Technical Field

[0001] The invention relates to electrical power generation. Embodiments provide methods for detecting loss of excitation in synchronous generators as well as generator systems comprising systems for detecting loss of excitation conditions.

Background

[0002] A synchronous generator produces an AC output having a frequency that depends on the speed of rotation of the generator. A typical synchronous generator has a rotor that is mounted on a shaft for rotation relative to a stator. The rotor is driven by an engine, steam turbine, water-driven turbine or other prime mover. A magnetic field is generated by a field winding. The magnetic field induces an AC voltage in armature windings. The field windings may be rotating or stationary. The armature windings may be stationary or rotating. [0003] DC current is supplied to the field winding by an excitation system. A wide range of excitation systems have been developed. The excitation system may regulate excitation current in response to generator load and/or other factors. Excitation systems are known to fail for various reasons. For example, a field winding may be come shorted or may break so that it is an open circuit, components that generate, regulate or carry excitation current to the field winding may break or malfunction etc.

[0004] If a generator experiences a loss of excitation (LOE) while the generator remains connected to an energized power system (such as a power grid) then the generator may be damaged. Such damage may occur, for example, because a generator that has lost excitation may absorb reactive power from the system. Damaging high currents in armature windings and end core heating can be produced in the generator when this happens. LOE can also result in mechanical damage to a generator. Such mechanical damage may result from torque pulsation in the shaft following a loss of synchronism. A LOE condition may cause permanent damage to a generator.

[0005] Various systems have been developed to detect LOE conditions and to automatically protect a generator by disconnecting the generator from an electrical power system in the event of an LOE condition. Such systems may monitor various parameters. A problem faced by the designers of such systems is that the values that such parameters may have in the event of a LOE may also occur during normal operation of a generator. A protective system configured to automatically take a generator offline when monitored parameters have certain values selected for indicating LOE may undesirably operate to take the generator offline when the generator is operating normally. For example, power swing conditions and certain external fault conditions could cause such a protective system to operate. [0006] The following references discuss various approaches that have been used to detect LOE.

[1] Mason, C.R.; "A New Loss-of-Excitation Relay for Synchronous Generators", AIEE Trans. Part III, vol. 68, pp. 1240-1245, 1949.

[2] Tremaine, R.L., and Blackburn, J.L.; "Loss-of-Field Protection for Synchronous Machines", AIEE Trans. Part III: Power Apparatus and Systems, vol. 73, No. 1, pp. 765- 772, 1954.

[3] Bisbee, J.M., Rosol, M.F,, et.al.; "A Partial Survey of Relay Protection ofSteam- Driven A-C Generators", AIEE Trans. Part III, vol. 80, pp. 954-957, 1962.

[4] Mackenzie, W.F., Imhof, J.A., et.al.; "Loss-oj "-Field Relay Operation During System Disturbances Working Group Report - June 1971", IEEE Trans, on Power Apparatus and Systems, vol. PAS-94, no. 5, pp. 1464-1472, Sep./Oct. 1975.

[5] Berdy, J.; "Loss of Excitation Protection for Modern Synchronous Generators", IEEE Trans, on Power Apparatus and Systems, vol. PAS-94, no. 5, pp. 1457-1463, 1975.

[6] Lee, D.C., Kundur, P., and Brown, R.D.; "A High Speed, Discriminating

Generator Loss of Excitation Protection" , IEEE Trans, on Power Apparatus and Systems, vol. PAS-98, no. 6, pp. 1895-1899, Nov/Dec. 1979.

[7] Herrman, H.J., and Smit, A., "Increased Sensitivity of Loss of Field Protection based on Admittance Measurement" , Western Protective Relay Conference, Washington State University, pp. 1-15, 2009

[8] Tambay, S.R., and Paithankar, Y.G.; "A New Adaptive Loss of Excitation Relay Augmented by Rate of Change of Reactance" , Proceedings of the IEEE Power Engineering Society General Meeting, PES-GM, vol. 2, pp. 1831-1835, June 2005. [9] Li, L., Caixin, S. and Daohuai, M.; "Study on the Excitation Protection and Control of Synchronous Generator Based on the delta and s", Proceedings of the IEEE PES Transmission and Distribution Conference & Exhibition: Asia and Pacific, pp. 1-4, August 2005.

[10] Usta, O., et.al., "A New Relaying Algorithm to Detect Loss of Excitation of Synchronous Generators", Turkey Journal of Electrical Engineering, Vol. 15, No. 3, pp. 339-349, 2007

[11] Yaghobi, H., et.al., "A Novel Flux-Based Method for Synchronous Generator Loss of Excitation Protection" , 25th International Power System Conference, PSC 2010, pp. 1- 14, 2010 [12] Shi, Z.P., et.al., "The Comparison and Analysis for Loss of Excitation Protection Schemes in Generator Protection" , International Conference on Developments in Power System Protection, DPSP, pp. 1-6, 2012

[13] Lee, C.H., et.al., "Lessons Learned from the Generator Loss of Field at a

Cogeneration Thermal Plant in Taiwan", IEEE Transactions on Power Systems, Vol. 26, No. 4, pp. 2093-2100, 2011

[14] Siwang, Y., et.al., "Discussion on Setting Calculation of Loss-of-Excitation Protection for Large Turbogenerator", International Conference on Electrical Machines and Systems, ICEMS, pp. 1413-1416, 2010

[15] Sharaf, A.M., and Lie, T.T.; "ANN Based Pattern Classification of Synchronous Generator Stability and Loss of Excitation" , IEEE Trans, on Energy Conversion, vol. 9, no. 4, pp. 753-759, Dec. 1994.

[16] So, K.H., Heo, J.Y., Aggarwal, R.K., and Song, K.B., "Out-of-step detection algorithm using frequency deviation of voltage" , IET Generation, Transmission and Distribution, Vol. 1, No. 1, , pp. 119-126, January 2007

[17] Bo,F., et.al., "The research UL-P of loss of excitation protection for generator based on the artificial neural networks", Asia Pacific Power and Energy Engineering Conference, APPEEC 2009, pp. 1-4, 2009

[18] Morais, A.P., Cardoso, G., Mariotto, L.; "A« Innovative Loss-of-Excitation Protection Based on the Fuzzy Inference Mechanism" , IEEE Trans, on Power Delivery, vol. 25, no. 4, pp. 2197-2204, Oct. 2010.

[19] Bi,T., et.al., "Adaptive Loss of Field Protection Based on Phasor Measurements" , Power and Energy Society General Meeting, pp. 1-4, 2011

[20] Seethalekshmi, K., Singh, S.N., and Srivastava, S.C., "A Classification Approach Using Support Vector Machines to Prevent Distance Relay Maloperation Under Power Swing and Voltage Instability", IEEE Transactions on Power Delivery, Vol. 27, No. 3, pp. 1124-1133, July 2012 [21] IEEE Std. C37.102, IEEE Guide for AC Generator Protection, 2006

[22] IEEE, Tutorial on the Protection of Synchronous Generators, Special Publication of the IEEE Power System Relaying Committee, Second Edition, 2011

[23] Mason, C.R., The Art and Science of Protective Relaying, General Electric Co., John Wiley and Sons, 1967 [24] Reimert, D.; Protective Relaying for Power Generation Systems, CRC Press, 2006

[25] IEEE Std. 421.5 , IEEE Recommended Practice for Excitation System Models for Power System Stability Studies - 2005, April 2006

[26] US20130250458A1 LEVERAGING INHERENT REDUNDANCY IN A

MULTIFUNCTION IED

[27] US 2008007481 OA 1 APPARATUS AND METHOD FOR PROVIDING

PROTECTION FOR A SYNCHRONOUS ELECTRICAL GENERATOR IN A POWER SYSTEM [28] US7710693B2 Apparatus and method for providing protection for a synchronous electrical generator in a power system

[29] US5453901A Detection and protection of excitation system from diode failure [30] US2917672A 1 Loss-of-field relays

[31] US2728071A Stability reserve indicator for synchronous dynamoelectric machines [32] US 2584765 A Distance relay protective system

[33] US2523771A Loss of excitation protection

[34] US2509025A Protective system for fault and out-of-step conditions

[35] US2425759A Impedance type distance relay

[0007] There is a need for practical methods for detecting LOE in synchronous generators and for generator systems and apparatus that can automatically detect LOE conditions that have improved selectivity for LOE.

Summary

[0008] This invention has a range of different aspects. One aspect provides methods for detecting loss of excitation conditions in synchronous generators. Another aspect provides systems for detecting loss of excitation conditions in synchronous generators. Another aspect provides generator protection systems that provide loss of excitation detection. Another aspect provides excitation control systems that provide loss of excitation detection.

[0009] One aspect of the invention provides a method for detecting loss of excitation in a synchronous generator connected to supply power to a power grid. The method comprises monitoring currents and voltages at outputs of the generator and processing the monitored currents and voltages to yield positive sequence current and voltage phasors. The method processes the positive sequence current and voltage phasors to determine a plurality of features. The processing yields complex power data and, in some embodiments, yields both complex power data and complex impedance data. The method performs pattern recognition on the plurality of features to determine whether or not the plurality of features correspond to a loss of excitation condition. At least one of the features characterizes a rate of change of the complex power within a time frame of 250 ms or less.

[0010] In some embodiments the plurality of features include at least a directness measure that compares a length of a trajectory of a complex power in the complex power plane over a time window to a magnitude of a total change of the complex power during the time window; and the maximum rate of change of the complex power within a time frame of 15 to 75 ms that occurs during the time window.

[0011] In some embodiment, the method generates at least the following features: a) a distance in the impedance plane between a monitored impedance and a specified point in the impedance plane; b) an average rate of change of the imaginary part of a monitored complex power (i.e. of the reactive power); c) a directness measure that compares a length of a trajectory of a complex power in the complex power plane over a time window to a magnitude of a total change of the complex power during the time window; and d) a rate of change measure that indicates a maximum rate of change of the complex power over the time frame within a time window. [0012] Another aspect provides apparatus for detecting loss of excitation in synchronous generators. The apparatus includes a data processor or other circuit elements configured to perform a method as described herein. The apparatus may be integrated into a generator protection system or an excitation system, applied as a stand-alone system or used in combination with another system, for example as a supervisory system. [0013] Another aspect provides generator systems and generator protection systems which incorporate apparatus and/or perform methods as described herein.

[0014] Another aspect provides a component, which may be a software plug-in component, for a power system simulation tool which incorporates apparatus and/or performs methods as described herein. [0015] Further aspects of the invention as well as features of example embodiments are described herein and/or illustrated in the accompanying drawings.

Brief Description of the Drawings

[0016] Exemplary embodiments are illustrated in referenced figures of the drawings. It is intended that the embodiments and figures disclosed herein are to be considered illustrative rather than restrictive.

[0017] Figure 1 is a block diagram showing an example generator connected to a power grid.

[0018] Figure 1 A is another block diagram showing an example generator connected to a power grid with details of an example generator protection system.

[0019] Figure IB is a flow chart showing a method according to an example embodiment.

[0020] Figures 1C, ID and IE show time evolutions of complex impedance and power for example simulated power swing conditions.

[0021] Figures 2A, 2B and 2C show time evolutions of complex impedance and power for example simulated loss of excitation conditions.

[0022] Figures 3A and 3B illustrate time evolutions of derived features for use in classification for a power swing event and a loss of excitation event respectively.

[0023] Figure 4 is a block diagram illustrating a generator system equipped with control systems that include a generator protector system.

[0024] Figure 5 is a block diagram showing an example generator system according to an embodiment of the invention.

[0025] Figures 6 to 11 are tables listing information relating to various simulations of fault conditions and the results of classifying feature vectors obtained from those simulations.

[0026] Figure 12 illustrates a location of the mho center for an example generator in the complex impedance plane.

[0027] Figure 13 is a network diagram illustrating a network modeled to test performance of an example prototype embodiment of the invention.

[0028] Figure 14 is a model of an excitation control.

[0029] Figure 15 is a table listing example parameters for a simulation using the excitation control model of Figure 14.

[0030] Figure 16 is a model of an under-excitation limiter (UEL).

[0031] Figure 17 is a table listing example parameters for a simulation using the UEL model of Figure 16. [0032] Figures 18A and 18B show a trajectory respectively in the PQ plane and in the impedance plane for a temporary system overvoltage condition.

[0033] Figure 19 shows parameter values for the parameters XI, X2, X3 and X4 (which are described below) during a simulation of a temporary system overvoltage condition.

Description [0034] Throughout the following description specific details are set forth in order to provide a more thorough understanding to persons skilled in the art. However, well known elements may not have been shown or described in detail to avoid unnecessarily obscuring the disclosure. Accordingly, the description and drawings are to be regarded in an illustrative, rather than a restrictive, sense. [0035] Figure 1 shows a generator 5 comprising an excitation system 5 A and output terminals 6 connected to deliver power to a power grid 7. Generator 5 is driven by a prime mover 4 (e.g. an engine, turbine, etc.). A protective system 8 is provided to disconnect generator 5 from power grid 7. Protective system 8 may be triggered by various conditions. In this example, protective system 8 is operated to disconnect generator 5 from power grid 7 in the event of a loss of excitation (LOE) in generator 5.

[0036] Figure 1 A is a block diagram showing a generator 5 connected to deliver electricity to power grid 7 with a more detailed example generator protection system. In Figure 1 A, generator protection system 8 comprises a controller 8 A having signal inputs 8B, 8C and 8D. Input 8B provides a measure of current at the generator terminals. Input 8C provides a measure of voltage at the generator terminals. Input 8D provides a measure of voltage and current provided by excitation system 5A to field coils of generator 5.

[0037] The example generator protection system 8 of Figure 1 A has control outputs 8E, 8F and 8G. Output 8E is connected to control high voltage breaker 8H which, when tripped, can disconnect generator 5 from power grid 7. Output 8F is connected to control breaker 8J which, when tripped, can cut off field current in generator 5. Output 8G is connected to control prime mover 4 (for example by triggering shutdown of prime mover 4).

[0038] As described above, loss of excitation (LOE) can arise for a variety of reasons and can result in serious problems unless LOE is detected and suitable remediation initiated in a timely fashion. One aspect of this invention provides a method that monitors a combination of parameters. The parameters include derived parameters that have been found by the inventors to be particularly effective for sensitive and selective detection of LOE. These parameters may be selected to allow LOE conditions to be distinguished from power swing conditions and other non-LOE conditions. The method may initiate suitable remediation when the parameters indicate a LOE condition. By way of non- limiting example, the method may be performed by a generator protection system 8 as shown in Figure 1 or 1A.

[0039] Figure IB illustrates a method 10 according to an example embodiment. Method 10 may be performed continuously. In block 12, method 10 acquires data from which the complex power and/or the complex impedance seen by the generator can be determined. Block 12 may, for example, comprise monitoring voltage and current at output terminals 6 of the generator. Block 12 yields a sequence of measurements. In block 14 the measurements are processed to yield a plurality of features. Examples of these features are set out below. In some embodiments the features include features that characterize changes of derived quantities (such as complex power or complex impedance) with time.

[0040] In block 16 the features produced in block 14 are processed by a pattern recognition or classification algorithm. If the algorithm of block 16 detects a LOE condition then a signal is generated in block 18. In block 19 an action is taken in response to the signal of block 18. A range of one or more actions may be taken. For example, block 19 may comprise one or more of: triggering a breaker to disconnect the generator from a power grid; reconfiguring an excitation system (for example by switching to a redundant set of field windings or switching in a redundant excitation generation or control circuit or device); signaling a grid control system; or the like. [0041] Some embodiments monitor voltage and current at output terminals of a generator. Values for voltage and current may be sampled at a suitable sampling rate. In some embodiments the sampling rate is selected to provide a convenient number of samples for each cycle of generated power. For example, where a generator operates to produce power having a frequency of approximately 60 Hz, a sampling rate of 15360 samples/sec yields approximately 256 samples per cycle. Having a number of samples that is a power of two is convenient for further processing as described below. Higher or lower sampling rates could be used. In general, it is desirable to provide an anti-aliasing filter prior to sampling the instantaneous voltage and current values. For example, where the sampling rate is 15360 Hz a low-pass anti-aliasing filter having a cutoff frequency of 7680 Hz (50% of 15360) or lower should be provided.

[0042] Where the generator is a poly-phase generator (most typically a three-phase generator) the measurements of current and voltage may be made for one or more phases. Measurements for a single phase will suffice in some applications. The instantaneous measurements may be processed to yield phasors. This may be done, for example, using a Fourier transform algorithm. The phasors may be generated by hardware devices or by software processes or a combination thereof. The discrete Fourier transform (DFT) is convenient. There are available DFT implementations for both hardware and software. In an example embodiment, current and voltage phasors V and / are respectively obtained by performing DFT processing using 256 samples of the voltage and current at the generator terminals. In some embodiments DFT is performed with angle normalization so that a steady state fundamental frequency condition produces a fixed phasor value.

[0043] In an example embodiment, voltage, V, and current, /, are monitored at output terminals of a three-phase generator and the discrete measurements of V and / are converted to phasors.

[0044] It can be desirable to sample the instantaneous voltage and current for multiple phases (for example, 3 phases in a three phase system) and to combine the results together to yield a positive sequence voltage and current. For example, the positive sequence voltage may be determined according to:

Vi = (Va + aVb + a ² Vc)/3 (1) where Vi is the positive sequence voltage, a is a complex operator that has the effect of rotating a phasor by 120 degrees and V _a , V _b , and V _c are voltage phasors for three phases as measured at an output of the generator. Current phasors can be combined to yield a positive sequence current in a similar manner. [0045] In some embodiments, three phase voltage and three phase current phasors are produced for every sample measured (e.g. 256 sets of phasors may be acquired per cycle). These phasors may then be combined to yield positive sequence voltage and current phasors, also at the rate of 256 per cycle.

[0046] It is typically not necessary to monitor phasors V and / continuously. Monitoring for LOE may be performed using information obtained at discrete times that are reasonably frequent in comparison to the typical time scale for onset of an LOE condition. In some of the examples presented below values for the phasors V and / are obtained at a rate of 20Hz. In such examples, in each 1 -second period 20 points (each comprising a voltage phasor Vu and a current phasor with k an index in the range of 0 to 19) are acquired. These points may be distributed evenly over a one second time window. This is equivalent to sampling the phasors.

[0047] It is generally desirable to provide suitable filtering to avoid aliasing. For example, The positive- sequence phasors may be filtered using a suitable low-pass filter. In an example embodiment a second order Butterworth low-pass filter is used. The filter may have a cutoff frequency low enough relative to the rate the phasors are considered to avoid aliasing (e.g. at a frequency that is lower than or equal to ½ the sampling frequency). For example, where the phasors are sampled at a rate of 20 Hz, the filter may have a cutoff frequency of 10 Hz. The real and imaginary components of the positive sequence phasors are filtered separately in some embodiments. [0048] A window of approximately 1 second is appropriate because one second is a typical time used in the coordination of an LOE function with power swing conditions. The window may be longer or shorter than one second. The number of phasors obtained in each window may be more or fewer than twenty. The window length and number of phasors used is a matter of design choice. However, it is generally desirable and sometimes necessary to detect and take action in response to a LOE condition before loss of synchronism happens. Features that are useful for detecting LOE may no longer be present after loss of synchronization occurs. Therefore, the window should not be too long and the interval between acquiring phasors should not be too long.

[0049] Complex impedance, Zu, may be determined according to: where φ is the phase angle between the voltage and current phasors.

[0050] Complex power, ¾ may be determined according to:

S _k = I _k V _k e-w. (3)

[0051] When a generator experiences LOE, the complex impedance and complex power, as determined from values of V and / measured at the generator terminals will evolve along trajectories in the complex impedance plane and the complex power plane respectively. It is common and convenient to scale the power and impedance values according to generator ratings so that the power and impedance values are expressed 'per unit' (pu). This is not required, however.

[0052] In an example embodiment the parameters monitored include: a) a distance in the impedance plane between a monitored impedance and a specified point in the impedance plane; b) an average rate of change of the imaginary part of a monitored complex power (i.e. of the reactive power); c) a directness measure that compares a length of a trajectory of a complex power in the complex power plane over a time window to a magnitude of a total change of the complex power during the time window; and d) a rate of change measure that indicates a maximum rate of change of the complex power within the time window. Each of these parameters is described in more detail below. Other embodiments monitor only a subset of these parameters or monitor one or more of these parameters in combination with one or more other parameters.

[0053] One monitored parameter is a distance in the complex impedance plane between a monitored impedance and a specified point in the impedance plane. During a LOE condition, the complex impedance Z tends to follow a trajectory that passes through a segment of the negative imaginary axis. The specified point is preferably on or near to this segment. In an example embodiment, the specified point is on the negative imaginary axis at a midpoint of the segment. The segment may be the segment extending between the value -j¾ and the value -jX where -X'd is the subsynchronous reactance and -Xd is the synchronous reactance. In such embodiments, the specified point may be at the location (0, X (- -Xy--X r _f )) · This location may be called the 'mho center' .) Figure 12 shows the location of the mho center 88 in the complex impedance plane for an illustrative example.

[0054] A specific example of a way to derive a value, xj, representing the distance in the complex impedance plane between a monitored impedance and an appropriate specified point in the impedance plane is provided in Equation (4) in which Zi ₉ is the most recent complex impedance value in the current window (which may, for example, be a one- second window).

[0055] An average rate of change of the imaginary part of a monitored complex power may be determined by considering how much the imaginary part of the complex power (reactive power) changes during a time period (e.g. a time window of suitable length). This may be done, for example, by subtracting the imaginary part of one complex power measurement ¾ from another. For example, the difference in the imaginary part of the complex power at the beginning of a time window may be subtracted from the imaginary part of the complex power at the end of the time window. [0056] A specific example of a way to derive a value, x _{2 ,} representing a rate of change of the imaginary part of a monitored complex power is shown in Equation 5. In Equation 5, So and Sw are respectively the earliest and the most recent complex power measurement points in a window (which may, for example, be a one-second window). x ₂ = 3{S ₁₉ - S ₀ } (5) [0057] A directness measure that compares a length of a trajectory of a complex power in the complex power plane over a time window to a magnitude of a total change of the complex power during the time window exploits the fact that a trajectory of complex power during a LOE condition tends to follow a path that is more direct, i.e. closer to a straight line, than a trajectory of complex power during a power swing condition. As a result, the total length of a portion of a trajectory of complex power in a time window which occurs during an LOE condition will tend to be closer to the length of a straight line joining the complex power at the beginning and end of the time window than would be the case if the window coincided with a power swing event. Comparison of the path length along the complex power trajectory and a straight- line distance may be made, for example, by subtraction, taking a ratio or the like.

[0058] A specific example of a way to derive a value, ¾, suitable as a directness measure is indicated in Equation 6. The value of ¾ tends toward 1.0 as the trajectory becomes more direct.

[0059] A rate of change measure that indicates a maximum rate of change of the complex power within a time window can be obtained, for example, by finding a maximum of a time derivative of a function indicating the position of the complex power along its trajectory as a function of time. The rate of change measure can be useful because the complex power typically tends to traverse a trajectory relatively more slowly during a LOE condition (at least before loss of synchronism occurs) than occurs during a power swing condition. The rate of change measure is preferably sensitive to motions of the complex power that occur on a time scale of 250 ms or less and more preferably 100 ms or less. A rate of change measure may be obtained by determining differences between different complex power points. For example, the measure may be given by the magnitude of the maximum difference between any two consecutive complex power points in a time window. In an example embodiment, the two complex power points are separated in time by an interval on the order of 50 ms ±15ms. [0060] A specific example of a way to derive a value, ¼, suitable as a power rate of change measure is shown in Equation 7. In this example, is equal to the distance that the complex power point would travel along its trajectory during the time window if every consecutive complex power value was separated from adjacent complex power values by the maximum change. x ₄ = (N- l) - , N = 20, i e [l..(N- l)] (7)

[0061] The values x; to X4 may be assembled to provide a feature vector. The inventors have determined that such feature vectors which correspond to LOE events can be effectively discriminated from feature vectors corresponding to power swings and other events that may be encountered during normal operation of a generator. Any suitable method of discrimination may be used. Some example methods that may be constructed or trained to identify LOE conditions using such feature vectors as input include: artificial neural networks (ANN), fuzzy logic, support vector machines (SVM), pattern

classification techniques, sets of logical conditions, linear discriminant analysis (LDA) and the like.

[0062] Figures 1C to IE and 2A to 2C respectively illustrate a simulated power swing event and a simulated LOE event. These Figures illustrate that power swing events and LOE events have different patterns. These Figures illustrate how the features discussed above can be advantageously applied to detect LOE events in a manner that distinguishes the LOE events from power swing events.

[0063] One can see from inspecting Figures ID and 2B that, in both power swing events and LOE events, complex impedance follows a trajectory that approaches the mho center. Thus, feature x; or a suitable alternative distance measurement may be applied to identify events that should be considered as possible LOE events. [0064] In Figures 1C and 2A, the generator capability curve (GCC) is plotted as a reference at rated voltage conditions. The GCC is not constant and varies with generator terminal voltage. Figure 2A shows that the trajectory of complex power for an LOE event is directed generally in a negative imaginary direction. However, this behaviour is only valid up to the time the machine loses synchronism at around 3.5s as seen in Figure 2B. In contrast Figure 5 shows that for a power swing event (t > 0.216s) the trajectory of complex power tends to be directed in a positive imaginary direction (disregarding the prefault condition (t < 0 s), transition between prefault to fault condition (t=0 s to 16ms), fault condition (t=16ms to 0.2s), and the transition from fault into the power swing (t=0.2s to 0.216 s). Thus, feature X2 or a suitable alternative measure of the direction in which the imaginary component of complex power is changing can be useful for discriminating power swing events from LOE events.

[0065] It can be seen that during the time window 0.216 s to 1.216 s the trajectory of complex power in Figure 5. This time window includes the power swing condition. It can be easily seen that the value of ¾ is going to be much larger than 1.0 because the trajectory of the complex power follows a long trajectory during this time window and yet the complex power at the end of the time window is close to the complex power at the start of the time window. By contrast, from Figure 1C it can be seen that the complex power trajectory resulting from the LOE event is much closer to a straight line, resulting in a value of ¾ that is much closer to 1. Thus, feature ¾ or a suitable alternative measure of directness of the complex power trajectory can be useful for discriminating power swing events from LOE events.

[0066] It is notable that during transition periods from prefault to fault (e.g. 0.2 s to 0.225 s for the example simulated power swing event of Figure 1C) and from fault into the power swing (0.4 s to 0.425 s in Figure 1C) the monitored phasors are significantly influenced by the response of the full cycle DFT phasor estimation. For instance, the transition from prefault to fault is practically instantaneous in the time domain, i.e. a fraction of a millisecond. During this transition, the DFT processes both samples collected before occurrence of the fault and samples collected after occurrence of the fault. During this transition period the DFT consequently yields phasor values in between these two conditions. Feature can assist in distinguishing a LOE condition from other conditions during these transition periods in which the relatively slow response of the DFT may cause problems. It can be seen from Figure 1C that the power swing condition is characterized by rapid changes in complex power whereas changes in the complex power are more gradual during a LOE event.

[0067] Figures 3 A and 3B respectively show plots of features x; to X4 for the simulated power swing event of Figure 1C and the simulated LOE event of Figure 2 A. It can be seen that feature x; drops below 1.0 pu during an LOE condition before a loss of synchronism happens. Feature x; can also drop temporarily during a power swing condition.

Consequently, feature x; taken on its own cannot distinguish between power swings and LOE with high reliability. [0068] Feature X2 is negative during an LOE condition, although its value can be very small in cases where the generator is lightly loaded. During a power swing condition feature x ₂ also may become negative temporarily.

[0069] Feature ¾ has a value close to 1.0 for an LOE condition. Feature ¾ is larger than 1.0 during a power swing, The difference between these two events may be less marked while a fault is present ( e.g. in the period of t=0 to 0.2s).

[0070] Feature is clearly around 1.0 pu during the LOE condition. Feature X4 is larger than 10 pu for a stable power swing condition. Feature X4 can help to distinguish LOE from power swing especially during periods of rapid change in the variables considered. [0071] Some embodiments monitor all of the four features described above. It is thought that this combination of features provides an excellent basis for detecting LOE conditions. Alternative embodiments may apply a subset of the features described above, either alone or in combination with additional features. For example, some embodiments may monitor features that include a directness measure that compares a length of a trajectory of a complex power in the complex power plane over a time window to a magnitude of a total change of the complex power during the time window (e.g. ¾) ; and a rate of change measure that indicates a maximum rate of change of the complex power within the time window (e.g. X4) , possibly in combination with other features for detecting LOE or performing a supervisory or cross-checking function in relation to another LOE detection system.

[0072] Figure 4 illustrates a generator system 40 according to an example embodiment. Generator system 40 comprises a generator 41 driven by a prime mover 42 and connected to supply multi-phase power to a power grid 43. A prime mover controller 42A controls the speed and/or other aspects of operation of prime mover 42. Generator 41 comprises an exciter system 44 which is connected to energize field windings 41A of generator 41. An exciter controller 46 controls exciter system 44 based on measurements which may include current(s) and voltage(s) measured at one or more outputs 45 of generator 41 by current/voltage monitor 47.

[0073] A generation protection system 48 also receives data regarding current and voltage at the output of generator 41. Generator protection system 48 is configured to detect various fault conditions that may affect generator 41 and is connected to trip a disconnect (e.g. a circuit breaker) 49 that carries power from generator 41 to power grid 43 in the event that a serious fault condition is detected.

[0074] One or more systems for detecting LOE, as described herein, may be incorporated into a generator system 40 of the type illustrated in Figure 4. For example, generator protection system 48 may be configured to detect LOE conditions according to the methods described herein. A system for detecting LOE as described herein may also, or in the alternative, be incorporated into excitation controller 46. For example, in cases where excitation system 44 includes redundant components and/or generator 41 includes redundant field coils 41 A logic for determining whether to switch over to a backup part of excitation system 44 and/or to utilize backup field coils 41A may incorporate a system for detecting LOE as described herein.

[0075] As another example, a stand-alone or auxiliary LOE detection system may be provided (as indicated at 50). Outputs of auxiliary LOE detection system 50 may be put to various uses. Such a system may, for example, work in tandem with a generator protection system 48 and/or an excitation controller 46. The generator protection system and/or excitation controller 46 may receive an output signal from auxiliary LOE detection system 50 and may utilize that signal as an indication of the occurrence of an LOE condition. In some embodiments, an output of auxiliary LOE detection system may be connected to trip disconnect 49.

[0076] An LOE detection system, whether it is a stand-alone or auxiliary system or whether it is integrated with a generator protection system, circuit breaker, excitation controller or other part of generator system 40 may provide output to a grid control system 52. Such output may provide advanced warning of the possibility that generator 41 may be taken off-line, thus allowing grid control system 52 to take action or compensate.

[0077] A system as described herein may monitor one or more features in addition to or instead of some of the above-noted features. For example, a system may monitor complex impedance and may be configured to determine whether the complex impedance enters a mho zone or whether the complex impedance dwells in the mho zone for more than a threshold time period. The mho zone may, for example comprise a circle in the complex impedance plane. Figure 13 shows an example mho zone 55. In some embodiments, the system is configured to monitor whether or not the complex impedance enters each of two mho zones with one smaller than the other. The dwell-time of the complex impedance in each of the mho zones may be monitored. [0078] Additional features based on mho zones may be applied in a variety of ways. In some embodiments the mho zone features (e.g. dwell times in one or more mho zones) are used in combination with the above-noted features. In other embodiments the mho zone features and the features noted above are applied separately. For example, the mho zone features may be applied to detect a LOE condition (e.g. by determining that the complex impedance has dwelled for at least a first threshold amount of time in a larger mho zone and/or at least a second threshold amount of time in a smaller mho zone). A supervisory element may apply one or more of the above-noted features (for example, a combination of the third and fourth features above or all of the four features identified above) to improve the detection of LOE conditions. Such a supervisory element may also be used to add security to an existing one- or two- mho zone LOE detection method.

[0079] In an example embodiment, a system as described herein is deployed as a supervisory system in combination with another LOE detection system. When configured to operate in a supervisory mode the system as described herein may be arranged such that the generator is taken offline only if both the other LOE detection system and the system as described herein indicate a LOE condition.

[0080] In another example embodiment, a system as described herein is deployed to enhance dependability of another LOE detection system. In such a case the system as described herein may accelerate or inhibit generating a signal indicating a LOE condition. For example, consider the case where a system as described herein is used in conjunction with a LOE system that is triggered in response to a complex impedance dwelling in a mho zone for more than a threshold period of time as measured by a timer triggered when the complex impedance enters the mho zone. In such an embodiment the signal indicating the LOE condition may be generated in a case where the timer has been triggered and the system as described herein indicates a LOE condition (even before the timer reaches the threshold time). In some embodiments the generation of the signal indicating the LOE condition may be blocked or inhibited or the threshold time may be increased in response to the system as described herein determining a non-LOE condition.

[0081] As can be appreciated, a system for detecting LOE according to methods as described herein may be implemented by providing a data processor with software instructions that cause the data processor to execute the method. It is also possible to implement such systems using configurable logic (such as, for example, field- programmable gate arrays) and/or hard-wired logic circuits as well as combinations of any of these.

[0082] A system as described herein may be applied to detect LOE conditions in a simulation system that models a power grid. The input to the system may, for example, be modelled voltages and currents at terminals of a modelled generator in which the modelled voltages and currents are determined by the simulation system. For example, a system as described herein may be implemented in a software plug in for a simulation system such as PSS/E available from Siemens AG Energy Sector of Erlangen, Germany. [0083] Features as described herein may be input to a pattern classification system to detect LOE. The pattern classification system may be implemented in hardware and/or software. One example of a pattern classification method that may be used to evaluate whether a current feature vector does or does not correspond to a LOE event is SVM classification. [0084] The SVM classification is based on linear discriminant functions as described, for example, in Duda, R.O., Hart, P.E., and Stork, D.G.; Pattern Classification, John Wiley and Sons, 2001, 2nd Ed., which is hereby incorporated herein by reference for all purposes. A linear discriminant function is one that uses a linear decision boundary surface, i.e. a hyperplane, that separates two classes of data. In Fig. 5, hyperplanes H; and H2 separate the two classes of data while hyperplane H ₃ does not. The class of the data depends on which side of the hyperplane a given data vector Xk is located. A point x _p located in the hyperplane satisfies Equation 8 where w is the vector normal to the hyperplane. The distance from this plane to the origin is w ₀ /||w|| w ^T x _p + w ₀ (8) [0085] The data vector Xk is the pattern to be classified. This vector is also known as a feature vector because its coordinates are the values of key selected object features to help the classification process. In Figure 5, the coordinates x; and x ₂ are the values of the features used for classification. From Fig.5, it can be seen that the separation between the two sets of data vectors depends on the quality of the features chosen. For instance in Fig. 5, it is not possible to obtain good classification based on the x; coordinate alone, therefore the coordinate x ₂ is needed. The distance from a data vector to the hyperplane is given by (9) where b is the minimum distance from any data vector to this hyperplane. [0086] The hyperplane parameters w and wo are defined so that the data vector Xk satisfies one of the two inequalities (10) or (11), depending on which side of the hyperplane it is located. The category of a particular data vector Xk is given by the value of Zk. The two inequalities (10) and (11) can be combined into one resulting in (12). Combining (9) and (12) provides the relationship (13) between the minimum distance b and the normal vector w. w ^T x _k + w ₀ ≥+l => z _k = +l (10) w ^r x, + w ₀ ≤-l => z _k = -\ (11) ¾ (w ^r x, + w ₀ ) > +l V (12) [0087] A SVT classifier may be trained by finding a hyperplane that separates the two classes of data. The training is performed using vectors from both classes and whose category is known . A trained classifier produces zero classification error when the two classes of data are separable. In a separable case, there may be multiple hyperplanes that separate the two classes. The SVM method finds an optimum separating hyperplane that maximizes the distance b. In Fig. 5, both hyperplanes H; and H ₂ separate the two classes of data, but only H; is optimum that maximizes the distance b. From (13), this is equivalent to minimizing the vector norm w subject to the constraints (12). Using the Lagrange method to minimize ||w|| results in (14). Notice that the Lagrange multipliers a _k in (14) are restricted to values greater or equal to zero because the constraints in (12) are inequalities. Also, notice that n is the number of training vectors. From the duality principle, the problem in (14) becomes an optimization problem with respect to a instead of with respect to w and wo as described in (15). Applying the two conditions in the right side of (15) to the Lagrange function in (14) results in (16) and (17).

L(w, w ₀ , a) = ^||w|| ² -∑ajz,(w ^r x, + w ₀ ) - l] ; a _k ≥0 Vk (14)

^ k=l

V _w L = 0→w =∑<¾ (16) k=l

— = 0→¾ = 0 (17)

[0088] Substituting (16) and (17) into (14) gives (18); which is subject to the constraints given by (19). n n n

L{a) = _j a _k -- _{J j} a _k a _l z _k z _l * _k .* _l (18) k=l ^ k=l 1=1

∑¾¾ = 0 ; _k ≥0 Vk (19) k=l

[0089] The solution of (18) and (19) requires solving a quadratic programming problem of the form given in (20). This kind of problem can be solved using any suitable numerical optimization method. For example, a null-space active-set method may be used.

L(a) = - a ^r Ha + c ^r a (20) where

c ^r = [l 1 - l\xn

[0090] Once a is known the values of w and w ₀ are easily obtained using (16) and (12). The data vectors Xk located exactly at a distance b from the hyperplane are known as support vectors. The Lagrange multipliers a _k corresponding to the support vectors may be greater than zero while all other multipliers are zero. The number of support vectors is typically significantly fewer than the total number of training vectors. Training vectors may be selected to train the system to distinguish LOE conditions from non- LOE conditions. [0091] If the data are not separable in the original space by a hyperplane then mapping functions such as polynomial functions, Gaussian functions, etc. may be used to map the feature vectors to a space in which the two classes are linearly separable. In this new mapped space, the hyperplane is constructed and the methods described before can be applied to find the SVM classifier. Examples 1

[0092] The alternative electromagnetic transient program (ATP/EMTP) as described, for example, in Canadian / American EMTP User Group, Alternative Transients Program Rule Book, 2002 was used to obtain time domain solutions for a model of a generator system experiencing various faults. The model was of a generator connected to an infinite bus through a power line represented by a lumped impedance. The power line had impedance of 0.11 Z 84.3° pu in generator base units.

[0093] The generator modelled was a 104.4 MVA, 13.8 kV, 3600 rpm, wye connected synchronous generator. Parameters used in the simulations are as follows: Generator Data

MVA base 104.4 MVA

Inertia Constant H 3.09 KW-sec/KVA

Synchronous Xd 1.48 pu

Synchronous ¾ 1.42 pu

Transient X'd 0.193 pu

Transient X' _q 0.484 pu

Subtransient X" _d 0.136 pu

Subtransient X" _q 0.132 pu

Leakage Xl _m 0.14 pu

Transient Tdo ' 3.59 s

Transient ¹ qo 0.312 s

Subtransient Tdo " 0.033 s

Subtransient T ¹ qo " 0.084 s

Excitation System Data

IEEE standard excitation system model type STIA [28]

K _A 210

T _A O s

Tc 1.0 s

T _B 1.0 s

TBI O s

VRMAX 6.43 pu

VRMIN -6.0 pu

Kc 0.038

K _F 0

T _F O s

KLR 4.54

Iui 4.4 pu IEEE standard power system stabilizer PSS2A [28]

Ksi 20

Ks3 I

Ti=T ₃ 0.16 s

T ₂ =T ₄ 0.02 s

Twi=Tw2 ^: =Tw3 10 s

Tw4 O s

M 2

N 4

VsTMAX 0.20 pu

VsTMIN -0.066 pu

T ₆ O s

T ₇ 10 s

T ₈ 0.3 s

T ₉ 0.15 s

Governor Data

K 18

T ₂ O s

Ti 0.077 s

T ₃ 0.198 s

PMIN 0 pu

PMAX 0.95 pu

TcH 0.025 s

TRH 7 s

Tco 0.28 s

FHP 0.27

FJP 0.2555

FLP 0.4745

[0094] A time domain model of the synchronous machine using the dqO transformation was used to represent the generator. The mechanical behaviour of the generator was represented by a lumped mass. Prime mover dynamics were not considered in the model because time constants of prime mover dynamics are typically longer than the simulation time.

[0095] The simulations used a time step of 65.1 μ8 which is equivalent to 256 samples every power system cycle. This time step was short enough to reproduce a bandwidth of 0.00 to 7.68 kHz which is sufficient to model most relevant transients. The simulations were run for long enough (modelling parameters of the generator for 35 seconds) to clearly observe the response of the generator to both LOE and power swing disturbances and, in the case of power swing disturbances, to verify that the generator eventually returns to a stable condition.

[0096] Stable power swing conditions were simulated by modelling the application of a three phase short circuit, and removing the short circuit after a short period of time to allow the system to recover (i.e. return to a stable operating condition). The short circuit had a fault resistance of 0.005 pu connected at 50% of the power line. To produce stable simulated power swings, the three phase fault was applied at t=0s and removed at 200 ms in the simulation time scale.

[0097] LOE conditions were simulated by modelling the application of the following sequence: starting with an initial load, setting the field voltage to produce the desired load conditions in the PQ plane. The field voltage was then set to zero for the balance of the simulation. In the simulations discussed herein the modelled field voltage was reduced to zero at t=0 s.

[0098] The LOE conditions and stable power swings were modelled for a variety of starting conditions which differed from one another in terms of different loads as well as lagging, and unity power factor. Two parameters were varied to achieve these different initial conditions. These parameters were the voltage magnitude and the phase angle of the generator relative to the infinite bus.

[0099] Feature vectors {xj, ¾, ¾, *} were generated from the instantaneous voltages and currents yielded by the simulations using Foreign Models compiled objects in ATP. A traditional two mho zone LOE detector was also implemented as a Foreign Model in ATP for comparative purposes. [0100] 116 data vectors were used for training the prototype SVM. The simulation conditions used to produce these vectors are listed in Figure 6. The training vectors were used to solve the SVM equation. The parameters of the resulting classifier are given in Equations (22) and (23). w ^r = [17.6668 238.786 6.1026 22.7685] (22) w ₀ = -19.730 (23)

[0101] A first group of new cases for testing the classifier were obtained from simulations of the same disturbances that were used to generate the training cases. In Figure 7, the SVM method is compared with the traditional LOE detection. The SVM pickup time is the time it takes for the proposed method to confirm that it is an LOE condition. The SVM reset time also indicates the time when loss of synchronism happens.

[0102] A second group of new cases for testing the classifier was generated from simulations in which initial load conditions were different from those used to obtain the training cases. Two different new initial load conditions were used. For each of these initial load conditions, one LOE scenario and one power swing scenario was simulated. The results are listed in Figure 8.

[0103] A third group of new cases for testing the classifier was generated from

simulations in which unbalanced faults were simulated. Four more cases with the following fault types were simulated: AG, BG, ABG, and BCG. The four cases are listed in Figure 9. In order to bring the impedance during power swing closer or inside of the LOE characteristic, the worst case scenario of the six initial load conditions is used, i.e. a heavy load with leading power factor (0.78-j0.50 pu).

[0104] A fourth group of new cases for testing the classifier was generated from simulations in which power lines of different lengths are modelled. The electrical center of the power swing moves away from the generator when the line length is increased while the electrical centre moves closer to the generator impedance or may fall inside the generator impedance when the line length is reduced. If the electrical center is far from the generator impedance, then the power swing may not enter the LOE region. In this fourth group of cases the modelled power line connecting the generator to the infinite bus was increased from 0.11 pu to 0.33 pu. Beyond 0.33 pu line length, the power swing did not enter the LOE impedance zone. Simulations of four initial load conditions were used combining load and power factor variations: heavy/light load and leading/lagging power factor. The eight new cases are listed in Figure 10. [0105] To ascertain whether the classifier is stable in the presence of changes in the generator parameters, the cases for the most severe LOE event (case #6 in Figure 7) and corresponding power swing (case #12 in Figure 7) were used to study the sensitivity of the proposed method to changes in the parameters of the generator modelled in the simulation. A variation of either ±5% was used, depending on which one caused the operating point to enter the LOE mho zone, i.e. more risk of maloperation. The results are given in

Figure 11.

[0106] The parameter that affected the detection of an LOE condition the most (case #6 of Figure 7) by the SVM method proposed was an increase in the Xq' of 5%. This change caused a delay of 33ms in detection of the LOE event as indicated in Figure 11. For the same LOE condition, the traditional two zone mho impedance method was also affected by the parameter variation, but resulted in a faster trip time by 33 ms for an increase in the rated voltage of 5%. However, the overall decision time using the SVM classifier was still faster (453ms) compared to the traditional two mho zone method (1.73s).

[0107] On the other hand, for a power swing condition (case #12 of Figure 7) the SVM was not affected. The SVM classifier successfully and clearly identified all the instances where the impedance entered the LOE mho zone as a non LOE condition. However, the traditional two mho zone method was affected, because the simulated generator was operating very close to the transient stability limit. This limit depends on two variables: the point in the PQ plane relative to the GCC curve and the fault duration. The point in the PQ plane in this case is on the limit of the GCC curve in the leading reactive region and the fault duration used for all prior cases is 200ms. The parameters that caused the most impact on the performance of the traditional LOE detection method for this power swing condition were 5% variation of in the MVA rating, the kV rating and the Xd synchronous reactance, which made the machine lose synchronism for the fault duration of 200ms being used. It is unlikely a machine would be operated so close to the transient stability limit, thus the fault duration time was reduced from 200ms to 150 ms so that there was no loss of synchronism. Examples 2

[0108] A method for detecting LOE was tested in a prototype hardware DSP platform and RTDS (Real Time Digital Simulator) simulation. The results of this testing are summarized in Tables I and II below.

Table I: SVM method in DSP tested with RTDs - two MHO zone LOE

Two MHO zone

Large Small MHO MHO

Time, s Time, s

Case Load/PF Initial Type of Pickup Trip Pickup Trip No. P+jQ,pu case

1 LL/lag. 0.10+j0.68 LOE 8.54 9.53 - -

3 LL/lead. 0.03-j0.55 LOE 0.93 1.945 - -

4 HL/lag. 0.78+j0.41 LOE 2.564 3.556 3.424 3.816

6 HL/lead. 0.79-j0.49 LOE 0.771 1.763 1.618 2.013

Pickup Reset Pickup Reset

7 LL/lag. 0.10+j0.68 PS-3P - - - -

9 LL/lead. 0.03-j0.55 PS-3P 0.251 1.239 0.251 00.647

10 HL/lag. 0.78+j0.41 PS-3P 0.251 0.297 0.261 0.267

12 HL/lead 0.79-j0.49 PS-3P 0.226 0.767 0.226 0.571

LL: light load, HL: heavy load, PS: power swing, 3P: 3 phase fault used Table II: SVM method in DSP tested with RTDS - SVM Method

SVM Time, s

Case Load/PF Initial Type of Pickup Reset Loss of No. case Synch.

P+jQ,pu

time, s

1 LL/lag. 0.10+j0.68 LOE 0.558 -

3 LL/lead. 0.03-j0.55 LOE 0.345 -

4 HL/lag. 0.78+j0.41 LOE 0.188 4.578

6 HL/lead. 0.79-j0.49 LOE 0.406 2.766

LL: light load, HL: heavy load, PS: power swing, 3P: 3 phase fault used

[0109] The results of these tests provide a working example application of methods as described herein in the context of a real time embedded platform. Also, the similarity with the prior results shows that parameters for use in detecting LOE conditions may be developed using offline (non-real-time) methods and applied in actual real-time field applications.

Examples 3

[0110] The methods as described herein including the use of SVM for classification of LOE conditions were applied in a simulation of a real power network using data from a real generator in the network. The generator in question was a thermal unit of 483 MVA capacity at the Sundance Plant located near Edmonton, Alberta, Canada. Figure 13 shows the network in which the generator (G4 in Figure 13) is connected. The network was modelled using the PSS/E (Power System Simulator) tool from Siemens PTI (Power Technologies Inc), which is widely used by the electric industry for planning and operating studies.

[0111] The model was a dynamic model that did not assume fixed excitation. The excitation control for this generator was modelled using the IEEE ST5B model as recommended in the IEEE 421.5 standard. A characteristic of the ST5B model is that it considers different transient gains depending on which control action is currently active: a) Automatic Voltage Regulator (AVR), b) Under Excitation Limiter (UEL), or c)

Overexcitation Limiter (OEL). Another characteristic is that Power System Stabilizer (PSS) action is always active.

[0112] The excitation model is illustrated in Figure 14. This model can properly represent the takeover type of under-excitation limiter used in this system. In Figure 14: V _re f is reference or desired terminal voltage; Ec is terminal voltage measured; T _R is a voltage measurement delay time constant; EFD is excitation voltage output considering rectifier effect and ceiling voltage limits; IFD is field current measured; K _c represents equivalent internal resistance of rectifier; VOTHSG represents auxiliary signals, e.g. power system stabilizer (PSS); K _R ; Ti represent the main exciter control loop; T _cl ; T _C2 ; T _B1 ; T _B2 represent lead lag compensation for the AVR control loop; VUEL is the input signal from UEL control action; VOEL is the input signal from OEL control action; Tuci; Tuc ₂ ; TUBI ; TUB ₂ represent lead lag compensation for the UEL control loop; Toci; T ₀ c ₂ ; T ₀ BI ; T ₀ B ₂ represent lead lag compensation for the OEL control loop. Values for the parameters of Figure 14 are shown in Figure 15.

[0113] The UEL control loop model is illustrated in Figure 16. The terminal voltage signal VT and the active power signal PT are used as reference values. From these two reference values the current reactive power limit level Q ₀ by using a lookup table function Q =f(P). The voltage dependence is obtained using the blocks 1/V ki and V ^, considering that the lookup table Q =f(P) is defined at rated voltage level. The main regulating control loop is a proportional integral with parameters Kui and KUL- The controlled variable is the reactive power signal QT. The output signal VUEL from the UEL control loop is applied to the main AVR control loop in order to increase the excitation to the rotor field when the operating point of the machine moves below the UEL characteristic in the PQ plane.

[0114] Parameters used to model UEL behaviour are illustrated in Figure 17.

[0115] The SVM was trained for the above-described dynamic system and then tested for its ability to discriminate LOE conditions from other conditions such as temporary system overvoltage. The coordination was verified for a temporary system overvoltage condition. A system overvoltge condition causes the AVR to automatically reduce the excitation. The movement of the reactive power power Q in the PQ plane activates the UEL. The trajectory in the PQ plane for this condition is shown in Figure 18A. The impedance plane for this condition is shown in Figure 18B. [0116] In the illustrated case, the power swing trajectory stays at the edge of the LOE zone for the Blackburn large mho zone for about 200 ms, i.e. from 350 to 550 ms. There is increased risk of maloperation by the Blackburn mho zone. The approach to the

Berdy/Mason mho zone is safe and does not present risk of maloperation up to this point.

[0117] The operating point goes below the UEL characteristic so that the UEL is activated. In the PQ plane it appears that the operating point does not leave the GCC characteristic. However, in the impedance plane the impedance touches the large Berdy mho zone. The explanation for this apparent inconsistency is that voltage drops significantly during this disturbance, causing the impedance magnitude to reduce and touch the mho zone. [0118] Figure 19 shows coordination for the case of temporary system overvoltage. In Figure 19, consider the time interval from 350 to 550 ms. The first feature XI is reduced as the impedance approaches temporarily the mho center. The second feature X2 is reduced as well, as the reactive power Q goes temporarily below the UEL limit. The third feature X3 increases as the disturbance is not a true LOE condition, and the path in the PQ plane is not straight. The fourth feature X4 also increases as there are rapid changes within a 1.0 second window.

[0119] Correct operation of the SVM classifier was observed in all cases tested.

Coordination was also verified for the most severe stable power swing condition and for unstable power swing conditions. [0120] One advantage of some embodiments of the present invention as compared to certain prior LOE detection systems (such as, for example, systems which detect LOE using one or two mho zones) is that a system as described herein may operate more quickly than such prior art systems. This rapid operation may be exploited in various ways. In some embodiments rapid LOE detection may be applied to trip a breaker more quickly upon occurrence of an LOE event than could be done, for example, where LOE detection is reliant on dwell time in one or two mho zones. In other embodiments, rapid detection of a potential LOE condition provides more time to monitor the generator after the potential LOE condition has been detected to verify that the LOE condition is 'real' before tripping a breaker. Thus, methods for detecting LOE as described herein can be faster, more robust against false triggering or both as compared to such prior methods.

[0121] In some embodiments, a system as described herein can operate quickly enough to monitor a generator for a period of time after the potential LOE condition has been detected and still trip a breaker in time to prevent generator damage. In some

embodiments, during the period after the method initially detects a potential LOE condition and before the generator is taken off line a system as described herein initiates and/or controls steps to place the generator in a 'safe mode'. These steps may bring the generator toward a condition from which the generator may be more safely taken off line and/or comprise steps that tend to cause the generator to resume operating normally. Also, a system as described herein can optionally provide a signal to a network control system in advance of a generator being taken off line due to a LOE condition. The network control system may thereby have enough time to initiate steps to minimize disruption resulting from the generator being taken off line.

[0122] Certain implementations of the invention comprise computer processors which execute software instructions which cause the processors to perform a method of the invention. For example, one or more processors in a generator protection system or an excitation control system or a stand alone LOE detection system may implement methods as described herein by executing software instructions in a program memory accessible to the processors. The instructions comprise firmware in some embodiments. The data processor comprises an embedded processor in some embodiments. The invention may also be provided in the form of a program product. The program product may comprise any non-transitory medium which carries a set of computer-readable signals comprising instructions which, when executed by a data processor, cause the data processor to execute a method of the invention. Program products according to the invention may be in any of a wide variety of forms. The program product may comprise, for example, physical media such as magnetic data storage media including floppy diskettes, hard disk drives, optical data storage media including CD ROMs, DVDs, electronic data storage media including

ROMs, flash RAM, or the like. The computer-readable signals on the program product may optionally be compressed or encrypted.

[0123] Where a component (e.g. a software module, processor, assembly, device, circuit, etc.) is referred to above, unless otherwise indicated, reference to that component

(including a reference to a "means") should be interpreted as including as equivalents of that component any component which performs the function of the described component (i.e., that is functionally equivalent), including components which are not structurally equivalent to the disclosed structure which performs the function in the illustrated exemplary embodiments of the invention.

[0124] While a number of exemplary aspects and embodiments have been discussed above, those of skill in the art will recognize certain modifications, permutations, additions and sub-combinations thereof. It is therefore intended that the following appended claims and claims hereafter introduced are interpreted to include all such modifications, permutations, additions and sub-combinations as are within their true spirit and scope.

Interpretation of Terms [0125] Unless the context clearly requires otherwise, throughout the description and the claims:

"comprise", "comprising", and the like are to be construed in an inclusive sense, opposed to an exclusive or exhaustive sense; that is to say, in the sense "including, but not limited to".

"connected", "coupled", or any variant thereof, means any connection or coupling, either direct or indirect, between two or more elements; the coupling or connection between the elements can be physical, logical, or a combination thereof.

"herein", "above", "below", and words of similar import, when used to describe this specification shall refer to this specification as a whole and not to any particular portions of this specification.

"or", in reference to a list of two or more items, covers all of the following interpretations of the word: any of the items in the list, all of the items in the list, and any combination of the items in the list. • the singular forms "a", "an", and "the" also include the meaning of any appropriate plural forms.

• Words that indicate directions such as "vertical", "transverse", "horizontal", "upward", "downward", "forward", "backward", "inward", "outward", "vertical", "transverse", "left", "right", "front", "back"," "top", "bottom", "below", "above", "under", and the like, used in this description and any accompanying claims (where present) depend on the specific orientation of the apparatus described and illustrated. The subject matter described herein may assume various alternative orientations. Accordingly, these directional terms are not strictly defined and should not be interpreted narrowly.

[0126] Specific examples of systems, methods and apparatus have been described herein for purposes of illustration. These are only examples. The technology provided herein can be applied to systems other than the example systems described above. Many alterations, modifications, additions, omissions and permutations are possible within the practice of this invention. This invention includes variations on described embodiments that would be apparent to the skilled addressee, including variations obtained by: replacing features, elements and/or acts with equivalent features, elements and/or acts; mixing and matching of features, elements and/or acts from different embodiments; combining features, elements and/or acts from embodiments as described herein with features, elements and/or acts of other technology; and/or omitting features, elements and/or acts from described embodiments.

[0127] It is therefore intended that the following appended claims and claims hereafter introduced are interpreted to include all such modifications, permutations, additions, omissions and sub-combinations as may reasonably be inferred. The scope of the claims should not be limited by the preferred embodiments set forth in the examples, but should be given the broadest interpretation consistent with the description as a whole.