J Supercond Nov Magn (2014) 27:701–709DOI 10.1007/s10948-013-2365-3

O R I G I NA L PA P E R

Modeling of Shield-Type Superconducting Fault-Current-LimiterOperation Considering Flux Pinning Effect on Flux andSupercurrent Density in High-Temperature SuperconductorCylinders

Arsalan Hekmati

Received: 21 July 2013 / Accepted: 23 August 2013 / Published online: 6 September 2013© Springer Science+Business Media New York 2013

Abstract Superconducting fault current limiter, SFCL,forms an important category of fault-current-limiting de-vices which limit the short-circuit current levels in electri-cal networks. Therefore, modeling its operation and antici-pating its characteristic parameters are too important in itsdesign and optimization process. In this paper a novel in-tegrative method has been proposed which predicts, with agood accuracy, the behavior of inductive shield-type SFCLin different circumstances and approximates its main opera-tional characteristics, as the through current, the inductanceand the voltage-current characteristics. An algorithm is pre-sented to calculate the exact distribution of magnetic fluxand supercurrent density inside the superconductor bulk indifferent operational conditions using the well-known Beanmodel and for the first time the flux pinning effect has beentaken into account in SFCL operation modeling. For esti-mation of flux density distribution outside the supercon-ductor bulk, the FEM analysis has been utilized. An iter-ative method has been used, based on the numerical solu-tion of differential equations, to calculate the instant valueof the SFCL through-current and inductance. The proposedmethod of modeling has been studied on a specific designof shield-type SFCL and its through current in normal andfault conditions of a test circuit, variation of its inductancewith time and its voltage-current characteristic are calcu-lated theoretically. A prototype has been fabricated based onthe studied SFCL design and has been tested experimentally.The comparison of the experimental and theoretical resultsshows that this modeling predicts the SFCL operation witha good accuracy.

A. Hekmati (B)Electrical & Computer Engineering Department, Shahid BeheshtiUniversity, Tehran, 1983963113, Irane-mail: a_hekmati@sbu.ac.ir

Keywords Superconducting · Fault Current Limiter ·Shield-type · Modeling · Flux Pinning

1 Introduction

Because of its low nominal losses, reliable operation, veryshort reaction times to fault currents and an automatic re-sponse feature without the requirement of external trig-ger mechanism, the Superconducting Fault Current Limiter,SFCL, is one of the most promising fault-current-limitingdevices to be used in electrical transmission and distributionnetworks to limit short-circuit current levels [1, 2].

An inductive shield-type SFCL consists of a high-temperature superconductor, HTS, in the form of a bulkcylinder around an iron core and a primary copper wind-ing directly connected to the electric circuit, as shown inFig. 1. Under normal operation condition, the function ofthe HTS cylinder is to shield the flux generated by the pri-mary winding from entering the iron core. This function ismade possible through the induction of supercurrents in-side the cylinder and therefore the SFCL acts as a very lowinductance. While, during the fault condition, the ampere-turns balance between the primary winding and the HTScylinder fails to be satisfied and the flux passes through theiron core. The inductance seen from the primary windingrapidly increases and, consequently, the fault current of thecircuit is confined [2, 3]. This type of SFCL has been a topicof interest in recent years and research has been done onthe simulation and estimation of some of its operational andcharacteristic parameters such as the limitation impedance,losses, recovery characteristics, mechanical and transientproperties [4–13], and on its thermal, physical, numericaland circuit modeling and design [14–24]. However, therehas been a lack of relatively accurate integrative models to

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Fig. 1 Cross-cut view of the shield-type SFCL

predict the shield-type SFCL behavior in different circum-stances for its design and optimization, and to approximate,with a good accuracy, the main operational characteristicsof this type of SFCL as the through-current waveform undernormal and fault conditions, the SFCL inductance variationswith through current and time and the voltage-current char-acteristics of the SFCL. In this paper, for the calculationof the SFCL characteristic parameters, the detailed varia-tion of the magnetic flux density inside the superconductorbulk is investigated using the well-known Bean model andconsidering the flux pinning effect. The flux density distri-bution may be quite complicated due to the flux pinninginfluence which mystifies the analysis of the SFCL opera-tion and the calculation of its flux-based characteristics. Inthis work an algorithm is presented, to obtain the exact dis-tribution of the magnetic flux density and the supercurrentdensity inside the superconductor bulk. For the estimationof the flux density distribution outside the superconductorbulk, the simulation results of the finite element method,FEM, have been used through the FLUX2D developed byCEDRAT [25]. An iterative method has been utilized, basedon the numerical solution of the related governing differ-ential equations, to calculate the instant value of the SFCLinductance and through current. The proposed method ofmodeling has been applied to a specific design of the shield-type SFCL and the predicted through currents have beenobtained in normal and fault conditions in a test circuit. Themost important operational characteristics of the SFCL—the variation of the inductance with time and the voltage-current characteristic—are also brought forth by the pro-posed model. A prototype has been fabricated based on thestudied SFCL design and has been tested experimentally inthe same test circuit. The experimental results have shown agood consistency with the predicted results of the proposedmodeling method.

Fig. 2 A typical case of the distribution of (a) flux density and (b) su-percurrent density, inside the HTS cylinder

2 Proposed Method of Modeling

In the proposed algorithm, the magnetic flux density of thepoints within the interior radius of the copper winding arecalculated through the FEM analysis for a specific throughcurrent. Knowing the external magnetic flux density at thesuperconductor cylinder position, from the Bean’s CriticalState Model [26], the magnetic flux density and the super-current density inside the superconductor cylinder may becalculated. As the flux density and therefore the supercur-rent density are time-variant, complicated cases may be en-countered for the magnetic flux density and the supercurrentdistribution inside the superconductor cylinder, due to theflux pinning effect [26]; a typical case is shown in Fig. 2.The lines in the flux density distribution of Fig. 2(a) haveslopes of Jc, the critical current density of the superconduc-tor cylinder, or −Jc and the supercurrent density in Fig. 2(b)is either Jc (for positive slopes of magnetic flux density) or−Jc (for negative slopes of magnetic flux density), accord-

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Table 1 Main Characteristicsof the Fabricated Prototype Internal radius of the superconductor cylinders, ri 15 mm

External radius of the superconductor cylinders, ro 17 mm

Height of the superconductor cylinders, h 5 mm

Critical current density of the superconductor cylinders, Jc 30 × 106 A/m2

Radius of the core, rc 14 mm

Internal radius of the copper winding, rcu 18 mm

Diameter of the copper wire, dcu 0.8 mm

Total height of the copper winding, hcu 40 mm

Number of turns of the copper winding, N 240

Relative permeability of the core, μr 500

Fig. 3 The magnetic flux density at the SFCL cross-sectional area byFLUX2D

ing to the Bean model. To take these sophisticated variationsinto account, a procedure has been developed which is dis-cussed in Sect. 3. In this algorithm, all the possible modesof the magnetic flux density variation are considered, themost complicated modes being encountered in fault initia-tion cases.

Knowing the current density inside the superconductor,the magnetic flux density within the interior radius of thesuperconductor cylinder may be calculated from the FEManalysis, a typical case shown in Fig. 3, with the designparameters as in Table 1. Thus, the magnetic flux densitywould be known everywhere inside the copper winding, andthe magnetic flux linkage through the copper winding maybe calculated as in (1).

λ = N

∫ rc

0Bc(r)2πr dr + N

∫ ri

rc

Bg(r)2πr dr

+ N

∫ ro

ri

Bsc(r)2πr dr + N

∫ rcu

ro

Bext(r)2πr dr (1)

Fig. 4 Test circuit with an SFCL terminal fault

Here, N is the number of turns of the copper winding, rc,rcu, ri and ro are the core radius, the interior radius of cop-per winding, the interior and exterior radius of supercon-ductor cylinder, respectively, and Bc, Bg, Bsc and Bext arethe magnetic flux densities inside the iron core, in the gapbetween the core and superconductor cylinder, inside thesuperconductor cylinder and between the superconductorcylinder and the copper winding, respectively. The induc-tance of the copper winding at the through current of i wouldbe LSFCL = λ/i.

Two cases are encountered during the SFCL operation:the normal operation case and the fault case [27]. A typicalsingle-phase test circuit is utilized, as in Fig. 4. The sourcehas a peak voltage of Es = 30 V, frequency of f = 50 Hzand an internal inductance of Ls = 2 mH. The load is aresistive-inductive load with resistance and inductance ofRload = 10 � and Lload = 15 mH, respectively. The mostsevere fault case, a fault at the SFCL terminals, has beenconsidered.

2.1 Normal Operation Case

In this case, the voltage across the copper winding is too low.The governing differential equation for the current throughthe copper winding, i, should be calculated.

The voltage across the SFCL is as in (2).

VSFCL = dλ

dt+ RSFCLi = d(LSFCLi)

dt+ RSFCLi

= LSFCLdi

dt+ i

dLSFCL

dt+ RSFCLi (2)

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Here, RSFCL and LSFCL are the resistance and inductance ofthe copper winding, respectively.

For the test circuit of Fig. 4, using (2), the governing dif-ferential equation would be as (3).

(LSFCL + Ls + Lload)di

dt+ i

dLSFCL

dt+ (RSFCL + Rload)i = Es cos(2πf t + α) (3)

Here, α is the phase of the source voltage when the SFCL isconnected to the load.

2.2 Fault Case

The most severe fault case, the fault at the SFCL terminals,has been regarded, as shown in Fig. 4. In this case, the gov-erning differential equation for the current through the cop-per winding would be as (4).

(LSFCL + Ls)di

dt+ i

dLSFCL

dt+ RSFCLi

= Es cos(2πf t + β) (4)

Here, β is the phase of the source voltage at the fault incep-tion.

2.3 Numerical Solution of the Differential Equations

Assuming very small time intervals, the time derivatives ofthe current, i, and inductance, L, may be approximated us-ing a four-point stencil [28], as in (5) and (6).

di

dt= 11i(t) − 18i(t − �t) + 9i(t − 2�t) − 2i(t − 3�t)

6�t(5)

dL

dt= 11L(t) − 18L(t − �t) + 9L(t − 2�t) − 2L(t − 3�t)

6�t

(6)

An iterative method may be used to calculate the SFCLthrough-current and inductance from the differential equa-tions of (3) and (4). A new sample for the current is obtainedat each step and the process is repeated for the new sample.The first sample of the current at the SFCL startup is appar-ently zero. As initial samples of the current have low am-plitudes and no complication of the flux density distributionhas occurred due to the flux pinning effect, the initial sam-ples may be obtained utilizing lower order numerical differ-entiation approximations as in (7) and (8).

di

dt= i(t) − i(t − �t)

�t

di

dt= 3i(t) − 4i(t − �t) + i(t − 2�t)

2�t

(7)

dL

dt= L(t) − L(t − �t)

�t

dL

dt= 3L(t) − 4L(t − �t) + L(t − 2�t)

2�t

(8)

Fig. 5 The modeling process of SFCL

The initial inductance samples may be calculated by thefact that at the SFCL startup, apparently the current wouldtend to increase in one direction. Thus the variation of theflux linkage across the copper winding with its unidirec-tional through current may be simply obtained regardingthe single slope distribution of the magnetic flux inside thesuperconductor bulk. Finally, the inductance at initial in-stants is obtained from LSFCL = dλ/di for small currents.The summary of the overall modeling process is shown inthe flowchart in Fig. 5.

3 Flux and Current Density Inside the HTS Cylinder

The Critical State Model has been used for the determinationof the flux density and the current density distribution insidethe superconductor cylinder. The magnetic field varies withtime at the position of the superconductor cylinder; however,the flux pinning effect would cause the magnetic flux den-sity inside the superconductor cylinder not to vary instantlyby the variation of the external magnetic field; an example ofthis hesitation is shown in Fig. 2(a). This will make the cur-rent density inside the superconductor cylinder to show analternating behavior between +Jc and −Jc, as in Fig. 2(b).Thus to calculate a more precise value for the magnetic flux

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Fig. 6 The flow chart yielding the magnetic flux and the supercurrent density inside the superconductor cylinder in varying magnetic fields

linkage of the copper winding, this phenomenon should betaken into account. A procedure has been developed, withthe flow chart as in Fig. 6, through MATLAB-language oftechnical and numerical computing (MATrix LABoratory),developed by MathWorks [29], which yields the exact mag-netic flux density and the supercurrent density inside the su-perconductor cylinder at different radii. The output of theprocedure is a set of lines with slopes of +Jc or −Jc, repre-senting the magnetic flux density inside the superconductorcylinder. Calculating the intersection points of these lines,the current density inside the superconductor cylinder maybe obtained according to the Bean model.

The two sets of Bmax and Bmin, in the flow chart in Fig. 6,indicate two specific cases:

– If the magnetic flux density decreases after a rise and thenstarts to increase beyond the previous maximum, the twoprevious lines should be eliminated and a new maximummagnetic flux density, Bmax, is assigned, as demonstratedin Fig. 7(a).

– If the magnetic flux density increases after a decrementand then starts to decrease beyond the previous mini-mum, the two previous lines should be eliminated and anew minimum magnetic flux density, Bmin, is assigned, asdemonstrated in Fig. 7(b).

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Fig. 7 Sets of (a) maximum and (b) minimum magnetic flux densities

4 Fabricated Prototype SFCL

A prototype has been fabricated to verify the proposedmethod of SFCL modeling, and is shown in Fig. 8. Three su-perconductor cylinders fabricated from yttrium barium cop-per oxide (YBCO) powder through casting method [30] areset around an open iron core. The copper winding is splitinto two parts and each part is set between two supercon-ductor cylinders. This structure presents several advantagesas the more efficient cooling of the superconductor cylin-ders [27] and also the decrease in the impedance during thesteady-state operation without the large change on the per-formance in the current limiting operation [31].

The main structural characteristics of the prototype arepresented in Table 1.

5 Results

The fabricated prototype SFCL is tested in the circuit ofFig. 4, and its operation in both the normal operation caseand the fault case is analyzed.

5.1 The SFCL Through Current

The measured limited fault current is shown in Fig. 9, to-gether with the fault current without the SFCL. The peakstable fault current without the SFCL is approximately 70 Awhile it is 5 A with SFCL, thus a limitation factor of70/5 = 14 is obtained.

The calculated through current of SFCL, from the model-ing method, is as plotted in Fig. 10, before and after the faultoccurrence, where the experimental results are also pre-sented. As may be seen, the results of the proposed methodfor the modeling of the SFCL operation show a good con-sistency with the experimental results. The special form ofthe SFCL through current observed is recorded in previousworks too as in [3, 11, 30, 32], in which the proposed mod-eling provides verification for all.

Fig. 8 Fabricated prototype SFCL

5.2 The SFCL Inductance

The proposed method estimates the variation of the SFCLinductance, LSFCL, with the time as demonstrated in Fig. 11.Before the fault inception, the SFCL inductance, thoughvarying with time, is too small and the SFCL is rather in-visible to the system. After the fault, the flux penetrates theiron core, thus the inductance starts to increase rapidly. Afterthe entrance of the flux to the core, the inductance remainsrather constant until the fault current starts to decrease, butdue to the flux pinning effect the flux does not decrease withthe same rate of the current; thus, according to the relationL = λ/I , the inductance continues increasing. For a nega-tive fault current, the core loses its flux and the inductancedrops to the values before the fault. For negative fault cur-rents with larger amplitudes, the flux again penetrates thecore and the aforementioned cycle is repeated in the reversedirection yielding the same inductances.

5.3 The Voltage-Current Characteristics of the SFCL

The voltage-current, V–I, characteristics of the SFCL, beingan important characteristics showing the impedance varia-tion behavior of the SFCL, may be obtained by imposingsinusoidal voltages of different amplitude and the same fre-quency to the SFCL terminals and measuring the ampli-tude of the through currents. The V–I characteristics of theSFCL are accepted to be a verification and explanation forthe SFCL operation [14, 16, 18, 33, 34]. For the fabricatedprototype the measurement results are as shown in Fig. 12.

The proposed modeling method may be utilized to pre-dict the V–I characteristics of the SFCL, by numericallysolving the differential equation as (9).

VSFCL = LSFCLdi

dt+ i

dLSFCL

dt+ RSFCLi

= Es cos(2πf t + β) (9)

The modeling results are also shown in Fig. 12. The com-parison between these results and the experimental results

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Fig. 9 Limited fault current and the fault current without the SFCL

Fig. 10 The calculated and measured through current of the SFCL before and after the fault occurrence

Fig. 11 Variation of the SFCL inductance with time

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Fig. 12 The V–I characteristics of the SFCL

verifies the capability of the proposed method in the model-ing of the SFCL operation.

6 Summary and Conclusion

A modeling method has been proposed to predict the be-havior of the shield-type SFCL under different operationalcircumstances and to calculate its operational characteris-tics. A laboratory scale prototype has been fabricated andtested and the experimental results have verified the model-ing method. As a first step, only the SFCL voltage-currentcharacteristics and the variations of the SFCL through-current and inductance with the time have been consid-ered. The study may be extended to other SFCL characteris-tics such as losses, recovery characteristics and mechanicalproperties.

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