Lithium battery charge state estimation method based on self-adaptive double-fusion Kalman filtering

Abstract

The invention belongs to the technical field of state estimation, and relates to a lithium battery state-of-charge estimation method based on self-adaptive double-fusion Kalman filtering, which comprises the following steps: constructing an equivalent circuit model; acquiring battery parameters based on the equivalent circuit model; acquiring a battery measurement value and a measurement estimation value based on the battery parameters; based on an error absolute value of the measurement value and the measurement estimation value, performing estimation by adopting corresponding Kalman filtering to obtain a battery state-of-charge estimation value; and obtaining a battery state-of-charge estimation result based on the battery state-of-charge estimation value. According to the method, the advantages and disadvantages of EKF and XKF filtering are integrated, different estimation schemes are adopted in different time periods, and the estimation efficiency can be improved under the condition that the precision is guaranteed.

Description

Technical Field

[0001] The present invention belongs to the technical field of battery state estimation, and in particular relates to a lithium battery charge state estimation method based on an adaptive dual fusion Kalman filter. Background Art

[0002] With the continuous depletion of global fossil energy, environmental pollution and other issues have received much attention. Taking an electric car is a relatively energy-saving and environmentally friendly means of travel, and the production and sales of electric cars have received more attention. As an important power source for electric vehicles, power batteries are one of the most important components. The safety of power batteries has also become important at this moment. When using them, you need to pay attention to their operating temperature, current, and voltage at all times. The device that monitors and manages power batteries at all times is the battery management system (BMS). Through its interactive interface, users can clearly understand the current operating temperature, state of charge, remaining battery life and other information of the battery, while managing the charging and discharging of the battery. It can also help users determine whether the battery system is faulty and whether maintenance is required. Therefore, whether the battery management system is excellent is related to the efficiency and safety of electric vehicles. As the power source of electric vehicles, the performance of power battery packs has always been the focus of research. When using electric vehicle batteries, they need to work within a reasonable voltage, current, and temperature range. Therefore, it is necessary to effectively manage the use of electric batteries on electric vehicles. The level of the battery management system largely determines the performance of the power battery pack. Therefore, a real-time and efficient battery management system is very important. As an important power source for electric vehicles, power batteries are one of the most important components. After research by many scholars, it has become possible to estimate the state of charge of lithium batteries based on equivalent circuit models. First, through experiments, it is determined that the relevant lithium batteries have many filtering algorithms that have been born to improve the calculation accuracy of this method. These filters include extended Kalman filtering, unscented Kalman filtering, volumetric Kalman filtering, etc. and their many variants. From a mathematical point of view, these schemes still retain some problems of nonlinear systems, such as the extended Kalman filter is difficult to deal with the slow convergence speed and easy to fall into local misunderstandings. In addition, the contribution of these algorithms to the improvement of accuracy is unquestionable, but the overall computational efficiency of the estimation scheme is reduced, which is very unfriendly to the operation of the vehicle system and the information interaction between the vehicle-cloud platform. Therefore, there is an urgent need for a lithium battery state of charge estimation method based on an adaptive dual fusion Kalman filter. Summary of the invention

[0003] In order to solve the above technical problems, the present invention proposes a lithium battery state of charge estimation method based on adaptive dual fusion Kalman filtering, which can improve the estimation accuracy of SOC.

[0004] The present invention provides a method for estimating the state of charge of a lithium battery based on adaptive dual-fusion Kalman filtering, including:

[0005] Construct an equivalent circuit model;

[0006] Based on the equivalent circuit model, obtain battery parameters;

[0007] Based on the battery parameters, obtain battery measurement values and measurement estimated values;

[0008] Based on the absolute value of the error between the measurement value and the measurement estimated value, perform estimation using the corresponding Kalman filtering to obtain an estimated value of the state of charge of the battery;

[0009] Based on the estimated value of the state of charge of the battery, obtain an estimated result of the state of charge of the battery.

[0010] Optionally, based on the equivalent circuit model, obtaining battery parameters includes:

[0011] Use the least squares method to obtain the battery parameters of the equivalent circuit model, where the battery parameters include: the internal resistance, voltage, capacity, and charge and discharge efficiency of the battery.

[0012] Optionally, based on the battery parameters, obtaining battery measurement values and measurement estimated values includes:

[0013] Initialize the state parameters in the battery parameters and the battery equivalent circuit model to obtain the initialized state parameters;

[0014] Based on the initialized state parameters, initialize the error covariance;

[0015] According to the initialized error covariance, obtain the battery measurement estimated value.

[0016] Optionally, based on the absolute value of the error between the measurement value and the measurement estimated value, performing estimation using the corresponding Kalman filtering to obtain an estimated value of the state of charge of the battery includes:

[0017] Judge whether the absolute value of the error between the measurement value and the measurement estimated value is greater than a first voltage threshold. If it is not greater, use the extended Kalman filtering for estimation. If it is greater, use the exogenous Kalman filtering for iterative estimation.

[0018] Optionally, using the extended Kalman filtering for estimation to obtain an estimated value of the state of charge of the battery includes:

[0019] Perform a priori estimation on the battery parameters to obtain a priori estimated values;

[0020] Based on the a priori estimated values, update the Kalman gain;

[0021] Update the posterior error covariance according to the updated Kalman gain to obtain the current error covariance;

[0022] Obtain the state of charge (SOC) estimate of the battery according to the current error covariance.

[0023] Optionally, the current error covariance is also involved in the initialization of the error covariance at the next moment.

[0024] Optionally, using an exogenous Kalman filter for estimation, obtaining the state of charge (SOC) estimate of the battery includes:

[0025] Establish the observation equation and state equation of the battery;

[0026] Based on the observation equation and state equation, perform a priori estimation to obtain the a priori estimate;

[0027] Update the Kalman gain, posterior error covariance, and state estimate according to the a priori estimate;

[0028] Based on the updated information, determine whether the absolute value of the error between the battery measurement value and the measurement estimate value is greater than the second voltage threshold. If it is not greater than the second voltage threshold, perform local iteration; otherwise, directly obtain the error covariance;

[0029] Obtain the state of charge (SOC) estimate of the battery according to the error covariance.

[0030] Optionally, performing local iteration includes:

[0031] S1. Modify the noise covariance;

[0032] S2. Update the Kalman gain matrix based on the modified noise covariance;

[0033] S3. Update the posterior error covariance based on the updated Kalman gain matrix;

[0034] S4. Repeat steps S1 - S3 until the absolute value of the error between the battery measurement value and the measurement estimate value is less than the preset value, then stop the iteration.

[0035] Optionally, the observation equation is:

[0036]

[0037] The state equation is:

[0038]

[0039] where, x k+1 and x k represent the state variables of the system at times k + 1 and k respectively, l kThe input to the system at time k, v k ~(0, Q k );w k ~(0, R k ), R k and Q k represent the measurement noise covariance and the process noise covariance respectively, y k represents the measured state variable, A, B, C, D are system matrices, Φ k represents the data variable of the system, θ k represents the parameter variable of the system, a1, a2, a3, a4, a5 are the corresponding constant coefficients, y(k) is the output of the system at time k, y(k - 1) is the output of the system at time k - 1, y(k - 2) is the output of the system at time k - 2, l(k - 1) is the input of the system at time k - 1, l(k - 2) is the input of the system at time k - 2.

[0040] Optionally, based on the estimated value of the state of charge of the battery, the method for obtaining the estimated result of the state of charge of the battery is:

[0041]

[0042] U d,k =f(SOC k ) - U 1,k - U 2,k - R0I k + w k

[0043] where represents the state variable of the model; time constants τ1 = R1C1, τ2 = R2C2; R1 represents the activation polarization resistance, R2 represents the concentration polarization resistance, C1 represents the activation polarization capacitance, C2 represents the concentration polarization capacitance, η is the Coulomb efficiency; Q N is the maximum available capacity of the battery at the current temperature; I k-1 represents the current at time k - 1; f(SOC k ) represents the functional relationship between the open - circuit voltage U oc and the state of charge SOC; SOC k is the state of charge at time k; U 1,k is the electrochemical polarization voltage at time k; U 2,k is the concentration polarization voltage at time k, Δt is the time step, v k is the Gaussian process noise at time k, R0 is the internal resistance of the battery, I k is the current measured at time k, w k is the Gaussian measurement noise at time k, U d,k is the battery terminal voltage calculated at time k.

[0044] Compared with the prior art, the present invention has the following advantages and technical effects:

[0045] Based on considering the absolute value of the difference between the measured terminal voltage and the estimated terminal voltage, the present invention updates the error by involving an additional function, thereby improving the estimation accuracy of the SOC. Meanwhile, by integrating the advantages of the EKF and XKF filters and adopting different estimation schemes at different time periods, the estimation efficiency can be improved while ensuring the accuracy. It is of great significance for estimating and controlling the entire battery state, giving full play to the battery capacity, and improving the information transmission efficiency of the vehicle-cloud platform. BRIEF DESCRIPTION OF THE DRAWINGS

[0046] The drawings constituting a part of this application are used to provide a further understanding of this application. The schematic embodiments of this application and their descriptions are used to explain this application and do not constitute an improper limitation to this application. In the drawings:

[0047] Figure 1 is a schematic diagram of the second-order RC equivalent circuit model of the lithium-ion battery according to the embodiment of the present invention;

[0048] Figure 2 is a flowchart of the method for estimating the state of charge of a lithium battery based on adaptive dual-fusion Kalman filtering according to the embodiment of the present invention. DETAILED DESCRIPTION OF THE EMBODIMENTS

[0049] It should be noted that, without conflict, the embodiments in this application and the features in the embodiments can be combined with each other. The following will refer to the drawings and combine the embodiments to detail this application.

[0050] It should be noted that the steps shown in the flowchart of the drawings can be executed in a computer system such as a set of computer-executable instructions, and although the logical order is shown in the flowchart, in some cases, the steps shown or described can be executed in a different order than here.

[0051] The present invention proposes a method for estimating the state of charge of a lithium battery based on adaptive dual-fusion Kalman filtering, as Figure 2 shown, which specifically includes the following steps:

[0052] As Figure 1 shown, construct an equivalent circuit model;

[0053] Based on the equivalent circuit model, obtain battery parameters;

[0054] Based on the battery parameters, obtain battery measurement values and measurement estimated values;

[0055] Based on the absolute value of the error between the measurement value and the measurement estimated value, use the corresponding Kalman filter for estimation to obtain the estimated value of the state of charge of the battery;

[0056] Based on the estimated value of the battery state of charge, obtain the estimated result of the battery state of charge.

[0057] Specifically, as Figure 1 shown, establish an equivalent model, where U oc represents the open-circuit voltage of the battery, R0 represents the ohmic internal resistance, U d represents the battery terminal voltage, I represents the load current, U1 represents the activation polarization voltage of the battery, C1 represents the activation polarization capacitance, R1 represents the activation polarization resistance, U2 represents the concentration polarization voltage of the battery, C2 represents the concentration polarization capacitance, and R2 represents the concentration polarization resistance.

[0058] Based on Figure 1 the equivalent circuit model, establish the continuous-time state-space equation of the battery as follows:

[0059]

[0060] Furthermore, based on the equivalent circuit model, obtain the battery parameters including:

[0061] Adopt the least squares method to obtain the battery parameters of the equivalent circuit model. Among them, the battery parameters include: the internal resistance, voltage, capacity, and charge-discharge efficiency of the battery.

[0062] Specifically, parameter identification: Take the recursive least squares method with a forgetting factor as an example

[0063] Its basic calculation formula is as follows:

[0064] y k = Φ k θ k + ξ k #(2)

[0065] Among them, y k represents the output variable of the system, Φ k represents the data variable of the system, θ k represents the parameter variable of the system, and ξ k is a fixed zero-mean white noise. The system identification method is as shown in formula (3):

[0066]

[0067] Among them, λ is the forgetting factor, and its value range is generally 0.95 to 1. K Ls,k is the algorithm gain, and P Ls,k

[0068] is the error covariance matrix of the state estimation, and P Ls,k-1 is the error covariance matrix at the k-1 moment, is the transpose matrix of the data variable at time k, Φ Ls,k is the matrix of the data variable at time k, θ Ls,k is the parameter estimate at time k, θ Ls,k-1 is the parameter estimate at time k - 1, K Ls,k is the Kalman gain at time k. For formula (1)

[0069] Taking the Laplace transform gives:

[0070]

[0071] Its transfer function is:

[0072]

[0073] where U oc represents the open - circuit voltage of the battery, R0 represents the ohmic internal resistance, U d represents the terminal voltage of the battery, I represents the load current, C pa represents the value of the activation polarization capacitance, R pa represents the value of the activation polarization resistance, C pc represents the value of the concentration polarization capacitance, R pc represents the value of the concentration polarization resistance.

[0074] Performing a bilinear transformation on formula (5), let to obtain the discretized transfer function:

[0075]

[0076] where a1, a2, a3, a4, a5 are the corresponding constant coefficients, and the difference equation obtained from formula (6) is:

[0077] y(k) = U OC (k) - U t (k) = a1y(k - 1) + a2y(k - 2) + a3l(k) + a4l(k - 1) + a5l(k - 2)#

[0078] where y(k) is the system output and l(k) is the system input.

[0079] Then the identifiable battery model is:

[0080]

[0081] Furthermore, based on the battery parameters, obtaining the battery measurement value and measurement estimate value includes:

[0082] Initializing the state parameters in the battery parameters and the battery equivalent circuit model to obtain the initialized state parameters;

[0083] Initialize the error covariance based on the initial state parameters;

[0084] Obtain the battery measurement estimate according to the initialized error covariance.

[0085] Specifically, the measurement value of the battery is obtained by actual measurement, usually real-time data obtained by sensors during the operation of the battery. Among them, the role of noise and error covariance initialization is to reflect the initial uncertainty of the system. In the initialization stage, empirical values or larger initial covariances are usually used, indicating a greater uncertainty about the initial state. At subsequent moments, the error covariance will be updated according to the Kalman gain.

[0086] More specifically, state initialization: Initialize the parameters x0, P0, and R0 in the algorithm

[0087]

[0088] R0 = 200

[0089] Furthermore, based on the absolute value of the error between the measurement value and the measurement estimate, corresponding Kalman filtering is used for estimation. Obtaining the state of charge (SOC) estimate of the battery includes:

[0090] Judge whether the absolute value of the error between the measurement value and the measurement estimate is greater than the first voltage threshold. If it is not greater, extended Kalman filtering is used for estimation. If it is greater, exogenous Kalman filtering is used for iterative estimation.

[0091] Specifically, adaptively judge whether to perform iteration using the absolute value of the error between the measurement estimate and the measurement value. Set m as the first voltage error threshold, m = 0.01V. If Then iteration is not required, and extended Kalman filtering (EKF) is used to improve the calculation efficiency. If Indicates that the error exceeds the threshold, and exogenous Kalman filtering (XKF) needs to be used.

[0092] Furthermore, using extended Kalman filtering for estimation, obtaining the state of charge (SOC) estimate of the battery includes:

[0093] Perform a priori estimation of the battery parameters to obtain the a priori estimate value;

[0094] Update the Kalman gain based on the a priori estimate value;

[0095] Update the posterior error covariance according to the updated Kalman gain to obtain the current error covariance;

[0096] Obtain the state of charge (SOC) estimate of the battery according to the current error covariance.

[0097] Specifically, state a priori estimation and state covariance a priori estimation:

[0098]

[0099] where represents the prior estimate of the state variable at time k, represents the prior estimate of the state variable at time k - 1; represents the estimated value of the measurement state quantity at time k; P k-1 , respectively represent the posterior estimate covariance at time k - 1 and the prior estimate covariance at time k; represents the estimated value of the measurement state quantity at time k; F and H represent the state transition matrices; Q k-1 and R k-1 are respectively the process noise covariance and the measurement noise covariance at time k - 1.

[0100] Corrected estimate:

[0101]

[0102] Update the Kalman gain:

[0103]

[0104] Update the posterior estimate covariance:

[0105]

[0106] where represents the posterior estimate of the state variable at time k; P k represents the posterior estimate covariance at time k; E is the identity matrix; K k is the Kalman gain at time k under this scheme.

[0107] Furthermore, the current error covariance also needs to participate in the initialization of the error covariance at the next moment.

[0108] Furthermore, establish the observation equation and the state equation of the battery;

[0109] Based on the observation equation and the state equation, perform prior estimation to obtain the prior estimate value;

[0110] According to the prior estimate value, update the Kalman gain, the posterior error covariance, and the state estimate value;

[0111] Based on the updated information, determine whether the absolute value of the error between the battery measurement value and the measurement estimate value is greater than the second voltage threshold. If it is not greater than the second voltage threshold, perform local iteration; otherwise, directly obtain the error covariance;

[0112] Obtain the estimated value of the battery state of charge based on the error covariance.

[0113] Further, in S1, correct the noise covariance;

[0114] In S2, update the Kalman gain matrix based on the corrected noise covariance;

[0115] In S3, update the posterior error covariance based on the updated Kalman gain matrix;

[0116] In S4, repeat steps S1 - S3 until the absolute value of the error between the battery measurement value and the measurement estimated value is less than the preset value, and then stop the iteration.

[0117] Specifically, establish an observer: establish the observation equation of the battery and the state equation of the battery (Formula 8):

[0118] Observation equation:

[0119]

[0120] where: x k+1 and x k represent the state variables of the system at times k + 1 and k respectively; l k is the system input; v k ~(0, Q k ); w k ~(0, R k ), R k and Q k represent the measurement noise covariance and the process noise covariance respectively; y k represents the measured state quantity. Where A, B, C, and D are system matrices.

[0121] State prior estimate and state covariance prior estimate:

[0122]

[0123] where represents the prior estimate of the state variable at time k, represents the prior estimate of the state variable at time k - 1; represents the estimated value of the measured state quantity at time k; P k-1 , represent the posterior estimate covariance at time k - 1 and the prior estimate covariance at time k respectively; represents the estimated value of the measured state quantity at time k; F and H represent the state transition matrices; Q k-1 and R k-1 are the process noise covariance and the measurement noise covariance at time k - 1 respectively.

[0124] (1) Set n as the second voltage error threshold, where n = 0.02V.

[0125] If then IXKF degenerates into the XKF scheme and directly execute step (3).

[0126] If it indicates that the error exceeds the threshold and the accuracy needs to be improved. First, step (2) needs to be executed, and then step (3).

[0127] (2) Iterative update:

[0128]

[0129] After step (2) is completed, proceed to step (3).

[0130] (3) Kalman gain matrix:

[0131]

[0132] Error covariance correction:

[0133]

[0134] System state correction:

[0135]

[0136] Furthermore, the method for establishing a battery discrete state space model and solving for SOC based on the battery second-order RC equivalent circuit model and Kirchhoff's law is as follows:

[0137]

[0138] U d,k = f(SOC k ) - U 1,k - U 2,k - R0I k + w k

[0139] Where represents the state variable of the model; time constants τ1 = R1C1, τ2 = R2C2; R1 represents the activation polarization resistance, R2 represents the concentration polarization resistance, C1 represents the activation polarization capacitance, C2 represents the concentration polarization capacitance, η is the Coulomb efficiency; Q N is the maximum available capacity of the battery at the current temperature; I k-1 represents the current at time k - 1; f(SOC k ) represents the functional relationship between the open-circuit voltage U oc and the state of charge SOC; SOC k is the state of charge at time k; U1,k is the electrochemical polarization voltage at time k; U 2,k is the concentration polarization voltage at time k, Δt is the time step, v k is the Gaussian process noise at time k, R0 is the internal resistance of the battery in the second-order equivalent circuit of the battery, I k is the measured current at time k, w k is the Gaussian measurement noise at time k, U d,k is the terminal voltage of the battery calculated at time k.

[0140] The above are only the preferred specific embodiments of the present application, but the protection scope of the present application is not limited thereto. Any changes or substitutions that can be easily thought of by those skilled in the art within the technical scope disclosed in the present application should be covered by the protection scope of the present application. Therefore, the protection scope of the present application should be subject to the protection scope of the claims.

Claims

1. A lithium battery state of charge estimation method based on an adaptive dual fusion Kalman filter, characterized in that: include: Construct an equivalent circuit model; Based on the equivalent circuit model, obtaining battery parameters; Based on the battery parameters, obtaining battery measurement values ​​and measurement estimates; Based on the absolute value of the error between the measured value and the estimated value, a corresponding Kalman filter is used to perform estimation to obtain an estimated value of the battery state of charge; Based on the battery state of charge estimation value, a battery state of charge estimation result is obtained.

2. The method for estimating the state of charge of a lithium battery based on an adaptive dual fusion Kalman filter according to claim 1, characterized in that: Based on the equivalent circuit model, obtaining battery parameters includes: The least square method is used to obtain the battery parameters of the equivalent circuit model, wherein the battery parameters include: internal resistance, voltage, capacity and charge and discharge efficiency of the battery.

3. The method for estimating the state of charge of a lithium battery based on an adaptive dual fusion Kalman filter according to claim 2, characterized in that: Based on the battery parameters, obtaining battery measurement values ​​and measurement estimates includes: Initializing the state parameters and the battery equivalent circuit model in the battery parameters to obtain the initialized state parameters; Initializing the error covariance based on the initialization state parameters; According to the initialized error covariance, a battery measurement estimate is obtained.

4. The method for estimating the state of charge of a lithium battery based on an adaptive dual fusion Kalman filter according to claim 1, characterized in that: Based on the absolute value of the error between the measured value and the estimated value, a corresponding Kalman filter is used to perform estimation, and obtaining the estimated value of the battery state of charge includes: It is determined whether the absolute value of the error between the measured value and the measured estimated value is greater than a first voltage threshold; if not, an extended Kalman filter is used for estimation; if greater, an exogenous Kalman filter is used for iterative estimation.

5. The method for estimating the state of charge of a lithium battery based on an adaptive dual fusion Kalman filter according to claim 4, characterized in that: The extended Kalman filter is used for estimation to obtain the estimated value of the battery state of charge, including: Performing a priori estimation on the battery parameters to obtain a priori estimation value; Based on the a priori estimate, updating the Kalman gain; According to the updated Kalman gain, the posterior error covariance is updated to obtain the current error covariance; A battery state of charge estimation value is obtained according to the current error covariance.

6. The method for estimating the state of charge of a lithium battery based on an adaptive dual fusion Kalman filter according to claim 5, characterized in that: The current error covariance also needs to participate in the initialization of the error covariance at the next moment.

7. The method for estimating the state of charge of a lithium battery based on an adaptive dual fusion Kalman filter according to claim 4, characterized in that: Using exogenous Kalman filtering for estimation, the estimated value of battery state of charge includes: Establish the observation equation and state equation of the battery; Based on the observation equation and the state equation, a priori estimation is performed to obtain a priori estimation value; According to the prior estimate, the Kalman gain, the a posteriori error covariance and the state estimate are updated; Based on the updated information, determine whether the absolute value of the error between the battery measurement value and the measurement estimate value is greater than a second voltage threshold, if not, perform local iteration, otherwise directly obtain the error covariance; A battery state of charge estimation value is obtained according to the error covariance.

8. The method for estimating the state of charge of a lithium battery based on an adaptive dual fusion Kalman filter according to claim 7, characterized in that: Performing local iterations includes: S1, corrected noise covariance; S2, updating the Kalman gain matrix based on the corrected noise covariance; S3, updating the posterior error covariance based on the updated Kalman gain matrix; S4. Repeat steps S1-S3 until the absolute value of the error between the battery measurement value and the measurement estimation value is less than a preset value, and then stop iterating.

9. The method for estimating the state of charge of a lithium battery based on an adaptive dual fusion Kalman filter according to claim 7, characterized in that: The observation equation is: The state equation is: Among them, x k+1 and x k Represent the state variables of the system at time k+1 and time k respectively, l k is the input of the system at time k, v k ~(0,Q k );w k ~(0,R k ), R k and Q k Represent the measurement noise covariance and process noise covariance respectively, y k represents the measured state quantity, A, B, C, D are system matrices, Φ k represents the data variables of the system, θ k represents the parameter variables of the system, a1, a2, a3, a4, and a5 are the corresponding constant coefficients, y(k) is the output of the system at time k, y(k-1) is the output of the system at time k-1, y(k-2) is the output of the system at time k-2, l(k-1) is the input of the system at time k-1, and l(k-2) is the input of the system at time k-2.

10. The lithium battery state of charge estimation method based on adaptive dual fusion Kalman filtering according to claim 1, characterized in that: Based on the battery state of charge estimation value, a method for obtaining a battery state of charge estimation result is: U d,k =f(SOC k )-U 1,k -U 2,k -R0I k +w k in, represents the state variables of the model; time constants τ1=R1C1, τ2=R2C2; R1 represents the activation polarization resistance, R2 represents the concentration polarization resistance, C1 represents the activation polarization capacitance, C2 represents the concentration polarization capacitance, η is the coulomb efficiency; Q N is the maximum available capacity of the battery at the current temperature; I k-1 represents the current at time k-1; f(SOC k ) represents the open circuit voltage U oc Functional relationship with state of charge SOC; SOC k is the charge state at time k; U 1,k is the electrochemical polarization voltage at time k; U 2,k is the concentration polarization voltage at time k, Δt is the time step, v k is the Gaussian process noise at time k, R0 is the internal resistance of the battery, I k is the current measured at time k, w k is the Gaussian measurement noise at time k, U d,k is the battery terminal voltage obtained by calculation at time k.

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