1122 ieee transactions on smart grid, vol. 4, no. 2, june...

1122 IEEE TRANSACTIONS ON SMART GRID, VOL. 4, NO. 2, JUNE 2013

Identification of Critical Components forVoltage Stability Assessment Using Channel

Components TransformIraj Rahimi Pordanjani, Student Member, IEEE, Yunfei Wang, Student Member, IEEE, and Wilsun Xu, Fellow, IEEE

Abstract—Channel Components Transform (CCT) is a recentlydeveloped technique to decouple interconnected power networks.This paper aims to further explore the CCT and extend its appli-cations. Methods and algorithms are proposed to extend its appli-cation in identifying the critical generators and branches of a net-work from the voltage stability perspective. The proposed methodsare verified by case studies conducted on multiple test systems.This paper also demonstrates the capability of the CCT to workproperly when a limited number of phasor measurement units areavailable. For this purpose, a strategy is proposed to determinethe number and location of PMU installations that are sufficient totrack the modes of voltage collapse and associated critical compo-nents. The proposed allocation strategy is examined through casestudies of an actual power system.

Index Terms—Critical branch, critical generator, network de-coupling, phasor measurement unit (PMU), voltage stability.

I. INTRODUCTION

T HE PROBLEM OF maintaining power system voltagestability is one of the main concerns in planning and oper-

ating power systems. Many studies have been conducted duringthe past decades to develop methods for voltage stability anal-ysis and mitigation [1]–[3]. In recent years, the advent of PMUtechnology has opened new perspectives for voltage stabilitymonitoring and analysis [4]–[6]. In this respect, many attemptshave been made to either propose new methods or adapt ex-isting offline methods for the online monitoring and analysisof voltage stability [7]–[11]. One of the most recent attemptsis reported in [11]. In the paper, a network transform techniquecalled the Channel Components Transform (CCT) has been pro-posed to process the phasor data. New insights into the voltagestability characteristics of a power system have been revealed.The CCT technique treats a power network as a multi-node,

multi-branch Thevenin circuit connecting the loads to thegenerators. By applying eigen-decomposition to the Theveninimpedance matrix, the network is decoupled into a set ofsingle-node, single-branch equivalent circuits. These circuits

Manuscript received January 30, 2012; revised July 11, 2012; accepted Jan-uary 20, 2013. Date of publication March 22, 2013; date of current version May18, 2013. This work was supported by Natural Sciences and Engineering Re-search Council of Canada and a number of utility companies in Alberta. Paperno. TSG-00047-2012.The authors are with the Department of Electrical and Computer Engi-

neering, University of Alberta, Edmonton, Alberta, T6G 2V4, Canada (e-mail:[email protected]).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/TSG.2013.2242498

are much easier to analyze and they carry valuable informationof a power system.Reference [11] shows that the CCT can establish a new

framework for voltage stability analysis. The application of thisframework in identifying the critical load buses is proposedand verified in [11]. This current paper extends the findings in[11] and has two main objectives.• The CCT technique is further investigated, and its appli-cation is extended to other tasks of voltage stability anal-ysis. For this purpose, CCT-based methods are developedto identify the generators and branches that are critical withrespect to the voltage stability of a power system.

• Algorithms are proposed to determine the number andlocation of PMU installations that are sufficient to trackthe modes of voltage collapse. The proposed allocationstrategy will enable the CCT to be applied successfullyeven when only a limited number of PMUs are availablein a power system.

The remainder of this paper is organized as follows. InSection II, the basic concept of the CCT technique is brieflyreviewed first. A CCT-based method is then proposed for theidentification of the critical generator. In Section III, a methodis proposed for identifying the critical branch. Section IVpresents a strategy to allocate PMUs so that the modes ofvoltage collapse and the critical components can be identifiedusing a limited number of PMUs. Section V consists of detailedcase study results. The performance of the proposed methodsand algorithms are verified by using multiple case studies in-cluding three standard test systems and an actual large system.

II. IDENTIFICATION OF THE CRITICAL GENERATOR

Generators, through their reactive power support to powersystem voltages, have a significant impact on the voltage sta-bility of a power system [3]. Utility planners and operators havea strong desire to know the most important generators, calledthe critical generators, for a given voltage instability scenario orwhen voltage stability margins become too small. This need wasfirst recognized in [12] where the Jacobian matrix based modalanalysis technique was used to identify the critical load points.Since the reduced Jacobian matrix used by [12] provides no re-active power information for generators, empirical indices areproposed for generator ranking [13]. In order to establish a theo-retical basis for generator ranking, some references such as [14]propose to use the active power information contained in thereduced Jacobian matrix to determine the active (power) partic-ipation factors (APF) of the generators. As a result, the genera-

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PORDANJANI et al.: IDENTIFICATION OF CRITICAL COMPONENTS FOR VOLTAGE STABILITY ASSESSMENT 1123

Fig. 1. A general electric power network.

Fig. 2. Channel domain representation of the complex network.

tors are ranked in terms of their active power impacts on voltagestability. However, APF may fail to act as a proper index sincevoltage stability is usually associated with the reactive powerlimitations [3]. In other words, ranking the generators based onthe impacts of their reactive powers on voltage stability wouldbe very beneficial. This section will show that the CCT can beused to develop a new and effective method to do so.A general power system shown in Fig. 1 can be modeled as a

multi-port Thevenin circuit as follows:

(1)

In the above, is the terminal voltages or the internal voltages(if a generator’s is reached) of the generators and isthe nodal voltages at the load buses. Note thatis the open circuit voltage vector of the traditional multi-portThevenin equivalent circuit.Eigen-decomposition (i.e., Channel Components Transform)

can be performed on the matrix of the Thevenin circuit asfollows [11]:

(2)

where and are the eigenvalue and eigenvector matricesof , respectively. Applying the above to (1) yields

(3)

Denote as the transformed voltageas the transformed current

as the transformed voltage sourceThis leads to the following decoupled (modal) networks

whose circuit representations are shown in Fig. 2.

(4)

The significance of the above transform is that a complexnetwork has been transformed into a set of decoupled simple

one-source, one-load networks. In this paper, each mode (i.e.,eigenvalue) of is called a channel. It presents one pattern of(decoupled) power flow in a network.As shown in [11], among different channels of a system, there

is one critical channel which is most responsible for the voltagecollapse. Using the critical channel, useful information aboutthe voltage stability characteristics of the system can be ex-tracted. One example is the identification of critical loads [11].As presented in [11], the contribution of bus currents to thecritical channel current as shown in (5) can be used as anindex to rank the loads in terms of their impacts on voltagestability.

(5)

where is the contribution of load bus to the channelcurrent and is the angle difference between the channelcurrent and the term . The bus which has the highestcontribution to the critical channel is the critical bus.Similarly, we propose to use the critical channel information

to identify the critical generator. For this purpose, the genera-tors’ contributions to the critical channel are determined. Thegenerator with the highest contribution to the critical channelwill be the critical generator.The channel voltage sources can be expressed in terms of

generator voltages as follows.

(6)

Matrix represents the contribution of each generator voltageto the voltage . For example, the magnitude and angle ofworks like a weighting factor to indicate the weight of

on the channel voltage :

(7)

Using the above equation, the contribution of each generatorvoltage to the critical channel i can be calculated by

(8)

where is the contribution of generator to thechannel voltage , and is the angle difference between thechannel voltage and the term .Assume that generator has the highest contribution to

the critical channel. In this case, if the angles of generator volt-ages are assumed to remain constant, an increase in the voltagemagnitude of the generator will lead to the highest increase inthe magnitude of the channel voltage . This will result in thehighest increase in the stability margin of the critical channel.On the other hand, the increase in the magnitude of the generatorvoltage corresponds to an increase in its reactive power. As a re-sult, the proposed contribution index can be used to determinethe impact of the generators’ reactive powers on the stability ofthe critical channel and as a result, on the stability of the actualsystem.Note that even if some generators reach their reactive limits,

the above ranking method still works after changing the modelsof the generators. This model change is explained below [11].


Fig. 3. Modeling of the generators reaching reactive power limits. (a) PVmodel (before reaching the reactive limit). (b) PE model (after reaching thereactive limit). (c) Modified network.

Before reaching the reactive limit, each generator is modeledas a PV bus as shown in Fig. 3(a). A generator’s reactive limit isactually caused by the limit on the field current, i.e., the field cur-rent becomes a constant. Since the field voltage is in proportionto the field current, the generator thus behaves as a constant fieldvoltage (the maximum field voltage) behind its synchronousreactance . Therefore, the PE model presented in [26] canbe used to represent such a generator. In other words, the gener-ator is modeled as a PV bus behind the synchronous reactanceas shown in Fig. 3(b). As it can be seen from the model, the orig-inal PV bus (bus 1) now retreats to bus 1’. Its effect is equivalentto an increase of the electrical distance from the generator to thesystem.Therefore, when a generator hits its reactive limit, a new bus

(bus 1’) is created and is considered as a new PV bus. Bus 1is not a PV bus anymore. It should be treated as a network orload bus. The network needs to be modified for the purpose ofthe CCT application. As an example, bus 1 in Fig. 3(b) will beconsidered as a new network bus. This bus along with willbe moved into the network as shown in Fig. 3(c). The networkimpedance matrix and the transformation matrix are modified.The CCT-related computations [11] are then performed for themodified network.Note that when a generator’s exciter is reaching its thermal

limit, a dynamic phenomenon (such as the exciter’s temporaryoverloading) might be involved [27]. The CCT does not treatthose dynamics. Only after the dynamics have died out, andthe system has reached a steady-state operating point, the CCT(with the modified generator model shown in Fig. 3) is appliedto determine the critical components associated with the newoperating point. The same situation exists for any other systemtopology changes such a line/generator outage.

Fig. 4. Single line diagram of a radial network.

Fig. 5. Voltage phasor diagram of the radial system.

III. IDENTIFICATION OF THE CRITICAL BRANCH

Critical branches can be considered as the weakest links in agrid. So power system planners and operators have a keen in-terest in identifying such links for a given system configura-tion or scenario. This section aims to identify the critical branchbased on the information obtained by the CCT. The criticaltransmission path (CTP) is identified first, and then its criticalsegment is determined.In [18], an index called the transmission path stability index

(TPSI) was proposed which is adopted here to identify the crit-ical transmission path. The TPSI of the n-bus radial systemshown in Fig. 4 is defined as [18]:

(9)

where is called the total corrected voltage drop. This termis equal to the sum of the corrected voltage drops of all seg-ments. The corrected voltage drop of a segment is obtained byprojecting the direct-axis component of its voltage drop into thevoltage phasor of the supply bus. This concept is illustrated inFig. 5, which shows the phasor diagram of a radial system. Asan example, consider the segment , connecting bus tobus . As this figure shows, the corrected voltage dropof this segment is indeed the projection of the di-rect-axis voltage drop into the supply voltage .The corrected voltage drops of all segments are calculated andadded together to obtain the total corrected voltage drop(see Fig. 5).Therefore, the TPSI of a n-bus radial system can be calculated

using the voltage phasors as follows.

(10)


Since power systems are meshed networks, a different situa-tion exists. However, [18] has shown that TPSI can be applied tothe transmission paths of a meshed network as an approximatestability index. A transmission path is usually defined as a se-ries of buses with declining voltage magnitude [19]. A transmis-sion path always ends in a load bus, and there might be differentpaths supplying one load bus. In the method proposed in [18],all the transmission paths supplying all the system loads are de-termined, and their TPSI are calculated using system voltagephasors. The transmission path with the smallest TPSI is thecritical one. It is worthwhile to mention that as shown in [18],when the system is close to the collapse point, the critical trans-mission path might have a negative TPSI.The main problem of the above method is that it involves too

many transmission paths especially in actual power systems.Therefore, it requires too much calculation effort especially ifit is intended for online applications. The channel componentstransform (CCT) can be utilized to simplify this method andovercome its problem.• The CCT can identify the critical load bus. It has beenshown in the literature (such as in [18]) that the criticaltransmission path ends in the critical load bus.

• The CCT can also identify the critical generator. As ex-plained in the previous section, the critical generator hasthe highest impact on the voltage stability of the system.In other words, among all the generators, the critical gen-erator limits the system stability level the most. Therefore,the critical generator should be the one supplying the crit-ical transmission path.

Therefore, instead of considering every possible transmissionpath, it suffices to consider only those from the critical generatorto the critical load. If only one such path is present, it will bethe critical path. If there are more than one path (which willnot be too many), the TPSIs of these paths are determined andcompared. The critical path will be the path with the smallestTPSI.Once the critical path is identified, its segments can be com-

pared to determine which segment is the most responsible forthe low TPSI (or the high total corrected voltage drop) of thepath. For this purpose, the corrected voltage drops of the seg-ments are compared. The segment with the highest correctedvoltage drop is the weakest segment. The identified segment isthe critical branch in the system.The proposed method can be summarized as the following

steps. It is assumed that the CCT has already been applied tothe system and that the critical load and the critical generatorhave been identified.• Determine all the transmission paths which supply the crit-ical load bus.

• From the determined paths, consider only those which startfrom the critical generator.

• For each path, find the corrected voltage drop of its seg-ments, and calculate the total corrected voltage drop of thepath.

• Compute the transmission path stability index (TPSI) ofeach path.

• Compare the TPSI of all paths, and identify the critical pathwhich has the smallest TPSI (if there is only one path, it isthe critical one).

• Consider the critical transmission path, compare the cor-rected voltage drops of its segments, and identify thecritical segment. The critical segment is the one with thehighest corrected voltage drop.

Note that the TPSI [18] has been widely used in the literatureto form a voltage collapse proximity index (VCPI). In this ap-proach, the TPSIs of all transmission paths are computed, andthe minimum TPSI is considered as the VCPI. Based on thevalue of VCPI, the proximity to the voltage collapse is then de-termined in a system. Some improvements have also been pro-posed in the literature. For example, [23] proposes a two-busequivalent for each transmission path to be used in the TPSIcalculation. Reference [24] proposes a method to allocate thelosses of each transmission path among different generators, re-sulting in an improved TPSI. Reference [24] has also shown thatVCPI is not sufficient to determine the proximity to the voltagecollapse since a voltage collapse might occur as a consequenceof insufficient reactive power production while the maximumpower-transmission capabilities have not yet been reached. Anadditional index called the reactive power index (RPI) is pro-posed in [24] to account for the reactive power production avail-able to a particular load bus. In this approach, the VCPI and itsassociated RPI are determined. Both values are then examinedto determine the proximity to an upcoming voltage collapse.However, the TPSI is used in our method to determine the

critical transmission path only. It is not used as a voltage col-lapse proximity index. Therefore, the RPI or similar indiceswhich consider the limitation of reactive power production arenot required in the proposed method. Furthermore, the originalTPSI formulation proposed in [18] has been used in this paper.Themain feature of ourmethod is that it decreases the number oftransmission paths significantly, leading to a much faster iden-tification. Once the transmission paths are determined, one maychoose to use a different TPSI formulation such as those of [23],[24]. Similarly, RPI may also be calculated for the transmissionpaths.

IV. PMU ALLOCATION

The CCT transforms a power system into channel circuits.Useful information about the actual system can then be ex-tracted by monitoring and analyzing the channel circuits. Thechannel quantities, i.e., the channel source voltages , channelvoltages , and channel currents , can be computed byusing the generator bus voltages , load bus voltages , andload bus currents which are all available from the PMUs.Therefore, all channel quantities can be obtained and monitoredonline. In theory, PMUs are needed at all generator and loadbuses to provide the data required by the CCT analysis. Ourresearch has shown that this is not necessary as many channelsare not important for a given power system. The results of theprevious sections and [11] have shown the following:• For the voltage stability analysis, all the network chan-nels do not need to be analyzed and monitored. It suffices


to consider only a small number of the channels (criticalchannels).

• Only a small portion of system loads contribute signifi-cantly to a particular channel. The effects of other loadson that channel are small and can be ignored.

• Only a small portion of the system generators contributesignificantly to a particular channel. The effects of othergenerators on that channel are small and can be ignored.

The implication is the following. A small number of channels,which are the most critical, can be determined. These channelsare considered as the critical channels which are to be monitoredby using PMUs. The load and generator buses which have sig-nificant contributions to those channels are also found. Thesebuses can be considered as the locations for installing PMUs.Therefore, the following strategy is proposed to allocate PMUs.1) Stress the system to a point close to the collapse point,apply the CCT, and compute the channel margins.

2) Set a value for the maximum channel margin, and select the channels whose margins are

lower than this value as the critical channels.3) Consider each critical channel one at a time, and performthe following process:• Compute the contributions of the load buses to thechannel, as in [11], and find the maximum contribution

. Set a value for the minimum load factor, and compare the contribution of each

load bus with the value of . Ifthe contribution of the load is lower than this value, thatload bus is added to the insignificant buses associatedwith the channel.

• Compute the contributions of generator buses tothe channel, and find the maximum contribution

. Set a value for the minimum gener-ator factor , and compare the con-tribution of each generator bus with the value of

. If the contribution of thegenerator is lower than this value, that generator bus isadded to insignificant buses associated with the channel.

4) Compare the insignificant buses obtained for all the criticalchannels. Those buses which are common in all the crit-ical channels are selected as the final insignificant buses.There is no need to install any PMU at these buses. Theremaining load/generator buses are considered as the loca-tions in which PMUs need to be installed.

With the above PMU allocation, the voltage/current of in-significant buses will no longer be available. Therefore, the ap-plication of the transform matrix to get the channel quantities ismodified as follows:• For each insignificant load bus, consider the load current as

, where is the nominal load power (the loadpower at the base case). In fact, the voltage is approximatedby 1 p.u., and the power at the current operating conditionis approximated by the nominal power.

• For each insignificant generator bus, approximate the busvoltage by 1 p.u., i.e., .

• Calculate the channel currents and the channelsource voltages .

• Calculate the channel voltages .

V. CASE STUDY RESULTS

In this section, the methods proposed for identifying the crit-ical generator and the critical branch are verified by using sev-eral case studies. For this purpose, three standard test systemsincluding the 30-bus system from [17], the IEEE 30-bus system,and the IEEE 57-bus system along with an actual large system[the 2038-bus Alberta Integrated Electric System (AIES)] areconsidered. The proposed algorithm for allocation of PMUs isalso applied to the AIES, and the obtained results are discussedin detail.

A. Critical Generator IdentificationHow to verify the results obtained for the critical generator

is challenging because as mentioned in Section II, determiningthe impacts of the generators’ reactive powers on the voltagestability by using the currently available methods is difficult.However, to determine if the results are reasonable, a sensi-tivity-based method [21] is used, and its results are comparedwith those of the proposed method. This method is as follows:The sensitivity of the loadability margin with respect to

the generator voltages is used to rank the generators. For thispurpose, for each generator, a very small increase (say 1%) inits voltage magnitude is made, and the increased system marginis calculated using the continuation power flow (CPF) [22].This process is performed for all the generators and they areranked according to their associated increases in the margin.This method is based on the generator reactive power’s closerelationship with its voltage magnitude. The higher the voltagemagnitude is, the higher the produced reactive power is likelyto be.The proposed generator ranking method is applied to the case

studies, and the obtained results are compared with those of thesensitivity-based method. Since the critical load is also requiredlater on to find the critical branch, the load-ranking results arealso presented for the case studies.• IEEE 30-bus systemThe first case study is the IEEE 30-bus system. By applying

the CCT to this system and calculating the proposed index forthe generator ranking, Fig. 6(a) is obtained. Fig. 6(b) illustratesthe results obtained by the sensitivity-based method. As thesefigures show, the critical generator identified by the proposedindex, which is the generator connected to bus 8, is verified bythe sensitivity-based method.By applying the load ranking method proposed in [11] to the

IEEE 30-bus system, Fig. 7 is obtained. According to this figure,bus 30 is identified as the critical load in this system.• AIES 2038-bus systemThe Alberta Integrated Electric System (AIES), which is an

actual 2038-bus system, is considered as the next case study.This system’s buses are renumbered from 1 to 2038 for the easeof reference. Figs. 8(a) and (b) show the generator-ranking re-sults obtained for this system. According to these figures, thegenerator connected to bus 123 is identified as the critical gen-erator using both the proposed method and the sensitivity-basedmethod. Fig. 9 shows the load ranking results in this system. Asthis figure reveals, bus 630 is identified as the critical load.Two other test systems including the 30-bus system from

[17], and the IEEE 57-bus system have also been analyzed. To


Fig. 6. Generator ranking for IEEE 30-bus system. (a) Using the proposedmethod. (b) Using the sensitivity-based method.

Fig. 7. Load ranking for IEEE 30-bus system.

save space, the detailed results of these systems are not shownhere. The results are, however, summarized in Table I. This tableindicates that the generator rankings obtained by the proposedmethod are totally reasonable and acceptable. The results of theload identification are also presented in Table I as they will beused in Section V-B for the critical branch identification.As mentioned before, the sensitivity-based method is not a

standard method for determining the impacts of generators’ re-active powers on the voltage collapse. Due to the lack of sucha method, the active participation factor (APF), which is basedon the impacts of generators’ active powers, has been used inthe literature ([14]–[16]) for optimal reactive power planning.However, as was expected, by applying this ranking method tosome test systems, we have found out that it cannot be usedto determine the impacts of generators’ reactive powers. As anexample, Fig. 10 shows the results of this method in the IEEE30-bus system. The comparison of this figure with Fig. 6(b)clearly reveals that APF is not a good method for determiningthe impacts of generators’ reactive powers. The implication isthat by using the proposed ranking method instead of APF, theexisting methods of reactive power planning on the generationside may be improved significantly.

Fig. 8. Generator ranking for AIES 2038-bus system. (a) Using the proposedmethod. (b) Using the sensitivity-based method.

Fig. 9. Load ranking for AIES 2038-bus system.

TABLE ISUMMARY OF THE RESULTS FOR CRITICALGENERATOR AND LOAD IDENTIFICATION

B. Critical Branch Identification

Similar to the method used in the previous section, a sen-sitivity-based method is used to verify the results obtained bythe proposed method for the critical branch identification. Forthis purpose, the sensitivity of the loadability margin with re-spect to the branch reactance is used. In this method, the re-actance of a branch is reduced by a small percent and the in-creased loadability margin is obtained using the continuationpower flow. This process is carried out for all the branches and


Fig. 10. Generator ranking in IEEE 30-bus system obtained by APF.

Fig. 11. TPSI of the transmission paths.

they are ranked. The critical branch is the one which results inthe highest margin increase. This method is based on the find-ings of [20], which show that the sensitivity of the loadabilitymargin to system line reactances can be used to find the mostcritical transmission lines.Four systems are considered as case studies, and the perfor-

mance of the proposed method is investigated.• IEEE 30 bus systemAs seen in the previous section, by applying the CCT to the

IEEE 30-bus system, the critical load and generator are identi-fied as follows.— Critical load bus: bus 30— Critical generator bus: bus 8When the system is close to the maximum loadability point,

there are four transmission paths from the critical generator tothe critical bus:— Path 1: 8-28-27-30— Path 2: 8-6-28-27-30— Path 3: 8-28-27-29-30— Path 4: 8-6-28-27-29-30The TPSIs of the transmission paths are illustrated in Fig. 11.

According to this figure, path 2 (8-6-28-27-30) is identified asthe critical path. This path is shown on the system’s single linediagram in Fig. 12.The critical path has four segments: segment 1: 8-6, segment

2: 6-28, segment 3: 28-27, and segment 4: 27-30. The correctedvoltage drops of these segments are shown in Fig. 13. As thisfigure reveals, segment 3 (27-28) is identified as the criticalsegment.Fig. 14 illustrates the results of the verification method. Ac-

cording to this figure, the verification method identifies segment27-28 as the critical branch. This result is the same as the resultof the proposed method.

Fig. 12. Single line diagram of the IEEE 30-bus system.

Fig. 13. Corrected voltage drops of the critical path’s segments.

Fig. 14. Results of the verification method.

• AIES 2038-bus systemBy applying the CCT to this large system, the following re-

sults were obtained for the critical load and generator.— Critical load bus: bus 630— Critical generator bus: bus 123There is only one path from the critical generator to the crit-

ical load. So this path, which is shown below, is the critical path.Critical Path: 123-1049-1029-1063-1316-630The critical path consists of five segments including segment

1: 123-1049, segment 2: 1049-1029, segment 3: 1029-1063,segment 4: 1063-1316, and segment 5: 1316-630. Fig. 15 shows


Fig. 15. Corrected voltage drops of the critical path’s segments.

Fig. 16. Results of the verification method.

TABLE IISUMMARY OF THE RESULTS FOR THE CRITICAL BRANCH IDENTIFICATION

the corrected voltage drops of the segments. As this figure re-veals, segment 5 is the critical segment. Therefore, the criticalbranch in this system is the branch 1316-630.Fig. 16 shows the results of the sensitivity-based method.

According to this figure, branch 1316-630 leads to the highestsensitivity. As a result, this branch is the critical branch in thesystem, verifying the results obtained by the proposed method.The above results along with the results obtained for the

30-bus system and the IEEE 57-bus systems are summarizedin Table II. As this table shows, the results of the proposedmethod are verified by the verification method in all the casestudies. This finding indicates that the proposed method istotally acceptable, and can be used as a simple and fast methodfor the critical branch identification.Note that in order to apply the sensitivity-based method for

branch/generator ranking, the CPF needs to be performed manytimes (one time for each branch/generator). Each run of the CPFitself consists of multiple runs of power flow. Therefore, the sen-sitivity-based method might be very time-consuming. On thecontrary, the proposed generator/branch ranking methods arevery fast as we only need to perform power flow (or CPF) onlyonce, and apply the transformation procedure [11], which is a

TABLE IIICOMPUTATIONAL TIME FOR CRITICAL BRANCH IDENTIFICATION

MATLAB 7.6.0, CPU: I7-3.33 GHz, RAM: 12.0 GB (powerflow has been performed using MATPOWER3.2 [25]).

fast procedure. As an example, Table III compares the compu-tational time of the proposed critical line identification methodwith that of the sensitivity-basedmethod in the IEEE 30-bus andIEEE 57-bus system. Note that the system is scaled up to a pointclose to the last loadability point first and is considered as thebase case for both methods. The computational time providedfor the proposedmethod includes the whole CCT procedure [11]consisting of the eigen-decomposition, the power flow, and theidentification of critical channel, critical load, critical generator,and critical branch. In online applications of the CCT, the powerflow, which is the most time-consuming part of the above pro-cedure, is not required.

C. PMU Allocation

As an illustrative example, the strategy proposed for the PMUallocation is investigated in the 2038-bus AIES system. Thefirst step is to set some predetermined values for the parameters

, and . It is suggestedto use a relatively large value for . For instance,

is set to 40% in our example. In other words, allthe channels whose margins are less than 40% when the actualsystem is close to the nose point are considered as the criticalchannels. This large value assures that if the system topologychanges, the critical channel will not be missed.The values of , and significantly

affect the number of load and generator buses chosen forthe PMU locations. The larger the threshold values are, thesmaller the number of PMUs is. On the other hand, when thenumber of PMUs decreases, the accuracy might also decrease.Therefore, a trade-off occurs between the number of PMUsand the accuracy of the monitoring and analysis. To chooseproper values for these thresholds, a sensitivity analysis shouldbe performed. The effects of different threshold values on theaccuracy should be examined, and, with respect to the desiredaccuracy, a decision should be made. The sensitivity analysishas shown that the following thresholds work well for the AIESsystem: , and .By using the above predetermined values, the proposed

strategy is applied to the AIES system, and the results arediscussed below.Three channels (channels 384, 18, and 425) are determined as

the critical channels. This system has 684 load buses, and 205generator buses. By applying the PMU allocation strategy, thefollowing results are obtained.— Number of significant load buses: 108—Number of significant generator buses: 13—Number of insignificant load buses: 576—Number of insignificant generator buses: 192


Fig. 17. Comparison of the critical channel margins with and without the PMUallocation. (a) Critical channel margins. (b) Margin differences.

The implication is that 108 load buses and 13 generator busesare selected for the PMU locations. There is no need to installany PMU on the other (insignificant) buses.To check the effect of this allocation on the CCT, the CCT-

based offline analysis [11] is applied to the system, but the cur-rents and voltages of the insignificant buses are not used. In-stead, the modified procedure suggested in Section IV is usedto calculate the channel quantities. The obtained results showno noticeable error in the critical channels. The curvesof the critical channels are very close to those in the originalcase when all the buses are used. As an example, Fig. 17(a)shows the critical channels’ margins. In this figure, the solidlines are for the case in which all the load/generator bus data(currents and voltages) are used, and the dash lines are for thecase in which only the data of the significant buses are used. Asthis figure shows, the margins with and without the allocationstrategy are very close to each other for each critical channel.This result is confirmed by Fig. 17(b), which shows the differ-ence in channel margins. According to this figure, the error isalways below 0.6% indicating a high accuracy. Figs. 18 and 19illustrate the contribution of the load buses and generator busesto channel 18 for both cases. These figures reveal that when thePMU placement strategy is used, no change in the ranking ofthe top rank loads/generators is made. The implication is thatby using the allocation strategy, the CCT-based analysis can beaccurately applied.Note that as mentioned before, one may choose to use smaller

values for , and . The results will bemore accurate in this case, but a larger number of PMUs willbe required. On the other hand, larger values could be chosen

Fig. 18. Load bus contributions to channel 18 with and without using the PMUallocation strategy.

Fig. 19. Generator bus contributions to channel 18 with and without using thePMU allocation strategy.

for the thresholds. Doing so would further reduce the numberof PMUs but might increase the analysis error.

D. Effects of Contingencies on the Proposed Strategy

The proposed PMU allocation strategy is based on the cur-rent topology of the system. The system needs to be accuratelyanalyzed even if a change occurs in its topology. Therefore, dif-ferent contingencies need to be considered when PMUs are allo-cated. For this purpose, an approach is suggested in this section.First, a contingency scan should be performed to identify the

critical contingencies which have the highest impacts on thevoltage stability margin. For this purpose, each branch is setout of service, and the resulting system margin is calculatedusing CPF. This process is performed for all the branches oneby one, and they are ranked based on their associated margins.The lower the margin is, the more critical the branch is.The proposed PMU allocation strategy needs to be applied

to the base case first. The allocation strategy is then performedfor each critical contingency one at a time. The results of all thecontingencies are compared with those of the base case at theend. Two different outcomes might occur:• The results are very similar. In other words, the significantload/generator buses obtained for each contingency are al-most the same as those of the base case. In this case, the


Fig. 20. Significant generator buses.

significant buses obtained for the base case can be used asthe final PMUs locations.

• The results are different. In other words, the significantload/generator buses obtained for each contingency are dif-ferent from those of the base case. In this case, all the sig-nificant buses of all contingencies and the base case shouldbe considered as the final PMUs locations.

In theory, both of the above cases are possible. In the actualcases, however, the first one is more likely to occur. As an ex-ample, the above procedure is applied to the AIES system, andthe results are discussed in the following.The top eight critical contingencies are considered in the pro-

cedure. For each contingency, the contingency is applied to thesystem and the offline CCT-based analysis is performed. Theproposed allocation strategy is then applied to the system andthe significant generator/load buses are obtained. Figs. 20 and21 illustrate the significant generator and load buses for the basecase and for the contingencies. In these figures, each significantbus is displayed by a red square. Cases 1 to 8 refer to the eighttop contingencies. As these figures show, the buses obtained forthe base case cover almost all the buses for all the contingen-cies. In other words, most of the significant buses obtained forthe contingencies belong to those obtained for the base case.Note that the analyses of many more contingencies showed thesame result, but for the ease of illustration, the results of onlythe top eight contingencies have been shown in this paper. Theabove findings show that by allocating PMUs at the base case,the AIES system can bemonitored even if a contingency occurs.As mentioned before, it might be possible in another system

that the significant buses obtained for contingencies be differentfrom those of the base case. Although this result in unlikely, itmight happen, especially if a severe contingency changes thefirst critical channel to a channel which has not been selectedas a critical (monitored) channel in the second step of the PMUallocation strategy (See Section IV). If this happens, one of thefollowing alternatives may be followed.• The significant buses of the base case are considered as theinitial list for the PMU locations. Those buses which be-long to the significant buses of those different contingen-cies, and are not already in the initial list, are then addedto the PMU location list. Adding a relatively small numberof new buses to the list would not be a cause for concern.However, if we end up with a long list consisting of toomany buses, the obtained results might not be practical in

Fig. 21. Significant load buses.

the near future due to the limited number of PMUs avail-able in a power system.

• The PMU allocation strategy can be repeated again byusing a larger value for the . By doing so,more channels are selected as the critical channels whenallocating PMUs. In other words, the allocated PMUswill be able to monitor a larger number of channels.Obviously, this approach may result in a larger number ofrequired PMUs. However, the chance that a contingencycould change the main critical channel to a non monitoredchannel would decrease. In other words, the likelihoodthat the base case’s significant buses would cover thoseof the contingencies would increase. If this approach isto be followed, one may need to do a sensitivity analysisto check the effect of different values, anddetermine a proper value.

VI. CONCLUSIONS

The Channel Components Transform (CCT) is a networkdecoupling transform for analyzing interconnected powersystems. This paper further studied the CCT and extendedits applications. CCT-based methods to rank generators andbranches in terms of their impacts on the voltage stability wereestablished. Multiple case studies confirmed the validity of themethods. In addition, a strategy based on tracking the behaviorsof significant channels was proposed to allocate PMUs. Theresults have shown that by installing PMUs at limited numberof buses, the voltage stability modes can be accurately tracked,and the critical load/generator buses can be identified. Notethat the proposed method, like the PV curve technique, worksfor systems in steady-state. It cannot track voltage collapse andpredict the critical components when a system is experiencingdynamic variations (i.e., not in steady-state). More researchwork is needed to apply the transform for analyzing the dy-namic phasor data.


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Iraj Rahimi Pordanjani (S’09) received his B.Sc. degree (with First ClassHonors) and M.Sc. degree in Electrical Engineering from Amirkabir Universityof Technology (Tehran Polytechnic), Tehran, Iran in 2005 and 2008, respec-tively. He is currently pursuing his Ph.D. program in Electrical and ComputerEngineering at University of Alberta, Canada. His research interests are powersystems stability and power quality.

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 En-gineering, University of Alberta, Edmonton, Canada. His research interests arepower quality and power system stability.

Wilsun Xu (M’90–SM’95–F’05) received the Ph.D. degree from the Univer-sity of British Columbia, Vancouver, BC, Canada, in 1989. He was an Engi-neer with BC Hydro, Burnaby, BC, Canada, from 1990 to 1996. Currently, heis a Professor and a NSERC/iCORE Industrial Research Chair at the Univer-sity of Alberta, Edmonton, Canada. His research interests are power quality andvoltage stability.

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