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Modeling of the magnetic field diffusion in the high-Tc superconducting tube for fault current limitation
Physica C 387 (2003) 234–238 www.elsevier.com/locate/physc
Modeling of the magnetic field diffusion in the high-Tc superconducting tube for fault current limitation Paweł Surdacki
*
Institute of Electrical Engineering and Electrotechnology, Lublin University of Technology, Ulica Nadbystrzycka 38a, 20-618 Lublin, Poland
Abstract Operation of the superconducting fault current limiter of the inductive type is analyzed together with the magnetic diffusion process that occurs in the high-Tc superconducting tube. Due to the highly non-linear voltage–current relation of the superconductor, a critical state model based on the power law is used to compute resistivity changes during transition from the superconducting state to the resistive state. The influence of the penetration depth, for various operating modes, on the magnetic flux density distribution across the SC tube wall is investigated using the slab geometry and a linear approximation of the magnetic diffusion process. Ó 2003 Elsevier Science B.V. All rights reserved. Keywords: Superconducting fault current limiter; Magnetic diffusion; Critical state model
1. Introduction The superconducting fault current limiter (SFCL) is a potential application for high-Tc superconductivity in power engineering. It can improve power system capacity without replacing existing switchgear devices. In the case of a short circuit in the power system, when no limiting action takes place, the prospective fault current increases with a certain rate that depends on the circuit parameters (source voltage and impedance) and the phase angle of fault occurrence [1]. Without extra limitation a conventional circuit breaker
*
Fax: +48-81-5381-292. E-mail address: [email protected] (P. Surdacki).
breaks the current after a few periods of the current waveform. Increasing electric power production may drive power networks to the limits of their short circuit current capability; therefore, an effective fault current limitation is required. To limit the first peak of the current, an appropriate high voltage drop should be rapidly inserted into the circuit. A superconducting fault current limiter (SFCL) that employs a strongly non-linear voltage–current relation of the superconducting element and can develop a very high resistance in less than 1 ms can provide such an action [2]. In order to model a limiting operation of the inductive type of a SFCL device, when the resistive transition from a superconducting state occurs, it is necessary to investigate magnetic field diffusion in a high-Tc superconducting tube that is an essential part of the limiter.
0921-4534/03/$ - see front matter Ó 2003 Elsevier Science B.V. All rights reserved. doi:10.1016/S0921-4534(03)00677-4
P. Surdacki / Physica C 387 (2003) 234–238
2. Current limitation and magnetic field diffusion The inductive-type SFCL (Fig. 1) consists of a closed iron core inside a superconducting tube, and an external copper coil connected in series with a protected power line, generating a timedependent applied field. During normal operation, when a nominal load current flows in the primary winding, current induced in the SC tube is below the critical value. The SC tube shields the iron core from the magnetic field and keeps the inductance introduced to the line low. Due to the extremely low resistivity of a superconductor, magnetic diffusion practically occurs in the tube wall up to the penetration depth only. A dominant phenomenon is the magnetic coupling between the SC tube and the primary winding. The SFCL device operates in the inductive mode like a transformer with a short-circuited secondary winding (i.e. SC tube). Under fault conditions, the current in the primary winding abruptly rises and the excessive current induced in the SC tube destroys its superconductivity. The tube no longer shields the iron core and the very high impedance introduced to the electric power line substantially limits the fault current peak, even in the first half-wave. The superconductor goes to a resistive state and the
235
SFCL device operates like a transformer with the open secondary winding. Because the resistivity rapidly increases by several orders, the magnetic flux can deeply penetrate the tube wall. The dominant phenomenon is the resistive zone propagation, which generates heat locally in the tube. The magnetic diffusion in the tube depends on resistivity that changes in time. The resistivity of a superconductor is a function of both current density and temperature, qðJ ; T Þ. The local heating makes the temperature distribution in the superconductor highly non-uniform under adiabatic conditions. The temperature is much higher near the surface of the wall than near its center. Because resistivity increases rapidly with temperature, the magnetic diffusion rate will be increased, especially near the surface of the wall. In the fault limitation mode, the superconducting tube will continue to warm up under adiabatic thermal conditions. The coupled non-linear magnetic and thermal diffusion equations should, therefore, be solved in order to model transient phenomena in the superconducting tube. The present approach is based on magnetic field analysis only, because the thermal diffusivity is much lower than the magnetic diffusivity, and it assumes an isothermal condition accomplished by sufficient heat transfer to a coolant. Employing the Maxwell equations, we obtain the magnetic diffusion equation r ½qðJ Þr H ¼
oBðHÞ ot
ð1Þ
where resistivity qðJ Þ is a function of current density and the magnetic flux density B is explicitly expressed as a function of magnetic field intensity H. Type-II superconductors are characterized by the non-linear B–H relation B ¼ l0 ðH þ MÞ, where M is the magnetization. When the applied field is much larger than the magnetization of the superconductor, which is the case for the fault limitation mode, B becomes linearly proportional to H and we get the following non-linear magnetic diffusion equation
Fig. 1. Longitudinal cross-section of the SFCL structure.
r ½Dm ðJ Þr B ¼
oB ot
ð2Þ
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P. Surdacki / Physica C 387 (2003) 234–238
where the magnetic diffusivity is qðJ Þ Dm ðJ Þ ¼ ð3Þ l0 To solve Eq. (2), it is necessary to obtain resistivity qðJ Þ that can be calculated from the current–voltage characteristics of the superconductor.
3. Modeling of the superconductor Type-II superconductors have non-linear and smooth current–voltage characteristics and their E–J relation can be expressed with a power law [3] N J E ¼ Ec ð4Þ Jc where E is the electric field intensity, J is the current density, Jc ðBÞ is the critical current density, Ec is the electric field intensity at Jc , and the exponent N ðB; T Þ is a function of both the local magnetic flux density and temperature, and can vary widely for various types of superconductors. Most often, the Ec ¼ 1 lV/cm criterion is used for the definition of Ic for high-Tc superconductors. In the case of N ¼ 1 (Fig. 2), Eq. (4) reduces to the linear OhmÕs law (E ¼ qJ ) and when N ¼ 1, Eq. (4) corresponds to the BeanÕs critical state model, in which J equals either zero or Jc . For most high-Tc superconductors N is much lower than 50. For modeling purposes, the resistivity qðJ Þ may be written in the form
Fig. 3. Resistivity vs. current density plot.
qðJ Þ ¼ Ec
J Jc
N
1 : J
ð5Þ
For a BSCCO-2223 tube, which has a critical temperature of 108 K, a critical current in the tangential direction of 625 A (at 77 K), an inner diameter of 59 mm, a height of 50 mm, and a wall thickness of 2.5 mm, the resistivity has been computed at a current density operating range between 3 106 A/m2 and 7 106 A/m2 (Fig. 3) for different exponent values ranging from 1 to 30, depending on the magnetic flux density and temperature. The obtained resistivity values for the exponent values N ¼ 10 and 30 are in a very broad range between 1015 X m (superconducting mode of the tube) and 106 X m (resistive mode).
4. Linear diffusion model
Fig. 2. Power law E=Ec versus J =Jc for different exponent number N .
For the tube dimensions given, the wall thickness is very small when compared to its inner diameter and height. In order to obtain qualitative results for the magnetic field distributions, this approach concentrates on the simplified linear magnetic diffusion model with geometry based on a slab of thickness 2a in the x-direction and subjected to a sinusoidal applied field in the z-direction (Fig. 4). From Eqs. (2) and (3), the following simplified linear diffusion equation is derived
P. Surdacki / Physica C 387 (2003) 234–238
237
Fig. 4. Geometry of the superconductor slab subjected to sinusoidal field.
Dm
o2 B ox2
¼
oB ot
ð6Þ
where Dm is the magnetic diffusivity, B is the magnetic flux density in the z-direction, and t is time. For the initial condition B ¼ 0 at t 6 0 and the boundary conditions B ¼ B 0 sinðxtÞ at x ¼ a, the AC part of the analytical solution [4] has, respectively, a relative amplitude A and phase angle / cosh x ð1 þ jÞ d and A¼ cosh ad ð1 þ jÞ ( ) cosh dx ð1 þ jÞ ; / ¼ Arg ð7Þ cosh ad ð1 þ jÞ where the characteristic penetration depth is rffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffi 2Dm 2q ¼ : d¼ l0 x x
Fig. 5. Distribution of the AC component of the transient solution in the slab for various relative penetration depths d=a.
state. The magnetic flux distribution in this case is more uniform and the induced current is very low. The phase angle between the applied magnetic field and the AC component of the transient solution in the slab is shown in Fig. 6. In the case of the superconducting mode (relative penetration depth d=a below 0.1), there is a highly heterogeneous distribution of the magnetic field phase angle that has the highest absolute value in the slab center. As the slab becomes more resistive (d=a > 1), the phase angle distribution is fairly uniform.
ð8Þ
The relative amplitude distribution of the AC component of the transient solution in the slab is shown in Fig. 5. It is highly heterogeneous at low values of the relative penetration depth d=a below 0.01 that represent the normal, i.e. superconducting, operation mode of the SC tube. In this case, the magnetic field and induced current of the large value are restricted to a very thin layer near the surface of the slab. A much higher relative penetration depth (d=a P 1) corresponds to the limitation mode, when the SC tube goes to the resistive
Fig. 6. Phase angle distribution of AC component of the transient solution in the slab for various relative penetration depths d=a.
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P. Surdacki / Physica C 387 (2003) 234–238
5. Conclusions
References
The presented linear approach to magnetic field diffusion in the tube of the superconducting fault current limiter, based on the simplified slab geometry, gives only a qualitative explanation of both the superconducting and limiting operation modes. Non-linear diffusion, including the high-Tc superconductor model, should be considered along with heating diffusion for the resistive transition process.
[1] M. Steurer, K. Froehlich, Current limiters––state of the art, Fourth Workshop & Conference on EHV Technology, Indian Institute of Science, Bangalore, India, 15–16 July 1998. [2] W. Paul, M. Chen, M. Lakner, J. Rhyner, D. Braun, W. Lanz, Physica C 354 (2001) 27–33. [3] S. Stavrev, F. Grilli, et al., IEEE Trans. Magn. 38 (2) (2002) 849–852. [4] Y.S. Cha, Physica C 361 (2001) 1–12.
Modeling of the magnetic field diffusion in the high-Tc superconducting tube for fault current limitation Paweł Surdacki
*
Institute of Electrical Engineering and Electrotechnology, Lublin University of Technology, Ulica Nadbystrzycka 38a, 20-618 Lublin, Poland
Abstract Operation of the superconducting fault current limiter of the inductive type is analyzed together with the magnetic diffusion process that occurs in the high-Tc superconducting tube. Due to the highly non-linear voltage–current relation of the superconductor, a critical state model based on the power law is used to compute resistivity changes during transition from the superconducting state to the resistive state. The influence of the penetration depth, for various operating modes, on the magnetic flux density distribution across the SC tube wall is investigated using the slab geometry and a linear approximation of the magnetic diffusion process. Ó 2003 Elsevier Science B.V. All rights reserved. Keywords: Superconducting fault current limiter; Magnetic diffusion; Critical state model
1. Introduction The superconducting fault current limiter (SFCL) is a potential application for high-Tc superconductivity in power engineering. It can improve power system capacity without replacing existing switchgear devices. In the case of a short circuit in the power system, when no limiting action takes place, the prospective fault current increases with a certain rate that depends on the circuit parameters (source voltage and impedance) and the phase angle of fault occurrence [1]. Without extra limitation a conventional circuit breaker
*
Fax: +48-81-5381-292. E-mail address: [email protected] (P. Surdacki).
breaks the current after a few periods of the current waveform. Increasing electric power production may drive power networks to the limits of their short circuit current capability; therefore, an effective fault current limitation is required. To limit the first peak of the current, an appropriate high voltage drop should be rapidly inserted into the circuit. A superconducting fault current limiter (SFCL) that employs a strongly non-linear voltage–current relation of the superconducting element and can develop a very high resistance in less than 1 ms can provide such an action [2]. In order to model a limiting operation of the inductive type of a SFCL device, when the resistive transition from a superconducting state occurs, it is necessary to investigate magnetic field diffusion in a high-Tc superconducting tube that is an essential part of the limiter.
0921-4534/03/$ - see front matter Ó 2003 Elsevier Science B.V. All rights reserved. doi:10.1016/S0921-4534(03)00677-4
P. Surdacki / Physica C 387 (2003) 234–238
2. Current limitation and magnetic field diffusion The inductive-type SFCL (Fig. 1) consists of a closed iron core inside a superconducting tube, and an external copper coil connected in series with a protected power line, generating a timedependent applied field. During normal operation, when a nominal load current flows in the primary winding, current induced in the SC tube is below the critical value. The SC tube shields the iron core from the magnetic field and keeps the inductance introduced to the line low. Due to the extremely low resistivity of a superconductor, magnetic diffusion practically occurs in the tube wall up to the penetration depth only. A dominant phenomenon is the magnetic coupling between the SC tube and the primary winding. The SFCL device operates in the inductive mode like a transformer with a short-circuited secondary winding (i.e. SC tube). Under fault conditions, the current in the primary winding abruptly rises and the excessive current induced in the SC tube destroys its superconductivity. The tube no longer shields the iron core and the very high impedance introduced to the electric power line substantially limits the fault current peak, even in the first half-wave. The superconductor goes to a resistive state and the
235
SFCL device operates like a transformer with the open secondary winding. Because the resistivity rapidly increases by several orders, the magnetic flux can deeply penetrate the tube wall. The dominant phenomenon is the resistive zone propagation, which generates heat locally in the tube. The magnetic diffusion in the tube depends on resistivity that changes in time. The resistivity of a superconductor is a function of both current density and temperature, qðJ ; T Þ. The local heating makes the temperature distribution in the superconductor highly non-uniform under adiabatic conditions. The temperature is much higher near the surface of the wall than near its center. Because resistivity increases rapidly with temperature, the magnetic diffusion rate will be increased, especially near the surface of the wall. In the fault limitation mode, the superconducting tube will continue to warm up under adiabatic thermal conditions. The coupled non-linear magnetic and thermal diffusion equations should, therefore, be solved in order to model transient phenomena in the superconducting tube. The present approach is based on magnetic field analysis only, because the thermal diffusivity is much lower than the magnetic diffusivity, and it assumes an isothermal condition accomplished by sufficient heat transfer to a coolant. Employing the Maxwell equations, we obtain the magnetic diffusion equation r ½qðJ Þr H ¼
oBðHÞ ot
ð1Þ
where resistivity qðJ Þ is a function of current density and the magnetic flux density B is explicitly expressed as a function of magnetic field intensity H. Type-II superconductors are characterized by the non-linear B–H relation B ¼ l0 ðH þ MÞ, where M is the magnetization. When the applied field is much larger than the magnetization of the superconductor, which is the case for the fault limitation mode, B becomes linearly proportional to H and we get the following non-linear magnetic diffusion equation
Fig. 1. Longitudinal cross-section of the SFCL structure.
r ½Dm ðJ Þr B ¼
oB ot
ð2Þ
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P. Surdacki / Physica C 387 (2003) 234–238
where the magnetic diffusivity is qðJ Þ Dm ðJ Þ ¼ ð3Þ l0 To solve Eq. (2), it is necessary to obtain resistivity qðJ Þ that can be calculated from the current–voltage characteristics of the superconductor.
3. Modeling of the superconductor Type-II superconductors have non-linear and smooth current–voltage characteristics and their E–J relation can be expressed with a power law [3] N J E ¼ Ec ð4Þ Jc where E is the electric field intensity, J is the current density, Jc ðBÞ is the critical current density, Ec is the electric field intensity at Jc , and the exponent N ðB; T Þ is a function of both the local magnetic flux density and temperature, and can vary widely for various types of superconductors. Most often, the Ec ¼ 1 lV/cm criterion is used for the definition of Ic for high-Tc superconductors. In the case of N ¼ 1 (Fig. 2), Eq. (4) reduces to the linear OhmÕs law (E ¼ qJ ) and when N ¼ 1, Eq. (4) corresponds to the BeanÕs critical state model, in which J equals either zero or Jc . For most high-Tc superconductors N is much lower than 50. For modeling purposes, the resistivity qðJ Þ may be written in the form
Fig. 3. Resistivity vs. current density plot.
qðJ Þ ¼ Ec
J Jc
N
1 : J
ð5Þ
For a BSCCO-2223 tube, which has a critical temperature of 108 K, a critical current in the tangential direction of 625 A (at 77 K), an inner diameter of 59 mm, a height of 50 mm, and a wall thickness of 2.5 mm, the resistivity has been computed at a current density operating range between 3 106 A/m2 and 7 106 A/m2 (Fig. 3) for different exponent values ranging from 1 to 30, depending on the magnetic flux density and temperature. The obtained resistivity values for the exponent values N ¼ 10 and 30 are in a very broad range between 1015 X m (superconducting mode of the tube) and 106 X m (resistive mode).
4. Linear diffusion model
Fig. 2. Power law E=Ec versus J =Jc for different exponent number N .
For the tube dimensions given, the wall thickness is very small when compared to its inner diameter and height. In order to obtain qualitative results for the magnetic field distributions, this approach concentrates on the simplified linear magnetic diffusion model with geometry based on a slab of thickness 2a in the x-direction and subjected to a sinusoidal applied field in the z-direction (Fig. 4). From Eqs. (2) and (3), the following simplified linear diffusion equation is derived
P. Surdacki / Physica C 387 (2003) 234–238
237
Fig. 4. Geometry of the superconductor slab subjected to sinusoidal field.
Dm
o2 B ox2
¼
oB ot
ð6Þ
where Dm is the magnetic diffusivity, B is the magnetic flux density in the z-direction, and t is time. For the initial condition B ¼ 0 at t 6 0 and the boundary conditions B ¼ B 0 sinðxtÞ at x ¼ a, the AC part of the analytical solution [4] has, respectively, a relative amplitude A and phase angle / cosh x ð1 þ jÞ d and A¼ cosh ad ð1 þ jÞ ( ) cosh dx ð1 þ jÞ ; / ¼ Arg ð7Þ cosh ad ð1 þ jÞ where the characteristic penetration depth is rffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffi 2Dm 2q ¼ : d¼ l0 x x
Fig. 5. Distribution of the AC component of the transient solution in the slab for various relative penetration depths d=a.
state. The magnetic flux distribution in this case is more uniform and the induced current is very low. The phase angle between the applied magnetic field and the AC component of the transient solution in the slab is shown in Fig. 6. In the case of the superconducting mode (relative penetration depth d=a below 0.1), there is a highly heterogeneous distribution of the magnetic field phase angle that has the highest absolute value in the slab center. As the slab becomes more resistive (d=a > 1), the phase angle distribution is fairly uniform.
ð8Þ
The relative amplitude distribution of the AC component of the transient solution in the slab is shown in Fig. 5. It is highly heterogeneous at low values of the relative penetration depth d=a below 0.01 that represent the normal, i.e. superconducting, operation mode of the SC tube. In this case, the magnetic field and induced current of the large value are restricted to a very thin layer near the surface of the slab. A much higher relative penetration depth (d=a P 1) corresponds to the limitation mode, when the SC tube goes to the resistive
Fig. 6. Phase angle distribution of AC component of the transient solution in the slab for various relative penetration depths d=a.
238
P. Surdacki / Physica C 387 (2003) 234–238
5. Conclusions
References
The presented linear approach to magnetic field diffusion in the tube of the superconducting fault current limiter, based on the simplified slab geometry, gives only a qualitative explanation of both the superconducting and limiting operation modes. Non-linear diffusion, including the high-Tc superconductor model, should be considered along with heating diffusion for the resistive transition process.
[1] M. Steurer, K. Froehlich, Current limiters––state of the art, Fourth Workshop & Conference on EHV Technology, Indian Institute of Science, Bangalore, India, 15–16 July 1998. [2] W. Paul, M. Chen, M. Lakner, J. Rhyner, D. Braun, W. Lanz, Physica C 354 (2001) 27–33. [3] S. Stavrev, F. Grilli, et al., IEEE Trans. Magn. 38 (2) (2002) 849–852. [4] Y.S. Cha, Physica C 361 (2001) 1–12.