CN104007390A

CN104007390A - Battery state of charge tracking, equivalent circuit selection and benchmarking

Info

Publication number: CN104007390A
Application number: CN201410062166.6A
Authority: CN
Inventors: B·巴乐兴艾姆; Y·巴-莎伦; B·弗伦奇; B·派迪派缇; K·R·派迪派缇; 詹姆斯·米查姆; 特莱维斯·威廉斯; G·V·艾沃瑞; 黄泰植
Original assignee: Fairchild Semiconductor Suzhou Co Ltd; University of Connecticut; Fairchild Semiconductor Corp
Current assignee: Fairchild Semiconductor Suzhou Co Ltd; University of Connecticut; Fairchild Semiconductor Corp
Priority date: 2013-02-24
Filing date: 2014-02-24
Publication date: 2014-08-27
Anticipated expiration: 2034-02-24
Also published as: CN104007390B

Abstract

The present application relates to battery state of charge tracking, equivalent circuit selection and benchmarking methods and systems. The method includes calculating a first estimated state of charge (SOC) of a battery at a first time, receiving a voltage value representative of a measured voltage across the battery at a second time, calculating a filter gain at the second time, and at the second time The second time calculates a second estimated SOC of the battery based on the first estimated SOC, the voltage value, and the filter gain. Another method includes storing in memory a library of equivalent circuit models representing batteries, determining an operating mode of the battery based on a load associated with said battery, selecting one of said equivalent circuit models based on the determined operating mode, and calculating a state of charge (SOC) of the battery using the selected equivalent circuit model.

Description

Battery state of charge tracking, equivalent circuit selection and benchmarking method and system

技术领域technical field

实施例涉及计算电池的电荷状态。Embodiments relate to calculating the state of charge of a battery.

背景技术Background technique

电化学储能装置在未来能源战略中起到重要作用。实际上，电池是目前和不久的将来可行的储能技术。多种多样的设备例如便携式电子设备、移动家用电器、航空航天设备等日益由电池供电。使用已知系统和方法可能难以对例如电池的电荷状态进行准确估计。因此，需要用以解决目前技术的不足并提供其他新颖和创新特征的系统、方法和设备。Electrochemical energy storage devices play an important role in future energy strategies. In fact, batteries are a viable energy storage technology for the present and the near future. A wide variety of devices such as portable electronic devices, mobile home appliances, aerospace equipment, etc. are increasingly powered by batteries. Accurate estimation of, for example, the state of charge of a battery can be difficult using known systems and methods. Accordingly, there is a need for systems, methods and apparatus that address the deficiencies of current technology and provide other new and innovative features.

发明内容Contents of the invention

一个实施例包括一种方法。该方法包括在第一时间计算电池的第一估计电荷状态（SOC），在第二时间接收代表电池两端的测量电压的电压值，在所述第二时间计算滤波器增益，以及在所述第二时间基于第一估计SOC、电压值和滤波器增益计算电池的第二估计SOC。One embodiment includes a method. The method includes calculating a first estimated state of charge (SOC) of the battery at a first time, receiving a voltage value representative of a measured voltage across the battery at a second time, calculating a filter gain at the second time, and at the second time Second, a second estimated SOC of the battery is calculated based on the first estimated SOC, the voltage value and the filter gain.

另一个实施例包括一种系统。该系统包括电池和电池电量计模块，其被配置为使用降阶滤波器计算所述电池的估计电荷状态（SOC），所述降阶滤波器为单状态滤波器，所述单状态滤波器被配置为基于之前计算出的SOC估计值来递归计算所述估计SOC。Another embodiment includes a system. The system includes a battery and a battery fuel gauge module configured to calculate an estimated state of charge (SOC) of the battery using a reduced order filter, the reduced order filter being a single state filter, the single state filter being It is configured to recursively calculate the estimated SOC based on previously calculated SOC estimates.

又一个实施例包括一种计算机可读介质，其包括代码段。代码段在由处理器执行时使所述处理器计算电池的估计电荷状态（SOC），将估计SOC存储在缓冲器，使用降阶滤波器计算电池的更新后的估计SOC，所述降阶滤波器为单状态滤波器，所述单状态滤波器被配置为基于估计SOC来递归计算更新后的估计SOC。Yet another embodiment includes a computer readable medium including code segments. The code segments, when executed by the processor, cause the processor to calculate an estimated state of charge (SOC) of the battery, store the estimated SOC in a buffer, calculate an updated estimated SOC of the battery using a reduced order filter, the reduced order filter The filter is a single-state filter configured to recursively calculate an updated estimated SOC based on the estimated SOC.

再一个实施例包括一种方法。该方法包括在存储器中存储代表电池的等效电路模型的库，基于与所述电池相关的负载来确定电池的运行模式，基于所确定的运行模式选择等效电路模型之一，以及使用选择的等效电路模型来计算电池的电荷状态（SOC）。Yet another embodiment includes a method. The method includes storing in memory a library representing equivalent circuit models of the battery, determining an operating mode of the battery based on a load associated with the battery, selecting one of the equivalent circuit models based on the determined operating mode, and using the selected Equivalent circuit model to calculate the state of charge (SOC) of the battery.

另一个实施例包括一种系统。该系统包括被配置为存储代表电池的等效电路模型的库的数据存储器，被配置为基于电池的运行模式来选择等效电路模型的模型选择模块，以及被配置为基于选择的等效电路模型来计算电池的估计电荷状态（SOC）的滤波器模块。Another embodiment includes a system. The system includes a data store configured to store a library representing equivalent circuit models of the battery, a model selection module configured to select the equivalent circuit model based on the operating mode of the battery, and a model selection module configured to select the equivalent circuit model based on the selected Filter module to calculate the estimated state of charge (SOC) of the battery.

又一个实施例包括一种计算机可读介质，其包括代码段。代码段在由处理器执行时使所述处理器基于电池的运行模式从代表电池的等效电路模型的库中选择等效电路模型；以及使用选择的等效电路模型来计算所述电池的电荷状态（SOC）。Yet another embodiment includes a computer readable medium including code segments. The code segments, when executed by a processor, cause the processor to select an equivalent circuit model from a library of equivalent circuit models representing the battery based on the operating mode of the battery; and calculate a charge of the battery using the selected equivalent circuit model state (SOC).

附图说明Description of drawings

根据本文以下给出的具体实施方式和附图，将更全面理解示例性实施例，附图中类似的元件由类似的参考标号表示，这些参考标号仅以举例说明方式给出，因此不是对示例性实施例的限制，并且其中：A more complete understanding of the exemplary embodiments will be obtained from the detailed description given herein below and the drawings, in which like elements are indicated by like reference numerals, which are given by way of illustration only and are not intended to represent limitations of the embodiment, and where:

图1和2示出了根据至少一个示例性实施例的电池管理系统（BMS）的框图。1 and 2 illustrate block diagrams of a battery management system (BMS) according to at least one example embodiment.

图3示出了根据至少一个示例性实施例的用于选择电池等效模型的信号流的框图。FIG. 3 shows a block diagram of a signal flow for selecting a battery equivalent model, according to at least one example embodiment.

图4示出了根据至少一个示例性实施例的用于计算电池电荷状态（SOC）的信号流的框图。FIG. 4 illustrates a block diagram of a signal flow for calculating a battery state of charge (SOC), according to at least one example embodiment.

图5示出了根据至少一个示例性实施例的电池电量计（BFG）系统的框图。FIG. 5 illustrates a block diagram of a battery fuel gauge (BFG) system, according to at least one example embodiment.

图6示出了根据至少一个示例性实施例的用于BFG系统的参数模块的信号流的框图。Fig. 6 shows a block diagram of a signal flow of a parameter module for a BFG system according to at least one exemplary embodiment.

图7示出了根据至少一个示例性实施例的用于BFG系统的SOC模块的信号流的框图。FIG. 7 shows a block diagram of a signal flow of a SOC module for a BFG system according to at least one example embodiment.

图8示出了根据至少一个示例性实施例的SOC模块的框图。FIG. 8 illustrates a block diagram of a SOC module according to at least one example embodiment.

图9示出了根据至少一个示例性实施例的SOC模块的总体最小二乘（TLS）模块的框图。FIG. 9 illustrates a block diagram of a total least squares (TLS) module of an SOC module, according to at least one example embodiment.

图10示出了根据至少一个示例性实施例的SOC模块的递归最小二乘（RLS）模块的框图。FIG. 10 illustrates a block diagram of a recursive least squares (RLS) module of a SOC module, according to at least one example embodiment.

图11和12示出了根据至少一个示例性实施例的方法的流程图。11 and 12 illustrate flowcharts of methods according to at least one example embodiment.

图13A-13D示出了根据至少一个示例性实施例的电池等效模型的示意图。13A-13D illustrate schematic diagrams of battery equivalent models according to at least one example embodiment.

图14为示出了便携式锂离子电池的OCV-SOC特性曲线的示意图。FIG. 14 is a schematic diagram showing an OCV-SOC characteristic curve of a portable lithium ion battery.

图15A和15B为示出了负载曲线的图。15A and 15B are graphs showing load curves.

图16A和16B为示出了模拟负载曲线的图。16A and 16B are graphs showing simulated load curves.

图17为示出了示例性系统实现的示意图。Figure 17 is a schematic diagram illustrating an exemplary system implementation.

图18为示出了可与系统实现结合使用的用户界面的示意图。Figure 18 is a schematic diagram illustrating a user interface that may be used in conjunction with a system implementation.

图19A和19B包括示出了示例性放电电压/电流曲线的图。19A and 19B include graphs showing exemplary discharge voltage/current curves.

图20A和20B为示出了示例性库仑计数评估方法的图。20A and 20B are diagrams illustrating exemplary coulomb counting evaluation methods.

图21A和21B为示出了关闭时间（TTS）评估方法的图。21A and 21B are diagrams illustrating a time-to-close (TTS) evaluation method.

图22A、22B和22C为表示电量计读数的表。22A, 22B and 22C are tables showing fuel gauge readings.

应该指出的是，这些图旨在示出某些示例性实施例中所使用的方法和/或结构的一般特征，并补充如下提供的书面说明书。然而，这些附图未必按比例绘制并且可能未精确反映任何给定实施例的精确结构特征或性能特征，并且不应解释为限定或限制由示例性实施例所涵盖的值或性质的范围。例如，为清楚起见，可能减小或夸大结构元件的相对厚度和定位。It should be noted that these figures are intended to illustrate the general characteristics of methods and/or structures employed in certain exemplary embodiments and to supplement the written description provided below. These drawings, however, are not necessarily to scale and may not precisely reflect the precise structural or performance characteristics of any given embodiment, and should not be construed as defining or limiting the range of values or properties encompassed by example embodiments. For example, the relative thicknesses and positioning of structural elements may be reduced or exaggerated for clarity.

具体实施方式Detailed ways

虽然示例性实施例可包括各种修改形式和替代形式，其实施例在附图中以举例的方式示出并且将在本文中详细描述。然而，应当理解，并非意图将示例性实施例限于本发明所公开的特定形式，而是正相反，示例性实施例将涵盖落入权利要求书范围内的所有修改形式、等同形式和替代形式。在对附图的整个描述中，类似的数字是指类似的元件。Although the exemplary embodiments may include various modifications and alternative forms, embodiments thereof are shown by way of example in the drawings and will be described in detail herein. It should be understood, however, that there is no intent to limit example embodiments to the particular forms disclosed, but on the contrary, example embodiments are to cover all modifications, equivalents, and alternatives falling within the scope of the claims. Like numbers refer to like elements throughout the description of the figures.

电池状态例如电荷状态（SOC）、健康状态（SOH）和剩余使用寿命（RUL）的准确估计对于可靠、安全和广泛使用由电池供电的装置是至关重要的。估计这些数量称为电池电量计量（BFG）。与许多当今汽车中的烃类燃料不同，电池的存储容量不是常量。通常，电池容量随电池使用年限、使用模式和温度而变化，从而给BFG造成具有挑战性的自适应估计问题，需要在考虑温度变化、SOC变化和使用年限的基础上对电池特性进行建模和在线参数识别。Accurate estimation of battery states such as state of charge (SOC), state of health (SOH) and remaining useful life (RUL) is critical for reliable, safe and widespread use of battery-powered devices. Estimating these quantities is called battery fuel gauge (BFG). Unlike the hydrocarbon fuels in many of today's cars, the storage capacity of batteries is not constant. Typically, battery capacity varies with battery age, usage patterns, and temperature, thus posing a challenging adaptive estimation problem for BFG, requiring modeling and analysis of battery characteristics taking into account temperature variation, SOC variation, and age. Online parameter identification.

图1和2示出了根据至少一个示例性实施例的系统100的框图。如图1所示，系统100包括电池105、电池管理系统（BMS）110、显示器120、无限制电源125（如壁装电源插座、汽车充电站等）以及开关130。1 and 2 illustrate block diagrams of a system 100 according to at least one example embodiment. As shown in FIG. 1 , system 100 includes battery 105 , battery management system (BMS) 110 , display 120 , unlimited power source 125 (eg, wall outlet, car charging station, etc.), and switch 130 .

BMS110可被配置为管理电池105的利用和/或状态。例如，BMS110可被配置为使用开关130将无限制电源125与电池105连接或断开以对电池105进行充电。例如，BMS110可被配置为将负载（未示出）与电池105连接或断开。例如，BFG115可被配置为计算电池105的电荷状态（SOC）和/或健康状态（SOH）。SOC和/或SOH可显示（例如，以百分比的形式，以剩余时间的形式等）在显示器120上。BMS 110 may be configured to manage the utilization and/or status of battery 105 . For example, BMS 110 may be configured to use switch 130 to connect or disconnect unlimited power source 125 to battery 105 to charge battery 105 . For example, BMS 110 may be configured to connect or disconnect a load (not shown) from battery 105 . For example, BFG 115 may be configured to calculate a state of charge (SOC) and/or a state of health (SOH) of battery 105 . The SOC and/or SOH may be displayed (eg, as a percentage, as time remaining, etc.) on the display 120 .

如图2所示，BMS110至少包括模数转换器（ADC）205、220，滤波器210、225，数字放大器215和电池电量计（BFG）115。BFG115包括存储器230、处理器235和控制器240。ADC205、220，滤波器210、225，数字放大器215和BFG115中的至少一者可为，例如专用集成电路（ASIC）、数字信号处理器（DSP）、现场可编程门阵列（FPGA）和/或处理器，等等。或者，BMS110可为包括所示功能块的ASIC、DSP、FPGA和/或处理器，等等。或者，系统100可实现为存储在存储器上并由例如处理器执行的软件。As shown in FIG. 2 , BMS 110 includes at least analog-to-digital converters (ADCs) 205 , 220 , filters 210 , 225 , digital amplifier 215 and battery fuel gauge (BFG) 115 . BFG 115 includes memory 230 , processor 235 and controller 240 . At least one of ADCs 205, 220, filters 210, 225, digital amplifier 215, and BFG 115 may be, for example, an Application Specific Integrated Circuit (ASIC), a Digital Signal Processor (DSP), a Field Programmable Gate Array (FPGA), and/or processor, etc. Alternatively, BMS 110 may be an ASIC, DSP, FPGA, and/or processor, etc., including the functional blocks shown. Alternatively, system 100 may be implemented as software stored on memory and executed by, for example, a processor.

BMS110可被配置为使用数字转换器（ADC）205、220，滤波器210、225和数字放大器215的组合将模拟测量值（如，I_b和V_b）转换为数字值（例如，以供BFG115计算SOC和/或SOH）。例如，数字放大器215可为差分放大器，其根据电池105两端的电压降V_b（例如，正和负端子之间的电压值的差）生成（如，产生）模拟信号，然后使用ADC220和滤波器225将该模拟信号转换为经滤波的数字值。BMS 110 may be configured to convert analog measurements (e.g., I _b and V _b ) to digital values (e.g., for BFG 115 Calculate SOC and/or SOH). For example, digital amplifier 215 may be a differential amplifier that generates (eg, produces) an analog signal based on the voltage drop _Vb across battery 105 (eg, the difference in voltage value between the positive and negative terminals), and then uses ADC 220 and filter 225 Convert this analog signal to a filtered digital value.

系统100可为利用电池供电的任何系统或电子设备的子系统。在一些具体实施中，电子设备可为或可包括（例如）具有传统膝上型形式因素的膝上型设备。在一些具体实施中，电子设备可为或可包括（例如）有线设备和/或无线设备（例如，支持Wi-Fi的设备）、计算实体（例如，个人计算设备）、服务器设备（例如，web服务器）、玩具、移动电话、音频设备、电机控制设备、电源（例如，离线电源）、个人数字助理（PDA）、平板设备、电子阅读器、电视和/或汽车，等等。在一些具体实施中，电子设备可为或可包括（例如）显示设备（例如，液晶显示（LCD）监控器，用于向用户显示信息）、键盘、定点设备（例如，鼠标、触控板，通过该设备，用户可向计算机提供输入）。System 100 may be any system or subsystem of an electronic device that utilizes battery power. In some implementations, the electronic device can be or include, for example, a laptop device having a conventional laptop form factor. In some implementations, an electronic device can be or include, for example, a wired device and/or a wireless device (e.g., a Wi-Fi enabled device), a computing entity (e.g., a personal computing device), a server device (e.g., a web servers), toys, mobile phones, audio equipment, motor control equipment, power supplies (e.g. off-line power supplies), personal digital assistants (PDAs), tablet devices, e-readers, televisions and/or automobiles, etc. In some implementations, an electronic device can be or include, for example, a display device (e.g., a liquid crystal display (LCD) monitor for displaying information to a user), a keyboard, a pointing device (e.g., a mouse, a touchpad, A device through which a user provides input to a computer).

图3示出了根据至少一个示例性实施例的用于选择电池等效模型的信号流的框图300。如图3所示，模型选择块310接收输入320（如，来自电池和/或负载的电压和/或电流信号）并使用输入320（或其某些变型）从等效模型库305中选择代表（或对应于）电池的等效模型。然后由电荷状态计算器块315使用该等效模型来计算电荷状态（SOC）325。等效模型库305可包括至少一个代表电池的等效模型。每个等效模型可基于电池（或等效电池）的运行模式。运行模式可基于与电池相关的负载。例如，运行模式可基于负载两端的电压降。例如，运行模式可基于负载两端的电压降是相对较高还是较低、是相对恒定还是动态和/或它们的组合。FIG. 3 shows a block diagram 300 of a signal flow for selecting a battery equivalent model, according to at least one example embodiment. As shown in FIG. 3 , model selection block 310 receives input 320 (e.g., voltage and/or current signals from a battery and/or load) and uses input 320 (or some variation thereof) to select representative (or correspond to) an equivalent model of the battery. The equivalent model is then used by state of charge calculator block 315 to calculate state of charge (SOC) 325 . The equivalent model library 305 may include at least one equivalent model representing a battery. Each equivalent model may be based on the operating mode of the battery (or equivalent battery). The mode of operation can be based on the load associated with the battery. For example, the mode of operation may be based on the voltage drop across the load. For example, the mode of operation may be based on whether the voltage drop across the load is relatively high or low, relatively constant or dynamic, and/or combinations thereof.

等效模型（参见以下图13A-13D）可包括电阻器、电压（如，电压降或电压源）、电阻-电流（RC）电路和/或阻抗电路等等的任何组合。因此，可建立电池等效模型的数学（如，公式）等效形式。等效模型库305可存储与电池运行模式相关的数学等效形式。该数学等效形式可被电荷状态计算器块315用来进行SOC的计算。例如，数学等效形式可用于确定变量，作为用于计算SOC（或SOC的估计值）的公式的输入。因此，BFG系统可基于运行模式选择等效模型以提高计算效率和减少处理时间。下面针对图5-12提供更多细节。Equivalent models (see FIGS. 13A-13D below) may include any combination of resistors, voltages (eg, voltage drops or voltage sources), resistance-current (RC) circuits, and/or impedance circuits, among others. Therefore, a mathematical (eg, formula) equivalent of the battery equivalent model can be established. Equivalent model library 305 may store mathematical equivalents related to battery operating modes. This mathematical equivalent can be used by the state of charge calculator block 315 to perform SOC calculations. For example, mathematical equivalents can be used to determine variables as input to formulas for calculating SOC (or estimates of SOC). Therefore, the BFG system can select an equivalent model based on the operating mode to improve computational efficiency and reduce processing time. More details are provided below with respect to Figures 5-12.

图4示出了根据至少一个示例性实施例的用于计算电池电荷状态（SOC）的信号流的框图400。如图4所示，框图400包括扩展卡尔曼滤波器（EKF）块405、电荷状态（SOC）块410、420、滤波器增益参数块41、425和缓冲器430。EKF块505可被配置为基于之前计算的电荷状态410和滤波器增益参数415，计算SOC420并确定滤波器增益参数425（如，读数和/或计算的SOC方差、测得的电压、计算出的容量和/或与等效电路相关的变量，等等）。因此，缓冲器430可被配置为存储例如处理循环中的之前计算出的SOC和滤波器增益参数。换句话讲，可基于至少一个之前计算出的SOC计算当前（或下一个）SOC。换句话讲，在第一时间计算出的SOC可用于计算在第二（稍后）时间的SOC。FIG. 4 illustrates a block diagram 400 of signal flow for calculating a battery state of charge (SOC), according to at least one example embodiment. As shown in FIG. 4 , block diagram 400 includes Extended Kalman Filter (EKF) block 405 , State of Charge (SOC) blocks 410 , 420 , filter gain parameter blocks 41 , 425 and buffer 430 . EKF block 505 may be configured to calculate SOC 420 and determine filter gain parameters 425 (e.g., reading and/or calculated SOC variance, measured voltage, calculated capacity and/or variables related to the equivalent circuit, etc.). Accordingly, buffer 430 may be configured to store previously calculated SOC and filter gain parameters, eg, in a processing loop. In other words, the current (or next) SOC may be calculated based on at least one previously calculated SOC. In other words, the SOC calculated at a first time can be used to calculate the SOC at a second (later) time.

在示例性具体实施中，可使用至少两个SOC410和/或滤波器增益参数415的集合。因此，至少两个SOC的向量、至少两个SOC的数组、至少两个SOC的平均以及至少两个SOC的平均值和对应的滤波器增益参数可用于计算下一个SOC420（或第二时间的SOC）并确定/计算对应的滤波器增益参数425。因此，缓冲器430可被配置为存储多个之前计算出的SOC410和计算出的/确定的滤波器增益参数415。因此，BFG系统可利用之前计算出的SOC提高计算效率和减少处理时间。下面针对图5-12提供更多细节。In an exemplary implementation, at least two sets of SOC 410 and/or filter gain parameters 415 may be used. Therefore, a vector of at least two SOCs, an array of at least two SOCs, an average of at least two SOCs, and an average of at least two SOCs and corresponding filter gain parameters can be used to calculate the next SOC 420 (or the SOC at the second time ) and determine/calculate the corresponding filter gain parameter 425 . Accordingly, buffer 430 may be configured to store a plurality of previously calculated SOC 410 and calculated/determined filter gain parameters 415 . Therefore, the BFG system can utilize the previously calculated SOC to improve computational efficiency and reduce processing time. More details are provided below with respect to Figures 5-12.

图5示出了根据至少一个示例性实施例的电池电量计（BFG）115系统的框图。如图5所示，BFG115包括估计模块510、跟踪模块520、预测模块530、开路电压-电荷状态（OCV-SOC）表征模块540、离线参数估计模块545以及电池寿命表征模块550。此外，该系统包括离线数据收集模块555和电池建模模块560。FIG. 5 shows a block diagram of a battery fuel gauge (BFG) 115 system, according to at least one example embodiment. As shown in FIG. 5 , the BFG 115 includes an estimation module 510 , a tracking module 520 , a prediction module 530 , an open circuit voltage-state-of-charge (OCV-SOC) characterization module 540 , an offline parameter estimation module 545 and a battery life characterization module 550 . Additionally, the system includes an offline data collection module 555 and a battery modeling module 560 .

离线数据收集模块555可被配置为在相对受控的测试环境中测量电池特性。例如，可在测试实验室环境中收集电池105（或等效电池）的开路电压（OCV）测量值和SOC测量值。例如，电池105（或等效电池）可被初始化为完全充电（如，接近完全充电、基本上完全充电）、静置（rested）状态。可进行OCV和SOC测量。然后可使电池105（或等效电池）缓慢放电，同时每隔一定时间（如，定期、周期性、不定期、预定时间）进行OCV和SOC测量，直到电池105（或等效电池）完全（或基本上）放电。OCV和SOC测量值可用于确定、计算或估计电池参数（如，下文所述的OCV参数K_i∈{K₀；K₁；K₂；K₃；K₄；K₅；K₆；K₇}）。Offline data collection module 555 may be configured to measure battery characteristics in a relatively controlled testing environment. For example, open circuit voltage (OCV) measurements and SOC measurements of battery 105 (or an equivalent battery) may be collected in a test laboratory environment. For example, battery 105 (or an equivalent battery) may be initialized to a fully charged (eg, near fully charged, substantially fully charged), rested state. OCV and SOC measurements can be performed. The battery 105 (or equivalent battery) may then be slowly discharged while OCV and SOC measurements are taken at regular intervals (e.g., periodically, periodically, irregularly, at predetermined times) until the battery 105 (or equivalent battery) is fully ( or basically) discharge. OCV and SOC measurements can be used to determine, calculate or estimate battery parameters (e.g., OCV parameters K _i ∈ {K ₀ ; K ₁ ; K ₂ ; K ₃ ; K ₄ ; K ₅ ; K ₆ ; K ₇ }).

来自离线数据收集模块555的数据可用在电池建模模块560中以确定例如电池105（或等效电池）的等效模型和/或等效模型的数学等效形式。来自离线数据收集模块555的数据可用在离线参数估计模块545中以确定和/或计算与电池105（或等效电池）相关的参数（如，与上述等效模型相关的元件的值）。来自离线数据收集模块555的数据可用在OVC-SOC表征模块540中以确定和/或计算OCV和SOC电池参数（如，下文所述的OCV参数K_i∈{K₀；K₁；K₂；K₃；K₄；K₅；K₆；K₇}）。来自离线参数估计模块545的数据可用在电池寿命表征模块550中。例如，来自离线参数估计模块545的数据可用于由电池寿命表征模块550计算初始健康状态（SOH）特性（如，最大SOC）。Data from the offline data collection module 555 may be used in the battery modeling module 560 to determine, for example, an equivalent model of the battery 105 (or an equivalent battery) and/or a mathematical equivalent of the equivalent model. Data from the offline data collection module 555 may be used in the offline parameter estimation module 545 to determine and/or calculate parameters related to the battery 105 (or equivalent battery) (eg, values of elements related to the equivalent model described above). Data from the offline data collection module 555 can be used in the OVC-SOC characterization module 540 to determine and/or calculate OCV and SOC battery parameters (e.g., OCV parameters K _i ∈ {K ₀ ; K ₁ ; K ₂ ; K ₃ ; K ₄ ; K ₅ ; K ₆ ; K ₇ }). Data from the offline parameter estimation module 545 can be used in the battery life characterization module 550 . For example, data from offline parameter estimation module 545 may be used by battery life characterization module 550 to calculate initial state-of-health (SOH) characteristics (eg, maximum SOC).

显示器120被示出为具有SOC显示565、SOH显示570、关闭时间（TTS）显示575和剩余使用寿命（RUL）显示580。每个显示可例如为显示百分比的仪表。可由BFG115计算或确定每个显示的值。例如，TTS可显示为如由TTS模块532计算出的时间值（如，小时和/或分钟）。Display 120 is shown with SOC display 565 , SOH display 570 , time to shutdown (TTS) display 575 , and remaining useful life (RUL) display 580 . Each display may, for example, be a gauge showing a percentage. Each displayed value can be calculated or determined by the BFG115. For example, TTS may be displayed as a time value (eg, hours and/or minutes) as calculated by TTS module 532 .

估计模块510包括参数模块512和容量模块510。估计模块510可被配置为计算和/或确定电池105（或等效电池）所特定的值（如，参数和容量值）。在稳定环境（如，测试实验室）中，参数和容量值可能是固定的（如，不变化）。然而，在真实世界环境中，参数和容量值可能是动态的或变化的。例如，完整的SOC跟踪解决方案通常涉及（1）通过离线OCV表征对形成状态空间模型的一部分的OCV参数的估计。OCV-SOC表征关于温度变化和电池老化是稳定的。一旦估计出这些参数，这些参数就形成具有已知参数的状态空间模型的一部分。（2）动态等效电路参数的估计。已观察到这些参数随电池的温度、SOC和使用年限而变化，因此应当在BFG运行的同时进行自适应地估计。（3）电池容量的估计：虽然电池的标称容量由制造商所指定，但已知可用电池容量会因为制造工艺误差、温度变化、使用模式和老化而变化。以及（4）受模型参数约束的SOC跟踪。一旦已知模型参数，SOC跟踪就变成了非线性滤波问题。然而，已观察到所得状态空间模型包含相关过程和测量噪声过程。恰当处理这些相关性效应将得出更好的SOC跟踪精度。因此，在示例性具体实施中，为了计算参数和容量，跟踪模块520可将数据反馈到估计模块510。Estimation module 510 includes parameter module 512 and capacity module 510 . Estimation module 510 may be configured to calculate and/or determine values (eg, parameters and capacity values) specific to battery 105 (or an equivalent battery). In a stable environment (eg, a test laboratory), parameter and capacity values may be fixed (eg, do not change). However, in a real world environment, parameters and capacity values may be dynamic or change. For example, a complete SOC tracking solution typically involves (1) the estimation of OCV parameters forming part of the state-space model via offline OCV characterization. The OCV-SOC characterization is stable with respect to temperature changes and battery aging. Once estimated, these parameters form part of a state-space model with known parameters. (2) Estimation of dynamic equivalent circuit parameters. These parameters have been observed to vary with the battery's temperature, SOC, and age, and thus should be adaptively estimated while the BFG is running. (3) Estimation of battery capacity: Although the nominal capacity of a battery is specified by the manufacturer, it is known that the usable battery capacity will vary due to manufacturing process errors, temperature changes, usage patterns, and aging. and (4) SOC tracking constrained by model parameters. Once the model parameters are known, SOC tracking becomes a nonlinear filtering problem. However, the resulting state-space model has been observed to contain correlated processes as well as measurement noise processes. Proper handling of these correlation effects will result in better SOC tracking accuracy. Thus, in an exemplary implementation, tracking module 520 may feed data back to estimation module 510 in order to calculate parameters and capacities.

另外，典型的估计电池容量的方法忽视了滞后效应，并且假设静置的电池电压代表电池的真实开路电压（OCV）。然而，根据示例性实施例，估计模块510将滞后建模为电池105的OCV中的误差，并采用实时线性参数估计和SOC跟踪技术的组合来补偿OCV中的误差。Additionally, typical approaches to estimating battery capacity ignore hysteresis effects and assume that the resting battery voltage represents the true open-circuit voltage (OCV) of the battery. However, according to an exemplary embodiment, the estimation module 510 models hysteresis as an error in the OCV of the battery 105 and employs a combination of real-time linear parameter estimation and SOC tracking techniques to compensate for the error in the OCV.

跟踪模块520包括SOC模块522和SOH模块524。SOC指示电池105中的“电量”。如上所述，SOC为可用容量，其表示为某个基准（如，额定容量或电流容量）的百分比。根据示例性实施例，SOC模块522通过使用下文更详细描述的跟踪补偿OCV中的误差（与参数估计相结合）来计算SOC。SOH指示一电池与新或理想电池相比的状态。SOH可基于充电接收能力、内阻、电压和/或自放电，等等。Tracking module 520 includes SOC module 522 and SOH module 524 . The SOC indicates the “charge” in the battery 105 . As mentioned above, SOC is available capacity expressed as a percentage of some reference (eg, rated capacity or current capacity). According to an exemplary embodiment, the SOC module 522 calculates the SOC by compensating for errors in the OCV (in combination with parameter estimates) using tracking described in more detail below. SOH indicates the state of a battery compared to a new or ideal battery. SOH may be based on charge acceptance, internal resistance, voltage, and/or self-discharge, among others.

预测模块530包括TTS模块532和RUL模块534。TTS模块532和RUL模块534可被配置为基于SOC计算TTS和RUL。Prediction module 530 includes TTS module 532 and RUL module 534 . TTS module 532 and RUL module 534 may be configured to calculate TTS and RUL based on the SOC.

图6示出了根据至少一个示例性实施例的用于BFG115的参数模块412的信号流的框图。BFG系统可基于运行模式选择等效模型以提高计算效率和减少处理时间。如图6所示，参数模块412包括运行模式模块605和模型选择模块610。运行模式模块605可被配置为基于来自电池105的至少一个输入和/或来自负载615的至少一个输入确定（电池的）运行模式。所述至少一个输入可基于与电池105和负载615中至少一个相关的电流和电压中的至少一个。例如，负载615两端的电压降。例如，运行模式可基于负载615两端的电压降是相对较高还是较低、是相对恒定还是动态和/或它们的组合。模型选择模块610可基于确定的运行模式选择等效模型（或其数学等效形式）。例如，模型选择模块610可生成用于搜索等效模型库305的查询项。FIG. 6 shows a block diagram of a signal flow for the parameter module 412 of the BFG 115 according to at least one example embodiment. The BFG system can select equivalent models based on the operating mode to improve computational efficiency and reduce processing time. As shown in FIG. 6 , the parameter module 412 includes an operating mode module 605 and a model selection module 610 . Operating mode module 605 may be configured to determine an operating mode (of the battery) based on at least one input from battery 105 and/or at least one input from load 615 . The at least one input may be based on at least one of a current and a voltage associated with at least one of the battery 105 and the load 615 . For example, the voltage drop across load 615 . For example, the mode of operation may be based on whether the voltage drop across the load 615 is relatively high or low, relatively constant or dynamic, and/or combinations thereof. The model selection module 610 may select an equivalent model (or its mathematical equivalent) based on the determined operating mode. For example, model selection module 610 may generate query terms for searching equivalent model repository 305 .

在一些具体实施中，可定义或表征多个运行模式。在示例性具体实施中，下文描述了与电池和使用电池的系统相关的四个运行模式。In some implementations, multiple modes of operation may be defined or characterized. In an exemplary implementation, four modes of operation are described below in relation to batteries and systems using batteries.

在第一运行模式中，电池105可连接于重且变化的负载。换句话讲，负载615可联合使用相对较高电压与动态或可变电流消耗（或消耗可变电流的高电压负载）。例如，在移动电话中，第一运行模式可包括这样的使用环境，其中移动电话使用包括长时间的视频播放、多媒体和游戏应用程序等等。以下图13A中所示的等效电路可表示连接至重且变化的负载的电池。In a first mode of operation, the battery 105 may be connected to heavy and varying loads. In other words, load 615 may use a relatively high voltage in combination with dynamic or variable current consumption (or a high voltage load that consumes variable current). For example, in a mobile phone, the first mode of operation may include usage environments where mobile phone usage includes prolonged video playback, multimedia and gaming applications, and the like. The equivalent circuit shown below in Figure 13A may represent a battery connected to a heavy and varying load.

在第二运行模式中，电池105可连接至动态负载和/或可变电压负载。换句话讲，负载615可使用动态或变化的电压。例如，在移动电话中，第二运行模式可包括这样的使用环境，其中移动电话使用包括电话呼叫、web浏览和/或播放视频剪辑的常规使用。以下图13B中所示的等效电路可表示连接至动态负载的电池。In the second mode of operation, the battery 105 may be connected to a dynamic load and/or a variable voltage load. In other words, load 615 may use a dynamic or varying voltage. For example, in a mobile phone, the second mode of operation may include a usage environment where mobile phone usage includes regular use for phone calls, web browsing, and/or playing video clips. The equivalent circuit shown in Figure 13B below may represent a battery connected to a dynamic load.

在第三运行模式中，电池105可连接至或消耗恒定电流。换句话讲，负载615可正拖带恒定的（或基本上恒定的）负载。或者，电池105可正利用恒定电流被充电。例如，在充电周期，电池105可与负载615断开（如，可使用开关130将无限制电源125连接至电池105以对电池105进行充电）。以下图13C中所示的等效电路可表示连接至恒定电流的电池。In a third mode of operation, the battery 105 may be connected to or draw a constant current. In other words, load 615 may be dragging a constant (or substantially constant) load. Alternatively, the battery 105 may be being charged with a constant current. For example, during a charging cycle, battery 105 may be disconnected from load 615 (eg, switch 130 may be used to connect unrestricted power source 125 to battery 105 to charge battery 105 ). The equivalent circuit shown in Figure 13C below may represent a battery connected to a constant current.

在第四运行模式中，电池105可连接至相对较低电压的负载。或者，电池105可处于周期性静置状态，其中电池105经受轻负载，随后充电，然后静置、最低或无负载。换句话讲，负载615可很少使用最低电压。例如，在移动电话中，第四运行模式可包括这样的使用环境，其中移动电话使用包括在完全（或基本上完全）充电后，使用不频发的电话呼叫对基站进行常规联系。以下图13D中所示的等效电路可表示连接至动态负载的电池。In a fourth mode of operation, the battery 105 may be connected to a relatively low voltage load. Alternatively, the battery 105 may be placed in a periodic rest state where the battery 105 is subjected to a light load, then charged, and then rested, minimal or no load. In other words, load 615 may rarely use the lowest voltage. For example, in a mobile phone, the fourth mode of operation may include an environment in which mobile phone use includes regular contact with the base station using infrequent phone calls after a full (or substantially full) charge. The equivalent circuit shown below in Figure 13D may represent a battery connected to a dynamic load.

图7示出了根据至少一个示例性实施例的用于BFG115系统的SOC模块422的信号流的框图。如图7所示，SOC模块422包括缓冲器块705、模型估计块710、SOC跟踪块715和电压降预测模块或块720。FIG. 7 shows a block diagram of signal flow for the SOC module 422 of the BFG 115 system, according to at least one example embodiment. As shown in FIG. 7 , the SOC module 422 includes a buffer block 705 , a model estimation block 710 , an SOC tracking block 715 , and a voltage drop prediction module or block 720 .

在示例性实施例中，将滞后建模为电池105的OCV中的误差。电压降v_D[k]可表示内部电池模型元件R₀、R₁、R₂和x_h[k]两端的电压（参见图13A）。x_h[k]项可用于说明预测的SOC中的误差。换句话讲，x_h[k]可为“瞬时滞后”，其可通过调整计算或估计出的SOC来修正为零。若计算或估计出的SOC等于SOC，则计算或估计出的x_h[k]应等于零。换句话讲，计算或估计出的x_h[k]不等于零，则指示计算或估计出的SOC存在误差。电压降模型参数向量（b）包括对应于计算或估计出的x_h[k]的元素。In an exemplary embodiment, hysteresis is modeled as an error in the OCV of the battery 105 . The voltage drop v _D [k] may represent the voltage across internal battery model elements R ₀ , R ₁ , R ₂ and x _h [k] (see FIG. 13A ). The x _h [k] term can be used to account for errors in the predicted SOC. In other words, _xh [k] may be an "instantaneous lag" that can be corrected to zero by adjusting the calculated or estimated SOC. If the calculated or estimated SOC is equal to SOC, then the calculated or estimated x _h [k] shall be equal to zero. In other words, the calculated or estimated x _h [k] is not equal to zero, indicating that there is an error in the calculated or estimated SOC. The voltage drop model parameter vector (b) includes elements corresponding to the calculated or estimated x _h [k].

因此，在图7的流程中，来自SOC跟踪块715的当前计算或估计出的SOC用在电压块预测块720中以计算电压降v_D[k]。至少一个过往的电压降v_D[k]存储在缓冲器705中并用于参数向量b的估计。参数向量b中的对应计算或估计出的x_h[k]为非零值指示存在瞬时滞后。这意味着SOC估计误差。SOC跟踪块715的SOC跟踪算法被配置为在计算或估计出的x_h[k]为非零的任何时候修正SOC。下文（在数学上）描述了有关电压降v_D[k]、滞后、估计x_h[k]、电压降模型参数向量（b）和SOC跟踪的更多细节。因此，BFG系统可利用之前计算出的SOC和SOC误差准确估计SOC、提高计算效率和减少处理时间。Thus, in the flow of FIG. 7, the current calculated or estimated SOC from the SOC tracking block 715 is used in the voltage block prediction block 720 to calculate the voltage drop _vD [k]. At least one past voltage drop v _D [k] is stored in the buffer 705 and used for the estimation of the parameter vector b. A non-zero value for the corresponding calculated or estimated x _h [k] in the parameter vector b indicates the presence of an instantaneous lag. This means SOC estimation error. The SOC tracking algorithm of the SOC tracking block 715 is configured to correct the SOC whenever the calculated or estimated x _h [k] is non-zero. More details about voltage drop v _D [k], hysteresis, estimated x _h [k], voltage drop model parameter vector (b) and SOC tracking are described below (mathematically). Therefore, the BFG system can use the previously calculated SOC and SOC error to accurately estimate SOC, improve calculation efficiency and reduce processing time.

图8示出了根据至少一个示例性实施例的SOC模块422的框图。如图8所示，SOC模块422包括扩展卡尔曼滤波器（EKF）块805。EKF块可被配置为计算SOC845和SOC误差840。EKF块805可被配置为使用方程1计算SOC845作为估计SOC，并使用方程2计算SOC误差840作为估计SOC误差（或方差）。在下列方程的每一个中，k是指瞬时迭代，k+1|k是指上一个、前一个或之前迭代，并且k+1|k+1是指当前、更新、下一个或后续迭代。 $\hat{x} [k + 1 | k + 1] = \hat{x} [k + 1 | k] + G [k + 1] v_{k + 1} - - - (1)$ FIG. 8 shows a block diagram of the SOC module 422 according to at least one example embodiment. As shown in FIG. 8 , the SOC module 422 includes an Extended Kalman Filter (EKF) block 805 . The EKF block can be configured to calculate SOC 845 and SOC error 840 . EKF block 805 may be configured to calculate SOC 845 as estimated SOC using Equation 1, and calculate SOC error 840 as estimated SOC error (or variance) using Equation 2. In each of the following equations, k refers to the instantaneous iteration, k+1|k refers to the previous, previous, or previous iteration, and k+1|k+1 refers to the current, update, next, or subsequent iteration. $\hat{x} [k + 1 | k + 1] = \hat{x} [k + 1 | k] + G [k + 1] v_{k + 1} - - - (1)$

其中：in:

是当前或更新迭代的估计SOC； is the estimated SOC for the current or updated iteration;

是^上一个或预测迭代的估计SOC； is the estimated SOC of ^{the previous} or forecast iteration;

G[k+1]是上一个或预测迭代的滤波器增益；以及G[k+1] is the filter gain of the previous or predicted iteration; and

v_k+1是上一个或预测迭代的负载电压。vk ₊₁ is the load voltage from the previous or predicted iteration.

其中：in:

P_s[k+1|k+1]是当前或更新迭代的SOC估计误差或方差；P _s [k+1|k+1] is the SOC estimation error or variance of the current or updated iteration;

G[k+1]是上一个或预测迭代的滤波器增益；G[k+1] is the filter gain of the previous or predicted iteration;

H[k+1]是线性化的观测系数；H[k+1] is the linearized observation coefficient;

P_s[k+1|k]是上一个或预测迭代的SOC估计误差或方差；以及P _s [k+1|k] is the SOC estimation error or variance of the previous or prediction iteration; and

是在初始化时具有零平均值和相关性的电压降噪声。 is the voltage drop noise with zero mean value and correlation at initialization.

SOC模块422包括OCV参数块810。OCV参数块810可被配置为从OVC-SOC表征模块540存储和/或接收OCV参数{K_i}。OCV参数{K_i}为常数，因为它们是离线测量的并且在电池105使用寿命内的变化可忽略不计（或不存在）。OCV参数用于根据方程3以SOC计算OCV。The SOC module 422 includes an OCV parameter block 810 . The OCV parameters block 810 may be configured to store and/or receive OCV parameters {K _i } from the OVC-SOC characterization module 540 . The OCV parameters {K _i } are constant because they are measured off-line and have negligible (or non-existent) changes over the life of the battery 105 . The OCV parameters are used to calculate OCV with SOC according to Equation 3.

${V V}_{00} ((s the s [[k k]])) = = {K K}_{00} + + \frac{{K K}_{11}}{s the s [[k k]]} + + \frac{{K K}_{22}}{{s the s}^{22} [[k k]]} \frac{{K K}_{33}}{{s the s}^{33} [[k k]]} + + \frac{{K K}_{44}}{{s the s}^{44} [[k k]]} + + {K K}_{55} s the s [[k k]] + + {K K}_{66} ln ln ((s the s [[k k]])) + + {K K}_{77} ln ln ((11 - - s the s [[k k]])) - - - - - - ((33))$

其中：in:

s[k]是SOC；以及s[k] is the SOC; and

V_o(s[k])是开路电压（OCV）；V _o (s[k]) is the open circuit voltage (OCV);

SOC模块422包括电压降模型块825。电压降块825可被配置为使用根据方程4或5的电压降模型（上文所讨论）计算负载两端的电压降。The SOC module 422 includes a voltage drop model block 825 . The voltage drop block 825 may be configured to calculate the voltage drop across the load using the voltage drop model according to equation 4 or 5 (discussed above).

Z_v[k]＝V_o(x_s[k])+a[k]^Tb+n_D[k] (4)Z _v [k]＝V _o (x _s [k])+a[k] ^T b+n _D [k] (4)

${Z Z}_{v v} [[k k]] = = {V V}_{o o} (({x x}_{s the s} [[k k]])) + + \overset{^^}{a a} {[[k k]]}^{T T} \overset{^^}{b b} + + {n no}_{D D.} [[k k]] - - - - - - ((55))$

其中：in:

Z_v[k]是测量的电压；Z _v [k] is the measured voltage;

V_o(x_s[k])]是开路电压（OCV）；V _o (x _s [k])] is the open circuit voltage (OCV);

a[k]^T是电压降模型；a[k] ^T is the voltage drop model;

b是电压降模型参数向量；b is the voltage drop model parameter vector;

是估计电压降模型； is the estimated voltage drop model;

是估计电压降模型参数向量；以及 is the estimated voltage drop model parameter vector; and

n_D[k]是电压降观测噪声。n _D [k] is the voltage drop observation noise.

如上所述，电压降模型可基于所选等效电路模型而变化。所选等效电路模型和/或电压降模型可从数据存储器855读取。例如，数据存储器855可包括等效模型库305。As mentioned above, the voltage drop model may vary based on the selected equivalent circuit model. The selected equivalent circuit model and/or voltage drop model may be read from data storage 855 . For example, data store 855 may include equivalent model library 305 .

EKF（模块或）块805可被配置为使用方程1计算SOC845作为估计SOC，并将所得的SOC845存储在缓冲器850中。EKF块805可被配置为使用方程2计算SOC误差840计算作为估计SOC误差（或方差），并将所得的SOC误差840存储在缓冲器850中。存储的SOC和SOC误差可读取为SOC815和SOC误差820，即存储的SOC845和SOC误差840。因此，EKF块可递归地（如，在循环中）计算SOC845和SOC误差840，使得后续（在时间上为更新、下一个和/或之后的）SOC845和SOC误差840计算可基于至少一个前面的（在时间上为当前、上一个或之前的）SOC815和SOC误差820计算。The EKF (module or) block 805 may be configured to calculate the SOC 845 as the estimated SOC using Equation 1 and store the resulting SOC 845 in the buffer 850 . The EKF block 805 may be configured to calculate the SOC error 840 using Equation 2 as an estimated SOC error (or variance) and store the resulting SOC error 840 in a buffer 850 . The stored SOC and SOC error can be read as SOC 815 and SOC error 820 , ie stored SOC 845 and SOC error 840 . Accordingly, the EKF block may recursively (e.g., in a loop) calculate SOC 845 and SOC error 840 such that subsequent (newer, next, and/or subsequent in time) SOC 845 and SOC error 840 calculations may be based on at least one previous (Current, previous or previous in time) SOC 815 and SOC error 820 calculations.

如图8所示，递归最小二乘（RLS）块830和总体最小二乘（TLS）块835可生成到EKF块805的输入。RLS块可生成初始估计电压降模型参数向量（其可包括至少一个电压降模型参数），并且TLS块835可生成初始估计容量。可针对每个循环都生成初始估计电压降模型参数向量和初始估计容量。在示例性具体实施中，随着迭代次数（k）增加，初始估计电压降模型参数向量和初始估计容量的变化可变得忽略不计。As shown in FIG. 8 , a recursive least squares (RLS) block 830 and a total least squares (TLS) block 835 may generate inputs to the EKF block 805 . The RLS block can generate an initial estimated voltage drop model parameter vector (which can include at least one voltage drop model parameter), and the TLS block 835 can generate an initial estimated capacity. An initial estimated voltage drop model parameter vector and an initial estimated capacity may be generated for each cycle. In an exemplary implementation, as the number of iterations (k) increases, changes in the initial estimated voltage drop model parameter vector and initial estimated capacity may become negligible.

图9示出了根据至少一个示例性实施例的SOC模块422的总体最小二乘（TLS）块835的框图。如图9所示，TLS块635包括缓冲器910和TLS计算模块915。缓冲器910被配置为接收、存储和输出SOC数据，例如ΔSOC数据920（或SOC数据920的变化值），以供TLS计算模块915使用。缓冲器910还被配置为接收、存储和输出Δ库仑数据925（或库仑数据925的变化值），以供TLS计算模块915使用。缓冲器910可接收基于测得的与电池105相关的电流的电流数据905作为例如库仑计数数据。FIG. 9 shows a block diagram of a total least squares (TLS) block 835 of the SOC module 422 in accordance with at least one example embodiment. As shown in FIG. 9 , the TLS block 635 includes a buffer 910 and a TLS calculation module 915 . Buffer 910 is configured to receive, store and output SOC data, such as ΔSOC data 920 (or a variation of SOC data 920 ), for use by TLS calculation module 915 . Buffer 910 is also configured to receive, store and output delta Coulomb data 925 (or a change in Coulomb data 925 ) for use by TLS calculation module 915 . Buffer 910 may receive current data 905 based on the measured current associated with battery 105 as, for example, coulomb counting data.

TLS计算模块915可被配置为基于ΔSOC920和Δ库仑925来计算电池105的容量930。例如，TLS计算模块915可使用方程6计算容量930。方程6的推导在下文中更详细示出。TLS calculation module 915 may be configured to calculate capacity 930 of battery 105 based on ΔSOC 920 and ΔCoulomb 925 . For example, TLS calculation module 915 may calculate capacity 930 using Equation 6. The derivation of Equation 6 is shown in more detail below.

其中：in:

是估计容量； is the estimated capacity;

是增广观测矩阵的协方差；以及 is the covariance of the augmented observation matrix; and

Δ^k(2，2)是非负特征值的对角2×2矩阵；Δ ^k (2, 2) is a diagonal 2×2 matrix of non-negative eigenvalues;

图10示出了根据至少一个示例性实施例的SOC模块422的递归最小二乘（RLS）块830的框图。如图10所示，TLS块635包括缓冲器1005和RLC计算模块1010。缓冲器1005被配置为接收和存储SOC815、SOC误差820和作为来自电压降模型块825的输出的电压降数据（如，Z_v[k]或OCV）。缓冲器1005被配置为输出电压降1015以及电流和电容（I&C）矩阵1020。FIG. 10 shows a block diagram of a recursive least squares (RLS) block 830 of the SOC module 422 in accordance with at least one example embodiment. As shown in FIG. 10 , the TLS block 635 includes a buffer 1005 and an RLC calculation module 1010 . Buffer 1005 is configured to receive and store SOC 815 , SOC error 820 , and voltage drop data (eg, Z _v [k] or OCV) as output from voltage drop modeling block 825 . The buffer 1005 is configured as an output voltage drop 1015 and a current and capacitance (I&C) matrix 1020 .

RLC计算模块1010可被配置为基于电压降1015和（I&C）矩阵1020来计算初始化参数1025。例如，RLC计算模块1010可使用方程7计算初始化参数1025。方程7的推导在下文中更详细示出。RLC calculation module 1010 may be configured to calculate initialization parameters 1025 based on voltage drop 1015 and (I&C) matrix 1020 . For example, RLC computation module 1010 may compute initialization parameters 1025 using Equation 7. The derivation of Equation 7 is shown in more detail below.

其中：in:

α_j是R₁C₁电路中的电流衰减系数；α _j is the current attenuation coefficient in the R ₁ C ₁ circuit;

β_i是R₂C₂电路中的电流衰减系数；β _i is the current attenuation coefficient in the R ₂ C ₂ circuit;

是R₁的估计电阻值； is the estimated resistance value of _R1 ;

是R₂的估计电阻值； is the estimated resistance value of _R2 ;

是电池的估计滞后电压；以及 is the estimated hysteresis voltage of the battery; and

x_h[k]是瞬时滞后；x _h [k] is the instantaneous lag;

值得注意的是，如上针对图7所述，电池的估计滞后电压应为零。因此，在示例性具体实施中，由于使用SOC跟踪块715通过SOC跟踪去除了（或基本上去除了）因滞后产生的误差，因此方程7中的b(6)应为零。因此，SOC估计值更准确，因为可将滞后考虑在内。It is worth noting that, as mentioned above for Figure 7, the estimated hysteresis voltage of the battery should be zero. Therefore, in an exemplary implementation, b(6) in Equation 7 should be zero since errors due to hysteresis are removed (or substantially removed) by SOC tracking using SOC tracking block 715 . Therefore, the SOC estimate is more accurate because hysteresis can be taken into account.

在图8-10中，缓冲器1005长度可为用于参数估计的L_b，并且缓冲器905长度可为用于容量估计的L_c。EKF块805针对每个k进行迭代，而RLS830针对每个作为L_b,整数倍的k进行迭代，并且TLS835针对每个作为L_c整数倍的k进行迭代，其中k为时间索引。BFG估计SOC跟踪所需的所有所需模型参数和电池容量，但除OCV参数（其为离线估计的）以及来自测量仪表电路校准的电压和电流测量误差标准差σ_v,σ_i之外。RLS块不需要任何外部初始条件，只需设置λ=1，就能提供稳固LS估计值为初始值，即和其中和为批次编号。EKF块805的数学证明在下文描述。In FIGS. 8-10, the buffer 1005 length may be _Lb for parameter estimation, and the buffer 905 length may be _Lc for capacity estimation. EKF block 805 iterates for each k, while RLS 830 iterates for each k that is an integer multiple of _Lb , and TLS 835 iterates for each k that is an integer multiple of _Lc , where k is the time index. The BFG estimates all required model parameters and battery capacity required for SOC tracking, except for the OCV parameters (which are estimated off-line) and the voltage and current measurement error standard deviations σ _v , σ _i from the gauge circuit calibration. The RLS block does not require any external initial conditions, and only needs to set λ=1 to provide a solid LS estimate for the initial value, namely and in and for the batch number. The mathematical proof of EKF block 805 is described below.

图11和12示出了根据至少一个示例性实施例的方法的流程图。针对图11和12所述的步骤可由于软件代码的执行而进行，所述软件代码存储在与设备（如，图1和2中所示的BMS110）相关的存储器（如，存储器230）中并由与设备相关的至少一个处理器（如，处理器235）执行。然而，可设想替代实施例，例如具体体现为专用处理器的系统。虽然下述步骤被描述为由例如处理器执行，但这些步骤不必由同一的处理器执行。换句话讲，至少一个处理器可执行下文针对图11和12所述的步骤。11 and 12 illustrate flowcharts of methods according to at least one example embodiment. The steps described with respect to FIGS. 11 and 12 may be performed as a result of the execution of software code stored in a memory (eg, memory 230 ) associated with a device (eg, BMS 110 shown in FIGS. 1 and 2 ) and Executed by at least one processor (eg, processor 235) associated with the device. However, alternative embodiments are contemplated, such as a system embodied as a dedicated processor. Although the steps described below are described as being performed by, for example, a processor, these steps are not necessarily performed by the same processor. In other words, at least one processor may perform the steps described below with respect to FIGS. 11 and 12 .

图11描述了选择代表电池的等效模型以用于计算估计SOC的方法的流程图。如图11所示，在步骤S1105中，代表电池的等效电路模型的库存储在存储器中。例如，使用离线数据收集模块555，可收集与电池105（或等效电池）相关的数据。使用所述数据和通用电路工具，可生成代表电池的至少一个等效电路。该等效电路可包括至少一个等效电压、电阻、电容和/或等效阻抗的任何组合。参见例如以下的图13A-13D。还可生成每个等效电路的数学等效形式。等效电路和/或数学等效形式可存储在例如等效模型库305中。11 depicts a flowchart of a method of selecting an equivalent model representative of a battery for use in calculating an estimated SOC. As shown in FIG. 11 , in step S1105 , a library representing equivalent circuit models of batteries is stored in memory. For example, using offline data collection module 555, data related to battery 105 (or an equivalent battery) may be collected. Using the data and general circuit tools, at least one equivalent circuit representing a battery can be generated. The equivalent circuit may comprise any combination of at least one equivalent voltage, resistance, capacitance and/or equivalent impedance. See, eg, Figures 13A-13D below. A mathematical equivalent of each equivalent circuit can also be generated. Equivalent circuits and/or mathematical equivalents may be stored, for example, in equivalent model library 305 .

在步骤S1110中，基于与电池相关的负载来确定电池的运行模式。例如，每个等效模型可基于电池（或等效电池）的运行模式。运行模式可基于与电池相关的负载。例如，运行模式可基于负载两端的电压降。例如，运行模式可基于负载两端的电压降是相对较高还是较低、是相对恒定还是动态和/或它们的组合。因此，运行模式可基于与电池相关的电流和/或电压和/或与电池相关的负载来确定。In step S1110, the operating mode of the battery is determined based on the load related to the battery. For example, each equivalent model may be based on the operating mode of the battery (or battery equivalent). The mode of operation can be based on the load associated with the battery. For example, the mode of operation may be based on the voltage drop across the load. For example, the mode of operation may be based on whether the voltage drop across the load is relatively high or low, relatively constant or dynamic, and/or combinations thereof. Accordingly, the operating mode may be determined based on current and/or voltage associated with the battery and/or load associated with the battery.

在步骤S1115中，基于所确定的模式来选择用于所确定的模式的等效电路模型之一。例如，可基于所确定的运行模式来搜索等效模型库305。例如，代表电池的等效电路和/或数学等效形式可采用与运行模式识别（如，唯一名称或唯一识别编号）相对应的方式存储在等效模型库305中。因此，确定运行模式可包括确定运行模式识别，该运行模式识别随后用于搜索等效模型库305。选择等效电路可包括选择通过搜索等效模型库305所返回的等效电路或数学等效形式。In step S1115, one of the equivalent circuit models for the determined mode is selected based on the determined mode. For example, the library of equivalent models 305 may be searched based on the determined mode of operation. For example, an equivalent circuit and/or a mathematical equivalent representing a battery may be stored in the equivalent model library 305 in a manner corresponding to an operating mode identification (eg, unique name or unique identification number). Accordingly, determining an operating mode may include determining an operating mode identification, which is then used to search the equivalent model library 305 . Selecting an equivalent circuit may include selecting an equivalent circuit or a mathematically equivalent form returned by searching the equivalent model library 305 .

在步骤S1120中，使用所选择的等效电路模型来计算电池的电荷状态（SOC）或估计SOC。例如，如上所述，计算SOC可基于电压降模型参数向量（b）。电压降模型参数向量可具有基于电池的等效电路的参数（参见上述方程7）。因此，确定的电压降模型参数向量可基于等效电路具有或高或低的复杂度。例如，如下所述，等效电路可不包括RC电路元件，因为电容充电并绕过电阻。因此，b(3)可为唯一剩余的电压降模型参数向量元素。从而简化了SOC或估计SOC的计算。另外，电池105端子间的电压v[k]（可用于计算SOC或估计SOC）可基于等效电路模型。v[k]相关的方程、SOC和等效电路模型在下文中更详细描述。In step S1120 , a state of charge (SOC) or an estimated SOC of the battery is calculated using the selected equivalent circuit model. For example, calculating the SOC may be based on the voltage drop model parameter vector (b), as described above. The voltage drop model parameter vector may have parameters based on the battery's equivalent circuit (see Equation 7 above). Therefore, the determined voltage drop model parameter vector may have a higher or lower complexity based on the equivalent circuit. For example, as described below, the equivalent circuit may not include RC circuit elements because the capacitor charges and bypasses the resistor. Therefore, b(3) may be the only remaining voltage drop model parameter vector element. This simplifies the calculation of SOC or estimated SOC. Additionally, the voltage v[k] across the terminals of the battery 105 (which may be used to calculate SOC or estimate SOC) may be based on an equivalent circuit model. The equations related to v[k], SOC and equivalent circuit model are described in more detail below.

图12示出了使用递归滤波器计算估计SOC的方法的流程图。如图12所示，在步骤S1205中，从缓冲器读取所存储的电池的估计电荷状态（SOC）。例如，缓冲器850可在其中存储在针对该流程图所述的步骤的前一次迭代中所计算出的至少一个SOC误差和SOC。可从缓冲器850读取存储SOC值中的至少一个。FIG. 12 shows a flowchart of a method of calculating an estimated SOC using a recursive filter. As shown in FIG. 12 , in step S1205 , the stored estimated state of charge (SOC) of the battery is read from the buffer. For example, the buffer 850 may store therein at least one SOC error and the SOC calculated in a previous iteration of the steps described for the flowchart. At least one of the stored SOC values may be read from the buffer 850 .

在步骤S1210中，读取电池两端的测量电压。例如，可使用例如数字放大器215读取或确定电压（如，下图13A-13D中所示的v[k]）。在一个示例性具体实施中，电压存储在缓冲器中。因此，不同迭代可使用不同电压测量值。换句话讲，前一个（时间上）电压测量值可用于当前迭代或者v[k+1]可用于迭代k+2。In step S1210, the measured voltage across the battery is read. For example, a voltage (eg, v[k] as shown in FIGS. 13A-13D below) may be read or determined using, for example, digital amplifier 215 . In one exemplary implementation, the voltage is stored in a buffer. Thus, different iterations may use different voltage measurements. In other words, the previous (in time) voltage measurement can be used for the current iteration or v[k+1] can be used for iteration k+2.

在步骤S1215中，计算滤波器增益。例如，如上文简要描述和下文更详细描述，计算EKF块805的滤波器增益（如，G[k+1]）。滤波器增益可基于使用加权最小二乘算法计算出的至少一个容量值。例如，滤波器增益可基于使用加权递归最小二乘（RLS）算法和总体最小二乘（TLS）算法中的至少一者计算出的至少一个容量值。滤波器增益可基于使用加权RLS算法计算出的容量值，所述加权RLS算法基于SOC跟踪误差协方差和电流测量误差标准差。滤波器增益可基于估计SOC方差。滤波器增益可基于使用TLS算法计算出的容量值，所述TLS算法基于协方差矩阵的递归更新。滤波器增益可基于使用开路电压（OCV）查找计算出的容量值。SOC跟踪误差协方差、电流测量误差标准差、SOC方差、协方差矩阵和OCV中的每一个在下文更详细（如在数学上）描述。In step S1215, filter gain is calculated. For example, the filter gain (eg, G[k+1]) of the EKF block 805 is calculated as briefly described above and described in more detail below. The filter gain may be based on at least one capacity value calculated using a weighted least squares algorithm. For example, the filter gain may be based on at least one capacity value calculated using at least one of a weighted recursive least squares (RLS) algorithm and a total least squares (TLS) algorithm. The filter gain may be based on a capacity value calculated using a weighted RLS algorithm based on SOC tracking error covariance and current measurement error standard deviation. Filter gain may be based on estimated SOC variance. Filter gains may be based on capacity values calculated using a TLS algorithm based on recursive updates of the covariance matrix. Filter gain can be based on a capacitance value calculated using an open circuit voltage (OCV) lookup. Each of the SOC tracking error covariance, current measurement error standard deviation, SOC variance, covariance matrix, and OCV are described in more detail (eg, mathematically) below.

在步骤S1220中，基于电池的存储SOC、电池两端的电压和滤波器增益来计算电池的估计SOC。例如，估计的SOC可等于滤波器增益乘以数字电压值加上存储的估计SOC。在步骤S1225中，将计算出的估计SOC存储在缓冲器（如，缓冲器850）中。如果有必要和/或需要进一步计算估计SOC（S1230），则处理返回至步骤S1205。例如，如果电池105在持续使用中，如果SOC误差超过所需值且进一步迭代可减少误差，和/或如果电池测试正在进行，等等，那么可能有必要和/或需要进一步计算。In step S1220, an estimated SOC of the battery is calculated based on the stored SOC of the battery, the voltage across the battery and the filter gain. For example, the estimated SOC may be equal to the filter gain multiplied by the digital voltage value plus the stored estimated SOC. In step S1225, the calculated estimated SOC is stored in a buffer (eg, buffer 850). If it is necessary and/or required to further calculate the estimated SOC (S1230), the process returns to step S1205. For example, further calculations may be necessary and/or desirable if the battery 105 is in continuous use, if the SOC error exceeds a desired value and further iterations may reduce the error, and/or if battery testing is ongoing, etc.

图13A-13D示出了根据至少一个示例性实施例的电池等效模型的示意图。下面将视需要参照图13A-13D以描述一个或多个示例性具体实施。如图13A-13D中所示，代表电池的等效模型1300-1、1300-2、1300-3、1300-4可包括电阻器1315、1325、1340，电容器1330、1345和等效电压源1305、1310的任何组合。电压1355表示加载时电池两端的电压降。电流1320、1335和1350表示流过（或流至）等效模型的元件的电流。例如，电流1350表示流至负载的电流。13A-13D illustrate schematic diagrams of battery equivalent models according to at least one example embodiment. One or more exemplary implementations are described below with reference to Figures 13A-13D as appropriate. As shown in Figures 13A-13D, an equivalent model 1300-1, 1300-2, 1300-3, 1300-4 representing a battery may include resistors 1315, 1325, 1340, capacitors 1330, 1345 and an equivalent voltage source 1305 , any combination of 1310. Voltage 1355 represents the voltage drop across the battery when loaded. Currents 1320, 1335, and 1350 represent currents flowing through (or to) elements of the equivalent model. For example, current 1350 represents the current flowing to the load.

电阻器和电容器可定义一RC电路。例如，电阻器1325和1330定义一RC电路。在一些示例性具体实施中，电容器可被完全充电并短路，导致RC电路实际上从等效模型消失。例如，在代表电池的等效模型1300-2中，由电阻器1340和电容器1345定义的RC电路不在模型中，因为电容器1345被完全充电，形成了短路。在一些示例性实施例中，没有（或极少）与电池相关的滞后（如，电池处于静置或拖带极少负载）。因此，如代表电池的等效模型1300-4中所示，由于不存在滞后，等效电压源1310不在模型中。Resistors and capacitors define an RC circuit. For example, resistors 1325 and 1330 define an RC circuit. In some example implementations, the capacitor can be fully charged and shorted, causing the RC circuit to virtually disappear from the equivalent model. For example, in equivalent model 1300-2 representing a battery, the RC circuit defined by resistor 1340 and capacitor 1345 is not in the model because capacitor 1345 is fully charged, forming a short circuit. In some exemplary embodiments, there is no (or minimal) battery-related hysteresis (eg, the battery is resting or dragging very little load). Therefore, as shown in the equivalent model 1300-4 representing a battery, the equivalent voltage source 1310 is not in the model due to the absence of hysteresis.

本申请接下来描述了示例性具体实施的细节。所述细节可包括至少一个上述方程的建立（如，数学证明或简化）。为清楚起见，可重复这些方程，然而，这些方程将保留方括号（[]）中所示的方程编号。从实时模型识别开始，其可参考以下注释。The application follows to describe the details of an exemplary implementation. The details may include establishment (eg, mathematical proof or simplification) of at least one of the above equations. These equations may be repeated for clarity, however, the equations will retain the equation numbers shown in square brackets ([ ]). Start with real-time model recognition, which can refer to the following notes.

SOC跟踪算法的元素可包括：Elements of a SOC tracking algorithm may include:

a. OCV参数的估计：当通过使用年限和依赖于使用年限的电池容量进行归一化时，OCV-SOC表征随温度变化和电池老化是稳定的。a. Estimation of OCV parameters: When normalized by age and age-dependent battery capacity, the OCV-SOC characterization is stable over temperature and battery aging.

b. 动态等效电路参数的估计：已观察到这些参数随电池的温度、SOC和使用年限而变化，因此应当在BFG运行的同时自适应地估计。b. Estimation of dynamic equivalent circuit parameters: These parameters have been observed to vary with the temperature, SOC, and age of the battery, and thus should be adaptively estimated while the BFG is running.

c. 电池容量的估计：虽然电池的标称容量由制造商所指定，但已知可用电池容量会因为制造工艺误差、温度变化、加载模式和老化而变化。c. Estimates of battery capacity: Although the nominal capacity of a battery is specified by the manufacturer, usable battery capacity is known to vary due to manufacturing process tolerances, temperature variations, loading patterns, and aging.

d. 受模型参数约束的SOC跟踪：一旦已知模型参数，SOC跟踪就变成了非线性滤波问题。d. SOC tracking constrained by model parameters: Once the model parameters are known, SOC tracking becomes a nonlinear filtering problem.

示例性实施例允许对电池的动态等效电路参数进行实时线性估计。通过解决下列问题，在该示例性实施例中实现了改进现有的电池等效电路建模和参数估计的方法：Exemplary embodiments allow real-time linear estimation of dynamic equivalent circuit parameters of a battery. Improvements to existing battery equivalent circuit modeling and parameter estimation methods are implemented in this exemplary embodiment by addressing the following issues:

a. 一些模型仅考虑电阻，不适合动态负载。a. Some models only consider resistance and are not suitable for dynamic loads.

b. 它们采用非线性方法来进行系统识别。b. They employ nonlinear methods for system identification.

c. 需要用于模型识别方法的初始参数估计。c. Need for initial parameter estimates for model identification methods.

d. 假定单个动态等效模型代表所有电池运行模式。d. Assume that a single dynamic equivalent model represents all battery operating modes.

在该示例性具体实施中，解决了上述四个问题并总结如下：In this exemplary implementation, the above four problems are addressed and summarized as follows:

a. 用于模型参数估计的在线线性方法，无需估计电池等效电路的准确物理表示形式的参数。SOC跟踪状态空间模型利用了修改后的可线性估计的参数的估计。a. An online linear method for model parameter estimation without estimating the parameters of an exact physical representation of the battery equivalent circuit. The SOC tracking state-space model utilizes modified estimates of linearly estimable parameters.

b. 适用于各种各样的电池，无需任何初始值或校准：由于示例性状态空间模型的自适应性，所提出的SOC跟踪方法不需要模型参数的任何离线初始化。最小二乘（LS）方法提供了在需要的任何时候对参数的初始化（或重新初始化），块递归最小二乘（RLS）被用来持续跟踪模型参数。另外，已表明修改后的开路电压（OCV）模型在不同电池模型、不同温度和不同负载条件下是有效的。这就使示例性BFG可以即插即用方式应用于宽泛范围的电池，无需与其相关的任何其他另外信息。b. Applicable to a wide variety of batteries without any initialization or calibration: Due to the adaptive nature of the exemplary state-space model, the proposed SOC tracking method does not require any offline initialization of the model parameters. Least squares (LS) methods provide initialization (or reinitialization) of parameters whenever needed, and block recursive least squares (RLS) are used to keep track of model parameters. In addition, the modified open-circuit voltage (OCV) model has been shown to be effective under different battery models, different temperatures and different load conditions. This allows the exemplary BFG to be applied to a wide range of batteries in a plug-and-play manner without any other additional information associated with it.

c. 对不同电池模式进行无缝SOC跟踪的可能性。可识别四个不同电池等效模型以反映非常轻的负载或静置状态、恒定电流或低频率负载、动态负载和变化的重负载。还识别四个（稍微）不同的动态等效模型以最佳匹配这些模式。这些模型可用于无缝SOC跟踪，而不论电池运行的模式变化。c. Possibility of seamless SOC tracking for different battery modes. Four different battery equivalent models can be identified to reflect very light loads or resting conditions, constant current or low frequency loads, dynamic loads and varying heavy loads. Four (slightly) different dynamic equivalent models were also identified to best match these patterns. These models can be used for seamless SOC tracking regardless of changes in the mode of battery operation.

d. 滞后建模，其消除了滞后建模的需要：示例性具体实施认识到几乎不可能（完美地）离线进行滞后建模，因为滞后与SOC∈[01]和负载电流I∈R有关。因此，根据示例性实施例，在电压降模型中，将滞后建模为OCV中的误差，并且在线滤波方法持续尝试通过调整SOC（至修正值）来填补差距。d. Hysteresis modeling, which eliminates the need for hysteresis modeling: The exemplary implementation recognizes that hysteresis modeling is almost impossible (perfectly) offline because hysteresis is related to SOC ∈ [01] and load current I ∈ R. Therefore, according to an exemplary embodiment, hysteresis is modeled as an error in OCV in the voltage drop model, and the online filtering method continuously tries to fill the gap by adjusting the SOC (to a corrected value).

实时模型识别包括使用等效电路进行的实时模型参数估计。图13A为示例性电池（如，电池105）的等效电路。当电池处于静置时，V₀(s[k])是电池的OCV。OCV唯一取决于电池的SOC，s[k]∈[0，1]。当电池处于活跃状态时，例如当存在电流活动时，电池的行为通过动态等效电路表示，所述动态等效电路由滞后元件h[k]、串联电阻R₀以及串联连接的两个并联RC电路(R₁,C₁)和(R₂,C₂)组成。离散时间使用[k]指示。Real-time model identification includes real-time model parameter estimation using equivalent circuits. 13A is an equivalent circuit of an exemplary battery (eg, battery 105). V ₀ (s[k]) is the OCV of the battery when the battery is at rest. OCV only depends on the SOC of the battery, s[k]∈[0,1]. When the battery is active, i.e. when there is current flow, the behavior of the battery is represented by a dynamic equivalent circuit consisting of a hysteresis element h[k], a series resistance _R0 , and two parallel RCs connected in series Circuit (R ₁ , C ₁ ) and (R ₂ , C ₂ ) are composed. Discrete times are indicated using [k].

在图13A中，将流过电池的测量电流写成：In Figure 13A, the measured current through the battery is written as:

z_i[k]=i[k]+n_i[k] （8）z _i [k]=i[k]+n _i [k] (8)

其中i[k]是流过电池的真实电流，并且n_i[k]是电流测量噪声，假定电流测量噪声为零平均值并具有标准差（s.d.）σ_i.电池两端的测量电压为：where i[k] is the true current flowing through the battery, and n _i [k] is the current measurement noise, assumed to be zero mean and have standard deviation (sd) σ _i . The measured voltage across the battery is:

z_v[k]=v[k]+n_v[k] （9）z _v [k] = v [k] + n _v [k] (9)

其中v[k]是电池两端的真实电压，并且n_v[k]是电压测量噪声，假定电压测量噪声为具有s.d.的零平均值。σ_v。where v[k] is the true voltage across the battery, and _nv [k] is the voltage measurement noise, assumed to be zero mean with sd. σ _v .

按照如下形式书写内部元件R₀,R₁,R₂与h[k]两端的电池电压降：Write the battery voltage drop across the internal components R ₀ , R ₁ , R ₂ and h[k] as follows:

${v v}_{D D.} [[k k]] \overset{Δ Δ}{= =} {z z}_{V V} [[k k]] - - {V V}_{00} [[s the s [[k k]]]] = = i i [[k k]] {R R}_{00} + + {x x}_{{i i}_{11}} [[k k]] {R R}_{11} + + {x x}_{{i i}_{22}} [[k k]] {R R}_{22} + + {x x}_{h h} [[k k]] + + {n no}_{V V} [[k k]] - - - - - - ((1010))$

其中流过电阻器R₁和R₂的电流可按照如下形式书写where the current flowing through resistors _R1 and _R2 can be written as follows

${x x}_{{i i}_{11}} [[k k + + 11]] \overset{Δ Δ}{= =} {i i}_{11} [[k k + + 11]] = = {α α}_{11} {i i}_{11} [[k k]] + + ((11 - - {α α}_{11})) i i [[k k]] . . - - - - - - ((1111))$

${x x}_{{i i}_{22}} [[k k + + 11]] \overset{Δ Δ}{= =} {i i}_{22} [[k k + + 11]] = = {α α}_{22} {i i}_{22} [[k k]] + + ((11 - - {α α}_{22})) i i [[k k]] - - - - - - ((1212))$

其中，in,

${α α}_{11} \overset{Δ Δ}{= =} {e e}^{- -} \frac{Δ Δ}{{R R}_{11} {C C}_{11}};; - - - - - - ((1313))$

$α_{2} \overset{Δ}{=} e^{-} \frac{Δ}{R_{2} C_{2}};$ 以及（14） $α_{2} \overset{Δ}{=} e^{-} \frac{Δ}{R_{2} C_{2}};$ and (14)

Δ为取样间隔。Δ is the sampling interval.

通过用测量电流z_i[k]替换i[k]，可将（11）和（12）中的电流重新书写成如下形式：By replacing i[k] with the measured current z _i [k], the currents in (11) and (12) can be rewritten as follows:

${x x}_{{i i}_{11}} [[k k + + 11]] = = {α α}_{11} {x x}_{{i i}_{11}} [[k k]] + + ((11 - - {α α}_{11})) {z z}_{i i} [[k k]] - - ((11 - - {α α}_{11})) {n no}_{i i} [[k k]] - - - - - - ((1515))$

${x x}_{{i i}_{22}} [[k k + + 11]] = = {α α}_{22} {x x}_{{i i}_{22}} [[k k]] + + ((11 - - {α α}_{22})) {z z}_{i i} [[k k]] - - ((11 - - {α α}_{22})) {n no}_{i i} [[k k]] - - - - - - ((1616))$

现在使用（8）、（10）可以在z域中重新书写成如下形式：Now using (8), (10) can be rewritten in the z domain as follows:

${V V}_{D D.} [[z z]] = = {Z Z}_{i i} [[z z]] {R R}_{00} + + {X x}_{{i i}_{11}} [[z z]] {R R}_{11} + + {X x}_{{i i}_{11}} [[z z]] {R R}_{22} + + {x x}_{h h} [[z z]] + + {N N}_{v v} [[z z]] - - {R R}_{00} {N N}_{i i} [[z z]] - - - - - - ((1717))$

接下来，在z域中重新书写（15）：Next, rewrite (15) in the z domain:

$z z {X x}_{{i i}_{11}} [[z z]] = = {α α}_{11} {X x}_{{i i}_{11}} [[z z]] + + ((11 - - {α α}_{11})) {Z Z}_{i i} [[z z]] - - ((11 - - {α α}_{11})) {N N}_{i i} [[z z]] - - - - - - ((1818))$

得出inferred

${X x}_{{i i}_{11}} [[z z]] = = \frac{11 - - {α α}_{11}}{z z - - {α α}_{11}} (({Z Z}_{i i} [[z z]] - - {N N}_{i i} [[z z]])) - - - - - - ((1919))$

并且对于（16）类似地，and similarly for (16),

${X x}_{{i i}_{22}} [[z z]] = = \frac{11 - - {α α}_{22}}{z z - - {α α}_{22}} (({Z Z}_{i i} [[z z]] - - {N N}_{i i} [[z z]])) - - - - - - ((2020))$

通过将（19）和（20）代入（17）：By substituting (19) and (20) into (17):

$\begin{matrix} {V V}_{D D.} [[z z]] = = {Z Z}_{i i} [[z z]] {R R}_{00} + + \frac{11 - - {α α}_{11}}{z z - - {α α}_{11}} {Z Z}_{i i} [[z z]] {R R}_{11} + + \frac{11 - - {α α}_{22}}{z z - - {α α}_{22}} {Z Z}_{i i} [[z z]] {R R}_{22} + + {X x}_{h h} [[z z]] + + {N N}_{v v} [[z z]] \\ - - (({R R}_{00} + + \frac{11 - - {α α}_{11}}{z z - - {α α}_{11}} {R R}_{11} + + \frac{11 - - {α α}_{22}}{z z - - {α α}_{22}} {R R}_{22})) {N N}_{i i} [[z z]] \end{matrix} - - - - - - ((21 twenty one))$

重新整理（21）并将其转换回时域：Rearrange(21) and convert it back to the time domain:

其中，in,

α=α₁+α₂,（23）α=α ₁ +α ₂ , (23)

β=α₁α₂,（24）β=α ₁ α ₂ , (24)

$\overset{&OverBar; &OverBar;}{h h} [[k k]] = = {x x}_{h h} [[k k]] - - α α {x x}_{h h} [[k k - - 11]] + + β β {x x}_{h h} [[k k - - 22]],, - - - - - - ((2727))$

${\overset{&OverBar;}{n}}_{v} [k] = n_{v} [k] - α n_{v} [k - 1] + β n_{v} [k - 2],$ 以及（28） ${\overset{&OverBar;}{no}}_{v} [k] = {no}_{v} [k] - α {no}_{v} [k - 1] + β {no}_{v} [k - 2],$ and (28)

现在将（22）重新书写成下列形式：Now rewrite (22) in the following form:

v_D[k]=[k]^T+n_D[k] （30）v _D [k]=[k] ^T +n _D [k] (30)

其中观测模型a[k]^T和模型参数向量b给出如下：where the observation model a[k] ^T and the model parameter vector b are given as follows:

$a a {[[k k]]}^{T T} {= =}_{44} {[[k k]]}^{T T} \overset{Δ Δ}{= =} [[[\begin{matrix} {v v}_{D D.} [[k k - - 11]] & - - {v v}_{D D.} [[k k - - 22]] & {z z}_{i i} [[k k]] & {z z}_{i i} [[k k - - 11]] & - - {z z}_{i i} [[k k - - 22]] & 11 \end{matrix}] . . - - - - - - ((3131))$

其中下标4指示与图13A-13B中所示四个模型的模型4对应的上述模型。where the subscript 4 indicates the above model corresponding to model 4 of the four models shown in Figures 13A-13B.

（30）中的电压降观测的噪声书写成：The noise of the voltage drop observation in (30) is written as:

${n no}_{D D.} [[k k]] \overset{Δ Δ}{= =} {\overset{&OverBar; &OverBar;}{n no}}_{i i} [[k k]] + + {\overset{&OverBar; &OverBar;}{n no}}_{v v} [[k k]] - - - - - - ((3333))$

其具有如下给出的自相关性：It has an autocorrelation given by:

并且假定在长度L_b的时间间隔的批次期间滞后分量为常数，例如，and assume that the lag component is constant during batches of time intervals of length _Lb , e.g.,

$\overset{&OverBar; &OverBar;}{h h} [[l l]] = = {x x}_{h h} [[l l]] - - (({α α}_{11} + + {α α}_{22})) {x x}_{h h} [[l l - - 11]] + + {α α}_{11} {α α}_{22} {x x}_{h h} [[l l - - 22]] \approx \approx H h,, l l = = k k - - {L L}_{b b} + + 11,, . . . . . .,, k k - - - - - - ((3535))$

现在，可以描述的四个不同的电池运行“模式”以及匹配这些模式的适当电池等效模型。Now, four different "modes" of battery operation can be described and appropriate battery equivalent models to match these modes.

a. 模式1–轻负载或静置状态：当电池仅受到轻负载，然后被充电，然后静置时，滞后分量将小到可忽略不计。该模式的例子将是蜂窝电话，其在完全充电后，耗费几乎所有时间联系基站直至下一次充电事件，除了可能的少数电话呼叫外。单个电阻器（参见图13D）非常适配该模式。a. Mode 1 – light load or rest state: When the battery is only lightly loaded, then charged, and then rested, the hysteresis component will be negligibly small. An example of this pattern would be a cell phone which, after being fully charged, spends almost all of its time contacting the base station until the next charging event, except perhaps for a few phone calls. A single resistor (see Figure 13D) works well for this mode.

b. 模式2–恒定电流运行：当流过电池的电流恒定时，RC电路中的电容器变为完全充电。因此，从参数估计观点看，所得的电路可被视为单个电阻器和滞后/偏置元件（参见图13C）。电池的恒定电流充电是该模式的好例子。b. Mode 2 – Constant Current Operation: When the current flowing through the battery is constant, the capacitor in the RC circuit becomes fully charged. Therefore, from a parameter estimation point of view, the resulting circuit can be viewed as a single resistor and hysteresis/biasing element (see Figure 13C). Constant current charging of batteries is a good example of this mode.

c. 模式3–动态负载：当电池处于该模式时，存在不同大小的大量负载。实例：定期地用于电话呼叫、web浏览、视频剪辑等等的智能电话。图13B中所示的等效电池非常适配该场景。c. Mode 3 – Dynamic Loads: When the battery is in this mode, there are a large number of loads of different sizes. Example: Smartphone used regularly for phone calls, web browsing, video clips, etc. The equivalent battery shown in Figure 13B fits this scenario well.

d. 模式4–繁重且变化的使用：对于移动电话，繁重且变化的使用包括长时的视频播放、多媒体和游戏应用程序等等。图13A非常匹配该场景。d. Mode 4 – heavy and varied usage: For mobile phones, heavy and varied usage includes prolonged video playback, multimedia and gaming applications, etc. Figure 13A matches this scenario very well.

注意，动态等效电路的不同模型复杂性可通过仅改变[k]^T来表示。以下示出了针对每个模型的[k]^T的定义。对于上述模型复杂性中的每一个，噪声项n_D[k]按照如下形式以和表示：Note that different model complexities of the dynamic equivalent circuit can be represented by varying only [k] ^T . The definition of [k] ^T for each model is shown below. For each of the above model complexities, the noise term n _D [k] is given by and express:

其中，in,

α=α₁+α₂ （38）α=α ₁ +α ₂ (38)

β=α₁α₂ （39）β=α ₁ α ₂ (39)

${\overset{~ ~}{R R}}_{11} = = {α α}_{11} {R R}_{00} - - ((11 - - {α α}_{11})) {R R}_{11} - - - - - - ((4040))$

以下讨论涉及时不变动态模型参数的最小二乘估计。将时间k处的真实SOC表示为：The following discussion deals with least squares estimation of time-invariant dynamic model parameters. Denote the true SOC at time k as:

${x x}_{s the s} [[k k]] \overset{Δ Δ}{= =} s the s [[k k]] - - - - - - ((4141))$

建立的SOC跟踪算法可用于获得即x_s[k]的更新估计。现在，（10）中的电压降v_D[k]可书写成：The established SOC tracking algorithm can be used to obtain That is, the updated estimate of x _s [k]. Now, the voltage drop _vD [k] in (10) can be written as:

${v v}_{D D.} [[k k]] = = {z z}_{v v} [[k k]] - - {V V}_{00} (({\overset{^^}{x x}}_{s the s} [[k k | | k k]])) - - - - - - ((4242))$

其中表示电池的估计开路电压（OCV），其可被描述为估计SOC的函数。可采用以下OCV-SOC关系：in represents the estimated open circuit voltage (OCV) of the battery, which can be described as a function of the estimated SOC. The following OCV-SOC relationship can be used:

$\begin{matrix} \end{matrix} \begin{matrix} {V V}_{00} (({\overset{^^}{x x}}_{s the s} [[k k | | k k]])) = = {K K}_{00} + + \frac{{K K}_{11}}{{\overset{^^}{x x}}_{s the s} [[k k | | k k]]} + + \frac{{K K}_{22}}{{\overset{^^}{x x}}_{s the s} {[[k k | | k k]]}^{22}} + + \frac{{K K}_{33}}{{\overset{^^}{x x}}_{s the s} {[[k k | | k k]]}^{33}} + + \frac{{K K}_{44}}{{\overset{^^}{x x}}_{s the s} {[[k k | | k k]]}^{44}} \\ + + {K K}_{55} {\overset{^^}{x x}}_{s the s} [[k k | | k k]] + + {K K}_{66} ln ln (({\overset{^^}{x x}}_{s the s} [[k k | | k k]])) + + {K K}_{77} ln ln ((11 - - {\overset{^^}{x x}}_{s the s} [[k k | | k k]])) \end{matrix} - - - - - - ((4343))$

可通过如下详细描述的步骤对OCV参数K_i∈{K₀,K₁,K₂,K₃,K₄,K₅,K₆,K₇}进行离线估计。通过考虑L_b观测批次，可将（30）重新书写成如下形式：The OCV parameter K _i ∈ {K ₀ , K ₁ , K ₂ , K ₃ , K ₄ , K ₅ , K ₆ , K ₇ } can be estimated offline through the steps described in detail below. By considering _Lb observation batches, (30) can be rewritten as follows:

${v v}_{D D.}^{κ κ} = = {A A}^{κ κ} + + {n no}_{D D.}^{κ κ} - - - - - - ((4444))$

其中κ为批次编号，where κ is the batch number,

${v v}_{D D.}^{κ κ} = = [[{v v}_{D D.} [[κ κ {L L}_{b b} - - {L L}_{b b} + + 11]] {v v}_{D D.} [[κ κ {L L}_{b b} - - {L L}_{b b} + + 22]] . . . . . . {v v}_{D D.} {[[κ κ {L L}_{b b}]]]]}^{T T} - - - - - - ((4545))$

A^κ=[[κL_b-L_b+1][κL_b-L_b+2]…[κL_b]]^T （46）A ^κ = [[κL _b -L _b +1][κL _b -L _b +2]…[κL _b ]] ^T (46)

并且噪声具有以下协方差and noise with the following covariance

${Σ Σ}_{D D.}^{κ κ} = = E E. {{{n no}_{D D.}^{κ κ} {n no}_{D D.}^{κ κ}^{T T}}} - - - - - - ((4747))$

其中为五对角托普利兹矩阵，其中对角线元素、第一和第二非对角线元素分别通过和给出（见（32））。现在，动态模型参数向量可通过最小二乘（LS）优化由（42）估计如下：in is a five-diagonal Toeplitz matrix, in which the diagonal elements, the first and second off-diagonal elements pass through and given (see (32)). Now, the dynamic model parameter vector can be estimated by (42) via least squares (LS) optimization as follows:

${\overset{^^}{b b}}_{LS LS} [[κ κ]] = = {(({(({A A}^{κ κ}))}^{T T} {(({Σ Σ}_{D D.}^{κ κ}))}^{- - 11} {A A}^{κ κ}))}^{- - 11} {A A}^{κ κ}^{T T} {(({Σ Σ}_{D D.}^{κ κ}))}^{- - 11} {v v}_{D D.}^{κ κ} - - - - - - ((4848))$

LS估计量的协方差矩阵给出如下：The covariance matrix of the LS estimator is given as:

${P P}_{b b} [[κ κ]] = = {(({(({A A}^{κ κ}))}^{T T} (({Σ Σ}_{D D.}^{κ κ})) {A A}^{- - 11}^{κ κ}))}^{- - 11} - - - - - - ((4949))$

当获得新批次的测量值时，可通过重复（50）–（51）递归更新LS估计值When a new batch of measurements is obtained, the LS estimate can be recursively updated by repeating (50)–(51)

${P P}_{b b}^{- - 11} [[κ κ + + 11]] = = λ λ {P P}_{b b}^{- - 11} [[κ κ]] + + {(({A A}^{κ κ + + 11}))}^{T T} {(({Σ Σ}_{D D.}^{κ κ}))}^{- - 11} {A A}^{κ κ + + 11} - - - - - - ((5050))$

${\overset{^^}{b b}}_{RLS RLS} [[κ κ + + 11]] = = {P P}_{b b} [[κ κ + + 11]] ((λ λ {P P}_{b b}^{- - 11} [[κ κ]] {\overset{^^}{b b}}_{RLS RLS} [[κ κ]] + + {(({A A}^{κ κ + + 11}))}^{T T} {(({Σ Σ}_{D D.}^{κ κ}))}^{- - 11} {v v}_{D D.}^{κ κ + + 11})) - - - - - - ((5151))$

其中λ为遗忘（衰减记忆）因子，(·)^T表示转置，(·)^-1表示求逆，并且被称为信息矩阵，该信息矩阵可通过适当大小的单位矩阵乘以合适的常数来初始化。可以注意到，当λ=0时，变为可存在使近似的多种方法。可选择以下两种近似法用于比较：where λ is the forgetting (decaying memory) factor, ( ) ^T means transpose, ( ) ^-1 means inversion, and Known as the information matrix, this information matrix can be initialized by multiplying an identity matrix of appropriate size by an appropriate constant. It can be noticed that when λ=0, becomes can exist so that Various methods of approximation. The following two approximations can be chosen for comparison:

a. 可进行以下近似法：a. The following approximations can be made:

${Σ Σ}_{D D.}^{κ κ} \approx \approx {I I}_{{l l}_{b b}} - - - - - - ((5252))$

b_.使用之前的估计值以构建电流协方差矩阵，例如使用：b _. Use previous estimates to construct the current covariance matrix, for example using:

中的 ${\hat{b}}_{RLS} [κ - 1] - - - (53)$ middle ${\hat{b}}_{RLS} [κ - 1] - - - (53)$

以下讨论涉及时变的动态模型参数的最小均方误差（MMSE）估计。假设动态模型参数为发生下列缓慢变化维纳过程的随机变量：The following discussion involves minimum mean square error (MMSE) estimation of time-varying dynamic model parameters. Assume that the dynamic model parameters are random variables that undergo the following slowly varying Wiener process:

x_b[k+1]=x_b[k]+w_b[k] （54）x _b [k+1]=x _b [k]+w _b [k] (54)

其中w_b[k]为具有协方差∑_b的零平均值高斯白噪声。现在，使用（30）作为测量模型以及（53）作为过程模型，卡尔曼滤波器给出b的MMSE估计。SOC可用于确定v_D[k]（见（42）），SOC跟踪/平滑的迭代算法和通过足够长度的观测窗进行的基于卡尔曼滤波的参数估计可用于提高SOC跟踪和参数估计的精度。where w _b [k] is zero-mean white Gaussian noise with covariance ∑ _b . Now, using (30) as the measurement model and (53) as the process model, the Kalman filter gives the MMSE estimate of b. SOC can be used to determine _vD [k] (see (42)), and iterative algorithms for SOC tracking/smoothing and Kalman filter-based parameter estimation through observation windows of sufficient length can be used to improve the accuracy of SOC tracking and parameter estimation.

以下讨论涉及开路电压（OCV）参数估计。SOC估计可利用电池的开路电压（OCV）与SOC之间的独特且稳定的关系并允许针对测得的OCV计算SOC。然而，仅当电池处于静置时，才可直接测量OCV。当电池在使用中时，电池电压与电流之间的动态关系必须通过参数和状态估计方法进行说明。基于OCV-SOC的电荷状态估计方法包括与如下相关的误差：（1）电池的动态等效电模型的建模和参数估计的不确定性；以及（2）测量的电压和电流的误差。（43）中OCV-SOC表征的参数可按如下方式通过在样品电池上采集OCV表征数据来估计：The following discussion deals with open circuit voltage (OCV) parameter estimation. The SOC estimation can exploit the unique and stable relationship between the open circuit voltage (OCV) of the battery and the SOC and allow calculation of the SOC for the measured OCV. However, OCV can only be measured directly when the battery is at rest. When a battery is in use, the dynamic relationship between battery voltage and current must be accounted for by parameter and state estimation methods. OCV-SOC-based state-of-charge estimation methods include errors related to: (1) uncertainty in modeling and parameter estimation of the dynamic equivalent electrical model of the battery; and (2) errors in measured voltage and current. The parameters of OCV-SOC characterization in (43) can be estimated by collecting OCV characterization data on a sample cell as follows:

a. 从完全充电、完全静置的电池开始a. Start with a fully charged, fully rested battery

b. 记录其开路电压V_batt=V_full b. Record its open circuit voltage V _batt =V _full

c. 设定k=1c. Set k=1

d. 记录v[k]=V_batt；记录SOC[k]=1d. Record v[k]=V _batt ; record SOC[k]=1

e. 设定k=k+1e. Set k=k+1

f. 使用极少量的（通常C/30或C/40，其中C为以Ah表示的电池容量）恒定电流i[k]使电池连续放电，直到电池完全放电。一旦完全放电，就使电池保持静置，并且此后充电直到电池充满电。然后f. Using a very small amount (usually C/30 or C/40, where C is the battery capacity expressed in Ah) constant current i[k] to continuously discharge the battery until the battery is completely discharged. Once fully discharged, the battery is left at rest and thereafter charged until the battery is fully charged. Then

1.测量电池端电压，每Δ秒的V_batt 1. Measure the battery terminal voltage, V _batt every Δ second

2.记录v[k]=V_batt 2. Record v[k]=V _batt

g. 记录SOC[k]=SOC[k-1]+c_hi[k]Δg. Record SOC[k]=SOC[k-1]+c _h i[k]Δ

现在，OCV模型（43）对于所有测量值v[k]可以如下向量格式表示：Now, the OCV model (43) can be expressed in the following vector format for all measured values v[k]:

v=A_ocvk （55）v=A _ocv k (55)

其中in

v=[v[1],v[2]…,v[Nv]]^T （56）v=[v[1],v[2]...,v[Nv]] ^T (56)

A_ocv=[a_ocv(1),a_ocv(2),…,a_ocv(Nv)]^T （57）A _ocv =[a _ocv (1),a _ocv (2),…,a _ocv (Nv)] ^T (57)

k=[K₀ K₁ K₂ K₃ K₄ K₅ K₆ K₇ R₀]^T （58）k=[K ₀ K ₁ K ₂ K ₃ K ₄ K ₅ K ₆ K ₇ R ₀ ] ^T (58)

然后通过赋值s[k]=SOC[k]Then by assigning s[k]=SOC[k]

${a a}_{ocv ocv} [[k k]] = = [\begin{matrix} 11 & \frac{11}{s the s [[k k]]} & \frac{11}{{s the s}^{22} [[k k]]} & \frac{11}{{s the s}^{33} [[k k]]} & \frac{11}{{s the s}^{44} [[k k]]} & s the s [[k k]] & ln ln ((s the s [[k k]])) & ln ln ((11 - - s the s [[k k]])) & i i [[k k]] \end{matrix}] T T - - - - - - ((5959))$

现在，OCV参数的最小二乘估计值和电池内阻R₀给出如下：Now, the least squares estimates of the OCV parameters and the battery internal resistance _R0 are given as follows:

$\overset{^^}{k k} = = {(({A A}_{ocv ocv}^{T T} {A A}_{ocv ocv}))}^{- - 11} {A A}_{ocv ocv}^{T T} v v - - - - - - ((6060))$

以下讨论涉及四个示例性等效电路模型。The following discussion refers to four exemplary equivalent circuit models.

表1Table 1

模型编号model number 等效电路Equivalent Circuit 模型1model 1 图13DFigure 13D 模型2model 2 图13CFigure 13C 模型3model 3 图13BFigure 13B 模型4model 4 图13AFigure 13A

表1中示出的四个等效电路模型中每一个的电路元件两端的电压降可以如下形式书写：The voltage drop across the circuit elements of each of the four equivalent circuit models shown in Table 1 can be written as follows:

v_D[k]=[k]^T[k]+n_D[k] （61）v _D [k]=[k] ^T [k]+n _D [k] (61)

其中，in,

对于模型3：For model 3:

${\overset{~ ~}{R R}}_{11} = = {α α}_{11} {R R}_{00} - - ((11 - - {α α}_{11})) {R R}_{11} - - - - - - ((6464))$

$\overset{~ ~}{H h} = = h h [[k k]] - - {α α}_{11} h h [[k k - - 11]] - - - - - - ((6565))$

以及，对于模型4：and, for model 4:

α⁼α₁+α₂ （66）α ⁼ α ₁ + α ₂ (66)

β=α₁α₂ （67）β=α ₁ α ₂ (67)

以下涉及噪声相关性的推导。在本部分中，针对|l|=0,1,2和针对|l|>2的自相关性（27）可由（33）推导如下：The following deals with the derivation of the noise correlation. In this section, the autocorrelation (27) for |l|=0,1,2 and for |l|>2 can be derived from (33) as follows:

l=0： l=0 :

l=1： l=1 :

l=2： l=2 :

现在，上述可针对每个模型表示如下：Now, the above can be expressed for each model as follows:

该讨论继续进行实时容量估计，其可参考以下注释。The discussion continues with real-time capacity estimation, which can be found in the following notes.

电池的电荷状态（SOC），定义为：The state of charge (SOC) of the battery is defined as:

公式78提供有关电池状态的信息。SOC和电池容量的了解用于估计电池的关闭时间（TTS）或完全充电时间（TTF）。电池容量通常随温度变化并且其根椐使用模式和使用年限随时间推移而减弱。准确的电池容量跟踪是电池电量计量的关键要素。Equation 78 provides information on the state of the battery. Knowledge of the SOC and battery capacity is used to estimate the battery's time to shutdown (TTS) or time to full charge (TTF). Battery capacity typically varies with temperature and degrades over time based on usage patterns and age. Accurate battery capacity tracking is a key element of battery fuel gauging.

在该示例性具体实施中，在线容量估计可基于：In this exemplary implementation, the online capacity estimate may be based on:

a. 具有准确权重推导的容量的加权递归最小二乘（RLS）估计。用于在线容量估计的加权RLS方法包括基于在更新后的SOC跟踪的整个时间内的方差和协方差以及电流测量误差标准差推导权重的表达式。a. Weighted recursive least squares (RLS) estimation of capacity with accurate weight derivation. The weighted RLS method for online capacity estimation includes deriving expressions for weights based on variance and covariance and current measurement error standard deviation over the entire time of the updated SOC trace.

b. 用于实时跟踪电池容量的TLS方法。TLS方法给出用于容量估计的封闭式表达式。该方法可用于连续跟踪电池容量的变化。b. TLS method for real-time tracking of battery capacity. The TLS method gives closed-form expressions for capacity estimation. This method can be used to continuously track changes in battery capacity.

c. 基于静置电池的OCV查找的自适应容量估计。用于在线跟踪电池容量的TLS方法通过利用电池静置点进行基于OCV查找的SOC估计。c. Adaptive capacity estimation based on OCV lookup for stationary batteries. The TLS method for online tracking of battery capacity performs SOC estimation based on OCV lookups by utilizing battery resting points.

d. 通过不同方法获得的容量估计的融合。d. Fusion of capacity estimates obtained by different methods.

以下讨论涉及电池容量估计和融合。电池的瞬时电荷状态（SOC）可书写为以下过程模型，其也称为库仑计数公式，按照如下形式以测量电流表示：The following discussion deals with battery capacity estimation and fusion. The instantaneous state of charge (SOC) of a battery can be written as the following process model, also known as the Coulomb counting formula, expressed in terms of measured current as follows:

${x x}_{s the s} [[k k + + 11]] = = {x x}_{s the s} [[k k]] + + \frac{ηΔ ηΔ}{36003600 {C C}_{batt batt}} {z z}_{i i} [[k k]] + + {w w}_{s the s} [[k k]] - - - - - - ((7979))$

其中x_s[k]∈[0,1]表示电池的SOC，C_batt为以安培小时(Ah)表示的电池容量，并且z_i[k]为测量电流where x _s [k] ∈ [0,1] represents the SOC of the battery, C _batt is the battery capacity in Amp hours (Ah), and z _i [k] is the measured current

z_i[k]=i[k]+n_i[k] （80）z _i [k]=i[k]+n _i [k] (80)

其被具有标准差（s.d.）σ_i.的零平均值白噪声n_i[k]损坏。（79）中的过程噪声与（80）中的测量噪声相关，如It is corrupted by zero-mean white noise n _i [k] with standard deviation (sd) σ _i . The process noise in (79) is related to the measurement noise in (80), as

w_s[k]=-c_hΔn_i[k] （81）w _s [k]=-c _h Δn _i [k] (81)

并且为具有零平均值，并且s.d.为：and for has zero mean and s.d. is:

σ_s=c_hΔσ_i （82）σ _s = c _h Δσ _i (82)

其中库仑计数系数为where the Coulomb counting coefficient is

${c c}_{h h} &cong; &cong; \frac{η η}{36003600 {C C}_{batt batt}} - - - - - - ((8383))$

这里，η为取决于电池正在充电还是放电的常数，例如，Here, η is a constant depending on whether the battery is being charged or discharged, for example,

$η η = = \{\begin{matrix} {η η}_{c c} & i i [[k k]] > > 00 \\ {η η}_{d d} & i i [[k k]] < < 00 \end{matrix} - - - - - - ((8484))$

并且Δ为（常数）取样间隔。and Δ is the (constant) sampling interval.

以下讨论涉及使用递归最小二乘（RLS）的在线电池容量估计。估计SOC可基于电压和电流测量值。两个连续的SOC值x_s[k]和x_s[k+1]以其估计值书写为：The following discussion deals with online battery capacity estimation using recursive least squares (RLS). Estimating SOC can be based on voltage and current measurements. Two consecutive SOC values x _s [k] and x _s [k+1] are written in their estimated values as:

${x x}_{s the s} [[k k]] = = {\overset{^^}{x x}}_{s the s} [[k k | | k k]] + + {\overset{~ ~}{x x}}_{s the s} [[k k | | k k]] - - - - - - ((8484))$

${x x}_{s the s}]] k k + + 11]] = = {\overset{^^}{x x}}_{s the s} [[k k + + 11 | | k k + + 11]] + + {\overset{~ ~}{x x}}_{s the s} [[k k + + 11 | | k k + + 11]] - - - - - - ((8585))$

其中估计误差和分别具有零平均值和方差P_s[k|k]和P_s[k+1|k+1]。两个连续估计误差之间的协方差为：where the estimation error and have zero mean and variance P _s [k|k] and P _s [k+1|k+1], respectively. The covariance between two consecutive estimation errors is:

$E E. {{{\overset{~ ~}{x x}}_{s the s} [[k k | | k k {]] \overset{~ ~}{x x}}_{s the s} [[k k + + 11 | | k k + + 11]]}} = = ((11 - - G G [[k k + + 11]] H h [[k k + + 11]])) {P P}_{s the s} [[k k | | k k]] - - - - - - ((8686))$

其中G[k+1]为标量卡尔曼增益，以及H[k+1]为时间k+1处的标量线性化观测模型。现在，以如下形式重新书写（80）：where G[k+1] is the scalar Kalman gain, and H[k+1] is the scalar linearized observation model at time k+1. Now, rewrite (80) as follows:

${x x}_{s the s} [[k k + + 11]] - - {x x}_{s the s} [[k k]] = = \frac{ηΔ ηΔ}{36003600 {C C}_{batt batt}} {z z}_{i i} [[k k]] + + {w w}_{s the s} [[k k]] - - - - - - ((8787))$

将（84）和（85）代入（87）得到：Substituting (84) and (85) into (87) gives:

$Δ Δ {x x}_{s the s} [[k k + + 11,, k k]] &cong; &cong; {\overset{^^}{x x}}_{s the s} [[k k + + 11 | | k k + + 11]] - - {\overset{^^}{x x}}_{s the s} [[k k | | k k]] = = \frac{{η η}_{h h} Δ Δ}{{C C}_{batt batt}} {z z}_{i i} [[k k]] + + {\overset{~ ~}{w w}}_{s the s} [[k k]] - - - - - - ((8888))$

其中，in,

${η η}_{h h} = = \frac{η η}{36003600} - - - - - - ((8989))$

以及微分误差给出如下：and the differential error is given by:

${\overset{~ ~}{w w}}_{s the s} [[k k]] = = {\overset{^^}{x x}}_{s the s} [[k k | | k k]] + + {\overset{~ ~}{x x}}_{s the s} [[k k + + 11 | | k k + + 11]] + + - - {w w}_{s the s} [[k k]] - - - - ((9090))$

为具有零平均值，且方差如下：has zero mean and variance as follows:

$E E. {{{\overset{~ ~}{w w}}_{s the s} {[[k k]]}^{22}}} &cong; &cong; {R R}_{{\overset{~ ~}{w w}}_{S S} [[k k]]} = = G G {[[k k + + 11]]}^{22} S S [[k k + + 11]] - - - - - - ((9191))$

其中S[k+1]为卡尔曼滤波器的新息（innovation）协方差。通过考虑L_c样品批次，（88）可以向量形式书写成如下形式：where S[k+1] is the innovation covariance of the Kalman filter. By considering L _c sample batches, (88) can be written in vector form as follows:

${d d}_{s the s}^{κ κ} = = {C C}_{batt batt}^{- - 11} {d d}_{c c}^{κ κ} + + {\overset{~ ~}{w w}}_{s the s}^{κ κ} - - - - - - ((9292))$

其中，in,

κ为批次编号，κ is the batch number,

L_c为批次长度，L _c is the batch length,

$\begin{matrix} {d d}_{s the s}^{κ κ} = = [[Δ Δ {x x}_{s the s} [[κ κ {L L}_{c c} - - {L L}_{c c} + + 11,, κ κ {L L}_{c c} - - {L L}_{c c}]] Δ Δ {x x}_{s the s} [[κ κ {L L}_{c c} - - {L L}_{c c} + + 22,, κ κ {L L}_{c c} - - {L L}_{c c} + + 11]] . . . . . . \\ Δ Δ {x x}_{s the s} {[[κ κ {L L}_{c c},, κ κ {L L}_{c c} - - 11]]]]}^{T T} \end{matrix} - - - - - - ((9393))$

$\begin{matrix} {d d}_{c c}^{κ κ} = = [[{η η}_{h h} {Δ Δ}_{κ κ {L L}_{c c} - - {L L}_{c c} + + 11} {z z}_{i i} [[κ κ {L L}_{c c} - - {L L}_{c c} + + 11]] {η η}_{h h} {Δ Δ}_{κ κ {L L}_{c c} - - {L L}_{c c} + + 11} {z z}_{i i} [[κ κ {L L}_{c c} - - {L L}_{c c} + + 22]] . . . . . . \\ {η η}_{h h} {Δ Δ}_{κ κ {L L}_{c c}} {z z}_{i i} {[[κ κ {L L}_{c c}]]]]}^{T T} \end{matrix} - - - - - - ((9494))$

${\overset{~ ~}{w w}}_{s the s}^{κ κ} = = [[\overset{~ ~}{w w} [[κ κ {L L}_{c c} - - {L L}_{c c} + + 11]] \overset{~ ~}{w w} [[κ κ {L L}_{c c} - - {L L}_{c c} + + 22]] . . . . . . \overset{~ ~}{w w} {[[κ κ {L L}_{c c}]]]]}^{T T} - - - - - - ((9595))$

以及为具有如下协方差的高斯白噪声向量：as well as is a Gaussian white noise vector with the following covariance:

${Σ Σ}_{{\overset{~ ~}{w w}}_{s the s}}^{κ κ} = = E E. {{\begin{matrix} κ κ \\ s the s \end{matrix} {((\begin{matrix} κ κ \\ s the s \end{matrix}))}^{T T}}} - - - - - - ((9696))$

其为L_c×L_c对角矩阵，其第n对角元素给出如下：It is an L _c ×L _c diagonal matrix whose nth diagonal element is given as follows:

${(({Σ Σ}_{{\overset{~ ~}{w w}}_{s the s}}^{κ κ}))}_{nn nn} = = G G {[[κ κ {L L}_{c c} - - {L L}_{c c} + + n no]]}^{22} S S [[κ κ {L L}_{c c} - - {L L}_{c c} + + n no]] - - - - - - ((9797))$

现在，电池容量倒数的LS估计给出如下：Now, the LS estimate of the reciprocal of the battery capacity is given as:

${\overset{^^}{C C}}_{LS LS}^{- - 11} = = {(({(({d d}_{c c}^{κ κ}))}^{T T} {(({Σ Σ}_{{\overset{~ ~}{w w}}_{s the s}}^{κ κ}))}^{- - 11} {d d}_{c c}^{κ κ}))}^{- - 11} {(({d d}_{c c}^{κ κ}))}^{T T} {(({Σ Σ}_{{\overset{~ ~}{w w}}_{s the s}}^{κ κ}))}^{- - 11} {d d}_{s the s}^{κ κ} - - - - - - ((9898))$

以及LS容量倒数估计的方差为：and the variance of the reciprocal estimate of the LS capacity is:

${\overset{^^}{R R}}_{RLS RLS} [[κ κ]] = = {(({(({d d}_{c c}^{κ κ}))}^{T T} {(({Σ Σ}_{{\overset{~ ~}{w w}}_{s the s}}^{κ κ}))}^{- - 11} {d d}_{c c}^{κ κ}))}^{- - 11} - - - - - - ((9999))$

当获得新的一批对时，LS估计可递归更新如下：when a new batch is obtained In time, the LS estimate can be updated recursively as follows:

${\overset{^^}{R R}}_{RLS RLS}^{- - 11} [[κ κ + + 11]] = = λ λ {\overset{^^}{R R}}_{RLS RLS}^{- - 11} [[κ κ]] + + {(({d d}_{c c}^{κ κ + + 11}))}^{T T} {(({Σ Σ}_{{\overset{~ ~}{w w}}_{s the s}}^{κ κ + + 11}))}^{- - 11} {d d}_{c c}^{κ κ + + 11} - - - - - - ((100100))$

${\overset{^^}{C C}}_{RLS RLS}^{- - 11} [[κ κ + + 11]] = = {\overset{^^}{R R}}_{RLS RLS} [[κ κ + + 11]] ((λ λ {\overset{^^}{R R}}_{RLS RLS}^{- - 11} [[κ κ]] {\overset{^^}{C C}}_{RLS RLS}^{- - 11} [[κ κ]] + + {(({d d}_{c c}^{κ κ + + 11}))}^{T T} {(({Σ Σ}_{{\overset{~ ~}{w w}}_{s the s}}^{κ κ + + 11}))}^{- - 11} {d d}_{b b}^{κ κ + + 11})) - - - - - - ((101101))$

其中为用于容量估计的L_c×L_c信息矩阵，以及λ为衰减记忆常数。应该指出的是，（92）中的由已知有噪声的测量电流值构建，而上述LS和RLS估计方法假设是完全已知的。对于更实际的解决方案，应考虑中的不确定性。接下来，描述了基于总体最小二乘（TLS）优化的方法，其解决了中的误差。in is the L _c ×L _c information matrix for capacity estimation, and λ is the decay memory constant. It should be noted that in (92) the constructed from measured current values known to be noisy, whereas the above LS and RLS estimation methods assume is perfectly known. For a more practical solution, one should consider uncertainty in . Next, a method based on total least squares (TLS) optimization is described, which solves the error in .

以下讨论涉及使用自适应总体最小二乘（TLS）的在线电池容量估计。在本部分中，基于TLS建立了在线容量估计方法，其假定在（92）中的和中均存在不确定性。构建以下增广观测矩阵：The following discussion deals with online battery capacity estimation using adaptive total least squares (TLS). In this section, an online capacity estimation method is established based on TLS, which assumes that in (92) and There are uncertainties in both. Build the following augmented observation matrix:

${H h}^{κ κ} = = [\begin{matrix} {d d}_{s the s}^{κ κ} & {d d}_{c c}^{κ κ} \end{matrix}] - - - - - - ((102102))$

与增广观测矩阵相关的信息矩阵为：The information matrix associated with the augmented observation matrix is:

${S S}_{H h}^{k k} = = {(({H h}^{κ κ}))}^{T T} {H h}^{κ κ} - - - - - - ((103103))$

以如下形式书写的特征分解：written in the following form The eigendecomposition of :

${S S}_{H h}^{k k} = = {V V}^{κ κ} {Λ Λ}^{κ κ} {V V}^{κT κ T} - - - - - - ((104104))$

其中，in,

Λ^κ为按从最大到最小排列的非负特征值的对角2×2矩阵，即Λ^κ(1,1)表示最大特征值，而Λ^κ(2,2)表示最小特征值。Λ ^κ is a diagonal 2×2 matrix of non-negative eigenvalues arranged from largest to smallest, that is, Λ ^κ (1,1) represents the largest eigenvalue, and Λ ^κ (2,2) represents the smallest eigenvalue.

2×2矩阵的每列具有对应特征向量，即第一列为对应于最大特征值的特征向量，而第二列为对应于最小特征值的特征向量。2×2 matrix Each column of has a corresponding eigenvector, that is, the first column is the eigenvector corresponding to the largest eigenvalue, and the second column is the eigenvector corresponding to the smallest eigenvalue.

然后电池容量倒数的TLS估计通过分量的比率给出，即Then the TLS estimate of the reciprocal of the battery capacity is passed The ratio of the components is given, that is

${\overset{^^}{C C}}_{TLS TLS}^{- - 11} [[κ κ]] = = - - \frac{{v v}_{22}^{κ κ} ((11))}{{v v}_{22}^{κ κ} ((22))} = = \frac{{S S}_{H h}^{k k} ((1,2 1,2))}{{S S}_{H h}^{k k} ((1,1 1,1)) - - {Λ Λ}^{κ κ} ((2,2 2,2))} - - - - - - ((105105)) [[66]]$

其中为的第i元素，而为的第（i,j）元素。（105）的推导示出如下。in for The ith element of , while for The (i,j)th element of . The derivation of (105) is shown below.

对于平滑估计，（103）中的信息矩阵可采用衰减记忆更新，如下：For smooth estimation, the information matrix in (103) can be updated with attenuation memory as follows:

${S S}_{H h}^{k k} = = λ λ {S S}_{H h}^{k k - - 11} + + \frac{{(({H h}^{κ κ}))}^{T T} {H h}^{κ κ}}{{L L}_{c c} - - 11} - - - - - - ((106106))$

现在，基于[85]，TLS估计误差协方差（近似）为：Now, based on [85], the TLS estimation error covariance (approximately) is:

${\overset{^^}{R R}}_{TLS TLS} [[κ κ]] = = {((\frac{11}{{(({z z}^{κ κ}))}^{T T} {S S}_{H h}^{k k} {z z}^{κ κ}} {Σ Σ}_{i i = = 11}^{M m} {h h}_{i i}^{κ κ} {(({h h}_{i i}^{κ κ}))}^{T T}))}^{- - 11} - - - - - - ((107107))$

其中为H^κ的第i行，M为H^κ中的行数，并且in is the i-th row of H ^κ , M is the number of rows in H ^κ , and

${z z}^{κ κ} = = {[[- - \frac{{v v}_{22}^{κ κ} ((11))}{{v v}_{22}^{κ κ} ((22))} - - 11]]}^{T T} - - - - - - ((108108))$

以下讨论涉及基于开路电压（OCV）的电池容量估计。对于给定的静置电压z_v[k]，对应的SOC估计可通过对（33）取倒数获得。由于电池中的滞后，该SOC估计将不同于实际SOC x_s[k]，得到OCV查找误差OCV查找误差在放电期间将始终为负而在充电期间始终为正。电池的开路电压（OCV）可书写为SOC的非线性函数，如The following discussion deals with battery capacity estimation based on open circuit voltage (OCV). For a given resting voltage z _v [k], the corresponding SOC estimate It can be obtained by taking the inverse of (33). Due to the hysteresis in the battery, the SOC estimate will be different from actual SOC x _s [k], resulting in OCV lookup error The OCV lookup error will always be negative during discharge and positive during charge. The open circuit voltage (OCV) of the battery can be written as a nonlinear function of SOC, such as

$\begin{matrix} \end{matrix} \begin{matrix} {V V}_{00} (({x x}_{s the s} [[k k]])) = = {K K}_{00} + + \frac{{K K}_{11}}{{x x}_{s the s} [[k k]]} + + \frac{{K K}_{22}}{{x x}_{s the s} {[[k k]]}^{22}} + + \frac{{K K}_{33}}{{x x}_{s the s} {[[k k]]}^{33}} + + \frac{{K K}_{44}}{{x x}_{s the s} {[[k k]]}^{44}} + + \\ {K K}_{55} {x x}_{s the s} [[k k]] + + {K K}_{66} ln ln (({x x}_{s the s} [[k k]])) + + {K K}_{77} ln ln ((11 - - {x x}_{s the s} [[k k]])) \end{matrix} - - - - - - ((109109))$

其中可通过采集对电池进行缓慢充电然后放电得到的电压和电流测量值，来对系数K₀,K₁,K₂,K₃,K₄,K₅,K₆和K₇进行离线估计。无论电池是否充分静置，所得的OCV-SOC特性曲线都可用于获取SOC的测量值。对于给定的静置端电压（其也为开路电压）z_v[k]的电池的SOC书写为：The coefficients K ₀ , K ₁ , K ₂ , K ₃ , K ₄ , K ₅ , K ₆ and K ₇ can be estimated offline by collecting voltage and current measurements obtained by slowly charging and then discharging the battery. The resulting OCV-SOC characteristic curve can be used to obtain a measurement of SOC regardless of whether the battery has been sufficiently quiescent. The SOC of a battery for a given resting terminal voltage (which is also the open circuit voltage) z _v [k] is written as:

${\overset{^^}{x x}}_{s the s,, ocv ocv} [[k k]] = = {f f}_{ocv ocv - - soc soc}^{- - 11} (({z z}_{v v} [[k k]])) - - - - - - ((110110))$

可使用OCV-SOC表征通过计算（109）的倒数进行计算。有多个用于计算非线性函数的倒数的方法，例如牛顿法和二分查找。这可称为基于OCV查找的SOC估计。（110）中的SOC估计被滞后电压损坏如下：It can be calculated by calculating the inverse of (109) using the OCV-SOC characterization. There are several methods for computing the inverse of a nonlinear function, such as Newton's method and binary search. This may be referred to as SOC estimation based on OCV lookup. The SOC estimation in (110) is corrupted by the hysteresis voltage as follows:

${x x}_{s the s} [[k k]] = = {\overset{^^}{x x}}_{s the s,, pcv pcv} [[k k]] + + {\overset{~ ~}{x x}}_{s the s,, ocv ocv} [[k k]] - - - - - - ((111111))$

其中OCV查找误差由OCV中的滞后效应引起。应该指出的是，当电池在放电过程后于时间k处变为静置，OCV查找误差应为负。相似地，当电池在充电过程后于时间k处变为静置，OCV查找误差将始终为正。然而，误差的大小将随滞后量大小而改变，其为静置前电流大小、SOC和时间的函数。现在，以如下形式重新书写（79）：where OCV lookup error Caused by hysteresis effects in OCV. It should be noted that when the battery becomes resting at time k after the discharge process, the OCV lookup error should be negative. Similarly, when the battery becomes resting at time k after the charging process, the OCV lookup error will always be positive. However, the magnitude of the error will vary with the magnitude of the hysteresis, which is a function of current magnitude, SOC and time before rest. Now, rewrite (79) as follows:

x_s[k+1]=x_s[k]+c_hΔz_i[k]+w_s[k] （112）x _s [k+1]=x _s [k]+c _h Δz _i [k]+w _s [k] (112)

x_s[k+2]=x_s[k+1]+c_hΔ_k+1z_i[k+1]+w_s[k+1] （113）x _s [k+2]=x _s [k+1]+c _h Δ _k+1 z _i [k+1]+w _s [k+1] (113)

..

x_s[k+N]=x_s[k+N-1]+c_hΔ_k+N-1z_i[k+N-1]+w_s[k+N-1] （114）x _s [k+N]=x _s [k+N-1]+c _h Δ _k+N-1 z _i [k+N-1]+w _s [k+N-1] (114)

通过在两边分别相加（112）至（114），得到如下结果：By adding (112) to (114) on both sides, the following results are obtained:

其中，in,

为具有标准差的零平均值。 to have standard deviation zero mean.

假设电池在时间k和k+N处于静置，则（115）可书写为：Assuming the battery is at rest at times k and k+N, (115) can be written as:

${d d}_{s the s,, ocv ocv}^{k k} = = {C C}_{batt batt}^{- - 11} {d d}_{c c,, ocv ocv}^{k k} + + {\overset{~ ~}{w w}}_{s the s,, ocv ocv}^{k k} - - - - - - ((116116))$

其中，in,

${d d}_{c c,, ocv ocv}^{k k} = = \frac{η η {Σ Σ}_{j j = = k k}^{j j = = K K + + N N - - 11} Δ Δ {z z}_{i i} [[k k]]}{36003600} - - - - - - ((117117))$

${d d}_{s the s,, ocv ocv}^{k k} = = {\overset{^^}{x x}}_{s the s,, ocv ocv} [[k k + + N N]] - - {\overset{^^}{x x}}_{s the s,, ocv ocv} [[k k]] - - - - - - ((118118))$

应该指出的是，不管OCV查找误差的正负号偏向电池模式∈{充电，放电}这一事实，“微分误差”（在（119）中定义）可为正或负。通过考虑大量微分误差，假设近似为白。假设第k批次的微分分别在第一组静置点 $k = k_{1}, k_{2}, . . ., k_{L_{c}}$ 和第二组静置点 $k + N = k_{1} + N_{1}, k_{2} + N_{2}, . . ., k_{L_{c}} + N_{L_{c}},$ 之间采集，并给出如下：It should be noted that regardless of the OCV lookup error The sign of is biased towards the fact that the battery mode ∈ {charge, discharge}, "differential error" (defined in (119)) can be positive or negative. By considering a large number of differential errors, assuming Approximately white. Assuming that the k-th batch of differential at the first set of static points $k = k_{1}, k_{2}, . . ., k_{L_{c}}$ and the second set of resting points $k + N = k_{1} + N_{1}, k_{2} + N_{2}, . . ., k_{L_{c}} + N_{L_{c}},$ collected between and are given as follows:

${d d}_{s the s,, ocv ocv}^{κ κ} = = {C C}_{batt batt}^{- - 11} {d d}_{c c,, ocv ocv}^{κ κ} + + {\overset{~ ~}{w w}}_{s the s,, ocv ocv}^{κ κ} - - - - - - ((120120))$

其中，in,

${d d}_{s the s,, ocv ocv}^{κ κ} = = {[[{d d}_{s the s,, ocv ocv}^{{k k}_{11}},, {d d}_{s the s,, ocv ocv}^{{k k}_{22}},, . . . . . .,, {d d}_{s the s,, ocv ocv}^{{k k}_{{L L}_{c c}}}]]}^{T T} - - - - - - ((121121))$

${d d}_{c c,, ocv ocv}^{κ κ} = = {[[{d d}_{c c,, ocv ocv}^{{k k}_{11}},, {d d}_{c c,, ocv ocv}^{{k k}_{22}},, . . . . . .,, {d d}_{c c,, ocv ocv}^{{k k}_{{L L}_{c c}}}]]}^{T T} - - - - - - ((122122))$

${\overset{~ ~}{w w}}_{s the s,, ocv ocv}^{κ κ} = = {[[{w w}_{s the s,, ocv ocv}^{{k k}_{11}},, {w w}_{s the s,, ocv ocv}^{{k k}_{22}},, . . . . . .,, {w w}_{s the s,, ocv ocv}^{{k k}_{{L L}_{c c}}}]]}^{T T} - - - - - - ((123123))$

现在，可以看出（120）具有与（92）相同的形式，其中和分别替换了和从而，基于RLS和TLS的容量估计可针对基于OCV的观测推导出来，如下所示。基于OCV的容量的RLS和TLS估计分别表示为和应该指出的是，给定穿插有放电的给定N_r数量的静置状态，可进行N_r(N_r-1)/2微分观测。例如，对于N_r=4,，假设电池在时间点t₁,t₂,t₃和t₄.处于静置状态。Now, it can be seen that (120) has the same form as (92), where and respectively replaced and Thus, RLS- and TLS-based capacity estimates can be derived for OCV-based observations, as follows. The RLS and TLS estimates of OCV-based capacity are expressed as and It should be noted that given a given N _r number of resting states interspersed with discharges, N _r (N _r −1)/2 differential observations can be made. For example, for N _r =4, assume that the battery is at rest at time points t ₁ , t ₂ , t ₃ and t ₄ .

以下涉及通过融合进行的容量估计。在本部分中，描述了用于融合容量的TLS估计值的示例性具体实施。在本部分中，针对基于TLS的容量估计值，建立推导。这些推导也可应用于融合基于RLS的容量估计值。The following deals with capacity estimation by fusion. In this section, an exemplary implementation for TLS estimation of fusion capacity is described. In this section, a derivation is established for TLS-based capacity estimates. These derivations can also be applied to fuse RLS-based capacity estimates.

在线容量估计值被测量电流中的误差所造成的（参见（92））的不确定性和SOC跟踪算法中的误差所造成的的不确定性损坏。相似地，基于OCV的容量估计值被测量电流中的误差所造成的（参见（120））的不确定性和OCV查找微分误差所造成的的不确定性损坏。假定在线容量估计值e_t[κ]的误差和基于OCV查找的容量估计值e_to[κ]的误差不相关。基于这些假设，容量融合变成了两个独立航迹的融合。Online Capacity Estimates Errors in the measured current caused by (see (92)) and errors in the SOC tracking algorithm caused by Uncertainty damage. Similarly, OCV-based capacity estimates Errors in the measured current caused by (see (120)) and the OCV lookup differential error caused by Uncertainty damage. It is assumed that the error in the online capacity estimate e _t [κ] and the error in the OCV lookup based capacity estimate e _to [κ] are uncorrelated. Based on these assumptions, volume fusion becomes the fusion of two independent tracks.

首先，应该指出的是，容量倒数的估计值，例如容量估计值和各自为1/C_batt的估计值。相应地，各自的估计误差协方差和也对应于容量倒数估计值。基于泰勒级数展开，基于TLS的动态容量估计值的期望值和对应估计误差方差近似为：First, it should be noted that estimates of the reciprocal of capacity, such as capacity estimates and Each is an estimate of 1/C _batt . Correspondingly, the respective estimation error covariance and Also corresponds to the reciprocal capacity estimate. Based on Taylor series expansion, the expected value and corresponding estimation error variance of TLS-based dynamic capacity estimation are approximated as:

${\overset{^^}{C C}}_{TLS TLS} [[κ κ]] = = \frac{11}{{\overset{^^}{C C}}_{TLS TLS}^{- - 11} [[κ κ]]} + + \frac{{\overset{^^}{R R}}_{TLS TLS} [[κ κ]]}{{(({\overset{^^}{C C}}_{TLS TLS}^{- - 11} [[κ κ]]))}^{33}} - - - - - - ((124124))$

${R R}_{TLS TLS} [[κ κ]] = = \frac{{\overset{^^}{R R}}_{TLS TLS} [[κ κ]]}{{(({\overset{^^}{C C}}_{TLS TLS}^{- - 11} [[κ κ]]))}^{44}} - - - - - - ((125125))$

其中为基于动态数据的C_batt的估计值，并且R_TLS[κ]为估计误差方差。通过遵循相同的程序，可获得基于OCV的容量估计值C_TO[κ]和对应的估计误差协方差R_TO[κ]。现在，假定电池容量为随机变量，其经历如下缓慢变化的维纳过程：in is the estimated value of C _batt based on dynamic data, and R _TLS [κ] is the estimation error variance. By following the same procedure, the OCV-based capacity estimate C _TO [κ] and the corresponding estimation error covariance R _TO [κ] can be obtained. Now, assume that the battery capacity is a random variable that undergoes a slowly varying Wiener process as follows:

x_c[κ+1]=x_c[κ]+w_c[κ] （126）x _c [κ+1]=x _c [κ]+w _c [κ] (126)

其中w_c[κ]假定为具有方差Q_c[κ]的零平均值高斯白噪声。容量估计值和适配下列观测模型：where _wc [κ] is assumed to be zero-mean white Gaussian noise with variance _Qc [κ]. capacity estimate and Fit the following observation models:

z_c[κ]=x_c[κ]+n_c[κ] （127）z _c [κ]=x _c [κ]+n _c [κ] (127)

其中，in,

${z z}_{c c} [[κ κ]] &Element; &Element; {{{\overset{^^}{C C}}_{TLS TLS} [[{κ κ}^{' '}]],, {\overset{^^}{C C}}_{TO TO} [[{κ κ}^{' '' '}]]}} - - - - - - ((128128))$

κ′和κ″为根据对应算法（分别为TLS和TO）的最新估计值的时间索引，并且n_c[κ]假定为具有如下方差的零平均值白噪声：κ′ and κ″ are the time indices of the latest estimates according to the corresponding algorithms (TLS and TO, respectively), and n _c [κ] is assumed to be zero-mean white noise with the following variance:

${R R}_{c c} [[κ κ]] = = \{\begin{matrix} {R R}_{TLS TLS} [[κ κ]] & if if {z z}_{c c} [[κ κ]] = = {\overset{^^}{C C}}_{TLS TLS} [[κ κ]] \\ {R R}_{TO TO} [[κ κ]] & if if {z z}_{c c} [[κ κ]] = = {\overset{^^}{C C}}_{TO TO} [[κ κ]] \end{matrix} - - - - - - ((129129))$

现在，无论何时接收到新测量值其中κ'=κ或κ''=κ，融合的容量估计值以如下方式获得：Now, whenever a new measurement is received where κ' = κ or κ'' = κ, the fused capacity estimates were obtained as follows:

$\begin{matrix} {\overset{^^}{x x}}_{c c} [[κ κ | | κ κ]] = = {\overset{^^}{x x}}_{c c} [[κ κ - - 11 | | κ κ - - 11]] + + \\ \frac{{P P}_{c c} [[k k - - 11 | | k k - - 11]] + + {Q Q}_{c c} [[κ κ - - 11]]}{{P P}_{c c} [[κ κ - - 11 | | κ κ - - 11]] + + {Q Q}_{c c} [[κ κ - - 11]] + + {R R}_{c c} [[κ κ]]} (({z z}_{c c} [[κ κ]] - - {\overset{^^}{x x}}_{c c} ((k k - - 11 | | k k - - 11)))) \end{matrix} - - - - - - ((130130))$

其中为容量估计值的前一个更新，并且P_c[κ-1|κ-1]为前一个估计误差方差，其被更新为：in is the previous update of the capacity estimate, and P _c [κ-1|κ-1] is the previous estimate error variance, which is updated as:

${P P}_{c c} [[κ κ | | κ κ]] = = \frac{{R R}_{c c} [[κ κ]] (({P P}_{c c} [[κ κ - - 11 | | κ κ - - 11 + + {Q Q}_{c c} [[κ κ - - 11]]))}{{P P}_{c c} [[κ κ - - 11 | | κ κ - - 11]] + + {Q Q}_{c c} [[κ κ - - 11]] + + {R R}_{c c} [[κ κ]]} - - - - - - ((131131))$

上述融合方法可相似地用于融合基于RLS的容量估计值。The fusion method described above can be similarly used to fuse RLS-based capacity estimates.

以下涉及容量估计误差协方差的推导。在本部分中，推导出（90）中的微分误差的协方差。为了方便起见，微分误差（90）改写为如下形式The following deals with the derivation of the capacity estimation error covariance. In this section, the covariance of the differential error in (90) is derived. For convenience, the differential error (90) is rewritten as follows

${\overset{~ ~}{w w}}_{s the s} [[k k]] = = {\overset{~ ~}{x x}}_{s the s} [[k k | | k k]] - - {\overset{~ ~}{x x}}_{s the s} [[k k + + 11 | | k k + + 11]] + + {w w}_{s the s} [[k k]] - - - - - - ((9090))$

目的是计算方差：The purpose is to calculate the variance:

$E E. {{{\overset{~ ~}{w w}}_{s the s} [[k k]] {\overset{~ ~}{w w}}_{s the s} {[[k k]]}^{T T}}} &cong; &cong; {R R}_{{\overset{~ ~}{w w}}_{s the s} [[k k]]} - - - - - - ((132132))$

按照下列形式书写过程方程（79）：Write the process equation (79) in the following form:

x_s[k+1]=x_s[k]+c_hΔz_i[k]+w_s[k] （134）x _s [k+1]=x _s [k]+c _h Δz _i [k]+w _s [k] (134)

${\overset{^^}{x x}}_{s the s} [[k k + + 11 | | k k + + 11]] = = {\overset{^^}{x x}}_{s the s} [[k k | | k k]] + + {c c}_{h h} Δ Δ {z z}_{i i} [[k k]] + + G G [[k k + + 11]] v v [[k k + + 11]] - - - - - - ((135135))$

其中为x_s[k+1]的卡尔曼滤波器估计值，ν[k+1]为滤波器新息，并且G[k+1]为卡尔曼增益。（134）与（135）之间的差为：in is the Kalman filter estimate of x _s [k+1], ν[k+1] is the filter innovation, and G[k+1] is the Kalman gain. The difference between (134) and (135) is:

${\overset{~ ~}{x x}}_{s the s} [[k k + + 11]] = = {\overset{~ ~}{x x}}_{s the s} [[k k]] + + {w w}_{s the s} [[k k]] - - G G [[k k + + 11]] v v [[k k + + 11]] - - - - - - ((136136))$

其可重新整理为如下形式：It can be rearranged as follows:

${\overset{~ ~}{x x}}_{s the s} [[k k]] - - {\overset{~ ~}{x x}}_{s the s} [[k k + + 11]] + + {w w}_{s the s} [[k k]] = = {\overset{~ ~}{w w}}_{s the s} [[k k]] = = G G [[k k + + 11]] v v [[k k + + 11]] - - - - - - ((137137))$

从而，thereby,

${R R}_{{\overset{~ ~}{w w}}_{S S} [[k k]]} = = G G {[[k k + + 11]]}^{22} S S [[k k + + 11]] - - - - - - ((138138))$

其中，in,

S[k+1]为新息协方差。S[k+1] is the innovation covariance.

以下描述了确证上述推导的替代或第二方法。展开（132）：An alternative or second method of corroborating the above deduction is described below. expand(132):

其中，in,

E₁=P_s[k|k] （139）E ₁ = P _s [k|k] (139)

E₂=P_s[k+1|k+1] （140）E ₂ =P _s [k+1|k+1] (140)

${E E.}_{33} = = {σ σ}_{s the s}^{22} - - - - - - ((141141))$

E₅=0 （143）E ₅ =0 (143)

$\begin{matrix} {E E.}_{66} = = E E. {{{\overset{~ ~}{x x}}_{s the s} [[k k + + 11]] {w w}_{s the s} [[k k]]}} \\ = = E E. {{{\overset{~ ~}{x x}}_{s the s} [[k k | | k k]] {w w}_{s the s} [[k k]]}} + + E E. \\ = = {σ σ}_{s the s}^{22} \end{matrix}, {{{w w}_{s the s} [[k k]] {w w}_{s the s} [[k k]]}} - - G G [[k k + + 11]] H h [[k k + + 11]] E E. {{{\overset{~ ~}{x x}}_{s the s} [[k k | | k k]] {w w}_{s the s} [[k k]]}} - - - - - - ((144144))$

$\begin{matrix} E E. {{{\overset{~ ~}{w w}}_{s the s} [[k k]] {\overset{~ ~}{w w}}_{s the s} [[k k]]}} = = E E. 11 + + E E. 22 + + E E. 33 - - 22 E E. 44 + + 22 E E. 55 - - 22 E E. 66 \\ = = {P P}_{s the s} [[k k | | k k]] + + {P P}_{s the s} [[[[k k + + 11 | | k k + + 11]] + + {σ σ}_{s the s}^{22} - - 22 ((11 - - G G [[k k + + 11]] H h [[k k + + 11]])) {P P}_{s the s} [[k k | | k k]] - - 22 {σ σ}_{s the s}^{22} \\ = = {P P}_{s the s} [[k k | | k k]] + + {((11 - - G G [[k k + + 11]] G G [[k k + + 11]] H h [[k k + + 11]]))}^{22} {P P}_{s the s} [[k k | | k k]] + + {σ σ}_{s the s}^{22} + + G G [[k k + + 11]] {R R}_{D D.} [[00]] G G [[k k + + 11]] \\ + + {σ σ}_{s the s}^{22} - - 22 (([[11 - - G G [[k k + + 11]] H h [[k k + + 11]])) {P P}_{s the s}]] k k | | k k]] - - 22 {σ σ}_{s the s}^{22} \\ = = G G {[[k k + + 11]]}^{22} H h {[[k k + + 11]]}^{22} {P P}_{s the s} [[k k | | k k]] + + G G [[k k + + 11]] {R R}_{D D.} ((00)) G G [[k k + + 11]] \\ G G {[[k k + + 11]]}^{22} ((H h {[[k k + + 11]]}^{22} {P P}_{s the s} [[k k | | k k]] + + {R R}_{D D.} ((00)))) \\ = = G G {[[k k + + 11]]}^{22} S S [[k k + + 11]] \end{matrix} - - - - - - ((145145))$

以下涉及总体最小二乘（TLS）容量估计值的闭式推导。将2×2矩阵 $A \overset{Δ}{=} S_{H}^{k}$ 书写成：The following involves a closed-form derivation of total least squares (TLS) capacity estimates. The 2×2 matrix $A \overset{Δ}{=} S_{h}^{k}$ written as:

$A A = = [\begin{matrix} {σ σ}_{1111} & {σ σ}_{1212} \\ {σ σ}_{1212} & {σ σ}_{22 twenty two} \end{matrix}] - - - - - - ((146146))$

A的特征值满足The eigenvalues of A satisfy

|A-λI|=0 （147）|A-λI|=0 (147)

$| | \begin{matrix} {σ σ}_{1111} - - λ λ & {σ σ}_{1212} \\ {σ σ}_{1212} & {σ σ}_{22 twenty two} - - λ λ \end{matrix} | | = = 00 - - - - - - ((148148))$

其简化为which simplifies to

${λ λ}_{11} = = \frac{{σ σ}_{1111} + + {σ σ}_{22 twenty two} + + \sqrt{{(({σ σ}_{1111} - - {σ σ}_{22 twenty two}))}^{22} + + 44 {(({σ σ}_{1212}))}^{22}}}{22} - - - - - - ((149149))$

${λ λ}_{22} = = \frac{{σ σ}_{1111} + + {σ σ}_{22 twenty two} - - \sqrt{{(({σ σ}_{1111} - - {σ σ}_{22 twenty two}))}^{22} + + 44 {(({σ σ}_{1212}))}^{22}}}{22} - - - - - - ((150150))$

其中λ₁为最大特征值并且λ₂为最小特征值。对应于λ₂的特征值满足where λ ₁ is the largest eigenvalue and λ ₂ is the smallest eigenvalue. The eigenvalue corresponding to λ ₂ satisfies

$A A {v v}_{22}^{κ κ} = = {λ λ}_{22} {v v}_{22}^{κ κ} - - - - - - ((151151))$

其中in

${v v}_{22}^{κ κ} = = [\begin{matrix} \frac{- - {σ σ}_{1212}}{\sqrt{{σ σ}_{1212}^{22} + + {(({σ σ}_{1111} - - {λ λ}_{22}))}^{22}}} \\ \frac{{σ σ}_{1111} - - {λ λ}_{22}}{\sqrt{{σ σ}_{1212}^{22} + + {(({σ σ}_{1111} - - {λ λ}_{22}))}^{22}}} \end{matrix}] - - - - - - ((152152))$

例如，For example,

${v v}_{22}^{κ κ} ((11)) = = \frac{- - {σ σ}_{1212}}{\sqrt{{σ σ}_{1212}^{22} + + {(({σ σ}_{1111} - - {λ λ}_{22}))}^{22}}} - - - - - - ((153153))$

${v v}_{22}^{κ κ} ((22)) = = \frac{{σ σ}_{1111} - - {λ λ}_{22}}{\sqrt{{σ σ}_{1212}^{22} + + {(({σ σ}_{1111} - - {λ λ}_{22}))}^{22}}} - - - - - - ((154154))$

以及as well as

${\overset{^^}{C C}}_{TLS TLS}^{- - 11} [[κ κ]] = = - - \frac{{v v}_{22}^{κ κ} ((11))}{{v v}_{22}^{κ κ} ((22))}$

$= = \frac{{S S}_{H h}^{κ κ} ((1,2 1,2))}{{S S}_{H h}^{κ κ} ((1,1 1,1)) - - {Λ Λ}^{κ κ} ((2,2 2,2))} - - - - - - ((155155)) [[66]]$

以下涉及容量倒数估计的变换。示例性具体实施包括基于倒数估计值和倒数估计误差方差得出容量估计值和估计误差方差的方法。为倒数容量估计值和误差方差赋值简单变量，例如，The following deals with the transformation of the reciprocal estimate of capacity. Exemplary implementations include methods of deriving capacity estimates and estimation error variances based on the reciprocal estimates and reciprocal estimation error variances. Assign simple variables to the reciprocal capacity estimate and error variance, for example,

$x x &cong; &cong; {C C}_{TLS TLS}^{- - 11} [[κ κ]] - - - - - - ((156156))$

${x x}_{00} &cong; &cong; E E. {{x x}} = = {C C}_{TLS TLS}^{- - 11} [[κ κ]] - - - - - - ((157157))$

${R R}_{x x} &cong; &cong; E E. {{{((x x - - {x x}_{00}))}^{22}}} = = {\overset{^^}{R R}}_{TLS TLS} [[κ κ]] - - - - - - ((158158))$

定义：definition:

$y the y &cong; &cong; f f ((x x)) = = \frac{11}{x x} - - - - - - ((159159))$

我们的目的是找到E{y}和E{(y-E{y})²}的近似值。Our aim is to find approximate values of E{y} and E{(yE{y}) ² }.

以下涉及确定y的期望值。二阶泰勒级数近似值由以下给定：The following involves determining the expected value of y. The second-order Taylor series approximation is given by:

$y the y = = f f ((x x)) = = f f (({x x}_{00})) + + {f f}^{' '} (({x x}_{00})) ((x x - - {x x}_{00})) + + \frac{11}{22} {f f}^{' '' '} (({x x}_{00})) {((x x - - {x x}_{00}))}^{22} - - - - - - ((160160))$

E{y}的二阶近似值由以下给定：A second-order approximation to E{y} is given by:

以下涉及确定y的期望值的方差。在真实值x₀.周围将f(x)展开为一阶泰勒级数The following involves determining the variance of the expected value of y. Expand f(x) as a first-order Taylor series around the true value x ₀ .

y=f(x)=f(x₀)+f'(x₀)(x-x₀) （162）y=f(x)=f(x ₀ )+f'(x ₀ )(xx ₀ ) (162)

y的方差由以下给定：The variance of y is given by:

$\begin{matrix} E E. {{{((y the y - - E E. {{y the y}}))}^{22}}} = = E E. {{{(({f f}^{' '} (({x x}_{00})) ((x x - - {x x}_{00}))))}^{22}}} \\ = = \frac{{R R}_{x x}}{{x x}_{00}^{44}} \end{matrix} - - - - - - ((163163))$

现在，容量估计值的期望值及其估计误差方差由以下给定：Now, the expected value of the capacity estimate and its estimation error variance are given by:

${C C}_{TLS TLS} [[κ κ]] = = \frac{11}{{\overset{^^}{C C}}_{TLS TLS}^{- - 11} [[κ κ]]} + + \frac{{\overset{^^}{R R}}_{TLS TLS} [[κ κ]]}{{(({\overset{^^}{C C}}_{TLS TLS}^{- - 11} [[κ κ]]))}^{33}} - - - - - - ((164164))$

${R R}_{TLS TLS} [[κ κ]] = = \frac{{\overset{^^}{R R}}_{TLS TLS} [[κ κ]]}{{(({\overset{^^}{C C}}_{TLS TLS}^{- - 11} [[κ κ]]))}^{44}} - - - - - - ((165165))$

本公开内容接下来是电荷状态（SOC）跟踪，其可参考下列符号。This disclosure is followed by state of charge (SOC) tracking, which may be referenced by the following symbols.

在该示例性具体实施中，基于瞬时端电压、负载电流和温度测量值来跟踪电化学储能装置（电池）的电荷状态（SOC）。SOC跟踪算法使用上述模型参数估计和电池容量估计的了解。示例性SOC跟踪将滞后建模为开路电压（OCV）中的误差并采用参数估计和SOC跟踪技术的组合对其进行补偿。这消除了将滞后离线建模为SOC和负载电流的函数的需要。该示例性模型导致用于SOC跟踪的降阶（如，单个状态）滤波，其中不论电池等效模型的复杂性水平如何，均无需跟踪附加变量。识别相关性噪声的存在并将其用于改善SOC跟踪。与常规“一个模型适配所有”策略不同的是，识别了电池的四个不同等效模型，这些模型代表典型电池运行的四个独特模式并基于适当的模型建立了用于无缝SOC跟踪的框架。In this exemplary implementation, the state of charge (SOC) of an electrochemical energy storage device (battery) is tracked based on instantaneous terminal voltage, load current, and temperature measurements. The SOC tracking algorithm uses the above knowledge of model parameter estimates and battery capacity estimates. Exemplary SOC tracking models hysteresis as an error in open circuit voltage (OCV) and compensates for it using a combination of parameter estimation and SOC tracking techniques. This eliminates the need to model hysteresis offline as a function of SOC and load current. This exemplary model results in reduced order (eg, single state) filtering for SOC tracking, where no additional variables need to be tracked, regardless of the complexity level of the battery equivalent model. The presence of correlated noise is identified and used to improve SOC tracking. In contrast to the conventional "one model fits all" strategy, four different equivalent models of the battery are identified that represent four unique modes of typical battery operation and based on the appropriate model a method for seamless SOC tracking is established. frame.

包括SOC与其他冗余（多余）量的联合（递归）估计的典型降阶状态滤波方法涉及在计算上昂贵的矩阵运算并且降低了SOC估计的精度。在该示例性具体实施中，使用了不增加状态空间维数的降阶滤波，从而得到更好的SOC精度和降低的计算复杂性。通过将滞后建模为OCV中的误差，消除了滞后建模的需要，并且在线滤波方法持续尝试通过调整SOC（至修正值）来填补差距。从而，滞后被建模为时变偏置。应用噪声白化程序，并推导出修改后的状态空间模型，以确保SOC跟踪算法在最小均方误差意义上得出尽可能好的结果。使用电池的不同“模式”跟踪SOC。至少四个不同电池等效模型用于反映非常轻的负载或静置状态、恒定电流运行或低频率加载（如，充电）、动态负载和重负载。还识别四个（稍微）不同的动态等效模型以最佳匹配这些模式。所提供的降阶滤波方法确保了无缝SOC跟踪，而不论电池运行的模式变化。Typical reduced-order state filtering methods involving joint (recursive) estimation of SOC and other redundant (redundant) quantities involve computationally expensive matrix operations and reduce the accuracy of the SOC estimation. In this exemplary implementation, reduced-order filtering is used that does not increase the dimensionality of the state space, resulting in better SOC accuracy and reduced computational complexity. By modeling hysteresis as an error in the OCV, the need for hysteresis modeling is eliminated, and online filtering methods continually attempt to fill the gap by adjusting the SOC (to a corrected value). Thus, hysteresis is modeled as a time-varying bias. A noise whitening procedure is applied and a modified state-space model is derived to ensure that the SOC tracking algorithm yields the best possible results in the sense of minimum mean square error. Track SOC using different "modes" of the battery. At least four different battery equivalent models are used to reflect very light loads or resting conditions, constant current operation or low frequency loading (eg charging), dynamic loads and heavy loads. Four (slightly) different dynamic equivalent models were also identified to best match these patterns. The provided reduced-order filtering method ensures seamless SOC tracking regardless of battery operating mode changes.

以下讨论涉及示例性系统模型。本文所考虑的电池等效电路模型在图13A中示出。当电池处于静置时，V₀(s[k])是电池的OCV。OCV唯一取决于电池的SOC，s[k]。当电池处于活跃状态时，例如当存在电流活动时，电池的行为通过动态等效电路表示，所述动态等效电路由滞后元件h[k]、串联电阻R₀以及串联连接的两个并联RC电路、(R₁,C₁)和(R₂,C₂)组成。离散时间使用[k]指示。The following discussion refers to exemplary system models. The battery equivalent circuit model considered here is shown in Figure 13A. V ₀ (s[k]) is the OCV of the battery when the battery is at rest. OCV only depends on the SOC of the battery, s[k]. When the battery is active, i.e. when there is current flow, the behavior of the battery is represented by a dynamic equivalent circuit consisting of a hysteresis element h[k], a series resistance _R0 , and two parallel RCs connected in series circuit, (R ₁ , C ₁ ) and (R ₂ , C ₂ ). Discrete times are indicated using [k].

本部分所考虑的电池等效电路模型在图13A中示出。就电池等效电路中的元件而言的端电压v[k]由以下给定：The battery equivalent circuit model considered in this section is shown in Figure 13A. The terminal voltage v[k] in terms of elements in the battery equivalent circuit is given by:

v[k]=V₀(s[k])+i[k]R₀+i₁[k]R₁+i₂[k]R₂+h[k] （166）v[k]=V ₀ (s[k])+i[k]R ₀ +i ₁ [k]R ₁ +i ₂ [k]R ₂ +h[k] (166)

其中V₀(s[k])表示时间k处电池的开路电压（以伏特计），其在此处写成时间k处的SOC的函数，s[k]∈[0,1]；h[k]说明电池电压中的滞后；i₁[k]和i₂[k]分别为流过R₁和R₂的电流。where V ₀ (s[k]) represents the open circuit voltage (in volts) of the battery at time k, which is written here as a function of the SOC at time k, s[k]∈[0,1]; h[k] Account for the hysteresis in the battery voltage; i ₁ [k] and i ₂ [k] are the currents flowing through R ₁ and R ₂ , respectively.

存在若干非线性表示形式，其使OCV近似于SOC的函数。在该示例性具体实施中，用于以SOC表示OCV的倒数多项式对数线性模型：There are several non-linear representations that approximate OCV as a function of SOC. In this exemplary implementation, the reciprocal polynomial log-linear model used to express OCV in terms of SOC:

$\begin{matrix} \end{matrix} \begin{matrix} {V V}_{00} ((s the s [[k k]])) = = {K K}_{00} + + \frac{{K K}_{11}}{s the s [[k k]]} + + \frac{{K K}_{22}}{{s the s}^{22} [[k k]]} + + \frac{{K K}_{33}}{{s the s}^{33} [[k k]]} + + \frac{{K K}_{44}}{{s the s}^{44} [[k k]]} + + \\ {K K}_{55} s the s [[k k]] + + {K K}_{66} ln ln ((s the s [[k k]])) + + {K K}_{77} ln ln ((11 - - s the s [[k k]])) \end{matrix} - - - - - - ((167167)) [[33]]$

其中K₀,K₁,K₂,K₃,K₄,K₅,K₆和K₇可通过OCV-SOC表征离线估计。SOC的瞬时变化可书写成下列形式（引入x的下标以指示状态分量）：Among them, K ₀ , K ₁ , K ₂ , K ₃ , K ₄ , K ₅ , K ₆ and K ₇ can be estimated offline by OCV-SOC characterization. The instantaneous change of SOC can be written in the following form (the subscript of x is introduced to indicate the state component):

${x x}_{s the s} [[k k + + 11]] &cong; &cong; s the s [[k k + + 11]] = = s the s [[k k]] + + {c c}_{h h} Δi Δi [[k k]] - - - - - - ((168168))$

其中i[k]以安培数计；where i[k] is in amperage;

c_h=η/3600C_batt （169）c _h =η/3600C _batt (169)

为以安培^-1秒^-1计的库仑计数系数，C_batt为以安培小时（Ah）计的电池容量，Δ为以秒计的取样间隔，并且η为常数，其取决于电池是充电还是放电，例如，is the coulomb counting coefficient in ampere ^-1 sec ^-1 , C _batt is the battery capacity in ampere-hours (Ah), Δ is the sampling interval in seconds, and η is a constant that depends on whether the battery is charging or discharging ,For example,

$η η = = \{\begin{matrix} {η η}_{c c} & i i [[k k]] > > 00 \\ {η η}_{d d} & i i [[k k]] < < 00 \end{matrix} - - - - - - ((170170))$

应该指出的是，（168）得出电池的瞬时SOC。计算SOC的该技术被称为库仑计数和/或“预测的SOC”。库仑计数假定了解初始电荷状态并完全了解电池容量以便在考虑转移自/转移进电池的库仑量之后计算剩余电荷状态。库仑计数误差包括（1）对初始SOC的了解的不确定性；（2）对电池容量的了解的不确定性；以及（3）因为测量电流的误差和因定时振荡器不准确/漂移所致的时差的误差造成的测量库仑的误差。It should be noted that (168) derives the instantaneous SOC of the battery. This technique of calculating SOC is known as coulomb counting and/or "predicted SOC". Coulomb counting assumes knowledge of the initial state of charge and full knowledge of the battery capacity to calculate the remaining state of charge after accounting for the amount of coulombs transferred from/into the battery. Coulomb counting errors include (1) uncertainty in knowledge of initial SOC; (2) uncertainty in knowledge of battery capacity; and (3) due to errors in measured current and due to timing oscillator inaccuracies/drifts The error of the time difference causes the error of measuring Coulomb.

电流i[k]被测量并且电流测量值容易出现误差。将测量电流z_i[k]写成：The current i[k] is measured and the current measurement is prone to errors. Write the measured current z _i [k] as:

z_i[k]=i[k]+n_i[k] （171）z _i [k]=i[k]+n _i [k] (171)

其中n_i[k]为电流测量噪声，其被认为是具有白的零平均值且具有已知标准差（s.d.）σ_i.。可通过按照如下方式用z_i[k]替换i[k]来改写状态方程（168）：where n _i [k] is the current measurement noise, which is considered to have zero mean value with a known standard deviation (sd) σ _i . The equation of state (168) can be rewritten by substituting z _i [k] for i[k] as follows:

x_s[k+1]=x_s[k]+c_hΔz_i[k]-c_hΔn_i[k] （172）x _s [k+1]=x _s [k]+c _h Δz _i [k]-c _h Δn _i [k] (172)

可按照如下方式书写流过电阻器R₁和R₂的电流：The current flowing through resistors _R1 and _R2 can be written as follows:

${x x}_{{i i}_{11}} [[k k + + 11]] &cong; &cong; {i i}_{11} [[k k + + 11]] = = {α α}_{11} {i i}_{11} [[k k]] + + ((11 - - {α α}_{11})) i i [[k k]] - - - - - - ((172172))$

${x x}_{{i i}_{22}} [[k k + + 11]] &cong; &cong; {i i}_{22} [[k k + + 11]] = = {α α}_{22} {i i}_{22} [[k k]] + + ((11 - - {α α}_{22})) i i [[k k]] - - - - - - ((173173))$

其中in

${α α}_{11} = = {e e}^{- - \frac{Δ Δ}{{R R}_{11} {C C}_{11}}} - - - - - - ((174174))$

${α α}_{22} = = {e e}^{- - \frac{Δ Δ}{{R R}_{22} {C C}_{22}}} - - - - - - ((175175))$

通过用测量电流z_i[k]替换i[k]，可按照如下方式改写（172）和（173）中的电流：By replacing i[k] with the measured current z _i [k], the currents in (172) and (173) can be rewritten as follows:

${x x}_{{i i}_{11}} [[k k + + 11]] = = {α α}_{11} {x x}_{{i i}_{11}} [[k k]] + + ((11 - - {α α}_{11})) {z z}_{i i} [[k k]] - - ((11 - - {α α}_{11})) {n no}_{i i} [[k k]] - - - - - - ((176176))$

${x x}_{{i i}_{22}} [[k k + + 11]] = = {α α}_{22} {x x}_{{i i}_{22}} [[k k]] + + ((11 - - {α α}_{22})) {z z}_{i i} [[k k]] - - ((11 - - {α α}_{22})) {n no}_{i i} [[k k]] - - - - - - ((177177))$

滞后电压h[k]是电池的负载电流和SOC的非线性函数。滞后过程可写成：The hysteresis voltage h[k] is a nonlinear function of the battery's load current and SOC. The hysteresis process can be written as:

${x x}_{h h} [[k k]] &cong; &cong; h h [[k k]] = = {f f}_{h h} (({x x}_{s the s} [[k k]],, i i [[k k]])) {x x}_{h h} [[k k]] + + {n no}_{h h} [[k k]] - - - - - - ((178178))$

其中n_h[k]为滞后模型的过程噪声，假定其为零平均值高斯白噪声并具有s.d. σ_h。（166）中的电压为测量量并且测量电压z_v[k]容易出现误差。测量电压写成：where n _h [k] is the process noise of the hysteresis model, which is assumed to be zero-mean white Gaussian noise with sd σ _h . The voltage in (166) is a measured quantity and measuring the voltage z _v [k] is prone to errors. The measured voltage is written as:

z_v[k]=v[k]+n_v[k]=V₀(s[k])+i[k]R₀+i₁[k]R₁+i₂[k]R₂+h[k]+n_v[k] （179）z _v [k]=v[k]+n _v [k]=V ₀ (s[k])+i[k]R ₀ +i ₁ [k]R ₁ +i ₂ [k]R ₂ +h [k]+n _v [k] (179)

其中n_v[k]假定为具有零平均值和s.d. σ_v的高斯白噪声。现在，通过将（171）、（172）、（173）和（178）代入（179）中，推导出下列测量模型：where _nv [k] is assumed to be Gaussian white noise with zero mean and sd _σv . Now, by substituting (171), (172), (173) and (178) into (179), the following measurement model is derived:

${z z}_{v v} [[k k]] = = {V V}_{00} (({x x}_{s the s} [[k k]])) + + {z z}_{i i} [[k k]] {R R}_{00} + + {x x}_{{i i}_{11}} [[k k]] {R R}_{11} + + {x x}_{{i i}_{22}} [[k k]] {R R}_{22} + + {x x}_{h h} [[k k]] + + {n no}_{{z z}_{v v}} [[k k]] - - - - - - ((180180))$

其中，in,

${n no}_{{z z}_{v v}} [[k k]] = = {n no}_{v v} [[k k]] - - {R R}_{00} {n no}_{i i} [[k k]] - - - - - - ((181181))$

现在，给出瞬时电压和电流测量值z_v[k]和z_i[k]，BFG的目的是跟踪电池的瞬时SOC x_s[k].。观测模型（180）中“多余”变量和的存在造成联合估计问题，即SOC和这些变量必须联合估计。这可通过形成如（183）–（189）中所示的向量形式的多维过程和测量模型和/或通过应用贝叶斯非线性滤波技术以递归地估计来实现：Now, given the instantaneous voltage and current measurements z _v [k] and _zi [k], the purpose of the BFG is to track the instantaneous SOC x _s [k] of the battery. "Excess" variables in observational models (180) and The existence of causes a joint estimation problem, that is, SOC and these variables must be jointly estimated. This can be achieved by forming multidimensional process and measurement models in vector form as shown in (183)–(189) and/or by applying Bayesian nonlinear filtering techniques to recursively estimate:

$[[k k]] &cong; &cong; [\begin{matrix} {x x}_{s the s} [[k k]] \\ {x x}_{h h} [[k k]] \\ {x x}_{{i i}_{11}} [[k k]] \\ {x x}_{{i i}_{22}} [[k k]] \end{matrix}] - - - - - - ((182182))$

给出直到时间k的所有测量值，{[0],[1],[2],…,[k]}，其中由（171）和（180）组成。这可通过应用熟知的非线性滤波技术（例如扩展卡尔曼滤波器（EKF）、无迹卡尔曼滤波器（UKF）或粒子滤波器）来有效进行。可以向量形式将过程方程（172）、（176）、（177）和（178）书写成：Given all measurements up to time k, {[0],[1],[2],...,[k]}, where consists of (171) and (180). This can be efficiently done by applying well known nonlinear filtering techniques such as Extended Kalman Filter (EKF), Unscented Kalman Filter (UKF) or Particle Filter. The process equations (172), (176), (177) and (178) can be written in vector form as:

$\begin{matrix} [\begin{matrix} {x x}_{s the s} [[k k + + 11]] \\ {x x}_{h h} [[k k + + 11]] \\ {x x}_{{i i}_{11}} [[k k + + 11]] \\ {x x}_{{i i}_{22}} [[k k + + 11]] \end{matrix}] = = [\begin{matrix} 11 & 00 & 00 & 00 \\ 00 & {f f}_{h h} ((s the s [[k k]],, i i [[k k]])) & 00 & 00 \\ 00 & 00 & {α α}_{11} & 00 \\ 00 & 00 & 00 & {α α}_{22} \end{matrix}] [\begin{matrix} {x x}_{s the s} [[k k]] \\ {x x}_{h h} [[k k]] \\ {x x}_{{i i}_{11}} [[k k]] \\ {x x}_{{i i}_{22}} [[k k]] \end{matrix}] + + [\begin{matrix} {c c}_{h h} Δ Δ \\ 00 \\ 11 - - {α α}_{11} \\ 11 - - {α α}_{22} \end{matrix}] {z z}_{i i} [[k k]] \\ + + [\begin{matrix} - - {c c}_{h h} Δ Δ & 00 \\ 00 & 11 \\ {α α}_{11} - - 11 & 00 \\ {α α}_{22} - - 11 & 00 \end{matrix}] [\begin{matrix} {n no}_{i i} [[k k]] \\ {n no}_{h h} [[k k]] \end{matrix}] \end{matrix} - - - - - - ((183183))$

或，简写为：or, in short:

x[k+1]=F_kx[k]+u[k]+Γ_kw[k] （184）x[k+1]=F _k x[k]+u[k]+Γ _k w[k] (184)

其中，in,

$w w [[k k]] = = [\begin{matrix} {n no}_{i i} [[k k]] \\ {n no}_{h h} [[k k]] \end{matrix}] - - - - - - ((185185))$

为具有零平均值和如下协方差的白噪声向量：is a white noise vector with zero mean and covariance as follows:

${Σ Σ}_{w w} = = [\begin{matrix} {σ σ}_{i i}^{22} & 00 \\ 00 & {σ σ}_{h h}^{22} \end{matrix}] - - - - - - ((186186))$

相应地，测量方程（180）可书写成：Correspondingly, the measurement equation (180) can be written as:

${z z}_{v v} [[k k]] = = g g (([[k k]])) + + {z z}_{i i} [[k k]] {R R}_{00} + + {n no}_{{z z}_{v v}} [[k k]] - - - - - - ((187187))$

其中，in,

$g g (([[k k]])) = = {V V}_{00} (({x x}_{s the s} [[k k]])) + + {x x}_{h h} [[k k]] + + {x x}_{{i i}_{11}} [[k k]] {R R}_{11} + + {x x}_{{i i}_{22}} [[k k]] {R R}_{22} - - - - - - ((188188))$

以及具有零平均值和如下s.d.的n_z[k]噪声向量。and _nz [k] noise vectors with zero mean and sd as follows.

${σ σ}_{z z} = = \sqrt{{σ σ}_{v v}^{22} + + {R R}_{00}^{22} {σ σ}_{i i}^{22}} - - - - - - ((189189))$

此外，应该指出的是，状态空间表示形式（183）–（189）涉及需要通过系统识别技术来估计的下列模型参数，包括电池容量：C_batt，开路电压模型参数：K₀,K₁,K₂,K₃,K₄,K₅,K₆,K₇，动态等效电路模型参数：R₀,R₁,C₁,R₂,C₂，充电和放电效率：η_c,η_d，过程噪声方差：以及测量噪声方差 Furthermore, it should be noted that the state space representations (183)–(189) involve the following model parameters that need to be estimated by system identification techniques, including battery capacity: C _batt , open circuit voltage model parameters: K ₀ ,K ₁ ,K ₂ , K ₃ , K ₄ , K ₅ , K ₆ , K ₇ , dynamic equivalent circuit model parameters: R ₀ , R ₁ , C ₁ , R ₂ , C ₂ , charging and discharging efficiency: η _c , η _d , Process noise variance: and the measurement noise variance

对所有模型参数估计的要求使SOC跟踪问题更具挑战性。另外，电池的化学性质会因温度变化、老化和使用模式而变化，因此这些模型参数会随时间而变化。从而，随时间推移，必须重新估计模型参数。The requirement for all model parameter estimates makes the SOC tracking problem more challenging. Additionally, battery chemistry changes due to temperature changes, aging, and usage patterns, so these model parameters change over time. Thus, over time, model parameters must be re-estimated.

在示例性具体实施中，假定电池的OCV参数K₀,K₁,…,K₇是离线估计的。上文描述了估计这些参数的程序。假定电压和电流误差标准差，即分别为σ_v和σ_i，可从测量电路设计得到。假定充电和放电效率，即分别为η_c和η_d，通过校准而知。从而，目的是通过假定了解电池容量C_batt和电池的电等效模型参数R₀,R₁,R₂,C₁和C₂来建立在线SOC跟踪算法。In an exemplary implementation, it is assumed that the battery's OCV parameters K ₀ , K ₁ , ..., K ₇ are estimated offline. The procedure for estimating these parameters is described above. It is assumed that the voltage and current error standard deviations, namely _σv and _σi , respectively, can be obtained from the measurement circuit design. It is assumed that the charge and discharge efficiencies, ie, η _c and η _d , respectively, are known by calibration. Thus, the aim is to build an online SOC tracking algorithm by assuming knowledge of the battery capacity C _batt and the battery's electrical equivalent model parameters R ₀ , R ₁ , R ₂ , C ₁ and C ₂ .

下列讨论涉及SOC跟踪。降阶滤波的目的是跟踪x_s[k]，同时不必跟踪冗余变量和x_h[k]。首先，按照下列形式重写（172）：The following discussion relates to SOC tracking. The purpose of reduced-order filtering is to track x _s [k] without having to track redundant variables and _xh [k]. First, rewrite (172) in the following form:

x_s[k+1]=x_s[k]+c_hΔz_i[k]+w_s[k] （190）x _s [k+1]=x _s [k]+c _h Δz _i [k]+w _s [k] (190)

其中，in,

w_s[k]=-c_hΔn_i[k] （191）w _s [k]=-c _h Δn _i [k] (191)

为过程噪声，其为具有如下s.d.的白噪声：is the process noise, which is white noise with the following s.d.:

σ_s=c_hΔσ_i （192）σ _s = c _h Δσ _i (192)

现在，将电压测量值（180）改写为：Now, rewrite the voltage measurement (180) as:

z_v[k]=V₀(x_s[k])+a[k]^Tb+n_D[k] （193）[4]z _v [k]=V ₀ (x _s [k])+a[k] ^T b+n _D [k] (193)[4]

其中，in,

a[k]^Tb=[v_D[k-1]v_D[k-2]z_i[k]z_i[k-1]-z_i[k-2]1] （194）a[k] ^T b=[v _D [k-1]v _D [k-2] z _i [k] z _i [k-1]-z _i [k-2]1] (194)

并且电压降由如下给出：and the voltage drop is given by:

v_D[k]=z_v[k]-V₀(x_s[k]) （195）v _D [k]=z _v [k]-V ₀ (x _s [k]) (195)

其中b为待估计的参数向量，并且n_D[k]为测量噪声。应该指出的是，（194）中的[k]以电压降v_D[k-1]和v_D[k-2].定义。上文描述了基于电压降观测模型的估计参数b。将估计的参数按照图13A中的电池等效模型的参数推导为：where b is the parameter vector to be estimated and n _D [k] is the measurement noise. It should be noted that [k] in (194) is defined in terms of voltage drops v _D [k-1] and v _D [k-2]. The estimated parameter b based on the voltage drop observation model is described above. The estimated parameters are derived according to the parameters of the battery equivalent model in Figure 13A as:

$b b ((11)) &cong; &cong; α α ((k k)) = = {α α}_{11} + + {α α}_{22} - - - - - - ((196196))$

$b b ((22)) &cong; &cong; β β ((k k)) = = {α α}_{11} {α α}_{22} - - - - - - ((197197))$

b(3)=R₀ （198）b(3)=R ₀ (198)

$b b ((66)) &cong; &cong; \overset{^^}{h h} [[k k]] = = {x x}_{h h} [[k k]] - - α α ((k k)) {x x}_{h h} [[k k - - 11]] + + β β ((k k)) {x x}_{h h} [[k k - - 22]] - - - - - - ((201201))$

测量噪声n_D[k]为零平均值并具有由如下给定的自相关性 The measurement noise n _D [k] is zero-mean and has an autocorrelation given by

接下来，描述了电压降在估计（193）中的参数b的意义。使用（193），电压降（195）可写成：Next, the significance of the parameter b in the estimation (193) of the voltage drop is described. Using (193), the voltage drop (195) can be written as:

v_D[k]=a[k]^Tb+n_D[k] （203）v _D [k]=a[k] ^T b+n _D [k] (203)

给定电压降观测，上述模型（203）可用于线性估计b。然而，为了获得作为观测的电压降，可使用SOC的预测值为或更新后的SOC估计值为的SOCx_s[k]的了解，例如，Given voltage drop observations, the above model (203) can be used to linearly estimate b. However, to obtain the voltage drop as observed, the predicted value of SOC can be used as or the updated SOC estimate is SOCx _s [k] is known, for example,

${v v}_{D D.} [[k k | | k k]] = = {z z}_{v v} [[k k]] - - {V V}_{00} (({\overset{^^}{x x}}_{s the s} [[k k | | k k]])) - - - - - - ((204204))$

下面描述了如何获得预测和更新后的（分别参见（209）和（215））。BFG的现有方法使用电压和电流观测值z_v[k]和z_i[k]来进行模型识别和SOC跟踪。考虑常规的电压观测模型（180）。x_h[k]中的项表示滞后电压，其如（178）中所示，是电流i[k]、SOC x_s[k]和时间k.的函数。例如，当电池经历1A的负载（这在移动应用中是重负载）数秒时，与当负载持续30分钟为1A的情况相比，造成的滞后的幅度较小。此外，滞后的幅度在该时间也为SOC的函数。The following describes how to obtain the forecast and the updated (See (209) and (215) respectively). Existing methods for BFG use voltage and current observations _zv [k] and _zi [k] for model identification and SOC tracking. Consider the conventional voltage observation model (180). The term in x _h [k] represents the hysteresis voltage, which is shown in (178) as a function of current i[k], SOC x _s [k] and time k. For example, when a battery is subjected to a load of 1A for a few seconds (which is a heavy load in mobile applications), the magnitude of the hysteresis caused is smaller than when the load is 1A for 30 minutes. Furthermore, the magnitude of the hysteresis is also a function of SOC at this time.

由于使用用于模型识别的电池端子间的电压观测（180），还需要对滞后xh[k]建模并且必须估计模型参数。以SOC、电流和时间表示的滞后的模型为非线性的，并且尚未完全弄清楚。尝试对滞后建模和估计的另一个缺点是它使得在线模型识别几乎不可能。由于滞后为SOC的函数，模型识别需要横跨整个SOC范围的数据。这有时可能是不可能的，因为一些应用可能从不会将电池从满用到空。因为滞后还是电流的函数，所以模型识别需要横跨应用于各种持续时间的可能负载电流的使用数据。从而，完整的滞后建模和模型识别变得不现实。Due to the use of voltage observations across battery terminals ( 180 ) for model identification, the hysteresis xh[k] also needs to be modeled and model parameters must be estimated. The model of hysteresis in terms of SOC, current and time is non-linear and not yet fully understood. Another disadvantage of trying to model and estimate lags is that it makes online model identification nearly impossible. Since hysteresis is a function of SOC, model identification requires data across the entire SOC range. This may sometimes not be possible as some apps may never charge the battery from full to empty. Since hysteresis is also a function of current, model identification requires usage data across possible load currents applied for various durations. Thus, complete lag modeling and model identification becomes unrealistic.

还重要的是，应注意到，使用样品电池离线估计模型参数，再将这些参数用于电量计量可能不令人满意；某些电池参数已知会基于使用模式而变化。示例性实施例通过引入上述电压降模型避免了滞后建模。电压降v_D[k]表示内部电池模型元件R₀,R₁,R₂和x_h[k].两端的电压。项x_h[k]被有意引入以考虑用于推导电压降“测量值”的预测SOC中的误差。x_h[k]可被称为“瞬时滞后”，其根据示例性实施例，应通过调整SOC估计值来修正至零。如上文针对图7所述，电压降模型的使用可用于消除滞后效应。现有的SOC了解用于计算电压降。一批过往的电压降收集在缓冲器中并用于参数b的估计。（如在模型估计模块或块710中所确定的）的非零值指示瞬时滞后的存在，这意味着存在SOC估计误差。SOC跟踪算法设计成在估计的为非零的任何时候，（在SOC跟踪块715中）修正SOC。It is also important to note that using sample batteries to estimate model parameters offline may not be satisfactory for use in fuel gauging; some battery parameters are known to vary based on usage patterns. Exemplary embodiments avoid hysteresis modeling by introducing the voltage drop model described above. The voltage drop v _D [k] represents the voltage across the internal battery model elements R ₀ , R ₁ , R ₂ and x _h [k]. The term x _h [k] was intentionally introduced to take into account the derivation of the voltage drop "measurement" The predicted SOC of error in . x _h [k] may be referred to as "instantaneous hysteresis", which, according to an exemplary embodiment, should be adjusted by adjusting the SOC estimate to be corrected to zero. As described above for Figure 7, the use of a voltage drop model can be used to remove hysteresis effects. The existing SOC knowledge is used to calculate the voltage drop. A batch of past voltage drops is collected in a buffer and used for parameter b estimation. (as determined in the model estimation module or block 710) A non-zero value of indicates the presence of an instantaneous lag, which means that there is an SOC estimation error. The SOC tracking algorithm is designed to estimate the Anytime non-zero, the SOC is corrected (in SOC Tracking block 715).

OCV-SOC模型（167）代表OCV-SOC关系假定电压降（195）中（201）中的的估计值将为然而，意味着用于计算电压降观测值v_D[k]的SOC估计值存在误差。从而，SOC跟踪算法需要相应调整这通过采用下列修改后的观测模型取代（193）来实现。The OCV-SOC model (167) represents the OCV-SOC relationship assuming the voltage drop (195) in (201) The estimated value of will be However, means that there is an error in the SOC estimate used to calculate the voltage drop observation _vD [k]. Thus, the SOC tracking algorithm needs to be adjusted accordingly This is achieved by replacing (193) with the following modified observational model.

${z z}_{v v} [[k k]] = = {V V}_{00} (({x x}_{s the s} [[k k]])) + + \overset{~ ~}{a a} {[[k k]]}^{T T} \overset{~ ~}{b b} + + {n no}_{D D.} [[k k]] - - - - - - ((205205 [[55]]))$

其中，in,

$\overset{~ ~}{a a} {[[k k]]}^{T T} = = [[{v v}_{D D.} [[k k - - 11]] {v v}_{D D.} [[k k - - 22]] {z z}_{i i} [[k k]] {z z}_{i i} [[k k - - 11]] - - {z z}_{i i} [[k k - - 22]]]] - - - - - - ((206206))$

通过分别去除a[k]T和b中的最后一个元素而获得。换句话讲，去除滞后项。下面描述了修改后的观测模型的意义。另外，（190）中的过程噪声w_s[k]和（205）中的测量噪声n_D[k]之间存在下列协方差。Obtained by removing the last element in a[k]T and b, respectively. In other words, remove lag terms. The significance of the modified observation model is described below. In addition, the following covariance exists between the process noise w _s [k] in (190) and the measurement noise n _D [k] in (205).

$E E. {{{w w}_{s the s} [[k k]] {n no}_{D D.} [[k k]]}} &cong; &cong; U u [[k k]] = = {R R}_{00} {c c}_{h h} Δ Δ {σ σ}_{i i}^{22} - - - - - - ((208208))$

给定电荷状态的估计值和相关方差P_s[k|k]，下列EKF递归（参见图8）使用电压和电流测量值z_v[k+1],z_i[k],z_i[k+1]来得出的更新后的SOC估计值及其相关方差P_s[k+1|k+1].。这些步骤还确保了SOC估计值被最佳调整以考虑（208）的协方差。滤波递归由下列组成：Estimated value for a given state of charge and associated variance P _s [k|k], the following EKF recurrence (see Figure 8) uses voltage and current measurements z _v [k+1], z _i [k], z _i [k+1] to find The updated SOC estimate and its associated variance P _s [k+1|k+1]. These steps also ensure that the SOC estimate is optimally adjusted to account for the covariance of (208). Filtering recursion consists of the following:

${\overset{^^}{x x}}_{s the s} [[k k + + 11 | | k k]] = = {\overset{^^}{x x}}_{s the s} [[k k | | k k]] + + {c c}_{h h} [[k k]] Δ Δ {z z}_{i i} [[k k]] - - - - - - ((209209))$

${P P}_{s the s} [[k k + + 11 | | k k]] = = {P P}_{s the s} [[k k | | k k]] + + {σ σ}_{s the s}^{22} - - - - - - ((210210))$

$H h [[k k + + 11]] = = \frac{d d {z z}_{v v} [[k k]]}{d d {x x}_{s the s} [[k k]]} {| |}_{{\overset{^^}{x x}}_{S S} [[k k + + 11 | | k k]]} - - - - - - ((211211))$

${\overset{^^}{z z}}_{v v} [[k k + + 11]] = = {V V}_{00} (({\overset{^^}{x x}}_{s the s} [[k k + + 11 | | k k]])) + + {[[k k]]}^{T T} [[k k]] - - - - - - ((212212))$

$S S [[k k + + 11]] = = H h [[k k + + 11]] P P [[k k | | k k]] H h {[[k k + + 11]]}^{T T} + + {R R}_{{n no}_{D D.}} ((00)) + + 22 H h [[k k + + 11 U u [[k k]] - - - - - - ((213213))$

$G G [[k k + + 11]] = = \frac{P P [[k k + + 11 | | k k]] H h {[[k k + + 11]]}^{T T} + + U u [[k k]]}{S S [[k k + + 11]]} - - - - - - ((214214))$

${\overset{^^}{x x}}_{s the s} [[k k + + 11 | | k k + + 11]] = = {\overset{^^}{x x}}_{s the s} [[k k + + 11 | | k k]] + + G G [[k k + + 11]] (({z z}_{v v} k k + + 11]] - - {\overset{^^}{z z}}_{v v} [[k k + + 11]])) - - - - - - ((215215))$

$\begin{matrix} {P P}_{s the s} [[k k + + 11 | | k k + + 11]] = = ((11 - - G G [[k k + + 11]] H h [[k k + + 11]])) {P P}_{s the s} [[k k + + 11 | | k k]] {((11 - - G G [[k k + + 11]] H h [[k k + + 11]]))}^{T T} \\ + + G G {[[k k + + 11]]}^{22} {R R}_{{n no}_{D D.}} ((00)) \end{matrix} - - - - - - ((216216)) [[22]]$

其中c_h[k]和分别是库仑计数系数和模型参数向量的最新估计值。应描述使用状态空间模型（190）–（205）中的和进行SOC跟踪的重要性。滞后可被认为是OCV-SOC特征曲线中的误差。可能难以建模和准确估计滞后，因为其会随前一个电流和SOC而变化（参见（178））。然而，可估计真实OCV-SOC关系。实际上，（205）中的V₀(x_s[k])基于真实OCV-SOC模型。例如，假定估计的滞后为这意味着滤波器“感知的OCV”与电池的真实OCV相差10mV。对于BFG算法，感知的OCV，V₀(x_s[k])，与SOC估计值直接（且单调地）相关。换句话讲，如果滤波器的OCV不同于实际OCV，那么滤波器估计值也不同于电池的真实SOC。从而，当滤波器在（212）中发现其预测端电压下降10mV，其会在（215）中调整其SOC估计值使得“感知的OCV误差”（或估计的滞后H）（逐渐地）调整为零。从而，所提出的方法的正常运行的良好指示是估计的始终接近零。where c _h [k] and are the latest estimates of the Coulomb counting coefficient and model parameter vector, respectively. should describe the use of state-space models in (190)–(205) and The importance of doing SOC tracking. Hysteresis can be considered as an error in the OCV-SOC characteristic curve. Hysteresis can be difficult to model and accurately estimate because it varies with previous current and SOC (see (178)). However, the true OCV-SOC relationship can be estimated. Actually, V ₀ (x _s [k]) in (205) is based on the real OCV-SOC model. For example, suppose the estimated lag is This means that the "perceived OCV" of the filter is 10mV away from the true OCV of the battery. For the BFG algorithm, the perceived OCV, V ₀ (x _s [k]), is related to the estimated SOC are directly (and monotonically) related. In other words, if the OCV of the filter is different from the actual OCV, then the filter estimate Also different from the real SOC of the battery. Thus, when the filter in (212) finds its predicted terminal voltage 10mV drop, it adjusts its SOC estimate in (215) Such that the "perceived OCV error" (or estimated lag H) adjusts (gradually) to zero. Thus, a good indication of the proper functioning of the proposed method is to estimate Always close to zero.

不存在以期望方式验证电量计量算法的可靠设备和方法。使用模拟来评估电量计是不可行的，因为缺乏例如可允许模拟电池动态的可靠数学模型。例如，增强的自校正模型可不考虑电池老化的影响。由于电荷状态、电池容量和内阻的真值中的不确定性（全部都可为不断漂移的量），使用单一度量或验证方法来验证电量计可能很难。需要计算多个验证度量以理解电量计精度的全貌。There are no reliable devices and methods for validating fuel gauging algorithms in the desired manner. Using simulations to evaluate fuel gauges is not feasible due to the lack of reliable mathematical models that would allow simulation of battery dynamics, for example. For example, the enhanced self-correcting model does not take into account the effects of battery aging. Verifying a fuel gauge using a single metric or verification method can be difficult due to uncertainties in the true value of state of charge, battery capacity, and internal resistance, all of which can be constantly drifting quantities. Several verification metrics need to be calculated to understand the full picture of fuel gauge accuracy.

本公开内容接下来是基准测试，其从度量开始。在该示例性具体实施中，描述了用于验证电化学储能装置（电池）的电量计量算法的多个基准测试方法。相对准确的电量计量（FG）可延长电池的循环寿命。该具体实施方式还描述了精确和客观的电量计评估方案。本文所述的度量可用于在多个方面测量FG精度并且返回可指示电量计整体性能的至少一个数字。本文所述的基准测试可应用于多种电量计量算法。例如，包括在该具体实施方式中的细节可与名称为“Methods and Apparatus Related to Tracking Battery State of Charge:AReduced Order Filtering Approach（与跟踪电池电荷状态相关的方法和设备：降阶滤波方法）”的描述中所述的任何概念结合。Benchmarking follows this disclosure, which starts with metrics. In this exemplary implementation, a number of benchmarking methods for validating fuel gauging algorithms for electrochemical energy storage devices (batteries) are described. Relatively accurate fuel gauge (FG) can extend the cycle life of the battery. This detailed description also describes an accurate and objective fuel gauge evaluation scheme. The metrics described herein can be used to measure FG accuracy in a number of ways and return at least one number that can be indicative of the overall performance of the fuel gauge. The benchmarks described in this article can be applied to a variety of fuel gauging algorithms. For example, the details included in this detailed description can be compared to the article entitled "Methods and Apparatus Related to Tracking Battery State of Charge: AReduced Order Filtering Approach (Methods and Apparatus Related to Tracking Battery State of Charge: Reduced Order Filtering Method)" Any combination of concepts described in the description.

该具体实施方式中所述的基准测试可通过例如计算如下定义的三种度量中的一者或多者来进行：The benchmarking described in this detailed description can be performed, for example, by computing one or more of the three metrics defined below:

第一示例性度量为库仑计数误差。结合对实验的电池容量和起始的电荷状态（SOC）点的了解，库仑计数方法和/或设备可提供对电池电荷状态的准确估计。基于库仑计数的SOC估计和电量计经过一段时间的SOC估计之间的误差（如均方根（RMS））可用作基准测试的第一度量。基于FG的SOC和基于库仑计数的SOC之间的误差可能意味着与被验证的FG相关的一种或多种问题：A first exemplary metric is Coulomb counting error. Combined with knowledge of the experimental battery capacity and the starting state of charge (SOC) point, the coulomb counting method and/or device can provide an accurate estimate of the battery state of charge. The error (eg root mean square (RMS)) between the SOC estimate based on coulomb counting and the fuel gauge's SOC estimate over time can be used as a first metric for benchmarking. Errors between the FG-based SOC and the Coulomb-counting-based SOC could indicate one or more problems with the FG being verified:

-用于开路电压电荷状态（OCV-SOC）表征的模型可能不够准确（假设FG采用OCV-SOC表征）- The model used for OCV-SOC characterization may not be accurate enough (assuming FG is characterized by OCV-SOC)

-通过FG进行的电池容量估计可能不准确- Battery capacity estimation via FG may not be accurate

-动态等效电路模型可能在用于FG的模型选择和参数估计方案方面存在问题。- Dynamic equivalent circuit models can be problematic with model selection and parameter estimation schemes for FG.

第二示例性度量为OCV-SOC误差。（可使用一种或多种方法和/或设备进行的）电池的OCV-SOC表征可提供用于查找SOC的查找程序。从而，通过使电池处于完全（或至少部分）静置状态以及通过测量电池的电压，可将电量计在给定时间的SOC估计可与OCV-SOC表征进行比较以得到误差。OCV-SOC误差可指示电量计的一种或多种以下问题：A second exemplary metric is OCV-SOC error. OCV-SOC characterization of a battery (which may be performed using one or more methods and/or devices) may provide a lookup procedure for finding the SOC. Thus, by placing the battery in a fully (or at least partially) resting state and by measuring the voltage of the battery, the fuel gauge's SOC estimate at a given time can be compared to the OCV-SOC characterization for error. OCV-SOC errors can indicate one or more of the following problems with the fuel gauge:

-用于电池等效模型的动态模型可能无效- Dynamic models used for battery equivalent models may not be valid

-最小OCV-SOC误差可指示所用动态模型与电池的实际动态性质很好地匹配。- The minimum OCV-SOC error may indicate that the dynamic model used matches well with the actual dynamic properties of the battery.

第三示例性度量为预测达电压时间（TTV）误差。给出恒定负载/充电电流，电量计可（使用一种或多种方法和/或设备）预测达到某一电压所耗时间（TTV）。关闭时间（TTS）和完全充电时间（TTF）可为TTV估计的特殊情况。TTV估计中的误差可在达到所考虑的实际电压后计算。该TTV误差可指示与被评估的电量计相关的以下一者或多者：A third exemplary metric is predicted time to voltage (TTV) error. Given a constant load/charge current, the fuel gauge can predict (using one or more methods and/or devices) the time it takes to reach a certain voltage (TTV). Time to turn off (TTS) and time to full charge (TTF) may be special cases of TTV estimation. The error in the TTV estimation can be calculated after reaching the actual voltage considered. The TTV error may indicate one or more of the following associated with the fuel gauge being evaluated:

-电池的阻抗估计精度，- the impedance estimation accuracy of the battery,

-电量计的电池容量估计精度，- battery capacity estimation accuracy of the fuel gauge,

-电量计对电池SOC的了解（如信息），- Fuel gauge knowledge of battery SOC (eg info),

-OCV-SOC表征的精度。- Accuracy of OCV-SOC characterization.

电池响应于温度变化可表现出不同品质。例如，电池的阻抗可在低温下较高（并且从而可用功率可能较低）。响应于在较低SOC下的负载，与较高SOC水平的相同负载下相比，OCV变化率可能较大且高度非线性的。优质的电量计可具有在宽泛的温度和SOC水平范围内运行良好的能力。本文所述的基准测试可被配置为确保在性能评估期间将这些要素中的至少一些纳入测试中。Batteries can exhibit different qualities in response to temperature changes. For example, the impedance of the battery may be higher at low temperatures (and thus the available power may be lower). In response to load at lower SOC, the rate of change of OCV may be larger and highly non-linear than at the same load at higher SOC levels. A good fuel gauge can have the ability to perform well over a wide range of temperatures and SOC levels. The benchmarks described herein can be configured to ensure that at least some of these elements are included in the test during performance evaluation.

以下讨论涉及基于OCV-SOC表征的度量。电池的电荷状态可唯一地与其开路电压（OCV）有关。此类关系的一个例子在图14中示出为曲线图。The following discussion deals with metrics based on OCV-SOC characterization. A battery's state of charge can be uniquely related to its open circuit voltage (OCV). An example of such a relationship is shown as a graph in FIG. 14 .

多种方法可用于获得OCV表征数据。一种示例性方法总结如下：Various methods are available for obtaining OCV characterization data. An exemplary method is summarized as follows:

1）从完全充电、完全静置的电池开始1) Start with a fully charged, fully rested battery

2）记录其开路电压V_BAT=V_full 2) Record its open circuit voltage V _BAT =V _full

3）设定i=13) Set i=1

4）记录OCV(i)=V_BAT 4) Record OCV(i)=V _BAT

记录SOC(i)=1Record SOC(i)=1

5）设定i=i+15) Set i=i+1

6）使用恒定电流I使电池放电达ΔT的持续时间6) Discharge the battery for a duration of ΔT using a constant current I

7）使电池充分静置（如至少2小时）7) Let the battery rest fully (eg at least 2 hours)

8）测量电池端电压，V_BAT 8) Measure the battery terminal voltage, V _BAT

9）记录OCV(i)=V_BAT 9) Record OCV(i)=V _BAT

计算SOC(i)=SOC(i-1)+c_hI(i)ΔT，其中c_h=η/(3600C_batt)，η为指示充电/放电效率的常数，C_batt为以安培小时(Ah)表示的电池容量以及I(i)为流入电池的电流。Calculate SOC(i)=SOC(i-1)+c _h I(i)ΔT, where c _h =η/(3600C _batt ), η is a constant indicating charge/discharge efficiency, and C _batt is expressed in ampere hours (Ah ) represents the battery capacity and I(i) is the current flowing into the battery.

10）重复步骤5至9直到OCV(i)达到电池关闭电压VSD。10) Repeat steps 5 to 9 until OCV(i) reaches the battery shutdown voltage VSD.

现在，使用{OCV(i)；SOC(i)}对，可针对SOC∈[0；1]获得OCV表征。Now, using the pair {OCV(i); SOC(i)}, an OCV representation can be obtained for SOC∈[0;1].

附注：Note:

1）OCV-SOC表征可因温度不同而变化。（可使用若干方法来计算一定使用范围内的任何温度处的OCV表征）。1) OCV-SOC characterization can vary with temperature. (Several methods can be used to calculate the OCV characterization at any temperature within a range of use).

2）在一些具体实施中，电池容量C_batt可得自制造商数据表或其可估计。2) In some implementations, the battery capacity C _batt can be obtained from the manufacturer's data sheet or it can be estimated.

3）OCV-SOC表征可以无变化，而不论电池的使用年限。3) The OCV-SOC characterization can be unchanged regardless of the age of the battery.

4）使用上述OCV-SOC表征，可计算给定SOC s的电池的OCV，写成υ=OCV（s）。4) Using the above OCV-SOC characterization, the OCV of a battery with a given SOC s can be calculated, written as υ=OCV(s).

5）使用上述OCV-SOC表征，可计算针对给定的静置端电压υ_r的电池的SOC，写成s=OCV^-1（υ_r）。如果在时间k处的某电量计量算法的SOC估计值报告为则对应的误差可计算为5) Using the above OCV-SOC characterization, the SOC of the battery for a given resting terminal voltage υ _r can be calculated, written as s=OCV ^-1 (υ _r ). If the SOC estimate for a fuel gauging algorithm at time k is reported as Then the corresponding error can be calculated as

${&Element; &Element;}_{OCV OCV} = = {\overset{^^}{s the s}}_{FG FG} [[k k]] - - {OCV OCV}^{- - 11} (({&upsi; &upsi;}_{r r} [[k k]])) - - - - - - ((217217))$

其中υ_r[k]为时间k处（电池静置之后）的端电压。where υ _r [k] is the terminal voltage at time k (after the battery has rested).

可使用基于OCV-SOC表征的验证来计算整个温度和/或SOC区域的OCV误差（217）。The OCV error for the entire temperature and/or SOC region may be calculated ( 217 ) using verification based on the OCV-SOC characterization.

相对较高SOC区域中的验证可通过如下进行：首先从全充电电池开始并施加时变负载达一段足以消耗（大约）小于被测试电池的1/2容量的时间。相似地，低SOC区域中的验证可通过如下进行：首先从全充电电池或高SOC验证之后的电池开始，并施加足以将电池带到较低SOC区域的动态负载。Verification in the relatively high SOC region can be performed by first starting with a fully charged battery and applying a time-varying load for a time sufficient to drain (approximately) less than 1/2 the capacity of the battery under test. Similarly, verification in the low SOC region can be done by first starting with a fully charged battery or battery after high SOC verification, and applying a dynamic load sufficient to bring the battery to the lower SOC region.

平均OCV-SOC误差（以%计）可计算为The average OCV-SOC error (in %) can be calculated as

${\overset{&OverBar; &OverBar;}{&Element; &Element;}}_{OCV OCV} = = [[\frac{11}{88} {Σ Σ}_{i i = = 11}^{44} {&Element; &Element;}_{OCV OCV} (({s the s}_{L L},, {T T}_{i i})) + + {&Element; &Element;}_{OCV OCV} (({s the s}_{H h},, {T T}_{i i}))]] 100100 - - - - - - ((218218))$

其中∈OCV(s_L，T_i)表示在低SOC区域处和在温度T_i下计算出的误差。在一些具体实施中，越低，FG算法越佳。where εOCV(s _L , T _i ) denotes the calculated error at the low SOC region and at temperature T _i . In some implementations, The lower the value, the better the FG algorithm.

以下讨论涉及基于相对库仑计数误差的度量。电池SOC可按照如下方式通过库仑计数（CC）计算：The following discussion deals with metrics based on relative Coulomb counting error. Battery SOC can be calculated by coulomb counting (CC) as follows:

${\overset{^^}{s the s}}_{CC CC} [[k k]] = = {\overset{^^}{s the s}}_{CC CC} [[k k - - 11]] + + {c c}_{h h} {&Integral; &Integral;}_{{t t}_{k k - - 11}}^{{t t}_{k k}} I I [[t t]] dt dt - - - - - - ((219219))$

假定电池容量C_batt和足够准确的起始是已知的。从而，与库仑计数相关的FG误差（以%计）可定义为：Assuming a battery capacity C _batt and a sufficiently accurate initial is known. Thus, the FG error (in %) associated with Coulomb counting can be defined as:

${\overset{&OverBar; &OverBar;}{&Element; &Element;}}_{CC CC} = = \frac{\sqrt{{Σ Σ}_{k k = = 11}^{T T} {(({\overset{^^}{s the s}}_{CC CC} [[k k]] - - {\overset{^^}{s the s}}_{FG FG} [[k k]]))}^{22}}}{T T} 100100 - - - - - - ((220220))$

其中T是执行验证测试的持续时间（以秒计）。where T is the duration (in seconds) for which the verification test is performed.

1）在一些具体实施中，电池计量设备和/或方法中的一些可包括库仑计数作为其组成部分。然而，上述度量仍可视为验证工具，这是由于已假定了解电池容量以及验证的起始而FG方法可能不假定该了解。1) In some implementations, some of the battery gauging devices and/or methods may include coulomb counting as part of it. However, the above metric can still be considered as a verification tool, since knowledge of the battery capacity is assumed and the starting point of the verification Whereas the FG approach may not assume this knowledge.

2）可通过将电池从满（或基本上满）完全放电至空（或基本上空），来预先估计用于验证的电池容量。或者，可从满（或基本上满）至空（或基本上空）进行验证测试，并且可用最新估计的电池容量Cbatt来更新在一些具体实施中，可以在这种方式的电池容量估计中考虑滞后和松弛因子。在一些具体实施中，可禁止FG算法这样做。2) Can be pre-estimated for verification by fully discharging the battery from full (or substantially full) to empty (or substantially empty) battery capacity. Alternatively, a proof test can be performed from full (or substantially full) to empty (or substantially empty) and updated with the latest estimated battery capacity Cbatt In some implementations, hysteresis and relaxation factors may be considered in battery capacity estimation in this manner. In some implementations, the FG algorithm may be prohibited from doing so.

3）在一些具体实施中，正评估的电池的温度在整个（如，基本上整个）验证过程中可保持恒定。3) In some implementations, the temperature of the battery being evaluated can be kept constant throughout (eg, substantially throughout) the verification process.

以下讨论涉及基于达电压时间（TTV）的度量。若在时间k处的电量计算法的SOC估计值为s_FG[k]，则其达到电压υ需要的时间可书写成如下形式：The following discussion refers to time to voltage (TTV) based metrics. If the estimated SOC value of the electricity calculation method at time k is s _FG [k], the time required to reach the voltage υ can be written as follows:

${T T}_{FG FG,, &upsi; &upsi;} = = \frac{[[{OCV OCV}^{- - 11} ((&upsi; &upsi;)) - - {s the s}_{FG FG} [[k k]]]]}{I I} - - - - - - ((221221))$

其中电流I（充电过程中I>0并且放电过程中I<0）保持恒定（或基本上恒定）直到达到电压υ。where the current I (I>0 during charging and I<0 during discharging) remains constant (or substantially constant) until the voltage υ is reached.

一旦在运行过程中达到端电压υ，即可记录达到该电压时的实际时间。当在特定时间达到电压υ时，可计算下列TTV验证度量Once the terminal voltage ν is reached during operation, the actual time when this voltage is reached can be recorded. When the voltage υ is reached at a specific time, the following TTV verification metrics can be calculated

${&Element; &Element;}_{TTV TTV} = = \frac{11}{T T - - k k} \sqrt{{Σ Σ}_{i i = = k k}^{T T} {((\frac{{T T}_{EG EG,, &upsi; &upsi;} [[i i]] - - {T T}_{&upsi; &upsi;} [[i i]]}{{T T}_{&upsi; &upsi;} [[i i]]}))}^{22}} - - - - - - ((222222))$

其中T_v[i]=T–i是从时间i达到电压υ需要的实际时间。∈_TTV值可以分钟计。在一些具体实施中，可计算下列度量（以%计）：where _Tv [i]=T–i is the actual time required to reach voltage υ from time i. ∈ _TTV values can be measured in minutes. In some implementations, the following metrics (in %) can be calculated:

${\overset{&OverBar; &OverBar;}{&Element; &Element;}}_{TTV TTV} = = \frac{{&Element; &Element;}_{TTV TTV}}{T T} 100100 - - - - - - ((223223))$

组合的度量可定义为：The combined measure can be defined as:

${\overset{&OverBar; &OverBar;}{&Element; &Element;}}_{FG FG} = = \frac{{\overset{&OverBar; &OverBar;}{&Element; &Element;}}_{OCV OCV} + + {\overset{&OverBar; &OverBar;}{&Element; &Element;}}_{CC CC} + + {\overset{&OverBar; &OverBar;}{&Element; &Element;}}_{TTV TTV}}{33} - - - - - - ((224224))$

其中以%计。在一些具体实施中，该值越低，电量计越佳。in In %. In some implementations, the lower the value, the better the fuel gauge.

在一些具体实施中，基准测试程序可包括为被评估电池加载反映电池使用（如，典型使用）的一个或多个不同电流负载，并记录电量计报告的SOC和TTV读数。该方法可包括在不同温度下重复这些步骤，直到例如表I、II和III被填充。In some implementations, the benchmarking procedure may include loading the battery being evaluated with one or more different current loads reflecting battery usage (eg, typical usage) and recording the SOC and TTV readings reported by the fuel gauge. The method may include repeating these steps at different temperatures until, for example, Tables I, II and III are filled.

在一些具体实施中，可在验证过程中使用模拟以及实际负载曲线。模拟负载曲线的优点是可计算从电池获得的库仑的量（如，准确量），从而可避免由于取样和电流感测所致的一个或多个误差。这可基于这样的假定：负载模拟装置可能不会引入一个或多个显著误差。可创建各种实际负载曲线和模拟负载曲线。In some implementations, simulated as well as actual load curves may be used in the validation process. An advantage of simulating a load profile is that the amount of coulombs drawn from the battery can be calculated (eg, the exact amount), so that one or more errors due to sampling and current sensing can be avoided. This may be based on the assumption that the load simulator may not introduce one or more significant errors. Various actual load curves and simulated load curves can be created.

实际负载曲线：可使用例如智能电话作为负载，来创建实际负载曲线（如图15A和15B中所示）。当负载连接至被验证的电池时，可执行下列活动：电话呼叫（15分钟）、web冲浪、读取电邮、玩游戏等等（20分钟）、发短信（10条消息）、使用扬声器听音乐或看视频（30分钟视频）、待机–启用蜂窝无线电联系基站（1小时）。Actual Load Profile: An actual load profile (as shown in Figures 15A and 15B) can be created using, for example, a smartphone as a load. When the load is connected to the certified battery, the following activities can be performed: phone calls (15 minutes), web surfing, reading emails, playing games, etc. (20 minutes), sending text messages (10 messages), listening to music using the speaker Or watch video (30 minutes video), standby - enable cellular radio to contact base station (1 hour).

图15A和15B中所示的负载曲线示出了这样的场景，其中可在单个试验中计算三种类型度量（如，OCV度量1505、TTV度量1510和CC度量1515）中每一个的一个输入。该试验从全充电电池开始，并且动态使用负载施加约3小时又15分钟。之后电池处于休闲状态2小时。5小时标记提供了计算OCV-SOC误差度量∈_OCV(s_H，T_i)的机会。在试验即将结束时的恒定电流负载允许计算∈_TTV(s_L，T_i)。可根据整个数据计算库仑计算数度量∈_CC(T_i)。The load curves shown in FIGS. 15A and 15B illustrate a scenario where one input for each of the three types of metrics (eg, OCV metric 1505 , TTV metric 1510 , and CC metric 1515 ) can be calculated in a single trial. The test starts with a fully charged battery and the dynamic use load is applied for about 3 hours and 15 minutes. The battery is then at rest for 2 hours. The 5-hour mark provides an opportunity to compute the OCV-SOC error metric ∈ _OCV (s _H , T _i ). The constant current load towards the end of the test allows calculation of ε _TTV (s _L , T _i ). The coulomb count measure ∈ _CC (T _i ) can be calculated from the entire data.

可使用图15A和15B所示的负载曲线计算温度T_i下的下列度量：The following metrics at temperature _Ti can be calculated using the load curves shown in Figures 15A and 15B:

·在高SOC区域处的OCV-OSC误差度量∈_OCV(s_H，T_i)OCV-OSC error metric ∈ _OCV (s _H , T _i ) at high SOC regions

·在低SOC区域处的TTC度量∈_TTV(s_L，T_i)· TTC metric ∈ _TTV (s _L , T _i ) at low SOC region

·库仑计数度量∈_CC(T_i)Coulomb counting measure ∈ _CC (T _i )

模拟负载曲线：可使用短持续时间Δs内的不同幅度的分段常数电流负载I_m来创建模拟负载曲线。这些分段负载可被混合且接在一起，从而获得图16A和16B所示的模拟负载曲线。请注意，模拟负载可在测试的大约3.5小时至6.5小时之间出现。该负载曲线可使用例如Δs=2秒且I_m={40、120、130、160、300、400、440、520、600、640、800、880}（单位：mA）的Kikusui可编程负载装置来模拟。Simulated load curve: A simulated load curve can be created using a piecewise constant current load I _m of different amplitudes over a short duration Δs. These segmented loads can be mixed and tied together to obtain the simulated load curves shown in Figures 16A and 16B. Note that simulated load can occur between approximately 3.5 hours and 6.5 hours of testing. The load curve can use Kikusui programmable load device such as Δs=2 seconds and _Im ={40, 120, 130, 160, 300, 400, 440, 520, 600, 640, 800, 880} (unit: mA) to simulate.

图16A和16B中所示的模拟负载曲线示出了这样的场景，其中可在单个试验中计算三种类型度量（如，OCV度量1605、TTV度量1610和CC度量1615）中每一个的一个输入。该试验从全充电电池开始，并且施加恒定500mA负载约1.5小时。之后电池处于休闲状态2小时，然后施加动态负载曲线3小时。3小时又15分钟标记提供了计算OCV-SOC误差度量∈_OCV(s_H，T_j)的机会。在试验即将结束时的恒定电流负载允许计算∈_TTV(s_L，T_j)。可根据整个数据计算库仑计数度量∈_CC(T_j)。 The simulated load curves shown in Figures 16A and 16B illustrate a scenario where one input for each of the three types of metrics (e.g., OCV metric 1605, TTV metric 1610, and CC metric 1615) can be calculated in a single trial . The test was started with a fully charged battery and a constant 500 mA load was applied for about 1.5 hours. Afterwards the battery is left at rest for 2 hours and then a dynamic load profile is applied for 3 hours. The 3 hour and 15 minute mark provides an opportunity to calculate the OCV-SOC error metric ∈ _OCV (s _H , T _j ). The constant current load towards the end of the test allows calculation of ε _TTV (s _L , T _j ). The coulomb counting measure ∈ _CC (T _j ) can be calculated from the entire data.

在一些具体实施中，可使用图16A和16B所示的模拟负载曲线来计算温度T_j下的下列度量：In some implementations, the following metrics at temperature _Tj can be calculated using the simulated load curves shown in Figures 16A and 16B:

·库仑计数度量∈_CC(T_i)Coulomb counting measure ∈ _CC (T _i )

上述示例性具体实施描述了适用于电池供电装置（如，便携式移动装置）的SOC跟踪。所述示例性实施例实现了线性方法，该线性方法在计算上低成本并且性能效率优于现有的用于在线模型识别的方法。描述了用于参数估计的加权最小二乘方法。描述了参数估计的LS方法中的权重（基于方差）和参数估计中经证实的显著改善。对于电池的不同运行模式的适用性包括识别代表典型电池运行模式的电池的四个不同等效模型，并建立无缝SOC跟踪的框架。所述方法将滞后建模为开路电压（OCV）中的误差，从而消除了将滞后建模为SOC和负载电流的函数的需要。该方法还有助于从错误的SOC初始化快速恢复。The above exemplary implementations describe SOC tracking suitable for battery powered devices such as portable mobile devices. The exemplary embodiments implement a linear approach that is computationally inexpensive and performance efficient over existing methods for online model identification. A weighted least squares method for parameter estimation is described. The weighting (variance-based) in the LS method of parameter estimation and the proven significant improvement in parameter estimation are described. The applicability to different operating modes of the battery includes identifying four different equivalent models of the battery representing typical battery operating modes and establishing a framework for seamless SOC tracking. The method models hysteresis as an error in open circuit voltage (OCV), thereby eliminating the need to model hysteresis as a function of SOC and load current. This approach also facilitates fast recovery from erroneous SOC initialization.

上述示例性具体实施描述了用于电池容量估计以促进电池电量计量进步的特征。具有准确权重推导的容量的加权递归最小二乘（RLS）估计。权重的公式可基于SOC跟踪误差协方差和电流测量误差标准差来计算。描述了用于电池容量实时跟踪的TLS方法。TLS估计值以封闭式推导出并且可用于通过用衰减记忆更新协方差矩阵来自适应估计。基于静置电池的OCV查找的自适应容量估计的技术。考虑了推导中的OCV查找误差（滞后）的源并描述了通过OCV查找的自适应容量估计的方法。基于容量估计值和估计误差协方差描述了通过不同方法获得的容量估计值的最优融合的方法，所提出的方法使用卡尔曼滤波器来进行自适应最优融合。The above example implementations describe features for battery capacity estimation to facilitate battery fuel gauging advancements. Weighted recursive least squares (RLS) estimation of capacity with accurate weight derivation. The formula for the weights can be calculated based on the SOC tracking error covariance and the current measurement error standard deviation. A TLS method for real-time tracking of battery capacity is described. TLS estimates are derived in closed form and can be used to adapt estimates by updating the covariance matrix with a decay memory. Techniques for Adaptive Capacity Estimation Based on OCV Finding for Resting Batteries. Sources of OCV lookup error (hysteresis) in the derivation are considered and a method for adaptive capacity estimation via OCV lookup is described. A method for optimal fusion of capacity estimates obtained by different methods is described based on capacity estimates and estimation error covariance, and the proposed method uses a Kalman filter for adaptive optimal fusion.

上述示例性具体实施描述了适用于电池供电装置（如，便携式移动装置）的SOC跟踪。用于通过将另外的冗余参数连同SOC一起堆叠在状态向量上来估计另外的冗余参数的常规技术在计算上高昂且性能效率低下。为了避免这些问题，示例性实施例描述了降阶滤波模型，用于通过新的状态空间模型进行SOC跟踪。描述了具有去相关噪声模型的状态空间模型。SOC跟踪问题涉及两个测量量，即电压和电流，这就使得SOC跟踪问题的状态和测量噪声模型之间具有相关性。描述了具有不相关状态和测量噪声过程的修改后的状态空间表现形式。示例性实施例描述了电池的不同运行模式，并识别代表典型电池运行模式的电池的至少四个不同等效模型，并建立无缝SOC跟踪的框架。示例性实施例描述了一种方法，该方法将滞后建模为开路电压（OCV）中的误差，从而消除了将滞后建模为SOC和负载电流的函数的需要。该方法还有助于从错误的SOC初始化快速恢复。The above exemplary implementations describe SOC tracking suitable for battery powered devices such as portable mobile devices. Conventional techniques for estimating additional redundant parameters by stacking them along with the SOC on the state vector are computationally expensive and performance inefficient. To avoid these problems, exemplary embodiments describe a reduced-order filtering model for SOC tracking via a new state-space model. A state-space model with a decorrelated noise model is described. The SOC tracking problem involves two measurement quantities, namely voltage and current, which makes there is a correlation between the state of the SOC tracking problem and the measurement noise model. A modified state-space representation with uncorrelated states and measurement noise processes is described. Exemplary embodiments describe different operating modes of a battery, and identify at least four different equivalent models of the battery representing typical battery operating modes, and establish a framework for seamless SOC tracking. Exemplary embodiments describe a method that models hysteresis as an error in open circuit voltage (OCV), thereby eliminating the need to model hysteresis as a function of SOC and load current. This approach also facilitates fast recovery from erroneous SOC initialization.

上述示例性具体实施描述了通过若干策略实现的SOC跟踪。首先，通过最小电池建模。所提出的方法唯一需要对电池的开路电压（OCV）特性进行离线建模。所有其他所需参数通过稳固手段估计。由于配备有OCV参数的单个集合，所提出的方法能够在任何温度下执行SOC跟踪，无需任何另外的参数。第二，电压降观测模型。观测的电压降模型允许在线SOC跟踪，而不必担心对电池的滞后元件建模。这就使得所提出的方法可得到更好的精度和稳健性。第三，通过稳固参数估计。识别了用于参数估计的最小二乘模型中的相关噪声结构的效应。对于参数估计算法，这就得到了明显更好的精度并增强了稳健性。第四，通过电池容量估计。所提出的用于容量估计的总体最小二乘（TLS）方法确保了容量估计的优异精度。以及最后，通过使用滤波，降阶EKF方法考虑到状态空间模型中噪声过程的相关性（被推导出以用于SOC跟踪）并应用适当的去关联滤波器以使SOC跟踪中的误差最小化。The above exemplary implementations describe SOC tracking through several strategies. First, through minimal battery modeling. The proposed method uniquely requires offline modeling of the open-circuit voltage (OCV) characteristics of the battery. All other required parameters are estimated by robust means. Equipped with a single set of OCV parameters, the proposed method is able to perform SOC tracking at any temperature without any additional parameters. Second, the voltage drop observation model. The observed voltage drop model allows online SOC tracking without having to worry about modeling the lagging elements of the battery. This leads to better accuracy and robustness of the proposed method. Third, by stabilizing parameter estimates. The effect of correlated noise structure in least squares models for parameter estimation is identified. This results in significantly better accuracy and increased robustness for parameter estimation algorithms. Fourth, through battery capacity estimation. The proposed total least squares (TLS) method for capacity estimation ensures excellent accuracy for capacity estimation. And finally, by using filtering, the reduced-order EKF method takes into account the correlation of the noise process in the state-space model (derived for SOC tracking) and applies appropriate decorrelation filters to minimize the error in SOC tracking.

在具体实施方式中，描述了基于至少三个评估度量的电池电量计量算法的基准测试方法。第一评估度量可基于电池的开路电压（OCV）表征。第二评估度量可基于电量计的相对库仑计数误差，并且第三基准可基于电池达到特定电压所需的时间的计算。每个验证度量可包括在各种SOC水平、不同温度和/或电压区域，诸如此类处计算若干度量。In a specific embodiment, a method for benchmarking a battery fuel gauging algorithm based on at least three evaluation metrics is described. The first evaluation metric may be based on an open circuit voltage (OCV) characterization of the battery. The second evaluation metric can be based on the relative coulomb counting error of the fuel gauge, and the third benchmark can be based on a calculation of the time it takes for the battery to reach a particular voltage. Each verification metric may include calculating several metrics at various SOC levels, different temperature and/or voltage regions, and the like.

一些SOC跟踪方法包括至少如下不足之处：（1）一些模型仅考虑电阻，不适合动态负载；（2）它们采用非线性方法来进行系统识别；（3）需要用于模型识别方法的初始参数估计；（4）假定单个动态等效模型代表所有电池运行模式；（5）未解决在线容量估计的重要性；（6）现有的在线电池容量估计技术受到SOC和参数估计误差的影响，即它们不稳固；（7）除SOC外，它们还采用许多冗余量的在线跟踪（这导致增大了计算复杂性并降低了SOC跟踪精度）；（8）它们需要对电池滞后进行单独建模，电池滞后是SOC和负载电流的函数（因此为无限模型），只可能对滞后进行近似建模；（9）现有方法中任何一种均未认识到该过程与测量噪声过程存在相关性；以及（10）现有方法中任何一种均未认识到由于温度、老化、SOC和负载变化导致的电池特性中的变化以及单个等效模型可能不适配所有这些条件的事实。Some SOC tracking methods include at least the following deficiencies: (1) some models only consider resistance and are not suitable for dynamic loads; (2) they use nonlinear methods for system identification; (3) require initial parameters for model identification methods estimation; (4) a single dynamic equivalent model is assumed to represent all battery operating modes; (5) the importance of online capacity estimation is not addressed; (6) existing online battery capacity estimation techniques suffer from SOC and parameter estimation errors, namely They are not robust; (7) they employ online tracking of many redundant quantities in addition to SOC (which leads to increased computational complexity and reduced SOC tracking accuracy); (8) they require separate modeling of battery lag , the battery hysteresis is a function of SOC and load current (hence an infinite model), and only approximate modeling of the hysteresis is possible; (9) none of the existing methods recognize that this process is correlated with the measurement noise process; And (10) none of the existing methods recognize the changes in battery characteristics due to temperature, aging, SOC and load changes and the fact that a single equivalent model may not fit all these conditions.

因此，本文所述的具体实施可具有短设计时间（数天内），可具有相对较快的算法收敛，并且在“真实世界”使用条件中可具有大约1%的SOC和电池容量报告精度的精度。在一些具体实施中，可不（或很少）需要定制电池模型或数据，并且可包括具有相对较快SOC跟踪收敛的自适应学习算法。一些具体实施可包括自动温度、使用年限和负载补偿。Therefore, the implementation described herein can have a short design time (within days), can have relatively fast algorithm convergence, and can have an accuracy of about 1% of the reported accuracy of SOC and battery capacity in "real world" use conditions . In some implementations, no (or little) custom battery model or data may be required, and an adaptive learning algorithm with relatively fast SOC tracking convergence may be included. Some implementations may include automatic temperature, age and load compensation.

一些具体实施可基于例如降阶扩展卡尔曼滤波器、相关测量噪声去耦、在线电模型参数估计和实时容量估计。Some implementations may be based on, for example, reduced-order extended Kalman filters, correlated measurement noise decoupling, on-line model parameter estimation, and real-time capacity estimation.

降阶卡尔曼滤波器可包括涉及共同估计四个不同参数的准确SOC估计（跟踪）：SOC、流过动态等效模型中的两个不同电阻器的电流、以及滞后，所有这些在电池处于负载/充电时会变化。这涉及通常称为递归滤波的复矩阵运算。该降阶滤波方法以这样的方式进行了简化：通过递归滤波程序仅估计SOC。三个其他参数通过数学运算被边缘化。所得的一个或多个SOC跟踪算法现在在电量计SOC中在计算上是可行的。A reduced-order Kalman filter can include accurate SOC estimation (tracking) that involves jointly estimating four different parameters: SOC, current through two different resistors in a dynamic equivalent model, and hysteresis, all while the battery is under load / changes while charging. This involves complex matrix operations commonly known as recursive filtering. The reduced-order filtering method is simplified in such a way that only the SOC is estimated by a recursive filtering procedure. Three other parameters are marginalized through mathematical operations. The resulting SOC tracking algorithm or algorithms are now computationally feasible in the fuel gauge SOC.

相关测量噪声去耦可包括卡尔曼滤波器，其基于过程噪声和测量噪声不相关的假设而运行。在电量计应用中，电流测量中固有的测量噪声可在降阶EKF中耦合至SOC和电压变量的测量噪声和过程噪声两者。已采用了一种独特方法，该方法将电流感应噪声与卡尔曼滤波器的过程噪声去耦。Correlated measurement noise decoupling may include a Kalman filter that operates on the assumption that process noise and measurement noise are uncorrelated. In fuel gauge applications, the measurement noise inherent in current measurements can couple to both the measurement noise and process noise of the SOC and voltage variables in a reduced order EKF. A unique approach has been taken that decouples the current induced noise from the process noise of the Kalman filter.

在线电模型参数估计可包括模型参数（系数）的动态估计。模型参数（系数）的动态估计可包括当其随SOC、时变电流负载曲线、温度、充电-放电循环和/或诸如此类变化时的估计。EKF滤波器是适用的，前提条件是电池的动态等效电路的模型参数是已知的，然而，等效电路代表电池的内部元件；这些模型参数也可使用可得自电池的测量值来估计：电压和电流。本文所述的解决方案以实时方式估计模型参数，从而能够使用卡尔曼滤波器方法。On-line model parameter estimation may include dynamic estimation of model parameters (coefficients). Dynamic estimation of model parameters (coefficients) may include estimation as they vary with SOC, time-varying current load profile, temperature, charge-discharge cycles, and/or the like. The EKF filter is applicable provided that the model parameters of the dynamic equivalent circuit of the battery are known, however, the equivalent circuit represents the internal components of the battery; these model parameters can also be estimated using measurements available from the battery : Voltage and current. The solution described here estimates model parameters in real-time, enabling the use of Kalman filter methods.

实时或在线容量估计可包括基于实际使用条件、负载、温度、使用年限来更新（如，持续更新）可用容量的一个或多个算法。一些具体实施可包括库仑计数方法，该方法将FG报告的SOC与基于库仑计数（簿记法）计算出的SOC进行比较。一些具体实施可包括TTE（电力用尽时间）方法，该方法可使用FG来预测TTE并与实际情况进行比较。一些具体实施可包括SOC/OCV曲线查找方法，该方法可将FG报告的SOC与SOC/OCV曲线进行比较。Real-time or online capacity estimation may include one or more algorithms that update (eg, continuously update) available capacity based on actual usage conditions, load, temperature, age. Some implementations may include a coulomb counting method that compares the SOC reported by the FG with the SOC calculated based on coulomb counting (bookkeeping). Some implementations may include a TTE (time to exhaust) method that uses FG to predict TTE and compare to actual conditions. Some implementations may include a SOC/OCV curve lookup method that compares the SOC reported by the FG to the SOC/OCV curve.

在一些具体实施中，可支持各种各样的电池，其具有多种特定电池模型和每种电池的化学成分、电池制造商和/或老化数据，诸如此类。在一些具体实施中，电量计评估程序可变化。例如，电荷状态精度评估方法和测试程序、动态负载和/或详细的测试要求，诸如此类可变化。在一些具体实施中，对要求规格（例如关键系统参数和精度要求和/或系统集成要求，诸如此类）的反馈可变化。在一些具体实施中，操作系统驱动要求可变化。In some implementations, a wide variety of batteries may be supported with a variety of specific battery models and each battery's chemical composition, battery manufacturer and/or aging data, and the like. In some implementations, the fuel gauge evaluation procedure may vary. For example, state of charge accuracy evaluation methods and test procedures, dynamic loads and/or detailed test requirements, etc. may vary. In some implementations, feedback on required specifications (eg, key system parameters and accuracy requirements and/or system integration requirements, and the like) can vary. In some implementations, operating system driver requirements may vary.

图17为示出了示例性系统实现的示意图。系统1700包括电量计评估模块1705、电池1710、电池电量计1715和计算装置1715。电量计评估模块1705可实现为BMS110内的软件模块或ASIC。换句话讲，电量计评估模块1705可以为代码，该代码存储在存储器230中并由处理器235和/或与BMS110相关的另一个模块执行。计算装置1715可从电量计评估模块1705接收信息并将该信息显示在图形用户界面上（如图18所示）。Figure 17 is a schematic diagram illustrating an exemplary system implementation. System 1700 includes fuel gauge evaluation module 1705 , battery 1710 , battery fuel gauge 1715 , and computing device 1715 . Fuel gauge evaluation module 1705 may be implemented as a software module or ASIC within BMS 110 . In other words, fuel gauge evaluation module 1705 may be code stored in memory 230 and executed by processor 235 and/or another module associated with BMS 110 . Computing device 1715 may receive information from fuel gauge evaluation module 1705 and display the information on a graphical user interface (as shown in FIG. 18 ).

图18为示出了可与系统实现结合使用的用户界面的示意图。图19A和19B包括示出了示例性放电电压/电流曲线以图示说明SOC查找验证1905和TTS测试1910的图。图20A和20B为示出示例性CC评估方法以图示说明电量计算法2005与库仑计数2010的接近性的图。图21A和21B为示出TTS评估方法的示意图，其图示说明电量计2105和实际SOC2110的明显重叠以及TTS误差2115。Figure 18 is a schematic diagram illustrating a user interface that may be used in conjunction with a system implementation. 19A and 19B include graphs showing exemplary discharge voltage/current curves to illustrate SOC lookup verification 1905 and TTS testing 1910 . 20A and 20B are graphs showing an exemplary CC evaluation method to illustrate the proximity of coulometry 2005 to coulomb counting 2010 . FIGS. 21A and 21B are schematic diagrams illustrating a TTS evaluation method, illustrating significant overlap of fuel gauge 2105 and actual SOC 2110 and TTS error 2115 .

上述示例性实施例中的一些被描述为以流程图示出的过程或方法。虽然流程图将操作描述为顺序过程，但许多操作可并行、并发或同时执行。此外，可重新排列操作次序。这些过程可在完成其操作时终止，但还可具有图中未包括的另外步骤。这些过程可对应于方法、功能、程序、子例程、子程序等等。Some of the above-described exemplary embodiments are described as procedures or methods shown in flowcharts. Although a flowchart describes operations as a sequential process, many operations can be performed in parallel, concurrently, or simultaneously. Also, the order of operations can be rearranged. These processes may terminate when their operations are complete, but may also have additional steps not included in the figure. These procedures may correspond to methods, functions, procedures, subroutines, subroutines, and the like.

上述方法（其中一些由流程图示出）可通过硬件、软件、固件、中间件、微码、硬件描述语言或它们的任何组合来实现。当以软件、固件、中间件或微码实现时，执行必要任务的程序代码或代码段可存储在机器或计算机可读介质例如存储介质中。可由一个或多个处理器执行必要的任务。The methods described above, some of which are illustrated by flowcharts, can be implemented by hardware, software, firmware, middleware, microcode, hardware description languages, or any combination thereof. When implemented in software, firmware, middleware or microcode, the program code or code segments to perform the necessary tasks may be stored in a machine or computer readable medium such as a storage medium. Necessary tasks may be performed by one or more processors.

本文所公开的具体结构和功能细节仅是代表性的，以便描述示例性实施例。然而，示例性实施例具体体现为许多替代形式并且不应理解为仅限于本文给出的实施例。Specific structural and functional details disclosed herein are merely representative for purposes of describing example embodiments. Exemplary embodiments may, however, be embodied in many alternative forms and should not be construed as limited to only the embodiments set forth herein.

应当理解，虽然术语第一、第二等等可在本文中用于描述各种元件，但这些元件不应受这些术语的限制。这些术语仅用于使一个元件与另一个元件区分开。例如，第一元件可以称为第二元件，并且相似地，第二元件可以称为第一元件，而不脱离示例性实施例的范围。如本文所用，术语“和/或”包括列出的相关项中的一个或多个的任何和所有组合。It will be understood that, although the terms first, second, etc. may be used herein to describe various elements, these elements should not be limited by these terms. These terms are only used to distinguish one element from another. For example, a first element could be termed a second element, and, similarly, a second element could be termed a first element, without departing from the scope of example embodiments. As used herein, the term "and/or" includes any and all combinations of one or more of the associated listed items.

应当理解，当元件被称为“连接”或“耦合”至另一元件时，其可直接连接或耦合至该另一元件或可存在中间元件。相反，当元件被称为“直接连接”或“直接耦合”至另一元件时，不存在中间元件。用于描述元件之间的关系的其他词应以类似方式解释（如，“介于...之间”与“直接介于...之间”，“相邻的”与“直接相邻的”等等）。It will be understood that when an element is referred to as being "connected" or "coupled" to another element, it can be directly connected or coupled to the other element or intervening elements may be present. In contrast, when an element is referred to as being "directly connected" or "directly coupled" to another element, there are no intervening elements present. Other words used to describe the relationship between elements should be interpreted in a similar fashion (e.g., "between" versus "directly between," "adjacent" versus "directly adjacent of" and so on).

还应该指出的是，在一些替代具体实施中，指出的功能/动作可不按图中所指出的次序发生。例如，以连续方式示出的两个图可实际上并发执行或有时可以倒序执行，具体取决于所涉及的功能/动作。It should also be noted that in some alternative implementations, the functions/acts noted may occur out of the order noted in the figures. For example, two figures shown in succession may in fact be executed concurrently or may sometimes be executed out of order, depending upon the functionality/acts involved.

本文所述的各种技术的具体实施可在数字电子电路中或在计算机硬件、固件、软件中或在它们的组合中实现。具体实施可实现为计算机程序产品，即有形体现于信息载体中，如机器可读存储装置（计算机可读介质、非临时性计算机可读存储介质、有形计算机可读存储介质）中或传播的信号中的计算机程序，用于通过数据处理设备（例如可编程处理器、计算机或多台计算机）处理，或控制所述数据处理设备的操作。计算机程序例如上述一个或多个计算机程序可以任何编程语言形式（包括编译或解释语言）编写，并且可以任何形式部署，包括独立程序或适用于计算环境的模块、组件、子例程或其他单元。Embodiments of the various techniques described herein may be realized in digital electronic circuitry or in computer hardware, firmware, software, or in combinations thereof. The specific implementation can be implemented as a computer program product, that is, a signal tangibly embodied in an information carrier, such as a machine-readable storage device (computer-readable medium, non-transitory computer-readable storage medium, tangible computer-readable storage medium) or propagated A computer program for processing by a data processing device (such as a programmable processor, a computer or multiple computers), or controlling the operation of said data processing device. A computer program, such as one or more of the above computer programs, can be written in any programming language, including compiled or interpreted languages, and can be deployed in any form, including a stand-alone program or as a module, component, subroutine or other unit suitable for a computing environment.

上述示例性实施例和对应的具体实施方式中的各部分按照对计算机存储器内的数据位上的操作的软件或算法和符号表示来提出。算法（当该术语在此处使用且当其以一般方式使用时）被设想为得出所需结果的自洽步骤序列。这些步骤是要求对物理量进行物理操纵的步骤。通常，虽然不是必要的，这些量采用能够存储、转移、组合、比较和以其他方式操纵的光信号、电信号或磁信号的形式。已证实有时较方便的是，特别是出于通用的原因，将这些信号称为比特、值、元素、符号、字符、项、数字等等。Portions of the above-described exemplary embodiments and corresponding detailed descriptions are presented in terms of software or algorithms and symbolic representations of operations on data bits within a computer memory. An algorithm (as the term is used here and when it is used in a generic way) is conceived to be a self-consistent sequence of steps leading to a desired result. The steps are those requiring physical manipulations of physical quantities. Usually, though not necessarily, these quantities take the form of optical, electrical or magnetic signals capable of being stored, transferred, combined, compared and otherwise manipulated. It has proven convenient at times, particularly for reasons of common usage, to refer to these signals as bits, values, elements, symbols, characters, terms, numbers, or the like.

在以上示例性实施例中，提及的可实现为程序模块或功能过程的操作（如，以流程图的形式）的动作和符号表示包括执行具体任务或实现具体抽象数据类型的例程、程序、对象、组件、数据结构等等，并且可使用现有结构元件处的现有硬件来描述和/或实现。一些现有硬件可包括一个或多个中央处理单元（CPU）、数字信号处理器（DSP）、专用集成电路、现场可编程门阵列（FPGA）计算机等等。In the exemplary embodiments above, references to actions and symbolic representations of operations (eg, in the form of flowcharts) that can be implemented as program modules or functional processes include routines, programs that perform specific tasks or implement specific abstract data types , objects, components, data structures, etc., and can be described and/or implemented using existing hardware at existing structural elements. Some existing hardware may include one or more central processing units (CPUs), digital signal processors (DSPs), application specific integrated circuits, field programmable gate array (FPGA) computers, and the like.

然而，应记住所有这些和类似术语与适当的物理量相关，并且仅仅是适用于这些量的方便标签。除非另外特别指明，否则如从讨论中可明显看出，术语例如“处理”或“计算”（computing）或“计算”（calculating）或“确定”或“显示”等等是指计算机系统或类似电子计算装置的动作和过程，将以计算机系统寄存器和存储器内的物理、电子量表示的数据操作和变换成类似地以计算机系统存储器或寄存器或其他此类信息存储、传输或显示装置内的物理量表示的其他数据。It should be borne in mind, however, that all of these and similar terms are to be related to the appropriate physical quantities and are merely convenient labels applied to these quantities. As is apparent from the discussion, terms such as "processing" or "computing" or "calculating" or "determining" or "displaying" or the like refer to computer systems or similar Actions and processes of electronic computing devices that manipulate and transform data represented by physical, electronic quantities in computer system registers and memories into physical quantities that are similarly stored, transmitted, or displayed in computer system memory or registers or other such information Other data represented.

还应注意，示例性实施例的由软件实现的方面通常编码于某种形式的非临时性程序存储介质上或在某种类型的传输介质上实现。程序存储介质可为磁性的（如，软盘或硬盘驱动器）或光学的（如，光盘只读存储器，或“CDROM”），并且是只读的或随机存取的。相似地，传输介质可为双绞线、同轴电缆、光纤或一些其他合适的传输介质。示例性实施例不受任何指定具体实施的这些方面的限制。Note also that the software implemented aspects of the exemplary embodiments are typically encoded on some form of non-transitory program storage medium or implemented over some type of transmission medium. The program storage medium may be magnetic (eg, a floppy disk or hard drive) or optical (eg, a compact disc read-only memory, or "CDROM"), and be read-only or random-access. Similarly, the transmission medium may be twisted pair wire, coaxial cable, fiber optics or some other suitable transmission medium. The exemplary embodiments are not limited in these aspects by any given implementation.

虽然所述具体实施的某些特征已被示出为如本文所述，但本领域的技术人员现将可以想到许多修改、替代、变更和等效方案。因此，应当理解，所附权利要求书旨在涵盖落入具体实施的范围内的所有此类修改形式和变更形式。应当理解，所述实施例仅以举例的方式而不是以限制的方式呈现，并且可在形式和细节方面进行各种变更。本文所述的装置和/或方法的任一部分可以以任何组合加以组合，但相互排斥的组合除外。本文所述的具体实施可包括所述不同具体实施的功能、部件和/或特征的各种组合和/或子组合。While certain features of the described implementations have been illustrated as described herein, many modifications, substitutions, changes, and equivalents will now occur to those skilled in the art. It is, therefore, to be understood that the appended claims are intended to cover all such modifications and changes as fall within the scope of particular implementations. It should be understood that the described embodiments have been presented by way of example only, not limitation, and that various changes in form and details may be made. Any portion of the devices and/or methods described herein may be combined in any combination, except mutually exclusive combinations. Implementations described herein can include various combinations and/or sub-combinations of the functions, components, and/or features of the different implementations described.

Claims

1. a method, comprising:

In the very first time, calculate first of battery and estimate state of charge (SOC);

In the second time, receive the magnitude of voltage of the measuring voltage that represents described battery two ends;

At described the second Time Calculation filter gain; And

In described the second time, based on described first, estimate that SOC, described magnitude of voltage and described filter gain calculate second of described battery and estimate SOC.

2. method according to claim 1, also comprises:

By described second, estimate that SOC is stored in impact damper; And

Before calculating the described second estimation SOC of described battery, from described impact damper, read described first and estimate SOC.

3. method according to claim 1, also comprises:

By described first, estimate that SOC is stored in impact damper, calculate described battery described second estimate SOC before, from described impact damper, read described first and estimate that SOC is as one of following:

The vector of at least two SOC,

The array of at least two SOC,

At least two SOC average and

The average of at least two SOC.

4. method according to claim 1, also comprises: from impact damper, read and estimate SOC variance, wherein said filter gain is based on described estimation SOC variance.

5. method according to claim 1, wherein, calculating filter gain comprises: use at least one in weighting recurrence least square (RLS) algorithm and total least square (TLS) algorithm to calculate at least one capability value.

6. method according to claim 1, wherein, calculating filter gain comprises: use the weighting RLS algorithm based on SOC tracking error covariance and current measurement errors standard deviation to carry out calculated capacity value.

7. method according to claim 1, wherein, calculating filter gain comprises: the TLS algorithm of using the recurrence based on covariance matrix to upgrade carrys out calculated capacity value.

8. method according to claim 1, wherein, calculating filter gain comprises: use open-circuit voltage (OCV) to search calculated capacity value.

9. method according to claim 1, wherein

Described second estimates that SOC equals described filter gain and is multiplied by described magnitude of voltage and adds the above and first estimate SOC.

10. method according to claim 1, also comprises:

At least one in voltage time (TTV) error metrics that reach based on coulomb counting error metrics, OCV-SOC error metrics and prediction, estimates that to described second SOC carries out benchmark test.

11. 1 kinds of systems, comprising:

Battery; And

Battery electric quantity meter module, it is configured to use Reduced Order Filter to calculate the estimation state of charge (SOC) of described battery, described Reduced Order Filter is single state filter, and the SOC estimated value that described single state filter is configured to based on calculating before to estimate SOC described in recursive calculation.

12. systems according to claim 11, wherein, described battery electric quantity meter comprises:

Voltage drop prediction module, it is configured to come calculating voltage to fall based on described estimation SOC;

Model estimation module, its equivalent model being configured to based on representing described battery calculates at least one voltage drop model parameter; And

Tracking module, it is configured to make described at least one voltage drop model parameter to change to remove instantaneous lagging.

13. systems according to claim 11, wherein, described battery electric quantity meter comprises capacity module, total least square (TLS) algorithm that described capacity module is configured to use the recurrence based on covariance matrix to upgrade calculates the capability value of described battery.

14. systems according to claim 11, wherein, described battery electric quantity meter comprises capacity module, described capacity module is configured to use open-circuit voltage (OCV) to search to calculate the capability value of described battery.

15. systems according to claim 11, wherein

Described battery electric quantity meter comprises impact damper, and described impact damper is configured to store at least one and estimates SOC value, and

The SOC estimated value calculating before described reads from described impact damper.

16. systems according to claim 11, also comprise data-carrier store, described data-carrier store is configured to store OCV parameter, voltage measurement error standard deviation and current measurement errors standard deviation, and wherein at least one voltage drop model parameter relevant to described Reduced Order Filter defined by one of described OCV parameter, described voltage measurement error standard deviation and described current measurement errors standard deviation.

17. systems according to claim 11, also comprise: display, described display is configured to show described estimation SOC.

18. 1 kinds of computer-readable mediums, comprise code segment, and described code segment, when being carried out by processor, makes described processor:

Calculate the estimation state of charge (SOC) of battery;

Described estimation SOC is stored in to impact damper; And

Use Reduced Order Filter to calculate the estimation SOC after the renewal of described battery, described Reduced Order Filter is single state filter, the estimation SOC after described single state filter is configured to upgrade described in recursive calculation based on described estimation SOC.

19. computer-readable mediums according to claim 18, also comprise code segment, and described code segment makes described processor:

Based on described estimation SOC calculating voltage, fall;

Equivalent model based on representing described battery calculates at least one voltage drop model parameter; And

Change open-circuit voltage (OCV) error that described at least one voltage drop model parameter lags behind to reduce indication.

20. computer-readable mediums according to claim 18, also comprise: code segment, and described code segment makes described processor:

Capacity based on described battery and at least one voltage drop model parameter are carried out calculating filter gain, and wherein said Reduced Order Filter calculates the estimation SOC after described renewal based on described filter gain and described estimation SOC.

21. 1 kinds of methods, comprising:

In storer, storage represents the storehouse of the equivalent-circuit model of battery;

Based on the load relevant to described battery, determine the operational mode of battery;

Based on determined operational mode, select one of described equivalent-circuit model; And

With selected equivalent-circuit model, calculate the state of charge (SOC) of described battery.

22. methods according to claim 21, wherein

The high voltage load of described operational mode based on consuming variable current,

Described operational mode comprises corresponding equivalent-circuit model, and described equivalent-circuit model comprises the first resistance-electric current (RC) circuit and the 2nd RC circuit.

23. methods according to claim 21, wherein

Described operational mode is based on variable voltage load,

Described operational mode comprises corresponding equivalent-circuit model, and described equivalent-circuit model comprises resistance-electric current (RC) circuit.

24. methods according to claim 21, wherein

Described operational mode is based on variable voltage load,

Described operational mode comprises corresponding equivalent-circuit model, and described equivalent-circuit model comprises resistor and biasing element.

25. methods according to claim 21, wherein

Described operational mode is based on low-voltage load,

Described operational mode comprises corresponding equivalent-circuit model, and described equivalent-circuit model comprises resistor.

26. methods according to claim 21, wherein

At least one in described equivalent-circuit model comprises lag element, and

Described lag element is modeled as open-circuit voltage (OCV) error of calculation.

27. methods according to claim 21, also comprise:

At battery eliminator interdischarge interval, measure the terminal voltage of described battery eliminator;

Terminal voltage based on measured is determined linear equation; And

Based on described linear equation, with weighted least square algorithm, calculate at least one parameter, to the calculating of the described SOC of described battery based on described at least one parameter.

28. methods according to claim 21, also comprise:

At least one in voltage time (TTV) error metrics that reach based on coulomb counting error metrics, OCV-SOC error metrics and prediction, carries out benchmark test to described SOC.

29. 1 kinds of systems, comprising:

Data-carrier store, it is configured to the storehouse that storage represents the equivalent-circuit model of battery;

Model Selection module, its operational mode being configured to based on described battery is selected equivalent-circuit model; And

Filter module, it is configured to calculate based on selected equivalent-circuit model the estimation state of charge (SOC) of described battery.

30. systems according to claim 29, also comprise:

Operational mode module, its be configured in electric current based on relevant to described battery and the voltage relevant with described battery at least one determine the operational mode of described battery.

31. systems according to claim 29, wherein

32. systems according to claim 29, wherein

Described operational mode is based on variable voltage load,

33. systems according to claim 29, wherein

Described operational mode is based on variable voltage load,

34. systems according to claim 29, wherein

Described operational mode is based on low-voltage load,

35. according to the method for claim 29, wherein

At least one in described equivalent-circuit model comprises lag element, and

36. systems according to claim 29, also comprise:

Model estimation module, the selected equivalent model that described model estimation module is configured to based on representing described battery calculates at least one voltage drop model parameter, and wherein said estimation SOC calculates based on described at least one voltage drop model parameter.

37. according to the system of claim 29, also comprises display, and described display is configured to show described estimation SOC.

38. 1 kinds of computer-readable mediums, comprise code segment, and described code segment, when being carried out by processor, makes described processor:

Operational mode based on described battery is selected equivalent-circuit model from represent the storehouse of equivalent-circuit model of battery; And

39. according to the computer-readable medium described in claim 38, also comprises code segment, and described code segment makes described processor:

Based on selected equivalent model, calculate at least one voltage drop model parameter, wherein said SOC calculates based on described at least one voltage drop model parameter.

40. according to the computer-readable medium described in claim 38, and at least one in wherein said equivalent-circuit model comprises lag element, and described code segment also makes described processor:

Based on selected equivalent model, calculate at least one voltage drop model parameter; And