Enhanced sample entropy-based health management of Li-ion battery for electrified vehicles

Energy 64 (2014) 953e960

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Energy journal homepage: www.elsevier.com/locate/energy

Enhanced sample entropy-based health management of Li-ion battery for electrified vehicles Xiaosong Hu a, *, Shengbo Eben Li b, Zhenzhong Jia c, Bo Egardt a a

Department of Signals and Systems, Chalmers University of Technology, Gothenburg 41296, Sweden State Key Laboratory of Automotive Safety and Energy, Tsinghua University, Beijing 100084, China c Department of Naval Architecture and Marine Engineering (NAME), The University of Michigan, Ann Arbor, MI 48109, United States b

a r t i c l e i n f o

a b s t r a c t

Article history: Received 26 June 2013 Received in revised form 18 November 2013 Accepted 23 November 2013 Available online 20 December 2013

This paper discusses an ameliorated sample entropy-based capacity estimator for PHM (prognostics and health management) of Li-ion batteries in electrified vehicles. The aging datasets of eight cells with identical chemistry were used. The sample entropy of cell voltage sequence under the well-known HPPC (hybrid pulse power characterization) profile is adopted as the input of the health estimator. The calculated sample entropy and capacity of a reference Li-ion cell (randomly selected from the eight cells) at three different ambient temperatures are employed as the training data to establish the model by using the least-squares optimization. The performance and robustness of the estimator are validated by means of the degradation datasets from the other seven cells. The associated results indicate that the proposed health management strategy has an average relative error of about 2%. Ó 2013 Elsevier Ltd. All rights reserved.

Keywords: Health management Li-ion battery Electrified vehicle Sample entropy

1. Introduction Electrified vehicles, including BEVs (battery electric vehicles), HEVs (hybrid electric vehicles) and PHEVs (plug-in hybrid electric vehicles), are important ingredients of a clean, efficient, and sustainable transportation system [1e3]. Traction battery packs, as a vital energy source in electrified vehicles, are still the main technological and cost bottlenecks after decades of investigation [4,5]. In order to safely and efficiently utilize battery packs in practice, a reliable and effective BMS (battery management system) is indispensable. One of its main tasks is to provide accurate knowledge of battery internal states, such as SOC (State of Charge) and SOH (State of Health) [6e9]. SOC is a meter of the remaining charge in a battery, resembling a fuel gauge in traditional internal combustion engine-based vehicles [10,11], while SOH characterizes the health status of the battery that is often manifested as capacity loss or power loss [12,13]. The power loss that causes declined vehicle acceleration and braking-regeneration capabilities is not very challenging to depict, because the increasing internal resistance can often be recalibrated using short-term current pulses [14]. In contrast, the capacity loss that induces a reduced vehicle driving range is more difficult to be accurately measured or estimated [15], * Corresponding author. E-building, Hörsalsvägen 11, Gothenburg 41296, Sweden. Tel.: þ46 31 772 1538; fax: þ46 31 772 1748. E-mail addresses: [email protected], [email protected] (X. Hu). 0360-5442/$ e see front matter Ó 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.energy.2013.11.061

as the battery packs of electrified vehicles are seldom fully charged or discharged in realistic operations. Moreover, a direct capacity recalibration is often time-consuming, even sometimes impossible, and has an adverse influence on the battery lifecycle. Therefore, it is significant and valuable to develop an accurate and robust capacity estimator for a rapid and reliable battery health management in electrified vehicles. In this study, the battery SOH is thus defined as the ratio of the current capacity to the nominal capacity when the battery is fresh. Many approaches to estimating the battery capacity are based on a direct analysis of battery capacity with respect to aging cycle. For example, Rezvani et al. made a comparative study of techniques to predict the Li-ion battery capacity [16]. Several black-box modeling methods were applied to establish the prediction models by using a portion of the capacity-cycle data pairs. Then, the prediction models were validated and compared at other different aging cycles. A capacity estimation model based on the DempstereShafer theory and Bayesian Monte Carlo methodology was also proposed [17]. In the model, the DempstereShafer theory was used to combine sets of the capacity-cycle pairs from multiple cells (training cells) for initializing the model. Given a portion of the capacity-cycle pairs of another cell (validation cell), the Bayesian Monte Carlo methodology was employed to readjust the initial model parameters and to realize the capacity indication at other cycles. To use this category of models, we need to exactly know the aging cycles of the battery. It is, however, quite difficult to know and record the aging cycles in actual

X. Hu et al. / Energy 64 (2014) 953e960

BMSs, particularly for HEVs. Furthermore, like the model developed in Ref. [17], in order to ensure good robustness against other cells of the same chemistry and the similar aging, certain new capacity values should be added to update the model. Nevertheless, how to attain the new information in practical operations of electrified vehicles is subject to query. Sample entropy is a useful tool for exploring complexity and predictability of a signal [18]. As the battery fades, the measurable voltage response under the same excitation accordingly alters. This alteration, in terms of complexity and fluctuation, can be properly captured by calculating the sample entropy of the voltage response. As a result, the sample entropy-based approaches were deployed to diagnose the battery capacity. A sample entropy-based health estimator was established for a lead acid battery unit [19]. In this method, the sample entropy values of the measured sequences of voltage and current under a discharging pulse were firstly calculated and then applied to qualitatively analyze the battery health status. This model had an advantage of being simple enough for on-board applications, whereas it was incapable of yielding numerical capacity estimates, i.e., realizing a quantitative analysis. A sample entropy-based capacity estimator for a Li-ion battery was built in Ref. [20]. The sample entropy of the voltage sequence collected in a complete constant-current discharge process was used as the input of the estimator. Despite offering numerical capacity estimates, the input acquirement was enormously costly, thanks to the long-time voltage sequences. Additionally, the full discharge is detrimental to the battery life. Moreover, it is worth pointing out that the estimator was trained and validated using different capacity-sample entropy data pairs over the lifetime of the same Li-ion cell. This modeling/validation scenario based on a single battery is not appropriate to actual BMSs, since the battery has failed, and its capacity estimation is useless. For a more practical scenario, the capacity estimator may be firstly constructed by using the data pairs over the lifetime of a reference battery, and then applied to estimate other batteries from the same batch that undergo a similar fade. In this paper, we propose an enhanced sample entropy-based capacity estimator for Li-ion battery health management in electrified vehicles. The proposed estimator effectively overcomes the shortcomings of the two forgoing sample entropy-based models by adding three important original contributions to the related literature. First, the sample entropy of the measured voltage sequence under the HPPC (hybrid pulse power characterization) profile is calculated to be the input of the estimator. Since the HPPC profile only lasts 60 s, the attainment of the input is quite easy and convenient, as well as has negligible harmful effect on the battery lifespan. Further, the HPPC profile comprising a discharging pulse, a rest, and a charging pulse is able to excite the battery better than does a single discharge or charge pulse. Second, the sample entropy

Static Capacity Test Characterization Test (10 ºC)

Rest

Hybrid pulse test Rest

Rest

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Characterization Test (35 ºC)

Repeat every 2 weeks

954

Rest

DST test Rest

Rest

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Characterization Test (22 ºC) Rest

Impedance Test (Ambient ~ 22 ºC)

Static Capacity Test Rest

Rest

CC-charge & CC-discharge

Aging Cycle (22 ºC)

Repeat Fig. 2. Flowchart of the test schedules [24].

and capacity of a reference Li-ion battery (arbitrarily selected from eight batteries) at three different temperatures are adopted to train the estimator by nonlinear least-squares optimization. The developed estimator is thus temperature-conscious. Finally, the estimator is applied to predict the capacities of the other seven Li-ion batteries at the three temperatures, so that its performance, usefulness, and robustness can be adequately examined. The remainder of this paper is structured as follows: the Li-ion battery test is briefly introduced in Section 2; the improved sample entropy-based capacity estimator is elaborated in Section 3; the validation results are elucidated in Section 4 followed by conclusions presented in Section 5. 2. Li-ion battery test Eight LiNMC (lithium nickelemanganeseecobalt) oxide UR14650P cells from Sanyo were chosen for experimentation in University of Michigan, Ann Arbor, USA. These cells were placed in cell holders (on the top layer) in a thermal chamber and independently tested using 8 channels of the battery tester, as shown in Fig. 1. Note that the same loading profile was applied to the eight cells. The test schedules shown in Fig. 2 were designed to excite and degrade the Li-ion cells. Each

0.95 Channel Channel Channel Channel Channel Channel Channel Channel

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Table 1 Parameters of the sample entropy algorithm.

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Fig. 4. HPPC current profile for the LiNMC cells.

experimental procedure consists of three characterization tests at 10  C, 22  C and 35  C, the impedance test at 22  C, and the degradation test (aging cycles) at 22  C. A static capacity test, a hybrid pulse test, a dc (direct-current) resistance test, a DST (Dynamic Stress Test) test, and a FUDS (Federal Urban Dynamic Schedule) test are consecutively conducted in each characterization test. The hybrid pulse test is a sequence of pulse cycles composed of the standard HPPC profile defined in Refs. [21e23] and a self-designed discharging/charging pulse profile. The dc resistance test uses the standard testing program from Arbin to calibrate the internal resistance. The DST and FUDS tests are to emulate the battery load in driving cycles of electrified vehicles. It took approximately two weeks to finish each procedure. Please refer to our previous work [24] to see more details on testing facility, battery specifications, loading schedules, and database establishment. Here, we only use the measured capacities in the static capacity tests and the voltage sequences under the HPPC profiles in the hybrid pulse tests at the three temperatures. For example, the measured capacities of the eight LiNMC cells at 10  C in the aging process are shown in Fig. 3. It can be seen that there is a capacity deviation among the eight cells. The deviation is slight before aging, but seems to become increasingly large, as the cells wear. Overall, the deviation is, however, small in the

well-controlled experimental condition. The similar results are observed for the capacities at 22  C and 35  C. The cell-to-cell or packto-pack variations can also be found in actual battery operations of modern electrified vehicles [25]. With continually improved battery chemistry, more mature manufacturing techniques, enhanced cooling systems and balancing circuitry, it is possible that the variations in batteries of the future electrified vehicles will be controlled to a small extent [24,26]. In this sense, a model identified by the data of a reference battery is more credible to be used for other batteries of the same chemistry and from the same batch. The HPPC current profile and the corresponding voltage responses of the LiNMC cell (Channel 17) at around 90%SOC under 22  C after different aging cycles are shown in Figs. 4 and 5. Note that the discharge current is herein positive, and the charge current is negative. It is evident that the voltage response under the HPPC profile alters in the aging process of the cell. As the battery degrades, the voltage fluctuation become larger, including the instantaneous voltage drop/rise governed by the dc resistance and the dynamic voltage variation (relaxed voltage in the rest period) mainly determined by the imaginary part of the battery impedance. 3. Improved sample entropy-based capacity estimator 3.1. Sample entropy algorithm The sample entropy SampEn(m, r, N) is the negative natural logarithm of an estimate of the conditional probability that windows of length m (subseries of a time series of length N) that remain similar within a tolerance r also match at the next point [18,27]. Self-matches are excluded in calculating the conditional probability. Therefore, the sample entropy value can be used to quantify the regularity and complexity of the time series of length N. The sample entropy algorithm [18] is summarized as follows:

Calibrated capacities HPPC voltage sequences 4.2

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Fig. 6. Multi-battery-based capacity estimation scenario.

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Fig. 9. Capacities and sample entropy values for the reference cell at 35  C.

(1) For a time series of length N, fsðjÞ : 1  j  Ng, form the N  m þ 1 vectors as follows:

Um ðiÞ ¼ fsði þ kÞ : 0  k  m  1g; i ¼ 1; 2:::; N  m þ 1:

(1)

(2) The distance between two such vectors is defined as the maximum absolute difference of their scalar elements:

Am i ðrÞ ¼

1 W mþ1 ðiÞ; i ¼ 1; 2:::N  m; Nm1

(4)

where W m ðiÞ is the number of vectors U m ðjÞ which make d½U m ðiÞ; U m ðjÞ  r for j ¼ 1; 2:::N  m and jsi. Similarly, W mþ1 ðiÞ is the number of vectors U mþ1 ðjÞ within r of U mþ1 ðiÞ, in which j ranges from 1 to N  m (jsi). (5) Define

d½Um ðiÞ; Um ðjÞ ¼ maxfjsði þ kÞ  sðj þ kÞj : 0  k  m  1g:

(2)

(3) The first N  m vectors of length m are considered such that for i ¼ 1; 2:::N  m, both U m ðiÞ and U mþ1 ðiÞ can be defined in the time series of length N. (4) Define

Bm i ðrÞ ¼

1 W m ðiÞ; i ¼ 1; 2:::N  m; Nm1

(3)

Bm ðrÞ ¼

X 1 Nm Bm ðrÞ; N  m i¼1 i

(5)

Am ðrÞ ¼

Nm X 1 Am ðrÞ; N  m i¼1 i

(6)

where Bm ðrÞ is the probability that two sequences will match for m points, Am ðrÞ is that for m þ 1 points. (6) The sample entropy of the time series of length N is estimated using the following equation:

Capacity/Ah

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The sample entropy of cell voltage sequence under HPPC at around 90%SOC is taken as the input of the capacity estimator. The parameters of the sample entropy algorithm are assigned as shown in Table 1. Since the HPPC profile is a short-time pulse, the input can

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Table 2 Capacity estimators at three temperatures.

0.015 0.01 0.005

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(1) 10  C

C ¼ 45311.522z3  3748.420z2 þ 82.707z þ 0.383 where C is the capacity estimate in Ah, and z is the sample entropy. C ¼ 40294.924z3  2585.393z2 þ 38.964z þ 0.757 C ¼ 103957.256z3  4584.918z2 þ 48.080z þ 0.801

(2) 22  C (3) 35  C

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Fig. 18. Capacity estimation for the cell in Channel 23 at 35  C.

be rapidly and efficiently obtained by using the aforementioned algorithm. Moreover, the HPPC excitation is possible to be easily implemented when recharging EVs/PHEVs. For future HEVs, there might be a special on-board loading unit for battery health monitoring, such as the impedance measurement box being designed by Idaho National Laboratory in USA [28]. In the envisioned situation, HPPC also applies to HEVs batteries in order to conveniently attain the input of the capacity estimator. Instead of using only one Li-ion battery, multi-battery-based capacity estimation scenario is used, as shown in Fig. 6. The capacity estimator is first offline calibrated based on the representative aging datasets of a reference battery, and then deployed to other batteries. Since the input (the sample entropy of a short-term voltage sequence) calculation is highly efficient, the capacity estimate can be rapidly achieved. This scenario is very beneficial to the battery capacity estimation in actual electrified vehicles, in which a vehicle prototype can be used to establish the capacity estimator that acts as a manual for health monitoring of user vehicles. Here, the LiNMC cell in Channel 17 is arbitrarily chosen as the reference cell. Fig. 7 shows the capacities and sample entropy values for the reference cell at 10  C. Those at 22  C and 35  C are shown in Figs. 8 and 9. It can be seen that the capacity decrease of the battery at 10  C can be roughly divided into

three phases with different rates; in contrast, the capacity changes at 22  C and 35  C are relatively simple (only two change phases are observed). At each temperature, the associated sample entropy can capture the dynamics of the capacity trajectory very well. Overall, as the cell capacity reduces, the sample entropy increases. The correspondence between the capacity and sample entropy of the reference cell at each temperature is modeled using a third-degree polynomial function optimized by the nonlinear least-squares numerical optimization algorithm. In order to avoid overfitting that aggravates the model generalization, a low-order polynomial is herein used. The optimization results are shown in Table 2. The capacity calibrated at an elevated ambient temperature is higher than that at a low temperature, owing to more active electrochemical reactions. Thus, the nonlinear least-squares optimization algorithm accordingly provides discrepant coefficients for the equations at the three different temperatures to reflect the aforementioned temperature effect on the capacity. The performance of the estimators at the three temperatures for the reference cell is shown in Figs. 10e12. It is clear that the capacity estimate at 10  C is able to accurately simulate the real three-phase capacity change. The corresponding maximum relative error is less than 2%. The relatively large error occurs at the phase-switching points. A similar

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X. Hu et al. / Energy 64 (2014) 953e960

estimator has good robustness against this cell. It is clear that the three-phase capacity trajectory at 10  C is modeled well by the proposed sample entropy-based method. Although the capacity trajectories at 22  C and 35  C (with only two phases) are very different from that at 10  C, the method is also able to precisely capture the aging dynamics, thereby yielding credible capacity estimates. All the relative errors at the three temperatures for this cell are less than 4%. To more clearly show the validity and robustness of the estimators, the probabilistic distribution of the relative capacity errors for all the eight cells at each temperature is calculated. The results are shown in Figs. 19e21. It can be seen that the average and standard deviation of the estimator error at 10  C are 1.6% and 1.5%, respectively. Those of the estimator at 22  C are 1.7% and 1.4%; 2.1% and 1.8% are for 35  C. Despite a capacity deviation among these cells as indicated in Fig. 3, the estimators at the three temperatures exhibit good performance.

45 40 Mean:1.7%

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Fig. 20. Distribution of the relative capacity errors for all the eight cells at 22

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result can be found at 22  C and 35  C: the maximum relative errors are less than 3.5% and 6.0%, respectively. 4. Validation results It is quite important and useful to investigate the robustness of the acquired estimators against the other seven cells. Given voltage sequences under the HPPC profile, the estimator inputs for the seven cells are efficiently calculated by the sample entropy algorithm. Then, the estimators are employed to indicate the capacities. For example, the estimation results for the cell in Channel 18 are shown in Figs. 13e15. It is obvious that the estimator developed by using the data of the reference cell (in Channel 17) is still effective to characterize the capacity fade of the “unseen” cell. The estimator at 10  C can also describe the three-phase change of the capacity of the cell in Channel 18 very well. The estimation error displays good randomness, which in turn verifies the validity and good generalization ability of the estimator from a system identification point of view. Further, it can be seen that the relatively large error appears in the second phase between 400 and 800 cycles. Analogous outcomes are observed at 22  C and 35  C. The estimation results for the cell in Channel 23 are illustrated in Figs. 16e18. Again, the

35

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In this paper, we propose an improved sample-entropy based capacity estimator for lithium ion batteries used in electrified vehicles. First, the sample entropy of the cell voltage sequence under the HPPC(hybrid pulse power characterization ) profile is used as the input of the estimator for characterizing the capacity loss. The sample entropy can be rapidly and efficiently calculated in realistic operations of electrified vehicles. Second, rather than a single cell, the aging datasets of eight cells are used to establish and evaluate the estimator. To be more useful in actual BMS applications, the estimator developed based on the aging datasets (capacities and corresponding sample entropy values) of a reference Li-ion cell (arbitrarily selected from the eight cells) is used to monitor the capacities of the other 7 Li-ion cells. Third, multiple temperatures are considered. Evaluation results demonstrate that all the estimators at the three different temperatures are satisfactorily robust.

Acknowledgments This work was in part supported by National Natural Science Foundation of China (No. 51205228). The authors would like to acknowledge Prof. Huei Peng at The University of Michigan, Ann Arbor, USA, for substantial help and many enlightening discussions on lithium-ion battery tests, data analysis, and modeling.

References

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[1] Chu S, Majumdar A. Opportunities and challenges for a sustainable energy future. Nature 2012;488(7411):294e303. [2] Juul N. Battery prices and capacity sensitivity: electric drive vehicles. Energy 2012;47(1):403e10. [3] Densing M, Turton H, Bäuml G. Conditions for the successful deployment of electric vehiclesea global energy system perspective. Energy 2012;47(1): 138e49. [4] Rahn CD, Wang C. Battery systems engineering. West Sussex: John Wiley & Sons; 2013. [5] Plett GL. Extended Kalman filtering for battery management systems of LiPBbased HEV battery packs Part 1. Background. J Power Sourc 2004;134(2):252e 61. [6] Andrea D. Battery management systems for large lithium-ion battery packs. Boston: Artech House; 2010. [7] Sun F, Hu X, Zou Y, Li S. Adaptive unscented Kalman filtering for state of charge estimation of a lithium-ion battery for electric vehicles. Energy 2011;36(5):3531e40. [8] Hu X, Sun F, Zou Y. Estimation of state of charge of a lithium-ion battery pack for electric vehicles using an adaptive Luenberger observer. Energies 2010;3(9):1586e603. [9] Zhang J, Lee J. A review on prognostics and health monitoring of Li-ion battery. J Power Sourc 2011;196(15):6007e14.

960

X. Hu et al. / Energy 64 (2014) 953e960

[10] Kim JH, Cho BH. Screening process-based modeling of the multi-cell battery string in series and parallel connections for high accuracy state-of-charge estimation. Energy 2013;57:581e99. [11] He H, Zhang X, Xiong R, Xu Y, Guo H. Online model-based estimation of stateof-charge and open-circuit voltage of lithium-ion batteries in electric vehicles. Energy 2012;39(1):310e8. [12] Fernández IJ, Calvillo CF, Sánchez-Miralles A, Boal J. Capacity fade and aging models for electric batteries and optimal charging strategy for electric vehicles. Energy 2013;60:35e43. [13] Krieger EM, Cannarella J, Arnold CB. A comparison of lead-acid and lithiumbased battery behavior and capacity fade in off-grid renewable charging applications. Energy 2013;60:492e500. [14] Hu X, Li S, Peng H, Sun F. Robustness analysis of state-of-charge estimation methods for two types of Li-ion batteries. J Power Sourc 2012;217:209e19. [15] Xing Y, Ma EWM, Tsui KL, Pecht M. Battery management systems in electric and hybrid vehicles. Energies 2011;4(11):1840e57. [16] Rezvani M, AbuAli M, Lee S, Lee J, Ni J. A comparative analysis of techniques for electric vehicle battery prognostics and health management (PHM). SAE Technical Paper 2011-01-2247. [17] He W, Williard N, Osterman M, Pecht M. Prognostics of lithium-ion batteries based on DempstereShafer theory and the Bayesian Monte Carlo method. J Power Sourc 2011;196(23):10314e21.

[18] Richman JS, Moorman JR. Physiological time series analysis using approximate entropy and sample entropy. Am J Physiol 2000;278(6):H2039e49. [19] Sun Y, Jou H, Wu J. Auxiliary diagnosis method for lead-acid battery health based on sample entropy. Energy Convers Manag 2009;50(9):2250e6. [20] Widodo A, Shim MC, Caesarendra W, Yang BS. Intelligent prognostics for battery health monitoring based on sample entropy. Expert Syst Appl 2011;38(9):11763e9. [21] US Department of Energy. PNGV battery test manual. revision 3, DOE/ID10597; 2001. [22] US Department of Energy. FreedomCAR battery test manual for power-assist hybrid electric vehicles. DOE/ID-11069; 2003. [23] US Department of Energy. Battery test manual for plug-in hybrid electric vehicles, revision 2. INL/EXT-07e12536; 2010. [24] Hu X, Li S, Peng H. A comparative study of equivalent circuit models for Li-ion batteries. J Power Sourc 2012;198:359e67. [25] Moore SW, Schneider PJ. A review of cell equalization methods for lithium ion and lithium polymer battery systems. SAE Technical Paper 2001-01-0959. [26] Mi C, Masrur MA, Gao D. Hybrid electric vehicleseprinciples and applications with practical perspectives. West Sussex: John Wiley & Sons; 2011. [27] Lake DE, Richman JS, Griffin MP, Moorman JR. Sample entropy analysis of neonatal heart rate variability. Am J Physiol 2002;283(3):R789e97. [28] Christophersen J. Impedance measurement box working toward an energy storage monitoring system. Technical note. USA: Idaho National Laboratory; 2011.