A robust multi-conductor transmission line model to simulate EM transients in underground cables

Transcription

1 A robust multi-coductor trasmissio lie model to simulate EM trasiets i udergroud cables H.M.J. De Silva, L.M. Wedepohl ad A.M. Gole Abstract--This paper itroduces a ew mesh-domai model to simulate time-domai trasiets i udergroud cables accurately cosiderig frequecy depedet effects. The model is formulated i mesh domai usig mesh voltages ad currets. The advatages of the mesh domai is that mesh currets ad voltages i co-axial cables are aturally decoupled at high frequecies, so that the mesh domai fuctios (characteristic admittace, propagatio fuctios etc.) behave smoothly compared with phase domai fuctios. The trasformatio matrices betwee mesh ad phase domai are real ad costat. The paper validates the proposed model by comparig the time domai simulatios with solutio obtaied usig iverse Laplace trasform method for simple liear termiatios. Keywords: trasmissio lie models, udergroud cables, direct phase domai, mesh domai. T I. INTRODUCTION IME domai trasmissio lie ad cable models for electromagetic trasiet simulatio are required to be accurate over a very wide frequecy rage from few Hertz to several tes of kilohertz []. The approach for time domai modellig begis with the frequecy domai represetatio. The the frequecy domai parameters (etries of the characteristic admittace ad propagatio matrices) are approximated usig ratioal fuctios (curve-fittig). The covetioal phase domai formulatio, which uses coductor voltages ad currets, ca be umerically challegig due to the superpositio of idepedet travelig waves arisig from the couplig of modes. This results i oscillatory behaviour i the frequecy respose of elemets of propagatio matrix, which makes delay extractio ad low order ratioal fuctio approximatio difficult [], [8]. This problem ca be sigificatly reduced by reformulatig the phase domai trasmissio lie expressios ito mesh domai expressios ivolvig mesh voltages ad currets. The mesh domai aalysis for cables is ot a ew idea. For example, the EMTDC theory book formulates mesh domai H.M.J. De Silva is with the Departmet of Electrical ad Computer Egieerig, Uiversity of Maitoba, CANADA ( of correspodig author: jeewath@ee.umaitoba.ca) A.M. Gole ad L.M. Wedepohl are with the Departmet of Electrical ad Computer Egieerig, Uiversity of Maitoba, CANADA ( s: gole@ee.umaitoba.ca, wedepohl@shaw.ca). J. Nordstrom is with the Maitoba HDC Research Cetre ( joh@hvdc.ca). Paper submitted to the Iteratioal Coferece o Power Systems Trasiets (IPST29) i Kyoto, Japa Jue 3-6, 29 equatios as a itermediate step to obtai parameters, but immediately coverts them to phase domai [5]. The resultig system is the simulated usig a modal domai or phase domai approach. However this paper shows that there are advatages i tryig to solve the problem directly i the mesh domai. The mesh equatios are aturally decoupled at high frequecies i a maer similar to modes. Also the characteristics of propagatio ad characteristic admittace fuctios are well behaved ad easy to fit usig ratioal fuctios. The resultig trasformatio matrix betwee phase ad mesh domais is frequecy idepedet. Oce mesh domai formulatio is made, kow methods ca be applied. This model exploits a feature i the ector Fittig algorithm [3], i which two or more fuctios ca be curve-fitted usig commo set of poles. The etries of each colum of the propagatio matrix are approximated usig ratioal fuctios with a commo set of poles; hece the umerical efficiecy of the model is improved. The modal order reductio techique [4] further improves the solutio by removig poles ad residues cotributig isigificatly to the accuracy of the approximated ratioal fuctio. Ulike the phase domai, the oscillatory behavior of mesh propagatio fuctio is less. However sometimes there is still a problem ad this problem ca be overcome by employig multiple delays (modal delays) i the fuctioal form of the fitted propagatio fuctio [2], [7]. The multiple delay approach otably improves the accuracy of the fitted propagatio fuctio, compared with the sigle commo delay assumptio (particularly i case of highly frequecy depedet cable systems). Fially time domai simulatios ivolvig example multicoductor udergroud cables are preseted i order to verify the validity of the proposed model. The time domai results of the proposed model are compared with the solutio obtaied via Numerical Iverse Laplace trasform techique. II. PHASE DOMAIN MODELLING This sectio briefly describes the basics of traditioal phase domai modellig ad the same equatios are applicable to mesh domai modellig. I the discussio to follow, the term lie refers to both the overhead lie ad the udergroud cable systems, as the treatmet developed is commo to both. For a -phase trasmissio lie havig legth l, the frequecy domai solutio of the travelig wave equatio ca be expressed by the well kow matrix-vector equatios at each ed-of the lie

2 give by [], I = Y - A( Y + I ) () k c k c m m I = Y - A( Y + I ) (2) m c m c k k I the above equatios, ad I are dimesioal voltage ad curret vectors ad subscripts k ad m deote sediged ad receivig-ed of the lie. Y ad Z are ( ) shut admittace ad series impedace matrices per uit legth respectively. The ( ) characteristic admittace matrix Yc ad the ( ) propagatio matrix A are calculated as below usig matrix fuctios []: ad armour, ad armour ad groud. Similarly the currets are selected from coductor to coductor, istead of coductor to groud. Later explaied i this paper, there are advatages i formulatig trasmissio lie equatios i mesh domai usig mesh voltages ad currets tha i covetioal phase domai. Yc = ( YZ ) - Y (3) A = e - YZl (4) I order to implemet the model i the time domai, the elemets of Yc ad A are approximated with ratioal fuctios of suitable orders M ad N [] i the form show below i (5) ad (6). Such forms ca easily be coverted ito differetial equatios which ca be umerically itegrated. Figure : Phase voltages ad currets i a coaxial cable system A i, j ( s) = N p= ce sτ p s a M q Yci, j( s) = + d q= s aq c p (5) (6) The ukow coefficiets, c p ad a p (p = : N) i equatios (5) ad a q, c q (q = : M) ad d i (6) are calculated usig a efficiet robust techique called ector Fittig [3]. Note that the time delay (τ) i equatio (5) is estimated before the fittig procedure. Sometimes for accurate curve fittig additioal terms are added to (5) with differet time delays (see equatio (7)). For most practical trasiet simulatio studies, it is sufficiet to cosider frequecies from zero Hz to MHz for the fittig procedure. III. MESH DOMAIN FORMULATION The phase domai approach for the previous sectio uses phase voltages ad currets, i.e. the voltages i equatios () & (2) are defied as voltages of each coductor with respect to groud ad similarly currets are defied as currets through each coductor with retur path through earth as show i figure. I this mesh domai approach, the currets ad voltages of a cable are defied i the mesh domai as show i figure 2. Istead of selectig ier coductor to groud, sheath to groud, armour to groud voltages, the ew set of voltages are chose as voltage betwee ier coductor ad sheath, sheath Figure 2: Mesh voltages ad currets i a coaxial cable system I. SERIES IMPEDANCE AND SHUNT ADMITTANCE MATRICES This sectio briefly describes the derivatio of impedace ad admittace matrices i mesh domai. The self ad mutual impedaces ad admittaces betwee the cable coductors ca be represeted usig the per uit legth equivalet circuit show i figure 3. Here, Z jj is the self impedace of j th loop. Ad Z ij is the mutual impedace betwee i th ad j th loops. Z = impedaces of iteral coductor + isulator betwee first sheath ad coductor + ier first sheath. Z mi = mutual impedace of i th sheath Z ii = sum of impedaces of outer (i-) th sheath + (i-) th

3 isulator + ier i th sheath. Z = sum of impedaces of outer (-) th sheath + the (- ) th isulator + self earth retur. Y ii = shut admittace betwee two adjacet coductor layers. fuctio approximatio of the propagatio fuctio. A low order fit ca be always foud ad this leads to a efficiet model. This is particularly true for the elemets of characteristic admittace matrix as well. A. Phase domai Z ad Y matrices The series impedace ad shut admittace matrices are traditioally formulated i the phase domai choosig the coductor currets -g, 2-g, 3-g...-g as the variables. Phase currets ad phase voltages (coductor to groud) have sigificatly mutual couplig withi the frequecy rage from Hz to MHz. This results i oscillatory behavior i propagatio ad characteristic admittace fuctios, whe plotted as a fuctio of frequecy. A higher order fit is usually required for the ratioal fuctio approximatio due to the oscillatory ature of elemets of propagatio matrix both i magitude ad phase. Figure 3: Equivalet circuits for uit legth admittace ad impedace This difficulty ca be sigificatly reduced by choosig differet formulatio for Z ad Y matrices. The ew curret meshes or meshes are selected as -2, 2-3, 3-4, ad (-)-. The Z ad Y matrices formulated i this ew mesh domai are, Z Zm. Zm Z22. Z =... Zmm ( ) Zmm ( ) Zmm Y Y. Y.... Y 22 = The mutual impedaces ted to zero as frequecy icreases. Thus the matrix formulated i the ew mesh domai becomes icreasigly diagoal at high frequecies. Sice Y is already a diagoal matrix, the currets (as well as voltages) defied i the ew mesh domai become decoupled at high frequecies. I other words the system supports almost idepedet travelig waves. This is a good approximatio to a pure modal domai system at very high frequecies. Both magitude ad phase of the frequecy respose of elemets of propagatio matrix (A) behave smoothly compared with that of direct phase domai formulatio (phase to groud). This is a sigificat advatage for ratioal (7) (8) Trasfer impedace ad trasfer admittace i mesh domai The electromagetic field peetratio through cable shields ca be explaied usig the trasfer impedace ad admittace. I traditioal phase domai approach, the trasfer impedace cosists of outer sheath iteral impedace, impedace due to time varyig flux i the outer isulatio, self impedace of the earth retur path ad sheath mutual impedace [9]. As the frequecy icreases, all the sub-impedace terms icrease except sheath mutual impedace. Hece, the trasfer impedace icreases with frequecy, resultig i a sigificat couplig i phase voltages ad currets. The phase domai trasfer admittace is the admittace of the isulator betwee two adjacet coductors (e.g. betwee the ier coductor ad the sheath) [9]. I the mesh domai approach, the trasfer Z (see impedace oly cotais sheath mutual impedace ( ) mi equatio (7)), which teds to zero at higher frequecies. The trasfer admittace i mesh domai is zero (see equatio (8)).. TRANSFORMATION MATRICES The goal of mesh domai model is to develop a time domai equivalet circuit compatible with emtp-type software. Sice fial quatities are the phase quatities, at the highest level the time domai model must be expressed i phase domai. It is ecessary to trasform voltages ad currets i phase domai to ew mesh domai ad vice versa. The voltage trasformatio matrix ca be derived as follows. g = ( ) 2g = ( ) (9)... = g ( )

4 Note that suffix g, phase, mesh deote groud, traditioal phase domai ad the ew proposed mesh domai respectively. = K phase mesh The voltage trasformatio matrix K.. =..... K is defied as, () () Similarly the curret trasformatio matrix ca be derived by cosiderig phase to mesh trasformatio for currets. The curret trasformatio matrix K is defied as, K I.. =..... I (2) Aother advatage of the mesh method is that curret ad voltage trasformatio matrices are real ad costat avoidig the difficulties with frequecy depedet trasformatio matrices. It ca be easily show that, T ( ) I (3) K = K I. RATIONAL FUNCTION APPROXIMATION OF YC AND A Oce mesh domai Z ad Y matrices are computed (as discussed i sectio I), mesh A ad mesh Yc are the calculated usig equatios (3) ad (4). Time domai models require ratioal fuctio approximatio (curve-fittig) of etries of the characteristic admittace (Yc) ad propagatio (A) matrices. A. Ratioal fuctio approximatio of Yc Although the discussio below is strategic to mesh domai formulatio, these methods have bee applied to phase domai approaches by previous researchers [2], [4]. For the curve-fittig of mesh Yc, the techique used to fit phase Yc i the Uiversal Lie Model [2] is employed. The elemets of mesh Yc matrix are approximated usig ratioal fuctios (i the form (6)) with a commo set of poles. The commo poles are idetified by curve-fittig the trace of the matrix [2]. B. Time delay estimatio for A The fuctioal form of the fitted propagatio fuctio (as show i equatio (7)) has a sigle delay for each etry. However the use of multiple delays (modal delays) for the fitted propagatio fuctio gives better accuracy [2]. The modal delays are the delays attributed to the modes of propagatio matrix (A). These modes are computed i the stadard maer usig a frequecy depedet trasformatio matrix as for the case of the phase domai Uiversal Lie Model [2]. The modal delays are estimated usig Bode s gaiphase formula from modes of mesh A matrix [2]. The ratioal form of the propagatio fuctio with multiple delays (τ i ) is, c c A ( s) e + e +... (4) i, j N N2 sτ sτ 2 2 = s a 2= s a 2 Here, τ i is the delay correspodig to the i th mode. The ukow residues ad poles c i ' s ad a i ' s are obtaied usig a least squares method. C. Idetificatio of poles ad residues i equatio (4) This model exploits a feature i the ector Fittig algorithm, i which two or more fuctios ca be simultaeously curve-fitted usig a commo set of poles [3]. Each colum of the mesh propagatio matrix is approximated with a commo set of poles usig the modified ector fittig algorithm (i.e. the ukow residues ad poles i (4) are calculated by curve-fittig each colum of the propagatio matrix ad the etries i a colum share same set of poles); hece the umerical efficiecy of the model is improved. The modal order reductio techique further improves the solutio by removig poles ad residues cotributig isigificatly to the accuracy of the approximated ratioal fuctio [4]. II. TIME DOMAIN EQUIALENT CIRCUIT The fial objective of mesh domai model is to develop a time domai equivalet circuit, which ca be implemeted i emtp-type software. The mesh domai trasmissio lie equatios are, I = Y - A ( Y + I ) (5) m, mesh c, mesh m, mesh mesh c, mesh k, mesh k, mesh I = Y - A ( Y + I ) (6) k, mesh c, mesh k, mesh mesh c, mesh m, mesh m, mesh Although core of the method is the mesh domai solutio, the ultimate real quatities are the phase quatities, at the highest level the time domai model must be expressed i phase domai. If equatio (8) is multiplied by curret

5 trasformatio matrix where, K I, KI I m, mesh = KY I c, meshm, mesh - KF I k (7) F = A ( Y + I ) (8) k m esh c,m esh k,m esh k,mesh Sice I, = K I, The equatio (7) becomes, m phase I m mesh Im, phase = KIYc, meshm, mesh - KI FK (9) The time domai form of (9) is, () = () () () i t K y t v t K f t m, phase I c,m esh m,m esh I k (2) where, the term " " deotes covolutio ad, ( ) () = () () () + ( ) f t a t y t v t i t mesh c,m esh k,m esh k,m esh (2) The lower case variables are the correspodig time domai form of the upper case variables. The above covolutios ca be efficietly evaluated by recursive covolutio [] ad the equatio (2) ow becomes, Figure 4: Time domai equivalet circuit III. APPLICATION EXAMPLE I order to explai the advatages of the proposed mesh domai method, a example udergroud cable system is cosidered. Figure 5 shows a three-cable system (each has a ier coductor ad sheath) with data show i Table I [22]. The typical plots of etries of the propagatio matrix formulated i traditioal phase domai are show i Figure 6, clearly idicatig the oscillatory behavior of phase domai elemets whe plotted as a fuctio of frequecy. ( ) ( ) im, phase t = yeq _ mvm, mesh t + ihist _ m() t (22) The term i _ () t is calculated usig past values of voltages hist m T ad currets. Sice yeq _ m = yeq _ m KI, the equatio (22) becomes, ( ) ( ) im, phase t = yeqvm, phase t + ihist _ m() t (23) Similarly for the other ed of the trasmissio lie (see equatio (6)), ( ) ( ) i t = y v t + i () t (24) k, phase eq k, phase hist_ k Equatios (23) ad (24) are the mathematical represetatio of a time domai equivalet circuit, which ca be realized i popular electromagetic trasiet programs as two curret sources i parallel with two coductaces as show i figure 4 for sigle coductor case. Figure 5: Three cable system TABLE I CABLE DATA RESISTIITY OF THE INNER CONDUCTOR ( Ω m).68e-8 RESISTIITY OF THE OUTER CONDUCTOR ( Ω m) 2.2E-7 RELATIE PERMITTIITY OF INNER AND OUTER INSULATORS 4./3.2 EARTH RESISTIITY ( Ω m) LENGTH OF THE CABLE SYSTEM (KM) 3

6 Magitude [p.u.] Frequecy [Hz] Magitude [p.u.] Figure 6: First colum of the phase propagatio fuctio If the propagatio matrix is formulated i mesh domai, the frequecy domai respose of elemets of mesh propagatio matrix (see Figure 8) show smooth behaviour i cotrast to respose of the phase domai fuctios. The fitted first colum of mesh domai propagatio matrix is show i Figure 8 as a fuctio of frequecy. Also show i the graphs are the theoretical (actual) curves (the solid lies i the same figure) idicatig that the fittig is accurate. I compariso with phase domai elemets, mesh domai elemets are smooth ad hece easy to curve-fit usig low order ratioal fuctios. Table II compares the order of ratioal fuctio for mesh domai with phase domai formulatio. For the propagatio fuctio, order of the trasfer fuctio is reduced oticeably by 3%, if mesh formulatio is used ad hece leads to a umerically efficiet model Time domai simulatios are coducted to validate the proposed mesh domai model. A short circuit test is coducted for the same cable system. The ier coductor of cable is eergized with step, while all other coductors are coected to the groud. Figure 9 shows the sedig-ed curret waveform (solid coductor) through ier coductor of cable ad the theoretical solutio (dotted lie) obtaied via umerical iverse Laplace trasform method. The curret waveform from mesh domai model is i a close agreemet with theoretical solutio, idicatig that simulatio results from mesh domai model are accurate..5 Frequecy [Hz] Figure 8: Actual ad fitted mesh propagatio fuctios FD solutio Simulatio output Magitude [p.u.] Frequecy [Hz] Sedig-ed Curret [A] Figure 7: First colum of the mesh propagatio fuctio Time [s] TABLE II ORDER COMPARISON (PER ELEMENT) ORDER OF PHASE YC ORDER OF MESH YC ORDER OF PHASE A 43 ORDER OF MESH A 33 Figure 9: Sedig-ed curret Aother simulatio is coducted for the same cable system. A step is applied to the ier coductor of cable ad all other termiatios are kept ope. Figure shows the iduced voltage (solid lie) at the sedig-ed of the ier coductor of cable 2. Also show i the graph (dotted lie) is the theoretical solutio. This cofirms that the iduced voltage is i a close agreemet with theoretical results.

7 Receivig-ed oltage [] Time [s] IX. CONCLUSION FD solutio Simulatio output Figure : Iduced voltage i the secod coductor This paper proposed a ew mesh-domai method to simulate time-domai trasiets i udergroud cables accurately cosiderig frequecy depedet effects. Oe advatage of this approach is that the resultig trasformatio matrix betwee phase ad mesh domais is frequecy idepedet. The mesh currets ad voltages i co-axial cables are aturally decoupled at high frequecies hece the propagatio fuctio shows relatively smooth behaviour. The ratioal fuctio approximatio is relatively easier tha i basic phase domai methods. [2] Ake Bjorck, Numerical Methods for Least Squares Problems, ISBN , SIAM publicatio. [3] Charles L. Lawso, Richard J. Haso, Solvig least squares problems, ISBN , Society for Idustrial Mathematics (Jauary, 987). [4] S.A Schelkuoff, The electromagetic theory of coaxial trasmissio lies ad cylidrical sheaths, the bell system techical joural volume XIII 934. [5] J.R Carso, Wave propagatio i overhead wires with groud retur Bell Syst. Tech, ol 5,pp ,926 [6] B. Gustavse, Adam Semlye, Simulatio of trasmissio lie trasiets usig ector Fittig ad modal decompositio, IEEE tras., Power Delivery, ol 3,No 2,April 998. [7] Douavis, A.; Xi Li; Nakhla, M.S.; Achar, R Passive closed-form trasmissio-lie model for geeral-purpose circuit simulators, IEEE Trasactios o Microwave Theory ad Techiques olume 47, Issue 2, Dec. 999 Page(s): [8] Coelho, C.P.; Phillips, J.; Silveira, L.M A covex programmig approach for geeratig guarateed passive approximatios to tabulated frequecy-data, IEEE Trasactios o Computer-Aided Desig of Itegrated Circuits ad Systems, olume 23, Issue 2, Feb. 24 Page(s): [9] Gustavse, B Computer Code for Passivity Eforcemet of Ratioal Macromodels by Residue Perturbatio, IEEE Trasactios o Advaced Packagig, olume 3, Issue 2, May 27 Page(s): [2] Chiea, A.; Grivet-Talocia, S A Passivity Eforcemet Scheme for Delay-Based Trasmissio Lie Macromodels, Microwave ad Wireless Compoets Letters, IEEE olume 7, Issue 8, Aug. 27 Page(s): [2] Grivet-Talocia, S Passivity eforcemet via perturbatio of Hamiltoia matrices, IEEE Trasactios o Circuits ad Systems olume 5, Issue 9, Sept. 24 Page(s): [22] Library example from PSCAD/EMTDC X. REFERENCES [] H.. Nguye, H.W. Dommel, J.R. Marti, Direct Phase Domai Modellig of Frequecy Depedet Overhead Trasmissio lies, IEEE Trasactios o Power Delivery, ol. 2, No 3, July 997. [2] Atef Morched, Bjor Gustavse, Maoocher tartibi, A Uiversal Model for Accurate Calculatio of Electromagetic Trasiets o Overhead Lies ad Udergroud cables, IEEE Trasactios o Power Delivery, ol 4, No 3, July 999. [3] Bjor Gustavse, Adam Semlye, Ratioal Approximatio of Frequecy Domai Resposes by ector Fittig, IEEE Trasactios o Power Delivery, ol 4, No 3, July 999. [4] Bjor Gustavse.; Joh Nordstrom, Pole Idetificatio for the Uiversal Lie Model Based o Trace Fittig, IEEE Trasactios o Power Delivery, olume 23, Issue, Ja. 28 Page(s): [5] H. W. Dommel, EMTP Theory Book, B.P.A., 986. [6] J.R Marti, Accurate modelig of frequecy depedet trasmissio lie i electromagetic trasiet simulatios IEEE Trasactios o Power Apparatus ad Systems, PAS -, No, Jauary 982 [7] B. Gustavse, Adam Semlye Combied phase ad modal domai calculatio of trasmissio lie trasiets based o ector Fittig, IEEE tras., Power Delivery, ol 3,No 2,April 998. [8] L. Marti, Simulatio of trasiets i udergroud cables with frequecy depedet modal trasformatio matrices, IEEE trasactios o Power delivery, ol 3, No 3, April 998. [9] L.M Wedepohl, D.J. Wilcox, Trasiet Aalysis of Udergroud Power Trasmissio Systems,Proc. IEE, ol.2, No 2, February 973. [] L.M Wedepohl, Applicatio of Matrix Methods to the Solutio of Travelig Wave Pheomea i Polyphase Systems, Proc IEE 963,, pp [] L.M Wedepohl, H. Nguye, G.D Irwi, Frequecy Depedet Trasformatio Matrices for U-trasposed Trasmissio lies usig Newto- Raphso method, IEEE Trasactios o Power Systems, ol, No 3, August 996.