A Nonlinear Modeling Method for Lithium-Ion Batteries under Complex Working Conditions

Abstract

The present invention provides a non-linear modeling method for lithium-ion batteries under complex working conditions, belonging to the technical field of lithium-ion batteries, and solving the technical problems of poor identification effect of least squares algorithms and low accuracy of traditional equivalent circuit models under complex working conditions. The technical solution is as follows: It includes the following steps: Step 1) Test the battery under DST and FUDS working conditions; Step 2) Establish a lithium-ion battery NL-ECM model; Step 3) Construct the algorithm flow of GWO; Step 4) Make improvements on the basis of GWO to construct IGWO; Step 5) Use IGWO to identify the model parameters and use the identification results to predict the terminal voltage under various working conditions. The beneficial effects of the present invention are as follows: The present invention introduces a block structure on the basis of ECM to construct a NL-ECM model, uses IGWO for parameter identification under DST working conditions, the algorithm has a fast convergence speed, the identification result is accurate, and the accuracy of the constructed model is significantly higher than that of ECM.

Description

Technical Field

[0001] The present invention relates to the technical field of lithium-ion batteries, and in particular, to a non-linear modeling method for lithium-ion batteries under complex working conditions. Background Art

[0002] With the continuous development of industrialization, the problems of global environmental damage and energy shortage have become increasingly serious, and the research on clean and renewable energy has become the focus at present. Electric vehicles with lithium-ion batteries as the core have received more and more attention, and their actual operating conditions are closely related to the data feedback by the Battery Management System (BMS). The State of Charge (SOC) of the battery is an important parameter of the BMS. Since the operating conditions of the vehicle are unpredictable and the operating environment is diverse, it is difficult to directly measure the SOC. Estimating the SOC through a battery model is the mainstream method. However, accurately modeling the battery model is an important problem.

[0003] The Equivalent Circuit Model (ECM) is one of the most widely used models, which describes the electrochemical polarization effect and concentration difference polarization effect of the battery through basic electrical components such as capacitors and resistors. However, in the actual process, the battery is not a linear system, and both the input and output will show non-linearity due to external disturbances, thus affecting the model accuracy. How to perform non-linear modeling on it has gradually become a research hotspot. At the same time, the traditional identification method of the battery model is mainly the least squares type identification algorithm, which constructs an identification model by writing the parameter vector and the information vector to perform identification. This method has strong applicability, but the model prediction error is large under complex working conditions. Using a swarm intelligence optimization algorithm to identify the parameters of the battery model is a novel identification method. The intelligent algorithm constructed according to the survival mode of biological populations in nature can effectively find the optimal solution of the problem. However, when dealing with different problems, the algorithm effect is not always excellent, and it is easy to fall into local optimum during the iteration process, thus affecting the convergence speed and accuracy of the algorithm. For different problems and models, it is necessary to improve the original algorithm to achieve the best identification effect.

[0004] How to solve the above technical problems is the subject faced by the present invention. Summary of the Invention

[0005] The object of the present invention is to provide a non - linear modeling method for lithium - ion batteries under complex working conditions. In the actual modeling process of lithium - ion batteries, the input and output of the model do not show linearity, but rather show non - linearity due to various disturbances. Considering the above factors, the present invention is based on a block structure, improves on the basis of the ECM, represents the model input with a static non - linear function, and adds colored noise disturbance to the model output to construct a non - linear equivalent circuit model of lithium - ion batteries (Nonlinear Equivalent Circuit Model, NL - ECM) so as to simulate the non - linear characteristics of lithium - ion batteries in actual situations and achieve accurate modeling. After compensating the model output with the open - circuit voltage, a more accurate terminal voltage value can be obtained. At the same time, in order to accurately identify the model parameters, the present invention uses the Grey Wolf Optimization (GWO) algorithm, a typical swarm intelligence optimization algorithm, and introduces various improvements to propose an improved Grey Wolf Optimization (IGWO) algorithm, which improves the optimization ability and convergence speed of the algorithm. The results show that all indicators of the improved algorithm are better than the original algorithm. The IGWO is used to segmentally identify the complex working condition of the Dynamic Stress Test (DST), and the model is verified under two working conditions of DST and the Federal Urban Driving Schedule (FUDS). The results show that under these two complex working conditions, the accuracy of the NL - ECM is significantly higher than that of the ECM.

[0006] The present invention is realized by the following measures: a non - linear modeling method for lithium - ion batteries under complex working conditions, which specifically includes the following steps:

[0007] Step 1) Conduct DST and FUDS working condition tests on the battery;

[0008] Step 2) Establish the NL - ECM model of the lithium - ion battery;

[0009] Step 3) Construct the algorithm flow of GWO;

[0010] Step 4) Improve on the basis of GWO to construct IGWO;

[0011] Step 5) Use IGWO to identify the model parameters and predict the terminal voltage using the identification results under various working conditions;

[0012] As a non - linear modeling method for lithium - ion batteries under complex working conditions of the present invention, the specific content of the said step 2) includes the following:

[0013] The operating mechanism of a lithium-ion battery is based on chemical reactions and exhibits non-linear characteristics. The present invention proposes an NL-ECM modeling method, which introduces a block structure based on the traditional lithium-ion ECM model to simulate the non-linearity existing in the battery charging and discharging process and thus achieve accurate modeling.

[0014] The derivation of the battery ECM model is as follows:

[0015] Let U oc and U represent the open-circuit voltage and terminal voltage of the battery, and the voltages across C 1 and C 2 are represented by U 1 and U 2 respectively. R 0 is the ohmic internal resistance. R 1 and C 1 characterize the electrochemical polarization effect during the rapid voltage change process; R 2 and C 2 characterize the concentration difference polarization reaction during the slow and stable voltage change process.

[0016] Establish a functional relationship:

[0017]

[0018] where I(t) is the current at time t, and SOC is the remaining capacity of the battery, which is defined as follows:

[0019]

[0020] Q n is the rated capacity of the battery.

[0021] Establish the discrete state-space expression of the system:

[0022]

[0023]

[0024] According to Kirchhoff's law, establish the transfer function for the ECM model:

[0025]

[0026] By using bilinear transformation to project the function from the s-plane to the z-plane, we can obtain:

[0027]

[0028]

[0029] where: τ 1 = R 1 C 1, τ 2 = R 2 C 2 , a = R 0 , b = τ 1 τ 2 , c = τ 1 + τ 2 , d = R 0 + R 1 + R 2 , e = R 0 (τ 1 + τ 2 ) + R 1 τ 1 + R 2 τ 2 。

[0030] Considering that the input current is non - linear due to disturbances during the measurement process, it is represented by a static non - linear function f(·) (in polynomial form):

[0031]

[0032] where I(z) is the input current, is a polynomial of order p, representing the input current affected by disturbances, and α t is the polynomial coefficient. In the present invention, p = 2.

[0033] There is an external disturbance w(z) at the battery output terminal voltage, i.e., colored noise. Assuming that the external disturbance is stationary and has a rational spectral density, the pole polynomial of the disturbance channel is the same as that of the process channel, and the zero polynomial has the same order as the pole polynomial, i.e.:

[0034]

[0035] h(·) is a non - linear function, where v(z) is white noise. Thus, the output expression of the battery NL - ECM model can be established:

[0036]

[0037] Assuming b 0 = 1, expand the above formula:

[0038]

[0039] Define the parameter vector θ and the information vector

[0040] θ = [a 1 , a 2 , b 1 , b 2 , γ 1 , γ 2,d 1 ,d 2 T ∈R n (12)

[0041]

[0042] Establish an identification model:

[0043]

[0044] As a non-linear modeling method for lithium-ion batteries under complex working conditions of the present invention, the specific content of step 3) is as follows:

[0045] GWO is a classic swarm intelligence optimization algorithm. It utilizes the hierarchical system existing in the group and simulates the hunting mechanism of the alpha wolf in the gray wolf group and behaviors such as encircling and hunting during the predation process, so as to achieve the purpose of optimization.

[0046] The alpha wolf is regarded as the gray wolf with the highest level. The beta wolf is the subordinate of the alpha wolf, and the delta wolf is subordinate to the beta and alpha. The remaining individuals are denoted as w. In the algorithm, the gray wolf alpha, as the optimal solution, the sub-optimal solution beta, and the re-optimal solution delta guide the w gray wolves to complete the hunting behavior, thereby realizing the iterative optimization of the algorithm. The hunting behavior includes two parts: encircling and attacking.

[0047] 1) Encircling

[0048] Let the size of the gray wolf population be set as N, and the search dimension be d. The process of encircling the prey is as follows:

[0049] D = |C·X p (t) - X(t)| (14)

[0050] X(t + 1) = X p (t) - A·D (15)

[0051] In the formula, t is the current iteration number, X p is the position vector of the prey, X is the position vector of the gray wolf, and D is the distance between the individual of the wolf pack and the prey; A is the attack coefficient of the gray wolf on the hunting object, and C is the cooperation coefficient. The calculation formulas are as follows:

[0052] A = 2a·r 1 - a (16)

[0053] C = 2·r 2 (17)

[0054] In the formula, r 1 and r 2 are random vectors, a is the convergence factor, a = 2 - 2t / t max , t max ​is the maximum number of iterations.

[0055] 2) Attack

[0056] After surrounding the prey, the grey wolves will attack the prey, and there is a position update during this process. The optimal wolf α will lead the wolf pack to approach the prey, and β and δ play an auxiliary role. The position update rules during the attack process are as follows:

[0057]

[0058]

[0059] where X a , X β and X δ are the position vectors of three leading wolves; D a , D β and D δ are the distance vectors between the leading wolves and the prey; X 1 , X 2 and X 3 represent the position vector updates of three wolves; X is the position vector update of the w grey wolf; A 1 , A 2 and A 3 are the attack coefficients of the grey wolves, and C 1 , C 2 and C 3 are the cooperation coefficients of the grey wolves. With the iterative update, the three optimal wolves in the population will replace the original leading wolves and continue to lead the rest of the individuals to approach the prey through the new leading wolves.

[0060] As a non-linear modeling method of lithium-ion batteries under complex working conditions of the present invention, the step 4) specifically includes the following contents:

[0061] Improvement strategy 1) Construct a new expression for the convergence factor. The coefficient A is used as the attack coefficient. When A≥1, the wolf pack expands the surrounding range, and the algorithm conducts a global search. When A<1, the wolf pack attacks the prey within the surrounded area, and the algorithm conducts a local search at this time. The coefficient A determines the optimization ability of the algorithm, and the convergence factor a directly affects the value of A. As the number of iterations increases, the convergence factor a decreases linearly continuously, but this change method does not adapt to the actual situation. During the optimization process in the early stage of the algorithm, a should decrease slowly to expand the search range of the grey wolf group. In the later stage, the convergence factor should decrease rapidly to prompt the group to narrow the search range. The present invention constructs a new expression for the convergence factor:

[0062] a = 2cos(2tπ / t max )(20)

[0063] As the number of iterations continues to increase, a shows a non-linear change, effectively balancing the shrinking trends in the early stage and the middle and late stages, thereby effectively improving the optimization ability of the algorithm.

[0064] Improvement strategy 2) introduces dynamic weights. In the original algorithm, the guiding degrees of wolves α, β, and δ are the same, but the position of the leading wolf is not necessarily the optimal solution and may also fall into a local optimum, resulting in a slow convergence rate of the algorithm. The present invention introduces dynamic weights and conducts dynamic weighted averaging during the optimization process to distinguish the contribution rates of the leading wolves and guide the position update of gray wolf individuals. The formula is as follows:

[0065] f = |f α +f β +f δ | (21)

[0066]

[0067] X(i + 1) = w 1 ·X 1 (i) + w 2 ·X 2 (i) + w 3 ·X 3 (i) (23)

[0068] f α 、f β and f δ represent the fitness values corresponding to three wolves, f is an intermediate variable, w 1 、w 2 and w 3 represent the influence weights of three gray wolves on position update. During the iteration process, the weights will change according to the actual situation, effectively avoiding the algorithm from falling into a local optimum.

[0069] Improvement strategy 3) introduces random perturbation. As the algorithm iterates, the algorithm gradually tends to converge, and it is difficult to jump out if it falls into a local optimum. For general evolutionary algorithms, an effective solution is to maintain the diversity of the population. The present invention introduces random perturbation to enhance the population diversity in the later stage of algorithm iteration, thereby jumping out of the local optimum. The expression is as follows:

[0070]

[0071] where r 3 is a random number in [0, 1], M is the perturbation probability, ub and lb are the upper and lower bounds of the gray wolf individual, G is the dimension of the gray wolf individual, and f(H(i + 1)) and f(X(i + 1)) represent the fitness function values corresponding to the perturbed individual H and the gray wolf X in the (i + 1)-th iteration process.

[0072] As a non - linear modeling method of lithium - ion batteries under complex working conditions of the present invention, the specific content of step 5) is as follows:

[0073] There are deviations in the model parameters of lithium - ion batteries under different SOCs. Therefore, the present invention performs segmented identification for each working condition. The SOC values of the DST working condition measured in step 1) are calculated by the ampere - hour integration method, and the entire charge - discharge process is segmented. The initial SOCs are 1, 0.9, 0.8, …, 0.1 respectively, and it is divided into 10 segments in total. Taking the discharge process with an initial SOC = 0.7 as an example, the GWO and IGWO are used for parameter identification to compare the effects before and after the algorithm improvement. The fitness function F(x) is established by using the sum of the squares of the errors between the experimentally measured terminal voltage and the actual terminal voltage:

[0074]

[0075] In the formula, y(k) is the measured voltage, is the predicted voltage. By performing parameter identification on 10 groups of charge - discharge processes, the model parameters under different initial SOCs can be obtained. Then, through polynomial fitting, the variation curve of the model parameters with SOC can be obtained. Substituting the fitted parameter results into the DST and FUDS working conditions for terminal voltage prediction, the accuracy of the model can be verified.

[0076] Compared with the prior art, the beneficial effects of the present invention are as follows:

[0077] (1) The present invention uses professional equipment to measure and study the two complex working conditions of DST and FUDS of the battery, with higher applicability and closer to the actual situation.

[0078] (2) The present invention considers the non - linear characteristics presented due to disturbances during the measurement process of lithium - ion batteries and the external disturbances existing in the output terminal voltage, introduces a block structure, and establishes an NL - ECM model.

[0079] (3) The present invention improves on the basis of the GWO algorithm, constructs a new expression of the convergence factor, and introduces a dynamic weight and random perturbation to propose the IGWO. It effectively enhances the diversity of the gray - wolf population and the ability of the algorithm to jump out of local extrema, and improves the optimization effect of the algorithm.

[0080] (4) The present invention uses the IGWO to perform segmented identification on the charge - discharge process of lithium - ion batteries, realizes the accurate modeling of the NL - ECM model of lithium - ion batteries, and the model accuracy is significantly higher than that of the ECM. BRIEF DESCRIPTION OF THE DRAWINGS

[0081] The drawings are used to provide a further understanding of the present invention, and constitute a part of the specification. They are used together with the embodiments of the present invention to explain the present invention, and do not constitute a limitation to the present invention.

[0082] Figure 1 Schematic diagram of the terminal voltage and current variation curves under DST conditions of the present invention.

[0083] Figure 2 Schematic diagram of the terminal voltage and current variation curves under FUDS conditions of the present invention.

[0084] Figure 3 Schematic diagram of the ECM model diagram of the present invention.

[0085] Figure 4 Schematic diagram of the NL-ECM model diagram of the present invention.

[0086] Figure 5 Schematic diagram of the GOW and IGWO iteration curves of the present invention.

[0087] Figure 6 Schematic diagram of the terminal voltage prediction curve and error curve under DST conditions of the present invention.

[0088] Figure 7 Schematic diagram of the terminal voltage prediction curve and error curve under FUDS conditions of the present invention. Detailed implementation manners

[0089] In order to make the objectives, technical solutions and advantages of the present invention clearer and more understandable, the present invention will be further described in detail below with reference to the accompanying drawings and embodiments. Of course, the specific embodiments described herein are only used to explain the present invention and are not used to limit the present invention.

[0090] See Figures 1 to 7 , this embodiment provides a non-linear modeling method for lithium-ion batteries under complex conditions, including the following steps:

[0091] Step 1) Test the battery under DST and FUDS conditions; the battery operates in an intermittent constant current discharge and rest mode, repeating the discharge until the voltage drops to the cut-off voltage, and perform two dynamic condition tests of DST and FUDS, and record the terminal voltage and current data. The results are as Figure 1 , Figure 2 shown.

[0092] Step 2) Establish an NL-ECM model for the lithium-ion battery;

[0093] Step 3) Construct the algorithm flow of GWO;

[0094] Step 4) Improve on the basis of GWO to construct IGWO;

[0095] Step 5) Use IGWO to identify the model parameters, and use the identification results to predict the terminal voltage under various conditions.

[0096] As a non - linear modeling method of lithium - ion batteries under complex working conditions in this embodiment, step 2) specifically includes the following steps:

[0097] The operating mechanism of lithium - ion batteries is based on chemical reactions and has non - linear characteristics. In this embodiment, an NL - ECM modeling method is proposed. By introducing a block structure on the basis of the traditional lithium - ion ECM model, the non - linearity existing in the battery charge - discharge process is simulated, thus achieving accurate modeling.

[0098] The derivation of the battery ECM model is as follows:

[0099] The ECM structure is as Figure 3 shown. Use U oc , U to represent the open - circuit voltage and terminal voltage of the battery, and the voltages at both ends of C 1 , C 2 are represented by U 1 , U 2 respectively. R 0 is the ohmic internal resistance. R 1 , C 1 characterize the electrochemical polarization reaction, the process of rapid voltage change; R 2 , C 2 characterize the concentration - difference polarization effect, the process of slow and stable voltage change.

[0100] Establish a functional relationship:

[0101]

[0102] where I(t) is the current at time t, and SOC is the remaining capacity of the battery, which is defined as follows:

[0103]

[0104] Qn is the rated capacity of the battery.

[0105] Establish the discrete - state - space expression of the system:

[0106]

[0107]

[0108] According to Kirchhoff's law, establish a transfer function for the ECM model:

[0109]

[0110] By using bilinear transformation to project the function from the s - plane to the z - plane, we can get:

[0111]

[0112]

[0113] where: τ 1 = R 1 C 1 , τ 2 = R 2 C 2 , a = R 0 , b = τ 1 τ 2 , c = τ 1 + τ 2 , d = R 0 + R 1 + R 2 , e = R 0 (τ 1 + τ 2 ) + R 1 τ 1 + R 2 τ 2 .

[0114] Considering that the input current is non - linear due to disturbances during the measurement process, it is represented by a static non - linear function f(·) (in polynomial form):

[0115]

[0116] where I(z) is the input current, is a polynomial of order p, representing the input current affected by disturbances, and α t is the polynomial coefficient. In this embodiment, p = 2.

[0117] There is an external disturbance w(z) at the battery output terminal voltage, i.e., colored noise. Assuming that the external disturbance is stationary and has a rational spectral density, the pole polynomial of the disturbance channel is the same as that of the process channel, and the zero polynomial has the same order as the pole polynomial, i.e.:

[0118]

[0119] h(·) is a non - linear function, where v(z) is white noise. Thus, the output expression of the battery NL - ECM model can be established:

[0120]

[0121] The model structure is as Figure 4 shown. After compensating y(z) with the open - circuit voltage U oc , the terminal voltage output value U(z) can be obtained. Assuming b 0 = 1, expand the above formula:

[0122]

[0123] Define the parameter vector θ and the information vector

[0124] θ = [a 1 , a 2 , b 1 , b 2 , γ 1 , γ 2 , d 1 , d 2 T ∈R n (12)

[0125]

[0126] Establish an identification model:

[0127]

[0128] As a non - linear modeling method of lithium - ion batteries under complex working conditions in this embodiment, step 3) specifically includes the following steps:

[0129] GWO is a classical swarm intelligence optimization algorithm. It utilizes the hierarchical system existing in the group and simulates the hunting mechanism of the alpha wolf in the gray wolf group and behaviors such as encirclement and hunting during the predation process, so as to achieve the purpose of optimization.

[0130] The alpha wolf is regarded as the gray wolf with the highest level. The beta wolf is a subordinate of the alpha wolf, and the delta wolf is subordinate to the beta and alpha. The remaining individuals are denoted as w. In the algorithm, the gray wolf alpha, as the optimal solution, together with the sub - optimal solution beta and the re - sub - optimal solution delta, guides the w gray wolves to complete the hunting behavior, thereby realizing the iterative optimization of the algorithm. The hunting behavior includes two parts: encirclement and attack.

[0131] 1) Encirclement

[0132] Set the scale of the gray wolf population as N and the search dimension as d. The process of encircling the prey is as follows:

[0133] D = |C·X p (t) - X(t)| (14)

[0134] X(t + 1) = X p (t) - A·D (15)

[0135] In the formula, t is the current iteration number, X p is the position vector of the prey, X is the position vector of the gray wolf, D is the distance between the individual of the wolf pack and the prey; A is the attack coefficient of the gray wolf on the hunting object, and C is the cooperation coefficient. The calculation formulas are as follows:

[0136] A = 2a·r 1 - a (16)​

[0137] C = 2·r 2 (17)

[0138] where r 1 and r 2 are random vectors, a is the convergence factor, a = 2 - 2t / t max , t max is the maximum number of iterations.

[0139] 2) Attack

[0140] After surrounding the prey, the gray wolves will attack the prey, and there is a position update during this process. The optimal wolf α will lead the wolf pack to approach the prey, and β and δ play an auxiliary role. The position update rules during the attack process are as follows:

[0141]

[0142]

[0143] where X a , X β and X δ are the position vectors of three leading wolves; D a , D β and D δ are the distance vectors between the leading wolves and the prey; X 1 , X 2 and X 3 represent the position vector updates of three wolves; X is the position vector update of the w gray wolf; A 1 , A 2 and A 3 are the attack coefficients of the gray wolves, C 1 , C 2 and C 3 are the cooperation coefficients of the gray wolves. As the iteration updates, the three wolves with the best population will replace the original leading wolves and continue to lead the rest of the individuals to approach the prey through the new leading wolves.

[0144] As a non - linear modeling method of lithium - ion batteries under a complex working condition in this embodiment, step 4) specifically includes the following steps:

[0145] Improvement Strategy 1) Construct a new expression for the convergence factor. Coefficient A is used as the attack coefficient. When A≥1, the wolf pack expands the encirclement range and the algorithm conducts global search. When A<1, the wolf pack attacks the prey within the encirclement and the algorithm conducts local search at this time. Coefficient A determines the optimization ability of the algorithm, and the convergence factor a directly affects the value of A. As the number of iterations increases, the convergence factor a decreases linearly. However, this change method does not adapt to the actual situation. During the early optimization process of the algorithm, a should decrease slowly to expand the search range of the gray wolf group. In the later stage, the convergence factor should decrease rapidly to prompt the group to narrow the search range. In this embodiment, a new expression for the convergence factor is constructed:

[0146] a = 2cos(2tπ / t max ) (20)

[0147] As the number of iterations continues to increase, a shows non-linear changes, effectively balancing the shrinking trends in the early stage and the middle and late stages, thus effectively improving the optimization ability of the algorithm.

[0148] Improvement Strategy 2) Introduce dynamic weights. In the original algorithm, the guiding degrees of wolves α, β, and δ are the same. However, the position of the leading wolf is not necessarily the optimal solution and may also fall into a local optimum, resulting in a slow convergence speed of the algorithm. In this embodiment, dynamic weights are introduced, and dynamic weighted averaging is performed during the optimization process to distinguish the contribution rates of the leading wolves and guide the gray wolf individuals to update their positions. The formula is as follows:

[0149] f = |f α +f β +f δ | (21)

[0150]

[0151] X(i + 1) = w 1 ·X 1 (i) + w 2 ·X 2 (i) + w 3 ·X 3 (i) (23)

[0152] f α 、f β and f δ represent the fitness values corresponding to three wolves. f is an intermediate variable, and w 1 、w 2 and w 3 represent the influence weights of three gray wolves on position update. During the iteration process, the weights will change according to the actual situation, effectively avoiding the algorithm from falling into a local optimum.

[0153] Improvement Strategy 3) Introduce random perturbation. As the algorithm iterates, the algorithm gradually converges. If it falls into a local optimum, it is difficult to jump out. For general evolutionary algorithms, an effective solution is to maintain the diversity of the population. In this embodiment, random perturbation is introduced to enhance the population diversity in the later stage of algorithm iteration, so as to jump out of the local optimum. The expression is as follows:

[0154]

[0155] where r 3 is a random number in [0, 1], M is the perturbation probability, ub and lb are the upper and lower bounds of the gray wolf individual, G is the dimension of the gray wolf individual, and f(H(i + 1)) and f(X(i + 1)) represent the fitness function values corresponding to the perturbed individual H and the gray wolf X in the (i + 1)-th iteration process.

[0156] As a non-linear modeling method of lithium-ion batteries under complex working conditions in this embodiment, step 5) specifically includes the following steps:

[0157] The lithium-ion battery model parameters deviate under different SOCs. Therefore, in this embodiment, each working condition is segmented and identified. The SOC value of the DST working condition measured in step 1) is calculated by the ampere-hour integration method, and the entire charge-discharge process is segmented. The initial SOCs are 1, 0.9, 0.8, …, 0.1 respectively, and a total of 10 segments are divided. Taking the discharge process with the initial SOC = 0.7 as an example, the GWO and IGWO are used for parameter identification, and the effects before and after the algorithm improvement are compared. The fitness function F(x) is established by using the sum of the squares of the errors between the experimentally measured terminal voltage and the actual terminal voltage:

[0158]

[0159] where y(k) is the measured voltage, is the predicted voltage. The iteration curve is as Figure 5 shown. Compared with GWO, IGWO has a faster convergence speed and higher accuracy, and is not easily trapped in a local optimum. It can obtain the optimal solution in a shorter time and is suitable for the parameter identification of lithium-ion battery models. By performing parameter identification on 10 groups of charge-discharge processes, the model parameters under different initial SOCs can be obtained. After polynomial fitting, the variation curve of the model parameters with SOC can be obtained. Substitute the fitted parameter results into the DST and FUDS working conditions for terminal voltage prediction, so as to verify the accuracy of the model. Use the same method to perform parameter identification on the ECM model of the lithium-ion battery and perform terminal voltage prediction. Compare the prediction results of ECM and NL-ECM, and the results are as Figure 6 、 Figure 7As shown. Under the premise of the same identification algorithm, NL-ECM obviously has higher accuracy. The terminal voltage prediction error is basically maintained within 2%, and the identification results are still accurate at the initial and final stages of discharge, without large deviations.

[0160] It can be seen that by using IGWO to segmentally identify the DST working condition and then using the obtained parameters to establish the NL-ECM model, and verifying it under the DST and FUDS working conditions, the model accuracy is significantly higher than that of ECM, realizing accurate modeling of lithium-ion batteries and having engineering value.

[0161] The above are only the preferred embodiments of the present invention and are not intended to limit the present invention. Any modifications, equivalent replacements, improvements, etc. made within the spirit and principle of the present invention shall be included within the protection scope of the present invention.

Claims

1. A non-linear modeling method for lithium-ion batteries under complex working conditions, characterized in that, it includes the following steps: Step 1) Conduct DST and FUDS working condition tests on the battery; Step 2) Establish a lithium-ion battery NL-ECM model; Step 3) Construct the algorithm flow of GWO; Step 4) Improve on the basis of GWO to construct IGWO; The said Step 4) includes the following steps: Improvement strategy 1) Construct a new expression for the convergence factor. Coefficient A is used as the attack coefficient. When A≥1, the wolf pack expands the encirclement range and the algorithm conducts global search. When A<1, the wolf pack attacks the prey within the encirclement and the algorithm conducts local search at this time. Coefficient A determines the optimization ability of the algorithm, and the convergence factor a directly affects the value of A. As the number of iterations increases, the convergence factor a decreases linearly, but this change method does not adapt to the actual situation. In the early stage of the algorithm optimization process, a should decrease slowly to expand the search range of the gray wolf population. In the later stage, the convergence factor should decrease rapidly to prompt the population to narrow the search range. The present invention constructs a new expression for the convergence factor: a = 2cos(2tπ / t max ) (20) As the number of iterations increases continuously, a shows non-linear changes, effectively balancing the shrinking trends in the early stage and the middle and late stages, thus effectively improving the optimization ability of the algorithm; Improvement strategy 2) Introduce dynamic weights. In the original algorithm, the guiding degrees of wolves α, β and δ are the same, but the position of the leading wolf is not necessarily the optimal solution and may also fall into the local optimum, resulting in a slow convergence speed of the algorithm. The present invention introduces dynamic weights and conducts dynamic weighted averaging during the optimization process to distinguish the contribution rates of the leading wolves and guide the position update of gray wolf individuals. The formula is as follows: f = |f α + f β + f δ | (21) X(i + 1) = w 1 ·X 1 (i) + w 2 ·X 2 (i) + w 3 ·X 3 (i) (23) f α 、f β and f δ represent the fitness values corresponding to three wolves. f is an intermediate variable, and w 1 、w 2 and w 3 represent the influence weights of three grey wolves on position update. During the iteration process, the weights will change according to the actual situation, effectively avoiding the algorithm falling into local optimum; Improvement strategy 3) Introduce random perturbations. As the algorithm iterates, the algorithm gradually tends to converge. If it falls into the local optimum, it is difficult to jump out. For general evolutionary algorithms, an effective solution is to maintain the diversity of the population. The present invention introduces random perturbations to enhance the population diversity in the later stage of the algorithm iteration, so as to jump out of the local optimum. The expression is as follows: where r 3 is a random number in [0, 1], M is the perturbation probability, ub and lb are the upper and lower bounds of the gray wolf individual, G is the dimension of the gray wolf individual, and f(H(i + 1)) and f(X(i + 1)) represent the fitness function values corresponding to the perturbed individual H and the gray wolf X in the (i + 1)-th iteration process; Step 5) Use IGWO to identify the model parameters and conduct terminal voltage prediction using the identification results under various working conditions.

2. A non-linear modeling method for lithium-ion batteries under complex working conditions according to claim 1, characterized in that, the said Step 2) includes the following contents: The derivation of the battery ECM model is as follows: Use U oc and U to represent the open-circuit voltage and terminal voltage of the battery, and C 1 and C 2 The voltages across both ends are represented by U 1 and U 2 respectively. R 0 is the ohmic internal resistance, and R 1 and C 1 characterize the electrochemical polarization effect and the rapid voltage change process; R 2 and C 2 characterize the concentration difference polarization reaction and the slow and stable voltage change process; Establish a functional relationship: where I(t) is the current at time t, and SOC is the remaining capacity of the battery, which is defined as follows: Q n is the rated capacity of the battery; Establish the discrete state space expression of the system: According to Kirchhoff's law, establish a transfer function for the ECM model: Adopt bilinear transformation to project the function from the s-plane to the z-plane to obtain: where: τ 1 = R 1 C 1 τ 2 = R 2 C 2 a = R 0 b = τ 1 τ 2 c = τ 1 + τ 2 d = R 0 + R 1 + R 2 e = R 0 (τ 1 + τ 2 ) + R 1 τ 1 + R 2 τ 2 ; Considering that the input current is non-linear due to disturbances in the measurement process, use the static non-linear function f(·) to represent it in polynomial form: where I(z) is the input current, is a polynomial of order p, representing the input current after the influence of the disturbance, and α t is the polynomial coefficient. In the present invention, p = 2; There is an external disturbance w(z) at the battery output terminal voltage, that is, colored noise. Assume that the external disturbance is stationary and has a rational spectral density. The pole polynomial of the disturbance channel is the same as that of the process channel, and the zero polynomial has the same order as the pole polynomial, that is: h(·) is a non - linear function, where v(z) is white noise, thus establishing the output expression of the battery NL - ECM model: Assume b 0 = 1, expand the above formula: Define the parameter vector θ and the information vector θ = [a 1 , a 2 , b 1 , b 2 , γ 1 , γ 2 , d 1 , d 2 T ∈R n (12)​ Establish an identification model:

3. According to a non - linear modeling method for lithium - ion batteries under complex working conditions described in claim 1, characterized in that, the step 3) includes the following content: GWO is a swarm intelligence optimization algorithm. It utilizes the hierarchical system existing in the group and simulates the hunting mechanism of the alpha wolf in the gray wolf group and the encircling and hunting behaviors during the predation process to achieve the purpose of optimization; The alpha wolf is regarded as the gray wolf with the highest level, the beta wolf is the subordinate of the alpha wolf, the delta wolf belongs to the beta and alpha wolves, and the remaining individuals are denoted as w. In the algorithm, the gray wolf alpha, as the optimal solution, together with the sub - optimal solution beta and the re - optimal solution delta, guides the w gray wolves to complete the hunting behavior, thus realizing the iterative optimization of the algorithm. The hunting behavior includes two parts: encirclement and attack; 1) Encirclement Let the scale of the gray wolf population be set as N and the search dimension be d. The process of encircling the prey is as follows: D = |C·X p (t) - X(t)| (14) X(t + 1) = X p (t) - A·D(15) where t is the current iteration number, X p is the position vector of the prey, X is the position vector of the grey wolf, and D is the distance between the individual wolf and the prey; A is the attack coefficient of the grey wolf on the hunting target, and C is the cooperation coefficient. The calculation formulas are as follows: A = 2a·r 1 -a (16) C＝2·r 2 (17) where r 1 and r 2 are random vectors, a is the convergence factor, a = 2 - 2t / t max , t max is the maximum number of iterations; 2) Attack After encircling the prey, the gray wolf will attack the prey. There is a position update in this process. The optimal wolf alpha will guide the wolf pack to approach the prey, and beta and delta play an auxiliary role. The position update rule during the attack process is as follows: Where X a , X β and X δ are the position vectors of three leading wolves; D a , D β and D δ are the distance vectors between the leading wolves and the prey; X 1 , X 2 and X 3 represent the updates of the position vectors of three wolves; X is the update of the position vector of the w gray wolf; A 1 , A 2 and A 3 are the attack coefficients of the gray wolves, C 1 , C 2 and C 3 are the cooperation coefficients of the gray wolves. As the iteration updates, the three wolves with the best population will replace the original leading wolves, and the remaining individuals will be led by the new leading wolves to approach the prey continuously.

4. According to a non - linear modeling method for lithium - ion batteries under complex working conditions described in claim 1, characterized in that, the step 5) includes the following steps: The model parameters of the lithium - ion battery are deviated under different SOCs. Therefore, the present invention conducts segmented identification for each working condition. Calculate the SOC value of the DST working condition measured in step 1) by the ampere - hour integration method, segment the entire charge - discharge process. The initial SOCs are 1, 0.9, 0.8, …, 0.1 respectively, and it is divided into 10 segments in total. Take the discharge process with the initial SOC = 0.7, and use GWO and IGWO for parameter identification, compare the effects before and after the algorithm improvement, and establish the fitness function F(x) using the sum of the squares of the errors between the experimentally measured terminal voltage and the actual terminal voltage: In the formula, y(k) is the measured voltage, and ŷ(k) is the predicted voltage. Through parameter identification for 10 groups of charge - discharge processes, the model parameters under different initial SOCs are obtained. Then, through polynomial fitting, the variation curve of the model parameters with SOC is obtained. Substitute the fitted parameter results into the DST and FUDS working conditions for terminal voltage prediction, so as to verify the accuracy of the model; Combine the obtained parameters with the actual data for terminal voltage prediction to verify the accuracy of the model under the intermittent constant - current discharge experiment.

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