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Acm Tech Talk - Decomposition Paradigms for Large Scale Systems

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ACM Tech Talk - November 2008 Edition

ACM Tech Talk - November 2008 Edition

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  • 1. 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
  • 2. Honeywell Technology Solutions I.I.T. Bombay, India 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.
  • 3. Honeywell Technology Solutions I.I.T. Bombay, India 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.
  • 4. Honeywell Technology Solutions I.I.T. Bombay, India 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
  • 5. Honeywell Technology Solutions I.I.T. Bombay, India 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)
  • 6. Honeywell Technology Solutions I.I.T. Bombay, India 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
  • 7. Honeywell Technology Solutions I.I.T. Bombay, India 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.
  • 8. Honeywell Technology Solutions I.I.T. Bombay, India 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
  • 9. Honeywell Technology Solutions I.I.T. Bombay, India 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’
  • 10. Honeywell Technology Solutions I.I.T. Bombay, India 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)
  • 11. Honeywell Technology Solutions I.I.T. Bombay, India 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.
  • 12. Honeywell Technology Solutions I.I.T. Bombay, India 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
  • 13. Honeywell Technology Solutions I.I.T. Bombay, India 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
  • 14. Honeywell Technology Solutions I.I.T. Bombay, India 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 …………
  • 15. Honeywell Technology Solutions I.I.T. Bombay, India 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.
  • 16. Honeywell Technology Solutions I.I.T. Bombay, India Collaborative problem solving PlatformA Optimizer PlatformB Optimizer ,Exit if common cons traints s atis fied Initialize Optimizer 1 Optimizer 2 Decomposed
  • 17. Honeywell Technology Solutions I.I.T. Bombay, India 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
  • 18. Honeywell Technology Solutions I.I.T. Bombay, India 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
  • 19. Honeywell Technology Solutions I.I.T. Bombay, India 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.
  • 20. Honeywell Technology Solutions I.I.T. Bombay, India 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
  • 21. Honeywell Technology Solutions I.I.T. Bombay, India 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
  • 22. Honeywell Technology Solutions I.I.T. Bombay, India 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.
  • 23. Honeywell Technology Solutions I.I.T. Bombay, India 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
  • 24. Honeywell Technology Solutions I.I.T. Bombay, India 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
  • 25. Honeywell Technology Solutions I.I.T. Bombay, India 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
  • 26. Honeywell Technology Solutions I.I.T. Bombay, India 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
  • 27. Honeywell Technology Solutions I.I.T. Bombay, India 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
  • 28. Honeywell Technology Solutions I.I.T. Bombay, India 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
  • 29. Honeywell Technology Solutions I.I.T. Bombay, India 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
  • 30. Honeywell Technology Solutions I.I.T. Bombay, India 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
  • 31. Honeywell Technology Solutions I.I.T. Bombay, India 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
  • 32. Honeywell Technology Solutions I.I.T. Bombay, India Case study: high purity distillation Switching Function
  • 33. Honeywell Technology Solutions I.I.T. Bombay, India 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.
  • 34. Honeywell Technology Solutions I.I.T. Bombay, India Thanks for your attention, Questions ?

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