Adaptive warburg-impedance-based control system

Abstract

A method includes obtaining a set of complex impedance measurements along a Warburg impedance for an energy storage device. The method includes determining, based on the set of complex impedance measurements, a slope and a real-axis intercept in a Nyquist plot for the Warburg impedance. The method includes configuring, based on the determined slope and the determined real-axis intercept, a configuration of a series of resistor-capacitor circuits to model a finite-space approximation of the Warburg impedance.

Description

ADAPTIVE WARBURG-IMPEDANCE-BASED CONTROL SYSTEMCROSS-REFERENCE TO RELATED APPLICATION

[0001] This application claims the benefit of U.S. Provisional Patent Application No. 63 / 504,822, filed May 30, 2023, the disclosure of which is hereby incorporated by reference in its entirety herein.FIELD

[0002] The present disclosure relates to battery management, and in particular, to predicting battery health and performance parameters using Warburg impedance.BACKGROUND

[0003] The health and performance of electrochemical or energy storage devices (for example, high-capacity batteries) can be estimated using an equivalent circuit model (ECM) of the battery. An ECM uses circuit components to characterize the battery’s operation and internal state. Common circuit components used in ECMs include voltage sources, resistors, capacitors, and constant phase elements (CPEs). Mathematical representations of many of these components can be translated into a discrete time-space, which is configurable for embedded microcontroller firmware that provides real-time monitoring and control. The mathematical representations for CPEs, however, cannot be translated into the discrete time-space, meaning that ECMs that incorporate CPEs generally cannot be expressed in a form suitable for real-time microcontroller use. Health and performance estimation of energy storage devices is especially useful in automotive applications to determine the instantaneous status of a vehicle battery. For example, battery sag, impedance, and / or heat can affect available energy and therefore vehicle control and performance parameters.

[0004] The background description provided here is for the purpose of generally presenting the context of the disclosure. Work of the presently named inventors, to the extent it is described in this background section, as well as aspects of the description that may not otherwise qualify as prior art at the time of filing, are neither expressly nor impliedly admitted as prior art against the present disclosure.SUMMARY

[0005] A method includes obtaining a set of complex impedance measurements along a Warburg impedance for an energy storage device. The method includes determining, based on the set of complex impedance measurements, a slope and a real-axis intercept in a Nyquist plot for the Warburg impedance. The method includes configuring, based on the determined slope and the determined real -axis intercept, a configuration of a series of resistor-capacitor circuits to model a finite-space approximation of the Warburg impedance.

[0006] In other features, the method includes, managing the energy storage device based on an equivalent circuit model that incorporates the finite-space approximation of the Warburg impedance. In other features, the series of resistor-capacitor circuits have respective time constants with frequency values at different intervals along a logarithmic scale. In other features, the series of resistor-capacitor circuits are each tangent to a linear fit of the set of complex impedance measurements.

[0007] In other features, the respective time constants of the series of resistor-capacitor circuits include at least two of 0.1, 0.01, or 0.001 Hertz. In other features, a right-hand side of the finite- space approximation of the Warburg impedance ends at a resistor-capacitor circuit with a smallest time constant. In other features, the finite-space approximation of the Warburg impedance and the equivalent circuit model for the energy storage device is updated at regular intervals.

[0008] In other features, the equivalent circuit model or the finite-space approximation of the Warburg impedance are stored in an index along with at least two of an age of the energy storage device, a state-of-charge of the energy storage device, a state-of-health of the energy storage device, or an ambient temperature to generate a multidimensional map of the energy storage device.

[0009] In other features, the index further includes the set of complex impedance measurements and an age or timestamp associated with each entry. In other features, the index further includes a confidence parameter associated with each equivalent circuit model or finite- space approximation of the Warburg impedance based on the age or the timestamp associated with each entry.

[0010] In other features, the method includes determining an updated finite-space approximation of the Warburg impedance for the energy storage device by interpolating two or more finite-space approximations of the Warburg impedance in the index based on a current ambient temperature and the age of the two or more finite-space approximations. In otherfeatures, the multidimensional map is stored in a nonvolatile memory. In other features, the method includes reassigning the energy storage device to a different use application based on the multidimensional map. In other features, the method includes determining a recyclability of electrochemical elements in the energy storage device based on the multidimensional map.

[0011] In other features, the multidimensional map is stored in a remote nonvolatile memory along with other multidimensional maps associated with similar energy storage devices. In other features, the set of complex impedance measurements are obtained during a charging session of the energy storage device, during a discharge event of the energy storage device, or by selectively switching on and off one or more loads electrically connected to the energy storage device.

[0012] A system includes memory hardware configured to store instructions. The system includes processing hardware configured to execute the instructions stored by the memory hardware. The instructions include obtaining a set of complex impedance measurements along a Warburg impedance for an energy storage device. The instructions include determining, based on the set of complex impedance measurements, a slope and a real-axis intercept in a Nyquist plot for the Warburg impedance. The instructions include configuring, based on the determined slope and the determined real-axis intercept, a configuration of a series of resistor-capacitor circuits to model a finite-space approximation of the Warburg impedance.

[0013] In other features, the instructions include managing the energy storage device based on an equivalent circuit model that incorporates the finite-space approximation of the Warburg impedance. In other features, the series of resistor-capacitor circuits have respective time constants with frequency values at different intervals along a logarithmic scale. In other features, the series of resistor-capacitor circuits are each tangent to a linear fit of the set of complex impedance measurements.

[0014] In other features, the instructions include updating the finite-space approximation of the Warburg impedance and the equivalent circuit model for the energy storage device at regular intervals. In other features, the instructions include storing the equivalent circuit model or the finite-space approximation of the Warburg impedance in an index along with the set of complex impedance measurements, an age or timestamp associated with each entry, and at least two of an age of the energy storage device, a state-of-charge of the energy storage device, a state-of-health of the energy storage device, or an ambient temperature to generate a multidimensional map of the energy storage device.

[0015] In other features, the instructions include determining an updated finite-space approximation of the Warburg impedance for the energy storage device by interpolating two or more finite-space approximations of the Warburg impedance in the index based on a current ambient temperature and the age of the two or more finite-space approximations.

[0016] In other features, the set of complex impedance measurements are obtained during a charging session of the energy storage device, during a discharge event of the energy storage device, or by selectively switching on and off one or more loads electrically connected to the energy storage device.

[0017] A non-transitory computer-readable storage medium stores processor-executable instructions. The instructions include obtaining a set of complex impedance measurements along a Warburg impedance for an energy storage device. The instructions include determining, based on the set of complex impedance measurements, a slope and a real-axis intercept in a Nyquist plot for the Warburg impedance. The instructions include configuring, based on the determined slope and the determined real-axis intercept, a configuration of a series of resistor-capacitor circuits to model a finite-space approximation of the Warburg impedance.

[0018] In other features, the instructions include managing the energy storage device using an equivalent circuit model that incorporates the finite-space approximation of the Warburg impedance.

[0019] Further areas of applicability of the present disclosure will become apparent from the detailed description, the claims, and the drawings. The detailed description and specific examples are intended for purposes of illustration only and are not intended to limit the scope of the disclosure.BRIEF DESCRIPTION OF THE DRAWINGS

[0020] The present disclosure will become more fully understood from the detailed description and the accompanying drawings.

[0021] FIG. 1 illustrates an example configuration of systems for an adaptive Warburg impedance calculator that can be used in an ECM to control and manage a battery of a vehicle.

[0022] FIGS. 2 and 3 illustrate an example ECM and corresponding Nyquist plot, respectively, that approximate the performance of an example battery.

[0023] FIGS. 4 and 5 illustrate an example ECM and corresponding Nyquist plot, respectively, that provide a more complex approximation of the performance of another example battery.

[0024] FIG. 6 illustrates an example Nyquist plot that exemplifies a method for calculating an adaptive Warburg impedance for a battery.

[0025] FIG. 7 illustrates an example method of an adaptive Warburg impedance calculator that can be used in an ECM to control and manage an energy storage device.

[0026] In the drawings, reference numbers may be reused to identify similar and / or identical elements.DETAILED DESCRIPTIONOVERVIEW

[0027] As described above, the health and performance of batteries and energy storage devices can be modeled using an ECM. An ECM uses circuit components (e.g., voltage sources, resistors, capacitors, CPEs) to characterize the battery operation and internal state. Such circuit components directly relate to the impedance of the battery, which includes a real impedance, an imaginary impedance that forms a semicircle on the imaginary plane, and a straight line on the imaginary plane that represents the Warburg impedance of the battery. The Warburg impedance is generally a semi-infinite linear diffusion impedance for the battery. Greater fidelity for an ECM model can be achieved by cascading additional circuit elements to represent the battery’s operation. This combination of resistors, capacitors, and CPEs can result in increasingly complex-shaped impedance curves. As the number of circuit elements in the ECM and the complexity of the impedance curve increase, the experimental requirements, cost, equipment, and time to parametrize the circuit elements also increases.

[0028] The ECMs can often be expressed mathematically as differential equations. The differential equations for resistors and capacitors can be translated into a discrete time-space using the Z-transform, which renders them suitable for embedded microcontroller firmware. As a result, RC networks or models are an attractive approach to predicting a battery’s behavior in real-time monitoring solutions. CPEs, however, cannot be translated into the discrete time-space (e.g., the z-domain), which makes them difficult to embody in real-time sampling solutions. This characteristic of CPEs limits the ability of ECMs to accurately represent impedances at time constants that exceed the first RC portion of the characteristic impedance of the battery (e.g., in the low-frequency domain). The Warburg impedance (which is represented by one or more CPEs) is an important attribute of the battery’s characteristic impedance, especially in the low- frequency domain. The inability to implement CPEs in discrete-time calculations suitable for microcontroller code severely limits the model’s low-frequency accuracy and usefulness.

[0029] This document describes techniques and systems for an adaptive Warburg impedance calculator that can be used to manage (including configuring and / or controlling) a battery or other energy storage device. As one example, a battery management system measures or obtains multiple complex impedance measurements along a Warburg impedance for the battery. Using at least two of the complex impedance measurements, a slope and real-axis intercept (in a Nyquist plot) for the Warburg impedance can be determined. Using the slope and real-axis intercept, the battery management system can then determine a series of RC circuits to model a finite-space approximation of the Warburg impedance. In some implementations, the approximation of the Warburg impedance is optimized. In some implementations, an optimized approximation meets an accuracy threshold (for example above 50, 70, 80, or 90 percent accuracy). The battery management system can then use an ECM that incorporates the finite- space approximation of the Warburg impedance to manage and control the battery. In this way, accurate and real-time ECMs can be generated to more efficiently manage and control energy storage devices throughout their lifetime.

[0030] This example is just one example of the described techniques and systems for an adaptive Warburg impedance calculator that can be used to control and manage batteries or other energy storage devices. This document describes other examples and implementations.VEHICLE CONFIGURATION

[0031] FIG. 1 illustrates an example configuration of systems for an adaptive Warburg impedance calculator that can be used in an ECM to control and manage a battery 104 of a host vehicle 102. The host vehicle 102 may include the battery 104, one or more sensors 106, one or more processors 108, computer-readable storage media (CRM) 110, and one or more communication devices 120. The host vehicle 102 may be a motorized vehicle (e.g., a car, an automobile, a motorcycle, a passenger truck, a bus, a tractor, a semi-trailer truck), a watercraft (e.g., a boat), or an aircraft (e.g., an airplane). The host vehicle 102 also includes a battery management system 112 and a battery indexer 118. The battery management system 112 estimates a real-time Warburg impedance and ECM. The battery indexer 118 saves the Warburg impedance and ECM over time along with other data to generate a multidimensional map of the battery’s health and performance under a variety of conditions.

[0032] The battery 104 is an energy storage device of the host vehicle 102. For example, the battery 104 may be a rechargeable Lithium-ion battery used in an electric vehicle. The battery 104 may also be another storage device (e.g., a battery associated with an airbag system or starter battery) located in the host vehicle 102. The term “battery” or “energy storage device” may be used interchangeably in this document to refer to one or more electrochemical cells andother energy storage devices configured to provide electrical energy to a connected energy resource or load during discharge and accept and store electrical energy during charging from an energy source.

[0033] Batteries and other energy storage devices are used in many personal, commercial, and automotive applications. These devices are available in various sizes and chemistries including, for example, Zinc-Carbon, Alkaline, Lead Acid, Nickel Cadmium, Nickel-Zinc, Nickel Metal Hydride, Lithium-Ion, solid-state, and Sodium-Ion. In addition, batteries usually must perform in a variety of conditions, including severe environments (for example, with extreme temperatures). Common applications include computers, medical devices, communication devices, data centers, aircraft, grid storage, grid ancillary services, hybrid vehicles, and electric vehicles. High cost and long-life requirements for many of these applications drive the need for a quick, accurate, and robust battery management system. In order to efficiently integrate batteries in these applications, it is useful to characterize their internal condition including state- of-charge (SOC), state-of-health (SOH), remaining-useful-life (RUL), and end-of-life (EOL).

[0034] The sensors 106 provide measurement data to assist in estimating and predicting the health and performance of the battery 104. The sensors 106 may provide impedance spectra, temperatures, offload and underload conditions, voltages, and currents for the battery 104. The sensors 106 may also provide data regarding the environment (e.g., outside temperature) around the host vehicle 102. The sensors 106 periodically collect data and provide the data to the CRM 110 or another storage device to facilitate the calculation of an adaptive and real-time Warburg impedance and ECM for the battery 104.

[0035] The processors 108 (e.g., energy processing units, electronic control units, or microcontrollers) may be a microprocessor or a system-on-chip. The processors 108 execute instructions stored in the CRM 110, on one or more disks, memories, or other non-transitory computer-readable storage media. For example, processors 108 process sensor data from sensors 106 and execute instructions loaded from the CRM 110 that cause the processors 108 to adaptively estimate the Warburg impedance and determine an ECM to improve battery management and control. For example, the instructions may cause the processors 108 to be configured to collect sensor data regarding battery 104 and determine an updated Warburg impedance and ECM for battery 104.

[0036] The battery management system 112 can be stored in the CRM 110. The battery management system 112 may include a Warburg impedance estimator 114 and an equivalent circuit modeler 116. The Warburg impedance estimator 114 periodically estimates an adaptive, real-time Warburg impedance for the battery 104. As described in greater detail with respect toFIG. 6, the Warburg impedance estimator 114 can approximate the Warburg impedance of the battery 104 using a series of RC networks. An initial Warburg impedance can be computed using laboratory measurements and pre-programmed into the Warburg impedance estimator. During operation and the lifetime of the battery 104, a real-time Warburg impedance can be computed by collecting at least two complex impedance measurements in the battery’s low- frequency domain. The complex impedance measurements allow the Warburg impedance calculator to determine a slope and real -axis intercept for the Warburg impedance and calculate, using the slope, a series of RC networks to approximate the Warburg impedance. The equivalent circuit modeler 116 uses the approximated Warburg impedance to periodically generate or update an ECM for the battery 104.

[0037] The battery indexer 118 can also be stored in the CRM 110 or another CRM. As described above, the battery indexer 118 indexes the complex impedance measurements, the approximated Warburg impedance, and / or the ECM according to the lifetime, state of charge, state of health, outside temperature, and / or other data to create a multidimensional map of the battery’s health and performance under a variety of conditions. The battery indexer 118 may also maintain a knowledge of the age of each index in the multidimensional map, which is continually updated with new entries throughout the life of the battery.

[0038] The communication devices 120 may include a sensor interface and a vehicle-based system interface. The sensor interface and the vehicle-based system interface transmit data (e.g., sensor data, state of charge, outside temperature, Warburg impedance approximation, ECM) over a communication bus of the host vehicle 102, for example, when the individual components of the sensors 106, battery management system 112, and / or the battery indexer 118 are integrated within the host vehicle 102. The communication devices 120 may also facilitate the exchange of data, including outside temperature and weather forecasts, over a wireless link between a remote system and the host vehicle 102. The communication devices 120 may include hardware, software, and / or firmware required to communicate via Wi-Fi, wireless local area networks (WLANs), Dedicated Short-Range Communication, Vehicle-to-Everything (V2X), or cellular communication.ECMs AND NYQUIST PLOTS

[0039] FIGS. 2 and 3 illustrate an example ECM 200 and corresponding Nyquist plot 310, respectively, that approximate the performance of an example battery. The ECM 200 includes a first resistor (Ri) 202-1, a second resistor (R2) 202-2, a capacitor (Ci) 204, and a CPE 206. The Nyquist plot 310 illustrates the complex impedance of the example battery with the horizontalaxis (e.g., the real axis) representing the real impedance (Zreai) and the vertical axis (e.g., the imaginary axis) representing the imaginary impedance (Zimag).

[0040] As illustrated in the Nyquist plot 310, the battery’s impedance includes a semicircle 312 and a Warburg impedance 314. The semicircle 312 is represented in the ECM 200 as the first resistor 202-1 and the capacitor 204 (e.g., a first RC circuit). The Warburg impedance 314 is represented in the ECM 200 as the CPE 206 and manifests in the Nyquist plot 310 as an ideal forty-five-degree (45°) line. The Warburg impedance 314 represents a semiinfinite linear diffusion impedance in the low-frequency domain of the example battery.

[0041] FIGS. 4 and 5 illustrate an example ECM 400 and corresponding Nyquist plot 512, respectively, that provide a more complex approximation of the performance of another example battery (e.g., the battery 104 of FIG. 1). The ECM 400 includes a first RC circuit (RiCi), which is a combination of a first resistor 402-1 and a first capacitor 404-1, and a second RC circuit (R2C2), which is a combination of a second resistor 402-2 and a second capacitor 404-2. The ECM 400 also includes a third resistor 402-3, a CPE 406, a voltage source 408, and a voltage terminal 410.

[0042] The Nyquist plot 512 illustrates the complex impedance of the example battery with the real axis representing the real impedance (Zreai) and the imaginary axis representing the imaginary impedance (Zimag). As illustrated in the Nyquist plot 512, the battery’s impedance includes a high-frequency impedance 514 (e.g., 1 Hz - 5,000 Hz), a semicircle 516 in the middle-frequency range (e.g., 0.05 Hz - 1 Hz), and a Warburg impedance 518 in the low- frequency range (e.g., 0.0005 Hz - 0.05 Hz). The high-frequency impedance 514 is represented in the ECM 400 as the third resistor 402-3. Semicircle 516 is represented in the ECM 400 as the combination of the first and second RC circuits. The Warburg impedance 518 is represented in the ECM 400 as the CPE 406 and manifests in the Nyquist plot 512 as a nearly forty-five-degree (45°) line.

[0043] FIG. 6 illustrates an example Nyquist plot 600 that exemplifies a method for calculating an adaptive Warburg impedance for a battery. Similar to FIGS. 3 and 5, the Nyquist plot 600 illustrates the complex impedance of the example battery with the horizontal axis (e.g., the real axis) representing the real impedance (Zreai) and the vertical axis (e.g., the imaginary axis) representing the imaginary impedance (Zimag). As illustrated in the Nyquist plot 600, the battery’s impedance includes a semicircle 602 and a Warburg impedance 604.

[0044] The Nyquist plot 600 also illustrates a series of cascading RC circuits 606 including a first RC circuit (R1C1) 606-1, a second RC circuit (R2C2) 606-2, and a third RC circuit (R3C3)606-3 that are calculated using a slope and real-axis intercept of the Warburg impedance 604. The slope and real-axis intercept are determined using at least two of the impedance measurements 608 (e.g., a first impedance measurement 608-1, a second impedance measurement 608-2, and a third impedance measurement 608-3). The impedance measurements are taken at different frequency values to the right of the semicircle 602. Once the slope and real-axis intercept of the Warburg impedance 604 are determined, the cascading RC circuits 606 are determined with real-axis intercepts to the right of the real-axis intercept of the Warburg impedance 604 and a point tangential to the Warburg impedance 604. In the Nyquist plot 600, the time constants for the first RC circuit 606-1, the second RC circuit 606-2, and the third RC circuit 606-3 are taken at 0.1 Hz, 0.01 Hz, and 0.001 Hz, respectively. In the illustrated example, the RC circuits are spaced apart with a logarithmic spacing of time constants. In other implementations, the time constants of the RC circuits can be spaced apart in a different manner.

[0045] The cascading RC circuits 606 provide an equivalent CPE in a time-sampling system to approximate the Warburg impedance 604. The approximation is generally valid up to the time constant of the last RC circuit of cascading RC circuits 606 used (e.g., illustrated here as the third RC circuit (R3C3) 606-3).

[0046] The Warburg impedance 604 is illustrated as having a forty-five-degree (45°) angle. In general, the state of charge, temperature, or aging of the battery 104 may result in a different angle to the Warburg impedance 604. An initial Warburg impedance 604 or its angle for a particular battery can be computed using laboratory measurements and pre-programmed into the Warburg impedance estimator 114.

[0047] During its operation, a real-time Warburg impedance 604 can be computed by measuring at least two points along the Warburg impedance 604 and determining its slope. The slope of the Warburg impedance 604 can be computed using any two or more of the impedance measurements 608. The impedance measurements 608 can be initiated by using a controllable or programmable battery charger, selectively switching on and off certain loads to the battery 104, or frequency information harvested from the normal operation of the host apparatus. In the case of the host vehicle 102, the normal operations of acceleration and deceleration cause discharge and recharge events for the battery. Once the slope of the Warburg impedance 604 is determined, the RC circuits can be automatically calculated to approximate the Warburg impedance 604, thus generating a finite-space Warburg impedance approximation. The end of the Warburg impedance approximation is equal to a frequency of the / / th RC circuit 606-n.METHODS

[0048] FIG. 7 illustrates an example method 700 of an adaptive Warburg impedance calculator that can be used in an ECM to control and manage an energy storage device. Method 700 is shown as operations (or acts) performed, but not necessarily limited to the order or combinations in which the operations are shown herein. Further, any one of one or more of the operations may be repeated, combined, or reorganized to provide other methods. In portions of the following discussion, reference may be made to the host vehicle 102 of FIG. 1, and entities detailed in FIGS. 1-6 reference to which is made for example only. The techniques are not limited to performance by one entity or multiple entities.

[0049] At step 702, a set of (for example, at least two) complex impedance measurements along a Warburg impedance for an energy storage device are obtained. For example, the battery management system 112 can obtain or take multiple complex impedance measurements of the battery 104 in a low-frequency domain along the battery’s Warburg impedance. The complex impedance measurements may be obtained during a charging session or a discharge event of the battery 104. The charging session may be programmable or controllable to facilitate obtaining the complex impedance measurements. The discharge event may include acceleration of the host vehicle 102. Alternatively, the complex impedance measurements may be obtained by selectively switching on and off one or more loads electrically connected to the battery 104.

[0050] At step 704, a slope and a real -axis intercept in a Nyquist plot for the Warburg impedance are determined using the complex impedance measurements. For example, the Warburg impedance estimator 114 uses the complex impedance measurements to determine the slope and real-axis intercept for the real-time Warburg impedance of the battery 104.

[0051] At step 706, a series of resistor-capacitor circuits are configured based on the slope and real-axis intercept to model a finite-space approximation of the Warburg impedance. For example, the Warburg impedance estimator 114 uses the slope and real-axis intercept to determine multiple resistor-capacitor circuits to model a finite-space approximation of the Warburg impedance. Each resistor-capacitor circuit has a semicircle in the Nyquist plot that is tangent to a linear fit of the complex impedance measurements or a line formed by the already- determined slope and real-axis intercept.

[0052] The series of resistor-capacitor circuits may have respective time constants with frequency values at different intervals along a logarithmic scale. For example, the time constants of the series of resistor-capacitor circuits may include at least two of 0.1, 0.01, or 0.001 Hertz. A right-hand side of the finite-space approximation ends at a resistor-capacitor circuit with the smallest time constant.

[0053] At step 708, the energy storage device is managed (for example controlled and / or configured), based on an equivalent circuit model (ECM) that incorporates the finite-space approximation of the Warburg impedance. For example, the equivalent circuit modeler 116 can use the finite approximation of the Warburg impedance to generate an ECM for the battery 104. Because the ECM does not use CPEs to model the Warburg impedance, the ECM can be translated into the z-domain or discrete time-space. The battery management system 112 or another system of the host vehicle 102 can then use ECM to control or manage the battery 104.

[0054] The battery management system 112 may also update the ECM and the finite-space approximation of the Warburg impedance at regular intervals during the lifetime of the battery 104. At step 710, control determines whether the time interval has elapsed. If the interval has elapsed, the Warburg impedance is updated by returning to step 702. If the interval has not elapsed, management of the energy storage device continues (returning to step 708). In some implementations, the management of the energy storage device continues while the Warburg impedance is updated (for example by using the previous impedance and / or ECM). In some implementations, the Warburg impedance is updated at frequency of 1000 Hz - 0.01 Hz.

[0055] The battery indexer 118 may store or save the ECM and / or the finite-space approximation of the Warburg impedance in an index. The index entries may also include the complex impedance measurements, the age of the battery 104, the state-of-charge of the battery 104, the state of health of the battery 104, the ambient temperature, or the age or a timestamp associated with each entry to generate a multidimensional map. The index or multidimensional map may also include a confidence parameter associated with each equivalent circuit model or finite-space approximation of the Warburg impedance based on its respective age or timestamp. For example, the battery indexer 118 or the battery management system 112 may determine an updated finite-space approximation of the Warburg impedance for the battery 104 by interpolating multiple finite-space approximations of the Warburg impedance in the index based on a current ambient temperature and the age of the respective entries.

[0056] The battery indexer 118 may store the multidimensional map in a nonvolatile memory (e.g., the CRM 110) or in a remote nonvolatile memory (e.g., on a remote server or computer system). If the multidimensional map or index is stored in a remote memory, it may also be stored with data from other vehicles with similar batteries. The battery indexer 118 may continually update a stored version of the multidimensional map or index.

[0057] At the end of a battery’s lifetime, the multidimensional map may be used to reassign the battery 104 to a different use application (e.g., solar energy storage or grid energy storage). Similarly, the recyclability of electrochemical elements in battery 104 may be determined basedon the multidimensional map. In this way, the lifetime index of the battery’s performance and health can improve the selection of subsequent use for the battery 104.EXAMPLES

[0058] In the following section, examples are provided.

[0059] Example 1. A method comprising: obtaining at least two complex impedance measurements along a Warburg impedance for an energy storage device; determining, using the at least two complex impedance measurements, a slope and a real -axis intercept in a Nyquist plot for the Warburg impedance; determining a series of resistor-capacitor circuits based on the slope and real-axis intercept to model a finite-space approximation of the Warburg impedance; and managing or controlling the energy storage device using an equivalent circuit model that incorporates the finite-space approximation of the Warburg impedance.

[0060] Example 2. The method of Example 1, wherein the series of resistor-capacitor circuits: have respective time constants with frequency values at different intervals along a logarithmic scale; and are each tangent to a linear fit of the at least two complex impedance measurements.

[0061] Example 3. The method of Example 2, wherein the respective time constants of the series of resistor-capacitor circuits include at least two of 0.1, 0.01, or 0.001 Hertz.

[0062] Example 4. The method of Example 2 or 3, wherein a right-hand side of the finite- space approximation of the Warburg impedance ends at a resistor-capacitor circuit with a smallest time constant.

[0063] Example 5. The method of any one of the preceding Examples, wherein the finite-space approximation of the Warburg impedance and the equivalent circuit model for the energy storage device is updated at regular intervals.

[0064] Example 6. The method of any one of the preceding Examples, wherein the equivalent circuit model or the finite-space approximation of the Warburg impedance are stored in an index along with at least two of an age of the energy storage device, a state-of-charge of the energy storage device, a state-of-health of the energy storage device, or an ambient temperature to generate a multidimensional map of the energy storage device.

[0065] Example 7. The method of Example 6, wherein the index further includes the at least two complex impedance measurements and an age or timestamp associated with each entry.

[0066] Example 8. The method of Example 7, wherein the index further includes a confidence parameter associated with each equivalent circuit model or finite-space approximation of the Warburg impedance based on the age or the timestamp associated with each entry.

[0067] Example 9. The method of Example 7 or 8, the method further comprising: determining an updated finite-space approximation of the Warburg impedance for the energy storage device by interpolating two or more finite-space approximations of the Warburg impedance in the index based on a current ambient temperature and the age of the two or more finite-space approximations.

[0068] Example 10. The method of any one of Examples 6 through 9, wherein the multidimensional map is stored in a nonvolatile memory.

[0069] Example 11. The method of Example 10, wherein the method further comprises: reassigning the energy storage device to a different use application based on the multidimensional map; or determining a recyclability of electrochemical elements in the energy storage device based on the multidimensional map.

[0070] Example 12. The method of any one of Examples 6 through 9, wherein the multidimensional map is stored in a remote nonvolatile memory along with other multidimensional maps associated with similar energy storage devices.

[0071] Example 13. The method of any one of the preceding Examples, wherein the at least two complex impedance measurements are obtained: during a charging session of the energy storage device; during a discharge event of the energy storage device; or by selectively switching on and off one or more loads electrically connected to the energy storage device.

[0072] Example 14. A system comprising one or more processors configured to perform the method of any one of Examples 1 through 13.

[0073] Example 15. A non-transitory computer-readable medium that stores computerexecutable instructions that, when executed by a processor, cause the processor to perform the method of any one of Examples 1 through 13.CONCLUSION

[0074] The foregoing description is merely illustrative in nature and is in no way intended to limit the disclosure, its application, or uses. The broad teachings of the disclosure can be implemented in a variety of forms. Therefore, while this disclosure includes particular examples, the true scope of the disclosure should not be so limited since other modifications will become apparent upon a study of the drawings, the specification, and the following claims. In the written description and claims, one or more steps within a method may be executed in a different order (or concurrently) without altering the principles of the present disclosure. Similarly, one or more instructions stored in a non-transitory computer-readable medium may be executed in a different order (or concurrently) without altering the principles of the present disclosure. Unless indicatedotherwise, numbering or other labeling of instructions or method steps is done for convenient reference, not to indicate a fixed order.

[0075] Further, although each of the embodiments is described above as having certain features, any one or more of those features described with respect to any embodiment of the disclosure can be implemented in and / or combined with features of any of the other embodiments, even if that combination is not explicitly described. In other words, the described embodiments are not mutually exclusive, and permutations of one or more embodiments with one another remain within the scope of this disclosure.

[0076] Spatial and functional relationships between elements (for example, between modules) are described using various terms, including “connected,” “engaged,” “interfaced,” and “coupled.” Unless explicitly described as being “direct,” when a relationship between first and second elements is described in the above disclosure, that relationship encompasses a direct relationship where no other intervening elements are present between the first and second elements as well as an indirect relationship where one or more intervening elements are present (either spatially or functionally) between the first and second elements.

[0077] As noted below, the term “set” generally means a grouping of one or more elements. However, in various implementations a “set” may, in certain circumstances, be the empty set (in other words, the set has zero elements in those circumstances). As an example, a set of search results resulting from a query may, depending on the query, be the empty set. In contexts where it is not otherwise clear, the term “non-empty set” can be used to explicitly denote exclusion of the empty set — that is, a non-empty set will always have one or more elements.

[0078] A “subset” of a first set generally includes some of the elements of the first set. However, in various implementations, a subset of the first set may not be a proper subset: in certain circumstances, the subset may be coextensive with (equal to) the first set. In contexts where it is not otherwise clear, the term “proper subset” can be used to explicitly denote that a subset of the first set must exclude at least one of the elements of the first set. Further, in various implementations, the term “subset” does not necessarily exclude the empty set. As an example, consider a set of candidates that was selected based on first criteria and a subset of the set of candidates that was selected based on second criteria; if no elements of the set of candidates met the second criteria, the subset may be the empty set. In contexts where it is not otherwise clear, the term “non-empty subset” can be used to explicitly denote exclusion of the empty set.

[0079] In the figures, the direction of an arrow, as indicated by the arrowhead, generally demonstrates the flow of information (such as data or instructions) that is of interest to theillustration. For example, when element A and element B exchange a variety of information but information transmitted from element A to element B is relevant to the illustration, the arrow may point from element A to element B. This unidirectional arrow does not imply that no other information is transmitted from element B to element A. Further, for information sent from element A to element B, element B may send requests for, or receipt acknowledgements of, the information to element A.

[0080] In this application, including the definitions below, the term “module” can be replaced with the term “controller” or the term “circuit.” In this application, the term “controller” can be replaced with the term “module.”

[0081] The term “module” may refer to, be part of, or include processor hardware (shared, dedicated, or group) that executes code coupled with memory hardware (shared, dedicated, or group) that stores code executed by the processor hardware.

[0082] The module may include one or more interface circuit(s). In some examples, the interface circuit(s) may implement wired or wireless interfaces that connect to a local area network (LAN) or a wireless personal area network (WPAN). Examples of a LAN are Institute of Electrical and Electronics Engineers (IEEE) Standard 802.11-2020 (also known as the WIFI wireless networking standard) and IEEE Standard 802.3-2018 (also known as the ETHERNET wired networking standard). Examples of a WPAN are IEEE Standard 802.15.4 (including the ZIGBEE standard from the ZigBee Alliance) and, from the Bluetooth Special Interest Group (SIG), the BLUETOOTH wireless networking standard (including Core Specification versions 3.0, 4.0, 4.1, 4.2, 5.0, and 5.1 from the Bluetooth SIG).

[0083] The module may communicate with other modules using the interface circuit(s). Although the module may be depicted in the present disclosure as logically communicating directly with other modules, in various implementations the module may actually communicate via a communications system. The communications system includes physical and / or virtual networking equipment such as hubs, switches, routers, and gateways. In some implementations, the communications system connects to or traverses a wide area network (WAN) such as the Internet. For example, the communications system may include multiple LANs connected to each other over the Internet or point-to-point leased lines using technologies including Multiprotocol Label Switching (MPLS) and virtual private networks (VPNs).

[0084] In various implementations, the functionality of the module may be distributed among multiple modules that are connected via the communications system. For example, multiple modules may implement the same functionality distributed by a load balancing system. In afurther example, the functionality of the module may be split between a server (also known as remote, or cloud) module and a client (or, user) module. For example, the client module may include a native or web application executing on a client device and in network communication with the server module.

[0085] The term code, as used above, may include software, firmware, and / or microcode, and may refer to programs, routines, functions, classes, data structures, and / or objects. Shared processor hardware encompasses a single microprocessor that executes some or all code from multiple modules. Group processor hardware encompasses a microprocessor that, in combination with additional microprocessors, executes some or all code from one or more modules. References to multiple microprocessors encompass multiple microprocessors on discrete dies, multiple microprocessors on a single die, multiple cores of a single microprocessor, multiple threads of a single microprocessor, or a combination of the above.

[0086] The memory hardware may also store data together with or separate from the code. Shared memory hardware encompasses a single memory device that stores some or all code from multiple modules. One example of shared memory hardware may be level 1 cache on or near a microprocessor die, which may store code from multiple modules. Another example of shared memory hardware may be persistent storage, such as a solid state drive (SSD) or magnetic hard disk drive (HDD), which may store code from multiple modules. Group memory hardware encompasses a memory device that, in combination with other memory devices, stores some or all code from one or more modules. One example of group memory hardware is a storage area network (SAN), which may store code of a particular module across multiple physical devices. Another example of group memory hardware is random access memory of each of a set of servers that, in combination, store code of a particular module. The term memory hardware is a subset of the term computer-readable medium.

[0087] The apparatuses and methods described in this application may be partially or fully implemented by a special purpose computer created by configuring a general purpose computer to execute one or more particular functions embodied in computer programs. Such apparatuses and methods may be described as computerized or computer-implemented apparatuses and methods. The functional blocks and flowchart elements described above serve as software specifications, which can be translated into the computer programs by the routine work of a skilled technician or programmer.

[0088] The computer programs include processor-executable instructions that are stored on at least one non-transitory computer-readable medium. The computer programs may also include or rely on stored data. The computer programs may encompass a basic input / output system(BIOS) that interacts with hardware of the special purpose computer, device drivers that interact with particular devices of the special purpose computer, one or more operating systems, user applications, background services, background applications, etc.

[0089] The computer programs may include: (i) descriptive text to be parsed, such as HTML (hypertext markup language), XML (extensible markup language), or JSON (JavaScript Object Notation), (ii) assembly code, (iii) object code generated from source code by a compiler, (iv) source code for execution by an interpreter, (v) source code for compilation and execution by a just-in-time compiler, etc. As examples only, source code may be written using syntax from languages including C, C++, C#, Objective-C, Swift, Haskell, Go, SQL, R, Lisp, Java®, Fortran, Perl, Pascal, Curl, OCaml, JavaScript®, HTML5 (Hypertext Markup Language 5th revision), Ada, ASP (Active Server Pages), PHP (PHP: Hypertext Preprocessor), Scala, Eiffel, Smalltalk, Erlang, Ruby, Flash®, Visual Basic®, Lua, MATLAB, SIMULINK, and Python®.

[0090] The term non-transitory computer-readable medium does not encompass transitory electrical or electromagnetic signals propagating through a medium (such as on a carrier wave). Non-limiting examples of a non-transitory computer-readable medium are nonvolatile memory circuits (such as a flash memory circuit, an erasable programmable read-only memory circuit, or a mask read-only memory circuit), volatile memory circuits (such as a static random access memory circuit or a dynamic random access memory circuit), magnetic storage media (such as an analog or digital magnetic tape or a hard disk drive), and optical storage media (such as a CD, a DVD, or a Blu-ray Disc).

[0091] The term “set” generally means a grouping of one or more elements. The elements of a set do not necessarily need to have any characteristics in common or otherwise belong together. The phrase “at least one of A, B, and C” should be construed to mean a logical (A OR B OR C), using a non-exclusive logical OR, and should not be construed to mean “at least one of A, at least one of B, and at least one of C ” The phrase “at least one of A, B, or C” should be construed to mean a logical (A OR B OR C), using a non-exclusive logical OR.

Claims

CLAIMS1. A method comprising: obtaining a set of complex impedance measurements along a Warburg impedance for an energy storage device; determining, based on the set of complex impedance measurements, a slope and a real- axis intercept in a Nyquist plot for the Warburg impedance; and configuring, based on the determined slope and the determined real-axis intercept, a configuration of a series of resistor-capacitor circuits to model a finite-space approximation of the Warburg impedance.

2. The method of claim 1, further comprising managing the energy storage device based on an equivalent circuit model that incorporates the finite-space approximation of the Warburg impedance.

3. The method of claim 1, wherein the series of resistor-capacitor circuits: have respective time constants with frequency values at different intervals along a logarithmic scale; and are each tangent to a linear fit of the set of complex impedance measurements.

4. The method of claim 3, wherein the respective time constants of the series of resistorcapacitor circuits include at least two of 0.1, 0.01, or 0.001 Hertz.

5. The method of claim 3, wherein a right-hand side of the finite-space approximation of the Warburg impedance ends at a resistor-capacitor circuit with a smallest time constant.

6. The method of claim 2, wherein the finite-space approximation of the Warburg impedance and the equivalent circuit model for the energy storage device are updated at regular intervals.

7. The method of claim 2, wherein the equivalent circuit model or the finite-space approximation of the Warburg impedance are stored in an index along with at least two of an age of the energy storage device, a state-of-charge of the energy storage device, a state-of-health of the energy storage device, or an ambient temperature to generate a multidimensional map of the energy storage device.

8. The method of claim 7, wherein the index further includes the set of complex impedance measurements and an age or timestamp associated with each entry.

9. The method of claim 8, wherein the index further includes a confidence parameter associated with each equivalent circuit model or finite-space approximation of the Warburg impedance based on the age or the timestamp associated with each entry.

10. The method of claim 8, further comprising determining an updated finite-space approximation of the Warburg impedance for the energy storage device by interpolating two or more finite-space approximations of the Warburg impedance in the index based on a current ambient temperature and the age of the two or more finite-space approximations.

11. The method of claim 7, wherein the multidimensional map is stored in a nonvolatile memory.

12. The method of claim 11, further comprising at least one of: reassigning the energy storage device to a different use application based on the multidimensional map; or determining a recyclability of electrochemical elements in the energy storage device based on the multidimensional map.

13. The method of claim 7, wherein the multidimensional map is stored in a remote nonvolatile memory along with other multidimensional maps associated with similar energy storage devices.

14. The method of claim 1, wherein the set of complex impedance measurements is obtained at least one of: during a charging session of the energy storage device; during a discharge event of the energy storage device; or by selectively switching on and off one or more loads electrically connected to the energy storage device.

15. A system comprising: memory hardware configured to store instructions; andprocessing hardware configured to execute the instructions stored by the memory hardware, wherein the instructions include: obtaining a set of complex impedance measurements along a Warburg impedance for an energy storage device; determining, based on the set of complex impedance measurements, a slope and a real-axis intercept in a Nyquist plot for the Warburg impedance; and configuring, based on the determined slope and the determined real-axis intercept, a configuration of a series of resistor-capacitor circuits to model a finite-space approximation of the Warburg impedance.

16. The system of claim 15, wherein the instructions include managing the energy storage device based on an equivalent circuit model that incorporates the finite-space approximation of the Warburg impedance.

17. The system of claim 15, wherein the series of resistor-capacitor circuits: have respective time constants with frequency values at different intervals along a logarithmic scale; and are each tangent to a linear fit of the set of complex impedance measurements.

18. The system of claim 16, wherein the instructions include updating the finite-space approximation of the Warburg impedance and the equivalent circuit model for the energy storage device at regular intervals.

19. The system of claim 16, wherein the instructions include: storing the equivalent circuit model or the finite-space approximation of the Warburg impedance in an index along with the set of complex impedance measurements; and storing an age or timestamp associated with each entry and at least two of: an age of the energy storage device, a state-of-charge of the energy storage device, a state-of-health of the energy storage device, or an ambient temperature to generate a multidimensional map of the energy storage device.

20. The system of claim 19, wherein the instructions include determining an updated finite- space approximation of the Warburg impedance for the energy storage device by interpolating two or more finite-space approximations of the Warburg impedance in the index based on a current ambient temperature and the age of the two or more finite-space approximations.

21. The system of claim 15, wherein the set of complex impedance measurements are obtained at least one of: during a charging session of the energy storage device; during a discharge event of the energy storage device; or by selectively switching on and off one or more loads electrically connected to the energy storage device.

22. A non-transitory computer-readable storage medium comprising processor-executable instructions, the instructions including: obtaining a set of complex impedance measurements along a Warburg impedance for an energy storage device; determining, based on the set of complex impedance measurements, a slope and a real- axis intercept in a Nyquist plot for the Warburg impedance; and configuring, based on the determined slope and the determined real-axis intercept, a configuration of a series of resistor-capacitor circuits to model a finite-space approximation of the Warburg impedance.

23. The non-transitory computer-readable storage medium of claim 22, wherein the instructions include managing the energy storage device using an equivalent circuit model that incorporates the finite-space approximation of the Warburg impedance.

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