Technical Field

The present disclosure relates to the technical field of battery health management, particularly to a method for estimating Li-ion batter}' capacity degradation, Technical Background

The high density and the light weight of Li-ion batteries have sparked interest in Li-ion batteries and resulted in a remarkably high number of studies aimed at improving the performance of Li-ion batteries. Considering the rate of capacity loss that highly depends on operating conditions and permanent capacity loss over time, accurate estimation of the available batter} _' capacity is often desired for reliability and better management of energy use.

Battery modeling and simulation have undergone significant advancements in the past decades because of significant improvements in software capability and modern experimental techniques. Few attempts have been made to estimate Li-ion battery capacity. Zhang X,, Ross P., Jr. Kostecki R., Kong F., Sloop 8., Kerr B., Striebel K., Cairns E.J. and McLamon F. focused on characterizing the electrochemical and physical properties of anode, cathode, electrolyte, and current collectors in Diagnostic Characterization of High-power Lithium-ion Batteries for Use in Hybrid Electric Vehicles, J Electrochem Soc 2001; 148: A463-70. Fuller T.F., Doyle M. and Newman J. used a "first-principles" electrochemical model to estimate Li-polymer cell capacity in Simulation and Optimization of the Dual Lithium ion insertion Cell. J Electrochem Soc 1994: 141(1): 1-10. Spotnitz R. incorporated SEI growth into Fuller's model and investigated the correlation of impedance change with capacity fade in Simulation of Capacity Fade in Lithium-ion Batteries, J Power Sources 2003; 113 (1): 72-80. In Modeling Capacity Fade in Lithium-ion Cells, J Power Sources 2005; 140: 157-61 by Liaw B.Y., Jungst R.G., Nagasubramanian G., Case H. L. and Doughty D. FL, an equivalent-circuit model was used to simulate ceil performance, particularly the capacity fade phenomenon as influenced by thermal aging, which is one of the most influential factors affecting battery calendar life during storage, standby, or operational periods. An equation was proposed in A Method for Online Capacity Estimation of Lithium Ion Battery Cells Using the State of Charge and the Transferred Charge, IEEE ICSET 2010, Kandy, Sri Lanka, December 6-9, 2010 by Einhorn M., Conte F.V., Krai C. and Fleig J., where two accurate state-of-charge (SOC) values are regarded as functions of the open circuit voltage (OCV), and the integrated current between these two values are sufficient for estimating the capacity of the battery cell. Chan C.C, Lo EWC and Weixiang S. in The Available Capacity Computation Model Based on Artificial Neural Network for Lead-acid Batteries in Electric Vehicles, J Power Sources 2000; 87 (1 ): 201-4 applied an artificial neural network with single input and single output to build a correlation between discharge current and capacity for lead-acid batteries. They assumed that aging and degradation of the battery do not significantly affect capacity estimation. However, this assumption does not apply to Li-ion batteries. An extended Kalman Filter in Extended Kalman Filtering for Battery Management Systems of LiPB-based HEV Batter}' Packs, J Power Sources 2004; 134: 277-92 by Plett G.L. has also been used for capacity estimation based on specific state/parameter models involved. In US 20020193953 A l, a multivariate linear model is established for the relationship between capacity and a multitude of inputs, including internal DC resistance, open circuit voltage (OCV), and temperature.

So far, most of the abovementioned models have largely contributed to accurate capacity estimation. However, some issues should be dealt with before battery capacity estimation models can be fully applied to real-world applications:

(1) Dependence on accurate models representing the dynamic behavior of batteries, which have been proven difficult to establish (see Tang X.D., Mao X.F., Lin I. and Koch B., Capacity Estimation for Li-ion Batteries, 2011 American Control Conference on O'Farrell Street, San Francisco, CA, USA, June 29-July 01 , 2011.14);

(2) Electrochemical parameters and properties of ba tteries are required;

(3) Reliance on accurate SOC values, which are also a significant and difficult research field; (4) OCV values are needed, which always require considerable time of rest;

(5) Being inappropriate for different operating conditions.

All of these issues can be divided into three main parts: 1) complex electrochemical mechanisms and corresponding models; 2) poor data conditions, i.e., significant for many existing methods but very difficult to obtain; and 3) different operating conditions, i.e., various factors affecting battery capacity estimation.

Summary of the invention

There mainly exist the following problems in the prior art. At the outset, complex electrochemical mechanisms and corresponding models are lacked. Besides, data conditions, which are significant for many existing methods are poor and very difficult to obtain. Furthermore, available battery capacity estimation methods are not applicable to different operating conditions, i.e., various factors affecting battery capacity estimation. Accordingly, the present disclosure discloses a method for estimating Li-ion battery capacity degradation, thereby solving the abovementioned problems.

The method for estimating Li-ion battery capacity degradation as disclosed by the present disclosure comprises: Step I: extracting geometrical features of a reference battery and an estimated battery under different operating conditions; Step II: establishing an intrinsic manifold based on both the Lap!acian Eigenmap (LE) method and the geometrical features extracted in Step I about the reference battery and the estimated batter} _' respectively; and Step III: achieving geometrical metric and estimation of battery capacity with a geodesic on the intrinsic manifold established in Step II about the reference battery and the estimated battery. As can be seen, the present disclosure proposes a capacity estimation process based on geometrical features completely.

In one embodiment, said geometrical features comprise length of constant voltage stage of the current curve in a charging process, maximum radius of curvature in the current curve of the constant voltage stage, areas under constant voltage curves in the charging processes and maximum slope of a discharge voltage curve during the early stage of the reference battery and those of the estimated battery. These geometrical features are sensitive to capacity degradation and can effectively reflect the intrinsic degradation or health state of a Li-ion batter} _' .

In one embodiment, in Step I, geometrical feature series can be derived from full life cycles of the reference battery and the estimated battery respectively, with any element of the series being extracted from the corresponding charging/discharging cycle; and in Step II, the geometrical feature series after linear normalization are used as inputs of the LE method,

In one embodiment, said different operating conditions in Step I comprise different ambient temperatures, different discharge rates and different end-of-discharges. Such being the case, the method of the present disclosure is adaptable for battery capacity estimation under different operating conditions.

In one embodiment, full life data of the reference battery can be considered as being obtained from continuously charging and discharging cycles.

In one embodiment, the parameters of the intrinsic dimensionality, nearest neighboring point , sigma and alpha are configured in Step II.

In one embodiment, geodesic distances between data points on the established low dimensional intrinsic manifold of the reference battery', and that of the estimated batteiy are calculated in Step III respectively, which as a geometrical metric of batter}' capacity can be further applied to batter capacity estimation.

In one embodiment, on the established intrinsic manifold of the reference battery and the estimated batter}', the geodesic distance between the first data point representing the battery capacity state at the end of a first charging-discharging cycle, and an estimation data point representing the battery capacity state at the end of the charging-discharging cycle when estimating is calculated using graph theory in Step III. In one embodiment, the estimated capacity in Step III can be expressed as wherein c is an estimated battery capacity value of the batter} _' Bj to be estimated; ^Lm represents the initial capacity of Bi; ^Ceol represents a specified engineeringly acceptable capacity to the end of the charging/discharging cycle of Bi; ^{&e o} * denotes a geodesic distance between the first point and a point to be estimated on the intrinsic manifold of B-,; and ^ge ^ indicates a geodesic distance between the first point and the last point on the intrinsic manifold of the reference battery B ₀ . As can be concluded, the law of battery performance degradation can be well described and recognized in the manifold space by the method of the present disclosure, which mvoles low cost of computation time and few requirements on capacity data conditions.

In one embodiment, the charging process is carried out in a constant current mode at the beginning until the battery voltage reaches a predetermined value and then continues in a constant voltage mode until the charge current drops to a predetermined current value.

The method according to the present disclosure is advantageous over the prior art in the following aspects. It is sensitive to slight changes in performance degradation and thus can effectively determine the intrinsic degradation or health state of a Li-ion battery; it is completely based on geometrical features for capacity estimation; with the method of the present disclosure, the law of battery capacity degradation can be well described and recognized in the manifold space; the present method is adaptable for battery capacity estimation under different operating conditions; it is less time consuming in computation; and the present method requires less in battery data conditions.

So long as the object of the present disclosure can be achieved, the above technical features can be combined in any technically applicable manner to produce new embodiments. Brief Description of Drawings

In the following, the present disclosure will be described in detail based on merely non-restrictive embodiments with reference to appended drawings, wherein,

Fig. 1 is an operating flow chart of the method according to the present disclosure;

Fig. 2 shows geometrical features of the current curve in the charging process;

Fig. 3 shows discharging curves with a very early stage zoomed in;

Figs. 4a and 4b are the original charging/discharging curves of battery #5, wherein Fig. 4a in an upper position indicates current curves of charging throughout batter}' life and Fig. 4b in a lower postiion indicates voltage curves of discharging throughout battery life;

Figs. 5a to 5d show values and tendencies of the four extracted and normalized features, wherein Fig, 5a on the top left indicates the values and tendencies of geometrical Feature 1 (length of constant voltage stage of the current curve in the charging process). Fig. 5b on the top right indictes the values and tendencies of geometrical Feature 2 (the maximum radius of curvature in the current curve of the constant voltage stage), Fig. 5c on the bottom left indicates the values and tendencies of geometrical Feature 3 (the areas under constant voltage curves in the charging processes) and Fig. 5d on the bottom right indicates the values and tendencies of geometrical Feature 4 (the maximum slope of a discharge voltage curve during the early stage);

Fig. 6 shows battery performance degradation on the intrinsic manifold established by the LE method: and

Figs. 7a to 7d show comparisons between the measured and estimated capacities, wherein Fig. 7a on the top left, Fig. 7b on the top right, Fig. 7c on the bottom left and Fig, 7d on the bottom right indicate battery data #BG005, #B0007, #BG029 and #B0054 respectively.

In the figures, the same component is indicated by the same reference sigh and the figures are not drawn in accordance with an actual scale.

Detailed Description of Embodiments

In the following, the present disclosure will be explained in detail with reference to the figures and embodiments.

The present disclosure discloses a method for estimating Li-ion battery capacity degradation. Fig. 1 shows the main operating process of the method with specific steps as follows.

Step I: extracting geometrical features of a reference battery and an estimated battery under different operating conditions.

Features or parameters that reliably represent the actual performance or degradation of Li-ion batteries must first be determined to accurately estimate the Li-ion battery capacity. Considering the different aforementioned operating conditions, these features must be highly adaptive to changing conditions.

In one embodiment, the following four geometrical features extracted from the current curves in the charging processes and voltage curves in the discharging processes are presented for use in Li-ion battery capacity estimation.

Fig, 2 depics current curves in the charging processes of the 10th, 60th, 110th and 160th cycles, from which three geometrical features are extracted. Generaly, the charging process can be distinctly separated into two stages, namely, the constant current and constant voltage stages. The charging process and its corresponding conditions are similar or the same for one application. Thus, regarding an estimation battery Bj and a full life reference battery Bo that have gone through N cycles of charging/discharging processes, since the reference battery Bo and the estimated battery Bi are similarly treated except some slight differences in the subsequent steps, Bi can be taken as an example for explaination. As to the reference battery Bo, the full life data thereof can be considered as being obtained from continuously charging and discharging cycles. The geometrical features can be described as follows.

(1) Geometrical feature 1 : length of constant voltage sta ge of the current curve in the charging process

Regarding an application, differences in two charging stages caused by regular use are reflected in the length of the constant current stage, as shown in Fig. 2, which adequatey shows the last discharging process state of the battery. However, variations in length of the constant voltage stage are largely caused by degradation of battery capacity. Fig. 5a shows the values and tendencies of Feature 1 in the charging/discharging cycle life of battery Bj. Therefore, when this geometrical feautre is extracted, lengths of constant voltage stages of the current curves in the charging processes under different operating conditions in the N cycles of the battery should be successively extracted to constitute a feature series Fl (fl -. , fl A ).

(2) Geometrical feature 2: the maximum radius of curvature in the current curve of the constant voltage stage

In Fig. 2, the curves of constant voltage stage are similar to one another. Nevertheless, the maximum radius of curvature in each curve is of fine distinction, which can be adopted to refer to the battery peformance degradation. Figure 5b shows the values and tendencies of Feature 2 in the charging/discharging cycle life of battery B . Therefore, when this geometrical feautre is extracted, the maximum radius of curvatures in the curve of the constant voltage stages under different operating conditions in the N cycles of the battery should be successively extracted to constitute a feature series F2 (ί¾, f2 ₂ , .. . , 2N).

(3) Geometrical feature 3: the areas under constant voltage curves in the charging processes

Like Features 1 and 2, the areas under the constant voltage curves are indicators of battery peformance degradation over time, Fig. 5c shows the values and tendencies of Feature 3 in the charging/discharging cycle life of battery Bj . Therefore, when this geometrical feautre is extracted, the areas under constant voltage curves in the charging processes under different operating conditions of the N cycles of the batter}' should be successively extracted to constitute a feature series F3 (f3i, f3 ₂ ,...,

(4) Geometrical teature 4: the maximum slope of a discharge voltage curve during the early stage Fig, 3, depicting the discharge curves of the 10th, 60th, 110th and 160th cycles, shows that unlike features 1, 2, and 3, feature 4 is extracted from a discharging curve during different cycles instead of a charging curve. Feature 4 can also demonstrate the performance degradation process, and in Fig. 5d, it shows the values and tendencies of Feature 4 in the cycle life of battery Eh . Therefore, when this geometrical feautre is extracted, the maximum slopes of a discharge curve during the early stage under different operating conditions of the N cycles of the battery should be successively extracted to constitute a feature series F4 (f4j , f4 ₂ ,.. ,, f4 _N ),

All these four geometrical features are utilized to estimate the real life Li-ion battery capacity,

Step II: establishing an intrinsic manifold based on the LE method.

The four geometrical feature series extracted in Step I, Fl, F2, F3 and F4, are linear normalized and the normalized four geometrical feature series NFl , NF2, NF3 and NF4 serve as inputs of the LE method. These normalized feature series are configured with parameters of intrinsic dimensionality, nearest neighboring point K, sigma and alpha, so as to construct a low-dimensional manifold embedded in a high-dimensional space, which is the intrinsic manifold, whereby representation of high geometrical features towards the low-dimensional manifold is achieved and data points on the low-dimensional manifold space can be obtained. Thus, battery capacity and information contained in the original four geometrical features can be well expressed in the low-dimensional manifold. Step ill: achieving geometrical metric and estimation of battery capacity with a geodesic on the intrinsic manifold.

In the present disclosure, when low-dimensional data points on the low-dimensional manifold are obtained, a geodesic on the intrinsic manifold thereof can be uti lized as a geometrical metric of battery health to estimate battery capacity since these data points can represent batter}' capacity degradation on the intrinsic manifold. In addition, the geodesic distance between the first point (the battery capacity state at the end of the first charging/discharging cycle) and an estimation data point (battery capacity state at the end of the charging/discharging cycle when estimating) on the intrinsic manifold is calculated using graph theory. The corresponding geodesic distance of the reference battery B ₀ is also calculated. The estimated capacity can be expressed as geo _s

AO

e°EOL (I) wherein < is an estimated battery capacity value of the battery' Bi to be estimated;

^A0 represents the initial capacity of Bi (the original capacity and nominal capacity are not always equal in actual projects and are thus determined accroding to specific conditions); ^c «* represents a specified engineeringly acceptable capacity to the end of the charging/discharging cycle of B ₃ ; ^geo < denotes a geodesic distance between the first point and a point to be estimated on the intrinsic manifold of Bi ; and ^ ^eo ∞t indicates a geodesic distance between the first point and the last point on the intrinsic manifold of the reference batter}' Bo.

In the following, it verifies and explains the efficiencyof the method according to the present disclosure and demonstrates how the method can be used through specific embodiments. Validation data are collected from the Prognostics CoE at NASA Ames, The experiments are conducted through three different operational profiles (charge, discharge, and impedance). Charging is carried out in a constant current (CC) mode at 1 .5 A until the battery voltage reaches 4.2 V and then continues in a constant voltage (CV) mode until the charge current drops to 20 niA. The discharge runs are stopped at different end-of-discharges (EODs). The experiments are carried out until the capacity decreases to specified end-of-life (EOL) criteria.

To validate the efficiency of the method according to the present disclosure, the typical data (#5, #7, #29, and #54) shown in Table 1 have been selected. Generally, these data exhibit different ambient temperatures (ATs °C), charging currents (Cs; ampere), discharging currents (DCs; ampere), EOD criteria (voltage), and EOL criteria (ratio of faded capacity to initial capacity) for comparison. These data are all full life data. In order to analyze the efficiency of the processes and results of the method according to the present disclosure, the full life data are considered as being obtained from continuously performing charging and discharging cycles. Each estimation result is that at the end when the corresponding charging discharging cycle is completed. This embodiment further proves the utility values of the implementing processes and projects of the present disclosure in detail.

Table 1 shows typical data obtained under different operating conditions of the batteries.

Step I: extracting geometrical features of a reference battery and an estimated battery under different operating conditions.

To illustrate the method according to the present disclosure, battery #5 is selected for demonstration. Figs. 4a and 4b show the original current curves in the charging processes (Fig. 4a) and the voltage curves in the discharge runs (Fig. 4b). The four geometrical features of the present disclosure are extracted from the original current curves of battery #5 and those of reference battery B ₀ . Figs. 5a to 5d show the tendencies of the four geometrical features extracted from the original curves of battery #5 after linear normalization, i.e., length of constant voltage stage of the current curve in the charging process, the maximum radius of curvature in the cerrent curve of the constant voltage stage, the areas under constant voltage curves in the charging processes and the maximum slope of a discharge voltage curve during the early stage.

Step II: establishing an intrinsic manifold based on the LE method.

Fig, 6 illustrates a trace line on the low-dimensional manifold generated by applying the normalized four geometrical feature series NFl, NF2, NF3 and NF4 as inputs of the LE method and configuring the parameters of the intrinsic dimensionality as 2, nearest neighboring point K as 8, sigma and alpha respectively as 10.0 and 1.0, wherein the battery performance (capacity) degradation in the 1 st, 50th, 100th and the last charging/discharging cycles are successively scattered on the intrinsic manifold with good visualization effect.

Step III: achieving geometrical metric and estimation of battery capacity with a geodesic on the intrinsic manifold.

The geodesic distance between the first data point (corresponding to the battery capacity state at the end of the first charging/discharging cycle), and an estimation data point (corresponding to the battery capacity state at the end of the charging/discharging cycle when estimating) is calculated using graph theory. All the data required in Equation (1) are obtained through calculation. The estimated capacity based on the geodesic distance computation on the intrinsic manifold using Equation (1) and the measured capacity from all testing data (#5, #7, #29, and #54) were compared. Figs. 7a to 7d demonstrate the close tracking of the measured capacity under different operating conditions. Table 2 shows that under different operating conditions, the estimated capacity maintains a high level of accuracy. The maximum and minimum of the error ¹ between estimated capacity and measured capacity are 4.48% and 1.85%, respectively. For the error ^"1 , the calculated values are 3.84% and 1.06%, respectively. Error l==measured capacity-estimated capacity; Error2Hmeasured capacity- estimated capacity !/measured capacity. Table 2 shows estimation accuracy of available capacity based on the method according to the present disclosure.

Table 2

The tests have shown that even with limited data and changing operating conditions, Li-ion battery capacity can be accurately estimated under different operating conditions with the geometrical method according to the present disclosure based on the treatment on the data generated from Li-ion batteries in use rather than establishing a complex electrochemical model.

What has not been explained in the disclosure belongs to common knowledge in the art, whereby confusion is avoided. Although the present disclosure has been discussed with reference to preferable embodiments, it extends beyond the specifically disclosed embodiments to other alternative embodiments and/or use of the disclosure and obvious modifications and equivalents thereof. The scope of the present disclosure herein disclosed should not be limited by the particular disclosed embodiments as described above, but encompasses any and all technical solutions following within the scope of the following claims.