Parabola and Catenary Equations for Conductor Height Calculation

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1 ELECTROTEHNICĂ, ELECTRONICĂ, AUTOMATICĂ, 6 (), nr. 3 9 Prbol nd Ctenr Equtions for Condutor Height Clultion Alen HATIBOVIC Abstrt This pper presents new equtions for ondutor height lultion bsed on the known miml sg. The equtions n be diretl pplied for plotting the ondutor urve fter ompleted sg-tension lultion for plnning nd designing overhed lines. The bsi differenes between the prbol nd the tenr urves re lso disussed. The vlidit of the shown formuls hs lso been proved b some numeril emples. Kewords: trnsendentl funtions, lgebri funtions, sg, overhed lines, leveled spn, inlined spn. Introdution The origin of the - oordinte sstem for sg-tension lultion is generll put t the top of the ondutor urve []. However, it is more dvntgeous to set the origin to the bottom of the left-hnd side support of the spn for defining the eqution for the ondutor height. B this w the -oordinte presents the height of the ondutor urve relted to -is. The distne of ondutor s rbitrr point from the -is is then the distne from the left-hnd side support. This pper shows both the tenr nd the prbol equtions under this ondition. The tul ondutor urve n be desribed b tenr funtion, i.e. b the hperboli osine funtion, whih belongs to the group of the trnsendentl funtions. The prbol urve n be desribed b qudrti funtion, whih belongs to the group of the lgebri funtions. The bsi differene between the lgebri nd trnsendentl funtions is in their eponent. While the eponent of the lgebri funtions is permnent, it is vring in the se of trnsendentl funtions. Despite the ft tht the the prbol nd the tenr funtions re mthemtill quite different, their urves n be ver similr. Therefore, when plnning overhed eletril lines the tenr is often pproimted b the prbol, sine it results in signifint simplifition of the lultion. It is eptble, beuse in most of the ses the differene between the tenr nd the prbol is negligible []. It is generll epted ft in the literture tht the ondutor urve n be pproimted b prbol for spns up to bout metres. For longer spns the et tnr bsed lultion shll be used, beuse the differene between the tenr nd the prbol urves nnot be ignored.. Ctenr eqution (;) Figure. Grphs of tenr urves osh(/) osh(/) - The top of the tenr urve is loted t the point (,) s it is shown in Figure, see urve. Its bsi eqution is the following [3]: ( ) h / () Alen HATIBOVIC, Senior engineer for eletril network development, EDF DÉMÁSZ Hálózt, Szeged, Hungr

2 ELECTROTEHNICĂ, ELECTRONICĂ, AUTOMATICĂ, 6 (), nr. 3 For the determintion of the tenr eqution it is neessr to know its prmeter ( >). It n be obtined b using the sg-tension lultion for plnning overhed lines. When the origin of the oordinte sstem is t the top of the tenr urve, the eqution beomes: ( / ) - h () After this step the origin shll be moved to the bottom of the left-hnd side support of the spn ording to Figure. This figure shows n inlined spn with verte t point. Figure. Ctenr urve in n inlined spn The eqution for ondutor height will be defined b the following dt: prmeter of the tenr urve oordinte of the verte point oordinte of the verte point The finl tenr eqution for ondutor height is (3). Its eponentil form is given b (). The intervl is lws [,], where is the spn length. (3) h e e Both of the two equtions re universl, sine these re vlid in the se of n tpe of inlined spn (h <h or h >h ) nd in the se of leveled spn (h h h), too. So b (3) or () it is possible to lulte the ondutor height t n point of the spn. It n be seen in () tht the vrible is loted t the eponent, whih is n importnt feture of trnsendentl funtions. In the se of leveled spns the equtions (3) nd () get simpler forms given in (5) nd (6) on () the bsis of the known miml sg b m. The height of the two supports is denoted with h. / h h b (5) e / e / m h b 3. Prbol eqution There is ver importnt defferene between the prbol nd the tenr onerning the miml sg of the ondutor. Sine the miml sg of the prbol is lws loted t midspn, both in the se of leveled nd inlined spns, the miml sg of the tenr in n inlined spn is slightl moved towrd the higher suspension point of the ondutor. This is one of the resons of the simpliit of the prbol bsed lultion in omprison to the tenr bsed lultion. When the miml sg of the prbol is known it is possible to obtin the prbol eqution for the ondutor height, sine the prbol is defined b n three points of its urve. In the se of the tenr it is more diffiult, beuse it is neessr to know both the prmeter of the tenr nd the oordintes of the verte point ording to the bse eqution (3). So the knowledge of the miml sg is not enough for the determintion of the tenr. The stndrd eqution of the prbol is (7), A B C where A, B nd C re the oeffiients of the prbol. The oeffiient A defines the shpe of the prbol urve. If A>, the urve hs the minimum [], [5], nd if A<, the urve hs the mimum. It will be mthemtill proved lter tht in the se of the eqution for the ondutor height the oeffiient A is positive. m 3.. Prbol eqution b three points Figure 3 demonstrtes the pplied method for n inlined spn with h <h. Points A nd B re the suspension points of the ondutor, while C is the ondutor point t mid-spn. The prmeters shown in Figure 3 re the following: spn length h height of the left-hnd side suspension point h height of the right-hnd side susp. point oordinte of the verte point oordinte of the verte point (6) (7)

3 ELECTROTEHNICĂ, ELECTRONICĂ, AUTOMATICĂ, 6 (), nr. 3 h h A(;h ) C / Figure 3. Prbol urve in n inlined spn with h <h B( ;h ) Sine the urve is prbol, the miml sg b m is loted t /. The left-hnd side nd the right-hnd side suspension points A(;h ) nd B(;h ) re lws known points. The third neessr point C is obtined b the known miml sg (8). The vlue of the miml sg n be obtined from the sg-tension lultion. h h C ; b m Bsed on the three points of the prbol the sstem of three lgebri equtions (9)-() n be written b utilizing of the stndrd eqution of the prbol (8): C h (9) A B C A (8) h () ( / ) B ( / ) C ( h h )/ bm () Writing these equtions in the mtri form P Q nd using the Crmer s rule [6], [7] to find the solution, the unknown oeffiients of the prbol A nd B n be obtined b: j j det( P )/det( P) ( j,, 3 ) () h h h h bm b A (3) m B h h b h h h h b m m () After the substitution of the oeffiients A, B, C into (7), the eqution for the ondutor height gets its finl form (5): bm h h bm h (5) This is universl eqution of the prbol, sine it is usble for both leveled nd inlined spns. In the se of the leveled spn the eqution (5) hnges into (6) nd the oeffiients hnge into (7). bm bm h bm bm A B C h (6) (7) There is ver importnt onsequene from (5) nd (6): beside the sme spn length nd miml sg both in leveled nd inlined spns, the oeffiient A of the prbol does not hnge. The vlidit nd usbilit of (5) nd (6) will be proved in the following three emples, one for leveled nd two for inlined spns. Emple. Leveled spn (h h h) m; h8m; b m 8m; ()? (8) [m] [m] Figure. Prbol urve from emple. (leveled spn)

4 ELECTROTEHNICĂ, ELECTRONICĂ, AUTOMATICĂ, 6 (), nr. 3 Emple. Inlined spn (h <h ) m; h 8m; h 3m; b m 8m; ()? (9) [m] [m] Figure 5. Prbol urve from emple. (inlined spn with h <h ) Emple 3. Inlined spn (h >h ) m; h 3m; h 8m; b m 8m; ()? 8 3 () [m] [m] Figure 6. Prbol urve from emple 3. (inlined spn with h >h ) The verte of the urve is shown in eh lst three figures nd the will be needed in the net prgrph 3.. With the help of the previous three emples it hs been proved tht the eqution for the ondutor height (5) is orret nd universl, sine it n be used in eh tpe of the spn. Therefore, we do not p ttention to either h <h or h >h, beuse the input dt is the sme in eh se of the tsks. We hve lso seen tht in the se of the eqution for the ondutor height, the oeffiient A of the prbol is lws positive, sine the spn length nd the miml sg of the prbol re lso positive. Therefore, the ondutor urve hs the minimum. Let us mention tht the verte of the prbol nd the lowest point of the ondutor re generll the sme point (point ) nd [,], like in eh of the previous figures. Figure 7. shows one rre se of the inlined spn when the verte point is out of the spn, i.e. [,]. In this se the lowest point of the ondutor (point M) is equl with the lower suspension point of the spn. For the pproprite presenttion of this rre se [8] the prbol urve is shown in n intervl [, ]. Figure 7. Inlined spn with M 3.. Prbol eqution in the verte form Beside the stndrd eqution of the prbol (7) its verte form is lso often used (). Eqution (7) n be trnsformed into form () b using (): ( ) A B A A AC B A () () Aording to () the oordintes of the verte point n be defined s (3) nd (): B h h A bm (3) AC B h h h bm () A bm Substituting (3) nd () into (), the verte form of the eqution for the ondutor height (5) is obtined: b m h h h bm b m h h bm (5)

5 ELECTROTEHNICĂ, ELECTRONICĂ, AUTOMATICĂ, 6 (), nr. 3 3 In the se of the leveled spn the verte oordintes beome / nd h b m, so the previous eqution hnges into (6): b m h b (6) B using eqution (5) it is es to lulte the oordintes of the verte point. So, b equtions (6) nd (5) we n now write the equtions in the verte form of the prbol from emples nr., nd 3, s it is shown below: m Emple. Leveled spn (h h h) Verte point (;) ( ) 8 (7) Emple. Inlined spn (h <h ) Verte point (5;6) ( 5) 6 8 (8) Emple 3. Inlined spn (h >h ) Verte point (5;6) ( 5) 6 8 (9) Sine the verte point nd its oordintes re lred shown in Figures., 5. nd 6., it is es to hek the results of the numeril lultion for defining nd. It n be onluded tht the results re orret Prbol eqution with prmeter p Hereunder shll be shown how the eqution for the ondutor height shll be defined b the prmeter of the prbol p insted of the miml sg b m. Knowing the mthemtil onnetion (3) between the oeffiient A of the prbol nd its prmeter p, the identit (3) is obtined. p A (3) A p bm bm p 8p (3) The prmeter p lws hs positive sign. Generll, the oeffiient A of the prbol n be positive or negtive, but sine in our se it is lws positive, there re not n problems with signs in (3). Using identit (3), the eqution for the ondutor height n be written both in stndrd form (3) nd in verte form (33) of the prbol eqution: h h h (3) p p p p p p ( h h ) ( h h ) h (33) In speil se when the suspension points on two supports re on the sme elevtion, the previous two equtions get simple forms s (3) nd (35): h p p p h 8p (3) (35). Usge of the prbol equtions for the ondutor height An of the bove presented equtions re useful for lulting the ondutor height nd for plotting the grph of the ondutor line s well. For solving other tsks the pproprite one hs to be hosen from the shown equtions depending on the tul tsk. Eqution (5) s the stndrd form of the prbol eqution is ver prtil for quik finding of the first derivtive () nd the seond derivtive () of funtion [9]. The oordinte of the verte n be obtined b solving (). After tht it is es to define the oordinte of the verte b solving ( ). This method will be shown numerill in eqution () from the emple 3. We lred know tht the verte point is (5;6), but it n be heked with the method below. ( ) 8 ' ( ) ' ( ) ( )

6 ELECTROTEHNICĂ, ELECTRONICĂ, AUTOMATICĂ, 6 (), nr. 3 ( ; ) (5;6) As it n be seen the sme results prove the vlidit of the shown method for using eqution (5) for defining the verte point. Eqution (5) s the verte form of the prbol eqution hs n even bigger usbilit thn the eqution (5). Beside the determintion of the verte point nd the oeffiient A, it n be used to reple the ondutor urve within the - oordinte sstem. A onrete emple of suh n pplition is the following formul [] for the determintion of the ondutor length in inlined spns on the bsis of the known miml sg of the prbol. L 6b rsh m m m m rsh m ( ) m ( ) ( ) m (36) For deriving formul (36) the urve hd to be ppropritel repled within the - oordinte sstem in order to mke possible the integrl lulus for the ondutor length. In the se of the leveled spn the previous formul hs muh simpler form: m L (37) bm bm b rsh m 5. Conlusions The eqution for ondutor height defined b the prbol eqution hs more vritions thn the one obtined b the tenr eqution. The simpliit of defining the prbol eqution is prtl due to the ft tht the miml sg of the prbol is lws loted t mid-spn, either it is leveled or n inlined spn. In the se of tenr this rule is not vlid. As it is shown, the prbol gives possibilit to find solution in different ws, so it ensures n effetive w of heking the results. These re some of the resons for the frequent usge of the prbol, when the differene between the prbol nd the tenr is insignifint. Through the numeril emples we hve seen the w of obtining the eqution for the ondutor height in the se of n tpe of the spn. It hs lso been shown how to define the oordinte of the verte point. The verte is ver often equl with the lowest point of the ondutor nd in the se of the inlined spn it is one ritil point of the ondutor, so its heking is highl reommended. The speil se of the inlined spn hs lso been disussed, when the verte point is out of the spn. Using different vritions of the prbol eqution for the ondutor height the vlidit of shown equtions hs been proved through numeril emples. A ver importnt result of the shown 3 emples is tht in the se of the sme spn length nd the vlue of the miml sg b m, both in leveled nd inlined spn, the oeffiient A of the prbol does not hnge. This pper highlighted this importnt hrteristi of the prbol. Referenes [] CIGRÉ 3, Sg-tension lultion methods for overhed lines, CIGRÉ 7 [] HATIBOVIC A., Usge of Prbol Clultion for Plnning of Eletril Overhed Network, ENELKO onferene, Kolozsvár [3] PANSINI A., Eletril Distribution Engineering, The Firmont Press, In., 7 [] GUSTAFSON D., FRISK P. nd HUGHES J., College Algebr, CENGAGE Lerning [5] OBÁDOVICS GY., Mtemtik, SCOLAR Budpest [6] TURKINGTON D. A., Mátri Clulus & Zero- One Mtries, CAMBRIDGE 5 [7] GENTLE J. E., Mtri Algebr, Springer 7 [8] HATIBOVIC A., TOMIC M., Determintion of the lowest point of ondutor for inlined spns, CIGRÉ onferene, Srjevo [9] OBÁDOVICS GY., Felsőbb mtemtiki feldtgűjtemén, SCOLAR Budpest [] HATIBOVIC A., Integrl Clulus Usge for Condutor Length Determintion on the Bsis of Known Miml Sg of Prbol, Periodi Poltehni Eletril Engineering, Budpest Universit of Tehnolog nd Eonomis Alen HATIBOVIC Senior engineer for eletril network development, EDF DÉMÁSZ Hálózt, Szeged, Hungr

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