Integral calculus usage for conductor length determination on the basis of known maximal sag of a parabola

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1 Ŕ periodic polytechnic Electricl Engineering nd Computer Science 56/ 01) doi: /PPee periodicpolytechnic.org/ ee Cretive Commons Attribution RESEARCH ARTICLE Integrl clculus usge for conductor length determintion on the bsis of known mximl sg of prbol Alen Htibovic Received , ccepted Abstrct This pper shows mthemtic solution for clcultion of conductor length using integrl clculus on prbol. The presented method cn be used for spns pproximtely up to 400 metres. In tht cse the ctenry my be pproximted by prbol qudrtic function), while the difference between the ctenry nd the prbol is negligible. The necessry dt re the spn length, the heights of the suspension points nd the vlue of the mximl sg. The lgorithm is prepred for inclined spns, but it is pplicble for levelled spns too, becuse it is just one specil cse when the conductor suspension points on two supports re on the sme height level. Nturlly, the conductor length chnges with temperture, so it is possible to clculte it for chosen temperture, using conductor sg for tht temperture, chosen tension nd conductor type. Keywords prbol ctenry spn sg conductor length eqution Acknowledgement The uthor would like to express his thnk to EDF DÉMÁSZ, especilly to its brnches Prímvill Kft. nd DHE Kft. where he hs been working 15 yers up to now s designer engineer thn s senior engineer for electric network development. Hving opportunity to pln nd design kilometres of overhed lines nd underground cbles nd gining lot of expirience, the uthor could lso provide mny new methods nd clcultions nd then to check them in prctice. As result of tht work the uthor hs written 15 scientific publictions nd hs shown them on respectble professionl conferences nd in scientific journls both in Hungry nd brod. This pper presents the prt of EDF DHE Kft. project clled Designer Progrmme, which entered the 19th Hungrin Innovtion Awrd Competition. The Designer Progrmme s recognised innovtion got into the Innovtion Awrd 010 book under number 1. Alen Htibovic EDF DÉMÁSZ Hálózti Elosztó Kft, H-670 Szeged, Hungry e-mil: htibovic.len@gmil.com 1 Introduction To clculte the conductor length in our cse the prbol length) we hve to define the eqution for the conductor height. In the presented method the origin of the coordinte system is put t the bottom of the left-hnd side support of the spn, so the conductor height t ny point of the spn is relted to x-xis. It mkes the usge of the presented method esy nd nturl in rel work. Knowing the eqution for the conductor height, it is possible to clculte the conductor length. Generlly, in scientific literture only the method for levelled spns [1] cn be found, on the bsis of the known spn length nd the mximl sg. L b mx 1) For inclined spns new lgorithm is needed bsed on the integrl clculus, which is shown in this pper. The finl lgorithm is useful both for the levelled nd the inclined spns s well. Moreover, it mkes possible to clculte the conductor length of ny prt of the spn mking the lgorithm universl. Generlly, the rc length of the curve y f x) over [x 1, x ], is defined by the following formul []: x L x1,x f x) dx ) x 1 So the conductor length within the whole spn is the length of the prbol over the intervl [0, ], ccording to the following formul: L 0 f x) dx 3) It is obvious tht the clcultion would be long nd very complicted by using the stndrd form of the prbol eqution. To reduce the clcultion, it is recommended to replce the prbol curve t x y coordinte system, so tht the vertex MIN point) of the prbol should be in the origin [3]. At first, it is necessry to trnsform the stndrd eqution of the prbol to its vertex form. By tht wy the coordintes of the vertex of the prbol shll lso be defined. After tht it is esy to replce the prbol in the coordinte system nd chnge ppropritely the integrl limits for length clcultion. Integrl clculus usge for conductor length determintion

2 Fig. 1. Inclined spn Eqution for the conductor height At first, the eqution for the conductor height hs to be defined [4]. It is solved on the bsis of the three known points of the conductor curve, becuse the prbol is completely defined when ny three points of its curve re known. The suspension points of the spn re lwys known points, while the third necessry point is defined by the known mximl sg. In mthemtic literture the prbol of the conductor is known s regulr verticl) prbol. The generl form of the eqution of the prbol is the following: y Ax + Bx + C 4) Two suspension points re A0; h 1 ) nd B; h ). In the cse of the prbol the mximl sg is lwys locted in the mid-spn, x c /, which defines the third point Cx c ; y c ). y C h h h 1 b mx h 1 + h h 1 b mx h h b mx 5) So the third point of the prbol is determined: C ; h ) h b mx Let us write the system of three equtions with three unknown vribles using the generl form of the eqution of the prbol 4) : h h 6) h 1 C 7) h A + B + C 8) b mx A + B + C ) 9) The system cn be written nd solved in the following wy : A h 1 1 B h 10) / / 1 C h h )/ b mx A 4b mx / B h h 1 4b mx )/ C h 1 11) So we hve the A, B, C coefficients of the prbol: A 4b mx 1) B h h 1 4b mx 13) C h 1 14) Substituting these into 4), the eqution for the conductor height is completely obtined: y 4b mx x + h h 1 4b mx x + h 1, x [0, ] 15) This is stndrd eqution form of the prbol. Let us emphsize tht the defined eqution is universl, since it is vlid both for levelled h 1 h ) nd inclined h 1 < h or h 1 > h ) spns, despite the fct tht the inclined spn hs two types. Of course, in ech cse the pproprite vlue of the mximl sg hs to be used, which hs to be obtined by some progrmme for sg clcultion, or tken from sg-tension tbles, becuse the determintion of the mximl sg is not the tsk of this pper. 3 Vertex form of the prbol In order to reduce the clcultion the prbol curve hs to be replced in x y coordinte system, so tht the vertex MIN point) of the prbol should be in the origin. At first, it is necessry to trnsform the stndrd eqution of the prbol 4) to its vertex form 16): y Ax x MIN + y MIN 16) The wy of the prbol eqution trnsformtion is shown in the following lines: y 4b mx 4b mx 4b mx x + h h 1 4b mx x + h 1 x + h h 1 4b mx 4b mx [ ) ] x h h x + h 1 4b mx ) x + h 1 36 Per. Pol. Elec. Eng. nd Comp. Sci. Alen Htibovic

3 y 4b mx + [ 4b mx { x 1 h ) h 1 x+ 1 h h 1 4b mx + h 1 4b [ mx [ x 4b mx )] 1 h h 1 4b mx 1 h h 1 4b mx + h 1 b mx 1 h h 1 4b mx )] )], x [0, ] 17) Eqution 17) is the vertex form eqution of the prbol. It is completely equl with eqution 15), but more pplicble for the prbol replcement nd integrl clculus. A 4b mx 18) x MIN 1 h ) h 1 19) 4b mx y MIN h 1 b mx 1 h h 1 0) 4b mx x MIN nd y MIN re the coordintes of the vertex. Since A > 0, y MIN f x MIN ) is the minimum [5] of prbol. In Figure 1 the vertex MIN point) is the lowest point of the conductor [6]. Prbol curve with vertex being replced into origin hs the following eqution: y 4b mx x, x [, x MIN ] 1) The prbol eqution 1) is much more pplicble thn the eqution 15) for the length integrl ccording to 3). The prbol replcement will not result chnge in its length in the cse of the pproprite chnge of the intervl, from [0, ] to [, x MIN ]. 4 Conductor length The length of the prbol 1) over the intervl [, x MIN ] is given by the following formul: xmin L y dx ) Using the bsic derivtion rule x n ) n x n 1 the first derivtive of 1) is: Squring 3) results in: y 4b mx y 8bmx x 8b mx x 3) x A 1 x 4) To compute esily the integrl we use the substitution method: A 1 x sinht5) A 1 dx cosht) dt 6) dx 1 A 1 cosht) dt 7) L Substituting 5) nd 7) into ) nd evluting the integrl xmin t t A 1 x dx sinh A 1 cosht) dt cosht) 1 A 1 t 1 A 1 1 A 1 cosht) dt cosh t) dt 1 1 A 1 t + 1 ) t sinh t cosh t 1 A 1 t + sinh t sinh t 1 [ rsinha 1 x) + A 1 x A 1 1 8b mx 16b mx rsinh 8b mx t + ) t ) sinh t t A 1 x ] x + 8b mx x 8bmx [ rsinh 8b mx x MIN ) + rsinh 8b mx + 8b mx x MIN ) + 8b mx x MIN 8bmx 8bmx x MIN) x MIN x x MIN 8) Eqution 8) is the finl formul for the conductor length within the whole spn. 5 Conductor length of ny prt of the spn Integrl clculus gives us n opportunity to determine not only the conductor length within the whole spn, but lso the conductor length t ny prt of the spn, L x1 x over the intervl [x 1, x ]. In this cse the pproprite clcultion is the following: L x1 x x x 1 [ 16b mx y dx rsinh 8b mx x x MIN ) rsinh 8b mx 8bmx x x MIN + 8b mx x x MIN ) 8b mx x 1 x MIN ) 8bmx x 1 x MIN x 1 x MIN ) 9) The lst formul is quite universl for the prbol. The clcultion of the conductor length of the spn prt is not usul tsk, but the formul 9) is bsic, importnt one. For instnce in the cse of the following integrl limits x 1 0 nd x, the result is the conductor length within the whole spn, very common tsk. Furthermore, the 9) formul solves the levelled spn, too. For tht cse 9) will be trnsformed into significntly simpler formul. Integrl clculus usge for conductor length determintion

4 Fig.. Exmple 1 - Determintion of the conductor length within the levelled spn Fig. 3. Exmple - Determintion of the conductor length within the inclined spn 38 Per. Pol. Elec. Eng. nd Comp. Sci. Alen Htibovic

5 6 Conductor length of the levelled spn The levelled spn is specil cse when the suspension points on two supports re on the sme elevtion. It is the simplest cse for ny kind of clcultion, so for the conductor length too. The finl formul is shorter thn the other ones shown in this pper. L / 1 / / y dx y dx 0 8b mx 8b mx rsinh 8b mx rsinh 4b mx x + 8b mx x + 4b mx 4bmx 8bmx x / 0 30) Eqution 30) is the finl formul for conductor length of the levelled spn. The lst eqution hs lso n nother form. By using identity 31) in the eqution 30), it yields L 8b mx ln rsinhx) ln x + x ) 31) 4b mx + 4bmx + 4b mx + 4bmx 3) Since the formul for the rdius of the curvture t vertex point for the prbol [7] is given by 33) it results : L p ln p + r p 8b mx 33) p + p p 34) 7 Prcticl exmples The prepred lgorithm cn esily be built in Excel for solving rel tsks. Two concrete exmples re given in the following, one for the levelled spn nd the other for the inclined spn. For the given input dt the conductor length within the whole spn, nd the conductor length within the spn prt hve to be clculted. The user hs to write in, h 1, h, b mx, x 1 nd x input vlues into the progrmme. The x MIN is lso necessry for the length clcultion, but the progrmme clcultes it itself. On the bsis of the ppliction of the prepred lgorithm the user does not hve to tke cre bout the type of spn levelled or inclined, h 1 < h or h < h 1, the lowest point is identicl with vertex or not, etc.) t ll. Moreover, it is not necessry to cre bout replcing prbol vertex into the origin. In ny cse the progrmme solves this tsk itself. For clcultion of the conductor length within the whole spn the x 1 0 nd x input vlues shll be given. In the cse of inclined spns the vertex of the prbol nd the lowest point of the conductor re not lwys locted t the sme point, i.e. the vertex is not lwys the point of the conductor [5]. In tht cse x MIN < 0 or x MIN >, i.e. x MIN [0, ]. This fct does not cuse ny problems for the lgorithm, becuse in ech cse the necessry dt for the length clcultion is x MIN x coordinte of vertex), but not the lowest point of the conductor. The ltter is not n input dt, but it is often equl with the top of the prbol vertex point). Since we hve used the sme spn length nd sg in both exmples, the results cn be compred to ech other. In the exmple Figure 3, inclined spn with h 1 h ) the conductor length is longer by 0,5 metres thn it is in the exmple 1 Figure, levelled spn, h 1 h ). The length difference is not negligible t ll. But let us check the result obtined by using the formul 1). L + 8 b mx , 7 401, 075m Since the formul 1) is prepred for levelled spns, the result is lmost identicl with the result from the exmple Nr. 1. The difference is only millimetres, nd it cn be neglected. But formul 1) should not be used for inclined spns. Since the new formul is pplicble in the cse of levelled nd inclined spns s well, its use is lwys recommended. 8 Comprison of the lengths bsed on prbol nd ctenry Quite nturlly comes the question: how big is the difference in the previous clcultions when the prbol is used insted of the ctenry. To nswer this question, let us first determine the ctenry eqution from the exmple Nr. 1 nd plot the difference between the prbol nd the ctenry within the spn. Then we compute the ctenry length nd compre it to the length got by the prbol. This is the wy to determine how big the length difference is due to using prbol in the given exmple. Prbol eqution : Ctenry eqution : y pr x) x , 8 35) y ct x) 1569, , 9158 cosh x , ) Prbol nd ctenry curves from the exmple Nr. 1 look quite identicl with ech other in Figure 4) within the spn, so it is necessry to clculte the difference between them numericlly, i.e. y y pr x) y ct x). The result is shown in Figure 5. Of course, the differences re zero t points A, B nd C. The biggest difference is pproximtely 4,3 millimetres, so it is very smll vlue. We hve used big sg on purpose; for smller sg the difference would be smller. Since y y pr Integrl clculus usge for conductor length determintion

6 Fig. 4. Levelled spn of 400 kv Fig. 5. Difference between the heights of the prbol nd the ctenry in the exmple 1. y ct 0 L pr < L ct within the intervl [0, ]. The finl tsk is to clculte L L ct L pr. Tht will be the difference between the length clcultion using prbol nd ctenry curve. Since L pr is lredy clculted L pr 401, 0768m), we only hve to clculte L ct. In the cse of the exmple Nr. 1, the formul 37) cn be used [8, 9]: L ct c sinh c 37) From the eqution 36) the constnt of the ctenry is c 1576, Since 400 m, the length of the ctenry is L ct 401, 0735 m nd L L ct L pr 0, 57 mm. So in the cse of the exmple Nr.1 in Figure, levelled spn with big sg) the length difference is negligible. In the cse of the exmple Nr. in Figure 3, inclined spn with big sg) the difference between the lengths got by the prbol nd by the ctenry could be few centimetres nd L could be few millimetres. So the conductor curve cn be freely considered s prbol for spns up to 400 metres. The clcultions nd conclusions presented in this section relte to the specil cse, when the conductor sg b mx is ssumed to be identicl, both of prbolic nd ctenry methods re used. 9 Conclusions As it cn be seen in the exmples, the finl lgorithm mkes it possible to determine not only the conductor length within the whole spn, but the length between ny two points of the conductor within the spn, i.e. the conductor length of ny spn prt. The lgorithm is pplicble in the cse of levelled nd inclined spns s well. So it is concrete solution useful for conventionl tsks nd lso for some specil tsks. The presented exmples prove the simplicity of the lgorithm in rel pplictions. It is prtly due to the fct tht the origin is put to the bottom of the left-hnd side support; this is n unusul pproch in comprison to existing scientific literture for similr topics. The subproducts of the lgorithm re the vertex point coordintes x MIN nd y MIN. When x MIN [0, ], these re the coordintes of the lowest point of the conductor t the sme time. The lowest point is one chrcteristic point of the inclined spn, but very often the most criticl point, too. Furthermore, the lgorithm defines the prbol coefficient A, so it is esy to determine the eqution of the conductor height in vertex form of the prbol in ny concrete cse of the tsks. With the help of the shown lgorithm the conductor curve nd the lowest point re not difficult tsks t ll. Finlly, it is shown tht for the inclined spn the conductor length differs significntly from tht of the levelled spn hving the sme spn length nd sg. This shows the importnce of the new formul, menwhile the existing formul 1) is pplicble only for the levelled spns. The ppliction of the new formul is recommended up to 400 metres. With simple numericl solution successive pproximtion 40 Per. Pol. Elec. Eng. nd Comp. Sci. Alen Htibovic

7 method) the limittion mentioned in the pper for the equtions which is in fct correct tht the sg must be known) could esily be by-pssed resulting in inverse function when the conductor length is given nd the sg is clculted - without the need for new equtions. References 1 Pnsini A, Electricl Distribution Engineering, The Firmont Press, Inc., 007. Stewrt J, Single Vrible Clculus, CENGAGE Lerning, Gustfson D, Frisk P, Hughes J, College Algebr, CENGAGE Lerning, Htibovic A, Mthemticl clcultion of the prbol wire nd sg for spns up to 400 meters, Bosnskohercegovck elektrotehnik, 5, 01). 5 Htibovic A, A vezeték legmélyebb pontjánk meghtározás, MEE 58. Vándorgyűlés, 011). 6 Htibovic A, Tomic M, Determintion of the lowest point of conductor for inclined spns, CIGRÉ conference, 011). 7 Perneczky G, Szbdvezetékek feszítése, Műszki Könyvkidó; Budpest, Grigsby LL, Electric Power Engineering Hndbook, Tylor & Frncis Group, LLC, CIGRÉ 34, Sg-tension clcultion methods for overhed lines, 007. Integrl clculus usge for conductor length determintion