Acm Tech Talk - Decomposition Paradigms for 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

SystemCauses

(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

SystemCauses

(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 disturbances

MVC2 G2

Gd2

y3 yd3

MVC1 G1

Gd

y Yd

Gd1

u2 u1

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 3MVC 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

*1Wismer, “Optimization methods for large scale systems

0x)y,G(m, ..

),,( ,,

ts

xymPMinxym


Model co-ordination method

First level

Choose z to minimizeH(z) = H1(z) + H2(z)

minm1,y1

P1(m1,y1,z1) H1(z) =

subjected to G1(m1,y1,z1,z2) = 0

Determine

minm2,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

zm1,y1


Flow shop scheduling problem

A1A2A3……………………AnA

Platform A

b1

b2

…………bnb

Platform B

C1C2C3

Cnc

Platform C

D1D2D3…………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

Platform A Optimizer

Platform B Optimizer

Exit, if common constraints satisfied

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:

1) Cost functions were exactly the same using both approaches for each case.

2) Decomposition and co-operation based solving is seen to be vastly superior to monolith approach.

3) Co-operative approach is definitely more scalable.

Problem Type Time Iterations

Monolith 68 33702

Co-operative 12 9403

Problem Type Time Iterations

Monolith 71 34367

Co-operative 12.2 9234

Scenario 1 Scenario 2


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

xfMinx

)(

0)( ..

)( Difficult constraints

Problem relaxation

0)( ..

])([)(

xgts

bxhxfMinx

Relaxed problem easy to solve


Lagrangian Relaxation methods

Tighten the relaxation

0)( ..

])([)(

xgts

bxhxfMinx

Max

For convex problems, the solution of the above relaxed problem is the same as that of the original problem.


Goal co-ordination method

x1z1

m1 y1

G1 (m1,y1,x1,z2) = 0 G2 (m2,y2,x2,z1) = 0

m2 y2

x2z2

x1 z1

Interaction balance principle : Require xi = zi as a result of goal co-ordination


Goal co-ordination method

0),,,( ..

),,(

21111

22

11

111

,,, 2111

zxymGts

zxxymPMinzxym

0),,,( ..

),,(

12222

22

11

222

,,, 1222

zxymGts

xzxymPMinzxym

0),,,(

0),,,( ..

)( )x,y,P2(m )x,y,(mP),,,,(

12222

21111

2221111

zxymG

zxymGts

zxzxymPMin


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 1622

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 ik 1 i C i C

i j j i

ˆ 1 x x 1 x j 1..n

j

k k

jj N

*j j

j N

ii E

ii N

C

j j j i i ik 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

Optimization formulationFailure 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

Approach Time Taken

MINLP SBB 50 secs*

Brute Force Enumeration 2.5 hours

Constraint Programming 500 secs

*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

CD

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

Plantcontrolleryd

+

-

y

u

disturbance

d

+

+

Plantu y Model 1u y

Model 2

Nonlinear plant Locally linear models


Case study: high purity distillation

Local Modely(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

SwitchingFunction


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 ?