BMS in the Bayesian Paradigm:

Part I: SOC Estimation

I. (Haran) Arasaratnam, McMaster U,

J. Tjong, U. of Windsor

& R. Ahmed, McMaster U

() June 17th, 2014 1 / 15

Introduction

Battery innovation occurs in 3 major areas:I Battery chemistryI Thermal managementI Battery control

State Estimator

V

I

T

SOC

SOP

SOH

Figure 1 :

The State-of-Charge (SOC) is defined as the ratio of remaining charge

capacity to nominal capacity

OEMs target: SOC error ≤ 3% throughout the service life

Why more precise SOC estimation?I To Improve usable energy/fuel economyI To prolong battery life

() June 17th, 2014 2 / 15

State Estimation (Cont’d)

3 major battery modeling techniques

I Black box/ model free (Coulomb counting, OCV)I Grey box (Equivalent circuit models)I White box (Electrochemical models)

Coulomb counting (CC) is the traditional method to estimate the SOC:

SOC[t] = SOC[0] +η

Q

∫ t

0

I(t)dt (1)

where Q is the nominal capacity, η is the Coulombic efficiency, and I is

current

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Time and Space Complexity

Pre

dic

tio

n A

ccu

racy

OCV-R-Model

FOM/ECM

ROM/ECM

OCV-R-RC-Model

OCV-R-2RC-Model

Figure 2 :

() June 17th, 2014 4 / 15

Equivalent Electrical Circuit Modeling

Vt

Rp

Cp

Ro

Voc = fcn(SOC)

IL

Vp

(a) OCV-R-RC equivalent circuit model

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 13.4

3.5

3.6

3.7

3.8

3.9

4

4.1

SOC

OC

V [

V]

DischargingChargingAveraged OCV

(b) OCV-SOC relationship

Figure 3 :

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Figure 4 : Dual Bayesian Estimation

EKBF

SRPF

Battery

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Battery Parameter Estimation

Governing equations:

Vp = −1

τVp +

1

CpIL

⇒ Vp,k+1 = e−Ts/τVp,k +Rp(1− e−Ts/τ )IL,k, (2)

Vt,k+1 = Voc(SOCk+1) + Vp,k+1 +R0IL,k+1 (3)

where the time constant τ = RpCp.

Eliminating Vp from (2)-(3) yields an ARX model (1,2,0):

yk+1 = a0yk + b0IL,k + b1IL,k+1, (4)

where

a0 = e−Ts/τ

b0 = −e−Ts/τRo + (1− e−Ts/τ )Rp

b1 = Ro

yk+1 = Vt,k+1 − Voc(SOCk+1)

() June 17th, 2014 7 / 15

Battery Parameter Estimation....

the SR-RLS algorithm is chosen to estimate the battery parameters,

which are related to a0, b0 and b1 as follows:

Ro = b1 (5)

Rp =b0 + a0b1

1− a0(6)

Cp =(a0 − 1)Ts

(a0b1 + b0) log(a0)(7)

The sequential RLS algorithm with a forgetting factor (λ) minimizes the

sum of error-squared cost function:

J(θ) =1

2

N∑i=1

λN−i(Vt,i − θTΦi

)2(8)

where λ: real, positive,< 1, and → 1

λ trades off the tracking capability for noise filtering; For our battery

experiment, λ of 0.995 was found to be sufficient

() June 17th, 2014 8 / 15

Battery State Estimation

State-space model:(˙SOC

Vp

)=

(0 0

0 − 1τ

)(SOC

Vp

)+

(1

Qbatt1Cp

)IL

(9)

Vt = Voc(SOC) + Vp +RoIL, (10)

Since the state space model is a continuous-time nonlinear model, the

extended Kalman-Bucy filter (EKBF) is applied to estimate the states.

EKBF steps:

C =∂Vt∂x

P = AP + PAT −KRKT + Q,P(0) = P0

K = PCR−1,

ˆx = Ax + BIL + K(Vt − Vt)

() June 17th, 2014 9 / 15

Computer Experiment

Figure 5 : OCV-R-RC type Battery Model Implemented in Matlab/Simulink

SOC estimators:I Coulomb Counting (CC)I Regressed Voltage (RV) Estimator

Assumed the initial SOC for both of these two estimators to be 64%

whereas the true initial SOC was set to be 80%

A current profile from a UDDS velocity profile was obtained by

simulating a mid-size EV in Matlab/Simulink() June 17th, 2014 10 / 15

Computer Experiment (Cont’d)

0 200 400 600 800 1000 12000

10

20

30

40

50

Sp

eed

[M

PH

]

Time [s]

UDDS Drive Cycle

(a) UDDS Speed Profile

200 400 600 800 1000 1200

−10

−5

0

5

10

Cu

rren

t [A

]

Time [s]

(b) Current Profile

Figure 6 : Speed and current profiles for one UDDS drive cycle

() June 17th, 2014 11 / 15

0 1000 2000 3000 4000 5000 6000 7000 800020

30

40

50

60

70

80

Time (s)

SO

C (

%)

TrueCCRV

3000 3500 40000

1

2

3

|SO

C E

rro

r| (

%)

(a)

25

30

35

Ro (

mΩ

)

5

10

Rp (

mΩ

)

200

400

600

Cp (

F)

0 1000 2000 3000 4000 5000 6000 7000 8000

0.51

1.5

τ =

RpC

p (s)

Time (s)

(b)

Figure 7 :

Parameter Ro [mΩ] Rp [mΩ] Cp [F] Max. SOC Err. [%]

True 25 5 300

RV Estimator 26.1 4.8 286.8 3.1

Table 1 :

() June 17th, 2014 12 / 15

Virtues of the Dual Estimator

Unlike the CC which uses the current measurement only, the dual method

systematically fuses the current measurement with the voltage

measurement

Applicable to any other types of battery chemistries whose SOC and

OCV relationship s one-to-one and monotonically increasing.

Modular: The parameter identification block can be re-used for other

battery control tasks such as battery SOP and SOH.

() June 17th, 2014 13 / 15

Open Questions

Table 2 :

Issues Root Causes Solution

Covariance Poor input excitation Upper bound covariance

Wind-up Time varying forgetting factor

Switch on/off RLS based on input’s

statistics

Convergence Model mismatch Open question?

Time varying parameter Voc Separate VocSOC error ≤ 3% ROM

at low temperature Voting/weighted average of multiple

methods (CC, RV, Data-driven)

() June 17th, 2014 14 / 15

Thank you!

() June 17th, 2014 15 / 15