Research on Resistance Enhancement Coefficient and Thermal Dissipation of Stator Strands in Huge Synchronous Generator

For large generators, accurate assessment for the resistance enhancement coefficient (REC) and thermal dissipation is of great significance. With the development of design and manufacturing technology for large generators, the different cross section of the upper and lower strand becomes an effective method to reduce the heat losses. In current electromagnetism design, the eddy current analysis does not take the strand structures into account. In this paper, the REC of strands with special structures of a 150MW turbo-generator is derived by an analytic algorithm. Besides that, a mathematical model of the stator slot is established and the heat losses distribution and REC of strands are calculated by finite element method (FEM). On this basis, the stator REC and thermal dissipation of a 1000MW power turbo-generator are investigated with hollow and solid strands. The calculated results match the experimental data. All of these could provide an important theoretical basis and reference for the design of stator windings in high-power generators.


I. INTRODUCTION
For traditional high-power turbo-generators, double layer windings with lap coils are often adopted for the stators. The strands of the upper and lower layers usually use the same structure, which have the same number of strands and strand cross section area. In this traditional structure, due to the transverse leakage of the magnetic field, a clear difference in eddy current distribution between the upper and lower strands appears. This could lead to significant temperature gradients along the radial direction. Under long operation processes of the generator, being affected by the action of thermal stress, strands and insulation of the windings may lead to wear, which may cause short-circuit fault and other accidents. Hence, the winding structure should be optimized The associate editor coordinating the review of this manuscript and approving it for publication was Kan Liu .
to reduce the eddy current losses and improve its distribution to reduce the temperature gradients in the stator slot.
With these factors considered, several special strand structures, such as the strands number and the cross section being different for the upper and lower windings are proposed and applied to the stator windings of large turbo-generators to make the eddy current tend uniform distribution.
A vertically placed strip winding which can limit interturn short-circuit fault current, ac resistance of a litz wire, and current sharing analysis of parallel strands have been researched [1]- [3]. Copper losses with form-wound windings, online measuring power factor in ac resistance spot welding, and different tapped windings for flux adjustment have been investigated [4]- [6]. To reduce the harmonic contents of the rotor winding, a ''double-sine'' wound rotor is proposed for the brushless doubly fed generator [7]. In order to reduce the copper losses at high frequency, some experts have proposed to use rectangular litz wires in form-wound stators [8], [9]. The proximity effect depends on the strength and frequency of magnetic field in the position where the conductor is located. Its impact on the copper losses is presented in [10]- [12]. For some complex structures, FEA could provide a desired solution [13]- [16]. Researchers have studied eddy current losses and ac resistance for the windings, but few do the research on REC of strands with special construction and thermal dissipation in large power turbo-generators. In this paper, a 150MW air-cooled turbo-generator is taken as an example. Eddy current losses of winding strands under rated load operation with special structures, where both the strands number and the cross section area are not the same, are derived by analytic algorithms. Based on the FEM, the physical and mathematical models of the stator slot are established with special strands structures. The magnetic field distribution of the slot is calculated. The real and imaginary currents are analyzed for each strand. The REC is obtained by real and imaginary currents. Comparing analytic and numerical algorithms, it can be shown that the deviation of REC gained by different algorithms is minor. Besides, the stator REC and thermal dissipation of a 1000MW power turbo-generator are investigated with hollow and solid windings. The heat sources of the hollow and solid windings are obtained and the RECs that are received from the magnetic field calculation are taken into account. The stator temperature is calculated based on a fluid and heat transfer principle [17]- [20]. The calculated temperature results and the test data are matched.

II. ANALYSIS FOR REC WITH DIFFERENT STRUCTURES
A. ANALYTIC AND NUMERICAL ALGORITHMS FOR REC WITH SAME STRANDS AREA For an AC electrical machine, the stator current and leakage magnetic flux are alternating. Other than the load current, the eddy current could be induced in windings that are in the alternating magnetic field. The eddy current could lead the upper strands current density to increase, which results in a skin effect of the current. Both copper losses and effective resistance are increased [21]- [24].
A stator slot with n strands is shown in Fig. 1. The stator slot is an open rectangular slot. The current in each strand is i = I m coswt. This is a sinusoidal magnetic field problem with two mediums. Some assumptions are listed as follows.
(1) The stator slot in a large ac machine is usually deep and narrow. Thus, the leakage magnetic flux crossing the slot is parallel to the bottom. The circulating current effect is ignored. (2) The magnetic field intensity H x is just the function of the coordinate y. (3) The stator core permeability is infinite and magnetic motive force drops is ignored. (4) The insulation matter has the same permeability as the vacuum. They are both 4π × 10 −7 H/m. The relative permeability of the insulating material is 1. Besides that, the insulation material is non-conductive material, and its conductivity is 0. The insulation material itself does not be induced eddy current. The insulation matter does not change the distribution of the magnetic field in the slot. Based on the assumption above, the magnetic flux that is generated by a winding with width b and current density J z is equivalent to a magnetic flux that is generated by a winding with width bs and current density bJ z /b S . Hence, where H is magnetic field intensity, J z is current density in zdirection, E is electric field intensity, and B x is flux density in the x-direction. Only a one dimensional eddy equation should be solved as follows.
The solution for (2) is wherė where is the depth of penetration, A 1 and A 2 are arbitrary constants, which are determined by the boundary conditions. f is the frequency, µ 0 is the permeability of vacuum, and σ is the conductivity.
According to the law of ampere loop, there iṡ Hence, it could be expressed as follow.
40358 VOLUME 8, 2020 A 1 and A 2 can be gained, then theḢ x can be determined further. (7) For there is only one strand in a slot, the energy flow is in the y direction, so the plural power that flows in the slot conductor could be expressed by (8) where 0 < ξ < 1, ξ = a/ , a is the height of strand, and l is the conductor length.
As there are n strands in one slot, the AC resistance of strand m could be expressed as The REC of strand m k rm can be obtained. where Fig. 2 (a) gives the two function curves of equation (11). These two functions are both increasing function. It can be seen that ϕ(ξ ) ∈ [1, 5] and ψ(ξ ) ∈ [0, 10] for ξ ∈ [0, 5].
In order to analyze the relationship of REC with the total number of strands in one slot, analytical and numerical calculations are conducted. Fig. 2 (b) shows the maximun REC with the total number of strands range from 1 to 28 when b/bs=0.8 with analytical and numerical algorithms. The analytical and numerical algorithms match well. It can be seen that the maximum REC of the strand first increases with the increase of the total number of strands, and then decreases with the further increase of total number of strands. As the total number of strands is 3, the maximum REC of a strand is the maximum.

B. ANALYTIC AND NUMERICAL ALGORITHMS FOR REC WITH SPECIAL STRAND CONSTRUCTION
Aiming at the different number and cross section area between the upper and lower winding strands, the REC of strands are derived by the following process. The structure of strands is shown in Fig. 3. The total number of upper strands is m+x, and lower strands is m-x.
In Fig. 3, where S represent the strand area. The assumptions are the same as those in Section II A. VOLUME 8, 2020 For the upper strand, the REC of strand i (k uri ) can be expressed as where I 1 is the total current from strands 1 to i-1, and I i is the current of strand i. The strand current could be expressed as where i h is the current of the lower winding strand, and i c is the strand current when the upper and lower strands have the same number and areas. Hence, it can be obtained that The average REC of the upper winding (k urav ) can be expressed as Hence, (16) can be further expressed as For further consolidation, (17) can be expressed as (18) where (m + x)a u is the total height of upper strands, b × a u is upper the strand area, and a u is the height of one upper strand. For the lower winding, the derivation process for the average REC is the same as the upper winding. The average REC of the lower winding (k drav ) can be expressed as (19) where (m − x)a d is the total height of lower strands, b × a d is the lower strand area, and a d is the height of one lower strand. It can be seen that the winding average additional losses of the upper one is much greater than that of the lower one. The winding average additional losses of the upper one is seven times of that in the lower one when their total areas are the same. The average REC is direct proportion to the total height and strand area, and is inversely proportional to the slot width. The winding current is irrelevant.
For the hollow strand and the relations of structure size, which are shown in Fig.4, the REC of the hollow strand can also be derived by formulas (20)- (24).
where I 1 = I ma + I , I 2 = I ma + I mb + I . I ma , I mb , and I mc are the current of part a, b, and c, respectively. I is the current below the strand m.
For the 150MW air-cooled turbo-generator, the upper and lower winding strands have different areas. For the numerical calculation, the boundary conditions of the electro-magnetic field equations could be written as where A z is magnetic vector potential, ω is the angular frequency, J z is the strand current density, µ 0 is air relative permeability, σ is conductivity of the strand, and n is the normal unit vector. The magnetic flux density and flux line are show in Fig. 5. Seen from Fig. 5, leakage magnetic flux density is relatively high at the top of slot and low at the bottom of slot. Due to the stator core permeability being considerably larger than that of the windings and insulation, the leakage magnetic flux crossing the slot is perpendicular to the two slot walls.
The eddy current consists of real and imaginary parts. The REC of strands (k r ) is equal to the ratio of resistance that is considered with the induced eddy current and resistance regarding no eddy current. Based on the magnetic  filed solution, the REC could be expressed as where R ac is the resistance that is considered with the induced eddy current, R dc is the resistance regarding no eddy current, J z is the current density that contains the eddy current, J sz is the source current density, J eR and J eI are the real and imaginary parts of the current density in element, S b is the strand area, n is the total element number in one strand, and e is the element area. The current density of real and imaginary parts of the top strand is shown in Fig. 6. The imaginary current density is lower in the center of the strand, and higher at the boundary of the strand. Fig. 7 shows the REC of strands from strands 1 to 84. The analytic and numerical calculation results of REC are matched. Due to the upper and lower winding strands that are not within the same area, the REC is not monotonically increasing from the bottom to the top of the slot. There is an inflection point from the top strand of the lower winding to the under strand of the upper winding. VOLUME 8, 2020

III. THERMAL DISSIPATION IN LARGE POWER TURBO-GENERATOR BASED ON REC CALCULATION
For the 1000MW turbo-generator that is discussed in this paper, solid and hollow windings are adopted for the stator. The hollow strands are water-cooled. Fig. 8 gives the stator structure of this turbo-generator. The prototype structure is given in Fig. 8 (a). The upper winding and lower winding are connected by a conductive connection sleeve in the end part. The hollow strands are combined with water pipes by a connector. The connection type of upper and lower windings is shown in Fig. 8 (b).
The rated parameters are listed in Table 1. For the 1000MW power generator, there are 28 hollow strands in the upper winding, and 24 hollow strands in the lower winding. For the upper and lower winding strands, the solid and hollow strands are numbered from the bottom to the top as shown in Fig. 9 (a)  and (b). Under rated operation, the calculated results of flux leakage and imaginary current density are shown in Fig. 9(c) and (d). It can be seen that the imaginary current density in the upper winding strands are obviously higher than those in the lower winding strands. For the hollow strands, their imagniary current are obviousely higher than those of the solid strands. Fig. 10 gives the REC of the upper and lower windings. The RECs of the solid and hollow strands of the lower winding are lower than 1.5. The maximum REC of the hollow strands of the upper winding strand has exceeded 3.0.
According to the REC of the solid and hollow strands obtained by the calculation of the magnetic filed, using the additional losses as the heat source, the thermal field of the turbo-generator stator strands is investigated. The temperature and fluid coupled model of the stator of 1000MW power turbo-generator is established. It is shown in Fig. 11 (a). The thermal and fluid coupled model consists of teeth, yoke, upper and lower windings, insulation, water pipes, hydrogen duct, and the slot wedge. The stator core is hydrogen-cooled, and the stator windings are water-cooled inside. There are 463,040 elements and 533,312 nodes in total in the solution region. The mesh plot of the thermal and fluid coupled model is shown in Fig. 11 (b) and (c).
The research on the flow and heat transfer process of the cooling medium involves the coupling analysis of temperature and fluid fields. During the calculation, the laws of  conservation of mass, conservation of momentum, and conservation of energy should be satisfied. The calculation control equations for fluid-thermal coupling analysis are shown as follow [25]. where ρ is the fluid density, t is the time, u is the velocity vector, u, v, w are components of the x, y, z directions, µ is dynamic viscosity, P is fluid pressure, S u , S v , S w are the general source term, S h is the volumetric rate of heat generation, λ is thermal conductivity, c is specific heat, and T is temperature. The 1000MW turbo-generator is water-hydrogen cooled. There is a ventilation duct between two stator core block. Fig. 12 (a) and (b) show the stream lines and velocity distribution in the hydrogen ventilation duct. It can be seen that the fluid velocity is high around the teeth, and lower around the yoke. The stator windings have a spoiler role on the hydrogen in the ventilation duct. The velocity is rather low at the bottom of the stator slot. The calculated results of the stator temperature of the rated operation condition is shown in Fig. 12 (c). The highest temperature is 90.3 • C, which appears on the stator teeth. Due to the windings are cooled by water, its temperature is not higher than that of the stator VOLUME 8, 2020  teeth. The strand max. temperature is 77.4 • C that appears on the top solid strand. Fig. 12 (d) gives the thermal sensor position at the layer insulation of the stator winding. The termal sensors, which are marked from #A to #F, are set and evenly distributed. Due to the fluid velocity being lower at the central part in the duct, the temperature of yoke is high at the extended position of slot central line than that of the two side as shown in Fig. 12(c). In order to illustrate the relationship, the temperature distribution on different trajectories of the yoke part is shown in Fig. 13.  In order to analyze the temperature distribution in the winding strands, Fig. 14 (a) and (b) give the temperature distribution of the upper and lower windings under rated operation condition.
Although the heat source of the hollow strands are higher than that of the solid strands due to the high eddy current density in the hollow strands, as the hollow strands are watercooled, however their temperature are much lower than that of the solid ones. The winding temperature at the position of the stator core inside and the ventilation duct inside are shown in Fig. 14 (c) and (d).
To verify the solution results, the temperature in the layer insulation and the outlet of the water pipe were measured under rated load condition. The temperature testing method is the same as the one referenced in [21]. The comparison of the calculated and test results of the temperature are listed in Table 2. The calculated temperature results of the 1000MW generator that is considered the strand's REC well match the test data.
In order to reduce the REC, a new winding configuration shown in Fig. 15 is discussed. Different from the orignial structure, the new structure of the winding is with the hollow and solid strands being staggered. For the new assemblies, the maximum REC of the upper solid strand is 1.37, which is 1.42 for the orignial structures. The maximum REC of the upper hollow strand is 3.03, which is 3.1 for the orignial structures. The average REC of the up strands is 1.567, which is 1.672 for the orignial structures. The average REC of the down strands is 1.085, which is 1.167 for the orignial structures. Compared with the orignial structure, the winding temperature could be reduced by 1.6 • C with staggered structure for solid and hollow strands.
It can be seen that the REC is reduced by adopting the staggered struture. However, the slot construction and the dimension of each strand are not changed, which doesn't increase the manufacturing cost of the generator.

IV. CONCLUSION
1. For large turbo-generators, due to the stator upper and lower windings with different structures, the REC does not monotonically increase from the bottom to the top of the slot. There is an inflection point from the top strand of the lower winding to the bottom strand of the upper winding.
2. The average additional winding loss of the upper one is seven times that in the lower one when their total areas are the same. The average REC is directly proportion to the total height and strand area, and is inversely proportional to the slot width. It is irrelevant to the winding current value.
3. The 1000MW power turbo-generator windings are water-cooled from the inside, which results in the winding temperature being lower in the stator part. The maximum temperature appears on the stator teeth.
4. Due to the high eddy current density in the hollow strands, the heat source of the hollow strands are higher than that of the solid strands. Meanwhile, as the hollow strands are water-cooled, the temperature of the hollow strands are much lower than that of the solid ones.