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Active constraint regions for economically optimal operation of distillation columns

Sigurd Skogestad and Magnus G. Jacobsen

Department of Chemical EngineeringNorwegian University of Science and Tecnology (NTNU)Trondheim, Norway

AIChE Annual Meeting, Minneapolis18 Oct. 2011

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Question: What should we control (c)? (primary controlled variables y1=c)

• Introductory example: Runner

What should we control?

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– Cost to be minimized, J=T

– One degree of freedom (u=power)

– What should we control?

Optimal operation - Runner

Optimal operation of runner

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Sprinter (100m)

• 1. Optimal operation of Sprinter, J=T– Active constraint control:

• Maximum speed (”no thinking required”)

Optimal operation - Runner

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• 2. Optimal operation of Marathon runner, J=T• Unconstrained optimum!• Any ”self-optimizing” variable c (to control at

constant setpoint)?• c1 = distance to leader of race

• c2 = speed

• c3 = heart rate

• c4 = level of lactate in muscles

Optimal operation - Runner

Marathon (40 km)

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Conclusion Marathon runner

c = heart rate

select one measurement

• Simple and robust implementation• Disturbances are indirectly handled by keeping a constant heart rate• May have infrequent adjustment of setpoint (heart rate)

Optimal operation - Runner

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Conclusion: What should we control (c)? (primary controlled variables)

1. Control active constraints!

2. Unconstrained variables: Control self-optimizing variables!

– The ideal self-optimizing variable c is the gradient (c = J/ u = Ju)

– In practice, control individual measurements or combinations, c = H y– We have developed a lot of theory for this

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Distillation columns: What should we control?

• Always product compositions at spec? NO

• This presentation: Change in active constraints

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Optimal operation distillation column

• Distillation at steady state with given p and F: N=2 DOFs, e.g. L and V

• Cost to be minimized (economics)

J = - P where P= pD D + pB B – pF F – pV V

• ConstraintsPurity D: For example xD, impurity · max

Purity B: For example, xB, impurity · max

Flow constraints: min · D, B, L etc. · max

Column capacity (flooding): V · Vmax, etc.

Pressure: 1) p given (d) 2) p free: pmin · p · pmax

Feed: 1) F given (d) 2) F free: F · Fmax

• Optimal operation: Minimize J with respect to steady-state DOFs (u)

value products

cost energy (heating+ cooling)

cost feed

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Example column with 41 stages

u = [L V]

for expected disturbances d = (F, pV)

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Possible constraint combinations (= 2n = 23 = 8)

1. 0*

2. xD

3. xB*

4. V*

5. xD, V

6. xB, V*

7. xD, xB

8. xD, xB, V (infeasible, only 2 DOFs)

*Not for this case because xB always optimally active (”Avoid product give away”)

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Constraint regions as function of d1=F and d2=pV

3 regions

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5 regions

Only get paid for main component (”gold”)

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I: L – xD=0.95, V – xB? Self-optimizing?! xBs = f(pV)II: L – xD=0.95, V = VmaxIII: As in I

Control, pD independent of purity

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I: L – xD?, V – xB? Self-optimizing? II: L – xD?, V = VmaxIII: L – xB=0.99, V = Vmax ”active constraints”

No simple decentralized structure. OK with MPC

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2 Distillation columns in seriesWith given F (disturbance): 4 steady-state DOFs (e.g., L and V in each column)

DOF = Degree Of FreedomRef.: M.G. Jacobsen and S. Skogestad (2011)

Energy price: pV=0-0.2 $/mol (varies)Cost (J) = - Profit = pF F + pV(V1+V2) – pD1D1 – pD2D2 – pB2B2

> 95% BpD2=2 $/mol

F ~ 1.2mol/spF=1 $/mol < 4 mol/s < 2.4 mol/s

> 95% CpB2=1 $/mol

N=41αAB=1.33

N=41αBC=1.5

> 95% ApD1=1 $/mol

25 = 32 possible combinations of the 5 constraints

17 DOF = Degree Of FreedomRef.: M.G. Jacobsen and S. Skogestad (2011)

Energy price: pV=0-0.2 $/mol (varies)Cost (J) = - Profit = pF F + pV(V1+V2) – pD1D1 – pD2D2 – pB2B2

> 95% BpD2=2 $/mol

F ~ 1.2mol/spF=1 $/mol < 4 mol/s < 2.4 mol/s

> 95% CpB2=1 $/mol

1. xB = 95% BSpec. valuable product (B): Always active!Why? “Avoid product give-away”

N=41αAB=1.33

N=41αBC=1.5

> 95% ApD1=1 $/mol

2. Cheap energy: V1=4 mol/s, V2=2.4 mol/sMax. column capacity constraints active!Why? Overpurify A & C to recover more B

2 Distillation columns in series. Active constraints?

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Active constraint regions for two distillation columns in series

[mol/s]

[$/mol]

CV = Controlled Variable

Energyprice

BOTTLENECKHigher F infeasible because all 5 constraints reached

8 regions

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Active constraint regions for two distillation columns in series

[mol/s]

[$/mol]

CV = Controlled Variable

Assume low energy prices (pV=0.01 $/mol).How should we control the columns?

Energyprice

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Control of Distillation columns in series

Given

LC LC

LC LC

PCPC

Assume low energy prices (pV=0.01 $/mol).How should we control the columns? Red: Basic regulatory loops

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Control of Distillation columns in series

Given

LC LC

LC LC

PCPC

Red: Basic regulatory loops

CC

xB

xBS=95%

MAX V1 MAX V2

CONTROL ACTIVE CONSTRAINTS!

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Control of Distillation columns in series

Given

LC LC

LC LC

PCPC

Red: Basic regulatory loops

CC

xB

xBS=95%

MAX V1 MAX V2

Remains: 1 unconstrained DOF (L1):Use for what? CV=xA? •No!! Optimal xA varies with F •Maybe: constant L1? (CV=L1)•Better: CV= xA in B1? Self-optimizing?

CONTROL ACTIVE CONSTRAINTS!

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Active constraint regions for two distillation columns in series

CV = Controlled Variable

3 2

01

1

0

2

[mol/s]

[$/mol]

1

Cheap energy: 1 remaining unconstrained DOF (L1) -> Need to find 1 additional CVs (“self-optimizing”)

More expensive energy: 3 remaining unconstrained DOFs -> Need to find 3 additional CVs (“self-optimizing”)

Energyprice

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Conclusion

• Generate constraint regions by offline simulation for expected important disturbances– Time consuming - so focus on important disturbance

range

• Implementation / control– Control active constraints!

– Switching between these usually easy

– Less obvious what to select as ”self-optimizing” CVs for remaining unconstrained degrees of freedom