Investigation on the nonlinear thermal-electrical properties coupling performance of converter transformer

Applied Thermal Engineering 148 (2019) 846–859

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Research Paper

Investigation on the nonlinear thermal-electrical properties coupling performance of converter transformer

T

Chi Chenga, , Gao Binga, , Yang Fana, , Liao Ruijina, Cheng Lia, Zhang Liangxianb ⁎

⁎

⁎

a

State Key Laboratory of Power Transmission Equipment & System Security and New Technology, School of Electrical Engineering, Chongqing University, Chongqing 400044, China b Xi’an XD Transformer Co., Ltd, Xi’an 710077, China

HIGHLIGHTS

A three-electrode system is constructed to test thermal-electrical dependence. An electro-thermal-fluid model is built to study uneven temperature distribution. A bilateral coupling is adopted to calculate the temperature and losses. The effect of nonlinear thermal-electric coupling on insulation is studied.

ARTICLE INFO

ABSTRACT

Keywords: Nonlinear thermal-electrical coupling Converter transformer Electro-thermal-fluid field Non-uniform temperature

Thermal state and insulation ability of converter transformer are significant to evaluate condition, while the impacts of nonlinear thermal-electric coupling impact are often ignored in the design analysis, and might result in misestimating its actual performance. In this paper, a bilateral thermal-electric coupling method is proposed to investigate the thermal and insulation performance of converter transformer on basis of an actual size. In addition, the influence of non-uniform temperature on the overall winding losses and the nonlinear thermalelectric coupling of insulation system are considered. Firstly, the thermal-electrical parameters coupling characteristics of insulation system in converter transformer are investigated based on the built experimental platform. Results indicate that temperature distribution would change the electrical performance all the time, while electrical-dependent characteristic of oil increases with electric field in U type curve. Then, the impact of nonuniform temperature is discussed based on the built bilateral thermal-electric coupling model, and it is proved that temperature would decrease obviously in considering the losses calculated by bilateral coupling. Finally, the nonlinear thermal-electric coupling performance of converter transformer is studied. It indicates that comprehensive factors of temperature dependence and electric filed dependence have great influences on the insulation properties. Significantly, the electric field of pressboard would decrease by 12.8% under hybrid voltage condition.

1. Introduction Converter transformer acts as the connection of electric power network in high voltage direct voltage (HVDC) system and alternating current system, it differs greatly with traditional transformers: (1) nonsinusoidal alternating current-direct current (AC-DC) hybrid voltage is suffered in the valve winding, including: DC voltage, fundamental AC voltage, and many other higher order harmonics [1]; (2) the polarity of DC voltage in valve winding might reverse fast, and electric field would change dramatically in a short time [2]; (3) larger concentrated electric

⁎

field and higher temperature are expected in insulation materials, and the stronger coupling phenomenon would cause nonlinear characteristics of insulation properties. These factors might cause a series of problems, including losses rising, overheating and insulation failure. As a result, supplement work has been done to improve its reliability in terms of numerical calculation and faults detection/assessment [3–5], and the former method is often adopted to ensure the converter transformer performance in advance. As the thermal rise and insulation ability are the two main factors to indicate the reliability of transformer, most attentions have been paid

Corresponding authors. E-mail addresses: [email protected] (C. Chi), [email protected] (B. Gao), [email protected] (F. Yang).

https://doi.org/10.1016/j.applthermaleng.2018.11.041 Received 12 July 2018; Received in revised form 7 November 2018; Accepted 10 November 2018 Available online 12 November 2018 1359-4311/ © 2018 Elsevier Ltd. All rights reserved.

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Nomenclature E γoil γ0oil β sT N I T Se S σ ω d B ρ v u

μ p z r fz fr λ cp ε n t φ Γ1 Γ2 Tamb h Tavg Tmax

electric field the resistivity of oil the resistivity of oil at benchmark the fitting coefficient heat source number of turns phase current temperature cell area the area of winding the conductivity of winding angular frequency wiresize flux density density radial velocity axial velocity

on the thermal rise and insulation ability of converter transformer. Ref. [6] researched the temperature of converter transformer in terms of average power losses method in considering of the ring, shunt, spacers and external cooling circuit in detail. Refs. [7,8] compared winding temperature caused by the uniform and non-uniform heat losses cases, and temperature difference is as expected, with the maximum value over 380 K. During this process, non-uniform heat losses could be calculated with high harmonic voltages and can provide more accurate temperature field than the unchanged losses [1]. Ref. [9] investigated the temperature influenced by ohmic power losses, and pointed out that the temperature different is over 50 K. Consequently, a bilateral coupling of thermal-electric field is proposed, which considers the dynamic balance of power losses and thermal field. In addition, it is proved that the nonlinear characteristic of insulation system indeed has great impact on insulation ability. Ref. [10] discussed the difference of linear conductivity and nonlinear conductivity for polarity reversal test, and found that the electric field of oil-immersed pressboard varies greatly under nonlinear condition, and the similar phenomenon is also verified by Refs. [11,12]. However, the temperature field is always assumed as uniform for these various cases, Refs. [13,14] studied the change of

dynamic viscosity pressure axial direction radial direction axial external force density radial external force density thermal conductivity heat capacity at constant pressure dielectric constant normal direction time potential Dirichlet boundary condition Neumann boundary condition ambient temperature convective heat-transfer coefficient average temperature maximum temperature

maximum electric field at different uniform temperatures, and Refs. [15,16] reveals that the conductivity of transformer oil and oil-immersed pressboard varied with temperature distinctively. Therefore, it can be concluded that the thermal performance plays an important role in the electric field distribution, and thereby the thermal-electric field nonlinear coupling characteristic should be applied to the thermal and insulation design. To include the impact of temperature on the nonlinear thermal-electrical coupling characteristic of converter transformer, this paper has constructed a numerical model under hybrid voltage condition and polarity reverse voltage condition. On the basis of the previous work, a bilateral coupling has been built to approximate the actual operation, in which the mutual relationship of temperature and winding losses has been considered. On the other hand, example of an actual converter transformer, combing with the measured thermal performance and insulation properties of dielectric materials, is adopted to investigate the nonlinear characteristics of thermal-electric coupling. The format of the paper is organized as followed: experimental platform is constructed to test actual physical parameters in Part 2, and the basis of the method is described in Part 3. Part 4 compares

Fig. 1. Schematic diagram of experimental system. 847

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voltage electrode can be regulated precisely by the double nuts and the lock washer. As shown in Fig. 1, the three-electrode system consists of measuring electrode, high voltage and guard electrode, which are all designed according to the national standards in order to prevent from distorted electric field. Besides, electrode system should be sealed in the polytetrafluoroethylene (PTFE) cylindrical tank, and thermostat can control ambient temperature from 273 K to 393 K evenly for test cell. The leakage current was measured by Keithly, and the schematic structure of experimental system is shown in Fig. 1. Transformer oil and oil-immersed pressboard are important dielectrics in converter transformer, and the resistivity of which is systematically investigated. The type of oil is KI50X mineral oil from Kunlun, the type of pressboard is T4 transformer-board made by Taizhou Weidmann Co. Ltd. and the fundamental parameters show in Table 1 [17]. All samples are measured within temperature ranging from 290 K to 360 K under electric field ranging from 0.1 kV/mm to 15 kV/mm. Before experiments on physical parameters, the test oil was filtered and heated according to standard proceed [18], and transformer-board was pretreated, including heat, dehydration and immersion, based on IEC standard [19].

Table 1 Fundamental parameters of testing samples. Parameters

Transformer-oil

Parameters

Pressboard

Type ρf/kg m−3 (298 K)

KI50X 882

Type Raw materials

Surface tension/mN m Acid-value/mgKOH g−1 Pour Point/K Flash Point/K Freezing point/K

40 < 0.03 < 228 > 413 < 223

ρp/kg m−3 (298 K)

T4 Unbleached kraft pulp 1200

Antioxidant content

0.31%

Tightness/g cm−3 Moisture content Ash content Oil adsorption ability Shrinkage

1.25 g/cm3 < 6% < 1.0% < 11.0% < 0.7%

2.2. Analysis of experimental results The measured conductivity under different temperature and electric field conditions is shown in Fig. 2. It is obvious that the conductivity decreases drastically with electric field (E) within the range E < 2 kV/ mm, then, the oil conductivity increases with the electric field gradually, and the conductivity is about three times larger than the initial value when electric field increases about 10 kV/mm. In addition, the conductivity increases with temperature, with the slope of 11.5e−13. Hence, the thermal-electric coupling phenomenon should be considered in the thermal and insulation ability evaluation. In previous work, the oil conductivity is assumed to increase with the electric field strength in exponential form, just as expressed by Eq. (1).

Fig. 2. The fitting function of KI50X conductivity with thermal-electric coupling properties.

oil

=

0oil exp(

E)

(1)

Obviously, the measured result differs from the previous assumption, it varies with the electric field in U type, and it has good accordance with the test results of CIGRE report in 2016 [20]. The U type change can be explained by the dynamics balance of ions production and dissipation, and charge injection should be considered when electric field keeps in higher level [21,22]. In addition, the probability density of field strength of converter transformer is shown in Fig. 3. It can be seen that the electric field in converter transformer varies from 0 to 6 kV/mm, which means that the electric field critical point E = 2 kV/ mm lies in the range, meanwhile, the low electrical field strength part accounts for a certain proportion. Consequently, it is the previous assumption that field intensity-conductivity in exponential function will result in unreasonable results. The relationship among temperature, electric field and the conductivity of T4 transformer-board is shown in Fig. 4, and the similar thermal-electric coupling as observed in oil conductivity can also be obtained in the pressboard condition. The conductivity of pressboard increases over 10 times with the electric field within the range from 0.1 kV/mm to 15 kV/mm. On the other hand, the conductivity fluctuation is lower than 1e−14 kV/mm when the changing scope of temperature is about 10 degrees. Therefore, it can be found clearly that both temperature and electric field have great contribution to the conductivity of pressboard. As a consequence, the novel relationship between oil conductivity and electric field, as well as the nonlinear thermal-electric field characteristics could lead to a totally different calculation result, which would mislead the insulation and thermal rise design otherwise. Therefore, the nonlinear electrical-thermal coupling model is built to investigate the temperature performance and insulation properties on basis of the experimental measured results.

Fig. 3. The probability density of field strength in converter transformer.

temperature distribution between bilateral coupling and unilateral coupling, and Part 5 studies nonlinear thermal-electrical properties coupling effect. Finally, Part 6 concludes the work of this paper. 2. The experimental thermal-electrical coupling characteristics of insulation properties 2.1. Experimental system of thermal-electrical characteristic To study the thermal-electric coupling characteristics of insulation materials in converter transformer, experimental platform was built with the three-electrode system which provides DC power ranging from 100 V to 20,000 V, and the gap between measuring electrode and high848

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transformer with directed-oil circulation and forced-air cooling (ODAF) mode. A schematic view of the part of the converter transformer is shown in Fig. 5. The flow patterns of converter transformer have two kinds of oil-paths as shown in Fig. 5(b) and (c). Firstly, the adiabatic side walls on both sides of the winding make the transformer oil only flow out from the end of winding, and the angle ring at the end of transformer changing the direction of oil-flow make temperature more even, preventing from local overheating. Secondly, duct spacers installed between the horizontal disks and vertical pressboard could divide the winding into several parts, and offer a zigzag oil-flow path to cool the internal winding. Besides the direction of velocity would change when the transformer oil flows through the interior winding and spacer, and the next part repeats like this process. 3.2. Governing equation of bilateral model An operating temperature suitable for insulation property could improve the performance of converter transformer and elongate its lifetime. Iron losses and winding losses are the main part to produce heat, and iron losses are studied by losses separated method in this paper [23], while winding losses are set as non-uniform heat source and the effect of temperature on losses is taken into consideration [24]. The winding losses including ohmic losses and eddy losses can be affected by temperature as Eq. (2) and σ = 108−24545 × T (S·m−1).

Fig. 4. The exponential relationship of T4 transformer-board.

3. Mathematical model and numerical method 3.1. Mathematical model The converter transformer under investigation is a 321MVA 530 kV

Fig. 5. Geometrical model of converter transformer. 849

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Table 2 Material parameters of converter transformer. Item

Parameters

Transformer-oil

Density (kg m ) Thermal conductivity (W (m K)−1) Specific Heat (J (kg K)−1) Dynamic viscosity (Pa s) Relative dielectric constant Density (kg m−3)

Iron core

Thermal conductivity (W (m K)−1) Specific Heat (J (kg K)−1) Density (kg m−3) Thermal conductivity (W (m K)−1) Specific Heat (J (kg K)−1)

Winding

Density (kg m−3) Thermal conductivity (W (m K)−1) Specific Heat (J (kg K)−1) Relative dielectric constant

Oil-immersed Pressboard

N

sT = e=1

Value −3

ρf = 1055.04607–0.581753034 × T kf = 0.134299084–8.04973822 × 10−5 × T cpf = −13408.1491 + 123.044152 × T − 0.335401786 × T2 + 3.125 × 10−4 × T3 uf = 91.4524999–1.33227058 × T + 0.00777680216 × T2-2.27271368 × 10−5 × T3 + 3.32419673 × 10−8 × T4 −1.94631023 × 10−11 × T5 ε f = 2.2 ρfe = 7650 kfe = 45 cpfe = 460 ρcu = 8900 kcu = 338 cpcu = 390 ρp = 1200 kp = 0.03 cpp = 2700 εp = 4

N 2I 2 1 Se + ( dB )2Se S2 24

·( / t + g (T , E )) | 1 = u (t )

(2)

| n 2

To investigate the flow and temperature field, computational fluid dynamic is adopted in geometrical converter transformer model. The steady state Navier-Strokes equations (i.e. the conservation of mass, momentum and energy) are for axisymmetric incompressible flow given as follows:

( u) 1 ( rv ) + =0 z r r ( vu) = fz r

p u 1 u + (µ ) + (µr ) z z z r r r

(4)

( uv ) + z

( vv ) = fr r

p v 1 v + (µ ) + (µr ) r z z r r r

(5)

T 1 v )+ ( r ) z r r r

(6)

( ucp T ) z

+

( vcp T ) r

= sT +

z

(

(t )

|t = 0 =

(0)

1

+

2

= (7)

3.3. Material properties and boundaries The transformer oil and other materials used in the converter transformer numerical model have the properties as shown in Table 2, where T is temperature in K. A nonlinear thermal-electrical coupling analysis model for converter transformer is established in COMSOL Multiphysics software. In order to reduce the calculating cost and obtain accurate results, a multi-level meshing process is executed in the finite element modeling (FEM). Multi-level meshing has been adopted to the critical components (angle ring, insulation layer and spacer), finer meshing is processed rather than other layers like the windings and iron core. The total number of nodes is 157,652 and total number of body elements is 315181, and the mesh illustration at the end of valve winding is also shown in Fig. 6. Besides, a grid independent study is done on a grid size of 0.12, 0.15 and 0.19 million nodes, the maximum winding temperature is found to change less than 0.2% compared to the result with the finest grid, and the location of hot spot is always in 7th disc. The ambient temperature is set as 298 K and a defined heat transfer coefficient is applied to the oil tank to simulate the heat convection of air, which can be described as followed.

(3)

( uu ) + z

=

= 0,

Eqs. (3)–(5) are flow governing equations, and Eq. (6) is utilized to calculate heat transfer. When the velocity both in radial and axial direction equals zero, the total equation can be simplified as solid steadystate heat dissipation governing equation [25]. The nonlinear effect on insulation property can be reflected on calculating the electric field of insulation materials, and electric field in converter transformer is thought as an electro-quasi-electric field [26]. As experiments in Part 2, γ = g(T,E) is an coupling parameter with nonlinear characteristics of temperature and electric factor, which plays an important role in calculating electric field, especially resistive electric field.

T = h (T n

Tamb)

(8)

The velocity of inlet oil is a specified constant, while an average pressure condition (Pavg = 0 Pa) is set at the domain outlet, and all the 850

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Fig. 6. Schematic view and boundary conditions for simulation. Table 3 Boundary condition of nonlinear thermal-electric coupling. Item

Boundary

Line#1 Line#2 Line windings with its electrostatic ring Valve windings with its electrostatic ring Tank

Inlet Outlet Voltage source Voltage source Heat transfer coefficient

solid surfaces are set in no-slip condition in contact with fluid. Windings are excited by voltage sources to produce magnetic flux and heat loss, and the detailed boundaries are illustrated in Fig. 6 and Table 3. The exciting voltage for calculating electric field is got in DC transmission system model by PSCAD, and the Y0-Y converter transformer in HVDC system is selected to study the nonlinear coupling phenomena. The voltage source of line winding is a sinusoidal function, and the hybrid voltage exciting valve winding includes direct current, fundamental frequency alternating current and higher harmonic component, as shown in Fig. 7.

Fig. 7. The voltage curves of line winding and valve winding.

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Fig. 8. The impact of bilateral coupling.

4. Non-uniform temperature distribution of converter transformer

oil of unilateral coupling flows fast slightly, with the maximum velocity of transformer oil under two conditions being 0.1354 m/s and 0.1362 m/s respectively. The main reason is that lower temperature of bilateral coupling is negative proportional with the dynamic viscosity and density, which could lower the velocity of transformer oil, and then the oil flow would affect temperature in turn until dynamic balance. Fig. 9a and b are temperature contours from unilateral and bilateral coupling, and temperature from unilateral coupling ranges from ambient temperature to 355.1 K, while the maximum temperature of bilateral coupling is 349.9 K, the enlarged pictures have the same colorbars which only display temperature over 342 K to emphasize the hot spot. As shown, the temperature rise can be up to 5 K with consideration of bilateral coupling. The main reason is that the less winding

Firstly, the uneven temperature distribution is discussed to explore the corresponding influence. The average power loss method is often adopted in the previous researches, and the winding losses are reckoned as unchanged. The general justification behind this approach is that the voltage of distribution transformers is relatively small, and temperature distribution under this condition does not significantly perturb the losses. Actually, the bilateral coupling of thermal-electric could affect the losses, velocity and temperature distribution directly in converter transformer. The velocity contour and the scatter plot of discs from top to bottom curve at the top oil ducts are depicted in Fig. 8(a) and (b). As can be seen that the values on the two conditions are almost equal, but

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Fig. 9. Temperature contours in converter transformer by unilateral and bilateral coupling.

losses would be produced under the temperature influence, and the line winding losses are 158.8 kW and 120.98 kW by unilateral coupling and bilateral coupling respectively. On another aspect, the hot spot emerges on the seventh discs of line windings, instead of the top side of windings, because the oil of the end region has more thermal energy and the area above the top disk could not produce heat continuously. Simultaneously, in every part of windings, the maximum value locates at the upper-middle position of each part, mainly because the locations of spacers would drive the oil flow faster at the bottom and top of part, which could dissipate more heat, and lead the gradual decrease of temperature until the next part. Fig. 10(a) provides the temperature data in the center of discs from Part 1 to Part 3. Obviously, the process of temperature rise can be divided into three stages, and it waves with the disc number in every stages. In addition to variation law of every part, the temperature differences of two methods for three parts are shown in Fig. 10(b). It can be seen that, from Part 1 to Part 3, the maximum temperature difference of two methods are 5.4 K, 3.9 K, 3.2 K respectively, due to higher temperature at the end which has a greater influence on windings losses. The temperature of winding discs has been discussed in many literatures, yet the temperature of insulating materials, including transformer-oil and oil-immersed pressboard, should be studied intensively. Fig. 11 shows the temperature gradient contours of transformer-oil and oil-immersed pressboard, it’s clear that temperature differences exists obviously in these insulation materials, even temperature gradient of some pressboard can be up to 6 K/mm, and Table 4 gives the maximum value and ambient temperature. Temperature difference is uneven obviously in interior converter transformer, if the uniform temperature or unchanged conductivity parameter is adopted to analyze the insulation margin, the potential hazard would threaten operation safety. In fact, insulating oil and oil-immersed pressboard always operate in high temperature condition. It’s well known that when temperature rises every 10 K, the speed of oil oxidation process would double, and the polarization characteristic of the inner interface on pressboard might be

Fig. 10. Temperature comparisons. (a) Central temperature of each disk by unilateral coupling and bilateral coupling. (b) Temperature difference of each disc. 853

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Fig. 11. Temperature gradient contours of transformer-oil and oil-immersed pressboard at the end of windings. Table 4 The temperature of insulating materials by bilateral coupling. Item

Tavg (K)

Tmaxt (K)

Tmaxt − Tamb (K)

Transformer-oil Oil-immersed pressboard

309.0 307.2

349.8 345.0

51.8 47.0

strengthened seriously when temperature is over 343 K. Therefore, the great effect of temperature on conductivity would affect insulation condition, and the nonlinear coupling would be analyzed detailly in the next chapter. 5. Effect of nonlinear coupling on insulation property 5.1. Numerical models at different uniform temperature In the design of converter transformer, the insulation ability is a prominent factor related to safety, the transient electric field from nonsinusoidal hybrid voltage in a period is conducted to track the practical insulation ability of transformer-oil and pressboard. The maximum electric field has always been taken into consideration as a significant factor when designing the model structure and insulation margin, Fig. 12 shows the variation range of field intensity when ambient uniform temperature changes from 290 K to 350 K. Obviously, the field intensity of pressboard has a greater change range compared to that of oil and the largest difference of pressboard electric field can be up to 0.67 kV/mm. The explanation is that the conductivity of pressboard rises faster with temperature than that of oil, and has a larger range of variation within the same temperature interval, as experiments measured in Part 2. 5.2. Insulation ability of nonlinear electric dependence

Fig. 12. (a) The variation range of oil field intensity at different uniform temperature from 290 K to 350 K. (b) The variation range of pressboard field intensity at different uniform temperature from 290 K to 350 K.

As stated above, both electric field and temperature have great influences on the insulation. Therefore, the nonlinear electric dependence of converter transformer at different uniform temperature is investigated at first, and the measured conductivity of insulation properties is also adopted in the simulation. Fig. 13 records the maximum

field intensity in converter transformer in a period. In Fig. 13(a), the periodic curve is similar to sine curve, which implies that the point location of oil maximum electric field is next to line winding though the valve winding relies on higher voltage. The explanation is that there is

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Fig. 13. (a) Electric field dependence of transformer-oil at different uniform temperature from 290 K to 350 K. (b) Electric field dependence of oil-immersed pressboard at different uniform temperature from 290 K to 350 K.

no direct current component in line winding, and the dielectric constant of transformer oil is lower, as a result, the oil near line winding has larger electric field than oil at valve winding. However, the maximum field intensity of oil changes slightly with different uniform temperature. It can be seen from Fig. 13(b), one can see that the pressboard point with maximum field intensity should locate next to the valve winding, and the maximum field intensity occurs at the same time when voltage peak takes place. With the temperature increasing from 290 K to 350 K, maximum field intensity of pressboard decreases from 9.57 kV/mm to 9.33 kV/mm, which illustrates electric field dependence of pressboard is affected by temperature considerably.

curves recording electric field of the sampling points are depicted in Fig. 15. It should be noted that the changing tendency of oil and pressboard electric field changes under two conditions are similar to the voltage curve from valve winding all the time. From Fig. 15(a), the electric field of oil under condition (2) is higher than that of condition (1). It is considered that the conductivity of pressboard grows faster at the same interval of temperature and electric field, so oil stands more voltage. In contrast, the comprehensive factors lead a drop to the field intensity which oil-immersed pressboard stands as Fig. 15(b). Periodic curves of maximum oil electric field under two conditions are shown in Fig. 16(a). For a circle time, the curve of oil electric field is similar to sine curve, which implies that the point location with maximum electric field is next to line winding. The oil maximum electric field under condition (2) is usually larger than that under condition (1) as expected, but the differences between two conditions are diminished. To sum up, taken nonlinear factors into consideration, maximum electric field of oil would increase subtly, because the point locates near line winding would be affected by dielectric constant rather than conductivity. Fig. 16(b) show that the maximum value curves of oil-immersed pressboard under two conditions are similar to the changing tendency of valve winding voltage, revealing that the point with maximum electric intensity located near valve winding. In a period, the electric field of pressboard can be up to 13.46 kV/mm under condition (1), however the field intensity under condition (2) has a sharp decrease during the entire period. Because valve winding has higher temperature and larger field intensity, and the mutual coupling phenomena is so

5.3. The nonlinear thermal-electric coupling characteristic of converter transformer As discussed above, the thermal rise and insulation ability are determined by temperature and electric field. To compare the comprehensive factors, numerical simulations could be performed in two different conditions: (1) Without none thermal-electric coupling case; (2) Nonlinear thermal-electric coupling considered case. As observed from the potential contours in Fig. 14, it is obvious that the thermal-electric coupling characteristics would lead to a great change in potential at 0.0005 s and 0.0015 s, furthermore, and more equipotential lines are concentrated on pressboard. To avoid the influence of alternating current component, the sampling points are selected at the end of valve winding. The varying

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Fig. 14. The potential contours at some time points. (a) Condition (1). (b) Condition (2).

Fig. 15. Transient electric field of sample points under condition (1) and condition (2).

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5.4. The nonlinear coupling phenomenon at polarity reversal Besides hybrid voltage of the operation condition, the coupling phenomena at polarity reversal (PR) is studied with the comprehensive nonlinear factors, and the aim of PR test is to simulate the reversal voltage waveforms of valve winding at instant with the line winding grounded. According to IEC standard [27], the polarity voltage inside valve windings shifts to negative DC voltage after positive DC voltage being applied for 90 min, then shifts back to the initial voltage, and the time of reversal process is 120 s. Fig. 17 shows the potential contours under two conditions at the same time of second reversal. With the nonlinear impacts, the range and distribution of electric field both varied modestly, and the specific changes are reflected in field intensity and the reaction rate. Fig. 18 shows the changing curves of field intensity under two conditions in a period. Under condition (2), the tendency and the varying range are slightly different. The maximum electric field decreases from 10.72 kV/mm to 6.7 kV/mm, and the time of maximum value occurs before the third reversal. In Fig. 18(a), the changing curve of pressboard drops gradually after a sharp increase, however, this phenomena doesn’t occur in Fig. 18(b). Fig. 19 shows the amplified figure of pressboard curves under two conditions when second reversal, the times of reverse points are 5620 s and 5590 s respectively under condition (1) and condition2. This difference is caused by charge relaxation time constant, because the conductivity of pressboard would rise dramatically with the nonlinear impacts, and the reaction rate would be fast. 6. Conclusion This paper presents a study of nonlinear thermal-electrical properties coupling characteristics in converter transformer, which derives from temperature dependence and electric field dependence of insulation materials. Thermal-electrical coupling experiments were done for transformer-oil and oil-immersed pressboard, and temperature distribution obtained by bilateral coupling was utilized to analyze the nonlinear thermal-electrical characteristics of insulation properties. The following conclusions can be drawn:

Fig. 16. Periodic curves of maximum electric field.

Table 5 The Maximum difference of two conditions. Item

Time (s)

Condition (1) (kV mm−1)

Condition (2) (kV mm−1)

Increment (%)

Transformer-oil Oil-immersed pressboard

0.0051 0.0396

8.1 12.4

8.3 7.8

2.5 −12.8

(1) Through experiments from three-electrode measuring system, the U type conductivity curves of transformer-oil varying with electric field are obtained at different temperatures, and tests show that the conductivity of both transformer-oil and oil-immersed pressboard have a close nonlinear relationship with temperature and electric field. (2) The bilateral coupling is adopted to estimate the dynamic balance between losses and temperature field. The line winding losses calculated by bilateral coupling decrease from 158.8 kW to 120.98 kW, while temperature drops by 5 K. In addition, temperature distribution is fluctuated that the temperature gradient can be up to 6 kV/mm in oil-immersed pressboard. (3) Nonlinear thermal-electric coupling properties have significant impacts on insulation properties. Taking temperature dependence and electric field dependence into consideration, the electric field of pressboard would decrease by 12.8% in hybrid voltage condition, while reaction rate would be faster in PR condition.

strong that it affects the conductivity of pressboard dramatically. In this way, the drop of field intensity is always over 3 kV/mm with the comprehensive factors of temperature and electric field, and specific impacts are shown in Table5. The comprehensive factors have a great influence on oil near valve windings as stated above, but oil area next to line winding with capacitive characteristic has a larger field intensity. As for pressboard, the field intensity of pressboard decreases by 12.8% with the comprehensive factors.

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Fig. 17. The potential contours at 5520 s under two conditions.

Fig. 19. Curves of pressboard at second reversal under two conditions.

Acknowledgement This work was supported by the National Key R&D Program of China (grant numbers 2017YFB0902703). Appendix A. Supplementary material Supplementary data to this article can be found online at https:// doi.org/10.1016/j.applthermaleng.2018.11.041. References [1] J. Smajic, J. Hughes, T. Steinmetz, et al., Numerical computation of ohmic and eddy-current winding losses of converter transformers including higher harmonics of load current, IEEE Trans. Magn. 48 (2012) 827–830. [2] K. Wen, Y. Zhou, J. Fu, et al., A calculation method and some features of transient field under polarity reversal voltage in HVDC insulation, IEEE Trans. Power Delivery 8 (1993) 223–230. [3] J. Zheng, H. Huang, J. Pan, Detection of winding faults based on a characterization of the nonlinear dynamics of transformers, IEEE Trans. Instrum. Measur. 99 (2018) 1–9.

Fig. 18. The changing tendency of electric field under two conditions in a period.

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