## Adv-radio-sci.net

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
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**Figure 8. **Statistic analysis of positive classification rates for the

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to update the hyperplanes in an enhanced machine learning
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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**
Source: http://www.adv-radio-sci.net/13/127/2015/ars-13-127-2015.pdf

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