Calculation Method for dynamic Stiffness of multi-span overhead transmission lines

The dynamic motion of a span is coupled to the other spans in transmission lines. From the continuity conditions, force equilibrium conditions, and dynamical equations of every span conductor and insulator string, a close-form expression for the dynamic stiffness of a coupled two spans in harmonic motion is presented. Unlike some existing theories of single span conductor, the effect between the conductors and insulator strings is considered here. By means of example calculations, the validity of the dynamic stiffness of two-span is demonstrated by the consistency of the results determined by the ABAQUS. Meanwhile, the dynamic stiffness and natural frequency are discussed under variation of the span lengths ratio, Irvine parameter, and insulator string length. Moreover, the modal shape function corresponding natural frequency is derived, and the contribution of localization mode is studied. The results show that the contribution is either sensitive to or independent of transmission line parameters in certain parameter ranges. Finally, generalizing the work of two-span, the dynamic stiffness of arbitrary span number is obtained.


I. INTRODUCTION
Using insulator string to connect a span with its adjacent span and forming multi-span structure are widely used in electricity transmission lines. Galloping is a well-known phenomenon for iced electricity transmission lines. Galloping of multi-span conductors can reach high amplitude causing high tension on conductors, and the periodic change of the tension can be leaded to damage of sub-conductors, spacers, and towers (Kermani et al., 2013;Chen et al., 2017), which usually give rise to disruption of power supply, as shown in Fig.1. In January 2018, a serious cold snap swept through much of northern and central china, knocking down the regional power lines due to galloping [3] .  For the analysis of the galloping and other dynamic response of transmission lines, the first basic and most important aspect is to obtain vibratory characteristic of transmission lines as low-sag cables. Initially, the linear theory of free vibration of single-span horizontal cable with small sag was studied by Irvine (1974). Meanwhile, an important geometric-elastic parameter λ was raised, and the natural frequencies and corresponding symmetrical in-plane modes depended on the parameter. Irvine (1978) generalized the theory of horizontal cable and applied it to an inclined cable. Yamaguchi (1979) considered the weight component parallel to the chord of an inclined cable and studied natural frequencies and modal shapes of the inclined cable. Triantafyllou (1984Triantafyllou ( ,1986 found that for inclined cable an "avoided crossing" on frequency curves occurs. Afterward, in order to avoid complicated calculation, Wu et al. (2005) and Zhou et al. (2011) presented a simplified method by using two characteristic parameters λ and ε. In order to study the dynamic response of cable-tower, cable-bridge, and other cable-supported structures, Veletsos et al. (1983) proposed the dynamic stiffness of single span cable. The dynamic stiffness can evaluate the deformation resistance (Kim et al.,2001;Han et al.,2017), and can identify the frequencies of corresponding deferent modes (Li et al.,2017;Dai et al.,2017;Kari et al.,2017). Because the cable is a flexible structure, the cable exhibits very rich nonlinear dynamic response. In particular, if two or more linear natural frequencies are nearly commensurable, these commensurable relationships of frequencies can cause the multiple modes to be strongly coupled, and an internal resonance occurs (Rega et al.,1999(Rega et al., , 2004.
Most of the interesting internal resonances occur at the crossover points in the spectrum of cable natural frequencies. Based on the obtained modes and corresponding frequencies, a multimode discretization of the continuum formulations is adopted, then a simplified asymptotic analysis is applied by using any perturbation technique (Lacarbonara et al.,2003;Srinil et al.,2007). In recent decades, a lot of results, which include internal resonance and multiple internal resonances ( . In overhead transmission lines, the bare wire conductors are suspended from towers via insulator strings. Overhead transmission lines between two anchoring towers, which is connected to tangent towers in a row by a freely movable insulator string, is called a strain section. Due to the swing of suspension insulator strings along a transmission line, it is possible for coupling to occur between different spans in a strain section (Rienstra, 2005). Initially, in order to simplify the calculation, a single-span model with spring boundary conditions, in which suspension insulator string and adjacent span are represented by equivalent linear spring, is established (Yu et al.,1993;Desai et al., 1995;Guo et al., 2018). Afterward, the effect of the spring boundary conditions on galloping amplitude is compared to the effect of the previous fixed boundary conditions by using FEM (Wang et al.,2009(Wang et al., ,2010. The results showed that the boundary conditions of a single span can significantly influence the galloping amplitude. The dynamic interaction between the adjacent spans is reflected in the equivalent stiffness spring at both ends of single span. In fact, the equivalent stiffness represented adjacent spans not only depends on the cable geometrical parameters, but also depends on the frequency of external excitation. In addition, in order to understand the interaction between the multiple spans, Xie et al. Further investigation on the interaction between different spans in a strain section is important to deeply understand the nonlinear galloping characteristics of transmission lines, such as multi-span internal resonance conditions, frequency, mode of multi-span vibration, and the effect of the multi-span conductors on the towers. This paper is intended to be responsive to these needs. Since the two-span transmission lines is a simplest multi-span transmission lines, we introduce a dynamic stiffness model of two-span transmission lines that right end is fixed and the left end is subjected to a harmonically varying horizontal displacement. The configuration of every span conductor is considered to have a parabolic profile at static equilibrium position under self-weight, the displacement at the left end is presumed to be small. The effect of span ratio, insulator string length, and external exciting frequency on the dynamic stiffness is discussed, respectively. Special attention is paid to the identification of frequencies through dynamic stiffness peaks. The accuracy of obtained frequencies and corresponding modes is discussed through comparison of theoretical and FEM results. Finally, the two-span dynamic stiffness is generalized to the case of arbitrary multi-span transmission lines. The results can provide a theoretic foundation for the development of the galloping research.

A. THEORETICAL FORMULA TO CALCULATE THE DYNAMIC STIFFNESS
This article has been accepted for publication in IEEE Access. This is the author's version which has not been fully edited and content may change prior to final publication.  A transmission line suspended from end supports located at equal elevation is considered, as shown in Fig.2 , the dynamic configuration of the cable is the solid line as shown in Fig.2. The transmission line dynamics is described by the horizontal u and vertical w displacement components measured from the static configuration. The amplitude of additional force ΔF is presumed to be small, so that the displacement of the left end is also small. The vibration of two-span cable with respect to its position of static equilibrium can be expressed by the partial differential equations [3] where m is mass per unit length; c is the coefficient of viscous damping per unit of chord length; Δhτ2 is the dynamic additional tension of H. The effects of the axial components of the inertia forces on the dynamic stiffness are small compared with those of the y-direction components, and may be neglected. The ΔF is equal to Δhτ1 when the axial inertia force of a cable is neglected. By substituting (1) into (2), the (2) can be simplified, as 22 1 The insulator string shown in Fig.1 is suspended from a smooth bearing at O. The insulator string which oscillates with a small amplitude executes harmonic motion. The insulator string has an angular displacement of θ measured from the equilibrium position at the instant. The sun of the moments about O of all the external forces is equal to J , the moment of inertia J with the weight W relative to O is 3 W a g and  is the angular acceleration. Thus, were a is the length of the insulator string, Δx2 is the displacement of the lower end B. Tr1 and Tl2 are vertical forces exerted by the two-span conductors at the lower end B.
In conditions of moderately small vibration amplitudes and assuming small sag, the additional tension becomes [3] Accounting for the cable boundary conditions and compatibility of the resulting strains and displacements, the additional tensions of two-span conductors are given Le1 and Le1 are the effective lengths of two-span conductors.
For harmonically excited two-span conductors, the values of the displacement increment and additional tension are of the form: x e   =  (7c) Apply (7), the system equations (3) can be written as: When the angular displacement of θ is small, theθ is equal approximately to Δx2/a. With this approximation and (7) the (4) lead to the following expression as The solution of (8), with the given boundary conditions, is This article has been accepted for publication in IEEE Access. This is the author's version which has not been fully edited and content may change prior to final publication.
. The additional tensions, hτ1 and hτ2, may now be evaluated from (6). Substituting the (10) into the (6), the following expressions of additional tensions are obtained by definite integral 12 The dynamic stiffness, Kh, is the ratio of the applied force to the corresponding displacement. From (11) and (9), the dynamic stiffness function would then be given by the expression The remaining quantities are defined as Ks is equivalent static stiffness of two-span conductors. A typical 500kV transmission line section, which consists of two-span quad bundle conductors, suspension clamp, and an insulator string, is selected to obtain the dynamic stiffness. The span lengths of first and second spans are the same as 200m. Each sub-conductor of quad bundle conductors is LGJ-400/50, mass per unit length 1.511kg/m, Young's modulus E is 7.0×10 4 MPa, and the dimension of the conductor is 27.63×10 -3 m. The suspension clamp is LX-4245, and its total mass is 89kg. The spacer is JZF-400, and its mass is 7.5kg. A suspension insulator consists of 28 ball-and-socket porcelain insulators of model XP-16 with self-weight 6.0kg, the length of suspension insulator string is 6.47m, and its Young's modulus is set to 200GPa. The tension stress at its lowest point in the conductor under self-weight is 30.3Mpa(λ1=π). It is noted that for quad bundle conductors the mass per unit length m in (13) is four times larger than a single conductor.
The variation of dynamic stiffness, Kd , with respect to the dimensionless frequency, 1 /  , is shown in Fig.3 for the cable having two equal spans. The dynamic stiffness in each figure is normalized with respect to the corresponding static stiffness value. The static stiffness value calculated by (15) is constant, and the dynamic stiffness, which is calculated according to (13), is variable with frequency. In Fig.3(a) the peak corresponds to the frequency of first symmetrical mode, and in Fig.3(b) the first peak corresponds to the frequency of second symmetrical mode. The first peak in Fig.3(b) is much larger than the peak in Fig.3(a), it is shown from (13) that the effect of second symmetrical mode on the tension is much larger than the first symmetrical mode. This phenomenon of two-span conductors is different from single span.
In Fig3, the red line corresponds to a dynamic stiffness of quad bundle conductors, and the blank line corresponds to a dynamic stiffness of single conductor. It is shown from Fig.3 that the frequency of single-conductor is very close to that of quad bundle conductors when the geometrical and material parameters of single conductor and bundle conductors are the same. This conclusion provides a way to simplify the calculation of bundle conductors. In order to verify the validity of (13), the corresponding finite element model is established in ABAQUS. The frequencies of first two symmetrical modes of single conductor and bundle conductors obtained by ABAQUS are respectively 0.3536Hz, 0.4741Hz, 0.3515Hz and 0.4727Hz, respectively. The corresponding dimensionless frequencies are respectively 1.000, 1.340, 0.9937, and 1.336. The results obtained by ABAQUS showed that the frequency of single-conductor is very close to that of quad bundle conductors. The natural frequencies of single-conductor and quad bundle conductors, which are identified from Fig.3, are close to those obtained by the ABAQUS.  In order to simplify the theoretical formula for two-span conductors, the effect of insulator strings mass on the dynamic stiffness is studied. The black line corresponds to the dynamic stiffness of two equal spans considering insulator string mass, and the red line corresponds to the dynamic stiffness of two equal spans under the assumption of ignoring insulator string mass. The comparison of the frequency identified from Fig.4 is shown that the effect of insulator string mass on the frequency of two-span conductors is very small. Therefore, the J in (13) can be ignored when this (13) is used to calculate the frequency of two-span conductors.  Next, applying some examples, which have the same material parameters, insulator string length, and horizontal tension under self-weight as the previous examples, we will investigate the effects of the span ratio l1/l2 on dimensionless natural frequencies. In particular, first assuming that a constant first span length of l1 is 200m, the second span length of l2 is changed. Fig.5(a) shows the variation of the dynamic stiffness of the two-span transmission lines with different span ratios when first span length of l1 is larger than the second span length of l2. In particular, these peaks of dynamic stiffness represent the dimensionless natural frequencies of first symmetrical mode. It is seen that the dimensionless frequencies of corresponding first five peaks are greater than 1. It is possible to note that the dimensionless natural frequency of first symmetrical mode is decreasing as the ratio of l1/l2 decreases. When the ratio of is reduced to 1 the dimensionless natural frequency of first symmetrical mode is close to 1 by Combining the results of Fig.2. Fig.5(b) shows the variation of the dynamic stiffness of the two-span transmission lines with different span ratios when the first span length of l1 is less than the second span length of l2. It is seen that the dimensionless natural frequency of first symmetrical mode is increasing as the ratio of l1/l2 is increases, and when the ratio of l1/l2 increases to 1 the dimensionless natural frequency is closed to 1.  The dynamic stiffness formula can also be used to evaluate the natural frequencies of corresponding symmetrical modes by the peak of a dynamic stiffness. When first span length is a constant of 200m, the variation of the natural frequencies with span length ratio of l1/l2 is obtained from the peak of the dynamic stiffness of two spans at λ1/π=1 and a=6.47 as shown in Fig.6. It is seen that the dimensionless natural frequencies of first two symmetrical modes are increasing as the ratio of span lengths increases. A scatter plot in Fig.6 presents the natural frequency obtained by ABAQUS. It is indicated that the natural frequency obtained by the dynamic stiffness is consistent with the ABAQUS result. On the other hand, the comparison between two methods shows that the dynamic stiffness formula is reasonable. The modes in Fig.6 shown that the ratio of span lengths has a significant effect on the mode shape of two spans. For the mode of two spans with 200m-50m, the modal displacement of the second span is close to 0. The first two symmetrical modes show that the modal displacement of a span having longer span length dominates when the length of a span in two span transmission lines is much greater than the length of another span. In order to obtain a full understanding of the dynamic stiffness, the variation of the dynamic stiffness with dimensionless natural frequency is carried out on two-span conductors of 200m-250m, see Fig.7. Fig.7(a) shows the variation of the dynamic stiffness with dimensionless natural frequency for different insulator string lengths when the span lengths are 200m-250m and the λ1/π is 1. For λ1/π=1, the insulator string length strongly influences the peak of dynamic stiffness. The dimensionless natural frequency of first symmetrical mode decreases from 0.90 to 0.89, when the insulator string length increases from 1m to 50m. It is shown that the insulator string length has few effect on dimensionless natural frequency when λ1/π=1. Fig.7(b) reports the variation of the dynamic stiffness with dimensionless natural frequency for different insulator strings when the span lengths are 200m-250m and the λ1/π is 7. The calculated results also show that the insulator length has a remarkable effect on dynamic stiffness. In particular, the minimum peak occurs at the dimensionless natural frequency close to 1. This conclusion is in accord with what we discussed in Fig.6. As the insulator string length increases from 1m to 20m, the dimensionless natural frequency decreases from 1.15 to 0.92. For the two span transmission liens with λ1/π=1, the insulator string length has a remarkable effect on the dimensionless natural frequency by comparison with Fig.7(a). Fig.7(a) and (b) highlights a similar trend in the dimensionless natural frequency as a function of the insulator string length. This article has been accepted for publication in IEEE Access. This is the author's version which has not been fully edited and content may change prior to final publication.  The solid lines in Fig.8 show that the variation of dimensionless natural frequencies with insulator string length obtained through dynamic stiffness peak in Fig.7. The calculated results show approximate constant natural frequency for λ1/π=1, whereas a dependency on insulator string length is found for λ1/π=7. In Fig.8 the scatter plots show that dimensionless natural frequencies are obtained by ABAQUS for different insulator string length. The theoretical results are generally consistent with those by ABAQUS. The maximum percent error between theory and ABAQUS results of 2.2% occurs at λ1/π=7 when insulator string length is 20m. The Fig.8 also shows that insulator string length has a few effect on symmetrical mode shape.

B. THEORETICAL FORMULA TO CALCULATE THE SYMMERTRICAL MODE
Substituting (13) into (11) and ignoring Δx1, the i-th symmetrical modal shape can be written as where the Bi is determined by the normalization condition. In order to verify the validity of the new theoretical formulas, we use an example for comparison. We choose the geometrical parameters of two-span overhead transmission lines as follows: l1=200m, l2=250m, a=6.47m. The material property settings are same with the first example for the insulators string and cable. The finite element model based on ABAQUS is established as follows:  In order to better understand the application range of (17) and (16), Fig.11 reports the comparison between the results of the two new theories and FEM method. It is interesting to observe that the results by the two methods agree well with that of ABAQUS when λ1/π=1, and only the result by (16) agrees with that of ABAQUS when λ1/π=7. It is shown that the simplified formula of (17) can be applied only to two-span conductors with a small sag, and the exact formula of (16) is also suitable for two-span conductors with a larger sag. In order to understand the modal properties better, a factor βi, which represents the ratio between the kinematic energy stored in a span conductor and the total kinematical energy of the two-span transmission lines having mono-frequent free linear vibration, is suggested. Based on the obtained closed-form solution of the mode of two-span transmission lines, the factor βi for the j-th mode is defined as where βi1 and βi2 reflect the contribution of the first and second span to the mode shape of the total system, respectively. In the following two examples, the effect of transmission line parameters on the localization factor is illustrated. When the insulator lengths are, respectively,1m and 7m in the two-span transmission lines, in which the span lengths are 200m and 250m, the variations of the localization factor of the first symmetrical mode with the λ1/π are reported in Fig.12(a). It can be seen that when λ1/π is in the range of 0~2, the localization factor is decreases and when λ1/π is in the range of 2~10, the localization factor has a constant trend. The calculated results also show that when the two span lengths are 200m and 250m, the effect of the insulator string length on the localization factor is very small.
When the insulator lengths are, respectively,1m and 7m in the two-span transmission lines, in which the λ1/π is equal to 1, the variation of the localization factor of the first symmetrical mode with the span ratio l2/l1 are in Fig.12(b). It is noted that when the value of l2/l1 is large, the span ratio and insulator string length have a significant effect on the localization factor.

A. THEORETICAL FORMULA TO CALCULATE THE DYNAMIC STIFFNESS
Similarly, for multi-span transmission lines with arbitrary span number, the dynamical stiffness can be also obtained by referring to the calculation method of two-span conductors. Multi-span transmission lines with N spans are reported in Fig.13. All suspension points of N-1 insulator strings are located at the same level. The support A1 is a smooth roller, and the support AN+1 is a hinge. Each span can be considered as a substructure, and local coordinate system, which is attached to each span at its left end, is established on the substructure. For example, the xj-axis is taken along the chord of the j-th span, and the yj-axis is taken along the vertical direction. wj is the transverse displacement in the yi direction. If the left end A1 of the transmission lines is subjected to an additional harmonically varying horizontal force it F Fe   =  , the dynamic configuration of the transmission lines is the dotted line as shown in Fig.13. Neglecting the conductor inertia component parallel to the chord, The vertical vibration of j-th span conductor with respect to its position of static equilibrium can be expressed by the partial differential equation [3] 2 2 2 where the hτj is the additional tension of j-th span conductor. yj is the static equilibrium configuration of j-th span conductor with small sag, which is represented by (1). Solution to the partial differential can be obtained using the technique known as separation of variables. For j-th span conductor, we substitute (1) and (7) into the (19) to obtain The solution of (20), with the given boundary condition of . Considering the left end A1 of multiple span transmission lines is subjected to a harmonically varying horizontal force, the following relation is obtained from the differential equation governing j-th insulator string rotation: were Gj-1 is the self-weight of (j-1)-th span conductor, and Gj is the self-weight of j-th span conductor.  This article has been accepted for publication in IEEE Access. This is the author's version which has not been fully edited and content may change prior to final publication.
Were ( 1) ( 1) In order to verify the validity of the dynamic stiffness of multi-span transmission lines, we use some examples for comparison. The material parameters for the following two examples are same as for the first example in this paper. For the transmission lines with three spans of 100m-140m-180m that the insulator string length is 6.47m, and λ1/π is 1 under self-weight, the variation of the dynamic stiffness with 1 /  is reported in Fig.14(a). The dimensionless natural frequencies can be identified by these peaks of the dynamic stiffness. It is shown in Fig.14(a) that the natural frequencies identified by the dynamic stiffness are close to those obtained by ABAQUS method. The Fig.14(a) also shows that the values of the second, third and fourth peaks are significantly greater than those of other peaks. It is indicated that the energy is easily transferred to an adjacent structure when the resonance of the transmission lines includes the modes that correspond to the natural frequencies of the second, third and fourth peaks.
For the transmission lines with five spans of 100m-140m-180m-220m-260m having the insulator string length of 6.47m and λ1/π of 1, the variation of the dynamic stiffness with respect to the dimensionless frequency of 1 /  is reported in Fig.14(b). A good agreement between the dimensionless natural frequencies by the two methods in Fig.14(b) further provides proof of the correctness of the dynamic stiffness. Similar to three spans, the Fig.14(b) shows the maximum peak value of the fourth peak, and the energy is easily transferred to an adjacent structure when the resonance of the five-span transmission lines includes the fourth symmetrical mode.

B. THEORETICAL FORMULA TO CALCULATE THE SYMMETRICAL MODE
For multi-span transmission lines having arbitrary span number, the modal function can be obtained by referring to the calculation method of two-span transmission lines. According to the (28), the symmetrical mode of N-span transmission lines can be obtained as follows: 1 2 (1 tan( ) sin( ) cos( )) 2 () 1 ( ) (1 tan( / 2) /( / 2)) x  … are unknown quantities and can be obtained by the following equation: In order to verify to the validity of Eq.(29), according to the preceding of three-span transmission lines with 100m-140m-180m having the insulator string length of 6.47m and λ1/π of 1, the dimensionless natural frequency corresponding first symmetrical mode is obtained from Fig.14(a) and is equal to 0.629. Substituting 0.629 into (29), we obtain 32 / xx  of 5.8. According to corresponding geometrical and material parameters of this three-span transmission lines, 2 x  and 3 x  in first symmetrical mode are obtained from ABAQUS and are equal to -0.0132m and -0.0777m, respectively. The 32 / xx  obtained by ABAQUS is 5.9 and is close to that with (29). Substituting the parameters of three-span transmission lines and 32 / xx into (29), the first symmetrical mode of the three-span conductors is obtained as shown in Fig.15. It is indicated that the two first symmetrical modes obtained by two methods are in good agreement.

IV. CONCLUSIONS
In this paper, a theoretical method for the calculation of equivalent dynamic stiffness is presented. Unlike previous dynamic stiffness of a single span, the effect of the adjacent span and swing of insulator string are considered. The parametric study reveals that the previously ignored factor, as the insulator string length, is indeed very important for the dynamic stiffness. The sizes of the peaks that correspond to different modes are not the same.
The natural frequencies are identified based on the peak-to-peak of the dynamic stiffness curve with varying external exciting frequency, and corresponding modes are obtained. The validity and accuracy of the proposed frequencies and the associated modes are verified by using finite element method. It is shown that the method is accuracy, and more convenient than the finite element method. At the same time, a parametric study is performed to examine the sensitivity of the natural frequencies and associated modes to the span ratio, insulator length, Irvine parameters, and span number. In particular, the mode localization is discussed.
In this paper the main contribution is a closed-form expression for the dynamic stiffness of arbitrary multi-span transmission lines. The dynamic stiffness and natural frequencies obtained by the dynamic stiffness can be used to guide the design of overhead transmission lines so as to avoid the coupling between towers and liens or coupling between different spans, which may cause severe damage of transmission lines.