### Ic Inhomogeneity in FCL Modelling Work

We present here a modelling work done in the frame of the ECCOFLOW FP7 project.

For Fault Current Limiter (FCL) applications a very good homogeneity of the critical current (I_{c}) is required. In case of inhomogeneity, the heat dissipation during a fault is very inhomogeneous too and may lead to a local overheating of the tape, called "hot spot". This phenomenon is particularly important in the case of high impedance faults leading to low faults currents which are just slightly above I_{c}.

In this modelling work, we estimate the influence of the inhomogeneity of I_{c} along the tape calculating temperature maxima as a function of tape stabilization and fault impedance.

Critical current inhomogeneity can be represented by a Gaussian distribution, function of the average critical current (I_{c};av) and the standard deviation (s). The Gaussian distribution reported here is based on experimental data provided by the courtesy of SuperPower-Inc

By discretizing the Gaussian distribution it is possible to generate an array of inhomogeneity like the one represented in the figure a) above. The relative position of each element implies a different amount of thermal conduction between each block’s neighbour. Therefore, we used random permutations of these terms like the one represented in part b) of the figure above.

The transient limiting performance is influenced by the grid where the limiter will operate. Therefore it s necessary to study the FCL in the equivalent circuit of the electrical network that will host the device like represented in the figure above. The external shunt impedance (Zs) determines the limited current allowed after the disconnection of the superconducting part of the FCL which occurs after 80 ms. Its presence and value is imposed by the utility companies which often require a long fault duration used by their protecting scheme.

When considering a clear three phases short-circuit (fault resistance RF=0 Ω) which corresponds to optimize the device to the maximum energy dissipated in the circuit, we cannot exclude thermal instability of the FCL under low values of fault current (RF >> 0 Ω). Therefore, a correct analysis of the thickness stabilizer should consider the whole fault current range.

The thickness of the silver stabilizer has been varied from 1.8 μm up to 4.8 μm. For each value, the single conductor length has been chosen in order to respect the maximum allowable temperature criteria of 360 K under clear three phases short-circuit (fault resistance RF=0 Ω). The figure above represents the maximum temperature reached by the weakest part of the tape as a function of the fault impedance. We clearly see that the most "dangerous" situation arise with quite high impedance faults which means low fault currents.

This work is necessary in order to determine optimum stabilisation, remembering that higher stabilization is safer but implies higher tape length therefore higher costs.