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 CVs) 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 CVs) ? 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 0s and some 1s (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 Jmin: Forces us to be nonoptimal (two steady states: may require strange operation) u J J min J>J min J 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 Sigurds 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)