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Archive of SID A Comparative Study of Taguchi and Methods in Design Optimization of Current Transformers

M.B. Moghadam Assistant Professor, Deptartment of , Allameh Tabatabaee University, bamenimoghadam @atu.ac.ir

Sh. Darbandi Research Scholar, Amirkabir University,[email protected]

(Received: June 29, 2004- Accepted in Revised Form: May 7, 2005)

Abstract In this paper a cost effective method of optimization, namely, Taguchi Method is used to optimize system design of current transformers (100/5 Ampere C;;:rs).However, it is shown that due to heavy-tailed distribution of generated from inner-outer arrays which tend to generate outliers, design analysis by M-Estimation of Robust Regression Leads to better results compared to customary analysis method used in TM. The results show that with better performance, the market price of proposed current transformer can be much less than one twentieth of the price of one of the most famous current transformer brands in the market.

Key Words Robust Parameter Design, M-Estimations of Robust Regression

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1. INTRODUCTION of confidence to correctly make inferences from the produced data. It is shown that under the The product design optimization method used here iid is application of Robust Parameter Design (RPD) condition &i ~ N(O, (2) , data analysis of RPD method in engineering which is also called optimization method is almost similar to least TaguchiMethod (TM) , and introduced by Taguchi square (LS) method which minimizes the sum of and Wu (1985), Kackar (1985), Taguchi squared residuals of the model under study. (1986,1987), Phadke (1989), Bisgard(1996), and However, we seldom come across with data that many other researchers in subsequent years. To satisfy the whole assumptions in random variables facilitate the RPD optimization, Taguchi and Wu &;'s [1]. Failure of these assumptions may give suggest the use of experimental design methods rise to biased estimators of the parameters and their and, in particular, orthogonal arrays laid out as corresponding , and therefore the inner and outer array designs to Produce generate statistical inferences based on such estimators are proper data. In this regard, some assumptions like no longer valid. These failures may arise from normalityshould be met with an acceptable degree

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several sources. One of these sources is the case in logical phase is concerned with the optimization of which the produced data follows a distribution that the system found in the previous phase. Usually, has longer or heavier tails than the normal. If this the objective of parameter design is to optimize non-normality is not overcome with any type of variation in the products output caused by variation transformation in hand, it tends to generate outliers in components or in the environment by which may have a strong influence on the method manipulation of the design parameters under vast of analysis and in turn it leads to inaccurate and manufacturing and environment conditions. The unreliable inferences. designer is then asked to provide tolerances that In such cases, we are required to use some other balance the desire to reduce the variability in the methods of analysis in which it is to be robust to output with the need to allow for variability in the this type of violation [2]. One of the most input. This is the third and final step called prominent methods of analysis which is robust to tolerance design [5,6, 7, 8, and 9]. heavy tailed data distribution is M-Estimation(ME) It is now widely recognized that the quality and method of robust regression(RR) , introduced by cost of a product is, to a large degree, determined Huber[3] in 1964. It is in this context that TM and by its quality and cost designed into the product M-Estimation method of robust regression have and process design in the first phases of off-line been used for product design optimization of 100 activities [10]. On the other hand, in to 5 Ampere current transformers (CTs) and results the context of most non-industrialized countries, were compared through confirmation . the off-line quality control activities of system It appears that there is nothing notable in published and/or tolerance designs in product and process literature about optimization of product design of design stages are negligible if not absent. So, CTs comparing design analysis of TM and ME whatever quality control activities there may be in method of RR. Therefore, the present comparative these countries are confined mainly to conduct of study seems to be a pioneer work in its own in parameter design in product and/or process design, respect of using these combined approaches in and on-line quality control activities. Therefore, it product design optimization of CTs. This paper has is in this context that product design optimization been organized in to five sections. In the first of 100 to 5 Ampere current transformer is taken section, the problem under study is illustrated. In under study. section 2 and 3, TM of optimization and ME To facilitate the parameter design optimization, method of RR are introduced. The application of Taguchi and Wu suggest the use of experimental mentioned methods is illustrated in the context of design methods and, in particular, orthogonal system design optimization of current transformer arrays laid out as inner - and outer - array designs (CTs) in the fourth section. Finally, the result of [4]. In fact, a complete Taguchi - type parameter the analysis and their comparisons through design consists of two parts: a design confirmation of experiments are given in the fifth parameter matrix and a noise factor matrix. The section. design parameter matrix specifies the test settings of design parameters. Its columns represent the design parameters and its rows show different combinations of test settings. The noise factor 2. TAGUCHI METHOD matrix specifies the test levels of noise factors. Its columns represent the noise factors and its rows Off-line quality control (optimization method) as represent different combinations of noise levels. introduced by Taguchi and Wu [4], provides The complete experiment consists of a quality engineers with a comprehensive system combination of the design parameter and the noise approach to the product design process which factor matrices (See Figure 1). Each test run of the consists of three steps, namely, system design, design parameter matrix is crossed with all rows of parameter design, and tolerance design. System the noise factor matrix. So that in the example in design is the phase of product design process figure 1, there are four trials in each test run- one where general layout of the perspective product is for each combination of noise levels in the noise established. Parameter design which is the next factor matrix. The performance characteristic is

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www.SID.ir Archive of SID evaluated for each of the four trials in each of the sense that they "pull" the regression equation too nine test runs. Thus, the variation in multiple much in their direction [11, ]2]. The most common values of the performance characteristic minimize general method of robust regression to such a case the product (or process) performance variation at is M-Estimation (ME), introduced by Huber the given design parameter settings. (1964)[3].To illustrate the method consider the following :

Yj = 130+ f31xn + f32Xj2 + + f3kXjk + E:j = X:f3 + E:j 3. M-ESTIMATION of ROBUST REGRESSION

When the observations y in the The general ME minimizes the objective function model Y =xf3 + 8 is normally distributed, the n n method of is a good parameter LP(ei)= LP(Y; -X;b) estimation procedure in the sense that it produces ;=1 ;=1 an estimator of the parameter vector f3 that has good statistical properties. However there are where the function p gives the contribution of many situations where we have evidence that the each residual to the objective function. That is, the distribution of the response variable is not normal function p is related to the and/or there are outliers that affect the regression model. A case of considerable practical interest is for an appropriate choice of the error distribution. one in which the observations follow a distribution In this regard let \f = p' be the derivativeofp . that has a longer or heavier tails than a normal. Differentiating the objective function with respect These heavy-tailed distributions tend to generate to the coefficients, b, and setting the partial outliers, and these outliers may have a strong derivatives to 0, produces a system of k+] influence on the method of least squares in the estimating equations for the coefficients:

ParameterDesign Matrix Noise Factor Matrix Quality Characteristic Performance Statistics 0 W y Z(o) - Run1 81 I(J2 I83 I84 Wt W2 W3 1 1 1 1 1 1 1 1 2 2 2 """' Z( )I 2 2 2 2 1 2 ~i i 2 2 1 i 3 1 3 3 3 f }-- Ii 4 2 1 2 3 i j ! 5 2 2 3 1 I ! 6 2 3 1 2 Wt W2 W3 I 7 3 1 3 2 1 1 1 Z(B)9 8 3 2 1 3 1 2 2 ' 2 1 2 9 3 3 2 1 II 2 2 1 f: }------

Figure 1. A complete Experiment Consists of Design Parameter and Noise Factor

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n

I \!fCYj - X;b)X; = 0 4. APPLICATION 1=1 . . \!fee) Two of the most quality issues in current Define the weIght functIOn w(e) =- and e transformers (CTs) are reduction of output errors in amplitude and phase which are called ratio and let Wj = w(ej) . Then the estimating equations may n phase errors, respectively. These are errors caused by not exactly reproducing the primary current (lP) be written as I Wj (Yi - X;b)X; =O. 1=1 in CTs. This study focuses on robust optimization Solving the estimating equations is a weighted in product design of a particular current least squares problem which is minimizing transformer (100/5 A. CT) in respect of ratio error n quality characteristic (le). Since CTs are used for I wi2ei2. The weights, however, depend upon the two main purposes in industries, namely, electrical 1=1 measuring and system protection, residuals, the residuals depend upon the estimated Malfunctioning of CTs leads to incorrect coefficients, and the estimated coefficients depend measuring of designed parameters and destruction upon the weights. An iterative solution (called of electrical equipments in a system. Therefore, iteratively reweighed least square (IRLS)) [11] is minimization of ratio error quality characteristic in therefore required: CTs is important from quality and cost point of I-Select initial estimates b (0), such as the least view. As it is mentioned before, a complete squares estimates. parameter design experiment consists of two matrices, namely, Design Parameter Matrix (lnner 2-At each iteration t, calculate residuals e;r-I) and Array) and Noise Factor Matrix (Outer Array) associated weights W}'-I) = w[e;H)] from the which take controllable and specific uncontrollable previous iteration. parameters into consideration, respectively. 3-Solve for new weighted least squares estimate Base upon theoretical findings in the study, between core width (CW) and core bet) =[XW(t-l) Xrl XW(t-I) Y where X is the material (CM) is suspected to exist, and therefore, model matrix, with x; as its row, and it is taken into consideration. On the other hand, if interaction of at least one factor be tested in the W (H) = diag{ wt1)} is the current weight experiment, the other interaction can be matrix. Steps 2 and 3 are repeated until the determined through severity Index (SI). The estimated coefficients converge.

Table 1. Controllable factors and their levels

No Factor Code Level I Level 11 Level III

1 Core Width CW CWl CW2 CW3 2 Core Material CM CMl CM2 -

3 Turn of Winding TW TWl TW2 TW3 4 Cross Sectional Area of Wire CSAW CSAWl CSAW2 CSAW3

5 Core Structure CS CSl CS2 -

6 Winding Arrangement WA WAl WA2 WA3 7 Path Location of Primary Winding PLPW PLPWl PLPW2 - 8 Core Cross- SectionalShape CCSS CCSSl CCSS2 -

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severity index calculates the strength of presence Since one interaction effect (CW x CM), four 3 of interaction through magnitude of angle between level and four 2 level factors effects are to be two plotted lines in factors interaction graph. The studied, the Standard orthogonal array L27 is term severity index (SI) is defined such that SI = considered. Therefore, the design matrix of 100%when the angle between the lines is 900, and controllable factors is as below: SI = 0 when the angle is zero. In this way, there will be considerable reduction in number of 4.2. Specific Noise Parameters in the CT and its interaction to be tested, and cost and time savings Design Matrix In the country of study, there are in conduction of experiment. Base upon above two important functional and environmental findings, table 1 shows the controllable factors and factors, namely, Network (NF) and theIr levels: Temperature (T) which affect the performance of CTs. Since dispersion of network frequency around 50 Hz is high and there is a vast of environmental temperature in the country of study, these two factors. are considered as specific uncontrollable factors for which the optimum CT Table 2. Design Matrix Controllable Factors should be insensitive to these factors also. Colunms Therefore, three levels for network frequency 4 5 6 7 8 9 10 11 12 13 (49.5, 50, 50.5 Hz) as NFl, NF2, and NF3, and I I 1 1 1 I I I I 1 0 0 0 three levels for temperature (270, 450, and 550C) I I I I 2 2 2 2 2 2 0 0 0 as T1, T2, and T3 are considered in the study. Base 1 I 1 1 3 3 3 3 3 3 0 0 0 upon the findings, and in order to have sufficient 1 2 2 2 1 1 I 2 2 2 0 0 0 number of data points in the experiment, the Tnal, 1 2 2 2 2 2 2 3 3 3 0 0 0 standard orthogonal array L9 is considered. 2 2 2 3 ., 3 ! I I 0 0 0 Therefore, the design matrix of uncontrollable 1 Trial6. ! (noise) factors is as below: ITrial" 1 j 3 3 3 I I I 3 3 3 0 0 0 I ,TrialS I I 3 3 3 2 2 2 I 1 ! 0 0 0 3 3 3 3 3 3 2 2 2 0 0 0 ITnal91T£1al101 2i 1 2 3 1 2 3 1 2 3 0 0 0 Table3. Design Matrix of Uncontrollable Factors ITrialll1 2 I 2 3 2 3 I 2 3 I 0 0 0 Trial 12' 2 1 2 3 3 1 2 3 I 2 0 0 0 Triali3 2 2 3 1 I 2 3 2 3 1 0 0 0 Columns 2 I () Trial14 2 2 3 1 3 3 I 2 0 0 2 3 4 ! ITrial15 2 2 3 j 3 1 2 I 2 3 0 0 0 I Trial I I I I 0 0 , I 2 3 I 2 j 2 3 3 1 2 0 0 () 1 ;Trial 16 ; () I 0 0 0 Trial 2 I I 2 0 3 I 2 2 3 1 1 2 3 i 3 1 2 3 1 2 2 3 I 0 0 0 Trial 3 I I 3 () O ,! Trial19 3 I 3 2 I 3 2 1 3 2 0 0 0 I 0 () I TriallO 3 1 3 2 2 I 3 2 I 3 0 0 0 Trial 54/ 2 2 0 0 Trial 21 3 I 3 2 3 2 I 3 2 I 0 () 0

3 2 I 3 I 3 2 2 I 3 0 0 0 2 3 0 0 iTrial22, ,Trial 23 3 2 I 3 2 I 3 3 2 I 0 0 0 Trial 6/7 3 I 0 0 ITrial 24 3 2 I 3 3 2 I ] 3 2 0 0 0 2 () 0 3 3 2 ] ] 3 2 3 2 I 0 0 0

ITri"125Trial26 3 3 2 I 2 J 3 I 3 2 0 0 0 Trial 981 3 3 0 0

ITrialn 3 3 2 1 3 2 I 2 1 3 0 0 0

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4.3. Results of Experiment As it can be observed from design matrix of controllable and uncontrollable factors, a observations for each of 27 test runs can be obtained. Therefore, conduction of the experiment led to 27 x 9 = 243 data points which is given in table 4.

C.CS C." C.ZC C.JC cos Table 4. Results of Experiment , Figure 9. Density plot ofresponse variable

0.07 0.10 0.16 0.04 0.07 0.16 0.05 0.10 0.12 0.08 0.12 0.21 0.06 0.14 0.19 0.05 0.11 0.14 to estimate the effect of general and 0.11 0.13 0.16 0.10 0.11 0.14 0.09 0.11 0.12 particular noise factors on the performance 4 0.18 0.19 0.21 0.14 0.16 0.18 0.10 0.11 0.13 characteristic. The noise factors mentioned here 5 0.09 0.12 0.13 0.08 0.10 0.12 0.07 0.09 0.12 6 0.10 0.12 0.13 0.08 0.09 0.10 0,07 0.09 0.10 may be due to network frequency (NF) and 7 0.17 0.18 0.18 0.09 0.14 0.15 0.05 0.14 0.14 temperature (T) and unspecific different sources of 8 0.15 0.15 0.16 0.10 0.11 0.12 0.10 0.10 0.11 variation. Therefore, if we are able to find a 9 0.07 0.12 0.13 0.08 0.18 0.22 0.18 0.17 0.19 specific performance statistic which reflects the 10 0.07 0.09 0.10 0.01 0.02 0.05 0.06 0.06 0.07 effects of a particular class of noise factors and 11 0.15 0.15 0.17 0.07 0.07 0.09 0.13 0.14 0.18 then transform their effects into a data point for 12 0.12 0.13 0.14 0.08 0.09 0.10 0.06 0.07 0.09 each of the conducted tests in the experimental 13 0.12 0.13 0.15 0.11 0.12 0.12 0.]0 0.11 0.12 desigI1,we wil! be able to find what factors and 14 0.]6 0.17 0.18 0.18 0.25 0.30 0.16 0.16 0.18 what levels are to overcome the effects of that 15 0.13 0.13 0.15 0.19 0.15 0.14 0.11 0.13 0.]3 particular class of noise factors upon the 0.10 0.06 0.07 0.07 0.06 0.06 0.07 ]6 0.08 0.09 performance characteristic (ratio errors) in this 17 0.09 0.10 0.05 0.06 0.08 0.05 0.06 0.08 0.06 study. 18 0.09 0.10 0.11 0.08 0.09 0.09 0.07 0.08 0.09 ]9 0.14 0.15 0.15 0.09 0.10 0.12 0.13 0.12 0.13 One such performance statistic which can reflect 20 0.05 0.10 0.11 0.06 0.09 0.09 0.05 0.06 0.08 the effects of categorized sources of variation (i.e., 21 0.07 0.12 0.17 0.08 0.15 0.19 0.05 0.09 0.10 due to external, internal, and unit to unit variation) 22 0.21 0.23 0.27 0.20 0.22 0.26 0.15 0.17 0.18 is the performance statistic Z (B) given below. For 23 0.14 0.18 0.19 0.12 0.13 0.14 0.12 0.13 0.15 this, the performance characteristic Y with a non 24 0.25 0.20 0.30 0.02 0.22 0.20 0.12 0.15 0.18 negative values, a target value equal to zero 25 0.12 0.14 0.14 0.09 0.10 0.11 0.08 0.]0 0.10 ( T = 0), and a lossfunctionL(Y)whichincreases 0.]2 26 0.11 0.19 0.20 0.12 0.25 0.17 0.05 0.10 as Y increases from zero, reflects the present study. 0.15 0.]6 0.11 0.13 0.14 0.10 0.12 0.13 27 0.12 In this case the expected loss, i.e. , l(y)=k. E[y2], is proportional to MSD (B) = E [ (Y-0)2] = E [y2 ] and Taguchi recommends using the performance measure 4.4. Analysis of Experiment The figure 9 below ~(B) =-lOlogMSD(B). is a density function of response variable. It may be observed that the plot has a heavier tail than The larger the performance measure, the smaller is normal density plot [13]. Therefore, it seems the the squared deviations (MSD). Let Yh Y2,... use of Robust Regression is more appropriate than Yn approximate a random sample from the Ordinary Least Square method for estimating distribution of Y for a given design parameters parameters of Regression Model settings (B). The above performance measure can To study the possibility of minimizing ratio error then be estimated by the performance statistic in design analysis of TM, we use a performance Z (B) = SIN = - 10 log [ I y; / n ] where the

388 - Vo!. 17, No. 4, December 2004 HE Transactions B: Applications

www.SID.ir Archive of SID performance statistic Z (B) is moments estimator Table 5. SIN Ratio of Experimental Test Results of It is in this context that the obtained data ~(B). Trial # SIN Ratio of each test in the experiment is transformed ] 19.564 through performance statistic Z (B) and is given 2 17.544 below as SIN ratio [14]. 3 18.373 4 15.94 Base upon the transformed data, the analysis of 5 19.647 6 20.058 variancetables are calculated (Tables 6 and 7). 7 16.856 8 18.106 From ANOVA and its pooled table appear that 9 16.102 only one factor, namely, Core Material, out of 8 ]0 12.355 11 17.482 considered design parameters with 9.76% of 12 19.904 13 18.363 contribution may be able to counteract the effect of 14 14.035 15 16.98 the mentioned noise factors on the performance 16 22.552 characteristic. The decision about the presence of ]7 22.849 18 20.957 interactions can also be made by comparing the 19 17.919 20 21.991 slopes of the lines drawn for interaction between 21 18.277 22 13.417 factors. The strength of presence of interaction 23 16.691 may be calculated by degrees magnitude at angles 24 14.094 25 19.12 of the lines which range between zero (00) and 26 16.1 27 17.712 ninety (900). The term severity index (SI) is

Table 6. Analysis of (ANOVA)

Factors Sums Of I F-Ratio Pure Sum Percent nares Variance I I CW ! 2 5.964 2.982 I 0.405 0.000 0.000 2 CM I 2 25.772 I.753 11.072 6.046 3 INTER COL 2 x I 2 4.245 12.88612.122 0.288 0.000 0.000 4 INTER COL 2 x I 2 70.482 35.241 4.794 55.782 30.460 5 TW 2 4.235 2.ll7 0.288 0.000 0.000 6 CSAW 2 2.676 1.338 0.182 0.000 0.000 7 cs 2 2.978 1.489 0.202 0.000 0.000 I 8 WA 2 I 3.437 1.718 0.233 0.000 0.000 9 PLPW 2 ! 7.562 3.781 0.514 0.000 0.000 10 CCSS 2 11.672 5.836 0.794 0.000 0.000 ,.------.------..- '--'--'-----, OtherlError IT] 44.100 i-~~-_J 1- 63.494i Total: 26 183.129 100.000"10

Table 7. (Pooled)

Factors I 0 I SumsOf . , . S [D F [Squares Vanancc [ F-RallO , Pure urn I ' I CW (2): (5.964), 'p 0 0 L E D(CL= "NC") 0.000 :

2 CM I 2: 25.772: 12.886 3.263! 1i.875 I 9.761 : 3 INfER COL 2 x I I (2)! (4245): 00 L E DKCL="NC") ! 0.000 [ 4 INTER COL2 x I ! 2 i 70.482 i 35.241 8.924! 62.584 I 34.175 5 TW i (2) i (4.235): POOL E D(CL= "NC") I 0.000 6 CSAW POOLEDKcL="NC") i 0.000 7 CS i (2). (2.676) 1 .i (2) (2.978). 1 P 0 0 LED (CL= "NC") ! 0.000 , 8 WA (2), (3437) j POOL E DkCL="NC") ! 0.000 9 PLPW (2) I , ' (7.562)i IP0 0LEDkcL=61.78%)!0.000 10 CCSS .., L_0...i_J.!..~7~)_L ~ P0 0 LEDkcL=76.45%)! 0.000 I OtherlError '-'22-r--86.873r---:i.948. .. -- ...L=:":=:==I:::~~o.~~J Total: 26 183.129 100.000%

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defined such that SI = 100% when the angle Table 9. Optimum Condition and Performance between the lines is 900 and SI = 0 when the angle is zero. Base upon above findings interaction effect of Core Width and Core Material which Factors Level Dese. :Level iConlribution theoretically was suspected to exist is quite 1 CW CW2 0.497 significant in ANOVA tables and least significant 2 CM 3 1039 :; INTER COL 2, ! 'INTER* I 0560 in compare with other interacting factor pairs 4 INTER COL 2, ! 2 2.132 (Table 8). 5 TW TW2 2 0.383 6 CSAW CSAWI I 0.442 As it is observed from above tables, the optimum 7 CS CS2 2 0.288 condition obtained from main effects (Tables 9) 8 WA WA3 0.495 are quite conformable with optimum levels 9 PLPW PLPWI 0.704 to CCSS CeSS! 0.846 achieved in interacting factor pairs [14]. ------....-....---.-.... Tolal Contribution From AIl Factors. 7J&6 Current Grand Average Of Performance.. 17.88& ExpCC1edResnll At Optimum Condition.. 25.274

Table 8. Interacting Factor Pairs Base upon design analysis of TM, the theoretical estimate of expected results at optimum condition

SitBiiJt"..ft;;;tf¥t;;i;\'~(j;{D;d.'T;rs1;;;~5\.Coi;';;ii;:GrS[(",$»,,'(;ol'... Opt.' (25.281 decibel) can be calculated as below: 1 CW.WA 1. 8 6635 9 [1.1] SIN = -10 Log (MSD) = 25.281 or 2 CM.CSAW 2. 6 5654 4 [1,1] . 3 CM.PLPW 2. 9 55.1311 [1,1] MSD = 10 [- (SIN) /10] = 0.002964 4 TW.PLPW 5. 9 53.9512 [3,1] Where 5 CM.WA 2. 8 4898 10 [3,3) 6 CWxCSAW 1 .6 4839 7 [2,1] MSD = [(Yl)2+ (Y2)2+ ,.. + (Yn)2] / n 7 CWxCCSS 1. 10 '10.7211 [1,1] = Average (yi 8 CM.CCSS 2. 10 3927 8 [3,1] 9 CSxCCSS 7. 10 29.8913 [2,1] = [E (Y)f 10 CSAWxCCSS 6. ID 25.7312 [1,1] or 11 SxWA 7. 8 2109 15 [2,3] 12 CMxCS 2. 7 21.ot 5 [3,2J E(Y) = ~MSD 13 CMxTW 2. 5 2!J.937 [3,1] 14 CW x PLPW I . 9 19.93 8 [1,1] 15 TW.CS 5. 7 18.27 2 [3,2] Therefore, expected performance in QC unit (ratio 16 PLPW .CCSS 9. 10 1698 3 [I,IJ error in ampere) based on theoretical results of SIN 17 WA.CCSS 8. ID 12.47 2 [1,1] = 25.281 decibel at optimum condition is: 18 CW.TW I. 5 11.87 4 [2,3] 19 WAxPLPW 8. 9 10.45 1 [1,1] 20 TW.WA 5. 8 962 13 [1,3] E (Y) = 0.054 Ampere 21 CS x PLPW ,.7. 9 9.09 14 [3,1] 22 TWxCCSS 5. 10 9.06 15 [2,3] 23 CSAWxWA 6. 8 866 14 [3,1] As it was mentioned and shown before (figure 9), 24 CSAWxPLPW 6. 9 593 15 [1,1] the heavier tail of response distribution indicated 25 Cw.CS 1. 7 346 6 [2,2J 26 TWxCSAW 5. 6 2.54 3 [3,IJ that the RR method of analysis should be more 27 CSAWxCS 6. 7 171 1 [1,2] appropriate than ANOVA method of design I. 2 1.19 3 28 CW.CM [2,3] analysis used in TM [12]. For this, a program by using S-Plus Package was prepared to fit the OLS and EM method of RR to generated data obtained from design layout of TM. Table 10 shows a comparison between application of OLS and RR methods in estimating the coefficients of explanatory variables in the model.

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Table 10.Coefficients estimate from OLS and Robust Regression methods

Robust Estimators Ordinary Least Square Estimators T T Factor Value Std.Error Value Std.Error Value Pr(> ItI) Value Pr(> ItI) Intercept -0.0124826 0.108142 -.1154 0.909384 -.0521868 0.2052851 -.25422 0.80221 Xl 0.00009358 0.000300 0.3119 0.758680 0.0004632 0.0005615 0.82485 0.42025 X2 0.01519006 0.008741 1.7379 0.099309 0.0274856 0.0155621 1.76618 0.09432 I X3 0.00396956 0.005012 0.7921 0.438633 0.0016656 0.0089848 0.18538 0.85501 X4 -0.0020533 0.010254 -.2002 0.843532 -.0034842 0.0188370 -.18497 0.85532 X5 -0.0211473 0.009008 -.3477 0.030523 0.0171678 0.0155621 -.10318 0.28448 X6 -0.0064487 0.003349 1.9167 0.071292 0.0009183 0.0058819 0.15611 0.87768 X7 0.03045487 0.009093 3.3493 0.003571 0.0010189 0.0155621 '0.06547 0.94852 X8 0.01231599 0.008488 1.4509 0.164006 0.0137822 0.0155621 0.88563 0.38750

As it is observed from the above table, the effects in the literature that inferences obtained from using of X5 (CS), X6 (WA) and X7 (PLPW) are OLS method for heavy tailed normal response significant and these of others are not significant distribution [13] are not accurate enough to depend enough on the response variable in the RR model. on it. These results may be clearly observed from A comparison of table 7 and 10 shows that the residuals and their qq plots (Figure 10 and 11) obtained results of design analysis of TM and OLS obtained from OLS and RR models. method [IS, 16] are similar. However, it is proved

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Figure 11. A comparison of QQ-plots of OLS and RR Models under study

5. CONCLUSION procedures given better performance of CTs in confronting with such data cases .As ratio and Based upon findings in OLS or TM and RR phase errors are two of the most quality issues in methods used for analysis of Taguchi's robust CTs. ,it is suggested that these two quality engineering layout of design which were obtained characteristics should be considered and optimized under vast environmental conditions (network simultaneously in future research. frequency and temperature variations as specific noise factors), two hand-made CTs have manufactured were manufactured in laboratory to have a fair comparison between proposed optimum conditions of the procedures. Then, the mentioned Table 11. Ratio Error Comparison between above CTs in our market, namely Mag and Landa different CTs with precision of 0.5 and 1 respectively, were compared for different input current under Proposed Mag CT Landa Proposed confirmation experiment. The results of the Input with CT with CTby Current CT by TM TM and comparison study which were obtained under the O.5p LOp and OLS same conditions in laboratory is given in Table 11, ME below: 20 0.5 1.8 0.5 0.5 The comparison of ratio error' (Ampere) in current transformers under study shows that the proposed 50 0.5 5 CT by TM and ME which is a hand made, has 0.7 0.5 quite better performance than Landa CT with 80 18 precision one, and is quite competitive with respect 0.5 0.8 0.5 to Mag CT. From cost point of view, the customer price of proposed CT in our market can be less 90 0.5 23.3 0.9 0.6 than 1 (U.S.) dollar whereas the customer prices of Landa and MAG CTs in the market are more than 22 and 25 (U.s.) dollars, respectively .As it is observed from results of the confirmation experiment, the proposed CT by TM and ME

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