Analysis of the DC Bias E ﬀ ects on the UHV Autotransformer with Rated Load Based on a Reduced-Scale Model Experiment

Featured Application: This paper proposes a new equivalent technique, the DC bias current equivalent relationship between the UHV transformer and the reduced-scale model (RSM) is established by means of the identical saturation degree, the proposed RSM is capable to reﬂect the actual operation of the UHV transformer under DC bias. Abstract: The DC bias of transformers as a result of geomagnetically induced currents (GIC), monopole operation of high voltage direct current (HVDC) transmission and the increasing power electronic devices in the power grid, creates a high magnitude of fundamental and harmonic components in winding currents, reactive power, and vibro-acoustic noise. For transformers with large capacity, it is very expensive and di ﬃ cult to carry out an in-depth DC bias e ﬀ ects investigation on a practical equipment, especially for the ultra-high voltage (UHV) transformer. In this paper, a reduced-scale model (RSM) based on the principle of identical saturation degree is proposed, and the equivalent relationship of DC bias current is obtained. Furthermore, the DC bias platform with the RSM is established, experiments have been conducted to analyze the characteristics of the transformer under DC bias. It turns out from the experiments that the current of high voltage side and excitation are less tolerated to the DC bias current, the distortion occurs during the ﬁrst half of the cycle and peaking at the quarter. Besides, the no-load loss and total harmonic distortion (THD) increase signiﬁcantly with an increasing DC bias. The results of the characteristics of RSM with load under DC bias may provide great services in protecting and reducing the DC bias e ﬀ ects on the UHV transformer in actual operation.


Introduction
Due to the geomagnetically induced current (GIC) caused by periodic solar storms, monopole operation of the large-scale construction of high voltage direct current (HVDC) transmission and harmonics caused by the increasing nonlinear loads in the power grid, the direct current (DC) flows through the earthed neutrals of the transformer and cause a DC bias to the magnetizing current of the transformer [1][2][3][4][5]. This results in the asymmetric magnetic core saturation during a sinusoidal half-period, the serious distortion of excitation current, increased reactive power absorption, partial overheating and increase of vibration and noise [6][7][8][9][10][11][12]. Because of depletion of fossil fuel reserves and increasing demand for energy consumption, the ultra-high voltage UHV transmission projects have been put into service for solving the energy crisis. The (UHV) autotransformer is the key equipment of the UHV transmission systems, mostly with self-coupling

Experiment Setup
The DC bias test platform is shown in Figure 1. The voltage regulator is connected to a voltage source, providing a 360 V 50 Hz sinusoidal excitation to the arrangement. The isolation transformer is utilized for avoiding the DC current effects on the supply source. The DC source with adjustable resistance are used to obtain various DC current while the eight parallel capacitors of 1194 µF are used to reduce the AC current passing through the DC source. The electrical quantities are measured by the power analyzer. The equipment parameters in the test platform are shown in Table 1.
In order to build the equivalent relationship with the UHV transformer, the core structure and winding connection mode of RSM are consistent with the UHV transformer, as shown in Figure 2. The RSM is a single-phase four-column three-winding autotransformer built with an approximate scale of 1/12 relative to the UHV transformer. The capacity and rated voltage of the RSM are 5 kVA and 360 V, whose length, width and height are 661 mm, 100 mm and 356 mm, respectively. The flux density at rated voltage is 1.65 T, which is consistent with the UHV transformer. Figure 3 shows the silicon steel type of UHV transformer and reduced-scale model, which are 27 ZH095 and B23R075, respectively. It has to be noted that the measured B-H data are only provided to 1.91 T, the linear curve fitting method is used to extend the curve. The parameters of the UHV transformer and the RSM are presented in Table 2.

Theory of Reduced-Scale Model
The principle to establish the relationship of RSM to UHV transformer under DC bias is to obtain the identical saturation degree of the core. How to accurately reflect the saturation degree of the transformer is the key to building the RSM.
It is generally known that even harmonics occur with the introduction of the DC bias current and rise greatly with the increase of the DC bias current. Besides, the second harmonic is a significant indicator to detect the DC bias current flowing into the transformers. In order to establish the equivalent saturation degree under DC bias of the UHV transformer and RSM, the ratio of the second harmonic to the fundamental was selected. In this paper, a time-domain field-circuit coupling FEM was utilized to calculate the electromagnetic quantities of UHV transformer and RSM. The ratio of second harmonic to fundamental was selected to show the saturation degree of the core. By means of this ratio index, the equivalent relationship of the reduced-scale model to the UHV transformer was established. The field-circuit coupling FEM was utilized to obtain this ratio indicator. Afterwards, the equivalent relationship of DC bias current of RSM to UHV transformer was obtained.

Time-Domain Field-Circuit Coupling FEM
The field-circuit coupling 3-D FEM was utilized to calculate the ratio of second harmonic to fundamental of UHV transformer and RSM, which was used to establish their equivalent relationship. The number of turns and geometry of the FEM model of the UHV and RSM are exactly followed by their own parameters, respectively. The core structure and windings of RSM were at an approximate scale of 1/12 relative to a UHV transformer's full size. There were low voltage winding, common winding and series winding from inside to outside, respectively. It has to be noted that the tank, the tie bars and the clamping plate which were made of massive steel and carry eddy currents were neglected. The core and the tank shielding were laminated and were assumed to be free of eddy currents. The eighth magnetic model of RSM and UHV transformer is shown in Figure 4a.
The FEM differential equation uses vector magnetic potential A and field equation according to Maxwell [13] is as follows where µ is permeability of magnetic material, A is vector magnetic potential, J is current density. The circuit diagram is shown in Figure 4b, and the transient differential equations of the circuit model is shown as follows where U m and U DC are AC and DC voltage, respectively. r, L and M present the resistance, dynamic self and mutual inductance, respectively. R L is the load resistance. The subscript 1, 2, 3 present the series, common and low voltage winding, respectively.
common winding and series winding from inside to outside, respectively. It has to be noted that the tank, the tie bars and the clamping plate which were made of massive steel and carry eddy currents were neglected. The core and the tank shielding were laminated and were assumed to be free of eddy currents. The eighth magnetic model of RSM and UHV transformer is shown in Figure 4a. The FEM differential equation uses vector magnetic potential A and field equation according to Maxwell [13] is as follows where µ is permeability of magnetic material, A is vector magnetic potential, J is current density. The circuit diagram is shown in Figure 4b, and the transient differential equations of the circuit model is shown as follows where Um and UDC are AC and DC voltage, respectively. r, L and M present the resistance, dynamic self and mutual inductance, respectively. RL is the load resistance. The subscript 1, 2, 3 present the series, common and low voltage winding, respectively.

Equivalent Relationship of Reduced-Scale Model to UHV Transformer
By utilizing the field-circuit coupling model mentioned above, the electromagnetic quantities of the UHV transformer and RSM can be calculated, as shown in Figures 5-8. Figure 5 show the excitation current waveform of the UHV and RSM under various DC bias current, respectively. Discrete Fourier transform (DFT) is applied to the waveform of the excitation current of the UHV transformer under DC bias. The harmonic magnitudes of excitation current under various DC bias currents are shown in Figure 6. Due to the distortion mainly concentrated below the 6th harmonic and less sensitive of high order harmonics to DC bias current, the figure only shows the harmonics under 6th order. As can be seen from Figure 6, the fundamental and second harmonic rose almost linearly with the DC bias current, whereas, the growth of higher harmonics slowed down gradually. Figure 7 shows the ratio of the second harmonic to the

Equivalent Relationship of Reduced-Scale Model to UHV Transformer
By utilizing the field-circuit coupling model mentioned above, the electromagnetic quantities of the UHV transformer and RSM can be calculated, as shown in Figures 5-8. Figure 5 show the excitation current waveform of the UHV and RSM under various DC bias current, respectively. Discrete Fourier transform (DFT) is applied to the waveform of the excitation current of the UHV transformer under DC bias. The harmonic magnitudes of excitation current under various DC bias currents are shown in Figure 6. Due to the distortion mainly concentrated below the 6th harmonic and less sensitive of high order harmonics to DC bias current, the figure only shows the harmonics under 6th order. As can be seen from Figure 6, the fundamental and second harmonic rose almost linearly with the DC bias current, whereas, the growth of higher harmonics slowed down gradually. Figure 7 shows the ratio of the second harmonic to the fundamental of excitation current with various DC bias current of the UHV transformer and RSM. By means of the same ratio, the equivalent relationship of DC bias current of the UHV transformer and RSM was obtained, as can be seen in Figure 8. Table 3 presents the results of the equivalent relationship of DC bias current under the same saturation degree.
Appl. Sci. 2020, 10, x FOR PEER REVIEW 6 of 14 fundamental of excitation current with various DC bias current of the UHV transformer and RSM. By means of the same ratio, the equivalent relationship of DC bias current of the UHV transformer and RSM was obtained, as can be seen in Figure 8. Table 3 presents the results of the equivalent relationship of DC bias current under the same saturation degree.    fundamental of excitation current with various DC bias current of the UHV transformer and RSM. By means of the same ratio, the equivalent relationship of DC bias current of the UHV transformer and RSM was obtained, as can be seen in Figure 8. Table 3 presents the results of the equivalent relationship of DC bias current under the same saturation degree.    fundamental of excitation current with various DC bias current of the UHV transformer and RSM. By means of the same ratio, the equivalent relationship of DC bias current of the UHV transformer and RSM was obtained, as can be seen in Figure 8. Table 3 presents the results of the equivalent relationship of DC bias current under the same saturation degree.     RSM Ratio of 2nd to 1st UHV 0 0 0

Experimental Data
A series of DC bias test of the RSM was carried out, which is presented in Table 4. The DC bias effects on the RSM with and without load are presented, respectively. Figure 9a shows five cycles of the no-load excitation current without the DC bias current, Figure 9b-d show five cycles of the no-load excitation current with 1 A, 3 A and 5 A DC bias current, respectively. As can be seen from the figure, the distortion of excitation current rose sharply as the DC bias current increased. The distortion occurred mainly in the first half of the cycle, peaking in the quarter cycle. For instance, the peak value of excitation current without DC bias current was 2.5 A, and the peak value increased to 8.5 A, 25.8 A, and 42.6 A with 1 A, 3 A and 5 A DC bias current, respectively. The distortion under 5 A DC bias current was 17 times higher than the rated excitation current. By contrast, the excitation current in last half of the cycle decreased with the DC bias current due to the opposite direction.
DFT was then applied to the waveform of excitation and winding currents, respectively. Figure 10 show that the harmonics increased obviously as the DC bias current increased. Although the DC bias current brought distortion to all the current waveform, the cause of these current distortions was not the same. The harmonics magnitudes of excitation current and the high voltage side current showed an identical tendency. The even harmonics occurred due to the DC bias current, especially the second harmonic, which turned into the biggest harmonic component. The occurrence of even harmonics caused the distortion of the waveform of the excitation current and high voltage side current. Whereas the middle and low voltage side current showed identical tendency, the DC bias effects on even harmonic was small, the main reason for distortion of the common and low voltage current was the odd harmonics.
The winding current at each side of the RSM with load under various DC bias are shown in Figure 11. As can be seen from figure, the distortion of the waveform increased obviously as the DC bias current increased. The distortion occurred when DC bias current flowed into the windings and the distortion showed the same tendency as the excitation current, which mainly occurred in the first half cycle, the current maximum was located close to the moment of a quarter. Besides, the middle and low voltage side current seemed less affected by the DC bias, whereas the high voltage side current waveform showed an obvious distortion. The total harmonic distortion (THD) was utilized to present the distortion degree of current under DC bias current, which is defined as the ratio of the root mean square (RMS) amplitude of a set of higher harmonic frequencies to the fundamental [8], expressed by THD = I 2 2 I 3 2 I 4 2 + . . .
The THD of the current at each side are shown in Figure 12. The excitation and high voltage wingding current were less tolerant of DC bias current, the distortion degree increased obviously as the DC bias current increased, whereas, the current of the middle and low voltage side remained almost the same under DC bias current.
The no-load loss and the active power at rated operation were measured by a power analyzer, as shown in Figure 13. The DC bias current caused a significant increase in no-load loss. Therefore, active power of each side decreased as the DC bias current increased.
wingding current were less tolerant of DC bias current, the distortion degree increased obviously as the DC bias current increased, whereas, the current of the middle and low voltage side remained almost the same under DC bias current.
The no-load loss and the active power at rated operation were measured by a power analyzer, as shown in Figure 13. The DC bias current caused a significant increase in no-load loss. Therefore, active power of each side decreased as the DC bias current increased.

Performed test With and without DC bias current
No load Excitation current ( Figure 9); No-load loss (Figure 13a)

Performed test With and without DC bias current
No load Excitation current ( Figure 9); No-load loss (Figure 13a) With load Winding current of each side ( Figure 11); Active power of each side (Figure 13b) (c) (d)

Performed test With and without DC bias current
No load Excitation current ( Figure 9); No-load loss (Figure 13a) With load Winding current of each side ( Figure 11); Active power of each side (Figure 13b

Simulation Results
This paper conducted simulation analysis for the RSM. The no-load excitation current and the winding current with load under various DC bias currents were calculated by the time domain field-circuit coupling FEM and these results will be presented in Section 4.3 for making a comparison with the experimental data. The flux distribution in Figure 14 clearly shows that the core was severely saturated under DC bias current, especially in the main leg and the T-joint area.

Simulation Results
This paper conducted simulation analysis for the RSM. The no-load excitation current and the winding current with load under various DC bias currents were calculated by the time domain field-circuit coupling FEM and these results will be presented in Section 4.3 for making a comparison with the experimental data. The flux distribution in Figure 14 clearly shows that the core was severely saturated under DC bias current, especially in the main leg and the T-joint area.

Comparison between Experiment and Simulation
The measured value of no-load excitation current without DC bias current was compared with simulation results, as shown in Figure 15. Figure 16 shows the comparison between the simulated and measured values of winding current with rated operation under various DC bias currents.
As can be seen from Figures 15 and 16, the measured data showed a good accuracy with the calculated by field-circuit coupling FEM.
It should be noted that two errors can be noticed easily in Figures 15 and 16b, which are mainly caused by: neglecting the hysteresis effects, which caused the error when excitation current was close to 0, as shown in Figure 15a.
Due to the actual data provided only to 1.9 T, linear curve fitting was utilized for the deep saturation situation of BH curve, therefore the actual saturation condition was more severe than the simulated one, which caused the calculation error about peak value, as shown in Figures 15b and  16b.
However, the relative deviations of peak value and effective value of excitation current without DC bias were 2.2% and 4.2% respectively. The relative deviations of peak excitation current and effective value under 5 A DC bias were 7.9% and 3.7%, respectively. These errors are within the acceptable range in engineering.

Comparison between Experiment and Simulation
The measured value of no-load excitation current without DC bias current was compared with simulation results, as shown in Figure 15. Figure 16 shows the comparison between the simulated and measured values of winding current with rated operation under various DC bias currents.
As can be seen from Figures 15 and 16, the measured data showed a good accuracy with the calculated by field-circuit coupling FEM.
It should be noted that two errors can be noticed easily in Figures 15 and 16b, which are mainly caused by: neglecting the hysteresis effects, which caused the error when excitation current was close to 0, as shown in Figure 15a.
Due to the actual data provided only to 1.9 T, linear curve fitting was utilized for the deep saturation situation of BH curve, therefore the actual saturation condition was more severe than the simulated one, which caused the calculation error about peak value, as shown in Figures 15b and 16b.
However, the relative deviations of peak value and effective value of excitation current without DC bias were 2.2% and 4.2% respectively. The relative deviations of peak excitation current and effective value under 5 A DC bias were 7.9% and 3.7%, respectively. These errors are within the acceptable range in engineering. as shown in Figure 15a.
Due to the actual data provided only to 1.9 T, linear curve fitting was utilized for the deep saturation situation of BH curve, therefore the actual saturation condition was more severe than the simulated one, which caused the calculation error about peak value, as shown in Figures 15b and  16b.
However, the relative deviations of peak value and effective value of excitation current without DC bias were 2.2% and 4.2% respectively. The relative deviations of peak excitation current and effective value under 5 A DC bias were 7.9% and 3.7%, respectively. These errors are within the acceptable range in engineering.

Conclusions
In this paper, a novel reduced-scale model based on an identical saturation degree is proposed to reflect the actual operation condition of the UHV autotransformer under DC bias. The proposed equivalent technique is applicable for all the reduced-scale model only if they shared the same structure and identical saturation degree. The load test results of the RSM under various DC bias current have been presented at rated operation. The experimental results show that the distortion of excitation current rises sharply as the DC bias current increases, and mainly occurs on the first half cycle. The even harmonics occur due to the DC bias current, the second harmonic almost grows linearly with DC bias current, whereas high harmonics growth rate decreases with the deepening of DC bias. The high voltage side current shows less tolerance to the DC bias, the THD is much higher than the middle and low voltage side current. As the DC bias current increases, the flux distribution cloud map shows the main leg and T-joints are highly saturated, which causes the increases of no-load loss. By good use of the proposed equivalent reduced-scale model, the rated load operation of the UHV transformer under DC bias can be predicted, and the experimental data of the RSM can play a significant role in guiding the protection of the UHV transformer under DC bias.

Conclusions
In this paper, a novel reduced-scale model based on an identical saturation degree is proposed to reflect the actual operation condition of the UHV autotransformer under DC bias. The proposed equivalent technique is applicable for all the reduced-scale model only if they shared the same structure and identical saturation degree. The load test results of the RSM under various DC bias current have been presented at rated operation. The experimental results show that the distortion of excitation current rises sharply as the DC bias current increases, and mainly occurs on the first half cycle. The even harmonics occur due to the DC bias current, the second harmonic almost grows linearly with DC bias current, whereas high harmonics growth rate decreases with the deepening of DC bias. The high voltage side current shows less tolerance to the DC bias, the THD is much higher than the middle and low voltage side current. As the DC bias current increases, the flux distribution cloud map shows the main leg and T-joints are highly saturated, which causes the increases of no-load loss. By good use of the proposed equivalent reduced-scale model, the rated load operation of the UHV transformer under DC bias can be predicted, and the experimental data of the RSM can play a significant role in guiding the protection of the UHV transformer under DC bias.