Finite element joint simulation model applied to fast mechanical switching of controlled resonance combination circuit breakers

Abstract

Fast mechanical switches (FMSs) are critical components of DC circuit breakers (DCCBs), which require the switch action time to break to a sufficient distance within 3 ms in the DC line breaking scenario, while ensuring a long service life. The breaking mechanism significantly affects the current interruption capability of DCCBs. The operation of the repulsion mechanism, along with the morphology of the arc and its transformation within the interrupter chamber, collectively influence the breaking performance of the FMSs. This paper presents a comprehensive analysis of the FMSs, which serves as the pivotal component of controlled resonance combination circuit breakers (CRCBs). This study establishes a multi physics coupling simulation analysis method based on the equivalent circuit of repulsion mechanism discharge, combined with electromagnetic and solid mechanics fields. By constructing a full cycle magnetohydrodynamic particle arc (MHP) model and using a combined simulation of Finite Element joint model (FEJM), the evolution law of arc characteristics during the superimposed current interruption process was systematically explored. The focus was on analysing the physical process of the zero zone of the superimposed arc, the multi physics field coupling relationship of the arc, and the interaction mechanism with external characteristic parameters. Further combining with optimisation design methods, the effectiveness of the model was verified through experiments, FEJM provides comprehensive technical support for effectively reflecting the stress issues of core components during the breaking process of FMS and can provide accurate theoretical references for the optimisation design of mechanical motion components in FMS. It also accurately represents the arc extinguishing process during the breaking of FMS and provides a convenient method for the selection and design of circuit parameters for the entire circuit breaker.

INTRODUCTION

In long-distance transmission, overhead lines are the primary choice in DC grid design due to factors such as terrain and economic costs. However, overhead lines are frequently exposed to atmospheric conditions, increasing the likelihood of faults. Among these challenges, the short circuit problem is considered a significant challenge for DC transmission technology. In power systems, circuit breakers (CBs) are regarded as the primary protection and control-switching devices and must be highly reliable to ensure the safe and stable operation of the system after a local short-circuit fault occurs in the DC line.

To meet the growing power demand [1–3], the optimised design and development of high-capacity and high-performance CBs have become a prominent research topic [4–9]. The operation of an artificial current zero DC circuit breaker (DCCB) to interrupt current involves a superposition of DC and reverse mid-frequency currents, differing from the interruption of DC alone. The fast mechanical switch (FMS) is a pivotal component of the DCCBs, and its breaking performance significantly influences the current breaking capacity of the DCCBs. The breaking performance of the FMS is influenced by factors such as the rapid breaking characteristics of the repulsion mechanism, the morphology of the arc and other factors. Therefore, research on modelling technology for the FMS in flexible DCCBs holds significant theoretical and engineering importance.

The electromagnetic repulsion mechanism, known for its rapid breaking characteristics and operational reliability, can swiftly separate the movable and static contacts within a few hundred microseconds [10, 11] Thus, the fault removal can be effectively accomplished. In the topology optimisation design of the FMSs Yu et al. [12] discuss the simulation optimisation of the coil skeleton and shell bracket. Wang et al. [13] utilised 3-D simulation software to optimise the electromagnetic repulsion mechanism and enhance its output efficiency. Jiang et al. [14] employed a particle swarm multi-objective optimisation method, coupling physical and theoretical models with multi-objective optimisation joint computation, to optimise various parameters of the repulsion mechanism. Regarding deformation stress analysis, Yang et al. [15] accurately determined the deformation stress distribution of crucial components using a flexible body dynamics model. Zhou et al. and Vilchis-Rodriguez et al. [16, 17] employed the finite element simulation calculation method to analyse the influence of component parameters of various repulsion mechanisms on electromagnetic repulsion. In the aforementioned studies, most focus on single electromagnetic field analysis, with fewer integrating other fields for calculation.

In high current switching, the discharge recovery phase is significantly influenced by the arc-firing process [18–20]. The widespread use of spectrometers and fast cameras with various detectors allows for insights into the thermal plasma physical processes. However, in a closed vacuum interrupter system, the measurable parameters of the vacuum arc are limited. Additionally, the experimental environment is harsh. Therefore, simulation modelling is required to clarify the mechanism and properties of the vacuum arc. Yan [21] conducted a detailed modelling and simulation analysis of the characteristics of the small-current steady-state vacuum arc under a longitudinal magnetic field. Xiang [22] takes into account the energy radiated by vacuum arc electrons to simulate the vacuum arc at various current levels. Sarrailh et al. [23] developed a hybrid model, assuming that electrons obey the Boltzmann distribution, thereby eliminating the need to solve for electron distribution. However, this method has the disadvantage of the highly non-linear Poisson equation, making it challenging to solve. Soloot et al. [24] employed a fluid model, considering the equations of motion of electrons and mass conservation equations to calculate the distributions of ions and electrons. This enabled the analysis of the motion process of microscopic particles in the zero region. The arc plasma evolution process involves coupled electric, magnetic, thermal, and particle flow fields, making it a complex process influenced by the interplay of these fields. Therefore, the simplified multi-physics field coupling simulation calculation of arc can be a more efficient and accurate expression of the change mechanism of arc.

This paper conducts an in-depth study on the FMSs, the core component in the controlled resonance combination circuit breaker (CRCB), encompassing the following: (1) Using the method of multi-physical field coupling simulation and analysis, we propose a model of the repulsive mechanism based on the field-circuit coupling of the electromagnetic field and the solid mechanics field with the discharge circuit, and analyse the motion process of the repulsive mechanism. (2) We propose a full-cycle magnetohydrodynamic particle (MHP) arc model and establish the joint simulation model Finite Element joint model (FEJM) of FMS repulsion mechanism and interrupter chamber. The analysis includes the characteristics of the arc-firing of the FMS for breaking the superimposed current. It involves the study of the physical process of the zero zone of the superimposed arc. Additionally, it encompasses research on the arc in various physical fields. Furthermore, it explores the relationship between the physical fields and the arc's external characteristics. This can serve as the FMS of CRCB design and reverse superimposed the current frequency range selection method. (3) Based on the results of the FMS optimisation design method, the experimental research on the arc-firing process of the FMS is conducted to verify the correctness of the FMS model.

In summary, assessing the breaking capacity of the FMS relies heavily on understanding the action process of the electromagnetic repulsion mechanism, the principle of arc plasma change, and particle dynamic behaviour. In this study, a DCCB breaking experimental platform was developed based on the simulation and analysis of the FEJM. Intermediate frequency-breaking experiments were conducted, and the simulation experimental data were analysed to provide reasonable mechanism parameters for breaking requirements of the FMS. The research results presented in this article can serve as a reference for the design and evaluation of fast mechanical switches, ensuring the rapid and reliable disconnection of high-voltage DC circuit breakers and the dependable operation of power systems.

STUDY OF MECHANICAL SWITCHING BREAKAGE OF DC CIRCUIT BREAKERS

Controlled resonance combination circuit breakers

CRCB can quickly interrupt the fault current in DC systems, as illustrated in Figure 1, which depicts the topology of the CRCB layout in DC transmission systems. In this paper, an in-depth study is conducted on the FMS, which is the core component of the CRCB. The breaking capacity of medium-frequency currents in the frequency range of kilohertz level is investigated.

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The topology of the combined high-voltage DCCB, depicted in Figure 2, comprises a fast-breaking switch (FBS) main breaker (MB), a fast-closing switch (FCS) auxiliary breaker (AB) in parallel, an H-bridge power electronic module comprising 4 IGCTs (T₁ – T₄) and a capacitor C₁ with a pre-charge voltage of U₀, a set of resonant capacitors/inductors C_H, L_H, and a surge arrester MOV in series. Meaning of other markings: RCB is the system side switch, LDC is the DC line inductance where I_K is the FMS current, I_line is the line current, I_C is the commutation injection current, and I_MOV is the surge arrester current.

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Operating principle of CRCB

Figure 3 illustrates the current and voltage waveforms during the breaking process, using a line fault as an example. Among them, I_C is the commutation resonance current.

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At time t₀, a fault occurs on the line, leading to the detection of current and issuance of the breaking instruction.

At time t₁, the switch receives the instruction, initiating its action. The FMS MB starts to break, while the FCS AB closes.

At time t₂, the AB is in the just-closed position (less than 2 ms from time t₁). The control and protection system initiates the resonance of T₁, T₄, C_H and L_H first. When the resonance current reaches zero, T₁ and T₄ are turned off, followed by triggering the resonance of T₂, T₃, C_H, and L_H.

At time t₃, the line current i_k reaches zero, and the MB extinguishes the arc, diverting part of the current to facilitate transfer. After the FMS MB extinguishes the arc and disconnects, T₁ and T₄ remain conducting, with the fault current charging the resonant branch capacitor C.

At time t₄, the voltage across both ends of MOV reaches the threshold voltage, resulting in the disconnection of T₁ and T₄. The fault current is entirely diverted to the arrester branch, dissipating energy.

At time t₅, the energy absorption process is completed, and the CRCB finishes breaking.

MULTI-PHYSICS FIELD SIMULATION MODEL FOR FMS

The primary breaking performance of a DCCB is directly manifested in the breaking action time and speed of the movable and static contacts. The FMS breaks intermediate-frequency current with a frequency on the order of kilohertz, amplitude on the order of tens of kiloamperes, and arc ignition time on the order of milliseconds. Additionally, the transient restoration voltage rise rate is significant after the break, highlighting the urgent need to enhance the intermediate-frequency arc breaking capability. Understanding the action process of the electromagnetic repulsion mechanism, the principle of arc plasma change, and particle dynamic behaviour is crucial for evaluating the breaking capacity of the FMS.

Figure 4 illustrates the construction process of the mechanical switch FEJM in this study, which includes the discharge circuit of the repulsive mechanism, the electromagnetic field, the solid mechanics field coupling model, and the MHP model of the vacuum arc extinguishing chamber. The physical field results obtained by jointly simulating the two important components of the switch through FEJM will be more consistent with the actual situation.

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Model of electromagnetic repulsion mechanism based on eddy current field

The FMS mainly consists of a metal repulsion disk, coils, a vacuum interrupter, insulated rods, and other components. The principle of the FMS process is as follows: At the initial moment, the repulsion disk and the excitation coil are in the closed position, with insulators and the upper tie rod connecting parts placed on compression stacked springs. This arrangement is designed to overcome the reaction force of the spring and the gravity of the moving parts and to prevent short-circuit fault currents caused by electrodynamic forces or mechanical vibrations resulting from misclosing. Hence, effective retention force is required to maintain the closing position. The magnetic force, balanced by the combination of the permanent magnet and armature, is generated. During operation, the thyristor control operation causes the energy storage capacitor to discharge the tap coil, generating pulse current. This current induces eddy currents in the repulsion disk, resulting in the generation of a reverse current with the tap coil due to the eddy current effect. This generates electromagnetic repulsion, causing the repulsion disk to drive the connected moving parts in the direction of the tap. The principal structure of the FMS is depicted in Figure 5. The red markings in the figure indicate the movement parts of the mechanism.

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The prototype-breaking process involves the electromagnetic field and solid mechanics field. To completely and accurately simulate this process, it is necessary to conduct field-circuit coupling simulation calculations. The general idea of the simulation involves establishing the FMS repulsion force mechanism model, the discharge equivalent circuit model, and analysing and solving the electromagnetic field and solid mechanics field coupling model.

The excitation circuit, as depicted in Figure 6, comprises a capacitor and a parallel current-continuing diode, which are integrated with the electromagnetic field domain. The external circuit part supplies current to the switching coil, while the electromagnetic field part performs electromagnetic field simulation calculations. The Lorentz force generated by the electromagnetic field in the model acts as the primary load on the metal repulsive disk, causing it to move. This process generates induced eddy currents due to the magnetic field produced by the pulsed current of the switching coil, resulting in unevenly distributed Lorentz forces in the metal repulsion disk. Equation (1) represent the governing equations for the generation of these Lorentz forces. 1 $$\left\{\begin{array}{@{}l@{}}\sigma \cdot E=J\\ \nabla \times A=B\\ \nabla \times B=\mu \cdot J\\ J\times B={F}_{\mathrm{z}}\end{array}\right.$$ where σ and μ are the electric and magnetic permeability of the repulsion disk respectively; E is the electric field strength; B is the magnetic induction strength; A is the vector magnetic potential; J is the current density; F_z is the Lorentz force.

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Analytical and physical field mixed model of repulsive mechanism

The motion process of an independent repulsive mechanism can be simulated using the analytical and physical field mixed model (APM), and the construction method of this model is intuitive and straightforward.

Set the upper tie bar in which the moving contact is connected to the total body load force as the axial stacked spring force F_{p_contact}. Set the lower tie bar and insulator as the total body load force as the combined force of the axial stacked spring force F_{p_bar} and the holding armature magnetic force F_B. The repulsion disk sets the total body load force to the Lorentz force F_z analysed above. 2 $${F}_{\mathrm{p}{\_}\text{bar}}=\left\{\begin{array}{@{}ll@{}}\,\mbox{--}{k}_{\mathrm{p}}\times \left({s}_{\mathrm{p}}+{s}_{\mathrm{p}0}\,\right)& {s}_{\mathrm{c}}=0,{s}_{\mathrm{p}} > \mbox{--}{s}_{\mathrm{p}0}\\ \,\mbox{--}{k}_{\mathrm{p}}\times \left({s}_{\mathrm{p}}+{s}_{\mathrm{p}0}\mbox{--}{s}_{\mathrm{c}}\right)& {s}_{\mathrm{c}} > 0,{s}_{\mathrm{c}}\mbox{--}{s}_{\mathrm{p}}< {s}_{\mathrm{p}0},\\ {k}_{\mathrm{p}}\times {s}_{\mathrm{p}0}\mbox{--}{k}_{\mathrm{r}}\times \left({s}_{\mathrm{p}}+{s}_{\mathrm{p}0}\mbox{--}{s}_{\mathrm{c}}\right)& {s}_{\mathrm{c}}\mbox{--}{s}_{\mathrm{p}} > {s}_{\mathrm{p}0}\end{array}\right.$$ 3 $${F}_{\mathrm{p}\_\text{contact}}=\left\{\begin{array}{@{}ll@{}}\,0& {{s}_{\mathrm{c}}=0,s}_{\mathrm{p}} > \mbox{--}{s}_{\mathrm{p}0}\,\\ {k}_{\mathrm{p}}\times \left({s}_{\mathrm{p}}+{s}_{\mathrm{p}0}\mbox{--}{s}_{\mathrm{c}}\right)& {s}_{\mathrm{c}} > 0,{s}_{\mathrm{c}}\mbox{--}{s}_{\mathrm{p}}< {s}_{\mathrm{p}0},\\ {\mbox{--}k}_{\mathrm{p}}\times {s}_{\mathrm{p}0}+{k}_{\mathrm{r}}\times \left({s}_{\mathrm{p}}+{s}_{\mathrm{p}0}\mbox{--}{s}_{\mathrm{c}}\right)& {s}_{\mathrm{c}}\mbox{--}{s}_{\mathrm{p}} > {s}_{\mathrm{p}0}\end{array}\right.$$ 4 $${F}_{\mathrm{B}}=\frac{{k}_{\mathrm{B}}}{{{s}_{\mathrm{p}}}^{2}},$$ where s_p0 represents the overtravel of the disc spring compression, s_p represents the displacement of the upper pull rod, and s_c represents the displacement of the insulator and lower pull rod. k_p is the elastic coefficient of the disc spring, k_r is the rigidity coefficient, and k_B is the magnetic coefficient.

Finite element joint model of repulsive mechanism

The actual process of mechanical switch breaking is not only reflected in the mechanical motion of the independent repulsive mechanism but also requires the dynamic contact inside the arc extinguishing chamber as an important component of the entire switch breaking. The energy loss caused by collisions during the breaking process between the connecting rod and the insulator of the moving contact as well as the influence of arc electric force on the motion results under load breaking is significant. Therefore, a multiphysics field hybrid model for fully moving switch components was proposed.

When the repulsive mechanism completes the switch overtravel, the insulator collides with the upper pull rod, connecting the moving contact to drive the moving contact to move. If the initial overtravel is small during the breaking process, the upper pull rod and insulator will collide multiple times, causing energy loss and resulting in a smaller actual breaking distance of the switch. As shown in Figure 7, this model is set up in a solid mechanics field with insulators and upper tension rods in collision contact. The contact method adopts a penalty function, and the disc spring is set under the condition of an elastic thin layer. The red dashed line represents the position of the elastic thin layer, and the red double-headed arrow represents the contact between the model and the object.

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The repulsive disk motion excitation of the model is set to F_z, thus constructing a complete simulation model of the mechanical switch moving parts, which includes the impact of contact collision loss and partial load arc electric force of the moving contact during the load-breaking process on the overall motion results. Compared with the mixed model of analytical and physical fields, it is more consistent with reality.

Model of full-cycle magnetohydrodynamic particle arc

This section presents the construction of a full-cycle MHP arc model through joint simulation analysis of multi-physical field coupling during the arc-firing stage and continuous transition afterwards. This model serves as a reliable method for understanding arc-firing characteristics, mechanisms, and optimising CB circuit parameters.

When studying the macroscopic motion of an arc plasma, it can be regarded as a conducting fluid. The arc plasma magneto-fluid equations consist of the conventional hydrodynamic equations and Maxwell's equations describing the electromagnetic field.

Mass continuum equation: 5 $$\frac{\partial \rho }{\partial t}+\rho \nabla \cdot \overrightarrow{v}=0,$$ where ρ is the mass density and $$\overrightarrow{v}$$ is the velocity of the plasma.

Momentum conservation equation: 6 $\begin{align*}\rho \frac{\partial \overrightarrow{v}}{\partial t}=\overrightarrow{J}\times \overrightarrow{B}-\nabla P-\eta \nabla \times \nabla \times \overrightarrow{v}\\ +\left(\frac{4}{3}\right)\eta \nabla \left(\nabla \cdot \overrightarrow{v}\right)+{\rho }_{\text{e}}\overrightarrow{E}+\rho g\end{align*}$ where $$\overrightarrow{J}$$ is the arc current density, $$\overrightarrow{B}$$ is the arc magnetic field strength, $$\overrightarrow{J}\times \overrightarrow{B}$$ is the electromagnetic force on the arcing plasma, P is the plasma pressure, η is the viscous coefficient, ρ_e is the charge density, $$\overrightarrow{E}$$ is the electric field strength $${\rho }_{\mathrm{e}}\overrightarrow{E}$$ is the electric field term, g is gravitational acceleration, ρg is the gravity term.

Energy conservation equation: 7 $$\rho \frac{\mathrm{d}h}{\mathrm{d}t}=\overrightarrow{J}\cdot \overrightarrow{E}+\nabla \cdot (k\nabla T)+Q-{e}_{\text{tr}}+\frac{\mathrm{d}P}{\mathrm{d}t}+\Phi ,$$ where k is the heat conduction coefficient, Q is the heat of reaction term, e_tr is the radiation term, Φ is the dissipated work term, h = U + Pρ is the enthalpy per unit mass, U is the internal energy of matter, and T is the temperature.

Maxwell's system of equations: 8 $$\left\{\begin{array}{@{}l@{}}\nabla \times \overrightarrow{E}=-\frac{\partial \overrightarrow{B}}{\partial t}\\ \nabla \cdot \overrightarrow{E}=\frac{{\rho }_{\text{e}}}{{\varepsilon }_{0}}\\ \nabla \times \overrightarrow{B}={\mu }_{0}\overrightarrow{J}+{\mu }_{0}{\varepsilon }_{0}\frac{\partial \overrightarrow{E}}{\partial t}\\ \nabla \cdot \overrightarrow{B}=0\end{array}\right.,$$ where μ₀ is the vacuum permeability and ε₀ is the vacuum permittivity.

Ohm's law: 9 $$\overrightarrow{J}=\sigma [\overrightarrow{E}+\overrightarrow{v}\times \overrightarrow{B}+{\left({n}_{\text{e}}e\right)}^{-1}\nabla {P}_{\text{e}}-{\left({n}_{\text{e}}e\right)}^{-1}J\times B],$$ where e is the charge, n_e is the electron density, and ∇P_e is the electron pressure gradient.

The above includes the process of the current burning arc. The attenuation of metal vapour after the arc is also an important process for vacuum CB insulation recovery. After the arc is extinguished, there are still a lot of neutral metal vapour particles among the gaps. The metal vapour in the post-arc stage is ionised, which will cause an increase in the probability of rekindling.

The growth of the post-arc sheath layer is determined by the density, velocity distribution, and metal vapour density of the plasma near the moment of arc quenching as the initial conditions result in a variety of microscopic particles migrating and diffusing, which leads to attenuation.

The growth process of the post-arc sheath layer can be described by using a continuous transition model as follows: 10 $${l}^{2}=\frac{4{\varepsilon }_{0}{U}_{0}}{9eZ{N}_{\text{i}}}\left[{\left(1+\frac{u(t)}{{U}_{0}}\right)}^{\tfrac{3}{2}}+\frac{3u(t)}{{U}_{0}}-1\right],$$ 11 $${U}_{0}=\frac{{M}_{\text{i}}}{2e}{\left({v}_{\text{i}}-\frac{\text{d}l}{\text{d}t}\right)}^{2},$$ 12 $$i(t)=\frac{{\uppi }{D}^{2}Z{N}_{\text{i}}e}{4}\left({v}_{\text{i}}-\frac{\mathrm{d}l}{\mathrm{d}t}\right),$$ 13 $${N}_{\text{i}0}=\frac{{I}_{1}}{Z\mathit{\cdot }\left({\uppi }\cdot {D}^{2}/4\right)\cdot {v}_{\text{i}}},$$ 14 $${N}_{\text{i}}={N}_{\text{i}0}{e}^{-\tfrac{t-{t}_{1}}{\tau }}\left(1+{\delta }_{\text{AMP}}\frac{{l}^{2}}{{l}_{\text{gap}\,}^{2}}\right),$$ where l is the sheath length. ε₀ is the vacuum permittivity. e is the electron charge; Z is the average charge of metallic copper ions. u(t) is the voltage between the vacuum electrodes, which can be expressed by the transient recovery voltage TRV. U₀ is the sheath potential. M_i is the mass of the ions. v_i is the velocity of the ion motion. i(t) is the after-arc current after the beginning of the development of sheath. I₁ is the initial after-arc current value. l_gap is the electrode spacing; τ is the diffusion decay time constant of the ions. δ_AMP is the spatial distribution coefficient of the ions in the electrode gap. N_i0 is the initial ion density. N_i is the ion density at the boundary of the sheath layer after the arc.

The metal vapour decay after passing the zero zone is defined as follows: 15 $${N}_{n}(t)=\frac{{S}_{\text{m}}}{\sqrt{{\omega }^{2}+{\beta }^{2}}}\left[\sin \left(\mathrm{arctan}\frac{\omega }{\beta }\right)\mathrm{exp}\left(-\beta \frac{\pi }{\omega }\right)+\sin \left(\pi -\mathrm{arctan}\frac{\omega }{\beta }\right)\right]\cdot \mathrm{exp}(-\beta t),$$ where $$\beta =\frac{1}{2R}\sqrt{\frac{8kT}{{\uppi }M}}$$, $$S(t)=\frac{\int {\Gamma }_{\text{ev}}\cdot \mathrm{d}S}{V}=\frac{\sqrt{2}{K}_{e}\gamma I}{MV}\sin \omega t={S}_{\text{m}}\,\sin \omega t$$, β denotes the decay coefficient that indicates the density of the metal vapour. Γ_ev is the evaporative flux. S_m denotes the coefficient of influence of the ablation rate of the contact; ω is the angular frequency. K_e is the influence coefficient considering the evaporation of electrode material caused by the heating of the electrode by the arc. γ is the electrode corrosion rate, taking the value range from 30 to 110 μg/C. M is the atomic mass of the metal vapour. V = πR²l_gap is the interstitial volume. k is the Boltzmann constant.

The criterion for arc re-ignition based on particle properties is that breakdown is caused by a metal vapour density exceeding the critical value. The transient recovery voltage (TRV), neutral metal vapour density, and gap sheath layer length collectively determine whether instantaneous rekindling or delayed rekindling occurs. In the basin curve depicted in Figure 8, n denotes the neutral metal vapour density, whereas, d denotes the sheath length. The product n × d represents the multiplication of neutral metal vapour density and the of the gap sheath length. The product (n × d)_crit represents the breakdown limit value.

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The FEJM can be calculated as the product of neutral metal vapour density and gap sheath length. This, combined with the above-mentioned Barshen curves, can be used to judge the breaking limit of the arc. It can also serve as a method to understand the arc ignition mechanism and to optimise the selection of DCCB circuit element parameters.

ESTABLISHMENT OF FMS MODEL

This section presents a simulation and analysis of FEJM arc ignition. The breaking current limit under superimposed circuit parameters is determined through model calculation and analysis, followed by the optimisation of circuit element parameters under fixed current. This data is valuable for designing repulsion mechanisms, selecting the frequency range of reversing current, and optimising the selection of circuit element parameters for CRCB.

Mechanical strength testing and optimisation of materials for repulsion mechanism components

FMS is utilised in scenarios where CRCBs interrupt substantial currents, and expansive breaking distances serve as a crucial indicator of reliable interruption. Expansive breaking distances necessitate considerable electromagnetic repulsive forces. Assessing the mechanical strength of the moving components of the repulsion mechanism is essential prior to the comprehensive testing of the CRCB.

Keep the electromagnetic repulsion mechanism storage capacitance capacity unchanged. Change the charging voltage to obtain the electromagnetic repulsion mechanism tripping motion characteristic data, and analyse the effect of driving voltage on the contact time and breaking action time: FMS repulsion mechanism no-load test. The repulsion mechanism discharge capacitance is 4 mF. The coil has 30 turns with a wire diameter of 1 mm × 3.5 mm. The aluminium disc has a diameter of 120 mm and a thickness of 10 mm. Figure 9 shows the relevant components of the experiment.

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After building the structural model of the mechanical switch, based on the actual FMS dynamic process, the repulsion mechanism boundary conditions of the model are as follows:

• (1)

  The breaking discharge circuit conducts the coil by simulating the circuit module as the coil excitation source, setting the coil boundary conditions in the electromagnetic field module, selecting the corresponding objects in the model, setting the number of turns and coil size, setting the material types of the repulsive disk and coil, and adding the ‘force calculation’ boundary condition.

• (2)

  In the solid mechanics module, the repulsive disk load is set as the result of the force calculation in condition (1), which is analysed through the magnetic field dynamics model.

The Resin material properties are shown in Table 1. According to the results of Figure 10 from the FMS repulsion mechanism simulation, it is shown that the stress at the connection between the insulator and tie-rod reached a maximum of 1.01 × 10³ MPa. The following figure depicts the fracture diagram of the test insulator tie-rod.

TABLE 1 Resin material properties.

Properties	Densities (kg/m3)	Young's modulus (GPa)	Poisson's ratio
Value	2230	2	0.4

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FMS breaking test results as shown in Figure 11.

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Table 2 shows the statistics of the experimental results.

TABLE 2 Displacement result of repulsion disk for 2 ms.

Capacitor voltage (V)	600	650	700	700
Measuring distance (mm)	2.84	3.28	3.77	/

The insulator tie rod broke during the breaking experiment when the voltage of the discharge capacitor was 700 V, resulting in a failure of breaking.

Although the resin material insulator tie rods are less massive, their strength is suboptimal, making it challenging to withstand the stresses associated with significant breaking distances. Consequently, it is essential to undertake mechanical strength tests and breaking distance mapping simulations of alternative material ties (steel and fibreglass ties) to refine the repulsion mechanism.

High-strength steel metal tie rods were used to replace resin insulator tie rods for the breaking simulation. The steel material properties are shown in Table 3.

TABLE 3 Steel material properties.

Properties	Densities (kg · m−3)	Young's modulus (GPa)	Poisson's ratio
Value	7820	206	0.25

Figure 12 shows that the stress at the connection between the insulator and tie rod is 0.541 × 10³ MPa. Because of the sufficiently high strength of the fibreglass composites, no deformation or damage to the fibreglass insulated tie rod occurs.

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Fibreglass tie rods were used to replace resin insulator tie rods for the breaking simulation. The Fibreglass material properties are shown in Table 4.

TABLE 4 Fibreglass material properties.

Property	Densities (kg · m−3)	Young's modulus (GPa)	Poisson's ratio
Value	2450	85	0.22

Figure 13 shows that the stress at the connection between the insulator and tie rod is 0.972 × 10³ MPa.

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Comparison of the simulation results of breaking experiment with different materials (high-strength steel and fibreglass).

Figure 14 and Table 5 show the statistics of the simulation results. Table 6 shows the quality of insulators under various materials.

• (1)

  Using artificial resin for the material of the insulating tie rod results in poor stress resistance and increases the likelihood of deformation or even fracture, leading to breaking failure. Adopting steel or fibreglass material can address the above problems.

• (2)

  However, if high-strength steel material is used as the insulating tie rod, its higher density compared to artificial resin results in an increase in the mass of the moving part, which leads to a decrease in switching speed and a 20%–30% reduction in the 2 ms repulsion disk moving distance.

• (3)

  From the simulation results, the FMS breaks faster if fibreglass is used as the insulating tie rod. However, with the gradual increase in discharge capacitor voltage, the fibreglass rod progressively reduces the efficiency of the repulsion disk's moving distance compared with that of the steel rod. Thus, selection of smaller mass and higher material strength fibreglass rods as insulator tie rods.

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TABLE 5 Repulsion disk displacement results.

Simulation result	Capacitor voltage
650 V	700 V
Simulation results (steel tie rod 2 ms) (mm)	2.49	2.86
Simulation results (fibreglass tie rod 2 ms) (mm)	2.74	3.02

TABLE 6 The masses of the three tie rods.

Material	Resin	Steel	Fibreglass
Mass (kg)	0.5	1.4	0.8

The higher the withstand voltage of the FMS, the larger the breaking distance required between the movable and static contacts when in the breaking position. Concurrently, the current injection branch is activated 2 ms after the FMS interruption, necessitating a specified breaking distance between the contacts at this juncture. The contact gaps during the breaking process in this model are delineated as follows:

As shown in Figure 15, the 2 ms breaking distance is 3.02 mm when the insulating tie rod is fibreglass material and the voltage of the tap storage capacitor is 700 V. The contact breaking distance at 2 ms is the same as the repulsion disk displacement.

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Selection of current injection branch resonance parameters

FMSs can be effectively constructed in the form of multiple breaks in series [22, 23] to diminish the withstand voltage of a single-break and thus minimise the travel of a single-break FMS.

In this paper, the FMS implemented in the CRCB employs a double-break tandem structure. The two FMSs selected are of identical type, and the electromagnetic repulsion actuator used in them exhibits minimal dispersion of action; that is, the double-break is synchronised with the same gap breaking [25, 26]. The ratio of the breaking capacity of the double-break FMS to that of the single-break FMS is 1.08 [27]. The interaction of the double-break FMS can realise a synergistic gain effect.

Equation (16) mathematically describes the double-break FMS dynamic recovery. The dielectric recovery rate of gain post arc process is 1.38. 16 $\begin{align*}{\nu }_{\text{str}}^{\text{multi}}(t)=478.032\left(\frac{{\uppi }{X}_{\text{m}}}{{T}_{\text{full}}}\right)\left\{\frac{\sin \left[\frac{{\uppi }\left(t\mbox{-}{t}_{\text{off}}\right)}{{T}_{\text{full}}}\right]}{1+(5.75)\left(\frac{{X}_{\text{m}}}{2}\right)\left[1\mbox{-}\cos \frac{{\uppi }\left(t\mbox{-}{t}_{\text{off}}\right)}{{T}_{\text{full}}}\right]}\right\},\end{align*}$ where the double-break FMS post-arc dielectric recovery speed is denoted by $${\nu }_{\text{str}}^{\text{multi}}$$, X_m is the breaking distance of each FMS, T_full is the full breaking time, and t_off is the moment of contact breaking [28].

In this section, we investigate the distribution and motion characteristics of the vacuum arc under different current frequencies. Therefore, the parameter design of the circuit follows the mid-frequency range (1000–3000 Hz) already utilised by the CRCB. The 2 ms total breaking distance of the double-break FMS is 5 and 6 mm as an example for analysis. Subsequently, we construct a dynamic model for the full-cycle MHP arc to obtain the distribution and physical field parameters under different frequencies. Figure 16 shows the transverse and longitudinal magnetic contacts.

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Vacuum arc extinguishing chamber: (1) Set the arc extinguishing chamber in a laminar flow state, the arc in a weakly compressible flow state, and add ‘volume force calculation’ to consider the effect of arc electric force (Lorentz force) in the entire fluid domain. (2) In the fluid heat transfer module, the electrodes and fluid domain are processed using formulae corresponding to their respective states. (3) In an electric field, the entire computational region follows the law of conservation of current. Set up electrical insulation (arc extinguishing chamber shell) and normal current density (electrode terminal). Except for the central axis, the boundary between the two electrodes, including the boundary used, is set as magnetic insulation. (4) Electrodes in the anode and electrode interface temperature coupling as well as the cathode and the electrode interface temperature coupling. (5) In the fluid flow particle tracking module, it is necessary to set the wall conditions for rebound as well as the material properties of the particles, the area of dielectric swimming force, and the electric field.

In constructing the FEJM, the intermediate moving contact that bridges these two parts must account for not only the tensile force imposed by the repulsive mechanism model but also the electrodynamic force exerted by the arc plasma in the interrupter chamber under loaded conditions. The Lorentz volume force from boundary condition (1) of the arc extinguishing chamber is used as the electric force for the partial body load in the dynamic contact area of the model.

Transverse and longitudinal magnetic field contacts apply both longitudinal and transverse magnetic fields to improve the arc breaking capability owing to their ability to effectively inhibit the formation of arc agglomerations.

Given that the contacts of the MHP arc model are chosen to be modelled as two-dimensional axisymmetric, it is necessary to account for the effect of diffusion arcs of such contacts, expressed in terms of volumetric forces in the fluid. 17 $${F}_{\mathrm{L}}=J\times {B}_{\mathrm{q}},$$ where B_q is the plasma magnetic field strength, F_L represents the volumetric force generated by the interaction of the current and the magnetic field in the fluid. To couple the computed velocity field back to electromagnetism, we utilise the velocity (Lorentz term) feature at the interface of the magnetic and electric fields as well as recognise the fluid velocity field as an input to this feature to achieve the coupling between the two physical fields.

Motion grid setup: The repulsion force mechanism will be combined with the switch moving contact to carry out the breaking operation. This requires that the simulation calculation must link the material coordinate system with the geometric coordinate system through mathematical expressions to occur at the same time. That is, the motion of the repulsion force mechanism during the tripping process is synchronised and fed back to the moving contact part.

According to the structural parameters of the repulsion mechanism and the arc extinguishing chamber, a two-dimensional axisymmetric geometric model is established as shown in Figure 17.

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To analyse based on the maximum current breaking cycle limit of the FMS such as the FMS extinguishes the arc when the current crosses zero within 2.5 cycles, considering a resonance current of 2500 Hz.

The results of the physical field parameters at different frequencies with a current amplitude of 35 kA are depicted in Figures 18 and 19.

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As shown in Figures 18 and 19, the particle velocity parameters of the MHP arc model can be obtained at different frequencies in their respective arc quenching zero zones (for a resonant current of 2500 Hz, the arc is quenched past zero after 2.9 ms of the arc ignition process).

The results of the full-cycle MHP arc model provide input parameters, which can serve as the re-ignition criterion under different superimposed current frequencies.

The parameters of the calculated example, δ_AMP is 5, τ is 10 μs, and Ω is collision cross-section. Ω takes the range 10–20. Arc reignition discrimination was performed for single-break FMS breaks of 2 ms with 5 and 6 mm breaking distances, as well as for single-break FMS and double-break FMS breaks of 2 ms with a total breaking distance of 6 mm.

The Figure 20 illustrates the effect of changing the breaking current frequency on the decay characteristics of the metal vapour density, with the product of the gap sheath lengths gradually increasing over time. Although the curves exhibit the same trend, higher frequencies of the FMS breaking current result in an earlier attainment of the product of the gap sheath length reaching 3 × 10¹⁹ m⁻², indicating a higher likelihood of reignition. Table 7 shows the criterion-specific results

• (1)

  For FMS superimposed current frequency selection:

  According to arc full-cycle MHP arc modelling calculations, it is observed that in the kHz frequency range, increasing frequency results in a more uneven distribution of cathode spots, leading to higher ion density before arc extinction. This slower growth of the sheath layer increases the likelihood of arc reignition. Conversely, if the frequency is too low, the peak value of the reverse superimposed current needs to be increased. So the pre-charge capacitor capacity and voltage must be increased, at which point the equipment economy needs to be considered. Simulation results indicate that arc reignition is less likely when the superimposed current frequency is 2500 Hz or lower.

• (2)

  Requirements for safe breaking distance of the FMS:

  Increasing the FMS breaking speed reduces the likelihood of arc reignition. Enlarging the contact spacing of the vacuum interrupter during current injection branch conduction can be achieved by increasing the FMS breaking speed, albeit at the expense of increased performance requirements and volume. Alternatively, extending the FMS breaking time increases the fault current at the moment of breaking, necessitating comprehensive analysis and trade-offs between economy and feasibility. Simulation results demonstrate that arc reignition is less likely at a 2 ms safe breaking distance of 6 mm for the FMS.

• (3)

  This section examines the synergistic gain effect of a double-break tandem connection. Thus, for an equivalent breaking duration of 2 ms, a double-break tandem connection with a total breaking distance of 6 mm exceeds the performance of a single-break configuration with a breaking distance of 6 mm. From the results of the arc breakdown criterion, the fracture capacity of the double fracture is 10% higher than that of the single fracture, which aligns with the findings discussed in this section. Considering economic and practical factors, the selection of double-break FMSs with a 6 mm total breaking distance (i.e. a 3-mm-breaking distance for each FMS) and an injected current frequency not exceeding 2500 Hz promotes the formation of sheath layers and lowers the metal vapour density in the gap. This contributes to the reduction of metal vapour density in the gap and aids the recovery of the post-arc medium in the CRCB vacuum interrupter.

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TABLE 7 Arc reignition criterion results for each resonant current frequency.

f (Hz)	n × d
2 ms × 5 mm	2 ms × 6 mm	2 ms × 6 mm
Single-break FMS	Double-break FMS
1000	1.03 × 1017	2.47 × 1016	2.26 × 1016
1500	2.62 × 1018	9.63 × 1017	8.86 × 1017
2000	1.68 × 1019	5.84 × 1018	5.42 × 1018
2500	3.02 × 1019	1.91 × 1019	1.74 × 1019
3000	5.35 × 1019	4.06 × 1019	3.64 × 1019

The joint model enables the determination of CRCBs with FMS design, optimal reverse current frequency range selection, and accurate current breaking limit calculation methods under superimposed circuit parameters.

CRCB BREAKING EXPERIMENT

Experimental system setup

To verify the performance of the FMS in the CRCB, a sub-module comprising a single FMS and a single module is utilised to create a prototype of a CRCB. This setup aims to verify correctness of the parameters of the FMS components.

Figure 21 shows the CRCB breaking experiment diagram. The primary objective of the experiment is to ascertain whether FMS K₁ possesses the capability to interrupt faulty high currents.

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The experiment control logic:

• (1)

  Initially, the FMS K₁ is closed, followed by closing the charging switch KM₁ to charge the test resonant capacitor bank C. Once the charging reaches the expected voltage value, KM₁ is disconnected. Subsequently, the start switch KM₂ is closed, allowing the test capacitor bank to discharge to the test circuit breaker through the test resonant reactor L, generating the discharge current.

• (2)

  Upon detecting that the amplitude of line current CT₁ reaches the expected value, issue a breaking command to the FMS, the high-speed camera start signal is triggered to start capturing the image of the breaking K₁ at the same time. After a delay of 2 ms, start the resonant switch to generate resonant current.

• (3)

  If the current CT₂ in the FMS remains zero continuously for 100 μs (below the set value of 180 A), the FMS is determined to extinguish the arc. Subsequently, the resonant switch is blocked when the resonant current passes zero. The test current automatically flows through the LC of the transfer branch to charge the capacitor until the arrester action consumes energy and the breaking is successful.

• (4)

  If within 2.5 cycles, the current in the FMS does not pass the zero point (less than 180 A for 100 consecutive us), the resonant switch will be blocked when the current passes the zero point after 2.5 resonant cycles. The FMS breaking fails.

Tables 8 and 9 present the experimental conditions and component parameters.

TABLE 8 Experiment parameters.

Item	Model specification parameters
Breaking energy storage capacitor	4 mF
Breaking energy storage capacitor voltage	700 V
Fibreglass tie bar quality	0.8 kg
Resonant capacitor C	140 μF
Resonant inductor L	26.4 μH
Submodule capacitance	18,000 μF

TABLE 9 Equipment parameters.

Item	Model specification parameters	Remarks
Charger	5 kV, 1 A	Constant current and voltage limiting charging
Charge switch KM1	10 kV, 630 A
Start switch KM2	10 kV, 630 A
Experiment resonant capacitor bank	30,000 μF,5 kV	7500 V/2.8 kV capacitor 2 series 8 parallel, 16 pcs.
Experiment resonant reactor	0.35 mH/50 kA
Line current CT2	Current 40 kA
Main circuit current CT1	Current 40 kA
Experiment lightning arrester F	3 pieces	YH25W-4/7.5 × 15
High-speed camera frame rate	8000

Experimental results

The CB breaking experiment was carried out according to the above experiment circuit. The experiment results of no-load and fault current were obtained. The result of high-speed camera shooting is shown in Figure 22.

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Image processing operations are required to analyse the motion of the repulsion disk. Inter-frame differencing is employed to extract the target contour of a moving marker point through a differencing operation between two successive frames in a video sequence. The absolute value of the pixel value differences at corresponding positions within the frames is calculated to analyse the motion characteristics of the object in the video.

After image processing, as shown in Figure 23, the average results of the no-load circuit breaking experiment and the motion results of the repulsion plate during the fault currents of 5, 6.4 and 18 kA circuit breaking experiments. The FMS K₁ applied in this experiment is a double-break structure with a total displacement of twice the results of the single-break structure. Table 10 presents the 2-ms displacement data.

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TABLE 10 Displacement of the FMS repulsion disk (2 ms).

Item	No-load (mean) experiment	Fault current 5 kA experiment	Fault current 6.4 kA experiment	Fault current 18 kA experiment
2 ms displacement single-break FMS (mm)	3.05	3.03	3.00	2.98
2 ms displacement double-break FMS (mm)	6.10	6.06	6.00	5.96

Comparing the results of no-load and loaded tests, there is a slight difference of 2.3% in the total displacement of the repulsive disk, and the electric force of the arc has a minor impact on the breaking of the FMS. Therefore, to demonstrate the impact of the characteristics of the arc inside the extinguishing chamber on the overall motion results of the FMS, FEJM can be used to more accurately reflect it.

To verify the correctness of the joint model, the following figure shows the comparison between the simulation model results and the experimental results. Table 11 shows the displacement of the repulsive disk at the moment of breaking.

TABLE 11 Comparison of displacement disconnection completion.

Item	Single-break FMS displacement (moment of breaking)	Double-break FMS displacement (moment of breaking)
Current 5 kA FEJM	4.087 mm (2.63 ms)	8.174 mm (2.63 ms)
Current 5 kA experiment	4.063 mm (2.63 ms)	8.126 mm (2.63 ms)
Current 6.4 kA FEJM	4.048 mm (2.47 ms)	8.096 mm (2.47 ms)
Current 6.4 kA experiment	4.039 mm (2.47 ms)	8.078 mm (2.47 ms)
Current 18 kA FEJM	4.025 mm (2.50 ms)	8.050 mm (2.50 ms)
Current 18 kA experiment	4.037 mm (2.50 ms)	8.074 mm (2.50 ms)

Figure 24 shows the correlation curves after the moment of test control logic (2). As shown in Figure 24, when disconnecting a fault current of 5 kA, the circuit breaker did not complete the disconnection at the first zero-crossing point. After half a resonance cycle, it reached the zero-crossing point again to complete the disconnection, with a disconnection time of 2.63 ms. When breaking the fault current of 6.4 kA, the injection current at the first zero crossing is utilised to achieve interruption, with a break time of approximately 2.47 ms. At this juncture, the interruption distances are 3.90 mm for the single-break FMS and 7.80 mm for the double-break FMS. When breaking the fault current of 18 kA, the injection current at the first zero crossing is utilised to achieve interruption, with a break time of approximately 2.51 ms. At this point, the interruption distances are 3.98 mm for the single-break FMS and 7.96 mm for the double-break FMS. When breaking the 5 kA fault current, due to the small fault current, the breaking distance condition for the first zero-crossing of the main branch current is not conducive to arc extinguishing due to the large slope of the superimposed current (di/dt) at this time. When the second zero-crossing occurs, the arc extinguishing and breaking are completed after the breaking distance reaches the condition. Therefore, to ensure the effectiveness of switch disconnection, it is necessary to obtain more accurate results of the motion process of the repulsive mechanism.

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Figure 25 shows the comparison of motion results between the APM and FEJM models. The APM construction process is an independent repulsive mechanism model that obtains the breaking characteristic data of the switch by using the motion result as the motion excitation of the independent arc extinguishing chamber's moving contact.

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Table 12 lists the motion results of the two models and the experimental results. The manual zero-crossing occurred 2 ms after the switch started to open. The root mean square errors (RMSE) between the two models and the experiment after 2 ms can be compared. Through comparison, it can be seen that the data error between APM and the experiment (RMSE_APM) is greater than that between FEJM and the experiment (RMSE_FEJM). The data obtained by APM is the result of independent calculations of various components of FMS, and the poor linkage of the whole machine results in the failure to reflect the energy loss caused by component collisions.

TABLE 12 Displacement root mean square error of repulsive disk.

Current (kA)	RMSEAPM	RMSEFEJM
5	0.529	0.263
6.4	0.391	0.205
18	0.452	0.251

Table 13 lists the reliability discrimination results of the model's breaking. Through comparison, it can be seen that the discrimination result n × d of APM in the 5 kA breaking condition deviates more from the critical state of arc reignition compared to FEJM, which is consistent with the experimental results. However, the error in APM arc reignition criterion results is due to the lack of contact collision loss and the influence of arc electric force, resulting in a larger breaking distance, which will cause misjudgement on the success or failure of CRCB breaking and is not conducive to the pre-operation investigation and prediction of the whole machine.

TABLE 13 Arc reignition criterion results for each fault current.

Current (kA)	n × d (APM) double-break FMS (m−2)	n × d (FEJM) double-break FMS (m−2)
5	2.21 × 1019	2.89 × 1019
6.4	1.93 × 1019	2.35 × 1019
18	1.84 × 1019	1.97 × 1019

The parameters of the mechanical switch-related components of the circuit breaker-breaking test are based on the parameter selection method in Section 4, and the joint model simulation results are obtained to be consistent with the test results, and the circuit breaker successfully breaks the 18 kA high current.

FEJM provides comprehensive technical support for effectively reflecting the stress issues of core components during the breaking process of FMS and can provide accurate theoretical references for the optimisation design of mechanical motion components in FMS. It also accurately represents the arc extinguishing process during the breaking of FMS and provides a convenient method for the selection and design of circuit parameters for the entire circuit breaker.

In the experiments, the CRCB switched off within 3 ms. These experiments demonstrate the fast breaking capacity of the developed CB and FMS, which is essential for the control and protection of DC systems.

CONCLUSION

In this paper, the breaking process of FMS in a CRCB is investigated by experiment and simulation. The repulsion mechanism in the FMS and the arc characteristics in the interrupter chamber are analysed in detail. The following main conclusions are drawn.

• 1)

  By establishing a two-dimensional simulation model of the repulsion mechanism and conducting simulation analyses of different material component stresses, it is evident that the resin material electromagnetic repulsion mechanism generates excessive impact force, leading to component damage. Conversely, the use of high-strength alloy steel to reduce the repulsion disk-moving distance and the mechanical switch breaking distance is not conducive to the FMS breaking. Therefore, fibreglass insulated ties with a material quality between resin and steel and sufficient material strength were selected. Effectively reflecting the stress issues of core components during the breaking process of FMS, providing comprehensive technical support and accurate theoretical references for the optimisation design of mechanical motion components in FMS.

• 2)

  A full-cycle MHP arc joint model was established under different superimposed currents, reflecting the arc motion process through the results of particle motion. The results demonstrate that, under comprehensive considerations of economy and reliability, increasing the contact breaking distance of the double-break FMS to 6 mm and above and referencing the superimposed current frequency within 2500 Hz accelerate the sheath layer development speed and reduce the metal vapour density in the gap. This is beneficial to the post-arc medium recovery process of the vacuum interrupter chamber of the CRCB. The accurate representation of the arc extinguishing process during the FMS breaking provides a convenient method for the selection and design of circuit parameters for the entire CB.

• 3)

  Both experimental and numerical results indicate that the experimental study of the arc-firing process of FMSs, based on the results of the above switch optimisation design method, provides certain simulation methods and theoretical references for the optimal design of FMSs. At the same time, it also reflects the accuracy of FEJM in the CRCB breaking process, confirming that FEJM has the reliability to predict the overall operation results of CRCB. The control and protection of DC systems are crucial.

ACKNOWLEDGEMENTS

This work was supported by State Grid Corporation Headquarters Science and Technology Project (5500-20220110A-1-1-ZN).

CONFLICT OF INTEREST STATEMENT

The authors declare no conflicts of interest.

DATA AVAILABILITY STATEMENT

The data that support the findings of this study are available from the corresponding author upon reasonable request.

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