Fault analysis and simulation of ground-fault accident on transformer in 500 kV AC/DC grids

Abstract

An actual occurred ground-fault accident of 500 kV transformers in HVDC converter station is presented. A 2D transient electromagnetic-thermal-mechanical finite-element coupling model of 500 kV transformer is developed to analyze the transient process of failure, with consideration of the resistance-temperature relationship and plasticity displacement in the windings’ thermal and mechanical characters. To verify the proposed finite-element model, the model is calibrated by pre-fault field test of the transformer in steady state, which shows good agreement with measurement. The short-circuit current adopted in simulation is based on actual on-site measurement of the ground-fault accident. By using the proposed model, simulation results reflect the transient process and explain the post-fault inspected damage well in term of over burn regions in thermal analysis and deformation tendency in mechanical analysis when compared with field post-fault inspection photos.

Keywords

Electromagnetic analysis finite-element transformer ground-fault accident and DC grid

1. Introduction

Transformer is one of the most important primary devices in power grid, especially for HVDC system where transformers are adopted as the key equipment in AC/DC energy converting systems. Grounding fault is a major failure for high voltage transformers, which leads to huge outage losses. There are lots of existing publications focusing on electromagnetic and force analysis of transformers under short circuit current. The calculations of electromagnetic and stress character are presented in analytical way in [1,2], in 3D finite-element (FE) analysis in[3–8] and verified by experiment in[9,10]. In order to reduce calculating time, simplified 2D FE model is also investigated in [11], verified with 3D FE models under inrush current and short circuit current in [12], and applied in transient analysis in [13]. However, the power level shown in most of the published literatures are low in electromagnetic force and stress analysis, which is sufficient for low voltage apparatus investigation, but gives limited reference to high voltage fault analysis with huge current, temperature rise, stress and deformation. On the other hand, most the thermal analysis of conventional transformers in publications focus on steady state heat transfer and fluid dynamics [14,15] instead of temperature rise under short circuit. [16] presents a thermal analysis taking into consideration the resistance-temperature relationship, however the calculation case is just an ideal winding with ideal source which is quite different from an actual fault incident of transformer in terms of temperature rise level. Since actual fault seldom occurred and carrying out full-scale tests under actual operation situation is almost impossible due to the tremendous cost, few academic literature represented the actual fault case of 500 kV transformers.

In this paper, a 2D transient electromagnetic-thermal-mechanical FE coupling model is presented, along with an actual occurred grounding fault of 500 kV transformers in HVDC converter station as an industry application case. In order to analyze the completed transient electromagnetic performance as well as the thermal and stress process under the huge pulsed short-circuit current, the FE model takes into account the coil resistance-temperature varying relationship as well as the yield strength check and plasticity displacement analysis.

2. Basic theory for the analysis

2.1. Electromagnetic process

A single-phase grounding fault leads to huge impulse current on the windings of transformer. When current flows into the windings of the transformer, a magnetic field is created and the electromagnetic equation in cylindrical coordinate system can be written as below: $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle {\nabla}\times ({\mu}^{-1}{\nabla}\times \overrightarrow{A})=\overrightarrow{J}-{\sigma}_{e}\frac{\partial \overrightarrow{A}}{\partial t}\end{eqnarray}$ (1) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle {\nabla}\times \overrightarrow{A}=\overrightarrow{B},\end{eqnarray}$ (2) Where $\vec{A}$ is the magnetic vector potential, $\vec{B}$ is the magnetic flux density, $\vec{J}$ is the current density, 𝜇 is the permeability and σ_e is the electrical conductivity of the windings. When magnetic flux density $\vec{B}$ is obtained, the magnetic force density $\vec{f}$ generated in the conductive windings can be obtained as. $\begin{eqnarray}\displaystyle \overrightarrow{f}=\overrightarrow{J}\times \overrightarrow{B}. & & \displaystyle\end{eqnarray}$ (3)

In the transient process with pulsed current flowing into the windings of the transformer, the electrical conductivity of the windings cannot be treated as constant. In fact it varies as a function of temperature due to the Joule heating in the grounding process and thus the following equation taking into consideration the effect of temperature is adopted to calculate the electrical conductivity σ(T): $\begin{eqnarray}\displaystyle {\sigma}(T)=\displaystyle \frac{1}{{\rho}(T_{ref})(1+{\alpha}{\Delta}T)}. & & \displaystyle\end{eqnarray}$ (4) Where 𝜌(T _ref) is the conductor resistance at reference temperature (T _ref = 20 °C), 𝛼 is known as the temperature coefficient of copper windings, and ΔT is the temperature difference between transient temperature and reference temperature.

2.2. Thermal process

The impact of transient thermal process on the transformer is performed as the varying resistance of windings via temperature. For the normal used single-phase/ three-limb transformer in 500 kV converter station, the thermal diffusion equation is written as: $\begin{eqnarray}\displaystyle {\rho}C\frac{\partial T}{\partial t}+{\nabla}\cdot (-k{\nabla}T)=Q & & \displaystyle\end{eqnarray}$ (5) where 𝜌 presents the density of materials, C presents the heat capacity, k presents the heat conductivity and Q means the heat source.

In the performed analysis, the heat source Q refers to the Joule heating of the conductors, which can be written as [17], $\begin{eqnarray}\displaystyle Q=\frac{1}{{\sigma}}|\overrightarrow{J}|^{2}. & & \displaystyle\end{eqnarray}$ (6)

2.3. Stress and displacement process

During the grounding fault transient process, the stress is induced on the coils due to the interaction between impulse current and magnetic field, leading to corresponding displacement on the windings. The relationship between stress and electromagnetic force on the coil can be written as, $\begin{eqnarray}\displaystyle -{\nabla}\cdot {\sigma}_{f}=\overrightarrow{f} & & \displaystyle\end{eqnarray}$ (7) where σ_f is the stress tensor, and $\vec{f}$ is the magnetic force density which can be obtained in Eq. (3).

During the mechanical process under huge current, the relationship between stress and strain can be linear or nonlinear, which is determined due to the maximum stress induced in the windings. Corresponding to different stress level, the stress-strain relationship can be expressed as [17], $\begin{eqnarray}\displaystyle {\sigma}_{f}=\left\{\begin{array}{@{}ll@{}}E{\varepsilon} & {\sigma}_{f}<{\sigma}_{f0}\\[3.0pt] {\sigma}_{f0}+A{\varepsilon}_{p}^{B} & {\sigma}_{f}\geq {\sigma}_{f0}\end{array}\right. & & \displaystyle\end{eqnarray}$ (8) where E means Young’s modulus with a value of 115 GPa for copper, ϵ is the strain, σ_f0 means the initial yield stress with a value of 250 MPa, ϵ_p means the plastic strain which can be expressed as ${\varepsilon}-\frac{{\sigma}_{f0}}{E}$ , A and B are fitting coefficients corresponding to stress-strain curve of copper when the stress is over the yield stress point.

3. Fault analysis and finite-element modeling

3.1. Pre-fault operation status and data

As shown in Fig. 1, a transformer is in operation in a 500 kV converter station. Bases on on-site measurement system, the input and output power of the transformer before the fault occurred are 151 MVA and 143 MVA, respectively. The 550 kV high voltage (HV) winding and 242 kV middle voltage (MV) winding are in operation and no power transferred through 35 kV low voltage (LV) winding. The on-site line voltage and line current on the HV side of the transformers are 532.1 kV and 164 A, while the MV voltage and current by on-site measurement are 228 kV and 383 A, respectively. The grounding fault occurs at the MV side of the transformer at the beginning. Based on the field test results of the transformer before the fault occurred, the equivalent resistor R and inductor L on MV side are 0.304 Ω and 45.97 mH, respectively.

Fig. 1.

The local power networks for the proposed 500 kV transformer.

3.2. Finite-element modeling and verification in steady state before grounding fault

FE methods are used to solve the transient electromagnetic-thermal-mechanical response of the proposed 500 kV transformer under grounding fault. The main parameters of the transformer are shown in Table 1.

Table 1
Main parameters of the 500 kV transformer

Parameters Value

Structure Single-Phase, Three-Limb

Frequency 50 [Hz]

Rated Power of HV/MV/LV Winding 250/250/80 [MVA]

Rated Voltage of HV/MV/LV Winding 550 / 242 ± 8 × 1.25% /35 [kV]

Connection Type Y/Y/Δ

Rated Current of HV/MV/LV Winding 787.3 /1789.3 /2285.7 [A]

Turns of HV/MV/LV Winding 753 / 349 / 83

Reference resistivity at 20 °C 1.72e-8 [Ω ⋅ m]

Reference temperature coefficient 0.0039 [1/K]

Density of Windings 8700 [kg/m3]

Heat capacity of Windings 385 [J/(kg ⋅ K)]

Thermal conductivity of Windings 400 [W/(m ⋅ K)]

Young’s modulus of Windings 115 [GPa]

Yield stress of Windings 250 [MPa]

Parameters	Value
Structure	Single-Phase, Three-Limb
Frequency	50 [Hz]
Rated Power of HV/MV/LV Winding	250/250/80 [MVA]
Rated Voltage of HV/MV/LV Winding	550 / 242 ± 8 × 1.25% /35 [kV]
Connection Type	Y/Y/Δ
Rated Current of HV/MV/LV Winding	787.3 /1789.3 /2285.7 [A]
Turns of HV/MV/LV Winding	753 / 349 / 83
Reference resistivity at 20 °C	1.72e-8 [Ω ⋅ m]
Reference temperature coefficient	0.0039 [1/K]
Density of Windings	8700 [kg/m3]
Heat capacity of Windings	385 [J/(kg ⋅ K)]
Thermal conductivity of Windings	400 [W/(m ⋅ K)]
Young’s modulus of Windings	115 [GPa]
Yield stress of Windings	250 [MPa]

Owing to the axial symmetry character of single-phase, three-limb transformers, which is widely used in 500 kV and higher voltage power system in China and abroad, 2D FE model is preferred than 3D for reducing the calculating loads, especially when nonlinear thermal and displacement transient are coupled with electromagnetic analysis. In FE analysis, a feasible simplified 2D axisymmetric model is adopted which has been recommended in [11], and verified against 3D models in [12] in both electromagnetic and force analysis under short circuit current. A specially shaped yokes and outer limbs with the same cross-sectional area and the same reluctance as actual 3D structure is adopted as shown in Fig. 2. The correctness of this 2D model has been verified against 3D simulation and field test in previous work published in [18].

Fig. 2.

The simplifying principle from 3D to 2D axisymmetric model.

According to the parameters shown in Table 1, the MV RMS line-to-line voltage and current of 228 kV and 383 A with an output power flow of 143 MVA are added as loads in the proposed FE model to simulate the actual operation point. HV line-to-line RMS voltage and current are calculated in coupled circuit by FE model as 535.3 kV and 162.7 A with the magnetic field distribution shown in Fig. 3. Owing to the small current of 383 A in operation (1789.3 A in rated value), only 1.17–1.22 T magnetic density is created in the core region of transformer according to simulation shown in Fig. 3, which is much smaller than the magnetic distribution at rated power that is presented and verified against field test data in previous work published in [18,19].

Fig. 3.

The magnetic field distribution before grounding fault.

The equivalent inductor L on MV side can also be calculated by FE model as, $\begin{eqnarray}\displaystyle E=\frac{1}{2}L\cdot I^{2} & & \displaystyle\end{eqnarray}$ (9) where E is the magnetic energy generated by the transformer. So, by using the Eq. (9), the equivalent inductor on the MV side can also be calculated as 43.3 mH by FE model. Comparing these simulated results with measured data shown in Section 3.1, the simulation results agree well with the on-site measurement as shown in Table 2, which verified the proposed FE model.

Table 2

Comparison of the simulated results and measurement

Parameters	Simulated results	On-site measurement
HV voltage	535.3 kV	532.1 kV
HV current	162.7 A	164 A
Equivalent inductor on the MV side	43.3 mH	45.97 mH

3.3. Fault current transient by online measurement

The grounding fault occurs on the MV side of the transformer because of lighting. Figure 4 shows the transient waveform of the fault current on the MV and HV side by online measurement. According to the on-site measurement by fault recorder, the peak value of the transient grounding current is 27.85 kA on the MV side and 11.752 kA on the HV side starts at the beginning in 15.2 ms. However, due to the overvoltage and flashover occurred inside the HV windings during the grounding fault transient with lighting, a high current with a peak of 47.272 kA was created and observed later at 83.4 ms on the HV windings according to the online measurement, which is 4 times higher than the initial short circuit current and totally damaged the transformer. Since the HV winding or the core of the transformer is badly damaged due to the huge fault current, the MV current drops to only 9.668 kA after 77 ms, indicating that the voltage ratio between MV and HV is totally damaged and no longer exists.

Fig. 4.

Measured transient waveform of the fault current on the MV and HV of transformer.

Owing to the overvoltage and flashover occurred with lighting, the short circuit current waveform measured shown in Fig. 4 is totally different from the waveform presented in [5,20] and [21], so the conventional short circuit current equations cannot be adopted directly in this fault case. In order to analyze the transient performance of the transformer and evaluate the damage occurred on windings, the fault current expression fitting the waveform shown in Fig. 4 can be approximately rewritten as below, $\begin{eqnarray}\displaystyle I_{f}=\left\{\begin{array}{@{}ll@{}}I_{p1}[\cos {\alpha}e^{-\frac{r}{L}t}-\cos ({\omega}t+{\alpha})]\quad & 15.2∼\text{ms}<t\leq 67.4∼\text{ms}\\[3.0pt] I_{p2}\sin ({\omega}t+{\beta})\quad & 67.4∼\text{ms}<t\leq 77.4∼\text{ms}\\[3.0pt] I_{p3}\sin ({\omega}t+{\beta})\quad & 77.4∼\text{ms}<t\leq 149.6∼\text{ms}\end{array}\right. & & \displaystyle\end{eqnarray}$ (10) where the transient fault current is divided into 3 sub processes corresponding to the 3 expressions in Eq. (10). The sub process 1 is treated as conventional short circuit current referred to equations presented in [20]. r and L are the equivalent resistor and inductor on the HV side of the transformer, which is determined corresponding to the time constant (4L∕r) of sub process 1 shown in Fig. 4. The ω is 314.15 rad/s due to 50 Hz power system. The 𝛼 is set as −4.77 rad and I _p1 is set as 6.339 kA to fit the initial phase angle and peak of the waveform corresponding to sub process 1. As for the sub process 2 and 3, sinusoid functions are adopted based on the waveform shown in Fig. 4. In calculation, I _p2, I _p3, and 𝛽 are set as 37.7 kA, 48.5 kA and −17.9 rad, respectively, for fitting the curve.

Since the voltage ratio relationship between MV and HV doesn’t work during and after the fault transient, so the magnetic field distribution calculation with fault current also doesn’t make sense, which is not presented in this paper.

3.4. Thermal analysis under grounding fault

Temperature rise and winding deformation are the main reasons for the transformer destroy, which should be clearly analyzed in order to study the transient process. During the grounding fault, the rising temperature during the transient process leads to a rise of resistance, which ultimately affects the temperature back. Figure 5 shows the temperature-time curve calculated according to Eq. (5) and (6) with the fault current referred to Eq. (10). The simulation is carried out with and without taking into account the resistance-temperature relationship as a comparison. Since the thermal time constant of winding is much larger than that of the short circuit current [16], the heat transfer between oil and winding is negligible and it is assumed in this paper that the heat development during the transient process is retained in the coils itself. In simulation, 105 °C is set as initial temperature of winding in operation [16]. As shown in Fig. 5, resistance-temperature curve does lead to an additional rise of temperature when compared to the curve without taking into account the resistance-temperature relationship shown in dash line in Fig. 5. It is proved that the resistance-temperature curve should be taken into consideration during the transient process when the short circuit current is huge.

Fig. 5.

The transient temperature rise curve of the HV windings.

3.5. Stress and displacement analysis under grounding fault

Apart from the temperature rising effects, electromagnetic force, stress and displacement will also occur due to the short circuit current. Figure 6 shows the electromagnetic force versus height of the HV windings when peak current occurred. The 0 mm height in Fig. 6 presents the horizon medium plane of windings. It is observed that the axial electromagnetic force, shown in Fig. 6 as blue dash line, increases with the distance to the horizon medium plane, which is due to the interaction between leak magnetic field and huge current on the upper/lower terminal of the windings, corresponding to Eq. (3) and (7) mentioned in previous section. To the contrary, the radial force researches its peak value on the horizon medium plane, as shown in black solid line in Fig. 6. The value of electromagnetic force will be added as loads in the following stress and deformation analysis.

Fig. 6.

Radial and axial electromagnetic force versus height of the HV windings.

Figure 7 (a) and (b) shows the stress and displacement distribution in the HV windings when peak current occurred. As shown in Fig. 7(a), the peak von Mises stress of 167 MPa on the copper winding is less than its initial yield stress of 250 MPa, which will not lead to destroy according to Eq. (8) since the winding is still working below the yield stress point and the impact of nonlinear stress-strain curve is negligible for this fault case. The windings can recover after the transient at this stress level. On the other hand, the displacement tendency shown in Fig. 7(b) matches well with the axial and radial force distribution shown in Fig. 6, which indicates the deformation direction of windings. Note that the displacement and stress on MV winding is pretty small due to the small MV current which is analyzed in previous Section 3.3.

Fig. 7.

(a) Stress distribution and (b) displacement tendency in MV and HV windings.

4. On-site post-fault inspection

Figure 8 shows the post-fault on-site photo of the HV winding of the damaged 500 kV transformer. Note that the spacer installed in actual transformers are not modelled in FE models since the elasticity modulus of the spacer is 10 times smaller than that of the windings (8 GPa to 115 GPa) [20], which is negligible in mechanical analysis. In Fig. 8, it is visible that the deformation trend of the outer frame and spacer of the HV windings matches well with the simulated mechanical character shown in Figs 7(b) and 6. Stress simulated results shown in Fig. 7(a) indicates that the peak stress is less than windings’ initial yield stress, so the windings are still working in linear stress-strain region under 48 kA short circuit current, and after the fault transient, it can still return to the original status shown in Fig. 8, although with most of the spacers damaged.

Fig. 8.

Post-fault inspection of the damaged HV windings.

From Fig. 8, there are also some over burn regions spotted at the windings due to the temperature rise of coils under the short circuit process. Simulation result in Fig. 5 shows that the transient temperature is 175.1 °C, which is near the limitation of insulation of the coils (the transformer works for more than 15 years with insulation aging) and clearly explain the reason for the over burn regions shown in Fig. 8.

5. Conclusion

In this paper, an actual occurred fault of 500 kV transformers in HVDC converter station is present. An electromagnetic-thermal-mechanical coupling FE model is developed to provide an effective way to analyze the transient grounding fault process of 500 kV transformers. With the huge short circuit current, resistance-temperature curve of the windings in transient thermal analysis and stress character in mechanical analysis are taken into consideration. The resistance-temperature curve character affects the transient performance of transformers and should be considered when short circuit current is huge. The proposed simulation model, which is verified in steady state, reflects the transient process and explains the post-fault inspected damage well in term of over burn regions in thermal analysis and deformation tendency in mechanical analysis.

Due to the tremendous cost, it is normally impossible to carry out full-scale tests with sensors installed in the windings on 500 kV transformers to verify the simulated thermal and mechanical results in transient. But with the pre-fault field tests as verification and post-fault inspection photos as comparison, the proposed simulation method still shows acceptable accuracy in industry application. Pre-fault field tests, On-fault transient simulation and post-fault inspection tests are generally becoming a feasible and effective way for fault analysis and grid recover by electric power companies.

Footnotes

Acknowledgements

This work was supported by State Grid Hubei Electric Power Company. Dr. Bo Zhang was a senior engineer with Key Laboratory of High Voltage Field Test Technique of SGCC when dealing with fault analysis of the transformer, and now he is with Rensselaer Polytechnic Institute.

References

Adly

, Computation of inrush current forces on transformer windings, IEEE Transactions on Magnetics 37(4) (2001), 2855–2857.

Wong

Wang

and Tang

, Numerical simulation of transient force and eddy current loss in a 720-MVA power transformer, IEEE Transactions on Magnetics 40(2) (2004), 687–690.

Fonseca

Lima

Soeiro

and Nunes

, Analysis of electromagnetic-mechanical stresses on the winding of a transformer under inrush currents conditions, International Journal of Applied Electromagnetics and Mechanics 50(4) (2016), 511–524.

Fonseca

Lima

Nunes

Bezerra

and Soeiro

, Analysis of structural behavior of transformer’s winding under inrush current conditions, IEEE Transactions on Industry Applications 54(3) (2018), 2285–2294.

Ahn

Lee

Kim

Jung

and Hahn

, Finite-element analysis of short-circuit electromagnetic force in power transformer, IEEE Transactions on Industry Applications 47(3) (2011), 1267–1272.

Faiz

Ebrahimi

and Noori

, Three- and two-dimensional finite-element computation of inrush current and short-circuit electromagnetic forces on windings of a three-phase core-type power transformer, IEEE Transactions on Magnetics 44(53) (2008), 590–597.

Zhang

, 3-D nonlinear transient analysis and design of eddy current brake for high-speed trains, International Journal of Applied Electromagnetics and Mechanics 40(3) (2012), 205–214.

Wang

Zhang

Wang

and Yuan

, Cumulative deformation analysis for transformer winding under short-circuit fault using magnetic–structural coupling model, IEEE Transactions on Applied Superconductivity 26(7) (2016), 0606605.

Geißler

and Leibfried

, Short-circuit strength of power transformer windings-verification of tests by a finite element analysis-based model, IEEE Transactions on Power Delivery 32(4) (2017), 1705–1712.

10.

Ahn

Kim

Song

and Hahn

, Experimental verification and finite element analysis of short-circuit electromagnetic force for dry-type transformer, IEEE Transactions on Magnetics 48(2) (2012), 819–822.

11.

Steurer

and Frohlich

, The impact of inrush currents on the mechanical stress of high voltage power transformer coils, IEEE Transactions on Power Delivery 17(1) (2002), 155–160.

12.

Kumbhar

and Kulkarni

, Analysis of short-circuit performance of split-winding transformer using coupled field-circuit approach, IEEE Transactions on Power Delivery 22(2) (2007), 936–943.

13.

Ahn

Kim

Kwak

and Hahn

, Computation of transient electromagnetic force on power transformer windings by inrush current, International Journal of Applied Electromagnetics and Mechanics 45(1-4) (2014), 605–612.

14.

Susa

and Lehtonen

, Dynamic thermal modeling of power transformers: Further development - Part I, IEEE Transactions on Power Delivery 21(4) (2006), 1961–1970.

15.

Djamali

and Tenbohlen

, Hundred years of experience in the dynamic thermal modelling of power transformers, IET Generation Transmission & Distribution 11(11) (2017), 2731–2739.

16.

Rahman

Lie

and Prasad

, Computation of the thermal effects of short circuit currents on HTS transformer windings, IEEE Transactions on Applied Superconductivity 22(6) (2012), 5501211.

17.

Cao

, Analysis and reduction of coil temperature rise in electromagnetic forming, Journal of Materials Processing Technology 225 (2015), 185–194.

18.

Zhang

, Field-circuit cosimulation of 500-kV transformers in AC/DC hybrid power grid, IEEE Transactions on Applied Superconductivity 26(4) (2016), 5500405.

19.

Zhang

Electromagnetic transient modeling and simulation of large-scale HVDC power grid with all primary devices, in: Proceedings of 2016 IEEE International Conference on Power and Renewable Energy, 2016, pp. 43–47.

20.

Zhang

, Dynamic deformation analysis of power transformer windings in short-circuit fault by FEM, IEEE Transactions on Applied Superconductivity 24(3) (2014), 0502204.

21.

Moradi

and Madani

, Technique for inrush current modelling of power transformers based on core saturation analysis, IET Generation Transmission & Distribution 12(10) (2018), 2317–2324.