Analytical Modeling of Rotary Electromagnetic Fault Current Limiter

Rotary electromagnetic fault current limiter (FCL) is an emerging technology for limiting fault currents in the power network. The device consists of two movable air-core spherical reactor rings (i.e., inner and outer rings). The movable feature allows the reactor rings to rotate, and form a proper mutual positional displacement. This can limit the fault current since the effective inductance of the device is dependent to the mutual angular position of the reactor rings. Due to the unconventional structure, to this date, the design process has solely been based on finite element (FE) method. This paper aims to analytically validate the theory and the overall functionality of the device in both the steady state and transient regime. For this propose, the resistance and the inductance of the device are first computed at normal operating condition using the basic theory of engineering electromagnetics. Next, performance characteristics such as electromagnetic torque, rotational displacement of the reactors, effective inductance of the device, and the network current are calculated at the faulty condition. For verification purposes, analytical results are compared with via FE simulation and experimental test.

To reduce the manufacturing and the operational costs associated with conventional electromagnetic FCLs, a rotary air-core electromagnetic FCL is introduced by authors in [23]. The presented device is characterized by small size, light-weight, simple structure, (since it does not The associate editor coordinating the review of this manuscript and approving it for publication was Philip Pong . require detection circuit, control system, and ferromagnetic core), low maintenance cost, acceptable response time, and reversible operation. However, the design process and the performance analysis of the presented device is only carried out numerically via Finite Element (FE) method. Despite the precision of the FE modeling, design procedure solely based on FE could get computationally costly, particularly during the optimization phase. In addition, it is often difficult to gain a physical insight and create a meaningful connection between design parameters, the performance of the device and its effect on the network solely based on the FE. Analytical and/or semi-analytical modeling techniques are relatively faster and also could serve as a reliable tool for design validation purposes. In the literature, various analytical techniques such as mesh or filament method [24] are available for calculating the self and the mutual inductance of coaxial and non-coaxial coils with arbitrary mutual position. In filament method, the cross sectional area of the coil is divided into multiple subsections, with each subsection being considered as a current filament in its center. In other words, the coil's cross section includes a set of current filaments. The self-inductance, mutual inductance and the magnetic force and torque between two current filaments can be formulated in terms of elliptical integrals of the first, second and third kind, Heuman's Lambda function, Bessel functions, and Legendre functions, and computed based on fundamental principle of engineering electromagnetics such as Maxwell equations, Neumann integral and Bio-savart law [25], [26]. Next, these relations are extended and applied to the entire cross section of the coil in order to compute the inductance and the torque between two coils [24], [25], [27], [28], [29], [30], [31], [32], [33], [34], [35], [36], [37]. In particular, in [33] the magnetic force between circular coils with nonparallel axes and rectangular cross section are computed by adding magnetic forces between all individual filaments. Using Grover's formula [38] and filament method Babic and Akyel obtained the torque between two filamentary circular coils with inclined axes [35]. However, existing models have mostly been applied/reported to co-axial and/or leveled, stationary coils with simple/standard geometrical shape (e.g., solenoid coil, disk-shape coil). To make the model compatible with the presented FCL device, proper adjustments must be made based on the geometrical shape of the two coils (i.e., spherical ring-shape coils). In addition, the inductance value and the electromagnetic between the coils should be continuedly updated in accordance with the angular displacement of the two coils.
Motivated by the merits of analytical modeling, and based on the fundamental concept of engineering electromagnetics this paper presents an analytical framework to validate the operational theory and to facilitate the design process of the newly developed rotary electromagnetic FCL. To fully capture the performance of the device, analytical modeling shall describe the main design parameters (e.g., resistance and inductance of coils), and the respective performance characteristic (e. g., torque between the coils, and the angular displacement of the coils, and etc). For this purpose, detailed performance characteristics of the device are computed at a faulty scenario, and compared against FE results. The presented model can be generalized and applied to coils with different structures and with arbitrary positions. This paper is organized as follows. Section II briefly describes the structure and the operating principle of the rotary electromagnetic FCL. Section III presents the analytical framework, followed up by related results and discussion in section IV. Finally, section V presents experimental validation.

II. STRUCTURE AND THE OPERATING PRINCIPLE
The structure of the presented air-core FCL includes two concentric spherical reactor rings as shown in Fig. 1. Both reactors are movable along polar angle (θ) in spherical coordinate system (Fig. 1), but to simplify explaining the concept, the outer reactor is assumed fixed/stationary here. The two reactor rings are connected in series (with reverse excitation current flow) from one end, and to the input/output terminal of the device from the other end. The portable feature of the inner reactor ring allows the mutual angular displacement between the two reactors. Since the effective inductance of the device changes in correlation with the mutual angular displacement between the reactor rings, the device behaves like a variable inductance. Initially, the electromagnetic torque between the two coils is negligible, and the reactor rings are kept almost aligned with the help of an external force (i.e., spring force). In other words, in normal condition, the reactor rings are aligned (with minimal angular displacement), and thus the reactors' mutual inductance is relatively high. Considering the reverse excitation current flow of the reactor rings, the mutual and self-inductances have an opposite polarity. Therefore, the overall inductance/impedance of the device is initially low. Once the fault occurs in the network, it exerts a significant repulsive torque to the moving reactor, which results in a steady increase of the positional/angular displacement between the reactor rings as illustrated in Fig. 1. During this transition, the negative mutual inductance keeps declining until the inner and outer reactor rings are displaced by 90 degrees. At this stage, the negative mutual inductance becomes zero. Beyond this point (when the angular displacement reaches slightly above 90 degrees), the current flow of the inner reactor reverses and flow in a same direction as the current in the outer reactor. This converts the initial repulsive torque between the reactors to an attraction torque, and thus, the mutual displacement between the two reactor rings keeps rising up to 180 degrees. It is noted that during this transition (i.e., from 90 degrees to 180 degrees angular displacement) the polarity of the mutual inductance also reverses, and becomes positive and similar to the self-inductance. Therefore, the total inductance of the device keeps rising up from 0 degrees to 180 degrees displacement, with the maximum inductance occurring at 180 degrees displacement. Once the fault is cleared and the electromagnetic attraction or repulsive torque is removed, and the external/spring force returns the movable reactor back to its initial/aligned position. To ensure that the device operates properly, initially the reactor rings must be slightly displaced to create sufficient amount of repulsive electromagnetic torque. If the angular displacement between the reactor rings is initially zero, the torques may cancel out and the equipment may not react to the short circuit current. The structure and the operating principle of the device is extensively explained in [23]. Fig. 2(a) illustrates the single circuit diagram of the power network with the FCL being connected to the network. At normal conditions, the circuit includes source impedance, FCL impedance, line impedance and the load impedance. As discussed in section II, the reactor rings are almost aligned and device exhibits a minimal inductance (L 1FCL ).

III. ANALYTICAL MODELING A. EQUIVALENT CIRCUIT OF THE POWER NETWORK AT NORMAL AND FAULTY CONDITIONS
At the transient stage of the faulty condition, the angular displacement between the reactors is continuously rising up from the initial angle (e.g., 15 degrees) to 180 degrees, which results in rapid and steady increase of the inductance. Therefore, the inductance of the device is represented by a variable inductance in the circuit diagram of Fig. 2(b). During the transient fault period, the rotary motion of the inner reactor induces a moving/rotational voltage, which is represented by e(θ) in the circuit diagram. After the transient period has passed and the two reactors are displaced by 180 degrees, the inductance of the device will be maximum.
The circuit model of the distribution network during this transient period can be described by the following set of relations [39]: and e are the source/network voltage, source inductance, source resistance, FCL flux linkage, FCL inductance, FCL resistance, angular position of the rotating (inner) coil with respect to the stationary (outer) coil, angular velocity of the rotating coil, and the induced rotational voltage (dependent voltage source) respectively.
The circuit model implies that the short circuit current is suppressed not only with the rise of the FCL inductance (L FCL ), but with the moving/rotational voltage (e) induced by the rotary motion of the inner reactor ring. The induced voltage is proportional to the angular velocity of the inner coil as expressed in (4). Inserting (2) into (1) yields: The short circuit current is given by: where, V m , Z tn , δ, ϕ and ω are the peak value of the network phase voltage, equivalent network impedance at short circuit conditions, phase angle of the network voltage, phase angle of the network impedance, and the network frequency respectively. To accurately capture the response of the network to the fault, it is necessary to consider the behavior of the FCL device from the electrical, magnetic, and mechanical perspectives. For this purpose, electrical parameters (i.e., resistance, and inductance), the current and the torque are calculated within one operational cycle of the inner coil (i.e., from 15 degrees to 180 degrees angular displacement).

B. ELECTRICAL PARAMETERS OF THE FCL (RESISTANCE, SELF AND MURUAL INDUCTANCE)
The overall resistance of a coil is simply obtained by collecting the conductor's resistance at each turn as expressed in (4).
where R coil , R i , N , ρ, l i , A, and r i represent resistance of coil, resistance of each turn, turn number of each coil, special resistance of conductor, length of turn, cross section of conductor, and the radius of each turn, respectively. In electromagnetics, the mutual inductance between two circular current filaments with arbitrary positions with respect to each other (Fig. 3) can be calculated by (12) and [27].
where R p , R s , and θ are radius of primary filament, radius of secondary filament, and angle between z and z ′ axis, respectively. These parameters are visually illustrated in Fig. 3. The horizontal distance and vertical distance between two centers of filaments are shown by parameters c and d respectively. K (k), E (k), and Q1/2(x) are complete elliptic integral of the first kind, complete elliptic integral of the second kind and Legendre function of the second kind and halfintegral degree respectively. µ o is magnetic permeability of vacuum.
Based on the filament method, the cross sectional areas of two neighboring coils are modeled and considered by a set of current filaments as shown (with small white pixels) in Fig. 4. The mutual inductance between every two individual current filaments in coil# 1 and coil# 2 is obtained via (12), and the mutual inductance between the whole set of current filaments in coil# 1 and coil # 2 is obtained via Babic's relation (13) [27]. It is noted that Babic relation (13) is only valid for coils with rectangular shape cross section. However, in case of the presented FCL structure, the cross section of the reactor coils is spherical-shape (Fig. 5). Thus, the Babic's expression is modified and updated to be compatible with the spherical structure of the coils in the presented device. Applying proper adjustments on (13), the mutual inductance between two spherical coil rings with arbitrary positions with respect to each other takes the form (14):   where (14-a), as shown at the bottom of the next page, M n wp .n tp .n ws .n ts = µ o R p (n wp .n tp )R s (n ws .n ts ) π  The self-inductance is calculated by the filament method as a special case of the mutual inductance between two coils. In other words, the self-inductance of a coil can be regarded as the mutual inductance between two identical and coincided coils. The presented method is general and is valid irrespective of wiring arrangement and/or the size/dimension of the coil. Therefore, the self-inductance is modeled and calculated as the mutual inductance between two filaments located along the same axis as shown in Fig. 6, when the angle between z and z ′ axis (θ), and the distance between the centers of filaments are both set to zero (15). In fact, relation (15) is a special case of relation (12) when parameters θ and c are set to zero.
where a, b, and D are radius of primary filament, radius of secondary filament, and distance between center of two filaments, respectively. For this purpose, the cross-sectional area of the coil is divided into multiple segments and the mutual inductance value between every two filament segments is calculated using (15). Finally, the self-inductance of the coil is obtained by adding the mutual inductance of all filament segments as expressed in (16).
where Mij is the mutual inductance between filament i and j which is calculated by (15). N and n are turn number of coil and number of segment (mesh) respectively.

C. ELECTOMAGNETIC TORQUE
The electromagnetic torque between the inner and outer reactor rings can be computed via meshing method (similar to inductance calculation) by segmenting the coil into multiple current carrying filaments. Suppose that current I 1 and I 2 pass through two filaments with the mutual inductance of M . The torque between the two filaments is obtained by the product of the two currents and the derivative of their mutual inductance as expressed in (17). Inserting proper form of mutual inductance in (17), the torque between two coils with rectangular cross section is obtained by (18), and for spherical cross section is obtained by (19).
where (19-a), as shown at the bottom of the next page, R s (n ws .n ts ) = R ms cos θ s 2N ws + 1 n ws + T s (2N ts + 1) n ts    180 degrees angular displacement with respect to the stationary reactor). As a result, the overall inductance of the device rises up to a level that can effectively suppress the short circuit current. The mechanical operation of the moving reactor can be realized via the following formulations: T spring = k θ (21) where, T mag , T spring , I, α, k, o , and θ o are electromagnetic torque, spring torque, the moment of inertia of the inner coil, angular acceleration, spring constant, initial angular speed, and the initial angle with respect to Z axis, respectively. Since the mass and the radius of the two reactor rings are different, the polar angle θ is calculated separately for each of the two coils as expressed in (23) and (24). Therefore, the overall angle between the reactors is equal to the sum of the angle of the moving reactors (25).

IV. RESULT AND DISCUSSION
For verification purposes, analytical results are compared with those obtained from 3-D FE software package. Specifications of the presented case study (i.e., FCL and the distribution network) are adopted based on existing FCLs in practical/realistic power networks as listed in Table 1 and  Table 2, respectively.
The initial angle θ between the z axis of the two reactor rings is initially set to 15 • before the fault. This angle rises up to 180 • within 20 ms (i.e., equivalent to one fault cycle). At this stage of operation (when the two reactor rings are   displaced by 180 o ), the overall inductance of the FCL device does reach to its peak value. Although, it takes 20 ms for the two reactors to arrive at the final/optimal position for maximum inductance, the FCL device starts limiting the short circuit current from the very beginning as soon as the fault occurs. Fig. 7 presents the calculated and simulated short circuit current waveform within three fault cycles (i.e., 0 to 60 ms). For a more detailed observation, the amplitude of the fault current is presented at different time instants in Table 3. As observed, the discrepancy between the calculated and the FE simulated fault current values is very negligible, and is less than 3% in most instances. This validates the accuracy of the presented analytical model. In addition, Fig. 7 illustrates that the presented FCL can successfully reduce the fault current in the network by approximately up to 45%. This confirms the functionality of the presented FCL.
To further evaluate the accuracy of the presented model, the self-inductance, mutual inductance and total inductance of the presented FCL are calculated within the first fault cycle (i.e., 0 to 20 ms), and compared against the FE simulation results (Figs 8 and 9). For a more detailed observation, the          Fig. 11 presents the calculated and simulated electromagnetic torque between the two coils within the same time period. The torque is initially small due to a relatively low current amplitude in the first few milliseconds. However, the torque rises rapidly (proportional to the current amplitude) and separates the two coils away  Both the modeling and simulation are performed on a computer with the specifications of: Intel(R) Core(TM) i7-7700HQ CPU @ 2.80GHz, RAM 16.0 GB.
For the given computer workstation computing power, the processing time of the presented model per time step varies between 37 s to 50 s (depending upon the angle between the   coils), whereas the processing time of the FEM model varies between 450 s to 520 s.

V. EXPREMENTAL RESULT
A down-scale prototype of the proposed FCL is built and tested to verify the functionality of the proposed device in the laboratory environment (Fig. 12). The coils are shaped in a plastic frame and mixed with an epoxy resin to produce void-free insulation between conductors and to improve the mechanical strength. Once the resin is dried and solid the coils are removed from the plastic frame. The data and specification of the prototyped model is presented in Table 9. The self and mutual inductance of the coils are measured and compared with the calculated values in Tables 10 and 11.   The profile of the total FCL inductance at different angular displacmenets between the coils is plotted in Fig. 13.
To emulate the fault current in the FCL, the prototyped model is supplied by a 50 Hz and 100 V (Phase-to-Neutral) source voltage and the reaction of the inner coil is videoed and recorded. Consistent with the theory discussed in section II, the moving(inner) reactor reacted/rotated quickly and eventually came into rest at 180 degrees positional displacement with the outer reactor. In this test, the outer coil is kept stationary. The FCL current is measured by a digital oscilloscope and shown in Fig. 14 along with the calculated current waveform. As seen, calculated and measured current waveforms are in a good agreement. Since the supply voltage of the laboratory test is scaled down to 100V, the FCL response time is higher than the desireable time, but if the prototyped device was to get excited by higher voltage levels (e.g., 1000 V), the response time would be as low as 15 ms.

VI. CONCLUSION
The coreless electromagnetic FLC with rotational motion is an emerging technology for limiting fault currents in the power network. FE results reported in [23] showed a swift and effective response from the presented FCL to the fault in the network. In this paper, the theory and the functionality of the device was validated analytically using the fundamental concept of engineering electromagnetics. For this purpose, detailed electromagnetic parameters (i.e., resistance and inductance), the performance metrics of the device (i.e., torque between the coils, the angular displacement), and the amplitude of the fault current were computed and compared against FE simulation and measurement results. The calculated, simulated and measured results are in good agreement, indicating that the presented model features the operational excellency both in terms of computational speed, and also precision. In particular, for the given computer workstation computing power, the processing time of the presented model per time step was in the range of 37 s to 50 s, whereas the processing time of the FE model was in the range 450 s to 520 s.
It is noted that the simulation results presented in the paper and the 20 ms response time of the FCL are all captured when the FCL equipment was applied to a practical power grid with realistic power level, voltage level (i.e., 20 kV) and current level. Only the size of the prototyped model and the power and voltage levels of the experimental section are scaled down to laboratory levels. Although the FE simulation is carried out in medium voltage ranges (i.e., 20 kV), the use of composite material in between the coils (thickness of 10 mm) can provide insulation at higher voltage levels. It is also noted that the presented device in this article has a basic design. Research work to optimize the design and maximize the operating speed is ongoing and will be reported in a separate paper. The optimal design not only includes optimized size/dimensions but also incorporates light composite material and aluminum conductors (as opposed to copper conductors) to lower the weight and reduce the operating time.