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Adv. Radio Sci., 13, 2015www.adv-radio-sci.net/13/127/2015/doi:10.5194/ars-13-127-2015 Author(s) 2015. CC Attribution 3.0 License.
Impedance spectra classification for determining the state of charge
on a lithium iron phosphate cell using a support vector machine
P. Jansen, D. Vergossen, D. Renner, W. John, and J. Götze
1Audi Electronics Venture GmbH, Gaimersheim, Germany2SiL GmbH – Paderborn/Leibniz Universität Hannover, Hanover, Germany3Technische Universität Dortmund (AG DAT), Dortmund, Germany Correspondence to: P. Jansen ([email protected]) Received: 12 December 2014 – Revised: 11 April 2015 – Accepted: 21 April – Published: 3 November 2015 Abstract. An alternative method for determining the state of
energy is still left in the battery, and how much energy can be charge (SOC) on lithium iron phosphate cells by impedance charged back. This essential knowledge of the energy level in spectra classification is given. Methods based on the electric the storage device defines the whole operational strategy of equivalent circuit diagram (ECD), such as the Kalman Fil- the EMS. The SOC indicates critical states such as deep dis- ter, the extended Kalman Filter and the state space observer, charge or overcharge. These levels of extremely high or low for instance, have reached their limits for this cell chemistry.
SOC can lead to irreversible damage in the battery The new method resigns on the open circuit voltage curve The task of the EMS is to avoid these critical states and the parameters for the electric ECD. Impedance spectra in any case to enable a high endurance.
classification is implemented by a Support Vector Machine In this context a specific definition of the SOC is needed.
(SVM). The classes for the SVM-algorithm are represented The most common definition for the SOC is the ratio between by all the impedance spectra that correspond to the SOC (the the difference of the rated capacity Cn and the charge balance SOC classes) for defined temperature and aging states. A di- Qb to the rated capacity Cn. The SOC is 1 when state of vide and conquer based search algorithm on a binary search charge FULL is reached and 0 after a net discharge of the tree makes it possible to grade measured impedances us- ing the SVM method. Statistical analysis is used to verify the concept by grading every single impedance from each impedance spectrum corresponding to the SOC by class with different magnitudes of charged error.
This definition ignores the problem of the battery aging, as the capacity that can be delivered by a battery may changein the course of its life due to problems such as the loss ofcharge acceptance of the active materials on either of the electrodes, changes in the physical properties of the elec-trolyte or corrosion on the current conductors The exact determination of the state of charge (SOC) of a This aging behavior is called state of health (SOH).
cell, especially of a lithium iron phosphate cell, is a challeng- Together the temperature behavior and the SOH have the ing task in signal processing. The requirements as regards the biggest influence on the SOC.
accuracy of the determined SOC are very significant in the The aim of this work is to propose a method to determine automotive industry to ensure that the electrochemical stor- the SOC of a lithium iron phosphate cell, in an automotive age device operates in a reliably and efficiently mode.
application under load conditions, with a specific SOH and The exact SOC in automotive applications, for hybrid as a defined temperature from the frequency domain data. The well as conventional electrical supply systems on board a ve- reference cell used for the impedance spectroscopy was in hicle, is a very important information for the energy man- mint condition, so the SOH was approximately 100 % when agement system (EMS). The EMS needs to know how much the impedance spectroscopy was carried out.
Published by Copernicus Publications on behalf of the URSI Landesausschuss in der Bundesrepublik Deutschland e.V.
P. Jansen et al.: SOC determination using a support vector machine
Methods for determining the state of charge
The following are existing methods for determining the SOC independent of the battery type: – discharge test,
– Ah balance,
– open circuit voltage,
– Kalman filter,
Figure 1. Equivalent circuit diagram.
– state space observer,
– artificial neuronal network,
– machine learning.
Some of the methods are limited in their range of func- tionality. The discharge test and Ah balancing for instance are not suitable solutions for an automotive on-board appli- cation. The most common time domain based methods for on-board SOC determination are equivalent circuit diagram(ECD) (see Fig. based methods such as the Kalman Filter, the extended Kalman Filter and the state space observer. The mathematical basis of those methods is a state space model Figure 2. Open circuit voltage curve.
several different time based chemical reactions such as diffu- sion or the charge carrier movement.
The ECD based methods are suitable for many different types of chemistries but they reach their limits when it comes to lithium iron phosphate chemistry. There are two main rea- sons for this, in particular for lithium iron phosphate cells: the extremely smooth plateau of the OCV curve in the mid- dle SOC range (see Fig. and the accuracy of the ECD. The OCV calculated from the terminal voltage highly depends on the accuracy of the ECD parameters. For that reason OCV based methods are only capable of estimating the SOC in themiddle range of the OCV curve with a monadic percentage precision with very substantial effort in terms of measure-  + Ri · IBatt Recent theoretical work on methods in the field of ma- chine learning, use support vector Regression or support vec- tor machine (SVM) on time domain based data such as volt- In this model based on the ECD, the open circuit voltage age, current and temperature to determine the SOC (OCV) can be calculated to correct the SOC of the state space Ampere hour integrator to compensate for untraceable ca- other approach in this case is to use the frequency domain pacity losses. The quality of the model's dynamic behavior data to determine the SOC. The basis of this method will be is important when it comes to the accuracy of the calculated provided in this paper.
OCV. That dynamic behavior depends on the number of RC-circuits of the ECD (see Fig. The RC-circuits represent Adv. Radio Sci., 13, 2015
P. Jansen et al.: SOC determination using a support vector machine
Principle and methodology
The SVM is a binary classifier from the field of machinelearning theory. The SVM is a support vector learning al-gorithm for pattern recognition, with the aim of classifyingquantities with certain attributes and grade unknown sam-ples to one of two classes. There are other methods for clas-sification problems such as the nearest neighbor decision(NND) and its derivatives such as the kn-nearest neighbordecision and the distance-weighted kn-nearest-neighbor de-cision. NND-methods use euclidean metrics to evaluate the distance between the sample and its classified single-nearest neighbor or kn-nearest neighbors. An a priori assumption ofthe underlying statistics of the training data as for a Bayes classifier is not necessary. Therefore the big advantages of this method are the simplicity and performance. A disadvan- tage in this regard is the fact that, as sets of training data increase, the classification probability decreases Compared to the SVM method, NND-methods have higher costs in terms of memory spaceto store the entire volume of training data and in terms ofruntime because all the training data has to be evaluated tograde a single sample. SVMs on the other hand resign on thestorage of the training data. Their aim is to detect a patternwithin the training data via its class-specific attributes so that Figure 3. Support vector machine in 2-D space
the data can be intersected by a hyperplane based on supportvectors and the training data can be separated without errors The training data is required as the basis for the classifica- tion, where every element x of a quantity is affiliated to one Support vector machine
class by its class label yi.
, yl) ∈ <N × {±1} A binary support vector classifier such as the SVM is based on a class of linear hyperplanes, The class label is defined by ±1, so that all elements of one class are labeled by +1, and all elements of the other class w ∈ <N , b ∈ < are labeled by −1. Based on this information in a first stepthe SVM classification function, based on a linear kernel, is to separate a number of elements into two specific classes, capable of evaluating the optimal hyperplane to separate the based on class specifying attributes – for instance color, two classes.
shape or other metadata – using a hyperplane. The hyper- The corresponding SVM decision function for linear sep- plane is the shortest orthogonal line connecting the convex hulls of this two classes, and intersects them half way. The optimum hyperplane has a symmetrical maximum margin (x) = sgn((w · x) + b) to both convex hulls. The normal vector w with a thresh- leads in a second step to a grading of an new unknown ele- old b defines the linear hyperplane and its margin of the two ment x to one of the classes with the return of its affiliation classes, so that a grading of a new unknown element x is pos- sible. The so-called support vectors x are specific objects of In case of non linear separable data the normal vector w the training data. That are the elements of the convex hulls is a representation of a linear combination of support vectors closest to the margin (see Fig. The SVM is also applicable x from the training data, the corresponding class labels y to non linear separable data, by using the so-called "Kernel and the lagrange multipliers vk Trick" to transform the data into a high-dimensional featurespace where the data is linear separable.The kernel depends on several usable funktions, for instance a polynomial or a radial basis function, to evaluate the hyperplane that sepa- rates the data in the feature space. A suitable kernel function As this efficient learning algorithm has simple correspon- has to be chosen specifically for the training data.
dence to a linear method in a high-dimensional feature space Adv. Radio Sci., 13, 2015
P. Jansen et al.: SOC determination using a support vector machine
i=bn− n2 c+1 {ci} bn− n2 c+1 {ci} bn− n2 c+1 {ci} Figure 4. Impedance spectra frequency domain cutout 80–0.1 Hz.
Figure 5. Binary search tree.
that is non-linearly related to its input space, it is straight-forward to analyze it mathematically. We are dealing herewith a classification algorithm, where a superset of elements sented by their specific linear combinations, is separated into two power sets or classes where single new elements are graded to one of those classes. A SVM is only capable of a single binary decision regarding whether the ap- plicable element belongs to one power set or the other.
of lagrange multipliers v ek,med , corresponding class labels yk and the support vectors x Impedance spectra classification
ek,med of the represented SOC class Determining the SOC via impedance spectra classification using SVM is an alternative method to achieve this aim. This method resigns an electric ECD with components such as the OCV curve and the element parameters of the electric The two new nodes of the graph represent the two roughly equal power sets of the above superset. The separation of The task of determining the SOC of a lithium iron phos- the generated power sets can be repeated recursively down phate cell can be achieved with an optimal classifier such to the power set elements representing a single impedance as the SVM by grading measured impedances to a certain spectrum or SOC class {ci}. The resulting graph corresponds class. The classes for the SVM are represented by all the to a binary search tree, where the root is the whole superset impedance spectra for different SOC levels, corresponding of all elements from the impedance spectra with the nodes as to defined temperature and aging states, generated by an a power set of its parent superset and the leaves representing impedance spectroskopy, are the foundation of this classifica- the single SOC classes (see Fig. tion method (see Fig. The data of those spectra represents This binary search tree can easily be parsed by a binary the training data of the SVM classification function – based tree search algorithm where the edges of the graph repre- on a polynomial kernel function – that is used to calculate sent the binary decisions of the SVM decision function. So the hyperplanes that separate all impedance spectra to their a binary tree search algorithm such as the divide and con- nearest neighbors. As noted above, the SVM is only capable quer search algorithm, applied to the afore mentioned binary of a binary decision, therefore with more than two classes a search tree, makes it possible to grade measured impedances separation of every impedance spectra to its nearest neighbor Zi using the SVM decision function, has to be realized by a hyperplane via SVM. So n classeswill yield to n − 1 hyperplanes to be evaluated by the SVM to separate all spectra from each other.
The SVM decision function can only make binary deci- sions so that all the SVM decisions have to be rated and con- where d indecates the degree of the used polynomial kernel textualized. The most efficient way to do so is to create a graph to arrange the hyperplanes. The whole quantity of the impedance spectra elements or the superset, the root of the graph, is separated by the median of the hyperplanes repre- Adv. Radio Sci., 13, 2015
P. Jansen et al.: SOC determination using a support vector machine
Class {+1} Support Vectors {−1} Support Vectors Class {+1} Graded Impedance Figure 7. Statistic analysis of positive classification rates for single
Figure 6. Impedance grading using SVM.
impedances depending on polynomial degree and impedance error.
distributed random noise X (ω) ∼ N µ, σ 2
charged with dif- ferent magnitudes m, The SOC of the cell can now be determined by grading at least one on-vehicle measured impedance Z from the cell under load conditions. The impedance spectra of the rele- vant defined SOC levels therefore have to be classified, asdescribed above. The measured impedance now has to be graded to a single SOC class – SOC specific impedance spec-  = X(ω) · m, m ∈ < 10−7 ≤ m ≤ 10−4o , trum – to determine the SOC of the cell. The binary decisionof the SVM decision function can only prove for two classes where ω is a random value, the mean value of µ = 0 and to which class, separated by the hyperplane, the measured the variance of σ 2 = 1, where added to each impedance of a impedance belongs (see Fig. By using a divide and con- impedance spectrum. These error charged impedance objects quer search algorithm on the binary search tree with the SVM are the specification for the SOC determination algorithm, decision function as a key criterion, the SOC can be deter- to clarify that they would be graded correctly to their origin mined by multiple binary decisions along the search tree.
impedance spectrum.
This divide and conquer algorithm starts at the median hy- The classification rate of this trial is highly dependent on perplane of all separated impedance curves, that separates two different factors. The first important factor is the polyno- this superset into two roughly equal power sets. The binary mial degree of the kernel, which defines the separation accu- decision, of the SVM decision function, whether the mea- racy of the hyperplanes between the impedance spectra. An- sured impedance belongs to the power set on one side of other important aspect is the magnitude of the charged error the hyperplane or the other decreases the quantity of relevant of the original impedances of the spectra.
SVM decisions by half. The remaining power set after the A linear representation of the hyperplanes is highly ineffi- decision containing the measured impedance will therefore cient, because of the very low classification rates (see Fig. also be divided by its median hyperplane into two subsidiary The optimal polynomial degree would be around 5–8 as a power sets. By continuing this recursive structure the SVM trade-off between accuracy and execution time. Another fea- decisions on the binary search tree ultimately grade the mea- ture is the high tolerance to the variance of error. The bi- sured impedance to a single dedicated SOC class. This class nary SVM based tree search algorithm is capable of grading represents the SOC of the cell for the measured impedance.
impedances charged with a variance of error up to 10 %, with This implementation of this binary SVM based tree search an accuracy of 60 %, for a single impedance. The statistical algorithm makes an optimal execution time of O(logn) for accuracy with 30 graded impedance objects of one SOC class the grading of one measured impedance possible.
rises up to 80 %. A classification rate of 90 % can then beachieved by decreasing the variance of error below 1 % (see Statistical evaluation demonstrates that the concept of a Trials with measured impedance spectra have demonstrated binary SVM based tree search is capable of determining the that the new concept for grading impedances using SVM SOC of a lithium iron phosphate cell in the middle SOC is effective for determining the SOC. After classifying the impedance spectra with the SVM classification function, anerror , in both directions (= and <), consisting of normally Adv. Radio Sci., 13, 2015
P. Jansen et al.: SOC determination using a support vector machine
Starter- und Bordnetzbatterien) which is funded by the BMWi (Bundesministerium für Wirtschaft und Energie) under the grant number 03 ET6003 I. The responsibility for this publication is held by the authors only.
Edited by: D. Killat Reviewed by: two anonymous referees Ávarez Antón, J. C., García Nieto, P. J., Viejo, C. B., and Vilán Vilán, J. A.: Support Vector Machines Used to Estimate the Bat- Figure 8. Statistic analysis of positive classification rates for the
tery State of Charge, IEEE Trans. Power Electro., 28, 5919– SOC determination.
Codeca, F., Savaresi, S., and Rizzoni, G.: On battery State of Charge estimation: A new mixed algorithm, IEEE Intl. Conf Contr., 102– Conclusions and future work
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one measured impedance, and a maximum error variance of Dudani S. A.: The Distance-Weighted k-Nearest-Neighbor Rule, 10 %. The accuracy of the classification rate can increase to IEEE T Syst. Man. Cyb., SMC-13, 325–327, 1976.
90 % depending on two factors, the variance of error of the Hearst, M., Dumais, S., Osuna, E., Platt, J., and Schölkopf, B.: Sup- measured impedance and the polynomial degree of the hy- port vector machines, IEEE Intell. Syst. App., 13, 18–28, 1998.
perplane function. Therefore the requirements for a on-board Hu, J., Hu, J., Lin, H., Li, X., Jiang, C., Qiu, X., and Li, W.: State-of- impedance spectrum measurement can be circumvented. For Charge Estimation for Battery Management System Using Opti- an in-vehicle application, it is important to be able to iden- mized Support Vector Machine for Regression, J. Power Sour.,269, 682–693, 2014.
tify different impedances at certain frequencies. To calculate Klotz, D., Schönleber, M., Schmidt, J., and Ivers-Tiffée, E.: New impedances for several frequencies out of the time domain approach for the calculation of impedance spectra out of time by a Fast Fourier Transformation, would be one method for domain data, Electrochim. Acta, 56, 8763–8769, 2011.
this application Taking the results of Lee, J., Nam, O., and Cho, B.: Li-ion battery SOC estimation the impedance grading method into account, it is possible method based on the reduced order extended Kalman filtering, to identify the requirements for the impedance calculation.
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Future topics for research in this regard are the analysis Piller, S., Perrin, M., and Jossen, A.: Methods for state-of-charge of impedances from time domain data for classification pur- determination and their applications, J. Power Sour., 96, 113– poses to determine the on-board SOC and the comparison with other methods such as the ECD based Kalman Filter Plett, G. L.: Extended Kalman filtering for battery management sys- or the state space observer, NND and its derivatives and ar- tems of LiPB-based HEV battery packs – Part 1: Background, J.
Power Sour., 134, 252–261, 2004.
tificial neuronal networks for SOC determination to iden- Sauer, D. U., Bopp, G., Jossen, A., Garche, J., Rothert, M., tify the advantages and disadvantages of the different meth- and Wollny, M.: State of Charge – What do we really speak ods to combine them into a hybrid SOC determination algo- about?, International Telecommunications Energy Conference rithm. It will also be important to incorporate aging detection (INTELEC), Copenhagen, Denmark, 6–8 June, 1999.
to update the hyperplanes in an enhanced machine learning Weng, C., Cui, Y., Sun, J., and Peng, H.: On-board state of health method based on the above binary SVM search tree algo- monitoring of lithium-ion batteries using incremental capacity rithm. Updating the hyperplanes will ensure that the SOC is analysis with support vector regression, J. Power Sour., 235, 36– correctly determined as the cell ages over its lifetime.
Acknowledgements. This contribution was developed within thescope of the project Drive Battery 2015 (Intelligente Steuerungs-und Batteriesysteme zur Steigerung der Effizienz und Sicherheit sowiezur Senkung der Systemkosten – AEV-Subproject: Optimierungdes Energiemanagements von Fahrzeugen mit Lithium-Ionen Adv. Radio Sci., 13, 2015

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