Power System Model Reduction. Fall 2014 CURENT Course

Transcription

1 Power System Model Reduction Fall 04 CURENT Course October 9, 04 Joe H. Chow Rensselaer Polytechnic Institute, Troy, New York, USA

2 Model Reduction of Large Power Systems. Simulation of power system dynamics for stability analysis on a digital computer: needs the most comprehensive power system model so that i. the relevant dynamics can be accurately simulated given the computing resources ii. the simulation can be completed in a reasonable amount of time.. One of the decisions is the geographic extent of the power system data set: disturbance always in the study region, but external system(s) also needed. Rensselaer Polytechnic Institute October 04 JHC

3 Model Reduction of Large Power Systems Internal and external systems Gen + Controls Gen + Controls V b V b V bk I I I k Gen + Controls Gen + Controls External System Study System Disturbance Travelling through the external system Gen + Controls Gen + Controls V b V b P P Gen + Controls A B Gen + Controls External System Study System Rensselaer Polytechnic Institute October 04 JHC 3

4 Model Reduction Approaches. Coherency some machines swinging together after a disturbance i. Time simulation (R. Podmore) ii. Modal coherency (R. Schleuter) iii. Slow coherency (J. Chow, et. al) coherency with respect to the slow interarea modes iv. Weak-link method slow coherent areas are weakly connected (N. Rao, J. Zaborzsky). Aggregation obtaining an equivalent power system model i. Generator aggregation (R. Podmore) ii. Singular perturbations provides improvements to equivalent generator reactances (J. Chow, et. al) Rensselaer Polytechnic Institute October 04 JHC 4

5 Model Reduction Approaches 3. Linear model analysis of external systems Gen + Controls V b V b I I i. Modal truncation (J. Undrill, W. W. Price) V bk Gen + Controls External System ii. Selective modal analysis (SMA) (G. Verghese, I. Peres-Arriaga, L. Rouco) iii. Krylov method moment matching (D. Chaniotis, M.A. Pai) iv. Balanced truncaton best reachibility (controllability) - observability (Shanshan Liu) 4. Artificial Neural Networks (F. Ma, V. Vittal) 5. Synchrophasor data (A. Chakrabortty, Chow) I k Rensselaer Polytechnic Institute October 04 JHC 5

6 Monographs Rensselaer Polytechnic Institute October 04 JHC 6

7 . Slow coherency Topics. Reduced order modeling of dominant transfer paths in large power systems using synchrophasor data (work with Dr. Aranya Chakrabortty) Rensselaer Polytechnic Institute October 04 JHC 7

8 Slow Coherency A large power system usually consists of tightly connected control regions with few interarea ties for power exchange and reserve sharing The oscillations between these groups of strongly connected machines are the interarea modes These interarea modes are lower in frequency than local modes and intra-plant modes Singular perturbations method can be used to show this time-scale separation c J. H. Chow (RPI-ECSE) Coherency October 9, 04 / 9

9 Klein-Rogers-Kundur -Area System Gen Gen Local mode Local mode Gen Gen Load 3 Interarea mode Load 3 c J. H. Chow (RPI-ECSE) Coherency October 9, 04 3 / 9

10 Coherency in -Area System Disturbance: 3-phase fault at Bus 3, cleared by removal of line between Buses 3 and Gen Gen Gen Gen.005 Machine Speed (pu) Time (sec) c J. H. Chow (RPI-ECSE) Coherency October 9, 04 4 / 9

11 Power System Model An n-machine, N-bus power system with classical electromechanical model and constant impedance loads: m i δi = P mi P ei = P mi E iv j sin(δ i θ j ) x di = f i (δ,v) () where machine i modeled as a constant voltage E i behind a transient reactance x di m i = H i /Ω, H i = inertia of machine i Ω = πf o = nominal system frequency in rad/s damping D = 0 δ = n-vector of machine angles P mi = input mechanical power, P ei = output electrical power c J. H. Chow (RPI-ECSE) Coherency October 9, 04 5 / 9

12 Power System Model the bus voltage V j = ( ) Vjre +V jim, θ j = tan Vjim V jre () V jre and V jim are the real and imaginary parts of the bus voltage phasor at Bus j, the terminal bus of Machine i V = N-vector of real and imaginary parts of load bus voltages c J. H. Chow (RPI-ECSE) Coherency October 9, 04 6 / 9

13 Power Flow Equations For each load bus j, the active power flow balance ( ) N Vjre + jv jim P ej Real (V jre + jv jim V kre jv kim ) R k=,k j Ljk + jx Ljk V j G j = g j = 0 (3) and the reactive power flow balance ( ) N Vjre + jv jim Q ej Imag (V jre + jv jim V kre jv kim ) R k=,k j Ljk + jx Ljk V j B j +Vj B Ljk = g j = 0 (4) R Ljk, X Ljk, and B Ljk are the resistance, reactance, and line charging, respectively, of the line j-k P ej and Q ej are generator active and reactive electrical output power, respectively, if bus j is a generator bus G j and B j are the load conductance and susceptance at bus j Note that j denotes the imaginary number if it is not used as an index. c J. H. Chow (RPI-ECSE) Coherency October 9, 04 7 / 9

14 Electromechanical Model M δ = f(δ,v), 0 = g(δ,v ) (5) M = diagonal machine inertia matrix f = vector of acceleration torques g = power flow equation c J. H. Chow (RPI-ECSE) Coherency October 9, 04 8 / 9

15 Linearized Model Linearize (5) about a nominal power flow equilibrium (δ o,v o ) to obtain M δ = f(δ,v) δ + f(δ,v) δo,v o V = K δ+k V (6) δo,v o 0 = g(δ,v ) δ + g(δ,v) δo,v o V = K 3 δ +K 4 V (7) δo,v o δ = n-vector of machine angle deviations from δ o V = N-vector of the real and imaginary parts of the load bus voltage deviations from V o K, K, and K 3 are partial derivatives of the power transfer between machines and terminal buses, K is diagonal K 4 = network admittance matrix and nonsingular. the sensitivity matrices K i can be derived analytically or from numerical perturbations using the Power System Toolbox c J. H. Chow (RPI-ECSE) Coherency October 9, 04 9 / 9

16 Linearized Model Solve (6) for to obtain where V = K 4 K 3 δ (8) M δ = K K K 4 K 3 δ = K δ (9) K ij = E i E j (B ij cos(δ i δ j ) G ij sin(δ i δ j )) δo,v o, i j (0) and G ij +jb ij is the equivalent admittance between machine i and j. Furthermore, n K ii = K ij () j=,j i Thus the row sum of K equal to zero. The entries K ij are known as the synchronizing torque coefficients. c J. H. Chow (RPI-ECSE) Coherency October 9, 04 0 / 9

17 Slow Coherent Areas Assume a power system has r slow coherent areas of machines and the load buses that interconnect these machines Define δ α i = deviation of rotor angle of machine i in area α from its equilibrium value m α i = inertia of machine i in area α Order the machines such that δ α i from the same coherent areas appears consecutively in δ. c J. H. Chow (RPI-ECSE) Coherency October 9, 04 / 9

18 Weakly Coupled Areas We attribute the slow coherency phenomenon to be primarily due to the connections between the machines in the same coherent areas being stiffer than those between different areas, which can be due to two reasons: The admittances of the external connections Bij E much smaller than the admittances of the internal connections Bpq I ε = BE ij B I pq () where E denotes external, I denotes internal, and i,j,p,q are bus indices. This situation also includes heavily loaded high-voltage, long transmission lines between two coherent areas. c J. H. Chow (RPI-ECSE) Coherency October 9, 04 / 9

19 The number of external connections is much less than the number of internal connections ε = γe γ I (3) where γ E = max α {γe α }, γi = min{γ I α α }, α =,...,r (4) γα E = (the number of external connections of area α)/nα γα I = (the number of internal connections of area α)/n α where N α is the number of buses in area α. c J. H. Chow (RPI-ECSE) Coherency October 9, 04 3 / 9

20 Internal and External Connections For a large power system, the weak connections between coherent areas can be represented by the small parameter Separate the network admittance matrix into ε = ε ε (5) K 4 = K I 4 +ǫk E 4 (6) where K4 I = internal connections and KE 4 = external connections. The synchronizing torque or connection matrix K is where K = K K (K I 4 +ε(ki 4 )) K 3 = K K (K I 4 (I +ε(ki 4 ) K E 4 )) K 3 = K I +ǫk E (7) K I = K K (K I 4) K 3, K E = K K E 4εK 3 (8) In the separation (7), the property that each row of K I sums to zero is preserved. c J. H. Chow (RPI-ECSE) Coherency October 9, 04 4 / 9

21 Slow Variables Define for each area an inertia weighted aggregate variable y α = n α i= m α i δα i /mα, α =,,...,r (9) where m α i = inertia of machine i in area α, n α = number of machines in area α, and m α = n α i= is the aggregate inertia of area α. m α i, α =,,...,r (0) c J. H. Chow (RPI-ECSE) Coherency October 9, 04 5 / 9

22 Denote by y = the r-vector whose αth entry is y α. The matrix form of (9) is where y = C δ = M a U T M δ () U = blockdiag(u,u,...,u r ) () is the grouping matrix with n α column vectors [ ] T u α =..., α =,,...,r (3) M a = diag(m,m,...,m r ) = U T MU (4) is the r r diagonal aggregate inertia matrix. c J. H. Chow (RPI-ECSE) Coherency October 9, 04 6 / 9

23 Fast Variables Select in each area a reference machine, say the first machine, and define the motions of the other machines in the same area relative to this reference machine by the local variables z α i = δα i δ α, i =,3,...,n α, α =,,...,r (5) Denote by z α the (n α )-vector of z α i and conside z α as the αth subvector of the (n r)-vector z. Eqn. (5) in matrix form is z = G δ = blockdiag(g,g,...,g r ) δ (6) where G α is the (n α ) n α matrix G α = (7) c J. H. Chow (RPI-ECSE) Coherency October 9, 04 7 / 9

24 Slow and Fast Variable Transformation A transformation of the original state δ into the aggregate variable y and the local variable z [ ] y [ ] C z = G δ (8) The inverse of this transformation is ( )[ y δ = U G + z ] (9) where is block-diagonal. G + = G T (GG T ) (30) c J. H. Chow (RPI-ECSE) Coherency October 9, 04 8 / 9

25 Slow Subsystem Apply the transformation (8) to the model (9), (7) where M a ÿ = ǫk a y +ǫk ad z M d z = ǫk da y +(K d +ǫk dd )z (3) M d = (GM G T ), K a = U T K E U K da = U T K E M G T M d, K da = M d GM K E U K d = M d GM K I M G T M d,k dd = M d GM K E M G T M d (3) K a, K ad, and K da are independent of K I System (3) is in the standard singularly perturbed form ε is both the weak connection parameter and the singular perturbation parameter Neglecting the fast dynamics, the slow subsystem is M a ÿ = εk a y (33) c J. H. Chow (RPI-ECSE) Coherency October 9, 04 9 / 9

26 Formulation including Power Network Apply (8) to the model (6)-(7) where M a ÿ = K y +K z +K 3 V M d z = K y +K z +K 3 V 0 = K 3 y +K 3 z +(K I 4 +εk E 4 ) V (34) K = U T K U, K = U T K G +, K 3 = U T K, K = (G + ) T K U K = (G + ) T K G +, K 3 = (G + ) T K, K 3 = K 3 U, K 3 = K 3 G + (35) Eliminating the fast variables, the slow subsystem is M a ÿ = K y +K 3 V 0 = K 3 y +K 4 V (36) This is the inertial aggregate model which is equivalent to linking the internal nodes of the coherent machines by infinite admittances. c J. H. Chow (RPI-ECSE) Coherency October 9, 04 0 / 9

27 -Area System Example Connection matrix K = Decompose K into internal and external connections K I = εk E = (37) (38) (39) c J. H. Chow (RPI-ECSE) Coherency October 9, 04 / 9

28 EM Modes in -Area System λ(m K) = 0, 4.787, , 6.53 (40) with the corresponding eigenvector vectors v = 0.5, v = 0.567, v = 0.94, v = Interarea mode: 4.79 = ±j3.779 rad/s Local modes: = ±j7.795 rad/s and 6.53 = ±j7.890 rad/s (4) c J. H. Chow (RPI-ECSE) Coherency October 9, 04 / 9

29 Slow and Fast Subsystems of -Area System Slow dynamics: M a = π 60 [ ] [ , εk a = ] (4) The eigenvalues of Ma K a are 0 and 4.56 an interarea mode frequency of 4.56 = ±j3.86 rad/s. Fast local dynamics: [ ] [ ] M d =, K π d = (43) The eigenvalues of M d K d are and local modes of ±j and ±j7.393 rad/s. c J. H. Chow (RPI-ECSE) Coherency October 9, 04 3 / 9

30 Finding Coherent Groups of Machines Compute the electromechanical modes of an N-machine power Select the (interarea) modes with frequencies less than Hz 3 Compute the eigenvectors (mode shapes) of these slower modes 4 Group the machines with similar mode shapes into slow coherent groups 5 These slow coherent groups have weak or sparse connections between them c J. H. Chow (RPI-ECSE) Coherency October 9, 04 4 / 9

31 Use V s to form a new coordinate system c J. H. Chow (RPI-ECSE) Coherency October 9, 04 5 / 9 Grouping Algorithm Plot rows of the slow eigenvector V s of the interarea modes 0 0 Use Gaussian elimination to select the r most separated rows as reference vectors and group them into V s and reorder V s as [ ] [ ] [ ] [ ] [ ] Vs Vs I I 0 V s = Vs = = + L O(ε) V s V s L g (44)

32 Coherent Groups in -Area System V s = Then Gen Gen Gen Gen, V s = 0 V s V s 0 = [ Gen Gen Gen Gen ] Gen Gen (45) (46) c J. H. Chow (RPI-ECSE) Coherency October 9, 04 6 / 9

33 Coherency for Load Buses V θ = Bus Bus Bus 3 Bus 0 Bus Bus Bus 3 Bus 0 Bus 0 Bus 0 Bus 0 V θ V s = (47) Bus 0 c J. H. Chow (RPI-ECSE) Coherency October 9, 04 7 / 9

34 7-Area Partition of NPCC 48-Machine System c J. H. Chow (RPI-ECSE) Coherency October 9, 04 8 / 9

35 EQUIV and AGGREG Functions for Power System Toolbox L group: grouping algorithm coh-map, ex group: tolerance-based grouping algorithm Podmore: R. Pormore s algorithm of aggregating generators at terminal buses i agg: inertial aggregation at generator internal buses slow coh: slow-coherency aggregation, with additional impedance corrections These functions use the same power system loadflow input data files as the Power System Toolbox. c J. H. Chow (RPI-ECSE) Coherency October 9, 04 9 / 9

36 Power Transfer Paths/Interfaces Rensselaer Polytechnic Institute October 04 JHC 8

37 Power Transfer Paths/Interfaces. The integrity of the power transfer interfaces is crucial for a large interconnected system to function properly. A disruption of a major transfer path may have a spillover effect into parallel power transfer paths in neighboring regions.. Questions: a. How can PMU data be used to monitor the integrity of the power transfer interfaces, for both internal and external visibility? b. How many PMUs do we need and where should they be located? c. Are there analytical concepts and approaches to extract stability margin information from PMU data? Rensselaer Polytechnic Institute October 04 JHC 9

38 Power Transfer Paths/Interfaces Power transfer between two areas with multiple lines and PMU monitoring PMU PMU P Zone A Exports Power P P 3 Zone B Imports Power PMU 3 Some US Interconnection interfaces may consist of a limited number of transmission lines (CA, NY, NE) many transmission lines Rensselaer Polytechnic Institute October 04 JHC 0

39 Inter-area Model Estimation -Area Power System -Machine Equivalent 0 G G G L4 L4 G 4 0 V3 r V3 e jx V ~ e 0 re jxe Ga jx jx G a E, ~ ~ Problem: Given synchronized voltage phasors (magnitude and phase) at Buses 3 and 3 and current phasor between the two buses in the -area power system, develop an equivalent -machine model to represent the power transfer between the two areas. Variations: Voltage phasors at Bus 0 also available. Voltage phasor at Bus 3, the current phasor going from Bus 3 to Bus 0, and the impedance between Buses 3 and 3. Difficulty is how to look outward into the importing and exporting areas. Need dynamic data, i.e., can t do it with State Estimator solution. In particular, need interarea mode oscillations in the voltage and current phasors. I E, Rensselaer Polytechnic Institute October 04 JHC

40 Inter-area Model Estimation Measurement-based model reduction approach Use bus voltage and line current phasors Develop new concepts and calculation methods: Exact extrapolation equations can be developed for an ideal -machine system Dynamic Model Estimation (DME) algorithm Extend DME to -area system Interarea Model Estimation (IME) algorithm: verify with disturbances Additional research and customization needed: transfer paths with intermediate voltage support, transfer interfaces with multiple transmission lines, impact of disturbances, Rensselaer Polytechnic Institute October 04 JHC

41 Dynamic Model Estimation DME Problem Formulation for a -machine system Two machine power system I ~ ~ ~ ~ E G r r z T jx T T Classical model representation ~ ~ V V z I z jx r jx V V z e re jxe z e E ~ E r E E e jx ~ e z Ti = transformer impedance z i = transformer impedance + direct-axis transient reactance r z T jx T T E G Two generators G and G with inertias H and H are connected to Buses and G supplies power to G (load) Dynamic model swing equations H where P P P m loss e H r z m P m m H EE z P H m H s E H P P H e m H s P E loss s (i =, ) cos( ) H cos( ) H Rensselaer Polytechnic Institute October 04 JHC 3

42 Dynamic Model Estimation DME Problem Formulation for a -machine system (Contd.) ~ ~ V V z I z r jx r jx E ~ E r E E e jx ~ e Assume PMUs are located at Buses and Available Phasor Variables V V I,,,,, I DME Problem : Given the available phasor variables, that exhibit a few cycles of oscillations, and assuming E and E to be constant, compute E E z z z H, H,,,,,, e, to completely characterize the dynamic behavior of the -machine system. Because z ~ ~ ~ e ( V V ) / I and E ~ V ~ jz I ~, the problem reduces to the estimation of z e ~ E ~ V jz z z H, H ~ I,, Rensselaer Polytechnic Institute October 04 JHC 4

43 Reactance Extrapolation Dynamic Model Estimation Key idea : Amplitude of voltage oscillation at any point on the transfer path is a function of its electrical distance from the two fixed voltage sources. A B z C ~ V ( x, r) [ E( a) E( acos( ) bsin( ))] j [ E ( bcos( ) asin( )) be where E, Voltage Magnitude at B a rr ' xx ~ V( x, r) V ( x, r) ~ I V x ' b xr ' rx e e e e re ' xe ' re ' xe ' c EE ( a a b ) cos( ) bsin( ) ' E, 0 ] where c E ( b ( a) ) E ( b a ) Rensselaer Polytechnic Institute October 04 JHC 5

44 Dynamic Model Estimation Reactance Extrapolation Linearize model about (δ 0, ω 0 = 0, V ss ) : where the Jacobian is V x) J( a, b, ) ( 0 V ( a, b, 0 ) EE J( a, b, 0 ) : [( a a b )sin( 0 ) bcos( 0 )] V ( a, b, 0 ) 0 x Assume uniform impedance along transfer path a and b 0 EE sin( 0) J( a,0, 0) a ( a) (*) V ( a, b, ) (*) 0 x e Note : J(x,δ0) in (*) has a numerator varying with x, and a denominator equal to steady state voltage magnitude at B. Rensselaer Polytechnic Institute October 04 JHC 6

45 Dynamic Model Estimation Reactance Extrapolation J(x,δ0) in (*) can be used to estimate x and x if voltage oscillations are measured or calculated at an additional intermediate bus between Bus and ~ ~ V V ~ V 3 z z e z e z E, 3 E, Following a perturbation, let V V im iss Amplitudeof Voltage Swing at Bus i Steady - statevoltage at Bus i Computed from PMU measurements From (*), V in V im V iss A( a ) a, i,, i i 3 with A EEsin( o The reactances x and x, and the unknown constant A can be solved numerically (3 nonlinear equations with 3 unknowns) Rensselaer Polytechnic Institute October 04 JHC 7

46 Rensselaer Polytechnic Institute October 04 JHC 8 Dynamic Model Estimation Inertia Estimation From linearized model H H H H H ( 0) r e where f s is the measured swing frequency and is the aggregate Inertia For a second equation in H and H, use conservation of angular momentum 0 ) ( ) ( dt P P P P dt H H H H e m e m H H However, ω and ω are not available from PMU data, Estimate ω and ω from the measured frequencies ξ and ξ at Buses and ) ( ) cos( 0 x x x H E E f e s

47 Rensselaer Polytechnic Institute October 04 JHC 9 Dynamic Model Estimation Again use linearization to derive bus frequency and machine speed expressions ) cos( ) )cos( ( c b a c b a ) cos( ) )cos( ( c b a c b a where ), (, ) ( i i i i i i i r E c r r E E b r E a x e x x x r e e x x x x x r (i=, ) ξ and ξ are measured, and a i, b i, c i are known from reactance extrapolation Hence, we can calculate ω /ω and solve for H and H and, Normalized Reactance r Frequency Oscillation Magnitude (r/s) Inertia Estimation

48 Example Dynamic Model Estimation Illustrate DME on classical -machine model (re = 0) Disturbance is applied to the system and the response simulated in MATLAB V V V m ss n V V V m ss n V V V 3m 3ss 3n DME Algorithm Voltage Oscillations at 3 buses x = pu x = pu Exact values: x = 0.34 pu, x = 0.39 pu Rensselaer Polytechnic Institute October 04 JHC 0

49 Example (cont.) Dynamic Model Estimation Jacobian curve fit for reactance extrapolation Rensselaer Polytechnic Institute October 04 JHC

50 Example (cont.) Dynamic Model Estimation Inertia estimation Find the bus frequencies by passing the bus voltage angles through a derivative filter with bandwidth higher than the inter-area mode frequency (0.9 Hz) G( s) 0.0 s s DME Algorithm H = 6.48 pu, H = 9.49 pu Exact values: H = 6.5 pu, H = 9.5 pu Rensselaer Polytechnic Institute October 04 JHC

51 Example (cont.) Dynamic Model Estimation Frequency curve fit for inertia estimation Rensselaer Polytechnic Institute October 04 JHC 3

52 Inter-area Model Estimation Consider inter-area dynamics across the transfer interface of a multimachine power system Extend the DME algorithm to estimate an equivalent two-machine model Assumption: Interface exhibits a single dominant mode of inter-area oscillation A new approach to power system model reduction based on PMU measurements Rensselaer Polytechnic Institute October 04 JHC 4

53 Inter-area Model Estimation -Area Power System -Machine Equivalent 0 G G 3 0 ~ ~ V3 r V3 e jx V ~ e 0 re jxe Ga jx jx G a E, E, G L4 L4 G 4 I IME Problem : Given the phasor measurements V that exhibit a 3, 3, V3, 3, I, I few cycles of inter-area oscillations, and assuming E and E to be constant, compute E E z z z H, H,,,,,, e, a a to completely characterize the dynamic behavior of the reduced -machine system. Similar derivations as DME but in this case local modes are present in system response Apply a modal estimation method such as ERA to isolate inter-area mode Rensselaer Polytechnic Institute October 04 JHC 5

54 Inter-area Model Estimation Disturbance applied to a -area system Bus 3 Voltage Bus 3 Voltage Bus 0 Voltage With 300 MW power transfer the inter-area oscillation frequency is Hz, and the local mode frequencies are.93 Hz and.308 Hz. IME estimates: x = pu, x = pu H a = 8.98 pu, H a =.55 pu Rensselaer Polytechnic Institute October 04 JHC 6

55 Inter-area Model Estimation Disturbance applied to a -area system Frequency at Bus 3 Frequency at Bus 3 Rensselaer Polytechnic Institute October 04 JHC 7

56 Inter-area Model Estimation Frequency Oscillation Magnitude (r/s) x x e x Normalized Reactance r Reactance Curve Fit Frequency Curve Fit Rensselaer Polytechnic Institute October 04 JHC 8

57 Inter-area Model Estimation Disturbance plots comparing full model to reduced models Voltage Oscillations at Bus 3 Voltage Oscillations at Bus 3 Angular Difference Speed Difference Rensselaer Polytechnic Institute October 04 JHC 9

58 Inter-area Model Estimation Some important observations Inter-area model estimation is independent of the severity of the disturbance event resulting in the same post-fault condition (robustness) Inter-area model is dependent on the amount of inter-area power transfer. In the -area system example, larger amount of power transfers results in: i. Higher angular separation between the sending end and the receiving end ii. Slower frequency of oscillation iii. Smaller reactance at the sending end and larger reactance at the receiving area iv. Equivalent inertias of the two areas also change Rensselaer Polytechnic Institute October 04 JHC 30

59 Conclusions and Future Research New techniques in using synchronized phasor data to identify power transfer paths/interfaces and hence for PMU siting. Establish power-angle curves and energy function analysis on each power transfer path. Monitor the power-angle curves as the system evolves through a sequence of disturbances. Monitor energy levels in the transfer paths to determine stability (loss of synchronism or negative damping) and to determine which transfer path is under the most stress. Improve system transmission planning and operations planning studies Different types of studies Different ways to see and evaluate results Extend calculations to dynamic voltage stability. Rensselaer Polytechnic Institute October 04 JHC 3

60 Acknowledgements This work was supported primarily by the ERC Program of the National Science Foundation and DOE under NSF Award Number EEC Other US federal and state government agencies and industrial sponsors of CURENT research are also gratefully acknowledged. Rensselaer Polytechnic Institute October 04 JHC 3