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No Load Simulation and Downscaled Experiment of UHV Single-Phase Autotransformer Under DC Bias

The time-domain field-circuit coupled finite element method (FEM) is widely used to analyze the DC effects on transformers. However, the long transient process of calculation will encounter difficulties due to the tremendous requirement of computer memory and CPU time, especially when it comes to ultra-high-voltage (UHV) autotransformers, which has a large time constant due to their less resistance and larger inductance. Besides, compared to the AC voltage, the DC voltage is much smaller, miscalculations usually occur as the component of DC bias can be easily neglected in the simulation. In this paper, a time-domain field-circuit coupling FEM combined with the series resistance and voltage compensation method (SRVCM) is proposed, a large resistance is in series in the circuit model to speed up the convergence of transient process and amplify DC component. Afterwards, the voltage supply is increased to compensate for the reduced voltage drop due to series resistance. The excitation waveform and its frequency characteristics of UHV autotransformer under various DC bias are discussed under no-load condition. Its advantages over conventional decrease the numerical miscalculation and the iterations calculation of the transient process. An experimental verification of the proposed method is carried out by a downscaled autotransformer.


I. INTRODUCTION
The increasing global demand for energy consumption and environmental impact become increasingly prominent, the serious energy crisis is becoming a key focus worldwide [1]- [3]. Thus, eight 1000-kV ultra-high voltage (UHV) AC power transmission projects have been put into operation in China and over 100 units of UHV transformers have been put into service for solving the energy crisis [4]. UHV transformer is one of the key equipment of UHV transmission system, their safe and stable operation is directly related to the safety of UHV transmission system. UHV transformers mostly use self-coupling structure, and the transformer group composed of the single-phase autotransformer has less tolerance to the DC bias current [5].
DC bias of transformers is an abnormal working condition. The main causes of DC bias current flowing in transformer windings are mainly from monopole operation of The associate editor coordinating the review of this manuscript and approving it for publication was Mehmet Alper Uslu.
HVDC transmission [6], nonlinear loads such as AC drives, switch-mode power supplies and grid-connected converters, which can cause, the generation of unwanted current harmonics and a DC current component injection into the grid [7], and geomagnetically induced current (GIC) caused by solar magnetic storms, which could attain up to 100 A per phase for 1 min and 50 A per phase for 5 min during the most severe magnetic storm [8]. The main effect of a DC current flowing into a transformer windings is the asymmetric magnetic core saturation during a sinusoidal half-period (half-cycle saturation), as a result, a series of problems occur, such as the serious distortion of excitation current, increased reactive power absorption, partial overheating and increase of vibration and noise [4], [9]- [14]. Thus, precise and in-depth analysis of DC bias for UHV autotransformers is necessary for the safe operation of equipment and transmission systems.
Nowadays, many in-depth research and analysis have been studied on the DC bias problems of transformers by simulations. FEM are wildly used in the analysis of simulations [15]- [21]. However, FEM for the simulation can be extremely time-consuming with the massive mesh and long transient process, especially in the calculation of UHV autotransformer, consists of large inductance and small resistance, which means a bigger time constant and longer transient process. Due to the large inductance to small resistance as well as the large 1000 kV AC voltage to small DC component in UHV autotransformer, a long transient process of the minimal time step and error criterion is required for avoiding miscalculation by conventional calculation. Although there are many proposed algorithms to shorten the calculation time of FEM, it still needs hours even days to solve the large capacity transformers [22], not mention the DC bias condition, and some research has a really district standard to set up an initial value, which may not suitable for other models [23].
For the purpose of increasing the calculation accuracy and speeding up the convergence of the transient process, this paper proposes a new fast solution by making good use of series resistance and voltage compensation method (SRVCM) based on time-domain filed-circuit coupling FEM. By using this method, (i) we can shorten the transient process and obtain the precise value with acceptable time and memory consumption. (ii) The excitation current waveform, its frequency characteristics and flux distribution of UHV autotransformer without load under DC bias are analyzed, respectively. (iii) An experimental verification of the proposed method is carried out by a downscaled autotransformer.

II. COMPUTATION MODEL COMBINED WITH SRVCM
Nowadays, the time-domain field-circuit coupling FEM is wildly used to calculate electromagnetic quantities of transformer. Based on the time-domain field-circuit coupling model, the electromagnetic calculation of the transformer is divided into field and circuit model, which can obtain dynamic inductance and transient current, respectively.
In order to reduce the transient process, a large series resistance is utilized in the circuit model to reduce the time constant of UHV autotransformer and shorten the transient process. At the same time, voltage compensation is used to make up the voltage drop caused by series resistance, which ensures the winding voltages are consistent with the rated voltage.

A. FIELD-CIRCUIT COUPLING MODEL WITH SERIES RESISTANCE
The capacity of UHV autotransformer has reached 1000 MVA, single-phase four-limb or single-phase fivelimb are mainly adopted. This paper focuses on single-phase four-limb UHV autotransformer, which consists of two main limbs and two side limbs. There are 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 are made of massive steel and carry eddy currents are neglected. The core and the tank shielding are laminated and are assumed to be free of eddy currents. The eighth magnetic model is shown in Fig. 1 (a). The B-H curve of the core used in the calculation is shown in Fig. 1 (b). It should be noted that the measured data are only provided to 1.91 T, the linear curve fitting method is used to extend the curve. Thus, the actual results are more saturated than the simulation under DC bias. The magnetic field based on the edge FEM is utilized for its good accuracy of boundary problem of high permeability material [24]. The edge FEM differential equation uses vector magnetic potential A and field equation according to Maxwell is as follows where µ is permeability of magnetic material, A is vector magnetic potential, J is current density.

VOLUME 8, 2020
Edge element interpolation function can be expressed as where {M n , k=1, 2, . . . , n} is weight function sequences, n is the number of base function sequence, n n is the total number of edge.
Applying the Green theorem, the Galerkin weighted residual equation is where M m is weight function sequences which is the same as basis function, with the known current density J, which is supposed to be obtained by circuit model, magnetic vector A can be calculated by substituting the weight function in (3).
In order to accurately calculate the variation of excitation current, the dynamic inductance parameters (the slope of the BH curve) of winding coil are needed to reflect the saturation degree of core while calculating the field-circuit coupling model of DC bias problem. The voltage of dynamic inductance u L can be obtained by where ψ is coil flux linkage, i is winding current, L is dynamic inductance. According to the principle of the disturbance energy, the magnetic field energy change can be expressed as The magnetic field energy changes in the transformer with winding current changes i is shown as Based on the principle of energy conservation, W 1 = W 2 , the dynamic inductance L can be obtained. The circuit diagram with a series resistance is shown in Fig. 1 (c), and the transient differential equations of the circuit model is shown as follows where u AC and u DC are AC and DC voltage, respectively, R is series resistance, L and M present the dynamic self and mutual inductance, respectively. The fourth-order Runge-Kutta method is selected as the numerical method to solve the voltage differential equation obtained in the field-circuit coupling method. The classical fourth-order Runge-Kutta formula is shown as where h is time step size K 1 ∼ K 4 are the four slope in a time step, i k and i k+1 are the current values at time t k and t k+1 .
In conclusion, the steps of the non-linear magnetic field with time-domain field-circuit coupling FEM are as follows: (i) Input coil current i k to the transformer magnetic field model, based on edge FEM energy disturbance principle, dynamic inductance L can be calculated.
(ii) L is then substituted to the differential equation of the circuit model and i k+1 can be calculated by the fourth-order Runge-Kutta method.
(iii) i k+1 is used as a new input current to solve the new dynamic inductance at the next moment.
(iv) When the error between the excitation current i k and i k+1 is less than 0.1%, it is considered that the current has reached a steady state and the iteration terminates.

B. SERIES RESISTANCE AND VOLTAGE COMPENSATION METHOD (SRVCM)
Unlike with the general transformer, the resistance of the UHV autotransformer is extremely small compared to its large inductance, especially when inductive load imposed at low voltage side. The long transient process will encounter difficulties due to the tremendous requirements of computer memory and CPU time. Beyond that, this characteristic of large inductance and small resistance could also easily lead to the miscalculations as the component of DC bias can be easily neglected by the ultra-high AC voltage in the simulation. By means of series resistance, the DC voltage can be amplified to a content, which is not easy to be neglected, as shown in Fig. 2. Therefore, the accuracy of DC bias calculation can be increased by series resistance.
In order to speed up the convergence of the transient process, a large resistance R s is series in the circuit, as shown in Fig. 1 (c). The time to steady state of excitation current with various series resistance are shown in Fig. 3, it can be seen  that the series resistance obviously speeds up the convergence of the transient process, from 0.2892s of 200 to 0.1286s of 1000 , which reduced almost 55% transient time.
The series resistance does shorten the calculation time obviously by reducing the time constant. However, as the increase of the series resistance the accuracy grow rate gradually slows down. The series resistance value should be selected according to various situations. On the one hand, the series resistance is determined by the DC bias current. As can be seen in Table. 1, with the same series resistance, different DC bias current owns different miscalculation. When series resistance is 200 , the miscalculation is 23.8% and 7.0% with 20 A and 100 A, respectively. On the other hand, the side effects of the series resistance are also needed to be taken into account. The larger series resistance will cause a bigger voltage drop, which means more voltage compensation, whereas, the effects by increasing the series resistance are getting smaller. Such as when I DC is 100 A, the miscalculation can drop to 2.5% with 500 and only get 1.8% decrease with another 500 .
In order to revise the voltage drop caused by series resistance, voltage compensation is required, the compensation steps are as follows: (i) Solving the k th calculated winding current i k of a full cycle in steady state.
(ii) The value of compensation voltage is equal to series resistance multiplied the difference between winding current i k and DC component I kDC .
(iii) The k+1 th voltage compensation can be obtained by adding the compensation voltage to the original AC and DC voltage source, as shown in (9).
(iv) The compensation ends until the voltage u 1 meets the criterion, as presented in (10). u k+1 = U m cos(ωt)+U DC +R(i k −I kDC ) k = 1, 2... (9) max |u 1 − U m cos (ωt)| ≤ 1%U m (10) where u k+1 is the (k + 1) voltage supply, U m is the amplitude of the rated AC voltage on the original side, U DC is the inserted DC voltage, i k is k th current in steady state, I kDC is DC component in steady state. The winding voltage u 1 after series resistance with and without compensation are shown in Fig. 4, respectively. As shown in Fig. 4, a big voltage variation occurs before the compensation, which is the side effects of the series resistance. By the use of voltage compensation, the voltage waveform almost restored to the rated voltage. Table 1 shows the miscalculation of DC bias calculation under various series resistance, 200 , 500 and 1000 are in series at high voltage side under various DC bias current, respectively. In theory, the DC component of winding current should be equal to the imposed DC bias current when the winding current reached the steady state. Therefore, the accuracy of the calculation can be verified by comparing the DC component of winding current with the DC bias current. Taking I DC = 100 A as an example, with the increase of series resistance, the DC component in the high voltage winding gradually approximates to the added DC bias current value, and its miscalculation gradually drop from 7.0% of 200 to 2.5% of 500 and 0.7% of 1000 . It can be concluded that the series resistance can effectively improve the calculation accuracy, as shown in Table 1. And the comparison of  miscalculation between conventional and the SRVCM under various critical error criterion are present in Table 2. The miscalculation is obtained by comparing the DC components in the excitation current with the imposed DC bias current. As can be seen in the Table. 2, even under a strict critical error criterion like ε = 1.0e-06, the miscalculation could still reach 17.6% with conventional method, whereas, the miscalculation reaches 94.8% with the critical error criterion ε = 1.0e-02. However, by means of SRVCM, the miscalculation with 1000 can drops to 3.7% with the critical error criterion ε = 1.0e-02.

III. SIMULATION OF DC BIAS EFFECTS ON UHV AUTOTRANSFORMER
This paper conducts simulation analysis for 1000 kV UHV autotransformer, and its specific parameters are presented in Table 3. By utilizing the SRVCM, the electrical quantities can be calculated with acceptable time and accuracy. The flux distribution, excitation current, and its harmonic spectrum under various DC bias are shown in Fig. 5 and Fig. 6. During the first half cycle in which the AC flux and the DC flux are in the same direction, hence the core will be saturated. As can be seen from Fig. 5 (a), the flux density of the main limb of the core is arising from 1.65T to 2.0T under 5A DC bias current. Due to the saturation the excitation current will arise greatly. Whereas, in last half cycle, the AC and DC flux are in the opposite direction, hence, the excitation current will reduce rapidly to a small value, as shown in Fig. 5 (b). As can be seen from Fig. 5 (c), the peak value of excitation current without DC bias current is 1.84 A, and the peak value increase to 194 A, 393A, 545A and 628A with 20 A, 50 A, 80 A and 100 A DC bias current, respectively. The distortion under 100 A DC bias current is 340 times higher than the rated excitation current.
Discrete Fourier transform (DFT) is applied to the waveform of excitation current under DC bias. The harmonic magnitudes of excitation current under various DC bias currents are shown in Fig. 6. Among the harmonic frequencies, the number 0 represents the DC component, no.1 represents the fundamental component, and the other numbers represent each harmonic order. It can be seen from Fig. 6, harmonics occur and arise greatly as the DC bias current increase, and the distortion mainly concentrated before 6th harmonic. High order harmonics are less affected by the DC bias current. By calculating and analyzing the DC bias effects on UHV transformer under no load condition, three aspects are notable: (i) Distortion occurs mainly in the first half of the cycle, peaking in the quarter cycle, the distortion under DC bias current can be hundreds of times higher than the rated one. The excitation current in last half of the cycle is less affected by the DC bias current due to the opposite direction.
(ii) Harmonics increase greatly as DC bias current increase, even harmonics appear in excitation current harmonics, mainly the 2nd, 4th and 6th order, high order harmonics increase much slower than the low order harmonics.
(iii) The growth rate of second harmonic almost remains unchanged, whereas, the growth rate of higher harmonics gradually slows down as the DC bias current goes. This means the second harmonic could be selected as an indicator sign for monitoring the DC bias current introduced into the transformer windings.

IV. EXPERIMENTAL VALIDATION
To verify the correctness of the model, a downscaled singlephase four-column three-winding autotransformer was built with parameters of 5000 VA and 360V, which is shown in Fig. 7 (a). In order to ensure the downscaled model has the same behavior as the UHV autotransformer, its core structure and connection mode are consistent with UHV autotransformer, the downscaled model's size is 1/12 of the UHV autotransformer, whose length is 661 mm, width is 100 mm and height is 356 mm, respectively. The flux density in rated operation is 1.65 T, which is the same as the UHV autotransformer. And the ratio of turns, voltage, and capacity of each side are consistent with the UHV autotransformer. The DC bias test platform is shown in Fig. 7 (b) and (c). The voltage regulator is connected to a voltage source, providing a 360V 50Hz 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 1194uF are used to reduce the AC current passing through the DC source. The electrical quantities are measured by a power analyzer. The equipment parameters in the test platform are shown in Table 4.
The DC bias no-load test of a downscaled transformer model was carried out. The measured value of no-load excitation current without DC bias current and winding current under rated operation are compared with simulation results, as shown in Fig. 8. And the comparison between the simulated and measured values of excitation current and winding current under various DC bias current is shown in Fig. 9.  As can be known in Fig. 9, the excitation current under DC bias current of the downscaled transformer shows the same tendency as the excitation current of UHV transformer. The results show that the relative deviations of peak value and effective value of excitation current without DC bias are 2.2% and 4.2% respectively. Taking excitation current under 5A DC bias current as an example, the relative deviations of peak excitation current and effective value under DC bias are 7.9% and 3.7%, respectively. This measured data shows a good agreement with the simulation results.
It should be noted that there are two errors can be noticed easily in Fig. 8 and Fig. 9, which are mainly caused by: (i) Neglecting the hysteresis effects, which cause the error when excitation current close to around 0, as shown in Fig. 8. (ii) Linear curve fitting is utilized for the deep saturation situation, as shown in Fig. 1 (b), thus the actual saturation condition is severer than the simulated one, which causes the calculation error about peak value, as shown in Fig. 9.
The comparison between the simulation results and the experimental data shows that the proposed field-circuit coupled FEM combined with SRVCM has good accuracy in calculating DC bias effects on transformer.

V. CONCLUSION
In this paper, a fast and accurate solution based on the time-domain field-circuit coupled FEM, combined with SRVCM is proposed. By good use of this method, the DC bias effects on UHV autotransformer without load is calculated and the distortion of excitation current under various DC bias are analyzed. By utilizing the SRVCM, it can not only speed up the convergence of the transient process due to the reduction of the time constant but also decrease the miscalculation by amplifying the order of magnitudes of DC component to the level of AC voltage source. As the increase of DC bias current, the distortion of excitation current rises sharply, mainly occurs on the first half cycle, even harmonics appear. The second harmonic almost keeps linear with DC bias current, and other harmonics decrease with the deepening of DC biasing. The calculation error can be reduced if more data can be provided in deep saturation of the BH curve. The correctness of the proposed model is verified by a downscaled transformer.