Battery states online estimation based on exponential decay particle swarm optimization and proportional-integral observer with a hybrid battery model

Journal Pre-proof Battery states online estimation based on exponential decay particle swarm optimization and proportional-integral observer with a hybrid battery model

Xiaolong Yang, Yongji Chen, Bin Li, Dong Luo PII:

S0360-5442(19)32204-2

DOI:

https://doi.org/10.1016/j.energy.2019.116509

Reference:

EGY 116509

To appear in:

Energy

Received Date:

10 July 2019

Accepted Date:

06 November 2019

Please cite this article as: Xiaolong Yang, Yongji Chen, Bin Li, Dong Luo, Battery states online estimation based on exponential decay particle swarm optimization and proportional-integral observer with a hybrid battery model, Energy (2019), https://doi.org/10.1016/j.energy.2019.116509

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Journal Pre-proof

Battery states online estimation based on exponential decay particle swarm optimization and proportional-integral observer with a hybrid battery model Xiaolong Yang*, Yongji Chen, Bin Li, Dong Luo State Key Laboratory of Advanced Design and Manufacture for Vehicle Body, Hunan University, Changsha, Hunan 410082, China

Journal Pre-proof HIGHLIGHTS 

Hybrid battery model is used to reflect the dynamic behavior of battery capacity.



An improved PSO is introduced to realize accurate online parameters identification.



Battery capacity is estimated online using accumulated charge and OCV variation.



The estimated capacity could converge quickly with error less than 2%.



The proposed method can result in accurate SOC estimation with error less than 1%.

ABSTRACT The rational design of the battery management system requires a high-fidelity battery state estimation method. However, the nonlinear varieties of battery parameters, external interference and the dynamic behaviors of battery capacity will cause big problems for battery state estimation. A new state estimation method for lithium-ion batteries is proposed. A hybrid battery model is firstly established to better reflect the dynamic behaviors of battery capacity and voltage. Then the model parameters are identified online using the exponential decay particle swarm optimization (EDPSO). Finally, a proportional-integral observer (PIO) is designed for battery state-of-charge (SOC) estimation. In addition, the battery maximum available capacity (Cmax) is online estimated using the accumulated charge and variation of battery open-circuit voltage (OCV), which helps to update SOC estimation at different aging cycles. A verifying experiment is carried out based on the urban dynamometer driving schedule (UDDS) cycles. The results indicate that the proposed method has good performance and high accuracy. The online estimated parameters consist well with experimental data, the error of the terminal voltage is less than 0.02 V. The error of the estimated SOC can be controlled within 1%. Moreover, the estimated capacity could converge in 12 minutes with an error less than 2%. Keywords: Lithium-ion battery Hybrid battery model State of Charge estimation Maximum available capacity estimation Exponential decay particle swarm optimization Proportional-integral observer

Journal Pre-proof 1. Introduction As the main power of electric vehicles, lithium-ion batteries have been rapidly developed [1]. However, there are still some key issues needed to be solved for the practical application. For example, due to the strong non-linear characteristics and time-varying parameters of lithium batteries, it is difficult to estimate internal states from the external characteristics, such as state of charge (SOC) and state of health (SOH) [2,3]. An efficient battery management system of electric vehicles should achieve an accurate and real-time SOC estimate, which can improve the energy efficiency, avoid over-charge and over-discharge to prolong the longevity of battery cells [4–6]. A variety of proposed SOC estimation techniques could be divided into non-model based methods and model-based methods [7,8]. The most widely used model-based SOC estimation methods require a high-accuracy battery model. At present, the static equivalent circuit model is adopted for the SOC estimation, such as Rint, RC, Thevenin and PNGV models, etc [9]. Some studies show that the discharging process of lithium-ion batteries is greatly affected by the temperature and current rate. Usually the maximum available capacity (Cmax) cannot be released completely [10,11]. This dynamic behavior of the battery capacity is generally neglected in traditional circuit models. J. Manwell suggested a kinetic battery model (KiBaM) [12]. Owing to the rate capacity effect and the recovery effect, the KiBaM could better reflect the dynamic behavior of the battery capacity. But it cannot describe the current–voltage (I-V) characteristics which are very important for the electrical circuit simulation [13]. Therefore, KiBaM is combined with an equivalent circuit model called the hybrid battery model here in order to reveal the actual working state of lithium-ion batteries. For above battery model, there are some parameters needed to be identified. The traditional methods for identifying model parameters usually are offline based methods. A series of battery experiments could be used to obtain model parameters, which is time-consuming and difficult to operate accurately for complex real-world applications [14]. Recently, the online identification strategy has been widely used to obtain battery parameters. Plett [15] proposed a dual extended Kalman filter (DEKF) to online estimate battery parameters and SOC simultaneously; Xiong Rui et al [16]. used a H∞ filter to receive the model parameter online estimation and then applied unscented Kalman filter (UKF) to evaluate SOC; Zhang Li et al [17]. used a memory-redirected least squares (MRLS) method to adaptively identify model parameters and designed a PI observer to realize the SOC estimation; Min Ye et al. [18] proposed an adaptive unscented Kalman filter (AUKF) algorithm for the online estimation of battery parameters and SOC. However, for the parameter identification, the accuracy of the least square method is poor. Even the H∞ filter and UKF can meet the accuracy requirement,

Journal Pre-proof those complicated algorithms result in poor real-time performance and are difficult for the large-scale application. Furthermore, although some high order algorithms such as the sliding mode observer, PI observer and Kalman filter can compensate certain estimation errors of SOC, those algorithms ignore the problem of the capacity fade with the battery aging. The accuracy of SOC for used batteries is not comparable to new batteries. It is known that SOH is used to describe the degree of battery aging, which is usually indicated by the actual capacity fade. The accurate SOH estimation is helpful for the rational utilization of batteries, aiming to prolong cycle life and avoid sudden failure [19,20]. Usually, most researches use chemical analysis or electrical circuit for the SOH estimation. Chemical analysis methods based on incremental capacity analysis (ICA), differential voltage analysis (DVA) [21] and electrochemical impedance spectroscopy (EIS) [22] can estimate SOH accurately. However, those methods cannot realize online identification. Moreover, the testing process is complex and only applicable to the laboratory environment. Considering that, the electrical circuit based SOH analysis methods have become a research hotspot. Chao Hu et al. [23] proposed a multiscale framework with EKF for the SOC and capacity estimation; Akram et al. [24] estimated and monitored the battery SOH using a recurrent neural network (RNN). Ning B et al. [25] presented parameter adaptive sliding mode observers (PASMO) to evaluate battery SOC and SOH online. J.G Wei et al. [26] proposed a new SOH estimation method, which was based on a novel support vector regression with capacity and impedance as state variable. Although the battery aging effect is taken into account in the SOC estimation, these modelbased cyclic states estimation methods still have drawbacks. Firstly, the equivalent circuit model can only reflect the dynamic behavior of the battery voltage but ignore the nonlinear characteristics of the battery capacity. Secondly, the error of the SOC itself is ignored when it is applied to calculate the model parameters and SOH. Thereafter, the error will be transferred to the next step estimation, which will certainly decrease the estimation accuracy. Therefore, it is particularly crucial to maintain the accuracy of the battery model at different aging stages and working conditions. To solve those above problems, a new state estimation method is rationally designed based on a hybrid battery model. The flow chart of the new algorithm is shown in Fig.1. Firstly, the EDPSO algorithm is proposed to identify the open-circuit voltage and electrical impedance of the circuit model online. Furthermore, the battery maximum available capacity is estimated and updated online. Thereafter, a PI observer is designed for the SOC estimation. Compared with previous research, the proposed method in this study has many outstanding features. 1) Considering the dynamic behavior of the battery capacity, the characteristics of the battery can

Journal Pre-proof be better simulated; 2) The battery capacity directly estimated via online identified model parameters can reduce the system error influence; 3) EDPSO algorithm has remarkable local search capability to obtain more accurate battery parameters, which is easy to be conducted. 4) The PI observer has favorable real-time performance and strong robustness to disturbance. The following parts of thus paper are organized as follows: Section 2 describes the experiment setup. In section 3, the hybrid battery model and online parameter identification method based on EDPSO are discussed in detail. In section 4, the online state estimation method is described. In section 5, UDDS cycles are applied to the hybrid battery model and state estimation algorithms to verify the new method. Finally, some conclusions are given in section 6. i(t),(t-1)…

Start Loading profiles 2

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Fig. 1. Flowchart of the co-estimation algorithm. 2. Experiment Setup A systematic experiment is carried out to test and verify the new algorithm. The Samsung 18650-35E lithium-ion batteries are used as the research objects. The basic specifications of battery cells are given in Table 1. As shown in Fig. 2(a), the battery test bench consists of a battery test instrument with 8 independent channels, a thermal chamber, and a hosting computer. Fig. 2(b) illustrates the battery test projects, including the characterization test and aging test. Their loading profiles are displayed in Fig. 3. Table 1. Basic specifications of the battery cells. Nominal voltage

Cut off voltage

Nominal capacity

Actual capacity

3.6 V

2.65 / 4.2 V

3.5 Ah

3.45 A

Journal Pre-proof

Every 50 aging cycles

Capacity test at 25℃ Thermostat

PC

Battery test instrument

Rate discharge test at 25℃ HPPC test at 25℃ UDDS test at 25℃ for aging cycles

(a)

(b)

Fig. 2. Battery experiments: (a) test bench, (b) test schemes. As shown in Fig. 3(a), the capacity test is usually executed to measure the battery capacity. The battery is firstly charged at the CC (constant current)-CV (constant voltage) mode and then discharged with a constant current of 1 A until the terminal voltage reaches the discharge cutoff voltage. In Fig. 3(b), the rate discharge test is conducted by measuring the battery discharge capacity at different rates (0.5C, 1C, 1.5C, 2C), which is used to calculate the parameters of the kinetic battery model. Fig. 3(c) shows the hybrid power pulse characterization (HPPC) test Current / A

profile, in which a series of dynamic current pulses are applied to the battery cell. The HPPC 2

test is usually 0used for the offline parameter identification. The UDDS cycles are employed to -2

-4 verify the proposed battery model and the states estimation algorithm. ItsRate current test profile is -6 -8

shown in Fig. 30(d).

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Journal Pre-proof aging cycles. In addition, the OCV-SOC curves almost keeps unchanged at different aging cycles except at low SOC region, which is consist with other people’s experiments [27]. The maximum OCV difference for all cycles is less than 0.03V in the 10%-90% SOC region. Capacity / Ah

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Fig. 4. The experimental capacity and OCV-SOC at different cycle numbers: (a) capacity, (b) OCV-SOC. 3. The Hybrid Battery Model As shown in Fig. 1, in order to reflect the dynamic behavior of the battery in a real situation, a hybrid model is proposed, which includes a kinetic battery model (KiBaM) and an equivalent circuit model. The KiBaM is applied to reflect the dynamic behavior of the battery capacity, and the equivalent circuit model is used to depict the dynamic behavior of the battery voltage. 3.1. The Kinetic Battery Model The kinetic battery model is suitable to capture the dynamic behavior of the battery capacity. KiBaM was first applied in Lead-acid battery [12] and is expanded to Lithium-ion battery here. It assumes that a battery has two charge wells, where the whole electric charge is distributed with a capacity ratio c (0 < c < 1) between the two wells, as shown in Fig. 5. c

h2

y2

h1

y1

k

Fig. 5. Structure of KiBaM.

i (t )

Journal Pre-proof The right part y1 is called the available capacity well, which can deliver charges to the external load directly. While the left part y2 is called the bound capacity well, which supplies charges only to the available charge well via a valve k. The charge flow rate from the bound charge well to the available charge well is depended on k and the height difference between two wells h1 and h2. Assuming that the battery discharges at a constant current I, the dynamic flow equations of the KiBaM can be established based on the above description, as shown in Eq. (1) and Eq. (2):  ( y (0)k c  I )(1  e  k t ) Ic(k t  1  e  k t ) y ( t )  y (0)   1 1   k k  t  1  e  k t ) I (1  c )( k  y (t )  y (0)  y (0)(1  c)(1  e  k t )  2 2  k 

(1)

 (t )  h2 (t )  h1 (t )  y2 (t ) / (1  c)  y1 (t ) / c

(2) '

Where 𝛿(𝑡) denotes the height difference between two wells, and 𝑘 = 𝑘/[𝑐(1 ― 𝑐)] is a constant related to the diffusion rate. The parameters like c and k could be obtained by a series of rate discharge tests [28]. Considering that the current is piecewise, y1 and y2 can be calculated using Eq. (1) section-by-section. The unusable capacity 𝐶𝑛𝑜𝑡 = (1 ― 𝑐)𝛿(𝑡) can be expressed as Eq. (3): I 1  e  k (t t0 ) Cnot (t )  (1  c)  [ (t0 )e  k (t t0 )   ] c k

(t0  t  td )

(3)

Since the unusable capacity is considered, the traditional Ampere-Hour (AH) method cannot be applied directly to calculate the battery SOC. Therefore, an enhanced Ampere-Hour method is given for the battery SOC estimation, as shown in Eq. (4): SOCEA (t ) 

Cavailable (t ) 1  t  SOCinit  i (t )dt  Cnot (t )   Cmax Cmax  t0

(4)

For better understanding the relationship between the total capacity Cmax, available capacity Cavailable and unusable capacity Cnot, one may consider the following example. At some conditions, the total capacity Cmax of the battery is not completely discharged when it reaches the cut-off condition during the discharging process. However, its available capacity Cavailable is close to 0 at this time (Fig. 5 right well) due to polarization or temperature. The remained capacity of the battery becomes the unusable capacity Cnot (Fig. 5 left well). According to the definition of SOC in Eq. (4), SOC can be regarded as 0 in this situation. For numerical calculation, usually a discretized form of the Eq. (4) is needed, the expression of SOC can be rewritten as follows: SOC (t  1)  SOC (t ) 

1  FCnot (t  1)  i (t  1) Cmax

FCnot (t  1)  Cnot (t  1)  Cnot (t )  / i (t  1)

(5) (6)

Journal Pre-proof 3.2. The Equivalent Circuit Model Because the online parameter identification can effectively reduce the model error caused by time-varying parameters, there is no high requirements for the accuracy of the circuit model [29]. So the Thevenin model is adopted, which consists of a voltage source (E0), an ohmic resistor (R1), a parallel polarization capacitor (C2) and resistor (R2). According to Kirchhoff's law, the terminal voltage V0 and polarization voltage V2 can be expressed as Eq. (7): V0  E0  IR1  V2  V2  V2 / ( R2 C2 )  I / C2

(7)

Under normal working conditions, the change of the open-circuit voltage E0 of the battery is small in a short time. Therefore, E0, R1, R2 and C2 are selected as slow time-varying parameters [25]. The derivation of V0 can be expressed as Eq. (8): V0  R1 I  ( R1  R2 ) I / ( R2 C2 )  V0 / ( R2 C2 )  E0 / ( R2 C2 )  [ R1 ( R1  R2 ) / ( R2 C2 ) 1 / ( R2 C2 ) E0 / ( R2 C2 )]  [ I I V0 1]T

(8)

 [1  2  3  4 ]  [ 1  2 3  4 ]

T

Where, 𝜃(𝑅1,𝑅2,𝐶2,𝐸0) = [𝜃1 𝜃2 𝜃3 𝜃4] is the parameter matrix to be identified and 𝜇(𝐼,𝑉0) = [𝜇1 𝜇2 𝜇3 𝜇4] is the system input. 3.3. The Online Parameter Identification by EDPSO For the above hybrid battery model, there are some parameters needed to be identified. Usually, a set of experiments are carried out to obtain those parameters. However, the static battery model with parameters offline identified in the ideal condition is unable to satisfy various actual applications. To obtain an accurate and real-time battery model, an improved PSO algorithm is adopted to identify parameters online, which is easy to implement and is not affected by the model linearity [30]. The PSO is a global search algorithm that obtains the optimal solution by iteration. Each particle continually updates its speed and position to bring the entire population close to the optimal solution [30]. The standard PSO sets up a group of random particles. Assuming that the total number of particles is n, the position 𝑥𝑖 and velocity 𝑣𝑖 of the 𝑖𝑡ℎ particle in the Ddimensional space can be expressed as Eq. (9) and Eq. (10). xi  ( xi1 , xi 2 , , xiD ) i  1, 2, , n

(9)

vi  (vi1 , vi 2 , , viD ) i  1, 2, , n

(10)

During each iteration, particles will update their position and velocity according to an individual optimal solution 𝑃𝑗 and global optimal solution 𝑃𝑔, as shown in Eq. (11) and Eq. (12):

Journal Pre-proof vik 1    vik  c1   k ( Pi ,kj  xik )  c2  k ( Pgk  xik )

(11)

xik 1  xik  vik 1

(12)

Where k denotes the number of iterations. ζ and η are random numbers between 0 and 1. 𝑐1 and 𝑐2 are learning factors. ω is the inertia coefficient, which can effectively improve the convergence speed of the algorithm. Further research shows that the dynamic inertia coefficient is more efficient than the fixed inertia coefficient. M. Clerc et al. [31] used a global contraction factor to control the particle velocity, which can considerably improve the convergence speed. In order to better balance the global and local search capability of particles, a new exponential decay strategy for particle velocity is introduced as follows: vik 1   k 1  vik  c1   k ( Pi ,kj  xik )  c2  k ( Pgk  xik )

(13)

 k  0  e    k / N

(14)

Where 𝜔0 is the initial value of the inertia coefficient. N is the maximum number of iterations. α is a regulatory factor used to adjust the decay rate of the exponential function. Because of the inherent character of an exponential function, the decay step will become lager in the early procedure than that of the later procedure, which realizes a lager search region to ensure the global exploration ability. While at the final stage, a small step will keep the particle velocity in a small range, which is helpful to improve the local exploitation performance. Another important issue for the PSO algorithm to solve practical problems is the particle boundary. In the traditional PSO method, particles that do not satisfy constraints are discarded directly, damaging the integrity and diversity of the particle swarm. Moreover, numerous discarded particles may increase the number of iterations, resulting in a slow solution speed or even failure of optimization [30]. Herein, the concept of the particle tempering is introduced. If the particle position exceeds the boundary, the particle will fall back to its historical optimum position and re-participate in the next round of evolution instead of abandoning it directly. This method can not only remain the diversity and integrity of particles but also avoid the resampling process. The convergence speed is faster, and the probability of successful optimization is also improved. The flow table of the new EDPSO is shown in Table 2. Table 2. Flow table of the EDPSO method. Step 1: Determine the identified parameters 𝜃(𝑅1,𝑅2,𝐶2,𝐸0) = [𝜃1 𝜃2 𝜃3 𝜃4] and input data 𝜇(𝐼,𝑉0) = [𝜇1 𝜇2 𝜇3 𝜇4]. Step 2: Obtain the battery input current 𝑖(𝑘) and the actual terminal voltage 𝑢(𝑘). Step 3: Parameters estimation using the EDPSO method. 𝑘 𝑘 a) Randomly generate N initial particles 𝜃0(𝑘 = 1,2,…,𝑁) for parameters, and give an initial weight 𝜔0(𝑘 = 1,2,…,𝑁)

Journal Pre-proof for each particle. For k=1, 2 …, N 𝑘 ―𝛼 ∙ 𝑘/𝑁 b) Update weight coefficient: The weight of each particle is defined as 𝜔 = 𝜔0 ∙ 𝑒 . 𝑇

𝑘 𝑘 𝑘 𝑘 c) Calculate fitness: The fitness of each particle 𝑓(𝜃𝑖 ) = 𝐸[𝑒(𝜃𝑖 ) 𝑒(𝜃𝑖 )]) is calculated by 𝑒(𝜃𝑖 ) = 𝑦(𝑘) ― 𝑦(𝑘). Where

y(k) and 𝑦(𝑘) are the measured and predicted battery terminal voltage, respectively. c) Evaluate particle individual optimal solution Pj and global optimal solution Pg. 𝑘 d) If the stopping criteria are satisfied, as |𝑓(𝜃𝑖 )| < 𝑒𝑚𝑖𝑛 or 𝑘 > 𝑁𝑚𝑎𝑥. Go to End.

e) Update the position and velocity of each particle: vik 1   k 1  vik  c1   k ( Pi ,kj  xik )  c2  k ( Pgk  xik )   k 1 k k 1   xi  xi  vi

f) Particle tempering and re-participate in the calculation if the physical constraint is not gratified, as follows: Exceed constraints   Pi , j xi (t )     xi (t  1)  vi (t ) Satisfy constraints

End Step 4: Output the optimal parameters of Pg. Step 5: After a certain sampling interval, the algorithm goes back to Step 3 to revise the model parameters online.

Using the modified PSO algorithm above, the parameters, such as E0, R1, R2 and C2, of the hybrid battery model can be identified online. Among those parameters, the open-circuit voltage is a very important parameter for the battery model and state estimation. Usually it can be evaluated by a piecewise function such as 𝐸0 = 𝑎 ∙ 𝑆𝑂𝐶 + 𝑏 [17], which may introduce the SOC error into the battery model. To avoid this problem, E0 is identified online directly here. However, the sampling error and model uncertainty would greatly affect the accuracy of online identification. Therefore, a weighted result in Eq. (15) is taken as the value of E0. This treatment may simultaneously weaken the influence of uncertain errors in these two methods. E0   (a  SOC  b)   E0, online  aˆ  SOC  bˆ

(15)

Where λ and β are weighting factors, 𝐸0,𝑜𝑛𝑙𝑖𝑛𝑒 is the online identified open-circuit voltage. 4. Battery State Estimation 4.1. The SOC Estimation While using particle swarm optimization to identify battery parameters, a real-time online observer is also needed to estimate the battery SOC. A PI Observer is used for the state estimation, which is robust to withstand the disturbance and model uncertainty. And the steadystate error can also be eliminated [32]. Owing to the harsh working environment and inherent nonlinearity of lithium-ion batteries, it could not be sufficient to reflect the actual battery states with a linear system. Considering the influence of external disturbance d(t), the state equation of the battery can be described as

Journal Pre-proof follows [33]:  x  Ax  Bu  Fd (t )   y  Cx  Du x   SOC V2 

T

y  V0  bˆ

0  0   A 0  1  R2 C2   C   aˆ 1

(16) uI

 (1  FCnot )  B   Cmax   1 / C2 

D  R1

The structure diagram of the SOC estimation method based on the PI observer is shown in Fig. 6. i(t)

y(t)

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PI Observer

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y (t )

D

Fig. 6. Structure diagram of a PI observer. Where 𝐾𝑃 and 𝐾𝑖 are the proportion and integral factor used to adjust the feedback gain of the SOC observer. And F is used to describe the influence of the system nonlinearities, which could be obtained by experiments and some attempts [34]. The PI observer is designed as follows:  x  Ax  Bu  Fd (t )  K p  ( y  y )  K i      y  y

(17)

Considering the actual applications of Lithium-ion batteries, the disturbance d(t) could be mainly caused by temperature variation and sensor noise. The variation rate of temperature is slow. In addition, the sensor error is generally considered to be white Gaussian noise. Hence, it can be assumed that 𝑑(𝑡) = 0. When 𝑒𝑥 = 𝑥 ― 𝑥, 𝑒𝑑 = 𝑑(𝑡) ―𝜔 and 𝐹 = 𝐾𝑖 are defined, the error equations can be written as Eq. (18):  ex   A  K p C K i   ex  e   Ae  x  e    C    0  ed   d  ed 

(18)

Ae could be arbitrarily configured only if the following matrix pair is observable, which is equivalent to the following equation:

Journal Pre-proof   A Ki   rank   , C 0   3  0 0  

(19)

Substituting the relevant state equation into the matrix pair, as follows: 0 0 K i1      A K i    rank     rank   0 1 / R2 C2 K i 2    3   C 0     aˆ 1 0   

(20)

In Eq. (20), the matrix Ae is Hurwitz, enabling it to be arbitrarily assigned. The gain matrix 𝐾𝑃 and 𝐾𝑖 for the PI observer could be selected by a pole assignment method and linear quadratic method. Based on the above analysis, 𝑒𝑥→0 and 𝑒𝑑→0 as 𝑡→∞ means that the estimated result of the system state such as the battery SOC would converge the real state when 𝑡→∞. 4.2. The Capacity Estimation From the above analysis, it can be seen that the maximum available capacity is one of the most important parameters for battery SOC estimation. However, with the continuous usage of batteries, the maximum available capacity will decrease and no longer equal to its rated capacity due to the aging. Therefore, it is necessary to estimate the battery capacity online. By rewriting Eq. (4), Cmax can be expressed as Eq. (21): t2

Cmax 

 i(t )  Cnot (t2 )  Cnot (t1 ) t t 1

(21)

SOC (t2 )  SOC (t1 )

Where t1 and t2 are the beginning time and ending time of the capacity estimation period, respectively. However, there is definitely a difference between the current Cmax and the last updated value, which results in a certain SOC estimation error (Eq. (4)). In return, the estimated Cmax will be affected by the inaccurate SOC value. Therefore, the SOC here is obtained via an OCV-SOC look-up table using the online identified open-circuit voltage. Because the nonlinear monotonic relationship between the open-circuit voltage and SOC has no obvious difference with the temperature and battery aging, as shown in Fig. 4 and other research [35,36], the SOC obtained by the OCV-SOC lookup table is applicable. As the reference OCV is identified based on measurable data, rather than the conventional observer with an assumed constant capacity, the estimated capacity could converge to the actual value. Considering the slow change of the battery capacity (Fig. 4), the maximum capacity could be estimated and updated at regular intervals. In addition, the battery SOH can be expressed as Eq. (22): SOH  Cmax /Cnew

(22)

Journal Pre-proof Where Cnew is the maximum available capacity for new cells. 5. Experimental Verification 5.1. The Verification of Hybrid Battery Model In order to verify the ability of KiBaM to reflect the battery dynamic capacity behavior, a series of constant current discharge test are carried out under different current rates from 0.5C to 2C. The battery parameters are identified offline by the HPPC test. The responses of the unusable capacity and terminal voltage are shown in Fig. 7.

Cnot / mAh

(a) 100 60 40 20 0 4.2

2C 1.5C 1C 0.5C

80

0

20

40

Terminal voltage / V

(b) 4.2

4 3.8 3.6 3.4

60

80

100

120

Time / min Measured value Hybrid model Thevenin model

3.8 3.4 3.0 2.6

2C 0

20

1.5C 40

1C 60

0.5C 80

100

120

Time / min 3.2

Fig. 7. Responses of the unusable capacity and terminal voltage (a) Cnot (b) V0. 3

Fig. 7(a) shows the variation of unusable capacity, which will gradually raise with the

2.8

increasing discharge depth and current rate. Fig. 7(b) compares the terminal voltage responses obtained from the Thevenin model and proposed hybrid model. As it can be seen, the simulated

2.6

0

1000

2000

3000

4000

5000

6000

7000

results of the hybrid model can better match experimental results, especially when the battery is close to full discharged. However, the terminal voltage errors of the Thevenin model are obvious and increase significantly as the increasing discharge current due to the neglect of the unavailable capacity. Since KiBaM can determine the unusable capacity from the total capacity, the proposed hybrid model can simulate the actual working states of batteries more accurately. The proposed adaptive hybrid battery model with online identified parameters is further verified through the UDDS cycles. The model parameters are online identified using the EDPSO algorithm. Initial parameters and coefficients required for the EDPSO algorithm are listed in Table 3. Table 3. Initial parameters and coefficients. Parameter

Value

Journal Pre-proof n N C1 C2 ω0 α E0 R1 R2 C2 (a) 4.8

E0 Reference E0 Estimated

4.4 3.6

3.9

3.2

3.8

3.4

2.8

3.7

3.2

2.4

3.6 60

2.0

0

20

80

40

3.0 161

100

60

80

(b) 0.3

R1 Reference R1 Estimated

0.08

R1 / Ω

E0 / V

4.0

30 30 2 2 0.8 1.43 3.75V 0.2Ω 0.2Ω 500F

163 165

0.2

0.06

0.1

0.04 60

0

100 120 140 160

0

20

80

40

Time / min R2 Reference R2 Estimated

0.03

R2 / Ω

0.2 0.025 0.1

0.02 60

80

(d) 6000

100

0 -0.10

0

20

40

60

80

60

80

100 120 140 160

Time / min

C2 / F

(c) 0.3

100

1500

4000

1200 60

2000 0

100 120 140 160

C2 Reference C2 Estimated

1800

0

20

80

40

Time / min

60

100

80

100 120 140 160

Time / min

Fig. 8. Online parameter estimation curves: (a) E0; (b) R1; (c) R2; (d) C2.

3.8 3.2

3.4

0.04

-0.02

3.0 160

0

Thevenin model Hybird model

0.06 0.02 Reference Thevenin model Hybrid0 model

3.1

3.0 2.6

(b) 0.08 Error / V

Reference Thevenin model Hybrid model

Voltage / V

Voltage / V

(a) 4.2

20

161

162

Time / min

40

60

80

100 120 140 160

Time / min

-0.04

0

20

40

60

80

100 120 140 160

Time / min

Fig. 9. Comparison of the terminal voltage under UDDS cycles. The online estimated parameters (E0, R1, R2 and C2) are shown in Fig. 8, where the reference results (red points in Fig. 8) are obtained via the electrochemical impedance spectroscopy [22]. As it can be seen, the estimated parameters with initial errors can quickly converge to reference curves and then fluctuate with a small error. Fig. 9 compares the model terminal voltage and experimental result under UDDS cycles. The terminal voltage of the adaptive hybrid model is substantially coincident with the experimental result. The error increased at the end of discharge

Journal Pre-proof also can be controlled within 0.02 V, which may be caused by the modeling error of the oneorder RC model. The results indicate that the adaptive hybrid battery model may have better reliability for reflecting the working characteristics of lithium batteries under complex working conditions. 5.2. The Verification of SOC Estimation In order to verify the applicability of the proposed SOC estimation method based on the PI observer, the full-charged battery was cyclically discharged using the UDDS test until the battery terminal voltage reached the discharge cut-off voltage. The SOC estimation results for the new battery are shown in Fig. 10. 1.0

1.0 Reference SOC-init=0.6 SOC-init=0.8

0.6 0.4

0

0

20

40

60

80

100 120 140 160

0.02 0 -0.02

0

25

50

Time / s

20

40

60

0

20

126

128

Time / min

40

60

80

100 120 140 160

Time / min

0.02

DEKF MRLS-PIO EDPSO-PIO

SOC-init=0.8 SOC-init=0.6 ±1% bound

0.01 0

-0.01 -0.02

0

0.21 124

0.03

Error / V

0.1 0.1 0 0 -0.05 -0.1

SOC-init=0.6 SOC-init=0.8 ±1% bound

0.04

0.2 0.2

Reference DEKF MRLS-PIO EDPSO-PIO

0.23

0.04

0.4 0.3 0.3

0.25

0.4 0.2

Time / min

Error Error

0.6

0.2 0

Reference DEKF MRLS-PIO EDPSO-PIO

0.8

SOC

SOC

0.8

80

100 120 140 160

Time / min

(a)

0

20

40

60

80

100 120 140 160

Time / min

(b)

Fig. 10. SOC estimation: (a) results with initial SOC error (b) results of different methods. Fig. 10(a) shows the SOC estimation result with uncertain initial SOC value (SOC0). Two initial values are tested, 0.8 and 0.6, respectively. It can be seen that the estimated SOC converges to the reference value quickly no matter what the initial SOC is. When the initial SOC sets to 0.8 and 0.6, it takes 400 and 600 s to converge to the reference, respectively. After convergence, the error can be controlled within 1%. It can be concluded that the new algorithm has good convergence and does not require high accuracy of initial SOC. But the more accurate the initial SOC is, the faster the simulation results converge to the reference value. Fig. 10(b) shows the comparison between the proposed EDPSO-PIO method and the methods described in Ref [15] (DEKF) and Ref [17] (MRLS-PIO). Compared with the experiment data (shown in Fig. 10(b) as reference value), all methods obtain good results. The EDPSO-PIO shows the best result which can control the SOC estimation error within 1%, while DEKF and MRLS-PIO methods can confine most errors within 2%. However, there are

Journal Pre-proof relatively large errors (red line in Fig. 10(b), but still less than 1%) at the end of discharge for EDPSO-PIO, which can be attributed to the terminal voltage error caused by the aforementioned modeling errors. The quantitative comparison results of different methods are shown in Table 4. The mean square error and the maximum error of the SOC based on EDPSOPIO are the smallest, indicating that this new method is the most accurate. The average singlestep time of the DEKF algorithm is the longest, revealing the worst real-time performance of DEKF algorithm. Meanwhile, the average single-step time of EDPSO-PIO is slightly higher than MRLS-PIO algorithm but much better than DEKF algorithm. Therefore, the EDPSO-PIO method may be more suitable for real-time SOC estimation. Table 4. Comparison of different SOC estimation algorithms. algorithm

Maximum error

Mean square error

single-step time /s

DEKF

1.51%

1.742×10-3

9.344×10-4

MRLS-PIO

2.08%

3.416×10-3

5.272×10-4

EDPSO-PIO

0.95%

1.254×10-3

5.847×10-4

5.3. The Verification of Capacity Estimation In order to verify the accuracy of the proposed capacity estimation method at different aging stages, a 400-cycles aging test is carried out. For each cycle the battery cell is firstly charged via the CC (constant current)-CV (constant voltage) method. After one hour standing, the UDDS cycles are operated until the battery terminal voltage reaches the cut-off voltage. The online estimated results of 100th cycle and 400th cycle are displayed in Fig. 11. The gray curve represents the actual estimation result, and the blue curve refers to the result after the median filtering. Note the calculation period (t1-t2 in Eq. (22)) is taken as 10 seconds considering computation cost and precision. Since the capacity is estimated based on the accumulated charge and OCV variation, there is also a converging period for the capacity estimation at the beginning. The estimated battery capacity could converge and tend to be stable within 12 minutes. In addition, the estimated capacity has a relatively large error at the end of discharging, which is caused by the estimated OCV error. Hence, the averaged value of capacity in the middle SOC range (0.3~0.7) is taken as the final Cmax for the current cycle, as marked by the red line in Fig. 11.

Journal Pre-proof (b) 3.2 3.08Ah

3.36Ah

3.4 3.1

Cmax / Ah

Cmax / Ah

3.2 (a) 3.5

3.3 3.0 3.2 2.9 3.1 2.8

Cmax Estimated Average Cmax

0

20

40

60

80

3.1 3.0 2.9 2.8

100 120 140 160

Cmax Estimated Average Cmax

0

20

Time / min

40

60

80

100 120 140 160

Time / min

Fig. 11. Battery capacity estimation: (a) 100th cycle (actual capacity 3.38Ah), (b) 400th cycle (actual capacity 3.13Ah) (a) 3.5

Absolute error / %

3.4

Cmax / Ah

(b) 1.5

Cmax Reference Cmax Estimated

3.3 3.2 3.1 3.0

0

50 100 150 200 250 300 350 400

Cycle times

1 0.5 0

0

50 100 150 200 250 300 350 400

Cycle times

Fig. 12. Capacity estimation at different aging degrees: (a) Estimation results. (b) Estimation error. Fig. 12 compares the estimated Cmax and measured capacity at different aging degrees. The estimated value consists well with experimental data at each aging stage. The estimated error of Cmax can be controlled within 2%, further demonstrating this is an accurate and effective capacity estimation method. 6. Conclusion A new battery state online estimation method based on the EDPSO-PIO method has been proposed in this study. Lithium-ion batteries are firstly modeled by the hybrid battery model, which combines the Thevenin model, reflecting the dynamic behavior of the voltage and KiBaM, reflecting the dynamic behavior of the capacity. Model parameters like the open-circuit voltage are identified online using the exponential decay particle swarm optimization. The battery maximum capacity is estimated and updated based on the online identified parameters. Finally, a PI observer is used to estimate the battery SOC. The validity of the proposed method was investigated through the UDDS cycles. The results indicate that the adaptive hybrid battery model is more accurate than the Thevenin model due to the accurate prediction of the battery dynamic capacity behavior. The online identified parameters via EDPSO show good results compared with experiment data. Among them, the error of terminal voltage is less than 0.02V. For cells at different aging degrees, the estimated capacity could converge in 12 minutes and the capacity error can be controlled within 2%. Compared with previous reports, the proposed

Journal Pre-proof SOC estimation algorithm in this study features a simple structure, real-time and strong robustness. The absolute error of SOC is less than 1% for our test cases. Additionally, this new method is easy to implement without high hardware requirements. Funding This work was supported by the National Natural Science Foundation of China [grant numbers 51775179]. References [1]

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Journal Pre-proof Declaration of Interest Statement We would like to submit the enclosed manuscript entitled “Battery states online estimation based on exponential decay particle swarm optimization and proportional-integral observer with a hybrid battery model”, which we wish to be considered for publication in “Energy”. We declare that none of the authors of this paper has a financial or personal relationship with other people or organizations that could inappropriately influence or bias the content of the paper. It is to specifically state that “No Competing interests are at stake and there is No Conflict of Interest” with other people or organizations that could inappropriately influence or bias the content of the paper. Yours sincerely, Xiaolong Yang State Key Laboratory of Advanced Design and Manufacture for Vehicle Body, Hunan University, Changsha, Hunan 410082, China [email protected].