CA2588856C - Method and system for battery state and parameter estimation - Google Patents

Abstract

Methods and systems for estimating values descriptive of a battery's present operating condition, comprising: estimating state-of-charge in a battery where said state-of-charge is comprised one of the internal states; and estimating state-of-health in a battery where said state-of-health is comprised one of the internal parameters. In particular, methods for estimating state-of-charge in a battery, comprising: making an internal states prediction of said battery where said state-of-charge is one of said internal states; making an uncertainty prediction of said internal states prediction; correcting said internal states prediction and said uncertainty prediction; and applying an algorithm that iterates said making an internal states prediction, said making an uncertainty prediction and said correcting to yield an ongoing estimation to said state-of-charge and an ongoing uncertainty to said state-of-charge estimation. And methods and systems for estimating present parameters of an electrochemical cell system comprising: making an internal parameter prediction of the cell; making an uncertainty prediction of the internal parameter prediction; correcting the internal parameter prediction and the uncertainty prediction; and applying an algorithm that iterates the internal parameter prediction, and the uncertainty prediction and the correction to yield an ongoing estimation to the parameters and an ongoing uncertainty to the parameters estimation.

Description

METHOD AND SYSTEM FOR BATTERY STATE AND PARAMETER ESTIMATION
Technical Field The present invention relates to methods and apparatus for estimation of battery pack system states and parameters using digital filtering techniques. In particular, Kalman filtering and extended Kalman filtering. Battery management systems in battery packs must estimate values descriptive of the pack's present operating condition, which include battery state-of-charge (SOC) and power-fade, capacity-fade, and instantaneous available power. The power-fade and capacity-fade are often lumped under the description state-of-health (SOH). The present invention provides an advanced methods and apparatus for estimating values descriptive of the pack's present operating condition including battery SOC and SOH.
Background Art Batteries are used in a wide variety of electronic and electrical devices. In each application, it is often useful and necessary to measure how much charge is left in the battery. Such a measurement is called the state-of-charge (SOC) . It is useful, for example, for a cell phone user to know how much longer he can talk on his phone. On the other hand, recharging devices need to know how much charge is in a battery to prevent overcharging. Many types of battery are sensitive to overcharging as well as undercharging.
Overcharging and undercharging can erode the effectiveness of batteries and even damage them.
Currently there are many techniques that measure the remaining charge of a battery. Each of these SOC

determination techniques has drawbacks. Some such as Ampere-hour counting are sensitive to measurement errors. Others such as Coup de fouet work for only One type of battery.
Still other techniques such as Impedance Spectroscopy are constrained by battery conditions such as rapidly changing temperature. Also many do not give an uncertainty range in their estimation of the SOC. In applications such as HEV and EV batteries, the uncertainty range associated with the SOC
measurement is very critical. Vehicles can lose power on the road and cause danger if the uncertainty range is unknown and the battery is erroneously undercharged. Knowing the uncertainty range can prevent this. For example if the battery SOC is determined to be within 10% of the minimum charge threshold and the uncertainty range is known to be 15%, the system will know to charge the battery because the uncertainty range is greater than the distance to the threshold.
Existing Techniques Presented here is an overview of the existing techniques and some of their shortcomings. One technique called the discharge test is an accurate form of testing. It involves completely discharging the battery to determine the SOC under controlled conditions. However, the complete discharge requirement renders this test impractical for real-life application. It is too time consuming to be useful and interrupts system function while the test is being performed.
Another SOC determination technique is called Ampere-hour counting. This is the most common technique for determining the SOC because of its ease of implementation. It measures the current of the battery and uses the measurement

2 to determine what the SOC is. Ampere-hour counting uses the following:

SOC = SOC0 + 0 (jbwt - Ross) dI
C'v (1) where CN is the rated capacity of the battery, Ibatt is the battery current, and Iloss is the current consumed by the loss reactions. The equation determines the SOC based on an initial SOCo starting point. Ampere-hour counting is essentially an "open loop" method that is easily confused.
Measurement error accumulates over time to degrade the accuracy of SOC determination. There are methods to improve current measurement but they are expensive.
Electrolyte Measurement is another common technique. In lead-acid batteries, for example, the electrolyte takes part in reactions during charge and discharge. Thus, a linear relationship exists between the change in acid density and the SOC. Therefore measuring the electrolyte density can yield an estimation of the SOC. The density is measured directly or indirectly by ion-concentration, conductivity, refractive index, viscosity, etc. However, this technique is only feasible for vented lead-acid batteries. Furthermore it is susceptible to acid stratification in the battery, water loss and long term instability of the sensors.
An open-circuit voltage measurement may be performed to test the SOC of the battery. Although the relationship between the open circuit voltage and the SOC is non-linear, it may be determined via lab testing. Once the relationship is determined, the SOC can be determined by measuring the open circuit voltage. However the measurement and estimation

3 are accurate only when the battery is at a steady state, which can be achieved only after a long period of inactivity.
This makes the open-circuit voltage technique impractical for dynamic real time application.
Impedance Spectroscopy is another technique used to determine the SOC. Impedance spectroscopy has a wide variety of applications in determining the various characteristics of batteries. Impedance Spectroscopy exploits a relationship between battery model parameters derived from impedance spectroscopy measurements and the SOC. However the drawback of this technique is that impedance curves are strongly influenced by temperature effects. Thus its application is limited to applications where temperature is stable.
Internal resistance is a technique related to impedance spectroscopy. Internal resistance is calculated as the voltage drop divided by the current change during the same time interval. The time interval chosen is critical because any time interval longer than 10 ms will result in a more complex resistance measurement. Measurement of internal resistance is very sensitive to measurement accuracy. This requirement is especially difficult to achieve in Hybrid Electric Vehicle (HEV) and Electric Vehicle (EV) applications.
Some techniques use non-linear modeling to estimate SOC
directly from measurements. An example is artificial neural networks. Artificial neural networks operate on any system and predict the relationship between input and output. The networks have to be trained repeatedly so that it can improve its estimation. Because the accuracy of the data is based on the training program for the networks, it is difficult to

4 determine the error associated with the SOC prediction given by artificial neural networks.
There is another group of SOC estimation techniques called the interpretive techniques. Interpretive techniques do not give SOC directly. Instead they use electrical discharge and charge characteristics to determine the SOC. As such, the SOC must be inferred from the calculated values.
One of these techniques is called the Coup de fouet. Coup de fouet describes the short voltage drop region occurring at the beginning of discharge following a full charge of lead-acid battery. Using a special correlation between the voltage parameters occurring in this Coup de fouet region, the SOC
can be inferred. One limitation of the Coup de fouet technique is that it works for lead-acid batteries only.
Moreover it is effective only in cases where full charge is frequently reached during battery operations.
The Kalman Filter One SOC determination technique involves mathematically modeling the behavior of the battery and predicting the SOC
based on the model. One such model is the Kalman filter. It has mathematical basis in statistics, probabilities and system modeling. The main purpose of the Kalman filter is to predict recursively the internal states of a dynamic system using only the system's outputs. In many instances this is very useful because the internal states of the system are unknown or cannot be directly measured. As such, the Kalman filter can work on all types of batteries and addresses a limitation of many aforementioned techniques.
The Kalman filter has been widely used in fields such as aerospace and computer graphics because it has several

5 advantages over many other similar mathematical system models.
In particular, the Kalman filter takes into account both measurement uncertainty and estimation uncertainty when it updates its estimation in successive steps. The Kalman filter corrects both uncertainties based on new measurements received from sensors. This is very important for two reasons.
First, sensors often have a noise factor, or uncertainty, associated with its measurement. Over time, if uncorrected, the measurement uncertainty can accumulate. Second, in any modeling system the estimation itself has inherent uncertainty because the internal dynamic of the system may change over time. The estimation of one time step may be less accurate than the next because the system may have changed internally to behave less similarly to the model. The correction mechanism in the Kalman filter minimizes these uncertainties at each time step and prevents them from degrading accuracy over time.
FIG. 1. shows the basic operation of the Kalman filter.
There are two main components in the Kalman filter-predict component 101 and correct component 102. To start, a set of initial parameters are fed into predict component 101.
Predict component 101 predicts the internal states of the system at a particular point in time using a set of input parameters. Besides predicting the internal states, it also gives the uncertainty of its prediction. Thus, as shown in FIG. 1, the two outputs of predict component 101 are the predicted internal state vector (which encompasses the internal states) and its uncertainty.
The role of correct component 102 is to correct the predicted internal states and uncertainty it receives from

6 predict component 101. The correction is made by comparing the predicted internal states and predicted uncertainty with new measurements received from sensors. The result are the corrected internal states and corrected uncertainty, both of which are then fed back as parameters to predict component 101 for the next iteration. At the next iteration, the cycle repeats itself over again.
Mathematical Basis of Kalman Filter FIG. 1A and FIG. 1B show the equations used within both predict and correct components of the Kalman Filter. To understanding the origin of equations used, consider a dynamic process described by an n-th order difference equation of the form yk+1=ao,k~'~k+ . . +Llrx-1,kvk-rt+l+ ka 1f.=~D, (2) where uk is a zero-mean white random process noise. Under some basic conditions, this difference equation can be re-written as Yk+l ao,k al,k an-2,k an-1,k Yk 1 Yk 1 0 ... 0 0 Yk-1 0 J4+1 Yk-l = 0 1 ... 0 0 Yk-2 + 0 uk Yk-n+2 0 0 ... 1 0 Yk-n+1 0 A -~-yk B (3) in which "k+l represents a new state modeled by a linear combination of the previous state xk and input Uk. Note the notation of matrices A and B. This leads to the state-space model X k+1 Ak X k+Bkuuk (4)

7

8 PCT/KR2004/003101 Y t+ 0 ... 0] x (5) or the more general form X k-t-l=Ak X k+Rk1'1k (6) Y k-=Ck x k+Dkuk (7) which is the basis of many linear estimation models.
While equations (3) to (5) assume a system with a single input and a single output, the general form in equations (6), (7) and the following equations allow multiple inputs and outputs if B has multiple columns and C has multiple rows.
Building on equations (6) and (7), the Kalman filter is governed by the equations Xk=AXk-1+BUk-1+Wk-1 (8) ykCXk+DUk+Vk . (9) Equation (9) is in a more general form, though D is often assumed to be 0. The matrices A and B in equation (8) relate to matrices Ak and Bk in equation (6), respectively.
The matrices C and D in equation (9) relate to matrices Ck and Dk in equation (7). As equation (8) governs the estimation of the dynamic system process, it is called the process function. Similarly since equation (9) governs the estimation of the measurement uncertainty, it is called the measurement function. The added random variables Wk and vk in equation (8) and (9) represent the process noise and measurement noise, respectively. Their contribution to the estimation is represented by their covariance matrices EN, and EV in FIGS. 1A and lB.

Referring again to FIG. 1A (which shows equations in predict component 101), equation 151 is based on equation (8) and equation 152 is based on in part on equation (9).
Equation 151 takes closely after the form of equation (8) but the necessary steps to transform equation (9) into the form shown in equation 152 are not shown here. Equation 151 predicts the internal states of the system in the next time step, represented vector by using parameters from the current time step. The minus notation denotes that the vector is the result of the predict component. The plus notation denotes that the vector is the result of the correct component. Hence in equation 151, the result of the correct component in the current time step is used to predict the result for the next time step. Equation 152 predicts the.
uncertainty, which is also referred to as the error covariance. As such, the matrix EN, in equation 152 is the process noise covariance matrix.
FIG. 1B shows the equations within correct component 102.
These three equations are executed in sequence. First, equation 161 determines the Kalman Gain factor. The Kalman Gain factor is used to calibrate the correction in equations 162 and 163. The matrix C in equation 161 is from that of equation (9), which relates the state to the measurement yk.
In equation 162, the Kalman Gain factor is used to weight between actual measurement Yk and predicted measurement CXx(-).
As shown in equation 161, matrix Etõ the actual measurement-noise covariance, is inversely proportional to the Kalman Gain factor Lk. As Er, decreases, Lk increases and gives the actual measurement Yk more weight. However, if matrix Ee,k(+), the predicted uncertainty, decreases, Lk decreases and gives

9 more weight to predicted measurement CXkO. Thus Kalman Gain factor favors either the actual measurement or predicted measurement, depending on which type of measurement has a smaller uncertainty.
Using this method of weighing measurement, equation 162 computes the corrected internal. state vector Ilk(-) based on predicted internal state vector Xk(-) (from predict component 101), new measurement yk and predicted measurement C&(-) .
Finally in the last equation of correct component 102, equation 163 corrects the predicted uncertainty, or the state-error covariance. Matrix I in equation 163 represents the identity matrix. The output of equations 162 and 163 are fed to predict component 101 for the next iteration. More specifically, the calculated value U0-) in equation 162 is substituted into equation 151 for the next iteration and the calculated value Ee,k(+) in equation 163 is substituted into equation 152 for the next iteration. The Kalman filter thus iteratively predicts and corrects the internal states and its associated uncertainty. It must be noted that in practice, both A, B, C, D, EN, and EV might change in each time step.
Extended Kalman Filter Whereas the Kalman filter uses linear functions in its model, the Extended Kalman filter was developed to model system with non-linear functions. Aside from this distinction, the mathematical basis and operation for the Extended Kalman filter are essentially the same as the Kalman filter. The Extended Kalman filter uses an approximation model similar to the Taylor series to linearize the functions to obtain the estimation. The linearization is accomplished by taking the partial derivatives of the now non-linear process and measurement functions, the basis for the two equations in the predict component.
The Extended Kalman filter is governed by the following equations Xk+1 = f(Xk, Uk, Wk) (10) and Yk+1 = h(Xk, Uk, Vk) (11) where random variables wk and vk represent process noise and measurement noise, respectively. Non-linear function f in equation (10) relates the internal state vector Xk at the current step k to the internal state vector Xk+1 at the next time step k+1. Function f also includes as parameters both the driving function Uk and the process noise wk. The non-linear function h in equation (11) relates the internal state vector Xk and input Uk to the measurement Yk.
FIG. 2A and FIG. 2B show the equations of the Extended Kalman Filter. The sequence of operation remains the same as the Kalman filter. There are still two components-predict component 201 and correct component 202. The equations are slightly different. Specifically, matrices A and C now have a time step sub-script k meaning that they change at each time step. This change is needed because the functions are now non-linear. We can no longer assume that the matrices are constant as in the case of the Kalman filter. To approximate them, Jacobian matrices are computed by taking partial derivatives of functions f and h at each time step. The Jacobian matrices are listed below.
A is the Jacobian matrix computed by taking the partial derivative of f with respect to x, that is __8ful A[~ .ilk a, U1 xk -k (12) The notation means "with Xk evaluated as, or replaced by, Xk in final result."
C is the Jacobian matrix computed by taking the partial derivative of h with respect to x, that is d hf,l C[i,Jlk =
UX Ul xk -Xk (13) Aside from these additional steps of taking partial derivatives of functions, the operation of the Extended Kalman filter remains essentially the same as the Kalman filter.
Using Kalman Filter to Determine the SOC in Batteries Because it only has to measure the battery output, the Kalman filter has an advantage in that it works on all types of batteries system, including dynamic applications such as HEV and EV. There are existing applications that use the Kalman filter to determine SOC of batteries. However, none of them uses SOC as an internal state of the model. Thus the uncertainty associated with the SOC estimation cannot 'be determined. The defect is particularly important in HEV and EV batteries where the uncertainty range is needed to prevent undercharging of battery or loss of vehicle power. Also none of the existing methods uses the Extended Kalman filter to model battery SOC non-linearly.
It is important to note that as the Kalman filter is only a generic model. Each application of the Kalman filter still needs to use a good specific battery model and initial parameters that accurately describe the behavior of the battery to estimate the SOC. For example, to use the Kalman filter to measure the SOC as an internal state, the filter needs to have a specific equation describing how the SOC
transitions from one time step to the next. The determination of such an equation is not trivial.
Using Kalman Filter to Determine the SOH in Batteries Moreover, in the context of rechargeable battery pack technologies, it is desired in some applications to be able to estimate quantities that are descriptive of the present battery pack condition, but that may not be directly measured.
Some of these quantities may change rapidly, such as the pack state-of-charge (SOC), which can traverse its entire range within minutes. Others may change very slowly, such as cell capacity, which might change as little as 20% in a decade or more of regular use. The quantities that tend to change quickly comprise the "state" of the system, and the quantities that tend to change slowly comprise the time varying "parameters" of the system.
In the context of the battery systems, particularly those that need to operate for long periods of time, as aggressively as possible without harming the battery life, for example, in Hybrid Electric Vehicles (HEVs), Battery Electric Vehicles (BEVs), laptop computer batteries, portable tool battery packs, and the like, it is desired that information regarding slowly varying parameters (e.g., total capacity) be available to determine pack health, and to assist in other calculations, including that of state-of-charge (SOC).

There are a number of existing methods for estimating the state-of-health of a cell, which are generally concerned with estimating two quantities: power-fade, and capacity-fade (both slowly time varying). Power fade may be calculated if the present and initial pack electrical resistances are known, and capacity fade may be calculated if present and initial pack total capacities are known, for example, although other methods may also be used. Power- and capacity-fade are often lumped under the description "state-of-health" (SOH). Some other information may be derived using the values of these variables, such as the maximum power available from the pack at any given time. Additional parameters may also be needed for specific applications, and individual algorithms would typically be required to find each one.
The prior art uses the following different approaches to estimate SOH: the discharge test, chemistry-dependent methods, Ohmic tests, and partial discharge. The discharge test completely discharges a fully charged cell in order to determine its total capacity. This test interrupts system function and wastes cell energy. Chemistry-dependent methods include measuring the level of plate corrosion, electrolyte density, and "coup de fouet" for lead-acid batteries. Ohmic tests include resistance, conductance and impedance tests, perhaps combined with fuzzy-logic algorithms and/or neural networks. These methods require invasive measurements.
Partial discharge and other methods compare cell-under-test to a good cell or model of a good cell.
There is a need for a method to continuously estimate the parameters of a cell, such as the cell's resistance and capacity. Furthermore, there is a need for tests that do not interrupt system function and do not waste energy, methods that are generally applicable (e.g., to different types of cell electrochemistries and to different applications), methods that do not require invasive measurements, and more rigorous- approaches. There is a need for a method that will work with different configurations of parallel and/or series cells in a battery pack.
Disclosure of the Invention The present invention relates to an implementation of estimating the values descriptive of the packs present operating condition including battery state-of-charge (SOC) and state-of-health (SOH) for any battery-powered application.
The batteries may be either primary type or secondary (rechargeable) type. Moreover, the invention may be applied to any battery chemistry. It addresses the problems associated with the existing implementations such as high error uncertainty, limited range of applications (i.e. only one type of battery) and susceptibility to change in temperature.
Embodiments of the present invention use a Kalman Filter, a linear algorithm, with a battery model that has SOC as an internal system state. Embodiments of the present invention use an Extended Kalman Filter, a non-linear algorithm, with a battery model that has SOC as an internal system state. Having SOC as an internal state allows the invention to provide an uncertainty associated with its SOC estimation. Embodiments of the present invention do not take battery temperature as a parameter in its SOC estimation. Other embodiments of the present invention use battery temperature as a parameter to adjust its SOC estimation. This is important to keep the accuracy of the SOC estimation from being affected by changing temperature.
One embodiment has the option of allowing different modeling parameters during battery operation to accommodate highly dynamic batteries used in Hybrid Electric Vehicle (HEV) and Electric Vehicle (EV) where such previous implementations were difficult.
Further the present invention relates to methods and apparatus for estimating the parameters of an electrochemical cell. More particularly, for example, estimating parameter values of a cell.
Another aspect of the invention is a method for estimating present parameters of an electrochemical cell system comprising: making an internal parameter prediction of the cell; making an uncertainty prediction of the internal parameter prediction; correcting the internal parameter prediction and the uncertainty prediction; and applying an algorithm that iterates the making an internal parameter prediction, the making an uncertainty prediction and the correcting to yield an ongoing estimation to the parameters and an ongoing uncertainty to the parameters estimation.
Also disclosed herein in an exemplary embodiment is a system to estimate present parameters of an electrochemical cell system comprising: a means for making an internal parameter prediction of the cell; a means -for making an uncertainty prediction of the internal parameter prediction; a means for correcting the internal parameter prediction and the uncertainty prediction; and a means for applying an algorithm that iterates the making an internal parameter prediction, the making an uncertainty prediction and the correcting to yield an ongoing estimation to the parameters and an ongoing uncertainty to the parameters estimation.
Further, disclosed herein in another exemplary embodiment is a storage medium encoded with a machine-readable computer program code, wherein the storage medium includes instructions for causing a computer to implement a method for estimating present parameters of an electrochemical cell comprising:
making an internal parameter prediction of the cell; making an uncertainty prediction of the internal parameter prediction;

correcting the internal parameter prediction and the uncertainty prediction; and applying an algorithm that iterates making an internal parameter prediction, making an uncertainty prediction and correcting to yield an ongoing estimation to the parameters and an ongoing uncertainty to the parameters estimation.
BRIEF DESCRIPTION OF THE DRAWINGS
These and other features, aspects and advantages of the present invention will become better understood with regard to the following description, appended claims and accompanying drawings wherein like elements are numbered alike in the several Figures:
FIG. 1 shows the operation of a generic Kalman Filter.
FIG. 1A shows the equations of a predict component of a generic Kalman Filter.
FIG. 1B_ shows the equations of a correct component of a generic Kalman Filter.
FIG. 2A shows the equations of the predict component of a generic Extended Kalman Filter.
FIG. 2B shows the equationsof the correct component of a generic Extended Kalman Filter.
FIG. 3A shows the components of the SOC estimator according to an embodiment of the present invention.
FIG. 3B shows the components of the SOC estimator according to another embodiment of the present invention.
FIG. 4A shows the equations of the predict component of an implementation of the Extended Kalman Filter according to an embodiment of the present invention.
FIG. 4B shows the equations of the correct component of an implementation of the Extended Kalman Filter according to an embodiment of the present invention.
FIG. 5A shows the equations of the predict component of an implementation of the Extended Kalman Filter according to an embodiment of the present invention.

FIG. 5B shows the equations of the correct component of an implementation of the Extended Kalman Filter according to an embodiment of the present invention.
FIG. 6 shows the operation of an Extended Kalman Filter according to an embodiment of the present invention.
FIG. 7 shows the operation of an Extended Kalman Filter according to another embodiment of the present invention.
FIG. 8A shows the equations of the predict component of an implementation of the Kalman Filter according to an embodiment of the present invention.
FIG. 8B shows the equations of the correct component of an implementation of the Kalman Filter according to an embodiment of the present invention.
FIG. 9 shows the operation of a Kalman Filter according to an embodiment of the present invention.
FIG. 10 shows the operation of an embodiment of the present invention that dynamically changes the modeling equations for the battery SOC.
FIG. 11 is a block diagram illustrating an exemplary system for parameter estimation in accordance with an exemplary embodiment of the invention.
FIG. 12 is a block diagram depicting a method of filtering for parameter estimation, in accordance with an exemplary embodiment of the invention.

}

Best Mode for Carrying Out the Invention Implementation of a Battery State-of-Charge (SOC) Estimating Embodiments of the present invention relate to an implementation of a battery state-of-charge (SOC) estimator for any battery-powered application.
The present invention may be applied to batteries of primary type or secondary (rechargeable) type. The invention may be applied to any battery chemistry. Embodiments of the present invention work on dynamic batteries used in Hybrid Electric Vehicle (HEV) and Electric Vehicle (EV) where previous implementations were difficult. It has the advantage of giving both the SOC estimate and the uncertainty of its estimation. It addresses the problems associated with the existing implementations such as high error uncertainty, limited range of applications and susceptibility to temperature changes.
Temperature-Independent Model FIG. 3A shows the components of the SOC estimator according an embodiment of the present invention. Battery 301 is connected to load circuit 305. For example, load circuit 305 could be a motor in an Electric Vehicle (EV) or a Hybrid Electric Vehicle (HEV). Measurements of battery terminal voltage are made with voltmeter 302. Measurements of battery current are made with ammeter 303. Voltage and current measurements are processed with arithmetic circuit 304, which estimates the SOC. Note that no instrument is needed to take measurements from the internal chemical components of the battery. Also note that all measurements are non-invasive;
that is, no signal is injected into the system that might interfere with the proper operation of load circuit 305.
Arithmetic circuit 304 uses a mathematical model of the battery that includes the battery SOC as a model state. In one embodiment of the present invention, a discrete-time model is used. In another embodiment a continuous-time model is used. In one embodiment, the model equations are Xk+1 = f(Xkr -1k, Wk) (14) and Yk+1 = h (Xk, lk, Vk) (15) where Xk is the model state at time index k (Xk may either be a scalar quantity or a vector), ik is the battery current at time index k, and wk is a disturbance input at time index k. The function f(Xk, ik, wk) relates the model state at time index k to the model state at time index k+1, and may either be a linear or nonlinear function. Embodiments of the present invention have the battery SOC as an element of the model state vector Yk.
In equation (15), the variable Vk is the measurement noise at time index k, and Yk is the model's prediction of the battery terminal voltage at time index k. The function h(xk, ik, wk) relates the model's state, current and measurement noise to the predicted terminal voltage at time index k. This function may either be linear or nonlinear. The period of time that elapses between time indices is assumed to be fixed, although the invention allows measurements to be skipped from time to time.
Temperature-Dependent Model FIG. 3B shows the components of the SOC estimator according another embodiment of the present invention.
Battery 351 is connected to load circuit 355. For example, load circuit 355 could be a motor in an Electric Vehicle (EV) or Hybrid Electric Vehicle (HEV). Measurements of battery terminal voltage are made with voltmeter 352. Measurements of battery current are made with ammeter 353. Battery temperature is measured by temperature sensor 356. Voltage, current and temperature measurements are processed with arithmetic circuit 354, which estimates the SOC.
Arithmetic circuit 354 uses a temperature dependent mathematical model of the battery that includes the battery SOC as a model state. In one embodiment of the present invention, a discrete-time model is used. In another embodiment a continuous-time model is used. In one embodiment, the model equations are Xk+1 = f (Xk, ik, Tk, Wk) (16) and Yk+1 = h(Xk, lk, , Tk Vk) (17) where Xk is the model state at time index k (Xk may either be a scalar quantity or a vector), Tk is the battery temperature at time index k measured at one or more points within the battery pack, ik is the battery current at time index k, and wk is a disturbance input at time index k. The use of battery temperature as a dependent parameter is important to keep the accuracy of the. SOC estimation from being affected by changing temperature. The function f(Xk, ik, Tk, wk) relates the model state at time index k to the model state at time index k+1, and may either be a linear or nonlinear function. Embodiments of the present invention have the battery SOC as an element of the model state vector xk.
In equation (17), the variable Vk, is the measurement noise at time index k, and Yk is the model's prediction of the battery terminal voltage at time index k. The function h(Xk, ik, , Tk vk) relates the model's state, current and measurement noise to the predicted terminal voltage at time index k. This function may either be linear or nonlinear. The period of time that elapses between time indices is assumed to be fixed, although the invention allows measurements to be skipped from time to time.
Applying the Models to Kalman Filter and Extended Kalman Filter In one embodiment of the present invention, the temperature-independent mathematical battery model of equations (14) and (15) is used as the basis for a Kalman filter to estimate the battery SOC as the system operates.
The functions f and h in this embodiment are linear. In another embodiment of the present invention, the temperature-dependent mathematical battery model of equations (16) and (17) is used as the basis for a Kalman filter to estimate the battery SOC as the system operates. The functions f and h in this embodiment are also linear.
In another embodiment of the present invention, the temperature-independent mathematical battery model of equations (14) and (15) is used as a basis for an Extended Kalman filter. The functions f and h in this embodiment are non-linear. In another embodiment of the present invention, the temperature-dependent mathematical battery model of equations (16) and (17) is used as a basis for an Extended Kalman filter. The functions f and h in this embodiment are also non-linear. Those skilled in the art will recognize that other variants of a Kalman filter may also be used, as well as any Luenberger-like observer.

The Operation of the Extended Kalman Filter FIG. 4A and FIG. 4B show an embodiment with an Extended Kalman filter. In this embodiment, equations (14) and (15) from the temperature-independent model is used as the basis of the Extended Kalman filter. Within the two figures, the equations within both the predict and correct components retain the generic form of the Extended Kalman Filter as shown in FIG. 2. However, in this embodiment there is some variation in the variable names. The differences reflect the use of equations (14) and (15) and the variables used in the battery SOC measurement. !k(-) now represents the predicted vector representing the internal states of the battery while Ee,k(-) is now the predicted state-error covariance (uncertainty). The functions f and h are the same as those described in equations (14) and (15) . Note also that in equation 462 of correct component 402, the actual measurement term is now denoted by Mk-FIG. 5A and FIG. 5B show another embodiment with an Extended Kalman filter. In this embodiment, equations (16) and (17) from the temperature-dependent model is used as the basis of the Extended Kalman filter. All the equations are the same as those in FIG. 4A and FIG. 4B except that equation 551 and 562 now have an extra temperature term Tk. Thus at every iteration of the Extended Kalman filter in this embodiment, the temperature of the battery is used to determine the estimation. Since battery capacity is sometimes affected by the temperature, this extra term allows the equations to model the battery more accurately.
FIG. 6. shows the operation of the Extended Kalman filter according to an embodiment of the present invention that uses the temperature-independent model. In block 600, an algorithm is initialized with prior estimates of !A-) and Ee,k(-) . xk(-) is from function f in equation (14) while Ee,k(-) is from function h in equation (15) . Upon the completion of block 600, with the estimates of 5k(-) and Ee,k (-) the algorithm enters correct component of the Extended Kalman filter. The estimates 1k(-) and Ee,k(-) serve as the output from the predict component needed by the correct component. In block 601, the partial derivative of the equation h with respect to x is computed, yielding matrix C. In block 602, the Kalman gain Lk is computed using matrix C, 4H and Ee,k (-) . This corresponds to the first equation (equation 461) of correct component 402 in FIG. 4B. Then in block 603, the predicted internal state vector Xk(-) , the Kalman gain Lk and the measurement from terminal voltage Mk are used to calculate a corrected state vector . This corresponds to the second equation of the correct component in the Extended Kalman filter. In block 604, the predicted state-error covariance Ze,k(-) is used to compute a corrected state-error covariance Ee,k(+) This corresponds to the third equation of the correct component.
In block 605, both of the equations of the predict component are computed. The matrix A is computed by taking the partial derivative of the function f with respect to x Then the prediction for the next iteration is computed, namely and Y1e,k+1(-) - In block 606 the time index k is incremented and the operation begins in block 601 again with the next time step.
FIG. 7. shows the operation of the Extended Kalman filter according to another embodiment of the present invention that uses the temperature-dependent model. In block 700, an algorithm is initialized with prior estimates of 4H
and Ee,k(-) is from function f in equation (16) while Ee,k(-) is from function h in equation (17) . Upon the completion of block 700, with the estimates of Xk(-) and Ee,k(-), the algorithm enters correct component of the Extended Kalman filter. The estimates Xk(-) and Ee,k.(-) serve as the output from the predict component needed by the correct component. In block 701, the partial derivative of the equation h with respect to x is computed, yielding matrix C. In block 702, the Kalman gain Lk is computed using matrix C, Ck(-) and Ee,k (-) This corresponds to the first equation (equation 561) of correct component 502 in FIG 5B. Then in block 703, the predicted internal state vector Xk(-) , the Kalman gain Lk and the measurement from terminal voltage Mk are used to calculate a corrected state vector Xk(+). This corresponds to the second equation of the correct component in the Extended Kalman filter. In block 704, the predicted state-error covariance Ee,k(-) is used to compute a corrected state-error covariance Ee,k(+). This corresponds to the third equation of the correct component.
In block 705, both of the equations of the predict component are computed. The matrix A is computed by taking the partial derivative of the function f with respect to x.
Then the prediction for the next iteration is computed, namely xk+=(-) and Ee,k+l (-) . In block 706 the time index k is incremented and the operation begins in block 701 again with the next time step.
The Operation of the Kalman Filter FIG. 8A and FIG. BE show an embodiment with a Kalman filter. In one embodiment, equations (14) and (15) from the temperature-independent model is used as the basis of the Kalman filter. In another embodiment, equations (16) and (17) from the temperature-dependent model is used as the basis of 'the Kalman filter. Within the two figures, the equations within both the predict and correct components retain the generic form of the Kalman Filter as shown in FIG.1. However, in this embodiment there is some variation in the variable names. In one embodiment, the differences reflect the use of equations (14) and (15) and the variables used in the battery SOC measurement. In another embodiment, the differences reflect the use of equations (16) and (17) and the variables used in the battery SOC measurement. Xk(-) now represents the predicted vector representing the internal states of the battery while Ee,k(-) is now the predicted state-error covariance (uncertainty). Note also that in equation 862 of correct component 802, the actual measurement term is now denoted by Mk-FIG. 9 shows the operation of the Kalman filter according to an embodiment of the present invention. In block 900, an algorithm is initialized with prior estimates of CJ-) and Ze,k (-) . In one embodiment, Xk(-) is from function f in equation (14) while Ee,k(-) is from function h in equation (15). This embodiment is temperature-independent. In another embodiment, !A-) is from function f in equation (16) while Ze,k(-) is from function h in equation (17). This embodiment is temperature-dependent. Upon the completion of block 900, with the estimates of Xk(-) and Ee,k(-), the algorithm enters correct component of the Kalman filter. The estimates Xk(-) and Ee,k(-) serve as the output from the predict component needed by the correct component. In block 901 the Kalman gain Lk is computed using matrix C, M-) and This corresponds to the first equation (equation 861) of correct component 802 in FIG. 8B. Then in block 902 the predicted internal state vector Xk(-) , the Kalman gain Lk and the measurement from terminal voltage Mk are used to calculate a corrected state vector Xk(+). This corresponds to the second equation of the correct component in the Kalman filter. In block 903, the predicted state-error covariance Fe,k(-) is used to compute a corrected state-error covariance Ee,k(+). This corresponds to the third equation of the correct component. In block 904, both of the equations of the predict component are calculated.
Then the prediction for the next iteration is computed, namely and Ee,k+i (-) . In block 905 the time index k is incremented and the operation begins in block 901 again with the next time step.
Specific Equations In one embodiment, the following specific form of function f is used. The internal state vector Xk is:

SOCk FILTk xk IF]k 112k (18) and the governing equation for each state is:
SOC,+1 =SOCk -r)(Ik)IIkI"At/Cp(temp...) FILT,+1 = SOCk + k, FILTH + k5 al a2l (0 IFk+1 = [ -a z 11Fk + L 1 CP(temp...).
2 1 J (19) The batterySOC is the first element of the state vector.
The variables are defined as follows: Ik is the instantaneous current, At is the interval between time instants, Cp (temp . . ) is the "Peukert" capacity of the battery adjusted to be temperature-dependent, n is the Peukert exponent related to the Peukert. capacity, and rl(Ik) is the battery coulombic efficiency as a function of current. The state variables FILT and IF are filter states that capture most of the smooth slow dynamics of the battery.
In one embodiment, the following specific form of function h is used:

vk=kOF'LTk+kl+k2fk+k3'(SoCk+k,5)+k4SOC.k+[C1c?]IFk_ (20) where Yk is the terminal voltage. All other variables (ko,k1retc) are coefficients of the model, which may be determined a priori from lab tests and may be adjusted during system operation using mechanisms not discussed here. These coefficients vary in the present invention so that the coefficients used for an instantaneous discharge of 10 Amps would be different from those used for an instantaneous charge of 5 Amps, for example. This allows the invention to more precisely model the current-dependence of the model.
In another embodiment, the following form of function f is used. The internal state vector Xk is SOCk SOCk-1 SOCk-a Ck ik-1 xk Zk Yk-1 Yk-2 Yk (21) where SOCk is the present SOC estimate, SOCk_1 is the previous SOC estimate (and so forth), ik is the present current measurement (and so forth) and Yk-1 is the previous battery voltage estimate. a, (3 and y are positive constants chosen to make an acceptable model with a parsimonious number of state variables. The governing equation for the SOC state is:

S(~C'k+1-SOCI, 1(Ik) IlkI"At/CC(temp ... ) (2 2 In the embodiment, the specific form of function h is used Yk=h (Xk, TO r (23) where h is implemented as a nonlinear function fit to measured data. For example, h may be implemented using a neural network.
In one embodiment a neural network may be used to estimate the internal states of the battery. The difference between this embodiment and prior neural networks is as follows. In prior art neural networks, the estimated SOC is the output of the neural networks. This embodiment indirectly measures the SOC by first modeling the battery cell using a neural network with SOC as one of its states, and then uses a Kalman filter with the neural network to estimate SOC. This approach has two main advantages. First it can be trained on-line while it is in operation. Second, error bounds on the estimate may be computed.
Changing Parameters FIG. 10 shows the operation of an embodiment of the present invention that dynamically changes modeling equations for the battery SOC. In this embodiment, the arithmetic circuit can accommodate changing behaviors of the battery to use different parameters for different time periods. In block 11000, a change in battery current level is detected. For example, in a Hybrid Electric Vehicle (HEV), a sudden drain in the battery power is caused by the vehicle going uphill.
The sudden change in condition triggers the arithmetic circuit to use a different set of modeling equations to more accurately estimate the SOC in the new condition. In block 1010 a new set of modeling equations are used. In 1020, the new equations are used to determine the SOC. This adaptive modeling behavior is useful in highly dynamic applications such as in Hybrid Electric Vehicles (HEV) and Electric Vehicles (EV).
Further, disclosed herein and various embodiments are methods, systems and apparatus for the estimation of parameters of an electrochemical cell using filtering are disclosed. Referring now to FIG. 11 and FIG. 12, in the following description, numerous specific details are set forth in order to provide a more complete understanding of the present invention. It will be appreciated that while the exemplary embodiments are described with reference to a battery cell, numerous electrochemical cells hereinafter referred to as a cell, may be employed, including, but not limited to, batteries, battery packs, ultracapacitors, capacitor banks, fuel cells, electrolysis cells, and the like, as well as combinations including at least one of the foregoing. Furthermore, it will be appreciated that a battery or battery pack may include a plurality of cells, where the exemplary embodiments disclosed herein are applied to one or more cells of the plurality.
One or more exemplary embodiments of the present invention estimate cell parameter values using a filtering method. One or more exemplary embodiments of the present invention estimate cell parameter values using Kalman filtering. Some embodiments of the present invention estimate cell parameter values using extended Kalman filtering. Some embodiments estimate cell resistance. Some embodiments estimate cell total capacity. Some embodiments estimate other time-varying parameter values. It will further be appreciated that while the term filtering is employed for description and illustration of the exemplary embodiments, the terminology is intended to include methodologies of recursive prediction and correction commonly denoted as filtering, including but not limited to Kalman filtering and/or extended Kalman filtering.
Implementation of a Battery State of Health (SOH) Estimating FIG. 11 shows the components of the parameter estimator system 10 according an embodiment of the present invention.
Electrochemical cell pack 20 comprising a plurality of cells 22, e.g., battery is connected to a load circuit 30. For example, load circuit 30 could be a motor in an Electric Vehicle (EV) or a Hybrid Electric Vehicle (HEV). An apparatus for measuring various cell characteristics and properties is provided as 40. The measurement apparatus 40 may include but not be limited to a device for measurement of cell terminal voltage such as a voltage sensor 42, e.g. a voltmeter and the like, while measurements of cell current are made with a current sensing device 44, e.g., an ammeter and the like. Optionally, measurements of cell temperature are made with a temperature sensor 46, e.g., a thermometer and the like. Additional cell properties, such as internal pressure or impedance, may be measured using (for example) pressure sensors and/or impedance sensors 48 and may be employed for selected types of cells. Various sensors may be employed as needed to evaluate the characteristics and properties of the cell(s). Voltage, current, and optionally temperature and cell-property measurements are processed with an arithmetic circuit 50, e.g., processor or computer, which estimates the parameters of the cell(s). The system may also include a storage medium 52 comprising any computer usable storage medium known to one of ordinary skill in the art.
The storage medium is in operable communication with arithmetic circuit 50 employing various means, including, but not limited to a propagated signal 54. It should be appreciated that no instrument is required to take measurements from the internal chemical components of the cell 22 although such instrumentation may be used with this invention. Also note that all measurements may be non-invasive; that is, no signal must be injected into the system that might interfere with the proper operation of load circuit 30.
In order to perform the prescribed functions and desired processing, as well as the computations therefore (e.g., the modeling, estimation of parameters prescribed herein, and the like), arithmetic circuit 50 may include, but not be limited to, a processor(s), gate array(s), custom logic, computer(s), memory, storage, register(s), timing, interrupt(s), communication interfaces, and input/output signal interfaces, as well as combinations comprising at least one of the foregoing. Arithmetic circuit 50 may also include inputs and input signal filtering and the like, to enable accurate sampling and conversion or acquisitions of signals from communications interfaces and inputs. Additional features of arithmetic circuit 50 and certain processes therein are thoroughly discussed at a later point herein.
One or more embodiments of the invention may be implemented as new or updated firmware and software executed in arithmetic circuit 50 and/or other processing controllers.
Software functions include, but are not limited to firmware and may be implemented in hardware, software, or a combination thereof. Thus a distinct advantage of the present invention is that it may be implemented for use with existing and/or new processing systems for electrochemical cell charging and control.
In an exemplary embodiment, Arithmetic circuit 50 uses a mathematical model of the cell 22 that includes indicia of a dynamic system state. In one embodiment of the present invention, a discrete-time model is used. An exemplary model for the cell 22 in a (possibly nonlinear) discrete-time state-space ' form -has the form:

xk+1 J (Xk,uk,Ok) +wk (24) Yk -g(xk,uk,ek) +Vk, where xk is the system state, k is the set of time varying model parameters, Uk is the exogenous input, Ykis the system output, and wk and vk are "noise" inputs-all quantities may be scalars or vectors. f(.,=,=) and g(.,.,.) are functions defined by the cell model being used. Non-time-varying numeric values required by the model may be embedded within f(=;,=) and and are not included in 0k The system state xk includes, at least, a minimum amount of information, together with the present input and a mathematical model of the cell, needed to predict the present output. For a cell 22, the state might include: SOC, polarization voltage levels with respect to different time constants, and hysteresis levels, for example. The system exogenous input uk includes at minimum the present cell current ik , and may, optionally, include cell temperature (unless temperature change is itself modeled in the state).
The system parameters Bk are the values that change only slowly with time, in such a way that they may not be directly determined with knowledge of the system measured input and output. These might include, but not be limited to: cell capacity, resistance, polarization voltage time constant(s), polarization voltage blending factor(s), hysteresis blending factor(s), hysteresis rate constant(s), efficiency factor(s), and so forth. The model output yk corresponds to physically measurable cell quantities or those directly computable from measured quantities at minimum for example, the cell voltage under load.
There are a number of existing methods for estimating the state of a cell including, but not limited to the state charge of a cell 22. SOC is a value, typically reported in percent, which indicates the fraction of the cell capacity presently available to do work. A number of different approaches to estimating SOC have been employed: a discharge test, ampere-hour counting (Coulomb counting), measuring the electrolyte, open-circuit voltage measurement, linear and nonlinear circuit modeling, impedance spectroscopy, measurement of internal resistance, coup de fouet, and some forms of Kalman filtering. Each of these methodologies exhibits advantages as well as limitations.

According to the above embodiments of the present invention related to an implementation of a battery State of Charge (SOC) estimator, a filter, preferably a Kalman filter is used to estimate SOC by employing a known mathematical model of cell dynamics and measurements of cell voltage, current, and temperature. Advantageously, this method directly estimates state values for the cell where SOC is at least one of the states. However, it should be appreciated that there are numerous well-known methodologies for computing SOC.
Continuing with FIG. 12, a mathematical model of parameter dynamics is also utilized. An exemplary model has the form:

0k+1 k + rk / (25) dk ` g(xk,ukIOk)+ ek.
The first equation states that the parameters 01, are primarily constant, but that they may change slowly over time, in this instance, modeled by a "noise" process denoted, rk.
The "output" dk is a function of the optimum parameter dynamics modeled by g(=;;) plus some estimation error ek . The optimum parameter dynamics g(=,=,=) being a function of the system state xk, an exogenous input Uk, and the set of time varying parameters k.
With a model of the cell system, requirements for state dynamics, and model of the parameter dynamics defined, in an exemplary embodiment, a procedure of filtering is applied.
Once again, alternatively, a Kalman filter may be employed, or an extended Kalman filter. Table 1 identifies an exemplary implementation of the methodology and system utilizing an extended Kalman filter 1100. Once again, it should be appreciated that while the cell model and parameter estimation models employ the state Xk of the cell 22, the state is not necessarily predicted as part of the parameter estimation process. For example, in one exemplary embodiment, the state Xk of the cell 22, is computed by another process with the resulting state information supplied to the parameters model. Continuing with an exemplary implementation of Table 1, the procedure is initialized by setting a parameter estimate, denoted 9 to the best guess of the true parameters, e . g . , = E[ o] . While not required or defined for the state estimate, the state estimate denoted z may be set to the best estimate of the cell state, e.g., i = E[xo] . An estimation-error covariance matrix B is also initialized. For example, an initialization of state, and particularly, SOC might be estimated/based on a cell voltage in a look-up table, or information that was previously stored when a battery pack/cell was last powered down. Other examples might incorporate the length of time that the battery system had rested since power-down and the like.
Table 1: Extended Kalman filter for parameter update.
State{-(space models:

xk+l -J //(xk,uk,Ok) +wk and 0k+1 9k +rk Yk g(xkukIek)+Vk dk g(xkIukIek)+ek where wk , Vk , rk and ek are independent, zero-mean, Gaussian noise processes of covariance matrices E,,, E,, Zr and Ee, respectively.
Definition:
C dg(xk ~uk~e) k dO
B=Bk Initialization: For k=0, set Bo = E[00 ], Y_+'0 = E[( o - Bo )(eo -L9+) T ]
Computation: For k=1,2,...,compute:

Time update 0k - ek-1 k Y- B,k-1 + F'r Measurement update Lk = Y- B k `Ck)T [C o,k \CkB)T +Y-el Ok =0k +Lk[yk g(J (xk-l,uk-1IBkl ),uk,9k )~
EB1 =(I-LBCB)> 1.
In this example, several steps are performed in each measurement interval. First, the previous parameter estimate is propagated forward in time. The new parameter estimate is equal to the old parameter estimate 0k = Bki , and the parameter error uncertainty is larger due to the passage of time (accommodated for in the model by the driving noise r,.).
It should be readily appreciated that various possibilities exist for updating the parameter uncertainty estimate, the table provides an illustrative example. A measurement of the cell output is made, and compared to the predicted output based on the state estimate, i (however estimated or provided) and parameter estimate, 0; the difference is used to update the values of B. Note also, the state estimate i may be propagated forward by the parameter estimate or may be supplied via an external means as identified above. Ck may be computed using the following recurrence relationship:

dg(xk, uk, 0) = 5g(xk, Uk, 0) +'t(xk, uk, 0) dxk dO d0 OXk d0' ~r q~ / (26) dxk = "./ (xk-l,uk-1,G) + "firJ (xk-1,uk-1,e) dxk-1 d0 dO &k-1 d0 The derivative calculations are recursive in nature, and evolve over time as the state xk evolves. The term dxo/d0 is initialized to zero unless side information yields a better estimate of its value. It may readily be appreciated that the steps outlined in the table may be performed in a variety of orders. While the table lists an exemplary ordering for the purposes of illustration, those skilled in the art will be able to identify many equivalent ordered sets of equations.
Turning now to FIG. 12 as well, an exemplary implementation of an exemplary embodiment of the invention is depicted. A recursive filter 1100 adapts the parameter estimate, B . The filter has a time update or prediction 1103 aspect and a measurement update or correction 1104 aspect. Parameter time update/prediction block 1103 receives as input the previous exogenous input Uk_l, the previous time varying parameters estimate k1 and a corrected parameter uncertainty estimate Y-B,k_1 Parameter time update/prediction block 1103 outputs predicted parameters Bk and predicted parameter uncertainty Eek to the parameter measurement update/correction block 1104. Parameter measurement update block 1104, which provides current parameter estimate Ok and parameter uncertainty estimate, Eek receives the predicted parameters k and predicted parameter uncertainty Eek as well as the exogenous input Uk and the modeled system output yk.
It will also be appreciated that a minus notation denotes that the vector is the result of the prediction component 1103 of the filter 1100, while the plus notation denotes that the vector is the result of the correction component 1104 of the filter 1100.

Embodiments of this invention require a mathematical model of cell state and output dynamics for the particular application. In the exemplary embodiments, this is accomplished by defining specific functions for f(.,-,-) and g(-,.,.) to facilitate estimation or receipt of the various states and estimation of the various parameters of interest. An exemplary embodiment uses a cell model that includes effects due to one or more of the open-circuit-voltage (OCV) for the cell 22, internal resistance, voltage polarization time constants, and a hysteresis level. For the purpose of example, parameter values are fitted to this model structure to model the dynamics of high-power Lithium-Ion Polymer Battery (LiPB) cells, although the structure and methods presented here are general and apply to other electrochemistries. For example, in an exemplary embodiment, the states and parameters of interest are embedded in f(,,) and g(=,-;), and examples follow:

k - [~i,k, Ck, al k ... an f k, gl,k~...gnf-1 kl A-1 Rk, Mk IT. (27) where 7/f,k is an efficiency factor(s) such as Coulombic efficiency, Ck is the cell capacity/capacities, alk,...anfk are polarization voltage time constant(s), 91,k'...9,, f-l,k are the polarization voltage blending factor (s) , Rk is the cell resistance(s), Mk is the hysteresis blending factor(s), and yk is the hysteresis rate constant(s).

In this example, SOC is captured by one state of the model as part of function f(=,=,=). This equation is:

Zk+1 -Ck -(7i,kAt/Ck)lk (28) where At represents the inter-sample period (in seconds), Ck represents the cell capacity (in ampere-seconds), zk is the cell SOC at time index k, ik is the cell current, and riik is the Coulombic efficiency of a cell at current level ik.
In this example, the polarization voltage levels are captured by several filter states. If we let there be of polarization voltage time constants, then fk+l -Affk+Bfik= (29) The matrix Af e ?Ifx,zf may be a diagonal matrix with real-valued polarization voltage time constants al,k===aõf,k . If so, the system is stable if all entries have magnitude less than one. The vector Bf eTnfxl may simply be set to n f '11"s. The entries of Bf are not critical as long as they are non-zero.
The value of n f entries in the ,4f matrix are chosen as part of the system identification procedure to best fit the model parameters to measured cell data. The Al , and Bf matrices may vary with time and other factors pertinent to the present battery pack operating condition.
In this example, the hysteresis level is captured by a single state h ex '7i,klkrkzt h + 1- exph7i,klkrkI t s (i (30) k+1 - P - k - tk)~
Ck Ck where yk is the hysteresis rate constant, again found by system identification.
In yet another exemplary embodiment, the overall model state is a combination of the above examples as follows:

T T
xk=[fk hk Zk] , (31) where other orderings of states are possible.
In this example, the output equation that combines the state values to predict cell voltage is Vic = OCV(Zk)+ Gkfk -Rkik +Mkhk, (32) where Gk E 93lx"f is a vector of polarization voltage blending factors gt,k===gõf,k that blend the polarization voltage states together in the output, Rk is the cell resistance (different values may be used for discharge/charge), and Mk is the hysteresis blending factor. Note, Gk may be constrained such that the dc-gain from ik to Gkfk is zero, which results in the estimates of Rk being accurate.
Some embodiments of the present invention may include methods to constrain the parameters of the model to result in a stable system. In an exemplary embodiment, the state equation may include terms for polarization voltage time constants in the form fk+1Affk+Bfik, where the matrix Af E=-91"""' =

is diagonal matrix with real-valued polarization voltage time constants al,k===a, Pk . These time constants may be computed as ask=tanh(a;k), where the parameter vector of the model contains the a1k values and not directly the ark values. The tanh ( ) function ensures that the ask are always within 1 (i.e., stable) regardless of the value of ask.
Some embodiments of the present invention include constraints to the model to ensure convergence of a parameter to its correct value. An exemplary embodiment using the model herein described constrains Gk so that the dc-gain from ik to Gkfk is zero, which results in the estimates of Rk being accurate. This is done by enforcing that the last element of Gk be computed using other elements of Gk and the polarization voltage time constants g,,Pk =-~Zf1 gik(1-a, fk)ai,k) .
This also requires more care when computing elements of CB
~(xk, uk, gq") 1-anf,k relating to Gk =fk,i fk,nf+1<i<n If the ai,k 'ki,k 1- ai,k values are always within 1 (for example, by using the method described in the previous paragraph), then there will never be a divide-by-zero problem with the derivative computation.
Another exemplary embodiment includes methods for estimating important aspects of SOH without employing a full filter 1100. The full filter 1100 method may be computationally intensive. If precise values for the full set of cell model parameters are not necessary, then other methods potentially less complex or computationally intensive might be used. The exemplary methodologies determine cell capacity and resistance using filtering methods. The change in capacity and resistance from the nominal "new-cell" values give capacity fade and power fade, which are the most commonly employed indicators of cell SOH.
In this example, to estimate cell resistance using a filtering mechanism, we formulate a model:

Rk+1 - Rk + rk (33) Yk = OCV(Zk) - ikRk + ek where R. is the cell resistance and is modeled as a constant value with a fictitious noise process rk allowing adaptation.
Yk is an estimate of the cell's voltage, ik is the cell current, and ekmodels estimation error. If an estimate of zk that may be externally generated and supplied is employed, then a filter 1100 may be applied to this model to estimate cell resistance. In the standard filter 1100, the model's prediction of yk is compared with the true measured cell voltage. Any difference resultant from the comparison is used to adapt Rk.
Note that the above model may be extended to handle different values of resistance for a variety of conditions of the cell 22. For example, differences based on charge and discharge, different SOCs, and different temperatures. The scalar Rk may then be established as a vector comprising all of the resistance values being modified, and the appropriate element from the vector would be used each time step of the filter during the calculations.
In this example, to estimate cell capacity using a filter 1100, we again formulate a cell model:

Ck+l -Ck+rk (34) 0- Zk - Zk-1 + l7i,ktk- lAt / Ck-1 + ek.
Again, a filter is formulated using this model to produce a capacity estimate. As the filter 1100 runs, the computation in the second equation (right-hand-side) is compared to zero, and the difference is used to update the capacity estimate.
Note that good estimates of the present and previous states-of-charge are desired, possibly from a filter estimating SOC.
Estimated capacity may again be a function of temperature (and so forth), if desired, by employing a capacity vector, from which the appropriate element is used in each time step during calculations.
Industrial Applicability Thus, a method for estimation of cell parameters has been described in conjunction with a number of specific embodiments. One or more embodiments use a Kalman filter 1100. Some embodiments use an extended Kalman filter 1100.
Further, some embodiments include a mechanism to force convergence of one or more parameters. One or more embodiments include a simplified parameter filter 1100 to estimate resistance, while some embodiments include a simplified parameter filter 1100 to estimate total capacity.
The present invention is applicable to a broad range of applications, and cell electrochemistries.
The disclosed method may be embodied in the form of computer-implemented processes and apparatuses for practicing those processes. The method can also be embodied in the form of computer program code containing instructions embodied in tangible media 52, such as floppy diskettes, CD-ROMs, hard drives, or any other computer-readable storage medium, wherein, when the computer program code is loaded into and executed by a computer, the computer becomes an apparatus capable of executing the method. The present method can also be embodied in the form of computer program code, for example, whether stored in a storage medium, loaded into and/or executed by a computer, or as data signal 54 transmitted whether a modulated carrier wave or not, over some transmission medium, such as over electrical wiring or cabling, through fiber optics, or via electromagnetic radiation, wherein, when the computer program code is loaded into and executed by a computer, the computer becomes an apparatus capable of executing the method. When implemented on a general-purpose microprocessor, the computer program code segments configure the microprocessor to create specific logic circuits.

It will be appreciated that the use of first and second or other similar nomenclature for denoting similar items is not intended to specify or imply any particular order unless otherwise stated. Furthermore, the use of the terminology "a" and "at least one" of shall each be associated with the meaning "one or more" unless specifically stated otherwise.
While the invention has been described with reference to an exemplary embodiment, it will be understood by those skilled in the art that various changes may be made and equivalents may be substituted for elements thereof without departing from the scope of the invention. In addition, many modifications may be made to adapt a particular situation or material to the teachings of the invention without departing from the essential scope thereof. Therefore, it is intended that the invention not be limited to the particular embodiment disclosed as the best mode contemplated for carrying out this invention, but that the invention will include all embodiments falling within the scope of the appended claims.

Claims (9)

Claims

1. A method for estimating present parameters of an electrochemical cell system comprising:
receiving a state estimate and ;
determining a current measurement;
determining a voltage measurement;
making an internal parameter prediction by using said state estimate, said current measurement and said voltage measurement in a mathematical model;

making an uncertainty prediction of said internal parameter prediction by using said state estimate, said current measurement and said voltage measurement in said mathematical model; and correcting said internal parameter prediction and said uncertainty prediction;

applying an algorithm that iterates said making an internal parameter prediction, said making an uncertainty prediction and said correcting to yield an ongoing estimation to said parameters and an ongoing uncertainty to said parameters estimation.

2. The method of claim 1 wherein said correcting comprises:

computing a gain factor;

computing a corrected internal parameter prediction using said gain factor, said voltage measurement and said internal parameter prediction; and computing a corrected uncertainty prediction using said gain factor and said uncertainty prediction.

3. The method of claim 2 wherein said applying comprises using said corrected internal parameter prediction and said corrected uncertainty prediction to obtain predictions for a next time step where said algorithm repeats.

4. The method of claim 3 wherein said algorithm is at least one of a Kalman Filter and an extended Kalman Filter.

5. The method of claim 4 wherein said internal parameter includes one or more of: a resistance, a capacity, a polarization voltage time constant, a polarization voltage blending factor a hysteresis blending factor, a hysteresis rate constant, and an efficiency factor.

6. The method of claim 1 wherein said making an uncertainty prediction further comprises determining a temperature measurement, and wherein said making an uncertainty prediction comprises using said state estimate and said current measurement, said voltage measurement, and said temperature measurement in a mathematical model.

7. The method of claim 1 wherein said making an internal parameter prediction further comprises:
determining a temperature measurement; and using said state estimate, said temperature measurement, said current measurement and said voltage measurement in a mathematical model to make said internal parameter prediction.

8. The method of claim 1 wherein a filter pole is computed using a tanh function to ensure that its magnitude is less than 1.

9. The method of claim 1 further including an initial state value and an initial parameter value associated with the electrochemical cell.