A reliable estimation method for mining lithium-ion battery

Abstract

Power battery SOC (state of charge, SOC) is one of the important decision-making factors of energy management. Accurate estimation plays an important role in optimizing vehicle energy management and improving the utilization of power battery energy. The key to accurate estimation of SOC is to determine circuit model parameters and estimation methods. The research object of this article is lithium manganese oxide battery for mining (LiMn2O4). The experiments of multiplying power, temperature and HPPC (hybrid pulse power characteristic, HPPC) are carried out. A self-tuning calculation method of dynamic system is proposed, and the dynamic self-tuning model based on second-order RC is established. At the same time, in view of the shortcoming that the UKF (Unscented Kalman Filter, UKF) algorithm cannot estimate the noise in real time, In order to improve the accuracy of battery SOC estimation, an adaptive square root unscented Kalman filter (ASR-UKF) algorithm is proposed, which can make the noise statistical characteristics follow the estimation results for adaptive adjustment. Finally, the constant current and dynamic conditions are tested. The results show that the maximum change rate of model parameters with magnification is 76%, and the maximum change rate with temperature is 73.7%. The analysis of dynamic characteristics is a key factor to improve the accuracy of SOC estimation; ASR-UKF Compared with the UKF algorithm, the error is reduced by 78% under constant current conditions and 85.7% under dynamic conditions. The reliability and real-time performance of the algorithm can be obtained by comparing the simulation data with the actual data. The conclusions of this paper can be used as a theoretical basis, which can be used for model analysis of lithium batteries for mining and estimation of internal state variables.

Keywords

Dynamic self-correction model SOC estimation mining lithium-ion battery adaptive square root unscented Kalman filter

1. Introduction

The rapid development of industry has brought new problems, such as energy shortages and environmental pollution. This is already a problem that the world urgently needs to solve [1]. Therefore, various countries have accelerated the research and development of electric vehicles and hybrid vehicles, and at the same time, power battery technology has also been rapidly developed, and major breakthroughs have been made [2]. Since 2008, the power density of power batteries has reached 3 KW $\cdot$ h/L, and the durability has reached 10,000 hours [3]. However, in terms of mining power batteries, it is still at a relatively backward level, mainly because mining power batteries need to meet certain special work requirements, such as large power requirements, high pressure and low temperature environments in the Northwest, and strong squeezing of ore and other requirements [4]. Lithium manganate battery (LiMn2O4) is widely used in mine power system due to its excellent low-temperature performance, common resources, no pollution, small volume, light weight, puncture non explosion, high cycle life, long endurance and large output power [5]. In the cause of meet the power demand of mining transport vehicles, hundreds or even thousands of single cells are usually connected in series and parallel to form a large-scale lithium-ion battery pack. Generally speaking, the larger the battery pack size, the more storage capacity, but the higher the purchase and maintenance costs, the more prominent the potential dangers [6]. How to ensure the efficient, reliable and Safety on the operation of electricity has become the focus of attention of the whole industry. The important way to solve the above problems is to monitor and manage the state of charge (SOC) and state of health (SOH) through the battery management system (BMS) [7, 8]. However, only the battery port voltage and current can be obtained by BMS, and the SOC status inside the battery must be estimated [9]. The dynamic characteristics between SOC and port voltage and current are due to the nonlinear characteristics of lithium-ion batteries [10]. How to accurately represent the relationship between SOC and port voltage and current is a difficult task. These factors use estimation algorithms to accurately model the battery and design a reliable SOC, which is of great significance and engineering application value. Such as, accurately describe the relationship between power batteries, develop an optimized management system, and ensure the safety of transportation vehiclesduring driving.

2. State of the art

According to the “Technical Requirements for Security of Lithium-Ion Batteries for Mining”, the accurate computation of SOC of lithium-ion power batteries for mining and the design of safety management system can refer to the standard of lithium-ion batteries for electric vehicles (GB 31241-2014). Therefore, the current research status of SOC computation of lithium-ion batteries for mining can refer to power batteries for automobiles. At present, the methods of studying the SOC estimation of automotive power batteries at home and abroad can be roughly divided into three categories: ampere-hour integration, characterization parameter-based and model-based [11, 12, 13].

The ampere-hour integration method is also called the Coulomb counting method, which is convenient to use, easy to implement, and easy to cause accumulated errors when the current accuracy is not high. Zhang et al. proposed an improved model of the SOC initial value estimation of the dynamic multi-parameter method, considering the relaxation effect [14]. The dependence of the initial value of SOC on the ampere-hour integration method is effectively reduced. The error caused by charge accumulation is not described.

The methods based on characterization parameters can be divided into impedance spectroscopy and open circuit voltage, mostly by establishing the correlation between characterization parameters and SOC. This method is considered to be the most direct method to determine the SOC of power battery, but the relative relationship is difficult to determine, generally with the model Online identification methods are used in combination. Wang et al. used the constant current mode electrochemical impedance spectroscopy method to obtain the ohmic impedance, charge transfer impedance and ion diffusion impedance of the battery, and carried out the research of the SOC and temperature on the impedance features of the lithium iron phosphate battery, and concluded that the battery is in different SOC and the impedance characteristic under temperature effectively reflects the internal polarization state of the battery, so that the functional state of the energy storage battery can be judged according to the above three parameters; Xing et al. established a temperature-based SOC estimation model [11], combined with open circuit voltage (OCV) and SOC, a table of three variables is made, and the model parameters of each sampling step are optimized by the method of unscented Kalman filtering (UKF), which explains that the method may effectively build up the SOC estimation; Krishna’s method of estimating the state of charge of a battery is to combine Coulomb counting and open circuit voltage, and verified the whole system in real time with a semi-physical experimental device. The results showed that the method is efficientand accurate.

The estimation algorithm based on models such as electrochemical model, equivalent circuit model, and black box model is a fusion of multiple algorithms. These algorithms include ampere-hour integration method, open circuit voltage method, neural network method. Kalman filter based on the model, the state equation is established, and the filtering method and observer are used to complete the SOC estimation. Hu et al. proposed a robust Kalman filter method to solve the uncertainty and non-linear error involved in the battery modeling process of the extended Kalman filter method to compensate, so that the root mean square error of the SOC estimation is reduced, and the noise statistics are reduced [15]. Shows strong robustness; Bizeray et al. used an orthogonal configuration method to solve the thermal-electrochemical P2D model [16], and used an improved extended Kalman filter algorithm to achieve SOC estimation, which ensures accuracy and reduces computational costs; Luo et al. proposed a temperature-corrected estimation model for SOC estimation at lower temperatures, and used Sigma-based unscented Kalman filtering to accomplish SOC estimation [8]; Zhang et al. had access to a nonlinear complementary model to capture the open circuit voltage the hysteresis effect [17]. One method uses the time to modify the integral and multi-channel voltage combined battery state SOC, and solves and completes the SOC accumulation problem caused by the initial value through the battery capacity and Coulomb growth temperature change. Established an EKF uses the open circuit voltage method to determine the initial value of SOC when estimating SOC, so that it can converge to the true state more quickly [18, 19]. It is considered that the capacity loss of the battery and arranged the ADKF (Adaptive Double Kalman Filter) algorithm based on the fractional-order model, which realized the accurate estimation of the SOC of the battery under the condition of capacity loss; The hysteresis neural network algorithm realizes the joint estimation of the SOC and SOH of the battery [20]. This algorithm compensates for the non-linear part of the battery and does not need to know the external parameters of the battery in advance; Chen et al. proposed a method based on RBF (Radial Basis Function), which estimated the battery SOC by the nonlinear observer of the neural network [21, 22, 23, 24, 25]. This method uses the neural network to estimate the uncertainty of the battery model online [26, 27], and proves that the SOC estimation error is ultimately bounded and the error can be arbitrarily small.

From the above research, it can be seen that the ampere-hour integration method is simple to operate, but cumulative errors will occur with the accumulation of charge, and it is more dependent on the initial value of SOC, which is not suitable for working conditions the battery; Choosing an accurate model and filtering algorithm has become the research focus. Literature uses different equivalent models, identification methods and SOC estimation algorithms under different environmental conditions to analyze measures to further the accuracy of SOC estimation. However, there is no relevant literature to carry out research on mining LiMn2O4 under the special environment of Northwest coal mines. Therefore, this paper takes LiMn2O4 as the research object, carries out HPPC experiments with charge-discharge rate and temperature as the research content, establishes a dynamic self-correcting model based on second-order RC, uses adaptive square root unscented Kalman filter method to estimate SOC, and the influence of adaptive correction model on the accuracy of real-time SOC estimation.

The third part expounds the battery model and parameter identification process, determines the dynamic self-correcting model parameters of the second-order RC; expounds the idea and calculation process of the ASR-UKF algorithm. The fourth part is based on the model and algorithm of this paper to conduct tests under constant current and dynamic conditions, and analyze the test results.

3. Methodology

In this experiment, a 50AH/4.2V LiMn ${}_{2}$ O ${}_{4}$ battery produced by AVIC Lithium Battery (Luoyang) Company was used, and its characteristic parameters are shown in Table 1. The test uses the BTT-331C three-layer independent temperature control high and low temperature test box to ensure that it is in a constant temperature state during the test. Use BTS5-120 type power battery single test system to conduct battery charge and discharge test, complete battery capacity test, internal resistance test, charge and discharge characteristics test.

Table 1
Characteristic parameters of LiMn ${}_{2}$ O ${}_{4}$ battery

Longwidehigh (mm)	Nominal capacity	Working current			Working voltage	Working temperature
		Nominal	Maximum charge	Maximum discharge
1482798	50Ah	1/3C	3C	4C (CC-CV)	4.2 V (charge)	0–45 ${}^{\circ}$ C (charge)
				12C (Hppc)	2.75 V (discharge)	$-$ 20–60 ${}^{\circ}$ C (discharge)

The recent rate and temperature of the battery have an impact, internal resistance, terminal voltage and SOC of the battery. The experiment analyzes its influence on the SOC by grasping the battery’s current rate and temperature change characteristics, and lays a solid foundation for the determination of the parameters of the second-order RC dynamic self-correcting model. The test results are shown in Figs 1 and 2.

Figure 1.

R0 under different discharge rates.

Figure 2.

R0 under different temperatures.

Figure 1 shows the variation of the internal resistance Ro of the second RC model with SOC under different magnification conditions; Fig. 2 shows the variation of the internal resistance Ro of the second-order RC model with SOC under different temperature conditions. Similarly, the open circuit voltage Uoc, polarization resistance Rs and Ru, polarization capacitance Cs and Cu, and energy storage capacitance Cb can be plotted against SOC.

Dynamic working condition method, HPPC test, self-discharge test and energy efficiency test, etc. Among several test methods, HPPC measures the cost. Low, simple measurement equipment used, high measurement accuracy, mainly used to test battery dynamic characteristics. In this test, according to the requirements of HPPC, the LiMn ${}_{2}$ O ${}_{4}$ battery is tested at 25 ${}^{\circ}$ C. Cyclic test process: 1C current discharge 10 seconds -put it aside for 40 seconds $-$ 1C current charge 10s-put it aside for 40 seconds. The corresponding battery voltage response curve is shown in Fig. 3. In the whole HPPC test, the pulse test was performed when the battery SOC was 1, 0.9, …, 0. A set of pulse tests are carried out when the SOC is 1, and then the battery is discharged with 1C current for 6 minutes, that is, 5Ah of electricity is discharged, the battery SOC drops by 0.1, and the HPPC test is performed for the next SOC point. The voltage and current of each SOC point.

Figure 3.

Battery voltage curve in HPPC test process.

Figure 4.

Circuit diagram of second-order RC model.

Figure 4 is a second-order RC model. The test of 3.1.2 is designed for this model, and the meaning of the parameters Uoc, Ro, Cb, Rs, Ru, Cs and Cu in the circuit are described, UL is the terminal voltage. The following is the identification of parameters based on the obtained HPPC curve (Fig. 3).

Uoc is the stable voltage of the positive and negative poles of the battery when the battery is in a static state. After measuring the Uoc of the battery when the SOC is 1, discharge the battery at a constant current rate of 1C for 6 minutes to reduce the SOC by 0.1 to obtain the Uoc value in different states. After the value of the voltage is stable, repeat the measurement to get the average value of Uoc. The purpose of leaving the battery for a long time is to completely eliminate the influence of polarization effects on battery performance and ensure the accuracy and reliability of the measurement.

Ohm internal resistance: The polarization of the battery has not yet occurred at the beginning and end of discharge. The battery voltage suddenly drops or rises due to the ohmic internal resistance, as shown in the U1–U2 and U3–U4 segments in Fig. 3. It uses Ohm’s theorem to get the value of internal resistance, and the equation is shown in Eq. (1):

$\displaystyle R_{O}=\frac{(U4-U3)+(U1-U2)}{2I}$ (1)

Energy storage capacitor: $C_{b}$ is a large energy storage capacitor, which is closely related to the size of the open circuit voltage $U_{oc}$ . The relationship is as shown in Eq. (2), where $\textit{Energy}{\_}C_{b}$ represents the energy stored by the energy storage capacitor $C_{b}$ , and $U_{oc}$ represents the open circuit of the battery corresponding to different SOC values the voltage value, $Q_{N}$ represents the rated capacity of the battery. The test obtains the $U_{oc}$ of the lithium battery when the SOC is 0% and 100%, and the corresponding $C_{b}$ value under different SOC states can be obtained according to Eq. (2):

$\displaystyle\left\{{\begin{array}[]{l}\textit{Energy}\_C_{b}=\frac{1}{2}C_{b}% U^{2}=\frac{1}{2}C_{b}(U_{\text{100\% SOC}}^{2}-U_{\text{0\% SOC}}^{2})\\ \textit{Energy}\_C_{b}=Q_{N}\ast U_{oc}\\ \end{array}}\right.$ (2)

Polarized internal resistance and polarized capacitance: The polarization phenomenon reflects how fast the battery reaches a steady state after charging and discharging. To identify the resistance and capacitance in the two RC links in Fig. 4, first identify the discharge direction. From Fig. 4, the model output equation can be obtained as Eq. (3):

$\displaystyle U_{L}=U_{oc}-IR_{o}-U_{S}-U_{U}-\frac{1}{C_{b}}\int{Id_{t}}$ (3)

Figure 3 is the U4–U5 section of the voltage change curve 40 s after the end of the discharge pulse. The current input is 0. At this time, it is the zero input response of RC.

$\displaystyle U_{L}(t)=U_{oc}-U_{S}(0)e^{-t/\tau s}-U_{U}(0)e^{-t/\tau u}$ (4)

where $\tau=$ RC. According to Eq. (4), using the least squares method to fit the curve can find $U_{s}(0)$ , $U_{u}(0)$ , $\tau s$ , $\tau u$ .

In the U2–U3 section, since the U2 point has been put on hold for a long time, its internal polarization effect has basically disappeared, and the polarization voltage can be considered to be zero. The U2–U3 section is regarded as the zero-state response of the RC link, and the voltage across the polarizing capacitor is:

$\displaystyle U_{S}(t)=IR_{S}(1-e^{{-t}\mathord{\left/{\vphantom{{-t}{\tau s}}% }\right.\kern-1.2pt}{\tau s}})$ (5) $\displaystyle U_{u}(t)=IR_{u}(1-e^{{-t}\mathord{\left/{\vphantom{{-t}{\tau u}}% }\right.\kern-1.2pt}{\tau u}})$ (6)

The instantaneous battery polarization voltage from point U3 to point U4 is basically unchanged, which can be obtained:

$\displaystyle U_{S}(0)=IR_{S}(1-e^{{-tv}\mathord{\left/{\vphantom{{-tv}{\tau s% }}}\right.\kern-1.2pt}{\tau s}})$ (7) $\displaystyle U_{u}(0)=IR_{u}(1-e^{{-tv}\mathord{\left/{\vphantom{{-tv}{\tau u% }}}\right.\kern-1.2pt}{\tau u}})$ (8)

where $t v$ is the time from U2 to U3 discharge pulse.

Substituting the already calculated $U_{s}(0)$ , $U_{u}(0)$ , $\tau s$ , $\tau u$ into Eqs (7) and (8), $R_{s}$ and $R_{u}$ can be obtained. Then calculate $C_{s}$ and $C_{u}$ based on the time constant expression $\tau=$ RC.

Considering the identification accuracy and the computational complexity, the 7th fitting is finally selected, and the fitting polynomial is as follows: $f(x)=$ p1 $x^{7}$ $+$ p2 $x^{6}$ $+$ p3 $x^{5}$ $+$ p4 $x^{4}$ $+$ p5 $x^{3}$ $+$ p6 $x^{2}$ $+$ p7 $x$ $+$ p ${}^{8}$ . Where $x$ is the SOC, $f(x)$ is the parameter curve to be fitted, and the coefficients p1, p2, p3, p4, p5, p6, p7, p8 of the identification parameters are shown in Table 2.

Table 2

Parameter fitting results

	UOC/V	R0/m $\Omega$	Cb/KF	Rs/m $\Omega$	Ru/m $\Omega$	Cs/KF	Cu/KF
p1	$-$ 22.19	$-$ 6.375	$-$ 1653	880.8	$-$ 1138	2.103e $+$ 06	$-$ 1.89e $+$ 06
p2	136.45	93.72	7426.5	$-$ 3294	4525	$-$ 7.99e $+$ 06	7.499e $+$ 06
p3	$-$ 308.01	$-$ 290.3	$-$ 1245	4963	$-$ 7304	1.225e $+$ 07	$-$ 1.21e $+$ 07
p4	340.21	415	1108.7	$-$ 3843	6136	$-$ 9.676e $+$ 06	1.003e $+$ 07
p5	$-$ 198.01	$-$ 326.7	$-$ 5464	1611	$-$ 2861	4.175e $+$ 06	$-$ 4.64e $+$ 06
p6	59.71	145.6	1547	$-$ 350.5	732.7	$-$ 9.582e $+$ 05	1.172e $+$ 06
p7	$-$ 7.82	$-$ 34.28	$-$ 153	33.87	$-$ 95.62	1.047e $+$ 05	$-$ 1.47e $+$ 05
p8	3.77	5.732	5.4	$-$ 0.707	5.309	$-$ 3936	7107

The operating conditions of mining batteries are very complicated, and their state of charge is affected by many factors, mainly including charge and discharge rate, temperature, self-discharge, cycle times and aging, etc., among which the charge and discharge current rate and temperature are more important.

Therefore, according to the experimental data of different magnification and temperature in 3.1.2, combined with the identification parameters, the improvement of the dynamic self-correcting model of the second-order RC that changes with the current magnification is proposed. The specific method is as follows: Set Ro under 1C magnification as the reference parameter, and the ratio of Ro to the reference parameter under other magnifications is called Ro correction coefficient, and Dro is used to represent the correction coefficient. In the same way, the correction coefficients of Uoc, Rs, Ru, Cs, Cu, and Cb can be obtained, which are represented by Duoc, Drs, Dru, Dcs, Dcu, Dcb in Table 3. It can be seen from the table that the maximum change rate of model parameters with magnification is 76%.

Table 3

Correction coefficients of dynamic system parameters under different multiplying powers

		1	0.9	0.8	0.7	0.6	0.5	0.4	0.3	0.2	0.1
Dro	0.5C	0.975	0.952	1.000	0.953	0.976	0.976	0.954	0.977	0.957	0.914
	0.3C	0.950	0.952	0.975	0.930	0.952	0.976	0.931	0.954	0.936	0.936
	0.2C	0.975	0.904	0.975	0.883	0.928	0.976	0.909	0.931	0.914	0.936
Duoc	0.5C	1.010	1.004	1.004	1.004	1.002	1.001	1.001	1.001	1.001	1.001
	0.3C	1.050	1.014	1.013	1.013	1.008	1.004	1.002	1.003	1.003	1.002
	0.2C	1.099	1.024	1.019	1.019	1.013	1.006	1.004	1.005	1.005	1.003
Drs	0.5C	1.516	0.925	0.992	1.024	0.986	0.992	1.029	1.042	0.956	0.996
	0.3C	4.080	0.988	0.921	1.088	0.968	0.963	1.089	1.160	0.859	0.982
	0.2C	1.754	1.533	0.828	1.129	0.970	0.937	1.122	1.255	0.799	0.970
Dru	0.5C	0.666	1.045	0.994	1.014	1.072	1.072	0.932	0.975	1.003	0.968
	0.3C	1.313	1.122	1.005	1.025	1.236	1.305	0.795	0.909	1.009	0.896
	0.2C	1.053	1.048	1.046	1.016	1.336	1.517	0.724	0.856	1.014	0.849
Dcs	0.5C	1.204	1.070	1.086	1.111	1.017	0.988	1.059	0.055	0.577	1.033
	0.3C	2.071	1.312	1.293	1.429	1.118	0.948	1.181	2.926	0.350	1.096
	0.2C	1.215	1.617	1.442	1.691	1.250	0.917	1.248	2.210	0.922	1.128
Dcu	0.5C	1.292	0.996	1.083	1.137	0.970	1.002	1.497	0.963	1.107	0.919
	0.3C	2.897	1.023	1.238	1.547	0.965	0.970	2.600	0.850	1.362	0.739
	0.2C	5.389	1.194	1.296	1.888	1.039	0.923	3.280	0.751	1.537	0.625
Dcb	0.5C	1.215	1.525	0.823	0.992	1.032	1.051	1.064	1.078	1.108	0.972
	0.3C	1.849	3.117	0.242	0.936	1.100	1.177	1.229	1.283	1.402	0.894
	0.2C	2.419	4.590	0.331	0.856	1.137	1.269	1.356	1.445	1.645	0.824

It can be seen from Fig. 2 that Ro changes at the same SOC and at different temperatures, while the fitting result is fixed. Therefore, an adaptive improvement with temperature changes is proposed. The specific method is as follows: Set Ro at 25 ${}^{\circ}$ C as the reference parameter, the ratio of Ro to the reference parameter at other temperatures is called the Ro correction coefficient, and Tro represents the correction coefficient. In the same way, the correction coefficients of Uoc, Rs, Ru, Cs, Cu, Cb can be obtained, which are represented by Toc, Trs, Tru, Tcs, Tcu, Tcb in Table 4. It can be seen from the table that the maximum change rate of model parameters with temperature is 73.7%.

Table 4

Correction coefficients of dynamic system parameters under different temperatures

		1	0.9	0.8	0.7	0.6	0.5	0.4	0.3	0.2	0.1
Tro	0 ${}^{\circ}$ C	1.04	1.045	1.069	1.091	1.112	1.138	1.178	1.215	1.258	1.337
	10 ${}^{\circ}$ C	1.008	1.032	1.024	1.053	1.072	1.0638	1.132	1.121	1.156	1.15
	40 ${}^{\circ}$ C	0.975	0.984	0.978	0.978	0.988	0.968	0.97	0.963	0.955	0.9325
Toc	0 ${}^{\circ}$ C	0.952	0.959	0.961	0.972	0.985	0.992	0.991	0.989	0.990	0.993
	10 ${}^{\circ}$ C	0.972	0.981	0.979	0.984	0.991	0.996	0.996	0.995	0.995	0.997
	40 ${}^{\circ}$ C	1.025	1.008	1.008	1.008	1.005	1.002	1.002	1.002	1.002	1.001
Trs	0 ${}^{\circ}$ C	0.633	1.295	0.823	1.015	1.177	0.904	0.640	0.866	1.554	0.995
	10 ${}^{\circ}$ C	0.445	1.361	0.915	0.945	1.101	0.995	0.825	0.858	1.259	1.009
	40 ${}^{\circ}$ C	2.408	0.893	0.969	1.053	0.975	0.980	1.058	1.094	0.910	0.990
Tru	0 ${}^{\circ}$ C	0.984	0.928	0.914	0.567	0.411	1.179	1.850	1.139	1.027	1.399
	10 ${}^{\circ}$ C	1.132	0.894	1.029	0.832	0.641	0.885	1.412	1.096	0.997	1.183
	40 ${}^{\circ}$ C	1.047	1.092	0.994	1.023	1.150	1.169	0.865	0.946	1.006	0.934
Tcs	0 ${}^{\circ}$ C	0.386	0.418	0.344	0.571	1.151	0.828	0.249	5.515	6.024	0.435
	10 ${}^{\circ}$ C	0.561	0.699	0.596	0.638	1.052	0.989	0.639	4.826	3.427	0.772
	40 ${}^{\circ}$ C	1.521	1.164	1.181	1.248	1.052	0.971	1.119	1.314	0.136	1.064
Tcu	0 ${}^{\circ}$ C	0.616	0.481	0.293	0.735	1.343	0.371	2.186	1.004	0.117	2.067
	10 ${}^{\circ}$ C	0.712	0.845	0.551	0.645	1.252	0.822	1.711	1.089	0.482	1.479
	40 ${}^{\circ}$ C	1.829	0.995	1.163	1.312	0.954	0.993	2.020	0.915	1.225	0.834
Tcb	0 ${}^{\circ}$ C	1.021	1.393	1.488	0.792	0.624	0.543	0.485	0.419	0.278	1.362
	10 ${}^{\circ}$ C	0.939	0.739	1.478	0.942	0.815	0.753	0.711	0.664	0.561	1.146
	40 ${}^{\circ}$ C	1.483	2.191	0.586	0.974	1.065	1.109	1.138	1.168	1.236	0.938

The previous article analyzed the parameters and SOC influencing factors in the battery equivalent circuit model, and then combined the model to establish a space equation to study the SOC estimation method.

According to Fig. 4, select SOC and three-dimensional variables as the state variables of the system, and the battery terminal voltage as the observed variables of the system.

$\displaystyle\left\{{\begin{array}[]{l}\left({\begin{array}[]{l}\textit{SOC}_{% k}\\ U_{s,k}\\ U_{u,k}\\ \end{array}}\right)=\left({{\begin{array}[]{*{20}c}1&0&0\\ 0&{e^{-\frac{T_{s}}{C_{s}R_{s}}}}&0\\ 0&0&{e^{-\frac{T_{s}}{C_{u}R_{u}}}}\\ \end{array}}}\right)\left({\begin{array}[]{l}\textit{SOC}_{K-1}\\ U_{S,K-1}\\ U_{U,K-1}\\ \end{array}}\right)+\left({\begin{array}[]{l}\frac{\eta_{k}T_{S}}{Q_{N}}\\ R_{s}(1-e^{-\frac{T_{s}}{C_{s}R_{s}}})\\ R_{u}(1-e^{-\frac{T_{S}}{C_{u}R_{u}}})\\ \end{array}}\right)\left({I_{k-1}}\right)\\ \quad{}+\left({\begin{array}[]{l}W_{1,k-1}\\ W_{2,k-1}\\ W_{3,k-1}\\ \end{array}}\right)\\ U_{L,k}=U_{OC}(\textit{SOC}_{k})-U_{s,k}-U_{u,k}-R_{o}I_{k-1}-\frac{T_{s}}{C_{% b}}I_{k-1}+v_{k}\\ \end{array}}\right.$ (9)

$Q_{N}$ represents the rated capacity of the battery, and $T_{S}$ is the sampling period. $U_{L,k}$ represents the voltage value of the terminal at time $K$ . $W_{1,k-1}$ represents the state error value at the moment $k-1$ . $v_{k}$ represents the output error value at time $k$ .

Initialization: The initial value belongs to the state variable and the error covariance is determined. $S_{0}$ is the Cholesky factor of covariance, and the initial value is determined as follows:

$\displaystyle\left\{{\begin{array}[]{l}\hat{x}_{0}=E(x_{0})\\ P_{0}=E((x_{0}-\hat{x}_{0})(x_{0}-\hat{x}_{0})^{T})\\ S_{0}=\textit{chol}(P_{0})\\ \end{array}}\right.$ (10)

UT transformation: Approximate the Gaussian distribution with a fixed number of parameters.

$\displaystyle\left\{{\begin{array}[]{l}x_{k-1}^{i}=\hat{x}_{k-1},i=0\\ x_{k-1}^{i}=\hat{x}_{k-1}+\sqrt{(n+\lambda)}S_{k-1}^{i},i=1\ldots n\\ x_{k-1}^{i}=\hat{x}_{k-1}-\sqrt{(n+\lambda)}S_{k-1}^{i-n},i=n+1\ldots 2n\\ \end{array}}\right.$ (11) $\displaystyle\left\{{\begin{array}[]{l}\omega_{m}^{0}=\frac{\lambda}{n+\lambda% }\\ \omega_{c}^{0}=\frac{\lambda}{n+\lambda}+1-\alpha^{2}+\beta\\ \omega_{m}^{i}=\omega_{c}^{i}=\frac{1}{2\ast(n+\lambda)},i=1\ldots 2n\\ \end{array}}\right.$ (12)

where $S^{i}_{k}$ represents the ith column of the Cholesky factor of the covariance of the state variable at time $k$ . $\omega_{m}$ is the mean weight, and $\omega_{c}$ is the variance weight.

Time update: According to the state $U_{oc}$ variable and the value of the input variable at $k-1$ , one-step prediction of the state variable through the state equation, such as Eqs (13) and (14).

$\displaystyle x_{k|k-1}^{i}=f(x_{k-1|k-1}^{i},u_{k-1})$ (13) $\displaystyle\hat{x}_{k|k-1}=\sum\nolimits_{i=0}^{2n}{\omega_{m}^{i}x_{k|k-1}^% {i}}$ (14)

According to the one-step prediction of the sampling point, the error covariance of the state variable is QR decomposed, as shown in Eq. (15).

$\displaystyle S_{xk}^{-}=qr\left\{{\left[{\sqrt{\omega_{c}^{1:2n}}\left({x_{k|% k-1}^{1:2n}-\hat{x}_{k|k-1}}\right),\sqrt{Q_{k}}}\right]}\right\}$ (15)

Considering that the values of $\alpha$ and $k$ are different, which may lead to negative values of $\omega_{c}^{0}$ , Eq. (16) is used to ensure the semi-positive definiteness of the matrix. $S_{xk}$ represents the update value of the square root of the error covariance of the state variable at time $k$ .

$\displaystyle S_{xk}=\textit{cholupdate}\left\{{S_{xk}^{-},\sqrt{\textit{abs}(% {\omega_{c}^{0}})}\left({x_{k|k-1}^{0}-\hat{x}_{k|k-1}}\right),\textit{sign}({% \omega_{c}^{0}})}\right\}$ (16)

Equations (17) to (20). $S_{yk}$ represents the square root update value of the error covariance of the observed variable at time $k$ .

$\displaystyle y_{k|k-1}^{i}=h\left({x_{k|k-1}^{i},u_{k}}\right)$ (17) $\displaystyle\hat{y}_{k|k-1}=\sum\nolimits_{i=0}^{2n}{\omega_{m}^{i}}y_{k|k-1}% ^{i}$ (18) $\displaystyle S_{yk}^{-}=qr\left\{{\left[{\sqrt{\omega_{c}^{1:2n}}\left({y_{k|% k-1}^{1:2n}-\hat{y}_{k|k-1}}\right),\sqrt{R_{k}}}\right]}\right\}$ (19) $\displaystyle S_{yk}=\textit{cholupdate}\left\{{S_{yk}^{-},\sqrt{\textit{abs}(% {\omega_{c}^{0}})}\left({y_{k|k-1}^{0}-\hat{y}_{k|k-1}}\right),\textit{sign}({% \omega_{c}^{0}})}\right\}$ (20)

State update: The cross-covariance between the state variable and the observed variable is shown in Eq. (21), and its value directly affects the size of Kalman gain, and the accuracy of Kalman gain will affect the estimation effect of SOC. Its calculation method is as Eq. (22), the system state variable update and its error covariance update are shown in Eq. (23), $y_{k}$ is the experimental measurement value at time $k$ .

$\displaystyle P_{xy,k}=\sum\nolimits_{i=0}^{2n}{\omega_{c}^{i}}\left[{x_{k|k-1% }^{i}-\hat{x}_{k|k-1}}\right]\left[{y_{k|k-1}^{i}-\hat{y}_{k|k-1}}\right]^{T}$ (21) $\displaystyle K_{k}=P_{xy,k}({S_{yk}S_{yk}^{T}})^{-1}$ (22) $\displaystyle\left\{{{\begin{array}[]{l}{\hat{x}_{k|k}=\hat{x}_{k|k-1}+K_{k}({% y_{k}-\hat{y}_{k|k-1}})}\\ {S_{k}=\textit{cholupdate}({S_{xk}^{-},K_{k}S_{yk,}-1})}\\ \end{array}}}\right.$ (23)

This algorithm introduces noise adaptive covariance matching, and automatically cyclically updates and transmits the noise covariance matrix to make it closer to the real noise situation. In the algorithm, the innovation of the observed variable at time $k$ is defined as shown in Eq. (24). The innovation is mainly determined by the error, and the innovation covariance can well reflect the influence of the error at the current time. Take the weighted average of the variance of the previous $M$ innovations to get $H_{k}$ , the expression is shown in Eq. (25). $H_{k}$ is the covariance function of the real-time estimation of innovation obtained by the windowing estimation principle, and $M$ represents the window size. Equation (26) shows the update process of the noise value, which comes from the system and observation. Obviously, the calculation of system noise and observation noise is inseparable from $H_{k}$ . In order to reduce the computational complexity as much as possible while improving the estimation accuracy, take the first 3 innovations for calculation, that is, take $M=$ 3. Since $P_{k}$ gradually decreases with time and eventually tends to 0, the $C_{k}P_{k}C_{k}^{T}$ part can be ignored.

$\displaystyle e_{k}=y_{k}-\hat{y}_{k|k-1}$ (24) $\displaystyle H_{k}=\frac{1}{M}\sum\nolimits_{i=k-M+1}^{k}{e_{k}e_{k}^{T}}$ (25)

Noise update is presented as below:

$\displaystyle\left\{{{\begin{array}[]{l}{Q_{k}=K_{k}H_{k}K_{k}^{T}+H_{k}}\\ {R_{k}=H_{k}-C_{k}P_{k}C_{k}^{T}}\\ \end{array}}}\right.$ (26)

The SOC estimation flowchart of ASR-UKF algorithm is shown in Fig. 5.

Figure 5.

SOC estimation flowchart of ASR-UKF algorithm.

4. Test results and analysis

In the ASR-UKF algorithm, the observation equation plays a vital role, and accurate observations can ensure the accuracy of SOC. In the cause of verify the accuracy, reliability and real-time performance of the model, the second-order RC model is selected for comparison, and experimental analysis is carried out under constant current and dynamic conditions.

Constant current working condition: Fully discharge LiMn2O4 at a constant current of 1C at a discharge rate of 1C in an environment of 10 ${}^{\circ}$ C. Through the previous experiments on the internal parameters of battery models at different temperatures, the terminal voltages were estimated using the model in this paper and the second-order RC, and the results are shown in Fig. 6. The error between the two methods to estimate the terminal voltage and the true value is shown in Fig. 7. The curve Error1 represents the error of the second-order RC model, and Error2 represents the error of the model in this paper. It can be seen from Figs 6 and 7 that the errors of the parameters of the two models are basically the same in the initial and mid-stage discharge. At the end of discharge, the model proposed in this paper takes into account the influence of temperature on the internal polarization of the battery, and the maximum error is reduced by 62.5%. It shows that the model has strong adaptive ability and good real-time performance.

Figure 6.

Comparison of terminal voltage under constant-current discharge.

Figure 7.

Terminal voltage error under constant-current discharge.

Dynamic working conditions: Charge and discharge the battery under 10 ${}^{\circ}$ C environment, including battery charging, discharging and shelving. Experiments were carried out using the model in this paper and the second-order RC model, and the results are shown in Fig. 8. The error between the estimated terminal voltage and the true value of the two methods is shown in Fig. 9. The curve Error1 represents the error of the second-order RC model, and Error 2 represents the error of the model in this paper. It can be seen from Figs 8 and 9 that two RC parallel circuits can effectively characterize the polarization of the battery. In the case of the same temperature, the error during charging and shelving is basically the same. During the discharge process, the model proposed in this paper allow the effect of the current rate on the internal polarization of the battery and the relative error is reduced by 52%. It shows that the model in this paper is closer to the real battery change state than the second-order RC.

Figure 8.

Comparison of terminal voltage under dynamic working condition.

Figure 9.

Terminal voltage error under dynamic working condition.

The condition is about constant current: Full discharge of LiMn2O4 at a constant current of 1C at a discharge rate of 1C in an environment of 10 ${}^{\circ}$ C. The test values are imported into the simulation model, and the algorithm in this paper and the UKF algorithm be accustomed to estimate the battery SOC respectively. The results are shown in Fig. 10. And find the error between the estimated value of the two methods and the experimental value as shown in Fig. 11. From Figs 10 and 11, it can be seen that as time is updated, the error of the UKF algorithm gradually increases with the change of discharge time, and does not converge; while this algorithm tends to converge quickly, and the maximum error after stability is 0.06 %, compared with the UKF algorithm reduced by 78%.

Figure 10.

Comparison between the two algorithms in SOC estimation.

Figure 11.

Error oscillograms of the two algorithms.

Figure 12.

Comparison between the two algorithms in SOC estimation under dynamic working condition.

Figure 13.

Comparison between the two algorithms in SOC estimation error under dynamic condition.

Dynamic operating conditions: Charge and discharge the battery under a 10 ${}^{\circ}$ C environment, including battery charging, discharging and shelving. The changes in this process are more complicated. Therefore, the SOC of the battery is approximated in conjunction with the simulated charge and discharge test. Apply the algorithm in this paper and UKF is an algorithm for estimating SOC under dynamic conditions, and the results are shown in Fig. 12. Figure 13 shows the error between the estimated results of the two methods and the experimental values. From Figs 12 and 13, it can be seen that in the dynamic condition test, the maximum error of UKF estimation is 1.05%, and the fluctuation is large, and the stability is poor; the stability of the algorithm in this paper is good, and the maximum error is only 0.15%. Reduced by 85.7%.

5. Conclusions

Mining power batteries are more and more widely used in mining transportation vehicles. In order to further study the performance of power batteries, this paper selects the commonly used mining power battery LiMn ${}_{2}$ O ${}_{4}$ as the research object to establish an equivalent circuit model and realize it through experiments and simulations SOC is estimated and compared and analyzed. Finally, this article draws the following conclusions:

(1)

Determine the second-order RC dynamic adaptive model based on the internal characteristics of the battery such as ohmic polarization, steady state change, active polarization, concentration polarization, etc., combined with the external change characteristics of the battery such as temperature and rate;

(2)

Compared with the second-order RC model, the error of this model is reduced under constant current conditions and dynamic conditions. Shows that the model has a strong self-correction function.

(3)

Based on the square root unscented Kalman filter algorithm, combined with the characteristics of the model, the three-dimensional state variable is used as the state equation, the battery terminal voltage is used as the observation variable, and the accuracy of battery SOC estimation is improved by real-time prediction and correction of noise, and the adaptive square root unscented Kalman filter algorithm is finally determined.

(4)

Compared with the Unscented Kalman Filter algorithm, the error of the algorithm in this paper is reduced under constant current conditions and under dynamic conditions. The result of this algorithm shows the characteristics of fast convergence speed and high accuracy.

In general, the algorithm in this paper has the advantages of simple structure and strong adaptability of parameters; the proposed SOC algorithm is easy to implement and has high estimation accuracy, which lays a foundation for further analysis of the application of lithium-ion batteries in mining power systems. However, the SOH the battery is not considered in the SOC estimation process, and the joint estimation algorithm of SOC and SOH will be studied in the later stage.

Footnotes

Acknowledgments

This work was supported by the Natural Science Foundation of Education Department of Anhui Province (No. KJ2019A0692, No. KJ2020A0640) and the Natural Science Foundation of Huainan Normal University (No. 2018xj17zd).

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A reliable estimation method for mining lithium-ion battery

Abstract

Keywords

1. Introduction

2. State of the art

3. Methodology

Table 1 Characteristic parameters of LiMn 2 O 4 battery

Footnotes

Acknowledgments

References

Table 1
Characteristic parameters of LiMn ${}_{2}$ O ${}_{4}$ battery