A Nonlinear Regression based Approach for Multilayer Soil Parameter Estimation

Transcription

1 Internatonal Journal of Control and Automaton Vol.7, No. (04), pp A Nonlnear Regresson based Approach for Multlayer Sol Parameter Estmaton Mn-Jae Kang, Chang-Jn Boo, Ho-Chan Km and Jace M. Zurada Department of Electrcal Engneerng, Jeju Natonal Unversty, Korea Department of Electrcal Engneerng, Unversty of Lousvlle, USA {mnj, boo004, hcm}@jejunu.ac.r, j.zurada@eee.org Abstract The estmaton of sol parameters of multlayer structure leads to useful nformaton for desgnng a safe groundng system. Ths paper presents a nonlnear regresson based estmaton scheme to extract sol parameters from the ernel functon of apparent earth resstvty. The ernel functon of apparent earth resstvty can be obtaned from the measured apparent earth resstvty data. The performance of the proposed method has been verfed by carryng out a numercal example. Keywords: Estmaton, sol parameters, groundng system, nonlnear regresson, sol resstvty, ernel functon, apparent earth resstvty. Introducton It s mportant to now the earth structure n the gven area when the groundng system s desgned. Because badly desgned groundng system cannot ensure the safety of equpment and personnel []. The sol s modeled as a unform medum n early researches and the smplfed formula s used to estmate the resstance of groundng system. For smplfyng the problem, n a host of engneerng applcaton, multlayer sols are modeled by N horzontal layers wth dstnct resstvty and depths []. A Wenner confguraton method s well nown to measure the sol resstvty for ths smplfed earth model. The nverson of sol parameter s an unconstraned nonlnear mnmzaton problem [3, 4]. Supposng there are N dfferent layers below the ground surface then N- parameters need to be determned, because there exsts dfferent N- thcnesses and N resstvty n the Wenner confguraton model. Therefore, many dfferent optmzaton methods have been carred out to nvert sol parameters n the hope of mprovng the performance. However, there exst two dffcultes n nvertng the sol parameters usng optmzaton methods. On one hand, t s hard to obtan the dervatves of the optmzed expresson. On the other hand, the computng tme s hugely consumed. These dffcultes can be solved effcently by usng the proposed method n ths paper. Ths paper presents a regresson scheme to estmate sol parameters from the ernel functon of the apparent earth resstvty. Ths regresson based estmaton scheme s composed of two steps: frst, t s needed to obtan the ernel functon of apparent earth resstvty from the measured apparent resstvty data, and secondly from ths ernel functon, the regresson method s appled to estmate sol parameters. The obtanng ernel functon method has been showed n J. Zou et al. []. In ths paper J. Zou s method s modfed more effcently usng lnearzaton method for obtanng the ernel. A three-layer earth structure has been used n a numercal example for smplcty to examne the proposed method, however a mult-layer structure more than three can be appled ISSN: IJCA Copyrght c 04 SERSC

2 Internatonal Journal of Control and Automaton Vol.7, No. (04) to use ths method. The rest of ths paper s organzed as follows: Secton gves a bref overvew of the apparent sol resstvty for a mult-layer earth model usng Wenner method. Secton 3 presents an analytc formulaton to nvert sol parameters usng the ernel functon and a numercal example result, and the conclusons are presented n Secton 4.. Wenner 4-pont Test and Apparent Resstvty A schematc dagram of the apparent resstvty measurement setup has been shown n Fgure. h (,,, N ) and (,,, N) are the thcness and the resstvty of the th layer respectvely for an N layer sol structure. A current I s njected nto the sol by applyng the power between electrodes A and B and the potental dfference V between electrodes C and D s measured. By changng the test A V +I -I A B C D h ρ a a a h ρ hn- ρ N- ρ N Fgure. Wenner confguraton for measurng apparent sol resstvty of N- layer earth structure electrode span a, a set of apparent resstvty curves varyng wth electrode span can be obtaned. If a pont current source enters at a pont A on the surface, the Laplace dfferental equaton can be used to descrbe the potental dstrbuton by usng cylndrcal coordnates, r,, z. In ths case, Laplace dfferental equaton becomes that there s cylndrcal symmetry and so s elmnated, V V V 0 r r r z Wth approprate boundary condtons, the surface potental at z=0 becomes ( ) I V r f ( ) J ( r ) d 0 0 Where s the sol resstvty of the frst layer, J ( r) s the zero order Bessel s functon 0 of the frst nd and the ernel functon f ( ) s as follows [] () () f ( ) ( ) (3) 66 Copyrght c 04 SERSC

3 Internatonal Journal of Control and Automaton Vol.7, No. (04) Ke h ( ) K ( ) h Ke Ke h 3 3 ( ) K ( ) h Ke 3 3 (4) K e ( ) ( ). hn n n n n K h n n Kn e n n If the potental dfference V between potental electrodes C and D n Fgure s tested, the apparent resstvty ( a) CD can be expressed by V CD [ V ( a) V ( a)] ( a) a a I I Where Vaand ( ) V( a ) are the potental of electrodes C and D generated by current test electrodes A and B respectvely. By substtutng () nto (5), one can obtan ( ) a a f ( )[ J ( a ) J ( a )] d (5) (6) 3. Estmaton of Sol Parameters K As seen n (4), wth the assumed N-layer earth sol parameters, ( ) n s calculated frst, then ( ) n K, ( ) n,. K ( ), ( ) are calculated n reverse order. Wth the last calculated ( ), the ernel functon f ( ) can be formulated as n (3). Ths ernel functon f () s used n () to evaluate the apparent resstvty (a). The assumed N-layer earth sol parameters are changed contnuously and the apparent resstvty (a) s calculated untl ths evaluated (a) s as close as possble to the measured apparent resstvty by Wenner method. Ths secton wll descrbe an analytcal formulaton of the nverse soluton to the sol parameter estmatng problem usng Wenner method. The analytcal formulaton s developed by analyzng the ernel functon of the ntegral equaton of apparent resstvty. Frst, the ernel functon s obtaned from the data of measured apparent resstvty usng Wenner method. Second, usng the ernel functon obtaned n the frst step, the proposed formulaton n ths paper provdes the estmaton of thcness and resstvty of the earth layers. The structure of earth layers can then be determned from the respectve sol parameter estmates. 3.. Inverson of the ernel functon usng the measured apparent resstvty J. Zou et al. [] showed that the ernel functon of apparent earth resstvty can be obtaned from the data measured by Wenner confguraton method. In hs method, the ernel functon f ( ) s assumed to be a contnuous and smooth functon wth respect to, and the property of ts asymptotc functon s much smlar to the dampng exponental functon [, 5]. So the ernel functon f ( ) can be expressed by a seres of exponental functon as Copyrght c 04 SERSC 67

4 Internatonal Journal of Control and Automaton Vol.7, No. (04) c f ( ) b e (7) where b, c are constant. Usng the followng Lpschtz ntegratng (8) c e J0( l) d 0 c l the apparent resstvty ( a) n (6) can be approxmated as follows N ( a) a b c a c 4a To fnd b and c from the measured apparent resstvty data, rearrangng (9) and replacng ( a) wth the measured m ( a) gves N m c 4 a a c a ( a ) b,(,,..., M ) Where ( a ) ( M ) s the measured apparent resstvty wth the test electrode span a ( M ) of the Wenner confguraton method and c s the th decay constant correspondng to the th samplng of ntegral []. 3.. Determnng Kernel Functon by Lnearzaton method In ths secton the new algorthm has been suggested how to lnearze J. Zou s method for obtanng the ernel functon effcently. Equaton (0) s a nonlnear system and ths nd of nonlnear equaton can be solved by versatle teratve methods ncludng Newton-Raphson method [7, 8]. However, ths case s dffcult to use general teratve methods because a large number of b and c are requred for fndng the correct ernel functon f ( ) n (7). It has been nown that t s dffcult to fnd a precse soluton of a large number varables nvolvng nonlnear system. Assumng f ( ) n (7) can be transformed as follows f ( ) N b e d Then () becomes a lnear system because left hand sde bracet n (0) s determned. Where d s very small constant, 0. s good enough for d accordng to the author s experence. Then (0) can be expressed as a lnear system as follows (9) (0) () 68 Copyrght c 04 SERSC

5 Internatonal Journal of Control and Automaton Vol.7, No. (04) where m ( a ) a A A A N b m ( a) A A A N b a A A A b M M MN N m ( am ) A a M d a d 4a () (3) Because a s constant whch s the -th test electrode span, A s also constant. Nevertheless, the number of equatons(n) s not same as the number of varables(m), n other words ( M N ), therefore the unque soluton s not possble. After trals and errors, t s found that the ernel functon f ( ) can be obtaned correctly only n the case of N M. Ths system s an underdetermned system where many possble solutons exst. QR factorzaton s one of many methods to solve an underdetermned system. QR factorzaton s used to solve n ths paper Inverson Sol Parameters usng a Nonlnear Regresson Method The proposed formulaton provdes the analytc method for estmatng thcness and resstvty of the earth layers. The characterstc of the ernel functon s analyzed to develop the analytc method for estmatng sol parameters. As seen n (4), K ( ) s a functon of, and ( ). And because ( ) converges to as ncreases, K ( ) converges as follows K lm K ( ) ( ) ncrease By rearrangng the equaton of left hand sde n (4), also K ( ) can be derved as follows where (4) h e (5) K ( ) ( ) ( ) ( ) ( ) To mae K ( ) n (5) converge to same value as (4), ( ) has to be the exponentally decreasng form cancelng out the exponental part n (5) as follows Copyrght c 04 SERSC 69

6 Internatonal Journal of Control and Automaton Vol.7, No. (04) Then c becomes as lm ( ) c e h (6) ncrease c Therefore, the th layer thcness h and c can be estmated from ( ) usng nonlnear regresson method. Eq. (6) can be lnearzed by tang ts natural logarthm to yeld ln ( ) ln c h (7) Thus, the plot of ln ( ) versus wll yeld a straght lne wth a slope of h and an ntercept of ln c. Therefore, the th layer thcness h and c can be estmated from the slop and ntercept of the plot of ln ( ). As seen n (3), s obtaned from the determned ernel functon f ( ), therefore c and h can be obtaned whch are the rato of the frst and second layer s resstvty and the frst layer s thcness respectvely. At the same tme when h s determned, also the correct K ( ) s determned from (5). And then can be obtaned from the rght hand sde equaton of (4) as follows ( ) K( ) K ( ) In ths manner, all K ( ) and ( ) can be obtaned teratvely and so are all the sol parameters ( h, h,, h n, 3.4. Calculaton Results,,, ). n For the numercal example, a three-layer earth model wth the sol parameters h, h,, and ) was consdered as shown n Table. 3 ( Table. Parameters of a three-layer earth structure Layer No Resstvty(Ω.m) Thcness(m) (8) 70 Copyrght c 04 SERSC

7 f() Internatonal Journal of Control and Automaton Vol.7, No. (04) Fgure shows that the ernel functons f ( ) of a three layer earth model where the sold lne s the exact ernel functon determned from (3) and (4), and the dotted lne s obtaned by () whch s a lnearzaton method. The exact ernel functon can be obtaned wth the nown sol parameters n Table. As shown n Fgure, the obtaned ernel functon by the proposed method s very close almost same to the exact one Exact value Calculated value by () Fgure. The ernel functons of 3-layer earth structure Next step s to obtan the sol parameters from the ernel functon f ( ). To estmate the frst layer depth h, and the rato of frst and second layer s resstvty c, the natural logarthm of s needed as follow ( ) f ( ) ln ln ln ( ) f ( ) (9) The sold lne n Fgure 3 s the natural logarthm of and the dotted lne s the best ftlne. As shown n Fgure 3, the best ft-lne yeld a straght lne wth a slope of -4.43,whch s h, and an ntercept of -0., whch s ln c. From these values, h and c are estmated as.5 and 0.8 respectvely. Those values are really close to the exact ones of h and c, whch are. and 0.80 respectvely. Wth ths same manner, K ( ), ( ) obtaned teratvely and so are all the sol parameters ( h, h,,, 3 ). and ( ) can be Copyrght c 04 SERSC 7

8 ln() Internatonal Journal of Control and Automaton Vol.7, No. (04) 0 X: 0 Y: X: 0.8 Y: In() the best ft -lne X:.6 Y: Concluson Fgure 3. The natural logarthm of and the best ft-lne The determnaton of the sol parameters from Wenner s test data s an nverse problem. For the three-layer model, t s a fve-varable optmzaton problem. In ths paper, the analytc method rather than a general optmzaton algorthm s presented to nvert the sol parameters. Ths proposed method s formulated based on the characterstc of the ernel functon of apparent resstvty. Also, ths method s composed of two steps whch are obtanng the ernel functon and then estmatng the sol parameters from the ernel functon. In ths paper, the frst step has been mproved by lnearzng the nonlnear system and n the second step, a nonlnear regresson method has been suggested for estmatng sol parameters from the ernel functon. The robustness of the proposed algorthm has been verfed by carryng out numercal smulaton for three-layer model. The results show a promsng performance of the method. Acnowledgements Ths research was supported by Basc Scence Research Program through the Natonal Research Foundaton of Korea (NRF) funded by the Mnstry of Educaton, Scence and Technology ( ). References [] H. R. Seedher and J. K. Arora, Estmaton of two-layer sol parameters usng fnte Wenner resstvty expresson, IEEE Trans. on Power Delvery, vol. 7, (99), pp [] J. Zou, J. L. He, R. Zeng, W. M. Sun and S. M. Chen, Two-Stage Algorthm for Invertng structure Parameters of the Horzontal Multlayer Sol, IEEE Trans. on Magnetcs (do:0.09/tmag ), vol. 40, (004), pp Copyrght c 04 SERSC

9 Internatonal Journal of Control and Automaton Vol.7, No. (04) [3] J. L. Del Alamo, A comparson among eght dfferent technques n two-layered earth, IEEE Trans. on Power Delvery, vol. 8, (993), pp , do:0.09/ [4] F. P. Dawalb, Electromagnetc Felds generated by overhead and bured short conductors, IEEE Trans. on Power Delvery, vol., (986), pp. 05-9, do:0.09/tpwrd [5] B. Zhang, X. Cu, L. L and J. L. He, Parameter Estmaton of Horzontal Multlayer Earth by Complex Image Method, IEEE Trans. on Power Delvery, vol. 0, (005), pp , do:0.09/tpwrd [6] F. P. Dawalb, Electromagnetc Felds generated by overhead and bured short conductors, IEEE Trans. on Power Delvery, vol., (986), pp [7] S. C. Chapra, Appled Numercal Methods wth MATLAB for Engneerng and Scentsts, nd ed, McGrow-Hll, NewYor, (008). [8] H. S. Yazd, M. Arghan and E. Nemat, Nonlnear Regresson Model of a Human H and Volume: A Nondestructve Method, Internatonal Journal of Control and Automaton, vol. 4, no., (0) June, pp. -4. Mn-Jae Kang Authors He receved hs B.S. degree n Electrcal Engneerng from Seoul Natonal Unversty, Korea, n 98, M.S. and Ph.D. degrees n Electrcal Engneerng from Unversty of Lousvlle n 989 and 99, respectvely. Snce 99, he has been wth the Department of Electronc Engneerng at Jeju Natonal Unversty, where he s currently a professor. He was a Vstng Scholar at the Unversty of Illnos at Urbana-Champagn n 003. Hs research nterests nclude neural networs, groundng systems, and wnd power control. Chang-Jn Boo He receved hs B.S., M.S., and Ph.D. degrees n Electrcal Engneerng from Jeju Natonal Unversty n 00, 003 and 007, respectvely. Snce 0 he has been worng at the Research Insttute of Advanced Technology at Jeju Natonal Unversty. Hs research nterests nclude groundng system desgn and power system control. Ho-Chan Km He receved hs B.S., M.S., and Ph.D. degrees n Control and Instrumentaton Engneerng from Seoul Natonal Unversty n 987, 989, and 994, respectvely. He was a research staff member from 994 to 995 at the Korea Insttute of Scence and Technology (KIST). Snce 995, he has been wth the Department of Electrcal Engneerng at Jeju Natonal Unversty, where he s currently a professor. He was a Vstng Scholar at the Pennsylvana State Unversty n 999 and 008. Hs research nterests nclude smart grd, electrcty maret analyss, and control theory. Copyrght c 04 SERSC 73

10 Internatonal Journal of Control and Automaton Vol.7, No. (04) Jace M. Zurada He receved hs MS and PhD degrees (wth dstncton) n electrcal engneerng from the Techncal Unversty of Gdans, Poland. Snce 989 he has been a Professor wth the Electrcal and Computer Engneerng Department at the Unversty of Lousvlle, Kentucy. He was Department Char from 004 to 006. Hs nterests nclude extenson of complex-valued neurons to assocatve memores and percepton networs, senstvty concepts appled to multlayer neural networs, applcaton of networs for clusterng. Dr. Zurada has served the professon and the IEEE n varous elected capactes, ncludng as Presdent of IEEE Computatonal Intellgence Socety n He has been member and char of varous IEEE CIS and IEEE TAB commttees, ncludng the IEEE TAB Perodcals Commttee and IEEE TAB Perodcals Revew Commttee. He was the Foundng Char of the NNC Neural Networs Techncal Commttee. In 0-3 he serves as IEEE TAB Perodcals Revew and Advsory Commttee Char and n 03 as VP-Techncal Actvtes Elect 74 Copyrght c 04 SERSC