CN109977501B - Estimation Method of Supercapacitor Stored Energy Based on Fractional Calculus - Google Patents

Abstract

The invention discloses a method for estimating stored energy of a supercapacitor based on fractional calculus, comprising: S10 constructing a fractional model of the supercapacitor based on the theory of fractional calculus and determining its model parameters; S20 applying voltage to the fractional model of the supercapacitor Excite the step signal to obtain the dynamic characteristics of the supercapacitor; S30 establishes the constrained minimization problem of supercapacitor terminal voltage estimation based on the dynamic characteristics of the supercapacitor and its fractional order model, and obtains the estimated value of the supercapacitor terminal voltage by solving it based on the genetic algorithm ; S40 establishes a stored energy model of the supercapacitor based on the terminal voltage of the supercapacitor, and then obtains an estimated value of the stored energy of the supercapacitor. The method for estimating stored energy of the supercapacitor further accurately estimates the stored energy of the supercapacitor on the basis of accurately estimating the terminal voltage of the supercapacitor.

Description

Estimation Method of Supercapacitor Stored Energy Based on Fractional Calculus

technical field

The invention relates to the technical field of supercapacitors, in particular to a method for estimating energy stored in a supercapacitor.

Background technique

Nowadays, supercapacitors have been widely used in pulse power supply systems, electric energy recovery systems, electric vehicle energy storage systems, hybrid electric vehicle energy storage systems and other fields. Compared with traditional batteries, the advantage of supercapacitors is that their strong ability to accumulate charge without any chemical reaction increases the number of charge/discharge cycles by hundreds of times; and the high charge/discharge rate makes it possible to Output or absorb energy to the greatest extent in a short period of time to meet the high power density requirements of the application object. However, the premise of ensuring the high reliability of supercapacitors in practical applications is to be able to obtain the energy accumulation information of supercapacitors in time, that is, the stored energy information.

Generally speaking, the energy E of the supercapacitor is calculated and confirmed according to the formula E=1/2CU ² , that is, it is only related to the quantitative rated capacity C and the variable terminal voltage U. However, in practical applications, there is also a diffusion process associated with charge redistribution inside the supercapacitor, so the existing method of determining the energy E based only on the variable of the terminal voltage U is not accurate.

Contents of the invention

Aiming at the deficiencies of the above-mentioned prior art, the present invention provides a method for estimating stored energy of a supercapacitor based on fractional calculus, which effectively solves the technical problem in the prior art that the stored energy of a supercapacitor cannot be accurately estimated.

In order to achieve the above object, the present invention is achieved through the following technical solutions:

A method for estimating energy stored in supercapacitors based on fractional calculus, comprising:

S10 Construct a fractional-order model of a supercapacitor based on fractional-order calculus theory and determine its model parameters;

S20 applies a voltage excitation step signal in the fractional order model of the supercapacitor to obtain the dynamic characteristics of the supercapacitor;

S30 establishes a constrained minimization problem of supercapacitor terminal voltage estimation according to the dynamic characteristics of the supercapacitor and its fractional order model, and obtains an estimated value of the supercapacitor terminal voltage by solving it based on a genetic algorithm;

S40 establishes a stored energy model of the supercapacitor based on the terminal voltage of the supercapacitor, and then obtains an estimated value of stored energy of the supercapacitor.

In the method for estimating stored energy of a supercapacitor based on fractional calculus provided by the present invention, a fractional model of the supercapacitor is established according to the theory of fractional calculus, and a systematic description of complex physical phenomena and dynamic working processes inside the supercapacitor is realized. In addition, based on this theory, the constrained minimization problem of supercapacitor terminal voltage estimation is established to accurately estimate the supercapacitor terminal voltage; finally, the storage capacity of the supercapacitor can be accurately estimated according to the estimated supercapacitor terminal voltage and the established storage energy model. energy.

Description of drawings

A more complete understanding of the invention, and its accompanying advantages and features, will be more readily understood by reference to the following detailed description, taken in conjunction with the accompanying drawings, in which:

Fig. 1 is the schematic flow chart of supercapacitor stored energy estimation method based on fractional calculus in the present invention;

Fig. 2 is the supercapacitor fractional order model that only includes component current-limiting resistance and fractional order capacitor among the present invention;

Fig. 3 adds the supercapacitor fractional order model of series resistance on the basis of supercapacitor fractional order model shown in Fig. 2 in the present invention;

FIG. 4 is a fractional-order model of a supercapacitor with parallel resistance added on the basis of the fractional-order model of the supercapacitor shown in FIG. 3 in the present invention.

Detailed ways

In order to make the content of the present invention clearer and easier to understand, the content of the present invention will be further described below in conjunction with the accompanying drawings. Of course, the present invention is not limited to this specific embodiment, and general replacements known to those skilled in the art are also covered within the protection scope of the present invention.

如式(1)：Fractional calculus is an extension of classical integer-order calculus to the set of real numbers R at fractional α order, α∈R. Assuming that the function f(t) can be differentiable and integrable many times, then the calculus operator of the function f(t) in the range [a,t]

Such as formula (1):

Among them, d ^α /dt ^α is the α-order differential operator.

For the calculus operator in formula (1), according to the definition of Grunwald-Letnikov (GL), it can be further expressed as formula (2):

如式(3)：Among them, h is the sampling time interval, j is an imaginary number, and the binomial

Such as formula (3):

In order to obtain the fractional order model at discrete time, the definition of GL in discrete form is simplified to formula (4):

Using the backward difference method, the fractional derivative of equation (4) at discrete time can be expressed as equation (5):

Wherein, k is a sampling point, and k=0, 1, 2....

In practical systems, due to limited memory and limited computing time, the total samples must be limited to a finite value, and the finite-length discrete time of GL can be approximated as Equation (6):

Wherein, when l=k-j<0, f(l)=0; L is the length of the model.

Based on the above fractional calculus theory, the present invention provides a method for estimating stored energy of a supercapacitor based on fractional calculus to solve the technical problem that the stored energy of a supercapacitor cannot be accurately estimated in the prior art. As shown in Figure 1, in this supercapacitor energy storage estimation method based on fractional calculus includes:

S10 Construct a fractional-order model of a supercapacitor based on fractional-order calculus theory and determine its model parameters;

S20 applies a voltage excitation step signal in the fractional order model of the supercapacitor to obtain the dynamic characteristics of the supercapacitor;

S30 establishes a constrained minimization problem of supercapacitor terminal voltage estimation according to the dynamic characteristics of the supercapacitor and its fractional order model, and obtains an estimated value of the supercapacitor terminal voltage based on the genetic algorithm solution;

S40 establishes a stored energy model of the supercapacitor based on the terminal voltage of the supercapacitor, and then obtains an estimated value of the stored energy of the supercapacitor.

In the fractional-order model of supercapacitor shown in Figure 2, only the component current-limiting resistor and fractional-order capacitor are included, where the supercapacitor current i _C (t) is limited by the current-limiting resistor connected in series with the fractional-order capacitor, and the supercapacitor The terminal voltage u _C (t) is estimated from the model's response to a step signal excited by the input voltage. In this model, choosing an appropriate order α can explain the physical phenomenon of the diffusion process related to the charge redistribution of the supercapacitor in the charge and discharge process under this model. The relationship between the supercapacitor terminal voltage and its current can be expressed as formula (7):

Wherein, R is the resistance value of the current limiting resistor, C _α is the capacity of the fractional order capacitor, and the unit is F/sec ^1-α .

The fractional-order model of the supercapacitor is a first-order inertial system. Applying a voltage excitation step signal to the model can provide energy for the supercapacitor and at the same time test the dynamic characteristics of the supercapacitor. In the Laplace domain (s domain) The fractional order transfer function G(s ^α ) of this model is as formula (8):

Among them, s ^α is the fractional Laplacian operator, T=RC _α .

Considering the low-capacity situation of the supercapacitor, add a series resistor in series with the fractional capacitor in the fractional order model of the supercapacitor shown in Figure 2, as shown in Figure 3. The fractional-order model of the supercapacitor is a phase delay correction system, and the fractional-order transfer function G(s ^α ) in the s domain is as follows:

Wherein, T ₁ =C _α (Rr _S /r _P +R+r _S ), T ₂ =r _S C _α , and r _S is the resistance value of the series resistor.

Considering the impact of the leakage current _IL on the fractional-order model of the supercapacitor in Fig. 3, in the fractional-order model of the supercapacitor shown in Fig. 3, further add a parallel resistance on the branch formed by the fractional-order capacitor and the series resistance in series, Establish the model of the leakage current, as shown in Figure 4. The fractional order transfer function G(s ^α ) of this model in the s domain is as formula (10):

Wherein, K=R/r _P +1, and r _p is the resistance value of the parallel resistor.

In order to fully describe the complex physical phenomenon and dynamic working process inside the supercapacitor, the present invention takes the fractional-order model of supercapacitor with series-parallel resistance as shown in Figure 4 as the research object, and its model parameters are: order α, fractional-order capacitor capacity C _α , the resistance value of the series resistance r _S and the resistance value of the parallel resistance r _p .

In the process of estimating the stored energy of the supercapacitor, set the model parameter vector θ=[α,C _α ,r _S ,r _P ], in order to obtain the model parameter vector θ, the fractional order transfer function G( s ^α ) is inversely transformed by Laplace to obtain the time-domain response of the fractional-order model of the supercapacitor under the voltage excitation step signal, that is, the dynamic characteristics of the supercapacitor, as shown in formula (11):

Among them, u _C (t) is the supercapacitor terminal voltage at time t, and u(t) is the voltage excitation step signal at time t.

After that, design the minimization optimization criterion as formula (12):

为超级电容器分数阶模型在电压激励阶跃信号u(k)下的超级电容器端电压的第k次采样值。Among them, u _C (k) is the kth sampling value of the supercapacitor terminal voltage measured in the experiment,

is the k-th sampling value of the supercapacitor terminal voltage under the voltage excitation step signal u(k) of the supercapacitor fractional model.

After minimizing the initial error of formula (12) by using the least square method, find a model parameter vector θ∈Θ _ad to minimize the square criterion J, and obtain its minimum value Y, as shown in formula (13):

Among them, Θ _ad is the set of allowable parameter values of the model parameters, and N is the total number of samples.

Based on this, the estimated value of the terminal voltage of the supercapacitor is obtained based on the genetic algorithm in the MATLAB simulation environment.

The energy stored in the supercapacitor varies depending on the power supplied per unit time, as in Equation (14):

dE(t)=P(t)dt(14)

The power of a supercapacitor can also be expressed as the product of terminal voltage and current, that is, the energy change within a given time _t can be rewritten as equation (15):

dE(t)＝ _uC (t) _iC (t)dt(15)

Therefore, within the time interval [t ₁ , t ₂ ], the total stored energy E _tot of the supercapacitor can be obtained by integrating the energy change during this time, as shown in formula (16):

According to the relationship between the supercapacitor terminal voltage and its current in formula (7), the total stored energy can be expressed as formula (17):

在时间间隔[0,t]内超级电容器中的总存储能量E(t)如式(18)：In equation (17), it is assumed that t ₁ =0, and

The total stored energy E(t) in the supercapacitor in the time interval [0,t] is as formula (18):

Thus, after estimating the terminal voltage of the supercapacitor, based on the fractional calculus theory, the stored energy of the supercapacitor can be further estimated according to this formula.

It is worth noting that in formula (18), when α=1, the classical energy calculation formula as in formula (19) is obtained:

Claims (2)

1. the supercapacitor energy storage estimation method based on fractional calculus is characterized by comprising the following steps of:

s10, constructing a fractional order model of the supercapacitor based on a fractional order calculus theory and determining model parameters of the model;

s20, applying a voltage excitation step signal to the fractional order model of the supercapacitor to obtain the dynamic characteristics of the supercapacitor;

s30, establishing a constrained minimization problem of supercapacitor terminal voltage estimation according to the dynamic characteristics of the supercapacitor and a supercapacitor fractional order model, and solving to obtain an estimated value of the supercapacitor terminal voltage based on a genetic algorithm;

s40, establishing a storage energy model of the super capacitor based on the terminal voltage of the super capacitor, and further obtaining an estimated value of the storage energy of the super capacitor;

in step S10, the supercapacitor score order model includes: the device comprises a current limiting resistor, a fractional capacitor, a series resistor and a parallel resistor, wherein the fractional capacitor is connected with the series resistor in series and then further connected with the parallel resistor in parallel to obtain a parallel branch, and the current limiting resistor is connected with one end of the parallel branch in series;

the model parameters of the supercapacitor fractional order model are as follows: order alpha, fractional order capacitor capacity C _α Resistance r of series resistor _S Resistance r of parallel resistor _p ；

In step S20, after applying a voltage excitation step signal to the supercapacitor fractional order model, the transfer function G (S ^α ) The method comprises the following steps:

wherein k=r/R _P +1，T ₁ ＝C _α (Rr _S /r _P +R+r _S )，T ₂ ＝r _S C _α R is the resistance value of the current-limiting resistor, R _p The resistance value of the parallel resistor, C _α R is the capacity of fractional order capacitor _S Is the resistance value of series resistance s ^α Is a fractional order Laplacian;

in the time domain, the dynamic characteristics of the supercapacitor are:

wherein d ^α /dt ^α As an alpha-order differential operator, u _C (t) is the supercapacitor terminal voltage at time t, u (t) is the voltage excitation step signal at time t;

in step S30, the constrained minimization problem of supercapacitor-side voltage estimation is to find a model parameter vector θεΘ _ad With the least squares criterion J, the minimum value Y of the least squares criterion J is obtained:

wherein, model parameter vector θ= [ α, C _α ,r _S ,r _P ]，Θ _ad Allowing a set of parameter values for model parameters, N being the total number of samples;

u _C (k) For the kth sampling value of the experimentally measured supercapacitor-side voltage, < >>

The kth sampling value of the supercapacitor terminal voltage at the voltage excitation step signal u (k) is used as the supercapacitor fractional model.

2. The supercapacitor energy storage estimate of claim 1The counting method is characterized in that in step S40, at time interval [ t ] ₁ ,t ₂ ]Total stored energy E of inner supercapacitor _tot The method comprises the following steps:

wherein i is _C (t) is the supercapacitor current at time t, and

carry it into the post-consumer total energy storage E _tot The method comprises the following steps:

the power of a supercapacitor is expressed as the product of the terminal voltage and current, i.e. the energy change over a given time t:

dE(t)＝u _C (t)i _C (t)dt。

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