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Self-optimizing control Theory 1 Self-optimizing control Theory 1

Outline • Skogestad procedure for control structure design I Top Down • Step S Outline • Skogestad procedure for control structure design I Top Down • Step S 1: Define operational objective (cost) and constraints • Step S 2: Identify degrees of freedom and optimize operation for disturbances • Step S 3: Implementation of optimal operation – What to control ? (primary CV’s) – Active constraints – Self-optimizing variables for unconstrained, c=Hy • Step S 4: Where set the production rate? (Inventory control) II Bottom Up • Step S 5: Regulatory control: What more to control (secondary CV’s) ? • Step S 6: Supervisory control • Step S 7: Real-time optimization 2

Step S 3: Implementation of optimal operation • Optimal operation for given d*: minu Step S 3: Implementation of optimal operation • Optimal operation for given d*: minu J(u, x, d) subject to: Model equations: Operational constraints: → uopt(d*) f(u, x, d) = 0 g(u, x, d) < 0 Problem: Usally cannot keep uopt constant because disturbances d change 3 How should we adjust the degrees of freedom (u)? What should we control?

“Optimizing Control” y 4 “Optimizing Control” y 4

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

Definition of self-optimizing control Acceptable loss ) self-optimizing control “Self-optimizing control is when we Definition of self-optimizing control Acceptable loss ) 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). ” 6 Reference: S. Skogestad, “Plantwide control: The search for the self-optimizing control structure'', Journal of Process Control, 10, 487 -507 (2000).

Optimal operation - Runner J=T Marathon runner copt c=heart rate select one measurement c Optimal operation - Runner J=T Marathon runner copt c=heart rate select one measurement c = heart rate 8 • 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 (cs)

Unconstrained optimum Optimal operation Cost J Jopt copt 11 Controlled variable c Unconstrained optimum Optimal operation Cost J Jopt copt 11 Controlled variable c

Unconstrained optimum Optimal operation Cost J d Loss 1 Jopt Loss 2 n copt Unconstrained optimum Optimal operation Cost J d Loss 1 Jopt Loss 2 n copt 12 Controlled variable c Two problems: • 1. Optimum moves because of disturbances d: copt(d) • 2. Implementation error, c = copt + n

Unconstrained degrees of freedom The ideal “self-optimizing” variable is the gradient, Ju c = Unconstrained degrees of freedom The ideal “self-optimizing” variable is the gradient, Ju c = J/ u = Ju – Keep gradient at zero for all disturbances (c = Ju=0) – Problem: Usually no measurement of gradient cost J Ju<0 Ju=0 uopt Ju 0 13 u

J J>Jmin J<Jmin ? u Unconstrained optimum: NEVER try to control a variable that J J>Jmin J Jmin: Forces us to be nonoptimal (two steady states: may require strange operation)

Parallel heat exchangers What should we control? Th 1, UA 1 T 1 α Parallel heat exchangers What should we control? Th 1, UA 1 T 1 α T 0 T m [kg/s] T 2 1 -α Th 2, UA 2 15 • 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 T 1=T 2? Not quite • Optimal: Control Ju=0

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

17 17

c = JÄSCHKE TEMPERATURE 18 c = JÄSCHKE TEMPERATURE 18

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

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

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

CV=Measurement combination Nullspace method (HF=0) gives Ju=0 Proof: • 23 Proof. Appendix B in: CV=Measurement combination Nullspace method (HF=0) gives Ju=0 Proof: • 23 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) ,

CV=Measurement combination Example. Nullspace Method for Marathon runner u = power, d = slope CV=Measurement combination Example. Nullspace Method for Marathon runner u = power, d = slope [degrees] y 1 = hr [beat/min], y 2 = v [m/s] F = dyopt/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!) 24

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

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

Example: CO 2 refrigeration cycle J = W s (work supplied) DOF = u Example: CO 2 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? 27 p. H

CO 2 refrigeration cycle Step 1. One (remaining) degree of freedom (u=z) Step 2. CO 2 refrigeration cycle Step 1. One (remaining) degree of freedom (u=z) Step 2. Objective function. J = Ws (compressor work) Step 3. Optimize operation for disturbances (d 1=TC, d 2=TH, d 3=UA) • Optimum always unconstrained Step 4. Implementation of optimal operation • No good single measurements (all give large losses): – ph, Th, z, … • Nullspace method: Need to combine nu+nd=1+3=4 measurements to have zero disturbance loss • Simpler: Try combining two measurements. Exact local method: – c = h 1 ph + h 2 Th = ph + k Th; k = -8. 53 bar/K • Nonlinear evaluation of loss: OK! 28

CO 2 cycle: Maximum gain rule 29 CO 2 cycle: Maximum gain rule 29

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

Importance of optimal operation for CO 2 capturing process Dependency of equivalent energy in Importance of optimal operation for CO 2 capturing process Dependency of equivalent energy in CO 2 capture plant verses recycle amine flowrate An amine absorption/stripping CO 2 capturing process* *Figure from: Toshiba (2008). Toshiba to Build Pilot Plant to Test CO 2 Capture Technology. http: //www. japanfs. org/en/pages/028843. html. 31

Case study Control structure design using self-optimizing control for economically optimal CO 2 recovery* Case study Control structure design using self-optimizing control for economically optimal CO 2 recovery* Step S 1. Objective function= J = energy cost + cost (tax) of released CO 2 to air 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 k. W (nominal +20%) Step S 2. (a) 10 degrees of freedom: 8 valves + 2 pumps 4 levels without steady state effect: absorber 1, stripper 2, make up tank 1 Disturbances: flue gas flowrate, CO 2 composition in flue gas + active constraints (b) Optimization using Unisim steady-state simulator. Mode I = Region I (nominal feedrate): No inequality constraints active 2 unconstrained degrees of freedom =10 -4 -4 Step S 3 (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. 32 *M. Panahi and S. Skogestad, ``Economically efficient operation of CO 2 capturing process, part I: Self-optimizing procedure for selecting the best controlled variables'', Chemical Engineering and Processing, 50, 247 -253 (2011).

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

Mode II: large feedrates of flue gas (+30%) Feedrate flue gas (kmol/hr) Self-optimizing CVs Mode II: large feedrates of flue gas (+30%) Feedrate flue gas (kmol/hr) Self-optimizing CVs in region I CO 2 recovery % Temperature tray no. 16 °C Reboiler duty (k. W) Cost (USD/ton) Optimal nominal point 219. 3 95. 26 106. 9 1161 2. 49 +5% feedrate 230. 3 95. 26 106. 9 1222 2. 49 +10% feedrate 241. 2 95. 26 106. 9 1279 2. 49 +15% feedrate 252. 2 95. 26 106. 9 1339 2. 49 +19. 38%, when reboiler duty saturates 261. 8 95. 26 106. 9 1393 (+20%) 2. 50 +30% feedrate (reoptimized) 285. 1 91. 60 103. 3 1393 2. 65 region II max 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 34

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

Conditions for switching between regions of active constraints (“supervisory control”) • Within each region 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 ca = c, a, constraint 2. Control self-optimizing variables at cso = c, so, optimal • Define in each region i: • Keep track of ci (active constraints and “self-optimizing” variables) in all regions i • Switch to region i when element in ci changes sign 36

Example – switching policies CO 2 plant (”supervisory control”) • Assume operating in region Example – switching policies CO 2 plant (”supervisory control”) • Assume operating in region I (unconstrained) – with CV=CO 2 -recovery=95. 26% • When reach maximum Q: Switch to Q=Qmax (Region II) (obvious) – CO 2 -recovery will then drop below 95. 26% • When CO 2 -recovery exceeds 95. 26%: Switch back to region I !!! 37

Conclusion optimal operation ALWAYS: 1. Control active constraints and control them tightly!! – Good 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 38

Sigurd’s rules for CV selection 1. 2. Always control active constraints! (almost always) Purity Sigurd’s rules for CV selection 1. 2. Always control active constraints! (almost always) 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 – – 39 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 = J - Then setting J < Jmin: Gives infeasible operation (cannot meet constraints) - and setting J > Jmin: Forces us to be nonoptimal (which may require strange operation; see Exercise on recycle process)