Short-Circuit Fault Analysis of the Sen Transformer Using Phase Coordinate Model

Abstract

The Sen Transformer (ST) provides an economical solution for power flow control and voltage regulation. However, fault analysis and evaluation of the performance of the transmission protection system in the presence of a ST have not been investigated. Hence, a short-circuit model of the ST using the phase coordinate method is proposed in this paper. Firstly, according to the coupled-circuit ST model, the nodal admittance matrix between the sending end and receiving end of the ST was deduced. Subsequently, a fully decoupled mathematical model was established that can reflect three characteristics, including its winding connection structure, electrical parameters, and ground impedance. Thus, with the help of the phase-coordinate-based solving methodology, a short-circuit ST model may be built for various short-circuit faults. The MATLAB and PSCAD/EMTDC software were employed to carry out simulated analyses for an equivalent two-bus system. The short-circuit currents obtained from the time-domain simulation and the analytic calculation utilizing the proposed model reached an acceptable agreement, confirming the simulation’s effectiveness. Moreover, the variation of the fault currents with the variation of the compensating voltage after single-phase-to-ground and three-phase short-circuit faults was demonstrated and used to analyze the effect of the ST on the fault currents.

1. Introduction

With the rapidly growing penetration of renewable energy sources and the ever-increasing complexity of electrical systems, transmission lines are becoming overloaded and experiencing reduced stability, increased voltage variation, and loop-flow of power. This fact is even more pronounced in large-scale power systems [1,2,3,4]. The demands for better utilization of existing power systems and to increase power transfer capability by installing flexible AC transmission system (FACTS) devices have become increasingly urgent. The Sen Transformer (ST), which can provide an independent and bidirectional active and reactive power flow control in a transmission line, has been proven to be reliable and cost-effective when compared with the emerging technology of VSC [5,6,7]. However, the complicated operational modes of the ST and the unique winding connections create some challenges for the fault analysis of power systems with STs. Therefore, it is necessary to establish a short-circuit model for the ST to properly evaluate its influence on fault currents.

From the perspective of the modeling of the ST and its variants, a power flow model of a ST was developed to study the impact of the ST for congestion management and ATC enhancement [8,9]. A steady-state model of a power-transistor-assisted ST(TAST) was established in [10,11], and power flow control for congestion management using TAST was performed in [12]. In addition, a power flow model of the extended ST(EST) was proposed in [13] for power flow analysis in a large-scale power system. However, these models are only suitable for steady-state analysis of power systems involving STs, and fault analysis cannot be performed with the help of these models. For the transient analysis of the ST, a ST digital simulation model was developed using the PSCAD/EMTDC software package [14] and a new algorithm capable of selecting the best combination of tap settings was implemented in it. A detailed electromagnetic transient model was developed in [15] for the ST embedded in a network, but the detailed flux paths in the core were not considered in these models. Furthermore, an electromagnetic transient model for the ST based on the real-time high-fidelity magnetic equivalent circuit was proposed in [16] and estimated on the field-programmable gate array (FPGA) for hardware-in-the-loop applications. Meanwhile, in order to explore the internal characteristics of the ST, an analytical electromagnetic model considering the multi-winding coupling in the ST with a three-phase, three-limb structure was presented in [17]. Additionally, the application of high-power electronic on-load tap-changers in ST was introduced in [18] and a switching transient model was proposed for studying the commutation process. In [19], a quasi-steady-state model of a three-phase five-limb ST (TPFLST) is proposed, applying the principle of duality to evaluate the performance of a TPFLST with unbalanced load conditions. Nevertheless, there has been not much study on the fault analysis of a power system with a ST reported in the published literature. Although the electromagnetic transient ST models proposed in the existing literature can be used for short-circuit analysis, the electromagnetic transient calculations are tedious and time-consuming, especially in large-scale and complex power systems. There is an urgent need to develop a suitable method for short-circuit analysis of power systems containing STs.

From the perspective of short-circuit fault analysis methods, the symmetrical component-based method has been widely applied to perform fault analysis for multiphase distribution networks [20,21]. Moreover, regarding electrical equipment such as the transformer, superconducting fault current limiter, and permanent-magnet synchronous machine, the relevant fault models for carrying out short-circuit studies via symmetrical components are also presented in [22,23,24]. Although symmetrical components afford an admirable degree of unbalanced design and system operation, the assembly of sequence networks and the solution of sequence equations are not always easy, particularly in the case of a system as distinct from load unbalance when mutual coupling exists between the sequence networks [25,26,27]. Meanwhile, as for the three-phase, three-limb model of the ST, it should be noted that the magnetic coupling among all windings of the ST results in its unbalanced three-phase operation [17]. The method of symmetrical sequences may not be as accurate as expected in this case. The phase coordinate method was considered to be an attractive methodology to solve this problem, having been previously employed to model the conventional transformer, three-phase autotransformer, and synchronous machine [28,29,30,31] for short-circuit fault analysis. Hence, it may be an effective way to accurately calculate the short-circuit currents when a ST is installed in a power system, with the aid of the phase coordinate method.

Therefore, this paper proposes a short-circuit model of the ST, utilizing the phase coordinate method according to the different fault types. The contributions of this paper are:

• A fully decoupled mathematical model of the ST is deduced based on the coupled circuit, considering the mutual inductances among all windings for the ST and the ground impedance on the secondary side of the ST.
• Short-circuit models involving single ST faults and combined ST faults are proposed.
• Variations of short-circuit current after the occurrence of single-phase-to-ground and three-phase-to-ground faults in a power system with a ST are presented, and the reasons for these variations are analyzed.

This paper is organized as follows. Section 2 presents a fully decoupled mathematical model of a ST for the phase coordinate method, and the ground impedance on the primary side of the ST is extended in this model. The short-circuit calculation for an electric power network using the phase coordinate method is described in Section 3. In Section 4, case studies are implemented to illustrate the effectiveness of the proposed short-circuit ST model and analyze the variations of short-circuit current magnitude for single-phase-to-ground and three-phase short-circuit faults caused by the ST. Finally, the conclusions of this work are drawn in Section 5.

2. A Fully Decoupled Mathematical Model of ST for Phase Coordinate Method

2.1. The Working Principle of the ST

The basic topology structure of the ST is shown in Figure 1a. The voltages U_sA, U_sB, and U_sC at any point in the transmission lines supply to the Y-connected primary windings of the ST in phases A, B, and C, respectively. These primary windings constitute the exciting unit. A total of nine secondary windings with tap changers constitute the compensating voltage unit. Each limb of the core has three of the secondary windings, i.e., a1, a2, and a3 are placed on the limb of phase A; b1, b2, and b3 are placed on the limb of phase B; and c1, c2, and c3 are placed on the limb of phase C. The vector diagram for the ST’s operation is shown in Figure 1b. As for phase A, the series-connected compensating voltage is as follows: U_ss′A = U_a1 + U_b1 + U_c1. Since there is a 120° phase shift between the induced voltages U_a1, U_b1, and U_c1, the sum of three voltage vectors can be varied by adjusting the tap changers in the secondary windings. Consequently, the obtained U_ss′A becomes variable in magnitude and in the phase angle from 0° to 360°. In this way, the series-connected compensating voltages U_ss′B and U_ss′C in phases B and C can also be derived. Furthermore, the four-quadrant adjustment of the sending end voltage from U_s to U_s′ can be realized, i.e., U_s′ = U_s + U_ss′.

By varying the magnitude of U_ss′ and the phase angle, β, of the injected compensating voltage, the three operational modes for the ST can be achieved: voltage regulation, phase angle regulation, and independent power-flow regulation [6,7].

• Voltage regulation mode.
  If the ST works as a voltage regulator, the phase voltage U_s is added to U_s′ by injecting the compensating voltage U_ss′, and U_ss′ is either in phase (β = 0°) or out of phase (β = 180°) with U_s.
• Phase angle regulation mode.
  When the ST is operating in the phase angle regulation mode, the series-connected compensating voltage U_ss′ is added to the transmission line in quadrature with U_s (β = ±90°).
• Independent power-flow regulation mode.
  If the ST is used as a power flow regulator, the active and the reactive power flows are regulated independently by changing U_s to U_s′ by varying the magnitude and phase angle of the injected voltage U_ss′. The required U_ss′ can be derived from the desired active and reactive power.

2.2. Modeling of ST for Phase Coordinate Method

The coupled circuit of the ST is displayed in Figure 2. L_a, L_b, and L_c denote the self-inductances for the primary windings in phases A, B, and C, respectively. L_a1, L_a2, L_a3; L_b1, L_b2, L_b3; and L_c1, L_c2, and L_c3 denote the self-inductances for the ST secondary windings in phases A, B, and C, respectively. M_ij represents the mutual inductance between windings i and j, where i, j = A, B, C, a1, a2, a3, b1, b2, b3, c1, c2, and c3, and it should be noted that i≠j. U_pA, U_pB, and U_pC represent the exciting voltages for the ST in phases A, B, and C, respectively. U_ss’A, U_ss’B, and U_ss’C stand for the series-connected compensating voltages for the transmission line in phases A, B, and C, respectively. i_pA, i_pB, and i_pC represent the exciting currents for the ST in phases A, B, and C, respectively. i_s’A, i_s’B, and i_s’C stand for the currents of the transmission line at the receiving end of the ST in phases A, B, and C, respectively.

According to the ST coupled circuit shown in Figure 2, the branch voltage U_branch and current i_branch for the ST are defined as follows:

$${\mathit{U}}_{branch}={\left[\begin{array}{cccccc}{U}_{pA}& U_{pB}& U_{pC}& U_{ss\prime A}& U_{ss\prime B}& U_{ss\prime C}\end{array}\right]}^T$$

(1)

$${\mathit{i}}_{branch}={\left[\begin{array}{cccccc}{i}_{pA}& i_{pB}& i_{pC}& i_{s\prime A}& i_{s\prime {\rm B}}& i_{s\prime C}\end{array}\right]}^T$$

(2)

where U_pA, U_pB, and U_pC denote the voltages of the primary windings for the ST, respectively. i_pA, i_pB, and i_pC denote the currents of the primary windings for the ST, respectively. i_s′A, i_s′B, and i_s′C represent the currents of the secondary windings for the ST, respectively. U_ss′A, U_ss′B, and U_ss′C represent the voltages of the secondary windings for the ST, and they are given by

$$\{\begin{array}{c}{U}_{ss\prime A}=U_{a1}+U_{b1}+U_{c1}\\ U_{ss\prime B}=U_{a2}+U_{b2}+U_{c2}\\ U_{ss\prime C}=U_{a3}+U_{b3}+U_{c3}\end{array}$$

(3)

The relationship between the branch voltage and branch current can be established as follows:

$${\mathit{U}}_{branch}={\mathit{Z}}_{branch}{\mathit{i}}_{branch}$$

(4)

where Z_branch denotes the branch impedance matrix for ST, and it is described as follows:

$${\mathit{Z}}_{branch}=\left[\begin{array}{cccccc}{Z}_{sA\_A}& Z_{sA\_B}& Z_{sA\_C}& Z_{sA\_a}& Z_{sA\_b}& Z_{sA\_c}\\ Z_{sB\_A}& Z_{sB\_B}& Z_{sB\_C}& Z_{sB\_a}& Z_{sB\_b}& Z_{sB\_c}\\ Z_{sC\_A}& Z_{sC\_B}& Z_{sC\_C}& Z_{sC\_a}& Z_{sC\_b}& Z_{sC\_c}\\ Z_{s\prime a\_A}& Z_{s\prime a\_B}& Z_{s\prime a\_C}& Z_{s\prime a\_a}& Z_{s\prime a\_b}& Z_{s\prime a\_c}\\ Z_{s\prime b\_A}& Z_{s\prime b\_B}& Z_{s\prime b\_C}& Z_{s\prime b\_a}& Z_{s\prime b\_b}& Z_{s\prime b\_c}\\ Z_{s\prime c\_A}& Z_{s\prime c\_B}& Z_{s\prime c\_C}& Z_{s\prime c\_a}& Z_{s\prime c\_b}& Z_{s\prime c\_c}\end{array}\right]$$

(5)

Indices M, N, m, and n are defined in order to clarify each element in the branch impedance matrix, among M, N = A, B, and C; m, n = a, b, and c, respectively. Equation (5) can then be rewritten as follows:

$${\mathit{Z}}_{branch}=\left[\begin{array}{cc}{\mathit{Z}}_{sM\_N}& {\mathit{Z}}_{sM\_n}\\ {\mathit{Z}}_{s\prime m\_N}& {\mathit{Z}}_{s\prime m\_n}\end{array}\right]$$

(6)

where Z_{sM_N} represents the mutual inductance between the primary winding in phase M and primary winding in phase N and Z_{sM_n} denotes the sum of the mutual inductances between the primary winding in phase M and the secondary windings series-connected in the transmission line for phase n. Z_{s’m_N} denotes the sum of mutual inductances between primary winding in phase N and the secondary windings series-connected in the transmission line for phase m. Z_{s’m_n} represents the sum of the mutual inductances between the secondary windings series-connected in the transmission line for phase m and the secondary windings series-connected in the transmission line for phase n. Here, Z_{sA_A,B,andC}, Z_{sA_a,b,andc}, Z_{s’a_A,B,andC}, and Z_{s’a_a,b,andc} are taken as examples for illustration in Equation (7). The other specific elements in the impedance matrix are shown in Appendix A.

$$\begin{array}{l}\{\begin{array}{l}{Z}_{sA\_A}=Z_{AA}\\ Z_{sA\_B}=Z_{AB}\\ Z_{sA\_C}=Z_{AC}\end{array} \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em} \left(\mathrm{a}\right) \{\begin{array}{l}{Z}_{sA\_a}=Z_{Aa1}+Z_{Ab1}+Z_{Ac1}\\ Z_{sA\_b}=Z_{Aa2}+Z_{Ab2}+Z_{Ac2}\\ Z_{sA\_c}=Z_{Aa3}+Z_{Ab3}+Z_{Ac3}\end{array} \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em} \left(\mathrm{b}\right)\\ \{\begin{array}{l}{Z}_{s\prime a\_A}=Z_{a1A}+Z_{b1A}+Z_{c1A}\\ Z_{s\prime a\_B}=Z_{a1B}+Z_{b1B}+Z_{c1B}\\ Z_{s\prime a\_C}=Z_{a1C}+Z_{b1C}+Z_{c1C}\end{array} \left(\mathrm{c}\right) \{\begin{array}{l}{Z}_{s\prime a\_a}=Z_{a1a1}+Z_{a1b1}+Z_{a1c1}+Z_{b1a1}+Z_{b1b1}+Z_{b1c1}\\ \hspace{1em}\hspace{1em}\hspace{1em}+Z_{c1a1}+Z_{c1b1}+Z_{c1c1}\\ Z_{s\prime a\_b}=Z_{a1a2}+Z_{a1b2}+Z_{a1c2}+Z_{b1a2}+Z_{b1b2}+Z_{b1c2}\\ \hspace{1em}\hspace{1em}\hspace{1em}+Z_{c1a2}+Z_{c1b2}+Z_{c1c2}\\ Z_{s\prime a\_c}=Z_{a1a3}+Z_{a1b3}+Z_{a1c1}+Z_{b1a3}+Z_{b1b3}+Z_{b1c3}\\ \hspace{1em}\hspace{1em}\hspace{1em}+Z_{c1a3}+Z_{c1b3}+Z_{c1c3}\end{array} \left(\mathrm{d}\right)\end{array}$$

(7)

Because the expression in Equation (4) cannot fully reflect the mathematical characteristics between the terminals of the ST and the external electrical network, it is necessary to further convert the branch voltage and the branch current into the nodal voltage and the nodal current. Hence, U_branch and i_branch are replaced by the nodal voltage U_node and the nodal current i_node, and the latter are defined as follows:

$${\mathit{U}}_{node}={\left[\begin{array}{cccccc}{U}_{sA}& U_{sB}& U_{sC}& U_{s\prime A}& U_{s\prime B}& U_{s\prime C}\end{array}\right]}^T$$

(8)

$${\mathit{i}}_{node}={\left[\begin{array}{cccccc}{i}_{sA}& i_{sB}& i_{sC}& i_{s\prime A}& i_{s\prime B}& i_{s\prime C}\end{array}\right]}^T$$

(9)

According to the coupled circuit in Figure 2, U_node can be expressed by U_branch, and the relationship between i_branch and i_node is also obtained as follows:

$$\begin{array}{l}{\mathit{U}}_{node}={\mathit{N}}_1{\mathit{U}}_{branch}\\ {\mathit{i}}_{node}={\mathit{N}}_2{i}_{branch}\end{array}$$

(10)

where N₁ and N₂ are the conversion matrices for converting U_branch and i_branch into U_node and i_node, respectively; they are shown as follows:

$${\mathit{N}}_1=\left[\begin{array}{cccccc}1& 0& 0& 0& 0& 0\\ 0& 1& 0& 0& 0& 0\\ 0& 0& 1& 0& 0& 0\\ 1& 0& 0& -1& 0& 0\\ 0& 1& 0& 0& -1& 0\\ 0& 0& 1& 0& 0& -1\end{array}\right], {\mathit{N}}_2=\left[\begin{array}{cccccc}1& 0& 0& 1& 0& 0\\ 0& 1& 0& 0& 1& 0\\ 0& 0& 1& 0& 0& 1\\ 0& 0& 0& 1& 0& 0\\ 0& 0& 0& 0& 1& 0\\ 0& 0& 0& 0& 0& 1\end{array}\right]$$

(11)

By combining Equations (4) and (8)–(11), the relationship between i_node and U_node can be established:

$${\mathit{i}}_{node}=N_2{\mathit{Z}}_{branch}^{-1}{\mathit{N}}_1^{-1}{\mathit{U}}_{node}$$

(12)

The nodal admittance matrix G_node for the ST is then defined and expressed as follows:

$${\mathit{G}}_{node}={\mathit{N}}_2{\mathit{Z}}_{branch}^{-1}{\mathit{N}}_1^{-1}$$

(13)

Consequently, a fully decoupled mathematical model for the ST can be constructed according to the nodal admittance matrix G_node in Equation (13) and it is shown in Figure 3. The specific elements in Figure 3 are demonstrated in Appendix B.

2.3. ST Phase Coordinate Model Incorporating Ground Impedance on Its Primary Side

The established ST model ignores the ground impedance on its primary windings, although this can be an important part of the analysis of a grounding system. Thus, in order to calculate the short-circuit current flowing through the ST in a grounding system precisely, it is necessary to consider the ground impedance of the primary windings for the ST.

The ground impedance is considered to be Z_sG, and the modified relationship between the branch voltage and branch current can be obtained as follows:

$$\left[\begin{array}{c}{\mathit{U}}_{\mathit{branch} }\\ {\mathit{U}}_{\mathit{sG}}\end{array}\right]=\underset{{{\mathit{Z}}^{\prime }}_{\mathit{branch}}}{\underbrace{\left[\begin{array}{cc}{\mathit{Z}}_{\mathit{branch} }& 0\\ 0& {\mathit{Z}}_{sG}\end{array}\right]}}\left[\begin{array}{c}{\mathit{i}}_{\mathit{branch} }\\ {\mathit{i}}_{sG}\end{array}\right]$$

(14)

where Z_branch represents the branch impedance matrix for the ST, and U_sG and i_sG denote the voltage and current of ground impedance, respectively.

Furthermore, the modified branch admittance matrix G′_branch is derived as follows:

$$\left[\begin{array}{c}{\mathit{i}}_{\mathit{branch} }\\ {\mathit{i}}_{\mathrm{s}G}\end{array}\right]=\underset{{{\mathit{G}}^{\prime }}_{\mathit{branch}}}{\underbrace{\left[\begin{array}{cc}{\mathit{Z}}_{\mathit{branch} }^{-1}& 0\\ 0& {\mathit{Z}}_{\mathit{sG}}^{-1}\end{array}\right]}}\left[\begin{array}{c}{\mathit{U}}_{\mathit{branch}}\\ {\mathit{Z}}_{\mathit{sG}}\end{array}\right]$$

(15)

The conversion matrix N₁ in Equation (10) may be rewritten as N_1′:

$$\begin{array}{c}\left[\begin{array}{c}{U}_{s\mathrm{A}}\\ U_{s\mathrm{B}}\\ U_{s\mathrm{C}}\\ U_{s^{\prime }\mathrm{A}}\\ U_{s^{\prime }\mathrm{B}}\\ U_{s^{\prime }}\mathrm{C}\\ {\ddot{U}}_G\end{array}\right]=\underset{N_1^{\prime }}{\underbrace{[\begin{array}{ccccccc}1& 0& 0& 0& 0& 0& 1\\ 0& 1& 0& 0& 0& 0& 1\\ 0& 0& 1& 0& 0& 0& 1\\ 1& 0& 0& -1& 0& 0& 1\\ 0& 1& 0& 0& -1& 0& 1\\ 0& 0& 1& 0& 0& -1& 1\\ 0& 0& 0& 0& 0& 0& 1\end{array}]}}\left[\begin{array}{c}{U}_{p\mathrm{A}}\\ U_{p\mathrm{B}}\\ U_{p\mathrm{C}}\\ U_{s{s}^{\prime }\mathrm{A}}\\ U_{s{s}^{\prime }\mathrm{B}}\\ U_{s{s}^{\prime }\mathrm{C}}\\ {\ddot{U}}_{sG}\end{array}\right]\end{array}$$

(16)

In this way, the conversion matrix N₂ in Equation (11) can be modified as N_2′:

$$\begin{array}{c}\left[\begin{array}{c}{i}_{s\mathrm{A}}\\ i_{s\mathrm{B}}\\ i_{s\mathrm{C}}\\ i_{s^{\prime }\mathrm{A}}\\ i_{s^{\prime }\mathrm{B}}\\ i_{s^{\prime }}\mathrm{C}\\ i_G\end{array}\right]=\underset{N_2^{\prime }}{\underbrace{\left[\begin{array}{ccccccc}1& 0& 0& 1& 0& 0& 0\\ 0& 1& 0& 0& 1& 0& 0\\ 0& 0& 1& 0& 0& 1& 0\\ 0& 0& 0& 1& 0& 0& 0\\ 0& 0& 0& 0& 1& 0& 0\\ 0& 0& 0& 0& 0& 1& 0\\ 0& 0& 0& 0& 0& 0& 1\end{array}\right]}}\left[\begin{array}{c}{i}_{p\mathrm{A}}\\ i_{p\mathrm{B}}\\ i_{p\mathrm{C}}\\ i_{s^{\prime }\mathrm{A}}\\ i_{s^{\prime }\mathrm{B}}\\ i_{s^{\prime }\mathrm{C}}\\ i_{sG}\end{array}\right]\end{array}$$

(17)

Therefore, the modified nodal admittance matrix ${\mathit{G}}_{node}^{\prime }$ involving the ground impedance on the ST’s primary windings can be acquired as follows:

$${\mathit{G}}_{node}^{\prime }={\mathit{N}}_2^{\prime }{\mathit{G}}_{branch}^{\prime }{\mathit{N}}_1^{\prime }{}^{-1}$$

(18)

3. Short-Circuit Calculation for Electric Power Network Based on the Phase Coordinate Method

3.1. Modeling of Electric Power Network Using the Phase Coordinate Method

An electric power network is usually represented by a nodal admittance network. Therefore, the fault location is added as a new bus to convert the line fault into a bus fault. In this paper, the bus fault is considered in the phase domain.

The nodal network equation of a power system can be described as follows:

$$\left[\begin{array}{ccc}{\mathit{Y}}_{11}& \cdots & {\mathit{Y}}_{1n}\\ ⋮& & ⋮\\ {\mathit{Y}}_{n1}& \cdots & {\mathit{Y}}_{nn}\end{array}\right]\left[\begin{array}{c}{\mathit{U}}_1\\ ⋮\\ {\mathit{U}}_n\end{array}\right]=\left[\begin{array}{c}{\mathit{i}}_1\\ ⋮\\ {\mathit{i}}_n\end{array}\right]$$

(19)

where Y_ij denotes the mutual admittance between the ith bus and the jth bus. i_i denotes current injected at the ith bus, and U_i is the voltage at the ith bus. The specific elements in Equation (19) can be demonstrated as follows:

$${\mathit{Y}}_{ij}=\left[\begin{array}{ccc}{Y}_{ij}^{\mathrm{AA}}& Y_{ij}^{\mathrm{AB}}& Y_{ij}^{\mathrm{AC}}\\ Y_{ij}^{\mathrm{BA}}& Y_{ij}^{\mathrm{BB}}& Y_{ij}^{\mathrm{BC}}\\ Y_{ij}^{\mathrm{CA}}& Y_{ij}^{\mathrm{CB}}& Y_{ij}^{\mathrm{CC}}\end{array}\right], {\mathit{U}}_i=\left[\begin{array}{c}{U}_{iA}^{}\\ U_{iB}^{}\\ U_{iC}^{}\end{array}\right], {\mathit{i}}_i=\left[\begin{array}{c}{i}_{i\mathrm{A}}^{}\\ i_{i\mathrm{B}}^{}\\ i_{i\mathrm{C}}^{}\end{array}\right]$$

3.2. Single-Phase-to-Ground Fault

When an A-phase-to-ground fault occurs, the voltage at the additional ith bus $U_{i\mathrm{A}}^{\prime }=0$. Meanwhile, a current source i_f is added as the equivalent short-circuit current at the ith bus, i.e., ${\mathit{i}}_f={\left[\begin{array}{ccc}{i}_{fA}& 0& 0\end{array}\right]}^T$. Equation (19) can then be modified as follows:

$$\left[\begin{array}{cccccc}{\mathit{Y}}_{11}& {\mathit{Y}}_{12}& \cdots & {\mathit{Y}}_{1i}& \cdots & {\mathit{Y}}_{1n}\\ {\mathit{Y}}_{21}& {\mathit{Y}}_{22}& \cdots & {\mathit{Y}}_{2i}& \cdots & {\mathit{Y}}_{2n}\\ ⋮& ⋮& ⋮& ⋮& ⋮& ⋮\\ {\mathit{Y}}_{i1}& {\mathit{Y}}_{i2}& \cdots & {\mathit{Y}}_{ii}& \cdots & {\mathit{Y}}_{in}\\ ⋮& ⋮& ⋮& ⋮& ⋮& ⋮\\ {\mathit{Y}}_{n1}& {\mathit{Y}}_{n2}& \cdots & {\mathit{Y}}_{ni}& \cdots & {\mathit{Y}}_{nn}\end{array}\right]\left[\begin{array}{c}{\mathit{U}}_1\\ {\mathit{U}}_2\\ ⋮\\ {\mathit{U}}_i\\ ⋮\\ {\mathit{U}}_n\end{array}\right]=\left[\begin{array}{c}{\mathit{i}}_1\\ {\mathit{i}}_2\\ ⋮\\ {\mathit{i}}_i\\ ⋮\\ {\mathit{i}}_n\end{array}\right]+\left[\begin{array}{c}0\\ 0\\ ⋮\\ {\mathit{i}}_f\\ ⋮\\ 0\end{array}\right]$$

(20)

Equation (20) can be further rewritten as

$$\left[\begin{array}{cccccc}{\mathit{Y}}_{11}& {\mathit{Y}}_{12}& \cdots & {\mathit{Y}}_{1i}^{\prime }& \cdots & {\mathit{Y}}_{1n}\\ {\mathit{Y}}_{21}& {\mathit{Y}}_{22}& \cdots & {\mathit{Y}}_{2i}^{\prime }& \cdots & {\mathit{Y}}_{2n}\\ ⋮& ⋮& ⋮& ⋮& ⋮& ⋮\\ {\mathit{Y}}_{i1}& {\mathit{Y}}_{i2}& \cdots & {\mathit{Y}}_{ii}^{\prime }& \cdots & {\mathit{Y}}_{in}\\ ⋮& ⋮& ⋮& ⋮& ⋮& ⋮\\ {\mathit{Y}}_{n1}& {\mathit{Y}}_{n2}& \cdots & {\mathit{Y}}_{ni}^{\prime }& \cdots & {\mathit{Y}}_{nn}\end{array}\right]\left[\begin{array}{c}{\mathit{U}}_1\\ {\mathit{U}}_2\\ ⋮\\ {\mathit{U}}_i^{\prime }\\ ⋮\\ {\mathit{U}}_n\end{array}\right]=\left[\begin{array}{c}{\mathit{i}}_1\\ {\mathit{i}}_2\\ ⋮\\ {\mathit{i}}_i\\ ⋮\\ {\mathit{i}}_n\end{array}\right]$$

(21)

where

$${\mathit{U}}_i^{\prime }={\left[\begin{array}{ccc}{i}_{fA}& U_{i\mathrm{B}}^{\prime }& U_{i\mathrm{C}}^{\prime }\end{array}\right]}^T$$

(22)

It should be noted that the admittances Y_ij and Y_ii are modified as follows:

$${\mathit{Y}}_{ij}^{\prime }=\left[\begin{array}{ccc}0& Y_{ij}^{\mathrm{AB}}& Y_{ij}^{\mathrm{AC}}\\ 0& Y_{ij}^{\mathrm{BB}}& Y_{ij}^{\mathrm{BC}}\\ 0& Y_{ij}^{\mathrm{CB}}& Y_{ij}^{\mathrm{CC}}\end{array}\right], {\mathit{Y}}_{ii}^{\prime }=\left[\begin{array}{ccc}-1& Y_{ii}^{AB}& Y_{ii}^{\mathrm{BC}}\\ 0& Y_{ii}^{\mathrm{BB}}& Y_{ii}^{\mathrm{BC}}\\ 0& Y_{ii}^{\mathrm{CB}}& Y_{ii}^{\mathrm{CC}}\end{array}\right]$$

(23)

In order to apply this calculation method to other short-circuit faults, it is defined that ${\mathit{Y}}_{ij}^{\prime }={\mathit{Y}}_{ij}{\mathit{T}}_1$ and ${\mathit{Y}}_{ii}^{\prime }={\mathit{Y}}_{ii}{\mathit{T}}_1+{\mathit{T}}_2$, and the coefficient matrices T₁ and T₂ can be deduced:

$${\mathit{T}}_1=\left[\begin{array}{ccc}0& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right], {\mathit{T}}_2=\left[\begin{array}{ccc}-1& 0& 0\\ 0& 0& 0\\ 0& 0& 0\end{array}\right]$$

(24)

Meanwhile, the voltage and the short-circuit current at the ith bus can be concluded:

$${\mathit{U}}_i={\mathit{T}}_1{\mathit{U}}_i^{\prime }$$

(25)

$${\mathit{i}}_f=-{\mathit{T}}_2{\mathit{U}}_i^{\prime }$$

(26)

3.3. Other Short-Circuit Faults

For a fault between phase A and B, the voltage at the ith bus and the short-circuit current are given as follows:

$$\{\begin{array}{l}{U}_{i\mathrm{A}}^{}=U_{i\mathrm{B}}^{}\\ i_{f\mathrm{A}}=-i_{f\mathrm{B}}\end{array}$$

(27)

$${\mathit{U}}_i^{\prime }={\left[\begin{array}{ccc}{U}_{i\mathrm{A}}& i_{f\mathrm{A}}& U_{i\mathrm{C}}\end{array}\right]}^T$$

(28)

In accordance with the calculation method in the previous subsection, the modified admittance matrices Y_ii’ and Y_ij’ can be deduced as follows:

$${\mathit{Y}}_{ij}^{\prime }=\left[\begin{array}{ccc}{Y}_{ij}^{\mathrm{AA}}+Y_{ij}^{\mathrm{AB}}& 0& Y_{ij}^{\mathrm{AC}}\\ Y_{ij}^{\mathrm{BA}}+Y_{ij}^{\mathrm{BB}}& 0& Y_{ij}^{\mathrm{BC}}\\ Y_{ij}^{\mathrm{CA}}+Y_{ij}^{\mathrm{CB}}& 0& Y_{ij}^{\mathrm{CC}}\end{array}\right], {\mathit{Y}}_{ii}^{\prime }=\left[\begin{array}{ccc}{Y}_{ii}^{\mathrm{AA}}+Y_{ii}^{\mathrm{AB}}& -1& Y_{ii}^{\mathrm{AC}}\\ Y_{ii}^{\mathrm{BA}}+Y_{ii}^{\mathrm{BB}}& 1& Y_{ii}^{\mathrm{BC}}\\ Y_{ii}^{\mathrm{CA}}+Y_{ii}^{\mathrm{CB}}& 0& Y_{ii}^{\mathrm{CC}}\end{array}\right]$$

(29)

The coefficient matrices T₁ and T₂ are obtained:

$${\mathit{T}}_1=\left[\begin{array}{ccc}1& 0& 0\\ 1& 0& 0\\ 0& 0& 1\end{array}\right], {\mathit{T}}_2=\left[\begin{array}{ccc}0& -1& 0\\ 0& 1& 0\\ 0& 0& 0\end{array}\right]$$

(30)

Similarly, the coefficient matrices T₁ and T₂ for different short-circuit faults can be deduced. U_i’, T₁, and T₂ for different short-circuit faults are listed in

Table 1. The coefficient matrices for different short-circuit faults.

Faults	Ui’	T1	T2
Phase-A-to-ground fault	$\left[\begin{array}{c}{i}_{f\mathrm{A}}\\ U_{i\mathrm{B}}\\ U_{i\mathrm{C}}\end{array}\right]$	$\left[\begin{array}{ccc}0& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]$	$\left[\begin{array}{ccc}-1& 0& 0\\ 0& 0& 0\\ 0& 0& 0\end{array}\right]$
Phase-A-to-phase-B fault	$\left[\begin{array}{c}{U}_{i\mathrm{A}}\\ i_{f\mathrm{A}}\\ U_{i\mathrm{C}}\end{array}\right]$	$\left[\begin{array}{ccc}1& 0& 0\\ 1& 0& 0\\ 0& 0& 1\end{array}\right]$	$\left[\begin{array}{ccc}0& -1& 0\\ 0& 1& 0\\ 0& 0& 0\end{array}\right]$
Both phase-A- and phase-B-to-ground fault	$\left[\begin{array}{c}{i}_{f\mathrm{A}}\\ i_{f\mathrm{B}}\\ U_{i\mathrm{C}}\end{array}\right]$	$\left[\begin{array}{ccc}0& 0& 0\\ 0& 0& 0\\ 0& 0& 1\end{array}\right]$	$\left[\begin{array}{ccc}-1& 0& 0\\ 0& -1& 0\\ 0& 0& 0\end{array}\right]$
Three-phase fault	$\left[\begin{array}{c}{U}_{i\mathrm{A}}\\ i_{f\mathrm{A}}\\ i_{f\mathrm{B}}\end{array}\right]$	$\left[\begin{array}{ccc}1& 0& 0\\ 1& 0& 0\\ 1& 0& 0\end{array}\right]$	$\left[\begin{array}{ccc}0& -1& 0\\ 0& 0& -1\\ 0& 1& 1\end{array}\right]$
Three-phases-to-ground fault	$\left[\begin{array}{c}{i}_{f\mathrm{A}}\\ i_{f\mathrm{B}}\\ i_{f\mathrm{C}}\end{array}\right]$	$\left[\begin{array}{ccc}0& 0& 0\\ 0& 0& 0\\ 0& 0& 0\end{array}\right]$	$\left[\begin{array}{ccc}-1& 0& 0\\ 0& -1& 0\\ 0& 0& -1\end{array}\right]$
Phase-A-to-ground and phase-B-to-phase C fault	$\left[\begin{array}{c}{i}_{f\mathrm{A}}\\ U_{i\mathrm{B}}\\ i_{f\mathrm{B}}\end{array}\right]$	$\left[\begin{array}{ccc}0& 0& 0\\ 0& 1& 0\\ 0& 1& 0\end{array}\right]$	$\left[\begin{array}{ccc}-1& 0& 0\\ 0& 0& -1\\ 0& 0& 1\end{array}\right]$

. Therefore, the admittance matrix can be modified to calculate the short-circuit currents for various short-circuit faults.

4. Case Studies

4.1. Validation of the Proposed Model

In order to verify the effectiveness of the proposed short-circuit fault model for the ST, case studies were carried out on the equivalent two-bus system shown in Figure 4, involving all possible short-circuit faults. The parameters of the electrical system originate from [14] and are listed in

Table 2. The main parameters and settings for the case studies.

Parameters	Value
Base values	160 MVA, 138 kV
Sending end line-to-line voltage	1∠0°
Receiving end line-to-line	1∠−20°
Source impedance at the sending end	1.0053 Ω, 19.17 mH
Source impedance at the receiving end	0 Ω, 0 mH
Transmission line impedance	3.0159 Ω, 59.19 mH
Rating of the ST transformer	30 MVA
Single-transformer leakage reactance of the ST	15.73 mH
Ground impedance	4 Ω

. The model introduced in [14] is also adopted for comparison in this paper.

Firstly, the number of secondary winding turns for ST was acquired according to the series-connected compensating voltage. The compensating voltage U_ss’ of 0.2 p.u. at an angle β of 120° was set in this case. Furthermore, the self and mutual inductances of a ST can be calculated based on its unified magnetic equivalent circuit (UMEC), as shown in [17]. Thus, the admittance matrix for the fully decoupled mathematical model of the ST was obtained. Subsequently, the fault voltage vector U_i’ and the coefficient matrices T₁ and T₂ can be obtained based on the types of short-circuit fault, and the admittances Y_ii and Y_ij in (19) are then modified as Y_ii’ and Y_ij’. Consequently, the short-circuit current i_f and voltage U_f for different short-circuit faults can be calculated, as shown in

Table 3. The comparison of fault analysis between analytic and simulated studies for different short-circuit faults.

Faults	Simulation Results		Analytic Results		Relative Errors
	$\left[\begin{array}{c}{\mathit{U}}_{\mathit{s}\prime \mathbf{A}}\\ {\mathit{U}}_{\mathit{s}\prime \mathbf{B}}\\ {\mathit{U}}_{\mathit{s}\prime \mathbf{C}}\end{array}\right]/\mathbf{k}\mathbf{V}$	$\left[\begin{array}{c}{\mathit{i}}_{\mathit{f}\mathbf{A}}\\ {\mathit{i}}_{\mathit{f}\mathbf{B}}\\ {\mathit{i}}_{\mathit{f}\mathbf{C}}\end{array}\right]/\mathbf{k}\mathbf{A}$	$\left[\begin{array}{c}{\mathit{U}}_{\mathit{s}\prime \mathbf{A}}\\ {\mathit{U}}_{\mathit{s}\prime \mathbf{B}}\\ {\mathit{U}}_{\mathit{s}\prime \mathbf{C}}\end{array}\right]/\mathbf{k}\mathbf{V}$	$\left[\begin{array}{c}{\mathit{i}}_{\mathit{f}\mathbf{A}}\\ {\mathit{i}}_{\mathit{f}\mathbf{B}}\\ {\mathit{i}}_{\mathit{f}\mathbf{C}}\end{array}\right]/\mathbf{k}\mathbf{A}$	$\left[\begin{array}{c}\Delta {\mathit{U}}_{\mathit{s}\prime \mathbf{A}}\\ \Delta {\mathit{U}}_{\mathit{s}\prime \mathbf{B}}\\ \Delta {\mathit{U}}_{\mathit{s}\prime \mathbf{C}}\end{array}\right]/\%$	$\left[\begin{array}{c}\Delta {\mathit{i}}_{\mathit{f}\mathbf{A}}\\ \Delta {\mathit{i}}_{\mathit{f}\mathbf{B}}\\ \Delta {\mathit{i}}_{\mathit{f}\mathbf{C}}\end{array}\right]/\%$
Phase-A-to-ground fault	$\left[\begin{array}{c}0\\ 77.28\\ 78.59\end{array}\right]$	$\left[\begin{array}{c}12.63\\ 0\\ 0\end{array}\right]$	$\left[\begin{array}{c}0\\ 77.53\\ 78.17\end{array}\right]$	$\left[\begin{array}{c}12.58\\ 0\\ 0\end{array}\right]$	$\left[\begin{array}{c}0\\ 0.32\\ -0.53\end{array}\right]$	$\left[\begin{array}{c}-0.40\\ 0\\ 0\end{array}\right]$
Phase-A-to-phase B fault	$\left[\begin{array}{c}36.27\\ 36.27\\ 72.53\end{array}\right]$	$\left[\begin{array}{c}12.67\\ 12.67\\ 0\end{array}\right]$	$\left[\begin{array}{c}36.28\\ 36.28\\ 72.43\end{array}\right]$	$\left[\begin{array}{c}12.73\\ 12.73\\ 0\end{array}\right]$	$\left[\begin{array}{c}0.03\\ 0.03\\ -0.14\end{array}\right]$	$\left[\begin{array}{c}0.47\\ 0.47\\ 0\end{array}\right]$
Both phase-A- and phase-B-to-ground fault	$\left[\begin{array}{c}0\\ 0\\ 81.24\end{array}\right]$	$\left[\begin{array}{c}13.95\\ 13.71\\ 0\end{array}\right]$	$\left[\begin{array}{c}0\\ 0\\ 81.18\end{array}\right]$	$\left[\begin{array}{c}13.99\\ 13.78\\ 0\end{array}\right]$	$\left[\begin{array}{c}0\\ 0\\ -0.07\end{array}\right]$	$\left[\begin{array}{c}0.29\\ 0.51\\ 0\end{array}\right]$
Three-phase-to-ground fault	$\left[\begin{array}{c}0\\ 0\\ 0\end{array}\right]$	$\left[\begin{array}{c}14.62\\ 14.62\\ 14.62\end{array}\right]$	$\left[\begin{array}{c}0\\ 0\\ 0\end{array}\right]$	$\left[\begin{array}{c}14.65\\ 14.57\\ 14.73\end{array}\right]$	$\left[\begin{array}{c}0\\ 0\\ 0\end{array}\right]$	$\left[\begin{array}{c}0.21\\ -0.34\\ 0.75\end{array}\right]$
Phase-A-to-ground and phase-B-to-phase-C fault	$\left[\begin{array}{c}0\\ 46.15\\ 46.15\end{array}\right]$	$\left[\begin{array}{c}12.63\\ 12.66\\ 12.66\end{array}\right]$	$\left[\begin{array}{c}0\\ 45.37\\ 45.37\end{array}\right]$	$\left[\begin{array}{c}12.59\\ 12.61\\ 12.61\end{array}\right]$	$\left[\begin{array}{c}0\\ -1.69\\ -1.69\end{array}\right]$	$\left[\begin{array}{c}-0.32\\ -0.39\\ -0.39\end{array}\right]$

. Meanwhile, the time-domain simulation results using the PSCAD/EMTDC are also listed in Table 3.

It can be seen from Table 3 that the analytic results including the short-circuit current i_f and the voltage U_f at the receiving end of the ST were basically consistent with the simulation results using PSCAD/EMTDC. The range of difference was −1.7–0.027%, which proves the effectiveness of the proposed model. The cause of the difference may be the multi-winding coupling of the ST and the ground impedance on the ST’s primary windings, which was ignored in [14]. At the same time, the short-circuit currents were unbalanced for phase A, phase B, and phase C when the symmetrical faults occurred in transmission lines such as the three-phase-to-ground fault. The reason is that the mutual inductances between all windings of the ST were considered in the phase coordinate model for the ST. The mutual inductances between phases were not equal due to the asymmetrical core, resulting in the unbalanced system operation.

4.2. Short-Circuit Current Analysis of ST for Different Operational Modes

Since a ST can inject a variable compensating voltage into the power system, it may have a non-negligible effect on the fault currents flowing through the ST. Therefore, in order to fully analyze the influence of the ST on short-circuit currents in various operational modes, the calculation is carried out under the following conditions: the angle β was varied at a discrete step of 30° in the range of 0° to 360°, and the series-connected compensating voltage of the ST was kept constant at 0.1 p.u. or 0.2 p.u. The variations of short-circuit currents for the single-phase-to-ground and three-phase-to-ground faults at varying phase angles are shown in Figure 5 and Figure 6, respectively.

It can be seen from Figure 5 that the short-circuit current for the compensated mode was less than that for the uncompensated mode during the single-phase-to-ground faults, and the fault current for U_ss’ = 0.2 p.u. was smallest among them. The trajectories of the short-circuit currents are approximately symmetric about 0°. When the angle of the injected voltage varied from 0° to 180°, the short-circuit current magnitude reached the maximum at 120° and the minimum at 180°. The variation of the short-circuit currents for the compensated mode can be generalized as a trend of decreasing in the region of 0° to 60° and 120° to 180° and increasing when β was varied from 60° to 120°. The maximum reduction of short-circuit currents reached 27.04% for U_ss’ = 0.2 p.u. and 14.31% for U_ss’ = 0.1 p.u. The reasons for the above phenomena in the single-phase-to-ground faults can be described as follows:

• The dominant components of the series-injected voltage in phase A are the induced voltages from the series windings b1 and c1 when a single-phase-to-ground fault occurs. Because the phase difference between the compensating voltage and the sending-end voltage source in phase A is over 120°, it has a negative effect on the voltage magnitude of phase A and further results in lower fault currents.
• In the process of increasing β from 0° to 180°, the number of series winding turns showed a trend of increase in the ranges of 0° to 60° and 120° to 180°, and decreased gradually from 60° to 120°. The equivalent impedance brought by the ST followed the same trend as the number of series winding turns. This can explain the opposite phenomenon that happened on the fault currents for the compensated mode.

The short-circuit currents for the three-phase-to-ground faults, as shown in Figure 6, were also symmetric about 0°. However, there were some features different from those shown in Figure 5. In the range of 0° to 180°, the fault currents for the compensated mode were less than those for the uncompensated mode when the angle of injected voltage was less than 90°, and the fault current for U_ss’ = 0.2 p.u. was less than that for U_ss’ = 0.1 p.u. The opposite happened when β is greater than 90°. Additionally, both increased gradually in the operational region of 0° to 180°. For the compensating voltage at 0.2 p.u., the short-circuit currents reached a maximum of 12.32 kA and a minimum of 8.87 kA. The short-circuit current reached a maximum of 11.86 kA and a minimum of 9.85 kA when the compensation voltage was equal to 0.1 p.u.

When a three-phase-to-ground fault occurs, a system with a ST can be simplified as shown in Figure 7 by ignoring the asymmetrical mutual inductances between phases in ST. In this situation, the ST can be equivalent to a combination of an ideal ST without internal resistance and an equivalent impedance in series. The fault current flowing through the ST can be approximately calculated as follows:

$$i_{L,ST}=\frac{U}{k{Z}_S+\frac{Z_{ST}}{k}}=\frac{kU}{{\left|k\right|}^2{Z}_S+Z_{ST}}$$

(31)

where $k=1+\xi e^{j\beta }$, which is the voltage ratio of the ST. ξ stands for the magnitude of the series-connected compensating voltage U_ss’. Z_ST and Z_S represent the equivalent impedances of the ST and the sending end, separately. U denotes the voltage of the sending-end resource.

It can be seen from Equation (31) that the magnitude of the short-circuit current is mainly affected by $1/\left|k\right|$ if Z_ST is considered to be 0. The variation of $1/\left|k\right|$ with the compensation phase angle is shown in Figure 8.

It can be observed from Figure 8 that $1/\left|k\right|$ gradually increased in the range of 0° to 180°, which may have caused the same trend of the fault current. Moreover, when the compensation angle was greater than 90°, $1/\left|k\right|$ for ξ = 0.1 p.u. was greater than that for ξ = 0.2 p.u. while it was opposite in the range of 0° to 90°. This may lead to the same phenomenon as shown in Figure 6.

From the above cases, the effect of the ST on the fault current is revealed. The effect was variable with the different compensating voltage and fault types. Moreover, it also highlights the potential ability of ST in restraining fault current. In addition, the adjustment of system protection with the addition of a ST is worthy of study.

4.3. Transient Behavior of Different Faults for ST

In order to further study the transient processes of the ST when a short-circuit occurred, the short-circuit model of ST in the phase domain was transformed into the time-domain model. Figure 9 shows the detailed conversion processes of short-circuit model.

In this case, the short-circuit faults were set at the receiving end of the ST and the fault time was set at 0.1 s. The compensating voltage U_ss’ of 0.2 p.u. was set with the phase angle β of 120°.

In accordance with the proposed method, the short-circuit currents of ST after different faults were obtained as shown in Figure 10.

As shown in Figure 10, the current peak values for the four types of short-circuit fault reached over 20 kA, and decayed to steady-state after 0.18 s. The peak value for the three-phase-to-ground fault was the highest among them, exceeding 25 kA. When a single-phase fault happened, the currents of the other two phases (healthy phases) were also obviously affected. Due to the coupling between the windings of the ST, a fault in any one phase will affect the compensation voltage in other two phases, resulting in the current changes presented in Figure 10a.

5. Conclusions

In this paper, a short-circuit model of a ST considering the mutual inductance between all windings of the ST and the ground impedance on its primary side for the phase coordinate method is proposed. The validity of the proposed model was verified by comparison between the results from the analytic calculation utilizing MATLAB and the time-domain simulation using PSCAD/EMTDC. Moreover, additional findings or perceptions were as follows:

• The results calculated by the proposed method are in good agreement with the simulation results determined using PSCAD/EMTDC, with an error range of −1.69% to 0.75%. Moreover, the proposed phase coordinate model of ST accurately accounted for the mutual inductances between the windings of the ST. The influence of the unbalanced system parameters and ground impedance was also considered in the short-circuit calculation. These factors introduced in the proposed model make it possible to interconnect a ST with the associated grounding system for a proper evaluation of current distribution in the case of ground faults.
• The effect of the ST on the short-circuit currents was evaluated. It was found that the main influencing factors were the magnitude and phase angle of the compensation voltage and the equivalent impedance, which varied with the number of winding turns. In most cases, it had a negative effect on the short-circuit currents. After a single-phase-to-ground fault occurred, the maximum reduction of short-circuit currents reached 27.04% for the compensating voltage U_ss’ of 0.2 p.u. and 14.31% when U_ss’ = 0.1 p.u.
• The ability of the ST to limit the short-circuit current was identified. This may be a point of concern when designing control strategies for STs. In addition, the impact of this ability on the design and setting of transmission protection systems is worth exploring further.