Hybrid modelling and model reduction for control & optimisation

Transcription

1 Hybrid modelling and model redction for control & optimisation based on research done by RWTH-Aachen and TU Delft presented by Johan Grievink

2 Models for control and optimiation market and environmental conditions Model state and distrbance estimates (slow Dynamic Real Time Optimisation Model measrements Estimation Time Scale Separation state and distrbance estimates (fast Model Predictive Control optimal reference trajectories Model control setpoints Plant (inclding base control Large-scale rigoros model

3 Contents * Large scale physical modeling * Hybrid components * Physical based compleity redction * Mathematical compleity redction * Application aspects for INCOOP tasks * Conclsions

4 Large Scale chemical plant Models Plant operations modelling: process with sensors and actators conventional controllers eternal distrbances operating window constraints - economical objectives - Aspects of modelling: technology of modelling mainly generic aspects specific aspects: see Bayer and Shell applications work processes ndereposed

5 LSM: plant featres Feed preparation section Chemical conversion section Prodct prification section mltiple modes of prodction + switching distrbances mltiple nits / section & many compartments / nit recycling & heat integration non-linear behavior (nits + plant wide integration very wide range of relevant time scales (µs - days several thermodynamic phases with many species > Large scale dynamic process models ( DAE s

6 LSM: Systematic model development Reqirements Synthesis abstraction model eqations Analysis eistence soltion(s soltion method(s Model based design of eperiments Validation parameter estimation model discrimination Implementation coding verification Docmentation & transfer

7 LSM: fnctional reqirements Aim: accrate representation of physical behavior: relevant time scales > how small? feasible operational window: * physical ranges of inpts states otpts for realistic distrbance scenarios Trade-off s: compleity vs costs of development & validation compleity vs maintenance (costs in closed loop: compleity vs accracy

8 LSM: Synthesis of modelling space Abstraction: from reality to a modelling space with bonds: objects: spatial resoltion: time resoltion: chemical species: thermodynamic phases: V L L S S.. chemical reactions: contents and walls of eqipment streams sensors controllers actators.. mied compartments distribted in n-d. continos discrete discrete continos discrete continos; states of particles: sie stress charge.... Key isse: how to find relevant level of detail

9 LSM: Strctre of nit models Ψ µ k (α µ k (β (m.a/ t n F (n.a (n + Ψ (dif. (a + R(a F Model eqations: conservation / change: thermodynamic states: transfer rate: sorce/sink rates phase eqilibria trly first principles aa(ptc Ψ Ψ ( P T c FF( P RR(PTc µ (α k µ (β k T (α T (β

10 LSM: Physical properties & thermo Eqations for: thermodynamic variables per species and per phase: density specific heat entropy enthalpy free enthalpy transport coefficients: viscosity heat diffsion species diffsion srface tension Needed information: variables: P k (p k (0 PT;par k 0 and p (α P (α (PT p (0 gradients: p / q P(PT / q with q { P T } Isses: models in closed software bo comptational speed & accracy of soltion?

11 LSM: Transitions between models Trajectory in D Model domain : M ( p 0 D {g ( > 0} bondary Model domain : M ( p 0 D {g ( > 0} Trajectory in D Transition: e.g. phase split State event: g ( 0 if approaching from D g ( 0 if approaching from D Continity: T( 0

12 LSM: Hybrid model & components Hybrid models comprise first principles as well as empirical components to deal with ncertainty or compleity. Parallel strctre: ad-hoc model error compensation v Empirical model residal + y First principles model + prior

13 LSM: Seqential hybrid model Seqential strctre: physically motivated etension v Empirical model (- First principles model y Empirical modelling: static (non-linear regression model; e.g. ANN (π v y dynamic trend model: (t (π v y with π 0; pdate of π

14 LSM: Hybrid empirical components Reqirements to empirical components in combination with physical modelling: continos and differentiable (bonded in all variables comply with correct physical asymptotic behavior: - / t 0 for 0 when 0 ; e.g. not dc / dt k.c - a (a>0; - no fles remain when forces vanish; eqilibrim conditions no introdction of sprios roots - e.g. rate k.c /(+k.c + k 3.c - second order term in denominator may enhance fitting accracy bt it creates a maimm in the rate allowing for mltiple soltions.

15 LSM: Wiener / Hammerstein hybrid models static fndamental model available (process simlators plant dynamics accessible by plant tests (linear models Wiener strctre: Linear dynamics Non-linear static model y Hammerstein strctre: Non-linear static model Linear dynamics y

16 LSM: Linking of nit models Unit y y Unit Connectivity conditions: f y - y 0 Physical conseqences of copling (mass / heat: recycles increase I/O time constants: T i τ i / ( - i K i can indce non-linear positive feedback: > mltiple stationary states can occr

17 LSM: Wrap - p Focs has been on model synthesis Strctred approach is essential to redce modelling errors Decomposition / aggregation at the finer scales? Model synthesis more an art than a science Not covered: many other important aspects differential inde problems scaling and initialisations of DAE s global sensitivity analysis eperimental validation aspects Large scale plant model will now serve as sorce model for redction to models sitable for on-line se in control and optimisations

18 Models for control and optimiation market and environmental conditions Model state and distrbance estimates (slow Dynamic Real Time Optimisation Model measrements Estimation Time Scale Separation state and distrbance estimates (fast Model Predictive Control optimal reference trajectories Model control setpoints Plant (inclding base control Large-scale rigoros model

19 Redction of model compleity Options for redction: redce the modelling space - lmping of species reactions phases - combining compartments lmping by OCFE - simplify eqations (strctre: - by linearisation; - (non-linear approimations of comple epressions; e.g for physical properties and kinetics redce nmber of eqations / order redction - remove trivial linear connectivity eqations and variables - order redction

20 Model simplification: elimination connectivities Modlar modeling (e.g. in gproms large nmber of redndant eqations of type X Y (e.g. connection of different trays in distillation colmn increased model sie withot additional physical information Idea: Determine redndant eqations and variables by atomatic analysis of the system s incident matri Perform atomatic mapping of variables in simlation /optimiation software Prototype Implemented in seqential approach dynamic optimiation software

21 Reslts of elimination Application to distillation colmn I Simplification procedre Model sie redced from 3496 to 039 Redndant eqations Comptation time in optimiation redced by 53%

22 Singlar pertrbation: ( ( f f & & ( 0 ( f f & ( ( f f & & ( c c f & c 0 & Model redction by projection Trncation:

23 Eample of singlar pertrbation Batch reactions: (slow: B + C <> D (fast: A + D <> B + E Model: c A - r r k (c B.c C - c D / K c B - r + r r k (c A.c D - c B.c E / K c C - r k k / e c D + r - r c E + r Transform model to separate fast and slow modes ξ r c A c A 0 - ξ c B c B 0 - ξ + ξ ξ r > e. ξ r c C c C 0 - ξ c D c D 0 + ξ - ξ c E c E 0 + ξ Singlar pertrbation: ξ r (ξ ξ 0 r (ξ ξ

24 Model redction by projection (I Generic procedre. Transform original state space into a state space better revealing important process dynamics. Decomposition into two complementary sbspaces * U [ U ] U U U + U U Sbstittion into original DAE system and decomposition yields & & U U (0 U 0 T T (0 U T T g( f ( ( ( * 0 0 * f ( * * * + U + U + U + U + U + U y y y * original trajectory

25 Model redction by projection (II 3. Dedction of a redced model a Trncation ( 0 ( (0 ( * * 0 * y U g U y U f U T T + + & ( 0 lower nmber of eqations and variables than original model not steady-state accrate b Residaliation 0 ( & ( 0 ( (0 ( 0 ( * * 0 * * y U U g U y U U f U y U U f U T T T & same sie as original model bt less differential and more algebraic eqations and variables steady-state accrate

26 Model redction by projection Problem How to compte this transformation matri? Options Physical based lmping Gramian based inpt otpt balancing transformations Proper orthogonal decomposition Properties Redces the nmber of differential eqations Linear control problems are redced with n-cbed Increases the compleity of the model Effort nmerical integration will not be redced if strctre was eploited by the solver

27 Model redction by Gramian based trncation Dynamics of plant with basic control LC 0 CC 0 F 0 0 T 0 D L T d Reactor: eothermic reaction d d A B TC 0 T F LC 0 Q CC 03 LC 03 Colmn: constant molar overflow constant relative volatility 4 trays V' B b

28 Model redction by Gramian based trncation F 0 [mol/s] [-] B [-] D [-] time [hr] time [hr] legend: original 45 th order linearied 45 th order redced 4 th order.

29 Proper orthogonal decomposition (POD Starting point: Representative trajectory for given 0 and (t. Uniformly sample trajectory: snapshot matri X [ ( t ( t... ( t with * ( t ( p k t k ]. Singlar vale decomposition of X yields basis U [ d d... d ] p 3. a Select p << p left singlar vectors of X associated with largest singlar vales captre dominant dynamics corresponds to trncation b Use all p singlar vectors of X as basis can apply residaliation

30 Case stdy optimiation Test plant Operated in semi-batch mode degree of freedom: refl ratio R L/D objective: minimie operation time t f endpoint constraints fied amont of prodct N p (t f 000 mol prodct composition p (t f 0.0 path constraint reactor hold-p 350 mol < N(t < 600 mol

31 Soltion with trncated model (I Apply projection and trncation to colmn model Original colmn model: 4 trays states 5 levels of redction: n { } model order integrations iterations obj. fn. vale CPU time [s] nominal trncation problem infeasible Projection partly destroys sparsity strongly increased CPU time Projection deteriorates optimality

32 Soltion with residalied model Apply projection and residaliation to colmn model Original colmn model: 4 trays states 3 levels of redction: n {30 6 8} model order integrations iterations obj. fn. vale CPU time [s] nominal residaliation Soltion almost identical to nominal case Matri fill-p independent of redction level CPU time in the same order of magnitde bt mch higher as nominal

33 Nonlinear Estimation I Problem Need state to initialie model Given measred data prodce state-estimate Soltion Least sqares horion estimation LTV approimation instead of tre nonlinear Single step LTV gives Etended Kalman Filter (EKF Featres: Mlti-rate measrements Constraints on process variables Delayed measrements Primitive line-search

34 Nonlinear Estimation II Problem Large nmber of parameters > n+h*nw Ill-conditioning of estimation problem Tning estimation fnction difficlt Soltion Select only relevant inpt/otpt behavior Redce nmber of differential variables Uncertain feeds/energy flows to measred process variables I/O balanced model-redction Properties Well conditioned small problem Physical interpretation in dominant modes Online feasible / reliable from certain nmber of states downwards

35 Model redction for Estimation III Compte Jacobian of the model ( 0 ( y g y f & D C y B A + + & D C y B A + + ~ ~ ~ & D C y B A b b b + + ˆ ˆ ˆ & Derive linear model First redction linear model with fied projection (e.g. remove connectivity Second redction on redced linear model by balanced trncation (online Significant comptational savings obtained de to n 3 effect in control comptations

36 Available models for control and optimiation Rigoros model Physical redced model Linearied model Linearied redced model Available models for control and optimiation Dynamic Real Time Optimiation Model Predictive Control Estimation

37 Discssion Is it possible to redce the model while retaining strctre? Can one redce the nmber of eqation significantly? Can the application aspect be directly considered in the redction procedre?

38 Conclsions Model synthesis: systematic approach needed Model redction for ChE DAE systems: mied reslts Redcing nmber of eqations helps better than shifting the balance between ODE s and AE s in DAE problems. If algorithms eploit sparsity order redction is less effective. Modelling and model compleity redction for integrated d-optimisation & control largely open Key isses: - combination of fndamental and empirical model components - physical based lmping (species phases reactions - model compleity redction - real time adaptation of strctre of redced models - consistency of redced models for varios tasks - closed loop model validation

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