1 Introduction For a long time, China has installed secondary equipment such as relay protection equipment in the main control building of substations. Moving these secondary equipment to the switch yard of substations can not only enable local information processing of current and voltage signals, but also has important economic value. In the Three Gorges power transmission and transformation project, a number of secondary equipment of 500 kV substations will be moved to the switch yard. An important technical problem restricting the decentralization of secondary equipment is electromagnetic interference (EMI). Some developed countries, especially the Electric Power Research Institute (EPRI) in the United States, have conducted relatively in-depth research on this problem in the past two decades [1]. However, domestic research is still in its infancy. According to domestic and foreign experience, the most serious sources of interference for substation secondary equipment are: substation switch operation, short circuit faults, radiated electromagnetic fields, and the effects of lightning in nature. Among them, the EMI generated by cutting and closing short unloaded busbars with circuit breakers and disconnecting switches is the most serious. At this time, the electromagnetic wave process generated on the busbar is coupled to the secondary equipment not only through conduction coupling, but also through the radiation coupling of the spatial electromagnetic field. Therefore, in order to study the radiated electromagnetic field generated by the wave process of the unloaded busbar, the wave process on the busbar should be calculated first. The analysis and calculation methods for wave processes in transmission lines are nearly mature both domestically and internationally [2], but there are very few research reports on wave process calculation methods for unloaded buses in substations. This paper establishes a calculation model for the wave process of unloaded buses in substations using the multi-conductor transmission line (MTL) theory, discusses the finite-difference time-domain (FDTD) method for solving MTL, and conducts numerical analysis on the wave process of unloaded buses in substations without loss or load. The calculation results can be used for the study of EMI problems of secondary equipment in substation switchyards. [b]2 Basic Equations of Multi-Conductor Transmission Lines[/b] The uniform MTL model is shown in Figure 1 [3]. For non-uniform MTLs, they can generally be segmented and homogenized, and then processed according to the uniform MTL method [4] or processed according to the convolution-characteristic line method [5]. The MTL model can be described by time-domain telegraph equations as follows [4] [img=355,116]http://zszl.cepee.com/cepee_kjlw_pic/files/wx/zgdjgcxb/2000-6/40-1.jpg[/img] where the z-axis is the transmission direction; V and I are the voltage and current column vectors at time t at point z on the line, respectively; VF and IF are the excitation voltage source and current source column vectors at time t at point z, respectively; L, C, R, and G are the unit-length inductance, capacitance, resistance, and conductance matrices of the MTL, respectively. These distributed parameter matrices can be calculated using electromagnetic field methods based on the geometry and medium parameters of the MTL. [img=347,99]http://zszl.cepee.com/cepee_kjlw_pic/files/wx/zgdjgcxb/2000-6/40-2.jpg[/img] [b]3 FDTD Method for MTL[/b] FDTD is a widely used numerical calculation method for electromagnetic fields [6]. It requires little memory, can easily handle complex objects, and has a simple algorithm, making it particularly suitable for time-domain analysis of electromagnetic fields. Professor Paul first applied the FDTD method to calculate the branchless MTL and compared it with actual tests, obtaining good calculation results [7]. The MTL was discretized as shown in Figure 2, with the voltage along the line discretized into NDZ+1 points and the current along the line discretized into NDZ points. The time-domain telegraph equations (1) and (2) are approximately discretized by spatial-temporal difference as follows: [img=389,200]http://zszl.cepee.com/cepee_kjlw_pic/files/wx/zgdjgcxb/2000-6/40-3.jpg[/img] (4) Where Δz is the spatial step size along the line and Δt is the time step size; the subscript indicates the spatial location point, k in equation (3) takes integers from 1 to NDZ, and k in equation (4) takes integers from 1 to NDZ+1; the superscript is the time series and n is a natural number. When the superscript of voltage is zero and the superscript of current is 1/2, it indicates the initial value, and the initial boundary condition I0=0, INDZ+1=0. Obviously, equations (3) and (4) are a set of spatial-temporal difference equations, which can be solved by iterative method. The iterative process adopts the leapfrog method shown in Figure 3 [7], and calculates the current and voltage sequentially. To ensure the stability of the algorithm, Δt ≤ Δz/v is required. Where v is the maximum mode velocity of electromagnetic wave propagation in MTL, which can be obtained by MTL mode analysis [3]. [img=343,365]http://zszl.cepee.com/cepee_kjlw_pic/files/wx/zgdjgcxb/2000-6/40-4.jpg[/img] Figure 3 FDTD Iterative Process [b]4 FDTD Algorithm Verification[/b] In the analysis of transmission line wave processes, Bergeron's method is often used. Since this method equates the transmission line to a Norton circuit or Thevenin circuit and calculates according to circuit theory [2, 4], it can only solve the voltage and current wave processes at nodes. For a certain discrete point on the transmission line, an imaginary node should be set up, so it is difficult to solve the voltage and current wave processes at all discrete points on the transmission line. To verify the effectiveness of the FDTD method, the crosstalk problem of the three-conductor lossless transmission line shown in Figure 4 was calculated using both the FDTD method and the Bergeron method. Here, VS2 is a unit step voltage source, RS1, RS2, RL1, and RL2 are all 50Ω, the line length is 0.5m, and the initial condition is zero. The unit length inductance and capacitance matrix can be calculated as follows: [img=344,212]http://zszl.cepee.com/cepee_kjlw_pic/files/wx/zgdjgcxb/2000-6/40-5.jpg[/img] Figure 4 Crosstalk problem of two lines. Figure 5 shows the calculation results of the FDTD method and the Bergeron method, where curves A, B, C, and D are the voltage waveforms at the corresponding four points in Figure 4. As can be seen from Figure 5, the results obtained by the FDTD method and the Bergeron method are basically consistent. The FDTD method produces some glitches, but these do not lead to large errors. [img=378,147]http://zszl.cepee.com/cepee_kjlw_pic/files/wx/zgdjgcxb/2000-6/41-1.jpg[/img] Figure 5 Comparison of FDTD and Bergeron method calculation results 5 Calculation of wave process of unloaded busbar in substation Figure 6 shows a simplified model of a 500kV unloaded busbar in a substation, which considers only one set of busbars and leads, and the busbars are unloaded, ignoring the influence of lines and structures. L2 and L3 are busbars, L1 is a lead, L1 = 50 m, L2 = 90 m, L3 = 50 m, h = 16 m. The busbar spacing is 6.5 m, and the lead spacing is 8 m. The busbars use LGJQT-1400 type special lightweight steel-cored aluminum stranded wire. In the model, the proximity effect between buses and the edge effect at both ends are ignored, and the buses are assumed to be lossless. The ground is an infinitely large and perfectly conductive plane. The unit length inductance and capacitance matrices of the buses are respectively [img=388,339]http://zszl.cepee.com/cepee_kjlw_pic/files/wx/zgdjgcxb/2000-6/41-2.jpg[/img] Obviously, unlike Figure 1, the MTL in Figure 6 has a branching structure. Therefore, the three MTL segments are discretized separately, making branch point 2 a voltage discrete point. Using the voltage and current boundary conditions of branch point 2, the FDTD method iterative formula for the following two points can be derived. [img=354,32]http://zszl.cepee.com/cepee_kjlw_pic/files/wx/zgdjgcxb/2000-6/41-3.jpg[/img] [img=390,278]http://zszl.cepee.com/cepee_kjlw_pic/files/wx/zgdjgcxb/2000-6/41-4.jpg[/img] Figure 6 Simplified model of substation busbar. The equations (5) and (6) of branch point 2 and the iterative equations (3) and (4) of each MTL segment are combined, and the wave process numerical analysis of the MTL with branch structure can be carried out using the initial conditions and boundary conditions. Assume that phases A, B, and C are unit cosine symmetrical three-phase voltage sources with zero internal resistance, and the initial state of the unloaded busbar is zero. Power is applied to the unloaded busbar at time t = 0. The FDTD method is used to calculate the voltage and current waveforms at discrete points on the line. Figures 7, 8, and 9 show the voltage waveforms at points 1, 2, 3, and 4 on the phases A, B, and C busbars and leads, respectively. Figure 10 shows the current waveforms at four points on the phase A busbar and leads. [img=360,434]http://zszl.cepee.com/cepee_kjlw_pic/files/wx/zgdjgcxb/2000-6/41-5.jpg[/img] Figure 9: Voltage waveforms at four points on phase C. As can be seen from the figure, the voltage and current waveforms are extremely complex due to the effects of multiple reflections and transmissions. The voltage waveforms of phases B and C are almost identical within the studied time period. Because the inter-line coupling effects are not significantly different at this point, the voltage sources of phases B and C are essentially the same within the short time period studied. In Figure 7, since the peak voltage of phase A is 1 V, the voltage at point 1 remains constant within the short time period when its internal resistance is 0. When charging the bus, as time changes, each point on L1 is successively charged with a voltage of 1 V according to the speed of light. When it reaches branch point 2, due to impedance mismatch, wave reflection and transmission occur, with a reflection coefficient of -0.33. The transmission coefficients to L2 and L3 are both 0.66. At this time, the voltages of phases B and C also propagate to branch point 2, which also affects phase A, but the effect is small. Thus, the voltages on L2 and L3 become 0.66 V, and the time is 0.1668 μs. Then, the voltage propagates on L2 and L3 with a voltage of 0.66 V, while the reflected wave voltage on L1 is -0.33 V. At 0.3337 μs, the voltage wave on L3 reaches terminal 3 and is reflected, with a reflection coefficient of 1, so the voltage value becomes twice its original value. Simultaneously, the wave on L1 reaches point 1; due to the voltage source limitation, its voltage should be 1 V, so the reflection coefficient is -1. Then, at 0.467 μs, terminal 4 of L2 is reflected, with a reflection coefficient of 1, and the voltage value becomes twice its original value. At 0.50 μs, the waves on L1 and L2 reach point 2 again; due to reflection and transmission, the voltage changes, becoming twice its original value. This cycle continues. Within the studied time period, the maximum voltage on the line can reach 2.9 times the original voltage source amplitude, i.e., 2.9 V, and the duration is extremely short, approximately 0.066 μs. [img=359,116]http://zszl.cepee.com/cepee_kjlw_pic/files/wx/zgdjgcxb/2000-6/42-1.jpg[/img] Figure 10 Current waveforms at four points in phase A (b, c, and d represent points 2 on the L1, L2, and L3 sides, respectively). It should be noted that the above calculation results are obtained under the assumption that the MTL is lossless and the ground is an ideal conductor surface. Although the results cannot represent the actual situation, the wave process should be similar to the actual situation. By performing a fast Fourier transform on the voltage wave process, the voltage and current wave process spectra at each discrete point on the line can be further obtained. Figure 11 shows the spectrum of line 4 in Figure 7. It can be seen that at the moment the 500kV unloaded busbar of the substation is energized, the electromagnetic interference spectrum is mainly concentrated in the frequency range of 10MHz, which is consistent with the actual measurement results on site [8]. [img=333,152]http://zszl.cepee.com/cepee_kjlw_pic/files/wx/zgdjgcxb/2000-6/42-2.jpg[/img] Figure 11 Voltage spectrum of phase point 4 [b]6 Conclusion[/b] It is feasible to model the unloaded busbar of the substation using a multi-conductor transmission line. The time-domain finite difference formula of the multi-conductor transmission line with branch structure proposed in this paper can be used to perform effective numerical calculations on the wave process of the unloaded busbar of the substation. The calculation results can be further applied to the study of electromagnetic interference problems after the secondary equipment of the substation is decentralized. [b]References:[/b] ［1］ EPRITR102006: Electro magnetic transients in substations［R］. Vol I, II, III, Palo Alto, CA, 1993. ［2］ Wu Weihan, Zhang Fangliu, et al. Numerical calculation of overvoltage in power system［M］. Beijing: Science Press, 1989. [3] Tesche FM, Ianoz MV, Karlsson T. EMC analysis methods and co-mputation models［M］． New York: John Wiley & Sons Press, 1996. ［4］ Paul CR. Analysis of multicondutor transmission lines［M］． New York: John Wiley & Sons Press, 1994. [5] Mao Jun-ma, Li Zheng-Fan． Analysis of the time response of MTL with frequency-dependent losses by the method of convolution- characteristics［J］． IEEEon MTT, 1992, 40(4): 637～644. [6] Kunz KS, Luebbers RJ. The Finite difference time do mainmethod in electro magnetics［M］． Boca Raton, FL: CRC Press, 1993. [7] Paul CR. Finite-Difference Time-Domain analysis of lossy transmission lines［J］． IEEE on EMC, 1996, 38(1): 15-23. [8] He Bin. Electromagnetic compatibility issues of secondary equipment in power systems [J]. China Electric Power,