optimal dynamic operation of chemical processes ...jbr dynamic operation of chemical processes:...
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Optimal dynamic operation of chemical processes:Assessment of the last 20 years and current research
opportunities
James B. Rawlings
Department of Chemical and Biological Engineering
April 13, 2010Department of Chemical Engieering
Carnegie Mellon University
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Outline
1 The last 20 years — what tools have researchers developed
2 Industrial impact of these ideas
3 Have all the questions been answered?Control of large-scale systemsOptimizing economics
4 Conclusions and future outlook
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The power of abstraction
process
sensorsactuators
dx
dt= f (x , u)
y = g(x , u)
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The model predictive control framework
Measurement
MH Estimate
MPC control
Forecast
t time
Reconcile the past Forecast the future
sensorsy
actuatorsu
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Predictive control
Measurement
MH Estimate
MPC control
Forecast
t time
Reconcile the past Forecast the future
sensorsy
actuatorsu
minu(t)
∫ T
0|ysp − g(x , u)|2Q + |usp − u|2R dt
x = f (x , u)
x(0) = x0 (given)
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State estimation
Measurement
MH Estimate
MPC control
Forecast
t time
Reconcile the past Forecast the future
sensorsy
actuatorsu
minx0,w(t)
∫ 0
−T|y − g(x , u)|2R + |x − f (x , u)|2Q dt
x = f (x , u) + w (process noise)
y = g(x , u) + v (measurement noise)
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Industrial impact of the research
Validation
Planning and Scheduling
Reconciliation
Model UpdateOptimizationSteady State
Plant
Controller
Two layer structure
Steady-state layerI RTO optimizes steady-state
modelI Optimal setpoints passed to
dynamic layer
Dynamic layerI Controller tracks the setpointsI Linear MPC
(replaces multiloop PID)
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Large industrial success story!
Linear MPC and ethylene manufacturing
Number of MPC applications in ethylene: 800 to 1200
Credits 500 to 800 M$/yr (2007)
Achieved primarily by increased on-spec product, decreased energy use
Eastman Chemical experience with MPC
First MPC implemented in 1996
Currently 55-60 MPC applications of varying complexity
30-50 M$/year increased profit due to increased throughput (2008)
Praxair experience with MPC
Praxair currently has more than 150 MPC installations
16 M$/year increased profit (2008)
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Impact for 13 ethylene plants (Starks and Arrieta, 2007)
Hydrocarbons AC&O 17
Advanced ControlAdvanced Control& Optimization& Optimization
We’re Doing it For the Money
$0
$10,000,000
$20,000,000
$30,000,000
$40,000,000
$50,000,000
$60,000,000
1Q 2
000
3Q 2
000
1Q 2
001
3Q 2
001
1Q 2
002
3Q 2
002
1Q 2
003
3Q 2
003
1Q 2
004
3Q 2
004
1Q 2
005
3Q 2
005
1Q200
6
3Q200
6
$0
$100,000,000
$200,000,000
$300,000,000
$400,000,000
$500,000,000
$600,000,000
Cumulative Quarterly
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Are all the problems solved?
Some questions to consider
How do we best decompose large-scale systems into manageableproblems?
How do we optimize dynamic economic operation?
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Electrical power distribution
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Chemical plant integration
Material flow
Energy flow
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MPC at the large scale
Decentralized Control
Most large-scale systems consist of networks ofinterconnected/interacting subsystems
I Chemical plants, electrical power grids, water distribution networks, . . .
Traditional approach: Decentralized controlI Wealth of literature from the early 1970’s on improved decentralized
control a
I Well known that poor performance may result if the interconnectionsare not negligible
a(Sandell Jr. et al., 1978; Siljak, 1991; Lunze, 1992)
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MPC at the large scale
Centralized Control
Steady increase in available computing power has provided theopportunity for centralized control
Most practitioners view centralized control of large, networkedsystems as impractical and unrealistic
A divide and conquer strategy is essential for control of large,networked systems (Ho, 2005)
Centralized control: A benchmark for comparing and assessingdistributed controllers
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Nomenclature: consider two interacting units
Objective functions V1(u1, u2), V2(u1, u2)
and V (u1, u2) = w1V1(u1, u2) + w2V2(u1, u2)
decision variables for units u1 ∈ Ω1, u2 ∈ Ω2
Decentralized Control minu1∈Ω1
V1(u1) minu2∈Ω2
V2(u2)
Noncooperative Control minu1∈Ω1
V1(u1, u2) minu2∈Ω2
V2(u1, u2)
(Nash equilibrium)
Cooperative Control minu1∈Ω1
V (u1, u2) minu2∈Ω2
V (u1, u2)
(Pareto optimal)
Centralized Control minu1,u2∈Ω1×Ω2
V (u1, u2)
(Pareto optimal)
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Noninteracting systems
-2
-1
0
1
2
-2 -1 0 1 2
u1
u2
V2(u)
V1(u)
b
a
n, d , p
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Weakly interacting systems
-2
-1.5
-1
-0.5
0
0.5
-2 -1.5 -1 -0.5 0 0.5 1 1.5 2
u1
u2
V2(u)
V1(u)
b
a
pn, d
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Moderately interacting systems
-1
-0.5
0
0.5
1
1.5
2
-2 -1.5 -1 -0.5 0 0.5 1 1.5 2
u1
u2
V2(u)
V1(u)
n
b
a
p
d
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Strongly interacting (conflicting) systems
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
-2 -1.5 -1 -0.5 0 0.5 1 1.5 2
u1
u2
V2(u)
V1(u)
b
a
p
d
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Strongly interacting (conflicting) systems
-20
0
20
40
60
80
100
120
140
160
-10 0 10 20 30 40 50
u1
u2
V2(u)V1(u)
n
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Geometry of cooperative vs. noncooperative MPC
-10
-5
0
5
10
-10 -5 0 5 10
u1
u2 V1(u) V2(u)
n
a
b
p
01
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Two reactors with separation and recycle
F0, xA0
Q
Fpurge
D, xAd, xBd
Hr Hm
B→ CA→ BA→ B
B→ C
Hb
F1, xA1
Fm, xAm, xBm
Fb, xAb, xBb,T
Fr, xAr, xBr
MPC3
MPC1 MPC2
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Two reactors with separation and recycle
-0.25-0.2
-0.15-0.1
-0.050
0.050.1
0 5 10 15 20 25 30 35 40Time
Hm
setpoint
Cent-MPC
Ncoop-MPC
Coop-MPC (1 iterate)
-0.35-0.3
-0.25-0.2
-0.15-0.1
-0.050
0.05
0 5 10 15 20 25 30 35 40Time
Hb
setpoint
Cent-MPC
Ncoop-MPC
Coop-MPC (1 iterate)
-0.04
-0.02
0
0.02
0.04
0 5 10 15 20 25 30 35 40Time
F1
Cent-MPC
Ncoop-MPC
Coop-MPC (1 iterate)
-0.2
-0.1
0
0.1
0.2
0 5 10 15 20 25 30 35 40Time
D
Cent-MPC
Ncoop-MPC
Coop-MPC (1 iterate)
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Two reactors with separation and recycle
Performance comparison
Cost (×10−2) Performance loss
Centralized MPC 1.75 0Decentralized MPC ∞ ∞Noncooperative MPC ∞ ∞Cooperative MPC (1 iterate) 2.2 25.7%Cooperative MPC (10 iterates) 1.84 5%
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Traditional hierarchical MPC
Coordinator
MPCMPC MPC
1s 1s5s 3s 0.5s
Setpoints
2min1min
1hr
Data
Plantwide coordinator
Coordinator
MPC MPC
Multiple dynamical time scales in plant
Data and setpoints are exchanged on slower time scale
Optimization performed at each layer
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Cooperative MPC data exchange
MPCMPC MPC
1s 1s5s 3s 0.5s
Data storageData storage
Read
Write
5s
MPC MPC
All data exchanged plantwide
Slowest MPC defines rate of data exchange
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Cooperative hierarchical MPC
MPCMPC MPC
1s 1s5s 3s 0.5s
Data storage
1min
Read
Write2min
1hr
Plantwide data storage
Data storage
MPC MPC
Optimization at MPC layer only
Only subset of data exchanged plantwide
Data exchanged at slower time scale
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The big(ger) picture — What is the goal?
The goal of optimal process operations is to maximize profit.— Helbig, Abel, and Marquardt (1998) . . . (−10 years)
Thus with more powerful capabilities, the determination ofsteady-state setpoints may simply become an unnecessaryintermediate calculation. Instead nonlinear, dynamic referencemodels could be used directly to optimize a profit objective.— Biegler and Rawlings (1991) . . . (−20 years)
In attempting to synthesize a feedback optimizing controlstructure, our main objective is to translate the economicobjective into process control objectives.— Morari, Arkun, and Stephanopoulos (1980) . . . (−30 years)
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Optimizing economics: Current industrial practice
Validation
Planning and Scheduling
Reconciliation
Model UpdateOptimizationSteady State
Plant
Controller
Two layer structure
DrawbacksI Inconsistent modelsI Re-identify linear model as
setpoint changesI Time scale separation may not
holdI Economics unavailable in
dynamic layer
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Motivating the idea
-4 -2 0 2 4-4
-20
24
Profit
Input (u)
State (x)
Profit
-4 -2 0 2 4-4
-20
24
Profit
Input (u)
State (x)
Profit
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Economics controller
minu(t)
∫ T
0L(x , u)dt subject to:
x = f (x , u)y = g(x , u)
Target tracking (standard)
L(x , u) = |ysp − g(x , u)|2Q + |usp − u|2R
Economic optimization (new)L is the negative of economic profit function
L(x , u) = −P(x , u)
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Strong duality and asymptotic stability
Strong Duality
If there exists a λ such that the the following problems have the samesolution
minx ,u
L(x , u) minx ,u
L(x , u)− λ(f (x , u))
f (x , u) = 0 h(x , u) ≤ 0
h(x , u) ≤ 0
Asymptotic stability of the closed-loop economics controller with astrictly convex cost and linear dynamics (Rawlings et al., 2008)
Asymptotic stability of the closed-loop economics controller withstrong duality in the steady-state problem (Diehl et al., 2010)
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Example
xk+1 =
[0.857 0.884−0.0147 −0.0151
]xk +
[8.565
0.88418
]uk
Input constraint: −1 ≤ u ≤ 1
Economics
Leco = α′x + β′u
α =[−3 −2
]′β = −2
Tracking
Ltarg = |x − x∗|2Q+|u − u∗|2RQ = 2I2 R = 2
x∗ =[60 0
]′u∗ = 1
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x2x2
targ-MPC
x2
targ-MPC60 65 70 75 80 85
x1
-2
0
2
4
6
8
10
x2x2x2
targ-MPC eco-MPC
x2
targ-MPC eco-MPC60 65 70 75 80 85
x1
-2
0
2
4
6
8
10
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55
60
65
70
75
80
85
90
0 2 4 6 8 10 12 14
State
targ-MPCeco-MPC
-2
0
2
4
6
8
10
0 2 4 6 8 10 12 14
State
targ-MPCeco-MPC
-1
-0.5
0
0.5
1
0 2 4 6 8 10 12 14
Input
Time
targ-MPCeco-MPC
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Conclusions
Optimal dynamic operation of chemical processes has undergone atotal transformation in the last 20 years. Both in theory and inpractice.
The currently available theory splits the problem into state estimationand regulation. Both are posed and solved as online optimizationproblems. Basic properties have been established. Lyapunov functionsare the dominant theoretical tool for analysis and design.
Industrial implementations and vendor software are basically keepingpace with the best available theory and algorithms. That is asurprising and noteworthy outcome!
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Critiquing the research enterprise
The abstraction level is high and barrier to entry is significant.
But the barrier is no higher than any other mathematically intensiveresearch field in chemical engineering. Fluid mechanics, statisticalmechanics, molecular dynamics, . . .
Researchers in this community have not done a good jobcommunicating the significant advances in this field to theircolleagues outside the field.
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Future directions — Current research in MPC
Distributed versions of MPCI Controlling large-scale systems composed of many small-scale MPCsI How to structure the small-scale MPCs so they cooperate on plantwide
objectives
Optimizing economics with MPCI The optimal economic point is not necessarily a steady stateI Allows removal of the steady-state economic optimization layerI Dynamic economic optimization subject to settling at the optimal
steady state
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New MPC graduate textbook
576 page text
214 exercises
335 page solution manual
3 appendices on web (133pages)
www.nobhillpublishing.com
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Further reading I
L. T. Biegler and J. B. Rawlings. Optimization approaches to nonlinear model predictivecontrol. In Y. Arkun and W. H. Ray, editors, Chemical Process Control–CPCIV,pages 543–571. CACHE, 1991.
M. Diehl, R. Amrit, and J. B. Rawlings. A Lyapunov function for economic optimizingmodel predictive control. IEEE Trans. Auto. Cont., 2010. Accepted for publication.
A. Helbig, O. Abel, and W. Marquardt. Structural concepts for optimization basedcontrol of transient processes. In International Symposium on Nonlinear ModelPredictive Control, Ascona, Switzerland, 1998.
Y.-C. Ho. On Centralized Optimal Control. IEEE Trans. Auto. Cont., 50(4):537–538,2005.
J. Lunze. Feedback Control of Large Scale Systems. Prentice-Hall, London, U.K., 1992.
M. Morari, Y. Arkun, and G. Stephanopoulos. Studies in the synthesis of controlstructures for chemical processes. Part I: Formulation of the problem. Processdecomposition and the classification of the control tasks. Analysis of the optimizingcontrol structures. AIChE J., 26(2):220–232, 1980.
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Further reading II
J. B. Rawlings, D. Bonne, J. B. Jørgensen, A. N. Venkat, and S. B. Jørgensen.Unreachable setpoints in model predictive control. IEEE Trans. Auto. Cont., 53(9):2209–2215, October 2008.
N. R. Sandell Jr., P. Varaiya, M. Athans, and M. Safonov. Survey of decentralizedcontrol methods for large scale systems. IEEE Trans. Auto. Cont., 23(2):108–128,1978.
D. D. Siljak. Decentralized Control of Complex Systems. Academic Press, London,1991. ISBN 0-12-643430-1.
D. M. Starks and E. Arrieta. Maintaining AC&O applications, sustaining the gain. InProceedings of National AIChE Spring Meeting, Houston, Texas, April 2007.
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