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

Supervisory long-term prediction of state of available power for lithium-ion batteries in electric vehicles

T

⁎

Lin Yanga, , Yishan Caia,b, Yixin Yangc, Zhongwei Denga a

School of Mechanical Engineering, Shanghai Jiao Tong University, Shanghai 200240, China Shanghai Jieneng Automobile Technology Co., Ltd. of SAIC Motor, Shanghai 201804, China c School of Electronic and Information Engineering, Xi’an Jiao Tong University, Xi’an 710049, China b

H I GH L IG H T S

model incorporated with dynamic open circuit voltage is established. • AThebattery adaptive two step ﬁlter is introduced into battery state estimation. • A novel supervisory long-term battery SOAP prediction approach is put forward. • The robustness the proposed approaches is systematically evaluated. • The experimentofresults verify the long-term SOAP prediction error reduced by 85.9%. •

A R T I C LE I N FO

A B S T R A C T

Keywords: Lithium–ion battery State of available power prediction Parameter identiﬁcation State estimation Battery management system

The battery state of available power (SOAP) is crucial to improve the energy management of electric vehicles (EVs) and protect batteries from damage. This paper proposes a novel supervisory long-term prediction scheme of SOAP for lithium-ion batteries in electric vehicles. The supervisory long-term prediction denotes that the SOAP is online predicted under the supervision of the EV’s future long-term driving conditions, instead of the traditional approaches under the constant working limitations. Firstly, to accurately capture the battery dynamics, a battery model incorporated with multi-parameters dynamic open circuit voltage is established, and the least square approach with an adaptive forgetting factor is applied to online identify the battery parameters. A new battery state estimation algorithm based on an adaptive two step ﬁlter is then proposed to improve the accuracy of the state estimation. A battery’s long-term power demand (LTPD) prediction model is also established for EVs. Based on the improved battery model and predicted battery states, especially under the supervision of the predicted LTPD, the novel supervisory long-term battery SOAP prediction approach is ﬁnally put forward to make the prediction practical and accurate. The long-term state of charge (SOC) and SOAP of battery are online co-predicted by the derived algorithms. The robustness of the proposed approach against erroneous initial values, diﬀerent battery aging levels and ambient temperatures is systematically evaluated by experiments. The experimental results verify the long-term battery SOAP prediction error reduced by 85.9% when compared with that by traditional approaches.

1. Introduction To deal with the growing concern over oil shortage and environmental issues, electric vehicles (EVs) are becoming increasingly popular in the global market [1]. As the core power source of an EV, the lithiumion battery needs to be well monitored by the battery management system (BMS). The accurate estimation and prediction of the battery states, including the state of charge (SOC) [2], state of health (SOH) [3]

⁎

and state of available power (SOAP) [2] etc., is one of the most primary functions of the BMS. Among these states, the battery SOAP represents the available charging or discharging power capability of the batteries [2]. For EV applications, it is used as a boundary by the vehicle control unit for limiting the battery operation. So it not only is essential in determining the performance of an EV such as the maximum acceleration, regenerating braking and gradient climbing [4], but also is the key to

Corresponding author. E-mail address: [email protected] (L. Yang).

https://doi.org/10.1016/j.apenergy.2019.114006 Received 5 August 2019; Received in revised form 2 October 2019; Accepted 14 October 2019 0306-2619/ © 2019 Elsevier Ltd. All rights reserved.

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Nomenclature AEKF AFFLS ATSF AUKF BMS DEKF ECM EKF EMS EV KF LS

LTPD MAE ME OCV RBF-NN RC RELS RLS SOAP SOC UKF UDDS WRLS WVUSUB

adaptive extended Kalman ﬁlter least square approach with adaptive forgetting factor adaptive two step ﬁlter adaptive unscented Kalman ﬁlter battery management system dual extended Kalman ﬁlter equivalent circuit model extended Kalman ﬁlter energy management strategy electric vehicle Kalman ﬁlter least squares

long-term power demand mean absolute error maximum error open circuit voltage radial basis function neural network resistance and capacitance recursive extended least squares recursive least squares state of available power state of charge unscented Kalman ﬁlter urban dynamometer driving schedule weighted recursive least squares west Virginia suburban driving schedule

variations. X. Zhao et al. [11] established a dual-polarization-resistance model to make the ECM robust under diﬀerent current load, and then applied an extended Kalman particle ﬁlter to estimate the battery SOC. L. Pei et al. [19] proposed a dual EKF (DEKF) to directly estimate the ECM parameters and battery SOC. Z. Deng et al. [20] developed a dual adaptive EKF (AEKF) to estimate the battery parameters and SOC against varying degradations. The battery parameters and / or states can be estimated under diﬀerent temperatures and aging conditions. However, the EKF algorithm or its variants perform a ﬁrst-order Taylor expansion of the battery state-space model to approximate the nonlinear characteristic of lithium-ion battery. So the linearization errors in the EKF-based approaches are inevitable, which may harm the accuracy of battery parameters identiﬁcation and states estimation. To solve this problem, the approaches based on unscented Kalman ﬁlter (UKF) [21] or adaptive UKF (AUKF) [22] were proposed. However, the robustness of UKF/AUKF might decrease with uncertain distribution [23]. Therefore, for accuracy improvement of battery parameters identiﬁcation and states estimation in real applications, the contradiction between the linearization errors and robustness in these existing approaches needs to be settled. Compared with KF-based approaches, the LS-based approach can avoid the complex matrix operations such as inversions. It can be implemented on a low cost microcontroller. Therefore, LS-based approaches are commonly used in BMS applications for battery parameter identiﬁcation, especially the recursive least squares (RLS)-based approaches due to their no requirement of storage of a signiﬁcant amount of data. S. Wang et al. [24] applied a weighted recursive least squares (WRLS)-based approach to regress the R-RC circuit ECM model parameters. T. Feng et al. [25] proposed a novel ECM by adding a moving average noise to the one RC circuit model, and applied the recursive extended least squares (RELS)-based approach to online identify the model parameters. The battery parameters can be eﬃciently identiﬁed online by RLS-based approaches. However, for accuracy improvement, two issues should be solved. Firstly, the dynamic characteristics of battery open circuit voltage (OCV) are not incorporated in these ECMs. So the hysteresis eﬀects of battery are unable to be well reﬂected, which may harm the ECM accuracy in real applications. Moreover, the impedance characteristic of the battery depends signiﬁcantly on the battery current rate [26]. But the RLS approach does not consider this current-dependence [27]. As a result, the accuracy of battery parameter identiﬁcation might decrease in diﬀerent current in the real working conditions. The second signiﬁcant challenge of battery SOAP prediction is how to realize the accurate long-term SOAP prediction. For safe and durable operation of battery and EMSs of EVs, researchers have studied on the long-term SOAP prediction. R. Xiong et al. [4] used AEKF-based approach to jointly estimate the battery SOC and SOAP, and realized a long-term SOAP prediction. S. Wang et al. [24] introduced a timevarying charge-transfer resistance to describe the diﬀusion eﬀect,

protect the battery from damage for safety and service life [5]. Thus, an accurate prediction of battery SOAP is crucial for battery and vehicle management. Furthermore, many researchers have pointed out that the predictive energy management strategy (EMS) can remarkably reduce the energy consumption of various EVs (for example, of the pure electric vehicles [6], fuel cell hybrid electric vehicles [7] and hybrid electric vehicles [8]). To improve energy eﬃciency or to minimize energy consumption of an EV, the future available power of the battery is required. Accordingly, an accurate online long-term prediction of battery SOAP is becoming more and more essential to BMS. 1.1. Literature review The SOAP prediction approaches used in BMS can be divided into characteristic-maps (CMs)-based approaches and equivalent-circuitmodel (ECM)-based approaches [9]. The CMs-based approach uses the relationship between the battery parameters and the battery SOAP. The ECM-based approach uses the parameters of battery ECM to calculate the battery SOAP. There are two signiﬁcant challenges of battery SOAP prediction. The ﬁrst one is how to accurately obtain the battery parameters and states online, since the accurate prediction of battery SOAP strongly depends on them [9]. However, the internal process of battery is time-variable, nonlinear and unmeasurable in ﬁeld; and the values of battery parameters and states will vary with such random working conditions as driving loads and operating environment of battery [10]. Accordingly, an adaptive approach, which can identify the battery parameters and estimate the battery states adaptively in real-time operation, is useful in obtaining the battery parameters and states. And many approaches have been proposed in recent research. The battery parameters and states can be estimated based on a model with certain ﬁltering or data-driven algorithms. Exemplary approaches include the Kalman ﬁlter (KF)-based approaches, particle ﬁlter-based approaches [11], H∞ ﬁlter-based approaches [12], sliding mode observer-based approaches [13], the least squares (LS)-based approaches, other adaptive ﬁlters and observer-based approaches [14], and the fuzzy logic [15] and machine-learning-based approaches [16], etc. C. Burgos-Mellado et al. [15] proposed a fuzzy battery model, and used the particle ﬁltering algorithm to online estimate the battery SOC. E. Chemali et al. [16] used a recurrent neural network (RNN) with long short-term memory (LSTM) to estimate battery SOC. These data-driven approaches do not require a very in-depth understanding of the battery, but suﬃcient and rich data from prior tests must be collected to train the estimation model, which are the main weaknesses of these methods [17]. Therefore, many researchers have focused on the ﬁlter-based approaches. The ECM and / or electrochemical model [18] can be employed in these approaches. And the extended Kalman ﬁlter (EKF)based approach is the most widely studied ﬁlter-based approach in literature. To improve the accuracy, it can be used in diﬀerent 2

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approach cannot.

which helps improve the accuracy of SOAP prediction. However, the above researches only consider the voltage limitation when calculating the battery SOAP, but omit the restriction of battery current. So, T. Feng et al. [25] considered both the voltage and current restrictions of the battery to predict the long-term battery SOAP by iteration. A. Farmann et al. [9] have made a comprehensive review of on-board SOAP prediction techniques for lithium-ion batteries in EVs. However, most of the previous algorithms treat short-term prediction and long-term prediction equally, which does not make full use of the updated information. And in all the previous long-term SOAP prediction systems, the batteries are assumed to always work under their extreme conditions for all the future time. The battery states are also assumed to remain unchanged, or to change always under current and voltage limits of battery for all the future time. In fact, these assumptions do not meet the actual working conditions of EVs, because the EV operation is actually dynamic. Hence, it is sure that the batteries will not always work under their extreme conditions. And the battery states will also always vary along the future real dynamic operation of the EV, rather than remain unchanged, or change always under the current and voltage limits of battery. Therefore, the long-term battery SOAP predicted by the traditional approaches is very impractical and inconsistent with its actual value as will be shown later in this paper.

1.3. Organization of this paper The remainder of this paper is organized as follows. Section 2 proposes the new concept of the supervisory long-term battery SOAP, and presents the approaches for the battery SOAP prediction, including the battery model and parameter identiﬁcation, the ATSF-based battery state estimation approach and the supervisory long-term battery SOAP prediction approach. Section 3 describes the battery test bench and the test schedules for experimental veriﬁcation of the proposed approaches. Section 4 demonstrates the veriﬁcation results. Conclusions are drawn in Section 5. 2. The proposed approach for supervisory long-term SOAP prediction 2.1. The supervisory long-term battery SOAP As mentioned in Section 1, the battery SOAP represents the available peak power of the battery during charging or discharging process, it strongly depends on the battery states and parameters [9]. The vehicle power demand is actually dynamic. Hence, it is impractical to predict the long-term (e.g., ≥10 s) battery SOAP based on the assumption that the batteries always work under their extreme conditions (i.e., always at their maximum available peak power), like the traditional approaches such as those in papers [4,5,9,24,25] etc.. It is also impractical to predict the long-term battery SOAP based on unvaried battery states, or based on battery states varied always under the voltage and current limits. We ﬁnd that they are the essence of the huge deviations between the long-term SOAPs predicted by traditional approaches and their actual values. Thus, we propose a new concept, the supervisory long-term battery SOAP, to predict the battery SOAP based on the actual power demand of vehicle in the future. The supervisory long-term SOAP is predicted under the supervision of the long-term power demand (LTPD) of battery over a future longterm time horizon, and based on the online estimated and updated battery parameters and states. In this way: (1) unlike the traditional approaches for long-term SOAP prediction, the battery states will be updated according to the predicted battery LTPD at each prediction time step; (2) unlike the traditional approaches which assume that batteries always work in their extreme conditions, battery power demand will be online predicted to make batteries always work at their future actual working conditions at each prediction time step. It is expected that by realizing this supervisory long-term SOAP prediction, the drawbacks of the traditional long-term SOAP prediction approaches can be eﬀectively overcome, making the predicted results practical and accurate. Because the SOAP prediction needs to be based on battery model, parameters and states, the battery model and parameter identiﬁcation are ﬁrstly improved in Section 2.2. Then, a new battery state estimation approach is proposed in Section 2.3 to improve the state estimation accuracy. The novel supervisory long-term battery SOAP prediction approach is put forward in Section 2.4, including the prediction scheme and algorithms.

1.2. Contributions of the work Focusing on those problems, we devoted our attention to the proposal of a novel supervisory long-term prediction system to make the SOAP prediction (especially, the long-term SOAP prediction) practical and accurate for lithium-ion batteries in EVs. And the robustness and accuracy of the proposed approach against erroneous initial values, diﬀerent battery aging levels and ambient temperatures are systematically evaluated by experiments. Three main contributions have been made in this paper: (1) A battery model incorporated with multi-parameters dynamic OCV is established, and the least square approach with adaptive forgetting factor (AFFLS) is employed to identify the model parameters. The polarization and hysteresis eﬀects in the charging or discharging process are eﬀectively reﬂected in the model. The accuracy improvement of the established model is veriﬁed by experiment. And the model parameters can be accurately updated in diﬀerent working conditions of battery. (2) A new battery state estimation algorithm based on adaptive two step ﬁlter (ATSF) is proposed. This is a new model-based adaptive approach for battery state estimation, into which the ATSF is ﬁrst introduced. It is diﬀerent from the commonly used EKF-based or UKF-based approaches. It can avoid linearizing the cost function, and concurrently provide adaptability to noise. The experiments on batteries under diﬀerent working conditions and aging levels prove that: the maximum error of state estimation can be remarkably reduced by more than 45% on average, even when compared with those by the advanced AEKF approach and AUKF approach. (3) A new concept, the supervisory long-term battery SOAP, is proposed. And a novel supervisory long-term battery SOAP prediction approach is put forward. This is a new idea to predict the long-term SOAP under the supervision of the predicted vehicle’s future power demand, instead of the traditional approaches under the constant working limitations and conditions. It is based on the improved battery model and predicted battery states, especially under the supervision of the battery’s future long-term power demand predicted by a model. The battery long-term SOC and SOAP are ﬁrst co-predicted online by the derived algorithms. It is proved by experiments that: the error of the long-term SOAP prediction can be signiﬁcantly reduced by 85.9% compared with that by the traditional approach; more importantly, the proposed approach can make the predicted long-term SOAP practical, while the traditional

2.2. The battery model and parameter identiﬁcation 2.2.1. The improved equivalent-circuit model For battery SOAP prediction, the approaches based on an ECM are suitable for online application [28,29 14]. Usually, with a larger number of RC networks, the battery ECM is more accurate, but the computational complexity is higher [18]. To improve the model accuracy and retain low computational complexity, an improved ﬁrst-order RC model with dynamic OCV characteristics is established, as shown in Fig. 1. A controllable voltage source Uoc is introduced to represent the 3

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2.2.2. Adaptive model-parameter identiﬁcation Fig. 1 illustrates the improved ECM. In this Section, a solution is presented to online identify the improved ECM parameters. To overcome the drawback of the widely used approach in identiﬁcation of battery parameters based on RLS (which does not consider the current dependence of the battery parameters [27], so the identiﬁcation accuracy might decrease in diﬀerent current), we note that the AFFLS approach [33] can improve the identiﬁcation accuracy by using a current-dependent forgetting factor for second-order ECM. Here, we adopt AFFLS to online identify the battery parameters of our improved ﬁrst-order RC model. The battery RC model in Fig. 1 can be discretized using a linear discrete form as follows:

Fig. 1. The schematic diagram of the improved RC model with dynamic OCV.

Uoc = Uoc, CB (SOC , T , ς ) ψ + Uoc, DB (SOC , T , ς )(1 − ψ)

Ut , k = Uoc, k − Up, k − Ik R 0

(4)

(

3.1

0.1

0.2

0.3

Uoc,DB

40

Uoc,DB

0.4

0.5

25 40 0.6

0.7

)

where we add an extra item ek to account for the error generated in the approximate process. In order to improve the ability of anti-noise interference and parameter tracking, a forgetting factor needs to decrease with the increase of battery current. The adaptive forgetting factor λk at time step k can be deﬁned as

λk = 1 −

|Ik | ζ Cn

(7)

where Cn is the capacity of the battery, ζ is used to control the average value of λk . Note that: a smaller ζ can make the AFFLS more adaptive to

(2)

Uoc,CB Uoc,CB

3.25 3.2

0

3.15

Uoc,CB 0.8

(6)

(

3.25

25

(5)

Ek = Uoc, k − Ut , k ⎧ ⎪ φk = [Ek − 1, Ik − 1, Ik , 1] T ⎨ Δt Δt Δt ⎪ θk = [a1, a2 , a3, ek ]T = ⎡e− τ , 1 − e− τ Rp − e− τ R 0, R 0, ek ⎤ ⎣ ⎦ ⎩

3.3

3.15

)

)

where

3.3

0

Δt

Ek = φk θk

3.35

Uoc,DB

(

Δt

Because the sampling interval is very short (< 0.1 s), the OCV can be assumed not to change during a sampling interval [25]. Accordingly, we can get Uoc, k ≈ Uoc, k − 1. In order to identify the parameters of the battery model with AFFLS method, we rewrite Eq. (5) as

OCV / V

OCV / V

(3)

3.35

0

)

Ut , k = Uoc, k − e− τ Up, k − 1 + 1 − e− τ Rp Ik − 1 − Ik R 0

where Uoc, CB and Uoc, DB are the OCV boundaries of the charge step and of the discharge step respectively, as shown in Fig. 2 taking as an example the OCV boundaries of a LiFePO4 battery measured in this paper. In previous studies, the battery aging factor ς could be deﬁned using battery capacity, internal resistance, or remaining battery life. The capacity, i.e., the maximum available charge of a battery, can be accurately determined with a battery tester via the Coulomb count. Therefore, we use it to represent ς , which can be identiﬁed based on fast wavelet transform and cross D-Markov machine. We have described this identiﬁcation algorithm in our another paper [31], so do not repeat it here. ψ is the hysteresis factor, which represents the close degree of the real OCV to the OCV boundaries under a charge step, and can be calculated by the approach proposed by M.A. Roscher et al. [32]. According to Eq. (1), the derivative of the dynamic OCV about battery SOC can be expressed as

3.2

Δt

Up, k = e− τ Up, k − 1 + 1 − e− τ Ik − 1 Rp

where Δt is the sampling interval, k is the discrete time index, τ = Rp Cp is the time constant of the parallel RC network, and Uoc, k , Up, k , Ik and Ut , k are the discretized values of Uoc , Up , I and Ut at time step k respectively. We ﬁrstly need to eliminate the polarization voltage Up, k from Eq. (4). By replacing Up, k in Eq. (4) with Eq. (3), Ut , k can be rewritten as

(1)

dUoc, DB (SOC , T , ς ) dUoc, CB (SOC , T , ς ) dUoc (1 − ψ) = ψ+ dSOC dSOC dSOC

(

Δt

dynamic OCV of the battery to improve the accuracy of the model,and R 0 is employed as the ohm internal resistance of the battery. The RC network is used to describe the charge transfer reaction of the battery. Here, Rp is the polarization resistance, Cp is the polarization capacitance and Up is the voltage of the RC network. Ut is the terminal voltage of battery. I is the battery current, which is deﬁned as negative for charging and positive for discharging in this paper. The OCV after a charge step is higher than the OCV after a discharge step, there is a hysteresis loop caused by the cycling history of the lithium-ion battery. G. Pérez et al. [30] built a dynamic OCV model with a hysteresis factor. While the OCV is also a function of the battery SOC, temperature and aging state. Hence, to improve the accuracy of the model in Fig. 1, we establish a multi-parameters dynamic OCV model, and describe it as a function of the battery SOC, temperature T, aging factor ς and hysteresis factor ψ:

3.1

0.9

SOC

0.1

0.2

0.3

0

Uoc,DB

Uoc,CB

25

Uoc,DB

25

Uoc,CB

40

Uoc,DB

40

Uoc,CB

0.7

0.8

0.4

0.5

0.6

0.9

SOC

(a)

(b)

Fig. 2. The measured OCV boundaries of a LiFePO4 battery under diﬀerent temperatures at diﬀerent aging levels: (a) for battery aging factor ς = 1; (b) for battery aging factor ς = 0.8. 4

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noise interference when the current rate |Ik | Cn is small, while a bigger ζ can improve the parameter tracking ability of AFFLS when the current rate |Ik | Cn is big. In order to improve the accuracy of parameter identiﬁcation in the whole current range of battery, we choose ζ as 30 according to the analyzed data and calibration. Once θ has been identiﬁed by AFFLS, the battery parameters can be obtained by

a a + a2 −Δt ⎤ [R 0 , Rp , Cp] = ⎡a3, 1 3 , ⎢ 1 − a1 Rp ln(a1) ⎥ ⎣ ⎦

the battery state estimation problem is to achieve an optimal solution of ∂J = 0 , where J deﬁned in Eq. (14) is the cost function [37]. ∂x

J=

1 2

N

∑ [zk − F (xk) − Dk uk ]T R−1[zk − F (xk) − Dk uk]

(14)

k=1

where N is the sum of the time steps. 2.3.2. ATSF-based estimation of battery states To solve this nonlinear optimization problem in the above Eqs. (10)–(14), we introduce the ATSF approach to estimate the state vector x k in this paper. By splitting the cost function minimization by deﬁning a set of new states, the approximation can be moved out of the ﬁlter measurement update. It is implemented in two steps: Step I, ﬁrstly deﬁne a new state vector yk as

(8)

According to Eq. (5), Eq. (6) and Eq. (7), it can be seen that: the AFFLS is applied to the improved ﬁrst-order ECM; the adaptive forgetting factor can be time-varying adaptively with the battery current; and the error from the approximate process of the model can also be reduced by adding the extra item ek . Thus, this approach can online identify and update the battery parameters with high accuracy at different working conditions. The detail process of AFFLS method can be found in the paper by S. Schwunk et al. [33].

Then, Eq. (14) can be rewritten as a linear measurement cost function, given by

2.3. The ATSF-based battery state estimation

Jy =

yk = f (x k) = [Uoc, k , Up, k ]T

To predict the battery SOAP, the battery states should be accurately estimated, which include the battery SOC and its polarization voltage Up in Fig. 1. Currently, numerous studies have used EKF/AEKF to estimate battery SOC. However, the linearization errors of EKF/AEKF have an impact on the estimation accuracy [34]. So, the UKF/AUKF are studied for the estimation accuracy improvement [21,22]. But the robustness of UKF/AUKF decreases with uncertain distribution [23]. We note that the adaptive two step ﬁlter (ATSF) was used as an integrated approach of a two-step optimal estimator by N.J. Kasdin et al. [35], and was used as a measurement noise statistical estimator with application to bearings by D. Zhou et al. [36]; it can avoid linearizing the cost function and provide adaptability to noise. Thus, in this section, we ﬁrst introduce the ATSF trying to solve these problems. And a novel battery state estimation approach based on ATSF is proposed to improve the estimation accuracy of the battery states.

∫t

t

ηI (t ) dt

0

N

∑ [zk − Hk yk − Dk uk ]T R−1[zk − Hk yk − Dk uk]

Jx =

1 [y ̂ − f (x k)]T P−y 1 [yk̂ − f (x k)] 2 k

1) The ﬁrst-step measurement update:

k]−1 k = 1, ⋯, N K = M yk HTk [Hk M yk HTk + R

(18)

yk̂ = yk + K (zk − Hk yk − Dk uk − rk̂ )

(19)

Pyk = [I − KHk ] M yk

(20)

(9) where M yk = E [(yk − yk)(yk − yk)T ], Pyk = E [(yk − yk̂ )(yk − yk̂ )T ], yk is the predicted vector of the state yk , the measurement noise is estimated by

x k + 1 = Φk x k + Bk uk + Γk wk

(10)

rk̂ = (1 − dk ) rk̂ − 1 + dk (zk − Hk yk̂ − Dk uk)

zk = F (x k) + Dk uk + vk

(11)

k = (1 − dk ) R k − 1 + dk [(I − R

where x k = [ SOCk Up, k ]T is the state vector at the time index k , uk = Ik is the system input, wk = [ w1, k w2, k ] is the noise vector that describes the estimation uncertainty of the states, zk = Ut , k is the output of the system, and vk models the “sensor noise” that aﬀects the measurement of the output. The coeﬃcients in Eq. (10) and Eq. (11) are

)

(21)

− Hk Kk

)T

+

Hk Pyk HTk (22)

(12)

2) The ﬁrst-step time update:

and the nonlinear function F (x k) is deﬁned as follows:

F (x k) = Uoc, k − Up, k

Hk Kk) εk εkT (I

where dk is a time dependent forgetting factor which is deﬁned as dk = (1 − b) (1 − bk + 1), 0 < b < 1, εk = zk − Hyk̂ − Dk and −k −1 1. Note that: a smaller b makes the estimator uk − rk̂ , Kk = Pyk HTk R more adaptive to a fast time-varying noise, but is also prone to induce a ﬁltering divergence. In order to ensure a stable ﬁltering, we choose b as 0.96.

ηΔt

(

(17)

where Py is the state estimate error covariance from step I. A GaussNewton iterative algorithm can be applied to get the state estimation x k̂ . Therefore, the cost function (Eq. (16)) can be accurately minimized. That is to say, without linearizing the cost function and with adaptability to noise, the battery state estimation can be the optimal result of the original problem by introducing the ATSF. The detailed process of this estimation can be summarized in the four steps:

where t0 is the initial time, and η is the current eﬃciency during the charging and discharging process. According to Eq. (3), Eq. (4) and Eq. (9), the battery model system can be described as the following state- space model:

− C ⎡ ⎤ 1 0 ⎤ n ⎢ ⎥, Γk = I2 × 2 , Dk = −R 0 = Φk = ⎡ B , k Δ t Δt −τ ⎥ ⎢ − ⎢ e 0 ⎦ ⎣ 1 − e τ Rp ⎥ ⎣ ⎦

(16)

k=1

where F (x k) = Hk yk , Hk = [1 − 1]. For the linear optimization problem, the KF can be used to obtain the optimal state vector estimate yk̂ . The measurement noise mean rk and covariance R k , which are prior unknown time-varying, can be adaptively updated by a Sage-Husa estimator [36]. Step II, solve the desired state estimate x k̂ by the cost function

2.3.1. The state-space model and cost function The battery SOC is deﬁned as the ratio of the remaining battery capacity to the maximum available capacity, which can be calculated as follows:

1 SOC (t ) = SOC (t0) − Cn

1 2

(15)

yk + 1 ≈ yk̂ + f (x k + 1) − f (x k̂ ) k = 1, ⋯, N − 1 (13)

According to the principle of least square approach, the objective of 5

(23)

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L. Yang, et al.

M yk + 1 ≈ Pyk +

∂f ∂x k + 1

∂f Pxk ∂x k

M xk + 1 xk + 1= xk + 1

∂f ∂x k + 1

T

− xk + 1= xk + 1

∂f ∂x k

linearity for lithium-ion battery. The ΔSOCk, i in the iteration step Δx k̂ , i = [ ΔSOCk, i ΔUp, k, i ]T may sometimes be too large, making the algorithm unstable and even the results to diverge. So, to improve this algorithm, we give a step constraint ΔSOCmax on the ﬁrst element of Δx k̂ , i which is deﬁned as

xk = xk

T

(24)

xk = xk

where M xk + 1 is the covariance of the predicted state x k + 1.

ΔSOCk, i |ΔSOCk, i| ⩽ ΔSOCmax Δx k̂ , i (0) = ⎧ (Δ sign SOC ⎨ k , i ) × ΔSOCmax |ΔSOCk , i | > ΔSOCmax ⎩

3) The second-step measurement update:

LGk, i =

∂f ∂x k

T xk = xk, i

P−yk1

∂f ∂x k

(25)

Δx k̂ , i = −L−Gk1, i qTk, i

(26)

x k̂ , i + 1 = x k̂ , i + Δx k̂ , i

(27)

q k, i = −[yk̂ − f (x k̂ , i)]T P−yk1

Pxk =

where sign (ΔSOCk, i ) represents to be positive for ΔSOCk, i ⩾ 0 and negative for ΔSOCk, i < 0 . And because the range of SOC is only 0 ~ 1, ΔSOCmax can be set to 0.1. The ﬂow chart of battery states estimation algorithm based on ATSF is shown in Fig. 3.

k = 1, ⋯, N xk = xk, i

(32)

2.4. The supervisory long-term SOAP prediction

∂f ∂x k

(28)

xk = xk, i

L−Gk1, i

The batteries in an EV do not always work in their extreme conditions, and the battery states will vary with the working loads. In the long-term prediction of battery SOAP, the battery states are the crucial factors. Knowledge of battery power demand in the future can be a reference for battery states prediction in long-term predictive time window. Therefore, battery LTPD prediction model for EVs is established. Supervisory long-term prediction algorithm of battery states is derived to accurately obtain the battery states in the future based on the predicted battery LTPD. And supervisory SOAP prediction approach, which takes the predicted long-term battery states as a reference, is put forward to make the long-term SOAP prediction practical and improve its accuracy remarkably.

(29)

where i is the iterative step, Eq. (27) is iterated until Δx k̂ , i → 0 or reaches the maximum iterative number Nmax . 4) The second-step time update:

x k + 1 = Φk x k̂ + Bk uk k = 1, ⋯, N − 1

(30)

M xk + 1 = Φk Pxk ΦTk + Γk Qk ΓTk

(31)

It should be noted that the Gauss-Newton iterative algorithm can achieve quadratic convergence, while the OCV function has a large non-

Initialization: k 0, y 0 , M y 0 , xˆ 0 , rˆ0 , Rˆ 0 , b , Q , H k

>1

[email protected]

The second-step measurement update: for i 1: N max

The first-step measurement update: K

ˆ ]1 M yk H [H k M yk H R k

yˆ k

y k K (z k H k y k Dk u k rˆk )

Pyk

[I KH k ]M yk

T k

T k

'x k ,i

rˆk (1 d k )rˆk 1 d k (z k H k yˆ k Dk u k ) ˆ ˆ d [(I H K )İ İT (I H K )T R k (1 d k )R k 1 k k k k k k k

wf wx k

M xk 1

Pxk

xk 1 xk +1

Pxk xk xˆk

wf wx k

k

wf wx k 1

xk xˆk ,i

wf wx k

xk xˆk ,i

wf wx k

xk xˆk ,i

LG1k ,i qTk ,i

if 'xk ,i o 0 end end

y k 1 | yˆ k f ( xk 1 ) f (xˆ k ) wf wx k 1

Pyk1

xˆ k ,i 1 xˆ k ,i 'xk ,i

T k

The first-step time update: M yk 1 | Pyk

T

[yˆ k f (xˆ k ,i )]T Pyk1

q k ,i

measurement noise estimation: dk (1 b) / (1 bk 1 )

H k Pyk H

wf wx k

LG k ,i

T

LG1k ,i

The second-step time update: xˆk xˆk ,i

xk 1 xk +1

xk 1

T

M xk 1

xk xˆk

ĭ k xˆ k B k u k ĭ k Pxk ĭ Tk ī k Q k ī Tk

k 1

Fig. 3. The ﬂow chart of battery SOC estimation algorithm based on ATSF. 6

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realize the step (3), step (4) and step (5) is explained in the following subsections.

2.4.1. The proposed scheme of supervisory long-term SOAP prediction Fig. 4 shows the supervisory long-term prediction scheme of battery SOAP proposed in this paper. The main steps can be summarized as follows: (1) Capture the battery dynamics accurately by the improved battery ECM, and online identify the battery parameters by AFFLS, and the battery aging factor by fast wavelet transform and cross D-Markov machine. (2) Estimate the battery states online by the new adaptive estimation algorithm based on ATSF. (3) Predict the battery’s future LTPD by a model established for EVs. (4) Predict the battery’s future long-term states under the supervision of the predicted LTPD and based on the estimated battery states and identiﬁed battery parameters. (5) Predict the long-term battery SOAP (meanwhile, the short-term SOAP) under the voltage and current constraints by applying the identiﬁed battery parameters, especially by taking the predicted battery‘s future long-term states as a reference. At every time step, the ATSF estimates the states using the currently identiﬁed model parameters from the AFFLS while the AFFLS identiﬁes the model parameters using the currently estimated states from the ATSF. The battery aging factor is identiﬁed at the current time step by using the current and voltage series sampled online. The time-rolling iteration is applied to online update the prediction. In this way, the battery parameters and states can be co-updated accurately, and the long-term battery states and SOAP can be co-predicted under the supervision of the battery’s future LTPD. The prediction of long-term SOAP of battery is based on the actual working conditions and states of battery in the future, rather than based on those assumed in the traditional approaches as described in Section 1. Hence, the aforementioned drawbacks of the traditional long-term SOAP prediction approaches can be overcome. For step (1) and step (2), the model and approaches have been presented in Section 2.2 and Section 2.3, respectively. The approach to

2.4.2. Long-term power demand prediction of battery in the future Because the future driving proﬁle of a vehicle is unknown, we need a model to predict the long-term power demand of battery in the future time window. As the power source of an EV, the batteries not only provide power to the electric motor to drive the vehicle, but also receive the power regenerated by the braking energy recovery system, moreover power the electric accessories in an EV [38]. Usually, in a battery system, batteries are connected in series. Hence, considering the inconsistency of batteries in battery system, the power demand of a battery in an EV can be calculated by

Pdem =

1 − sgn(Pem) ⎞ ⎞ Ubatt ⎛ ⎛ 1 1 + sgn(Pem) Pea + ⎜ + ηf ηem ⎟ Pem⎟ Upack ⎜ ηf ηem 2 2 ⎝ ⎠ ⎠ ⎝ (33)

where Pdem is the battery power demand, Upack is the total voltage of the battery system sampled by BMS, Ubatt is the voltage of a battery in the battery system sampled by BMS, ηf is the transmission mechanical efﬁciency of the EV, ηem is the electric motor eﬃciency, Pea is the power of the electrical accessories, sgn(Pem ) represents the sign of Pem (which is 1 when Pem ≥ 0 and −1 when Pem < 0). Here, Pem is the vehicle power demand and can be calculated as follows:

Pem = Ft v ⎧ 1 dv ⎨ where Ft = μmg cos α + mg sin α + 2 ρair CD Af v 2 + δm dt ⎩

(34)

where μ is the rolling resistance coeﬃcient between the tires and road surface, m is the mass of the EV, g is the acceleration of gravity in

Fig. 4. Supervisory long-term prediction scheme of SOAP. 7

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9.8 m·s−2, α is the road grade, CD is the aerodynamic drag coeﬃcient, ρair is the air density in 1.22 kg·m−3, Af is the frontal area of the vehicle, v is the vehicle velocity, and δ is the inertia coeﬃcient. In Eq. (33), the electric motor eﬃciency ηem can be extracted from empirical map ηem = ψm (Tem, wem) , where Tem = Ft r (ηfsgn (Ft ) i f ) and ωem = vi f r are the motor torque and rotation speed, respectively. Here, r is the wheel rolling radius, and i f is the speed ratio of transmission of the EV. The transmission mechanical eﬃciency ηf can be extracted from empirical map ηf = ψf (i f , Ft r , v r ) . From Eq. (33) and Eq. (34), we can see that the power demand of a battery is a function of vehicle velocity, acceleration and road grade. Here, the road grade α in the future can be known based on the onboard satellite navigation system and electronic map information (including the vehicle current location, future route and road grade). And the acceleration can be calculated from the time-velocity sequence. Thus, the sequence of the battery LTPD in the future can be deduced from the predictive vehicle velocity sequence. Intuitively, the future driving behavior is related to the historical velocity proﬁle. So, the future velocity sequence can be predicted from the historical velocity sequence. C. Sun et al. [39] proposed a velocity prediction approach based on a radial basis function neural network (RBF-NN) with an accurate velocity prediction of 60 s. Thus, this paper applies the RBF-NN to predict the vehicle velocity. Compared with the velocity prediction technique [40] based on the vehicle to everything (V2X) and the intelligent transportation system (ITS), no telemetric equipment is required, so the hardware cost can be reduced.

ηΔt

dem dem SOCkdem ⎧ + j = SOCk + j − 1 − Cn Ik + j − 1 ⎪ dem − Δt ⎪ U dem = e− Δτt U dem p, k + j − 1 + 1 − e τ Rp Ik + j − 1 ⎪ p, k + j ⎪ ⎞ dem UOC ⎛SOCkdem ⎪ + j , Tk + j, ςk , ψk + j − U p, k + j ⎝ ⎠ ⎪ ⎪ 2 ⎛ ⎞ dem ⎞ dem − ⎜UOC ⎛SOCkdem + j , Tk + j, ςk , ψk + j − U p, k + j ⎟ − 4R 0 Pk + j ⎨ ⎝ ⎠ ⎝ ⎠ ⎪ Ikdem +j = 2R 0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ where j = 1, 2, …… ⎩

(

⎜

⎟

⎜

⎟

(36)

U pdem ,k+j

SOCkdem +j

is the predicted battery SOC, is the predicted where battery polarization voltage, Ikdem + j is the predicted battery current. They are updated alternatively. The battery temperature Tk + j in the third equation of Eq. (36) can be calculated by Eq. (37) derived by C. Zhang et al. [42]. But With practical EV current rates, the term in Eq. (37) Tk + j − 1 Ikdem + j − 1 (∂UOC ∂T ) , which is the heat generated or consumed because of the reversible entropy change within the battery, is generally negligible [43] compared with dem dem the term Ikdem + j − 1 (Ik + j − 1 R 0 + U p, k + j − 1) . In fact, because the battery in EV is alternately charged and discharged, the net reversible heat eﬀect will be near zero[44], which eﬀect on SOAP will also be near zero. So, we can ignore this term to simplify the calculation of Tk + j . dem dem dem Ikdem + j − 1 (Ik + j − 1 R 0 + U p, k + j − 1) + Tk + j − 1 Ik + j − 1

2.4.3. Supervisory long-term prediction of battery states The battery current is measured by sensor at current time, and the battery states can be estimated by the approach in Section 2.3. However, in the case of future time, the battery current cannot be acquired directly. So, we take the predicted LTPD of battery in the future as a supervisor, to calculate the battery current and states in the future time window. According to the battery power deﬁnition, the battery power demand can be expressed as dem dem Pkdem + j = Ut , k + j Ik + j

)

Tk + j = Tk + j − 1 +

∂UOC ∂T Tk + j − 1

− hA (Tk + j − 1 − Tamb ) mB Ch (37)

where mB is the battery mass, Ch is the battery heat capacity, h is the battery heat transfer coeﬃcient, A is the total surface of battery, and Tamb is the measured ambient temperature of battery. It should be noted that the iteration for the long-term prediction starts at the predictive time step j = 1, so it begins at the current time step k . At the current time step: Ikdem and Tk are measured by sensors, the battery parameters (R 0 , Rp , τ ) and aging factor ςk are identiﬁed by the approach in Section 2.2, and the battery states (SOCkdem and U pdem , k ) are estimated by the approach in Section 2.3. The battery parameters and states have been updated online for the next SOAP prediction. Especially, as the iteration progresses by j = 1, 2, 3, …, the battery states dem (SOCkdem + j and U p, k + j ) will vary according to the forecasted battery power dem demand Pk + j of EV in the future. It is diﬀerent from the traditional approach, which assumes that the battery states remain unchanged or change always under current and voltage limits for all the time in the future, so ensures to make the long-term SOAP prediction practical and remarkably improve the prediction accuracy.

(35)

where k is the current time step, j is the predictive time step in the dem dem future time window from k , Pkdem + j , Ik + j and Ut , k + j are the battery power demand, the terminal voltage and the battery current at time step k + j respectively. In Eq. (35), Pkdem + j has been predicted in Section 2.4.2 (i.e., the value of the (k + j )th point of the predicted battery LTPD sequence). In order to calculate the Ikdem + j , the OCV at this time step is required. So according to Eq. (1), we ﬁrstly need to determine the battery hysteresis factor ψk + j and aging factor ςk + j . Because the time interval from time step k + j − 1 to time step k + j is very short, the OCV can be assumed not to change during a time interval [25]. Accordingly, ψk + j can be calculated using

2.4.4. Supervisory long-term prediction of SOAP According to the original deﬁnition of battery SOAP, the SOAP strongly depends on the battery states. Therefore, we take the battery dem future long-term states (SOCkdem + j , U p, k + j ), which have been predicted above at the current time step k according to the predicted LTPD of battery in the future, as a reference (in other words, as a supervisor) to predict the long-term SOAP. Because one-time step from k + j to k + j + 1 in battery SOAP calculation is very short, the inﬂuence of the change of battery current on SOAP is very small during this one-time step. Accordingly, based on the battery model in Fig. 1, the terminal voltage at time step k + j + 1 for SOAP calculation can be expressed as

the battery current Ikdem + j − 1 by the approach proposed by M.A. Roscher et al. [32]. Moreover, considering the battery in EVs can be used for years [41], the change of its aging state will obviously be very little in the next time window for SOAP prediction. So, we can assume ςk + j equals to ςk during such future time window. Accordingly, according to Eq. (3), Eq. (4), Eq. (9) and Eq. (35), the battery states and current in the future can be calculated as follows:

SOAP SOAP SOAP UtSOAP , k + j + 1 = Uoc, k + j + 1 − U p, k + j + 1 − Ik + j R 0

is the battery current at the where IkSOAP +j culation, UtSOAP , k + j + 1 is the terminal voltage 8

(38) time step k + j for SOAP calunder IkSOAP at the time step +j

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L. Yang, et al. SOAP SOAP at k + j + 1, UocSOAP , k + j + 1 and U p . k + j + 1 are the battery states also under Ik + j the time step k + j + 1 which can be deduced by

According to Eq. (38) and Eq. (39), the battery terminal voltage can be expressed by current IkSOAP as +j

UtSOAP ,k+j+1

SOAP

ηIk + j Δt dem ⎧ SOCkSOAP + j + 1 = SOCk + j − Cn ⎪ dem ⎪ ∂Uoc ,k+j dem SOAP dem ⎪UocSOAP , k + j + 1 ≈ Uoc, k + j + ∂SOC dem (SOCk + j + 1 − SOCk + j ) k+j ⎨ SOAP ⎪ − Δt dem − Δt U pSOAP , k + j + 1 = e τ U p, k + j + 1 − e τ Rp Ik + j ⎪ ⎪ where j = 1, 2, …… ⎩

(

dem Δt ⎛ ηΔt ∂Uoc, k + j ⎞ − Δt dem = Uocdem + 1 − e− τ Rp + R 0 ⎟ IkSOAP , k + j − e τ U p, k + j − ⎜ +j dem C ⎝ n ∂SOCk + j ⎠ (42)

(

)

reaches the maximum value Idch,max at discharging (or the If IkSOAP +j minimum value Icha,min at charging), the battery terminal voltage is limited by the current. Therefore, the battery SOAP under the current constraint can be expressed as

(39)

SOAP where SOCkSOAP at the time step + j + 1 is the battery SOC under Ik + j k + j + 1. It should be noted that by Eq. (38) and Eq. (39), the SOAP at the each future time step k + j + 1 can be calculated always based on the dem battery’s future long-term states (SOCkdem + j , U p, k + j ) at future time step k + j , which have been predicted at the current time step k . This guarantees that the long-term SOAP prediction is always under the supervision of the predicted actual LTPD of battery in the future, rather than under the assumption that batteries always work in their extreme conditions in the traditional approaches. For safe and durable operation, the working current and voltage of lithium-ion batteries must be restricted in a window, and the battery available power will be limited by the minimum value of the two constraints [25]. The following explains how to predict the long-term SOAP based on Eq. (38) and Eq. (39).

I SOAP I ⎧ SOAPdch, k + j + 1 = Ut , k + j + 1 |IkSOAP + j = Idch,max dch,max ⎪ I SOAP SOAPcha, k + j + 1 = Ut , k + j + 1 |IkSOAP I + j = Icha,min cha,min ⎨ ⎪ where j = 0, 1, ⋯, L − 1 ⎩

I SOAPdch ,k+j+1

C. Final battery SOAP and other factors limiting SOAP Finally, under the two constraints, the battery SOAP can be calculated as V I ⎧ SOAPdch, k + j + 1 = min(SOAPdch, k + j + 1, SOAPdch, k + j + 1) ⎪ V I SOAPcha, k + j + 1 = max(SOAPcha, k + j + 1, SOAPcha , k + j + 1) ⎨ ⎪ where j 0, 1, , L 1 = ⋯ − ⎩

According to Eq. (38) and Eq. (39), the battery current can be expressed by terminal voltage UtSOAP , k + j + 1 as dem SOAP − Uocdem , k + j − e τ U p, k + j − Ut , k + j + 1 dem ηΔt ∂Uoc, k + j Cn ∂SOC dem k+j

(

Δt

)

+ 1 − e− τ Rp + R 0

(40)

UtSOAP ,k+j+1

When reaches the minimum value Ut,min at discharging (or the maximum value Ut,max at charging), the battery current is limited by the voltage. Therefore, the battery SOAP under the voltage constraint can be expressed as V SOAP ⎧ SOAPdch, k + j + 1 = Ut ,min Ik + j |UtSOAP , k + j + 1= Ut ,min ⎪ V SOAP SOAPcha, k + j + 1 = Ut ,max Ik + j |UtSOAP , k + j + 1= Ut ,max ⎨ ⎪ = ⋯ − where j 0, 1, , L 1 ⎩

(44)

where SOAPdch, k + j + 1 and SOAPcha, k + j + 1 are the battery SOAP at discharging and the battery SOAP at charging, respectively, for time step k + j + 1. At each current time step k , the short and long-term SOAPs at each of the future time steps up to L are predicted. Besides the current and voltage constraints, there are some other factors to be considered for the correction of the battery SOAP.

Δt

=

(43)

I SOAPcha ,k+j+1

where and are the battery SOAP at discharging and the battery SOAP at charging, respectively, for time step k + j + 1 under the current constraint.

A. Supervisory prediction of SOAP based on voltage constraint

IkSOAP +j

)

1) When the battery temperature is too high, the battery SOAP should be reduced to a low level to protect the battery from damage and extend the battery life [45]. 2) When the battery insulation resistance is too low, the battery SOAP must be set as zero for safety. 3) When the connection resistances among batteries are too large, the battery SOAP should be reduced because the connection resistances will generate a large voltage drop. 4) For a battery pack, there is an inconsistency among the batteries. Its SOAP at discharging is determined by the battery with minimum voltage, and at charging is determined by the battery with maximum voltage. And they can be determined directly by BMS through the Ubatt of each battery. The SOAP of a battery pack at discharging

(41)

V V where SOAPdch , k + j + 1 and SOAPcha, k + j + 1 are the battery SOAP at discharging and the battery SOAP at charging, respectively, for time step k + j + 1 under the voltage constraint; L is the length of the future time window.

B. Supervisory prediction of SOAP based on current constraint

Communication bus

BMS

Sampling bus

Battery tester

Battery

Chamber Cable Cable

Computer Fig. 5. Battery test bench. 9

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accuracy and adaptability of the proposed long-term SOAP prediction approach for diﬀerent working conditions, the tests are conducted not only under the UDDS used in the battery state estimation veriﬁcation, but also under another practical dynamic cycle (WVUSUB, the west Virginia suburban driving schedule). Considering the accurate battery state estimation of aged battery is more diﬃcult than that of fresh battery, the battery with SOH = 0.8 is specially selected in the tests. Additionally, in order to further verify the accuracy and robustness of the proposed prediction approach against diﬀerent initial SOC levels, the tests are conducted on batteries with initial real SOC = 0.95, 0.7, 0.3 and 0.05 respectively. Both the UDDS cycle and WVUSUB cycle used in the tests are the power proﬁles that are deduced from the velocity proﬁles by the battery power calculation model in Section 2.4.2. Here, considering that without aﬀecting the test of the proposed approach, we use 1 Nb instead of Ubatt Upack in Eq. (33), where Nb is the number of the battery cells. And the road grade is assumed to be zero, the transmission mechanical eﬃciency is set to a typical value. The vehicle speciﬁcations are given in Table 2. And the electric motor eﬃciency is illustrated in Fig. 7. In the tests, a long-term predictive cycle section of 60 s is used. To measure the real charging/discharging SOAP, the charging/discharging pulses every 10 s interval are performed on the battery. For battery SOAP under the voltage constraint, we interrupt the cycling process by setting the battery voltage to the uppermost threshold or lowermost threshold, and then monitor the battery current against time via the BMS. And for battery SOAP under current constraint, we interrupt the cycling process by setting the battery current to the uppermost threshold or lowermost threshold, and then monitor the battery voltage against time via the BMS. The maximum discharging and charging capabilities under voltage constraint can be obtained by multiplying the measured battery currents with the threshold values of the battery voltage [24], and under current constraint can be obtained by multiplying the measured battery voltages with the threshold values of the battery current [25]. There are seven test points and twenty-eight SOAP tests during a cycling process. After each SOAP test, we ﬁrst use the battery tester and thermal chamber to adjust the tested battery to its states (SOC, voltage and temperature) before this SOAP test, then conduct the next SOAP test or load the rest of the deduced power proﬁle on the battery to next test point. According to Eq. (44), for each test point, the ﬁnal battery SOAP for discharging takes the minimum value of the SOAPs under voltage and current constraints, and for charging takes the maximum value. Through this test procedure, the inﬂuence of SOAP test on the battery states can be avoided. As a result, the real SOAPs at diﬀerent points along the long-term predictive cycle are measured by experiments, so can be as the reference of real long-term SOAP.

and at charging can be calculated by multiplying the SOAP of the minimum-voltage battery and the SOAP of the maximum-voltage battery, respectively, with the number of the batteries in the battery pack of the EV. Therefore, for the SOAP prediction of a battery pack, the key is to predict the SOAPs of these typical batteries in the pack. 3. Experiment 3.1. Battery test bench To evaluate the proposed supervisory long-term SOAP prediction approach comprehensively, the 50 A h commercial LiFePO4 batteries with diﬀerent aging levels are used to conduct all the experimental tests. And a test bench is purposely established as shown in Fig. 5, including a battery tester, a thermal chamber, a computer, a BMS board and the tested LiFePO4 lithium-ion batteries. The battery tester is used to load current proﬁles or power proﬁles on the battery and collect the test data. The thermal chamber is used to provide the temperature environment for the tested battery. The BMS is connected in parallel to record the battery current, voltage and temperature, and to run the programs to test the performance of the approaches. The BMS board is with a 200 MHz MCU controller of Freescale MPC5746. The accuracy of the cell voltage measurement of the BMS is 0.8 mV, and of the current measurement is 0.1A. All the data are transmitted to the host computer through a communication bus. The battery parameters are listed in Table 1. 3.2. Test schedules A test schedule is designed to verify the accuracy and robustness of the proposed approach against erroneous initial values, diﬀerent ambient temperatures, diﬀerent loading proﬁles and aging levels. The accelerated aging experiments of batteries are conducted, and the aged batteries with diﬀerent SOH = 0.9 and 0.8 are prepared at ﬁrst. Also, the fresh batteries with SOH = 1 are prepared. 3.2.1. Battery model test In order to test the improved battery model, we design a special current proﬁle as shown in Fig. 6. The current proﬁle includes two parts of dynamic conditions and one part of pulse-modulated charging process, which is aimed to simulate the real working conditions of the battery in EV and the current proﬁle of an EV charger, respectively. The battery with SOH = 1 and initial SOC = 0.56 is tested at 25 °C. The common ﬁrst-order RC model is also tested at the same time to evaluate our improved ﬁrst-order RC model. 3.2.2. Battery state estimation test In order to verify the accuracy and robustness of the proposed battery state estimation approach in dynamic operations, urban dynamometer driving schedule (UDDS) cycles are used to represent the practical proﬁles for batteries. And a series of tests are conducted under these cycles. Considering the SOH requirements for batteries for EVs, the tests are implemented on batteries with diﬀerent SOH = 1, 0.9, 0.8 respectively, and at diﬀerent temperatures of 0 °C, 25 °C and 40 °C by thermal chamber respectively. Additionally, to further evaluate the proposed battery state estimation approach, the tests are also carried out for other diﬀerent battery state estimation approaches, respectively. Here, the EKF, AEKF, AUKF-based approaches are selected to compare, considering that they are the widely studied approaches for battery state estimation up to now [14]. And in all the tests, the initial SOC in the algorithms is purposely set with the initial error of 20% to evaluate the robustness of the approaches against erroneous initial values.

4. Veriﬁcation and discussion 4.1. Battery model veriﬁcation The tracking accuracy comparison of diﬀerent battery equivalentcircuit models is shown in Fig. 8, where the traditional ﬁrst-order RC model is named as 1RC model. By observing Fig. 8 (a) and (b), the mean absolute error (MAE) of the output voltage of the improved battery RC model in this paper can Table 1 Tested battery parameters.

3.2.3. Supervisory long-term SOAP prediction test After verifying the battery state estimation, in order to verify the 10

Parameters

value

Parameters

value

Rated capacity Charge limit current Discharge limit current Charge limit voltage

50 A h 210 A 300 A 3.65 V

Discharge limit voltage Mass Heat capacity Total surface

2.5 V 1.726 kg 1.12 × 103 J/(kg·K) 865.8 cm2

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-50 0

200

400

600

800

1000

0.84

100

-200

2

0.88

0

1200

0.8

0.7

0 -100

8

0.84 0.8 0.76 0.72

Motor Torque T em (Nm)

0

0.8

200

0.8.76 0

Current / A

50

Time / s

2000

4

0.8 0.92 0.88 0.72 0.84 0.8 0.76 0.72 0.76 0.80.84 0.92 8 0.8 4 0.8 0.8

4000 6000 8000 Motor Speed w em (rpm)

10000

Fig. 6. Input current in the test of the battery model.

Fig. 7. Eﬃciency map of the electric motor.

be limited to 0.9 mV and the maximum error (ME) 13.1 mV, while of the traditional 1RC model are 1.1 mV and 17.4 mV respectively. The reduced error of 3.3 mV is very important for such batteries like LiFePO4 due to their very ﬂat voltage platform especially in the SOC range from about 0.3 to 0.85, and the voltage hysteresis. The result shows that compared with the traditional 1RC model, the improved RC model can more accurately capture the battery dynamics. This will help to improve the subsequent parameter identiﬁcation, state estimation and SOAP prediction.

by 27% and 19% respectively compared with those by other approaches.) (2) Comparison with other state estimation approaches To compare with the other battery state estimation approaches, the same tests in the above ﬁve cases are also carried out for the approaches based on EKF, AEKF or AUKF. For fair comparison, each of the approaches is based on the same improved ECM and AFFLS-based parameter identiﬁcation presented in this paper. So we name them as AFFLS-EKF, AFFLS-AEKF and AFFLS-AUKF, respectively. The comparison results are also summarized in Table 3. And the detailed results are shown in Fig. 10, taking the UDDS cycles on the battery with SOH = 1 at 25 °C as an example. It can be seen that all of the approaches can track the battery SOC trajectory of the battery, but the errors are different. At battery diﬀerent aging levels and temperatures, among these three other approaches, the estimation accuracy of the AUKF-based approach is the highest. While our proposed ATSF-based approach, which is named as AFFLS-ATSF, can further reduce the SOC estimation MAE by 27% on average, and reduce the estimation ME by 45% on average, even when compared with the AUKF-based battery state estimation approach. These results indicate that the proposed approach in this paper can more accurately identify the battery parameters and estimate the battery states than other approaches. It can ensure the accuracy of battery state estimation under dynamic conditions against diﬀerent battery aging levels, temperatures and erroneous initial values. Moreover, it should be pointed out that the tests of battery SOC estimation covered the working temperature range and aging range of batteries in real applications of EVs. Therefore, the proposed approach can provide a solid foundation for subsequent long-term SOAP prediction in real applications.

4.2. Battery state estimation results To verify the accuracy and robustness of the proposed battery state estimation approach, tests on batteries with diﬀerent SOH and at different temperatures are conducted according to the test schedules described in Section 3.2.2. (1) Veriﬁcation of the state estimation The battery parameters (R 0 , Rp and τ ) identiﬁcation and states (SOC and Up) estimation are implemented in ﬁve diﬀerent cases (a), (b), (c), (d) and (e) at diﬀerent battery aging levels and temperatures, as shown in Fig. 9. In all the ﬁve cases, the initial SOC is purposely set to 0.8 while the actual initial SOC is 1, with the initial SOC error of 20%. The results are depicted in Fig. 9 and summarized in Table 3. Since the used current sensor is highly accurate, the referenced SOC calculated by coulombic counting can be reasonably regarded as the “real SOC”. The parameter identiﬁcation accuracy is not given due to the lack of real value. From Fig. 9 and Table 3, it can be seen that the SOC estimation can quickly converge to its actual value after 300 s in all these ﬁve cases. After that, the MAE and ME of the SOC estimation in the case (c) are about 3% and 7% respectively, while in all the other four cases can be always less than 2% and 5% respectively. It should be pointed out that in the case (c), the battery SOH is 0.8, which means the battery has been completely aged for EV applications according to the policy from Chinese government. It is already very satisfactory for the SOC estimation with such errors in BMS for such aged batteries (As summarized in Table 3, for the battery with SOH = 0.8, the MAE and ME are reduced

4.3. Supervisory long-term battery SOAP prediction results The veriﬁcation in Section 4.2 indicates that battery state estimation is very accurate and robust under UDDS cycles. The proposed long-term SOAP prediction is then ﬁrstly veriﬁed. The results show that the MAE of 60 s long-term SOAP prediction under UDDS is less than 3%. In order

Table 2 Vehicle speciﬁcations. Parameters

value

Parameters

value

Accessory power, Pea Number of the battery cells, Nb Mass of the vehicle, m Rolling resistance coeﬃcient, µ Frontal area of the vehicle, A

1000 W 110 1380 kg 0.0086 2.08 m2

Drag coeﬃcient of the vehicle, CD Inertia coeﬃcient, δ Wheel rolling radius, r Speed ratio of ﬁnal drive, if Transmission mechanical eﬃciency, ηf

0.31 1.03 0.316 m 8.928 0.96

11

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0.02

3.4

0.01

Ut error / V

3.35

Ut / V

1RC model Improved model

3.3 Measurement

0 -0.01

1RC model

3.25

Improved model 3.2

0

200

400

600

800

1000

-0.02

1200

0

200

400

600

800

Time / s

Time / s

(a)

(b)

1000

1200

Fig. 8. Tracking accuracy comparison of diﬀerent battery RC models: (a) battery terminal voltage, (b) voltage error.

established for EVs is veriﬁed. Fig. 11 (a) illustrates the predicted velocity of 60 s by the approach based on RBF-NN, taking a very dynamic segment from the 680th second in WVUSUB cycle as an example. And the predicted power demand of single battery is shown in Fig. 11 (b). The real power demand calculated from the real velocity is also depicted in Fig. 11 (b). Since the predicted velocity has some diﬀerence with the real velocity, the corresponding acceleration and deceleration

SOC

(e)

-0.05

T=25 °C SOH=1

-0.1 2000 4000 6000 8000 10000 12000 14000

Time / s SOC real SOC estimation SOC error

0.15

0 T=25 °C SOH=0.8

1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

-0.1 2000 4000 6000 8000 10000 12000 14000

0.1 0.05 0 -0.05

T=25 °C SOH=0.9

-0.1 2000 4000 6000 8000 10000 12000 14000

0

1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

Time / s

0.15 SOC real SOC estimation SOC error

0.1 0.05 0

T=0 °C SOH=1 0

-0.05

-0.1 2000 4000 6000 8000 10000 12000 14000

0.1 0.05 0

T=40 °C SOH=1

0.15 SOC real SOC estimation SOC error

Time / s

0.15 SOC real SOC estimation SOC error

0

(d)

-0.05

Time / s

1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

0.1 0.05

0

SOC

0

SOC error

0.1 0.05

0

(b)

SOC

1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

0.15 SOC real SOC estimation SOC error

SOC error

SOC

(c)

1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

SOC error

SOC

(a)

SOC error

4.3.1. Long-term battery power demand prediction veriﬁcation The LTPD of battery is ﬁrst introduced to the long-term SOAP prediction in this paper. Here, the battery LTPD prediction model

SOC error

to further verify the adaptability of the prediction approach to diﬀerent working conditions, this Section demonstrates the prediction under WVUSUB cycles in detail.

-0.05

-0.1 2000 4000 6000 8000 10000 12000 14000

Time / s Fig. 9. Test of the SOC estimation accuracy and robustness: (a) result with SOH = 1 at 25 °C, (b) result with SOH = 0.9 at 25 °C, (c) result with SOH = 0.8 at 25 °C, (d) result with SOH = 1 at 0 °C, (e) result with SOH = 1 at 40 °C. 12

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Table 3 Battery SOC estimation errors and analysis of diﬀerent approaches. SOH

1

Temperature (°C)

Battery SOC error

0

1

25

1

40

0.9

25

0.8

25

MAE Maximum MAE Maximum MAE Maximum MAE Maximum MAE Maximum

error error error error error

Average of MAE Average of Maximum error

Ratio of error reduction

EKF

AEKF

AUKF

ATSF

ATSF to EKF

ATSF to AEKF

ATSF to AUKF

0.0392 0.1022 0.0254 0.0893 0.0202 0.0829 0.0357 0.0936 0.0442 0.1017 0.0329 0.0939

0.0278 0.0936 0.0196 0.081 0.0155 0.0748 0.0253 0.0827 0.0397 0.0931 0.0256 0.0850

0.0229 0.0908 0.0151 0.0865 0.0139 0.0701 0.0239 0.0794 0.0376 0.0825 0.0227 0.0819

0.0153 0.0405 0.0102 0.0456 0.011 0.0375 0.0187 0.0419 0.0275 0.0668 0.0165 0.0465

60.97% 60.37% 59.84% 58.57% 45.54% 54.76% 47.62% 55.24% 37.78% 34.32% 50.35% 52.65%

44.96% 56.73% 47.96% 54.32% 29.03% 49.87% 26.09% 49.33% 30.73% 28.25% 35.75% 47.70%

33.19% 55.40% 32.45% 57.23% 20.86% 46.50% 21.76% 47.23% 26.86% 19.03% 27.02% 45.08%

traditional approach. A. Farmann et al. [9] have carried out a comprehensive review of the on-board SOAP prediction techniques for lithium-ion batteries in EVs. The previous long-term SOAP prediction approaches in literature all have the same two basic characteristics: (1) not updating the battery states according to the battery power demand in the future; and (2) assuming that the battery always works in its extreme conditions over the long-term prediction time horizon. Except that battery model, parameter identiﬁcation or state estimation used are diﬀerent, the long-term SOAP prediction process under current and voltage constraints in traditional approaches is similar, and can be found in any of the related papers such as [4,5,24,25] ect. But as presented above, the accuracy of the improved battery model, battery parameter identiﬁcation and state estimation approaches proposed in this paper can be greatly improved. For fair comparison, they are also used in the traditional approach. To further analyze the eﬀect of battery state estimation on battery SOAP prediction, tests are also conducted for the proposed supervisory long-term SOAP prediction idea-based approaches using other diﬀerent battery state estimation approaches respectively. Here, we also select the most widely studied EKF, AEKF or AUKF-based estimation approaches [14] to compare. And for fair comparison, except for the

changes are not completely the same as the real changes. The predicted power variation is then less than the actual variation. However, it should be noted that: although there exist some deviations (the root mean squared error (RMSE) of predicted velocity is 1.8105), the trend and average of predicted power demand are the same as the real power demand. Hence, as shown in the following Section 4.3.2, the MAE of long-term SOAP prediction can be reduced by 85.9% compared with that by the traditional approach, beneﬁting from the supervision of the predicted LTPD. This indicates that the prediction is already good enough to provide another solid foundation for subsequent long-term SOAP prediction. The eﬀects of the velocity prediction accuracy on the long-term SOAP prediction will be analyzed in Section 4.3.2. 4.3.2. Supervisory long-term battery SOAP prediction veriﬁcation After the tests and veriﬁcations presented above, this Section is aimed at testing and verifying the proposed supervisory long-term SOAP prediction approach. The test schedules have been described in Section 3.2.3. To compare with the traditional approaches, the same tests are performed on both the new approach proposed in this paper and the

(a)

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Fig. 10. Accuracy comparison of diﬀerent SOC estimation approaches: (a) battery current, (b) battery voltage, (c) SOC estimation comparison, (d) SOC estimation error comparison. 13

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150

15

100

10

Power / W

velocity / m.s -1

Real velocity Predicted velocity

5

0 680

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(b)

(a)

Fig. 11. Vehicle velocity prediction and battery power demand prediction: (a) velocity prediction, (b) power demand prediction of a battery.

EVs. These results indicate that due to the supervision of the predicted LTPD, at diﬀerent SOC levels of battery: the accuracy of the long-term SOAP prediction can be remarkably improved compared with that by the traditional approach; more importantly, the proposed approach in this paper can make the predicted long-term SOAP practical, while the traditional approach cannot.

battery state estimation, everything else in the scheme proposed in Fig. 4 is the same. The results output at the 695th second (see Fig. 11) in the tests conducted under WVUSUB cycle are shown in Fig. 12, and summarized in Table 4. In Fig. 12, “CPSOAP” denotes the long-term SOAP predicted by the traditional approach, “SPSOAP with X” represents the long-term SOAP predicted by the new supervisory SOAP (SPSOAP) prediction idea with the battery state estimation approach named as “X” (here, “X” is the EKF, AEKF, AUKF or the ATSF introduced in this paper), “SOAP measurement” denotes the real long-term SOAP measured by experiments according to the test procedure described in Section 3.2.3.

B. Analysis when diﬀerent state estimation approaches are used From Fig. 12 and Table 4, it can be also seen that unlike the traditional approach, all of the long-term SPSOAP curves are similar in trends and close in values to the measured actual SOAP curve. This is the common feature, whether the proposed supervisory prediction ideabased approaches use EKF, AEKF or AUKF-based approaches or the ATSF-based approach proposed in this paper to estimate the battery states. Among all these four approaches, for long-term SOAP prediction of 60 s, the ME of discharging SOAP can be limited to a band of 13.3%, and of charging SOAP to a band of 7.5%, which are still signiﬁcantly smaller than those (36% and 33% respectively) by the traditional approach. The average MAE of all the cases is about 2.2%, also remarkably reduced by 80% compared with the MAE (10.8%) by the traditional approach. These results indicate that: for making the predicted long-term SOAP practical and accurate in real applications, predicting the long-term SOAP under the supervision of the predicted battery LTPD is the most important foundation. However, four diﬀerent battery state estimation approaches also result in diﬀerent battery SOAP prediction results. Because the contradiction between the linearization errors and robustness is settled, the introduced ATSF-based approach achieves the highest accuracy of battery state estimation (Table 3). Therefore, for SOAP prediction, the best is the proposed approach with the ATSF-based battery state estimation:

A. Veriﬁcation of the proposed supervisory long-term SOAP prediction From Fig. 12 and Table 4, it is apparent that the long-term SOAP predicted by the traditional approach (CPSOAP) is quite diﬀerent with the measured actual SOAP. While the long-term SOAP predicted by the proposed new supervisory prediction approach (SPSOAP with ATSF) can always be very close to the measured actual SOAP. It can be seen that (1) In all the tests at the diﬀerent four initial SOC levels, for the longterm prediction of 60 s: by traditional approach, the mean error and maximum error (ME) of discharging SOAP are up to −30% and 36% respectively, and of charging SOAP are up to −27% and 33% respectively; while by the proposed approach in this paper, the mean error and ME of discharging SOAP are only −5% and 7% respectively, and of charging SOAP are only −3.5% and 5% respectively. The maximum errors are reduced by more than 80%. (2) At diﬀerent initial SOC levels, the MAE of total by the traditional approach is 10.8%, while of the proposed approach in this paper is only 1.5%. The proposed approach reduces the MAE of total by 85.9%. (3) In particular, the trend of the SOAP curve predicted by traditional approach is obviously diﬀerent from the trend of the measured actual SOAP curve; while the SOAP curve predicted by the proposed approach in this paper is very similar to the measured actual SOAP in trend.

(1) From Table 4, among the ﬁrst three approaches at diﬀerent initial SOC levels, the approach with AUKF-based battery state estimation has the highest accuracy for the long-term SOAP prediction of 60 s with MAE of 2.1%; while the proposed approach in this paper can further reduce the MAE by 26.9%. (2) By observing the ﬁrst points on all the curves in Fig. 12, it can be seen that for the short-term SOAP prediction of 1 s, the approach with AUKF-based battery state estimation also shows a higher accuracy than the approaches with EKF or AEKF-based battery state estimation. At diﬀerent initial SOC levels, the MAE of the shortterm SOAP predicted by the approach with AUKF-based battery state estimation is 2.5%, while by the proposed approach with the proposed ATSF-based battery state estimation is 1.8%. This MAE can be further reduced by 28.3% by the proposed approach in this paper.

In addition, the inﬂuence of battery aging factor identiﬁcation on SOAP prediction is analyzed. The mean error of the aging factor identiﬁcation is within 0.042. And this reduces the long-term SOAP prediction MAE of total by 9.52% (from 1.68% to 1.52%). But compared with reducing the MAE of total from 10.76% to 1.52% (Table 4), we can see that its contribution to improving the accuracy of long-term SOAP prediction is obviously much smaller than that from the supervision of the predicted LTPD. The tests covered the SOC range of battery in real applications in 14

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SPSOAP with EKF

SPSOAP with AUKF

SPSOAP with AEKF (b) -400

SPSOAP with ATSF

840

Charging SOAP / W

Discharging SOAP / W

(a)

SOAP measurement CPSOAP

820 800 780 760

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Fig. 12. Long-term SOAP prediction results of diﬀerent approaches at diﬀerent initial SOC levels: (a) and (b) initial SOC = 0.95, (c) and (d) initial SOC = 0.7, (e) and (f) initial SOC = 0.3, (g) and (h) initial SOC = 0.05.

To analyze the eﬀect of the velocity prediction accuracy on the longterm SOAP prediction accuracy, the MAEs in three situations are compared. The ﬁrst situation is the supervisory SOAP prediction approach proposed in this paper, in which the future velocity proﬁle is predicted based on RBF-NN. The second situation is also the supervisory SOAP prediction approach, but in which the future velocity proﬁle is assumed to be perfectly known in advance. Except for the future

These results indicate that: besides the most important foundation of the proposed supervisory long-term SOAP prediction approach, the ATSF-based battery states estimation proposed in this paper is also very helpful to further improve the accuracy of both the short-term and longterm battery SOAP prediction at diﬀerent SOC levels of battery. C. Analysis of the eﬀect of the velocity prediction accuracy 15

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Table 4 The errors analysis of the long-term battery SOAP predicted by diﬀerent approaches. SOC estimated by

Mean Error

MAE of total MaximumError

Supervisory prediction

SOAPdch SOAPcha SOAPdch SOAPcha SOAPdch SOAPcha SOAPdch SOAPcha

Initial SOC = 0.95

SOAPdch SOAPcha SOAPdch SOAPcha SOAPdch SOAPcha SOAPdch SOAPcha

Initial SOC = 0.95

Initial SOC = 0.7 Initial SOC = 0.3 Initial SOC = 0.05

Initial SOC = 0.7 Initial SOC = 0.3 Initial SOC = 0.05

EKF −1.14% −5.61% −0.57% −0.38% −0.67% −0.40% −9.67% −1.42% 2.48% 1.31% 7.49% 0.71% 0.53% 0.81% 0.55% 13.22% 1.65%

Traditionalprediction

AEKF −1.02% −4.63% −0.76% −0.33% −0.83% −0.34% −11.08% −1.24% 2.53% 1.24% 7.41% 0.94% 0.49% 1.01% 0.48% 12.93% 1.45%

velocity proﬁle, everything else in the ﬁrst and second situations is the same. The third situation is the traditional approach. For the ﬁrst and third situations, the long-term SOAPs have been predicted above. For the second situation, the long-term SOAP of 60 s is predicted here. Then respectively for each of the SOAP measurement points, the MAE of the charging and discharging SOAPs predicted at diﬀerent initial SOC levels is calculated. Fig. 13 compares the MAE results in these three situations. As it can be seen from Fig. 13, except for the short-term SOAP (at the ﬁrst point on the curves), all the MAEs in the ﬁrst and second situations are much smaller than the MAEs in the third situation. In particular, the MAEs in the ﬁrst situation are very close to the MAEs in the second situation. The diﬀerence between the MAEs in the ﬁrst and second situations is less than 0.73%. The reason is that: although there exist some deviations between the predicted and real velocities, their trend and average are the same as shown in Fig. 11 (a). It is good enough to make the absolute deviations between the predicted and real power demands much smaller than the absolute values of the actual SOAP, as can be seen from the results in Fig. 11 (b) and Fig. 12 (SOAP measurement). It is also noteworthy that, because it is not yet possible to make the error of battery parameter identiﬁcation and state estimation zero, the error still exists at the ﬁrst point in Fig. 13. Therefore, even assuming the velocity proﬁle is perfectly known in advance in the second situation, the MAE of the predicted long-term SOAP is still not zero. Additionally, in Table 4, the maximum error of SOAP predicted by our approach is 7.25%. This is mainly because it is diﬃcult to completely avoid the deviation between the predicted and real transient velocities of vehicle, due to the random working loads and running environment of vehicle. Some deviations between the predicted and real power demands are unavoidable. However, beneﬁting from the supervision of the predicted LTPD of battery, this error is reduced by 80% compared with that (31.65%) by the traditional approach.

AUKF −0.74% −4.67% −0.63% −0.28% −0.68% −0.29% −8.19% −1.16% 2.08% 0.93% 7.05% 0.81% 0.44% 0.85% 0.45% 10.56% 1.34%

ATSF −0.91% −3.45% −0.57% −0.36% −0.58% −0.24% −5.33% −0.75% 1.52% 1.05% 4.88% 0.72% 0.48% 0.73% 0.36% 7.25% 0.97%

ATSF −6.48% −26.86% −5.90% 2.58% −5.96% 2.68% −30.37% 5.21% 10.76% 7.83% 32.55% 6.96% 3.23% 7.02% 3.33% 36.15% 6.87%

improvement can be expected by improving the battery LTPD prediction in the future. (2) The initial deviations of each curve of SPSOAP with diﬀerent battery state estimation approaches are caused by the diﬀerence of the battery SOC, Up and R0 estimated by these approaches. And the diﬀerent change rates of the SPSOAP curves in the future time window are inﬂuenced by battery Rp and Cp. The diﬀerent results indicate that the improved ECM, the AFFLS-based parameter identiﬁcation and the new ATSF-based battery state estimation approach proposed in this paper, also help improve the SOAPs (including the shortterm and long-term SOAP) prediction accuracy evidently as presented above. In addition, the real-time performance has been veriﬁed on the real BMS board during the testing process. The step execution time of battery state estimation by ATSF-based approach is 19 ms, which is at the same level as the shortest of the four state estimation approaches tested (7 ms by EKF-based approach). And the execution time of 60 s supervisory long-term SOAP prediction is 1.913 s. Since the long-term SOAP prediction can be updated every several seconds, the real-time requirement is met for BMS. In the control of EVs, the SOAP for the next sampling interval is critical to restrict the input and output powers of the battery in realtime operation. The proposed approach provides an accurate short-term SOAP prediction for it. Particularly, the battery needs to input or output an available peak power continuously in some cases [4]; the predictive EMS also needs the future available peak power of the battery to minimize the vehicle energy consumption [6–8]. And the proposed approach oﬀers a practical and very accurate long-term SOAP prediction for the vehicle controller, which contributes to fully exploit the

15

MAE / %

D. Discussion Above all, we can see that: (1) For making the long-term battery SOAP prediction practical and accurate, using the battery LTPD as a supervisor is the most important foundation. This is why, although there exist some deviations between the predicted battery LTPD and the actual values, the prediction error can also be drastically reduced by more than 80% in all the tested cases when compared with that by the traditional approach. More importantly, due to the overcoming of the drawbacks of the traditional long-term SOAP prediction approaches, the proposed approach can make the predicted long-term SOAP practical, while the traditional ones cannot. The further accuracy

10 Supvisory approach assuming VsP fully kown Supvisory approach with VsP prediction Traditional approach

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Time / s Fig. 13. MAE comparison of SOAP prediction under WVUSUB cycle, where: “VsP” is short for the velocity proﬁle. 16

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potential of the battery within the safe operation window and for vehicle performance optimization. Applying the proposed long-term SOAP (meanwhile, the short-term SOAP) prediction, the available power capability of the battery in the EV can be eﬀectively monitored and managed. Therefore, the proposed approach is promising for the real applications of EVs.

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5. Conclusion In this paper, a novel approach has been proposed to predict the long-term SOAP of lithium-ion batteries in electrical vehicles. Trying to make the prediction practical and accurate, it incorporates the inﬂuence of battery states and parameters, especially the long-term power demand (LTPD) of battery on long-term SOAP prediction. To this end, the ﬁrst-order RC battery model is improved by considering polarization and hysteresis eﬀects, the model parameters are online identiﬁed by AFFLS algorithm. And to improve the estimation accuracy of battery states, a new battery state estimation approach by introducing ATSF is proposed. Furthermore, a model is established to predict the LTPD of a battery in EV in the future. The long-term battery SOAP is ﬁnally predicted by the approach put forward in this paper to adapt the future working conditions of an EV, taking the predicted battery LTPD as a supervisor. To verify the proposed approach, comprehensive tests were carried out. The accuracy and robustness of the proposed approach against erroneous initial battery states, diﬀerent aging levels, initial SOC-levels, working conditions and ambient temperatures are systematically veriﬁed by experiments. Experimental results show that the improved battery model can describe the battery dynamics accurately; thanks to the ﬁrstly introduced ATSF, the battery state estimation approach can evidently improve the estimation accuracy; especially, the proposed supervisory long-term SOAP prediction approach could make the predicted long-term SOAP practical and with the prediction error reduced by 85.9%. For making the long-term battery SOAP prediction practical and accurate, using a reasonable battery LTPD as a supervisor is the most important foundation, while accurate battery model and state estimation provide very useful help. Moreover, the execution time of the algorithm on BMS board shows that the proposed approach meets the real-time requirements and can be used online for real applications. Declaration of Competing Interest The authors declare that they have no known competing ﬁnancial interests or personal relationships that could have appeared to inﬂuence the work reported in this paper. Acknowledgements This work is supported by the National Natural Science Foundation of China (Grant no. 51741707 and 51875339) and the National Key Technology R&D Program (Grant no. 2013BAG03B01). The authors express their appreciation for the funds. References [1] Abbasi MH, Taki M, Rajabi A, Li L, Zhang J. Coordinated operation of electric vehicle charging and wind power generation as a virtual power plant: A multi-stage risk constrained approach. Appl Energy 2019;239:1294–307. https://doi.org/10. 1016/j.apenergy.2019.01.238. [2] Sun F, Xiong R, He H. Estimation of state-of-charge and state-of-power capability of lithium-ion battery considering varying health conditions. J Power Sources 2014;259:166–76. https://doi.org/10.1016/j.jpowsour.2014.02.095. [3] Xiong R, Tian J, Mu H, Wang C. A systematic model-based degradation behavior recognition and health monitoring method for lithium-ion batteries. Appl Energy 2017;207:372–83. https://doi.org/10.1016/j.apenergy.2017.05.124. [4] Xiong R, He H, Sun F, Liu X, Liu Z. Model-based state of charge and peak power capability joint estimation of lithium-ion battery in plug-in hybrid electric vehicles. J. Power Sources 2013;229:159–69. https://doi.org/10.1016/j.jpowsour.2012.12. 003.

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