Cage rotor MMF: winding function approach - IEEE Xplore

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Keywords: Induction motor, MMF, cage rotor, rotor slot harmon- ics, winding ..... [5] G. Joksimovic, “Analysis and simulation of faults in squirrel cage induction ...
Cage Rotor MMF: Winding Function Approach

Cage Rotor MMF: The winding functions of multiturn coils placed in the slots described by angular coordinates 1 and 2 are defined by α     , θ1 ≤ θ ≤ θ2 N ⋅ 1−   2π  N (θ ) =  α  −N ⋅ , for rest of θ 2π 

G. Joksimovic, M. Djurovic, J. Penman Author Affiliation: Department of Electrical Engineering, University of Montenegro, Montenegro, Yugoslavia; Department of Engineering, University of Aberdeen, Scotland, U.K. Abstract: An analytical expression for the cage rotor magnetomotive force (MMF) is derived using the winding function approach. This approach enables the analysis of stator current spectra in the cage rotor induction machine. It is demonstrated that in stator current spectra one can observe only harmonic components at high frequencies, known as the rotor slot harmonics. The position of these components in stator current spectra depends on the number of rotor bars, number of poles of the induction machine, and slip (i.e., on the actual speed of rotor). Keywords: Induction motor, MMF, cage rotor, rotor slot harmonics, winding function. Introduction: A noninvasive technique used often for condition monitoring of cage rotor induction machines is current signature analysis. This method is based on the monitoring of the stator current spectra in real time. The appearance of stator current components or increase of magnitude of some components at characteristic frequencies indicates a fault condition in a machine. These characteristic frequencies for common faults in cage rotor machines such as broken rotor bar(s), static or dynamic eccentricity, etc., are widely known in the literature [1]. On the other hand, the rotor slot harmonics, whose position in the stator current spectra depends on actual rotor speed, have found application in modern speed detection techniques [2]. In this letter an analytical equation is derived for the cage rotor MMF by using the winding function approach, [3]. In addition, the stator current spectrum in the cage rotor induction machine is analyzed in relation to the nature of cage rotor MMF. MMF of Symmetrical Three-Phase Stator Winding: The equation for this MMF is known to be [4] M s ( t, θ ) = ∑ M sµ cos(ω 1 t − µpθ ); µ = 6 g + 1; g = 0, ±1, ±2,... µ

where N is the number of turns and α is a coil pitch, α = θ 2 − θ 1 . Two near-by bars and ring segments between them make the rotor loop. Therefore, the rotor loop can be observed as a one-turn coil with pitch α = 2 π / R, where R is the number of rotor bars. Hence, the winding function of the rotor loop 1, whose magnetic axis is in the center of the reference frame fixed to the rotor, is given by π π   1  1 − , − ≤ θr ≤   R R R N loop 1 (θ r ) =  1 − , for rest of θ r .  R

where p is the number of pole pairs and M sµ is amplitude of the µth harmonic. It is clear that beside the fundamental MMF wave ( g = 0, µ = 1) there exist waves with 5p, 7p, 11p, 13p, ... pair of poles, even in the case of symmetrical machines. By assuming uniform air-gap length, the flux density waves in the air-gap produced by stator windings will be of the same waveform.

(3)

This function can be resolved in Fourier series as follows: N loop1 (θ r ) =



2 π ∑ νπ sin  ν R  cos( νθ ). r

(4)

ν=1

The rotor current, which flows in this loop as a result of µth stator flux density space harmonic, produces the following MMF: ∞

π

∑ νπ sin  ν R  I

M loop1 ( t, θ r ) =

2

ν=1

rµm

cos( s µω 1 t ) cos( νθ r )

(5)

where s µ = 1 − µ(1 − s ), and s is the slip. Equation (5) can be resolved as M loop1 ( t, θ r ) =

(1)

(2)



∑ K [ cos( s ω t + νθ ) + cos( s ω t − νθ )]. µν

ν=1

µ

1

r

µ

1

r

(6)

In the next rotor loop, which is in space displaced for 2π / R, flows current of the same magnitude and frequency but shifted in phase by µ ⋅ p ⋅ 2 π / R. This rotor loop produces MMF M loop2 ( t, θ r ) =



∑K ν=1

µν

2π   cos s µω 1 t + νθ r − ( ν + µp )  R   2π  + cos s µω 1 t − νθ r + ( ν − µp )  .  R  

(7)

It can be shown that by summing MMFs of all rotors loops that the resultant MMF of the cage rotor is R− 1 ∞

2π  M r ( t, θ r ) = ∑ ∑ K µν cos s µω 1 t + νθ r − i ⋅ ( ν + µp )    R i=0 ν=1 2π  + cos s µω 1 t − νθ r + i ⋅ ( ν − µp )  .  R   (8)

Figure 1. (a) Cage rotor MMF at an instant of time (p=2, R=32, ␮=1) (top); (b) Spectral content of MMF from Figure 1(a) (bottom) 64

Discussion: The case when rotor currents are due to µth stator flux density space harmonic is considered. By inspection of (8), the following rotor MMF waves exist: for ν = ± µp as well as for ν + µp = ± λR and ν − µp = ± λR, i.e., for ν = ± λR − µp and ν = ± λR + µp, where λ = 1, 2, 3... Since ν can be only a positive integer, it follows that the cage rotor MMF waves exist only for ν =| µ| p, ν =| λR + πp|, and ν =| λR − µp|. Therefore, in addition to the fundamental cage rotor MMF wave ( ν =| µ| p ), which is the armature reaction on the stator MMF wave, there is also the so-called rotor slot harmonics of order | λR ± µp|, λ = 1, 2, 3... These harmonics are the direct consequence of the space distribution of rotor bars, i.e., of placement of rotor cage in

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IEEE Power Engineering Review, April 2001

slots. Therefore, the rotor currents caused by the µth stator flux density space harmonic produce the following MMF waves: M rµ ( t, θ r ) = M rµ 1 ( t, θ r ) + M rµsh 1 ( t, θ r ) + M rµsh 2 ( t, θ r )

(9)

M rµ 1 ( t, θ r ) = M rµ 1 cos( s µω 1 t − µpθ r )

(10)

M rµsh 1 ( t, θ r ) = M rµsh 1 cos( s µω 1 t + ( λR − µp )θ r )

(11)

M rµsh 2 ( t, θ r ) = M rµsh 2 cos( s µω 1 t − ( λR + µp )θ r )

(12)

where

However, the main conclusion can be derived from (13), (14), and (15). The cage rotor of induction machine reflects all MMF space harmonics from stator side at the fundamental frequency ( f1 ) and at frequencies (1 ± λR(1 − s ) / p ) f1 . By taking into account only the constant term of permeance of the air-gap in induction machines, the flux density waves will have the same form as MMF waves. Thus, the induced emfs, as well as currents, in a stator winding can be expected to appear at discussed frequencies. The magnitudes of these emfs (currents) depend on the number of pole pairs of the reflected flux density waves. It

or, observed from the stator side, using transformation θ = θ r + ω 1 t ⋅ ((1 − s ) / p ), M rµ 1 ( t, θ ) = M rµ 1 cos(ω 1 t − µpθ )

(13)

   λR  M rµsh 1 ( t, θ ) = M rµsh 1 cos 1 − (1 − s ) ω 1 t + ( λR − µp )θ  p  (14)    λR  M rµsh 2 ( t, θ ) = M rµsh 2 cos 1 + (1 − s ) ω 1 t − ( λR + µp )θ .  p   (15) Figure 1(a) shows the waveform of the fundamental cage rotor MMF wave (µ = 1) at a time instant assuming that the amplitude of the rotor loop current is 1A. Rotor has R = 32 bars and it is assumed that Figure 2. (a) Cage rotor MMF at an instant of time (p=2, R=32, m=(5) (top); machine has two pair of poles, p = 2. Figure 1(b) shows spectral con- (b) Spectral content of MMF from Figure 2(a) (bottom) tents of MMF wave from Figure 1(a). It is obvious that beside fundamental harmonic (µp = 2 nd ), there exist rotor slot harmonics, R − p and R + p, 2R − p and 2R + p, etc. (i.e., 30th, 34th, 62nd, 66th). This follows from (11) and (12). Figure 2(a) shows the waveform of the cage rotor MMF, produced by rotor currents that are the result of µ = −5 stator flux density space harmonic for the same machine and, once again, under assumption that the amplitude of rotor loop current is 1 A. In this case, the number of rotor bars per one pole of harmonic field is less than before, so as a result, more rugged waveform is obtained. The spectral content of the waveform from Figure 2(a) is shown in Figure 2(b). It is obvious that in addition to the “fundamental” harmonic (now, ΜP=10th), the rotor slot harmonics of the order (R᎐5⭈p), (R+5⭈p), (2R᎐5⭈p), (2R+5⭈p), i.e., 22nd, 42nd, 54th, 74th will appear. It is in accordance with (11) and (12) as well. It is obvious that now the amplitude of rotor slot harmonics are more pronounced relative to the “fundamental” harmonic when compared to the situation in Figure 1(b). Figure 3. Stator current spectra for experimental motor (S=36, R=32, p=2, s=5.25%) IEEE Power Engineering Review, April 2001

65

follows from (14) and (15) that the most significant impact on the stator emfs (currents) at discussed frequencies will have waves for which λR ± µp = ± p satisfied. Figure 3 shows the stator current spectra for an experimental motor 3 kW, p = 2, with S = 36 and R = 32, stator winding connections ∆, at slip s = 5.25%. S is the number of stator slots. The most prominent harmonic in the spectra is the higher rotor slot harmonic, at 808 Hz. In [4], Say points out that motors that fulfill the condition S ± R = 2 p have very prominent rotor slot harmonics. The lower rotor slot harmonic at 708 Hz is of very small magnitude. The reason is the fact that lower rotor slot harmonic is triplen harmonic (from (14) follows for s = 0 and λ = 1: |1 − R / p| =|1 − 32 / 2| = 15) and since the stator winding is delta connected, the current of this frequency flows in delta. The fact that all space harmonics from the stator side are reflected by cage rotor at only two frequencies (for λ = 1) is the main limiting factor for condition monitoring of various fault conditions such as interturn short circuits or static eccentricity condition of an induction machine [5]. Namely, in fault condition such as interturn short circuit in stator winding, all MMF and flux density space harmonics will appear in the air gap. But a cage rotor will reflect all of these harmonics at only two frequencies (for λ = 1) in stator current spectra, which exists in spectra even in a healthy machine. Conclusions: As the result of the nature of the rotor cage winding, all MMF (flux density) space harmonics from the stator side will be reflected by the rotor at the fundamental frequency f1 and at the frequencies (1 ± λR(1 − s ) / p ) f1 . As a consequence, harmonic components will exist at these frequencies in the stator current spectra. The position of these components in the stator current spectra depends on actual rotor speed. This fact is employed for sensorless measurement of rotor speed. The fact that all space harmonics from the stator side are reflected by the cage rotor at only two frequencies (for λ = 1) is the main limiting factor for condition monitoring of various fault conditions of an induction machine such as interturn short circuits or static eccentricity conditions. References: [1] P. Vas, Parameter Estimation, Condition Monitoring and Diagnosis of Electrical Machines. Monographs in Electrical and Electronic Engineering. New York: Oxford Science Publications, 1993. [2] K.D. Hurst and T.G. Habetler, “A comparison of spectrum estimation techniques for sensorless speed detection in induction machines,” IEEE Trans. Ind. Applicat., vol. 33, pp. 898-905, July/Aug. 1997. [3] G. Joksimovic, M. Djurovic, and A. Obradovic, “Skew and linear rise of MMF across slot modeling-winding function approach,” IEEE Trans. Energy Conversion, vol. 14, pp. 315-320, Sept. 1999. [4] M..G. Say, Alternating Current Machines, 5th ed. New York: Pitman 1984. [5] G. Joksimovic, “Analysis and simulation of faults in squirrel cage induction motors by multiple coupled circuit model,” Ph.D. dissertation, University of Montenegro, Yugoslavia, 2000. Copyright Statement: ISSN 0282-1724/01/$10.00  2001 IEEE. Manuscript received 10 March 2000; revised 18 September 2000. This paper is published herein in its entirety.

2001 European EMTP-ATP User Group Meeting and Conference Call for Papers

Application of Unified Power Flow Controller to Available Transfer Capability Enhancement Y. Xiao, Y.H. Song, Y.Z. Sun Author Affiliation: Brunel Institute of Power Systems (BIPS), Brunel University, Uxbridge, London UB8 3PH, U.K.; Tsinghua University, Beijing, China. Abstract: From the point view of operational planning, a methodology for improving available transfer capability (ATC) by the unified power flow controller (UPFC) is presented. It is highly recognized that FACTS devices, especially the UPFC, can be applied to redistribute load sharing and support node voltage, therefore, to mitigate the critical situation that results from the increase of system loading or occurrence of inherent uncertainties in power systems. Thus, it is essential to take the impact of FACTS devices as an important issue for reinforcement of ATC. In this letter, with respect to the effect of UPFC, ATC is modeled as the maximum value of unused transfer power that causes no thermal or voltage limit violations. In the model formulated, three major uncertainties of power systems that affect the ATC value considerably are also considered. Case studies with a reduced practical system are presented. The technical merits of the UPFC for reinforcement of the ATC are demonstrated clearly. Introduction: The requirement of the shared use of the existing network resources intensively and reliably have motivated the development of effective methods to enhance ATC. Various approaches have been proposed, such as the Lagrangian multiplier and the generalized reduced gradient method [3] and direct interior point algorithm [4]. Traditionally, rescheduling active power of generators, adjusting terminal voltage of generators, and taps changing of on-load tap changer (OLTC) are considered as major control variables for optimization. In a privatized electricity market, however, all of the parameters may not be centrally dispatched by the transmission network owner. On the other hand, in terms of steady-state power flow, since circuits do not normally share power in proportion to their ratings, and in most situations the voltage profile can not be smooth, ATC value will be limited ultimately by heavily loaded circuits and (or) nodes with relatively low voltage. Meanwhile, as stated, FACTS technology makes it possible to use circuit reactance, voltage magnitude, and angle as controls to redistribute line flow and regulate nodal voltage. Therefore, it can offer a convenient and promising alternative to conventional methods for the enhancement of ATC. Nevertheless, so far, from the view of operational planning, there are few reports about using FACTS to improve ATC of interconnected networks. A new approach is proposed in this letter, which focuses on the application of the UPFC on transmission corridors to enhance ATC. To maintain the symmetry of the admittance matrix, the power injection model (PIM) is employed to derive control parameters of the UPFC(s). Furthermore, taking into consideration the main uncertainties of power systems associated with the determination of ATC value, a stochastic optimization model is formulated, thereby, providing more robust and practical information. ATC Optimization Model Formulated: Based on PIM, which is shown in Figure 1, active and reactive power injections of the UPFC are taken as independent control variables for ATC optimization. For the

Abstract deadline: 31 May 2001 The European EMTP-ATP User Group Meeting and Conference (EEUG) will be held 3-5 September 2001 in Bristol, UK. The conference seeks papers on all aspects of power engineering using ATP-EMTP. For more information, refer to the conference Web site, http://www.uwe. ac.uk/facults/eng/EEUG2001, on contact Hassan Nouri, +44 117 344 2631, fax +44 117 344 3800, email Hassan. [email protected]. Figure 1. Power injection model of the UPFC 66

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IEEE Power Engineering Review, April 2001

Table 1. Results of ATC optimization with and without the UPFC Parameters of the UPFC (p.u.) Scenario

VT

ϕT

Without UPFC

-

-

With the UPFC

0.6378

5.1374

λ*

ATC (MW)

Iteration

-

1.350

4594.88

3

66.1712

1.472

6020.29

7

Iq

UPFC installed on line I − J near node I, since the two back-to-back converter-based synchronous voltage source (SVS) of the UPFC can generate or absorb reactive power independently, essentially there are two reactive power injections to node I involved in power flow control. One is injected by the shunt SVS directly to node I for regulating voltage as QII ( inj ) . Generated by the series SVS, the other flows via line L for reactive line flow control, which is represented as QIL ( inj ) . That is, QI ( inj ) = QII ( inj ) + QIL ( inj ) . Thus, as a matter of fact, there are five power injections involved in PIM of the UPFC: PI ( inj ) , QIL ( inj ) , QII ( inj ) , PJ ( inj ) , and QJ ( inj ) . Considering no loss in the device, active power injection to node J, PJ ( inj ) is equal to that to node I, PI ( inj ) . Since the power flow PL + jQL from node I side is considered as line flow of L, QJ ( inj ) is not taken as a control variable. Therefore, control vector S = [ PI ( inj )QIL ( inj )QII ( inj ) ] is applied to enhance the ATC value by implementing the corresponding active, reactive line flow control and nodal voltage control, respectively. As the power injections are only an interim result, once the required power injections are obtained, the original control parameters can be derived easily according to the corresponding one-voltage, one-current source model of the UPFC. Moreover, since the UPFC possesses the functions of both series controller and shunt controller, the control models for series controller and shunt controller can be inferred conveniently from the UPFC model described above. The ATC optimization model is formulated to enable the maximum transfer of the specified interface by controlling the relevant power injections of installed UPFCs, meanwhile, increasing all the complex loads of current situation using loading factor λ * , until critical situation occurs; that is, line thermal limits or nodal voltage limits are attained. Furthermore, with respect to the uncertainties with adverse effects on ATC, a stochastic model is formulated. In the model, the availability of generators and circuits are considered as discrete random variables fol~ lowing binomial distributions, which are represented by ξ, ~ η, respec~ is taken into consideration as a tively. In addition, fluctuation of load ω

h = 1, 2,⋅⋅⋅H. L denotes operating constraints, which include nodal voltage limits, circuit thermal limits, and capacity limits of the UPFC(s). Subscript min and max in each equation represent the minimum and maximum limits of the variable, respectively. Equations (2) and (3) are power flow equations of the current situation and critical situation under loading factor λ * , respectively. In (3), impact of the UPFCs can be considered conveniently by modifying the right-hand side of the relevant power flow equations using the related power injections as shown in (5)-(8). For node I (suppose a UPFC is installed on line I − J near node I),

normally distributed variable. The developed model is described as follows: HF

(

)

Max:F ( X , λ * , X * , S * ) = ∑ PL h * ( X h * , S h * ) − PL h ( X h ) h= 1

+

H

∑ (P ( X ) − P ( X ))

h= HF + 1

Lh *

h*

Lh

h

(1) Figure 2. Flow chart of the proposed methodology

(

)

~ ~ s.t.:G X, ξ, ~ η =ω

(

)

~ ~ G * λ * , X * , S * , ξ, ~ η = ωλ *

Lmin ≤ L( X ), L* ( λ * , X * , S * ) ≤ Lmax

(2)

(3) (4)

where variables with subscript * represent those at the critical equilibrium point, while variables without subscript * denote those at the current operating point. X is state vector, which includes magnitude and angle of nodal voltage. H is the total number of the investigated interconnected transmission lines across the studied interface, in which the active powers share the same prescribed direction. HF denotes the number of UPFCs installed. PLh represents the active power flow of line h, IEEE Power Engineering Review, April 2001

Figure 3. The reduced system 67

N−1 ~ ~ +P ~ ξ I PIG * − λ *ω =0 IP 1 ( inj ) − VI * ∑ Vj * ηIJ G Ij cos θ Ij * + BIj sin θ Ij * j=1 (5)

(

)

ξ I QIG * − λ *ω IQ + QIL ( inj ) + QII ( inj ) 1

(

)

−VI * ∑ ηIjVj * GIj sin θ Ij * − BIj cosθ Ij * = 0. j=1

(6)

For node J, N−1 ~ ~ −P ~ ξ J PJG * − λ *ω =0 JP I ( inj ) − VJ * ∑ ηJjVj * G Jj c os θ Jj * + BJj sin θ Jj * j=1 (7)

(

)

N−1 ~ ~ +Q ~ ξ J QJG * − λ *ω =0 JQ J ( inj ) − VJ * ∑ ηJjVj * G Jj sin θ Jj * − BJj cos θ Jj * j=1 (8)

(

)

Figure 4. The iteration of lines thermal burden of the studied interface without FACTS device

Figure 5. Voltage profile of the studied interface under current and critical situation (note: mode 50 is PV node)

Figure 6. The iteration of lines thermal burden of the studied interface with FACTS device 68

where V and θ are the magnitude and phase angle of nodal voltage. Gij + jBij is the impedance of line i − j. N denotes the total number of nodes. PiG andQiG are active and reactive power generations of nodei. And ω~iP and ω~iQ represent active and reactive loads of node i, respectively. Mathematical Solution: For the stochastic model developed with both discrete and continuous variables, a novel hybrid stochastic methodology, which takes advantage of both two-stage stochastic programming with recourse (SPR) and chance constrained programming (CCP), is proposed to deal with this problem and is described in detail in [1]. Since heavily loaded systems and the introduction of FACTS devices will aggravate the ill condition of power systems, optimal multiplier Newton Raphson (OMNR) method is applied in power flow calculation involved. In addition, a primal-dual interior point linear programming (IPLP) is used to solve the problem of UPFC control for high-speed and reliability. The overall procedure is briefly illustrated in Figure 2. Case Studies and Result Interpretation: A reduced system of a real-world network with 29 nodes and 69 branches is utilized to test the validity of the proposed method. The corresponding high-voltage (HV) network is shown in Figure 3. The situation of generation center and load centre located in the north and the south results in a huge amount of power transferred along the studied interface. In order to demonstrate the effectiveness of the UPFC on enhancement of the ATC, thermal limits of the lines are modified to result in a seriously unbalanced power sharing among the lines across the studied interface, which is clearly illustrated as the initial situation in Figure 4. It is also perceivable that among all the interconnected lines, circuit 25 has the lightest thermal burden. Consequently, with the increase of the loading factor other lines may have already reached full load, while further capability still remains on this line. On the other hand, with the system stressed more heavily, node voltage will also be pulled down. All HV nodal voltages are stipulated to be within the range 0.90-1.10 p.u. under intact and contingent situations. From the voltage profile of the studied interface under critical situations without UPFC depicted in Figure 5, the ATC value is constrained by the voltage limit of node 90. Therefore, in order to improve the ATC value, a UPFC is installed on line 25 near the side of node 90. The comparison of the results of ATC optimization with and without the UPFC are shown in Table 1, in which the considerable difference between the two ATC values highlights the good performance of the UPFC on ATC improvement. In Figure 5, the voltage profile of the studied interface under critical situation with the UPFC is demonstrated. It is evident that with the reactive power injection of the UPFC to node 90, the weakest point is shifted from node 90 to node 120. The effect of the series part of the UPFC on reinforcement of the ATC is also illustrated from the lines thermal burden profiles of the studied interface with and without the UPFC, which are shown in Figures 4 and 6. Clearly, with the UPFC, line thermal burden of the studied interface, especially that of line 25, is ultimately balanced. It should be pointed out that in this case the ATC value is limited by violation of the nodal voltage. While there are undoubtedly other cases in which further increases of transfer capability are prevented by line thermal limits, which can be reinforced by the UPFC with the function of load flow redistribution. Therefore, FACTS devices, particularly the UPFC, can play an important role in enhancing the ATC of the interconnected system. Conclusion: From the point view of short-term operational planning, this paper focuses on the steady-state optimization of ATC of interconnected systems by control of the UPFC. A stochastic model for ATC enhancement is presented. The case study demonstrates that the use of the UPFC, which has the characteristics of balancing line flow and regulating node voltage simultaneously, can improve ATC considerably. Therefore, FACTS technologies can offer competitive solutions to the complex challenges in the operation of modern power systems. Acknowledgments: The work is partly supported by the National Science Foundation of China for overseas youth R&D cooperation. References: [1] Y. Xiao, Y.H. Song, and Y.Z. Sun, “Application of stochastic programming for available transfer capability enhancement using (continued on page 72) IEEE Power Engineering Review, April 2001