Decentralized Robust Control for Damping Inter-area Oscillations in Power Systems

Transcription

1 Decentralzed Robust Control for Dampng Inter-area Oscllatons n Power Systems Janmng Lan, Shaobu Wang, Rusheng Dao and Zhenyu Huang arxv: v [cs.sy] 8 Jan 07 Abstract As power systems become more and more nterconnected, the nter-area oscllatons has become a serous factor lmtng large power transfer among dfferent areas. Underdamped (Undamped) nter-area oscllatons may cause system breakup and even lead to large-scale blackout. Tradtonal dampng controllers nclude Power System Stablzer (PSS) and Flexble AC Transmsson System (FACTS) controller, whch adds addtonal dampng to the nter-area oscllaton modes by affectng the real power n an ndrect manner. However, the effectveness of these controllers s restrcted to the neghborhood of a prescrbed set of operatng condtons. In ths paper, decentralzed robust controllers are developed to mprove the dampng ratos of the nter-area oscllaton modes by drectly affectng the real power through the turbne governng system. The proposed control strategy requres only local sgnals and s robust to the varatons n operaton condton and system topology. The effectveness of the proposed robust controllers s llustrated by detaled case studes on two dfferent test systems. Index Terms Small sgnal stablty, nter-area oscllaton, oscllaton mtgaton, decentralzed robust control. I. INTRODUCTION The nter-area oscllatons of low frequency are nherent phenomena between synchronous generators that are nterconnected by transmsson systems. It has been ponted out n [] that these oscllatons often become poorly damped wth heavy power transfer between dfferent areas over weak connectons of long dstance. Because sustaned oscllatons could cause system breakup and even lead to large-scale blackout, t s extremely mportant to mantan the stablty of these oscllatons for the system securty. However, envronmental constrants and economc pressures often push power systems close to ther operatonal lmts [], whch substantally reduces the stablty margn of the normal operaton. One of the promnent events caused by undamped oscllatons s the notable breakup and blackout n the western North Amercan Power System on August 0, 996 [3]. To ensure the secure system operaton, the amount of power transfer on te-lnes between dfferent areas are often lmted due to the nter-area oscllatons that are poorly damped. Hence, new control strateges that can mprove the dampng of these nter-area oscllatons are needed to ncrease the transmsson capacty for a better utlzaton of the exstng transmsson network. The well-known dampng controller s the power system stablzer (PSS) [4]. It adds a supplementary control sgnal nto the generator exctaton system through the automatc voltage regulator (AVR) to apply an addtonal electrc torque J. Lan, S. Wang, R. Dao and Z. Huang are wth Pacfc Northwest Natonal Laboratory, Rchland, WA 9935, USA (e-mal: {janmng.lan, shaobu.wang, rusheng.dao, zhengyu.huang}@pnnl.gov). on the generator rotor. However, the classcal desgn of the PSS, whch reles on the lnearzaton of a sngle-machne nfnte-bus (SMIB) model, has two major shortcomngs. Frst, the SMIB model s a reduced-order model by neglectng those mportant dynamcs assocated wth network nterconnectons. It cannot adequately descrbe the nter-area oscllaton modes due to network nterconnectons. Hence, the classcal PSS s effectve n dampng local oscllaton, but t may not provde enough dampng to the nter-area oscllatons unless carefully tuned. Furthermore, the local controller desgn s ndependent of one another wthout any proper coordnaton among them. Second, the lnearzaton based controller desgn greatly restrcts the valdty of the controller to the neghborhood of the operatng pont under consderaton. When the system loadng or network topology drastcally changes durng the normal system operaton, the resultng PSS may no longer yeld satsfactory dampng effect. Another approach of dampng the nter-area oscllatons s to ntroduce varous supplementary modulaton controllers to flexble AC transmsson system (FACTS) devces (see, for example, [5] [8]). Although FACTS controllers can acheve satsfactory dampng effect for nter-area oscllatons [9], t s not cost-effectve to use them for the sole purpose of dampng control. On the other hand, FACTS controllers share the same shortcomng wth the classcal PSS due to the lnearzaton based controller desgn. Furthermore, the nteractons between FACTS controllers and the PSSs could even potentally degrade the dampng wthout proper coordnaton [0]. Recently, there have been a lot of efforts nto the development of coordnated and robust PSSs for power systems [] [4]. In [], robust tunng based on egenvalue analyss was proposed to select the parameters of PSSs over a prescrbed set of operatng condtons. In [], a novel method based on modal decomposton was proposed to elmnate the nteractons among dfferent modes for tunng PSSs. In [3] and [4], robust PSSs were desgned by utlzng robust control technques such as µ-synthess and lnear matrx nequalty (LMI). In addton to robust PSSs, robust exctaton control strateges have also been developed for dampng control and the resultng controllers have dfferent structure from that of the classcal PSS. In [5], a novel methodology was proposed to desgn robust exctaton controller based on the polytopc model for mproved robustness over varyng operatng condtons. Although these newly developed robust dampng controllers are more robust than the classcal PSS, ther robustness s stll lmted to a fnte number of operatng condtons. There have been also persstent efforts on determnng approprate global sgnals as the control nput of classcal

2 PSSs and FACTS controllers as n [6] and [7]. Because the feedback of global sgnals actually establshes the correlaton among ndependent local controllers, t s more effectve than the feedback of local sgnals only. These efforts have been partcularly facltated by the technology of wde area measurement system (WAMS) enabled by the development of Phasor Measurement Unt (PMU). Many WAMS-based dampng controllers have been developed (see, for example, [8] []). However, the robustness of WAMS-based dampng controllers to the varatons n operatng condton and system topology stll requres further nvestgated. Moreover, the destablzng factors of communcaton network such as tme-varyng delay and random packet dropout has to be explctly taken nto account for the relablty purpose [], whch greatly complcates the correspondng controller desgn. In most cases, the nter-area oscllatons are usually caused by transferrng real power between nterconnected areas. Thus, t s a straghtforward way to control the real power n the system for the oscllaton mtgaton. However, the majorty of exstng dampng controllers ntroduce addtonal dampng to the nter-area oscllaton modes through ether generaton exctaton system (e.g., PSS) or transmsson lne (e.g., FACTS), whch actually affects the real power n an ndrect manner. Some of the work that drectly affects the real power to damp the nter-area oscllatons has been reported n [3] [5]. In [3], robust dampng controllers were desgned for superconductng magnetc energy storages to enhance the dampng of multple nter-area modes by njectng real power nto the system. It s shown that real power modulaton can be an effectve way to damp power flow oscllaton, although the resultng controller theren s only robust to the prescrbed set of operatng condtons. In [4], two dampng control systems usng ultracapactor based energy storages for real power njecton were desgned for areas expected to oscllate aganst one another. Because the frequency measurement of one area s requred by the controller n the other area, the communcaton effects could greatly degrade the expected controller performance. In [5], a Modal Analyss for Grd Operatons (MANGO) procedure was establshed to enable grd operators to mmedately mtgate the nter-area oscllatons by adjustng operatng condtons, where the detecton of low dampng s acheved by the modal analyss wth real-tme phasor measurements. The MANGO procedure can enact as part of a remedal acton scheme. However, the nteractons among dfferent oscllaton modes make t challengng to determne unversal dspatch patterns that are effectve for all the nter-area oscllaton modes. In ths paper, the decentralzed robust control strategy developed n [6] s adopted heren to mprove the dampng ratos of the nter-area oscllaton modes. In [6], a dstrbuted herarchcal control archtecture was proposed for large-scale power systems to mprove the transent stablty and frequency restoraton. However, the capablty of ths control strategy n enhancng the small sgnal stablty of multarea power systems has not been analyzed yet. Hence, t s the man focus of ths paper to show the effectveness of ths control strategy n mtgatng the nter-area oscllatons. Ths robust controller ntroduces an auxlary control sgnal nto the turbne governng system through the governor to apply an addtonal mechancal torque on the generator rotor. Ths prncple s smlar to that of the PSS, whereas the PSS provdes an addtonal electrc torque on the generator rotor through the generator exctaton system. The proposed control strategy has several appealng advantages ascrbed to the underlyng coordnated controller desgn. Frst, t s robust to the varatons n operatng condton and system topology. Ths s because the correspondng controller desgn does not requre system lnearzaton over a specfc operatng condton, nor does t depend on a specfc network topology ether. Such robustness s extremely crucal for the secure operaton of power systems, whch constantly experence changes n operatng condtons and transmsson networks. Second, t has no destructve nterference wth exstng PSSs n the system regardng dampng control, whch elmnates the need of proper coordnaton. Ths s because the turbne governng system s weakly coupled wth the exctaton system. Thrd, t s decentralzed requrng only local sgnals that are easly obtaned for controller mplementaton, whch greatly mproves the relablty of the controller by elmnatng the use of communcaton. Lastly, t has smple controller desgn of low complexty and does not nvolve addtonal costly equpments such as power electroncs and energy storages, whch make t easy for practcal mplementaton. Detaled case studes are performed on both small and medum-szed test systems to llustrate the effectveness and robustness of the proposed robust controllers n mprovng the dampng ratos of the nter-area oscllaton modes. The remander of ths paper s organzed as follows. In Secton II, the dynamc model of mult-area power systems s gven. In Secton III, the desgn of decentralzed robust controllers s descrbed wth the dscusson of practcal mplementaton. Then detaled case studes are presented n Secton IV. Conclusons are found n Secton V. II. SYSTEM MODELING The mult-area power systems usually consst of a number of generators that are nterconnected through a transmsson network. In ths paper, robust controllers are proposed for generators wth steam valve governors. Although only steam valve governors are consdered heren, the proposed control strategy can be extended to other types of governors such as hydraulc governors. However, the resultng controller structures could vary due to dfferent turbne governng system dynamcs to be consdered, whch s left for future research. As llustrated n Fg., the generator dynamcs wth steam valve governor can be descrbed by the followng dfferental equatons: Rotor Dynamcs: δ = ω r = ω ω o, () ω r = D H ω r ω o H (P m P e ) ; () Governor Control: Ẋ e = K e ω r X e P c, (3) T e R ω o T e T e 0 X e ;

3 P e P m P m ωr δ ω 0 P e Dω 0 H s s D H s s ω 0 P c ωr δ D(a) H Rotor P c sdynamcs T max s = T max = ω r K e KR e ω 0 T max = P m ω r ω r P e ωr δ X P X e c e R ω 0 T e s T e s K e X T mn = e R T mn = ω 0 T e s T mn = (b) Governor control X e X m P m X e T T 3 s 3 s m T T 4 s 4 s m X e T T m s m T 3 s s m T T 5 s T 4 s 5 s m T m s (c) Turbne T dynamcs 5 s Fg.. Generator dynamcs wth steam valve governor control [6]. Turbne Dynamcs: K et 3 Ẋ m = ω r X m T m T e R ω 0 T m ( T ) 3 X e T 3 P c, (4) T m T e T m T e P m = K et 3 T 4 ω r P m T m T e T 5 R ω 0 T 5 ( T ) 4 X m T 3T 4 P c T 5 T m T m T e T 5 T ( 4 T ) 3 X e ; (5) T m T 5 T e In the above dynamcs, δ s the generator rotor angle (rad), ω s the generator rotor speed (rad/sec), ω r s the relatve rotor speed (rad/sec), ω 0 s the nomnal rotor speed (rad/sec), P m s the mechancal power (p.u.) provded by turbne, P e s the electrcal power (p.u.) resulted from network nterconnectons, X e s the valve openng (p.u.), and P c s power control nput (p.u.). For the normal operaton of power systems, P c = P ref c, whch s the prescrbed generator reference power receved from economcal dspatch. When power systems do not have enough dampng, any varatons n P e resulted from the changes n ether loadng condton or network topology can cause poorly damped oscllatons of low frequency. In ths work, the proposed decentralzed robust controllers wll modulate the mechancal power P m to ntroduce addtonal dampng to the poorly damped oscllaton modes. In the followng secton, the controller desgn wll be brefly presented. The nterested readers are referred to [6] for more detals. III. DECENTRALIZED ROBUST CONTROL To proceed, suppose there are N generators n the system. Let x = [δ ω r P m X m X e ] and x = [x... x N ]. Then the dynamcs of the -th generator can be represented by the followng state space model, ẋ = A x B P c G P e, (6) where A, B and G can be derved from () through (5) and are gven by D ω 0 H H 0 0 A = 0 KeT3T4 T mt et 5R ω 0 T m T 4 T 4(T e T 3), T 5 T 5T m T mt 5T e 0 KeT3 T mt er ω 0 0 T e T 3 T m T mt e 0 Ke T er ω T e [ ] B = T T mt e T, e G = T 3T 4 T mt et 5 [ 0 ω0 H ]. For each generator, the orgnal power control nput P ref c wll be augmented wth a local decentralzed control u so that P c = P ref c u. Ths addtonal control nput u s defned as u = k ( x x d ), (7) where x d = [δd ωd r P m d Xd m Xd e ] s an arbtrary operatng pont, and k s the lnear feedback gan matrx. In practce, x d can be always chosen as the current equlbrum state of the system before any dsturbances. Remark : The state space model (6) s derved wthout lnearzaton snce all the nonlneartes resulted from network nterconnectons are encompassed by the unknown nput P e. In the followng controller desgn, the varatons n P e wll be treated as external dsturbances to ndvdual generators and then rejected by the proposed controllers. The determnaton of controller gan k n (7) s ndependent of any specfc operatng condton or network topology n order to guarantee the controller robustness. The operatng pont x d n (7) only serves as the reference from whch the control nput should be calculated. If the proposed controllers are actvated durng the normal operaton, x d wll be the current equlbrum operatng pont. If they are actvated durng the dsrupted operaton, x d wll be the last equlbrum operatng pont before the dsrupton occurs. Once actvated n the system, the proposed controllers wll contnue to work regardless of the operatng condton and network topology. A. Controller Desgn 3 Let x e = [δe ωr e P m e Xe m Xe e ] denote the equlbrum state the system reaches after dsturbances. It follows from (6) and (7) that x e satsfes the followng algebrac equaton, ( ) A x e B P ref c u e G Pe e = 0, (8) where u e = k (x e xd ). To study the stablty of each generator (6) drven by the controller (7) after dsturbances, the perturbed system dynamcs about the equlbrum state x e s consdered. Let x = [ x x N ], where x = x x e s the vector of the devatons of δ, ω r, P m, X m, and X e, respectvely, from ther equlbrum values after

4 4 dsturbances. Then by subtractng (8) from (6), the perturbed dynamcs of the -th generator can be derved as ẋ = A x B u G h ( x), (9) where u = u u e = k x and h ( x) = P e Pe e. The unknown nput h ( x) = P e Pe e represents external dsturbances resulted from network nterconnectons. Gven the two-axs generator model descrbed n [7], the dsturbance h ( x) can be equvalently represented as h ( x) = h qq wth h qq ( x) hqd( x) h dq ( x) h dd ( x) (0) N ( x) = E qe qj (G j C j B j S j ), h qd ( x) = h dq ( x) = h dd ( x) = j= N E qe dj (B j C j G j S j ), j= N E de qj ( B j C j G j S j ), j= N E de dj (G j C j B j S j ), j= where E q s the q-axs transent EMF (p.u.), E d s the dqaxs transent EMF (p.u.), C j cos δ j cos δj e, and S j sn δ j sn δj e wth δ j = δ δ j and δj e = δe δe j. Applyng the same argument as n [8], t gves h qq ( x)h qq where y = [ y y N ] wth and D qq y j = δ j δ e j ( x) y D qq y = δ δ j, = dag [ d qq d qq N ] wth d qq j = ( 4Ē qē qj G j Bj) N k= Ē qk ( G k Bk), where Ē represents the allowable maxmum absolute value of transent EMF (p.u.). Smlarly, the same arguments can be appled, as above, to h qd ( x), h dq ( x) and h dd ( x). Then, t follows from (0) that h ( x)h ( x) 4y D y, () where D = D qq D qd D dq D dd. As shown n [8], the nequalty () can be further represented as or, more generally, h ( x)h ( x) x H H x, h ( x)h ( x) β x H H x, where β and H s drectly determned from D. Now, the lnear feedback gan matrx k can be obtaned by solvng a LMI-based optmzaton problem as formulated n [8]. It can be shown usng smlar arguments as n [8] that local decentralzed controllers can stablze the perturbed system (9) f the followng optmzaton problem s feasble, N mnmze = (γ κ Y κ L ) subject to Y D > 0 W D G D Y D H Y D H N G D I O O H Y D O γ I O < [ H N Y D O ] O [ γ N I ] κl I L Y < 0, I > 0 L I I κ Y I γ β < 0, =,..., N, () where γ = /β, β, W D = A D Y D Y D A D B D L D L DB D wth A D = dag [ ] A A N and B D = dag [ ] B B N, and κy and κ L are prescrbed postve lmts mposed on the magntude of L and Y. Once the above optmzaton problem s solved, the gan matrces of local decentralzed controllers can be calculated as K D = L D Y D, where K D = dag [ ] k k N. When solvng the above optmzaton problem, the system s tolerance to nterconnecton uncertantes s maxmzed. B. Practcal Implementaton There are two of many possble ways to practcally mplement the proposed robust controllers for dampng the nter-area oscllatons. One way s to keep them actvated n the system as a pre-emptve strategy to mprove the dampng ratos of the nter-area oscllaton modes. The other way s to nstall them n the system wthout actvaton untl poorly damped nterarea oscllatons are detected, whch can be acheved by the approach of real-tme mode estmaton as proposed n [9]. In ths way, the actvaton of these pre-nstalled robust controllers becomes part of the remedal acton scheme as an emergency measure to prevent potental cascadng power falures. The followng smulaton results confrm the effectveness under both ways of controller mplementaton. IV. CASE STUDIES In ths secton, the effectveness of the proposed decentralzed robust control n mprovng the small sgnal stablty s frst demonstrated under both normal and dsrupted system operatons. In partcular, the robustness of the proposed control wth respect to the changes n the operatng condton and system topology s llustrated through comparatve case studes that are smulated by the Power System Toolbox (PST) [30]. Then the proposed control strategy s also nvestgated for the case when the power system has only a very lmted number of generators equpped wth steam valve governors. Case : In ths case, the performance of the proposed robust controllers s compared to that of PSSs under dfferent normal operatons by varyng the operatng condtons. The selected test system s an IEEE -area, 4-machne test system as shown n Fg., whose prototype was orgnally ntroduced n [4]. Two PSSs are nstalled on generators G and G3, respectvely, and a statc VAR compensator (SVC) s connected

5 5 dampng rato (%) Hz te lne flow (MW) (a) Wthout robust controllers Fg.. Sngle-lne dagram of IEEE -area, 4-machne test system [30]. TABLE I OSCILLATION MODE OF MINIMUM DAMPING RATIO IN BASE CASE FOR IEEE -AREA, 4-MACHINE TEST SYSTEM Robust controllers none G-G4 G,G4 G,G3 Oscllaton mode nter-area local local nter-area Natural frequency 0.60 Hz.30 Hz.0 Hz 0.66 Hz Dampng rato 6.96% 3.08% 8.0% 6.09% dampng rato (%) Hz.0 Hz 0.66 Hz G, G, G3 and G4 G and G4 G and G3 4 to bus 0. The base case has a 400 MW te-lne flow from the left area to the rght area, where the real power of two loads at bus 4 and bus 4 s 976 MW and 757 MW, respectvely. The modal analyss performed n the PST ndcates that the base case wth PSSs only has a major nter-area oscllaton mode around 0.60 Hz wth a dampng rato of 6.96%. Ths nter-area oscllaton mode has the mnmum dampng rato among all the oscllaton modes of the base case wth PSSs only. Snce all the generators are equpped wth steam valve governors, the proposed decentralzed robust controllers can be desgned for all the generators. When robust controllers are mplemented on generators G to G4, there s an nterarea oscllaton mode around 0.70 Hz wth a greatly mproved dampng rato of 40.6%. However, the oscllaton mode of mnmum dampng rato s a local mode assocated wth generators G and G. When robust controllers are mplemented on generators G and G4, there s an nter-area oscllaton mode around 0.65 Hz wth a greatly mproved dampng rato of 4.0%. However, the oscllaton mode of mnmum dampng rato s agan a local mode assocated wth generators G and G. When robust controllers are mplemented on generators G and G3, the oscllaton mode of mnmum dampng rato s an nter-area oscllaton mode around 0.66 Hz wth a greatly mproved dampng rato of 6.09%. The mnmum dampng rato among all the oscllaton modes, whch s the most used ndex for evaluatng the small sgnal stablty [3], s lsted n Table I for each mplementaton of robust controllers. Therefore, dfferent controller mplementatons wll lead to dfferent mprovements n the small sgnal stablty. The greatest mprovement of the mnmum system dampng occurs when robust controllers are mplemented on all the generators. In order to examne how the mnmum dampng rato s correlated wth the te-lne flow, varous system stress levels te lne flow (MW) (b) Wth robust controllers Fg. 3. Correlaton of the mnmum dampng rato wth the te-lne flow for the IEEE -area, 4-machne test system. The red sold lne n (a) represents the 5% dampng margn. are consdered by varyng the real and reactve power of two loads at buses 4 and 4 wth the same percentage. At the same tme, the power output of generators G to G4 s also changed accordngly by the same percentage. Ths adjustment pattern creates a number of operatng ponts wth dfferent te-lne flows, and t also mnmzes the locatonal effect of generaton and load. The correspondng varatons of the mnmum dampng rato wth and wthout robust controllers are shown n Fg. 3. When the system has PSSs only, the dampng rato of the major nter-area oscllaton mode s very senstve to the system stress level, and decreases sgnfcantly wth the ncrease of te-lne flow as shown n Fg. 3(a). However, t can be seen from Fg. 3(b) that the mnmum dampng rato of the system s not only greatly mproved but also much less senstve to the system stress level, whch leads to a hgher power transfer capacty. Thus, the effectveness of the proposed control n mprovng the small sgnal stablty s robust to varyng operatng condtons because no lnearzaton s nvolved n the proposed controller desgn. Case : In ths case, the performance of the proposed control s compared to that of the PSS under the dsrupted system operaton by changng the system topology, where the same test system used n Case s consdered agan. The power demand of two loads and the power output of four generators are ncreased by 5.58% over the base case. That s, the real power of two loads at bus 4 and bus 4 s ncreased to MW and 855 MW. As pervously determned, ths

6 6 test system wth PSSs only has a major nter-area oscllaton mode around 0.55 Hz wth a dampng rato of 6.34%. Snce the dampng rato s larger than the 5% dampng margn, any oscllatons resulted from small load fluctuatons are expected to be well damped. However, the mpact of the topology changes due to contngences on the test system wth PSSs only remans to be examned. Thus, n ths case, the system topology s changed by trppng a sngle lne between buses 3 and 0 wthout fault. After removng one of the lnes 3-0, the test system wth PSSs only s dentfed to have a major oscllaton mode around 0.44 Hz wth a dampng rato of 0.46%. It turns out that the selected topology change has a very large mpact on the test system wth PSSs only, whch causes a 5.88% dampng reducton. Although there are stll PSSs on generators G and G3, the resultng system s expected to exhbt a hghly oscllatory response, whch s dangerous snce t may lead to system breakup and large-scale blackouts. Now consder the test system wth robust controllers. When robust controllers are mplemented on G to G4, the test system after topology change has a local oscllaton mode around.30 Hz wth the mnmum dampng rato of 3.4%. When robust controllers are mplemented on generators G and G4, the test system after topology change has a local oscllaton mode around.8 Hz wth the mnmum dampng rato of 8.0%. When robust controllers are mplemented on generators G and G3, the test system after topology change has an nter-area oscllaton mode around 0.48 Hz wth the mnmum dampng rato of 6.90%. Unlke the PSSs, the proposed control mantans adequate system dampng even when the system topology changes. In order to better llustrate the robustness of the proposed control wth respect to the topology changes, the dynamc response of the test system s smulated, respectvely, wth and wthout robust controllers. In the frst dynamcal smulaton, the robust controllers are assumed to be actvated n the test system all the tme, and the correspondng response of the relatve rotor angle between generators G3 and G s shown n Fg. 4(a). In the second dynamcal smulaton, the robust controllers are not onlne ntally. They are only actvated after 0 sec as the reacton to the detecton of sustaned oscllatory system response. It can be seen from Fg. 4(b) that the robust controllers can quckly damp out the oscllatons once they are actvated. Case 3: In ths case, the performance of the proposed control s nvestgated when the power system has only a very small number of generators wth steam valve governors. In such a case, the effectveness of the proposed control could be lmted due to the restrctve placement of robust controllers. The selected test system s the 7-machne system developed n [3], whch s a modfed verson of the one orgnally developed n [33] as a smplfed model of the western North Amercan power grd. Ths test system as shown n Fg. 5 conssts of generator buses 7 through 4 and 45, and load buses 3 through 4. All the generators except generator G7 are set to work n pars. Each par conssts of a base generator (G to G8) wthout speed governor and a loadfollowng generator (G9 to G6) wth speed governor. Only four of the load-followng generators (G, G, G3 and G5) are equpped wth steam valve governors, whle the rest Fg. 4. angle (deg) angle (deg) PSSs only G, G, G3 and G4 G and G4 G and G tme (sec) (a) Robust controllers actvated all the tme PSSs only G, G, G3 and G4 G and G4 G and G tme (sec) (b) Robust controllers actvated at 0 sec Relatve angle δ 3 -δ n response to the topology change. 684 IEEE TRANSAC Fg. 3. Example smulaton system. Fg. 5. Sngle-lne dagram of 7-machne test system [3]. TABLE I INTERAREA MODES OF EXAMPLE 7-MACHINE SYSTEM are equpped wth hydraulc governors. Generator G7 s a base generator wthout speed governor. All the generators are equpped wth fast voltage regulators and PSSs. The modal analyss ndcates that ths test system wth PSSs only has four major nter-area oscllaton modes around TABLE II 0.30 Hz, 0.40 Hz, 0.60 Hz and 0.80 Hz, where the 0.40 Hz INTERAREA MODES OF EXAMPLE 6-MACHINE SYSTEM mode has the mnmum dampng rato. In order to nvestgate the correlaton between the dampng ratos of these nterarea oscllaton modes wth the system stress level, the power output of generator G and the power demand at bus 35 and reactve loads s obtaned by passng ndependent Gaussan whte nose through a flter. It has been hypotheszed that such a flter s approprate for load modelng. Fg. 4. Power spectrum 0.38-Hz and 0.4-Hz system states. Fg. phase angle sgnal perodogram avera ablty of the 0.3 sgnals good cand The power spectru condton are very 0.4 Hz combne The key to select observablty of the tral plot as a dom peak at the mode(s frequences. The s to an extent typca VII. DATA PR To obtan optma gorthm, the data m parameters must be dards and optmal ductng, many Mo system [7], result

7 7 dampng rato (%) PSSs only G, G, G3 and G stress level (%) Fg. 6. Correlaton of the dampng rato of the 0.40 Hz mode wth the system stress level for the 7-machne test system. are adjusted by the same percentage, whch s 00% for the base case. It can be seen from Fg. 6 that there exsts a consstent correlaton between the dampng rato of the 0.40 Hz mode and the system stress level when there are only PSSs n the system. Smlar correlaton also exsts for all the other nter-area oscllaton modes. When robust controllers are mplemented on generators G, G, G3 and G5, the dampng ratos of the 0.30 Hz, 0.60 Hz and 0.80 Hz modes are greatly mproved and mantaned above the 5% dampng margn at varous system stress levels. However, t can be seen from Fg. 6 that the effectveness of the proposed control n mprovng the dampng rato of the 0.40 Hz mode s not that sgnfcant because the requred dampng margn cannot be acheved when the system s hghly stressed. Ths s manly because there are no robust controllers mplemented n the areas nterconnected by the te-lnes between buses 7 and 8. The low dampng rato of the 0.4 Hz mode s the result of large power flow over the lne 7-8. Therefore, the effectveness of the proposed control requres that robust controllers be mplemented n the areas that contrbute to the poorly damped nter-area oscllaton modes. Snce t s not feasble for the gven 7-machne test system to satsfy ths requrement, other dampng controllers should be deployed n the areas around buses 7 and 8 to complement the exstng four robust controllers n mprovng the small sgnal stablty. For example, selectve FACTS controllers can be consdered to be nstalled on the lne 7-8. In order to verfy the above mplementaton requrement for the guaranteed effectveness of the proposed control, the dampng rato of the 0.40 Hz mode s examned for a modfed 7-machne system as shown n Fg. 7. In ths modfed system, the orgnal generator G n Fg. 5 s splt nto two generators, G and G8. The new G s stll equpped wth the hydraulc governor, whle the new G8 s equpped wth the steam valve governor. Ther machne parameters are carefully selected such that the dampng rato of the 0.4 Hz mode for the base case keeps the same after the system modfcaton. As shown n Fg. 8, when an addtonal robust controller mplemented on generator G8, the dampng rato of 0.4 Hz mode s greatly mproved and the requred dampng margn s guaranteed at varous system stress levels. It confrms agan the robustness of the proposed control to varyng operatng condtons. Fg. 7. Sngle-lne dagram of modfed 7-machne test system. dampng rato (%) PSSs only G, G, G3, G5 and G stress level (%) Fg. 8. Correlaton of the dampng rato of the 0.40 Hz mode wth the system stress level for the modfed 7-machne test system. Wth the modfed 7-machne system n Fg. 7, the mpact of topology changes resulted from N contngences on the dampng rato of the 0.40 Hz mode for the base case s also evaluated. Total 5 topology changes are consdered wth each assocated wth the trppng of one transmsson lne. The resultng dampng rato of the 0.40 Hz mode after contngency s shown n Table II, where there are 5 cases wth converged power flow after smple lne trppng. It s confrmed agan that the effectveness of the proposed control n mprovng the small sgnal stablty s robust to the topology change. V. CONCLUSIONS In ths paper, decentralzed robust controllers have been developed for generators wth steam valve governors to mprove the dampng ratos of the nter-area oscllaton modes by drectly affectng the real power n the system. The proposed dampng control strategy ntroduces an auxlary control sgnal nto the governor, creatng an addtonal mechancal torque on the generator rotor. It has several mportant advantages over the exstng dampng strateges. The most valuable advantage s the robustness wth respect to dfferent operatng condtons and system topologes. The controller performance has been demonstrated by detaled case studes on both small and medum-szed test systems. It has been shown that the

8 8 TABLE II DAMPING RATIO OF THE 0.40 HZ MODE AFTER SIMPLE LINE TRIPPING IN MODIFIED 7-MACHINE TEST SYSTEM Lne trppng PSSs only G,G,G3,G5,G % 8.% % 7.7% 7-3.7% 8.33% -3.85% 7.99% % 8.4% % 7.58% % 6.56% % 8.5% % 8.50% % 8.47% -7.95% 8.3% % 8.8% % 8.3% % 8.8% % 6.56% small sgnal stablty of power systems can be effectvely enhanced by the proposed robust control to allow larger power exchange between dstnct areas, whch n turn leads to hgher transmsson capacty and better network utlzaton. Currently, the proposed control strategy s beng extended to nclude generators wth hydraulc governors as well, whch are partcularly prevalent n the Pacfc Northwest. Furthermore, the approach of mtgatng the nter-area oscllatons drectly through real power control s also under nvestgaton for other energy resources such as energy storage or wnd generators n addton to conventonal generators. REFERENCES [] M. Klen, G. J. Rogers, and P. 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