Calculation of AC Losses in a 500 kJ/200 kW Multifilamentary MgB₂ SMES Coil

Abstract

SMES technology based on MgB${}_2$ superconductor and cryogenic-free cooling can offer a viable solution to power-intensive storage in the short term. One of the main obstacles to the development of dry-cooled SMES systems is the heat load removal due to the AC loss of the superconductor during fast charging and discharging cycles at high power. Accurate knowledge of the amount and distribution of AC loss in the coil is of paramount importance for the sizing and the design of the cooling system and for assessing the possible operational limits of the technology. In this manuscript, the AC loss of a 500 kJ/200 kW multifilamentary MgB${}_2$ SMES during charge–discharge cycling at full power is numerically investigated. The methodology and assumptions of the calculation are briefly resumed, and numerical results are reported and discussed in detail. In particular, the time profile and the distribution of the dissipated power all over the coil are reported. An average dissipation of 85.5 mW/m is found all over the coil during one charge–discharge cycle at full power, with a peak of 150.1 mW/m in the turns lying at the ends of the coil.

1. Introduction

Superconducting magnetic energy storage (SMES) is an established technology based on low-temperature superconductor (LTS) materials. Several LTS SMES systems with rating up to 10 MW have been developed in the past and integrated into the grid [1,2,3,4]. In these systems, the superconductor (Ni-Ti) operates at 4.2 K and the cooling is obtained utilizing a liquid helium (LHe) bath. Despite the technical success, LTS SMES technology has not been widespread in the market due to the complexity, the cost and the safety concerns related to LHe usage. The use of new high-temperature superconductors enables new prospects for SMES technology, today available at the industrial level and able to operate (for the required field level) at much higher temperatures and in the range of 20–50 K compatible with LHe-free cooling [5,6,7,8,9,10,11]. In particular, using MgB${}_2$ with dry cooling offers a viable short-term and low-cost alternative for developing SMES technology [11,12,13,14,15,16,17]. In this context, the national DRYSMES4GRID research project has been funded by the Italian Minister of Economic Development (MISE) and aimed at demonstrating the viability of MgB${}_2$ SMES technology with no use of liquid cryogens [18,19].

One of the main obstacles to the development of dry-cooled SMES systems is the heat load removal due to AC loss of the superconductor during fast charging and discharging cycles at high power. Accurate knowledge of the amount and distribution of AC loss in the coil is of paramount importance for the sizing and the design of the cooling system and for assessing the possible operational limits of the technology, while substantial work has been conducted for predicting AC loss of SMES coils (or fast ramping coils in general) made of 2G HTS tapes (ReBCO coated conductors) [10,20,21,22,23,24], little work has been conducted for calculating the AC loss of SMES coils made of multifilamentary MgB${}_2$ conductors. This paper investigates the space- and time-distribution of AC loss of an MgB${}_2$ SMES coil during rated operation at 200 kW. The coil is made of a composite six-filament MgB${}_2$ tape. We also discuss the cooling requirements for continuous operation (charge–discharge cycling) of the coil at rated power. The analysis was made employing the THELMA code, which is an in-house numerical environment for the coupled thermal-hydraulic (TH) and electromagnetic (EM) analysis of superconductor cables subject to transport current and applied field initially developed for fusion systems [25,26]. The work was carried out in the frame of the DRYSMES4GRID project aimed at developing a dry-cooled MgB${}_2$ SMES system with a rating of 500 kJ/200 kW. A reference six-filament MgB${}_2$ conductor was first selected, and the executive design of the 200 kW coil was carried out [27]. The project was recently concluded with successful results concerning the possibility of fast ramping operation of the dry-cooled MgB${}_2$ coil, thanks to the realization and test of a demonstrator with a 7 kW power rating. This paper focuses on the original full-scale coil with a 200 kW power rating. The details of the scaled-down demonstrator and the testing results will be the subject of a separate publication.

The paper is organized as follows. First, the characteristics of the MgB${}_2$ conductor and the 200 kW SMES coil are resumed in Section 2, along with the considered operating conditions. The THELMA model used to carry out the calculation is briefly resumed in Section 3, along with the main modeling assumptions and parameters as well as the description of the modeled geometry. Numerical results of the space- and time- distribution of the AC loss during SMES operation are reported and discussed in Section 4. Total cooling power is analyzed, and possible operation limits regarding standby time for thermal recovery before the next charge–discharge cycle are also discussed. Concluding remarks are finally drawn in Section 5.

2. Reference MgB${}_2$ Conductor Main Characteristics and Coil Operating Conditions

A composite MgB${}_2$ tape with a 2.05 × 1.1 mm${}^2$ rectangular cross-section was initially selected to develop the 200 kW SMES coil. The tape comprises six MgB${}_2$ filaments embedded in a Nickel matrix and surrounded by a Monel sheath. The filaments are twisted with a 600 mm twist pitch. The filling factor of the superconductor is 29%. The micrography of the described conductor is shown in

Table 1. Main characteristics of the MgB${}_2$ conductor.

Parameter	Value
Micrography
Number of filaments	6
MgB${}_2$ filling factor	29%
Thickness	1.1 mm
Width	2.05 mm
Twist pitch	600 mm
Critical current at 16.2 K and 1.63 T	772 A
Thickness of the tin-soldering copper strip	500 $\mathsf{\mu }$m
Thickness of electrical insulation (wrapped polyester)	125 $\mathsf{\mu }$m

. A copper strip with 500 $\mathsf{\mu }$m thickness is co-laminated and tin-soldered onto the tape before electrical insulation to improve the coil’s stability and quench protection. The tape’s critical current is 461 A at 22 K and 1.8 T. The $I_c$ vs. B performance of the tape in the temperature range from 22 K to 30 K is shown in Figure 1. A practically isotropic behavior was observed in all the considered temperature and field ranges with no appreciable impact on the magnetic field direction. The main characteristics of the MgB${}_2$ tape conductor used for the model are resumed in Table 1.

The dry-cooled MgB${}_2$ coil with 500 kJ deliverable energy and 200 kW power, forming the core of the SMES system, was designed based on the selected tape. The coil is a solenoid with an inductance $L=6.8$ H, made of 10 layers of 522 turns, each corresponding to a total length of tape of 10.1 km. The main characteristics of the coil are listed in

Table 2. Main characteristics of the MgB${}_2$ SMES.

Parameter	Value
Inner radius	300 mm
Height	1200.6 mm
Number of layers	10
Number of turns per layer	522
Length of cable	10.1 km
Voltage of the DC-bus	750 V
Min Current $I_{min}$	266.6 A
Max current $I_{max}$	467 A
Field on the conductor (at $I_{max}$)	1.63 T
$I/I_c$ ratio (at $I_{max}$)	0.6
Inductance	6.80 H
Total energy (at $I_{max}$)	741 kJ
Deliverable energy	500.4 kJ
Operating temperature	16.2 K

. The design details, including mechanical, thermal and 3D quench analysis, are reported in [28]. The coil is connected to the three-phase power grid using a DC/DC chopper and a DC/AC inverter connected to a common DC bus, forming the power conditioning systems (PCSs) of the SMES. An operating voltage $V_{dc}=750$ V is chosen for the DC bus based on the design consideration of the converters. To withstand this voltage and the corresponding electric stress, particularly during the fast commutation of the DC/DC converter, a 125 $\mathsf{\mu }$m thick insulation is applied to the tape through the polyester wrapping. Since a deliverable/absorbed power $P=200$ kW is to be guaranteed, the SMES cannot be discharged below a minimum current $I_{coilmin}=267$ A ($I_{coilmin}=P/V_{dc}$). Hence, the energy $W_e$ that can be extracted, at a power rate not lower than 200 kW, when the coil operates with a current $I_{coil}$ is given by Equation (1).

$$W_e=\frac{1}{2}L\left(I_{coil}^2-I_{coilmin}^2\right)$$

(1)

In order to ensure the target deliverable energy of 500 kJ, the operating current of the coil must reach the maximum value $I_{coilmax}=467$ A. The total energy stored in the coil at the maximum operating current is 741 kJ. It is pointed out that a residual energy of 241.7 kJ is still stored in the coil when the minimum current of 266.6 A is reached. This energy can still be extracted at a power rate lower than 200 kW and continuously decreases as the discharge proceeds. Hence, similar to all other storage systems, the complete discharge of the SMES cannot be considered for practical applications where a minimum power must be guaranteed for providing the required service.

The temperature T and the coil’s current $I_{coil}$ (via the magnetic field) affect the conductor’s critical current. Hence, a different $I_{coil}/I_c$ is obtained depending on the operating conditions. The curves with the designed coil’s constant $I_{coil}/I_c$ ratio are shown in Figure 2 as a function of the operating temperature and current. The curves with constant deliverable energy are also shown in the same plot. Since, according to Equation (1), the deliverable energy does not depend on the operating temperature but only on the current, these latter curves appear as horizontal lines in the T vs. $I_{coil}$ plane. A target value of $I_{coil}/I_c=0.6$ was chosen during the coil design as a compromise between an appropriate safety margin and a reasonable usage of a superconductor. It can be seen from Figure 2 that in order to not exceed the chosen $I_{coil}/I_c$ margin of 0.6 in the most severe operating condition, when the SMES is fully charged at 467 A, the operating temperature of the coil must not exceed 16.2 K.

The total AC loss increases with the ramp rate, hence, with the power delivered or absorbed by the coil. For calculating the AC loss in the most severe conditions, a complete discharge/charge cycle at full power is considered in this paper. The discharge and the charge at 200 kW last 2.5 s, and the cycle lasts 5 s. We now consider a generic time instant $t_0$, in which the SMES current is $I_{coil0}$, and assume that starting from $t_0$, the SMES delivers a constant (positive or negative) power P to the grid. By stating the energy balance and neglecting the losses, time evolution of the SMES current is obtained as reported in Equation (2).

$$I_{coil}\left(t\right)=\sqrt{I_{coil0}^2\mp \frac{2}{L}P(t-t_0)}$$

(2)

Equation (2) is generally valid and holds regardless of the power converters’ architecture and the DC bus’s voltage. In Equation (2), the generator convention is assumed, whereby the coil delivers a positive power and corresponds to a decrease in the SMES current. The time profile of the SMES current during a complete discharge–charge cycle at full power obtained utilizing Equation (2) is shown in Figure 3 and will be used as the input for the AC loss calculation in the following sections.

3. Calculation Model and Simulated Cases

The AC loss of the designed MgB${}_2$ coil during a discharge–charge cycle is calculated employing the electromagnetic module of the THELMA model. This model was initially developed, in cooperation with the Polytechnic University of Turin and the University of Udine, for the coupled thermal–hydraulic (TH) and electromagnetic (EM) analysis of superconductor cables subject to transport current and applied field in the frame of an EFDA initiative aimed at the analysis of fusion systems [28]. The electromagnetic module of the model, developed by the University of Bologna and used in this research [25,26], has been successfully used in a variety of problems involving different materials and architectures, including cable-in-conduit conductor (CICC) [29,30], Rutherford cables [31] and multi-filamentary wires [32]. The main features of the THELMA model are described in this Section. Further information related to the numerical implementation is reported in Appendix A.

3.1. Geometrical Model and Simulated Cases

For calculating the current distribution and AC loss, the MgB${}_2$ conductor forming a specified portion (a set of turns) of the SMES coil is subdivided into a finite number $N_E$ of 3D elements. Only the longitudinal conducting filaments of the conductor (i.e., the six MgB${}_2$ filaments and the copper strip) are considered in this phase. The matrix materials (nickel and Monel) are taken into account using the transverse conductance (as explained in the next Section) and are not directly included in the geometrical model. The 2D model of the conducting elements shown in Figure 4a, comprising $N_F=7$ elements, is first introduced. The 3D geometrical model, shown in Figure 4b, is generated by the extrusion of the 2D subdivision along the helix pattern of the coil. To account for twisting, a rotation perpendicular to the helix pattern is added to the filaments upon the extrusion. The length of the extrusion step is assigned and is a sub-multiple of the twist pitch so that an entire number of subdivisions is included in one twisting length. The extrusion is continued until the desired length of the conductor to be modeled is reached. In the process, $N_S$ sections, separated by one extrusion length, are introduced. The first and the last of these sections lay on the terminals of the modeled length of the conductor, whereas the remaining $(N_S-2)$ lay at the interior. Overall, $N_E=N_F(N_S-1)$ 3D elements and $N_{NP}=N_F{N}_S$ nodes are generated. These nodes, referred to as primal nodes, correspond to the centers of the faces shared by two longitudinally adjacent elements or laying on the terminal sections of the domain and are used for assembling the solving system of the numerical problem. The interior sections are formed by the faces shared by two adjacent elements and lying at the same position along the axial length of the modeled conductor. Each of the interior sections includes $N_F$ nodes. In this paper, we have modeled ten consecutive turns of conductors, for which the detailed space- and time- distribution of current and AC loss during one discharge and charge cycle is looked for. A uniform current density is assumed in the remaining part of the coil, which acts as a time-varying magnetic field source for the domain under investigation. The ten modeled turns belong to the same layer as shown in Figure 4c and correspond to a total conductor length of about 19 m. A total of six subdivisions per twist pitch were used, corresponding to 190 sections and a total of 1323 3D elements ($N_S=190$, $N_E=1323$, $N_{NP}=1330$). The modeled turns can occupy different positions within the coil, giving rise to different AC losses depending on the field condition at the considered position. In this manuscript, we investigated twenty different cases in which the turns were placed at the middle and the bottom of each of the ten layers, as it is schematically shown in Figure 4d for the innermost layer. The AC loss distribution and the overall energy injected in the coil during the discharge–charge cycle are finally obtained based on the results of the individual simulations, and the required cooling power and operation limits are deduced accordingly. A convergence analysis was run to investigate the impact of the number of simulated turns and the number of subdivisions per twist pitch on the numerical results. It was found that no appreciable change was produced in the calculated distributions of current and AC loss by increasing the number of simulated turns and/or by increasing the number of subdivisions per twist pitch. In every longitudinal subdivision of the discretization, each filament (superconductor or copper strip) is modeled by one element only. Hence, the total current of one longitudinal conducting element is uniformly spread all over the cross-section, and no detail of the current distribution can be obtained. Nevertheless, the coupling currents and overall current of SC filaments and strips, as well as their distribution along the conductor’s length, can be correctly reproduced through the model.

3.2. Mathematical Model

The arc length x of the coil helix, shown in Figure 4b,c, is used to state the problem. At any point of the modeled domain, any information on the discretized domain (axis of elements, vertexes, cross-section and so forth) is deduced from the x coordinate. A uniform current density is assumed in the cross-section of the longitudinal element (superconductor or strip). Any longitudinal element h of the discretization can exchange coupling current in the transverse direction (perpendicular to x) through the matrix material. The current balance of an element with infinitesimal length is schematized in Figure 5, where I is the longitudinal current of the terminal sections of the element and is expressed in A and K is the transversal coupling current per unit length that leaves or enters the element all along the lateral surface and is expressed in A/m. By stating the charge conservation, Equation (3) relation is obtained between the longitudinal current I and the transverse current K per unit length of the element.

$$\frac{dI(x,t)}{dx}=K(x,t)$$

(3)

The electric field $\mathbf{E}$ at any point of the domain is related to the current density $\mathbf{J}$ using the constitutive law of the material. The following $\mathbf{E}$ vs. $\mathbf{J}$ power law is used for modeling the superconductor material, yielding a nonlinear resistivity $\rho$

$$\mathbf{E}-\rho \left(J\right)\mathbf{J}\mathrm{with}\rho \left(J\right)=\frac{E_c}{J_c}{\left(\frac{J}{J_c}\right)}^{n-1}$$

(4)

where $E_c$ and n are the critical electric field and the power law exponent, respectively. A liner resistivity is assumed instead for the copper, the Monel and the matrix’s nickel. By using Faraday’s law, the electric field is expressed using the vector magnetic potential $\mathbf{A}$ and the scalar electric potential v as

$$\rho \left(J\right)\mathbf{J}=-\frac{\partial }{\partial t}({\mathbf{A}}_I+{\mathbf{A}}_k+{\mathbf{A}}_{coil})-\nabla v$$

(5)

where the non-linearity of the resistivity applied to the superconductor is pointed out. In Equation (5), the total magnetic vector potential $\mathbf{A}$ is split in a contribution ${\mathbf{A}}_I$ due to the longitudinal currents following in the filaments (superconductor or strip), a contribution ${\mathbf{A}}_k$ due to the transverse coupling currents flowing in the matrix and a contribution ${\mathbf{A}}_{coil}$ due to the current circulating in the remaining part of the coil not included in the modeling domain. A solution of the longitudinal current distribution problem is looked for in the weak form by imposing that Equation (5) is satisfied at any of the $N_{NP}$ nodes of the subdivision, supplemented by imposing that the overall current at any cross-section of the conductor coincides with the transport current $I_{coil}$. In order to account for the coupling currents that develop all over the modeled length, the longitudinal problem is interleaved with the transverse problem, as it is schematized in Figure 6. In particular, it is imposed that Equation (5) is satisfied on the further set of $N_{ND}$ dual nodes obtained by interleaving the original (primal) $N_{NP}$ node. These further nodes are $N_F{N}_S$ in total ($N_{NP}=N_F{N}_S$) and coincide with the centers of the longitudinal elements. Furthermore, two additional nodes are involved in the model to account for the connection of the modeled domain with the rest of the coil (see Appendix A). It is assumed that the inductive contribution of the transverse currents to the longitudinal and transverse electric fields is negligible. In other words, it is assumed that the time derivative $\partial {\mathbf{A}}_k/\partial t$ of the vector potential produced by the transverse current is much smaller than the terms $\partial {\mathbf{A}}_I/\partial t$ and $\partial {\mathbf{A}}_{coil}/\partial t$ produced by the longitudinal current and the transport current of the coil, respectively. Based on this assumption, Equation (5) yields

$$\rho \left(J\right)\mathbf{J}=-\frac{\partial }{\partial t}({\mathbf{A}}_I+{\mathbf{A}}_{coil})-\nabla v$$

(6)

In this way, a non-dynamic circuit is associated with the transverse problem that allows for solving the scalar electric potential v. More in particular, since due to neglecting $\partial {\mathbf{A}}_k/\partial t$ no time derivative of transverse currents is involved, the scalar electric potential v of all nodes can be related, via the matrix of the mutual conductances between the elements, to the transversal currents and, afterward, to the longitudinal currents via Equation (3). The matrix of conductances needed for solving the transverse problem is calculated using commercial software during pre-processing. In this way, the solving system of the longitudinal problem (Equation (6) at the primal nodes) only involves the time derivative of the longitudinal currents and forms the state equation of the dynamic problem that can be expressed as

$$\left\{\begin{array}{c}\mathbf{M}\frac{d\mathbf{I}}{dt}=-\mathbf{f}\left(\mathbf{I}\right)-\mathbf{U}\frac{d{I}_{coil}}{dt}-\mathbf{V}{I}_{coil}\hfill \\ \mathbf{I}\left(0\right)={\mathbf{I}}_0\hfill \end{array}\right.$$

(7)

in which $\mathbf{I}$ is the vector of currents in the first ($N_F-1$) filaments at the internal primal nodes (i.e., not lying on the terminal sections), $\mathbf{M}$ is the matrix of the mutual inductance coefficients, $\mathbf{U}d{I}_{coil}/dt$ is the vector of the electromotive forces induced by the time change of the field produced by the coil, $\mathbf{f}\left(\mathbf{I}\right)$ is the vector of the resistive voltage drops and vector $\mathbf{V}{I}_{coil}$ takes the effect of the applied coil current into account.

Since in the investigated cases, the coil operates in steady state in the initial instant of the simulation, a uniform current in the superconductor filaments and no current in the copper strip is assumed as the initial condition by assigning vector $I_0$ accordingly. The details on the deduction of Equation (7) and the precise definition of the terms involved are reported in Appendix A.

Once the longitudinal currents are obtained, at any instant, by solving the state Equation (7), the transverse current can be obtained by taking the numerical space derivative according to Equation (3). Hence, the profile of current density J can be reconstructed, at any instant, all over the modeled length of the conductor, both in the conducting filaments and in the matrix material. Finally, the distribution of AC loss, in W/m${}^3$, can be obtained by taking $\rho J^2$ and the overall AC loss, in W, of the modeled domain can be obtained from the volume integral. In particular, the overall loss can be split into the two components $P_{filaments}$ and $P_{matrix}$, occurring in the longitudinal filaments (superconductor and Cu strip) and in the matrix, respectively, and calculated as

$$\begin{array}{cc}\hfill P_{filaments}\left(t\right)& =\underset{{\tau }_{filaments}}{\int }\rho {\mathbf{J}}^2({\mathbf{r}}^{\prime },t)d^3{r}^{\prime }\hfill \\ \hfill P_{matrix}\left(t\right)& =\underset{{\tau }_{matrix}}{\int }\rho {\mathbf{J}}^2({\mathbf{r}}^{\prime },t)d^3{r}^{\prime }\hfill \end{array}$$

(8)

It is finally stressed that the matrix $\mathbf{M}$ of the differential Equation (7) is dense and may require prohibitive storage resources and inversion time by increasing the number of elements. A method for replacing Equation (7) with a sparse and, hence, more manageable problem, obtained by involving only the deviation of the elements’ current from the value that it would reach in case of uniform distribution among the filaments, is discussed in Appendix A. In Section 4, the results of the calculation of the AC loss of the MgB${}_2$ coils are reported. The parameters assumed to carry out the calculation are listed in

Table 3. Parameters used for the numerical calculation.

Parameter	Value
Operating temperature	16.2 K
Critical current density Jc of the MgB${}_2$ at 16.2 K and 1.63 T 1	1.18 × 10${}^5$ A/cm${}^2$
Power law n-exponent	30
Critical electric field of the power law	1.0 mV/cm
Resistivity of Copper at 16.2 K	0.0178 m$\mathsf{\Omega }$cm
Resistivity of Nickel at 16.2 K	1.1 m$\mathsf{\Omega }$cm
Resistivity of Monel at 16.2 K	42 m$\mathsf{\Omega }$cm

¹ The data are deduced from the critical current and filling factor reported in Table 1. In the calculation, the dependence of J_c on the magnetic field, deduced from the critical current data of Figure 1, is used.

. In all simulations, isothermal operation at the design value of 16.2 K was assumed.

4. Numerical Results and Discussion

The AC loss of the MgB${}_2$ coil during one complete discharge–charge cycle at full power (200 kW) is calculated in this section. The main loss data obtained are finally summarized in

Table 4. Main loss of the SMES coil during one discharge–charge cycle at full power.

Average AC loss per unit length of conductor at the center and at the bottom of all layers (mJ/m)
Layers	Center	Bottom	Layers	Center	Bottom
layer 1	122.3	150.1	layer 6	39.9	130.7
layer 2	103.1	141.9	layer 7	29.1	128.0
layer 3	91.0	156.4	layer 8	15.7	117.9
layer 4	74.9	139.0	layer 9	10.2	109.6
layer 5	59.3	134.4	layer 10	20.2	114.6
Overall energy loss of the coil during one cycle (J)					4319

. The coil’s current during the cycle is shown in Figure 3 and is assigned as the model domain’s transport current in the calculation. The discharge phase starts at $t=0$ when the coil operates at the maximum current $I_{coilmax}=467$ A and ends at the time $t=2.5$ s when it reaches the minimum current $I_{coilmin}=267$ A. Afterward, the recharge starts that ends at $t=5$ s when the coil reaches the maximum current again. Strict steady-state operation at constant current is assumed before ($t\le 0$ s) and after ($t\ge 5$ s) the cycle. In the following, different time instants during the discharge, the recharge and the steady state are considered to describe the current’s time behavior and the loss distribution. Superconductor filament currents are denoted from $I_1$ to $I_6$; $I_7$ refers to the copper strip one.

In Figure 7, the current of the filaments in ten turns placed at the bottom of the inner layer at $t=0$ s, immediately before the start of the discharge, is shown. All over the length of the conductor forming the turns (about 19 m), the total current of the coil (467 A) is uniformly distributed among the superconductor filaments that carry 77.8 A each. No current circulates in the copper strip. Furthermore, no transverse current circulates in the nickel matrix, creating the coupling of the superconductor filaments in this condition. In Figure 8a, the current distribution in the ten turns at $t=1.3$ s, when the coil’s current reaches 376.3 A and 35% of the stored energy (200 kJ) is discharged, is shown. In Figure 8b, the detail of the distribution over a conductor length of three twist pitches is shown. During this fast-ramping regime, the current is not uniform. In particular, due to the fast-ramping magnetic field investing the conductor, continuous (along the length) circulation of transverse currents occurs which creates the coupling of the filaments. Due to the high resistivity of the Monel sheath surrounding the superconductor filaments, no substantial current is observed in the copper strip. The distribution is periodical with the twist pitch. The coupling currents can be envisioned as current loops crossing the nickel matrix and reclosing in the superconductor filaments, where they sum or subtract to the transport current. As a result, the filaments experience a continuously changing (along the length) current that goes much beyond the uniform current share (that in this case would be 62.7 A per filament, giving the total transport current of 376.3 A), reaching about 200 A. The current is even reversed in the positions where the loop current subtracts from the transport current reaching $-180$ A. The same behavior discussed above regarding the discharge is also obtained during the recharge. After the ramping cycle, when the coil’s current reaches the steady state again, the induced electromotive force no longer exists, and the transverse current flowing in the resistive matrix relaxes. The profiles of the filaments’ current gradually return to the uniform distribution holding before the ramping (Figure 7): Figure 9a,b shows the current distribution in the ten turns at $t=55$ s when very substantial relaxation has already occurred and at $t=155$ s when relaxation is practically completed, respectively.

As discussed, due to the transverse coupling currents, the current of the filaments goes far beyond the critical value, bringing the superconductor to the dissipative state. This is the main reason for the loss occurring in the conductor. The transverse currents through the resistive nickel matrix also contribute to the total loss, but their impact is much lower. In Figure 10a, the local current of the first filaments ($I_1$) is compared with its critical current ($I_{c1}$) over a conductor length of three twist pitches. The local value of magnetic flux density acting on the conductor ($B_1$) is taken into account for calculating the critical current and is also shown in the figure. The corresponding loss per unit length of the conductor is shown in Figure 10b. It can be seen that the loss in the matrix materials (nickel and Monel sheath) due to transverse current is much lower that the loss in the filaments. It is also pointed out that since negligible current flows in the copper strip and in the Monel matrix, no significant loss is produced therein; thus, practically all of the matrix loss is produced in the nickel.

Figure 11 shows the time profile of loss in a turn at the bottom of the coil during the discharge–charge cycle. It can be seen that the most dissipative phase is at about the middle of the recharge, where a peak power of about 540 mW is reached in the turn. No appreciable dissipation occurs anymore at $t=6$ s, which is one second after the end of the discharge. The conductor’s average dissipation per unit length during the cycle is about 150.1 mW/m, corresponding to 284 mW for the whole turn (1.9 m). Similar power profiles are obtained for the turns in different radial or axial positions along the coil. Figure 12 shows the map of energy loss per unit volume of conductor in one cycle. The data at each point are obtained by taking the time integral of the power of Figure 11 and dividing it by the volume of the conductor (cross-section x length). Linear interpolation in the radial direction and quadratic interpolation in the axial direction were used for obtaining energy loss that was not directly included in the simulations (see Figure 4d and the related discussion in Section 3.1). As expected, higher losses are obtained at the innermost end of the coil due to the higher component of the magnetic field perpendicular to the long side of the conductor. The total loss of the SMES coil in one cycle can be calculated by integrating the distribution of Figure 12; it amounts to 4.32 kJ.

5. Conclusions

A method for calculating the AC loss in dynamic conditions of superconducting coils made of multifilamentary MgB${}_2$ tapes with external copper stabilizer was set up. The AC loss of a 500 kJ/200 kW SMES during charge–discharge cycling at full power was numerically investigated by the method. An average dissipation of 85.5 mW/m was found all over the coil during one charge–discharge cycle at full power, with a peak of 150.1 mW/m in the turns lying at the ends of the coil.

Future developments will try to reduce AC losses employing MgB${}_2$ tape or wire with a higher number of filaments (e.g., 19 filaments, currently under development) while at the same time increasing critical current value ($+30\%$ expected).