50_5_269_276

7/27/2019 50_5_269_276

1/8

Strojarstvo 50 (5) 269-276 (2008) H. WOLF et. al., Using the Energy Balance Method in Estimation... 269

CODEN STJSAO ISSN 0562-1887

ZX470/1351 UDK 62(05)=862=20=30

Original scientic paper

This paper considers using the Energy Balance Method for estimationof overhead transmission line Aeolian vibrations. Since the dampersefciency strongly depends on its position, the procedure of determiningthe optimum position of the damper is described. It is observed that theoptimal damper position depends on the span length also and that the bestresults can be achieved by determining the damper position for each spaninside the line section.In order to examine the accuracy of the developedcomputational program, as well as correctness of the estimation of thedata used (wind power, data on the dampers mechanical impedance, and

mechanical characteristics of the conductor), numerical results generatedusing the developed computer program and data from eld measurements(before and after the application of Stockbridge dampers) are compared. Avery high correlation between these data has been observed.

Primjena metode ravnotee energija u procjeni eolskih

vibracija dalekovoda

Izvornoznanstveni lanak

U ovome radu razmatrana je primjena metode ravnotee energija zaprocjenu eolskih vibracija dalekovoda. Kako ekasnost priguivaavibracija u velikoj mjeri ovisi o njegovu poloaju, u radu je opisan

postupak za odreivanje optimalnog poloaja priguivaa. Uoeno je danjegov optimalni poloaj takoer ovisi o duini raspona, te da se najboljirezultati postiu odreivanjem poloaja priguivaa za svaki raspon unutarrazmatranog rasponskog polja. Radi provjere preciznosti razvijenograunalnog programa i ispravnosti procjene koritenih podataka (podaci osnazi vjetra, mehanikoj impedanciji priguivaa i mehanikim svojstvimavodia) usporeeni su numeriki podaci koji su dobiveni razvijenimraunalnim programom s podacima koji su prikupljeni tijekom terenskihmjerenja (prije i poslije montae priguivaa tipa Stockbridge na vodidalekovoda). Uoeno je vrlo dobro poklapanje numerikih i izmjerenih

podataka.

Hko WOlF1), Bors AduM2),damr SeMenSKi1) anddraga PuSTAi1)

1) Fakultet strojarstva i brodogradnje, Sveuiliteu Zagrebu (Faculty of Mechanical Engineeringand Naval Architecture, University of Zagreb),Ivana Luia 5, HR - 10000 ZagrebRepublic of Croatia

2) Dalekovod d.d.,Marijana avia 4,HR - 10000 ZagrebRepublic of Croatia

Keywords

Energy balance methodMeasuring of Aeolian vibrations

Overhead transmission line Aeolian vibrations

Kljune rijei

Eolske vibracije dalekovoda

Metoda ravnotee energija

Mjerenje eolskih vibracija

Received (primljeno): 2008-01-30

Accepted (prihvaeno): 2008-07-15

Using the Energy Balance Method in Estimation of

Overhead Transmission Line Aeolian Vibrations

[email protected]

1. Introduction

Aeolian vibrations of overhead transmissionline conductors (wind-induced vibrations due to theshedding of Von Karman vortices) are usually noticed atfrequencies 5-50 Hz. These vibrations can cause damageand break the conductor due to material fatigue, and cansignicantly shorten its lifetime. Even though Stockbridgevibration dampers have been used for damping of Aeolianvibrations relatively successfully since 1930, a vibrationmodel that would adequately describe this phenomenon isstill under development. The mechanism of forming and

shedding of Von Karman vortices is a very complicatedaero-elastic phenomenon, and the empirical data on the

inuence of the wind force on the conductor, as well ason the power thus entering the system differ signicantlyfrom source to source [1, 2]. In practice, this is even morecomplicated, since the data on the characteristics of theair ows on any given location (the velocity and directionof the air ow, turbulence in the uid ow) are, as a rule,unknown [2]. Moreover, the mechanical characteristicsof the conductor (bending stiffness, the characteristicbending diameter and conductor self-damping) dependon a number of parameters (the construction of theconductor, conductor tension, vibration frequency andamplitudes, temperature), and are difcult to estimate

[3, 4]. Overhead transmission line Aeolian vibrationsare most often estimated in practice by using the Energy

7/27/2019 50_5_269_276

2/8

270 H. WOLF et. al., Using the Energy Balance Method in Estimation... Strojarstvo 50 (5) 269-276 (2008)

Balance Method [4-9] which is based on a non-linearalgebraic equation of power balance among the power ofaerodynamic forces brought into the system (for a givenwind velocity), the power dissipated by the vibrationdamper and the power dissipated by the conductor due

to conductors self-damping. The damper efciencystrongly depends on the interaction between the damper(characterized by damper impedance and damperlocation on the conductor) and the oscillating conductor(characterized by its tensile load and mass per unitlength). This paper considers using the Energy BalanceMethod for estimation of overhead transmission lineAeolian vibrations, as well as determining the optimumStockbridge-type damper position.

2. Vibration model

An overhead transmission line conductor with aStockbridge-type vibration damper is shown in Figure 1.

Figure 1. Overhead transmission line conductor with aStockbridge-type damper

Slika 1. Vodi dalekovoda s priguivaem tipa Stockbridge

Conductors are often modeled as beams of bendingstiffnessEI, with signicant tensile forces Tat the ends[7,10]. Transverse vibrations of the system are thendescribed by using a non-homogeneous non-linear partialdifferential equation of the fourth order

EI u x t Tu x t m u x t

q x t d u u t x l

IV

L

K

, '' , ,

( , ) ( , , ) ,

( ) ( ) + ( ) =

= +

1

(1)

where u is the transverse displacement of the conductor at

locationx at time t,EIis the conductor bending stiffness,Tis the conductor tension force, mL

is the conductor massper unit length, q(x,t) is the wind force imparted on theconductor due to Von Karman vortex shedding, and d

K(

) is the member representing the conductors self-damping. The ' signs denote differentiation with respectto coordinate x, while dots denote differentiation withrespect to time t. The equation (1) is valid only wherex l 1 , because force andtorque imparted on the conductorby the damper must be taken into account at pointx=l

1

(the position of the damper). The bending stiffnessEIofthe conductor can be determined only empirically [3,4],

due to the complex structure of the conductor. The actualbending stiffness of the conductor lies between EImax

(when there would be no movement of one conductorstrand in relation to the other) andEI

min(when conductor

strands would freely glide in relation to one another).The transverse string vibration model is also often used[8]. Since the bending stiffness of a string is zero, the

vibrations of a cable are in this case described by non-homogeneous non-linear partial differential equation ofthe second order

( ) + ( ) = + Tu x t m u x t q x t d u u t x lL K'' , , ( , ) ( , , ), 1 . (2)

The bending stiffness of the conductor is small and itsinuence on the resonant frequencies in the consideredfrequency range (5-50 Hz) can be ignored, so the resonantfrequencies resulting from both models are very similar[10]. All resonant frequencies of the string vibrationmodel are integer multiples of the resonant frequency

corresponding to the rst mode shape.

n

L

n

L

T

mn= =, , , , ...1 2 3 (3)

Equations (1) and (2) are non-linear partial differentialequations which due to their complexity are solved as arule using numerical methods. The rst resonant frequencyof typical overhead transmission line conductors is of theorder of magnitude of 0,1 Hz. Therefore, the frequencyrange of 5 to 50 Hz corresponds to the interval from the50th to the 500th mode shape. When solving the non-

linear equations using numerical methods, all thesemode shapes must be appropriately modeled. In lieu ofthe numerically demanding solving of the original non-homogeneous non-linear partial differential equations ofthe fourth and second order, the energy balance methodis often used in determining the conductor vibrationamplitudes.

3. Energy balance method

This method is based on an algebraic equation

of power balance in the case of stationary conductorvibrations

P A P A P AW D C( ) = ( ) + ( ) . (4)

PW

(A) is the power of aerodynamic forces brought intothe system (for a given wind velocity and correspondingfrequency of Von Karman vortices shedding),P

D(A) is the

power dissipated by the vibration damper, whilePC(A) is

the power dissipated by the conductor due to conductorself-damping. The conductor vibration amplitude A canbe determined using the algebraic equation (4), for any

given frequency or wind velocity.

,

7/27/2019 50_5_269_276

3/8

7/27/2019 50_5_269_276

4/8


Power dissipated in the damper is calculated usingequations given in [6,12,13]

P Tc kh g

h gD

A

DD w=

+( )+ +

1

4

1

1

2

2 2

2 2

2

2

, (8)

where

hkl kl

kl kl kl =

+( )+ + +( )

sin sin sin

sin sin sin

2

1 1

2

1

2

1 1

2 2

2

,

gkl kl kl

kl kl kl =

+ ++ + +

sin cos sin sin

sin sin sin

2

1 1

2

1

2

1

2

1 1

2 2

2

( )

,

k f m T L

= 2 / is the wave number, cW

is the wave

velocity, ( c T mw L2

= ), , Z is the absolutevalue of damper impedance, is the phase angle of thelag between the force and velocity, and l

1is the distance

between the damper and the suspension clamp.

3.4. Conductor bending strain

Although the bending stiffness of the conductor issmall and does not signicantly impact on the resonantfrequencies, it is necessary to know its value to determinethe bending strain . The conductor strain cannot bedirectly determined from the results of equation (2).

Expressions for conductor strain at characteristic pointsalong the span (middle of the span, at the suspension clampand at the damper clamp) are derived using perturbation

Table 1.Comparison of conductor self-damping rules [1]

Tablica 1. Usporedba eksponenata za odreivanje snagepriguenja vodia

Investigators / Istraivai l m nSpan length / Duina

raspona,m

Method / Metoda

a) Tompkins et al. 2,3 5,0 1,9 36 ISWr

b) Claren & Diana 2,0 4,0 2,5 46 PT

c) Sepp 2,5 5,75 2,8 36 ISWr

d) Kraus & Hagedorn 2,47 5,38 2,80 30 PT

e) Noiseux 2,44 5,63 2,76 63 ISWr

f) Mcks & Schmidt 2,45 5,38 2,4 30 PT

g)Mech. Lab Politecnico di

Milano2,43 5,5 2 46 ISWr

h) Rawlins 2,2 5,4 1 36 ISWr ISWR: Inverse Standing Wave Method, PT: Power Method

techniques given in [6]. The comparison of resultsobtained for a beam with bending stiffness EIand thoseobtained using the perturbation approach conrms thatthe differences are very small [8]. The general expressionfor bending strain is:

x td

u x t, ,( ) = ( )2

, (9)

where dis the so called bending diameter, which is notequal to the conductor diameterD, due to the conductorscomplex structure, but is determined as follows:

d KDS

= . (10)

KS

is the slippage coefcient, determined empiricallyand depends on whether the mid-span portion or theportion of the conductor at the clamp is being analyzed.It also depends on the construction and diameter of the

conductor, as well as on the ratio of the conductor tensionTand the conductor tensile strength P. According to [6]conductor strain in the middle of the span (point A) is

A

k Ad

22

, (11)

the conductor strain at the suspension clamp, with adamper near it (point B), is

B

kA

h g

T

EI

d

g h

klh kl g kl

=+ +( )

+ +

+ + (

2 1 2

14 2 1

2 2

2 2

2

1

1

2

1sincot cot ))

, (12)

7/27/2019 50_5_269_276

5/8


and the conductor strain at the damper clamp (point C)is:

CdkU b c b c

T

EI= ( ) + ( )

21 1

2

2 2

2

,(13)

whereb kl g kl h kl1 1 1 1= sin cos sin ,

b kl g kl h kl2 1 1 1= + cos sin cos ,

c gkl

klh kl1

2

1

1

11= ( ) +cos

sincos ,

c g kl hkl

kl2 1

2

1

1

1= +( ) +coscos

sin.

4. Field measurements

Field measurements of Aeolian vibrations areperformed by vibration recorder VIBRECTM 400. Therecorder is intended for automated measuring of Aeolianvibrations of conductors and earth-wires of overheadtransmission lines at all voltage levels. Within thedevice, there is a microprocessor that records the data(amplitudes, frequencies, wind velocity and surroundingtemperature) into its memory. A characteristic position ofthe VIBrECTM 400 is shown in Figure 4. Figure 5 showsthe device fastened on the suspension clamp of the cable.The so-called Poffenberger-Swart formula [14]:

b pf b

K Y= , (14)

relates the stress in outer cable layer at the last pointof contact between conductor and clamp (

b) and the

measured shift of the cable Yb

(the so-called bendingamplitude) at a certain distance from the clamp edge(Figure 6).K

pfis the conversion factor that depends only

on the characteristics of the conductor, conductor tensionand the distance of the displacement transducer to thesupporting clamp edge:

KE d p

e pc

pf

a S

px=

+( )

2

4 1

, (15)

where Yb

(mm) is the so-called bending amplitude (peakto peak), E

a(N/mm2) is Youngs modulus of a strand in

outer cable layer, b

(N/mm2) is stress (zero to peak) inouter cable layer (at the last point of contact betweenconductor and clamp), d

S(mm) is diameter of outer layer

strand, T (N) is tension force in the cable at averagetemperature,E

cIc(N/mm2) is the sum of exural rigidities

of the individual conductor strands, c (mm) is distance atthe point of measurement from the last point of contactbetween the clamp and the conductor (Figure 4), and

p

T

E Ic c

=(1/mm).

The vibration recorder uses a classifying approachto store the measured data. This is done to reduce theamount of memory needed to store the measurement datafor measuring over a long period of time (usually a fewmonths). When a measurement value has to be classied

it is compared with the upper limit of every class (forexample up to 32 amplitude classes), beginning with therst (the lowest) class. As soon as the value is smalleror equal than the upper limit of a class, the number ofevents of this class is increased by one. These numbersof vibration events are stored in a 3-dimensional matrixwhere the dimensions are amplitude, frequency and windvelocity [15].

Figure 4. Vibration recorder VIBRECTM 400

Slika 4. Ureaj za mjerenje vibracija

Figure 5. Measuring of aeolian vibrations of transmission lineby VIBRECTM 400

Slika 5. Mjerenje eolskih vibracija dalekovoda s ureajem

Figure 6. Stress versus distance from clamp edge

Slika 6. Naprezanje u ovisnosti o udaljenosti od ruba stezaljke

7/27/2019 50_5_269_276

6/8


5. Results

The developed computer program enables theestimation of Aeolian vibrations of overhead transmissionlines, i.e. conductor vibration amplitude and alternating

bending stress and strain at characteristic points alongthe span, i.e. at the middle of the span, at the suspensionclamp and at the damper clamp. The highest stressesmostly occur at the suspension clamp and much less oftenat the damper clamp. Bending stresses at the suspensionclamp (at point B), and at the damper clamp (at pointC) are shown in Figure 7 as a function of vibrationfrequencies and distance from the damper to the clampedge l

1(conductor ACSR 490/65, span length 310 m, 25

% UTS, two Stockbridge dampers per span). Damperwill be the most efcient if it is placed at the position atwhich the maximum bending stress (at suspension anddamper clamp) is minimal for all considered frequencies.Figure 8 shows stress at suspension and damper clamp independence of distance from the damper to the clamp edgel1

for all considered frequencies (each curve represents

stress at one frequency). Figure 9 shows the optimalposition of the damper which corresponds to the positionof the minimum of the envelope of the curves shown inFigures 8a and 8b. In this example, the optimal damperposition is at distance l

1=1,106 m from the suspension

clamp edge. Inside the one line section dampers areusually placed at the same distances (installation reasons)from the suspension clamp regardless of the single spanlength.

One can see (Figure 10) that the difference betweenthe damper optimal positions of span length l=310 mand l=200 m is 10,6 %. If we place damper at l

1=1 m

(what is optimal damper position for 200 m lengthspan) at 310 m length span, stress at suspension clampwill be 29 % higher than if the damper is placed at l

1=

1,106 m, what is optimal position for 310 m length span(Figure 9). On the other side, if we place damper at l

1

= 1,106 m (optimal damper position for 310 m lengthspan) at 200 m length span, stress at the suspensionclamp will be 74 % higher than if the damper is placedat l

1= 1 m (Figure 11). One can conclude that stress

Figure 7. Alternating stress versus frequency and damperdistance from the clamp edge: a) stress at suspension clamp, b)stress at damper clamp

Slika 7. Naprezanje u ovisnosti o frekvenciji i udaljenostipriguivaa od nosne stezaljke: a) naprezanje kod nosnestezaljke, b) naprezanje kod stezaljke priguivaa

a)

b)

Figure 8. Alternating stress versus damper distance from theclamp edge: a) stress at suspension clamp, b) stress at damperclamp

Slika 8. Naprezanje vodia u ovisnosti o udaljenostipriguivaa od nosne stezaljke: a) naprezanje kod nosnestezaljke, b) naprezanje kod stezaljke priguivaa

a)

b)

7/27/2019 50_5_269_276

7/8


level in overhead transmission line conductors can beconsiderably reduced if we determine optimal positionsof the dampers for each span inside the line section.

However, if all the dampers inside the one linesection should be placed at the same distance from the

clamp edge, the damper position should be determinedwith respect to the longest span in the considered linesection.

Figure 9. Maximum stress versus damper distance from theclamp edge (l= 310 m)

Slika 9. Maksimalno naprezanje u ovisnosti o udaljenostipriguiva od nosne stezaljke

Figure 10. Optimal damper position versus span length

Slika 10. Optimalni poloaj priguivaa u ovisnosti o duiniraspona

Very large differences among the numerical resultsobtained by different computational programs arefrequently observed [5]. Two main reasons were identiedfor the large differences among the results. First, theaforementioned programs are based on several differentmodels, i.e. different assumptions that are built into themodels. Second, these programs use data from differentsources that can differ signicantly (wind power, dataon the dampers mechanical impedance, and mechanicalcharacteristics of the conductor). In order to examine the

accuracy of the developed computational program, as wellas correctness of the estimation of the data used (wind

power, data on the dampers mechanical impedance,mechanical characteristics of the conductor: bendingstiffness, characteristic bending diameter, conductorself-damping), results generated using the developedcomputer program and data from eld measurements

using device VIBRECTM

400 are compared in Figure 12(conductor ACSR 490/65, span length 520 m, 25 % UTS,two Stockbridge dampers at each suspension clamp).

Figure 11. Maximum stress versus damper distance from theclamp edge (l=200 m)

Slika 11. Maksimalno naprezanje u ovisnosti o udaljenostipriguivaa od nosne stezaljke

Figure 12. Comparison of numerical results (EBM) and datafrom eld measurements

Slika 12. Usporedba numerikih rezultata (EBM) i podatakadobiveni terenskim mjerenjima

The above mentioned span was considered before andafter the application of Stockbridge dampers. The spanis in the valley of the river bed, which indicates a highprobability of intense Aeolian vibrations. Measurementsof the span vibrations without the damper were conducted

for 41 days. The measuring device was activated every15 minutes and gathered data for 10 s. The total device

7/27/2019 50_5_269_276

8/8


working time was 10.9 hours. Measurements of thespan vibrations with dampers were conducted two yearslater and lasted for 38 days with the same rhythm ofmeasurements. Figure 12 shows that alternating bendingstress was signicantly reduced after the dampers were

applied. As mentioned in chapter 4, given the memorycapacity limitations of the measurement device, thedevice doesnt record the amplitude and frequency of eachobserved vibration. Given the amplitude and frequencyof each vibration, the observed data is classied into theappropriate class of the frequency-amplitude matrix.Each class encompasses an interval of frequencies andamplitudes. Classes with the highest observed stressamplitudes (for each frequency class) are shown in Figure12 with rectangles. It should be noted that vibrations withthese large amplitudes are in nature relatively rare. As canbe seen in Figure 12, a high similarity between measured

eld data and data generated with the developed computerprogram (EBM) has been obtained.

6. Conclusion

This paper considers using the Energy BalanceMethod for estimation of overhead transmission lineAeolian vibrations. Since the dampers efciency stronglydepends on its position, the procedure of determiningthe optimum position of the damper is described.Based on the obtained results one can conclude thatmaximum stresses in the conductor at the suspension

clamp and at the damper clamp can be signicantlyreduced if the damper position is determined by usingthe described optimization procedure. Since the optimaldamper position, among other parameters, depends onthe span length also, the best results can be achieved bydetermining the damper position for each span insidethe line section. In order to examine accuracy of thedeveloped computational program, as well as correctnessof the estimation of the data used (wind power, data onthe dampers mechanical impedance, and mechanicalcharacteristics of the conductor), results generated usingthe developed computer program and data from eld

measurements are compared. A very high correlationbetween these data has been observed.

From the considered eld measurements data, onecan see that the observed span before the applicationof Stockbridge type damper wasnt satisfactory eitherunder the EPRI Endurance Limit criteria (maximumdynamic stress is signicantly larger than the allowed8,5 MPa), or under the Minimum lifetime estimationcriteria (the estimated life time of conductors under theMiners rule [4] was 18,4 years). After the applicationof the dampers, both criteria were satised. Dynamicstress was greatly reduced, and the estimated life span is

more than 500 years.

Acknowledges

The data shown are a result of a scientic projectNumerical and Experimental Research of Non-linearMechanical Systems that was conducted with the

support of the Ministry of Science, Education and Sportsof the Republic of Croatia.

The data was collected using Dalekovod d.d.equipment and resources.

REFERENCES

[1] CIGRE TF 22.11.1: Modeling of Aeolian Vibration ofSingle Conductors: Assessment of the Technology, Electra181 (1998) 53-68.

[2] KrAUS, M.; HAGEDORN, P.: Aeolian Vibrations:Wind Energy Input Evaluated from Measurements on

an Energized Transmission Line, IEEE Transactions onPower Delivery 6 (1991) 3, 1264-1270.

[3] CLAREN, R.; DIANA, G.:Dynamic Strain Distributionon Loaded Stranded Cables, IEEE Transactions on PowerApparatus and Systems 88 (1969) 11, 1678-1687.

[4] CIGRE TF 22.01: Report on Aeolian Vibration, Electra124(1989) 41-77.

[5] CIGRE TF B2.11.01: Modelling of Aeolian Vibrationsof a Single Conductor Plus Damper: Assessment of

Technology, Electra 223 (2005) 28-36.

[6] HAGEDORN, P.: Ein einfaches Rechenmodellzur Berechnung winderregter Schwinungen an

Hochspannungsleitungen mit dampfern, Ingenieur-Archiv49 (1980) 161-177.

[7] HAGEDORN, P.: On the Computation of Damped Wind-Excited Vibrations of Overhead Transmission Lines,

Journal of Sound and Vibration 83 (1982) 2, 253-271.

[8] HAGEDORN, P.:Wind-Excited Vibrations of TransmissionLines: a Comparison of Different Mathematical Models,Mathematical Modeling 8 (1987) 352-358.

[9] MARKIEWICZ, M.: Optimum Dynamic Characteristicof Stockbridge Dampers for Dead-end-Spans, Journal ofSound and Vibration 188 (1995) 2, 243-256.

[10] CLAREN, R.; DIANA, G.: Mathematical Analysis ofTransmission Line Vibration, IEEE Transactions on Power

Apparatus and Systems, 88 (1969) 12, 1741-1771.[11] CIGRE TF 22.11.1: Modeling of Aeolian Vibration of

Single Conductors: Assessment of the Technology, Electra181 (1998), APPENDIX A

[12] DIN VDE 0212 1986: Teil 51 (1986) 1-7.

[13] IEC 61897: Overhead Transmission Lines-Requirementsand Tests for Stocbridge Type Aeolian Vibration Dampers

(1998), 25-33.

[14] CIGRE TF 22.11.2: Guide to Vibration Measurements onOverhead Lines, Electra 163 (1995) 124-137.

[15] VIBrECTM 400 Users Manual, SEFAG EXPORT AG,Switzerland, 2002.

50_5_269_276

Documents