Method for calculating safety service reliability of power transmission line under deicing protection measures

Abstract

The invention relates to a method for calculating the safety service reliability of a power transmission line under deicing protection measures. The method comprises the following steps: constructing a hybrid random variable distribution function of icing thickness and line body temperature, constructing a joint cumulative distribution function of continuous service reliability of the power transmission line under a deicing protection measure, constructing a continuous service performance function of the power transmission line under the deicing protection measure, and performing multi-dimensional Lebesgue decomposition of the joint cumulative distribution function to obtain a continuous service reliability degree of the power transmission line. Determining an integral domain of the decomposed cumulative distribution function, solving a subitem failure probability, calculating a total failure probability, and calculating the safety service reliability of the power transmission line under the deicing protection measure, thereby achieving the precise calculation of the safety service reliability of the power transmission line under the deicing protection measure. According to the method, the hybrid random variable distribution function of the icing thickness and the line body temperature under the intervention of the deicing measures can be accurately constructed, the sub-item failure probability and the total failure probability can be accurately solved, and a scientific basis is provided for reasonable setting of a deicing discriminant value.

Description

TECHNICAL FIELD

[0001] The present application relates to a method for calculating the safe service reliability of a power transmission line under deicing protection, belonging to the technical field of power transmission line safety evaluation. BACKGROUND

[0002] The power transmission line is the core infrastructure of the power system, and its safe service directly determines the stability of power supply. In high-latitude and high-altitude areas, heavy icing is one of the core meteorological disasters that threaten the safe and stable operation of power transmission lines. Under the meteorological conditions of low temperature, high humidity and light wind, ice such as glaze and rime is easily formed on the surface of the power transmission line conductor, resulting in a sharp increase in the weight of the conductor and the icing load. On the one hand, the increase in load significantly increases the tension of the conductor, which may exceed its tensile limit, causing structural damage such as conductor breakage and tower collapse. On the other hand, icing leads to an increase in conductor sag, which easily causes insufficient safety clearance between the line and the ground or cross-spanning objects (such as roads, railways, and other lines), thereby causing electrical faults such as interphase short circuit and flashover. According to statistics, in China, the annual average number of accidents of power transmission lines caused by icing in heavy icing areas accounts for more than 30% of the total number of meteorological disaster accidents, and in severe cases, it can cause regional power grid shutdown, causing significant losses to the national economy and social livelihood.

[0003] To cope with heavy icing disasters, various deicing protection measures have been developed and applied in the industry, among which thermal ice melting (such as using high current to heat the conductor to melt ice) and mechanical deicing (such as mechanical ice scraping and manual ice knocking) are the mainstream technical means. These measures control the ice thickness by active intervention to reduce the damage risk of icing load on the line, but the effectiveness of the deicing measures depends on the accurate safe service reliability evaluation. If the reliability calculation is biased too much, it will lead to unreasonable setting of deicing discrimination value. If the discrimination value is too loose, it will lead to icing thickness exceeding the limit without starting deicing, increasing the risk of line failure. If the discrimination value is too strict, it will lead to frequent deicing even if the icing does not reach the danger threshold, causing waste of resources such as electric energy and manpower, and reducing the economic efficiency of power grid operation. Therefore, under the action of deicing protection measures, it is necessary to build a reliability calculation method that adapts to the actual working conditions, which is a key technical requirement to ensure the safe and economic operation of power grids in heavy icing areas.

[0004] Currently, the reliability calculation of transmission lines is mainly based on the traditional structural reliability theory. The core idea is to construct the limit state function of "load effect-structure resistance", combine the probability distribution of each random variable (such as ice thickness, wind speed, wire tensile strength, etc.), and solve the failure probability (i.e. the probability that the line cannot safely serve) by integration or approximation method. However, in the scenario of deicing protection measures intervention, the existing method is out of touch with the deicing working condition. The key random variables such as ice thickness and line temperature are treated as continuous probability distribution (such as ice thickness using extreme value type I distribution, line temperature using Gaussian distribution) under natural icing condition, without considering the intervention characteristics of deicing measures. The multi-dimensional distribution decomposition method is not optimized for the variable characteristics of transmission lines. Either directly ignoring the discrete part of mixed variables and only decomposing according to continuous distribution, or unable to effectively separate the probability contribution of discrete components and continuous components, resulting in that the joint distribution function cannot accurately describe the probability correlation between multi-variables. In addition, the existing method usually takes "wire breakage" as the limit state (i.e. wire breakage failure), but according to the "Technical Code for Design of Overhead Transmission Lines with Heavy Ice Cover DL / T5440-2020", the plastic elongation of the wire before breakage will cause imbalance of sag, insufficient safety distance and other hidden dangers. The specification clearly requires that the tension at the maximum sag point should not exceed 60% of the wire breaking force, and the tension at the suspension point should not exceed 66% of the breaking force. The existing method mostly does not fully combine the specification requirements to construct the limit state function, still taking the breaking force as the only criterion, resulting in the disconnection between the limit state and the actual safe service boundary. At the same time, the existing method does not determine the integral domain for the distribution characteristics of mixed variables under deicing measures. For example, the discrete points of ice thickness are not included in the integral range, or the high-temperature discrete interval of line temperature is not associated with the tension limit, resulting in that the integral domain cannot cover the real failure scenario, further amplifying the error of reliability calculation.

[0005] In summary, the existing transmission line reliability calculation method in the scenario of deicing protection measures is difficult to accurately evaluate the safe service reliability of the line due to the problems of mixed random variable modeling, multi-variable joint distribution decomposition, and integral domain determination according to the specification. SUMMARY

[0006] To solve the above problems existing in the prior art, the present application provides a method for calculating the safe service reliability of a transmission line under deicing protection measures. The present application can accurately construct the mixed random variable distribution function of ice thickness and line temperature under the intervention of deicing measures, and accurately solve the partial failure probability and the total failure probability, providing a scientific basis for the reasonable setting of deicing discrimination value.

[0007] To achieve the above purpose, the technical scheme provided by the present application is a method for calculating the safe service reliability of a transmission line under deicing protection measures, comprising the following steps:

[0008] (1) Constructing a mixed random variable distribution function of ice thickness and line temperature: Based on the impact of de-icing protection measures on the ice thickness and line temperature of transmission lines, a random variable of ice thickness is constructed respectively. and line body temperature random variable The distribution function of a mixed random variable, which includes a discrete probability part and a continuous distribution part;

[0009] (2) Constructing the joint cumulative distribution function of the reliability of transmission lines under de-icing protection measures: Let the random vector involved in the reliability calculation of transmission lines under de-icing protection measures be... Construct random vectors Joint cumulative distribution function ;in For discrete random variables, subvectors The number of discrete variables; For mixed-type random variable subvectors, The number of mixed-type variables, and Including the icing thickness in step (1) and line temperature ; For continuous random variables, subvectors The number of continuous variables must include at least the tensile stress and cross-sectional area of ​​the transmission line.

[0010] (3) Constructing the function function for the continuous service of transmission lines under de-icing protection measures: Constructing a function function with the safe service of the transmission line as the ultimate state, the function function for the continuous service of the transmission line is obtained as follows: ;

[0011] In the formula, The tensile strength of the conductor or ground wire; For load effect; The safety operating tension limit coefficient for the conductor is set at 60% for the point of maximum sag and 66% for the suspension point. The tensile stress that breaks the conductor or ground wire; The cross-sectional area of ​​the conductor; The tension of the transmission line under comprehensive load;

[0012] (4) Multidimensional Lebesgue decomposition of joint cumulative distribution function: Icing transmission lines under de-icing mechanism in the failure domain Failure probability below Based on the multidimensional Lebesgue theorem, the joint cumulative distribution function in step (2) is... Decompose the equation into the following formula: ;in, , , and are all bounded non-decreasing functions, respectively called type I, II, III and IV bounded non-decreasing functions, ; ignoring the singular function term F III , then ; ;

[0013] When there is no continuous random variable , i.e. , F(V) contains the term when decomposed, and the decomposition is:

[0014] ;

[0015] When there is continuous random variable , i.e. , F(V) is decomposed into:

[0016] ;

[0017] For the transmission line under the deicing mechanism, is the ice thickness or the line temperature , the cumulative distribution function when discrete values are obtained, is the ice thickness or the line temperature , the cumulative distribution function when continuous values are obtained;

[0018] (5) Determine the integral domain of the decomposed cumulative distribution function: according to the continuous service function function in step (3), determine the failure domain , the integral domain of , and respectively obtained by decomposition in step (4);

[0019] (6) Solve the partial failure probability: calculate the failure probability of , and in the corresponding integral domain, denoted as , and ;

[0020] where, the failure probability of under the failure domain is obtained by summing the probability values of the discrete points that satisfy the continuous service function function, and the calculation formula is as follows:

[0021] ;

[0022] In the formula, The probability of failure in the failure domain The probability of the element in the point set satisfying the condition, represents the number of point sets satisfying the condition;

[0023] The failure probability in the failure domain The calculation formula is as follows:

[0024] ;

[0025] Where, ;

[0026] ;

[0027] In the formula, subscript "D" represents the variable taking the discrete part in the mixed variable, and subscript "C" represents the variable taking the continuous part in the mixed variable; The probability of the first point set in the point set composed of discrete combinations The probability of the first point set in the point set composed of discrete combinations and is obtained by joint; The combination of variables in the mixed random variable There are , is an integer, ; is a step function; ; The distribution function of the random variable under the condition of the point set is obtained by combination, that is:

[0028]

[0029] In the formula, is the conditional probability distribution function of the random variable under the point set , is the complement symbol;

[0030] The random variable does not satisfy the normalization condition, and its density function is , is calculated by the following formula:

[0031] ;

[0032] In the formula, is the point set​​ The quantity is The product of the number of discrete elements of each discrete variable;

[0033] In the failure domain Failure probability below The calculation formula is as follows:

[0034] ;

[0035] In the formula, for The derivative;

[0036] right and Normalization is performed to obtain and normalized expression and Then we have:

[0037] ;

[0038] ;

[0039] In the formula, and They are respectively and The normalized coefficient; and This is the normalized probability density function;

[0040] (7) Calculate the overall failure probability: Based on the individual failure probabilities obtained in step (6) , and Calculate the overall failure probability for safe operation of transmission lines under de-icing protection measures. ;

[0041] Final component failure probability:

[0042] ;

[0043] ;

[0044] ;

[0045] Overall failure probability of transmission lines under de-icing protection measures for safe service:

[0046] ;

[0047] (8) Calculate the reliability of the transmission line under the de-icing protection measures: Based on the overall failure probability of the transmission line under the de-icing protection measures obtained in step (7), Calculate the reliability of power transmission lines under de-icing protection measures. ;

[0048] ;

[0049] In the formula, It is the inverse function of the cumulative function of the standard normal distribution.

[0050] The further improvement to the above technical solution is as follows:

[0051] The random variable of ice thickness mentioned in step (1) The distribution function of the mixed random variable is as follows:

[0052] ;

[0053] ;

[0054] In the formula, and For random variables upper and lower boundaries, For corresponding The probability value, and They are random variables The original cumulative distribution function and probability density function.

[0055] The random variable of line temperature mentioned in step (1) The distribution function of the mixed random variable is as follows:

[0056] ;

[0057] ;

[0058] In the formula, Temperature random variable discrete points, For corresponding The probability, and They are random variables The original cumulative distribution function and probability density function.

[0059] The random variable of line temperature mentioned in step (1) The distribution function of the mixed random variable is as follows:

[0060] ;

[0061] ;

[0062] In the formula, Temperature random variable discrete points, For corresponding The probability, and They are random variables The original cumulative distribution function and probability density function.

[0063] The tension of the transmission line under the comprehensive load described in step (3) The calculation methods include:

[0064] 1) Define two meteorological states:

[0065] State 1 (Annual Average Temperature State): Temperature Wind speed Ice thickness Allowable stress ;

[0066] State 2 (Thermal Melting State): Temperature Wind speed Ice thickness ;

[0067] 2) Calculate the specific load for the two states. , Wind deflection angle , :

[0068] ;

[0069] ;

[0070] ;

[0071] ;

[0072] ;

[0073] ;

[0074] ;

[0075] ;

[0076] ;

[0077] In the formula, Mass per unit length of the power transmission line; This refers to the cross-sectional area of ​​the power transmission line; For power transmission line density; It is the acceleration due to gravity; The icing thickness is for state 1. The icing thickness is for state 2; The outer diameter of the power transmission line; This is the wind load adjustment factor for transmission lines; This is the coefficient for wind speed non-uniformity. The shape factor of the overhead line; This represents the wind speed in state 1. This represents the wind speed in state 2. The angle between the wind direction and the line direction; and These are the horizontal and vertical load ratios for state 1, respectively. The overall load ratio for state 1; and These are the horizontal and vertical load ratios for state 2, respectively. The overall load ratio for state 2; This refers to the allowable stress of the power transmission line under average annual temperature and meteorological conditions. This refers to the horizontal stress component of the overhead line along the line direction.

[0078] 3) Solve for the horizontal stress components in state 2. : Horizontal stress component of state 1 Based on this, by solving cubic equations in one variable δ 02 ;

[0079] in:

[0080] ;

[0081] ;

[0082] The above cubic equation is solved iteratively using Newton's method. Let:

[0083] ;

[0084] Its derivative is:

[0085] ;

[0086] Then Newton's alternative expression is:

[0087] ;

[0088] Provide initial values ​​for iteration , calculate and The above formula is used to iteratively obtain the result. Repeat until Next time until, Take 1×10 -6 ;

[0089] The axial stress of the transmission line at any point in the wind deflection plane under state 2 is:

[0090] ;

[0091] In the formula: The horizontal stress component of the overhead line along the line direction under state 2; The coefficient of thermal linear expansion of the overhead power line; Young's modulus; The overall load ratio under state 2; This refers to the wind deflection angle under state 2. It is the elevation difference angle; The coordinates of any point; For power transmission line span;

[0092] 4) Calculate the tension at the suspension point and the tension at the maximum sag point :

[0093] Coordinates of the suspension point of a power transmission line and Substitute Obtain the power transmission line Axial stress at suspension point and Axial stress at suspension point :

[0094] ;

[0095] ;

[0096] The tension at the suspension point is obtained through comparison. :

[0097] ;

[0098] For the axial stress of the transmission line at the location of the maximum sag point within a span, the equation of the catenary shape is found. The coordinates of the point are obtained by finding the minimum value. The equation for the shape of the catenary suspended between two towers. for:

[0099] ;

[0100] in:

[0101] ;

[0102]

[0103] ;

[0104] ;

[0105] x max Substitution The maximum sag point stress is obtained , Multiply by the cross-sectional area of ​​the transmission line The maximum sag point tension is obtained. ;

[0106] In the formula, The elevation difference of one span of the transmission line; Transmission line span.

[0107] The specific process of multidimensional Lebesgue decomposition in step (4) includes:

[0108] 1) For discrete random variables Let the first Each component The number of elements is ,but There are discrete variables. A combination of points, components The probability value of the discrete point is ;

[0109] 2) For mixed random variables , each component The absolutely continuous part of the set of real numbers into which it is decomposed and discrete part ;

[0110] 3) For continuous random variables , its first The cumulative distribution function of each component is: The probability density function is .

[0111] The rule for determining the integration domain in step (5) is as follows:

[0112] for The failure domain, the first A list of points should meet the following conditions:

[0113] ;

[0114] for ,Mode The integration domain in is:

[0115] ;

[0116] for ,Mode The integration domain in is:

[0117] .

[0118] The steps described in step (6) and The calculation methods include:

[0119] 1) Variable transformation: In random variables The function is Below, for the continuous portion of continuous or mixed variables. Perform transformation ,make Transform to standard normal space; transform to ordinary linear transformation, Rosenblatt transformation, or Nataf transformation;

[0120] 2) Transform into a reliability index optimization problem: Transform the failure probability calculation into the Hasofer-Lind reliability index. Solving for;

[0121] After transformation, it becomes The number of random variables that need to be transformed depends on and Let the number of random variables in the joint probability density function of the calculation formula be denoted as . 1, calculate first. and The reliability index corresponding to the calculation formula and ,use represent and ,Right now:

[0122] ;

[0123] In the formula, As a constraint, ensure that the point On a hypersurface in standard normal space; and The first Upper and lower bounds of a random variable;

[0124] 3) Constraint optimization transformation: Add icing thickness variable In the range For unequal constraints within the range, the interior point penalty function method is used to transform the unequal-constrained optimization problem into an unconstrained optimization problem; the calculation formula is as follows:

[0125] ;

[0126] In the formula, λ is the penalty coefficient corresponding to the equality constraint; it is used to control the influence of the penalty function, λ>0; λg[G -1 [u] represents the control parameter; n C Let n be the number of variables in a mixed-type variable that take the continuous portion and the number of continuous variables. C =h+er;μ i Let be the penalty coefficient for the i-th random variable; For random variables The lower bound;

[0127] 4) Unconstrained optimization solution: The particle swarm optimization algorithm (PSO) is used to solve the above unconstrained optimization problem;

[0128] For the failure probability calculation formula P f,II and P f,IV The dimension of the search space is determined by the number of random variables in its continuous part. C Suppose there exists m C There are particles, where the position of the i-th particle is... The speed is The fitness value is β i ,according to The calculation shows that the position where an individual obtains its optimal fitness value is... The position where the group obtains the optimal fitness value is i=1,2,...,m C During the algorithm's search for the optimal solution, the particles continuously optimize. When they evolve to the j-th generation, the velocities, velocity boundaries, and positions of each particle are as follows:

[0129] ;

[0130] ;

[0131] ;

[0132] In the formula, It is called the inertia factor, with a value of 0.75, and is used to control the performance of model optimization. and A number randomly distributed between 0 and 1; and Let be the acceleration constant. ; and The evolution of particles to the th generation and first The position of the generation; and The evolution of particles to the th generation and first The speed of generation; For the time interval, take ; Indicates the maximum speed;

[0133] 5) Failure probability calculation: Calculated using the Particle Swarm Optimization (PSO) algorithm. The fitness function that yields the minimum value. Then based on the two generations The difference is less than 1×10 -6 The iteration termination condition is used to calculate the reliability index. Then, based on the relationship between reliability index and failure probability, the corresponding failure probability is calculated;

[0134] The calculation formula is as follows:

[0135] ;

[0136] Based on the above calculation of β HL The calculation method is used to calculate P. f,II and P f,IV Reliability index corresponding to the calculation formula and Then, calculate their failure probabilities separately, using the following formulas:

[0137] ;

[0138] ;

[0139] Final failure probability for:

[0140] ;

[0141] In the formula, These are the optimal position coordinates obtained by the particle swarm optimization algorithm; It is the cumulative function of the standard normal distribution.

[0142] As can be seen from the above technical solution, the method for calculating the reliability of transmission lines under de-icing protection measures provided by this invention achieves accurate calculation of the reliability of transmission lines under de-icing protection measures by constructing a mixed random variable distribution function of ice thickness and line temperature, constructing a joint cumulative distribution function of the continuous service reliability of transmission lines under de-icing protection measures, performing multidimensional Lebesgue decomposition of the joint cumulative distribution function, constructing a continuous service function of transmission lines under de-icing protection measures, determining the integration domain of the decomposed cumulative distribution function, solving for the failure probability of individual terms, calculating the overall failure probability, and calculating the reliability of transmission lines under de-icing protection measures. This provides a scientific basis for the reasonable setting of de-icing discrimination values ​​and balances the safe service of lines with operational economy. This invention has the following advantages over existing technologies:

[0143] (1) The technical solution adopted in this invention treats the ice thickness and line temperature under the intervention of de-icing measures as a mixed random variable containing discrete probability part and continuous distribution part, and constructs the corresponding distribution function, which accurately adapts to the actual distribution characteristics of the variables under the de-icing condition, solves the model distortion problem caused by the pure continuous distribution in the traditional method, and lays an accurate variable foundation for subsequent reliability calculation.

[0144] (2) The technical solution adopted in this invention decomposes the joint cumulative distribution function containing discrete, mixed and continuous variables based on the multidimensional Lebesgue theorem, effectively separating the probability contributions of different types of variables, breaking through the technical bottleneck of traditional decomposition methods that are difficult to handle the joint distribution of multiple types of variables, and ensuring the accuracy of the joint distribution function.

[0145] (3) The technical solution adopted in this invention strictly follows the "Technical Specification for Design of Overlapping Overhead Transmission Lines DL / T5440-2020" to construct a limit state function with "the maximum sag point tension does not exceed the breaking force by 60% and the suspension point tension does not exceed the breaking force by 66%" as the core, and determines the integral domain of each decomposition function in the failure domain accordingly, so that the integral domain is highly consistent with the actual safety boundary of the project, avoiding the integral deviation caused by the disconnect between the limit state and the specification in the traditional method.

[0146] (4) The technical solution adopted in this invention combines the interior point penalty function method to transform the constrained optimization problem into an unconstrained optimization problem, and uses the particle swarm algorithm to solve the Hasofer-Lind reliability index to calculate the failure probability of each item. Finally, the overall failure probability is obtained by superimposing the results. This not only improves the accuracy and efficiency of failure probability calculation, but also provides a quantitative basis for setting the de-icing discrimination value, effectively balancing the safe service of the line and the economic efficiency of operation, and avoiding the risks and waste of resources caused by overly lenient or overly strict discrimination values. Attached Figure Description

[0147] Figure 1Flowchart of the method of this invention;

[0148] Figure 2 A schematic diagram of the distribution of a mixed random variable;

[0149] Figure 3 Comparison chart of failure probabilities under different operating conditions. Detailed Implementation

[0150] The present invention will now be described in detail with reference to the accompanying drawings and specific embodiments, but the scope of protection of the present invention is not limited to the following embodiments.

[0151] The present invention provides a method for calculating the reliability of power transmission lines under de-icing protection measures, the flowchart of which is shown below. Figure 1 As shown, it includes the following steps:

[0152] (1) Constructing a mixed random variable distribution function of ice thickness and line temperature: Based on the impact of de-icing protection measures on the ice thickness and line temperature of transmission lines, a random variable of ice thickness is constructed respectively. and line body temperature random variable The distribution function of a mixed random variable, which includes a discrete probability part and a continuous distribution part;

[0153] The random variable of ice thickness The distribution function of the mixed random variable is as follows:

[0154] ;

[0155] ;

[0156] In the formula, and For random variables upper and lower boundaries, For corresponding The probability value, and They are random variables The original cumulative distribution function and probability density function.

[0157] The random variable of the line temperature The distribution function of the mixed random variable is as follows:

[0158] ;

[0159] ;

[0160] In the formula, Temperature random variable discrete points, For corresponding The probability, and They are random variables The original cumulative distribution function and probability density function.

[0161] (2) Constructing the joint cumulative distribution function of the reliability of transmission lines under de-icing protection measures: Let the random vector involved in the reliability calculation of transmission lines under de-icing protection measures be... Construct random vectors Joint cumulative distribution function ;in For discrete random variables, subvectors The number of discrete variables; For mixed-type random variable subvectors, The number of mixed-type variables, and Including the icing thickness in step (1) and line temperature ; For continuous random variables, subvectors The number of continuous variables must include at least the tensile stress and cross-sectional area of ​​the transmission line.

[0162] (3) Constructing the function function for the continuous service of transmission lines under de-icing protection measures: Constructing a function function with the safe service of the transmission line as the ultimate state, the function function for the continuous service of the transmission line is obtained as follows: ;

[0163] In the formula, The tensile strength of the conductor or ground wire; For load effect; The safety operating tension limit coefficient for the conductor is set at 60% for the point of maximum sag and 66% for the suspension point. The tensile stress that breaks the conductor or ground wire; The cross-sectional area of ​​the conductor; The tension of the transmission line under comprehensive load;

[0164] The tension of the transmission line under the combined load The calculation methods include:

[0165] 1) Define two meteorological states:

[0166] State 1 (Annual Average Temperature State): Temperature Wind speed Ice thickness Allowable stress ;

[0167] State 2 (Thermal Melting State): Temperature Wind speed Ice thickness ;

[0168] 2) Calculate the specific load for the two states. , Wind deflection angle , :

[0169] ;

[0170] ;

[0171] ;

[0172] ;

[0173] ;

[0174] ;

[0175] ;

[0176] ;

[0177] ;

[0178] In the formula, Mass per unit length of the power transmission line; This refers to the cross-sectional area of ​​the power transmission line; For power transmission line density; It is the acceleration due to gravity; The icing thickness is for state 1. The icing thickness is for state 2; The outer diameter of the power transmission line; This is the wind load adjustment factor for transmission lines; This is the coefficient for wind speed non-uniformity. The shape factor of the overhead line; This represents the wind speed in state 1. This represents the wind speed in state 2. The angle between the wind direction and the line direction; and These are the horizontal and vertical load ratios for state 1, respectively. The overall load ratio for state 1; and These are the horizontal and vertical load ratios for state 2, respectively. The overall load ratio for state 2; This refers to the allowable stress of the power transmission line under average annual temperature and meteorological conditions. This refers to the horizontal stress component of the overhead line along the line direction.

[0179] 3) Solve for the horizontal stress components in state 2. : Horizontal stress component of state 1 Based on this, by solving cubic equations in one variable δ 02 ;

[0180] in:

[0181] ;

[0182] ;

[0183] The above cubic equation is solved iteratively using Newton's method. Let:

[0184] ;

[0185] Its derivative is:

[0186] ;

[0187] Then Newton's alternative expression is:

[0188] ;

[0189] Provide initial values ​​for iteration , calculate and The above formula is used to iteratively obtain the result. Repeat until Next time until, Take 1×10 -6 ;

[0190] The axial stress of the transmission line at any point in the wind deflection plane under state 2 is:

[0191] ;

[0192] In the formula: The horizontal stress component of the overhead line along the line direction under state 2; The coefficient of thermal linear expansion of the overhead power line; Young's modulus; The overall load ratio under state 2; This refers to the wind deflection angle under state 2. It is the elevation difference angle; The coordinates of any point; For power transmission line span;

[0193] 4) Calculate the tension at the suspension point and the tension at the maximum sag point :

[0194] Coordinates of the suspension point of a power transmission line and Substitute Obtain the power transmission line Axial stress at suspension point and Axial stress at suspension point :

[0195] ;

[0196] ;

[0197] The tension at the suspension point is obtained through comparison. :

[0198] ;

[0199] For the axial stress of the transmission line at the location of the maximum sag point within a span, the equation of the catenary shape is found. The coordinates of the point are obtained by finding the minimum value. The equation for the shape of the catenary suspended between two towers. for:

[0200] ;

[0201] in:

[0202] ;

[0203]

[0204] ;

[0205] ;

[0206] x max Substitution The maximum sag point stress is obtained , Multiply by the cross-sectional area of ​​the transmission line The maximum sag point tension is obtained. ;

[0207] In the formula, The elevation difference of one span of the transmission line; Transmission line span.

[0208] (4) Multidimensional Lebesgue decomposition of joint cumulative distribution function: Icing transmission lines under de-icing mechanism in the failure domain Failure probability below Based on the multidimensional Lebesgue theorem, the joint cumulative distribution function in step (2) is... Decompose the equation into the following formula: ;in, , , and All Bounded non-decreasing functions of type I, II, III, and IV are respectively called bounded non-decreasing functions. Ignore the singular function term F III ,but ; ;

[0209] When there are no continuous random variables At that time, that is F(V) decomposes to contain The term, factored as:

[0210] ;

[0211] When there are continuous random variables At that time, that is F(V) is then decomposed into:

[0212] ;

[0213] For power transmission lines under de-icing mechanisms, Ice thickness or line temperature The cumulative distribution function when obtaining discrete values, Ice thickness or line temperature The cumulative distribution function when continuous values ​​are obtained;

[0214] The specific process of the multidimensional Lebesgue decomposition includes:

[0215] 1) For discrete random variables Let the first Each component The number of elements is ,but There are discrete variables. A combination of points, components The probability value of the discrete point is ;

[0216] 2) For mixed random variables , each component The absolutely continuous part of the set of real numbers into which it is decomposed and discrete part ;

[0217] 3) For continuous random variables , its first The cumulative distribution function of each component is: The probability density function is .

[0218] (5) Determine the integration domain of the cumulative distribution function after decomposition: Determine the failure domain based on the continuous service function in step (3). The following is the decomposition obtained in step (4) , and Their respective integration domains;

[0219] The rule for determining the integration domain is as follows:

[0220] for The failure domain, the first A list of points should meet the following conditions:

[0221] ;

[0222] for ,Mode The integration domain in is:

[0223] ;

[0224] for ,Mode The integration domain in is:

[0225] .

[0226] The rule for determining the integration domain is as follows:

[0227] for The failure domain, the first A list of points should meet the following conditions:

[0228] ;

[0229] for ,Mode The integration domain in is:

[0230] ;

[0231] for ,Mode The integration domain in is:

[0232] .

[0233] The and The calculation methods include:

[0234] 1) Variable transformation: In random variables The function is Below, for the continuous portion of continuous or mixed variables. Perform transformation ,make Transform to standard normal space; transform to ordinary linear transformation, Rosenblatt transformation, or Nataf transformation;

[0235] 2) Transform into a reliability index optimization problem: Transform the failure probability calculation into the Hasofer-Lind reliability index. Solving for;

[0236] After transformation, it becomes The number of random variables that need to be transformed depends on and Let the number of random variables in the joint probability density function of the calculation formula be denoted as . 1, calculate first. and The reliability index corresponding to the calculation formula and ,use represent and ,Right now:

[0237] ;

[0238] In the formula, As a constraint, ensure that the point On a hypersurface in standard normal space; and The first Upper and lower bounds of a random variable;

[0239] 3) Constraint optimization transformation: Add icing thickness variable In the range For unequal constraints within the range, the interior point penalty function method is used to transform the unequal-constrained optimization problem into an unconstrained optimization problem; the calculation formula is as follows:

[0240] ;

[0241] In the formula, λ is the penalty coefficient corresponding to the equality constraint; it is used to control the influence of the penalty function, λ>0; λg[G -1 [u] represents the control parameter; n CLet n be the number of variables in a mixed-type variable that take the continuous portion and the number of continuous variables. C =h+er;μ i Let be the penalty coefficient for the i-th random variable; For random variables The lower bound;

[0242] 4) Unconstrained optimization solution: The particle swarm optimization algorithm (PSO) is used to solve the above unconstrained optimization problem;

[0243] For the failure probability calculation formula P f,II and P f,IV The dimension of the search space is determined by the number of random variables in its continuous part. C Suppose there exists m C There are particles, where the position of the i-th particle is... The speed is The fitness value is β i ,according to The calculation shows that the position where an individual obtains its optimal fitness value is... The position where the group obtains the optimal fitness value is i=1,2,...,m C During the algorithm's search for the optimal solution, the particles continuously optimize. When they evolve to the j-th generation, the velocities, velocity boundaries, and positions of each particle are as follows:

[0244] ;

[0245] ;

[0246] ;

[0247] In the formula, It is called the inertia factor, with a value of 0.75, and is used to control the performance of model optimization. and A number randomly distributed between 0 and 1; and Let be the acceleration constant. ; and The evolution of particles to the th generation and first The position of the generation; and The evolution of particles to the th generation and first The speed of generation; For the time interval, take ; Indicates the maximum speed;

[0248] 5) Failure probability calculation: Calculated using the Particle Swarm Optimization (PSO) algorithm. The fitness function that yields the minimum value. Then based on the two generations The difference is less than 1×10 -6 The iteration termination condition is used to calculate the reliability index. Then, based on the relationship between reliability index and failure probability, the corresponding failure probability is calculated;

[0249] The calculation formula is as follows:

[0250] ;

[0251] Based on the above calculation of β HL The calculation method is used to calculate P. f,II and P f,IV Reliability index corresponding to the calculation formula and Then, calculate their failure probabilities separately, using the following formulas:

[0252] ;

[0253] ;

[0254] Final failure probability for:

[0255] ;

[0256] In the formula, These are the optimal position coordinates obtained by the particle swarm optimization algorithm; It is the cumulative function of the standard normal distribution.

[0257] (7) Calculate the overall failure probability: Based on the individual failure probabilities obtained in step (6) , and Calculate the overall failure probability for safe operation of transmission lines under de-icing protection measures. ;

[0258] Final component failure probability:

[0259] ;

[0260] ;

[0261] ;

[0262] Overall failure probability of transmission lines under de-icing protection measures for safe service:

[0263] ;

[0264] (8) Calculate the reliability of the transmission line under the de-icing protection measures: Based on the overall failure probability of the transmission line under the de-icing protection measures obtained in step (7), Calculate the reliability β of the power transmission line under de-icing protection measures. O ;

[0265] ;

[0266] In the formula, Ф -1 It is the inverse function of the cumulative function of the standard normal distribution.

[0267] Example: The transmission conductor is model JL / G1A-400 / 50, with a mass per unit length of 1511 kg / km and an outer diameter d. c The diameter is 27.63 mm, and the cross-sectional area A is 451.55 mm. 2 The line is designed to have an icing thickness of 25 mm. The statistical distribution of the basic random variables is shown in Table 1.

[0268] Table 1 Statistical parameters of random variables

[0269] Random variable Statistical distribution Mean value Coefficient of variation Ice thickness D (mm) Extreme value type I 20.00 0.18 Power line temperature T (°C) Gaussian distribution -5 0.16 Wind speed V (m / s) Extreme value type I 5.00 0.50 Cross-sectional area A C (mm 2 )]]> Gaussian distribution 451.55 0.05 Tensile strength F y (N / mm 2 )]]> Gaussian distribution 273.28 0.10

[0270] Regarding the distribution of icing thickness, when the icing thickness is less than the de-icing discrimination value d... ice It follows the existing probability distribution, in d ice Obtain a fixed probability Its value is:

[0271] ;

[0272] In the formula, F D (·) represents the original cumulative distribution function of the ice thickness.

[0273] The temperature variable T of the line body is at a temperature t that is significantly greater than the original temperature distribution value. ice Probability of obtaining Its temperature value is:

[0274] ;

[0275] In the formula, I ice For the de-icing current, d ice This is the de-icing discrimination value.

[0276] Then, the safe service reliability β of the transmission line under de-icing protection measures is calculated using the method of the present invention for calculating the safe service reliability of the transmission line under de-icing protection measures. O , ;

[0277] By changing the de-icing discrimination value d ice Figure 3 shows the failure probabilities of the random variable of icing thickness under different discriminant values ​​and various processing methods. Under de-icing protection measures, the failure probabilities of the transmission lines differ significantly depending on the processing method of the random variable of icing thickness. Figure 3 In this study, the failure probabilities of Failure Probability Without the De-icing Measure (FPWoDM), Failure Probability With the De-icing Measure (FPWDM), Failure Probability of Discrete Part (FPDP), and Failure Probability of Continuous Part (FPCP) were compared. FPCP corresponds to treating the icing thickness distribution considering de-icing as a truncated distribution, and FPWDM is equal to the sum of FPDP and FPCP. The maximum sag point condition was also investigated. Figure 3 (a) and (b), and the suspension point condition, Figure 3 The failure probabilities of (c) and (d) are given by the following working conditions: a 100 m span and a 0-degree elevation difference angle, and a 500 m span and a 25-degree elevation difference angle, respectively.

[0278] Comparing the curves of the de-icing mechanism failure probability and the continuous part failure probability in the figure, the physical meaning is that the truncated distribution, which does not consider the tail probability aggregation, is treated as a mixed distribution. As shown in the figure, FPWDM is always greater than FPCP, and the difference is reflected in FPDP, exhibiting the shape of a Gaussian probability density function curve. As the de-icing discrimination value increases, FPCP and FPWDM approach the same value, which is FPWoDM, but in different ways. For FPCP, FPWoDM is its upper limit, and it approaches it upwards. For FPWDM, it first surpasses FPWoDM and then approaches it downwards. The reason for this phenomenon is that the mixed distribution has more discrete probabilities than the truncated distribution, and as the discrete points increase, i.e., the de-icing discrimination value increases, the probability of the discrete points falling into the failure domain decreases, and the impact on the failure probability becomes smaller.

[0279] Comparing the curves showing the failure probability of de-icing protection measures and the curves without considering the failure probability of de-icing protection measures, there is a critical discrimination value between FPWDM and FPWoDM. To the left of the critical discrimination value, FPWDM is less than FPWoDM, and to the right of the critical discrimination value, FPWDM is greater than FPWoDM. Furthermore, after widening the difference, FPWDM tends to converge to FPWoDM. For the suspension point condition, the minimum de-icing discrimination value corresponding to the critical discrimination value is 31.8 mm (obtained through linear interpolation); for the maximum sag point condition, the minimum is 32.7 mm, both of which are greater than the icing design value of 25 mm for this line (see Table 2). If a de-icing discrimination value is set for this line, it will also be much smaller than the critical discrimination value. The analysis reveals the following reasons for the above phenomena: When the de-icing discrimination value is less than the critical discrimination value, meaning the ice thickness is small, the line is relatively safe, and the safety domain for calculating the probability of de-icing mechanism failure is large. Conversely, when the de-icing discrimination value is greater than the critical discrimination value, the ice thickness is large, and the safety domain for calculating the probability of de-icing mechanism failure begins to be smaller than the safety domain without considering the probability of de-icing protection measures failure. Therefore, it is necessary to consider the ice distribution of transmission lines under de-icing protection measures as a mixed probability distribution. In extreme cases, when the de-icing discrimination value is large, the probability of this value occurring is extremely small, and the line can be considered to have no de-icing protection measures. In this case, FPWDM tending towards FPWoDM is inevitable, and FPCP tending towards FPWoDM is for the same reason.

[0280] Table 2 De-icing thickness (mm) of FPWDM exceeding FPWoDM

[0281]

[0282] In summary, the method for calculating the reliability of iced transmission lines under de-icing protection measures proposed in this invention has significant practical engineering application value. The ice thickness distribution under de-icing protection measures is processed into a truncated distribution, which underestimates the probability of continued service failure of iced transmission lines, thus setting a high-risk de-icing threshold. When the de-icing threshold is less than the critical threshold, processing the continued service reliability of transmission lines under the de-icing mechanism according to the original distribution is conservative; conversely, when it is greater than the critical threshold, processing the safe service of transmission lines under the de-icing mechanism according to the original distribution is detrimental.

Claims

1. A method of calculating the reliability of safe service of a power transmission line under de-icing protection, characterized in that, The method comprises the following steps: (1) Constructing a mixed random variable distribution function of ice thickness and line temperature: Based on the impact of de-icing protection measures on the ice thickness and line temperature of transmission lines, a random variable of ice thickness is constructed respectively. and line body temperature random variable The distribution function of a mixed random variable, which includes a discrete probability part and a continuous distribution part; (2) Construct the joint cumulative distribution function of the transmission line reliability under the de-icing protection: Let the random vector involved in the calculation of the transmission line reliability under the de-icing protection be , construct the joint cumulative distribution function of the random vector ; wherein is a discrete random variable sub-vector, is the number of discrete variables; is a mixed random variable sub-vector, is the number of mixed variables, and includes the ice thickness and the line temperature in step (1); is a continuous random variable sub-vector, is the number of continuous variables, including at least the transmission line breaking stress and the cross-sectional area; (3) The function function of the power transmission line under the deicing protection measure is constructed: the function function of the power transmission line under the deicing protection measure is constructed, and the function function of the power transmission line under the deicing protection measure is obtained as ; In the formula, is the tensile strength of the conductor or ground wire; is the load effect; is the safe operating tension limit coefficient of the conductor, the maximum sag point being 60% and the suspension point being 66%; is the tensile strength of the conductor or ground wire; is the cross-sectional area of the conductor; is the tension of the power transmission line under the comprehensive specific load; (4) Multidimensional Lebesgue decomposition of joint cumulative distribution function: failure probability of iced transmission line under de-icing mechanism in failure domain ; based on multidimensional Lebesgue theorem, decompose joint cumulative distribution function in step (2) as ; where , , and are all dimensional bounded non-decreasing functions, respectively called I, II, III and IV type bounded non-decreasing functions, ; ignoring singular function term F III , then ; ;​​ When there is no continuous random variable , i.e. , F(V) contains term in the decomposition, the decomposition is: ； When there are continuous random variables , i.e. , F(V) is then decomposed into: ； for an ice shedding mechanism of a power transmission line, for ice thickness or wire temperature cumulative distribution function when discrete values are obtained, for ice thickness or wire temperature cumulative distribution function when continuous values are obtained; (5) Determine the integral domain of the decomposed cumulative distribution function: according to the function of the continuous service function in step (3), determine the failure domain Next, the integral domain of each of the decomposed cumulative distribution functions obtained in step (4) , and is determined respectively; (6) Solving the sub-item failure probability: respectively calculate , and the failure probability in the corresponding integral domain, denoted as , and ; wherein, the failure probability under the failure domain The failure probability is calculated by summing the probability values of the discrete points that satisfy the continuous service function, and the calculation formula is as follows:​ ； wherein to meet the failure domain the probability of the element in the point set of denotes the number of point lists that meet the condition;​ The failure probability under the failure domain The calculation formula is as follows:​ ； wherein ; ； where subscript "D" represents the variable taking discrete part in the mixed variable, and subscript "C" represents the variable taking continuous part in the mixed variable; for discrete combination the first group point sequence the probability of and jointly obtained; by the combination of mixed random variables in variables, total is an integer, ; is a step function; ; is a random variable under the distribution function of point sequence , obtained by combination, namely: wherein is a random variable is a conditional probability distribution function under the point set is a complement symbol Not satisfying the random variable normalization condition, let its density function be Then It is calculated by the following formula: ； wherein is the number of points is the number of points is the product of the number of discrete elements of each of the discrete variables The failure probability under the failure domain The calculation formula is as follows:​ ； wherein is the derivative of is the derivative of To and normalized, we get and normalized expression and then we have: ； ； wherein and are the normalized coefficients of and ; and and are the normalized probability density functions. (7) Calculate the overall failure probability: According to the sub-failure probability obtained in step (6) , and calculate the overall failure probability of the power transmission line under the ice-melting protection measure ; Final sub-item failure probability: ； ； ； Overall failure probability of the power transmission line in safe service under the ice-melting protection measure: ； (8) calculating the reliability of the power transmission line in safe service under the de-icing protection measure: according to the total failure probability of the power transmission line in safe service under the de-icing protection measure obtained in step (7) calculating the reliability of the power transmission line in safe service under the de-icing protection measure ; ； In the formula, is the inverse function of the cumulative function of the standard normal distribution.

2. The method of claim 1, wherein, The ice thickness random variable in step (1) The mixed random variable distribution function is specifically: ； ； wherein and are upper and lower bounds of a random variable are upper and lower bounds of a random variable are upper and lower bounds of a random variable are upper and lower bounds of a random variable are upper and lower bounds of a random variable are upper and lower bounds of a random variable are upper and lower bounds of a random variable 3. The method of claim 1, wherein, The wire body temperature random variable in step (1) The mixed random variable distribution function is specifically: ； ； wherein is a temperature random variable is a discrete point, is a probability corresponding to is a probability corresponding to and are the original cumulative distribution function and the probability density function of the random variable respectively.

4. The method of claim 1, wherein, The tension of the power transmission line under the comprehensive specific load in step (3) The calculation method comprises: 1) Two meteorological states are defined: State 1 (annual average temperature state): temperature , wind speed , ice thickness , allowable stress ; State 2 (Thermal Ice Melt State): Temperature , Wind Speed , Ice Thickness ; 2) Calculate the ratio of the two states , and the wind yaw angle , : ； ； ； ； ； ； ； ； ； wherein, is the mass per unit length of the transmission line; is the cross-sectional area of the transmission line; is the density of the transmission line; is the acceleration of gravity; is the ice thickness in state 1 ; is the ice thickness in state 2; is the outer diameter of the transmission line; is the wind load adjustment coefficient of the transmission line; is the wind speed unevenness coefficient; is the body shape coefficient of the overhead line; is the wind speed in state 1 ; is the wind speed in state 2; is the included angle between the wind direction and the line direction; and are the horizontal and vertical specific loads in state 1, respectively, is the comprehensive specific load in state 1 ; and are the horizontal and vertical specific loads in state 2, respectively, is the comprehensive specific load in state 2; is the allowable stress of the transmission line under the annual average temperature meteorological condition, is the horizontal stress component of the overhead line along the line direction. 3) solving the horizontal stress component of state 2 : based on the horizontal stress component of state 1 , by solving a cubic equation δ 02 is obtained; Wherein: ； ； The derivative is: ； The Newton iteration formula is: ； The axial stress of the power transmission line at any point in the wind deflection plane under state 2 is: ； The iteration initial value is given , the calculation and , the iteration is obtained by using the above formula , repeated until times, , Take 1 x 10 -6 ; Wherein: ； wherein: is the horizontal stress component along the line direction for overhead lines in state 2; is the temperature linear expansion coefficient of the overhead line; is the Young's modulus; is the comprehensive specific load in state 2; is the wind yaw angle in state 2; is the height difference angle; is the coordinate value of any point; is the span of the transmission line; 4) Calculate the suspension point tension and the maximum sag point tension : Coordinates of the suspension point of a power transmission line and Substitute Obtain the power transmission line Axial stress at suspension point and Axial stress at suspension point : ； ； By comparing the obtained suspension point tension : ； For the axial stress of the transmission line at the location of the maximum sag point within a span, the equation of the catenary shape is found. The coordinates of the point are obtained by finding the minimum value. The equation for the shape of the catenary suspended between two towers. for: ； The specific process of the multi-dimensional Lebesgue decomposition in step (4) comprises: ； ； ； Substitute x max into to obtain the stress at the point of maximum sag , Multiply by the cross-sectional area of the transmission line to obtain the tension at the point of maximum sag ; In the formula, The height difference of the first span of the transmission line; The span of the transmission line.

5. The method of claim 1, wherein, The determination rule of the integral domain in step (5) is: 1) For discrete random variables , let the number of elements of the th component be , then discrete variables exist point column combinations, and the probability value of the discrete point of the th component is ; 2) for mixed random variables each component is decomposed into an absolutely continuous part and a discrete part ; 3) for continuous random variables whose first component cumulative distribution function is and whose probability density function is .

6. The method of claim 1, wherein, 4) Unconstrained optimization solution: the particle swarm optimization (PSO) is used to solve the above unconstrained optimization problem; For a failure domain of , the th point column should satisfy the following condition: ； For , the integral domain in the formula is: ； For , the integral domain in the formula is: 。 7. The method of claim 1, wherein, The calculation method of the above-mentioned and comprises: 1) Variable transformation: In random variables The function is Below, for the continuous portion of continuous or mixed variables. Perform transformation ,make Transform to standard normal space; transform to ordinary linear transformation, Rosenblatt transformation, or Nataf transformation; 2) Transformation into a reliability index optimization problem: transformation of the failure probability calculation into a Hasofer-Lind reliability index solution; ; After transformation becomes The number of random variables that need to be transformed depends on and The number of random variables in the joint probability density function in the calculation formula is set to , first calculate and The reliability index corresponding to the calculation formula and , use to represent and , that is: ； wherein is a constraint, guaranteeing the point on a hypersurface in the standard normal space; and are the upper and lower bounds, respectively, of the th random variable 3) Constrained optimization transformation: adding ice thickness variable Inequality constraints in the interval The inequality constraints are transformed into equality constraints by using the interior point penalty function method. The calculation formula is as follows: ； where λ is a penalty coefficient corresponding to the equality constraint; λ > 0 is used to control the influence of the penalty function; λg[G -1 (u)] is a control parameter; n C is the number of variables taken from the continuous part of the mixed variable and the continuous variable, n C = h + e - r; μ i is the penalty coefficient of the ith random variable; is the lower bound of the random variable . The calculation formula is as follows: For the failure probability calculation formula P f,II and P f,IV , the search space dimension is determined as n C according to the number of random variables of the continuous part C , it is assumed that there are m i particles, wherein the position of the i-th particle is , the speed is , and the fitness value is β C , the position of the individual obtaining the best fitness value is calculated according to , the position of the population obtaining the best fitness value is , i=1,2,...,m -6 ; in the process of solving the optimal solution by the algorithm, the particles are constantly optimized, and when the evolution reaches the j-th generation, the speed, the speed boundary and the position of each particle are respectively: ； ； ； wherein, called inertia factor, is equal to 0.75 and is used to control the performance of the model optimization; and is a number randomly distributed between 0 and 1 ; and is an acceleration constant, ; and are the positions of the particle evolved to the th and th generation, respectively; and are the velocities of the particle evolved to the th and th generation, respectively; is the time interval, taken ; indicates the maximum value of the velocity; 5) Failure probability calculation: the fitness function of the particle swarm optimization (PSO) formula is calculated to obtain the minimum , and then the iteration is ended according to the iteration end condition that the difference between the two generations is less than 1×10 -6 , the reliability index is calculated, and the corresponding failure probability is calculated according to the relationship between the reliability index and the failure probability; ​ ； According to the above calculation of β HL The calculation method of P f,II and P f,IV The calculation formula of the corresponding reliability index and After that, the failure probability of each is calculated, and the calculation formula is as follows: ； ； Final failure probability is: ； wherein is the best position coordinate obtained by the particle swarm algorithm; is the cumulative function of the standard normal distribution.

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