DETERMINATION OF TWO LAYER EARTH STRUCTURE PARAMETERS

Transcription

1 DETERMINATION OF TWO LAYER EARTH STRUCTURE PARAMETERS Ioannis F. GONOS, Vassiliki T. KONTARGYRI, Ioannis A. STATHOPULOS, National Tehnial University of Athens, Shool of Eletrial and Coputer Engineering, High Voltage Lab. 9, Iroon Politehniou Str. GR Zografou Capus, Athens, GREECE Antonios X. MORONIS, Anastasios P. SAKARELLOS, Nikolaos I. KOLLIOPOULOS Tehnologial Eduational Institute of Athens, Departent of Energy Tehnology, Ag. Spyridonos & Milou Str. GR 22 0, Egaleo, Athens, GREECE Suary: In this paper, a ethodology has been proposed, aording to whih one an opute the paraeters of the two-layer earth struture after arrying out a set of soil s resistivity easureents. On that purpose, different optiization funtions have been used. These optiization funtions are the relative error, the absolute error, the square error, and the L-infinity nor. By oparing the results of eah optiization funtion, the relative error is being proved to be the ost suitable optiization funtion that gives the best fitting urve to the experiental data. Keywords: Geneti Algoriths, grounding syste, soil resistivity easureents, two-layer earth struture.. Introdution The soil type signifiantly affets the behavior of a grounding syste. The effet of earth struture an be studied out through easureents of soil resistivity. The analysis of the results oing fro suh easureents is an iportant proedure for the aurate analysis of grounding systes [-3]. The easureents of the soil resistivity have shown that the soil has to be siulated as a two-layer struture (at least) [-9]. The alulation of the paraeters of the earth s twolayer struture is transfored into a proble of iniization. A ethodology based on a Geneti Algorith (GA) is proposed in this paper, whih alulates the paraeters of the earth s two-layer struture using the easureents of soil resistivity. The paraeters (soil resistivity and thikness) of the earth struture are the neessary input data for the iruital or field siulations of grounding systes. Hene, by the use of the suggested ethodology, it is possible to aurately alulate the two-layer earth paraeters for a speifi ground. These paraeters will be thereafter used as essential input data for the siulation of the behaviour of the grounding syste that will be installed in this ground. On that purpose, different optiization funtions have been used. These optiization funtions are the relative error, the absolute error, the square error and the L-infinity nor. By oparing the results of the GA for eah optiization funtion, the ost suitable funtion derives. 2. Soil resistivity easureents It has been found that, usually, the ground inorporates a ultilayer struture [3-9]. The deterination of the soil s struture paraeters (soil resistivity and thikness of eah soil s layer) is essential to the design of grounding systes. These paraeters an be alulated through the analysis of data oing fro soil resistivity easureents. After that, the ost suitable earth struture is hosen (aording to the thikness of the layers) in order to proeed to the design of the grounding syste [6]. Fig.. The four point ethod For an eletrode pair (Fig. ) with urrent I at eletrode A, and I at eletrode B, the potential at a point is given by the algebrai su of the individual ontributions [2]: ρ I U = 2 π r A r B () where r A and r B are the distanes fro a point to eletrodes A and B, respetively. 0.-

2 Between urrent eletrodes A and B, an additional pair of eletrodes M and N is plaed, where the potential differene V ay be easured. Aording to eq. the potential differene V is given by the forula [2]: V = U M U N ρ I = 2 π AM BM AN BN = (2) where U M and U N are the potentials at points M and N. (AM), (BM), (AN), (BN) are distanes between the eletrodes AM, BM, AN and BN, respetively. The ipat of soil resistivity on ground resistane is substantial. Many tehniques have been developed for the easureent of soil resistivity [-3]. The ost oonly used are Wenner, Shluberger, and dipoledipole. An eletrode array with onstant spaing is used to investigate lateral hanges in apparent resistivity refleting lateral geologi variability or loalized anoalous features. To investigate hanges in resistivity with depth, the size of the eletrode array is varied. The apparent resistivity is affeted by soil s oposition at inreasingly greater depths (hene larger volue) as the eletrode spaing is inreased. In any ase, the geoetri fator for any four-eletrode syste an be alulated fro eq. 2 and an be developed for ore opliated systes by using the rule illustrated by eq. [2]. The Wenner ethod [-3] is the ost aurate for the easureent of the average soil resistivity. Four eletrodes (rods) are driven in a depth d in four points of the soil, whih are in the sae line and they have the sae distane α=l =L 2 =L 3 fro eah other (Fig. ). The quotient V/I gives the apparent resistane R (in Ohs). Aording to eq. 2 the apparent soil resistivity of the soil ρ is given by the following equation [-3]: A ore aurate forula [3] is: ρ 2 π R α (3) 4 π R a ρ = (4) + 2 a a a + 4 d a + d The Shluberger ethod [-3] uses four eletrodes in the sae line, with the distane L =L 3 =b (Fig. ) and L 2 =a. In usual field operations, the inner (potential) eletrodes reain fixed, while the outer (urrent) eletrodes are adjusted to vary the distane b. The spaing b ust be larger than 3a. Also, the a spaing ay soeties be adjusted with b held onstant in order to detet the presene of loal inhoogeneities or lateral hanges in the neighborhood of the potential eletrodes. Aording to eq. 2 the apparent soil resistivity of the soil ρ is given by the following equation [2]: ( a + b) b ρ = π R (5) a The dipole-dipole ethod [2] is one eber of a faily of arrays using dipoles (losely spaed eletrode pairs) to easure the urvature of the potential field. If the separation between both pairs of eletrodes (Fig. ) is the sae α=l =L 3, and the separation between the enters of the dipoles is restrited to L 2 =a(n+), the apparent resistivity is given by: ( n + ) ( + 2) ρ = π R a n n (6) The dipole-dipole ethod is espeially useful for easuring lateral resistivity hanges and has been inreasingly used in geotehnial appliations. 3. Estiation of two-layer earth struture paraeters The paraeter alulation of the two-layer earth struture is an optiization proble whih involves the alulation of three paraeters (the soil resistivity ρ of the upper layer, the soil resistivity ρ 2 of the lower layer and the depth h of the upper layer) and the iniization of the objetive funtion [6, 7]. In this paper four different optiization funtions (F, F 2, F 3, and F 4 ) have been used. F represents the relative error between the easured and the optiized data. The oputation of the paraeters requires the iniization of the funtion F. F is given by the following equation: ρai F = N i= ρ ρ where is the i th easureent of the soil resistivity of the soil, using the Wenner ethod [3] for a distane between two sequentially auxiliary eletrodes equal to α, and ρai is the alulated value of the soil resistivity at distane α between the auxiliary eletrodes orresponding to the i th pair of easureents. N is the total nuber of soil resistivity easureents. The alulation of the soil resistivity is ade using equations (8-) [3, 6]: ρ (7) n ρ α = ρ + 4 K (8) n A B where n =.... K is the refletion oeffiient: ρ2 ρ K = (9) ρ + ρ 2 A and B are two paraeters, whih are given by the following equations: 0.-2

3 2 2 n h + A = (0) α B = A + 3 () Another objetive funtion, whih will be used by the GA, is the absolute error, whih is given by the forula: F N 2 = ρ i= ρ (2) An extra objetive funtion, whih will be used, is the square error, whih is given by the forula: F 3 = N i= ρ ρ ρ 2 (3) Also, the L-infinity nor has been used as an objetive funtion for the optiization of the values of the paraeters of two-layer earth struture paraeters. for i=,,n. i F4 = ax ρ ρ (4) 4. Appliation of geneti algorith Geneti algoriths are adaptive algoriths widely applied in siene and engineering for solving pratial searh and optiization probles. Many probles an be effiiently takled by using a GA approah beause orrelation between the variables is not a proble. The basi GA does not require extensive knowledge of the searh spae, suh as solution bounds or funtional derivatives [0, ]. This paper proposes a ethodology, whih uses the developed GA for the optiization of the paraeters of the soil struture paraeters. This GA has been developed using Matlab. The sae GA produes exellent results in several optiization probles [6, 2-5]. It has been applied for the oputation of earth struture paraeters [6, 7], fatorization of ultidiensional polynoials [2], filter design [3], alulation of ar paraeters at polluted insulators [4] and estiation of eletrostati disharge urrent paraeters [5]. The applied GA starts with a randoly generated population of P s =30 hroosoes. It generates 30 rando values for the first layer resistivity (0<ρ <500), 30 rando values for the seond layer resistivity (0<ρ 2 <500) and 30 rando values for the thikness of first layer (0<h <6). Eah paraeter is onverted to a 20-bit binary nuber. Eah hroosoe has variables =3 so 60-bits are required for the hroosoe. Eah pair of parents with rossover generates N =4 hildren. The rossover begins as eah hroosoe of any parent is divided into N p =6 parts, the pair of parents interhange their geneti aterial. After rossover there is a P =0% probability of utation. Through reprodution the population of the parents is enhaned with the hildren. By applying the proess of natural seletion only 30 ebers survive. These are the ebers with the lower values of the objetive funtion, sine a iniization proble is solved. In this paper four different optiization funtions have been used. By repeating the iterations of reprodution under rossover, utation and natural seletion, GAs an find the iniu error. The best values of the population onverge at this point. The terination riterion is fulfilled when the ean value of the optiization funtion in the P s -ebers population is no longer iproved or the nuber of iterations is greater than the axiu defined nuber of iterations N ax = Results Experiental results, whih have been published by Del Alao [4], are presented in Tab.. Table. Experiental results [4] α i [] ρ i [Ω] α i [] ρ i [Ω] Experiental results, whih have been published by Seedher and Arora [5], are presented in Tab. 2. Table 2. Experiental results [5] α i [] ρ i [Ω] α i [] ρ i [Ω] Table 3. Our experiental results (4//2006 & 8/2/2007) α i [] ρ i [Ω] α i [] ρ i [Ω] The easureents, whih are presented in Tab. 3, have been arried out at the Tehnologial Eduation Institute of Athens Capus [7] applying the Wenner ethod. Four eletrodes of 50 in length have been used, whih were plaed on a straight line. The ondutors that onneted the auxiliary eletrodes with the ground eter had a ross setion of The distane between two suessive eletrodes was α and it was varying with disrete steps. 0.-3

4 The GA was applied on the experiental data (Tabs. - 3), whih were obtained by the experiental setup desribed in seond setion. The α i and ρ i values were used as input data in eah GA appliation. In Tabs 4-9 the optiized values of the two-layer earth struture paraeters and the errors for eah objetive funtion are presented. Useful onlusions about the auray of the equation an be drawn fro the values of the errors F, F 2, F 3 and F 4. Both first oluns of Tab. are the input data of the GA for the alulation of the earth paraeters presented at Tab. 4, while both last oluns of Tab. are the input data of the GA for the alulation of the earth paraeter of Tab. 5. Table 4. Geneti algorith results (Tab. ) Table 8. Geneti algorith results (Tab. 3) ρ [Ω] ρ 2 [Ω] h [] Error Table 9. Geneti algorith results (Tab. 3) ρ [Ω] ρ 2 [Ω] h [] Error An exaple of the onvergene of two-layer earth paraeters is presented in Fig. 2. ρ [Ω] ρ 2 [Ω] h [] Error Table 5. Geneti algorith results (Tab. ) ρ [Ω] ρ 2 [Ω] h [] Error Both first oluns of Tab. 2 are the input data of the GA for the alulation of the earth paraeters presented at Tab. 6, while both last oluns of Tab. 2 are the input data of the GA for the alulation of the earth paraeter of Tab. 7. Table 6. Geneti algorith results (Tab. 2) ρ [Ω] ρ 2 [Ω] h [] Error Fig. 2. Convergene of two-layer earth struture paraeters. In Figs. 3-8 oon graphs of the experiental data of the apparent soil resistivity and the apparent soil resistivity for the optiized paraeters values are illustrated. Table 7. Geneti algorith results (Tab. 2) ρ [Ω] ρ 2 [Ω] h [] Error Both first oluns of Tab. 3 are the input data of the GA for the alulation of the earth paraeters presented at Tab. 8, while both last oluns of Tab. 3 are the input data of the GA for the alulation of the earth paraeter of Tab. 9. Fig. 3. Coparison between the experiental data of soil resistivity and the soil resistivity for the optiized paraeter values of Tab

5 Fig. 4. Coparison between the experiental data of soil resistivity and the soil resistivity for the optiized paraeter values of Tab. 5. Fig. 7. Coparison between the experiental data of soil resistivity and the soil resistivity for the optiized paraeter values of Tab. 8. Fig. 5. Coparison between the experiental data of soil resistivity and the soil resistivity for the optiized paraeter values of Tab. 6. Fig. 8. Coparison between the experiental data of soil resistivity and the soil resistivity for the optiized paraeter values of Tab. 9. Coparison of the graphs provides lear evidene that soil resistivity easureents are best approahed when the relative error is being used as an objetive funtion. Besides that, the L-infinity nor funtion gives the worst fitting at ost experiental data. In the ase of Fig. 5 all objetive funtions give siilar results. Coparing the urves of Figs. 3-8 for eah equation it an be onluded that the equations an be sorted as follows: F, F 3, F 2 and F 4, with F giving the best result. Coparing the runtie for eah different objetive funtions (eq. 7 and eqs. 2-4) it an be onluded that the equations an be sorted as follows: F 4, F 2, F, and F 3, with F 4 giving the fastest result. Fig. 6. Coparison between the experiental data of soil resistivity and the soil resistivity for the optiized paraeter values of Tab Conlusions In this paper the GA has been suessfully applied for the estiation of the paraeters of the two-layer earth struture. By using different objetive funtions (eq. 7 and eqs. 2-4) for the appliation of the GA algorith 0.-5

6 at experiental data oing fro soil paraeter easureents, it has been shown that, in ost ases, the relative error funtion (eq.7) provides the best fit to the experiental data. The square error, given by eq. 3, is the seond preferable funtion. 7. Aknowledgeent This work was supported by the National Researh Progra, EPEAEK ARCHIMIDIS II. This Projet is o-funded by the European Soial Fund (75%) and National Resoures (25%). We would like to thank all the ebers of our researh group for their help in this projet. 8. Referenes [] G. F. Tagg: Earth Resistanes, George Newnes Liited, London, 964. [2] US Ary Corps of Engineers titled Geophysial Exploration for Engineering and Environental Investigations, Report nuber EM , 995. [3] IEEE Std 8-983, IEEE guide for easuring earth resistivity, ground ipedane, and earth surfae potentials of a ground syste, Marh 983. [4] Del Alao J.L.: A oparison aong eight different tehniques to ahieve an optiu estiation of eletrial grounding paraeters in two-layered earth, IEEE Transations on Power Delivery, Vol. 8, No 4, Otober 993, pp [5] Seedher H.R., Arora J.K.: Estiation of two layer soil paraeters using finite wenner resistivity expressions, IEEE Transations on Power Delivery, Vol. 7, No 3, July 992, pp [6] Gonos I.F., Stathopulos I.A.: Estiation of Multi- Layer Soil Paraeters using Geneti Algoriths, IEEE Transations on Power Delivery, Vol. 20, No, January 2005, pp [7] Gonos I.F., Moronis A.X., Stathopulos I.A.: Variation of Soil Resistivity and Ground Resistane during the Year, Proeedings of the 28 th International Conferene on Lightning Protetion (ICLP 2006), Kanazawa, Japan, Septeber 8-22, 2006, pp [8] Lagae P.J., Fortin J., Craini E.D.: Interpretation of resistivity sounding easureents in N-layer soil using eletrostatis iages, IEEE Transations on Power Delivery, Vol., No 3, July 996, pp [9] Yang H., Yuan J. and Zong W.: Deterination of three-layer earth odel fro wenner four-probe test data, IEEE Transations on Magnetis, Vol. 37, No. 5, Septeber 200, pp [0] Holland H.: Adaptation in natural and artifiial systes, University of Mihigan Press. Reprinted in MIT Press, 992. [] Goldberg D.E.: Geneti algoriths in searh, optiisation, and ahine learning, Addison- Wesley, 989. [2] Gonos I.F., Mastorakis N.E. and Sway M.N.S.: A Geneti Algorith Approah to the Proble of Fatorization of General Multidiensional Polynoials, IEEE Transations on Ciruits and Systes, Part I, Vol. 50, No., January 2003, pp [3] Mastorakis N. E., Gonos I. F. and Sway M.N.S.: Design of two diensional reursive filters using Geneti Algoriths, IEEE Transations on Ciruits and Systes, Part I, Vol. 50, No. 5, May 2003, pp [4] Gonos I.F., Topalis F.V. and Stathopulos I.A: A geneti algorith approah to the odelling of polluted insulators, IEE Proeedings Generation, Transission and Distribution, Vol. 49, No. 3, May 2002, pp [5] Fotis G.P., Gonos I.F., Stathopulos I.A.: Deterination of the Disharge Current Equation Paraeters of ESD using Geneti Algoriths, IEE, Eletronis Letters, Vol. 42, No 4, July 2006, pp