Research on the Thermal-Electric Coupling Behavior Under Compound AC-DC Voltage Within Saddle-Like Electric-Stress Dependence

This paper proposes a nonlinear electromagnetic-thermal-velocity coupling model to study the insulation performance of converter transformer, in which the saddle-like electric-stress dependence is considered under compound alternating current–direct current (AC-DC) voltage situation. Firstly, the conductance of transformer oil and oil-immersed pressboard have been measured through constructed three-electrode experimental system. An interesting electric-stress characteristic, which is saddle-like electric-stress dependence is obtained and counted in the thermal-insulation analysis in the proposed model. Then, the high-order harmonics of compound AC-DC voltage have been considered in the built electromagnetic-thermal-velocity coupling model. Additionally, the impact of non-uniform temperature result has been mapped to the electric field domain through interpolation method, due to different meshes between the flow-thermal and electric fields in accordance with distinct computational characteristics. Finally, the performance of nonlinear thermal-electric coupling properties, including saddle-like electric-stress dependence, have been considered in the proposed coupling model. Results determine that saddle-like thermal-electric dependent properties would considerably change insulation capability, compared with results from the traditional method. Even intensity field of some sampling points could jump more than 1.6 kV/mm. Therefore, it is essential to evaluate the insulation design by practical saddle-like electric-dependent properties.


I. INTRODUCTION
Converter transformer in high voltage direct current (HVDC) system is one of the most significant equipment for ensuring electric network security, and its operation condition is different from that of the traditional transformer. Specifically, the direct current (DC) component and high-order harmonic current of voltage source have a great influence on insulation state and thermal condition, respectively. In reality, the valve winding has to suffer a compound alternating current-direct current (AC-DC) voltage, which includes a DC voltage component, fundamental sinusoidal voltage, and high-order The associate editor coordinating the review of this manuscript and approving it for publication was Guiwu Wei .
harmonics. This hybrid-exciting source generates a resistive and capacitive electric field [1]; however, the production test, such as DC or polarity reversal test, cannot archive to predict the compound field.
Under the compound AC-DC condition of converter transformer, the coupling phenomenon is severe due to larger temperature and electric field. Recent studies show that insulation conditions of transformer oil and oil-immersed pressboard (PB) can be easily affected by field intensity [2], temperature [3], moisture [4], and mechanical stress, and the conductivity of insulation materials markedly increases with the temperature [5]. Therefore, thermal study is an important part in coupling analysis, which would greatly affect the electric filed distribution. Scholars have also conducted thermal research of converter transformer. Reference [6] stated that the highest temperature is approximately 50 K higher than the ambient temperature and the hot spot is located near the top of line winding. Reference [8], [9] considered nonuniform losses in analyzing temperature distribution, and demonstrated that high-order harmonic, which could enlarge losses, should be considered.
Additionally, field intensity would also have influence on conductance, and the latest report of CIGRE joint working group [10] demonstrated that the trend curve of oil electric-dependent conductivity is saddle-like, which is different from the traditional exponential manner [11]. However, in previous studies, the electric-stress dependence has been ignored or assumed as an exponential relationship for designing the insulation margin, which would deviate from the actual operation condition, leading to safety hazard.
In this paper, an electromagnetic-thermal-velocity coupling model is constructed on the basis of the measured insulation properties to test nonlinear thermal-electric coupling behavior within the saddle-like electric-stress dependence. The three-electrode experimental platform is constructed to test the electro-thermal coupling characteristic and electricstress dependence in Section II. Section III describes the coupling model in considering of the high order harmonic and the impact of non-uniform temperature distribution through the proposed fast mapping method. Section IV depicts the entire model and sub-model, which are utilized in temperature and electric field, respectively, and discusses the boundary conditions and meshing information in detail. Section V studies the effect of saddle-like electric-stress dependence on nonlinear thermal-electric performance of converter transformer. Finally, Section VI concludes the entire work.

II. MEASURING SYSTEM AND EXPERIMENTAL DATA A. THERE-ELECTRODE SYSTEM
This study adopted a three-electrode method [12] with higher accuracy to test the physical parameter of transformer oil and oil-immersed pressboard. The experimental platform is shown in Fig.1, and the test cell consists of high-voltage, guard, and test electrodes. A high voltage can provide a DC voltage from 0.1 kV to 20 kV, and measuring electrode connects with Keithley electrometer. Besides, all electrodes are designed according to the national standard to diminish electric field distortion, and the electrode system sets in the PTFE tank achieves a uniform temperature from 273 K to 393 K in the thermostat. The conductivity should be obtained on the basis of the structure and size of the electrode, as follows: where V is the volume conductivity (S/m), due to the leakage current going through the insulation material; R is the measured volume resistance ( ) when the measuring current is stable; A is the effective area of the measuring electrode (m 2 ), which can be calculated on the basis of the national standard [13]; and h is the sample thickness (m), which is equal to the electrode gap when the transformer oil is measured.

B. CONDUCTIVITY MEASURED RESULTS
The KI50X mineral oil produced by Kunlun Lubricant, which is widely utilized in HVDC project, has been selected as the oil sample after filter and heat. The measured temperature ranges from 300 K to 360 K under an electric field ranging from 0.1 kV/mm to 15 kV/mm. Fig.2 shows that the conductivity increases with the temperature at a slope of 11.5×10 −13 . The changing tendency of electric-stress dependence is saddle-like, and the knee point is approximately 2 kV/mm. This variation is different from the traditional exponential manner, but it accords well with the latest report from the CIRGRE joint working group. The saddlelike tendency greatly affects the electric field, which may mislead the previous insulation evaluation and cause potential hazard. The adopted transformer-board type is T4, which is provided by Weidman Company Ltd., and it should be pretreated on the basis of the national standard before measurement. Fig. 3 demonstrates the relationship of temperature and electric field. It shows that when the electric field ranges from 0.1 kV/mm to 10kV/mm, the conductivity rises more than twice of initial value, and the conductivity fluctuation is lower than 1e −14 S/m while the changed scope of temperature is 10K approximately.

C. ELECTRIC-STRESS DEPENDENCE OF OIL
Obviously, transformer-board conductivity varies with field intensity in exponential manner, whereas oil conductivity changes with electric field in the form of a saddle-like curve. The obtained experimental results differ from the previous assumption, which affects the insulation capability, because the electric field component for E <2 kV/mm is dominate in the transient electric field of converter transformer. Fig. 4 shows the probabilistic density of electric filed at some moments. The wave tendency of the transformer oil conductivity is caused by two reasons. First, the production and dissipation of free ion directly affect oil conductivity. If the dissipation rate is larger than the producing rate, then the density of oil ions decreases, thereby decreasing oil conductivity. The main reason is that the external electric field breaks the original dynamic balance of dissociation and recombination and make oil approach to electrolyze completely. Equation (2) demonstrates the effect of applied electric field on dissociation and recombination, which influences the conductivity [14].
where σ is the conductivity, E is the electric field, and k D , k R are the coefficients of dissociation and recombination, respectively. Second, when the number of produced ions becomes larger than that of dissipated ions, oil conductivity would increase. The effect of Wien and Onsager theory performs an important role in conductivity when the electric field is maintained in a high level; thus, oil conductivity increases with the influence of charge injection. Equation (3) depicts the relationship between conductivity and applied electric field with a charge injection [15], [16].
where σ is the conductivity, E is the electric field, F(b) is the Bessel equation, and k es is the coefficient. The changing conductivity is the coupling parameter to be utilized in the electric sub-model.

III. MULTI-PHYSICS FIELDS MODEL OF CONVERTER TRANSFORMER A. COMPOUND AC-DC VOLTAGE
The exciting voltage [17] for the calculating electric field is obtained through the built HVDC transmission system model by PSCAD, as shown in Fig.5. The voltage source of line winding is a sinusoidal function, whereas the hybrid voltage exciting valve winding includes direct current component, fundamental frequency alternating current and higher harmonic component. Fig. 6 presents the voltage curves. Fourier series analysis is applied to decompose compound AC-DC voltage. Fig. 7 depicts the amplitudes of every harmonic. The voltage is composed of DC current, fundamental frequency alternating current and higher harmonic component. In this study, an Y0-converter transformer in a HVDC system is selected to investigate the nonlinear coupling phenomena.

B. ELECTROMAGNETIC-FLOW-THERMAL COUPLING
Heat sources include iron [18] and winding losses, which are calculated within the consideration of high-order harmonics. The magnetic flux density can be calculated on the basis of Maxwell equations, and the magnetic vector potential vector can be described as follows: where A is the magnetic vector potential;v and σ are the reluctivity and conductivity, respectively; J s represents the source current density of winding; and z and r stand for the axial and radial directions, respectively. The type of silicon steel sheets is 30RG120, and the influence of frequency is considered in calculating iron losses. Firstly, the B-P and B-H curves should be measured at 50 Hz on the basis of the selected silicon steel sheets. After the maximum magnetic flux density of every element B m is obtained, mass loss P of every element, excited by fundamental frequency AC, is obtained by the B-P curve.
where P is the iron loss of every element at a fundamental frequency, ρ is the density, h j is the thickness of the iron core in the jst class, N is the total number of elements, and S i is the area of every element.
Considering the influence of high-order harmonics, as analyzed in Fig. 7, the losses have been studied by using loss separation method. The core losses [19] mainly include the hysteresis, eddy current, and additional losses, which can be expressed as: where the first item on the right is the sum of the hysteresis and excess losses, the second item is the eddy current loss, f is the frequency, and B m is the maximum magnetic flux density. K h , K e , α and β are the constants obtained from losses curve.
Winding losses [20] consist of ohmic and eddy losses as Eq. (7). Ohmic losses are dominant, which can be affected by high-order harmonics. The coefficient is the ratio of currents, as expressed in Eq. (8).
where P W is the winding loss; I and N are the current and number of turns, respectively; S is the area of the winding model; d is the wire size; ω is the angular frequency; B is the magnetic flux density in one element; and S e is the area of one element.
where P is the ohmic loss at a fundamental frequency, I f is the fundamental current, and I h is the current of the h th highorder harmonics. Through the thermal convection of transformer oil, winding losses are transferred to an oil tank, and the temperature is increased. The Mach number of transformers is small, and the steady-state mass conservation equation [21] of incompressible flow is as follows: The momentum conservation equations are as follows: The energy conservation equation is shown as follow: where z and r are the axial and radial directions, respectively; u and v are the axial and radial velocities, respectively; and f z and f r are the axial and radial external force densities, respectively, which are caused by gravity in this study. ρ is the density, µ is the dynamic viscosity, λ is the thermal conductivity, and c p is the heat capacity at a constant pressure. S T is the heat source, including the iron and winding losses introduced previously. In addition, the relationship between kinematic viscosity and temperature is also counted, since the kinematic viscosity decreases with temperature [5].
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 follows [22]: where n is the normal direction, h is the convective heat transfer coefficient, and T amb is the ambient temperature.

C. NONLINEAR THERMAL-ELECTRIC COUPLING
In the nonlinear thermal-electric coupling calculation process, the mapped temperature data to electric field calculation model should be utilized, and the electric-stress dependence of the conductivity should be considerable, as measured in Part II. The electric quasi-static field can be measured as follows: where is computational region and ϕ is potential; ε is dielectric constant, and g is conductivity which is the function of temperature and field intensity here. 1 and 2 are Dirichlet boundary condition and Neumann boundary condition respectively. And t is time, E is electric field.

D. TEMPERATURE IN SUB-MODEL
The transformer oil flows from the inlet at the bottom to the outlet at the top; thus, the thermal-velocity coupling model is adopted to analyze the temperature distribution of the entire model. While the middle of winding is grounded in the electric research, the sub-model is adopted with the symmetry axis being the Neumann boundary condition. Moreover, the best fitting grids between the temperature and electric fields have different shapes and sizes, and fine grids at regions with large temperature gradient should be generated [23]. Therefore, the fast mapping method is applied to interpolate temperature results into the sub-model, which could call these temperature data in thermal-electric coupling calculation [24].
As shown in Fig. 8, (x, y) is the coordinate of node D in the sub-model, and the temperature date of node D can be obtained by a shape function.
If the shape function N m (m = i, j, k) is less than 1, then the temperature data of the sub-model can be described: (18) where T D i , T D j , and T D k are node data of the temperature field. Fig. 9 shows the model structures and mesh conditions of two physical fields.  Fig. 10 concludes the entire calculation processes. Before the coupling analysis, the time constants of the transformer oil and PB insulation system are relatively large, generally reaching thousands of seconds, whereas the period of excited voltage is only 0.02 s. Therefore, temperature can be thought as unchanged in electric analysis. Iron and winding losses with high-order harmonics are studied in the first step. In the temperature analysis process, the inlet velocity and outlet pressure are set as boundary conditions, and the losses are generated heat sources. Then, thermal-velocity coupling calculation is applied to the entire model. Finally, the nonlinear thermal-electric coupling is studied after the non-uniform temperature field mapped partly to an electric sub-model by interpolation. Conductivity affected by thermal-electric condition is seen as coupling parameter, which is the function of temperature and field intensity in electric field calculation. The whole process is done by finite element method and iteration method.  Fig. 11 shows the computational model, a quarter of a converter ±500 kV transformer. Multi-level meshing has been adopted to the critical components, and fine meshing is processed rather than other layers, such as windings and iron core. All grids are close to the ideal shape, and fine grids at regions with large temperature gradient are generated. The total number of nodes is 157652, the total number of body elements is 315181, and the average element quality measured by skew is 0.65, which meets the criteria of meshing quality. A grid independent study is conducted on grid sizes of 0.12, 0.15, and 0.19 million nodes. The maximum winding temperature changes by less than 0.2% compared with the result with the finest grid, and the location of the hot spot is always in the 7th disc. Thus, the grid size of 0.15 million nodes is considered sufficient and used in all present simulations. Fig. 12 shows the sub-model with the Neumann boundary condition of electric field due to grounded middle turns of winding. To analyze the hybrid electric field, the valve winding stands compound AC-DC voltage, as displayed in Fig. 7, and the fundamental sinusoidal voltage excites the line winding. Moreover, AB and BC are the walls of oil tank, whereas FIGURE 12. Sub-model of the electric field. VOLUME 8, 2020 the AD line refers to one boundary of the iron core; these three lines are grounded [25].

V. EFFECT OF SADDLE-LIKE ELECTRIC-STRESS DEPENDENCE A. COMPARISON WITH EXPONENTIAL VARIATION
The electric-stress dependence of oil conductivity varies with the voltage in exponential manner. By measuring the physical parameter, a saddle-like tendency occurs between  the field intensity and conductivity, as shown in Section II.
To study the effect of saddle-like electric-stress dependence, two types of conductivity relationship are applied and compared. In Fig. 13, the fitting relationship of experimental data has saddle-like changing tendency as the black curve shows. Another traditional changing tends in exponential polynomial manner as red curve shows. To study the effect of saddle-like type explicitly, data after 2kV/mm are approximately equal in two curves to avoid other interference terms. Fig. 14(a) shows the location and temperature of sampling points, Fig. 14(b) displays the changing curve of oil and PB under two different conditions. Two numerical simulations are performed on the basis of the properties of oil conductivity: condition 1, stereotype manner in exponential fitting function; condition 2, saddle-like electric-stress dependence (as experimented in Part II). The field intensity of PB sampling points stays unchanged with the consideration of saddle-like properties, but the field intensity of the oil sampling point increases gradually, even more than 0.5 kV/mm. The rising variation of field intensity indicates that the saddlelike electric-stress dependence may lead to a relatively safe point to face the potential hazard and even cause insulation failure. Hence, practical saddle-like properties must be considered in insulation design.

B. SADDLE-LIKE CHANGE IN THERMAL-ELECTRIC COUPLING
To count the influence of temperature, uneven loss distributions are studied as Eq. (4)(5)(6)(7)(8), and thermal-velocity coupling is calculated as Eq. (9-12) by using finite element method.   15 shows the temperature contour of converter transformer, and the enlarged figure on the left shows a high temperature of over 342 K to observe the location of the hot spot. It's obvious that temperature differences exist markedly in the insulation system, and the temperature gradient of some pressboards can be up to 6 K/mm, and these thermal data are listed in Table 1. Therefore, the nonlinear thermalelectric coupling characteristic should be counted in electric calculation. Based on the steady nonuniform temperature distribution in Fig.15, Fig. 16 shows that the potential contours at 0.005 and 0.015 s under two conditions. In considering saddle-like properties, the pressboard stands at a high field intensity, and the maximum value is close to 9 kV/mm at 0.015s. Additionally, the electric distribution of the transformer oil becomes uneven, especially at the end of horizontal pressboard. Fig. 17 displays the field intensity of sampling points, as located in Fig. 14(a). The difference between conditions (condition 2 considers saddle-like change) becomes more obvious due to non-uniform temperature distribution. The pressboard field intensity with saddle-like thermal-electric properties jumps by 1.3 kV/mm compared with that under condition 1, whereas the oil electric filed increases by more than 1.6kV/mm. This changing trend is similar to that in Fig. 14(b). Markedly, the insulation state of oil sampling point is more severe within the influence of saddle-like properties, and the maximum electric field is an important factor for evaluating the insulation capability of the transformer oil. Fig. 18 depicts the maximum value at every moment. The changing tendency is similar to a sinusoidal curve, which indicates that the oil point with maximum value is next to the line winding. However, the variation affected by saddle-like properties is FIGURE 18. Maximum value of transformer oil in one period. VOLUME 8, 2020 not evident. The reason is that the electric field distribution near the line winding depends on the dielectric constant rather than conductivity.

VI. CONCLUSION
Based on the found saddle-like electric-stress dependence of insulation system in converter transformer, the thermalelectric coupling performance under compound AC-DC voltage is investigated, and some comparisons of different insulation material properties are also discussed. The obtained conclusions are summarized as follows: (1) Three-electrode experimental system is constructed to test the conductance of transformer oil and oil-immersed pressboard. It is found that the electricdependent conductivity of transformer oil shows a variation trend with saddle-shape. (2) The voltage of valve winding has been simulated in HVDC transmission system, and the hybrid voltage is analyzed by spectrum analysis to study iron and winding losses with high-order harmonics. In addition, the non-uniform temperature indeed has great impact on the insulation ability through the built electromagneticthermal-velocity coupling model, and the highest temperature is 350 K, with temperature gradient is up to 6K/mm. (3) The saddle-shape property has important impact on the electric field distribution. The field intensity of some points has changed and even increased by more than 1.6 kV/mm due to saddle-like change. The increase will cause additional points in the transformer to face the safety hazard, which may paralyze the normal operation.