A Critical Look at Coulomb Counting Towards Improving the Kalman Filter Based State of Charge Tracking Algorithms in Rechargeable Batteries
Kiarash Movassagh, Sheikh Arif Raihan, Balakumar Balasingam, Krishna Pattipati
aa r X i v : . [ ee ss . S Y ] J a n A Critical Look at Coulomb Counting Towards Improving theKalman Filter Based State of Charge Tracking Algorithms inRechargeable Batteries
Kiarash Movassagh † , Sheikh Arif Raihan † , Balakumar Balasingam ⋆ † , Senior Member, IEEE and Krishna Pattipati ‡ , Fellow,IEEE
Abstract —In this paper, we consider the problem of state of chargeestimation for rechargeable batteries. Coulomb counting is one of thetraditional approaches to state of charge estimation and it is consideredreliable as long as the battery capacity and initial state of charge areknown. However, the Coulomb counting method is susceptible to errorsfrom several sources and the extent of these errors are not studied inthe literature. In this paper, we formally derive and quantify the state ofcharge estimation error during Coulomb counting due to the followingfour types of error sources: (i) current measurement error; (ii) currentintegration approximation error; (iii) battery capacity uncertainty; and(iv) the timing oscillator error/drift. It is shown that the resulting stateof charge error can either be of the time-cumulative or of state-of-charge-proportional type. Time-cumulative errors increase with time and hasthe potential to completely invalidate the state of charge estimationin the long run. State-of-charge-proportional errors increase with theaccumulated state of charge and reach its worst value within onecharge/discharge cycle. Simulation analyses are presented to demonstratethe extent of these errors under several realistic scenarios and the paperdiscusses approaches to reduce the time-cumulative and state of charge-proportional errors.
Index Terms —Battery management system, state of charge, Coulombcounting, battery capacity, measurement errors, battery impedance,equivalent circuit model.
I. I
NTRODUCTION
Rechargeable batteries are becoming an integral part of the futureenergy strategy of the globe. The use of rechargeable batteries aresteadily on the rise in a wide ranging applications, such as, electricand hybrid electric vehicles, household appliances, robotics, powerequipment, consumer electronics, aerospace, and renewable energystorage systems. Accurate estimation of the state of charge (SOC) ofa battery is critical for the safe, efficient and reliable management ofbatteries [3], [4], [5], [6].There are three approaches to estimate the SOC of a battery [7]: (i)voltage-based approach, (ii) current-based approach, and (iii) fusionof voltage/current based approaches. The fusion-based approachesseek to retain the benefits of both voltage-based and current-basedapproaches by employing non-linear filters, such as the extendedKalman filter, in order to fuse the information obtained through thevoltage and current measurements.
Some preliminary findings were published in [1], [2]Submitted to
IEEE Transactions on Control Systems Technology in Jan.2021. ⋆ Balakumar Balasingam is the corresponding author [email protected] † Department of Electrical and Computer Engineering, Universityof Windsor, 401 Sunset Ave., Office { movassa,raihans,singam } @uwindsor.ca. ‡ Department of Electrical and Computer Engineering, University of Con-necticut, 371 Fairfield Rd, Office
In its simplest form, the voltage-based approach serves as atable look-up method — the measured voltage across the batteryterminals is matched to its corresponding SOC in the OCV-SOCcharacterization curve [8]. In more generic terms, we havevoltage-measurement = f (SOC) | {z } OCV-SOC model + g ( parameters, current ) | {z } voltage drop (1)where the function f ( · ) refers to the open circuit voltage model thatrelates the OCV of the battery to SOC and g ( · ) accounts for thevoltage-drop within the battery-cell due to hysteresis and relaxationeffects. Challenges in voltage-based SOC estimation arise due to thefact that the functions f ( · ) and g ( · ) are usually non-linear and thatthere is a great amount of uncertainty as to what those functionsmight be [8], [9]. For instance, the parameters can be modelledthrough electrical equivalent circuit models (ECM) [10], [11] orelectrochemical models [12] each of which can result into numerousreduced-order approximations. The voltage based approach suffersfrom the following three types of errors:(i) OCV-SOC modeling error.
The OCV-SOC relationship of abattery can be approximated through various models [8]: linearmodel, polynomial model and combine models are few exam-ples. Reducing the OCV-SOC modeling error is an ongoingresearch problem — in [13] a new modeling approach wasreported that resulted in the “worst case modeling error” ofabout 10 mV. It must be mentioned that the OCV modelingerror is not identical at all voltage regions of the battery.(ii)
Voltage-drop modeling error.
Voltage-drop models accounts forthe hysteresis and relaxation effects in the battery. Variousapproximations were proposed in the literature in order torepresent these effects [14], [9].(iii)
Voltage measurement error.
Every voltage measurement systemcomes with errors; this translates into SOC estimation error.In order to reduce the effect of uncertainties in voltage-based SOCestimation, it is often suggested to rest the battery before taking thevoltage measurement for SOC lookup [11] — when the current iszero for sufficient time the voltage-drop also approaches to zero.However, all the other sources of errors mentioned above (OCV-SOCmodeling error, hysteresis, and voltage measurement error) cannot beeliminated by resting the battery.The current-based approach, also known as the
Coulomb countingmethod [11], computes the amount of Coulombs added/removedfrom the battery in order to compute the SOC as a ratio betweenthe remaining Coulombs and the battery capacity that is assumedknown. The Coulomb counting approach to SOC estimation can beapproximated as follows (see Section II for details)
SOC( k ) = SOC( k −
1) + ∆ k i ( k )3600C batt (2) where SOC( k ) indicates the SOC at time k , ∆ k is the samplingtime, i ( k ) is the current through the battery at the at time k , and C batt is the battery capacity an Ampere hours (Ah). Assumingthe knowledge of the initial SOC, the Coulomb counting methodcomputes the effective change in Coulombs in/out of the battery basedon the measured current and time in order to compute the updatedSOC. The important advantage of the Coulomb counting approachis that it does not require any prior characterization, such as theOCV-SOC characterization [8] that is required for the voltage-basedSOC estimation method. However, the Coulomb counting method canresult in SOC estimation errors due to the following five factors:1) Initial SOC.
Coulomb counting approach assumes the knowl-edge of the initial SOC before it starts counting Coulombs inand out of the battery based on the measured current. Anyerror/uncertainty in the initial SOC will bias the Coulombcounting process.2)
Current measurement error.
Current sensors are corrupted bymeasurement noise [15], [16]; simple, inexpensive currentsensors are likely to be more noisy.3)
Current integration error.
Coulomb counting methods employ asimple, rectangular approximation for current integration. Suchan approximation results in errors that increase with samplingas the load changes rapidly.4)
Uncertainty in the knowledge of battery capacity [17].
Coulomb counting method assumes perfect knowledge of thebattery capacity, which is known to vary with temperature,usage patterns and time (age of the battery) [18], [19].5)
Timing oscillator error.
Timing oscillator provides the clock for(recursive) SOC update, i.e., the measure of time comes fromthe timing oscillator. Any error/drift in the timing oscillator willhave an effect on the measured Coulombs.The fusion-based approach seeks to retain the best features of bothvoltage and current-based approaches. This is achicved by creatingthe following state-space model
SOC( k ) = SOC( k −
1) + ∆ k i ( k )3600C batt + n s ( k ) (3) z v ( k ) = f (SOC( k )) | {z } OCV-SOC model + g ( parameters , i ( k )) | {z } voltage drop + n z ( k ) (4)where (3) is the process model that is derived from the Coulombcounting equation (2), (4) is the measurement model that is is derivedfrom the voltage measurement equation (1), n s ( k ) is the processnoise, z v ( k ) is the measured voltage across the battery terminals, f (SOC( k )) is the OCV characterization function [8] that representsthe battery voltage as a function of SOC, g ( parameters , i ( k )) is thevoltage drop due to impedance and hysteresis within the battery, and n z ( k ) is the measurement noise corresponding to the model (4). Thegoal from the above state-space model is to recursively estimate theSOC given the voltage and current measurements.The state-space model described in (3)-(4) is non-linear due tothe OCV-SOC model and the different approximate representationsfor voltage-drop models [9]. If the models are known, a non-linearfilter, such as the extend Kalman filter [20], can yield near-accurateestimate of SOC in real time. The filter selection is based on themodel assumptions:(I) Kalman filter:-
Here, the following assumptions need to be met:the state-space model is known and linear, i.e., the functions f ( · ) and g ( · ) in (4) are linear in terms of SOC, the ’parameter’ andthe current i ( k ) in (4) are known with negligible uncertainty inthem. and that the process and measurement noises, n s ( k ) and n v ( k ) , respectively, are i.i.d. Gaussian with known mean andvariance.(II) Extended/unscented Kalman filter:-
Here, only the linearityassumption is relaxed, i.e., the functions f ( · ) and g ( · ) in (4)can be non-linear in terms of SOC. All other assumptions forthe Kalman filter need to be met, i.e., the model parametersand the noise statistics need to be perfectly known and that theprocess and measurement noises need to be i.i.d. Gaussian withknown mean and variance.(III) Particle filter:-
Compared to the Kalman filter assumptions, par-ticle filter allows to relax both linear and Gaussian assumptions;here, the f ( · ) and g ( · ) can be non-linear and both process andmeasurement noise statistics can be non-Gaussian. It needs to bere-emphasized that, similar to the cases in (I) and (II) above, themodels f ( · ) and g ( · ) and the parameters of the noise statisticsneed to be perfectly known.It is important to note that the recursive filters discussed above allassume that the model, which consists of the functions f ( · ) , g ( · ) andthe parameters of the noise statistics, is perfectly known. However,we have discussed several ways earlier in this section in whichthe known-model assumptions can be violated. Indeed, the “knownmodel” assumptions can be violated through any of the following tenways:(a) Five sources of error in defining the process model (3) : namely,the initial SOC error, current measurement error, current inte-gration error, battery capacity error, and timing oscillator error.(b) Three sources of error in defining the measurement model (4) : namely, OCV-SOC modeling error, voltage-drop modeling andits parameter estimation error, and voltage measurement error.(c) The process noise n s ( k ) . The statistical parameters of theprocess noise should be computed based on the knowledge aboutthe statistics of the five error sources in (a) above.(d)
The measurement noise n v ( k ) . The statistical parameters of themeasurement noise should be computed based on the knowledgeabout the statistics of the three error sources in (b) above.The focus of the present paper is to develop detailed insights aboutthe error sources (a) and (c) above. In a separate work [21], wediscuss the noise sources (b) and (d) in detail.
A. Background
The classical estimation theory [20] states that when the linear-Gaussian conditions and the known model assumptions (stated under(I) in Section I) are met the SOC estimate will be efficient , i.e., thevariance of the SOC estimation error will be equal to that of the posterior Cram´er-Rao lower bound (PCRLB ) which is proved to bethe theoretical bound; under non-Bayesian conditions this limit isknown just as the Cram´er-Rao lower bound (CRLB). That is, thePCRLB or CRLB can be used as a gold standard on performance. Inthis regards, some prior works in the literature have [22], [23], [24]derived the CRLB as a measure of performance evaluation. Theseapproaches were developed to estimate the ECM parameters of thebattery; the ECM parameters are involved in the measurement modelfor the SOC in (1). In addition to the use in SOC estimation, ECMparameter estimation has other important applications in a batterymanagement system.Several other approaches attempted to theoretically derive the errorbound on SOC estimation separately and jointly with ECM parameteridentification. In [25], a RLS-based parameter identification techniquewith forgetting factor was presented in which a sinusoidal currentexcitation made of two sinusoid component was used. According to the results, the CRLB of resistance decrease with the increase offrequencies and thus the large frequency components are preferablefor higher accuracy in parameter estimation; similar observationswere reported in [26]. Influence of voltage noise, current amplitudeand frequency on parameter identification has been illustrated in [27]where a sinusoidal excitation current was used. Here, the CRLB of thebattery equivalent circuit model was derived using Laplace transform;the authors round no influence of the frequency of the excitationsignal on the single-parameter identification of ohmic resistance andreported that reducing the voltage measurement noise and increasingcurrent amplitude improves the identification accuracy. A posteriorCRLB was developed to quantify accuracy for EKF based ECMparameter identification in which a second order battery ECM wasadopted in [28]. The CRLB was determined numerically with thehelp of sinusoidal current excitation. It showed that the CRLB ofohmic resistance estimation decreases with the increase of currentamplitude and frequency as well. Unlike [28], the CRLB was derivedin analytic expression in discrete time and Laplace transform in [29]in which a (known) sinusoidal current input was considered. A non-linear least-square based electrode parameter (e.g. electrode capacity)identification method was presented in [22] in which only the terminalvoltage was considered to contain measurement noise. This CRLBwas derived and used to quantify the error bound of the estimator todetermine the uncertainty of the parameter estimation. The parameterestimates were interpreted with the help of analytically derived confidence levels . Here, the noise was assumed to be Gaussian whitenoise with standard deviation 10 mV in the demonstrations. In [23],battery SOC estimation error was derived theoretically as a functionof sensor noises; the proposed approach considers measurement noisein both current and voltage. Effect of different components involvedin SOC estimation were demonstrated using a parameter sensitivityanalysis in [30] and the effect of bias and noise were reported in [31]as well.The five sources of error in Coulomb counting have been recog-nized in the literature and some remedies were proposed. In [32],the initial SOC is modeled as a function of the terminal voltage,temperature and the relaxation time. The authors in [33] proposedthe use of neural networks to gain a better estimate of the initialSOC. In [34] a data fusion approach is proposed where a H-infinityfilter is used to minimize the error in the initial SOC estimate. Inthe battery fuel gauge evaluation approach proposed in [19] theuncertainty in initial SOC error was taken into account and theOCV lookup method [8], [35] is introduced as a performance metric.It was pointed out in [36] that the accuracy of the OCV lookupmethod might be affected with battery age. The effect of currentintegration error was also recognized in the literature and remedieswere proposed: in [37], [33] a model based approach was proposedto reduce current integration error; in [38], it was proposed to resetCoulomb counting when the present SOC is known when the batteryis fully charged/discharged where the “fully-charged” and “fully-empty” conditions were declared based on measured voltage acrossbattery terminals; here, the authors propose a way to minimize theerror due to voltage-only based declaration of these two conditions.Many articles recognize the imperfect knowledge of battery capacityand ways to estimate them; a neural network based approach tobattery capacity estimation was proposed in in [33]; an approachbased on the charge/discharge currents and the estimated SOC forbattery capacity estimation was proposed [39]; the authors in [38]propose estimating the battery capacity when the battery is fullycharged/discharged, which can be known easily when the terminalvoltage reach the max./min. voltage respectively; in [17] a state-space model was introduced to track battery capacity where measurementscan be incorporated using multiple means, including when the batteryis at rest. None of the existing works explored the effect of timingoscillator error in the estimated SOC.In summary, the importance of theoretical performance derivationand analysis is recognized in the literature, particularly in the abovedetailed publications. Considering the nature of the complexity ofthe real-world measurement model, the existing literature representsonly a small fraction of what needs to be done for a completeunderstanding of the battery SOC estimation problem. For example,even though the effect of some of the five sources of Coulombcounting error (summarized earlier in this section) were noted in theliterature, it was not fully incorporated into the fusion-based SOCtracking approaches. In other word, the process noise n s ( k ) in (3)was not accurately defined in the literature. Table I summarizes howthe process noise is defined in some notable works in the literature.Setting arbitrary values to a process noise will have the followingadverse effect on the filter outcome: • Too small process noise:
When the process noise is smallerthan the reality, the filter will compute the weights such thatthe measurements are ignored. • Too large process noise:
When the process noise is larger thanthe reality, the variance of the filtered estimates will be high –effectively the benefits of using a filter will be lost.Based on Table I, it is clear that there is a knowledge gap about theprocess noise in recursive-filtering approach to SOC tracking. Thefocus of this paper is to derive accurate models for SOC tracking;particularly we focus on the process model only. Similar discussionabout one of the possible measurement models can be found in [21].Model validation strategies and analyses using practical data are leftfor a future discussion.
B. Summary of Contributions
A large portion of the existing work related to battery SOCestimation in the literature lack theoretical validations. Almost allthe work that employ some form of theoretical validation are sum-marized in Subsection I-A — the number of papers in this sectionis insignificant compared to the number of publication in SOCestimation in the past year alone. This indicates the need to focusmore in theoretical performance analyses and to understand wherethe remaining challenges in battery SOC estimation.In this paper, we develop a mathematical model to theoreticallycompute the accumulated SOC error as a result of current mea-surement error, current integration approximation, battery capacityuncertainty, and timing oscillator error. These four sources of errorare identified in [51]. In this paper, we provided the formulas forexact statistical error parameters (mean and standard deviation) thatcan be used to improve all existing SOC estimation methods. Assuch, the contributions of this paper are summarized as follows: • Exact computation of Coulomb counting error.
With realisticnumerical examples, we demonstrate the errors and their severityduring Coulomb counting. Further, we derive mathematicalformulas to determine these errors such that the statisticalconfidence in the SOC estimates can be explicitly stated. • Five different error sources in Coulomb counting are ana-lyzed.
We derive the exact mean and standard deviation ofthe error (with time) due to all five possible sources of errorsduring Coulomb counting: current measurement error, currentintegration error, battery capacity uncertainty, charge, dischargeefficiency uncertainty, and timing oscillator error. It is demon-strated that the resulting error will fall into one of the following
TABLE I:
Process noise in SOC tracking
Paper Filtering Method Definition of process noise [40, page 279] Extended Kalman filter “small”[41, page 1370] Unscented Kalman filter “stochastic process noise or disturbance that models some unmeasured input whichaffects the state of the system”[42, page 7] Kalman filter “process noise”[43, page 334] Frisch scheme based bias “zero-mean white noise with variance σ i ”compensating recursiveleast squares[44, page 8954] Extended Kalman filter “zero-mean white Gaussian process noise”[45, page 13205& 13206] Adaptive unscentedKalman filter “zero-mean Gaussian white sequence”; “In practice, the mean and covariance ofprocess noise is frequently unknown or incorrect”[46, page 4610] Extended Kalman filter “The EKF assumes knowledge of the measurement noise statistics. Moreover,any uncertainty in the system’s model will degrade the estimator’s performance”[47, page 10] Correntropy unscentedKalman filter “The process noise covariance and measurement noise are assumed to be known inCUKF. However, they are real time in general and may not be obtained prior inpractice. Therefore, they should be updated with changes in time on the basis of someobtained prior knowledge.”“ w k ∼ N (0 , Q k ) ” where Q k is the covariance matrix[48, page166660] Adaptive weightingCubature particle filter “In the process of practical application, the statistical characteristics of the processnoise and measurement noise of the system are highly random and vulnerable toexternal environmental factors.”[49, page 8614] Extended Kalman filter “Model bias is the inherent inadequacy of the model for representing the real physicalsystems due to the model assumptions and simplifications.”[50, page 5,8] Adaptive square-rootsigma-point Kalman filter “ w k refers to process noise, which represents unknown disturbances that affect thestate of the system”; “Usually, covariance matrices are constant parameters determinedoffline before the estimation process begins. In practice, the characteristics of noisesvary depending on the choice of sensors and the operating conditions.” two categories: time-proportional errors and SOC-proportionalerrors. • Time proportional errors increase indefinitely.
We demonstratethat the standard deviation of the time-proportional error ap-proaches to infinity as the number of samples reaches to infinity. • State of charge proportional errors reach worst case within onecycle.
It is shown that the errors due to battery capacity uncer-tainty and timing oscillator drifts reach their peak values withinone discharge/charge cycle. In addition, the standard deviationof these errors vary with the accumulated SOC. The proposedexact model can be used to improve the SOC estimation byincorporating them in state space models, e.g., the proposedmodel can be used to improve the extended Kalman filter basedSOC estimation techniques [5]. • Accurate state-space models for real-time state of charge estima-tion.
The models were presented in a way that their applicabilityin state-space models is explicit. The proposed models can beused to improve the accuracy of virtually all online filtering ap-proaches, i.e., those based on extended Kalman filter, unscentedKalman filter, particle filter etc., that have been employed forreal-time SOC estimation.The effect of initial SOC error will remain as a bias in the Coulombcounting process, and as such it does not require any further analysisin this paper. Some initial versions of the derivations presented in thispaper were reported in [1]; the present papers expands all derivationspresented [1] towards a generalized state-space model.It must be noted that all the contributions listed above will translateinto an accurate process noise model in the state-space model forrecursive SOC tracking. It will be shown later in this paper thatthe process noise variance is a significantly time-varying quantity —something never considered in the literature before. Further, eventhough Coulomb counting is considered an outdated approach toSOC estimation, it is still widely used in practical implementations[52], [18], [19]. For example, whenever the fusion based approachesencounter failures, due to unexpected measurements and errors etc., the battery management systems are usually programmed to fall backto the Coulomb counting method as an alternative. Hence, the paperis written in a way that quantifies the error in computed SOC fromCoulomb counting. Later, we discuss how the findings in this paperwill be used to derive an accurate model for voltage-current fusionbased SOC tracking using recursive filters.
C. Organization of the Paper
The remainder of this paper is organized as follows: SectionII formally introduces Coulomb counting and identifies the fourdifferent error sources. The accumulated error in SOC due to currentmeasurement error, current integration approximation, battery capac-ity uncertainty, and timing oscillator drift are derived and analyzedin Sections III-A, III-B, III-C and III-E, respectively. A summaryof individual uncertainties and their effect on the counted Coulombsis presented in in Section IV. In Section V, some practical waysare discussed into how individual effects can be combined into theprocess model of a recursive filter implementation for SOC tracking.Finally, the paper is concluded in Section VII.L
IST OF A CRONYMS
CRLB . . . . . . . Cramer-Rao lower bound
ECM . . . . . . . . Equivalent circuit model
EKF . . . . . . . . Extended Kalman filter
OCV . . . . . . . . Open circuit voltage
PCRLB . . . . . Posterior Cramer-Rao lower bound
RLS . . . . . . . . . Recursive least squares
SOC . . . . . . . . State of chargeL
IST OF N OTATIONS
List of notations used in the remainder of this paper are summa-rized below. C true . . . . . . . . True battery capacity (see (48)) C batt . . . . . . . . Assumed battery capacity (5) C ∆ . . . . . . . . . . Battery capacity uncertainty (48) δ I ( k ) . . . . . . . . Current integration error at time k (29) ∆ k . . . . . . . . . . Sampling duration at time k (8) ∆ . . . . . . . . . . . Sampling time that is assumed constant (13) ∆ true . . . . . . . . True sampling time (75) ∆ ǫ . . . . . . . . . . Timing oscillator error (75) η . . . . . . . . . . . . Coulomb counting efficiency (5) η c . . . . . . . . . . . Charging efficiency (6) η d . . . . . . . . . . . Discharging efficiency (6) i ( t ) . . . . . . . . . . Current through battery at time t (5) i ( k ) . . . . . . . . . Sampled current through battery at time instant k (7) n i ( k ) . . . . . . . . Current measurement noise (10) n s ( k ) . . . . . . . . Process noise (3) n z ( k ) . . . . . . . . Measurement noise (4) κ . . . . . . . . . . . . Integration error constant (35) ρ i . . . . . . . . . . . Current measurement noise coefficient (27) ρ I . . . . . . . . . . . Current integration noise coefficient (45) ρ C . . . . . . . . . . Capacity uncertainty coefficient (64) ρ η c . . . . . . . . . . Charging uncertainty coefficient (72) ρ η d . . . . . . . . . . Discharging uncertainty coefficient (72) ρ ∆ . . . . . . . . . . Timing error coefficient (76) s ( t ) . . . . . . . . . SOC at time t (5) s (0) . . . . . . . . . Initial SOC (5) s ( k ) . . . . . . . . . SOC at discretized time instance k (7) s CC ( n ) . . . . . . Change in SOC over n samples (15) σ i . . . . . . . . . . . Std. deviation of current measurement error (11) σ L . . . . . . . . . . Std. deviation of load current changes (32) σ batt . . . . . . . . Std. deviation of battery capacity uncertainty(49) σ η c . . . . . . . . . . Std. deviation of charging uncertainty (74) σ η d . . . . . . . . . . Std. deviation of discharging uncertainty (74) σ ∆ . . . . . . . . . . Std. deviation of timing uncertainty (93) σ s , i ( n ) . . . . . . . Std. deviation of w i ( n ) (24) σ s , I ( n ) . . . . . . Std. deviation of w I ( n ) (43) σ s , C ( n ) . . . . . . Std. deviation of w C ( n ) (62) σ s ,η ( n ) . . . . . . Std. deviation of w η ( n ) (74) σ s , ∆ ( n ) . . . . . . Std. deviation of w ∆ ( n ) (83) σ s ( n ) . . . . . . . . Std. deviation of w ( n ) (93) w i ( n ) . . . . . . . . SOC error due to current measurement error (15) w I ( n ) . . . . . . . SOC error due to current integration error (39) w C ( n ) . . . . . . . SOC error due to battery capacity uncertainty(55) w η ( n ) . . . . . . . SOC error due to the uncertainty in c/d efficiency(70) w ∆ ( n ) . . . . . . . SOC error due to timing oscillator uncertainty(78) w ( n ) . . . . . . . . SOC error due combined uncertainties (91) z i ( k ) . . . . . . . . Measured current at time k (10) z v ( k ) . . . . . . . . Measured voltage at time k (95)II. P ROBLEM D EFINITION
The traditional Coulomb counting equation to compute the stateof charge (SOC) of a battery at time t is given below [10] s ( t ) = s (0) + η batt Z t i ( t ) dt (5)where η is the Coulomb counting efficiency defined as η = (cid:26) η c i ( t ) > (charging efficiency) η d i ( t ) < (discharging efficiency), (6) the unit of time t is in seconds, s ( t ) denotes the SOC at time t , s (0) denotes the initial SOC at time t = 0 , i ( t ) is the current in Amperes(A) through the battery at time t , and C batt is the battery capacity inAmpere hours (Ah). There are different approaches to compute theinitial SOC s (0) ; the error/uncertainty involved in computing s (0) will remain the same for any value of t . In this paper, we do notdelve into the error associated with computing s (0) and assume that s (0) is perfectly known.The Coulomb counting equation (5) is written in continuous-time domain. Considering that i ( t ) is not mathematically defined,a discretized Coulomb counting form needs to be adopted in order toperform the integration of (5). Widely adopted version of the discrete-time, recursive Coulomb counting equation is given below: s ( k ) = s ( k −
1) + η batt Z t ( k ) t ( k − i ( τ ) dτ (7)where s ( k ) is the SOC of the battery at time t ( k ) , and i ( τ ) is themeasured current at time τ . By approximating the integration in (7)using a rectangular (backward difference) method as Z t ( k ) t ( k − i ( τ ) dτ ≈ ∆ k i ( t ( k )) = ∆ k i ( k ) (8)where ∆ k = t ( k ) − t ( k − is the sampling duration betweentwo adjacent samples. Now, the widely known form of the Coulombcounting equation can be written as follows [10], [11] s ( k ) = s ( k −
1) + η ∆ k i ( k )3600C batt (9)The Coulomb counting equation (9) is only approximate due tothe following sources of errors:1) Measurement error in the current i ( k )
2) Error due to the approximation of the integration in (8)3) Uncertainty in the knowledge of battery capacity C batt
4) Uncertainty in the knowledge of the Coulomb counting effi-ciency η
5) The error in the measure of sampling time ∆ In the next four sections of this paper, we mathematically quantifythe effect of the above four sources of error in the computed SOC s ( k ) in (9). In each section, simulation examples are employed toverify the mathematically derived error quantities.III. I NDIVIDUAL U NCERTAINTY A NALYSIS
A. Effect of Current Measurement Error
The current through the battery is measured using a current sensorthat is prone to errors. The measured current z i ( k ) can be modeledas follows z i ( k ) = i ( k ) + n i ( k ) (10)where i ( k ) is the true current though the battery and n i ( k ) is themeasurement error in the current that can be assumed to be zero-mean with standard deviation σ i , i.e., E { n i ( k ) } = 0 E { n i ( k ) } = σ (11)Let us substitute the measured current (10) in (9) and re-write theCoulomb counting equation that considers the current measurementerror as follows: s ( k + 1) = s ( k ) + η ∆ k z i ( k )3600C batt = s ( k ) + η ∆ k i ( k )3600C batt + η ∆ k n i ( k )3600C batt | {z } SOC error (12)
Now, assuming that the sampling time is perfectly known and fixedas ∆ , ∆ k (13)the SOC at time step k = 1 , , . . . can be written as, s (0) = initial SOC estimation s (1) = s (0) + η ∆ i (1)3600C batt + η ∆ n i (1)3600C batt s (2) = s (1) + η ∆ i (2)3600C batt + η ∆ n i (2)3600C batt = s (0) + η ∆[ i (1) + i (2)]3600C batt + η ∆[ n i (1) + n i (2)]3600C batt (14)Considering n consecutive samples, the SOC at time k = n can beshown to be s ( n ) = s (0) + η ∆3600C batt n X k =1 i ( k ) | {z } s CC ( n ) + η ∆3600C batt n X k =1 n i ( k ) | {z } w i ( n ) = s (0) + s CC ( n ) + w i ( n ) (15) where, s CC ( n ) indicates the change in SOC from time k = 0 until k = n and w i ( n ) is the error in the computed SOC at time k = n .It can be noticed that the change in SOC can be decomposed intocharging Coulombs and discharging SOC as follows s CC ( n ) = s CCc ( n ) + s CCd ( n ) (16)where s CCc ( n ) = η c ∆3600C batt n X k =1 i ( k ) × [ i ( k ) > (17) s CCd ( n ) = η d ∆3600C batt n X k =1 i ( k ) × [ i ( k ) < (18)where the logical quantity [ i ( k ) > is defined as [ i ( k ) >
0] = (cid:26) i ( k ) > i ( k ) < (19)and the logical quantity [ i ( k ) > is defined as [ i ( k ) <
0] = (cid:26) i ( k ) < i ( k ) > (20)Similarly, the error in the computed computed SOC can be splitinto two terms corresponding to charging and discharging, i.e., w i ( n ) = w ic ( n ) + w id ( n ) (21)where w ic ( n ) = η c ∆3600C batt n X k =1 n i ( k ) × [ i ( k ) > (22) w id ( n ) = η d ∆3600C batt n X k =1 n i ( k ) × [ i ( k ) < (23)It must be noted that the current measurement noise n i ( k ) ∼N (0 , σ ) has the same characteristics during charging and discharg-ing.Now, it can be verified that the SOC error w i ( n ) has the followingproperties E { w i ( n ) } = 0 E { w i ( n ) } = σ s , i ( n ) = ∆ σ C ( η c n c + η d n d ) (24) where n c is the number of current charging current samples and n d is the number of current discharging current samples that satisfy n c + n d = n (25)It can be noted that as n → ∞ , the noise variance of the computedSOC error also approaches infinity. Let us write the SOC noise dueto current measurement error in a simplified form as follows: σ s , i ( n ) = (cid:18) ∆ ρ i (cid:19) √ η c n c + η d n d (26)where the ratio between the measurement noise standard deviationand battery capacity (in Ah), denoted in this paper as the currentmeasurement noise coefficient (which has a unit of hour − ), isdefined as ρ i = σ i C batt (27)It must be noted that since the SOC s ( n ) is defined within [0 , .However, SOC is usually displayed in percentage. As such, thestandard deviation of the SOC error in (26) is given in percentage asfollows: σ s , i ( n ) in % = (cid:18) ∆ ρ i (cid:19) √ η c n c + η d n d % (28)Table II shows the standard deviation (s.d.) in the SOC error dueto current measurement error for different sampling intervals overdifferent durations of time under the above assumptions. Here it isassumed that the battery capacity is C batt = 1 . and the currentmeasurement error standard deviation is σ i = 10 mA . TABLE II: SOC error s.d. (%) due to current measurement error ∆ = 0 . ∆ = 1 s ∆ = 10 s It must be noted that the SOC error shown in Table II is com-puted assuming zero uncertainties in all the other sources of error(integration, capacity, timing oscillator) and the initial SOC s (0) .The variance of the SOC error (28) due to current measurementerror keeps increasing with time. As such, we denote this as a time-cumulative error. For time-cumulative errors, the standard deviationof the error keeps increasing with time – if it is not reset, it willcompletely corrupt the estimated SOC. A possible approach to reducetime-cumulaitive error is by resetting the Coulomb count to s ( k ) = s (0) once in a while. Considering that the reset value of SOC alsocomes with errors (that is not considered in this paper) it is importantto select an instant where the uncertainty in the reset SOC will besmaller than the uncertainty derived in (28). B. Effect of Approximating Current Integration
The Coulomb counting approach summarized in the previoussection approximates the integration of current over time using asimple first-order (rectangular) approximation (see (8)). A genericrectangular approximation to integration is illustrated in Figure 1. Forsuch rectangular approximation, the integration error δ I ( k ) is definedas the difference between the true integral and the approximation, i.e., Z τ ( k ) τ ( k − i ( τ ) dτ | {z } true integration = ∆ i ( k ) | {z } approximation + δ I ( k ) | {z } integration error (29) Fig. 1:
A generic illustration of the current integration error.
The integration error δ I ( k ) is shown in shade. It can be noticed thatthe integration error can be both positive and negative – the darkshade indicates positive error and the light shade indicates negativeerror. Based on this observation, the integration error is assumed tobe zero-mean.The nature of the integration error δ I ( k ) is of specific interest. Itcan be observed that, for rectangular approximation, the integrationerror is proportional to the sampling duration ∆ [53], i.e., δ I ( k ) ∝ ∆ (30)Further, the integration error is proportional to the difference in theadjacent samples of measured current, i.e., δ I ( k ) ∝ [ i ( k ) − i ( k − (31)Since, [ i ( k ) − i ( k − in (31) is a time varying quantity, we canapproximately write δ I ( k ) ∝ σ L (32)where σ L is the standard deviation of the load (or charging) current(e.g., if the current is constant then σ L = 0 and so is the integrationerror). In addition, the sign of the integration error is both positiveand negative when there is variance in the magnitude of the current i ( k ) – see Figure 1 for an illustration of this. Using this observation,we can write E { δ I ( k ) } ≈ (33)That is, considering a large number of samples, we can assume theerror due to the rectangular approximation of current-integration tobe zero-mean.Based on the discussion so far, the integration error has thefollowing (approximate) properties. E { δ I ( k ) } = 0 E { δ I ( k ) } = σ (34)where σ is the variance of the current integration error. From (30)and (32), we can write σ I ∝ ∆ σ L = κ ∆ σ L (35)where κ is a constant, ∆ is the sampling time, and σ L is the standarddeviation of the load current. Figure 2 shows two different load current profiles from practicalapplications. It supports the assumption made in (31) that the currentdifference [ i ( k ) − i ( k − indeed is zero mean. Time (min) -1-0.500.51 C u rr en t ( A ) -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 Current (A) F r equen cy (a) Smart Phone [19], σ L = 0 . , C batt = 1 . Time (k) -2002040 C u rr en t ( A ) -30 -20 -10 0 10 20 30 40 50 Current (A) H i s t og r a m (b) Electric Vehicle [11], σ L = 8 . , C batt ≈ Fig. 2:
Current difference in realistic loads.
In both (a) and (b), thetop plot shows the current difference [ i ( k ) − i ( k − in typical loadprofile in Amperes and the plot at the bottom shows the magnitudeof the current difference as a histogram.By following the same approach of Section III-A, we can writethe computed SOC in recursive form as s ( k + 1) = s ( k ) + η (∆ i ( k ) + δ I ( k ))3600C batt = s ( k ) + η ∆ i ( k )3600C batt + ηδ I ( k )3600C batt | {z } Integ . Error (36)where the integration error is incorporated based on (29).Now, let us write the SOC at time step k = 0 , , . . . as s (0) = initial SOC estimation s (1) = s (0) + η ∆ i (1)3600C batt + ηδ I (1)3600C batt (37) s (2) = s (1) + η ∆ i (2)3600C batt + ηδ I (2)3600C batt = s (0) + η ∆[ i (1) + i (2)]3600C batt + η ( δ I (1) + δ I (2))3600C batt (38)Considering n consecutive samples, the computed SOC at time k = n , can be shown to be s ( n ) = s (0) + η ∆3600C batt n X k =1 i ( k ) | {z } s CC ( n ) + η batt n X k =1 δ I ( k ) | {z } w I ( n ) = s (0) + s CC ( n ) + w I ( n ) (39) where w I ( n ) is SOC error due to the approximation of integration.Similar to (56)–(58), the SOC error w I ( n ) can be decomposed,corresponding to charging and discharging, as follows w I ( n ) = w Ic ( n ) + w Id ( n ) (40)where w Ic ( n ) = η c batt n X k =1 δ I ( k ) × [ i ( k ) > (41) w Id ( n ) = η d batt n X k =1 δ I ( k ) × [ i ( k ) < (42)It can be noted that the SOC error due to integration has thefollowing properties E { w I ( n ) } = 0 E { w I ( n ) } = σ s , I ( n ) = κ ∆ σ C ( η c n c + η d n d ) (43)The standard deviation of integration error is σ s , I ( n ) = κ ∆ ρ I √ η c n c + η d n d (44)where the integration error coefficient is defined as ρ I = σ L C batt (45)Considering that the SOC s ( n ) is defined within [0 , , the standarddeviation of the SOC in (43) ranges between σ s , I ( n ) ∈ [0 , . Usually,SOC is displayed in percentage. As such, the standard deviation ofthe SOC error in (44) can be displayed in percentage as follows σ s , I ( n ) / year (in %) = κ ∆ ρ I √ η c n c + η d n d % (46)Now, let us make some realistic assumptions in order to simplifythe above expression further. Based on the data shown in Figure 2,we have ρ I = (cid:26) . Smart Phone . Electric Vehicle (47)Table III and Table IV show the computed SOC error standarddeviation due to the current integration error for different samplingintervals over longer periods of time. These two tables are madebased on the values shown in (47) and by assuming κ = 1 .TABLE III: S.D. of SOC Error (%) - Smart Phone Data ∆ = 0 . ∆ = 1 s ∆ = 10 s TABLE IV: S.D. of SOC Error (%) - EV Data ∆ = 0 . ∆ = 1 s ∆ = 10 s C. Effect of the Uncertainty in Battery Capacity
Battery capacity is the amount of coulombs that can be charged to(or discharged from) the battery. The battery capacity fades over time[54] and the rate of capacity fade depends on calendar life as wellas environmental and usage patterns the battery has experienced overlong periods of time [55]. Thus, true value of the battery capacity C batt is not precisely known. Usually a measure of the batterycapacity, denoted C batt , is used to estimate the battery SOC. Such acapacity measure is not exact and it relates to the true battery capacityas follows C batt = C true + C ∆ (48)where C ∆ represents the uncertainty in the knowledge about the truebattery capacity C true . For instance, it was argued in [17] that thisuncertainty can be modeled as a zero-mean Gaussian distribution,i.e., C ∆ ∼ N (0 , σ ) (49)where σ batt is the standard deviation of the capacity estimation error.The first order Taylor series approximation of a function f ( x ) around a point x is given by f ( x ) = f ( x ) + ( x − x )∆ f ′ ( x ) (50)using the above Taylor series approximation and the relationship (48)the inverse capacity can be approximated as follows batt ≈ true − C ∆ C (51)With the above approximation to the inverse capacity, let us re-write the Coulomb counting equation as follows s ( k + 1) = s ( k ) + η ∆ i ( k )3600C batt = s ( k ) + (cid:18) η ∆ i ( k )3600 (cid:19) × (cid:18) true − C ∆ C (cid:19) = s ( k ) + η ∆ i ( k )3600C true − η ∆ i ( k )C ∆ (52)Now, SOC at time step k = 0 , , , . . . can be written as s (0) = initial SOC estimation s (1) = s (0) + η ∆ i (1)3600C true − η ∆ i (1)C ∆ (53) s (2) = s (1) + η ∆ i (2)3600C true − η ∆ i (2)C ∆ = s (0) + η ∆[ i (1) + i (2)]3600C true − C ∆ η ∆[ i (1) + i (2)]3600C (54)Considering n consecutive samples the computed SOC at time k = n ,can be shown to be s ( n ) = s (0) + η ∆3600C true n X k =1 i ( k ) | {z } s CC ( n ) − η ∆C ∆ n X k =1 i ( k ) | {z } w C ( n ) = s (0) + s CC ( n ) + w C ( n ) (55) where w C ( n ) is the SOC error due to the uncertainty in batterycapacity. Similar to (56)–(58), w C ( n ) above can be decomposedinto the following two terms: w C ( n ) = w Cc ( n ) + w Cd ( n ) (56)where w Cc ( n ) = η c ∆C ∆ n X k =1 i ( k ) × [ i ( k ) > (57) w Cd ( n ) = η d ∆C ∆ n X k =1 i ( k ) × [ i ( k ) < (58)(59)Now, w C ( n ) becomes w C ( n ) = (cid:18) C ∆ C true (cid:19) ( s CCc ( n ) + s CCd ( n ))= (cid:18) C ∆ C true (cid:19) s CC ( n ) (60)Now, we can write the following about the SOC error w C ( n ) dueto the uncertainty in the knowledge of the battery capacity E { w C ( n ) } = 0 (61) E { w C ( n ) } = σ s , C ( n ) = E { C } C s CC ( n ) (62) = σ C s CC ( n ) = ρ s CC ( n ) (63)where the dimensionless capacity uncertainty coefficient is definedas ρ C = σ batt C true (64) D. Effect of the Uncertainty in Charging Efficiency
Let us assume the uncertainty in charging efficiency as follows η c = η ct + η c∆ (65) η d = η dt + η d∆ (66)In summary, we may write η = η t + η ∆ (67)where η t = (cid:26) η ct if i ( k ) > η dt if i ( k ) < η ∆ = (cid:26) η c∆ if i ( k ) > η d∆ if i ( k ) < (68)Let us substitute the measured current (10) in (9) and re-write theCoulomb counting equation that considers the current measurementerror as follows: s ( k + 1) = s ( k ) + η t ∆ i ( k )3600C batt + η ∆ ∆ i ( k )3600C batt (69) Considering n consecutive samples, the SOC at time k = n can beshown to be s ( n ) = s (0) + η t ∆3600C batt n X k =1 i ( k ) | {z } s CC ( n ) + η ∆ ∆3600C batt n X k =1 i ( k ) | {z } w η ( n ) = s (0) + s CC ( n ) + w η ( n ) (70) where the SOC error w η ( n ) can be expressed as the following w η ( n ) = η c∆ η ct η ct ∆3600C batt n X k =1 i ( k ) × [ i ( k ) > ! + η d∆ η dt η dt ∆3600C batt n X k =1 i ( k ) × [ i ( k ) < ! = η c∆ η ct s CCc ( n ) + η d∆ η dt s CCd ( n )= ρ η c s CCc ( n ) + ρ η d s CCd ( n ) (71)where ρ η c = η c∆ η ct and ρ η d = η d∆ η dt (72)are defined as the charging uncertainty coefficient and the discharginguncertainty coefficient , respectively. Let us model these two coef-ficients as ρ η c ∼ N (0 , σ η c ) and ρ η d ∼ N (0 , σ η d ) . With thisassumption, it can be shown that the SOC error w η ( n ) has thefollowing properties E { w η ( n ) } = 0 (73) E { w η ( n ) } = σ s ,η ( n ) = σ η c s CCc ( n ) + σ η d s CCd ( n ) (74) E. Effect of the Uncertainty in Timing Oscillator
The timing oscillator Hence, for this approach we have ∆ = ∆ true + ∆ ǫ (75)where ∆ ǫ is the timing oscillator error which is not a randomparameter. The timing oscillator error ∆ ǫ acts like a bias — weconsider it to be a constant over long periods of time. Also, let usquantify the timing error coefficient as follows ρ ∆ = ∆ ǫ ∆ true (76)Let us assume that a timing oscillator is off by three minutes in onemonth (30 days); in this case the constant ρ ∆ will be ρ ∆ = 330 × ×
60 = 6 . × − ≈ × − (77)Using (75) in main Coulomb counting equation (9) we have s ( k + 1) = s ( k ) + η ∆3600C batt n X k =1 i ( k )= s ( k ) + η ∆ true batt n X k =1 i ( k ) + η ∆ ǫ batt n X k =1 i ( k )= s (0) + s CC ( n ) + w ∆ ( n ) (78) The SOC estimation error can be simplified as w ∆ ( n ) = η batt n X k =1 i ( k )∆ ǫ ! = ρ ∆ η batt n X k =1 i ( k )∆ true ! = ρ ∆ s CC ( n ) (79)Assuming that the initial SOC s (0) is zero, it can be said that ≤ s CC ( n ) ≤ (80)Hence, the SOC error varies between ≤ w ∆ ( n ) ≤ ρ ∆ (81) The SOC error w ∆ is a deterministic quantity for a given batteryprovided that ρ ∆ is known. However, a realistic assumption is thatthe knowledge of ρ ∆ is only probabilistic. Let us assume that ρ ∆ ∼N (0 , σ ) . Under this scenario, the SOC error w ∆ has the followingproperties E { w ∆ ( n ) } = 0 (82) E { w ∆ ( n ) } = σ s , ∆ ( n ) = σ s CC ( n ) (83)Considering that ρ ∆ is a very small number, see (77), the error inSOC due to timing oscillator error can be considered to be negligible.IV. S UMMARY OF I NDIVIDUAL E RRORS
In this paper, we present a critical look at Coulomb countingmethod that is employed to estimate the state of charge of a battery.The Coulomb counting approach computes the present SOC as s ( t ) = s (0) |{z} initial SOC + Z t i ( τ )3600C batt dτ | {z } change in SOC where i ( t ) is the instantaneous current through the battery and C batt is the battery capacity in Ampere hours. That is, the presentSOC is the summation of initial SOC and the change in SOCthat is computed through the above integration. The SOC can beapproximately computed in a recursive manner as follows s ( n ) = s (0) + ∆3600C batt n X k =1 i ( k )= s (0) |{z} initial SOC + s CC ( n ) | {z } change in SOC where s ( k ) denotes the SOC at time instance k , i ( k ) is the measuredcurrent at time instance k , and ∆ is the sampling time in seconds.That is, the SOC at time n is the summation of the initial SOC s (0) and the accumulated SOC s CC from time n = 0 until n .In this paper, we showed that the above (discrete) recursiveapproximation to computing SOC suffers from four sources of error:current measurement error, current integration error, battery capacityuncertainty and the timing oscillator error. Particularly, we computedthe exact amount of the resulting SOC uncertainty as a result of theabove four types of errors. Those results areA) Current measurement error:
Considering that the current mea-surement error is zero-mean with standard deviation σ i , thecomputed SOC at time n can be written as s ( n ) = s (0) + s CC ( n ) + w i ( n ) where s (0) is the initial SOC and s CC ( n ) is the accumulatedSOC from the start at n = 0 . The SOC error w i ( n ) is shownto be zero mean with standard deviation (see (24)) σ s , i ( n ) = (cid:18) ∆ ρ i (cid:19) √ n % (84)It must be noted that the variance of the Coulomb countingerror due to current measurement noise is accumulative withtime . As the time increases, i.e., n → ∞ , so does the standarddeviation of the SOC error.B) Current integration error:
Considering that the current integra-tion is approximated using a rectangular method, the resultingapproximation error is shown to be zero-mean with standard deviation σ I . As a result, the computed SOC at time n can bewritten as s ( n ) = s (0) + s CC ( n ) + w I ( n ) where the SOC error w I ( n ) is shown to be zero mean withstandard deviation σ s , I ( n ) = κ ∆ ρ I √ n % (85)Once again, it can be noticed that the variance of the Coulombcounting error due to current integration approximation erroris accumulative with time.C) Uncertainty in the knowledge of battery capacity:
Consideringthat the uncertainty in the knowledge of battery capacity iszero-mean with standard deviation σ , the SOC at time n isderived as s ( n ) = s (0) + s CC ( n ) + w C ( n ) where the SOC error w C ( n ) is shown to be zero mean withstandard deviation σ s , C ( n ) = ρ s CC ( n ) where ρ C is defined as the capacity uncertainty coefficient. Itmust be noted that the variance of the capacity uncertainty erroris not accumulative with time, rather, it is proportional to theaccumulated SOC s CC ( n ) ∈ [0 , . In other words the SOCerror due to uncertainty in the knowledge of battery capacity, w C ( n ) , alternates between zero and ρ C . However, depending on the value of ρ C (the ratio betweenthe s.d. of the uncertainty and the assumed battery Capacity C batt ) the error could be anywhere between zero and 100%.For example, let us assume that ρ C = 0 . and let us assumethat the computed SOC at time n is s ( n ) = 40% . Thestandard deviation of the uncertainty in the computed s ( n ) is . s ( n ) = 0 . ×
40 = 4% . That is, the true SOC can beanywhere between and with confidence. Thiscan be extended to different levels of confidence as follows:
Where true SOC is? Confidence −
68 % −
95 % − Charging efficiency error:
The charging and discharging effi-ciencies are denoted η c and η d , respectively. The uncertaintiesin charging and discharging efficiencies are denoted η c∆ and η d∆ , respectively. The SOC at time n is written as s ( n ) = s (0) + s CC ( n ) + w η ( n ) where w η ( n ) = ρ η c s CCc ( n ) + ρ η d s CCd ( n ) (86)where is the error in the computed SOC due to the uncertaintyin the charging/discharging efficienc. Similar to w C , w η ( n ) does not accumulate with time, rather it accumulates with theaccumulated Coulombs.E) Timing oscillator error:
Considering an error of ρ ∆ (ratio ofclocked time vs. true time) in the timing oscillator, the SOC attime n is derived as s ( n ) = s (0) + s CC ( n ) + w ∆ ( n ) where the SOC error w ∆ ( n ) is a deterministic value given by w ∆ ( n ) = ρ ∆ s CC ( n ) Similar to the error due to capacity uncertainty, w ∆ ( n ) is notaccumulative with time and it is proportional to the accumu-lated SOC. Further, it is shown that practical value of η is verysmall number. For example, a timing oscillator that is slower(or faster) by 3 minutes in a month has η = 69 × − . Hence,the contribution of timing oscillator error can be considered tobe negligible in the computed SOC.In summary, the resulting four types of error can be grouped intotwo categories: time-accumulative and
SOC-proportional . The SOCerrors due to current measurement error and integration approxima-tion fall under the category of time accumulative errors. The SOCerrors due to the uncertainty in battery capacity and timing oscillatorerror fall under the category of SOC-proportional errors. Next, webriefly discuss the nature of these errors and possible ways to mitigatethem.
Mitigating Time-Accumulative Errors
It must be stressed that the best way to mitigate Coulomb countingerrors is to employ a state-space filter, such as the Kalman filter, withcorrectly derived model parameters — as briefly discussed in SectionV. However, practical battery management systems are implementedthrough complex state diagrams [19] where at some stages Coulombcounting is the best way to compute the SOC. Some strategiesdiscussed below can be useful when the SOC is computed basedon Coulomb counting only.The following strategies can be looked at to reduce time-accumulative errors. • Over sampling.
It can be noted that both σ s , i ( n ) and σ s , I ( n ) , in(84) and (85), respectively, are proportional to ∆ √ n where ∆ and n are related by n = T ∆ (87)where T is the total time duration. Now, both σ s , i ( n ) and σ s , I ( n ) can be written as σ s , i ( n ) = ρ i √ ∆ T % (88) σ s , I ( n ) = κρ I √ ∆ T % (89)Now, one must realize that the integration error coefficient ρ I reduces with oversampling, i.e., as ∆ decreases so does ρ I . How-ever, the current measurement noise coefficient is unaffected bysampling time. The conclusion is that both σ s , i ( n ) and σ s , I ( n ) reduce with higher sampling rate — however, σ s , I ( n ) reducesat a higher rate compared to σ s , i ( n ) with oversampling. • Reinitialization.
Time-accumulative errors increase with time.Hence, the accumulation of error can be prevented by re-initializing the SOC intermittently. For example, the SOC canbe reset by OCV-lookup method [18], [19] where the measuredvoltage across the battery terminals is used on the OCV-SOCcharacterization curve in order to find the OCV — the OCVlookup can be done only when the battery is at rest.
Mitigating SOC-Proportional Errors
Here, the SOC error is shown to be a fraction of the accumulatedSOC over time. Intermittent re-initialization — within a singlecharge-discharge cycle — will help to minimize this error. However,in most practical cases, there may not be many opportunities (a restedbattery) for frequent reset within a single cycle. The knowledge of the uncertainty in battery capacity σ batt will be very useful in theSOC error management. For example, if it is known that σ batt issignificantly high, then the SOC can be computed solely based onthe voltage approach.Finally, it must be emphasized that the focus of this paper isexclusively about the Coulomb counting approach. As such, wedid not delve into other types of approaches that are shown to beuseful in improving the SOC estimates, such as the voltage/currentbased approaches through the use of nonlinear filters [5], [41]. Theresults reported in this paper, such as the standard deviation of theCoulomb counting error for various scenarios, will help to improvethe voltage/current based SOC estimations as well.V. C OMBINED E FFECT AND THE S TATE -S PACE M ODEL D ERIVATION
So far, the Coulomb counting uncertainty is computed only basedon individual sources of errors. In this section, we discuss how thecombined effect due to all sources of error can be approximated usinga naive combination approach. Exact derivation of the combinedeffect can be quite lengthy due to the non-linear relationships involved— this is left for a future work. Under the naive combinationapproach, the SOC at time n is written as s ( n ) = s (0) + s CC ( n ) + w ( n ) (90)where w ( n ) = w i ( n ) + w I ( n ) + w C ( n ) + w η ( n ) + w ∆ ( n ) (91)Under the above naive assumption, it can be shown that E { w ( n ) } = 0 (92) E { w ( n ) } = σ s ( n ) = ∆ σ C ( η c n c + η d n d )+ κ ∆ σ C ( η c n c + η d n d )+ ρ s CC ( n ) + σ η c s CCc ( n ) + σ η d s CCd ( n ) + σ s CC ( n ) (93)With the combined noise derived above, now we are ready to redefinethe state-apace model (3)-(4).Based on the detailed derived about the Coulomb counting error,the process model (3) can be written as s ( k ) = s ( k −
1) + ∆ z i ( k )3600C batt + n s ( k ) (94)where n s ( k ) is the process noise that has zero-mean and variancegiven by (93) when n is set to 1.Based on the notations introduced in [9], the measurement equationin (4) can be written in detail as follows z v ( k ) = V ◦ ( s ( k )) + a ( k ) T b + n z ( k ) (95)where V ◦ ( s ( k )) the open circuit voltage model, a ( k ) T b approxi-mates the voltage drop in the relaxation elements of the battery, b the parameter vector of the relaxation elements, and n z ( k ) is themeasurement noise. VI. N
UMERICAL A NALYSIS
A. Effect of Current Measurement Error
The objective in this section is to validate — using a Monte-Carlo simulation approach — the standard deviation of the SOCerror due to current measurement error that was derived in (28).For this experiment, errors from all the other possible sources ofuncertainties (current integration error, battery capacity uncertainty,timing oscillator error as well as initial SOC error) are assumed to bezero. In order to do this, a special current profile, shown in Figure 3,is created. For this profile, the amount of Coulombs can be perfectlycomputed using geometry. Once the Coulombs are computed, thetrue SOC can be computed by making use of the knowledge of thetrue battery capacity and other noise-free quantities. The followingprocedure details the Monte-Carlo experiment:a) Generate a perfectly integrable current profile, similar to theone shown in Figure 3. The generated current profile denotes i ( k ) in (10).– First 40 seconds of the true current profile generated for theexperiment is shown Figure 4.b) Compute the true SOC at time k , s true ( k ) , using the Geometricapproach illustrated Figure 3 for the entire duration of theprofile, i..e, for k = 1 , . . . , n where n denotes the numberof samples in the entire current profile.c) Set m = 1 , where m denotes the index of the Monte-Carlorun.d) Generate current measurement noise n i ( k ) as a zero-meanGaussian noise with standard deviation σ i = 10 mA . Usingthis, generate the measured current profile z i ( k ) = i ( k )+ n i ( k ) .– Figure 4 shows the true current profile i ( k ) along with themeasured current profile z i ( k ) for a duration of 40 seconds.e) Compute the (noisy) SOC, s m ( k ) using traditional Coulombcounting equation given in (69), i.e., s m ( k ) = s m ( k −
1) + ∆ k z i ( k )3600C batt where the subscript m denotes the m th Monte-Carlo run.– Figure 5 shows the true SOC s true ( k ) and the computed noisySOC s m ( k ) . The top plot (a) shows the SOC at the start of thecurrent profile and the plot (b) at the bottom shows the SOCtowards the end of applying 3.5 hours of load profile.f) If m = M , where M denotes the maximum number of Monte-Carlo runs, go to step g); otherwise, set m ← m + 1 and goto step d)g) End of simulation (all the data generated during the above stepsneeds to be stored for analysis).After M = 1000 Monte-Carlo runs, the standard deviation of theSOC error due to current measurement error is computed as ˆ σ s , i ( k ) = vuut M M X m =1 ( s true ( k ) − s m ( k )) (96)Figure 6 shows the standard deviation of the SOC error computedusing the theoretical formula (28) and the standard deviation of theSOC error computed using the Monte-Carlo method detailed in (96).As expected, the theoretical derivation matches with the SOC errorstandard deviation obtained through 1000 Monte-Carlo simulations. B. Effect of Current Integration Error
The objective in this section is to validate the standard deviationof the SOC error due to integration that we derived in (44). For this Fig. 3:
Generic illustration to computing the true amount ofCoulombs.
Computing true Coulombs is challenging. Here, we as-sume the true current to take the above pattern; under this assumption
Total Coulombs = A + A + A + A + A . Time (Sec) -1.5-1-0.50 C u rr en t ( A ) TrueMeasured
Fig. 4:
Current measurement error.
True vs. measured current thatwas simulated by assuming a current measurement error standarddeviation of σ i = 10 mA .experiment, errors from all the other possible sources of uncertainties(current measurement error, battery capacity error, timing oscillatorerror as well as initial SOC error) are assumed to be zero. In orderto do this, similar to previous analysis, a special current profileshown in Figure 7 is made up of constant current signals of differentamplitudes. For this profile, the amount of Coulombs can be perfectlycomputed using geometry similar to the example illustrated in Figure3. Once the Coulombs are computed, the true SOC can be computedby making use of the knowledge of the true battery capacity. Thefollowing procedure details the Monte-Carlo experiment to validatethe standard deviation of the SOC error due current integration error:a) Generate a perfectly integrable current where the generatedcurrent allows one to perfectly compute R k +1 k i ( k ) dk shownin (29).– First 18 seconds of the noiseless current profile i ( k ) is shownin red Figure 7. Note that the true current profile is thedownsampled version — this emulates the fact that discretelymeasured current is always a downsampled version and it willnever be the same as the real current (shown in blue). Firstfour minutes of the current profile along with the true SOC(assuming initial SOC =1) is shown in Figure 8.b) Let the true battery capacity to be C true = 1 . .c) Assuming the knowledge of the true capacity, compute the Time (Sec) S O C ( % ) TrueCoulomb Count (a) Start of load profile
Time (Hr) S O C ( % ) TrueCoulomb Count (b) End of the load profile in 3.5 hours
Fig. 5:
Effect of current measurement error in SOC. (a) At thestart of the experiment, the true SOC and the computed SOC throughCoulomb counting are nearly identical. (b) Within 3.5 hours, thetrue SOC and the computed SOC are slightly different.
SimulationParameters: current measurement error s.d. σ i = 10 mA andsampling time ∆ = 200 ms . Time (Hr) S O C E rr o r S . D . ( % ) -3 SimulationFormula
Fig. 6:
Standard deviation of the SOC error due to current mea-surement error.
Simulated value is plotted in comparison with thetheoretical value derived in (28) shown against time that correspondsto n . true SOC at time k , s true ( k ) , using the geometric approachillustrated Figure 3 for the entire duration of the profile, i..e,for k = 1 , . . . , n where n denotes the number of samples inthe entire current profile.– The second plot in Figure 8 shows the true SOC.d) Set m = 1 where m denotes the index of the Monte-Carlo run.e) Compute the (noisy) SOC s m ( k ) using traditional Coulombcounting equation given in (69), i.e., s m ( k ) = s m ( k −
1) + ∆ k i ( k )3600C batt where i ( k ) are the ‘measured current’ indicated by red linesin Figure 7, and the subscript m denotes the m th Monte-Carlorun.f) If m = M , where M denotes the maximum number of Monte-Carlo runs, go to step g); otherwise, set m ← m + 1 and goto step e)g) End of simulation (all the data generated during the above stepsneeds to be stored for analysis).After M = 1000 Monte-Carlo runs, the standard deviation of theSOC error due to current measurement error is computed as ˆ σ s , I ( k ) = vuut M M X m =1 ( s true ( k ) − s m ( k )) (97)Figure 9 shows the standard deviations of error computed throughthe theoretical approach, σ s , I ( n ) in (46), and through the Monte-Carlo simulation approach, ˆ σ s , I ( k ) (97). The constant κ for thetheoretical approach in (46) is found to be κ = 0 . throughempirical means (i.e., different values for κ was used until thetheoretical curve in red aligned well with the simulation curve inblue). It must be noted that κ will be different for different currentprofiles. Time (Sec) -0.9-0.8-0.7-0.6-0.5-0.4-0.3-0.2-0.10 C u rr en t ( A ) True CurretMeasured Current
Fig. 7:
Perfectly integrable current profile.
The blue curve showsa perfectly integrable current that is made of rectangular pulsesof different amplitude; it can be integrated using the geometricapproach detailed in Figure 3. The measured current, shown inred, is a downsampled version of the true current profile – thisemulates the way in which discrete measurement systems measurethe voltage/current in BMS.
C. Effect of Battery Capacity Uncertainty
The objective in this section is to validate the standard deviationof the SOC error due to battery capacity uncertainty that we derivedin (62) using Monte-Carlo simulation approach. For this experiment, Time (Min.) -1.5-1-0.50 C u rr en t ( A ) Time (Min.) S O C ( % ) Fig. 8:
Current profile and corresponding SOC.
First four minutesof the true current profile and the corresponding true SOC that iscomputed using the geometric approach detailed in Figure 3. Only4 minutes of profiles are shown; true profile lasted for 4 hours (seeFigure 9).
Time (Hr) S O C E rr o r S . D . ( % ) SimulationFormula
Fig. 9:
The Standard deviation of SOC error due to currentintegration error.
The red curve is the theoretical value of the s.d. σ s , I ( k ) derived in (46); the blue curve shows ˆ σ s , I ( k ) , the s.d. obtainedthrough Monte-Carlo simulation as shown in (97). The constant κ iscomputed through empirical methods to be κ = . . It must be notedthat κ varies for different types of current profiles.errors from all the other possible sources of uncertainties (currentmeasurement error, current integration error, timing oscillator error aswell as initial SOC error) are assumed to be zero. In order to do this,similar to previous analysis, a special current profile that is shown inFigure 10 is created. The current profile in Figure 10 is made of lowfrequency (constant current) signals of different amplitudes. For thisprofile, the amount of Coulombs can be perfectly computed usinggeometry similar to the example illustrated in Figure 3. Once theCoulombs are computed, the true SOC can be computed by makinguse of the knowledge of the true battery capacity. The followingprocedure is followed to perform the Monte-Carlo experiment tovalidate the standard deviation of the SOC error due to uncertaintyin battery capacity:a) Generate a perfectly integrable current where the generatedcurrent profile denotes i ( k ) in (10).– The entire true current profile generated for the experiment isshown at the top plot Figure 10.b) Let the true battery capacity to be C true = 1 . . c) Assuming the knowledge of the true capacity, compute thetrue SOC at time k , s true ( k ) , using the geometric approachillustrated Figure 3 for the entire duration of the profile, i..e,for k = 1 , . . . , n where n denotes the number of samples inthe entire current profile.– The second plot in Figure 10 shows the accumulated Coulombs s CC ( n ) . From this, the true SOC can be computed as s true ( n ) = s (0) + s CC ( n ) . d) Set m = 1 where m denotes the index of the Monte-Carlo run.e) Assuming capacity estimation error s.d. of σ batt = 0 . usethe capacity uncertainty model of (48) to compute the estimatebattery capacity C batt = C true + C ∆ where is a zero-meanrandom number with standard deviation σ batt . – Figure 11 shows all the C batt values generated for m =1 , . . . , M in the form of a histogram.f) Compute the (noisy) SOC s m ( k ) using traditional Coulombcounting equation given in (69), i.e., s m ( k ) = s m ( k −
1) + ∆ k i ( k )3600C batt where the subscript m denotes the m th Monte-Carlo run.– Figure 12 shows the true SOC s true ( k ) and the computed noisySOC s m ( k ) for different Monte-Carlo runs.g) If m = M , where M denotes the maximum number of Monte-Carlo runs, go to step h); otherwise, set m ← m + 1 and goto step e)h) End of simulation (all the data generated during the above stepsneeds to be stored for analysis).After M = 1000 Monte-Carlo runs, the standard deviation of theSOC error due to current measurement error is computed as ˆ σ s , C ( k ) = vuut M M X m =1 ( s true ( k ) − s m ( k )) (98)Figure 13 shows the SOC error standard deviation obtained throughthe theoretical equation (62) as well as the Monte-Carlo simulationapproach summarized through (98). It can be noticed that the theo-retical value and the simulated values slightly differ — this can beattributed to the approximation made in (51) in order to derive thetheoretical value. Time (Hour) -0.4-0.200.20.4 C u rr en t ( A ) Time (Hour) -1.5-1-0.50 C A cc u m u l a t ed ou l o m b s ( A h ) Fig. 10:
Simulated current profile and corresponding true SOC.
This figure is showing the difference between the true SOC and theSOC with battery capacity uncertainty after 100 runs of Monte Carlo. Assumed capacity, C batt (Ah) P M F Battery Capacity Uncertainty (C true = 1.5 Ah)
Fig. 11:
The histogram of C batt generated during 1000 Monte Carlosimulations. This graph is showing that the battery capacity error thatwe are using in our Monte Carlo runs is reasonable.Fig. 12: SOC error due to battery capacity uncertainty.
This figureis showing the difference between the true SOC and the SOC withbattery capacity uncertainty for different simulation. The true SOC iscomputed using the true battery capacity of C true = 1 . ; EachMonte Carlo run assumes a different battery C batt that is distributed N (C true , σ ) . Figure 11 all the C batt during different runs.VII. C ONCLUSIONS AND D ISCUSSIONS
In this paper, we developed an in-depth mathematical analysis ofCoulomb counting method for state of charge estimation in recharge-able batteries. Particularly, we derived the exact statistical values ofthe state of charge error as a result of (i) current measurement error,(ii) current integration error, (iii) battery capacity uncertainty, and(iv) timing oscillator error. It was shown that the state of chargeerror due to current measurement error and current integration errorgrow with time whereas the state of charge error due to batterycapacity uncertainty and timing oscillator error are proportional tothe accumulated state of charge that ranges between 0 and 1. Themodels presented in this paper will be useful to improve the overallstate of charge estimation in majority of the existing approaches.A
CKNOWLEDGMENTS
B. Balasingam acknowledges the support of the Natural Sciencesand Engineering Research Council of Canada (NSERC) for financialsupport under the Discovery Grants (DG) program [funding referencenumber RGPIN-2018-04557]. B. Balasingam acknowledges the help
Time (Hour) S t d . de v i a i t on o f S O C e rr o r ( % ) SimulationTheory
Fig. 13:
The Standard deviation of SOC error due to batterycapacity uncertainty.
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