The FDTD method for lightning surge propagation in 115-kV power transmission systems of PEA s Thailand

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1 The FDTD metho for lightning surge propagation in 5-kV power transmission systems of PEA s Thailan * Kokiat Aosup ) an Thanatchai Kulworawanichpong ) ), ) Power System Research Unit, School of Electrical Engineering Institute of Engineering, Suranaree Uniersity of Technology, Nakhon Ratchasima, THAILAND ) Kokiat_a@hotmail.co.th an ) Thanatchai@gmail.com ABSTRACT This paper escribes a simulation of lightning surge propagation in power transmission lines of proincial electricity authority (PEA) Thailan by using finiteifference time-omain (FDTD) metho. Numerical computation of soling the Telegraphist s equations is etermine an inestigate. A source of lightning surge wae on the line is moele by using Heiler s surge moel. The propose metho was teste against 5-kV power transmission systems in comparison with the solution obtaine by using Bewley lattice iagram. As a result, the calculation showe that the FDTD metho is one of accurate methos to analyze transient lightning surge wae in power transmission lines.. INTRODUCTION Lightning an switching surge are any isturbance on a transmission of steay-state conition. These phenomena can prouce high oltage leel in a ery short time that can amage insulation or can cause serer flashoer escribe in (Bewley 95), (Hileman 990). Lightning or surge protection of electrical an electronic systems from isturbances has been increasingly important. Because the inuce lightning surges can cause significant amages to electric power components, telecommunication equipment, computer networks, etc. These result in seere amages of equipment, interruption of serices, increase operation an maintenance cost. For insulation esign of the power transmission system, it is ital to exhibit the inuce oltage behaiors propagate along the transmission lines. Consequently, both theoretical an experimental stuies of lightning inuce electromagnetic fiels hae been conucte continuously (Kokiat 0). In theory, characteristics of lightning surge propagation in transmission lines can be escribe mathematically in forms of partial ifferential equations (PDEs) as the wellknown Telegraphist s equations (Benesoa 006). These equations are linear seconorer partial ifferential equations with constant coefficients. These equations fall into three basic categories: parabolic, elliptic an hyperbolic. These equations are ) Grauate Stuent ) Professor 90

2 hyperbolic. In case of lossless lines where series resistance of lines an shunt conuctance representing insulation losses are neglecte, the system equations can be simplifie into the wae equations which the lightning surge can propagate along the line without any line attenuation. Although the wae equations as hyperbolic PDEs hae an exact equations in some circumstances, further inestigation such as appearance of lightning surge arresters somewhere in transmission lines can raise complexity an nonlinearity in the goerning system equations. Solutions of these equations were obtaine seeral ecaes ago by Heaisie in Englan an Poincare in France (Bewley 95). The FDTD metho is basically a numerical tool an can be aapte in associating with surge arrester moels in the future work.. POWER TRANSMISSION LINE. Mathematical moel of a power transmission line The stuy of transmission line surges regarless of their origin is ery complex. Although the long-line moel is recommene for lines more than approximately 50 mi long (Granger 994), the lightning surge wae propagation is a ery short-time impulse wae-shape therefore the long-line moel is a goo representation of power lines for a high-frequency impulse of lightning surges. Fig. Distribute line moel for power transmission line wae propagation Fig. shows the frame an the equialent circuit of a ery small section of a singlephase power transmission line. Assuming that the line conuctors are parallel to the groun an uniformly istribute, the time-omain characteristics in form of partial ifferential equations of the single-conuctor line can be expresse as follows. 9

3 x x i t x, t Ri x, t L x, t () t x, t G x, t C ix, t () Where i(x,t) is a current surge wae function (x,t) is a oltage surge wae function R, L, C an G are per-unit length line parameters Consier the istance x along the transmission line from the sening en (rather than the receiing en) to the ery small ifferent element of length x shown in the aboe figure. The oltage (x,t) an current i(x,t) are both a function of space an time so that they are in form of partial eriaties. Since it assumes that the transmission line is a lossless line, R an G will be equal to zero to gie the following expressions. x x i t x, t L i x, t (3) t x, t C ix, t (4) Now either current i(x,t) can be eliminate by taking the partial eriaties of both terms in Eq.(3) with respect to x an in Eq. (4) with respect to t, or oltage (x,t) can be eliminate by taking the partial eriaties of both terms in Eq. (3) with respect to t an in Eq. (4) with respect to x. This will prouce a linear secon-orer partial ifferential equation in form of hyperbolic PDEs as shown in Eq. (5) for the oltage wae equation or calle traelling wae equation. LC x t x, t x, t (5). Heiler s Surge Function The moel base on the traelling-wae source was introuce by Heiler (Bewley 95). in which the surge wae propagates at infinitely large spee while the returnstroke spee (front spee) is still finite. The equation for surge function introuce by Heiler satisfies the two basic requirements neee for the lightning surge simulation, i.e. the current oes not hae iscontinuity at t = 0 s an the current eriatie also oes not hae a iscontinuity at t = 0 s proie that k >. At present time, Heiler representation of the lightning surge wae is one of the most wiely-use surge moel for the lightning surge propagation in transmission line. The Heiler s surge function can be escribe by the following equation. f F k t t 0 t e e (6) 9

4 3. FINITE-DIFFERENCE TIME-DOMAIN (FDTD) METHOD The stanar example of a hyperbolic PDE is the one-imensional wae equation as escribe follows. c,, x t x t x t (0) Initial conitions are gien for (x,0) an also its eriatie. The bounary conitions are gien at x = 0 an x = L where L is the maximum limit of x. Accoring to the explicit metho of soling the wae equation, replacing the space eriatie in the wae equation by finite ifference formula at the λ th time step, Eq. () is obtaine. In the same manner, replacing the time eriatie by the finite ifference formula at the λ th space step, Eq. () is forme. By substituting Eq. () an Eq. () into Eq. (0), it gies the upate oltage wae solution as summarize in Eq. (3). x x, t,,, x () t x, t,,, () t,,,,, (3) Where t t c x x LC is the aspect ratio 4. REFLECTION OF TRAVELING WAVES When a traelling wae on a transmission line reaches a transition point at which there is an abrupt change of line parameters a part of the wae is reflecte back on the incoming line an the rest may pass through other line section. The traelling wae before reaching the transition point is calle the incient wae. The incient wae may be ecompose into two component waes calle the reflecte wae an the transmitte wae. This relation is a oltage-wae solution of Eq. (5) an it can be expresse as in Eq. (7). The transmitte wae, (x,t), is a wae portion traelling towar the next line section while the reflecte wae, (x,t), is a wae portion traelling backwar to the source. These waes can be illustrate by the equialent circuit shown in Fig.. x, t x, t x t (7), 93

5 i i i Fig. Waes reflecte an transmitte at the junction If the line section # has the surge impeance of Z an the line section # has the surge impeance of Z, the transmitte an the reflecte portions of the traelling wae can be represente in terms of the refraction coefficient (β) an the reflection coefficient (α), respectiely, as gien in Eq. (8) an Eq. (9). Z Z Z (8) Z Z (9) Z Z Where L Z an C LC L C Z L C u is the wae spee of line u is the wae spee of line L is per-unit inuctance of line section L is per-unit inuctance of line section C is per-unit capacitance of line section C is per-unit capacitance of line section In practical power network, many line sections are typical. This leas to multiple reflections among line junctions to exhibit complicate resulting waes. Howeer, in a lateral line case, both the reflection an the refraction occur from the left to the right or from the right to the left, coefficients of reflection an refraction can be pre-calculate an then use repeately when any incient wae has arrie. The component waes calculate at any time an any position by using this pre-calculation of all coefficients at eery junction can be rawn as the so-calle Bewley lattice iagram (Bewley 95)as illustrate in Fig

6 u Fig. 3 Example of Bewley lattice iagram 4. SIMULATION RESULT AND DISCUSSION The stuy of successie reflection of traelling waes cause by either irect or inirect lightning stroke can be inestigate through a test in power transmission lines of proincial electricity authority (PEA) Thailan as shown in Fig. 4. This example consists of two transmission line sections with the open an short circuit line termination. The line parameters of each section are gien as follows. Fig. 4 The transmission line systems The parameter of transmission line systems is : Line Transmission line from substations of Uthaitani to a connection point. Line length: km, L = 0.00 H/m, C = F/m Line Transmissions line from a connection point to substations of Chainat. Line length: km, L = µh/m, C = 0.04 F/m The Heiler s surge moel of the lightning inuce oltage can be characterize by the waeform in Fig. 5. The Heiler s surge wae has the 0-kV peak an /30-µs of the rise an ecay time constants. 95

7 0 Lightning surge Voltage (kv) Time (us) Fig. 5 Heiler s surge wae for test the transmission line systems With the help from MATLAB programming, lightning surge waes propagation on transmission lines can be simulate numerically. In this paper, this simulation use the time step of 0. µs an the step length of 5 m. Assume that the lightning surge was inuce at the sening en of the transmission line. The incient wae can trael along the line section with a constant spee an without attenuation approach line junction A as shown in Fig. 6. After the incient wae hitting the junction, the incient wae of 0-kV peak was separate into the reflecte wae of -.57-kV peak an the transmitte wae of 8.44-kV peak. These two wae components can be epicte as shown in Fig. 7 0 Trael of lightning surge along the transmission line 5 0 Incient wae 0 kv Voltage (kv) Connection point Position (km) Fig. 6 Incient wae before arriing connection point. 96

8 0 5 Trael of lightning surge along the transmission line Transmitte wae towar 8.44 kv 0 Voltage (kv) Reflecte wae from connection point -.57 kv Connection point Position (km) Fig. 7 Transmitte an reflecte waes at connection point When transmitte wae on a transmission line reaches a line terminal at which there is open an short circuit. Transmitte wae components can be epicte as shown in Fig. 8 an Fig.9 for a line terminal is open an short circuit respectiely 0 Trael of lightning surge along the transmission line 5 0 Transmitte waes at a line terminal kv Voltage (kv) Connection point Position (km) Fig. 8 Transmitte waes at a line terminal is open circuit 97

9 0 Trael of lightning surge along the transmission line 5 0 Voltage (kv) Transmitte waes at a line terminal kv Connection point Position (km) Fig. 9 Transmitte waes at a line terminal is short circuit In aition, the full simulation of the whole system which consists of two line sections haing a total of 4-km line length an the total time span of 500 µs can be plotte in 3D surface for open an short circuit at terminal line as shown in Fig. 0 an Fig. respectiely. Fig. 0 Lightning surge propagation along the transmission lines of the test systems at open circuit line terminal after.5 ms 98

10 Fig. Lightning surge propagation along the transmission lines of the test systems at short circuit line terminal after.5 ms The FDTD metho is comparison with the solution obtaine by using Bewley lattice iagram as shown in Table. Table The FDTD metho comparison with the solution by using Bewley lattice iagram. Reflaction Refraction Bewley lattice iagram -.4 kv kv FDTD metho -.57 kv 8.44 kv 6. CONCLUSIONS In this paper, the finite-ifference time-omain (FDTD) metho to analyze lightning surge propagation in power transmission lines of proincial electricity authority (PEA) Thailan. Numerical computation of soling the Telegraphist s equations is etermine an inestigate. A source of lightning surge wae on the line is moele by using Heiler s surge moel. The propose metho was teste against 5-kV power transmission systems in comparison with the solution obtaine by using Bewley lattice iagram. As a result, the calculation showe that the effectieness an the accuracy of the solutions obtaine by the FDTD metho are confirme 99

11 REFERENCES Bewley Caappa, L. V. (95), Traelling Waes on Transmission Systems, Doer Publication. Hileman, A. R. (999), Insulation Coorination for Power Systems, Marcel Dekker. Granger, J. J. an Stephenson, W. D. (994), Power System Analysis, McGraw-Hill. Benesoa, Z. an Kotlan, V. (006), Propagation of surge waes on nonhomogeneous transmission lines inuce by lightning stroke, Aances in Electrical an Electronic Engineering, Vol. 5, no., Fausett, L. V. (999), Applie Numerical Analysis using MATLAB. Prentice-Hall. Aosup, K. an Kulworawanichpong, T. (0), Simulation of Lightning Surge Propagation in Transmission Lines Using the FDTD Metho, Worl Acaemy of Science, Engineering an Technology, Issue 7,