Finite-element calculation of leakage inductances of a saturated salient-pole synchronous machine with damper circuits

It is important to understand the relationship between leakage flux distributions and machine characteristics for better design of synchronous machines. This paper presents a calculation method for leakage inductances of saturated salient-pole machines with damper circuits. All leakage inductances are divided into the self-leakage, gap leakage, and winding-differential leakage inductances. The leakage inductances correspond directly to the leakage flux distributions in the machines. Cross-magnetizing inductances are also calculated. The method is applied to a 300-MVA class generator. The relationship between the winding distributions, flux distributions, saturation, and inductances is discussed. Weak magnetic influence of the damper circuits on the armature in the d-axis is quantitatively illustrated through values of the winding-differential leakage inductances. The cross-magnetizing inductances, except for the d-axis damper circuits, are relatively large. Variations of the armature self-leakage inductances with saturation are small, and variations of the field and damper self-leakage inductances are larger.     

A new model of saturated synchronous machines for power systemtransient stability simulations

This paper presents a new method to express the main flux saturation in synchronous machines. In the new method, the saturation is expressed by auxiliary currents and unsaturated magnetizing inductances instead of the saturated inductances. The new model using the currents contains only constant coefficients defined in terms of the unsaturated magnetizing inductances     

Methods for determining the intermediate-axis saturation Characteristics of salient-pole synchronous Machines from the measured D-axis Characteristics

In the analysis of the steady-state performances of saturated synchronous machines using the classical two-axis (d- and q-axis) frame model, the accurate calculation of the machine performances depends to a large extent on their saturation conditions. However, the effect of saturation depends not only on the saturation level in the axis of the resultant machine ampere-turns (intermediate-axis) but also on the phase angle between the resultant ampere-turns and the resultant magnetic flux. This work presents four analytical methods for determining the intermediate-axis saturation characteristics of salient-pole synchronous machines from the measured d-axis saturation characteristics. The accuracies of these four methods have been verified by comparing the measured field currents and load angles of two laboratory salient-pole synchronous machines of different designs with those calculated using an innovated approach which uses the intermediate-axis saturation characteristics directly in the modeling of the saturated synchronous machines. Moreover, the calculated intermediate-axis saturation characteristics have been compared with the measurable ones in the case of one of these machines.     

Magnetic saturation effects on the control of a synchronous reluctance machine

Experimental results on the magnetic saturation effects on the control of a small synchronous reluctance machine (600 W) are presented in this paper. A new model of the machine, including saturation effects and cross magnetization is first developed. An approach based on total and mutual inductances is followed instead of the traditional approach through magnetizing and leakage inductances. All of the electrical and mechanical parameters measurements required for the simulations are given. A rotor-oriented vector control of the synchronous reluctance machine is achieved with a digital-signal-processor (DSP) board (TMS320C31) and experimental results are presented. When the magnetic circuit is saturated, the simulation with the developed model shows good, accurate results when compared to the experimental ones.     

Modelling of magnetic saturation in smooth air-gap synchronousmachines

Main flux saturation in d-q axis representation of synchronous machines is at present modelled by selecting either all the winding currents or all the winding flux linkages as state-space variables. However, these two available models are just a tiny portion of the complete set of models that can be obtained by selecting other combinations of state-space variables. This paper presents a general procedure for main flux saturation modelling in smooth air-gap synchronous machines, that is applicable for most selections of state-space variables. The method relies on the concept of `generalised flux' and `generalised inductance', that has been successfully applied in modelling of saturated single-cage and double-cage induction machines. The concept is extended to saturation modelling in smooth air-gap synchronous machines. A number of models, that result from the application of the method for different selections of state-space variables, are presented in detail     

Double winding, high-Voltage cable wound Generator: steady-State and fault analysis

This paper describes a detailed circuit representation of synchronous machines equipped with two similar or dissimilar armature windings. The equivalent circuit at steady state is presented. The self and mutual leakage inductances of the two sets of three-phase stator windings are evaluated by the finite element method as well as an analytical method. The simulation results from a steady-state analysis as well as a fault analysis are verified experimentally on the world's first high-voltage generator Powerformer installed in Porjus, Sweden, rated at 45 kV and 11 MVA.     

Steady-state magnetic circuit analysis of salient-pole synchronous machines considering cross-magnetization

A simple magnetic circuit of salient-pole synchronous machines is presented. A method for calculating steady-state electrical performances using the circuit is described. Magnetic saturation including the cross-magnetizing phenomenon is considered in the method. The saturated reactances, field currents, and load angles are calculated easily from the method without much computational effort as compared with the finite element method. Despite the simplicity of the magnetic circuit, the calculated reactances at various load conditions using the method are close to the measured and FEM values.     

A novel method for modeling dynamic air-gap eccentricity insynchronous machines based on modified winding function theory

This paper presents the modeling of synchronous machines under eccentric rotors. The winding function theory accounting for all space harmonics and presented by earlier researchers has been modified to adopt a nonsymmetric air-gap for the calculation of machine winding inductances. The effect of dynamic air-gap eccentricity on the inductances of a salient-pole synchronous machine using the modified winding function approach (MWFA) has been discussed. The coupled magnetic circuits approach has been used for simulating the machine behavior under healthy and eccentric rotor conditions. The simulation results are in close agreement with the experimental results     

Monitoring of magnetic saturation level of a synchronous generatorunder load

This paper presents a new method to monitor magnetic saturation level of a synchronous generator operating in an electric power system. The proposed method makes use of air-gap permeance distributions, which can be derived from air-gap flux distributions measured by a search coil wound around a stator tooth of a generator. Tests were conducted on a laboratory-model power system to verify the validity of the proposed monitoring method. Test results under various operating conditions are presented to show a wide range of change in saturation level inside the generator. Its effects on generator performance in the power system and the benefits of the monitoring method are also discussed     

Impact of dynamic cross-saturation on accuracy of saturatedsynchronous machine models

Cross-saturation effect in synchronous machines has been a subject of considerable attention. Inclusion of the cross-saturation in the machine model has two consequences. The first one, called here “steady-state cross-saturation”, consists of dependence of the steady-state d-q axis magnetizing inductances on the currents in both axes. The second one, termed here “dynamic cross-saturation”, is the existence of nonzero elements in the system matrix, that describe cross coupling between d- and q-axis. Dynamic cross-saturation appears in all the saturated machine models, regardless of the selected set of state-space variables, with the exception of the winding flux linkage state-space model. The aim of this paper is to compare behavior of various models when dynamic cross-saturation is neglected. It is shown that the impact of dynamic cross-saturation on accuracy depends on the selected set of state-space variables. In the majority of cases omission of dynamic cross-saturation leads to very inaccurate results. However, it is found that for one particular class of models, omission of dynamic cross-saturation has practically no impact on accuracy. These models therefore fully describe the complete saturation effect by means of only continuous variation of the d-q axis magnetizing inductances     

Three dimensional magnetic fields in extra high speed modifiedLundell alternators computed by a combined vector-scalar magneticpotential finite element method

A three-dimensional (3-D) finite-element (FE) approach was developed and implemented for computation of global magnetic fields in a 14.3 kVA modified Lundell alternator. The essence of the method is the combined use of magnetic vector and scalar potential formulations in 3-D FEs. This approach makes it practical, using state-of-the-art supercomputer resources, to globally analyze magnetic fields and operating performances of rotating machines which have 3-D magnetic flux patterns. The 3-D FE computed fields and machine inductances as well as various machine performance simulations of the 14.3-kVA machine are presented     

Methods for determining the q-axis saturation characteristics of salient-pole synchronous machines from the measured d-axis characteristics

For the accurate analysis of salient-pole synchronous machines using the two-axis frame models, the direct- (d-) and quadrature-axis (q-axis) saturation characteristics are needed. Usually, the d-axis saturation characteristics can be obtained easily by the conventional open-circuit test with the machines excited from their field winding. On the other hand, the q-axis saturation characteristics of synchronous machines cannot be measured applying simple, conventional methods, and thus, they are usually not available. In this paper, four different methods for calculating the q-axis saturation characteristics of salient-pole synchronous machines from the measured d-axis saturation characteristics are explored. In these methods, the q-axis saturation characteristics can be calculated from the readily available test data, namely the d-axis saturation characteristics and the d- and q-axis unsaturated magnetizing reactances. Finally, a comparison between these methods is made.     

An Accelerated Newton-Raphson Technique for Finite Element Analysis of Magnetic Fields in Electrical Machines

An accelerated (modified) Newton-Raphson technique is presented. The new method reduces the computational (CPU) time required to carry out a finite element magnetic field analysis in saturated electrical machines by up to about 60% of typical CPU times required by present state of the art methods. In order to illustrate practical use of the method, the technique is applied to the problem of field determination in a 6-pole, 18 slot, samarium cobalt permanent magnet brushless dc motor system rated at 120 Volt, 15 hp (11.2 KW). The investigated cases include overloads of more than 4 times the rated armature current, thus insuring a substantial degree of magnetic saturation throughout the flux paths in the stator (armature) teeth and yoke. Numerical results of the field solution and armature winding inductances obtained using this method are compared with those obtained using the standard state of the art Newton-Raphson method. It is found that practically no numerical errors are introduced as a result of use of this accelerated Newton-Raphson process. Meanwhile, substantial savings can be made in the computation cost incurred for such analysis, particularly in computer-aided design situations in which large numbers of field solution cases are often required.     

Coupled finite-element/state-space modeling of turbogenerators inthe abc frame of reference-the no-load case

This first of two companion papers centers on applying a coupled finite-element/state-space technique to the determination of self and mutual winding inductances of a 733 MVA turbogenerator and computing its open-circuit characteristic, in the natural abc frame of reference. In this method, the apparent self and mutual inductance profiles of the armature and field windings, expressed as functions of rotor position angle, are computed from a series of magnetic field solutions performed at uniformly-distributed samples of rotor positions covering the entire 360° electrical cycle, using the energy perturbation method. These inductances, which are obtained at no-load for three different excitation levels, include the full effect of space harmonics introduced by the magnetic circuit geometry, winding layouts and magnetic circuit saturation. The abc-frame/finite-element computed open-circuit characteristic is in excellent agreement with the test results. This computed no-load set of parameters forms the initial data for simulation of the full-load performance given in the companion paper, including the full impact of space harmonics and saturation on the flux linkage, current and voltage waveforms, and other performance parameters     

Practical detailed model for generators

The Parks equation is currently used for mathematical expressions of the state of synchronous generators connected to power systems. The calculation results using the Parks equation, however, differ greatly from the results of measurement on the real generator when the Parks equation is applied to an event where the flux linking the generator undergoes a very large change as when the generator is suddenly disconnected from a load during normal operation. The difference between the field test result and the calculation result is larger for a turbogenerator with a solid iron rotor than for a salient synchronous machine. To deal with this problem, this paper introduces a mathematical expression of generating machines, which is an extension of the Parks equation and which is practically applicable to an event in which the flux within the generating unit changes sharply as when it experiences a load rejection. The newly introduced expression takes the following considerations: (1) a magnetic flux saturation coefficient k is applied to d- and q-axis; and (2) T'_q0 that corresponds to T'_d0 of d-axis is introduced to q-axis. A generator model incorporating the magnetic flux saturation coefficient k and another parameter T'_q0 has proved capable of providing a good simulation representing a load rejection whose calculation results coincide satisfactorily with the measurements of actual generators     

Stray losses in large synchronous machines

The results of an investigation of the components of stray losses in large salient pole synchronous machines are presented. Stray losses are caused by the magnetic leakage and fringing flux in the rotating machine, and their prediction is subject to uncertainties in the machine design. A 750 HP experimental machine has been designed, constructed, and extensively instrumented and tested for direct measurement of the loss components. The nature and distribution of the stray losses in various parts of the machine are described     

Online Parameter Estimation of Round Rotor Synchronous Generators Including Magnetic Saturation

Round rotor synchronous generator parameters are identified from online data measured at the terminals of the machine. Parameter estimation results from two generators are presented. Data sets at different operating points are considered to examine the consistency of the proposed method at varying operating conditions. A magnetic saturation model is implemented in the estimation procedure to account for the effect of saturation on generator inductances. Inclusion of saturation in the machine model is found to improve the estimation results. Noise filtering and bad data detection and rejection techniques are employed to increase the reliability of the estimates. An observer is used to estimate the damper winding currents that are otherwise unknown. The estimated parameters are compared to available manufacturer data and the accuracy of the method is assessed.     

Coupled finite-element/state-space modeling of turbogenerators inthe abc frame of reference-the short-circuit and load cases includingsaturated parameters

In this paper, a coupled finite-element/state space modeling technique is applied in the determination of the steady-state parameters of a 733 MVA turbogenerator in the abc frame of reference. In this modeling environment, the forward rotor stepping-finite element procedure described in a companion paper is used to obtain the various machine self and mutual inductances under short-circuit and load conditions. A fourth-order state-space model of the armature and field winding flux linkages in the abc frame of reference is then used to obtain the next set of flux linkages and forcing function currents for the finite-element model. In this process, one iterates between the finite-element and state-space techniques until the terminal conditions converge to specified values. This method is applied to the determination of the short-circuit, and reduced- and rated-voltage load characteristics, and the corresponding machine inductances. The spatial harmonics of these inductances are analyzed via Fourier analysis to reveal the impact of machine geometry and stator-to-rotor relative motion, winding layout, magnetic saturation, and other effects. In the full-load infinite-bus case, it is found that, while the three-phase terminal voltages are pure sinusoidal waveforms, the steady-state armature phase currents are nonsinusoidal and contain a substantial amount of odd harmonics which cannot be obtained using the traditional two-axis analysis     

Three dimensional magnetic field computation by a coupledvector-scalar potential method in brushless DC motors with skewedpermanent magnet mounts-the no-load and load results

The coupled magnetic vector potential-magnetic scalar potential (CMVP-MSP) method of computation of 3D magnetic fields by finite elements (3D-FE) is applied here to a brushless DC motor with skewed permanent magnet mounts on its rotor. Results of the CMVP-MSP and 3D-FE computation of the magnetic field and associated motor parameters (EMFs and armature inductances) are detailed in this paper. These results demonstrate vividly the three dimensional nature of the computed flux distributions, caused by the torque ripple reduction design employing skewed magnet mounts on the rotor. Experimental evidence supporting the validity of the BD-FE field computations, through comparison between computed and measured armature EMF waveforms is also provided in this paper     

Analysis of saturated synchronous reactances of large turbinegenerator by considering cross-magnetizing reactances using finiteelements

This paper describes a method to calculate saturated reactances of synchronous machines at various loads under steady-state operation by considering influence of cross-magnetizing phenomenon between d- and q-axis circuits. In the method, mutual reactances x_dq due to the cross-magnetizing phenomenon can be calculated on the basis of two-dimensional magnetic field analysis. The validity of analysis is shown by good agreement between measured and calculated field current values of a 592 MVA turbine generator at various loads. It is clear that quantitative accuracy requires the magnetic coupling between d- and q-axis circuits be considered. Subsequently, steady-state reactances at various loads are analyzed by considering mutual reactances x_dq due to cross-magnetizing phenomenon. The relation between the load angle δ and mutual reactance x_dq is clarified, and the difference between equivalent synchronous reactances that are obtained by considering x_dq and conventional defined synchronous reactances that do not consider x_dq is shown numerically