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Internal faults in synchronous machines. I. The machine model
Abstract
This paper discusses the construction of a mathematical model of a
large synchronous machine suitable for analyzing internal phase and
ground faults in stator windings. The method employs a direct phase
representation, and uses conventional, and readily available, machine
data. The methodology was validated by comparison with results obtained
from independent finite element analyses
... Defining the electrical quantities of the new circuit with the subscript ''f'' (fault), we revise the motor voltage equation as (1). As some of the parameters and variables will change their values after the fault, we use a prime ('' 0 '') for these post-fault quantities in (1) to distinguish them from the corresponding variables in healthy motor operation [16]. According to [17], where an interturn short up to 23% in a single phase of the motor has been considered under the worst case full-load operating conditions using FEM analysis, an over current in a stator winding would not cause significant demagnetization for a PM motor composed of Nd-Fe-B material. ...
... The key problem now is the calculation of the magnetic system inductance, L s , composed of the self and mutual inductances between the shorted portion of the fault winding and the other healthy portions. Several studies have discussed this problem, e.g., [14][15][16]. Bastard proposed a method for transformer fault studies in [15], which involves only a few assumptions and provides good accuracy while maintaining the method's mathematical simplicity. The principles of consistency, leakage and proportionality therein are applicable to the PMSM. ...
... In (17), n is a positive number that is larger than 1, and the definitions of F(Á) and G(Á) in (16) and (17) are the same. Because the time constant of (17) is 1/n that of (16), the system response of (17) is n times faster than (16). Given the same initial conditions and n times faster input, (17) can provide n times faster response than (16). ...
Because of its high efficiency, robustness, and high power density, a permanent magnet synchronous machine (PMSM) is a desirable choice for high-performance applications, such as naval shipboard power systems. The stator winding fault is the most common electrical fault in PMSM; thus, detection of this type of fault is very important. The objective of this paper is to model and detect the location and severity of the stator winding fault of PMSM. To achieve this objective, a mathematical model that can describe both healthy and fault conditions is developed. Simulation results match the observations of this type of fault in the literature. According to the fault model, two parameters associated with fault location and fault severity must be identified in order to detect the fault. Because of the complex distribution of these two parameters in the fault model, the identification problem is extremely difficult for nonlinear identification techniques. To overcome this difficulty, the detection/identification problem is first transformed into a corresponding optimization problem and then solved using particle swarm optimization (PSO). By simulating a modified model instead of the original model, the PSO-based identification algorithm is able to identify the fault in real time. The real-time PSO-based identification algorithm also can be applied to many other identification problems.
... After implementing the winding partitioning, both VBR and direct-phase methods could be used. A partitioned winding direct-phase model, which is given in [19] and [20], is more appropriate for the protection scheme proposed in this paper than the partitioned winding VBR model, which is given in [21]. The reason is that in [19] and [20], the applicable faults are not limited to symmetrical faults. ...
... A partitioned winding direct-phase model, which is given in [19] and [20], is more appropriate for the protection scheme proposed in this paper than the partitioned winding VBR model, which is given in [21]. The reason is that in [19] and [20], the applicable faults are not limited to symmetrical faults. Therefore, in this paper, the machine model is obtained from [19] and [20], and a backup overcurrent protection scheme for the stator winding of a synchronous generator is proposed to operate in conditions that the main protection scheme, which is the high-speed differential protection, fails to operate. ...
... The reason is that in [19] and [20], the applicable faults are not limited to symmetrical faults. Therefore, in this paper, the machine model is obtained from [19] and [20], and a backup overcurrent protection scheme for the stator winding of a synchronous generator is proposed to operate in conditions that the main protection scheme, which is the high-speed differential protection, fails to operate. In this protection scheme, a fuzzy controller is used as the final decision maker and sets the plug setting of the overcurrent relay. ...
SUMMARY The most common protection scheme for synchronous generators against stator windings fault is the differential protection scheme. In this paper, a new voltage-controlled overcurrent protection scheme is proposed as a backup protection method for the differential protection method. This scheme is designed to operate in conditions in which differential protection fails to do so and is implemented by fuzzy controllers. The fuzzy controllers set the plug setting of the overcurrent relay. In the proposed method, the fault conditions can be determined regardless of variations of voltage and the current of terminal in different operating states of the synchronous generator. This is realized by measuring not only the terminal voltages and currents of the generator, which are usually used in conventional voltage-controlled overcurrent protection schemes but also other variables that promote the accuracy of the proposed scheme. For presenting this protection method, a synchronous generator with internal fault model is used. The fault model is based on the direct-phase representation that uses the conventional and readily available machine data. Simulations for various types of stator faults of synchronous generator validate the proposed method. Copyright © 2013 John Wiley & Sons, Ltd.
... Because of lack of data on winding distributions, the following example considers the case for initially sinusoidally distributed windings. This sinusoidal winding distribution has been considered earlier by others [5,7,8] and thus the results of the un-faulted case can be cross-checked against these papers. Note that even with this assumption, the faulted windings are no longer sinusoidally distributed, and so the example serves to demonstrate the validity of the proposed approach. ...
... This section compares the approach of this paper with that of others [5,7,8] and then shows real-time simulation results for winding faults. For comparison, the calculated inductances obtained with the proposed method are compared with those from [5,7,8]. ...
... This section compares the approach of this paper with that of others [5,7,8] and then shows real-time simulation results for winding faults. For comparison, the calculated inductances obtained with the proposed method are compared with those from [5,7,8]. To simulate the machine in real time, an embedded approach [1], is used. ...
This paper presents the development of a real-time digital simulator model for the simulation of arbitrary internal faults in synchronous machines. The model is an extension of the embedded phase-domain model of the synchronous machine (1). To represent a fault, the winding or windings involved in the fault are considered as a set of split windings(2,3,4,5) with their terminal nodes connected by a suitable fault impedance (or short circuit), which is switched in when the fault is applied. From the machine and winding geometry, values are calculated for the resulting set of mutually coupled inductances for this new winding arrangement. The proposed method takes into account the actual geometry of the slots and the number of turns in each coil and uses an off- line procedure to obtain the magneto-motive force (MMF) distribution due to each winding for a unit injection of current. This MMF along with the air-gap geometry information is used to calculate the flux linkages, and hence the self and mutual inductances of the windings. Thus in contrast with earlier approaches, it is able to calculate the inductances of the machine when the windings are arbitrary distributed. Since this model is developed for real-time digital simulator, it has the unique feature of being a tool to test the relays designed to protect the synchronous machines from internal faults.
... However, they all have similar fault types and distribution [8]. Amongst all the faults of these wind generators, the winding fault, as the second most frequent fault, has attracted significant interest in the past 30 years [8][9][10][11][12][13][14][15]. It has been reported by the authors in [8] and the Electrical Apparatus Service Association (EASA) that there are five major winding faults: (1) inter-turn (turn-to-turn) short circuit (ITSC), (2) coil-to-coil short circuit, (3) open circuit of one phase, (4) phase-to-phase short circuit, and (5) coil-to-ground short circuit. ...
... A cost-effective fault model is especially useful for large-power electrical machines because performing fault tests on them is costly, difficult and often destructive. For these reasons, some prior work on ITSC fault modelling of different electrical machines has been carried out by researchers [10][11][12][19][20][21][22][23][24][25]. In the literature, there are three well-established methods to model ITSC faults, i.e., the analytical approach, magnetic equivalent circuit, and finite element method (FEM). ...
This paper proposes a general analytical model for large-power surface-mounted permanent magnet (SPM) wind generators under inter-turn short-circuit (ITSC) faults. In the model, branch currents rather than phase currents are used as state variables to describe the electromagnetic behavior of the faulty machine. In addition, it is found that the multiphase Clarke transformation can be used to simplify the proposed fault model with the inductances calculated analytically or numerically using finite element analysis. With the latter, both linear and nonlinear inductances can be obtained, and the non-linear inductances are used for the fault modelling of large power rating machines due to larger electrical loading and heavier magnetic saturation. With the developed fault model, studies of scaling effects (different power ratings such as 3 kW, 500 kW and 3 MW) and the influence of fault location on the electromagnetic performance of SPM generators with series-parallel coil connections have been carried out. The simulation results show that large-power SPM wind generators are vulnerable to ITSC faults when a relatively small number of turns are short-circuited and a single-turn short-circuit fault at the top of the slot is found to be the worst case.
... This presents hurdles in the widespread use of such models. The authors in [17] and [18] proposed a method in which the need for information about the distribution of the windings is eliminated. In this approach, it is assumed not only that the healthy windings create a perfect, sinusoidally distributed MMF, but also that the MMF due to the subwindings is sinusoidal. ...
... Based on the above description, the method of [17] and [18] with certain enhancements is used in this paper to calculate the inductances of subwindings. ...
Correct operation of generator protection is critical to avoid forced outages and to minimize damage during internal faults and other abnormal events. Testing security for external faults and system disturbances has been carried out in the past using real-time systems or transient simulation software. Scaled physical models have been employed to simulate internal faults. However, these machines are restricted in the types of faults that can be applied and the variety of systems that can be modeled.
This paper describes a new synchronous generator model that has been developed in a real-time digital simulator. The model can be configured with a wide range of electrical and mechanical parameters and can simulate various types of faults both on the rotor and on the stator.
The paper also describes how the new model was used to validate a new multifunction generator protection relay. The relay incorporates several novel protection elements, so comprehensive validation of these elements was very important. A wide range of faults were applied, including external faults, power swings, stator winding faults, field short-circuits, and faults during static starting.
This paper focuses on stator winding protection for ground faults, phase faults, turn-to-turn faults, and series faults. The performance of the protection was measured for these events. Application guidance for stator ground, split-phase, and negative-sequence directional protection for generators is provided.
... To simulate the synchronous generator internal fault, the stator winding at which the phase-to-ground fault occurs should be divided into two parts, including exterior and interior windings [12], [13] and [14]. As illustrated in Fig. 1, the faulty phase winding is partitioned into a 1 as the interior winding, which is terminated at the neutral (N), and a 2 as the exterior winding, which is adjacent to the generator terminal (T). ...
... The generator model consists of electrical, magnetic, and mechanical equations of the generator and its associated prime-mover [12], [13], and [14]. A comprehensive generator model including the internal stator earth fault is developed and implemented for studies of this paper, which is not scrutinized here due to the number of pages restriction. ...
Small-scale synchronous generators (SSSGs) are widely utilized as distributed generators in power systems. A large fault current due to the SSSG phase-to-ground short circuit may result in excessive damages to the stator core and windings, in a way that major repairs, such as core disassembling and rewindings, become necessary. Thus, the fault current should be properly restricted to prevent such damages and reduce the required capital and time for SSSG repairs. In this research work, a model to calculate the SSSG internal fault current is developed. Meanwhile, an appropriate index to quantify the damages to the faulty SSSG is presented. Afterward, influencing parameters on reduction of the damages are investigated when either frequently used TT or TN grounding systems are applied for SSSGs. Studies show that the stator damages are mainly caused by the SSSG fault current rather than that of the system. Accordingly, to reduce the SSSG vulnerability to the stator earth fault, not only some applicable strategies are proposed for the both grounding systems, but also the favorable grounding system is determined.
... A technique, used to implement the internal fault for modeling the synchronous generator, is provided in [6]. A partitioned winding direct-phase model, which is given in [7] and [8], is used for the protection scheme proposed in this paper instead of the partitioned winding (VBR) model, which is given in [9]. The reason is that in [7] and [9] the possible occurring faults are not limited to symmetrical faults. ...
... A partitioned winding direct-phase model, which is given in [7] and [8], is used for the protection scheme proposed in this paper instead of the partitioned winding (VBR) model, which is given in [9]. The reason is that in [7] and [9] the possible occurring faults are not limited to symmetrical faults. A Backup Overcurrent Protection scheme for the stator winding of a synchronous generator is proposed to operate in conditions that the main protection scheme, which is the high-speed differential protection [11], is unable to detect the fault. ...
A new voltage controlled overcurrent protection scheme is proposed which is a backup protection method for differential protection. This scheme which is realized by fuzzy controllers is considered to be mounted beside a differential protection system, which operates in the cases that differential protection is unable to detect the fault. This novel protection scheme operates by adjusting the plug setting of the overcurrent relay that sends the trip signal to three-phase breaker. In the proposed method fault occurrences can be determined regardless of variations of voltage and current of terminal in different working states of synchronous generator. This is done by measuring other variables than those used in conventional protection methods. To demonstrate the proposed method, a synchronous generator with internal fault model is used. Numerical simulations validate the functionality of this novel scheme under different conditions.
... Muthumuni et al. [4] used the physical arrangement of the conductors, but only the fundamental component of the MMF of the faulted section was considered. Peter et al. [5][6][7] used a straightforward method for partitioning the sinusoidally distributed stator windings. However, the actual stator windings were never perfectly sinusoidally distributed in space. ...
... where K 0 is given as in Eq. (6). The stator-to-rotor mutual inductances are evaluated using Eqs. ...
When an internal fault occurs in the lap-connected windings of turbogenerators, the symmetry between the parallel windings is broken, and different currents then flow because unsymmetrical magnetic linkage exists between the stator windings. The aim of this article is to present a real-time simulation model to investigate the internal fault currents in large turbogenerators with lap windings. This model is based on a modified winding function approach, where the machine inductances are calculated directly from the machine winding distribution. The calculation of the machine inductances are made easier by the use of machine electrical parameters instead of geometrical parameters. The simulation results have been obtained for different cases of internal faults. By using simulated internal fault data, a suitable numerical protection scheme for turbogenerators can be developed.
... Besides, the dq0 model has constant inductance matrices [13]. The PDM uses voltage and flux-linkage equations in a reference phase to derive the fault model in the SGs [14][15][16]. ...
Synchronous generator (SG) plays a vital and critical role in the power system by supplying electric power to consumers. Various faults in SGs can cause some catastrophic events such as power disruption or blackout. These faults can be classified into two electrical and mechanical faults. Short circuit in stator windings and field winding are electrical fault while bearing, static/dynamic eccentricity, and broken damper bars faults are mechanical one. Unlike the induction machines, there are no much researches in SGs condition monitoring owing to its complex behaviour against the faults. Herein, the SG modelling approaches are presented briefly to elaborate shortcoming and challenging issues in the modelling, and then a comprehensive review of various electrical and mechanical fault detection methods is presented.
... Therefore, ignoring higher harmonics will affect the accuracy of the simulation analysis. In [5][6][7][8], the direct domain approach is used for modelling and analysing the internal fault of synchronous machines. The short-circuit winding is divided into two parts: one adjacent to the neutral point and the second adjacent to the load, and assuming that the inductances of fault windings are proportional to the effective turns of windings. ...
Fractional pole‐ratio winding is a new type of AC winding, consisting of coils with different pitches. The application of fractional pole‐path ratio windings in synchronous generators will bring new problems to the modelling and simulation of internal faults. It is important to establish a mathematical model for the fractional pole‐path ratio synchronous generators with internal faults and accurately calculate the fault currents. In this study, the multi‐loop model of fractional pole‐path ratio synchronous generators is first proposed. The method for calculating mutual inductances between stator coils with arbitrary pitch is given, and all the space harmonics, including the fractional ones, are considered in the inductance calculation. In order to improve the simulation accuracy of turn‐to‐turn faults, the effect of core localised saturation is modelled by modifying the air gap function of fault coils. A 300 MW fractional pole‐path ratio synchronous generator is set as an example, and three types of internal faults are simulated. The comparisons of simulation results are made between the multi‐loop model and the finite element model to verify the validity of the multi‐loop model proposed in this study.
... II. STATE SPACE MODEL OF PMSM WITH STATOR TURN FAULT Several works have been done on modeling the induction machine and permanent magnet synchronous machine with 978-1-7281-8873-7/20/$31.00 ©2020 IEEE inter turn fault in [12], [13]. The three-phase armature winding of a PMSM is the same as the armature of a three-phase induction motor [14]. ...
... As the core of thermal power generation and nuclear power generation, the safe and stable operation of turbo-generators is very important to the power grid. However, many internal faults can seriously affect the safe and stable operation of these generators [1][2][3][4][5][6][7]. The rotor of the generator is in high-speed rotating state for a long time, the excitation winding is embedded in the rotor groove and rotates with the rotor, and the interturn insulation is easily destroyed by centrifugal force, which causes excitation winding interturn short circuit (EWISC) faults. ...
Excitation winding interturn short circuit (EWISC) is a common fault in turbo-generators. Once the fault occurs, if not handled in time, it will result in significant security risks to the power system. Using the multifield characteristics of fault generators for a comprehensive diagnosis can make the diagnostic results more accurate and credible. In this paper, taking a TA-1100-78 type, two pole pairs turbo-generator as the research object, the two-dimensional finite element electromagnetic model of stator/rotor and the three-dimensional finite element heat transfer model of rotor were established. The electromagnetic field, temperature field, and stress field of the generator were simulated and analyzed. At the same time, the air gap magnetic field, three-dimensional temperature field, and stress field distribution of the rotor were calculated for EWISC faults in different fault degrees and positions. The results showed that the EWISC fault weakened the air gap magnetic field and caused unbalanced electromagnetic distribution. At the same time, it caused a distortion of the rotor temperature field, resulting in an unbalanced distribution of the temperature field. The stress field was affected by the distortion of temperature field, and the local thermal stress increased but did not exceed the yield limit of the material. Restorable elastic deformation occurred when the rotor was heated, which caused the thermal bending of the rotor. The method adopted in this paper can provide a reference for the calculation of multiphysical field after a generator fault. It is also pointed out that the thermal unbalance influence should not be neglected in the study of generator vibration characteristics.
... In this case, the differential relay may not operate. The windings subjected to an internal fault are split into two parts with a connection point available for the insertion of fault branches [12][13][14]. To simulate an internal fault, it is necessary to use split-phase winding in the affected stator winding(s). ...
Background
Differential relay is normally used to detect faults in Synchronous Generator (SG) stator windings. Nevertheless, detection of ground fault depends on the generator grounding type. For high impedance grounding, the ground faults near the neutral terminal of the stator windings are not detectable by the differential relay. So, the ability to identify the internal fault of SG is a very important task for stable and safe operation of power systems.
Methods
Accurate algorithms for fault detection and classification based on Recurrent Neural Network (RNN) are suggested in this paper. RNNs are trained using different data available from SG MATLAB/ SIMULINK model. Simulation of different fault scenarios based on LabVIEWTM program is discussed. The studied fault scenarios include; fault type, location, resistance and fault inception angle. The RNN based algorithm is experimentally tested using an actual SG. Practical design and implementation of the proposed fault detector and classifier are presented. The hardware system is designed and built to acquire the currents at both ends of SG terminals.
Results
The presented results confirm the effectiveness of the proposed algorithm to detect minor ground faults near the neutral terminal (less than 5% of stator winding).
Conclusion
The experimental analysis shows that the proposed RNN detects and classifies the internal faults correctly, fastly and remain stable after the faults occur.
... Pre-multiplying the circuit equation by the transformation matrix results in (11) and (12). ...
Stator winding interturn, interbranch, and series faults can result in large circulating currents in the faulted coils. Generator protection elements may not be sensitive enough to detect these fault conditions until the fault evolves into a phase-to-phase or phase-to-ground fault. Large machines have been severely damaged by delayed or failed protection system operation. Determining fault quantities for the various possible internal faults is not trivial and requires the aid of numerical models. Protection element models can then be used to determine the protection coverage provided by these elements.
Certain machine modeling methods are useful for analyzing external faults or power system transients but are not appropriate for analyzing internal faults. The most commonly used machine models use dq0 transformation and assume an ideal equivalent model of the machine derived from its normal operating mode using lumped winding parameters. Consequently, these models ignore the effect of the strong harmonics that result from the internal machine asymmetry during internal faults. Alternate methods, such as symmetrical component analysis or phase-coordinate methods, are simplified models that introduce large errors during internal asymmetric conditions.
The multi-loop method treats a machine as a set of loops in relative motion. The method involves a permeance analysis of the machine to calculate the time-variant electric parameters of the stator branches and rotor loops. The stator branches (including fault branches) are converted to loops via a transformation matrix corresponding to the state of the machine. The model is then solved using a numerical method. Because the multi-loop method uses machine geometry and winding design information, it preserves the harmonics that result from internal faults. The transformation matrix provides a simple and intuitive mechanism to apply internal faults in the fractional winding.
In this paper, we validate the multi-loop method using test data from a scale-model machine in a lab. We then use the fault quantities obtained from the multi-loop method to determine the sensitivity and coverage provided by various generator internal-fault protection algorithms for the lab machine.
... où £ q est l'inductance propre d'une phase, ¥ la mutuelle entre phase, ¤ le rapport entre le nombre de spires considérées en défaut et le nombre de spires total d'une phase. Ces approches sont aussi appliquées pour les machines synchrones à aimants permanents [Rei00], [You07], [Far08] ou à rotor bobiné. [Xia05], et [Liu06] proposent des formules d'adaptation pour tenir compte partiellement de certains bobinages mais l'approche reste non généralisable. ...
Les systèmes électriques embarqués dans l'aéronautique doivent satisfaire à des cahiers des charges de plus en plus exigeants portant sur le poids, les performances et la fiabilité, d'où l'utilisation des Machines Synchrones à Aimants Permanents (MSAP). Vu les contraintes imposées, les prototypes sont parfois assez éloignés d'une MSAP classique. La surveillance en ligne de ces systèmes est alors plus délicate mais représente un enjeu considérable vu l'aspect critique des applications (ailerons d'avion, freinage...) et a pour objectif d'éviter un incident majeur en le détectant puis en basculant sur un système identique redondé. Ce document propose un travail de modélisation de MSAP saine et en présence de défaut inter-spires ayant pour objectif de définir des méthodes de détections de défauts inter-spires en ligne, sans capteurs supplémentaires. Deux approches sont présentées pour modéliser les MSAP en présence de défauts inter-spires et sont comparées à des essais expérimentaux réalisés sur un prototype aéronautique. L'une d'entre elles, reposant sur une approche par Réseaux De Perméances (RDP), permet d'obtenir le meilleur compromis. La partie suivante propose de tester deux approches utilisant un indicateur de défaut basé sur un modèle d'Onduleur-MSAP sain et montre l'intérêt de ce type d'approche comparé à des approches plus classiques. L'ensemble des indicateurs développés est ensuite analysé à l'aide d'un outil utilisant la Reconnaissance de Formes (RDF)
... The effect of the air gap on winding function theory is studied in [27]. The authors in [28][29][30] presented the modelling of synchronous machines in fault mode considering the harmonic components. ...
Sinusoidal voltage waveform in synchronous generators is of great significance in power systems. Generally speaking, the output voltage of a synchronous generator is semi-sinusoidal including some disturbance frequencies (known as harmonics) rather than the fundamental frequency. The harmonic contents cause them to increase in iron loss and reduction in efficiency of electrical machines. For this reason, the harmonic contents of synchronous generators should be carefully taken into consideration. In this study, a new winding pattern for stator of the synchronous generators is proposed to eliminate the harmonic contents of output voltage and so makes it more sinusoidal. In this method, the harmonic contents of output voltage are almost eliminated by selecting the optimal number of conductors per slot. Afterward, through several analytical and simulation studies the suggested winding pattern is compared with those of conventional ones. Moreover, the experimental setup is carried out based on a 3 kW, 1500 RPM synchronous generator. The conventional and proposed winding patterns are separately tested in this setup, and then the harmonic-based behaviour of output voltage is studied for each case under different load conditions. The obtained results prove the superiority of the proposed winding pattern over the conventional patterns.
... Bearing this in mind, in this paper, a methodology has been proposed to incorporate stator faults in phasor-based dynamic analysis algorithms and a simplified method is developed to incorporate stator earth fault calculation to be used in engineering applications. From the works has been done so far on the subject, experimental tests and extensive time-domain simulation has been of interest [1, 2]. Most popular synchronous machine analysis has been established based on a so called two-reaction theory [3]. ...
This paper presents a new approach for simulating the internal faults of synchronous machines using distributed computing and Large Change Sensitivity (LCS) analysis. LCS analysis caters for a parallel solution of 3-phase model of a faulted machine within the symmetrical component-based model of interconnected network. The proposed method considers dynamic behavior of the faulty machine and connected system and tries to accurately solve the synchronous machine’s internal fault conditions in the system. The proposed method is implemented in stand-alone FORTRAN-based phasor software and the results have been compared with available recordings from real networks and precisely simulated faults by use of the ATP/EMTP as a time domain software package. An encouraging correlation between the simulation results using proposed method, ATP simulation and measurements was observed and reported. The simplified approach also enables engineers to quickly investigate their particular cases with a reasonable precision.
... Such models often require strong hypotheses. Core saturation, leakage inductances are often neglected and winding arrangements are simplified [5]- [9]. This leads to a lack of accuracy on results. ...
... The physical arrangement of the conductors have been considered (Muthumuni et al., 2001), but only the fundamental component of MMF of faulty section has been used. A straight forward method for partitioning of the sinusoidally distributed stator windings has been used (Reichmeider et al., 2000a(Reichmeider et al., , 2000b(Reichmeider et al., , 2000c. However, the actual stator windings are never perfectly sinusoidally distributed in space. ...
When an internal fault occurs in a synchronous generator, the symmetry between the parallel paths of the winding is broken and different currents flow in them, due to unsymmetrical magnetic linkage between the stator windings. The aim of this paper is to present a simulation model to investigate the effect of internal fault on the parallel path currents of a large synchronous generator using direct phase quantities. This model is based on a modified winding function approach where the machine inductances are calculated directly from the machine winding distribution using machine electrical parameters instead of the geometrical ones. The simulation results for different cases of internal faults in salient-pole and non-salient-pole synchronous machines have been obtained. Salient-pole synchronous generator has wave winding distribution while the non-salient-pole generator has lap winding arrangement. Due to different stator winding arrangements, the two machines have been simulated individually. By using the simulated fault data, a suitable numerical protection scheme for synchronous generators can be developed.
... The steady-state and transient behaviour of large generators have been studied under a variety of representative fault conditions in references [1,2,78910. Based on the different mathematical models, simulation results have indicated that when an internal fault occurs on a stator winding, not only the airgap field harmonics are very strong, but also fractional harmonics are yielded, except for both odd and even harmonics . ...
This paper proposes a digital computer technique based on wavelet transform for generator incomplete differential protection scheme. Exploitation of the fault-generated high frequency currents, the new scheme can provide fault detection with high sensitivity and is also capable of discriminating between internal and external faults. The effectiveness of the proposed scheme was verified both in the experiment and in the field. The results show that the scheme can detect the generator fault with high sensitivity and selectivity during all operation conditions.
... [87]. Recently, several new models were proposed to describe PMSM fault condition operations for FDD[76][79]. ...
Faults in engineering systems are difficult to avoid and may result in serious consequences. Effective fault detection and diagnosis (FDD) can improve system reliability and avoid expensive maintenance. FDD is especially important for some special applications, such as Navy ships operating in hostile environments. So far, FDD for nonlinear systems has not been fully explored. There is still a big gap between FDD theories and applications. This dissertation makes an effort to fill the gap by developing an integrated FDD system structure and a series of algorithms for FDD of Permanent Magnet Synchronous Motors (PMSM).
A fault model is proposed for the stator winding turn-to-turn fault of PMSM. The model provides a good compromise between computational complexity and model accuracy and is versatile for both the healthy and the fault condition. Simulation studies demonstrate a good correspondence with both the theoretical analysis and the experimental observations in the existing literature. The model is especially important for the design of model based FDD algorithms.
Based on the fault model, a series of algorithms are proposed for the fault detection and diagnosis of PMSM. Since the reliability of sensors is the basis of FDD and control systems, a nonlinear parity relation based algorithm is proposed for sensor fault detection. The algorithm can successfully detect single faults in the currents and speed sensors. To track the parameter variations, which are symptoms of system internal changes and faults, an adaptive synchronization based parameter estimation algorithm is proposed. Simulation and experimental studies demonstrate that the algorithm can estimate not only constant parameters but also slowly time varying and abruptly changing parameters in a fast manner. Besides the detection of PMSM internal faults, the algorithm can also provide accurate parameters for the sensor fault detection algorithm. Based on the fault data, a Particle Swarm Optimization based fault diagnosis approach is proposed to find the fault location and severity information of a stator winding turn-to-turn fault. Finally, the proposed algorithms are integrated into a general FDD system structure. The integration provides a FDD system with enhanced robustness to system parametric uncertainties, as shown by extensive simulation studies.
... The physical arrangement of the conductors and only the fundamental component of the MMF of the faulted section were considered in [4]. A straight-forward method for partitioning the sinusoidally distributed stator windings was used in [5][6][7]. However, the actual stator windings are never perfectly sinusoidally distributed in space. ...
When an internal fault occurs in the wave-connected windings of salient-pole synchronous generators, the symmetry between the parallel windings is broken, and different currents flow in them since unsymmetrical magnetic linkage exists between the stator windings. The aim of this article is to present a model to investigate the internal fault currents in large hydrogenerators with wave windings. This model is based on a modified winding function approach, where the machine inductances are calculated directly from the machine winding distribution, and the space harmonics produced by them are also taken into account. The calculation of the machine inductances are made easier by the use of machine electrical parameters instead of geometrical parameters. The fast Fourier transform analysis of the simulated results has been tabulated for different cases of internal faults. By using the simulated internal fault data, suitable numerical protection schemes for hydrogenerators can be developed.
... Parameters N 1 , N 2 , γ 1 , γ 2 depend on the location where the short circuit has happened. In [2][3][4][5][6][7][8], the formulations of these parameters have been derived. ...
In this paper, we have modelled a synchronous generator with internal one phase to ground fault and then the performance of this machine with internal one phase to ground fault have been analyzed. The results show that if the faults occur in vicinity of machine's terminal, then we would have serious damages. To protect the machine from this kind of faults we have suggested integrating a SFCL (superconducting fault current limiter) into the machine's model. The results show that the fault currents in this case will reduce considerably without influencing the normal operation of the machine.
... Inductance profiles of the windings are the most important characteristics in the modeling and analyzing the synchronous generators. To investigate the performance of the generator under different fault conditions, the inductances of the rotor and stator circuits must be calculated precisely [25, 26]. Due to the symmetrical distribution of the windings in the generators slots, the inductance profile of stator windings will have identical variation with a phase shift in the presence of the eccentricity. ...
In this paper, two-dimensional time-stepping finite-element (TSFE) method is performed for modeling and analyzing of a salient pole synchronous generator with different degree of dynamic eccentricity (DE) fault. TSFE analysis is used to describe the influence of DE fault on the flux distribution within the generator and no-load voltage profiles at low and high field current is obtained for healthy and faulty cases. Comparing the magnetic flux distribution of healthy and faulty generators helps to detect the influence of DE fault. Also, it can be seen at no-load condition with low excitation current, the effect of the eccentricity is considerable compared to that of the rated excitation current. Since the calculation of inductances of the machine is the most Corresponding author: M. Babaei (mba babaei@yahoo.com).
The modeling of electrical machines typically employs an analytical method based on the winding function or its magnetic equivalent circuit. Other numerical techniques, such as finite difference, boundary element, and finite element (FEM) methods, have also been adopted. This chapter provides a better understanding of FEM, synchronous machine modeling in FEM, and fault modeling in FEM. Generally, iron losses consist of hysteresis and eddy current losses. The electricity‐conducting materials of the generator are exposed to an alternating magnetic flux, which induces eddy currents. The band object must be defined in order to separate the moving objects from the stationary objects. Traditional methods for modeling a synchronous generator can provide the approximate magnetic field within the generator. These techniques generally offer reasonable results for steady‐state operation, but they are unreliable for transient operation.
In this paper, a comprehensive mathematical model for synchronous generators under stator winding faults near the star point using the Modified Winding Function is described. The model includes space harmonics and their effects on windings inductances. The model is verified using electromagnetic analysis through 2D Finite Element Analysis simulation and experimental testing. The results prove the accuracy of the proposed mathematical model in reflecting the features of synchronous generators under stator windings ground faults in different parts of the winding and specifically near the star point. The developed modeling technique can be used for designing protections schemes for the very challenging 100% stator winding ground fault protection. The model enables protection designers to obtain rich set of data for protection relays designing and testing.
A new internal fault detection and classification technique for synchronous generators is presented. First, the presence of an abnormal/fault condition is detected based on the Negative Sequence Component (NSC) of neutral end current. Thereafter, the internal fault and abnormal/external-fault conditions are discriminated by utilizing the instantaneous phase angle between the NSCs of the terminal voltage and current. Subsequently, the internal fault is classified based on phase angle between NSCs of terminal voltage and neutral end current along with the zero sequence component of the neutral end current. Efficacy of the proposed scheme has been verified on a Phase Domain Synchronous Machine model developed on the Real Time Digital Simulator (RTDS®). The results derived from various fault data sets show that the proposed technique is able to effectively distinguish between internal and external faults including special cases such as current transformer saturation, overloading and unbalance condition. Further, it provides added capability of detecting and identifying inter-turn faults irrespective of the winding configuration of the generator. Comparative assessment of the proposed scheme in terms of detection time and incorporation of different types of faults with the existing techniques proves its superiority.
This paper offers methods to analyze the performance of three-phase synchronous machines operating with an unsymmetrical stator winding caused by cutting out and bypassing coils. The resulting unsymmetrical current distribution in the stator winding tends to produce overheating in parts of the stator and damper windings at full load as well as torque pulsations. Thus, the main objective of this study is to determine the maximal load, under which the unsymmetrical machine may operate without exceeding the permissible temperature limit. Based on the multi-loop method, a mathematical model of a synchronous machine with missing coils in the stator winding is developed. In order to determine the current distribution, the machines' system of differential equations is solved by using a numerical integration method (Runge-Kutta method). As an alternative method, the system of differential equations in the steady state is reduced to a system of algebraic equations, which can be solved with the Gaussian elimination method. Both methods are validated by the measurement results of a 25-kW salient pole synchronous machine. Furthermore, both the circulating current within the parallel branches and the line currents of the different phases as a result of coil or coil group cut-outs are analyzed for a large 41-MW cylindrical-rotor synchronous motor.
A precise simulation of the internal faults of synchronous generators is crucial for the design of the main protection scheme; therefore, an accurate model is necessary. In this paper, based on the circuit-coupled finite element method (CCFEM), an improved model of a synchronous generator with internal faults is proposed, wherein, the localized model of each stator coil is built as per its actual structure. Using this model, the internal fault occurring inside the coil can be accurately simulated, better approximating the location of the actual fault point; thus, the fault currents can be calculated more accurately. The accuracy of the improved model is verified by a dynamic simulation experiment. Moreover, the electromagnetic magnitudes, such as the air gap flux density, fault currents, field current, and all the damper currents are investigated in detail, under fault conditions, thereby, revealing the fault characteristics more clearly and providing a basis for designing protection schemes.
This paper proposes an original, simple and fast permanent magnet synchronous generator model, implemented with a new conception on the graphic interface of Matlab/Simulink environment. The obtained physical model suggests a new way to easily carry out different types of stator faults such as inter-turns short circuit faults, phase to phase faults and phase to ground faults as well as simultaneous multi-faults. This is handled by connecting the two desired points in the stator, on the graphic interface of Simulink, exactly as they can be carried out in the experimental tests. This model has the originality to take into account the arrangement of the stator windings of a multi-pole machine by differentiating between the windings under the same pole-pair and those under different pole-pairs. A comparison between the simulated and the experimental results is done to verify the behavior of the proposed model in both healthy and faulty modes under different operating speeds. The good comparison results lead to validate the correct behavior of the proposed model with a satisfactory accuracy.
This paper investigates the stator winding short circuit characteristics in synchronous generators under rated operating conditions by combing the single-winding analytical method and the finite element method (FEM), which is still the effective method to determine currents and torques under specific fault conditions. Some special measures have to be considered in case of the electromagnetic transient process under the condition of inner faults, e.g., the correct interconnection between sub-strands and the grid is of the particular importance for inner fault simulations. The previous results show that the traditional calculation method based on the equivalent networks fails in case of the inner asymmetry magnetic field. Based on the systematic investigation of a turbo generator with inter-turn short circuits, the results of this paper show the short circuit currents, phase currents, torques and the radial forces have strong correlations with the location and the turn number of faults. In some cases of inter-turn short circuits, the fault currents exceed that of the outer faults. Furthermore, strong radial forces acting on the magnet wheel might lead to the damages of the total shaft train.
In order to analyze the fault feature of the field winding inter-turn short circuits, this paper proposed multi-loop mathematical model of turbine generator with field winding inter-turn short circuits fault considering eddy effect of the solid rotor. The inductance related to the field windings was given, the normal and the short circuits fault operation conditions of generator under no load had been simulated based on the model, which was verified by the simulation and the experiment results. A summary of fault features were obtained, such as, the harmonics circumfluence of stator parallel branches emerges and the field current increases gradually to a new constant value under the constant field voltage after the field windings short fault. The model and results provide the theoretic basis for fault analysis and fault diagnosis of turbine generator with field winding inter-turn short circuits.
In order to analyze the fault feature of large turbo-generator with field winding short circuits in simulation method, the loop selecting principle was introduced in multi-loop method, and the multi-loop mathematical model of turbo-generator with field winding short circuits under infinite network load was established. In air-gap permeance method the single coil's inductance was deduced, the inductance formulas in the model were given as follows: stator winding branch's self-mutual inductance, damping winding loop's self-mutual inductance, mutual inductance between stator winding branch and damping winding loop, field winding loop's self-mutual inductance, mutual inductance between stator winding branch and field winding loop, and mutual inductance between field and damping winding loop. The mathematical model and inductance formulas were verified by comparison of inductance results and by the simulation and experiment results under the short circuit fault.
For the problem of hydro-generator stator winding internal faults transient current computational, the FEM coupled with electric circuit method, which effectively combined multi loop method and FEM analysis method, was adopted to analyze this problem detailed. SF600-42/1308 large hydro-generator was used as the object of study. In the no-load condition, sudden single-phase and two-phase and three-phase short circuit fault current were calculated, and the results was verified by classical analytic formula method;In the generator with three phase symmetric resistor load condition, turn to turn short-circuit and branch to branch short-circuit and branch to branch short-circuit with different phase transient process were analyzed, and the laws governing the internal faults of the hydro-generator were investigated. The result shows that the position of internal faults and the moment of internal faults occurring have great influence on the current after occurring internal faults. The different faults are also compared to find their influence on the internal faults of generator. It provides the theoretical support for good configuring main protection of hydro-generator.
In this study, an automatic voltage regulator fed by photovoltaic (PV) cells was designed for use in small power alternators. The design aimed to increase the energy output of the alternator and to ensure necessary current output from the PV cells to support the synchronous generator excitation system. For energy transformation from small renewable energy sources, designers have favored the use of alternators that produce higher quality electric power, rather than asynchronous generators. We also address the designed system's suitability for real-life application, examining the steady-state performance of the photovoltaic cell excited alternator (PVCEA) with various loads and its behavior in sudden load and fault situations.
In this paper, first, turn-to-turn fault in a synchronous machine stator winding is investigated, and its inductances are derived with the winding function method. Then, using the nonlinear control theory, the first-order Poincaré map of the synchronous machine is computed, and a new extended Poincaré map of the machine is established. Also, the characteristic multipliers of the synchronous machine are calculated by the Poincaré map. This new map is capable of analyzing electrical machines in the nonlinear and unbalanced cases for evaluation of stability, bifurcation, and chaos phenomena. The results show that the new map is an effective technique for modeling and analyzing any AC electrical machines. The characteristic multipliers and the Lyapunov exponents of the extended map indicate the system's condition. Chaotic phenomena are observed in some machines with internal faults. Since chaotic systems have a continuous spectrum and various kinds of frequencies in their spectrum, identification of the internal fault location is not easy with harmonic analyses such as Fourier spectrum method on the excitation current in synchronous machines. © 2013 Institute of Electrical Engineers of Japan. Published by John Wiley & Sons, Inc.
SUMMARY This paper presents a model for the analysis of internal faults of synchronous generators with multiple parallel connection paths for phase. Several kinds of faults between different portions of the stator windings (and ground, if the case) can be described in the subtransient time frame. Data necessary for the model derive from the electromagnetic design of the machine, as well as from details on equivalent parameters of the external grid and from the neutral connection characteristics. Such modeling procedure allows accurate insight of currents in windings and terminals, exploring each possible fault configuration. In the second part of this work, the developed model will be applied to assess the validity and to define possible improvements of conventional protection configurations against internal faults. Copyright © 2013 John Wiley & Sons, Ltd.
When an internal fault occurs in a synchronous generator, the symmetry between the parallel paths of the winding is broken and different currents flow in them, due to unsymmetrical magnetic linkage between the stator windings. The aim of this paper is to present a simulation model to investigate the effect of internal fault on the parallel path currents of a large synchronous generator using direct phase quantities. This model is based on a modified winding function approach where the machine inductances are calculated directly from the machine winding distribution using machine electrical parameters instead of the geometrical ones. The simulation results for different cases of internal faults in salient-pole and non-salient-pole synchronous machines have been obtained. Salient-pole synchronous generator has wave winding distribution while the non-salient-pole generator has lap winding arrangement. Due to different stator winding arrangements, the two machines have been simulated individually. By using the simulated fault data, a suitable numerical protection scheme for synchronous generators can be developed.
This paper presents the generator's internal fault modeling and analysis based on the one-machine infinite-bus power system for the simulation of internal faults in a synchronous generator. The model applied is a low-order synchronous generator model, and is based on the so-called fourth-order representation with the mechanical mode from the second-order system and the electrical mode from the second-order system. The test cases for internal faults demonstrates the output waveforms for id, iq, vd, vq, ω, Efd, Vt, Te, ia, ib, ic, va, vb, and vc using the MATLAB tool. Furthermore, the fault perturbations for internal windings and for an exciter terminal are also carried out and are compared by the results of four cases of simulations.
In this study, the winding function method is modified and used to calculate the machine inductances in two different cases, namely, a healthy case and a short-circuited stator winding case. For the first time, a numerical air-gap function is used in addition to a model of the air gap of salient-pole machines which considers the stator slot effects. The modified winding function (MWF) method introduced here is more precise when compared with previous methods. This MWF enables one to compute the air-gap function more accurately. Also a mathematical model that allows taking stator and rotor core saturation into account in both healthy and faulty conditions is presented in this study. Stator and rotor winding currents in healthy and faulty condition, obtained from experimental results, are compared with simulation results to verify the accuracy of the proposed method.
Inter-turn short circuit of field windings is a common electrical fault of generators. Simulation is an important method of investigating the fault and providing data support for fault monitoring. However, huge numbers of pole pairs and damper loops in large hydro-generators would lead to lengthy calculation time, hindering scientific research and engineering application. To deal with this problem, we analyze a theoretical basis for a damper winding simplified model and then propose an equivalent treatment method. Through the analysis of steady-state current harmonic characteristics of generators with different stator winding configurations during the fault, the simplified models suitable for steady-state calculation are derived from two aspects, namely, additional rotor harmonic current frequency characteristics and the relationship of the amplitude as well as the phase of each branch current of the stator. The calculation and experimental results of the two simplified models are then compared to verify the models’ correctness. A calculation example of the Three Gorges left bank VGS generator shows few deviations between the calculation results of the simplified model and the original model. Moreover, the calculation time using the simplified model is 1/1500 that using the original model, which provides a more effective tool for on-line fault monitoring. Finally, the sensitivity-verification application of the fault-monitoring scheme based on the stator steady-state unbalanced current RMS is depicted. The result shows that the scheme can monitor two-turn short circuits of field windings in the Three Gorges generator and provide high sensitivity.
The inter-turn short circuits fault of field windings is a common fault in generators. According to the multi-loop method of AC machines, this paper proposes multi-loop mathematical model of turbine generator with inter-turn short circuits fault of field windings. Based on the model, the normal operation and the short circuits fault operation of generator with no load have been simulated combining with the actual structure of a 300MW turbine generator. The time-varying mutual inductances between stator and rotor winding are calculated, from simulation results, the field current and stator voltages show the model reasonable under normal operation, After the field windings short faults, the field current increases gradually to a new constant value under the constant field voltage, the transient feature of stator voltage are obtained, which provides the basis for further fault analysis and fault diagnosis of turbine generator with field winding inter-turn short circuits.
The inter-turn short circuits fault of field windings is one common fault in generators. It will cause serious damages if such faults are not handled properly in time?For this reason, according to the multi-loop method of AC machines this paper proposes multi-loop mathematical model of turbine generators with inter-turn short circuits fault of field windings. The inductance parameters are calculated by air-gap permeance method and the calculated wave-form of inductance parameters shows reasonable. Combining with the actual structure of a 300MW turbine generator, the normal rated load operation and the short circuits fault operation of generator have been simulated based on the model. The normal stator winding currents and the fault currents difference in stator winding parallel branches of same phase were obtained. The currents of normal and fault field winding are calculated. The model considers the different structure and the harmonics in air gap magnetic field, which provides the basis for further fault analysis for turbine generator with inter-turn short circuits of field winding.
This paper presents a theoretical approach as far as faulty permanent magnet synchronous machines (PMSM) are concerned. The knowledge of PMSM's behavior under inter-turn fault is very useful in order to check drive performances under fault conditions. Control, stability and fault detection often require models related to electrical or mechanical faults. The aim of this paper is to develop a dynamic model of a PMSM under fault conditions without using circuit models coupled with finite element analysis (FEA). The method exposed in this paper only needs some preliminary FE computations that can be used in order to include winding distributions and leakage fluxes in the slots.
When synchronous machine transients in an extensive system are to be accurately calculated, many problems arise for which the information required to resolve them is not available. The resulting accuracy will depend upon the assumptions used not only in the development of the machine model but also in the way in which the incomplete model is solved. The primary problems involve the manner in which the machine terminal relations are related to those of the system and in the handling of the various torques produced in the transient operation. The state of the art of machine analysis is now such that synchronous machine transients are fairly well understood and calculations may be reasonably comprehensive. The system representation is, however, not capable of accepting the information available from the machine with the result that a satisfactory marriage is difficult. Some of the inadequacies of large scale stability analysis as it now exists and some of the precautions and planning that should be observed before embarking upon extensive studies or in drawing conclusions from such studies are covered here, in a philosophic and general manner.
Test methods for derivation of model parameters and validation of synchronous machine model structures for use in stability studies were investigated. The period of performance was April 1977 to December 1979. The conclusions are: (1) load rejection tests are practical and provide good data for derivation of machine model parameters, and for verification of adequacy of model structures; (2) state of the art second order model structures for both the d and q axis are adequate for the representation of machine flux dynamics in response to stator currents and modifications of the d axis model are needed to properly represent field current transients that are affected by rotor iron effects; and (3) improvements in saturation representation seem to offer the greatest potential for gains in machine model accuracy.
The paper describes a mathematical model for the simulation of a 3-phase synchronous machine using direct-phase quantities, thus obviating the need for any transformation. Numerical solution using a digital computer has also been described, and compared with digital simulation in transformed d-q-0- and ¿-Ã-0-axes models of a synchronous machine. The proposed model in direct-phase quantities enables a unified approach to be adopted in the study of both symmetrical and asymmetrical conditions. Since the constraints to be imposed are direct operating conditions, asymmetrical operating conditions can be studied very easily. Modifications required in the model to simulate various types of faults are described. Versatility of the proposed model is illustrated by the study of a single-line-earth fault with single-phase opening and automatic reclosure. It is shown that this type of fault can be studied as simply as, say, a 3-phase fault.
This paper discusses application of MATLAB(R)-Simulink(R) to
transient analysis of large synchronous machines, with specific
reference to the analysis of internal faults, and integration with the
computer program GENSIM
A synchronous machine with a fault in its armature winding may be represented by an equivalent circuit in three symmetrical components. The line and neutral terminals of the equivalent circuit are connected to a system network in each sequence. The fault point of the positive sequence is connected to the neutral of the negative sequence, and so forth, representing a single phase- to-ground, or turn-to-turn, fault. Electromotive forces behind the corresponding reactance are introduced in every branch of the positive-sequence network of the faulty machine and an equivalent electromotive force of the system is considered. The resulting network consists of three loops with four electromotive forces. The calculation of the current flow in this network is programmed for solution on the IBM-7090 digital computer.
This paper discusses a technique for partitioning the stator
windings of large synchronous machines, for application to internal
fault analysis, and determining corresponding winding inductances. The
method employs a direct phase representation, which is shown to reduce
to the classical phase representation when the partitioned windings are
collapsed
For pt.I see ibid., vol.15, no.4, p.376-9 (2000). This paper
applies techniques for analyzing internal phase and ground faults in the
stator windings of large synchronous machines. A variety of internal
fault conditions are considered, with results for a 75% ground fault
presented, along with comprehensive data for a test machine. The
methodology is validated by comparison with results obtained from
independent finite element analyses
An internal fault in the armature winding of a synchronous
generator occurs due to the breakdown of the winding insulation. In this
paper, a method for simulating internal faults in synchronous
generators, using direct phase quantities, is described. Simulation
results showing the fault currents, during a single phase to ground
fault and a two phase to ground fault, are presented
A three-phase synchronous machine model is developed for
unbalanced harmonic load flow analysis and for initializing EMTP
(electromagnetic transient program) transient simulations. Two nonlinear
effects, the frequency conversion and saturation, are represented in
conjunction with the machine load flow constraints. The model is in the
form of a frequency-dependent three-phase equivalent circuit. It can
therefore be easily incorporated into existing harmonic programs for
systemwide harmonic analysis. Thus, the generation of harmonies by
synchronous machines under various load flow conditions and the harmonic
interaction between machines and other harmonic sources can readily be
analyzed
Per-unit systems with special reference to electric machines
- M R Harris
- P J Lawrenson
- J M Stephenson
Chapter 8 Three-phase synchronous machine
- H W Dommel
- V Brandwajn