1. 1 Outline Skogestad procedure for control structure design I Top Down Step S1: Define operational objective (cost) and constraints Step S2: Identify degrees of freedom and optimize operation for disturbances Step S3: Implementation of optimal operation –What to control ? (primary CV’s) (self-optimizing control) Step S4: Where set the production rate? (Inventory control) II Bottom Up Step S5: Regulatory control: What more to control (secondary CV’s) ? Step S6: Supervisory control Step S7: Real-time optimization

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 Solution I (“obvious”): Optimal feedforward Problem: UNREALISTIC! 1.Lack of measurements of d 2.Sensitive to model error

4. 4 Solution II (”obvious”): Optimizing control Estimate d from measurements y and recompute u opt (d) Problem: COMPLICATED! Requires detailed model and description of uncertainty y

5. 5 Solution III (in practice): FEEDBACK with hierarchical decomposition When disturbance d: Degrees of freedom (u) are updated indirectly to keep CVs at setpoints y CVs: link optimization and control layers

6. 6 “Self-Optimizing Control” = Solution III with constant setpoints Self-optimizing control: Constant setpoints give acceptable loss y Issue: What should we control?

7. 7 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

8. 8 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

9. 9 Step S3…. What should we control (c)? Simple examples What should we control?

10. 10 –Cost to be minimized, J=T –One degree of freedom (u=power) –What should we control? Optimal operation - Runner Optimal operation of runner

11. 11 Self-optimizing control: Sprinter (100m) 1. Optimal operation of Sprinter, J=T –Active constraint control: Maximum speed (”no thinking required”) Optimal operation - Runner

12. 12 2. Optimal operation of Marathon runner, J=T Optimal operation - Runner Self-optimizing control: Marathon (40 km)

13. 13 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

14. 14 Solution 2 Marathon – Feedback (Self-optimizing control) –What should we control? Optimal operation - Runner

15. 15 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)

16. 16 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

17. 17 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

18. 18 Further examples self-optimizing control Marathon runner Central bank Cake baking Business systems (KPIs) Investment portifolio Biology Chemical process plants: Optimal blending of gasoline Define optimal operation (J) and look for ”magic” variable (c) which when kept constant gives acceptable loss (self- optimizing control)

19. 19 More on further examples 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

20. 20 Step S3. Implementation of optimal operation. Summary so far. Main issue: What should we control (c)? Equivalently: What should H be? c = H y c – controlled variables (CVs) y – available measurements (including u’s) H – selection or combination matrix 1.Control active constraints! * 2.Unconstrained variables: Control self-optimizing variables! * Of course, the active constraints may be viewed as the obvious self-optimizing variables

21. 21 Outline Skogestad procedure for control structure design I Top Down Step S1: Define operational objective (cost) and constraints Step S2: Identify degrees of freedom and optimize operation for disturbances Step S3: Implementation of optimal operation –What to control ? (primary CV’s) (self-optimizing control) Step S4: Where set the production rate? (Inventory control) II Bottom Up Step S5: Regulatory control: What more to control (secondary CV’s) ? Step S6: Supervisory control Step S7: Real-time optimization

22. 22 Step S3, in more detail….. 1. CONTROL ACTIVE CONSTRAINTS!

23. 23 1. CONTROL ACTIVE CONSTRAINTS! Active input constraints: Just set at MAX or MIN –Example: Max. heat input (boilup) distillation Active output constraints: Need back-off Step S3, in more detail….. The backoff is the “safety margin” from the active constraint and is defined as the difference between the constraint value and the chosen setpoint Backoff = |Constraint – Setpoint|

24. 24 a)If constraint can be violated dynamically (only average matters) Required Back-off = “bias” (steady-state measurement error for c) b)If constraint cannot be violated dynamically (“hard constraint”) Required Back-off = “bias” + maximum dynamic control error J opt Back-off Loss c ≥ c constraint c J Active output constraints Need back-off Want tight control of hard output constraints to reduce the back-off. “Squeeze and shift”-rule The backoff is the “safety margin” from the active constraint and is defined as the difference between the constraint value and the chosen setpoint Backoff = |Constraint – Setpoint|

25. 25 Example. Optimal operation = max. throughput. Want tight bottleneck control to reduce backoff! Time Back-off = Lost production Rule for control of hard output constraints: “Squeeze and shift”! Reduce variance (“Squeeze”) and “shift” setpoint c s to reduce backoff

26. 26 Hard Constraints: «SQUEEZE AND SHIFT» © Richalet SHIFT SQUEEZE

27. 27 Example back-off Speed limit = 80 km/h (hard constraint) –Measurement error (bias): 8 km/h –Control error (variation due to poor control): 5 km/h –Total implementation error = 13 km/h (= Backoff) –Setpoint [km/h] = 80 – 13 = 67 –Can reduce backoff with better control (“squeeze and shift”) Speed limit = 80 km/h (average over 10 km distance) (soft constraint) –Total implementation error = 8 km/h (= Backoff) –Setpoint [km/h] = 80 – 8 = 72 –Do not need to include the control error because it averages out

28. 28 SUMMARY ACTIVE CONSTRAINTS –c constraint = value of active constraint –Implementation of active constraints is usually “obvious”, but may need “back-off” (safety limit) for hard output constraints C s = C constraint - backoff –Want tight control of hard output constraints to reduce the back-off “Squeeze and shift”

29. 29 Outline Skogestad procedure for control structure design I Top Down Step S1: Define operational objective (cost) and constraints Step S2: Identify degrees of freedom and optimize operation for disturbances Step S3: Implementation of optimal operation –What to control ? (primary CV’s) (self-optimizing control) Step S4: Where set the production rate? (Inventory control) II Bottom Up Step S5: Regulatory control: What more to control (secondary CV’s) ? Step S6: Supervisory control Step S7: Real-time optimization

30. 30 Step S3. Self-optimizing control 1.OBVIOUS: CONTROL ACTIVE CONSTRAINTS, c=c constraint 2.NOT OBVIOUS: REMAINING UNCONSTRAINED, c=? “Self-optimizing”: Acceptable loss with c s =constant What should we control? c = Hy H Unconstrained variables

31. 31 Unconstrained variables H measurement noise steady-state control error disturbance controlled variable / selection

32. 32 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

33. 33 Optimal operation Cost J Controlled variable c c opt J opt Unconstrained optimum

34. 34 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

35. 35 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

36. 36 Guidelines for CV Selection Rule 1: The optimal value for CV c should be insensitive to disturbances d (minimizes effect of setpoint error) Rule 2: c should be sensitive to changes in u (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) Rule 3: c should be easy to measure and control (small implementation error n) Rule 4: 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). Unconstrained variables

37. 37 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

38. 38 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

39. 39 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

40. 40 EXAMPLE: Recycle plant (Luyben, Yu, etc.) 1 2 3 4 5 Assume constant reactor temperature. Given feedrate F 0 and column pressure: Dynamic DOFs: N m = 5 Column levels: N 0y = 2 Steady-state DOFs:N 0 = 5 - 2 = 3 Feed of A Recycle of unreacted A (+ some B) Product (98.5% B)

41. 41 Recycle plant: Optimal operation mTmT 1 remaining unconstrained degree of freedom, CV=?

42. 42 Control of recycle plant: Conventional structure (“Two-point”: CV=x D ) LC XC LC XC LC xBxB xDxD Control active constraints (M r =max and x B =0.015) + x D

43. 43 Luyben rule Luyben rule (to avoid snowballing): “Fix a stream in the recycle loop” (CV=F or D)

44. 44 Luyben rule: CV=D (constant) LC XC

45. 45 A. Maximum gain rule: Steady-state gain Luyben rule: Not good economically Conventional: Looks good

46. 46 How did we find the gains in the Table? 1.Find nominal optimum 2.Find (unscaled) gain G 0 from input to candidate outputs:  c = G 0  u. In this case only a single unconstrained input (DOF). Choose at u=L Obtain gain G 0 numerically by making a small perturbation in u=L while adjusting the other inputs such that the active constraints are constant (bottom composition fixed in this case) 3.Find the span for each candidate variable For each disturbance d i make a typical change and reoptimize to obtain the optimal ranges  c opt (d i ) For each candidate output obtain (estimate) the control error (noise) n The expected variation for c is then: span(c) =  i |  c opt (d i )| + |n| 4.Obtain the scaled gain, G = |G 0 | / span(c) 5.Note: The absolute value (the vector 1-norm) is used here to "sum up" and get the overall span. Alternatively, the 2-norm could be used, which could be viewed as putting less emphasis on the worst case. As an example, assume that the only contribution to the span is the implementation/measurement error, and that the variable we are controlling (c) is the average of 5 measurements of the same y, i.e. c=sum yi/5, and that each yi has a measurement error of 1, i.e. nyi=1. Then with the absolute value (1-norm), the contribution to the span from the implementation (meas.) error is span=sum abs(nyi)/5 = 5*1/5=1, whereas with the two-norn, span = sqrt(5*(1/5^2) = 0.447. The latter is more reasonable since we expect that the overall measurement error is reduced when taking the average of many measurements. In any case, the choice of norm is an engineering decision so there is not really one that is "right" and one that is "wrong". We often use the 2-norm for mathematical convenience, but there are also physical justifications (as just given!). IMPORTANT!

47. 47 B. “Brute force” loss evaluation: Disturbance in F 0 Loss with nominally optimal setpoints for M r, x B and c Luyben rule: Conventional

48. 48 B. “Brute force” loss evaluation: Implementation error Loss with nominally optimal setpoints for M r, x B and c Luyben rule:

49. 49 Conclusion: Control of recycle plant Active constraint M r = M rmax Active constraint x B = x Bmin L/F constant: Easier than “two-point” control Assumption: Minimize energy (V) Self-optimizing

50. 50 Cooment on Luyben’s rule Luyben law no. 1 (“Plantwide process control”, 1998, pp. 57): “A stream somewhere in all recycle loops must be flow controlled. This is to prevent the snowball effect” This is not a general rule (and certainly not a law). It is true that recycle generally introduces an extra degree of freedom, and fixing a flow (R) in the loop may be reasonable for low- throughput operation with a fixed throughput However, consider throughput changes (F increases) 1.Unconstrained region: Clearly all flows should be increases in proportion to F to remain optimal, so R/F (and not R) should be constant. 2.F increases further and we reach new constraints: This consumes degrees of freedom, and eventually we need to adjust also R to remain feasible. Also: Importance of “Snowballing” is much overrated. Only important for a few sensitive recycle processes.

51. 51

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. 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 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 region 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 (region I)

56. 56 Region 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 (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 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

62. 62 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.

63. 63 Optimal measurement combination Candidate measurements (y): Include also inputs u H measurement noise control error disturbance controlled variable CV=Measurement combination

64. 64 Nullspace method No measurement noise (n y =0) CV=Measurement combination

65. 65 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 Want: To achieve this for all values of  d: To find a F that satisfies HF=0 we must require 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

66. 66 H /selection ”Exact local method” (with measurement noise) Problem definition CV=Measurement combination

67. 67 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 c u d CV=Measurement combination

68. 68 Special cases No noise (n y =0, W ny =0): Optimal is HF=0 (Nullspace method) But: If many measurement then solution is not unique No disturbances (d=0; W d =0) + same noise for all measurements (W ny =I): Optimal is H T =G y (“control sensitive measurements”) Proof: Use analytic expression c s = constant + + + + + - K H y c u CV=Measurement combination

69. 69 c s = constant + + + + + - K H y c 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

70. 70 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

71. 71 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

72. 72 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!

73. 73 CO2 cycle: Maximum gain rule

74. 74 Refrigeration cycle: Proposed control structure Control c= “temperature-corrected high pressure” CV=Measurement combination