Application of Piecewise Catenary Method in Length Calculation of Soft Busbar in Ultra-high Voltage Substation

Abstract: According to actual setting-up operation condition of the soft busbar in the ultra-high voltage substation, the nonlinear piecewise catenary equations are established including weight of the insulator string, elastic elongation of the busbar, weight load of the electrical fitting etc. several influence factors on basis of the elastic catenary equation. The initial value selection plan is also given out for Newton iteration method of the nonlinear equations. The method sets both the insulator string and the busbar as the catenary lines, and considers the weight of fitting of multiple split conductors in the final cable shape, which greatly improves calculation accuracy. Reliability of the method is validated through comparison between calculation result of the piecewise catenary method and actual data. The piecewise catenary calculation forwarded in this paper is simple in formation and small in calculation quantity, which is liable to realize by program. The sags corresponding to the different blanking length of the busbar are obtained according to calculation results. The method can provide the effective mean for the length calculation and engineering construction of the soft busbar in the ultra-high voltage substation.


Introduction
At present, construction of the ultra-high voltage power transmission line is developing in great scale, and the corresponding construction technology of the busbar in the substation is changed greatly from the conventional 500kV and 750kV substation.In engineering process, the key factor affecting construction quality of the busbar is the blanking length of the conductor [1,2].
In the conventional substation, length and weight of the insulator string are far less than the corresponding parameters of the conductor during construction of the busbar, which makes blanking calculation of the busbar conductor as the simple suspension cable issue.Length of the insulator string is neglected in the busbar calculation method of the conventional substation, the busbar is considered as one catenary line, the single catenary equation or its similar solution is applied to calculate blanking length [3].
In the ultra-high voltage substation, influence of the insulator string on the sag of the conductor is increased because length and weight of the insulator strings in the busbar are increased greatly; error is great when the catenary equation is directly applied to calculate the blanking length of the conductor, which can't meet with design requirement [4,5].Therefore Hu Shenghui et al. put forward the combined model method [6], which considers the insulator string as the suspension catenary, and calculates the busbar as the parabola or the straight line.This method doesn't obtain the accurate cable shape of the soft conductor; it also doesn't consider the influence caused by weight or centralized loads on the sag.
For the above issue, a piecewise catenary method is put forward in this paper for the blanking length calculation of the conductor during construction of the busbar in the ultra-high voltage substation.This method can analyse the elastic elongation of the busbar and the influence of fitting and centralized loads on the sag.Through comparison calculation with the engineering calculation sample in the reference document, it is seen the results are in accordance with engineering design requirements, which have high engineering practicability.

Piecewise analysis of busbar
Lengths of the insulator string and the fitting in the 1000 kV ultra-high voltage substation are very great (about 13m), and weights are also greater.Therefore influence of the insulator string on the whole sag can't be neglected, in particularly influence is more obvious when the span is very small.
Because difference of the conductor and the insulator string is very great in line density and there is the spacer.fittings in the conductor, the space cable shape of the busbar is generally divided into several segments, such as the conductor sections and conductor sections.The conductor section is split up by the fittings between the insulator strings at two ends, shown as Figure 1.The insulator string section and the conductor section are the catenary lines which only bear weight.For the split conductor, length changes of the conductors at upper and lower can be calculated according to connection conditions at two ends of the conductor.

Piecewise equation set of elastic busbar
In the busbar, both the insulator string section and the conductor section are catenary lines, set q as the cable density which is distributed along length of the arc, K is elastic constant of material, K = 0 for the insulator string (i.e., elastic constant is 0), for the conductor K = 1/ (EA 0 ), E is elastic module of the conductor, A 0 is section area.Shown as Figure 2, according to forcing conditions of the suspension cable micro-body, elastic catenary equations of height difference h and horizontal span l are obtained [7,8]: In which: H is horizontal component of tangential tension; a = sinh -1 (V A /H), b = sinh -1 (V B /H), V A , V B are vertical components of tangential tension at end points; s 0 is initial cable length, s is length of the cable after elastic elongation.For the above equation set, horizontal tension H in the cable and vertical force V A , V B at end points can be calculated when l, h and s 0 are known.We will construct the cable equation set of the busbar which is applicable to different material and several concentrated loads according to this equation set, and calculate its horizontal tension H and the sag f in the cable.
The whole busbar can be considered as a combination of the Piecewise catenaries which are divided according to change point of the material quality, the centralized load action point, the suspension cable at every section only bears dead weight action.Assuming the suspension cable consists of n sections, the end points of every section are A i , B i , dead weight density is q i , elastic constant is K i , initial arc length is s i , horizontal span is l i , height difference is h i (i = 1, 2, …, n).There are n-1 centralized loads on conductor sections, which are p i (i = 1, 2, …, n-1) respectively.
Establish the local coordinate system for every section of the suspension cable, and establish the non-linear equation set of the piecewise busbar under several centralized loads in every local coordinate system, shown as Figure 3.

Continuity condition of tension
For the connection point between every cable section, according to the force balance conditions at horizontal direction and vertical condition in Figure 4, there are …, n-1).Therefore it is learnt horizontal tension H i of every cable section is same, which is marked as H.
Establish the force balance equation of every load point according to V A /H=sinh (a) and V B /H=sinh (b):

Section length
Length of the insulator string is generally a determined value during construction of the busbar in the substation, size of the sag is determined by the blanking length of the conductor.The sag which meets with design condition is reversely reckoned through adjusting the blanking length side of the conductor.The length equation of every section is obtained according to the equation ( 3): [sinh( ) sinh( )]

Compatibility condition
Every section of the busbar shall be in accordance with the whole compatibility equation of the horizontal span There is deformation compatibility equation at height direction , the following equation is obtained according to the equation (2):

Nonlinear equation set
Summarizing the above equation, the non-linear equation set of the elastic busbar under action the multiple centralized loads are obtained: This equation set includes 2n+1 nonlinear equations, unknown numbers are ai (i = 1, 2,…, n), bi (i = 1, 2, …, n), H, there are 2n+1 unknown number.Therefore the nonlinear equation set is enclosed, and it can be expressed as: In which Newton iteration method is applied to solve this nonlinear equation set [9].When X is calculated, place into the basic catenary equation ( 1) and ( 2) of every section, and obtain every position coordination of the busbar, and then the sag is obtained.

Initial value selection plan of Newton iteration method
Because calculation of Newton iteration method is very sensitive to selection of initial value, the iteration initial value X 0 shall approach true solution as possible in order to ensure calculation convergence and calculation speed.This paper calculates approximate vector height f according to the parabola equation: And horizontal component H and a 1 of tension under action of centralized load are further solved: In which, q is average density of the section busbar: H ] of the nonlinear equation set F (X) = 0 are calculated by the equations ( 5) and( 6).

Validation of calculation sample
In order to validate accuracy of the piecewise catenary calculation of the busbar, data in [4] and [10] are compared.
During actual construction engineering of the busbar in the substation mentioned in [4], actual measured span L = 57.58m;lengths of the insulator and the metal string at upper layer L 1 = 13.42m,quality of the single string is 503.15kg; the conductor select the four split heat resistant aluminum alloy diameter expansion conductor, actual measured quality of every meter of the conductor is 4.47kg.Carry out actual measurement of the sag according to the blanking length of the conductor.Seen from comparison data in Table 1, the sag obtained through calculation of the piecewise catenary calculation is very close the actual measurement value, maximum error is 3.07%, which meets with design requirements.Rope shape is shown as Figure 5 when conductor length is 32.804m.The calculation sample in [10] is the contour line span of 5 loads, and refers to table 2 for basic data.
Seen from comparison data in table 3, error of horizontal tension is 4.65%, error of distance of the lowest point to right end is 0.21%, which meet with design requirement.It is seen that shape calculation of the busbar with multiple centralized loads is more accurate by the piecewise catenary.

Conclusion
In the ultra-high voltage AC substation engineering, the insulator string of the busbar is long and weight is great, which affects the sag of the conductor greatly.This paper takes the whole busbar as combination of the multiple piecewise catenaries which are classified as change point of the material and centralized load action point.And tension continuity condition, height difference compatibility condition and the span compatibility conditions of every section of the catenaries are analyzed, and the balance equation set of the elastic piecewise catenary under action of the multiple centralized loads is established.Formation of the equation set is simple, structure is clear, which can be solved and calculated by Newton iteration method.Calculation quantity is small, and result accuracy is high.The forwarded method is applicable to the cable shape calculation of the busbar with different height difference and considering weights of several fittings and length of the different insulator string, and accurate sag of the busbar is obtained.Accuracy of this calculation method is validated through 2 calculation samples.This calculation method provides theoretic basis for blanking length calculation of the busbar in the ultra-high voltage substation, which is helpful to improve construction efficiency of the busbar in the substation, which reduces engineering waste.

Figure 4 .
Figure 4. Forcing analysis of centralized load action location.