International Journal of Scientific & Engineering Research Volume 8, Issue 7, July-2017 ISSN

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1 Iteratioal Joural of Scietific & Egieerig Research Volume 8, Issue, July- ISSN Estimatio of two layer soil parameters usig traditioal ad evolutioary methods Dr. Abraham George, M.S. Egieerig College, Bagalore, Idia. Soi.M, HKBK College of Egieerig, Bagalore, Idia. Abstract I this paper, the optimum values of the parameters of a two-layer soil model, are estimated by a traditioal method ad a evolutioary method. Both the methods start with the well-kow Weer resistivity measuremet data. I Weer method of measurig soil resistivity, a set of apparet resistivity measuremets are made with differet electrode spacig. For each measured value of apparet resistivity, the same is computed i terms of soil parameters, which are the upper layer soil resistivity, the lower layer soil resistivity ad the height of the upper layer. The optimal values of soil parameters are iteratively estimated. Optimal values are those values of soil parameters for which the sum of absolute values of ormalised error is the miimum. The differece betwee measured ad computed values of soil resistivity is termed as error. Particle Swarm Optimizatio is used as the evolutioary method i this paper. The search process is accelerated by a search space reductio techique to obtai the results i miimum umber of iteratios ad time. The algorithms have bee tested o ie sets of test data ad the results obtaied are better tha all the results published so far. Keywords Particle Swarm Optimizatio, Search space reductio, Soil parameter estimatio, Soil resistivity, Weer method, Evolutioary Computatio, Apparet resistivity. u. INTRODUCTION As soil resistivity is foud to vary with probe spacig, a set The Weer four probe method of measurig soil of measuremets are made with differet electrode spacig resistivity is very popular amog groud mat desig for estimatig the parameters of the two-layer soil model. egieers. Soil resistivity is rarely uiform ad a two-layer Usig the measuremet data, apparet resistivity model is widely accepted as the best for groud mat desig correspodig to each measured value are computed usig purpose. I Weer method, four probes are used for soil resistivity measuremet. They are placed alog a straight lie ad drive ito earth with equal spacig amog them. The depth of peetratio of a probe ito earth is adjusted approximately to te percet of the probe spacig. The outer probes are meat for passig a curret through earth ad the ier probes are meat for measurig the potetial differece betwee them. (). r I () é éæ k ö æ k öùù() ç ( h / a) ç 4 ( h / a) è + ø è + ø ë ûû a = r + 4å - = ë ρ is the upper layer soil resistivity, k is the reflectio factor give by (ρ - ρ)/( ρ + ρ), ρ is the lower layer soil resistivity, h is the height of the upper layer ad a is the probe spacig. ρ, ρ ad h are the three parameters of the two-layer soil model. Fig. Weer four probe method of measurig apparet resistivity The ifiite term give i () is computed a fiite umber of times depedig o the accuracy requiremet; is a quite reasoable value for the upper limit of as suggested by researchers. If a set of m measuremets are made, each measuremet correspodig to particular electrode spacig, the

2 Iteratioal Joural of Scietific & Egieerig Research Volume 8, Issue, July- ISSN apparet resistivity values are computed for each measured value. As computed value of apparet resistivity is a fuctio of ρ, ρ ad h, the optimizatio problem is stated as locatig values for ρ, ρ ad h such that the objective fuctio F is miimum ad stated as (). B T is the traspose of matrix B ad I is a idetity matrix of size m x m. α is a factor which decides the rate of covergece. The choice of α is ormally by trial ad error. But sufficiet care should be take as a wrog choice of α results i failure of covergece. m å meas comp meas miimize F = (( abs ( rai -rai )) / rai ) here i= r a meas i is the i th measured value of apparet resistivity ad apparet resistivity. () r a comp i is the i th computed value of Research has bee goig o i this filed for decades together ad plety of literature is already available o estimatio of two-layer soil parameters. Additioal iformatio o determiatio of soil parameters usig traditioal ad evolutioary methods ca be obtaied from [] - []. Now the elemets of A ad B are to be determied. If there are m measuremets, the for each measured value of apparet resistivity, the same is computed usig (). The differece betwee measured ad computed values is foud i all the m cases to form vector A. For determiig the elemets of B, the equatios give Fig. are made use of which give the partial derivatives of ρa with respect to ρ,ρ ad h. To evaluate these equatios, a iitial set of values of ρ,ρ ad h are essetial. These values ca be suitably assumed. A better practice is to take the iitial guess value of ρ as the measured value of apparet resistivity with miimum electrode spacig ad the iitial guess value of ρ as the measured value of apparet resistivity with imum electrode spacig. Iitial guess value of h is take as half of the imum. TRADITIONAL ALGORITHM electrode spacig. The traditioal method used here is a ucostraied, oliear optimizatio algorithm i which F, as suggested by = + 4å æ ö ç - - r é a æ ( -k ) ö k k ù (), is miimized without subjectig it to ay costrait. F is r = k ç ( ha / ) 4 ( ha / ) ë è ø è + + ø û zero whe measured ad computed values of apparet resistivity are the same i all the m cases. As apparet r é a æ ö k k ù = resistivity values are to be computed i m cases, there is a å ( k ) æ ö ç - - r = k ç ( h/ a) 4 ( h/ a) set of m o-liear equatios which are split ito a set of ë è øè + + ø û m liear equatios ad a correctio part. The equatios éæ ö æ öù ra 6r h k k are solved iteratively to obtai the optimum values of soil = å ç -ç.5.5 h a = ç parameters. ( 4 + ( h / a) ) ç ( + ( h / a) ) ëè ø è øû The liear equatios are of the form A=B*C where A is the vector of differece betwee measured ad computed values of apparet resistivity, B is the matrix of partial derivatives of ρa give i () with respect to ρ, ρ ad h C is the correctio vector cosistig of elemets ρ, ρ ad h. A is of size m x, B is of size m x 3 ad C is of size 3 x. Whe the set of liear equatios are solved, the correctio vector C ca be foud usig (3). Fig. Partial derivates of ρa with respect to ρ, ρ ad h. Structure of A,B ad C A elemet of vector A is the differece betwee the measured ad computed values of apparet resistivity. Vector A is of size m x. Matrix B cotais elemets which are the partial derivatives of ρa with respect to ρ, ρ ad h. There are m rows, where each row correspods to a measuremet. The matrix is of size m x 3. Vector C has correctio elemets to update the values of ρ, ρ ad h i iteratio. It is of size 3x. The structures are show i Fig.3. C BIB BIA a = (3) T - T ( ) ( )

3 Iteratioal Joural of Scietific & Egieerig Research Volume 8, Issue, July- ISSN meas comp éra -r ù a meas comp ra -ra A =... meas comp ëram -r am û B é r r r r r h r r r a a a ù a a a = r r h... ram ram ram ë r r h û C = h Fig.3. Structure of A.B ad C. Step-by-step procedure for estimatig optimal values of soil parameters by traditioal algorithm. Assig suitable iitial values for ρ, ρ ad h. Start iteratio cout. Specify tolerace.. Evaluate elemets of A usig the most recet values of ρ, ρ ad h. 3. Evaluate elemets of B usig the equatios give i Fig. with the most recet values of ρ, ρ ad h. Iitial guess value 4. Compute ( T - T C = B I B) ( B I A) a 5. If ρ, ρ ad h, the elemets of C, are less tha the specified tolerace, go to step. 6. Modify the preset values of ρ, ρ ad h to (ρ+ ρ), (ρ+ ρ) ad (h+ h). Advace iteratio cout ad go to step.. Display the fial values of ρ, ρ ad h. 8. Stop 3. THE PSO ALGORITHM The PSO method elimiates most of the mathematical computatios ivolved i the traditioal algorithm. The basic PSO algorithm for obtaiig the optimal values of soil parameters ρ, ρ ad h is the followig. A populatio of L umber of trial values is geerated for each soil parameter. With oe trial value for a parameter from each populatio, apparet resistivity values are computed for m measuremets ad the objective fuctio stated i () is evaluated. The process is repeated for the ext set of trial values ad so o till the ed of populatio. For each set of trial values persoal best is preserved ad at the ed of populatio global best is idetified. Now the populatio elemets are modified ad the process is repeated with the modified populatio elemets. The objective fuctio value keeps reducig ad the process is termiated after a defiite umber of iteratios. 3. Search Space Reductio The ormal PSO algorithm explores a large space for locatig optimal values of ρ, ρ ad h. The search space ca be reduced ad the process ca be expedited usig the followig techique. I ay traditioal iterative method, iitial guess values of the soil parameters ρ, ρ ad h are madatory for solvig the problem; also these guess values should be close to the fial coverged values. This aspect ca be icorporated i the PSO algorithm to reduce the search space. Thus the iitial search space for each soil parameter is defied by coverig a rage above ad below its iitial guess value. For example, at start, the upper limit of a soil parameter ca be set as thrice the iitial guess value ad the lower limit as zero. The iitial guess value is foud i the traditioal algorithm. Fig. 4. Iitial search space for a soil parameter Hece to reduce the search space, the ucostraied optimizatio problem is coverted to a costraied problem by imposig limits o variatio of the values of soil parameters ρ, ρ ad h. The modified optimizatio problem is stated as (4). miimize F = mi mi mi m i= meas comp meas (( abs( rai -rai ))/ rai ) subject to the imposed costraits r r r r r r h h h å The objective fuctio metioed above is costraied by upper ad lower limits of soil parameters ρ, ρ ad h at ay (4)

4 Iteratioal Joural of Scietific & Egieerig Research Volume 8, Issue, July- ISSN poit of time durig the iteratio process. Iitial guess values are foud as i the traditioal algorithm. Now the ρ, ρ ad h ad the fourth colum, to retai computed fitess values. The elemets of the first upper ad lower limits of ρ, ρ ad h ca be fixed. The three colums are iitialized with radom upper limit of a parameter ca be k times the iitial guess umbers mapped to the iitial search rage usig value ad the lower limit ca be zero. Normally k is stadard mappig rule. assiged a value greater tha. Due to the imposed costraits, the search for a parameter is carried out oly i the rage defied.. Copy the elemets of A to aother matrix B. I due course, B will retai the best persoal values of ρ, ρ, h ad the correspodig fitess value. As the search progresses, the upper ad lower limits of soil 3. Cosider the set of trial values for ρ, ρ ad h from parameters are reset about their most recet global best the first row of A. values istead of iitial guess values. The upper limits of 4. Compute the apparet resistivity the parameters are set as k times the most recet global best value of that parameter ad the lower limits are set as measured value of resistivity usig (). for each (-k) times the global best value. The limits are fixed as give i (4) with startig value for k equal to. This meas that the upper limit of the search space of a soil parameter has become twice its most recet global best value with Compute fitess usig (6) ad store the value i A as the fourth colum elemet of the same row. Compare the fitess values i A ad B. If the fitess value i A is higher, copy the row elemets lower limit remaiig zero. Subsequetly, say, whe k is to the correspodig positios i B. reduced to.5 o improvemet i global best fitess value,. Extract the set of trial values of ρ, ρ ad h from the upper limit ad lower limit of a parameter become.5 times ad.5 times its most recet global best value. The the ext row of A. value of k ca be gradually reduced to.. This meas that, 8. If ot ed-of-matrix, go to step 4. at ay poit of time durig the search, we are tryig to 9. Locate the highest fitess value i B ad the locate the optimal value of a parameter i a arrow rage correspodig global best values of ρ, ρ ad h. If above ad below its most recet global best value. Fig.5 fitess is more tha.999 or iteratio cout exceeds shows the reductio i search space as iteratio progresses. 5, stop the process. mi. Modify the search space usig (5) by reducig the r = (- k )* r gbest, r = k* rgbest value of k depedig o the curret best global mi r = (- k )* r gbest, r = k* rgbest fitess value. Modify all the values of ρ, ρ ad h mi h = (- k )* hgbest, h = k * hgbest (5) i matrix A without violatig limits. Go to step Modifyig the values of ρ, ρ ad h For modifyig the trial values of ρ, ρ ad h i matrix A, the followig well kow equatio is used. Fig. 5. Reductio i search space Fitess = / (+F). (6) 3. Step-by-step procedure for estimatig optimal values of soil parameters. Iitialize a matrix A of size L x 4, the first three colums of which are for retaiig trial values of v modified =w*v +c * rad * (pbest p) + c * rad * (gbest p) () The modified value of a soil parameter is the sum of three terms which are the followig. its preset value multiplied by a weightig fuctio the product of three parameters: a costat, the differece betwee its persoal best value ad the preset value ad a radom umber less tha oe the product of three parameters: a costat, the differece betwee its global best value ad the preset value ad a radom umber less tha oe While modifyig the soil parameter values i matrix A, their persoal best values ca be extracted from the same

5 Iteratioal Joural of Scietific & Egieerig Research Volume 8, Issue, July- ISSN row of matrix B ad the global best values as obtaied from step 9 of the step-by-step procedure. I case the modified value of a parameter violates the upper limit or lower limit, its value is fixed as the limitig value it has violated. This check retais the trial values i the search space. c, c ad w ca be assiged values as i ay stadard PSO algorithm. As this is a search for local optimum because of the ature of the problem it is ideal to choose a value of.4 for w. 4 EXECUTION AND RESULTS Table I ad Table II give the Weer resistivity measuremet data. Table I gives electrode spacig for ie differet set of measuremet data ad Table II gives the correspodig measured apparet resistivity values. Note that the first three sets of data are theoretically geerated. 4. Test data TABLE I ELECTRODE SPACING 4. Programmig tips for PSO algorithm Recommeded values: c=, c=, w=.4, iitial value of k=, fial value of k=., i ()= Covergece criterio: fitess >. 999/iteratio cout >5 4.3 Programmig tips for Traditioal algorithm The program is sesitive to the choice of α metioed i step 4 of the step-by-step procedure. I this algorithm a value of. give for α Covergece criterio: All the elemets of C are less tha tolerace, Executio results Executio results are show i Table III. Executio is doe usig a 3.5 GHz, AMD FX -83 Eight-Core processor with 4GB RAM. TABLE III EXECUTION RESULTS Set Electrode spacig (m) Set ρ ρ h Iterat Executi F Meth (Ω-m) (Ω-m) m io o time od cout (s) PSO Trad PSO Trad PSO Trad PSO Trad PSO Trad. TABLE II PSO Trad. MEASURED VALUES OF APPARENT RESISTIVITYCORRESPONDING PSO TO THE ELECTRODE SPACING GIVEN IN TABLE I Trad PSO Se Measured apparet resistivity (Ω-m) Trad. t PSO Trad Commets o results Data courtesy: [], [] Compariso of results obtaied by the two algorithms shows that PSO stads superior as far as simplicity ad miimizatio are cocered. Durig executio it has bee foud that PSO does t eed 5 iteratios for optimal results, i most of the cases covergece has bee obtaied withi 3 iteratios. Moreover PSO algorithm elimiates all complex mathematical steps ad the eed of derivatives. Objective fuctio values show that accuracy is more for PSO results. 4.6 Compariso of PSO results with GA solutios of Ioais [3] Table IV shows the compariso of PSO solutios with GA solutios of Ioais [3]. I all the six cases PSO solutios

6 Iteratioal Joural of Scietific & Egieerig Research Volume 8, Issue, July- ISSN are foud superior to GA solutios as far as accuracy is cocered. Also PSO works faster comparig with GA. [3] Del Alamo J. L.993. A compariso of eight differet techiques to achieve a optimum estimatio of electrical groudig parameter i two-layered earth. IEEE trasactio o power Delivery : TABLE IV COMPARISON OF PSO SOLUTION WITH GA SOLUTIONS OF IOANNIS [3] Set of data ρ (Ω-m) ρ (Ω-m) h m F Method GA PSO GA PSO GA PSO GA PSO GA PSO GA PSO [4] Ioais F. Goos., ad Ioais AStathopulos.5. Estimatio of multilayer soil parameters usig Geetic algorithm. IEEE Trasactio o Power Delivery:-6. [5] IEEE Guide for Measurig Earth Resistivity, Groud Impedace, ad Earth Surface Potetials of a Groudig System, IEEE Std [6] CalixtoW.P., Neto L.M.,YamaakaK.,ad Da Paz MoreiraE.. Parameter Estimatio of a Horizotal Multilayer Soil usig Geetic Algorithm. IEEE Trasactio o Power Delivery:5-5. [] IEEE Guide for Safety i AC Substatio groudig IEEE Std. 8-. [8] Soi. M ad Dr. Abraham George.5.. Cost Effective Groudig Grid Desig for Substatio. Iteratioal Joural of Scietific & Egieerig Research: Dr. Abraham George: He did his graduatio ad post 5 CONCLUSION graduatio i Egieerig from Calicut Uiversity ad PhD i Basically a PSO algorithm elimiates the eed of solvig Power system Optimizatio from Dr. MGR Uiversity. He has over 3 years of experiece i teachig. His area of iterest is a set of o-liear equatios for optimal values of soil power system related topics. parameters. A ormal PSO algorithm ca yield such accurate results oly with a large umber of iteratios. But, Soi M: Received B.E ad M.E degree from Bagalore the accelerated algorithm yields highly accurate results Uiversity. She is ito teachig sice 5 years ad presetly with the least umber of iteratios i the PSO eviromet. workig as associate prof. i EEE Dept. of HKBK College of The algorithm developed has bee tested o ie differet Egieerig, Begaluru. Her area of iterest is power system related topics. sets of data with three trials o each set of data. The stregth of this algorithm lies i reducig the search space gradually about the most recet global best values of soil. parameters. The comparative study of the PSO algorithm with the traditioal algorithm proves the superiority of the PSO algorithm i respect of simplicity ad accuracy. REFERENCES [] Has R. Sreedhar, ad J.K. Arora.99. Estimatio of two layer soil parameters usig fiite Weer resistivity equatios. IEEE Trasactio o Power Delivery:3-.. [] Del Alamo J. L. 99. A secod order gradiet techique for a improved estimatio of soil parameters i a two-layer earth. IEEE Trasactio o Power Delivery:66-.