Design of Process
Operations using Hybrid
Dynamic Optimization
Paul I. Barton and Cha Kun Lee
Department of Chemical Engineering
Massachusetts Institute of Technology
77 Massachusetts Avenue
Cambridge, MA 02139
FOCAPO
14th January 2003
Outline
• Modeling, Simulation and Sensitivity Analysis of Process
Operations as Hybrid Systems
• Global Dynamic Optimization of Hybrid Systems
• Examples
Hybrid Systems
• Hybrid systems exhibit both discrete state and continuous
state dynamics
Impulse
All
Vapor
Two
Phase
All
Liquid
P > Pburst
Intact
Burst
Pressure Vessel
with Rupture Disk
Tank with
Impulsive Input
Flash Vessel with
Phase Transitions
• Accurate dynamic models of process operations, e.g., start-ups
and shut-downs are best analyzed using hybrid systems
Modeling Process Operations
• Increasing need for efficient, safe and environmentally friendly
process operations
• How will these improvements in process operations be
achieved?
• Detailed physical models, and rigorous analysis and
optimization frameworks
• Continuous process dynamics familiar, but as perturbation
from steady-state becomes larger, discrete aspects become
more and more important too
• Hybrid Systems is the rigorous modeling paradigm for
process operations
Tank-Changeover Example
min tf
u(t),tf
t
t∗
Feasible
Path
CH4 (P1 )
N2 (P2 )
O2 (P3 )
N2
Explosive
Region
(P4 )
Pressure Vessel
CH4
O2
Mole fraction space
• Objective: Minimize the time needed to change from a tank
full of methane to one full of oxygen
Tank-Changeover Example
f k = N k − C v Ak
q
Pk +P
P
√k −P
2
b+ Pk −P
Laminar/
Turbulent
P
Pk
f4 = N 4 − C v A4
(uk < 0.5) ∨ ( PPk ≥ 1)
P4
P
> 0.53
P
Pk
Choked
Flow
q
P
P +P4
√−P4
2
b+ P −P4
Laminar/
Turbulent
> 0.53
P4
P
(u4 < 0.5) ∨ ( PP4 ≥ 1)
≤ 0.53
≤ 0.53
(uk > 0.5) ∧ ( PPk < 1)
Zero
Flow
Choked
Flow
(u4 > 0.5) ∧ ( PP4 < 1)
uk < 0.5
Zero
Flow
u4 < 0.5
fk = N k
Pk
f k = N k − C v Ak √
0.85
2
f4 = N 4
f4 = N4 − Cv A4 √P2 0.85
Valve k ∈ {1, 2, 3}
Valve 4
For all i ∈ I,
f(4+i) = Mi0 − N1 y1i − N2 y2i − N3 y3i + N4 y4i
f(7+i) = Mi − MT yi
f(10+i) = y4i − yi
f14 = y1 + y2 + y3 − 1
f15 = P V − MT RT
Tank
Tank-Changeover Example
• Controls: on/off signals to valves as a function of time
– infinite dimensional optimal control problem
• Control parameterization leads to the following:
– Initial positions of valves known
– Allow position to switch at finite number of switching times
– Continuous switching times decision variables in NLP
– Embeds optimal control if sufficient number of switches
allowed
Control parameterization
∂x(p,t)
∂p
Master NLP
sets p
Parametric
Sensitivities
Gradients
Objective
function,
Constraints
t
x(p, t)
u(p, t)
Control
Variables
Hybrid
System
IVP
t
State
Variables
t
Parametric Sensitivities of Dynamic Systems
Given a dynamic system expressed in terms of a vector of time
invariant parameters p:
ẋ(p, t) = f (x(p, t), p, t)
• then there exists a nx by np matrix of partial derivatives
determined by the related matrix differential equations:
∂f ∂x ∂f
∂ ẋ
=
+
∂p
∂x ∂p
∂p
• extension to DAEs straightforward
∂x
∂p
Parametric Sensitivities of Hybrid Systems
• [Galán et al. (1999)]
• Existence and uniqueness for sensitivities of hybrid systems
embedded with:
– nonlinear ODEs
– linear time invariant DAEs
• In general, sensitivities of hybrid systems exist almost
everywhere
• Critical parameter values at which (for example) sequence of
events changes qualitatively
– theorems break down
– typically associated with discontinuity or
nondifferentiability in the objective function
Direct Stochastic Search
Objective
function,
Constraints
NLP Solver
sets p
x(p, t)
u(p, t)
Control
Variables
Hybrid
System
IVP
t
State
Variables
t
Results from Stochastic Search
Valve Valve
Signal
Profiles
Signal Profiles
State
Space Plot
State Space Plot, Composition Space
Values x 10 -3
Values
Valve 3 Signal
Explosive Limit
1.00
Valve 2 Signal
950.00
0.95
Valve 1 Signal
900.00
0.90
850.00
0.85
800.00
0.80
750.00
0.75
700.00
0.70
650.00
0.65
600.00
0.60
Y(N2)
550.00
0.55
500.00
0.50
450.00
0.45
400.00
0.40
350.00
0.35
300.00
0.30
250.00
0.25
200.00
0.20
0.15
150.00
0.10
100.00
0.05
50.00
0.00
0.00
-0.05
Time
0.00
50.00
100.00
150.00
200.00
250.00
Y(CH4)
0.00
0.20
0.40
0.60
0.80
1.00
Results from Stochastic search
Pressures
Flow Through Valves
Pressures
Flows Through Valves
Values
Values
P [BAR]
12.00
P3 [BAR]
11.50
P2 [BAR]
60.00
N3 [MOL/S]
N2 [MOL/S]
N1 [MOL/S]
55.00
P1 [BAR]
11.00
P4 [BAR]
10.50
10.00
N4 [MOL/S]
50.00
45.00
9.50
9.00
40.00
8.50
35.00
8.00
7.50
30.00
7.00
6.50
25.00
6.00
5.50
20.00
5.00
15.00
4.50
4.00
10.00
3.50
3.00
5.00
2.50
2.00
0.00
Time
0.00
50.00
100.00
150.00
200.00
250.00
Time
0.00
50.00
100.00
150.00
200.00
250.00
Verification Problems
The direct stochastic search approach
• can deal with nonsmoothness in the master problem
• cannot guarantee the global minimum, can be computationally
expensive and can encounter difficulties with constraints
These methods cannot be used for problems where location of the global
minimum is crucial, e.g., formal verification of logic based controllers.
LSH
3
LSH
4
Tank A
Tank B
ZSL
1
ZSH
1
ZSL
2
HS
1
pump
on
HS
7
Tank A
level high
LSH
3
start filling HS
Tank A
1
ZSH
2
HS
2
stop filling
Tank A
HS
1
S
R
HV
1
ZSH
1
A
A
open valve
valve open
HV
2
ZSL
1
HV
2
HV
2
HS
7
start filling HS
Tank B
2
S
stop filling
Tank B
HS
2
R
Tank B
level high
LSH
4
pump
off
HS
7
Feed Pump
PSL
5
pump suct. PSL
5
press. low
ZSH
2
open valve
valve open
HV
1
operate
pump
R
stop
pump
OR
close valve
valve closed
HV
1
S
ZSL
1
A
close valve
valve closed
OR
DI
5s
Other Approaches
• Discrete time horizon [Bemporad, Borrelli and Morari (2000)]
• Total discretization [Avraam, Shah and Pantelides (1998)]
• Partial discretization
– Solve directly as an NLP with hybrid system embedded
∗ Stochastic methods [Barton, Banga and Galán (2000)]
∗ Nonsmooth optimization methods
– Reformulation of problem as a global Mixed-Integer
Dynamic Optimization problem
• Two stage decomposition of switched systems [Xu and
Antsaklis (2001)]
• Reachability analysis of hybrid automata [Alur et al. (1995)],
[Shakernia, Pappas and Sastry (2000)]
DAEPACK and ABACUSS II
Software packages are available that can robustly handle simulation
and sensitivity analysis of hybrid systems:
• ABACUSS II / ABACUSS 3
– Advanced equation based modeling environment
– Features include robust sensitivity analysis, sparsity pattern
generation, interface to NLP solvers, etc.
• DAEPACK
– All structural symbolic and numerical information required
by modern algorithms for hybrid systems can be generated
automatically from a FORTRAN source file
– Works with very general code: functions, subroutines,
common blocks, iterative procedures, etc.
• Very easy to get qualitatively wrong sensitivities using
standard codes - must use our symbolic tools
Outline
• Modeling, Simulation and Sensitivity Analysis of Process
Operations as Hybrid Systems
– Hybrid systems framework is modeling paradigm for process
operations
– Interesting problems motivate development of global
optimization methods
• Global Dynamic Optimization of Hybrid Systems
• Examples
Motivation
Problems that require the global minimum to be located motivate
the development of tools for the deterministic global optimization
of hybrid systems. A key tool is the ability to
• construct convex relaxations for general, nonlinear Bolza type
functions [Singer and Barton (2001)]:
´ Z tf ¡
³
¢
f ẋ(p, t), x(p, t), p, t dt
F (p) = φ ẋ(p, tf ), x(p, tf ), p +
t0
where x(p, t) is given by the solution of an embedded dynamic
system described by linear time varying ODEs:
ẋ(p, t) = A(t)x(p, t) + B(t)p + q(t)
x(p, t0 ) = Ep + k
• this has recently been extended to nonlinear ODEs embedded
Hybrid Trajectories
The time horizon is split into contiguous intervals called epochs:
• Tµ , the sequence of modes, is called the hybrid mode trajectory
• Tτ , the sequence of epochs, is called the hybrid time trajectory
Mode 1
Mode 1
ẋ = −2x + p
ẋ = −2x + p
ẋ = x − 2p
Mode 2
t
t0
Epoch 1
I1 = [t0 , t1 ]
t1
Epoch 2
I2 = [t01 , t2 ]
Tµ = 1, 2, 1
t2
Epoch 3
I3 = [t02 , tf ]
Tτ = I 1 , I 2 , I 3
tf
Classes of Hybrid Optimization Problems
• Tµ unchanged and vector field does not jump at events:
– parametric sensitivities continuous (no jumps)
– smooth Master NLP
• Tµ unchanged:
– parametric sensitivities discontinuous functions of time
– smooth Master NLP
• Tµ to be determined:
– nonsmooth nonconvex Master NLP
– decomposition approach - MIDO: Master problem searches
over different sequences
Convexity theory for fixed Tµ and Tτ
• With Tµ and Tτ fixed, convexity theory holds and convex
relaxations can be constructed.
• Well-known methods for obtaining convex relaxations on
Euclidean spaces can be employed, e.g., McCormick’s method,
αBB. For example:
Z 1
−x2 (p, t) dt
F (p) =
0
U (p) =
Z
1
(xL (t) + xU (t))(xL (t) − x(p, t)) − xL (t)2 dt
0
• This enables standard global optimization algorithms, such as
Branch and Bound, nonconvex Outer Approximation, etc., to
be applied.
Example
min
F (p) =
p∈[−4,4]
Z
3
−x2 (p, t) dt,
0
where x(p, t) is given by the solution to the following hybrid system,
Mode 1 : ẋ(p, t) = −2tx(p, t) + p,
x(p, t) + p
,
t + 10
= {1, 2, 1}, and state continuity is enforced at t1 = 1,
Mode 2 : ẋ(p, t) =
x(p, 0) = 1, Tµ
t2 = 2.
0
4
0
xL (t)
−5
Value
Value
Value
F (p)
−5
xU (t)
2
0
−10
U (p)
−15
−15
−2
0
1
2
3
−20
−4
−2
0
−10
2
4
−20
−4
−2
0
Time t
Parameter p
Parameter p
(a) Implied State Bounds
for p = [−4, 4]
(b) Objective Function
and Convex Underestimator
(c) Branch
and Bounding
2
4
Varying Sequence of Modes
• Binary decision variables are introduced to represent the
sequence of modes:
Z 5
−x2 (p, y, t) dt,
min F (p, y) =
p,y
s.t. y11 + y21 = 1,
0
y12 + y22 = 1,
y13 + y23 = 1,
where x(p, y, t) is given by the solution to the following hybrid system,
µ
¶
x+p
Mode 1 : ẋ(p, y, t) = y11 (−2tx + p) + y21
,
t + 10
µ
¶
x+p
,
Mode 2 : ẋ(p, y, t) = y12 (−2tx + p) + y23
t + 10
¶
µ
x+p
,
Mode 3 : ẋ(p, y, t) = y13 (−2tx + p) + y23
t + 10
x(p, 0) = 1,
p ∈ P = [−4, 4], y ∈ Y binary,
Tµ = 1, 2, 3 and state continuity is enforced at t1 = 1 and t2 = 2.
Hybrid Superstructure
y11
y12
y1ne
Mode 1
Mode 1
Mode 1
..
.
..
.
..
.
yi1
z1
Mode i
yi2
z2
Mode i
..
.
yine
z3
zn e
Mode i
..
.
..
.
ynm 1
ynm 2
ynm ne
Mode nm
Mode nm
Mode nm
t
t0
Epoch 1
n
m
P
i=1
yi1 = 1
t1
Epoch 2
n
m
P
i=1
yi2 = 1
t2
Epoch ne
n
m
P
i=1
yine = 1
tf
MIDO Reformulation
• Bilinear terms destroy LTV structure of hybrid system
• Introduce nm × ne dynamic LTV systems:
ẋmi (p, Z, t) = A(m) (t)xmi (p, Z, t) + B(m) (t)p + q(m) (t),
∀ m ∈ M, i = 1, . . . , ne ,
M is the set of nm modes, and ne is the total number of epochs
• Introduce nx × ne additional parameters to represent initial
conditions for each epoch:
z1 = E 0 p + k 0 ,
zi = xmi (p, Z, t0i ),
∀ m ∈ M, i = 1, . . . , ne .
• Bilinearities shifted to constraints:
!
Ãn
m
X
ymi xmi (p, Z, ti ) +Ei p+ki , ∀ i = 1, . . . , ne −1.
zi+1 = Di
m=1
MIDO Reformulation
• Reformulated problem is a MIDO with a set of LTV ODE
systems
• Using nonconvex OA, MIDO is solved as a nonconvex MINLP
• Ability to construct convex lower bounding MINLP is key,
resulting in the following subproblems in OA:
– Primal: noncovex NLP, provides upper bound
– Primal Bounding: convex NLP, potentially reduces number
of primal problems to solve
– Relaxed Master: MILP, provides lower bound
Nonconvex Outer Approximation
[Kesavan and Barton (1999)]
1
k = 1, y given
UBD = +∞
Primal Bounding
Problem
Convex NLPB(yk )
Is Primal
Bounding
Feasible?
Yes
New Integer
Realization, yk
k =k+1
No
Integer Cut
Primal Problem
Nonconvex NLP(yk )
Solve for UBD
Solve
Feasibility
Problem
Derive
Linearizations
Yes
Is Relaxed
Master
Feasible?
No
Global solution = UBD
Relaxed Master
Problem
(MILP)
NLP(yk ) and NLPB(yk )
can be solved in parallel
Solving the MIDO
• Exact linearizations can be obtained for bilinear/trilinear
constraints introduced in the reformulation
• Additional linearizations in the relaxed Master problem only
have to be constructed for nonlinear terms in the objective
function and constraints
• Mode assignment constraints constitute S0S1 sets and can be
exploited when solving the relaxed Master problem
• It suffices to add only linearizations of the active constraints in
the primal bounding problem to the relaxed Master problem not all nm × ne sets of dynamic systems have to be solved
Outline
• Modeling, Simulation and Sensitivity Analysis of Process
Operations as Hybrid Systems
– Hybrid systems framework is modeling paradigm for process
operations
– Interesting problems motivate development of global
optimization methods
• Global Dynamic Optimization of Hybrid Systems
– Construct convex relaxations for LTV ODE hybrid system
with Tµ and Tτ fixed
– Reformulation into MIDO when Tµ is optimization
parameter
• Examples
Catalyst Loading Problem
• Given
– an isothermal PFR at steady state,
– 2 loading bays,
– 3 possible kinds of catalyst on support,
Section 1
Section 2
Isothermal, steady state PFR
l=0
l=1
determine the optimal catalyst loading profile for the PFR that
will maximize the profit.
• This can be viewed as an optimization problem with a hybrid
system embedded.
• Optimal Tµ corresponds to the optimal choice of catalyst
loading.
Catalyst Loading Problem
Section 1
Section 2
Isothermal, steady state PFR
l=0
l=1
Reaction scheme
A+B
1
I
2
4
W1
W2
Kinetics
3
P
ẋA = −(k1 + k2 )xA
ẋB = −(k1 + k2 )xB
ẋW1 = k2 xA
ẋI = k1 xA − (k3 + k4 )xI
ẋW2 = k4 xI
ẋP = k3 xI
Catalyst Loading Problem
xA (0) = 1000
xP (1) − 0.01xW1 (1) − 0.1xW2 (1)
Section 1
Section 2
Catalyst
k1
k2
k3
k4
1
2.098
1.317
0.021 0.033
2
29.53
110.2
3
182.6
2325
k1 /k2
k3 /k4
1
1.594
0.627
0.295 0.079
2
0.268
3.728
1.826 0.143
3
0.079
12.729
max xP (1) − 0.01xW1 (1) − 0.1xW2 (1)
Tµ
Tµ = 1, 3
F = 290.4
• For 10 catalyst sections, we get Tµ = 1, 1, 1, 1, 3, 3, 3, 3, 3, 3 with
F = 296.5
Batch Reaction and Distillation
• Building on the previous example, we consider a batch reaction
and distillation problem:
R1
R2
F1
Raw Material
F2
D1
D2
Product and By-Product
• The objective is to minimize raw material and waste treatment
costs while maintaining a desired level of production of product.
• Amount of raw material, recycle flows, and choice of catalyst
are optimization variables. Each reactor’s run is for 1.5h.
Batch Reaction and Distillation
• Components B, W1 and P form a ternary and 2 binary
azeotropes:
B
Region 3
Region 4
Azeotrope Composition
W1 -P B-W1 -P B-W1
B
Region 2
Region 5
0.72
0.35
0.65
W1
0.15
0.06
P
0.85
0.22
Region 1
W1
P
Product Cut Sequence
1 : { A, W1 -P, W1 , I, B-W1 , W2 }
4 : { B, A, B-W1 -P, I, P, W2 }
2 : { A, W1 -P, B-W1 -P, I, B-W1 , W2 }
5 : { A, W1 -P, B-W1 -P, I, P, W2 }
3 : { B, A, B-W1 -P, I, B-W1 , W2 }
Batch Reaction and Distillation
• Process constraints:
– molar solvent to reactant ratio (B, W1 , W2 : A, I) ≥ 15
– at least 2 times B in excess of A
– 99% pure P
• Optimal solution: (Cost of $5.061M to produce 300 kmol P)
540.5 I
1458.3 B-W1 -P
8107.2 B-W1
5.4 A
4634.3 W1
R1
R2
F1
F2
894.2 A
R1: Catalyst 3
R2: Catalyst 1
D1
D2
1680.0 B
D1: Region 1
D1: Region 4
445.2 B-W1
59.9 W2
875.0 B-W1 -P
20.0 W1 -P
280.0 P
Batch Reaction and Distillation
• Postulating a superstructure with 4 trains:
R1
F1
D1
R2
Raw Material
F2
D2
R3
F3
D3
Product
By-Product
R4
Fixed Points
F4
D4
Batch Reaction and Distillation
• Optimal solution: (Cost of $1.905M to produce 300 kmol P)
3.2 A
1565.0 B-W1
6061.1 W1
205.7 W1
12.9 W2
R1
541.2 A
541.2 B
F1
Catalyst 1
D1
Region 1
311.4 I
22.6 W2
R2
F2
Catalyst 3
20.0 W1 -P
D2
Region 4
280.0P
17.1 I
8.8 P
4928.2 W2
Conclusion
• Modeling, Simulation and Sensitivity Analysis of Process
Operations as Hybrid Systems
– Hybrid systems framework is modeling paradigm for process
operations
– Interesting problems motivate development of global
optimization methods
• Global Dynamic Optimization of Hybrid Systems
– Construct convex relaxations for LTV ODE hybrid system
with Tµ and Tτ fixed
– Reformulation into MIDO when Tµ is optimization
parameter
• Examples
– Catalyst loading problem
– Batch reaction and distillation problem
Future Work
• Varying Tτ
• Development of convexity theory for nonlinear dynamic
systems (done)
• Extension of results for nonlinear hybrid systems
Acknowledgments
• This work was supported by the DoD Multidisciplinary
University Research Initiative (MURI) program administered
by the Army Research Office under Grant DAAD19-01-1-0566
• The National Science Foundation under Grant CCR-0208956
• The Engineering Research Program of the Office of Basic
Energy Sciences at the U.S. DoE under Grant
DE-FG02-94ER1444
• Adam B. Singer, supported by the National Science
Foundation under Grant CTS-0120441