1. 1 From process control to business control: A systematic approach for CV-selection Sigurd Skogestad Department of Chemical Engineering Norwegian University of Science and Technology (NTNU) Trondheim PIC-konferens, Stockholm, 29. mai 2013 CV = Controlled variable Porsgrunn, 12 June 2013

2. 2 Sigurd Skogestad 1955: Born in Flekkefjord, Norway 1978: MS (Siv.ing.) in chemical engineering at NTNU 1979-1983: Worked at Norsk Hydro co. (process simulation) 1987: PhD from Caltech (supervisor: Manfred Morari) 1987-present: Professor of chemical engineering at NTNU 1999-2009: Head of Department 170 journal publications Book: Multivariable Feedback Control (Wiley 1996; 2005) –1989: Ted Peterson Best Paper Award by the CAST division of AIChE –1990: George S. Axelby Outstanding Paper Award by the Control System Society of IEEE –1992: O. Hugo Schuck Best Paper Award by the American Automatic Control Council –2006: Best paper award for paper published in 2004 in Computers and chemical engineering. –2011: Process Automation Hall of Fame (US)

3. 3 Optimal operation of systems (outline) «System» = Chemical process plant, Airplane, Business, …. General approach 1.Classify variables (MV=u, DV=d, Measurements y) 2.Obtain model (dynamic or steady state) 3.Define optimal operation: Cost J, constraints, disturbances 4.Find optimal operation: Solve optimization problem 5.Implement optimal operation: What should we control (CV=c=Hy) ? Need one CV for each MV Applications –Runner –KPI’s –Process control –...

4. 4 Optimal operation of a given system 1. Classify variables –Manipulated variables (MVs) = Degrees of freedom (inputs): u –Disturbance variables (DVs) = “inputs” outside our control: d –Measured variables: y (information about the system) –States = Variables that define initial state: x Question: How should we set the MVs (inputs u) 3. Quantitative approach: Define scalar cost J + define constraints + define expected disturbances System u d x y 2. Model of system (typical): dx/dt = f(x,u,d) y = f y (x,u,d)

5. 5 3. Define optimal operation (economics) What are we going to use our degrees of freedom u (MVs) for? 1.Define scalar cost function J(u,x,d) 2.Identify constraints –min. and max. flows –Product specifications –Safety limitations –Other operational constraints J = cost feed + cost energy – value products [$/s]

6. 6 Optimal operation distillation column Distillation at steady state with given p and F: 2 DOFs, e.g. L and V (u) 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 (u): p min · p · p max Feed: 1) F given (d) 2) F free (u): F · F max Optimal operation: Minimize J with respect to steady-state DOFs (u) value products cost energy (heating+ cooling) cost feed

7. 7 4. Find optimal operation: Solve optimization problem to find u opt (d) Find optimal constraint regions (as a function of d) Optimize operation with respect to u for given disturbance d (usually steady-state): min u J(u,x,d) subject to: Model equations: f(u,x,d) = 0 Constraints: g(u,x,d) < 0

8. 8 Energy price Example: optimal constraint regions (as a function of d)

9. 9 5. Implement optimal operation Our task as control engineers! Theoretical (centralized): Reoptimize continouosly Practical (hierarchical): Find one CV for each MV

10. 10 Centralized implementation: Optimizing control (theoretically best) Estimate present state + disturbances d from measurements y and recompute u opt (d) Problem: COMPLICATED! Requires detailed model and description of uncertainty y Implementation of optimal operation

11. 11 Hierarchical implementation (practically best) Direktør Prosessingeniør Operatør Logikk / velgere / operatør PID-regulator u = valves Implementation of optimal operation

12. 12 PID RTO MPC Implementation of optimal operation Hierarchical implementation (practically best) u = valves

13. 13 What should we control? y 1 = c =Hy? (economics) y 2 = H 2 y (stabilization)

14. 14 Self-optimizing Control Self-optimizing control is when acceptable operation can be achieved using constant set points (c s ) for the controlled variables c (without re-optimizing when disturbances occur). c=c s

15. 15 Implementation of optimal operation Idea: Replace optimization by setpoint control Optimal solution is usually at constraints, that is, most of the degrees of freedom u 0 are used to satisfy “active constraints”, g(u 0,d) = 0 CONTROL ACTIVE CONSTRAINTS! –Implementation of active constraints is usually simple. WHAT MORE SHOULD WE CONTROL? –Find variables c for remaining unconstrained degrees of freedom u. u cost J

16. 16 –Cost to be minimized, J=T –One degree of freedom (u=power) –What should we control? Optimal operation - Runner Optimal operation of runner

17. 17 Self-optimizing control: Sprinter (100m) 1. Optimal operation of Sprinter, J=T –Active constraint control: Maximum speed (”no thinking required”) Optimal operation - Runner

18. 18 2. Optimal operation of Marathon runner, J=T Optimal operation - Runner Self-optimizing control: Marathon (40 km)

19. 19 Solution 1 Marathon: Optimizing control Even getting a reasonable model requires > 10 PhD’s … and the model has to be fitted to each individual…. Clearly impractical! Optimal operation - Runner

20. 20 Solution 2 Marathon – Feedback (Self-optimizing control) –What should we control? Optimal operation - Runner

21. 21 Optimal operation of Marathon runner, J=T 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 Self-optimizing control: Marathon (40 km)

22. 22 Ideal “self-optimizing” variable The ideal self-optimizing variable c is the gradient: c =  J/  u = J u –Keep gradient at zero for all disturbances (c = J u =0) –Problem: Usually no measurement of gradient, that is, cannot write J u =Hy Unconstrained degrees of freedom u cost J J u =0 JuJu

23. 23 Unconstrained variables H measurement noise steady-state control error disturbance controlled variable / selection Ideal: c = J u In practise: c = H y

24. 24 Candidate controlled variables c for self-optimizing control Intuitive: 1.The optimal value of c should be insensitive to disturbances 2.Optimum should be flat ( ->insensitive to implementation error). Equivalently: Value of c should be sensitive to degrees of freedom u. Unconstrained optimum BADGood

25. 25 Nullspace method No measurement noise (n y =0) CV=Measurement combination Ref: Alstad & Skogestad, 2007

26. 26 Example. Nullspace Method for Marathon runner u = power, d = slope [degrees] y 1 = hr [beat/min], y 2 = v [m/s] F = dy opt /dd = [0.25 -0.2]’ H = [h 1 h 2 ]] HF = 0 -> h 1 f 1 + h 2 f 2 = 0.25 h 1 – 0.2 h 2 = 0 Choose h 1 = 1 -> h 2 = 0.25/0.2 = 1.25 Conclusion: c = hr + 1.25 v Control c = constant -> hr increases when v decreases (OK uphill!) CV=Measurement combination

27. 27 Nullspace method (HF=0) gives J u =0 Proof. Appendix B in:Jäschke and Skogestad, ”NCO tracking and self-optimizing control in the context of real-time optimization”, Journal of Process Control, 1407-1416 (2011).

28. 28 Ref: Halvorsen et al. I&ECR, 2003 Alstad et al,, JPC, 2009 ”Exact local method” (with measurement noise) u J Loss Controlled variables, c s = constant + + + + + - K H y cmcm u d CV=Measurement combination Analytic solution for the case of “full” H With measurement noise)

29. 29 Conclusion Marathon runner c = heart rate Simplest: 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

30. 30 Further examples Central bank. J = welfare. c=inflation rate (2.5%) Cake baking. J = nice taste, c = Temperature (200C) Business, J = profit. c = ”Key performance indicator (KPI), e.g. –Response time to order –Energy consumption pr. kg or unit –Number of employees –Research spending Optimal values obtained by ”benchmarking” Investment (portofolio management). J = profit. c = Fraction of investment in shares (50%) Biological systems: –”Self-optimizing” controlled variables c have been found by natural selection –Need to do ”reverse engineering” : Find the controlled variables used in nature From this identify what overall objective J the biological system has been attempting to optimize

31. 31 Sigurd’s rules for CV selection 1.Always control active constraints! (almost always) 2.Purity constraint on expensive product always active (no overpurification): (a) "Avoid product give away" (e.g., sell water as expensive product) (b) Save energy (costs energy to overpurify) 3.Unconstrained optimum: NEVER try to control a variable that reaches max or min at the optimum –For example, never try to control directly the cost J Will give infeasibility

32. 32 Sigurd’s pairing rule: “Pair MV that may (optimally) saturate with CV that may be given up” Reason: Minimizes need for reassigning loops Important: Always feasible (and optimal) to give up a CV when optimal MV saturation occurs. –Proof (DOF analysis): When one MV disappears (saturates), then we have one less optimal CV.

33. 33 Example: Optimal blending of gasoline Stream 1 Stream 2 Stream 3 Stream 4 Product 1 kg/s Stream 199 octane0 % benzenep 1 = (0.1 + m 1 ) $/kg Stream 2105 octane0 % benzenep 2 = 0.200 $/kg Stream 395 → 97 octane 0 % benzenep 3 = 0.120 $/kg Stream 499 octane2 % benzenep 4 = 0.185 $/kg Product> 98 octane< 1 % benzene Disturbance m 1 = ? ( · 0.4) m 2 = ? m 3 = ? m 4 = ?

34. 34 Optimal solution Degrees of freedom u = (m  m 2 m 3 m 4 ) T Optimization problem:Minimize J =  i p i m i = (0.1 + m 1 ) m 1 + 0.2 m 2 + 0.12 m 3 + 0.185 m 4 subject to m 1 + m 2 + m 3 + m 4 = 1 m 1 ¸ 0; m 2 ¸ 0; m 3 ¸ 0; m 4 ¸ 0 m 1 · 0.4 99 m 1 + 105 m 2 + 95 m 3 + 99 m 4 ¸ 98 (octane constraint) 2 m 4 · 1 (benzene constraint) Nominal optimal solution (d * = 95): u opt = (0.26 0.196 0.544 0) T ) J opt =0.13724 $ Optimal solution with d=octane stream 3=97: u opt = (0.20 0.075 0.725 0) T ) J opt =0.13724 $ 3 active constraints ) 1 unconstrained degree of freedom

35. 35 Implementation of optimal solution Available ”measurements”: y = (m 1 m 2 m 3 m 4 ) T Control active constraints: –Keep m 4 = 0 –Adjust one (or more) flow such that m 1 +m 2 +m 3 +m 4 = 1 –Adjust one (or more) flow such that product octane = 98 Remaining unconstrained degree of freedom 1.c=m 1 is constant at 0.126 ) Loss = 0.00036 $ 2.c=m 2 is constant at 0.196 ) Infeasible (cannot satisfy octane = 98) 3.c=m 3 is constant at 0.544 ) Loss = 0.00582 $ Optimal combination of measurements c = h 1 m 1 + h 2 m 2 + h 3 m a From optimization:  m opt = F  d where sensitivity matrix F = (-0.03 -0.06 0.09) T Requirement: HF = 0 ) -0.03 h 1 – 0.06 h 2 + 0.09 h 3 = 0 This has infinite number of solutions (since we have 3 measurements and only ned 2): c = m 1 – 0.5 m 2 is constant at 0.162 ) Loss = 0 c = 3 m 1 + m 3 is constant at 1.32 ) Loss = 0 c = 1.5 m 2 + m 3 is constant at 0.83 ) Loss = 0 Easily implemented in control system

36. 36 Example of practical implementation of optimal blending Selected ”self-optimizing” variable: c = m 1 – 0.5 m 2 Changes in feed octane (stream 3) detected by octane controller (OC) Implementation is optimal provided active constraints do not change Price changes can be included as corrections on setpoint c s FC OC m tot.s = 1 kg/s m tot m3m3 m 4 = 0 kg/s Octane s = 98 Octane m2m2 Stream 2 Stream 1 Stream 3 Stream 4 c s = 0.162 0.5 m 1 = c s + 0.5 m 2 Octane varies

37. 37 Conclusion. Optimal operation of systems «System» = Chemical process plant, Airplane, Business, …. General approach 1.Classify variables (MV=u, DV=d,) 2.Obtain model (dynamic or steady state) 3.Define optimal operation: Cost J, constraints, disturbances 4.Find optimal operation: Solve optimization problem Identify active constraint regions 5.Implement optimal operation: What should we control (CV=c) Need one CV (KPI) for each MV 1.Control active constraints 2.Control «self-optimizing» variables, c=Hy ≈ J u