CN116702674A - Power inductance fractional order model parameter measurement method - Google Patents
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Abstract
The invention provides a power inductance fractional order model parameter measurement method, which comprises the following steps: 1. constructing a fractional order model of the target power inductor, and expressing the total impedance of the power inductor by using model parameters; 2. designing an experimental current step response of the experimental acquisition power inductor; 3. calculating theoretical current time domain step response of the power inductor by numerical value; 4. collecting experimental currents and theoretical currents at n discrete moments to form n groups of sample data; 5. and establishing an objective function of the fitting parameters based on the sample data, and optimizing the objective function through a PSO particle swarm algorithm to obtain an optimal solution of the inductance parameters. The method provided by the invention has low requirements on experimental equipment and strong operability, and can realize accurate measurement of the fractional order parameters of the inductor, so that the fractional order model can more accurately describe the dynamic characteristics of the power inductor under medium-low frequency, long-term transient and step response.
Description
Technical Field
The invention relates to the technical field of power inductors, in particular to a method for measuring impedance parameters of a fractional order model of a power inductor.
Background
The power inductor is an inductor device which is applied to a power electronic converter and can be used for energy transmission and can transmit high power. The high-power energy transfer station has the function of realizing high-power energy transmission and moving through energy storage and energy release of the power inductor, and is widely applied to the fields of communication, medical treatment, industry, household appliances, automobile electronics and the like. There is no inductance of integer order in the actual converter, and the inductance in actual application has fractional order characteristics. In a power electronic converter, the order of the inductor affects the ripple of the inductor current, and the order of the inductor affects the dynamic performance and stability of the system. In the analysis of power electronic converters and the design of controllers, neglecting fractional differential characteristics of the converters affects ripple of the output voltage and brings errors to the research system. Fractional order impedance models are more realistic in describing power inductances than integer order impedance models. However, it is difficult to obtain accurate parameters of fractional impedance models of power inductances in practical applications.
The existing research at home and abroad mainly focuses on theoretical analysis of a fractional inductance model, but the research on the actual fractional parameter of the inductance in practical application is very few. When fractional modeling is carried out on an actual circuit, the existing research is only stopped in the fractional modeling stage because the fractional parameter of the actual inductance is unknown, and the fractional model theory and the actual model of the converter in actual application cannot be corresponding. Many studies have only performed fractional modeling of the transducer, but the modeling of the transducer is not related to faults such as actual application, design of the controller, etc. Therefore, the real fractional order parameter value of the power inductor designed in the experiment is measured and then substituted into the fractional order theoretical model to obtain a complete real model, and in the application of the power electronic converter, the system parameter can be analyzed more accurately, and the quantitative design of the controller and the like has higher theoretical and application values.
Disclosure of Invention
The invention aims at: the invention provides a fractional order model impedance parameter measurement method of a power inductor, which is based on the characteristic that an actual capacitor and an actual inductor are fractional orders, and effectively solves the technical problem of accurately measuring the fractional order model parameter of the power inductor in practical application by measuring the corresponding curve of an input voltage source step of a power inductor and resistor series circuit.
In order to achieve the above purpose, the present invention provides the following technical solutions: a power inductance fractional order model parameter measurement method comprises the following steps:
step 1, constructing a fractional order model of a power inductor based on a target power inductor, and analyzing the relation between the total impedance of the power inductor and each parameter of the fractional order model to obtain a total impedance expression of the power inductor;
step 2, designing a series experimental circuit of a target power inductor and a resistor R; and input voltage source V in Experimental current step response of a target power inductor is acquired experimentally;
step 3, based on the total impedance expression of the power inductor and the input voltage source V in Obtaining a theoretical current time domain step response of the target power inductor under the voltage source through mathematical calculation;
step 4, obtaining experimental currents and theoretical currents at n discrete moments in acquisition time according to a preset sample number n to form n groups of sample data;
and 5, establishing an objective function of the power inductance fractional order model parameter fitting process based on n groups of sample data, and obtaining an optimal solution of each parameter of the power inductance fractional order model through iterative computation of a PSO particle swarm algorithm.
Further, the foregoing step 1 specifically includes:
constructing a fractional order model of the target power inductance as an internal resistance R 0 And a constant phase angle element lβ; the relationship between the voltage and the current of the constant phase angle element is:
wherein L is β The inductance value of the power inductor, and beta is the order of the power inductor;
the Laplace transform of the formula (1) can be obtained:
U(S)＝L β S β ·I(S) (2)
wherein S is β Is a fractional order operator;
therefore, the total impedance of the fractional order model's power inductance is: z is Z L (S)＝R 0 +L β S β (3)
Further, the step 2 specifically includes:
the voltage source step excitation is connected to two ends of a series circuit of the target power inductor and the resistor R;
measuring a current step response curve at two ends of the inductor by using a current probe of an oscilloscope;
and storing the current step response curve into a CSV format, and introducing the CSV format into MATLAB to obtain the experimental current step response of the target power inductor.
Further, the specific process of the numerical calculation in the step 3 is as follows:
according to the total impedance expression of the power inductor of the fractional order model, under the Laplace domain, the relation between the inductor current and the inductor impedance and the order can be obtained:
the functions commonly used in fractional calculus are employed: and carrying out inverse Laplace transformation on the inductance current in the Laplace domain by using a Mittag-Leffle function and a Gamma function, thereby obtaining a theoretical current time domain step response:
wherein V is cc For inputting electricityAmplitude of pressure source, L β Is the inductance value of the power inductor, R is the resistance value of the series resistor, R 0 Is the internal resistance of the power inductor, beta is the order of the power inductor, E a,b () Is a Mittag-Leffle function.
Further, the objective function of establishing the inductance fraction order parameter fitting in the step 5 is:
wherein i is L Beta (j) is theoretical current data of each discrete moment, i data (j) For each discrete time experimental current data, n is a preset sample number.
Further, the iterative computation process of the PSO particle swarm algorithm in the step 5 specifically includes:
respectively setting particle groups with preset sizes for parameters to be measured of the inductor, and respectively initializing random positions and random speeds of the particle groups;
for the initialized particle swarm, the steps for obtaining the optimal solution of the parameter to be measured are as follows:
s1, calculating the fitness of initial particles, namely the size of a current objective function f (t), and recording the best position pbest of each particle passing by the particle and the best position gbest of each particle group passing by the particle;
s2, updating the speed and the position of the particles according to the following formula, and iterating;
v i ＝w×v i +c 1 ×rand()×(pbest i -x i )+c 2 ×rand()×(gbest-x i ) (7)
x i ＝x i +v i (8)
wherein W is an inertia factor; c (C) 1 ，C 2 As learning factors, C is usually taken 1 ＝C 2 =2; rand () is a random number between 0 and 1;
s3, calculating the fitness of the particles, and recording the best position pbest of each particle passing by and the best position gbest of each particle group passing by;
s4, judging whether the minimum adaptation requirement of a preset objective function is met, and if so, outputting the searched optimal position and the current adaptation degree of the particle swarm, wherein the optimal position is the optimal solution of the corresponding parameter; if not, returning to S2.
Further, the inertia factor W adopts a dynamic value, and a specific calculation formula is as follows:
W(g)＝(W ini -W end )(G k -g)/G k +W end
wherein G is the current iteration number, G k For the preset iteration number upper limit value, the initial value W ini =0.9, final value W end ＝0.4。
Compared with the prior art, the fractional order model impedance parameter measurement method of the power inductor has the following technical effects:
1. the method has low requirement on experimental equipment and strong operability; only a direct-current voltage source is needed to be used as step voltage input, and a current probe is used for measuring the waveform of the inductance output current on an oscilloscope;
2. the method can realize accurate measurement of the inductance fractional order parameter, so that the fractional order model can more accurately describe the dynamic characteristics of the power inductance under medium-low frequency, long-term transient and step response.
Drawings
FIG. 1 is a block diagram showing steps of a power inductor fractional order parameter measurement method according to the present invention;
FIG. 2 is a schematic diagram of a fractional impedance model of a power inductor according to the present invention;
FIG. 3 is a schematic diagram of a series experimental circuit based on a fractional order model of power inductance in the present invention;
FIG. 4 is an experimental current step response of a target power inductor in accordance with the present invention;
FIG. 5 is a flowchart of a PSO particle swarm algorithm according to the present invention;
FIG. 6 is a graph of the optimal individual fitness change in the PSO particle swarm algorithm;
FIG. 7 shows a PSO particle swarm algorithmMedium parameter R 0 Is a optimization curve of (1);
FIG. 8 is an optimized curve of the parameter Lβ in the PSO particle swarm algorithm;
FIG. 9 is an optimized plot of parameter β in the PSO particle swarm algorithm;
FIG. 10 is a comparison of the fitting degree of the experimental current step response curve and the theoretical current time domain step response curve in the present invention.
Detailed Description
For a better understanding of the technical content of the present invention, specific examples are set forth below, along with the accompanying drawings.
Aspects of the invention are described herein with reference to the drawings, in which there are shown many illustrative embodiments. The embodiments of the present invention are not limited to the embodiments described in the drawings. It is to be understood that this invention is capable of being carried out by any of the various concepts and embodiments described above and as such described in detail below, since the disclosed concepts and embodiments are not limited to any implementation. Additionally, some aspects of the disclosure may be used alone or in any suitable combination with other aspects of the disclosure.
As shown in fig. 1, the method for measuring parameters of fractional order model of power inductance provided in this embodiment adopts the following 5 steps to realize the parameter measurement of fractional order model of power inductance:
1. constructing a fractional order model of the target power inductor, and expressing the total impedance of the power inductor by using model parameters;
2. designing an experimental current step response of the experimental acquisition power inductor;
3. calculating theoretical current time domain step response of the power inductor by numerical value;
4. collecting experimental currents and theoretical currents at n discrete moments to form n groups of sample data;
5. and establishing an objective function of the fitting parameters based on the sample data, and optimizing the objective function through a PSO particle swarm algorithm to obtain an optimal solution of the inductance parameters.
Firstly, constructing a fractional order model of a target power inductor, and expressing the total impedance of the power inductor by using model parameters;
as shown in fig. 2, a fractional order model of the power inductance is constructed as an internal resistance R 0 And a Constant Phase Element (CPE). Wherein L is β The voltage and current of a constant phase element are expressed as the following relation:
wherein L is β The inductance value of the power inductor, and beta is the order of the power inductor.
The Laplace transform of the formula (1) can be obtained:
U(S)＝L β S β ·I(S)
wherein S is β Is a fractional order operator.
The total impedance of the fractional order model power inductor is:
Z L (S)＝R 0 +L β S β
secondly, designing an experimental current step response of the experimental acquisition power inductor;
as shown in fig. 3, a fractional order model of the power inductor is connected in series with a resistor R to form a series experimental circuit, and in this embodiment, in order to ensure that the rising time of the power inductor current is suitable, the resistor R selects a power resistor of 10Ω. And (3) connecting voltage sources with the amplitude of 10V at two ends of the series experimental circuit, measuring current step response curves at two ends of the inductor by using a current probe of an oscilloscope, storing the current step response curves in a CSV format, and introducing the current step response curves into MATLAB.
As shown in fig. 4, experimental current step response data waveforms for power inductance were obtained in MATLAB.
Then, calculating theoretical current time domain step response of the power inductor in a numerical value mode; according to the total impedance formula of the power inductor of the fractional order model and the input voltage source, the relation between the inductance current and the inductance impedance as well as the order can be written in the Laplace domain, and the relation is as follows:
the functions commonly used in fractional calculus are employed: and carrying out inverse Laplace transformation on the inductance current in the Laplace domain by using a Mittag-Leffle function and a Gamma function, thereby obtaining a time domain step response relation of the inductance current, and facilitating the later identification of inductance fractional order model parameters.
The Gamma function of Euler is factorial n-! The extension in the real number domain and the complex number domain can meet the use of fractional calculus. The definition formula is as follows:
the Mittag-Leffle function has 2 definition forms, and the expression with only one parameter definition is as follows:
when a=1, it is possible to obtain:
the Mittag-Leffle function has two parameter-defined expressions:
when a=b=1, it is possible to obtain:
the Mittag-Leffle function Laplacian transform with two parameters is as follows
Since the input voltage is a step response of 10V in amplitude, equation (4) can be written as:
performing reverse pull conversion on the formula (11) according to the formula (10), thereby obtaining a time domain step response relation of the inductance current:
then, 1000 experimental currents and theoretical currents at discrete moments are collected to form 1000 groups of sample data; 1000 discrete moments are randomly selected from the acquisition time, and experimental current and theoretical current at each discrete moment are recorded respectively. The experimental current and the theoretical current at each discrete time form 1 group of sample data, and 1000 groups of sample data are obtained in total.
Finally, establishing an objective function of the fitting parameters based on the sample data, and optimizing the objective function through a PSO particle swarm algorithm to obtain an optimal solution of the inductance parameters, wherein in the embodiment, the inductance parameters comprise an inductance value Lbeta and an internal resistance R 0 An order beta.
The objective function for establishing the inductance fractional order parameter fit is as follows:
wherein i is L Beta (j) is numerical analysis current data, i data (j) For the experimental record of current data, n is the total number of samples, n being 1000 in this example.
And (3) searching a parameter optimal solution meeting the minimum of the objective function f (t) by adopting a PSO particle swarm algorithm.
The inductance to be measured parameter is divided into 3 groups of particle swarms, the size of each group of the 3 groups of particle swarms is set to be 100, and the random position and the random speed of the 3 groups of particle swarms are initialized.
The fitness of each particle, i.e. the size of the objective function f (t), was evaluated for the initialized 3-group population. The best position pbest (individual optimum) passed by each particle itself is recorded, and the best position gbest (global optimum) passed by each particle group is recorded.
The velocity and position of the particles are updated according to equations (14) (15) and iterated.
v i ＝w×v i +c 1 ×rand()×(pbest i -x i )+c 2 ×rand()×(gbest-x i ) (14)
x i ＝x i +v i (15)
Wherein, the left part of the equal sign of the speed formula is the sum of the memory term, the self-recognition term and the group-recognition term respectively. C (C) 1 ，C 2 As learning factors, C is usually taken 1 ＝C 2 =2. The rand () is a random number between 0 and 1, w is an inertia factor, and if w is large, the global optimization is strong and the local optimization is weak; if W is small, global optimization is weak and optimization is strong.
In this embodiment, the dynamic value is adopted by w, and the specific calculation formula is as follows:
W(g)＝(W ini -W end )(G k -g)/G k +W end
wherein G is the current iteration number, G k For the preset iteration number upper limit value, the initial value W ini =0.9, final value W end ＝0.4。
And after each iteration, calculating the fitness of the primary particles, and if the minimum adaptation requirement of the objective function is met, outputting the searched optimal position and the current fitness of the particle swarm, and finishing the measurement of the power inductance fractional order model parameters.
The measurement results are shown in the formulas of fig. 6,7,8 and 9, and the internal resistance of the power inductor is 0.0243 Ω, the inductance is 196.27uH and the order is 0.9729 when the adaptation value is 0.3633. And comparing the experimental inductor current step response and the numerical analysis inductor current curve fitting degree, as shown in fig. 10, it can be seen that the method for measuring the power inductor fractional order model parameter provided by the invention has higher accuracy.
The method provided by the invention realizes accurate measurement of the inductance fractional order parameter, so that the fractional order model can more accurately describe the dynamic characteristics of the power inductance under medium-low frequency, long-term transient and step response.
While the invention has been described in terms of preferred embodiments, it is not intended to be limiting. Those skilled in the art will appreciate that various modifications and adaptations can be made without departing from the spirit and scope of the present invention. Accordingly, the scope of the invention is defined by the appended claims.
Claims (7)
1. A method for measuring parameters of a fractional order model of a power inductor, which is used for measuring parameters of a fractional order impedance model of a target power inductor, and is characterized by comprising the following steps:
step 1, constructing a fractional order model of a power inductor based on a target power inductor, and analyzing the relation between the total impedance of the power inductor and each parameter of the fractional order model to obtain a total impedance expression of the power inductor;
step 2, designing a series experimental circuit of a target power inductor and a resistor R; and input voltage source V in Experimental current step response of a target power inductor is acquired experimentally;
step 3, based on the total impedance expression of the power inductor and the input voltage source V in Obtaining a theoretical current time domain step response of the target power inductor under the voltage source through mathematical calculation;
step 4, obtaining experimental currents and theoretical currents at n discrete moments in acquisition time according to a preset sample number n to form n groups of sample data;
and 5, establishing an objective function of the power inductance fractional order model parameter fitting process based on n groups of sample data, and obtaining an optimal solution of each parameter of the power inductance fractional order model through iterative computation of a PSO particle swarm algorithm.
2. The method for measuring parameters of fractional order model of power inductance according to claim 1, wherein the step 1 specifically comprises:
constructing a fractional order model of the target power inductance as an internal resistance R 0 And a constant phase angle element L β Is a series of (1); the relationship between the voltage and the current of the constant phase angle element is:
wherein L is β The inductance value of the power inductor, and beta is the order of the power inductor;
the Laplace transform of the formula (1) can be obtained:
U(S)＝L β S β ·I(S) (2)
wherein S is β Is a fractional order operator;
therefore, the total impedance of the fractional order model's power inductance is: z is Z L (S)＝R 0 +L β S β (3)。
3. The method for measuring the fractional order model parameters of the power inductor according to claim 1, wherein the step 2 specifically comprises:
two ends of a series circuit of the target power inductor and the resistor R are connected with voltage source step excitation;
measuring a current step response curve at two ends of the inductor by using a current probe of an oscilloscope;
and storing the current step response curve into a CSV format, and introducing the CSV format into MATLAB to obtain the experimental current step response of the target power inductor.
4. The method for measuring the fractional order model parameters of the power inductor according to claim 1, wherein the specific process of the mathematical calculation in the step 3 is as follows:
according to the total impedance expression of the power inductor of the fractional order model, under the Laplace domain, the relation between the inductor current and the inductor impedance and the order can be obtained:
the functions commonly used in fractional calculus are employed: and carrying out inverse Laplace transformation on the inductance current in the Laplace domain by using a Mittag-Leffle function and a Gamma function, thereby obtaining a theoretical current time domain step response:
wherein V is cc For the amplitude of the input voltage source, lbeta is the inductance value of the power inductor, R is the resistance value of the series resistor, R 0 Is the internal resistance of the power inductor, beta is the order of the power inductor, E a,b () Is a Mittag-Leffle function.
5. The method for measuring parameters of fractional order model of power inductance according to claim 1, wherein the objective function of establishing the fitting of the fractional order parameters of inductance in step 5 is:
wherein i is L Beta (j) is theoretical current data of each discrete moment, i data (j) For each discrete time experimental current data, n is a preset sample number.
6. The method for measuring parameters of fractional order model of power inductance according to claim 1, wherein the iterative calculation of PSO particle swarm algorithm in step 5 specifically comprises:
respectively setting particle groups with preset sizes for parameters to be measured of the inductor, and respectively initializing random positions and random speeds of the particle groups;
for the initialized particle swarm, the steps for obtaining the optimal solution of the parameter to be measured are as follows:
s1, calculating the fitness of initial particles, namely the size of a current objective function f (t), and recording the best position pbest of each particle passing by the particle and the best position gbest of each particle group passing by the particle;
s2, updating the speed and the position of the particles according to the following formula, and iterating;
v i ＝w×v i +c 1 ×rand()×(pbest i -x i )+c 2 ×rand()×(gbest-x i ) (7)
x i ＝x i +v i (8)
wherein W is an inertia factor; c (C) 1 ，C 2 As learning factors, C is usually taken 1 ＝C 2 =2; rand () is a random number between 0 and 1;
s3, calculating the fitness of the particles, and recording the best position pbest of each particle passing by and the best position gbest of each particle group passing by;
s4, judging whether the minimum adaptation requirement of a preset objective function is met, and if so, outputting the searched optimal position and the current adaptation degree of the particle swarm, wherein the optimal position is the optimal solution of the corresponding parameter; if not, returning to S2.
7. The method for measuring the fractional order model parameters of the power inductor according to claim 5, wherein the inertia factor W adopts a dynamic value, and a specific calculation formula is as follows:
W(g)＝(W ini -W end )(G k -g)/G k +W end
wherein G is the current iteration number, G k For the preset iteration number upper limit value, the initial value W ini =0.9, final value W end ＝0.4。
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