1. 1 ECONOMIC PLANTWIDE CONTROL: Control structure design for complete processing plants using self-optimizing control Sigurd Skogestad Department of Chemical Engineering Norwegian University of Science and Tecnology (NTNU) Trondheim, Norway Feb. 2015 TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: A A A

2. 2 Abstract and bio ECONOMIC PLANTWIDE CONTROL: Control structure design for complete processing plants. Sigurd Skogestad, Department of Chemical Engineering, Norwegian University of Science and Technology (NTNU), Trondheim, Norway Abstract: A chemical plant may have thousands of measurements and control loops. By the term plantwide control it is not meant the tuning and behavior of each of these loops, but rather the control philosophy of the overall plant with emphasis on the structural decisions. 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. Bio: Sigurd Skogestad received his Ph.D. degree from the California Institute of Technology, Pasadena, USA in 1987. He has been a full professor at Norwegian University of Science and Technology (NTNU), Trondheim, Norway since 1987 and he was Head of the Department of Chemical Engineering from 1999 to 2009. He is the principal author, together with Prof. Ian Postlethwaite, of the book "Multivariable feedback control" published by Wiley in 1996 (first edition) and 2005 (second edition). He received the Ted Peterson Award from AIChE in 1989, the George S. Axelby Outstanding Paper Award from IEEE in 1990, the O. Hugo Schuck Best Paper Award from the American Automatic Control Council in 1992, and the Best Paper Award 2004 from Computers and Chemical Engineering. He was an Editor of Automatica during the period 1996-2002. His research interests include the use of feedback as a tool to make the system well-behaved (including self- optimizing control), limitations on performance in linear systems, control structure design and plantwide control, interactions between process design and control, and distillation column design, control and dynamics. He is a Fellow of AIChE and IFAC.

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 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 –Step 3: Select primary controlled variables c=y 1 (CVs) –Step 4: Where set the production rate? (Inventory control) 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 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

12. 12 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)

13. 13 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)

14. 14 Theory: Optimal operation Objectives Present state Model of system Theory: Model of overall system Estimate present state Optimize all degrees of freedom Problems: Model not available Optimization complex Not robust (difficult to handle uncertainty) Slow response time Process control: Excellent candidate for centralized control (Physical) Degrees of freedom CENTRALIZED OPTIMIZER

15. 15 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: on-off + PI-control + nonlinear fixes + some feedforward Same in biological systems

16. 16 Practical operation: Hierarchical structure Manager Process engineer Operator/RTO Operator/”Advanced control”/MPC PID-control u = valves Our Paradigm

17. 17 c s = y 1s MPC PID y 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

18. 18 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» CV = controlled variable

19. 19 “Self-optimizing control”: Translate optimal operation into control objectives: What should we control? CV 1 = c ? (economics) CV 2 = ? (stabilization)

20. 20 Control structure design procedure I Top Down 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 Step 3: Select primary controlled variables c=y 1 (CVs) Step 4: Where set the production rate? (Inventory control) II Bottom Up 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

21. 21 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

22. 22 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

23. 23 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

24. 24 …. 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

25. 25 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 )

26. 26 –Cost to be minimized, J=T –One degree of freedom (u=power) –What should we control? Optimal operation - Runner Optimal operation of runner

27. 27 1. Optimal operation of Sprinter –100m. J=T –Active constraint control: Maximum speed (”no thinking required”) CV = power (at max) Optimal operation - Runner

28. 28 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

29. 29 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)

30. 30 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

31. 31 Step 3. What should we control (c)? (primary controlled variables y 1 =c) Selection of controlled variables c 1.Control active constraints! 2.Unconstrained variables: Control self-optimizing variables!

32. 32 Expected active constraints distillation Both products (D,B) generally have purity specs Valuable product: Purity spec. always active –Reason: Amount of valuable product (D or B) should always be maximized Avoid product “give-away” (“Sell water as methanol”) Also saves energy Control implications: 1.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

33. 33 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 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

34. 34 Operation of Distillation columns in series With given F (disturbance): 4 steady-state DOFs (e.g., L and V in each column) DOF = Degree Of Freedom Ref.: M.G. Jacobsen and S. Skogestad (2011) Energy price: p V =0-0.2 $/mol (varies) Cost (J) = - Profit = p F F + p V (V 1 +V 2 ) – p D1 D 1 – p D2 D 2 – p B2 B 2 > 95% B p D2 =2 $/mol F ~ 1.2mol/s p F =1 $/mol < 4 mol/s < 2.4 mol/s > 95% C p B2 =1 $/mol N=41 α AB =1.33 N=41 α BC =1.5 > 95% A p D1 =1 $/mol QUIZ: What are the expected active constraints? 1. Always. 2. For low energy prices. Example /QUIZ 1 = ==

35. 35 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

36. 36 Control of Distillation columns in series Given LC PC QUIZ. Assume low energy prices (pV=0.01 $/mol). How should we control the columns? HINT: CONTROL ACTIVE CONSTRAINTS Red: Basic regulatory loops QUIZ 2

37. 37 Control of Distillation columns in series Given LC PC Red: Basic regulatory loops CC xBxB x BS =95% MAX V1 MAX V2 SOLUTION QUIZ 2 UNCONSTRAINED CV=?

38. 38 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

39. 39 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

40. 40 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%

41. 41 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:

42. 42 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

43. 43 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

44. 44 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)

45. 45 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

46. 46 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

47. 47 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

48. 48 Example. Optimal operation = max. throughput. Want tight bottleneck control to reduce backoff! Time Back-off = Lost production CV = Active constraint

49. 49 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%

50. 50 Unconstrained optimum: Control “self-optimizing” variable. Which variable is best? Often not obvious (marathon runner)

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 Control of Distillation columns in series Given LC PC Red: Basic regulatory loops CC xBxB x BS =95% MAX V1 MAX V2 CC xBxB x AS =2.1% Cost (J) = - Profit = p F F + p V (V 1 +V 2 ) – p D1 D 1 – p D2 D 2 – p B2 B 2

53. 53 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

54. 54 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

55. 55 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 ?

56. 56 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

57. 57 c = T J1 – T J2 T J = JÄSCHKE TEMPERATURE

58. 58 c = T J1 – TJ2 T J = JÄSCHKE TEMPERATURE

59. 59 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

60. 60 H Ideal: c = J u In practise: c = H y. Task: Determine H!

61. 61 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

62. 62 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

63. 63 Optimal operation Cost J Controlled variable c c opt J opt Unconstrained optimum

64. 64 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

65. 65 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

66. 66 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

67. 67 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

68. 68 Optimal measurement combination H Candidate measurements (y): Include also inputs u

69. 69 Optimal measurement combination: Nullspace method Want optimal value of c independent of disturbances  Δc opt = 0 ∙Δ d 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 Comment: Nullspace method is equivalent to J u =0

70. 70 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!)

71. 71 More general (“exact local method”) With measurement noise -Can evaluate specific H -Can find optimal single measurement (use code by Kariwala and..) -Find optimal combination, analytical formula -No measurement error: HF=0 (nullspace method) -With measuremeng error: Minimize GF c -Maximum gain rule

72. 72 “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

73. 73 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

74. 74 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!

75. 75 CO2 cycle: Maximum gain rule

76. 76 Refrigeration cycle: Proposed control structure Control c= “temperature-corrected high pressure”. k = -8.5 bar/K

77. 77 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

78. 78 Proposed control structure with given feed

79. 79 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

80. 80 Production rate set at inlet : Inventory control in direction of flow* * Required to get “local-consistent” inventory controlC TPM

81. 81 Production rate set at outlet: Inventory control opposite flow TPM

82. 82 Production rate set inside process TPM Radiating inventory control around TPM (Georgakis et al.)

83. 83 “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!

84. 84 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

85. 85 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 = ?

86. 86 “ 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

87. 87 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

88. 88 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)

89. 89 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)

90. 90 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.

91. 91 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

92. 92 ”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

93. 93 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

94. 94 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

95. 95 Summary: Rules for plantwide control Here we present a set of simple rules for economic plantwide control to facilitate a close-to-optimal control structure design in cases where the optimization of the plant model is not possible. the rules may be conflicting in some cases and in such cases, human reasoning is strongly advised.

96. 96 Rule 1: Control the active constraints. In general, process optimization is required to determine the active constraints, but in many cases these can be identified based on a good process knowledge and engineering insight. Here is one useful rule: Rule 1A: The purity constraint of the valuable product is always active and should be controlled. This follows, because we want to maximize the amount of valuable product and avoid product “give away” (Jacobsen and Skogestad, 2011). Thus, we should always control the purity of the valuable product at its specification. For “cheap” products we may want to overpurify (purity constraint may not be active) because this may reduce the loss of a more valuable component. In other cases, we must rely on our process knowledge and engineering insight. For reactors with simple kinetics, we usually find that, the reaction and conversion rates are maximized by operating at maximum temperature and maximum volume (liquid phase reactor). For gas phase reactor, high pressure may increase the reaction rate, but this must be balanced against the compression costs. Rules for Step S3: Selection of primary (economic) controlled variables, CV1

97. 97 Rule 2: (for remaining unconstrained steady-state degrees of freedom, if any): Control the “self-optimizing” variables. This choice is usually not obvious, as there may be several alternatives, so this rule is in itself not very helpful. The ideal self-optimizing variable, at least, if it can be measured accurately, is the gradient of the cost function. Ju, which should be zero for any disturbance. Unfortunately, it is rarely possible to measure this variable directly and the “self-optimizing” variable may be viewed as an estimate of the gradient Ju The two main properties of a good “self-optimizing” (CV1=c) variable are: 1.Its optimal value is insensitive to disturbances (such that F = dc opt /dd is small) 2.It is sensitive to the plant inputs (so the process scaled gain G = dc/du is large). The following rule shows how to combine the two desired properties: Rule 2A: Select the set CV1=c such that the ratio G -1 F is minimized. This rule is often called the “maximum scaled gain rule”.

98. 98 Rule 3: (for remaining unconstrained steady-state degrees of freedom, if any): Never try to control the cost function J (or any other variable that reaches a maximum or minimum at the optimum) First, the cost function J has no sensitivity to the plant inputs at the optimal point and so G = 0 which violates Rule 2A. Second, if we specify J lower than its optimal value, then clearly, the operation will be infeasible Also, specifying J higher than its optimal value is problematic, as we have multiplicity of solutions. As mentioned above, rather controlling the cost J, we should control its gradient, Ju. u J J min J>J min J<J min ?

99. 99 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)

100. 10 0 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.

101. 10 1 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.

102. 10 2 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).

103. 10 3 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).

104. 10 4 Rule 9: “Pair-close” rule: The pairings should be selected such that, effective delays and loop interactions are minimal.

105. 10 5 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.

106. 10 6 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.

107. 10 7 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

108. 10 8 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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