Unsupervised Reduce Order Modeling of Lead- Acid Battery Using Markov Chain Model

Size: px

Start display at page:

Download "Unsupervised Reduce Order Modeling of Lead- Acid Battery Using Markov Chain Model"

Ann McDowell
5 years ago
Views:

1 Unsupervised Reduce Order Modeling of Lead- Acid Battery Using Markov Chain Model Ali Akbar Shahbazi PhD Candidate of Mechanical Eng. Department of University of Tehran Vahid Esfahanian Professor of Mechanical Eng. Department of University of Tehran June

2 OUTLINE Introduction Theory Snapshot Data Collection Construction of Reduced Order Subspace Dynamical modeling Projection method Markov chain Results and Discussion Conclusion 2

3 Introduction 3

4 Introduction LEAD-ACID BATTERY (LAB) Lead-Acid Battery (LAB hereafter) is one of the most common energy storage devices 4

5 Introduction LAB SIMULATION AND MODELING Design Decisionmaking tasks Efficient simulation Optimizatio n Monitoring Control 5

6 Introduction LITERATURE REVIEW Author (Year) Description J. Newman and W. Tiedemann (1975) First review of flooded porous electrodes theory H. Gu, T.V. Nguyen, R.E. White (1987) First charge, rest and discharge model for battery simulation W.B. Gu, C.Y. Wang, B.Y. Liaw (1997) First use of Finite Volume Method (FVM) V. Esfahanian, F. Torabi (2006) Using Keller-Box method for 1-D modeling F. Torabi, V. Esfahanian (2011, 2013) Thermal runaway study of lead-acid batteries L. Cai and R. E. White (2009) Implement POD-based ROM for simulation of lithium-ion battery J. Burkardt, M. Gunzburger, and H.C. Lee (2006) Introduction of Clustering concept In ROM A.B. Ansari, V. Esfahanian, F. Torabi (2016) 1-D POD-based ROM of lead-acid battery during a cycle V. Esfahanian, A.B. Ansari, F. Torabi (2015) 1-D POD-based ROM of lead-acid battery during discharge E. Kaiser et al. (2014) Introduce Markov chain dynamical modeling in ROM 6

7 Introduction REDUCED ORDER MODELING (ROM) OF LAB Lead-Acid Battery Simulation CFD simulation Other methods (ENG) Time Consuming Good Accuracy Fast Simulation Less Accuracy Reduced Order Modeling (ROM) 7

8 Introduction REDUCED ORDER MODELING (ROM) OF LAB Reduced Order Modeling Steps Snapshot data collection Construction of Reduced Order Subspace Dynamical Modeling Experiment Simulation Orthogonality Similarity Projection System identification (Markov Chain) 8

9 Theory 9

10 Theory Snapshot data collection SNAPSHOT DATA COLLECTION High-fidelity model FVM solution of 1-D leadacid cell model 10

11 Theory Snapshot data collection CHEMICAL REACTIONS Negative Electrode Positive Electrode discharge 4 charge 4 Pb+HSO PbSO +H 2e discharge 2 4 charg e 4 2 PbO +HSO +3H 2e PbSO +2H O 11

12 Theory Snapshot data collection GOVERNING EQUATIONS (1D ASSUMPTION) Conservation of Charge in Solid eff. s Aj 0 s x0 eff V, 0 s x s x0, L xl I Conservation of Charge in Liquid eff eff. l. D ln c Aj 0 l x x0, L 0 Conservation of Species c eff Aj c. D c a2 t 2F x x 0, L 0 The above equations is solved using Finite Volume Method (FVM) to collect the snapshot data 12

Theory Construction of Reduced Order Subspace CONSTRUCTION OF REDUCED ORDER SUBSPACE ROM field Approximation N (n), t a t x v x n 1 n POD POD Based ROM 1 M (1) (2) ( ) (n),,.

13 Theory Construction of Reduced Order Subspace CONSTRUCTION OF REDUCED ORDER SUBSPACE ROM field Approximation N (n), t a t x v x n 1 n POD POD Based ROM 1 M (1) (2) ( ) (n),,..., N M N v 2 m 1 m a n 1 n t x,,..., argmin,,..., (1),opt (2),opt ( N),opt (1) (2) ( N) POD (i) subjected to, (j) 1 0 i j i j RO n m m1 M Cluster Based ROM C v, n 1,..., N C C, i j i j N M n1cn vm m1 C v : v v (n) n m m m (1) (2) ( N) N ( n) v n1 C m M,,..., v (1),opt (2),opt ( ),opt (1) (2) ( ),,..., N N argmin ROM,,..., (i) m n 2 13

14 Theory Dynamical Modeling DYNAMICAL MODELING Goal Determining time variation of modes amplitude Dynamical Modeling Projection method System identification methods like Markov chain model System identification refers to extracting information or building a mathematical model of a dynamical system from measured data 14

Theory Dynamical Modeling PROJECTION METHOD Concept This method projects the governing equations onto the reduced order subspace Like tracking a dynamic shadow of

15 Theory Dynamical Modeling PROJECTION METHOD Concept This method projects the governing equations onto the reduced order subspace Like tracking a dynamic shadow of a trajectory vx, t t da t x Projection Method, ts x, t v x N N n (n) n x n1 dt n1 (n) x a t x S( x, t) d a dt a t, (i) (j) ij (i) (j) ij x, i, j 1,..., (i) i, S N 15

16 Theory Dynamical Modeling MARKOV CHAIN MODEL Previous Works At the first time Eckhardt group uses Markov model for Statistical analysis of coherent structures pipe flow. (2004 & 2007) Kaiser et al. (2014) used Markov chain model for dynamical modeling in ROM Advantages of Markov Model The dynamical behavior of the system directly from snapshot data The computational time is reduced comparing with the projection technique The model could be handled in an unsupervised manner unsupervised means that it can be used for any physics with different governing equation 16

17 Theory Dynamical Modeling MARKOV CHAIN MODEL Definition Markov chain is a stochastic model describing a random process that has Markov property Markov property (memoryless property ) property of a random process in which the probability of next event depends only on the present event and conditionally is independent of previous events X i X i, X i,, X i X i X i n1 n n n n1 n1 n n 17

Theory Dynamical Modeling MARKOV CHAIN MODEL Transition Matrix The element P ij in the matrix P denotes the probability of moving from state i to state j or P j

18 Theory Dynamical Modeling MARKOV CHAIN MODEL Transition Matrix The element P ij in the matrix P denotes the probability of moving from state i to state j or P j i in the state space. The transition matrix P ij is defined as the probability of moving from cluster C i to cluster C j in one forward time-step P ij o o ij i 18

19 Theory Dynamical Modeling MARKOV CHAIN MODEL Markov Model in ROM The coefficient a in the ROM approximation is obtained from transition matrix multiplication. ROM field Approximation n 1 n 1N N1 N (n), t a t x a v x k1 k a P a k1 k 0 a P a 19

20 Results and Discussion 20

21 Results and Discussion VALIDATION AND VERIFICATION Test-Case Gu et al. (1987) Reproduced Gu et al. (1997) Esfahanian and Torabi (2006) Figure 3: Cell voltage during discharge (grid size=64) 21

22 Results and Discussion VALIDATION AND VERIFICATION Table 1: Performance of different dynamical models Method Snapshot Dim. Run time (s) Speed up factor RMSAE 1 FVM Ref Ref CROM + Projection E-2 OCROM + Projection E-2 CROM + Markov E-1 OCROM + Markov E-1 FVM Ref Ref CROM + Projection E-3 OCROM + Projection E-3 CROM + Markov E-2 OCROM + Markov E-2 22

23 CONCLUSIONS ROM of LAB Snapshot data: 1D FVM solution Basis Construction: Similarity approach (Clustering) Dynamical modeling: Projection and Markov chain Results The results show good agreement with previous results Markov model is about 2-4 times faster than projection technique 23

24 Thanks for your attention 24

Three-Dimensional Numerical Simulation of Lead-Acid Battery

Three-Dimensional Numerical Simulation of Lead-Acid Battery 1 Three--Dimensional Numerical Simulation Three of Lead Lead--Acid Battery Vahid Esfahanian Hamid Afshari Arman Pouyaei Amir Babak Ansari Vehicle, Fuel and Environment Research Institute (VFRI) Department