1. 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

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

3. Sigurd Skogestad 1955: Born in Flekkefjord, Norway
1978: MS (Siv.ing.) in chemical engineering at NTNU
: Worked at Norsk Hydro co. (process simulation)
1987: PhD from Caltech (supervisor: Manfred Morari)
1987-present: Professor of chemical engineering at NTNU
: 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)

4. 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

5. 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
...

6. 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

7. 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]

8. 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: ) 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

9. 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)

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

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

12. 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

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

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

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

16. 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

17. 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

18. 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?

19. 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”)

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

21. 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

22. 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

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

24. 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

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

26. 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 = [ ]’
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 v
Control c = constant -> hr increases when v decreases (OK uphill!)

27. ”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

28. 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)

29. 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

30. 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

31. 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 = $/kg
Stream 3
95 → 97 octane
p3 = $/kg
Stream 4
2 % benzene
p4 = $/kg
Product
> 98 octane
< 1 % benzene
Disturbance

32. Optimal solution Degrees of freedom u = (m1 m2 m3 m4 )T
Optimization problem: Minimize
J = i pi mi = (0.1 + m1) m m m m4
subject to
m1 + m2 + m3 + m4 = 1
m1 ¸ 0; m2 ¸ 0; m3 ¸ 0; m4 ¸ 0
m1 · 0.4
99 m m m m4 ¸ 98 (octane constraint)
2 m4 · (benzene constraint)
Nominal optimal solution (d* = 95):
uopt = ( )T ) Jopt= $
Optimal solution with d=octane stream 3=97:
uopt = ( )T ) Jopt= $
3 active constraints ) 1 unconstrained degree of freedom

33. 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 ) Loss = $
c=m2 is constant at ) Infeasible (cannot satisfy octane = 98)
c=m3 is constant at ) Loss = $
Optimal combination of measurements
c = h1 m1 + h2 m2 + h3 ma
From optimization:  mopt = F  d where sensitivity matrix F = ( )T
Requirement: HF = 0 )
-0.03 h1 – 0.06 h 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 ) Loss = 0
c = 3 m1 + m3 is constant at ) Loss = 0
c = 1.5 m2 + m3 is constant at ) Loss = 0
Easily implemented in control system

34. Example of practical implementation of optimal blendingFC 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 =
0.5
m1 = cs 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

35. 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

36. 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