WO2006057468A1 - Method and system for battery state and parameter estimation - Google Patents
Method and system for battery state and parameter estimation Download PDFInfo
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- WO2006057468A1 WO2006057468A1 PCT/KR2004/003101 KR2004003101W WO2006057468A1 WO 2006057468 A1 WO2006057468 A1 WO 2006057468A1 KR 2004003101 W KR2004003101 W KR 2004003101W WO 2006057468 A1 WO2006057468 A1 WO 2006057468A1
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- H—ELECTRICITY
- H01—ELECTRIC ELEMENTS
- H01M—PROCESSES OR MEANS, e.g. BATTERIES, FOR THE DIRECT CONVERSION OF CHEMICAL ENERGY INTO ELECTRICAL ENERGY
- H01M10/00—Secondary cells; Manufacture thereof
- H01M10/42—Methods or arrangements for servicing or maintenance of secondary cells or secondary half-cells
- H01M10/48—Accumulators combined with arrangements for measuring, testing or indicating the condition of cells, e.g. the level or density of the electrolyte
- H01M10/486—Accumulators combined with arrangements for measuring, testing or indicating the condition of cells, e.g. the level or density of the electrolyte for measuring temperature
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- G—PHYSICS
- G01—MEASURING; TESTING
- G01R—MEASURING ELECTRIC VARIABLES; MEASURING MAGNETIC VARIABLES
- G01R31/00—Arrangements for testing electric properties; Arrangements for locating electric faults; Arrangements for electrical testing characterised by what is being tested not provided for elsewhere
- G01R31/36—Arrangements for testing, measuring or monitoring the electrical condition of accumulators or electric batteries, e.g. capacity or state of charge [SoC]
- G01R31/389—Measuring internal impedance, internal conductance or related variables
-
- H—ELECTRICITY
- H01—ELECTRIC ELEMENTS
- H01M—PROCESSES OR MEANS, e.g. BATTERIES, FOR THE DIRECT CONVERSION OF CHEMICAL ENERGY INTO ELECTRICAL ENERGY
- H01M10/00—Secondary cells; Manufacture thereof
- H01M10/42—Methods or arrangements for servicing or maintenance of secondary cells or secondary half-cells
- H01M10/48—Accumulators combined with arrangements for measuring, testing or indicating the condition of cells, e.g. the level or density of the electrolyte
-
- Y—GENERAL TAGGING OF NEW TECHNOLOGICAL DEVELOPMENTS; GENERAL TAGGING OF CROSS-SECTIONAL TECHNOLOGIES SPANNING OVER SEVERAL SECTIONS OF THE IPC; TECHNICAL SUBJECTS COVERED BY FORMER USPC CROSS-REFERENCE ART COLLECTIONS [XRACs] AND DIGESTS
- Y02—TECHNOLOGIES OR APPLICATIONS FOR MITIGATION OR ADAPTATION AGAINST CLIMATE CHANGE
- Y02E—REDUCTION OF GREENHOUSE GAS [GHG] EMISSIONS, RELATED TO ENERGY GENERATION, TRANSMISSION OR DISTRIBUTION
- Y02E60/00—Enabling technologies; Technologies with a potential or indirect contribution to GHG emissions mitigation
- Y02E60/10—Energy storage using batteries
Definitions
- the present invention relates to methods and apparatus for estimation of battery pack system states and parameters using digital filtering techniques.
- 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.
- SOC battery state-of-charge
- the power-fade and capacity- fade are often lumped under the description state-of-health (SOH) .
- SOH state-of-health
- 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. .
- 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 System on Chip
- 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 to determine what the SOC is. Ampere-hour counting uses the following:
- Electrolyte Measurement is another common technique.
- the electrolyte takes part in reactions during charge and discharge.
- 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.
- 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 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
- Electric Vehicle EV
- 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 determine the error associated with the SOC prediction given by artificial neural networks.
- 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 advantages over many other similar mathematical system models.
- 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.
- the Kalman filter-predict component 101 There are two main components in the Kalman filter-predict component 101 and correct component 102.
- 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.
- 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 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.
- FIG. IA and FIG. IB 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
- ⁇ k C ⁇ x k +D k u k ( 7 )
- 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 A k and Bjt in equation (6), respectively.
- the matrices C and D in equation (9) relate to matrices C* and D*; in equation (7).
- equation (8) governs the estimation of the dynamic system process, it is called the process function.
- equation (9) governs the estimation of the measurement uncertainty, it is called the measurement function.
- the added random variables vr k and v k in equation (8) and (9) represent the process noise and measurement noise, respectively. Their contribution to the estimation is represented by their covariance matrices ⁇ w and ⁇ v in FIGS. IA and IB.
- 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 * « ⁇ i ( - ) . 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.
- Equation 152 predicts the. uncertainty, which is also referred to as the error covariance.
- the matrix ⁇ w in equation 152 is the process noise covariance matrix.
- FIG. IB shows the equations within correct component 102. These three equations are executed in sequence.
- 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 y*.
- the Kalman Gain factor is used to weight between actual measurement y ⁇ and predicted measurement cx * ⁇ .- ) .
- matrix ⁇ v the actual measurement- noise covariance, is inversely proportional to the Kalman Gain factor L*. As ⁇ v decreases, L ⁇ increases and gives the actual measurement ⁇ k more weight.
- Equation 162 computes the corrected internal state vector ⁇ +) based on predicted internal state vector * * ( - ) (from predict component 101) , new measurement ⁇ k and predicted measurement C M-) .
- equation 163 corrects the predicted uncertainty, or the state-error covariance.
- Matrix I in equation 163 represents the identity matrix.
- Equation 162 and 163 are fed to predict component 101 for the next iteration. More specifically, the calculated value * * M in equation 162 is substituted into equation 151 for the next iteration and the calculated value ⁇ e , ⁇ (+) 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, ⁇ w and ⁇ v might change in each time step.
- the Extended Kalman filter uses linear functions in its model
- the Extended Kalman filter was developed to model system with non-linear functions.
- 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 basi s for the two equations in the predict component .
- Non-linear function f in equation (10) relates the internal state vector x k at the current step k to the internal state vector x k +i at the next time step k+1.
- Function f also includes as parameters both the driving function u k and the process noise w k .
- the non ⁇ linear function h in equation (11) relates the internal state vector x k and input u k to the measurement y k .
- FIG. 2A and FIG. 2B show the equations of the Extended Kalman Filter.
- the sequence of operation remains the same as the Kalman filter.
- 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
- the notation means "with Xk evaluated as, or replaced by, X f in final result.”
- C is the Jacobian matrix computed by taking the partial derivative of h with respect to x, that is
- the operation of the Extended Kalman filter remains essentially the same as the Kalman filter.
- the Kalman filter has an advantage in that it works on all types of batteries system, including dynamic applications such as HEV and EV.
- 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.
- none of the existing methods uses the Extended Kalman filter to model battery SOC non-linearly.
- 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.
- 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.
- SOH state-of-health
- 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.
- Partial discharge and other methods compare cell-under-test to a good cell or model of a good cell.
- FIG. 1 shows the operation of a generic Kalman Filter.
- FIG. IA shows the equations of a predict component of a generic Kalman Filter.
- FIG. IB 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 equations of the correct component of a generic Extended Kalman Filter.
- FIG. 3A shows the components of the SOC estimator according an embodiment of the present invention.
- FIG. 3B shows the components of the SOC estimator according 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 an embodiment of the present invention.
- FIG. 7 shows the operation of an Extended Kalman Filter according 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 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. 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.
- 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.
- 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 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.
- Another aspect is an apparatus configured to estimate present parameters of an electrochemical cell comprising: a component configured to make an internal parameter prediction of the cell; a component configured to make an uncertainty prediction of the internal parameter prediction; a component configured to correct the internal parameter prediction and the uncertainty prediction; and a component configured to iterate steps taken by the component configured to make an internal parameter prediction, the component configured to make an uncertainty prediction and the component configured to correct to yield an ongoing estimation to the parameter and an ongoing uncertainty to the parameter estimation.
- Also disclosed herein in an exemplary embodiment is a system for estimating present parameters of an electrochemical cell 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 said parameters and an ongoing uncertainty to said parameters estimation.
- a storage medium encoded with a machine- readable computer program code including instructions for causing a computer to implement the abovementioned method for estimating present parameters of an electrochemical cell.
- computer data signal embodied in a computer readable medium.
- the computer data signal comprises code configured to cause a computer to implement the abovementioned method for estimating present parameters of an electrochemical cell.
- Embodiments of the present invention relate to an implementation of a battery State of Charge (SOC) estimator for any battery-powered application.
- SOC Battery State of Charge
- 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.
- HEV Hybrid Electric Vehicle
- EV Electric Vehicle
- FIG. 3A shows the components of the SOC estimator according an embodiment of the present invention.
- Battery 301 is connected to load circuit 305.
- load circuit 305 For example, load circuit
- 305 could be a motor in an Electric Vehicle (EV) or a Hybrid
- EV Electric Vehicle
- Hybrid Hybrid
- HEV Electric Vehicle
- 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.
- a discrete-time model is used.
- a continuous-time model is used.
- the function f ⁇ x k/ i-k r w k ) 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 xj t .
- the variable v k is the measurement noise at time index k
- y ⁇ is the model's prediction of the battery terminal voltage at time index k.
- the function h[x k , i f c, W k ) 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.
- FIG. 3B shows the components of the SOC estimator according another embodiment of the present invention.
- Battery 351 is connected to load circuit 355.
- 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.
- a discrete-time model is used.
- a continuous-time model is used.
- x ⁇ is the model state at time index k (x ⁇ may either be a scalar quantity or a vector)
- T* is the battery temperature at time index k measured at one or more points within the battery pack
- x k is the battery current at time index k
- w* 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[x k , i kr T k/ W k ) 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 x ⁇ .
- the variable v ⁇ is the measurement noise at time index k
- y ⁇ is the model's prediction of the battery terminal voltage at time index k.
- the function h ⁇ x k , i kr , T k 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.
- 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.
- 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.
- 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.
- 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.
- FIG. 4A and FIG. 4B show an embodiment with an Extended Kalman filter.
- equations (14) and (15) from the temperature-independent model is used as the basis of the Extended Kalman filter.
- equations (14) and (15) from the temperature-independent model is used as the basis of the Extended Kalman filter.
- the equations within both the predict and correct components retain the generic form of the Extended Kalman Filter as shown in FIG. 2.
- X *H now represents the predicted vector representing the internal states of the battery while ⁇ e,*( ⁇ ) is now the predicted state-error covariance (uncertainty) .
- the functions f and h are the same as those described in equations (14) and (15) .
- equation 462 of correct component 402 the actual measurement term is now denoted by m*.
- FIG. 5A and FIG. 5B show another embodiment with an Extended Kalman filter.
- 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 T*.
- T* 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.
- an algorithm is initialized with prior estimates of * * ( - ) and ⁇ e, /c ( ⁇ ) • * * ( "" ) is from function f in equation (14) while ⁇ e ⁇ (-) is from function h in equation (15) .
- the algorithm enters correct component of the Extended Kalman filter.
- the estimates x * ( ⁇ ) and ⁇ e ,j t (-) serve as the output from the predict component needed by the correct component.
- the partial derivative of the equation h with respect to x is computed, yielding matrix C.
- the Kalman gain L* is computed using matrix C, * * ( - ) and ⁇ e ,*(-) - This corresponds to the first equation (equation 461) of correct component 402 in FIG. 4B.
- the predicted internal state vector ** ( - ) the Kalman gain L* and the measurement from terminal voltage m*; are used to calculate a corrected state vector ** ( ⁇ * •) .
- the predicted state-error covariance ⁇ e ,k( ⁇ ) is used to compute a corrected state-error covariance ⁇ e ,*(+) This corresponds to the third equation of the correct component.
- 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 ⁇ e ,A + i( ⁇ ) -
- 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.
- an algorithm is initialized with prior estimates of * * ( - ) and ⁇ e , f c(-) is from function f in equation (16) while ⁇ e ,jc(-) is from function h in equation (17) .
- the algorithm enters correct component of the Extended Kalman filter.
- the estimates ** ( - ) and ⁇ ⁇ ,*.(-) serve as the output from the predict component needed by the correct component.
- the partial derivative of the equation h with respect to x is computed, yielding matrix C.
- the Kalman gain L ⁇ is computed using matrix C, M- ) and ⁇ erk ⁇ -) . This corresponds to the first equation (equation 561) of correct component 502 in FIG 5B. Then in block 703, the predicted internal state vector M- ) , the Kalman gain L k and the measurement from terminal voltage m ⁇ are used to calculate a corrected state vector * * W . This corresponds to the second equation of the correct component in the Extended Kalman filter. In block 704, the predicted state-error covariance ⁇ e ,k( ⁇ ) is used to compute a corrected state-error covariance ⁇ ⁇ rk ⁇ +) . This corresponds to the third equation of the correct component.
- 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 ⁇ e ,jc + i(-") •
- the time index k is incremented and the operation begins in block 701 again with the next time step.
- FIG. 8A and FIG. 8B show an embodiment with a Kalman filter.
- equations (14) and (15) from the temperature-independent model is used as the basis of the Kalman filter.
- equations (16) and (17) from the temperature-dependent model is used as the basis of 'the Kalman filter.
- the equations within both the predict and correct components retain the generic form of the Kalman Filter as shown in FIG.l.
- the differences reflect the use of equations (14) and (15) and the variables used in the battery SOC measurement.
- the differences reflect the use of equations (16) and (17) and the variables used in the battery SOC measurement.
- X *H now represents the predicted vector representing the internal states of the battery while ⁇ e ,/c( ⁇ ) 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 m ⁇ .
- FIG. 9 shows the operation of the Kalman filter according to an embodiment of the present invention.
- an algorithm is initialized with prior estimates of * * H and ⁇ e ,k ⁇ -) •
- **0 ⁇ ) is from function f in equation (14) while ⁇ e ,j t (-) is from function h in equation (15) .
- This embodiment is temperature-independent.
- * «H is from function f in equation (16) while ⁇ e , ⁇ (-) is from function h in equation (17) .
- This embodiment is temperature-dependent.
- the estimates * * ( ⁇ ) and ⁇ e , k ( ⁇ ) serve as the output from the predict component needed by the correct component.
- the Kalman gain L ⁇ is computed using matrix C, **H and ⁇ e ,*(-) . This corresponds to the first equation (equation 861) of correct component 802 in FIG. 8B.
- the predicted internal state vector ** ( - ) , the Kalman gain L ⁇ and the measurement from terminal voltage m* are used to calculate a corrected state vector *ftW . This corresponds to the second equation of the correct component in the Kalman filter.
- the predicted state-error covariance ⁇ e ,*(-) is used to compute a corrected state-error covariance ⁇ e , / t(+) • This corresponds to the third equation of the correct component.
- both of the equations of the predict component are calculated. Then the prediction for the next iteration is computed, namely %-i ( - ) and ⁇ ⁇ , A+I ( ⁇ ) •
- the time index k is incremented and the operation begins in block 901 again with the next time step.
- the internal state vector x* is: soc k
- FILTM SOC k +Ic 1 FILT 1 , + Ic 5
- the battery SOC is the first element of the state vector.
- the variables are defined as follows: I ⁇ is the instantaneous current, ⁇ t 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 ⁇ (I ⁇ ) 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.
- y fc is the terminal voltage.
- All other variables (k o ,ki,etc) 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.
- SOC ⁇ is the present SOC estimate
- SOC ⁇ 1 is the previous SOC estimate (and so forth)
- Lk is the present current measurement (and so forth)
- y ⁇ -i is the previous battery voltage estimate
- ⁇ , ⁇ and ⁇ are positive constants chosen to make an acceptable model with a parsimonious number of state variables.
- the governing equation for the SOC state is :
- h may be implemented using a neural network.
- a neural network may be used to estimate the internal states of the battery.
- 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.
- the arithmetic circuit can accommodate changing behaviors of the battery to use different parameters for different time periods.
- 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.
- HEV Hybrid Electric Vehicle
- EV Electric Vehicles
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- firmware may be implemented in hardware, software, or a combination thereof.
- 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.
- Arithmetic circuit 50 uses a mathematical model of the cell 22 that includes indicia of a dynamic system state.
- 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:
- ⁇ « ( ⁇ .. « . ⁇ )+v * »
- x k is the system state
- ⁇ k is the set of time varying model parameters
- u k is the exogenous input
- y k is the system output
- w k and v k are "noise" inputs—all quantities may be scalars or vectors.
- /(-,-,•) and g(-,y) are functions defined by the cell model being used. Non-time-varying numeric values required by the model may be embedded within /(•,-, ⁇ ) and g(-,v) , and are not included in ⁇ k
- the system state x k 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.
- the state might include: SOC, polarization voltage levels with respect to different time constants, and hysteresis levels, for example.
- the system exogenous input u k includes at minimum the present cell current i k , and may, optionally, include cell temperature
- the system parameters ⁇ k 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 y k corresponds to physically measurable cell quantities or those directly computable from measured quantities at minimum for example, the cell voltage under load.
- 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.
- 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.
- this method directly estimates state values for the cell where SOC is at least one of the states.
- An exemplary model has the form:
- d k g(x k ,u k , ⁇ k )+e k .
- the first equation states that the parameters ⁇ k are primarily constant, but that they may change slowly over time, in this instance, modeled by a "noise" process denoted, r k .
- the "output" d k is a function of the optimum parameter dynamics modeled by g(-,v) plus some estimation error e k .
- the optimum parameter dynamics g-(-.v) being a function of the system state x k , an exogenous input u k , and the set of time varying parameters ⁇ k .
- a procedure of filtering is applied.
- 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.
- An estimation-error covariance matrix ⁇ is also initialized.
- SOC 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.
- the previous parameter estimate is propagated forward in time.
- the new parameter estimate is equal to the old parameter estimate & k - @ k - ⁇ > anc * tne parameter error uncertainty is larger due to the passage of time (accommodated for in the model by the driving noise r k ) .
- 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, x (however estimated or provided) and parameter estimate, ⁇ ; the difference is used to update the values of ⁇ .
- the state estimate x may be propagated forward by the parameter estimate or may be supplied via an external means as identified above.
- C k may be computed using the following recurrence relationship: dg(x k ,u k , ⁇ ) _ dg ⁇ x k ,u k , ⁇ )
- dg ⁇ x k ,u k , ⁇ ) dx k d ⁇ d ⁇ 3c k d ⁇ ' ( 2 6 ) chic #X*fr-i» M iM»fl) , ⁇ f Oft-i>%-i>ff) tekr-i d ⁇ d ⁇ d ⁇ k _ x d ⁇
- the derivative calculations are recursive in nature, and evolve over time as the state x k evolves.
- the term dx o /d ⁇ 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.
- a recursive filter 1100 adapts the parameter estimate, ⁇ .
- 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-i, the previous time varying parameters estimate ⁇ £_ ⁇ and a corrected parameter uncertainty estimate ⁇ A _ j • Parameter time update/prediction
- block 1103 outputs predicted parameters ⁇ and predicted parameter uncertainty ⁇ - ⁇ to the parameter measurement
- Parameter measurement update block 110 which provides current parameter estimate ⁇ and parameter uncertainty estimate, ⁇ /c receives the predicted
- 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 /(,,)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.
- OCV open-circuit-voltage
- 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.
- LiPB Lithium-Ion Polymer Battery
- the states and parameters of interest are embedded m /(,,) and g(,,), and examples follow:
- C k is the cell capacity/capacities
- a xk ,...a n k are polarization voltage time constant (s)
- g lk ,...g n _ lk are the polarization voltage blending factor (s)
- R k is the cell resistance (s)
- M k is the hysteresis blending factor (s)
- ⁇ k is the hysteresis rate constant (s) .
- SOC is captured by one state of the model as part of function /(-,-,•) .
- This equation is: where At represents the inter-sample period (in seconds), C 4 represents the cell capacity (in ampere-seconds), z k is the cell SOC at time index k, i k is the cell current, and ⁇ ik is the Coulombic efficiency of a cell at current level i k .
- the polarization voltage levels are captured by several filter states. If we let there be n f polarization voltage time constants, then
- the matrix A j e y t ma y be a diagonal matrix with real- valued polarization voltage time constants a lk ---a n ⁇ k . If so, the system is stable if all entries have magnitude less than
- the vector B j e$R" /X may simply be set to n f "l"s.
- the entries of B f are not critical as long as they are non-zero.
- the value of W x entries in the ⁇ matrix are chosen as part of the system identification procedure to best fit the model parameters to measured cell data.
- the A j - , and B j - 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
- the overall model state is a combination of the above examples as follows:
- G k e 5R xn/ is a vector of polarization voltage blending factors g ⁇ yg n tk that blend the polarization voltage states together in the output
- R ⁇ is the cell resistance (different values may be used for discharge/charge)
- M k is the hysteresis blending factor.
- G k may be constrained such that the dc-gain from i k to G k f k is zero, which results in the estimates of R k being accurate.
- Some embodiments of the present invention may include methods to constrain the parameters of the model to result in a stable system.
- the state equation may include terms for polarization voltage time
- time constants a xk ---a n k are diagonal matrix with real-valued polarization voltage time constants a xk ---a n k .
- time constants may be computed as a ik -tanh(a ik ), where the parameter vector of the model contains the a ik values and not directly the a ik values.
- the tanh() function ensures that the a ik are always within +1 (i.e., stable) regardless of the value of a ik .
- 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 G k so that the dc-gain from i k to G k f k is zero, which results in the estimates of R k being accurate. This is done by enforcing that the last element of G k be computed using other elements of G k and the n ,- ⁇ polarization voltage time constants g n k - ⁇ " f g ⁇ k 0--a n k )l( ⁇ -a k ).
- 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.
- R ⁇ R k+r k (33)
- y k OCV(Z k )-i k R k +e k
- R k is the cell resistance and is modeled as a constant value with a fictitious noise process r k allowing adaptation.
- y k is an estimate of the cell's voltage
- i k is the cell current
- e k models estimation error. If an estimate of z k 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 y k is compared with the true measured cell voltage. Any difference resultant from the comparison is used to adapt R k .
- 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 R k 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.
- a filter is formulated using this model to produce a capacity estimate.
- 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
- 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.
- the computer program code segments configure the microprocessor to create specific logic circuits.
Abstract
Description
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CA2588856A CA2588856C (en) | 2004-11-29 | 2004-11-29 | Method and system for battery state and parameter estimation |
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