Estimation of Reduced Order Equivalent Circuit Model Parameters of Batteries from Noisy Current and Voltage Measurements

Identification of reduced order equivalent circuit battery model from current and voltage measurements allows modeling, classification and monitoring of batteries, and these tasks are very essential for battery management systems. This study presents a theoretical study to investigate performance of computer-aided identification of the reduced order equivalent circuit battery model from noisy current and voltage measurement data. The battery model is expressed by fractional order differential equation and time domain solution of this model is numerically calculated according to Grunwald-Letnikov definition of fractional-order derivative. Paper demonstrates an application of this numerical solution in order to fit noisy current and voltage measurement data by using particle swarm optimization (PSO) method. Then, parameters of the equivalent circuit battery model are estimated. Performance of the parameter estimation method is investigated for various noise levels of the synthetically generated current and voltage profiles.


I. INTRODUCTION
UE TO growing demand for mobility in daily life, battery utilization in real applications increase and hence battery monitoring and management methods are becoming an essential component of mobile systems.In fact, practical performance of many battery powered systems e.g.mobile devices, electrical vehicles depends on battery management performance [1,2].Nowadays, monitoring battery health and optimal control of battery packs are major topics of battery research studies, and these studies has focused on modeling and real-time identification of battery model parameters, which is useful for evaluating battery status and health [3].Therefore, online control and the management of batteries are performed based on model parameter estimations, which can provide the meaningful information related to conditions of batteries.
Manuscript received July 30, 2018; accepted October 28, 2018.DOI: 10.17694/bajece.449265 There are several models that were utilized for modeling batteries in different levels of complexity [3,4].High complexity models can be useful for more proper representation of the behavior or properties of batteries in simulation environment, however increasing model complexity can heavily complicate real-time identification problem and model parameter estimation efforts.One of the widely utilized models for battery monitoring is the reduced order equivalent circuit model.This model can be used by battery monitoring algorithms and identified either in time domain by measuring current and voltage profiles or in the frequency domain by applying impedance spectroscopy measurements [2]. Figure 1(a) shows a reduced order equivalent circuit model based on the impedance spectrum data [2,5].Figure 1(b) illustrates an s-domain transfer function representation of this model [6].The fractional order capacitor element makes this model a fractional order system model [6] and time-domain analysis of this model requires application of fractional calculus.Since recent developments in fractionalorder system analysis tools [7], the s-domain representation of this system can be preferable for battery parameter estimation and simulation purposes.
Grünwald-Letnikov (GL) definition of fractional order differentiation has been widely utilized for the numerical solution of fractional order differential equations, solution of fractional order system models in state space form and calculation of time response of fractional order transfer functions [7,8,9,10,11,12].Numerical calculation methods based on Grünwald-Letnikov definition are commonly used to develop fractional-order system analysis tools [7,10].
In previous studies, Grünwald-Letnikov based numerical solution of this battery model were utilized in many studies for estimating state of charge, parameter sensitivity analysis etc. [1,13,14,15].Some of them were utilized metahereustic optimization methods, such as hybrid multi-swarm particle swarm optimization and genetic algorithm, because model fitting problem is not so easy for fractional-order systems.In addition to model coefficients (component values), the estimation of fractional-orders of the differential equation models is required, and this significantly increases complexity of numerical solution of the problem compared to fixed order differential equation models, namely integer-order system models.Main reason of this complication comes from the long memory effect, that is, fractional order derivative depends on all past values of the fractionally derived function [7] and hence fractional-order derivative is not a local operator.Performance of real-time model parameter estimation algorithms depends on a number of factors: (i) Limitation of computational resources such as the speed of processor, memory capacity can decrease practical performance.Computational complexity of the estimation method should be low enough so that it can be implemented on control cards by using embedded programming techniques.(ii) Negative impacts of measurement noise on the accuracy of parameter estimation can decrease practical performance.The estimation accuracy should be enough robust against measurement noises.(ii) For online analysis of model parameters, estimation method should be independent of waveform of the measured data because, in real applications, measured current and voltage can be in any waveforms.Real-time operating model identification algorithms should deal with waveform uncertainty of the measured data.
The current study discusses performance of a time-domain model identification approach that may be utilized for realtime estimation of parameters of the reduced order equivalent circuit model.This method performs fitting Grünwald-Letnikov based numerical solutions of the equivalent circuit model to noisy current and voltage data profiles that are measured from battery terminals.To achieve approximation of the model response to this measured data, an average of squared difference of numerical voltage and measured voltage values is used as the objective function that should be minimized to achieve identification.Here, PSO algorithm is employed to solve this optimization problem.To improve estimation of model parameters, this objective function, namely mean square error (MSE), is modified by taking logarithm of MSE function.This modification to objective function can improve minimization performance of PSO at very low values of MSEs because of the logarithmic stretching of MSE values at very low levels.Here, the logarithm of MSE can make objective function more distinguishable at very low values.

II. PRELIMINARIES AND PROBLEM STATEMENTS
Impedance spectroscopy is a main technique that has been applied for identification of sophisticated equivalent circuit models [2,5,16].This method is performed by matching real or complex parts of frequency domain models or magnitude and phase responses for the each measured frequency [2].In this manner, the equivalent circuit model shown in Figure 1(a) is expresses in the complex impedance form, The last term of the equation (1) expresses a fractional order behavior, which was also known as constant phase element [2,17].This term is also referred to as ZARC or Cole-Cole circuit element [17,18].Parameters We aim to perform time domain identification of the model from a given current and voltage data set.By considering  j s  relation, the transfer function model of the reduced order equivalent circuit model was written in the form [6], The s-domain equivalent circuit representation was shown in Figure 1(b) and this representation implies that fractional order system analysis methods can be applied to obtain time domain solutions of this model.Considering current and voltage in impedance function, by considering the equation ( 2), a fractional order system model is written as To facilitate time domain solution of battery voltage, the equation ( 4) can be decomposed as the sum of three voltage elements as, where the voltage terms are for constant phase element.By taking inverse Laplace transform of these voltage components, time domain solution of the components can be written as follows, where ) (t i is the battery current.To obtain time domain , and by taking inverse Laplace transform and using the properties of ) Numerical solutions of this fractional-order differential equation can be obtained by using Grünwald-Letnikov definition of fractional-order derivative operator, which is written by [7], where a parameter is a lower bound for time derivative and it limits calculations from the time a to current time.Although it reduces need of memory elements to store the past values, it results in a truncation error in calculations.For more accurate calculation of fractional derivatives in system analysis, a can be zero.Parameter h is step size (sampling period) for computations.
The equation ( 11) is used in equation ( 8), the numerical solution of this differential equation can be found by solving, This equation can be rewritten as . (13) By rearranging the equation ( 13), the solution ) The weights ) ( j w is calculated recursively according to equation (10).By considering equation ( 5), a battery voltage solution with respect to a given current ) (t i can found by By considering equation ( 6), ( 7) and ( 14), ) where h is the sampling period and the battery voltage can be calculated by ) to time domain solution of the equivalent circuit battery model, which is calculated by equation (15), a mean square error to minimize is commonly written by [10,[13][14][15] where, p is the total number of the sampled data.The ) (n v a is calculated according to the equation (19).To increase accuracy in the estimation of model parameters, MSE, given by equation (20), can be modified by taking logarithm of MSE function.This makes very low error values more distinguishable as a result of stretching effect of logarithmic at very low values.The logarithmic error function can be written by, To estimate the reduced order equivalent model parameters of battery, which are        This accuracy level can be acceptable for some battery parameter estimation applications.To test algorithm for a severe noisy measurement case, the current waveform was noised at 30.05  SNR dB level and ground truth voltage waveform was noised at 7.59  SNR dB as illustrated in Figure 8. Figure 9   The performance data in Table 1 reveals that the estimation of resistance values,   .This simulation confirms that the method can be applied for any measurement waveforms, and therefore it can be used for real-time monitoring of batteries.

V. CONCLUSION
This study theoretically discussed performance of a scheme for online estimation of reduced order equivalent circuit model parameters of batteries from noisy and irregular waveform current and voltage measurement data.Battery model parameter estimation is needed for modeling, classification and monitoring of batteries in real applications.Temporal change of battery model parameters can be utilized for detection of battery status, diagnoses of aging effect, faults and defects.Besides, real-time identification of battery models can allow optimally control of battery charge and discharge processes.Therefore, properly estimation of battery model parameter is very essential for battery management systems and algorithms.Nowadays, increase of battery usage as a result of growing demands of electrical vehicles and mobile applications places battery modeling and parameter estimation efforts at the center of battery management system researches.The presented solution for battery parameter estimation can estimates the reduced order equivalent circuit model parameters by fitting Grünwald-Letnikov based numerical solutions of the battery model to current and voltage profiles of the batteries.This fitting problem was expressed in the form of minimization of a logarithmic MSE function of battery voltages and solved by employing PSO algorithm.The numerical studies, which were carried out for various levels of random noise addition to input and output of Li-ion battery models, reveal that this method can work effectively in case of noisy measurements.Therefore, we conclude that the model parameter estimation of this method may be applied in real battery management applications.A future study should be conducted for experimental validation of performance of this method in experimental systems.This program performs equation ( 19) to obtain GL solution of battery voltage.This code can be used in optimization process as shown bellow.

L
are output resistance and inductance of the model, respectively.
Fig. 2. Fitting of the model voltage ) (n v a

For
Fig. 3. Fitting of the model voltage ) (n v a shown subsampled in the figure.

R 1 C and 0 L 1 C and 0 L
are static elements that are mainly effective in determining DC level of the measured signal.Smoothing effect of fitting operation can reduce negative impacts of irregular pattern of noise on the estimation of 1 R and 0 R , and this makes their estimations more robust against random noise signal.The dynamic components mostly act on transient parts of the measured signal.Therefore, irregular motif of noise can easily interfere predictions of these parameters, and this may reduce accuracy of estimations of parameters .In particular, since 0 L parameter is directly associated with derivative of the measured current, 0 L estimations are very sensitive to the noise of current measurements.Therefore, in case of noisy data, parameter search range of 0 L should be confined to its realistic ranges in order to avoid misleading parameter estimation results.In our analyses, we restricted ]

Appendixes
An example Matlab code to perform for numerically calculation of voltage of the reduced order equivalent circuit battery model is given, below.This code calculates battery voltage

Fig. 12 .
Fig. 12.A basic block diagram that shows application of PSO algorithm in this problem 19)An example Matlab code, which performs the numerical calculation of the reduced order equivalent circuit battery model is presented in Appendix section.
L in Table1are mainly results of this restriction.

TABLE I A
LIST OF PARAMETER ESTIMATIONS FOR VARIOUS NOISE CONDITIONS AND ABSOLUTE RELATIVE ERRORS IN PARAMETER ESTIMATION