Dynamic Model of Medium Voltage Vacuum Circuit Breaker and Induction Motor for Switching Transients Simulation Using Clark Transformation

Abstract

A derivation of the dynamic model of a medium voltage vacuum circuit breaker and induction motor in space vectors in coordinates $\alpha \beta 0$ allow us to model switching transients in various dynamic states of the motor. In the case of the Clark transformation, the corresponding numerical integration technique can be selected including variable time-step integration techniques to avoid numerical instabilities due to the stiffness of the system. Assymetrical operations such as switching cause the power system to become unbalanced and the transformed equations $\alpha$, $\beta ,$ and $0$ are not uncoupled. Therefore, it is necessary to derive a coupling matrix between circuit breaker voltages and currents in the coordinate system $\alpha \beta 0$. The subject of our interest is switching overvoltages that arise when turning off small inductive currents by a vacuum circuit breaker. When deriving the model of a vacuum circuit breaker, all its properties encountered during this action are taken into account, i.e., current chop, virtual current chop, dielectric barrier in the circuit breaker and its recovery rate, and the ability of the vacuum circuit breaker to extinguish high frequency currents. Simulation results are compared with the measured results on a medium voltage motor as well as with the simulation results of the mathematical model of the test circuit according to IEC 62271-110 resolved using the nodal method (EMTP algorithm). Models are implemented in the MATLAB/Simulink programming environment.

1. Introduction

The nodal method is nowadays used for the simulation of electromagnetic transients like medium voltage motor switching by vacuum circuit breaker. The nodal equations are assembled after discretizing all circuit devices with a trapeziodal method of numerical integration (EMTP algorithm). In the case of the simulation of motor switching, the motor is represented by a parallel $RLC$ circuit or Universal Machine model in the ATP-EMTP program. Literature concerning the numerical simulations of transients originating from motor switching is vast [1,2,3,4,5,6,7,8].

In [1,2,3,4,5] is an equivalent circuit of the motor shown in Figure 1. In Figure 1, a test circuit is shown for lab tests, which is defined in standard IEC 62271-110 ed. 4. that specifies the requirements for medium voltage circuit breakers that are used for switching medium voltage motors. In Figure 1, $R_{GN}$ is the resistance between the grid node and ground, $u\left(t\right)$ is the supply voltage, $L_S$ is the grid-side inductance, $L_{B1}$ is the inductance of the capacitors and connection lines, $C_S$ is the grid-side capacitance, $L_{B2}$ is the inductance of the connection lines, $R_{CB}$ is the resistence of the circuit breaker, and $R_{CBP}$ is the parasitic resistance of the circuit breaker, $L_{CBP}$ is the parasitic inductance of the circuit breaker, $C_{CBP}$ is the parasitic capacitance of the circuit breaker, $C_{PI\_S}$ is the capacitance of the cable line (supply-side) ($C_{PI\_S}=C_L/2$), $R_L$ is the cable line resistance, $L_L$ is the inductance of the cable line, $C_{PI\_L}$ is the capacitance of the cable line (load-side) ($C_{PI\_L}=C_L/2$), $C_L$ is the cable line capacitance, $C_P$ is the equivalent parallel capacitance of the motor, $L$ is the equivalent inductance of the motor, $R$ is the equivalent resistance of the motor, and $R_P$ is the equivalent parallel resistance of the motor. By changing the circuit parameters $R$, $L$ of the equivalent connection of the induction motor, we can define its various operating states characterized by the size of the current and power factor.

In [6,7,8], it is an equivalent circuit of the motor shown in Figure 2. The test circuit in Figure 1 is derived from this equivalent circuit of the motor. In Figure 2, $R_G$ is the grid-side resistance, $L_G$ is the grid-side inductance, $C_G$ is the grid-side capacitance, $R_S$ is the stator winding resistance, $L_{S\sigma }$ is the stray inductance of the stator winding, $L_m$ is the magnetizing inductance, $R_R$ is the resistance of the rotor winding (converted to the stator), $L_{R\sigma }$ is the stray inductance of the rotor winding (converted to the stator), and $s$ is the rotor slip. By changing the circuit parameters of the equivalent connection of the induction motor, we can define its various operating states characterized by the size of the current, power factor, transient recovery voltage ($TRV$), and frequency of the high frequency current.

The equivalent circuit of the motor in Figure 2 is derived from the dynamic (mathematical) model of induction motor in Figure 3, where $C_M$ is the motor capacitance, $R_{SN}$ and $R_{RN}$ are the resistance between the stator and rotor winding node, respectively, and the ground. The EMTP programs use the Universal Machine model or a current injection into the network to represent the induction motor shown in Figure 3 [4,10,11]. In this article, the Clark transformation (space vector transformation) is used for power systems modelling in Figure 3 in steady-state and to analyze transients [12,13]. In the case of the Clark transformation, the corresponding numerical integration technique can be selected, including variable time-step integration techniques to avoid numerical instabilities due to the stiffness of the system [14]. Clark transformation, or any other modal transformation, decouples symmetric power system equations through matrix diagonalization. Asymmetrical operations such as switching cause the power system to become unbalanced and the transformed equations $\alpha$, $\beta$, and $0$ are not uncoupled [15]. Therefore, the straightforward use of the Clark transformation is prevented, and to solve power systems involving switching device, it is necessary to derive a coupling matrix between the circuit breaker voltage and current in the coordinate system $\alpha \beta 0$. Values in the coupling matrix depend on resistances values in the circuit breaker phases. Finally, the results of the solved equations are inversed, transformed into the three-phase system $abc$.

The main goal of this article is the derivation of a dynamic model in space vectors for the simulation of a vacuum breaker switched medium voltage motor. The subject of our interest will be the switching overvoltages that arise when turning off small inductive currents by a vacuum circuit breaker, therefore, when deriving the model of a vacuum circuit breaker, all its properties encountered during this action will be taken into account, i.e., current chop, virtual current chop, dielectric barrier in the circuit breaker and its recovery rate, and the ability of the vacuum circuit breaker to extinguish high frequency currents [6,7,8,16]. Simulations results will be compared with the measured results on a medium voltage motor as well as with the simulation results of the mathematical model of the test circuit according to IEC 62271-110 resolved using the EMTP algorithm in three cases: interruption during normal loading, interruption during inrush current without reignition, and interruption during inrush current with multiple reignition.

This paper is organized as follows. In Section 2, a dynamic model for the simulation of switching transients is derived, involving derivation of state equations of the electrical grid, cable line, and induction motor. In Section 3, a numerical solution of the dynamic model of medium voltage vacuum circuit breaker and motor for a simulation of switching transients using the Clark transformation is discussed regarding a stiff system of ordinary differential equations. In Section 4, a nodal analysis of the test circuit (equivalent motor circuit) according to IEC 62271-110 ed. 4. and a numerical solution of nodal equations are shown. In Section 5, the state and nodal equations are parametrized by specific distribution system data and a comparison of the simulation results with measured results is discussed. Conclusions are shown in Section 6.

2. Derivation of a Dynamic Model for the Simulation of Switching Transients

When deriving a dynamic model in space vectors, we start from the circuit diagram in Figure 3. We compiled the equations using the method of loop currents, which we transformed into the $\alpha \beta 0$ coordinate system.

The Park transformation of general variables in phases $a, b,c$ from a three-phase system (coordinate system $abc$) into a general rotating coordinate system $b$, rotating at angular velocity ${\omega }_b=\frac{d{\theta }_b\left(t\right)}{dt}$, where ${\theta }_b\left(t\right)$ is the time-varying rotation angle of the coordinate system $b$ and $t$ is time, with a zero (homopolar) component, is defined by the equation [12]

$${\mathit{x}}_{dq0}\left(t\right)={\mathit{T}}_{dq0}\left({\theta }_b\right) · {\mathit{x}}_{abc}\left(t\right),$$

(1)

where

$${\mathit{T}}_{dq0}\left({\theta }_b\right)=\frac{2}{3}\left[\begin{array}{ccc}\cos \left(-{\theta }_b\right)& \cos \left(-{\theta }_b+\frac{2\pi }{3}\right)& \cos \left(-{\theta }_b-\frac{2\pi }{3}\right)\\ \sin \left(-{\theta }_b\right)& \sin \left(-{\theta }_b+\frac{2\pi }{3}\right)& \sin \left(-{\theta }_b-\frac{2\pi }{3}\right)\\ \frac{1}{2}& \frac{1}{2}& \frac{1}{2}\end{array}\right],$$

(2)

where ${\mathit{x}}_{abc}\left(t\right)$ is a vector of general phase variables $a, b,c$, ${\mathit{x}}_{dq0}\left(t\right)$ is a vector of transformed variables in components $d, q,0$, and ${\mathit{T}}_{dq0}\left({\theta }_b\right)$ is a transformation matrix $dq0$. For the inverse Park transformation, then

$${\mathit{x}}_{abc}\left(t\right)={\mathit{T}}_{abc}\left({\theta }_b\right) · {\mathit{x}}_{dq0}\left(t\right),$$

(3)

where

$${\mathit{T}}_{abc}\left({\theta }_b\right)=\left[\begin{array}{ccc}\cos \left(-{\theta }_b\right)& \sin \left(-{\theta }_b\right)& 1\\ \cos \left(-{\theta }_b+\frac{2\pi }{3}\right)& \sin \left(-{\theta }_b+\frac{2\pi }{3}\right)& 1\\ \cos \left(-{\theta }_b-\frac{2\pi }{3}\right)& \sin \left(-{\theta }_b-\frac{2\pi }{3}\right)& 1\end{array}\right],$$

(4)

where ${\mathit{T}}_{abc}\left({\theta }_b\right)$ is an inverse transformation matrix. Clark transformation is the special case of the Park transformation into the coordinate system $dq0$ rotating at angular velocity ${\omega }_b=0$ (${\theta }_b=0$, stationary system) [17]. From relation (1) results for Clark transformation:

$${\mathit{x}}_{\alpha \beta 0}\left(t\right)={\mathit{T}}_{dq0}\left(0\right) · {\mathit{x}}_{abc}\left(t\right),$$

(5)

where ${\mathit{x}}_{\alpha \beta 0}\left(t\right)$ is a vector of transformed variables in components $\alpha , \beta , 0$. For the inverse Clark transformation from the coordinate system $\alpha \beta 0$ into a three-phase system $abc$, from relation (3), we obtain the following:

$${\mathit{x}}_{abc}\left(t\right)={\mathit{T}}_{abc}\left(0\right) · {\mathit{x}}_{\alpha \beta 0}\left(t\right).$$

(6)

Furthermore, when transforming into a coordinate system $\alpha \beta 0$, $x_{\alpha }=x_a$, i.e., the value of the general variable in phase $a$ is equal to the value of the transformed variable in the component $\alpha$. The corresponding space vector of the transformed general variable in components $\alpha , \beta$ is defined as follows:

$$x^{-S}\left(t\right)=x_{\alpha }\left(t\right)+j{x}_{\beta }\left(t\right).$$

(7)

In the following text, space vectors in a general rotating coordinate system $b$/ or in the $\alpha \beta 0$ coordinate system will be marked with letter $-b$/ or $-S$ in superscript.

2.1. Dynamic Model of Electrical Grid

The electrical grid dynamic model, written in space vectors, in a general rotating coordinate system $b$, with a zero component, according to the diagram in Figure 3, has the following form:

$$\frac{d}{dt}{\mathit{x}}_G\left(t\right)={\mathit{A}}_G\left({\omega }_b\right){\mathit{x}}_G\left(t\right)+{\mathit{B}}_G,$$

(8)

where

$${\mathit{x}}_G\left(t\right)={\left[i_G^{-b}\left(t\right),i_{G0}\left(t\right),u_{C_G}^{-b}\left(t\right),u_{C_{G}0}\left(t\right)\right]}^{\mathit{T}},$$

(9)

$${\mathit{A}}_G\left({\omega }_b\right)=\left[\begin{array}{cccc}-\frac{R_G}{L_G}-j{\omega }_b& 0& -\frac{1}{L_G}& 0\\ 0& -\frac{\left(R_G+3R_{GN}\right)}{L_G}& 0& -\frac{1}{L_G}\\ \frac{1}{C_{G }}& 0& -j{\omega }_b& 0\\ 0& \frac{1}{C_{G }}& 0& 0\end{array}\right],$$

(10)

$${\mathit{B}}_G={\left[\frac{u^{-b}\left(t\right)}{L_G},\frac{u_0\left(t\right)}{L_G},-\frac{i_B^{-b}\left(t\right)}{C_G},-\frac{i_{B0}\left(t\right)}{C_{G }} \right]}^{\mathit{T}},$$

(11)

where ${\mathit{x}}_G\left(t\right)$ is the column vector of state variables, ${\mathit{A}}_G\left({\omega }_b\right)$ is the electrical grid system dynamics matrix, ${\mathit{B}}_G$ is the electrical grid inputs matrix, $i_G^{-b}\left(t\right)$ is the space vector of the electrical grid current, $i_{G0}\left(t\right)$ is the zero component of the electrical grid current, $u_{C_G}^{-b}\left(t\right)$ is the space vector of voltage at the electrical grid connection point (space vector of voltage at the electrical grid capacity $C_G$), $u_{C_{G}0}\left(t\right)$ is the zero voltage component at the electrical grid connection point (zero voltage component at the electrical grid capacitance $C_G$), $u^{-b}\left(t\right)$ is the space vector of the supply voltage, $u_0\left(t\right)$ is the zero component of the supply voltage, $i_B^{-b}\left(t\right)$ is the space vector of the connection line current, and $i_{B0}\left(t\right)$ is the zero component of the connection line current. By transforming the state Equation (8) into the coordinate system $\alpha \beta 0$, we obtain the following state equation:

$$\frac{d}{dt}{\mathit{x}}_G\left(t\right)={\mathit{A}}_G\left(0\right){\mathit{x}}_G\left(t\right)+{\mathit{B}}_G.$$

(12)

During transformation, equations of the zero component do not change.

2.2. Dynamic Model of Medium Voltage Vacuum Circuit Breaker and Connection Line

The medium voltage circuit breaker and connection line dynamic model, written in space vectors, in a general rotating coordinate system $b$, with a zero component, according to the diagram in Figure 3, has the following form:

$$\frac{d}{dt}{\mathit{x}}_{BVCB}\left(t\right)={\mathit{A}}_{BVCB}\left({\omega }_b\right){\mathit{x}}_{BVCB}\left(t\right)+{\mathit{B}}_{BVCB},$$

(13)

where

$${\mathit{x}}_{BVCB}\left(t\right)={\left[\begin{array}{cccccccc}{i}_B^{-b}\left(t\right),& i_{B0}\left(t\right),& u_{C_{PI\_S}}^{-b}\left(t\right),& u_{C_{PI\_S}0}\left(t\right),& i_p^{-b}\left(t\right),& i_{P0}\left(t\right),& u_{C_{CBP}}^{-b}\left(t\right),& u_{C_{CBP}0}\left(t\right)\end{array}\right]}^{\mathit{T}},$$

(14)

$${\mathit{A}}_{BVCB}\left({\omega }_b\right)=\left[\begin{array}{cccccccc}-j{\omega }_b& 0& -\frac{1}{L_{B2}}& 0& 0& 0& 0& 0\\ 0& 0& 0& -\frac{1}{L_{B2}}& 0& 0& 0& 0\\ \frac{1}{C_L}& 0& -j{\omega }_b& 0& 0& 0& 0& 0\\ 0& \frac{1}{C_L}& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& -\frac{R_{CBP}}{L_{CBP}}-j{\omega }_b& 0& -\frac{1}{L_{CBP}}& 0\\ 0& 0& 0& 0& 0& -\frac{R_{CBP}}{L_{CBP}}& 0& -\frac{1}{L_{CBP}}\\ 0& 0& 0& 0& \frac{1}{C_{CBP}}& 0& -j{\omega }_b& 0\\ 0& 0& 0& 0& 0& \frac{1}{C_{CBP}}& 0& 0\end{array}\right],$$

(15)

$${\mathit{B}}_{BVCB}={\left[\begin{array}{cccccccc}\frac{u_{C_G}^{-b}\left(t\right)-u_{CB}^{-b}\left(t\right)}{L_{B2}},& \frac{u_{C_{G}0}\left(t\right)-u_{CB0}\left(t\right)}{L_{B2}},& -\frac{i_L^{-b}\left(t\right)}{C_L},& -\frac{ i_{L0}\left(t\right)}{C_L},& \frac{u_{CB}^{-b}\left(t\right)}{L_{CBP}},& \frac{u_{CB0}\left(t\right)}{L_{CBP}},& 0,& 0\end{array}\right]}^{\mathit{T}},$$

(16)

where ${\mathit{x}}_{BVCB}\left(t\right)$ is the column vector of state variables, ${\mathit{A}}_{BVCB}\left({\omega }_b\right)$ is the medium voltage vacuum circuit breaker and the connection line system dynamics matrix, ${\mathit{B}}_{BVCB}$ is the medium voltage vacuum circuit breaker and the connection line inputs matrix, $u_{C_{PI\_S}}^{-b}\left(t\right)$ is the space vector of the voltage of the cable line at the point of connection of the cable line (the space vector of voltage at the capacitance of the cable line $C_{PI\_S}$ (supply-side)), $u_{C_{PI\_S}0}\left(t\right)$ is the zero voltage component of the cable line at the cable line connection point (the zero component of the voltage at the capacitance of the cable line $C_{PI\_S}$ (supply-side)), $i_P^{-b}\left(t\right)$ is the space vector of the parasitic current of the circuit breaker, $i_{P0}\left(t\right)$ is the zero component of the parasitic current of the circuit breaker, $u_{C_{CBP}}^{-b}\left(t\right)$ is the space vector of the voltage at the parasitic capacitance of the circuit breaker $C_{CBP}$, $u_{C_{CBP}0}\left(t\right)$ is the zero component of the voltage at the parasitic capacitance of the circuit breaker $C_{CBP}$, $u_{CB}^{-b}\left(t\right)$ is the space vector of the voltage on the circuit breaker, $u_{CB0}\left(t\right)$ is the zero component of the voltage on the circuit breaker, $i_L^{-b}\left(t\right)$ is the space vector of the cable line current, and $i_{L0}\left(t\right)$ is the zero component of the cable line current. By transforming the state Equation (13) into the coordinate system $\alpha \beta 0$, we obtain the following state equation:

$$\frac{d}{dt}{\mathit{x}}_{BVCB}\left(t\right)={\mathit{A}}_{BVCB}\left(0\right){\mathit{x}}_{BVCB}\left(t\right)+{\mathit{B}}_{BVCB}.$$

(17)

For the voltage on the circuit breaker, ${\mathit{u}}_{C{B}_{\alpha \beta 0}}\left(t\right)$ as a function of circuit breaker current ${\mathit{i}}_{C{B}_{\alpha \beta 0}}\left(t\right)={\mathit{i}}_{B_{\alpha \beta 0}}\left(t\right)-{\mathit{i}}_{P_{\alpha \beta 0}}\left(t\right)$, in the coordinate system $\alpha \beta 0$ we derive the following relationship that shows coupling among $\alpha , \beta , 0$ components during assymetrical transients (circuit breaker switching):

$${\mathit{u}}_{C{B}_{\alpha \beta 0}}\left(t\right) =\left[\begin{array}{ccc}\frac{2}{3}{R}_a+\frac{1}{6}{R}_b+\frac{1}{6}{R}_c& -\frac{\sqrt{3}}{6}{R}_b+\frac{\sqrt{3}}{6}{R}_c& \frac{2}{3}{R}_a-\frac{1}{3}{R}_b-\frac{1}{3}{R}_c\\ -\frac{\sqrt{3}}{6}{R}_b+\frac{\sqrt{3}}{6}{R}_c& \frac{1}{2}{R}_b+\frac{1}{2}{R}_c& \frac{\sqrt{3}}{3}{R}_b-\frac{\sqrt{3}}{3}{R}_c\\ \frac{1}{3}{R}_a-\frac{1}{6}{R}_b-\frac{1}{6}{R}_c& \frac{\sqrt{3}}{6}{R}_b-\frac{\sqrt{3}}{6}{R}_c& \frac{1}{3}{R}_a+\frac{1}{3}{R}_b+\frac{1}{3}{R}_c\end{array}\right]{\mathit{i}}_{C{B}_{\alpha \beta 0}}\left(t\right),$$

(18)

where $R_a$ and $R_b$ and $R_c$ are the resistance of the circuit breaker at phase $a$ and $b$ and $c$, respectively. In this paper, we focus on turning off small inductive currents with a vacuum circuit breaker. The magnitude of the resistance of the vacuum circuit breaker ($cbPhResistance=R_X=R_{CB\_ON}$ (phase x of the circuit breaker is on, variable x representing phase a, b, or c), $cbPhResistance=R_X=R_{CB\_OFF}$ (phase x of the circuit breaker is off)) in the various phases is then controlled by the algorithm below, implementing the physical processes that occur when turning off small inductive currents; the algorithm is further extended by the possibility of turning off by a current chop in the oncoming half-period as far as the turning off was not successful [1,7,8].

At the point in time $time\_open$ of moving the circuit breaker contacts apart, see line 1 in Figure 4, an electric arc will occur due to the existence of inductances in the turned-off electric circuit. This state will last until the current is chopped by the circuit breaker, i.e., the current is interrupted before the current passes through its natural zero. The magnitude of the chopped current ($I_{chop}$) is a random variable having a normal distribution with a relative standard deviation equal to $15\%$, whose mean value, for a grid frequency of $50 \mathrm{Hz}$, can be estimated using the relation [16]:

$$\overline{I_{chop}}={\left(\omega \cdot \widehat{i }\cdot \alpha \cdot \beta \right)}^q,$$

(19)

where $\omega$ is the angular velocity of the grid frequency and $\widehat{i }$ is the load current amplitude. Parameters $\alpha ,\beta , q$ are a function of the material, of which the circuit breaker contacts are made. In the case of modern vacuum circuit breakers, whose contacts are made of a copper-chromium alloy, $\alpha =6.2\times {10}^{-16} s,\beta =14.3, q={\left(1-\beta \right)}^{-1}$ applies for their values. By moving the contacts of the circuit breaker apart, a dielectric barrier is formed between them, whose breakdown voltage ($U_b$) is a random variable having a normal distribution with a relative standard deviation equal to $15\%$, whose mean value can be estimated using the relation [1,8] (the cold gap breakdown is considered in this work):

$$U_B=A\left(t-t_{OPEN}\right)+B,$$

(20)

where $A$ or $B$, is the dielectric barrier parameter, whose unit is $\mathrm{V}/\mathsf{\mu }\mathrm{s}$ or $\mathrm{V}$, respectively. After the arc is extinguished, at the moment in time $time\_chop$, when the resulting circuit breaker current ($i_{B_x}\left(t\right)$) drops below the level $I_{chop}$, see line 18 in Figure 4, a $TRV$ will start to rise between the circuit breaker contacts. Depending on its magnitude and the magnitude of the breakdown voltage ($U_b$), in a moment of time $time\_reignition$, a breakdown of the incepting dielectric barrier may occur and thus the arc will be reignited, see line 13 in Figure 4. The resulting high frequency arc current will be extinguished again, in a moment of time $time\_noArc$, when the arc current passes through zero, and its time derivative at this moment will be less than the value calculated using the relation [1,8]:

$$\frac{di\left(t\right)}{dt}=C\left(t-t_{OPEN}\right)+D,$$

(21)

where $C$ and $D$ is a parameter, whose unit is $\mathrm{A}/{\mathsf{\mu }\mathrm{s}}^2$ and $\mathrm{A}/\mathsf{\mu }\mathrm{s}$, respectively, characterizing the circuit breaker’s ability to extinguish high frequency currents, see lines 4, 5, 6 in Figure 4. The value calculated by (21) is the mean value of the random variable of the vacuum circuit breaker quenching capability having a normal distribution with a relative standard deviation equal to $15\%$. If—after the interruption of the current at the moment in time $time\_noArc$—a breakdown (reignition) occurs again, see line 13 in Figure 4, the process described above will be repeated until the magnitude of the breakdown voltage ($U_b$) of the dielectric between the circuit breaker contacts is permanently greater than the resulting $TRV$. The vacuum circuit breaker model for turning off small inductive currents, implemented in the algorithm in Figure 4, is a deterministic model, and for parameter values $\overline{I_{chop}}, A, B, C$, and $D$ we will choose their mean values in the algorithm in Figure 4 [18]. Typical parameter values $A, B, C$, and $D$, see [7,8].

2.3. Dynamic Model of Cable

The cable line dynamic model, written in space vectors, in a general rotating coordinate system $b$, with a zero component, according to the diagram in Figure 3, has the following form:

$$\frac{d}{dt}{\mathit{x}}_{PI}\left(t\right)={\mathit{A}}_{PI}\left({\omega }_b\right){\mathit{x}}_{PI}\left(t\right)+{\mathit{B}}_{PI},$$

(22)

where

$${\mathit{x}}_{PI}\left(t\right)={\left[i_L^{-b}\left(t\right),i_{L0}\left(t\right),u_{C_M}^{-b}\left(t\right),u_{C_{M}0}\left(t\right) \right]}^{\mathit{T}},$$

(23)

$${\mathit{A}}_{PI}\left({\omega }_b\right)=\left[\begin{array}{cccc}-\frac{R_L}{L_L}-j{\omega }_b& 0& -\frac{1}{L_L}& 0\\ 0& -\frac{R_L}{L_L}& 0& -\frac{1}{L_L}\\ \frac{1}{C_{PI\_L}+C_M}& 0& -j{\omega }_b& 0\\ 0& \frac{1}{C_{PI\_L}+C_M}& 0& 0\end{array}\right],$$

(24)

$${\mathit{B}}_{PI}={\left[\begin{array}{cccc}\frac{u_{C_{PI\_S}}^{-b}\left(t\right)}{L_L},& \frac{u_{C_{PI\_S}0}\left(t\right)}{L_L},& -\frac{i_S^{-b}\left(t\right)}{C_{PI\_L}+C_M},& -\frac{ i_{S0}\left(t\right)}{C_{PI\_L}+C_M}\end{array}\right]}^{\mathit{T}},$$

(25)

where ${\mathit{x}}_{PI}\left(t\right)$ is the column vector of state variables, ${\mathit{A}}_{PI}\left({\omega }_b\right)$ is the cable line system dynamics matrix, ${\mathit{B}}_{PI}$ is the cable line inputs matrix, $u_{C_M}^{-b}\left(t\right)$ is the space vector of the voltage of the cable line at the point of connection of the load (motor) (the space vector of the voltage at the capacitance of the cable line $C_{PI\_L}$ (load-side) and capacitance of the motor $C_M$, respectively), $u_{C_{M}0}\left(t\right)$ is the zero voltage component of the cable line at the load (motor) connection point (the zero component of the voltage at the capacitance of the cable line $C_{PI\_L}$ (load-side) and capacitance of the motor $C_M$, respectively), $i_S^{-b}\left(t\right)$ is the space vector of the stator current, and $i_{S0}\left(t\right)$ is the zero component of the stator current. By transforming the state Equation (22) into the coordinate system $\alpha \beta 0$, we obtain the following state equation:

$$\frac{d}{dt}{\mathit{x}}_{PI}\left(t\right)={\mathit{A}}_{PI}\left(0\right){\mathit{x}}_{PI}\left(t\right)+{\mathit{B}}_{PI}.$$

(26)

For the simplification in the derivation of the cable line dynamic model in this section, we used one PI model. In the case study in Section 5, we use five PI models to represent the cable line shown in Figure 3. This travelling wave model is accurate to one frequency; we tuned this model to the main frequency associated to a switching transient.

2.4. Dynamic Model of Induction Motor

Neglecting the effects of magnetic saturation, eddy currents, non-sinusoidal magnetomotive force distribution, and slotting, with the assumption that rotor parameters and variables are recalculated to the stator side, the induction motor dynamic model, written in space vectors, in a general rotating coordinate system b with a zero component, according to the diagram in Figure 3, has the following form:

$$\frac{d}{dt}{\mathit{x}}_{M,\Psi }\left(t\right)={\mathit{A}}_M\left({\omega }_b\right){\mathit{x}}_{M,i}\left(t\right)+{\mathit{B}}_M,$$

(27)

where

$${\mathit{x}}_{M,\Psi }\left(t\right)={\left[ {\Psi }_S^{-b}\left(t\right),{\Psi }_{S0}\left(t\right),{\Psi }_R^{-b}\left(t\right),{\Psi }_{R0}\left(t\right)\right]}^{\mathit{T}},$$

(28)

$${\mathit{x}}_{M,i}\left(t\right)={\left[\begin{array}{cccc}{i}_S^{-b}\left(t\right),& i_{S0}\left(t\right),& i_R^{-b}\left(t\right),& i_{R0}\left(t\right)\end{array}\right]}^{\mathit{T}},$$

(29)

$${\mathit{A}}_M\left({\omega }_b\right)=\left[\begin{array}{cccc}-R_S-j{\omega }_b{L}_S& 0& -j{\omega }_b{M}_M& 0\\ 0& -\left(R_S+3R_{SN}\right)& 0& 0\\ -j\Delta \omega M_M& 0& -R_R-j\Delta \omega L_R& 0\\ 0& 0& 0& -\left(R_R+3R_{RN}\right)\end{array}\right],$$

(30)

$${\mathit{B}}_M\left(t\right)={\left[\begin{array}{cccc}{u}_{C_M}^{-b}\left(t\right),& u_{C_{M}0}\left(t\right),& 0,& 0\end{array}\right]}^{\mathit{T}},$$

(31)

where ${\Psi }_S^{-b}\left(t\right)=\left(L_{\sigma S}+L_M\right)i_S^{-b}\left(t\right)+M_M{i}_R^{-b}\left(t\right)$ is the stator flux-linkage space vector, ${\Psi }_{S0}\left(t\right)$ is the zero component of the stator flux-linkage, ${\Psi }_R^{-b}\left(t\right)=\left(L_{\sigma R}+L_M\right)i_R^{-b}\left(t\right)+M_M{i}_S^{-b}\left(t\right)$ is the rotor flux-linkage space vector, ${\Psi }_{R0}\left(t\right)$ is the zero component of the rotor flux-linkage, $\Delta \omega ={\omega }_b-{\omega }_{er}\left(t\right)$, ${\omega }_{er}\left(t\right)$ is the angular velocity of the rotor in electrical degrees, $L_S=L_{\sigma S}+L_M$, $L_R=L_{\sigma R}+L_M$, ${\mathit{x}}_{M,\Psi }\left(t\right)$ and ${\mathit{x}}_{M,i}\left(t\right)$ are the column vectors of state variables, ${\mathit{A}}_M\left({\omega }_b\right)$ is the induction motor system dynamics matrix, ${\mathit{B}}_M\left(t\right)$ is induction motor inputs matrix, $M_m$ is the mutual inductance between stator and rotor windings, while furthermore $L_m=M_m$ applies; $i_R^{-b}\left(t\right)$ is the space vector of the rotor current, and $i_{R0}\left(t\right)$ is the zero component of the rotor current. By transforming the induction motor state model Equation (27) into a system of axes rigidly tied to the stator ($\alpha \beta 0$), we obtain the following state equation:

$$\frac{d}{dt}{\mathit{x}}_{M,\Psi }\left(t\right)={\mathit{A}}_M\left(0\right){\mathit{x}}_{M,i}\left(t\right)+{\mathit{B}}_M.$$

(32)

The motor torque is calculated according to the following relation:

$$T=\frac{3}{2}{p}_p{L}_m\left(i_{R\alpha }^{-S}{i}_{S\beta }^{-S}-i_{R\beta }^{-S}{i}_{S\alpha }^{-S}\right),$$

(33)

where $p_p$ is the number of motor pole pairs, $i_{S\alpha }^{-S}$ is the α-component of the stator current, $i_{S\beta }^{-S}$ is the β-component of the stator current, $i_{R\alpha }^{-S}$ is the α-component of the rotor current, $i_{R\beta }^{-S}$ is the β-component of the rotor current. By adding the motion equation of the drive

$$\frac{d{\omega }_{er}}{dt}=\frac{p_p}{J}\left(T-T_L\right),$$

(34)

where $J$ is the moment of inertia and $T_L$ is the load torque, we obtain a complete dynamic model of the induction motor written in the form of space vectors in the coordinate system $\alpha \beta 0$, which is formed by the system of Equations (32)–(34).

3. Numerical Solution of a Dynamic Model of Medium Voltage Vacuum Circuit Breaker and Motor for a Simulation of Switching Transients Using Clark Transformation

The dynamic model for the simulation of switching transients derived in the previous chapter formed by Equations (12), (17), (26), (32)–(34) written in the form of space vectors is in components $\alpha , \beta , 0$ represented by a system of 32 (56 for five PI elements to represent cable in Figure 3; also see Section 2.3) ordinary differential equations. The given system represents a stiff system of ordinary differential equations, for the solution of which we will use Rosenbrock’s (A-stable) method (ODE23s) [14]. The final version of the mathematical model was implemented in the MATLAB/Simulink program environment.

4. Mathematical Model for the Simulation of Switching Transients according to IEC 62271-110 ed. 4. and Its Numerical Solution

We will compare the simulation results of the dynamic model derived in Part 2 not only with the measured results on a motor, but also with the simulation results of the mathematical model of the test circuit (equivalent motor circuit) according to IEC 62271-110 ed. 4. The numerical solution of the mathematical model of the test circuit will be performed using the EMTP algorithm. The algorithm is based on the transformation of differential circuit equations of passive dipoles, inductor, and capacitor, into algebraic equations using the trapezoidal method of numerical integration [10]. We will then compile the circuit equations for the given circuit using the nodal voltage method and calculate the voltage in grid nodes for the selected time-step of the calculation. In our case, the mathematical model of the test circuit is represented by a system of 32 (56 for five PI elements to represent cable in Figure 1; also see Section 2.3) algebraic equations. Similar to the above, the mathematical model was implemented in the MATLAB program environment. In this case, it is also possible to use one of the ATP-EMTP, PSCAD/EMTDC, etc., program packages.

5. Case Study

The subject of the case study is switching overvoltages when turning off the motor $IM 1$ with a rated power of $P_n=200 \mathrm{kW}$, supplied from a distribution grid with a rated voltage of $6.3 \mathrm{kV}$ from switchgear R6. To validate the dynamic model discussed in Section 1 and Section 2, the simulation results were compared with measured results and the simulation results of the mathematical model discussed in Section 4.

A simplified single line diagram of the distribution system powering the motor $IM 1$ is shown in Figure 5 including the parameters of the circuit elements; the induction motor parameters are shown in

Table 1. Induction motor parameters.

Parameter	Value	Description
$U_n$	$6 \mathrm{kV}$	Rated voltage
$P_n$	$200 \mathrm{kW}$	Rated power
$I_n$	$24.5 \mathrm{A}$	Rated current
$I_{LR}$	$159.3 \mathrm{A}$	Locked rotor current
$I_{LR}/I_n$	$6.5$	Ratio of the locked rotor current and the rated current
$cos{\phi }_n$	$0.84$	Rated power factor
${\omega }_n$	$1551.51 \mathrm{rad}/\mathrm{s}$	Rated speed
Stator windings connection	Y	Star connected
$p_p$	$2$	Number of pole pairs
$J$	$5.8{ \mathrm{kgm}}^2$
$T_n$	$1286 \mathrm{Nm}$	Rated torque
$R_S$	$1.722 \mathsf{\Omega }$	Stator resistance
$L_{\sigma S}$	$0.0355 \mathrm{H}$ 1	Stator leakage inductance
$R_R$	$1.1045 \mathsf{\Omega }$ 1,2	Rotor resistance
$L_{\sigma R}$	$0.0355 \mathrm{H}$ 1,2	Rotor leakage inductance
$L_m$	$1.6 \mathrm{H}$ 1	Magnetizing inductance

¹ Estimated value. ² Rotor parameter value recalculated to the stator side.

, and the calculated short-circuit powers where ${Sk3}^{″}max$ is the maximum initial symmetrical three-phase short-circuit power, $Sk3max$ is the maximum steady-state symmetrical three-phase short-circuit power calculated for the cable temperature equal to $30°$, $Sk3min$ is the minimum steady-state symmetrical three-phase short-circuit power, and $R/X$ is the ratio of resistance and reactance component of the short-circuit impedance.

The switching transients will be analysed in three different situations. The first one is the motor switching operation under medium load condition (current chop). The second and third situation are the switching operation of a motor under starting condition without reignition and with multiple reignition, respectively.

5.1. Switching Operation under Medium Load Condition (Current Chop)

In this section, we derive vacuum circuit breaker parameters by comparing measured and calculated data and show overall voltage waveform $u_{C_L}\left(t\right)$ on the terminals of the induction motor and its angular velocity when turning off the motor under medium load condition. Derived vacuum circuit breaker parameters will be in the further analysis in Section 5.2 and Section 5.3.

The measurement results for the vacuum circuit breaker are shown in Figure 6a and for the oil circuit breaker in Figure 6b. In both cases, a successful turning off was achieved without reignitions. Load current amplitude was equal to $18 \mathrm{A}$ (the motor was loaded to $52\%$) with a corresponding power factor equal to $cos\phi =0.72$ at the time of measurement. Most motors fed from a switchgear R6 were turned off at the time of measurement. The CB8 medium voltage circuit breaker is in a withdrawable design, and it was thus possible to perform measurements for both the vacuum circuit breaker and the oil circuit breaker. The current measurement was carried out by the indirect method of connecting the measuring device to the secondary circuit of the measuring current transformer for protection. The measurement was carried out by the maintenance staff of the given distribution system and the measured data were made available to us for further analysis. The phase represented by the red current waveform is the first phase, in which the current is interrupted in the case of a vacuum circuit breaker (the first pole to clear). The waveforms of the currents in the remaining two phases imply that the circuit breaker contacts move apart, and the arc occurs only after the current in the first pole to clear has passed through its natural zero. In case of an oil circuit breaker, the first phase in which the current is interrupted is the phase represented by the green current waveform. As can be seen from the waveforms of the currents in the remaining two phases, the distance between the contacts, the formation of an arc, and the current chop occurs before passing through its natural zero. Figure 6a,b further show that current chop by the circuit breaker occurs both in the case of a vacuum circuit breaker and in the case of an oil circuit breaker [5,19]. To simulate the switching transients of an oil circuit breaker when turning off small inductive currents in the event of current chop without reignitions, we can use the model of a vacuum circuit breaker see below. Simulation results using the EMTP algorithm or Clark transformation are shown in Figure 7a,b and Figure 8a,b, respectively. The shapes of current waveforms shown in Figure 7a, Figure 8a and Figure 7b, Figure 8b are similar to current waveforms in Figure 6a,b, respectively. The simulation results shown in Figure 7b and Figure 8b were calculated assuming that at the moment of current chop, the magnitude of the breakdown voltage $U_B$ between the circuit breaker contacts was equal to its maximum value, in which case it was possible to use a vacuum circuit breaker model with the corresponding maximum breakdown voltage $U_B$ between the contacts of the circuit breaker and the mean value of the cut-off current $I_{chop}$ as an oil circuit breaker model. The magnitude of the current chop in the simulation calculations in Figure 7b and Figure 8b was derived from the time waveforms of the currents in the phases, which are not the first pole to clear in Figure 6b. Vacuum circuit breaker parameters used in simulations are shown in

Table 2. Vacuum circuit breaker parameters.

Parameter	Value	Description
$A$	$50 \mathrm{V}/\mathsf{\mu }\mathrm{s}$	Dielectric barrier parameter
$B$	$1.2 \mathrm{kV}$	Dielectric barrier parameter
$TR{V}_{Limit}$	$80 \mathrm{kV}$ 1	Maximum dielectric strength
$C$	$0$ 1	High frequency quenching capability parameter
$D$	$600 \mathrm{A}/\mathsf{\mu }\mathrm{s}$ 1	High frequency quenching capability parameter

¹ Estimated value.

and were derived by comparing simulations results with measured results. In this case, corresponding modeling of the initial rate of rise of $TRV$ is important. It is determined by the $TRV$ component with the highest frequency. Derived vacuum circuit breaker parameters shown in Table 2 will also be used in next simulations. In these calculations, the used value of the current chop will be equal to 5 A.

Voltage waveforms on the load $u_{C_P}\left(t\right)$ and $u_{C_M}\left(t\right)$ for the case of turning off the vacuum circuit breaker, corresponding to the calculated currents shown in Figure 7a or Figure 8a, are further shown in Figure 9a,b and Figure 10a,b, respectively. Frequency of the oscillation of voltage $u_{C_P}\left(t\right)$ in Figure 9a is around $1 \mathrm{kHz}$ and its magnitude corresponds to the resonant frequency of the load circuit. Voltage $u_{C_P}\left(t\right)$ has also a power frequency part. The simulation results using the Clark transformation shown in Figure 9b and Figure 10a,b show a gradual voltage drop $u_{C_M}\left(t\right)$ on the terminals of the induction motor due to motor braking after turning off by the vacuum circuit breaker, where the motor torque $T$ drops at a constant load torque $T_L$. Figure 10a,b then show part of the overall voltage waveforms $u_{C_M}\left(t\right)$ and the entire voltage waveforms $u_{C_M}\left(t\right)$ on the terminals of the induction motor from the moment of turning off by the circuit breaker. Voltage waveforms $u_{C_M}\left(t\right)$ consists of two different frequencies. The first voltage part has the frequency of around $2.9 \mathrm{kHz}$ and its magnitude corresponds to the resonant frequency of the load circuit; this part is quickly damped. The frequency of the second part is equal to power frequency $50 \mathrm{Hz}$ and this part drops gradually. The switching overvoltage appears in phase 1 in Figure 9a,b and Figure 10a,b with the peak value around $15 \mathrm{kV}$ and $9.4 \mathrm{kV}$, respectively, which is below the BIL level. For the insulation coordination, the IEC 60034-15 standard suggests a BIL of $29 \mathrm{kV}$ peak value (rated lightning impulse withstand voltage ($1.2/50 \mathsf{\mu }\mathrm{s}$ wave)) for a $6 \mathrm{kV}$ motor [20]. Overvoltage peaks in Figure 9a,b and Figure 10a,b depend on the value of the current chop. In steady-state running, the interruption is not associated with significant $TRV$. Since the mechanical time constants are much larger than the electrical ones, a running motor does not lose speed during the $TRV$ period and maintains a load side electromotive force during the slowing down of the rotation as show in Figure 9b, Figure 10a,b and Figure 11 where the motor’s angular velocity in electrical degrees drop is shown.

5.2. Switching Operation under Starting Condition without Reignition

Figure 12a, Figure 13a and Figure 12b, Figure 13b show the simulation results of the voltage waveforms on load $u_{C_P}\left(t\right)$ and $u_{C_M}\left(t\right)$, respectively, when using the EMTP algorithm or Clark transformation for the case of turning off the vacuum circuit breaker during the start-up of the induction motor ${\mathrm{I}}_{\mathrm{LR}}=225 \mathrm{A}$, $\cos \mathsf{\phi }=0.13$. The turning off occurs without reignitions and the voltage waveforms in Figure 12a,b and Figure 13a,b are consistent with the simulation results or measured values shown in Figure 14a,b, respectively [1]. Simulation results and measured values shown in Figure 14a or Figure 14b were calculated and measured for the test circuit according to IEC 62271-110 ed. 4., powered from a grid with a rated voltage of $12 \mathrm{kV}$ [1,9]. The magnitude of oscillations of voltage on load $u_{C_L}\left(t\right)$ in phases 1 and 3 at the time of the first current chop in phase 2, shown in Figure 12a,b, depends on the magnitude of impedance connected between the node of the power source (grid) and the earth, i.e., the method of earthing of the given distribution system. Simulation results shown in Figure 12a,b and Figure 13a,b were calculated for the case where the grid node is grounded through a resistor $R_{GN}$ of $100 \mathrm{M}\mathsf{\Omega }$ and $40 \mathsf{\Omega }$, respectively. The switching overvoltage appears in phase 2 in Figure 12a,b and Figure 13a,b with the peak value around $8.5 \mathrm{kV}$ and $10 \mathrm{kV}$, respectively, which is below the BIL level. In Figure 12a,b and Figure 13a,b, the frequency of oscillation of load voltage is around $2.9 \mathrm{kHz}$ and $2.4 \mathrm{kHz}$, respectively, at the time of the first current chop in phase 2, otherwise the frequency of oscillation of voltage $u_{C_P}\left(t\right)$ or $u_{C_M}\left(t\right)$ is aprox. $2.9 \mathrm{kHz}$. Voltages $u_{C_P}\left(t\right)$ and $u_{C_M}\left(t\right)$ have a power frequency part, too. The difference in the magnitude of the frequency of oscillation of voltage $u_{C_P}\left(t\right)$ in Figure 9a, Figure 12a, and Figure 13a is due to the change of circuit parameters of the equivalent connection of the inductive motor, which newly define the motor starting operating state, see Section 1.

5.3. Switching Operation under Starting Condition with Multiple Reignition

Simulation results of the voltage waveforms on load $u_{C_P}\left(t\right)$ and $u_{C_M}\left(t\right)$, in case the vacuum circuit breaker is turned off during the start-up of the induction motor under same conditions as above, i.e., $I_{LR}=225 \mathrm{A}$, $cos\phi =0.13$, when reignitions occur during turning off, are shown in Figure 15, Figure 16 and Figure 17a,b, respectively. Figure 17a,b, respectively, then show the entire voltage waveforms $u_{C_P}\left(t\right)$ or $u_{C_M}\left(t\right)$ on the terminals of the induction motor from the moment of turning off by the circuit breaker. Compared to the above case, the moment of time of moving the contacts apart is closer to the moment of time of reaching the natural zero of current in phase 2, in which the current is chopped first. As shown in Figure 15 and Figure 16, the peak value of switching overvoltage in the motor terminals is around $66 \mathrm{kV}$ and $40 \mathrm{kV}$ (phase 2), respectively, which is above the BIL level. That is a result of the occurrence of reignitions during turning off. The difference in peak values of switching overvoltages is caused by the different magnitude of reflected waves. The virtual current chopping does not occur in this case.

Virtual current chopping causes high switching overvoltages. Virtual current chopping is always initiated by multiple reignitions. This is determined by arcing time (short arcing time increases the reignition probability), $TRV$ frequency (high $TRV$ frequency increases reignition probability), and chopping current (high chopping current increases reignition probability) [5]. In general, virtual current chopping is a rather remote possibility [5].

$TRV$ waveforms and dielectric barrier between circuit breaker contacts are shown in Figure 18a,b, respectively. $TRV$ waveforms consist of three main frequencies: frequency of the source and load side and power frequency. The frequency of oscillation of load side voltages $u_{C_P}\left(t\right)$ and $u_{C_M}\left(t\right)$ is around $2.9 \mathrm{kHz}$ and its magnitude is lower than the frequency of oscillation of line side voltages. Breakdown of the dielectric barrier between the circuit breaker contacts causes the formation of a high frequency current see Section 2.2. The waveform of the high frequency current is shown in Figure 19a,b. For comparison, simulation results shown in Figure 16, Figure 17b, Figure 18b, and Figure 19b with waveforms in Figure 15, Figure 17a, Figure 18a, and Figure 19a shows that simulation results are similar.

The worst case occurs when the quenching capability of the vacuum circuit breaker is minimal with other circuit parameters unchanged [8]. Simulation results of the voltage waveforms on load $u_{C_P}\left(t\right)$ and $u_{C_M}\left(t\right)$, respectively, in case of decreasing the value of high frequency quenching capability parameter to $D=150 \mathrm{A}/\mathsf{\mu }\mathrm{s}$ are shown in Figure 20 and Figure 21, respectively. The shapes of voltage waveforms on load $u_{C_M}\left(t\right)$ in Figure 21 are similar to voltage waveforms on load $u_{C_P}\left(t\right)$ in Figure 20, also concerning the amplitudes of the steep voltage peaks. The peak value of switching overvoltage in the motor terminal in Figure 20 and Figure 21 is around $130 \mathrm{kV}$ (phase 3) and $115 \mathrm{kV}$ (phase 2), respectively, which is above the BIL level. Peak values of switching overvoltages are different due to the different magnitude of reflected waves. In Figure 20 and Figure 21, the frequency of oscillation of voltages $u_{C_P}\left(t\right)$ and $u_{C_M}\left(t\right)$ is same as above.

Around this time, $0.1835 \mathrm{s}$ (phase 1) or $0.1838 \mathrm{s}$ (phase 3) and $2.003 \mathrm{s}$ (phase 1) or $2.004 \mathrm{s}$ (phase 3) virtual current chopping occurs. Virtual current chopping appears after the BIL level was reached in phase 2.

As shown in Figure 15, Figure 16, Figure 17a,b, Figure 20, and Figure 21, reignitions may stress the motor insulation. The comparison between the simulated overvoltages and the standard withstand voltage is done not only with magnitude criteria but also with time rise. The overvoltage curve obtained by the simulation must be within the envelope defined by IEC 60034-15. Protection is recommended, especially when currents are below $600 \mathrm{A}$ [5]. A surge arrester is applied as a form of protection against switching overvoltages, which limits the size of the phase to earth overvoltage but do not protect against steep fronts. An RC snubber slows down the rate of rise of the voltage wave front, reduces the probability the occurrence of reignitions and thus also the virtual current chop, and limits the size of the overvoltage [21]. Do not protect against high overvoltages.

6. Conclusions

Derivation of the dynamic model of medium voltage vacuum circuit breaker and induction motor in space vectors in coordinates $\alpha \beta 0$ allows us to model switching transients in various dynamic states of the motor (steady-state or transient conditions). A dynamic (space-vector) model simplifies the system representation, provides a better interpretation of the polyphase circuit, and is a useful tool to model power systems when power electronics converters are present. A case study is presented to illustrate the performance and advantages of the derived dynamic model. The subject of the case study is switching overvoltages that arise when turning off small inductive currents by a vacuum circuit breaker. Asymetrical operations such as switching cause that power system to become unbalanced and the transformed equations $\alpha$, $\beta$, and $0$ are not uncoupled. Therefore, a coupling matrix between the circuit breaker voltages and currents in the coordination system $\alpha \beta 0$ has to be derived. To validate the dynamic model, simulation results were compared with measured results and the simulation results of the mathematical model of the test circuit according to IEC 62271-110, resolved by the EMTP algorithm. A comparison confirms the abilities of the dynamic model to investigate critical distribution system configurations numerically.

Future work should focus on extending the model with the frequency dependent cable and protection measures against overvoltages such as a surge arrester and RC snubber. The next step should be the derivation of a vacuum circuit breaker model for the case of turning on, and model implementation into more complex distrubution systems including power electronics converters.