Lithium battery state of charge estimation method based on nonlinear singular cell model

Abstract

The application provides a lithium battery charge state estimation method based on a nonlinear singular battery model, which firstly establishes a singular model of the battery based on a second-order RC equivalent circuit of the lithium battery, then judges whether the established singular model is observable, for the singular model which is observable, designs each auxiliary parameter matrix in a given structure H∞ observer system according to each matrix coefficient, and constructs a linear matrix inequality for solving, so as to finally calculate each parameter of the observer, and finally fuses calculation, estimates the battery SOC based on the solved observer model and the current size and open circuit voltage input in the battery operation process. The application models the lithium battery second-order RC physical model through the nonlinear generalized state space model, reduces the modeling error of the model, and has a better response to the pulse current input; and based on the model, a battery SOC estimation method is provided, and the robustness of the estimation algorithm is improved through the H∞ method.

Description

Technical Field

[0001] This invention belongs to the field of lithium battery design technology, specifically relating to a method for estimating the state of charge of a lithium battery based on a nonlinear singular battery model. Background Technology

[0002] As the energy source for new energy vehicles, lithium batteries are widely used due to their advantages in energy storage efficiency, charging and discharging speed, and lifespan. The State of Charge (SOC) measures the percentage of remaining battery capacity relative to its rated capacity, allowing for accurate measurement of remaining battery capacity and thus estimation of the remaining driving range of new energy vehicles. However, the SOC cannot be directly obtained from sensors; sensors can only directly measure the open-circuit voltage (OCV), which is indirectly related to SOC. A simple method is to linearize the relationship between OCV and SOC to establish a linear state-space model. However, this method suffers from significant errors as SOC approaches zero due to increased nonlinearity. Therefore, most existing SOC estimation methods are based on the nonlinear relationship between OCV and SOC.

[0003] Singular system models, possessing both differential equations and algebraic constraint equations, can accurately represent physical models and their responses. Compared to traditional state-space models, their algebraic constraints provide impulse responses, making them widely used in establishing chemical and circuit models. Currently, the common approach to estimating nonlinear singular systems is through sliding mode state observers. While the sliding mode method is applicable to many fields due to its strong robustness and structural simplicity, its drawbacks are also significant. The jitter introduced by the sliding mode term adds unnecessary oscillations to the system, especially when the system is disturbed or the modeling is inaccurate, which can exacerbate the oscillations, significantly reducing the performance of the state estimation algorithm and even causing it to fail to converge. Therefore, there is an urgent need for a practical, simple, and accurate SOC estimation method based on nonlinear battery generalized systems that balances system stability, response characteristics, robustness, and performance. Summary of the Invention

[0004] Based on the above problems, the purpose of this invention is to propose a lithium battery state of charge estimation method based on a strange battery model. This method can accurately estimate the SOC value of the battery under various complex operating conditions, and has certain robustness, convergence speed and accuracy.

[0005] A method for estimating the state of charge of a lithium battery based on a singular battery model includes:

[0006] Step S1: Establish a singular model of the battery based on the second-order RC equivalent circuit of the lithium battery;

[0007] Step S2: Determine whether the established singular model is observable. If the observability condition is met, proceed to step S3.

[0008] Step S3: Based on the matrix coefficients of the singular model in Step S1, design the auxiliary parameter matrices in the given structure H∞ observer system, and construct linear matrix inequalities to solve them, thereby finally calculating the parameters of the observer.

[0009] Step S4: Fusion calculation, based on the observer model solved in step S2 and the current magnitude and open-circuit voltage input during battery operation, to estimate the battery SOC.

[0010] The singular model in step S1 is as follows:

[0011]

[0012] Where z represents the battery's state of charge (SOC); i represents the battery's charging and discharging current; v c The polarization effect capacitance C c Voltage across terminals; v oc Indicates the battery open-circuit voltage; v t ω represents the measurable terminal voltage; ω represents the modeling error and the uncertainty caused by disturbances; D ω The coefficient matrix represents the uncertainties; the superscript symbol ⊥ indicates matrix transpose; v(z) represents the nonlinear relationship between OCV and SOC based on experimental data fitting; according to Kirchhoff's laws, the matrices in the formula are calculated as follows:

[0013]

[0014] Among them, R e R represents the conduction resistance. c R represents the diffusion resistance. t Indicates terminal resistance, C c The capacitance C represents the polarization effect. n Battery open circuit voltage v oc The nominal capacitance.

[0015] Step S2 is specifically described as follows:

[0016] The observer structure used to determine whether the established singular model is observable is as follows:

[0017]

[0018] Where ξ1 and ξ2 represent the dynamic parameters inside the observer; These represent the state estimates in the singular battery system model; v t The measured battery terminal voltage is represented by N, J, F, P, and Q, which are the coefficient matrices to be solved, where N is the system matrix of the observer, J is the input matrix of the observer, P is the output matrix, F is the dynamic feedback matrix, and Q is the output feedback matrix; the input matrix J needs to satisfy the condition ΦE=0; T is the transformation matrix to be solved, which is the transformation matrix of the system's singular model to the observer (that is, by multiplying this matrix on the right by the singular model shown in formula (1), the singular model of the system can be transformed into the observer dynamic shown in formula (4); Ψ is the auxiliary parameter matrix, calculated as follows:

[0019]

[0020] B,D,D u Same as in the singular system equation (1) of the battery.

[0021] Determine whether the rank of the following observability matrices is equal to the system dimension n, i.e.

[0022]

[0023] The value of the auxiliary output matrix is If the equation holds true, proceed to step S3; if the equation does not hold true, the singular model does not satisfy the observability requirement, and its state variables cannot be estimated using this method.

[0024] Step S3 is specifically described as follows:

[0025] Step S3-1: To ensure the stability of the observer, the established linear matrix must have a feasible solution. The following linear matrix inequality is established:

[0026]

[0027] Where X, X1, X2, Y3 are the unknowns to be solved, and X is a symmetric positive definite matrix; σ>0 is a constant less than 1; γ is the upper bound of the desired L2 gain of the noise; (*) is a matrix block generated by the symmetry of the linear matrix inequality; to stabilize the estimation error of the observer, a stable threshold matrix is ​​taken. Where λ represents the Lipschitz coefficients of the OCV-SOC nonlinear relationship, and the matrices are calculated as follows:

[0028]

[0029] In the formula, O 2×3 I1 represents a 2×3 zero matrix; I2 and I3 represent identity matrices with subscript dimensions. R is the parameter matrix to be solved, R has a dimension of 2×3, and satisfies The rank is n;

[0030] Step S3-2: If the linear matrix inequality has no feasible solution, you can try changing the candidate parameter matrix R or try increasing the noise gain γ and substituting it back into formula (5)-(6) to solve until the linear matrix inequality shown in formula (5) has a feasible solution.

[0031] Step S3-3: Based on the above linear matrix inequalities, if a feasible solution exists, the values ​​of the unknown matrices X, X1, X2, Y3 are obtained through a semidefinite programming solver (such as SDPT3 or MOSEK solvers), thereby further obtaining Y1 = X -1 X1,Y2=X -1 X2;

[0032] Step S3-4: Calculate the observer parameter matrix as follows:

[0033]

[0034] The intermediate parameters are calculated as follows:

[0035]

[0036] In the formula, K represents the solution weight coefficient matrix, and the superscript + indicates the Moore–Penrose generalized inverse operation;

[0037] Step S4 includes the following steps:

[0038] Step S4-1: For the state variables ξ = [ξ1, ξ2] in the observer (3) ⊥ Perform initialization;

[0039] Step S4-2: Using ξ = [ξ1, ξ2] ⊥ The battery state variables are estimated using the measured input current value i to obtain the SOC value at that moment. The derivatives of the state variables are then calculated based on the measured current input and the battery open-circuit voltage OCV.

[0040] Step S4-3: Using numerical methods such as the Euler method, determine the value of the internal state ξ output by the observer at the previous time step (i.e., the observer determined by formula (3)) and its derivative. The value of ξ at the next time step is then calculated.

[0041] Step S4-4: Repeat steps S4-2 and S4-3 to update the battery state values ​​and obtain the real-time SOC.

[0042] The beneficial effects of this invention are:

[0043] This invention models the second-order RC physical model of a lithium battery using a nonlinear generalized state-space model, reducing modeling errors and providing better response to pulse current inputs. Based on this model, a battery SOC estimation method is proposed, which improves the robustness of the estimation algorithm through the H∞ method: that is, it can still maintain the accuracy of SOC estimation even under various external disturbances and certain modeling errors. Attached Figure Description

[0044] Figure 1 This is a flowchart illustrating the implementation of a lithium battery charge state estimation method based on a singular battery model in this invention.

[0045] Figure 2 This is a second-order RC equivalent circuit model of a lithium battery in this invention;

[0046] Figure 3 The fitted OCV-SOC relationship curve in this embodiment of the invention.

[0047] Figure 4 This is a dynamic discharge current curve in an embodiment of the present invention;

[0048] Figure 5 In the embodiments of the present invention, when adding Figure 4 Battery charge estimation curve under discharge current conditions;

[0049] Figure 6 In the embodiments of the present invention, when adding Figure 4 Battery charge estimation error curve under discharge current conditions and with 0.03V gain random noise;

[0050] Figure 7 In the embodiments of the present invention, when adding Figure 4 Battery charge estimation error curve under discharge current conditions and with a 10% OCV-SOC modeling error. Detailed Implementation

[0051] The invention will be further explained below with reference to the accompanying drawings and specific implementation examples.

[0052] like Figure 1 As shown, a method for estimating the state of charge of a lithium battery based on a singular battery model includes:

[0053] Step S1: Establish a singular model of the battery based on the second-order RC equivalent circuit of the lithium battery;

[0054] The singular model in step S1 is as follows:

[0055]

[0056] Where z represents the battery's state of charge (SOC); i represents the battery's charging and discharging current; v c The polarization effect capacitance C c Voltage across terminals; v oc Indicates the battery open-circuit voltage; v t ω represents the measurable terminal voltage; ω represents the modeling error and the uncertainty caused by disturbances; D ω The coefficient matrix represents the uncertainties; the superscript symbol ⊥ indicates matrix transpose; v(z) represents the nonlinear relationship between OCV and SOC based on experimental data fitting; according to Kirchhoff's laws, the matrices in the formula are calculated as follows:

[0057]

[0058] Among them, R e R represents the conduction resistance. c R represents the diffusion resistance. t Indicates terminal resistance, C c The capacitance C represents the polarization effect. n Battery open circuit voltage v oc The nominal capacitance.

[0059] Second-order RC model of lithium battery, such as Figure 2 As shown, the battery parameters in this battery model are selected as follows:

[0060] C n =18000F,C c =200F,R e =0.003Ω,R c =0.003Ω,R t =0.001Ω

[0061] By Kirchhoff's laws, we can obtain the following continuous singular state-space model:

[0062]

[0063] v t =[0 0.5 0.5][zv c v oc ] T +0.0025i

[0064] To obtain the SOC-OCV relationship, it is necessary to obtain the actual parameters and OCV-SOC data of the battery through charge and discharge experiments, and then obtain the OCV-SOC fitting curve of the battery through polynomial interpolation, thereby establishing a nonlinear singular system model of the battery.

[0065] Specifically, there is a non-linear relationship between OCV and SOC. For the SOC-OCV relationship curve, after obtaining the relationship data experimentally, the curve can be obtained using interpolation methods such as Lagrange interpolation. In this embodiment, a method such as... Figure 3 The nonlinear fitting function shown is in the following form:

[0066] v(z) = 0.1689z 4 +0.07667z 3 -1.433z 2 +2.54z +2.81

[0067] Since SOC is bounded and lies between 0 and 1, its Lipschitz coefficient can be calculated as λ = 2.54.

[0068] Step S2: Determine whether the established singular model is observable. If the observability condition is met, proceed to step S3. Specifically:

[0069] The observer structure used to determine whether the established singular model is observable is as follows:

[0070]

[0071] Where ξ1 and ξ2 represent the dynamic parameters inside the observer; These represent the state estimates in the singular battery system model; v t The measured battery terminal voltage is represented by N, J, F, P, and Q, which are the coefficient matrices to be solved, where N is the system matrix of the observer, J is the input matrix of the observer, P is the output matrix, F is the dynamic feedback matrix, and Q is the output feedback matrix; the input matrix J needs to satisfy the condition ΦE=0; T is the transformation matrix to be solved, which is the transformation matrix of the system's singular model to the observer (that is, by multiplying this matrix on the right by the singular model shown in formula (1), the singular model of the system can be transformed into the observer dynamic shown in formula (4); Ψ is the auxiliary parameter matrix, calculated as follows:

[0072]

[0073] B,D,D u Same as in the singular system equation (1) of the battery.

[0074] Determine whether the rank of the following observability matrices is equal to the system dimension n, i.e.

[0075]

[0076] The value of the auxiliary output matrix is If the equation holds true, proceed to step S3; if the equation does not hold true, the singular model does not satisfy the observability requirement, and its state variables cannot be estimated using this method.

[0077] For the generalized battery system model, there exists an input parameter matrix Φ = [0 0 1] such that ΦE = 0. 1×3 Since n=3, the auxiliary output matrix can be calculated. Right now Since the system is established, it is observable, and the next step, S3, can be performed.

[0078] Step S3: Based on the matrix coefficients of the singular model in Step S1, design the auxiliary parameter matrices in the given structure H∞ observer system, and construct linear matrix inequalities to solve them, thereby finally calculating the parameters of the observer; specifically:

[0079] Step S3-1: To ensure the stability of the observer, the established linear matrix must have a feasible solution. The following linear matrix inequality is established:

[0080]

[0081] Where X, X1, X2, Y3 are the unknowns to be solved, and X is a symmetric positive definite matrix; σ>0 is a constant less than 1; γ is the upper bound of the desired L2 gain of the noise; (*) is a matrix block generated by the symmetry of the linear matrix inequality; to stabilize the estimation error of the observer, a stable threshold matrix is ​​taken. Where λ represents the Lipschitz coefficients of the OCV-SOC nonlinear relationship, and the matrices are calculated as follows:

[0082]

[0083] In the formula, O 2×3 I1 represents a 2×3 zero matrix; I2 and I3 represent identity matrices with subscript dimensions. R is the parameter matrix to be solved, R has a dimension of 2×3, and satisfies The rank is n;

[0084] Step S3-2: If the linear matrix inequality has no feasible solution, you can try changing the candidate parameter matrix R or try increasing the noise gain γ and substituting it back into formula (5)-(6) to solve until the linear matrix inequality shown in formula (5) has a feasible solution.

[0085] Step S3-3: Based on the above linear matrix inequalities, if a feasible solution exists, the values ​​of the unknown matrices X, X1, X2, Y3 are obtained through a semidefinite programming solver (such as SDPT3 or MOSEK solvers), thereby further obtaining Y1 = X -1 X1,Y2=X-1 X2;

[0086] Step S3-4: Calculate the observer parameter matrix as follows:

[0087]

[0088] The intermediate parameters are calculated as follows:

[0089]

[0090] In the formula, K represents the solution weight coefficient matrix, and the superscript + indicates the Moore–Penrose generalized inverse operation;

[0091] For the nonlinear singular battery system obtained in steps S1 and S2, Ψ can be calculated as the auxiliary parameter matrix.

[0092] Setting parameter matrix Make the matrix The ranks are full.

[0093] The intermediate parameter matrices obtained by solving are as follows:

[0094]

[0095] Choosing the maximum noise gain as γ = 1.15, we substitute it into the solution of the linear matrix inequality and finally obtain the result through equations (6) and (7):

[0096]

[0097] Step S4: Fusion calculation, based on the observer model solved in step S2 and the current magnitude and open-circuit voltage input during battery operation, to estimate the battery SOC; including the following steps:

[0098] Step S4-1: For the state variables ξ = [ξ1, ξ2] in the observer (3) ⊥ Perform initialization;

[0099] Step S4-2: Using ξ = [ξ1, ξ2] ⊥ The battery state variables are estimated using the measured input current value i to obtain the SOC value at that moment. The derivatives of the state variables are then calculated based on the measured current input and the battery open-circuit voltage OCV.

[0100] Step S4-3: Using numerical methods such as the Euler method, determine the value of the internal state ξ output by the observer at the previous time step (i.e., the observer determined by formula (3)) and its derivative. The value of ξ at the next time step is then calculated.

[0101] Step S4-4: Repeat steps S4-2 and S4-3 to update the battery state values ​​and obtain the real-time SOC.

[0102] The method of the present invention will be numerically simulated below.

[0103] The initial conditions of the battery system are set as follows: The initial state is [ξ1,ξ2]=[0,0]. Figure 4 The dynamic stress test current (DST current) input shown is used to verify the stability of the method of the present invention, such as... Figure 5 As shown.

[0104] To verify the anti-interference capability of the method of this invention, simulation results were obtained after incorporating random measured voltage noise with a maximum gain of 0.03V, as shown below. Figure 6 As shown; since the OCV-SOC characteristics are obtained through fitting, there may be a large fitting error. Based on this issue, Figure 7 The performance curves of the battery SOC estimation algorithm were verified under the condition that there is a maximum random error of 10% in the OCV-SOC fitting curve.

[0105] Simulation results show that, within a certain range of interference and a large modeling error, the method of the present invention can achieve good robustness and response speed.

Claims

1. A method for estimating the state of charge of a lithium battery based on a nonlinear singular battery model, characterized in that, include: Step S1: Establish a singular model of the battery based on the second-order RC equivalent circuit of the lithium battery; Step S2: Determine whether the established singular model is observable. If the observability condition is met, proceed to step S3. Step S3: Design the given structure based on the matrix coefficients of the singular model in Step S1. The auxiliary parameter matrices of the observer system are obtained, and linear matrix inequalities are constructed and solved to finally calculate the parameters of the observer. Step S4: Fusion calculation, based on the observer model solved in step S2 and the current magnitude and open-circuit voltage input during battery operation, to estimate the battery SOC; Specifically, step S3 is described as follows: Step S3-1: To ensure the stability of the observer, the established linear matrix must have a feasible solution. The following linear matrix inequality is established: (5) in, Let be the unknown quantity to be solved, and It is a symmetric positive definite matrix; Coefficient matrix representing uncertain factors; superscript symbol Indicates matrix transpose; A constant less than 1; For the desired noise Gain upper bound; A block of matrices generated by the symmetry of linear matrix inequalities; ,in The Lipschitz coefficients represent the nonlinear relationship between OCV and SOC, where each matrix is ​​calculated as follows: (6) In the formula, express A zero matrix of dimension; , The identity matrix representing the dimension of the subscripts; , Let be the parameter matrix to be solved. The dimension is And satisfy The rank is ; Step S3-2: If the linear matrix inequality has no feasible solution, then change the parameter matrix to be selected. Or amplify noise gain Substitute the solutions back into equations (5) and (6) until a feasible solution exists for the linear matrix inequality shown in equation (5). Step S3-3: Based on the above linear matrix inequalities, if a feasible solution exists, the unknown matrix is ​​obtained through a positive semidefinite programming solver. The value of is obtained, thus further obtaining ; Step S3-4: Calculate the observer parameter matrix as follows: (7) Among them, The system matrix of the observer, The input matrix of the observer, For the output matrix, For dynamic feedback matrix, To output the feedback matrix, The intermediate parameters for the transformation matrix to be solved are calculated as follows: (8) In the formula, Represents the solution weight coefficient matrix, with superscript... This represents the Moore–Penrose generalized inverse operation; Specifically, step S4 is described as follows: Step S4-1: For the state variables in the observer shown in formula (3) Perform initialization; Step S4-2: Through Compared with the input current measurement value The battery state variables are estimated to obtain the SOC value at that moment, and the derivatives of the state variables are calculated based on the measured current input and the battery open-circuit voltage OCV. ; Step S4-3: Based on the internal state output by the observer at the previous time step The value of its derivative Value for the next time step The value is solved; Step S4-4: Repeat steps S4-2 and S4-3 to update the battery state values ​​and obtain the real-time SOC.

2. The method for estimating the state of charge of a lithium battery based on a nonlinear singular battery model according to claim 1, characterized in that, The singular model in step S1 is as follows: (1) in, Indicates the battery's State of Charge (SOC). Indicates the battery charging and discharging current; Indicates polarization effect capacitance Voltage at both ends; Indicates the battery open-circuit voltage; This indicates that the terminal voltage can be measured; This represents the errors introduced by modeling and the uncertainties caused by disturbances; Coefficient matrix representing uncertain factors; superscript symbol Indicates matrix transpose; This represents the nonlinear function of OCV and SOC; according to Kirchhoff's laws, the matrices in the formula are calculated as follows: (2) in, Indicates conduction resistance. Indicates diffusion resistance. Indicates terminal resistance. Capacitance representing polarization effect, Battery open circuit voltage The nominal capacitance.

3. The method for estimating the state of charge of a lithium battery based on a nonlinear singular battery model according to claim 1, characterized in that, Step S2 is specifically described as follows: The observer structure used to determine whether the established singular model is observable is as follows: (3) in, Represents the dynamic parameters inside the observer; represent the state estimates in the battery singular system model; This indicates the measured battery terminal voltage; , , , , Let be the coefficient matrix to be solved, where The system matrix of the observer, The input matrix of the observer, For the output matrix, For dynamic feedback matrix, The output feedback matrix; the input matrix. need Meet the conditions ; Let be the transformation matrix to be solved; The auxiliary parameter matrix is ​​calculated as follows: (4) Determine whether the rank of the following observability matrices is equal to the system dimension. ,Right now The value of the auxiliary output matrix is... If the equation is true, proceed to step S3; if the equation is false, it means that the singular model does not satisfy the observability requirement, and the operation ends.

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