Probabilistic Voltage Fault Correction Method for Lithium-Ion Batteries Using a Decentralized Cell Voltage Measurement Approach

This article proposes a bias detection method in the voltage measurement of lithium-ion (Li-ion) battery cells to identify faulty sensor(s). The proposed method is based on a Bayesian probabilistic approach that detects possible measurement bias in any battery cell in real-time. A hypothesis bank is constructed for possible bias magnitudes in each cell. Subsequently, the fault detection algorithm computes the probability associated with all hypotheses. Once the probability of a certain hypothesis converges to unity, the faulty sensor and its associated bias are identified. The quantified bias can then be compensated in the measurement model of the associated cell. Details on the proposed method followed by experimental verification using commercial lithium-ion battery datasets are provided.

Most reported monitoring techniques of Li-ion batteries are found to work well at the cell level.However, many of these techniques fail or have a reduced performance when implemented at the pack level.For example, while many high-performance SOC estimation methods were proposed in recent years for Li-ion battery cells, limited work has been reported on estimating the SOC for a battery pack.The issue with the SOC concept of a battery pack is the lack of a clear definition, unlike for a cell where it is well defined [3].In addition, SOC estimation techniques totally depend on the cell voltage, where any uncertainty in the voltage measurement will adversely affect the SOC estimation accuracy.Besides the SOC estimation issues, which have been widely addressed in recent years, a new emerging area that has undergone extensive research lately is the sensorless temperature estimation of batteries.Traditionally, to measure the surface temperature of an individual battery cell, a temperature sensor is attached to its surface.For a battery pack, having a single sensor attached to its surface does not indicate the actual temperature of the pack, as temperature variations between cells are significant and cannot be neglected.Given the voltage and current measurements of each cell in a pack, the temperature of each individual cell can be estimated as demonstrated in [4], [5], [6], [7].For an ideal battery pack with identical and fully balanced cells, the pack and the cell will both have the same SOC and temperature.Practically, these ideal battery packs do not exist in the real-world as cells will always have variations in their internal parameters and balancing issues.The balancing issues can be mitigated by implementing a balancing mechanism that monitors the voltage of each cell and performs regular dis/charge for the cells to maintain the same voltage across all cells [8], [9], [10].However, like other battery mentoring techniques, balancing methods are only effective when the voltage measurement of the cells is accurate, which cannot be guaranteed in a practical application [11].
This article proposes a method for bias detection in the voltage measurement of the cells in a Li-ion battery pack that is based on a decentralized voltage measurement approach, unlike conventional measurement systems where all measurements are taken by a single voltmeter making the measurement prone to a single-point failure.The proposed fault identification method is based on the Bayesian probabilistic approach.Conceptually, filtering algorithms such as Kalman-based filters and particle filters, to name a few, fuse the measurements and dynamics of a system under the assumption of accurate models with known measurement and dynamics noise statistics.Under these conditions, these methods achieve accurate state estimation regarding its expected value and the associated error covariance matrix [12], [13], [14], [15], [16], [17].Practically, however, the assumption of a sustained and known accurate system's model is not valid.The system's dynamics and measurement noise are assumed Gaussian white noise sequences with zero mean and a nominal covariance matrix.These assumptions of unbiased noise mean and known covariance magnitudes may not be realistic in a practical application, which explains why many of the proposed methods for battery SOC estimation have significantly lower performance compared to the results obtained at the lab using brand-new, recently calibrated, measuring equipment.The battery's sensors and operational environment are prone to different types of biases and/or uncertainties [18].Since current and voltage measurements are routinely used in SOC estimation, sensors' measurement quality dramatically affects the accuracy of estimation errors.Current and voltage sensors could incur bias and/or increased-magnitude noise variance in comparison to those given in the sensor's datasheets.
Sensors' failure and performance evaluation of these measurement devices have been widely addressed in various applications such as power systems and unmanned aerial vehicles (UAVs) [19], [20], [21], [22].The problem of a Li-ion battery cell experiencing unknown dynamics and measurements noise statistics was addressed in [22], [23], [24], [25].A change in the dynamics noise covariance matrix and/or the measurement noise covariance matrix assumed in the filter was detected/estimated.The needed adjustment results from a change in the operating environment of the Li-ion battery or a degradation in the quality of the measurement sensor due to aging or malfunctioning.Sensor faults, aging, or calibration errors of sensors are possible causes of measurement bias.This bias could be fixed throughout the system's operation or vary depending on the battery's operational environment or health condition.There are mainly two ways to detect the bias fault; hardware redundancy and analytical redundancy [26].In the hardware redundancy approach, redundant sensors are used to measure one variable, and a fault is detected if there is a notable difference between the measurements from different sensors.In the analytical redundancy approach, the system is analyzed using multiple model-based techniques.In [27], the author estimated the effect of the sensor's measurements bias and noise covariance on the battery SOC estimation accuracy.The bias and noise statistics are modeled as slowly varying.It was found that the tuning of the filtering approach cannot adequately compensate for the sensor's biases.A bias correction-based method considering the temperature and polarization inconsistency is proposed in [28].The neural network and extended Kalman filter (EKF) are used for SOC estimation based on the corrected model, which may increase the system's complexity.In [29], the authors presented a model-based fault detection and isolation scheme based on sliding mode observes.Their method assumed the knowledge of the constant parameters that define the thermal dynamics of the battery given a small SOC range.Researchers propose an online model-and entropy-based fault detection scheme for detecting sensor faults, connection faults, and short-circuit faults in Li-ion battery systems in [30].A special sensor measurement topology was implemented to allow for the isolation of different faults.Data-driven correlation-based Li-ion short circuit fault detection methods are proposed in [31], [32], [33].
In this article, a novel technique is proposed to detect and identify a possible bias in the measurement of any of the battery cells out of the battery pack.A hypothesis bank is initialized for the possible bias magnitudes on each battery cell.The proposed method propagates the probability of each hypothesis as the measurements are acquired.The probability associated with the correct hypothesis will approach one, and this hypothesis is subsequently declared as the fault bias in the battery pack.The hypothesis bank can then be fine-tuned to determine the magnitude of the fault more accurately.This process is periodically performed to identify and remove any bias that could affect any battery cell.The proposed approach can significantly improve the battery management system (BMS) performance.For example, it can enhance the effectiveness of the balancing mechanism used by the BMS by identifying faulty sensor(s) and compensating for the fault.It can also improve the SOC estimation accuracy, which has a short-term benefit of increasing the battery's runtime by setting confident SOC cutoffs and a long-term benefit of extending the battery's cycle-life by preventing destructive overcharge or over discharge.Moreover, in sensorless temperature estimation systems where the voltage, which is SOC-dependent, is used as an input to the temperature model [4], [5], [6], [7], the BMS will not make the right decision if the voltage measurement is inaccurate due to a false estimation of the temperature.In this scenario, the BMS will terminate the battery operation due to false detection of a temperature rise.Alternatively, it may allow the battery to operate above the upper-temperature limit when a real temperature rise is not detected, leading to fire hazards, reduced service-life, or in the best-case scenario, reduced performance.In this work, the SOC estimation application is demonstrated as an example to show the effectiveness of the proposed method in improving the estimation accuracy of the SOC monitoring algorithms.The contributions of this article are as follows: r A decentralized measurement approach for the cell volt- ages in a battery pack is proposed, and r A novel fault detection algorithm is proposed for identify- ing faulty sensors in a battery pack and compensating for faults in the SOC estimation algorithm.In summary, the proposed method has twofold benefits; first, it improves the BMS reliability when bias exists in a sensor, and second, it reduces the maintenance and replacement costs of sensors in inaccessible locations or when the battery pack utilizes a massive number of sensors in which case the maintenance and/or replacement of sensors are inconvenient and costly.
This article is organized as follows: Section II describes the cell voltage measurement system.In Section III, the battery dynamic model and the SOC estimation method are presented.In Section IV, the proposed fault detection method is given.Experimental results are presented in Section V to verify the accuracy of the proposed algorithms.Finally, Section VI concludes the paper.

II. MEASUREMENT SYSTEM DESCRIPTION
A common practice in measuring a cell voltage in a battery pack is to utilize a single voltmeter to measure all cell voltages.A simplified conventional setup for cell voltage measurement in a battery string consisting of four cells is demonstrated in Fig. 1 [41].The voltmeter measures the voltage at nodes a, b, c and d (V a , V b , V c, and V d , respectively).Using this setup, the cell voltages can be calculated as: If a bias exists in the measurement of one of the nodal voltages (V a , V b , V c, or V d ), the bias will impact the calculated cell voltages for the two cells connected to that node.
Alternatively, the cell voltages can be directly measured by connecting each cell (or a group of cells) to a multiplexer or a differential Op-Amp, as demonstrated in Fig. 2.This setup is one step above the one in Fig. 1, as the error in one nodal voltage measurement does not propagate to adjacent cells.
While the setups in Figs. 1 and 2 are simple and demand low implementation cost, they will have reduced accuracy if the voltmeter's measurement is biased.This will impact all the voltage measurements.The issue with these setups is that, unless the voltmeter is calibrated frequently (an uneasy requirement to meet in many applications), it is almost impossible to automatically detect the bias in the voltage measurement as a bias in the voltmeter will consistently impact all the cells.
To address this issue, a decentralized measurement system is proposed and adopted to improve the reliability of the cell voltage measurements.Instead of using one centralized voltmeter to measure the cell voltages as in Figs. 1 and 2, the proposed setup decentralizes the voltage measurement by utilizing one voltmeter for each cell.This approach was inspired by micro-inverter based photovoltaic (PV) systems where each PV module is connected to a micro-inverter instead of a centralized or string inverter where several modules are connected.In PV systems, having an independent path to the grid for each PV module increases the overall system's reliability if one of the inverters fails and increases the total power extracted by optimizing the operation of each individual PV module.In the proposed measurement setup, instead of utilizing one centralized voltmeter to perform all voltage measurements, a completely independent measurement system for each cell is utilized.With this setup, when one of the cell voltage measurements has a bias, it will not affect the other cells as each cell has its own independent measurement system.The proposed measurement setup is shown in Fig. 3.
In Fig. 3, any bias in one of the measured cell voltages will be immediately auto-detected and compensated for by the BMS, which is difficult, if not impossible, to achieve using the systems in Figs. 1 and 2. Despite its increased cost, this setup will improve the reliability of the cell voltage measurement mechanism, which will directly improve other functionalities of the BMS such as SOC estimation, as will be demonstrated in Section V, and will overweigh the increased implementation requirements.

A. Battery Cell
A battery cell model is used to estimate the SOC of a battery.The dynamics equation describes the time rate of change of the SOC as a function of the input and the SOC at the previous time step.Additionally, the measurement equation describes the sensors' measurements as a function of the current state of charge and current input.Both equations are assumed to be corrupted by additive Gaussian noise.Literature is rich with SOC models, ranging from electrochemical models, mathematical models, and electrical models [34], [35], [36], [37].The model considered in this article is the Improved Hysteresis Model.This model offers an improved transient response during transient times and relaxation by adding the exponential block and the Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.Fig. 4. Improved hysteresis model adopted from [37].
The discretized dynamics of the SOC can be modeled by discretely integrating the current over time, leading to the following dynamics equation [36]: where is the SOC at instant k, η is the charging/discharging efficiency and is assumed to be 100% for a brand-new Li-ion battery, C n is the nominal capacity of the battery in ampere-hours (Ah) and ω k is the zero-mean white Gaussian process noise sequences modeled as ω k ∼ (0, Q ω ).
The measured output voltage, z k , is a function of the SOC and can be expressed as [37]: The parameters of the measurement model are obtained experimentally.The OCV is the open-circuit voltage, which is obtained from a lookup table for a given SOC.The internal resistance, R, is modeled to be the cell's average internal voltage drop, which is found at different currents from the following equation: where N is the number of true voltage data points, and V cell k is the measurement acquired from battery cell j.
The parameter β, found experimentally, is a multiplier to improve the transient response.The parameter τ represents the time constant of the cell state transition and is equivalent to the time constant of the diffusion capacitance.It is obtained empirically by forcing the model to follow the true voltage.The parameter f k is an additive correction term, which is a linear function of relaxation time and is given in a general form in (4) as: where T max is the time at which the voltage settles to a steadystate value during relaxation (assumed to be one hour), and ε V is some small user-defined voltage threshold.
The relationship between OCV and SOC was obtained by charging and discharging the battery at C 30 rate and averaging the results.v k is the zero-mean white Gaussian measurements noise sequences modeled as The hysteresis voltage is a function of the SOC.It is given in (5), where α is a correction factor determined empirically.It represents the cell's internal voltage drop during the charging/discharging at C 30 [31], and V c /V d represent the terminal voltage battery during charging/discharging.

B. Battery Pack
A simplified battery cell arrangement is shown in Fig. 5, where an arbitrary number of cells is connected in series.It is assumed that the battery cells used are nearly identical.Therefore, given that the input current is the same, the SOC of each cell is approximately equal.This assumption mandates that the battery management system keeps the battery cells balanced throughout the operation in the series-connected cells.The terminal voltage measurement across each cell can be written, utilizing (2), as: where ) , and j is an index representing any battery cell in the battery pack, and v k is the zero-mean white Gaussian measurements noise sequences modeled as Therefore, for a three-cell battery where the cells are series-connected, the following vector measurement equation is formed: Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.
Above, the assumption is made that the battery cells inside the pack are similar enough to one another.Throughout testing this was challenged by injecting white Gaussian noise into the measurements sequences to force them to slightly differ from one another.
Next, we describe the algorithm used to estimate the SOC under the assumption of known statistics of the dynamics and measurement noise sequences.

C. SOC Estimation
In this section, the EKF is used to estimate the SOC at the cell level.This knowledge is fundamental for estimating the SOC at the pack level.Equations ( 1) and ( 2) are used to formulate the EKF estimation algorithm [37].The EKF is first initialized with an initial state estimate, x0 , and associated initial state estimate error covariance, P0 .Next, the prediction step is used to propagate the estimate of the state.Therefore, the a priori state estimate, xk+1 , and the associated propagated state estimate's covariance, Pk+1 , are determined.These are described in ( 8) and ( 9) as: where xk is the updated estimate of the SOC at time k given measurements up to time k and Pk is its associated covariance.
The linearized measurement matrix is obtained as: To obtain the updated SOC estimate at time k+1, the Kalman gain at time k+1 is obtained as: Finally, the estimate of the state at time k+1 and its covariance are updated as follows: In addition to the EKF algorithm presented above, we show that the proposed bias-correcting algorithm enhances the performance of estimation algorithms that compensates for highly nonlinear system models such as our Li-ion measurement model.Namely, the Cubature Kalman Filter (CKF) [38] is used to estimate the SOC with the proposed algorithm integrated to correct for any measurements' biases.The CKF selects 2n Cubature points based on the third-order spherical-radial cubature rule to approximate an n dimensional Gaussian weighted integral values [38].It is worthy to note that the square-root version of the CKF is chosen over the normal version for its numerical robustness during testing.
After initializing the filter with a state estimate, x0 , and associated state estimate error covariance, P0 , the time propagation step is performed: where S k−1|k−1 is the matrix square root calculated by any triangularization algorithm, ξ i = m 2 [1] i denotes the i th cubature point, [1] i is a vector of zeros except for the i th place having a value of one, i = 1, 2, . . ., m, m = 2n, n is the dimension of the state vector, X * i is the evaluated cubature point X i , and the other terms are the same as those in the EKF formulation.
The time update step is then performed: where Z i is the propagated measurement cubature points, P zz is the innovation covariance, P xz is the cross covariance, W is the Kalman gain, and the other terms are the same as those in the EKF formulation.

IV. PROPOSED FAULT DETECTION METHOD
A nonlinear method that sequentially samples and processes and measurement equations of the battery pack to detect possible bias in one of the cells is presented in this section.The proposed method detects the faulty sensor and identifies the bias error incorporated in its measurement.By subtracting the identified bias from the sensor's measurements, the estimator will ensure unbiased measurements, which will ensure a sustained highaccuracy SOC estimate.
A sequential hypothesis test to estimate the mean of a random variable with known covariance among two possible hypotheses was presented in [39], [40].By utilizing an independent and Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.
identically distributed measurement sequence, r k , an algorithm was proposed to identify the unknown mean of the measurement.This mean was obtained by computing the probability ratio of the considered two hypotheses.The ratio of the probability of having hypothesis 1 as the correct hypothesis, P 1 (k), to the probability of hypothesis 0 as the correct hypothesis, P 0 (k), given all the measurements up to time k is constructed as: If the ratio in ( 28) is smaller than a specified lower threshold, hypothesis 0 is accepted.On the other hand, hypothesis 1 is accepted if the probability ratio is greater than a chosen upper threshold.Probabilities of Type 1 and Type 2 fault detection errors are used when choosing the lower and upper thresholds [39].
The algorithm proposed in this article extends the number of hypotheses from 2 to a multi-hypothesis test.For the three battery cells in a battery pack, each cell is assumed to either have a healthy, unbiased measurement or alternatively have two possible values of incurred bias.The algorithm will subsequently evaluate the probability of each bias hypothesis.The possible bias hypotheses can be increased, or the algorithm can be repeated to converge on the correct bias in the cell's measurement.Now, we target the evaluation of the probability that a certain hypothesis, a possible bias value, exists in each battery cell given the sampled measurements of the battery pack.The probability of a bias hypothesis H i to occur given sampled measurements up to time k+1 can be described as a function of the probability of that bias hypothesis to occur given sampled measurements up to time k as follows: where the fault hypothesis H i represents a measurement bias μ i , μ i ∼ N (μ i , σ 2 M ), in one of the battery cell's measurements.The measurement residual, r k , will be described later.Defining the probability of hypothesis i, given measurements up to time k as: 13) can be formulated as: Or, To form the measurement residual, (7) can be linearized about the predicted state and written in vector form as: where μ k is the possible bias fault on the measurements of the battery cells.With R k as the covariance matrix of the measurement noise v k ∼ N (v k , σ 2 ), a least-squares estimate of state can be written as: Therefore, the measurement residual can be written as: The above residual can be written as: Therefore, given (31), the residual can be expressed as: Equation ( 34) follows from the fact that the matrix P is in the null space of H k .Therefore, the statistics of the residual process are a function of only the possible bias fault and the measurement noise sequence.
The matrix P , being in the null space of the H k matrix, is not invertible.Accordingly, a representation, based on singular value decomposition, is written as: where U T 1 U 1 = I.Utilizing the above representation, ( 34) is written as: Equation ( 36) is multiplied by U T 1 to obtain the following modified residual: Note that the statistics of r is a function of the possible bias and the measurement noise statistics.Given a specific bias hypothesis, the conditional mean of r can be represented as: where μ is the possible constant bias incurred on one of the sensors.In (38), the bias is assumed on battery cell 2 as an example.It is assumed that this bias will be constant until identified and corrected.We define: Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.where in (39), the mean of the measurement noise is assumed to equal zero.From (39), assuming that the measurement noise and the bias noise are independent, the conditional covariance can be found to be: where V i 1 is the i th column of V 1 that corresponds to the battery number on which a hypothesized bias magnitude exists.Therefore, utilizing (38) and (39), the probability density of the residual process conditioned on a certain bias hypothesis is given as As shown in Fig. 6, (41) will be utilized to obtain the conditional probability density for all assumed hypotheses as measurements of the battery pack are sampled.Subsequently, the obtained densities will be substituted in (30) to propagate the probability of each hypothesis sequentially.The correct hypothesis will have a probability that approaches 1, while the probability of other hypotheses will approach zero.
The detected bias will subsequently be removed from the measurements, and the fault detection and identification algorithm will be restarted with possibly finer bias hypotheses.
An additional component to make the SOC algorithm more robust is to reject sudden outlier measurements that can adversely interfere with the probabilistic methodology proposed above.A straightforward approach can be formulated using the innovation sequence ϑ k+1 = z k+1 − z k+1 (x k+1 ), following the definition of the variables in Section II.Note that the innovation sequence is appropriate as it is a white noise sequence with zero mean.Therefore, a χ 2 test is proposed for the statistic below where S is the innovation covariance from the Kalman filter equations.The statistic is subject to the χ 2 distribution with degrees of freedom equal to the size of the innovation vector ϑ k+1 .By selecting a level of significance α as P {χ 2 > χ 2 α } = α where 0 < α < 1, the threshold value χ 2 α for disregarding measurement can be found.For an epoch when the hypothesis that the measurement is faulty is true, the following adjustment is applied to the Kalman filter equations: where λ is a small positive number.

V. EXPERIMENTAL VERIFICATION
In this section, the performance of multiple fusion algorithms that incorporate the proposed fault detection algorithm is evaluated and compared.Multiple discharge cycles performed at the cell and pack levels using a 3.6-V/1100-mAh Li-ion battery cell and a 12.8-V/150-Ah Li-ion battery pack are used with simulated faults introduced in a hypothetical battery.In any given scenario, only one fault is assumed to occur in the system before the onset of faults in other battery cells.As will be shown in the results, since the algorithm quickly detects and identifies the bias, this assumption is realistic.The tests show the SOC estimation results for the hypothetical battery pack consisting of several series-connected cells.For every fault assumption magnitude, b, the following fault hypothesis bank is defined: In other words, every fault magnitude, b, introduces 6 possible hypotheses for which the likelihood must be evaluated.In addition to the null hypothesis H 1 = [0 0 0] T , a total of two other fault magnitudes are assumed, which are 0.005 V and 0.05 V.This makes for a total of 13 hypotheses populating the hypothesis bank at any given time.
As mentioned in the previous section, the fault magnitudes can be fine-tuned to increase the accuracy of the SOC estimation algorithm, which is demonstrated later in the article.
In all cases, given x 0 ∼ (x_true, P 0 ), the initial SOC is randomly sampled within the 95% (±2 σ) region around the true value.The initial state estimate's covariance is assumed as Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.

A. Constant Pulse Discharge Test
A constant pulse discharge test was performed on a fully charged 3.6-V Li-ion battery cell from SOC of 100% down to 0%.
Hypothesis 11 was simulated by injecting a bias of H 11 = [0 −0.005 0] T into the measurements vector.Fig. 7 shows the SOC estimation performance using the faulty measurement vector.The estimation performance of the proposed algorithm is not greatly affected by the small fault magnitude of −0.005 V in battery cell 2. In contrast, the traditional EKF, marked as traditional in this figure and the ones that follow, does not get compensated for possible measurement bias and therefore slowly diverges away from the true SOC.It is noticed that the proposed algorithm quickly converges to the true hypothesis, even with a small fault magnitude.The probability associated with hypothesis 11 converges to 1 while the probability associated with other wrong hypotheses converges to 0. The algorithm takes around 2 minutes or 8 epochs to detect and quantify the fault magnitude.The proposed adaptation routine detects the faulty measurements and corrects for it in the covariance by increasing its corresponding component, thereby decreasing the weight given to the faulty measurement in the SOC calculation.
The results show a Mean Absolute Error (MAE) level of 1.47% and 4.68% for the proposed CKF and EKF approaches, respectively, and 4.777% for the traditional SOC estimation algorithm in this test.Results also show maximum Absolute Error (AE max ) levels of 3.81%, 7.81%, and 8.01% for the CKF, EKF, and traditional algorithms, respectively.These errors were computed using ( 46) and (47).

B. Variable Pulse Discharge Test
A variable pulse discharge test was performed on a fully charged 3.6-V Li-ion battery cell from 100% down to 0% SOC.This test consists of a discharge current load that varies between 10 different levels.It serves to test the proposed algorithm against a more dynamic load.
Hypothesis 3 was simulated by injecting a bias of H 3 = [0 0 0.05] T into the measurements vector.Fig. 8 shows the SOC estimation performance using the faulty measurements vector.Again, the proposed fault-tolerant estimator's performance is not affected by the fault magnitude of 0.05 V in battery cell 3.In comparison, the traditional approach slowly diverges before recovering towards the end of the test.Fig. 9 displays the propagation of the probabilities in this test.While it is evident that the algorithm is effective even with a more dynamic test, the varying current profile seems to slow down convergence to the true hypothesis.It takes the algorithm around 4 minutes or 24 epochs to correctly identify the fault and its magnitude.Another reason for this is the smaller order of magnitude of this fault compared with the magnitude of the previous simulated fault.Nevertheless, the adaptation routine overcomes the existence of the faulty cell to report accurate overall battery SOC.
The results show the proposed CKF approach leading with an MAE of 0.86% while the traditional SOC estimation algorithm yielded an MAE of 7.45% for this test.The level of maximum AE was 5.48% and 11.42% for the proposed CKF and traditional approaches, respectively.Additionally, with the proposed approach, the SOC tracking is more consistent throughout the test, contributing to a better SOC estimation accuracy.

C. Dynamic Stress Test
This test comprises aggressive 115 DST cycles performed using a 3.6 V Li-ion battery cell covering the entire SOC range from 100% down to 0%.The purpose of this test is to evaluate the proposed technique when exposed to highly dynamic and close to realistic charge or discharge power pulses that would stress a battery cell.Hypotheses 5, 6, 1, and 2 are examined in this test; that is, the data is processed four times with different fault magnitudes injected into the measurements vector.
This realistic test cycle validates the performance of the proposed approach in an even more dynamic environment than the previous two tests.The estimation performance is shown in Fig. 10 .SOC estimation performance still retains its accuracy with the proposed algorithm using faulty measurements.The traditional approach also seems to suffer drastically in this highly dynamic test.
As the fault magnitude grows larger, the estimation performance will degrade even more, and it might even diverge.The algorithm takes less than one minute or just a few epochs to identify the fault and the battery cell where it occurs.Impressively, in that amount of time, the battery will not have lost more than 1% of charge in this test.This, consequently, leads to the early identification of sensor and/or battery faults.A similar trend with SOC estimation is noticed here, but the discrepancy is slightly smaller than with the other tests.This can be attributed to the availability of a better battery cell model for this dataset as opposed to the other cell models.
The results show an MAE ranging between 0.71% and 1.47% for the proposed CKF approach, a consistent 0.99% for the proposed EKF approach, and 2.76% for the traditional SOC estimation algorithm in this test.The performance of the algorithms also exhibits a similar trend with the maximum AE levels.

D. Discussion on Results
In all the cases presented in this section, the initial test bank included the true hypothesis, and convergence was shown to be quick.However, this information is not available in an uncontrolled environment.Here, we present a routine to resample a bank with a constant number of hypotheses online till convergence to the true hypothesis is achieved.
The algorithm runs with some predefined bank of hypotheses until convergence to one hypothesis at a steady state, hyp ss , is observed.Then, the hypothesis bank is reset and resampled with new hypothesis magnitudes hy p new = [ 2 3 * hyp ss hyp ss 3 2 * hyp ss ] .The algorithm will converge to another hypothesis from which a new bank can be redefined.This process repeats until convergence to one of the hypotheses in the test bank signaling it as the true hypothesis.
To showcase the proposed routine, the variable pulse discharge test is run with the true fault hypothesis H = [0.0330 0] T , while the test bank was initialized with the same fault magnitudes as the other tests; 0.005 V and 0.05 V.The algorithm will always settle to the closest fault magnitude available in the hypothesized bank.Subsequently, the converged hypothesis will be used to hypothesize new fault magnitudes around that value.This process will repeat until the new sampled hypothesis reaches some steady state.
Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.The estimation performance is shown in Fig. 11, and the hypotheses probabilities are shown in Fig. 12.The SOC estimation accuracy of the proposed approach still outperforms traditional estimators, even though the true fault magnitude was not present in the hypothesis test bank.Fig. 12 details the evolution of the probability of every hypothesis in time.It also depicts the instances when the bank is resampled and when the probabilities are reinitialized.When the true fault magnitude is not in the test bank, the algorithm converges to the fault magnitude closest to the true fault, it then undergoes resampling, and the process repeats.The algorithm still performs well even though it needs to resample the bank of hypotheses five different times in this test case.In terms of MAE, the proposed CKF yields 0.91% followed by the proposed EKF with 1.25%, while the traditional approach yields 1.99%.
The estimation performance in all test cases is documented in Table I where the Mean Absolute Error (MAE) and the Maximum Absolute Error (AE) are shown for the proposed and traditional approaches.Results confirm the added benefit of the proposed algorithm, especially when it comes to minimizing swing and maximum estimation errors during runtime.

E. Multi-Cell Packs
One more test is presented in this subsection where the proposed algorithm fault detection performance is tested against a battery pack with a larger number of cells.This test is to validate the approach with larger battery packs that contain more than the previously tested number of cells.The test is  performed on a battery pack that contains a total number of 6 cells, a more challenging fault hypotheses bank of 0.001 V and 0.01 V is assumed, and the variable pulse discharge test with hypothesis H = [0.010.01 0 0 0 0 ] T injected into the measurements vector is tested.
The results are shown in Figs. 13 and 14 where the performance of the proposed approach is validated once more.The trend seen before continues with the proposed CKF leading with a MAE of 1.58%, followed by the proposed EKF with 2.78%, and finally the traditional EKF with 5.86%.In testing, it was noticed that the larger number of available measurements allowed for the detection of smaller fault magnitudes.The evolution of hypotheses probabilities is also shown to converge to the true hypothesis in little over 8 minutes in this test case with the larger number of hypotheses and the smaller fault magnitude.

VI. SUMMARY AND CONCLUSION
The operational requirements of Li-ion BMSs have been ever increasing since the invention of Li-ion battery technology.This is due to the escalating demand for the technology in different applications and the increase in the emerging markets of EVs and grid storage.This article proposes a method to improve the voltage-dependent monitoring algorithms at the pack level.The  proposed method identifies the faulty cell voltage measurement using a decentralized measurement setup in a Li-ion battery pack consisting of several series-connected cells.The proposed method guarantees improving several functions of a BMS such as cell balancing, SOC estimation and temperature monitoring.To experimentally verify the proposed method, the SOC estimation results of commercial Li-ion battery cells were evaluated with and without the proposed method.The SOC was selected in specific since it is a fundamental quantity that affects other state(s) and parameters of the battery.
The method was tested on multiple discharge cycles that have varying levels of runtime dynamics.Testing confirmed that the method is independent of the battery model and can quickly detect and identify biases with varying orders of magnitudes.Future related research must be directed toward the following: r Integrating cell-level monitoring algorithms in a battery pack and implementing these algorithms onboard the BMS.
r Addressing the variation in the state-of-health and aging rates among the cells in a battery pack and compensating for these variations by the BMS.
r Optimizing the BMS design considering the software and hardware requirements needed to achieve the aforementioned tasks.

Fig. 7 .
Fig. 7. Constant pulse discharge test SOC estimation performance when hypothesis #11 is simulated using the tested battery cell.

Fig. 8 .
Fig. 8. Variable pulse discharge test SOC estimation performance when hypothesis #3 is simulated using the tested battery cell.

Fig. 9 .
Fig. 9. Probability associated with all assumed hypotheses throughout the variable pulse discharge test when hypothesis # 3 is simulated.

Fig. 11 .
Fig. 11.Variable pulse discharge test SOC estimation performance for the tested battery cell when true hypothesis H = [0.0330 0] T is simulated.

Fig. 12 .
Fig. 12. Probability associated with all assumed hypotheses throughout the variable pulse discharge test with the bias not in the initial hypothesis bank.

Fig. 14 .
Fig. 14.Probability associated with all assumed hypotheses throughout the variable pulse discharge test Variable pulse discharge test SOC estimation performance for the multicell battery pack when true hypothesis H = [0.010.01 0 0 0 0] T is simulated.

TABLE I QUANTITATIVE
PERFORMANCE OF THE PROPOSED ALGORITHM RELATIVE TO THE TRADITIONAL APPROACH