Optimization Design of Oil-Immersed Iron Core Reactor Based on the Particle Swarm Algorithm and Thermal Network Model

In this paper, according to the design parameters of oil-immersed iron core reactor, the thermal network model of windings is established by the thermo-electric analogy method, and the temperature distribution of the windings can be obtained. Meanwhile, a ﬂuid-thermal coupled ﬁnite element model is established, the temperature and ﬂuid velocity distribution are extracted, and the simulation results show that the error coeﬃcient of temperature is less than 3% compared with the thermal network model, so the correctness of thermal network model has been veriﬁed. Taking the metal conductor usage and loss of windings as the optimization objects, the optimization method based on the particle swarm algorithm and thermal network model is proposed, and the Pareto optimal solutions between the metal conductor usage and loss of windings are given. The optimization results show that the metal conductor usage is reduced by 23.05%, and the loss is reduced by 20.25% compared with the initial design parameters, and the maximum temperature of winding does not exceed the expected value. Thus, the objects of low metal conductor usage and loss of windings are conﬂicted and cannot be optimized simultaneously; the optimization method has an important guiding signiﬁcance for the design of oil-immersed iron core.


Introduction
e oil-immersed iron core reactor is an indispensable equipment in the power system, which plays the role of reactive power compensation, limiting the fault current and improving the power factor of the system [1]. However, the reactor encounters overheating and even burns out in the operation process, and the relevant studies show that the excessive temperature of the windings is the main reason [2]. e oil flow of the oil-immersed reactor enters the oil channels of the disc-type windings; the hot spot is formed as the convection heat dissipation capacity of each disc which is different; when the hot spot exceeds the limit, it will directly affect the safe and stable operation of the reactor. In order to reduce the temperature of reactor, the methods, such as increasing the cross-sectional area of conductor and oil channel width, are adopted; they can reduce the loss and temperature of the windings, but the metal conductor usage is increased. e loss and the metal conductor usage of windings are two key parameters in the design of reactor; in order to realize the low loss and the metal conductor usage simultaneously, the temperature calculation is needed and the optimization design method about the reactor is essential.
Currently, the relevant studies mainly include (1) the temperature calculation of the reactor. e finite-difference method is used to calculate the temperature of the reactor [3,4], and it is merely suitable for the situations that the temperature and heat flux distribution of the coil surface are the same, which limits its practical application. e finite element model is adopted [5][6][7], and the detailed temperature and fluid velocity distribution of the reactor can be obtained, but the calculation process is complex and calculation time is long. e thermal network model is put forward, and it was applied to the transformer windings [8]. On the basis, the optimization model of transformer windings is proposed [9][10][11][12]. In [13], the complex model of oil channels in windings is simplified to an approximate matrix of hydraulic channel, and the correctness has been verified. (2) e optimization design of the reactor. In [14], the finite element method is adopted to improve the heat dissipation efficiency of the windings by adjusting the oil channel width and the thickness of the coils, but the calculation time is large and the principle is not clear. In [15,16], the thermal-electromagnetic combined optimization method is proposed to reduce the metal conductor, but the influence of air duct's width on the heat dissipation efficiency is not considered. In [17][18][19][20][21], the genetic algorithm and particle swarm optimization are applied to minimize the metal conductor usage or the loss of the transformer, but it can only realize the optimization of a single target, the overall optimization of the reactor cannot be achieved.
In this paper, taking the metal conductor usage and loss of windings as the optimization objects, the thermal network model of windings is established by the thermo-electric analogy method, the temperature distribution of the windings can be obtained, and the correctness has been verified by the fluidthermal coupled finite element method. Meanwhile, the optimization method based on particle swarm algorithm and thermal network model is proposed, and the Pareto optimal solutions between the metal conductor usage and loss of windings are given. e optimization results show that the metal conductor usage is reduced by 23.05%, and the loss is reduced by 20.25% compared with the initial design parameters, and the maximum temperature of windings does not exceed the expected value, and the correctness has been verified.

The Structure and Equivalent
Model of Windings 2.1. e Structure and Parameters of Core Reactor. e oilimmersed core reactor with a rated voltage of 26.4 kV and rated capacity of 66 MVA is selected as a research object. e disc-type windings are made of wires with insulating layer, which are surrounded on the iron core column. In the windings, the horizontal oil channels are formed between disc-type windings in the axial direction; the inside and outside of the windings are the vertical insulation paper tubes. Meanwhile, a baffle for oil is set up at intervals of a certain number of discs, so as to lead the oil flow in the vertical direction to the horizontal direction, which can enhance the cooling effect of windings. According to the structure characteristic of iron reactor, the influence of iron column and iron yoke on the heat dissipation of windings can be ignored; thus, the windings of iron reactor are only considered in this paper, and the basic structure of windings is given in Figure 1.

e Equivalent Model of Windings.
e total windings consist of 60 layers of disc, and they were divided into 4 channels by the oil guiding washer. Each channel consists of 15 layers of disc, each layer of disc consists of 23 turns of conductor with insulating layers, and the thickness of the insulating layer is 0.4 mm. Considering the temperature and fluid velocity distribution in each channel of the windings are basically the same, thus, one of the channels is selected to research the temperature distribution of windings, and the equivalent model and parameters are given in Figure 2.
e characteristics of metal conductor and insulating materials are given in Table 1.

The Temperature Calculation of Windings
Based on the Thermal Network Model

e Establishment of ermal Network
Model. e heat transfer model of windings can be equivalent to the circuit node network, which adopts the analogy method between the circuit and the heat transfer system. e basic idea of thermo-electric analogy is to compare the transfer of heat between media to the transfer of electric current in a conductor. e related physical essence of the temperature field is analogized to the heat transfer parameters, such as temperature difference, heat flux, heat resistance, and heat capacity. So, the topology of the thermal network can be established, and it is similar to the circuit [22]. In order to obtain the temperature distribution of each conductor and oil flow node, a thermal network model with each conductor as the basic unit is established based on the equivalent model of windings, as shown in Figure 3.
In Figure 3, T disc and T oil are the node temperature of conductor and oil flow, respectively, R cir is the conduction thermal resistance between conductors and insulating layers, R oor is the conduction thermal resistance of oil flow, R ior is the convection thermal resistance between the insulation layers and oil, and q is the heat source of the conductor.

e Calculation of ermal Resistance.
In the actual operation process of oil-immersed iron core reactor, the influence of the leakage magnetic on the eddy current loss of each turn is basically the same, so it is considered that each turn conductor has the same heat source. Considering the temperature difference between the adjacent conductors is  small, the radiation heat transfer can be ignored, so the heat transfer modes of heat flux in the four directions of the conductor are merely the heat conduction and heat convection. In order to establish the thermal network model, the calculation of thermal resistance is essential.
(1) e conduction thermal resistance: in the inner of conductors and insulating material, the conduction thermal resistance depends on the geometric structure and material properties. When the heat flux is transferred to the oil channel, the heat transfer will occur in the oil flow, and the conduction thermal resistance of the oil flow is related to the geometric shape of the oil channel and physical properties of the oil. e conduction thermal resistance in the conductors, insulation layers and oil can be calculated, as shown in the following equation: where L q is the thickness of the conductor in the direction in which the heat flows, k m is the thermal conductivity of material, S is the surface area, which is perpendicular to heat flow direction, R cir is the conduction thermal resistance of metal conductor and insulating layers, and R oor is the conduction thermal resistance of oil. (2) e convection thermal resistance: the convection heat transfer is formed between the discs and the horizontal and vertical oil channel, and the thermal resistance is related to the fluid velocity and temperature of oil.
According to the Newton's law of cooling, the convection thermal resistance between the surfaces of the insulating layers and the oil channels can be expressed, as shown in the following equation: where R ior is the convection thermal resistance between the insulation layers and oil, A is the area of the insulating layer, and α is the convection coefficient, which is related to the geometric structure, material, and fluid characteristics of oil, and it can be written as follows: where L io is the geometric characteristic length of the convection heat transfer surface, k oil is the thermal conductivity of transformer oil, and N u is the Nusselt number, and the expression is [23] N u � 1.86 R e P r D L where L is the horizontal length, D is the diameter of oil channel, μ is the kinetic viscosity of oil with the ambient temperatures, μ s is the kinetic viscosity of

Materials
Attribute Value Copper Insulating materials where ρ oil and c p are the density and specific heat capacity of oil, respectively, and v i is the fluid velocity of horizontal oil channel in the i th layer.

e Temperature Calculation of Windings.
According to the thermal network model in Figure 3, it contains N nodes, so the n-1 node equations can be listed [21], which is similar to the node voltage equation in the circuit. e temperature equations of each node can be written [24], as shown in the following equation: where R self−impedance is the node self-thermal resistance, R mutual−impedance is the node mutual thermal resistance, T disc,i,j and T oil,i,j are the conductor and oil flow nodes of the ith row and jth column in the model, respectively, m i and n j are the numbers of adjacent nodes, m i � (i − 1, i + 1), n j � (j − 1, j + 1). e above equation can be solved by the MATLAB software, and the node temperature can be expressed as where T is the temperature of node, Q is the heat source matrix of the conductor, and G is the heat conductivity matrix, which is composed by the self-thermal resistance and mutual thermal resistance, as shown in the following equation: where, (1/R ma1 i ) and (1/R ma2 j ) are the sum of mutual admittance of the ith and jth nodes, respectively, and the parameter, (1/R sa i ), is the sum of self-admittance of the node.
According to the structure and electrical parameters of windings, combined with the above calculation method, the temperature of each node can be obtained by solving equation (7).

e Calculation Results Based on ermal Network Model.
e settings of thermal network model are as follows: the loss of each turn conductor is 34.5 W, the temperature in the inlet of the oil channel is 303.15 K, and the velocity is 0.3 m/s; the boundary conditions of the upper and lower surfaces of each channel are heat insulation as the low thermal conductivity of the oil guide baffle. In the thermal network model, there exist 437 conductor nodes and 460 oil channel nodes; the number of the discs is sorted from top to bottom; combined with the thermal resistance calculation method and node temperature equation, the temperature of each node can be obtained, and the maximum temperature of each disc is given, as shown in Table 2.
In Table 2, it can be seen that the maximum temperature of discs is gradually increased from the bottom to the top, but there is a downward trend when nearing the top, the maximum temperature is 62.65°C, and the maximum temperature area is mainly concentrated in the upper middle position of winding; the main reason is that the fluid velocity of oil is large in the inlet and outlet, and the convection heat transfer degree is high. When the oil flows in the long channel, the temperature is increased and the fluid velocity of oil is decreased, which leads to the accumulation of heat in the middle of windings, so the temperature of the middle discs is higher than the upper and lower discs.

e Finite Element Simulation Calculation.
According to the structure parameters and material characteristics of the windings, the two-dimensional model is established based on the COMSOL, as shown in Figure 4. e governing equations and boundary conditions settings are the key steps of simulation calculation.

Governing Equation.
e heat conduction and heat convection are the main heat transfer processes in the windings region, and the governing equation of heat conduction can be expressed by the following equation [25]: e heat convection between windings and oil channels satisfies mass, momentum, and energy conservation, as shown in the following equation [26]: where u and v are the radial and axial fluid velocities of oil, respectively, F r and F z are the radial and axial components of fluid gravity, and p is the pressure of fluid.

e Boundary Conditions
Setting. e boundary conditions are set as follows: (1) the heat transfer between the insulating paper and the outer transformer oil can be ignored, and it sets as an adiabatic boundary. (2) e discs surface is regarded as the stationary, the axial and radial velocity are zero, the inlet velocity of oil is 0.3 m/s, and the temperature is 303.15 K. (3) e outlet boundary condition of oil flow is natural pressure outlet boundary and the axis of rotation is axisymmetric boundary condition. e heat source of each turn conductor is setting the same; the value is 34.5 W. Meanwhile, the characteristic parameters of oil are given in Table 3.

e Simulation Results.
e transient field is applied in the simulation, the total calculation time is 10 h, and the time step is 0.1 h, which can be considered that it has reached the steady state. e temperature of windings and fluid velocity distribution of oil are given in Figure 5.
In Figure 5, the temperature of discs in the inlet is lower than in the middle, and the maximum temperature reaches 66.20°C in the 8th disc; the fluid velocity of oil is relatively large in the inlet and outlet. In order to analyze the heat transfer process of discs, the radial and axial direction paths are extracted, as shown in Figure 6.
According to the paths selected and simulation results, the temperature distribution with the different paths is plotted, as shown in Figures 7-9. From Figures 7-9, it can be seen that the temperature of discs is increased first and then decreased in axial and radial directions. On the contrary, the fluid velocity along the axial direction is decreased first and then increased, which is opposite to the trends of temperature. Meanwhile, the maximum temperature locates the eighth disc, and the fluid velocity is the lowest on the both sides of the discs with the maximum temperature, the main reasons are that the fluid velocity in lower channels is relatively large when the oil passes the inlet, and the temperature is low. e heat Mathematical Problems in Engineering dissipation conditions are weak because of the high temperature and low velocity of the oil in the middle discs; the temperature of the middle discs is the highest. In the upper discs, the oil converges in the outlet; it leads to the increase of the fluid velocity, so the temperature of the upper discs is lower compared with the middle discs.

Model Validation.
In order to verify the accuracy of the thermal network model, the maximum temperature of each disc is obtained in both the thermal network model and finite element method, as shown in Figure 10.
In Figure 10, it can be seen that the maximum temperature is basically the same based on the thermal network model and finite element method, and the maximum error coefficient of the disc is only 5.30%, which locates the 8th disc, so the correctness of the thermal network model is verified.
e error coefficient is defined as follows: where ε is the error coefficient, T hot is the temperature based on the thermal network model, and T COM is the temperature based on the finite element method. Meanwhile, the thermal network model is a multivariate linear equation related to thermal resistance, and the calculation speed is fast compared with the finite element method; thus, the thermal network model is adopted to optimize the windings in this paper.

The Optimization Design of Windings Based on Particle Swarm Algorithm and Thermal Network Model
According to actual engineering requirements of the oilimmersed iron core reactor, the metal conductor usage and loss of windings are the two major optimization objects; however, these goals are in conflict with each other, the performance improvement of one object results in the reduction of the other object. In order to achieve the optimal design parameters of windings, the optimization design of iron reactor is needed.

e Particle Swarm Optimization
Algorithm. e particle swarm optimization is one of the most influential multiobjective algorithms, including the prominent characteristics: the optimal solution of multiobjective can be easily obtained for its efficient search ability, and the algorithm has good multipeak search ability, which means it has a good ability to search for a global optimal value and it converges quickly. e flow chart of particle swarm optimization method is given in Figure 11. e formulas of particle update in particle swarm optimization are as follows [27]: where w is the inertia weight, r 1 , r 2 are the random numbers distributed in [0, 1], k is the current iteration number, P i is the individual optimal particle position, P g is the global particle position, c 1 , c 2 are the acceleration constants, V is the velocity of the particle, and X is the position of the particle. e optimization object is to obtain the minimum metal conductor usage and loss of windings; according to the structural characteristics of the windings, the size of the conductor W coil , the height of horizontal channels H coil , and the thickness of insulation layer L ins are selected as the optimization variables. e object functions of metal conductor usage M cu and loss P loss can be written as where ρ is the density of metal conductor, m is the number of discs, n disc is the number of conductors in each disc, d i is the average diameter of the i th conductor, I is the current of the conductor, and c is the resistance of the conductor. e inequality constraints are where T Hot max is the maximum temperature of discs by calculation methods, T max is the maximum temperature constraints of the discs, L is the inductance of the reactor, and W coil min , W coil max , H coil min , H coil max , H duct min , and H duct max are the minimum and the maximum values of variables.

e Optimization Results.
In particle swarm optimization algorithm, the variable parameters are set as follows: the radial width of the conductor is [1.6, 2.6] mm, the axial  Mathematical Problems in Engineering 7 height of the conductor is [12, 14.6] mm, the thickness of the insulation layer is [0.25, 0.5] mm, and the height of horizontal channels is [3.9, 5.4] mm. e expected inductance difference is less than 2%, and the maximum temperature of winding is 75°C. e settings of particles in particle swarm optimization are as follows: the acceleration constants c 1 and c 2 are 0.8, the iteration time is 200, and the inertia weight value is 0.8-1.2. e Pareto optimum results are shown in Figure 12, e change rate of metal conductor usage and loss, K M and K loss , is defined as the following equation:   where x 0 is the initial design parameter and x is the design parameter with the optimization method. In the initial parameters, the metal conductor usage and loss of windings are 90.98 kg and 11.9 kW before optimization, respectively.
In Figure 12, it can be seen that there exists an opposite trend between the metal conductor usage and loss of windings. Combined with the particle swarm optimization algorithm and thermal network model, the representative results can be obtained, as shown in Table 4. e inductance change rate is less than 1% compared with the initial parameters.
In order to verify the correctness of the optimization results, taking the groups 2, 9, 17, and 20 in Table 4 as an example, the maximum temperatures of windings are 54.91°C, 67.74°C, 68.56°C, and 73.40°C, respectively, based on the thermal network model. Meanwhile, the simulation results based on the finite element method are given in Figure 13; the maximum temperatures of windings are 55.10°C, 68.20°C, 70.00°C, and 72.63°C respectively; it can be seen that the temperature distribution laws of the four groups are basically the same, and the error coefficient is less than 3%; thus, the correctness of the optimization method has been verified.
In Table 4, it can be seen that the metal conductor usage of windings is reduced by 23.05%, and the loss of the windings is reduced by 20.25% compared with the initial design parameters. As the purpose of low metal conductor

Conclusions
According to the structure and parameters of oil-immersed iron core reactor, a multiobjective optimization method based on particle swarm optimization and thermal network model is proposed, and the following conclusions can be obtained: (1) e maximum temperature error of discs is less than 3% between the thermal network model and finite element method, and the position of the temperature is basically the same; the correctness of thermal network model has been verified (2) e maximum temperature region is mainly concentration of the middle of windings; the reasons are given by analyzing the fluid velocity distribution of oil, so this area is the main focus to monitoring temperature for the windings (3) e optimization design of windings based on particle swarm algorithm and thermal network model is proposed, the metal conductor usage is decreased by 23.05%, and the loss of the windings is reduced by 20.25% compared with the initial design parameters, so the optimization method has an important guiding significance for the design of oil-immersed iron core

Data Availability
e data used to support the findings of this study are included in this paper and available without any restriction.

Conflicts of Interest
e authors declare no conflicts of interest.