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.

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.

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

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)

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, ...

System description 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 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 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. 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 disturbances 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)

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 .

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 * Model co-ordination method *1 Wismer, “Optimization methods for large scale systems G 1 (m 1 ,y 1 ,x 1 ,x 2 ) = 0 G 2 (m 2 ,y 2 ,x 1 ,x 2 ) = 0 m 1 y 1 m 2 y 2 x 1 x 2

Model co-ordination method First level Choose z to minimize H(z) = H 1 (z) + H 2 (z) m 2 ,y 2 z Second Level Multilevel solution using model coordination z m 1 ,y 1 min m 1 ,y 1 P 1 (m 1 ,y 1 ,z 1 ) H 1 (z) = subjected to G 1 (m 1 ,y 1 ,z 1 ,z 2 ) = 0 Determine min m 2 ,y 2 P 2 (m 2 ,y 2 ,z 2 ) H 2 (z) = Determine subjected to G 2 (m 2 ,y 2 ,z 1 ,z 2 ) = 0

Flow shop scheduling problem A 1 A 2 A 3 ………… ………… A nA Platform A b 1 b 2 ………… b nb Platform B C 1 C 2 C 3 C nc Platform C D 1 D 2 D 3 ………… D nd Platform D n A – number of A lines; n B – number of B lines; n c – number of C lines; n D – number of D lines …………

Collaborative problem solving 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. 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

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: 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. Scenario 1 Scenario 2 Iterations Time Problem Type 9403 12 Co-operative 33702 68 Monolith Iterations Time Problem Type 9234 12.2 Co-operative 34367 71 Monolith

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.

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 . Difficult constraints Problem relaxation Relaxed problem easy to solve

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 .

Problem relaxation

Lagrangian Relaxation methods Tighten the relaxation For convex problems, the solution of the above relaxed problem is the same as that of the original problem.

Tighten the relaxation

Goal co-ordination method x 1 z 1 m 1 y 1 G 1 (m 1 ,y 1 ,x 1 ,z 2 ) = 0 G 2 (m 2 ,y 2 ,x 2 ,z 1 ) = 0 m 2 y 2 x 2 z 2 x 1 z 1 Interaction balance principle : Require x i = z i as a result of goal co-ordination

Goal co-ordination method

Combinatorial Complexities: Sensor location in steam metering flowsheet of methanol plant 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 . 5 7 11 9 8 4 3 1 6 2 10 1 3 15 24 25 12 27 6 9 13 4 17 28 14 7 8 20 21 26 18 19 10 11 16 22 23 2 5

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 Optimization formulation Failure rate expression

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

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

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 Choice Point & Failure Choice Point & Solution

Constraint programming – Results on steam metering * No guarantee of global optimality 500 secs Constraint Programming 2.5 hours Brute Force Enumeration 50 secs * MINLP SBB Time Taken Approach

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

Operator level inventory scheduling :

Individual Tank Assignments

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 Increasing model granularity Specialized solvers could be used at each levels to fulfil goals at that level Level-1 Level-2 Level-3

Operator level inventory scheduling :

Individual Tank Assignments

Spatial decomposition: Model Identification for Control Plant controller y d + - y u disturbance d + + Plant u y Model 1 u 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.

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 ?