Acm Tech Talk - Decomposition Paradigms for Large Scale Systems

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ACM Tech Talk - November 2008 Edition

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Acm Tech Talk - Decomposition Paradigms for Large Scale Systems

  1. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 32. Honeywell Technology Solutions I.I.T. Bombay, India Case study: high purity distillation Switching Function
  33. 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. 34. Honeywell Technology Solutions I.I.T. Bombay, India Thanks for your attention, Questions ?

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