An Adaptive Three-bus Power System Equivalent forEstimating Voltage Stability Margin fromSynchronized Phasor MeasurementsFengkai Hu, Kai SunUniversity of TennesseeKnoxville, TN, [email protected]@utk.eduAbstract—This paper utilizes an adaptive three-bus powersystem equivalent for measurement-based voltage stabilityanalysis. With that equivalent identified online, a measurementbased approach is developed to estimate real-time voltagestability margin for a load-rich area supported by remotegeneration via multiple tie lines. Compared with traditionalThevenin equivalent based approach, this new approach is ableto provide more accurate voltage stability margin for eachindividual tie line. This approach is validated on a three-bussystem and the IEEE 39-bus system.Index Terms—parameter estimation; phasor measurement unit;Thevenin equivalent; voltage stability monitoringI.INTRODUCTIONVoltage instability is a major concern for power systemsoperation. Usually, it initiates from a local bus or region butmay develop to a wide-area or even system-wide stabilityproblems. Online Voltage Stability Assessment (VSA) is akey function in system operations to help operators foreseepotential voltage insecurity. Traditional VSA is based on asimulation-based approach. By either power-flow analysis ortime-domain simulation, it employs power system models tosimulate a list of contingencies on a State Estimator solutionthat represents the current operating condition. However, thesimulation based approach has limitations: it is modeldependent and it requires a convergent State Estimationsolution for the current operating condition.Different measurement-based VSA approaches have beenstudied to directly estimate real-time voltage stability marginfor a monitored load bus or area [1]-[9] or predict potentialpost-contingency voltage insecurity by means of data miningtechniques [10]. A majority of the measurement-basedapproaches are based on Thevenin’s Theorem. For instance,local measurements at the monitored buses are used toapproximate the rest of the system as an impedance connectedThis work is supported by the Electric Power Research Institute and theUS Department of Energy (under award DE-OE0000628)978-1-4799-6415-4/14/ 31.00 2014 IEEEAlberto Del Rosso, Evangelos Farantatos,Navin BhattElectric Power Research InstituteKnoxville, TN, ri.comto a voltage source, i.e. the Thevenin equivalent. The powertransferred to the bus reaches its voltage stability limit whenthat external Thevenin impedance has the same magnitude asthe load impedance at the bus [1]. Based on Theveninequivalent, the voltage stability or reactive-power reserveindices can be obtained [2]-[5]. A modified model with twoequivalent voltage sources is studied in [6] to predict thestability limit. Paper [7] applies the Thevenin equivalent basedapproach to an actual EHV network. Some other worksconsider load tap changers and over-excitation limiters in theirmodels for better detection of voltage instability [8][9]. Theabove methods work well on a radially-fed load bus ortransmission corridors. EPRI developed a Theveninequivalent-based method for load center areas, which requiressynchronized measurements on boundary buses [4][5]. Asillustrated by Fig. 1, the method merges all boundary busesand tie lines to one fictitious boundary bus connected by onetie line with the external system such that the Theveninequivalent can be applied. An ongoing project isdemonstrating this method in the real-time environment [11].Figure 1. Load area and its Thevenin equivalentHowever, the Thevenin equivalent-based approach cannotprovide voltage stability margin for each individual tie linewhen the monitored load is fed by multiple tie lines. For sucha case, transfer limits of various tie lines may be reached atdifferent times, or in other words, voltage instability may startnear one of the boundary buses sooner and then progress to

the others. However, by monitoring the total transfer limitthrough a single equivalent, the Thevenin equivalent basedapproach may not detect the time variability across theinterface associated with voltage instability.In this paper, an adaptive three-bus power networkequivalent is proposed for estimating voltage stability marginfor a load-rich area fed by multiple tie lines. A real-timevoltage stability monitoring method is then developed basedon that new equivalent. It is explained and demonstrated laterthat such a three-bus equivalent, if applied to a load-rich areafed by two or more tie lines, is able to estimate the real-timepower transfer limit in terms of voltage stability for eachindividual tie line if synchronized measurements are availableon all boundary buses. This new method is tested on a threebus system and the IEEE 39-bus system.II.A THREE-BUS EQUIVALENTpotentials in reflecting significant changes on the generationside, e.g. voltage drops due to a generator limit being met.III.APPROACH FOR VOLTAGE STABILITY MARGINCALCULATIONBased on the three-bus equivalent, an approach forcalculating voltage stability limits and margins for N tie linesof a load area is presented in this section. The approachassumes that time-synchronized voltage phasor data V1 V Nat boundary buses and current phasor data I 1 I N of tie linesare available. The data may be from synchrophasors, e.g.phasor measurement units (PMUs) at 30-60 samples persecond or a state estimator at a slower rate, e.g. 20s to severalminutes, depending on the speed requirements for voltagestability monitoring. This paper uses synchrophasor data as anexample. The approach conducts the following steps:i)Determine the number of three-bus equivalents,depending on how many lines need to be monitored indetail for power transfer limits and margins. For eachequivalent, use measurements to calculate voltage phasordata of V1 and V2 and complex power-flow data of S1 andS2 on two load buses as indicated by Fig. 2. For instance,if the 1st bus is selected vs. the others, V1, V2, S1 and S2are calculated by (1)S1 V1 I1*V1 V1 ,NV2 Vi I i *i 2Figure 2. Propsed 3-bus equivalentFigure 3. Strategies for three-bus equivalencing – N equivalentsAs shown in Fig. 2, a three-bus power network equivalentis proposed to monitor voltage stability for a load-rich area. Itsthree buses include a voltage source and two interconnectedload buses representing the load center. The voltage sourcerepresents the external system, whose generators are assumedto be strongly coherent without risking any angular instability.The two load buses represent either actual or fictitiousboundary buses, depending on the requirement of voltagestability monitoring. For example, if it is required to estimatethe transfer limit for each of the N tie lines of a load area, anytie line versus the rest can be studied to create N three-busequivalents, as illustrated by Fig. 3. Then, voltage stabilityanalyses on all such equivalents provide comprehensiveresults on all tie lines. In practice, usually only one or veryfew tie lines are most vulnerable to voltage instability, so it isunnecessary to study all N equivalents. Since this equivalentdoes not model generator VAR limit, it focuses on detectingor predicting the saddle-point bifurcation type voltage collapseon the load side [12]. However, since the equivalent will beestimated in real time from measurement data, it also hasN Ii 2*i,NS 2 Vi I i *(1)i 2ii)At any time when estimation of stability limits ormargins is expected, use the data of V1, V2, S1 and S2 overa latest time window to estimate the other parameters ofthe three-bus equivalent including those of the externalsystem, i.e. E, Z1, Z2, and those of the load area, i.e. ZL1,ZL2 and ZT. Details are presented in subsections A and B.iii)Find the maximum limit of the active power transferredto each of the two load buses, denoted by P1max andP2max. An exhaustive or heuristic searching algorithmmay be employed to find the limit. Since the searchingspace is not large for the three-bus equivalent, subsectionC gives an algorithm for exhaustive searching.A. External System Parameters EstimationAssume that E, Z1 are Z2 are constant over the timewindow. Thus, similar to [1], a least-square method may beadopted to give estimates for E, Z1 are Z2. Note that theThevenin equivalent has 4 real unknowns while this newequivalent has 6 real unknowns to solve, i.e.:[E rEiR1X1R2X 2 ] ( H T H ) 1 H T Z(2)where Er jEi E, R1 jX1 Z1, R2 jX2 Z2, and matrices H and Zare formed based on measurement data at n time points of thetime window. V1r,k and V1i,k respectively denote the real andimaginary parts of bus 1 voltage at the k-th time point. p1,k andq1,k are respectively the active and reactive powers received bybus 1 at the k-th time point. Similarly, V2r,k, V2i,k, p2,k and q2,kare data of bus 2.

p 1 ,1 q 1 ,100 q 1 ,1p 1 ,10000 p 2 ,1 q 2 ,100 q 2 ,1p 2 ,1MV 1i , n V 1r ,nV 2i,n V 2 r ,nM p 1, n q 1, n00M q 1, np 1, n00M00 p 2 ,n q 2,nM00 q 2 ,np 2 ,n[]ZT V12r,1 V1i2,1 0 V22r,1 V22i,1 0 L V12r,n V1i2,n 0 V22r,n V22i,n 0 (3)22 V1*aV 2 a )Y12 S 1*a22 V1*bV 2 b )Y12 S 1*bV1b Y11 b ( V1b22 V 2*a V1 a )Y12 S 2*a22 V 2*bV1b )Y12 S 2*bV 2 a Y 22 a ( V 2 aV 2 b Y 22 b ( V 2 bA three-bus system and the IEEE 39-bus system are usedto test the proposed approach. Simulations are conducted andthe simulation results at the boundary buses are treated assynchrophasor data.A. Three-bus SystemThe three-bus system has the same structure as theequivalent shown in Fig. 2 with all parameters given below inper unit:(4)B. Load Area Parameters EstimationTo estimate the load area parameters, at least two timepoints (denoted by ta and tb) of measurement data are needed.The following equations could be obtained:V1 a Y11 a ( V1aCASE STUDIESIV.(5)Symbols labelled “a” or “b” are linked to thecorresponding time ta or tb. For example, V1a denotes the bus 1voltage phasor at time ta. Y11a denotes the admittance of theload connected to bus 1. Y12 represents the transfer admittancebetween the two load buses, which is assumed constant.Another assumption is that each load impendence has aconstant impedance angle, i.e. constant power factor. Thus,G11 a / B11 a G11b / B11b , G 22 a / B 22 a G 22 b / B 22 b (6)Equations (5) and (6) actually correspond to 10 real equations,which are solved for 10 real unknowns, i.e. real and imaginaryparts of complex unknowns Y12, Y11a, Y11b, Y22a and Y22b. Theabove constant power factor assumption can toleratereasonably slight changes in the impedance angles over a shorttime window based on our studies.C. Finding the Power Transfer LimitsBased on the current operating condition, which dependson the estimated ZL1 and ZL2, the maximum limits of the activepower transferred to two load buses need to be solved. It isassumed that ZL1 and ZL2 vary in a zone and then an exhaustivesearch is conducted to check power-flow solutions of allmeshed representative points in that space. The goal is to findthe maximum power flows delivered to the two load buseswithout causing voltage insecurity.Since the dimension of the space is not high, those pointsmay have a very high density. Also, when solving the powerflow solution at each point, the power injected by the slackbus is limited within a range around its original value to avoidunrealistically large changes at the slack bus. A heuristicalgorithm may also be applied to utilize the gradientinformation from two successive points during the searchingto speed on the process.E 1.0475 0, Z 1 0.002 j 0.02, Z 2 0.03 j 0.1,Z L1 1 .9953 j 0.5915 , Z L 2 1.2746 j 0.3405Time-domain simulations are conducted to continuouslydecrease the magnitudes of two load impedances by 1% everysecond to simulate a load area with increasing load until twolines meet the maximum power transfer limits. The purposesof this case are to demonstrate: 1) how different the limits oftwo lines may be, and 2) differences between the results ofthis new approach and those from a traditional Theveninequivalent based approach.1.2Bus 1 PVBus 2 PV1Voltage(p.u.)V 1 i ,1 V 1 r ,1V 2 i ,1 V 2 r , Power(MW)Figure 4. PV curves of two load buses with tight interconnection1.4Bus 1 PVBus 2 PV1.1Voltage(p.u.) V 1 r ,1 V 1 i ,1 V 2 r ,1 V 2 i ,1H M V 1r ,n V 1i , n V 2 r , n V 2i,n0.80.50.204812Active Power(MW)1620Figure 5. PV curves of two load buses with weak interconnectionTwo cases are simulated with two different values of thetransfer impedance Z12, i.e. 0.00003 j0.0005pu and0.03 j0.5pu, which respectively represent a tightinterconnection and a relatively weak interconnection betweenthe two load buses (corresponding to the boundary buses ofthe load area). Fig. 4 and Fig. 5 give the PV curves fromsimulation results at two load buses. Each curve is about thebus voltage magnitude and the active power transferred to thebus. When the two load buses are more weakly connected, the

two P-V curves are more different, indicating the need of theestimating stability limits for individual buses.The proposed approach is performed every 1 second over asliding time window of 1 second. For tight and weakinterconnections, Fig. 6 gives the active line flows P1 and P2,and total interface flow P1 P2, and their limits calculated bythe new approach, i.e. P1max, P2max, and Pmax(new) P1max P2max. For comparison purposes, the total interfaceflow from the Thevenin equivalent based approach is alsogiven as Pmax (Thevenin) in the figures. Based on the results, itcan be observed that when the two interface buses have tighterinterconnection, the transfer limits of the two lines are met atthe same time [around t 380s in Fig. 6(a)], which means thetwo buses can be reasonably merged into one bus withoutlosing accuracy. That is the basic assumption of the traditionalThevenin equivalent based approach, so the Theveninequivalent based approach also estimates the total interfacelimit to be met almost at the same time as individual lines. Asshown by Fig. 6(b), when the two buses are weakly connected,the limits of two lines are met at different time instants, att 470s and t 260s, respectively. On the other hand, theThevenin equivalent based approach estimates that the totalinterface flow limit is met at around t 370s, i.e. not muchdifferent from the tight interconnection case. The resultsillustrate that if only the total transfer limit for the entireinterface of a load area is estimated, detection of voltageinstability may be delayed since some tie line may be morestressed and voltage instability may occur there first.For the IEEE 39-bus system, a load area is defined asindicated by Fig. 7. It has three interface buses, i.e. buses 4, 8and 14. The system can be simplified into a three-busequivalent system, and utilize the approach proposed in thispaper. The following contingency is simulated to create avoltage instability scenario: Starting from t 0s, keep increasing the total load ofthe area from 1898 MW at a speed around 1.3MW(with slight randomization) per second to create slowdecay in the voltage level of the area. At t 439s, trip the generator on bus 32, i.e. one of thetwo local generators of the load area. Keep increasing the load of the area at the same speeduntil voltage collapse around t 539s.Fig. 8 shows all bus voltage magnitudes, in which thehighlighted curves are those inside the load area.Figure 8. New England system bus voltage magnitude1000P 34(b)Figure 6. Line flow limits for tight (a) and weak (b) interconnectionsbetween two busesB. IEEE 39-bus SystemFigure 7. IEEE 39-bus system diagramVoltage Magnitude (p.u.)Active Power (MW)(a)1P 98800P 151460040020000200400Time (s)6000.90.80.7V m80.60.5V m4V m140200400600Time (s)Figure 9. Active power and voltage magnitude of buses on the boundaryFig. 9 gives the active power flows and the bus voltagemagnitudes of three boundary buses. Bus 4 and bus 14 haveclose voltage curves, so they can be merged into a singlefictitious bus, named bus E. The two lines connected, i.e. 1514 and 3-4, are also merged to an equivalent line, named lineE. Thus, the three-bus equivalent is applied. Fig. 10 gives theactive power flows of line 9-8 and line E and the voltagemagnitudes at bus 8 and bus E. Fig. 11 gives the P-V curvesfrom the simulation results on the two buses. It shows that twocurves have different shapes and their “nose” points may bereached at different times in the simulation.Parameters of the external system and load area areestimated at each time step of 0.25s using measurements (i.e.from simulation results) on the three boundary buses over thelatest 5s time window. Let P98 and PE denote the active powers

1P 98Active Power (MW)Voltage Magnitude 0600Figure 10. Bus equivalent resultPV 8Voltage Magnitude (p.u.)0.95PV E4140.850.750.65300500700Active Power (MW)9001100Figure 11. PV curves of two buses in England New system2500P 981600P 98max14002000PEP EmaxActive Power (MW)Active Power (MW)120015001000100080060040050020000200400Time (s)6000200150100500-50-1000200400600Time (s)Figure 13. Comparison of the percentage active power margins of two linesV.CONCLUSIONThis paper proposed a new three-bus equivalent for realtime estimation of voltage stability margin using synchronizedmeasurements at the boundary buses of a load area. Thedetailed approach was compared to the traditional Theveninequivalent based approach by case studies. The comparisonindicates that the new approach is able to assess voltagestability limits of a load area served by multiple lines moreaccurately than the traditional approach.[1]Time (s)100PEmax250VI. REFERENCES0.6Time (s)0.55P98max3000.70.5600350Active Power Margin (%)in the line 9-8 and line E, whose real-time values are directlyfrom the measurements. At each time step, in the plane aboutP98 and PE, a rectangular region of 50% around the pointcorresponding to their real-time values is considered. Powerflow solutions are studied for representative points in theregion at a density. The real power change at the slack bus foreach time step is restricted to 20% in solving the power flows.The maxima of P98 and PE, i.e. P98max and PEmax, among allsolved power flows are identified as the limits of the two linesin terms of voltage stability. Fig. 12 gives the identified activepower limits. Fig. 13 compares the percentage active powermargins, i.e. (P98max-P98)/P98 100% and (PEmax-PE)/PE 100%.At the beginning, the percentage margin of P98 is larger thanthat of line E. Two margins become closer after the generatortrip at t 439s. Finally, the voltages at two buses almostcollapse at the same time. The generator trip has more impacton the voltage stability margin of line E because the trippedgenerator is closed to bus 4 and bus 14. Such information isnot available from a traditional Thevenin equivalent basedapproach. This new approach offers more accurate monitoringof voltage stability margins for individual tie lines.0200400600Time (s)Figure 12. Active powers of two lines and their voltage stability limitsK. Vu, M. Begovic, D. Novosel, and M. Saha, “Use of localmeasurements to estimate voltage-stability margin,” IEEE Trans.Power Systems, vol. 14, no. 3, pp. 1029–1035, Aug. 1999.[2] B. Milosevic and M. Begovic, "Voltage-stability protection and controlusing a wide-area network of phasor measurements," IEEE Trans.Power Systems, vol. 18, pp. 121-127, 2003.[3] Smon, et al, "Local Voltage-Stability Index Using Tellegen'sTheorem," IEEE Trans. Power Systems, Vol. 21, No. 3, Aug. 2006.[4] P. Zhang, L. Min, et al, “Measurement based voltage stabilitymonitoring and control”, US Patent # 8,126,667, Feb 2012.[5] K. Sun, P. Zhang, L. Min, “Measurement-based Voltage StabilityMonitoring and Control for Load Centers”, EPRI Technical Report No.1017798, 2009.[6] M. Parniani, et al, "Voltage Stability Analysis of a Multiple-InfeedLoad Center Using Phasor Measurement Data," IEEE PES PowerSystems Conference and Exposition, Nov 2006[7] S. Corsi and G. Taranto, “A real-time voltage instability identificationalgorithm based on local phasor measurements,” IEEE Trans. PowerSystems. vol. 23, no. 3, pp. 1271–1279, Aug. 2008.[8] C. D. Vournas and N. G. Sakellaridis, "Tracking Maximum LoadabilityConditions in Power Systems," in 2007 iREP Symposium Bulk PowerSystem Dynamics and Control - VII. Revitalizing OperationalReliability, 2007.[9] M. Glavic and T. Van Cutsem, "Wide-Area Detection of VoltageInstability from Synchronized Phasor Measurements. Part I: Principle,"IEEE Trans. Power Systems, vol. 24, pp. 1408 - 1416, 2009.[10] R. Diao, K. Sun, V. Vittal, et al, "Decision Tree-Based Online VoltageSecurity Assessment Using PMU Measurements", IEEE Trans. PowerSystems, vol. 24, pp.832-839, May 2009.[11] F. Galvan, A. Abur, K. Sun, et al, "Implementation of SynchrophasorMonitoring at Entergy: Tools, Training and Tribulations", IEEE PESGeneral Meeting, 23-26 July 2012, San Diego[12] C. Canizares, et al, Voltage Stability Assessment: Concepts, Practicesand Tools, IEEE PES Power System Stability Subcommittee SpecialPublication, IEEE, Aug 2002

bus system and the IEEE 39-bus system. II. A THREE-BUS EQUIVALENT Figure 2. Propsed 3-bus equivalent Figure 3. Strategies for three-bus equivalencing – N equivalents As shown in Fig. 2, a three-bus power network equivalent is proposed to monitor voltage stability for a load-rich a