1. Introduction

2. Low–Frequency Oscillation and Overvoltage Characteristics of the Contact Line

3. Vehicle–Line–Institute Model and Oscillation Characteristics

3.1. Research on the model of train passing through the neutral zone

In the tested Beijing–Zhangjiakou railway line, the AT power supply mode is adopted, and an autotransformer is installed every 10–15km. One terminal is connected to the contact line, the other terminal is connected to the positive feeder, and the neutral point is directly connected to the rail. As shown in Figure 5, the AT power supply mode is equipped with a protection wire (PW), which is directly connected in parallel with the rail (R); a connector of protective wire (CPW) is added to connect the rail and the midpoint of the autotransformer with the protection wire. In order to reduce the impedance of the traction power system, increase the voltage at the end of the feeding section, and reduce the power loss, the downstream contact lines, positive feeders, and steel rails of the double–track traction line are connected in parallel at each AT.

Table 4. Equivalent circuit distributed capacitance calculation results.

Table 4. Equivalent circuit distributed capacitance calculation results.

c	T1	R1	P1	F1	T2	R2	P2	F2
T1	13.12	–0.83	–1.74	–2.04	–2.00	–0.41	–0.36	–0.56
R1	–0.83	20.12	–0.63	–0.23	–0.41	–0.44	–0.12	–0.09
P1	–1.74	–0.63	8.42	–1.12	–0.36	–0.12	–0.08	–0.11
F1	–2.04	–0.23	–1.12	8.28	–0.56	–0.09	–0.11	–0.26
T2	–2.00	–0.41	–0.36	–0.56	13.12	–0.83	–1.74	–2.04
R2	–0.41	–0.44	–0.12	–0.09	–0.83	20.12	–0.63	–0.23
P2	–0.36	–0.12	–0.08	–0.11	–1.74	–0.63	8.42	–1.12
F2	–0.56	–0.09	–0.11	–0.26	–2.04	–0.23	–1.12	8.28

The electrical parameters of the voltage transformer are shown in Table 5

Table 5. Voltage transformer inductance parameters.

Table 5. Voltage transformer inductance parameters.

Serial Number	JDZXW5–25J	JDZXW5A –25J	JDZXW7–25D
EMU Type	CRH3X	CRH5X	CR400XF
Primary DC Resistance 20℃	43160Ω	46488Ω	43160Ω
Primary inductance	11000H–12000H	11000–12000H	11000H–12000H

In the entire EMU, only the pantograph and the roof voltage transformer are connected to the contact line and the articulated electrical split. This main circuit can be equivalent to a high–level circuit composed of resistance, inductance and capacitance. The equivalent circuit is shown in Figure 6.

In Figure 6, Ua and Ub are traction power supplies; RS and LS are the resistance and inductance of the traction transformer converted according to the Thevenin circuit equivalent; R1 and L1 are the equivalent resistance and reactance of the contact line feeding section; C1 is the feeding section ground capacitance; C2 is the neutral zone–to–ground capacitance; C12 is the coupling capacitance between the neutral zone and the power supply zone; PT is the roof voltage transformer; Cm is the EMU pantograph–to–ground capacitance; K1 and K2 are equivalent switch. The initial state is the off state, which is used to simulate the circuit conversion process of the over–phase division of the EMU.

When the EMU enters the neutral zone, the equivalent switch K₂ remains open, and the equivalent switch K₁ quickly changes from the open state to the closed state. The neutral zone voltage in this transient process can be determined by the differential equation. Since the core inductances L_m and R_m of the roof voltage transformer are relatively large, and C₁ and C_m are relatively small compared to C₂, the neutral zone and the power supply zone coupling capacitor C₁₂ are short–circuited after the switch K₁ is closed, so in the differential equation expression, the influence of the parameters L_m, R_m, C₁, C_m, and C₁₂ can be ignored. If L=L_s+L₁, R=R_s+R₁, C=C₂ according to the equivalent circuit of the EMU after the equivalent switching action, the loop equation can be listed:

$$u_L+u_C+u_R=u_s$$

(1)

Among them us is the power supply voltage, the inductor voltage uL=Ldi/dt, the resistance voltage uR=R·i, and the loop current i=CduC/dt. Substituting them into equation (1), the differential equation can be obtained:

$$LC\frac{d^2{u}_C}{d{t}^2}+RC\frac{d{u}_C}{dt}+u_C=u_s$$

(2)

Since the impedance of the contact line in the neutral zone is very small, the voltage drop of the contact line impedance in the neutral zone can be ignored. Then the voltage uc at both ends of the neutral zone–to–ground capacitance C2 in the equation (2) is the voltage of the neutral zone on the joint–type electrical split.

Equation (2) is a second–order linear non–homogeneous differential equation with constant coefficients, and its full response can be decomposed into zero input response and zero state response.

The induced voltage of the neutral zone–to–ground capacitance C2 on the electric neutral zone, after the switch K1 is closed, the capacitance C2 discharges to the R–L circuit. The differential equation is:

$$LC\frac{d^2{u}_C}{d{t}^2}+RC\frac{d{u}_C}{dt}+u_C=0$$

(3)

The zero input expression of the effective circuit is:

$$\begin{array}{l}{u}_C=-\frac{p_2{U}_0}{p_1-p_2}{e}^{p_{1}t}+\frac{p_1{U}_0}{p_1-p_2}{e}^{p_{2}t}\\ =\frac{U_0}{p_1-p_2}\left(p_1{e}^{p_{2}t}-p_2{e}^{p_{1}t}\right)\end{array}$$

(4)

Due to the different circuit parameters, the characteristic root may have three different situations. The neutral zone–to–ground voltage in these three situations will be discussed:

1) When α=ω₀, that is$R>2\sqrt{L/C}$, p₁ and p₂ are two unequal negative real numbers.

$$u_C=\frac{U_0}{p_1-p_2}\left(p_1{e}^{p_{2}t}-p_2{e}^{p_{1}t}\right)$$

(5)

This situation is called aperiodic discharge or non–oscillating discharge process.

2) When α=ω₀, that is R<2(L/C)^0.5, p₁ and p₂ are equal, then the equivalent circuit has only one frequency, and the neutral voltage to ground is:

$$u_C=U_0\left(1+\alpha t\right)e^{-\alpha t}$$

(6)

It can be seen from equation (6) that there is no oscillating change, and it has a non–oscillating nature, but this is the dividing line between an oscillating circuit and a non–oscillating circuit, so the situation when R<2(L/C)^0.5 is called a critical non–oscillating process.

3) When α < ω₀, that is R<2(L/C)^0.5, p₁ and p₂ are a pair of conjugate complex roots, and the neutral voltage to ground is:

$$u_C=\frac{U_0{\omega }_0}{{\omega }_d}{e}^{-\alpha t}\sin \left({\omega }_{d}t+\beta \right)$$

(7)

This is an oscillating discharge situation. During the whole process, the waveform will periodically change direction, and the energy storage element will also periodically exchange energy.

When u_s=U_msin(ωt+Φ) in formula (2), where $\omega$is the angular frequency of the power supply and Φ is the initial phase angle of the power supply voltage. At this time equation (2) becomes:

$$LC\frac{d^2{u}_C}{d{t}^2}+RC\frac{d{u}_C}{dt}+u_C=U_m\sin \left(\omega t+\varphi \right)$$

(8)

1) If it is a non–oscillating circuit, the neutral voltage to ground can be expressed as:

$$u_C=\frac{U_m}{Z}{X}_C\sin \left(\omega t+\varphi -\phi -\frac{\pi }{2}\right)+A_1{e}^{p_{1}t}+A_2{e}^{p_{2}t}$$

(9)

Substituting formula (5) into the formula, the expression of U_c and i is:

$$\begin{array}{l}{u}_C=\frac{U_m}{Z}{X}_C[\sin \left(\omega t+\varphi -\phi -\frac{\pi }{2}\right)+\frac{\cos \left(\varphi -\phi \right)}{p_1-p_2}\left(p_1{e}^{p_{2}t}-p_2{e}^{p_{1}t}\right)\\ +\frac{\sin \left(\varphi -\phi \right)}{p_2-p_1}\left(p_1{e}^{p_{1}t}-p_2{e}^{p_{2}t}\right)]\end{array}$$

(10)

2) If the circuit is in a critical state, in the same initial state, the same as above can be obtained:

$$u_C=\frac{U_m}{Z}{X}_C\{\sin \left(\omega t+\varphi -\phi -\frac{\pi }{2}\right)+e^{-\alpha t}[\left(1+\alpha t\right)\cos \left(\varphi -\phi \right)+t\omega \sin \left(\varphi -\phi \right)]\}$$

(11)

3) If it is an oscillating circuit, at this time $p_1$and $p_2$are a pair of conjugate complex numbers, p₁=–α+jω_d, p₂=–α–jω_d, which can be obtained under the same initial state:

$$\begin{array}{l}{u}_C=\frac{U_m}{Z}{X}_C\{\sin \left(\omega t+\varphi -\phi -\frac{\pi }{2}\right)+\frac{e^{-\alpha t}}{{\omega }_d}[{\omega }_0\sin \left({\omega }_{d}t+\beta \right)\cos \left(\varphi -\phi \right)\\ +t\omega \sin {\omega }_d\sin \left(\varphi -\phi \right)]\}\end{array}$$

(12)

When the EMU enters the neutral zone and low–frequency oscillation occurs, the equivalent circuit is obviously oscillating. According to the principle of circuit superposition, add equations (7) and (6) with equations (12) and (11) respectively, and then we can get the mathematical expression of electric current when the EMU enters the electrical neutral zone.

$$\begin{array}{l}{u}_C\left(t\right)=\frac{U_m}{Z}{X}_C\sin \left(\omega t+\varphi -\phi -\frac{\pi }{2}\right)+\frac{U_m{e}^{-\alpha t}}{Z{\omega }_{d}C}\sin {\omega }_{d}t\sin \left(\varphi -\phi \right)\\ +\left[\frac{U_0{\omega }_0}{{\omega }_d}+\frac{U_m{X}_C}{Z\sin \beta }\cos \left(\varphi -\phi \right)\right]e^{-\alpha t}\sin ({\omega }_{d}t+\beta )\end{array}$$

(13)

Combining equation (13) at the moment when the EMU enters the articulated electrical neutral zone, the magnitude of the neutral zone's ground voltage is related to the phase angle of the contact line power supply of the traction substation. The neutral zone structure affects the ground capacitance parameter C, it will also determine the voltage oscillation process in the neutral zone. Since the time when the pantograph enters the neutral zone is random, the instantaneous phase angle of the system power supply voltage is different each time the EMU enters the neutral zone. Therefore, when the EMU enters the electrical neutral zone at different times, different electromagnetic transient processes will be generated, resulting in different excitation levels for the roof voltage transformer. The oscillation angular frequency of the circuit ω₀ is higher than the oscillation frequency of the free component ω_d. The oscillation frequency of the free component ω_d decays exponentially. The speed of the decay depends on the attenuation coefficient α=R/2L. Obviously, the larger the value α, the faster the amplitude decays, and the shorter time it takes to decay to zero. Taking into account the contact line impedance

$$Z=0.05+j0.145\lg \frac{D}{d}$$

(14)

The resistance part is 0.05Ω/km, and the resistance value is about 0.01Ω when the length of the neutral zone is 200m.

Comparing the inductance and resistance parameters of the oscillating system, it can be seen that α=0.001/10000, the damping in the oscillation process is extremely small, and there is almost no attenuation process.

4. Conclusions

References

1. Chen L, Lu X, Min Y, et al. Optimization of governor parameters to prevent frequency oscillations in power systems. IEEE Transactions on Power Systems 2018, 33, 4466–4474. [Google Scholar] [CrossRef]
2. Chen G, Tang F, Shi H, et al. Optimization strategy of hydro–governors for eliminating ultra–low frequency oscillations in hydro–dominant power systems. IEEE Journal of Emerging and Selected Topics in Power Electronics, 2017, 6, 1086–1094. [Google Scholar]
3. Demello F P, Koessler R J, Agee J, et al. Hydraulic turbine and turbine control models for system dynamic studies. IEEE Transactions on Power Systems, 1992, 7, 167–179. [Google Scholar] [CrossRef]
4. Yu X, Zhang J, Fan C, et al. Stability analysis of governor turbine hydraulic system by state space method and graph theory. Energy 2016, 114, 613–622. [Google Scholar] [CrossRef]
5. Liu Z, Yao W, Wen J, et al. Effect analysis of generator governor system and its frequency mode on inter–area oscillations in power systems. Int J Elect Power Energy Syst, 2018, 96, 1–10. [Google Scholar] [CrossRef]
6. Schleif F R, Martin G E, Angell R R, et al. Damping of system oscillations with a hydrogenerating unit. IEEE Transactions on Power Apparatus and Systems, 1967, 86, 438–442. [Google Scholar]
7. Martinez JA, Mork BA. Transformer Modeling for Low– and Mid–Frequency Transients—A Review. IEEE Transactions on Power Delivery. 2005, 20, 1625–1632. [Google Scholar] [CrossRef]
8. Leon F, Farazmand A, Joseph P. Comparing the T and pi Equivalent Circuits for the Calculation of Transformer Inrush Currents. IEEE Transactions on Power Delivery. 2012, 27, 2390–2398. [Google Scholar] [CrossRef]
9. Jazebi S, Zirka SE, Lambert M, Rezaei–Zare A, Chiesa N, Moroz Y, et al. Duality Derived Transformer Models for Low–Frequency Electromagnetic Transients—Part I: Topological Models. IEEE Transactions on Power Delivery. 2016, 31, 2410–2419. [Google Scholar] [CrossRef]
10. Mork BA, Gonzalez F, Ishchenko D, Stuehm DL, Mitra J. Hybrid Transformer Model for Transient Simulation—Part I: Development and Parameters. IEEE Transactions on Power Delivery. 2007, 22, 248–255. [Google Scholar] [CrossRef]
11. Monteiro TC, Martinz FO, Matakas L, Komatsu W. Transformer Operation at Deep Saturation: Model. M: Transformer Operation at Deep Saturation.
12. and Parameter Determination. IEEE Transactions on Industry Applications. 2012, 48, 1054–1063.
13. Casoria S, Sybille G, Brunelle P. Hysteresis modeling in the MATLAB/Power System Blockset. Mathematics and Computers in Simulation. 2003, 63, 237–248. [Google Scholar] [CrossRef]
14. Moses PS, Masoum MAS, Toliyat HA. Dynamic Modeling of Three–Phase Asymmetric Power Transformers With Magnetic Hysteresis: No–Load and Inrush Conditions, IEEE Transactions on Energy Conversion. 2010, 25, 1040–1047.
15. Hassani V, Tjahjowidodo T, Do TN. A survey on hysteresis modeling, identification and control. Mechanical Systems and Signal Processing. 2014,49(1–2):209–233.
16. Leite Jv, Benabou A, Sadowski N, da Luz Mvf. Finite Element Three–Phase Transformer Modeling Taking Into Account a Vector Hysteresis Model, IEEE Transactions on Magnetics. 2009, 45 (3): 1716 – 1719.
17. Sun J, Mehrotra V. Orthogonal winding structures and design for planar integrated magnetics, In: Proc IEEE APEC; USA: Anaheim 2004, 933 – 938.
18. Sun Y, Zheng W, Xu, W, A new method to model the harmonic generation characteristics of the thyristor controlled reactors, Proceedings of the Power Tech, Lausanne, 1–5 July, 2007, New York: IEEE, 2007, 1785–1790.
19. TYLAVSKY D J, BROWN K A,MA T T Closed form solution for underground impedance calculations, Proceedings of the IEEE, 1986, 74, 1290–1292.
20. TYLAVSKY D, J. Conductor impedance approximations for deep underground mines, IEEE Transactions on Industry Applications, 1987, 23, 723–730.
21. WAITJR, Quasi–Static Limit for the Propagating Mode along a Thin Wire in a Circular Tunnel, IEEE Transactions on Antennas and Propagation, 1977, 25, 441–443.
22. Zhou Hongyi，Liu Zhigang，Cheng Ye，et al．Extended black–box model of pantograph arcing considering varying pantograph detachment distance, Proceedings of 2017 IEEE Transportation Electrification Conference and Expo, Asia–Pacific. Harbin: IEEE, 2017.
23. Liu Y J, CHang G W, Huang H M. Mayr’s equation–based model for pantograph arc of high–speed railway traction system, IEEE Transactions on Power Delivery, 2010, 25, 2025–2027.
24. Qu Zhijian, Liu Yuxin, Zhou Min, et al. Over–voltage suppression of electric railway articulated neutral insulator, Advanced Materials Research, 2013, 791–793, 1837–1840.
25. Wang Ying, Liu Zhigang, Mu Xiuqing, et al. Research on electromagnetic transient process in articulated split–phase insulator of high–speed railway considering viaduct’s electrical coupling, International Transactions on Electrical Energy Systems, 2017, 27, e2376.
26. Zhang Ruifeng, Bai Yihui, Huang Ruyun, Fu Yu. Cause Analysis and preventive measures of low frequency oscillation in power supply side of power supply terminal, Guizhou Electric Power Technology. 2013, 16, 15–17.
27. Wang Yun, Chang Peak, Chang Xi, Guo Xiaolong. Study on the key technology of identifying the type of low frequency oscillation and locating the disturbance source in power grid, Xinjiang Electric Power Technology. 2015,2:1–7.
28. Li Yanghai, Pan Jian, Yang Tao, Huang Shuhong, Gao Wei. Status and development of low frequency oscillation monitoring and suppression technology in Power Grid. Hubei Electric Power. 2014, 38, 8–16. [Google Scholar]
29. Zhang Xinyu, Chen Jie, Zhang Gang, Wang Lei, Qiu Ruichang, Liu Zhigang. An active oscillation compensation method to mitigate high–frequency harmonic instability and low–frequency oscillation in railway traction power supply system, IEEE. 6:70359–70367.
30. Chen Feng, Yang Xiaonan, Liu Liang, Ma Xiaochen, Zhang Lei, Yu Yiping. Study on the influence of HVDC to low frequency oscillation in interconnected power system[C]. 2018 2nd IEEE Conference on Energy Internet and Energy System Integration (EI2), Beijing, China, 2018, 1–5.
31. Fan Lingling, Li Naihu, Wang Haifeng. Analysis of low frequency oscillation of damping power system with controllable series capacitor compensation device, Power Grid Technology, 1998, 22, 35–39.
32. Zhang Fang, Fang Dazhong, Song Wennan, Chen Jiarong. Study on a novel two–stage control method for UPFC with damping tie–line low–frequency Oscillation. Proceedings of the twenty–one th annual conference of power system automation, 2005, 1039–1044.
33. Li Hairong, Penk, Chen Yu, Nick Clegg, Allen Lee, Li Xialin. Analysis of low frequency oscillation mechanism of DC voltage control time scale in DC distribution system, High voltage technology, 2021, 1–16.
34. Chen He, Huang He, Zhang Yong, Su Yinsheng. Researches on DC modulation to damp low frequency oscillation in China Southern Power Grid. IEEE Power & Energy Society General Meeting, 2015, 1–6.
35. Gao Hongliang, Zhan Xisheng, Jiang Xiaowei, Wu Jie, Yang Qingsheng. Research on mechanism of power system low frequency oscillation based on damping coefficient, 2017 4th International Conference on Electrical and Electronics Engineering, 2017, 99–103.
36. Yang Jie, Hu Haitao, Zhou Yi, Tao Haidong, Zhao Chaopeng, he zhengyou. Analysis of low frequency Oscillation suppression in Traction Power Supply System [ J ]. Electrified Railways, 2018, 29, 15–28. [Google Scholar]
37. Zhou Yi, Hu Haitao, Yang Xiaowei, he zhengyou. Railway Electrification System, Chinese Journal of Electrical Engineering, 2017, 37, 72–80.
38. Zha Wei, Yuan Yue. Mechanism of active–power–PSS low–frequency oscillation suppression and characteristic of anti–regulation[C]. 2011 Third International Conference on Measuring Technology and Mechatronics Automation, Shanghai, China, 2011, 538–541. 2011.
39. Lu Shengyang, Zhang Wuyang, Wang Tong, Cai Yupeng, Li Huan, Zhu Tianyi, Gang Yining, Yu Yongliang. Parameter Tuning and simulation analysis of PSS function in excitation system with suppression of low frequency oscillation[C]. 2019 IEEE 8th International Conference on Advanced Power System Automation and Protection , Xi'an, China, 2019, 474–479.
40. Pan Yiwei, Yang Yongheng, He Jinwei, Ariya Sangwongwanich, Frede Blaabjerg. Low–frequency oscillation suppressi–bregeon in series resonant dual–active–bridge converters under fault tolerant operation, 2019 IEEE energy conversion congress and exposition, Baltimore, MD, USA, 2019, 1499–1505.
41. Zhou Yi, Hu Haitao, Lei Ke, Meng Zhaofei, he zhengyou. Low frequency constant amplitude oscillation mechanism of railway electrification system, Chinese Journal of Electrical Engineering, 2020, 1–13.
42. Zhou Yi, Hu Haitao, Jiang Xiaofeng, he Zhengyou, Zhao Chaopeng. Study on voltage resonance of Low Frequency Power Supply Network for Traction Power Supply. Grid Technology, 2016, 40, 1830–1838. [Google Scholar]
43. Yao Chao, Wang Xiaojun, Bi Chengjie, Zhang Yizhi, Jin Cheng. An approach to suppress low–frequency oscillation in electrification railway based on TCSC impedance control, 2018 IEEE 2nd International Electrical and Energy Conference (CIEEC), Beijing, China, 2018, 110–113.
44. Bi Chengjie, Wang Xiaojin, Yao Chao, Pang Kexin, Jin Cheng. Analysis and evaluation of suppression methods on low–frequency oscillation in electric railways, 2018 IEEE 2nd International Electrical and Energy Conference (CIEEC), Beijing, China, 2018, pp. 56–63.
45. Zhou Guohua, Tan Wei, Zhou Shuhan, Wang Yue, Ye Xin. Analysis of Pulse Train Controlled PCCM Boost Converter With Low Frequency Oscillation Suppression, IEEE, 2018, 6, 68795–68803.