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

Citation: Pígl, J.; Cipín, R. Dynamic Model of Medium Voltage Vacuum Circuit Breaker and Induction Motor for Switching Transients Simulation Using Clark Transformation. Energies 2023,16, 1020. https://doi.org/ 10.3390/en16031020 Academic Editor: Sheldon Williamson Received: 26 October 2022 Revised: 15 January 2023 Accepted: 15 January 2023 Published: 17 January 2023 Copyright: © 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/). energies Article Dynamic Model of Medium Voltage Vacuum Circuit Breaker and Induction Motor for Switching Transients Simulation Using Clark Transformation Jan Pígl * and Radoslav Cipín Department of Power Electrical and Electronic Engineering, Faculty of Electrical Engineering and Communication, Brno University of Technology, 616 00 Brno, Czech Republic *Correspondence: [email protected] Abstract: A derivation of the dynamic model of a medium voltage vacuum circuit breaker and induction motor in space vectors in coordinates αβ 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 α , β , and 0 are not uncoupled. Therefore, it is necessary to derive a coupling matrix between circuit breaker voltages and currents in the coordinate system αβ 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. Keywords: switching transients; vacuum circuit breaker; induction motor; clarke transformation; space vectors; ODE23s; EMTP 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 [1–8]. In [ 1 – 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, RGN is the resistance between the grid node and ground, u(t) is the supply voltage, LS is the grid-side inductance, LB1 is the inductance of the capacitors and connection lines, CS is the grid-side capacitance, LB2 is the inductance of the connection lines, RCB is the resistence of the circuit breaker, and RCBP is the parasitic resistance of the circuit breaker, LCBP is the parasitic inductance of the circuit breaker, CCBP is the parasitic capacitance of the circuit breaker, CPI_S is the capacitance of the cable line (supply-side) ( CPI_S=CL/ 2), RL is the cable line resistance, LL is the inductance of Energies 2023,16, 1020. https://doi.org/10.3390/en16031020 https://www.mdpi.com/journal/energies Energies 2023,16, 1020 2 of 22 the cable line, CPI_L is the capacitance of the cable line (load-side) ( CPI_L=CL/ 2), CL is the cable line capacitance, CP 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 RP 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. Energies 2023, 16, x FOR PEER REVIEW 2 of 22 capacitance of the cable line (supply-side) (𝐶_=𝐶/2), 𝑅 is the cable line resistance, 𝐿 is the inductance of the cable line, 𝐶_ is the capacitance of the cable line (load-side) (𝐶_=𝐶/2), 𝐶 is the cable line capacitance, 𝐶 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 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. 3 u(t) L S L B1 C S ~ ~ L B2 Cable L R C P R CBP C CBP R CB L CBP R GN 14 25 ~ 6 1(2)(3) 4(5)(6) R P 7 8 9 10 11 12 C PI_S R L L L C PI_L 7(8)(9) 10(11)(12) PI model Figure 1. Test circuit for the simulation of switching transients according to IEC 62271-110 ed. 4. Source: [9] and own modification. Numbers refer to the interconnection points among components drawings in Figure. In [6–8], 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 is the gridside resistance, 𝐿 is the grid-side inductance, 𝐶 is the grid-side capacitance, 𝑅 is the stator winding resistance, 𝐿 is the stray inductance of the stator winding, 𝐿 is the magnetizing inductance, 𝑅 is the resistance of the rotor winding (converted to the stator), 𝐿 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. u(t) ~ R G L G R S R R /s L Rσ C G L m L Sσ Figure 2. Equivalent circuit of an induction motor for simulating switching transients. Source: [6] and own modification. The equivalent circuit of the motor in Figure 2 is derived from the dynamic (mathematical) model of induction motor in Figure 3, where 𝐶 is the motor capacitance, Figure 1. Test circuit for the simulation of switching transients according to IEC 62271-110 ed. 4. Source: [ 9 ] and own modification. Numbers refer to the interconnection points among components drawings in Figure. In [ 6 – 8 ], it is an equivalent circuit of the motor shown in Figure 2. The test circuit in Figure 1is derived from this equivalent circuit of the motor. In Figure 2, RG is the grid-side resistance, LG is the grid-side inductance, CG is the grid-side capacitance, RS is the stator winding resistance, LSσ is the stray inductance of the stator winding, Lm is the magnetizing inductance, RR is the resistance of the rotor winding (converted to the stator), LRσ 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. Energies 2023, 16, x FOR PEER REVIEW 2 of 22 capacitance of the cable line (supply-side) (𝐶_=𝐶/2), 𝑅 is the cable line resistance, 𝐿 is the inductance of the cable line, 𝐶_ is the capacitance of the cable line (load-side) (𝐶_=𝐶/2), 𝐶 is the cable line capacitance, 𝐶 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 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. 3 u(t) L S L B1 C S ~ ~ L B2 Cable L R C P R CBP C CBP R CB L CBP R GN 14 25 ~ 6 1(2)(3) 4(5)(6) R P 7 8 9 10 11 12 C PI_S R L L L C PI_L 7(8)(9) 10(11)(12) PI model Figure 1. Test circuit for the simulation of switching transients according to IEC 62271-110 ed. 4. Source: [9] and own modification. Numbers refer to the interconnection points among components drawings in Figure. In [6–8], 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 is the gridside resistance, 𝐿 is the grid-side inductance, 𝐶 is the grid-side capacitance, 𝑅 is the stator winding resistance, 𝐿 is the stray inductance of the stator winding, 𝐿 is the magnetizing inductance, 𝑅 is the resistance of the rotor winding (converted to the stator), 𝐿 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. u(t) ~ R G L G R S R R /s L Rσ C G L m L Sσ Figure 2. Equivalent circuit of an induction motor for simulating switching transients. Source: [6] and own modification. The equivalent circuit of the motor in Figure 2 is derived from the dynamic (mathematical) model of induction motor in Figure 3, where 𝐶 is the motor capacitance, Figure 2. Equivalent circuit of an induction motor for simulating switching transients. Source: [ 6 ] and own modification. The equivalent circuit of the motor in Figure 2is derived from the dynamic (mathematical) model of induction motor in Figure 3, where CM is the motor capacitance, RSN and RRN are the resistance between the stator and rotor winding node, respectively, and Energies 2023,16, 1020 3 of 22 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 3in 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 α , β , 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 αβ 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. Energies 2023, 16, x FOR PEER REVIEW 3 of 22 𝑅 and 𝑅 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 𝛼, 𝛽, 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 𝛼𝛽0. 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. ~ ~ ~ u(t) RGLGRSLSσRRLRσ CG RGN RSN RRN ~ ~ ~ u(t) RGLGCableLB2 RSLSσLmRRLRσLm CGCM RGN RSN RRN 14 25 36 7 8 9 10 11 12 RCBP CCBP RCB LCBP 1(2)(3) 4(5)(6) CPI_S RLLL CPI_L 7(8)(9) 10(11)(12) PI model Figure 3. Circuit diagram of induction motor. Numbers refer to the interconnection points among components drawings in Figure. 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 [6–8,16]. 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 Figure 3. Circuit diagram of induction motor. Numbers refer to the interconnection points among components drawings in Figure. 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 – 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 Energies 2023,16, 1020 4 of 22 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 αβ0 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 ωb=dθb(t) dt , where θb(t) 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 ] xdq0(t) = Tdq0(θb)·xabc(t), (1) where Tdq0(θb)=2 3  cos(−θb)cos−θb+2π 3cos−θb−2π 3 sin(−θb)sin−θb+2π 3sin−θb−2π 3 1 21 21 2  , (2) where xabc(t) is a vector of general phase variables a , b , c , xdq0(t) is a vector of transformed variables in components d , q , 0, and Tdq0(θb) is a transformation matrix dq 0. For the inverse Park transformation, then xabc(t) = Tabc(θb)·xdq0(t), (3) where Tabc(θb)=  cos(−θb)sin(−θb)1 cos−θb+2π 3sin−θb+2π 31 cos−θb−2π 3sin−θb−2π 31 , (4) where Tabc(θb) is an inverse transformation matrix. Clark transformation is the special case of the Park transformation into the coordinate system dq 0 rotating at angular velocity ωb=0 (θb=0, stationary system) [17]. From relation (1) results for Clark transformation: xαβ0(t) = Tdq0(0)·xabc(t), (5) where xαβ0(t) is a vector of transformed variables in components α , β , 0. For the inverse Clark transformation from the coordinate system αβ 0 into a three-phase system abc , from relation (3), we obtain the following: xabc(t) = Tabc(0)·xαβ0(t). (6) Furthermore, when transforming into a coordinate system αβ 0, xα=xa , i.e., the value of the general variable in phase a is equal to the value of the transformed variable in the component α . The corresponding space vector of the transformed general variable in components α,βis defined as follows: x−S(t) = xα(t) + jxβ(t). (7) In the following text, space vectors in a general rotating coordinate system b / or in the αβ0 coordinate system will be marked with letter −b/ or −Sin 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: d dtxG(t) = AG(ωb)xG(t) + BG, (8) Energies 2023,16, 1020 5 of 22 where xG(t) = hi−b G(t),iG0(t),u−b CG(t),uCG0(t)iT , (9) AG(ωb)=      −RG LG−jωb0−1 LG0 0−(RG+3RGN) LG0−1 LG 1 CG0−jωb0 01 CG0 0       , (10) BG="u−b(t) LG ,u0(t) LG ,−i−b B(t) CG ,−iB0(t) CG#T , (11) where xG(t) is the column vector of state variables, AG(ωb) is the electrical grid system dynamics matrix, BG is the electrical grid inputs matrix, i−b G(t) is the space vector of the electrical grid current, iG0(t) is the zero component of the electrical grid current, u−b CG(t) is the space vector of voltage at the electrical grid connection point (space vector of voltage at the electrical grid capacity CG ), uCG0(t) is the zero voltage component at the electrical grid connection point (zero voltage component at the electrical grid capacitance CG ), u−b(t) is the space vector of the supply voltage, u0(t) is the zero component of the supply voltage, i−b B(t) is the space vector of the connection line current, and iB0(t) is the zero component of the connection line current. By transforming the state Equation (8) into the coordinate system αβ0, we obtain the following state equation: d dtxG(t) = AG(0)xG(t) + BG. (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: d dtxBVCB(t) = ABVCB(ωb)xBVCB(t) + BBVCB, (13) where xBVCB(t) = hi−b B(t),iB0(t),u−b CPI_S(t),uCPI_S0(t),i−b p(t),iP0(t),u−b CCBP (t),uCCBP0(t)iT ,(14) ABVCB(ωb)=                  −jωb0−1 LB20 0 0 0 0 0 0 0 −1 LB20 0 0 0 1 CL0−jωb0 0 0 0 0 01 CL0 0 0 0 0 0 0 0 0 0 −RCBP LCBP −jωb0−1 LCBP 0 0 0 0 0 0 −RCBP LCBP 0−1 LCBP 0 0 0 0 1 CCBP 0−jωb0 0 0 0 0 0 1 CCBP 0 0                  ,(15) BBVCB =u−b CG(t)−u−b CB(t) LB2,uCG0(t)−uCB0(t) LB2,−i−b L(t) CL,−iL0(t) CL,u−b CB(t) LCBP ,uCB0(t) LCBP , 0, 0T ,(16) where xBVCB(t) is the column vector of state variables, ABVCB(ωb) is the medium voltage vacuum circuit breaker and the connection line system dynamics matrix, BBVCB is the medium voltage vacuum circuit breaker and the connection line inputs matrix, u−b CPI_S(t) Energies 2023,16, 1020 6 of 22 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 CPI_S (supply-side)), uCPI_S0(t) 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 CPI_S (supply-side)), i−b P(t) is the space vector of the parasitic current of the circuit breaker, iP0(t) is the zero component of the parasitic current of the circuit breaker, u−b CCBP (t) is the space vector of the voltage at the parasitic capacitance of the circuit breaker CCBP , uCCBP0(t) is the zero component of the voltage at the parasitic capacitance of the circuit breaker CCBP , u−b CB(t) is the space vector of the voltage on the circuit breaker, uCB0(t)is the zero component of the voltage on the circuit breaker, i−b L(t) is the space vector of the cable line current, and iL0(t) is the zero component of the cable line current. By transforming the state Equation (13) into the coordinate system αβ0, we obtain the following state equation: d dtxBVCB(t) = ABVCB(0)xBVCB(t) + BBVCB. (17) For the voltage on the circuit breaker, uCBαβ0(t) as a function of circuit breaker current iCBαβ0(t) = iBαβ0(t)−iPαβ0(t) , in the coordinate system αβ 0 we derive the following relationship that shows coupling among α , β , 0 components during assymetrical transients (circuit breaker switching): uCBαβ0(t) =    2 3Ra+1 6Rb+1 6Rc−√3 6Rb+√3 6Rc2 3Ra−1 3Rb−1 3Rc −√3 6Rb+√3 6Rc1 2Rb+1 2Rc√3 3Rb−√3 3Rc 1 3Ra−1 6Rb−1 6Rc√3 6Rb−√3 6Rc1 3Ra+1 3Rb+1 3Rc    iCBαβ0(t), (18) where Ra and Rb and Rc 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 =RX=RCB_ON (phase x of the circuit breaker is on, variable x representing phase a, b, or c), cbPhResistance =RX=RCB_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 ( Ichop ) 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 Hz , can be estimated using the relation [16]: Ichop =ω·ˆ i·α·βq, (19) where ω is the angular velocity of the grid frequency and ˆ i is the load current amplitude. Parameters α , β , 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, α= 6.2 × 10 −16 s , β= 14.3, q=(1−β)−1 applies for their values. By moving the contacts of the circuit breaker apart, a dielectric barrier is formed between them, whose breakdown voltage ( Ub ) 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): UB=A(t−tOPEN)+B, (20) Energies 2023,16, 1020 7 of 22 where A or B , is the dielectric barrier parameter, whose unit is V/µs or V , respectively. After the arc is extinguished, at the moment in time time_chop , when the resulting circuit breaker current ( iBx(t) ) drops below the level Ichop , 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 ( Ub ), 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]: di(t) dt =C(t−tOPEN)+D, (21) where C and D is a parameter, whose unit is A/µs2 and A/µs , 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 ( Ub )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 Ichop , 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]. Energies 2023, 16, x FOR PEER REVIEW 8 of 22 1 if time > time_open 2 if phFirstChop == true 3 if ignition == true 4 if (((Ib(time) * Ib(time - deltaTime)) < 0) && 5 ((abs(Ib(time)-Ib(time - deltaTime)) / deltaTime) 6 < didt(time, cbOpenTime, C, D))) 7 || 8 ((abs(Ib(time)) <= Ichop) && ((abs(Ib(time) – Ib(time-deltaTime)) / deltaTime) 9 < (2 * Irated * (2 * pi * frequency)))) 10 cbPhResistance = Rcb_off; ignition = false; 11 end 12 else 13 if abs(TRV) >= Ub(time, cbOpenTime, A, B) 14 cbPhResistance = Rcb_on; ignition = true; 15 end 16 end 17 else 18 if abs(Ib(time)) <= Ichop 19 cbPhResistance = Rcb_off; phFirstChop = true; 20 end 21 end 22 end Figure 4. Circuit breaker turning off algorithm. 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:  𝒙(𝑡)= 𝑨 (𝜔)𝒙(𝑡)+𝑩, (22) where 𝒙(𝑡)=𝑖(𝑡),𝑖(𝑡),𝑢 (𝑡),𝑢(𝑡) 𝑻, (23) 𝑨(𝜔)=⎣ ⎢ ⎢ ⎢ ⎢ ⎢ ⎡ − − 𝑗 𝜔0−  0 0−  0−   _0−𝑗𝜔 0 0  _ 00 ⎦ ⎥ ⎥ ⎥ ⎥ ⎥ ⎤ , (24) 𝑩=_  () ,_() ,− ()  _ ,− ()  _ 𝑻, (25) where 𝒙(𝑡) is the column vector of state variables, 𝑨(𝜔) is the cable line system dynamics matrix, 𝑩 is the cable line inputs matrix, 𝑢 (𝑡) 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_ (load-side) and capacitance of the motor 𝐶, respectively), 𝑢(𝑡) 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_ (load-side) and capacitance of the motor 𝐶, respectively), 𝑖(𝑡) is the space vector of the stator current, and 𝑖(𝑡) is the zero component of the stator current. Figure 4. Circuit breaker turning off algorithm. Energies 2023,16, 1020 8 of 22 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: d dtxPI(t) = API(ωb)xPI(t) + BPI, (22) where xPI (t) = hi−b L(t),iL0(t),u−b CM(t),uCM0(t)iT , (23) API (ωb)=      −RL LL−jωb0−1 LL0 0−RL LL0−1 LL 1 CPI_L+CM0−jωb0 01 CPI_L+CM0 0       , (24) BPI =u−b CPI_S(t) LL,uCPI_S0(t) LL,−i−b S(t) CPI_L+CM,−iS0(t) CPI_L+CMT , (25) where xPI (t) is the column vector of state variables, API (ωb) is the cable line system dynamics matrix, BPI is the cable line inputs matrix, u−b CM(t) 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 CPI_L (load-side) and capacitance of the motor CM , respectively), uCM0(t) 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 CPI_L (load-side) and capacitance of the motor CM , respectively), i−b S(t) is the space vector of the stator current, and iS0(t) is the zero component of the stator current. By transforming the state Equation (22) into the coordinate system αβ 0, we obtain the following state equation: d dtxPI(t) = API(0)xPI(t) + BPI. (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 bwith a zero component, according to the diagram in Figure 3, has the following form: d dtxM,Ψ(t) = AM(ωb)xM,i(t) + BM, (27) where xM,Ψ(t) = hΨ−b S(t),ΨS0(t),Ψ−b R(t),ΨR0(t)iT , (28) xM,i(t) = hi−b S(t),iS0(t),i−b R(t),iR0(t)iT , (29) AM(ωb)=    −RS−jωbLS0−jωbMM0 0−(RS+3RSN)0 0 −j∆ωMM0−RR−j∆ωLR0 0 0 0 −(RR+3RRN)     , (30) Energies 2023,16, 1020 9 of 22 BM(t) = hu−b CM(t),uCM0(t), 0, 0iT , (31) where Ψ−b S(t) = (LσS+LM)i−b S(t) + MMi−b R(t) is the stator flux-linkage space vector, ΨS0(t) is thezerocomponentofthe stator flux-linkage, Ψ−b R(t) = (LσR+LM)i−b R(t) + MMi−b S(t) is the rotor flux-linkage space vector, ΨR0(t) is the zero component of the rotor fluxlinkage, ∆ω=ωb−ωer(t) , ωer(t) is the angular velocity of the rotor in electrical degrees, LS=LσS+LM , LR=LσR+LM , xM,Ψ(t) and xM,i(t) are the column vectors of state variables, AM(ωb) is the induction motor system dynamics matrix, BM(t) is induction motor inputs matrix, Mm is the mutual inductance between stator and rotor windings, while furthermore Lm=Mm applies; i−b R(t) is the space vector of the rotor current, and iR0(t) 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 ( αβ 0), we obtain the following state equation: d dtxM,Ψ(t) = AM(0)xM,i(t) + BM. (32) The motor torque is calculated according to the following relation: T=3 2ppLmi−S Rαi−S Sβ−i−S Rβi−S Sα, (33) where pp is the number of motor pole pairs, i−S Sα is the α -component of the stator current, i−S Sβ is the β -component of the stator current, i−S Rα is the α -component of the rotor current, i−S Rβis the β-component of the rotor current. By adding the motion equation of the drive dωer dt =pp J(T−TL), (34) where J is the moment of inertia and TL 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 αβ0, 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 α , β , 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 Energies 2023,16, 1020 16 of 22 66 kV and 40 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. Energies 2023, 16, x FOR PEER REVIEW 16 of 22 first current chop in phase 2, otherwise the frequency of oscillation of voltage 𝑢(𝑡) or 𝑢(𝑡) is aprox. 2.9 kHz. Voltages 𝑢(𝑡) and 𝑢(𝑡) have a power frequency part, too. The difference in the magnitude of the frequency of oscillation of voltage 𝑢(𝑡) in Figures 9a, 12a, and 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. (a) (b) Figure 12. Simulation results—𝑅 = 100 MΩ (a) Load voltage (EMTP algorithm); (b) Load voltage (Clark transformation). Source: own calculations. (a) (b) Figure 13. Simulation results—𝑅 = 40 Ω (a) Load voltage (EMTP algorithm); (b) Load voltage (Clark transformation). Source: own calculations. Figure 13. Simulation results— RGN = 40 Ω ( a ) Load voltage (EMTP algorithm); ( b ) Load voltage (Clark transformation). Source: own calculations. Energies 2023, 16, x FOR PEER REVIEW 17 of 22 (a) (b) Figure 14. (a) Simulation results—Load voltage; (b) Measurement results—Load voltage. Source: [1]. 5.3. Switching Operation under Starting Condition with Multiple Reignition Simulation results of the voltage waveforms on load 𝑢(𝑡) and 𝑢(𝑡), 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=225 A, 𝑐𝑜𝑠𝜑=0.13, when reignitions occur during turning off, are shown in Figures 15, 16 and 17a,b, respectively. Figure 17a,b, respectively, then show the entire voltage waveforms 𝑢(𝑡) or 𝑢(𝑡) 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 Figures 15 and 16, the peak value of switching overvoltage in the motor terminals is around 66 kV and 40 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]. Figure 14. ( a ) Simulation results—Load voltage; ( b ) Measurement results—Load voltage. Source: [ 1 ]. 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]. Energies 2023,16, 1020 17 of 22 Energies 2023, 16, x FOR PEER REVIEW 18 of 22 Figure 15. Simulation results—Load voltage 𝑢(𝑡) (EMTP algorithm). Source: own calculations. Figure 16. Simulation results—Load voltage 𝑢(𝑡) (Clark transformation). Source: own calculations. (a) (b) Figure 17. Load voltage (a) EMTP algorithm; (b) Clark transformation. Source: own calculations. U [V] U [V] Figure 15. Simulation results—Load voltage uCP(t)(EMTP algorithm). Source: own calculations. Energies 2023, 16, x FOR PEER REVIEW 18 of 22 Figure 15. Simulation results—Load voltage 𝑢(𝑡) (EMTP algorithm). Source: own calculations. Figure 16. Simulation results—Load voltage 𝑢(𝑡) (Clark transformation). Source: own calculations. (a) (b) Figure 17. Load voltage (a) EMTP algorithm; (b) Clark transformation. Source: own calculations. U [V] U [V] Figure 16. Simulation results—Load voltage uCM(t) (Clark transformation). Source: own calculations. Energies 2023,16, 1020 18 of 22 Energies 2023, 16, x FOR PEER REVIEW 18 of 22 Figure 15. Simulation results—Load voltage 𝑢(𝑡) (EMTP algorithm). Source: own calculations. Figure 16. Simulation results—Load voltage 𝑢(𝑡) (Clark transformation). Source: own calculations. (a) (b) Figure 17. Load voltage (a) EMTP algorithm; (b) Clark transformation. Source: own calculations. U [V] U [V] Figure 17. Load voltage (a) EMTP algorithm; (b) Clark transformation. Source: own calculations. 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 uCP(t) and uCM(t) is around 2.9 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 Figures 16,17b, 18b, and 19b with waveforms in Figures 15,17a, 18a, and 19a shows that simulation results are similar. Energies 2023, 16, x FOR PEER REVIEW 19 of 22 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(𝑡) and 𝑢(𝑡) is around 2.9 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 Figures 16, 17b, 18b, and 19b with waveforms in Figures 15, 17a, 18a, and 19a shows that simulation results are similar. (a) (b) Figure 18. TRV (a) EMTP algorithm; (b) Clark transformation. Source: own calculations. (a) (b) Figure 19. High frequency current (a) EMTP algorithm; (b) Clark transformation. Source: own calculations. 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(𝑡) and 𝑢(𝑡), respectively, in case of decreasing the value of 0.18 0.182 0.184 0.186 0.188 0.19 t [s] -1 -0.5 0 0.5 1 U [V] 105 trv1(t) trv2(t) trv3(t) diel. barr.(t) diel. barr(t) 2 2.002 2.004 2.006 2.008 t [s] -1 -0.5 0 0.5 1 U [V] 105 trv1(t) trv2(t) trv3(t) diel. barr.(t) diel. barr(t) Figure 18. TRV (a) EMTP algorithm; (b) Clark transformation. Source: own calculations. 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 uCP(t) and uCM(t) , respectively, in case of decreasing the value of high Energies 2023,16, 1020 19 of 22 frequency quenching capability parameter to D= 150 A/µs are shown in Figures 20 and 21, respectively. The shapes of voltage waveforms on load uCM(t) in Figure 21 are similar to voltage waveforms on load uCP(t) in Figure 20, also concerning the amplitudes of the steep voltage peaks. The peak value of switching overvoltage in the motor terminal in Figures 20 and 21 is around 130 kV (phase 3) and 115 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 Figures 20 and 21, the frequency of oscillation of voltages uCP(t)and uCM(t)is same as above. Energies 2023, 16, x FOR PEER REVIEW 19 of 22 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(𝑡) and 𝑢(𝑡) is around 2.9 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 Figures 16, 17b, 18b, and 19b with waveforms in Figures 15, 17a, 18a, and 19a shows that simulation results are similar. (a) (b) Figure 18. TRV (a) EMTP algorithm; (b) Clark transformation. Source: own calculations. (a) (b) Figure 19. High frequency current (a) EMTP algorithm; (b) Clark transformation. Source: own calculations. 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(𝑡) and 𝑢(𝑡), respectively, in case of decreasing the value of 0.18 0.182 0.184 0.186 0.188 0.19 t [s] -1 -0.5 0 0.5 1 U [V] 105 trv1(t) trv2(t) trv3(t) diel. barr.(t) diel. barr(t) 2 2.002 2.004 2.006 2.008 t [s] -1 -0.5 0 0.5 1 U [V] 105 trv1(t) trv2(t) trv3(t) diel. barr.(t) diel. barr(t) Figure 19. High frequency current ( a ) EMTP algorithm; ( b ) Clark transformation. Source: own calculations. Energies 2023, 16, x FOR PEER REVIEW 20 of 22 high frequency quenching capability parameter to 𝐷=150 A/μs are shown in Figures 20 and 21, respectively. The shapes of voltage waveforms on load 𝑢(𝑡) in Figure 21 are similar to voltage waveforms on load 𝑢(𝑡) in Figure 20, also concerning the amplitudes of the steep voltage peaks. The peak value of switching overvoltage in the motor terminal in Figures 20 and 21 is around 130 kV (phase 3) and 115 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 Figures 20 and 21, the frequency of oscillation of voltages 𝑢(𝑡) and 𝑢(𝑡) is same as above. Around this time, 0.1835 s (phase 1) or 0.1838 s (phase 3) and 2.003 s (phase 1) or 2.004 s (phase 3) virtual current chopping occurs. Virtual current chopping appears after the BIL level was reached in phase 2. Figure 20. Simulation results—Load voltage 𝑢(𝑡) (EMTP algorithm). Source: own calculations. Figure 21. Simulation results—Load voltage 𝑢(𝑡) (Clark transformation). Source: own calculations. As shown in Figures 15, 16, 17a,b, 20, and 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 A [5]. A surge arrester is applied as a form of protection against switching overvoltages, which limits the U [V] U [V] Figure 20. Simulation results—Load voltage uCP(t)(EMTP algorithm). Source: own calculations. Energies 2023,16, 1020 20 of 22 Energies 2023, 16, x FOR PEER REVIEW 20 of 22 high frequency quenching capability parameter to 𝐷=150 A/μs are shown in Figures 20 and 21, respectively. The shapes of voltage waveforms on load 𝑢(𝑡) in Figure 21 are similar to voltage waveforms on load 𝑢(𝑡) in Figure 20, also concerning the amplitudes of the steep voltage peaks. The peak value of switching overvoltage in the motor terminal in Figures 20 and 21 is around 130 kV (phase 3) and 115 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 Figures 20 and 21, the frequency of oscillation of voltages 𝑢(𝑡) and 𝑢(𝑡) is same as above. Around this time, 0.1835 s (phase 1) or 0.1838 s (phase 3) and 2.003 s (phase 1) or 2.004 s (phase 3) virtual current chopping occurs. Virtual current chopping appears after the BIL level was reached in phase 2. Figure 20. Simulation results—Load voltage 𝑢(𝑡) (EMTP algorithm). Source: own calculations. Figure 21. Simulation results—Load voltage 𝑢(𝑡) (Clark transformation). Source: own calculations. As shown in Figures 15, 16, 17a,b, 20, and 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 A [5]. A surge arrester is applied as a form of protection against switching overvoltages, which limits the U [V] U [V] Figure 21. Simulation results—Load voltage uCM(t) (Clark transformation). Source: own calculations. Around this time, 0.1835 s (phase 1) or 0.1838 s (phase 3) and 2.003 s (phase 1) or 2.004 s (phase 3) virtual current chopping occurs. Virtual current chopping appears after the BIL level was reached in phase 2. As shown in Figures 15,16,17a,b, 20, and 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 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 αβ 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 α , β , and 0 are not uncoupled. Therefore, a coupling matrix between the circuit breaker voltages and currents in the coordination system αβ 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. Energies 2023,16, 1020 21 of 22 Author Contributions: Conceptualization, R.C. and J.P.; methodology, R.C. and J.P.; software, J.P.; validation, J.P.; formal analysis, J.P.; investigation, J.P.; writing—original draft preparation, J.P.; writing—review and editing, R.C. and J.P.; supervision, R.C. All authors have read and agreed to the published version of the manuscript. Funding: This research work has been carried out in the Centre for Research and Utilization of Renewable Energy (CVVOZE). Authors gratefully acknowledge financial support from the Ministry of Education, Youth and Sports under institutional support and BUT specific research programme (project No. FEKT-S-20-6379). Data Availability Statement: The data presented in this study are available on request from the corresponding author. Acknowledgments: I would like to thank Ing. Jan Nytra for providing us measurement data. Conflicts of Interest: The authors declare no conflict of interest. References 1. Novak, P.; Haim, M.; Beer, P.; Kaltenborn, U.; Melquiond, S. Switching of Small Inductive Currents Using Vacuum CircuitBreakers. In Proceedings of the 21st International Conference on Electricity Distribution, Frankfurt, Germany, 6–9 June 2011. 2. Penkov, D.; Vollet, C.; De Metz-Noblat, B.; Nikodem, R. Overvoltage Protection Study on Vacuum Breaker Switched MV Motors. 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