The rotor configuration of the brushless doubly fed induction generator (BDFIG) plays an important role in its performance. In order to make the magnetomotive force (MMF) space vector in one set rotor windings to couple both magnetic fields with different polepair and have low resistance and inductance, this paper presents a novel wound rotor type for BDFIG with low space harmonic contents. In accordance with the principles of slot MMF harmonics and unequal element coils, this novel rotor winding is designed to be composed of threelayer unequalpitch unequalturn coils. The optimal design process and rules are given in detail with an example. The performance of a 700kW 2/4 polepair prototype with the proposed wound rotor is analyzed by the finite element simulation and experimental test, which are also carried out to verify the effectiveness of the proposed wound rotor configuration.
1. Introduction
Brushless doublyfed induction generator (BDFIG), regarding to be an alternative of doublyfed induction generator (DFIG), has gained broad appeal to be ideal future generators in wind turbines. The advantages of the BDFIGbased wind turbines includes eliminating brush and slip rings, lower operation and maintenance fees, taking fractional rated capacity of converters and superior crowbarless fault ridethrough (FRT) capability
[1

3]
. Owing to these clear superiorities, the commercial potential of BDFIG is enormous.
In order to couple both of the two magnetic fields between two stator windings in BDFIG, some configurations of specific rotor winding types have been proposed in the literatures. A cagetype rotor of the BDFIG, which is known as the “nestedloop”, was originally proposed and investigated by Broadway and Burbridge
[4]
. To reduce the rotor spatial harmonics, a novel cage rotor configuration comprising loops connected in series was presented in
[5]
, and it was shown that the rotor harmonics have a direct impact on the referred rotor leakage reactance and the effective performance of the machine. Compare with the conventional wound rotor windings, large cagetype rotor winding designs are deemed to create higher space harmonic contents. A comparison analysis of nestedloop rotor windings and seriesloop wound rotor winding was presented in
[6]
. Focus on the electrical and manufacturing, it shows the latter may be preferable to large machines, but they did not propose methods to reduce harmonic content which cannot be ignored for BDFIG.
The purpose of this paper is to present the general method to design a wound rotor winding for BDFIG. Moreover, the rules to reduce the harmonic content are also given. In this paper, a doublelayer unequal element coils wound rotor structure of the BDFIG is presented to reduce the harmonic contents and improve the winding factors. A detailed analysis and design principles of the proposed rotor winding is given in section 2. Thereafter, an analytical model is given to calculate the proposed unequal element coil in section 3. Optimization and selection of winding elements is implemented base on this model. On the basis of previous analysis, both simulations and experimental results are presented to illustrate the performance of a prototype with the proposed rotor winding type in section 4. Experimental testes were implemented on a D560 framesize 700kW 2/4 polepair wound rotor BDFIG.
2. Winding Design
The stator of BDFIG has two independent windings. In general, stator winding 1 acts as an electrical terminal for directly generating power and is therefore named as the power winding (PW). For another, stator winding 2, named as the control winding (CW), acts as an electrical terminal to connect with a variable voltage variable frequency converter which owns only a partial rated power capacity. These two stator windings can be distinguish from each other in pole pairs for generating two fundamental magnetic fields. The number
p
_{1}
and
p
_{2}
represent the polepair of PW and CW respectively. A BDFIGbased wind turbine system is shown in
Fig. 1
.
Wind turbine system based on BDFIG
The rotor of the BDFIG is required to be designed to couple both of the two magnetic fields,
p
_{1}
and
p
_{2}
, which were generated by the two windings in the stator. However, the other order of harmonic contents in rotor winding (RW) should be as little as possible. Due to any order of harmonic except for
p
_{1}
and
p
_{2}
will not induce voltage in the stator, from the BDFIG standpoint, it manifests as harmonic reactance.
Approximately, the relation of the air gap peak flux densities for two fundamental,
B
_{δ1}
and
B
_{δ2}
, is given by
[4]
where
k
_{Nr1}
and
k
_{Nr2}
are the winding factors for two fundamental in RW. This is a critical operation rule of BDFIG and it indicates that the winding factors of two fundamental,
p
_{1}
and
p
_{2}
, have a inherent relation with the relevant two air gap flux densities, therefore, the design goal of the RW is to achieve high winding factors for two fundamentals and low winding factors for other harmonics.
With the above conclusion in mind, we should set constraints for the design optimizations of the RW firstly. They are: (1) the winding factors of
p
_{1}
and
p
_{2}
polepair both should be above 0.7; (2) the resultant MMF percent of the highest harmonic content should be kept less than 3%. These constraints are obtained considering the performance of the initial design models and the previous prototypes.
 2.1 Theory of slot MMF harmonics
In practical induction machines, the coils of the windings are always placed in slots along with the air gap. Unless the winding distribution is sinusoidal, under the action of currents, the polyphase windings will create a fundamental MMF with a wide range of harmonics. The harmonic contents with
ν
=
kZ/p
±1=2
kmq
±1 (
k
=1, 2, 3,...) orders are the slot MMF harmonics, and
Z
is the number of slots,
m
is the number of the phases,
p
and
q
is the number of polepair and slots per pole per phase respectively. All slot MMF harmonics have the same distribution factor with the fundamental, so they may not be destroyed
[7]
.
In the following analysis, it must to clarify that only space harmonics on account of distributed winding are included, while permeance harmonics on account of slotting effects will be intentionally ignored. In practice, the geometrical profile of the air gap, rather than the size and arrangement of the coils, determines the permeance harmonics. Based on the same consideration, harmonics due to magnetic field saturation effect are also ignored.
For the
v
th order slot MMF harmonic,
where
f
is the 3phase MMF generated by the
A
,
B
 and
C
phase currents,
θ
is the angular displacement,
ω
is the angular speed and
t
is the time.
When
ν
=1, the resultant MMF of the fundamental is,
When
ν
=2
mq
1, the resultant MMF of
ν
th harmonic is,
Comparing (3) and (4), it is shown that the rotating direction of
ν
=2
mq
1th slot MMF harmonic is reverse with that of the fundamental. As shown in
Fig. 2
, voltage phasor diagrams for threephase symmetrical winding with
Z
=6,
p
_{1}
=2 and
p
_{2}
=4 are given. The first order slot MMF harmonic for
p
_{1}
polepair is
v
= 2, which is also the fundamental component for
p
_{2}
polepair. Meanwhile, the first order slot MMF harmonic for
p
_{2}
polepair is
ν
=0.5, which is also the fundamental component for
p
_{1}
polepair.
Voltage phasor diagrams for (a) Z_{r}=6, p_{1}=2 and (b) Z_{r}=6, p_{2}=4
Summarize the analysis above: (1) The MMFs of
p
_{1}
polepair and
p
_{2}
polepair appear simultaneously; (2) the MMFs rotating directions of
p
_{1}
polepair and
p
_{2}
polepair are reverse; (3) The winding factors of
p
_{1}
polepair and
p
_{2}
polepair are the same.
In consideration of the basic operation rule of the BDFIG, the RW should have the ability to generate two opposite rotating MMFs which also differ in polepair. By using the principle of slot MMF harmonic, the design of the RW can satisfy the rotor structure requirement of the BDFIG and the number of the rotor slots should be choose as,
 2.2 Method of slotnumber phasor diagram
The phase distribution of the induced electromotive force (EMF) for each coil sides is illustrated with a voltage phasor diagram in
Fig. 2
, which is presented in electrical degrees and is an effective tool to analysis winding
[8]
. However, it is cumbersome to draw the voltage phasor diagram with a large number of slots. In this paper, a new analysis method, slotnumber phasor diagram, is adapted to explain the design process of the RW in BDFIG. The slotnumber phasor diagram develops as an imaginary process of “cutting” and “unrolling” the rotary voltage phasor diagram to a linear counterpart.
The threephase coils for the RW with
Z
=6,
p
_{1}
=2 and
p
_{2}
=4, which mentioned above, can be found from the slotnumber phasor diagrams as
Fig. 3
shown. The slot numbers demonstrate the space vectors of the EMFs or MMFs produced by the coils (or the upper layer coils) inserted in the slots, the sign “–” in front of the slot numbers demonstrate the reverse polarity coils (or the upper layer coils) in 360 electric degrees.
Slotnumber phasor diagrams for (a) Z_{r}=6, p_{1}=2 and ) Z_{r}=6, p_{2}=4
 2.3 Slot division and discard
According to (5), when
p
_{1}
=2 and
p
_{2}
=4, the number of the rotor slots
Z_{r}
should be chosen as 6 (named as Case 1), however, an amount of MMFs harmonic contents except for that of
p
_{1}
polepair and
p
_{2}
polepair still exists. Since less rotor slots leads to a relatively high referred rotor harmonic leakage inductance, consequently, next step should be taken to effectively expand the number of slots. In order to maintain the MMFs of
p
_{1}
polepair and
p
_{2}
polepair appear simultaneously, an integral multiple of
Z_{r}
could be adopted and expressed as following,
where
k
is a positive integer. The number of the rotor phase
m_{r}
should be deliberately designed to (
p
_{1}
+
p
_{2}
)/
m_{k}
, where
m_{r}
and
m_{k}
also should be positive integers. The slotnumber phasor diagram of
Z_{r}
=84 and
p
_{1}
=2 is plotted as shown in
Fig. 4
, make a comparison with
Fig. 3
, the number of the rotor phase is 6, the slot number in each phase seems to be divided to 14 (named as Case 2).
Slotnumber phasor diagram for describing slot division (Z_{r}=84, p_{1}=2)
After slot division, we should discard some usefulness coils in the next step. Taking phase A1 for example, firstly, we should discard some coils which exceed 180 electrical degrees in the slotnumber phasor diagram. As shown in
Fig. 5
, slot 12, 13 and 14 exceed the boundary of the 180 electrical degrees in the slotnumber phasor diagram with
p
_{2}
polepair (The winding arrangement after this step was named as Case 3). Secondly, in order to improve the winding factor of the two fundamentals, slotnumber 9 and 10 can be discarded. At this moment, the slotnumber of phase A1 is constituted of 1, 2, 3, 4, 5, 6, 7, 8 and 9 (named as Case 4).
Slotnumber phasor diagram for describing slot discard (Z_{r}=84, p_{2}=4)
The proportional relation of the fundamental resultant MMF and
ν
th harmonic resultant MMF that created by
m_{r}
phase symmetrical currents can be expressed as
[7]
,
where
p_{v}
and
k_{Nv}
is the polepair and winding factor for
ν
th harmonic, respectively. To analyze the above cases, the winding factors and the resultant MMFs calculated from (7) are shown in
Table 1
, where signs “+” and “–” represent the opposite rotation of the resultant MMFs in forward direction and reverse direction. Make a comparison with Case 1 to Case 4, the effect of the above design processes is obvious. Case 4 can basically satisfy the design requirements of RW as shown in
Table 1
, but at the same time some harmonic contents are still high and will affect the performance.
Winding harmonic analysis of Case 1~Case 4
Winding harmonic analysis of Case 1~Case 4
3. Optimal Design by Unequal Element Coils
In this section, to guarantee the better performance of the machine, the winding arrangement was optimal design by unequalpitch unequalturn coils, which is also called unequal element coils as
Fig. 5
shown. The goal of the optimization is to make the space harmonic content as less as possible and the MMF space vector of a phase winding along with the air gap close to sinusoidal distribution by choosing and regulating the coil turns and coil pitches reasonably. An analytical model is given hereinafter to calculate and choose the unequal element coils.
For a general winding with unequal element coils, the number of coils with
N
_{1}
turns is
c
_{1}
, the number of coils with
N
_{2}
turns is
c
_{2}
, the vector sum of all the coil electrical potentials for phase A1 can be expressed as,
where
y
_{1}
and
y
_{2}
is the short pitch of the two element coils, respectively;
c
_{1}
and
c
_{2}
is the number of upper conductors for the two element coils,
c
is the pitch between them, respectively;
v
is the harmonic order;
α
is the slot pitch, which is 2
π
/84 in this example. The complex numbers are used to represent the phase differences. The winding factor of the
v
th harmonic of the unequal element coils can be express as,
Using the introduced winding factor, the harmonic spectra of the MMF of unequal element coils can be investigated.
The pole pitches of
p
_{1}
polepair and
p
_{2}
polepair are,
To enlarge the winding factors of
p
_{1}
polepair and
p
_{2}
polepair and lessen the winding factors of the other harmonic,
y
_{1}
and
y
_{2}
should be set in the range of τ
_{1}
and τ>
_{2}
, pitch
y
_{1}
was set to 12. Meanwhile, the size of the slot should be considered, the layer of the RW has better to be limited to threelayer, therefore pitch
y
_{2}
was set to be 10.
The coil turns of the two element coils,
N
_{1}
and
N
_{2}
, are needed to be calculated and regulated by using the design program according to (8) and (9). In this paper, the turns of the two element coils were set to be 4 and 3. The winding arrangement of phase A1 was describe in
Fig. 5
(named as Case 5). In the slotnumber phasor diagram, when the upper and lower conductors were both represented, the MMF produced by the unequal element coils was a good match for quasisinusolidal distribution as
Fig. 6
shown. The comparison of winding factors and the resultant MMFs before and after optimization was shown in
Table 2
, Case 5 satisfies the design requirements and had been chosen to implement in the prototype.
Winding arrangement for unequal element coils of phase A1
Winding harmonic analysis of Case 4 and Case 5
Winding harmonic analysis of Case 4 and Case 5
4. Results and Discussion
The winding configuration proposed and analyzed hereinbefore is applied to design of a 700kW 2/4 polepair wound rotor BDFIG. The crosssection of the rotor type is shown in
Fig. 7
, which will be used to demonstrate the performance with a finite element (FE) method.
Slotnumber phasor diagram for unequal element coils of phase A1 (Z_{r}=84, p_{2}=4)
To calculate and analyze the air gap flux density in the proposed RW structure, a magnetostatic FE simulation is built firstly to present a noload operating condition. Only the threephase RW are energized with the currents:
The obtained air gap density with the corresponding harmonics spectrum is provided in
Fig. 8
. There is a good agreement between the space harmonic spectra from magnetostatic FE simulation and that from the predictions in
Table 2
.
BDFIG crosssection with proposed rotor winding. Different colors are used to distinguish rotor threephase
A 2D timeharmonic field application analysis is carried out later to model the BDFIG and study the air gap magnetic fields. The design parameters and main dimensions of the prototype are listed in
Table 3
. The flux lines of the generator are shown in
Fig. 9
. In line with (1), the flux density of 4 polepair is nearly twice as large as that of 2 polepair. It is predicted that the magnetic fields of 4 and 2 polepair are induced in the RW simultaneously to ensure that the magnetic fields generated by two stator windings are indirect coupling.
Diagrams obtained from magnetostatic FE simulation (a) Airgap flux density (b) relevant space harmonic spectra
Main machine dimensions
Flux distributions in timeharmonic FE simulation of the prototype BDFIG
It is worth comprising the electromagnetic performance of each winding configurations.
Fig. 11
compare the values of the predicted flux linkage for the winding configurations, it is clear that Case 5 has the maximum flux linkage, this means that the rotor winding of Case 5 obtain more cross coupling than other cases. The torque curves for the four winding configurations are presented in
Fig. 12
. It can be seen that Case 5 has the highest value of output torque and the lowest value of ripple torque.
Diagrams obtained from timeharmonic FE simulation: (a) Air gap flux density; (b) relevant space harmonic spectra
Comparison of the rotor winding flux linkage for each winding configurations
The experiments have been implemented on a 700kW prototype BDFIG as
Fig. 14
shown. The CW is supplied with a threephase converter, while the PW is in open circuit situation and measured by a voltmeter. The PW induced voltage was kept at 450Vrms/60Hz, the CW excited voltage was regulated with the rotor speed and PW voltage. The FE calculated and experimental measured CW voltage versus rotating speed for the prototype BDFIG is presented in
Fig. 13
. It is evident that, intermediate the stator two windings, the optimized winding arrangement (Case 5) owns a more sufficient ability for creating cross coupling.
Comparison of torque curves for each winding configurations (T_{avg} is the average torque and T_{rip} is the peak to peak value of the torque ripple divided by T_{avg})
CW excited voltage against rotor speed (the induced voltage of the open circuited PW is 450Vrms, 60Hz)
Pictures of the proposed wound rotor BDFIG
Before the proposed wound rotor type BDFIG construction, the most widely used type is the nestedloop cage rotor BDFIG which makes a great contribution to BDFIG analysis
[2

6]
. The nestedloop cage rotor contains multiple loops and can be regarded as a concentrated winding, thus, it is difficult to obtain high winding factors for both two pole pairs. Under the conditions of the consistent rotor slot area, the consistent number of rotor conductors and excited by the same value current, compare with the nestedloop cage rotor, the proposed wound rotor BDFIG will obtain higher copper usage efficiency due to the higher winding factors of the two pole pairs.
Another issue is the copper loss. Compared with the same size BDFIG with nestedloop cage rotor, the higher value of the referred rotor resistance for wound rotor may leads to higher rotor copper losses
[7]
. Nonetheless, large reduction of the rotor harmonics and the referred rotor leakage inductance are more effective in improving performance of the generator and counteracts the negative influence of the increased rotor copper loss.
From the viewpoint of manufacture and generator performance, the specially fabricated nestedloop cage rotor is made of copper alloys and brass, which needs more manufacturing complexity and higher cost, and this special welded construction also needs safety evaluation for wind turbines. In addition, the bars of the nestedloop cage rotor must be insulated which is hard to cast. However, the manufacturing process required for the proposed wound rotor type BDFIG is as straightforward as the traditional wound rotor DFIG design. It dose not change much, except for needing two types of coils with different number of turns. Therefore, upgrading the production line from DFIG to wound rotor type BDFIG is very convenient for wind turbines manufacturers.
5. Conclusion
The space harmonics of air gap flux density has a strong impact on the performance of a BDFIG. Based on slot MMF harmonic principle and unequal element coils, a low space harmonic content wound rotor type BDFIG is described for the first time in this paper. The proposed wound rotor with unequal element coils shows a good performance and the manufacturing process required for the proposed wound rotor BDFIG construction is similar with the traditional wound rotor DFIG design. Hence, this research has significant meaning in advancing the commercial use of BDFIG in wind turbines.
Acknowledgements
This work was supported by the China National Key Technology Support Program (No. 2012BAG03B01).
BIO
Xin Chen He was born in Jingzhou, China, on February 17, 1989. He received the Bachelor’s degree from Huazhong University of Science and Technology (HUST), in 2010. He is presently pursuing the Ph.D. degree at HUST. His researches include wound rotor brushless doublyfed machine design and control.
Xuefan Wang He was born in Wuhan, China, on December 26, 1954. He received the Ph.D. degree in electrical engineering from Huazhong University of Science and Technology, China, in 1989. He is presently a Professor in the same university. His research interests include electrical machines and power electronics, particularly for wound rotor brushless doublyfed machine design and application.
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