Theoretical Performance Evaluation of Battery Equivalent Circuit Model Parameter Estimators

As the main source of power for pure electric vehicles, new energy vehicle power lithium batteries are also a key technology that restricts the development of pure electric vehicles, and can directly affect the driving performance of vehicles. A reliable and efficient management system (Battery Management System, BMS) can allow the lithium-ion battery to output its best performance stably, while ensuring that the battery has a long enough service life. State of charge (SOC) estimation plays an important role in Li-ion battery management systems. The establishment of the battery working model is an important part of the state of charge estimation. Therefore, higher requirements are put forward for the establishment of the battery model. At the same time, the error of the subsequent data processing and estimation results is smaller, and the Kalman filter is often used for processing. This paper starts from the current situation of model research and improvement, focuses on the equivalent circuit model, analyzes the advantages and disadvantages of each model, and summarizes the comparative research on the analysis and processing effect of Kalman filter. And further experiment verification of SOC estimation through battery test experiments and algorithms, use Thevenin model, use least squares method for parameter identification, and finally use Kalman filter for SOC estimation, according to the analysis of the results, the fitting accuracy is 1.69%, provided by battery research a strong theoretical foundation.

Estimating the parameters of lithium-ion (Li-ion) batteries under dynamic working conditions is a critical challenge in the health management of electrical energy storage systems. This paper estimates the equivalent circuit model (ECM) parameters and analyzes the influence of different factors on the Li-ion batteries impedance using the electrochemical impedance spectroscopy (EIS) technique. Firstly, the influence of the temperature, state of charge (SOC) and number of charging/discharging cycles on the impedance spectrum was studied. Nyquist plots were used to perform this analysis. Subsequently, ECMs were used to characterize the Li-ion batteries under study at different operating conditions. Specifically, the ECM parameters were estimated using the gaining-sharing knowledge (GSK) metaheuristic algorithm, allowing the consequent estimation of the impedance spectra obtained by the EIS technique. Experimental results included 15 datasets measured experimentally at three temperature levels for five SOCs and different number of charging/discharging cycles. To investigate the variation of Li-ion battery parameters with cyclic aging 350 datasets from the literature were also used. The results showed that the impedance and model parameters depend considerably on the considered factors. The estimation of the ECM parameters demonstrated high accuracy and reliability (mean RMSE values of 4.32E-02 and 2.33E-02 and mean STD values of 3.38E-16 and 5.82E-13 with measured and literature datasets, respectively), allowing the robust determination of the numerous impedance spectra under study.

This paper discusses the problem of Non Data Aided (NDA) Signal to Noise Ratio (SNR) estimation of Binary Phase Shift keying (BPSK) modulated signals using the Expectation Maximization (EM) Algorithm. In addition, the Cramer-Rao Lower Bounds (CRLB) for the estimation of Data Aided (DA) and Non Data Aided (NDA) Signal to Noise Ratio (SNR) estimation is derived. Multiple Input Single Output (MISO) channels with Space Time Block Codes (STBC) is used. The EM algorithm is a method that finds the Maximum Likelihood (ML) solution iteratively when there are unobserved (hidden or missing) data. Extension of the proposed approach to other types of linearly modulated signals in estimating SNR is straight forward. The performance of the estimator is assessed using the NDA CRLBs. Alamouti coding technique is used in this paper with two transmit antennas and one receive antenna. The authors’ assumption is that the received signal is corrupted by additive white Gaussian noise (AWGN) with unknown variance, and scaled by fixed unknown complex channel gain. Monte Carlo simulations are used to show that the proposed estimator offers a substantial improvement over the conventional Single Input Single Output (SISO) NDA SNR estimator due to the use of the statistical dependences in space and time. Moreover, the proposed NDA SNR estimator works close to the NDA SNR estimator over Single Input Multiple Output (SIMO) channels.

Linear inverse problems with uncertain measurement matrices appear in many different applications. One of the standard techniques for solving such problems is the total least squares (TLS) method. Recently, an alternative approach has been suggested, based on maximizing an appropriate likelihood function assuming that the measurement matrix consists of random Gaussian variables. We refer to this technique as the total maximum likelihood (TML) method. Here we extend this strategy to the case in which the measurement matrix is structured so that the perturbations are not arbitrary but rather follow a fixed pattern. The resulting estimate is referred to as the structured TML (STML). As we show, the STML can be viewed as a regularized version of the structured TLS (STLS) approach in which the regularization consists of a logarithmic penalty. In contrast to the STLS solution, the STML always exists. Furthermore, its performance in practice tends to be superior to that of the STLS and competitive to other regularized solvers, as we illustrate via several examples. We also consider a few interesting special cases in which the STML can be computed efficiently either by reducing it into a one-dimensional problem regardless of the problem size or by a decomposition via a discrete Fourier transform.

The real-timely estimation of the SOC (state of charge) is the key technology in Li-ion battery management system. In this paper, to overcome the error of the SOC estimation of Extended Kalman filter (EKF), a new estimation method based on modified-strong tracking filter (MSTF) is applied to SOC estimation of Li-ion battery, based on the second-order RC equivalent circuit model. Experiments are made to compare the new filter with the EKF and Coulomb counting approach (Ah). The simulation results demonstrate that the new filter algorithm MSTF used in this paper has higher filtering accuracy under the same conditions.

An equivalent circuit model (ECM) is a highly practical tool for simulating Li-ion battery behavior. There are many relevant studies which compare different ECM variants or suggest algorithms to extract model parameters from the experimental data. However, little attention has been given to the battery tests used for identification of the ECM parameters. Therefore, here the influence of experimental test pulse characteristics on the parameterized ECM accuracy was systematically studied. The test pulse duration was varied in a wide range from 9 s to about 2.5 min. The portion of the relaxation phase data used by the parameter optimization algorithm was also varied in an even wider range. Total 168 ECM parameter sets were obtained. Each parameter set was validated using nine diverse current profiles representing different battery operation conditions, including one based on Urban Dynamometer Driving Schedule (UDDS). The validation results prove that the impact of the test pulse choice on the parameterized ECM accuracy is great to the point that it can overshadow the use of a higher-order Thevenin model. By choosing the optimal parameter set, the simulated voltage root mean square error (RMSE) was reduced to as low as 3.0 mV and 1.2 mV for first- and second-order ECM, respectively, while the second-order model based on arbitrary chosen test pulse on average yields RMSE value above 5 mV.

Introduction: Phosphorus magnetic resonance spectroscopy ( 31 P-MRS) allows insight into cardiac energetics in vivo , but is a technique with an intrinsically low signal to noise ratio (SNR). As ultra-high magnetic field MRI scanners become more widely available this limitation will lessen, allowing detection of smaller changes in metabolite concentrations or measurement of changes in smaller patient groups, which will further the understanding of cardiac energetics. Hypothesis: We hypothesised that the 2.4x increase in SNR, predicted by theory at 7T compared to 3T, will allow more precise metabolite concentration measurements in dilated cardiomyopathy (DCM) patients. Methods: 26 idiopathic DCM patients (69% male, mean age 55±2 yr, mean LVEF 35±2%) underwent 31 P-MRS at 3T and 7T field strengths. Patients were NYHA Class I (46%), II (50%) or III (4%). Results: The mean SNR increased from 6.5 ± 0.5 to 16.2 ± 1.5 for PCr (p <0.0001, paired t-test) and from 3.4 ± 0.2 to 11.4 ± 0.8 for γATP (p <0.0001) as the field strength increased from 3T to 7T. The PCr/ATP ratio was consistent between the 2 field strengths; at 3T PCr/ATP was 1.5 ± 0.1 and at 7T 1.5 ± 0.1; there was as expected no statistically significant difference (p=0.67) (fig a). The Cramér-Rao lower bounds (CRLB) quantify the minimum precision of fitted parameters. A desirable decrease in the PCr/ATP CRLB was seen from 3T to 7T (47 ± 3% vs 25 ± 2% p<0.0001)(fig b), i.e.-metabolites are indeed quantified more precisely at 7T compared to 3T. Conclusion: This study represents the first cardiac 31 P-MRS patient data acquired at a 7T field strength. Compared to a 2.8x SNR increase from 3T to 7T previously reported in normal volunteers, we observed a 2.5x increase in this patient population. There is no significant bias at 7T as the PCr/ATP ratios are similar. In conclusion 7T 31 P-MRS is feasible in DCM patients and more accurate than at 3T, which in future should allow more accurate quantification and understanding of myocardial energetics.

Considering the influence of acceleration and the Gaussian envelope for a laser Doppler velocimeter (LDV), parameter estimation of a Doppler signal with a Gaussian envelope was investigated based on introducing acceleration. According to the theory of mathematics statistics, the Cramer-Rao lower bounds (CRLBs) of Doppler circular frequency and its first order rate were analyzed, formulas of CRLBs were given, and the power spectrum estimation with adjustment was discussed. The results of theory and the simulation show that the CRLBs are related to the data length, the signal-to-noise ratio (SNR), and the width of the Gaussian envelope, and they can be decreased by increasing the data length or improving the SNR; the larger the acceleration is and the narrower the Gaussian envelope is, the larger the CRLBs of Doppler circular frequency and its first order rate are; the gap between the variances of the measuring results and the CRLBs narrows when the SNR of the signal is improved, and is almost eliminated when the SNR is higher than 6 dB. It is concluded that the model presented is much more suitable for a LDV than that acquired by Rife and Boorstyn [IEEE Trans. Inform. Theory 20, 591 (1974)].

It has recently been found that by jointly processing multiple (sum, azimuth-, and elevation-difference) matched filter samples, it is possible to extract and localize several (more than two) targets spaced more closely than the classical interpretation of radar resolution. This paper derives the Cramer-Rao lower bound (CRLB) for sampled monopulse radar data. It is worthwhile to know the limits of such procedures; and in addition to its role in delivering the measurement accuracies required by a target tracker, the CRLB reveals an estimator's efficiency. We interrogate the CRLB expressions for cases of interest. Of particular interest are the CRLB's implications on the number of targets localizable: assuming a sampling-period equal to a rectangular pulse's length, five targets can be isolated between two matched filter samples given that the target's signal-to-noise ratios (SNRs) are known. This reduced to three targets when the SNRs are not known, but the number of targets increases back to five (and beyond) when a dithered boresight strategy is used. Further insight to the impact of pulse shape and of the benefits of oversampling are given.

Li-ion batteries experience mechanical stress evolution due in part to Li intercalation into and de-intercalation out of the electrodes, ultimately resulting in performance degradation. In situ measurements of electrode stress can be used to analyze stress generation factors, verify mechanical deformation models, and validate degradation mechanisms. They can also be embedded in Li-ion battery management systems when stress sensors are either implanted in electrodes or attached on battery surfaces. This paper reviews in situ measurement methods of electrode stress based on optical principles, including digital image correlation, curvature measurement, and fiber optical sensors. Their experimental setups, principles, and applications are described and contrasted. This literature review summarizes the current status of these stress measurement methods for battery electrodes and discusses recent developments and trends.

A joint multi-carrier frequency offset (CFO) and multi- time of arrival (TOA) estimation algorithm for multiuser single-input multiple-output (SIMO) orthogonal frequency division multiplexing (OFDM) systems is proposed. With carefully designed pilots, multiple CFOs and TOAs of K users are separated jointly, dividing a complex 2K-dimensional estimation problem into 2K low-complexity mono-dimensional estimation problems. Two CFO estimation approaches, including a low-complexity closed-form solution and a high-accuracy null-subcarrier assisted accurate estimation approach, are proposed, where the integer and fractional parts of each CFO are estimated as a whole rather separately. Each TOA is estimated regardless of CFO by exploring the features of the inter-carrier interference matrix. The Cramer-Rao lower bounds (CRLBs) of multi-CFO and mutli-TOA estimation are derived for the first time for SIMO OFDM systems. Simulation results show that the proposed CFO and TOA estimators provide higher estimation accuracy than the existing approaches. They also achieve performances close to the CRLBs especially at high signal-to-noise-ratios (SNRs).

A low complexity, open-loop, discrete-time, delay-multiply-average (DMA) technique for estimating the frequency offset for digitally modulated MPSK signals is investigated. A nonlinearity is used to remove the MPSK modulation and generate the carrier component to be extracted. Theoretical and simulated performance results are presented and compared to the Cramer-Rao lower bound (CRLB) for the variance of the frequency estimation error. For all signal-to-noise ratios (SNRs) above threshold, it is shown that performance very close to the CRLB can be achieved with linear complexity.

Lithium-ion (Li-ion) battery necessitates an accurate state of charge (SOC) estimation technique to control charging and discharging for the safety of electric vehicle (EV). SOC is an indicator to display the remaining capacity of the battery. To achieve an accurate SOC, model-based SOC estimation algorithm is developed in this paper. Due to the self-correction nature of Kalman filter (KF), KF family algorithm is most promising for SOC estimation when the system is running. As battery modelling has a direct impact on SOC estimation, an equivalent circuit model (ECM) is preferred due to its balance between accuracy and complexity. In this paper, a closed loop approach with the combination of ECM and extended Kalman filter (EKF) algorithm is implemented on Li-ion battery to estimate an accurate SOC. For ease of implementation, model parameters are considered as univariant of SOC. The obtained discharge characteristic curves at different C-rate are close matches with manufacturer’s catalogue. The simulation results show minimum estimation error and good convergence rate of EKF.KeywordsBattery management systemC-rateEquivalent circuit modelElectric Vehicle

A method is provided for determining necessary conditions on sample size or signal to noise ratio (SNR) to obtain accurate parameter estimates from remote sensing measurements in fluctuating environments. These conditions are derived by expanding the bias and covariance of maximum likelihood estimates (MLEs) in inverse orders of sample size or SNR, where the first-order covariance term is the Cramer-Rao lower bound (CRLB). Necessary sample sizes or SNRs are determined by requiring that (i) the first-order bias and the second-order covariance are much smaller than the true parameter value and the CRLB, respectively, and (ii) the CRLB falls within desired error thresholds. An analytical expression is provided for the second-order covariance of MLEs obtained from general complex Gaussian data vectors, which can be used in many practical problems since (i) data distributions can often be assumed to be Gaussian by virtue of the central limit theorem, and (ii) it allows for both the mean and variance of the measurement to be functions of the estimation parameters. Here, conditions are derived to obtain accurate source localization estimates in a fluctuating ocean waveguide containing random internal waves, and the consequences of the loss of coherence on their accuracy are quantified.

Maximum-likelihood (ML) estimation is an established paradigm for the assessment of imaging system performance in nonlinear quantitation tasks. At high signal-to-noise ratio (SNR), ML estimates are asymptotically Gaussian-distributed, unbiased and efficient, thereby attaining the Cramer–Rao bound (CRB). Therefore, at high SNR the CRB is useful as a predictor of the variance of ML estimates and, consequently, as a basis for measures of estimation performance. At low SNR, however, the achievable parameter variances are often substantially larger than the CRB and the estimates are no longer Gaussian-distributed. These departures imply that inference about the estimates that is based on the CRB and the assumption of a normal distribution will not be valid. We have found previously that for some tasks these effects arise at noise levels considered clinically acceptable. We have derived the mathematical relationship between a new measure, χ2pdf-ML, and the expected probability density of the ML estimates, and have justified the use of χ2pdf-ML-isocontours in parameter space to describe the ML estimates. We validated this approach by simulation experiments using spherical objects imaged with a Gaussian point spread function. The parameters, activity concentration and size, were estimated simultaneously by ML, and variances and covariances calculated over 1000 replications per condition from 3D image volumes and from 2D tomographic projections of the same object. At low SNR, where the CRB is no longer achievable, χ2pdf-ML-isocontours provide a robust prediction of the distribution of the ML estimates. At high SNR, the χ2pdf-ML-isocontours asymptotically approach the analogous χ2pdf-F-contours derived from the Fisher information matrix. The χ2pdf-ML model appears to be suitable for characterization of the influence of the noise level and characteristics, the task, and the object on the shape of the probability density of the ML estimates at low SNR. Furthermore, it provides unique insights into the causes of the variability of estimation performance.