PLANTWIDE CONTROL
How to design the control system for a
complete plant in a systematic manner
Sigurd Skogestad
Department of Chemical Engineering
Norwegian University of Science and Tecnology (NTNU)
Trondheim, Norway
Petrobras, March 2010
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Summary.
Systematic procedure for plantwide control
1.
Start “top-down” with economics:
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2.
Optimize steady-state operation
Identify active constraints (should normally be tightly controlled to
maxize profit)
For remaining degrees of freedom: Select controlled variables c based on
self-optimizing control.
Regulatory control I: Decide on how to move mass through the plant:
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3.
Regulatory control II: “Bottom-up” stabilization of the plant
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Control variables to stop “drift” (sensitive temperatures, pressures, ....)
Pair variables to avoid interaction and saturation
Finally: make link between “top-down” and “bottom up”.
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“Advanced control” system (MPC):
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Where to set the throughput (usually: feed)
Propose “local-consistent” inventory (level) control structure.
CVs: Active constraints and self-optimizing economic variables +
look after variables in layer below (e.g., avoid saturation)
MVs: Setpoints to regulatory control layer.
Coordinates within units and possibly between units
cs
Summary and references
• The following paper summarizes the procedure:
– S. Skogestad, ``Control structure design for complete chemical plants'',
Computers and Chemical Engineering, 28 (1-2), 219-234 (2004).
• There are many approaches to plantwide control as discussed in the
following review paper:
– T. Larsson and S. Skogestad, ``Plantwide control: A review and a new
design procedure'' Modeling, Identification and Control, 21, 209-240
(2000).
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S. Skogestad ``Plantwide control: the search for the self-optimizing control structure'', J. Proc. Control, 10, 487-507 (2000).
S. Skogestad, ``Self-optimizing control: the missing link between steady-state optimization and control'', Comp.Chem.Engng., 24, 569-575 (2000).
I.J. Halvorsen, M. Serra and S. Skogestad, ``Evaluation of self-optimising control structures for an integrated Petlyuk distillation column'', Hung. J.
of Ind.Chem., 28, 11-15 (2000).
T. Larsson, K. Hestetun, E. Hovland, and S. Skogestad, ``Self-Optimizing Control of a Large-Scale Plant: The Tennessee Eastman Process'', Ind.
Eng. Chem. Res., 40 (22), 4889-4901 (2001).
K.L. Wu, C.C. Yu, W.L. Luyben and S. Skogestad, ``Reactor/separator processes with recycles-2. Design for composition control'', Comp. Chem.
Engng., 27 (3), 401-421 (2003).
T. Larsson, M.S. Govatsmark, S. Skogestad, and C.C. Yu, ``Control structure selection for reactor, separator and recycle processes'', Ind. Eng.
Chem. Res., 42 (6), 1225-1234 (2003).
A. Faanes and S. Skogestad, ``Buffer Tank Design for Acceptable Control Performance'', Ind. Eng. Chem. Res., 42 (10), 2198-2208 (2003).
I.J. Halvorsen, S. Skogestad, J.C. Morud and V. Alstad, ``Optimal selection of controlled variables'', Ind. Eng. Chem. Res., 42 (14), 3273-3284 (2003).
A. Faanes and S. Skogestad, ``pH-neutralization: integrated process and control design'', Computers and Chemical Engineering, 28 (8), 1475-1487
(2004).
S. Skogestad, ``Near-optimal operation by self-optimizing control: From process control to marathon running and business systems'', Computers and
Chemical Engineering, 29 (1), 127-137 (2004).
E.S. Hori, S. Skogestad and V. Alstad, ``Perfect steady-state indirect control'', Ind.Eng.Chem.Res, 44 (4), 863-867 (2005).
M.S. Govatsmark and S. Skogestad, ``Selection of controlled variables and robust setpoints'', Ind.Eng.Chem.Res, 44 (7), 2207-2217 (2005).
V. Alstad and S. Skogestad, ``Null Space Method for Selecting Optimal Measurement Combinations as Controlled Variables'', Ind.Eng.Chem.Res,
46 (3), 846-853 (2007).
S. Skogestad, ``The dos and don'ts of distillation columns control'', Chemical Engineering Research and Design (Trans IChemE, Part A), 85 (A1), 1323 (2007).
E.S. Hori and S. Skogestad, ``Selection of control structure and temperature location for two-product distillation columns'', Chemical Engineering
Research and Design (Trans IChemE, Part A), 85 (A3), 293-306 (2007).
A.C.B. Araujo, M. Govatsmark and S. Skogestad, ``Application of plantwide control to the HDA process. I Steady-state and self-optimizing
control'', Control Engineering Practice, 15, 1222-1237 (2007).
A.C.B. Araujo, E.S. Hori and S. Skogestad, ``Application of plantwide control to the HDA process. Part II Regulatory control'', Ind.Eng.Chem.Res,
46 (15), 5159-5174 (2007).
V. Kariwala, S. Skogestad and J.F. Forbes, ``Reply to ``Further Theoretical results on Relative Gain Array for Norn-Bounded Uncertain systems''''
Ind.Eng.Chem.Res, 46 (24), 8290 (2007).
V. Lersbamrungsuk, T. Srinophakun, S. Narasimhan and S. Skogestad, ``Control structure design for optimal operation of heat exchanger
networks'', AIChE J., 54 (1), 150-162 (2008). DOI 10.1002/aic.11366
T. Lid and S. Skogestad, ``Scaled steady state models for effective on-line applications'', Computers and Chemical Engineering, 32, 990-999 (2008). T.
Lid and S. Skogestad, ``Data reconciliation and optimal operation of a catalytic naphtha reformer'', Journal of Process Control, 18, 320-331 (2008).
E.M.B. Aske, S. Strand and S. Skogestad, ``Coordinator MPC for maximizing plant throughput'', Computers and Chemical Engineering, 32, 195-204
(2008).
A. Araujo and S. Skogestad, ``Control structure design for the ammonia synthesis process'', Computers and Chemical Engineering, 32 (12), 29202932 (2008).
E.S. Hori and S. Skogestad, ``Selection of controlled variables: Maximum gain rule and combination of measurements'', Ind.Eng.Chem.Res, 47 (23),
9465-9471 (2008).
V. Alstad, S. Skogestad and E.S. Hori, ``Optimal measurement combinations as controlled variables'', Journal of Process Control, 19, 138-148 (2009)
E.M.B. Askoe and S. Skogestad, ”Consistent inventory control”, Submitted (2009)
Plantwide control intro course: Contents
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Overview of plantwide control
Selection of primary controlled variables based on economic : The link
between the optimization (RTO) and the control (MPC; PID) layers
- Degrees of freedom
- Optimization
- Self-optimizing control
- Applications
- Many examples
Where to set the production rate and bottleneck
Design of the regulatory control layer ("what more should we
control")
- stabilization
- secondary controlled variables (measurements)
- pairing with inputs
- controllability analysis
- cascade control and time scale separation.
Design of supervisory control layer
- Decentralized versus centralized (MPC)
- Design of decentralized controllers: Sequential and independent design
- Pairing and RGA-analysis
Summary and case studies
Outline
• Control structure design (plantwide control)
• A procedure for control structure design
I Top Down
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Step 1: Degrees of freedom
Step 2: Operational objectives (optimal operation)
Step 3: What to control ? (primary CV’s) (self-optimizing control)
Step 4: Where set the production rate? (Inventory control)
II Bottom Up
• Step 5: Regulatory control: What more to control (secondary CV’s) ?
• Step 6: Supervisory control
• Step 7: Real-time optimization
• Case studies
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Main message
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1. Control for economics (Top-down steady-state arguments)
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Primary controlled variables c = y1 :
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2. Control for stabilization (Bottom-up; regulatory PID control)
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Secondary controlled variables y2 (“inner cascade loops”)
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Control active constraints
For remaining unconstrained degrees of freedom: Look for “self-optimizing”
variables
Control variables which otherwise may “drift”
Both cases: Control “sensitive” variables (with a large gain)!
Idealized view of control
(“Ph.D. control”)
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Practice: Tennessee Eastman challenge
problem (Downs, 1991)
(“PID control”)
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How we design a control system for a
complete chemical plant?
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Where do we start?
What should we control? and why?
etc.
etc.
• Alan Foss (“Critique of chemical process control theory”, AIChE
Journal,1973):
The central issue to be resolved ... is the determination of control system
structure. Which variables should be measured, which inputs should be
manipulated and which links should be made between the two sets?
There is more than a suspicion that the work of a genius is needed here,
for without it the control configuration problem will likely remain in a
primitive, hazily stated and wholly unmanageable form. The gap is
present indeed, but contrary to the views of many, it is the theoretician
who must close it.
• Carl Nett (1989):
Minimize control system complexity subject to the achievement of accuracy
specifications in the face of uncertainty.
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Control structure design
• Not the tuning and behavior of each control loop,
• But rather the control philosophy of the overall plant with emphasis on
the structural decisions:
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Selection of controlled variables (“outputs”)
Selection of manipulated variables (“inputs”)
Selection of (extra) measurements
Selection of control configuration (structure of overall controller that
interconnects the controlled, manipulated and measured variables)
– Selection of controller type (LQG, H-infinity, PID, decoupler, MPC etc.).
• That is: Control structure design includes all the decisions we need
make to get from ``PID control’’ to “Ph.D” control
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Process control:
“Plantwide control” = “Control structure
design for complete chemical plant”
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Large systems
Each plant usually different – modeling expensive
Slow processes – no problem with computation time
Structural issues important
– What to control? Extra measurements, Pairing of loops
Previous work on plantwide control:
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•Page Buckley (1964) - Chapter on “Overall process control” (still industrial practice)
•Greg Shinskey (1967) – process control systems
•Alan Foss (1973) - control system structure
•Bill Luyben et al. (1975- ) – case studies ; “snowball effect”
•George Stephanopoulos and Manfred Morari (1980) – synthesis of control structures for chemical processes
•Ruel Shinnar (1981- ) - “dominant variables”
•Jim Downs (1991) - Tennessee Eastman challenge problem
•Larsson and Skogestad (2000): Review of plantwide control
• Control structure selection issues are identified as important also in
other industries.
Professor Gary Balas (Minnesota) at ECC’03 about flight control at Boeing:
The most important control issue has always been to select the right
controlled variables --- no systematic tools used!
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Main objectives control system
1. Stabilization
2. Implementation of acceptable (near-optimal) operation
ARE THESE OBJECTIVES CONFLICTING?
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Usually NOT
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Different time scales
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Stabilization doesn’t “use up” any degrees of freedom
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Stabilization fast time scale
Reference value (setpoint) available for layer above
But it “uses up” part of the time window (frequency range)
Dealing with complexity
Main simplification: Hierarchical decomposition
The controlled variables (CVs)
interconnect the layers
Process control
OBJECTIVE
Min J (economics);
MV=y1s
RTO
cs = y1s
MPC
y2s
PID
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Follow path (+ look after)
CV=y1 (+ u); MV=y2s
Stabilize + avoid drift
CV=y2; MV=u
u (valves)
Hierarchical decomposition
Example: Bicycle riding
Note: design starts from the bottom
• Regulatory control:
– First need to learn to stabilize the bicycle
• CV = y2 = tilt of bike
• MV = body position
• Supervisory control:
– Then need to follow the road.
• CV = y1 = distance from right hand side
• MV=y2s
– Usually a constant setpoint policy is OK, e.g. y1s=0.5 m
• Optimization:
– Which road should you follow?
– Temporary (discrete) changes in y1s
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Summary: The three layers
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Optimization layer (RTO; steady-state nonlinear model):
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Supervisory control (MPC; linear model with constraints):
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Identifies active constraints and computes optimal setpoints for primary controlled variables (y1).
Follow setpoints for y1 (usually constant) by adjusting setpoints for secondary variables (MV=y2s)
Look after other variables (e.g., avoid saturation for u’s used in regulatory layer)
Regulatory control (PID):
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Stabilizes the plant and avoids drift, in addition to following setpoints for y2. MV=valves (u).
Problem definition and overall control objectives (y1, y2) starts from the top.
Design starts from the bottom.
A good example is bicycle riding:
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Regulatory control:
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Supervisory control:
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First you need to learn how to stabilize the bicycle (y2)
Then you need to follow the road. Usually a constant setpoint policy is OK, for example, stay
y1s=0.5 m from the right hand side of the road (in this case the "magic" self-optimizing variable
self-optimizing variable is y1=distance to right hand side of road)
Optimization:
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Which road (route) should you follow?
Stepwise procedure plantwide control
I. TOP-DOWN
Step 1. DEGREES OF FREEDOM
Step 2. OPERATIONAL OBJECTIVES
Step 3. WHAT TO CONTROL? (primary CV’s c=y1)
Step 4. PRODUCTION RATE
II. BOTTOM-UP (structure control system):
Step 5. REGULATORY CONTROL LAYER (PID)
“Stabilization”
What more to control? (secondary CV’s y2)
Step 6. SUPERVISORY CONTROL LAYER (MPC)
Decentralization
Step 7. OPTIMIZATION LAYER (RTO)
Can we do without it?
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Control structure design procedure
I
Top Down
• Step 1: Identify degrees of freedom (MVs)
• Step 2: Define operational objectives (optimal operation)
– Cost function J (to be minimized)
– Operational constraints
• Step 3: Select primary controlled variables c=y1 (CVs)
• Step 4: Where set the production rate? (Inventory control)
y1
y2
II Bottom Up
• Step 5: Regulatory / stabilizing control (PID layer)
– What more to control (y2; local CVs)?
– Pairing of inputs and outputs
• Step 6: Supervisory control (MPC layer)
• Step 7: Real-time optimization
MVs
Process
Understanding and using this procedure is the most important part of this course!!!!
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Step 1. Degrees of freedom (DOFs) for
operation (Nvalves):
To find all operational (dynamic) degrees of freedom
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Count valves! (Nvalves)
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“Valves” also includes adjustable compressor power, etc.
Anything we can manipulate!
Steady-state degrees of freedom (DOFs)
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IMPORTANT! No. of steady-state CVs = No. of steady-state DOFs
Three methods to obtain no. of steady-state degrees of freedom (Nss):
1. Equation-counting
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Nss = no. of variables – no. of equations/specifications
Very difficult in practice (not covered here)
2. Valve-counting (easier!)
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Nss = Nvalves – N0ss – Nspecs
N0ss = variables with no steady-state effect
3. Typical number for some units (useful for checking!)
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CV = controlled variable (c)
Steady-state degrees of freedom (Nss):
2. Valve-counting
• Nvalves = no. of dynamic (control) DOFs (valves)
• Nss = Nvalves – N0ss – Nspecs : no. of steady-state DOFs
• N0ss = N0y + N0,valves : no. of variables with no steady-state effect
– N0,valves : no. purely dynamic control DOFs
– N0y : no. controlled variables (liquid levels) with no steady-state effect
• Nspecs: No of equality specifications (e.g., given pressure)
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Distillation column with given feed and pressure
Nvalves = 6 , N0y = 2 , Nspecs = 2, NSS = 6 -2 -2 = 2
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Heat-integrated distillation process
Nvalves = 11 (w/feed), N0y = 4 (levels), Nss = 11 – 4 =
7
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Heat-integrated distillation process
Nvalves = 11 (w/feed), N0y = 4 (levels), Nss = 11 – 4 = 7
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Heat exchanger with bypasses
CW
Nvalves = 3, N0valves = 2 (of 3),
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Nss = 3 – 2 = 1
Heat exchanger with bypasses
CW
Nvalves = 3, N0valves = 2 (of 3),
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Nss = 3 – 2 = 1
Steady-state degrees of freedom (Nss):
3. Typical number for some process units
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each external feedstream: 1 (feedrate)
splitter: n-1 (split fractions) where n is the number of exit streams
mixer: 0
compressor, turbine, pump: 1 (work/speed)
adiabatic flash tank: 0*
liquid phase reactor: 1 (holdup-volume reactant)
gas phase reactor: 0*
heat exchanger: 1 (duty or net area)
column (e.g. distillation) excluding heat exchangers: 0* + no. of sidestreams
pressure* : add 1DOF at each extra place you set pressure (using an extra
valve, compressor or pump), e.g. in adiabatic flash tank, gas phase reactor or
column
* Pressure is normally assumed to be given by the surrounding process and is then not a degree of
freedom
Heat exchanger with bypasses
CW
“Typical number heat exchanger”
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Nss = 1
Distillation column with given feed and pressure
“Typical number”,
Nss= 0 (distillation) + 2*1 (heat exchangers) = 2
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Heat-integrated distillation process
Typical number, Nss = 1 (feed) + 2*0 (columns) + 2*1
(column pressures) + 1 (sidestream) + 3 (hex) = 7
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HDA process
Purge (H2 + CH4)
Compressor
H2 + CH4
Toluene
Mixer
FEHE
Furnace
PFR
Quench
Separator
Toluene
Benzene
Toluene
Column
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Diphenyl
Cooler
CH4
Benzene
Column
Stabilizer
HDA process: steady-state degrees of freedom
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7
feed:1.2
hex: 3, 4, 6
3
splitter 5, 7
1
compressor: 8
2
4
5
distillation: 2 each
column
6
37
13
11
9
14
12
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Assume given column pressures
Conclusion: 14
steady-state
DOFs
• Check that there are enough manipulated variables (DOFs) - both
dynamically and at steady-state (step 2)
• Otherwise: Need to add equipment
– extra heat exchanger
– bypass
– surge tank
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Outline
• Introduction
• Control structure design (plantwide control)
• A procedure for control structure design
I Top Down
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Step 1: Degrees of freedom
Step 2: Operational objectives (optimal operation)
Step 3: What to control ? (self-optimizing control)
Step 4: Where set production rate? (inventory control)
II Bottom Up
• Step 5: Regulatory control: What more to control ?
• Step 6: Supervisory control
• Step 7: Real-time optimization
• Case studies
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Step 2. Define optimal operation (economics)
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What are we going to use our degrees of freedom for?
Define scalar cost function J(u0,x,d)
– u0: degrees of freedom
– d: disturbances
– x: states (internal variables)
Typical cost function:
J = cost feed + cost energy – value products
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Optimal operation for given d:
minuss J(uss,x,d)
subject to:
Model equations:
Operational constraints:
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f(uss,x,d) = 0
g(uss,x,d) < 0
Optimal operation distillation column
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Distillation at steady state with given p and F: N=2 DOFs, e.g. L and V
Cost to be minimized (economics)
cost energy (heating+ cooling)
J = - P where P= pD D + pB B – pF F – pV V
value products
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cost feed
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,
2) p free: pmin · p · pmax
Feed:
1) F given
2) F free: F · Fmax
Optimal operation: Minimize J with respect to steady-state DOFs
Optimal operation
minimize J = cost feed + cost energy – value products
Two main cases (modes) depending on marked conditions:
1. Given feed
Amount of products is then usually indirectly given and J = cost energy.
Optimal operation is then usually unconstrained:
“maximize efficiency (energy)”
2. Feed free
Control: Operate at optimal
trade-off (not obvious how to do
and what to control)
Products usually much more valuable than feed + energy costs small.
Optimal operation is then usually constrained:
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“maximize production”
Control: Operate at bottleneck (“obvious”)
Comments optimal operation
• Do not forget to include feedrate as a degree of freedom!!
– For LNG plant it may be optimal to have max. compressor power or max.
compressor speed, and adjust feedrate of LNG
– For paper machine it may be optimal to have max. drying and adjust the
feedrate of paper (speed of the paper machine) to meet spec!
• Control at bottleneck
– see later: “Where to set the production rate”
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QUIZ
Purge
some B, trace C)
Purge (mostly
(mest A, A,
litt B)
Gas phase process
(e.g. ammonia, methanol)
Kompressor
Reaktor
A+B
C
Separator
Føde
Feed
ca 51%A, 49% B
Flash
51%A, 49%B
Oppvarming
Heating
Kjøling
Cooling
LiquidProdukt
Product(C)
(C)
1.
2.
3.
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Degrees of freedom (Dynamic, steady-state)?
Expected active constraints?
Proposed control structure?
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
How should we adjust the degrees of freedom (u)?
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Solution I: Optimal feedforward
With availability of perfect model and
measurement of all disturbances d,
degrees of freedom u can be continuously
updated using online optimizer
Problem: UNREALISTIC!
Feedforward problems:
1. Lack of measurements of d
2. Sensitive to model error
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Solution II: Optimizing feedback control
y
Estimate d from measurements y
and recompute uopt(d)
Problem:
TOO COMPLICATED!
Requires detailed model and
description of uncertainty
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Solution III (Practical!): Hierarchical
decomposition with separate layers
y
Controlled variables that
link the optimization and
control layers
When disturbance d:
Degrees of freedom (u)
are updated indirectly to
keep controlled variables
at setpoints
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“Self-Optimizing Control”
= Solution III with constant setpoints
y
When constant
setpoints are OK
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Formal Definition
Controller
e
+
-
cs
Acceptable loss )
self-optimizing control
cm
u(d)
+
n
c = f(y)
d
Process
Self-optimizing control is said to occur when we can achieve an acceptable loss (in comparison with
truly optimal operation) with constant setpoint values for the controlled variables without the need
to reoptimize when disturbances occur.
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Reference: S. Skogestad, “Plantwide control: The search for the self-optimizing control structure'',
Journal of Process Control, 10, 487-507 (2000).
How does self-optimizing control (solution
III) work?
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When disturbances d occur, controlled variable c deviates from setpoint cs
Feedback controller changes degree of freedom u to uFB(d) to keep c at cs
Near-optimal operation / acceptable loss (self-optimizing control) is achieved if
– uFB(d) ≈ uopt(d)
– or more generally, J(uFB(d)) ≈ J(uopt(d))
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Of course, variation of uFB(d) is different for different CVs c.
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We need to look for variables, for which J(uFB(d)) ≈ J(uopt(d)) or Loss = J(uFB(d)) J(uopt(d)) is small
Remarks
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Self-optimizing control provides a trade-off between complexity of control system
and optimality.
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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.”
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The term “self-optimizing control” is similar to “self-regulation”, where acceptable
dynamic behavior is attained by keeping manipulated variables constant.
Relation with Similar Techniques
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Broadly, self-optimizing control can be seen as a measurement-based (or feedback-based)
optimization technique.
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Some related ideas are:
–
–
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Control of necessary conditions of optimality (Profs. Dominique Bonvin, B. Srinivasan and coworkers)
Extremum seeking control (Profs. Miroslav Krstic, Martin Guay and co-workers)
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In these techniques, u is updated to drive the gradient of Lagrange function (obtained
analytically or estimated) to zero.
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Gradient of Lagrange function is a possible self-optimizing variable!
Step 3. What should we control (c)?
(primary controlled variables y1=c)
What should we
control?
• Introductory example: Marathon runner
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Optimal operation – Runner
– Cost: J=T
– One degree of freedom (u=power)
– Optimal operation?
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Optimal operation - Runner
Solution 1: Optimizing control
• Even getting a reasonable model
requires > 10 PhD’s  … and
the model has to be fitted to each
individual….
• Clearly impractical!
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Optimal operation - Runner
Solution 2 – Feedback
(Self-optimizing control)
– What should we control?
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Optimal operation - Runner
Self-optimizing control: Sprinter (100m)
• 1. Optimal operation of Sprinter, J=T
– Active constraint control:
• Maximum speed (”no thinking required”)
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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)?
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c1 = distance to leader of race
c2 = speed
c3 = heart rate
c4 = level of lactate in muscles
Optimal operation - Runner
Conclusion Marathon runner
select one measurement
c = heart rate
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• Simple and robust implementation
• Disturbances are indirectly handled by keeping a constant heart rate
• May have infrequent adjustment of setpoint (heart rate)
Example: Cake Baking
• Objective: Nice tasting cake with good texture
Disturbances
Measurements
d1 = oven specifications
y1 = oven temperature
d2 = oven door opening
y2 = cake temperature
d3 = ambient temperature
y3 = cake color
d4 = initial temperature
u1 = Heat input
u2 = Final time
Degrees of Freedom
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Further examples self-optimizing control
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Marathon runner
Central bank
Cake baking
Business systems (KPIs)
Investment portifolio
Biology
Chemical process plants: Optimal blending of gasoline
Define optimal operation (J) and look for ”magic” variable
(c) which when kept constant gives acceptable loss (selfoptimizing control)
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More on further examples
•
•
•
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” :
•
•
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Find the controlled variables used in nature
From this possibly identify what overall objective J the biological system has
been attempting to optimize
Step 3. What should we control (c)?
(primary controlled variables y1=c)
Selection of controlled variables c
1. Control active constraints!
2. Unconstrained variables: Control “magic” selfoptimizing variables!
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1. CONTROL ACTIVE CONSTRAINTS!
Optimal solution is usually at constraints, that is, most of the
degrees of freedom are used to satisfy “active constraints”,
g(u,d) = 0 -> c=cconstraint
J
c ≥ cconstraint
Jopt
copt
1.
If c = u = manipulated input (MV):
•
2.
Implementation trivial: Keep u at constraint (umin or umax)
If c = y = output variable (CV):
•
•
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c
Use u to control y at constraint (feedback)
BUT: Need to introduce back-off (safety margin)
Back-off for active output constraints
c ≥ cconstraint
J
Loss
Back-off
Jopt
copt
a)
If constraint can be violated dynamically (only average matters)
•
b)
Required Back-off = “bias” (steady-state measurement error for c)
If constraint cannot be violated dynamically (“hard constraint”)
•
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c
Required Back-off = “bias” + maximum dynamic control error
Example. Optimal operation = max. throughput.
Want tight bottleneck control to reduce backoff!
Back-off
= Lost
production
Time
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Rule for control of hard output constraints: “Squeeze and shift”!
Reduce variance (“Squeeze”) and “shift” setpoint cs to reduce backoff
Hard Constraints: «SQUEEZE AND SHIFT»
COST FUNCTION
N Histogram
LEVEL 0 / LEVEL 1
2
Sigma 1 -- Sigma 2
Sigma 2
OFF
SPEC
LEVEL 2
Q1 -- Q2
W2
1.5
DELTA COST (W2-W1)
W1
1
Sigma 1
0.5
SQUEEZE
Q2
0
68
0
© Richalet
50
100
150
Q1
200
SHIFT
QUALITY
250
300
350
400
450
• SUMMARY ACTIVE CONSTRAINTS
– cconstraint = value of active constraint
– Implementation of active constraints is usually “obvious”, but may need
“back-off” (safety limit) for hard output constraints
• Cs = Cconstraint - backoff
– Want tight control of hard output constraints to reduce the back-off
• “Squeeze and shift”
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Optimal operation distillation
•
Cost to be minimized (economics)
J = - P where P= pD D + pB B – pF F – pV V
cost energy (heating+ cooling)
value products cost feed
•
Constraints
Purity D: For example
Purity B: For example,
Flow constraints:
Column capacity (flooding):
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xD, impurity · max
xB, impurity · max
0 · D, B, L etc. · max
V · Vmax, etc.
Expected active constraints distillation
•
Valueable product: Purity spec. always active
– Reason: Amount of valuable product (D or B)
should always be maximized
• Avoid product “give-away”
(“Sell water as methanol”)
• Also saves energy
•
valuable
product
methanol
+ max. 0.5%
water
Control implications valueable product: Control
purity at spec.
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methanol
+ water
cheap product
(byproduct)
water
+ max. 0.1%
methanol
Expected active constraints distillation:
Cheap product
•
Over-fractionate cheap product? Trade-off:
–
–
•
Yes, increased recovery of valuable product (less loss)
No, costs energy
Control implications cheap product:
1. Energy expensive: Purity spec. active
→ Control purity at spec.
2. Energy “cheap”: Overpurify
(a) Unconstrained optimum given by trade-off between energy
and recovery.
In this case it is likely that composition is self-optimizing
variable
→ Possibly control purity at optimum value
(overpurify)
(b) Constrained optimum given by column reaching capacity
constraint
→ Control active capacity constraint (e.g. V=Vmax)
–
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Methanol + water example: Since methanol loss anyhow is low (0.1% of water), there is
not much to gain by overpurifying. Nevertheless, with energy very cheap, it is probably
optimal to operate at V=Vmax.
methanol
+ water
valuable
product
methanol
+ max. 0.5%
water
cheap product
(byproduct)
water
+ max. 0.1%
methanol
Summary: Optimal operation distillation
•
Cost to be minimized
J = - P where P= pD D + pB B – pF F – pV V
•
•
N=2 steady-state degrees of freedom
Active constraints distillation:
–
–
•
Purity spec. valuable product is always active (“avoid giveaway of valuable product”).
Purity spec. “cheap” product may not be active (may want to
overpurify to avoid loss of valuable product – but costs energy)
Three cases:
1. Nactive=2: Two active constraints (for example, xD, impurity = max. xB,
impurity = max, “TWO-POINT” COMPOSITION CONTROL)
WHAT SHOULD WE
2. Nactive=1: One constraint active (1 unconstrained DOF)
CONTROL (TO SATISFY
3. Nactive=0: No constraints active (2 unconstrained DOFs) UNCONSTRAINED DOFs)?
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Can happen if no purity specifications
(e.g. byproducts or recycle)
Solution:
Often compositions
but not always!
QUIZ (again)
Purge
some B, trace C)
Purge (mostly
(mest A, A,
litt B)
Gas phase process
(e.g. ammonia, methanol)
Kompressor
Reaktor
A+B
C
Separator
Føde
Feed
ca 51%A, 49% B
Flash
51%A, 49%B
Oppvarming
Heating
Kjøling
Cooling
LiquidProdukt
Product(C)
(C)
1.
2.
3.
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Degrees of freedom (Dynamic, steady-state)?
Expected active constraints? (1. Feed given; 2. Feed free)
Proposed control structure?
2. UNCONSTRAINED VARIABLES:
WHAT MORE SHOULD WE CONTROL?
• Intuition: “Dominant variables” (Shinnar)
• Is there any systematic procedure?
A. Senstive variables: “Max. gain rule” (Gain= Minimum singular value)
B. “Brute force” loss evaluation
C. Optimal linear combination of measurements, c = Hy
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