A Study on the Optimization Design of Power System Winding Structure Equipment Based on NSGA-II

Abstract

As a key component for maintaining the efficient and stable operation of flexible DC transmission systems, the arm reactor often suffers from uneven loss distribution and localized overheating in its windings due to the superimposed AC and DC currents, which adversely affects its operational lifespan. Furthermore, arm reactors are frequently deployed in offshore environments for long-distance, high-capacity power transmission, imposing additional requirements on energy utilization efficiency and seismic resistance. To address these challenges, this study proposes an optimization design method for arm reactors based on a triple-constraint mechanism of “equal resistive voltage–equal loss density–equal encapsulation temperature rise,” aiming to achieve “low loss–low temperature rise–low weight.” First, an equivalent electromagnetic model of the arm reactor under combined AC and DC operating conditions is established to analytically calculate the self- and mutual-inductance-distribution characteristics between winding layers and the loss distribution across windings. The calculated losses are then applied as heat sources in a fluid–thermal coupling method to compute the temperature field of the arm reactor. Next, leveraging a Kriging surrogate model to capture the relationship between the winding temperature rise in the bridge-arm reactor and the loss density, encapsulation width, encapsulation height, and air duct width, the revised analytical expression reduces the temperature rise error from 43.74% to 11.47% compared with the traditional empirical formula. Finally, the triple-constraint mechanism of “equal resistive voltage–equal loss density–equal encapsulation temperature rise” is proposed to balance interlayer current distribution, suppress total loss generation, and limit localized hotspot formation. A prototype constructed based on the optimized design demonstrates a 44.51% reduction in total loss, a 39.66% decrease in hotspot temperature rise, and a 24.83% reduction in mass while maintaining rated inductance, validating the effectiveness of the proposed design algorithm.

1. Introduction

With the accelerated advancement of the “dual-carbon” strategy, China is undergoing a large-scale energy structure transition. Flexible DC transmission technology, by virtue of its flexibility, controllability, rapid response, and adaptability to complex scenarios, has assumed a pivotal role in long-distance renewable energy integration, grid interconnection, and urban distribution network upgrades. As a critical component in flexible DC converter stations, the arm reactor performs multiple functions, including suppressing circulating currents in the arms, balancing power transmission, and stabilizing voltage fluctuations [1]. Its operational conditions feature superimposed AC and DC currents—requiring it to withstand high steady-state currents on the DC side while responding to dynamic harmonic components on the AC side. This unique electrical characteristic poses severe challenges for arm reactors in scenarios such as offshore wind power transmission and islanded power supply. On the one hand, to ensure energy utilization efficiency, the arm reactor must maintain low losses during operation; on the other hand, uneven loss distribution caused by superimposed AC and DC currents can lead to localized overheating, potentially triggering insulation aging or even equipment fires, thereby severely threatening grid safety. Furthermore, arm reactors often operate in harsh offshore environments, necessitating lightweight designs to meet seismic performance requirements. Consequently, addressing the operational characteristics of arm reactors by reducing their losses, temperature rise, and weight while preserving electromagnetic performance has become a critical issue in their structural design.

Although current research on reactor optimization design has formed a systematic framework, it primarily focuses on conventional devices under AC operating conditions, lacking specificity for the unique requirements of arm reactor structural design. In terms of structural optimization methods, scholars such as Zhu Dongbai proposed the equal resistive voltage method, which minimizes total losses by balancing interlayer resistance distribution in windings. However, this approach considers only a single-objective constraint, providing merely a preliminary design for reactor structures and failing to comprehensively address structural design challenges [2]. Zhang Meng et al. employed the equal temperature and equal terminal voltage method to design the coil body of large-capacity dry-type air-core reactors, but their calculation method considered only temperature rise as a single objective, neglecting other critical factors in arm reactor structural design [3]. Liu Quanfeng’s approach involving encapsulated equal height and temperature rise improved heat dissipation spacing but limited optimization to local parameters [4]. Yu Zhenyang adopted a dual-constraint strategy of equal resistive voltage and equal current density, successfully reducing AC losses in dry-type air-core reactors. However, his model did not account for the distortion effect of DC components on current density distribution, rendering the solution inapplicable to arm reactor design [5]. Although scholars like Hou Qixin attempted to optimize winding parameters through an electromagnetic–thermal coupling model and proposed an axially segmented cooling structure, their weighted-coefficient hybrid constraint method required iterative parameter adjustments, resulting in low computational efficiency and limited scalability to reactors of different specifications [6].

More critically, existing temperature rise calculations predominantly rely on empirical formulas derived from transformer winding temperature rise analysis, overlooking the differences in cooling mechanisms and dimensional characteristics between bridge-arm reactors and transformers. This leads to significant deviations between theoretically calculated temperature fields and actual measurements, thereby introducing substantial errors in optimization designs based on such formulas [7]. Furthermore, most existing studies use hotspot temperature rise as the sole evaluation metric, ignoring localized hotspots caused by superimposed AC/DC currents. Such extreme temperature gradients not only accelerate thermal aging of epoxy resin insulation materials but also induce partial discharges and dendritic carbonization channels, drastically reducing equipment lifespan.

Table 1. Contributions and limitations of existing related studies.

Equipment	Design Method	Main Contribution	Limitations
AC Hollow Reactor	① Equal Resistance Voltage ② Equal Temperature Rise ③ Equal Current Density	Proposes a systematic design process for AC hollow reactors, significantly reducing total losses and suppressing temperature rise	① Current density constraints apply only to pure AC conditions, failing to address the AC/DC superimposed conditions of bridge-arm reactors; ② Temperature rise calculations rely on transformer empirical formulas, unable to accurately predict temperature rise under narrow gap conditions; ③ Optimization focuses solely on temperature rise, neglecting efficiency and weight metrics; ④ Weighting coefficients require repeated adjustments, resulting in low computational efficiency and poor generalizability
Bridge-Arm Reactor	① Equal Resistance Voltage ② Equal Temperature Rise ③ Equal Current Density ④ Weighted Coefficient Method	Constructs an electromagnetic model under AC/DC superimposed conditions and uses a weighted coefficient method for multi-scheme trade-offs

summarizes the main contributions and limitations of existing related studies. Thus, current design methods exhibit deficiencies in three dimensions: First, the designs lack specificity, targeting only conventional AC reactors without considering the impact of superimposed AC/DC conditions in bridge-arm reactors. Second, the physical models are idealized, and the temperature rise empirical formulas employed in conventional optimization design fail to accurately reflect the temperature rise behavior of bridge-arm reactor windings. Third, the optimization objectives are oversimplified, neglecting the comprehensive integration of efficiency, temperature rise, and seismic performance as core design goals for bridge-arm reactors under practical operating conditions.

This study employs a Kriging surrogate model to characterize the encapsulation temperature rise in the bridge-arm reactor and refines the calculation method for its winding temperature rise. By employing the triple constraints of “equal interlayer resistive voltage—equal loss density—equal encapsulation temperature rise”, the design objectives of low loss, low temperature rise and lightweight are achieved for the arm reactor. First, an equivalent electromagnetic model under combined AC/DC operating conditions is established. Analytical methods are employed to determine the self-inductance and mutual inductance distribution characteristics between winding layers, which are used to calculate the rated inductance in optimization design schemes and screen out designs that fail to meet inductance requirements. The winding loss distribution is then calculated and serves as the heat source for the fluid–thermal coupling method to compute the temperature field of the arm reactor. Second, a Kriging surrogate model is employed to characterize the relationship between the winding temperature rise in the bridge-arm reactor and factors such as loss density, winding width, winding height, and air duct width. This approach overcomes the applicability limitations of traditional empirical formulas in analytically calculating the temperature rise in bridge-arm reactors. The refined temperature rise calculation method is then adopted as the governing equation for the equal-temperature-rise constraint of the bridge-arm reactor. Finally, based on the aforementioned work, a triple-constraint mechanism of “equal resistive voltage—equal loss density—equal encapsulation temperature rise” is proposed. The equal resistive voltage constraint balances interlayer current distribution to suppress total loss generation; the equal loss density constraint limits localized hotspot formation; and the equal encapsulation temperature rise constraint achieves temperature field homogenization. The NSGA-II algorithm is employed to construct the Pareto optimal solution set. With the design objectives of “low loss—low temperature rise—lightweight”, crossover, mutation and elite retention strategies are introduced to enhance global search capability and improve convergence efficiency. Finite element simulations and prototype temperature rise experiments demonstrate that compared to the pre-optimized arm reactor, the proposed solution achieves a 44.51% reduction in total losses, a 39.66% decrease in hotspot temperature rise, and a 24.83% reduction in mass, thereby validating the effectiveness of the proposed design methodology.

2. Temperature Field Calculation of Arm Reactors Based on Fluid-Thermal Coupling

2.1. Equivalent Electromagnetic Model of the Bridge-Arm Reactor

Bridge-arm reactors typically adopt a multi-layer encapsulation structure, with windings arranged inside the encapsulations and cast with epoxy resin to ensure structural stability. The encapsulations are separated by insulating spacers to form cooling air channels, resulting in a parallel structural characteristic for the entire arm reactor. Therefore, each encapsulation can be regarded as an equivalent helical-tube winding, whose self-inductance, mutual inductance between windings, and winding resistance constitute the main electromagnetic parameters of the equivalent electromagnetic model. The equivalent circuit structure of the bridge-arm reactor is shown in Figure 1, which reflects its electromagnetic behavior specifically under AC operating conditions applicable to both the fundamental and harmonic components.

In Figure 1, $R_{\mathit{zi}}$ and $L_{ii}$ represent the resistance and self-inductance parameters of the i-th winding, $M_{ij}$ denotes the mutual inductance parameter between the i-th and j-th windings, and ${\dot{I}}_i$ is the current in the i-th winding branch, and the currents of all branches are assumed to enter from the same-named terminal of the windings. $\dot{U}$ is the voltage across the arm reactor, ${\dot{I}}_N$ is the rated current of the arm reactor, and n is the number of winding branches.

Under fundamental-frequency operating conditions (50 Hz), and based on the equivalent circuit shown in Figure 1, the skin effect and the influence of temperature variations on the resistivity of the winding material (aluminum) are neglected. With the winding resistance assumed to remain constant, based on Kirchhoff’s voltage and current laws the relationship between the winding voltage and current can be established, as shown in Equation (1) [8]:

$$\left\{\begin{array}{c}(\mathrm{j}{\omega }_{(1)}{L}_{11}+R_{z1}){\dot{I}}_{1(1)}+\mathrm{j}{\omega }_{(1)}{M}_{12}{\dot{I}}_{2(1)}+\cdots +\mathrm{j}{\omega }_{(1)}{M}_{1n}{\dot{I}}_{n(1)}={\dot{U}}_{(1)}\\ \mathrm{j}{\omega }_{(1)}{M}_{21}{\dot{I}}_{1(1)}+(\mathrm{j}{\omega }_{(1)}{L}_{22}+R_{z2}){\dot{I}}_{2(1)}+\cdots +\mathrm{j}{\omega }_{(1)}{M}_{2n}{\dot{I}}_{n(1)}={\dot{U}}_{(1)}\\ ⋮\\ \mathrm{j}{\omega }_{(1)}{M}_{n1}{\dot{I}}_{1(1)}+\mathrm{j}{\omega }_{(1)}{M}_{n2}{\dot{I}}_{2(1)}+\cdots +(\mathrm{j}{\omega }_{(1)}{L}_{nn}+R_{zn}){\dot{I}}_{n(1)}={\dot{U}}_{(1)}\\ {\dot{I}}_{1(1)}+{\dot{I}}_{2(1)}+{\dot{I}}_{3(1)}+\cdots \cdots +{\dot{I}}_{n(1)}={\dot{I}}_{\mathrm{N}(1)}\end{array}\right.$$

(1)

where ${\omega }_{(1)}$ denotes the angular frequency of the fundamental component; ${\dot{I}}_{i(1)}$ refers to the current flowing through the i-th winding under fundamental frequency conditions; ${\dot{U}}_{(1)}$ represents the terminal voltage of the bridge-arm reactor at the fundamental frequency; and ${\dot{I}}_{N(1)}$ denotes the rated operating current corresponding to the fundamental frequency.

For the mutual inductance parameters $M_{ik}$ of the windings in the equation set, they can be equivalently treated as the mutual inductance between two coaxial helical tubes of finite length. Based on Neumann’s formula and applying the Bartky transformation to the expression, the mutual inductance calculation formula is given as [8]:

$$M_{ik}={\mu }_0{n}_{pi}{n}_{pk}{R}_i{R}_k[F(R_i,R_k,Z_1)-F(R_i,R_k,Z_2)+F(R_i,R_k,Z_3)-F(R_i,R_k,Z_4)]$$

(2)

where $F\left(R_i,R_k,Z_j\right)=R_i{R}_k{\displaystyle {\int }_0^{\pi }\frac{\sqrt{R_i^2+R_k^2-2R_i{R}_k\cos \theta +Z_j^2}}{R_i^2+R_k^2-2R_i{R}_k\cos \theta }}{\sin }^2\theta d\theta$, and $\theta$ denote the angular difference between the two winding elements.

The parameters are defined as: $Z_1={\displaystyle \frac{H_i}{2}}+{\displaystyle \frac{H_k}{2}}+S$, $Z_2={\displaystyle \frac{H_i}{2}}-{\displaystyle \frac{H_k}{2}}+S$, $Z_3=-{\displaystyle \frac{H_i}{2}}-{\displaystyle \frac{H_k}{2}}+S$, and $Z_4=-{\displaystyle \frac{H_i}{2}}+{\displaystyle \frac{H_k}{2}}+S$.

Where ${\mu }_0$ is the permeability of free space, $n_{pi}$ and $n_{pk}$ are the number of turns per unit height of the two windings, $H_i$ and $H_k$ are the heights of the two windings, $R_i$ and $R_k$ are the radii of the two windings, and S is the axial center-to-center height difference between the two windings.

Furthermore, when calculating the self-inductance $L_{ii}$ of the i-th winding, it can be regarded as a special case of the mutual inductance between windings. Specifically, when two windings are completely coincident in both geometric parameters and spatial position, their mutual inductance can be equivalently treated as self-inductance. Under this condition, by setting $H_i=H_k$ and S = 0, the following expression can be obtained [8]:

$$L_{ii}=2{\mu }_0{n}_i^2{R}_i^2[F(R_i,R_i,H_i)-F(R_i,R_i,0)]$$

(3)

Under power-frequency conditions, since the reactance of each layer winding is much greater than its resistance, the influence of $R_{zi}$ can be neglected, and Equation (1) can be simplified as:

$$\left\{\begin{array}{c}{N}_1{N}_1{k}_{11}{I}_{1(1)}+N_1{N}_2{k}_{12}{I}_{2(1)}+\cdots +N_1{N}_n{k}_{1n}{I}_{n(1)}=U_{(1)}/{\omega }_{(1)}\\ N_2{N}_1{k}_{21}{I}_{1(1)}+N_2{N}_2{k}_{22}{I}_{2(1)}+\cdots +N_2{N}_n{k}_{2n}{I}_{n(1)}=U_{(1)}/{\omega }_{(1)}\\ ⋮\\ N_n{N}_1{k}_{n1}{I}_{1(1)}+N_n{N}_2{k}_{n2}{I}_{2(1)}+\cdots +N_n{N}_n{k}_{nn}{I}_{n(1)}=U_{(1)}/{\omega }_{(1)}\\ I_{1(1)}+I_{2(1)}+I_{3(1)}+\cdots +I_{n(1)}=I_{N(1)}\end{array}\right.$$

(4)

where $I_{i(1)}$ denotes the scalar value of the fundamental frequency current in the i-th encapsulated winding; $U_{(1)}$ represents the scalar value of the terminal voltage of the encapsulated winding under fundamental frequency conditions; $I_{N(1)}$ is the scalar value of the rated operating current of the reactor at the fundamental frequency; and $k_{ij}={\displaystyle \frac{M_{ij}}{N_i\cdot N_j}}$ denotes the geometric mutual inductance coefficient between the i-th winding and the j-th winding, where $N_i$ and $N_j$ are the numbers of turns of the i-th and j-th windings, respectively.

Under the influence of the m-th harmonic component, the circuit equations can be expressed as:

$$\left\{\begin{array}{c}{N}_1{N}_1{k}_{11}{I}_{1(m)}+N_1{N}_2{k}_{12}{I}_{2(m)}+\cdots +N_1{N}_n{k}_{1n}{I}_{n(m)}=U_{(m)}/{\omega }_{(m)}\\ N_2{N}_1{k}_{21}{I}_{1(m)}+N_2{N}_2{k}_{22}{I}_{2(m)}+\cdots +N_2{N}_n{k}_{2n}{I}_{n(m)}=U_{(m)}/{\omega }_{(m)}\\ ⋮\\ N_n{N}_1{k}_{n1}{I}_{1(m)}+N_n{N}_2{k}_{n2}{I}_{2(m)}+\cdots +N_n{N}_n{k}_{nn}{I}_{n(m)}=U_{(m)}/{\omega }_{(m)}\\ I_{1(m)}+I_{2(m)}+I_{3(m)}+\cdots +I_{n(m)}=I_{N(m)}\end{array}\right.$$

(5)

where $I_{i(m)}$, $U_{(m)}$, and $I_{N(m)}$ represent the branch current, terminal voltage, and total current under the m-th harmonic component, respectively; and ${\omega }_{(m)}$ denotes the angular frequency corresponding to the m-th harmonic.

Under the influence of the DC component, the circuit relationship of the arm reactor can be expressed as:

$$\left\{\begin{array}{c}{R}_{z1}{I}_{1(d)}=U_{(d)}\\ R_{z2}{I}_{2(d)}=U_{(d)}\\ R_{z3}{I}_{3(d)}=U_{(d)}\\ ⋮\\ R_{zn}{I}_{n(d)}=U_{(d)}\\ I_{1(d)}+I_{2(d)}+I_{3(d)}+\cdots +I_{n(d)}=I_{N(d)}\end{array}\right.$$

(6)

where $I_{i(d)}$, $U_{(d)}$, and $I_{N(d)}$ denote the DC current, DC terminal voltage, and total branch current, respectively, and $R_{zi}={\displaystyle \frac{L_i}{\gamma {\sigma }_i}}={\displaystyle \frac{8R_i{N}_i}{\gamma d_i^2}}$ is the DC resistance of the i-th winding layer, where $L_i$ is its length, ${\sigma }_i$ is its cross-sectional area, $d_i$ is its wire diameter, and γ is the conductivity of the conductor material.

Since the amplitude of AC harmonic components is much lower than that of the power-frequency component, the subsequent strand number design process is based on satisfying the circuit equations under power-frequency operating conditions.

Based on the above electromagnetic model, the current $I_i$ of each winding layer can be obtained, and the branch loss $P_{Ri}$ of each winding in the arm reactor can be calculated as the heat source term in the temperature field analysis:

$$P_{Ri}=I_i^2\cdot R_{zi}=I_i^2\cdot {\displaystyle \frac{8R_i{N}_i}{\gamma d_i^2}}$$

(7)

Based on the above formula, the self-inductance L, mutual inductance M, and branch losses $P_R$ of each winding can be calculated according to the basic structural parameters of the arm reactor (such as the number of branches n, winding mean radius R, wire diameter d, and number of turns N), providing a foundation for the calculation of objective functions and constraints in subsequent structural optimization design.

2.2. Heat Dissipation of the Arm Reactor

The thermal dynamic process of the arm reactor can be divided into three stages, Initial stage: The body temperature rises rapidly, forming a significant temperature difference with the surrounding environment; Transition stage: Heat exchange occurs with the surrounding air through thermal conduction, natural convection, and radiation, resulting in a gradual increase in ambient temperature and a gradual reduction in temperature difference; Steady stage: The system reaches thermal equilibrium, where the internal heat generation rate equals the external heat dissipation rate. Under thermal equilibrium conditions, the temperature field can be described by the three-dimensional steady-state heat conduction equation as follows:

$${\displaystyle \frac{\partial }{\partial x}}\left(k{\displaystyle \frac{\partial T}{\partial x}}\right)+{\displaystyle \frac{\partial }{\partial y}}\left(k{\displaystyle \frac{\partial T}{\partial y}}\right)+{\displaystyle \frac{\partial }{\partial z}}\left(k{\displaystyle \frac{\partial T}{\partial z}}\right)=-q$$

(8)

where q denotes the heat source intensity per unit volume, and k represents the thermal conductivity between the encapsulation and the air.

Natural convection and radiation heat transfer occur simultaneously between the arm reactor and the external air.

The heat conduction due to convective heat transfer can be calculated by the following equation:

$$\mathsf{\Phi }=Ah(z)\left(t_w-t_f\right)$$

(9)

where $t_w$ is the surface temperature of the arm reactor encapsulation, $t_f$ is the temperature of the contacting air, and $h(z)$ is the convective heat transfer coefficient between the encapsulation and the air, A is the effective heat dissipation area of the encapsulation, and Φ is the convective heat transfer power.

Radiative heat transfer can be expressed as:

$$q=\epsilon \sigma \left(t_w^4-t_f^4\right)$$

(10)

where σ is the Stefan–Boltzmann constant, and ε is the surface emissivity between the encapsulation and the air, typically taken as 0.9.

2.3. Temperature Field Simulation of the Arm Reactor

This study takes a decommissioned 21.25 mH bridge-arm reactor from a certain factory as the research object. Its rated current is 360 A _(DC) + 635 A _(AC) + 12 A (2nd), and its key structural parameters are listed in

Table 2. Structural parameters of the decommissioned factory bridge-arm reactor.

Encapsulation No.	Turns	Inner Diameter (mm)	Outer Diameter (mm)	Wire Diameter (mm)	Height (mm)
1	140	3379.62	3442.6	8.32	836
2	122	3506.16	3556.2	8.06	845
3	110	3617.14	3660	7.86	843
4	103	3718.82	3758	7.83	830
5	87	3817.81	3851.5	7.40	836
6	85	3907.99	3941.9	7.48	836
7	80	3999.475	4031.4	7.48	835
8	72	4092.827	4119.4	7.24	835
9	67	4177.215	4200.9	7.16	836
10	61	4259.136	4279.8	7.01	831
11	60	4337.599	4358.3	7.06	832
12	59	4415.818	4437.6	7.12	840
13	58	4495.553	4517.6	7.28	834
14	57	4578.67	4601	7.09	835

, while the material properties are shown in

Table 3. Material properties of the bridge-arm reactor.

Material	Electrical Conductivity (S/m)	Thermal Conductivity (W/m·K)	Density (kg/m3)	Specific Heat Capacity (J/kg·K)	Kinematic Viscosity (m2/s)
Epoxy Resin	1 × 10−14	0.28	1749	971	—
Aluminum	3.774 × 107	237	2731	881	—
Air	1 × 10−15	0.0257	1.205	1005	1.51 × 10−5

. Based on the previously established equivalent electromagnetic model, the losses of each encapsulated winding are first calculated and used as heat sources in the temperature field analysis. Subsequently, using a finite element multiphysics simulation platform, the internal temperature distribution of the arm reactor is simulated and analyzed under flow–thermal coupling conditions.

To ensure the accuracy of the simulation and the rationality of airflow distribution within the fluid domain, the bottom boundary of the fluid region is set to match the support height of the reactor base, and the top boundary height is approximately equal to the reactor body height to simulate air exchange at the top; The right boundary is set at a distance approximately equal to the mean radius of the outermost winding of the reactor, providing the necessary lateral heat dissipation space. Meanwhile, the environmental temperature is set to 20 °C in the finite element simulation, and the top and right boundaries are applied with fluid and thermal open boundary conditions. The bottom boundary is set as ground with wall boundary conditions, and its temperature is fixed at ambient temperature; Thermal radiation boundary conditions are applied to the surfaces of all encapsulations, with a uniform emissivity of 0.9.

As shown in Figure 2, which presents the computed temperature field, the maximum winding temperature reaches 71.1 °C, with the hotspot located in the upper region of the 10th winding, and significant temperature rise is also observed in the 5th and 9th layers. The radial temperature distribution indicates that windings with higher loss density exhibit more significant temperature rise, whereas those with lower loss density show relatively lower temperature rise. This result demonstrates that the distribution of loss density is the main factor influencing the temperature rise distribution of the windings. Therefore, in the structural optimization design of the arm reactor, in addition to controlling the overall loss level, it is also necessary to further balance the loss density among the windings to effectively suppress local overheating and achieve uniform temperature rise across the entire structure.

As shown in the axial temperature rise distribution in Figure 3, the temperature rise along the height direction exhibits a trend of “low at the bottom—high in the middle—decreasing at the top.” This distribution characteristic is mainly attributed to hot air rising rapidly after being heated at the bottom of the windings, carrying away some heat from the bottom region and resulting in lower temperature rise at the bottom. In the middle region, the continuously rising heat flow gradually accumulates, causing local heat retention and resulting in the peak temperature rise. In the top region, airflow is further enhanced, improving heat dissipation conditions and leading to a moderate decrease in temperature. Therefore, the internal windings of the arm reactor exhibit a significant axial temperature rise gradient.

Overall, the current structure shows large temperature rise differences among windings, with the maximum temperature difference reaching 49.38 °C, indicating severe non-uniformity in temperature distribution. Therefore, it is necessary to optimize the structure of the arm reactor to reduce the temperature rise gradient and enhance the thermal stability and reliability of the equipment.

3. Multi-Objective Optimization Model for Arm Reactor Design

Non-dominated Sorting Genetic Algorithm II (NSGA-II) is an efficient multi-objective optimization algorithm with strong global search capability, widely applied in the parameter design of complex equipment. In the structural design of the arm reactor, the NSGA-II algorithm can optimize multiple objective functions—such as mass, loss, and temperature rise—through mechanisms like non-dominated sorting, crossover, and mutation, thereby providing a set of uniformly distributed Pareto-optimal solutions and offering designers more options and trade-off space.

3.1. Optimization Variables

Focusing on the triple optimization objectives of “low loss—low temperature rise—low weight,” this study selects four key parameters as optimization variables: the number of branches n, winding mean radius R, number of turns N, and wire diameter d. These core parameters not only directly determine the performance indicators and current distribution of the reactor, but are also key factors affecting the mass of the arm reactor.

Specifically, the selection of branch number must balance inductance characteristics and mass—although increasing n can enhance inductance, it may also lead to an increase in the overall mass. The winding mean radius R is directly related to the overall size of the device, and an optimal balance must be sought between spatial utilization and heat dissipation performance. The number of turns N critically affects both inductance and overall structural mass, with variations in N significantly impacting the rated inductance and total weight of the arm reactor. The wire diameter d influences both electrical conductivity and device weight; a larger wire diameter can effectively reduce resistive loss, but also increases the total mass.

Given the complex interdependencies among these design variables, global optimization must be performed using the NSGA-II multi-objective algorithm to achieve a comprehensive optimal configuration of the arm reactor in terms of electrical performance, thermal stability, and structural lightweighting.

3.2. Optimization Objectives and Model Construction

The optimization objectives are achieved by adjusting the above design variables, enabling the arm reactor to meet performance requirements while fulfilling the following goals:

(1) Minimize the mass of the reactor $W_{Al}$: By rationally designing the number of branches, number of turns, and wire diameter, material usage is reduced, thereby decreasing the total mass of the reactor. The corresponding expression is:

$$W_{\mathrm{Al}}={\displaystyle \frac{{\pi }^2\rho }{2}}{\displaystyle \sum _{i=1}^n{N}_i}{R}_i{d}_i^2$$

(11)

where $\rho$ is the density of the winding material and n is the number of winding branches.

(2) Minimize the reactor loss $P_{loss}$: By optimizing the wire diameter and number of turns, resistive losses are reduced to improve reactor efficiency. The corresponding expression is:

$$P_{\mathrm{loss}}={\displaystyle \sum _{i=1}^n{P}_{Ri}}$$

(12)

(3) Minimize the temperature rise ΔT of the arm reactor: By reasonably designing structural parameters such as the number of winding branches n, mean radius R, wire diameter d, and number of turns N, the winding temperature rise during operation can be effectively reduced, thereby slowing the thermal aging of insulation materials and enhancing the thermal stability and reliability of the equipment.

It should be noted that the temperature rise is influenced by multiple coupled factors such as structural dimensions, current distribution, and heat dissipation conditions. However, traditional temperature rise formulas widely used in engineering are mostly derived from transformer design experience and fail to accurately capture the natural convection heat transfer characteristics in the cavity and multi-encapsulation structure of arm reactors. To improve the applicability and prediction accuracy of the model, this paper derives an analytical expression for temperature rise under narrow-gap convection conditions, based on the thermal-structural characteristics of the arm reactor. The relevant formulas will be detailed in the later section on the isothermal temperature rise constraint.

Based on the above three optimization objectives, a multi-objective optimization mathematical model is established, incorporating design variables and constraints in a unified framework.

3.3. Constraint Conditions

For the structural design of the arm reactor, this study adopts the design objectives of “low loss–low temperature rise–low weight” and proposes a triple-constraint strategy of “equal resistive voltage–equal temperature rise–equal loss density.”

(1) Equal resistive voltage constraint: This constraint requires the resistive voltage across each winding to be equal, thereby balancing the current distribution among branches and minimizing total resistive losses. The constraint is expressed as:

$$R_{zi}{I}_i=R_{zj}{I}_j(i\ne j)$$

(13)

That is:

$${\displaystyle \frac{\pi D_i}{\gamma {\sigma }_i}}{n}_i{I}_i={\displaystyle \frac{\pi D_j}{\gamma {\sigma }_j}}{n}_j{I}_j(i\ne j)$$

(14)

where ${\sigma }_i$ is the cross-sectional area of the i-th winding, $D_i$ is the diameter of the i-th encapsulation layer.

Let $k_{ij}={\displaystyle \frac{{\sigma }_i{D}_i}{{\sigma }_j{D}_j}}$, by combining the steady-state circuit equations with the equal resistive voltage constraint, the distribution rules for winding turns $N_i$ and current $I_i$ can be obtained:

$$N_i={\displaystyle \frac{\sqrt{L_N{\displaystyle \sum _{m=1}^n\frac{{\sigma }_m}{D_m}}{\displaystyle \sum _{k=1}^n\frac{f_{mk}{\sigma }_k}{D_k}}}}{{\displaystyle \sum _{j=1}^n\frac{f_{ij}{\sigma }_j}{D_j}}}}(i=1,2,3,\cdots ,n)$$

(15)

$$I_i={\displaystyle \frac{I_N{\displaystyle \sum _{k=1}^n{K}_{ki}}{f}_{ik}}{{\displaystyle \sum _{j=1}^n{K}_{ji}^2}{\displaystyle \sum _{k=1}^n{K}_{kj}}{f}_{jk}}}(i=1,2,3,\cdots ,n)$$

(16)

(2) Equal temperature rise constraint: The temperature rise in different encapsulated windings is required to be approximately equal to ensure uniform thermal distribution, avoid local hotspots, and enhance the thermal reliability of the device.

At present, the temperature rise in multilayer encapsulated parallel arm reactors is commonly calculated based on empirical formulas for the average temperature rise in naturally air-cooled transformers, as expressed in [9,10,11]:

$$\Delta T=0.266H^{0.2}{q}^{0.8}=0.266H^{0.2}{({\displaystyle \frac{P_i}{S_i}})}^{0.8}$$

(17)

where $\Delta T$ is the winding temperature rise (°C); H is the air duct (encapsulation) height (m); $P_i$ is the electrical loss of the i-th encapsulation (W); $S_i$ is its effective cooling area (m²); and $q$ is the corresponding heat-dissipation density (W/m²). The coefficient 0.266 is an empirical constant obtained from the regression of experimental and numerical data in SI units and can therefore be treated as dimensionless within this unit system.

However, since bridge-arm reactors are mostly of dry-type air-core structure, their cooling mechanisms and dimensional characteristics differ significantly from those of transformers. As a result, traditional empirical formulas fail to accurately reflect their thermal behavior. It is therefore necessary to develop a dedicated temperature rise characterization method for bridge-arm reactors to improve the accuracy of design approaches based on equal temperature rise constraints.

In this study, a Kriging response surface modeling approach is introduced to enhance the accuracy of winding temperature rise prediction for bridge-arm reactors. The Kriging method is well-suited for capturing complex nonlinear relationships between multidimensional inputs and response variables, especially in small sample, highly nonlinear problems. It demonstrates high precision in surrogate modeling and optimization analysis during electrical equipment design. Essentially, Kriging treats the target response as a stochastic process and performs optimal linear unbiased estimation based on known observations across the input space, and it can be expressed as:

$$\widehat{y}(x)=\mu +Z(x)$$

(18)

In the Kriging model, $\widehat{y}(x)$ represents the predicted value, μ is the global mean, and Z(x) is a zero-mean stochastic process characterized by spatial correlation. The core idea of the Kriging approach is to establish a correlation-based mapping between input variables and output responses using sample data.

In this study, key influencing parameters in the design of the bridge-arm reactor are selected as input variables, including winding loss density ($P_d$), encapsulation width ($W_e$), encapsulation height ($H_e$), and air duct width ($\delta$). Specifically, the winding loss density directly determines the internal heat generation intensity of the device, while the encapsulation width, height, and air duct width significantly affect the convective heat dissipation capability and airflow efficiency around the windings.

To efficiently and comprehensively capture the influence of these factors on winding temperature rise, the Latin Hypercube Sampling (LHS) method is employed to generate 40 sample points in a four-dimensional design space. This ensures good spatial coverage and non-overlapping distribution among sample points, thereby enhancing the predictive accuracy of the Kriging model.

For each sampled geometry, a three-dimensional steady-state thermal model is established in COMSOL Multiphysics 6.3 to simulate the temperature field of the winding region in detail. The resulting temperature rise at key observation points is then used as training data for the Kriging model. To ensure the fidelity of the simulation results, reasonable boundary conditions are applied, including natural convection at the air gap boundaries and surface radiation at the encapsulation surface (with an emissivity of 0.9). The ambient temperature is set to 20 °C. The material properties are consistent with those listed in Table 2, ensuring consistency of the thermal-field calculation model.

Table 4. Sampling Results of Bridge-Arm Reactor Winding Temperature Rise.

No.	Pd (W/m3)	He (mm)	We (mm)	δ (mm)	ΔT (°C)
1	13,000	850	17	70	34.2
2	6500	1100	18	55	28.1
3	14,000	1050	20	45	31.5
…	…	…	…	…	…
38	20,000	950	12	50	35.8
39	11,500	1400	10	50	27.7
40	17,500	1100	13	65	34.5

summarizes the sampling results of the winding temperature rise.

The sampled data are substituted into the Kriging response surface model to perform fitting and prediction of the response values at other design points within the defined parameter space. The Kriging response surfaces corresponding to the individual design variables are shown in Figure 4. According to the evaluation results, the error between the Kriging model surface and the sampled data is small, with all sample points located on the fitted surface. Furthermore, the Kriging surrogate model is capable of extrapolating and predicting the response at unsampled points within the design space. As a locally constructed nonlinear interpolation model, the Kriging method demonstrates high accuracy in small-sample prediction scenarios.

Based on a full-scale prototype of a bridge-arm reactor, the winding temperature rise is calculated using both the traditional empirical formula for reactor windings (Equation (17)) and the Kriging-based response surface model. The results are compared with the experimentally measured temperature rise, as shown in Figure 5. It can be observed that the temperature rise distribution obtained using the proposed method closely matches the measured distribution, with an average error of only 1.043 °C, accounting for 11.47% of the average measured temperature rise. In contrast, the traditional empirical formula yields an average error of 3.978 °C, corresponding to 43.74% of the average value, indicating significantly lower accuracy compared to the method proposed in this study.

The uniform temperature rise constraint for encapsulated windings can be expressed as:

$$\Delta T_1=\Delta T_2=\cdots =\Delta T_n$$

(19)

(3) Uniform loss density constraint: This constraint ensures that the power loss per unit volume of each winding is approximately equal. It aims to regulate the spatial distribution of heat sources at the source level, improve the uniformity of temperature rise, prevent local overheating and thermal aging of insulation materials, and thereby extend the service life of the equipment.

The uniform loss density for encapsulated windings can be expressed as:

$$P_1/V_1=P_2/V_2=\cdots =P_n/V_n$$

(20)

where $V_i$ denotes the volume of the i-th winding.

3.4. NSGA-II Optimization Procedure

When applying the NSGA-II multi-objective optimization algorithm to optimize the structure of the bridge arm reactor, the core design variables are the number of branches n, the number of turns N, the mean winding radius R, and the conductor diameter d. Through mechanisms such as sorting, crossover, and mutation, the algorithm evolves the population generation by generation. Ultimately, the optimization process satisfies the triple constraints of “equal resistance voltage, uniform loss density, and encapsulated uniform temperature rise,” while simultaneously pursuing the three objectives of “low loss, low temperature rise, and low weight.” The specific optimization steps are as follows. The NSGA-II–based optimization workflow for the bridge-arm reactor is illustrated in Figure 6.

(1) Initialization of the Population

First, an initial population is randomly generated within the allowable range of the design parameters. Each individual contains key structural parameters such as the number of branches n, the number of turns N, the mean winding radius R, and the wire diameter d. Based on Equations (15) and (16), the turn distribution and current parameters for each winding are calculated. Derived parameters such as winding height, inductance characteristics, and air gap width are then computed to form a complete initial population set.

(2) Objective Function Evaluation and Constraint Assessment

For all individuals in the population, performance indicators such as winding weight, temperature rise, and loss density are analytically calculated based on Equation (11), the Kriging model, and Equation (20). Individuals that do not meet the constraint conditions are screened using constraint penalty functions.

(3) Population Evolution Mechanism

The parent and offspring populations are merged, and a Pareto front hierarchy is established using fast non-dominated sorting. A crowding distance comparator is applied to maintain diversity of the solution set. During crossover and mutation, a differentiated mutation strategy is applied to the number of branches n, number of turns N, wire diameter d, and mean radius R, ensuring the rated inductance of the reactor remains essentially unchanged.

(4) Iterative Optimization Strategy

Through iterative evolution of multiple generations, the algorithm gradually approaches the Pareto optimal front. In each generation, individuals with higher non-dominated ranks are retained with priority, supported by an elitism strategy to enhance convergence speed and solution uniformity. Ultimately, the final solution is selected from the Pareto set to ensure that loss, temperature rise, mass, and loss density all meet design requirements, while minimizing their relative deviation, resulting in the optimal bridge arm reactor structural configuration.

4. Algorithm Validation

To address the multi-objective optimization problem of the reactor, this study adopts the NSGA-II-based optimization method to obtain structural designs that satisfy the imposed constraints. Regarding algorithm parameter settings, a population size of 300 is adopted to ensure sufficient sampling across the solution space, and the maximum number of generations is set to 150, with convergence tests conducted to verify its rationality. The crossover and mutation probabilities are set to 0.8 and 0.3, respectively, aiming to enhance global search capability and prevent convergence to local optima. By incorporating elitism and non-dominated sorting mechanisms, the convergence speed of the algorithm and the distribution quality of the Pareto solution set are improved, ensuring efficient convergence to the Pareto optimal front.

Specifically, the number of branches n ranges from 9 to 16, the mean radius R ranges from 600 mm to 1200 mm, the wire diameter d ranges from 7 mm to 10 mm, and the number of turns N ranges from 80 to 140. Based on these variables, three optimization objectives are defined: total mass $W_{Al}$ (kg), power loss $P_{loss}$ (kW), and temperature rise ΔT (°C), which are optimized simultaneously according to Pareto optimality theory. Throughout the parameter evaluation, the operating conditions of the optimized bridge-arm reactor are kept identical to those of the pre-optimization baseline to ensure comparability. After 150 generations of evolution, a well-distributed Pareto front solution set is obtained, as shown in Figure 7. Representative optimized design results are selected from the Pareto solution set, and the specific structural parameters of the bridge-arm reactor are listed in

Table 5. Optimized Structural Parameters of the Arm Reactor.

Encapsulation No.	Turns	Mean Radius Meter (m)	Wire Diameter (mm)	No. of Internal Windings	Loss Density (kW/m3)	Height (m)
1	129.8	1.131	6.61	1	60.47	1.172
2	116.2	1.192	6.72	1	67.01	1.172
3	106.4	1.251	6.86	1	72.46	1.172
4	98.8	1.310	7.08	1	76.71	1.172
5	92.9	1.369	7.38	1	79.44	1.172
6	88.5	1.427	7.15	1	80.58	1.172
7	84.8	1.491	7.82	1	80.37	1.172
8	82.3	1.555	7.79	1	78.51	1.172
9	80.9	1.613	7.21	1	75.49	1.172
10	80.3	1.670	6.63	1	71.62	1.172
11	80.1	1.730	7.22	1	66.91	1.172
12	80.7	1.791	7.76	1	61.49	1.172
13	82.4	1.851	6.87	1	55.71	1.172
14	84.8	1.908	6.88	1	50.41	1.172

.

According to the optimization design scheme, a prototype of the bridge-arm reactor was constructed, as shown in Figure 8. The test results show that the rated inductance of the optimized prototype is 21.82 mH, which is close to the pre-optimization value of 21.25 mH, thus verifying the effectiveness and accuracy of the equivalent electromagnetic model and the optimization method for the bridge-arm reactor.

Based on flow-thermal coupling calculation, the temperature field distribution of the optimized bridge-arm reactor was recalculated, as shown in Figure 9. After optimization, the total loss of the bridge-arm reactor decreased from 61.65 kW to 34.21 kW, representing a reduction of 44.51%. Meanwhile, with the application of equal resistive loss density distribution, the winding losses became more uniform, with maximum and minimum values of 80.58 kW/m³ (Winding 6) and 50.41 kW/m³ (Winding 14), and a range ratio of 37.44%. The reduction in heat source intensity further lowered the hot-spot temperature from 71.1 °C to 42.9 °C, a reduction of 39.66%. In addition, the temperature distribution among the windings became more uniform, with the hot-spot temperature rise ranging from 34.57 °C to 42.94 °C, and a maximum temperature difference of only 8.37 °C. In terms of mass, the total mass of the windings decreased from 1733.2 kg to 1302.7 kg after optimization, a reduction of 24.83%, significantly reducing the equipment weight, which helps lower manufacturing costs and improve seismic performance.

5. Conclusions

This paper addresses key design concerns of bridge-arm reactors, such as non-uniform loss distribution, energy efficiency, and seismic performance, and proposes an NSGA-II-based optimization method. By introducing a threefold constraint mechanism—equal resistance voltage, uniform loss density, and equal enclosure temperature rise—the comprehensive goals of “low loss, low temperature rise, and lightweight” are achieved. The effectiveness of the proposed optimization method is validated through simulation analysis and prototype testing. The specific conclusions are as follows:

(1) An equivalent electromagnetic model applicable to AC–DC hybrid conditions was established, enabling accurate calculation of winding self-inductance, mutual inductance, DC resistance, and current distribution, thus providing theoretical support for inductance accuracy in the optimization process.

(2) A Kriging surrogate model is employed to characterize the relationship between the winding temperature rise in the bridge-arm reactor and parameters such as loss density, encapsulation width, encapsulation height, and air duct width. Compared with conventional temperature rise formulas, the proposed method reduces the temperature rise error from 43.74% to 11.47%, significantly improving the effectiveness and accuracy of the analytical temperature rise calculation.

(3) An NSGA-II-based optimization framework was proposed, incorporating a threefold constraint strategy—equal resistance voltage, uniform loss density, and equal enclosure temperature rise—with the objectives of minimizing loss, temperature rise, and overall mass. A prototype reactor was constructed, and experimental results showed that total loss was reduced by 44.51%, hot-spot temperature rise decreased by 39.66%, and total mass dropped by 24.83%, all while maintaining the rated inductance. These results fully demonstrate the engineering applicability of the proposed method.