Multi-Objective Structural Optimization of a 10 kV/1 MVar Superconducting Toroidal Air-Core Reactor

Abstract

With the increase in urban cableization rate and cable length, the overvoltage problem caused by the capacitive effect becomes more and more serious. To limit overvoltage and achieve regional reactive power balance, shunt reactors are installed in substations. Based on a series of previous research, a type of superconducting toroidal air-core reactor is presented in this paper. The aim is to improve the power density of reactive power compensation and reduce magnetic leakage and noise pollution. In this paper, the structural optimized design of a 10 kV/1 MVar reactor is carried out based on COMSOL and MATLAB. In consideration of the usage of high-temperature superconducting tapes and AC loss of the reactor, combined with critical current, this paper uses corresponding finite element method (FEM) models and the optimal solution set is obtained via multi-objective genetic algorithm (MOGA). Finally, the solutions are analyzed economically and the set of solutions with the lowest cost is obtained, which provides a reference for the actual fabrication of a toroidal reactor in Shanghai, and can be used in the design of superconducting reactors at higher voltage levels.

1. Introduction

With the growth of electricity demand, the capacity of the power system continues to increase, and the scale of power grids is expanding. In urban power grids, the proportion of underground lines is continuously increasing, and the effects of capacitance of underground lines are increasingly significant [1]. The issues of reactive power imbalance and overvoltage caused by long-distance, high-voltage power transmission have become prominent. Especially during the low-load period, there is an insufficient reactive power compensation for inductive loads and a relatively high voltage level, serious problems which pose a threat to the safe operation of the power grid. The shunt reactor provides an effective path to solve these problems. It can weaken the voltage rise caused by the capacitance effect, improve the voltage and reactive power distribution in the system, and reduce the line loss [2,3].

There are two main types of reactors: iron-core reactors and air-core reactors. The iron-core reactor, which is put into use first, can reduce magnetic leakage and provide a stable inductance value because of the high magnetic permeability of the iron core. However, iron-core reactors inevitably suffer from high noise and vibration due to magnetostriction. Moreover, the presence of the iron core carries the risk of potential saturation and inrush current. In addition, their large footprint is a significant factor limiting their application in urban substations [4]. The air-core reactor has a simple structure, low noise, and high inductance linearity. However, it has poor uniformity of magnetic field distribution, which can lead to high magnetic leakage. Moreover, its insulation performance is easily affected by climate, requiring high standards for installation [5].

Due to their volume, loss, and noise problems, the aforementioned conventional reactors are difficult to install and promote in central urban areas [6,7]. This is especially true for high-voltage level shunt reactors, where copper losses caused by winding resistance are especially significant. Owing to their high current-carrying density and low loss characteristics, high-temperature superconducting (HTS) tapes offer new possibilities for addressing these issues [8].

At present, there are relatively few studies on HTS reactors. Ref. [9] proposes a HTS controlled reactor prototype mainly composed of a cylindrical iron-core, copper working winding, and HTS control winding. The AC magnetic flux generated by the copper coil is orthogonal to the DC magnetic flux produced by the HTS coil, significantly reducing system losses. Ref. [10] designs an improved orthogonal-core HTS controllable reactor (HTS-OCR) which consists of copper working winding and HTS control winding. It incorporates an air gap in the control section to reduce harmonics and enhance the stability of the total inductance, offering advantages in adjusting the range of inductance and the content of harmonics. However, these works mainly focus on small-capacity iron-core reactors; little research has been carried out on the optimization of air-core reactors applied in large-capacity systems. Because of the toroidal air-core topology, this type of reactor can effectively reduce magnetic leakage and noise pollution, making the reactor more compact and flexible, which makes it more suitable to install in central urban areas.

In this paper, a 10 kV/1 MVar superconducting toroidal air-core reactor is studied, and the electromagnetic optimized design of the reactor is completed by COMSOL v6.2 and MATLAB co-simulation. In consideration of the usage of high-temperature superconducting (HTS) tapes and AC loss of the reactor, combined with critical current, this paper obtains the optimal solution set via multi-objective genetic algorithm (MOGA). During the optimization process, the inductance of the reactor is calculated using a 3D simplified model and the AC loss is calculated using an equivalent 2D axisymmetric stacked model. Finally, combined with the actual engineering cost of HTS tapes and cooling systems, an economic analysis is conducted on the optimal solution set and the solution set with the lowest cost is obtained.

2. Operating Principle and Basic Parameter Index

2.1. Operating Principle of Shunt Reactor

The underground line has a large capacitance to ground, making the capacitive effect more significant. Figure 1 is a simplified circuit of a power transmission system. The steady-state solution for the line is as follows:

$$\left[\begin{array}{c}\stackrel{•}{U_S}\\ \stackrel{•}{I_S}\end{array}\right]=\left[\begin{array}{cc}\cos (\omega \sqrt{L_0{C}_0}l)& j{Z}_c\sin (\omega \sqrt{L_0{C}_0}l)\\ j{\displaystyle \frac{1}{Z_c}}\sin (\omega \sqrt{L_0{C}_0}l)& \cos (\omega \sqrt{L_0{C}_0}l)\end{array}\right]\left[\begin{array}{c}\stackrel{•}{U_{LD}}\\ \stackrel{•}{I_{LD}}\end{array}\right]$$

(1)

where $\stackrel{•}{U_S}$ and $\stackrel{•}{U_{LD}}$ are the source voltage and the load voltage, $\omega =2\pi f$ is the angular frequency, $l$ is the length of the transmission line, and $L_0$ and $C_0$ are the inductance and capacitance per unit length of the line.

The load in the power system is changing at any given time. When the load in the line (LD) is zero, the voltage at the end of the line ($\stackrel{•}{U_{LD}}$) is as follows:

$$\left\{\begin{array}{c}\stackrel{•}{U_{LD}}={\displaystyle \frac{\stackrel{•}{U_S}}{\cos (\omega \sqrt{L_0{C}_0}l)}}\\ \cos (\omega \sqrt{L_0{C}_0}l)<1\end{array}\right.,$$

(2)

The line voltage is higher than the source voltage. As the length of the line (l) increases, $\cos (\omega \sqrt{L_0{C}_0}l)$ gradually decreases, with the voltage reaching its maximum at the end of the line. To mitigate this overvoltage, a shunt reactor (L_S) needs to be introduced into the system. Leveraging the inherent phase opposition between inductance and capacitance, the reactor compensates for the line’s capacitive effect, thus restraining the voltage rise. By choosing an appropriate inductance, the capacitive effect is neutralized, which simultaneously achieves overvoltage suppression and restores reactive power balance.

2.2. Site Selection and Capacity Determination of Shunt Reactor

The power system in Shanghai is characterized by significant peak–valley load differences, a common feature in large cities. This challenge is exacerbated during long holidays, when a sharp decline in load leads to insufficient inductive reactive power compensation and consequent voltage rise. For instance, in the Far East Zone of the Shanghai Grid, the 500 kV busbar voltage reached 521.4 kV during the 2023 Spring Festival, when the load hit its annual minimum. Additionally, on New Year’s Day 2023, the reverse reactive power transmission reached 741.27 MVar—an increase of about 80 MVar compared to 2022, indicating a growing trend.

To address the issue of reactive power reversal in the Shanghai power grid, installing shunt reactors within 35 kV substations for compensation has been considered. The 35 kV power substation, Gulong Station in Shanghai, was selected as the demonstration site. It is located near residential areas and thus places higher importance on the noise of the reactor. Conventional iron-core reactors produce excessive noise and thus do not meet this requirement. Compared with conventional air-core reactors, the superconducting reactor operates in a liquid nitrogen (LN2) environment. LN2 is safe, nonflammable, and non-polluting to the environment. It is used instead of insulating oil as a cooling and insulating medium to avoid the environmental pollution caused by oil leakage, and has no potential fire hazards. Therefore, the superconducting air-core reactor is more suitable for installation near residential areas, as well as reducing noise, vibration, and coverage area.

During the Spring Festival, the reactive power reversal at Gulong Station is 1 MVar, so the rated capacity of the reactor is set to 1 MVar and the reactor will be installed on the 10-kV side of the transformer. Therefore, the final design specification for the shunt reactor is 10 kV/1 MVar.

The parameters of the reactor are calculated using the following formulas:

$$Q=\sqrt{3}{U}_L{I}_L=3U_{\phi }{I}_{\phi }$$

(3)

$$U_{\phi }=I_{\phi }X=2\pi f{L}_s\times I_{\phi }$$

(4)

where Q = 1 MVar is the reactive power capacity, $U_L$ = 10 kV is the root-mean-square (RMS) value of line voltage, $I_{\phi }$ = 57.7 A is the RMS value of phase current, and L_s = 0.318 H is the designed inductance of the shunt reactor.

2.3. Basic Structure of the HTS Toroidal Air-Core Reactor

Due to its compact structure and low leakage field, the toroidal structure was chosen in the toroidal air-core reactor design.

Figure 2 shows the structure of a three-phase toroidal air-core reactor. Each single-phase reactor is composed of multiple circular double-cake superconducting coils uniformly arranged along the toroidal circumference of a certain inner diameter. In Figure 2, R is the surrounding radius, r is the inner radius of single coil, th is the thickness of superconducting tape, nt is the number of turns, nc is the number of coils, and D is the diameter of the entire reactor.

In addition, the parallel branches are used to ensure that the operating current is less than the critical current of the magnet as well as to reduce the AC loss. As shown in Figure 2b, b₁, b₂, and b₃ indicate that there are three parallel branches. The number of parallel branches is an important parameter of the shunt reactor. The current in each phase is evenly distributed to different branches and the coils in the same branch are in series. Since the total current remains constant, the more parallel branches there are, the smaller the current in each coil and each tape, and the smaller the AC loss of the reactor.

3. Optimization Method and Procedure

3.1. Objective Function and Optimization Constraint

Five independent structural parameters can be extracted as the variables of optimized design: r, R, nt, nc, and the number of parallel branches (b).

The total cost of the reactor is primarily governed by the consumption of HTS tapes and the cooling system expenditure, the latter being determined by the AC loss of the coils. However, these two cost factors present a trade-off: reducing the length of HTS tapes typically leads to an increase in AC loss. To address this conflict, this paper employs a multi-objective optimization approach aimed at striking a balance to minimize the overall cost.

The least total length of HTS tapes (TL) and the least AC loss (Loss) are used as the following objective functions:

$$\left\{\begin{array}{l}T{L}_{\min }=f_1(r,nt,nc)\hfill \\ Los{s}_{\min }=f_2(r,R,nt,nc,b)\hfill \end{array}\right.$$

(5)

Specifically, the total length (TL) is calculated using the following formula:

$$TL=3\cdot nc\cdot 2\pi nt\cdot (r+(nt-1)\cdot th)$$

(6)

Because there are two objective functions, the optimal solution is not a single point, but a set of solutions consisting of multiple points, known as the Pareto optimal set [11]. In this problem, the Pareto optimal solution is a set of solutions where no objective function (TL) can be optimized without sacrificing another objective function (Loss).

To achieve a total capacity (Q) of 1 MVar, the single-phase inductance (L) needs to be equal to 0.318 H. When the voltage remains constant, the smaller the inductance, the larger the reactive capacity, but the current will increase. With comprehensive consideration, the designed value of inductance can be slightly reduced by 1%, which means the inductance needs to be in the range of 0.314 H to 0.318 H [12].

Taking full consideration of engineering factors, such as assembly, space, and operational stability, the above structural parameters of the toroidal reactor need to be set constraints

$$\left\{\begin{array}{l}di{s}_{\min }>0.03m\hfill \\ I_{OP}<\alpha I_C\hfill \end{array}\right.$$

(7)

where ${dis}_{\min }$= $(R-r-nt\cdot th)\cdot 2\pi /nc$ is the minimum distance between coils, $I_C$ is the critical current of the circular coil, $I_{OP}$ is the operating current in the coil, and $\alpha$ = 0.7 is the safety factor.

3.2. Procedure of Design Optimization

The genetic algorithm toolbox in MATLAB and finite element simulation in COMSOL are used for co-calculation to optimize the structural parameters of the reactor. The optimization algorithm employs multi-objective genetic algorithm (MOGA) [13,14].

Figure 3 shows the procedure of design optimization. The electromagnetic design optimization program is controlled by MATLAB. The specific process is as follows:

Step 1: Variable Initialization

Five independent optimization variables (r, R, nc, nt, b) are initialized, and their range of values are defined.

Step 2: Adjusting the Number of Turns

The number of turns (nt) is adjusted to make the inductance of the reactor meet design requirements. In this step, the nt adjustment is divided into coarse tuning and fine tuning. Coarse tuning refers to quickly calculating and adjusting the inductance based on a simplified Neumann formula in MATLAB [15,16], as in (8). After this, the inductance is adjusted to a range between 0.3 H and 0.35 H. Fine tuning refers to precisely calculating and adjusting the inductance via a magnetic field energy method in a 3D simplified model in COMSOL.

$$M={\displaystyle \frac{\varphi }{I}}={\displaystyle \frac{{\displaystyle \int \mathit{A}d{l}^{\prime }}}{I}}={\displaystyle \frac{{\mu }_0}{4\pi }}{\displaystyle \underset{l}{\int }{\displaystyle \underset{l^{\prime }}{\int }\frac{dld{l}^{\prime }}{\left|r-r^{\prime }\right|}}}$$

(8)

where M is the mutual inductance between different units and $\mathit{A}$ is the magnetic vector potential.

Step 3: Constraint Condition Judgment

After the critical current, inductance, and minimum distance are calculated, it is judged whether this set of variables meet the constraint conditions; if not, the optimization variables will be revalued, and the program will go back to the previous steps until the constraint conditions are met. If met, proceed to the next step to calculate the objective functions.

Step 4: Calculating the Objective Functions

After the set of variables is determined, the total length and AC loss of the reactor will be calculated by an equivalent 2D axisymmetric model in COMSOL.

Step 5: Optimal Solution Set Output

After multiple calculations of the objective functions, it is determined whether the population generation has reached the set value. If not reached, the design process will go on. If reached, the process will end and the optimal Pareto set will be output.

4. Electromagnetic Modeling and Analysis

The tape parameters used in electromagnetic simulation were all sourced from the Shanghai Superconductor Technology Co., Ltd., (SST), Shanghai, China, including the width and thickness of HTS tapes and the function between critical current of HTS tapes and magnetic field. The width of the tape is 4.8 mm. The thickness of the HTS tape is 0.22 mm and its thickness after being insulated with polyimide (kapton) tapes is 0.42 mm.

4.1. Inductance and Critical Current Evaluation

The magnetic field for each coil is the same in the toroidal reactor. Therefore, when calculating the magnetic field in the 3D model, one of the coils can be selected to simplify the electromagnetic modeling analysis by setting perfect magnetic boundary conditions, as shown in Figure 4. When crossing the set boundary surface, the magnetic field is perfectly mirrored. This boundary condition also means that the magnetic field is along the normal direction of the boundary [17]

$$\mathit{n}\times \mathit{H}=0$$

(9)

where $\mathit{n}$ is a normal vector, and $\mathit{H}$ is magnetic field intensity.

Figure 5 shows the magnetic field distribution on single coil and toroidal coils when the operating current ($I_{OP}$) is 20.4 A. The simplified model and the toroidal model have the same magnetic field distribution. The single calculation times are 10 s and 60 s, respectively, which indicates that the simplified model can significantly reduce computation time while ensuring accuracy.

The accurate inductance of the reactor is calculated by the magnetic field energy method in the simplified 3D model using the following equation:

$$L={\displaystyle \frac{{\displaystyle {\int }_{\mathsf{\Omega }}\mathit{B}\cdot \mathit{H}}}{I_{O{P}^2}\cdot b^2}}\cdot nc$$

(10)

where B is the magnetic flux density, and b is the number of parallel branches.

Superconducting toroidal reactors with different structural parameters exhibit significant differences in critical current due to the varying magnetic field distributions. The relationship between critical current of HTS tapes and magnetic field is as follows [18,19]:

$$J_C(B_{per},B_{par})=J_{C0}{\left(1+{\displaystyle \frac{\sqrt{k^2{B}_{par}^2+B_{per}^2}}{B_0}}\right)}^{-\alpha }$$

(11)

where $J_{C0}$ is the critical current density without applied magnetic field, $B_{par}$ is the parallel magnetic field, $B_{per}$ is the perpendicular magnetic field, k = 0.332, $B_0$ = 0.342, and $\alpha$ = 2.11.

As shown in Figure 6, this paper employs the load line method to calculate the critical current of the superconducting reactor. This method requires comparing the curve of the magnetic field generated by the reactor with the curved surface of the HTS tape’s critical current under an external magnetic field. The current at the intersection point is the critical current of the reactor.

4.2. AC Loss Evaluation

Due to the complex structure of the toroidal reactor, it is difficult to directly calculate its AC loss. To quickly estimate the AC loss of the toroidal reactor, this paper uses an equivalent 2D axisymmetric stacked model to simplify the 3D toroidal model, as shown in Figure 7. The distance between coils in the toroidal model can be calculated by the following optimization variables:

$$\left\{\begin{array}{l}{d}_1=(R-r-nt\cdot th)\cdot 2\pi /nc\hfill \\ \begin{array}{l}{d}_2=(R-r)\cdot 2\pi /nc\\ d_3=(R+r)\cdot 2\pi /nc\end{array}\hfill \end{array}\right.$$

(12)

where $d_1$, $d_2$, and$d_3$ represent different distances, which will affect the coil arrangement in the 2D axisymmetric stacked model.

The AC loss of the 2D axisymmetric stacked model is calculated via the T-A homogeneous approach, which allows simulating the AC loss of large-scale HTS systems while keeping a high accuracy in the current distribution [20,21]. The homogeneous approach assumes that a stack of HTS tapes can be modeled as a homogeneous anisotropic bulk.

The governing equation of the T-A homogeneous formulation in 2D axisymmetric model [22] is as follows:

$$\left\{\begin{array}{l}{\nabla }^2{A}_{\phi }=-\mu {\displaystyle \frac{\delta }{\mathsf{\Lambda }}}\cdot {\displaystyle \frac{T_r}{\partial z}}\hfill \\ {\displaystyle \frac{\partial }{\partial z}}({\rho }_{HTS}\cdot {\displaystyle \frac{\partial T_r}{\partial z}})={\displaystyle \frac{\partial B_r}{\partial t}}\hfill \end{array}\right.$$

(13)

where $\mu$ is the magnetic permeability, $\delta$ is the thickness of a tape, and $\mathsf{\Lambda }$ is the thickness of the unit cell, including the insulating layer. ${\rho }_{HTS}$ is the superconducting resistivity, derived from the E-J power law, as shown in (14).

$${\rho }_{HTS}={\displaystyle \frac{E_C}{J_C(\mathit{B})}}\cdot {\left|{\displaystyle \frac{\mathit{J}}{J_C(\mathit{B})}}\right|}^{n-1}$$

(14)

where E_C = 1 $\mu$V/cm, J is the current density, J_C(B) represents the critical current density of HTS tapes, as shown in (11), and n = 21 is the exponent of E-J power law.

As shown in Figure 8, the current vector potential T is only defined over the superconducting tapes, and the magnetic vector potential A is defined over the entire bounded universe. $T_1$ and $T_2$ are the boundary conditions, defined in (15). Thus, a different transport current can be impressed by modifying $T_1$ and $T_2$. Additionally, the boundary conditions in the other two edges of the bulk are Neumann boundary conditions.

$$I=(T_1-T_2)\delta$$

(15)

The equivalent stacked model generates a uniform magnetic field in the center coil, which is used to equivalently simulate the toroidal magnetic field in the toroidal reactor. In particular, as shown in Figure 8, only the center coil of the stacked model is defined as superconducting, while other coils are defined as copper coils. This approach aims to increase computational speed while keeping the magnetic field unchanged.

The average AC loss of the center coil (Q_av) is defined as (16). Loss is the total AC loss of the reactor, which is equal to the loss of the center coil multiplied by the number of coils.

$$\left\{\begin{array}{l}{Q}_{av}={\displaystyle \frac{2}{T}}{\displaystyle {\int }_{T/2}^T{\displaystyle {\int }_{\mathsf{\Omega }}(\mathit{E}\cdot \mathit{J})d\mathsf{\Omega }dt}}\hfill \\ Loss=nc\cdot Q_{av}\hfill \end{array}\right.$$

(16)

where E is the electric field, J is the current density, and T is the period of an AC cycle.

Figure 9 shows the current distribution of different simulation methods in a 2D axisymmetric double-cake coil, when nt = 100, r = 0.18 m, R = 0.4 m, and I_OP = 20.4 A. The current distribution of the two simulation methods is consistent. The AC loss calculated by the H formulation is 4 W, while the calculating time is 180 s. The AC loss calculated by T-A homogeneous formulation is 4.06 W, while the calculating time is only 15 s, 90% faster than the H formulation. The result shows that T-A homogeneous formulation is a faster simulation method while keeping a high accuracy in the current distribution. Moreover, the T-A homogeneous model has a simpler geometric structure which is conducive to accelerating the iteration of the following structural optimization.

The number of stacked coils and the distance between different coils are important variables in the 2D axisymmetric model. They will affect the magnetic field distribution in space, thereby affecting the calculation of AC loss.

Figure 10 shows the AC loss and magnetic field distribution of the center coil in the 2D axisymmetric stacked model, when nt = 100, r = 0.18 m, R = 0.4 m, and I_OP = 20.4 A. With the distance between coils (d = 3 cm) remaining constant, the AC loss of the center coil decreases as the number of stacked coils increases. And as the number of coils increases, the parallel component of the magnetic field increases and gradually reaches saturation when the number is equal to 25. The results show that perpendicular magnetic field is the main factor affecting the AC loss, and the central magnetic field of 25 stacked coils is sufficient to simulate the magnetic field of toroidal coil.

Figure 10b indicates that as the distance between coils increases, the perpendicular magnetic field and the AC loss of the center coil increase. Therefore, during the optimization process, the maximum distance (d₃) of the toroidal structure is selected as the distance for the stacked model shown in Figure 7, which aims to calculate the upper bound of the AC loss of the entire reactor.

Based on the parameters mentioned above (nt = 100, r = 0.18 m, R = 0.4 m, I_OP = 20.4 A, number of stacked coils = 25, distance between coils = d₃), the cross-sectional magnetic distribution on the outside of the toroidal coils and stacked coils is shown in Figure 11. The two models have generated similar uniform magnetic fields at the center of multi-coils, and their directions follow the red arrow in the figure. The magnetic field distribution at the outside of the coils is essentially the same, verifying the effectiveness of the 2D axisymmetric stacked model to calculate the AC loss.

5. Analysis of Optimization Result

5.1. Optimization Result of MOGA

Based on the former procedure of design optimization and two-part electromagnetic model, the structural parameters of the superconducting toroidal air-core reactor have been optimized. The time for a single optimization run is approximately 12 h and the optimization results of the superconducting toroidal air-core reactor by using MOGA are shown in Figure 12. The coordinate axes represent the two optimization-objective functions, and each point represents a feasible solution. The feasible solutions first appear in the middle and gradually develop towards the lower left corner (shorter length of tapes and lower AC loss) until reaching the optimal solution set. Ultimately, the feasible solutions are mainly concentrated in three areas, which are in different red frames. Different numbers indicate different parallel branches (b = 2, 3, 4). The more parallel branches there are, the smaller the AC loss of the reactor, but the greater the usage of superconducting tapes. The red points are the Pareto optimal solution set and the lowest-cost solution will be in the Pareto set.

The AC losses of different solutions in Figure 12 are all upper bounds, used to provide more cooling margin. After obtaining the optimal solution set, the minimum distance (d₂) of the toroidal structure is then used as the distance for the 2D axisymmetric stacked model to calculate the lower bound of AC loss of the reactor. In this way, the upper and lower bound of AC loss of the reactor under optimal parameters can be obtained, as shown in Figure 13. The AC losses of superconducting reactors with different structural parameters will be within the upper and lower bounds.

5.2. Economic Analysis of the Shunt Reactor

After obtaining the optimized parameter sets and ensuring both the self-sufficiency in tape supply and the technical reliability of the refrigeration system, this paper will proceed with the final design of the superconducting reactor based on the total cost.

Based on the current costs of the HTS tapes and cooling machine [23], an economic analysis is conducted on the optimal solution set for the structural parameters of the reactor. The upper bound of AC loss is selected to calculate Loss for more cooling margin. The total cost includes the cooling cost and the cost of HTS tapes, defined by the following equation:

$$Cost={\sigma }_1\cdot \alpha \cdot Loss+{\sigma }_2\cdot \beta \cdot TL$$

(17)

where $\alpha$ = 1000 CNY/W (137.3 USD/W) is the cooling cost factor, and $\beta$ = 150 CNY/m (20.6 USD/m) is the current tape cost factor. ${\sigma }_1$ and ${\sigma }_2$ are the margin factors for the cooling system and HTS tapes, respectively. Considering the thermal losses of the cooling machine itself and other components in the system, ${\sigma }_1$ is chosen to be 2. And due to the extra usage of HTS tapes during the manufacturing process, ${\sigma }_2$ is chosen to be 1.1.

Based on this, the structural parameters of the 10 kV/1 MVar toroidal air-core reactor with the lowest total cost are shown in

Table 1. Parameters of reactor with the least total cost.

Items	Value
Number of coils nc	$30 \times$3
Number of turns nt	$92 \times$2
Number of parallel branches b	3
Inner radius of coil r (m)	0.228
Surrounding radius R (m)	0.452
Diameter D (m)	1.44
Total length TL (m)	25,712
$\mathrm{Operating} \mathrm{current} I_{\phi ,peak}$ (A)	27.2
$\mathrm{Critical} \mathrm{current} I_C$ (A)	63.7
Inductance L (H)	0.3173
Lower-bound of AC loss (W)	681
Upper-bound of AC loss (W)	955
Cost of HTS tapes (USD)	582,493
Cost of cooling system (USD)	261,670
Total cost (USD)	844,163

, and the corresponding optimal solution is shown in Figure 13. Three different colors represent three parallel branches.

The total cost is USD 844,163, including the tape cost (USD 582,493) and the cooling cost (USD 261,670). Among them, the cost of superconducting tapes is the main cost. However, with the increasing demand for HTS tapes in various fields, the cost of HTS tape is gradually decreasing and will eventually be below 6.85 USD/m [24]. Therefore, under the same parameters, the final cost of the superconducting shunt reactor will be less than USD 450,000.

5.3. Performance Analysis of the Reactor with the Least Cost

The basic structure of the superconducting toroidal air-core reactor is shown in Figure 14a, including the three-phase reactor and cryogenic system. The dewar is a two-layer structure with vacuum interlayer between the two layers. To leave enough installation allowance for the reactor and ensure the insulation distance of more than 20 mm [25], the inner diameter of the dewar is 1.76 m. Moreover, the vacuum interlayer of the dewar needs a certain space to install multilayer insulation materials and pipes. The outer diameter of the dewar is 2.03 m, which means the superconducting reactor covers an area of 3.24 m².

Figure 14b shows the magnetic field distribution of the three-phase superconducting shunt reactor. The maximum magnetic field in normal operation is 0.154 T and the line graph is the radial magnetic field distribution at the center of the reactor. The results show that the magnetic flux leakage at the radial 1.015 m (the outer radius of the dewar) is only 0.002 mT, far lower than 1 mT, indicating that the toroidal structure can effectively restrict the magnetic field and reduce the space magnetic flux leakage.

The inductance matrix of the three-phase reactor is as follows:

$$\left[\begin{array}{ccc}{L}_A& M_{AB}& M_{AC}\\ M_{BA}& L_B& M_{BC}\\ M_{CA}& M_{CB}& L_C\end{array}\right]=\left[\begin{array}{ccc}0.31731& -1\times {10}^{-4}& -1\times {10}^{-5}\\ -1\times {10}^{-4}& 0.31731& -1\times {10}^{-4}\\ -1\times {10}^{-5}& -1\times {10}^{-4}& 0.31731\end{array}\right]\mathrm{H}$$

(18)

Both mutual inductances are very small numerically, and the coupling between different phases is low, which also shows that the magnetic field shielding effect of this structure is good.

6. Conclusions

Based on the finite element software COMSOL and genetic algorithm toolbox of MATLAB, the electromagnetic optimized design of a 10 kV/1 MVar superconducting toroidal air-core reactor is completed by MOGA. Furthermore, this paper uses a 2D axisymmetric stacked model to equivalently simulate a 3D toroidal model, thereby rapidly calculating the AC loss of large-scale toroidal structures. Combined with the current costs, considering both cooling cost and HTS tape usage, a set of solutions with the lowest cost is obtained in this paper.

This design optimization will provide a reference for the actual fabrication of the superconducting shunt reactor and the grid-connected operation in Gulong Station in Shanghai, China. And the method proposed in this paper can be applied to the design of subsequent 35 kV or higher-voltage, larger-capacity superconducting reactors.