Mathematical model analysis of new-type magnetic integrated CRT considering magnetic leakage

WANG Tiange , TIAN Mingxing, ZHANG Huiying

(1. Shaanxi Railway Institute, Weinan 714000, China；2. School of Automation and Electrical Engineering, Lanzhou Jiaotong University, Lanzhou 730070, China)

Abstract： A novel magnetic integrated controllable reactor of transformer type (CRT) has the advantages of simple structure, flexible assembly, convenient maintenance and practicability. To analyze its operation characteristics accurately, we establish corresponding equivalent mathematical model considering magnetic leakage based on magnetic circuit and circuit dualistic transformation method. The distribution of magnetic leakage field of each winding is analyzed qualitatively, and the analytical calculation formulas of magnetizing inductance and leakage inductance of each winding are derived. Based on this, the analytical calculation formulas of short-circuit impedance and winding current of CRT under different working conditions are derived. The field-circuit coupling finite element model of the magnetic integrated CRT is established to simulate the current of each winding under different working conditions. The results show that the analytical calculation results of each winding current have good consistency with the finite element calculation results, indicating the validity of CRT equivalent mathematical model and correctness of the analytical formulas of leakage inductance, short-circuit impedance and winding current of CRT. The working winding current of CRT is increasing gradually with the operation of control winding in turn to realise the transition of CRT compensation capacity from zero to a rated value.

Key words： controllable reactor of transformer type (CRT)； magnetic integration technology； equivalent mathematical model； leakage inductance； winding current

0 Introduction

As a widely used secondary energy, electrical energy has the characteristics of cleanliness and efficiency and plays a vital role in the development of social economy[1]. The structure of extra-high voltage (EHV) and large-capacity transmission network is becoming complex to meet the power demand of different regions and optimize the allocation of power resources nationwide[2]. With the increasing number of non-linear users, adaptation to drastic changes in load, limited line power frequency overvoltage and the improvement of power flow distribution have become urgent problems to be solved in power systems[3]. To ensure the safe, reliable, economic and high-quality operation of power grids, researchers should develop a large-capacity and fast-regulating reactive power source to realize reactive power compensation and voltage control of power systems[4]. Controlled reactor of transformer type (CRT) is a novel thyristor-controlled reactor, which is in the state of graded short circuit and is essentially a multi-winding transformer[5，6]. As a new type of reactive power compensation device, CRT has the advantages of small harmonic current, stepless continuous capacity adjustment and fast response. CRT is an important guarantee to deal with the drastic changes in the power flow of EHV AC transmission lines and improve the power quality of power systems.

Magnetic integration technology involves two or more discrete magnetic components (such as transformer, inductor, etc.) in electromagnetic equipment wound on the same magnetic core to integrate them structurally[7]. With the development of magnetic integration technology, lightweight and miniaturization of CRT can be realized. The basic theory of CRT has been deeply studied in Ref.[8]. The regulation mode and control strategy of CRT have also been analyzed in detail in Ref.[9]. In Ref.[10], array magnetic integrated CRT was proposed based on magnetic integration technology. This case is the first time that magnetic integration technology has been introduced into the structure design of CRT. In Ref.[11], a magnetic integration CRT with multiple magnetic materials was proposed by setting different magnetic materials at different positions of the iron core. In Ref.[12], the discus was set in the middle of the iron core to improve the short-circuit impedance, and the split magnetic integrated CRT was proposed. However, the above-mentioned magnetic integrated CRTs have complex structures and high manufacturing process requirements, thus indicating their poor practicability. To solve such problems, Ref.[13] replaced the discus in Ref.[12] by setting the air gap column in the core column, and the single control-winding basic-unit magnetic integrated CRT was proposed. This CRT has a simple structure and is easy to manufacture. However, given the characteristics of its single control winding, material waste is produced, which increases not only the manufacturing cost, but also the volume and weight of the equipment. Therefore, Refs.[14-15] proposed a dual control-winding basic-unit magnetic integrated CRT, which integrates two control winding units in the same core structure to reduce the cost of CRT. Dual control-winding basic-unit magnetic integrated CRT has the advantages of simple structure, low cost, small volume and small weight, and flexible assembly and maintenance, which is the most superior new-type magnetic integrated CRT[14]. However, in the process of CRT structural analysis, Ref.[15] ignored its magnetic flux leakage. In fact, the normal working state of CRT is short-circuit operation, which leads to a large magnetic flux leakage. Therefore, the neglect of magnetic flux leakage will inevitably lead to the inaccurate calculation of short-circuit impedance, harmonic analysis, control characteristics and other working characteristics of the dual control-winding basic-unit magnetic integrated CRT.

To sum up, we consider the magnetic integrated CRT, that is, the dual control-winding basic-unit magnetic integrated CRT, as the research object and analyze its basic working principle. Based on the magnetic circuit and circuit dualistic transformation method, the equivalent mathematical model of the CRT considering leakage magnetic field is established. The distribution of leakage magnetic field of each winding is analyzed qualitatively, and the analytical calculation formulas of magnetizing inductance and leakage inductance of each winding of the magnetic integrated CRT are deduced. Based on the results, the analytical calculation formulas of short-circuit impedance and winding current of CRT under different working conditions are derived. A calculation example is designed, in which the field-circuit coupling simulation model of the magnetic integrated CRT is established, and the winding current under different working conditions is simulated to verify the correctness of CRT equivalent mathematical model and analytical calculation formulas. This study aims to provide a theoretical basis for structural design and optimization of the magnetic integrated CRT to analyze its working characteristics.

1 Basic theory

Fig.1 shows the structural diagram of dual control-winding basic-unit magnetic integrated CRT. For the dual control-winding basic-unit magnetic integrated CRT withs-level control winding, the magnetic integrated CRT is composed of several independent dual control-winding basic units. Each dual control-winding basic unit contains a working winding and two control winding units. For each control winding unit, in addition to one control winding, it has an air gap column in parallel with the iron core of the control winding.

In Fig.1,BWkis the working winding of thekth dual control-winding basic unit;CW2k-1andCW2kare the first control winding and the second control winding of thekth dual control-winding basic unit, respectively；i0kis the working winding current of thekth dual control-winding basic unit;i2k-1andi2kare the current of the (2k-1)th control winding and the (2k)th control winding, respectively;T2k-1andT2kare the anti-parallel thyristors connected in series with the (2k-1)th control winding circuit and the (2k)th control-winding circuit, respectively.CW1,…,CWsare all levels of the control winding of CRT；T1,T2,…,Tsare the anti-parallel thyristors in series in the control-winding circuits at all levels.

Fig.1 Structural diagram of dual control-winding basic-unit magnetic integrated CRT

Fig.2 displays the working-principle diagram of the dual control-winding basic-unit magnetic integrated CRT.

Fig.2 Working principle diagram of dual control-winding basic-unit magnetic integrated CRT

In the structural design of CRT, the design principle of “high impedance, weak coupling” should be met[19]. In Fig.2, the working windings of each dual control-winding basic unit are connected in parallel with each other, and the parallel terminalsAandxare connected in parallel with the high-voltage transmission line；i0is the working winding current of the CRT. The short-circuit impedance between the working winding and the control winding of CRT should be 100%.Mij(i≠j) refers to the mutual inductance between control windingsCWiandCWj. Given that the principle of “weak coupling” should be realized between control windings at all levels, thenMij≈0. To achieve the continuous and smooth adjustment of compensation capacity, CRT has single branch regulation modes of sequential single branch, fixed single branch and transfer single branch, and multi-branch regulation modes[9]. This study introduces the basic working principle of CRT by using sequential single branch regulation mode as an example. In the regulation mode of sequential single branch, by adjusting the turn-on of anti-parallel thyristorsT1,T2,…,Ts, the control windingsCW1,CW2,…,CWsare short-circuited in turn, so as to realize the gradual increase of working currenti0and meet the transition of CRT from no-load to full-load condition. It should be noted that the single-branch regulation mode of CRT is based on the principle of harmonic dilution, that is, the harmonic current generated by one regulation winding is diluted through the short circuit of multiple control windings to suppress the harmonic current of CRT[9]. Therefore, the number of control winding stages of CRT should be greater than 1, or a considerable amount of harmonic will be injected into the power grid under light load[8].

2 Equivalent mathematical model

The dual control-winding basic-unit magnetic integrated CRT is composed of several independent dual control-winding basic units. Any one of the dual control-winding basic unitkis selected, and its structure diagram is shown in Fig.3.

Based on the structural characteristics of dual control-winding basic unit shown in Fig.3, the equivalent magnetic circuit model considering magnetic flux leakage is established, as shown in Fig.4.

Fig.3 Structure diagram of dual control-winding basic unit

Fig.4 Equivalent magnetic circuit model of dual control-winding basic unit

In Figs.3 and 4, the magnetomotive force and leakage magnetoresistance of working windingBWkof dual control-winding basic unitskareF1andRmδ1, respectively. The magnetomotive force and leakage magnetoresistance of control windingCW2k-1areF2andRmδ2, respectively. The magnetomotive force and leakage magnetoresistance of control windingCW2kareF3andRmδ3, respectively. In addition, the magnetoresistances of magnetic circuits ‘ab’, ‘ahgf’, ‘af’, ‘fe’, ‘be’ and ‘bcde’ areRm1,Rm2,Rm3,Rm4,Rm5andRm6, respectively.

According to the magnetic circuit and circuit dualistic transformation method[16]as well as the equivalent magnetic circuit diagram shown in Fig.4, the equivalent circuit model of the ‘transformer-inductor’ of dual control-winding basic unit is established, as shown in Fig.5. It should be emphasized that Fig.5 shows the equivalent circuit model of the dual control-winding basic unit, which is reduced to the working winding side.

Fig.5 Equivalent circuit model of dual control-winding basic unit

In Fig.5,N0is the turns of working windingBWkof any dual control-winding basic unitk;N1andN2are the turns of control windingsCW2k-1andCW2k, respectively;Lδ1is the leakage inductance of working windingBWk;Lδ2andLδ3are the leakage inductance of control windingsCW2k-1andCW2k, respectively.L1-L6are the magnetizing inductances corresponding to the magnetoresistancesRm1-Rm6, respectively.

3 Calculation of inductance

Fig.6 shows the schematic of the structural parameters and the calculation principle diagram of the leakage inductance for dual control-winding basic unitk.

Fig.6 Diagram of structural parameters and winding leakage inductance calculation principle of dual control-winding basic unit

3.1 Calculation of magnetizing inductance

In accordance with the principle of magnetic circuit and circuit dualistic transformation, the calculation formula of the magnetizing inductanceLi(i=1,2,…,6) is given by

(1)

(2)

whereRδis the magnetoresistance of the air gap magnetic circuit, which has been described in Ref.[17] in detail, andRmiis the magnetoresistance of the iron core magnetic circuit, which is calculated by

(3)

whereμis the permeability of the iron core material. Given that the CRT runs in the linear phase of the ferromagnetic material, the nonlinearity of its magnetisation characteristics can be ignored and is regarded as a constant. In addition,Aiis the cross-sectional area of the iron core; andliis the length of the iron core magnetic circuit, which is calculated by

(4)

whereδ1andδ2represent the air gap lengths of magnetic circuits ‘af’ and ‘be’ respectively.

3.2 Calculation of winding leakage inductance

To reduce the eddy current loss of winding, researchers can use a reasonable winding arrangement and calculation of iron core parameters to reduce the crosslink between windings and air gap flux leakage. Given the small crosslink between air gap flux leakage and windings, it is ignored in this study.

The magnetic flux leakage of CRT winding is distributed in the air near windings, and its calculation surface is the annular surface[18]. Given that CRT runs in the linear phase of magnetization characteristics of the iron core, its magnetoresistance can be regarded as a constant. Assuming that the winding current is uniformly distributed in the conductor section, according to the Ampere circuital theorem, with the increase in winding turns, the magnetic field intensity in the winding increases linearly. And then, the leakage flux density is linearly distributed in the air in which the winding is located[18]. Therefore, the calculation of magnetic flux should be divided based on the region of magnetic flux density. If the CRT winding is completely surrounded by the calculated ring, the leakage flux density reaches the maximum.

Taking the working winding of dual control-winding basic unitkas an example, we analyze the calculation principle of leakage inductance. As shown in Fig.6, if the inner side of the working winding is considered the reference point for leakage inductance calculation, the leakage magnetic density at distancexfrom the reference point is recorded asBx. According to the above analysis, combined with the structural characteristics of the dual control-winding basic unit, the calculation of winding leakage inductance can be divided into two parts. One is the calculation region that includes a part of turns of the working winding only, that is, 0≤x≤w0； the other is the calculation region that includes all turns of the working winding, that is,w0

When 0≤x≤w0, the formula for calculating the leakage magnetic density at distancexfrom the reference point is expressed as

(5)

whereBmis the maximum magnetic density of the winding, which is calculated by

(6)

By introducing Eq.(6) into Eq.(5), the calculation formula of leakage flux densityBxcan be obtained as

(7)

The calculation formula of magnetic field energyWmis expressed as

(8)

Taking the calculation surface of leakage flux distribution for the working winding of the dual control-winding basic unit as the annular surface, the calculation formula for the unit volume dVof space occupied by winding radial unit length dxis obtained by

dV=

(9)

wherer0is the distance between the centre lines of the working winding and iron core column at which the working winding is located.

Combining Eqs.(7) to (9), when 0≤x≤w0, the calculation formula for the maximum magnetic field energy of the working winding of the dual control-winding basic unit can be deduced as

(10)

The calculation of self-induced magnetic energy is given by

(11)

By combining Eqs.(10) and (11), under the condition of 0≤x≤w0, the calculation formula for the working winding leakage inductanceLδ11of dual control-winding basic unit can be obtained as

(12)

Whenw0

According to the above analysis, similarly, whenw0

(13)

By combining Eqs.(11) and (13), whenw0

(14)

By summing Eqs.(12) and (14), the leakage inductancesLδ1of the working winding of the dual control-winding basic unit can be obtained as

Lδ1=

(15)

Similarly, the leakage inductance of control windingsCW1andCW2of the dual control-winding basic unit can be deduced, as shown in Eqs.(16) and (17), respectively.

Lδ2=

(16)

Lδ3=

(17)

whereN1andN2are the turns of control windingsCW1andCW2, respectively.

4 Calculation of CRT impedance and winding current under different working conditions

According to the equivalent circuit model of dual control winding basic unit shown in Fig.5 and the inductance calculation principle in Section 3, the calculation formula (reduced to working winding side) of the impedance between the windings under different working conditions can be derived as

(18)

whereω=2πfis the angular frequency of CRT.

(19)

When control windingCW1is in short circuit, whileCW2is in open circuit, that is, the dual control-winding basic unit is running at half load, we have

(20)

(21)

where

When control windingsCW1andCW2are all in short circuit, that is, the dual control-winding basic unit is in full load, we can get

(22)

(23)

Fig.1 shows that the dual control-winding basic-unit magnetic integrated CRT withs-level control winding is composed of several independent dual control-winding basic units in parallel.

Assuming that CRT haskindependent dual control-winding basic units, the stages of control winding of CRT iss=2k. If the control windingsCW1-CWt(1≤t≤s) are in short circuit in turn, the impedanceZdof the dual control-winding basic unit magnetic integrated CRT is computed by

(24)

Under different working conditions, with the control windingsCW1-CWt(1≤t≤s) in short circuit in turn, the working winding currentI(reduced to working winding side) of dual control-winding basic-unit magnetic integrated CRT can be calculated by

(25)

whereUNis the rated voltage of working winding.

5 Example analysis

Given the dual control-winding basic-unit magnetic integrated CRT has two independent dual control-winding basic units, that is,k=2, its control numver of winding stages iss=4. For the convenience of analysis and calculation, assuming that the working frequency of CRT is 50 Hz, the rated voltage is 220 V, the turn ratio between the working winding and each control winding is 1∶1, and the rated current of each control winding is 10 A, in accordance with the electromagnetic design principle of electromagnetic equipment, the basic and structural parameters of dual control-winding basic-unit magnetic integrated CRT can be calculated, as listed in Table 1. Given that the rated parameters of each winding of CRT are the same, the structural parameters of the dual control-winding basic unit can be designed to be the same. With the small capacity of CRT, we select the square section with side lengthDto design the iron core structure.

Table 1 Structural parameters of dual control-winding basic-unit magnetic integrated CRT

5.1 Calculation of winding current

By introducing the structural parameters of magneticintegrated CRT in Table 1 into Eqs.(1)-(4), (15)-(17) and (19)-(24), the analytical calculation results of each winding current under different working conditions can be obtained, as shown in Table 2.

Table 2 Winding current calculation under different loads in analytical method

5.2 Winding current calculation with finite element method

Based on the finite element method, the three-dimensional finite element model and external circuit model are established based on the structural parameters of the dual control-winding basic-unit magnetic integrated CRT shown in Table 1, as shown in Fig.7. Fig.7(a) shows the 3D finite element field model and Fig.7(b) shows the external circuit model when the CRT is in full load (all control windings are in short-circuit).

(a) 3D finite element field model

(b) External circuit model of CRT in full load

The finite element model of dual control-winding basic-unit magnetic integrated CRT shown in Fig.7 is used for field-circuit coupling co-simulation, and the finite element calculation results of each winding current under different working conditions can be obtained, as shown in Table 3.

Table 3 Winding current calculation under different loads in FEM

5.3 Results and discussion

Tables 2 and 3 show that the analytical calculation results of each winding current of dual control-winding basic-unit magnetic integrated CRT under different working conditions are consistent with the finite element calculation results. These findings prove the validity of the equivalent circuit model of dual control-winding basic-unit magnetic integrated CRT considering winding leakage inductance and the correctness of the calculation formulas of winding impedance and winding current. In addition, the working winding current of CRT increases gradually with the operation of control winding in turn to realize the transition of CRT compensation capacity from zero to a rated value. However, Tables 2 and 3 show that with the operation of control windingCW2k(k=1,2,3,…), the current of control windingCW2k-1, which has been put into operation, will still be slightly reduced. Therefore, although the dual control-winding basic-unit magnetic integrated CRT meets the basic design principle of “high impedance and weak coupling”， control windingsCW2k-1andCW2kare not completely decoupled.

6 Concusions

Based on the magnetic circuit and circuit dualistic transformation method, we establish the equivalent mathematical model of dual control-winding basic-unit magnetic integrated CRT considering magnetic leakage, qualitatively analyze the distribution of magnetic leakage field of each winding, and deduce the principle, the analytical calculation formulas of magnetizing inductance and leakage inductance of each winding of dual control-winding basic-unit magnetic integrated CRT as well as the short-circuit impedance and winding current of CRT under different working conditions. The finite element method is used to simulate the current of each winding under different working conditions. The conclusions are as follows：

1) With the gradual operation of the control winding, the analytical calculation results of the winding current of the dual control-winding basic-unit magnetic integrated CRT are in good agreement with the finite element calculations. This agreement shows the effectiveness of the equivalent mathematical model of the dual control-winding basic-unit magnetic integrated CRT considering magnetic leakage, the correctness of the analytical calculation formulas of the magnetizing inductance and leakage inductance of the CRT and the correctness of the impedance of the CRT and winding current under different working conditions.

2) The working winding current of the dual control-winding basic-unit magnetic integrated CRT gradually increases with the operation of control winding in turn to realize the transition of CRT compensation capacity from zero to rated value.

3) Although the dual control-winding basic-unit magnetic integrated CRT meets the basic design principles of “high impedance and weak coupling”, no complete decoupling occurs between control windingsCW2k-1andCW2k.