Patent Publication Number: US-10324135-B2

Title: Methods and systems for data-driven battery state of charge (SoC) estimation

Description:
FIELD 
     The present disclosure relates to methods and systems for a data-driven battery state of charge (SoC) estimation. More particularly, the present disclosure relates to estimating the state of charge of a rechargeable battery. 
     BACKGROUND 
     State of charge (SoC) is defined as the percentage of available charge remaining in the battery. The SoC gives an indication when the battery should be recharged, which can enable battery management systems to improve the battery life by protecting the battery from over-discharge and over-charge events. Therefore, there is of great importance to accurately measure the SoC for proper battery management. 
     Rechargeable batteries store energy through a reversible chemical reaction. Conventionally, rechargeable batteries provide a lower cost of use and result in supporting Green initiatives toward impacting the environmental than non-rechargeable batteries. For example, Lithium-ion (Li-ion) rechargable batteries have been widely deployed as a major energy storage component in numerous applications including consumer electronics, residential rooftop solar photovoltaic systems, electric vehicles, smart grid systems and etc. At least some main advantages of Li-ion batteries over other types of batteries with different chemistries are low self-discharge rate, high cell voltage, high energy density, lightweight, long lifetime, and low maintenance. 
     However, a Li-ion battery and other types of batteries are a chemical energy storage source, and this chemical energy cannot be directly accessed. This issue makes the estimation of the SoC of a battery difficult. Accurate estimation of the SoC remains very complex and is difficult to implement, because battery models cannot capture physics-based non-linear dynamics and associated parametric uncertainties. Many examples of poor accuracy and reliability of the estimation of the SoC of batteries are found in practice. 
     Conventional SoC battery estimation techniques are usually classified into model-based and data-driven based methods. Model-based methods exploit models capturing battery&#39;s chemical and physical processes. Data-driven methods use training data to map the measurements of physical quantities of the battery to corresponding values of SoC. However, the processes in the battery are very complex, because a battery is an interconnected system of many subsystems representing physical and chemical processes happening in the battery. The output of each subsystem additively contributes to the overall SoC. Such a complexity allows using only overly simplified models or simplified mapping preventing an accurate estimation of the SoC of the battery. 
     Therefore, there is a need for improved methods and systems for estimating the SoC of a battery. 
     SUMMARY 
     Some embodiments are based on recognition that conditions in the environment where the battery operates, such as outer temperature, humidity, air motions vary in an unpredictable manner and cause the outputs of the subsystems of the battery to vary in an unpredictable, e.g., a random, way. Thus, the subsystem outputs can be modeled as random variables and consequently, the resulting SoC is also random. 
     To that end, some embodiments are based on realization that by invoking the Central Limit Theorem (CLT), the resulting SoC given the input measurements can be modeled as a Gaussian distribution. Similarly, the SoC values corresponding to different inputs, given those inputs, are jointly Gaussian distributed according to the CLT. 
     Accordingly, some embodiments determine, e.g., during a training phase given the inputs and corresponding outputs in the training data, parameters of a first joint Gaussian distribution of the outputs given the inputs. During an estimation phase, some embodiments determine, using parameters of the first joint Gaussian distribution, a second joint Gaussian distribution of the values of SoC and a current value of the SoC of the battery given the set of measurements and the current measurement. In such a manner, the SoC of the battery can be determined probabilistically, e.g., a mean and a variance of the current value of the SoC of the battery can be determined from the second joint Gaussian distribution. 
     According to another method of the disclosed subject matter, a method for estimating a state of charge (SoC) of a battery while the battery is in communication with at least one processor connected to a memory. The method including determining a first joint Gaussian distribution of values of the SoC of the battery from a set of historical measured physical quantities of the state of the battery and a corresponding set of historical values of the SoC of the battery. Further, determining a second joint Gaussian distribution of values of the SoC of the battery using the set of historical measured physical quantities of the state of the battery and the corresponding set of historical values of the SoC of the battery, current measured physical quantities of the battery, and the determined first joint Gaussian distribution. Finally, determining a mean and a variance of a current SoC of the battery from the second joint Gaussian distribution, wherein the mean is an estimate of the current SoC of the battery, and the variance is a confidence of the estimate, wherein steps of the method are determined using the at least one processor. 
     According to another method of the disclosed subject matter, a method for estimating a state of charge (SoC) of a rechargeable battery while the battery is in communication with at least one processor connected to a memory. The method including selecting a first joint Gaussian distribution determined based upon, a set of historical measured physical quantities of the state of the battery and a corresponding set of historical values of the SoC of the battery. Further, determining a second joint Gaussian distribution of values of the SoC of the battery using the set of historical measured physical quantities of the state of the battery, the corresponding set of historical values of the SoC of the battery, the current measured physical quantities of the battery, and the determined first joint Gaussian distribution. Finally, determining, a mean and a variance of a current SoC of the battery from the second joint Gaussian distribution, wherein the mean is an estimate of the current SoC of the battery, and the variance is a confidence of the estimate, wherein steps of the method are determined using the at least one processor. 
     According to a system of the disclosed subject matter, a sensor system for estimating a state of charge (SoC) of a battery. The system including a memory having stored therein information about data related to estimating the SoC of the battery. A processor operatively connected to the memory and outputs of sensors measuring physical quantities of the battery. The processor is configured to select from the memory a first joint Gaussian distribution of values of the SoC of the battery given a set of historical measured physical quantities of a state of the battery and the corresponding set of historical values of the SoC of the battery. The processor is further configured to determine a second joint Gaussian distribution of SoC of the battery using the set of historical measured physical quantities of the state of the battery and the corresponding set of historical values of the SoC of the battery, the current measured physical quantities of the battery, and the first joint Gaussian distribution. The processor is also configured to determine a mean and a variance of the current value of the SoC of the battery from the second joint Gaussian distribution, wherein the mean is an estimate of the current SoC of the battery, and the variance is a confidence of the estimate. 
     Further features and advantages will become more readily apparent from the following detailed description when taken in conjunction with the accompanying drawings. 
    
    
     
       DESCRIPTION OF THE DRAWINGS 
       The present disclosure is further described in the detailed description which follows, in reference to the noted plurality of drawings by way of non-limiting examples of exemplary embodiments of the present disclosure, in which like reference numerals represent similar parts throughout the several views of the drawings. The drawings shown are not necessarily to scale, with emphasis instead generally being placed upon illustrating the principles of the presently disclosed embodiments. 
         FIG. 1A  is a block diagram of a method for estimating a state of charge (SoC) of a battery while the battery according to some embodiments; 
         FIG. 1B  illustrates an application of one embodiment for determining the SoC for a battery powering an electric vehicle; 
         FIG. 1C  is a schematic of an exemplar instrumental panel of the vehicle for displaying the estimated SoC of the embodiment of  FIG. 1B ; 
         FIG. 2A  and  FIG. 2B  are block diagrams estimating state of charge (SoC) of a battery that includes the training stage and the estimation stage, according to embodiments of the disclosure; 
         FIG. 3  illustrates a block diagram that further defines the estimation stage, according to embodiments of the disclosure; 
         FIG. 4  illustrates processes regarding embodiments of the disclosure; 
         FIG. 5  illustrates three examples of probability distributions of  FIG. 4 , according to embodiments of the disclosure; 
         FIG. 6  illustrates a block diagram that further defines the training stage, according to embodiments of the disclosure; 
         FIG. 7  illustrates a block diagram of an SoC estimation method based on a combination of GPR and Kalman Filter, according to another embodiment of the disclosure; 
         FIG. 8  illustrates a block diagram a feedback of a certain number of previous SoC estimates to the input of the estimation stage, according to another embodiment of the disclosure; 
         FIG. 9  illustrates a block diagram of a training stage of  FIG. 8 , according to embodiments of the disclosure; 
         FIG. 10  illustrates graphs of an experimental data set including voltage, current, temperature and SoC of the battery vs. time during five charging-discharging cycles, according to embodiments of the disclosure; 
         FIGS. 11A-11C  illustrate graphs analyzing the performance of SoC estimation using GPR in terms of RMSE and MAE, wherein  FIGS. 11A-11C  display the actual SoC, the estimated SoC values and 95% confidence interval for four covariance functions:  FIG. 11A  corresponding to the squared exponential (SE) covariance function;  FIG. 11B  corresponds to the Matèrn covariance function;  FIG. 11C  corresponds to the rational quadratic (RQ) covariance function; and  FIG. 11D  corresponds to the sum of Matèrn and RQ covariance functions, according to embodiments of the disclosure; 
         FIGS. 12A-12C  illustrate graphs evaluating the performance of SoC estimation method based on the combination of the GPR and Kalman filter, wherein  FIGS. 12A-12C  show plots of the actual SoC, the estimated SoC values and 95% confidence interval for different covariance functions:  FIG. 12A  corresponds to the squared exponential (SE) covariance function;  FIG. 12B  corresponds to the Matèrn covariance function;  FIG. 12C  corresponds to rational quadratic (RQ) covariance function; and  FIG. 12D  corresponds to sum of Matèrn and RQ covariance functions, according to embodiments of the disclosure; and 
         FIG. 13  illustrates shows a block diagram of at least system configured for determining the SoC of the battery, according to some embodiments of the disclosure. 
     
    
    
     While the above-identified drawings set forth presently disclosed embodiments, other embodiments are also contemplated, as noted in the discussion. This disclosure presents illustrative embodiments by way of representation and not limitation. Numerous other modifications and embodiments can be devised by those skilled in the art which fall within the scope and spirit of the principles of the presently disclosed embodiments. 
     DETAILED DESCRIPTION 
     According to some embodiments of the disclosure is to provide methods and systems for estimating a state of charge (SoC) of a battery. A state of charge (SoC) may be understood as the percentage of remaining charge in a battery relative to the full battery capacity. “State of charge values” can refer to the specific percentages. For example, SoC gives an indication when the battery should be recharged and can be at least one key parameter for assessing a battery&#39;s state. Further, SoC can assist battery management systems to improve the battery life by protecting the battery from over-discharge and over-charge events. 
     According to some other embodiments of the disclosure is to provide a method that quantifies uncertainty of the estimate, which can be important for evaluating the reliability of SoC estimates. In particular, a battery can be viewed as an interconnected system of many subsystems, where each subsystem represents one of many physical and chemical processes happening in the battery, and where the output of each subsystem additively contributes to the overall battery&#39;s SoC. Further the term battery may include a device consisting of one or more electrochemical cells that convert stored chemical energy into electrical energy. The definition of battery can include a rechargeable battery. 
     The conditions in the environment where the battery operates, such as outer temperature, humidity, air motions, which vary in an unpredictable manner and which cannot be measured or controlled, cause the outputs of the subsystems to vary in an unpredictable way. Thus, the subsystem outputs may be modeled as random variables and consequently, the resulting SoC is also random. By invoking a Central Limit Theorem (CLT), some embodiments are based on realization that the SoC of the battery can be determined probabilistically, because the SoC given the input measurements can be modeled as a Gaussian distribution. 
     Additionally, one embodiment is based on recognition that because the SoC is, by definition, a value between 0 and 1 (i.e., 0% and 100%), the mean and variance of the Gaussian distribution above are such that the probability that the SoC is outside this range is vanishingly small. Alternatively, the Gaussian distribution can be bounded between 0 and 1, in which case the Gaussian distribution becomes a truncated Gaussian distribution. 
     According to principles of the Gaussian distribution the SoC values corresponding to similar inputs are not very different from each other. In other words, the SoC values corresponding to similar inputs, should also be similar. Consequently, one embodiment embeds a similarity measure into a definition of the way the covariance matrix of the joint Gaussian distribution. For example, the embodiment determines the covariance between two outputs based on a measure of similarity between the corresponding inputs. In various implementation of this embodiment, the measure of similarity can be a Euclidean distance between the inputs, an inner product between the inputs or some more complicated function with parameters. 
       FIG. 1A  shows a block diagram of a method for estimating a state of charge (SoC) of a battery while in communication with the battery according to some embodiments. The method can be implemented using a processor  155  operatively connected to a memory (not shown) having stored therein information about data related to estimating the SoC of the battery and operatively connected to outputs of sensors (not shown) measuring physical quantities of the battery. It is contemplated processor  155  may be used for calculating the state of charge (SoC) of a battery, among other things, and may be integrated into a device in which the battery is used, or may be an external system. Processor  155  may alternatively be a component of a separate device, and may determine other aspects when the battery is inserted into this separate device and may be an electrical circuit. Information about the SoC of the battery may be transmitted wirelessly or via wires from a separate device to the processor and from the processor to a display or the separate device. Further, processor  155  can be more than one processor, such that each processor may have at least one memory. It is contemplated that more than one processor may be in communication with another processor. 
     Still referring to  FIG. 1A , the method may be implemented using a battery management application  150  running on the processor  155 , and can implement and execute various battery management methods. The method can determine  110  a first joint Gaussian distribution  115  of values of the SoC of the battery from a set of historical measured physical quantities  112  of the state of the battery and a corresponding set of historical values  114  of the SoC of the battery. The method determines  120  a second joint Gaussian distribution  125  of values of the SoC of the battery using the set of historical measured physical quantities  112  of the state of the battery and the corresponding set of historical values  114  of the SoC of the battery, current measured physical quantities  117  of the battery, and the determined first joint Gaussian distribution  115 . 
     The second joint Gaussian distribution  125  of values of the SoC of the battery is the probabilistic distribution of the values of the SoC given current measurements  117 . In such a manner, the SoC of the battery is determined probabilistically. For example, the method determines  130  a mean and a variance  135  of the current value of the SoC of the battery from the second joint Gaussian distribution is the probabilistic measurements of the SoC. Specifically, the mean is an estimate of the current SoC of the battery, and the variance is a confidence of the estimate. 
       FIG. 1B  shows an exemplar determination of the SoC of the battery  161  installed at an electric vehicle  160  according to some embodiments of the disclosure. For example, some embodiments enable a driver of the vehicle  160  to manage the battery system to ensure enough power is available. In general, a separate device  173 , e.g., including the processor and memory, is connected to the battery  171  and/or the sensors of the battery  171  via connectors  172  can perform the necessary measurements and estimates the SoC. It is contemplated the separate device  173  can include a battery management application, among other things. 
       FIG. 1C  shows an exemplar instrumental panel  162  of the vehicle  160 . The instrumental panel  162  can include one or several displays  163  and  164  for displaying the results of the SoC estimation to the driver of the vehicle. The SoC estimation can be shown on the displays  163 ,  164  automatically or when the critical level of the SoC is reached. Additionally, or alternatively, the SoC can be displayed on demand, e.g., via controls  165  and/or controls  167 ,  168  and/or  169  located on a steering wheel  166 . It is contemplated the display may be a wireless device separate from the instrumental panel  162 . 
       FIG. 2A  shows a block diagram of a multi-stage method for determining SoC of the battery according to one embodiment of the disclosure. The embodiment determines the SoC in two stages, i.e., a training stage  220  and an estimation stage  240 . It is contemplated when estimating a state of charge (SoC) of a battery the battery is in communication with at least one processor having a memory. 
     The method of  FIG. 2A  uses measurements of physical quantities  201 , which are obtained from measurement devices such as sensing devices coupled to an electronic circuit structure, attached to a battery of interest. The measured physical quantities  201  can include voltage, current and temperature of the battery of interest. Contemplated is the measured physical quantities  201  that may also include ambient temperature, volume of the battery, and gas leak measurement from the battery. Furthermore, in another embodiment, that the measured physical quantities may include current measurements in the current as well as in previous time instants. Note, in this other embodiment, the input at time t can be measurements at time t, t−1, . . . , t−T, for example. 
     Training data is gathered and is used to infer mapping between physical quantities and the battery&#39;s SoC. For example, training data input  210  is retrieved from a memory of the processor. In particular, the training data input  210  is obtained offline, usually in a lab, and the measurements are taken by exposing a battery of the same type as the battery of interest, to a wide range of different temperatures, loads, etc. For example, the training data input  210  can be a collection of measurements of different physical quantities such as previous SoC, current, voltage, temperature, volume, etc., or some combination thereof, that is of the same type of battery as the battery of interest. Contemplated is the training data input  210  that may also include ambient temperature and gas leak measurement from the battery and also previous temporal values of the current, voltage and volume. The training data output  212  is the corresponding SoC values of the training data input  210  which was previously determined and saved in the memory of the processor. 
     Training Stage 
     Still referring to  FIG. 2A , the training stage  220 , the processor offline utilizes the training data (training data input  210  and training data output  212 ), and performs training, in which, optimal hyper-parameters  222  are determined such that a chosen covariance function reasonably models the properties of the training data  210 ,  212 ; and wherein the optimized hyper-parameters  222  are stored in memory. 
     Learning Optimized Hyper-Parameters During Training Stage 
     At least one method in determining the optimized hyper-parameters  222  in the training stage  220 , includes using a Gaussian Process Regression (GPR) framework, which is a probabilistic, nonparametric machine learning method, to accurately estimate the SoC of Li-ion batteries. It is noted the term regression means can be an expression of an output variable of interest in terms of a function or a combination of functions of one or more input parameters. 
     The GPR framework can be used as a nonparametric machine learning to model the relationship between the voltage, current and temperature and the SoC. GPR is a very useful due the GPR&#39;s ability to represent a wide variety of models and to provide accurate SoC estimation and a measure of estimation uncertainty, which will be discussed later. Further, GPR can be trained offline by using voltage, current and temperature measurements of the battery, and then used to infer the SoC values. One of the main advantages of GPR is analytically tractable inference with elegant closed-form expressions. Based upon a review of the technology within this space, this is the first time this method is being used to investigate the use of a GPR learning method to estimate the SoC of Li-ion batteries. For example, GPR is further discussed below: 
     Gaussian Process Regression (GPR) 
     Using a training data set,  210 ,  212  D=(X,y) comprising D-dimensional N input vectors X={x n } n=1   N    210 , where x n ∈R D , and the corresponding outputs y={y n } n=1   N    212 , where y n ∈R. In this setting, the input-output relationship is written as
 
 y   n =ƒ( x   n )+ε n ,  (1)
 
where ƒ(.) is the underlying latent function and ε n  denotes zero-mean additive Gaussian noise with variance σ n   2 , i.e., ε n :N(0, σ n   2 ). Assumed is that {ε n } n=1   N  form an independent and identically distributed (i.i.d) sequence. The main objective is to model the underlying function ƒ(.) which maps the inputs, X to their corresponding output values, y. The key assumption in GPR is that any set of function values follow a multivariate Gaussian distribution
 
 p ( f|x   1   ,x   2   , . . . ,x   n )= N (0, K ).  (2)
 
     Above, f=[ƒ(x 1 ), ƒ(x 2 ), . . . , ƒ(x n )] T  and 0 is a N×1 vector whose elements are all 0. In addition, K is a covariance matrix, whose entries K ij =k(x i , x j ) are the values of covariance function evaluated for all pairs of training inputs. Covariance functions, also called kernels, play an important role in GPR since they encode assumptions about the smoothness, periodicity, non-stationary and other properties of the latent function that we are trying to model. Such processes can be previously stored as one or more distribution selection procedure on the processor. 
     At least three covariance functions that can be adopted in this work are briefly described: 
     Squared Exponential Covariance Function: 
     A squared exponential (SE) covariance function may be used. For D-dimensional inputs  210 , the SE covariance function takes the following form 
                         k   s     ⁡     (       x   i     ,     x   j       )       =       σ   0   2     ⁢     exp   ⁡     [       -     1   2       ⁢       ∑     d   =   1     D     ⁢           ⁢       (         x   id     -     x   jd         l   d       )     2         ]           ,           (   3   )               
where the use of the subscript s will become clear later. Above, σ 0   2 &gt;0 represents the signal variance, which determines the magnitude of the variation of the underlying function from its respective mean, l D &gt;0 denotes the characteristic length scale for the input dimension D. The characteristic length scales quantify the relative importance of corresponding input variables to the target output. More specifically, a smaller value of the characteristic length scale implies that the corresponding input dimension has more impact on the output, hence the smaller value of the characteristic length scale is more relevant. The covariance function is parameterized by this set of unknown parameters Θ=[σ 0 , l 1 , l 2 , . . . , l D ] T  called hyperparameters of the GPR. Respectively, the hyperparameters need to be determined from the training data set such that the resulting covariance function reasonably well models the properties of the data.
 
Matèrn Covariance Function:
 
     A Matèrn covariance function for D-dimensional inputs  210  is given by 
                         k   s     ⁡     (       x   i     ,     x   j       )       =       σ   1   2     ⁢         1       Γ   ⁡     (   v   )       ⁢     2     v   -   1           ⁡     [         2   ⁢           ⁢   v       ⁢       ∑     d   =   1     D     ⁢     (         x   id     -     x   jd         ρ   d       )         ]       v     ×       K   v     ⁡     (         2   ⁢           ⁢   v       ⁢       ∑     d   =   1     D     ⁢     (         x   id     -     x   jd         ρ   d       )         )           ,           (   4   )               
where Θ=[σ 1 , ν, ρ 1 , . . . , ρ D ] T  denotes the hyperparameters of the above covariance function. More specifically, σ 1   2 &gt;0 and ν&gt;0 denote the signal variance and the smoothness parameter, respectively, and ρ D &gt;0 represents the characteristic length scale for each input dimension. In addition, Γ(.) is the Gamma function and K ν (.) is the modified Bessel function of the second kind. As the value of the smoothness parameter ν increases, the function becomes more smooth. We consider a specific case where the value of ν is not too high, i.e., ν=3/2, such that the covariance is given by
 
                       k   s     ⁡     (       x   i     ,     x   j       )       =       (     1   +       3     ⁢       ∑     d   =   1     D     ⁢     (         x   id     -     x   jd         ρ   d       )           )     ⁢       exp   ⁡     (       -     3       ⁢       ∑     d   =   1     D     ⁢     (         x   id     -     x   jd         ρ   d       )         )       .               (   5   )               
Rational Quadratic Covariance Function:
 
     A rational quadratic (RQ) covariance function for D-dimensional inputs  210  is defined as 
                       k   s     ⁡     (       x   i     ,     x   j       )       =           σ   2   2     ⁡     (     1   +       1     2   ⁢           ⁢   α       ⁢       ∑     d   =   1     D     ⁢       (         x   id     -     x   jd         η   d       )     2           )         -   α       .             (   6   )               
The hyperparameters are given by Θ=[σ 2 , α, η 1 , . . . , η D ] T . In particular, σ 2   2 &gt;0 represents the signal variance, α&gt;0 determines the smoothness and η d &gt;0 corresponds to a characteristic length scale for the input dimension D.
 
     Different structures of the dataset may be represented by combining covariance functions. At least one approach is to add together covariance Functions, which results in a valid new covariance function. 
     Recall that the output in (1) is assumed to be corrupted by additive Gaussian noise with variance σ n   2 . Therefore this noise term can be incorporated into the aforementioned covariance functions as follows:
 
 k ( x   i   ,x   j )= k   s ( x   i   ,x   j )+σ n   2 δ ij ,  (7)
 
where δ ij  denotes the Kronecker delta, which takes value 1 if and only if i=j and 0 otherwise. In this setting, the distribution of y, given the latent function values f and the input X, is written as
 
 p ( y|f,X )= N ( f,σ   n   2   I ),  (8)
 
where I is an N×N identity matrix. By using (2) and (8), the marginal distribution of y can be found to be
 
 p ( y|X )=∫ p ( y|f,X ) p ( f|X ) df=N (0, K+σ   n   2   I ).  (9)
 
     Based on (9), the marginal log-likelihood of y can be written as 
                       log   ⁢           ⁢     p   ⁡     (       y   ❘   X     ,   Θ     )         =       -     1   2       ⁢         y   T     ⁡     (     K   +       σ   n   2     ⁢   I       )         -   1       ⁢   y   ⁢     1   2     ⁢   log   ⁢          K   +       σ   n   2     ⁢   I            ⁢     N   2     ⁢   log   ⁢           ⁢   2   ⁢           ⁢   π       ,           (   10   )               
where |.| is the determinant of a matrix. The hyperparameters are optimized by maximizing the marginal log-likelihood function in (10). In this regard, the gradient of (10) with respect to the ith element of Θ is calculated as
 
                           ∂   log     ⁢           ⁢     p   ⁡     (       y   ❘   X     ,   Θ     )           ∂     θ   i         =                 -       1   2     ⁢     tr   ⁡     (         (     K   +       σ   n   2     ⁢   I       )       -   1       ⁢       ∂     (     K   +       σ   n   2     ⁢   I       )         ∂     θ   i           )       ⁢           ⁢               ⁢     1   2     ⁢           ⁢         y   T     ⁡     (     K   +       σ   n   2     ⁢   I       )         -   1       ⁢           ⁢       ∂     (     K   +       σ   n   2     ⁢   I       )         ∂     θ   i         ⁢       (     K   +       σ   n   2     ⁢   I       )       -   1       ⁢   y         ,           (   11   )               
which allows the use of any gradient-based optimization method to find the optimal values of the hyperparameters that maximize the marginal log-likelihood function in (10). Note that the objective function is not necessarily convex, so that the gradient based method may converge to a local optimum. A possible approach to alleviate this problem may be to initialize multiple gradient based searches and then to choose the optimal point which yields the largest marginal log-likelihood. Further, such processes may be previously stored as one or more distribution selection procedure on the processor.
 
Estimation/Testing Stage
 
       FIG. 2A  further illustrates an estimation stage  240  according to embodiments of the disclosure. Unlike conventional methods which discard training data after the training stage, as noted above, the present disclosure further utilizes the training data  210 ,  220  in the estimation stage  240 . For example,  FIG. 2A  shows the estimation stage  240  that takes inputs including: (1) training data input  210 ; (2) training data output  220 ; (3) the measurements of the physical quantities  201 ; and (4) the determined hyper-parameters  222  evaluated in the training stage  220 , so that the estimation stage  240  outputs an SoC estimate  250 . 
       FIG. 2B  illustrates aspects of  FIG. 2A , for example, in the estimation stage  240 , after determining the optimal hyperparameters in the training stage  220 , the joint distribution of y and y* can be expressed as the following 
                       p   ⁡     (     y   ,       y   *     ❘   X     ,     x   *     ,   Θ     )       =     N   ⁡     (       [         0           0         ]     ,     [           K   +       σ   n   2     ⁢   I             k   *               k   *   T             k   **     +     σ   n   2             ]       )         ,           (   12   )               
where k*=[k(x 1 , x*), . . . , k(x N , x*)] T  and k**=k(x*, x*). The main goal of GPR is to find the predictive distribution for a new input vector, x*. In this regard, by marginalizing the joint distribution (12) over the training data set output y, we obtain the predictive distribution of test output, y*, which is a Gaussian distribution, i.e., p(y*|X, y, x*, Θ)=N(μ*, Σ*) with the mean and covariance given by
 
μ*= k*   T ( K+Υ   n   2   I ) −1   y   (13)
 
Σ*=σ n   2   +k**−k*   T ( K+σ   n   2   I ) −1   k*.   (14)
 
     Observed from the equation in (13) that the mean μ* of the predictive distribution is obtained as a linear combination of the noisy outputs, stored in the vector y, which is effectively the estimate of the test output. In addition, the variance of the predictive distribution in (14) is a measure of the uncertainty. By using (13) and (14), the 100(1−α)% confidence interval is computed as
 
[μ*− z   (1-α)/2   Σ*,μ*+z   (1-α)/2 Σ*],  (15)
 
where α∈[0,1] represents the confidence level and z (1-α)/2  is the critical value of the standard normal distribution. The confidence interval provides a range of values which is likely to contain the true value of the test output. In particular, smaller variance results in a narrower confidence interval, and hence indicates more precise estimates of the test output. Noted is that the GPR provides not only the estimated test output, but also gives a predictive probability distribution which is one of the practical advantages of GPR over SVM, NN and other non-probabilistic machine learning methods.
 
     Still referring to  FIG. 2B , the online estimation stage  240  estimates  241 A battery SoC based on the current measurements of physical quantities, hyper-parameters learned in the training stage and, unlike conventional methods in the prior art, training dataset. For example, the estimation stage  240  includes determining, a first joint Gaussian distribution of values  220  of the SoC of the battery given a set of historical measured physical quantities  210  of the state of the battery and a corresponding set of values  212  of the SoC of the battery. Determining a second joint Gaussian distribution of SoC of the battery using the set of historical measured physical quantities  210  of the state of the battery and the corresponding set of values  212  of the SoC of the battery, current measurement physical quantities of the battery and the determined first joint Gaussian distribution. Finally, determining  241 B, a mean and a variance of the current value of the SoC of the battery from the second joint Gaussian distribution, wherein the mean is an estimate of the current SoC of the battery, and the variance is a confidence of the estimate, wherein steps of the method are determined using the at least one processor. 
       FIG. 3  shows a block diagram of the estimation stage  240  of  FIG. 2A  that further includes an initial processing step  302  of the estimation stage  240 , that computes some form of similarity measures between all pairs of an input measurement tuple  201  and measurement tuples from the training data input dataset  210 . This is achieved by employing an appropriately selected kernel function which maps a pair of measurements into a real number representing similarity between them. The kernel function depends on hyper-parameters  222  computed in the training stage  220  and stored in the memory. The similarity measurements and output training data  212  are then used in the following step  303  to evaluate probability density function of SoC. Alternatively, the SoC estimate  250  is given as a weighted combination of the SoC values from the training data output  212 , where the weights are computed based on similarity measures. 
     Measuring Similarity Between Measured Data and Training Data Points 
     This intuition is illustrated in  FIG. 4  and  FIG. 5 , wherein the training dataset contains two data points,  401  and  402 .  FIG. 4  illustrates processes regarding embodiments of the disclosure.  FIG. 5  illustrates three examples of probability distributions of  FIG. 4 . Referring to  FIG. 4 , the SoC corresponding to the measured data  420  is estimated as a combination of the SoC&#39;s corresponding to the training data points  401 ,  402 . The weights in this combination are obtained based on how similar the measured data point is to each of the training data points  401 ,  402 . If a measured data point  420  is very similar to one of the training data points  402 , then the SoC corresponding to that training data point appears in the combination with a large weight. In contrast, if there is almost no similarity between the measured data point  430  and one of the training data points  401 , then the SoC corresponding to the training data point appears in the combination with the weight that is very close to 0. 
     Still referring to  FIG. 4  and  FIG. 5 , along with the estimated SoC, the methods and systems also reports the confidence interval of the estimated SoC. The methods and systems inherently evaluates the probability distribution of the SoC corresponding to the measured data based on the historical data. Three examples of probability distributions are shown in  FIG. 5 . For example, if the distribution is peaked around its mean  501 , this implies there is large confidence in the estimated SoC. Further, if the distribution is spread around its mean  502 , this implies there is small confidence in the estimated SoC. Finally, if the distribution is very spread around its mean  503 , then this implies there is very small confidence in the estimated SoC. 
     Intuitively, more peaked output distribution  501  implies more confidence in the estimated SoC. Referring again to  FIG. 4 , if the measured data point  420  is very similar to one of the points in the training data, then the reported SoC is almost the same as the SoC corresponding to that training data point  402 , and the corresponding probability distribution is very peaked  501  around that SoC. In contrast, if the measured data point  430  is equally similar to both training data points  401 , then the probability distribution of the resulting SoC is spread around the midpoint  502  between the SoC&#39;s corresponding to the training data points. 
     Training Stage Continued 
       FIG. 6  illustrates a block diagram of the training stage  220  in further detail of  FIG. 2A . The SoC estimation  250  according to embodiments of the disclosure is tied to a kernel function  601  that the user needs to choose. There are many possible kernel functions, see examples above for some of the kernel functions. The training data input  210  and selected kernel function  601  are inputs to  642  which computes similarities between pairs of measurements in the input training dataset  210 . These similarity measurements are functions of hyper-parameters  222  to be determined. The similarity measures are then used to specify the likelihood function  643  of the training data output  212 . A likelihood function of the training dataset is casted as an objective function in the optimization routine  645  which finds values of the hyper-parameters which maximize the likelihood of the training dataset. The optimization routine  645  is started with some initial values of the hyper-parameters  644  which may be completely random or selected according to a chosen kernel function. 
     SoC Estimation Method Based on GPR 
     As noted above,  FIG. 2A  illustrates the novel SoC estimation method based on GPR, which provides a probabilistic, nonparametric model to estimate the SoC of lithium-ion batteries as a function of voltage, current and temperature of the battery. For example, a definition of the SoC of the battery includes the following. 
     The SoC of the battery at time t is defined as a percentage of the residual capacity of the battery Q(t) with respect to its rated capacity, Q r , 
                     SoC   ⁡     (   t   )       =         Q   ⁡     (   t   )         Q   r       ×   100   ⁢     %   .               (   16   )               
Note that Q(t)∈[0, Q r ]. Above, Q r  is defined as the maximum amount of charge that can be drawn from a new battery under certain conditions specified by the manufacturer in terms of the ampere-hours (Ah). The fully discharged battery has an SoC of 0% and SoC increases while the battery is being charged. Consequently, the fully charged battery reaches 100% SoC.
 
     As noted above,  FIG. 2A  illustrates the disclosed method consisting of two parts, i.e., the training stage  220  and the estimation stage  240 . First, as noted above, the training stage  220  is performed in which the optimal hyperparameters of the chosen covariance function are determined by using conjugate gradient method. Noted is that the SoC values in the training data are normalized to have zero mean by subtracting their sample mean. Then, the online SoC estimation of the battery is carried out based on voltage, current and temperature measurements of the battery, also noted above. More specifically, the mean of the predictive distribution corresponds to the SoC estimate. 
     At least one process example is provided of SoC estimation method using GPR of  FIG. 2A , which is non-limiting in scope, and is illustrated merely for the purpose of understanding aspects of the disclosure. Aspects of steps of the process of SoC estimation method using GPR of  FIG. 2A  may include, among other things:
         Step 1: Obtain the training data set, D=(X,y), where x contains voltage, current and temperature measurements of the battery, and y are the corresponding SoC values.   Step 2: Initialize the hyperparameters of the given covariance function:   for SE covariance function, Θ=[σ 0 , l 1 , l 2 , l 3 , σ n ]=[0,0,0,0,0] T ,   for Matèrn covariance function, Θ=[σ 1 , ν, ρ 1 , ρ 2 , ρ 3 , σ n ]=[0,0,0,0,0,0] T ,   for RQ covariance function, Θ=[σ 2 , α, η 1 , η 2 , η 3 ] T =[0,0,0,0,0] T ,   for the sum of Matèrn and RQ covariance functions,   Θ=[σ 2 , α, η 1 , η 2 , η 3 , σ 1 , ν, ρ 1 , ρ 2 , ρ 3 , σ n ]=[0,0,0,0,0,0,0,0,0,0] T .   Step 3: Find the optimal hyperparameters that minimize the negative marginal log-likelihood function (equivalently maximize the marginal log-likelihood function) by using the conjugate gradient method.   Step 4: Estimation part:   Step 5: Obtain the predictive distribution given optimal hyperparameters, training data set,  D , test input x*. The mean of the predictive distribution corresponds to the SoC estimate.
 
SoC Estimation Method Based on a Combination of GPR and Kalman Filter
       

     In another embodiment of the disclosure,  FIG. 7  illustrates an SoC estimation method based on a combination of GPR and filtering of the GPR-based estimates. For example, in a situation where the estimated SoC  250  abruptly varies as a result of noise overfitting in the estimation stage  240 , then an instructive to filter or smoothen out the consecutive SoC estimates can be used. Then, as another embodiment of the disclosure as shown in  FIG. 7 , the SoC estimates can be processed through a filter  701  which smoothens possible noise-induced variations in the SoC estimates. This can be achieved with, for example, moving average filtering. Alternatively, noting that in the output of the estimation stage  240  there may be access to the probability distribution of the SoC,  701  to implement the Kalman filter. 
     More specifically, at least one motivation behind incorporating the Kalman filtering of the GPR outputs is to reduce the estimation error, and hence obtain more accurate estimates. As illustrated in  FIG. 7 , where the output of the GPR, i.e. estimation stage  240  is fed into the Kalman filter  701 , the state-space representation for the Kalman filter is given by
 
State equation:
 
                     SoC   ⁡     (     k   +   1     )       =       SoC   ⁡     (   k   )       -           I   c     ⁡     (   k   )         Q   r       ⁢   Δ   ⁢           ⁢   t     +   ψ             (   17   )               Measurement equation:   y *( k+ 1)=SoC( k+ 1)+ξ.  (18)
 
     Above, k is the time index, I c  is the current at time k, Δt is the sampling period, y*(k+1) is the SoC estimate of GPR at time k+1, and ψ represents the process noise assumed to be Gaussian distributed with zero mean and covariance, Q≥0, which is an adjustable parameter. In addition, ξ denotes the measurement noise, which also follows Gaussian distribution with zero mean and covariance R&gt;0, i.e., N(0, R). The state equation (17) is based on Coulomb counting, which calculates the SoC by integrating the measured current over time [16]. Also, the measurement in (18) is the SoC output of the GPR model. The two-step iterative process of the Kalman filter is given in the Algorithm below, where K denotes the Kalman gain,  (k+1) and P 1 (k+1) represent the prior estimate and the prior error covariance at time k+1, respectively. 
     Accordingly, the Algorithm is disclosed as the following: 
     
       
         
           
               
               
             
               
                   
               
             
            
               
                   
                 Set the parameters Q, R 
               
               
                   
                 Initialize P(0), k = 0 
               
               
                   
                 State prediction: 
               
               
                   
               
               
                   
                 
                   
                     
                       
                         
                           
                             
                               SoC 
                               ⩓ 
                             
                             1 
                           
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             ( 
                             
                               k 
                               + 
                               1 
                             
                             ) 
                           
                         
                         = 
                         
                           
                             
                               SoC 
                               ⩓ 
                             
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               ( 
                               k 
                               ) 
                             
                           
                           - 
                           
                             
                               
                                 
                                   I 
                                   c 
                                 
                                 ⁡ 
                                 
                                   ( 
                                   k 
                                   ) 
                                 
                               
                               
                                 Q 
                                 r 
                               
                             
                             ⁢ 
                             Δt 
                           
                           + 
                           ψ 
                         
                       
                     
                   
                 
               
               
                   
               
               
                   
                 P 1 (k + 1) = P(k) + Q 
               
               
                   
                 Measurment update: 
               
               
                   
               
               
                   
                 
                   
                     
                       
                         
                           
                             
                               
                                 
                                   K 
                                   ⁡ 
                                   
                                     ( 
                                     
                                       k 
                                       + 
                                       1 
                                     
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                                 = 
                                 
                                   
                                     
                                       P 
                                       1 
                                     
                                     ⁡ 
                                     
                                       ( 
                                       
                                         k 
                                         + 
                                         1 
                                       
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                                         P 
                                         1 
                                       
                                       ⁡ 
                                       
                                         ( 
                                         
                                           k 
                                           + 
                                           1 
                                         
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                                     + 
                                     R 
                                   
                                 
                               
                             
                           
                           
                             
                               
                                 
                                   
                                     SoC 
                                     ⩓ 
                                   
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   
                                     ( 
                                     
                                       k 
                                       + 
                                       1 
                                     
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                                 = 
                                 
                                   
                                     
                                       
                                         SoC 
                                         ⩓ 
                                       
                                       1 
                                     
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     
                                       ( 
                                       
                                         k 
                                         + 
                                         1 
                                       
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                                   + 
                                   
                                     
                                       K 
                                       ⁡ 
                                       
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                                           k 
                                           + 
                                           1 
                                         
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                                     [ 
                                     
                                       
                                         
                                           y 
                                           * 
                                         
                                         ⁡ 
                                         
                                           ( 
                                           
                                             k 
                                             + 
                                             1 
                                           
                                           ) 
                                         
                                       
                                       - 
                                       
                                         
                                           
                                             SoC 
                                             ⩓ 
                                           
                                           1 
                                         
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                                           ( 
                                           
                                             k 
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                                             1 
                                           
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                 P(k + 1) = [1 − K(k + 1)]P 1 (k + 1) 
               
               
                   
               
            
           
         
       
     
     In another embodiment of the disclosure,  FIG. 8  illustrates a feedback of a certain number of previous SoC estimates  801  to the input of the estimation stage  240 . In particular, with that, the noise-induced variation in the SoC estimates can be further reduced and hence the estimation accuracy improves. Noted is that the training stage  220 , which delivers hyper-parameters  222 , slightly changes to accommodate the existence of the feedback loop. 
       FIG. 9  illustrates a training stage of  FIG. 8 , wherein the step of  942  computes similarities between training data points (as similar to step  642  of  FIG. 6 ), where each data point consists of measured physical quantities  210  and a predefined number of previously recorded SoC values  911 . The likelihood function of the training data is computed in  943  based on similarity measures (as similar to step  643  of  FIG. 6 ). The hyper-parameters  922  (as similar to step  222  of  FIG. 6 ) are obtained as an argument which optimizes likelihood function using some optimization routine implemented in  945  (as similar to step  645  of  FIG. 6 ) and initialized with  944  (as similar to step  644  of  FIG. 6 ). 
     Sparse GPR 
     Instead of using all training dataset as in regular GPR, a subset of training data points, called inducing points are used for training the regression model. Therefore, the proposed method significantly reduces the computational complexity when the size of the training dataset exceeds a few thousand. The computational cost of a regular GPR is reduced by introducing inducing variables and modifying the joint prior distribution, p(ƒ*, f). Let u=[u 1 , . . . , u m ] T  denote the inducing variables which correspond to a set of input locations X u  called inducing points. The inducing points are chosen as a subset of the data points. Given the inducing points, the joint prior distribution, p(ƒ*, f) can be rewritten as
 
 p (ƒ*, f )=∫ p (ƒ*, f|u ) p ( u ) du,   (19)
 
where p(u)=N(0, K u,u ). It is assumed that ƒ* and f are conditionally independent given u for the approximation of p(ƒ*, f) in the following
 
 p (ƒ*, f )≈ q (ƒ*, f )=∫ q (ƒ*| u ) q ( f|u ) p ( u ) du.   (20)
 
     Subsequently, it is assumed that the training conditional q(f|u) is fully independent and the test conditional remains exact as 
                       q   ⁡     (     f   ❘   u     )       =         ∏     n   =   1     N     ⁢           ⁢     p   ⁡     (       f   n     ❘   u     )         =     N   ⁡     (         K     f   ,   u       ⁢     K     u   ,   u       -   1       ⁢   u     ,     diag   ⁡     [       K     f   ,   f       -     Q     f   ,   f         ]         )           ,           (   21   )                   q   ⁡     (       f   *     ❘   u     )       =     p   ⁡     (       f   *     ❘   u     )         ,           (   22   )               
where diag[A] denotes the diagonal matrix in which all of the diagonal elements equal the corresponding elements of A and other elements are zero. By inserting above distributions into (2) and integrating over u, the joint prior is given by
 
                     q   ⁡     (     f   ,     f   *       )       =     N   ⁡     (     0   ,     [             Q     f   ,   f       -     diag   ⁡     [       Q     f   ,   f       -     K     f   ,   f         ]               Q     f   ,   *                 Q     *     ,   f               K     *     ,   *               ]       )               (   23   )               
where Q a,b =K a,u K u,u   −1 K u,b  is a low-rank matrix (i.e., rank M). Using the above joint prior distribution, the predictive distribution is obtained as
 
 q ( y*|X,y,x *,Θ)= N ({tilde over (μ)}*,{tilde over (Σ)}*)  (24)
 
where
 
{tilde over (μ)}*= K   *,u   ΩK   u,f Λ −1   y   (25)
 
{tilde over (Σ)}*=σ n   2   +K   *,*   −Q   *,*   +K   *,u   ΩK   u,* .  (26)
 
     Above, Ω=(K u,u +K u,f Λ −1 K f,u ) −1  and Λ=diag[K f,f −Q f,f +σ n   2 I]. It is seen that the only matrix requiring inversion is the N×N diagonal matrix Λ, which yields a significant reduction in computational complexity. The computational cost of training becomes O(NM 2 ) that is linear in N and a larger M leads to better accuracy at the expense of increased computational requirements. Also, testing time complexity is O(M) and O(M 2 ) for calculating the mean and the variance, respectively. 
     Performance Evaluation 
     According to aspects of the disclosure, the SoC estimation methods and systems of the disclosure for Li-ion batteries are validated based on the methods of GPR and a combination of GPR and Kalman filter, with data obtained from testing the battery under constant charge and discharge current. Also identified, is the impact of covariance function selection on the estimation performance for both methods. The root mean square error (RMSE) and maximum absolute error (MAE) are chosen as the main performance metrics, which are respectively defined as follows 
                       R   ⁢           ⁢   M   ⁢           ⁢   S   ⁢           ⁢   E     =         1     N   t       ⁢       ∑     i   =   1       N   t       ⁢           ⁢       (       y     *     ,   i       true     -       y   ^       *     ,   i       est       )     2             ,           (   27   )                   M   ⁢           ⁢   A   ⁢           ⁢   E     =     max   ⁢            y   *   true     -       y   ^     *   est                ,           (   28   )               
where N t  denotes the size of test data, y* true  is a 1×N t  vector including SoC values of the test data and ŷ* est  is a 1×N t  vector containing the estimated SoC values. In the following subsections, is first described a battery dataset, and then presented the SoC estimation results of the proposed methods.
 
     Dataset 
       FIG. 10  illustrates a data set including voltage, current, temperature and SoC of the battery vs. time during five charging-discharging cycles. The dataset was collected from a LiMn2O4/hard-carbon battery with a nominal capacity of 4.93 Ah in the Advanced Technology R&amp;D Center, Mitsubishi Electric Corporation. In particular, five consecutive cycles of charging and discharging at 10 C-rates were performed using a rechargeable battery test equipment produced by Fujitsu Telecom Networks. The battery voltage, temperature and current were measured during the experiment. The sampling period was chosen to be 1 second. 
     Specifically,  FIG. 10  shows the dataset, where the negative values of the current indicate that the battery is being discharged. The GPR model is trained offline in which the optimal hyperparameters are determined for a given covariance function using the first samples of voltage  1005 , temperature  1010  and current measurements  1015 . The remaining 900 samples are used to verify the performance of the proposed SoC estimation methods  1020 . 
     Example 1: Performance of SoC Estimation Method Based on GPR 
     Referring to  FIG. 11A ,  FIG. 11B ,  FIG. 11C  and  FIG. 11D , in regard to analyzing the performance of SoC estimation using GPR in terms of RMSE and MAE. Specifically,  FIGS. 11A-11C  display the actual SoC, the estimated SoC values and 95% confidence interval for four covariance functions, wherein,  FIG. 11A  shows the squared exponential (SE) covariance function  1103 ,  FIG. 11B  shows the Matèrn covariance function  1106 ,  FIG. 11C  shows rational quadratic (RQ) covariance function  1109  and  FIG. 11D  shows sum of Matèrn and RQ covariance functions  1111 . The shaded area (in Blue) represents the 95% confidence interval. The corresponding RMSE and MAE values are listed in Table 1. 
     
       
         
           
               
               
               
               
             
               
                   
                 TABLE 1 
               
               
                   
                   
               
               
                   
                 Covariance Functions 
                 RMSE (%) 
                 MAE (%) 
               
               
                   
                   
               
             
            
               
                   
                 Squared Exponential (SE) (a) 
                 2.5369 
                 7.4829 
               
               
                   
                 Matern (b) 
                 0.7273 
                 2.1796 
               
               
                   
                 Rational Quadratic (RQ) (c) 
                 1.1233 
                 3.6897 
               
               
                   
                 Sum of Matern and RQ (d) 
                 0.4588 
                 1.5502 
               
               
                   
                   
               
            
           
         
       
     
     Upon review, the SoC estimation performance appears to heavily depend on the choice of the covariance function. For instance, GPR with SE covariance function does not adequately represent the data, hence results in less accurate SoC estimates with higher RMSE=2.5369% and MAE=7.4829%. On the other hand, GPR with Matèrn and RQ covariance functions give reasonable SoC estimates with RMSE (%)=0.7273, MAE (%)=2.1796, and RMSE (%)=1.1233, MAE (%)=3.6897, respectively. Thus, this leads one to consider the sum of Matèrn and RQ covariance functions. Observed from  FIGS. 11A-11C , is that the GPR with the sum of Matèrn and RQ covariance functions provides a better fit to the data, when compared to that attained with the other three covariance functions. In particular, RMSE is 0.4588% and MAE is 1.5502%, which implies good accuracy. 
     Also observed from the  FIGS. 11A-11C , is that there is a higher uncertainty, hence larger confidence interval when the difference between the actual and the estimated SoC values is higher. Whereas accurate SoC estimates result in lower uncertainty, thus smaller confidence interval. Such that this uncertainty characterization is at least one of the key advantages of the GPR-based methods over non-probabilistic machine learning methods such as SVM, NN, among other things. 
     The optimal hyperparameters associated with each input variable enable one to infer the relative importance of the inputs. For example, in the case of GPR with SE covariance function ( FIG. 11A ), smaller values of the characteristic length scales imply that the corresponding input dimension is more important and relevant. The optimal values of the characteristic length scales for voltage, current and temperature are 0.1670, 54.3450 and 5.9635, respectively, which indicates that voltage has more impact than temperature, and temperature has more impact than current on the SoC estimate. Noted is that the same relative importance order is observed for the other three covariance functions. However, noted is that the optimal values of the corresponding hyper parameters have not been included, merely for the sake of brevity. 
     Example 2: Performance of SoC Estimation Method Based on a Combination of GPR and Kalman Filter 
     Referring to  FIG. 12A ,  FIG. 12B ,  FIG. 12C  and  FIG. 12D , in regard to evaluating, the performance of SoC estimation method based on the combination of the GPR and Kalman filter, and in comparison with the above section, the output of the GPR is fed into Kalman filter. For example,  FIGS. 12A-12C  show plots of the actual SoC, the estimated SoC values and 95% confidence interval for different covariance functions, wherein,  FIG. 12A  shows the squared exponential (SE) covariance function  1204 ,  FIG. 12B  shows the Matèrn covariance function  1208 ,  FIG. 12C  shows rational quadratic (RQ) covariance function  1212  and  FIG. 12D  shows sum of Matèrn and RQ covariance functions  1216 . The resulting RMSE and MAE values are shown in Table 2. 
     
       
         
           
               
               
               
               
             
               
                   
                 TABLE 2 
               
               
                   
                   
               
               
                   
                 Covariance Functions 
                 RMSE (%) 
                 MAE (%) 
               
               
                   
                   
               
             
            
               
                   
                 Squared Exponential (SE) (a) 
                 1.0824 
                 2.6305 
               
               
                   
                 Matern (b) 
                 0.2931 
                 1.1443 
               
               
                   
                 Rational Quadratic (RQ) (c) 
                 0.4126 
                 2.1418 
               
               
                   
                 Sum of Matern and RQ (d) 
                 0.2070 
                 0.9802 
               
               
                   
                   
               
            
           
         
       
     
     Specifically, Kalman filter is an algorithm that can be implemented wherein substantial improvements in terms of the RMSE and MAE, for all of the covariance functions are made by applying the Kalman filter. In particular, RMSE is below 1.1% and MAE is below 2.7%. The choice of the sum of Matèrn and RQ covariance functions again gives the best accuracy, i.e., RMSE=0.2070% and MAE=0.9802%, which is almost a perfect fit to the actual SoC values. 
     Based upon the results the accuracy of the disclosed method, i.e., RMSE is less than 0.46% and MAE is less than 1.56% when the sum of Matern and RQ covariance functions is used can be confirmed. Also, in view of the effects of the covariance functions on the estimation performance, observed is that GPR with the sum of Matern and RQ covariance functions represents the data soundly. Also presented is the uncertainty representation through 95% confidence interval, which enables one to evaluate the reliability of the SoC estimation. Moreover, having identified the relative importance of the input variables on the estimation performance, or more specifically, that voltage is found to have more impact than temperature, and temperature has more impact than current on estimating the SoC. By further incorporating the Kalman filter into the GPR, more accurate estimation results are obtained. In particular, when GPR with the sum of Matern and RQ covariance functions is applied, there is an achievement of RMSE below 0.21% and MAE below 0.99%. 
       FIG. 13  shows a block diagram of an exemplary system  1300  configured for determining the SoC of the battery according to some embodiments of the disclosure. The system  1300  can be implemented integral with the battery or machinery having the battery. Additionally or alternatively, the system  1300  can be communicatively connected to the sensors measuring physical quantities of the battery. 
     The system  1300  can include one or combination of sensors  1310 , an inertial measurement unit (IMU)  1330 , a processor  1350 , a memory  1360 , a transceiver  1370 , and a display/screen  1380 , which can be operatively coupled to other components through connections  1320 . The connections  1320  can comprise buses, lines, fibers, links or combination thereof. 
     The transceiver  1370  can, for example, include a transmitter enabled to transmit one or more signals over one or more types of wireless communication networks and a receiver to receive one or more signals transmitted over the one or more types of wireless communication networks. The transceiver  1370  can permit communication with wireless networks based on a variety of technologies such as, but not limited to, femtocells, Wi-Fi networks or Wireless Local Area Networks (WLANs), which may be based on the IEEE 802.11 family of standards, Wireless Personal Area Networks (WPANS) such Bluetooth, Near Field Communication (NFC), networks based on the IEEE 802.15x family of standards, and/or Wireless Wide Area Networks (WWANs) such as LTE, WiMAX, etc. The system  400  can also include one or more ports for communicating over wired networks. 
     In some embodiments, the system  1300  can comprise sensors for measuring physical quantities of the battery, which are hereinafter referred to as “sensor  1310 ”. For example, the sensor  1310  can include a voltmeter for measuring voltage of the battery, an ammeter for measuring current of the battery, and a thermometer for measuring temperature of the battery. 
     The system  1300  can also include a screen or display  1380  rendering information about the SoC of the battery. In some embodiments, the display  1380  can also be used to display measurements from the sensor  1310 . In some embodiments, the display  1380  can include and/or be housed with a touchscreen to permit users to input data via some combination of virtual keyboards, icons, menus, or other GUIs, user gestures and/or input devices such as styli and other writing implements. In some embodiments, the display  480  can be implemented using a liquid crystal display (LCD) display or a light emitting diode (LED) display, such as an organic LED (OLED) display. In other embodiments, the display  480  can be a wearable display. 
     In some embodiments, the result of the fusion can be rendered on the display  1380  or submitted to different applications that can be internal or external to the system  1300 . For example, a battery management application  1355  running on the processor  1350  can implement and execute various battery management methods. 
     Exemplary system  1300  can also be modified in various ways in a manner consistent with the disclosure, such as, by adding, combining, or omitting one or more of the functional blocks shown. For example, in some configurations, the system  1300  does not include the IMU  1330  or the transceiver  1370 . 
     The processor  1350  can be implemented using a combination of hardware, firmware, and software. The processor  1350  can represent one or more circuits configurable to perform at least a portion of a computing procedure or process related to sensor fusion and/or methods for further processing the fused measurements. The processor  1350  retrieves instructions and/or data from memory  1360 . The processor  1350  can be implemented using one or more application specific integrated circuits (ASICs), central and/or graphical processing units (CPUs and/or GPUs), digital signal processors (DSPs), digital signal processing devices (DSPDs), programmable logic devices (PLDs), field programmable gate arrays (FPGAs), controllers, micro-controllers, microprocessors, embedded processor cores, electronic devices, other electronic units designed to perform the functions described herein, or a combination thereof. 
     The memory  1360  can be implemented within the processor  1350  and/or external to the processor  1350 . As used herein the term “memory” refers to any type of long term, short term, volatile, nonvolatile, or other memory and is not to be limited to any particular type of memory or number of memories, or type of physical media upon which memory is stored. In some embodiments, the memory  1360  holds program codes that facilitate SoC estimation, and other tasks performed by the processor  1350 . For example, the memory  1360  can store the measurements of the sensors as well as the estimation determined during the training stage. 
     In general, the memory  1360  can represent any data storage mechanism. The memory  1360  can include, for example, a primary memory and/or a secondary memory. The primary memory can include, for example, a random access memory, read only memory, etc. While illustrated in  FIG. 13  as being separate from the processors  1350 , it should be understood that all or part of a primary memory can be provided within or otherwise co-located and/or coupled to the processors  1350 . 
     Secondary memory can include, for example, the same or similar type of memory as primary memory and/or one or more data storage devices or systems, such as, for example, flash/USB memory drives, memory card drives, disk drives, optical disc drives, tape drives, solid state drives, hybrid drives etc. In certain implementations, secondary memory can be operatively receptive of, or otherwise configurable to a non-transitory computer-readable medium in a removable media drive (not shown). In some embodiments, the non-transitory computer readable medium forms part of the memory  1360  and/or the processor  1350 . 
     All patents, patent applications, and published references cited herein are hereby incorporated by reference in their entirety. Emphasized is that the above-described embodiments of the present disclosure are merely possible examples of implementations, merely set forth for a clear understanding of the principles of the disclosure. Many variations and modifications may be made to the above-described embodiment(s) without departing substantially from the spirit and principles of the disclosure. Appreciated is that several of the above-disclosed and other features and functions, or alternatives thereof, may be desirably combined into many other different systems or applications. All such modifications and variations are intended to be included herein within the scope of this disclosure, as fall within the scope of the appended claims.