A Study on Excitation Inrush Current and Overvoltage Mitigation Strategies Utilizing Phase Selection Control

Abstract

To address the challenges of system failures and equipment damage caused by excitation inrush currents and overvoltages during no-load energization of high-speed locomotive transformers, a simulation model was developed utilizing PSCAD electromagnetic transient simulation software. This study establishes a no-load switching simulation model for rolling stock transformers within PSCAD, analyzing variations in overvoltage and excitation inrush current amplitudes across different phase angles. Additionally, it compares excitation inrush current amplitudes under varying residual magnetism conditions. A phase-selective control strategy is proposed, integrating the hysteresis characteristics of the transformer core. The model’s accuracy is validated against empirical data obtained from a city train. Employing the Jiles–Atherton hysteresis model, the residual magnetism of the transformer core is quantified. Based on measured data, a relationship curve between switching phase and residual magnetism is fitted, enabling calculation of the optimal closing angle through the phase selection procedure. This approach effectively mitigates overvoltage and excitation inrush current hazards, thereby enhancing the operational safety of the train system.

1. Introduction

During frequent phase-splitting and closing operations of onboard circuit breakers in moving train sets, electromagnetic energy conversion within the inductive–capacitive circuit induces operating overvoltages on the primary side of onboard transformers. The high-frequency oscillatory components inherent in these transient overvoltages can precipitate faults such as turn-to-turn insulation breakdowns in transformers and contact erosion in vacuum circuit breakers [1,2,3]. Furthermore, due to the ferromagnetic hysteresis properties of transformer cores, residual magnetism persists following switching operations [4]. If the closing phase angle aligns with the direction of this residual magnetism during subsequent no-load energization, the core flux transient can reach 2.0 to 2.2 times the saturation flux density, resulting in peak excitation inrush currents up to six to eight times, or even exceeding, the rated transformer current [5,6]. Such phenomena may trigger false differential protection operations and adversely affect train scheduling efficiency [7,8,9]. Moreover, these nonlinear transient currents exhibit significant non-periodic components and second harmonic characteristics, thereby degrading the power quality of the contact network [10,11,12].

Previous studies and mitigation techniques have predominantly treated switching overvoltage and magnetizing inrush current as distinct phenomena. Common approaches to suppressing inrush current include the use of closing resistors [13,14] and enhancing magnetizing inductance through modifications in winding configurations [15,16]. For instance, some scholars have investigated the influence of the timing and resistance value of closing resistor insertion on the attenuation of magnetizing inrush current [17]. However, series-connected closing resistors are limited in their capacity to dissipate the energy associated with inrush currents and thus cannot fully suppress them. Furthermore, the reliability of closing resistors in terms of materials, structural design, and thermal management is critical; inadequate design may lead to overheating, burnout, or even catastrophic failure. The integration of mechanical switching devices with closing resistors introduces additional mechanical failure points and compromises electrical insulation integrity, which not only prolongs the switching process but may also induce secondary inrush currents. Alterations to winding configurations necessitate physical modifications to the primary circuit, thereby increasing system complexity and cost, which often renders such solutions impractical in engineering applications. In the context of overvoltage suppression, metal oxide surge arresters are widely employed; however, their suppression capabilities are limited, and they are ineffective against magnetizing inrush currents [18,19]. Strategies based on the rational distribution of line parameters may lack general applicability [20]. Some researchers have proposed the installation of RC protection devices to mitigate overvoltage [21], primarily targeting high-frequency oscillatory overvoltages, with limited efficacy against power-frequency overvoltages. Regarding phase selection control technologies applied to moving train sets [22], existing research tends to address switching overvoltage and magnetizing inrush current independently, which presents a somewhat fragmented perspective. The fundamental challenge lies in the fact that both transient phenomena originate from the same switching event within the EMU; however, their potential interactions and the necessity for coordinated suppression strategies are frequently neglected. Isolated mitigation measures may inadvertently exacerbate the alternate issue or fail to effectively resolve either problem.

In contrast to conventional single-objective phase selection control strategies [23,24,25], this study introduces a comprehensive phase selection control approach designed to concurrently mitigate both magnetizing inrush current and switching overvoltage during transformer no-load energization. The key contributions of this work are as follows: First, a coupled simulation model that integrates electromagnetic transient phenomena with magnetic hysteresis characteristics is developed to quantitatively assess the influence of varying residual magnetism and switching phases on the two targeted transient events. Second, the Jiles–Atherton hysteresis model is utilized and rigorously calibrated through empirical hysteresis loop measurements. A fitness function aimed at minimizing the root mean square error is formulated, and particle swarm optimization is employed to achieve global optimization of the model parameters. The transient magnetic flux density, derived via the voltage integration method, serves as the initial condition for determining the core’s residual magnetism state following switching operations. Finally, a relationship curve between residual magnetism and switching phase is established. Leveraging this curve, a closed-loop phase selection control procedure is developed to identify the optimal closing phase that effectively reduces both inrush current and overvoltage.

Section 2 presents a theoretical examination of the generation mechanisms underlying switching overvoltage and excitation inrush current, elucidating the critical roles of the closing phase angle and core residual magnetism. In Section 3, a transient simulation model is developed to represent the phase-segregated process of the Electric Multiple Unit (EMU), and the quantitative relationship between the closing phase angle and transient amplitude is established and validated through empirical data. Section 4 introduces a phase selection control strategy and corresponding procedure designed to concurrently mitigate overvoltage and inrush current phenomena. Section 5 presents accurate computation of residual magnetism by integrating the Jiles–Atherton (J-A) hysteresis model with optimization algorithms and voltage integration techniques; it further characterizes the relationship between the opening phase angle and residual magnetism and identifies optimal opening and closing phase angles via multi-objective optimization, demonstrating the efficacy of this approach within a continuous feasible domain. Finally, Section 6 offers a comprehensive summary of the research findings.

2. Introduction to Excitation Inrush Current and Operating Overvoltage

2.1. Mechanism Analysis of Transformer Excitation Inrush Current Generation

During the closing operation of the vacuum circuit breaker (VCB) in a moving train set, the secondary side of the transformer is unloaded. The majority of the magnetic flux traverses the transformer core. Given that the magnetic flux lags the network voltage phase by 90°, if the breaker closes at the instant when the network voltage reaches its maximum, the transient flux in the transformer core is zero, allowing the transformer to enter steady-state operation immediately without excitation current on the primary side. Conversely, if the breaker closes when the network voltage is zero, the transient flux amplitude in the transformer core attains its maximum. Since magnetic flux cannot change abruptly, the core generates a non-periodic transient flux component opposite to the steady-state flux to maintain the pre-closing zero-flux condition. Upon breaker opening, the residual magnetism in the transformer core remains approximately constant due to its decay time being significantly longer than the switching interval. Consequently, at the moment of closing, the total magnetic flux in the transformer core is the superposition of steady-state flux, transient flux, and residual magnetism. This total flux often exceeds the core’s saturation flux density, causing a substantial reduction in excitation inductance and a rapid increase in primary-side current, thereby producing excitation inrush current. The operational principle of the no-load transformer in dynamic train groups employing single-phase transformers is illustrated in Figure 1.

Voltage on the transformer primary side upon closing the VCB:

$$u_1\left(t\right) = \sqrt{2}{U}_1\sin (\mathit{\omega }t + \mathit{\alpha })$$

(1)

where $\mathit{\alpha }$—closing phase.

Assuming negligible leakage flux, the voltage equation for the transformer primary-side circuit can be expressed as

$$i_1{R}_1+{ N}_1{\displaystyle \frac{d\mathit{\phi }}{dt}}=\sqrt{2}{U}_1\sin (\mathit{\omega }t+\mathit{\alpha })$$

(2)

• where $i_1$—excitation current on the transformer primary side;
• $R_1$—equivalent resistance of the transformer primary winding.
• By substituting $i_1$ with the magnetic flux $\mathit{\phi }$, the equation becomes

  $$i_1={\displaystyle \frac{N_1\mathit{\phi }}{L_1}}$$

  (3)

  where $L_1$—self-inductance of the primary winding in a transformer.

Substituting Equation (3) into Equation (2) yields

$$N_1{\displaystyle \frac{R_1}{L_1}}\mathit{\phi }+{ N}_1{\displaystyle \frac{d\mathit{\phi }}{dt}}=\sqrt{2}{U}_1\mathit{sin}(\mathit{\omega }t+\mathit{\alpha })$$

(4)

The above formula is a first-order ordinary differential equation, which can be obtained after solving

$$\mathit{\phi }=-{\mathit{\phi }}_m\mathit{cos}\left(\mathit{\omega }t+\mathit{\alpha }\right) +({\mathit{\phi }}_m\mathit{cos} \mathit{\alpha }+{ \mathit{\phi }}_r)e^{-\mathit{\tau }t}$$

(5)

• where ${\mathit{\phi }}_r$—the residual magnetism of the transformer core before closing;
• $\mathit{\tau }$—transient flux time decay coefficient.

The initial term represents the steady-state flux component, whereas the subsequent term corresponds to the transient flux component. An examination of Equation (5) reveals that the total magnetic flux within the transformer core is influenced by both the initial phase angle at which the VCB closes and the residual magnetism present in the onboard transformer at the moment of VCB closure.

Considering the relationship between magnetic flux and current, the expression for the excitation inrush current can be simplified as shown in Equation (6). It is important to note that the inductance L varies dynamically in response to changes in magnetic flux linkage and current. During normal operation, the iron core operates within the linear region, where L is equal to ${\mathrm{L}}_1+{\mathrm{L}}_{\mathrm{m}}$, with ${\mathrm{L}}_{\mathrm{m}}$ treated as a constant parameter. However, once the core reaches saturation, the value of ${\mathrm{L}}_{\mathrm{m}}$ decreases substantially.The relationship between magnetic flux and excitation inrush current is illustrated in Figure 2.

$$i_m(t)\approx {\displaystyle \frac{N_1\left[-{\phi }_m\mathit{cos}\left( \omega t+\alpha \right)+({\phi }_{m}cos\alpha +{\phi }_r)e^{-\tau t}\right]}{L\left(t\right)}}$$

(6)

As illustrated in Figure 3, during the VCB closing phase, $\mathit{\alpha } = 0°$, and the transformer core residual magnetism ${\mathit{\phi }}_{\mathrm{r}}$ > 0. Neglecting the transient attenuation component, the transformer core flux attains a maximum value of ${2\mathit{\phi }}_{\mathrm{m}} + {\mathit{\phi }}_{\mathrm{r}}$ within half a cycle of VCB closing. Typically, the normal transformer saturation flux of a transformer ranges between 1.15${\mathit{\phi }}_{\mathrm{m}}$ and 1.4${\mathit{\phi }}_{\mathrm{m}}$; however, at this juncture, the core flux significantly exceeds the saturation flux, resulting in core saturation. Consequently, the excitation inductance exhibits nonlinear characteristics, leading to a rapid increase in excitation current and a pronounced excitation current surge. Specifically, when the VCB closing phase $\mathit{\alpha } = 90$°, if the residual magnetism ${\mathit{\phi }}_{\mathrm{r}}$ is smaller, the transformer core flux transitions directly into steady-state operation without magnetic saturation. Conversely, if the residual magnetism ${\mathit{\phi }}_{\mathrm{r}}$ is larger before VCB closing, the core flux during the half-cycle of VCB closing may reach ${\mathit{\phi }}_{\mathrm{m}} +{ \mathit{\phi }}_{\mathrm{r}}$, potentially inducing magnetic saturation and associated inrush currents, albeit with amplitudes considerably lower than those generated at 0° closing phase.

2.2. Mechanism Analysis of Operating Overvoltage Generation

When the transformer is unloaded during VCB closure, its equivalent circuit schematic is represented in Figure 4.

Here,$U_s$ is the equivalent power supply of the traction substation, $R_d$ and $L_d$ are the equivalent resistance and inductance of the the traction network to the pantograph, $C_T$ is the equivalent capacitance to the ground of the primary side of the no-load transformer, and $L_T$ is the excitation inductance of the no-load transformer; due to the inductance, current has the characteristic of not being able to be changed abruptly, so $L_T$ is regarded as an open circuit. At the closing moment, the traction substation equivalent power supply $U_s$ charges the transformer equivalent capacitance $C_T$. Due to the small value of the capacitance, the circuit will produce a high-frequency oscillation phenomenon, generating the closing overvoltage. The loop equation can be written as follows:

$$L_d{C}_T{\displaystyle \frac{d^{2}U}{t^2}} +{ R}_d{C}_T{\displaystyle \frac{\mathit{dU}}{t}} + U = U_s$$

(7)

The equivalent supply voltage at the traction substation at the closing phase $\mathit{\alpha } = 0°$ is

$$U_s=U_m\mathit{sin}{\mathit{\omega }}_{0}t$$

(8)

The frequency of high-frequency oscillation of the system at this time is

$${\mathit{\omega }}_{0 }={\displaystyle \frac{1}{\sqrt{L_d{C}_T}}}$$

(9)

The primary transformer voltage is calculated as

$$U\left(t\right)={ U}_m\left(1-e^{-{\displaystyle \frac{R_d}{2L_d}}t}\mathit{cos}{\mathit{\omega }}_{0}t\right)$$

(10)

From these expressions, it is evident that the high-frequency oscillation voltage during the closing process comprises both steady-state and transient components. The superposition of these components yields the transient operating overvoltage experienced by the transformer’s primary winding. The amplitude of this overvoltage is predominantly influenced by the closing phase and can reach up to twice the steady-state operating voltage before gradually decaying over time. The decay rate is determined by the equivalent resistance and inductance present in the line.

3. Simulation Study

To investigate the effects of residual magnetism and closing phase on excitation inrush current and operating overvoltage, an equivalent circuit model of transformer no-load closing during the split-phase initiation of a moving train set was developed using the electromagnetic transient simulation software PSCAD v46. The accuracy of this simulation model was validated by comparing its results with empirical measurements.

3.1. Simulation Model

Based on the equivalent circuit model described in Section 2, the simulation circuit was constructed within PSCAD. The saturated transformer module from the PSCAD library was employed to accurately and effectively simulate transformer operating characteristics. Figure 5 illustrates the simulation circuit diagram for the split-phase initiation of the moving train set, with relevant transformer parameters detailed in

Table 1. Transformer parameters.

Winding	Net Side Winding (Primary Coil)	Traction Winding (Secondary Coil)
Rated capacity/kVA	2100	2 × 1050
Rated voltage/V	25,000	2 × 970
Rated current/A	84	2 × 1082.4
Short-circuit impedance (%) HV-traction leakage inductance/mH	1.36 (1 ± 10%)
Frequency/Hz	50
Load loss/kW	67
No-load loss/kW	15

.

In Figure 5, R1, R1′, R2, and R2′ represent the equivalent resistances of the catenary system; L1, L1′, L2, and L2′ represent the equivalent inductances of the catenary system; C1 and C2 represent the equivalent capacitances of the catenary system to ground; R3 and R3′ are the equivalent resistances of the neutral section; L3 and L3′ are the equivalent inductances of the neutral section; C3 is the equivalent capacitance of the neutral section to ground; C12 is the coupling capacitance between the neutral section and the catenary system; CT represents the capacitance from the primary side of the transformer to ground; R0 and L0 are the equivalent resistance and inductance of the voltage transformer, respectively; C0 is the equivalent capacitance between the high-voltage roof lead and the pantograph to ground; US1 and US2 are the equivalent power sources of the traction substation; E1 is the measured primary-side voltage of the transformer; and I1 is the measured primary-side current of the transformer.

The simulation duration was set to 0.1 s, with residual magnetism initialized at zero. The switch BRK1 was closed at 0.02 s to simulate the split-phase closing of the train set, corresponding to a closing phase of 0°, during which no overvoltage was observed. As the closing time was varied, the closing phase increased gradually, resulting in an increase in the magnitude of the closing operating overvoltage, peaking at a closing phase of 90°. Figure 6 presents the voltage waveform at the transformer’s primary side, while Figure 7 depicts the relationship between closing phase and overvoltage magnitude. Analysis indicates that when the closing phase lies approximately between −45° and 45°, the overvoltage amplitude remains below the maximum steady-state operating voltage. Therefore, if the objective is solely to suppress operating overvoltage, a closing phase of 0° is optimal.

Simultaneously, the transformer’s primary side experiences excitation inrush current during operating overvoltage generation. This inrush current is influenced by both the closing phase and residual magnetism. When the residual magnetism of the transformer is set to 0.5 p.u., 0 p.u., or −0.5 p.u., according to the different residual magnetism, the relationship between the closing phase and the inrush current is shown in Figure 8. It is observed that when residual magnetism is zero, the excitation inrush current magnitude is minimized at a closing phase of 90°, whereas it is maximized at 0°.

3.2. Model Verification

To validate the simulation model’s accuracy, voltage and current data were collected from a city train. The data acquisition system comprised a PC, a pico data acquisition card, a high-voltage divider, a current probe, and associated equipment. For safety considerations, the high-voltage divider was installed downstream of the roof circuit breaker. Operating overvoltage and excitation inrush were induced by manually operating the circuit breaker while the train was stationary. Voltage signals were obtained from the pantograph of carriage no.2 via a 10,000:1 high-voltage divider located at the circuit breaker’s rear end. Current signals were measured on the primary side of the traction transformer in carriage no.2 using a current transformer and a current probe. Figure 9 depicts the test equipment setup.

Multiple datasets of operating overvoltage and excitation inrush current were acquired. Figure 10 presents a comparison between simulated and measured voltage and current waveforms at a closing angle of approximately 240°. At this point, the peak voltage on the transformer’s primary side is around 43 kV, with the peak overvoltage rapidly oscillating and decaying to a steady-state voltage of 39 kV. The excitation inrush current reaches its peak value of 443 A half a cycle after closing and gradually diminishes over subsequent cycles.

A comparison between test and simulation results revealed that at closing phases of ±90°, the maximum simulated voltage amplitude is 53.07 kV, while the maximum measured voltage is 55.29 kV. For closing phases of 0° and 180°, the maximum excitation inrush current in tests is 610 A, while simulations yield 580 A. Discrepancies are attributed to the simplified nature of the electrical equivalent model and the presence of residual magnetism in the circuit breaker core during opening, which is challenging to measure precisely and introduces variability in inrush current values. Overall, the strong agreement between experimental and simulation data confirms the reliability of the simulation model.

4. Research on Phase Selection Control Strategy

The preceding analysis demonstrates that operating overvoltage is influenced by the closing phase, while excitation inrush current depends on both closing phase and residual magnetism. To concurrently mitigate operating overvoltage and excitation inrush current, a phase-selective control strategy is proposed. The corresponding control flowchart is presented in Figure 11.

Section 2.1 indicates that when the circuit breaker closes within a phase range of approximately −45° to 45°, the overvoltage amplitude remains below the maximum steady-state operating voltage, and the transient flux ${\mathit{\phi }}_m\mathit{cos} \mathit{\alpha }$ lies around 0.7${\mathit{\phi }}_m$ to ${\mathit{\phi }}_m$. Given that transformer cores are composed of soft magnetic materials with typically low residual magnetism (generally not exceeding 0.7${\mathit{\phi }}_m$), the residual magnetism and transient flux cannot fully offset each other. Therefore, optimizing the transient flux value (x-value) to minimize flux amplitude is a viable approach to suppress excitation inrush current. The ideal condition is approached as the value of variable x nears zero. This is attributed to the fact that a value close to zero signifies a reduced transient flux, which ensures that the transformer experiences only steady-state flux during the closing operation, thereby preventing core saturation. The lower bound of x is established by the maximum absolute residual flux observed in the relationship between the opening phase and residual flux. Conversely, defining the upper bound of x is more intricate, as it necessitates a thorough evaluation of the upper limit imposed on the inrush current, along with the dynamic interaction between the closing phase and inrush current under varying flux conditions.

A critical aspect of this process involves establishing the relationship between the switching phase and the residual magnetism based on measured data. From this relationship, a curve can be obtained to meet the formula ${\mathit{\phi }}_m\mathit{cos} \mathit{\alpha } +{ \mathit{\phi }}_r<x$, allowing for the determination of the residual magnetism ${\mathit{\phi }}_r$ corresponding to the switching phase range. At the same time, by applying this to the train set of the split-phase circuit breaker, the residual magnetism of the transformer core is rapidly obtained. Combined with the formula ${\mathit{\phi }}_m\mathit{cos} \mathit{\alpha } + {\mathit{\phi }}_r < x$, the optimal closing phase ${\mathit{\alpha }}_{\mathit{best}}$ can be calculated. By optimizing the switching phase, both the operating overvoltage and excitation inrush current can be suppressed at the same time.

5. Transformer Core Residual Magnetism Calculation

To derive the breaking phase-residual magnetism relationship curve, it is necessary to compute the residual magnetism following circuit breaker breaking. This section develops the Jiles–Atherton (J-A) hysteresis model, with model parameters identified via the particle swarm optimization (PSO) algorithm. The hysteresis loop is obtained by inputting flux density data, derived from voltage integration, into the model. The terminal point of the hysteresis loop corresponds to the residual magnetism of the core.

5.1. Jiles–Atherton Hysteresis Model

The J-A hysteresis model, introduced by D. Jiles and D. Atherton in 1984 [26], is grounded in energy conservation principles and characterizes the hysteresis loop using five parameters: saturation magnetization strength $M_s$, anisotropy coefficient a, coupling coefficient $\mathit{\alpha }$, hysteresis loss coefficient k, and reversible magnetization coefficient c.

To obtain the hysteresis curve representative of the actual transformer core, cold-rolled oriented silicon steel sheets (model 30Q130) were employed as test samples. The hysteresis loop of the silicon steel was measured using the Epstein frame methodology, with the test setup depicted in Figure 12.

An inverse J-A model was established, wherein magnetic flux density B serves as the input and magnetic field strength H as the output. The inverse model is expressed as

$${\displaystyle \frac{\mathit{dM}}{\mathit{dB}}}={\displaystyle \frac{{\delta }_M(1-c\left)\right(M_{\mathit{an}}-M)+k\eta c·\frac{d{M}_{\mathit{an}}}{\mathit{dH}}}{{\mu }_0{\delta }_M(1-\mathit{\alpha }\left)\right(1-c\left)\right(M_{\mathit{an}}-M)+{\mu }_{0}k\eta +{\mu }_{0}k\eta c·\frac{d{M}_{\mathit{an}}}{\mathit{dH}}}}$$

(11)

where $M$ is the actual magnetization intensity; η is the direction coefficient, η = 1 when dH/dt > 0, η = −1 when dH/dt < 0; ${\delta }_M$ is the coefficient to prevent non-physical understanding, which needs to satisfy the following equation:

$${\delta }_{M }=\left\{\begin{array}{c}1 sgn\left({\displaystyle \frac{\mathit{dH}}{\mathit{dt}}}\right)·sgn(M_{\mathit{an}}-M) > 0\\ 0 sgn\left({\displaystyle \frac{\mathit{dH}}{\mathit{dt}}}\right)·sgn(M_{\mathit{an}}-M) < 0\end{array}\right.$$

(12)

The expression for the hysteresis-free magnetization strength $M_{\mathit{an}}$ is

$$M_{\mathit{an}}={ M}_S\left[\mathit{cot}\mathrm{h}({\displaystyle \frac{H+\mathit{\alpha }{M}_{\mathit{an}}}{a}}\right)-{\displaystyle \frac{a}{H+\mathit{\alpha }{M}_{\mathit{an}}}}]$$

(13)

Hysteresis return line calculation can be obtained by combining Equation (11) with Equation (13):

$$B ={ \mu }_0(H + M)$$

(14)

To enhance output accuracy, the PSO algorithm was employed to identify model parameters by minimizing the root mean square error between measured and calculated magnetic field strengths, as defined by the objective function:

$$\mathit{fitness} = \sqrt{{\displaystyle \frac{{\sum }_{i=1}^Z{\left(H_m\right(i)-H_c(i\left)\right)}^2}{M}}}$$

(15)

where $H_m$ is the measured magnetic field strength, $H_c$ is the calculated magnetic field strength, i is the sampling point, and Z is the total number of sampling points.

Table 2. Identified parameters of the saturated hysteresis loop model.

Parameter	Value
$M_s$/$(\mathrm{A}·{\mathrm{m}}^{-1})$	1,432,244
$a$/$(\mathrm{A}·{\mathrm{m}}^{-1})$	10.414
$\mathit{\alpha }/{10}^{-5}$	3.128
$k$/$(\mathrm{A}·{\mathrm{m}}^{-1})$	33.918
c	0.1608

summarizes the identified parameters of the J-A hysteresis model, and Figure 13 compares the fitted hysteresis loop with measured data, demonstrating good agreement.

Since the parameters of the J-A hysteresis model, except for the saturation magnetization strength, change exponentially with the change in the flux density, in order to obtain an accurate value of the residual magnetism, the fitted parameter model is shown in Equation (16) [27]:

$$\left\{\begin{array}{c}\mathit{\alpha } = {\mathit{\alpha }}_{\mathit{max}}\left[10.89·e^{0.0434·(B/B_{\mathit{max}})}-10.36\right]\\ a = a_{\mathit{max}}\left[10.27·e^{-0.0695·(B/B_{\mathit{max}})}-8.601\right]\\ k = k_{\mathit{max}}\left[11.98·e^{0.0447·(B/B_{\mathit{max}})}-11.52\right]\\ c = c_{\mathit{max}}\left[137·e^{-0.0349·(B/B_{\mathit{max}})}-131.3\right]\end{array}\right.$$

(16)

Using magnetic flux density at 50 Hz and 1.5 T as an example, parameters were calculated and substituted into the J-A model. Figure 14 compares the resulting local hysteresis loop with measured data, confirming the model’s accuracy.

By integrating the data presented in Figure 13 and Figure 14, it is evident that the computed hysteresis loops correspond closely with the experimental measurements. The application of this dynamic parameter approach enables the J-A model to accurately replicate the observed data not only within the saturation region but also across certain low magnetic flux intervals. This outcome underscores the efficacy of the model parameter calibration technique and highlights its robustness across different operating conditions.

The residual magnetism in the iron core after circuit breaker opening is the key to phase selection control. This paper uses the voltage integration method to calculate the transient magnetic flux density B(t), which serves as the input to the J-A model to track the magnetic state. The principle is as follows:

$$B\left(t\right)={\displaystyle \frac{1}{N_{1}S}}\int u_1\left(t\right)dt+B_0$$

(17)

where $B_0$—integral initial magnetic flux.

To reduce errors in voltage measurement and mitigate the influence of the integration procedure on accuracy, as well as to ensure the integrity of input data and the dependability of integration outcomes, this study adopted a systematic methodology. Initially, calibrated high-voltage dividers alongside high-precision data acquisition systems were employed to measure the voltage signal u₁(t), thereby guaranteeing both amplitude and phase accuracy. Subsequently, the integration start time was aligned with a voltage zero-crossing point immediately preceding breaker operation. Under the justified assumption that the core flux is zero at this instant, the integration constant B₀ was set to zero to eliminate associated uncertainties. Additionally, the recorded voltage signals underwent digital filtering to attenuate high-frequency noise, and the trapezoidal numerical integration technique was applied to improve robustness against cumulative computational errors. Finally, the flux waveform obtained from the integration served as input to a calibrated J-A model, whose output magnetic field strength H was then compared with values derived from the primary side current measurements. This comparison provided an indirect validation of the entire computational procedure.

5.2. Measurement Results

By applying the aforementioned methodology to the voltage data recorded during multiple breaker operations, a series of transient magnetic flux trajectories at the instants of breaker opening were derived. These trajectories were computed utilizing the adaptive parameter J-A model described previously, with the terminal point of each hysteresis loop trajectory representing the residual magnetism (Br) of the iron core following each breaker operation. Due to the limited number of measured data points, a fourth-order Fourier series was employed to fit the discrete residual magnetism data, thereby establishing a continuous functional relationship between the breaker phase and residual magnetism. The fitting performance was demonstrated to be excellent, as evidenced by a sum of squared errors (SSE) of 0.008693 and a coefficient of determination R-squared of 0.9926. The resulting quantitative curve depicting the relationship between breaker phase and residual magnetism is presented in Figure 15. This high-precision fitting outcome substantiates the reliability of the entire data processing chain—from voltage measurement and integration calculations to hysteresis model computations—thereby ensuring accurate inputs for subsequent phase selection control strategies.

This quantitative relationship enables correlation of the VCB breaking phase with transformer residual magnetism via the fitted equation. Determining the optimal breaking and closing phases involves balancing objectives related to operating overvoltage and excitation inrush current, which are interdependent. To address this complexity, the PSO algorithm was utilized to identify optimal phases by minimizing the objective function:

$$F ={ \mathit{\omega }}_V·{\displaystyle \frac{V_{\mathit{ov}}}{V_{\mathit{const}}}} + {\mathit{\omega }}_I·{\displaystyle \frac{I_{inrus\mathrm{h}}}{I_{\mathit{const}}}}$$

(18)

• where $V_{\mathit{ov}}$—operating overvoltage amplitude;
• $I_{\mathit{inrush}}$—excitation inrush current magnitude;
• $V_{\mathit{const}}$, $I_{\mathit{const}}$—constraint upper limits for overvoltage and inrush current;
• ${\mathit{\omega }}_V$, ${\mathit{\omega }}_I$—the adaptive weighting factor, which satisfies ${\mathit{\omega }}_V +{ \mathit{\omega }}_I = 1$.

It is known that the closing phase is $\mathit{\alpha }$ and the splitting phase is $\beta$, with:

$$\left\{\begin{array}{c}{V}_{\mathit{ov}} = f\left(\mathit{\alpha }\right)\\ I_{inrus\mathrm{h}} = f\left({\mathit{\phi }}_{\mathit{trans}}\right)\\ {\mathit{\phi }}_{\mathit{trans}} = {\mathit{\phi }}_{m}cos \mathit{\alpha } + {\mathit{\phi }}_r\\ {\mathit{\phi }}_r = f\left(\mathit{\beta }\right)\end{array}\right.$$

(19)

The initial population size was set to 40, and the maximum number of iterations was set to 100. Constraints were set with an overvoltage upper limit of 38 kV (steady-state maximum) and an inrush current limit of 138 A (considering 1.5 times overload and a 1.1 safety margin). The optimal breaking phase was found to be 130°, and the optimal closing phase was 135°, yielding an overvoltage magnitude of 33 kV and an excitation inrush current of 121 A. These results were validated through simulation, with closing voltage and current waveforms depicted in Figure 16.

Simulation outcomes confirm that operating overvoltage remains below the transformer’s normal peak voltage, and inrush current stays within prescribed limits, demonstrating the efficacy of the proposed phase-selective control strategy in mitigating both phenomena.

A comprehensive analysis was conducted by iterating through all switching phases (0–360°) with a 5° step size.

Table 3. Phase combinations meeting overvoltage and inrush current constraints.

Split Phase/(°)	Closing Phase/(°)	Overvoltage Amplitude/kV	Excitation Inrush Current Amplitude/A
105	128	37.9766	−128.0567
110	131	35.7813	−131.2287
115	134	33.6287	−132.1741
120	135	33.1412	−131.7983
125	136	32.5732	−129.0383
130	135	33.1412	−121.6727
135	133	34.3809	−124.2538
140	130	36.4904	−125.0408
200	310	−35.8854	122.2541
205	313	−34.1095	125.8901
210	315	−34.0121	123.2456
215	314	−34.0627	129.5134
220	313	−34.1095	132.2451
225	310	−35.8854	124.2135
230	307	−36.0187	127.4356

summarizes phase combinations satisfying the imposed constraints.

The data indicate that the range of acceptable breaking phases is broader than that of closing phases. Considering that breaker opening times are generally more stable than closing times [28], the overvoltage and excitation inrush current amplitudes were evaluated within ±10° of the optimal closing phase (135°) while maintaining the breaking phase at 130°. Figure 17 illustrates these variations. Despite isolated extreme cases (e.g., peak overvoltage of 41.4 kV corresponding to low inrush current of 18.1 A and peak inrush current of 211.6 A corresponding to low overvoltage of 25.4 kV), the optimized scheme effectively controls excitation inrush current magnitude while ensuring overvoltage remains below the safety threshold of 38 kV. The feasible phase domain satisfying both criteria is continuous, underscoring its practical engineering significance. This approach guarantees an insulation safety margin of 9.5% and maintains an inrush current safety margin of 44.8%, complying with standards such as GB/T 311.1 [29] and DL/T 572 [30].

It is important to emphasize that the practical efficacy of this strategy is largely contingent upon the precision of the circuit breaker’s closing timing. Empirical studies have demonstrated that vacuum circuit breakers inherently exhibit variability in their closing times, with deviations on the order of milliseconds, corresponding to approximately ±45° in electrical phase angle [31]. Such variability directly induces control inaccuracies during the closing phase, potentially causing the actual closing instant to diverge from the theoretically optimal phase or even fall outside the continuous feasible region established through simulation. Consequently, the suppression effect may be diminished, and in extreme scenarios, the strategy may temporarily lose effective control over certain transient phenomena. To improve the engineering robustness of this approach, future research should address two key areas: firstly, the utilization of vacuum circuit breakers with enhanced precision and reduced timing variability; secondly, the integration of a closed-loop feedback control mechanism that incorporates closing time prediction and real-time phase adjustment to dynamically mitigate phase errors arising from mechanical inconsistencies.

5.3. MOA Combined with Closing Resistor Method Analysis

To evaluate the efficacy of the phase selection control strategy in suppression, a comparative analysis was conducted against the suppression performance of a closing resistor employed alongside the arrester. The simulation model utilized for this comparison is illustrated in the Figure 18 below. In this model, the closing resistor is assigned a resistance value of 100 Ω, while the arrester parameters are detailed in

Table 4. Technical specifications of the surge arrester.

System Nominal Voltage/kV	Rated Voltage RMS/kV	Continuous Operating Voltage/kV	DC 1 mA Reference Voltage/kV	Current Carrying Capacity of a 2 ms/A Square Wave
27.5	42.0	34.0	65.0	400

. Notably, the arrester possesses a rated root mean square (RMS) voltage of 42 kV and a maximum voltage rating of 59.4 kV.

Figure 19 illustrates the closing overvoltage at a 90° closing angle when the transformer core exhibits no residual magnetism. Following the installation of the Metal Oxide Arrester, the operating overvoltage on the transformer’s primary side measures 54.3 kV. A comparison between Figure 6 and Figure 19 reveals that when the operating overvoltage remains within the maximum voltage threshold safeguarded by the arrester, the arrester remains inactive, and the amplitude of the operating overvoltage does not experience a significant reduction. Furthermore, the introduction of a closing resistor ensures that the primary side of the onboard transformer retains its pre-suppression condition, thereby preventing the generation of magnetizing inrush current.

Figure 20 illustrates the relationship between the phase angle and the excitation inrush current when a Metal Oxide Arrester (MOA) is employed in conjunction with a 100 Ω closing resistor, within the network voltage phase range of 0° to 180°. A comparison between Figure 8 and Figure 20 reveals that the integration of the MOA and the closing resistor substantially reduces the peak excitation inrush current on the primary side of the onboard transformer, indicating effective suppression capabilities. Nevertheless, this approach necessitates the use of surge arresters and multiple circuit breakers, leading to a complex array of devices that complicate operational procedures. Furthermore, it does not achieve significant mitigation of overvoltages within the surge arresters’ protection voltage range.

6. Conclusions

The closure of split-phase circuit breakers in moving train sets induces operating overvoltage and excitation inrush current due to alterations in line parameters. This study theoretically analyzed the influence of the closing phase and residual magnetism on these phenomena during transformer no-load closing, supported by simulation and experimental validation. The relationships between the closing phase and operating overvoltage, as well as the closing phase and excitation inrush current under varying residual magnetism, were established.

Furthermore, a comprehensive suppression system encompassing split-phase selection, residual magnetism calculation, and closing phase optimization was developed. Utilizing the Jiles–Atherton hysteresis model, a nonlinear mapping between breaker breaking phase angle and transformer core residual magnetism was constructed. This mapping, integrated with the phase selection control process, facilitated calculation of the optimal closing phase, accounting for transformer core saturation characteristics. Simulation results confirmed that this strategy effectively reduces both operating overvoltage amplitude and peak excitation inrush current, achieving synergistic suppression of transient impacts. The findings provide a theoretical foundation for advancing phase-controlled closing technologies and optimizing engineering parameters.