Determination of the battery pack capacity considering the estimation error using a Capacity–Quantity diagram

Applied Energy 177 (2016) 384–392

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Applied Energy journal homepage: www.elsevier.com/locate/apenergy

Determination of the battery pack capacity considering the estimation error using a Capacity–Quantity diagram Minggao Ouyang a,⇑, Shang Gao a, Languang Lu a,b, Xuning Feng a, Dongsheng Ren a, Jianqiu Li a, Yuejiu Zheng c, Ping Shen a a b c

State Key Laboratory of Automotive Safety and Energy, Tsinghua University, Beijing 100084, China Collaborative Innovation Center of Electric Vehicles, Beijing 100084, China College of Mechanical Engineering, University of Shanghai for Science and Technology, Shanghai 200093, China

h i g h l i g h t s

g r a p h i c a l a b s t r a c t

a r t i c l e

i n f o

Article history: Received 28 February 2016 Received in revised form 9 May 2016 Accepted 21 May 2016

Keywords: Battery pack capacity Capacity–Quantity diagram Estimation error State-of-charge State-of-health

Cell A Cell B Esmated Baery Pack Real Baery Pack Line of equal RCQ Line of equal RDQ Line of equal SOC

QA

CA

α%

β% 1− β % β% 1+ β %

QB

Electric Quanty

minimum RDQ

Cell A Cell B Baery Pack Line of full charged state Line of equal RCQ Line of equal RDQ Line of equal SOC

minimum RCQ

errors of pack capacity estimation is proposed.
The C–Q diagram is utilized to analyze the estimation error of pack capacity.
An equation calculating error of pack capacity estimation is proposed.
The analysis can help choose a proper accuracy of SOC estimation for BMS.
The analysis can help choose a proper accuracy of SOH estimation for BMS.

Electric Quanty


Quantitative analysis on estimation

Cpack

CB

Capacity

Capacity

The Capacity-Quanty diagram used to define the states of baery pack

Using the Capacity-Quanty diagram to determine the pack capacity considering esmaon errors

a b s t r a c t Accurate estimation of the capacity of a battery pack is essential for the battery management system (BMS) in electric vehicles. The SOCs and capacities of individual cells are the prerequisites for accurately estimating the capacity of a battery pack. This paper proposes quantitative analysis on how the estimation errors of individual cells’ SOCs and capacities influence the estimation error of the battery pack capacity using an approach named Capacity–Quantity diagram (C–Q diagram). The analysis concludes that the estimation error of cell SOC has more influence on the estimation error of pack capacity than the estimation error of cell capacity does. The theoretical analysis is further validated by an experiment using six NCM batteries connected in series with different initial SOC variations. The results help to guide the determination of specifications, e.g., the estimation error of the SOC and that of the capacity, during the design process of a BMS. Ó 2016 Elsevier Ltd. All rights reserved.

1. Introduction ⇑ Corresponding author. E-mail address: [email protected] (M. Ouyang). http://dx.doi.org/10.1016/j.apenergy.2016.05.137 0306-2619/Ó 2016 Elsevier Ltd. All rights reserved.

Lithium-ion batteries have been widely used as the power source of electric vehicles (EVs) in recent years [1,2]. Generally,

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M. Ouyang et al. / Applied Energy 177 (2016) 384–392

the battery system for EVs is composed of numerous single cells, because the voltage and capacity of a single battery are insufficient [3]. Consequently, the battery management system (BMS) in EVs requires accurate estimation of the battery pack capacity, to monitor the capacity fade and avoid abusive conditions such as overcharge and over-discharge [4–6]. Variations of individual cells within the pack always exist due to inconsistencies during manufacturing and inhomogeneity of the working conditions [7]. Two typical approaches for estimating the SOC and capacity of the battery pack are available considering cell variations [8,9]. The first approach utilizes the ‘‘Mean+Difference Model” [8,10–12]. This kind of approach introduces a ‘‘Mean Model” to represent the mean state, which takes advantage of similar states among single cells and guarantees the accuracy. To capture cell variations, a model named the ‘‘Difference Model” was appended to the ‘‘Mean Model” [8]. Plett [8] firstly introduced the concept of the ‘‘Mean+Difference Model” in 2009 and fulfilled an estimation of the cell SOCs within a battery pack using the bar-delta filter. Dai et al. [10] developed a method to estimate the cell SOC using EKF (Extended Kalman Filtering). Zheng et al. [11] developed the ‘‘Mean+Difference Model” to estimate the cell SOCs and resistances of a battery pack. Sun and Xiong [12] screened the cell parameters as the mean model parameters and estimated the cell SOCs of a battery pack. However, the ‘‘Mean+Difference Model” approach cannot estimate the pack capacity and is only used for estimating the SOCs and resistances of individual cells. The second approach converts the pack state estimation problem into a 2-cell estimation problem. To be specific, the first fully charged cell and the first completely discharged cell are the worst cells that are used to determine the state of the battery pack [9,13,14]. However, approaches on the identification of the two worst cells remains unknown. In other words, the full estimation, i.e. estimation of the states for all of the cells in the battery pack, must be performed to select the two worst cells. However, full estimation, which requires huge amount of computing resources, is hardly available to be applied in BMS [15–19]. In conclusion, the battery pack capacity estimation is based on the estimation of the cell SOCs and capacities of the battery pack. The errors, which exist in the estimation of individual cells’ SOCs and capacities, influence the accuracy of the state estimation for the battery pack. This paper provides an original investigation on the determination of the battery pack capacity considering the estimation error using a Capacity–Quantity diagram. Previous works have been proposed to estimate the battery pack capacity, and the SOCs and capacities of individual cells, separately. However, there is still no literature that has discussed the relationship between the estimation of the battery pack capacity and the estimation of the cell SOCs and capacities. Therefore, how the estimation errors in the SOCs and capacities of individual cells affect the accuracy of the battery pack capacity estimation requires further investigation. An approach named the ‘‘Capacity–Quantity diagram” (C–Q diagram) was used to determine the battery pack capacity [8,9]. However, the C–Q diagram has not been utilized to analyze the estimation error of the battery pack capacity yet. Hence, this paper provides quantitative analysis on how the estimation errors of the cell SOCs and capacities influence the estimation error of the battery pack capacity. The error analysis is conducted with the help of the C–Q diagram [8,9]. The analysis can provide guidance to choose proper accuracy of the estimation algorithms for cell SOC and capacities during BMS design, and broaden the horizon of the BMS technology. The paper is organized as follows. In Section 2, the definition of pack capacity is introduced, and the relationship between the capacity of the battery pack and the SOCs and capacities of individ-

ual cells are analyzed. The C–Q diagram is briefly introduced and used to describe the relationship between the state of individual cells and the state of battery pack. In Section 3, quantitative analysis on the capacity estimation error of the battery pack are conducted based on the C–Q diagram. In Section 4, experiments using six NCM batteries connected in series with different initial SOC variations are performed to validate the theoretical analysis.

2. The C–Q diagram 2.1. The capacity of the battery pack A battery pack connected in series reaches the end of charge (EOC) once the maximum cell voltage reaches the charge cut-off voltage. Similarly, the battery pack reaches the end of discharge (EOD) when the minimum cell voltage decreases to the discharge cut-off voltage, as shown in Fig. 1. The capacity of the battery pack is defined as the electric quantity released from EOC to EOD. Considering that the coulombic efficiency is approximately 100% for lithium ion batteries, the capacity of a battery pack can be calculated by Eq. (1).

C pack ¼ min fRCQ i g þ min fRDQ i g 16i6n

16i6n

ð1Þ

where Cpack is the capacity of the battery pack, n is the number of series-connected cells in the pack, and RCQi is the remaining charging electric quantity of the ith cell, while RDQi represents the remaining discharging electric quantity of the ith cell, as shown in Fig. 1. The capacity of a battery pack is determined by the two worst cells, the cell (Cell A) with the minimum remaining charging electric quantity (RCQ) determines the EOC, whereas the cell (Cell B) with the minimum remaining discharging electric quantity (RDQ) determines the EOD. Furthermore, Eq. (1) can be rewritten as

C pack ¼ min fSOC i  C i g þ min fð1  SOC i Þ  C i g 16i6n

16i6n

ð2Þ

where SOCi and Ci are the SOC (state of charge) and capacity of the ith cell, respectively.

2.2. The C–Q diagram A Capacity–Quantity diagram as in [20] is a graphic illustration of Eq. (2), and can be utilized to determine the capacity of the battery pack (Cpack). As shown in Fig. 2, the x axis of the C–Q diagram denotes the cell capacity, whereas the y axis is the electric quantity. The orange dot in Fig. 2 denotes the C–Q state of Cell A, whereas the green dot denotes that of Cell B. Cpack is the x-value of the red dot, which is the intersection of the orange line and the blue line. The green dots represent the state of other cells in the battery pack. Moreover, the equal SOC line can be presented using the gray dotted line in Fig. 2 [21]. The scheme of the capacity estimation of the battery pack in a BMS is depicted in Fig. 3. Correlated algorithms estimate the SOCs and capacities of each of the cells of the battery pack with inputs of the voltage (V) and current (I) measure from the battery pack using sensors. The two worst cells can be selected to estimate the Cpack according to the C–Q diagram. The voltage and current have measurement errors, and the algorithms have estimation errors. Hence the SOCs and capacities of the two worst cells are not accurate. The Cpack determined by the C–Q diagram considers the effects of the SOC and capacity estimation accuracies.

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The C–Q diagram evolves from Figs. 2–4 considering the errors in the SOC estimation under this condition. The dots in the C–Q diagram in Fig. 4 denote the states of the individual cells. The confidence intervals grown from the dots in the vertical direction definitely reflect the estimation error in the SOC defined in Eqs. (5) and (6). Interestingly, the boundary of the confidence interval for Cell A (B) forms a region of equal RCQ (RDQ). The intersection of the region of equal RCQ and equal RDQ is a parallelogram marked by shadow lines in Fig. 4. The red1 dot representing the state of the battery pack can be located at any point within the parallelogram. The parallelogram can be interpreted by the interval defined using Eq. (7):

Maximum cell voltage Charge cut-off voltage RCQ EOC

RDQ

Charge

Discharge

RCQ

EOD

C pack  ½C A  ð1  SOC A ð1  SOC A

RDQ

 a%Þ þ C B  ðSOC B

est

est þ a%Þ

 a%Þ; C A ð7Þ

Furthermore, the absolute error (Eabs,SOC) of the pack capacity is defined in Eq. (8).

Discharge cut-off voltage Minimum cell voltage

Eabs;SOC ¼ maxfjC pack  C pack

Fig. 1. The definition of the EOC and the EOD of the battery pack.

est jg

¼ a%  ðC A þ C B Þ

ð8Þ

where Cpack_est = CA(1-SOCA_est)+CB  SOCB_est, is the estimated pack capacity. The relative error (Erel,SOC) of the pack capacity is calculated in Eq. (9).

Cell A Cell B Baery Pack Line of full charged state Line of equal RCQ



C pack Erel;SOC ¼ max

1  C

minimum RCQ

Line of equal RDQ Line of equal SOC



CA þ CB  a%
¼ C

est


pack

ð9Þ

pack

Erel,SOC equals the product of thea% and CCA þC B . For convenience of pack

further error analysis, assume:

Erel;SOC ¼ RC  a%

ð10Þ

where, RC is the ratio between the sum of the capacities of the two worst cells and the capacity of the battery pack.

minimum RDQ

Electric Quanty

est

est þ a%Þ þ C B  ðSOC B

RC ¼

CA þ1 CA þ CB CB ¼C A C pack  ð1  SOC A Þ þ SOC A  ðSOC A  SOC B Þ C

ð11Þ

B

According to Eq. (11), RC is determined by the capacity deviation   CA and the SOC deviation (SOCA  SOCB) of the battery pack under CB

Capacity

different conditions. Given a certain capacity deviation and SOC deviation, RC changes in a specific range from EOC to EOD of the battery pack, as shown in Table 1. Fig. 5 collects the results listed in Table 1. When the capacity   deviation CCAB increases, the range of RC is extended, whereas the

Fig. 2. The Capacity–Quantity diagram.

3. Error analysis of the C–Q diagram 3.1. The error effect caused by the SOCs estimation error First, a discussion on how the estimation error of the cell SOC individually affects the state estimation accuracy of the battery pack is presented. Given that the estimation error of the SOC for individual cells is a%, the interval of the SOC (SOCA, SOCB) for Cell A and Cell B can be described by Eqs. (3) and (4), respectively.

SOC A 2 ½SOC A

est

 a%; SOC A

est

þ a%

ð3Þ

SOC B 2 ½SOC B

est

 a%; SOC B

est

þ a%

ð4Þ

where SOCA_est and SOCB_est is the estimated SOC for Cell A and Cell B, respectively. Similarly, the range of the electric quantities (QA, QB) for Cell A and Cell B are shown in Eqs. (5) and (6), respectively.

Q A 2 ½C A  ðSOC A

est

 a%Þ; C A  ðSOC A

est

þ a%Þ

ð5Þ

Q B 2 ½C B  ðSOC B

est

 a%Þ; C B  ðSOC B

est

þ a%Þ

ð6Þ

where CA and CB are the real capacity of Cell A and Cell B, respectively.

capacity deviation has little influence on the average value of RC. The SOC deviation (SOCA  SOCB) has a significant influence on RC. As the SOC deviation increases, the average RC becomes larger in specific condition. According to Eq. (10), the SOC deviation has a more significant influence on Erel,SOC. For a well-balanced pack, the SOC deviation is usually less than 10% in real application. RC is less than 2.5 at this time, as shown in Fig. 5. The Erel,SOC is less than 2.5a%, according to Eq. (10). 3.2. The error effect during the estimation of the capacities The cell capacity fades throughout its lifespan [22,23]. The capacities of individual cells of the battery pack are supposed to be estimated by the BMS online [24–27]. It is necessary to study how the estimation of the single cell capacity affects the accuracy of the estimated pack capacity. Note that the estimation error of the capacity for an individual cell is b%, the range of the real capac1 For interpretation of color in Figs. 4 and 13, the reader is referred to the web version of this article.

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M. Ouyang et al. / Applied Energy 177 (2016) 384–392

Fig. 3. Scheme of the capacity estimation of the battery pack in a BMS.

ity (CA and CB) for Cell A and Cell B can be described by Eqs. (12) and (13), respectively.

Cell A Cell B

 CA 2

Esmated Baery Pack  CB 2

α%

Line of equal RCQ Line of equal RDQ

QA

Electric Quanty

Real Baery Pack



C A est C A est ; 1 þ b% 1  b%



C B est C B est ; 1 þ b% 1  b%

QB

QA 2 

Cpack

QB 2

Capacity Fig. 4. Determining the pack capacity using the Capacity–Quantity diagram, given that errors exist in the SOC estimation.

Table 1 The RC of the battery pack under different conditions. SOCA  SOCB

CA CB

1

0

1

RC 2 [1.9,2.11]



9 10 10 9 4 5 5 4

2

10%

1

[1.8,2.25] 2.2222 [2.11,2.35]



9 10 10 9 4 5 5 4

3

20%

1

[2,2.5] 2.5 [2.38,2.64]



9 10 10 9 4 5 5 4

4

40%

1

10

9 10 9 4 5 5 4

4

Capacity deviaon( 1 9  ˄ ˅ 10 9 4 5 ˄ ˅ 5 4

RC

3.5 3

C B est C B est  SOC B ;  SOC B 1 þ b% 1  b%

 ð14Þ

 ð15Þ

The C–Q diagram considering the errors in the capacity estimation is illustrated in Fig. 6. The dots in the C–Q diagram in Fig. 6 denote the states of the individual cells, and the confidence intervals extending from the dots in Line of equal SOC definitely reflect the estimation error in the capacity defined in Eqs. (14) and (15). Similarly, the intersection of the region of equal RCQ and equal RDQ is a parallelogram. The dot representing the battery pack can be located at any point within the parallelogram. The parallelogram in Fig. 6 can be interpreted by Eq. (16):



C pack 2

C A est C B est C A est  ð1  SOC A Þ þ  SOC B ; 1 þ b% 1 þ b% 1  b%  C B est  SOC B ð1  SOC A Þ þ 1  b%

ð16Þ

[2.25,2.81] 3.33 [3.17,3.52]

Eabs;CAP ¼ maxfjC pack  C pack

[3,3.75]

b% ¼  ½C A 1  b%

est

est jg

 ð1  SOC A Þ þ C B

est

 SOC B 

ð17Þ

The relative error of the pack capacity, Erel,CAP, is defined in Eq. (18). Erel,CAP is approximately equal to b%.

CA ) CB



C pack Erel;CAP ¼ max

1  C




¼ b%

est


pack

ð18Þ

3.3. Combined effect considering both the estimation errors in SOC estimation and capacity estimation

2 1.5 0.1

C A est C A est  SOC A ;  SOC A 1 þ b% 1  b%

Furthermore, the absolute estimation error of the pack capacity considering capacity estimation error b% is defined in Eq. (17).

2.5

0

ð13Þ

where CA_est and CB_est is the estimated capacity for Cell A and Cell B, respectively. The range of the electric quantities for Cell A and Cell B are described in Eqs. (14) and (15), respectively.



Condition

ð12Þ

0.2

0.4

SOC deviaon ( SOC A-SOC B) Fig. 5. Analysis of the RC of the battery pack in different inconsistent conditions.

The independent effects of the SOCs estimation and the capacity estimation on the pack capacity estimation have been separately analyzed. The influence of the estimation errors of the cell SOC and the cell capacities on the pack capacity estimation is discussed in this section. Assume that the SOC estimation error is a% and the capacity estimation error is b%, then the real electric quantities (QA, QB) of Cell A and Cell B are described in Eqs. (19) and (20), respectively.

M. Ouyang et al. / Applied Energy 177 (2016) 384–392

Cell A Cell B

Cell A Cell B

Esmated Baery Pack

Esmated Baery Pack

CA

Line of equal SOC

Line of equal RCQ Line of equal RDQ

CB

Capacity

Fig. 6. Determining the pack capacity using the Capacity–Quantity diagram, given that errors exist in the capacity estimation.



C A est  ðSOC A 1 þ b%



C B est QB 2  ðSOC B 1 þ b%

est  a%Þ;

C A est  ðSOC A 1  b%

C B est  ðSOC B est  a%Þ; 1  b%

est þ a%Þ

est

þ a%Þ

ð19Þ

 ð20Þ



C A est C B est  ð1  SOC A est  a%Þ þ 1 þ b% 1 þ b% C A est  ð1  SOC A est þ a%Þ ðSOC B est  a%Þ; 1  b%  C B est þ  ðSOC B est þ a%Þ 1  b%

ð21Þ

Furthermore, the absolute error of the pack capacity is calculated in Eq. (22).

Eabs ¼ maxfjC pack  C pack ¼

a%

1  b%  ½C A

est

est jg

b% 1  b%

þ CB

est Þ

þ

 ð1  SOC A

est Þ

þ CB

 ðC A

est

est

 SOC B

est 

ð22Þ

The relative error of the pack capacity is given by Eq. (23).



C pack Erel ¼ max

1  C



C A est þ C B
¼ C pack

est


pack

 RC  a% þ b% ¼ Erel;SOC þ Erel;CAP

est

Fig. 7. Determining the pack capacity using the Capacity–Quantity diagram, given that errors exist in both the SOC and capacity estimation.



The C–Q diagram considering errors in the SOC and the capacity estimation is shown in Fig. 7. The dots in the C–Q diagram in Fig. 7 denote the states of the individual cells. The confidence regions displayed as a parallelogram represent the estimation errors in both the SOC and capacity, as defined in Eqs. (19) and (20). Similarly, the intersection of the region of equal RCQ and equal RDQ is a parallelogram. The red dot representing the battery pack can be located at any point within the shaded parallelogram. The confidence interval of the capacity (Cpack) for shaded parallelogram in Fig. 7 can be interpreted by Eq. (21):

C pack 2

CB

Cpack

Capacity

QA 2

β% 1− β % β% 1+ β %

Line of equal SOC

QB

Line of equal RDQ

α%

Line of equal RCQ

Cpack

CA

Real Baery Pack

β% 1− β % β% 1+ β%

Electric Quanty

Electric Quanty

Real Baery Pack

QA

388

Table 2 The cell capacities. Num.

1

2

3

4

5

6

Capacities (Ah)

25.86

25.33

26.02

26.02

26.17

26.57

4. Experiment validation 4.1. Experimental settings To validate the impact of the estimation errors of single cell states on the battery pack capacity estimation error discussed above, a battery pack is assembled with six cells connected in series in experiment validation. The battery is a rechargeable LiNixCoyMnzO2 lithium ion battery manufactured by HY Energy Co. Ltd. with a normal capacity of 25 A h. The cell capacities are listed in Table 2. Fig. 8(a) shows the scheme of the test platform including a Digatron test bench, a Current/Voltage Meter, a BMS sampling board, a BMS controller, a CAN data logger and a personal computer (PC). The battery pack is bicycled using a test bench manufactured by DigatronÒ, whereas the BMS components are supported by KeypowerÒ. The test platform is shown in Fig. 8(b). The Digatron test bench is used to charge/discharge the batteries under a designed test profile. The Current/Voltage Meter monitors the current passing the battery pack. The BMS sampling board monitors each of the cell voltages. The BMS controller obtains the information from the meter and the sampling board, sends the cell parameters data to the PC, and sends the maximum and minimum voltage of cells to the Digatron test bench every 1 s. The CAN data logger is used to connect the controller to the PC. The PC is responsible for monitoring and recording the data. Table 3 lists the accuracy for the measured signals. 4.2. Verification of the C–Q diagram

 a% þ b% ð23Þ

By synthesizing the Eqs. (10), (18), and (23), the battery pack capacity estimation error Erel considering both the estimation errors of SOC and capacity is the sum of two estimation errors (Erel,SOC + Erel,CAP).

The pack test was conducted to verify the C–Q diagram. Experiments were conducted with 4 different SOC distributions, as shown in Fig. 9. The (SOCA  SOCB) of the battery pack is {0.4, 0.2, 0.1 to 0}, respectively. Condition (a) represents the worst inconsistent condition in this study. Condition (b) and Condition (c) is the normal inconsistent condition between Condition (a)

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M. Ouyang et al. / Applied Energy 177 (2016) 384–392

CAN2_H CAN2_L BMS sampling board

BMS controller CAN1_L

CAN1_H





CAN Data logger PC

Digatron Test bench

Current/Voltage Meter

(a) Scheme of the test platform.

(b) Test platform. Fig. 8. The test platform for experimental validation.

Table 3 The accuracy of the measured signals.

Table 4 The test profile to measure the pack capacity.

Signals

Accuracy

Step

Step name

Duration

Cell voltages Current

±2 mV with a resolution 1 mV 0.1 A(<30 A) ±0.5%(>30 A)

1 2

Rest Constant current charge Rest Constant current discharge Rest Circle step 2–5 twice End

10 min

3 4

0.5

SOC Cell

5 6 7

1

2

3

4

5

6

1

2

3

(a)

5

0.5 1

2

3

4

(c)

5

6

1

2

3

4

5

8.33A(1/3C) until the maximum cell voltage reaching 4.2 V 90 min 8.33A(1/3C) until the minimum cell voltage dropping to 2.5 V 90 min

6

(b)

SOC Cell

4

Condition

6

(d)

Fig. 9. The SOC distributions of the experiment.

and Condition (d). Condition (d) represents the perfect inconsistent condition of the pack. All of the conditions have a same average SOC of 0.5. The test profile to measure the pack capacity is listed in Table 4. Fig. 10(a) shows the voltage curves of the pack capacity test for Condition (a). The cyan line represents the voltage curve of Cell 6, which has the smallest RCQ and highest voltage. The voltage of Cell

6 reached 4.2 V first. The blue line shows the voltage curve of Cell 1, which has the least RDQ. The voltage of the Cell 1 reached 2.5 V first. Cell 6 and Cell 1 are the worst cells, which determines the pack capacity in this case. Fig. 10(b) presents the pack voltage curves of the 4 different pack capacity tests with EOC aligned together. The EOD is delayed as the (SOCA  SOCB) increases, indicating that the capacity of the battery pack increases. The results of the 4 different pack capacity tests are listed in Table 5. The cells were adjusted to a specific SOC before the experiments, and the RDQ and RCQ of the battery pack can thus be calculated. The capacities of the battery pack are estimated using Eq. (2). The error between the estimated capacity and measured capacity are always less than 1%. Fig. 11 illustrates the capacity estimation of the battery pack using the C–Q diagram. Different colors indicate the battery pack at different conditions as presented in Fig. 9. Fig. 11 shows that there is little difference between the measured results and the estimated results, which verifies the C–Q diagram in Section 2.2. The errors between the estimated capacity and the measured capacity

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M. Ouyang et al. / Applied Energy 177 (2016) 384–392

(a) Cell voltages of Condition(a).

(b) Pack voltage under different conditions.

Fig. 10. The voltages curves for the pack capacity tests.

Table 5 The measured and estimated pack capacity in the experiments. Condition

RDQ (A h)

RCQ (A h)

Estimated capacity (A h)

Measured capacity (A h)

Error

(a) (b) (c) (d)

7.60 9.63 10.90 12.41

8.34 10.95 12.25 12.92

15.94 20.58 23.14 25.33

16.08 20.50 22.94 25.20

0.83% 0.38% 0.86% 0.50%

Electric Quanty(Ah)

25 23 21 19

Cell A Cell B Esmated Pack Measured Pack Line of equal RCQ Line of equal RDQ Condion(a) Condion(b) Condion(c) Condion(d)

Fig. 12. Schematic plot of the Dynamic Stress Test (DST).

17 15

13

15

17

19

21

23

25

27

Capacity(Ah) Fig. 11. The pack capacity determination using the C–Q diagram under 4 different SOC distributions.

may be caused by the measured error or negligence of the coulombic efficiency.

4.3. Verification of the error analysis in Section 3 Dynamic Stress Test (DST) were conducted to estimate the SOCs and capacities of individual cells within the battery pack to verify the pack capacity estimation error analysis discussed in Section 3. As shown in Fig. 12, the battery pack is constantly current charged with 8.33 A until the maximum voltage reaches the limit voltage 4.2 V. After a pause of 1.5 h, the DST working condition is repeated

six times, and then, the pack is constantly current discharged with 8.33 A until the minimum voltage reaches 2.5 V. Although DST tests were conducted upon all conditions in Fig. 9, only Condition (c) is selected as the representative condition to validate the error analysis to save spaces, because DST test results for other conditions lead to similar conclusion. The SOC estimation is fulfilled using extended Kalman filter referring to [5,28–32]. The SOH estimation is fulfilled by Eq. (24) as in [33]. The SOH estimation algorithm estimates two different SOCs first, then uses the transferred charge between these two states to calculate the battery capacity.

R tb C a;b ¼

ta

Icell ðtÞdt

SOCðOCVðt a ÞÞ  SOCðOCVðt b ÞÞ

ð24Þ

Two settings (T1 and T2) of Condition (c) are artificially made to verify the error analysis. Table 6 shows the measured and estimated SOC and capacity of the two worst cells; the errors and RC are calculated as well. The estimation errors of the SOCs and the capacities are less than 1.5% and 2.0%, respectively. The pack capacity is determined using the C–Q diagram given that errors exist in the estimation of the SOC and the capacity of T1 and T2 as shown in Fig. 13. The red dot representing the measured state of the battery pack is located in the parallelogram, which validates the error analysis. 4.3.1. Guidance for the SOCs and capacities estimation in BMS From the former discussion, it is derived that the battery pack capacity estimation is based on the estimation of the cell SOCs

391

M. Ouyang et al. / Applied Energy 177 (2016) 384–392 Table 6 The measured and estimated SOC and capacity of the two worst cells in Condition (c). Time

Cell

Estimated SOC

Measured SOC

SOC estimation error

Estimated capacity (A h)

Measured capacity (A h)

Capacity estimation error

RC

T1

A B

84.3% 96.8%

85.9% 96.5%

1.2% 0.3%

25.45 26.07

25.86 26.57

1.6% 1.9%

2.303

T2

A B

73.3% 85.9%

74.1% 85.0%

0.8% 0.9%

25.45 26.07

25.86 26.57

1.6% 1.9%

2.305

(a) T1

(b) T2

Fig. 13. Determining the pack capacity in Condition (c) using the Capacity–Quantity diagram.

2/5E

The C–Q diagram is utilized to analyze the estimation error of the battery pack capacity, which expands the function of the C–Q diagram. The analysis using the C–Q diagram reveals that the range of the real pack state is confined by a parallelogram region determined by individual cell states considering errors. In simplicity, the pack capacity estimation error Erel can be calculated as Eq. (26).

α/ %

E rel =E%

0

β /%

E

Fig. 14. Relationship between the estimation error of the pack capacity and the estimation error of the cell states.

Erel  a%  RC þ b% ¼ Erel;SOC þ Erel;CAP

ð26Þ

According to Eq. (25), with a fixed objective of the pack capacity estimation error, the estimation errors of the cell SOCs and capacities will be limited. Fig. 14 shows that when the estimation error of the pack capacity is supposed to be E% at most (Erel = E%), the cell SOCs estimation error (a%) and the cell capacity estimation error (b %) should be both located in the shadow triangle region below. The result provides guidance to choose proper accuracy of the estimation algorithms for cell SOC and capacities during BMS design.

which indicates that the estimation error of the cell SOCs has a greater influence on the pack capacity estimation error than that of the cell capacities. The conclusion of the analysis was verified by experiment using battery pack with 6-series-connected NCM batteries. The dot representing the measured state of the battery pack is located in the parallelogram in the C–Q diagram, which validates the error analysis. Finally, guidance for selecting proper estimation error for cell SOC and capacity during BMS design was given. The numerical relationship between the single cell state estimation accuracy and the pack capacity estimation accuracy is firstly set up, which broadens the horizon of the BMS technology. Future work is being conducted on cell balance based on the theory proposed in this paper, including: (1) analyzing different cell balance methods considering the errors in Eq. (26) with the C–Q diagram and (2) utilizing the test platform in Fig. 8 to verify the cell balance research.

5. Conclusion

Acknowledgment

Quantitative analysis on how the estimation errors of the cell SOCs and capacities determine the battery pack estimation error using a C–Q diagram was presented in this paper.

This work on the theory of the C–Q diagram was supported by the National Natural Science Foundation of China under the contract of U1564205 and 51507102.

and capacities of the battery pack. Estimation algorithm with high precision requires huge computation resources, which is hard to be applied in BMS. Selecting proper estimation accuracy of the SOCs and capacities is essential during BMS design. From Eq. (23) and the previous analysis, for a well-balanced pack (Condition (c)), the battery pack capacity estimation error can be described as below.

Erel  a%  RC þ b%  2:5a% þ b%

ð25Þ

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M. Ouyang et al. / Applied Energy 177 (2016) 384–392

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