Order Reduction of Large Scale DAE Models. J.D. Hedengren and T. F. Edgar Department of Chemical Engineering The University of Texas at Austin

Transcription

1 Order Reducion of Large Scale DAE Models J.D. Hedengren and T. F. Edgar Deparmen of Chemical Engineering The Universiy of Teas a Ausin 1

2 Ouline Moivaion Two Sep Process for DAE Model Reducion 1. Reducion of differenial equaions 2. Reducion of algebraic equaions wih ISAT Eamples 2

3 Single Currenly DAE model size Small 1-1 variables process unis e.g., reacor models Real-ime NMPC calculaions are feasible Medium 1-1, variables Muliple process unis Mulicomponen modeling, reacion neworks Real-ime NMPC applicaions very difficul Large 1,+ variables Plan wide dynamic models opimized a his level only wih seady sae models RTO 3

4 Moivaion Planwide NMPC conrol Large scale DAE sysems Sorage and rerieval of opimal conrol rajecories small and medium scale Reveal underlying srucure of he model Deermine dynamic degrees of freedom Find source of DAE iniializaion / convergence problems Auomae model reducion 4

5 Adapive DAE Model Reducion 1. Reducion of differenial equaions 2. Reducion of algebraic equaions 5

6 Proper ODE Model Reducion Opimally reduce he number of model variables Linear combinaion of saes ha reain he mos imporan dynamics Mehods Orhogonal Decomposiion POD Balanced Covariance Marices BCM 6

7 Predicing DDOF Can model reducion be made adapive? Possible error conrol sraegies: Singular values poor predicor Solve non-reduced model a check poins inefficien Equaion residuals new approach 7

8 Model Reducion Error Variable error consrain ε ROM Conrolling variable error model order, variable error model order, variable error DOF ol Toal degrees of freedom DOF = model order Dynamic degrees of freedom DDOF = reduced model order ha saisfies he variable error consrain 8

9 9 Predicing DDOF Linearized sysem Galerkin projecion Subsiue Predicor r = variable error, R = equaion residual Bu A + = ɺ ~ r P T + = ~ r P T ɺ ɺ ɺ + = ~ ~ Bu r Ar A P P T T + + = ɺ ɺ Ar r Ar R = ɺ 1 R A r Variable error predicor linearized

10 Adapive ODE Model Reducion Variable error consrain 1 ROM = r A R ε ol Open equaion forma f ɺ, = Soluion obained by finding roos f ~ T ~ T P ɺ, P R = Conrolling variable error ieraive approach 1 When A R ε, model order 1 When A R > ε, model order ol ol Soluion obained by minimizing residuals Variable error predicor can also be used o improve reduced model accuracy 1

11 Eample: Adapive ODE Reducion 1-D unseady hea conducion T 1 T 2 ρc T = k T Discreized he PDE o give a se of 2 ODEs Simulaion: Aluminum slab wih hickness 1 m Iniially a 25 ºC A = he lef boundary is changed o 1 ºC Tolerance se ε ol = 1 ºC 11

12 Eample Resuls Afer 1 minues he emperaure profile approaches seady sae 4 38 K r e36 u r a e p34 m e T 32 2 saes 3 saes 2 saes 1 sae Disance m Variable error predicor indicaes ha a leas 3 saes are required o mee error olerance of 1 ºC 12

13 Eample Resuls Variable error predicor can also be used o improve he reduced model accuracy 1 sae required wih correcion C º 35 r e u 34 r a e p m33 e T Disance m 2 saes 3 saes 2 saes 1 sae Ecellen predicion because he model is nearly linear and approaching seady sae 13

14 Adapive DAE Model Reducion 1. Reducion of differenial equaions model reducion 2. Reducion of algebraic equaions wih ISAT 14

15 Obain Pariioning and Precedence Ordering DAE model f z ɺ, z, DAE = = ODE sae; y = algebraic sae Sparsiy mari f f ODE AE ɺ,, y,, y, = = J ij = 1 if y j or oherwise appears in DAE i Pairing equaions and variables ɺ j f a maimum ransversal larges diagonal via rearrangemen Zero-free diagonal means ha each variable is uniquely paired wih an equaion 15

16 16 Pariioning and Precedence Ordering Lower riangular block form Each successive block of variables and equaions can be solved independenly Invering he sparsiy mari shows global variable dependencies Binary disillaion eample sysem: sa B sa A L V L V L A A P P h h n n T y h ɺ ɺ ɺ ɺ or h n n h y h P P T A L V L V A L sa B sa A ɺ ɺ ɺ ɺ or Original sparsiy Lower riangular block form

17 Scalabiliy o Large Sysems n number of algebraic equaions τ number of non-zeros in he sparsiy mari The maimum ransversal algorihm has a wors case bound of Onτ alhough ypical eamples are more like On + Oτ Duff, 1981 The lower riangular block algorihm also ehibis ecellen scaling for large problems wih an upper bound of On + Oτ Duff and Reid, 1978 Similar o approaches for solving process design equaions 197s 17

18 Reducion of Algebraic Equaions Eplici ransformaion of algebraic equaions Transform model equaions ino an eplici form Apply Tarjan s algorihm for precedence ordering Model equaions can be proprieary no available o user, e.g. commercial simulaor Neural neworks Erapolaion problems No reliable error conrol sraegy In siu adapive abulaion ISAT Dynamic daabase wih error conrol Replacemen for neural nes? 18

19 Eample: Flowshee Modeling and Model Reducion Mulicomponen, muliphase objec-oriened simulaor FORTRAN 9 rouines for fas eecuion DIPPR daabase wih properies for >17 compounds DASPK 3. for numerical inegraion and sensiiviy analysis Curren models are a compressor, splier, mier, vessel, hea echanger, and flash column 19

20 Eample: Flowshee Model Blending and separaion Feed sreams: buane, penane, heane, hepane, and ocane DAE model 12 differenial equaions 217 algebraic equaions Feed 1 Feed 2 Produc 1 Mier Produc 2 Holding Tank Spli valve Hea Echanger F l a s h Produc 3 2

21 Eample: Reduced Flowshee Model Resuls Algebraic equaion decomposiion 22 successively independen ses of variables and equaions One implici se: 16 equaions flash column Model reduced from 229 o 28 saes 12 ODEs / 16 AEs Feed 1 Feed 2 Produc 1 Mier Produc 2 Holding Tank Spli valve Hea Echanger F l a s h Produc 3 21

22 Eample: ISAT vs. Neural Nes Nonlinear funcion es case 2 independen variables 1 s eigenfuncion of an L-shaped membrane 2 nd and 3 rd eigenfuncions also appear on Mahworks publicaions Linear and nonlinear regions Poins ha are no coninuously differeniable ISAT also handles funcion disconinuiies, alhough ha capabiliy is no demonsraed here 22

23 ISAT Principal uning parameer ε ol Se o ε ol =.5 eremely coarse Inuiive adjusable parameer in his case lile accuracy is required ISAT creaed 12 linear regions 1, 2, f 23

24 ISAT Principal uning parameer Se o ε ol =.1 Moderae accuracy is required ISAT creaed 48 linear regions 24

25 ISAT Principal uning parameer Se o ε ol =.1 High accuracy is required ISAT creaed 26 linear regions 25

26 Principal uning parameers Srucure: 2 layers Neural Ne Hidden layer: 4 neurons, angen funcion Oupu layer: 1 neuron, linear funcion Opimizaion olerances Generaed wih MATLAB s neural ne oolbo 26

27 Eample: Conclusions ISAT advanages Fewer uning parameers More inuiive uning parameers Approimaes disconinuous and non coninuously differeniable funcions Builds in siu, wih no global opimizaions 27

28 Conclusions Two sep DAE model reducion process 1. Reducion of differenial equaions Adapive ODE reducion Predicor can also be used as a correcor from eample: 3 saes 1 sae Subsanial decreases in he number of ODEs are possible 28

29 Conclusions Two sep DAE model reducion process 2. Reducion of Algebraic Equaion wih ISAT Eample demonsraes ~1 imes reducion in number of variables Successful reducion of objec-oriened flowshee models wih mulicomponen processes ISAT eplicily ransforms ses of nonlinear equaions wih a given error olerance ISAT suggesed as a replacemen for neural neworks 29