Method for calculating activation energy of XLPE submarine cable based on fractional poynting-thomson model

Abstract

The application discloses a method for calculating the activation energy of XLPE submarine cables based on a fractional Poynting-Thomson model, which comprises the following steps: step 1: first, define a fractional element, and connect the fractional elements in series and in parallel to obtain a fractional Poynting-Thomson model; step 2: establish a relationship formula suitable for the fractional Poynting-Thomson model and the complex dielectric capacity of the XLPE submarine cable; step 3: construct a target function for parameter identification, and identify each parameter in the fractional Poynting-Thomson model by using a grey wolf algorithm; step 4: according to the parameter identification result in step 3, select appropriate parameters as characteristic quantities for calculating the activation energy of the XLPE submarine cable; analysis on the parameter identification result can show that the relaxation time satisfies the Arrhenius empirical formula; and step 5: establish a linear relationship between the parameters and the relaxation time, and calculate the activation energy of the XLPE submarine cable by fitting the slope of the straight line. The method applies the fractional order theory to the analysis of the relaxation process of the XLPE submarine cable, and effectively solves the problems that the operation steps of the traditional frequency-temperature method for calculating the activation energy are complicated and are greatly influenced by human factors.

Description

Technical Field

[0001] This invention relates to the field of submarine cable insulation aging assessment technology, specifically to a method for calculating the activation energy of XLPE submarine cables based on the fractional Poynting–Thomson model. Background Technology

[0002] Cross-linked polyethylene (XLPE) is used as the insulation layer for electrical and submarine cables due to its superior electrical, heat resistance, and mechanical properties. XLPE cables are widely used in onshore and offshore power transmission and distribution lines. With the development of offshore wind power technology, offshore wind power has become an important direction for national clean energy development. Unlike the transmission lines of onshore wind farms, offshore wind farms generally use submarine cables to transmit electrical energy. Submarine cables are affected by many factors during operation, including mechanical damage, chemical corrosion, and thermal aging. Among these, thermal aging is one of the main factors causing a decline in the insulation performance of submarine cables, and can even lead to insulation failure. Therefore, it is of great significance to conduct research on the insulation aging characteristics of submarine cables and evaluate their aging state.

[0003] Compared to conventional insulation testing methods, the frequency-temperature translation method can achieve non-destructive testing, providing an effective technical approach for inspecting the insulation condition of electrical equipment. It has become a research hotspot for many scholars, and progress has been made in methods for calculating activation energy, modeling reaction mechanism functions, and predicting the lifespan of insulation materials. Activation energy is an intrinsic characteristic of aging in insulation materials, reflecting some potential defects, assessing the insulation state, and evaluating the physicochemical properties of the material.

[0004] Currently, the main methods for calculating the activation energy of cables are thermogravimetric analysis (TGA) and frequency-temperature shifting (FTS). TGA is commonly used to calculate the activation energy of chemical reactions. However, this method only measures the activation energy of a portion of the cable's insulation aging sample, failing to accurately reflect the true aging state of the entire cable. Furthermore, it is destructive to the cable itself and cannot be used to measure the activation energy of cables currently in operation, thus having certain limitations. The FTS method for calculating the activation energy of insulation materials is a relatively new approach that offers possibilities for field applications. It enables non-destructive testing and provides an effective technical method for inspecting the insulation condition of electrical equipment. However, its calculation steps are complex, requiring multiple shifts of the frequency domain dielectric curve, making practical operation cumbersome. Summary of the Invention

[0005] To address the lack of a simple and effective method for calculating the activation energy of XLPE submarine cables in existing technologies, this invention applies fractional-order theory to analyze the relaxation process of XLPE submarine cables, proposing a method for calculating the activation energy of XLPE submarine cables based on the fractional Poynting–Thomson model. This method effectively solves the problems of cumbersome operation steps and significant influence from human factors in the traditional frequency-temperature method for calculating activation energy.

[0006] The technical solution adopted in this invention is as follows:

[0007] The method for calculating the activation energy of XLPE submarine cables based on the fractional Poynting–Thomson model includes the following steps:

[0008] Step 1: First, using fractional calculus theory, define a component with dielectric properties between resistance and capacitance—the fractional element. Connect the fractional elements in series and parallel to obtain the fractional Poynting-Thomson model.

[0009] Step 2: Establish the relationship between the fractional Poynting-Thomson model and the complex dielectric capacitance of XLPE submarine cables;

[0010] Step 3: Construct the objective function for parameter identification, and use the Grey Wolf algorithm to identify each parameter in the fractional Poynting-Thomson model;

[0011] Step 4: Based on the parameter identification results in Step 3, select appropriate parameters as characteristic quantities for calculating the activation energy of the XLPE submarine cable; analysis of the parameter identification results shows that the relaxation time τ satisfies the Arrhenius empirical formula.

[0012] Step 5: Establish a linear relationship between parameters 1 / τ and 1 / T, and calculate the activation energy of the XLPE submarine cable by fitting the slope of the straight line.

[0013] In step 1, the derivation process of the fractional Poynting-Thomson model is as follows:

[0014] Using fractional calculus theory, the concept of fractional elements is proposed by combining the electrical characteristics of resistors R and capacitors C. Ideal capacitors and resistors are circuit elements that satisfy the following equations.

[0015] For a capacitance C under ideal dielectric conditions, the following equation holds:

[0016]

[0017] For a resistor R, the following equation applies:

[0018]

[0019] The fractional element Z is a circuit device that satisfies the formula:

[0020]

[0021] In the formula: U C (t), U R (t), U Z (t) represents the voltage across the capacitor, resistor, and fractional element, respectively; I(t) is the current flowing through the element; Q C Q R Q Z The component represents the electrical charge; τ is the relaxation time; d α Q(t) / dt α Let f be the fractional derivative of the charge in the time domain. When the shape parameter α is 0 or 1, the fractional element represents the ideal capacitor and the resistor, respectively.

[0022] Different models can be formed by connecting fractional elements in series and parallel to describe the dielectric relaxation process of a medium. Figure 1 It is a diagram of the fractional Poynting-Thomson model, which is composed of two independent fractional elements connected in parallel, and then connected in series with a third fractional element.

[0023] In step 2, the fractional dynamic equations of the fractional Poynting-Thomson model are obtained based on the current distribution principle of series-parallel circuits in step 1:

[0024] Figure 1 This is a fractional Poynting-Thomson model diagram, which consists of two independent fractional elements Z1(α,C1,τ1) and Z2(β,C2,τ2) connected in parallel, followed by a series fractional element Z3(γ,C3,τ3). The voltages across the two parallel fractional elements (α,C1,τ1) and (β,C2,τ2) are the same, and the sum of the partial charges flowing through the two elements equals the total charge of the entire series-parallel structure. Therefore... Figure 1 The relationship between U and Q in the parallel section of the circuit shown can be obtained as follows:

[0025]

[0026] U Z (t) and Q Z (t) represents the voltage and charge across the fractional Poynting-Thomson model, respectively;

[0027] Based on the principle of equal current in series circuits, the fractional dynamic equations of the fractional Poynting-Thomson model are as follows:

[0028]

[0029] In the formula: α, β, and γ are shape parameters related to the relaxation time distribution, and τ=(C2τ1) α / C1τ2 β ) 1 / (α-β) C a =C1(τ1 / τ) α C b =C3(τ3 / τ) -γ .

[0030] By performing a Fourier transform on the fractional Poynting-Thomson model, the complex capacitance and complex permittivity of the model can be obtained.

[0031]

[0032] In the formula: α, β, and γ are shape parameters related to the relaxation time distribution, and τ=(C2τ1) α / C1τ2 β ) 1 / (α-β) C a =C1(τ1 / τ) α C b =C3(τ3 / τ) -γ ;C1, C2, C3, τ1, τ2, τ3, α, β, γ are as follows Figure 1 The model parameters are obtained from the three fractional elements Z1, Z2, and Z3. These three fractional elements are connected in parallel and then in series to form a fractional Poynting-Thomson model.

[0033] From the two-segment electrical expressions of a single fractional element in the above equation, and based on the fundamental relationship between voltage and current in a circuit, the voltage U across the fractional Poynting-Thomson model is derived. Z The expression for (t) can be obtained by using the Fourier transform to obtain the complex capacitance C of the fractional Poynting-Thomson model. * (ω).

[0034] τ and C in the fractional Poynting-Thomson model a C b The parameters in the model are all composed of the parameters of the three fractional elements and are derived from the formula. However, the parameters in the fractional Poynting-Thomson model have no practical significance.

[0035] The parameter α mainly determines the height and width of the loss peak in the mid-frequency part of the real part C′ and the imaginary part C″ of the dielectric capacitance;

[0036] Parameters β and γ are frequency domain parameters that determine the high-frequency characteristics and low-frequency relaxation characteristics of the complex capacitor, respectively.

[0037] Parameter C a C b This is a constant that affects the high and low frequency limits of C′.

[0038] In step 3, the parameter identification process is as follows:

[0039] S3.1: To analyze the relationship between the parameters of the fractional Poynting-Thomson model and the degree of degradation of the submarine cable sample, the experimentally measured complex dielectric capacitance C... * Substitute the -f data into the complex capacitance formula in step 3;

[0040] S3.2: To ensure the accuracy of parameter identification results, establish the relationship between measured values ​​C′, C″ and fitted value C′. m C m The objective function K is constrained, and the least squares method is used to minimize K.

[0041]

[0042] C′ and C″ are the real and imaginary parts of the complex capacitance of the XLPE submarine cable sample measured using the frequency domain dielectric spectrum testing platform of this invention, respectively.

[0043] C′ m C m "These are the real and imaginary parts of the complex dielectric capacitance obtained by fitting various model parameters after parameter identification based on the fractional Poynting-Thomson model parameters."

[0044] The parameters of the fractional Poynting-Thomson model should conform to the actual physical meaning. After repeated experiments, the constraints for each parameter were determined as follows: 0 < α, β, γ < 1, 10. -11 <C a C b <10 -9 0 < τ < 8000.

[0045] Since there are many model parameters in actual parameter identification, the selection of initial parameter values ​​is crucial to the convergence of the optimization algorithm's iterative calculations and the accuracy of the fitting results. By combining the physical meanings associated with different model parameters, multiple runs are performed to verify the constraints on the model parameters, ensuring the accuracy of this part of parameter identification in S3.2.

[0046] In step 4, the experimental results are analyzed, and it can be seen that the relaxation time τ satisfies the Arrhenius empirical formula, such as... Figures 5(a) to 5(c)As shown in the figure, the following equation is the Arrhenius empirical formula. From the equation, it can be seen that there is a linear relationship between the reaction rate K and the reciprocal of temperature 1 / T. At the same time, the reciprocal of relaxation time 1 / τ obtained after parameter identification also shows a linear relationship with the reciprocal of temperature 1 / T. Therefore, the reaction rate K in the Arrhenius empirical formula can be equivalently represented by the reciprocal of relaxation time 1 / τ, which verifies that the relaxation time 1 / τ satisfies the Arrhenius empirical formula.

[0047]

[0048] Figure 5 shows the functional relationship between 1 / τ and 1 / T (the reciprocal of temperature), which is as follows:

[0049] Aging for 144 hours: ln(1 / τ) = -4002.253(1 / T) + 6.509

[0050] Aging time 288h: ln(1 / τ)=-7018.891(1 / T)+14.341

[0051] Aging for 576 hours: ln(1 / τ) = -5560.312(1 / T) + 10.745.

[0052] Equating ln(1 / τ) to k in the Arrhenius empirical formula, the activation energy of the XLPE submarine cable is calculated as follows: Based on the Eling transition state theory, Arrhenius proposed the Arrhenius empirical formula, which is the relationship between the reaction rate constant and temperature:

[0053]

[0054] In the formula: A is the pre-exponential factor, also known as the frequency factor; E is the material activation energy; k is the reaction rate constant; R is the ideal gas constant, R = 8.314 J / (mol·K); T is the absolute temperature.

[0055] Taking the logarithm of both sides of the above equation, we get:

[0056]

[0057] In the above formula, lnk and 1 / T have a linear relationship. Parameter identification of the measured dielectric spectra of each group of aged samples at different temperatures revealed that the relaxation time τ satisfies the Arrhenius empirical formula, i.e., there is a linear relationship between 1 / τ and 1 / T:

[0058]

[0059] In the formula: A τ E is the frequency factor. τ It is the activation energy.

[0060] Taking the logarithm of both sides of the above equation, we get:

[0061]

[0062] From the above equation, we can see that there is a linear relationship between ln(1 / τ) and 1 / T. By fitting a straight line between ln(1 / τ) and 1 / T, we can obtain the slope of this line as E. τ / R, where R is the ideal gas constant, is a constant. Multiplying the slope of the fitted line by R gives the activation energy E of the sample. τ .

[0063] This invention discloses a method for calculating the activation energy of XLPE submarine cables based on the fractional Poynting–Thomson model, with the following technical advantages:

[0064] 1) In the method of this invention, the frequency domain dielectric spectrum of XLPE submarine cables with different aging degrees is measured at different test temperatures using frequency domain dielectric spectroscopy. After parameter identification, the relationship between the parameters of each fractional Poynting-Thomson model and the test temperature is analyzed. It is found that the relaxation time 1 / τ in the fractional model is equivalent to k in the Arrhenius empirical formula. Therefore, a new method for calculating the activation energy of XLPE submarine cables is proposed, which effectively solves the problems of cumbersome operation steps and large influence of human factors in the traditional frequency-temperature method.

[0065] 2) In the method of this invention, fractional-order theory is applied to the relaxation process of dielectrics, a new activation energy calculation method is proposed and its accuracy is verified, providing a new approach for calculating the activation energy of XLPE submarine cables. Attached Figure Description

[0066] Figure 1 This is a structural diagram of the fractional Poynting-Thomson model.

[0067] Figure 2 This is the frequency domain dielectric spectrum testing platform in this invention.

[0068] Figure 3(a) shows the frequency domain dielectric level shift curve of the XLPE submarine cable sample after aging for 144 hours;

[0069] Figure 3(b) shows the frequency domain dielectric shift curve of the XLPE submarine cable sample after aging for 288 hours.

[0070] Figure 3(c) shows the frequency domain dielectric level shift curve of the XLPE submarine cable sample after aging for 576 hours.

[0071] Figure 4(a) is a fitting calculation diagram of the activation energy of the XLPE submarine cable sample after aging for 144 hours;

[0072] Figure 4(b) is a fitting calculation diagram of the activation energy of the XLPE submarine cable sample after aging for 288 hours;

[0073] Figure 4(c) is a fitting calculation diagram of the activation energy of the XLPE submarine cable sample after aging for 576 hours.

[0074] Figure 5(a) shows the linear relationship between ln(1 / τ) and 1 / T after 144 hours of aging;

[0075] Figure 5(b) shows the linear relationship between ln(1 / τ) and 1 / T after aging for 144 hours.

[0076] Figure 5(c) shows the linear relationship between ln(1 / τ) and 1 / T after 144 hours of aging.

[0077] Figure 6 This is a flowchart of the XLPE submarine cable activation energy calculation method of the present invention.

[0078] Figure 7 Flowchart for parameter identification. Detailed Implementation

[0079] The method for assessing the insulation status of submarine cables based on the fractional Poynting-Thomson model includes the following steps:

[0080] Step 1: Using fractional calculus theory, the concept of "fractional elements" is proposed. Fractional elements are connected in series and parallel to obtain the fractional Poynting-Thomson model. The fractional elements are connected in series and parallel to form the fractional Poynting-Thomson model, as shown below. Figure 1 As shown.

[0081] At this point, based on the current distribution principle of series and parallel circuits, the PT modulus fractional dynamic equation can be obtained.

[0082]

[0083] Step 2: Establish the relationship between the parameters of the fractional Poynting-Thomson model applicable to XLPE submarine cable insulation and the complex dielectric capacitance, a characteristic quantity of the frequency domain dielectric spectrum.

[0084] In step one, the complex capacitance formula of the fractional Poynting-Thomson model can be obtained by Fourier transformation of the fractional Poynting-Thomson model.

[0085]

[0086] In the formula: τ=(C2τ1) α / C1τ2 β ) 1 / (α-β) C a=C1(τ1 / τ) α C b =C3(τ3 / τ) -γ ;

[0087] Parameter α primarily determines the height and width of the loss peak in the mid-frequency portion of the real part C′ and the imaginary part C″ of the dielectric capacitance; parameters β and γ are frequency domain parameters that determine the high-frequency characteristics and low-frequency relaxation characteristics of the complex dielectric capacitance, respectively; parameter C a C b This is a constant that affects the high and low frequency limits of C′.

[0088] Step 3: The parameters of the fractional Poynting-Thomson model should conform to the actual physical meaning. After repeated experiments, the constraints for each parameter are determined as follows: 0 < α, β, γ < 1, 10. -11 <C a C b <10 -9 0 < τ < 8000.

[0089]

[0090] After the initial values ​​of the parameters are set, an optimization objective function is established to optimize each model parameter. The least squares method is used to minimize the constraint function K. The optimal solution of the parameters in the fractional Poynting-Thomson model is obtained by the Grey Wolf Optimization Algorithm.

[0091] The Grey Wolf Optimization (GWO) algorithm is an intelligent algorithm that simulates the predation behavior and prey allocation of grey wolf packs in the animal kingdom. This algorithm represents the entire hunting activity of the wolf pack, including three intelligent behaviors (searching, encircling, and attacking) and the "winner-takes-all" alpha wolf selection rule. Based on the wolf pack's predation behavior and hierarchy, the dielectric model of the GWO algorithm establishes a wolf with the optimal fitness value as alpha wolf a, followed by wolves b and c, with wolf d defined as the remaining individual in the pack. Under the leadership of wolf a, the optimal solution of the model is continuously searched. The mathematical model of the distance between the wolf and prey during the wolf pack's hunting and encirclement process is shown in the following formula.

[0092] D = |CX P (n)-X(n)|

[0093] The gray wolf position update model is shown in the following formula.

[0094] X(n+1)=X P (n)-A·D

[0095] in:

[0096] A = 2qr1-q

[0097] C = 2r²

[0098] In the formula, D represents the distance between the wolf and its prey, and X... P X(n) represents the current position of the prey (i.e., in the nth iteration), X(n) represents the position of the gray wolf in the nth iteration, A and C represent constant variables, q is the convergence factor, and r1 and r2 are random numbers uniformly distributed within the range [0,1]. By updating the values ​​of A, C, and q, gray wolves a, b, and c can reach the target area around the prey. By changing r1 and r2, the gray wolves can move arbitrarily within the target area. When the gray wolf pack identifies the location of the target, the nearest gray wolves a, b, and c make a reasonable judgment, and then, under the leadership of the alpha wolf a, they surround and hunt the target. The above process can be expressed by a mathematical model, as shown in the following formula:

[0099] D a =|C1X a (n)-X(n)|

[0100] D b =|C2X b (n)-X(n)|

[0101] D a =|C3X c (n)-X(n)|

[0102] X1 = X a -A1·D a

[0103] X2 = X b -A2·D b

[0104] X3 = X c -A3·D c

[0105] X(n+1)=(X1+X2+X3) / 3

[0106] D a D b D c , where , represents the distance between gray wolves a, b, and c and their prey, and C is... a C b C c Let X(n) be a constant variable, representing the positions of gray wolves a, b, and c in the nth iteration. a (n), X b (n), X c (n) represents the position of the prey at the nth iteration when the gray wolves a, b, and c are in their positions. X1, X2, and X3 are the position update models for gray wolves a, b, and c.

[0107] The following formula can be used to determine the direction in which the prey escapes, and by continuously searching for prey, the optimal solution can be found from the global perspective.

[0108] X(n+1)=(X1+X2+X3) / 3

[0109] In the formula: X(n+1) is the position used to determine the prey's escape.

[0110] This invention applies chaotic sequences to improve the traditional Grey Wolf algorithm, enabling it to more accurately identify the parameters of the fractional Poynting-Thomson model. Chaotic sequences represent irregular motions occurring in a deterministic system, often exhibiting simple patterns behind random phenomena and possessing strong ergodicity. The original GWO algorithm suffers from low population diversity and uneven distribution within the search region during initialization, significantly impacting subsequent iterations and easily leading to local optima. Therefore, introducing the Logistic mapping into the GWO algorithm improves population diversity during initialization, effectively suppressing local optima traps in the early iteration stages and accelerating the optimization process.

[0111] The CGWO (Chaotic Gray Wolf) algorithm is used to identify parameters of the Poynting-Thomson model. The algorithm flow is as follows: Figure 7 As shown. First, 100 gray wolves are generated in the initial range, and the top 3 gray wolves are named gray wolves a, b, and c. They lead the remaining gray wolves d towards the target area (optimal solution). An initial population is generated using a chaotic sequence, and the positions of gray wolves are randomly generated. This process is repeated 99 times, generating 99 chaotic parameters, and finally obtaining the positions of 100 gray wolves. The positions of the gray wolves are updated, and gray wolves a, b, and c are updated again in the d wolves according to the calculated K, until the number of iterations is reached, the optimal position of the gray wolves is found, and the optimal solution with the score of Poynting-Thomson is output.

[0112] Step 4: Based on the results of the fractional Poynting-Thomson model identification in Step 3, it was found that the relaxation time τ satisfies the Arrhenius empirical formula, that is, there is a linear relationship between 1 / τ and 1 / T. By fitting the slope of the straight line, the activation energy of the XLPE submarine cable was calculated.

[0113] Step 5: Calculate the activation energy using the frequency-temperature method, and compare the calculated activation energy with the calculation results in this invention to verify the accuracy of the method of this invention.

[0114] The frequency-temperature shift method describes the relationship between the aging rate and aging temperature of insulating materials based on the time-temperature equivalence principle. That is, for molecular motion, increasing temperature and extending time are equivalent. Researchers have experimentally discovered that the dielectric curves in the frequency-domain dielectric spectrum at different temperatures can be shifted to a single principal curve, with the amplitude remaining unchanged before and after the shift. In the frequency-domain dielectric spectrum, the frequency and temperature before and after the shift follow the Arrhenius equation, i.e.:

[0115]

[0116] In the formula: a T Temperature shift coefficient, f0 is the frequency of a point in the dielectric spectrum at temperature T0 before shift; f is the frequency of that point in the dielectric spectrum at temperature T after shift; E a is the activation energy of the insulating material; k is the Boltzmann constant, k = 1.38 × 10⁻⁶. -23 J / K. T is the measured temperature of the curve, and T0 is the reference temperature (Kelvin temperature), which is selected as 40℃ in this embodiment. The frequency domain dielectric spectrum of the sample is measured at different temperatures, and each measured dielectric curve is shifted to the reference temperature to solve for α at each aging period. T And by fitting a T The slope is obtained from the linear relationship between 1 / T0 and 1 / T, and its slope is E. a / k, multiplied by k, yields the activation energy. This is then compared with the method of this invention to verify the accuracy of the method presented herein. Example:

[0117] In this embodiment, a 220kV XLPE submarine cable was selected as the raw material for sample preparation. First, the insulation layer and inner and outer semi-conductive layers at both ends of the submarine cable sample were stripped until the cable core was exposed. Next, the protective sheath in the middle section of the submarine cable was stripped away, exposing the insulating shielding layer and the submarine cable insulation layer (XLPE layer). A cable slicer was used to cut the submarine cable insulation layer into sheet-like samples with a thickness of 0.5±0.05mm. The submarine cable samples were pretreated: after treatment, the samples were placed in a dry environment at 40℃ and left to stand for 24 hours, followed by 48 hours of degassing and drying. Dust and other contaminants on the sample surface were then removed with alcohol.

[0118] According to the standard IEC 60216-2013 "Heat Resistance of Electrical Insulation Materials", 135℃ should be used as the aging temperature in the accelerated thermal aging test of insulation materials. This temperature was selected as the accelerated aging temperature for this test, and the 401B type thermal aging test chamber was used as the aging equipment in this test. Based on the standard IEC 60811-1-4, the accelerated aging times were set to 0h, 144h, 288h, and 576h. Four test temperatures were selected in this test: 40℃, 50℃, 60℃, and 70℃, which were achieved by setting the temperature of the constant temperature chamber.

[0119] Frequency domain dielectric response testing was performed using an IDAX-300 insulation tester manufactured by Megger Corporation, USA. The output AC voltage was set to 140V, and the test frequency range was 10. -3 ~10 3 Hz. The measuring electrode (e.g.) Figure 2 The blue line (shown in the middle) connects to the submarine cable insulation shield, and the high-voltage electrode (red line) connects to the cable core. Copper rings were added to the cross-linked polyethylene insulation surface at both ends of the submarine cable sample and grounded to prevent leakage current from affecting the test results. Discharge was initiated for two hours before each group of experiments. Following the above test procedures, the frequency domain dielectric spectrum of each group of aging submarine cable samples was measured at 40℃, 50℃, 60℃, and 70℃. The test results are as follows: Figures 5(a) to 5(c) As shown.

[0120] The curves of various submarine cable samples at different temperatures show that the overall trend of the frequency domain dielectric spectrum curves did not change significantly with the change of test temperature. However, temperature had a greater impact on the dielectric properties of the samples in the low-frequency range, resulting in obvious low-frequency dispersion. With increasing temperature, the dielectric loss gradually increased, reaching a certain level at 10... -3 ~10 -1 This is especially noticeable in the Hz frequency range.

[0121] The reason for the above phenomenon is that various polarizations are established in the high-frequency range, while in the low-frequency range, dielectric loss is mainly conductivity loss, with relaxation loss contributing less, and the relaxation process is not manifested. As the temperature rises, the thermal motion of molecules in the dielectric intensifies, the number of collision ionizations increases, and the conductivity of the submarine cable sample is enhanced. Therefore, as the temperature rises, the dielectric loss gradually increases.

[0122] This invention provides a method for measuring the complex dielectric capacitance of samples with different aging degrees at different temperatures using frequency-domain dielectric spectroscopy. After parameter identification, it was found that the relaxation time τ satisfies the Arrhenius empirical formula, that is, there is a linear relationship between 1 / τ and 1 / T. By fitting the slope of the straight line, the activation energy of the XLPE submarine cable is calculated, and the frequency-temperature method is applied for verification. The specific steps are as follows:

[0123] 1. First, using fractional calculus theory, the concept of "fractional element" is proposed, and the fractional elements are connected in series and parallel to obtain the fractional Poynting-Thomson model;

[0124] 2. Establish the relationship between the fractional Poynting-Thomson model and the complex dielectric capacitance of XLPE submarine cables;

[0125] 3. Construct the objective function for parameter identification, use the least squares method for parameter fitting, and establish the relationship between the measured values ​​C′ and C″ and the fitted value C′. m C″m The constraint function K for the fitted values ​​is determined, and K is minimized. The gray wolf algorithm is then used to identify the parameter values ​​of the fractional Poynting-Thomsonr model.

[0126] 4. After identification, the parameters of each model are shown in Table 1.

[0127] Table 1. Relationship between relaxation constant τ and temperature 1 / T

[0128]

[0129] Based on the Ering transition state theory, Arrhenius proposed the Arrhenius empirical formula, which is the relationship between the reaction rate constant and temperature:

[0130]

[0131] In the formula: A is the pre-exponential factor, also known as the frequency factor; E is the material activation energy; k is the reaction rate constant; R is the ideal gas constant, R = 8.314 J / (mol·K); T is the absolute temperature.

[0132] Taking the logarithm of both sides of the above equation yields:

[0133]

[0134] In the above formula, lnk and 1 / T have a linear relationship. Parameter identification of the measured dielectric spectra of each group of aged samples at different temperatures revealed that the relaxation time τ satisfies the Arrhenius empirical formula, i.e., there is a linear relationship between 1 / τ and 1 / T:

[0135]

[0136] In the formula: A τ E is the frequency factor. τ It is the activation energy.

[0137] pass Figures 4(a) to 4(c) The fitting results show a goodness of fit of 0.97, indicating a linear relationship between ln(1 / τ) and 1 / T, suggesting that ln(1 / τ) satisfies the Arrhenius empirical formula. ln(1 / τ) can be equated to the reaction rate k in the formula. The activation energy of each group of aged samples can be obtained from the slope of the fitted line, such as... Figures 4(a) to 4(c) The activation energy of each sample is calculated as shown.

[0138] 5. Compare the activation energy obtained by the frequency-temperature method with the calculation results in this invention to verify the accuracy of the method: The frequency-temperature shift method describes the relationship between the aging rate of insulating materials and the aging temperature based on the time-temperature equivalence principle. That is, for molecular motion, increasing the temperature and extending the time are equivalent. Researchers have found through experiments that the dielectric curves in the frequency domain dielectric spectrum at different temperatures can be shifted to a main curve, and the amplitude remains unchanged before and after the shift. In the frequency domain dielectric spectrum, the frequency and temperature before and after the shift follow the Arrhenius equation, that is:

[0139]

[0140] In the formula: a T Temperature shift coefficient, f0 is the frequency of a point in the dielectric spectrum at temperature T0 before shift; f is the frequency of that point in the dielectric spectrum at temperature T after shift; E a is the activation energy of the insulating material; k is the Boltzmann constant, k = 1.38 × 10⁻⁶. -23 J / K. T is the measured temperature of the curve, and T0 is the reference temperature (Kelvin temperature), which is selected as 40℃ in this embodiment. The frequency domain dielectric spectrum of the sample is measured at different temperatures, and each measured dielectric curve is shifted to the reference temperature to solve for α at each aging period. T And by fitting a T The slope is obtained from the linear relationship between 1 / T0 and 1 / T, and its slope is E. a / k, multiplied by k, gives the activation energy.

[0141] like Figures 3(a) to 3(c) As shown, the low-frequency band curves of the dielectric spectrum at 50℃, 60℃, and 70℃ are shifted and normalized to 40℃, and the measured curves at each temperature are shifted onto the main curve. For example... Figures 3(a) to 3(c) As shown, the points after the translation remain basically on the same curve as the main curve, achieving good coincidence. The translation frequency and temperature data were fitted, and Table 2 shows the translation factors of submarine cable samples with different aging degrees.

[0142] Table 2 Translation factors of submarine cable samples with different aging levels

[0143] Translation factor 40℃ 50℃ 60℃ 70℃ <![CDATA[α T(144h) ]]> 0 0.539 1.215 2.111 <![CDATA[α T(288h) ]]> 0 0.438 0.955 1.805 <![CDATA[α T(576h) ]]> 0 0.296 0.712 1.236

[0144] By fitting the relationship between the translation factor and temperature, the activation energy of each group of aged submarine cable samples can be calculated. As shown in Table 3, the activation energy values ​​of each group of submarine cable samples obtained by the frequency-temperature method and the τ parameter method are roughly the same, verifying the accuracy of the activation energy calculation by the τ parameter method and providing a new solution for solving the activation energy of insulation materials and evaluating the insulation performance of XLPE submarine cables.

[0145] Table 3. Solving activation energy using two methods.

[0146] Aging time (h) 144 288 576 Frequency-frequency method (eV) 0.600 0.495 0.355 τ parameter method (eV) 0.605 0.479 0.345

Claims

1. A method for calculating the activation energy of XLPE submarine cables based on the fractional Poynting–Thomson model, characterized in that... Includes the following steps: Step 1: First, define fractional elements, and connect the fractional elements in series and parallel to obtain the fractional Poynting-Thomson model; Step 2: Establish a fractional Poynting-Thomson model applicable to the complex capacitance of XLPE submarine cables. Relationships; Step 3: Construct the objective function for parameter identification, and use the Grey Wolf algorithm to identify each parameter in the fractional Poynting-Thomson model; Step 4: Based on the parameter identification results in Step 3, select appropriate parameters as characteristic quantities for calculating the activation energy of XLPE submarine cables; Step 5: Establish parameters and The linear relationship between them is used to calculate the activation energy of the XLPE submarine cable by fitting the slope of the straight line; where, Indicates the relaxation time. This indicates the measured temperature.

2. The method for calculating the activation energy of XLPE submarine cables based on the fractional Poynting–Thomson model according to claim 1, characterized in that: In step 1, the derivation process of the fractional Poynting-Thomson model is as follows: using fractional calculus theory, the electrical characteristics of both resistor R and capacitor C are combined to propose the concept of fractional element. Ideal capacitor and resistor devices are circuit elements that satisfy the following equations respectively. For capacitance under ideal dielectric conditions C Satisfy the following formula: ； For resistance R Satisfy the following formula: ； Fractional elements Z For circuit devices that satisfy the formula: ； In the formula: , , These are the voltages across the capacitor, resistor, and fractional element, respectively. The current flowing through the component; , , Indicates the component's electrical charge; d α Q ( t ) / d t α The fractional derivative of the charge in the time domain, and its shape parameter. α When the value is 0 or 1, the fractional element represents the ideal capacitor and the resistance, respectively. Different models can be formed by connecting fractional elements in series and parallel to describe the dielectric relaxation process of a medium.

3. The method for calculating the activation energy of XLPE submarine cables based on the fractional Poynting–Thomson model according to claim 2, characterized in that: In step 2, the fractional dynamic equations of the fractional Poynting-Thomson model are obtained based on the current distribution principle of series-parallel circuits in step 1: The fractional Poynting-Thomson model consists of two independent fractional elements. Z 1( α , C 1, τ 1) Z 2( β , C 2, τ 2) After parallel connection, connect fractional elements in series. Z 3( γ , C 3, τ 3) Composition; two fractional elements in parallel ( α , C 1, τ 1) and ( β , C 2, τ 2) When the voltages at both ends are the same, the sum of the partial charges flowing through the two units equals the total charge of the entire series-parallel structure. The parallel portion... U and Q The relationship can be obtained as follows: ； and These represent the voltage and charge at both ends of the fractional Poynting-Thomson model, respectively. Based on the principle of equal current in series circuits, the fractional dynamic equation of the fractional Poynting-Thomson model can be derived as follows: ； By performing a Fourier transform on the fractional Poynting-Thomson model, the complex capacitance of the model can be obtained. and complex permittivity; ； In the formula: α , β , γ Shape parameters related to the relaxation time distribution; ； , , , , , , α , β , γ It consists of three fractional elements. Z 1 Z 2 Z The model parameters in 3 are used to form a fractional Poynting-Thomson model by first connecting them in parallel and then in series. parameter Determine the complex capacitor Real part The mid-frequency part and the imaginary part The height and width of the loss peak; parameter , These are the frequency domain parameters that determine the high-frequency characteristics and low-frequency relaxation characteristics of the complex capacitor, respectively. parameter , To influence Constants that limit the high and low frequency bands.

4. The method for calculating the activation energy of XLPE submarine cables based on the fractional Poynting–Thomson model according to claim 3, characterized in that: In step 3, the parameter identification process is as follows: S3.1: To analyze the relationship between the parameters of the fractional Poynting-Thomson model and the degree of degradation of the submarine cable sample, the complex capacitance measured in the experiment was used. Substitute the data into the complex capacitance formula in step 3; S3.2: To ensure the accuracy of parameter identification results, establish the measured values. , with fitted value , Constrained objective function Using the least squares method to make To reach the minimum; ； , These are the real and imaginary parts of the complex capacitance of the XLPE submarine cable sample, respectively. , These refer to the real and imaginary parts of the complex capacitance obtained by fitting various model parameters after identifying the parameters of the fractional Poynting-Thomson model.

5. The method for calculating the activation energy of XLPE submarine cables based on the fractional Poynting–Thomson model according to claim 4, characterized in that: In step 4, the relaxation time Satisfying the following empirical formula of Arrhenius, ； The reaction rate can be derived from the Arrhenius empirical formula. and the reciprocal of temperature The relationship between them is linear, and the parameters obtained after identification are... and the reciprocal of temperature The relationship between them is linear, thus... The reaction rate in the equivalent Arrhenius empirical formula This can be verified. It satisfies the Arrhenius empirical formula.

6. The method for calculating the activation energy of XLPE submarine cables based on the fractional Poynting–Thomson model according to claim 5, characterized in that: Will Equivalent to the Arrhenius empirical formula The activation energy of the XLPE submarine cable is calculated as follows: (Formula omitted for brevity) Taking the logarithm of both sides, we get: ； In the above formula, and The relationship between them is linear; by analyzing the dielectric spectra of each group of aged samples at different temperatures, the relaxation time was found to be... Satisfying the Arrhenius empirical formula, i.e. and There is a linear relationship between them: ； In the formula: For frequency factor, Activation energy; Taking the logarithm of both sides of the above equation, we get: ； From the above formula, we can obtain and The relationship between them is linear, and by fitting... and From the straight line between them, we can obtain: the slope of this straight line is... , For the ideal gas constant, multiply the slope of the fitted line by... The activation energy of the sample can then be obtained. .

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