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From process control to business control:A systematic approach for CV-selection

Sigurd Skogestad

Department of Chemical Engineering

Norwegian University of Science and Technology (NTNU)

Trondheim

PIC-konferens, Stockholm, 29. mai 2013

CV = Controlled variable

Porsgrunn, 12 June 2013

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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)

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Optimal operation of systems (outline)• «System» = Chemical process plant, Airplane, Business, ….

General approach

1. Classify variables (MV=u, DV=d, Measurements y)

2. Obtain model (dynamic or steady state)

3. Define optimal operation: Cost J, constraints, disturbances

4. Find optimal operation: Solve optimization problem

5. Implement optimal operation: What should we control (CV=c=Hy) ?• Need one CV for each MV

• Applications– Runner– KPI’s– Process control– ...

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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

Systemu

d x

y 2. Model of system (typical):dx/dt = f(x,u,d)

y = fy(x,u,d)

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3. Define optimal operation (economics)

What are we going to use our degrees of freedom u (MVs) for?1.Define scalar cost function J(u,x,d)

2.Identify constraints– min. and max. flows – Product specifications– Safety limitations– Other operational constraints

J = cost feed + cost energy – value products [$/s]

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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)

value products

cost energy (heating+ cooling)

cost feed

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4. Find optimal operation:Solve optimization problem to find uopt(d)

Find optimal constraint regions (as a function of 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) = 0Constraints: g(u,x,d) < 0

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Energyprice

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

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5. Implement optimal operation

• Our task as control engineers!

• Theoretical (centralized): Reoptimize continouosly

• Practical (hierarchical): Find one CV for each MV

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Centralized implementation: Optimizing control (theoretically best)

Estimate present state + disturbances d from measurements y and recompute uopt(d)

Problem:COMPLICATED!Requires detailed model and description of uncertainty

y

Implementation of optimal operation

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Hierarchical implementation (practically best)

Direktør

Prosessingeniør

Operatør

Logikk / velgere / operatør

PID-regulator

u = valves

Implementation of optimal operation

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PID

RTO

MPC

Implementation of optimal operation

Hierarchical implementation (practically best)

u = valves

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What should we control?

y1 = c =Hy? (economics)

y2 = H2y (stabilization)

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Self-optimizing Control

Self-optimizing control is when acceptable operation can be achieved using constant set points (c

s)

for the controlled variables c

(without re-optimizing when disturbances occur).

c=cs

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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

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– Cost to be minimized, J=T

– One degree of freedom (u=power)

– What should we control?

Optimal operation - Runner

Optimal operation of runner

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Self-optimizing control: Sprinter (100m)

• 1. Optimal operation of Sprinter, J=T– Active constraint control:

• Maximum speed (”no thinking required”)

Optimal operation - Runner

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• 2. Optimal operation of Marathon runner, J=T

Optimal operation - Runner

Self-optimizing control: Marathon (40 km)

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• 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

Optimal operation - Runner

Self-optimizing control: Marathon (40 km)

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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

Unconstrained degrees of freedom

u

cost J

Ju=0

Ju

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Unconstrained variables

H

measurement noise

steady-statecontrol error

disturbance

controlled variable

/ selection

Ideal: c = Ju

In practise: c = H y

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Candidate controlled variables c for self-optimizing control

Intuitive:

1. 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.

Unconstrained optimum

BADGoodGood

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Nullspace method

No measurement noise (ny=0) CV=Measurement combination

Ref: Alstad & Skogestad, 2007

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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!)

CV=Measurement combination

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Ref: Halvorsen et al. I&ECR, 2003

Alstad et al, , JPC, 2009

”Exact local method” (with measurement noise)

u

J

( )opt ou d

Loss

'd

Controlled variables,c yH

ydG

cs = constant +

+

+

+

+

- K

H

yG y

'yn

cm

u

dW nW

d

optu

CV=Measurement combination

Analytic solution for the case of “full” H

With measurement noise)

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Conclusion Marathon runner

c = heart rate

Simplest: select one measurement

• Simple and robust implementation• Disturbances are indirectly handled by keeping a constant heart rate• May have infrequent adjustment of setpoint (heart rate)

Optimal operation - Runner

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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 spendingOptimal 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

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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

–For example, never try to control directly the cost J• Will give infeasibility

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Example: Optimal blending of gasoline

Stream 1

Stream 2

Stream 3

Stream 4

Product 1 kg/s

Stream 1 99 octane 0 % benzene p1 = (0.1 + m1) $/kg

Stream 2 105 octane 0 % benzene p2 = 0.200 $/kg

Stream 3 95 → 97 octane

0 % benzene p3 = 0.120 $/kg

Stream 4 99 octane 2 % benzene p4 = 0.185 $/kg

Product > 98 octane < 1 % benzene

Disturbance

m1 = ? (· 0.4)

m2 = ?

m3 = ?

m4 = ?

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Optimal solution

• Degrees of freedom

u = (m m2 m3 m4 )T

• Optimization problem: MinimizeJ = i pi mi = (0.1 + m1) m1 + 0.2 m2 + 0.12 m3 + 0.185 m4

subject tom1 + 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

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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 freedom1. c=m1 is constant at 0.126 ) Loss = 0.00036 $

2. c=m2 is constant at 0.196 ) Infeasible (cannot satisfy octane = 98)

3. c=m3 is constant at 0.544 ) Loss = 0.00582 $

• Optimal combination of measurementsc = 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 = 0This 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

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Example of practical implementation of optimal blending

• 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

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

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Conclusion.Optimal operation of systems • «System» = Chemical process plant, Airplane, Business, ….

General approach

1. Classify variables (MV=u, DV=d,)

2. Obtain model (dynamic or steady state)

3. Define optimal operation: Cost J, constraints, disturbances

4. Find optimal operation: Solve optimization problem• Identify active constraint regions

5. Implement optimal operation: What should we control (CV=c) • Need one CV (KPI) for each MV

1. Control active constraints

2. Control «self-optimizing» variables, c=Hy ≈ Ju