## 3 Theory of Reduced-Scale Model
### 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.  
\begin{table}
\begin{tabular}{l l} \hline
**Equipment** & **Parameter** \\ \hline Voltage & Rated capacitor: 10 kVA \\ \hline Regulator & Output voltage: 0 V–300 V \\ \hline Isolation & Rated capacitor:10 kVA, \\ \hline Transformer & Input voltage: 220 V, Output voltage: 360 V \\ \hline DC & Tektronix PWS 2326, Output DC current: 0 A–6 A, voltage: 0 V–32 V \\ \hline Capacitor & BSMJ-0.4-60-1, Capacitance: 8*1194 \(\mu\)F (8 capacitors in parallel, impedance is 0.3 \(\Omega\)) \\ \hline Power & Fluke Norma 5000 \\ Analyzer & \\ \hline Rated & Resistor at middle voltage side: 6.72 \(\Omega\) \\ Load & Resistor at low voltage side: 2.16 \(\Omega\) \\ \hline \end{tabular}
\end{table}
Table 1: Equipment configuration of experimental platform.  
\begin{table}
\begin{tabular}{c c c} \hline
**Parameters** & **UHV** & **RSM** \\ \hline Type & ODFPS-1000000/1000 & DG-5/0.36 \\ Rated capacity (kVA) & \(1\times 10^{6}\)/\(1\times 10^{6}\)/\(3.34\times 10^{5}\) & 5/5/1.67 \\ Frequency (Hz) & 50 & 50 \\ Rated voltage (kV) & \(\frac{1050}{\sqrt{5}}\Big{/}\Big{(}\frac{525}{\sqrt{5}}\pm 4\times 1.25\% \Big{)}/110\) & 0.36/0.18/0.06 \\ Rated current (A) & 1649.57/3299.14/3036.3 & 13.89/27.78/27.78 \\ High/middle/low voltage winding turns & 1356/678/246 & 120/60/20 \\ Core structure & Two main columns, two side columns & Two main columns, two side columns \\ \hline \end{tabular}
\end{table}
Table 2: Parameters of the ultra-high voltage (UHV) transformer and reduced scale model (RSM).  
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 \(\mathbf{A}\) and field equation according to [PERSON] [13] is as follows  
\[\triangledown\times\frac{1}{\mu}\triangledown\mathbf{\times}\mathbf{A}=\mathbf{J} \tag{1}\]  
where \(\mu\) is permeability of magnetic material, \(\mathbf{A}\) is vector magnetic potential, \(\mathbf{J}\) is current density.  
The circuit diagram is shown in Figure 4b, and the transient differential equations of the circuit model is shown as follows  
\[\left[\begin{array}{cccc}L_{1}+M_{12}&L_{2}+M_{12}&M_{13}+M_{23}\\ M_{12}&L_{2}&M_{23}\\ M_{13}&M_{23}&L_{3}\end{array}\right]\left[\begin{array}{c}\frac{\mathrm{d}i_ {1}}{\mathrm{d}t}\\ \frac{\mathrm{d}j_{2}}{\mathrm{d}t}\\ \frac{\mathrm{d}i_{3}}{\mathrm{d}t}\end{array}\right]+ \tag{2}\] \[\left[\begin{array}{cccc}r_{1}&r_{2}&0\\ -R_{\mathrm{L2}}&r_{2}+R_{\mathrm{L2}}&0\\ 0&0&r_{3}+R_{\mathrm{L3}}\end{array}\right]\left[\begin{array}{c}i_{1}\\ i_{2}\\ i_{3}\end{array}\right]=\left[\begin{array}{c}U_{\mathrm{m}}\cos(\omega t)+U_{ \mathrm{DC}}\\ 0\\ 0\end{array}\right]\]  
where \(U_{\mathrm{m}}\) and \(U_{\mathrm{DC}}\) are AC and DC voltage, respectively. \(r\), \(L\) and \(M\) present the resistance, dynamic self and mutual inductance, respectively. \(R_{\mathrm{L}}\) is the load resistance. The subscript 1, 2, 3 present the series, common and low voltage winding, respectively.