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A systematic state-of-charge estimation framework for multi-cell battery pack in electric vehicles using bias correction technique q Fengchun Sun, Rui Xiong ⇑, Hongwen He National Engineering Laboratory for Electric Vehicles, School of Mechanical Engineering, Beijing Institute of Technology, No. 5 South Zhongguancun Street, Haidian District, Beijing 100081, China Collaborative Innovation Center of Electric Vehicles in Beijing, Beijing Institute of Technology, No. 5 South Zhongguancun Street, Haidian District, Beijing 100081, China

h i g h l i g h t s Bias correction for single cells based on an average pack model is proposed. An integrated method for battery modeling and parameter identiﬁcation is proposed. A RBF neural network based uncertainty quantiﬁcation algorithm is proposed. A systematic SoC estimation framework for multi-cell battery pack is proposed.

a r t i c l e

i n f o

Article history: Received 4 November 2014 Received in revised form 30 November 2014 Accepted 9 December 2014 Available online 31 December 2014 Keywords: Electric vehicles Lithium-ion polymer battery Uncertainty Bias correction Response surface approximate model State-of-charge

a b s t r a c t In order to maximize the capacity/energy utilization and guarantee safe and reliable operation of battery packs used in electric vehicles, an accurate cell state-of-charge (SoC) estimator is an essential part. This paper tries to add three contributions to the existing literature. (1) An integrated battery system identiﬁcation method for model order determination and parameter identiﬁcation is proposed. In addition to being able to identify the model parameters, it can also locate an optimal balance between model complexity and prediction precision. (2) A radial basis function (RBF) neural network based uncertainty quantiﬁcation algorithm has been proposed for constructing response surface approximate model (RSAM) of model bias function. Based on the RSAM, the average pack model can be applied to every single cell in battery pack and realize accurate terminal voltage prediction. (3) A systematic SoC estimation framework for multi-cell series-connected battery pack of electric vehicles using bias correction technique has been proposed. Finally, three cases with twelve lithium-ion polymer battery (LiPB) cells series-connected battery pack are used to verify and evaluate the proposed framework. The result indicates that with the proposed systematic estimation framework the maximum absolute SoC estimation error of all cells in the battery pack are less than 2%. Ó 2014 Elsevier Ltd. All rights reserved.

1. Introduction Under the world wide demand for reduction in greenhouse gas emissions and PM2.5 production, advanced battery systems powered electric vehicles (EVs) have earned widespread respect and recognition. Lithium-ion batteries (LiBs), as one of most popular

q This paper is included in the Special Issue of Energy innovations for a sustainable world edited by Prof. S.k Chou, Prof. Umberto Desideri, Prof. Duu-Jong Lee and Prof. Yan. ⇑ Corresponding author at: National Engineering Laboratory for Electric Vehicles, School of Mechanical Engineering, Beijing Institute of Technology, No. 5 South Zhongguancun Street, Haidian District, Beijing 100081, China. Tel./fax: +86 (10) 6894 0589. E-mail addresses: [email protected] (F. Sun), [email protected] (R. Xiong).

http://dx.doi.org/10.1016/j.apenergy.2014.12.021 0306-2619/Ó 2014 Elsevier Ltd. All rights reserved.

candidates due to their ability to provide fast response to power/ energy demand, have been widely used in various kinds of EVs [1]. Due to limitations of electrode potential and materials, capacity and energy stored in every single cell hardly can satisfy the requirements of energy storage system in EVs. Hence, larger numbers of battery cells are used in series and parallels. Unfortunately, the more number of battery cells would be in worse of reliability and consistent characteristic of battery packs and battery systems [2]. Thus, accurate battery cell state-of-charge (SoC) estimations are essential in making more rational allocation of battery energy. 1.1. Review of the SoC estimation approach A variety of methods has previously been reported to realize accurate estimation of battery cells [3–20], each one having its

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relative merits, as reviewed by Refs. [3,15]. On the whole, apart from the archived data based black-box methods [4–6], reduced single particle electrochemical battery model based methods [7,8] and several nonlinear observers [20], most of the remaining methods are developed for overcoming the shortcoming of the ampere–hour counting method. For instance, all sorts of ﬁlters and observers, such as extended Kalman ﬁlter [9], sigma-point Kalman ﬁlter [10], unscented Kalman ﬁlter [11], particle ﬁlter [12], robust extended Kalman ﬁlter [13], adaptive extend Kalman ﬁlter (AEKF) [14–16] and other nonlinear observers [3,17–19], are employed to calibrate the erroneous initial SoC and accumulated estimation error. The above approaches have achieved desired estimation accuracy for battery cells in their own ﬁelds. However, due to the inhabited inconsistent characteristics of cell parameters, SoC of battery pack is very different from battery cell. It can hardly be estimated using the same method as battery cell unless all cells having consistent characteristic. Or else, it will lead to security issues of over-charge or over-discharge caused by the erroneous cell SoC estimations in battery pack. As a result, estimating the SoC of battery pack remains problematic and challenging. Recently, great efforts have been made to estimate battery pack SoC through the estimation of battery cells [2,21–24]. Dr. Xiong proposed a ﬁltering process based method for calculating the SoC of battery pack. It used the selected cell to calculate the SoC of each cell in battery pack, where the cells of battery pack have similar capacities and resistances [2]. Dr. Chen proposed methods to estimate the SoC of battery pack base don several representative cells, such as ﬁrst over-discharged cell and the ﬁrst over-charged cell [21]. Dr. Liaw calculated the average SoC of battery pack [22]. Dr. Dai proposed cell SoC estimation approach based on the divergence between the average pack model and cell model [23]. Dr. Xiong proposed a cell SoC estimation approach based on an online bias correction model, the parameter of the bias correction model were identiﬁed in real-time [24]. The above approaches have made valuable contributions in the implementation of battery pack SoC estimation. However, the two ends of terminal voltage cannot decide that the SoC has reached its lowest or highest level. Additionally, the two ends of cell terminal voltage are widely used for avoiding over-charge or over-discharge instead of calibrating cell SoC. It is because that the cell resistances greatly affect its terminal voltage. What is more, the duplication of cell SoC estimation for battery pack would yield huge computation cost for battery management system (BMS). On the other hand, corrections based on the average pack model or selected cell model for every single cell in battery pack are complex, error-prone and time-consuming. Last but not least, the uncertainties from model and parameter have been ignored. Thus, the performance and accuracy of the proposed approach are limited for EVs application.

1.3. Organization of the paper Section 2 ﬁrstly describes model and parameter uncertainties, and then proposes an integrated battery system identiﬁcation method for model order determination and parameter identiﬁcation. Section 3 presents the SoC estimation approach with model bias correction technique. The case study and evaluation of the proposed method are reported in Section 4 before conclusions are drawn in Section 5. 2. Model and parameter uncertainties In order to implement battery SoC estimation, we ﬁrstly should have an accurate battery model. This section focuses on the qualiﬁcation of model and parameter uncertainties for maximizing the agreement between model prediction and the experimental data. 2.1. Lumped parameters battery model with N–RC networks Battery model can be roughly divided into three groups: electrochemical models, empirical models and equivalent circuit models. Equivalent circuit models are widely used to battery management system and vehicular energy management system for its simple model structure [25,26]. For different anode–cathode material system, battery shows different properties. Thus, for LiPB battery cell with graphite anodes and nickel–manganese–cobalt oxide (NMC) cathodes, we should determine its model structure ﬁrstly. Fig. 1 presents an equivalent circuit model with n RC networks, named the NRC model hereafter. The model contains three parts: (i) The ﬁrst part is the voltage source. It uses UocOCV (open circuit voltage) to denote battery voltage source. (ii) The second part is the equivalent ohmic resistance Ri. It represents the electrical resistance from various battery components or with the accumulation and dissipation of charge in the electrical double layer. (iii) The last part is the mass transport effects and dynamic voltage performances and the elements of component RD and CD are accordingly described as the diffusion resistance and diffusion capacitance. CDi is the ith equivalent diffusion capacitance and RDi is the ith equivalent diffusion resistance, UDi is the voltage across CDi, i = 1, 2, 3, 4, . . . n. Where iL is the load current (assumed positive for discharge, negative for charge), Ut is the terminal voltage. The electrical behavior of the NRC model can be expressed by Eq. (1).

8 1 1 _ > > < U Di ¼ RDi CDi U Di þ C Di iL n X > U Di > : U t ¼ U oc iL Ri

ð1Þ

i¼1

1.2. Contribution of the paper Based on our previous research experience on system identiﬁcation and state estimation, a key contribution of this study is that a systematic SoC estimation framework for multi-cell series connected battery pack used in electric vehicles considering model and parameter uncertainties has been proposed. This paper proposes a novel systematic framework to quantify battery model and parameter uncertainties for battery pack SoC estimation. An integrated battery system identiﬁcation method for model order determination and parameter identiﬁcation is proposed for quantifying battery and model uncertainties. To reduce the computational cost, the average battery pack model has been built and a radial basis function (RBF) neural network based uncertainty quantiﬁcation algorithm has been proposed for approximating model and parameter uncertainties to improve the prediction precision of the average pack model.

RDi

iL

OCV + −

RDn

CDi

CDn

+ U Di −

+ U Dn −

Ri

×

Ut

− • Fig. 1. Schematic diagram of the NRC equivalent circuit model.

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(i) In case of n = 0, the NRC model is simpliﬁed as a Rint model and model equation presented in Eq. (1) can be rewritten as Eq. (2), where k denotes the discretization step with a sample interval of Dt, k = 1, 2, 3, . . .

U t;k ¼ U oc;k iL;k Ri

ð2Þ

(i) In case of n P 1, a discretization form of the NRC model can be simpliﬁed by Eq. (3) [2]. 8 si ¼ RDi C Di > > > > < U Di;k ¼ iDi;k RDi ð1expðDt=si ÞÞ Dt=si ÞÞ > exp ðDt=si Þ þ iL;k1 ð1expð > iDi;k ¼ iL;k 1 Dt=si Dt=si > > : þiDi;k1 expðDt=si Þ

ð3Þ

y ¼ /h þ w

ð4Þ

where y denotes the vector of output values, w denotes the white noise information mainly from the measurement, h and / denote the parameter vector and the data matrix, respectively. Then the discretization equation of the NRC equivalent circuit model can be rewritten as the form of Eq. (4).

/ ¼ ½ 1 iL

iD1

h ¼ ½ U oc

RD1

Ri

Om ðvÞ ¼

M X

wm qm ðvÞ

ð8Þ

j¼1

For a dynamic system:

(

between cell j and average value of battery pack. It is noted that the superscript j is used to denote the cell number in battery pack. The determination of model and parameter uncertainties is a recognized problem [27,28]. In this study Radial Basis Function (RBF) neural network is used to develop a response surface approximate method for the determination of the bias function. RBF neural network has been independently proposed by many researchers and it is good at modeling nonlinear data. RBF is one of the most widely used neural network architecture in literature for classiﬁcation or regression problems. It is a robust classiﬁer with the ability to generalize for imprecise input data [29,30]. The ﬁnal output of the network O(v) can be expressed as Eq. (8).

. . . . . . iDi . . . . . . RDi T

ð5Þ

The recursive least square (RLS) algorithm has been employed to identify the parameters of NRC battery model. The detailed computation processes of RLS algorithm are presented in Eq. (6).

8 1 T T > l > > K Ls;k ¼ PLs;k1 /k h/k PLs;k1 /k þ < i ^hk ¼ ^hk1 þ KLs;k y / ^hk1 k k > > > : P ¼ 1 ½I K / P Ls;k Ls;k k Ls;k1 l

ð6Þ

qm ðvÞ ¼ exp

kv 1m k 2r2m

ð9Þ

where fm and rm denote the center and spread width of the m-th node, respectively. The weights are optimized using least mean square (LMS) algorithm once the centers of RBF units are determined [29]. Thus, for each deterministic cell j, the model and parameter uncertainties djm C jrate ; zj can be approximated through the RBF neural network. Lastly, the uncertainties for all cells in battery pack can be approximated considering the variable of DQ. 2.3. An integrated battery system identiﬁcation method

where KLs,k and PLs,k denote the gain and error covariance matrixes of the RLS algorithm at the k-th sampling interval. I denotes the unit matrix, y denotes the system output and it is the measured terminal voltage in this study, ^ h denotes the identiﬁed parameter, l denotes the forgetting factor l 6 1 and it is used to diminish the weight of older data relative to data that was obtained more recently. Typically l is set to a value between 0.95 and 1 depending on the desired rate of adaptation. In off-line applications of this study, l is generally set to 0.995. 2.2. Model and parameter uncertainties In our study, the reference model is built on the basis of the average terminal voltage of battery pack with the system identiﬁcation method presented in Section 2.3. Due to the inevitable inconsistent properties in battery pack, although the average model can track voltage behavior of battery pack accurately, it cannot accurately tack the dynamic behavior of each cell in battery pack. The deterministic difference between the predicted terminal voltage and the measured terminal voltage of battery cell is deﬁned as model and parameter uncertainties, which indicates that the model is inadequacy for describing the actual physical system under various battery operating conditions and cell inevitable inconsistencies properties. Thus, a stochastic difference between the predicted terminal voltage from the reference model and measured cell terminal voltage can be deﬁned by a model bias d, which can be deﬁned in Eq. (7).

U jt ¼ U oc U D1 . . . U Dn iL Ri þ dðC jrate ; zj ; DQ j Þ

where M denotes the number of bias functions, v denotes the input data vector, wm represents a weighted connection between the basis function and output layer and qm() denotes the radial basis function of the m-th hidden node, which is typically a Gaussian of the form:

ð7Þ

where uncertainty d is the function of cell discharge/charge rateC jrate ; cell SoC–zj, maximum available capacity difference DQj

Based on the discretization equation of the NRC model in Eq. (1), a detailed ﬂowchart of integrated system identiﬁcation method for model order determination and parameter identiﬁcation has been proposed and presented in Fig. 2, which can be divided into ﬁve steps: (i) Extract archived operation data. It ﬁrstly extracts the measurements of battery current, voltage and SoC from the archived operation data stored in battery management system. With the 5% of the SoC interval of adjacent parameter group and the whole SoC range, we can calculate the number N of data group for parameter identiﬁcation. Where m denotes the order of the data group, m = 1, 2, , N. (ii) Initialization. It initializes @ and k. @ represents the SoC range of each group, its initial value is set to [0.5%, 0.5%]. k represents the allowable times for enlarging the data range during parameter identiﬁcation, its initial value is 1. For instance, if parameter identiﬁcation result under @ = [0.5%, 0.5%] cannot meet the demand of ﬁtting factor, the SoC range @ will be enlarged to [0.5% k 0.1%, 0.5% + k 0.1%], and k will be increased by 1 later. (iii) Parameter identiﬁcation. Provided the RC order n (n = 0, 1, 2, 3, 4, 5), we can identify the model parameters under different RC network numbers on the basis of Eq. (6). Then the coefﬁcient of determinationR2 is used to statistic measures how successful the ﬁt is in explaining the variation between the model and the data. Herein the minimal value of R2 is set to 0.98, namely, if R2 6 0.98, we will enlarge the SoC range. However, the maximum allowable times k is 5, namely the widest SoC range is [1%, 1%]. If all the R2 in the 5 times are less than 0.98, the parameter group with the biggest R2

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F. Sun et al. / Applied Energy 162 (2016) 1399–1409

Start

Serach SoC data, locate m and N

Y N

m>N

Initialization Prediction error

widen search scope

λ =λ +1

Locate identified data

N

δ <30 mV?

unfeasible model

m=m+1

Y Identify parameters Determine model order N

N

λ >5 Y

R 2>0.98 Quantify model parameter

Y

Use the best group

Save parameters End

Step i

Step ii

Step iii

Step iv

Step v

Fig. 2. A combined model order determination and parameter identiﬁcation approach.

will be selected as model parameter. Afterwards, it judge whether m > N or not, namely when all the SoC points are identiﬁed, it will transfer to next step. (iv) Model prediction error. With model parameters under different RC numbers, we can calculate the model prediction error. It is worth noting that battery model with more than 30 mV error would be removed as an unfeasible model. The remaining battery models will be ready for order evaluation. (v) Quantify model parameter uncertainty. The best order of NRC battery models are selected from the remaining orders with the balance between model prediction accuracy and model complexity. At this time, the most suitable battery model for describing the average voltage behavior of battery pack will be located. It is deﬁned as reference model. With the reference model and model-bias correction function, the terminal voltage behavior of every single cell in battery pack can be accurately tracked.

SoC estimator. To begin with the AEKF algorithm, we should have a state space for describing battery model. From Eqs. (1) and (3), we can obtain the discretization form of battery state-space equation for the average pack voltage:

82 3 2 U D1;k expðDt=s1 Þ 0 > > >6 > 7 6 .. > . > . 7 6 .. > . 0 >6 6 . 7 ¼ 6. > > 7 6 6 > > 5 4 4 0 expðDt=si Þ U > Di;k > > > > > 0 0 z k > > 3 2 > < ð1 expðDt=s1 ÞÞRD1 7 6 .. > 7 6 > > . 7iL;k 6 > þ > 7 6 > > 4 > ð1 expðDt=si ÞÞRDi 5 > > > > > gDt=Q n > > > > n X > > > U t;k ¼ U oc iL;k Ri > U Di :

3 U D1;k1 76 7 .. 7 6 07 . 76 7 76 7 0 54 U Di;k1 5 0

1

32

zk1

i¼1

ð10Þ

3. Cell SoC estimation for battery pack This section works for the SoC estimation approach based on our previous research experience on battery cell SoC estimation under the speciﬁc battery model.

where g denotes the coulomb efﬁciency, Qn denotes the maximum available capacity. Let us deﬁne a discrete-time state-space form for ﬁltering use. Speciﬁcally, we assume the form:

3.1. Cell SoC estimation approach From Refs. [2,3], adaptive extended Kalman ﬁlter (AEKF) algorithm has been successfully employed to develop an adaptive

xk ¼ f ðxk1 ; uk1 Þ þ xk1 yk ¼ hðxk ; uk Þ þ tk

ð11Þ

where xk is the system state vector at discrete-time index k, wk denotes the unmeasured ‘‘process noise’’ that affects the system

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F. Sun et al. / Applied Energy 162 (2016) 1399–1409 Table 1 Summary of the AEKF algorithm based SoC estimation approach. Based on Eqs. (20) and (21), let us deﬁne ; Ck ¼ @hðx;uÞ Ak ¼ @f ðx;uÞ @x @x x¼xk

For

(12)

x¼xk

(I) Initialization ^þ k ¼ 0; set : x 0 ¼ E½x0 ;

h i ^þ ^þ T Pþ 0 ¼ E ðx0 x0 Þðx0 x0 Þ ; Q 0 ; R0 :

(13)

(II) Computation: time update-prior estimation (from (k 1)+ to (k)). For k = 1,2,. . .. . ., compute ^þ State prior estimate : ^ x k ¼ f ðxk1 ; uk Þ T Error covariance prior estimate : P k ¼ Ak Pk1 Ak þ Q k (III) Measurement update-posteriori estimation (from (k) to (k)+) Error innovation : ek ¼ yk hðxk ; uk Þ

(15)

1

T T Kalman gain matrix update : Kk ¼ P k Ck ðCk Pk Ck þ Rk1 Þ Pk T T 1 Adaptive law-covariance matching : Hk ¼ M i¼kMþ1 ei eTi ; Rk ¼ Hk Ck P k Ck ; Q k ¼ Kk Hk Kk ^þ ^ State estimate measurement update : x ¼ x þ K e k k k k Error covariance measurement update : Pþ k ¼ ðI Kk Ck ÞPk (IV) Time update: return to step II

2

0

0

ð21Þ

1

The output matrix C and the state vector x can be calculated by:

(

x ¼ ½ U D1;k1

U Di;k1

C ¼ ½ 1 1

zk1

dOCV dz

(16) (17) (18) (19) (20)

state, vk denotes the measurement noise. Then, the AEFK algorithm can be employed to develop an adaptive model-based SoC estimation approach. The detailed computation process is listed in Table 1. Where Q and R denote the covariance matrices of independent, zero-mean, Gaussian noise of w and v. P denotes the covariance matrix of state estimation error, M denotes the covariance matching widow, H denotes the innovation covariance matrix based on the innovation sequence inside a moving estimation window of size M. K denotes the Kalman gain matrix, e denotes the innovation ^ ^þ þ formation, x k ; Pk and xk ; Pk are for the priori estimate before the measurement is taken into account and the posteriori estimate after the measurement is taken into account respectively. From Eq. (10), the state matrix A can be calculated by:

3 expðDt=s1 Þ 0 0 7 6 .. 6 ... . 0 07 7 A¼6 7 6 4 0 expðDt=si Þ 0 5

(14)

ð22Þ

3.2. Model-based SoC estimation for multi-cell in battery pack In terms of SoC estimation for multi-cell series connected battery pack, we do not want to repeat the SoC estimation with the number of the cells in battery pack. It is because that the cells connected in battery system used in EVs are always more than 100, thus the computational burden would be very huge. In considering that the cell model is established based on the reference model, the initialization and prior estimation parts of the AEKF algorithm are same for SoC estimation of all cells. Namely, we do one time computation for the initialization and prior estimation. The detailed calculation process of the SoC estimation approach for battery pack is listed in Table 2. It is worth noting that all cells use the same state and observer matrixes during the SoC estimation, but the bias functions are different if they experience different capacity degradation status. Namely, in the proposed method, large numbers of matrix calculations are avoided for reducing the computational time.

3.3. A framework of battery SoC estimation using bias correction technique Based on the description of modeling and state estimation presented in above sections, the ﬂowchart of the proposed systematic framework for battery SoC estimation considering model and parameter uncertainties is illustrated in Fig. 3. The presented implementation ﬂowchart can be divided into two categories: off-line data train, on-line estimation. Each category contains four steps, the detailed steps are following. (i) Extract archived operation data. It can be seen at Section 2.3. (ii) Quantify model uncertainty. When the characteristic described by a battery model is deviated from the actual physical system, the model prediction error can hardly be reduced to the idea level even if provided that the most suitable parameter groups. Thus, the structure of the battery model is very important for dynamic voltage behavior modeling. In this study, an integrated system identiﬁcation method is used to obtain the most suitable model structure for battery pack. (iii) Quantify parameter uncertainty. The RBF neural network is employed to develop a response surface approximate method for the determination of the parameter uncertainty. (iv) Correct battery cell model. With the reference model and model bias function, the terminal voltage behavior of every single cell in battery pack can be accurately tracked. (v) AEKF algorithm based SoC estimator. An adaptive extended Kalman ﬁlter algorithm has been employed to develop an adaptive model-based SoC estimation approach for multicell in battery pack. Its detailed process is shown in Section 3.1. (vi) Loading proﬁles. The dynamic driving cycles are used to verify the proposed method. (vii) Predict cells terminal voltage. The terminal voltage of each cell in battery pack can be predicted by Eq. (7), it is the sum of the terminal voltage predicted by reference model and model-bias correction function. (viii) Estimate SoC for all cells. Through the detailed calculation process is listed in Table 2, the SoC of all cells in series-connected battery pack can be estimated orderly in real-time. 4. Case study This section presents three cases to demonstrate the effectiveness of the proposed framework for cell voltage prediction and SoC estimation.

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F. Sun et al. / Applied Energy 162 (2016) 1399–1409

Table 2 Calculation process of the SoC estimation approach for battery pack. Calculate the reference model (I) Initialization (II) Computation: time update-prior estimation(from (k 1)+ to (k)). For k = 1, 2, . . .. . ., compute State prior estimate-Eq. (14) Error covariance prior estimate-Eq. (15) j; For each cell j; ^ xj; ¼^ x k ; Pk ¼ Pk k Calculate the reference model and bias function d for each cell (III) Measurement update-posteriori estimation (from (k) to (k)+)

(23)

Error innovation : ejk ¼ yjk hðxk ; uk Þ dðC jrate ; zj ; DQ j Þ Kalman gain matrix update-Eq. (17) Update adaptive law-covariance matching-Eq. (18) Update state estimate measurement-Eq. (19) Update error covariance measurement-Eq. (20) (IV) Time update: return to step II

(24)

Extract archived

Quantify model

Quantify parameter

Correct battery cell

operation data

uncertainty

uncertainty

model

Off-line data train

cells

voltage

Loading profiles (a)100 0

-100 0

AEKF algorithm

(b) 48 voltage(V)

Predict cells terminal

current(A)

Estimate SoC for all

5

10 15 time(h)

20

25

46

based SoC estimator

44 42 40 38 0

5

10 15 time(h)

20

25

On-line SoC estimation Fig. 3. Flowchart of the proposed framework for battery SoC estimation.

4.1. Experiments Ref. [2] has built a test bench and designed a comprehensive test scheme for battery packs. The test bench setup consists of Arbin BT2000 cycler, a thermal chamber to regulate the operation temperature, a computer to the program and store experimental data. The measurement inaccuracy of the current and voltage sensors inside the Arbin BT2000 system is less than 0.05%. Twelve LiPB cells series-connected battery pack has been used to verify and evaluate the proposed approach. The basic parameters of the tested LiPB cells are listed in Table 3. The capacities of cells are plotted in Fig. 4. Two types of current proﬁles have been loaded on battery pack. It contains hybrid pulse test and dynamic stress test (DST) cycles. The hybrid pulse testis used for quantifying the uncertainties of model and parameter, one of proﬁles for battery pack current and voltage under the hybrid pulse test are plotted in Fig. 5. The DST cycles are used for verifying and evaluating the proposed approach and their voltage and current proﬁles are plotted in Fig. 6.

RC network. More importantly, battery model with double RC networks has the best performance. It well meets the balance between model complexity and prediction precision. In terms of battery OCV parameter, Fig. 8 presents the identiﬁed battery OCV at different SoC points. Fig. 8(a) plots the identiﬁed OCV considering different numbers of RC network. Fig. 8(b) plots the OCV differences between double RC network and other four numbers of RC networks. It shows that the differences of the identiﬁed OCVs at same SoC point are very small, especially for selected battery model the differences are less than 1 mV except n = 0.Thus, a double RC networks based lumped parameters battery model has been selected in this study ﬁnally. Its electrical equation is presented in Eq. (25).

8 > U_ ¼ C D11RD1 U D1 þ C1D1 iL > < D1 U_ D2 ¼ C D21RD2 U D2 þ C1D2 iL > > : U t ¼ U oc U D1 U D2 iL Ri

ð25Þ

The analytic function of battery OCV is shown in Eq. (26). 4.2. Case01 for parameter identiﬁcation Through the integrated battery system identiﬁcation method, we can obtain the prediction errors of battery models under different RC network orders. Statistical analysis of the prediction errors are illustrated in Fig. 7. It shows that all the mean absolute errors (MAE) of the predicted terminal voltage are less than 4 mV and mean squared errors (MSE) are less than 20 mV2. It indicates that the proposed parameter identiﬁcation approach can capture the dynamic voltage behavior of battery pack accurately. On the other hand, Fig. 7 also shows that the battery model without RC network would lead to worst performance comparing with battery models considering

2 ! 2 ! z 1:56 z 0:42 þ 2:2 exp gðzÞ ¼ 4:7exp 0:95 0:53 2 ! 2 ! z 0:02 z 0:06 þ 1:4exp þ 0:4exp 0:097 0:298

ð26Þ

Table 3 The basic parameters of LiPB cell. Material

Nominal capacity

Nominal voltage

Lower cut-off voltage

Upper cut-off voltage

Li(NiMnCo)O2

32 A h

3.7 V

3.0 V

4.05 V

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F. Sun et al. / Applied Energy 162 (2016) 1399–1409

32

Mean=30.52Ah, STD. =0.85Ah

Capacity (Ah)

31.5 31 30.5 30 29.5 29

0

1

2

3

4

5

6

7

8

9

10

11

12

Cell number Fig. 4. Measured maximum available capacities of the twelve LiPB cells.

(b)

Current (A)

(a) 100

Voltage (V)

50 0 -50

48 46 44 42

-100 650

655

660

665

670

675

650

655

660

Time(min)

665

670

675

Time (min)

Current (A)

(a)

80

(b)

49

Voltage (V)

Fig. 5. Proﬁles of battery pack current and voltage under the hybrid pulse test: (a) current; (b) terminal voltage.

48

60 40 20 0

47 46

-20 -40

0

300

600

900

45

1200

0

200

400

Time (sec) #01

#02

#03

600

800

1000

1200

Time (sec) #04

#05

#06

(c)

#07

(d)

#08

#09

#10

#11

#12

1 0.98

3.9

SoC

Voltage (V)

4

3.8

0.96

0.94

3.7 0

300

600

900

1200

Time (sec)

0

300

600

900

1200

Time (sec)

Fig. 6. Proﬁles of battery pack current and voltage under the ﬁrst 1200 s of the DST cycles: (a) current; (b) terminal voltage of battery pack; (c) cell terminal voltage; (d) cell SoC.

4.3. Case02 for uncertainty quantify To investigate the predicted performance of the reference model for all cells in battery pack, Fig. 9 plots the statistical analysis result of the absolute terminal voltage errors.

It shows that maximum absolute prediction error is close to 300 mV, which is ten times of the allowable prediction error set in the proposed parameter identiﬁcation method. Moreover, the mean absolute errors of cell #3 and cell #11 are more than 50 mV. It indicates that the reference model has a persistent

F. Sun et al. / Applied Energy 162 (2016) 1399–1409

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Fig. 7. Statistical analysis of prediction errors for models with different RC network order: (a) MAE; (b) MSE.

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Fig. 8. Identiﬁed OCV at different SoC points: (a) OCV; (b) discrepancy between two RC network and others.

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predicted error for these two cells. Namely, the reference model can hardly work well for all cells in battery pack. On the other hand, although cell #6 and cell #11 having biggest capacity differences learned from Fig. 4, the terminal voltage prediction error of cell #6 is much smaller than cell #3 which has much smaller capacity difference. It indicates that there is a stochastic difference existed between the predicted terminal voltage and measurements. The stochastic difference can be deﬁned by a model bias. To improve the prediction precision of the reference model for all cells in battery pack, the model and parameter uncertainties inhabited in battery model should be quantiﬁed ﬁrstly. RBF neural network is used to quantify model and parameter uncertainties through determination of the model bias function. Four cells having different capacity differences are selected to build the response surface approximate model (RSAM). Fig. 10 presents the RSAM of cell #6, cell #08, cell #09 and cell #11. They are functions of cell SoC and discharge/charge rate. All R2 of the goodness of ﬁtting are more than 0.998. It indicates that the RBF neural network can accurately

approximate and determine model and parameter uncertainties. With the four bias functions, we can build the bias function for all cells. Statistical analysis of the absolute prediction errors using the model bias correction technique are illustrated in Fig. 11. As expected, voltage prediction errors of all cells in battery pack are reduced obviously. All the maximum absolute prediction errors are less than 30 mV. Especially for cell #11, its prediction error has been reduced to the 10% of the origin value. What is more, all the mean absolute prediction errors are less than 5 mV. It indicates that the persistent model error has been corrected effectively. Thus, the proposed model bias correction technique can improve the model prediction performance greatly. 4.4. Case03 for SoC estimation In considering that model with double RC networks has been selected, thus the system state equation presented in Eq. (10) can be rewritten as the following equation.

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Fig. 10. Model bias of four typical cells.

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3 2 32 3 82 U D1;k expðDt=s1 Þ 0 0 U D1;k1 > > > 6 7 6 76 7 > > expðDt=s2 Þ 0 54 U D2;k1 5 4 U D2;k 5 ¼ 4 0 > > > > > 0 0 1 zk1 > < 2 zk 3 ð1 expðDt=s1 ÞÞRD1 > > 6 7 > > þ4 ð1 expðDt=s2 ÞÞRD2 5iL;k > > > > > gDt=Q n > > : U t;k ¼ U ock U D1;k U D2;k iLk Ri

where dg(z)/dz can be calculated by:

ð27Þ

Then the state vector, state matrix and output matrix can be obtained from Eq. (27).

8 T xk ¼ ½ U D1;k U D2;k zk > > > 2 3 > > expðDt=s1 Þ 0 0 > > < 6 7 expðDt=s2 Þ 0 5 A ¼ 40 > > > 0 0 1 > > h i > > : l ¼ 1 1 dgðzÞ dz

ð28Þ

2 ! dgðzÞ z 1:56 z 1:56 z 0:42 4:4 ¼ 9:4 exp dz 0:95 0:95 0:53 2 ! 2 ! z 0:42 z 0:02 z 0:02 exp exp 0:8 0:53 0:097 0:097 ! 2 z 0:06 z 0:06 2:8 exp ð29Þ 0:298 0:298 Fig. 12 plots the statistical analysis for cell absolute SoC estimation errors with the uncorrected battery model. It is worth noting that all the initial SoCs of this study are set to 50%, namely they contain 50% initial errors, which are used for evaluating the robust performance of the adaptive SoC observer. It shows that most of maximum SoC estimation errors are bigger than 2%, some of them even more than the 5%. What is worse, most of mean SoC estimation errors are bigger than 1%, some of

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SoC estimation with the uncorrected battery model

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Cell number Fig. 12. Error analysis of estimated cell SoC without model bias correction.

Maximum SoC error (%)

them even more than the 3%. It indicates that cell SoC estimation with average battery pack model can hardly guarantee the estimation accuracy. On the contrary, the maximum estimation error would be more than 8% if a worse inconsistency characteristic happened in battery pack. To evaluate the contribution of the proposed model-bias correction approach for cell SoC estimation, a comparative analysis for cell SoC estimation between the uncorrected battery model

6

(a)

and corrected battery model has been done accordingly. Fig. 13 plots the comparative analysis of the cell absolute SoC estimation errors. It shows that all the maximum absolute SoC estimation errors are less than 2% when using the model bias correction technique. What is more, all the mean absolute SoC estimation errors and standard variations are less than 0.5% when using the model bias correction technique.

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F. Sun et al. / Applied Energy 162 (2016) 1399–1409

Based on the above three veriﬁcation cases, the performance of the proposed systematic SoC estimation framework for multi-cell battery pack using bias correction technique has been effectively demonstrated. It has the potential that accurately estimating the terminal voltage and SoC of all cells in battery pack with lower computational burden. 5. Conclusions This paper has proposed a systematic SoC estimation framework for multi-cell series-connected battery pack in electric vehicles using bias correction technique. (1) In order to reduce the computation burden, the average pack model has been proposed. For locating an optimal balance between model complexity and prediction precision, an integrated battery system identiﬁcation method for model order determination and parameter identiﬁcation is proposed. Additionally, the detailed computational steps are summarized. (2) In order to use the reference model to predict terminal voltage of all singe cells in battery pack, a model bias correction technique has been employed based on the consideration of model and parameter uncertainties exercised in battery cells. Then, a RBF neural network has employed to develop a response surface approximate method for the determination of the bias function. With the bias correction technique and adaptive extended Kaman ﬁlter algorithm, an operation ﬂowchart of the proposed systematic framework is developed. Additionally, its detailed computational steps are summarized. (3) Three cases with twelve LiPB cells series-connected battery pack are used to demonstrate the proposed framework. The results shows that with the proposed systematic estimation framework the maximum cell terminal voltage prediction error is less than 30 mV and the maximum cell SoC estimation error is less than 2%.It reveals that the bias correction technique has the potential to improve the extension ability of battery model and greatly reduce the computational cost of battery management system in electric vehicles.

Acknowledgments This work was supported by the National Natural Science Foundation of China (51276022), the National Science & Technology Pillar Program (2013BAG05B00) and Beijing Institute of Technology Research Fund Program for Young Scholars. We would like to thank Dr. Zhimin Xi, in University of Michigan-Dearborn for many helpful discussions. References [1] Tsuyoshi S, Yoshio U, Petr NovÃ K. Memory effect in a lithium-ion battery. Nat Mater 2013;12(6):569–75. [2] Xiong R, Sun F, Gong X, He H. Adaptive state of charge estimator for lithiumion cells series battery pack in electric vehicles. J Power Sources 2013;242:699–713. [3] Xiong R, Sun F, Gong X, Gao C. A data-driven based adaptive state of charge estimator of lithium-ion polymer battery used in electric vehicles. Appl Energy 2014;113:1421–33.

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