Unsupervised Reduce Order Modeling of Lead- Acid … · Unsupervised reduce order modeling of lead acid battery using Markov chain model Ali Akbar Shahbazi, ... At the first time

Unsupervised reduce order modeling of lead acid battery using Markov chain model Ali Akbar Shahbazi, Vahid Esfahanian 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

June 2017

Vahid Esfahanian

Professor of Mechanical Eng. Department of University of Tehran

Unsupervised reduce order modeling of lead acid battery using Markov chain model Ali Akbar Shahbazi, Vahid Esfahanian

OUTLINE

Introduction

Theory

Snapshot Data Collection

Construction of Reduced Order Subspace

Dynamical modeling

● Projection method

● Markov chain

Results and Discussion

Conclusion

2


Introduction

3


LEAD-ACID BATTERY (LAB) Lead-Acid Battery (LAB hereafter) is one of the most common

energy storage devices

Introduction

4


LAB SIMULATION AND MODELING

Introduction

5

Efficient simulation

Design

Optimization

Control Monitoring

Decision-making

tasks


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

Introduction

6


REDUCED ORDER MODELING (ROM) OF LAB

Lead-Acid Battery Simulation

CFD simulation

Time Consuming

Good Accuracy

Other methods (ENG)

Fast Simulation

Less Accuracy

Introduction

7

Reduced Order Modeling (ROM)


REDUCED ORDER MODELING (ROM) OF LAB

Introduction

8

Reduced Order Modeling Steps

Snapshot data collection1Experiment

Simulation

Construction of Reduced Order Subspace

2Orthogonality

Similarity

Dynamical Modeling3Projection

System identification (Markov Chain)


Theory

9


SNAPSHOT DATA COLLECTION

Snapshot data collectionTheory

10

High-fidelity model

FVM solution of 1-D lead-acid cell model


CHEMICAL REACTIONS


11

Negative Electrode

discharge

charge4 4Pb+HSO PbSO +H 2e

Positive Electrode

discharge

charg2 4 2e 4PbO +HSO +3H 2e PbSO +2H O


GOVERNING EQUATIONS (1D ASSUMPTION)


12

The above equations is solved using Finite Volume Method (FVM) to collect the snapshot data

Conservation of Charge in

Solid eff. 0s Aj

Conservation of Charge in Liquid

eff eff. . ln 0l D c Aj

Conservation of Species

eff

2.2

c AjD c a

t F

0,

0x L

c

x

0

eff

0,

, 0

s sx x L

s

x L

V

Ix

0,

0l

x Lx


CONSTRUCTION OF REDUCED ORDER SUBSPACE

Construction of Reduced Order SubspaceTheory

13

ROM field Approximation

(n)

1,

N

nnt a t

v x x

POD Based ROM

2

(1) (2) ( ) (n)

1 1POD

1, ,...,

M NN

m nm na t

M

v x

POD(1),opt (2),opt ( ),opt (1) (2) ( )

(i) (j)

, ,..., argmin , ,...,

1subjected to ,

0

N N

i j

i j

Cluster Based ROM

2

(1) (2) ( )

R

(

O

n)

1M , ,...,m n

NN

mn C

v

v

(1),opt (2),opt ( ),opt

R

(1

O

)

M

) (2 ( ), ,..., argmin , ,...,N N

1

1 1

(n) (i)

, 1,...,

,

:

M

n m m

i j

MN

n n m m

n m m m

C n N

C C i j

C

C

v

v

v v v


DYNAMICAL MODELING

Dynamical ModelingTheory

14

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


PROJECTION METHOD

Concept

This method projects the governing equations onto the reduced order subspace

Like tracking a dynamic shadow of a trajectory


15

Projection Method

,, ,

tt S t

t

x

v xv x x

(n) (n)

1 1

( , )N N

n

nn n

da ta t S t

dt

xx x x

d

a a tdt

(i) (j)

(i) (j)

(i)

,

, , 1, ...,

,

ij

ij

i

i j N

S

x


MARKOV CHAIN MODEL


16

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


MARKOV CHAIN MODEL


17

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

1 1 0 0 1 1 1 1| , , , | n n n n n n n nX i X i X i X i X i X i


MARKOV CHAIN MODEL

Transition Matrix

The element 𝑃𝑖𝑗 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 𝑃𝑖𝑗 is defined as the probability of

moving from cluster Ci to cluster Cj in one forward time-step


18

ij

ij

i

oP

o


MARKOV CHAIN MODEL

Markov Model in ROM

The coefficient a in the ROM approximation is obtained from transition matrix multiplication.


19

1

1 0

k k

k k

a P a

a P a

ROM field Approximation

(n)

1 11,

N

n N Nnt a t a

v x x


Results and

Discussion

20


VALIDATION AND VERIFICATION

Test-Case

Gu et al. (1987)

Reproduced

Gu et al. (1997)

Esfahanian and Torabi (2006)


21

Figure 3: Cell voltage during discharge (grid size=64)


VALIDATION AND VERIFICATION

Method Snapshot Dim. Run time

(s)

Speed up

factor

RMSAE1

FVM 64 × 25 1252.27 Ref Ref

CROM + Projection 64 × 25 148.14 8.45 3.94E-2

OCROM + Projection 64 × 25 159.03 7.87 1.36E-2

CROM + Markov 64 × 25 94.26 13.29 1.50E-1

OCROM + Markov 64 × 25 92.66 13.52 1.21E-1

FVM 128 × 80 2349.49 Ref Ref

CROM + Projection 128 × 80 244.11 9.62 5.43E-3

OCROM + Projection 128 × 80 259.89 9.04 2.15E-3

CROM + Markov 128 × 80 144.95 16.21 1.96E-2

OCROM + Markov 128 × 80 138.52 16.96 1.78E-2


22

Table 1: Performance of different dynamical models


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


Thanks for your

attention

24

Documents

Unsupervised Reduce Order Modeling of Lead- Acid … · Unsupervised reduce order modeling of lead acid battery using Markov chain model Ali Akbar Shahbazi, ... At the first time