Patent Publication Number: US-7710693-B2

Title: Apparatus and method for providing protection for a synchronous electrical generator in a power system

Description:
CROSS-REFERENCE TO RELATED APPLICATIONS 
   None 
   BACKGROUND OF THE INVENTION 
   The present invention generally relates to synchronous generators, and more specifically, to an apparatus and method for providing generator protection for a synchronous electrical generator in a power system. 
   Synchronous electrical generators (“synchronous generators”) are used in many applications requiring alternating current (AC) power generation. For example, electric utility systems or power systems include a variety of power system elements such as synchronous generators, power transformers, power transmission lines, distribution lines, buses, capacitors, etc. to generate, transmit and distribute electrical energy to loads. A synchronous generator operates to, for example, convert mechanical rotation via a prime mover (e.g., shaft rotation provided by a coal powered steam turbine) into AC current via electromagnetic principles. After suitable conditioning, the alternating electrical current is transmitted and distributed as three-phase electric power to a variety of loads. 
   As is known, synchronous generator design is based on Faraday&#39;s law of electromagnetic induction and includes a rotational portion for inducing an electromotive force (EMF) in a stationary portion. The rotational portion is driven by the prime mover. More specifically, the rotational portion, or rotor, includes a field winding wrapped around a rotor body, and the stationary portion includes a stator having an armature winding. The rotor body, typically made of steel, may have a salient pole structure (i.e., poles protruding from a shaft) or a cylindrical structure. 
   In operation, EMFs are induced in the armature windings of the stator upon application of DC current to the field winding of the rotor. That is, direct current is made to flow in the field winding. This results in a magnetic field, and when the rotor is made to rotate at a constant speed, the magnetic field rotates with it. Accordingly, as the moving magnetic field passes through the stator winding(s), an EMF is induced therein. If the stationary armature includes, for example, three stationary armature windings, they experience a periodically varying magnetic field, and three EMFs are induced therein. These three EMFs conform a three-phase system of voltages. Thus, for 60-Hz AC systems, in a two-pole machine, the rotor has to rotate at 3600 revolutions per minute with three armature windings displaced equally in space on the stator body to generate three-phase electric power. 
   As the generator electric load increases, the generator demands more mechanical power from its prime mover and more current flows through the stator winding, and therefore more electric active power is delivered from the synchronous generator to the power system. By increasing the current to the rotor winding, the synchronous generator produces more reactive power, also called reactive volt-amperes (VARs), which, in effect, can raise the power system voltage. Conversely, by decreasing the current in the rotor winding, VARs are absorbed by the generator, effectively lowering the power system voltage. As is known, we express in Watts or Megawatts the active power delivered to or consumed by a load, while VAR or MVAR is the imaginary counterpart of the Watt or Megawatt and represents the reactive power consumed or generated by a reactive load (i.e., a load having a phase difference between the applied voltage and the current). 
   Generator capability curves (“capability curves”) are typically provided by a generator manufacturer to define the operating or thermal limits of a particular synchronous generator at different cooling pressures. Each capability curve represents the synchronous generator capability limit for a pressurized coolant (e.g., hydrogen) circulating to cool the stator and rotor windings. More cooling enables more armature current to flow during synchronous generator operation, while less cooling enables less current to flow. Additionally, over excitation limiter (OEL) curves and minimum excitation limiter (MEL) curves are typically included with the manufacturer-provided capability curves. Steady state stability limit (SSSL) curves may further be determined with generator impedance data and power system parameters. 
   Because there are limits to the amount of current that can flow through the stator and rotor winding, the operating limits reflected in the capability curves are imposed on the amount of Watts and VARs that the synchronous generator can deliver to the power system. There is also a minimum value of current that must flow in the rotor field to maintain generator stability, and this imposes a limit on the amount of VARs that the synchronous generator can absorb for each delivered active power value. Thus, the operating limits graphically illustrated by the capability curve(s) include an active power component “P” expressed in Megawatts (MW) and a reactive power component “Q” expressed in Mega VARs (MVARs). As long as the P, Q operating point of the synchronous generator (i.e., as long as the amount of Watts and VARs flowing out of or into the generator) is within its safe operating limits, or inside its capability curve, the synchronous generator will operate within safe limits. 
   Although the operating limits defined by capability curves are utilized by power generating station operators to ensure safe synchronous generator operation, it has been suggested to utilize these curves to influence excitation control of a synchronous generator in real time. For example, U.S. Pat. No. 5,264,778, entitled “Apparatus Protecting a Synchronous Machine from Under Excitation,” issued on Nov. 23, 1993, describes a microprocessor based voltage regulator system that provides a minimum limit on excitation that is defined using one or more straight line segments approximating the associated machine capability curves. Such a minimum limit on excitation prevents the excitation of the synchronous generator from falling below a predetermined P-Q characteristic. The microprocessor based voltage regulator system of the U.S. Pat. No. 5,264,778 is included in a control system of the synchronous generator. 
   It has also been suggested that synchronous generator operation may be improved via use of a visual display that reflects synchronous generator operation with respect to its capability curves. U.S. Pat. No. 5,581,470, entitled “Apparatus for Visually Graphically Displaying the Generator Point of a Generator in Reference to its Capability Curve Including Digital Readouts of Watts, VARs and Hydrogen Pressure,” issued on Dec. 3, 1996, describes a computer-based meter that provides a real time graphical display which visually indicates an operating point in relation to a capability curve(s) of a synchronous generator during operation. The operating point(s) and capability curves are defined and displayed based on measurement signals from Watt, VAR and hydrogen pressure transducers. 
   Synchronous generator outages or failures due to power system faults, abnormal operating conditions, and the like, can be some of the costliest in the power system. Accordingly, protective devices are operatively coupled to the synchronous generators and their outputs in order to measure currents and voltages indicative of synchronous generator operation. Such protective devices are referred to hereinafter as protective relays, and typically include a variety of protective functions or elements. 
   SUMMARY OF THE INVENTION 
   According to an embodiment of the invention, a method enables protection for a synchronous generator. The method includes deriving a plurality of generator safe operating boundary data expressions from a plurality of power system data. The power system data may be supplied by a manufacturer of the synchronous generator. The power system data may include generator impedance. The power system data may include power system parameters such as equivalent power system impedance. The generator safe operating boundary data expressions may be used by a protective relay. During operation, the protective relay utilizes at least one of the plurality of generator safe operating boundary data expressions to enable the protection for the synchronous generator. Each of the plurality of generator safe operating boundary data expressions is selected from the group consisting of quadratic equations, circle equations, look-up tables, linear equations and combinations thereof. 
   The generator safe operating boundary data expressions may be set in relation to a generator capability curve, a steady-state stability limit curve, a minimum excitation limiter curve, or an over excitation limiter curve. It is contemplated that the user may manually set the generator safe operating boundary data expressions (e.g., in relation to a manufacturer provided generator capability curve, by providing generator safe operating boundaries such as the capability curve and user-defined loss-of-field element) or, in some cases, the generator safe operating boundary data expressions may be derived based on generator and/or power system operating limits (e.g., in relation to a steady-state stability limit curve, a minimum excitation limiter curve, or an over excitation limiter curve). In one embodiment, the protective relay determines the generator safe operating boundary data expressions automatically using generator capability curve data, generator impedance data and power system impedance data. 
   According to another embodiment of the invention, the generator safe operating boundary data expression is set for loss-of-field protection. In one example, a loss-of-field element is provided which is set with respect to the generator capability curve. In another example, a loss-of-field element is provided which is set with respect to an SSSL curve. In another embodiment, one or more active power elements, and/or an undervoltage element may further be provided. 
   According to another embodiment of the invention, in a protective relay, a method provides protection for a synchronous generator. The method includes selecting at least one of a plurality of generator safe operating boundary data expressions based on a predetermined user programmable input and a generator operating indication. The plurality of generator safe operating boundary data expressions is derived from either an SSSL curve, a plurality of generator safe operating boundaries supplied by a manufacturer of the synchronous generator, or other similar means. The method also includes calculating an active power value sum and a reactive power value sum based on measured three-phase currents and voltages associated with synchronous generator operation. The method further includes comparing the active power value sum and the reactive power value sum to the at least one of the plurality of generator safe operating boundary data expressions, and providing the protection for the synchronous generator based on this comparison. 
   An apparatus provides protection for a synchronous generator in a power system. The apparatus comprises a means for deriving a plurality of digitized signals representative of measured three-phase secondary currents and voltages associated with synchronous generator operation, and a microcontroller operatively coupled to the means for deriving the plurality of digitized signals. The microcontroller includes a microprocessor and a memory operatively coupled to the microprocessor. The microcontroller is programmed to, based on a plurality of predetermined user programmable inputs and generator operating indications, select at least one of a plurality of generator safe operating boundary data expressions. The plurality of generator safe operating boundary data expressions may be derived from a plurality of generator safe operating boundaries supplied by a manufacturer of the synchronous generator. The plurality of generator safe operating boundary data expressions may be derived from user-defined values such as coordinates of a loss-of-field element. The microcontroller is also programmed to calculate an active power value sum and a reactive power value sum based on the plurality of digitized signals, compare the active power value sum and the reactive power value sum to the at least one of the plurality of generator safe operating boundary data expressions, and provide protection for the synchronous generator based on this comparison. 
   A computer readable medium having program code recorded thereon provides protection for a synchronous generator in a power system. The computer readable medium includes a first program code for selecting at least one of a plurality of generator safe operating boundary data expressions based on a generator operating indication and a predetermined user programmable input. The plurality of generator safe operating boundary data expressions is derived from a plurality of generator safe operating boundaries supplied by a manufacturer of the synchronous generator. The computer readable medium also includes a second program code for calculating an active power value sum and a reactive power value sum based on measured three-phase currents and voltages associated with synchronous generator operation, a third program code for comparing the active power value sum and reactive power value sum to at least one of the plurality of generator safe operating boundary data expressions, and a fourth program code for providing the protection for the synchronous generator based on the comparison. 
   It should be understood that the present invention includes a number of different aspects and/or features which may have utility alone and/or in combination with other aspects or features. Accordingly, this summary is not an exhaustive identification of each such aspect or feature that is now or may hereafter be claimed, but represents an overview of certain aspects of the present invention to assist in understanding the more detailed description of preferred embodiments that follow. The scope of the invention is not limited to the specific embodiments described below, but is set forth in the claims now or hereafter filed. 

   
     BRIEF DESCRIPTION OF THE DRAWINGS 
       FIG. 1  is a single line schematic of a power system that may be utilized in a typical wide area network. 
       FIG. 2   a  is an exemplary functional block of a generator protective relay of  FIG. 1 , according to an embodiment of the invention. 
       FIG. 2   b  is an exemplary functional block of the generator operating boundary function of generator protection relay of  FIG. 2   a.    
       FIG. 2   c  is an exemplary functional block of the generator operating boundary function adapted for providing positive-sequence values of generator protection relay of  FIG. 2   a.    
       FIG. 3  is an exemplary set of generator capability curves that may be provided by a synchronous generator manufacturer to define the operating limits of a synchronous generator of  FIG. 1 . 
       FIG. 4  is one of the capability curves of the exemplary set of generator capability curves of  FIG. 3 . 
       FIG. 5   a  is a flowchart of a method for synchronous generator protection in the power system of  FIG. 1 , according to an embodiment of the invention. 
       FIG. 5   b  is a flowchart of a method for asserting either an alarm condition or trip condition for synchronous generator protection in the power system of  FIG. 1 , according to an embodiment of the invention. 
       FIG. 6  is a generator capability curve generated based on generator safe operating boundary data expressions derived from the capability curve of  FIG. 4  for use by the generator protection relay of  FIG. 2   a , according to an embodiment of the invention. 
       FIG. 7  is a graphical representation of an estimation curve for the boundary for field winding heating associated with the field winding current limit of a generator capability curve. 
       FIG. 8  is a graphical representation of an estimation curve for the boundary for armature heating associated with the armature current limit of a generator capability curve. 
       FIG. 9  is a graphical representation of an estimation curve for the boundary for a stator core temperature associated with the stator end region heating limit of a generator capability curve. 
       FIG. 10  is a generator capability curve including an arrangement for loss-of-field protection further illustrating the protection zone whereupon a trip signal would be asserted if a condition were to fall therein. 
       FIG. 11  is another generator capability curve including an arrangement for loss-of-field protection further illustrating the protection zone whereupon a trip signal would be asserted if a condition were to fall therein. 
       FIG. 12  is a generator capability curve including an arrangement for issuing an alarm signal and showing the alarming zone. 
       FIG. 13  is a generator capability curve including an arrangement for loss-of-field protection, and an arrangement for issuing an alarm signal, and also showing the generator normal operation zone, the protection zone, and the alarming zone. 
   

   DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS 
     FIG. 1  is a single line schematic diagram of a power system  10  that may be utilized in a typical wide area system. As illustrated in  FIG. 1 , the power system  10  includes, among other things, three synchronous generators  11 ,  12  and  13 , configured to generate three-phase voltage sinusoidal waveforms such as 12 kV sinusoidal waveforms, three step-up power transformers  14   a ,  14   b  and  14   c , configured to increase the generated voltage sinusoidal waveforms to higher voltage sinusoidal waveforms such as 138 kV sinusoidal waveforms and a number of circuit breakers  18 . The step-up power transformers  14   a ,  14   b ,  14   c  operate to provide the higher voltage sinusoidal waveforms to a number of long distance transmission lines such as the transmission lines  20   a ,  20   b  and  20   c . In an embodiment, a first substation  16  may be defined to include the two synchronous generators  11  and  12 , the two step-up power transformers  14   a  and  14   b  and associated circuit breakers  18 , all interconnected via a first bus  19 . A second substation  35  may be defined to include the synchronous generator  13 , the step-up power transformer  14   c  and associated circuit breakers  18 , all interconnected via a second bus  25 . At the end of the long distance transmission lines  20   a ,  20   b , a third substation  22  includes two step-down power transformers  24   a  and  24   b  configured to transform the higher voltage sinusoidal waveforms to lower voltage sinusoidal waveforms (e.g., 15 kV) suitable for distribution via one or more distribution lines  26  to loads such as a load  32 . The second substation  35  also includes two step-down power transformers  24   c  and  24   d  on respective distribution lines  28  and  29  to transform the higher voltage sinusoidal waveforms, received via the second bus  25 , to lower voltage sinusoidal waveforms suitable for use by respective loads  30  and  34 . 
   As discussed above, one or more protective relays are operatively coupled to the synchronous generators  11 ,  12  and  13  to measure currents and voltages indicative of synchronous generator operation. Based on the measured currents and/or voltages, one or more protective elements (e.g., an over-voltage element) of the protective relay may operate to actuate a trip action in the event of an abnormal condition. In the illustrated example of  FIG. 1 , protection of the generator  12  is provided by a protective relay  100 . While not separately shown, it should be understood that additional protective relays  100  may be included in the power system  10 . 
     FIG. 2   a  is an exemplary functional block diagram of the protective relay  100 , according to an embodiment of the invention. It should be understood that functional block diagram of  FIG. 2   a  is only one example of a protective relay implementation of the instant invention, and that other implementations are possible. 
   As illustrated, the protective relay  100  includes a number of inputs  101 - 106  configured to receive secondary current waveforms I A , I B , and I C  and secondary voltage waveforms V A , V B , and V C  from corresponding voltage and current transformers operatively coupled to each of the A-, B- and C-phases provided by the generator  12 . Although illustrated as being received via individual inputs, it should be understood that the secondary current waveforms I A , I B , and I C  and secondary voltage waveforms V A , V B , and V C  may be received via a combination of phase-input current transformers, current transformers, voltage transformers, and non-conventional current and voltage sensors. In general, the secondary current waveforms I A , I B , and I C  and secondary voltage waveforms V A , V B , and V C  are processed to determine whether an alarm and/or trip signal should be issued by the protective relay  100 . 
   More specifically, each of the secondary current and voltage waveforms I A , I B , and I C  and V A , V B , and V C  is further transformed into corresponding scaled sinusoidal waveforms via current transformers  111 - 113  and voltage transformers  114 - 116  respectively, and resistors (not separately illustrated). The scaled sinusoidal waveforms are filtered via (hardware) analog low pass filters  121 - 126 . A multiplexer  128  then selects each of the filtered scaled sinusoidal waveforms, one at a time, and provides the selected filtered scaled sinusoidal waveforms to an analog-to-digital (A/D) converter  130 . The A/D converter  130  samples and digitizes each of the selected filtered scaled sinusoidal waveforms to form corresponding digitized signals  131 - 136 . The corresponding digitized signals  131 - 136  are representative of the A-, B- and C-phase secondary current and voltage waveforms I A , I B , and I C  and V A , V B , and V C , respectively. 
   The corresponding digitized signals  131 - 136  are received by a microcontroller  138  (or digital signal processor (DSP) or personal computer ((PC))) for signal processing, where they are digitally filtered via, for example, Cosine filters to eliminate DC and unwanted frequency components. In the illustrated example of  FIG. 2   a , the digital filtering is provided by digital band pass filters (DBPFs)  141 - 146  where DBPFs  141  and  144  perform digital filtering for digitized signals  131  and  134  to form filtered digital signals  161  and  164  representative of the A-phase secondary current and voltage waveforms I A  and V A , where DBPFs  142  and  145  perform digital filtering for digitized signals  132  and  135  to form filtered digital signals  162  and  165  representative of the B-phase secondary current and voltage waveforms I B  and V B , and where DBPFs  143  and  146  perform digital filtering for digitized signals  133  and  136  to form filtered digital signals  163  and  166  representative of the C-phase secondary current and voltage waveforms I C  and V C . In an embodiment, the filtered digital signals  161 - 166  are further converted into phasor form to enable subsequent calculations by the microcontroller  138 . 
   An indication input  180  is also provided to receive generator operating indications such as, for example, pressure transducer inputs indicative of generator operating parameters. Other generator indications include cooling pressure, excitation or field current, stator temperature, gearing temperature, ambient temperature and the like. 
   The microcontroller  138  includes a generator operating boundary function  148 , and additional generator protection functions  156  that comprise one or more protective elements. Using the filtered digital signals  161 - 166  from the DBPFs  141 - 146 , the additional generator protection functions  156  performs one or more typical protection functions. For example, the additional generator protection functions  156  may include a differential protection element, a stator ground fault protection element, a rotor ground fault protection element, a motoring protection element, an over-excitation protection element, a thermal protection element, an under-frequency protection element, an over-current protection element, an over-voltage protection element, and/or an out-of-step protection element. Based on magnitudes and phase angles of the phasors representing the filtered digital signals  161 - 166 , the additional generator protection functions  156  may actuate a trip and/or an alarm indication. 
   As illustrated in  FIG. 2   b , the generator operating boundary function  148  includes an A-phase P, Q calculator  150 , a B-phase P, Q calculator  152  and a C-phase P, Q calculator  154 . Each of the A-phase P, Q calculator  150 , the B-phase P, Q calculator  152  and the C-phase P, Q calculator  154  includes two inputs for receiving corresponding filtered digital signals, and at least two outputs. The generator operating boundary function  148  also includes a phase sum P, Q calculator  160  and a curve function  158 . The phase sum P, Q calculator  160  is coupled to receive the outputs of the A-phase P, Q calculator  150 , the B-phase P, Q calculator  152  and the C-phase P, Q calculator  154 . The curve function  158  includes a first ( 177 ) and second ( 178 ) inputs for receiving outputs from the phase sum P, Q calculator  160 , and a third input for receiving user programmable inputs  182 . While discussed in terms of P, Q calculators, it should be understood that the generator operating boundary function  148  may be implemented in one of any number of suitable ways, for example, as software executed via operation of the microcontroller  138 . 
   More specifically, the A-phase P, Q calculator  150  includes a first and a second input for receiving the filtered digital signals  161  and  164 , and a first and second output for providing an A-phase P value  171  and an A-phase Q value  174 , respectively, to the phase sum P, Q calculator  160 . During operation, the A-phase P, Q calculator  150  calculates the A-phase P value  171  and the A-phase Q value  174  based on corresponding A-phase secondary current and voltage waveforms I A , V A    101 ,  104 . The A-phase P value  171  represents a calculated active power operating point of the synchronous generator  12 , while the A-phase Q value  174  represents a calculated reactive power operating point of the synchronous generator  12  for the A-phase. 
   Similarly, the B-phase P, Q calculator  152  includes a first and a second input for receiving the filtered digital signals  162  and  165 , and a first and second output for providing a B-phase P value  172  and a B-phase Q value  175 , respectively, to the phase sum P, Q calculator  160 . During operation, the B-phase P, Q calculator  152  calculates the B-phase P value  172  and the B-phase Q value  175  based on B-phase secondary current and voltage waveforms I B , V B    102 ,  105 . The B-phase P value  172  represents a calculated active power operating point of the synchronous generator  12  and the B-phase Q value  175  represents a calculated reactive power operating point of the synchronous generator  12  for the B-phase. Likewise, the C-phase P, Q calculator  154  includes a first and a second input for receiving the filtered digital signals  163  and  166 , and a first and second output for providing a C-phase P value  173  and a C-phase Q value  176 , respectively, to the phase sum P, Q calculator  160 . During operation, the C-phase P, Q calculator  154  calculates the C-phase P value  173  and the C-phase Q value  176  based on the C-phase secondary current and voltage waveforms I C , V C    103 ,  106 . The C-phase P value  173  represents a calculated active power operating point of the synchronous generator  12  and the C-phase Q value  176  represents a calculated reactive power operating point of the synchronous generator  12  for the C-phase. 
   Each of the A-phase P value  171 , the A-phase Q value  174 , the B-phase P value  172 , the B-phase Q value  175 , C-phase P value  173  and the C-phase Q value  176  are received by the phase sum P, Q calculator  160  where the A-phase P value  171 , the B-phase P value  172  and the C-phase P value  173  are added together to form a P value sum  177 , and the A-phase Q value  174 , the B-phase Q value  175  and the C-phase Q value  176  are added to form a Q value sum  178 . The P value sum  177  represents a sum of three-phase active power, while the Q value sum  178  represents a sum of three-phase reactive power. 
   As illustrated in  FIG. 2   c , positive-sequence calculators  181 ,  183  may further be included to provide positive-sequence values for power calculation. In this embodiment, the following equations may be utilized therein. 
   Positive-sequence Voltage (Output  185 ) 
   
     
       
         
           
             
               
                 
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   Positive-sequence Current (Output  184 ) 
   
     
       
         
           
             
               
                 
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   Positive-sequence Apparent Power
 
 S   1 =3 ·V   1 ·conj( I   1 )  (3)
 
   Positive-sequence Active Power (Output  177 )
 
 P   1 =real( S   1 )  (4)
 
   Positive-sequence Reactive Power (Output  178 )
 
 Q   1 =imag( S   1 )  (5)
 
   Where:
 
a:=e j·120°   (6)
 
   Now referring concurrently to  FIGS. 2   a  and  2   b , as discussed above, the curve function  158  includes a first input for receiving the P value sum  177 , a second input for receiving the Q value sum  178 , and a third input for receiving user programmable inputs  182 . The curve function  158  further includes two outputs; a first output for enabling transmission of a binary alarm bit and a second output for enabling transmission of a binary trip bit. This arrangement may further be adapted to include additional outputs for assertion of alarm and/or tripping. For example, additional alarm bits may be included for assertion of alarm conditions associated with the field winding current limit, the armature current limit, the stator end region heating limit, a generator motoring condition, or a loss-of-field condition as will be discussed in further detail below. 
   In an embodiment, the curve function  158  is mathematically derived from, and is therefore representative of, a manufacturer-provided set of specific capability curves. The curve function  158  may be implemented as at least one set of derived curve expressions (e.g., polynomial equations of the second order; ax 2 +bx+c=0 or otherwise circle equations as discussed in greater detail below), one or more look-up tables, one or more linear equations, or equivalent means, collectively referred to herein as generator safe operating boundary data expressions. Each of the generator safe operating boundary data expressions may be derived from power system data such as manufacturer-provided sets of specific capability curves, SSSL curves, MEL curves, OEL curves or the like, collectively referred to herein as generator safe operating boundaries. As used herein, “power system data” may further include power system parameters such as power system equivalent impedance, power system component impedance, and the like. 
   Thus, in an embodiment, the apparatus and method for synchronous generator protection of the instant invention utilizes derived curve expressions to perform a portion of the protective functions of the protective relay  100 . In another embodiment, the apparatus and method for synchronous generator protection of the instant invention may utilize MEL look-up tables, SSSL linear equations, or any combination of capability curves, SSSL curves MEL curves, and/or OEL curves suitably expressed in the form of quadratic equations, circle equations, look-up tables, linear equations, or equivalent means, to perform protective functions of the protective relay  100 . 
   In one embodiment, the curve function  158  is implemented as a set of three quadratic equations derived from plotted P, Q coordinates of associated manufacturer-provided capability curves. In another embodiment, the curve function  158  is implemented through separate circle equations which provide a graphical representation of an estimation curve for the boundary for field winding heating associated with the field winding current limit of a generator capability curve; the boundary for armature heating associated with the armature current limit of a generator capability curve; and the boundary for stator core temperature associated with the stator end region heating limit. It should be understood however, that other implementations of the capability curves may be used for the curve function  158  (e.g., look-up tables). Further, it is contemplated that the curve function  158  may be derived from P, Q coordinates of SSSL, MEL curves, or OEL curves. For example, a loss-of-field element characteristic may be provided in relation to an SSSL curve, or the stator end region heating limit curve of the capability curve, as will be discussed in greater detail below. 
   While illustrated as functional blocks in  FIG. 2   a , the microcontroller  138  may be implemented via one of any number of suitable means. For example, in an embodiment, the microcontroller  138  may include a CPU, or a microprocessor, a program memory (e.g., a Flash EPROM) and a parameter memory (e.g., an EEPROM). Alternatively, the microcontroller  138  may be implemented as a field programmable gate array (FPGA), a digital signal processor (DSP) or a PC-based platform, to name a few. 
     FIG. 3  is an exemplary set of generator capability curves  200  that may be provided by a synchronous generator manufacturer to define the thermal, or heating, operating limits of the synchronous generator  12 . In the illustrated example, the set of generator capability curves  200  represent actual capability curves for a 312 MW, 3600 RPM, inner-cooled turbine generator, where the active power component “P” is expressed in MW on the horizontal axis and the reactive power component “Q” is expressed in MVAR on the vertical axis. 
   An overexcitation region (VARs are being supplied by the synchronous generator  12 ), defined above the zero MVAR point on the vertical axis, may also be referred to as lagging power factor region. An underexcitation region (VARs are being consumed by the synchronous generator  12 ), defined below zero MVAR on the vertical axis, may also be referred to as the leading power factor region. The capability curve  205  corresponds to a higher cooling system hydrogen pressure (e.g., 3 kg/cm 2 ) than the capability curve  203  (e.g., 2 kg/cm 2 ). Accordingly,  FIG. 3  illustrates that the effectiveness of the cooling and hence the allowable generator loading depends on the cooling pressure, and that the synchronous generator  12  can provide increased power output when the cooling system pressure is increased, provided that the prime mover has the ability to provide the additional power. 
   For ease of discussion,  FIG. 4  is the capability curve  205  of the exemplary set of generator capability curves  200 . In general, synchronous generators are rated in terms of maximum MVA output at a specified voltage and power factor (pf) (e.g., 0.85 lagging) which they can carry continuously without overheating. The active power output is limited by the prime mover capability to a value within the MVA rating of the generator. The continuous reactive power output capability is limited by the three factors; an armature current limit, a field winding current limit and an end region heating limit. The armature current limit associated with an RI 2  power loss is the maximum current that can be carried by the armature without exceeding heating limitations. The field current limit is associated with an R fd i fd   2  power loss. The localized heating in the end region of the stator imposes a third limit on the synchronous generator  12  which affects the capability of the generator in the underexcited condition. 
   Referring to  FIG. 4 , for each cooling pressure, a first curve portion  206  of the capability curve  205  represents a boundary for field winding heating associated with the field winding current limit. This is also generally referred to as the rotor current limit. The rotor-current limit on the generator field current generally results from copper power losses in the rotor winding. A second curve portion  210  of the capability curve  205  represents a boundary for armature heating associated with the armature current limit. The armature current limit on the generator field current generally results from stator copper power losses, wherein there is generally a maximum current that a generator can carry continuously without exceeding the allowable operating temperature. A third curve portion  208  of the capability curve  205  represents a boundary for a stator core temperature associated with the stator end region heating limit. This is generally referred to as the stator end heating limit. 
   An interior region  240  bounded by the first, second and third curve portions  206 ,  210 ,  208  is referred to as a safe operation region  240  indicating normal generator operation, while an exterior region  242  outside of the interior region is referred to as an unsafe operation region  242  indicating abnormal generator operation. 
   Typically, the manufacturer-provided generator capability curves are utilized by a power station operator to determine the operational capability of an associated synchronous generator and to determine whether additional MW can be obtained from the synchronous generator under various conditions. For example, referring to  FIG. 4 , when operated at a lagging power factor (pf) of 0.9 (point  218 ), the synchronous generator  12  will generate 312.5 MW of active power and 150 MVAR of reactive power. When operated at a leading pf of 0.9 (point  226 ), the synchronous generator  12  will generate 330 MW of active power and absorb 112.5 MVAR of reactive power. When operated at a pf of 1.0 (point  222 ), the synchronous generator  12  will generate a maximum of 345 MW of active power and no MVAR of reactive power. 
   An SSSL curve  244  and an MEL curve  246  are also illustrated in  FIG. 4 . The SSSL curve  244  represents a power limit to maintain system stability. The SSSL curve will vary with the synchronous generator and with the power system connected, as well as with voltage. 
   All synchronous generators connected to the power system operate at the same average speed. The generator speed governors maintain the machine speed close to its nominal value. There is a balance between generated and consumed active power under normal power system operating conditions. Random changes in load and system configuration constantly take place and impose small disturbances to the power system. The property of a power system to keep the normal operating condition under these small slow changes of system loading is generally known as steady-state stability or system stability for small perturbations. 
   For the two-machine power system the active power transfer P e  is given by: 
   
     
       
         
           
             
               
                 
                   P 
                   e 
                 
                 = 
                 
                   
                     
                       
                         E 
                         q 
                       
                       ⁢ 
                       
                         E 
                         s 
                       
                     
                     
                       
                         X 
                         d 
                       
                       + 
                       
                         X 
                         s 
                       
                     
                   
                   ⁢ 
                   sin 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   δ 
                 
               
             
             
               
                 ( 
                 7 
                 ) 
               
             
           
         
       
     
   
   wherein the generator internal voltage and synchronous reactance are E q  and X d  respectively; the power system voltage and reactance are E s  and X s  respectively; and the system power angle δ is the angle between E q  and E s . 
   Referring back to  FIG. 4 , the center position and radius of the SSSL circle are expressed by the following equations: 
   
     
       
         
           
             
               
                 
                   
                     Center 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       ( 
                       
                         P 
                         , 
                         Q 
                       
                       ) 
                     
                   
                   = 
                   0 
                 
                 , 
                 
                   
                     
                       V 
                       t 
                       2 
                     
                     2 
                   
                   ⁢ 
                   
                     ( 
                     
                       
                         1 
                         
                           X 
                           s 
                         
                       
                       - 
                       
                         1 
                         
                           X 
                           d 
                         
                       
                     
                     ) 
                   
                 
               
             
             
               
                 ( 
                 8 
                 ) 
               
             
           
           
             
               
                 Radius 
                 = 
                 
                   
                     
                       V 
                       t 
                       2 
                     
                     2 
                   
                   ⁢ 
                   
                     ( 
                     
                       
                         1 
                         
                           X 
                           d 
                         
                       
                       - 
                       
                         1 
                         
                           X 
                           s 
                         
                       
                     
                     ) 
                   
                 
               
             
             
               
                 ( 
                 9 
                 ) 
               
             
           
         
       
     
   
   Wherein V t  is the generator terminal voltage. Typically when the power system is strong (X s  is low) the SSSL locus is outside the generator capability curve. However, on weak systems, the manual SSSL can be more restrictive than the generator capability in the underexcited region. 
   Under automatic operation, the automatic voltage regulator (AVR) rapidly varies the field current in response to system operating conditions. This changes the maximum value of the power angle curve upwards or downwards as required by the system. This dynamic response improves the SSSL as compared to that resulting from manual regulator operation. The effect of AVR on SSSL depends on the voltage regulator gain, the regulator time constant and the field time constant. 
   MEL is a control function included in the automatic voltage regulator that acts to limit reactive power flow into the generator. During normal operation, the AVR keeps generator voltage at a preset value. When system conditions require the generator to absorb reactive power in excess of the MEL set point, the MEL interacts with the AVR to increase terminal voltage until reactive power inflow is reduced below the setting. The MEL curve  246  represents a boundary below which the MEL included in the AVR of the synchronous generator  12  operates to restrict generator reactive power inflow. The MEL curve  246  is situated just above the SSSL curve  244 . 
   Overexcitation limiter (OEL) is a control function included in the AVR that protects the generator from overheating resulting from prolonged field overcurrent. OEL detects the field-overcurrent condition and acts with time delay to ramp down the excitation to a preset value. The OEL operating characteristic (not shown in  FIG. 4 ) plots as a line in the P-Q plane, placed below the field winding current limit curve  206 . 
   The generator safe operating boundary expressions may be derived from generator data and/or power system data (e.g., in relation to the stator end region heating limit boundary  208 , MEL curve  246 , OEL curve (not shown), or SSSL curve  244 ). For example, as will be discussed in greater detail below, loss-of-field protection may be provided by situating a generator safe operating boundary expression in relation to the stator end region heating limit boundary  208 , or SSSL curve  244 . 
   Referring again to  FIG. 2   a , the user programmable inputs  182  include pre-programmed user inputs that may be selected/set during commissioning of the protective relay  100 . In an embodiment, each of the user programmable inputs  182  corresponds to one of a number of sets of derived curve expressions selectable by the microcontroller  138  upon occurrence of specified generator operating conditions. However, other selection arrangements are contemplated. 
   As described above, each set of curve expressions is derived from plotted P, Q coordinates of a manufacturer-provided capability curve. For example, one set of curve expressions is derived from the P, Q coordinates of the first, second and third curve portions  206 ,  208 ,  210  of the capability curve  205 . Similarly, a different set of curve expressions may be derived from the P, Q coordinates of the capability curve  203 . 
   In general, during relay operation, the microcontroller  138  selects a particular set of derived curve expressions for the curve function  158  based on its user programmable input(s)  182 , the relay operating conditions determined from the measured secondary currents and voltages, and the generator operating indications received via the indication input  180 . As the generator operating conditions change, so do the sets of derived curve expressions utilized by the microcontroller  138  when performing the generator operating boundary function  148 . For example, using manufacturer-provided capability curves as a basis, the user may specify, via the user programmable inputs  182 , that the microcontroller  138  utilize a first set of derived curve expressions upon detecting a generator operating pressure of 3 Kg/cm 3 , and utilize a second set of derived curve expressions upon detecting a generator operating pressure of 2 Kg/cm 3 . Other generator operating conditions such as generator temperature measurements from associated temperature transducers, etc., may be used as a basis for the user programmable inputs  182  and subsequent microcontroller selection of derived curve equation sets. 
   Although illustrated as including only one user programmable input  182 , it is contemplated that the curve function  158  may include more or less user programmable inputs  182 , depending on the implementation. Further, although illustrated as one curve, the curve function  158  may include a number of sets of curve expressions derived from multiple manufacturer-provided capability curves, and therefore represent multiple operating limits at different cooling system pressures. 
   Additionally, one curve function may be assigned to determine an alarm bit output, and another curve function may be assigned to determine a trip bit output. Assertion of an alarm and/or trip output bit may be delayed via operation of one or more timers such as a first timer  191  and a second timer  192 . For example, an output of a comparison of the P value sum  177  and the Q value sum  178  to one set of derived curve expressions may actuate an alarm action after a 10 second timeout of the timer  191 , while an output of a comparison of the P value sum  177  and the Q value sum  178  to another set of derived curve expressions may actuate a trip action after a 0.2 second timeout of the timer  192 . 
   Thus, during relay operation, in addition to providing protective functions such as differential protection, ground fault protection, etc., illustrated as the additional generator protection functions  156 , the microcontroller  138  compares the P value sum  177  and the Q value sum  178  to a selected set(s) of derived curve expressions to determine whether operating conditions of the synchronous generator  12  are inside or outside of the generator safe operating boundaries, and to actuate subsequent alarm and/or trip actions when warranted. Such P and Q value sums  177 ,  178  reflect the “P, Q operating point” of the synchronous generator  12 . 
     FIG. 5   a  is a flowchart of a method  300  for providing synchronous generator protection in the power system  10 , according to an embodiment of the invention. In general, for each manufacturer-provided generator capability curve, the method  300  includes deriving sets of curve expressions or curve elements from any of generator capability curve data, generator impedance characteristic, and/or power system data for use by the protective relay  100  during operation. While implemented as sets of derived curve expressions, it should be understood that the expressions may be linear, quadratic or otherwise circle equations, and that other generator safe operating boundary data expressions (e.g., look-up tables) are contemplated. Other generator safe operating boundaries (e.g., situated in relation to SSSL curves, MEL curves, or OEL curves) may be used as a basis for deriving the generator safe operating boundary data expressions. 
   The method  300  begins when a number of sets of curve elements or expressions are derived from power system data such as generator capability curve data, generator impedance characteristic, and/or power system parameters (step  303 ). For example, close approximations of the first, second and third curve portions  206 ,  210 ,  208  of the capability curve  205  of  FIG. 4  can be expressed as one set of derived curve expressions. The set of derived curve expressions may be calculated using a curve-fitting algorithm such as one available in Matlab® (or similar curve-fitting algorithm) along with a number of plotted (P, Q) coordinates of the capability curve  205 . 
   For example, Table 1 illustrates a number of (P, Q) coordinates of the capability curve  205  that may be used to derive one set of curve expressions. 
   
     
       
         
             
             
             
           
             
               TABLE 1 
             
           
          
             
                 
             
             
               Curve 1 (206) 
               Curve 2 (210) 
               Curve 3 (208) 
             
          
         
         
             
             
             
             
             
             
          
             
               P (MW) 
               Q (MVAR) 
               P (MW) 
               Q (MVAR) 
               P (MW) 
               Q (MVAR) 
             
             
                 
             
          
         
         
             
             
             
             
             
             
          
             
               0 
               212.5 
               312.5 
               150 
               0 
               −168 
             
             
               137.5 
               200 
               343.7 
               50 
               100 
               −162.5 
             
             
               200 
               187.5 
               345 
               0 
               237.5 
               −137.5 
             
             
               312.5 
               150 
               343.7 
               −50 
               329.3 
               −108.2 
             
             
                 
                 
               329.3 
               −108.2 
             
             
                 
             
          
         
       
     
   
   Where, in  FIG. 4 : 
   0 MW and 212.5 MVAR is represented by reference number  212 , 
   137.5 MW and 200 MVAR is represented by reference number  214 , 
   200 MW and 187.5 MVAR is represented by reference number  216 , 
   312.5 MW and 150 MVAR is represented by reference number  218 , 
   343.7 MW and 50 MVAR is represented by reference number  220 , 
   345 MW and 0 MVAR is represented by reference number  222 , 
   343.7 MW and −50 MVAR is represented by reference number  224 , 
   329.3 MW and −108.2 MVAR is represented by reference number  226 , 
   237.5 MW and −137.5 MVAR is represented by reference number  228 , 
   100 MW and −162.5 MVAR is represented by reference number  230 , and 
   0 MW and −168 MVAR is represented by reference number  232 . 
   In one embodiment, using Table 1, a set of three derived curve expressions (10), (11) and (12) approximating the first, second and third curve portions  206 ,  210 ,  208  can be derived from the plotted (P, Q) coordinates reflected in Table 1, where the active power component “P” is expressed as a function of the reactive power component “Q”. In this embodiment, the curve expressions are in the form of a set of three quadratic equations derived from plotted P, Q coordinates of associated manufacturer-provided capability curves. Each quadratic equation provides a graphical representation of an estimation curve for the boundary for field winding heating associated with the field winding current limit (Curve  1 ); the boundary for armature heating associated with the armature current limit (Curve  2 ); and the boundary for a stator core temperature associated with the stator end region heating limit (Curve  3 ). Other arrangements are possible. 
   Curve  1  Quadratic Equation:
 
 P ( Q )=−0.0882 ·Q   2 −19.2898 ·Q− 726.1124  (10)
 
   Curve  2  Quadratic Equation:
 
 P ( Q )=−0.0011 ·Q   2 −0.0234 ·Q+ 340.1911  (11)
 
   Curve  3  Quadratic Equation:
 
 P ( Q )=−0.0404 ·Q   2 +10.6301 ·Q− 373.5395  (12)
 
     FIG. 6  is a generator capability curve  400  drawn based on the set of derived curve expressions defined by equations (10), (11) and (12), according to an embodiment of the invention. For each of the three derived curve expressions of  FIG. 6 , the active power component is expressed as a function of the reactive power component. Similarly, additional sets of three curve expressions may be derived from other manufacturer-provided capability curves. Each set represents generator operating (thermal) limits at a specific cooling pressure. As discussed above, approximations of the SSSL curves and MEL curves may also be derived in a suitable form, and then used by the protective relay  100  during generator operation. 
   In another embodiment, a set of three derived curve expressions (13), (17) and (21) approximating the first, second and third curve portions  206 ,  210 ,  208  can be derived from the plotted (P, Q) coordinates (e.g., reflected in Table 1), where the active power component “P” is expressed as a function of the reactive power component “Q”. In this embodiment, the curve expressions are in the form of a set of three circle equations derived from plotted P, Q coordinates of associated manufacturer-provided capability curves. Each circle equation provides a graphical representation of an estimation curve for the boundary for field winding heating associated with the field winding current limit (Curve  1 ); the boundary for armature heating associated with the armature current limit (Curve  2 ); and the boundary for a stator core temperature associated with the stator end region heating limit (Curve  3 ). Other arrangements are possible. 
   More specifically, the following circle equation (equation (13)) may provide the graphical representation of the estimation curve for the boundary for field winding heating associated with the field winding current limit (Curve  1 ) as illustrated in  FIG. 7 . 
   Curve  1  Circle Equation for Field Winding Current Limit: 
   
     
       
         
           
             
               
                 
                   S 
                   ⁡ 
                   
                     ( 
                     β 
                     ) 
                   
                 
                 = 
                 
                   
                     
                       R 
                       · 
                       
                         ⅇ 
                         
                           ⅈ 
                           · 
                           β 
                         
                       
                     
                     + 
                     
                       
                         ⅈ 
                         · 
                         C 
                       
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       for 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       ρ 
                     
                   
                   ≤ 
                   β 
                   ≤ 
                   
                     π 
                     2 
                   
                 
               
             
             
               
                 ( 
                 13 
                 ) 
               
             
           
         
       
     
   
   Where:
         R is the radius of the circle   C is the center of the circle   ρ is the circle lower limit       

   For Curve  1  the following equations (equations (14), (15), and (16)) may be solved in order to obtain the values for R, C, and ρ.
 
 R ·cos ρ= p   1   (14)
 
 R ·sin ρ+ C=q   1   (15)
 
 R+C=q   0   (16)
 
   The following circle equation (equation (17)) may provide the graphical representation of the estimation curve for the boundary for armature heating associated with the armature current limit (Curve  2 ) as illustrated in  FIG. 8 : 
   Curve  2  Circle Equation for Armature Current Limit:
 
 S (β)= R·e   i·β   +i·C  for −α≦β≦φ  (17)
 
   Where:
         R is the radius of the circle   C is the center of the circle   φ is the circle upper limit that corresponds to the minimum lagging power factor   −α is the circle lower limit that corresponds to the minimum leading power factor       

   For Curve  2  the following equations (equations (18), (19), and (20)) may be solved in order to obtain the values for R, α, and φ.
 
R=S nom   (18)
 
φ=cos −1 ( PF   Lag )=cos −1 ( P   1   /S   nom )  (19)
 
α=cos −1 ( PF   Lead )=cos −1 ( P   2   /S   nom )  (20)
 
   Where:
         PF Lag  is the minimum lagging power factor   PF Lead  is the minimum leading power factor   S nom  is the generator rated capacity       

   The following circle equation (equation (21)) may provide the graphical representation of the estimation curve for the boundary for a stator core temperature associated with the stator end region heating limit (Curve  3 ) as illustrated in  FIG. 9 : 
   Curve  3  Circle Equation for Stator End Region Heating Limit: 
   
     
       
         
           
             
               
                 
                   S 
                   ⁡ 
                   
                     ( 
                     β 
                     ) 
                   
                 
                 = 
                 
                   
                     
                       R 
                       · 
                       
                         ⅇ 
                         
                           ⅈ 
                           · 
                           β 
                         
                       
                     
                     + 
                     
                       
                         ⅈ 
                         · 
                         C 
                       
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       for 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         
                           3 
                           2 
                         
                         · 
                         π 
                       
                     
                   
                   ≤ 
                   β 
                   ≤ 
                   
                     - 
                     γ 
                   
                 
               
             
             
               
                 ( 
                 21 
                 ) 
               
             
           
         
       
     
   
   Where:
         R is the radius of the circle   C is the center of the circle   −γ is the circle upper limit       

   For Curve  3  the following equations (equations (22), (23), and (24)) may be solved in order to obtain the values for R, C, and γ.
 
 R ·cos γ= p   2   (22)
 
 C−R ·sin γ= q   2   (23)
 
 C−R=q   3   (24)
 
   Using circle equations (13), (17) and (21), a generator capability curve may be drawn similar to that shown in  FIG. 6 . As discussed above, approximations of the SSSL curves, MEL curves and OEL curves or user-entered curves may also be derived in a suitable form using circle equation (21). For example and as will be discussed in greater detail below, using the circle equation of Curve  3  (equation (21)), the estimation curve for a loss-of-field protection element curve such that it is situated in relation to the stator end region heating limit curve, or in relation to SSSL may also be derived. In yet another embodiment, the operating region defined in order to provide for loss-of-field protection is located below a loss-of-field element characteristic (situated with respect to the stator end region heating limit curve, or in relation to an SSSL curve), and between two active power elements vertical straight lines. 
   In one embodiment, as shown in the P-Q plane of  FIG. 10 , a loss-of-field protection characteristic is provided, including a loss-of-field element characteristic and two active power element vertical straight lines. The use of the embodiment of  FIG. 10  with the aforementioned apparatuses, systems and methods of the present invention is generally beneficial where the SSSL characteristic is outside the capability curve. In this arrangement, the loss-of-field element characteristic is set coinciding with the stator end heating limit curve, and situated above the SSSL curve. Situating the loss-of-field element in this manner allows the capability curve to protect the generator from stator end core heating. This arrangement further allows using the full generator capability to absorb reactive power, beyond the MEL setting. 
   In another embodiment, as shown in the P-Q plane of  FIG. 11 , another loss-of-field protection characteristic is provided. The use of the embodiment of  FIG. 11  with the aforementioned apparatuses, systems and methods of the present invention is generally beneficial where the SSSL characteristic is inside the capability curve, which may occur in weak power systems. In this arrangement, the loss-of-field element characteristic is situated above the SSSL curve, and also inside the capability curve. However, in comparing the P-Q planes of  FIGS. 10 and 11 , the loss-of-field element of  FIG. 11  is situated generally closer to the SSSL curve. Situating the loss-of-field element in this manner limits the amount of reactive power that the generator can absorb. 
   In the arrangements of  FIGS. 10 and 11 , the loss-of-field element characteristic along with the capability curve (comprising curves representing a rotor current limit and armature current limit) define a generator normal operation zone, an alarming zone and a protection zone. The area bounded by the loss-of-field element characteristic and the capability curve defines the generator normal operation zone, whereas the area bounded by the two active power elements vertical straight lines and the loss-of-field element characteristic defines the protection zone. It is to be noted that the area outside the area bounded by the loss-of-field element and the capability curve defines an alarming zone as discussed in detail above. As discussed in full detail above, when the generator operating condition falls within the protection zone and/or the alarming zone, the apparatus of  FIG. 2   a  is adapted to assert an alarm and/or trip signal. 
   The active power elements serve as blinders which restrict coverage along the P axis of the P-Q plane. The left-side active power element may be set to any value. In one embodiment, the left side active power element is set to coincide with the Q axis. The right-side active power element may be set to any value. For example, the right-side active power element may be set such that it adapts to the generator load condition. In another example, the right-side active power element may be set to the measured pre-disturbance active power, plus 20% of the generator rated active power. In yet another example, the upper limit of the right-side active power element may be set to the generator MVA rating or, alternatively the turbine MW rating. 
   In yet another embodiment, the generator loss-of-field protection element may further include an undervoltage element or curve (not shown). In this arrangement, the undervoltage element operates to accelerate a trip and/or alarm assertion when a low voltage condition indicates that the system may collapse. For example, an undervoltage element may be provided and set to 0.8-0.9 of the generator nominal voltage. Once the generator voltage falls below this value, a trip and/or alarm signal is asserted. 
   In one embodiment, which may be referred to as the manual method, the techniques described herein relating to deriving a set of expressions to determine the generator capability curve and loss-of-field protective element characteristic may be performed by a protective relay automatically. In this embodiment, the user enters power system data that includes data based on the capability curve and the user-defined loss-of-field element characteristic. For example, the user may enter two coordinates corresponding with the rotor current limit  206  such as (p 0 , q 0 )  212  and (p 1 , q 1 )  218  of  FIG. 4 . For the loss-of-field element, the user may enter coordinates on the user-defined loss-of-field element, and select between a straight-line configuration and a curved-line configuration. The straight-line configuration may include one or more straight-line approximations. Also, the curved-line configuration may include one or more curved-line approximations. For example, the user-defined loss-of-field element may correspond with the stator end heating limit  208  of the capability curve  205 , and the user may enter coordinates on this curve such as (p 3 , q 3 )  232  and (p 2 , q 2 )  226  of  FIG. 4 . Alternatively, the user-defined loss-of-field element may lie above the stator end heating limit  208  as illustrated in  FIG. 11 . From these entered coordinates, the generator safe operating boundary data expressions may be derived by techniques described above from the entered power system data. 
   In another embodiment, which may be referred to as the automatic method, the user need not enter a user-defined loss-of-field element. However, the user does enter power system data including generator manufacturer data and power system parameters. The generator manufacturer data includes coordinates of points on the capability curve, as described above and in conjunction with  FIG. 4  and Table 1, and the generator impedance. The power system parameters includes the equivalent system impedance. From this, the capability curve and the loss-of-field element characteristic are derived automatically, for example, by a protective relay, as further described herein. Thus, the generator safe operating boundary data expressions are derived from a plurality of power system data. 
   Referring again to  FIG. 5   a , next a number of sets of derived curve expressions (e.g., quadratic, circle or any other suitable equations) are provided to the protective relay  100  for selection and use by the microcontroller  138  when performing the generator operating boundary function  148  (step  304 ). In general, during relay operation, the microcontroller  138  determines which set of derived curve expressions should be used as the curve function  158 . Such “selected sets of derived curve expressions” may vary depending on previously entered user programmable inputs  182  (step  311 ), on generator operating indications (step  305 ), or on generator terminal voltage and/or stator current (step  306 ). For example, if a generator operating indication is determined to be 2 Kg/cm 3 , a first set of derived curve expressions derived from a first generator-manufacturer capability curve is used, and if a generator operating indication is determined to be 3 Kg/cm 3 , a second set of derived curve expressions derived from a second generator-manufacturer capability curve is used. The selected sets of derived curve expressions may also vary depending on the configuration of the generator operating boundary function  148  (e.g., multiple user programmable inputs  182 , multiple sets of derived curve expressions used to provide a binary output to actuate an alarm and/or a trip bit, etc.) 
   Among other things, both secondary voltage and current waveforms V A , V B , V C , I A , I B , and I C  are processed by the protective relay  100  to form the P value sum  177  and the Q value sum  178  as described in connection with  FIG. 2   a . The P and Q value sums  177  and  178  determine the P, Q operating point of the synchronous generator  12  at a particular moment in time. Generator cooling pressures and the like however, are determined from generator operating indications received via the indication input  180  (see  FIG. 2   a ). In one embodiment, pressure transducers may provide generator cooling gas pressure measurements to the microcontroller  138  via the indication input  180 . Other generator operating indications may be used such as, for example, excitation or field current, stator temperature, gearing temperature, ambient temperature and the like. With the user programmable inputs, generator operating indications, and/or generator terminal voltage and/or stator current, the microcontroller  138  calculates the P and Q value sums and determines the P-Q operating point of the synchronous generator. (step  308 ). 
   After determining the P-Q operating point, the microcontroller  138  adapts the protective element characteristics of the relay to the generator operating conditions (step  309 ). The microcontroller  138  compares the P, Q operating point of step  308  to protective element characteristics adapted in step  309  to determine whether the P, Q operating point falls within the safe operation region  240  or the unsafe operation region  242  of a corresponding curve approximated by the selected set of derived curve expressions. 
     FIG. 5   b  illustrates an embodiment of a method for determining whether an alarm or trip condition should be asserted. If the P, Q operating point falls within the safe operation region  240  of the curve (e.g., as defined by the selected set of derived curve expressions, etc.), the microcontroller  138  concludes that the synchronous generator  12  is operating within its normal limits (i.e., the normal operating region) and no action is taken. 
   Another example of a generator normal operation zone is further shown as the unshaded region of  FIG. 12 . An example of a safe operating condition is point P A , Q A  in  FIG. 13 . It is to be noted that this safe operation region is bounded by the loss-of-field element characteristic along with the capability curve (comprising curves representing a rotor current limit and armature current limit), and it is also bounded by an active power element characteristic, which coincides with the Q axis of the P-Q plane. 
   Referring back to  FIG. 5   b , if the P, Q operating point is determined to fall within the unsafe operation region  242  (e.g., as defined by the selected set of derived curve expressions, etc.), the microcontroller  138  concludes that the synchronous generator  12  is not operating within its safe limits and causes an action. Another example of an alarming zone is further shown as the shaded region of  FIG. 12 , which corresponds to an armature current limit violation, a rotor current limit violation, a motoring condition, a loss-of-field condition or otherwise an under excitation condition.  FIG. 13  further illustrates the protection zone based on a loss-of-field element, wherein a trip signal is asserted if an operating condition would fall therein. 
   In the case of abnormal generator operation, the action may include actuating an audible alarm to indicate the unsafe operating conditions, actuating a trip signal to remove the synchronous generator  12  from service, notifying the power station operator via a mobile text message or a computer terminal display message, etc. Other notification or remedial actions are contemplated. 
   More specifically, in one example, the alarm characteristic in the P-Q plane may be formed by the upper and right side branches of the capability curve, by the loss-of-field element characteristic, and by an active-power characteristic that coincides with the Q axis. In one arrangement, the SSSL characteristic may be situated outside the capability curve. Accordingly, as shown in  FIG. 12  and similar to  FIG. 10 , the alarm characteristic fully coincides with the generator capability curve. Depending on the limit violated by the generator operating point (P, Q), the alarm element issues one of the following alarms, Armature-Current Limit Violation; Rotor-Current Limit Violation; Loss-of-field/Underexcitation Condition or Motoring Condition 
   In yet another embodiment, when the SSSL characteristic may be situated inside the capability curve, as shown in  FIG. 11 , the lower side of the alarm characteristic lies inside the capability curve, coinciding with the loss-of-field element characteristic. Depending on the limit violated by the generator operating point (P, Q), the alarm element issues one of the following alarms, Armature-Current Limit Violation; Rotor-Current Limit Violation; Loss-of-field/Underexcitation Condition or Motoring Condition. 
     FIG. 13  illustrates yet another embodiment, comprising both a loss-of-field protection characteristic and a capability curve violation alarming characteristic. In this embodiment, an alarm signal is asserted if an operating condition would fall out of the “generator normal operation zone”. Additionally, a trip signal is asserted after a time delay if an operating condition would fall within the “relay protection zone”. 
   For example, referring concurrently to  FIGS. 5   b ,  12  and  13 , if there is an armature current limit violation, a rotor current limit violation, a motoring condition, a loss-of-field condition or otherwise an under excitation condition, an alarm is asserted (for example, a rotor current limit violation is shown as point (P B , Q B ) of  FIG. 13 ). Additionally, referring concurrently to  FIGS. 5   b  and  13 , if there is a loss-of-field condition or otherwise an under excitation condition, a trip signal may also be asserted to trip the associated generator and/or field breaker (e.g., shown as point (P C , Q C ) of  FIG. 13 ). 
   The present method may be implemented as a computer process, a computing system or as an article of manufacture such as a computer program product or computer readable medium. The computer program product may be a computer storage media readable by a computer system and encoding a computer program of instructions for executing a computer process. The computer program product may also be a propagated signal on a carrier readable by a computing system and encoding a computer program of instructions for executing a computer process. 
   In one embodiment, the logical operations of the present method are implemented (1) as a sequence of computer implemented acts or program modules running on a computing system and/or (2) as interconnected machine logic circuits or circuit modules within the computing system. The implementation is a matter of choice dependent on the performance requirements of the computing system implementing the invention. Accordingly, the logical operations making up the embodiments of the present invention described herein are referred to variously as operations, structural devices, acts or modules. It will be recognized by persons skilled in the art that these operations, structural devices, acts and modules may be implemented in software, in firmware, in special purpose digital logic, and any combination thereof without deviating from the spirit and scope of the present invention as recited within the claims attached hereto. 
   While this invention has been described with reference to certain illustrative aspects, it will be understood that this description shall not be construed in a limiting sense. Rather, various changes and modifications can be made to the illustrative embodiments without departing from the true spirit, central characteristics and scope of the invention, including those combinations of features that are individually disclosed or claimed herein. Furthermore, it will be appreciated that any such changes and modifications will be recognized by those skilled in the art as an equivalent to one or more elements of the following claims, and shall be covered by such claims to the fullest extent permitted by law.