1. Introduction

As one of the important equipment in HVDC transmission project, the safe operation of converter transformer is directly related to the stability of the system [1]. Though the converter transformer is different from AC transformer in structure and performance, the oil-paper insulation is still the main insulation of high voltage bushing. When converter transformer in normal operation, the line side winding is under AC voltage, and the valve side winding is under AC-DC hybrid voltage, meanwhile they also withstand higher harmonic voltage caused by converter valve, thus oil paper in winding is subjected to DC, AC and higher harmonic voltage. The electric field distribution is decided by the resistivity and permittivity of oil and oil immersed paper, while the resistivity ratio of oil and oil immersed paper is generally between 1:10 - 1:300. However when the insulation aging, the ratio can be up to 1:1 or 1:1000. So it’s necessary to analyze AC- DC hybrid electric field affected by the insulation aging [2].

The polarity reversal electric field of converter transformer has been analyzed in some paper [3]-[15]. There is some analysis of the periodic non-sinusoidal steady hybrid electric field, yet the electric field is always assumed to be under the fundamental frequency AC and DC voltage [16]. Actually the valve side winding of converter transformer also withstands high harmonic electric field [17].

For simplicity, the anisotropy of the oil press-paper and the nonlinearity of the insulation between conductivity and electric field strength are not taken into consideration. A two-dimensional periodic non-sinusoidal steady hybrid electric field of a ±500 kV converter transformer with different resistivity ratio is calculated and compared.

2. FEM Equation of Frequency Domain

The periodic non sinusoidal steady hybrid electric field in converter transformer belongs to quasi-static field. In this condition, the initial boundary problem based on scalar potential φ is as follows [18]:

(1)

Ω is for computed area; Γ₁ and Γ₂ are the first kind of boundary condition and the second kind of boundary condition respectively. The area Ω is triangularly meshed and the unit potential is linear interpolated in each area. The semi-dis- crete differential equation with nodal potential by Galerkin finite element method:

(2)

,―finite element stiffness matrix,―potential column vector, F― second kinds of boundary conditions.

For the is constant without considering electric field strength, temperature or other factors, so it’s easy to obtain non sinusoidal periodic steady electric field by frequency finite element [19]. Equation (2) can be changed into the following form, by using the fast Fourier transform [20].

(3)

Then we can get:

(4)

The first kind of boundary conditions is also converted by FFT, and each frequency point k(k = 0,1, …, N/2) would be imposed boundary conditions according to Equations (4). After it is obtained, non-sinusoidal periodic steady results in the original time domain can be calculated by Fast Fourier inverse transform. It is worthy that: according to Fourier transform, Equations (4) only need to be calculated for k = 0,1, …, N/2, then the value of φ(nΔ_t) (n = 0,1, …, N) can be obtained based on the fact that φ(nΔ_t) is real number and has the characteristics of periodic function.

3. Period Non Sinusoidal Steady Voltage ±500 kV of Converter Transformer

According to the actual parameters of Three Gorges-Changzhou HVDC project, the model has been established with a AC system with converter transformer, converter valve, AC/DC filter, flat wave reactor, transmission line, control system by PSCAD/EMTDC. The period of the steady voltages on valve side winding are shown in the left of Figure 1 and Figure 2, and the spectrum analysis of their

voltage are illustrated in the right of Figure1 and Figure 2. Two time instant t₁ and t₂ are shown in the left of Figure 1 and Figure 2, and they are the local maximum over one period of Y₀-Δ and Y₀-Y connected valve side winding.

From frequency spectrum of the Figure 1 and Figure 2, the valve side winding includes not only DC voltage and the fundamental frequency AC voltage, but also a lot of high harmonics whose amplitude is large (for example the amplitude of the 3rd harmonic is 60 kV.) If the hybrid electric field only includes DC and fundamental frequency AC components, the results will be different from the actual one.

In this paper, the steady AC-DC hybrid electric field is obtained by the frequency domain FEM, which is to consider the DC, fundamental and high harmonic voltage components, and the inverse fast Fourier transform method.

4. Model of Converter Transformer

The two-dimensional simplified model of a ±500 kV converter transformer is triangularly meshed. There are 153,692 elements and 79,929 nodes. 4 sets of resistivity are given in Table 1, while the relative permittivity of transformer oil, oil immersed pressed-paper and insulation paper are 2.2, 4.5 and 3.5 [2], due to their small changing ranges.

5. Analysis of Compound Electric-Field of ±500 kV Converter Transformer

The hybrid electric-field potentials of the ±500 kV converter transformer at different resistivity ratio are given in Figure 3 and Figure 4. Figure 3 shows the potential contour at time t₁ as Figure 1(a) in Y₀-Δ connected converter transformer; Figure 4 shows the potential contour at t₂ as Figure 2(a) in Y₀-Y connected converter transformer.

More equi-potential lines focus on oil immersed paper when the oil resistivity ratio changing from 1:1 to 1:1000, because the electric field is proportional to the oil resistivity when the valve side winding contains DC voltage.

The maximum electric field with time of transformer oil, oil immersed paper and insulation layer is given in Figure 5 and Figure 6; the maximum electric field of various media is given in Table 2 and Table 3 in one period.

The maximum electric field curve with time from various media is comprised in shown as Figure 5 and Figure 6：The maximum electric field strength curve is not the result of DC and fundamental AC, but also caused by high harmonic voltage on valve side winding and the fundamental frequency AC on network side winding.

The maximum electric field curves with time vary in different media when the resistivity ratios of oil paper are equal, because the DC component in Y₀-Δ connected converter transformer(about 350 kV) is much larger than that of Y₀-Y connected one (about 120 kV). The curve of electric field with time is relatively smooth in some intervals, but at some instants the curve is similar to the voltage curve of the valve side winding. The maximum point on the smoother curve is corresponding to the line side winding, and some is corresponding to the valve side winding.

When the oil resistivity changes a lot, the maximum electric field strength of oil, oil paper and insulation layer alters greatly (as Figure 5, Figure 6, Table 2, Table 3). e.g. the maximum electric field strength of oil, insulating layer and oil paper is respectively 6.16 kV/mm, 31.06 kV/mm and 45.38 kV/mm while the resistivity rate is 1:500:1000; the maximum is respectively 9.07 kV/mm, 11.23 kV/mm and 5.19 kV/mm while the resistivity rate is 1:1:1. The change of the maximum electrical field is similar to that of the oil resistivity. In the initial operation, the resistivity of oil, insulation layer and paper is higher (e.g. the resistivity ratio of the oil, insulation layer and pressboard 1:500:1000), after running

for some time, the resistivity of aging media will reduced (e.g. the resistivity ratio of the oil, insulation layer and pressboard 1:1:1 in extreme cases). It is beneficial for the insulation of oil immersed paper when the maximum electric field

strength of oil and insulation layer decreases while the voltages of network and valve side winding are invariable. Actually it's harmful for the oil insulation when the maximum electric field of oil increases from 6.16 kV/mm to 9.07 kV/mm. Therefore the effects of the resistivity aging on electric field should be taken consider into the converter transformer design.

6. Conclusion

The finite element method in frequency domain and the fast inverse Fourier transform method are used to obtain the steady AC-DC hybrid electric field in this paper. The resistivity ratio of oil and oil immersed paper have been changed, meanwhile the relative permittivity kept unchanged, to simulate the aging process of oil immersed paper after long time operation. More equip-potential lines tend to locate at pressed paper with increasing ratio. However, the curve of the maximum electric field intensity with time is not the same within different media even the resistivity ratios of paper are unchanged, because the DC component in Y₀-Δ is not the same as in Y₀-Y connected one. Meanwhile the maximum electric field of insulation paper increases obviously and the maximum

electric field of oil reduces slightly, and both are important to the transformer insulation.

Acknowledgements

This work is sponsored in part by the National Natural Science Foundation of China under Grant 51407075, in part by the Hebei Province Natural Science Foundation under Grant E2015502004, and in part by the Fundamental Research Funds for the Central Universities of China under Grant 2015 MS79.