Fractional order lithium battery model parameter identification method based on improved chaos evolutionary algorithm

Abstract

The invention relates to a fractional order lithium battery model parameter identification method based on an improved chaos evolutionary algorithm, and belongs to the technical field of lithium battery model parameter identification. The method comprises the following steps: representing a lithium battery second-order fractional order equivalent circuit model parameter identification problem as a parameter optimization problem; wherein the parameter optimization problem comprises known parameters, to-be-identified parameters and a target function, the known parameters comprise a function relationship between the OCV and the SOC and equivalent internal resistance, and the target function is the sum of root-mean-square errors of the estimated terminal voltage and the actually measured voltage; an improved Latin hypercube sampling strategy is introduced, Givens transformation and Householder transformation are introduced, and a restart strategy based on reverse learning and element rearrangement is introduced to improve an original chaotic evolutionary algorithm; and performing parameter optimization on the parameter optimization problem based on the improved chaos evolutionary algorithm to obtain an optimal parameter combination, thereby completing parameter identification of the to-be-identified parameter. The method aims at solving the technical problems that in the prior art, parameter identification precision is poor, and local optimum is prone to occurring.

Description

Technical Field

[0001] This invention relates to a method for identifying parameters of a fractional-order lithium battery model based on an improved chaotic evolution algorithm, belonging to the field of lithium battery model parameter identification technology. Background Technology

[0002] With the increasing demand for sustainable development and clean energy, new energy technologies are continuously innovating and accelerating industrialization. Among them, lithium batteries, as a highly efficient energy storage medium, have advantages such as high energy density, long cycle life, and low energy loss, and have become one of the most widely used energy storage devices. In order to achieve an accurate description of its characteristics and state prediction, it is necessary to use lithium battery models and identify the model parameters.

[0003] Common lithium battery models include electrochemical models, data-driven models, and equivalent circuit models. Electrochemical models have clear physical meaning and high accuracy, but suffer from high computational complexity, a large number of parameters, and difficulties in parameter identification. Data-driven models excel at fitting the nonlinear and time-varying characteristics of batteries and offer high flexibility, but lack explicit physical meaning and are highly dependent on the quality of training data. Equivalent circuit models, on the other hand, strike a good balance between accuracy and complexity and offer some physical interpretability. Therefore, equivalent circuit models have high practical application value. Among these, fractional-order equivalent circuit models consider the fractional-order characteristics of batteries, better fitting their dynamic characteristics and approximating their physical essence. Regarding parameter identification, traditional least squares methods and their variants, while computationally efficient, heavily rely on initial estimates, easily leading to local optima and limiting the accuracy of the final solution. Metaheuristic algorithms, however, do not depend on initial value settings and possess good global optimization capabilities, contributing to more accurate parameter identification results.

[0004] Chaotic Evolution Optimization (CEO) is a novel metaheuristic algorithm proposed in recent years. This algorithm combines chaotic mapping with differential evolution, enabling efficient searching within the search space. However, it also faces drawbacks such as poor initial population diversity and the potential for getting trapped in local optima in later iterations. Therefore, to address the specific needs of lithium-ion battery model parameter identification, further exploration of improvement strategies for the chaotic evolution algorithm is necessary to enhance its application performance in lithium-ion battery model parameter identification. Summary of the Invention

[0005] The technical problem to be solved by this invention is to provide an improved chaotic evolutionary algorithm (Multi-Strategy Enhanced Chaotic Evolutionary Optimization, MECEO) that integrates multiple strategies to address the shortcomings of the original chaotic evolutionary algorithm. MECEO is then applied to the parameter identification of the second-order fractional equivalent circuit model of lithium batteries, aiming to solve the technical problems of poor parameter identification accuracy and easy getting trapped in local optima in the existing technology.

[0006] The technical solution of this invention is: a method for parameter identification of fractional-order lithium battery models based on an improved chaotic evolution algorithm, comprising the following steps: Step 1: Establish a second-order fractional-order equivalent circuit model of a lithium battery; Step 2: Obtain the functional relationship between OCV and SOC through the incremental OCV test data of the lithium battery, and calculate the equivalent internal resistance in the second-order fractional equivalent circuit of the lithium battery; Step 3: Represent the parameter identification problem of the second-order fractional-order equivalent circuit model of the lithium battery as a parameter optimization problem; The parameter optimization problem includes known parameters, parameters to be identified, and an objective function. The parameters to be identified are obtained based on the known parameters and the objective function. The known parameters include the functional relationship between OCV and SOC and the equivalent internal resistance. The objective function is the sum of the root mean square error between the estimated terminal voltage and the measured voltage. Step 4: Improve the original chaotic evolution algorithm to obtain the improved chaotic evolution algorithm, including: Improvement Strategy 1: Use an improved Latin hypercube sampling strategy to replace the random initialization of the population in the original algorithm, so as to improve the uniformity of the initial population distribution in the search space and the boundary coverage capability. Improvement Strategy 2: Introduce Givens transformation and Householder transformation as dynamic adjustment mechanisms and embed them at the end of each iteration of the algorithm to enhance the diversity of the population; Improvement Strategy 3: Introduce a restart strategy based on reverse learning and element rearrangement to enhance the algorithm's ability to escape local optima; Step 5: Based on the improved chaotic evolution algorithm, perform parameter optimization on the parameter optimization problem to obtain the optimal parameter combination, thereby completing the parameter identification of the parameter to be identified.

[0007] Optionally, the improved Latin hypercube sampling strategy is based on the traditional Latin hypercube sampling strategy but introduces boundary enhancement and uniformity optimization operations, specifically: Boundary augmentation operation: In the initial sample set generated within the search space, find the minimum and maximum coordinate points in each dimension, push the boundary points to the boundary with a 50% probability, and apply random perturbation with an additional 50% probability to complete the boundary augmentation. Uniformity optimization operation: Perform the following on the sample set after boundary enhancement operation: T o Each optimization cycle randomly selects a pair of sample points in each cycle. and And randomly select a dimension d Exchange sample points and exist d If the minimum Euclidean distance between population sample points increases after the exchange operation, the exchange is accepted; otherwise, the original sample points are retained.

[0008] Optionally, the expression for applying a random perturbation to achieve boundary reinforcement is:

[0009]

[0010] in, and Representing dimensions The minimum and maximum coordinates of the point; and Representing dimensions d The lower and upper bounds; and All are random numbers uniformly distributed on [0,1]. This indicates the magnitude of the disturbance that is proportional to the size of the interval. For population size, This represents the disturbance coefficient.

[0011] Optionally, the second improvement strategy is specifically as follows: First, we introduce the Givens transformation matrix, expressed as:

[0012] in, Here is the Givens transformation matrix. To optimize the parameter dimension of the parameter optimization problem, This is a variable that is randomly generated each time the Givens transformation is called; Then, the Householder transformation matrix is ​​introduced, with the expression:

[0013] in, For the Householder transformation matrix, A randomly generated unit vector. It is the identity matrix. T Represented as a transpose matrix; Finally, at the end of each iteration of the algorithm, with a 50% probability, one of the Givens transformation and the Householder transformation is randomly selected to act on the candidate solution vector in the population, thereby retaining the best-fitting vector from both the newly generated population and the original population. Individual.

[0014] Optionally, the third improvement strategy is specifically as follows: When the algorithm is continuous m If a better solution is not found in the next iteration, a restart mechanism is initiated. Specifically, this restart mechanism involves retaining 70% of the best individuals in the current population and processing the remaining 30% of individuals: each individual has a 50% probability of generating a new solution by randomly rearranging vector elements, and a 50% probability of employing a reverse learning strategy: for a given individual in the population... The opposing solution is generated through a reverse learning strategy, expressed as:

[0015] in, This represents the opposing solution generated through a reverse learning strategy. and These are the lower and upper bounds of the search space.

[0016] Optionally, step 5 specifically includes: Step 5.1: Based on population size Generate based on improved strategy one The parameter combination of the parameter optimization problem is used as the initial population; Step 5.2: Perform a mutation operation on the initialized population. The expression is:

[0017] in, and These are the current individual and the mutated individual, respectively. This is the optimal solution for the current population. a Indicates the search step size. Indicates the direction of evolution. rand (0,1) is a random number uniformly distributed on [0,1]. Step 5.3: Perform a crossover operation on the initial population, for each individual Generate after completing the crossover operation. N test vectors The crossover operation expression is:

[0018] in, This represents the test vector generated by the crossover operation. D This represents the dimension of the test vector. This indicates that the mutated individual is in the first... j Values ​​in dimensions Represents a set A randomly selected integer. This indicates that the individual before the mutation was in the 1st month. j Values ​​in each dimension, for each variable dimension, Used to generate random numbers uniformly distributed in the range [0,1] and cross over with a preset probability. Compare, if the random number is less than If the current dimension is a randomly selected forced crossover point, then the offspring inherits the corresponding variable value of the mutated individual; otherwise, the value of the original parent individual is retained, and the crossover probability is... The value of is updated with each iteration and follows a uniform distribution on the interval [0,1]. Step 5.4: Perform the selection operation, the expression is:

[0019] in, These are the individuals obtained after the selection operation. Individual generated N The vector with the best fitness among the trial vectors, the function For fitness evaluation function; Step 5.5: Apply Improvement Strategy 2 to the new population generated after the selection operation, and update the individuals in the population; Step 5.6: Implement improvement strategy three, and check whether to perform a restart operation after each iteration of the algorithm; Step 5.7: Repeat steps 5.2 to 5.6 until the number of iterations reaches the maximum number of iterations (Maxiter). Output the individual corresponding to the optimal fitness value to obtain the optimal parameter combination for the parameter optimization problem.

[0020] Optionally, the generation of the evolutionary direction is specifically as follows: First, individuals in the population are randomly paired up. For each paired individual... and Mapped to the ranges [-0.5, 0.5] and [-0.25, 0.25] respectively, the expressions are:

[0021] in, and Individual and The vector after mapping and These are the lower and upper bounds of the search space; Secondly, the two individuals generated and As the initial state variable of the chaotic mapping, it is applied to the following chaotic mapping for... N The next iteration, expressed as:

[0022] in, and Represents an individual and Chaotic samples generated during the iteration process, when n When =0, , , respectively, serve as the initial state variables for the chaotic mapping. k c It is a control parameter used to adjust the dynamic behavior of chaotic mapping; conduct N After the second iteration, 2 is generated. N A chaotic sample, expressed as:

[0023] in, and This represents the chaotic sample set generated after iteration, containing a total of 2 N One chaotic sample; Then, the generated chaotic samples are mapped back to the search space of the original optimization problem, expressed as:

[0024] in, and They represent and The vector after range mapping; Finally, for individuals and Based on and generate N There are several evolutionary directions, expressed as:

[0025] in, and Each represents an individual and The direction of evolution that is generated.

[0026] The beneficial effects of this invention are as follows: First, by using an improved Latin hypercube sampling strategy for population initialization, the algorithm can generate a more uniform population with stronger boundary coverage compared to random initialization, thereby helping the algorithm quickly locate the region near the optimal solution in subsequent iterations. Then, by introducing Givens transformation and Householder transformation onto the population after the selection operation, the diversity of the population can be enhanced, thus providing diverse search directions and improving the algorithm's global search capability. Finally, by introducing a restart strategy based on back learning and element rearrangement, the algorithm can escape local optima and obtain solutions with higher accuracy. Through the fusion and improvement of multiple strategies, this invention enables the improved chaotic evolutionary algorithm to achieve good parameter identification performance on the established second-order fractional-order equivalent circuit model of a lithium battery. Attached Figure Description

[0027] Figure 1 This is a flowchart of the steps of the present invention; Figure 2 This is a circuit diagram of the second-order fractional-order equivalent circuit model of the lithium battery of the present invention; Figure 3 These are the current and voltage curves of the incremental OCV test data of this invention; Figure 4 This is the lithium battery OCV-SOC function curve fitted by this invention; Figure 5 This is the average convergence curve of the present invention and other algorithms in the task of identifying parameters of a second-order fractional-order equivalent circuit model for lithium batteries; Figure 6 This is a box plot of the present invention and other algorithms in the task of identifying parameters of a second-order fractional-order equivalent circuit model for lithium batteries; Figure 7 This is a comparison of the convergence curves of the present invention and other algorithms under the benchmark function; Figure 8 This is a box plot of the present invention and other algorithms under a benchmark function. Detailed Implementation

[0028] The present invention will be further described below with reference to the accompanying drawings and specific embodiments.

[0029] Example 1: As Figure 1As shown, a parameter identification method for a fractional-order lithium battery model based on an improved chaotic evolutionary algorithm is presented. This method first uses an improved Latin hypercube sampling to initialize the population, replacing the original algorithm's random initialization method, thus improving the uniformity of the initial population distribution and boundary coverage in the search space. Secondly, Givens transformation and Householder transformation are introduced and applied to all individuals in the population after each iteration to enhance population diversity. Finally, a restart strategy based on back learning and element rearrangement is proposed. By detecting the fitness change of the optimal individual in continuous iterations, a restart mechanism is triggered if there is no improvement, thereby strengthening the algorithm's ability to escape local optima. After improving the chaotic evolutionary algorithm, this algorithm is used to identify the parameters of a second-order fractional-order equivalent circuit model of a lithium battery, thereby obtaining high-precision parameter identification results. The method includes the following steps: Step 1: Establish a second-order fractional-order equivalent circuit model of a lithium battery; Optionally, the second-order fractional equivalent circuit of the lithium battery includes two fractional branches connected in series. The fractional branches are composed of resistors and constant-phase elements connected in parallel. The circuit also includes an equivalent internal resistance and a voltage source, which are connected in series with the two fractional branches. Specifically, such as Figure 2 The diagram shown is a circuit diagram of the second-order fractional-order equivalent circuit model of a lithium battery constructed in this embodiment. For resistance, and It is a constant phase element. Let be the open-circuit voltage (OCV) of the battery, which is related to the battery's state of charge (SOC) and serves as the voltage source in the equivalent circuit. According to the Grünwald-Letnikov definition of fractional differentials, the discretized expression of the second-order fractional equivalent circuit model of the lithium battery is:

[0030] in, and Resistors exist k Voltage at time, Indicates the sampling time. express k The charging and discharging current at any given time, and These represent the fractional orders of the two constant-phase elements, and For the corresponding fractional capacitance value, Indicates memory length. and The coefficients are generalized binomial coefficients. express k The SOC value at time +1 Indicates Coulomb efficiency. This indicates the battery's nominal capacity. express k The terminal voltage of the circuit at any given time.

[0031] Step 2: Obtain the functional relationship between OCV and SOC through the incremental OCV test data of the lithium battery, and calculate the equivalent internal resistance in the second-order fractional equivalent circuit of the lithium battery; Optionally, the current and voltage curves of the incremental OCV test data used in this embodiment are as follows: Figure 3 As shown, the voltage values ​​at the end of all resting stages were extracted from the OCV test data as OCV values, and correlated with the SOC values ​​at those end stages to obtain a series of discrete OCV-SOC data points. Subsequently, an 8th-order polynomial function was used to fit the obtained data points to obtain the functional relationship between OCV and SOC. The fitted OCV-SOC function curve is shown in the figure. Figure 4 As shown, the equivalent internal resistance The value is obtained by calculating the ratio of the voltage change to the current pulse value at the moment of pulse discharge.

[0032] Step 3: Represent the parameter identification problem of the second-order fractional-order equivalent circuit model of the lithium battery as a parameter optimization problem; The parameter optimization problem includes known parameters, parameters to be identified, and an objective function. The parameters to be identified are obtained based on the known parameters and the objective function. The known parameters include the functional relationship between OCV and SOC and the equivalent internal resistance. The objective function is the sum of the root mean square error between the estimated terminal voltage and the measured voltage. Specifically, in this embodiment, the other unknown parameters in the second-order fractional-order equivalent circuit model of the lithium battery are: These unknown parameters are treated as parameters to be identified in the parameter optimization problem, and the root mean square error between the model-estimated terminal voltage and the measured voltage is used as the objective function. The expression is:

[0033] in, This represents a candidate solution in the algorithm's iterative process, i.e., an individual in the population in this embodiment, representing a possible combination of parameters. express k The measured value of the battery terminal voltage at any given time. Indicates in Under the model parameter combination represented, k The predicted terminal voltage value of the time-matter model. This indicates the number of data points.

[0034] Step 4: Improve the original chaotic evolution algorithm to obtain the improved chaotic evolution algorithm, including: Improvement Strategy 1: Use an improved Latin hypercube sampling strategy to replace the random initialization of the population in the original algorithm, so as to improve the uniformity of the initial population distribution in the search space and the boundary coverage capability. Optionally, the improved Latin hypercube sampling strategy is based on the traditional Latin hypercube sampling strategy but introduces boundary enhancement and uniformity optimization operations, specifically: Boundary augmentation operation: In the initial sample set generated within the search space, find the minimum and maximum coordinate points in each dimension, push the boundary points to the boundary with a 50% probability, and apply random perturbation with an additional 50% probability to complete the boundary augmentation. Uniformity optimization operation: Perform the following on the sample set after boundary enhancement operation: T o Each optimization cycle randomly selects a pair of sample points in each cycle. and And randomly select a dimension d Exchange sample points and exist d The coordinate values ​​of the dimension, after performing a swap operation, represent the minimum Euclidean distance between population sample points. If an increase is made, the exchange is accepted; otherwise, the original sample points are retained. and This represents two sample points in the population, after... T o The set of sample points obtained after each iteration is the initial population of the algorithm; in this embodiment, T o Set it to 100.

[0035] Optionally, this embodiment uses a traditional Latin hypercube sampling strategy to generate within the search space. An initial sample set; Optionally, the expression for applying a random perturbation to achieve boundary reinforcement is:

[0036]

[0037] in, and Representing dimensions The minimum and maximum coordinates of the point; and Representing dimensions d The lower and upper bounds; and All are random numbers uniformly distributed on [0,1]. This indicates the magnitude of the disturbance that is proportional to the size of the interval. For population size, The disturbance coefficient is 0.5 in this embodiment.

[0038] Improvement Strategy 2: Introduce Givens transformation and Householder transformation as dynamic adjustment mechanisms and embed them at the end of each iteration of the algorithm to enhance the diversity of the population; Optionally, the second improvement strategy is specifically as follows: First, we introduce the Givens transformation matrix, expressed as:

[0039] in, Here is the Givens transformation matrix. To optimize the parameter dimension of the parameter optimization problem, This is a variable that is randomly generated each time the Givens transformation is called; Then, the Householder transformation matrix is ​​introduced, with the expression:

[0040] in, For the Householder transformation matrix, A randomly generated unit vector. It is the identity matrix. T Represented as a transpose matrix; Finally, at the end of each iteration of the algorithm, with a 50% probability, one of the Givens transformation and the Householder transformation is randomly selected to act on the candidate solution vector in the population, thereby retaining the best-fitting vector from both the newly generated population and the original population. Individual.

[0041] Improvement Strategy 3: Introduce a restart strategy based on reverse learning and element rearrangement to enhance the algorithm's ability to escape local optima; Optionally, the third improvement strategy is specifically as follows: Detect the change in fitness of the best individual in the current iteration compared to previous iterations. When the algorithm continuously... mIf a better solution is not found in the next iteration, a restart mechanism is initiated. Specifically, the restart mechanism retains 70% of the best individuals in the current population to ensure that the currently obtained high-quality solutions are not lost. The remaining 30% of individuals are then processed: each individual has a 50% probability of generating a new solution by randomly rearranging the vector elements (i.e., randomly rearranging the order of all elements in the vector), and a 50% probability of using a reverse learning strategy: for a given individual in the population... The opposing solution is generated through a reverse learning strategy, expressed as:

[0042] in, This represents the opposing solution generated through a reverse learning strategy. and These are the lower and upper bounds of the search space. In this embodiment, m Set it to 20.

[0043] Step 5: Based on the improved chaotic evolution algorithm, perform parameter optimization on the parameter optimization problem to obtain the optimal parameter combination, thereby completing the parameter identification of the parameter to be identified.

[0044] Step 5.1: Based on the number of parameters to be identified, set the dimension of the parameter optimization problem and the upper and lower bounds of the search space, and then based on the population size... Generate based on improved strategy one The parameter combination of the parameter optimization problem is used as the initial population; Step 5.2: Perform a mutation operation on the initialized population. The expression is:

[0045] in, and These are the current individual and the mutated individual, respectively. This is the optimal solution for the current population. a Indicates the search step size. Indicates the direction of evolution. rand (0,1) is a random number uniformly distributed on [0,1]. Optionally, the generation of the evolutionary direction is specifically as follows: First, individuals in the population are randomly paired up. For each paired individual... and Mapped to the ranges [-0.5, 0.5] and [-0.25, 0.25] respectively, the expressions are:

[0046] in, and Individual and The vector after mapping and These are the lower and upper bounds of the search space; Secondly, the two individuals generated and As the initial state variable of the chaotic mapping, it is applied to the following chaotic mapping for... N The next iteration, expressed as:

[0047] in, and Represents an individual and Chaotic samples generated during the iteration process, when n When =0, , , respectively, serve as the initial state variables for the chaotic mapping. k c It is a control parameter used to adjust the dynamic behavior of the chaotic map. In this embodiment, k c Set to 2.66; conduct N After the second iteration, 2 is generated. N A chaotic sample, expressed as:

[0048] in, and This represents the chaotic sample set generated after iteration, containing a total of 2 N One chaotic sample; Then, the generated chaotic samples are mapped back to the search space of the original optimization problem, expressed as:

[0049] in, and They represent and The vector after range mapping; Finally, for individuals and Based on and generate N There are several evolutionary directions, expressed as:

[0050] in, and Each represents an individual and The direction of evolution that is generated.

[0051] Step 5.3: Perform a crossover operation on the initial population, for each individual Generate after completing the crossover operation. N test vectors The crossover operation expression is:

[0052] in, This represents the test vector generated by the crossover operation. D This represents the dimension of the test vector. This indicates that the mutated individual is in the first... j Values ​​in dimensions Represents a set A randomly selected integer. This indicates that the individual before the mutation was in the 1st month. j Values ​​in each dimension, for each variable dimension, Used to generate random numbers uniformly distributed in the range [0,1] and cross over with a preset probability. Compare, if the random number is less than If the current dimension is a randomly selected forced crossover point, then the offspring inherits the corresponding variable value of the mutated individual; otherwise, the value of the original parent individual is retained, and the crossover probability is... The value of is updated with each iteration and follows a uniform distribution on the interval [0,1]. Step 5.4: Perform the selection operation, the expression is:

[0053] in, These are the individuals obtained after the selection operation. Individual generated N The vector with the best fitness among the trial vectors, the function In this embodiment, the objective function of the parameter optimization problem is used as the fitness evaluation function. Step 5.5: Apply Improvement Strategy 2 to the new population generated after the selection operation, and update the individuals in the population; Step 5.6: Implement improvement strategy three, and check whether to perform a restart operation after each iteration of the algorithm; Step 5.7: Repeat steps 5.2 to 5.6 until the number of iterations reaches the maximum number of iterations, Maxiter. Output the individual corresponding to the optimal fitness value to obtain the optimal parameter combination for the parameter optimization problem, which is the parameter value to be identified for the established second-order fractional equivalent circuit model of the lithium battery.

[0054] Based on the specific implementation details, the effectiveness of the technical solution of the present invention will be demonstrated through experiments.

[0055] Specifically, the parameter identification results of a single run of the present invention are shown in Table 1.

[0056] Table 1. Parameter identification results of a single run of the present invention

[0057] Furthermore, to verify the performance of the improved Chaotic Evolutionary Algorithm (MECEO) compared to other optimization algorithms, this embodiment compares MECEO with 10 other optimization algorithms using parameter identification, including Particle Swarm Optimization (PSO), Gray Wolf Optimization (GWO), Whale Optimization (WOA), Parrot Optimization (PO), Aurora Optimization (PLO), Ivy Optimization (IVY), Unreal Wild Oats Optimization (AOO), Dream Optimization (DOA), Tian Ji Horse Racing Optimization (THRO), and the original Chaotic Evolutionary Algorithm (CEO). Population Size The maximum number of iterations (Maxiter) is set to 100, and the dimension of the optimization problem is set to 6 based on the number of parameters to be estimated in the established model. Each algorithm is run independently 30 times to reduce the impact of random factors on the results. The parameter settings for each algorithm are shown in Table 2.

[0058] Table 2 Algorithm Parameter Settings

[0059] In the PSO algorithm, and These are individual and group learning factors, It is the inertia weight, which adopts a linear decreasing strategy, and its value changes uniformly from 0.9 to 0.6; in the GWO algorithm, This is a control factor whose value decreases linearly from 2 to 0; in the WOA algorithm, This is a control factor whose value decreases linearly from 2 to 0. It is a constant used to define the shape of the logarithmic spiral. l For random numbers in the interval [-1, 1]; in the PO algorithm, It is a feature index; in the PLO algorithm, It is the particle mass. It is the damping factor, a random number in the interval [1, 1.5]; in the IVY algorithm, It is a control factor, a random number in the interval [1, 1.5]; in the AOO algorithm, For Lévy's flight parameters; in the DOA algorithm, It is the number of iterations that marks the boundary between the exploration and development phases. It is the maximum number of iterations. q For the index of the forgotten dimension, u It is a strategy proportional control parameter; in the THRO algorithm, It involves adjusting the control parameters for Levi's flight; in the CEO algorithm, These are the control parameters that affect chaotic mappings; in the MECEO algorithm, k c These are the control parameters that affect chaotic mapping. f p As a disturbance factor, T o It improves the number of iterations for Latin hypercube sampling.

[0060] Specifically, Figure 5 The average convergence curves of each algorithm under the experimental conditions are shown. It can be seen that MECEO's convergence curve is steeper in the early stages of iteration compared to other algorithms, indicating that it can effectively reduce the fitness function value in the initial stage. This is mainly due to the synergistic effect of MECEO's population initialization strategy and the chaotic evolution and bioorthogonal transformation mechanisms, which effectively improve the algorithm's population diversity and search efficiency, enabling it to quickly approach the optimal solution. On the other hand, MECEO also outperforms other algorithms in terms of the accuracy of the final solution. This algorithm can minimize the value of the objective function, indicating that it can more accurately locate the optimal parameter combination during the search process. Furthermore, the introduction of the restart mechanism effectively avoids the problem of the algorithm getting trapped in local optima, achieving high-precision parameter identification.

[0061] Furthermore, Figure 6 Box plots show the optimal fitness of each algorithm based on 30 independent experiments. The DOA algorithm has the widest range of its box and whisker, and also the highest median. This indicates that the algorithm has significant performance fluctuations and is prone to getting trapped in local optima. While the PSO and PO algorithms have narrower ranges of their box and whisker than DOA, both algorithms exhibit several outliers. These outliers are far from the main data and are relatively scattered. Therefore, these two algorithms perform poorly in extreme cases, making them unsuitable for practical applications. The remaining algorithms show more stable performance, with more concentrated results, good medians, and smaller deviations from the main data for outliers. The MECEO algorithm exhibits the most stable performance, with its box plot almost forming a straight line. Furthermore, its box plot is located at the bottom, with outliers very close to the main data. This indicates that the algorithm maintains high optimization accuracy while ensuring stable performance, outperforming other algorithms in most cases, even in the worst-case scenario.

[0062] Furthermore, Table 3 summarizes the average and standard deviation of the optimal fitness of each algorithm after 30 runs. Among them, the MECEO proposed in this invention has an average value of 9.8693E-03 and a standard deviation of only 3.3852E-06, both of which are superior to the other comparison algorithms.

[0063] Table 3. Mean and standard deviation of the optimal fitness of each algorithm after 30 runs.

[0064] Furthermore, this embodiment uses the CEC2022 (Congress on Evolutionary Computation 2022) benchmark function set to compare the comprehensive optimization performance of different algorithms and verify the superiority of the MECEO algorithm. A description of the CEC2022 benchmark function set is shown in Table 4.

[0065] Table 4 Overview of CEC2022 Benchmark Function Set

[0066] To ensure fairness and reliability of benchmark function testing, the population size of all algorithms is [not specified]. The set values ​​are uniformly set to 30, the maximum number of iterations (Maxiter) is set to 500, and the dimension of the baseline function is [not specified]. The value was set to 20. For each benchmark function, each algorithm was run independently 30 times to reduce the impact of random factors on the results. The parameters for all algorithms were set according to Table 2.

[0067] Table 5 presents three evaluation metrics for all algorithms on the CEC2022 benchmark function set, including the mean and standard deviation of 30 independent runs (denoted by Mean and Std, respectively), and the comparison results of each algorithm with MECEO on the mean (using Winner, where "+" indicates better than MECEO, "-" indicates worse than MECEO, and "=" indicates comparable to MECEO). Table 6 summarizes all the comparison results.

[0068] Table 5. Average and standard deviation of all algorithms after running 30 times on the CEC2022 benchmark function set.

[0069] Table 6 Summary of comparison data between MECEO and various algorithms on average value

[0070] As shown in Tables 5 and 6, MECEO achieved the best average value on test functions F1, F3, F4, F8, and F12. Additionally, DOA achieved the best average value three times on F6, F10, and F11; THRO achieved the best average value twice on F5 and F7; and PSO achieved the best average value twice on F2 and F9. Compared to other algorithms, MECEO achieved the most best average values ​​across the 12 test functions, indicating that its performance is superior to other comparative algorithms in most tests. Moreover, only the MECEO algorithm achieved optimal values ​​across all function types (unimodal function F1, multimodal functions F3 and F4, mixed function F8, and composite function F12). Regarding standard deviation, MECEO achieved the best standard deviation on F1, F3, F4, F8, and F9, also covering all benchmark function types. It has the second-best standard deviation on F5, F7, F10, and F11. This demonstrates that MECEO maintains superior search accuracy and stability across various optimization problems.

[0071] Furthermore, Figure 7 The convergence curves of different algorithms on 12 test functions are shown. It can be seen that in the early stages of iteration, the MECEO convergence curve is very steep, exhibiting extremely fast convergence speed. In the middle stages of iteration, the MECEO curve consistently remains near the lower end of the curve. This indicates that MECEO possesses superior global search capabilities, not only approaching the optimal solution faster but also maintaining a high level of solution accuracy even in the middle stages of iteration. Furthermore, thanks to the introduction of the restart mechanism, a certain degree of downward trend can still be observed in the MECEO curve in the later stages of iteration on the F2 and F6 test functions. This phenomenon verifies the algorithm's ability to escape local optima. Regarding convergence accuracy, after the algorithm iterations are completed, the MECEO curve converges to near the theoretical optimum on all test functions, demonstrating high convergence accuracy.

[0072] Furthermore, Figure 8 Box plots of MECEO and other algorithms on different test functions are shown. It can be seen that on most test functions, MECEO's box plots are positioned lower, with a lower median and a flatter box. Especially on functions F1, F3, F5, F6, F8, F9, F10, and F11, it almost forms a straight line. Furthermore, the outliers of the MECEO algorithm deviate from the main data to a very low degree. These characteristics indicate that the MECEO algorithm can generally achieve high-precision solutions on most optimization problems, with more controllable performance fluctuations, and good stability and robustness.

[0073] In summary, this invention first establishes a second-order fractional-order equivalent circuit model of a lithium battery, abstracting the parameter identification task of this model into a mathematical optimization problem. Secondly, it introduces multiple strategies to improve the original chaotic evolutionary algorithm to enhance its performance. Then, it uses the improved chaotic evolutionary algorithm to optimize the parameters of the parameter identification problem model. Finally, it obtains the optimal parameter combination for the fractional-order lithium battery model parameter identification task. By integrating multiple improvement strategies, this invention enables the algorithm to generate a high-quality initial population, enhancing its exploratory ability and its ability to escape local optima, thus achieving excellent results in the fractional-order lithium battery model parameter identification task.

[0074] The specific embodiments of the present invention have been described in detail above with reference to the accompanying drawings. However, the present invention is not limited to the above embodiments. Within the scope of knowledge possessed by those skilled in the art, various changes can be made without departing from the spirit of the present invention.

Claims

1. A method for parameter identification of fractional-order lithium battery models based on an improved chaotic evolutionary algorithm, characterized in that, The method includes the following steps: Step 1: Establish a second-order fractional-order equivalent circuit model of a lithium battery; Step 2: Obtain the functional relationship between OCV and SOC through the incremental OCV test data of the lithium battery, and calculate the equivalent internal resistance in the second-order fractional equivalent circuit of the lithium battery; Step 3: Represent the parameter identification problem of the second-order fractional-order equivalent circuit model of the lithium battery as a parameter optimization problem; The parameter optimization problem includes known parameters, parameters to be identified, and an objective function. The parameters to be identified are obtained based on the known parameters and the objective function. The known parameters include the functional relationship between OCV and SOC and the equivalent internal resistance. The objective function is the sum of the root mean square error between the estimated terminal voltage and the measured voltage. Step 4: Improve the original chaotic evolution algorithm to obtain the improved chaotic evolution algorithm, including: Improvement Strategy 1: Use an improved Latin hypercube sampling strategy to replace the random initialization of the population in the original algorithm, so as to improve the uniformity of the initial population distribution in the search space and the boundary coverage capability. Improvement Strategy 2: Introduce Givens transformation and Householder transformation as dynamic adjustment mechanisms and embed them at the end of each iteration of the algorithm to enhance the diversity of the population; Improvement Strategy 3: Introduce a restart strategy based on reverse learning and element rearrangement to enhance the algorithm's ability to escape local optima; Step 5: Based on the improved chaotic evolution algorithm, perform parameter optimization on the parameter optimization problem to obtain the optimal parameter combination, thereby completing the parameter identification of the parameter to be identified.

2. The method for parameter identification of fractional-order lithium battery models based on an improved chaotic evolutionary algorithm according to claim 1, characterized in that, The improved Latin hypercube sampling strategy is based on the traditional Latin hypercube sampling strategy but introduces boundary enhancement and uniformity optimization operations, specifically: Boundary augmentation operation: In the initial sample set generated within the search space, find the minimum and maximum coordinate points in each dimension, push the boundary points to the boundary with a 50% probability, and apply random perturbation with an additional 50% probability to complete the boundary augmentation. Uniformity optimization operation: Perform the following on the sample set after boundary enhancement operation: T o Each optimization cycle randomly selects a pair of sample points in each cycle. and And randomly select a dimension d Exchange sample points and exist d If the minimum Euclidean distance between population sample points increases after the exchange operation, the exchange is accepted; otherwise, the original sample points are retained.

3. The method for parameter identification of fractional-order lithium battery models based on an improved chaotic evolutionary algorithm according to claim 2, characterized in that, The expression for applying random perturbations to achieve boundary reinforcement is: ； ； in, and Representing dimensions The minimum and maximum coordinates of the point; and Representing dimensions d The lower and upper bounds; and All are random numbers uniformly distributed on [0,1]. This indicates the magnitude of the disturbance that is proportional to the size of the interval. For population size, This represents the disturbance coefficient.

4. The method for parameter identification of fractional-order lithium battery models based on an improved chaotic evolutionary algorithm according to claim 1, characterized in that, The second improvement strategy is specifically as follows: First, we introduce the Givens transformation matrix, expressed as: ； in, Here is the Givens transformation matrix. To optimize the parameter dimension of the parameter optimization problem, This is a variable that is randomly generated each time the Givens transformation is called; Then, the Householder transformation matrix is ​​introduced, with the expression: ； in, For the Householder transformation matrix, A randomly generated unit vector. It is the identity matrix. T Represented as a transpose matrix; Finally, at the end of each iteration of the algorithm, with a 50% probability, one of the Givens transformation and the Householder transformation is randomly selected to act on the candidate solution vector in the population, thereby retaining the best-fitting vector from both the newly generated population and the original population. Individual.

5. The method for parameter identification of fractional-order lithium battery models based on an improved chaotic evolutionary algorithm according to claim 1, characterized in that, The third improvement strategy is as follows: When the algorithm is continuous m If a better solution is not found in the next iteration, a restart mechanism is initiated. Specifically, this restart mechanism involves retaining 70% of the best individuals in the current population and processing the remaining 30% of individuals: each individual has a 50% probability of generating a new solution by randomly rearranging vector elements, and a 50% probability of employing a reverse learning strategy: for a given individual in the population... The opposing solution is generated through a reverse learning strategy, expressed as: ； in, This represents the opposing solution generated through a reverse learning strategy. and These are the lower and upper bounds of the search space.

6. The method for parameter identification of fractional-order lithium battery models based on an improved chaotic evolutionary algorithm according to claim 1, characterized in that, Step 5 specifically involves: Step 5.1: Based on population size Generate based on improved strategy one The parameter combination of the parameter optimization problem is used as the initial population; Step 5.2: Perform a mutation operation on the initialized population. The expression is: ； in, and These are the current individual and the mutated individual, respectively. This is the optimal solution for the current population. a Indicates the search step size. Indicates the direction of evolution. rand (0,1) is a random number uniformly distributed on [0,1]. Step 5.3: Perform a crossover operation on the initial population, for each individual Generate after completing the crossover operation. N test vectors The crossover operation expression is: ； in, This represents the test vector generated by the crossover operation. D This represents the dimension of the test vector. This indicates that the mutated individual is in the first... j Values ​​in dimensions Represents a set A randomly selected integer. This indicates that the individual before the mutation was in the 1st month. j Values ​​in each dimension, for each variable dimension, Used to generate random numbers uniformly distributed in the range [0,1] and cross over with a preset probability. Compare, if the random number is less than If the current dimension is a randomly selected forced crossover point, then the offspring inherits the corresponding variable value of the mutated individual; otherwise, the value of the original parent individual is retained, and the crossover probability is... The value of is updated with each iteration and follows a uniform distribution on the interval [0,1]. Step 5.4: Perform the selection operation, the expression is: ； in, These are the individuals obtained after the selection operation. Individual generated N The vector with the best fitness among the trial vectors, the function For fitness evaluation function; Step 5.5: Apply Improvement Strategy 2 to the new population generated after the selection operation, and update the individuals in the population; Step 5.6: Implement improvement strategy three, and check whether to perform a restart operation after each iteration of the algorithm; Step 5.7: Repeat steps 5.2 to 5.6 until the number of iterations reaches the maximum number of iterations (Maxiter). Output the individual corresponding to the optimal fitness value to obtain the optimal parameter combination for the parameter optimization problem.

7. The method for parameter identification of fractional-order lithium battery models based on an improved chaotic evolutionary algorithm according to claim 6, characterized in that, The generation of evolutionary directions is specifically as follows: First, individuals in the population are randomly paired up. For each paired individual... and Mapped to the ranges [-0.5, 0.5] and [-0.25, 0.25] respectively, the expressions are: ； in, and Individual and The vector after mapping and These are the lower and upper bounds of the search space; Secondly, the two individuals generated and As the initial state variable of the chaotic mapping, it is applied to the following chaotic mapping for... N The next iteration, expressed as: ； in, and Represents an individual and Chaotic samples generated during the iteration process, when n When =0, , , respectively, serve as the initial state variables for the chaotic mapping. k c It is a control parameter used to adjust the dynamic behavior of chaotic mapping; conduct N After the second iteration, 2 is generated. N A chaotic sample, expressed as: ； in, and This represents the chaotic sample set generated after iteration, containing a total of 2 N One chaotic sample; Then, the generated chaotic samples are mapped back to the search space of the original optimization problem, expressed as: ； in, and They represent and The vector after range mapping; Finally, for individuals and Based on and generate N There are several evolutionary directions, expressed as: ； in, and Each represents an individual and The direction of evolution that is generated.

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