CN114372233B - Super capacitor multi-fractional order model parameter identification method based on time domain discretization - Google Patents
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Abstract
The invention discloses a super capacitor multi-fractional order model parameter identification method based on time domain discretization, which comprises the following steps: 1) Determining a proper super capacitor multi-fractional order model and parameters to be identified; 2) Establishing a multi-fractional order transfer function; 3) Carrying out Laplace inverse transformation on fractional calculus operators in the transfer function; 4) Performing discrete approximation processing on a continuous multi-fractional calculus equation; 5) In order to acquire parameters to be identified in an equation, performing DST dynamic stress test on the super capacitor, and collecting actual measurement voltage and current; 6) Performing population optimization identification on parameters to be identified; 7) Substituting the optimal identification parameters into a multi-fractional order model, and calculating the model terminal voltage; 8) Comparing the calculated voltage with the actually measured voltage, if the error meets the requirement, identifying successfully, otherwise, reconstructing the super capacitor multi-fractional order model until the error meets the requirement. The invention can improve the modeling precision of the super capacitor and provide support for the precise charge and discharge control of the super capacitor.
Description
Technical Field
The invention relates to the technical field of super capacitors, in particular to a super capacitor multi-fractional order model parameter identification method based on time domain discretization.
Background
With the rapid development of global economy, fossil energy is rapidly consumed and environmental pollution is continuously worsened. There is an increasing demand for sustainable and renewable energy by humans, and intensive research into clean, efficient energy storage elements is being conducted. The super capacitor is used as a novel energy storage element, has the characteristics between a conventional capacitor and a chemical battery, has the advantages of short charge and discharge time, high power density, long cycle life, wide working temperature range and the like, and is widely used for various application backgrounds such as auxiliary peak power, standby power, storage and regeneration energy, alternative power supply and the like.
Because the porous active carbon of the super capacitor has a complex structure, electrochemical reactions such as an electric double layer, faraday and the like occur inside the porous active carbon. Therefore, in development and application, an accurate, reliable and complex model needs to be established to describe the electrochemical dynamic characteristics of the energy storage system, so as to realize the control of the charge and discharge energy of the energy storage system. In the existing research, the model of the super capacitor is mainly divided into: electrochemical models, equivalent circuit models, and neural network models. In recent years, researchers have found that fractional order elements more closely match the dynamic response characteristics of supercapacitor electrodes and electrolytes in electrochemical reactions. Compared with an integer order model, the fractional order element can bring extra degrees of freedom to the model in order, so that the accuracy of the model can be improved, and meanwhile, the complexity of the model can be reduced.
Compared with the integer-order element, the fractional-order element has more obvious physical significance, and can describe electrochemical dynamic characteristics of a certain stage inside the super capacitor, for example, a common-phase element CPE used for representing frequency dependence caused by roughness or uneven distribution characteristics of the surface of an irregular electrode; a Warburg-like element for exhibiting impedance characteristics caused by ion diffusion effects during linear semi-infinite diffusion; a bounded Warburg element for exhibiting a linear diffusion resistance characteristic in a limited thickness uniform electric double layer. Typically, to expand modeling capabilities and model applicability, super-capacitors are modeled using a fractional order element.
The multi-fractional order model is characterized in that the model comprises two or more fractional order elements, and a plurality of fractional order elements are selected, so that the complete electrochemical characteristics of the super capacitor can be more comprehensively covered, and the modeling precision is greatly improved. However, the modeling research of the multi-fractional element proposed for the super capacitor is relatively few, because the fractional elements with multiple free orders comprise a large number of complex fractional calculus operations, and a certain identification difficulty exists.
The electrochemical impedance spectrum is tested in the frequency domain, which is the most direct method for identifying the dynamic characteristics of the super capacitor, but the data acquisition in the frequency domain not only needs professional and expensive instruments, but also is not easy to detect and use on an application device, and the voltage and current curve in the time domain is convenient to acquire, so that the dynamic characteristics of the super capacitor can be described, and can be used as input data for model identification. Therefore, the invention provides a super capacitor multi-fractional order model parameter identification method based on a time domain, which is simple and easy to carry out.
Disclosure of Invention
The invention aims to overcome the defects and shortcomings of the prior art, and provides a super capacitor multi-fractional order model parameter identification method based on time domain discretization.
In order to achieve the above purpose, the technical scheme provided by the invention is as follows: a super capacitor multi-fractional order model parameter identification method based on time domain discretization comprises the following steps:
1) Determining a proper super capacitor multi-fractional order model and parameters to be identified;
2) Establishing a multi-fractional transfer function by the determined super capacitor multi-fractional model;
3) Carrying out Laplace inverse transformation on fractional calculus operators in the multi-fractional transfer function, and converting the fractional calculus operators into a continuous multi-fractional calculus equation;
4) Based on G-L fractional calculus definition, performing discretization approximation processing on a continuous multi-fractional calculus equation in a time domain;
5) In order to acquire parameters in an equation, performing DST dynamic stress test on the super capacitor monomer to be tested, and acquiring actual measurement voltage and current as data of parameter identification;
6) Formulating an evolution strategy of parameter identification, randomly generating an initial population, determining an objective function and an adaptability function, and carrying out population optimization identification on parameters to be identified to obtain optimal identification parameters;
7) Substituting the optimal identification parameters into the super capacitor multi-fractional order model, and calculating the model terminal voltage, namely calculating the voltage;
8) Comparing the calculated voltage with the actually measured voltage, if the error meets the requirement, successfully identifying the parameters, otherwise, reconstructing the super capacitor multi-fractional order model, and repeating the steps 1) -8) until the error meets the requirement.
Further, in step 1), the supercapacitor multi-fractional order model contains two or more fractional order elements.
Further, in step 1), the parameters to be identified are determined by constituent elements of a supercapacitor multi-fractional order model.
Further, in step 3), the laplace transform of the fractional calculus operator is as shown in formula (1-1):
in the method, in the process of the invention,for fractional calculus operator, alpha is fractional order, alpha E R, R is real number, when alpha > 0,/-is>For differential operation, when α < 0, < +.>For the integral operation, when α=0, +.>t is the current time, t 0 When t is the initial time 0 When=0, it can be omitted from the expression; j is a counting variable; λ is a natural number defined by α, N is a natural number; s is a complex variable; f (t) is a sampling function, and F(s) is a frequency domain function; after the Laplace transformation of the fractional calculus operator, the initial condition f (0) is contained, and as the multi-fractional transfer function of the super capacitor multi-fractional model is defined under the zero initial condition, the Laplace transformation of the fractional calculus operator is shown as the formula (1-2):
further, in step 4), the G-L fractional calculus definition is a limit approximation differential definition, and based on the short-term memory effect principle, discrete approximation processing is performed on the continuous multi-fractional calculus equation, where the expression is shown in the formula (1-3):
in the method, in the process of the invention,for fractional calculus operator, alpha is fractional order, alpha E R, R is real number, when alpha > 0,/-is>For differential operation, when α < 0, < +.>For the integral operation, when α=0, +.>t is the current time, t 0 When t is the initial time 0 When=0, it can be omitted from the expression; j is a counting variable; t is the sampling time; m is the sampling point number; f (·) is the sampling function; />The expression is shown as the following formula (1-4) for Newton's binomial coefficient based on Gamma function:
wherein Γ (·) is a Gamma function, and the expression is represented by the formula (1-5):
in the method, in the process of the invention,as an independent variable, x is an integral variable.
Further, in step 6), the evolution strategy of parameter identification includes: setting initial population scale, total iteration times, selection method and probability, crossover method and probability and mutation method and probability.
Further, in step 6), the objective function is a discretized fractional calculus equation, and the approximate solution is the identified calculated voltage; the fitness function is composed of a calculated mean square error of voltage and measured voltage and a penalty function, and the expression of the fitness function is shown in the formulas (1-6):
wherein Y is the fitness function, A is the penalty coefficient, Y is the error function as shown in the formula (1-7), Y p For penalty functionThe number is shown as the formula (1-8):
wherein j is a counting variable, m is a sampling point number, U p To calculate the voltage, U is the measured voltage.
Compared with the prior art, the invention has the following advantages and beneficial effects:
1. the method of the invention adopts two or more multi-fractional elements to form the super capacitor multi-fractional model. According to the definition of the G-L fractional calculus, a finite difference method is adopted to carry out discretization approximation processing on a multi-fractional calculus equation in a time domain, so that complex operation for solving the multi-fractional calculus is avoided.
2. Compared with an integer order model, the fractional order element can replace a plurality of RC networks, so that the model precision can be improved, and the model complexity can be reduced.
3. Compared with a single fractional order model, the super capacitor multi-fractional order model more comprehensively covers the structural characteristics and electrochemical characteristics of the super capacitor, and accurately reflects the dynamic voltage response characteristics of the super capacitor.
4. Compared with the traditional impedance spectrum test, the method only needs to perform one-time DST dynamic stress test on the super capacitor monomer to be tested, has simple experiment and easy operation, and does not need a professional and expensive electrochemical test instrument.
5. The method is based on a genetic algorithm, parameters are continuously optimized in the identification process, the global searching capability is strong, the convergence is quick, and the identification accuracy is high.
In summary, the invention avoids complex operations for solving the multi-fractional calculus, and is easy to be implemented in operation. The degree of freedom of the orders brought by the multi-fractional order element can obviously improve the identification precision, reduce the complexity of the model, provide support for the accurate charge and discharge control of the super capacitor, have practical application value and are worth popularizing.
Drawings
FIG. 1 is a flow chart of the method of the present invention.
Fig. 2 is a schematic diagram of a multi-fractional order model of a supercapacitor according to an embodiment.
Fig. 3 is a graph of DST dynamic stress testing.
Fig. 4 is a graph of calculated voltage versus.
Fig. 5 is a graph of relative error.
Detailed Description
The present invention will be described in further detail with reference to examples and drawings, but embodiments of the present invention are not limited thereto.
As shown in fig. 1, the invention discloses a super capacitor multi-fractional order model parameter identification method based on time domain discretization, which comprises the following steps:
step one: and determining a proper super capacitor multi-fractional order model and parameters to be identified, wherein the super capacitor multi-fractional order model comprises two or more fractional order elements, and the parameters to be identified are determined by the determined constituent elements of the super capacitor multi-fractional order model.
Step two: and establishing a multi-fractional transfer function by the determined super capacitor multi-fractional model.
Step three: and carrying out Laplace inverse transformation on fractional calculus operators in the multi-fractional transfer function, and converting the fractional calculus operators into a continuous multi-fractional calculus equation, wherein the Laplace transformation of the fractional calculus operators is shown as a formula (1-1):
in the method, in the process of the invention,for fractional calculus operator, alpha is fractional order, alpha E R, R is real number, when alpha > 0,/-is>For differential operation, when α < 0, < +.>For the integral operation, when α=0, +.>t is the current time, t 0 When t is the initial time 0 When=0, it can be omitted from the expression; j is a counting variable; λ is a natural number defined by α, N is a natural number; s is a complex variable; f (t) is a sampling function, and F(s) is a frequency domain function; after the Laplace transformation of the fractional calculus operator, the initial condition f (0) is contained, and as the multi-fractional transfer function of the super capacitor multi-fractional model is defined under the zero initial condition, the Laplace transformation of the fractional calculus operator is shown as the formula (1-2):
step four: based on G-L fractional calculus definition, performing discretization approximation processing on a continuous multi-fractional calculus equation in a time domain; the G-L fractional calculus definition is a limit approximate difference definition, and after a continuous multi-fractional calculus equation is discretized based on a short-term memory effect principle, the expression is shown as the formula (1-3):
in the method, in the process of the invention,for fractional calculus operator, alpha is fractional order, alpha E R, R is real number, when alpha > 0,/-is>In order to perform the differential operation,when alpha < 0, < ->For the integral operation, when α=0, +.>t is the current time, t 0 When t is the initial time 0 When=0, it can be omitted from the expression; j is a counting variable; t is the sampling time; m is the sampling point number; f (·) is the sampling function; />The expression is shown as the following formula (1-4) for Newton's binomial coefficient based on Gamma function:
wherein, f (·) is a Gamma function, and the expression is shown in the formula (1-5):
in the method, in the process of the invention,as an independent variable, x is an integral variable.
Step five: in order to acquire parameters in the equation, DST dynamic stress test is carried out on the super capacitor monomer to be tested, and actual measurement voltage and current are acquired and used as data for parameter identification.
Step six: formulating an evolution strategy of parameter identification, randomly generating an initial population, determining an objective function and an adaptability function, and carrying out population optimization identification on parameters to be identified to obtain optimal identification parameters; among other things, the evolution strategy includes: setting initial population scale, total iteration times, selection method and probability, crossover method and probability and mutation method and probability. The objective function is a discretized fractional calculus equation, the approximate solution of the objective function is the identified calculated voltage, the fitness function is composed of the mean square error of the calculated voltage and the measured voltage, and a penalty function is introduced, and the expression of the fitness function is shown in the formula (1-6):
wherein Y is the fitness function, A is the penalty coefficient, Y is the error function as shown in the formula (1-7), Y p The penalty function is shown in equations (1-8):
wherein j is a counting variable, m is a sampling point number, U p To calculate the voltage, U is the measured voltage.
Step seven: substituting the optimal identification parameters into the super capacitor multi-fractional order model, and calculating the model terminal voltage, namely the calculated voltage.
Step eight: comparing the calculated voltage with the actually measured voltage, if the error meets the requirement, the parameter identification is successful, otherwise, reconstructing the super capacitor multi-fractional order model, and repeating the steps one to eight until the error meets the requirement.
The following further describes the parameter identification method in this embodiment in detail with reference to a supercapacitor multi-fractional order model as shown in fig. 2, which is specifically as follows:
step one: determining a super capacitor multi-fractional order model as shown in FIG. 2, comprising a normal phase CPE element, a Warburg-like element, and a series resistance R s And a parallel resistor R p . Constant phase CPE element and parallel resistor R p Forming a parallel unit, a Warburg-like element and a series resistor R s The super capacitor multi-fractional order model is formed by connecting the super capacitor multi-fractional order models in series;
wherein, the normal phase CPE element is used for representing the frequency dependence caused by the roughness or uneven distribution characteristic of the irregular electrode surface, and can replace a multi-stage RC network in an integer-order equivalent circuit model, and the impedance expression is shown as the formula (1-1):
where C represents the faraday pseudocapacitive capacitive reactance characteristic of the constant phase CPE element, α represents fractional order, and s represents complex variable.
Wherein, the Warburg-like element describes diffusion dynamics for representing impedance characteristics caused by ion diffusion effect in a linear semi-infinite diffusion process, and an impedance expression is shown as formula (1-2):
where W represents the capacitive reactance coefficient of the Warburg-like element and β represents the capacitance distribution.
Wherein, the resistor R is connected in parallel p Transfer resistance for representing charge diffusion inside super capacitor and polarization resistance generated by oxygen reduction reaction, series resistance R s The super capacitor is used for representing the resistance of the super capacitor electrolyte and the current collector.
The parameters to be identified can be determined as [ alpha, beta, R ] according to the super capacitor multi-fractional order model shown in FIG. 2 p ,R s ,C,W]；
Step two: the impedance expression shown in the formula (1-3) is written by the determined super capacitor multi-fractional order model, and a multi-fractional order transfer function of an s domain shown in the formula (1-4) is further established;
wherein C represents Faraday pseudocapacitance capacitive reactance characteristic of the CPE element in normal phase, alpha represents fractional order, W represents capacitive reactance coefficient of the Warburg-like element, beta represents capacitance distribution, s represents complex variable, R s Represents the series resistance, R p Representing the parallel resistance, U(s) representing the super capacitor terminal voltage, and I(s) representing the current flowing through the super capacitor.
Step three: according to the Laplace transformation definition of the fractional calculus operator shown in the formula (1-5), carrying out Laplace inverse transformation on the fractional calculus operator in the transfer function of the formula (1-4), and converting the transfer function of the multi-fractional model into a continuous fractional calculus equation shown in the formula (1-6);
(WD β +CR p WD α+β )u(t)＝
(CR p D α +(R S +R p )WD β +R S CR p WD α+β +1)i(t) (1-6)
in the method, in the process of the invention,for fractional calculus operators, alpha is fractional order, alpha e R, R is a real set, when alpha > 0,for differential operation, when α < 0, < +.>For the integral operation, when α=0, +.>t is the current moment, s is a complex variable; f (t) is a sampling function, F(s) is a frequency domain function, C represents Faraday pseudocapacitive capacitive reactance characteristics of a constant phase CPE element, and W represents a Warburg-like elementBeta represents the capacitance distribution and s represents the complex variable. R is R s Represents the series resistance, R p The parallel resistance is represented, u (t) is the calculated voltage, and i (t) is the measured current.
Step four: performing discretization approximation processing on continuous fractional calculus equations in a time domain according to G-L fractional calculus definitions shown in formulas (1-7), wherein the formulas (1-6) are approximately converted into formulas (1-8);
in the method, in the process of the invention,the method comprises the steps of representing fractional calculus operators, T representing the current moment, alpha being fractional order, alpha epsilon R, R being a real number set, j being a counting variable, and T representing sampling time; m represents the number of sampling points, f (t) is a sampling function, C represents the Faraday pseudocapacitance capacitive reactance characteristic of the constant phase CPE element, W represents the capacitive reactance coefficient of the Warburg-like element, beta represents the capacitance distribution, and s represents the complex variable. R is R s Represents the series resistance, R p The parallel resistance, u, i represent the calculated voltage and measured current, respectively. />The Newton's binomial coefficient based on Gamma function is expressed as the following formula (1-9):
wherein Γ (·) is a Gamma function, the expression of which is shown in the formula (1-10):
in the method, in the process of the invention,as an independent variable, x is an integral variable.
Step five: in order to obtain parameters in the equation, DST dynamic stress test is carried out on the super capacitor monomer to be tested, and the acquired actual measurement voltage and current waveforms are shown in figure 3 and are used as data for parameter identification.
Step six: setting a parameter identification evolution strategy, setting the initial population scale as 100, randomly generating an initial population, ensuring a certain population diversity within an acceptable duration, and enabling each individual in the population to represent a parameter vector to be identified; setting the total iteration number as 100 as a condition for finishing identification; setting the crossover probability to be 0.8, and searching new individuals while maintaining the good structure formed in the population; the probability of variation is set to 0.025, and the ability to generate new individuals is maintained while avoiding sinking to local optima. The discretized fractional calculus equation (1-8) is an objective function of model identification, and the approximate solution of the objective function is the identified calculated voltage u. The fitness function is composed of a calculated voltage and a mean square error of an actually measured voltage, and a penalty function is introduced, the expression of the fitness function is shown as the formula (1-11), and population optimization identification is carried out on parameters to be identified, so that optimal identification parameters are obtained.
Wherein Y is the fitness function, A is the penalty coefficient, Y is the error function as shown in the formulas (1-12), Y p The penalty function is shown in equations (1-13):
wherein j is a counting variable, m is a sampling point number, U p To calculate the voltage, U is the measured voltage.
Step seven: substituting the optimal identification parameters in the following table 1 into the super capacitor multi-fractional order model, and calculating the model terminal voltage, namely the calculated voltage.
Step eight: comparing the calculated voltage with the measured voltage, as shown in fig. 4, the average error is 17.8mV, the maximum error is not more than 80mV, further carrying out relative error analysis as shown in fig. 5, the average relative error is-0.054%, and the maximum relative error is not more than 4%, thereby meeting the error requirement.
TABLE 1 identification parameter Table
The above examples are preferred embodiments of the present invention, but the embodiments of the present invention are not limited to the above examples, and any other changes, modifications, substitutions, combinations, and simplifications that do not depart from the spirit and principle of the present invention should be made in the equivalent manner, and the embodiments are included in the protection scope of the present invention.
Claims (5)
1. The super capacitor multi-fractional order model parameter identification method based on time domain discretization is characterized by comprising the following steps of:
1) Determining a proper super capacitor multi-fractional order model and parameters to be identified;
2) Establishing a multi-fractional transfer function by the determined super capacitor multi-fractional model;
3) Carrying out Laplace inverse transformation on fractional calculus operators in the multi-fractional transfer function, and converting the fractional calculus operators into a continuous multi-fractional calculus equation; the Laplace transformation of the fractional calculus operator is shown as a formula (1-1):
in the method, in the process of the invention,for fractional calculus operator, alpha is fractional order, alpha E R, R is real number, when alpha > 0,/-is>For differential operation, when α < 0, < +.>For the integral operation, when α=0, +.>t is the current time, t 0 When t is the initial time 0 When=0, it can be omitted from the expression; j is a counting variable; λ is a natural number defined by α, N is a natural number; s is a complex variable; f (t) is a sampling function, and F(s) is a frequency domain function; after the Laplace transformation of the fractional calculus operator, the initial condition f (0) is contained, and as the multi-fractional transfer function of the super capacitor multi-fractional model is defined under the zero initial condition, the Laplace transformation of the fractional calculus operator is shown as the formula (1-2):
4) Based on G-L fractional calculus definition, performing discretization approximation processing on a continuous multi-fractional calculus equation in a time domain;
5) In order to acquire parameters in an equation, performing DST dynamic stress test on the super capacitor monomer to be tested, and acquiring actual measurement voltage and current as data of parameter identification;
6) Formulating an evolution strategy of parameter identification, randomly generating an initial population, determining an objective function and an adaptability function, and carrying out population optimization identification on parameters to be identified to obtain optimal identification parameters; the objective function is a discretized fractional calculus equation, and the approximate solution is the identified calculated voltage; the fitness function is composed of a calculated mean square error of voltage and measured voltage and a penalty function, and the expression of the fitness function is shown in the formulas (1-6):
wherein Y is the fitness function, A is the penalty coefficient, Y is the error function as shown in the formula (1-7), Y p The penalty function is shown in equations (1-8):
wherein j is a counting variable, m is a sampling point number, U p For calculating the voltage, U is the measured voltage;
7) Substituting the optimal identification parameters into the super capacitor multi-fractional order model, and calculating the model terminal voltage, namely calculating the voltage;
8) Comparing the calculated voltage with the actually measured voltage, if the error meets the requirement, successfully identifying the parameters, otherwise, reconstructing the super capacitor multi-fractional order model, and repeating the steps 1) -8) until the error meets the requirement.
2. The method for identifying parameters of a super capacitor multi-fractional order model based on time domain discretization according to claim 1, wherein in step 1), the super capacitor multi-fractional order model contains two or more fractional order elements.
3. The method for identifying the parameters of the super capacitor multi-fractional order model based on the time domain discretization according to claim 1, wherein in the step 1), the parameters to be identified are determined by constituent elements of the super capacitor multi-fractional order model.
4. The method for identifying the super capacitor multi-fractional order model parameters based on time domain discretization according to claim 1, wherein in the step 4), the G-L fractional order calculus definition is a limit approximate difference definition, and the discretization approximation processing is performed on a continuous multi-fractional order calculus equation based on a short-term memory effect principle, wherein the expression is as shown in the following formula (1-3):
in the method, in the process of the invention,for fractional calculus operator, alpha is fractional order, alpha E R, R is real number, when alpha > 0,/-is>For differential operation, when α < 0, < +.>For the integral operation, when α=0, +.>t is the current time, t 0 When t is the initial time 0 When=0, it can be omitted from the expression; j is a counting variable; t is the sampling time; m is the sampling point number; f (·) is the sampling function;the expression is shown as the following formula (1-4) for Newton's binomial coefficient based on Gamma function:
wherein Γ (·) is a Gamma function, and the expression is represented by the formula (1-5):
in the method, in the process of the invention,as an independent variable, x is an integral variable.
5. The method for identifying the super capacitor multi-fractional order model parameters based on time domain discretization according to claim 1, wherein in step 6), the evolution strategy of the parameter identification comprises: setting initial population scale, total iteration times, selection method and probability, crossover method and probability and mutation method and probability.
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