Transmission Lines Modeling Based on Vector Fitting Algorithm and RLC Active/Passive Filter Design

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1 Tansmission Lines Modeling Based on Vecto Fitting Algoithm and RLC Active/Passive Filte Design Ahmed Qasim Tuki a,*, Nashien Fazilah Mailah b, Mohammad Lutfi Othman c, Ahmad H. Saby d Cente fo Advanced Powe And Enegy Reseach (CAPER) Depatment of Electical and Electonic Engineeing Faculty of Engineeing, Univesity Puta Malaysia Malaysia,43400 Sedang, Selango, Malaysia. a ahmed.q.t.91@gmail.com, b nashien@upm.edu.my, c lutfi@upm.edu.my, d ahmed_hsaby@yahoo.com Abstact In the modeling of the tansmission line, the electic enegy poduced at geneating stations is tanspoted ove highvoltage tansmission lines to utilization points, and the tend towad highe voltages is motivated by the inceased line capacity while educing line losses pe unit of powe tansmitted. An electic tansmission line is modeled using seies esistance, seies inductance, shunt capacitance, and shunt conductance. Fo some studies, it is possible to omit the shunt capacitance and conductance and thus simplify the equivalent cicuit consideably. Fequency Response Analysis (FRA) on the tansmission line application is utilized fo behavio pediction and fault diagnosis, but a need fo moe investigation is impotant fo the bases on which the diagnosis detemined. The utilities of the measued FRA data points need to be enhanced with suitable o developed modeling categoy to facilitate the modeling and analysis pocess. This eseach poposes a new method fo modeling the tansmission line based on a ational appoximation function which can be extacted though the Vecto Fitting (VF) method, which attempts on the fequency esponse measued data points. A set of steps needs to be implemented to achieve this by setting up the extacted patial faction appoximation, which esults fom a least squae RMS eo via VF, in such a way that would constuct as eal numbes, fist and second ode pats as well as the gain constant. Active and passive filte design cicuits ae attempting to constuct the model of that ational function of the tansmission line. RLC design epesentation has been implemented fo modeling physically the system with MATLAB, Simulink fo veifying the esults. Keywods - Tansmission lines modeling,vecto fitting method. I. INTRODUCTION The electical powe system mainly consists of thee standad divisions; Geneating Stations, Tansmission System and Distibution System. The tansmission line (TL) is the most impotant enegy coido in a powe system [1]. The pefomance of a powe system is mainly based on the pefomance of the tansmission lines in the system. In geneal, tansmission lines have thee paametes; esistance, inductance, and capacitance (RLC). All these paametes ae distibuted ove the length of the line. The examination of the tansmission lines can be divided into thee categoies: A shot tansmission line has a length not moe than 80 km. Medium tansmission line length is between 80 km and 50 km. A long tansmission line has a length beyond 50 km. The classification of tansmission lines depends upon the accuacy of the model and its simplicity. While the shot tansmission line model is quite simple, the longe line model is somewhat complex and the medium tansmission line is in-between the two. In the field of tansmission lines modeling, the autho pesented a technique to model tansmission lines using time domain synthesis appoach []. The paametes of each tansmission line ae detemined by using a constaint, nonlinea, least squaes optimization technique pefomed in the time domain. The poposed modeling technique has seveal applications in micowave measuements, mateial chaacteization and micowave device modeling. While in anothe wok, the authos pesented two highly accuate tansmission line models [3]. The fist model is paticulaly suitable fo ovehead lines and in the second model, is used fo popagation a modal decomposition with a constant tansfomation matix and an optional phase domain coection tem. Both models ae computationally highly efficient due to thei time domain ealizations based on vecto fitting. II. VECTOR FITTING METHOD Vecto Fitting (VF) method is an appoximation of a measued tansmission line fequency esponse (FR) that ecently adopted in this application which is based on complex ational function models. Fequency dependent tansmission line models have been used in the simulation of powe system since the 1970s [4][5]. The vecto fitting method used in tansmission line tansient mentioned in [6] poposed by the autho to be fast and obust method fo ational fitting of fequency domain esponses, well suited DOI /IJSSST.a ISSN: x online, pint

2 fo both scala and vecto tansfe functions (TF). Application of the new method esults in inceased computational efficiency fo tansmission line models using model decomposition with fequency dependent tansfomation matices. This is due to the fact that the method allows the fitted elements of each eigenvecto to shae the same set of poles, and that accuate fitting can be achieved with a elatively low numbe of poles. In [7] the time delay fo identification fo tansmission lines modeling was studied. It shows that to simply use the time delay of lossless popagation can lead to significant loss of accuacy of the ational appoximation. A pocedue is shown which optimizes the time delay togethe with the poles and esidues of the ational appoximation. This is achieved by combining Bent s method with Vecto Fitting. In the above liteatue, the elated studies eithe addessed the modeling of tansmission line in diffeent categoies o they used Vecto Fitting Algoithm (VFA) with powe tansfome only. In this wok, the fundamental method fo tansmission lines modeling fom measued data points based on RLC modeling of the Rational Appoximation Function (RAF) obtained fom the Vecto Fitting method was poposed. The RLC modeling in this eseach uses the concept of the passive and active filte epesentation of the patial factions of the system tansfe function. III. PASSIVE FILTER DESIGN Low pass, high pass, band pass and band eject (notch) filtes ae the basic filte types which ae shown in Fig. 1 [8]. A passive filte is consists of esistos, inductos, and capacitos. The numbe of capacitos and inductos is equal to the highest powe of fequency, and the ode depends on the ode of the filte, in the cases of Buttewoth and Chebyshev. In the Elliptic filte case, the numbe of capacitos indicates the ode of the filte. The lumped elements values fo each filte type fo the nomalized fequency have been computed and tabulated [9], 1ad as c and souce and load impedances ae Z in =Z Load Sec (Ω). In this wok, only the passive filte cicuits ae used and descibed in some details. A. Voltage Divide Cicuit The voltage divide cicuit consists of two seies esistos to simulate the attenuation of the input signal is shown in Fig.. o R i R1 R B. Low Pass Filte Cicuit Low Pass Filte cicuit (1 st ode tansfe function) is shown in Fig o RC i 1 S RC C. RLC nd ode Low Pass Filte Cicuit Low pass filte cicuit ( nd ode tansfe function) is shown in Fig o 1 Sc LC vi 1 S LC 1 SRC R 1 SL R S S Rc L LC D. Band Pass Filte Cicuit Band Pass Filte cicuit ( nd ode tansfe function) is shown in Fig.5. R S o R SRC L (4) vi 1 S LC 1 RSC R 1 SL R S S SC L LC (1) () (3) Figue 1. The effect of some common filte types on swept fequency input signals [8] Figue. Voltage Divide cicuit DOI /IJSSST.a ISSN: x online, pint

3 Eq. (5) consists of thee pats as follows:- Real numbes, fist ode pats S p Second ode pats esults fom the poduct of two as b complex conjugate poles pats S csd Figue 3. Low Pass Filte cicuit Constants K. Secondly, active and passive filte design cicuits ae conducted in ode to constuct the model that epesents the tansfe function of the tansmission line by:- Implementing the equied calculations to find out the values of the cicuits components. Using MATLAB Simulink to veify the esults. Figue 4. RLC nd ode Low Pass Filte cicuit Figue 5. Band Pass Filte cicuit As mentioned above, this eseach addessed some filtes cicuit that is used and satisfied the modeling veification. IV. METHODOLOGY The algoithm of the vecto fitting is descibed in diffeent aspects in [10][11]. In this eseach, MATLAB envionment has been used to implements the pocedue because MATLAB has suitable commands to obtain the paametes and the ational appoximation function which is the key equations to acquie. Afte the tansmission line ational appoximation function is obtained which is based on the fequency esponse analysis data points by utilizing Vecto Fitting method, a set of steps need to implement such as follows. Fistly, setting up the extacted patial faction appoximation, which esults have less RMS eo due to VF method and it is expessed as in eq. (5) H S 1 n ( )... S p1 S p S pn K (5) Based on the concept that mentioned in [1], the flowchat which descibes the algoithm of the eseach is shown in Fig. 6, whee the measued data points ae pesented as an input to the pogam. These data ae changed to Catesian fom to geneate a ow of complex matix fo n dimension, and the pocess is descibed clealy in the flowchat. Refeing to the flowchat of Fig. 6, the esultant ational appoximation function (RAF) needs to be pocessed by some numeical filtes to ignoe the vey low o the ineffective imaginay pat fom each element with espect to its combined eal pat. As an example, if one complex numbe has a value like pn= a*106+b*10-7j, then pn= a*106, then, eaanging eq. 1, all the ational patial pats of the system function again to sum each two pai that has complex conjugate numbes togethe to get second ode patial function elements without imaginay pat fo all system RAF, so, as an example, the eq. 5. could be changed to the following fom. H S as b 1 n ( )... S p1 S p S CS d S pn as b whee S CS d 3 4 S p S p 3 4 epesents the esult of summing and 3 and 4 has a(eal+j imaginay) K (6) Theefoe, the esultant RAF, of eq. 6, was not a complex numbes and can be passed to the next step which is the RLC modeling fo H(s) of eq. 6 by compaing each patial pat with its equivalent cicuits (Fig. 3, Fig. 4. and Fig. 5). DOI /IJSSST.a ISSN: x online, pint

4 Stat epesents the TL behavio and has been selected to be modeled with this poposed method such as shown in Fig. 8 togethe with the cuve fitting of the modeling pocedue. Input Fequency Response (FR) Measuement Paamete Setting in MATLAB S: (fequency-domain *pi*f) N:(Ode of appoximation) Poles:( complex stating poles) Weight: Ones(1. Ns); % No weighting Opts:( some mode paametes) Lte:( numbe of iteation) Figue 7. The adopted TL Model as a case study Apply VFA No Is the RMS eo < Yes Display of ( Resulting state space model : ) and the Rational Appoximation Function RAF (matix <1x4 tf >) Figue 8. Plotting the measued data with poposed VFA vs fequency Figue 6. Changing RAF to TF fom Is Imaginay pat small with espect to all eal RAF elements. Summing each pai of RAF matix that have a complex conjugate poles Disp (complex patial faction TFs ); ( nd ode Disp (Real patial faction TFs );(1 st ode) Modeling with RLC cicuit and bode plot veification. Output dawing and esults End Flowchat of the poposed VF Algoithm V. APPLICATION AND RESULTS In ode to validate the poposed method, a model mentioned in [13] as shown in Fig. 7 has been adopted to analyze the fequency esponse analysis (FRA). The modeling of the tansmission line is opted as a data to Refeing to eq. 5 and the poposed algoithm of [1], the esultant Rational Appoximation Function paametes can be listed as follows: = i i i i P = i i i i K = 4 So the patial faction expansion fo the above tansfe functions ae as follows:- 1 TF1= S TF= S TF3= ( i) S ( i) TF4= ( i) S ( i) DOI /IJSSST.a ISSN: x online, pint

5 Afte all tansfe functions ae obtained, the equation is eaanged to eq (5): ( i) ( i) HS () 4 S4.809 S1.309 S( i) S( i) Based on eq. (5), each of two complex conjugate poles must be added to each othe such as follows to obtain equation such as in eq. (6):- TABLE 1. ELEMENTS OF EQUATION 6 nd ode patial factions 1 st ode patial factions S S S 1.49 S S S H ( S) 4 S S S S 1.49 This pocedue is based on the RLC passive filte design. The block diagams descibed in Fig. 9 epesent the pocedue. Input signal Tansmission lines Output signal (sine wave of vaiable fequency) Figue 10. Tansmission line modeling in tems of tansfe function Fom Fig. 10, the tansfe functions ae conveted to RLC passive filte cicuit using eq., 3 and 4. The new tansmission lines modeling is shown in Fig. 11. Magnitude Phase Figue 9. Fequency Response (FR) Measuement As descibed in the flowchat of Fig. 6, a sequence of filteing and complex conjugate pai summation has been implemented to avoid the existence of the imaginay pat in all patial factions of the RAF and theefoe, the elements of eq. 6 ae listed in Table 1. Table 1 has the final tansfe functions of 1 st and nd ode goup that would be tanslated to thei espective RLC equivalent cicuit. Hence, accoding to Eq. 6, all these patial faction functions has the same input and accumulated output as epesented in Fig. 10. Figue 11. RLC passive cicuit modeling in MATLAB simulation fo tested tansmission line VI. DISCUSSION AND CONCLUSION In this eseach, two contovesial issues ae investigated and concluded. Fistly, it is essential fo any electomagnetic system, such as the tansmission line to be modeled in tems of the basic electical elements i.e. RLC, DOI /IJSSST.a ISSN: x online, pint

6 o a combination of mathematical modeling equation togethe with some RLC modeling cicuits. This poposed method shows a common esponse to electomagnetic inteaction between the tansmission line elements and that fo geometical changes fo the tansmission line active pat, diffeent TF s types of the extacted RAF have diffeent sensitivity towads vaious fequency esponse behaviou of the tansmission line. This method povides an uncomplicated way to divide not only the fequency ange, but the ovelap and effects of each fist o second ode patial faction of the attained RAF onto FRA chat to exploe moe facilities fo diagnosis pocess to be estimated fom the end-to-end tansfe function. The vecto fitting algoithm has been utilized hee to povide a suitable solution to find analytical expessions fo measued tansfe functions. The necessay accuacy of the appoximation can be contolled by the use accoding to the needs of the application. The main advantage of a mathematical explanation of RAF of the numeical data is that subsequent algoithms and assessment ules can be implemented fo moe analysis. This pape suggests deiving an electical equivalent model fom measued fequency esponse data points. The model is subdivided into two goups of patial factions 1 st and nd ode tansfe functions. Fo the 1 st ode, two pats of cicuits ae needed which ae voltage divide and RC low pass filte, equivalent esonance cicuits with physical meaning can be identified fom measued data points. Fo the nd ode tansfe functions goup, thee pats of cicuits need to be used and an electically equivalent cicuit can be identified using the analytical expession gained by RAF of vecto fitting. The poposed model can be used to ceate pecise electical epesentation fo tansmission line and help to know the ange that the FRA may succeed to be utilized fo physically modeling. [Online]. Available: [Accessed: 5-Feb- 016]. [9] H. M. and D. A. H. Matzne, S. Levy, RF Laboatoy Manual - Passive Filte Design, 008. [Online]. Available: _filte_design109.pdf. [Accessed: 5-Feb-016]. [10] Adam SEMLYEN ANDBjøn Gustavsen, Vecto_Fitting_By_Pole_Relocation.Pdf [11] B. Gustavsen and A. Semlyen, Rational appoximation of fequency domain esponses by vecto fitting, IEEE Tansactions on Powe Delivey, vol. 14, no. 3. pp , [1] Ahmad H. Saby, D Wan Zuha Wan Hasan, Pof. D. Mohd Zainal, D. Mohd. Aman, D. Suhaidi B. Shafie, 016. Mathematical Modeling of Powe Consumption Rate Pofile based on developing a new algoithm deived fom Vecto Fitting method. accepted, not published yet. [13] Compute RLC paametes of ovehead tansmission line fom its conducto chaacteistics and towe geomety - MATLAB powe_linepaam. [Online]. Available: inepaam.html. [Accessed: 0-Ma-016]. REFERENCES [1] M. R. Poumi and A. Akbakehkhajavan, Simulation Of Exta High Voltage Long Tansmission Lines, Int. Res. J. Appl. Basic Sci., vol. 7, no. 7, pp , 013. [] K. M. Fidanboylu, A Tansmission Line Modeling Technique Using Time Domain Synthesis, no. 1, pp. 5, 003. [3] B. G. M and A. Semlyen, Gustavsen" &I?), vol. 13, no., pp , [4] a. Semlyen and a. Dabuleanu, Fast and accuate switching tansient calculations on tansmission lines with gound etun using ecusive convolutions, IEEE Tans. Powe Appa. Syst., vol. 94, no., pp , [5] J. R. Mati, Accuate Modeling of Fequency-Dependent Tansmission Lines in Electomagnetic Tansient Simulations, IEEE Powe Eng. Rev., vol. PER-, no. 1, pp. 9 30, 198. [6] B. Gustavsen and A. Semlyen, Simulation of tansmission line tansients using vecto fitting and modal decomposition, IEEE Tans. Powe Deliv., vol. 13, no., pp , [7] B. Gustavsen, Time delay identification fo tansmission line modeling, IEEE Tans. Micow. Theoy Tech., vol. 58, no. 3, pp , 004. [8] E. Ramsden, An Intoduction to Analog Filtes Sensos, 001. DOI /IJSSST.a ISSN: x online, pint