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A low-complexity state of charge estimation method for series-connected lithium-ion battery pack used in electric vehicles Zhongkai Zhou, Bin Duan, Yongzhe Kang, Naxin Cui, Yunlong Shang, Chenghui Zhang * School of Control Science and Engineering, Shandong University, Jinan, 250061, China

H I G H L I G H T S

� Proposed a low-complexity SoC estimation method for series-connected battery pack. � Simplified the pack capacity and SoC calculations based on the probability theory. � A selection method for the representative cells is proposed. � Verified the feasibility of simplified method by two aging experiments. A R T I C L E I N F O

A B S T R A C T

Keywords: Lithium-ion battery pack Representative cell Probability theory State of charge

Practical and accurate state of charge estimation for battery pack is a challenging task due to the inconsistency among in-pack cells. In this paper, we propose a low-complexity state of charge estimation method for seriesconnected battery pack. Firstly, to reduce computation cost, the capacity and state of charge calculations of battery pack are effectively simplified based on the probability theory. Secondly, we propose a selection method for the representative cells, which is validated by the simulation under Dynamic Stress Test. Subsequently, the state of charge of each representative cell is estimated by the recursive least squares-adaptive extend Kalman filter algorithm successively. Finally, the aging experiments for LiNCM and LiFePO4 battery packs are carried out to validate the feasibility of the simplified method, and the complexity of the three estimation methods is compared. Experimental results indicate that the capacity of battery pack depends only on the representative cells at different cycles, and the proposed “representative cell” method can estimate the state of charge of battery pack with high accuracy and low complexity. The mean absolute errors and root mean square errors for state of charge estimation are less than 3% under Urban Dynamometer Driving Schedule test at different cycles.

1. Introduction Lithium-ion batteries have been widely used in electric vehicles (EVs) owing to their high power density, high energy density, long cycle life and low self-discharge rate [1]. To meet the vehicle requirements for power and energy, hundreds and thousands of cells are connected in parallel and in series to make up a big battery pack [2–4]. Considering the safety management and efficient application for a large number of cells, battery management system (BMS) is designed to monitor each cell and estimate the state parameters of battery pack. State of charge (SoC), as the basic state parameters, is as important as the remaining fuel of diesel and petrol vehicles. However, because of the intrinsic and extrinsic differences among the in-pack cells [5–7], SoC estimation for

battery pack is different from battery cell and needs to consider their inconsistency, which makes accurate and low-complexity SoC estima tion for battery pack always challenging. According to the definition of the battery pack SoC, the SoC of each in-pack cell need be estimated and then used to calculate the SoC of battery pack [2,3,8–11], which is called as “each cell” method in this paper. For instance, there is a complicated battery system composed of 96 series-connected battery modules in Tesla’s Model S. If the commonly filter based estimator is employed, this method need duplicate 96 filters and will produce a huge amount of computation cost, which is an extremely heavy task for the micro-controller in BMS. To solve this problem, Dai et al. [9] proposed a SoC estimation method for series-connected battery pack based on the “averaged cell’’ model, and

* Corresponding author. E-mail addresses: [email protected] (Z. Zhou), [email protected] (B. Duan), [email protected] (Y. Kang), [email protected] (N. Cui), [email protected] edu.cn (Y. Shang), [email protected] (C. Zhang). https://doi.org/10.1016/j.jpowsour.2019.226972 Received 17 May 2019; Received in revised form 22 July 2019; Accepted 2 August 2019 0378-7753/© 2019 Elsevier B.V. All rights reserved.

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the dual time-scale implementation was designed to estimate the SoCs of all cells. Sun et al. [10] developed a dual-scale cell SoC estimator for series-connected battery pack. The cells with “average capacity” and “average resistance” were selected to estimate the SoCs in the micro time scale, and the SoCs of the other unselected cells were estimated in macro time scale. Wei et al. [11] introduced a dual-scale extended Kalman filter (EKF) method to estimate the SoC of each cell in a battery pack. The highest and lowest SoCs were estimated in micro time scale, and the SoCs of the other cells were estimated in macro time scale. The above cases effectively reduce computational complexity by using the dual time scales. Additionally, there are two other types of methods used to reduce the complexity of the SoC estimation for series-connected battery pack. The one type is called as “big cell” method [12,13]. All the cells in the battery pack are simplified as one “big cell”. Therefore, the SoC estimation methods for single cell can be used to estimate the SoC for battery pack, such as fuzzy logic [14] and artificial neural networks [15] based esti mation, Kalman filter [16–18] and particle filter [19] based estimators and other observers [20,21]. This method is simple but ignores the differences among in-pack cells. Furthermore, environmental tempera ture and battery aging apparently increase the cell-to-cell variation along with the battery operating time. These factors cause some changes in the characteristics of the battery pack, such as internal resistance and capacity, which result in inaccurate SoC estimation results. Besides, compared with single cell, the voltage sensor used in battery pack has a wider range, which makes the measurement error larger. Therefore, it is difficult for this method to obtain accurate SoC of battery pack in the long term. The other is called as “representative cell” method [4,22]. The SoCs of representative cells are estimated and then used to calculate the SoC of battery pack. Compared with the “each cell” method, this method greatly reduces the computation cost. Furthermore, if the selected cells can always represent the whole behavior of battery pack, this method should be the best choice for application in EVs. Zhong et al. [4] selected the first over-discharged cell and the first over-charged cell as the representative cells, and the SoC of battery pack was accurately calcu lated based on their SoC estimates. In fact, the upper and lower limits of terminal voltage is usually to protect a battery from being overcharged and over discharged [23]. With the aging of battery pack and uneven heat distribution, the internal resistance of each in-pack cell increases and their consistency becomes worse, which makes the terminal voltage difficult to determine the SoC level. Therefore, it is necessary to find a new variable to select representative cells. This paper proposes a low-complexity SoC estimation method for series-connected battery pack and has three original contributions as follows. Firstly, taking practical applications into account, the capacity and SoC calculation of battery pack is simplified based on the proba bility theory analysis. Secondly, the quasi open circuit voltage (OCV) in the Rint model is used to select the representative cells instead of the terminal voltage. Finally, the cycle aging experiments for the LiNCM and the LiFePO4 battery packs are carried out to validate the feasibility of the proposed method. The rest of this paper is arranged as follows. Section 2 describes the simplified calculation for capacity and SoC of battery pack based on probability theory. Section 3 introduces the selection strategy of repre sentative cells and its feasibility is verified by the simulation under Dynamic Stress Test (DST) profile. Subsequently, the recursive least squares (RLS) algorithm and the adaptive extend Kalman filter (AEKF) algorithm are employed to extract the model parameters and estimate the SoC of the representative cells successively. Section 4 provides the experimental results and discussion for the proposed approach, and compares the complexity of the three SoC estimation methods. Finally, Section 5 presents some conclusions of this work.

2. Simplification of the SoC calculation method for battery pack For the battery pack with n cells connected in series, the SoC and capacity of each one are denoted by SoCk and Ck, respectively. And the remaining charging electric quantity and remaining discharging electric quantity of each one are denoted by RCQk and RDQk, respectively, where k ¼ 1, 2, …, n. The releasable capacity of a battery pack is the minimum remaining electric quantity, and the absorbable capacity is the minimum electric quantity that can be charged. Therefore, the capacity of battery pack (Cpack) can be considered as the sum of the minimum RCQ and the minimum RDQ [24,25], which can be expressed as Cpack ¼ min ðRCQk Þ þ min ðRDQk Þ ¼ min ðCk ð1 1�k�n

1�k�n

1�k�n

SoCk ÞÞ þ min ðCk SoCk Þ 1�k�n

(1) Similar to single cell, the SoC of battery pack (SoCpack) is usually defined as the ratio of the remaining discharging electric quantity to the total capacity [2–4], which can be expressed as the following equation. The numerator denotes the minimum RDQ of all in-pack cells, namely RDQ of battery pack, and the denominator denotes the Cpack. min ðRDQk Þ

SoCpack ¼

1�k�n

min ðRCQk ÞÞ þ min ðRDQk Þ

1�k�n

1�k�n

min ðCk SoCk Þ

¼

1�k�n

min ðCk ð1

1�k�n

SoCk ÞÞ þ min ðCk SoCk Þ

(2)

1�k�n

According to Eq. (2), only after the SoC and capacity of each in-pack cell are estimated, can the SoC of battery pack be acquired. Obviously, this method requires a lot of calculations and is difficult to apply in BMS. Therefore, to reduce the computational cost, it is a feasible method to select several in-pack cells that can represent the whole behavior of battery pack. Fig. 1 (a) and (b) present the natural SoC and capacity distributions of 95 series-connected cells without balancing, which come from a typical electric passenger car after three-year operation with the mileage of 32,500 km [25]. The nominal capacity of each cell is 60 Ah. The Capacity-Quantity diagram (C-Q diagram) is firstly proposed to calculate the Cpack in Ref. [26], and this diagram is used to analyze the relationship between the Cpack and in-pack cells in this paper. According to the above SoC and capacity distributions, the C-Q di agram is illustrated in Fig. 1 (c), where the blue dots denote the in-pack cells; the green dots denote the representative cells (i.e., the cells with SoCmin, SoCmax and Cmin); SoCmin and SoCmax denote the minimum and maximum SoCs, respectively; Cmin denotes the minimum capacity. The SoC is a constant on the black dotted line and equal to 1 on the diagonal. As the battery pack has been fully charged, the line of equal RCQ co incides with the black dotted line. The red dot is the intersection of the line of equal RCQ and the line of equal RDQ, whose horizontal coordi nate value is equal to Cpack. In this example, Cpack depends only on the cells with the minimum and maximum SoCs, and the cell with Cmin is very close to the line of equal RDQ. The accurate state parameters of each in-pack cell (such as capacity, RCQ and RDQ) are hardly obtained until the battery pack reach the end of life in EV, which makes the inconsistency evolution of in-pack cells difficult to be analyzed. Therefore, to verify theoretically that the Cpack depends only on the representative cells, the probability theory is used to simulate the evolution of these state parameters in the aging process. Specifically, Weibull distribution is used to approximate cell capacity distribution, and normal distribution is used to approximate cell SoC distribution [26,27]. In addition, the consistency of in-pack cells always tends to be worse [28]. The battery pack has reached the end of life in the above example, so the capacity and SoC consistencies for in-pack cells have been the worst. The means of capacity and SoC are 51.22 Ah and 91.76% respectively and the standard deviations (SDs) are 1.60 Ah and 8.1% respectively in the above example. By means of the above statistical characteristics, with the aging of the battery pack, we 2

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Fig. 1. SoC distribution, capacity distribution and Capacity-Quantity diagram for 95 series-connected battery cells of a typical electric passenger car after three-year operation: (a) SoC distribution; (b) capacity distribution; (c) Capacity-Quantity diagram.

assume that the means of SoC and capacity (MSoC and MCap) decrease, and the standard deviations of SoC and capacity (SDSoC and SDCap) in crease. But MSoC and MCap are larger than the ‘worst’ means, and SDSoC

and SDCap are smaller than the ‘worst’ SDs mentioned in the above example. As listed in Table 1, we set the means and the SDs of capacity and SoC at different aging stages. Then the Weibull distribution of cell capacity and the normal distribution of cell SoC are repeated a thousand times at every aging stage in Matlab. According to Bernoulli’s law of large numbers shown in Eq. (3), when the same experiment is repeated N times where N tends to be infinite, the frequency of random event L approximates its probability p. Parameter ε represents an arbitrary given positive real number. The event L is that Cpack depends on the repre sentative cells in this paper. Therefore, the simulation results reflected by the thousand sets of capacity and SoC data are considered to be convincing based on this law.

Table 1 Set means and SDs of capacity and SoC at different aging stages. Aging stages

Stage 1

Stage 2

Stage 3

MSoC MCap SDSoC SDCap

96.70% 56.49 Ah 1.6% 0.32 Ah

95.05% 54.73 Ah 3.3% 0.64 Ah

93.41% 52.98 Ah 5.7% 1.12 Ah

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� � � p�� < ε ¼ 1

�� �L lim P �� N→∞ N

of battery pack in the charge and discharge process, respectively [29]. However, the internal resistance difference increases along with battery aging [28], which result in the fact that the terminal voltage cannot determine the SoC level of in-pack cells. Considering simplicity and practicality, the quasi OCV based on the Rint model is used to select the representative cells with SoCmax and SoCmin in this paper. The Rint model was first proposed in Ref. [8], which included the OCV and an ohmic resistance in series shown in Fig. 4 (a). The quasi OCV (Uquasi ocv ) is calcu lated in Eq. (6). Strictly speaking, this model cannot describe the dy namic behavior of cell accurately due to the neglect of polarization characteristics, but the quasi OCV still accurately reflects the SoC level of cell.

(3)

Fig. 2 shows the probabilities of Cpack determined by representative cells (i.e., the cells with SoCmin, SoCmax and Cmin) at different aging stages. The purple area indicates the probability that the Cpack depends only on the representative cells (Err ¼ 0). The green area indicates the probability that the capacity estimation error is within 1% (0 < Err�1%) and the yellow area indicates the probability that the capacity estima tion error is between 1% and 2% (1% < Err�2%) if the representative cells are used to calculate the Cpack. The probabilities of Cpack determined by representative cells at three aging stages are 96.0%, 87.9% and 82.6% within 1% estimation error respectively, while the probabilities are 100%, 97.6% and 92.8% within 2% estimation error respectively. Therefore, within an acceptable error range, the Cpack depends almost exclusively on the representative cells when the inconsistency is weak, and the Cpack depends on the representative cells with high probability when the inconsistency become strong. Especially, as the equalization management is gradually applied to the BMS, the differences among inpack cells would be reduced and the Cpack may always depend on the representative cells with high probability in the long run. Based on the above analysis, it has been proved to be feasible in theory that Cpack depends on the cells with SoCmin, SoCmax and Cmin with high probability. Therefore, Eqs. (1) and (2) can be rewritten as follows. In contrast, Cpack and SoCpack depend only on the three representative cells, and the computational complexity is greatly reduced. �� � Cpack � min Ci ð1 SoCmax Þ; Cmin 1 SoCj þ min Ck SoCmin ; Cmin SoCj 1�i;j�n

U quasi ocv ¼ Ut þ i*R0

A small battery pack that consists of eight LiNCM cells in series is utilized to validate the above method in Matlab Simulink. The 1st-order RC equivalent circuit model is used to simulate battery output, which is shown in Fig. 4 (b). In addition, eight cells have different initial SoCs and ohmic resistances but the other parameters are the same. Their initial SoCs of Cells 1 to 8 are set to 0.88, 0.87, …, 0.81 respectively. Accordingly, their ohmic resistances are set to 4.8 mΩ, 5.0 mΩ, 5.4 mΩ, 5.6 mΩ, 4.2 mΩ, 5.2 mΩ, 4.6 mΩ, 4.4 mΩ. The capacity of each cell is set to 30 Ah. DST profile is operated on the battery model to validate the above selection strategy. Fig. 3 shows the simulation results for eight cells in series under DST. As shown in Fig. 3 (a), the SoCs of Cells 1 to 8 decrease in turn. However, from the enlarged part in Fig. 3 (b), the terminal voltage distribution of the eight cells is inconsistent with their SoC distribution. The terminal voltage is mainly affected by ohmic resistance. Therefore, the terminal voltage is not suitable for judging the SoC level. Unlike the terminal voltage, the quasi OCV distribution is exactly the same as the SoC dis tribution in Fig. 3 (c), which can reflect the SoC level of cell well. Moreover, this variable can be obtained by simple calculations and thus can be used in BMS. In addition, many methods for battery capacity estimation have been summarized in Ref. [30]. The capacity of each in-pack cell can be esti mated by these methods, and then the traversal algorithm can be used to find the cell with the minimum capacity. The selection method for the cell with minimum capacity is not described in this paper. The capacity of each cell can be obtained by experimental test.

1�k;j�n

(4) SoCpack �

� min Ck SoCmin ; Cmin SoCj �� � SoCmax Þ; Cmin 1 SoCj þ min Ck SoCmin ; Cmin SoCj 1�k;j�n

min Ci ð1

1�i;j�n

(6)

1�k;j�n

(5) where Ci and Ck represent the capacity of the cells with SoCmax and SoCmin, respectively; SoCj represents the SoC of the cell with Cmin. 3. Selection of the representative cells and their SoC estimation 3.1. A selection strategy for the representative cells

3.2. Parameter identification for the representative cells

The OCV is usually used to determine the battery SoC in model-based estimation. However, as the accurate OCV cannot be directly measured by the voltage sensor and can only be estimated, it can hardly be utilized to determine the SoC level of each in-pack cell considering the huge computation burden. To simplify the calculation, the cells with the highest and lowest terminal voltages were selected to estimate the SoC

Fig. 4 (b) shows the 1st-order equivalent circuit battery model, the transfer function of which can be expressed as GðsÞ ¼

Uoc ðsÞ UðsÞ R1 ¼ R0 þ 1 þ R1 C 1 s IðsÞ

Fig. 2. Probabilities of Cpack determined by representative cells at different aging stages. 4

(7)

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Journal of Power Sources 441 (2019) 226972

Fig. 3. Simulation results for eight cells in series under DST: (a) SoC; (b) terminal voltage; (c) quasi OCV.

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Table 2 Summary of the AEKF algorithm based SoC estimation approach. � � � 1 0 Definitions: Ak ¼ , Bk ¼ 0 expð T=ðR1 *C1 ÞÞ R1 *ð1 � � dUoc 1 and Dk ¼ R0 (24) jSoC¼SoCk dSoC Initialization: For k ¼ 1, set þ ,b xþ 1 ¼ E½x1 �; Px;1 ¼ E½ðx1

Pv ¼ Hk

Uoc ðzÞ UðzÞ a2 a3 z ¼ IðzÞ 1 a1 z 1

where 8 > > > a1 ¼ > > > > > < a2 ¼ > > > > > > > > : a3 ¼

1

(28)

Lk Ck ÞPk (29)

(13)

1Þ

Thus, Eq. (12) can be rewritten as UðkÞ ¼ ð1

a1 ÞUoc ðkÞ þ a1 Uðk

1Þ þ a2 IðkÞ þ a3 Iðk

1Þ

ϕðkÞ ¼ ½1 Uðk 1Þ IðkÞ Iðk 1Þ � θðkÞ ¼ ½ ð1 a1 ÞUoc ðkÞ a1 a2 a3 �T

(16)

In this paper, the RLS algorithm with a forgetting factor is employed to identify the model parameters. The recursive equations can be described as

(9)

(17)

yðkÞ ¼ ϕðkÞb θðkÞ þ eðkÞ T 2R1 C1 T þ 2R1 C1

eðkÞ ¼ UðkÞ

R0 T þ R1 T þ 2R0 R1 C1 T þ 2R1 C1

(10) KðkÞ ¼

R0 T þ R1 T 2R0 R1 C1 T þ 2R1 C1

R0, R1 and C1 can be solved according to Eq. (10), and the results are shown in Eq. (11). 8 a3 a2 > > R0 ¼ > > 1 þ a1 > > > > < 2ðR0 þ a2 Þ R1 ¼ (11) a1 1 > > > > > > Tð1 þ a1 Þ > > : C1 ¼ 4ðR0 þ a2 Þ

PðkÞ ¼

a1 Uoc ðk

1Þ þ a1 Uðk

1Þ þ a2 IðkÞ þ a3 Iðk

1Þ

(18)

1Þ

Pðk

1Þ

KðkÞϕT ðkÞPðk λ1

(19) 1Þ

1Þ þ KðkÞeðkÞ

(20) (21)

where y(k) denotes the output of the system at the kth sampling time; b θðkÞ is the estimate of the model parameter vector θðkÞ; eðkÞ denotes the

error between the predicted voltage and the true voltage; KðkÞ is the gain and PðkÞ is the symmetrical matrix; λ1 is the forgetting factor and it is typically valued between 0.95 and 1. 3.3. SoC estimation for the representative cells based on AEKF

(12)

Since Uoc changes very little between two adjacent sampling instants, we assume

8� � � � � � � SoCk 1 0 < SoCkþ1 * ¼ þ U U1;kþ1 R1 *ð1 0 expð T=ðR1 *C1 ÞÞ 1 ;k : Uk ¼ OCVðSoCk Þ U1;k R0 *Ik þ vk

ϕðkÞb θðk

Pðk 1ÞϕT ðkÞ λ1 þ ϕT ðkÞPðk 1ÞϕðkÞ

b θðkÞ ¼ b θðk

Eq. (9) can be rewritten as Eq. (12) after the discretization. UðkÞ ¼ Uoc ðkÞ

(15)

Eq. (14) can be expressed as UðkÞ ¼ ϕðkÞθðkÞ

1

(14)

If we define

where z is the discretization operator. GðzÞ ¼

Ck Pk ðCk ÞT ; Lk ¼ Pk ðCk ÞT ½Ck Pk ðCk ÞT þ Pv �

Uoc ðkÞ � Uoc ðk

(8)

1

T

b x k þ Lk εk ; Pw ¼ Lk Hk ðLk ÞT ; Pþ xþ k ¼ b k ¼ ðI

where Uoc denotes the open-circuit voltage related to SoC; R0 is the ohmic resistance; R1 and C1 denote the electrochemical polarization resistance and capacitance, respectively; U1 denotes the voltage across C1 and R1; I denotes the working current which is positive in the dis charging process and negative in the charging process; Ut denotes the terminal voltage of the battery model; s is the frequency operator. The discretization calculation of Eq. (7) can be obtained by a bilinear transform shown in Eq. (8) with the sample time T and the result is shown in Eq. (9). 1

b xþ 1 Þ � (25)

b xþ 1 Þðx1

Computation: For k ¼ 2, 3, …, compute: Time update: Time update for state estimator: b x kþ1 ¼ Ak b xþ ¼ Ak Pþ ATk þ Pbx ;w , (26) k þ Bk Ik ; Pb x;kþ1 bx ;k The adaptive law: 1 Xk εk ¼ yk hðxk ; uk Þ; Hk ¼ εk εT (27) M i¼k M¼1 k Measurement and noise covariance update for state estimator:

Fig. 4. Schematic diagram of equivalent circuit battery models: (a) Rint model; (b) 1st-order model.

21 z s¼ T 1þz

� T=Cmax Ck ¼ expð T=ðR1 *C1 ÞÞÞ

Lithium-ion battery is a typical nonlinear system [31,32], and its system equation and measurement equation can be expressed as follows based on the above 1st-order equivalent circuit model.

� T=Cmax *Ik þ wk expð T=ðR1 *C1 ÞÞÞ

6

(22)

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The above state-space equations can be described as �

xkþ1 ¼ f ðxk ; uk Þ þ wk yk ¼ hðxk ; uk Þ þ vk

(23)

where xk, yk and uk are the system state vector, output vector and input vector at the kth sampling time respectively; wk and vk denote the system noise and measurement noise at the kth sampling time respectively. Considering the application in BMS, the battery SoC is estimated by the AEKF algorithm in this paper, and the summary of the AEKF algorithm based SoC estimation approach is listed in Table 2. 4. Experimental results and analysis 4.1. Test bench To validate the proposed method, a test bench is built, which is shown in Fig. 5 (a). The test bench consists of Arbin battery tester (BTMP 100V-200A) to charge and discharge battery pack, a computer to design test schedule and store experimental data (including voltage, current and temperature), a data collector to collect experimental data, a thermal chamber to maintain constant test temperature with 25 � C and two series-connected battery packs separately with eight LiNCM cells and eight LiFePO4 cells. The main parameters of LiNCM and LiFePO4 cells are shown in Table 3. The characteristics of Arbin battery tester are specified in Table 4. 4.2. Experimental procedure The flowchart of battery experiment is illustrated in Fig. 5 (b), which consists of three main parts: cell-level test for cell characterization, packlevel test to validate the proposed method and cycle aging test. The celllevel test consists of capacity test and OCV test, which are conducted to obtain the available capacity and the relationship between OCV and SoC of each in-pack cell, respectively. The pack-level capacity test is con ducted to get the capacity of battery pack, and Urban Dynamometer Driving Schedule (UDDS) test is used to simulate the real operating scenarios of battery pack. These tests are conducted every 50 cycles until the capacity of battery pack is less than 80% of the rated capacity. The contents of the above tests are listed as follows. Capacity test: consistent with capacity test for battery pack, the single cell is fully charged in the multi-step constant current (CC) mode, whose current profile is shown in Fig. 5 (c). The fully discharged cell is charged to upper-limit voltage using the multi-step constant currents of 1/3C, 1/6C and 1/20C in succession. Then the cell is discharged to the lower-limit voltage using a constant current of 1/3C. The test is repeated three times and their average capacity is taken as the available capacity. OCV test: the cell is firstly charged in the multi-step CC mode. The cell is then gradually discharged in 5% of rated capacity steps using a 1/ 3C discharge current until the lower-limit voltage is reached. After every discharge step, the cell is rested for 1 h to reach the discharge open circuit voltage (OCVdis). Similarly, the charging open circuit voltage (OCVcha) is obtained following a similar procedure. The cell is gradually charged in 5% of rated capacity steps until the upper-limit voltage is reached. The cell rests for 1 h again after each step and the OCVcha is recorded. The average voltage of the sum of OCVdis and OCVcha is calculated as the OCV of cell. UDDS test: eight cells are connected in series to assemble a battery pack, and it is fully charged in the multi-step CC mode. Then UDDS test is operated on the battery pack to simulate the realistic scenario. The current and voltage profiles are shown in Fig. 5 (d) and (e). Cycle aging test: after the cell-level and pack-level tests are finished, the battery pack is charged and discharged at 1C rate for 50 cycles. It is worth noting that before the battery pack is reassembled every time, the

Fig. 5. (a) Battery test bench; (b) flowchart of battery experiment; (c) charging current profile; (d) current profile of one UDDS cycle; (e) voltage profiles under UDDS test.

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battery packs. For the LiNCM battery pack, the Cpack indicated by the blue triangle is always lower than that of each cell in Fig. 6 (a). More over, the consistency of cell capacity become worse and worse with the increase of cycle time, which makes the Cpack gradually away from the cell capacity. The increasing inconsistency of cell capacity accelerates the reduction of the Cpack. Unlike the LiNCM battery pack, the capacity change of the LiFePO4 battery pack is more complex in Fig. 6 (b). As the capacity of Cell 3 declines obviously faster than that of others, the Cpack depends totally on this cell after the 100 cycles from the zoom figure. However, after the 600 cycles, the Cpack is again less than the capacity of Cell 3. Moreover, since Cell 3 is fully charged or discharged for a long time, its capacity reduces obviously between 750th cycle and 800th cycle, which causes the Cpack to follow this change too. In order to analyze the relationship between the Cpack and the representative cells, the C-Q diagrams of LiNCM and LiFePO4 battery packs at different cycles are illustrated in Fig. 7. For the LiNCM battery pack, the Cpack is always determined by the cells with SoCmax and SoCmin in Fig. 7 (a). The other cells located inside between the two lines of equal RCQ and RDQ have no influence on the Cpack. In contrast, for the LiFePO4 battery pack, the Cpack depends totally on the cell with Cmin, namely Cell 3, at the 350th and 550th cycles in Fig. 7 (b), where the red dot coincides with the green dot. The Cell 3 is always fully charged and discharged between the two cycles due to its low capacity, which results in the fact that its capacity reduces more rapidly than the others. However, at the 750th and 900th cycles, the Cpack is determined by the cells with SoCmax and SoCmin, where the capacity and SoC of Cell 3 are the minimum.

Table 3 Main parameters of LiNCM and LiFePO4 cells.

LiNCM LiFePO4

Nominal capacity

Nominal voltage

End-of-charge voltage

End-of-discharge voltage

32.50 Ah 31.00 Ah

3.60 V 3.20 V

4.20 V 3.65 V

3.00 V 2.00 V

Table 4 Characteristics of the battery tester. Characteristic

Measurement accuracy

Measurement range

Voltage for single cell Voltage for battery pack Current Temperature

�0.02% FSR �0.1% FSR �0.1% FSR �1 � C

0 V–5 V 10 V–100 V 100 A-100 A 200 � C-þ200 � C

initial electric quantity of each cell should be recharged so as not to affect the consistency of the in-pack cells. 4.3. Verification and results 4.3.1. Verification of the simplified Cpack calculation Two small battery packs that consist of eight LiNCM cells and eight LiFePO4 cells in series respectively are employed to validate the lowcomplexity SoC and capacity calculation method. Fig. 6 shows the cell and pack capacities at different cycles for the LiNCM and LiFePO4

Fig. 6. Cell and pack capacities at different cycles: (a) LiNCM battery pack; (b) LiFePO4 battery pack. 8

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Fig. 7. C-Q diagrams of two battery packs at different cycles: (a) LiNCM battery pack; (b) LiFePO4 battery pack.

When the battery pack is discharged, the Cell 3 is firstly fully discharged. However, when the battery pack is charged, the Cell 3 is not fully charged due to its apparently low capacity, which makes the Cpack not totally dependent on the Cell 3.

Tables 5 and 6 list the error results of two battery packs for the three methods, namely “big cell” method, “each cell” method and the pro posed “representative cell” method, respectively. Overall, the SoC estimation errors in “big cell” method are signifi cantly higher than those in the other two methods at the same cycle for the two battery packs. For the LiNCM battery pack, the maximum mean absolute error (MAE) and root mean square error (RMSE) in “big cell” method are 0.025 and 0.033, respectively. In addition, the SoC estima tion errors in “representative cell” method are slightly higher than these in “each cell” method at the same cycle. The maximum MAE and RMSE in “representative cell” method are 0.016 and 0.023, respectively. For the LiFePO4 battery pack, the maximum MAE and RMSE in “big cell” method are 0.027 and 0.031, respectively. Additionally, the SoC esti mation errors in “representative cell” method are similar to these in

4.3.2. SoC estimation for battery pack In order to verify the accuracy of the proposed estimation method, the SoC of battery pack is estimated under UDDS test at different cycles. The model parameters of the representative cells are identified by the RLS algorithm online, and their SoCs are estimated by the AEKF algo rithm. Then the SoC of battery pack is calculated according to Eq. (5). The current integration method is used to calculate the true SoC of battery pack. Fig. 8 shows the estimated SoC and error results of the LiNCM and LiFePO4 battery packs under UDDS test at different cycles. 9

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Journal of Power Sources 441 (2019) 226972

Fig. 8. Estimated SoC and error results of two battery packs under UDDS test at different cycles: (a) LiNCM battery pack; (b) LiFePO4 battery pack. Table 5 SoC estimation errors of LiNCM battery pack at different cycles for the three methods.

Table 6 SoC estimation errors of LiFePO4 battery pack at different cycles for the three methods.

Methods

Errors

250th cycle

450th cycle

600th cycle

700th cycle

Methods

Errors

350th cycle

550th cycle

750th cycle

900th cycle

Big cell

MAE RMSE MAE RMSE MAE RMSE

0.018 0.022 0.008 0.010 0.009 0.011

0.023 0.026 0.010 0.014 0.011 0.016

0.021 0.029 0.012 0.016 0.014 0.019

0.025 0.033 0.015 0.022 0.016 0.023

Big cell

MAE RMSE MAE RMSE MAE RMSE

0.024 0.030 0.008 0.010 0.009 0.012

0.026 0.031 0.010 0.013 0.010 0.012

0.025 0.029 0.011 0.014 0.011 0.013

0.027 0.031 0.015 0.019 0.015 0.018

Each cell Representative cell

Each cell Representative cell

“each cell” method at the same cycle. The maximum MAE and RMSE in “representative cell” method are 0.015 and 0.018, respectively. In fact, the SoC estimation accuracy is greatly affected by the

acquisition accuracy of voltage sensor for battery pack in “big cell” method, which leads to large estimation errors and unstable estimation results with battery pack aging. In contrast, the cell SoC is accurately 10

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Journal of Power Sources 441 (2019) 226972

estimated based on the voltage sensor for single cell in the other two methods, and their MAEs and RMSEs increase with the battery pack aging. Additionally, it is worth noting that the SoC of LiNCM battery pack is estimated based on the cells with SoCmax and SoCmin in “repre sentative cell” method, and that of LiFePO4 battery pack is estimated just depending on the cell with Cmin at the 350th and 550th cycles, and the cells with SoCmax and SoCmin at the 750th and 900th cycles. Their MAEs and RMSEs at different aging cycles are less than 3% for the two battery packs, which shows that the proposed “representative cell” method can ensure an accurate estimation for pack SoC. The experimental results indicate that the Cpack depends on the representative cells, namely the cells with SoCmin, SoCmax or Cmin, regardless of LiNCM battery pack and LiFePO4 battery pack. For the LiNCM battery pack, the Cpack is determined only by the cells with SoCmin and SoCmax, and is not affected by the other cells from the C-Q diagrams at different cycles. For the LiFePO4 battery pack, the Cpack is determined only by the cell with Cmin between 100th cycle and 600th cycle owing to its faster capacity degradation. And it is determined only by the cells with SoCmin and SoCmax after the 600th cycle due to the apparently low capacity of the cell with Cmin. On the other hand, the SoC estimation results show that the proposed “representative cell” method can esti mate the pack SoC accurately at different cycles under UDDS test.

Table 7 Comparison of the complexity of the three methods at 250 cycle for LiNCM battery pack. Methods

Occupied memory size (KB)

Computational time (ms)

Each cell Big cell Representative cell

16.384 4.126 8.162

2065 797 369

results indicate that the pack capacity is determined by the repre sentative cells at different cycles for the LiNCM and LiFePO4 battery packs. 2. The quasi OCV can be used to select the representative cells. The feasibility of this method is verified by the simulation where eight series-connected cells with different initial SoCs and ohmic re sistances are operated on DST profile. 3. The three SoC estimation methods for battery pack, namely “big cell” method, “each cell” method and the proposed “representative cell” method, are compared under UDDS test at different cycles. The estimation results show that the proposed “representative cell” method can accurately estimate the pack SoC, and the MAEs and RMSEs are less than 3%. Moreover, considering the occupied mem ory size and computational time, the “representative cell” method is the best choice to estimate the pack SoC, which is suitable for the real application in BMS.

4.4. Complexity analysis The occupied memory size and computational time are used to analyze the complexity of the three methods, namely “each cell” method, “big cell” method and the proposed “representative cell” method. The computational time of three methods is evaluated by the Matlab functions tic and toc in a computer with i7-4790 CPU and 4.00 GB RAM. Table 7 lists the comparison of the complexity of the three methods at 250 cycle for LiNCM battery pack. The “each cell” method takes the most computational time and occupies the most memory size, 2065 m s and 16.384 KB respectively. In contrast, the proposed “repre sentative cell” method takes the least computational time and occupies about half of the memory size in “each cell” method, 369 m s and 8.162 KB respectively. Compared to the “big cell” method, although the “representative cell” method occupies more memory size, it takes less computational time. The reason for this result is that the voltage value of battery pack is much larger than that of single cell, which makes the algorithm execution, especially the exponential operation, take more time. Therefore, considering the occupied memory size and computational time, the “representative cell” method is the best choice to estimate the SoC of series-connected battery pack. Furthermore, as the number of series-connected cells increases, the “each cell” method requires more computational time and occupies more memory size, which is difficult to be used on a low-cost microcontroller in BMS. In addition, the “big cell” method occupies the small memory size but needs more computational time with the increase of the voltage value of battery pack. On the contrary, the complexity of the “representative cell” method does not increase with the number of cells, which only depends on the selected cells. Therefore, the proposed “representative cell” method is promising for vehicle applications.

Acknowledgment This work is supported by the National Natural Science Foundation of China (No. 61633015, 61527809, U1764258, U1864205), National key R&D program of China and Shandong Province (No. 2018YFB0104000, 2016ZDJS03A02). References [1] I. Baghdadia, O. Briata, P. Gyanb, J.M. Vinassa, State of health assessment for lithium batteries based on voltage-time relaxation measure, Electrochim. Acta 194 (2016) 461–472. [2] X. Zhang, Y. Wang, C. Liu, Z. Chen, A novel approach of remaining discharge energy prediction for large format lithium-ion battery pack, J. Power Sources 343 (2017) 216–225. [3] X. Zhang, Y. Wang, D. Yang, Z. Chen, An on-line estimation of battery pack parameters and state-of-charge using dual filters based on pack model, Energy 115 (2016) 219–229. [4] L. Zhong, C. Zhang, Y. He, Z. Chen, A method for the estimation of the battery pack state of charge based on in-pack cells uniformity analysis, Appl. Energy 113 (2014) 558–564. [5] B. Kenney, K. Darcovich, D.D. MacNeil, I.J. Davidson, Modelling the impact of variations in electrode manufacturing on lithium-ion battery modules, J. Power Sources 213 (2012) 391–401. [6] M. Dubarry, C. Truchot, M. Cugnet, B.Y. Liaw, K. Gering, S. Sazhin, D. Jamison, C. Michelbacher, Evaluation of commercial lithium-ion cells based on composite positive electrode for plug-in hybrid electric vehicle applications: Part I: initial characterizations, J. Power Sources 196 (2011) 10328–10335. [7] Y. Zheng, M. Ouyang, L. Lu, J. Li, X. Han, L. Xu, H. Ma, T.A. Dollmeyer, V. Freyermuth, Cell state-of-charge inconsistency estimation for LiNCM battery pack in hybrid electric vehicles using mean-difference model, Appl. Energy 111 (2013) 571–580. [8] G.L. Plett, Efficient battery pack state estimation using bar-delta filtering, in: CDROM Proc. 24th Electric Vehicle Symposium (EVS24), Stavanger, Norway, May 2009. [9] H. Dai, X. Wei, Z. Sun, J. Wang, W. Gu, Online cell SoC estimation of Li-ion battery packs using a dual time-scale Kalman filtering for EV applications, Appl. Energy 95 (2012) 227–237. [10] F. Sun, R. Xiong, A novel dual-scale cell state-of-charge estimation approach for series-connected battery pack used in electric vehicles, J. Power Sources 274 (2015) 582–594. [11] J. Wei, G. Dong, Z. Chen, Y. Kang, System state estimation and optimal energy control framework for multicell lithium-ion battery system, Appl. Energy 187 (2017) 37–49. [12] G.L. Plett, Extended Kalman filtering for battery management systems of LiPB based HEV battery packs: Part 2. Modeling and identification, J. Power Sources 134 (2004) 262–276. [13] R. Xiong, F. Sun, X. Gong, H. He, Adaptive state of charge estimator for lithium-ion cells series battery pack in electric vehicles, J. Power Sources 242 (2013) 699–713.

5. Conclusions This paper has proposed a low-complexity SoC estimation method for series-connected battery pack. The SoC of battery pack can be accurately estimated based only on the representative cells, namely the cells with SoCmax, SoCmin and Cmin. The primary conclusions of the study are as follows: 1. It is verified by the probability theory that the pack capacity depends on the representative cells with high probability, independent of the other cells. Based on the analysis of C-Q diagram, the experimental 11

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