Decomposition Paradigms for Solving Large Scale Systems

1
Honeywell Technology Solutions I.I.T. Bombay, India
Decomposition Paradigms for Large Scale Systems
Department of Chemical
Engineering,
IIT Bombay, India.
Consultant – Research
Honeywell Technology
Solutions, Bangalore.
Dr. Ravi Gudi
ACM Technology talk

Talk Outline
 Overview of general decomposition strategies
 Approaches to Decomposition – brief preliminaries
 Decomposition paradigms
 Model co-ordination
 Goal co-ordination
 PSE applications: Optimization, Identification & Control
 Illustrative examples & case studies
 Concluding remarks.

Decomposition based problem solving
 Systems engineering is posed with lots of challenging
problems from analysis, optimization, and control viewpoints.
 A number of elegant solutions to the above class of problems
have been proposed
 Generally successful for small to medium scale problems.
 Require additional effort for tailoring to large scale applications
 Complexity introduced by large scale systems needs to be
analyzed and decomposed for solvability.
 Nature of complexity and the application requirements
influences the choice of the decomposition methodology.

Complexity ⇔ Decomposition
 Complexity could be distributed across time-scales, spatial
directions, combinatorial nature, etc.
 Decompositions could be {hierarchical, spatial and coordinated},
{strategic, tactical, operational}.
 Typical applications:
 Modeling and Simulation: partitioning
 Identification: segregation and composition
 Optimization: relaxation and co-operation
 Control: Optimizing control, communication-based
 Fault Detection and Diagnosis: discrimination / classification

Motivation for decomposition
 Complex Systems: Challenges offered*
 Dimensionality
 Computation intensity grows faster than size
 Information Structure Constraints
 Distributed sources of data
 Uncertainty
 Interconnections between subsystems; Local relationships can be
modeled accurately.
 Typical Applications: Manufacturing systems, Power networks,
Traffic networks, Digital communication networks, ...
*
Siljak (1996), Backx et al. (1998), Lu, (2000)

System description
System
Causes
(deterministic)
Effect
(measured)
Disturbances/ drifts
 Cause-effect relationships could be complex (nonlinear and dynamic) and time varying (normal
versus abnormal situations, parameter shifts etc.).
 Modeling & Simulation
 Given a cause profiles, predict the effect profile
 Optimization
 Design the system (parameters) operation to maximize profit
 Identification
 Determine in an empirical manner the cause-and-effect relationship
 Control
 Facilitate a cause to regulate the effect in the presence of disturbances
 Fault detection and diagnosis
 Mine the data to reveal data dependencies

Approaches to decomposition
System
Causes
(deterministic)
Effect
(measured)
Disturbances/ drifts
 Represent the overall system in terms of smaller sub-systems that are
relatively easily solvable
 Issues of efficient partitioning that facilitates co-existence & solution ease
 Union of these solutions does not necessarily represent the overall
system solution
 Issues of interaction and solution degradation exist.
 Co-ordinate so as integrate the local solutions such that it is optimal for
the entire problem.

Illustrative example: Control
Slurry
LCO
Gasoline
LPG
Tail Gas
Reactor
Regenerator
Catalyst/
coke
Catalyst
Air
Steam/ Oil feed
Slurry recycle
Main
Column
and Gas
Plant

Illustrative example: Control
Loop 1
Loop 2
Noise and
unmeasured
disturbancesMVC2 G2
Gd2
y3yd3
MVC1 G1
Gd
yYd
Gd1
u2u1
u3
+
-
+
+
+
++
-
Need to evolve a strategy to ‘Think globally but act locally’

Issues in Decentralized Control
 Objective: Decentralize but seek centralized performance
through co-ordination*1
 Decomposition
 Controllability and Observability aspects
 Vertical or Horizontal decomposition
 Decentralized Controller Design*2
: Design independently on the
basis of local sub-system dynamics and the nature of the
interconnections.
*1
Marquardt, CPC-VI, (2002), *2
Siljak (1996)

Co-ordination based control
MVC 1
MVC 2
MVC 3
MVC 4
Each node receives a plan of the other nodes moves and based on the
interacting dynamics, the node decides on its moves towards optimizing a
global cost.

Broad paradigms for decomposition*
G1 (m1
,y1
,x1
,x2
) = 0 G2 (m2
,y2
,x1
,x2
) = 0
m1
y1
m2
y2
x1
x2
Model co-ordination method
*1
Wismer, “Optimization methods for large scale systems
0x)y,G(m,..
),,(
,,
=ts
xymPMin
xym

Model co-ordination method
First level
Choose z to minimize
H(z) = H1(z) + H2(z)
min
m1
,y1
P1(m1
,y1
,z1
)H1(z) =
subjected to
G1(m1
,y1
,z1
,z2
) = 0
Determine
min
m2
,y2
P2(m2
,y2
,z2
)H2(z) =
Determine
subjected to
G2(m2
,y2
,z1
,z2
) = 0
m2
,y2
z
Second Level
Multilevel solution using model coordination
z
m1
,y1

Flow shop scheduling problem
A1
A2
A3
…………
…………
AnA
Platform A
b1
b2
…………bnb
Platform B
C1
C2
C3
Cnc
Platform C
D1
D2
D3
…………
Dnd
Platform D
nA – number of A lines; nB – number of B lines;
nc – number of C lines; nD– number of D lines
…………

Collaborative problem solving
Platform A Platform B Platform D
Individual formulations are simpler and intuitive when compared with a “monolith”
structure.
May perhaps be easier to solve to optimality at the individual steps.
Specialized solvers depending on nature of the problem can be used.
Often times, “interaction elements” are rather sparse – related to connectivity
Each platform has its individual formulation (constraints and solution method)
but updates the constraint bounds on other platform elements with which it
interacts.

Collaborative problem solving
PlatformA
Optimizer
PlatformB
Optimizer
,Exit if common cons traints s atis fied
Initialize
Optimizer 1 Optimizer 2
Decomposed

Some results: Flow shop scheduling problem
Scheduling for the lines in Platform A and B was solved using co-operative problem
solving for two scenarios:
• Cost functions were exactly the same using both approaches for each case.
• Decomposition and co-operation based solving is seen to be vastly superior to
monolith approach.
• Co-operative approach is definitely more scalable.
IterationsTimeProblem Type
940312Co-operative
3370268Monolith
IterationsTimeProblem Type
923412.2Co-operative
3436771Monolith
Scenario1 Scenario2

Lagrangian Relaxation methods
 Broad philosophy:
 Relax the constraint space of the problem by augmenting the objective function
with the difficult constraint(s) and solve the relaxed problem
 A solution to the less constrained problem is as good as or better than the
constrained solution. For a minimization (maximization) problem therefore, this
relaxation gives a lower (upper) bound to the true solution.
bxh
xgts
xfMin
x
≤
≤
)(
0)(..
)(
Difficult constraints
 Problem relaxation
0)(..
])([)(
≤
−+
xgts
bxhxfMin
x
λ
Relaxed problem easy
to solve

Lagrangian Relaxation methods
 Tighten the relaxation
0)(..
])([)(
≤
−+
xgts
bxhxfMin
x
λ
λ
Max
For convex problems, the solution of the above relaxed problem is the same
as that of the original problem.

Goal co-ordination method
x1
z1
m1
y1
G1 (m1
,y1
,x1
,z2
) = 0 G2 (m2
,y2
,x2
,z1
) = 0
m2
y2
x2
z2
x1
z1
Interaction balance principle : Require xi
= zi
as a result of goal
co-ordination

Goal co-ordination method
0),,,(..
),,(
2111
1
2
2
1
1
111
,,, 2111
=
−+
zxymGts
zxxymPMin
zxym
λλ
0),,,(..
),,(
1222
2
2
2
1
1
222
,,, 1222
=
−+
zxymGts
xzxymPMin
zxym
λλ
0),,,(
0),,,(..
)()x,y,P2(m)x,y,(mP),,,,(
1222
2
2111
1
2221111
=
=
−++=
zxymG
zxymGts
zxzxymPMin λλ
λ

Combinatorial Complexities: Sensor location in steam metering
flowsheet of methanol plant
5
7
11
9
8
4
3 1
62 10
1 3 15 24
2512
2769
13
4 17
28
14
7
8
20
21
26
18
19
10 11 16
22
2325
Objective: Determine
Sensor locations that
minimize failure rate
subject to cost constraint
*
Serth and Heenan, AIChE (1986)
Problem features:
11 balance equations involving 28
variables.
This flowsheet has a total of 21,474,180 sensor
combinations.Of these, 1,243,845 combinations
form an observable network.

Modeling failure rates
 Measured Variable
 Equal to the failure rate of the sensor measuring the variable
 Unmeasured Variable
 Sum of the failure rate of the sensors used for estimating the variable
( ) ( )
j
k k
C
j j j i i i
k 1 i C i C
i j j i
ˆ 1 x x 1 x j 1..nλ λ λ
= ∈ ∈
≠ ≠
   
   = − + − ∀ =   
      
∑ ∑ ∏
( )
( )
( ) ( )
j
k k
j
j N
*
j j
j N
i
i E
i
i N
C
j j j i i i
k 1 i C i C
i j j i
ˆMin max
s.t c 1 x C
x S 1, S V
1 x n m
ˆ 1 x x 1 x j 1..n
λ
λ λ λ
∀ ∈
∈
′∈
∈
= ∈ ∈
≠ ≠
− ≤
≥ − ∀ ⊆
− = −
   
   = − + − ∀ =   
      
∑
∑
∑
∑ ∑ ∏
Optimization formulation
Failure rate expression

Optimization Approaches
 Brute Force enumeration
 Time Consuming
 Greedy Search Algorithms
 Robust but do not guarantee optimality
 Mathematical programming Techniques
 Do not guarantee Optimality for MINLP
 Needs an explicit optimization formulation
 Constraint Programming
 Needs an explicit optimization formulation
 Guaranteed global optima and realizations
 Easy to generate pareto fronts

Constraint programming – an illustration
Initial Constraint Propagation
{ }
1
, , 1,2,3
Solve y z
x y
x z
x y z
<
− =
≠
∈
Choice Point & Failure Choice Point & Solution

Constraint programming – Results on steam metering
500 secsConstraint Programming
2.5 hoursBrute Force Enumeration
50 secs*
MINLP SBB
Time TakenApproach
*
No guarantee of global optimality

Hierarchical decomposition : Flowshop facility
Line
1
Line
2
Stage
3
A A
A
B
B
B B
C C
C
Tanks Tanks
D
D D
Stage
2
A
B
C
D
Tanks
E
B
A
Illustration

Functional decomposition
 Planning over a multi-period horizon:
Order Redistribution
 Detailed scheduling in each period:
Overall Inventory Profiles
 Operator level inventory scheduling :
Individual Tank Assignments
Level-1
Level-2
Level-3

Model granularity
 Upper bounds on processing times:
Abstraction of total inventory
 Upper bounds on total inventory :
Abstraction of total available compatible
tank volumes
 Operator level inventory scheduling :
Individual Tank Assignments
Level-1
Level-2
Level-3
Increasing model granularity
Specialized solvers could be used at each levels to fulfil goals at that level

Spatial decomposition: Model Identification for Control
Plantcontroller
yd
+
-
y
u
disturbance
d
+
+
Plant
u y Model 1u y
Model 2
Nonlinear plant
Locally linear models

Case study: high purity distillation
Local Model
y(t)=au(t) +
by(t) +
cy(t)u(t)
Model
Parameters
Gain Time
Constant
1 a 0.0030
b 0.9842
0.19 62.5
2 a 0.0053
b 0.9502
0.1064 18.75
3 a 0.004
b 0.9986
c 0.3424
- -
4 a 0.0096
b 0.9963
2.59 260

Case study: high purity distillation
Switching
Function

Conclusions
 Complexity introduced due to combinatoriality can be reduced using
intelligent enumeration via constraint programming.
 Typical applications: problems involving large number of integer/ binary
decision making
 Partitioning of large scale problems using collaborative /
communicative approaches simplifies solution procedures without
compromising solution rigor.

Typical application: large scale optimization and control problems.
 Lagrangian relaxation methods help to work around difficult
constraints and gradually progress towards the optimal via bounding
and relaxation.
 Typical applications: integer programming problems and those
bound by nonlinear constraints.

Thanks for your attention, Questions ?

Decomposition Paradigms for Solving Large Scale Systems

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