Introduction
Compared with low temperature superconducting (LTS) tapes, second-generation high-temperature superconductor (2G HTS) coated conductors have higher critical transition temperature, critical current density, and critical magnetic field;thus 2G HTS-coated conductors have a broad range of application prospects. The magnets made using HTS-coated conductors are small and widely used because of their high magnetic field and energy density. In terms of energy storage, wound magnets composed of conductors exhibit a reduced inductance and a shorter reaction time than single tapes. It should be noted that HTS tapes are more likely to be damaged than LTS tapes. Superconducting magnetic energy storage(SMES) with extremely high magnetic fields and current density leads to tape deformation and current carrying capacity degradation, which can even result in the loss the superconductivity. Therefore, we must consider the mechanical stability of HTS tapes [1], [2], [3].
The relationship between the strain and the critical current of HTS tapes has been measured at 77 K. The critical current has been reported to decrease significantly only when the tape strain reaches 0.5%. For lower strains, the critical current of the tapes was found to be relatively stable. However, irreversible strain has not been considered in the previous works [4]. The stress strain relationship of HTS tapes has been studied at different temperatures. Tapes have a high yield strength and a high critical current density at low temperatures [5]. Therefore, a higher energy density can be obtained by lowering the operating temperature of the magnet.
When the magnet is being charged, the superconducting tape does not only support the generation of a conduction current but is also affected by the changing magnetic field. According to Lenz's law, the tapes induce a current in the opposite direction to the transmission current, which is named screening current. In recent years, the tape damage due to such a screening current has resulted in an uneven stress distribution, and more attention has been devoted to this issue. Experiments on small coils and numerical simulations of the hoop stress modification due to screening currents have shown that the screening currents lead to a large increase in the local stress and a more pronounced hysteresis phenomenon [6]. The National High Magnetic Field Laboratory(US) has examined tape switching via metallographic cross-section observations and Hall microscopy [7]. A previous work [8] has summarized the research progress on screening currents in recent years and has included the latest research results, focusing on experimental characterization, numerical simulations, and possible solutions.
At present, magnetic coils are mainly solenoidal or toroidal. Toroidal coils can be composed of a multiple short coil or a D-shaped coil, which have a low leakage magnetic field but not a high energy storage efficiency. Solenoidal coils have the advantages of a simple structure, a high material utilization rate, and a high energy storage efficiency. Furthermore, solenoidal coils are simpler to manufacture and permit an easier handling of the mechanical stresses imposed on the structure due to Lorentz forces. The current carrying capacity of a single tape is limited, and the inductance of a 10-MJ magnet is too large to meet the fast charge and discharge requirements. In addition, more tapes are needed to build magnets,thus greatly increasing the production costs [9], [10], [11].
In this work, through the finite element method, the changes in the stress distribution in a 10-MJ SMES magnet with quasi-isotropic strands (QISs) and stacked-tape conductors (STCs) during charge and discharge are analyzed. Under the influence of the screening current, both magnets are subjected to a huge local stress. The stress on the QIS magnet is lower and more uniform that on the STC.
Magnet parameters
In this work, two conductors, namely QIS and STC, were used for the SMES magnet design. The conductor section is shown in Fig. 1. The center of the QIS are four stacked sub-strands arranged in geometrical symmetry. these sub-strands are made from 20 HTS tapes stacked in parallel. The tapes stacked in the center are connected by soldering, and aluminum is filled in the space between the superconducting core and the copper sheath [12]. The STC consists of 40 4-mm-thick HTS tapes stacked vertically, resulting in the same square cross section as for the QIS. The main parameters are listed in Table 1.
Fig. 1. Cross sections. |
Table 1. Main parameters of the two conductors. |
| Parameter | Value |
|---|---|
| Tape width (QIS/STC)/mm | 2/4 |
| Tape thickness/mm | 0.1 |
| Number of tapes (QIS/STC) | 80/40 |
| Thickness of the copper sheath/mm | 0.4 |
| Outer diameter (QIS/STC)/mm | 6.457 |
The bending radius of the conductor is large. Toroidal magnets need more space, which makes the design of cryogenic vessels difficult. Therefore, a solenoidal magnet was used. The critical current density of the conductor in the magnet is not uniform. The critical current density of the conductor was calculated from the point where the critical current density is the smallest. Furthermore, the safety margin was set to 80%. The critical current of a magnet depends on the working current and structure. The structure of the magnet is adjusted according to its inductance. Setting the minimum amount of tape material as the goal and a stored energy of 10 MJ as the constraint, the structural parameters of the magnet can be obtained through iteration. The inductance is determined by the following expression:
$E=\frac{1}{2} I^{2} L$
where E is the stored energy, I is the operating current, and L is the inductance. The inductance can be calculated using the flux method.
Table 2. Main parameters of the two magnets. |
| Parameter | Value |
|---|---|
| Operating temperature/K | 20 |
| Number of coils | 13 × 26 |
| Height of the element coil/mm | 173 |
| Inner radius of the coil/mm | 432 |
| Outer radius of the coil/mm | 519 |
| Inductance/H | 0.1441 |
| Operating current/kA | 12 |
| Stored energy/MJ | 10.1 |
| Seed of charging/kA/s | 1.2 |
| Seed of discharging/kA/s | 1.2 |
Numerical calculation of the stress distribution
Magnetic field and current calculation
The structure of the magnet is complex, the parameters of each part are not the same, and the coupling between the electromagnetic field and structure is challenging, so it is difficult to obtain an exact analytical expression of stress distribution. The modelling method of the two magnets is the same. Therefore, in this work only the implementation of the QIS model is described in detail. The model adopts the direct coupling method, that is, the electromagnetic and mechanical parameters are calculated simultaneously in each finite element cell. This method can simplify the calculation while avoiding the cumulative error caused by multiple calculations. Before developing the solid mechanical model, it is necessary to build the electromagnetic model of the magnet first.
The calculation amount increases exponentially, and the existing computing power can hardly solve a three-dimensional (3D) magnet model. Assuming that the magnet is an ideal cylinder, the model can be simplified to a two-dimensional (2D) axisymmetric model that uses the cylindrical coordinate system. Fig. 2 shows the overall structure of the model. The outer insulation layer of the conductor is simplified and assumed to be a stainless-steel shell. The outermost air domain is not fully represented. The HTS tapes are regarded as a superconducting layer using the homogenization method. The tapes are soldered together, ignoring the relative sliding between them. Assuming that the current between the tapes can flow freely, and in order to further simplify the model, ten adjacent tapes in the same direction are reduced to one [15].
Fig. 2. Geometric model of the magnet. |
The critical current density of the HTS is related to the magnetic field and temperature. Assuming that the magnet temperature remains unchanged during operation, it is possible to consider only the effect of the magnetic field. The relationship between the critical current and the magnetic field of the superconducting tape is described by this empirical expressions:
$\boldsymbol{I}_{\boldsymbol{c}}(\boldsymbol{B}, \boldsymbol{\theta})=\frac{\mathbf{b}_{0}}{\left(\boldsymbol{B}+\beta_{0}\right)^{\alpha_{0}}}+\frac{\mathrm{b}_{1}}{\left(\boldsymbol{B}+\beta_{0}\right)^{\alpha_{1}}} \times\left[\omega_{1}^{2}(\boldsymbol{B}) \cos ^{2}\left(\boldsymbol{\theta}-\phi_{1}\right)+\sin ^{2}\left(\boldsymbol{\theta}-\boldsymbol{\phi}_{1}\right)\right]^{-\frac{1}{2}},$
$\omega_{1}(\boldsymbol{B})=\mathbf{c}\left[\boldsymbol{B}+\left(\frac{1}{\mathbf{c}}\right)^{\frac{5}{3}}\right]^{\frac{8}{5}},$
Table 3. Parameter values obtained from a nonlinear fit. |
| Parameter | Value |
|---|---|
| b0 | 8870 |
| b1 | 18,500 |
| α0 | 1.30 |
| α1 | 0.809 |
| β0 | 13.8 |
| φ1 | −0.180° |
| c | 2.15 |
The governing equation of the electromagnetic field is obtained according to Ampere's law and Faraday's law of electromag-netic induction. In cylindrical coordinates, the equation system is:
$\left\{\begin{aligned} \frac{\partial E_{\varphi}}{\partial r} & =\mu_{0} \frac{\partial H_{\lambda}}{\partial t} \\ \frac{\partial E_{\varphi}}{\partial z} & =-\mu_{0} \frac{\partial H_{r}}{\partial t} \end{aligned}\right.$
$J_{\varphi}=\frac{\partial H_{z}}{\partial r}-\frac{\partial H_{F}}{\partial z}$
where Hr and Hz are the dependent variables for solving the equation, which represent the radial and axial magnetic field intensities, respectively; Jφ is the φ-direction current density; μ0 is the vacuum permeability. Eφ is the φ-direction electric field intensity, which is obtained from the E-J characteristic in the superconducting region according to:
$E_{\varphi}=\mathrm{E}_{\mathrm{c}}\left(\frac{J_{\varphi}}{J_{c}(B)}\right)^{\mathrm{n}},$
and Ohm's law in the nonsuperconducting region according to:
$E_{\varphi}=\rho \cdot J_{\varphi}$
where Ec = 10−4 V/m is the critical electric field strength,n is usually set to 25, and ρ is the resistivity of the HTS.
The relative permeability is 1. Magnetic insulation is only set at the outer edge of the air domain. The input current is added by integrating the current density of the superconducting conductor, which is integrated separately for each conductor. The charging time is 1 s.
Stress calculation
When an electric current passes through the tapes, its force equals the Lorentz force on the tape interior. Ignoring the circumferential magnetic field, this force can be expressed as:
$\left\{\begin{array}{c} f_{r}=B_{z} \cdot J_{\varphi} \\ f_{z}=-B_{r} \cdot J_{\varphi} \end{array}\right.$
where fr and fz are the radial and axial Lorentz components, respectively, and Br and Bz are the radial and axial magnetic induction intensities, respectively. The tape is homogenized by ignoring the stripping behavior between adjacent layers. The tape can be approximated as an isotropic material. It is assumed that the material of the other part of the model is also an isotropic linear elastic material, and the stress is within the elastic range. The stress balance equation, geometric equation, and constitutive equation can be obtained by ignoring the inertia force. They can be expressed as:
$\left\{\begin{array}{c} \frac{\partial \sigma_{r r}}{\partial r}+\frac{\partial \sigma_{\phi r}}{\partial \varphi}+\frac{\partial \sigma_{z r}}{\partial z}+f_{r}=0 \\ \frac{\partial \sigma_{r \varphi}}{\partial r}+\frac{\partial \sigma_{\varphi \varphi}}{\partial \varphi}+\frac{\partial \sigma_{z \psi}}{\partial z}=0 \\ \frac{\partial \sigma_{r z}}{\partial r}+\frac{\partial \sigma_{\varphi z}}{\partial \varphi}+\frac{\partial \sigma_{z z}}{\partial z}+f_{z}=0 \end{array}\right.$
$\left\{\begin{array}{c} \varepsilon_{r r}=\frac{\partial u}{\partial r} \\ \varepsilon_{\varphi \varphi}=\frac{\partial v}{\partial \varphi} \\ \varepsilon_{z z}=\frac{\partial w}{\partial z} \\ \varepsilon_{\varphi z}=\varepsilon_{z \varphi}=\frac{1}{2}\left(\frac{\partial v}{\partial z}+\frac{\partial w}{\partial \varphi}\right) \\ \varepsilon_{z r}=\varepsilon_{r z}=\frac{1}{2}\left(\frac{\partial u}{\partial r}+\frac{\partial u}{\partial z}\right) \\ \varepsilon_{r \varphi}=\varepsilon_{\varphi r}=\frac{1}{2}\left(\frac{\partial u}{\partial \varphi}+\frac{\partial u}{\partial r}\right) \end{array}\right.$
$\left\{\begin{array}{c} \varepsilon_{r r}=\frac{1}{\mathrm{E}}\left(\sigma_{r r}-\nu\left(\sigma_{\varphi \varphi}+\sigma_{z z}\right)\right) \\ \varepsilon_{\varphi \zeta \varphi}=\frac{1}{\mathrm{E}}\left(\sigma_{\varphi \varphi}-\nu\left(\sigma_{r r}+\sigma_{z z}\right)\right) \\ \varepsilon_{z z}=\frac{1}{\mathrm{E}}\left(\sigma_{z z}-\nu\left(\sigma_{\varphi \varphi}+\sigma_{r r}\right)\right) \\ \varepsilon_{\varphi z}=\frac{(1+\nu)}{\mathrm{E}} \sigma_{\varphi z} \\ \varepsilon_{z r}=\frac{(1+\nu)}{\mathrm{E}} \sigma_{z r} \\ \varepsilon_{r \varphi}=\frac{(1+\nu)}{\mathrm{E}} \sigma_{r \varphi} \end{array},\right.$
where σij and εij are the components of the stress and strain tensors, respectively. Because of the 2D axisymmetric model, the current distribution in each vertical section of the magnet is the same. The effect of the shear stress is ignored in the calculation. u, v, and w are the radial, hoop, and axial displacement fields, respectively. E and υ are Young's modulus and Poisson's ratio, respectively, which are listed in Table 4 [17].
Table 4. Young's modulus and Poisson's ratio. |
| Young's modulus/GPa | Poisson's ratio | |
|---|---|---|
| HTS | 228 | 0.307 |
| Aluminum | 70 | 0.33 |
| Copper | 110 | 0.35 |
| Insulator | 22 | 0.15 |
| Steel | 205 | 0.28 |
The strain on each part of the magnet caused by the Lorentz force cannot be ignored, so there is no rigid domain in the model, and we can not set fixed constraints. The top and bottom of the magnet are symmetric structures, and the central radial magnetic field is almost 0. In the calculation, we ignore the axial stress on the magnet center and add a roller support along the radial direction.
Results and discussion
Stress without the screening current
The hoop stress on the magnet is more than twice the radial and axial stresses, which is the main reason for the degradation of the tape performance. The hoop stress on the magnet was obtained from the average current density model, and the results are shown in Fig. 3.
Fig. 3. Hoop stress obtained from the average current density model: (a) magnet with protective measures adopted; (b) magnet without any adopted protective measure. |
In the absence of protective measures, the stress on the magnet is far beyond its endurance limit. The aluminum filler can be in close contact with the superconducting tapes, while the copper sheath has a higher strength, so they need to be used together. The stress is reduced to a safe level when the protective measures are in place. The hoop stress on the magnet is entirely tensile stress. In the radial direction, the hoop stress on the inner wall of the coil is much greater than that on the outer wall. The stress on the magnet center is the lowest because the magnetic fields generated by the surrounding coils cancel each other out, and there is almost no magnetic field in the central area. In the axial direction, the stress distribution of the magnet is axisymmetric.
Stress during charging
The magnet was charged at a rate of 1.2 kA/s. In order to improve the convergence of the calculation, the charging speed was reduced upon approaching the working current. The maximum stress on the two magnets changes with time, as shown in Fig. 4.
Fig. 4. Comparison of the maximum hoop stress or the different magnetsduring charging. |
The evolution of the maximum stress on the two magnets with time is similar and exhibits an evident hysteresis. After charging is complete, the stress continues to rise to its maximum value and then hardly attenuates. The stress on the QIS magnet is less than that on the STC magnet. Under the influence of the screening current, the stress is more than twice that derived from the average current model. The maximum hoop strain for the QIS and STC magnet is 0.08% and 0.09%, respectively; these values are not enough to cause a decrease in the critical current.
Fig. 5 shows the distribution of the hoop stress at 2.6 s. The stress distribution is the same as the average current density distribution. In the radial direction, the stress is more concentrated in the inner wall of the coil, and the position at which the stress is maximum is shifted. In the axial direction, the stress distribution of the magnet is not completely axisymmetric, and the presence of the screening current affects the current density and the uniformity of the magnetic field.
Fig. 5. (a) and (b) are hoop stress distribution(unit:MPa).(c) and (d) are hoop strain distribution,and radial and axial displacements. The displacement is magnified by a factor of 20. |
According to the Weibull distribution statistics, the cumulative function can be written as:
$F(\sigma)=1-\exp \left[-\left(\frac{\sigma-\sigma_{0}}{\mathrm{~b}}\right)^{\mathrm{a}}\right] \text {, }$
where σ0 is the position at which the stress is minimum, a is the shape parameter, and b is the size parameter. The stress on each point is sorted from the smallest to the largest, and the probability value corresponding to each stress can be calculated using Bernard's formula:
$F(\sigma)=\frac{i-0.3}{\mathrm{n}+0.4}$
where i is the order, and n = 368082 is the total number of samples. By taking the logarithm of both sides of the Weibull accumulation function and linearizing it, Eq. (14) is obtained. According to Eqs. (13), (15), a set of linearized data points can be obtained. By performing a linear regression analysis on the data points, the values of a and b can be calculated. Fig. 5 was thus obtained, where a = 7.5451 and b = 144.9093 for the QIS case, while a = 8.2328 and b = 152.5258 for the STC case.
$\ln \left\{\ln \left[\frac{1}{1-F(\sigma)}\right]\right\}=\mathrm{a} \cdot \ln \left(\sigma-\sigma_{0}\right)-\mathrm{a} \cdot \ln (\mathrm{b})$
$\left\{\begin{array}{c} y=\ln \left\{\ln \left[\frac{1}{1-F(\sigma)}\right]\right\}. \\ x=\ln \left(\sigma-\sigma_{0}\right) \end{array}.\right.$
Fig. 6 shows the Weibull probability distribution of the hoop stress during discharge. It can be clearly seen that the stress amplitude variation for the QIS case is less than that for the STC case. The stress distribution in the QIS magnet is more uniform than that in the STC magnet. The magnetic field angle and amplitude at each position of the magnet result from the superposition of the magnetic field generated by the surrounding coils. The distribution of the magnetic field is uneven. The QIS current is less affected by the magnetic field angle. Therefore, there is a lower probability that stress concentration will occur in the QIS magnet.
Fig. 6. Weibull probability distribution of the hoop stress. |
Stress during discharge
The magnets were discharged at the same speed, and the stress distribution of the two magnets was calculated. Fig. 7 shows the screening current attenuation for the magnets during discharge and for a short period after discharge. The screening current density in the QIS magnet is much lower than that in the STC magnet, presumably due to the fact that the QIS uses narrower tapes. The maximum stress on the QIS and STC magnets is 4 and 10 MPa, respectively. The screening current does not decay as the running current decreases. When the running current is low, the screening current density has a great influence on the stress amplitude. After discharge, the screening current is the main source of the Lorentz force on the magnet. Therefore, it is necessary to take measures to eliminate the screening current after discharge.
Fig. 7. Screening current attenuation during discharge. At 1 s, the transport current is 0. The screening current cannot be completely eliminated for neither magnet; the QIS magnet exhibits a lower screening current. |
In order to intuitively compare the stress uniformity of the two magnets, we also calculated their Weibull distribution. Fig. 8 shows the Weibull distribution of the torus stress 4 s after the complete discharge of the two magnets. In the figure, a = 2.3488 and b = 2.2027 for the QIS magnet, while a = 2.9150 and b = 5.0610 for the STC magnet. The stress on the QIS magnet is significantly smaller than that on the STC magnet, and the distribution is more uniform.
Fig. 8. Weibull probability distribution of the hoop stress during the discharge. |
Conclusion
Magnets with coiled conductors made of stacked tapes have a higher transport current and a smaller inductance, which are beneficial for achieving faster charge and discharge processes.
The stress reaches the maximum value after the charging process is completed due to the hysteresis phenomenon. After protection measures are adopted, such as a copper sheath and an aluminum filler, the stress on the tapes is still within the safe range, and the magnets can satisfy the requirements of rapid charging. The stress amplitude in the QIS magnet is lower than that in the STC magnet, and the stress uniformity in the former is greater than that in the latter.
After discharge, there is a residual screening current in the magnets. Therefore, the magnets still bear a non-negligible stress. The magnitude of this stress is closely related to the screening current.
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgment
This work is supported in part by the National Natural Science Foundation of China under Grant No. 51977078.
