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PEEC simulation of lightning over-voltage surge with corona discharges on the over head wires
Electric Power Systems Research 180 (2020) 106118
Contents lists available at ScienceDirect
Electric Power Systems Research journal homepage: www.elsevier.com/locate/epsr
PEEC simulation of lightning over-voltage surge with corona discharges on the over head wires
T
Peerawut Yutthagowitha,*, Thang H. Tranb, Yoshihiro Babac, Akihiro Ametanid, Vladimir A. Rakove,f a
Department of Electrical Engineering, Faculty of Engineering, King Mongkut’s Institute of Technology Ladkrabang, Chalongkrung Road, Ladkrabang, Bangkok, 10520, Thailand b Department of Creative Engineering, National Institute of Technology, Tsuruoka College, Tsuruoka, Japan c Department of Electrical and Electronic Engineering, Graduate School of Science and Engineering, Doshisha University, Kyoto, Japan d Department of Electrical Engineering, Polytechnique Montreal, Montreal, Canada e Department of Electrical and Computer Engineering, University of Florida, Florida, USA f Institute of Applied Physics, Russian Academy of Sciences, Nizhny Novgorod, Russia
A R T I C LE I N FO
A B S T R A C T
Keywords: Corona model Lightning surge over-voltage Partial element equivalent circuit (PEEC) method Overhead lines
In this paper, a simplified corona discharge model is adopted with a partial element equivalent circuit (PEEC) method in the time domain for simulation of lightning over-voltage surge. Effects of corona discharge to the voltage distortion and the electromagnetic coupling of the overhead wires are presented. In addition, the effect on the voltage measuring system to the induced voltage on its nearby parallel wire is presented and discussed. The corona progression from an overhead wire, to which a high voltage impulse is applied, is represented by the radial expansion of the conducting region. Undesired oscillations of the computed waveforms found in the authors’ previous paper have been discarded by iteration process for calculation of the corona radius and the voltage of each PEEC element at each calculated time step. To confirm the validity of the presented model with the PEEC method, the calculated results are compared with experimental results. The calculated results agree well with the corresponding experimental ones. This shows that the PEEC method in the time domain with the corona model is of use in simulations of lightning surges propagating along overhead wires.
1. Introduction Corona discharge emanated from an overhead wire distorts the wavefront of lightning surge voltage propagating along the wire, and increases the magnitude of voltage induced on its nearby parallel wire because of the increase of coupling between the wires. A simple and sufficiently accurate approach, which can also consider corona discharge effects, is necessary for lightning surge simulations aiming at economical insulation design of power transmission and distribution systems. There are several models of corona discharges for surge simulations for lighting over-voltage surge having been proposed and confirmed their validities with the experimental results [1–9]. The proposed models were implemented with the distributed-constant circuit representation [10]. Also, a simplified corona model [6,7] has been proposed for lightning surge simulations using the finite different time domain (FDTD) method for solving discretized Maxwell’s equations
⁎
[11]. Additionally, the simplified corona model has been adopted with the PEEC method successfully [12]. However, undesired oscillations on the calculated waveforms were found due to the nonlinear characteristic of the corona model and the PEEC model neglecting electromagnetic retardation. Especially, in some cases with long overhead wires the corona model adopted with the PEEC method leads to unstable calculated results. In this paper being the extended version of the authors’ previous paper [12], the simplified corona model [6] to surge simulations using the partial element equivalent circuit (PEEC) method [13–15] in the time domain is presented. The PEEC method in the time domain neglects electromagnetic retardation between divided segments, but it is still sufficiently accurate for analysing relatively low-frequency surges. Also, instability coming with the proposed method has been solved by iteration process for calculation of the corona radius and the voltage of each PEEC element at each calculated time step. The accuracy of the presented method is tested by comparison with experimental results
Corresponding author. E-mail address: [email protected] (P. Yutthagowith).
https://doi.org/10.1016/j.epsr.2019.106118 Received 5 April 2019; Received in revised form 12 September 2019; Accepted 7 November 2019 Available online 03 December 2019 0378-7796/ © 2019 Elsevier B.V. All rights reserved.
Electric Power Systems Research 180 (2020) 106118
P. Yutthagowith, et al.
Fig. 1. Longitudinal and transversal currents on a thin wire structure in the PEEC model.
Fig. 3. Experimental set up and configurations of the 44-m conductor of Noda’s experiment [4].
found in Refs. [1–4].
ε Gc = Cc ⎛ 0 ⎞ ⎝σ ⎠
2. Modelling 2.1. Corona discharges
2.2. Adoption of corona model with the PEEC method
The corona streamer progression from an energized wire is represented by the radial expansion of cylindrical conducting region with a conductivity of 40 μS/m to 100 μS/m depended on the environmental condition. The critical electric field, E0, for corona discharge initiation on the surface of an overhead with radius, r0, was provided by Hartmann in 1984 [16], and it is given by Eq. (1).
0.1269 E0 = m⋅2.594 × 106 ⎜⎛1 + 0.4346 ⎟⎞ [V/m] r0 ⎝ ⎠
The PEEC method was derived from the mixed-potential integral equation (MPIE) for the free space by Ruehli [13] in 1974. Also, the PEEC method for the thin wire structure [14,15] as shown in Fig. 1 has been used in lightning surge analyses successfully. As illustrated in Fig. 2, the final results are interpreted Maxwell’s equations to a circuit model by inductance, capacitance, and resistance extraction including retardation in theory. The mutual electromagnetic coupling can also represented by the dependent voltage and current sources: Vij and Ii. For adoption of the corona model in the PEEC method, longitudinal currents (ILi) along the elements still flow inside the conductors during the corona discharge process, and thus it is estimated that the series self and mutual impedances of the elements are not changed and only the shunt admittances of the elements are affected by the corona discharge. During the corona discharge process, each corona radius of the element is varied when the transverse electric field on the surface of the element reaches the critical electric field (E0), and the corona radius is calculated by Eq. (2) at each calculated step time. As shown in Fig. 1, the equivalent circuit of the corona model of each element is represented by shunt admittance (Cci and Gci). This shunt admittance is connected in series with the shunt admittance (C'i and G'i) of the PEEC element with the corona radius (rc). Under the corona discharge process, the corona radius is progresses and expanded to have the off-set electric field intensity (Ec) on the surface of the corona shell. The corona radius is always greater than the radius of the over-head wire, and it leads to the shunt admittance (C'i and G'i) between the corona shell surface and ground is always greater than the shunt admittance (Ci and Gi) of the normal wire without corona discharge effect. Therefore, the connection in series of the admittance of the normal element (C'i and G'i) and that of the corona shell (Cci and Gci) in Figs. 1 and 2 leads to increase of the total capacitance and conductance. The effect of increasing of the total admittance can be noticed form the increase of the ratio of the charge and the applied voltage which will be presented in Section 3. The corona radius from Eq. (2) and the voltage of the PEEC element are
(1)
The constant m is set to be 0.5. It is noted that all parameters in the corona model are the same as those employed by Thang et al. in their FDTD computation [6,7]. The corona discharge will progress until the corona radius expands to have the off-set corona electric field intensity, Ec. In this paper, Ec is set to 0.5 MV/m for positive polarity and 1.5 MV/m for negative polarity [6,7]. The relation of the electric field and the corona radius (rc) are given by Eq. (2). When the electric filed stress on the conductor surface reaches the critical electric field (E0), the corona discharge is initiated and expanded to corona radius (rc) having the off-set electric field intensity (Ec). Using Eq. (2), the corona radius can be calculated by an iterative numerical method [17].
U = r0⋅⎡E0 − ⎢ ⎣
Ec (2h − rc ) rc ⎤ r (2h − rc ) rc r ⋅ln ⎛ c ⎞ + ⋅ln ⎛ c ⎞ (2h − r0 ) h ⎥ r 2 h r 0 ⎝ ⎠ ⎝ 0⎠ ⎦ ⎜
⎟
⎜
⎟
(2)
Based on the simplified corona model [6,7], the radial progression of the corona shell is assumed. The per-unit capacitance (Cc) and conductance (Gc) of the corona shell can be calculated from the capacitance and conductance of the coaxial cylinders as expressed in Eqs. (3) and (4).
Cc = 2πε0/ln
() rc r0
(4)
(3)
Fig. 2. Equivalent circuit of the PEEC element. 2
Electric Power Systems Research 180 (2020) 106118
P. Yutthagowith, et al.
Fig. 4. Applied positive voltages and corona currents collected from Noda’s experiment [4] and calculated by the proposed method.
Fig. 5. Applied negative voltages and corona currents collected from Noda’s experiment [4] and calculated by the proposed method.
calculated by iteration process to obtain the appropriate radius and the voltage of the PEEC element in each time step, and its radius is updated at each calculated time step for avoiding the instability in the simulation. The corona current is calculated by the sum of the transversal currents (ITi) which is the different longitudinal current of the adjoining elements of the conductor wire. The corona charge is calculated by a numerical integration of the corona current.
In the first case, it should be considered Noda’s experiment [4] with a short conductor length of 44 m. The Noda’s experiment configuration is illustrated in Fig. 3. The matching resistor of 460 Ω (the characteristic impedance of the bare wire) was connected at the receiving end of the conductor for avoiding the reflection wave from the receiving end. The lightning surge over-voltages with different voltage peaks and polarities were applied to a 5-mm radius conductor installed at 2 m above ground. The applied voltages and the corona discharge currents were recorded in the experiments. The corona current which was measured at the sending end or the voltage source side. The current was injected from the voltage source side and flow to the ground plane as the return path. The corona charge was calculated by a numerical integration of the
3. Validation of the presented model To confirm the validity of the presented model, the calculated results by the PEEC method with the corona discharge model are compared with the experimental results collected from Refs. [1–4]. 3
Electric Power Systems Research 180 (2020) 106118
P. Yutthagowith, et al.
Fig. 6. Corona charge-applied voltage curves collected from Noda’s experiment [4] and calculated by the proposed method.
Fig. 7. Computed corona radii by the proposed method.
Fig. 8. The computed voltage waveforms by the proposed method at the different points along the conductor in comparison with Inoue’s experimental results.
Fig. 9. The computed voltage waveforms by the proposed method at the different points along the conductor in comparison with Wagner’s experimental results.
the presented model agree quite well with the results collected from Noda’s experiment [4]. From the results shown in Figs. 4 and 5, the charges calculated by numerical integration of the measured corona discharge currents and applied voltages are plotted in Fig. 6. Due to the corona effect the corona radius (rc) is enlarged from the wire radius (r0) as shown in Figs. 2 and 3, and it follows the applied voltage until the electrical field intensity on the corona shell surface reaches to the off-set electric field intensity (Ec). The corona radius can be calculated by Eq. (2), and the
measured current. The PEEC method with the corona discharge model was utilized to calculate the corona currents, charges, and corona radii. For the numerical analysis, the conductors of the system were divided into cylindrical segments of 2 m in length. The calculated time step were set to be 10 ns. The conductivity of the corona discharge region was set to be 40 μS/m. Figs. 4 and 5 show the applied voltages and the corona currents collected from Noda’s experiment and computed by the proposed method. It appears from these results that the calculated results using 4
Electric Power Systems Research 180 (2020) 106118
P. Yutthagowith, et al.
for the negative polarity were applied to a 21-mm radius, and the conductor installed at 14 m above ground. The matching resistor of 430 Ω (the characteristic impedance of the bare wire) was connected at the receiving end of the conductor for avoiding the reflection wave from the receiving end. The voltages along the conductor at the distances of 6600 m, 1300 m, and 2200 m from the sending end were recorded. The PEEC method with the corona discharge model was utilized to calculate the voltage at different points (660 m, 1300 m, and 2200 m from the sending end) along the conductor. For the numerical analysis, the conductors of the system were divided into cylindrical segments of about 10 m in length. The calculated time step were set to be 10 ns. The conductivity of the corona discharge region was set to be 100 μS/m which is the same as that employed by Thang et al. [7] in the FDTD simulations due to the best match with the experimental results. Fig. 9 shows the comparison of the calculated voltages at the different point along the conductor and collected from Wagner’s experiment. It also appears from these results that the calculated results using the presented model agree quite well with the results collected from the experiment, which confirms the validity of the presented method. In Figs. 8 and 9, it appears that the significant increase of rise time and distortion of the voltage waveforms along the propagation distance due to the corona effect. Additionally, the execution time for one simulation with a few hundred PEEC elements of based on Matlab programing was only a few minutes. In the last case, it should be considered Noda’s experiment [4] of which configuration is shown in Fig. 10. The impulse voltages was applied to a 1.8-mm radius conductor installed at 2 m above ground. The applied and induced voltages on the parallel conductors were recorded. It is found that the calculated induced voltage is quite different from the experimental results. It is found that on the result in Fig. 11, the induced voltage is attenuated on the late time after 2 μs. However, the calculated induced voltage agrees well with the FDTD-computed result by Thang et al. [6]. The reason of the deviation might be from the effect of a voltage measuring system. Normally, a resistive voltage divider is employed to measure the impulse voltage, and its resistance is in the range from 10 kΩ to 50 kΩ. To confirm the effect of the measuring system to the recorded waveform in the experiment, a resistive voltage divider with its resistance of from 20 kΩ to 50 kΩ is connected at an end of the nearby conductor. It is also found that on the result in Fig. 11, the most appropriate divider’s resistance is 20 kΩ and the calculated induced voltages agree quite well with the results collected from Noda’s experiment.
Fig. 10. Experimental set up and configurations of two parallel conductors of Noda’s experiment [4].
calculated corona radii are expressed in Fig. 7. From the results in Fig. 6, it is noticed that under the corona discharge process the ratio of the corona charge and the applied voltage is greater than the geometrical capacitance of the over-head wires as shown in Fig. 3, and it is confirmed that the corona discharge affects increasing shunt admittance and power loss of the over-head line. In the second case, it should be considered Inoue’s experiment [1,2] with a long conductor length of 1.4 km. The lightning surge voltages with the peak voltage of 1.58 MV for the positive polarity and of 1.67 MV for the negative polarity were applied to a 12.65-mm radius conductor at the sending end, and the conductor was installed at 22.2 m above ground. The matching resistor of 490 Ω (the characteristic impedance of the bare wire) was connected at the receiving end of the conductor for avoiding the reflection wave from the receiving end. The voltages along the conductor at the distances of 350 m, 700 m, and 1050 m from the sending end were recorded. The PEEC method with the corona discharge model was utilized to calculate the voltage at different points (350 m, 700 m, and 1050 m from the sending end) along the conductor. For the numerical analysis, the conductors of the system were divided into cylindrical segments of about 10 m in length. The calculated time step were set to be 10 ns. For the sake of confirming the validity of the presented method, the conductivity of the corona discharge region was set to be 40 μS/m which is the same as that employed by Thang et al. [7] in the FDTD simulations due to the best match with the experimental results. Fig. 8 shows the comparison of the voltages at the different point along the conductor calculated by the proposed method and collected from Inoue’s experiment. It appears from these results that the calculated results using the presented model agree reasonably well with the results collected from the experiment. In the third case, it should be considered Wagner’s experiment [3] with a long conductor length of 2.2 km. The lightning surge voltages with the peak voltage of 1.65 MV for the positive polarity and 1.75 MV
4. Conclusions The simplified corona model has been adopted in surge simulations using the PEEC method in the time domain. Undesired oscillations of
Fig. 11. Applied voltages and the computed induced voltages in comparison with the experimental results. 5
Electric Power Systems Research 180 (2020) 106118
P. Yutthagowith, et al.
the computed voltage and current waveforms found in the results of the previous authors’ paper have been discarded by iteration process for calculation of the corona radius and the voltage of each PEEC element at each calculated time step. In each simulation, the off-set electric field intensity (Ec) for the corona propagation was set to be 0.5 MV/m for the positive voltage application, and 1.5 MV/m for the negative voltage application. For confirming the validity of the proposed method, some simulations based on the proposed method were carried out in comparison with the experimental results collected from Noda’s, Inoue’s, and Wagner’s experiments. Additionally, the effect of a voltage measuring system was also addressed in this paper. It is found that the voltage measuring system in the form of a resistive voltage divider represents as the loss in the system, and affects the attenuation of the measured voltage waveform on the late time after 2 μs. It appears that all computed results agree quite well with the experimental results, and the execution time of the proposed method is rather short when it is compared with that of other numerical electromagnetic analysis methods. The analysis and results in this paper show that the proposed method has fairly accuracy and efficiency, and it is an attractive choice in lightning over-voltage analyses including the corona discharge model.
lines, AIEE Trans. Power Appar. Syst. 73 (1954) 196. [4] T. Noda, Development of a Transmission-line Model Considering the Skin and Corona Effects for Power System Transient Analysis. Ph.D. Dissertation, Dept. Electr. Eng., Doshisha Univ., Kyoto, Japan, 1996. [5] T. Noda, T. Ono, H. Matsubara, H. Motoyama, S. Sekioka, A. Ametani, Chargevoltage curves of surge corona on transmission lines: two measurement methods, IEEE Trans. Power Deliv. 18 (January (1)) (2003) 307–314. [6] T.H. Thang, Y. Baba, N. Nagaoka, A. Ametani, J. Takami, S. Okabe, V.A. Rakov, A simplified model of corona discharge on overhead wire for FDTD computations, IEEE Trans. Electromagn. Compat. 54 (June (3)) (2012) 585–593. [7] T.H. Thang, Y. Baba, N. Nagaoka, A. Ametani, J. Takami, S. Okabe, V.A. Rakov, FDTD simulation of lightning surges on overhead wires in the presence of corona discharge, IEEE Trans. Electromagn. Compat. 54 (December (6)) (2012) 1234–1243. [8] J.L. He, X. Zhang, P.C. Yang, S.M. Chen, R. Zeng, Attenuation and deformation characteristics of lightning impulse corona traveling along bundled transmission lines, Electr. Power Syst. Res. 118 (January (SI)) (2015) 29–36. [9] P.C. Yang, S.M. Chen, J.L. He, Lightning impulse corona characteristic of 1000 kV UHV transmission lines and its influences on lightning overvoltage analysis results, IEEE Trans. Power Deliv. 28 (October (4)) (2013) 2518–2525. [10] C.A. Nucci, S. Guerrieri, M.T. Correia de Barros, F. Rachidi, Influence of corona on the voltages induced by nearby lightning on overhead distribution lines, IEEE Trans. Power Deliv. 15 (October (4)) (2000) 1265–1273. [11] K.S. Yee, Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media, IEEE Trans. Antennas Propag. 14 (May (3)) (1966) 302–307. [12] P. Yutthagowith, T.H. Tran, A. Ametani, Y. Baba, V.A. Rakov, Application of a simplified corona discharge model to a lightning surge simulation with the PEEC method, Proc. 2018 34th International Conference on Lightning Protection (ICLP 2018), September, 2018, pp. 1–4. [13] A.E. Ruehli, Equivalent circuit models for three-dimentional multiconductor systems, IEEE Trans. Microw. Theory Tech. MTT-22 (March (3)) (1974) 216–221. [14] P. Yutthagowith, A. Ametani, N. Nagaoka, Y. Baba, Application of the partial element equivalent circuit method to analysis of transient potential rises in grounding systems, IEEE Trans. Electromagn. Compat. 53 (August (3)) (2011) 726–736. [15] P. Yutthagowith, A. Ametani, F. Rachidi, N. Nagaoka, Y. Baba, Application of a partial element equivalent circuit method to lightning surge analyses, Electr. Power Syst. Res. 94 (January (4)) (2013) 30–37. [16] G. Hartmann, Theoretical evaluation of Peek’s law, IEEE Trans. Ind. Appl. 20 (November (6)) (1984) 1647–1651. [17] J.F. Guillier, M. Poloujadoff, M. Rioual, Damping model of travelling waves by corona effect along extra high voltage three phase lines, IEEE Trans. Power Deliv. 10 (September (4)) (1995) 1851–1861.
Acknowledgements Authors would like to acknowledge the Faculty of Engineering, King Mongkut’s Institute of Technology Ladkrabang, Thailand for the financial support of this research work. References [1] A. Inoue, Study on propagation characteristics of high-voltage traveling waves with corona discharge (in Japanese), CRIEPI Rep. 114 (1983) 123. [2] A. Inoue, Propogation analysis of overvoltage surges with corona based upon versus voltage curve, IEEE Trans. Power Appar. Syst. PAS-104 (3) (1985) 655–662. [3] C.F. Wagner, I.W. Gross, B.L. Lloyd, High-voltage impulse tests on transmission
6
Contents lists available at ScienceDirect
Electric Power Systems Research journal homepage: www.elsevier.com/locate/epsr
PEEC simulation of lightning over-voltage surge with corona discharges on the over head wires
T
Peerawut Yutthagowitha,*, Thang H. Tranb, Yoshihiro Babac, Akihiro Ametanid, Vladimir A. Rakove,f a
Department of Electrical Engineering, Faculty of Engineering, King Mongkut’s Institute of Technology Ladkrabang, Chalongkrung Road, Ladkrabang, Bangkok, 10520, Thailand b Department of Creative Engineering, National Institute of Technology, Tsuruoka College, Tsuruoka, Japan c Department of Electrical and Electronic Engineering, Graduate School of Science and Engineering, Doshisha University, Kyoto, Japan d Department of Electrical Engineering, Polytechnique Montreal, Montreal, Canada e Department of Electrical and Computer Engineering, University of Florida, Florida, USA f Institute of Applied Physics, Russian Academy of Sciences, Nizhny Novgorod, Russia
A R T I C LE I N FO
A B S T R A C T
Keywords: Corona model Lightning surge over-voltage Partial element equivalent circuit (PEEC) method Overhead lines
In this paper, a simplified corona discharge model is adopted with a partial element equivalent circuit (PEEC) method in the time domain for simulation of lightning over-voltage surge. Effects of corona discharge to the voltage distortion and the electromagnetic coupling of the overhead wires are presented. In addition, the effect on the voltage measuring system to the induced voltage on its nearby parallel wire is presented and discussed. The corona progression from an overhead wire, to which a high voltage impulse is applied, is represented by the radial expansion of the conducting region. Undesired oscillations of the computed waveforms found in the authors’ previous paper have been discarded by iteration process for calculation of the corona radius and the voltage of each PEEC element at each calculated time step. To confirm the validity of the presented model with the PEEC method, the calculated results are compared with experimental results. The calculated results agree well with the corresponding experimental ones. This shows that the PEEC method in the time domain with the corona model is of use in simulations of lightning surges propagating along overhead wires.
1. Introduction Corona discharge emanated from an overhead wire distorts the wavefront of lightning surge voltage propagating along the wire, and increases the magnitude of voltage induced on its nearby parallel wire because of the increase of coupling between the wires. A simple and sufficiently accurate approach, which can also consider corona discharge effects, is necessary for lightning surge simulations aiming at economical insulation design of power transmission and distribution systems. There are several models of corona discharges for surge simulations for lighting over-voltage surge having been proposed and confirmed their validities with the experimental results [1–9]. The proposed models were implemented with the distributed-constant circuit representation [10]. Also, a simplified corona model [6,7] has been proposed for lightning surge simulations using the finite different time domain (FDTD) method for solving discretized Maxwell’s equations
⁎
[11]. Additionally, the simplified corona model has been adopted with the PEEC method successfully [12]. However, undesired oscillations on the calculated waveforms were found due to the nonlinear characteristic of the corona model and the PEEC model neglecting electromagnetic retardation. Especially, in some cases with long overhead wires the corona model adopted with the PEEC method leads to unstable calculated results. In this paper being the extended version of the authors’ previous paper [12], the simplified corona model [6] to surge simulations using the partial element equivalent circuit (PEEC) method [13–15] in the time domain is presented. The PEEC method in the time domain neglects electromagnetic retardation between divided segments, but it is still sufficiently accurate for analysing relatively low-frequency surges. Also, instability coming with the proposed method has been solved by iteration process for calculation of the corona radius and the voltage of each PEEC element at each calculated time step. The accuracy of the presented method is tested by comparison with experimental results
Corresponding author. E-mail address: [email protected] (P. Yutthagowith).
https://doi.org/10.1016/j.epsr.2019.106118 Received 5 April 2019; Received in revised form 12 September 2019; Accepted 7 November 2019 Available online 03 December 2019 0378-7796/ © 2019 Elsevier B.V. All rights reserved.
Electric Power Systems Research 180 (2020) 106118
P. Yutthagowith, et al.
Fig. 1. Longitudinal and transversal currents on a thin wire structure in the PEEC model.
Fig. 3. Experimental set up and configurations of the 44-m conductor of Noda’s experiment [4].
found in Refs. [1–4].
ε Gc = Cc ⎛ 0 ⎞ ⎝σ ⎠
2. Modelling 2.1. Corona discharges
2.2. Adoption of corona model with the PEEC method
The corona streamer progression from an energized wire is represented by the radial expansion of cylindrical conducting region with a conductivity of 40 μS/m to 100 μS/m depended on the environmental condition. The critical electric field, E0, for corona discharge initiation on the surface of an overhead with radius, r0, was provided by Hartmann in 1984 [16], and it is given by Eq. (1).
0.1269 E0 = m⋅2.594 × 106 ⎜⎛1 + 0.4346 ⎟⎞ [V/m] r0 ⎝ ⎠
The PEEC method was derived from the mixed-potential integral equation (MPIE) for the free space by Ruehli [13] in 1974. Also, the PEEC method for the thin wire structure [14,15] as shown in Fig. 1 has been used in lightning surge analyses successfully. As illustrated in Fig. 2, the final results are interpreted Maxwell’s equations to a circuit model by inductance, capacitance, and resistance extraction including retardation in theory. The mutual electromagnetic coupling can also represented by the dependent voltage and current sources: Vij and Ii. For adoption of the corona model in the PEEC method, longitudinal currents (ILi) along the elements still flow inside the conductors during the corona discharge process, and thus it is estimated that the series self and mutual impedances of the elements are not changed and only the shunt admittances of the elements are affected by the corona discharge. During the corona discharge process, each corona radius of the element is varied when the transverse electric field on the surface of the element reaches the critical electric field (E0), and the corona radius is calculated by Eq. (2) at each calculated step time. As shown in Fig. 1, the equivalent circuit of the corona model of each element is represented by shunt admittance (Cci and Gci). This shunt admittance is connected in series with the shunt admittance (C'i and G'i) of the PEEC element with the corona radius (rc). Under the corona discharge process, the corona radius is progresses and expanded to have the off-set electric field intensity (Ec) on the surface of the corona shell. The corona radius is always greater than the radius of the over-head wire, and it leads to the shunt admittance (C'i and G'i) between the corona shell surface and ground is always greater than the shunt admittance (Ci and Gi) of the normal wire without corona discharge effect. Therefore, the connection in series of the admittance of the normal element (C'i and G'i) and that of the corona shell (Cci and Gci) in Figs. 1 and 2 leads to increase of the total capacitance and conductance. The effect of increasing of the total admittance can be noticed form the increase of the ratio of the charge and the applied voltage which will be presented in Section 3. The corona radius from Eq. (2) and the voltage of the PEEC element are
(1)
The constant m is set to be 0.5. It is noted that all parameters in the corona model are the same as those employed by Thang et al. in their FDTD computation [6,7]. The corona discharge will progress until the corona radius expands to have the off-set corona electric field intensity, Ec. In this paper, Ec is set to 0.5 MV/m for positive polarity and 1.5 MV/m for negative polarity [6,7]. The relation of the electric field and the corona radius (rc) are given by Eq. (2). When the electric filed stress on the conductor surface reaches the critical electric field (E0), the corona discharge is initiated and expanded to corona radius (rc) having the off-set electric field intensity (Ec). Using Eq. (2), the corona radius can be calculated by an iterative numerical method [17].
U = r0⋅⎡E0 − ⎢ ⎣
Ec (2h − rc ) rc ⎤ r (2h − rc ) rc r ⋅ln ⎛ c ⎞ + ⋅ln ⎛ c ⎞ (2h − r0 ) h ⎥ r 2 h r 0 ⎝ ⎠ ⎝ 0⎠ ⎦ ⎜
⎟
⎜
⎟
(2)
Based on the simplified corona model [6,7], the radial progression of the corona shell is assumed. The per-unit capacitance (Cc) and conductance (Gc) of the corona shell can be calculated from the capacitance and conductance of the coaxial cylinders as expressed in Eqs. (3) and (4).
Cc = 2πε0/ln
() rc r0
(4)
(3)
Fig. 2. Equivalent circuit of the PEEC element. 2
Electric Power Systems Research 180 (2020) 106118
P. Yutthagowith, et al.
Fig. 4. Applied positive voltages and corona currents collected from Noda’s experiment [4] and calculated by the proposed method.
Fig. 5. Applied negative voltages and corona currents collected from Noda’s experiment [4] and calculated by the proposed method.
calculated by iteration process to obtain the appropriate radius and the voltage of the PEEC element in each time step, and its radius is updated at each calculated time step for avoiding the instability in the simulation. The corona current is calculated by the sum of the transversal currents (ITi) which is the different longitudinal current of the adjoining elements of the conductor wire. The corona charge is calculated by a numerical integration of the corona current.
In the first case, it should be considered Noda’s experiment [4] with a short conductor length of 44 m. The Noda’s experiment configuration is illustrated in Fig. 3. The matching resistor of 460 Ω (the characteristic impedance of the bare wire) was connected at the receiving end of the conductor for avoiding the reflection wave from the receiving end. The lightning surge over-voltages with different voltage peaks and polarities were applied to a 5-mm radius conductor installed at 2 m above ground. The applied voltages and the corona discharge currents were recorded in the experiments. The corona current which was measured at the sending end or the voltage source side. The current was injected from the voltage source side and flow to the ground plane as the return path. The corona charge was calculated by a numerical integration of the
3. Validation of the presented model To confirm the validity of the presented model, the calculated results by the PEEC method with the corona discharge model are compared with the experimental results collected from Refs. [1–4]. 3
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Fig. 6. Corona charge-applied voltage curves collected from Noda’s experiment [4] and calculated by the proposed method.
Fig. 7. Computed corona radii by the proposed method.
Fig. 8. The computed voltage waveforms by the proposed method at the different points along the conductor in comparison with Inoue’s experimental results.
Fig. 9. The computed voltage waveforms by the proposed method at the different points along the conductor in comparison with Wagner’s experimental results.
the presented model agree quite well with the results collected from Noda’s experiment [4]. From the results shown in Figs. 4 and 5, the charges calculated by numerical integration of the measured corona discharge currents and applied voltages are plotted in Fig. 6. Due to the corona effect the corona radius (rc) is enlarged from the wire radius (r0) as shown in Figs. 2 and 3, and it follows the applied voltage until the electrical field intensity on the corona shell surface reaches to the off-set electric field intensity (Ec). The corona radius can be calculated by Eq. (2), and the
measured current. The PEEC method with the corona discharge model was utilized to calculate the corona currents, charges, and corona radii. For the numerical analysis, the conductors of the system were divided into cylindrical segments of 2 m in length. The calculated time step were set to be 10 ns. The conductivity of the corona discharge region was set to be 40 μS/m. Figs. 4 and 5 show the applied voltages and the corona currents collected from Noda’s experiment and computed by the proposed method. It appears from these results that the calculated results using 4
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for the negative polarity were applied to a 21-mm radius, and the conductor installed at 14 m above ground. The matching resistor of 430 Ω (the characteristic impedance of the bare wire) was connected at the receiving end of the conductor for avoiding the reflection wave from the receiving end. The voltages along the conductor at the distances of 6600 m, 1300 m, and 2200 m from the sending end were recorded. The PEEC method with the corona discharge model was utilized to calculate the voltage at different points (660 m, 1300 m, and 2200 m from the sending end) along the conductor. For the numerical analysis, the conductors of the system were divided into cylindrical segments of about 10 m in length. The calculated time step were set to be 10 ns. The conductivity of the corona discharge region was set to be 100 μS/m which is the same as that employed by Thang et al. [7] in the FDTD simulations due to the best match with the experimental results. Fig. 9 shows the comparison of the calculated voltages at the different point along the conductor and collected from Wagner’s experiment. It also appears from these results that the calculated results using the presented model agree quite well with the results collected from the experiment, which confirms the validity of the presented method. In Figs. 8 and 9, it appears that the significant increase of rise time and distortion of the voltage waveforms along the propagation distance due to the corona effect. Additionally, the execution time for one simulation with a few hundred PEEC elements of based on Matlab programing was only a few minutes. In the last case, it should be considered Noda’s experiment [4] of which configuration is shown in Fig. 10. The impulse voltages was applied to a 1.8-mm radius conductor installed at 2 m above ground. The applied and induced voltages on the parallel conductors were recorded. It is found that the calculated induced voltage is quite different from the experimental results. It is found that on the result in Fig. 11, the induced voltage is attenuated on the late time after 2 μs. However, the calculated induced voltage agrees well with the FDTD-computed result by Thang et al. [6]. The reason of the deviation might be from the effect of a voltage measuring system. Normally, a resistive voltage divider is employed to measure the impulse voltage, and its resistance is in the range from 10 kΩ to 50 kΩ. To confirm the effect of the measuring system to the recorded waveform in the experiment, a resistive voltage divider with its resistance of from 20 kΩ to 50 kΩ is connected at an end of the nearby conductor. It is also found that on the result in Fig. 11, the most appropriate divider’s resistance is 20 kΩ and the calculated induced voltages agree quite well with the results collected from Noda’s experiment.
Fig. 10. Experimental set up and configurations of two parallel conductors of Noda’s experiment [4].
calculated corona radii are expressed in Fig. 7. From the results in Fig. 6, it is noticed that under the corona discharge process the ratio of the corona charge and the applied voltage is greater than the geometrical capacitance of the over-head wires as shown in Fig. 3, and it is confirmed that the corona discharge affects increasing shunt admittance and power loss of the over-head line. In the second case, it should be considered Inoue’s experiment [1,2] with a long conductor length of 1.4 km. The lightning surge voltages with the peak voltage of 1.58 MV for the positive polarity and of 1.67 MV for the negative polarity were applied to a 12.65-mm radius conductor at the sending end, and the conductor was installed at 22.2 m above ground. The matching resistor of 490 Ω (the characteristic impedance of the bare wire) was connected at the receiving end of the conductor for avoiding the reflection wave from the receiving end. The voltages along the conductor at the distances of 350 m, 700 m, and 1050 m from the sending end were recorded. The PEEC method with the corona discharge model was utilized to calculate the voltage at different points (350 m, 700 m, and 1050 m from the sending end) along the conductor. For the numerical analysis, the conductors of the system were divided into cylindrical segments of about 10 m in length. The calculated time step were set to be 10 ns. For the sake of confirming the validity of the presented method, the conductivity of the corona discharge region was set to be 40 μS/m which is the same as that employed by Thang et al. [7] in the FDTD simulations due to the best match with the experimental results. Fig. 8 shows the comparison of the voltages at the different point along the conductor calculated by the proposed method and collected from Inoue’s experiment. It appears from these results that the calculated results using the presented model agree reasonably well with the results collected from the experiment. In the third case, it should be considered Wagner’s experiment [3] with a long conductor length of 2.2 km. The lightning surge voltages with the peak voltage of 1.65 MV for the positive polarity and 1.75 MV
4. Conclusions The simplified corona model has been adopted in surge simulations using the PEEC method in the time domain. Undesired oscillations of
Fig. 11. Applied voltages and the computed induced voltages in comparison with the experimental results. 5
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the computed voltage and current waveforms found in the results of the previous authors’ paper have been discarded by iteration process for calculation of the corona radius and the voltage of each PEEC element at each calculated time step. In each simulation, the off-set electric field intensity (Ec) for the corona propagation was set to be 0.5 MV/m for the positive voltage application, and 1.5 MV/m for the negative voltage application. For confirming the validity of the proposed method, some simulations based on the proposed method were carried out in comparison with the experimental results collected from Noda’s, Inoue’s, and Wagner’s experiments. Additionally, the effect of a voltage measuring system was also addressed in this paper. It is found that the voltage measuring system in the form of a resistive voltage divider represents as the loss in the system, and affects the attenuation of the measured voltage waveform on the late time after 2 μs. It appears that all computed results agree quite well with the experimental results, and the execution time of the proposed method is rather short when it is compared with that of other numerical electromagnetic analysis methods. The analysis and results in this paper show that the proposed method has fairly accuracy and efficiency, and it is an attractive choice in lightning over-voltage analyses including the corona discharge model.
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Acknowledgements Authors would like to acknowledge the Faculty of Engineering, King Mongkut’s Institute of Technology Ladkrabang, Thailand for the financial support of this research work. References [1] A. Inoue, Study on propagation characteristics of high-voltage traveling waves with corona discharge (in Japanese), CRIEPI Rep. 114 (1983) 123. [2] A. Inoue, Propogation analysis of overvoltage surges with corona based upon versus voltage curve, IEEE Trans. Power Appar. Syst. PAS-104 (3) (1985) 655–662. [3] C.F. Wagner, I.W. Gross, B.L. Lloyd, High-voltage impulse tests on transmission
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