial methane flow rate=10 SLPM steam to carbon ratio=3 excess air ratio=5 initial methane flow rate=10 SLPM steam to carbon ratio=4 excess air ratio=5

Development of State Space Model:

Changing Excess Air Ratio

100 200 300 400 500 600 700 800 900 400 500 600 700 800 900 1000 1100 1200 1300 1400 time (sec) Temperature (K) Tw =213.1884 Tg =487.5032 Tb =666.8927 Te =1124.999 Tr =907.7055 Tw Tg Tb Te Tr 100 200 300 400 500 600 700 800 900 400 500 600 700 800 900 1000 1100 1200 1300 time (sec) Temperature (K) Tw =151.937 Tg =368.1042 Tb =487.445 Te =993.785 Tr =754.7771 Tw Tg Tb Te Tr

initial methane flow rate=10 SLPM steam to carbon ratio=3.5 excess air ratio=4 initial methane flow rate=10 SLPM steam to carbon ratio=3.5 excess air ratio=5

Development of State Space Model:

Specified values

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

Development of State Space Model:

Output Temperatures

100 200 300 400 500 600 700 800 400 500 600 700 800 900 1000 1100 1200 time (sec) Temperature (K) Tw =145.5671 Tg =350.0323 Tb =460.524 Te =897.0203 Tr =700.003 Tw Tg Tb Te Tr

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

Development of State Space Model:

States, Inputs, and Outputs

States =

⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ − − − − − = ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡

Rs R Es E Bs B Gs G Ws W

T T T T T T T T T T x x x x x

5 4 3 2 1

Inputs = ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ = ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ nv ratio air excess u u ) (

2 1

λ Outputs =

⎥ ⎦ ⎤ ⎢ ⎣ ⎡ = ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ = ⎥ ⎦ ⎤ ⎢ ⎣ ⎡

R B

T T y y g g

2 1 2 1

Development of State Space Model:

Matrices

' ' ' ' ' ' Du Cx y Bu Ax x + = + = &

j i ij

x f A ∂ ∂ =

j i ij

u f B ∂ ∂ =

j i ij

x g C ∂ ∂ =

j i ij

u g D ∂ ∂ =

Development of State Space Model:

Matrices

⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ − × − − − − × − =

− −

007322 . 004675 . 10 4285 . 1 004436 . 00472 . 008232 . 058625 . 020455 . 034462 . 0018443 . 004911 . 002115 . 10 098 . 7 001593 .

4 4

A

⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ − − − − − − − − = 0119 . 80 075294 . 6894 . 57 05429 . 8766 . 2032 0898 . 2 0762 . 156 14687 . 888 . 25 02424 . B

⎥ ⎦ ⎤ ⎢ ⎣ ⎡ = 1 1 C

⎥ ⎦ ⎤ ⎢ ⎣ ⎡ = D

Development of State Space Model:

Final Subsystem

2 Tr 1 Tb Uniform Random Number3 Uniform Random Number2 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 temp methane flow rate excess air ratio Tb burner temp

lambda nv Tb Tr

Subsystem3

Model Development

1000 2000 3000 4000 5000 440 445 450 455 460 465 Time (sec) Tb (deg C) 1000 2000 3000 4000 5000 675 680 685 690 695 700 705 Time (sec) Tr (deg C) 1000 2000 3000 4000 5000 4.8 5 5.2 5.4 5.6 Time (sec) Excess Air Ratio 1000 2000 3000 4000 5000 5.5 6 6.5 7 7.5 8 Time (sec) Methane Flow Rate to Burner (SLPM)

Temperature responses to excess air ratio change of 0.5 Lead-lag First order

1000 2000 3000 4000 5000 440 445 450 455 460 465 Time (sec) Tb (deg C) 1000 2000 3000 4000 5000 675 680 685 690 695 700 705 Time (sec) Tr (deg C) 1000 2000 3000 4000 5000 4 4.5 5 5.5 6 Time (sec) Excess Air Ratio 1000 2000 3000 4000 5000 6.5 6.6 6.7 6.8 6.9 7 7.1 7.2 7.3 Time (sec) Methane Flow Rate to Burner (SLPM)

Temperature responses to a methane flow rate change of 0.5658 SLPM Lead-lag First order

Model Parameters

1 + = s kp g

p p

τ

⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ + + = 1 1 s s kp g

p n p

τ τ

u y kp Δ Δ = First order equation: Lead-lag equation:

u y kp Δ Δ =

• ccurs

change

• f

63.2% when time =

p

τ

p n τ

τ , find to iterate

( ) ( ) ( ) ( ) ( ) ( )

⎥ ⎦ ⎤ ⎢ ⎣ ⎡ = ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ + ° − + ° − + + ° − + ° − 2 1 2 1 1 sec 710 69 . 35 1 sec 730 506 . 45 1 sec 400 1 sec 520 39 . 19 sec 400 1 sec 500 548 . 28 y y u u s C s C s s C s s C

Process Transfer Functions

Process vs Model

1000 2000 3000 4000 440 445 450 455 460 465 Time (sec) Tb (deg C) 1000 2000 3000 4000 675 680 685 690 695 700 705 Time (sec) Tr (deg C) 1000 2000 3000 4000 4.5 5 5.5 6 Time (sec) Excess Air Ratio 1000 2000 3000 4000 5.5 6 6.5 7 7.5 8 Time (sec) Methane Flow Rate to Burner (SLPM)

Model and process responses to setpoint change in excess air ratio

Process vs Model

1000 2000 3000 4000 440 445 450 455 460 465 Time (sec) Tb (deg C) 1000 2000 3000 4000 675 680 685 690 695 700 705 Time (sec) Tr (deg C) 1000 2000 3000 4000 4 4.5 5 5.5 6 Time (sec) Excess Air Ratio 1000 2000 3000 4000 6.5 6.6 6.7 6.8 6.9 7 7.1 7.2 7.3 Time (sec) Methane Flow Rate to Burner (SLPM)

Model and process responses to setpoint change in methane flow rate

SISO Controller Development:

IMC-based PID control strategy

⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ + = s kc g

I c

τ 1 1

PI controller w/ disturbance rejection for first order transfer functions PI controller w/ filter term for lead-lag transfer functions λ λ τ kp kc

p −

= 2

p p I

τ λ λ τ τ

2

2 − =

⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ + ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ + = 1 1 1 1 s s kc g

F I c

τ τ λ τ kp kc

p

=

p I

τ τ =

n F

τ τ =

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

10 20 30 40 460 470 480 490 500 510 Time (min) Tb (deg C) 10 20 30 40 50 60 690 700 710 720 730 740 750 760 770 Time (min) Tr (deg C) 10 20 30 40 3.6 3.8 4 4.2 4.4 4.6 4.8 5 5.2 Time (min) Excess Air Ratio 10 20 30 40 5.5 6 6.5 7 7.5 8 Time (min) Methane Flow Rate to Burner (SLPM) lambda = 25 lambda = 75 lambda = 100 lambda = 150 setpoint

10 20 30 40 460 470 480 490 500 510 Time (min) Tb (deg C) 20 40 60 80 680 700 720 740 760 780 Time (min) Tr (deg C) 20 40 60 80 100 4 4.5 5 5.5 6 Time (min) Excess Air Ratio 20 40 60 80 100 4.5 5 5.5 6 6.5 7 Time (min) Methane Flow Rate to Burner (SLPM) lambda = 25 lambda =75 lambda =100 lambda = 150 setpoint

Control by excess air ratio Control by methane flow rate

Reformer Temperature Control

10 20 30 40 50 450 500 550 600 Time (min) Tb (deg C) 10 20 30 40 50 680 700 720 740 760 780 Time (min) Tr (deg C) 10 20 30 40 50 1 2 3 4 5 6 Time (min) Excess Air R atio 10 20 30 40 50 5.5 6 6.5 7 7.5 8 Time (min) Methane Flow Rate to Burner (SLPM) lambda = 400 lambda = 500 lambda = 600 lambda = 700 setpoint 10 20 30 40 50 440 460 480 500 520 540 560 580 Time (min) Tb (deg C ) 10 20 30 40 50 680 700 720 740 760 780 Time (min) Tr (deg C ) 10 20 30 40 50 4 4.5 5 5.5 6 Time (min) Excess Air R atio 10 20 30 40 50 1 2 3 4 5 6 7 Time (min) Methane Flow R ate to Burner (SLPM) lambda = 400 lambda = 500 lambda = 600 lambda = 700 setpoint

Control by excess air ratio Control by methane flow rate

SISO Controllers

y1-u1 y1-u2 y2-u1 y2-u2

⎟ ⎠ ⎞ ⎜ ⎝ ⎛ + ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ + − = 1 500 1 400 1 1 1401 .

11

s s gc

⎟ ⎠ ⎞ ⎜ ⎝ ⎛ + ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ + − = 1 520 1 400 1 1 2063 .

12

s s gc

⎟ ⎠ ⎞ ⎜ ⎝ ⎛ + − = s gc 85 . 706 1 1 0315 .

21

⎟ ⎠ ⎞ ⎜ ⎝ ⎛ + − = s gc 96 . 692 1 1 0383 .

22

Multivariable Control:

RGA Analysis

⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ − − − − − − = Λ

21 12 22 11 22 11 21 12 22 11 12 21 21 12 22 11 21 12 21 12 22 11 22 11

k k k k k k k k k k k k k k k k k k k k k k k k

⎥ ⎦ ⎤ ⎢ ⎣ ⎡ − − − − = 69 . 35 506 . 45 39 . 19 548 . 28 K

⎥ ⎦ ⎤ ⎢ ⎣ ⎡ − − = Λ 463 . 7 463 . 6 463 . 6 463 . 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 ratio Tb burner temperature Tbset burner setpoint

lambda nv Tb Tr

Subsystem6 Step6 Step13 PID PID 6.5856 5

Multivariable Control:

Setpoint changes in both temperatures

50 100 150 200 250 300 440 460 480 500 520 540 Time (min) Tb (deg C) 50 100 150 200 250 300 680 700 720 740 760 780 Time (min) Tr (deg C) 50 100 150 200 250 300 4.6 4.8 5 5.2 5.4 5.6 Time (min) Excess Air Ratio 50 100 150 200 250 300 3 4 5 6 7 Time (min) Methane Flow Rate to Burner (SLPM)

Multivariable Control:

Setpoint changes in only one temperature

50 100 150 200 250 300 350 455 460 465 470 475 480 485 Time (min) Tb (deg C) 100 200 300 400 695 700 705 710 715 720 Time (min) Tr (deg C) 100 200 300 400 1 2 3 4 5 6 Time (min) Excess Air Ratio 100 200 300 400 6 7 8 9 10 11 12 13 Time (min) Methane Flow Rate to Burner (SLPM) 50 100 150 200 250 300 455 460 465 470 475 480 485 Time (min) Tb (deg C) 100 200 300 400 690 700 710 720 730 740 Time (min) Tr (deg C) 100 200 300 400 4 5 6 7 8 9 10 Time (min) Excess Air Ratio 100 200 300 400 1 2 3 4 5 6 7 Time (min) Methane Flow Rate to Burner (SLPM)

Setpoint change in burner temperature Setpoint change in reformer temperature

Disturbance Rejection:

First-order controller differences

⎟ ⎠ ⎞ ⎜ ⎝ ⎛ + − = s gc 710 1 1 0332 .

⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ + = s kc g

I c

τ 1 1

PI controller w/ disturbance rejection for first order transfer functions λ λ τ kp kc

p −

= 2

p p I

τ λ λ τ τ

2

2 − =

⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ + = s kc g

I c

τ 1 1

PI controller w/o disturbance rejection for first order transfer functions λ τ kp kc

p

=

p I

τ τ =

⎟ ⎠ ⎞ ⎜ ⎝ ⎛ + − = s gc 96 . 692 1 1 0383 .

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

50 100 150 200 440 460 480 500 520 540 Time (min) Tb (deg C) 100 200 300 400 680 700 720 740 760 780 800 Time (min) Tr (deg C) 100 200 300 400 1 2 3 4 5 6 Time (min) Excess Air Ratio 100 200 300 400 3 4 5 6 7 8 9 10 Time (min) Methane Flow Rate to Burner (SLPM) w/ dist rejection w/o dist rejection

Directional Sensitivity:

Scaling the ranges

6.5846 13.1712 6.5846 u 2 (methane) 5 10 5 u 1 (excess air) 249.997 950 700.003 450.006 y 2 (reformer) 200 660.524 460.524 260.524 y 1 (burner) ½ Range Max Value Nominal Value Min Value

⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎣ ⎡ = ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ = 997 . 249 1 200 1 ) 2 ( 1 ) 1 ( 1

2 1 2 1

y range y range So ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎣ ⎡ = ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ = 5846 . 6 1 5 1 ) 2 ( 1 ) 1 ( 1

2 1 2 1

u range u range S I

Scaled Output Matrix Scaled Input Matrix

Directional Sensitivity:

Scaled gain matrix

1 − ∗

× × =

I O

S G S G

⎥ ⎦ ⎤ ⎢ ⎣ ⎡ − − − − = 9402 . 9101 . 6385 . 7137 . * G

Directional Sensitivity:

SVD analysis

T

V U G Σ =

∗

T

⎥ ⎦ ⎤ ⎢ ⎣ ⎡ − ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ − − − = ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ − − − − 7133 . 7009 . 7009 . 7133 . 0555 . 6206 . 1 5903 . 8072 . 8072 . 5903 . 9402 . 9101 . 6385 . 7137 .

strongest output direction weakest output direction strongest input direction weakest input direction

Directional Sensitivity:

Scaling back to the process

∗ − ×

= y S y

O 1

⎥ ⎦ ⎤ ⎢ ⎣ ⎡ − − = ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ 7976 . 201 06 . 118

2 1

y y

Strong Direction

⎥ ⎦ ⎤ ⎢ ⎣ ⎡− = ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ 5732 . 147 44 . 161

2 1

y y

Weak Direction

• 1
• 0.5

0.5 1

• 1
• 0.5

0.5 1 u1 u2

• 1

1

• 1.5
• 1
• 0.5

0.5 1 1.5 y1 y2

Input to Output Mapping

Directional Sensitivity:

Changes in the strong direction

50 100 150 200 250 300 440 445 450 455 460 465 Time (min) Tb (deg C) 50 100 150 200 250 300 675 680 685 690 695 700 705 Time (min) Tr (deg C) 50 100 150 200 250 300 4.9 5 5.1 5.2 5.3 Time (min) Excess Air Ratio 50 100 150 200 250 300 6.4 6.6 6.8 7 7.2 7.4 Time (min) Methane Flow Rate to Burner (SLPM)

Setpoint changes were only 10% of the total

Directional Sensitivity:

Changes in the weak direction

50 100 150 200 250 300 440 445 450 455 460 465 470 Time (min) Tb (deg C) 100 200 300 400 680 690 700 710 720 Time (min) Tr (deg C) 100 200 300 400 4 6 8 10 12 Time (min) Excess Air Ratio 100 200 300 400

• 2

2 4 6 8 Time (min) Methane Flow Rate to Burner (SLPM)

Setpoint changes were only 10% of total Negative flow rate

Conclusion

2 Tr 1 Tb Uniform Random Number3 Uniform Random Number2 x' = Ax+Bu y = Cx+Du State-Space1 Product

• C-

Constant4

• C-

Constant3

• C-

Constant2 5 Constant1

• C-

Constant 2 nv 1 lambda