What are the three methods used to verify the performance of the PLS model?

Five driving cycles, NEDC, UDDS, FTP, HWFET and NYCC are also used to verify the three SOC estimation methods in terms of the estimation accuracy and the execution time.

What is the advantage of the proposed method?

The main advantage of the proposed method is that the state space function of the battery model is dynamically linearized into only one element, which makes it possible for the application of the linear Kalman filter.

What are the three driving cycles used to verify the proposed PLS model?

Despite NEDC and UDDS, three more driving cycles, Federal Test Procedure (FTP), Highway Fuel Economy Cycle (HWFET) and New York City Cycle (NYCC) are also used to verify the proposed battery model.

What is the average MAE of the five driving cycles?

The average MAE of the five driving cycles is 0.0153 for the EKF with ECM, 0.0037 for the AUKF with LSSVM, and 0.0050 for the proposed method.

What is the effect of the proposed method on the measurement noise?

online identification of the parameters in the ECM may lead to numerical problems and is sensitive to the measurement noise.

What is the accuracy of the PLS model with moving window?

The absolute errors in Fig. 13(b) are less than 0.05 V in most conditions and the MAE is only 0.0052 V, which proves the accuracy of the PLS modeling with moving window.

How many samples are in the cycle?

The total MAE is calculated as,11 ˆ Ni MAE x x N (17) where x̂ is the predicted voltage of the PLS model, x is the voltage measurement, N is the number of samples in the entire cycle.

What is the difference between the proposed method and the AUKF with LSSVM?

Since the modeling method and the estimation method are both nonlinear in AUKF with LSSVM, a better accuracy in SOC estimation is received.

What is the convention for the voltage and current profiles of the LiFePO4 battery?

The battery voltage and current profiles measured during NEDC and UDDS are shown in Fig. 9 with the convention that positive current means discharging.

(Open Access) A Simplified Model-Based State-of-Charge Estimation Approach for Lithium-Ion Battery With Dynamic Linear Model (2019) | Jinhao Meng

Q: What is the simplest way to extract a YPLS?

In PLS, the independent variable XPLS and the response YPLS are decomposed into their projection and the orthogonal loading matrices as follows, TPLS PLS PLS PLSnX T P E (1) TPLS PLS PLS PLSnY U Q F (2)0278-0046 (c) 2018 IEEE.

Aalborg Universitet

A Simplified Model based State-of-Charge Estimation Approach for Lithium-ion Battery

with Dynamic Linear Model

Jinhao, Meng; Stroe, Daniel-Ioan; Ricco, Mattia; Guangzhao, Luo; Teodorescu, Remus

Published in:

I E E E Transactions on Industrial Electronics

DOI (link to publication from Publisher):

10.1109/TIE.2018.2880668

Publication date:

2019

Document Version

Accepted author manuscript, peer reviewed version

Link to publication from Aalborg University

Citation for published version (APA):

Jinhao, M., Stroe, D-I., Ricco, M., Guangzhao, L., & Teodorescu, R. (2019). A Simplified Model based State-of-

Charge Estimation Approach for Lithium-ion Battery with Dynamic Linear Model. I E E E Transactions on

Industrial Electronics, 66(10), 7717 - 7727. [8536907]. https://doi.org/10.1109/TIE.2018.2880668

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Abstract—The performance of model based

State-of-Charge (SOC) estimation method relies on an

accurate battery model. Nonlinear models are thus

proposed to accurately describe the external

characteristics of the Lithium-ion (Li-ion) battery. The

nonlinear estimation algorithms and online parameter

identification methods are needed to guarantee the

accuracy of the model based SOC estimation with

nonlinear battery models. A new approach forming a

dynamic linear battery model is proposed in this paper,

which enables the application of the linear Kalman filter for

SOC estimation and also avoids the usage of online

parameter identification methods. With a moving window

technology, Partial Least Squares (PLS) regression is able

to establish a series of piecewise linear battery models

automatically. One element state space equation is then

obtained to estimate the SOC from the linear Kalman filter.

The experiments on a LiFePO

battery prove the

effectiveness of the proposed method compared with the

Extended Kalman Filter (EKF) with two Resistance and

Capacitance (RC) Equivalent Circuit Model (ECM) and the

Adaptive Unscented Kalman Filter (AUKF) with Least

Squares Support Vector Machines (LSSVM).

Index Terms—State-of-charge estimation, partial least

squares regression, Kalman filter, Lithium-ion battery.

I. I

NTRODUCTION



ith the significant progress of the battery technology,

Lithium-ion (Li-ion) batteries have become a promising

choice for Electrical Vehicle (EV) [1] and Battery Energy

Storage System (BESS) [2], [3]. The extensive usage of the

Li-ion batteries is mainly because of their superior properties

including long lifespan, high energy density, low self-discharge



Manuscript received June 23, 2018; revised September 10, 2018;

accepted October 28, 2018. This work was supported in part by the Key

Program for International S&T Cooperation and Exchange Projects of

Shaanxi Province under Grant 2017KW-ZD-05, and in part by the

Fundamental Research Funds for Central Universities under Grant

3102017JC06004 and Grant 3102017OQD029. (Corresponding author:

Guangzhao Luo, phone: 0086-029-88431335; fax: 0086-029-88431310;

e-mail: guangzhao.luo@nwpu.edu.cn)

J. Meng and G. Luo are with the School of Automation, Northwestern

Polytechnical University, Xi’an 710072, China (e-mail:

scmjh2008@163.com; guangzhao.luo@nwpu.edu.cn). D.-I. Stroe, M.

Ricco, and R. Teodorescu are with the Department of Energy

Technology, Aalborg University, Aalborg 9220, Denmark (e-mail:

dis@et.aau.dk; mri@et.aau.dk; ret@et.aau.dk).

rate, etc [4]. State-of-Charge (SOC) reflects the amount of

energy available in a battery. In order to guarantee the

effectiveness of the battery pack in real-life application, each

cell needs to be balanced according to their SOCs. In addition,

SOC is also an indicator in the Battery Management System

(BMS) to help avoiding the overcharge and the over discharge

of the cells. Nevertheless, SOC is not an inherent parameter of

the battery. SOC has to be estimated because it is impossible to

be directly measured by sensors.

Many SOC estimation methods have been proposed recently

by researchers [5]. A straightforward way to estimate the SOC

is the integration of the current flowing through the battery,

which is known as the Coulomb counting method [6]. However,

Coulomb counting method needs an accurate knowledge of the

initial SOC, and the measurement errors from the current

sensor inevitably accumulate during the calculation process of

the Coulomb counting method. Open Circuit Voltage (OCV)

also has the potential to reflect the SOC, which exhibits a

monotonic relationship with SOC [7]. However, Li-ion battery

needs a long relaxation time to gradually reach its inner

equilibrium, which means accurate OCV measurement can

only be obtained after hours of relaxation time as illustrated in

[8]. The difficulty of the OCV measurement in real applications

deteriorates its extensive usage in the SOC estimation area.

Consequently, more advanced methods have been proposed to

avoid the drawbacks of the previous two methods.

With the fast development of the machine learning methods,

artificial intelligence-based techniques are used to estimate the

SOC. Recurrent Neural Network (RNN) [9], Support Vector

Machine (SVM) [10], and Multivariate Adaptive Regression

Splines (MARS) [11], [12] have been used to establish a SOC

estimator without the requirement of any previous knowledge

of the battery electrochemistry. After collecting enough

training samples in advance, those data driven based estimation

methods can establish the connection between the measured

signals (i.e., voltage, current, temperature) and the SOC.

However, it is impractical to have the datasets covered all the

working conditions of a real system since the actual conditions

are unpredictable. Hence, data driven methods have difficulty

in estimating an accurate SOC under the profiles that are

completely different from the training dataset.

Utilizing the feedback loop structure from the control field,

the model based estimation (Fig. 1) has been proposed to

A Simplified Model based State-of-Charge

Estimation Approach for Lithium-ion Battery

with Dynamic Linear Model

Jinhao Meng, Student Member, IEEE, Daniel-Ioan Stroe, Member, IEEE, Mattia Ricco,

Member, IEEE, Guangzhao Luo, Member, IEEE, and Remus Teodorescu, Fellow, IEEE

Transactions on Industrial Electronics

IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS

guarantee the robustness and accuracy of SOC estimation in

various conditions. In Fig .1, online parameter identification

aims at guaranteeing the accuracy of the battery model. The

estimation algorithm is able to calculate the gain L for SOC

estimation according to the voltage difference

U

Fig. 1. Framework of the model based estimation

Compared with traditional methods, the model-based

estimation methods do not require accurate knowledge of an

initial SOC. As shown in Fig.1, the model-based estimation

mainly contains the battery model and the estimation algorithm.

PI observer [13], sliding mode observer [14], H-infinity filter

[15], and Kalman filters [16]–[18], have been used as the

estimation algorithms. Among them, different kinds of Kalman

filters are the most frequently adopted methods in literature [5].

According to the principle of the model-based estimation, the

estimation algorithm cannot handle the errors from the battery

model [19]. One way to improve the performance of the battery

model is integrating more affecting factors into the model

equation, such as, the hysteresis behavior of OCV in LiFePO

battery [20]. Meanwhile, the complexity of the model is also

increased. It is also the main reason that the electrochemical

model [21] is not as popular as Equivalent Circuit Model (ECM)

in SOC estimation area. If an nonlinear model is utilized,

nonlinear Kalman filter has to be used to predict the SOC [22].

Extended Kalman Filter (EKF), Unscented Kalman Filter

(UKF), Particular Filter (PF) have already been applied to deal

with the nonlinear battery models [5]. EKF linearizes the

battery model by calculating the Jacobian matrix, but the

linearization process inevitably causes an increase of the errors.

Compared with EKF, more computing power is needed from

the hardware when UKF and PF are used. The complex

calculation of the nonlinear filters and the high order battery

model reduces their values for most real-time applications. The

other way to enhance the battery model is using additional

online parameter identification method as shown in Fig. 1,

because the parameters in the battery model change with

different working conditions [23]. It should be noted that the

parameters in the nonlinear battery model cannot always be

identified online [24], [25]. The nonlinearities and the time

scale separation in the battery model may cause some

parameters only identifiable at a specific frequency [24]. The

uncertainties of the results in the model-based estimation are

inevitably increased because of the aforementioned issues. A

suitable modeling approach, with good accuracy and less

complexity, is expected in the model-based estimation. The

features of the model-based estimation inspire us to simplify

the battery modeling process and the complexity of the entire

estimation structure while simultaneously ensuring the

estimation accuracy.

Ref. [18] attempts to simplify the battery modeling process

in the traditional model based estimation by using a Least

Squares Support Vector Machines (LSSVM) battery model and

an Adaptive Unscented Kalman Filter (AUKF). Although this

method reduces the complexity of the battery model to some

extent, large computational burden still remains because of the

calculation of AUKF and the LSSVM modeling process. Linear

Kalman filter combined with a generic Resistance and

Capacitance (RC) model has already been used to estimate the

SOC of a lead-acid battery in [26]. The method in [26]

estimates the voltage of a bulk capacitor to form the final SOC,

which essentially uses the nearly linear OCV-SOC curve of the

lead-acid battery. However, in Li-ion battery, the OCV-SOC

curve is nonlinear. Especially, the LiFePO

battery has a flat

OCV-SOC compared with other chemistries, which is a

challenge for an accurate SOC estimation. LiFePO

battery has

a higher power density and lower cost, which is a popular

choice for EVs and BESSs [27], [28]. Thus, LiFePO

battery is

also selected to validate the methods in this paper.

Partial Least Squares (PLS) regression [29], [30] is

connected with a linear Kalman filter to form a simple model

based SOC estimation approach in this paper. Different from

previous works, this paper establishes a linear piecewise battery

model online using the PLS and the moving window method.

As a result, the parameters in the PLS battery model are

automatically updated, and the order of the matrices in the state

space function is reduced to one. SOC is then estimated by the

linear Kalman filter with one order state space function. The

proposed method is compared with the EKF with two RC ECM

and the AUKF with LSSVM [18] in terms of estimation

accuracy and execution time. The experimental results on a

LiFePO

battery prove the validation of the proposed method.

This paper is organized as follows. The PLS modeling

method with moving window is introduced in Section II. A

simplified SOC estimation approach including a piecewise

linearized battery model and a linear Kalman filter is presented

in Section III. The performance of the proposed method in term

of estimation accuracy and execution time is validated in

Section IV, while conclusions are given in Section V.

II. PLS

REGRESSION BASED BATTERY MODEL

PLS regression is popular in the modeling of different

industrial applications, because it captures the crucial features

between the input and the output [31]. PLS has proved to be

more robust than other multiple linear regression methods [32].

Therefore, PLS is chosen to dynamically linearize the battery

model.

A. PLS regression

In PLS, the independent variable X

PLS

and the response Y

PLS

are decomposed into their projection and the orthogonal

loading matrices as follows,



LS PLS PLS PLS

XTP E 

(1)





LS PLS PLS PLS

YUQ F 

(2)

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where

PLS

= [t

, t

, …, t

] and U

PLS

= [u

, u

, …, u

] are the

score matrices,

PLS

= [ p

, p

, …, p

] and Q

PLS

= [q

, q

, …, q

]

are the loading matrices,

PLS

and

PLS

are the residual terms.

Let’s define

PLS

= [x

, x

, …, x

], Y

PLS

= [y

, y

, …, y

]. [t

, …, t

] are the dominant eigenvectors extracted from X

PLS

and [

, u

, …, u

] are the dominant eigenvectors of Y

PLS

. t

the first dominant eigenvector extracted from

PLS

, and u

extracted from

PLS

. In the process of PLS, the correlation

between

and u

is maximized at first. The regression function

between

PLS

and t

is then established as

11 1

PLS PLS

YtrF .

Afterwards, the process continues with more dominant

eigenvectors extracted from the residual of the regression

function. The PLS process will not terminate until the

regression function meets the desired precision. It’s known

from the calculation steps of PLS that it utilizes the advantages

of the principal component analysis and the linear regression.

In order to calculate the linear factors between

PLS

and Y

PLS

in PLS, different algorithms have been proposed. A simple

concept is introduced in [33] with few calculation steps and

lower computing burden. The steps are show in Fig. 2 [30]–[33].

The dominant eigenvector

is directly calculated from the

matrix (

PLS

)

PLS

)

PLS

. Then, the score matrix T

PLS

and

the loading matrices

PLS

and Q

PLS

are obtained. When the

regression function is accurate enough to meet the predefined

condition, the coefficient matrix

PLS

is obtained.

Fig. 2. The flowchart of PLS

B. Battery modeling with PLS and moving window

In order to explain the advantages of the PLS battery model

with moving window, a typical battery ECM should be firstly

introduced. ECM has become a popular choice in the model

based estimation because of its concise structure [34]. Two RC

ECM as shown in Fig. 3 has already proved to be a good

tradeoff between the complexity and the accuracy.

OCV =

f(SOC) represents the nonlinear relationship between OCV and

SOC.

is the internal resistance, and the two RC networks

represent the charge transfer (

, C

), the diffusion process (R

), etc.

Fig. 3. Two RC ECM

The expression of the two RC ECM is as follows.





12 0

112 2

bat bat

bat

UfSOCUUIR

UdUU dU

IC C

dt R dt

 





 





(3)

where

and U

are the voltages of the first and the second RC

network respectively. It should be noted that the RC elements in

the ECM are not always constant during the battery charge and

discharge [23]. Hence, online parameter identification has to be

used to guarantee the accuracy of the battery model. Compared

with two RC ECM, the parameters in the PLS model with

moving window can be automatically updated without the

necessary of online parameter identification.

In this paper, the current measurement

bat

and the SOC are

selected as the independent variable

PLS

, the terminal voltage

bat

is the response Y

PLS

. According to the description in the

previous subsection, the linear PLS battery model is expressed

as the following function,

12 3

PLS PLS

bat bat

UBX bbSOCbI  (4)

where

PLS

=[b

, b

] is the coefficient matrix following the

steps in Fig. 2.

The relationship between the voltage and the current is

nonlinear. However, it can be regarded as a linear model in a

short period, which indicates the nonlinear battery model can

be linearize into a series of linear PLS model. A moving

window modeling method [18] is used to dynamically linearize

the battery in this paper. Fig. 4 illustrates the process of

establishing the PLS battery model with moving window.

Fig. 4. The PLS based battery modeling process

The width of the moving window is defined as M in Fig.4,

which means that

M training samples are needed to calculate

the

PLS

. Hence, M training samples should be collected in

advance to calculate the first PLS model. For the rest PLS

models, the SOC in the training samples is actually the

estimated SOC. The samples for the next PLS model are

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collected during the estimation process in the previous moving

window. After another

M new training samples are collected,

the PLS battery model is updated. With the application of the

moving window, the PLS battery model is able to update itself

with a small number of initial training samples. In addition, the

PLS model is only related with the training dataset and not

limited to a certain chemistry. PLS model allows the linear

Kalman filter to estimate the SOC and also reduces the order of

the state space function. The expression of PLS model (Eq.(4))

is simpler than the two RC ECM (Eq.(3)). However, a specific

width of the moving window should be set to update the PLS

battery model. The selection of a suitable width of the moving

window will be discussed in Section IV.

III. SOC

ESTIMATION WITH A SIMPLIFIED STRUCTURE

Different Kalman filters have proved to be able to estimate

the battery SOC with good accuracy and robustness. Due to the

nonlinearity of the ECM, nonlinear Kalman filters have been

chosen to implement the SOC estimation. The computation

complexity of those nonlinear Kalman filters is usually much

larger than that of the linear Kalman filter. Moreover, the

parameters of the battery models cannot always be effectively

updated online. Instead of using nonlinear Kalman filters and

online parameter identification, this section shows how to

estimate SOC by linear Kalman filter and dynamic linear PLS

model with a simplified structure.

A. Kalman filter

The calculation of the standard Kalman filter is introduced at

first in this section as shown in Fig. 5.

Fig. 5 Standard Kalman filter

To avoid the misleading of the symbols, the matrices in the

state space equation of the Kalman filter are defined as

, G

, D

. In addition, the symbols related to EKF with two RC

ECM are defined with the superscript ECM as shown in Section

III.B, while the symbols in Section III.C are with the

superscript Proposed Battery Model (PBM).

TABLE

HE CALCULATION PROCESS OF KALMAN FILTER

Prediction

State Prediction

1|kk k k k k

FX Gu





Covariance Prediction

kk k k k k

FXF Q



 

Update

Kalman Gain Matrix





+1| +1|

kkkk kkkk k

KP EEP ER





State Estimation





1 +1| 1|kkkkkkkkkk

XX KyEX Du





Covariance Estimation



11|kkkkk

PIKEP



  

The gain of Kalman filter is updated to correct the estimated

SOC on the foundation of the new information from the

measurement and the output of the battery model. The detailed

steps of the Kalman filter are shown in TABLE. I.

The calculation of the Kalman filter contains five steps. The

first two steps are the state prediction and the covariance matrix

prediction. The calculation of the Kalman gain matrix in the

third step contains the calculation of the inverse matrix



+1|

kkk k k

EP E R





. The order of the state space function has

an effect on the calculation efficiency. In the typical two RC

ECM, the state space equation contains third-order matrices,

and the calculation of the inverse matrix is time consuming.

B. Extended Kalman filter with two RC ECM and

Recursive Least Squares (RLS)

A typical model based estimation with Kalman filter is

detailed in this subsection to help understanding the

improvement of the proposed method. EKF with two RC ECM

and RLS is chosen as an example in Fig. 6.

The discretized form of the Coulomb counting equation is,

  

bat

cap

SOC k SOC k I k





  

(5)

where



is the Coulomb efficiency, C

cap

is the battery capacity,

is the sample time. The charge current is negative and the

discharge current is positive.

According to Eq. (3) and Eq. (5), the discrete form of the

state space equation of the two RC ECM is,

ECM ECM

kk kkkk

ECM ECM

kk kkkk

FXGuQ

YEXDuR

















(6)

where

is a

33

matrix,

is a

31

matrix, the

13

matrix

is the Jacobian matrix. Q

and R

are the noise

variance of the model and the measurement respectively. The

details of the matrices

and

are shown

in Fig. 6. In Fig. 6, the two RC ECM is the battery model, RLS

is used as an additional parameter identification method, the

gain for correcting the SOC estimation is calculated by EKF.

Fig. 6 Extended Kalman filter with two RC ECM and RLS

A Simplified Model-Based State-of-Charge Estimation Approach for Lithium-Ion Battery With Dynamic Linear Model

Figures

Citations

Noise-Immune Model Identification and State-of-Charge Estimation for Lithium-Ion Battery Using Bilinear Parameterization

LSTM-Based Battery Remaining Useful Life Prediction With Multi-Channel Charging Profiles

Machine Learning-Based Lithium-Ion Battery Capacity Estimation Exploiting Multi-Channel Charging Profiles

Multiobjective Optimization of Data-Driven Model for Lithium-Ion Battery SOH Estimation With Short-Term Feature

Online joint-prediction of multi-forward-step battery SOC using LSTM neural networks and multiple linear regression for real-world electric vehicles

References

Partial least-squares regression: a tutorial

SIMPLS: an alternative approach to partial least squares regression

Extended Kalman filtering for battery management systems of LiPB-based HEV battery packs: Part 3. State and parameter estimation

Review of energy storage systems for electric vehicle applications: Issues and challenges

Potential of lithium-ion batteries in renewable energy

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Critical Review on the Battery State of Charge Estimation Methods for Electric Vehicles

An Overview and Comparison of Online Implementable SOC Estimation Methods for Lithium-Ion Battery

Long Short-Term Memory Networks for Accurate State-of-Charge Estimation of Li-ion Batteries

Estimation of State of Charge for Lithium-Ion Battery Based on Finite Difference Extended Kalman Filter

Adaptive parameter identification method and state of charge estimation of Lithium Ion battery

Frequently Asked Questions (15)

Q1. What are the three methods used to verify the performance of the PLS model?

Q2. What are the contributions mentioned in the paper "A simplified model based state-of-charge estimation approach for lithium-ion battery with dynamic linear model" ?

Q3. What is the advantage of the proposed method?

Q4. How many initial training samples are needed to update the PLS model?

Q5. What are the three driving cycles used to verify the proposed PLS model?

Q6. What is the average MAE of the five driving cycles?

Q7. What is the simplest way to extract a YPLS?

Q8. What is the effect of the proposed method on the measurement noise?

Q9. What is the accuracy of the PLS model with moving window?

Q10. What is the corresponding symbol for the PBM?

Q11. How to estimate SOC by linear Kalman filter?

Q12. How many samples are in the cycle?

Q13. What is the difference between the proposed method and the AUKF with LSSVM?

Q14. What is the dominant eigenvector of YPLS?

Q15. What is the convention for the voltage and current profiles of the LiFePO4 battery?