Pérdida Sincronismo PD Enero 2006

46 IEEE TRANSACTIONS ON POWER DELIVERY, VOL. 21, NO. 1, JANUARY 2006

Online Detection of Loss of SynchronismUsing Energy Function Criterion

K. R. Padiyar, Senior Member, IEEE, and S. Krishna

AbstractMaintaining dynamic security of a power system sub-jected to large disturbances is of utmost importance. Fast and ac-curate online detection of instability is essential in initiating certainemergency control measures. The techniques reported in the litera-ture involve mainly the application of global phasor measurementsand heuristic algorithms. In this paper, an accurate technique forthe online detection of loss of synchronism based on voltage andcurrent measurements in a line is presented. The technique makesuse of the concept of potential energy in a line. The conditions forthe system instability are derived from energy function analysis.However, no assumptions are made regarding the power-angle re-lationship in a line, nor are any data on the system equivalents nec-essary in implementing the detection scheme.

Index TermsCritical cutset, energy function, transient sta-bility.

NOMENCLATURE

Rotor angle in centre of inertia (COI) reference.Rotor speed in COI reference.Inertia constant.Mechanical power input to the generator.Electrical power output of the generator.Generator quadrature axis voltage.Direct axis open-circuit transient time constant.Direct axis reactance.Direct axis transient reactance.Generator direct axis current.Generator field voltage.Generator direct axis voltage.Quadrature axis open-circuit transient time constant.Quadrature axis reactance.Quadrature axis transient reactance.Generator quadrature axis current.Automatic voltage regulator gain.Exciter time constant.Generator voltage reference.Static active power load, assumed to be constant.Reactive power output at the generator terminal.

Manuscript received March 1, 2004; revised July 5, 2004. This work wassupported by the Department of Science and Technology, Government ofIndia, under the project Dynamic Security Assessment and Control of PowerGrids. The paper is based on a contribution presented by the authors at theIEEE Transmission and Distribution Conference and Exposition, Atlanta, GA,October 2001. Paper no. TPWRD-00104-2004.

K. R. Padiyar is with the Department of Electrical Engineering, Indian Insti-tute of Science, Bangalore 560012, India.

S. Krishna is with the Department of Electrical and Electronics Engineering,M.S. Ramaiah Institute of Technology, Bangalore 560054, India.

Digital Object Identifier 10.1109/TPWRD.2005.848652

Static reactive power load, assumed to be a functionof bus voltage.

th element of the network admittance matrix.Bus voltage.Bus angle in COI reference.

I. INTRODUCTION

THE occurrence of a large disturbance in a power systemmay lead to uncontrolled tripping of generators and cas-cading outages and may finally result in a blackout if properactions are not taken. There are many discrete control measures[1], [2] which can be initiated to maintain system stability. Someof the emergency measures, such as generator tripping and con-trolled system separation, should be exercised only when thereis an absolute necessity. Hence, a fast and accurate method ofdistinguishing between stable and unstable swings is necessary.

The conventional out-of-step relaying based on theimpedance measurement on a transmission line has many limi-tations. To overcome these limitations, an adaptive out-of-steprelaying scheme was proposed [3]; this scheme requires theknowledge of the equivalent system parameters that have to bereasonably accurate.

There are many techniques reported in the literature whichinvolve mainly application of global phasor measurements [4],[5]; heuristic algorithms [6]; and use of intelligent techniques,such as decision trees [7] and artificial neural networks (ANNs)[8].

In this paper, a method of online detection of the loss of syn-chronism based on voltage and current measurements in a lineis presented. The conditions for system instability are derivedfrom energy function analysis. The potential energy with gen-erators represented by a classical model can be expressed asthe sum of energies in the series elements (transmission lines,transformers, and generator reactances) [9]. In this paper, it isshown that such an expression is applicable even for the de-tailed (two-axis) generator model. Under certain assumptions, itis possible to express the potential energy as the sum of energiesin the lines belonging to a cutset and the kinetic energy as a func-tion of the rate of change of phase angle across a line belongingto the cutset. The proposed technique makes use of the poten-tial energy in the lines belonging to the cutset and the kineticenergy. The technique has been tested by simulation studies onthe New England 10-generator system and the IEEE 17-gener-ator system. The paper reports on these studies along with themethods of speeding up the detection of instability based on theprediction of system trajectories.

0885-8977/$20.00 2006 IEEE

PADIYAR AND KRISHNA: ONLINE DETECTION OF LOSS OF SYNCHRONISM USING ENERGY FUNCTION CRITERION 47

Fig. 1. Single-line diagram of a 10-generator 39-bus New England system.

Fig. 2. Angle across series elements for the critically unstable case.

II. CRITICAL CUTSET ASSOCIATED WITHTRANSIENT INSTABILITY

When a power system becomes unstable, it initially splits intotwo groups. It is observed in simulation studies that there is aunique cutset consisting of series elements (connecting the twoareas) across which the angle becomes unbounded. This is illus-trated by simulating a three-phase fault at bus 14 of the 10-gen-erator New England system. The single-line diagram of the NewEngland system is shown in Fig. 1. The data for this systemare given in [2]. The angle across all series elements (transmis-sion lines, transformers, and generator reactances) is boundedfor critically stable clearing time (0.270 s). For the critically un-stable fault clearing time (0.280 s), generator 2 separates fromthe rest of the system. The angle across all of the series ele-ments for the unstable case is plotted in Fig. 2. The angle acrosslines 18-19 and 11-12 becomes unbounded. Hence, for this case,these lines form the critical cutset across which the system sep-arates into two areas.

Fig. 3. Excitation system.

III. SYSTEM MODEL AND ENERGY FUNCTION

A power system with buses and generators representedby a detailed (1.1) model (with static exciter) is considered. Thesystem is assumed to be lossless. The equations governing thesystem are [2]

(1)

(2)

(3)

(4)

(5)(6)

(7)

(8)

(9)

(10)

The static exciter is represented by the block diagram shownin Fig. 3. An energy function for this system [2], [10] isdefined as

(11)


where is kinetic energy and is potential energy given by

where is the reactive power output at the internal bus of gen-erator . The energy function is defined for the postfault systemand the time derivative of is zero along a postfault trajectory.The subscript in the above expressions indicates quantities attime .

The potential energy can also be expressed as follows.The proof is given in the Appendix.

(12)

where is the power flow in the series element , is thesteady-state value of , is the phase-angle difference acrossthe element, and is the total number of series elements. Thisexpression for potential energy is identical to the one derived in[9] in which the classical model is assumed for generators.

IV. CRITERION FOR INSTABILITYThe system can be represented by two areas connected by

the critical cutset as shown in Fig. 4. The potential energy canbe decomposed into the energy within the two areas and theenergy along the critical cutset [9]. Assuming coherent areas,the potential energy within an area is zero as all of the buses inthat area have the same frequency ( n is zero for all serieselements within an area). Hence, the potential energy given by(12) can be written as follows:

(13)

Fig. 4. Coherent areas.

where is the number of elements in the critical cutset. It canbe shown that the variation of potential energy in all of the linesin the critical cutset is similar. If a series element (line or trans-former) in the critical cutset connects buses and

(14)(15)

where and are the voltage magnitudes at buses (in area I)and (in area II), respectively, and is the susceptance of theseries element. Due to the assumption of coherency, the varia-tions of and are similar for all of the elements in the criticalcutset. This is also true of the variations of and . Hence,the variation of potential energy can be monitored from the en-ergy in the individual lines in the cutset. Hence

(16)where is a constant and subscript refers to any element inthe cutset, and

(17)

The corrected kinetic energy to properly account for theportion of the kinetic energy that contributes to system separa-tion [11] is given by

(18)

where

By assumption of coherency, the rotor speeds of all the gener-ators in an area are equal and the derivative of the angle acrossall of the elements in the critical cutset are the same. Hence

(19)

where is the angle across any line in the critical cutset. Thecorrected kinetic energy is given by

(20)


Fig. 5. Variation of power flow and rate of change of angle of the lines 11-12and 18-19 for the critically stable case.

The criterion derived for the detection of instability is basedon energy function analysis. The power system gains kinetic andpotential energy due to a disturbance. For transient stability, thesystem must be capable of absorbing the kinetic energy com-pletely. If the kinetic energy is not completely converted to po-tential energy, the system becomes unstable. Therefore, for astable swing, kinetic energy is zero when potential energy at-tains a maximum, and for an unstable swing, kinetic energy isnot zero (positive) when potential energy attains a maximum.This criterion is used for the detection of instability. Since thecriterion checks whether kinetic energy is zero or positive whenpotential energy is maximum, it is adequate to monitorinstead of the kinetic energy, and the potential energy given by(17) can be used instead of (13).

The potential energy attains a maximum value whenor . For stable cases, ( reaches

a maximum value) when potential energy attains the first max-imum; for unstable cases, when potential energy at-tains the first maximum. Hence, the system is unstable if de-creases to before becomes zero. The detection cri-terion requires and . These two quantities can be obtainedby local measurements at one end of a line. is obtained fromthe measurement of voltage and current at one end of a line withthe knowledge of line impedance.

For the 10-generator system, the three-phase fault is simu-lated at bus 14. For this case, the angle across the lines 11-12 and18-19 becomes unbounded in case of instability. Fig. 5 showsthe variation of and for the lines 11-12and 18-19 for the critically stable fault clearing time (0.270 s).

and are plotted for the postfault system. The vari-ation of and for critically unstable fault clearingtime (0.280 s) is plotted in Fig. 6. , which is the angle acrossseries element connecting buses and , is defined as

; and are chosen such that increases during thefault and is the power flow from bus to bus . It can be seenthat decreases to zero before decreases to zero forboth the lines when the system is stable; decreases to zero

Fig. 6. Variation of power flow and rate of change of angle of the lines 11-12and 18-19 for the critically unstable case.

when is still positive for both the lines when the systemis unstable.

As the assumption of coherency within a group of generatorsswinging together is not strictly correct, the instability criterionis not satisfied for all of the lines within the cutset at the sameinstant. If the system is unstable, all of the lines in the criticalcutset satisfy the instability criterion over a duration of time. Ifit is observed that the lines satisfying the criterion do not forma cutset, then the system is stable. This implies that to check forsystem instability, it is essential to verify whether all of the lines(which satisfy the instability criterion) form a cutset.

The presence of intermachine oscillations within an area canaffect the accurate detection of loss of synchronism using the en-ergy-based instability criterion which neglects such oscillations.Hence, for practical implementation of the instability criterion,it is necessary to modify the criterion by choosing appropriatelythe lower and upper limits on . The errors due to false alarmscan be avoided by choosing a threshold value for so thatonly those lines across which the angle exceeds the thresholdvalue are checked for instability. The error of false dismissalscan be avoided by choosing a value such that instability isassumed if the angle across a line exceeds this value.

The criterion for instability is refined as follows: The systemis unstable if , or and decreases tozero when is still positive in all of the lines belonging toa cutset; otherwise, the system is stable.

V. IDENTIFICATION OF CRITICAL CUTSETThe critical cutset depends on the operating condition and the

disturbance, and is not known beforehand. Therefore, the con-dition for instability is checked in all of the lines across whichthe angle exceeds the threshold value . As soon as insta-bility is detected in a new line, it is checked whether the lines inwhich the condition for instability is met up to that instant, forma cutset. When the condition for instability is detected in a newline, the information is transmitted to the central computing sta-tion where the identification of the critical cutset is carried out.


Fig. 7. Flowchart for the identification of the critical cutset.

An algorithm based on the fusion of adjacent buses is used todetermine whether a given set of lines forms a cutset. The algo-rithm presented in this section is a modified version of the onegiven in [12]; the algorithm given in [12] is to check the con-nectedness of a graph. The network connectivity information isstored in the form of the adjacency matrix. The adjacency ma-trix of a network with buses is a by symmetric binarymatrix whose element is 1 if there is a line connecting the

and th buses and 0 if there is no line between the th andth buses. If there are two or more parallel lines connecting two

buses, then these lines are treated as a single line since the vari-ation of angle and power in these lines is similar and, hence, thedetection of instability in these lines is at the same instant.

The postfault network is assumed to be connected initially.Let be the adjacency matrix of the postfault network andbe an empty set. The flowchart for the identification of the crit-ical cutset is shown in Fig. 7. Whenever instability is detected ina line , this line is included in the set and the connectivity in-formation about this line is removed from the adjacency matrix

by setting ; then, it is checked whether a pathexists between buses and . This is accomplished by fusing allbuses adjacent to bus (of the adjacency matrix ) repeatedly.All of the buses with numbers equal to the column numbers (orrow numbers) of the elements of the row (or column),which have a value of 1, are adjacent to bus . If a path existsbetween buses and , then at some stage, bus is adjacent to

bus and, hence, the set is still not a cutset. When instability isdetected in the last line belonging to the critical cutset, then nopath exists between buses and , and then the set is a cutset.

Whenever instability is detected in a line, the matrix isupdated and a new matrix is defined. This is done becausethe fusion of buses reduces the number of buses and the orig-inal connectivity information is lost; the original connectivityinformation is again required when instability is detected in anew line. Therefore, the original connectivity information is pre-served in the matrix and the fusions are performed on the ma-trix .

The fusion of the th bus to the bus is accomplished byOR-ing, that is, logically adding the th row to the th row aswell as the th column to the th column of the matrix . Inlogical adding, and .Then all of the elements of the th row and the th column ofthe matrix are set to zero.

Whenever instability is detected in a line, the maximumnumber of fusions that may have to be performed in thisalgorithm in order to check whether is a cutset or not,is where is the number of buses. The maximumnumber of logical additions that may have to be performed is

. Therefore, the upper bound on the executiontime is proportional to .

VI. PREDICTION OF INSTABILITY

For faster detection of instability, the variation of and ispredicted by fitting a polynomial curve to the postfault-sampledmeasurements. The sampling period is chosen as one cycle.The measurements separated by two cycles are used for curvefitting. The algorithm for prediction of instability in a line is asfollows:

1) If measured at the current sampling instant is less thanthat measured at the previous sampling instant, stabilityis indicated in the line since reaches a maximumand crosses zero during this sampling period.If and , or measured at thecurrent sampling instant is greater than , instabilityis indicated in the line.

2) If measured at the current sampling instant is greaterthan , a quadratic curve is fit to the three sampledmeasurements of and a cubic curve is fit to the foursampled measurements of . The samples of are mea-sured at the instants , , , and

where is the current sampling instantand is the fault clearing time. is measured at theinstants , , and

(21)(22)

3) The following two equations are solved for real positivevalues to obtain the instant at which and theinstant at which :

(23)(24)


4) If and , or if (23) has a real positivesolution with , and (24) does not have areal positive solution, instability is indicated in the line;otherwise, a new set of measurements is obtained at thenext sampling instant and the procedure from step 1 isrepeated.

The procedure is stopped as soon as stability or instability isindicated in the line. It is to be noted that curve fitting and thesolution of the quadratic equations are required only if ex-ceeds . The condition is used to limit the errordue to extrapolation. As soon as instability is predicted in a line,the information is sent to the central computing station. Systeminstability is predicted when the lines in which instability is pre-dicted form a cutset.

VII. CASE STUDIES

The proposed detection criterion is tested on the New Eng-land 10-generator system and the IEEE 17-generator system.The single-line diagram of the New England system is shown inFig. 1. The data for this system are given in [2]. The data for theIEEE 17-generator system are given in [13], [14]. The singlediagram of this system showing only the major lines is givenin [13]. For both systems, network and generator losses are ne-glected, and the loads are treated as constant impedances. Theproposed detection criterion is tested by simulating three-phasefaults at different locations; the fault is cleared at such an instantthat the system is critically unstable. is chosen as 50 forthe 10-generator system and 55 for the 17-generator system.

is chosen as 200 .The angle across the critical lines at the instant of detec-

tion/prediction is given in Table I. The instant of instability de-tection/prediction is also indicated in the table. The instant of in-stability detection/prediction is the instant at which instability isdetected/predicted in the last line belonging to the critical cutset.The value in brackets, in the columns 4 and 5, is the time dura-tion in terms of the number of cycles from the instant at whichinstability is detected/predicted in the first line to the instant atwhich instability is detected/predicted in the last line.

The proposed instability criterion indicates instability in allof the cases studied. The average time taken for detection ofinstability from the instant of fault clearing is 1.004 s for the10-generator system and 0.333 s for the 17-generator system.The corresponding values for the prediction of instability are0.569 and 0.254 s. The average value of the angle across crit-ical lines at the instant of instability detection is 159.2 for the10-generator system and 128.0 for the 17-generator system.The corresponding values at the instant of instability predictionare 54.7 and 63.2 .

For a fault at bus 12 cleared without line tripping and faultat bus 37 cleared by tripping the line 37-27 (of the 10-generatorsystem with the classical model of generator), there is a false dis-missal with stability detection using the energy function. There-fore, instability is detected when a sampled measurement ofis greater than . However, for both cases, the predictionalgorithm indicates instability at an earlier instant. There is nofalse dismissal in these cases with prediction.

When a fault is cleared by opening the line, the steady-statevalue of power in the elements belonging to the criticalcutset is different from the prefault steady-state value .can be obtained from the results of online static security assess-ment. Since the variation in is very large compared toand , can be used in place of .

The use of prefault steady-state power in place of postfaultsteady-state power does not affect the determination of stabilityfor both critically stable and critically unstable cases. There isnot much difference in the results obtained if is used insteadof .

The results given in this section are for a critically unstablefault clearing time. It is also tested by simulation studies that theproposed method indicates stability for the critically stable faultclearing time for all of the cases.

VIII. DISCUSSION

Unlike other techniques, such as E-SIME [5], the proposedstability criterion does not require the knowledge of the modeof instability and no assumptions are made regarding the power-angle relationship.

The only assumption made is that the system instability re-sults in the initial separation into two areas. This assumption isused implicitly in all of the methods proposed earlier.

The main objective of the proposed instability detection crite-rion is system protection as distinct from equipment protection.If instability is detected, corrective actions need to be taken inorder to maintain system integrity.

A major feature of the detection criterion proposed in thispaper is that it is based on local measurements (of current andvoltage) within a line. Only logical data need to be transmittedto the central location where it is processed to check for systeminstability. This is different from techniques based on the use ofphasor measurements and their telemetering [3], [15]. The com-putational complexity in detection of instability can be reducedfrom simplification of the system by identifying coherent groupsand the use of dynamic equivalents (from offline studies) [15].

For a given mode of instability, there are many possible cut-sets connecting the two separating areas; but there is usuallya unique cutset across which the angle becomes unbounded incase of instability. For the 10-generator system, there are fewcritical cutsets. For many contingencies, generator 2 separatesfrom the rest of the system since its inertia constant is high com-pared to other generators. For this mode of instability, the crit-ical cutset consists of lines 18-19 and 11-12.

The detection criterion derived from the energy function anal-ysis is based on the assumptions of coherent areas and constantpower loads. The coherency assumption is made to neglect theoscillations within the areas and account only for interarea os-cillations which contribute to system separation. The detectioncriterion is effective even when the loads are modeled as con-stant impedances.

The main requirements of a method to detect instability areaccuracy (no false alarms and false dismissals) and speed. Theproposed method is capable of distinguishing between stableand unstable swings accurately. The stability criterion is accu-rate even for critically stable and critically unstable cases which


TABLE IANGLE ACROSS THE CRITICAL LINES AT THE INSTANT OF INSTABILITY DETECTION/PREDICTION

are considered in the case studies. Therefore, it is expected thatthe criterion is able to perform accurate stability classification

for more stable and more unstable cases. The detection is spedup by the extrapolation of system trajectories.


Fig. 8. Swing curves for a fault at bus 20 cleared by tripping the line 20-33.

The mode of instability is the same for both classical and de-tailed models for all of the cases studied. For a fault at buses 27and 28 of the 10-generator system, the critical cutset is differentfor classical and detailed models of the generator.

The values of and are decided based on simulationstudies. These values are the same for all of the cases for a givensystem and the actual values used are not very critical. The de-tection/prediction procedure starts as soon as the angle acrossa line exceeds . In most cases, instability is predicted assoon as the angle exceeds . The method given in [15] in-volves only checking whether the angle between two areas ex-ceeds , the value which is not specified.

Transient instability may lead to uncontrolled tripping of gen-erators and lines, and may finally result in the formation of is-lands. Controlled system separation is used as a protective mea-sure in order to retain as much of the system intact as possible.The proposed instability detection method can be used to ini-tiate this emergency control measure by tripping the elementsof the critical cutset at the instant of instability detection/pre-diction. Fig. 8 shows the swing curves for the fault at bus 20 (ofthe 10-generator system) cleared by tripping the line 20-33 at0.23 s (critically unstable clearing time). Generator 3 initiallyseparates from the rest of the system; instability is first detectedin the transformer 20-3 which forms the critical cutset. Con-trolled system separation is simulated by tripping generator 3.Fig. 9 shows the plot of swing curves of the group containingnine generators when the system is separated at the instant ofinstability detection/prediction (0.713 s). The system is stableafter tripping generator 3 though the swing curves show thatthere is further loss of synchronism without controlled systemseparation (Fig. 8).

The proposed algorithm can be readily applied for offlinestudies. In most cases, the critical cutset and mode of insta-bility can be detected/predicted within about a second. This isquite fast as the application of energy function methods for di-rect evaluation of transient instability (using the potential energyboundary surface method) involves about 1 s of simulation (un-less shortcuts are used). Further, the algorithm presented in this

Fig. 9. Swing curves after system separation at the instant ofdetection/prediction of instability (0.713 s) due to a fault at bus 20.

paper gives additional information about the mode of instabilityand the critical cutset.

IX. CONCLUSION

This paper presents a method of detection of instability basedon energy function analysis. The method requires computationof power flow and phase angle across lines belonging to the crit-ical cutset. A computationally efficient method is used to iden-tify the critical cutset. The stability detection method is accurateand the detection of instability can be sped up by predicting thevariation of power and angle. The proposed instability detectioncriterion can be useful in initiating emergency control measuressuch as controlled system separation.

APPENDIXPROOF OF EQUATION (12)

The right-hand side of (12) can be expressed as

(25)In a lossless system, power flows satisfy Kirchhoffs currentlaw and bus frequencies satisfy Kirchhoffs voltage law. Hence,an equivalent network can be obtained for the power system asshown in Fig. 10, with power being considered analogous tocurrent and bus frequencies to voltage [16].

For any electric network, where the branch voltages and cur-rents satisfy Kirchhoffs laws, Tellegens theorem can be ap-plied, which states that at any time, the sum of the power deliv-ered to each branch of the network is zero [17]. Under steady-state, is equal to , and is equal to ; the loadpowers are assumed to be constants. Tellegens theorem isalso valid when branch voltages of one network and the branchcurrents of another network are considered, provided the net-works have the same graph. Applying Tellegens theorem to the


Fig. 10. Equivalent network of the power system.

powers in steady-state and bus frequencies at any instant, thefollowing equation is obtained:

(26)

By integrating (26) with respect to time, the second term on theright-hand side of (25) can be expressed as follows:

(27)

The first term on the right-hand side of (25) can be separatedinto terms corresponding to power flows in the generator reac-tances and those corresponding to power flows in the transmis-sion lines/transformers as follows:

(28)where is the number of lines and transformers, is the powerflow in the series element connecting buses and . The firstterm on the right-hand side of (28) is evaluated by integratingthe expression for given by (9) with respect to byparts. It can be shown that [18]

(29)

The second term on the right-hand side of (28) is evaluated byintegrating with respect to by parts.It can be shown that [18]

(30)

The sum of the last two terms on the right-hand side of (29) and(30) gives half of the reactive power losses in the generator reac-tances and the network which is equal to . Substituting theright-hand side of (29) and (30) in (28), the following equationis obtained:

(31)

Equation (12) follows from (25), (27) and (31).

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K. R. Padiyar (SM91) received the B.E. degree in electrical engineering fromPoona University, Poona, India, in 1962, the M.E. degree from the Indian In-stitute of Science, Bangalore, India, in 1964, and the Ph.D. degree from theUniversity of Waterloo, Waterloo, ON, Canada, in 1972.

Currently, he is a Professor of Electrical Engineering at the Indian Instituteof Science, Bangalore, India. He was with the Indian Institute of Technology,Kanpur, India, from 1976 to 1987, prior to joining the Indian Institute of Sci-ence. His research interests include HVDC and flexible ac transmission systems(FACTS), system dynamics, and control. He has authored three books and manypapers.

Dr. Padiyar is a Fellow of the National Academy of Engineering (India).

S. Krishna received the B.E. degree in electrical engineering from BangaloreUniversity, Bangalore, India, in 1995, and the M.E. and Ph.D. degrees in elec-trical engineering from Indian Institute of Science, Bangalore, in 1999 and 2003,respectively.

Currently, he is an Assistant Professor with M.S. Ramaiah Institute of Tech-nology, Bangalore. He was with Kirloskar Electric Company, Bangalore, from1995 to 1997. His research interests include power systems.

tocOnline Detection of Loss of Synchronism Using Energy Function CrK. R. Padiyar, Senior Member, IEEE, and S. KrishnaN OMENCLATUREI. I NTRODUCTION

Fig.1. Single-line diagram of a 10-generator 39-bus New EnglandFig.2. Angle across series elements for the critically unstableII. C RITICAL C UTSET A SSOCIATED W ITH T RANSIENT I NSTABILITY

Fig.3. Excitation system.III. S YSTEM M ODEL AND E NERGY F UNCTIONIV. C RITERION FOR I NSTABILITY

Fig.4. Coherent areas.Fig.5. Variation of power flow and rate of change of angle of tFig.6. Variation of power flow and rate of change of angle of tV. I DENTIFICATION OF C RITICAL C UTSET

Fig.7. Flowchart for the identification of the critical cutset.VI. P REDICTION OF I NSTABILITYVII. C ASE S TUDIESVIII. D ISCUSSION

TABLEI A NGLE A CROSS THE C RITICAL L INES AT THE I NSTANT OF Fig.8. Swing curves for a fault at bus 20 cleared by tripping tFig.9. Swing curves after system separation at the instant of dIX. C ONCLUSIONP ROOF OF E QUATION (12)

Fig.10. Equivalent network of the power system.IEEE Committee Report, A description of discrete supplementary cK. R. Padiyar, Power System Dynamics Stability and Control, 2nd V. Centeno, A. G. Phadke, A. Edris, J. Benton, M. Gaudi, and G. S. Rovnyak, C.-W. Liu, J. Lu, W. Ma, and J. Thorp, Predicting fuL. Wehenkel, M. Pavella, and Y. Zhang, Transient and Voltage StaL. Wang and A. A. Girgis, A new method for power system transienS. Rovnyak, S. Kretsinger, J. Thorp, and D. Brown, Decision treeC.-W. Liu, M.-C. Su, S.-S. Tsay, and Y.-J. Wang, Application of K. R. Padiyar and K. Uma Rao, Discrete control of series compensK. R. Padiyar and K. K. Ghosh, Direct stability evaluation of poA. A. Fouad and V. Vittal, Power System Transient Stability AnalN. Deo, Graph Theory With Applications to Engineering and ComputIEEE Committee Report, Transient stability test systems for direC. Counan, M. Trotignon, E. Corradi, G. Bortoni, M. Stubbe, and K. R. Padiyar and P. Varaiya, A Network Analogy for Power SystemC. A. Desoer and E. S. Kuh, Basic Circuit Theory . New York: McGS. Krishna, Dynamic security assessment and control using unifie

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Pérdida Sincronismo PD Enero 2006