G.A. Bunin a , Z. Wuillemin b , G. François a , S. Diethelm b , A. Nakajo b , and D. Bonvin a

Model-Predictive Control (MPC) of an Experimental SOFC Stack:

A Robust and Simple Controller for Safer Load Tracking

G.A. Bunina, Z. Wuilleminb, G. Françoisa,S. Diethelmb, A. Nakajob, and D. Bonvina

a Laboratoire d’Automatique, EPFLb Laboratoire d’Énergétique Industrielle, EPFL

The Goal of This Talk

To demonstrate that the transient SOFC control problem can be handled very simply, yet robustly, while requiring little control knowledge and only a very basic model of the process.

The Goal of This Talk

To demonstrate that the transient SOFC control problem can be handled very simply, yet robustly, while requiring little control knowledge and only a very basic model of the process.

Outline of the Talk The System

Basic MPC Theory

Our “HC-MPC” Formulation

Experimental Validation

Concluding Remarks

The System Inputs nH2: H2 flux nO2: O2 flux I: current

Safety Constraints Ucell: cell potential ν: fuel utilization λ: air excess ratio

Performance πel: power demand η: electrical efficiency

FuelAir79% N2 21% O2

Power

Current

97% H2 3% H2O

Furnace

6-cellSOFCStack

2 2 2

Reaction:12

H O H O

nH2: H2 flux nO2: O2 flux I: current Ucell: potential ν: fuel utilization λ: air ratioπel: power demand η: efficiency

Control Objective

Track the specified power demand while maximizing the efficiency and honoring the safety constraints.


Basic MPC Theory



Concluding Remarks

nH2: H2 flux nO2: O2 flux I: current Ucell: potential ν: fuel utilization λ: air ratioπel: power demand η: efficiency

Basic MPC Principles

nH2: H2 flux nO2: O2 flux I: current Ucell: potential ν: fuel utilization λ: air ratioπel: power demand η: efficiency p: pred. horizon m: cont. horizon B: dyn. matrix

πel (old)

πel (new)

t0

I = 0 A

I = 30 A

t0 Δt

a1a2

a3a4 a5 a6 a7 a8 ap

t0+pΔt

B = f(a1,…,ap)

Basic MPC Principles


πel (old)

πel (new)

t0

I = 0 A

I = 30 A

t0 Δt

t0+pΔt

B = f(a1,…,ap)

t0+mΔt

implement! (…then do it all again)

πel = πel ,0 + BΔI + d

πel,0

d

MPC with Optimization MPC objective function

Constraints: Ucell ≥ 0.79V, ν ≤ 0.75, 4 ≤ λ ≤ 7


2 2

2 22 2 2 2( ) ( ) ( )el cell H OU n n IJ w w w w w w

2 2

newel el cell H Oπ π U .79 ν .75 Δn Δn ΔI

QP Transformation

2

2

2

T T

[ ]

, 2

,

,

1min 2

NmL s.t.: 3.14 1,...,min cm

4 2 7 1,...,

0A 30A

H i

O i

H i

i

n i p

ni p

n

I

H O2 2Δu Δn Δn ΔIΔu HΔu c Δu

1,...,i p

MPC with Optimization MPC objective function

Constraints: Ucell ≥ 0.79V, ν ≤ 0.75, 4 ≤ λ ≤ 7


2 2

2 22 2 2 2( ) ( ) ( )el cell H OU n n IJ w w w w w w

2 2

newel el cell H Oπ π U .79 ν .75 Δn Δn ΔI

πel (low)

πel (high)

efficiency limited by ν

efficiency limited by Ucell

0cellUw

0w πel (mid)


Basic MPC Theory



Concluding Remarks


The HC-MPC Formulation HC = “Hard Constraint”


nH20

InH2 = 3.14mL

nH2 = 10.0mL

I = 30A

Ucell = 0.79Vν = 0.75



nH20

InH2 = 3.14mL

nH2 = 10.0mL

I = 30A

Ucell = 0.79Vν = 0.75



nH20

InH2 = 3.14mL

nH2 = 10.0mL

I = 30A

Ucell = 0.79Vν = 0.75



nH20

InH2 = 3.14mL

nH2 = 10.0mL

I = 30A

Ucell = 0.79Vν = 0.75



nH20

InH2 = 3.14mL

nH2 = 10.0mL

I = 30A

Ucell = 0.79Vν = 0.75



nH20

InH2 = 3.14mL

nH2 = 10.0mL

I = 30A

Ucell = 0.79Vν = 0.75



nH20

InH2 = 3.14mL

nH2 = 10.0mL

I = 30A

Ucell = 0.79Vν = 0.75



nH20

InH2 = 3.14mL

nH2 = 10.0mL

I = 30A

Ucell = 0.79Vν = 0.75



nH20

InH2 = 3.14mL

nH2 = 10.0mL

I = 30A

Ucell = 0.79Vν = 0.75



nH20

InH2 = 3.14mL

nH2 = 10.0mL

I = 30A

Ucell = 0.79Vν = 0.75



nH20

InH2 = 3.14mL

nH2 = 10.0mL

I = 30A

Ucell = 0.79Vν = 0.75



nH20

InH2 = 3.14mL

nH2 = 10.0mL

I = 30A

Ucell = 0.79Vν = 0.75



nH20

InH2 = 3.14mL

nH2 = 10.0mL

I = 30A

Ucell = 0.79Vν = 0.75

The HC-MPC Formulation


4

6

8

10

510

152025

3035

0

5

10

15

20

25

30

nO2nH2

I

λ = 4λ =

7

ν = 0.75

Ucell = 0.79V



4

6

8

10

510

152025

3035

0

5

10

15

20

25

30

nO2nH2

I

λ = 4λ =

7

ν = 0.75

Ucell = 0.79V



4

6

8

10

510

152025

3035

0

5

10

15

20

25

30

nO2nH2

I

λ = 4λ =

7

ν = 0.75



4

6

8

10

510

152025

3035

0

5

10

15

20

25

30

nO2nH2

I

λ = 4λ =

7

ν = 0.75



4

6

8

10

510

152025

3035

0

5

10

15

20

25

30

nO2nH2

I

λ = 4λ =

7

ν = 0.75



4

6

8

10

510

152025

3035

0

5

10

15

20

25

30

nO2nH2

I

λ = 4λ =

7

ν = 0.75



4

6

8

10

510

152025

3035

0

5

10

15

20

25

30

nO2nH2

I

λ = 4λ =

7

ν = 0.75



4

6

8

10

510

152025

3035

0

5

10

15

20

25

30

nO2nH2

I

λ = 4λ =

7

ν = 0.75

Ucell = 0.79V

Side-by-Side Standard MPC Issues

Weight Tuning Only partially intuitive Requires a good model Need validation

Active Constraint? Must know πel (mid) Degradation!

πel (mid) changes

Violations Norms are directionless Constraints are “soft”


HC-MPC Solutions Weight Tuning

Completely intuitive Practically no tuning Minimal validation

Active Constraint? ν kept active Degradation?

Doesn’t matter Violations

Inequalities have direction Constraints are “hard”

Intuitive Weight Scheme Sufficient to normalize

weights into 3 categories High Priority (w = 10)

e.g.: power demand Standard Priority (w = 1.0)

e.g.: efficiency (tracking active constraint)

Low Priority (w = 0.1) e.g.: penalties on input

moves (controller behavior)


Bias Filter α

1 (1 ): convergence

criterion (0 to 1): sampling time

: time to converge

c

tt

c

cc

tt

Side-by-Side Standard MPC Issues

Weight Tuning Only partially intuitive Requires a good model Need validation

Active Constraint? Must know πel (mid) Degradation!

πel (mid) changes

Violations Norms are directionless Constraints are “soft”


HC-MPC Solutions Weight Tuning

Completely intuitive Practically no tuning Minimal validation

Active Constraint? ν kept active Degradation?

Doesn’t matter Violations

Inequalities have direction Constraints are “hard”


Basic MPC Theory



Concluding Remarks


0 10 20 300.25

0.3

0.35

0.4

0.45

Time (min)

el

(W/c

m2 )

0 10 20 3015

20

25

30

Time (min)

I (A

)

0 10 20 300.6

0.65

0.7

0.75

0.8

Time (min)

0 10 20 300

5

10

15

Time (min)

Flux

es (N

mL/

min

/cm

2 )

0 10 20 3035

40

45

50

55

Time (min)

0 10 20 300.75

0.8

0.85

Time (min)

Uce

ll (V)

H2

air



η ≈ 42%

η ≈ 42%

η ≈ 38%

0 10 20 300.25

0.3

0.35

0.4

0.45

Time (min)

el

(W/c

m2 )

0 10 20 3015

20

25

30

Time (min)

I (A

)

0 10 20 300.6

0.65

0.7

0.75

0.8

Time (min)

0 10 20 300

5

10

15

Time (min)

Flux

es (N

mL/

min

/cm

2 )

0 10 20 3035

40

45

50

55

Time (min)

0 10 20 300.75

0.8

0.85

Time (min)

Uce

ll (V)

H2

air

Standard MPC HC-MPC

0 10 20 300.6

0.62

0.64

0.66

0.68

0.7

0.72

0.74

0.76

0.78

0.8

Time (min)

0 10 20 300.6

0.62

0.64

0.66

0.68

0.7

0.72

0.74

0.76

0.78

0.8

Time (min)

standard

HC

0 10 20 300.25

0.3

0.35

0.4

0.45

Time (min)

el

(W/c

m2 )

0 10 20 3015

20

25

30

Time (min)

I (A

)

0 10 20 300.6

0.65

0.7

0.75

0.8

Time (min)

0 10 20 300

5

10

15

Time (min)

Flux

es (N

mL/

min

/cm

2 )

0 10 20 3035

40

45

50

55

Time (min)

0 10 20 300.75

0.8

0.85

Time (min)

Uce

ll (V)

H2

air


η ≈ 42%

η ≈ 42%

η ≈ 38%

0 10 20 300.25

0.3

0.35

0.4

0.45

Time (min)

el

(W/c

m2 )

0 10 20 3015

20

25

30

Time (min)

I (A

)

0 10 20 300.6

0.65

0.7

0.75

0.8

Time (min)

0 10 20 300

5

10

15

Time (min)

Flux

es (N

mL/

min

/cm

2 )

0 10 20 3035

40

45

50

55

Time (min)

0 10 20 300.75

0.8

0.85

Time (min)

Uce

ll (V)

H2

air

Standard MPC HC-MPC

0 10 20 300.6

0.62

0.64

0.66

0.68

0.7

0.72

0.74

0.76

0.78

0.8

Time (min)

0 10 20 300.6

0.62

0.64

0.66

0.68

0.7

0.72

0.74

0.76

0.78

0.8

Time (min)

0 10 20 300.75

0.76

0.77

0.78

0.79

0.8

0.81

0.82

0.83

0.84

0.85

Time (min)

U cell (V

)

0 10 20 300.75

0.76

0.77

0.78

0.79

0.8

0.81

0.82

0.83

0.84

0.85

Time (min)

U cell (V

)

input regionexpansion

input regioncontraction

standard

HC


Basic MPC Theory



Concluding Remarks


Concluding Remarks The proposed HC-MPC is very effective as it:

does NOT require a good model only four experimental step responses were used here

has only one decision variable for tuning which is very intuitive

minimizes oscillatory behavior and overshoot Potential Applications

The above should hold for more complex systems + gas turbine + steam reforming + heat-load following

Thank You!

Questions?

Extra Slides



0 5 10 15 20 25 30 35 40 45 50 55 600.29

0.3

0.31

0.32

0.33

0.34

0.35

0.36

Time (min)

el(W

/cm2 )

0 5 10 15 20 25 30 35 40 45 50 55 600.6

0.62

0.64

0.66

0.68

0.7

0.72

0.74

0.76

0.78

0.8

Time (min)

0 5 10 15 20 25 30 35 40 45 50 55 600.75

0.76

0.77

0.78

0.79

0.8

0.81

0.82

0.83

0.84

0.85

Time (min)

U cell (V

)

G.A. Bunin a , Z. Wuillemin b , G. François a , S. Diethelm b , A. Nakajo b , and D. Bonvin a

Documents

o2 fluxi

current ucell

h2 flux no2

fuel utilization

horizon b

air ratioel

specified power demand

efficiency p