Sag calculation method for overhead power line
By deducing the catenary equation in a custom coordinate system and expanding it, combining the height of the two towers, the maximum sag value of the overhead power line is calculated, which solves the problem of complex and low accuracy of calculating sags in the existing technology, and realizes efficient and accurate sag calculations.
Patent Information
- Application Number
- CN202411554646.4
- Authority / Receiving Office
- CN · China
- Patent Type
- Applications(China)
- Current Assignee / Owner
- Filing Date
- 2024-11-04
- Publication Date
- 2025-03-11
AI Technical Summary
In the calculation of the sag of overhead power lines, the prior art has problems such as high observation conditions, large data volume, cumbersome calculation process, and low calculation accuracy when the height difference between the two towers is too large.
Using a custom coordinate system, the catenary equation is derived, and it is expanded through the hyperbolic cosine function expansion formula. Combined with the height of the two towers, the maximum sag calculation formula is obtained. By measuring the coordinates of the top and lowest points of the two towers, the maximum sag value is calculated.
Simplifies the measurement and calculation process, improves calculation accuracy, reduces operation complexity, and is suitable for towers of different heights.
Smart Images
Figure CN119598059A_ABST
Abstract
Description
Technical Field
[0001] The invention relates to the technical field of electric power engineering surveying and mapping, and in particular to a method for calculating the sag of an overhead power line. Background Art
[0002] At present, when total stations are used to measure the sag of overhead power lines in the field, the side angle method is often used. This method requires 10 data, such as Figure 1 The side angle method shown in the figure. The data used for calculation are: slant distance NA, horizontal distance NA1, height AA1, slant distance NC, horizontal distance ND, slant distance NB, horizontal distance NB 1, height BB 1, horizontal angle ∠A1NB 1, horizontal angle ∠B 1ND. To measure these data, it is necessary to set up a total station on one side of the two towers, and the on-site line of sight conditions should be able to observe the bottom of the tower. The observation and calculation process is divided into three steps: (1) Observe tower A; (2) Observe tower B; (3) Calculate the lowest point of the sag. When calculating, first use the triangle cosine theorem to obtain the third side length A1B 1, which is the span L, from the three parameters of horizontal distance NA1, horizontal distance NB 1 and horizontal angle ∠A1NB. Then use the sine theorem to calculate the horizontal angle ∠NA1B 1. In △A1ND, A1D = L / 2, and the sine theorem gives ∠A1ND = α. At this time, set the zero direction of the total station to NA1, and then rotate the angle α to project the maximum sag point to the ground point, and then stake it out at this angle. Finally, rotate the telescope vertically to tangent the power line, and measure again to get the height CD, the sag distance f = OD-CD, and OD = 1 / 2*(AA1+BB 1).
[0003] There are the following disadvantages when calculating sag using the side angle method: (1) The observation condition requires a clear view of the tower bottom, which is difficult to achieve due to the high tower height; (2) Too much data is used; (3) The calculation process is cumbersome; (4) When the height difference between the two towers is too large, that is, h / L>10%, the calculation accuracy is low. Summary of the invention
[0004] The object of the present invention is to provide a method for calculating the sag of an overhead power line, which is simple and convenient to test and calculate and has high calculation accuracy, in view of the deficiencies in the prior art.
[0005] In order to achieve the above object, the technical solution adopted by the present invention is:
[0006] A method for calculating the sag of an overhead power line, comprising:
[0007] S1, custom coordinate system, obtain the catenary equation of the overhead power line in the custom coordinate system;
[0008] S2, obtain the sag calculation formula of any point of the overhead power line;
[0009] S3, expanding the catenary equation obtained in step S1 using a hyperbolic cosine function expansion;
[0010] S4, according to the heights of the two towers on which the overhead power lines are suspended, combining steps S2 and S3, obtaining a maximum sag calculation formula;
[0011] S5, measuring the coordinates of the tops of the two towers on which the overhead power lines are suspended, obtaining the lowest point of the overhead power lines and measuring its coordinates, substituting them into the calculation formula of step S4, and calculating the maximum sag value of the overhead power lines.
[0012] In some embodiments, in step S1, a custom coordinate system is defined with the lowest point of the overhead power line as the origin, the horizontal direction as the X-axis, and the tower height direction as the Y-axis. The catenary equation in the custom coordinate system is calculated based on mechanical balance and geometric trigonometric functions as follows:
[0013]
[0014] Among them, x and y are the coordinate values of any point P on the overhead power line, respectively; g is the self-weight load of the overhead power line, which is a constant; σ is the standard stress of the overhead power line, which is a constant.
[0015] In some embodiments, the process of calculating the catenary equation includes:
[0016] (1) Based on the static force equilibrium condition, the horizontal tension T0 and vertical force G at point P are obtained as follows:
[0017] T0=Tx*cosα=σ*S,
[0018] G=Tx*sinα=g*lx*S,
[0019] Thus, it is concluded Differentiating both sides of the equation yields:
[0020] Shift the items to get:
[0021] Taking the indefinite integral on both sides of the equation, we get:
[0022] Simplified to:
[0023] It follows that:
[0024] From the definition of the inverse hyperbolic sine function:
[0025] Taking the special value α=0, x=0, we get C1=0, thus we get:
[0026]
[0027] (2) From trigonometric functions, we get:
[0028] (3) From steps (1) and (2), we can obtain Integrating both sides yields:
[0029] Thus, it can be deduced that:
[0030]
[0031] Take the special values x=0, y=0, and we get Thus we get the catenary equation;
[0032] Among them, S is the cross-sectional area of the overhead power line, which is a constant; lx is the arc length of the conductor from the origin to point P, α is the height difference angle; C1, C2, and C3 are all constants.
[0033] In some embodiments, the sag calculation formula for any point P(x, y) on the overhead power line is:
[0034] fx=Yb-d-Yp,
[0035]
[0036] Among them, the tops of the two towers hanging overhead power lines are marked as A and B respectively, and the vertical coordinates of the endpoints A and B in the custom coordinate system are Ya and Yb respectively, and Yb is not less than Ya; Ha and Hb are the heights of the two towers hanging overhead power lines respectively, and Hb is not less than Ha; L is the span; lb is the horizontal distance between point B and the coordinate origin; Xp is the horizontal distance between point P and the coordinate origin.
[0037] In some embodiments, the hyperbolic cosine function in step S3 is defined as:
[0038] Find its derivatives of various orders at x = 0, and expand them according to Taylor to get:
[0039] The catenary equation obtained in step S1 is thus expanded to:
[0040] The simplified calculation formula is:
[0041]
[0042] In some embodiments, when the heights of the two towers suspending the overhead power lines are the same, the maximum sag point is located at the midpoint of the span, x=L / 2, and the maximum sag calculation formula in step S4 is obtained as follows:
[0043]
[0044] In some embodiments, when the heights of the two towers suspending the overhead power lines are different, the maximum sag calculation method in step S4 includes:
[0045] In the geometric figure, draw a straight line A'B' parallel to AB and tangent to the overhead power line at point Q. Point Q is the point of maximum sag value. The coordinates of point Q in the custom coordinate system are (Xq, Yq). Derivative the catenary equation obtained in step S4 to obtain the slope of the straight line A'B':
[0046]
[0047] The slope k of line AB is:
[0048]
[0049] Since y′=k, we can get
[0050] The calculation results are:
[0051] The catenary equation obtained from step S1 is obtained:
[0052]
[0053] The maximum sag calculation formula obtained according to step S2 is:
[0054]
[0055] In some embodiments, the coordinates of the top points A and B of the two towers measured in step S5 in the city coordinate system are ((XA, YA, ZA) and (XB, YB, ZB) respectively, and the coordinates of the lowest point O of the overhead power line in the city coordinate system are (XO, YO, ZO), and it can be obtained:
[0056]
[0057] Ha=ZA-H 塔底 ,
[0058] Hb=ZB-H 塔底 ,
[0059]
[0060] Among them, H 塔底 is the height of the tower bottom from the coordinate origin.
[0061] In some embodiments, when the ratio of the height difference between the two towers suspending the overhead power lines to the span is no more than 10%, the two towers are defined to have the same height; when the ratio of the height difference between the two towers suspending the overhead power lines to the span is greater than 10%, the two towers are defined to have different heights.
[0062] In some embodiments, in step S5, the coordinates of the tops of the two towers on which the overhead power lines are suspended are measured by a total station without a prism, and the overhead power lines are cut transversely by the crosshairs of the total station to find the lowest point of the overhead power lines and measure their coordinate values.
[0063] Due to the application of the above technical scheme, the present invention has the following advantages compared with the prior art: in the sag calculation method of the overhead power line of the present invention, it is only necessary to measure the coordinate values of three points, namely the tops of the two towers on which the overhead power line is suspended and the lowest point of the overhead power line, to calculate the maximum sag value. This maximum sag calculation method is simple and easy to implement, which greatly improves the working efficiency, and has high calculation accuracy and good engineering practical value. BRIEF DESCRIPTION OF THE DRAWINGS
[0064] Attached Figure 1 It is a schematic diagram of various parameters when calculating sag using the mid-range side angle method of the prior art;
[0065] Attached Figure 2 is a flow chart of the sag calculation method for overhead power lines of this embodiment;
[0066] Attached Figure 3 A schematic diagram of the forces at any point P on the overhead power line of this embodiment;
[0067] Attached Figure 4 Schematic diagram of various parameters in the sag calculation method for overhead power lines of this embodiment. DETAILED DESCRIPTION
[0068] The technical solution of the present invention will be described clearly and completely below in conjunction with the accompanying drawings. Obviously, the described embodiments are part of the embodiments of the present invention, rather than all of the embodiments. Based on the embodiments of the present invention, all other embodiments obtained by ordinary technicians in this field without creative work are within the scope of protection of the present invention.
[0069] The process of the sag calculation method of the overhead power line of the present invention is as follows: Figure 2 As shown, the specific steps include the following steps.
[0070] S1, custom coordinate system, obtains the catenary equation of the overhead power line in the custom coordinate system.
[0071] A custom coordinate system is created with the lowest point of the overhead power line as the origin, the horizontal direction as the X-axis, and the tower height direction as the Y-axis. Take any point P on the overhead power line, and record the coordinate value as P(x,y).
[0072] (1) Calculate the horizontal tension T0 and vertical force G at point P based on the static force equilibrium condition, as follows: Figure 3 shown.
[0073] The specific calculation formula is:
[0074] T0=Tx*cosα=σ*S (1)
[0075] G=Tx*sinα=g*lx*S (2)
[0076] Where g is the deadweight load of the overhead power line N / (m*mm 2 ) or (Mpa / m), which is a constant and can be obtained by looking up the table; σ is the standard stress of the overhead power line (N / mm 2 ) or (Mpa), which is a constant and can be obtained by looking up the table; S is the cross-sectional area of the overhead power line (mm 2 ), is a constant that can be obtained by looking up the table; lx is the arc length of the wire from the origin to point P; α is the height difference angle.
[0077] According to formula (1) and formula (2), we can get:
[0078]
[0079] Differentiating both sides of formula (3) yields:
[0080]
[0081] Use the differential arc length to calculate d(lx), specifically:
[0082]
[0083] And dy=dx*tanα(6)
[0084] Substituting formula (5) and formula (6) into formula (4), we obtain:
[0085]
[0086] By shifting the terms in formula (7), we can obtain:
[0087]
[0088] Taking the indefinite integral on both sides of formula (8), we get:
[0089]
[0090] Because d(tanα)=sec 2 α*dα, So we simplify formula (9) to:
[0091]
[0092] From formula (10), we can get:
[0093]
[0094] Among them, C1 and C2 are constants.
[0095] Since the inverse hyperbolic sine function is defined as:
[0096] sh -1 (tanα)=ln(secα+tanα)+C (12)
[0097] From formula (11) and formula (12), we can get:
[0098]
[0099] In formula (13), we take the special values α=0, x=0, and obtain C1=0, thus obtaining:
[0100]
[0101] (2) From trigonometric functions, we can get:
[0102]
[0103] (3) Obtain the catenary equation.
[0104] Since both steps (1) and (2) calculate tanα, they should be the same, that is, through formula (14) and formula (15), we can obtain:
[0105]
[0106] Integrating both sides of formula (16) yields:
[0107]
[0108] Thus, it can be deduced that:
[0109]
[0110] Among them, C3 is a constant.
[0111] In formula (18), we take the special values x=0, y=0 and obtain Thus, the catenary equation is obtained, which is:
[0112]
[0113] S2, obtain the sag calculation formula at any point of the overhead power line.
[0114] In the custom coordinate system, the specific calculation formula for the sag fx at any point of the overhead power line is:
[0115] fx=Yb-d-Yp (20)
[0116]
[0117] Among them, the tops of the two towers hanging overhead power lines are marked as A and B respectively, and the vertical coordinates of the endpoints A and B in the custom coordinate system are Ya and Yb respectively, and Yb is not less than Ya; Ha and Hb are the heights of the two towers hanging overhead power lines respectively, and Hb is not less than Ha; L is the span; lb is the horizontal distance between point B and the coordinate origin; Xp is the horizontal distance between point P and the coordinate origin; Yp is the height of point P from the coordinate origin. As shown in 4.
[0118] S3, expanding the catenary equation obtained in step S1 using a hyperbolic cosine function expansion.
[0119] The hyperbolic cosine function is defined as:
[0120]
[0121] Find its derivatives of various orders at x = 0, according to Taylor expansion:
[0122] 0th order reciprocal ch(0) = 1; 1st order reciprocal ch 1 (0) = sh(0) = 0; 2nd order inverse ch 2 (0) = 1: 3rd order inverse ch 3 (0) = 0; 4th order inverse ch 4 (0)=1…….
[0123] That is, the odd-order derivatives of ch(x) are all 0 at x=0, so its expansion contains only even-order power terms.
[0124] Therefore, formula (22) can be expanded by Taylor as follows:
[0125]
[0126] Substituting formula (23) into formula (19,) the catenary equation is expanded to:
[0127]
[0128] In formula (24), the calculated value starting from the second term is relatively small and can be discarded, so the first term in formula (24) is taken and formula (24) is simplified to the calculation formula:
[0129]
[0130] S4, according to the heights of the two towers suspending the overhead power lines, combined with steps S2 and S3, a maximum sag calculation formula is obtained.
[0131] (1) The two towers that hang overhead power lines have the same height
[0132] The height difference between the two towers that suspend the overhead power lines is Δh. When Δh / L≤10%, the two towers are defined as having the same height.
[0133] When the two towers hanging overhead power lines have the same height, the geometric properties of the parabola show that the lowest point is located in the middle of the two end points A and B, so the maximum sag point is located at the midpoint of the span, that is, the maximum sag point is the same as the lowest point of the overhead power line, which is the origin of the coordinate system in the custom coordinate system.
[0134] That is, x = L / 2, d = 0, Yp = 0. At this time, the maximum sag calculation formula can be obtained from formula (20) and formula (25), which is:
[0135]
[0136] (2) The heights of the two towers that hang overhead power lines are different
[0137] The height difference between the two towers that suspend the overhead power lines is Δh. When Δh / L>10%, it is defined that the two towers have different heights.
[0138] When the heights of the two towers hanging overhead power lines are different, draw a straight line A'B' in the geometric figure parallel to AB and tangent to the overhead power line at point Q. Point Q is the point of maximum sag value. The coordinates of point Q in the custom coordinate system are (Xq, Yq). The slope of the straight line A'B' obtained by differentiating the catenary equation obtained in step S4 is:
[0139]
[0140] The slope k of line AB is:
[0141]
[0142] Since A'B' is parallel to AB, y'=k, so we can get:
[0143]
[0144] Calculated by formula (28):
[0145]
[0146] Substituting formula (29) into formula (19) yields:
[0147]
[0148] That is, the coordinate value of point Q in the custom coordinate system is:
[0149]
[0150] Substituting the coordinates of point Q into formula (20) and formula (21), and according to formula (25), the maximum sag calculation formula is obtained as follows:
[0151]
[0152] S5, measuring the coordinates of the tops of the two towers on which the overhead power lines are suspended, obtaining the lowest point of the overhead power lines and measuring its coordinates, and calculating the maximum sag value of the overhead power lines.
[0153] The coordinates of the top points A and B of the two towers in the city coordinate system are measured by the total station without prism ((XA, YA, ZA) and (XB, YB, ZB) respectively. The lowest point of the overhead power line is found by cutting the overhead power line horizontally with the crosshairs of the total station. The coordinates of the lowest point O of the overhead power line in the city coordinate system are measured as (XO, YO, ZO), and the following can be obtained:
[0154]
[0155] Ha=ZA-H 塔底 ,
[0156] Hb=ZB-H 塔底 ,
[0157]
[0158] Δh=Hb-Ha,
[0159] Among them, H 塔底 is the height of the tower bottom from the coordinate origin.
[0160] According to the measured coordinate values of points A, B and O, query the overhead power line parameter table to obtain the self-weight load g and standard stress σ under the corresponding overhead power line type specifications, meteorological conditions, electrical parameters and mechanical conditions;
[0161] When Δh / L≤10%, the calculated L, the queried g and σ are substituted into formula (26) to calculate the maximum sag value of the overhead power line.
[0162] When Δh / L>10%, the calculated L, Ha, Hb, lb and the queried g and σ are substituted into formula (31) to calculate the maximum sag value of the overhead power line.
[0163] Since the calculation formula of the maximum sag value is complicated, manual calculation is slow and prone to errors, a program can be written to calculate it. When calculating through the program, first use the query function of the program to find the corresponding deadweight load g and standard stress σ, and then enter the coordinate values of points A, B and O in the program to obtain the maximum sag value of the overhead power line.
[0164] The above embodiments are only for illustrating the technical concept and features of the present invention, and their purpose is to enable people familiar with the technology to understand the content of the present invention and implement it accordingly, and they cannot be used to limit the protection scope of the present invention. Any equivalent changes or modifications made according to the spirit of the present invention should be included in the protection scope of the present invention.
Claims
1. A method for calculating the sag of an overhead power line, characterized in that: include: S1, custom coordinate system, obtain the catenary equation of the overhead power line in the custom coordinate system; S2, obtain the sag calculation formula of any point on the overhead power line; S3, expanding the catenary equation obtained in step S1 using a hyperbolic cosine function expansion; S4, according to the heights of the two towers on which the overhead power lines are suspended, combining steps S2 and S3, obtaining a maximum sag calculation formula; S5, measuring the coordinates of the tops of the two towers on which the overhead power lines are suspended, obtaining the lowest point of the overhead power lines and measuring its coordinates, substituting them into the calculation formula of step S4, and calculating the maximum sag value of the overhead power lines.
2. The method for calculating the sag of an overhead power line according to claim 1, characterized in that: In step S1, a custom coordinate system is defined with the lowest point of the overhead power line as the origin, the horizontal direction as the X-axis, and the tower height direction as the Y-axis. The catenary equation in the custom coordinate system is calculated based on mechanical balance and geometric trigonometric functions: Among them, x and y are the coordinate values of any point P on the overhead power line, respectively; g is the deadweight load of the overhead power line, which is a constant; σ is the standard stress of the overhead power line, which is a constant.
3. The method for calculating the sag of an overhead power line according to claim 2, characterized in that: The calculation process of the catenary equation includes: (1) Based on the static force equilibrium condition, the horizontal tension T0 and vertical force G at point P are obtained as follows: T0=Tx*cosα=σ*S, G=Tx*sinα=g*lx*S, So we get tan Differentiating both sides of the equation yields: Shift the items to get: Taking the indefinite integral on both sides of the equation, we get: Simplified to: It follows that: From the definition of the inverse hyperbolic sine function: Taking the special value α=0, x=0, we get C1=0, thus we get: (2) From trigonometric functions, we get: (3) From steps (1) and (2), we can obtain Integrating both sides yields: Thus, it can be deduced that: Take the special values x=0, y=0, and we get Thus, we get the catenary equation; Among them, S is the cross-sectional area of the overhead power line, which is a constant; lx is the arc length of the conductor from the origin to point P, α is the height difference angle; C1, C2, and C3 are all constants.
4. The method for calculating the sag of an overhead power line according to claim 2, characterized in that: The sag calculation formula for any point P(x,y) on the overhead power line is: fx=Yb-d-Yp, Among them, the tops of the two towers hanging overhead power lines are marked as A and B respectively, and the vertical coordinates of the endpoints A and B in the custom coordinate system are Ya and Yb respectively, and Yb is not less than Ya; Ha and Hb are the heights of the two towers hanging overhead power lines respectively, and Hb is not less than Ha; L is the span; lb is the horizontal distance between point B and the coordinate origin; Xp is the horizontal distance between point P and the coordinate origin.
5. The method for calculating the sag of an overhead power line according to claim 4, characterized in that: The hyperbolic cosine function in step S3 is defined as: Find its derivatives of various orders at x = 0, and expand them according to Taylor to get: The catenary equation obtained in step S1 is thus expanded to: The simplified calculation formula is:
6. The method for calculating the sag of an overhead power line according to claim 5, characterized in that: When the heights of the two towers suspending the overhead power lines are the same, the maximum sag point is located at the midpoint of the span, x = L / 2, and the maximum sag calculation formula in step S4 is obtained as follows:
7. The method for calculating the sag of an overhead power line according to claim 5, characterized in that: When the heights of the two towers suspending the overhead power lines are different, the maximum sag calculation method in step S4 includes: In the geometric figure, draw a straight line A'B' parallel to AB and tangent to the overhead power line at point Q. Point Q is the point of maximum sag value. The coordinates of point Q in the custom coordinate system are (Xq, Yq). Derivative the catenary equation obtained in step S4 to obtain the slope of the straight line A'B': The slope k of line AB is: Since y′=k, we can get The calculation results are: The catenary equation obtained from step S1 is obtained: The maximum sag calculation formula obtained according to step S2 is:
8. The method for calculating the sag of an overhead power line according to claim 7, characterized in that: The coordinates of the top points A and B of the two towers measured in step S5 in the city coordinate system are ((XA, YA, ZA) and (XB, YB, ZB) respectively, and the coordinates of the lowest point O of the overhead power line in the city coordinate system are (XO, YO, ZO), which can be obtained: Ha=ZA-H 塔底 , Hb=ZB-H 塔底 , Among them, H 塔底 is the height of the tower bottom from the coordinate origin.
9. The method for calculating the sag of an overhead power line according to claim 7, characterized in that: When the ratio of the height difference between the two towers hanging overhead power lines to the span is no more than 10%, the two towers are defined as having the same height; when the ratio of the height difference between the two towers hanging overhead power lines to the span is greater than 10%, the two towers are defined as having different heights.
10. The method for calculating the sag of an overhead power line according to claim 1, characterized in that: In step S5, the coordinates of the tops of the two towers on which the overhead power lines are suspended are measured by a total station without a prism, and the overhead power lines are cut transversely by the crosshairs of the total station to find the lowest point of the overhead power lines and measure their coordinate values.