Multi-model probabilities based state fusion estimation method of lithium-ion battery for electric vehicles: State-of-energy

Applied Energy xxx (2016) xxx–xxx

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

Multi-model probabilities based state fusion estimation method of lithium-ion battery for electric vehicles: State-of-energy q Cheng Lin, Hao Mu, Rui Xiong ⇑, Jiayi Cao Department of Vehicle Engineering, School of Mechanical Engineering, Beijing Institute of Technology, No. 5 South Zhongguancun Street, Haidian District, Beijing 100081, China

h i g h l i g h t s
A novel multi-model probability battery SoE fusion estimation approach was proposed.
The linear matrix inequality-based H1 technique is employed to estimate the SoE.
Performance of the method was verified by different batteries at various temperatures.
The results show that the proposed method can achieve accurate SoE estimation.

a r t i c l e

i n f o

Article history: Received 16 March 2016 Received in revised form 21 April 2016 Accepted 9 May 2016 Available online xxxx Keywords: Electric vehicles Batteries State of energy estimation Multi-model probabilities H-infinity robust state observers

a b s t r a c t State-of-energy (SoE) is an important index for batteries in electric vehicles and it provides the essential basis of energy application, load equilibrium and security of electricity. To improve the estimation accuracy and reliability of SoE, a novel multi-model fusion estimation approach is proposed against uncertain dynamic load and different temperatures. The main contributions of this work can be summarized as follows: (1) Through analyzing the impact on the estimation accuracy of SoE due to the complexity of models, the necessity of redundant modeling is elaborated. (2) Three equivalent circuit models are selected and their parameters are identified by genetic algorithm offline. Linear matrix inequality (LMI) based H-infinity state observer technique is applied to estimate SoEs on aforementioned models. (3) The concept of fusion estimation is introduced. The estimation results derived by different models are merged under certain weights which are determined by Bayes theorem. (4) Batteries are tested with dynamic load cycles under different temperatures to validate the effectiveness of this method. The results indicate the estimation accuracy and reliability on SoE are elevated after fusion. Ó 2016 Elsevier Ltd. All rights reserved.

1. Introduction Since the lithium-ion batteries (LIBs) own the high energy/ power density and superior cycle lifetime, they are regarded as the most promising energy candidate in automotive industry and have been widely employed in hybrid electric vehicles (HEVs), plug-in hybrid electric vehicles (PHEVs), and battery electric vehicles (BEVs) [1–3]. In terms of safety, high efficiency and sustainable application, especially after the batteries are connected in series or parallel forming packs, battery management system (BMS) is indispensable to monitor the state of every single cell and rule them in case of the abuse resulting in catastrophic issues. But it is still a tough task for BMS to acquire the accurate states by direct q The short version of the paper was presented at CUE2015 on Nov. 15–17, Fuzhou, China. This paper is a substantial extension of the short version. ⇑ Corresponding author. E-mail addresses: [email protected], [email protected] (R. Xiong).

measurements because the sophisticated electrochemical process is occurring inside the battery during working operation which contains the strong nonlinearity and complex time-varying characteristic [4–6]. Normally, people will pay more attention to two indicators of LIBs-state of charge (SoC) and state of health (SoH) [7–9]. SoC is relevant to the residual capacity of a battery and it protects the battery working within the safe operating area (SOA) from being overcharged or over discharged. SoH is used to manifest the level of ageing of the battery and it helps to recalibrate the estimation results on SoC which relates to the actual available capacity. In recent years, SoC estimation methods have been extensively and deeply studied. The ampere-hour (A h) counting approach [10,11] is the most common one as it is simple and low-cost. But the estimation accuracy is vulnerable to noise, uncertain initial SoC knowledge and current drift. Based on equivalent circuit models, many estimation techniques are investigated. Kalman filter technique which is utilized for lead-acid battery originally and

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Please cite this article in press as: Lin C et al. Multi-model probabilities based state fusion estimation method of lithium-ion battery for electric vehicles: State-of-energy. Appl Energy (2016), http://dx.doi.org/10.1016/j.apenergy.2016.05.065

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then some derivatives, such as extended Kalman filter (EKF) [12,13] and unscented Kalman filters (UKF) [14,15] are applied successively. Adaptive techniques [16,17] are explored to break the strict limitation with the noisy items of former methods. Besides, some other ways, including particle filter (PF) [18], and robust state observers [19–22] are studied as well, which yield the satisfactory estimation results and robustness. The data-driven methodologies [23–26] are also effective tools to address this nonlinear issue, but they are of over-dependency on the priori knowledge of experimental data. As for SoH estimation, particularly for capacity estimation, the model-based methods [27] are still the most practical because not only they can meet the demand of accuracy, but also the real-time capability is superior to others. Back to the motivation to estimate the state of the battery, on one hand, these indexes are able to prohibit the abuse and guarantee the operating safety. On the other hand, how much the remaining driving range (RDR) of electric vehicles left is another critical point that people should concern. As a matter of fact, the state of energy (SoE), which signifies the residual available energy in the battery, is more qualified than SoC to estimate the RDR [28]. Some studies have existed with respect to the estimation of SoE. Liu et al. [29] presented the Back-Propagation Neural Network (BPNN) based method to estimate the SoE, considering the influence of the energy loss on the internal resistance, electrochemical reactions, the decline of open circuit voltage (OCV), the discharge rate and temperature fluctuation. Although the estimation results are satisfactory through validations, the complexity of this method is a vital defect for practical applications. Wang et al. [30] proposed a joint estimation approach based on PF algorithm to obtain the SoC and SoE respectively. However, the estimation accuracy of SoE relies on the prerequisite that the SoC has been well computed. Xiong et al. [31] chose the central difference Kalman filter (CDKF) and Gaussian model to estimate the SoE and controlled the estimation error within 1% both for LiFePO4 and LiMnO2 batteries. Dong et al. [32] adopted dual filters-EKF plus PF-to estimate the SoE. The former filter is employed to update parameters of battery model on-line and the other filter is used to estimate the SoE. By experimental validation, the estimation accuracy could converge to the authentic values within the error of ±4% under constant current conditions and dynamic current conditions. Wang et al. [33] applied recursive least square (RLS) with forgetting factor method to identify the battery model and adaptive technique to estimate the SoE. The simulation results demonstrated the effectiveness of the proposed method. Barai et al. [28] clearly illustrated the strong relationship between the RDR and SoE rather than that with SoC and presented a novel SoE estimation method based on the short-term cycling history. Considering the uncertainty on equivalent circuit models (ECMs) that we have discussed in our previous work in Ref. [34], we have investigated the multi-model probabilities based fusion estimation method for the battery SoC estimation. There have three differences between the previous paper and this study. (1) In this paper, the goal to be estimated is the SoE rather than SoC; (2) Due to the different object, the corresponding systematic functions are slightly of discrepancy; (3) To emphasize the robustness of the algorithm against temperatures. Experiments are carried out with three temperatures and the method is verified by data which own distinct temperature characteristics. The single model-based methods are still used for counterparts. Through the comparison, the proposed method is revealed being superior to the single model ones and the estimated SoE has been improved no matter in the aspects of accuracy or reliability. The remainder of this paper is listed as follows: Section 2 introduces the battery model and LMI-based H-infinity (H1) state observer. Section 3 analyzes the reason why the estimated results should be fused about SoE, and subsequently the novel fusion

estimation method is presented. The test bench and the experimental verification are introduced in Section 4. Some conclusions are drawn in the final section.

2. Model-based SoE estimation 2.1. Definition of SoE SoE which is a percentage represents the residual energy of the total in a battery and its form is analogous to the SoC, which is shown as follows:

zðtÞ ¼ zðt0 Þ þ

1 Eact

Z

t

t0

PðsÞds

ð1Þ

where z(t) denotes the SoE at sample time t similarly hereinafter, z (t0) denotes the initial SoE value, Eact denotes the actual available energy of the battery, and P(s) denotes the instantaneous power of the battery. The continuous form can be written as:

z_ ðtÞ ¼

PðtÞ Eact

ð2Þ

2.2. Battery models Due to the complicated electrochemical process occurring inside the battery, it is difficult to evaluate the accurate performance by simple measurements. Consequently, equivalent circuit model, the combination of several electrical circuit components, is designed to simulate the dynamic characteristics of the battery. Using ECMs can solve many problems about states estimation, but the lack of physical explanation is its fatal weakness. Common ECMs contain the Rint model, Nth-RC networks model, RC model, and PNGV model. Among these ECMs, the Nth-RC networks model, seen in Fig. 1, occupies the dominant position. It basically consists of three parts, the potential resource Uoc to describe the open circuit voltage (OCV) which holds the monotonous relationship with SoC (SoE), the ohmic resistance Ro to mimic the direct current voltage drop, and several Resistance-Capacitor (RC) networks RDi, CDi forming the polarization voltage UDi to express the polarization and diffusion effect, where, i = 1, 2, 3, . . . , N, Ut denotes the terminal voltage. There still exists the trade-off between the complexity of model and estimation accuracy. Although the more RC pairs placed in the circuit can improve the accuracy [35], it sacrifices a large amount of calculation resource of the hardware at the cost. Therefore, usually the order of RC networks model is confined within three. According to electric knowledge, dynamic equations of Thevenin model, double polarization model and 3rd-RC model are listed in Table 1. The OCV of the battery has a monotonous relation with SoE shown in Fig. 2.

Fig. 1. The diagram of general RC networks ECM.

Please cite this article in press as: Lin C et al. Multi-model probabilities based state fusion estimation method of lithium-ion battery for electric vehicles: State-of-energy. Appl Energy (2016), http://dx.doi.org/10.1016/j.apenergy.2016.05.065

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the terminal voltage that can be measured directly, u = iL denotes

Table 1 The Schematic diagrams and dynamic equations of ECMs. Models

Dynamic equations  U D1 iL U_ D1 ¼  RD1 C D1 þ C D1 U t ¼ U oc þ iL Ro þ U D1 8 U i > < U_ D1 ¼  D1 þ L

N = 1 (Thevenin model) N = 2 (DP model)

RD1 C D1

C D1

U D2 iL U_ D2 ¼  RD2 C D2 þ C D2 > : U t ¼ U oc þ iL Ro þ U D1 þ U D2 8 U D1 iL U_ ¼  RD1 > C D1 þ C D1 > > D1 < U D2 iL U_ D2 ¼  RD2 þ C D2 C D1 > L > U_ D3 ¼  R UD3 þ CiD3 > D3 C D3 : U t ¼ U oc þ iL Ro þ U D1 þ U D2 þ U D3

N = 3 (3rd-RC model)

T the system input and U ¼ ½ x t n  denotes the noise vector including the process noise x, t and measuring noise n. Since the nonlinear relationship between OCV and SoE depicted as Fig. 2, the observation equation includes the nonlinear term Ut(SoE(t)). Aiming at the deployment of LMI-based H1 robust state observer technique, the observation equation should be transformed into the standard matrix form. Therefore, the Taylor expansion is used and the first order term is kept to extract the state to estimate SoE, seen in Eq. (6).

h y¼ 1

dU oc dz

i

x þ Du þ HU ¼ Cx þ Du þ HU

ð6Þ

Open circuit voltage (V)

Therefore, the system state-space Eq. (4) can be reformulated as:

4.2



4 3.8

ð7Þ

y ¼ Cx þ Du þ HU

3.6

h where C ¼ 1

3.4

20

40

60

80

dU oc dz

i

and the other matrices are still the same.

According to the observability criterion of control theory, this system is observable except for dUdzoc ¼ 0, because the rank of matrix

3.2 0

x_ ¼ Ax þ Bu þ GU

100

SoE (%)

½ C CA T is always full. In order to ensure the observability of the system, we adopted the first order derivative of Uoc with respect to SoE at the fixed 50% point.

Fig. 2. The open circuit voltage as a function of SoE.

2.3. Models identification In Table 1, if the parameters of models are known, then SoE can be estimated by some filtering techniques. The genetic algorithm (GA) is selected to identify those parameters off-line. The advantage of GA is searching the optimal solution no matter how complex the model is and it can achieve the desirable accuracy of parameters. The fitness function is set to the variance of the error of estimated terminal voltages, shown in Eq. (3):

n  o 8 ^ jg > min f v > < L L     2 X g > 1 ^ t;j v > ^j ¼ L ^ jg U t;j  U :f v

ð3Þ

(



value of Ut. 2.4. SoE estimation 2.4.1. Observability discussion In order to elaborate the principle of applying the LMI-based H1 state observer technique to estimate the SoE, Thevenin model is adopted as an example. According to Table 1, the system statespace function can be derived by:



x_ ¼ Ax þ Bu þ GU y ¼ U oc ðSoEðtÞÞ þ U D þ Du þ HU

ð4Þ

where the system matrices are

8   1=C D RD 0 > > A ¼ ; B ¼ ½ 1=C D U t ðtÞ=Eact  > < 0 0   > 1 0 0 > > ; H ¼ ½0 0 1 :G ¼ 0 1 0 T

ð5Þ

and x ¼ ½ U D z  is the state vector to be estimated, z denotes the SoE of the battery, y = Ut denotes the observable value, Ut denotes

ð8Þ

e_ x ¼ ðA  KCÞex þ ðG  KHÞU

ð9Þ

ey ¼ Cex þ HU

^ jg is the fitness function to be minimized, L is the length where f v of data derived by experiments, g is the generation of population v, j ^ t is the estimation is the j-th individual among population v, and U

^x_ ¼ A^x þ Bu þ Kðy  y ^Þ ^ ¼ C^x þ Du y

And meanwhile the error system can be yielded by Eqs. (7) and (8)



i¼1



2.4.2. LMI-based H1 state observer for SoE estimation The fundamental concept of observer technique is using the known observer to mimic the original system such that the states hardly to be measured can be derived through that in the observer. Fig. 3 shows the flow chart of LMI-based H1 state observer. According Eq. (7), the corresponding observer can be modeled as:

^, the coefficient matrices A, C, H and the where ex ¼ x  ^x; ey ¼ y  y noise vector U are the same as those mentioned above, matrix K is the control gain to be calculated. From the perspective of control theory, the observer technique is feasible if and only if Eq. (9) is asymptotically stable, which means the disturbance of noise to the state estimation can be controlled in the given level, shown in Eq. (10).

kex k < ckUk

ð10Þ

k  k denotes the H1 norm of terms which manifests the amplitude of signals and c > 0 is a given attenuation level. Apparently, it is difficult to compute the control gain K with Eq. (10). Utilizing the bounded real theorem [36], the stabilization problem is transformed to a convex optimization issue and thus linear matrix inequality method can be employed to address it, which has been revealed in Eq. (11). So an equivalent condition is used to change Eq. (10) to Eq. (11) which is the LMI form. Eventually, the gain K = P1X is solved via double solvers in LMI optimization toolbox.

"

AT P  CT XT þ PA  XC þ I PG  XH ðPG  XHÞT

c2 I

# <0

ð11Þ

where P is a symmetric and positive definite matrix, P = PT.

Please cite this article in press as: Lin C et al. Multi-model probabilities based state fusion estimation method of lithium-ion battery for electric vehicles: State-of-energy. Appl Energy (2016), http://dx.doi.org/10.1016/j.apenergy.2016.05.065

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Fig. 3. Flow chart of LMI-based H1 state observer for SoE estimation.

3. Multi-model probabilities SoE fusion estimation

3.2. MMPFE: multi-model probabilities fusion estimation

3.1. Problem description

3.2.1. Scheme of the fusion estimation The concept of fusion estimation has been well investigated in many field, such as maneuver target tracking, satellite positioning and so forth. In battery application field, some researchers try to use this idea to improve the estimation accuracy and robustness of SoC [37–39]. However, they pay more attention to description and variation of the OCV–SOC relationship rather than the ohmic resistance and polarization effects. In our paper, fusion estimation mainly refers to merging the SoEs gained by different models under specific rules and the framework of the algorithm is shown in Fig. 5. First, the collected data including the terminal voltage and current are transmitted into the ECMs block. Parameters of three models are identified off-line so as to establish observers and these information together with collected data is sent to the filters block. In this block three LMI-based H1 state observers are applied to estimate the SoEs of different models, yielding the estimated results, ^z1 , ^z2 and ^z3 . Meanwhile the estimated values of terminal ^ t1 , U ^ t2 , and U ^ t3 are gathered for calculating the weights voltages U

It is known that as for SoC estimation, the estimation accuracy varies with the topologies of ECMs and it is different for kinds of LIBs as well. If the ageing factor is taken into account, the situation will become even more complicated. To further elaborate this problem, we chose three ECMs and one method for a battery so as to acquire the SoE of it and the estimation results can be seen in Fig. 4. The tests were carried out on the test bench elaborated below at room temperature (22 ± 2 °C). Three models share the same estimation method: LMI-based H1 state observer technique. From Fig. 4, it is not hard to find out that during different time regions, the optimal estimation results belong to different models. In the zoom figure A, the 3rdRC model is much closer to the true SoE values, while in zoom figure B the best model is DP model instead. This situation will change with the battery material and level of ageing. So it gives rise to low credibility of single model based estimation issue. In order to improve the reliability and redundancy of SoE estimation of LIBs, novel multi-model probabilities fusion estimation is proposed in this paper. 100 Thevenin model DP model SoE (%)

60 70

20 0 0

SoE (%)

SoE (%)

80

40

rd

^zfs ¼ x1^z1 þ x2 ^z2 þ x3 ^z3

Zoom figure B

ð12Þ

where weights xi satisfy the following equation:

22.5 20 2.69

Zoom figure A

True SoE

3 -RC model

25

xi of models which represent the contributions of each model to estimate SoE correctly. Finally, the fused SoE ^zfs is a weighted sum of three estimated values (see Eq. (12)) and it goes back to the filters block as the initial values for the next time.

2.76 Time (h)

2.83

3 X

xi ¼ 1

ð13Þ

i¼1

67 64 1

1.1 Time (h)

0.5

1

1.2

1.5

2

2.5

3

Time (h) Fig. 4. Comparison of SoE estimation profiles based on various models.

3.5

3.2.2. Weights calculation The crucial process for multi-model fusion estimation is the determination of weights. In this paper, we use scaled ‘probability’ to express the degree that the estimated SoE from certain model being approximate to the true SoE. There are many methodologies

Please cite this article in press as: Lin C et al. Multi-model probabilities based state fusion estimation method of lithium-ion battery for electric vehicles: State-of-energy. Appl Energy (2016), http://dx.doi.org/10.1016/j.apenergy.2016.05.065

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Fig. 5. Systematic block of the multi-model fusion estimation method.

to compute the weights. The most direct way is through the residual errors of the terminal voltages (RETVs) that shown in Eq. (14). Normally, the large error of terminal voltages means the model could not effectively simulate the battery. Nonetheless, the instantaneous errors of terminal voltages are too superficial to reflect the adhesion between the model and the battery itself since the terminal voltages are prone to be contaminated by noise. Hence the probability of the model i.e., the weight of the model, should include the statistical characteristics of RETVs as well.

In this section, the fusion estimation method on SoE is verified on the nickel–manganese–cobalt oxide/graphite (NMC) cell. In order to illustrate the effectiveness of this method which is robust to the ambient temperature, the tests are executed under 10 °C, 25 °C and 40 °C, respectively, including hybrid power pulse characteristic (HPPC) test, OCV test and dynamic stress test (DST). The basic parameter of the cell is listed in Table 2.

^ ti ðkÞ resi ðkÞ ¼ U t ðkÞ  U

4.1. Test bench

ð14Þ

^ ti is the estimation where Ut is the measured terminal voltage, U value of terminal voltage by the observer i (i ¼ 1; 2; . . . ; N), and k is the sample time. Enlightened by the multi-model estimation with respect to Kalman filter [39,40], at the sampling time k and given the measured terminal voltage Ut(k), the conditional probability density function of observer i is described as:

f ðU t ðkÞjpi Þ ¼

1 ð2pÞq=2 Var 1=2 i ðkÞ    exp resTi ðkÞVar 1 i ðkÞresi ðkÞ=2

ð15Þ

where pi denotes the parameter set of the model, including the system transition and observation matrices for example A, B, C and D in Eq. (7), Vari(k) means the variance of the RETVs of observer i, and q is the number of measured values (q = 1 in this study). Moreover, according to the Bayes theorem, the probability:

f ðU t ðkÞjpj ÞPrðpj Þ Prðpj jU t ðkÞÞ ¼ PN i¼1 f ðU t ðkÞjpi ÞPrðpi Þ

ð16Þ

where Pr(pj) is the probability of the parameter set pj and j is the index (j = 1, 2 or 3). The conditional probability indicates the weight values of each model’s contribution for the fused results, which means the estimated value could express the authentic SoE. Because

Prðpi jU t ðk  1ÞÞ ¼

PrðU t ðk  1Þjpi ÞPrðpi Þ PrðU t ðk  1ÞÞ

ð17Þ

4. Results and discussion

The configuration of the test bench is shown in Fig. 6. It is made up of an Arbin BT-5HC cycler with 16 independent channels, a constant temperature and humidity chamber, a host computer and tested cells. The cycler is responsible for loading the programmed current or power profiles on the cells within a range of voltage, 0– 5 V, and current, 100 to 100 A, and the measuring errors of its sensors with respect to voltage and current are less than 0.05%. The thermal chamber with 40 to 100 °C wide temperature range is to provide a stable operating environment for the test cells, avoiding the disturbance due to ambient temperature fluctuation. The host computer communicates with the cycler through Ethernet cables so that sending the order to control the cycler and storing the information of batteries’ states, such as the voltage, current and temperature. 4.2. Parameters inventory Before the parameters identification procedure, the traditional HPPC test and the OCV-SoE test are conducted advanced to determine the model parameters and the relationship between the OCVSoE. The OCV-SoE test is similar with that of SoC. Discharge (charge) 10% of total energy with nominal current, rest for 1 h (2 h for LFP cell), record the steady voltage as the open circuit voltage and repeat the former process 10 times, the final results shown as Fig. 2. The curve in Fig. 2 can be approximated by 8th order polynomial function (see Eq. (19)), where its parameters can be identified by the least squares approach.

U oc ðzÞ ¼ a0 þ a1 z þ a2 z2 þ a3 z3 þ a4 z4 þ a5 z5 þ a6 z6 þ a7 z7 þ a8 z8 ð19Þ

and Pr(Ut(k  1)|pi) = Pr(Ut(k  1)) = 1, combining Eqs. (15) and (17), thus Eq. (16) can be expressed as:

f ðU t ðkÞjpj ÞPrðpj jU t ðk  1ÞÞ

xj ðkÞ ¼ Prðpj jU t ðkÞÞ ¼ PN

i¼1 f ðU t ðkÞjpi ÞPrðpi jU t ðk

 1ÞÞ

ð18Þ

where the numerator is a recursive form and j represents the observer number. According to Eq. (18), the weight values of three observers can be calculated separately, and the real-time fusion estimation of SoEs can be obtained through Eq. (13).

Table 2 Basic parameters of test cell. Attributes

NMC

Nominal capacity Nominal voltage Cut-off voltage Operational temperature

2.0 A h 3.6 V 4.1 V/3.0 V 20 to 60 °C

Please cite this article in press as: Lin C et al. Multi-model probabilities based state fusion estimation method of lithium-ion battery for electric vehicles: State-of-energy. Appl Energy (2016), http://dx.doi.org/10.1016/j.apenergy.2016.05.065

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Fig. 6. Setup of the battery test bench.

4.3. Analysis of fusion estimation results

Table 3 Parameters identification results of three models. Models

10 °C

25 °C

40 °C

Thevenin model

Ro ¼ 0:055 X RD ¼ 0:025 X C D ¼ 1:10e3 F

Ro ¼ 0:044 X RD ¼ 0:021 X C D ¼ 1:68e3 F

Ro ¼ 0:037 X RD ¼ 0:024 X C D ¼ 1:91e3 F

DP model

Ro ¼ 0:056 X RD1 ¼ 0:024 X C D1 ¼ 1:61e3 F RD2 ¼ 0:0023 X C D2 ¼ 1:87e3 F

Ro ¼ 0:044 X RD1 ¼ 0:0010 X C D1 ¼ 3:97e3 F RD2 ¼ 0:021 X C D2 ¼ 1:87e3 F

Ro ¼ 0:037 X RD1 ¼ 6:5e  4 X C D1 ¼ 1:06e3 F RD2 ¼ 0:024 X C D2 ¼ 1:94e3 F

3th-RC model

Ro ¼ 0:053 X RD1 ¼ 0:0019 X C D1 ¼ 1:12e3 F RD2 ¼ 0:0017 X C D2 ¼ 1:10e3 F RD3 ¼ 0:025 X C D3 ¼ 1:60e3 F

Ro ¼ 0:044 X RD1 ¼ 0:0014 X C D1 ¼ 2:13e3 F RD2 ¼ 0:025 X C D2 ¼ 1:99e3 F RD3 ¼ 5:8e  4 X C D3 ¼ 4:38e3 F

Ro ¼ 0:037 X RD1 ¼ 0:0010 X C D1 ¼ 1:26e3 F RD2 ¼ 0:022 X C D2 ¼ 2:39e3 F RD3 ¼ 0:0010 X C D3 ¼ 2:18e3 F

2 0

4.3.2. Case02 temperature 25 °C The similar situation happens with the data under 25 °C. The DST test is carried out under 25 °C as well and the estimated results are shown in Fig. 9. Fig. 9(a) reveals that estimated SoEs

(b)

100 75

SoE (%)

(a) Current (A)

where ai (i = 1, 2, 3, . . . , 8) is the fitting coefficient and z denotes the SoE. In this study, GA is used to identify the parameters of three models of batteries. In order to decrease the complexity and demonstrate the robustness of the fusion estimation, we adopt the HPPC test data for computing the parameters of three models at fixed SoE point and the results with respect to temperatures are displayed in Table 3.

4.3.1. Case01 temperature 10 °C The DST is carried out for the NMC lithium-ion battery cell under 10 °C, and the current and SoE profiles are shown in Fig. 7 and the same cycles will be undertaken in the next two cases. LMI-based H1 state observers are applied to three models and the estimation results are exhibited in Fig. 8(a). In order to indicate the robustness of this algorithm, the initial SoEs are set 50% offset with true values. From Fig. 8(a), no matter the single model based methods or the multi-model fusion method can converge to the true values in finite time. Fig. 8(c) shows the maximum absolute error (MAE) point of DP model appears in 2.05–2.08 h whereas the MAE points of other methods concentrate in 3.13–3.15 h. After fusion, the minimum estimation error (5.15% in Fig. 8(d)) has been derived and the reliability of estimation which is evaluated by the mean estimation error (MEE) and root mean square error (RMSE) has been improved as well, since those two indexes of fusion method are optimal comparing with others. The detail analysis data is presented in Table 4.

-2 -4

50 25

-6 0

0.5

1

1.5

2

Time (h)

2.5

3

3.5

0

0

0.5

1

1.5

2

2.5

3

3.5

Time (h)

Fig. 7. DST for the NMC cell: (a) current vs. time profile; (b) reference SoE vs. time profile.

Please cite this article in press as: Lin C et al. Multi-model probabilities based state fusion estimation method of lithium-ion battery for electric vehicles: State-of-energy. Appl Energy (2016), http://dx.doi.org/10.1016/j.apenergy.2016.05.065

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Thevenin model DP model 3 rd -RC model MMPFE method True SoE

SoE (%)

75 50 25 0

0

0.5

1

1.5

2

2.5

3

(b)

8 4

SoE error (%)

(a) 100

0

B -4 Thevenin model DP model

-8 -12

3.5

0

0.5

1

1.5

Time (h)

(c)

(d)

Thevenin model DP model

2.06

2.07

Zoom area B

2.5

3

3.5

6

Thevenin model DP model

MAE point of Thevenin model MAE=5.95%

8

3 rd-RC model MMPFE method MAE point of fusion method MAE=5.15%

MAE point of 3rd-RC model MAE=5.45%

4

3rd-RC model MMPFE method

Zoom area A -7.5 2.05

10

SoE error (%)

SoE error (%)

-5.5

-6.5

2

Time (h)

-4.5

MAE point of DP model MAE=5.82%

A

3 rd -RC model MMPFE method

2 3.13

2.08

3.14

Time (h)

3.15

Time (h)

Fig. 8. SoE estimation profiles and errors for the NMC cell under 10 °C: (a) comparison of estimation results of SoE before fusion and after; (b) estimation errors of SoE vs. time; (c) zoomed picture of area A; (d) zoomed picture of area B.

Table 4 Statistical data of SoE error under 10 °C.

Table 5 Statistical data of SoE error under 25 °C.

Methods

MAE (%)

MEE (%)

RMSE (%)

Methods

MAE (%)

MEE (%)

RMSE (%)

Thevenin model DP model 3rd-RC model MMPFE

5.95 5.82 5.45 5.15

2.08 2.23 2.09 1.99

2.51 2.76 2.58 2.40

Thevenin model DP model 3rd-RC model MMPFE

2.36 2.21 2.17 2.05

0.78 0.74 0.70 0.72

0.95 0.88 0.86 0.85

SoE (%)

(a) 100

Thevenin model DP model

75

rd

3 -RC model MMPFE method True SoE

50 25 0 0

0.5

1

1.5

2

2.5

3

fusion estimation result is superior to that value. Therefore, the best estimation result is still obtained by MMPFE method. From the statistical table, it can be seen that the estimation accuracy is promoted remarkably and the lower MEE and RMSE mean the integral estimation performance is elevated. The detail analysis result is presented in Table 5.

(b) SoE error (%)

by four methods are all convergent and the maximum error points locate in different time range seen in Fig. 9(b). Through the fusion process, the estimation error is lower than that of Thevenin model and DP model in the time range 3.11–3.17 h. Although it is higher than that of 3rd-RC model in this area, the actual MAE point of 3rdRC model appears in time range 1.72–1.90 h of Fig. 9(c) and the

B

2 0 Thevenin model DP model

-2

3.5

3rd-RC model MMPFE method

0

0.5

1

Time (h) -1

Zoomed area A

-1.5

-2

Thevenin model DP model 3rd-RC model MMPFE method

MAE point for 3rd-RC model MAE=2.17%

-2.5 1.72

1.81

Time (h)

1.5

2

2.5

3

3.5

Time (h)

1.90

(d) 2.5 SoE error (%)

SoE error (%)

(c)

A

Thevenin model DP model

MAE point for Thevenin model MAE=2.36%

2

3rd-RC model MMPFE method

MAE point for DP model MAE=2.21%

1.5 1 3.11

Zoomed area B

3.14

3.17

Time (h)

Fig. 9. SoE estimation profiles and errors for the NMC cell under 25 °C: (a) comparison of estimation results of SoE before fusion and after; (b) estimation errors of SoE vs. time; (c) zoomed picture of area A; (d) zoomed picture of area B.

Please cite this article in press as: Lin C et al. Multi-model probabilities based state fusion estimation method of lithium-ion battery for electric vehicles: State-of-energy. Appl Energy (2016), http://dx.doi.org/10.1016/j.apenergy.2016.05.065

C. Lin et al. / Applied Energy xxx (2016) xxx–xxx

Thevenin model DP model 3rd-RC model MMPFE method True SoE

SoE (%)

75 50 25

(b)

2 0 -2 -4

-1

SoE error(%)

(a) 100

SoE error (%)

8

Zoom figure

-6

-3 6400

0 0

0.5

1

1.5

2

2.5

3

3.5

Thevenin model DP model

-2

-8

0

0.5

Time (h)

6550 Time (s)

1

3rd-RC model MMPFE method

6700

1.5

2

2.5

3

3.5

Time (h)

Fig. 10. SoE estimation profiles and errors for the NMC cell under 40 °C: (a) comparison of estimation results of SoE before fusion and after; (b) estimation errors of SoE vs. time.

Table 6 Statistical data of SoE error under 40 °C. Methods

MAE (%)

MEE (%)

RMSE (%)

Thevenin model DP model 3rd-RC model MMPFE

2.94 2.62 2.35 2.24

0.93 0.83 0.79 0.76

1.15 1.07 0.97 0.94

analysis of results, it demonstrates the argument that the uncertainty of ECMs affects the estimation accuracy and thus the reliability of single model-based methods is questioned. To solve this problem, the concept of the multi-model fusion estimation is introduced and employing the Bayes theorem the weights of various models can be computed. The final estimation result on SoE is the weighted sum of results coming from different models. Experiments are carried out under different temperatures to verify the validation of this method on LIBs. The fused results are demonstrated to be superior to that of using single model and robust to the temperatures, meanwhile the reliability is guaranteed as well. In the future, the estimation with respect to the total available energy should be concerned and the effect of the ageing level to the estimation accuracy of different models should be studied as well. Acknowledgements

Fig. 11. Statistical analysis of the SoE estimation error under different temperatures.

4.3.3. Case03 temperature 40 °C The DST test is executed under 40 °C on the bench and the collected data are used to estimate the SoE shown in Fig. 10. But in this time the SoE profiles of various methods are different from the former two cases, since the MAE points distribute in the same time range 6400–6700 s, shown in Fig. 10(b). In this picture, the fusion procedure is capable of yielding better result than other method, because the MAE (2.23%) is the smallest. From the whole SoE error profile of MMPFE method, it owns satisfactory speed of convergence which is close to that of Thevenin model in the initial time region and then it approaches to the zero error line in the middle time range where it holds the best MAE. In Table 6, the MEE and RMSE indicate the reliability of SoE estimation has been improved. To exhibit the effect of fusion process under different temperature, the statistical information is summarized in Fig. 11. 5. Conclusions This paper has presented a multi-model probabilities fusion estimation approach to improve the reliability and redundancy of the SoE estimation. Three equivalent circuit models, Thevenin model, DP model and 3rd-RC model, are selected to estimate the SoE of a battery by LMI-based H1 state observers. Through the

This work was supported by the National Natural Science Foundation of China (Grant No. 51507012). The systemic experiments of the lithium-ion batteries were performed by the Advanced Energy Storage and Application (AESA) Group of the National Engineering Laboratory for Electric Vehicles, Beijing Institute of Technology. Any opinions expressed in this paper are solely those of the authors and do not represent those of the sponsor. References [1] Sun FC, Xiong R, He HW. A systematic state-of-charge estimation framework for multi-cell battery pack in electric vehicles using bias correction technique. Appl Energy 2016;162:1399–409. [2] Subburaj Anitha S, Pushpakaran Bejoy N, Bayne Stephen B. Overview of grid connected renewable energy based battery projects in USA. Renew Sustain Energy Rev 2015;45:219–34. [3] Sun F, Xiong R. A novel dual-scale cell state-of-charge estimation approach for series-connected battery pack used in electric vehicles. J Power Sour 2015;274:582–94. [4] Yang FF, Xing YJ, Wang D, Kwok-Leung Tsui. A comparative study of three model-based algorithms for estimating state-of-charge of lithium-ion batteries under a new combined dynamic loading profile. Appl Energy 2016;164:387–99. [5] Kim T et al. Model-based condition monitoring for lithium-ion batteries. J Power Sour 2015;295:16–27. [6] Xiong R, Sun F, Chen Z, He H. A data-driven multi-scale extended Kalman filtering based parameter and state estimation approach of lithium-ion polymer battery in electric vehicles. Appl Energy 2014;113:463–76. [7] Duong V, Bastawrous H, Lim K, See K, Zhang P, Dou S. Online state of charge and model parameters estimation of the LiFePO4 battery in electric vehicles using multiple adaptive forgetting factors recursive least squares. J Power Sour 2015;296:215–24. [8] Xing YJ, He W, Pecht M, Tsui KL. State of charge estimation of lithium-ion batteries using the open-circuit voltage at various ambient temperatures. Appl Energy 2014;113:106–15. [9] Weng CH, Sun J, Peng H. A unified open-circuit-voltage model of lithium-ion batteries for state-of-charge estimation and state-of-health monitoring. J Power Sour 2014;258:228–37.

Please cite this article in press as: Lin C et al. Multi-model probabilities based state fusion estimation method of lithium-ion battery for electric vehicles: State-of-energy. Appl Energy (2016), http://dx.doi.org/10.1016/j.apenergy.2016.05.065

C. Lin et al. / Applied Energy xxx (2016) xxx–xxx [10] Ng KS, Moo CS, Chen YP, Hsieh YC. Enhanced coulomb counting method for estimating state-of-charge and state-of-health of lithium-ion batteries. Appl Energy 2009;86:1506–11. [11] Yang NX, Zhang XW. State of charge estimation for pulse discharge of a LiFePO4 battery by a revised Ah counting. Electrochim Acta 2015;151:63–71. [12] Lee Seongjun, Kim Jonghoon. Discrete wavelet transform-based denoising technique for advanced state-of-charge estimator of a lithium-ion battery in electric vehicles. Energy 2015;83:462–73. [13] Peréz G, Garmendia M, Reynaud JF, Crego J, Viscarret U. Enhanced closed loop state of charge estimator for lithium-ion batteries based on Extended Kalman Filter. Appl Energy 2015;155:834–45. [14] He W, Williard N, Chen C, Pecht M. State of charge estimation for electric vehicle batteries using unscented Kalman filtering. Microelectron Reliab 2013;53:840–7. [15] Tian Y, Xia B, Sun W, Xu Z, Zheng W. A modified model based state of charge estimation of power lithium-ion batteries using unscented Kalman filter. J Power Sour 2014;270:619–26. [16] Zhang WG, Wei Shi, Zeyu Ma. Adaptive unscented Kalman filter based state of energy and power capability estimation approach for lithium-ion battery. J Power Sour 2015;289:50–62. [17] Xiong R, Sun F, Gong X, He H. Adaptive state of charge estimator for lithiumion cells series battery pack in electric vehicles. J Power Sour 2013;242:699–713. [18] Liu XT, Chen ZH, Zhang CB, Wu J. A novel temperature-compensated model for power Li-ion batteries with dual particle filter state of charge estimation. Appl Energy 2014;123:263–72. [19] Xu J, Mi C, Cao B, Deng J, Chen Z, Li S. The state of charge estimation of lithiumion batteries based on a proportional-integral observer. IEEE Trans Veh Technol 2014;63:1614–21. [20] Zhong FL, Li H, Zhong SM, Zhong QS, Yin C. An SOC estimation approach based on adaptive sliding mode observer and fractional order equivalent circuit model for lithium-ion batteries. Commun Nonlinear Sci Numer Simulat 2015;24:127–44. [21] Chen X, Shen W, Cao Z, Kapoor A. Adaptive gain sliding mode observer for state of charge estimation based on combined battery equivalent circuit model. Comput Chem Eng 2014;64:114–23. [22] Barillas JK, Li JH, Gunther C, Danzer MA. A comparative study and validation of state estimation algorithms for Li-ion batteries in battery management systems. Appl Energy 2015;155:455–62. [23] Hu JN et al. State-of-charge estimation for battery management system using optimized support vector machine for regression. J Power Sour 2014;269:682–93. [24] Dai H, Guo P, Wei X, Sun Z, Wang J. ANFIS (adaptive neuro-fuzzy inference system) based online SOC (state of charge) correction considering cell

[25] [26] [27]

[28]

[29]

[30] [31] [32]

[33]

[34]

[35] [36] [37]

[38] [39]

[40]

9

divergence for the EV (electric vehicle) traction batteries. Energy 2015;80:350–60. Anton J, Nieto P, Viejo C, Vilan J. Support vector machines used to estimate the battery state of charge. IEEE Trans Power Electron 2013;28(12):5919–26. Sheng H, Xiao J. Electric vehicle state of charge estimation: nonlinear correlation and fuzzy support vector machine. J Power Sour 2015;281:131–7. Farmann A, Waag W, Marongiu A, Sauer DU. Critical review of on-board capacity estimation techniques for lithium-ion batteries in electric and hybrid electric vehicles. J Power Sour 2015;281:114–30. Barai A, Uddin K, Widanalage WD, McGordon A, Jennings P. The effect of average cycling current on total energy of lithium-ion batteries for electric vehicles. J Power Sour 2016;303:81–5. Liu XT, Zhang CB, Chen ZH. A method for state of energy estimation of lithiumion batteries at dynamic current and temperatures. J Power Sour 2014;270:151–7. Wang YJ, Zhang CB, Chen ZH. A method for joint estimation of state-of-charge and available energy of LiFePO4 batteries. Appl Energy 2014;135:81–7. He HW, Zhang YZ, Xiong R, Wang C. A novel Gaussian model based battery state estimation approach: state-of-energy. Appl Energy 2015;151:41–8. Dong GZ, Chen ZH, Wei JW, Zhang CB, Wang P. An online model-based method for state of energy estimation of lithium-ion batteries using dual filters. J Power Sour 2016;301:277–86. Wang YJ, Zhang CB, Chen ZH. An adaptive remaining energy prediction approach for lithium-ion batteries in electric vehicles. J Power Sour 2016;305:80–8. Lin C, Mu H, Xiong R, Shen WS. A novel multi-model probability battery state of charge estimation approach for electric vehicles using H-infinity algorithm. Appl Energy 2016;166:76–83. Zou Y, Li S, Shao B, Wang BJ. State-space model with non-integer order derivatives for lithium-ion battery. Appl Energy 2016;161:330–6. Zhou K, Doyle J. Essentials of robust control. Englewood Cliffs (NJ): PrenticeHall; 1998. Duan X, Li Yan, Kai Xu, Hui Jiang, Hanxu Sun. Open-circuit voltage-based state of charge estimation of lithium-ion battery using dual neural network fusion battery model. Electrochim Acta 2016;188:356–66. Fang HZ et al. Improved adaptive state-of-charge estimation for batteries using a multi-model approach. J Power Sour 2014;254:258–67. Avvari GV, Balasingam B, Pattipati KR, Bar-Shalom Y. A battery chemistryadaptive fuel gauge using probabilistic data association. J Power Sour 2015;273:185–95. Simon D. Kalman filter generalizations. In: Optimal state estimation: Kalman, and nonlinear approaches. Hoboken (NJ, USA): John Wiley & Sons, Inc.; 2006.

Please cite this article in press as: Lin C et al. Multi-model probabilities based state fusion estimation method of lithium-ion battery for electric vehicles: State-of-energy. Appl Energy (2016), http://dx.doi.org/10.1016/j.apenergy.2016.05.065