1. 1 Self-optimizing control Theory

2. 2 Step S3: Implementation of optimal operation Optimal operation for given d * : min u J(u,x,d) subject to: Model equations: f(u,x,d) = 0 Operational constraints: g(u,x,d) < 0 → u opt (d * ) Problem: Usally cannot keep u opt constant because disturbances d change How should we adjust the degrees of freedom (u)? What should we control?

3. 3 y “Optimizing Control”

4. 4 What should we control? (What is c? What is H?) H y C = Hy H: Nonquare matrix Usually selection matrix of 0’s and some 1’s (measurement selection) Can also be full matrix (measurement combinations) “Self-Optimizing Control”: Separate optimization and control Self-optimizing control: Constant setpoints give acceptable loss

5. 5 Definition of self-optimizing control “Self-optimizing control is when we achieve acceptable loss (in comparison with truly optimal operation) with constant setpoint values for the controlled variables (without the need to reoptimize when disturbances occur).” Reference: S. Skogestad, “Plantwide control: The search for the self-optimizing control structure'', Journal of Process Control, 10, 487-507 (2000). Acceptable loss ) self-optimizing control

6. 6 Remarks “self-optimizing control” 1. Old idea (Morari et al., 1980): “We want to find a function c of the process variables which when held constant, leads automatically to the optimal adjustments of the manipulated variables, and with it, the optimal operating conditions.” 2. “Self-optimizing control” = acceptable steady-state behavior with constant CVs. “Self-regulation” = acceptable dynamic behavior with constant MVs. 3. Choice of good c (CV) is always important, even with RTO layer included 3. For unconstrained DOFs: Ideal self-optimizing variable is gradient, J u =  J/  u –Keep gradient at zero for all disturbances (c = J u =0) –Problem: no measurement of gradient

7. 7 Marathon runner c = heart rate select one measurement CV = heart rate is good “self-optimizing” variable Simple and robust implementation Disturbances are indirectly handled by keeping a constant heart rate May have infrequent adjustment of setpoint (c s ) Optimal operation - Runner c=heart rate J=T c opt

8. 8 Example: Cake Baking Objective: Nice tasting cake with good texture u 1 = Heat input u 2 = Final time d 1 = oven specifications d 2 = oven door opening d 3 = ambient temperature d 4 = initial temperature y 1 = oven temperature y 2 = cake temperature y 3 = cake color Measurements Disturbances Degrees of Freedom

9. 9 Central bank. J = welfare. u = interest rate. c=inflation rate (2.5%) Cake baking. J = nice taste, u = heat input. 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 possibly identify what overall objective J the biological system has been attempting to optimize Further examples self-optimizing control Define optimal operation (J) and look for ”magic” variable (c) which when kept constant gives acceptable loss (self- optimizing control) Unconstrained degrees of freedom

10. 10 The ideal “self-optimizing” variable is the gradient, J u c =  J/  u = J u –Keep gradient at zero for all disturbances (c = J u =0) –Problem: Usually no measurement of gradient Unconstrained degrees of freedom u cost J J u =0 J u <0 u opt JuJu 0

11. 11 Unconstrained optimum: NEVER try to control a variable that reaches max or min at the optimum –In particular, never try to control directly the cost J –Assume we want to minimize J (e.g., J = V = energy) - and we make the stupid choice os selecting CV = V = J Then setting J < Jmin: Gives infeasible operation (cannot meet constraints) and setting J > Jmin: Forces us to be nonoptimal (two steady states: may require strange operation) u J J min J>J min J<J min ?

12. 12 Parallel heat exchangers What should we control? T0 m [kg/s] Th2, UA2 T2 T1 T α 1-α 1 Degree of freedom (u=split ® ) Objective: maximize J=T (= maximize total heat transfer Q) What should we control? -If active constraint : Control it! -Here: No active constraint. Control T? NO! Optimal: Control J u =0 Th1, UA1

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15. 15 Controlling the gradient to zero, J u =0 A. “Standard optimization (RTO) 1.Estimate disturbances and present state 2.Use model to find optimal operation (usually numerically) 3.Implement optimal point. Alternatives: –Optimal inputs –Setpoints c s to control layer (most common) –Control computed gradient (which has setpoint=0) Complex C. Local loss method (local self-optimizing control) 1.Find linear expression for gradient, J u = c= Hy 2.Control c to zero Simple but assumes optimal nominal point Offline/ Online 1.Find expression for gradient (nonlinear analytic) 2.Eliminate unmeasured variables, J u =f(y) (analytic) 3.Control gradient to zero: –Control layer adjust u until J u = f(y)=0 Not generally possible, especially step 2 B. Analytic elimination (Jäschke method) All these methods: Optimal only for assumed disturbances Unconstrained degrees of freedom

16. 16 Controlled variable, c = Hy y: all measured variables H: Nonsquare n c *n y matrix –H selection matrix: Control single measurements –H full matrix: Control measurement combinations n c =dim(c) = dim (u) Methods for finding H (c): 1.Single variables: Intuition 2.Single variables: Large scaled gain 3.Combinations: Nullspace method (simple, but need n y =n u +n d ) 4.Combinations: Exact local method (general) Afterwards: Check using brute force evaluation Unconstrained degrees of freedom

17. 17 Linear measurement combinations, c = Hy c=Hy is approximate gradient J u Two approaches 1.Nullspace method (HF=0): Simple but has limitations –Need many measurements if many disturbances (ny = nu + nd) –Does not handle measurement noise 2.Generalization: Exact local method + Works for any measurement set y + Handles measurement error / noise + - Must assume that nominal point is optimal Unconstrained degrees of freedom

18. 18 Unconstrained variables H measurement noise steady-state control error disturbance controlled variable / selection Ideal: c = J u In practise: c = H y c J c opt

19. 19 WHAT ARE GOOD “SELF- OPTIMIZING” VARIABLES? Intuition: “Dominant variables” (Shinnar) Is there any systematic procedure? A. Sensitive variables: “Max. gain rule” (Gain= Minimum singular value) B. “Brute force” loss evaluation C. Optimal linear combination of measurements, c = Hy Unconstrained variables

20. 20 «Brute force» analysis: What to control? Define optimal operation: Minimize cost function J Each candidate variable c: With constant setpoints c s compute loss L for expected disturbances d and implementation errors n Select variable c with smallest loss

21. 21 Constant setpoint policy: Loss for disturbances Acceptable loss ) self-optimizing control

22. 22 Good candidate controlled variables c (for self-optimizing control) 1.The optimal value of c should be insensitive to disturbances 2.c should be easy to measure and control 3.The value of c should be sensitive to changes in the degrees of freedom Proof: Follows

23. 23 Optimal operation Cost J Controlled variable c c opt J opt Unconstrained optimum

24. 24 Optimal operation Cost J Controlled variable c c opt J opt Two problems: 1. Optimum moves because of disturbances d: c opt (d) 2. Implementation error, c = c opt + n d n Unconstrained optimum

25. 25 Candidate controlled variables c for self-optimizing control Intuitive: 1.The optimal value of c should be insensitive to disturbances (avoid problem 1): 2.Optimum should be flat (avoid problem 2, implementation error). Equivalently: Value of c should be sensitive to degrees of freedom u. “Want large gain”, |G| Or more generally: Maximize minimum singular value, Unconstrained optimum BADGood

26. 26 Guidelines for selecting measurements as CVs Rule 1: Optimal value for CV (c=Hy) is insensitive to disturbances d (minimizes effect of moving optimum) –F c =|dc opt /dd| is small Rule 2: c should be easy to measure and control (small implementation error n) Rule 3: “Maximum gain rule”: c should be sensitive to input changes (large gain |G| from u to c) or equivalently the optimum J opt should be flat with respect to c (minimizes effect of implementation error n) –|G| = |dc/du| is large –For case of multiple CVs, the selected CVs should not be correlated. Reference: S. Skogestad, “Plantwide control: The search for the self-optimizing control structure”, Journal of Process Control, 10, 487-507 (2000).

27. 27 Quantitative combined rule: Maximum gain rule Maximum gain rule (Skogestad and Postlethwaite, 1996): Look for variables that maximize the scaled gain  (G s ) (minimum singular value of the appropriately scaled steady-state gain matrix G s from u to c) G = HG y u c Unconstrained variables

28. 28 Optimizer Controller that adjusts u to keep c m = c s Plant cscs c m =c+n u c n d u c J c s =c opt u opt n u = G -1 n ) Want c sensitive to u (large gain G = dc/du) to get small variation in u (n u ) when c varies (n) Control sensitive variables n

29. 29 Why is Large Gain Good? u J, c J opt u opt c opt c-c opt Loss G With large gain G: Even large implementation error n in c translates into small deviation of u from u opt (d) - leading to lower loss Variation of u Unconstrained variables

30. 30 Maximum Gain Rule in words In words, select controlled variables c for which the gain G (= “controllable range”) is large compared to its span (= sum of optimal variation and control error) Select CVs that maximize  (G s ) Unconstrained variables

31. 31 Toy Example Reference: I. J. Halvorsen, S. Skogestad, J. Morud and V. Alstad, “Optimal selection of controlled variables”, Industrial & Engineering Chemistry Research, 42 (14), 3273-3284 (2003).

32. 32 Toy Example: Single measurements Want loss < 0.1: Consider variable combinations Constant input, c = y 4 = u

33. 33 Summary: Procedure selection controlled variables 1.Define economics and operational constraints 2.Identify degrees of freedom and important disturbances 3.Optimize for various disturbances 4.Identify active constraints regions (off-line calculations) For each active constraint region do step 5-6: 5.Identify “self-optimizing” controlled variables for remaining degrees of freedom 6.Identify switching policies between regions

34. 34 Comments. Analyzing a given CV choice, c=Hy Evaluation of candidates can be time-consuming using general non-linear (“brute force”) formulation –Pre-screening using local methods. –Final verification for few promising alternatives by evaluating actual loss Local method: Maximum gain rule is not exact* –but gives insight Alternative: “Exact local method” (loss method) –uses same information, but somewhat more complicated Lower loss: try measurement combinations as CVs (next) *The maximum gain rule assumes that the worst-case setpoint errors Δc i, opt (d) for each CV can appear together. In general, Δc i, opt (d) are correlated.

35. 35 Optimal measurement combination Candidate measurements (y): Include also inputs u H measurement noise control error disturbance controlled variable CV=Measurement combination

36. 36 Nullspace method No measurement noise (n y =0) CV=Measurement combination

37. 37 Amazingly simple! Sigurd is told how easy it is to find H Proof nullspace method Basis: Want optimal value of c to be independent of disturbances Find optimal solution as a function of d: u opt (d), y opt (d) Linearize this relationship:  y opt = F  d –F – optimal sensitivity matrix Want: To achieve this for all values of  d: Always possible if Optimal when we disregard implementation error (n) V. Alstad and S. Skogestad, ``Null Space Method for Selecting Optimal Measurement Combinations as Controlled Variables'', Ind.Eng.Chem.Res, 46 (3), 846-853 (2007). No measurement noise (n y =0) CV=Measurement combination

38. 38 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!)

39. 39 Toy Example

40. 40 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). Proof:

41. 41 H /selection Extension: ”Exact local method” (with measurement noise) CV=Measurement combination cmcm Problem definition (expected average with d normally distributed)

42. 42 Ref: Halvorsen et al. I&ECR, 2003 Kariwala et al. I&ECR, 2008 ”Exact local method” (with measurement noise) Loss with c=Hy m =0 due to (i) Disturbances d (ii) Measurement noise n y u J Loss Controlled variables, c s = constant + + + + + - K H y cmcm u d CV=Measurement combination

43. 43 Optimal H General analytical solution (“full” H): No disturbances (W d =0) + same noise for all measurements (W ny =I): Optimal is H=G yT (“control sensitive measurements”) Proof: Use analytic expression No noise (W ny =0): Cannot use analytic expression, but optimal is clearly HF=0 (Nullspace method) Assumes enough measurements: #y ¸ #u + #d If “extra” measurements (>) then solution is not unique c s = constant + + + + + - K H y c u CV=Measurement combination

44. 44 c s = constant + + + + + - K H y cm u “Minimize” in Maximum gain rule ( maximize S 1 G J uu -1/2, G=HG y ) “Scaling” S 1 -1 for max.gain rule “=0” in nullspace method (no noise) Relationship to max. gain rule and nullspace method CV=Measurement combination

45. 45 Non-convex optimization problem (Halvorsen et al., 2003) st Improvement 1 (Alstad et al. 2009) st Improvement 2 (Yelchuru et al., 2010) D : any non-singular matrix Have extra degrees of freedom Convex optimization problem Global solution - Full H (not selection): Do not need J uu - Q can be used as degrees of freedom for faster solution - Analytical solution when YY T has full rank (w/ meas. noise): CV=Measurement combination

46. 46 Toy example...

47. 47 Example: heat exchanger split

48. 48 Example: CO2 refrigeration cycle J = W s (work supplied) DOF = u (valve opening, z) Main disturbances: d 1 = T H d 2 = T Cs (setpoint) d 3 = UA loss What should we control? pHpH

49. 49 CO2 refrigeration cycle Step 1. One (remaining) degree of freedom (u=z) Step 2. Objective function. J = W s (compressor work) Step 3. Optimize operation for disturbances (d 1 =T C, d 2 =T H, d 3 =UA) Optimum always unconstrained Step 4. Implementation of optimal operation No good single measurements (all give large losses): –p h, T h, z, … Nullspace method: Need to combine n u +n d =1+3=4 measurements to have zero disturbance loss Simpler: Try combining two measurements. Exact local method: –c = h 1 p h + h 2 T h = p h + k T h ; k = -8.53 bar/K Nonlinear evaluation of loss: OK!

50. 50 CO2 cycle: Maximum gain rule

51. 51 Refrigeration cycle: Proposed control structure Control c= “temperature-corrected high pressure” CV=Measurement combination

52. 52 Control structure design using self-optimizing control for economically optimal CO 2 recovery * Step S1. Objective function= J = energy cost + cost (tax) of released CO 2 to air Step S3 (Identify CVs). 1. Control the 4 equality constraints 2. Identify 2 self-optimizing CVs. Use Exact Local method and select CV set with minimum loss. 4 equality and 2 inequality constraints: 1.stripper top pressure 2.condenser temperature 3.pump pressure of recycle amine 4.cooler temperature 5.CO 2 recovery ≥ 80% 6.Reboiler duty < 1393 kW (nominal +20%) 4 levels without steady state effect: absorber 1,stripper 2,make up tank 1 *M. Panahi and S. Skogestad, ``Economically efficient operation of CO2 capturing process, part I: Self-optimizing procedure for selecting the best controlled variables'', Chemical Engineering and Processing, 50, 247-253 (2011).``Economically efficient operation of CO2 capturing process, part I: Self-optimizing procedure for selecting the best controlled variables'', Step S2. (a) 10 degrees of freedom: 8 valves + 2 pumps Disturbances: flue gas flowrate, CO 2 composition in flue gas + active constraints (b) Optimization using Unisim steady-state simulator. Mode I = Region I (nominal feedrate): No inequality constraints active 2 unconstrained degrees of freedom =10-4-4 Case study

53. 53 Exact local method * for finding 2 self-optimizing CVs The set with the minimum worst case loss is the best * I.J. Halvorsen, S. Skogestad, J.C. Morud and V. Alstad, ‘Optimal selection of controlled variables’ Ind. Eng. Chem. Res., 42 (14), 3273-3284 (2003) J uu and F, the optimal sensitivity of the measurements with respect to disturbances, are obtained numerically

54. 54 39 candidate CVs - 15 possible tray temperature in absorber - 20 possible tray temperature in stripper - CO 2 recovery in absorber and CO 2 content at the bottom of stripper - Recycle amine flowrate and reboiler duty Best self-optimizing CV set in Mode I: c 1 = CO 2 recovery (95.26%) c 2 = Temperature tray no. 16 in stripper These CVs are not necessarily the best if new constraints are met Use a bidirectional branch and bound algorithm * for finding the best CVs * V. Kariwala and Y. Cao. Bidirectional Branch and Bound for Controlled Variable Selection, Part II: Exact Local Method for Self-Optimizing Control, Computers & Chemical Engineering, 33(2009), 1402-1412. Identify 2 self-optimizing CVs

55. 55 Proposed control structure with given nominal flue gas flowrate (mode I)

56. 56 Mode II: large feedrates of flue gas (+30%) Feedrate flue gas (kmol/hr ) Self-optimizing CVs in region IReboiler duty (kW) Cost (USD/ton) CO 2 recovery % Temperature tray no. 16 °C Optimal nominal point219.395.26106.911612.49 +5% feedrate230.395.26106.912222.49 +10% feedrate241.295.26106.912792.49 +15% feedrate252.295.26106.913392.49 +19.38%, when reboiler duty saturates 261.895.26106.91393 (+20%) 2.50 +30% feedrate (reoptimized) 285.191.60103.313932.65 Saturation of reboiler duty; one unconstrained degree of freedom left Use Maximum gain rule to find the best CV among 37 candidates : Temp. on tray no. 13 in the stripper: largest scaled gain, but tray 16 also OK region I region II max

57. 57 Proposed control structure with large flue gas flowrate (mode II = region II) max

58. 58 Conditions for switching between regions of active constraints (“supervisory control”) Within each region of active constraints it is optimal to 1.Control active constraints at c a = c,a, constraint 2.Control self-optimizing variables at c so = c,so, optimal Define in each region i: Keep track of c i (active constraints and “self-optimizing” variables) in all regions i Switch to region i when element in c i changes sign

59. 59 Example – switching policies CO2 plant (”supervisory control”) Assume operating in region I (unconstrained) –with CV=CO2-recovery=95.26% When reach maximum Q: Switch to Q=Qmax (Region II) (obvious) –CO2-recovery will then drop below 95.26% When CO2-recovery exceeds 95.26%: Switch back to region I !!!

60. 60 Example switching policies – 10 km 1.”Startup”: Given speed or follow ”hare” 2.When heart beat > max or pain > max: Switch to slower speed 3.When close to finish: Switch to max. power Another example: Regions for LNG plant (see Chapter 7 in thesis by J.B.Jensen, 2008)

61. 61 Conclusion optimal operation ALWAYS: 1. Control active constraints and control them tightly!! –Good times: Maximize throughput -> tight control of bottleneck 2. Identify “self-optimizing” CVs for remaining unconstrained degrees of freedom Use offline analysis to find expected operating regions and prepare control system for this! –One control policy when prices are low (nominal, unconstrained optimum) –Another when prices are high (constrained optimum = bottleneck) ONLY if necessary: consider RTO on top of this

62. 62 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 –In particular, never try to control directly the cost J –- Assume we want to minimize J (e.g., J = V = energy) - and we make the stupid choice os selecting CV = V = J - Then setting J Jmin: Forces us to be nonoptimal (which may require strange operation; see Exercise 3 on evaporators)