Influence of voltage fluctuations on core vibration of a UHV shunt reactor

Abstract

A new core vibration calculation method of UHV shunt reactors is proposed to deal with the problem that vibration parameters of the core of an ultra-high voltage (UHV) shunt reactor cannot be measured on site. A series of tests have validated that the method can calculate the vibration parameters of a UHV shunt reactor core with different voltage ratios. The method is adopted to study the influence of nodal voltage fluctuations of the UHV AC tie line on the core vibration characteristics of a UHV shunt reactor under normal and abnormal operating conditions. The conclusion is drawn as follows: in the three operating states assessed herein, no matter whether the power grid is in a normal state or not, the core of the UHV shunt reactor will not reach magnetic saturation, and the vibration parameters of the reactor always maintain a linear relationship with the operating voltage. The changes of power grid operation parameters are introduced to the research into reactor vibration, which is conducive to a more comprehensive understanding of the actual operating state of a UHV shunt reactor. It can also provide help for the design and operation of UHV shunt reactors.

Keywords

Shunt reactor core vibration voltage fluctuations magnetic saturation

1. Introduction

The reactor, which is mainly used to compensate for the inductive reactive power of the line and suppress voltage fluctuations, is an important item of equipment in AC transmission projects [1–4]. Excessive vibration and the resulting noise are the most important problems in the operation of a shunt reactor [5]. The vibration of the reactor can be divided into two parts: core vibration and winding vibration. The strength of the winding vibration is about 10% of that of the core vibration [6]. The study of the core vibration is the key to explore the reactor vibration. The operating voltage of the ultra-high voltage (UHV) shunt reactor is one of the main factors that affect the vibration characteristics of its core. UHV shunt reactors, the type with the highest operating voltage (1100 kV in China) among existing reactors, are installed in parallel at the node (substation) of the UHV AC tie line, and the operating voltage is consistent with the nodal voltage. It is necessary to study the influence of voltage fluctuations on the vibration of the UHV shunt reactor to ensure reliable operation thereof.

At the present, it is impossible to obtain the vibration parameters of the UHV shunt reactor core in operation by on-site measurement. The numerical calculation method based on finite element simulation is widely applied in the research into reactor core vibration to solve these problems [7]. The electromagnetic-mechanical multi-field coupling model proposed by Yan [8] and Li [9] has been widely used to study core vibration of small dry reactors; however, the unique radiation structure of the shunt reactor core discs is not considered in this method so there is a higher error when the method is used to calculate the core vibration of the UHV shunt reactor. Lü [10] proposed a new three-dimensional (3-d) reactor core vibration calculation method, which considers the radiation structure of the core discs; however, this method ignores the fluid-structure interaction phenomenon arising in UHV shunt reactor (oil-immersed) core vibration, which reduces the calculation accuracy of this method. In addition, it is taken as a default condition that the reactor is in its rated operating state in the existing reactor core vibration research. The impacts of the voltage fluctuations of the reactor access point on the core vibration of the reactor are not considered, which does not conform to the actual operating state of the UHV shunt reactor.

According to the causes of voltage fluctuations, the voltage fluctuations of the UHV AC power grid can be divided into two categories: (1) the voltage fluctuates slightly due to load changes with normal power grid operating conditions; (2) power grid voltage fluctuations caused by large disturbances with abnormal power grid operating conditions. There are large differences in the fluctuation characteristics of the grid voltage in each category, therefore, they should be considered separately when studying the influence of voltage fluctuations on the core vibration of the UHV shunt reactor.

The influence of voltage fluctuations at the nodes of the UHV AC tie line on the core vibration of the UHV shunt reactor have been studied in the present research. First, to tackle the problem that the vibration parameters of the UHV shunt reactor core cannot be measured on site, a new core vibration calculation method of the UHV shunt reactor is proposed. Then, a set of tests based on the equivalent model of the UHV shunt reactor are designed and conducted to verify the accuracy of the method to calculate the vibration parameters of the UHV shunt reactor core with different operating voltages. Finally, the typical voltage fluctuation forms of the UHV AC grid under normal and abnormal operating conditions are analysed separately. The influence of voltage fluctuations on the core vibration of a UHV shunt reactor are studied by using the calculation method proposed in the present work. It is claimed as innovative that the changes of grid operating parameters are introduced into the research into reactor vibration. This is helpful for a more comprehensive understanding of the actual operating status of the UHV shunt reactor and provides help for the design and operation and maintenance thereof.

2. Method

The magnetostriction of the reactor core silicon steel lamination in the magnetic field and the Maxwell force at the core cake-air gap boundary are the main reasons for the core vibration of the reactor. In addition, the UHV shunt reactor is an oil-immersed reactor, so the damping of transformer oil on the vibration of the core is much greater than the damping of air on the core of the dry-type reactor, and cannot be ignored. In summary, the vibration of the core of the UHV shunt reactor can be described as: the periodic vibration of the core in the transformer oil under the action of electromagnetic external force, where the electromagnetic external force mainly refers to the Maxwell force and magnetostrictive force: $\begin{eqnarray}\displaystyle \mathbf{M}_{\mathbf{c}}{\displaystyle \frac{\partial ^{2}\mathbf{u}_{\boldsymbol{ c}}}{\partial t^{2}}}+\mathbf{C}{\displaystyle \frac{\partial \mathbf{u}_{\mathbf{c}}}{\partial t}}+\mathbf{K}_{\mathbf{c}}\mathbf{u}_{\mathbf{c}}=\mathbf{F}=\mathbf{F}_{\mathbf{max}}+\mathbf{F}_{\mathbf{mag}}, & & \displaystyle\end{eqnarray}$ (1) where M _c represents the mass matrix of the core; u _c is the displacement matrix of the core; C is the damping matrix, as related to the fluid-structure interaction between the core and the transformer oil, and can be determined by fluid-structure interaction analysis; k _c is the stiffness matrix of the core that can be calculated from the material parameters and structural dimensions of the core; F is the external resultant force causing vibration, F _max is the Maxwell force matrix, F _mag denotes the magnetostrictive force matrix, and t is time.

According to (1), the vibration of the UHV shunt reactor core is divided into two parts: fluid-structure interaction vibration analysis and electromagnetic external force calculation (Fig. 1).

Fig. 1.

Calculation of UHV shunt reactor core vibration.

The fluid-structure interaction [11,12] vibration analysis is undertaken using a numerical analysis method based on the arbitrary Lagrange–Eulerian description. Silicon steel sheet used in the UHV shunt reactor core is a linear elastic material, and its motion is described by the dynamic elasticity equation in Lagrange coordinates. The equilibrium equation of the initial configuration is: $\begin{eqnarray}\displaystyle {\rho}_{c}{\displaystyle \frac{\partial ^{2}\mathbf{u}_{\boldsymbol{ c}}}{\partial t^{2}}}={\nabla}\cdot ({\sigma}_{0}\boldsymbol{{\sigma}}_{\mathbf{ca}}({\boldsymbol{{\sigma}}_{\mathbf{c}}}^{-1})^{\text{T}})+{\rho}_{c}g, & & \displaystyle\end{eqnarray}$ (2) where σ _c is the deformation gradient tensor of the core. $\begin{eqnarray}\displaystyle \boldsymbol{{\sigma}}_{\mathbf{c}}=\mathbf{I}+{\nabla}\mathbf{u}_{\mathbf{c}}, & & \displaystyle\end{eqnarray}$ (3) σ _ca is the Cauchy stress tensor. $\begin{eqnarray}\displaystyle \boldsymbol{{\sigma}}_{\mathbf{ca}}={\displaystyle \frac{1}{{\sigma}_{0}}}\boldsymbol{{\sigma}}_{\mathbf{c}}({\lambda}_{c}(tr(\boldsymbol{{\sigma}}_{\mathbf{g}}))\mathbf{I}+2{\mu}_{c}\boldsymbol{{\sigma}}_{\mathbf{g}}){\boldsymbol{{\sigma}}_{\mathbf{c}}}^{\text{T}}, & & \displaystyle\end{eqnarray}$ (4) 𝜆_c is the first Lamé constant. $\begin{eqnarray}\displaystyle {\lambda}_{c}={\displaystyle \frac{E_{c}{\alpha}_{c}}{(1+{\alpha}_{c})(1-2{\alpha}_{c})}}, & & \displaystyle\end{eqnarray}$ (5) μ_c is the second Lamé constants, which is the shear modulus herein. $\begin{eqnarray}\displaystyle {\mu}_{c}={\displaystyle \frac{E_{c}}{2(1+{\alpha}_{c})}}, & & \displaystyle\end{eqnarray}$ (6) σ _g is the strain tensor of the core. $\begin{eqnarray}\displaystyle \boldsymbol{{\sigma}}_{\mathbf{g}}=\frac{1}{2}\left({\boldsymbol{{\sigma}}_{\mathbf{c}}}^{\text{T}}\boldsymbol{{\sigma}}_{\boldsymbol{ c}}-\mathbf{I}\right). & & \displaystyle\end{eqnarray}$ (7)

In (2) to (7), 𝜌_c is the density of the core material; g is the local acceleration due to gravity; I denotes the unit tensor; σ₀ is the determinant of σ _c; E _c and 𝛼_c are the Young’s modulus and Poisson’s ratio of the core, respectively.

Each component in the oil tank of UHV shunt reactors is immersed in the transformer oil, which is an incompressible Newtonian fluid. Its continuity equation in Lagrange–Euler coordinates is as follows: $\begin{eqnarray}\displaystyle \displaystyle \int _{S}(\mathbf{u}_{\mathbf{o}}-\mathbf{u}_{\mathbf{g}})\cdot \mathbf{n}\text{d}S=0, & & \displaystyle\end{eqnarray}$ (8) where, S is the surface area of the core, u ₀ represents the velocity matrix of the transformer oil, and n is the normal vector of the core surface. In the finite element simulation software, the moving grid method is used to deal with the deformation problem in the fluid-structure interaction; u _g is the velocity matrix of the moving grid.

The momentum equation of transformer oil is presented as follows: $\begin{eqnarray}\displaystyle {\displaystyle \frac{\partial }{\partial t}}\displaystyle \int _{V}{\rho}_{o}\mathbf{u}_{\mathbf{o}}\text{d}V+\int _{S}{\rho}_{o}(\mathbf{u}_{\mathbf{o}}-\mathbf{u}_{\mathbf{g}})(\mathbf{u}_{\mathbf{o}}\cdot \mathbf{n})\text{d}S=\displaystyle \int _{S}\boldsymbol{{\sigma}}_{\mathbf{o}}\cdot \mathbf{n}\text{d}S+\int _{V}{\rho}_{o}g\text{d}V, & & \displaystyle\end{eqnarray}$ (9) where, V is the volume of the core, 𝜌₀ is the density of the transformer oil.

At the interface between the UHV shunt reactor core and transformer oil, the relationship between speed consistency and stress balance is as follows: $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle \mathbf{u}_{\mathbf{c}}=\mathbf{u}_{\mathbf{o}}\end{eqnarray}$ (10) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle \boldsymbol{{\sigma}}_{\mathbf{c}}\mathbf{n}=\boldsymbol{{\sigma}}_{\mathbf{o}}\mathbf{n}.\end{eqnarray}$ (11)

3. Laboratory research

Before the calculation method of UHV shunt reactor core vibration described in this manuscript is used for subsequent research, it is necessary to validate the accuracy of this method under fluctuating voltage. The UHV shunt reactor in operation cannot be used for internal vibration testing, so the equivalent model of a UHV shunt reactor developed by Baoding Tianwei BaoBian Electric Co., Ltd (Fig. 2) is used for this test, which is similar to other UHV shunt reactors in operation in terms of structure and material. The design and operation parameters of the equivalent model are listed in Table 1.

Table 1
Design and operation parameters of the shunt reactor equivalent model

Main parameter Value

Rated capacity 6.7 × 10⁶ var

Reactance 25.23 Ω

Rated voltage 22.52 kV

Model of silicon steel sheet B30P105

Young’s modulus of the air gap 5.5 × 10¹⁰ Pa

Poisson’s ratio of the air gap 0.45

Density of transformer oil 895 kg m⁻³

Density of the air gap 2600 kg m⁻³

Lamination coefficient of the core 0.983

Operating frequency 50 Hz

Dynamic viscosity of transformer oil 0.021 Pa s

Main parameter	Value
Rated capacity	6.7 × 10⁶ var
Reactance	25.23 Ω
Rated voltage	22.52 kV
Model of silicon steel sheet	B30P105
Young’s modulus of the air gap	5.5 × 10¹⁰ Pa
Poisson’s ratio of the air gap	0.45
Density of transformer oil	895 kg m⁻³
Density of the air gap	2600 kg m⁻³
Lamination coefficient of the core	0.983
Operating frequency	50 Hz
Dynamic viscosity of transformer oil	0.021 Pa s

In this manuscript, the voltage ratio is defined as follows: $\begin{eqnarray}\displaystyle {\eta}={\displaystyle \frac{U}{U_{0}}}, & & \displaystyle\end{eqnarray}$ (12) where, U is the operating voltage of the reactor and U ₀ is the rated voltage of the reactor.

The structure of the UHV shunt reactor equivalent model used in this test is shown in Fig. 3 and it includes a compression system, yokes (top, side, and bottom yokes), core discs, windings, air gaps, etc.

Fig. 2.

The equivalent model of the UHV shunt reactor.

Fig. 3.

Structure and measurement point (red box) location of the equivalent model for the UHV shunt reactor.

The test was conducted in the high-voltage test hall of Baoding Tianwei BaoBian Electric Co., Ltd and the vibration test system was developed by Jiangsu Donghua Testing Technology Co., Ltd with a sampling frequency of 5 kHz. The vibration sensor parameters are listed in Table 2. Since it is difficult to arrange the vibration sensor at the core column of the reactor and the accuracy of the test results cannot be guaranteed, the five test points of this test are arranged in the yoke (Fig. 3), and the specific locations of the test points are summarised in Table 3.

Table 2

The vibration sensor parameters

Main parameter	Inside	Outside
Model	YA22S	ICP
Sensitivity	0 ∼ 500 mV g⁻¹	0 ∼ 500 mV g⁻¹
Frequency response range	0.2 ∼ 15000 Hz	0.2 ∼ 25000 Hz
Range	0 ∼ 100 m s⁻²	0 ∼ 100 m s⁻²

Table 3

Locations of measurement points in the equivalent model of the shunt reactor

Measurement point	Locations
1	Upper end of the side yoke
2	Inclined joint at the upper part of the side yoke
3	Outside of the upper part of the yoke
4	Side in the middle of the yoke
5	Height-centre of the middle side of the side yoke

This test is divided into five groups, and each test lasts for 20 s. The test voltage ratios are 0.8, 0.9, 1.0, 1.1, and 1.2, respectively. The root mean square value of core displacement (d) is taken as the index parameter representing core vibration.

The measured values of core vibration parameters of the UHV shunt reactor under each of five voltage ratios can be derived experimentally. To verify whether the core vibration calculation method proposed in this manuscript is accurate, vibration parameters of the UHV shunt reactor core at five voltage ratios are calculated.

The finite element simulation software used in this manuscript is COMSOL Multiphysics: the governing equations are introduced in Section 2 and Appendix A. A tetrahedral mesh is used with 38,098 mesh elements. It takes 22 minutes and 31 seconds to undertake the analysis using the BiCGSTAB solver [15]. The calculation is based on the core vibration calculation method of a UHV shunt reactor proposed herein, and the parameters in Table 1 are applied. To compare the calculated values of vibration parameters with the measured value (Table 4), a comparison of simulated and test results (Fig. 4) is demonstrated: the red circle represents the calculated value of d, and the blue square represents the measured value of d. The distance between two shapes in the same coordinate can represent the difference between the calculated and measured values.

Table 4

Comparisons between calculated and measured values of d

𝜂		0.8	0.9	1.0	1.1	1.2	Average value
1	Measured value (mm)	0.1013	0.1657	0.2864	0.4524	0.6545	0.33206
	Calculated value (mm)	0.1209	0.1731	0.2859	0.4475	0.6093	0.32734
	Deviation (mm)	−0.0196	−0.0074	0.0005	0.0049	0.0452	0.00472
	Accuracy	80.64%	95.52%	99.81%	98.91%	93.08%	93.59%
2	Measured value (mm)	0.5104	0.5911	0.716	0.9354	1.2562	0.80182
	Calculated value (mm)	0.5001	0.5975	0.7177	0.9337	1.2006	0.78992
	Deviation (mm)	0.0103	−0.0064	−0.0017	0.0017	0.0556	0.0119
	Accuracy	97.99%	98.91%	99.76%	99.81%	95.57%	98.41%
3	Measured value (mm)	0.1896	0.3157	0.5126	0.7568	0.9948	0.5539
	Calculated value (mm)	0.2057	0.3279	0.5181	0.7414	0.9772	0.55406
	Deviation (mm)	−0.0161	−0.0122	−0.0055	0.0154	0.0176	−0.00016
	Accuracy	91.51%	96.14%	98.92%	97.96%	98.23%	96.55%
4	Measured value (mm)	0.2932	0.4022	0.552	0.7413	0.9317	0.58408
	Calculated value (mm)	0.3277	0.4246	0.5658	0.7177	0.9079	0.58874
	Deviation (mm)	−0.0345	−0.0224	−0.0138	0.0236	0.0238	−0.00466
	Accuracy	88.22%	94.42%	97.50%	96.82%	97.45%	94.88%
5	Measured value (mm)	0.3506	0.4342	0.5677	0.747	0.9503	0.60996
	Calculated value (mm)	0.3234	0.4014	0.5128	0.6824	0.8503	0.55406
	Deviation (mm)	0.0272	0.0328	0.0549	0.0646	0.1000	0.0559
	Accuracy	92.25%	92.44%	90.34%	91.35%	89.48%	91.17%
Average accuracy		90.12%	90.12%	95.49%	97.27%	96.97%	94.76%

In the table, the deviation is defined as: $\begin{eqnarray}\displaystyle {\Delta}d=d_{c}-d_{t}. & & \displaystyle\end{eqnarray}$ (13)

The definition of accuracy is: $\begin{eqnarray}\displaystyle {\delta}=\left(1-\left|{\displaystyle \frac{d_{c}-d_{t}}{d_{t}}}\right|\right)\times 100\%. & & \displaystyle\end{eqnarray}$ (14)

The simulation and test results are compared based on the data in Table 4 and Fig. 4. The average accuracy of the simulation results is 94.76%, the average accuracy at each measurement point exceeds 93%, and the average accuracy of each test voltage ratio exceeds 90%. These results show that the difference between simulated and test results is small, and the accuracy of the finite element method results is satisfactory.

Fig. 4.

Comparisons between calculated and measured values of d.

Based on the simulation and test results, the following conclusions can be obtained: For different voltage ratios, the calculation method of UHV shunt reactor core vibration proposed herein can obtain core vibration data that are close to measured values, which can be used to replace on-site measurement in the subsequent research into UHV shunt reactor core vibration.

There is an inconsistency between the experimental results and the simulated results: all of the measured value at point 5 is greater than the calculated value. The previous conclusions will not be affected by this inconsistency. The discussion of test results and the analysis of simulation results are shown in Appendix B.

In addition, single-column structure and double-column structure are two core structures of UHV shunt reactors used in China. The design standards and requirements of two types of UHV shunt reactors are identical, they contain the same type of components, and the materials and mode of operation used are also identical. Therefore, the core vibration calculation method of the UHV shunt reactor proposed in this manuscript can be applied to all UHV shunt reactors produced in China.

4. Influence of UHV AC tie-line voltage fluctuations on core vibration of a UHV shunt reactor

At present, the UHV shunt reactors in operation in China are not equipped with vibration sensors, and the on-site measurement of the reactor core is not allowed by the grid operating company. In the present research, the theoretical calculation method is used to study the effect of UHV AC tie-line voltage fluctuations on the core vibration of a UHV shunt reactor in actual operation. It is verified that the new core vibration calculation method of the UHV shunt reactor proposed herein can be used to replace the onerous on-site measurement, and the vibration parameters obtained by the two methods are similar.

UHV shunt reactors are installed in substations at both ends of the UHV AC tie line. At present, more than 20 UHV AC single projects have been completed in China. Among them, the Changzhi–Nanyang–Jingmen UHV AC test demonstration project is the first such system that was put into operation. It connects the two major power grids in north China and central China (Fig. 5). Eight UHV shunt reactors are installed in Changzhi UHV substation and Jingmen UHV substation. In this work, UHV shunt reactors in these two substations are selected as the research objects. The eight UHV shunt reactors are of double-column core structure (Fig. 6), and the design and operation parameters of a double-column UHV shunt reactor are listed in Table 5.

Fig. 5.

UHV AC tie line between NCPG and CCPG.

Fig. 6.

Internal structure of a double-column UHV shunt reactor.

Table 5

Design and operation parameters of the shunt reactor equivalent model

Main parameter	Value
Rated capacity	2.4 × 10⁸ var
Rated current	377.9 A
Young’s modulus of silicon steel sheet	1.95 × 10¹¹ Pa
Poisson’s ratio of silicon steel sheet	0.27
Density of silicon steel sheet	7650 kg m⁻³
Young’s modulus of the air gap	5.5 × 10¹⁰ Pa
Poisson’s ratio of the air gap	0.45
Density of transformer oil	895 kg m⁻³
Density of the air gap	2600 kg m⁻³
Lamination coefficient of the core	0.983
Operating frequency	50 Hz
Dynamic viscosity of transformer oil	0.021 Pa s

4.1. Influence of voltage fluctuations on core vibration of the reactor during normal operation of the power grid

During normal operation of the power grid, the voltage of the UHV AC power grid will fluctuate due to the change in load. It is assumed that the active power transmitted from the central China power grid to the north China power grid through Jingmen station is P, and the exchanged reactive power is Q. Then, the power flow through the system is equivalent to splitting two power grids in central and North China at the Jingmen node. The two power grids have loads of P, Q and -P, -Q at the Jingmen node, respectively. It is assumed that the changes in active power and reactive power exchanged by two power grids in the Jingmen station are ΔP and ΔQ respectively. The mode of operation of the two power grids after the power flow is as before. For the central China power grid, the voltage fluctuation at Jingmen station is as follows: $\begin{eqnarray}\displaystyle {\displaystyle \frac{{\Delta}V_{1}}{V}}\cong -{\displaystyle \frac{{\Delta}Q}{S_{c1}}}-{\displaystyle \frac{{\Delta}PP}{{S_{c1}}^{2}}}, & & \displaystyle\end{eqnarray}$ (15) where, S _c1 is the short-circuit capacity provided to the Jingmen node by the central China side, ΔV ₁ is the voltage fluctuation at Jingmen node, and V is the nodal voltage at the Jingmen station. For the north China power grid, the voltage fluctuation at Jingmen station is given by (16): $\begin{eqnarray}\displaystyle {\displaystyle \frac{{\Delta}V_{2}}{V}}\cong -{\displaystyle \frac{{\Delta}Q}{S_{c2}}}-{\displaystyle \frac{{\Delta}PP}{{S_{c2}}^{2}}}, & & \displaystyle\end{eqnarray}$ (16) where, S _c2 is the short-circuit capacity provided to the Jingmen node by the North China side, and ΔV ₂ is the voltage fluctuation at the Jingmen node. Equation (17) thus holds: $\begin{eqnarray}\displaystyle {\Delta}V_{1}={\Delta}V_{2}={\Delta}V. & & \displaystyle\end{eqnarray}$ (17)

By solving the aforementioned formulae, the following results can be acquired: $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle {\Delta}Q={\displaystyle \frac{{\Delta}PP(S_{c1}-S_{c2})}{S_{c1}S_{c2}}}\end{eqnarray}$ (18) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle {\displaystyle \frac{{\Delta}V}{V}}\cong -\frac{{\Delta}PP}{S_{c1}S_{c2}}.\end{eqnarray}$ (19)

It can be seen from (19) that the UHV voltage fluctuation in normal operation is mainly related to the active power fluctuation, the transmission power of the tie line and the short-circuit capacity of the bus. The greater the change in active power is, the larger the voltage fluctuation. The larger the basic power flow in the tie line, the greater the voltage fluctuation and the larger the short-circuit capacity, the smaller the voltage fluctuation.

When the power grid operates normally and the grid structure remains unchanged, the short-circuit capacity is unchanged. When 1 GW (P = 1 GW) is transmitted southward through the UHV tie line, the voltage fluctuation caused by active power fluctuations in the tie line is studied by way of the following example. The voltage fluctuation data from Jingmen station when the UHV tie line transmits 1 GW to the central China power grid are given in the literature [16]. With the help of the calculation method for UHV shunt reactor core vibration proposed herein, the core vibration parameters of the UHV shunt reactor under voltage fluctuations can be calculated (Fig. 7). 192,622 tetrahedral mesh elements are used and the analysis requires 27 hours 14 minutes 11 seconds to run.

According to the data shown in Fig. 7, (20) can be used to calculate the Spearman correlation coefficient between the core vibration parameter and the operating voltage ratio as 0.9997. $\begin{eqnarray}\displaystyle {\rho}({\eta},d)={\displaystyle \frac{\displaystyle \mathop{\sum }_{i}({\eta}_{i}-\overline{{\eta}})(d_{i}-\overline{d})}{\sqrt{\displaystyle \mathop{\sum }_{i}({\eta}_{i}-\overline{{\eta}})^{2}(d_{i}-\overline{d})^{2}}}}, & & \displaystyle\end{eqnarray}$ (20) where 𝜂 and d are the averages of 𝜂 and d, and 𝜌(𝜂, d) is the Spearman correlation coefficient of 𝜂 and d.

It can be seen from Fig. 7 that: (1) during normal operation of the power grid, the actual operating voltage of the UHV AC tie line will fluctuate due to load fluctuation, but the fluctuation amplitude is very small (less than 4 ‰). (2) During the normal operation of the power grid, the change in vibration intensity of the core of the UHV shunt reactor installed in the node of the AC tie line is similar to the fluctuation in nodal voltage. The vibration intensity of the core can reflect the state of the fluctuation of the nodal voltage.

4.2. Influence of voltage waveform on reactor core vibration during abnormal operation of the power grid

In the construction of the Changzhi–Nanyang–Jingmen UHV AC test demonstration project, various possible faults are considered and corresponding solutions formulated, therefore, no case of voltage fluctuations caused by the fault of the UHV AC tie line has been found to date; however, large disturbances such as line faults, generation unit trips, main transformer trips, DC blocking, and line short circuits may occur in the north or central China power grid when connected by the UHV AC tie line, which will also cause large fluctuations in the UHV tie line voltage. After the three-phase transient short-circuit fault occurs in the Sichuan power grid (part of the central China power grid), the fluctuation waveform for the Changzhi UHV substation voltage is as given elsewhere [17]. The waveform and the core vibration calculation method for the UHV shunt reactor proposed herein can be used to study the variations of core vibration parameters with voltage fluctuations under the abnormal operation of the power grid.

Before the fault, the mode of operation of the UHV AC tie line is such that 500 GW of active power is transmitted from the central China power grid to the North China power grid. The fault is a three-phase short-circuit fault with a duration of 0.1 s in Jianshan Power Station (on the Sichuan power grid). In the 9 s after the fault, the changes in voltage and vibration parameters of the reactor core in Changzhi station are calculated (Fig. 8). 192,622 tetrahedral mesh elements are used and 68 hours 17 minutes 19 seconds are required to run this model.

Fig. 7.

Variation of d and 𝜂 with time during normal grid operation.

Fig. 8.

Variation of d and 𝜂 with time during abnormal grid operation.

According to the data shown in Fig. 8, the Spearman correlation coefficient between the core vibration parameter and the operating voltage ratio can be calculated as 0.99. Based on the data in Fig. 8, the following conclusions can be drawn: (1) the short-circuit fault at Jianshan station on the Sichuan power grid causes a voltage sag at Changzhi station on the UHV AC tie line, and the lowest nodal voltage is only 10% of the rated value. (2) The fluctuation in nodal voltage results in the changes in vibration parameters of the UHV shunt reactor. The correlation between d and 𝜂 is 0.9999, so it can be considered that the correlation between them is very high. That is to say, the vibration parameters of the reactor core are still positively related to the nodal voltage when the grid fault causes the nodal voltage sag.

5. Analysis

According to the variations in reactor core vibration parameters (d) with voltage fluctuations in the above two cases, the following conclusions can be drawn: in the normal and abnormal operating states of the power grid, the vibration parameters of the UHV shunt reactor core have a high correlation with the voltage at the reactor installation node. The fluctuation in nodal voltage will cause the similar change in the vibration parameters. The reasons for this are as follows: for any UHV shunt reactor, the strength of its core vibration is closely related to the magnitude of the electromagnetic force. The damping effect of fluid-structure interaction on the vibration of core is related to the vibration speed, which cannot affect the electromagnetic force on the core [18]. According to Equation (1): the Maxwell force and magnetostrictive force are two components of the electromagnetic external force on the core. The Maxwell force imposes a surface stress at the interface between the core silicon steel sheet and the air gap. It is mainly affected by magnetic flux density and magnetic field strength at the interface between the core and the air gap [10]. The ratio of magnetic flux density to magnetic field strength is the magnetic permeability (μ = B∕H), and the reciprocal thereof is its magnetic reluctivity. The core of a UHV shunt reactor is mainly composed of an air gap (marble-filled) and core cake (silicon steel sheet). The magnetic permeability of marble is constant, and the magnetic permeability of the silicon steel sheet is not constant. The relationship between the magnetic flux density and magnetic field strength in the silicon steel sheet can be expressed using the magnetisation curve of silicon steel sheet. As shown in Fig. 9, the magnetisation curve of B30P105 silicon steel sheet is measured by using the German Brockhaus one-dimensional magnetic and magnetostrictive measurement system. From Fig. 9, it can be found that the magnetisation curve of the silicon steel sheet is a saturation curve. When the magnetic flux density in the core is less than 1.5 T, the relationship between B and H is quasi-linear, that is, the magnetoresistance of silicon steel sheet with core is a constant. The permeability of the air gap and the core cake is constant, and the magnetic flux density and the magnetic field strength at the interface between the core and the air gap correspond to each other. Therefore, when the magnetic flux density in the reactor core is less than 1.5 T, the magnetic flux density is the only one to be considered when calculating the Maxwell force. When the magnetic flux density in the core of the reactor is greater than 1.5 T, the saturation characteristics of the magnetisation curve should also be considered.

The magnetostrictive force is an equivalent volume stress, which is related to the magnetostrictive phenomenon. The magnetostrictive force is related to the magnetostrictive strain in the core, which can be calculated from the magnetic flux density of the core by using the magnetostrictive curve [10]. The measured magnetostriction curve of the B30P105 silicon steel sheet is shown in Fig. 10, and the magnetostriction strain in the core also varies in a quasi-linear manner with changing magnetic flux density.

In conclusion, when the magnetic flux density in the core is less than 1.5 T, the magnetisation curve of the core is not saturated, and the external electromagnetic force on the core is linearly related to the internal magnetic flux density. When the magnetic flux density in the core is greater than 1.5 T, the magnetisation curve of the core reaches saturation and the magnetic flux density of the core is no longer linearly related to the magnetic field strength, and the external electromagnetic force on the core no longer follows a linear relationship with the magnetic flux density.

Fig. 9.

Magnetisation curve of B30P105 silicon steel sheet.

Fig. 10.

Magnetostrictive curve of B30P105 silicon steel sheet.

The variation of magnetic flux density with voltage in the core under normal and abnormal operation of the power grid is calculated and shown in Figs 11 and 12: the average magnetic flux density in the core under these two conditions is much less than 1.5 T; B and 𝜂 are similar (Spearman correlation coefficients of 0.9995 and 0.9999). Therefore, it can be considered that the vibration intensity of a UHV shunt reactor core is linearly related to its internal magnetic flux density. The magnetic flux density is positively proportional to the winding current when the magnetic reluctivity of the core is a constant [10]. The winding current of the UHV shunt reactor is linearly related to its operating voltage, so the magnetic flux density of the core is linearly related to its operating voltage, and the vibration strength of the core is also linearly related to its operating voltage.

Fig. 11.

Variation in B and 𝜂 with time during normal grid operation.

Fig. 12.

Variation in B and 𝜂 with time during abnormal grid.

In the above cases, at no time does the grid voltage exceed the rated voltage: this is because China’s UHV AC project is designed to avoid the occurrence of overvoltage as much as possible. At time of writing, there have been no cases of UHV AC tie line overvoltage reported.

It is necessary to analyse the influence of more complex voltage fluctuations on the core vibration of the UHV shunt reactor. A 𝜂 = 1.3 virtual case is designed and analysed, because the maximum allowable power frequency overvoltage amplitude of the UHV power grid in China is 1.3 times the rated voltage. The magnetic flux density distribution in the core of the UHV shunt reactor is calculated with 𝜂 = 1.3, as shown in Fig. 13: when the operating voltage of the UHV shunt reactor is 1.3 times the rated voltage, the maximum magnetic flux density in the core is close to 1.5 T, the core remains in an unsaturated state, and the vibration displacement of the core will still follow a linear relationship with the operating voltage. Therefore, it can be concluded that the vibration intensity of the UHV shunt reactor core is linearly related to its operating voltage whether the power grid is in a state of normal operation or not.

Fig. 13.

Distribution of magnetic flux density with 𝜂 = 1.3.

6. Conclusion

This work addresses the shortcomings in the existing calculation methods for the structure and operation characteristics of an oil-immersed UHV shunt reactor core. Through experimental work, it is proved that this method can accurately calculate the vibration parameters of a UHV shunt reactor core. The influence of the nodal voltage fluctuations on the UHV AC tie line on the core vibration of the UHV shunt reactor are studied by way of the new method. The following conclusions are reached:

(1) During normal operation of the power grid, the actual operating voltage across the UHV AC tie line will fluctuate due to load fluctuation, but the amplitude of such fluctuations is very small and the change in vibration intensity of the core of the UHV shunt reactor installed at the node of the AC tie line follows a trend similar to that of the nodal voltage.

(2) During the abnormal operation of the power grid, the fluctuation of the nodal voltage of the UHV AC tie line is more severe than that under normal operation of the power grid, however, the variations of core vibration parameters of the UHV shunt reactor remain similar to that of the voltage fluctuations.

(3) In the three operating states considered herein (normal, short circuit, and power frequency overvoltage), whether the grid is in a normal working state or not, the actual operating voltage of the UHV shunt reactor will not exceed 1.3 times the rated voltage. The core will not reach magnetic saturation, and the vibration parameters of the core always follow a linear relationship with its operating voltage.

The changes of power grid operation parameters are introduced to the research into reactor vibration, which is conducive to a more comprehensive understanding of the actual operating state of such a UHV shunt reactor. It can also provide guidance during the design and operation of UHV shunt reactors.

Footnotes

Acknowledgements

This work was supported by Science and Technology Project of State Grid Corporation of China (Study on the improvement of the whole life cycle of the UHV reactor and the control measures of safe operation, GYB17201900166).

Electromagnetic force calculation

The electromagnetic mechanical multi-field coupling model based on the laminated coordinate system proposed in the literature [10] is used to calculate the electromagnetic external force. When the reactor is in a normal state, the 3-d electromagnetic equation inside the core is given by formula (A.1): (A.1) $\begin{eqnarray}\displaystyle \left\{\begin{array}{@{}l@{}}{\displaystyle \frac{\partial }{\partial y}}\left[{\upsilon}_{z}\left({\displaystyle \frac{\partial A_{y}}{\partial x}}-{\displaystyle \frac{\partial A_{x}}{\partial y}}\right)\right]-{\displaystyle \frac{\partial }{\partial z}}\left[{\upsilon}_{y}\left({\displaystyle \frac{\partial A_{x}}{\partial z}}-{\displaystyle \frac{\partial A_{z}}{\partial x}}\right)\right]=J_{x}\\[12.0pt] {\displaystyle \frac{\partial }{\partial z}}\left[{\upsilon}_{x}\left({\displaystyle \frac{\partial A_{z}}{\partial y}}-{\displaystyle \frac{\partial A_{y}}{\partial z}}\right)\right]-{\displaystyle \frac{\partial }{\partial x}}\left[{\upsilon}_{z}\left({\displaystyle \frac{\partial A_{y}}{\partial x}}-{\displaystyle \frac{\partial A_{x}}{\partial y}}\right)\right]=J_{y}\\[12.0pt] {\displaystyle \frac{\partial }{\partial x}}\left[{\upsilon}_{y}\left({\displaystyle \frac{\partial A_{x}}{\partial z}}-{\displaystyle \frac{\partial A_{z}}{\partial x}}\right)\right]-{\displaystyle \frac{\partial }{\partial y}}\left[{\upsilon}_{x}\left({\displaystyle \frac{\partial A_{z}}{\partial y}}-{\displaystyle \frac{\partial A_{y}}{\partial z}}\right)\right]=J_{z},\end{array}\right. & & \displaystyle\end{eqnarray}$ where A _x, A _y, and A _z are the components of the vector potential (A) in the x, y, and z-directions of the spatial rectangular coordinate system respectively; 𝜐_x, 𝜐_y, and 𝜐_z are the magnetic reluctivity of core laminations in the x, y, and z-directions; J _x, J _y, and J _z are the components of winding current density (J) in the x, y, and z-directions, respectively.

The relationship between magnetic flux density (B) and vector potential (A) in the core is given by formula (A.2). (A.2) $\begin{eqnarray}\displaystyle \mathbf{B}={\nabla}\times \mathbf{A}. & & \displaystyle\end{eqnarray}$

The magnetic field strength matrix (H) in the core can be calculated through the magnetic reluctivity matrix (𝜐) and the magnetic flux density matrix of silicon steel laminations. (A.3) $\begin{eqnarray}\displaystyle \mathbf{H}=\boldsymbol{{\upsilon}}\mathbf{B}. & & \displaystyle\end{eqnarray}$

The external electromagnetic force on the core is the resultant of the magnetostrictive force and Maxwell force. (A.4) $\begin{eqnarray}\displaystyle \mathbf{F}=\mathbf{F}_{\mathbf{mag}}+\mathbf{F}_{\mathbf{max}}. & & \displaystyle\end{eqnarray}$

The Maxwell force at the boundary of the core cake can be obtained by integrating the Maxwell stress tensor: (A.5) $\begin{eqnarray}\displaystyle \mathbf{F}_{\mathbf{max}}=\displaystyle \int _{s}\mathbf{T}_{\mathbf{max}}\text{d}S, & & \displaystyle\end{eqnarray}$ where, T _max is the Maxwell stress tensor. (A.6) $\begin{eqnarray}\displaystyle \mathbf{T}_{\mathbf{max}}=\mathbf{B}\mathbf{H}-{\displaystyle \frac{1}{2}}\boldsymbol{{\mu}}(\mathbf{H})^{2}\mathbf{I}, & & \displaystyle\end{eqnarray}$ where, μ ( $\boldsymbol{\mu}=\frac{1}{\boldsymbol{\upsilon}}$ ) is the magnetic permeability matrix [10].

The magnetostrictive force can be obtained from the divergence of magnetostrictive stress: (A.7) $\begin{eqnarray}\displaystyle \mathbf{F}_{\mathbf{mag}}=-{\nabla}\cdot \boldsymbol{{\sigma}}, & & \displaystyle\end{eqnarray}$ where, σ is the magnetostrictive stress, which can be calculated according to its magnetostrictive strain (ϵ): (A.8) $\begin{eqnarray}\displaystyle \boldsymbol{{\sigma}}=\mathbf{D}\boldsymbol{{\varepsilon}}, & & \displaystyle\end{eqnarray}$ where, D is its elastic tensor, which can be calculated by use of Eq. (I): (A.9) $\begin{eqnarray} \displaystyle \mathbf{D}={\displaystyle \frac{E(1-{\alpha})}{(1+{\alpha})(1-2{\alpha})}}\cdot \left[ \begin{array}{@{} cccccc@{}}1 & {\displaystyle \frac{{\alpha}}{1-{\alpha}}} & {\displaystyle \frac{{\alpha}}{1-{\alpha}}} & 0 & 0 & 0\\ [12.0pt] {\displaystyle \frac{{\alpha}}{1-{\alpha}}} & 1 & {\displaystyle \frac{{\alpha}}{1-{\alpha}}} & 0 & 0 & 0\\ [12.0pt] {\displaystyle \frac{{\alpha}}{1-{\alpha}}} & {\displaystyle \frac{{\alpha}}{1-{\alpha}}} & 1 & 0 & 0 & 0\\ [12.0pt] 0 & 0 & 0 & {\displaystyle \frac{1-2{\alpha}}{2(1-{\alpha})}} & 0 & 0\\ [12.0pt] 0 & 0 & 0 & 0 & {\displaystyle \frac{1-2{\alpha}}{2(1-{\alpha})}} & 0\\ [12.0pt] 0 & 0 & 0 & 0 & 0 & {\displaystyle \frac{1-2{\alpha}}{2(1-{\alpha})}} \end{array} \right], & & \displaystyle \end{eqnarray}$ where: 𝛼 is the Poisson’s ratio of the silicon steel laminations, E is the elastic modulus of the silicon steel laminations, and ϵ is the magnetostrictive strain in the silicon steel laminations, which can be calculated according to their magnetostrictive characteristic curve.

Discussion of the test results and analysis of the simulated results

From the test results, it can be found that the order of vibration parameters as measured at the chosen five measurement points (in descending order) is: 2, 3, 5, 4, then 1. The vibration at measurement point 2 is the strongest, the vibration parameters at points 3, 5, and 4 are similar, and that at point 1 is the weakest.

The following conclusions can be drawn: (1) the vibration at the mitre joint of the upper part of the side yoke is the most severe. (2) The vibration at the upper end of the yoke is the weakest. (3) The vibration parameters at the other three measurement points on the side yoke are similar, among which the strongest occurs at measurement point 3, and the vibration at measuring point 5 is greater than at point 4.

According to the theoretical analysis of the core vibration principle as applied to a UHV shunt reactor in this work, the difference in magnetic flux density distribution in the core is the reason for different vibration parameters found at each of the five measuring points. The damping effect of fluid-structure interaction on the vibration of core is related to the vibration speed, which cannot affect the electromagnetic force on the core. According to the design and operation parameters of the UHV shunt reactor equivalent model (Table 1), the magnetic flux density distribution of the UHV shunt reactor core under five voltage ratios can be calculated by the method of finite element simulation. Taking 𝜂 = 1 as an example, the magnetic flux density distribution of the UHV shunt reactor equivalent model is illustrated in Fig. 14.

It can be seen from Fig. 14 that: (1) the magnetic flux density of the upper end of the side yoke where measuring point 1 is located is the smallest among the five measuring points, and the vibration intensity of the measuring point is also the smallest. (2) The magnetic flux density distribution at the joint between the side yoke and the top yoke at measuring point 2 is uneven. The magnetic flux density near the internal position is greater than that at other measuring points, so the vibration parameters at point 2 are greater than those at the other four measuring points. (3) The magnetic flux density at measuring points 3, 4, and 5 is evenly distributed and the values are similar, so the magnetic flux density is not the reason for the relationship between the vibration parameters measured at these three measuring points.

By comparing the locations of measuring points 3, 4, and 5, it can be found that measuring point 3 is closest to the top yoke and measuring point 5 is closest to the bottom yoke. According to the internal schematic diagram of the UHV shunt reactor core (Fig. 3), the bottom yoke is connected to the bottom of the oil tank through two bases and fixed to the ground, therefore, the connection surface of the bottom yoke and the base can be considered to be a fixed boundary, that is, the vibration displacement of the connection surface is zero. Measuring point 3 is furthest from the bottom yoke, and the vibration parameter thereat is the largest. Measuring point 5 is closer to the bottom yoke than point 4, and the vibration parameter should be less than that at measuring point 4, which was verified to be true in other reports [10].

The vibration parameters measured at points 4 and 5 are inconsistent with those predicted by the theoretical analysis despite their mutual proximity: the random errors in the measurements and the displacement of sensor positions may cause this inconsistency. By analysing the vibration parameters at points 4 and 5 at each of the five voltage ratios, it can be found that the vibration parameters at point 5 are greater than those at point 4, so the probability of random error is less than that of displacement of the sensor. The vibration measurement results can be affected by the arrangement and the small-scale displacement of the chosen sensors. Measuring points 4 and 5 are close to each other, the difference of vibration parameters is small, and the small range of displacement of the sensor can change the relationship between measurement results. In the experiment, the data recorded at measuring point 4 are accurate, and those from point 5 are biased.

References

Cui

, Finite element analysis of shunt reactors with auxiliary windings, International Journal of Applied Electromagnetics and Mechanics 20(3–4) (2004), 133–140.

Bao

Zhang

, Combined passive shielding design of magnetic field generated by air-core reactor of power supply of EAST poloidal field, International Journal of Applied Electromagnetics and Mechanics 54(3) (2017), 289–299.

Zhao

and Yang

, Online detection of inter-turn short circuit faults in dry-type air-core reactor, International Journal of Applied Electromagnetics and Mechanics 39(1–4) (2012), 443–449.

Zhao

Chen

Kang

, Optimum design of dry-type air-core reactor based on the additional constraints balance and hybrid genetic algorithm, International Journal of Applied Electromagnetics and Mechanics 33(1–2) (2010), 279–284.

Gao

Kazuhiro

Muhd

J.H.

, Design of a reactor driven by inverter power supply to reduce the noise considering electromagnetism and magnetostriction, IEEE Transactions on Magnetics 46(6) (2010), 2179–2182.

Rossi

and Le Besnerais

, Vibration reduction of inductors under magnetostrictive and maxwell forces excitation, IEEE Transactions on Magnetics 51(12) (2015), 1–6.

Gao

Nagata

Muramatsu

, Noise reduction of a three-phase reactor by optimization of gaps between cores considering electromagnetism and magnetostriction, IEEE Transactions on Magnetics 47(10) (2011), 2772–2775.

Tong

Qingxin

Rongge

, Magnetically controlled saturable reactor core vibration under practical working conditions, IEEE Transactions on Magnetics 53(6) (2017), 1–4.

Zhang

and Li

, Vibration properties of two-stage magnetic-valve controllable reactor, IEEE Transactions on Magnetics 54(3) (2018), 1–4.

10.

Lü

Guo

Niu

Geng

, A new 3d method for reactor core vibration based on silicon steel lamination rules and application in UHV shunt reactors, Mathematical Problems in Engineering (2019), 7290536, 1--11.

11.

Liu

Zhang

and Sundaravadivelu

, A new fluid structure coupling approach for high frequency/small deformation engineering application, IEEE Transactions on Magnetics 47(7) (2011), 1886–1889.

12.

Yang

Bache

, In vivo IVUS-based 3-D fluid–structure interaction models with cyclic bending and anisotropic vessel properties for human atherosclerotic coronary plaque mechanical analysis, IEEE Transactions on Biomedical Engineering 56(10) (2009), 2420–2428.

13.

Lihua

Jingjing

Qingxin

, An improved magnetostriction model for electrical steel sheet based on Jiles–Atherton model, IEEE Transactions on Magnetics 3 (2020), 1–4.

14.

Zhu

Sha

Zhang

Wang

Research on split-core reactor vibration reduction method based on magnetostrictive effect, in: 2018 IEEE International Magnetics Conference (INTERMAG), Singapore, 2018, pp. 1–1.

15.

Moradnouri

Vakilian

Hekmati

, Optimal design of flux diverter using genetic algorithm for axial short circuit force reduction in HTS transformers, IEEE Transactions on Applied Superconductivity 30(1) (2020), 5500108, 1--8.

16.

Lun

Chen

, Analysis on power flow and voltage fluctuation of tie line in UHVAC power grid, Power System Technology 36(9) (2012), 56–60, (in Chinese).

17.

Zheng

Sheng

, Study on the oscillation coupling mechanism between interconnected, power system and provincial power grid, Proceedings of the CSEE 34(10) (2014), 1556–1565, (in Chinese).

18.

Zhang

Yan

, A novel approach to investigate the core vibration in power transformers, IEEE Transactions on Magnetics 54(11) (2018), 1–4.