1. 1 Devising control structures for complete chemical plants – from art to science Sigurd Skogestad Department of Chemical Engineering Norwegian University of Science and Tecnology (NTNU) Trondheim, Norway 02 Dec. 2015 TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: A A A

2. 2 Abstract A chemical plant may have thousands of measurements and control loops. By devising the control structure we do not meanmean the tuning and behavior of each of these loops, but rather the control philosophy of the overall plant with emphasis on the structural decisions.This is sometimes referred to as “plantwide control”. In practice, the control system is usually divided into several layers, separated by time scale: scheduling (weeks), site-wide optimization (day), local optimization (hour), supervisory (predictive, advanced) control (minutes) and regulatory control (seconds). Such a hiearchical (cascade) decomposition with layers operating on different time scale is used in the control of all real (complex) systems including biological systems and airplanes, so the issues in this section are not limited to process control. In the talk the most important issues are discussed, especially related to the choice of variables that provide the link the control layers, and the location of throughput manipulator (TPM). Some simple rules for plantwide control will also be presented, and demonstrated on case studies.

3. 3 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 2015- : Director Subsea Processing Research Center, NTNU (SUBPRO) 160 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)

4. 4 Trondheim, Norway

5. 5 Trondheim Oslo UK NORWAY DENMARK GERMANY North Sea SWEDEN Arctic circle

6. 6 NTNU, Trondheim

7. 7 Outline 1. Introduction plantwide control (control structure design) 2. Plantwide control procedure I Top Down –Step 1: Define optimal operation –Step 2: Optimize for expected disturbances Find active constraints –Step 3: Select primary controlled variables c=y 1 (CVs) Self-optimizing variables –Step 4: Where locate the throughput manipulator? II Bottom Up –Step 5: Regulatory / stabilizing control (PID layer) What more to control (y 2 )? Pairing of inputs and outputs –Step 6: Supervisory control (MPC layer) –Step 7: Real-time optimization (Do we need it?) y1y1 y2y2 Process MVs

8. 8 How we design a control system for a complete chemical plant? Where do we start? What should we control? and why? etc.

9. 9 Example: Tennessee Eastman challenge problem (Downs, 1991) TCPCLCACxSRC Where place??

10. 10 Plantwide control = Control structure design Not the tuning and behavior of each control loop, But rather the control philosophy of the overall plant with emphasis on the structural decisions: –Selection of controlled variables (“outputs”) –Selection of manipulated variables (“inputs”) –Selection of (extra) measurements –Selection of control configuration (structure of overall controller that interconnects the controlled, manipulated and measured variables) –Selection of controller type (LQG, H-infinity, PID, decoupler, MPC etc.).

11. 11 Main objectives control system 1.Economics: Implementation of acceptable (near-optimal) operation 2.Regulation: Stable operation

12. 12 Example of systems we want to operate optimally Process plant –minimize J=economic cost Runner –minimize J=time «Green» process plant –Minimize J=environmental impact (with given economic cost) General multiobjective: –Min J (scalar cost, often $) –Subject to satisfying constraints (environment, resources) Optimal operation (economics)

13. 13 In theory: Optimal control and operation Objectives Present state Model of system Approach: Model of overall system Estimate present state Optimize all degrees of freedom Process control: Excellent candidate for centralized control (Physical) Degrees of freedom CENTRALIZED OPTIMIZER

14. 14 Optimal centralized solution Sigurd Academic process control community fish pond

15. 15 In theory: Optimal control and operation Objectives Present state Model of system Approach: Model of overall system Estimate present state Optimize all degrees of freedom Process control: Excellent candidate for centralized control Problems: Model not available Objectives = ? Optimization complex Not robust (difficult to handle uncertainty) Slow response time (Physical) Degrees of freedom CENTRALIZED OPTIMIZER

16. 16 Practice: Engineering systems Most (all?) large-scale engineering systems are controlled using hierarchies of quite simple controllers –Large-scale chemical plant (refinery) –Commercial aircraft 100’s of loops Simple components: PI-control + selectors + cascade + nonlinear fixes + some feedforward Same in biological systems But: Not well understood

17. 17 Alan Foss (“Critique of chemical process control theory”, AIChE Journal,1973): The central issue to be resolved... is the determination of control system structure. Which variables should be measured, which inputs should be manipulated and which links should be made between the two sets? There is more than a suspicion that the work of a genius is needed here, for without it the control configuration problem will likely remain in a primitive, hazily stated and wholly unmanageable form. The gap is present indeed, but contrary to the views of many, it is the theoretician who must close it. Previous work on plantwide control: Page Buckley (1964) - Chapter on “Overall process control” (still industrial practice) Greg Shinskey (1967) – process control systems Alan Foss (1973) - control system structure Bill Luyben et al. (1975- ) – case studies ; “snowball effect” George Stephanopoulos and Manfred Morari (1980) – synthesis of control structures for chemical processes Ruel Shinnar (1981- ) - “dominant variables” Jim Downs (1991) - Tennessee Eastman challenge problem Larsson and Skogestad (2000): Review of plantwide control

18. 18 Main objectives control system 1.Economics: Implementation of acceptable (near-optimal) operation 2.Regulation: Stable operation ARE THESE OBJECTIVES CONFLICTING? Usually NOT –Different time scales Stabilization fast time scale –Stabilization doesn’t “use up” any degrees of freedom Reference value (setpoint) available for layer above But it “uses up” part of the time window (frequency range)

19. 19 Practical operation: Hierarchical structure Manager Process engineer Operator/RTO Operator/”Advanced control”/MPC PID-control u = valves Our Paradigm setpoints constraints, prices

20. 20 CV 1s MPC PID CV 2s RTO Follow path (+ look after other variables) Stabilize + avoid drift Min J (economics) u (valves) OBJECTIVE Dealing with complexity Plantwide control: Objectives The controlled variables (CVs) interconnect the layers CV = controlled variable (with setpoint)

21. 21 Practice: Process operation Practice: Hierarchical structure Each layer has its own goal (”default action”): Keep CV at given setpoint Boss: layer above (adjusts setpoint) Goal each layer: Action without boss is «Self-optimizing»

22. 22 “Self-optimizing control”: Translate optimal operation into control objectives: What should we control? CV 1 = c ? (economics) CV 2 = ? (stabilization) Goal each layer: Action without boss is «Self-optimizing»

23. 23 Control structure design procedure I Top Down (mainly steady-state economics, y 1 ) Step 1: Define operational objectives (optimal operation) –Cost function J (to be minimized) –Operational constraints Step 2: Identify degrees of freedom (MVs) and optimize for expected disturbances Identify Active constraints Step 3: Select primary “economic” controlled variables c=y 1 (CV1s) Self-optimizing variables Step 4: Where locate the throughput manipulator (TPM)? II Bottom Up (dynamics, y 2 ) Step 5: Regulatory / stabilizing control (PID layer) –What more to control (y 2 ; local CV2s)? –Pairing of inputs and outputs Step 6: Supervisory control (MPC layer) Step 7: Real-time optimization (Do we need it?) y1y1 y2y2 Process MVs S. Skogestad, ``Control structure design for complete chemical plants'', Computers and Chemical Engineering, 28 (1-2), 219-234 (2004).

24. 24 Step 1. Define optimal operation (economics) What are we going to use our degrees of freedom u (MVs) for? Define scalar cost function J(u,x,d) –u: degrees of freedom (usually steady-state) –d: disturbances –x: states (internal variables) Typical cost function: Optimize operation with respect to u for given d (usually steady-state): min u J(u,x,d) subject to: Model equations: f(u,x,d) = 0 Operational constraints: g(u,x,d) < 0 J = cost feed + cost energy – value products

25. 25 Step 1: Optimal operation distillation column Distillation at steady state with given p and F: N=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

26. 26 Step S2. Optimize (a) Identify degrees of freedom (b) Optimize for expected disturbances Need good model, usually steady-state Optimization is time consuming! But it is offline Main goal: Identify ACTIVE CONSTRAINTS A good engineer can often guess the active constraints

27. 27 …. BUT A GOOD ENGINEER CAN OFTEN GUESS THE SOLUTION (active constraints) What are the optimal values for our degrees of freedom u (MVs)? Minimize J with respect to u for given disturbance d (usually steady-state): min u J(u,x,d) subject to: Model equations (e,g, Hysys): f(u,x,d) = 0 Operational constraints: g(u,x,d) < 0 OFTEN VERY TIME CONSUMING –Commercial simulators (Aspen, Unisim/Hysys) are set up in “design mode” and often work poorly in “operation (rating) mode”. –Optimization methods in commercial simulators often poor We use Matlab or even Excel “on top” J = cost feed + cost energy – value products Step S2b: Optimize for expected disturbances

28. 28 Step S3: Implementation of optimal operation Have found the optimal way of operation. How should it be implemented? What to control ? (primary CV’s). 1.Active constraints 2.Self-optimizing variables (for unconstrained degrees of freedom )

29. 29 –Cost to be minimized, J=T –One degree of freedom (u=power) –What should we control? Optimal operation - Runner Optimal operation of runner

30. 30 1. Optimal operation of Sprinter –100m. J=T –Active constraint control: Maximum speed (”no thinking required”) CV = power (at max) Optimal operation - Runner

31. 31 40 km. J=T What should we control? CV=? Unconstrained optimum Optimal operation - Runner 2. Optimal operation of Marathon runner u=power J=T u opt

32. 32 Any self-optimizing variable (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)

33. 33 Conclusion 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

34. 34 Step 3. What should we control (c)? Selection of primary controlled variables y 1 =c 1.Control active constraints! 2.Unconstrained variables: Control self-optimizing variables! 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.”

35. 35 Step 4. Where set production rate? Where locale the TPM (throughput manipulator)? –The ”gas pedal” of the process Very important! Determines structure of remaining inventory (level) control system Set production rate at (dynamic) bottleneck Link between Top-down and Bottom-up parts

36. 36 Production rate set at inlet : Inventory control in direction of flow* * Required to get “local-consistent” inventory control TPM

37. 37 Production rate set at outlet: Inventory control opposite flow TPM

38. 38 Production rate set inside process TPM Radiating inventory control around TPM (Georgakis et al.)

39. 39 TPM QUIZ. Are these structures workable? Yes or No?

40. 40 Example: Tennessee Eastman TPM ?

41. 41 “Sellers marked” = high product prices: Optimal operation = max. throughput (active constraints) 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 Want tight bottleneck control to reduce backoff!

42. 42 LOCATE TPM? Conventional choice: Feedrate Consider moving if there is an important active constraint that could otherwise not be well controlled Good choice: Locate at bottleneck

43. 43 Control structure design procedure I Top Down (mainly steady-state economics) Step 1: Define operational objectives (optimal operation) –Cost function J (to be minimized) –Operational constraints Step 2: Identify degrees of freedom (MVs) and optimize for expected disturbances Identify Active constraints Step 3: Select primary “economic” controlled variables c=y 1 (CVs) Self-optimizing variables Step 4: Where locate the throughput manipulator (TPM)? II Bottom Up (dynamics) Step 5: Regulatory / stabilizing control (PID layer) –What more to control (y 2 ; local CVs)? –Pairing of inputs and outputs Step 6: Supervisory control (MPC layer) Step 7: Real-time optimization (Do we need it?) y1y1 y2y2 Process MVs S. Skogestad, ``Control structure design for complete chemical plants'', Computers and Chemical Engineering, 28 (1-2), 219-234 (2004).

44. 44 Example/Quiz: Optimal operation of two distillation columns in series Feed: Components A, B and C Three products D1, D2, B2 (>95% purity) Most valuable: B Energy is cheap Cost (J) = - Profit = p F F + p V (V 1 +V 2 ) – p D1 D 1 – p D2 D 2 – p B2 B 2 Constraints: 3 purity constraints, 2 capacity constraints (max V) Steady-state degrees of freedom (given pressure, given feed): 2 +2 = 4 Need to find 4 variables to control / keep constant D1,A D2,B B2,C

45. 45 Expected active constraints distillation Both products (D,B) generally have purity specs Optimal operation. Control implications: 1.To avoid product give-away ALWAYS Control valuable product at spec. (active constraint). 2.May overpurify (not control) cheap product valuable product methanol + max. 0.5% water cheap product (byproduct) water + max. 2% methanol + water Selection of CV 1

46. 46 Control of Distillation columns in series Given LC PC Red: Basic regulatory loops 4 steady-state degrees of freedom

47. 47 Operation of Distillation columns in series 4 steady-state DOFs 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 QUIZ: What are the expected active constraints? 1. Always. 2. For low energy prices. Example /QUIZ 1 = ==

48. 48 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 QUIZ: What are the expected active constraints? 1. Always. 2. For low energy prices. Operation of Distillation columns in series With given F (disturbance): 4 steady-state DOFs (e.g., L and V in each column) SOLUTION QUIZ 1

49. 49 Control of Distillation columns in series Given LC PC Red: Basic regulatory loops CC xBxB x BS =95% MAX V1 MAX V2 SOLUTION QUIZ1 + new QUIZ2 UNCONSTRAINED CV=?

50. 50 Control of Distillation columns. Cheap energy Given LC PC CC xBxB x BS =95% MAX V1 MAX V2 What is a good CV? Quiz 2.

51. 51 Control of Distillation columns. Cheap energy Given LC PC Overpurified: To avoid loss of valuable product B CC xBxB x BS =95% MAX V1 MAX V2 1 unconstrained DOF (L1): What is a good CV? Not: CV= x B in D1! (why? Overpurified, so x B,opt increases with F) Maybe: constant L1? (CV=L1) Better: CV= x A in B1? Self-optimizing? Example. Overpurified

52. 52 Given LC PC CC xBxB x BS =95% MAX V1 MAX V2 CC xBxB x AS =2.1% Control of Distillation columns. Cheap energy Solution.

53. 53 Control of Distillation columns in series Given LC PC Red: Basic regulatory loops CC xBxB x BS =95% MAX V1 MAX V2 1 unconstrained DOF (L1): Use for what?? CV=? Not: CV= x A in D1! (why? x A should vary with F!) Maybe: constant L1? (CV=L1) Better: CV= x A in B1? Self-optimizing? General for remaining unconstrained DOFs: LOOK FOR “SELF-OPTIMIZING” CVs = Variables we can keep constant WILL GET BACK TO THIS! SOLUTION QUIZ 2

54. 54 Comment: Distillation column control in practice 1.Add stabilizing temperature loops –In this case: use reflux (L) as MV because boilup (V) may saturate –T 1s and T 2s then replace L 1 and L 2 as DOFs. 2.Replace V1=max and V2=max by dpmax-controllers (assuming max. load is limited by flooding) See next slide

55. 55 Control of Distillation columns in series Given LC PC TC T1 s T2 s T1 T2 Comment: In practice MAX V2 MAX V1 CC xBxB x BS =95%

56. 56 More on: Optimal operation Mode 1. Given feedrate Mode 2. Maximum production minimize J = cost feed + cost energy – value products Two main cases (modes/regions) depending on marked conditions:

57. 57 Active constraint regions for two distillation columns in series CV = Controlled Variable 3 2 0 1 1 0 2 [mol/s] [$/mol] 1 Mode 1, Cheap energy: 1 remaining unconstrained DOF (L1) -> Need to find 1 additional CV Energy price Mode 2: operate at BOTTLENECK Higher F infeasible because all 5 constraints reached Mode 1 (expensive energy) Steay-state optimization

58. 58 How many active constraints regions? Maximum: n c = number of constraints BUT there are usually fewer in practice Certain constraints are always active (reduces effective n c ) Only n u can be active at a given time n u = number of MVs (inputs) Certain constraints combinations are not possibe –For example, max and min on the same variable (e.g. flow) Certain regions are not reached by the assumed disturbance set Distillation n c = 5 2 5 = 32 x B always active 2^4 = 16 -1 = 15 In practice = 8

59. 59 Amount of products is then usually indirectly given and Optimal operation is then usually unconstrained “maximize efficiency (energy)” Control: Operate at optimal trade-off NOT obvious what to control CV = Self-optimizing variable Mode 1. Given feedrate J = cost feed– value products + cost energy c J = energy c opt Often constant

60. 60 Assume feedrate is degree of freedom Assume products much more valuable than feed Optimal operation is then to maximize product rate Mode 2. Maximum production Control: Focus on tight control of bottleneck “Obvious what to control” CV = ACTIVE CONSTRAINT c J c max Infeasible region J = cost feed + cost energy – value products

61. 61 Active constraint regions for two distillation columns in series CV = Controlled Variable 3 2 0 1 1 0 2 [mol/s] [$/mol] 1 Mode 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 Distillation example: Not so simple Mode 2: operate at BOTTLENECK Higher F infeasible because all 5 constraints reached Mode 1 (expensive energy)

62. 62 a)If constraint can be violated dynamically (only average matters) Required Back-off = “measurement bias” (steady-state measurement error for c) b)If constraint cannot be violated dynamically (“hard constraint”) Required Back-off = “measurement bias” + maximum dynamic control error J opt Back-off Loss c ≥ c constraint c J More on: 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| CV = Active constraint

63. 63 Hard Constraints: «SQUEEZE AND SHIFT» © Richalet SHIFT SQUEEZE CV = Active constraint Rule for control of hard output constraints: “Squeeze and shift”! Reduce variance (“Squeeze”) and “shift” setpoint c s to reduce backoff

64. 64 Example. Optimal operation = max. throughput. Want tight bottleneck control to reduce backoff! Time Back-off = Lost production CV = Active constraint

65. 65 Example back-off. x B = purity product > 95% (min.) D 1 directly to customer (hard constraint) –Measurement error (bias): 1% –Control error (variation due to poor control): 2% –Backoff = 1% + 2% = 3% –Setpoint x Bs = 95 + 3% = 98% (to be safe) –Can reduce backoff with better control (“squeeze and shift”) D 1 to large mixing tank (soft constraint) –Measurement error (bias): 1% –Backoff = 1% –Setpoint x Bs = 95 + 1% = 96% (to be safe) –Do not need to include control error because it averages out in tank CV = Active constraint D1xBD1xB 8 xBxB x B,product ±2%

66. 66 Unconstrained optimum: What variable should be control? Often not obvious “Self-optimizing control”: Constant setpoint gives optimal operation for changing disturbances Idea: Move optimization into faster control layer

67. 67 Rule: Control “self-optimizing” variables 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.” Unconstrained degrees of freedom

68. 68 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

69. 69 Never try to control the cost function J (or any other variable that reaches a maximum or minimum at the optimum) Better: control its gradient, Ju. u J J min J>J min J<J min Infeasible ?

70. 70 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? -Here: No active constraint. Control T? NO! Optimal: Control J u =0 Th1, UA1

71. 71 c = J u = T J1 – T J2 T J = JÄSCHKE TEMPERATURE

72. 72 c = T J1 – TJ2 T J = JÄSCHKE TEMPERATURE

73. 73 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

74. 74 H Ideal: c = J u In practise: c = H y. Task: Determine H!

75. 75 Systematic approach: What to control? Define optimal operation: Minimize cost function J Each candidate c = Hy: ”Brute force approach”: With constant setpoints c s compute loss L for expected disturbances d and implementation errors n Select variable c with smallest loss Acceptable loss  self-optimizing control

76. 76 Example recycle plant. 3 steady-state degrees of freedom (u) Minimize J=V (energy) Active constraint M r = M rmax Active constraint x B = x Bmin L/F constant: Easier than “two-point” control Self-optimizing

77. 77 Optimal operation Cost J Controlled variable c c opt J opt Unconstrained optimum

78. 78 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 Loss1 Loss2

79. 79 Good candidate controlled variables c (for self-optimizing control) 1.The optimal value of c should be insensitive to disturbances Small F c = dc opt /dd 2.c should be easy to measure and control 3.Want “flat” optimum -> The value of c should be sensitive to changes in the degrees of freedom (“large gain”) Large G = dc/du = HG y BADGood

80. 80 Optimizer Controller that adjusts u to keep c m = c s Plant cscs c m = c + n u y nyny d u c J c s =c opt u opt n Want c sensitive to u (”large gain”) “Flat optimum is same as large gain” H

81. 81 Optimal measurement combinations “Optimal policy independent of disturbances” Need model or data for optimal response to disturbances –Marathon runner case –c = h 1 hr + h 2 v Extension to polynomial systems –Preheat train for energy recovery (Jäschke) –Patent pending

82. 82 Optimal linear measurement combination H Candidate measurements (y): Include also inputs u

83. 83 Optimal linear measurement combination: Nullspace method Find optimal solution as a function of d: u opt (d), y opt (d) Linearize this relationship: Δy opt = F ∙Δd F – optimal sensitivity matrix Want: To achieve this for all values of Δd (Nullspace method): Always possible if

84. 84 Nullspace method 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).

85. 85 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!)

86. 86 More general (“exact local method”) With measurement noise -No measurement error: HF=0 (nullspace method) -With measuremeng error: Minimize GF c -Maximum gain rule

87. 87 “Minimize” in Maximum gain rule ( maximize S 1 G J uu -1/2, G=HG y ) “Scaling” S 1 “=0” in nullspace method (no noise) Optimal measurement combination, c = Hy With measurement noise

88. 88 Control structure design procedure I Top Down (mainly steady-state economics) Step 1: Define operational objectives (optimal operation) –Cost function J (to be minimized) –Operational constraints Step 2: Identify degrees of freedom (MVs) and optimize for expected disturbances Identify Active constraints Step 3: Select primary “economic” controlled variables c=y 1 (CVs) Self-optimizing variables Step 4: Where locate the throughput manipulator (TPM)? II Bottom Up (dynamics) Step 5: Regulatory / stabilizing control (PID layer) –What more to control (y 2 ; local CVs)? –Pairing of inputs and outputs Step 6: Supervisory control (MPC layer) Step 7: Real-time optimization (Do we need it?) y1y1 y2y2 Process MVs S. Skogestad, ``Control structure design for complete chemical plants'', Computers and Chemical Engineering, 28 (1-2), 219-234 (2004).

89. 89 Optimal centralized solution Sigurd Academic process control community fish pond

90. 90 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

91. 91 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!

92. 92 CO2 cycle: Maximum gain rule

93. 93 Refrigeration cycle: Proposed control structure Control c= “temperature-corrected high pressure”. k = -8.5 bar/K

94. 94 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

95. 95 Proposed control structure with given feed

96. 96 Step 4. Where set production rate? Where locale the TPM (throughput manipulator)? –The ”gas pedal” of the process Very important! Determines structure of remaining inventory (level) control system Set production rate at (dynamic) bottleneck Link between Top-down and Bottom-up parts

97. 97 Production rate set at inlet : Inventory control in direction of flow* * Required to get “local-consistent” inventory controlC TPM

98. 98 Production rate set at outlet: Inventory control opposite flow TPM

99. 99 Production rate set inside process TPM Radiating inventory control around TPM (Georgakis et al.)

100. 10 0 “Sellers marked” = high product prices: Optimal operation = max. throughput (active constraints) 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 Want tight bottleneck control to reduce backoff!

101. 10 1 LOCATE TPM? Conventional choice: Feedrate Consider moving if there is an important active constraint that could otherwise not be well controlled Good choice: Locate at bottleneck

102. 10 2 Step 5. Regulatory control layer Purpose: “Stabilize” the plant using a simple control configuration (usually: local SISO PID controllers + simple cascades) Enable manual operation (by operators) Main structural decisions: What more should we control? (secondary CV’s, y 2, use of extra measurements) Pairing with manipulated variables (MV’s u 2 ) y 1 = c y 2 = ?

103. 10 3 “ Control CV2s that stabilizes the plant (stops drifting)” In practice, control: 1.Levels (inventory liquid) 2.Pressures (inventory gas/vapor) (note: some pressures may be left floating) 3.Inventories of components that may accumulate/deplete inside plant E.g., amount of amine/water (deplete) in recycle loop in CO2 capture plant E.g., amount of butanol (accumulates) in methanol-water distillation column E.g., amount of inert N2 (accumulates) in ammonia reactor recycle 4.Reactor temperature 5.Distillation column profile (one temperature inside column) Stripper/absorber profile does not generally need to be stabilized

104. 10 4 Degrees of freedom for optimization (usually steady-state DOFs), MVopt = CV1s Degrees of freedom for supervisory control, MV1=CV2s + unused valves Physical degrees of freedom for stabilizing control, MV2 = valves (dynamic process inputs) Optimizer (RTO) PROCESS Supervisory controller (MPC) Regulatory controller (PID) H2H2 H CV 1 CV 2s y nyny d Stabilized process CV 1s CV 2 Physical inputs (valves) Optimally constant valves Always active constraints

105. 10 5 Main objectives control system 1.Implementation of acceptable (near-optimal) operation 2.Stabilization ARE THESE OBJECTIVES CONFLICTING? Usually NOT –Different time scales Stabilization fast time scale –Stabilization doesn’t “use up” any degrees of freedom Reference value (setpoint) available for layer above But it “uses up” part of the time window (frequency range)

106. 10 6 Example: Exothermic reactor (unstable) u = cooling flow (q) y 1 = composition (c) y 2 = temperature (T) u TC y 2 =T y 2s CC y 1 =c y 1s feed product cooling LC L s =max Active constraints (economics): Product composition c + level (max)

107. 10 7 Step 5: Regulatory control layer Step 5. Choose structure of regulatory (stabilizing) layer (a) Identify “stabilizing” CV2s (levels, pressures, reactor temperature,one temperature in each column, etc.). In addition, active constraints (CV1) that require tight control (small backoff) may be assigned to the regulatory layer. (Comment: usually not necessary with tight control of unconstrained CVs because optimum is usually relatively flat) (b) Identify pairings (MVs to be used to control CV2), taking into account –Want “local consistency” for the inventory control –Want tight control of important active constraints –Avoid MVs that may saturate in the regulatory layer, because this would require either reassigning the regulatory loop (complication penalty), or requiring back-off for the MV variable (economic penalty) Preferably, the same regulatory layer should be used for all operating regions without the need for reassigning inputs or outputs.

108. 10 8 Example: Distillation Primary controlled variable: y 1 = c = x D, x B (compositions top, bottom) BUT: Delay in measurement of x + unreliable Regulatory control: For “stabilization” need control of (y 2 ): 1.Liquid level condenser (M D ) 2.Liquid level reboiler (M B ) 3.Pressure (p) 4.Holdup of light component in column (temperature profile) Unstable (Integrating) + No steady-state effect Variations in p disturb other loops Almost unstable (integrating) TC TsTs

109. 10 9 ”Advanced control” STEP 6. SUPERVISORY LAYER Objectives of supervisory layer: 1. Switch control structures (CV1) depending on operating region Active constraints self-optimizing variables 2. Perform “advanced” economic/coordination control tasks. –Control primary variables CV1 at setpoint using as degrees of freedom (MV): Setpoints to the regulatory layer (CV2s) ”unused” degrees of freedom (valves) –Keep an eye on stabilizing layer Avoid saturation in stabilizing layer –Feedforward from disturbances If helpful –Make use of extra inputs –Make use of extra measurements Implementation: Alternative 1: Advanced control based on ”simple elements” Alternative 2: MPC

110. 11 0 Why simplified configurations? Why control layers? Why not one “big” multivariable controller? Fundamental: Save on modelling effort Other: –easy to understand –easy to tune and retune –insensitive to model uncertainty –possible to design for failure tolerance –fewer links –reduced computation load

111. 11 1 Summary. Systematic procedure for plantwide control Start “top-down” with economics: –Step 1: Define operational objectives and identify degrees of freeedom –Step 2: Optimize steady-state operation. –Step 3A: Identify active constraints = primary CVs c. –Step 3B: Remaining unconstrained DOFs: Self-optimizing CVs c. –Step 4: Where to set the throughput (usually: feed) Regulatory control I: Decide on how to move mass through the plant: Step 5A: Propose “local-consistent” inventory (level) control structure. Regulatory control II: “Bottom-up” stabilization of the plant Step 5B: Control variables to stop “drift” (sensitive temperatures, pressures,....) –Pair variables to avoid interaction and saturation Finally: make link between “top-down” and “bottom up”. Step 6: “Advanced/supervisory control” system (MPC): CVs: Active constraints and self-optimizing economic variables + look after variables in layer below (e.g., avoid saturation) MVs: Setpoints to regulatory control layer. Coordinates within units and possibly between units cscs http://www.nt.ntnu.no/users/skoge/plantwide

112. 11 2 Rule 4: Locate the TPM close to the process bottleneck. The justification for this rule is to take advantage of the large economic benefits of maximizing production in times when product prices are high relative to feed and energy costs (Mode 2). To maximize the production rate, one needs to achieve tight control of the active constraints, in particular, of the bottleneck, which is defined as the last constraint to become active when increasing the throughput rate (Jagtap et al., 2013). Rules for Step S4: Location of throughput manipulator (TPM)

113. 11 3 Rule 5: (for processes with recycle) Locate the TPM inside the recycle loop. The point is to avoid “overfeeding” the recycle loop which may easily occur if we operate close to the throughput where “snowballing” in the recycle loop occurs. This is a restatement of Luyben’s rule “Fix a Flow in Every Recycle Loop” (Luyben et al., 1997). From this perspective, snowballing can be thought of as the dynamic consequence of operating close to a bottleneck which is within a recycle system. In many cases, the process bottleneck is located inside the recycle loop and Rules 4 and 5 give the same result.

114. 11 4 Rule 6: Arrange the inventory control loops (for level, pressures, etc.) around the TPM location according to the radiation rule. The radiation rule (Price et al., 1994), says that, the inventory loops upstream of the TPM location must be arranged opposite of flow direction. For flow downstream of TPM location it must be arranged in the same direction. This ensures “local consistency” i.e. all inventories are controlled by their local in or outflows. Rules for Step S5: Structure of regulatory control layer.

115. 11 5 Rule 7: Select “sensitive/drifting” variables as controlled variables CV2 for regulatory control. This will generally include inventories (levels and pressures), plus certain other drifting (integrating) variables, for example, –a reactor temperature –a sensitive temperature/composition in a distillation column. This ensures “stable operation, as seen from an operator’s point of view. Some component inventories may also need to be controlled, especially for recycle systems. For example, according to “Down’s drill” one must make sure that all component inventories are “self- regulated” by flows out of the system or by removal by reactions, otherwise their composition may need to be controlled (Luyben, 1999).

116. 11 6 Rule 8: Economically important active constraints (a subset of CV1), should be selected as controlled variables CV2 in the regulatory layer. Economic variables, CV1, are generally controlled in the supervisory layer. Moving them to the faster regulatory layer may ensure tighter control with a smaller backoff. Backoff: difference between the actual average value (setpoint) and the optimal value (constraint).

117. 11 7 Rule 9: “Pair-close” rule: The pairings should be selected such that, effective delays and loop interactions are minimal.

118. 11 8 Rule 10: : Avoid using MVs that may optimally saturate (at steady state) to control CVs in the regulatory layer (CV2) The reason is that we want to avoid re-configuring the regulatory control layer. To follow this rule, one needs to consider also other regions of operation than the nominal, for example, operating at maximum capacity (Mode 2) where we usually have more active constraints.

119. 11 9 Rule 11: MVs that may optimally saturate (at steady state) should be paired with the subset of CV11 that may be given up. This rule applies for cases when we use decentralized control in the supervisory layer and we want to avoid reconfiguration of loops. The rule follows because when a MV optimally saturates, then there will be one less degree of freedom, so there will be a CV1 which may be given up without any economic loss. The rule should be considered together with rule 10. Rules for Step S6: Structure of supervisory control layer.

120. 12 0 Summary. Systematic procedure for plantwide control Start “top-down” with economics: –Step 1: Define operational objectives and identify degrees of freeedom –Step 2: Optimize steady-state operation. –Step 3A: Identify active constraints = primary CVs c. –Step 3B: Remaining unconstrained DOFs: Self-optimizing CVs c. –Step 4: Where to set the throughput (usually: feed) Regulatory control I: Decide on how to move mass through the plant: Step 5A: Propose “local-consistent” inventory (level) control structure. Regulatory control II: “Bottom-up” stabilization of the plant Step 5B: Control variables to stop “drift” (sensitive temperatures, pressures,....) –Pair variables to avoid interaction and saturation Finally: make link between “top-down” and “bottom up”. Step 6: “Advanced/supervisory control” system (MPC): CVs: Active constraints and self-optimizing economic variables + look after variables in layer below (e.g., avoid saturation) MVs: Setpoints to regulatory control layer. Coordinates within units and possibly between units cscs http://www.nt.ntnu.no/users/skoge/plantwide

121. 12 1 Summary and references The following paper summarizes the procedure: –S. Skogestad, ``Control structure design for complete chemical plants'', Computers and Chemical Engineering, 28 (1-2), 219-234 (2004). There are many approaches to plantwide control as discussed in the following review paper: –T. Larsson and S. Skogestad, ``Plantwide control: A review and a new design procedure'' Modeling, Identification and Control, 21, 209-240 (2000).`` The following paper updates the procedure: –S. Skogestad, ``Economic plantwide control’’, Book chapter in V. Kariwala and V.P. Rangaiah (Eds), Plant-Wide Control: Recent Developments and Applications”, Wiley (2012). More information: http://www.nt.ntnu.no/users/skoge/plantwide

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