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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 I1 Rα j X α I1e j R2 jX 2 j120 U B I1e Rα j X α j ( 120 ) R2 jX 2 I1e 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 References 14 [1] Christian D, Dirk D, Jan H. 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Generators merging network conditions accounting 20 method based on the positive sequence fundamental voltage synthetic phasor. 21 Proceedings of the CSEE 2006; 16: 52-56. 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 Traditional 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 1.8110 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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