A Network Decoupling Transform for Phasor Data Based Voltage Stability Analysis and Monitoring

Transcription

1 A Network Decouplng Transform for Phasor Data Based Voltage Stablty Analyss and Montorng Wlsun Xu, Fellow, IEEE, Iraj Rahm Pordanjan, Student Member, IEEE, Yunfe Wang, Student Member, IEEE, Ebrahm Vaahed, Fellow, IEEE Abstract It s well known that a power network can be represented as a mult-node, mult-branch Thevenn crcut connectng loads to generators. Ths paper shows that egendecomposton can be performed on the Thevenn mpedance matrx, creatng a set of decoupled sngle-node, sngle-branch equvalent crcuts. The decoupled crcuts can reveal mportant characterstcs of a power system. By applyng the transform to calculated or measured voltage phasor data, a technque for trackng the modes of voltage collapse and for dentfyng areas vulnerable to voltage collapse has been developed. Case studes conducted on multple power systems have confrmed the effectveness of the proposed method. In addton to voltage stablty applcatons, the proposed transform presents a new approach for processng and nterpretng mult-locaton phasor data. Index Terms Egen-analyss, Thevenn crcut, Voltage stablty. D I. INTRODUCTION ue to the rapd advancement of measurement and telecommuncaton technologes, a large amount of data are avalable nowadays for power system montorng and control. One example s the synchronzed phasor data []. Varous research works have been conducted to develop applcatons for the phasor data []-[4]. For example, reference [5] has documented the latest attempts to create stuaton awareness for power system planners and operators usng the wde-area phasor data. The challenges of extractng new and unque nformaton from the phasor data may be partally due to the lack of a support theory for phasor data processng and nterpretaton. Ths stuaton may be understood by examnng the use of three-phase voltage phasor data, V a, V b and V c. One can process the data n varous ways. However, operatons such as (V a +V b +V c )/3 (=V zero-sequence ) or (V a +a V b +av c )/3 (=V negatvesequence) (where a= o ) have been recognzed as the best means to analyze the data. The symmetrcal components transform s the support theory for these operatons. Because of the theory, convertng abc phasors to sequences has become a standard approach to process three-phase phasor data and a number of montorng and protecton schemes have been developed. The wde-area montorng systems nowadays have made mult-locaton (postve sequence) voltage phasor Ths work was supported by Natural Scences and Engneerng Research Councl of Canada and a number of utlty companes ncludng BC Hydro, Alberta Electrc System Operator and other Alberta power companes. The frst three authors are wth the Department of Electrcal and Computer Engneerng, Unversty of Alberta, Edmonton, Alberta, T6G V4, Canada (emal: wxu@ualberta.ca). Dr. E. Vaahed s wth BC Hydro. data, V, V, V 3 V n avalable. Inspred by the success of the symmetrcal components theory, one would wonder f operatons such as T V +T V + +T n V n can be rgorously derved for the mult-locaton (.e. mult-bus) phasor data, and f such operatons can reveal unque characterstcs of a power system. Ths paper shows that such a transform can be derved. The transform s able to extract mportant nformaton about a power system from the phasor data. The transform s conceved from the followng observaton: a power network can be represented as a mult-node, mult-branch Thevenn crcut connectng the loads to the generators. If one apples egen-decomposton on the Thevenn mpedance matrx, the network can be decoupled nto a set of sngle-node, snglebranch equvalent crcuts. These crcuts are much easer to analyze and they carry valuable nformaton of a power system. Smlarly, f the varatons of the transformed varables can be evaluated, one may be able to predct the complex behavors of the actual network. Ths paper presents the basc theory and characterstcs of the proposed transform. The transform s ndependent of the orgn of the phasor data. The data can be ether calculated from load flow programs or measured by phasor measurement unts. In ths paper, the theory s appled to voltage stablty analyss and montorng. Algorthms for voltage nstablty mode dentfcaton are developed. The proposed applcaton s a sgnfcant mprovement over the Jacoban matrx based voltage stablty modal analyss technque [6]-[]. II. PROPOSED NETWORK DECOUPLING TRANSFORM A general power system s shown n Fg.. Ths system conssts of n loads and m generators. There are many transmsson lnes and other components such as transformers nsde the network. Ths network (n postve sequence form) can be represented usng a generalzed mult-node, multbranch Thevenn equvalent crcut model. The equvalent crcut has the followng form: V k k. k E Z Z. Z I m n V k... E Z... I = V k k. k E Z Z. Z I n n n nm m n n nn n Or [ V ] = [ K][ E] [ Z][ I ] = [ E '] [ Z][ I ] () In ths model, [E] s the termnal voltages or the nternal voltages (f a generator s Q max s reached) of the generators and [V] s the nodal voltages at the load buses. Note that

2 Fg. 3. Modal doman representaton of a complex network. Fg.. A general electrc power network. E E E n V E ' Z Z. Z n I V E' Z... I = V E ' Z Z. Z I n n n n nn n Fg.. Mult-node, mult-branch Thevenn crcut. P +jq P +jq P n +jq n [E ]=[K][E] s the open crcut voltage vector of the tradtonal mult-node Thevenn equvalent crcut. A general crcut model for the representaton of () s shown n Fg.. Egen-decomposton can be performed on the [Z] matrx of the Thevenn crcut as follows: [ Z] = [ T] [ Λ ][ T] () where [Λ] and [T] are the egenvalue and egenvector matrces of [Z], respectvely. Applyng the above to () yelds [ V] = [ K][ E] [ Z][ I] = [ K][ E] [ T] [ Λ ][ T][ I] [ T][ V] = [ T][ K][ E] [ Λ ][ T][ I] (3) Denote [ U] = [ T][ V] as the transformed voltage [ J] = [ T][ I] as the transformed current [ F] = [ T][ K][ E] as the transformed voltage source Ths leads to the followng decoupled (modal) networks whose crcut representatons are shown n Fg. 3. U F λ J U F λ J = (4) Un Fn λn Jn The sgnfcance of the above transform s the followng: a complex network has been transformed nto a set of decoupled smple one-source, one-load networks. By analyzng the characterstcs of ndvdual decoupled networks, one may extract mportant nformaton about the actual network. The above transform s essentally one form of modal (egen) decomposton that has been wdely used n lnear system analyss []. In the power system feld alone, at least three modal transforms have been developed: () modal analyss on the state matrx [A] of a lnearzed dynamc power system [], () modal analyss on the load flow Jacoban matrx [6], and (3) modal analyss on the mpedance matrx of multphase transmsson lnes []. In fact, the symmetrcal components transform s also a form of modal analyss. In order to avod confuson wth those transforms, we propose to use Channel Components Transform (CCT) to desgnate the transform of (). The transformed doman s called the Channel Doman. Ths s based on the consderaton that each modal crcut represents a vrtual power flow channel or (a) physcal network Fg. 4. The one-source three-load system. V E Z+ Z Z Z I a a a V b E Z Z Zb Z I = + b V c E Z Z Z+ Z c I c (b) network equaton path (.e. mode). A power network can be vewed as consstng of varous vrtual power flow channels (paths) defned by [U], [J] and [F]. A channel component represents one pattern of currents flowng n a network. A very smple power system shown n Fg. 4 s used to llustrate the meanngs of the channel components. If Z a =Z b =Z c, the system can be transformed to 3Z + Z 3 a E [ Λ ] = Z a, and [ F ] =. Z a The results reveal that only channel has a voltage source,.e. channel s the only channel responsble for power transfer n ths case. In fact, the channel mpedance λ =3(Z+Z a /3) represents the seres connecton of Z branch wth that of a parallel combnaton of Z a, Z b and Z c branches. Ths s exactly what one wll do when analyzng the crcut of Fg. 4 on the back of an envelope. There s no need to analyze the other channels. So, a four-branch power network has been smplfed nto a one-branch network. If Z a, Z b, and Z c are not equal, such as Z a =.35, Z b =., Z c =., Z=j.3, E =., S a =.3, S b =.5, and S c =.4, the CCT yelds the followng parameters..65j.764 [ Λ ] =.78j, and [ F ] = j.566 It can be seen that all channels have non-zero source voltages. The channel mpedances are also dfferent. Snce the power transfer capablty of a lne s proportonal to V and nversely to X, channel s the mode responsble for transferrng power to loads. Table I, whch shows the channel currents n each branch of the system, reveals more nformaton on how the power s transferred to the loads. Fg. 5 shows the patterns of channel currents n the system. In ths fgure, the thckness of an arrow on a branch s n proporton to the value of the channel current flowng through that branch. As seen n the table and fgure, the Channel currents are all n phase, flowng from the source to the loads. So ths s the channel responsble for power transfer. The channel current mostly flows from bus 3 to buses 4 and 5, and the channel 3 current

3 3 TABLE I CHANNEL CURRENTS IN EACH BRANCH OF THE TEST SYSTEM Branch Current From To Channel Channel Channel 3.<-.3.89<-.6.69< < <77.4.4< <-.3.37< < <-.3.47<-.6.56<-8.4 Channel Channel Channel 3 Fg. 5. Channel currents n each branch of the test system. mostly flows from buses 3 and 4 to bus 5. These are manly load-to-load power flows that don t help transferrng power from source to loads. In summary, the channel currents can help to dentfy the man paths (or patterns) of power transfer n a power system. Each channel transfers dfferent amount of power, defned as U Conj(J ) at the recevng end. One can calculate the channel power at any gven network operatng ponts establshed by the load flow results. Fg. 6 shows the channel power levels of an actual 38-bus power system. The results show that only a small number of channels are responsble for transferrng the majorty of power n a network. One may only need to analyze or montor a small number of channels to understand the characterstcs of a power system. The computatonal procedure to form the equvalent crcut [Z] and [K] matrces s shown below. The standard node equatons n the matrx notaton are expressed as: [ I] = [ Y][ V] (5) where, [V] s the bus voltages, [I] s the net njected current, and [Y] s the admttance matrx. All buses can be classfed nto three types: generator bus (G), load bus (L), and network bus (N) whch has no generator or load. As a result, equaton (5) can be parttoned as Fg. 6. Dstrbuton of channel powers n a large system (P base =MW). IG YGG YGL YGN VG IL = YLG YLL YLN VL I N YNG YNL Y NN V N Snce the net njected currents of the network buses are equal to zero, we can wrte IN = YNGVG + YNLVL + YNNVN = Or : VN YNN (7) = ( YNGVG + YNLVL) Substtutng the obtaned V N n (6) wll result n IL = YLGVG + YLLVL YLNYNN( YNGVG + YNLVL) (8) Or : IL = ( YLG YLNYNNYNG) VG + ( YLL YLNYNNYNL) VL Rearrangng (8) yelds VL = ( YLL YLNYNNYNL) ( YLG YLNYNNYNG) VG (9) ( YLL YLNYNNYNL) IL Comparng (9) wth () reveals that [ K] = ( YLL YLNYNNYNL ) ( YLG YLNYNNYNG ) () [ Z] = ( YLL YLNYNNYNL ) The above equaton gves the expresson of matrces [K] and [Z]. Standard egen-decomposton routnes can then be appled to the [Z] matrx. III. PV AND Pδ CURVES IN CHANNEL DOMAIN Ths paper presents the applcaton of CCT to voltage stablty analyss and montorng. Mantanng voltage stablty s a major objectve n power system plannng and operaton [3]. Although our understandng on the subject has ncreased sgnfcantly n recent years, new fndngs are stll emergng, especally wth the emergence of measured phasor data. Many research actvtes have been conducted on how to utlze the phase nformaton for voltage stablty montorng [4]-[5]. The technque proposed n ths paper attempts to help dentfyng weak areas n a planned or operatng power system. Before presentng the complete algorthms, the characterstcs of PV and Pδ curves n the channel doman are explaned frst. The process to map PV curves to the channel doman s as follows: A PV curve calculaton procedure such as the contnuaton power flow s appled to the study system. At each PV curve pont obtaned by ths process, the CCT s appled to calculate the channel varables correspondng to that pont. The process wll produce a set of data ponts n the channel doman. The channel voltage versus channel power (P~V) or channel power versus channel angle (P~δ ) are plotted. The curves are called channel doman PV or Pδ curves. A. Channel Doman PV Curve Snce the channel crcut s a very smple one-branch crcut, one can actually derve the entre PV curve n channel doman analytcally, as follows: For a channel shown n Fg. 7, the relatonshp between the (6)

4 4 channel voltage (U ), and the channel load power (P, and Q ) s gven as 4 λ U + U [( PR + QX ) F ] + ( P + Q ) = () Usng (), the channel voltage can be plotted for dfferent actve powers assumng a constant power factor for the channel load. The resultng curve wll be the channel PV curve. The channel PV curves and mapped physcal PV curve ponts for the conceptual case shown n Fg. 4 when Z a =.35, Z b =., and Z c =. s llustrated n Fg. 8. There are three groups of channel curves. Each group corresponds to one channel as ths s a three-channel system. The stars are the PV curve ponts of the actual system mapped to the channel doman. The reason that each channel has a group of curves s due to the followng phenomenon. Two varables, the source voltage and the branch mpedance, defne the shape of a PV curve for a sngle-branch network accordng to (). In the channel doman, the channel mpedance λ s constant but the source voltage F changes wth the physcal voltages of the system. Each channel PV curve therefore corresponds to dfferent source voltage F. The results shown n Fg. 8 confrmed the observaton earler that each channel carres dfferent amount of power. One can also notce that voltage collapse occurs when one of the channels (channel ) reaches ts maxmum power transfer lmt, whch can be called the crtcal channel. B. Channel Doman Pδ Curve Voltage stablty analyss s usually performed usng the PV curves. But t can also be examned from the Pδ curves, at least for the sngle-load system. Snce ndustry s ncreasngly nterested n the angle nformaton provded by the PMUs, t s useful to examne voltage nstablty from the Pδ curve. However, there are n (n-) angle dfferences for a n-bus power network. It s a bg challenge to examne the angle nformaton on a PV curve trajectory. The CCT offers a natural soluton to the problem snce one can easly study the angle nformaton n the channel doman. Consder the th channel crcut as shown n Fg. 7. For the sake of smplfyng equaton dervaton, let s assume that the channel mpedance s purely reactve.e. λ = j X. The followng power angle equaton can therefore be developed for ths channel crcut. FU P = snδ () X Snce U vares wth P whch n turn vares wth δ, () cannot be used as the Pδ relatonshp. However, ths equaton can be rewrtten as follows. F U λ = R + jx Fg. 7. The th channel crcut. S = P + jq.5.5 Channel Fg. 8. Channel PV curves of the case study. Channel Channel PX U = Fsnδ (3) FU U Snce Q = cos δ (4) X X Substtutng (3) nto (4) yelds: XP F snδcosδp + F sn δ Q = (5) Dvdng (5) by P yelds: XP F snδcosδ + F sn δtanθ = (6) where, θ s the power factor angle of the th channel load and s kept constant accordng to the PV curve approach. P as a functon of δ can be solved from the above equaton as: F = sn cos sn tan X P δ δ δ θ (7) Equaton (7) s the analytcal expresson of the Pδ curve for the th channel. Snce the channel crcuts are decoupled, establshng Pδ curve for each channel s straghtforward. Fg. 9 shows the channel Pδ curves of the same case study. As seen n ths fgure, when the actual system reaches the voltage collapse pont, the Pδ curve of channel (crtcal channel) reaches ts maxmum pont. The maxmum angle s about 5 o. It can be shown that for the system of Fg. 7 where the recevng end voltage s uncontrolled, the δ max can be determned as snθ snδmax- = (8) whch s always less than 9 o due to the lack of reactve power support at the recevng end. In summary, the channel Pδ curves can be used to nterpret the voltage stablty of a power system. Wthout the proposed transform, t s mpossble to obtan any meanngful Pδ curves snce there are many physcal bus angles n an actual power system. Whch bus angle dfferences to examne are dffcult to determne..5.5 Channel 5 Channel Delta (deg.).6.4. Channel 5 Channel Delta (deg.) Fg. 9. Channel Pδ curves of the case study Channel 3 5 Channel Delta (deg.)

5 5 IV. VOLTAGE STABILITY ANALYSIS USING CCT The proposed Channel Components Transform s appled to voltage stablty analyss n ths secton. A set of algorthms are proposed for dentfcaton of the crtcal channel and assocated load buses called crtcal buses. The crtcal buses are those whose load demands cause voltage collapse. A. Load couplng n channel doman The advantage of CCT s that t can decouple the supply system nto ndependent channel crcuts. When t s appled to the loads (whch are physcally decoupled), however, the channel loads become coupled. Ths s because the CCT s a lnear transform. A load (Z or S) s a nonlnear varable. Ths phenomenon can be further understood as follows: The physcal loads may be modeled as constant power loads or varable mpedance loads that produce the effects of constant power consumptons. Snce t s easer to explan the concepts, the mpedance load model s used. The physcal load at varous buses can be expressed as V ZL.. I V ZL.. I Load : = (9) Vn.. ZLn In where, Z L satsfes load s power demand constrants. In the channel doman, the loads become: [ V] = [ Z ][ I] [ U] = [ T][ Z ][ T] [ J] [ U] = [ Z ][ J] L L C U Z Z.. Z J C C Cn U ZC ZC.. ZCn J Channel loads : = U Z Z.. Z J n Cn Cn Cnn n () The above equaton shows that the loads are coupled n the channel doman. Ths wll create dffcultes n dervng useful nformaton from the channel PV or Pδ curves. For example, the mapped channel operatng pont may resde below the channel PV curve, whch makes t hard to determne the crtcal channel. Based on the analyss presented n Appendx A, the couplng term of the th channel load conssts of the F j, λ j components (where j ). As a result, each channel load s actually a Norton crcut, whch s shown n Fg.. For any gven operatng pont, the couplng can be represented as a Norton current source. The current source s determned from: JE = J YCU () where Y C s the th dagonal element of [Y C ] = [Z C ] -. In other words, the couplng has been modeled as a current source J E. Ths current source representaton s accurate for the operatng pont from whch t s derved. Ths known current source s then merged nto the supply channel crcut as shown n Fg.. Ths results n the change of the channel voltage source from F to Feq = F λ JE. In summary, the couplng effects have been represented as a modfcaton to the channel voltage source F. Ths s an accurate model for the pont from whch the parameters are derved. It s an approxmaton for other ponts. Extensve case studes presented later have shown that ths approxmaton works very well. One of the outcomes s that whenever a physcal system reaches ts nose pont, one of the channels wll approach ts nose pont as well. Ths channel becomes the crtcal channel. In summary, the proposed method to construct the fnal channel crcut s as follows: Step : Calculate the load mpedance for each physcal bus, whch s Z L =V /I. Step : Obtan [Y C ] usng (). Note that the computaton s mnmal as [T] - s already avalable. [ Y ] = [ Z ] = ([ T][ Z ][ T] ) = [ T][ Z ] [ T] () C C L L Step 3: Calculate the current source J E for each channel accordng to (). Step 4: Calculate the modfed channel voltage source Feq = F λ JE for each channel. The result s the equvalent channel crcut shown n Fg.. Wth the above treatment, all channels are decoupled from each other. Therefore, t becomes easy to examne the PV curves from the channel doman. For ths purpose, equaton () can be appled to each of the equvalent channel crcuts. B. Identfcaton of crtcal channel and crtcal buses Among dfferent channels of a system, there s a crtcal channel responsble for the voltage collapse. Ths channel can be easly dentfed based on the channel s margn,.e. the crtcal channel s the one that has the smallest channel margn. Snce each channel resembles a two-bus system as shown n Fg., the maxmum channel power can be determned analytcally as follows: S max = eq λ θ + θ Xcosθ Rsnθ ( sn cos ) F X R F λ U J Modfed equvalent crcut (3) where θ s the power factor angle of the th channel load. The channel margn can then be calculated as follows: Smax Soperartng Channel margn = (4) Soperartng where S operatng s the channel load power at the current operatng condton. J E Fg.. The th channel crcut wth the modfed load. F eq =F λ JE U Fg.. The equvalent crcut for the th channel. λ S C Channel load

6 6 Once the crtcal channel s known, one can use the nformaton to dentfy the crtcal buses or to rank the buses. The method proposed for ths task s as follows. Snce the low stablty margn of the crtcal channel corresponds to ts large channel voltage drop, one can therefore nfer that the crtcal bus s the one that s the most responsble for the voltage drop of the crtcal channel. Snce the channel voltage drop s caused by the channel current, the contrbuton of bus currents I to the crtcal channel current J can be used to rank the contrbuton of dfferent buses to the channel voltage drop. The crtcal channel current J s as follows: J = TI + TI + + T I (5)... N N Accordng to (5), the followng contrbuton ndex s proposed: TI k k cos( αk ) Contk = (6) J where Cont k s the contrbuton of load bus k to the channel current J and α k s the angle dfference between the channel current J and the term T k I k. To determne the crtcal bus, the contrbutons of all load bus currents to the crtcal channel current are calculated usng (6) and are ranked. The bus whose current has the hghest contrbuton to the crtcal channel current s the crtcal bus. C. Summary of the proposed algorthms The CCT-based (off-lne) voltage stablty analyss technque s summarzed n the flowchart of Fg.. As seen n ths fgure, the transformaton matrx [T] and the channel mpedances [Λ] are frst calculated based on the network 7 Scale up the loads and generators (PV curve calculaton procedure) Obtan [Z] and [K] based on network Y matrx Compute the transform matrx [T] and channel mpedances [Λ] Perform power flow Power flow Converged? Yes Apply the transform on [V], [I], and [E], and obtan the channel quanttes [U], and [F eq] Compute channel margns Identfy the crtcal channel No Obtan channel PV and Pδ curves (optonal) admttance matrx. These parameters wll reman constant unless the network mpedance or topology s changed (whch ncludes generators reachng Q lmts, see later). The PV curve calculaton procedure (whch can be the conventonal PV curve method or the contnuaton power flow method) s then appled to the system. At each PV curve pont, the CCT s appled to calculate the channel varables and crcuts correspondng to that pont. The channel margns wll be determned as well. The crtcal channel, and the crtcal loads are then dentfed. The generaton of channel PV or Pδ curves are optonal because channel margns are suffcent for the crtcal channel dentfcatons. It can be seen that the proposed algorthms can be consdered as a dagnostc tool for the PV curves. It uses PV curve results to extract more useful nformaton about the system. Ths procedure s smlar to that used by the Jacoban matrx modal analyss technque of [6]. If one of the generators reactve power lmts s reached at a PV curve pont, that generator shall be represented as a constant exctaton voltage behnd ts synchronous mpedance. The termnal of the exctaton voltage becomes a new (PV) bus. Thus the network topology s changed. The network Z matrx and CCT transform need to be recalculated accordngly. The above procedure could be mplemented onlne where each operatng pont s analyzed to determne the crtcal buses or areas assocated wth that operatng pont. The process s smlar to that of Fg. but blocks 3 and 7 are no longer needed. At any gven tme, the EMS wll provde the network topology and voltage phasor data to the onlne CCT module. If there s a change of network topology, the transform matrx [T] wll be recomputed. Otherwse, the same matrx wll be used. The voltage phasor data can come from the PMU drectly or from the state-estmator. After performng calculatons, the CCT module wll output results such as the crtcal buses, the pattern of crtcal channel power flow and channel margn etc.. Ths procedure works n theory but more research s needed n at least two areas. The frst s to determne the most useful buses to nstall PMUs for trackng the top channels as t s not practcal to measure the phasor voltages of all buses and generators. The second s how to consder generators httng reactve power lmts before they actually occur, as the procedure cannot predct ther happenng. Because of ths lmtaton, the crtcal buses dentfed above are vald for the current condton only. D. Case study results The CCT-based technque has been appled to several systems ncludng an actual large system. The results for the IEEE 57-bus system are presented frst. The analyses are performed accordng to the procedure shown n Fg.. The WECC PV curve methodology [6] s used to perform the PV curve calculatons. 9 Identfy the crtcal loads Fg.. Procedure of the analyss.

7 7 Fg. 3 llustrates the channel margns for dfferent load scalng factors obtaned for IEEE 57-bus system. Accordng to ths fgure, channel s dentfed as the crtcal channel. Fg. 4 and Fg. 5 show the channel PV and Pδ curves, respectvely. Only the top four channels are shown. Both of these fgures clearly verfy that channel s ndeed the crtcal channel because when the physcal system reaches ts nose pont, the operatng pont n channel approaches ts PV/Pδ curve nose pont. It s nterestng to note that the power transferred by channel s comparable wth those of channel and 5. The contrbuton of load bus currents to the crtcal channel current can be calculated usng the proposed contrbuton ndex to rank the load buses. The results are shown n Fg. 6. Fg. 6 (a) shows the contrbutons of buses n the bar chart format. The bubble chart shown n Fg. 6(b) provdes the same nformaton n a more vsualzed way wth the topology of the IEEE 57-bus system as the background. The sze of the bubble on a load bus s n proporton to the value of the contrbuton of that bus. Ths bubble chart clearly shows the Bus contrbuton Load bus (a) bar chart Channel margn (%) 5 5 Channel Load Scalng Factor Fg. 3. Channel margns n IEEE 57-bus system. 4 Channel.5.5 Channel Channel Channel Fg. 4. Channel PV curves of 4 top-ranked channels n IEEE 57-bus system..5 Channel Channel Delta (deg)..5 Channel 4 3 Channel Delta (deg) Channel 4 Channel Delta (deg) Channel 5 3 Channel Delta (deg) Fg. 5. Channel Pδ curves of 4 top-ranked channels n IEEE 57-bus system. (b) bubble chart Fg. 6. Bus rankng n IEEE 57-bus. locatons whch are weak wth respect to voltage stablty. Bus 3 s the bus most far away from the generators relatve to ts load sze. Ths bus has been ranked as the crtcal bus correctly. The results obtaned from an actual large system s shown n Fgs. 7 to. It s nterestng to note that both channels 384 and 8 have very small margns. However, channel 8 carres much hgher power. So t represents a system wde mode. Ths s confrmed by the bus rankng ndces shown n Fgs. 9 and. Fg. 9 shows that channel 384 nvolves very small number of buses whle channel 8 has a large number of buses partcpatng. Both channels dentfy Bus 63 as the crtcal bus, whch s known to ndustry as problematc. Several other test systems have also been studed. The results for these systems are summarzed n Table II. The results are compared wth those obtaned from the Jacoban matrx modal analyss (JM) method of [6]. The table confrms that the proposed technque can dentfy the crtcal buses for all systems. In a few cases, some of the top buses are dfferent. Senstvty studes ndcate that they are caused by two factors: () The ndex values used to rank buses (by ether the JM or the CCT methods) can be very close among top ranked buses n some cases, so the rank order can be affected by the small dfferences n the ndex values. () The CCT bus rank consders the mpact of both actve and reactve power loads whle the (reduced) Jacoban matrx modal analyss method only consders the mpact of reactve power. Ths subject s further dscussed next.

8 8 Channel margn (%) Channel 8 Channel Load Scalng Factor Fg. 7. Channel margns n AIES..5 Channel Channel Channel Channel Channel Channel Fg. 8. Channel PV curves of 6 top-ranked channels n AIES. Fg. 9. Bus rankng obtaned from channel 384. Fg.. Bus rankng obtaned from channel 8. System TABLE II SUMMARY OF THE RESULTS Crtcal channel Top ranked crtcal load buses Proposed method JM method WECC 9-bus 9, 5, 7 9, 5, 7 3-bus from [7] 4 8, 7, 4, 3 8, 7, 4, 3 IEEE 3-bus 3,, 4, 6 3, 9, 6, 4 IEEE 57-bus 3, 5, 33, 3 3, 3, 33, 3 AIES 38-bus 8 63, 393, 77, * * Obtaned by PSS/E [9] whch gves only the frst crtcal bus. The other cases of JM analyss are done usng MATPOWER [8]. E. Dscussons For voltage stablty analyss, the applcaton procedure of CCT technque s qute smlar to that of the Jacoban matrx modal analyss (JM) technque. It s therefore useful to compare the two technques. The man dfferences can be summarzed below:. Physcal meanng: the channel components and crcuts have clear physcal meanngs and models. It s therefore easy to nterpret the results n channel doman and extract useful nformaton. The JM method, on the other hand, s just a numercal technque. There s no crcut model for the modal results. A doman to map and smplfy the PV curves does not exst. The Pδ curves are out of reach for the JM method.. Computng effort: If the network mpedance matrx remans the same, the CCT egen-decomposton only needs to be performed once. The whole PV curves can be mapped nto the channel doman wth lttle computng effort. Onlne mplementaton s straghtforward. The movement of modes can thus be traced easly. The JM method needs to execute egen-decomposton at every PV curve pont f one wants to track the mode changes. Furthermore, a complex, yet-to-be developed mode-trace (.e. root-locus plottng) technque s needed to relate one egenvalue obtaned at a PV curve pont to another egenvalue obtaned at the next PV curve pont. 3. Robustness: A sgnfcant problem that has not been solved for the JM method s whch matrx, the reduced Jacoban matrx or the full Jacoban matrx, shall be used for the egen-analyss. In theory, the full Jacoban matrx shall be used. But case studes have shown that the full Jacoban matrx yelds dfferent ΔQ bus rankngs compared to those derved from the reduced Jacoban matrx. Furthermore, the full Jacoban matrx yelds two bus rankng lsts, one correspondng to ΔP and one for ΔQ. These two lsts are dfferent, whch creates addtonal ambguty n bus rankng. The JM results presented n ths paper are from the reduced Jacoban matrx accordng to [6]. The CCT method does not have ths problem. It ncludes the mpact of both actve and reactve power n the form of channel margn. Another practcal ssue s the

9 9 dffculty to dentfy the crtcal mode correctly usng the JM method snce the ndex to rank modes s the magntudes of egenvalues. Because the PV curve technque cannot reach the exact nose pont where λ=, the mode rankng can only be done at a PV curve pont close to the nose pont (the contnuaton power flow wll help to zero n the nose pont but numercal dfferences can stll exst). As a result, numercal errors and the exstence of multple small egenvalues can mask the crtcal mode. V. CONCLUSIONS Ths paper has proposed and demonstrated a new method to analyze the behavor of complex power systems. The method s n the form of a network-decouplng transform. The man feature of ths transform s that t can decouple a complex network nto a set of decoupled sngle-source, sngle-branch and sngle-load networks. The transform has been successfully appled to analyze power system voltage stablty n terms of dentfyng the crtcal buses (or loads) responsble to voltage collapse n a power system. Ths s done by projectng the PV curves nto the channel doman and by evaluatng the crtcalty of each channel n that doman. Wth the proposed transform, the PV curves can also be examned from the perspectve of power-angle relatonshp n the channel doman. The results help one to gan mproved understandng on the role of bus voltage angles n voltage collapse. Fnally, the proposed transform can also be consdered as a technque for processng or mappng the phasor data ([U]=[T][V]). From ths perspectve, the transform has been appled successfully to process the phasor data calculated from the PV curve algorthms. REFERENCES [] A. G. Phadke, Synchronzed phasor measurements n power systems, IEEE Comput. Appl. Power, vol. 6, no., pp. 5, Apr [] P. Pourbek and C. Rehtanz, conveners, Wde Area Montorng and Control for Transmsson Capablty Enhancement, Fn. Rep. CIGRE Workng Group C4.6, Jan. 7. [3] A. G. Phadke, H. Voslks, R. M. de Morales, T. B, R. N. Nayak, Y. K. Sehgal, S. Sen, W. Sattnger, E. Martnez, O. Samuelsson, D. Novosel, V. Madan, and Y. A. Kulkov, The wde world of wde-area measurements, IEEE Power Energy Mag., vol. 6, no. 5, pp. 5 65, Nov. 8. [4] A. Chakrabortty, J.H. Chow, and A. Salazar, "A Measurement-Based Framework for Dynamc Equvalencng of Large Power Systems Usng Wde-Area Phasor Measurements," IEEE Transactons on Smart Grd, vol., no., pp.68-8, March. [5] T. Overbye, P. Sauer, C. DeMarco, B. Leseutre, M. Venkatasubramanan, Usng PMU Data to Increase Stuatonal Awareness. PSERC Publcaton -6, Sep.. [6] B. Gao, G. K. Morson, and P. Kundur, "Voltage stablty evaluaton usng modal analyss," IEEE Trans. Power Syst., vol. 7, pp , Nov.99. [7] C.M. Affonso, L.C.P. da Slva, F.G.M. Lma, and S. Soares, "MW and MVar management on supply and demand sde for meetng voltage stablty margn crtera," IEEE Trans. Power Syst., vol.9, no.3, pp , Aug. 4. [8] M.M. Farsang, H. Nezamabad-pour, Yong-Hua Song, and K.Y. Lee, "Placement of SVCs and Selecton of Stablzng Sgnals n Power Systems," IEEE Trans. Power Syst., vol., no.3, pp.6-7, Aug. 7. [9] J.A.P. Flho, N. Martns, and D.M. Falcao, "Identfyng Power Flow Control Infeasbltes n Large-Scale Power System Models," IEEE Trans. Power Syst., vol.4, no., pp.86-95, Feb. 9. [] A.Z. Gamm, I.I. Golub, A. Bachry, and Z.A. Styczynsk, "Solvng several problems of power systems usng spectral and sngular analyses," IEEE Trans. Power Syst., vol., no., pp , Feb. 5. [] I. J. Perez-Arraga, G. C. Verghese, and F. C. Schweppe, Selectve modal analyss wth applcatons to electrc power systems, Part I: Heurstc ntroducton, IEEE Trans. Power App. Syst., vol. PAS-, pp , Sept. 98. [] H. W. Dommel, ElectroMagnetc Transents Program. Reference Manual (EMTP Theory Book). Portland, OR: Bonnevlle Power Admnstraton, 986. [3] T. Van Cutsem and C. Vournas, Voltage Stablty of Electrc Power Systems. Norwell, MA: Kluwer, 998. [4] B. Mlosevc, and M. Begovc, "Voltage-stablty protecton and control usng a wde-area network of phasor measurements," IEEE Trans. Power Syst., vol.8, no., pp. - 7, Feb 3. [5] M. Glavc, and T. Van Cutsem, "Wde-Area Detecton of Voltage Instablty From Synchronzed Phasor Measurements. Part I: Prncple," IEEE Trans. Power Syst., vol.4, no.3, pp.48-46, Aug. 9. [6] A. M. Abed, WSCC Voltage stablty crtera, undervoltage load sheddng strategy, and reactve power reserve montorng methodology, IEEE PES Summer Meetng 999, vol., pp. 9-97, Jul [7] O. Alsac, and B. Stott, Optmal Load Flow wth Steady State Securty, IEEE Trans. Power App. Syst., vol. PAS 93, no. 3, pp , 974. [8] R. D. Zmmermann, and D. Gan, "Matpower a Matlab power system smulaton package, " User s Manual,, Verson., Dec [9] Semens PTI, PSS/E 3. Program Operatonal Manual, Volume II, 5. [] S. Skogestad and I. Postethwate, Multvarable Feedback Control, Analyss and Desgn. New York: Wley, 996. [] A. G. Phadke, Synchronzed phasor measurements a hstorcal overvew, n Proc. IEEE Power Eng. Soc. Asa Pacfc Transmsson Dstrbuton Conf. Exhb., Oct. 6,, vol., pp APPENDIX A: CHARACTERISTICS OF CHANNEL LOADS Theoretcal dervaton n Secton IV.A suggests that all channel loads are coupled to each other. The channel crcuts are connected as shown n Fg. A.. As seen n ths fgure, the load seen by the channel crcut wll contan the F and λ components assocated wth other channels. Ths ndcates that the channel load can be represented by a Thevenn (or Norton) equvalent crcut. F λ F λ F n λ n [Z C ] Equvalent load seen by channel crcut Fg. A.. The channel crcuts. The degree of couplng among the channel loads has been nvestgated. Ths study s based on a crteron developed n system control feld []. The system model n () can be treated as a MIMO (multple nput multple output) system n whch the channel voltages are the nputs and the channel currents are the outputs. Wth ths approach, the outputs are obtaned by multplyng the nput matrx n a gan matrx [Y C ]=[Z C ] -. The control theory states that a Relatve Gan Array (RGA) can be formed for every MIMO system []. RGA wll determne how coupled the MIMO system s. It can also determne the optmal nput-output varable parngs. RGA can be formed usng the gan matrx [Y C ] as follows. RGA = [ Y ].*([ Y ] ) T (A.) C C

10 where (.*) ndcates the element-wse multplcaton. As an example, the RGA for the smple case study shown n Fg. 4 s as follows: RGA = Snce the dagonal elements of the RGA are much larger than the off-dagonal elements, one may conclude that the MIMO system assocated wth the channels loads s almost decoupled. The RGA has also been computed and examned for several power systems. Fg. A. llustrates the RGA computed for several standard test systems when they are at ther normal operatng condtons. As seen from Fg. A., most of the channel loads are decoupled. Ths s an nterestng fndng. It supports the approach of approxmatng the couplng as current sources. APPENDIX B: COMPARISON WITH THE SC TRANSFORM The CCT shares many common characterstcs wth the Symmetrcal Components transform (SCT). It s therefore useful to compare the two transforms so that more nsghts can be ganed for CCT. The voltage and current relatonshp of a n-conductor (or n-phase) power lne shown n Fg. B. s the smple voltage drop relatonshp n a matrx form: [ V ] = [ V ] [ Z][ I] (B.) RGA R..5 3 Row S Column 3 (a) WECC 9-bus system (b) 3-bus system [7] (c) IEEE 3-bus system (d) IEEE 57-bus system S S A B C [I] S S [V S ] A [V R ] B C Fg. B.. A mult-conductor (or mult-phase) power lne. where [Z] s the n-phase mpedance matrx of the lne. The smplest case of the above equaton s the 3-phase lne where [ V ]= T R VRa VRb V Rc and [ I]= I T a Ib I c. If a three-phase lne has two sheld wres, there are fve conductors nvolved T (Fg. B.),.e. [ VR] = [ VRa VRb VRc VRS VRS]. In the EMTP analyss, such a lne s called fve-phase lne []. Ths concept can be generalzed to n-phase lnes f there are n- conductors n a tower structure. When conductng EMTP smulatons or fndng the modes of wave propagaton n a lne, the [Z] matrx must be dagonalzed,.e., egen-decomposton shall be performed on [Z]. [] has presented varous forms of [Z] decomposton or transform. The symmetrcal components transform s the smplest one among them. The SCT yelds three sngle branches n the modal doman. These are the postve-, negatve- and zero-sequence branches. The correspondng doman s the well-known sequence doman. Let s now compare (B.) wth (). These two equatons have exactly the same form f we set [E ]=[V S ] and [V]=[V R ]. The mplcaton s the followng: one can treat a multgenerator, mult-load and mult-branch network shown n Fg. as a network that has one generator supplyng one load through one transmsson lne that has multple phases, as shown n Fg. B.. In other words, each of the real generators can be consdered as one phase of a n-phase generator and each of the loads as one phase of a n-phase load. The mutual couplngs of the mult-phase transmsson lne represent the mutual nteracton of varous branches nsde the actual network. Once an actual complex network s vewed as a smple mult-phase network, the egen-decomposton technques well studed n the EMTP theory can be appled to decouple t nto n smple modal networks. The proposed transform s essentally dentcal to the transform used by EMTP analyss. The CCT s channel doman thus shares many characterstcs wth the sequence doman. For example, the system shown n E P +jq E phase couplng P +jq E n P n +jq n (e) IEEE 8-bus system Fg. A.. RGA for dfferent test systems. One generator One mult-phase transmsson lne One load Fg. B.. A Thevenne crcut vewed as a mult-phase lne.

11 Fg. 4 can be decoupled usng the SCT when Z a =Z b =Z c. Channel corresponds to the zero sequence component. In fact, 3Z+Z a s the zero sequence mpedance of an equvalent three-phase lne. One may thnk that the sample system uses zero sequence mode to transfer power. Smlarly, the observaton that a three-phase power system has a much lower negatve sequence voltages than ts postve sequence voltages also helps one to understand why some channel voltages are way lower than other channel voltages n a (postve sequence) power system. In summary, f one consders the SCT as a method for processng three-phase phasors (measured or calculated), and the transforms documented n [] as tools for processng mult-phase phasors, the proposed CCT can be vewed as an operaton for processng mult-bus (.e. mult-locaton) phasors. It s nterestng to note that the PMU was created orgnally for determnng the sequence components needed by a power system protecton scheme []. Bography Wlsun Xu (M 9-SM 95-F 5) receved the Ph.D. degree from the Unversty of Brtsh Columba, Vancouver, BC, Canada, n 989. He was an Engneer wth BC Hydro, Burnaby, BC, Canada from 99 to 996. Currently, he s a Professor and a NSERC/CORE Industral Research Char at the Unversty of Alberta, Edmonton, AB, Canada. Hs research nterests are power qualty and voltage stablty. Iraj Rahm Pordanjan (S 9) receved hs B.Sc. degree (wth Frst Class Honors) and M.Sc. degree n Electrcal Engneerng from Amrkabr Unversty of Technology (Tehran Polytechnc), Tehran, Iran n 5 and 8, respectvely. He s currently pursung hs Ph.D. program n Electrcal and Computer Engneerng at Unversty of Alberta, Canada. Hs research nterests are power systems stablty and power qualty. Yunfe Wang (S 8) receved the B.Sc. and M.Sc. degrees n control scence and technology from Harbn Insttute of Technology, Harbn, Chna, n 3 and Tsnghua Unversty, Bejng, Chna, n 6, respectvely. He s currently pursung the Ph.D. degree n the Department of Electrcal and Computer Engneerng, Unversty of Alberta, Edmonton, Canada. Hs research nterests are power qualty and power system stablty. Ebrahm Vaahed (S 78-M 79-SM 87-F ) receved the Ph. D. degree from Imperal College of Scence and Technologes, London, U.K., n 979. Currently, he s the chef technology offcer at the Brtsh Columba Transmsson Company (BCTC). Before jonng BCTC, he was a senor manager at Perot Systems where he was engaged n the development of grd operaton and market operaton system nfrastructures as well as other energy related consultng servces. He s the char of IEEE Subcommttee on Operaton Methods and the edtor of IEEE Transactons on Power Systems.