Transient Stability Assessment Using Individual Machine Equal Area Criterion Part I: Unity Principle

Transcription

1 1 Transent Stablty Assessment Usng Indvdual Mahne Equal Area Crteron Part I: Unty Prnple Songyan Wang, Jla Yu, We Zhang Member, IEEE arxv: v1 [eess.sp] 4 De 2017 Abstrat Analyzng system trajetory from the perspetve of ndvdual mahnes provdes a dstntve angle to analyze the transent stablty of power systems. Ths two-paper seres propose a dret-tme-doman method that s based on the ndvdualmahne equal area rteron. In the frst paper, by examnng the mappng between the trajetory and power-vs-angle urve of an ndvdual mahne, the stablty property to haraterze a rtal mahne s larfed. The mappng between the system trajetory and ndvdual-mahne equal area rteron s establshed. Furthermore, a unty prnple between the ndvdual-mahne stablty and system stablty s proposed. It s proved that the nstablty of the system an be onfrmed by fndng any one unstable rtal mahne, thene, the transent stablty of a multmahne system an be montored n an ndvdual-mahne way n transent stablty assessment. Index Terms transent stablty, equal area rteron, ndvdual mahne energy funton, partal energy funton COI CCT CDSP CUEP DLP DSP EAC IEEAC IMEAC IMEF IVCS LOSP OMIB PEF TSA UEP A. Lterature Revew ABBREVIATION Center of nerta Crtal learng tme DSP of the rtal stable mahne Controllng UEP Dynam lberaton pont Dynam statonary pont Equal area rteron Integrated extended EAC Indvdual-mahne EAC Indvdual mahne energy funton Indvdual mahne-vrtual COI mahne system Loss-of-synhronsm pont One-mahne-nfnte-bus Partal energy funton Transent stablty assessment Unstable equlbrum pont I. INTRODUCTION In the last deades, great efforts have been made to the applaton of dret methods n the transent stablty analyss. In S. Wang s wth Department of Eletral Engneerng, Harbn Insttute of Tehnology, Harbn , Chna (e-mal: wangsongyan@163.om). J. Yu s wth Department of Eletral Engneerng, Harbn Insttute of Tehnology, Harbn , Chna (e-mal: yupwrs@ht.edu.n). W. Zhang s wth Department of Eletral Engneerng, Harbn Insttute of Tehnology, Harbn , Chna (e-mal: wzps@ht.edu.n). early work, Lyapunov method was proved to yeld onservatve results [1]. Hereafter, dret methods suh as CUEP method, sustaned fault method and IEEAC method had reeved onsderable attenton and aheved advanes [2] [5]. These methods montor transent behavors of all mahnes n the system n TSA, whh are also named as global methods. Unlke global methods, some transent stablty analysts nlne to observe the power system transent stablty from a dstntve ndvdualmahne angle as the nstablty of a mult-mahne system s determned by the moton of some unstable rtal mahnes f more than one mahne tends to lose synhronsm [6]. Stmulated by ths onept of ndvdual-mahne perspetve, Vttal and Fouad [7], [8] stated that the nstablty of the system depends on the transent energy of ndvdual mahnes and IMEF was proposed. Stanton [9], [10] performed a detaled mahne-bymahne analyss of a mult-mahne nstablty, and PEF was used to quantfy the energy of a loal ontrol aton. Later, EAC of the rtal mahne s appled n PEF method to dentfy system stablty [12]. Rastgoufard et al. [13] used an IMEF n synhronous referene to determne the transent stablty of a multmahne system. Haque [14] proposed an effent ndvdual-mahne method to ompute the CCT of the system under transents. Ando and Iwamoto [15] presented a potental energy rdge whh an be used to predt the sngle-mahne stablty. Among all these works of ndvdual-mahne methods, the PEF method s qute representatve and an be seen as a mlestone beause Ref. [10], [11] ntally explaned fundamental theores and also provded some valuable underlyng hypothess regardng the ndvdual-mahne methods. Although ndvdualmahne methods were proved to be effetve for the stablty analyss, these methods were at a standstll for deades. The reason s that some rual onepts of the ndvdual-mahne methods were mssng or were llustrated n an unsystemat and tutoral form, leavng onfusons unlarfed and ontroversal problems unsolved when they are used for TSA. B. Sope and ontrbuton of the paper In ths two-paper seres a dret-tme-doman method that s based on ndvdual-mahne equal area rteron (IMEAC) s proposed. The frst paper systematally larfes the mehansm of the proposed method to montor the transent stablty of a mult-mahne system. The ompanon paper apples the proposed method for TSA and CCT omputaton. In ths paper, based on the atual trajetory of the system trajetory durng transents, the Kmbark urve of a rtal mahne s frst analyzed, and then

2 2 IMEAC s proved to strtly hold for a rtal mahne. Seond, followng trajetory stablty theory the onept of ndvdualmahne trajetory (IMT) s proposed, and the mappng between system trajetory and Kmbark urve of an ndvdual mahne s establshed. In the end of the paper, the unty prnple between ndvdual-mahne stablty and system stablty s proposed. It shows that IMT of any one unstable rtal mahne an drve system trajetory to go unstable, thene, and the transent stablty of a mult-mahne system an be montored n an ndvdual-mahne way durng TSA. Contrbutons of ths paper are summarzed as follows: () Followng trajetory stablty theory, ths paper explans the mehansm of the transent stablty of a mult-mahne system from an ndvdual-mahne angle. The transent nstablty of the system an be determned by the nstablty of any one unstable rtal mahne; () Some mstakes about stablty-haraterzaton of a rtal mahne n Ref. [10], [11] are orreted n ths paper; () The unty prnple expltly explans the relatonshp between ndvdual-mahne stablty and system stablty s proposed. Ths may release the potental of the usage of ndvdualmahne montorng n TSA. In ths paper we only dsuss the frst-swng stablty of a rtal mahne, and swng stablty n ths paper only depts the stablty state of a rtal mahne when the veloty of the mahne reahes zero, rather than followng the onventonal global onept. Three test systems are appled n ths two-paper seres. Test System-1 (TS-1) s a modfed IEEE 39-bus system. In TS-1 the nerta onstant of Unt 39 s modfed to 200 p.u. from 1000 p.u.; Test System-2 (TS-2) s the standard IEEE 118-bus system. Test System-3 (TS-3) s a pratal 2766-bus nteronneted system. All faults are three phase short-ruts faults whh ourred at 0 s and they are leared wthout lne swthng. Fault types n ths two-paper seres are desrbed n the form of [test-system, fault loaton, fault-on tme]. The smulatons of TS-1 and that of TS-2 are fully based on the lassal model gven n [5]. The smulatons of TS-3 are based on omplated dynam models, and the parameters of ths system an be found n the ompanon paper. The remanng paper s organzed as follows. In Seton II, IMEAC of a rtal mahne s analyzed. In Seton III, the Kmbark urves of both rtal mahnes and non-rtal mahnes that rely on atual system trajetory n the mult-mahne system are analyzed. In Seton IV, the mappng between IMT and the Kmbark urve of a rtal mahne s establshed. In Seton V, the unty prnple of system stablty and stablty of a rtal mahne s depted n the sense of IMT. In Seton VI, an example about the applaton of the proposed method s demonstrated. Conlusons are provded n Seton VII. II. INDIVIDUAL-MACHINE EQUAL AREA CRITERION A. Equaton of Moton of an ndvdual mahne Conventonally, for a dret method that s based on COI referene, an ndvdual mahne should be presely expressed as an ndvdual mahne n COI referene. For a n-mahne Fg. 1: Trajetory of the vrtual COI mahne [TS-1, bus-34, 0.202s] system wth rotor angle δ and nerta onstant M, the moton of an ndvdual mahne n the synhronous referene s governed by dfferental equatons: { δ = ω M ω = P m P e (1) Poston of the COI of the system s defned by: n δ COI = 1 M T M δ =1 n ω COI = 1 M T M ω (2) =1 P COI = n (P m P e ) where M T = n M. =1 =1 From (2), the moton of COI s determned by: { δcoi = ω COI M T ω COI = P COI (3) Eqn. (3) ndates that COI an also be seen as a vrtual mahne wth ts own equaton of moton beng desrbed as the aggregated moton of all mahnes n the system.the trajetory of the vrtual COI mahne n synhronous referene s shown n Fg. 1. Followng (1) and (3), sne mahne and COI are two sngle mahnes wth nteratons, a two-mahne subsystem an be formed by usng these two mahnes, whh s defned as a SVCS, as shown n Fg. 1. Sne mahne and COI are two ndvdual mahnes wth nteratons, a two-mahne system whh s named as Indvdual mahne-vrtual COI mahne system (IVCS) an be formed by these two mahnes, as n Fg. 2. The relatve trajetory between a rtal mahne and the vrtual COI mahne n an IVCS s shown n Fg. 3.

3 3 Fg. 2: Two-mahne system formed by mahne and vrtual COI mahne. Fg. 3: The nstablty of the relatve trajetory between a rtal mahne and the vrtual COI mahne [TS-1, bus-34, 0.202s]. (a) an IVCS n synhronous referene. (b) an ndvdual mahne n COI referene Followng (1) and (3), the relatve moton between mahne and the vrtual COI mahne of the SVCS an be gven as: { θ = ω (4) M ω = f where f = P m P e M M T P COI θ = δ δ COI ω = ω ω COI Eq. (4) depts the separaton of an ndvdual mahne wth respet to the COI of all mahnes n the system, whh s substantally dental to the moton of an ndvdual mahne n COI referene. Therefore, eah ndvdual mahne n COI referene should be presely depted as an IVCS that s formed by a par of mahnes. Yet, n ths paper for hstoral reasons the ndvdual mahne or rtal mahne s stll used n default for smplfaton. B. Strt EAC haraterst of an ndvdual mahne Sne EAC only strtly holds n the OMIB system and the twomahne system, the above analyss gven n Seton II.A s of value beause t ndates that EAC strtly holds for an ndvdual mahne as the IVCS s a prese two-mahne system. Followng (4), we have: f dθ = M ω d ω (5) Along the atual fault-on trajetory untl fault learng, we have θ θ 0 ω (f (F) )dθ = M ω d ω (6) ω j 0 where f (F) orresponds to f durng fault-on perod. Eqn. (6) an be further expressed as: Fg. 4: Smulatons of the fault [TS-1, bus-34, 0.202s]. (a) Kmbark urve of Mahne 34. (b) System trajetory 1 2 M ω 2 = θ j θ 0 {0 ( f (F) )}dθ (7) Along the atual post-fault trajetory after fault learng, we have: θ θ ω f (PF) dθ = M ω d ω (8) ω where f (PF) orresponds to f durng post-fault perod. Eqn. (8) an be further expressed as: 2 θ 1 2 M ω () = θ Substtutng (7) nto (9) yelds: θ j θ 0 {0 ( f (F) )}dθ = 1 2 M ω 2 + { f (PF) 0}dθ M ω 2 θ θ (9) { f (PF) 0}dθ (10) A typal Kmbark urve [11] (power-vs-angle urve) of an unstable rtal mahne that s formulated wth the atual smulated system trajetory n the θ f spae s shown n Fg. 4 (a). The rotor angles of the system are shown n Fg. 4 (b).

4 4 Assume the Kmbark urve of the mahne reahes the lberaton pont (P3) as n Fg. 4. Under ths rumstane, both ntegral parts n (9) an be seen as areas, and the dfferene between the aeleraton area and the deeleraton area s the resdual K.E. of the mahne at the lberaton pont (P3). Therefore, the mahne an be judged as unstable as long as the aeleraton area s larger than the deeleraton area (.e., the resdual K.E. s postve at the lberaton pont), and the mahne an be judged as stable f the aeleraton area s equal to the deeleraton area (.e., the resdual K.E. s strtly zero at the lberaton pont). Ths fully proves that EAC strtly holds for an ndvdual mahne. In power system transent stablty analyss, rtal mahnes are those a few severely dsturbed mahnes, whh are most possble to separate from the system, whle non-rtal mahnes are those slghtly dsturbed mahnes and they osllate durng postfault perod. Therefore, the transent behavor and orrespondng Kmbark urve of a rtal mahne wll be qute dfferent from that of a non-rtal mahne, whh wll be analyzed n followng setons. In ths two-paper seres the dentfaton of the rtal mahnes s fully based on the methodologes that were proposed n Ref [6] and [14] whh onsder rtal mahnes as those severely dsturbed mahnes wth advaned angles [6] and hgh aeleraton ratos [14]. III. KIMBARK CURVE OF AN INDIVIDUAL MACHINE A. Kmbark Curve of an Unstable Crtal Mahn From numerous smulatons, the representatve Kmbark urve of a rtal mahne gong unstable s shown n Fg. 5 (a). The orrespondng system trajetory s shown n Fg. 5 (b). Mahnes 33, 34 and 39 are rtal mahnes for the fault [TS-1, bus-34, 0.219s]. From Fg. 5 (a), a rtal mahne frstly aelerates from P1 to P2 durng fault-on perod, then t deelerates from P2 to P3 after fault s leared. One system trajetory goes aross P3, the rtal mahne wll aelerate n COI referene and then separate from the system. Thus P3 an be defned as dynam lberaton pont (DLP) of ths unstable rtal mahne [10]. Spefally, some rtal mahnes may have negatve veloty after fault learng and fnally ant-aelerates wth tme. In ths ase the Kmbark urve of the rtal mahne would seem to be rotated, as shown n Fg. 6 (a). From Fgs. 4-6, the unstable ase of the rtal mahne an be haraterzed by the ourrene of the DLP wth f of the mahne beng zero: w 0,f = 0 (11) Eq.?? orrets the nstablty haraterzaton of the rtal mahne n Ref. [10], [11] beause these two papers negleted the ant-aeleratng ase as shown n Fg. 6 (a). Followng EAC, when ouple mahnes go unstable, we have: A ACCj > A DECj (12) Fg. 5: Smulatons of the fault [TS-1, bus-34, 0.219s]. (a) Kmbark urve of Mahne 34. (b) System trajetory where A ACCj = θ θ 0 A DECj = θ DLP θ ( f (F) )dθ = 1 2 M ( f (F) )dθ ω () 2 Note that only the msmath f of mahne s zero whle msmath of other mahnes are not zero at DLP. For more than one rtal mahnes that go unstable after fault learng, eah rtal mahne orresponds to ts unque DLP. From analyss above, the Kmbark urve of an unstable rtal mahne has a lear aeleratng-deeleratng-aeleratng haraterst. Sne the Kmbark urve s formulated from atual smulated system trajetory, the Kmbark urve of an unstable rtal mahne and ts orrespondng DLP vares wth the hange of the faults. For nstane, DLP 34 n Fg. 4 (a) s dfferent from that n Fg. 5 (a) f the system s subjet to a dfferent fault. The Kmbark urve of an ndvdual mahne n the proposed method s a bt smlar to that of the OMIB system as n the IEEAC method and SIME method. However, we emphasze that the EAC n the proposed method s strtly based on the Kmbark urve of an ndvdual mahne n COI referene, whh s qute

5 5 Fg. 7: Smulatons of the fault [TS-1, bus-34, 0.180s]. (a-) Kmbark urves of Mahnes 33, 34 and 39. (d) System trajetory From Fg. 7, the stable ase of the rtal mahne an be haraterzed by the ourrene of the DSP wth the veloty of the mahne beng zero: w = 0,f 0 (13) Fg. 6: Smulatons of the fault [TS-1, bus-3, 0.580s]. (a) Kmbark urve of Mahne 39. (b) System trajetory dfferent from that n IEEAC method and SIME method that s based on the equvalent OMIB system n a mult-mahne system. B. Kmbark Curve of a Stable Crtal Mahne From smulatons, representatve Kmbark urves of rtal mahnes beng stable are shown n Fgs. 7 (a-). The system trajetory s shown n Fg. 7 (d). Mahnes 33, 34 and 39 are rtal mahnes for the fault [TS-1, bus-34, 0.180s]. From Fgs. 7(a-), for a stable rtal mahne, the mahne frst aelerates from P1 to P2 durng fault-on perod, then t deelerates and the veloty of the mahne then reahes zero at P3. Therefore, the mahne never goes aross DLP,.e., f wll not nterset wth horzontal zero lne. Instead, f mght turn upward or turn downward beause the osllaton of other mahnes may mpede baktrakng of the system trajetory. In ths way the rtal mahne wll not separate from the system and s mantaned stable n the frst swng, thus P3 wth zero veloty and the maxmum angle an be defned as the dynam statonary pont (DSP) of the rtal mahne. Eq. (13) orrets the stablty haraterzaton of the rtal mahne n Ref. [10, 11] beause these two papers negleted the ant-aeleratng ase as n Fg. 7 (). Followng IMEAC, when a rtal mahne s stable, we have: where A ACCj = θ DSP θ A ACCj = A DECj (14) ( f (F) )dθ In the Kmbark urve of the stable rtal mahne, DSP s the nfleton pont where f turns upward or downward. DSP desrbes frst swng stablty of a rtal mahne. At DSP of the stable rtal mahne, only the veloty of the rtal mahne s zero whle the veloty of other mahnes are nonzero. For the ase that more than one rtal mahnes are stable after fault learng, eah rtal mahne orresponds to ts unque DSP. The Kmbark urve of a stable rtal mahne has a lear aeleratng-deeleratng haraterst before DSP ours, and the DSP of a stable rtal mahne wll vary wth the hange of the faults. C. Kmbark Curve of a Crtal-stable Crtal Mahne Followng stable and unstable haraterzaton of a rtal mahne, one the rtal mahne s rtal stable, the mahne s stll stable and f wll nflet at the DSP of the rtal stable mahne (CDSP). However, the CDSP s speal beause t

6 6 f t t t Fg. 8: Smulatons of the fault [TS-1, bus-34, 0.201s]. (a) Kmbark urve of the Mahne 34. (b) System trajetory just falls n the zero horzontal lne, whh desrbes the rtally stable state of the mahne. Therefore, the rtal stable ase of a rtal mahne s haraterzed by: w = 0,f = 0 (15) The Kmbark urve of a rtal stable mahne and orrespondng system trajetory are shown n Fgs. 8 (a) and (b), respetvely. In ths ase Mahne 34 s rtal stable whle Mahnes 33 and 39 are stable. D. Kmbark Curve of a Non-rtal Mahne Sne non-rtal mahnes are majortes that are slghtly dsturbed by faults, non-rtal mahnes generally mantan synhronsm durng post-fault perod. In other words, the non-rtal mahnes may osllate wth tme, and the Kmbark urves of nonrtal mahnes do not have a lear aeleratng-deeleratng haraterst ompared wth that of rtal mahnes. The dstntve feature of the Kmbark urve of the rtal mahne reveals the potental of usng IMEAC when judgng the Fg. 9: Rotor angles of the system [TS-1, bus-2, 0.430s] stablty of a rtal mahne. Therefore, the foremost rual work for the mult-mahne transent stablty analyss s to dept the relatonshp between the stablty of the system and that of rtal mahnes, whh wll be analyzed n the followng setons. IV. MAPPING BETWEEN TRAJECTORY AND KIMBARK CURVE OF AN INDIVIDUAL MACHINE A. Indvdual Mahne Trajetory Theoretally, the transent stablty of the system should be expltly expressed as the transent stablty of the system trajetory. If a system goes unstable, the separaton of mahnes n the system would our along tme horzon. In ths paper, the varaton of the rotor angle of an ndvdual mahne along tme horzon s defned as the ndvdual-mahne trajetory (IMT). A smulaton ase to demonstrate the system trajetory and IMTs s shown n Fgs. 9 and 10. Mahnes 37, 38 and 39 are rtal mahnes n ths ase. The IMTs n Fg. 10 reveal a ommonly observed phenomenon n the power transent stablty,.e., after fault learng the IMTs of rtal mahnes flutuate most severely, and they are most possble to separate from the system. Comparatvely, IMTs of non-rtal mahnes flutuate slghtly and these mahnes hardly separate from the system. Deptng the varaton of the IMT n a mathematal form, the IMT of a rtal mahne gong unstable s dental to the rotor angle of the mahne n COI referene gong nfnte wth tme, whh an be expressed as: θ,t = t t 0 ω dt = + t = + (16) Comparatvely, the IMT of a rtal mahne beng stable s dental to the rotor angle of the mahne n COI referene beng bounded wth tme, whh an be denoted as: t θ,t = ω dt < θ bound t (0,+ ) (17) t 0 where θ bound s the upper bound of θ,t. Based on the analyss above, the orgnal thnkng of trajetory stablty of the system an be merted as below:

7 7 f Fg. 10: IMTs of ndvdual mahnes [TS-1, bus-2, 0.430s]. (a-d) IMTs of Mahnes 37, 38, 39 and 32. () If IMTs of all rtal mahnes are bounded, the separaton of mahnes n the system s mpossble to our, and system an mantan stable (Fg 10 ()). () If IMTs of some rtal mahnes go nfnte along wth tme, the separaton of mahnes n the system s ertan to our, and the system would go unstable (Fgs 10 (a,b)). () The IMTs of non-rtal mahnes always flutuate slghtly and they hardly separate from the system, thus IMTs of non-rtal mahnes are unable to ause system to go unstable (Fg 10 (d)). The statements above an be seen as the foundaton of the proposed method. From the angle of the trajetory stablty, the slght flutuatons of the IMTs of the non-rtal mahnes have no effet to the nstablty of the system. Comparatvely, those severely flutuated IMTs of rtal mahnes are most possble to go nfnte and ause system to go unstable. Therefore, the followng rteron s proposed for TSA: The system operator may only montor stablty of IMTs of rtal mahnes durng post-fault transent perod. Furthermore, the prme objetve of the system operator s to fnd out the IMTs of unstable rtal mahnes among all rtal mahnes n the system, beause only IMTs of unstable rtal mahnes may ause system to go unstable. B. 3-dmensonal Kmbark Curve of a Crtal Mahne In transent stablty analyss, a sgnfant defet of observng IMT s that the transent behavor of the mahne s qute dffult to be depted. To solve ths problem, t s neessary to map the stablty analyss of the IMT n the t θ spae to the θ f spae wheren IMEAC an be used to analyze the ndvdualmahne stablty. In order to demonstrate the mappng between IMT and IMEAC, a 3-dmensonal Kmbark urve (3DKC) of a rtal mahne n the t θ f spae s proposed n ths paper, f t t Fg. 11: 3DKC of an unstable rtal mahne n a mult-mahne system [TS-1, bus-34, 0.202s]. as shown n Fg. 11. All parameters n the 3DKC of the mahne are fully formulated from the atual smulated system trajetory. From Fg. 11, by usng the 3DKC of the rtal mahne, the IMT of the rtal mahne n the t θ spae s mapped to the Kmbark urve of the mahne n theθ f spae, and the stablty of a rtal mahne an be easly measured by the ourrene of DLP or DSP n the Kmbark urve of ths mahne. Ths proves that the stablty of IMT of a rtal mahne an be dentfed va IMEAC. We extend the onept of the 3DKC of an ndvdual mahne to the stablty evaluaton of a system wth n mahnes. Followng the defnton of IMT, t s obvous that the system trajetory an be seen as the set that omprses of IMTs of all ndvdual mahnes n the system. The mappngs between system trajetory and 3DKCs of all ndvdual mahnes n the system are shown n Fg. 12. V. UNITY PRINCIPLE A. Only-one-mahne Montorng From analyss n Seton 4, sne the nstablty of the system s determned by IMTs of unstable rtal mahnes, the system operators an montor the IMT of eah rtal mahne n the system n parallel to dentfy the real unstable rtal mahne. Then one queston emerges: ould the nstablty of the system be evaluated f the system operator does not montor all rtal mahnes? We extend the trajetory montorng to an extreme only-onemahne way. Takng the ase n Fg. 10 for example, one an fnd that the IMTs of rtal mahnes 37 and 38 both go unstable. Assume under an extreme rumstane that the system operator knows ten mahnes are operatng n the system. However, he only fouses on Mahne 37 and does not observe all the other mahnes n the system, as shown n Fg. 13. Under ths extreme only-one-mahne montorng rumstane, ould the system stll be defned as unstable? Fg. 13 ntutvely demonstrates the mehansm of usng IMEAC for TSA. From the fgure, the system trajetory omprses

8 8 Fg. 12: Mappngs between system trajetory and 3DKCs of all ndvdual mahnes n the system [TS-1, bus-34, 0.202s] of n IMTs and eah mahne s IMT orresponds to ts unque 3DKC, thus the system trajetory an be easly mapped nto n 3DKCs. Among all 3DKCs, onsderng that only IMTs of rtal mahnes may ause the nstablty of the system, the system operators only need to observe 3DKCs of rtal mahnes by negletng that of non-rtal mahnes. Insde 3DKC of eah rtal mahne, the stablty of the mahne s evaluated va IMEAC (.e., the ourrene of DLP or DSP). One one or more rtal mahnes are found to go unstable, the system an be judged as unstable aordng to the trajetory stablty theory. From Fg. 13 one an see, n COI referene DLP 37 ours at 0.777s and θ 37 reahes 687 deg. at 1.500s. Under ths extreme ndvdual-mahne montorng rumstane, although IMTs of all the other mahnes n the system are not montored, t s qute obvous that the system annot be mantaned stable beause IMT37 keeps separatng from the system wth tme. Theorem: Transent nstablty of any one IMT determnes the transent nstablty of the system trajetory. Proof: Usng Reduto-ad-absurdum, assume the system s Fg. 13: Montorng IMT of only one unstable mahne [TS-1, bus-2, 0.430s]

9 9 Fg. 14: Multple LOSPs n a multmahne system [TS-1, bus-2, 0.430s] stable when an IMT n the system already goes nfnte wth tme. Followng trajetory stablty theory, ths assumpton s ontradt to the suffent and neessary ondton suh that the system should mantan stable,.e., IMTs of all mahnes n the system should be bounded along tme horzon (Seton IV), thus theorem holds. Followng analyss above, the unty of ndvdual-mahne stablty and system stablty an be expressed as: (I) The system an be judged as stable f all rtal mahnes are stable. (II) The system an be judged as unstable as long as any one rtal mahne s found to go unstable. Prnples I and II substantally llustrates the unty of ndvdual-mahne stablty and system stablty. Espeally, Prnple II s of nterest beause t reflets that the transent stablty of a mult-mahne system may be montored n an ndvdualmahne way, and the nstablty of the system an be determned by any one unstable rtal mahne wthout montorng all rtal mahnes n the system. Ths provdes a qute novel ndvdual-mahne angle for transent stablty analyss n TSA. B. Ourrene of Multple LOSPs of the System Followng unty prnple, sne eah unstable rtal mahne may ause system to go unstable, the DLP of eah unstable rtal mahne an be seen as the LOSP of the system. Therefore, multple LOSPs mght exst along post-fault system trajetory f more than one rtal mahnes goes unstable, and these LOSPs an be lassfed as below: Leadng LOSP: The leadng LOSP s defned as the frst ourred DLP along tme horzon; Laggng LOSP: The laggng LOSP s defned as the DLP that ours later than the leadng LOSP. From defntons above, the system an be judged as unstable one the leadng LOSP or the laggng LOSPs our. However, the leadng LOSP s ertanly the most valuable for the system operators beause the system starts separatng at ths pont, as shown n Fg. 14. C. Mahne-by-mahne Stablty Judgement In atual TSA envronment, the system operator may montor eah rtal mahne n parallel one fault s leared, and the stablty of a rtal mahne an be dentfed one the TABLE I: ACCELERATION POWER OF MACHINES AT DLP 3 7 Generator f (p.u.) Generator f (p.u.) orrespondng DLP or DSP ours n the Kmbark urve. Yet, sne DSPs and DLPs our one after another along post-fault system trajetory as analyzed n Seton II, the stablty of rtal mahnes n the system an only be dentfed n a mahne-bymahne way along tme horzon. Furthermore, durng ths proess the system an be judged as unstable mmedately one the leadng LOSP ours wthout watng for the stablty judgements of the rest of rtal mahnes. Detaled analyss wll be provded n the next seton. A. Parallel Montorng VI. CASE STUDIES The ase [TS-1, bus-2, 0.430s] s provded here to demonstrate the mahne-by-mahne stablty judgement when usng proposed method n TSA. The smulated system trajetory s shown n Fg. 15. In Fg. 15, one fault s leared, Mahnes 37, 38 and 39 are dentfed as rtal mahnes. Therefore, the system operator montors IMTs of these three rtal mahnes n parallel by negletng that of non-rtal mahnes. Along tme horzon the system operator an fous on followng nstants. DLP 38 ours (0.614s): Mahne 38 s judged as unstable. DSP 39 ours (0.686s): Mahne 39 s judged as stable. DLP 37 ours (0.777s): Mahne 37 s judged as unstable. The stablty of the system s judged as below: DLP 38 ours (0.614s): DLP 38 s defned as leadng LOSP, and the system s judged as unstable. DLP 37 ours (0.777s): DLP 37 s defned as laggng LOSP, yet the system has already gone unstable for a whle. From analyss above, DLP 38 and DLP 37 are the leadng LOSP and laggng LOSP, respetvely. f s of all mahnes n the system at the nstant of the ourrene of DLP 37 (0.777s) are shown n Table I. From Table I one an see, at 0.777s only f 37 s zero whle f of other mahnes are not zero, thus DLP 3 7 s only the aeleraton pont of Mahne 37, whh s meanngless to the stablty analyss of other rtal mahnes n the system. B. Only-one-mahne Montorng For the ase gven n Fg. 15, assume that the system operator montors only one rtal mahne and msses montorng the other two rtal mahnes. Under suh rumstane, three dfferent ases are shown as below: a) Only Mahne 38 s montored In ths ase the system an be judged as unstable by the only montored unstable Mahne 38 va unty prnple, and the leadng LOSP an be obtaned. b) Only Mahne 37 s montored

10 10 Fg. 15: Demonstraton of parallel montorng [TS-1, bus-2, 0.430s] In ths ase the system an also be judged as unstable by the only montored unstable Mahne 37. Yet, the leadng LOSP annot be obtaned. Only laggng LOSP an be obtaned. ) Only Mahne 39 s montored In ths ase the system operator annot onfrm that the system s stable or not by the only-montored stable Mahne 39, and the rest of rtal mahnes stll need to be montored to onfrm the nstablty of the system. The ndvdual-mahne montorng ases above ndate that eah rtal mahne s status for transent stablty analyss n the system s dfferent, whh wll be analyzed n the ompanon paper. VII. CONCLUSION AND DISCUSSION Through the analyss of ths paper, one an onlude followng: () EAC s proved to strtly hold for a rtal mahne. The Kmbark urve of a rtal mahne exhbts a strong aeleratng-deeleratng haraterst. () The rtal mahnes are most possble to separate from the system, and the system operator may only fous on analyzng the stablty of rtal mahnes. () A rtal mahne gong unstable n θ f spae s dental to the IMT of the rtal mahne gong unstable n t θ spae. (v) The unty prnple ndates that montorng the transent stablty of the mult-mahne system an be treated from an ndvdual-mahne angle, and transent nstablty of the multmahne system an be determned by any one unstable rtal mahne. In the ompanon paper, the applaton of the IMEAC and ndvdual-mahne stablty judgement wll be analyzed, whh may demonstrate the effetveness of usng proposed method n TSA. REFERENCES [1] A. A. Fouad, Stablty theory-rtera for transent stablty, n Pro. Conf. on System Engneerng for Power: Status and Prospets, Hennker, NH, [2] T. Athay, R. Podmore, and S. Vrman, A pratal method for dret analyss of transent stablty, IEEE Trans. Power Apparatus and Syst. vol. PAS-98, pp , [3] N. Kakmoto, Y. Ohsawa, and M.Hayash, Transent stablty analyss of eletr power system va Lur s type Lyapunov funton, Part I New rtal value for transent stablty, Trans. IEE of Japan, Vol. 98, no. 5-6, pp , [4] Y. Xue, Th. Van Cutsem, and M.Rbbens-Pavella, Extended Equal Area Crteron Justfatons, Generalzatons, Applatons, IEEE Trans. Power Syst. vol. 4, no. 1, pp , [5] C. Jakpattanajt, A. Yokoyama, and N. Hoonhareon, Enhanng Small- Sgnal and Transent Stablty Performanes n Power Systems wth Integrated Energy Funton and Funtonal Senstvty, IEEJ Trans, vol. 11, no. 1, pp , [6] A. A. Fouad and S.E. Stanton, Transent stablty of a mult-mahne power system Part I and II, IEEE Trans. Power Apparatus and Syst. vol. PAS-100, pp , [7] V. Vttal, Power system transent stablty usng rtal energy of ndvdual mahnes, Ph.D dssertaton, Iowa State Unversty, [8] A. Mhel, A. A. Fouad, and V. Vttal, Power system transent stablty usng ndvdual mahne energy funtons, IEEE Trans. on Cruts Syst. vol. CAS-30, no. 5, pp , [9] S. E. Stanton, Assessment of the stablty of a mult-mahne power system by the transent energy margn, Ph.D dssertaton, Iowa State Unversty, [10] S. E. Stanton and W. P. Dykas, Analyss of a loal transent ontrol aton by partal energy funtons, IEEE Trans. Power Syst. vol. 4, no. 3, pp , [11] S. E. Stanton, Transent stablty montorng for eletr power systems usng a partal energy funton, IEEE Trans. Power Syst. vol. 4, no. 4, pp , [12] S.E. Stanton, C. Slvnsky, K. Martn, and J. Nordstrom, Applaton of phasor measurements and partal energy analyss n stablzng large dsturbanes, IEEE Trans. Power Syst.,, vol. 10, no. 1, pp , [13] P. Rastgoufard, A. Yazdankhah, and R. A. Shlueter, Mult-mahne equal area based power system transent stablty measure, IEEE Trans. Power Syst. vol. 3, no. 1, pp , 1988.

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