Hybrid state of charge estimation for lithium-ion battery under dynamic operating conditions

Electrical Power and Energy Systems 110 (2019) 48–61

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Electrical Power and Energy Systems journal homepage: www.elsevier.com/locate/ijepes

Hybrid state of charge estimation for lithium-ion battery under dynamic operating conditions

T

Datong Liua, , Lyu Lib, Yuchen Songa, Lifeng Wuc, Yu Penga, ⁎

⁎

a

School of Electronics and Information Engineering, Harbin Institute of Technology, Harbin, 150080, China Department of Automatic Test and Control, Harbin Institute of Technology, Harbin 150080, China c Information Engineering College, Capital Normal University, Beijing 100048, China b

ARTICLE INFO

ABSTRACT

Keywords: Lithium-ion battery Dynamic conditions State of charge estimation Deep belief network Hybrid method

Lithium-ion battery is widely used in various industrial applications including electric vehicles (EVs) and distributed grids due to its high energy density and long service life. As an essential performance indicator, state of charge (SOC) reflects the residual capacity of a battery. To ensure the safe operation of systems, it is vital to obtain battery SOC accurately. However, as a parameter which cannot be directly measured, the battery SOC are influenced not only by the measurement noise but also the cell temperature. Focusing on these challenging issues, this paper proposes a hybrid model to estimate the lithium-ion battery SOC under dynamic conditions. This method consists of deep belief network (DBN) and the Kalman filter (KF). The battery electric current, terminal voltage and temperature are used as the input of the proposed model of which output is the SOC. With the powerful nonlinear fitting capability of the DBN, the model can extract relationship between the measurable parameters and battery SOC. The KF algorithm is utilized to eliminate the effects from measurement noise and improve the estimation accuracy. Experiments under different operation conditions are carried out with commercial lithium-ion batteries. The biggest average estimation error is less than 2.2% which indicates that the proposed method is promising for battery SOC estimation especially for the complex operation conditions.

1. Introduction The diverse batteries in the existing power source have received more and more attention [1,2]. Compared with other electrochemical cells, lithium-ion batteries are widely used in various fields by virtue of the high energy density, long service life, etc. [3–5]. A variety of devices including portable devices, power tools and electric vehicles require lithium-ion batteries as stable power supply to ensure their systems operating safety [6–9]. The state of charge (SOC) of a lithium-ion battery is defined as the percentage of the residual capacity Qcurrent in its maximum available capacity Qrate [10], which is shown as the Eq. (1).

SOC (t ) =

Qcurrent Qrate

(1)

Obtaining a precise SOC helps to use the battery reasonably, avoid damage to the battery, and extend the useful life of battery. Thus, in order to improve the stability of energy storage system, it is necessary to estimate accurate SOC. However, due to non-linear, time-varying

characteristics and electrochemical reactions, battery SOC cannot be observed directly. Considerable research efforts are paid to solve these problems. The commonly used methods include Coulomb Counting [11,12], looking-up table based methods [13,14], extend Kalman filter (EKF) [15–18], particle filter (PF) [19–21], unscented Kalman filter (UKF) [22–24], machine learning algorithms [25–29], etc. Coulomb Counting is the most common method. This method does not consider the internal structure of the battery and its external electrical characteristics, so it is suitable for a variety kinds of batteries. However, the cumulative error in the measure current may cause errors in estimation results. Moreover, the inaccurate initial SOC also make the estimations drifting from the true value. Looking-up table based methods mainly include open circuit voltage (OCV) and impedance, etc. Owing to the SOC is highly related to its open circuit voltage, impedance, etc., the SOC can be inferred based on the looking-up table consisting of those battery parameters. Although methods based on looking-up table have a satisfied performance, they are not suitable for the real-time application since the long rest time is needed before the open circuit voltage is available.

Corresponding authors. E-mail addresses: [email protected] (D. Liu), [email protected] (L. Li), [email protected] (Y. Song), [email protected] (L. Wu), [email protected] (Y. Peng). ⁎

https://doi.org/10.1016/j.ijepes.2019.02.046 Received 17 July 2018; Received in revised form 6 October 2018; Accepted 26 February 2019 0142-0615/ © 2019 Published by Elsevier Ltd.

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When lithium-ion batteries are applied in practice, the current, voltage and temperature change quickly under the dynamic loads from the host system. The measurement noise is inevitable while measuring the electric current, terminal voltage and cell temperature. A number of model-based methods with filters are proposed to reduce the impact of noises. In those methods, battery models, such as electrochemical model (EM) and equivalent circuit model (ECM), etc., are usually established as the state equations, and then filters include the KF, EKF, etc. are used to estimate the battery SOC. Ye et al. [19] applied an error analysis method to improve the robustness of adaptive particle filter which can eliminate the SOC estimation error. Xing et al. [22] established a UKF model to estimate the SOC under dynamic conditions and achieved good estimation results. Compared with Coulomb Counting methods and looking-up methods, model-based approaches have attracted more attention. However, some details of modeling should be considered. Firstly, due to the internal complex chemical reaction process and uncertain external operating conditions, it is difficult to establish an accurate battery model. Secondly, the fixed model cannot adapt to the dynamic operation conditions, since the parameters in the model should be identified according to some special test protocols. Data-driven approaches do not require detailed physical knowledge of lithium-ion batteries. These approaches focusing on the relationship between inputs and the outputs have been widely applied for battery SOC estimation. Chang [28] combined a radial basis function (RBF) neural network, an orthogonal least-squares (OLS) algorithm and an adaptive genetic algorithm (AGA) to estimate SOC. He et al. [27] developed a SOC estimation method based on neural network and UKF. However, when the battery working in extreme situation, the estimation accuracy of data-driven methods may decrease due to the unsuitable model structure and inadequate fitting ability. To fit to more complex working conditions, more powerful and adaptable methods need to be applied. In recent years, deep learning algorithms have been widely used in different domains due to their strong feature extraction and fitting ability. Compared with traditional neural networks or machine learning algorithms, deep learning methods have a multi-layer network which adapts to more complex system. Ren et al. [29] proposed an integrated deep learning approach for multi-bearing remaining useful life (RUL) collaborative prediction. Wu et al. [30] utilized vanilla Long Short-Term Memory (LSTM) to get good RUL prediction accuracy on the health monitoring of aircraft turbofan engines with different issues. Li et al. [31] used deep convolution neural networks ensemble method for RUL prognostics. These methods all achieved good results in different situations. In this paper, a hybrid method based on DBN-KF is utilized to estimate the SOC of lithium-ion batteries under dynamic operating conditions. The proposed method consists of two parts, namely, initial model and the estimation model. The initial model provides an initial value which solves the uncertainty of the estimation. The voltage, current and temperature and previous SOC are used as the input of estimation model, and the output of model is the battery SOC. As a deep learning method, DBN has strong non-linear fitting and feature extraction ability. Therefore, the DBN model can capture the implied relationship between the measurable parameters and the SOC. To improve the robustness of the model, the KF algorithm is applied to reduce the system noise. The parameters of the whole model are optimized by the particle swarm optimization (PSO) algorithm. The rest of paper is organized as follows. The methodologies including DBN model and KF algorithm are described in Section 2. In the Section 3, the proposed hybrid model is described in details. Section 4 shows the results of experiments. The performance is demonstrated through battery SOC estimation experiments and evaluations based on the dynamical stress test (DST) and Randomized Battery Usage Data Set (RBUDS). A summary of critical observations and conclusive remarks are stated in Section 5.

Output Regression layer

...

... Stacked RBM ...

RBM1

... Input Fig. 1. The structure of DBN.

2. Methodologies 2.1. Principle of deep belief networks (DBN) DBN is a kind of deep learning algorithm that has been widely used for non-linear system modeling [32], prognostics, anomaly detection [33], classification [34], etc. The structure of a DBN network is shown as Fig. 1. It consists of stacked restricted Boltzmann machine (RBM) and a logistic regression layer. As a data-driven method, the RBM extracts the input data feature and uses the feature as the input for the next RBM. After feature extraction, the regression layer is applied for estimation. 2.1.1. Principle of the RBM RBM is the basic element of DBN model. It is a generative stochastic network and energy-based model which provides a powerful tool for representing dependency structure between random variables [35]. Each RBM contains a visible layer and a hidden layer, and the connections between the visible layer and hidden layer have the following characteristics: there is full connection between different layers and no connection between neurons in the same layer. The structure of RBM is shown in Fig. 2. In Fig. 2, V = (v1, v2, ...,vm)T is the vector of visible layer and H = (h1, h2 , ...,hn)T is the vector of hidden layer. W = ( i, j ) Rm × n is the matrix of weight of the RBM and the i, j represents the weight connecting visible unit i and hidden unit j . The a = (a1, a2 , ...,am )T is the bias term of the visible layer, and the b = (b1, b2 , ...,bn)T is the bias term of the hidden layer. The general RBM is a kind of binary network that the visible vectors are set as V = (v1, v2, ...,vm)T {0, 1} and hidden vectors are set as H = (h1, h2 , ...,hn)T {0, 1} . The energy function of RBM is defined as Eq. (2),

Fig. 2. The structure of RBM. 49

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D. Liu, et al. m

E (V, H | ) =

n

m

ai vi

n

bi hi

i=1

j=1

hj

j, i vi

i=1 j=1

(2)

where = { ij , ai , bj} is the parameters’ vector in the RBM. The joint probability distribution between the visible layer and hidden layer is defined as Eq. (3), h

p (V , H | ) =

e

Fig. 3. Unsupervised training of DBN.

E (v , h | )

(3)

Z( )

where Z ( ) = v, h e E (v, h | ) . This partition function represents the all possible states of visible and hidden layer neurons. When the hidden layer H or visible layer V is known, the probability of the visible layer V or hidden layer H can be obtained. The likelihood function of visible layer V and hidden layer H are defined as Eqs. (4) and (5).

p (V |H, ) =

p (vi |h)

p (H | V , ) =

p (hj |v )

The probability distribution of vi , hj can be fetched based on this energy equation when the parameters’ vector is determined. The conditional probabilities of the RBM can be given by Eqs. (6) and (7), m

P (hj = 1 |V) =

bj +

vi

ij

(6)

i=1

vi(t ) )

ai = ai +

× (p (hj = 1 |v (0) )

(10)

p (hj = 1 |v (t ) ))

(11)

Each of the RBM in DBN is trained layer by layer with the above method, then the parameters in DBN are set with global optimal. The unsupervised training process is shown in Fig. 3. After the unsupervised training is completed, the DBN needs a supervised training to reduce the estimated error and improve the regression accuracy. The common supervised training method is back propagation algorithm. In the supervised training, the whole parameters in DBN are updated at the same time and the training error is getting smaller. When the RBM training and supervised training are finished, the DBN can be utilized to estimation.

(5)

j

× (vi(0)

5. Stop training the RBM when the number of iterations reaches the vi hj recon reaches maximum, or the reconstruction error vi hj data the threshold.

(4)

i

bj = bj +

n

P (vi = 1 |H) =

ai +

hj

ij

2.2. Kalman filter (KF)

(7)

j=1

There are different kinds of noise during the battery measurement which increase the uncertainty in estimation. The statistical filter algorithms such as KF are wildly used for state estimation from noisy data. KF is an recursive filter that uses a series of measurements observed over time, containing statistical noise and other inaccuracies, and produces estimation that tend to be more precise than those based on a single measurement alone [36]. In order to apply KF to estimate the internal states of systems, the critical part is establishing state-space model. The state-space model can describe the internal processes of systems. The state-space model is made up of the time-varying state sequence x k and the observation sequence zk . The state-space model of a linear discrete time system can be described as follows.

where (x ) = (1 + e x ) 1. If the parameters of the RBM are known, the value of hidden neurons according to the input of visible layer can be calculated. 2.1.2. Training of DBN With respect to DBN, the training process can be divided into pretrain learning with RBM training and a global fine–tuning stage with supervised training algorithm. The RBM training is used to provide the whole parameters in DBN with initial value which can avoid local optimum and over-fitting issues. The global fine-tuning is applied to improve the diversity of the whole network and get more adaptive results. The target of RBM training is to make sure that the trained RBM model can reflect the distributions of input data as close as possible. In order to maximize the log likelihood, the update rule for the weights is shown as Eq. (8),

L( ) i, j

= vi hj

data

vi hj

recon

1. Initialize the parameters of RBMs randomly such as = { ij , ai , bj} and set the maximum number of iterations P for the entire training. 2. Calculate the value of the hidden layer neurons by equation hit p (hi |v (t ) ) , then update the visible layer neurons by equation vj(t + 1) p (vj |h(t ) ) . 3. Calculate the weights ij by equation according to the reconstruction error.

=

ij

+

× (p (hj = 1 |v (0) )·vi(0)

p (hj = 1 |v (t ) )· vi(t ) )

(12)

zk = Hk x k + vk

(13)

here, x k is the system state x at time k , uk reflects the control vector of system, zk stands for the observation value of system, wk represents the process noise which is assumed to be zero mean Gaussian distribution, N , with covariance, Qk : wk N (0, Qk ) , vk is the observation noise which is assumed to be zero mean Gaussian noise with covariance Rk : vk N (0, Rk ) , Fk represents the state transition model of x k , Bk is the control-input model, Hk is the observation model. In the KF algorithm, the initial state x 0 is needed to define the beginning state of the system. The uk is used to update the prior state estimate x k and error covariance P k , then the true state x k is calculated from the joint of observation sequence zk and x k . By minimizing the mean square error, the effects of the system noise are reduced and the optimal state estimate x k can be inferred. The details of KF are shown in Fig. 4.

(8)

where · data and · recon represent the expectations of the reconstructed distributions generated by the data and model, respectively. Contrastive Divergence algorithm is used to adjust the parameters in the RBM. The detailed steps are as follows.

ij

x k + 1 = Fk xk + Bk uk + wk

(9)

3. A lithium-ion battery SOC estimation model based on DBN-KF

4. Update the parameters of RBMs by following equations where represents the learning rate.

3.1. Overall framework of proposed model In this paper, a hybrid model which is composed of DBN model and 50

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Fig. 4. The process of Kalman filter.

KF algorithm is proposed to get more precise SOC estimations under dynamic conditions. Firstly, due to the battery SOC cannot be measured directly and it is affected by many internal factors and external factors, it is difficult to get accurate estimation results. The multi-layer structure of DBN can improve the fitting ability of the whole model under complicated conditions. Thus, the strong feature extraction ability of DBN can draw the internal relationship between the measurable data and the SOC. This promotion is meaningful for the SOC estimation under complicated conditions. Secondly, since the measured noises and other uncertain factors are inevitably involved in the SOC estimation, the KF is applied to reduce the uncertain affects and obtain more precise results. The outputs of the DBN model are applied as the observation sequence in the state-space model, which avoids the problems of establishing a complex state equation. This fusion can make the approach more suitable for dynamic conditions. Thirdly, the initial value the battery SOC need to be confirmed. The appropriate initial value can accelerate the convergence of the algorithm, while the improper initial value may lead large error in estimation. Therefore, an initial value model is utilized to provide the SOC estimation model with a suitable SOC initial value. The framework of proposed model is shown in Fig. 5. The processes of proposed model are shown as follows. Step 1. Sample the battery data at equal intervals and then split them into X (1) k = {vk , …, vk m , ik , ...,ik n , tk , ...,tk p} , where vk is the

voltage at sampling time k , and ik is the current at sampling time k , tk is the temperature at sampling time k . The m , n , p represent the input dimension of the voltage, current and temperature signals respectively. Step 2. Enter the X (1) k 1 into the initial value model, then the initial value SOCk 1 can be inferred. The initial value SOCk 1 is used as the beginning state of the battery. Step 3. Enter the X (2) k = {vk , …, vk m , ik , ...,ik n , tk , ...,tk p, SOCk 1} into the SOC estimation model, then the accurate SOC estimation SOCk at the sampling time k can be inferred. Step 4. Update the X (2) k + 1 = {vk + 1, …, vk m + 1, ik + 1, ...,ik n + 1, tk + 1, ..., tk p + 1, SOCk } with the SOCk . Step 5. Repeat step 3 to step 4, the accurate SOC estimation can be calculated. The proposed model is composed of the initial value model and the SOC estimation model. The details of those models are described in the rest sections. 3.2. Initial value model An estimator based on DBN is introduced in this section. First of all, the inputs of the model are the current, voltage and temperature. Due to the capacitive resistance in the battery, the current and voltage affect the present battery state. Therefore, the inputs of the model are set as [vk , …, vk m, ik , ...,ik n , tk , ...,tk p], and the output is SOCk , where vk is the voltage at sampling time k , ik is the current at sampling time k , tk is the temperature at sampling time k , the output is SOC at sampling time k .

Fig. 5. The framework of proposed model based on DBN-KF. 51

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Electrical Power and Energy Systems 110 (2019) 48–61

The symbols m , n , p represent the input dimension of the voltage, current and temperature signals respectively. After determining the inputs of the model, the numbers of DBN layers, the number of neurons at each layer need to be confirmed. The excessive number of layers increases the amount of computation. In order to decrease the computation and guarantee its fitting ability, a 3 layers of DBN is selected as the initial value model. The particle swarm optimization (PSO) algorithm is one of common methods which is usually applied to optimize structure of model [37]. Since the input dimension of current, voltage and temperature signals would affect the performance of proposed model, the PSO is applied to achieve an optimal parameters by minimizing the estimation error. It has to be pointed out that, the sampling rate of signals need to be considered since they would be different. If the sampling rate is too high, estimating SOC makes no sense and if the sampling rate is too low, the SOC could not be calculated for lacking of valid data. As a result, interpolation methods need to be applied to the sampling rate within a proper range. The model is established based on DBN model. The voltage, current and temperature signals are set as input and initial SOC value is the output. The historical data are utilized for model training. The dimension of each input is determined by PSO. The detailed step of initial value model are as follows. Step 1. Offline model training and optimizing based on historical data. Step 2. Construct the input vector [vk , …, vk m, ik , ...,ik n , tk , ...,tk p] from online voltage, current and temperature samples. Step 3. Obtain the initial SOC estimation value SOCk from the model (see Fig. 6).

DBN Ik

... Vk

SOCDBN

... Tk

...

...

...

SOCk

SOCk-1

SOCinitial

Transition

SOCCoulomb

Fig. 7. SOC estimation model.

...,tk p, SOCk 1]. In the real applications, the output of DBN may be influenced due to the measurement noise. To reduce the effects of noise and increase the robust of the DBN, filter technologies need to be fused with DBN. The KF and extend Kalman filter (EKF) are both typical technical approaches. In order to decrease the computation complexity, the DBN model is fused with KF to decrease the influence of the measurement noise. The structure of SOC estimation model is shown as Fig. 7. The DBN model is used as the SOC measurement function, and the Coulomb counting model is used as the state function, shown as Eqs. (16) and (17). Transition:

ik

3.3. SOC estimation model based on the DBN-KF Compared with the initial model, the SOC estimation model not only use the voltage, current and temperature as inputs, but also used the previous SOC as the reference to improve the SOC estimation accuracy. By inputting the historic voltage and current signals, the initial SOC value can be obtained from the initial valued model. The inputs of SOC estimation model are set as [vk, …, vk m, ik , ...,

n , tk ,

SOCk = SOCk

1

+

Measurement:

Fig. 6. The framework initial value model. 52

k I dt k 1 k

Crate

+ wk

(16)

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Reference Charge and Discharge Cycle

Reference Charge and Discharge Cycle

Pulse Load

Pulse Charge

Random Walk

Random Walk

Fig. 10. The battery test schedule.

^ DBN (SOC k 1, Vk , Ik , Tk ) = Zk = SOCk + vk

(17)

where Crate is the rated capacity of battery, Ik is the current at sampling time k , wk and vk are noises of system. DBN ( ) is the output of the DBN. The detailed process is shown as follows.

^ 1. The initial value SOC k 1 is calculated by the initial value model. 2. Predict the SOC at the time k from the system state function Eq. (16) and named as SOCk which contains process noise. 3. Update the error covariance P and then compute the Kalman gain Kk . 4. Estimate the k time’s SOC from the system measurement function

Fig. 8. The configuration of the test bench.

Voltage(V) Current(A)

Voltage Current

2

4.2

4.2

1

4.1

0

4.0

-1

3.9

-2

-4

3.0 -6 2.7

3.8 0

5000

10000

15000

20000

25000

0

100

200

300

-3 500

400

Time(s)

Time(s)

(b) The first 500 seconds of battery data

(a) The battery data of DST experiment 1.00

SOC

SOC

1.00

SOC

SOC

0.75

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0.99

0.25 0.98

0.00

0

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10000

15000

20000

25000

0

100

200

300

400

Time(s)

Time(s)

(c) The battery SOC of DST experiment

(d) The first 500 seconds of SOC

Fig. 9. Battery DST testing profiles. 53

500

Current(A)

3.3

Voltage(V)

-2

3.6

Current(A)

0

3.9

Voltage(V)

2

4.3

4.5

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4.1. Data set description and evaluation criteria 4.1.1. Dynamical stress test (DST) experiments The NCM 18,650 lithium-ion cell is applied in the experiment. The rated capacity of the cell is 2200 mAh. The limited charge voltage is 4.2 V and the limited discharge voltage is 2.7 V. The test bench is established as shown in Fig. 8. A set of two lithium-ion batteries (identified as CELL-1 and CELL-2) are tested in the temperature chamber. The testing data are collected using a DST profile, as specified in the US Advanced Battery Consortium (USABC) testing procedures [3]. In this experiment, the sampling rate of the battery test devise is 1 Sa/s (1 s for each sample data point). The current and voltage of the DST experiment are shown in Fig. 9. As can be seen from the Fig. 9, the variation of the current is simplified to simulate the battery real applications in hybrid electric vehicles. The experiment data will be used in this paper to evaluate the accuracy and robustness of proposed model.

Fig. 11. Reference charge and discharge cycle.

4.1.2. Randomized battery usage data set (RBUDS) In order to verify the adaptability of the proposed model, the different type of battery dataset is applied in the experiment. The data set is provided by the NASA PCoE [38]. A set of four 18,650 lithium-ion battery cells (identified as RW9, RW10, RW11 and RW12) were continuously operated using a sequence of charging and discharging currents between −4.5 A and 4.5 A. This type of charging and discharging operation is referred to here as random walk (RW) operation. Each battery set provides four parameters including battery test time, voltage, current and temperature. The detailed test steps are shown as Fig. 10.

Table 1 The initial capacity of battery. Battery samples

RW9

RW10

RW11

RW12

Capacity (Ah)

2.1022

2.0996

2.0995

2.1006

Eq. (17) and named as SOCk which are not inaccurate enough. ^ at k time, combine the 5. In order to estimate the optimal value SOC k SOCk and SOCk , then update the error covariance P . ^ is considered 6. Carry out the Step2 to Step5 repeatedly, and the SOC k to be the optimal SOC estimation result.

• Reference Charge and Discharge Cycle: Charge in the mode of

4. Experimental results and discussion In order to evaluate the proposed method, we use dynamical stress test (DST) profiles and Randomized Battery Usage Data Set (RBUDS) to verify the accuracy and generalization of the proposed model. The dynamical stress test (DST) profiles are the common tests for verifying the battery SOC estimation under dynamic conditions [3]. The battery test profiles of Randomized Battery Usage Data Set are set randomly and unpredicted instead of being fixed [38]. Thus, the accuracy of proposed model could be assessed when the battery operating under more complexed conditions.

• • •

SOC

Voltage (V) Current (A) Temperature (°C)

4.5

constant current 2.0 A until the voltage of the battery reaches 4.2 V. Then charge in the mode of constant voltage 4.2 V until the current of the battery bellows to 0.01 A. Discharge in the mode of constant current 1.0 A until the voltage of the battery bellows to 2.5 V (see Fig. 11). Pulse Load: Rest for 20 min, then load at 1 A for 10 min until the voltage of the battery reaches 2.5 V. Random Walk: Charge the battery to 4.2 V, then select the charge or discharge current at random from {−4.5 A, −3.75 A, −3 A, −2.25 A, −1.5 A, −0.75 A, 0.75 A, 1.5 A, 2.25 A, 3 A, 3.75 A, 4.5 A}. Pulse Charge: Rest for 20 min, then charge at 1 A for 10 min until

45

1.0

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SOC (1)

40 2

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3.5

-2 25

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-4 3.0

0

5000

20

10000

0

2000

4000

6000

Time (s)

Time (s)

(b) SOC data

(a) Voltage, current and temperature data

Fig. 12. Battery Random Walk test data (a) Voltage, current and temperature (b) SOC.

54

8000

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4.4

2

4.2

3 Voltage(V) Current(A)

2

4.2

3.8

Voltage(V)

0

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Current(A)

Voltage(V)

1 4.0

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Current(A)

4.4

-2

15000

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Time(s)

7500

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15000

Time(s)

(a) Training data

(b) Test data

Fig. 13. The training data and test data of DST experiments. 8

100

Actual SOC Estimation

6

Absolute Error(%)

SOC(%)

90

Abs error

80

70

4

2

60 0

2500

5000

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10000

12500

0

15000

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2500

5000

Time(s)

7500

10000

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Time(s)

(a) SOC estimation results

(b) SOC error profiles

Fig. 14. Experiment results for CELL-1.

5

100

4

Absolute Error(%)

90

SOC(%)

Abs error

Actual SOC Estimation

80

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2

1 60 0

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Time(s)

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(b) SOC error profiles

(a) SOC estimation results Fig. 15. Experiment results for CELL-2.

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used to reflect the performance of the estimation model. The evaluating indicators are as follows:

Table 2 Statistical results of estimated SOC for DST cycles. Battery samples

RMSE

MAX

MAE

CELL-1 CELL-2

0.5032 0.6953

4.4033 2.4083

0.4349 0.5668

(1) Root mean square error Root mean square error (RMSE ) reflects the degree of difference between the estimation value and true value. The calculation Eq. (18) is as follows:

the voltage of the battery reaches 4.2 V. Repeat this steps 1500 times.

RMSE =

To estimate battery SOC, the capacity needs to be inferred according to the definition of SOC. From the reference charge and discharge cycle test, we can calculate the capacity of lithium-ion by using Coulomb Counting method. It is considered that SOC equals 0% when the voltage of battery reaches 2.7 V, the SOC equals 100% when the voltage of battery reaches 4.2 V. Table 1 shows the capacity of four battery sets. When the battery works in the Random Walk step, the battery condition can be considered as a dynamically changing condition because the current varies randomly between [−4.5 A, 4.5 A]. The sampling rate for the voltage, current and temperature signals is 1 Sa/s (1 s for each sample data point). Some of the data as shown in the Fig. 12.

(18)

n

The Max Absolute Error ( MAX ) can reflect the max error between the estimation value and true value. A MAX of a precise SOC estimation model usually bellows 10%. The calculation of MAX is shown as Eq. (19).

MAX = max(|Xest , i

Xtrue, i |)

(19)

i = 1, 2, …, n

(3) Mean Absolute Error The Mean Absolute Error ( MAE ) can judge the actual situation of estimation error. The calculation Eq. (20) is as follows:

Actual SOC Estimation

10

Abs error 8

Absolute Error(%)

40

30

SOC(%)

Xtrue, i ) 2

(Xest , i

(2) Max Absolute Error

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(h) Estimation error of RW12 Fig. 16. (continued)

Table 3 Statistical results of estimated SOC for RBUDS.

MAE =

Battery samples

RMSE

MAX

MAE

RW9 RW10 RW11 RW12

2.264 2.6442 2.6345 1.9277

6.792 6.926 8.561 8.907

1.8126 2.1243 1.9038 1.5439

CELL-1 CELL-2 RW9 RW10 RW11 RW12

DBN

DBN-KF

DBN

DBN-KF

DBN

DBN-KF

1.1298 1.2544 5.2552 6.3350 6.4507 6.4077

0.5032 0.6953 2.2640 2.6442 2.6345 1.9277

4.5151 4.1574 56.102 51.515 27.132 59.108

4.4033 2.4083 6.792 6.926 8.561 8.907

0.8508 0.9457 3.5987 4.6435 4.9163 4.8760

0.4349 0.5668 1.8126 2.1243 1.9038 1.5439

(|Xest , i

Xtrue, i|)

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(20)

We first evaluate the accuracy and robustness of the proposed method with DST data. Then, the RBUSD test can verify the generalization of the proposed method. After that, the basic DBN and KF methods are used as comparative experiments. 4.2.1. Battery SOC estimation based on DBN-KF 4.2.1.1. SOC estimation under DST test. In this experiment, the k-fold cross validation is applied to evaluate the accuracy and robustness of the proposed method. The battery datasets are divided into two parts. One set of the battery data was applied as training data, while the other added with noised was used to examine the proposed model. After adding the noises, the voltage and current signals are similar with the real conditions. The training data and test data are shown as follow (see Fig. 13). The initial value that initial value model provide with CELL-1 equals 103.31% and the initial value of CELL-2 equals 103.08%. In the DST test, the optimization results are that the initial value model: m = 5, n = 5, p = 1, the number of first hidden layer q = 5 and SOC estimation model: m = 5, n = 5, p = 1, the number of first hidden layer q1 = 8 and the number of second hidden layer q2 = 3. Processed by using the

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4.2. Experimental results and discussion

Table 4 Comparison of SOC estimation using DBN approach and the DBN-KF approach. Battery samples

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proposed methods, the estimated results and absolute errors are shown as follows. In Figs. 14(a) and 15(a), it can be seen that the estimated SOC are close to true values In Figs. 14(b) and 15(b), the maximum estimated errors of SOC is located around 8750 s. Due to the appropriate initial value, the estimated SOC can accurately track the true value. The relative statistical information of SOC estimation results are revealed in Table 2. From Table 2, the results of MAX are below to 5% which means the proposed method has a high performance on SOC estimation. The low RMSE and MAE show that the estimated SOC can accurately track true value.

The RMSE and MAE are also increased which means the deviation between the estimated SOC and true value is increased. 4.2.2. Comparison of different SOC estimation methods Eventually, in order to verify the superiority of the proposed method, the DBN-KF method, DBN and KF method are compared. 4.2.2.1. Methods with/without KF. The DBN method is used as the comparative experiment for verify model accuracy. The final comparative experiment results are shown as follows. The relative statistical information of SOC estimation results are revealed in Table 4. (1) Comparison results under DST test (see Fig. 17) (2) Comparison results under RBUDS test (see Fig. 18)

4.2.1.2. SOC estimation under RBUDS. As it can be seen from Fig. 12, the current profiles of RBUDS are set randomly and the operating conditions are complicated. The algorithm is verified under RBUDS test which proves the universality and robust of proposed methods. Similarly, the battery dataset is divided into training set and testing set. Three of the battery samples are used for model training, the other is used for testing. In the RBUDS test, the optimization results are that the initial value model: m = 10 , n = 10 , p = 5, the number of first hidden layer q = 10 and SOC estimation model: m = 7, n = 7 , p = 1, the number of first hidden layer q1 = 10 and the number of second hidden layer q2 = 5. The experiment results are displayed in Fig. 16(a)–(h). Table 3 shows the evaluation criteria of the proposed method. Although the estimated SOC are close to the true value, the max absolute error is increased compared with the results for DST profiles, because the current profiles of RBUDS changes more disorderly and complexly,

From remarkable comparison between the DBN and proposed method, the MAX of proposed method manifest that it has better robustness. The RMSE and MAE of the proposed model are smaller than DBN, which indicates the estimated SOC of proposed model is closer to true value. As it can be seen from the results of MAX between the DST profiles and RBUDS profiles, both the DBN-KF and DBN methods achieve a worse estimation results on RBUDS profiles. However, the estimation errors of DBN-KF is lower than DBN which means the proposed model can achieve good SOC estimation under more complicated working conditions. Therefore, the proposed method has a better performance on the battery SOC estimation under dynamic conditions. 4.2.2.2. Methods with/without initial value model. To further evaluate the performance of initial value model, some extra experiments are 58

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(a) DST experiment results

(b) RBUDS experiment results Fig. 19. SOC estimation results by comparing different initial SOC value.

certain margin of error in the initial value. However, the estimation error would gradually decrease as the SOC estimation model works. In can be concluded that the initial SOC model would provide more suitable results compared with a random initial SOC value. Thus, the proposed model with initial value model could achieve a higher performance.

Table 5 The initial SOC value of estimator model. Battery samples

Estimation initial value

True initial value

Absolute error

CELL-1 CELL-2 RW9 RW10 RW11 RW12

103.31% 103.08% 15.32% 13.18% 18.07% 47.02%

99.96% 99.97% 16.31% 20.95% 10.77% 42.84%

3.35% 3.11% 1.01% 7.77% 7.3% 4.18%

4.2.2.3. Battery SOC estimation based on EKF. In this section, to compare the accuracy of SOC estimation under dynamic conditions, the extended Kalman filter (EKF) method which is the most applicable candidate for adequate SOC estimation [15–18,39,40]. The first-order Thevenin model is utilized as the battery model. Using DBN-KF and EKF, the estimation results are shown as follows. As it can be seen from the Fig. 20, the estimated error of the DBN-KF is smaller than EKF. From the Table 6, the accuracy of SOC estimation is obviously elevated by the DBN-KF since the max MAX is reduced from 5.6873% to 2.4033%, and simultaneously the stability index of the estimated SOC, namely RMSE , is further decreased from 2.8690% to 0.6953%, indicating that the estimation results are more reliable.

conducted in the revision. Besides the initial SOC value provided by the initial value model, four different initial SOC ranged from [0.2, 0.4, 0.6, 0.8] were set to evaluate the capability of convergence. The estimation results are shown as follows. As it can be seen from Fig. 19, it would cost more time to convergence if initial SOC value is unknown. In other words, the proposed framework which contains initial SOC model has a faster convergence speed which is more beneficial in the real application. The initial SOC value were listed in the following Table 5. It can be seen from Table 5, the initial SOC values provided from the initial value model are close to the true initial value. The absolute error of RW10 and RW11 are higher than 7%, which means there is still a

5. Conclusion and future work In this paper, a fused data-driven method based on the DBN-KF 59

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Fig. 20. Comparison between the proposed model and EKF.

References

Table 6 Comparison of SOC estimation using EKF and the proposed approach. Battery Samples

CELL-1 CELL-2 RW9 RW10 RW11 RW12

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MAE

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RMSE EKF

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1.5490 2.8690 6.1746 3.3890 3.3088 5.9660

0.5032 0.6953 2.2640 2.6442 2.6345 1.9277

5.5935 5.6873 9.0056 6.3194 5.8796 8.9388

4.4033 2.4083 6.792 6.926 8.561 8.907

1.0265 2.0372 5.7915 2.9138 2.8894 5.4659

0.4349 0.5668 1.8126 2.1243 1.9038 1.5439

model is proposed to estimate the battery SOC under the dynamic conditions. Firstly, with the powerful non-linear fitting ability of DBN and noise reduction capacity of KF, the complex and implied relationship between measureable parameters and SOC can be fitted accurately. Then, an initial model is utilized to provide an appropriate SOC initial value which solves the uncertainty of the model. Therefore, the battery SOC can be obtained with high estimation accuracy and stability. The experimental results for DST profiles show that RMSE of the SOC estimation is lower than 0.7, the max absolute error is lower than 2.5%, and the mean absolute error is lower than 0.57. The results under more complex conditions also indicate that the proposed method has a better performance. As a result, this method can obtain good accuracy on the battery SOC estimation under dynamic conditions. In the future, we will find a way to improve the correlation between SOC and the inputs of model, because the proposed method has low operating efficiency on training DBN if the correlation is low. As the continuous charging and discharging of the battery, the battery performance degrades and its capacity decreases, the joint state estimation considering both SOC and SOH will be focused in our next work. Acknowledgements This work was partially supported by National Natural Science Foundation of China under Grant Nos. 61771157, 61571160 and Key Project B Class of Beijing Natural Science Foundation No. KZ201710028028. Appendix A. Supplementary material Supplementary data to this article can be found online at https:// doi.org/10.1016/j.ijepes.2019.02.046. 60

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