2.2.2. Modelling Methodology

The following Section presents the detailed equations that have been used for the SCs development and the subprocess that has been followed to calculate the resistance of each layer. The modelling method followed is based on the equations proposed by the authors in [40], in which the modelling of transformers with superconducting windings is presented. To describe the HTS and copper layers, the operation of the multilayer SCs has been divided into three modes (referred as three distinct stages for simplicity) with respect to the current distribution and the values of the equivalent resistance. Stage 1 refers to the superconducting mode, where the applied current is lower than the critical current *IC* and the temperature is considered to be below the critical temperature *TC*,

$$I\_{applied} \ll I\_C \tag{5}$$

$$T = T\_{\text{operating}} \tag{6}$$

where *Toperating* is the operating temperature of 70 K.

Stage 2 refers to the flux flow mode, when the quench starts, and is determined by the following boundary conditions:

$$I\_{applied} \gg I\_{\mathbb{C}} \tag{7}$$

$$T < T\_{\mathbb{C}} \tag{8}$$

At this stage the HTS tapes start to quench and their resistivity increases sharply as a function of the current density *JC* and the accumulated heat.

At the final mode, stage 3, which is described by the boundary conditions (9) and (10), the HTS layer completely loses its superconductivity and enters normal state.

$$I\_{\text{apylied}} > I\_{\text{C}} \tag{9}$$

$$T > T\_{\mathbb{C}} \tag{10}$$

The main parameters that affect the resistance value and the operation mode of the HTS tapes-layers are the critical current density *JC*, and the critical temperature *TC* [7]. The relationship between the temperature T, the current density *JC* and the critical current *IC* is given by the following equations,

$$J\_{\mathbb{C}}(T) = \begin{cases} J\_{\mathbb{C}0}, \frac{\left(\frac{(T\_{\mathbb{C}} - T(t))^{a}}{\left(T\_{\mathbb{C}} - T\_{0}\right)^{a}}\right)^{a}}{\mathbf{0} \cdot T > T\_{\mathbb{C}}} \\ \mathbf{0} \cdot T > T\_{\mathbb{C}} \end{cases} \tag{11}$$

$$J\_{C0} = \frac{I\_{C\\_initial}}{s\_{HTS}}\tag{12}$$

$$I\_{\mathbb{C}\\_initial} = 26\mathcal{T} \cdot \mathbf{n} \tag{13}$$

$$ts\_{HTS} = w\_{HTS} \cdot t\_{HTS} \cdot \mathbf{n} \tag{14}$$

where *JC*0 is the critical current density *A*/*m*<sup>2</sup>at the initial operating temperature *T*0 = 70 K; *TC* = 92 K is the critical temperature of the HTS superconducting tape; the density exponent *a* is 1.5; *IC* \_ *initial* corresponds to the initial value of the critical current and *sHTS* is the cross section area of the superconductor; *wHTS* is the width of the HTS material and *tHTS* is the thickness of the HTS material and *n* is the number of tapes. As can be seen from Equations (11)–(13), the value of the critical current density *JC*(*T*), and by extent the value of the critical current *IC*\_*initial* , decreases drastically as the temperature *T*(*t*) rises. The temperature dependence on the critical current density is known in literature as 'critical current density degradation' [41]. The effect of the resulting degradation must be taken into consideration for the design of large-current-capacity AC SCs and their cooling systems.

To better understand the operation of the HTS cable it is crucial to estimate the resistance of the HTS and the copper stabilizer layers and the equivalent resistance of the SCs at every stage. Initially, at stage 1, the HTS tape is in a superconducting state. The resistivity of the HTS tape is ρ0 = 0 (<sup>Ω</sup>·<sup>m</sup>) and therefore its total resistance equals approximately zero. The copper stabilizer resistance has been considered constant and the total equivalent resistance of the cable is equal to the HTS layer resistance, as the main current flows through it only. At stage 2, when the applied current exceeds the value of the critical current, the resistivity of the HTS tape increases exponentially as a function of the current density and the temperature, according to Equation (15),

$$\rho\_{HTS} = \frac{E\_{\mathbb{C}}}{f\_{\mathbb{C}}(T)} \cdot \left(\frac{J}{f\_{\mathbb{C}}(T)}\right)^{N-1}, \text{ if } > I\_{\mathbb{C}}, \ T < T\_{\mathbb{C}} \tag{15}$$

where *EC* = 1 μV/cm is the critical electric field; the coefficient N has been selected to be 25, while the YBCO tapes should be within the range of 21 to 30 [42]. The copper stabilizer resistance corresponds to a constant value, similar to that of stage 1. This approximation can be confirmed by the small variation of copper resistivity with the temperature rise at this stage. The total resistance of the superconductor is obtained by the equation for equivalent resistance of parallel electrical circuits,

$$R\_{\rm SC} = \frac{R\_{HTS} \cdot R\_{\rm Cu}}{R\_{HTS} + R\_{\rm Cu}} \tag{16}$$

where *RSC* is the total resistance of the SC during stage 2.

When *T* > *TC*, stage 3 has been initiated, which corresponds to the normal resistive mode. The HTS layer-tape resistance reaches values much larger than the copper stabilizer resistance. For modelling purposes, a maximum limit has been set for the HTS resistance value at stage 3. However, in this case the resistivity (<sup>Ω</sup>·<sup>m</sup>) of the copper changes with respect to temperature rise and is determined by Equation (17). The maximum value that copper resistivity can reach is calculated for *T* = 250 K, which has been selected as the upper temperature limit [43].

$$
\rho\_{\square u} = (0.0084 \cdot T - 0.4603) \cdot 10^{-8}, 250 \text{ K} \geq T > T\_{\mathbb{C}} \tag{17}
$$

During the normal mode, the equivalent resistance of the SFCLC is a ffected solely by the value of the copper stabilizer resistance, as the transient current is diverted into the copper layers.

#### 2.2.3. Thermal Transfer Analysis during the Quenching

Superconducting tapes are immersed in liquid nitrogen *LN*2, which is used as a refrigerant for cooling the SCs below a certain temperature. When the resistance of the HTS tapes is zero, (stage 1) the amount of the power dissipated is not considered significant. When a fault occurs, the resistance increases, and heat is generated by the superconductor. The generated heat increases the superconductor temperature and part of it is absorbed by the *LN*2 circulation system (the heat transfer with the external environment has been neglected). The power dissipated is a function of the fault current and can be calculated by Equation (18),

$$P\_{\rm diss} = i(t)^2 \cdot R\_{\rm SC} \tag{18}$$

where *t* is time and *RSC* is the equivalent resistance of the superconductor.

The cooling power that can be removed by the *LN*2 cooler is given by Equation (19),

$$P\_{\text{cooling}} = h \cdot A \cdot (T(t) - 70) \tag{19}$$

where *T*(*t*) is the temperature; *A* is the total area that is covered by the cooler; *h* is the heat transfer coe fficient. The heat transfer coe fficient is a function of the temperature and considered as the major factor which determines the cooling system e ffectiveness and the cable recovery, representing the heat transfer process between the superconducting tapes and the *LN*2. Equations (20)–(23) below present the calculation of *h* based on the temperature variation [44].

$$h = 125 + 0.069 \cdot \Delta T,\ 56.3 \le \Delta T \le 214\tag{20}$$

$$h = 12292.13 - 709.32 \cdot \Delta T + 14.735 \cdot \Delta T^2, \text{ 18.94} \le \Delta T \le 56.3 \tag{21}$$

$$h = 82.74 - 131.22 \cdot \Delta T + 37.64 \cdot \Delta T^2, \ 4 \le \Delta T \le 18.94 \tag{22}$$

$$h = 21.945 \cdot \Delta T, \; 0 \le \Delta T \le 4 \tag{23}$$

If Equation (19) is subtracted from Equation (18) then the net power *PSC* can be calculated. Equation (24) is the thermal equilibrium equation which gives the part of the dissipated power which leads to temperature rise in the superconductor during the quenching process.

$$P\_{\text{SC}}(t) = P\_{\text{diss}}(t) - P\_{\text{cooling}}(t) \tag{24}$$

Finally, Equation (25) gives the temperature *T*(*t*) of the superconducting tapes at each iteration step,

$$T(t) = T\_0 + \frac{1}{\mathbb{C}\_p} \cdot \int\_0^t P\_{\text{SC}}(t)dt\tag{25}$$

where *T*0 is the initial temperature of the HTS materials and *Cp* (J/K) is the heat capacity.

For stage 2, when the quenching starts, the current starts to flow through the copper layer. However, as the temperature rise is not very high at this stage, the copper heat capacity variation with the temperature is neglected. The heat capacity of the YBCO material can be calculated by Equation (26) and the volume of the cable by Equation (27),

$$
\mathbb{C}\_P = 2 \cdot T \cdot d \cdot v \tag{26}
$$

$$
\omega = l \cdot th \cdot w \cdot n \tag{27}
$$

where *d* is the density of the material, *T* is the temperature, *v* is volume and *l* is length; *th* is the thickness and *w* is the width and *n* is the number of tapes. At stage 3, when the resistance of the HTS tapes-layer has reached very high values due to the increased temperature, the fault current flows through the copper layers. In this case the heat capacity in Equation (28) is substituted by the total heat capacity of the superconductor Equation (29) gives the heat capacity of the copper layer,

$$\mathbf{C}\_{P} = \mathbf{C}\_{PHTS} + \mathbf{C}\_{Cu} \tag{28}$$

$$\mathbf{C}\_{\rm PC} = \mathbf{C}\_{\rm Cu} \cdot d\_{\rm Cu} \cdot \boldsymbol{\upsilon} \tag{29}$$

where *CCu* is the heat capacity of the copper and *dCu* is the density of the copper.

The classification of the quenching process and the corresponding characteristics of each stage are listed in Table 1.


**Table 1.** Quenching characteristics.

Matlab has been used to model Equations (11)–(29) in order to compute the resistance values of the HTS tapes *RHTS*, the copper stabilizer *RCu*, and the variation of the temperature Δ *T* of the superconductor. The calculation process is shown in Figure 5. *To*,ι, *IC*\_*initial*, *JCinitial* are the initial values of the operating temperature, critical current and critical current density for the first iteration, respectively. Once the *Irms* gives a value of current density *Ji*, which exceeds the critical value *JC*, the HTS tapes start to quench. During the quenching process the values of *Irms Pdiss*, *Pcooling*, *PSC*, and *Ti*+<sup>1</sup> are updated in each time step, *Tstep*. The calculation process terminates once the *Ti*+1, *RHTS*, and *RCu* reach their maximum values, indicating that superconductor has entered into normal state.

**Figure 5.** Flowchart corresponding to the calculation process for the resistance values of the HTS tapes *RHTS*, the copper stabilizer *RCu*, the equivalent *Req*, and the variation of the temperature Δ*T* of the superconductor.
