1 Self-optimizing control Theory

2 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) –Active constraints –Self-optimizing variables for unconstrained, c=Hy 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

3 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?

4 y “Optimizing Control”

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

6 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, (2000). Acceptable loss ) self-optimizing control

7 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

8 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

9 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

10 Central bank. J = welfare. u = interest rate. c=inflation rate (2.5%) Cake baking. J = nice taste, u = heat input. c = Temperature (200C) Business, J = profit. c = ”Key performance indicator (KPI), e.g. –Response time to order –Energy consumption pr. kg or unit –Number of employees –Research spending Optimal values obtained by ”benchmarking” Investment (portofolio management). J = profit. c = Fraction of investment in shares (50%) Biological systems: –”Self-optimizing” controlled variables c have been found by natural selection –Need to do ”reverse engineering” : Find the controlled variables used in nature From this possibly identify what overall objective J the biological system has been attempting to optimize Further examples self-optimizing control Define optimal operation (J) and look for ”magic” variable (c) which when kept constant gives acceptable loss (self- optimizing control) Unconstrained degrees of freedom

11 Optimal operation Cost J Controlled variable c c opt J opt Unconstrained optimum

12 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

13 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

14 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 ?

15 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! Control T1=T2? Not quite Optimal: Control J u =0 Th1, UA1

16 Jäsche temperature Can show that J u =0 is equivalent to (see exercise for simplified case) T J1 – T J2 = 0 where T J is «Jäschke temperature» for each branch: T J1 = (T 1 -T 0 ) 2 / (T h1 – T 0 ) T J2 = (T 2 -T 0 ) 2 / (T h2 – T 0 ) T h1, T h2 = Inlet tempetaure on other side («hot») of heat exchanger T 1, T 2 = outlet temperature T 0 = feed temperature

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18 c = JÄSCHKE TEMPERATURE

19 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

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

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

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

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

24 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 = [ ]’ 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 v Control c = constant -> hr increases when v decreases (OK uphill!) CV=Measurement combination

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

26 BADGood Good candidate controlled variables c=Hy (for self-optimizing control) 1.The optimal value of c should be insensitive to disturbances Want small dc opt /dd = HF 2.The value of c should be sensitive to changes in the degrees of freedom Want large dc/du = Hg y (same as wanting flat optimum) 2.c should be easy to measure and control Proof: This keeps the gradient close to zero since CV=Single measurements

27 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

28 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 = bar/K Nonlinear evaluation of loss: OK!

29 CO2 cycle: Maximum gain rule

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

31 An amine absorption/stripping CO 2 capturing process * Dependency of equivalent energy in CO 2 capture plant verses recycle amine flowrate *Figure from: Toshiba (2008). Toshiba to Build Pilot Plant to Test CO2 Capture Technology. Importance of optimal operation for CO 2 capturing process

32 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, (2011).``Economically efficient operation of CO2 capturing process, part I: Self-optimizing procedure for selecting the best controlled variables'', Step S2. (a) 10 degrees of freedom: 8 valves + 2 pumps Disturbances: flue gas flowrate, CO 2 composition in flue gas + active constraints (b) Optimization using Unisim steady-state simulator. Mode I = Region I (nominal feedrate): No inequality constraints active 2 unconstrained degrees of freedom = Case study

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

34 Mode II: large feedrates of flue gas (+30%) Feedrate flue gas (kmol/hr ) Self-optimizing CVs in region IReboiler duty (kW) Cost (USD/ton) CO 2 recovery % Temperature tray no. 16 °C Optimal nominal point % feedrate % feedrate % feedrate %, when reboiler duty saturates (+20%) % feedrate (reoptimized) 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

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

36 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

37 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 !!!

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

39 Sigurd’s rules for CV selection 1.Always control active constraints! (almost always) 2.Purity constraint on expensive product always active (no overpurification): (a) "Avoid product give away" (e.g., sell water as expensive product) (b) Save energy (costs energy to overpurify) 3.Unconstrained optimum: NEVER try to control a variable that reaches max or min at the optimum –In particular, never try to control directly the cost J –- Assume we want to minimize J (e.g., J = V = energy) - and we make the stupid choice os selecting CV = V = J - Then setting J Jmin: Forces us to be nonoptimal (which may require strange operation; see Exercise on recycle process)