Static and Dynamic Rotor Eccentricity On-Line Detection and Discrimination in Synchronous Generators By No-Load E.M.F. Space Vector Loci Analysis C. Bruzzese, A. Giordani, and E. Santini Department of Electrical Engineering, University of Rome “Sapienza” Via Sette Sale 12-B, 00184 Rome, (Italy)
[email protected],
[email protected],
[email protected] Abstract—In this paper a study about the different effects of static and dynamic rotor eccentricities on the external electric variables of a salient-pole synchronous generator is presented. Air-gap irregularities and related monitoring techniques have been studied in the past mainly about large hydro-generators (with practical applications), but similar problems have been recognized for on-board ship synchronous generators. These latter require a different approach, i.e. non-invasive monitoring. Static and dynamic rotor eccentricities were simulated in this work, for a shipapplication sized generator, by using a dynamic model including inductances computed by parametric 3D FEM analysis of the faulty machine. Current, voltage, and no-load e.m.f. steady-state waveforms were analyzed by FFT and space-vector approach. No-load e.m.f. space vector is a sensitive fault indicator since its amplitude largely increases with the level of absolute eccentricity; furthermore, it is possible to discriminate the static eccentricity from the dynamic eccentricity utilizing the space vector loci ovality. Index Terms—Finite element analisys, Matrix of inductances, Rotor air-gap eccentricity fault, Synchronous Generators.
I. INTRODUCTION In the electromechanical process of energy conversion synchronous generators can be subject to failures. This determines economic losses due to service interruption and machine damage. Electric machines can operate under asymmetrical conditions due to mechanical and/or electrical irregularities. These asymmetrical conditions can be summarized as follows: • Broken rotor bars or cracked end-rings. • Inter-turn fault resulting in the shorting or opening of one or more circuits of a stator phase winding. • Rotor eccentricities. A rub between the rotor and the stator core can result in serious damage to stator and rotor windings and cores. • Short in rotor field windings in the case of synchronous machines causing overheating which may also result in bending of the rotor. Classic research efforts in fault diagnosis include electrical, mechanical, chemical, and magnetic techniques
[1]. These techniques are the basis for developing on-line and/or off-line rotating electrical machine condition monitoring systems. Electrical and magnetic techniques include magnetic flux measurement, stator current/voltage analysis, rotor current/voltage analysis, partial discharges for evaluating stator insulation strength for high voltage motors, shaft induced voltages, etc. Mechanical techniques include machine bearing vibration monitoring, speed fluctuation analysis of induction machines and bearing temperature measurement. One of the chemical techniques used is the carbon monoxide gas analysis due to degradation of electric insulation for closed circuit air cooled motors with water cooled heat exchangers. Another chemical technique is the analysis of bearing oil. Among all these techniques, the MCSA (motor current signature analysis) has received large attention due to its straightforward features such as non-invasivity, easiness of measure (by using plant CT or ready-to-use clamp-on sensors), and continuous on-line operating capability. Fault signatures can be detected in phase voltages too, even if with more problems about voltage measuring. It is acknowledged [2] that about 60% of faults in electrical machines are caused by mechanical parts such as bearing, shaft and coupling. Nearly 80% of these faults results in the displacement of the axis of symmetry of the rotating axis of the rotor. Therefore, existing asymmetry between rotor and stator causes about 50% of industrial down-times. Many papers deal with mechanical abnormalities and particularly with eccentricity problems in induction machines and relatively fewer concern the same problem in synchronous machines. E.g., in [3] a combined time stepping FEM - FFT analysis method was proposed to study the functional dependence of current frequency components on static eccentricity level in induction motors. However, synchronous machines (SMs) can be affected by eccentricities problems as well. E.g., large vertical-axis hydro-generators [4,5] and other particular duty machines (such as ship generators) need particular attention to air-gap non-uniformity. Literature concerning SM rotor eccentricities subdivide between papers mainly devoted to industrial
gap measuring, detection practices, sensors and tool development [4,5,1], and papers involved with development of mathematical models suitable of faulty machine simulation and fault signature investigation. In the first category, paper [4] presents the development and application of a fibre-optics instrumentation system for accurate measurement and analysis of the dynamic air gap behaviour of large hydroelectric generators. The technique presented is very effective, but somewhat invasive due to sensor installation on the rotor. In paper [5], ai-gap measurement is achieved by measuring the modulated capacitive current between a passive sensor and the field poles. The measuring principle, the technique and the equipment used are discussed, as well as some case-histories. For the second category, in [6] a modified winding function approach (MWFA) was presented for calculation of machine winding inductances in a synchronous generator (SG) with dynamic rotor eccentricity. The same authors successfully applied this technique to faulty induction motor modelling [7]. In [6] it was shown that 17th and 19th stator current harmonics have the highest relative percentage increase. Papers [8], [9] use the MWFA for studying air gap eccentricity in SGs, too. In spite of these investigations, in the current literature it has not been yet set-up a criterion to detect and discriminate between different types of rotor eccentricity (i.e., static and dynamic), in particular by using MCSA or related techniques. Considering that the electrical machine inductances (in particular, the mutual stator-rotor parameters) depend on the geometry, a static eccentricity should produce an imbalance among the no-load e.m.f.s generated into the three phases and therefore an imbalance among the three currents. Differently, a pure dynamic eccentricity should modify the harmonic content of the three e.m.f.s and currents in the same way (anyone remaining equal to each other). A lot of importance in the study of electrical machines with asymmetries are the sub-harmonics. Generally, a static eccentricity should not produce sub-harmonics of e.m.f.s and currents (multiple of f/P, where f is the feeding frequency and P the polar pairs), and neither a dynamic pure eccentricity. Nevertheless, sub-harmonics should be recognized in the case of mixed eccentricity. So, by analysing the electrical quantities of the three phases, it is possible to go back to the kind of SG eccentricity. This paper presents possible ways for detecting and discriminating rotor static and dynamic SG rotor misalignments, by analysing the three no-load e.m.f. spectra, and by using the e.m.f. space vector analysis. SG analysis was performed by a MatLab script program, and the faulty inductances were computed by finite element method (FEM). The machine model was coupled with the load phase model, to obtain differential equations for the whole dynamic system, without neutral connection; then simulated inductances and no-load e.m.f.s were computed and analyzed by space-vector and FFT methods.
II. STATIC AND DYNAMIC ROTOR ECCENTRICITIES Rotor eccentricities can be classified as: 1) Static eccentricity when the rotor rotates around its geometric axis of symmetry (point R in Fig.1) which is distinct from the stator axis of symmetry (point S). Stator core ovality or incorrect axis alignment during manufacturing or installation can produce this fault. βm qm
R
dm C
αm
S
Figure 1. Geometric representation of rotor eccentricity
2) Pure dynamic eccentricity, when the rotor rotates around the point S but the rotor is out-of-centre (R is not on C). Bearing weariness, unbalanced magnetic pull (UMP), mechanical shaft resonances for critical speeds and bending, non-homogeneous rotor overheating, are possible causes. 3) Mixed eccentricity when the rotational centre C is distinct from both S and R. The mixed eccentricity (with both static and dynamic components) is the most general case. The minimum air-gap thickness has constant amplitude and fixed position in case of static eccentricity, it changes position (rotating) in case of dynamic eccentricity, and it changes both position and amplitude in case of mixed eccentricity. In this model, the geometrical rotation axis (point C) is considered fixed. Despite of this simplifying assumption, the model allows good correspondence with experimental data [6]. The fault can be characterized by four constant parameters, i.e. components of vectors SC (fixed in the stator frame) and RC (fixed in the rotor frame), as indicated in the equations (1), (2) and in Fig.2. qm
βm dm R C
S
αm
Fig. 2. Decomposition of the vectors SC and RC
SC = (α m , β m ) RC = (d m , qm )
(1)
&A ψ
RS
(2)
&B ψ
RS
&C ψ
RS
Note that frames (αm, βm) e (dm, qm) are defined as mechanical plans (not reduced to one polar pair). So, many kind of unbalanced condition can be represented.
RF VF
ψ& F
III. THE SG MATHEMATICAL MODEL AND LOAD The SG linear coupled-circuit electrical model is considered. With series-connected stator belts, three stator equations and one rotor equation are reported in (3) in matrix form.
[i ] = [i A
[ψ ] = [ψ A
iB
(4)
vF ]
T
(5)
iF ]
(6)
ψ B ψ C ψ F ]T
(7)
iC
T
Vectors [v], [i], [ψ] contain phase and excitation tensions, currents, and flux linkages; [R] and [L] are resistance and inductance matrices: ⎡ L AA ⎢L [ L] = ⎢ BA ⎢ LCA ⎢ ⎣ LFA
L AB LBB
L AC LBC
LCB
LCC
LFB
LFC
L AF ⎤ ⎡ RS ⎢0 LBF ⎥⎥ [ R] = ⎢ ⎢0 LCF ⎥ ⎥ ⎢ LFF ⎦ ⎣0
0
0
RS
0
0
RS
0
0
0⎤ 0 ⎥⎥ 0⎥ ⎥ RF ⎦
1 dω R TM − [i ]T G (θ )[i ] = J 2 dt
LL
By expanding terms in (3) we obtain (11). (11)
[v]R = −([ R]R + [G ] R ⋅ θ&) ⋅ [i ] R − [ L] R ⋅ p[i ] R
(9)
L L ⎤ ⎡i A ⎤ p 2 L L ⎥⎦ ⎢⎣i B ⎥⎦
(10)
where p is the operator of derivation. Only line-line voltages are considered, with no neutral connection.
(12)
where reduction matrices [P], [Q] and reduced vectors and matrices are as follows (note [P]·[Q]T = [Q]·[P]T = =[U] = 3x3 unity matrix): ⎡1 0 0 0 ⎤ [P ] = ⎢⎢0 1 0 0⎥⎥ ⎢⎣0 0 0 1⎥⎦
⎡1 0 − 1 0⎤ [Q] = ⎢⎢0 1 − 1 0⎥⎥ ⎢⎣0 0 0 1⎥⎦
(13)
(14)
[ v ] R = [Q ] ⋅ [ v ]
(15)
[i ] R = [ P ] ⋅ [i ]
(8)
where TM is the motor torque and G(θ) is dL (θ ) (θ dθ defines the rotor position). The inductance matrix takes in account winding constructive format (number of coils and disposition), geometric machine configuration and material magnetic proprieties; clearly [L] is directly function of rotor eccentricity degree. The SG is connected to an elementary inductiveresistive load (Fig.3). The reduced load model is (10): R L ⎤ ⎡i A ⎤ ⎡2 L L + ⋅ 2 R L ⎥⎦ ⎢⎣i B ⎥⎦ ⎢⎣ L L
LL
Fig3. Schematic representation of the configuration studied
[ R ] R = [Q ] ⋅ [ R ] ⋅ [Q ]
(l6)
[ L]R = [Q ] ⋅ [ L] ⋅ [Q ]T
(17)
[G ]R = [Q ] ⋅ [G ] ⋅ [Q ]T
(18)
T
The mechanical torque balance is (9):
⎡v AC ⎤ ⎡2 R L ⎢v ⎥ = ⎢ R ⎣ BC ⎦ ⎣ L
LL
The SG reduced model is the following:
[ψ ] = [ L] ⋅ [i ]
vC
RL
(3)
where:
vB
RL
[v] = −([ R] + [G] ⋅ θ&) ⋅ [i] − [ L] ⋅ p[i]
[v] = −[R ]⋅ [i] − d [ψ ] dt
[v ] = [v A
RL
The final state-space form can be obtained by combining (10) for the load and (12) for the SG: ⎧⎡ 0 ⎤ ⎡iA ⎤ ⎫ ⎡i A ⎤ ⎪ ⎥ ⎥ ⎢ −1 ⎪ ⎢ p ⎢iB ⎥ = [ B] ⎨⎢ 0 ⎥ − [ A]⎢⎢i B ⎥⎥ ⎬ ⎪ ⎢v ⎥ ⎢⎣iF ⎥⎦ ⎪⎭ ⎢⎣iF ⎥⎦ ⎩⎣ F ⎦
(19)
where: ⎡2 RL & [ A] = −[ R]R − [G ]R ⋅ θ − ⎢⎢ RL ⎢⎣ 0
⎡2 LL [ B ] = −[ L]R − ⎢⎢ LL ⎢⎣ 0
LL 2 LL 0
RL 2 RL 0 0⎤ 0⎥⎥ 0⎥⎦
0⎤ 0⎥⎥ 0⎥⎦
(20)
(21)
Model y& = f (t , y ) as in (19) was time-integrated by using the second-order Adam-Bashford’s formula in a ‘script’ MatLab: ⎡ 3 ⋅ f ( y k , t k ) − f ( y k −1 , t k −1 ) ⎤ . y k +1 = y k + ⎢ ⎥ ⋅ dt 2 ⎣ ⎦
(22)
IV. SG 3D-FEM MODEL Inductance parameters were evaluated for a machine with static or dynamic eccentricity by using a complete 3D FEM model, as depicted in Fig.4 and 5. The various components which make up the SG under study were accurately drawn, and eccentric machine was simulated by varying parameter ϑ on 360 degrees, by steps of 5o. Fig.6 shows the 3D meshing of half machine.
Fig.6. Mesh of the SG.
V. SIMULATION RESULTS: INDUCTANCES 3D-FEM-produced inductance matrix was interpolated by using cubic splines. Fig.7 compares stator – rotor mutual inductances LAF, LBF, LCF for the symmetrical machine and with 75% static eccentricity (SC = 75% of air gap length on the quadrature axis, RC equal to zero). In Fig.8, mutual inductances with static eccentricity show clearly an imbalance: LAF is smaller than LBF , LCF.
Fig.4. Exploded drawing of SG mechanical parts.
Fig. 7. Stator – rotor mutual inductances
Fig. 5. Exploded drawing of windings. Fig. 8. Details of the Figure 7
VI. SIMULATION RESULTS: NO-LOAD EMF Dynamic simulations obtained by a MATLAB script furnished electrical waveforms (currents, e.m.f.s, fluxes, etc.). Fig.9 shows the space vector loci of the stator induced no-load e.m.f.s eA(t), eB(t), eC(t), eq.(23) (α is exp(2π/3i)), with and without fault. e (t ) = e A (t ) + α ⋅ eB (t ) + α ⋅ eC (t ) = e(t ) ⋅ e 2
jγ e (t )
(23)
Fig. 9. E.m.f space vector loci. for 75% static eccentricity.
In case of 75% quadrature-axis static eccentricity an average increase of the e.m.f. space vector modulus of about 23% happens. There is also an ovalization of the space vector loci in the same direction of the eccentricity vector SC. Fig.10 shows the locum obtained as difference between the space vector and its average value. Space vector locum ovality is so enhanced. Four-lobe locum was obtained using the following formula: e ' (t ) = e (t ) − em ⋅ e jγ e (t )
In Fig.11, locum obtained as difference between the space vector e (t ) and its minimum value vector is shown. The correspondent formula is:
e ' ' (t ) = e (t ) − emin ⋅ e jγ e (t )
(25)
where emin is the minimum value of e (t ) module. Twolobe locum better represents static eccentricity direction.
Figure 11. Difference between the space vector in the case of static eccentricity and its minimum value (two-lobe locum).
In the case of machine with dynamic eccentricity of 75% there is an average increase of about 24% on the e.m.f. vector amplitude, but there is no ovality (Fig.12).
(24)
where em is the average module on 2π (mechanical).
Figure 12. E.m.f. space vector loci for dynamic eccentricity
VII. NO-LOAD EMF HARMONIC SPECTRA
Fig. 10. Difference between the space vector in the case of static eccentricity and its average value (four-lobe locum).
In Figs.13, 14, 15 e.m.f. spectra for symmetrical machine, and with static and dynamic eccentricity are shown. Very substantial increases in first and third harmonic for static and dynamic eccentricity happens, but spectral patterns are however different in the two cases. In the final paper, a more deeper analysis and discussion will be addressed on spectra, by comparing 3D-FEM based model with a Modified Winding Function Approach (MWFA) based model, as reported in the companion paper [10].
an third harmonics. The same results have been obtained about current spectra. Harmonics multiple of three are not zero in the faulty machine current, because of lack of symmetry among the three phases, due to the eccentricity. The two types of eccentricity analyzed are characterized by common and differential effects as well; this fact paves the way for fault assessment diversification. Static eccentricity produces an ovality of the e.m.f. and current space vector loci whereas dynamic eccentricity does not. The research work is currently in progress, and simulations and experiments will be addressed and discussed on concomitant space-vector and FFT analysis in future papers, by comparing 3DFEM and MWFA results, and by using an appositely prepared machine. Fig. 13.E.m.f. spectral analysis for symmetrical machine
ACKNOWLEDGMENT This work has been developed as part of a research program on diagnostics for on-board ship generators with the financial support of the Italian Ministry of Defence, General Direction of Naval Equipments - NAVARM. APPENDIX Machine details in 3D-FEM analysis were:
Fig. 14. E.m.f. spectral analysis for static eccentricity
• • • • • • • •
4 poles Rated power and frequency: 1.4 MVA, 50Hz 36 stator slots, double-layer lap-winding Phase winding: four series-connected polar belts 3 slots for phase and for pole; step shortened by 2 slots Excitation: 100 turns for each pole Forged iron for compass and polar bodies; Magnetic laminations for polar expansions and stator
REFERENCES
Fig. 15. E.m.f. spectral analysis for dynamic eccentricity
Early results suggest that the whole e.m.f. and current harmonic content must be evaluated as a spectral pattern characteristic for any kind and degree of eccentricity. This is in counter-trend with actual research efforts, where focus is concentrated on particular fault-related harmonics (e.g., 17th and 19th in [6]). VIII. CONCLUSIONS Static and dynamic rotor eccentricities both produce increases in e.m.f. space vector amplitude. By analyzing the e.m.f. spectra it’s also evident an increase of the first
[1] P. J. Tavner, and J. Penman, Condition Monitoring of Electrical Machines. Letchworth, England: Research Press Limited, 1987. [2] H. A.Tolyat and S. Nandi, “Condition Monitoring and fault diagnosis of electrical machines”, in Proc. IEEE-IAS 1999 Annual Meeting, Phoenix, AZ, Oct. 3-7, 1999, pp. 197-204. [3] W. T. Thomson, A. Barbour, “On line current monitoring and application of a finite element method to predict the level of static airgap eccentricity in three-phase induction motors”, IEEE Transaction on Energy Conversion, Vol.13, No. 4, Dec. 1998. [4] P. Talas and P. Toom, “Dynamic measurement and analysis of air gap variations in large hydroelectric generators”, IEEE Transactions on Power Apparatus and System, Vol. PAS-102, No.9, September 1983. [5] G. B. Pollock and J. F. Lyles, “Vertical hydraulic generators experience with dynamic air gap monitoring”, IEEE Transaction on Energy Conversion, Vol. 7, No. 4, December 1992. [6] H. A. Toliyat, N. A. Al-Nuaim, “Simulation and Detection of Dynamic Air-Gap Eccentricity in Salient-Pole Synchronous Machines”, IEEE trans. IAS, vol. 35, no. 1, Jan/Febr 1999. [7] P. J. Tavner, B. G. Gaydon and D. M. Ward, “Monitoring Generators and Large Motors,” IEE Proceedings, vol. 133, pt. B, no. 3, pp. 169-180, May 1986. [8] J. Faiz, I. Tabatabaei, “Extension of Winding Function Theory for Nonuniform Air Gap in Electric Machinery”, IEEE transactions on magnetics, vol. 38, no. 6, November 2002. [9] N. A. Al-Nuaim, H. A. Toliyat, “A Novel Method for Modeling Dynamic Air-Gap Eccentricity in Synchronous Machines Based on Modified Winding Function Theory”, Dept. of Electrical Engineering Texas A&M University. [10] C. Bruzzese, A. Giordani, A. Rossi, and E. Santini, “Synchronous Generator Eccentricities Modeling by Improved MWFA and Fault Signature Evaluation in No-Load E.M.F.s and Current Spectra,” submitted for review to SPEEDAM 2008 Symposium, Ischia, Italy, 18-20 June, 2008.
