Frequency Domain Analysis of Capacitor Transient Overvoltages

Transcription

1 Frequency Domain Analysis of Capacitor Transient Overvoltages PATRICIA ROMEIRO DA SILVA JOTA Electrical Engineering Department CEFET-MG Av. Amazonas 7675, Belo Horizonte, Minas Gerais BRAZIL Abstract: - Switching of capacitor banks is necessary and an integral part of a Power System, although it produces transients. These transients propagate through the distribution lines and they are magnified in shunt compensated loads. Magnifications of these surges were studied and it was found that if there are two L-C loops with same natural frequencies, the transient overvoltage produced by the switching of the capacitor in the first loop is magnified in the second one. The magnification depends on capacitance and frequency ratios. This paper analyses the transient response of two L-C loops system and shows how the presence of overhead lines can affect this magnification. It was showed too that the magnification could occur when the natural frequencies are close but not equal. The influence of the transmission lines was showed in the frequency domain. Key-Words:Power quality, Capacitor switching, Transient overvoltages magnification. CSCC'99 Proc.pp Introduction Capacitor switching is very effective in preventing voltage collapse and providing Var support in Power system, [1]. Switching of capacitor banks within utility substations may result in overvoltage, which can be as high as 2.0 pu at the capacitor location. This overvoltage can be magnified in the customer facility if the customer has low voltage capacitors for power correction. The magnification occurs when the frequency generated by the switching capacitor excite a series L-C circuit formed by the customer step down transformer and low voltage capacitor bank. This problem has been modeled as two L-C loops, and it was shown that the maximum overvoltage occurs when the natural frequency of each loop is near each other, [2,3]. In this case, the overvoltage in the customer capacitor bank can reach very high values. Fig. 1. Equivalent circuit This paper analyses how the transmission line in the first loop of the circuit can affect the maximum overvoltage in both capacitor banks due to substation capacitor bank switching. It analyses too the frequency response of the circuit and shows how the line length affects the frequency resonances of the 2 Description of the problem References [4,5] discuss the transient response of two L-C loops system. In both references, they showed how the frequency ratio, ω 2 /ω 1, and capacitance ratio, C 2 /C 1, affect the overvoltage. The maximum transient overvoltage occurs when the natural frequency of one loop is very close to the other and its value is strongly dependent on the C 1 /C 2 ratio [4,5] where C 1 is the switched capacitor and C 2 is the customer capacitor, as shown in Figure 1. They have used this simplified circuit to calculate the maximum overvoltage in the system. The effects of other circuit parameters were analyzed in [2,3]. In these papers, the equivalent inductance L 1, that represents all the impedances between the generator and the switched capacitor bank C 1, is not represented by a lumped inductance as used in [4,5]. Normally, there are transmission lines (between the source and the utility substation) as well as distribution lines (between the utility and the customer capacitor) and the overvoltage can be affected by them. Other analyzed factors were the existence of loads and other shunt capacitors. Figure 2 shows the depicted

2 customer capacitance occurs for ω 2 /ω 1 <1. The effect of the line is to reduce the maximum amplitude of the overvoltage and to shift the natural frequency of the first loop to greater values. It was showed that double line affects more than single lines. Fig. 2. Analyzed circuit Fig. 3. Equivalent circuits 3 The influence of Transmission and Distribution Lines The representation of the transmission line as a distributed line, contrary to a simple inductance, affects the overvoltage in the customer capacitor bank, [2,3]. In their analysis, the first L-C loop was composed by sub-transient reactances (of the generators), generator step-up transformers (represented by a lumped inductance, L s ), transmission lines (represented by their equivalent model (R T, L T and C T )) and a switched capacitor (C 1 ), as shown in Figure 3(a). The second L-C loop was composed by a distribution line and the substation and customer transformers (all represented by a lumped inductance, L 2 ) and the customer shunt compensating capacitor (C 2 ). To permit the comparison between the simulations, the total inductance L 1 =L s +L T became constant although the length of the line may vary, then as the L T increases (length of the line increases) the value of L s decreases. They has showed that, when a transmission line is represented by a distributed model, rather than only by its inductance, the maximum overvoltage on It was showed too that for a 60 km long single line the overvoltage is reduced by 11% (compared with the reference). If the line is a double circuit the effect is to reduce it by 28.6% for the same line length. For distribution line they made the comparison between the aerial and the coaxial line and showed that the aerial distribution line causes more reduction of the overvoltage due to its resistance. A 10km long aerial line reduces the overvoltage by 28.3% and a cable of the same length reduces it by 23.6%. 4 Frequency response analysis of the circuit As discussed above, the influence of the transmission line is evident. This section will present a comparison between the lumped model and the distributed model of a transmission line in the frequency response of the The models used were the ones presented in figure 1 and 3a [2,3]. The comparison made here was done using the frequency response and frequency spectrum of the circuit in both models. The circuits with lumped elements (figure 1) will be denominated lumped circuit and with distributed elements (figure 3a) will be denominated distributed 4.1 Frequency response of the circuit with lumped inductance The used circuit has two loops (L 1 -C 1 and L 2 -C 2 ), both with the same natural frequency (400Hz). The analysis of the frequency response has been made using as the output voltage V C1 (ω) and V C2 (ω). The frequency response of the lumped circuit (with both L-C loops) is presented in figure 4. It can be seen that the voltage in C 1 is almost zero in the frequency 400Hz because the second L-C is in ressoance for this frequency. The voltage in both capacitors was high for two frequencies near the resonant one. The values of these frequencies is dependent to the first loop resonance value. If the first loop was adjusted to natural frequency 400Hz these two oscillation frequencies are 357 and 447Hz.

3 If the resonant frequency of the first loop varies, these frequencies changes too. It can be seen how these frequencies changes when the L1 value changes from 50mH to 70mH. Fig. 6. Frequency response of a short-circuited line (5km and 60km) Fig. 4. Frequency response of the circuit with both L-C loops The frequency 15kHz is the one (f o ) that makes the line a quarter wavelength long. This frequency is called anti-resonant frequency because it causes a parallel resonance in the line (its impedance goes to ). The others anti-resonant frequencies are its odd harmonics (3f o, 5f o, etc). The even harmonics are resonant frequencies (2f o, 4f o, etc). Zline 0, λ l =, 2 v f = 2l Zline, λ l =, 4 v f = 4l Fig. 5. Frequency spectrum of V C1 and V C2, with capacitor bank switching The same effect can be observed in figure 5. This figure presents the frequency spectrum of the transient voltage after the capacitor switching bank C 1. The voltage across C 1 was almost zero in the frequency 400Hz and grows up for two frequencies which were: 345 and 425Hz. It can be seen that, the oscillation frequency magnitude in the second L-C loop was higher than in the first one, then the voltage magnitude in the second capacitor was magnified. 4.2 Frequency response of the distributed line For this model, two different lengths of the transmission line was used, 5km and 60km. Figure 6 shows the frequency responses of the circuit (with 5 and 60km lines, respectively) with a short-circuit in the end. It can be seen that, the transmission line has many resonant and anti-resonant frequencies. Table 1 Resonant and anti-resonant frequencies Using a 60km line, the anti-resonant frequencies are 1.25kHz and its odd harmonics and the resonant ones are its even harmonics. Table 1 shows how to calculate the resonant and anti-resonant frequencies of a distributed line. 4.3 Frequency response of the circuit with two L-C loops and a 5km line The frequency response of the distributed circuit is showed in figure 7. The analysis has been done looking for the voltage V C1 (ω) and V C2 (ω). As occurred with the lumped circuit, the voltage across C 1 is almost zero in 400Hz, because of the series resonance of the second L-C loop. This circuit presents high voltages values for the frequencies near the 400Hz. Using a 5km line, these frequencies are: 355 and 455Hz. When the length of the line grows, the first frequency became smaller. The frequency spectrum of the transient overvoltage in the capacitor banks C 1 and C 2 during the switching

4 of the capacitor bank C 1 is showed in Figure 8. Both transient overvoltages are composed by the frequencies 60, 355 and 445Hz. These high frequencies are the ones obtained in the frequency response. as showed in figure 9. The frequencies in the spectrum are the same that ones in the frequency response, 348 and 436Hz. Fig. 7. Frequency response of the circuit considering both L-C loops Fig. 8. Frequency spectrum of V C1 and V C2, with capacitor bank switching using 5km line The voltage magnitude in C 2 is bigger than in C 1 for these frequencies. The overvoltage during the transient is due to the presence of this two frequencies. 4.4 Frequency response of the circuit with two L-C loops and a 60km line The same analysis has been made in the circuit with a 60km line. As showed in figure 6 the 60km line has resonant and anti-resonant frequencies different from the 5km line. The effect of a longer transmission line was shown in figure 7. The bigger the line, the smaller the overvoltage and the first frequency in the frequency response. For 60km line, these frequencies are: 348 and 436Hz. The same effect of voltage magnification, which occurs with a 5km line, occurs with a 60km line. In this case, it can be seen that the voltage in C 2 is a little bit smaller than in the circuit with the 5km line, Fig. 9. Frequency spectrum of V C1 and V C2, with capacitor bank switching using 60km line 5 Conclusions This paper has showed that the transmission lines must be considered when magnification of the voltage in a customer capacitor is studied. Detail model, of transmission lines, reduces the transient overvoltage due to switching capacitor bank when the length varies and has to be considered in the analysis. The analysis of a capacitor installation has to be done with care because there are many resonant and antiresonant frequencies in the circuit when a transmission line is present. The circuit frequency response is a good tool to prevent the amplitude and frequency of the oscillations. A parametric analysis can be done in an easy way. The analysis has shown that the magnification of the oscillations during a capacitor switching can occur when the first loop is near the resonant frequency of the second loop but not only when they are equals. References: [1]GREENWOOD, A.: Electrical Transients in Power System, Wiley Interscience, 1991 [2]JOTA, P.R.S. and ISLAM, S.M.: Effect of a Parallel L-C Loop on Magnification of Switching Transients on Low Voltage Capacitors, IEEE Power Engineering Review, Vol. 18, No. 7, 1998, pp [3]JOTA, P.R.S. and ISLAM, S.M.: Effect of realistic system modelling on low-voltage capacitor transient overvoltages, IEE Proc.-Gener. Transm. Distrib., Vol.145, No.6, 1998, pp [4]DUNSMORE, D.M., TAYLOR, E.R., WIRTZ, B.F. AND YANCHULA, T.L.: Magnification of transients voltages in multi-voltage-level, shunt-

5 capacitor-compensated, circuits, IEEE Transactions on Power Delivery, Vol. 7, No. 2, 1992, pp [5] MCGRANAGHAN, M.F., ZAVADIL, R.M., HENSLEY, G. SINGH, T. AND SAMOTYJ, M.: Impact of Utility switched capacitors on Customer System-magnification at low voltage capacitors, IEEE Transactions on Power Delivery, Vol. 7, No. 2, 1992, pp