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```1
Research on magnetic field state analysis of non-salient pole synchronous
2
generator
3
Baojun GE, Pin LV*, Dajun TAO, Fang XIAO, Hongsen ZHAO
4
College of Electrical and Electronic Engineering, Harbin University of Science and
5
Technology, Harbin, China
6
*Correspondence: [email protected]
7
Abstract: This paper is based on the definition of minor asymmetric degree and the
8
relative position of the magnetic field. The relative position of the magnetic field is the
9
angle between the positive sequence magnetic field and the negative sequence magnetic
10
field. When the generator operates at minor asymmetry, the corresponding relationship
11
between the non-salient pole synchronous generator steady performance and the
12
magnetic field state has been discussed. To analyze this corresponding relationship, the
13
Taylor expansion formula with Lagrange remainder and symmetrical components has
14
been adopted. Then we study the relationships between the armature current and the
15
relative position of the magnetic field, deducing the armature current expressions with
16
second order remainders. The law of the armature voltage is also obtained by phasor
17
analysis. As an example, this paper executes two-dimensional finite element numerical
18
calculation on a non-salient pole synchronous generator. The result shows numerical
19
calculation agrees with the deduced expression, which verifies the correctness of the
20
expressions. In the meantime, during the process of deducing the expressions, a method
21
of calculating the positive sequence current and the negative sequence current is
22
proposed. Moreover, their effective range has been offered. When the generator works
23
at maximum over-voltage and maximum under-voltage, the specific value of the relative
1
1
position of the magnetic field has been presented. This paper has important implications
2
for the generator running and protection.
3
Key words: minor degree of asymmetry, the relative position of the magnetic field,
4
non-salient pole synchronous generator, the positive sequence current, the negative
5
sequence current
6
1.
7
The non-salient pole synchronous generator is important power supply equipment in the
8
power system. When the generator works with asymmetric load, the armature voltage
9
and the armature current become unbalanced. Normally, the finite element numerical
10
calculation method has been adopted to solve these asymmetric problems [1-4]. The
11
finite element method has been employed to analyze the magnetic field state and the
12
magnetic field distortion [5-8]. However, the theoretical analysis linked with the
13
magnetic field state has limited to symmetrical state [9-11]. The magnetic field
14
distortion in the DC machine and AC machine has been discussed [12-14]. Then, the
15
magnetic field state under asymmetrical condition is investigated in this paper. The
16
magnetic field has been divided into the positive sequence magnetic field and the
17
negative sequence magnetic field [15-18]. From the former documents, only the
18
symmetrical magnetic field state has been investigated. The theoretical method in this
19
paper expands the scope of the theoretical analysis of the magnetic field. It has
20
theoretical innovation, and it can also provide valuable experience for generator
21
protection.
22
Based on research about the magnetic field state in this paper, a new method for the
23
stator positive sequence current and the stator negative sequence current is proposed.
24
Even the error of this method is given by precise numerical value. Compared with the
Introduction
2
1
old methods, which need large scales of power electronic, this new method can save
2
money and easily accessible [19-23]. Moreover, regarding the error, in the low
3
asymmetrical degree, the method discussed by this paper has more accurate value.
4
However, in the high asymmetrical degree the former method has more accurate value.
5
The over-voltage and the under-voltage fault in the generator have produced
6
tremendous harm in the power system [24-26]. It is necessary to know the specific value
7
of the relative position of the magnetic field at the maximum over-voltage and
8
maximum under-voltage and the law of the armature voltage distortion THDU. Through
9
the magnetic field state analysis, the specific value of the relative position of the
10
magnetic field at the maximum over-voltage and maximum under-voltage are also
11
presented in this paper. With the asymmetrical current degree increasing, the law of the
12
armature voltage distortion THDU is calculated by the finite element method in the
13
paper, too. This paper takes a non-salient pole synchronous generator as an example,
14
and we execute two-dimensional finite element numerical computation. The result of
15
the numerical computation is compared with the result of the analytic algorithm that
16
could verify the correctness of the analytic method. At the same time, a few beneficial
17
conclusions are reached.
18
2.
The analysis of minor asymmetry and relative position of the magnetic field
19
2.1.
The definition of the minor asymmetry
20
Under asymmetric operation, the running states of the generator can be divided into the
21
fault condition and the non-fault condition. Generally speaking, a minor asymmetric
22
state of the generator, namely, non-fault unbalance condition, is decided by three factors.
23
Firstly, the negative current is below 8% of the rated current. Secondly, the deviation
24
between the positive current and the rated current is below 10% of the rated current.
3
1
Finally, the phase deviation between the positive current and the rated current is below
2
5 electrical angular degrees. It is shown in formula (1)
 I2
 I  8%
 N
 I1  I N
 10%

 IN
     5
n


3
(1)
4
Here, I2 is the effective value of the negative current, I1 is the effective value of the
5
positive current, IN is the effective value of the rated current, φ is the power factor of the
6
positive network, and φn is the rated power factor of the positive sequence network.
7
2.2.
8
2.2.1.
9
When the three-phase non-salient synchronous generator operating under symmetrical
10
state and A phase current being at maximum, instantaneous magneto-motive force
11
vectors can been viewed in Figure 1.
12
Here, FA1 、FB1 、FC1 is the positive magneto-motive force induced by the positive
13
current of the phase-A, phase-B and phase-C, and FA1、FB1、FC1 is the negative
14
magneto-motive force induced by the positive sequence current of the phase-A, phase-B
15
and phase-C. According to the relationships between the positive magneto-motive force
16
and the negative magneto-motive force, they can be given by
The analysis for the relative position of the magnetic field
The analysis under symmetrical state
4
 FA1  FB1  FC1  FA1  FB1  FC1

  Fi   Fi   3 F 1 cos    t 
i=A1,B1,C1
2
i=A1,B1,C1

 3 4 2 Nkdp1
I1 cos    t 

 2 2 p


F
 i  0
i=A1,B1,C1
1
(2)
2
Here, F 1 is the phase-fundamental magneto-motive force of the positive sequence, p is
3
the number of the pole pairs, and  is displayed in Figure 2, which is the electrical
4
angle between the winding axis of the phase A and the complex magnetic field
5
produced by the positive currents.
6
2.2.2.
7
When the current of the phase-A is at maximum, the angle between the complex
8
positive magneto-motive force and the complex negative magneto-motive force is
9
regarded as the relative position of the magnetic field, which is the angle between the
10
positive magnetic field and the negative magnetic field. It is illustrated in Figure 3 by χ.
11
The relationships between the positive magnetic field amplitude and the negative
12
magnetic field amplitude can be described by
The analysis under asymmetry state
5

 FA1  FB1  FC1  FA1  FB1  FC1

3
  Fi   Fi   F 1 cos    t 
2
i=A1,B1,C1
i=A1,B1,C1

 3 4 2 Nkdp1 I cos    t 
 2 2 p 1

 FA2  FB2  FC2  FA2  FB2  FC2

3
  Fi   Fi  F  2 cos      t 
2
i=A2,B2,C2
i=A2,B2,C2

 3 4 2 Nkdp1 I cos      t 
 2 2 p 2



  Fi  Fi   0
i=A2,B2,C2
i=A1,B1,C1
1
(3)
2
Here, FA2 、FB2 、FC2 is the positive magneto-motive force induced by the negative
3
current of the phase-A, phase-B and phase-C, and FA2、FB2、FC2 is the negative
4
magneto-motive force induced by the negative current of the phase-A, phase-B and
5
phase-C. I2 is the effective value of the negative current, and F  2 is the phase-
6
fundamental magneto-motive force of the negative sequence.   is the instantaneous
7
angle between the complex positive magneto-motive force vector and the complex
8
negative magneto-motive force vector, which changes over the time.
9
3.
10
Applying analytic method to analyze the influence of the relative position of
the magnetic field on the generator performance
11
3.1.
12
Define the asymmetrical degree of the current as   I 2 / I1 . Based on the superposition
13
formula of the same frequency sinusoidal quantity, the fundamental sinusoidal effective
14
value of the phase-A can be given by
15
The expressions of the armature current
g A (  )  I12  I 2 2  2 I1 I 2 cos(  )
(4)
6
1
In order to simply the process of derivation, make variable substitution for the
2
formula (4) above that  equals with cos(  ) , hence, the function is exhibited as
3
g A ( )  I 21  I 2 2  2 I1I 2
2
1
2
2
1
2
1
(5)
2
 I   I  2 I   I1 1    2
4
At the point of  ω   / 2 , Taylor formula with Lagrange remainder term expansion in
5
formula (5) can be shown by
g A ( )   II
2
1
6


 (  )
 2 (  ) 2
2
2
 I1 
I1 
I   II
3 1
2
1    2 ω
2(1   2  2 ) 2
2
1

 (  )2
2
(6)
2
 I1   ( 

1
) I1 
2
2
(1   2  2 )
I1   II
2
3
2
1
7
Among them, the value of   is between  / 2 and  .The formula (5) could be
8
approximately represented by
9
g A ( )  I1   I1
(7)
10
As the asymmetrical degree turns smaller, the substitution of formula (7) for formula (5)
11
will be more accurate. It is approximately showed as sinusoidal function by
I
12
A
IB
I C    I1
I2 
1
1
 1



 cos(  ) cos(   4  ) cos(   4  ) 
3
3 

1
1
 1



 I1 1  
 cos(  ) cos(   4  ) cos(   4  ) 
3
3 

13
Apply the formula (8) to compute the positive current and the negative current that I1
14
and I2 can be given by
(8)
7
IA  IB  IC

 I1 
3


2
2
2
2
 I  I A  I B  I C  3I1
 2
3
1
(9)
2
3.2.
3
Taking no account of the magnetic saturation, the rotating generator can be divided
4
into three systems, including the positive sequence system, the negative sequence
5
system and the zero sequence system.The parameters of the generator are shown in
6
Table 1.
7
Assume   R1 / Rα and   X 1 / X α . R1 is the external resistance of the positive
8
sequence network, Rα is the rated resistance, X 1 is the external reactance of the
9
positive sequence network, and X α is the rated reactance. According to the
10
The phasor expressions of the armature voltage
relationships between the three sequence networks, the voltages can be displayed by
U A  I1   Rα  j X α    I1e j   R2  jX 2 

 j120
U B  I1e
 Rα  j X α  

j (  120 )
 R2  jX 2 
 I1e

 j120   R  j X  
α
α
U C  I1e
  j (  120 )
 R2  jX 2 
 I1e
0
0
11
(10)
0
0
12
Here, R2 is the external resistance of the negative sequence network, and X 2 is the
13
external reactance of the negative sequence network. Assume the angle between the
14
positive sequence voltage and the negative sequence voltage of the phase-A, phase-B
15
and phase-C as ε1、ε2、ε3，therefore, the voltage of the phase-A, phase-B and phase-C
16
is exhibited by
8
U A  (U1  U 2 cos 1 ) 2  (U 2 sin  1 ) 2


2
2
U B  (U1  U 2 cos  2 )  (U 2 sin  2 )

2
2
U C  (U1  U 2 cos  3 )  (U 2 sin  3 )
1
(11)
2
Among them, 1       ，  2        120 ，  3        120 ，U1 is the
3
effective value of the positive voltage, U2 is the effective value of the negative voltage,
4
and  is the power factor of the negative sequence network. Providing the formula (10)
5
and the formula (11), we could achieve that the angle between the maximum
6
fundamental wave voltage and the minimum fundamental wave voltage is 180 electrical
7
degree in every phase. When the fundamental wave voltage of each phase is at its
8
maximum, the corresponding  A 、  B 、  C is shown as
A    



  B   A  120 ;  C   A  120
9
(12)
10
11
4.
Finite element simulation
12
4.1.
The building of the simulation model
13
When we build the model of the generator, the end effect of the magnetic field has been
14
ignored, moreover, the current density J and the magnetic vector potential A only have
15
component along the generator axis direction. It is supposed that the axis direction is
16
along the axis-Z. In the Cartesian coordinate system, the current density J and the
17
magnetic vector potential A is the function of (x,y). To simplify the process of the
18
computation, the two-dimension finite model for the magnetic field, which is the round-
19
section of the generator, has been established. Assume as follows that:
9
1
2
1) The material of the generator is isotropic, the magnetic permeability of the stator
core and the rotor core is constant, and the generator operates at unsaturated state.
3
2) Whereas, the frequency of the generator is relatively low, furthermore, the
4
displacement current is so tiny that compares with the conduction current, hence, the
5
impact of the displacement current can be neglected, namely,
D
=0.
t
6
3) The magnetic flux leakage does not exist outside the stator circle.
7
4) When the magnetic field changes or the temperature varies, the resistance R
8
keeps the same value, and the conditions (13) below has been satisfied in the solution
9
domain of the generator magnetic field.
10

  AZ    AZ 
 :  
  
  Jz
 x  x  y  y 
 : A  0

z
(13)
11
Here,  is the magnetic reluctivity of the material, A is the magnetic vector, and J z is
12
the current density.  is the complete solution domain of the generator magnetic field,
13
the physical model of generator after subdivision and  are shown in Figure 4, and the
14
magnetic field lines of the rated-load is illustrated in Figure 5.
15
The equivalent circuit diagram of the generator is displayed in Figure 6. Through the
16
adjusting for the impedance of phase-A, phase-B and phase-C, the asymmetrical degree
17
 will be regulated.
18
Here, LD is the reactance of the armature, Lδ is the leakage reactance of the armature,
19
RA、RB、RC is separately the load-resistance of the phase-A, phase-B and phase-C, and
20
LA、LB、LC is respectively the load-reactance of the phase-A, phase-B and phase-C .
Z
10
1
When   1% ,   2% and   10% , the magnetic field flux density cloud map for the
2
generator are shown in Figure 7. The saturation characteristics of the stator and the rotor
3
are shown in Figure 8.
4
4.2.
5
The branches of the armature is two, namely, as  2 .When the asymmetrical degree is
6
1%，10%，20%，50%, the simulation waves and the analytic waves are displayed in
7
Figure 9, Figure 10, Figure 11 and Figure 12 respectively, which records the
8
relationships between the armature current and the relative position of the magnetic
9
field χ . The analytic waves are computed by formula (8).
Computation of the armature currents through finite element simulation
10
While we take the analytic calculation method, the errors of the analytic method
11
discussed in this paper and the traditional method are shown as Table 2. If the analytic
12
formula (8) is employed, as the asymmetrical increases, the error grows greater. While
13
  6% , precision requirements in engineering will not be met. It turns out that when
14
applying the formula (9) to calculate I1 、I2 , the effect of the minor asymmetrical
15
degree is superior to larger ones. In low asymmetrical degrees, the precision of the old
16
method has priority over that of the new method. However, the precision of the new
17
method has priority over that of the traditional method in high asymmetrical degrees.
18
Relying on precision requirements in engineering, the effective range of this method is
19
  [0,6%].
20
4.3.
21
Figure 12a and Figure 12b is the maximum voltage U 
22
element method, when  is 1%，2%，10% and the relative position of the magnetic
23
field is  =0o、30o、60o、90o、120o、150o、180o、210o、240o、270o、300o、330o.
Computation of the armature voltages through finite element simulation
MAX
calculated by the finite
11
1
Given that when harmonic order is above 9,U  MAX is so little, so   9 , the harmonic
2
waves are not shown in Figure 13a and Figure 13b.
3
With the asymmetrical degree going deeper, as shown in Figure 12a and Figure 12b, the
4
voltage distortion UD of the harmonic  , as well as the total voltage distortion THDU,
5
presents the gradual increment trend. When generator operating at balanced state, 
6
equals zero and THDU =1.45%. Because of the increasing asymmetrical degree  ,
7
THDU increases gradually. When  reaches 10%, THDU=8.77%. Therefore, especially
8
at minor asymmetrical degree, the influence of the harmonic voltage   2 on the
9
effective value can be ignored. So the interference of the harmonic voltages   2 can
10
be neglected at the minor asymmetrical degree. It is feasible to use formula (11) to
11
calculate the relative position of the magnetic field  at maximum over-voltage and
12
maximum under-voltage. Supposing X=A，B，C, define the load coefficient of the
13
generator as  x  rx / Rα and rx is taken as the load resistance in the corresponding
14
phase.
15
5
Conclusion
16
1) Based on Lagrange analysis method, this paper proposes a new method to show
17
the armature current. The result of the new analytic method is consistent with two-
18
dimensional finite element numerical calculation.
19
2) The changing laws of the armature current and the armature voltage on the
20
magnetic field state is analyzed. At minor asymmetrical degree of the current, with the
21
alteration of the relative position of the magnetic field  , the current turns sinusoidal
22
changing law. With the increasing  , the voltage distortion UD of  order harmonic
23
wave and total voltage distortion THDU tend to increase gradually.
12
1
3) Relying on the Lagrange analysis and formula derivations, we exhibit a novel
2
calculation method of the armature positive current I1 and the armature negative current
3
I2, moreover, their effective ranges have been given.
4
4) The simulation of the two-dimensional finite element method verifies the
5
correctness of the armature voltage phasor expressions that is deduced by this paper.
6
When the generator runs at maximum over-voltage fault and maximum under-voltage
7
fault, the three-phase relative positions of the magnetic field, namely,  A 、  B 、  C
8
can be acquired by the positive sequence power factor  and the negative sequence
9
power factor  , which is the evidence of the generator protection.
10
Acknowledgement
11
Project Supported by National Natural Science Foundation of China(51407050)；
12
National Science and Technology Major Project of China(2009ZX060040)．
13
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22
Figures:
16
  0
A
 F 
B1

FC1


FA1



FA1
1
B
  120

FC1


FB1

C
  240
2
Figure 1. Magneto-motive force vectors under symmetrical operation
3
when A phase current is at maximum

F 1

4
5
Figure 2. The spatial location of β
17
  0
   A
 FB1 FC1
FA1


FA1

1
2
3


FA2


FB2


FC2


F
A2
B
  120

FB2

FC2



FB1

FC1
C
  240
Figure 3. Magneto-motive force vector under asymmetrical operation
when the relative position of the magnetic field is χ

4
5
Figure 4. Physical model of the generator after the subdivision
18
1
2
Figure 5. Magnetic field lines of rated-load
LD  2.107 mH
Lδ  0.019181mH RA
LA
LD  2.107mH
Lδ  0.019181mH RB
LB
LD  2.107 mH
Lδ  0.019181mH RC
LC
3
4
Figure 6. Equivalent circuit diagram of the generator
5
19
  1%
  10%
  2%
1
Figure 7. Magnetic field flux density cloud map for the generator
2
20
2.00
B (tesla)
1.50
1.00
0.50
0.00
0.00E+000
1
5.00E+004
1.00E+005
H (A_per_meter)
1.50E+005
Figure 8a. Stator staturation charateristics
2
2.00
B (tesla)
1.50
1.00
0.50
0.00
3
4
0.00
5000.00
H (A_per_meter)
10000.00
14000.00
Figure 8b. Rotor staturation charateristics
21
Effective value of the current/A
Simulation
Analytic expression
34200
34100
34000
33900
33800
33700
33600
33500
0
2
3
4
5
6
The relative position of the magnetic field/rad
1
2
1
Figure 9. The curve of the current when the degree of asymmetry is 1%
Simulation
38000
Effective value of the current/A
Analytic expression
37000
36000
35000
34000
33000
32000
31000
30000
0
3
4
1
2
3
4
5
6
The relative position of the magnetic field/rad
Figure 10. The curve of the current when the degree of asymmetry is 10%
22
Simulation
Effective value of the current/A
42000
Analytic expression
40000
38000
36000
34000
32000
30000
28000
26000
0
2
3
4
5
6
The relative position of the magnetic field/rad
1
2
1
Figure 11. The curve of current when the degree of asymmetry is 20%
Simualtion
Effective value of the current/A
55000
Analytic expression
50000
45000
40000
35000
30000
25000
20000
15000
0
3
4
1
2
3
4
5
6
The relative position of the magnetic field/rad
Figure 12. The curve of current when the degree of asymmetry is 50%
23
1%
2%
10%
Amplitude of the voltage/kV
0.020
0.018
0.016
0.014
0.012
0.010
0.008
0.006
0.004
0.002
0.000
2
6
4
8
Harmonic order
1
2
Figure 13a. even harmonic U  MAX when the degree of asymmetry is
3
1%，2%，10%
1%
2%
10%
1.2
Amplitude of the voltage/kV
1.1
1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0.0
3
5
7
9
Harmonic order
4
5
Figure 13b. odd harmonic U  MAX when the degree of asymmetry is
6
1%，2%，10%
24
1
2
Tables:
Rated voltage/kV
24
Rated current/A
33847
Power factor
0.9
The stator winding connection
Y
Per-phase resistance of the stator winding/ 
0.00098
Negative sequence reactance(n on-saturation)/ 
0.14165
Zero sequence reactance/ 
0.07942
3
Table 1. Design parameters of the generator
4
5
The asymmetrical degree
Proposed method
(%)
(A)
(A)
1
1.6923 10
2
3.3847 10
0
2
1.709 10
3
1.726 10
3
25
3
6.7694 10
4
2
1.743 10
3
1.0154 10
3
1.76 10
5
1.3539 10
3
1.777 10
6
1.6923 10
3
1.794 10
7
2.0308 10
3
1.8110
8
2.3693  10
3
1.827 10
10
3.0462 10
3
1.8416 10
20
6.4309 10
3
2.03 10
50
1.6585 10
4
2.538 10
3
3
3
3
3
3
3
3
1
2
Table 2. Error about degree of asymmetry
26
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