1. 1 Active constraint regions for economically optimal operation of distillation columns Sigurd Skogestad and Magnus G. Jacobsen Department of Chemical Engineering Norwegian University of Science and Tecnology (NTNU) Trondheim, Norway AIChE Annual Meeting, Minneapolis 18 Oct. 2011

2. 2 Question: What should we control (c)? (primary controlled variables y 1 =c) Introductory example: Runner What should we control?

3. 3 –Cost to be minimized, J=T –One degree of freedom (u=power) –What should we control? Optimal operation - Runner Optimal operation of runner

4. 4 Sprinter (100m) 1. Optimal operation of Sprinter, J=T –Active constraint control: Maximum speed (”no thinking required”) Optimal operation - Runner

5. 5 2. Optimal operation of Marathon runner, J=T Unconstrained optimum! Any ”self-optimizing” variable c (to control at constant setpoint)? c 1 = distance to leader of race c 2 = speed c 3 = heart rate c 4 = level of lactate in muscles Optimal operation - Runner Marathon (40 km)

6. 6 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

7. 7 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 = J u ) –In practice, control individual measurements or combinations, c = H y –We have developed a lot of theory for this

8. 8 Distillation columns: What should we control? Always product compositions at spec? NO This presentation: Change in active constraints

9. 9 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= p D D + p B B – p F F – p V V Constraints Purity D: For example x D, impurity · max Purity B: For example, x B, impurity · max Flow constraints: min · D, B, L etc. · max Column capacity (flooding): V · V max, etc. Pressure: 1) p given (d)2) p free: p min · p · p max Feed: 1) F given (d) 2) F free: F · F max Optimal operation: Minimize J with respect to steady-state DOFs (u) value products cost energy (heating+ cooling) cost feed

10. 10 Example column with 41 stages u = [L V] for expected disturbances d = (F, p V )

11. 11 Possible constraint combinations (= 2 n = 2 3 = 8) 1.0* 2.x D 3.x B * 4.V* 5.x D, V 6.x B, V* 7.x D, x B 8.x D, x B, V (infeasible, only 2 DOFs) * Not for this case because x B always optimally active (”Avoid product give away”)

12. 12 Constraint regions as function of d 1 =F and d 2 =p V 3 regions

13. 13 5 regions Only get paid for main component (”gold”)

14. 14 I: L – x D =0.95, V – x B ?Self-optimizing?! xBs = f(p V ) II: L – x D =0.95, V = Vmax III: As in I Control, p D independent of purity

15. 15 I: L – xD?, V – xB?Self-optimizing? II: L – xD?, V = Vmax III:L – xB=0.99, V = Vmax ”active constraints” No simple decentralized structure. OK with MPC

16. 16 2 Distillation columns in series With given F (disturbance): 4 steady-state DOFs (e.g., L and V in each column) DOF = Degree Of Freedom Ref.: M.G. Jacobsen and S. Skogestad (2011) Energy price: p V =0-0.2 $/mol (varies) Cost (J) = - Profit = p F F + p V (V 1 +V 2 ) – p D1 D 1 – p D2 D 2 – p B2 B 2 > 95% B p D2 =2 $/mol F ~ 1.2mol/s p F =1 $/mol < 4 mol/s < 2.4 mol/s > 95% C p B2 =1 $/mol N=41 α AB =1.33 N=41 α BC =1. 5 > 95% A p D1 =1 $/mol 2 5 = 32 possible combinations of the 5 constraints

17. 17 DOF = Degree Of Freedom Ref.: M.G. Jacobsen and S. Skogestad (2011) Energy price: p V =0-0.2 $/mol (varies) Cost (J) = - Profit = p F F + p V (V 1 +V 2 ) – p D1 D 1 – p D2 D 2 – p B2 B 2 > 95% B p D2 =2 $/mol F ~ 1.2mol/s p F =1 $/mol < 4 mol/s < 2.4 mol/s > 95% C p B2 =1 $/mol 1. x B = 95% B Spec. valuable product (B): Always active! Why? “Avoid product give-away” N=41 α AB =1.33 N=41 α BC =1. 5 > 95% A p D1 =1 $/mol 2. Cheap energy: V 1 =4 mol/s, V 2 =2.4 mol/s Max. column capacity constraints active! Why? Overpurify A & C to recover more B 2 Distillation columns in series. Active constraints?

18. 18 Active constraint regions for two distillation columns in series [mol/s] [$/mol] CV = Controlled Variable Energy price BOTTLENECK Higher F infeasible because all 5 constraints reached 8 regions

19. 19 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? Energy price

20. 20 Control of Distillation columns in series Given LC PC Assume low energy prices (pV=0.01 $/mol). How should we control the columns? Red: Basic regulatory loops

21. 21 Control of Distillation columns in series Given LC PC Comment: Should normally stabilize column profiles with temperature control, Should use reflux (L) in this case because boilup (V) may saturate. T1 S and T2 S would then replace L1 and L2 as DOFs…… but leave this out for now.. Red: Basic regulatory loops TC T1 s T2 s T1 T2 Comment

22. 22 Control of Distillation columns in series Given LC PC Red: Basic regulatory loops CC xBxB x BS =95% MAX V1 MAX V2 CONTROL ACTIVE CONSTRAINTS!

23. 23 Control of Distillation columns in series Given LC PC Red: Basic regulatory loops CC xBxB x BS =95% MAX V1 MAX V2 Remains: 1 unconstrained DOF (L1): Use for what? CV=x A ? No!! Optimal x A varies with F Maybe: constant L1? (CV=L1) Better: CV= x A in B1? Self-optimizing? CONTROL ACTIVE CONSTRAINTS!

24. 24 Active constraint regions for two distillation columns in series CV = Controlled Variable 3 2 0 1 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”) Energy price

25. 25 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