Economic plantwide control: A systematic approach for CV-selection Sigurd Skogestad Department of Chemical Engineering Norwegian University of Science and Technology (NTNU) Trondheim COPPE, July 2014 CV = Controlled variable PIC-konferens, Stockholm, 29. mai 2013

d CVs Controller K u = MV Plant y Selected CVs H c = CV = Hy

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 170 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)

CV-selection K Plant H d CVs u = MV y c = CV= Hy Controller K u = MV Plant y Selected CVs H c = CV= Hy Overall operational objective: Minimize economic cost J Goal for selecting CV=Hy: Move economic into control layer “Self-optimizing control”: Can keep CVs constrant

Optimal operation of systems (outline) «System» = Chemical process plant, Airplane, Business, …. General approach Classify variables (MV=u, DV=d, Measurements y) Obtain model (dynamic or steady state) Define optimal operation: Cost J, constraints, disturbances Find optimal operation: Solve optimization problem Implement optimal operation: What should we control (CV=c=Hy) ? Need one CV for each MV Applications Runner KPI’s Process control ...

Optimal operation of a given system 1. Classify variables Manipulated variables (MVs) = Degrees of freedom (inputs): u Disturbance variables (DVs) = “inputs” outside our control: d Measured variables: y (information about the system) States = Variables that define initial state: x Question: How should we set the MVs (inputs u) 3. Quantitative approach: Define scalar cost J + define constraints + define expected disturbances 2. Model of system (typical): dx/dt = f(x,u,d) y = fy(x,u,d) u System y d x

3. 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) Identify constraints min. and max. flows Product specifications Safety limitations Other operational constraints J = cost feed + cost energy – value products [$/s]

Optimal operation distillation column Distillation at steady state with given p and F: 2 DOFs, e.g. L and V (u) Cost to be minimized (economics) J = - P where P= pD D + pB B – pF F – pVV Constraints Purity D: For example xD, impurity · max Purity B: For example, xB, impurity · max Flow constraints: min · D, B, L etc. · max Column capacity (flooding): V · Vmax, etc. Pressure: 1) p given (d) 2) p free (u): pmin · p · pmax Feed: 1) F given (d) 2) F free (u): F · Fmax Optimal operation: Minimize J with respect to steady-state DOFs (u) cost energy (heating+ cooling) value products cost feed

4. Find optimal operation: Solve optimization problem to find uopt(d) Optimize operation with respect to u for given disturbance d (usually steady-state): minu J(u,x,d) subject to: Model equations: f(u,x,d) = 0 Constraints: g(u,x,d) < 0 Find optimal constraint regions (as a function of d)

Example: optimal constraint regions (as a function of d) Energy price

5. Implement optimal operation Our task as control engineers! Theoretical (centralized): Reoptimize continouosly Practical (hierarchical): Find one CV for each MV

Centralized implementation: Optimizing control (theoretically best) Implementation of optimal operation Centralized implementation: Optimizing control (theoretically best) y Estimate present state + disturbances d from measurements y and recompute uopt(d) Problem: COMPLICATED! Requires detailed model and description of uncertainty

Hierarchical implementation (practically best) Implementation of optimal operation Hierarchical implementation (practically best) Direktør Prosessingeniør Operatør Logikk / velgere / operatør PID-regulator u = valves

Hierarchical implementation (practically best) Implementation of optimal operation Hierarchical implementation (practically best) RTO MPC PID u = valves

What should we control? y1 = c =Hy? (economics) y2 = H2y (stabilization)

Self-optimizing Control Self-optimizing control is when acceptable operation can be achieved using constant set points (cs) for the controlled variables c (without re-optimizing when disturbances occur). c=cs

Implementation of optimal operation Idea: Replace optimization by setpoint control Optimal solution is usually at constraints, that is, most of the degrees of freedom u0 are used to satisfy “active constraints”, g(u0,d) = 0 CONTROL ACTIVE CONSTRAINTS! Implementation of active constraints is usually simple. WHAT MORE SHOULD WE CONTROL? Find variables c for remaining unconstrained degrees of freedom u. u cost J

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

Self-optimizing control: Sprinter (100m) Optimal operation - Runner Self-optimizing control: Sprinter (100m) 1. Optimal operation of Sprinter, J=T Active constraint control: Maximum speed (”no thinking required”)

Self-optimizing control: Marathon (40 km) Optimal operation - Runner Self-optimizing control: Marathon (40 km) 2. Optimal operation of Marathon runner, J=T

Self-optimizing control: Marathon (40 km) Optimal operation - Runner Self-optimizing control: Marathon (40 km) Optimal operation of Marathon runner, J=T Any self-optimizing variable c (to control at constant setpoint)? c1 = distance to leader of race c2 = speed c3 = heart rate c4 = level of lactate in muscles

Ideal “self-optimizing” variable Unconstrained degrees of freedom Ideal “self-optimizing” variable The ideal self-optimizing variable c is the gradient: c =  J/ u = Ju Keep gradient at zero for all disturbances (c = Ju=0) Problem: Usually no measurement of gradient, that is, cannot write Ju=Hy u cost J Ju=0 Ju 24

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

Candidate controlled variables c for self-optimizing control Unconstrained optimum Candidate controlled variables c for self-optimizing control Intuitive: The optimal value of c should be insensitive to disturbances 2. Optimum should be flat ( ->insensitive to implementation error). Equivalently: Value of c should be sensitive to degrees of freedom u. BAD Good

CV=Measurement combination No measurement noise (ny=0) CV=Measurement combination Nullspace method Ref: Alstad & Skogestad, 2007

Example. Nullspace Method for Marathon runner CV=Measurement combination Example. Nullspace Method for Marathon runner u = power, d = slope [degrees] y1 = hr [beat/min], y2 = v [m/s] F = dyopt/dd = [0.25 -0.2]’ H = [h1 h2]] HF = 0 -> h1 f1 + h2 f2 = 0.25 h1 – 0.2 h2 = 0 Choose h1 = 1 -> h2 = 0.25/0.2 = 1.25 Conclusion: c = hr + 1.25 v Control c = constant -> hr increases when v decreases (OK uphill!)

”Exact local method” (with measurement noise) CV=Measurement combination ”Exact local method” (with measurement noise) Controlled variables, cs = constant + - K H y cm u u J d Loss Analytic solution for the case of “full” H Ref: Halvorsen et al. I&ECR, 2003 Alstad et al, , JPC, 2009 30

Conclusion Marathon runner Optimal operation - Runner Conclusion Marathon runner Simplest: select one measurement c = heart rate Simple and robust implementation Disturbances are indirectly handled by keeping a constant heart rate May have infrequent adjustment of setpoint (heart rate)

Further examples Central bank. J = welfare. c=inflation rate (2.5%) Cake baking. J = nice taste, 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 identify what overall objective J the biological system has been attempting to optimize

Sigurd’s rules for CV selection 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) Unconstrained optimum: NEVER try to control a variable that reaches max or min at the optimum For example, never try to control directly the cost J Will give infeasibility

Example: Optimal blending of gasoline Stream 1 Stream 2 Stream 3 Stream 4 Product 1 kg/s m2 = ? m3 = ? m4 = ? Stream 1 99 octane 0 % benzene p1 = (0.1 + m1) $/kg Stream 2 105 octane p2 = 0.200 $/kg Stream 3 95 → 97 octane p3 = 0.120 $/kg Stream 4 2 % benzene p4 = 0.185 $/kg Product > 98 octane < 1 % benzene Disturbance

Optimal solution Degrees of freedom u = (m1 m2 m3 m4 )T Optimization problem: Minimize J = i pi mi = (0.1 + m1) m1 + 0.2 m2 + 0.12 m3 + 0.185 m4 subject to m1 + m2 + m3 + m4 = 1 m1 ¸ 0; m2 ¸ 0; m3 ¸ 0; m4 ¸ 0 m1 · 0.4 99 m1 + 105 m2 + 95 m3 + 99 m4 ¸ 98 (octane constraint) 2 m4 · 1 (benzene constraint) Nominal optimal solution (d* = 95): uopt = (0.26 0.196 0.544 0)T ) Jopt=0.13724 $ Optimal solution with d=octane stream 3=97: uopt = (0.20 0.075 0.725 0)T ) Jopt=0.13724 $ 3 active constraints ) 1 unconstrained degree of freedom

Implementation of optimal solution Available ”measurements”: y = (m1 m2 m3 m4)T Control active constraints: Keep m4 = 0 Adjust one (or more) flow such that m1+m2+m3+m4 = 1 Adjust one (or more) flow such that product octane = 98 Remaining unconstrained degree of freedom c=m1 is constant at 0.126 ) Loss = 0.00036 $ c=m2 is constant at 0.196 ) Infeasible (cannot satisfy octane = 98) c=m3 is constant at 0.544 ) Loss = 0.00582 $ Optimal combination of measurements c = h1 m1 + h2 m2 + h3 ma From optimization:  mopt = F  d where sensitivity matrix F = (-0.03 -0.06 0.09)T Requirement: HF = 0 ) -0.03 h1 – 0.06 h2 + 0.09 h3 = 0 This has infinite number of solutions (since we have 3 measurements and only ned 2): c = m1 – 0.5 m2 is constant at 0.162 ) Loss = 0 c = 3 m1 + m3 is constant at 1.32 ) Loss = 0 c = 1.5 m2 + m3 is constant at 0.83 ) Loss = 0 Easily implemented in control system

Example of practical implementation of optimal blending FC OC mtot.s = 1 kg/s mtot m3 m4 = 0 kg/s Octanes = 98 Octane m2 Stream 2 Stream 1 Stream 3 Stream 4 cs = 0.162 0.5 m1 = cs + 0.5 m2 Octane varies Selected ”self-optimizing” variable: c = m1 – 0.5 m2 Changes in feed octane (stream 3) detected by octane controller (OC) Implementation is optimal provided active constraints do not change Price changes can be included as corrections on setpoint cs

Conclusion. Optimal operation of systems «System» = Chemical process plant, Airplane, Business, …. General approach Classify variables (MV=u, DV=d,) Obtain model (dynamic or steady state) Define optimal operation: Cost J, constraints, disturbances Find optimal operation: Solve optimization problem Identify active constraint regions Implement optimal operation: What should we control (CV=c) Need one CV (KPI) for each MV Control active constraints Control «self-optimizing» variables, c=Hy ≈ Ju

CV-selection K Plant H d CVs u = MV y c = CV= Hy Controller K u = MV Plant y Selected CVs H c = CV= Hy Overall operational objective: Minimize economic cost J Goal for selecting CV=Hy: Move economic into control layer “Self-optimizing control”: Can keep CVs constrant