Use of Modal Sensitivity to Operating Conditions for Damping Control in Power Systems

Transcription

1 Proceeding of the 44th Hawaii International Conference on Sytem Science - Ue of Modal Senitivity to Operating Condition for Damping Control in Power Sytem Zhenyu Huang ing Zhou Franci Tuffner Daniel Trudnowki Pacific orthwet ational Laboratory Montana Tech zhenyu.huang@pnl.gov ning.zhou@pnl.gov franci.tuffner@pnl.gov dtrudnowki@mttech.edu Abtract Small ignal tability i an inherent characteritic of dynamic ytem uch a power ytem. Pole poitioning through power ytem tabilizer (PSS) i often ued for improving damping in power ytem. A well-deigned PSS can be very effective in damping ocillation, epecially local ocillation. However, deigning PSS for inter-area ocillation ha been a very challenging tak due to time-varying operating condition affecting the ocillation. Thi paper explore the enitivity relationhip between ocillation and operating condition, and employ the relationhip to derive recommendation for operator action to adjut operating condition for improving damping. Low damping i uually conidered to be a reult of heavy power tranfer in long ditance, while pecific location alo have ignificant impact on damping of ocillation. Therefore, it i important to conider location in deriving recommendation. Thi paper propoe the concept of relative modal enitivity and preent it application in deriving recommendation for operator action in damping control.. Introduction Small ignal tability problem have long been recognized a one of the major threat to power grid tability and reliability. An untable mode can caue large-amplitude ocillation and may reult in ytem breakup and large-cale blackout []. There have been everal incident of ytem-wide low-frequency ocillation. Of them, the mot notable i the Augut, 996 wetern orth American power ytem breakup involving undamped ytem-wide ocillation []. In the Augut, 996 event, the ytem deteriorated over time after the firt line wa tripped off. About ix minute later, undamped ocillation occurred and the ytem broke up into everal iland. Other ocillation event in the US and elewhere have been oberved [][4]. They all exhibited utained lowfrequency ocillation and have led to a great concern about the advere effect of ocillation on power ytem operation. In power ytem, low-frequency ocillation are a reult of electromechanical coupling between the tranmiion network and generator. Coniderable undertanding and literature have been developed over the pat everal decade of the cae where thee ocillation become very lightly damped, or even untable. Small-ignal tability tudie have been mainly baed on the eigenvalue analyi of the characteritic matrix derived from the linearized model of a power ytem [5]. Power ytem tabilizer (PSS) have been ued for damping control in common indutrial practice. While effective in damping ocillation, epecially local-mode ocillation, PSS tuning i a very challenging tak for inter-area ocillation mode becaue of time-varying operating condition. The practical feaibility of PSS for interarea mode i further limited a power ytem model have been found inadequate in decribing real-time operating condition [][6]. A deirable alternative to PSS would be to bring the ytem to a new operating condition through operator action. The new operating condition would have damping high enough to utain diturbance o ocillation would not occur. Thi paper aim to develop uch an operation-oriented approach to perform modal analyi for grid operation (MAGO). Becaue mall ignal tability i an inherent characteritic of a dynamic ytem, the low damping ituation can be detected from ambient meaurement before a diturbance trigger the ocillation [7]. ModeMeter technology ha been demontrated to have the real-time capability of etimating ytem mode from both ocillation ignal and ambient data [8]. The MAGO control can be applied a oon a low damping ituation i detected a a preventive meaure to improve damping and prevent ocillation. Different from PSS and other modulation-baed method, MAGO aim to improve damping through adjuting operating point. Traditionally, the modulation-baed method do not change the ytem operating point, but improve damping through 5-65/ $6. IEEE

2 Proceeding of the 44th Hawaii International Conference on Sytem Science - automatic feedback control [9][][]. Figure illutrate the difference of thee two type of damping improvement method. Availability of phaor meaurement enable wide-area modulation control for control device uch a the Pacific DC Intertie (PDCI) in the wetern US power ytem and Flexible AC Tranmiion Sytem (FACTS) device []. MAGO, hown in red, and modulation control, hown in magenta, are complementary toward the ame objective. Modulation-baed method are deigned to maintain poitive damping at expected operating condition, wherea MAGO-recommended adjutment can be ued to move the ytem to a more table operating point that damping would be ufficient in the event of a ytem diturbance. Figure. MAGO veru modulation control Phaor meaurement becoming widely available and ModeMeter technique maturing provide a olid foundation for the development of MAGO technology. It i alo a natural need for new method, uch a MAGO, to bring modal information from a monitoring tool to a deciion upport tool. MAGO i expected to bridge the gap between modal analyi and power grid operation. To achieve thi goal, a key tep in MAGO i to identify modal enitivity with repect to controllable operating parameter, uch a generation and load. Thi paper explore the fundamental relationhip between modal damping and operating parameter (Section ) and propoe the concept of relative modal enitivity and how it may be ued in damping control (Section 4). Section 5 dicue the practical feaibility in MAGO implementation, and Section 6 conclude the paper.. Relationhip of Modal Damping and Operating Condition The operating point of a power ytem determine the eigenvalue, i.e., the ocillation mode, of the ytem. A power ytem can be decribed a a et of non-linear differential algebraic equation: ( x, y, u) ( x, y, u) x f g where x i the tate vector, y i the algebraic vector, and u i the input vector. Linearizing thee non-linear equation at an operating point determine the characteritic matrix of the ytem. The non-real eigenvalue of the characteritic matrix are the ocillation mode of the power ytem. The characteritic matrix i hown a the A matrix in the following equation: Δx A( p) Δx + B( p) Δu () Δy C( p) Δx + D( p) Δu where p repreent operating parameter. Thee parameter may be value uch a generator output, load conumption, tranformer tap, capacitor MVAr, and DC power etting, which can be adjuted by operator in real-time power ytem operation. Equation () indicate that the A matrix i a function of parameter, p. Thu, the mode can be influenced by adjuting ome of the p parameter, i.e., changing the operating point of the power ytem.. Single-Machine-Infinite-Bu Sytem The Single-Machine-Infinite-Bu (SMIB) ytem (Figure ) provide a unique opportunity to oberve the effect of operating condition on mall-ignal tability becaue of it implicity. The primary operating parameter i the teady-tate generation output P, which i ame a the power tranfer on the tie-line to the infinite ytem. Another operating parameter i the teady-tate terminal voltage V. Thee two parameter can be adjuted through reference etting of the governor and the exciter, repectively. Figure. Single-Machine-Infinite-Bu (SMIB) ytem Decribing thi SMIB ytem with the claical generator model, the following wing equation reult: H d Tm Te D dt Pm Pe D () dδ dt

3 Proceeding of the 44th Hawaii International Conference on Sytem Science - where δ i the rotor angle in radian, and i peed in radian/econd, T m and T e are the mechanical torque and the electrical torque in p.u., P m and P e are the mechanical power input and the electrical power output in p.u., i the ynchronou peed in radian/econd, and H and D are the generator inertia contant and damping coefficient, repectively. The tie-line power P ( P e ) i a function of the rotor angle δ, the generator internal voltage E, infinite bu voltage V, and the total impedance X: EV P inδ (4) X Given contant E, V, and X, the electrical power follow the power-angle curve hown in Figure. ξ K Pe + KD σ. σ + 4K S P (7) Figure 4 how that both the modal frequency and damping decreae a the generator power output increae. Parameter for Figure 4 include 77 radian/econd, D, E p.u., V p.u., 77 radian/econd, H, and X.. Thi i conitent with (5) and (6). For damping ratio, Equation (7) how if P e increae, S P decreae, and thu the damping ratio decreae. It i intereting to point out that many textbook aume, and the effect on the damping i not correctly captured. 4 Damping -5 Mode freq (Hz) Freq - Mode damping (%) -5 Figure. Power-angle curve for the SMIB ytem Auming T m i contant, linearization of () and (4) give the characteritic matrix: Δ EV P coδ Δ e D Δ H H H X δ Δδ (5) K P Δ e KD K S P Δδ where K, and H S P coδ. S P max P i pecific to the operating point (Figure ). The ytem ocillation mode, obtained by the eigenvalue analyi of (5), are: K P e + KD ± λ σ ± j, K P e KD 4K S P (6) When the quare-root term in (6) i le than zero, the ytem exhibit ocillatory behavior, and the damping ratio i: Pe (pu) Figure 4. Frequency and damping ratio veru generator power output To further examine the effect of operating point on the mall-ignal tability of thi SMIB ytem, expand the generator model to include an exciter and power ytem tabilizer, randomly vary P and V within their repective operating range, and calculate the damping ratio for each operating point. In Figure 5, the damping ratio i hown to have a trong correlation with the generator power output P (or the tie-line power). Increaing the power tranfer gradually decreae the damping ratio, a indicated by the red arrow line. Thi i conitent with the finding pointed out in [] that one of the main reaon for ocillation problem i the effort to tranmit bulk power over long ditance. The mall deviation from the correlation are due to the variation in the voltage etting. It clearly how the effect of the voltage etting i econdary compared to that of the power etting. Adjuting operating point through generation re-dipatch hould be an effective mean to increae the damping ratio, and thu the mall-ignal tability.

4 Proceeding of the 44th Hawaii International Conference on Sytem Science - Damping Ratio (%) Tie-line Flow (MW) Figure 5. Correlation between the damping ratio and generator power in the SMIB ytem. Multi-Machine Sytem It i clear that in the SMIB ytem, damping ratio i affected by the operating point. In uch a imple ytem, the relationhip can be viewed a modal enitivity to the ytem tre level, generator output, or tie-line flow. From the damping control perpective, one may ugget reducing the tre level, generator output, or tie-line flow. They are equally effective a they are the ame for the SMIB ytem. However, in real-life power ytem, the relationhip i much more complicated becaue there are multiple machine in the ytem, each having variou effect on damping. Thi ection further examine the relationhip in multimachine ytem in the following three categorie of relationhip:. Modal damping to the ytem tre level: The tre level i defined a total load and generation in the ytem. In the tudie, the total load and generation are adjuted, but the relative proportion of each load and generation remain unchanged.. Modal damping to generation/load pattern: The generation/load pattern i defined a the change in relative proportion contributed by each generator/load. Under thi category, the total load and generation remain unchanged, o the tre level remain the ame, but the percentage contribution of each generator i adjuted to oberve the change in ocillation mode.. Modal damping to tie-line flow: tie-line flow i adjuted to evaluate it influence on ocillation mode. To account for ocillation problem, operating procedure, uch a Bonneville Power Adminitration (BPA) Dynamic Standing Order, have been etablihed to cut tie-line flow when lightly damped mode are oberved. Thi tudy i to invetigate how the tie-line flow influence an inter-area mode. Two multi-machine ytem are ued for the tudie: a two-area-four-machine (A4M) ytem [5] (Figure 6) and a 7-machine ytem. The latter i a implified wetern U.S. power ytem []. All the tudie were performed uing the Power Sytem Toolbox with MATLAB [4]. Figure 6. A two-area, four-machine ytem Figure 7 how the correlation of the damping ratio of the.55 Hz mode and the ytem tre level in the A4M ytem. All the red circle are treed baecae, which were created by uniformly adjuting load and generation. The blue dot repreent diturbed cae. For each tre baecae, diturbed cae were generated by randomly adjuting generation pattern. The treed cae alo repreent the variou level of tie-line flow P of Figure 6. The damping ratio decreae with the increae of the ytem tre level, a well a the tie-line flow. Thi i conitent with that of the SMIB ytem. Heavily treed ytem are prone to mall ignal tability problem. From the tability improvement perpective, reducing the tre level or the tie-line flow can effectively improve the damping. The tie-line in thi context i the ocillation path identified by the mode hape. Figure 7 alo how that at each tre level, the mode damping and frequency can change in a wide range with the change in the generation pattern. Thi indicate an opportunity of improving damping through generation re-dipatch without having to hed load. Mode Freq (Hz) Mode Damping Ratio(%) Senitivity of mode wrt tre level adjutment (with a wing bu) Streed baecae Diturbed Cae Stre level (%) Figure 7. Correlation of the damping ratio with the ytem tre level (A4M Sytem) 4

5 Proceeding of the 44th Hawaii International Conference on Sytem Science - The adjutment of tie-line flow or ytem tre level through uniformly adjuting generation and load in a large ytem, at many location, in real-time operation, would be very difficult. Shedding load i uually not a favorable option. It practicality i limited by wide-area coordination and load erving obligation. A more practical olution would be to adjut the mallet et of elected generator with the leat diturbance to cheduled tranfer to achieve the deired change in damping. In the A4M ytem, all four combination of generator pair have been teted. The reult are hown in Figure 8. For example, G & G4 denote that adjutment of G' power output i balanced by G4'. All the combination reult in tieline flow change, and the damping ratio i conitently correlated with the tie-lie flow level. However, Figure 8 alo reveal the locational effect of the adjutment. i.e., different pair of generator have a different effect on the damping ratio, even though they can be adjuted to achieve the ame tie-line flow level. Therefore, cutting tie-line flow may improve damping, but depending on how the cut i performed, the effect on damping can be vatly different. For thi two-areafour-machine ytem, the G & G4 pair i the mot effective, while the G & G4 pair the leat effective. Damping Ratio(%) G & G4 G & G G & G G & G Tie-line flow (MW) Figure 8. Correlation of the damping ratio with the tie-line flow in repone to generation redipatch of variou pair (A4M Sytem) The ame tudie were performed for the 7- machine ytem. Figure 9 plot the reult of damping with repect to the tie-line flow between Bu 7 and Bu of the ytem[]. The red quare repreent treed baecae with the ame generation pattern, and the blue dot are diturbed cae with different generation pattern. It can be een that cutting tie-line flow, but maintaining the generation pattern, would be effective in damping control. However, it may not be practical for large ytem, a pointed out earlier. At the ame tie-line flow level, the blue dot, i.e., different generation pattern, indicate the locational effect of the adjutment. It alo indicate the opportunity of damping control through generation re-dipatch a wa oberved for the A4M ytem. Figure 9. Correlation of damping with tie-line flow of Bu 7 - Bu in the 7-machine ytem. Modal Senitivity The tudie in the previou ection clearly indicate the opportunity of operating point adjutment for damping improvement, a well a the locational effect of the adjutment. A key tep in the method i to quantify the enitivity of damping to operating parameter uch a generation and load. Modal enitivity tudie have been ued to tudy mot influential factor in inter-area ocillation [5], a well a to identify the proper location for PSS [] and FACT device []. Thi ection explore the enitivity from the eigenvalue theory perpective and introduce the concept of relative modal enitivity. The objective i to identify the right operating parameter for adjutment. In ome cae, epecially for mall ytem, engineering experience can provide atifactory election of parameter for operation action. However, experience-baed election may not identify the mot effective option for large-cale complex ytem, uch a the wetern U.S. power grid. Sytematic method have yet to be developed for electing location for damping improvement... Modal Senitivity Derived from Eigenvalue Theory Eigenvalue analyi technique are ued on an explicit or approximate model of the ytem. The model-baed analyi i conducted in a imulation environment. Therefore, all of the tate variable are known and available. Thi allow the eigenvalue, and by extenion the frequency and damping ratio, to be calculated directly. It further allow the calculation of eigenvalue enitivity to operating parameter of interet. Following (), the baic eigenvalue enitivity i given by [5]: 5

6 Proceeding of the 44th Hawaii International Conference on Sytem Science - A Ψ Φ (8) p p where λ repreent the eigenvalue of the ytem, Φ and Ψ denote the right and left eigenvector, repectively, and p repreent the operating parameter of interet, e.g., generation or load. To reduce the computation time for a large power grid, [6] propoed a pare method to calculate the modal enitivity. In [7], a enitive pole method i propoe to identify the pole that are mot enitive to parameter change. To arrive at a practical olution, the enitivity mut be calculated a a numerical approximation through perturbation. Under thi aumption, the eigenvalue enitivity become: ΔA Ψ Φ (9) p Δp Thi baic modal enitivity method ha been applied to a 7-machine ytem []. The reult of the.4 Hz modal enitivity, with repect to generator power output, are hown in Figure. The two larget enitivitie are aociated with generator 7 and generator. Increaing generator power will decreae the damping ratio, while increaing generator 7 power will increae the damping ratio. Generator 7 and would be the mot effective pair to ue for generation re-dipatch to improve the.4 Hz mode damping. The enitivity information alo indicate how much change in damping can be expected by the generator power adjutment. Figure. Modal enitivity of the.4 Hz mode in the 7-machine ytem Though the enitivity reult match obervation from other tudie, a preented in [8], it ha one major hortcoming. During the perturbation proce, the change of generation or load i aborbed by the wing bu. The enitivity i affected by the choice of the wing bu. In the 7-machine ytem, generator i relatively cloe to the wing bu. If the wing bu wa moved to a further geographical bu location, the interaction would be different and the enitivity i expected to change a well. To overcome thi iue, the concept of relative modal enitivity i propoed... Relative Modal Senitivity The eigenvalue-theory-baed enitivity work well to characterize the impact of independent variable. However, in the context of operating point adjutment, it involve at leat two variable. For example, a generation increae at one location will need to be balanced by a generation decreae or a load increae at another location. The effect of the adjutment on modal damping depend on how the pair i choen. In another word, modal enitivity of one location i alway relative to how the other location i choen. The relative modal enitivity method till ue perturbation, but rather than perturbing a parameter and allowing the wing bu to aborb the difference, thi enitivity method attempt to dipere the difference acro other location. Thi effectively minimize the nonlinear effect of the fixed wing bu on the enitivity. That i, if the real power output of a generator i increaed by 5 MW, then all the other generator in the ytem will be decreaed lightly to balance the extra 5 MW of generation. For a general ytem with generator and load location, perturb real power by at one location. Allocating the perturbation evenly at all other location reult in a change in the eigenvalue that can be expreed a: P P The ame magnitude perturbation applied to a different location would reult in: () P () + P The difference of and () i ( ) () P P Relative enitivity between location and can then be derived a () P P () Thi equation clearly how the relative nature of the modal enitivity for operating condition adjutment. If all location are perturbed, applying () 6

7 Proceeding of the 44th Hawaii International Conference on Sytem Science - to all the perturbation can generate - relative enitivitie:,,,. (4) P P P P P P Other relative enitivity can be obtained directly by differentiating two related relative enitivitie. For example, P P P P P P (5) ote that an alternative way of computing relative enitivity i to form a baic et of generator pair, a in (4), and compute the relative enitivity directly. Thi alternative approach i imple and will reult in the ame relative enitivity etimation a the linearized enitivity equation, uch a and (). In reality, becaue of nonlinearity, an error term will appear on the right ide of the enitivity equation. Thu, it i propoed that the ytem be perturbed in uch a way that the eigenvalue change are mainly caued by only one generation change. The propoed approach help reduce the influence of non-linear effect from all other generator in that the perturbation are dipered, and thu are relatively mall in all other generator. Figure how the reult of the relative enitivity method a applied to both generator and load location in the 7-machine ytem. All the enitivitie are relative to the wing bu ued in the previou ection. Figure how the enitivitie are identical to thoe obtained in the previou ection in Figure. Becaue the wing bu aborb the power imbalance, the enitivitie are relative to the wing bu generator. Therefore, the enitivitie obtained in the previou ection are actually relative enitivitie between other generator and the wing bu generator. Figure. Relative modal enitivity of the.4 Hz mode in the 7-machine ytem The mot ignificant importance of the relative enitivity concept i that it provide the inight for guiding operating condition adjutment to improve damping. in other word, in order to improve damping, the mot effective pair are the location with the larget enitivity difference. For the 7-machine ytem, the pair i Generator (the larget negative lope) and the load at Bu 4 (the larget poitive lope). Decreaing p.u. generation at Generator and increaing p.u. load at Bu 4 would increae the damping ratio by.75% [.5% (.5%)]. Thi require increaing the load, which may not be practical. If load adjutment i not preferred, a in many ituation, the mot effective generator pair i the generator and 7. A p.u. adjutment (Generator decreae and Generator 7 increae) would reult in damping increae of.% [.5% (.5%)]. 4. Dicuion on Implementation of Modal Senitivity-baed Damping Control In on-line damping control application, model are uually not available or not accurate for the real-time condition. Perturbing a model to obtain relative enitivitie i not practical. It i deirable to etimate relative modal enitivitie from real-time meaurement. Equation can be expanded to formulate the relative enitivity etimation problem. In a general power ytem, firt aume power meaurement P i at the location of generator and load are available, and mode can be etimated from meaurement uing mode meter technique [7]. Mode change from one time intant to another can be characterized a a linear combination of the effect of power change at all the location: Δ λ ε P (6) Thi i a generalized verion of : power change i not evenly ditributed, and a noie term i included to account for noie in the meaurement. Given a time erie of meaurement, M equation can be formulated: () ( () M ) () () ote the power balance condition: P ε () + ε P ε (7) P (8) i Eliminating one power meaurement, ay P, from (7) reult in an independent et of equation: 7

8 Proceeding of the 44th Hawaii International Conference on Sytem Science - () () ( ) () ( ) () P P ε () + ε P ε (9) If M >, a leat quare method can be ued to etimated the - relative enitivitie. Thi proce only need to run once for one operating condition, and all relative enitivitie can be calculated according to (5). There i no need to enumerate all the (-)/ pair of generator and load. The larget relative enitivitie of the (-)/ pair can then derive the appropriate MAGO recommendation. The derivation and etimation proce doe not rely on a pecific topology, o conceptually it i applicable to any power ytem. With the increae of ytem ize, the dimenion of the matrix in (9) increae and the computation time increae, but the principle hold. Some implementation iue include the robutne of the method in rejecting meaurement noie, coordination for recommended adjutment, topology change impact on the effectivene, and multi-mode interaction. It i alo worth exploring the combination of three or more location, epecially when one pair of location i limited by adjutment capacity. All thee are topic that are worthy of further reearch and many of them are part of ongoing work. Relevant reult will be reported in future publication. 5. Concluion Thi paper ha examined the modal enitivity with repect to operating parameter, uch a generation and load, and explored the ue of the enitivity information to derive recommendation for operator action. The recommendation intend to change the ytem operating condition in order to improve damping to a ufficient level o that a diturbance would not reult in loing tability. The reulting MAGO approach i complementary to traditional modulation control, uch a PSS unit, in improving mall ignal tability. A concept of relative modal enitivity i propoed to characterize the pairing nature of the adjutment. For thi method, at leat two parameter have to be involved in order to change the operating condition through generation or load adjutment. Traditional eigenvalue enitivity i not applicable to thi ituation. The relative modal enitivity provide inight a to what parameter need to be adjuted, a well a how much effect on damping would be expected. Implementation iue are dicued and future work i identified to bring the MAGO approach to actual operating environment 6. Reference []. CIGRE Technical Brochure, "Analyi and Control of Power Sytem Ocillation," CIGRE Tak Force 8..7, December 996. []. Koterev, D.., C.W. Taylor, and W. A. Mitteltadt, "Model Validation for the Augut, 996 WSCC Sytem Outage," IEEE Tranaction on Power Sytem, vol. 4, no., pp , Augut 997. []. Hauer, J.F., H. Lee, J. Burn, and R. Baker, Preliminary Analyi of Wetern Sytem Ocillation Event on June 4, : BPA and Canada, Working ote for the WECC Diturbance Monitoring Work Group, July 7,. [4]. Hauer, J.F. Preliminary Examination of the Alberta Trip on Augut 4,, Working ote for the WECC Modeling and Validation Work Group, June,. [5]. Kundur, P. Power Sytem Stability and Control, McGraw-Hill Inc., ew York, Y, 994. [6]. Hauer, J.F. and J. R. Hunt, in aociation with the WSCC Sytem Ocillation Work Group, V Sympoium of Specialit in Electric Operational and Expanion Planning (SEPOPE), Recife (PE), Brazil, May 9-4, 996. [7]. Zhou,., D. Trudnowki, J. Pierre, and W. Mitteltadt, Electromechanical Mode Online Etimation uing Regularized Robut RLS Method, IEEE Tranaction on Power Sytem, vol., no. 4, pp 67-68, ov. 8. [8]. Hauer, J.F., D. J. Trudnowki, and J.D. DeSteee, WAMS Analyi Tool for Tracking of Ocillatory Dynamic, in Proceeding of the IEEE Power Engineering Society General Meeting 7, Tampa, FL, June 4-8, 7. [9]. Otojic, D.R., Stabilization of multimodal electromechanical ocillation by coordinated application of power ytem tabilizer, IEEE Tranaction on Power Sytem, vol. 6, no. 4, pp , ov. 99. []. Ilea, V. Berizzi, A.; Eremia, M., Damping Inter-area Ocillation by FACTS Device, 9 44th International Univeritie Power Engineering Conference (UPEC 9), p 5 pp., 9. []. Chung, C.Y.; Wang, K.W.; Te, C.T.; Bian, X.Y.; David, A.K, Probabilitic eigenvalue enitivity analyi and PSS deign in multimachine ytem, IEEE Tranaction on Power Sytem, vol. 8, no. 4, pp , ov.. 8

9 Proceeding of the 44th Hawaii International Conference on Sytem Science - []. Pai, M.A., D.P. Sen-Gupta, and K.R.Padiyar, Small Signal Analyi of Power Sytem, Alpha Science International, Ltd., Harrow, Middleex, UK, 4. []. Trudnowki, D., M. Donnely, and E. Lightner, "Power- Sytem Frequency and Stability Control uing Decentralized Intelligent Load," Proceeding of the 5/6 IEEE Power and Energy Society Tranmiion and Ditribution Conference and Expoition, Dalla, TX, pp , May 6. [4]. Chow, J.H.; Cheung, K.W., A toolbox for power ytem dynamic and control engineering education and reearch, IEEE Tranaction on Power Sytem, vol.7, no.4, pp , ov 99. [5]. Te, C.T.; and To, S.K., Modal and eigenvalue enitivity analye of power ocillation in interconnected ytem, APSCOM International Conference on Advance in Power Sytem Control, Operation and Management (Conf. Publ. o.48), p 88-4 vol., 99. [6]. Smed, T., "Feaible eigenvalue enitivity for large power ytem," IEEE Tranaction on Power Sytem, vol. 8, no., pp , May 99. [7]. Romme, J. and. Martin, "Computing Large-Scale Sytem Eigenvalue Mot Senitive to Parameter Change, with Application to Power Sytem Small- Signal Stability." IEEE Tranaction on Power Sytem, vol., no., p , May 8. [8]. Huang, Z.,. Zhou, F. Tuffner, Y. Chen, D. Trudnowki, W. Mitteltadt, J. Hauer, and J. Dagle, Improving Small Signal Stability through Operating Point Adjutment, in Proceeding of the IEEE Power and Energy Society General Meeting, Minneapoli, M, USA, July 5-9,. 9