Trajectory Sensitivity Theory in Non Linear Dynamical Systems: Some Power System Applications

Transcription

1 Engneerng Seres, Brkhauser Boston, 23. Trajectory Senstvty Theory n Non Lnear Dynamcal Systems: Some Power System Applcatons M. A. Pa and T. B. Nguyen Dept of Electrcal and Computer Engneerng Unversty of Illnos at Urbana-Champagn Abstract: Trajectory senstvty analyss (TSA) has been appled n control system problems for a long tme n such areas as optmzaton, adaptve control etc. Applcatons n power systems n conjuncton wth Lyapunov/transent energy functons frst appeared n the 8's. More recently, t has found applcatons on ts own by defnng a sutable metrc on the trajectory senstvtes wth respect to the parameters of nterest. In ths paper we present the theoretcal as well as practcal applcatons of TSA for dynamc securty applcatons n power systems. We also dscuss the technque to compute crtcal values of any parameter that nduces stablty n the system usng trajectory senstvtes.. Introducton Securty n power systems became an ssue after the Northeast blackout n 965 []. Snce then a lot of research has been done nvestgatng both statc and dynamc aspects of securty. Whle a lot of success has been acheved on the statc front [2], such s not the case wth dynamc securty. Dynamc securty assessment (DSA) n power systems comprses of the followng man tasks: contngency selecton/screenng, securty evaluaton, contngency rankng, and lmt computaton. Dynamc smulaton has hstorcally been the man tool, and currently n combnaton wth heurstcs and some form of learnng, t stll remans as the tool for DSA n the energy control centers. Intensve research snce the 6 s n applyng Lyapunov s drect method has resulted n useful algorthms n the form of transent energy functon (TEF) technque [3-5] and sngle machne equvalent (SIME) technque [6]. Artfcal neural network (ANN) and artfcal ntellgent (AI) based technques have also been appled [7, 8]. Of these the TEF and SIME technques are consdered the most promsng ones by the research communty. In the deregulated envronment, the exstng transmsson system often operates at ts lmt due to nadequate capacty and multlateral transactons. In addton, power systems must be operated to satsfy the transent stablty constrants for a set of contngences. In these stuatons, dynamc securty assessment plays a crucal role. There exsts a need to replace the repettve nature of the dynamc smulaton for DSA by a procedure where complex models can be handled easly n a more drect manner. The am of on-lne DSA s to assess the stablty of the system to a set of predefned contngences. These contngences are user specfed or are chosen automatcally through some procedure such as a flterng process. For each contngency f the system s stable, t can also provde a securty margn based on the technque used. For Present address: Energy Scence and Technology Drectorate, Pacfc Northwest Natonal Laboratory, Rchland, WA.

2 Engneerng Seres, Brkhauser Boston, 23. nstance, f crtcal clearng tme s computed, the t cr -t cl s the margn. On the other hand, f the transent energy functon (TEF) s used, then V cr -V cl s the margn. The securty margn can be used to provde the operators wth gudelnes to mprove system securty whle at the same tme mantanng economc operaton. Ths s known as securty-constraned optmzaton or preventve reschedulng (See Fg ). The lterature on preventve control s largely ted to the TEF method, namely, to enhance the stablty margn as defned by the dfference between crtcal energy V cr and energy at clearng V cl. However, the need for computaton of the unstable equlbrum pont and the absence of an analytcal closed form of the energy functon makes t dffcult to apply for larger and more complex models of machnes. Securty montorng Contngency flterng Contngency lst Severe No Dscard yes Lst of severe contngences Choose a contngency Detaled dynamcs and trajectory senstvty computaton Choose next contngency Stablty crteron test passed Yes End of severe contngency lst No No Yes Reschedulng of generaton usng trajectory senstvtes Stop 2

3 Engneerng Seres, Brkhauser Boston, 23. Fg.. DSA scheme usng trajectory senstvtes In ths paper we revew some recent results n applyng trajectory senstvty (TS) technques [9] to DSA and preventve control. As t wll be shown, ths new technque has several advantages over all the other technques:. No restrcton on complexty of the model. 2. Extenson to systems wth dscrete events s possble. 3. Informaton other than mere stablty can be obtaned. 4. Lmts to any parameter n the system affectng stablty can be studed. 5. Identfcaton of weak lnks n the transmsson network s possble. 6. Preventve strateges can be ncorporated easly. However, the above advantages are obtaned at the expense of ncreased computatonal cost. Ths queston wll also be addressed n ths paper. The paper outlne s as follows. In Secton 2 we wll explan the dervaton of the basc theory of trajectory senstvty analyss (TSA) for dfferental algebrac equaton (DAE) form of the system model. The overall approach to DSA usng TSA technque wll be explaned. In Secton 3 we wll use the TSA technque to compute crtcal parameter values such as clearng tme, mechancal nput power, and lne reactance. In Secton 4 we wll explan the dynamc securty constraned dspatch problem and ts applcaton []. In Secton 5 we wll use TSA technque to fnd weak lnks, vulnerable relays and the electrcal centers of the system for a gven fault. Ths nformaton s useful n proper slandng of the system n a self-healng way []. 2. TRAJECTORY SENSITIVITY ANALYSIS Senstvty theory n dynamc systems has a long and rch hstory well documented n the books by Frank [2] and Eslam [3]. It can be traced back to the work of Bode [4] n desgnng feedback amplfers where feedback s used to cancel the effect of unwanted dsturbances and parameter varatons. The concept of senstvty matrx usng state space analyss can be used [5]. Whle bulk of the work s n the area of lnear tme nvarant (LTI) systems, the fundamental theory s applcable to nonlnear systems as well. Applcatons of senstvty theory or more specfcally trajectory senstvty analyss (TSA) to nonlnear dynamc systems are few. The books of Tomovc [5], Tomovc and Vukobratovc [6], and Cruz [7] contan control-orented applcatons. There have been applcatons of senstvty theory to power systems but mostly for lnear systems such as egenvalue senstvty [8]. From a stablty pont of vew t has been appled for computng senstvty of the energy margn whle usng the transent energy functon method [3, 4]. Applcatons of trajectory senstvty to nonlnear models of power systems are somewhat recent [9]. The applcaton n the lnear system arses from the lnearzaton of a nonlnear system around an equlbrum pont. Stablty of the equlbrum pont s evaluated through egenvalues. In trajectory senstvty analyss, we lnearze around a nomnal trajectory and try to nterpret the varatons around that trajectory. It s a challenge to develop a metrc for those varatons and relate t to the stablty of the 3

4 Engneerng Seres, Brkhauser Boston, 23. nomnal trajectory. In [5] the author hnts at the ntmate connecton between trajectory senstvty analyss and Lyapunov stablty but t s not quantfed. In ths paper we make an attempt to do so. 3. Trajectory Senstvty Theory for Dfferental Algebrac Equaton (DAE) Model The development n ths secton follows Ref [2, 2] where the applcaton to hybrd systems s dscussed. A DAE system s a specal case of the more general hybrd systems. A farly accurate descrpton of the power system model s represented by a set of dfferental algebrac equaton of the form x& = f( x, y, λ) () g = g + ( x, y, λ) ( x, y, λ) s( x, y, λ) < s( x, y, λ) > (2) and a swtchng occurs when s ( x, y, λ) =. etc.; In the above model, x are the dynamc state varables such as machnes rotor angles, veloctes, y are the algebrac varables such as load bus voltage magntudes and angles; and λ are the system parameters such as lne reactances, generator mechancal powers, and fault clearng tme. Note that the state varables nstants. x are contnuous whle the algebrac varables The ntal condtons for ()-(2) are gven by where y satsfes g ( x, y, λ) = x ( t ) = x, y ( t ) = y For compactness of notaton, the followng defntons are used x x = λ f f = Wth these defntons, ()-(2) can be wrtten n a compact form as y can undergo step changes at the swtchng x& = f( x, y) (3) g = g + ( x, y) ( x, y) s( x, y) < s( x, y) > (4) The ntal condtons for ()-(2) are x( t y( t ) = x ) = y We dvde the tme nterval as consstng of non-swtchng subntervals and swtchng nstants for whch the senstvty model s now developed. (5) 4

5 Engneerng Seres, Brkhauser Boston, 23. Trajectory senstvty calculaton for non-swtchng perods Ths secton gves the analytcal formulae for calculatng senstvtes x x ( t) and y x ( t) on the non -swtchng tme ntervals as dscussed n [2]. On these ntervals, the DA systems can be wrtten n the form Dfferentatng (6) and (7) wth respect to the ntal condtons x yelds x& = f( x, y) (6) = g( x, y) (7) x & = f () t x + f () t y (8) x x x y x = g + (9) x ( t) x x g ( ) y t y x where f f, g, and are tme-varyng matrces and are calculated along the system trajectores., x y x g y Intal condtons for x are obtaned by dfferentatng (5) wth respect to x as x x x ( t ) = I () where I s the dentty matrx. Usng () and assumng that g y ( t ) s nonsngular along the trajectores, ntal condton for y x can be calculated from (9) as y x ) = [ g ( t )] g ( t ) () ( t y x Therefore, the trajectory senstvtes can be obtaned by solvng (6) and (7) smultaneously wth (8) and (9) usng any numercal method wth (5), (), and () as the ntal condtons. At swtchng nstants, t s necessary to calculate the jump condtons that descrbe the behavor of the trajectory senstvtes at the dscontnutes. Snce we are consderng tme nstants, whch do not depend on the states, the senstvtes of the states wll be contnuous whereas those of the algebrac are not. When the trajectory senstvtes are known, the perturbed trajectores can be estmated by frst-order approxmaton wthout redong smulaton as x() t x () t x (2) x yt () y () t x (3) x Computaton of crtcal values of parameters usng energy functon as a metrc [22] In the lterature, trajectory senstvtes have been used [4] to compute the energy margn senstvty wth respect to system parameters such as nterface lne flow, system loadng, etc. usng TEF methods. In these cases, the crtcal energy ν cr, whch s the energy at the controllng u.e.p., depends on 5

6 Engneerng Seres, Brkhauser Boston, 23. the parameters. Therefore, computaton of ν cr t cl s necessary whle usng the TEF method. Ths s computatonally a dffcult task. On the other hand, because the energy functon ν(x) s used here only as a metrc to montor the system senstvty for dfferent t cl, we can avod the computaton ofν cr and use ν(x) drectly. The process of estmatng crtcal values of parameters wll be llustrated usng the clearng tme t cl. However, the process s approprate for any parameter that can nduce nstablty such as mechancal power P M. We can use the senstvty ν to estmate tcr drectly. Wth the classcal model for machnes, t cl the energy functon ν (x) for a structure-preservng model s computed as follows. frame as [3] The post fault power system can be represented by the DAE model n the center of angle reference & θ = % + ω =,..., m n g M M = P B V V sn P =,..., m ( ) n &% ωg M n+, j n+ j θn + θ j COA j= MT n Pd + sn( ),..., n BjVV j θ θ j = = (6) j= (4) (5) n Q ( V) B VV cos( θ θ ) = =,..., n d j j j j= (7) where m s the number of machnes, n s the number of buses n the system, and m n PCOA = PM sn( ) BjVV j θ θ j = j= We assume constant real power loads and voltage dependent reactve power load as α s V Qd ( V ) = Qd V s (8) s Q s V where and are the nomnal steady state reactve power load and voltage magntude at the th bus, and α s the reactve power load ndex. The correspondng energy functon s establshed as [3] 6

7 Engneerng Seres, Brkhauser Boston, 23. ν ( % ω, θ, ) % ω ( θ θ ) ( θ θ ) m m n 2 s s g V = M g PM n+ n+ + Pd 2 = = = Q α ( ) + ( ) 2 n n n s 2 s2 d α s B V V V s V α = = αv n = j= + s s s B ( VV cosθ V V cos θ ) j j j j j (9) where θ = θ θ. j j The senstvty S of the energy functon ν (x) wth respect to clearng tme ( λ = tcl by takng partal dervatves of (9) wth respect to t cl as ) s obtaned ν % ω θ θ V S = = M P + P BV t t t t t m m n n g n + % ωg M d cl = cl = cl = cl = cl V V V V θ + ( cos + cos sn ) n α n n s j j Qd B j V s j j V j VV α θ θ j θj = V tcl = j= + tcl tcl tcl (2) The partal dervatves of %,, and V wth respect to t cl are the senstvtes obtaned by ntegratng ω g θ the dynamc system and the senstvty system as dscussed earler. The senstvty S = ν s computed for two dfferent values of t cl, whch are chosen to be less t cl than t cr. Snce we are computng only frst-order trajectory senstvtes, the two values of t cl must be less than t cr by at most 2%. Ths mght appear to be a lmtaton of the method. However, extensve experence wth the system generally wll gve us a good estmate of t cr. Because the system under consderaton s stable, the senstvty S wll dsplay larger excursons for larger t cl [9]. Snce senstvtes generally ncrease rapdly wth ncreases n t cl, we plot the recprocal of the maxmum devaton of S over the postfault perod as η =. A straght lne s then constructed through the two ponts max( S) mn( S) ( t, η ) and (, ) cl t η. The estmated crtcal clearng tme s the ntersecton of the constructed cl 2 2 t cr, est straght lne wth the tme-axs n the ( t, η) -plane as shown n Fg 2. As dscussed later, ths lnearty s vald for a small regon around t cr. cl 7

8 Engneerng Seres, Brkhauser Boston, 23. η η η 2 t c t cl2 t cr,est t cl Fg. 2. Estmate of t cr Drect Use of Trajectory Senstvtes to Compute Crtcal Clearng Tme [22] In ths secton we outlne an approach usng trajectory senstvty nformaton drectly nstead of va the energy functon to estmate the crtcal clearng tme. To motvate ths approach, let us consder a SMIB system descrbed by Mδ& + D & δ = PM < t tcl Mδ& + D & δ = P P snδ t > t M em cl The correspondng senstvty equatons are Mu& + Du& = where u = δ. Mu& + Du& = ( P t cl em < t t cosδ ) u cl t > t cl δ() = δ, & δ() = (2),, u() =, u& () = (22) If we plot the phase plane portrat of the system for two values of t cl, one small and the other close to t cr and montor the behavor of senstvtes n the ( uu&)-plane, we observe that the senstvty magntudes ncrease much more rapdly as t cl approaches t cr. Also, the trajectores n the ( uu&)-plane, can cross each other snce the system (22) s tme varyng, whereas that s not the case for the system (2), whch s an autonomous system. Qualtatvely, both trajectores n the ( δ, ω) -plane and the ( uu&)-plane, gve the same nformaton about the stablty of the system, but the senstvtes seem to be stronger ndcators because of ther rapd changes n magntude as t cl ncreases. Hence, we can assocate senstvty nformaton wth the stablty level of the system for a partcular clearng tme. When the system s very close to nstablty, the senstvty reflects ths stuaton much more quckly. Ths qualtatve relatonshp has been dscussed for the general nonlnear dynamc systems by Tomovc [5]. One possble measure of proxmty to nstablty may be through some norm of the senstvty vector. The Eucldean norm s one such possblty. For the sngle machne system, f we plot the norm 2 u + u& 2 as a functon of tme for 8

9 Engneerng Seres, Brkhauser Boston, 23. dfferent values of t cl, one can get a quck dea about the system stablty as shown n Fgs. 3 and 4. For a stable system, although the senstvty norm tends to become a small value eventually, t transently assumes a very hgh value when t cl s close to t cr. Thus, we assocate wth each value of t cl the maxmum value of the senstvty norm. The procedure to calculate the estmated value of t cr s the same as descrbed n the prevous secton but usng the senstvty norm nstead of the energy functon senstvty. Here, the senstvty norm for an m-machne system s defned as S N = m 2 2 δ δ j ω + t t t = cl cl cl where the jth-machne s chosen as the reference machne Norm t (s) Fg. 3. Senstvty norm for small t cl ( 5% of t cr ) Norm t (s) Fg. 4. Senstvty norm for t cl close to t cr ( 8% of t cr ) 9

10 Engneerng Seres, Brkhauser Boston, 23. The norm s calculated for two values of t cl < t cr. For each t cl, the recprocal η of the maxmum of the norm s calculated. A lne through these two values of η s then extrapolated to obtan the estmated value of t cr. If other parameters of nterest are chosen nstead, the technque wll gve an estmate of crtcal values of those parameters. Snce ths technque does not requre computaton of the energy functon, t can be appled to power systems wthout any restrcton on system modelng. Ths s a major advantage of ths technque. Numercal Examples We consder three systems and applcaton of both energy functon based and drect senstvty based metrc. These are the 3-machne, 9-bus [23, 24]; the -machne, 39-bus [3]; and the 5-machne, 45-bus [4] systems. Results are presented n Tables -2 and Fg. 5. Table. Results for the 3-machne system Faulted TEF Senstvty Senstvty Norm Actual Bus t cr,est (s) t cr,est (s) t cr (s) Table 2. Results for the -machne system Faulted bus Lne Trpped Senstvty Norm Actual t cr,est (s) t cr (s) For the 5-machne system, the estmated value of clearng tme for a self-clearng fault at bus 58 usng the senstvty norm technque s computed. The correspondng values of η for dfferent values of t cl are shown n Fg. 5. We note that from Fg. 5 that f the two values of t cl are chosen n the close range of t cr =.35 s, the estmated value of t cr wll be qute accurate. On the other hand, pckng arbtrary values of t cl may gve erroneous results. Snce computng senstvtes s computatonally extensve, choosng good values of t cl requres judgment and experence.

11 Engneerng Seres, Brkhauser Boston, η t cl (s) Fg. 5. Estmate t cr for fault at bus 58 usng norm senstvty Computaton of Crtcal Loadng of Generator Next, the senstvty norm technque s used to estmate the crtcal value of generator loadng, or equvalently, the mechancal nput power P M. Two smulatons for two values of P M are carred out. The change from normal operatng values n P M s dstrbuted unformly among all loads n the system, so that the loadng of the rest of the generators s unchanged. The senstvty norm s calculated for the two specfed values of P M and then extrapolated to obtan the estmated value of the crtcal P M for the chosen generator. The -machne system A fault s smulated n the system at bus 2 of the -machne system and cleared at t cl =. s by trppng the lne The estmated results for a few generators are shown n Table 3. Table 3. Estmated value of crtcal nput power P M vs. the actual value Machne Number Senstvty Norm Actual P Mcr,est (pu) P M,cr (pu)

12 Engneerng Seres, Brkhauser Boston, 23. The 5-machne system A self-clearng fault s smulated at bus 58 and cleared at t cl =.5 s. Applyng the proposed technque, the results obtaned for crtcal value of P M s and shown n Table 4. To valdate the results t was verfed that wth the crtcal value of P M the system goes unstable. Table 4. Estmated value of crtcal nput power P M vs. the actual value Machne Number Senstvty Norm Actual P M,est (pu) P M,act (pu) Computaton of Crtcal Impedance of a Transmsson Lne The norm senstvty technque s used to estmate the crtcal value of a lne reactance. The 3- machne system s used to llustrate the technque. A fault s smulated at bus 7 and cleared at t cl =.8 s by trppng the lne 5-7. Fgure 6 shows the correspondng values of η for dfferent values of reactance of the lne 8-9. The crtcal value of the reactance of lne 8-9 s.246 pu. It can be seen from Fg. 6 that the estmated value of the crtcal reactance s qute accurate f the two values of the lne reactance are pcked n the close range of the actual crtcal value η Reactance (pu) Fg.6. Estmate crtcal value of the reactance of the lne 8-9 2

13 Engneerng Seres, Brkhauser Boston, 23. Knowng the crtcal value of a lne reactance s very mportant n controllng power flow path n the system by the varable mpedance devces. Such devces belong to a type of devces called flexble ac transmsson systems (FACTS) that can be very useful n controllng the stablty of power systems [25]. 4. Stablty Constraned Optmal Power Flow Formulaton [] Whle Ref [26] formulates the problem n a dfferent way, here we use the relatve rotor angles to detect the system stablty/nstablty of the power system. To check the stablty of the system for a credble contngency, relatve rotor angles are montored at each tme step durng dynamc smulaton. The senstvtes are also computed at the same tme. Although senstvty computaton requres extensve computatonal effort, effcent method to compute senstvtes s avalable by makng effectve use of the Jacoban whch s common to both the system and senstvty equatons [27]. Ths wll reduce the computatonal burden consderably. We propose that when the relatve rotor angle δj = δ δ j > π for a gven contngency, the system s consdered as unstable. Ths s an extreme case as ponted out n [6], and one can choose an angle dfference less than π dependng on the system. Here and j refer to the most and the least advanced generators respectvely. The senstvtes of the rotor angles at ths nstant are used to compute the amount of power needed to be shfted from the most advanced generator (generator ) to the least advanced one (generator j) accordng to the formulae δ δ j j P, j= δj P maxδ j = π (23) new new P = P P j and Pj = Pj + P j (24),, where P P, and δ are the base loadng of generators and j, and the relatve rotor angle of the two, j j respectvely at the soluton of the OPF problem. δ j P s the senstvty of relatve rotor angle wth respect to the output of the th-generator; P n ths case s the parameter λ n equatons stated n Secton 3. After shftng the power from generator to generator j accordng to (24), the system s secure for that contngency but t s not an optmal schedule. We can mprove the optmalty by ntroducng new power constrants on generaton. The OPF problem wth new constrants s then re-solved to obtan the new operatng pont for the system. The detaled algorthm s dscussed n [] 5. Assessment of Transmsson Protecton System Vulnerablty to Angle Stablty Problems Power system protecton at the transmsson system level s based on dstance relayng. Dstance relayng serves the dual purpose of apparatus protecton and system protecton. Sgnfcant power flow 3

14 Engneerng Seres, Brkhauser Boston, 23. oscllatons can occur on a transmsson lne or a network due to major dsturbances lke faults and subsequent clearng, load rejecton, etc. They are related to the swngs n the rotor angles of synchronous generators. If the rotor angles settle down to a new stable equlbrum pont, the dsturbance s classfed as stable. Otherwse, t s unstable. Hence, for stable dsturbances, power swngs de down wth tme. In ths secton, we dscuss the applcaton of trajectory senstvty analyss to detect vulnerable relays n the system. For each contngency we compute a quantty called branch mpedance senstvty, whch wll be used to dentfy weak lnks n the system. Its man advantage s that t can handle systems of any degree of complexty n terms of modelng and can be used as an on-lne DSA tool. Dstance relays are used to detect swngs and take approprate acton (trppng or blockng) dependng on the nature of swng (stable or unstable). The reason s that a change n transmsson lne power flow translates nto a correspondng change n the mpedance seen by the relay. Relay operaton after fault clearng depends upon the power swng and proxmty of the relay to an electrcal center. P flow j j The apparent mpedance seen by a relay on a transmsson lne connectng nodes and j havng + jq s gven by Z app P Q j j = + j P 2 Q 2 P 2 Q j j j j As V s only a scalar, t cannot dfferentate between the quadrants n the R-X plane. Thus, the locaton of Z app depends on the drecton of P and Q flows. Clearly, swngs are severe when are large and V small. Under such crcumstance Zapp s small, and hence can cause a relay to trp. V 2 Pj and/or Work based upon Lyapunov stablty crteron has been reported n [28] to rank relays accordng to the severty of swngs. In [29] relay margn s used as a measure of how close a relay s from ssung a trp command. Bascally, t s the rato of the tme of longest consecutve stay of a swng n zone to ts tme dal settng (TDS). For relays that see swng characterstc outsde of ther zone settngs, the relay space margn s used. It s defned as the smallest dstance between the relay characterstc and the swng trajectory n R-X plane. To dentfy the most vulnerable relay, magntude of the rato of swng mpedance to lne mpedance s used as a performance parameter. The most vulnerable relay corresponds to one wth mnmum rato, where the search space extends over all the relays and tme nstants of smulaton. Reference [3] dscusses the challenges to relayng n the restructured power system operaton scenaro. In such a scenaro, there wll be varyng power flow patterns dctated by the market condtons. Durng the congeston perod, relay margns and relay space margns wll be reduced. Ths may pose challenges to system protecton desgn. The vulnerablty of a relay to power swngs s drectly dependent on severty of oscllatons n power flow observed n the prmary transmsson lne and adjacent transmsson lnes that are covered by backup zones. Hence, the problem of assessment of transmsson protecton system vulnerablty to power swngs translates nto assessment of oscllatons n power flow on a transmsson system due to dsturbance. (25) Qj 4

15 Engneerng Seres, Brkhauser Boston, 23. We wll show that branch mpedance trajectory senstvty (BITS) (that s, trajectory senstvty of rotor angles to branch mpedance), can be used to locate electrcal center [9] n the transmsson system and rank transmsson system dstance relays accordng to ther vulnerablty to trppng on swngs. Electrcal center and weakest lnk n a network For power systems that essentally behave as a two-area system under nstablty, the out-of-step relayng can be explaned by consderng the equvalent generators connected by a te lne. When the two generators fall out of step,.e., the angular dfference between the two generator voltage phasors s 8, they create a voltage zero pont on the connectng crcut. Ths s known as the electrcal center. A dstance relay perceves t as a sold 3-phase short-crcut and trps the lne. One way to locate the electrcal center [3] n a power system s to create a fault wth fault clearng tme greater than the crtcal clearng tme of the crcut breaker to make the resultng postfault system unstable. Through transent stablty smulatons, one dentfes the groups of acceleratng and deceleratng machnes n the system. The sub network nterconnectng such groups wll contan the electrcal center. For the relays contaned n the sub network, by smulatng the power swng on the R-X plane one can locate the relays for whch the power swng ntersects the transmsson lne mpedance. Ths lne contans the electrcal center of the system. The electrcal center wll also be observed by the backup relays dependng upon ther zone 2 and 3 settng. Usually, the relays near the electrcal center are hghly senstve to power swngs. A well-known property to characterze an electrcal center s that the network adjonng the electrcal center has low voltages. Such characterzaton s qualtatve and can be used as a screenng tool. Snce natural splttng of the system due to operaton of dstance relays takes place at the electrcal center, we refer to such a lne (or a transformer) as the weakest lnk n the network. To compress the nformaton of all rotor angles wth respect to a gven lne, for an m-machne system the followng norm frst ntroduced n [22] s used n the numercal examples secton. The BITS norm s computed as follows: S N = m 2 m 2 α ω + = x = x (26) where α = δ δ j and the jth-machne s chosen as the reference, and x s the transmsson lne reactance. Numercal Examples consdered. A -machne, 39-bus system [3] s used for llustratve examples. The followng two cases are Fault at bus 28 5

16 Engneerng Seres, Brkhauser Boston, 23. The fault s cleared at t cl =.6 s by trppng lne smultaneously from both ends. Table 5 captures the normalzed ndces for a subset of lnes, whch have hgh, medum, and low senstvtes. BITS senstvty norm s computed for all the lnes and normalzed wth respect to the maxmum one. Therefore, after normalzaton, the lne wth max BITS norm has value. For all other lnes, BITS norm s less than or equal to. We would characterze the lne as the most vulnerable lne because ts rank s, and ts absolute peak norm of s also very hgh. The normalzed ndces for all other lnes are much lower than that of the lne Hence, other lnes are less vulnerable to the swng. These nferences have been confrmed by usng the relay space margn (RSM) concept. Fgures 6 and 7 show the swng curves for relay on lnes and respectvely. Table 5. Normalzed BITS norm (nomnal or % loadng) Lne Normalzed Senstvty RSM Z mn As lne s the weakest lnk n the post fault system, t has a large possblty of developng the electrcal center n case a fault leads to nstablty (because of larger t cl or system loadngs). To confrm the locaton of the electrcal center, t cl was ncreased from.6 s to.7 s to create system nstablty. The small ncrease of. s n fault clearng tme nduced nstablty. Ths ndcates that exstng system s workng close to ts stablty lmts Reactance Resstance Fg. 6. Swng curve for relay on lne (stable) 6

17 Engneerng Seres, Brkhauser Boston, Reactance Resstance Fg. 7. Swng curve for relay on lne (stable) The swng trajectory for the relay on lne located near bus 29 for t cl =.7 s s shown n Fg. 8. As the trajectory cuts the transmsson lne characterstc n zone, the locaton of electrcal center on s thus establshed. Ths s clearly an unstable case. The relatve rotor angles n ths case are show n Fg. 9 where machne 9 s the unstable machne Reactance Resstance Fg. 8. Electrcal center locaton on lne (unstable) 7

18 Engneerng Seres, Brkhauser Boston, Relatve rotor angles (rad) t (s) Fg. 9. Relatve rotor angles for the unstable case Table 6 summarzes actual BITS norms as well as the results for varous loadng condtons (8%, 9%, and % of the system load). It can be seen that as system loadng ncreases, the trajectory senstvty also ncreases. As the system approaches a dynamcal stablty lmt, the trajectory senstvty for the crtcal lne jumps from 45.7 for 9 % loadng case to for % loadng case. Thus, a hgh peak value of maxmum senstvty can be used as an ndcator of reduced stablty margns. Also, note that the senstvty norm changes rapdly when the system loadng approaches the stablty boundary. Therefore, t can be used as an ndcator for system stablty margn. Table 6. Absolute norm for dfferent loadng condtons Lne 8% load 9% load % load Fault at bus 4 The fault s cleared at t cl =. s by trppng lne 4-5 smultaneously from both ends. For ths scenaro, results smlar to the case dscussed earler are summarzed n Tables 7 and 8. Electrcal center s located on lne -2. The results replcate the same pattern of behavor as dscussed n the prevous case. From Table 8 t can be seen that peak senstvty norm s qute low (6 n comparson to of Table 6), 8

19 Engneerng Seres, Brkhauser Boston, 23. and the senstvty norm does not change much wth varous loadng condtons. It ndcates that for ths fault, the loadng condtons are such that the system s stll far away from the stablty boundary. In fact, the crtcal clearng tme for ths fault wth the nomnal loadng condton s.29 s. Table 7. Normalzed BITS norm wth % load Lne Normalzed senstvty RSM Z mn Table 8. Absolute BITS norm for varous loadng condtons Lne 9% load % load % load Ths secton has developed the concept of maxmum rotor angle branch mpedance trajectory senstvty as a tool for assessng transmsson protecton system vulnerablty to angle stablty problems. It s used for applcatons n power systems such as. Rankng transmsson lnes and relays as vulnerablty to swngs. The rankng can help to mprove the relay coordnaton n the presence of power swngs.. Determnng locaton of electrcal center. Because transmsson lnes adjonng the electrcal center have low voltages, other devces such as FACTS could be used to strengthen system stablty.. Indcatng stablty margn. When BITS s gettng hgher, t ndcates that the operatng pont of system s movng closer to the system stablty boundary. 6. Conclusons Ths paper summarzes a recent approach taken for dynamc securty assessment of power systems usng trajectory senstvty analyss. It s ndependent of model complexty and requres some apror knowledge of the system crtcalty. In power systems, years of experence provde ths. Future research wll nvolve applyng ths technque to releve congeston n deregulated systems. Future research should extend the applcaton to explorng the connecton between stablty and senstvty theory n hybrd dynamcal systems whch s an emergng area of research n control [32]. 9

20 Engneerng Seres, Brkhauser Boston, 23. In power systems, examples nclude state dependent varaton of taps n tap changng transformers or swtchng of FACTS devces for whch the theory of trajectory senstvty analyss s already avalable [2]. 7. Acknowledgements Ths research was supported by NSF grant ECS -474 and the Granger Foundaton. References [] G. L. Wlson and P. Zarakas, Anatomy of a blackout, IEEE Spectrum, vol. 5, no. 2, pp , Feb.978. [2] O. Alsac and B. Stott, Optmal load flow wth steady state securty, IEEE Transactons on Power Apparatus and Systems, vol. PAS-93, pp , 974. [3] M. A. Pa, Energy Functon Analyss for Power System Stablty. Norwell, MA: Kluwer Academc Publshers, 989. [4] A. A. Fouad and V. Vttal, Power System Transent Stablty Analyss Usng the Transent Energy Functon Method. Englewood Clffs, NJ: Prentce-Hall, 99. [5] H. D. Chang, C. C. Chu, and G. Cauley, Drect stablty analyss of electrc power system usng energy functons: Theory, applcatons, and perspectve, Proceedngs of the IEEE, vol. 83, no., pp , Nov. 995 [6] M. Pavella, D. Ernst, and D. Ruz-Vega, Transent Stablty of Power Systems: A Unfed Approach to Assessment and Control. Norwell, MA: Kluwer Academc Publshers, 2. [7] K. W. Chan, A. R. Edwards, R.W. Dunn, and A. R. Danels, On-lne dynamc securty contngency screenng usng artfcal neural networks, IEE Proceddngs on Generaton, Transmsson, and Dstrbuton, vol. 47, no. 6, pp , Nov. 2. [8] Y. Mansour, E. Vaahed, and M. A. El-Sharkaw, Dynamc securty contngency screenng and rankng usng neural networks, IEEE Transactons on Neural Networks, vol. 8, no. 4, pp , July 997. [9] T. B. Nguyen, Dynamc securty assessment of power systems usng trajectory senstvty approach, Ph.D. dssertaton, Unversty of Illnos, Urbana-Champagn, 22. [] T. B. Nguyen and M. A. Pa, Dynamc securty-constraned reschedulng of power systems usng trajectory senstvtes, IEEE Transactons on Power Systems, to appear. [] H. You, V. Vttal, Z. Yang, Self-Healng n Power Systems: An approach usng slandng and rate of frequency declne based load sheddng, IEEE Transactons on Power Systems, to appear. [2] P M. Frank, Introducton to System Senstvty Theory. New York: Academc Press, 978. [3] M. Eslam, Theory of Senstvty n Dynamc Systems. New York: Sprnger-Verlag,994. [4] H.. W. Bode, Network Analyss and Feedback Amplfer Desgn. New York: Van H Nostrand, 945. [5] R. Tomovc, Senstvty Analyss of Dynamc Systems. New York: McGraw-Hll,

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