2154 ieee transactions on power systems,...

2154 IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 26, NO. 4, NOVEMBER 2011

Voltage Stability Monitoring Based on theConcept of Coupled Single-Port Circuit

Yunfei Wang, Student Member, IEEE, Iraj Rahimi Pordanjani, Student Member, IEEE, Weixing Li, Member, IEEE,Wilsun Xu, Fellow, IEEE, Tongwen Chen, Fellow, IEEE, Ebrahim Vaahedi, Fellow, IEEE, and

Jim Gurney, Senior Member, IEEE

Abstract—This paper reveals that the impedance match (or theThevenin circuit) based voltage stability monitoring techniqueshave problems to predict voltage stability limits when applied tomulti-load power systems. Power system loads are nonlinear anddynamic. They cannot be simply represented as Thevenin circuitparameters for impedance match analysis. To overcome thesedifficulties, a new concept called “coupled single-port circuit” isproposed in this paper. The concept decouples a meshed networkinto individual single generator versus single bus network and,as a result, a modified version of the impedance match theoremcan be used. This leads to a real-time voltage stability monitoringscheme without the need to estimate Thevenin parameters. Thescheme can estimate voltage stability margin and identify weakareas in a system based on the SCADA and PMU data. Casestudies conducted on several test systems have verified the validityof the proposed method.

Index Terms—Impedance matching, load shedding, single-port,Thevenin equivalent circuit, voltage stability margin.

I. INTRODUCTION

T HE VOLTAGE stability problem has become a seriousconcern for the power industry as several major blackouts

were related to voltage collapse in recent years. One methodto prevent voltage collapse requires an efficient online voltagestability monitoring tool in addition to a good offline systemplanning practice. Based on this concern, a number of onlinevoltage stability indicators have been proposed [1]–[8].Among these indices, techniques based on the impedance

match concept have attracted a lot of interest [7], [9]–[17]. Themain idea of these techniques can be summarized as follows:local voltage and current phasors at a bus (port) are measured,and then the Thevenin equivalent of the rest of the system seenfrom this bus is derived. Voltage instability occurs if the load

Manuscript received July 09, 2010; revised October 08, 2010, January 09,2011, and March 16, 2011; accepted May 05, 2011. Date of publication June13, 2011; date of current version October 21, 2011. This work was supported inpart by the Natural Sciences and Engineering Research Council of Canada andthe China Scholarship Council (20066035). Paper no. TPWRS-00546-2010.Y. Wang, I. R. Pordanjani, W. Xu, and T. Chen are with the Department of

Electrical and Computer Engineering, University of Alberta, Edmonton, ABT6G 2V4, Canada (e-mail: [email protected]; [email protected]).W. Li is with the Department of Electrical Engineering, Harbin Institute of

Technology, Harbin 15001, China.E. Vaahedi and J. Gurney are with the British Columbia Transmission Cor-

poration, Vancouver, BC V7W2S9, Canada.Color versions of one or more of the figures in this paper are available online

at http://ieeexplore.ieee.org.Digital Object Identifier 10.1109/TPWRS.2011.2154366

impedance “matches” the system Thevenin impedance. In otherwords, the impedance matching condition can be used to es-timate the maximum power deliverable to the measured loadpoint. Due to its simplicity, lots of researches have been done onthis technique and several implementation schemes have beenproposed [9], [10], [13].In spite of its elegance, the impedance match technique has

some major drawbacks. As discussed in [14], one of its prob-lems is the assumption that the equivalent Thevenin parametersare constant while they are being estimated. Such a requirementcan hardly be satisfied during a voltage collapse process as thesystem would experience continuous changes such as line trip-ping, shunt capacitor switching, generators reaching var outputlimits, etc. In addition to implementation issues, our researchhas shown that the technique has some theoretical problemswhen applied to multi-load systems, which supports the obser-vations of [18]–[20]. One of the purposes of this paper is to an-alyze and clarify these theoretical problems. Based on the find-ings, a new concept that is able to overcome the theoretical prob-lems is introduced and a voltage stability monitoring scheme isproposed.The rest of the paper is organized as follows: Section II

outlines the impedance match method and investigates itsperformance when applied to multi-bus systems. Section IIIstudies the causes of errors associated with the method andproposes a new concept to reduce the errors. Based on theconcept, three different models to overcome the difficultiesfaced by the impedance match method are presented andcompared in Section III. In Section IV, a real-time voltagestability monitoring scheme is presented. Section V shows thesimulation results, followed by the conclusions in Section VI.

II. SINGLE-PORT IMPEDANCE MATCH METHOD

Based on the electric circuit theory, the maximum powertransferable to a single port is reached under the condition ofimpedance match, . This approach hasbeen recommended in literatures for designing “distributed”load-shedding schemes to prevent voltage collapse, as follows:1) One of the load buses (the monitored bus) is separatedfrom the system and the rest of the system is treated as aThevenin equivalent circuit, as shown in Fig. 1.

2) The impedance match theory is applied to predict thevoltage stability margin (or load margin) at that bus.

3) Load shedding is triggered if the margin is below a certainthreshold and other alternative corrective options, such asblocking tap changers, are exhausted.

0885-8950/$26.00 © 2011 IEEE

WANG et al.: VOLTAGE STABILITY MONITORING BASED ON THE CONCEPT OF COUPLED SINGLE-PORT CIRCUIT 2155

Fig. 1. Thevenin equivalent circuits.

Fig. 2. Sample power system diagram.

The above scheme requires knowing the system’s Theveninequivalent parameters: and . Many methods have beenproposed to track these parameters. Almost all of the methodsare based on the theory in [9], which is as follows:

(1)

where and are the load voltage and load current phasorsmeasured at time , while and are measured at time . Theequivalent voltage source can be determined as

(2)

The above equations assume that the Thevenin equivalentdoes not change between and and are implemented over timeto track the impedance. The maximum power deliverableto the measurement point can then be determined according tothe impedance match condition , which gives

(3)

(4)

where is the maximum apparent power at the study bus,denotes the power factor angle of the load, and is the

apparent power of the load at the base case.The above single-bus Z-match method is valid for linear,

impedance based circuits. For power systems, however, theloads are nonlinear (constant power type) and dynamic (witha load recovery characteristic). When the Z-match techniqueis applied to such systems, it essentially reduces all such loadsexcept the study load into the equivalent system impedance.To our knowledge, the validity of such an approach has notbeen thoroughly examined. One of the goals of this paper is toinvestigate this concern through case studies. Only the constantpower type loads are considered in the studies, because thelinear part of loads (constant impedance and constant currentparts) can always be equivalenced into the system admittanceby using the method in [24].As a starting point, we consider a simple power system shown

in Fig. 2. The system parameters are ; ;; , where is the maximum active

power transferred to load 1 when .

Fig. 3. Estimated and the ideal impedance .

The Z-match technique is applied to estimate the voltage sta-bility margin at bus 2 using the voltage and current taken at bus2 only. We are interested in finding out how , a nonlinear anddynamic load, will affect the margin estimate as this load willbe equivalenced with the rest of the system as . When per-forming this analysis, it is important to note the following powersystem characteristics:• Voltage collapse is often caused by excessive loading inmultiple locations instead of a single location of a powersystem. This is why the WECC guideline [21] suggestsscaling up all loads in the study area to determine thesystem margin (the PV curve). So when one tries to esti-mate the maximum power deliverable to a single bus usingZ-match, the other loads equivalenced into needs to bescaled up accordingly, i.e., the Thevenin circuit is no longerfixed. This situation can be compared to the equal-area cri-terion based transient stability analysis technique. Equal-area criterion works well for one-generator versus infinitebus system. But it could not function for multi-machinesystems since the infinite bus no longer exists.

• The transient response of a load is different from its steady-state response [22]. It is the steady-state load with its con-stant power characteristic that collapses a power system.However, (1) uses the transient responses of a network toestimate the Thevenin parameters. As a result, the esti-mated Thevenin parameters are not able to represent thesteady-state behavior of the upstream system. This situa-tion is somewhat similar to the attempt of using generatorterminal voltage and current to estimate an equivalent “in-finite bus” and then applying the equal-area criterion to it.

For the proposed case study, we therefore assume that in-creases in proportion to and is also a dynamic load that hasa constant impedance transient characteristic. The steady-stateload characteristic of is still the constant power type.Fig. 3 shows the estimated in comparison with the ideal

impedance value that is the equivalent impedance of maximum. is the operating point of load 2 at which the maximumis to be predicted. It can be seen that a large difference ex-

ists, even at the maximum loading point. Such a phenomenon iseasy to be understood if we imagine as a constant impedanceload behind a tap changer that makes into a constant powerload in the steady state. Since it is impossible for the mea-surement algorithm to take into account the tap characteristics,is treated as a constant impedance load in . The result


Fig. 4. Comparison of maximum power for different base cases.

Fig. 5. Comparison of estimated with actual for all loads.

is an optimistic estimation of the system margin. Fig. 4 showsthe estimation results for different operating conditions: and

are the operating point of load 1 and load 2, respectively,where the estimation is taken. Again the load 1 is a dynamicload with constant impedance transient characteristic and con-stant power steady-state characteristic. The results indicate thatthe estimated is significantly higher than the true .This will result in a false sense of security and loads will not beshed sufficiently. This is especially true when load is large.A similar study was conducted on the IEEE 30-bus system

[23]. Fig. 5 shows the results. It can be seen from Fig. 5 thatthe estimated is significantly larger than the actual valueat some buses. It will lead to insufficient load shedding.In summary, case studies and analysis have shown that the

single-port impedance match method does not work properlyfor multi-load systems. The Z-match criterion to voltage sta-bility is like the equal-area criterion to transient stability. Bothare elegant when dealing with single-bus systems but encounterconsiderable difficulties when applied to the multi-bus systems.In the following sections, we propose a method that seems to beable to make use the Z-match concept without encountering theissues discussed earlier.

III. CONCEPT OF COUPLED SINGLE-PORT CIRCUIT

The fundamental flaws involved in applying the single-portZ-match technique to power systems are from the inclusionof all other loads into the system equivalence. These loadsare nonlinear and dynamic. It is theoretically impossible to

Fig. 6. Single-port equivalent versus multi-port network equivalent. (a) Single-port equivalent system. (b) Multi-port network equivalent.

Fig. 7. Multi-port network system model.

represent them as a single value even if their power levelsare constant. One potential direction to overcome this difficultyis to keep all loads outside of the equivalent system as shownin Fig. 6. This leads to the concept of multi-port networkequivalent.The multi-port network can be modeled as shown in Fig. 7.

All the generators and load buses are brought outside of the net-work. The transmission network is converted to an equivalentimpedance matrix .Then, the multi-port power system can be described by

where the matrix is known as the system admittance matrix,and stand for the voltage and current vectors, and the sub-

script , , and represent load bus, tie bus, and generator bus,respectively.Eliminate the tie buses, and the above equation can be ex-

pressed as follows:

(5)

where is an matrix obtained from system admittancematrix , and is an impedance matrix.From (5), for load bus , we can obtain

(6)

where is the Thevenin impedance of the network at bus ,again without the inclusion of the other loads. Note thatis the diagonal element of the impedance matrix , and isessentially equal to the short-circuit impedance seen at bus . It


Fig. 8. Coupled single-port system equivalent.

Fig. 9. Equivalent source voltages at different buses.

is a constant as long as the system topology remains the same.represents the impact of other loads on bus , called

the coupling effect in this paper.The circuit corresponding to (6) is shown in Fig. 8. This is a

single-port network and it can be applied to all load buses. Com-pared to the traditional single-port equivalent circuit, the differ-ence is the new term which represents the impact ofother loads on bus ’s equivalent circuit. Using this model, apower system can be broken down into a set of single-port cir-cuits that have the impact of other loads included explicitly. Thisnew equivalent circuit is called “coupled single-port circuit” inthis paper.In order to develop a voltage stability margin estimator based

on the concept of the coupled single-port circuit, the character-istics of and should be understood first. (is computed from the network matrix and it does not changewith the system power flow patterns; therefore, there is no needto investigate its characteristics).The investigation is done by using the IEEE 30-bus system.

All the loads are increased proportionally from their respectivebase levels with the same scaling factor following the stan-dard PV curve approach [21]. The equivalent source voltage

and the coupled voltages are calculated corre-sponding to each . Figs. 9 and 10 show the results of and

at different bus locations. Each curve represents one ofthe load buses.Fig. 9 shows that remains relatively constant from the

base case load level to the maximum load level. As a result,it is reasonable to assume that keeps constant from thereal-time operating point to the voltage collapse point. Thisapproximation will introduce an error. Test results shown inSection VI indicate that the approximation is acceptable sinceit has negligible effect on the accuracy of margin and criticalbuses prediction.

Fig. 10. Coupled voltages at different buses (each line for a different bus).

Fig. 11. Coupled single-port system approximated with a virtual load.

On the other hand, the coupled voltage increases dra-matically (see Fig. 10). The implication is that the main factorcausing voltage collapse is the reduction of the total voltage

seen by the load due to rapid increase of theterm. This finding indicates that the impact of other

loads on the study bus must be considered.In summary, the concept of the coupled single-port circuit

helps to reveal the interactions of various loads in a powersystem and how such interactions affect the voltage stabilitymargins of the individual buses. The results have shown itis essential to model the interactions (namely, the couplingeffects) for proper estimation of voltage stability margins.

IV. MODELING THE COUPLING EFFECTS

Three approaches have been studied on how to model thecoupling terms while maintaining the single-port structure.1) Modeling the Coupling Term as an Extra Power Demand:

This approach is to approximate the coupling effects as an extrapower demand at bus which is called virtual load and denotedas (Fig. 11). The virtual load is determined by (7). Thisvirtual load model is similar to the idea proposed in [25]:

(7)

Characteristics of the virtual loads are shown in Fig. 12. Thecurves are the percentage of the virtual loads with respect to thephysical loads. Fig. 12 shows that the virtual loads increase ina nonlinear manner when physical load increases. The reactivepower load increases much more significantly. This reveals thatthe model is not suitable to represent the coupling term.2) Modeling the Coupling Term as an Extra Voltage Source:

This approach is to approximate the coupling term as an extravoltage source which is denoted as (Fig. 13). The virtualvoltage source is determined by (8).


Fig. 12. Characteristics of the virtual loads. (a) Active power ratio(%). (b) Reactive power ratio (%).

Fig. 13. Coupled single-port system approximated with a virtual voltagesource.

Fig. 14. Coupled single-port system approximated with a virtual impedance.

The characteristics of the virtual voltage source are shownin Fig. 10. This virtual voltage increases significantly when in-creasing the physical loads. This feature makes it unacceptablefor the voltage stability margin estimation:

(8)

3) Modeling the Coupling Term as an Extra Impedance: Thisapproach is to represent the coupling term as extra impedancecalled virtual impedance and denoted as (Fig. 14). The vir-tual impedance is defined by (9), in which the ratio of theloads is constant. Moreover, the bus voltage also changes pro-portionally when scaling up the power system, which means theratios of the bus voltages remain approximately constant. Thus,the extra impedance model is nearly constant in theory.Fig. 14 shows the extra impedance model:

(9)

The characteristics of the virtual impedance are shown inFig. 15. Fig. 15 shows that the impedance remains relativelyconstant with increasing the physical loads. The results furtherverify the conclusions obtained from (9). It implies that the cou-pling effects of other loads are somewhat equivalent to insertingconstant impedance into the single-port network. This featuremakes the virtual impedance model very promising.In order to compare the performance of these three models,

they are used to estimate the system margin. The results (the

Fig. 15. Characteristics of the virtual impedance.

Fig. 16. Estimated stability margin from different methods.

calculation procedure is detailed in Section V) are shown inFig. 16. As we expected, the virtual impedance model has thebest performance. It can be seen that the estimated margin atload bus 30 (it is also the critical bus identified by the modalanalysis method) matches with the actual margin of the systemwith a little error. The loading level of the base case is assumedas 100%, which is the reference value of the voltage stabilitymargin shown in Fig. 16.

V. PROPOSED VOLTAGE STABILITY MONITORING SCHEME

Based on the above analysis, a voltage stability moni-toring scheme is proposed using the virtual impedance model.In the proposed scheme, the power system is decomposedinto a set of coupled single-port network (see Fig. 17). Theequivalent impedance is a combination of the virtual andthe self-impedance as shown in Fig. 14. Using this coupledsingle-port equivalent, the voltage stability margin at each buscan be calculated according to the impedance match criterion.The equation for calculating maximum load has been shown in(3) and (4).It is understandable that the calculated margins will be dif-

ferent for different coupled single-port circuits. However, sincethe collapse of one bus voltage represents the collapse of the en-tire system (i.e., Jacobianmatrix becomes singular), the smallestmargin among all the circuits can be used to represent the systemmargin (see (10)):

(10)

Upon finding the system margin, the weakness of the busescan be determined by comparing their margins, i.e., the weakest


Fig. 17. Coupled single-port network equivalent for multi-port power system.

bus is the one with the smallest margin. Since there are approxi-mations in creating the equivalent circuits, buses that have mar-gins very close to the smallest one shall also be considered asweak buses.Ideally, the proposed scheme should use the synchronized

phasor measurement unit (PMU) measurements from all thegenerators and all the load buses to obtain the entire multi-port network model. However, to be practical, the proposedscheme can also be implemented by using only the PMU mea-surements from all the generators and the interested load buses,since the virtual impedance can be obtained from the followingequations:

(11)

The proposed scheme is to be implemented in a control centerand to guide the load shedding actions. The procedure (Fig. 18)described below is run continuously, say, once for every 5 s. As-suming an outage event occurs, the schemewill work as follows:• Gather system real-time operation status: admittancematrix from SCADA, synchronous generator voltagephasors , and voltage and current measurements at theinterested buses from PMUs. At the same time, the controldevices will be considered in the equivalent network.For instance, the on-load tap changers (OLTCs) will begrouped in their corresponding loads. The generators willbe converted to voltage sources behind its saturated syn-chronous reactance when they reach the reactive poweroutput limit (impact of the Q-limit on the proposed methodis shown in Appendix B).

• Calculate the equivalent voltage and the equivalentimpedance of the coupled single-portmodel by using (6) and (11), respectively.

• Compute voltage stability margin for each single-portsystem by using impedance matching (3) and (4). In (3),the equivalent voltage and impedance should bereplaced by their corresponding and .

• Find the system voltage stability margin and the weakbuses based on the individual margin prediction results byusing (10).

• Load shedding is triggered based on the margin/critical re-sults and load shedding criteria.

The main computing effort involved in the above procedure isthe inverse of real time [Y] matrix. Fortunately, this matrix doesnot change at each execution cycle. It changes only when majorequipment is switched on/off. The 5-s execution interval also

Fig. 18. Implementation procedure of the proposed scheme.

Fig. 19. Estimated margin for the nine-bus system.

ensures that the transients associated with faults have died outand the system is in quasi-steady-state. This mode of operationis acceptable since voltage instability is mainly caused by theslow power recovery process of loads in about 15- to 30-s timeframe. The requirements on data communication can be also ac-commodated with the 5-s cycle. For cases where the PMU dataare not available, state estimator results can be used assumingthe state estimator can have 5-s turn around cycle.

VI. SIMULATION RESULTS

The proposed monitoring scheme is applied to several testsystems and the results are shown in this section. The actualvoltage stability margin is calculated using the continuation(repetitive) power flow technique by using the commercial soft-ware PSS/E and following the WECC PV curve methodology[21].Nine-bus system: Fig. 19 shows the estimated stability margin

of the WECC nine-bus system. As seen from this figure, bus 9has the smallest margin which is very close to the actual margin.If the modal analysis method [8] is used, it is verified that bus9 is indeed the weakest load bus (critical load) in the system.Therefore, using the proposed scheme, both the stability marginand the weakest load are found accurately.IEEE 39-bus system: Fig. 20 shows the estimated margin for

the IEEE 39-bus system. Based on this figure, bus 4 has thesmallest margin. However, the estimated margins for buses 7,8, and 15 are also very close to the margin of bus 4. In fact,they are only 1.05% larger than the margin of bus 4. Therefore,


Fig. 20. Estimated margin for IEEE 39-bus system.

Fig. 21. Estimated margin for the Alberta system.

TABLE IRESULTS OF THE WEAKEST BUS IDENTIFICATION

a group of buses including buses 4, 7, 8, and 15 are determinedas the weak buses. Modal analysis method reveals that one ofthese buses, bus 7, is indeed the weakest bus in the system.Alberta Integrated Electric System: An actual large system,

Alberta Integrated Electric System (AIES), is also used as a testcase. By using the proposed method, the system stability marginis estimated as shown in Fig. 21. The figure reveals that the es-timated margin at load bus 630 is very close to the actual value.Again, the modal analysis method verified that bus 630 is theweakest load bus.The above results are summarized in the Tables I and II.

Table I shows the results of the weak bus identifications for dif-ferent systems. As seen in this table, the modal analysis methodhas verified that the proposed method can accurately identifythe weakest bus. Note that this identification has been done byusing the system data when the system is at its normal opera-tion condition, indicating the unique advantage of the proposedmethod.Table II illustrates the error of margin estimation for different

systems. The estimation has been performed twice: one is at

TABLE IIRESULTS OF THE MARGIN ESTIMATION

the normal operation condition and the other one is at the con-dition when the system has 5% voltage stability margin. Thistable reveals that the proposed method has an acceptable ac-curacy which makes it suitable for real-time voltage stabilitymonitoring.As a summary, the proposed coupled single-port model has

the following useful features:• It only needs one snapshot of the interested load buses’voltage and current phasors and the generator terminalvoltage phasors to estimate the load margin of each loadbus. As a result, margin results are available at anytimeand continuous tracking of system or bus margins becomespossible. It avoids the need to estimate the unattainableThevenin impedance.

• The method takes advantage of both global network in-formation (network matrix from SCADA and generatorterminal voltages from PMU) and the local bus informa-tion (bus voltage, current, and power snapshots) to forma mixed calculation-measurement based voltage stabilitymonitoring system. If a reliable state estimator is available,the need for PMU data can be avoided.

The proposed method is a stability monitoring scheme thatreveals the system margin at the existing steady-state condition.Like the basic Z-match methods, it cannot predict the impact ofcontingencies before they occur. However, the proposed con-cept may help to speed up the contingency oriented margin pre-diction methods such as the continuation power flow. Anotherremaining issue is how to use the results to design an optimalload shedding and demand response scheme. This subject is cur-rently under investigation.

VII. CONCLUSIONS

This paper reveals that the basic impedance match techniquehas problems to predict voltage stability margin when ap-plied to multi-load systems. Power system loads are nonlinearand dynamic. They cannot be easily equivalent to Theveninimpedance. In order to preserve the elegance and simplicity ofthe impedance matching idea, this paper proposes the conceptof coupled single-port circuit. In this circuit, all the loads arebrought outside of the equivalent system. Therefore, the cou-pling effects among the loads can be dealt with explicitly. Threedifferent models are investigated for representing the couplingeffects. The virtual impedance model is found to be the mostacceptable, which leads to a mixed measurement-calculationbased online voltage stability monitoring scheme. The proposedcoupled single-port circuit can be obtained by collecting thephasor measurements at the generator buses and the load buses


of concern from PMUs. The validity of the proposed methodhas been verified on several power systems using continuationpower flow simulation of each stressed power system and itssteady-state snapshots.The most significant contributions of this research work are

the proposition of the coupled single-port network concept andits virtual impedance model. In addition to solving the problempresented in this paper, the concept has other potential applica-tions. Examples are speeding up the continuation power flowalgorithms and identifying critical buses without using modalanalysis.

APPENDIX

A. Derivations for Equation (3) in Section II

The maximum power is obtained when the impedancematches, i.e., . Therefore, (3) can be derivedas shown in the equation at the bottom of the page, whereis the power factor angle of the load, , and

.

B. Impact of Generator Reactive Power Limit

The IEEE 30-bus system is used to determine how the gen-erators’ reactive power limits will affect the proposed method.To avoid showing complicated figures, only the generators con-nected to bus 5, 8, 11, and 13 are considered with reactive powerlimit. The other two generators, which are connected to bus 1and bus 2, are assumed not to have reached their limits.The simulation shows that the generator at bus 8 (Gen 4)

reaches its limit at , the generator at bus 5 (Gen 3)reaches its limit at , the generator at bus 11 (Gen 5)reaches its limit at , and the generator at bus 13 (Gen6) reaches its limit at . The actual maximum scalingfactor is 2.41.

Fig. 22. Estimated maximum scaling factor.

Fig. 23. Equivalent impedance ( as defined in Fig. 14) of differentload buses.

Fig. 22 shows the estimated maximum scaling factor at bus26 and bus 30, where the smallest estimated values are ob-tained. The figure also shows the actual maximum scaling factorfor four other operating scenarios. These four operating sce-narios are four snapshots of the system with different topolo-gies (i.e., with different generators that reach their limits). The


Fig. 24. Virtual impedance [ as defined in (9)] of different load buses.

Fig. 25. Equivalent voltage for different load buses.

Fig. 26. Coupling voltage for different load buses.

actual scaling factor for each scenario is calculated based on theassumption that no more topology changes (i.e., no more gener-ators reach their limits). The result has confirmed that the pro-posed method is generally robust in dealing with the generatorreactive limits.Figs. 23 and 24 show the equivalent impedance and the

virtual impedance of different load buses, respectively. Theresults indicate that remains almost constant until the nextgenerator reaches reactive power limit.The equivalent voltage defined in (6) is also plotted in

Fig. 25. The results show that the changes of are around5% to 10%. Fig. 26 shows the coupled voltage ofindividual load buses. The curves indicate that the coupling ef-fects between the load buses can increase significantly with in-creasing the loading level.

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Yunfei Wang (S’08) received the B.Sc. and M.Sc. degrees in control scienceand technology from Harbin Institute of Technology, Harbin, China, in 2003and Tsinghua University, Beijing, China, in 2006, respectively. He is currentlypursuing the Ph.D. degree in the Department of Electrical and Computer Engi-neering, University of Alberta, Edmonton, AB, Canada.His research interests are power quality and power system stability.

Iraj Rahimi Pordanjani (S’09) received the B.Sc. (with First Class Honors)and M.Sc. degrees in electrical engineering from Amirkabir University of Tech-nology (Tehran Polytechnic), Tehran, Iran, in 2005 and 2008, respectively. Heis currently pursuing the Ph.D. degree in electrical and computer engineering atthe University of Alberta, Edmonton, AB, Canada.His research interests are power systems stability and power quality.

Weixing Li (M’09) received the B.Sc., M.Sc., and Ph.D. degrees from HarbinInstitute of Technology, Harbin, China, in 1999, 2001, and 2007, respectively.Currently, he is an Associate Professor in the Department of Electrical Engi-

neering at Harbin Institute of Technology, and he is also a Postdoctoral Fellowat the University of Alberta, Edmonton, AB, Canada. His research interests in-clude power systems reliability, distribution system, voltage stability, and powertransfer capability.

Wilsun Xu (M’90–SM’95–F’05) received the Ph.D. degree from the Universityof British Columbia, Vancouver, BC, Canada, in 1989.He was an Engineer with BC Hydro, Burnaby, BC, Canada, from 1990 to

1996. Currently, he is a Professor and a NSERC/iCORE Industrial ResearchChair at the University of Alberta, Edmonton, AB, Canada. His research inter-ests are power quality and voltage stability.

Tongwen Chen (S’86–M’91–SM’97–F’06) received the B.Eng. degree inautomation and instrumentation from Tsinghua University, Beijing, China, in1984, and the M.A.Sc. and Ph.D. degrees in electrical engineering from theUniversity of Toronto, Toronto, ON, Canada, in 1989 and 1991, respectively.Presently, he is a Professor in the Department of Electrical and Computer En-

gineering, University of Alberta, Edmonton, AB, Canada. His research interestsinclude computer- and network-based control systems, and their applications toindustrial problems.

Ebrahim Vaahedi (S’78–M’79–SM’87–F’00) received the Ph.D. degree fromImperial College of Science and Technologies, London, U.K., in 1979.Currently, he leads BC Hydro’s Operations Technology Department respon-

sible for developing and delivering Operations Technology strategy. He is anAdjunct Professor at the University of British Columbia and Sharif Universityof Technology.Dr. Vaahedi is the chair of the IEEE Subcommittee on Operation Methods

and an associate editor of the IEEE TRANSACTIONS ON POWER SYSTEMS.

Jim Gurney (SM’92) received the B.Sc. degree in electrical engineering fromthe University of British Columbia, Vancouver, BC, Canada.He manages the BC Transmission Corporation Research and Development

Program. He has been with BCTC and BC Hydro for 37 years in a number ofengineering and management roles.Mr. Gurney served on the IEEE Standards Board from 1998 to 2002, and was

Vice Chair in 2002. He now serves on the Canadian National Committee of theInternational Electrotechnical Commission (IEC). He is a Professional Engineerin the Province of BC.

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