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Dominant frequency characteristics analysis of reclosing transient overvoltage in 10 kV cable-overhead hybrid line
Electric Power Systems Research 178 (2020) 106052
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Dominant frequency characteristics analysis of reclosing transient overvoltage in 10 kV cable-overhead hybrid line
T
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Jiaming Li, Xi Chen, Long Xu, Aixuan Zhao, Junbo Deng , Guanjun Zhang State Key Laboratory of Electrical Insulation and Power Equipment, Xi’an Jiaotong University, Xi’an, 710049, China
A R T I C LE I N FO
A B S T R A C T
Keywords: Cable-OHL hybrid line Reclosing overvoltage Dominant frequency Modal theory ATP-EMTP
Studying the characteristics of reclosing overvoltage is of great importance for risk assessment of reclosing in the cable-overhead hybrid networks because transient overvoltage (TOV) at different dominant frequency can have quite different negative effect to cable insulation. In a hybrid line, when a cable connects with an overhead line (OHL), the surge propagation can become very complex. To clarify the frequency characteristics of the reclosing transient overvoltage in a 10 kV hybrid line, in this paper, the simulations are carried out using ATP-EMTP (Alternative Transients Program, Electro-Magnetic Transients Program). The traditional approach to estimate the dominant frequency is suitable for pure cable-based or overhead line-based networks. However, for a hybrid line, due to the difference in line parameters between the cable and OHL, the characteristics of dominant frequency is investigated based on the distributed-parameter theory with new boundary conditions at the junction. According to the variation of dominant frequency considering the source impedance and the wave propagation in the modal domain, the non-monotonous variation of TOV with cable proportion is analyzed. The frequency calculation and the TOV analysis are verified by ATP-EMTP simulation.
1. Introduction In recent years, the application of power cables in a 10 kV distribution network has resulted in plenty of cable-overhead hybrid lines in urban areas of China. In a hybrid line system, the use of reclosing devices requires careful risk assessments, mainly considering the fault current, cable proportion, and environmental conditions [1]. In some cities such as Shenzhen, Shanghai, and Guangzhou, auto-reclosing is employed in a 10 kV hybrid line to ensure the consistent power supply based on the power cable types, laying methods, and operating status, but without considering the effect of cable proportion [2]. This paper mainly focuses on the characteristics of the dominant frequency and variation of reclosing overvoltage with the cable proportion, which can provide important references for risk assessments of reclosing in a 10 kV grid. In general, the insulation level of a distribution network is strong enough to withstand the switching impulses. However, it has been reported by the CIGRE and power company that incipient defects may exist in a cable due to the manufacturing or installation process [3,4], and can further deteriorate under electrical and thermal stress during the system operation [5–9]. Multiple switching impulses at different dominant frequencies generated along the cables in service can accelerate the development of micro flaws, and thus lead to the accessory damage and insulation breakdown [7,8,10]. Therefore, ⁎
the transient overvoltage (TOV) generated during the line energization or re-energization has a significant impact on the 10 kV grids. Considering that the reclosing strategy and the insulation level in a hybrid line are strongly influenced by the electromagnetic transients (EMT), the characteristics of reclosing overvoltage have become a crucial issue [7,6–9]. The transient propagation characteristics of switching surges in both pure overhead lines (OHLs) and underground cables have been widely studied [11–21]. The dominant frequency represents the oscillation characteristic in the propagation process of switching transient [17]. Ohno et al. studied the approach for estimating the dominant frequency in pure cable lines [18], and the variation of overvoltage in pure cables and pure OHL [17]. Lafaia et al. studied the frequency and oscillating period of modal voltage [19]. In Ref. [20], the equation for estimating the dominant frequency of a very fast transient overvoltage (VFTO) in typical gas-insulated substation (GIS) was derived. To study the surge propagation and frequency characteristics in hybrid lines, it is of high importance to consider the influence of cable proportion [10–14,22–25]. In Refs. [23] and [24], the statistical studies were performed to analyze the TOV amplitude, considering the cable proportions and line configurations. Besides, the shunt compensation in a hybrid line was investigated in detail in Ref. [25]. However, the difference in frequency characteristics of the TOV between a hybrid line
Corresponding author. E-mail address: [email protected] (J. Deng).
https://doi.org/10.1016/j.epsr.2019.106052 Received 30 May 2019; Received in revised form 25 September 2019; Accepted 26 September 2019 Available online 07 October 2019 0378-7796/ © 2019 Elsevier B.V. All rights reserved.
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frequency in a hybrid line.
and a pure line has not been considered and still needs further investigation. In this paper, the difference in frequency characteristics of TOV between a hybrid line and a pure line is theoretically studied in Section 2. Considering the effect of source impedance, the effect of cable proportion on dominant frequency is further studied, and calculation results are verified by ATP-EMTP simulation. Then, based on the hybrid line model established in ATP-EMTP in Section 3, the statistical study of the TOV at the receiving end is performed to derive the variation of reclosing overvoltage in Section 4. Using ATP-EMTP simulation and modal transformation method, the frequency and propagation characteristics of TOV caused by re-energization in cable-OHL hybrid lines with different cable proportion are deeply investigated and some useful conclusions are achieved in Section 5. The analysis results agree well with the waveform features of the TOV.
2.1.1. Dominant frequency calculation of pure line In general, according to the theory of a distributed-parameter circuit, the pure single-phase distributed-parameter line can be presented using basic line parameters [14] as illustrated in Fig. 2(a). Applying the Kirchhoff’s law to the circuit and solving partial differential equations, voltage v and current i can be expressed as a combination of hyperbolic functions, and they are given by Eq. (1) and (2), respectively.
v = A⋅cosh Γx + B⋅sinh Γx
(1)
i = −Y0 (A⋅sinh Γx + B⋅cosh Γx )
(2)
where A and B are the coefficients of voltage v and current i at distance x, Γ is the propagation constant, and Y0 is the characteristic admittance. For an open-circuited finite line, based on the boundary conditions illustrated in Fig. 2(a), v and i for a lossless line can be respectively calculated by:
2. Analysis of dominant frequency In this section, the frequency characteristics of the TOV at the receiving end of a hybrid line are analyzed so that the influence factors can be observed directly. The analysis workflow of the dominant frequency is presented in Fig. 1. Firstly, the dominant frequency of pure lines and hybrid lines is derived combining with the consistent conditions of voltage and current at the joint point. Then, the comparison of dominant frequency between hybrid line and pure line is conducted. Moreover, the effect of the cable proportion and source impedance is researched.
v = E⋅cos[ω LC (l − x )] cos[ω LC l]
(3)
i = Y0⋅E⋅sin[ω LC (l − x )] cos[ω LC l]
(4)
where l is the total line length, ω is the angular frequency, E is the source voltage, and L and C are the inductance and capacitance per line unit length, respectively. It should be noted that the transmission line is in a resonant condition when the denominator cos[ω LC l] is equal to zero [14]. Thus, the basic dominant frequency of the TOV for a traditional opencircuited line can be derived, and it is given by:
2.1. Calculation of dominant frequency
fdominant = 1 4l LC = 1 4τ
Following the traditional approach for dominant frequency calculation, an equation for a hybrid line is derived so that the dominant frequency can be conveniently estimated. According to Ref. [18], the dominant frequency can be obtained from the resonant condition of a pure transmission line, which can also be used to derive the dominant
(5)
where τ is the propagation time of the voltage wave. 2.1.2. Dominant frequency calculation of hybrid line For a hybrid line depicted in Fig. 2(b), the propagation and
Fig. 1. The analysis workflow of the dominant frequency. 2
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Fig. 2. Circuit representation of a distributed-parameter line.
TOV can be obtained [14]. Visible differences in the dominant frequency caused by line parameters of the pure line and hybrid line can be observed by comparing Eqs. (3) and (6). If there are more than two segments in a hybrid line, the dominant frequency can be calculated by substituting the boundary condition at each joint point.
frequency characteristics of the voltage wave change due to the difference in line parameters between two segments, which leads to different coefficients A and B, and propagation constants Γ in each segment. As shown in Fig. 1, the continuity conditions of voltage and current at the joint point shown in Fig. 2(b), are substituted to voltage and current equations. The calculation is given in Appendix A, and the values of v and i in the second segment are given by Eqs. (6) and (7), respectively.
Y v = ⎡1 cosh(Γ1l1)⋅cosh(Γ2 l2) + 20 ⋅sinh(Γ1l1)⋅sinh(Γ2 l2)⎤ ⎥ ⎢ Y10 ⎦ ⎣ ⋅ E⋅cosh Γ2 (l1 + l2 − x ) = [1 F (ω)]⋅E⋅cosh Γ2 (l1 + l2 − x )
(6)
Y i = ⎡1 cosh(Γ1l1)⋅cosh(Γ2 l2) + 20 ⋅sinh(Γ1l1)⋅sinh(Γ2 l2)⎤ ⎥ ⎢ Y10 ⎦ ⎣ ⋅ Y20⋅E⋅sinh Γ2 (l1 + l2 − x ) = [1 F (ω)]⋅Y20⋅E⋅sinh Γ2 (l1 + l2 − x )
(7)
2.2. Characteristics of dominant frequency As explained above, the dominant frequency of the TOV in hybrid lines can be estimated theoretically. In the meantime, the distributedparameter line model with the same parameters is employed in ATPEMTP to derive the TOV for both cable and OHL, The geometric and electric parameters of 10 kV pipe-type cable and OHL are shown in Fig. 3. For phase A of the cable and OHL, the line parameters estimated at 10 kHz used in calculation are given in Table 1. Using Fourier analysis, the dominant frequency can be obtained to verify the results derived by Eq. (8). 2.2.1. Comparison between hybrid line and pure line In this part, the cable proportion of a hybrid line is set to 50 percent and the total line length is set to 2 km, 3 km and 4 km for comparison. Fig. 4(a) shows the denominator F(ω) in the hybrid line and pure line when the total line length is 2 km, from which the dominant frequency can calculated from the zero point of F(ω). Fig. 4(b) shows the dominant frequency in hybrid line of different length. In Fig. 4, the dominant frequencies of the hybrid lines differ greatly from that of the pure OHLs or cables. The significant difference in inductance and capacitance between the cable and OHL results in the change in the resonant conditions in the transmission lines, which further leads to the difference in the dominant frequency. Besides, for the line structure wherein the OHL is placed after the cable, the highest dominant frequency (40.58 kHz) can be clearly observed. On the contrary, the lowest frequency (12.17 kHz) is when the cable is placed after the OHL. The variation of the dominant frequency shows the same trend at different total line lengths. Assuming the length of the two line types is the same, Y20/Y10 in Eq. (8) becomes reciprocal when the position of cable and OHL changes, resulting in a difference in the dominant frequency in a hybrid line. Therefore, the dominant frequency is affected by the difference in the characteristic admittance between the cable and OHL, which is determined by the impedance and admittance. The maximum relative error between the results obtained by Eq. (8)
where Γ1 is the propagation constant in segment 1, and it is given by Γ1 = (R1 + jωL1)⋅(G1 + jωC1) , while Γ2 is the propagation constant in segment 2, and it is given by Γ2 = (R2 + jωL2)⋅(G2 + jωC2) . The characteristic admittance Y10 of segment 1 can be expressed as Y10 = (G1 + jωC1) (R1 + jωL1) , and the characteristic admittance Y20 of segment 2 can be expressed as Y20 = (G2 + jωC2) (R2 + jωL2) , where l1 and l2 are the length of segment 1 and segment 2, respectively. Further, in a lossless line, by ignoring the attenuation constant, the denominator F(ω) can be represented in the following form:
F (ω) = cosh (Γ1′ l1)⋅cosh (Γ2′ l2) Y + 20 ⋅sinh (Γ1′ l1)⋅sinh (Γ2′ l2) Y10 ′ ⎞ Y20 1⎛ L1 C1 l1 + ⎜1 + ⎟ cos [ω ( ′ ⎠ 2⎝ Y10 Y′ 1 + ⎛⎜1 − 20 ⎞⎟ cos [ω ( L1 C1 l1 − ′ ⎠ 2⎝ Y10 =
L2 C2 l2)] L2 C2 l2)]
(8)
where the denominator F(ω) represents the frequency dependence, Γ1′ = ω L1 C1 , and Γ2′ = ω L2 C2 . The characteristic admittances are ′ = C1 L1 and Y20 ′ = C2 L2 . given by Y10 Similarly, when the denominator F(ω) is equal to zero, the hybrid line is in the resonant condition and thus, the dominant frequency of 3
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Fig. 3. Geometric structure of a 10 kV hybrid line.
total line length. The lengths l1 and l2 should not be ignored during the derivation of the zero point of F(ω) according to Eq. (8). Therefore, the resonant condition of a hybrid line is affected by the cable proportion, which leads to the variation of the dominant frequency. However, considering the effect of source impedance, the variation of frequency turns to be monotonous, as illustrated in Fig. 7, which is investigated in detail in the following.
Table 1 Line parameters of phase A of cable and OHL in Fig. 3. Resistance (Ω/m) Cable OHL
−3
8.5666 × 10 8.5367 × 10−3
Inductance (mH/m) −4
3.5022 × 10 1.8228 × 10−3
Capacitance (μF/m) 8.5757 × 10−5 7.4151 × 10−6
and the ATP-EMTP is merely 1.5%; thus, the accuracy of Eq. (8) can be guaranteed. Since the difference between the parameters of each segment can be determined, Eq. (8) is more suitable than Eq. (5) to estimate the dominant frequency of a hybrid line.
2.2.3. Effect of source impedance Source impedance represents the influence of feeding network on frequency and amplitude characteristics of the TOV. When the lumpedparameter source impedance is considered, the new boundary condition at the sending end is as shown in Fig. 6. As given by Eq. (9), considering the effect of source impedance, the denominator function Fs(ω) can be seen as a superimposition of F(ω) and F0(ω); F(ω) is given by (8), and the additional item F0(ω) is given in (10), and it includes the source impedance Zs.
2.2.2. Effect of cable proportion The length of the cable and OHL in a hybrid line depends on the real environment so that various percentage of cable and OHL can be observed. Fig. 5 shows the variation of the dominant frequency at different cable proportions. Bergeron model and frequency-dependent J. Marti model are adopted in the ATP-EMTP. The maximum relative error is merely 2.3% between the results calculated by using F(ω) and Bergeron model, and 7.1% between the calculation results and results of J. Marti model. The relative error between the calculation and Bergeron results is small since both two methods are based on the constant line parameters. When J. Marti model is adopted, as the line parameters are frequency-dependent, the maximum relative error increases slightly. Also, the calculation results coincide well with the simulation results at different cable proportions. Therefore, the calculation accuracy is verified. As shown in Fig. 5, the dominant frequency shows a non-monotonous variation with the cable proportion in hybrid lines at different
Fs (ω) = F (ω) + F0 (ω) F0 (ω) = −Zs⋅Y10⋅sinh(Γ1l1)⋅cosh(Γ2 l2) − Zs⋅Y20⋅cosh(Γ1l1)⋅sinh(Γ2 l2)
(9) (10)
where Zs = jωLs. Different source impedance, i.e., 1 mH, 5 mH, and 20 mH, are used in the analysis. Results of Fs(ω) shown in Fig. 7 are verified because the maximum relative error is only 2.7%. Compared with the result in Fig. 5, it can be noticed that considering the effect of source impedance, the dominant frequency decreases monotonously with the cable proportion. This is mainly because when the source impedance Zs is considered, F0(ω) is added to F(ω), as given by (9), and the inductance Ls can strongly influence the resonant condition of a line, which affects the
Fig. 4. Calculation results of the dominant frequency. 4
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Fig. 5. Variation of the dominant frequency with cable proportion.
BCTRAN model. To achieve more severe TOV, the load capacity was considered as 20% of the transformer capacity with a power factor of 0.95. Thus, the load impedance was calculated as (1.088 + j0.3578) Ω. Three single-phase DC voltage sources with an amplitude of −8.165 kV were connected with the sending end of the hybrid line at the beginning of simulation, in order to represent the effect of residual charge during the reclosing process. At 0.001 s, the DC voltage sources were removed. Since the distribution transformer winding at 10 kV side is delta connected, there was no discharge path for the trapped charge. Therefore, the voltage was maintained at −8.165 kV. The switching operation in simulation is shown in Fig. 9. Since the statistical analysis was conducted, the statistic switch was adopted. Phase A was set as the master switch in which the uniform distribution in a power frequency period was assumed. The slave switch with the standard deviation of 1 ms was adopted in phase B and phase C. Since in this study, the shortest line section was 20% of 2 km, ΔT of 0.1 μs was enough for switching transient studies assuming the wave traveling speed is equal to the light speed.
dominant frequency. As for the multi-conductor coupled lines, the frequency characteristics of each independent circuit in the modal domain can be analyzed using modal theory [21]. In the following, in order to introduce the variation of dominant frequency to the switching transient analysis, the hybrid line system model is established in ATP-EMTP based on the distribution network in Shaanxi Province, China. 3. Simulation model The simulation model of an asymmetrical hybrid line was established to study the variation of reclosing overvoltage with cable proportion, as shown in Fig. 8. For transient studies, the calculation of impedance and admittance of a hybrid line was performed using the Cable Parameters and Line Constants subroutines in the ATP-EMTP. The line structures are depicted in Fig. 3. Considering the frequency range of reclosing overvoltage, the line parameters were calculated at the frequency of 10 kHz using the Bergeron model [1]. By using the Verify button in the LCC dialog box from ATPDraw [26], it was shown that the zero-sequence impedance had the best agreement with the parameters of the frequency-dependent Exact PI at 10 kHz. The total length of the hybrid line was set to 2 km, and the steel pipe of cable was grounded through a 5 Ω resistance. As the maximum short-circuit current at a 10 kV busbar is 21.6 kA, the feeding network was modeled as a source impedance of 0.85 mH. Besides, the statistic switch was employed for CB-1, and 100 simulations were performed to conduct the statistical analysis on the TOV. In distribution network, as the load was directly connected to the distribution line, so the transformer with load impedance could not be ignored. The 630 kVA transformer was adopted and simulated as a
4. Statistical study of TOV in hybrid line Using the EMT model illustrated above, 100 simulations were performed. The focus was mainly on the reclosing overvoltage at the receiving end since higher overvoltage could be observed. Fig. 10 shows the cumulative probability of reclosing overvoltage. The probability of hybrid lines varied greatly at different cable proportions. When cable proportion was 20%, the overvoltage of the hybrid line was even more severe than that of the pure OHL. Further, the maximum value, the mean value, and the 2% value were calculated, and the variation of the statistical values with the cable proportion is illustrated in Fig. 11(a). Apparently, the TOV varied non-monotonously with cable proportion in Cable-OHL hybrid line.
Fig. 6. Distributed-parameter line considering the source impedance. 5
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Fig. 7. Variation of dominant frequency when the source impedance is considered.
6
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Fig. 8. Simulation model of a 10 kV hybrid line.
5.2. Propagation of modal voltage Applying the eigenvalues/vectors theory, the coupled multi-phase cable system can be decoupled into several independent modal circuits. Therefore, the voltage and current in modal domain can be transformed from those in phase domain by
Fig. 9. Operation of switches during the simulation in the time domain.
vph = Tv⋅vm
(11)
i ph = Ti⋅i m
(12)
where vph and iph are voltage and current in phase domain. vm and im are respectively voltage and current in modal domain. Tv and Ti are voltage and current transformation matrix, respectively. Based on modal theory, the phase voltage can be regarded as a superimposition of the voltage wave of each mode [17,21]. The analysis of the minimal value of TOV is based on the variation of dominant frequency and propagation of modal voltage. In this section, firstly, the determination of dominant mode is presented, and then, the propagation constants of modal voltage are calculated. 5.2.1. Determination of dominant mode As it is well known, the dominant frequency can represent the oscillation period of the TOV. In the meantime, the oscillation period of the dominant mode voltage can be derived using the dominant frequency [17,21]. Moreover, it is well known that earth-return mode is the dominant mode of overvoltage propagating along OHL. For a hybrid line, it is convenient to derive the dominant mode by transforming the overvoltage into modal domain [16,21]. The dominant mode voltage of both pure and hybrid line is presented in Fig. 13, where the non-zero initial conditions can be observed due to the effect of the trapped charges. Apparently, the earth-return mode still dominates the TOV in the OHL section of a hybrid line, yet the oscillation period is affected. Thus, in the following research, the earth-return mode of TOV at the receiving end is regarded as a dominant mode, and the aerial mode is regarded as a superimposed mode. Since the focus is on the overvoltage at the receiving end, the modal analysis is concentrated on the OHL section of a hybrid line. The dominant mode of a cable section can be further derived by modal transformation.
Fig. 10. Cumulative probability of the TOV.
Also, when lcable:lOHL = 600:1400, minimal values could be clearly observed. The overvoltage was 2.45 p.u. for both maximum and 2% value. The same phenomenon was also observed in [23,25] for lines with different length for a 380 kV network. The influence of the proportion of cable and OHL on TOV at the receiving end is analyzed in the following text based on the dominant frequency and propagation of TOV.
5. Analysis of variation of statistical values In this section, the variation of statistical values shown in Fig. 11(a) is further analyzed, based on the variation of dominant frequency explained in Section 2, and the propagation of modal voltage. The results regarding the dominant frequency are illustrated in Section 5.1, and then the modal analysis is presented in Section 5.2.
5.2.2. Calculation of propagation constants The propagation constants of independent modal voltage obtained by application of the eigenvalues/vectors theory to multi-conductor system are presented in Table 2 [21]. The line parameters of a hybrid line used in modal transform are derived in Section 3.
5.1. Variation of dominant frequency As can be observed in Section 2, the dominant frequency decreases monotonously with the cable proportion considering the effect of the source impedance. According to the results presented in Fig. 12, the dominant frequency of the TOV in the simulation test decreased with the cable proportion increasing, which coincides with the variation shown in Fig. 5(b). Therefore, it can be concluded that the oscillation period increases with the cable proportion.
5.3. Investigation of non-monotonous variation Using the frequency characteristics and propagation of modal voltage described above, the appearing of the minimal value of statistical results can be explained. The overvoltage waveform in phase A of a hybrid line at the cable proportion of 20%, 30% and 40% is illustrated 7
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Fig. 11. Variation of the statistical values and TOV at the junction and receiving end.
(1) Cable 20%, OHL 80% As depicted in Fig. 11(b), the overvoltage at the junction started to rise at 40.0017 ms after the voltage wave passed through the cable section. Then, the propagation of the aerial mode voltage from the head end of the OHL to the receiving end lasted approximately 5.37 μs, and thus started at 40.0179 ms; the second wave of the aerial mode arrived at the receiving end again after 10.8 μs. Considering the oscillation period of 51.39 μs, the dominant mode voltage, therefore, reached the highest value when the second wave of the superimposed mode arrived at the receiving end, which resulted in higher overvoltage. (2) Cable 30%, OHL 70% Similarly, in this case, after the first arriving at receiving end, the aerial mode voltage was reflected back to the head end of the OHL after 4.69 μs. Due to the fact that the propagation velocity was not influenced by the line length, the propagation time became lower as the OHL proportion decreased. The voltage wave of superimposed mode reached the receiving end every 9.2 μs, as shown in Fig. 11(c). Since the oscillation period was 59.14 μs, when the second wave of the aerial mode voltage arrived at the receiving end, the dominant mode voltage still had not reached the highest value, and the distortion of the TOV could be clearly observed. When the dominant mode voltage reached the highest value at 40.0179 ms, the aerial mode was reflected back to the head end of the OHL. Thus, comparing with the waveform depicted in Fig. 11(b), the maximum value of the TOV at the receiving end was relatively lower. (3) Cable 40%, OHL 60% Applying the same approach described above, it is known that when
Fig. 12. The dominant frequency of the TOV at different cable proportions.
in Fig. 11(b)–(d), respectively. Using the propagation velocity of the aerial mode presented in Table 2, i.e., the velocity of 298 m/μs, together with the OHL section length, the propagation time of modal voltage wave can be calculated, and the obtained values are given in Table 3. Considering the dominant frequency of the TOV shown in Fig. 12, the oscillation period of the dominant mode voltage, i.e., the earth-return mode, can be estimated. The calculation results coincide well with the time intervals of waveforms obtained by the simulation which are presented in Fig. 11. Three cases shown in Fig. 11(b)–(d) are analyzed with the use of the propagation characteristics of modal voltage waves, so that the minimal value of statistical results can be explained. 8
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Fig. 13. The voltage of phase A and earth-return mode.
6. Conclusion
Table 2 Propagation constants of the modal voltage of the OHL and cable. Transmission line
Mode
OHL
Earth-return mode Aerial mode 1 Aerial mode 2 Earth-return mode Pipe-return mode Inter-conductor 1 Inter-conductor 2
Cable
This paper presents both theoretical analyses and simulation study on the dominant frequency characteristics and propagation characteristics of TOV in 10 kV cable-OHL hybrid lines. The main conclusions are as follows.
Propagation velocity (m/μs) 257.08 298.21 297.92 19.70 176.25 180.76 180.76
(1) Using the continuity conditions of voltage and current at the junction in a distributed-parameter circuit, based on theoretical derivation, a convenient method to estimate the dominant frequency in cable-OHL hybrid lines is achieved. (2) he calculation results show that the dominant frequency of a hybrid line differs greatly from that of a pure line. When the effect of source impedance is considered, the dominant frequency of TOV monotonously decreases with the cable proportion. Each influence factor can be clearly observed from the denominator function Fs(ω). (3) Since the peak value of TOV is strongly affected by the superimposition timing of modal voltage, the statistical values of the TOV at the receiving end varies non-monotonously with the cable proportion in hybrid lines but shows a decreasing trend.
Table 3 Propagation time and oscillation period of the TOV of hybrid line. Line configuration
Cable 20%, OHL 80% Cable 30%, OHL 70% Cable 40%, OHL 60%
Propagation time (μs)
Oscillation period (μs)
Calculation
Simulation
Calculation
Simulation
5.37 4.69 4.00
5.40 4.60 4.00
51.39 59.14 66.40
53.20 60.30 64.29
The characteristics of TOV in hybrid lines with different cable proportion and different configurations can be investigated clearly using the method presented in this paper, which can provide a reference for assessing the risk and designing the reclosing strategy of hybrid lines.
the moment started at 40.003 ms, the fourth wave of aerial mode arrived at the receiving end after 28 μs. At the same time, the dominant mode voltage reached the highest value at 40.031 ms, and thus the maximum value of TOV in this case was higher than that at the cable proportion of 30%. Considering the attenuation of aerial mode when the fourth wave arrived, the overvoltage was relatively lower than that at the cable proportion of 20%. Consequently, the analysis results of the wave propagation and frequency coincide well with the TOV waveforms obtained by ATPEMTP simulation. The approach used in this work can also be applied to the asymmetrical structures with the cable section behind the OHL, or a symmetrical hybrid line, as long as the dominant frequency is modified accordingly.
Conflict of interests The authors declare that they have no conflict of interest. Acknowledgment This work was supported by the National Natural Science Foundation of China (51577150).
Appendix A In this part, the calculation of voltage and current along a hybrid line is presented. First, the equations of voltage and current are derived. Based on the single-phase hybrid line shown in Fig. 2(b), the voltage v and current i in segment 1 and segment 2 are given by Eqs. (A.1)–(A.2) and Eqs. (A.3)–(A.4), respectively.
v1 = A1 ⋅cosh Γ1x + B1⋅sinh Γ1x
(A.1)
i1 = −Y10 (A1 ⋅sinh Γ1x + B1⋅cosh Γ1x )
(A.2)
v2 = A2 ⋅cosh Γ2 x + B2⋅sinh Γ2 x
(A.3)
i2 = −Y20 (A2 ⋅sinh Γ2 x + B2⋅cosh Γ2 x )
(A.4)
where v1 and i1 correspond to segment 1, and v2 and i2 correspond to segment 2. Then, the boundary conditions at the sending end, receiving end, and the joint point can be expressed as:
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⎧ v1 = E , x = 0 ⎪ v1 = v2, x = l1 ⎨i1 = i2, x = l1 ⎪ l = 0, x = l + l 1 2 ⎩2
(A.5)
By substituting the boundary conditions given by Eq. (A.5), the following equations can be derived:
E = A1
(A.6)
Y10 (A1 ⋅sinh Γ1 l1 + B1⋅cosh Γ1 l1) = Y20 (A2 ⋅sinh Γ2 l1 + B2⋅cosh Γ2 l1)
(A.7)
A1 ⋅sinh Γ1 l1 + B1⋅cosh Γ1 l1 = A2 ⋅sinh Γ2 l1 + B2⋅cosh Γ2 l1
(A.8)
0 = Y20 [A2 ⋅sinh Γ2 (l1 + l2) + B2⋅cosh Γ2 (l1 + l2)]
(A.9)
Lastly, the values of coefficients A1, B1, A2 and B2 can be calculated by Eqs. (A.6)–(A.9), and they are given by Eqs. (A.10)–(A.13), respectively. Thus, voltage and current along each segment can be derived. By substituting Eqs. (A.12) and (A.13) into Eqs. (A.3) and (A.4), the voltage and current at the receiving end are derived and given by Eq. (6) and (7), respectively. (A.10)
A1 = E
B1 = −
a⋅Y10 (dn − cm) − b⋅Y20 (cn − dm) ⋅E b⋅Y10 (dn − cm) − a⋅Y20 (cn − dm)
(A.11)
a ⋅ Y (dn − cm) − b ⋅ Y (cn − dm)
A2 =
n⋅[b − a⋅ b ⋅ Y10 (dn − cm) − a ⋅ Y20 (cn − dm) ] 10
20
dn − cm
⋅E
(A.12)
a ⋅ Y (dn − cm) − b ⋅ Y (cn − dm)
B2 = −
m⋅[b − a⋅ b ⋅ Y10 (dn − cm) − a ⋅ Y20 (cn − dm) ] 10
dn − cm
20
⋅E
(A.13)
where a = sinh(Γ1l1), b = cosh(Γ1l1), c = sinh(Γ2l1), d = cosh(Γ2l1), m = sinh[Γ2(l1+l2)], and n= cosh[Γ2(l1+l2)]; Γ1 and Γ2 are the propagation constants of two segments in a hybrid line respectively; l1 and l2 are the lengths of the two segments, as shown in Fig. 2(b); lastly, Y10 and Y20 represent the characteristic admittances.
Applications, second ed., CRC Press, New York, 2017. [15] A. Ametani, I. Lafaia, Y. Miyamoto, T. Asada, Y. Baba, N. Nagaoka, High-frequency wave-propagation along overhead conductors by transmission line approach and numerical electromagnetic analysis, Electr. Power Syst. Res. 136 (2016) 12–20. [16] I. Lafaia, M.T.C. Barros, J. Mahseredjian, A. Ametani, I. Kocar, Y. Fillion, Surge and energization tests and modeling on a 225kV HVAC cable, Electr. Power Syst. Res. 160 (2018) 273–281. [17] T. Ohno, A. Ametani, C.L. Bak, W. Wiechowski, T.K. Sorensen, Analysis of the statistical distribution of energization overvoltages of EHV cables and OHLs, IPST 2013, Vancouver, Canada, July 18–20, 2013. [18] T. Ohno, C.L. Bak, A. Ametani, W. Wiechowski, T.K. Sorensen, Derivation of theoretical formula for dominant frequency, IEEE Trans. Power Deliv. 27 (2) (2012) 866–876. [19] I. Lafaia, Y. Yamamoto, A. Ametani, J. Mahseredjian, M.T.C. Barros, I. Kocar, A. Naud, Propagation of intersheath modes on underground cables, Electr. Power Syst. Res. 138 (2016) 113–119. [20] B. Kang, Study of Very Fast Transient Overvoltage in GIS and Its Influence on Electronic Transformers (PhD Thesis), Wuhan University, Wuhan, China, 2016. [21] A. Ametani, T. Ohno, N. Nagaoka, Cable System Transients: Theory, Modeling and Simulation, Wiley & Sons, Singapore, 2015. [22] F.F. Silva, C.L. Bak, P.B. Holst, Switching restrikes in HVAC cable lines and hybrid HVAC cable/OHL lines, IPST 2011, Delft, Netherlands, June 14–17, 2011. [23] H. Khalilnezhad, M. Popov, J.A. Bos, J.P.W. Jong, L. Sluis, Investigation of statistical distribution of energization overvoltages in 380kV hybrid OHL-cable systems, IPST 2017, Seoul, Republic of Korea, June 26–29, 2017. [24] H. Khalilnezhad, M. Popov, L. Sluis, J.A. Bos, A. Ametani, Statistical analysis of energization overvotlages in EHV hybrid OHL-cable systems, IEEE Trans. Power Deliv. 33 (December (6)) (2018) 2765–2775. [25] H. Khalilnezhad, S. Chen, M. Popov, J.A. Bos, J.P.W. Jong, L. Sluis, Shunt compensation design of EHV double-circuit mixed OHL-cable connections, Proc. IET Int. Conf. Resilience of Transmission and Distribution Networks. (2015). [26] L. Prikler, H.K. Høidalen, ATPDraw version 5.6 for windows 9x/NT/2000/XP/Vista User’s manual, Norway, (2009).
References [1] CIGRE WG C4.502, Power system technical performance issues related to the application of long HVAC cables, CIGRE TB 556, October 2013. [2] N. Jiang, Y. Xie, J. Hou, L. Xu, Q. Wang, P. He, Reclosing mode of cable-overhead mixed lines, Autom. Electr. Power Syst. 34 (February (3)) (2010) 112–115. [3] CIGRE WG B1.04, Maintenance for HV cables and accessories, CIGRE TB 279, August 2005. [4] CIGRE WG 21.16, Partial discharge detection in installed HV extruded cable systems, CIGRE TB 182, April 2001. [5] K. Zhou, X. Li, H. Huang, C. Wei, K. Yang, Analysis of partial discharge characteristics for installing cutting defects in cable terminations, Power Syst. Prot. Control 14 (May (10)) (2013) 104–110. [6] G. Mazzanti, The combination of electro-thermal stress, load cycling and thermal transient and its effects on the life of high voltage ac cables, IEEE Trans. Dielectr. Electr. Insul. 16 (August (4)) (2009) 1168–1179. [7] L. Cao, Study of Accelerated Aging of 15 kV XLPE and EPR Cable Insulation by Switching Impulses and Elevated AC Voltage (PhD Thesis), Mississippi State University, Mississippi, USA, 2009. [8] L. Cao, A. Zanwar, S. Grzybowski, Electrical aging phenomena of power cables aged by switching impulse, High Voltage Eng. 39 (August (8)) (2013) 1911–1918. [9] L. Xu, A. Zhao, J. Li, J. Deng, G. Zhang, J. Liu, Effects of multiple switching impulses on the partial discharge characteristics of 10kV cable termination with knife incision, High Voltage Eng. (2018) 1–9. [10] F.F. Silva, C.L. Bak, Electromagnetic Transients in Power Cables, Springer, London, 2013. [11] A. Ametani, A general formulation of impedance and admittance of cables, IEEE Trans. Power App. Syst. PAS-99 (May/June (3)) (1980) 902–910. [12] A. Ametani, Wave propagation characteristics of cables, IEEE Trans. Power App. Syst. PAS-99 (March/April (2)) (1980) 499–505. [13] L.M. Wedepohl, D.J. Wilcox, Transient analysis of underground power-transmission systems. System model and wave-propagation characteristics, Proc. Inst. Electr. Eng. 120 (February (2)) (1973) 253–260. [14] A. Ametani, N. Nagaoka, Y. Baba, Power System Transients Theory and
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Dominant frequency characteristics analysis of reclosing transient overvoltage in 10 kV cable-overhead hybrid line
T
⁎
Jiaming Li, Xi Chen, Long Xu, Aixuan Zhao, Junbo Deng , Guanjun Zhang State Key Laboratory of Electrical Insulation and Power Equipment, Xi’an Jiaotong University, Xi’an, 710049, China
A R T I C LE I N FO
A B S T R A C T
Keywords: Cable-OHL hybrid line Reclosing overvoltage Dominant frequency Modal theory ATP-EMTP
Studying the characteristics of reclosing overvoltage is of great importance for risk assessment of reclosing in the cable-overhead hybrid networks because transient overvoltage (TOV) at different dominant frequency can have quite different negative effect to cable insulation. In a hybrid line, when a cable connects with an overhead line (OHL), the surge propagation can become very complex. To clarify the frequency characteristics of the reclosing transient overvoltage in a 10 kV hybrid line, in this paper, the simulations are carried out using ATP-EMTP (Alternative Transients Program, Electro-Magnetic Transients Program). The traditional approach to estimate the dominant frequency is suitable for pure cable-based or overhead line-based networks. However, for a hybrid line, due to the difference in line parameters between the cable and OHL, the characteristics of dominant frequency is investigated based on the distributed-parameter theory with new boundary conditions at the junction. According to the variation of dominant frequency considering the source impedance and the wave propagation in the modal domain, the non-monotonous variation of TOV with cable proportion is analyzed. The frequency calculation and the TOV analysis are verified by ATP-EMTP simulation.
1. Introduction In recent years, the application of power cables in a 10 kV distribution network has resulted in plenty of cable-overhead hybrid lines in urban areas of China. In a hybrid line system, the use of reclosing devices requires careful risk assessments, mainly considering the fault current, cable proportion, and environmental conditions [1]. In some cities such as Shenzhen, Shanghai, and Guangzhou, auto-reclosing is employed in a 10 kV hybrid line to ensure the consistent power supply based on the power cable types, laying methods, and operating status, but without considering the effect of cable proportion [2]. This paper mainly focuses on the characteristics of the dominant frequency and variation of reclosing overvoltage with the cable proportion, which can provide important references for risk assessments of reclosing in a 10 kV grid. In general, the insulation level of a distribution network is strong enough to withstand the switching impulses. However, it has been reported by the CIGRE and power company that incipient defects may exist in a cable due to the manufacturing or installation process [3,4], and can further deteriorate under electrical and thermal stress during the system operation [5–9]. Multiple switching impulses at different dominant frequencies generated along the cables in service can accelerate the development of micro flaws, and thus lead to the accessory damage and insulation breakdown [7,8,10]. Therefore, ⁎
the transient overvoltage (TOV) generated during the line energization or re-energization has a significant impact on the 10 kV grids. Considering that the reclosing strategy and the insulation level in a hybrid line are strongly influenced by the electromagnetic transients (EMT), the characteristics of reclosing overvoltage have become a crucial issue [7,6–9]. The transient propagation characteristics of switching surges in both pure overhead lines (OHLs) and underground cables have been widely studied [11–21]. The dominant frequency represents the oscillation characteristic in the propagation process of switching transient [17]. Ohno et al. studied the approach for estimating the dominant frequency in pure cable lines [18], and the variation of overvoltage in pure cables and pure OHL [17]. Lafaia et al. studied the frequency and oscillating period of modal voltage [19]. In Ref. [20], the equation for estimating the dominant frequency of a very fast transient overvoltage (VFTO) in typical gas-insulated substation (GIS) was derived. To study the surge propagation and frequency characteristics in hybrid lines, it is of high importance to consider the influence of cable proportion [10–14,22–25]. In Refs. [23] and [24], the statistical studies were performed to analyze the TOV amplitude, considering the cable proportions and line configurations. Besides, the shunt compensation in a hybrid line was investigated in detail in Ref. [25]. However, the difference in frequency characteristics of the TOV between a hybrid line
Corresponding author. E-mail address: [email protected] (J. Deng).
https://doi.org/10.1016/j.epsr.2019.106052 Received 30 May 2019; Received in revised form 25 September 2019; Accepted 26 September 2019 Available online 07 October 2019 0378-7796/ © 2019 Elsevier B.V. All rights reserved.
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frequency in a hybrid line.
and a pure line has not been considered and still needs further investigation. In this paper, the difference in frequency characteristics of TOV between a hybrid line and a pure line is theoretically studied in Section 2. Considering the effect of source impedance, the effect of cable proportion on dominant frequency is further studied, and calculation results are verified by ATP-EMTP simulation. Then, based on the hybrid line model established in ATP-EMTP in Section 3, the statistical study of the TOV at the receiving end is performed to derive the variation of reclosing overvoltage in Section 4. Using ATP-EMTP simulation and modal transformation method, the frequency and propagation characteristics of TOV caused by re-energization in cable-OHL hybrid lines with different cable proportion are deeply investigated and some useful conclusions are achieved in Section 5. The analysis results agree well with the waveform features of the TOV.
2.1.1. Dominant frequency calculation of pure line In general, according to the theory of a distributed-parameter circuit, the pure single-phase distributed-parameter line can be presented using basic line parameters [14] as illustrated in Fig. 2(a). Applying the Kirchhoff’s law to the circuit and solving partial differential equations, voltage v and current i can be expressed as a combination of hyperbolic functions, and they are given by Eq. (1) and (2), respectively.
v = A⋅cosh Γx + B⋅sinh Γx
(1)
i = −Y0 (A⋅sinh Γx + B⋅cosh Γx )
(2)
where A and B are the coefficients of voltage v and current i at distance x, Γ is the propagation constant, and Y0 is the characteristic admittance. For an open-circuited finite line, based on the boundary conditions illustrated in Fig. 2(a), v and i for a lossless line can be respectively calculated by:
2. Analysis of dominant frequency In this section, the frequency characteristics of the TOV at the receiving end of a hybrid line are analyzed so that the influence factors can be observed directly. The analysis workflow of the dominant frequency is presented in Fig. 1. Firstly, the dominant frequency of pure lines and hybrid lines is derived combining with the consistent conditions of voltage and current at the joint point. Then, the comparison of dominant frequency between hybrid line and pure line is conducted. Moreover, the effect of the cable proportion and source impedance is researched.
v = E⋅cos[ω LC (l − x )] cos[ω LC l]
(3)
i = Y0⋅E⋅sin[ω LC (l − x )] cos[ω LC l]
(4)
where l is the total line length, ω is the angular frequency, E is the source voltage, and L and C are the inductance and capacitance per line unit length, respectively. It should be noted that the transmission line is in a resonant condition when the denominator cos[ω LC l] is equal to zero [14]. Thus, the basic dominant frequency of the TOV for a traditional opencircuited line can be derived, and it is given by:
2.1. Calculation of dominant frequency
fdominant = 1 4l LC = 1 4τ
Following the traditional approach for dominant frequency calculation, an equation for a hybrid line is derived so that the dominant frequency can be conveniently estimated. According to Ref. [18], the dominant frequency can be obtained from the resonant condition of a pure transmission line, which can also be used to derive the dominant
(5)
where τ is the propagation time of the voltage wave. 2.1.2. Dominant frequency calculation of hybrid line For a hybrid line depicted in Fig. 2(b), the propagation and
Fig. 1. The analysis workflow of the dominant frequency. 2
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Fig. 2. Circuit representation of a distributed-parameter line.
TOV can be obtained [14]. Visible differences in the dominant frequency caused by line parameters of the pure line and hybrid line can be observed by comparing Eqs. (3) and (6). If there are more than two segments in a hybrid line, the dominant frequency can be calculated by substituting the boundary condition at each joint point.
frequency characteristics of the voltage wave change due to the difference in line parameters between two segments, which leads to different coefficients A and B, and propagation constants Γ in each segment. As shown in Fig. 1, the continuity conditions of voltage and current at the joint point shown in Fig. 2(b), are substituted to voltage and current equations. The calculation is given in Appendix A, and the values of v and i in the second segment are given by Eqs. (6) and (7), respectively.
Y v = ⎡1 cosh(Γ1l1)⋅cosh(Γ2 l2) + 20 ⋅sinh(Γ1l1)⋅sinh(Γ2 l2)⎤ ⎥ ⎢ Y10 ⎦ ⎣ ⋅ E⋅cosh Γ2 (l1 + l2 − x ) = [1 F (ω)]⋅E⋅cosh Γ2 (l1 + l2 − x )
(6)
Y i = ⎡1 cosh(Γ1l1)⋅cosh(Γ2 l2) + 20 ⋅sinh(Γ1l1)⋅sinh(Γ2 l2)⎤ ⎥ ⎢ Y10 ⎦ ⎣ ⋅ Y20⋅E⋅sinh Γ2 (l1 + l2 − x ) = [1 F (ω)]⋅Y20⋅E⋅sinh Γ2 (l1 + l2 − x )
(7)
2.2. Characteristics of dominant frequency As explained above, the dominant frequency of the TOV in hybrid lines can be estimated theoretically. In the meantime, the distributedparameter line model with the same parameters is employed in ATPEMTP to derive the TOV for both cable and OHL, The geometric and electric parameters of 10 kV pipe-type cable and OHL are shown in Fig. 3. For phase A of the cable and OHL, the line parameters estimated at 10 kHz used in calculation are given in Table 1. Using Fourier analysis, the dominant frequency can be obtained to verify the results derived by Eq. (8). 2.2.1. Comparison between hybrid line and pure line In this part, the cable proportion of a hybrid line is set to 50 percent and the total line length is set to 2 km, 3 km and 4 km for comparison. Fig. 4(a) shows the denominator F(ω) in the hybrid line and pure line when the total line length is 2 km, from which the dominant frequency can calculated from the zero point of F(ω). Fig. 4(b) shows the dominant frequency in hybrid line of different length. In Fig. 4, the dominant frequencies of the hybrid lines differ greatly from that of the pure OHLs or cables. The significant difference in inductance and capacitance between the cable and OHL results in the change in the resonant conditions in the transmission lines, which further leads to the difference in the dominant frequency. Besides, for the line structure wherein the OHL is placed after the cable, the highest dominant frequency (40.58 kHz) can be clearly observed. On the contrary, the lowest frequency (12.17 kHz) is when the cable is placed after the OHL. The variation of the dominant frequency shows the same trend at different total line lengths. Assuming the length of the two line types is the same, Y20/Y10 in Eq. (8) becomes reciprocal when the position of cable and OHL changes, resulting in a difference in the dominant frequency in a hybrid line. Therefore, the dominant frequency is affected by the difference in the characteristic admittance between the cable and OHL, which is determined by the impedance and admittance. The maximum relative error between the results obtained by Eq. (8)
where Γ1 is the propagation constant in segment 1, and it is given by Γ1 = (R1 + jωL1)⋅(G1 + jωC1) , while Γ2 is the propagation constant in segment 2, and it is given by Γ2 = (R2 + jωL2)⋅(G2 + jωC2) . The characteristic admittance Y10 of segment 1 can be expressed as Y10 = (G1 + jωC1) (R1 + jωL1) , and the characteristic admittance Y20 of segment 2 can be expressed as Y20 = (G2 + jωC2) (R2 + jωL2) , where l1 and l2 are the length of segment 1 and segment 2, respectively. Further, in a lossless line, by ignoring the attenuation constant, the denominator F(ω) can be represented in the following form:
F (ω) = cosh (Γ1′ l1)⋅cosh (Γ2′ l2) Y + 20 ⋅sinh (Γ1′ l1)⋅sinh (Γ2′ l2) Y10 ′ ⎞ Y20 1⎛ L1 C1 l1 + ⎜1 + ⎟ cos [ω ( ′ ⎠ 2⎝ Y10 Y′ 1 + ⎛⎜1 − 20 ⎞⎟ cos [ω ( L1 C1 l1 − ′ ⎠ 2⎝ Y10 =
L2 C2 l2)] L2 C2 l2)]
(8)
where the denominator F(ω) represents the frequency dependence, Γ1′ = ω L1 C1 , and Γ2′ = ω L2 C2 . The characteristic admittances are ′ = C1 L1 and Y20 ′ = C2 L2 . given by Y10 Similarly, when the denominator F(ω) is equal to zero, the hybrid line is in the resonant condition and thus, the dominant frequency of 3
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Fig. 3. Geometric structure of a 10 kV hybrid line.
total line length. The lengths l1 and l2 should not be ignored during the derivation of the zero point of F(ω) according to Eq. (8). Therefore, the resonant condition of a hybrid line is affected by the cable proportion, which leads to the variation of the dominant frequency. However, considering the effect of source impedance, the variation of frequency turns to be monotonous, as illustrated in Fig. 7, which is investigated in detail in the following.
Table 1 Line parameters of phase A of cable and OHL in Fig. 3. Resistance (Ω/m) Cable OHL
−3
8.5666 × 10 8.5367 × 10−3
Inductance (mH/m) −4
3.5022 × 10 1.8228 × 10−3
Capacitance (μF/m) 8.5757 × 10−5 7.4151 × 10−6
and the ATP-EMTP is merely 1.5%; thus, the accuracy of Eq. (8) can be guaranteed. Since the difference between the parameters of each segment can be determined, Eq. (8) is more suitable than Eq. (5) to estimate the dominant frequency of a hybrid line.
2.2.3. Effect of source impedance Source impedance represents the influence of feeding network on frequency and amplitude characteristics of the TOV. When the lumpedparameter source impedance is considered, the new boundary condition at the sending end is as shown in Fig. 6. As given by Eq. (9), considering the effect of source impedance, the denominator function Fs(ω) can be seen as a superimposition of F(ω) and F0(ω); F(ω) is given by (8), and the additional item F0(ω) is given in (10), and it includes the source impedance Zs.
2.2.2. Effect of cable proportion The length of the cable and OHL in a hybrid line depends on the real environment so that various percentage of cable and OHL can be observed. Fig. 5 shows the variation of the dominant frequency at different cable proportions. Bergeron model and frequency-dependent J. Marti model are adopted in the ATP-EMTP. The maximum relative error is merely 2.3% between the results calculated by using F(ω) and Bergeron model, and 7.1% between the calculation results and results of J. Marti model. The relative error between the calculation and Bergeron results is small since both two methods are based on the constant line parameters. When J. Marti model is adopted, as the line parameters are frequency-dependent, the maximum relative error increases slightly. Also, the calculation results coincide well with the simulation results at different cable proportions. Therefore, the calculation accuracy is verified. As shown in Fig. 5, the dominant frequency shows a non-monotonous variation with the cable proportion in hybrid lines at different
Fs (ω) = F (ω) + F0 (ω) F0 (ω) = −Zs⋅Y10⋅sinh(Γ1l1)⋅cosh(Γ2 l2) − Zs⋅Y20⋅cosh(Γ1l1)⋅sinh(Γ2 l2)
(9) (10)
where Zs = jωLs. Different source impedance, i.e., 1 mH, 5 mH, and 20 mH, are used in the analysis. Results of Fs(ω) shown in Fig. 7 are verified because the maximum relative error is only 2.7%. Compared with the result in Fig. 5, it can be noticed that considering the effect of source impedance, the dominant frequency decreases monotonously with the cable proportion. This is mainly because when the source impedance Zs is considered, F0(ω) is added to F(ω), as given by (9), and the inductance Ls can strongly influence the resonant condition of a line, which affects the
Fig. 4. Calculation results of the dominant frequency. 4
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Fig. 5. Variation of the dominant frequency with cable proportion.
BCTRAN model. To achieve more severe TOV, the load capacity was considered as 20% of the transformer capacity with a power factor of 0.95. Thus, the load impedance was calculated as (1.088 + j0.3578) Ω. Three single-phase DC voltage sources with an amplitude of −8.165 kV were connected with the sending end of the hybrid line at the beginning of simulation, in order to represent the effect of residual charge during the reclosing process. At 0.001 s, the DC voltage sources were removed. Since the distribution transformer winding at 10 kV side is delta connected, there was no discharge path for the trapped charge. Therefore, the voltage was maintained at −8.165 kV. The switching operation in simulation is shown in Fig. 9. Since the statistical analysis was conducted, the statistic switch was adopted. Phase A was set as the master switch in which the uniform distribution in a power frequency period was assumed. The slave switch with the standard deviation of 1 ms was adopted in phase B and phase C. Since in this study, the shortest line section was 20% of 2 km, ΔT of 0.1 μs was enough for switching transient studies assuming the wave traveling speed is equal to the light speed.
dominant frequency. As for the multi-conductor coupled lines, the frequency characteristics of each independent circuit in the modal domain can be analyzed using modal theory [21]. In the following, in order to introduce the variation of dominant frequency to the switching transient analysis, the hybrid line system model is established in ATP-EMTP based on the distribution network in Shaanxi Province, China. 3. Simulation model The simulation model of an asymmetrical hybrid line was established to study the variation of reclosing overvoltage with cable proportion, as shown in Fig. 8. For transient studies, the calculation of impedance and admittance of a hybrid line was performed using the Cable Parameters and Line Constants subroutines in the ATP-EMTP. The line structures are depicted in Fig. 3. Considering the frequency range of reclosing overvoltage, the line parameters were calculated at the frequency of 10 kHz using the Bergeron model [1]. By using the Verify button in the LCC dialog box from ATPDraw [26], it was shown that the zero-sequence impedance had the best agreement with the parameters of the frequency-dependent Exact PI at 10 kHz. The total length of the hybrid line was set to 2 km, and the steel pipe of cable was grounded through a 5 Ω resistance. As the maximum short-circuit current at a 10 kV busbar is 21.6 kA, the feeding network was modeled as a source impedance of 0.85 mH. Besides, the statistic switch was employed for CB-1, and 100 simulations were performed to conduct the statistical analysis on the TOV. In distribution network, as the load was directly connected to the distribution line, so the transformer with load impedance could not be ignored. The 630 kVA transformer was adopted and simulated as a
4. Statistical study of TOV in hybrid line Using the EMT model illustrated above, 100 simulations were performed. The focus was mainly on the reclosing overvoltage at the receiving end since higher overvoltage could be observed. Fig. 10 shows the cumulative probability of reclosing overvoltage. The probability of hybrid lines varied greatly at different cable proportions. When cable proportion was 20%, the overvoltage of the hybrid line was even more severe than that of the pure OHL. Further, the maximum value, the mean value, and the 2% value were calculated, and the variation of the statistical values with the cable proportion is illustrated in Fig. 11(a). Apparently, the TOV varied non-monotonously with cable proportion in Cable-OHL hybrid line.
Fig. 6. Distributed-parameter line considering the source impedance. 5
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Fig. 7. Variation of dominant frequency when the source impedance is considered.
6
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Fig. 8. Simulation model of a 10 kV hybrid line.
5.2. Propagation of modal voltage Applying the eigenvalues/vectors theory, the coupled multi-phase cable system can be decoupled into several independent modal circuits. Therefore, the voltage and current in modal domain can be transformed from those in phase domain by
Fig. 9. Operation of switches during the simulation in the time domain.
vph = Tv⋅vm
(11)
i ph = Ti⋅i m
(12)
where vph and iph are voltage and current in phase domain. vm and im are respectively voltage and current in modal domain. Tv and Ti are voltage and current transformation matrix, respectively. Based on modal theory, the phase voltage can be regarded as a superimposition of the voltage wave of each mode [17,21]. The analysis of the minimal value of TOV is based on the variation of dominant frequency and propagation of modal voltage. In this section, firstly, the determination of dominant mode is presented, and then, the propagation constants of modal voltage are calculated. 5.2.1. Determination of dominant mode As it is well known, the dominant frequency can represent the oscillation period of the TOV. In the meantime, the oscillation period of the dominant mode voltage can be derived using the dominant frequency [17,21]. Moreover, it is well known that earth-return mode is the dominant mode of overvoltage propagating along OHL. For a hybrid line, it is convenient to derive the dominant mode by transforming the overvoltage into modal domain [16,21]. The dominant mode voltage of both pure and hybrid line is presented in Fig. 13, where the non-zero initial conditions can be observed due to the effect of the trapped charges. Apparently, the earth-return mode still dominates the TOV in the OHL section of a hybrid line, yet the oscillation period is affected. Thus, in the following research, the earth-return mode of TOV at the receiving end is regarded as a dominant mode, and the aerial mode is regarded as a superimposed mode. Since the focus is on the overvoltage at the receiving end, the modal analysis is concentrated on the OHL section of a hybrid line. The dominant mode of a cable section can be further derived by modal transformation.
Fig. 10. Cumulative probability of the TOV.
Also, when lcable:lOHL = 600:1400, minimal values could be clearly observed. The overvoltage was 2.45 p.u. for both maximum and 2% value. The same phenomenon was also observed in [23,25] for lines with different length for a 380 kV network. The influence of the proportion of cable and OHL on TOV at the receiving end is analyzed in the following text based on the dominant frequency and propagation of TOV.
5. Analysis of variation of statistical values In this section, the variation of statistical values shown in Fig. 11(a) is further analyzed, based on the variation of dominant frequency explained in Section 2, and the propagation of modal voltage. The results regarding the dominant frequency are illustrated in Section 5.1, and then the modal analysis is presented in Section 5.2.
5.2.2. Calculation of propagation constants The propagation constants of independent modal voltage obtained by application of the eigenvalues/vectors theory to multi-conductor system are presented in Table 2 [21]. The line parameters of a hybrid line used in modal transform are derived in Section 3.
5.1. Variation of dominant frequency As can be observed in Section 2, the dominant frequency decreases monotonously with the cable proportion considering the effect of the source impedance. According to the results presented in Fig. 12, the dominant frequency of the TOV in the simulation test decreased with the cable proportion increasing, which coincides with the variation shown in Fig. 5(b). Therefore, it can be concluded that the oscillation period increases with the cable proportion.
5.3. Investigation of non-monotonous variation Using the frequency characteristics and propagation of modal voltage described above, the appearing of the minimal value of statistical results can be explained. The overvoltage waveform in phase A of a hybrid line at the cable proportion of 20%, 30% and 40% is illustrated 7
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Fig. 11. Variation of the statistical values and TOV at the junction and receiving end.
(1) Cable 20%, OHL 80% As depicted in Fig. 11(b), the overvoltage at the junction started to rise at 40.0017 ms after the voltage wave passed through the cable section. Then, the propagation of the aerial mode voltage from the head end of the OHL to the receiving end lasted approximately 5.37 μs, and thus started at 40.0179 ms; the second wave of the aerial mode arrived at the receiving end again after 10.8 μs. Considering the oscillation period of 51.39 μs, the dominant mode voltage, therefore, reached the highest value when the second wave of the superimposed mode arrived at the receiving end, which resulted in higher overvoltage. (2) Cable 30%, OHL 70% Similarly, in this case, after the first arriving at receiving end, the aerial mode voltage was reflected back to the head end of the OHL after 4.69 μs. Due to the fact that the propagation velocity was not influenced by the line length, the propagation time became lower as the OHL proportion decreased. The voltage wave of superimposed mode reached the receiving end every 9.2 μs, as shown in Fig. 11(c). Since the oscillation period was 59.14 μs, when the second wave of the aerial mode voltage arrived at the receiving end, the dominant mode voltage still had not reached the highest value, and the distortion of the TOV could be clearly observed. When the dominant mode voltage reached the highest value at 40.0179 ms, the aerial mode was reflected back to the head end of the OHL. Thus, comparing with the waveform depicted in Fig. 11(b), the maximum value of the TOV at the receiving end was relatively lower. (3) Cable 40%, OHL 60% Applying the same approach described above, it is known that when
Fig. 12. The dominant frequency of the TOV at different cable proportions.
in Fig. 11(b)–(d), respectively. Using the propagation velocity of the aerial mode presented in Table 2, i.e., the velocity of 298 m/μs, together with the OHL section length, the propagation time of modal voltage wave can be calculated, and the obtained values are given in Table 3. Considering the dominant frequency of the TOV shown in Fig. 12, the oscillation period of the dominant mode voltage, i.e., the earth-return mode, can be estimated. The calculation results coincide well with the time intervals of waveforms obtained by the simulation which are presented in Fig. 11. Three cases shown in Fig. 11(b)–(d) are analyzed with the use of the propagation characteristics of modal voltage waves, so that the minimal value of statistical results can be explained. 8
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Fig. 13. The voltage of phase A and earth-return mode.
6. Conclusion
Table 2 Propagation constants of the modal voltage of the OHL and cable. Transmission line
Mode
OHL
Earth-return mode Aerial mode 1 Aerial mode 2 Earth-return mode Pipe-return mode Inter-conductor 1 Inter-conductor 2
Cable
This paper presents both theoretical analyses and simulation study on the dominant frequency characteristics and propagation characteristics of TOV in 10 kV cable-OHL hybrid lines. The main conclusions are as follows.
Propagation velocity (m/μs) 257.08 298.21 297.92 19.70 176.25 180.76 180.76
(1) Using the continuity conditions of voltage and current at the junction in a distributed-parameter circuit, based on theoretical derivation, a convenient method to estimate the dominant frequency in cable-OHL hybrid lines is achieved. (2) he calculation results show that the dominant frequency of a hybrid line differs greatly from that of a pure line. When the effect of source impedance is considered, the dominant frequency of TOV monotonously decreases with the cable proportion. Each influence factor can be clearly observed from the denominator function Fs(ω). (3) Since the peak value of TOV is strongly affected by the superimposition timing of modal voltage, the statistical values of the TOV at the receiving end varies non-monotonously with the cable proportion in hybrid lines but shows a decreasing trend.
Table 3 Propagation time and oscillation period of the TOV of hybrid line. Line configuration
Cable 20%, OHL 80% Cable 30%, OHL 70% Cable 40%, OHL 60%
Propagation time (μs)
Oscillation period (μs)
Calculation
Simulation
Calculation
Simulation
5.37 4.69 4.00
5.40 4.60 4.00
51.39 59.14 66.40
53.20 60.30 64.29
The characteristics of TOV in hybrid lines with different cable proportion and different configurations can be investigated clearly using the method presented in this paper, which can provide a reference for assessing the risk and designing the reclosing strategy of hybrid lines.
the moment started at 40.003 ms, the fourth wave of aerial mode arrived at the receiving end after 28 μs. At the same time, the dominant mode voltage reached the highest value at 40.031 ms, and thus the maximum value of TOV in this case was higher than that at the cable proportion of 30%. Considering the attenuation of aerial mode when the fourth wave arrived, the overvoltage was relatively lower than that at the cable proportion of 20%. Consequently, the analysis results of the wave propagation and frequency coincide well with the TOV waveforms obtained by ATPEMTP simulation. The approach used in this work can also be applied to the asymmetrical structures with the cable section behind the OHL, or a symmetrical hybrid line, as long as the dominant frequency is modified accordingly.
Conflict of interests The authors declare that they have no conflict of interest. Acknowledgment This work was supported by the National Natural Science Foundation of China (51577150).
Appendix A In this part, the calculation of voltage and current along a hybrid line is presented. First, the equations of voltage and current are derived. Based on the single-phase hybrid line shown in Fig. 2(b), the voltage v and current i in segment 1 and segment 2 are given by Eqs. (A.1)–(A.2) and Eqs. (A.3)–(A.4), respectively.
v1 = A1 ⋅cosh Γ1x + B1⋅sinh Γ1x
(A.1)
i1 = −Y10 (A1 ⋅sinh Γ1x + B1⋅cosh Γ1x )
(A.2)
v2 = A2 ⋅cosh Γ2 x + B2⋅sinh Γ2 x
(A.3)
i2 = −Y20 (A2 ⋅sinh Γ2 x + B2⋅cosh Γ2 x )
(A.4)
where v1 and i1 correspond to segment 1, and v2 and i2 correspond to segment 2. Then, the boundary conditions at the sending end, receiving end, and the joint point can be expressed as:
9
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⎧ v1 = E , x = 0 ⎪ v1 = v2, x = l1 ⎨i1 = i2, x = l1 ⎪ l = 0, x = l + l 1 2 ⎩2
(A.5)
By substituting the boundary conditions given by Eq. (A.5), the following equations can be derived:
E = A1
(A.6)
Y10 (A1 ⋅sinh Γ1 l1 + B1⋅cosh Γ1 l1) = Y20 (A2 ⋅sinh Γ2 l1 + B2⋅cosh Γ2 l1)
(A.7)
A1 ⋅sinh Γ1 l1 + B1⋅cosh Γ1 l1 = A2 ⋅sinh Γ2 l1 + B2⋅cosh Γ2 l1
(A.8)
0 = Y20 [A2 ⋅sinh Γ2 (l1 + l2) + B2⋅cosh Γ2 (l1 + l2)]
(A.9)
Lastly, the values of coefficients A1, B1, A2 and B2 can be calculated by Eqs. (A.6)–(A.9), and they are given by Eqs. (A.10)–(A.13), respectively. Thus, voltage and current along each segment can be derived. By substituting Eqs. (A.12) and (A.13) into Eqs. (A.3) and (A.4), the voltage and current at the receiving end are derived and given by Eq. (6) and (7), respectively. (A.10)
A1 = E
B1 = −
a⋅Y10 (dn − cm) − b⋅Y20 (cn − dm) ⋅E b⋅Y10 (dn − cm) − a⋅Y20 (cn − dm)
(A.11)
a ⋅ Y (dn − cm) − b ⋅ Y (cn − dm)
A2 =
n⋅[b − a⋅ b ⋅ Y10 (dn − cm) − a ⋅ Y20 (cn − dm) ] 10
20
dn − cm
⋅E
(A.12)
a ⋅ Y (dn − cm) − b ⋅ Y (cn − dm)
B2 = −
m⋅[b − a⋅ b ⋅ Y10 (dn − cm) − a ⋅ Y20 (cn − dm) ] 10
dn − cm
20
⋅E
(A.13)
where a = sinh(Γ1l1), b = cosh(Γ1l1), c = sinh(Γ2l1), d = cosh(Γ2l1), m = sinh[Γ2(l1+l2)], and n= cosh[Γ2(l1+l2)]; Γ1 and Γ2 are the propagation constants of two segments in a hybrid line respectively; l1 and l2 are the lengths of the two segments, as shown in Fig. 2(b); lastly, Y10 and Y20 represent the characteristic admittances.
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