ECEN 667 Power System Stability Lecture 25: FFT, Energy Methods

Transcription

1 ECEN 667 Power System Stability Lecture 5: FFT, Energy Methods Prof. Tom Overbye Dept. of Electrical and Computer Engineering Texas A&M University, 1

2 Announcements Read Chapter 9 Final is as per TAMU schedule. That is, Friday Dec 8 from 3 to 5pm Closed book, closed notes. Calculators and two 8.5 by 11 inch note sheets allowed

3 Midterm Problem 4 A 60 Hz generator is supplying 400 MW and 0 Mvar to an infinite bus (measured at the infinite bus) with 1.0 per unit voltage at the infinite bus through two parallel transmission lines. Each transmission line has a per unit impedance (100 MVA base) of 0.09j. Assume the generator is represented with a classical model with per unit transient reactance of j (on a 100 MVA base), a per unit inertia constant (H) of 10 seconds, zero damping. At time = 0 one of the transmission lines experiences a balanced three phase short to ground one third (1/3) of the way down the line from the generator to the infinite bus (i.e., model the line with 1/3 its original impedance on the generator side and /3 on the infinite bus side). E 1. 0 j j j During the Fault from Generator Terminal Looking into Network Zth j j j0.03 Vth j0.03 j0.09 3

4 Midterm Problem 4 b) During the fault V =0.5, Z =j0.05 th th d pus pu 377 dt d pu PM PE 4 sin = sin (during the fault) dt H c) x, ( 0), t x pu 0 1 x( tt) x( t) k1 k with k1= tf ( x( t)), k = tf ( x( t) k1), k1( 0. 0), ( 0. 0) 0. 0) k x( k1( 0. 0), ( 0. 0) 0. 04) k x(

5 Fast Fourier Transform (FFT) Applications: Motivational Example The below graph shows a slight frequency oscillation in a transient stability run The question is to figure out the source of the oscillation (shown here in the bus frequency) Plotting all the frequency values is one option, but sometimes small oscillations could get lost A solution is to do an FFT 5

6 FFT Overview To understand the FFT, it is useful to start with a Fourier series, which seeks to represent any periodic signal, with frequency F=1/T, as a sum of sinusoidals with frequencies that are integer multiples of F, nf DC is n=0, fundamental is n=1, harmonics are n > 1 Often the complete representation requires an infinite number of terms Figure shows the Fourier series for a square wave, showing the first four terms Nonperiodic signals can be represented by letting T go to infinity this gives a continuous Fourier Spectrum Image Source: wikipedia.org/wiki/fourier_series 6

7 Sampled Signals Signals used in power system analysis (or from PMUs) are sampled A bandlimited signal can be reproduced exactly if it is sampled at a rate at least twice the highest frequency Sampling shifts frequency spectrum by 1/T Sampling that is too slowly causes frequency overlap Image: upload.wikimedia.org/wikipedia/commons/c/ca/fourier_transform,_fourier_series,_dtft,_dft.gif 7

8 Aliasing All practical signals are time-limited (and thus not bandlimited) and therefore have infinite bandwidth, so there is always frequency overlap The overlapping of the frequencies is known as aliasing; image shows how sampling the high frequency signal too slowly gives the appearance of a lower frequency signal Aliasing is reduced by faster sampling; anti-aliasing filtering (low pass filters) can also be used Image: upload.wikimedia.org/wikipedia/commons/thumb//8/aliasingsines.svg/000px-aliasingsines.svg.png 8

9 Fast Fourier Transform (FFT) Overview Discrete Fourier Transforms (DFTs) can be used to provide frequency information about sampled, nonperiodic signals The FFT is just a fast DFT with N 0 points its computational order is N 0 ln(n 0 ) This allows it to be applied to many signals PowerWorld quick access to an FFT is available in the transient stability time values (or plot) case information displays by selecting "Frequency Analysis" from the right-click menu 9

10 Frequency Analysis Display The frequency analysis display shows the original data, the FFT for each time result, and a frequency summary With about 840 time values, and more than 18,000 signals (bus frequencies in this example), the FFT takes about six seconds The various tabs provide access to the original data, sampled data, individual FFT results and summary results Frequency resolution is 1/(T end - T start ) Hz 10

11 Frequency Analysis Display The locations of unusual frequencies can be determined by viewing/plotting the frequency summary results Here the spike at 3.35 Hz is shown to be associated with bus

12 Frequency Summary Plot of Maximum and Average Values The maximum value summary, plotted with the average values, makes the source of the observed 3.35 Hz oscillation readily apparent (here at gen 19318) For this example the system was modified to deliberately introduce a problem at gen by artificially increasing its exciter gain 1

13 Gen Bus Frequency (150 MW Output) Graphs shows the bus frequency at and its generator field voltage 13

14 Previous Results Showing Aliasing When Sampled at 6 Hz The below results show the impact of aliasing when the sampling is too slow 14

15 Stability Phenomena and Tools Large Disturbance Stability (Non-linear Model) Small Disturbance Stability (Linear Model) Structural Stability (Non-linear Model) Loss of stability due to parameter variations. Tools Simulation Repetitive time-domain simulations are required to find critical parameter values, such as clearing time of circuit breakers. Direct methods using Lyapunov-based theory (Also called Transient Energy Function (TEF) methods) Can be useful for screening Sensitivity based methods. 15

16 Transient Energy Function (TEF) Techniques No repeated simulations are involved. Limited somewhat by modeling complexity. Energy of the system used as Lyapunov function. Computing energy at the controlling unstable equilibrium point (CUEP) (critical energy). CUEP defines the mode of instability for a particular fault. Computing critical energy is not easy. 16

17 Judging Stability / Instability Monitor Rotor Angles (a) Stable (b) Stable (c) Unstable (d) Unstable Stability is judged by Relative Rotor Angles. 17

18 Mathematical Formulation A power system undergoing a disturbance (fault, etc), followed by clearing of the fault, has the following model I x( t) f ( x( t)) t 0 ( 1) F x( t) f ( x( t)) 0 t t ( ) x( t) f ( x( t)) t t ( 3) cl cl (1) Prior to fault (Pre-fault) () During fault (Fault-on or faulted) (3) After the fault (Post-fault) T cl is the clearing time t 0 t tcl X 0 t t cl X Faulted t t cl Post-Fault (line-cleared) 18

19 Critical Clearing Time Assume the post-fault system has a stable equilibrium point x s All possible values of x(t cl ) for different clearing times provide the initial conditions for the post-fault system Question is then will the trajectory of the post fault system, starting at x(t cl ), converge to x s as t Largest value of t cl for which this is true is called the critical clearing time, t cr The value of t cr is different for different faults 19

20 Region of Attraction (ROA) All faulted trajectories cleared before they reach the boundary of the ROA will tend to x s as t (stable) x( t cl ) The region need not be closed; it can be open:.. x o x s 0

21 Methods to Compute RoA Had been a topic of intense research in power system literature since early 1960 s. The stable equilibrium point (SEP) of the post-fault system, x s, is generally close to the pre-fault EP, x 0 Surrounding x s there are a number of unstable equilibrium points (UEPs). The boundary of ROA is characterized via these UEPs xui,, i 1,... f ( x) 0 i. e. f ( x ) 0 i 1,... ui, 1

22 Characterization of RoA Define a scalar energy function V(x) = sum of the kinetic and potential energy of the post-fault system. Compute V(x u,i ) at each UEP, i=1,, Defined V cr as Vcr Min V ( x) x ui, RoA is defined by V(x) < V cr But this can be an extremely conservative result. Alternative method: Depending on the fault, identify the critical UEP, x u,cr, towards which the faulted trajectory is headed; then V(x) < V(x u,cr ) is a good estimate of the ROA.

23 Lyapunov s Method Defining the function V(x) is a key challenge Consider the system defined by x f ( x), f ( xs ) 0 Lyapunov's method: If there exists a scalar function V(x) such that 1) V ( x ) = 0 s s ) V( x) > 0 for all x around x 3) V ( x) 0 for all x around x Then x s is stable in the sense of Lyapunov s EP x is asympotically stable if V ( x) < 0 for x x around x s s s 3

24 Ball in Well Analogy The classic Lyapunov example is the stability of a ball in a well (valley) in which the Lyapunov function is the ball's total energy (kinetic and potential) UEP UEP SEP For power systems, defining a true Lyapunov function often requires using restrictive models 4

25 Power System Example Consider the classical generator model using an internal node representation (load buses have been equivalenced) m d i di M i Di Pi ( Cij sinij Dij cos ij ) dt dt j1 j i C ij are the susceptance Pi TMi Ei Gii terms, D ij the conductance Functionally terms d i di M i Di Pi Pei ( 1,..., m) i 1,..., m dt dt i i s 1 i P i P ei ( i,..., m) D i ( i s ) M i 5

26 Constructing the Transient Energy Function (TEF) The reference frame matters. Either relative rotor angle formulation, or COI reference frame. COI is preferable since we measure angles with respect to the mean motion of the system. TEF for conservative system (i.e., zero damping) o 1 m MT i 1 M i i m where M T i 1 M With center of speed as i o 1 m MT i 1 M. We then transform the variables to the COI variables as,. i i o i i o i i It is easy to verify i i o i o i 6

27 TEF We consider the general case in which all M i 's are finite. We have two sets of differential equations: di F M f ( ) 0 t t ( Faulted) di dt i dt i cl And, i 1,,..., m i di M f ( ) t t ( Post fault) di dt i dt i cl, i 1,,..., m i Let the post fault system has a SEP at This SEP is found by solving fi ( θ) 0, i 1,.., m s θ θ, ω 0 7

28 TEF Steps for computing the critical clearing time are: Construct a Lyapunov (energy) function for the post-fault system. Find the critical value of the Lyapunov function (critical energy) for a given fault Integrate the faulted equations until the energy is equal to the critical energy; this instant of time is called the critical clearing time Idea is once the fault is cleared the energy can only decrease, hence the critical clearing time is determined directly Methods differ as to how to implement steps and 3. 8

29 Potential Energy Boundary Surface Figure from course textbook 9

30 TEF Integrating the equations between the post-fault SEP and the current state gives 1 V M f d m m i (, ) i i ( ) s i i i i1 i1 1 ( ) [ (cos cos )] V KE m m m1 m s s M ii Pi i i Cij ij ij i1 i1 i1 ji1 i s i j s j D ( ) V ( ) cos d( )] ij ij i j PE C ij are the susceptance terms, D ij the conductance terms; the conductance term is path dependent 30

31 TEF V ( θω, ) contains path dependent terms. Cannot claim that V( θω, ) is p.d. If conductance terms are ignored then it can be shown to be a Lyapunov function Methods to compute the UEPS are Potential Energy Boundary Surface (PEBS) method. Boundary Controlling Unstable (BCU) equilibrium point method. Other methods (Hybrid, Second-kick etc) (a) and (b) are the most important ones. 31

32 Equal Area Criterion and TEF For an SMIB system with classical generators this reduces to the equal area criteria TEF is for the post-fault system Change notation from T m to P m d max M Pm Pe sin dt ( 1) max EE 1 Pe sin I X ( ) X X F ( Faulted ) X P e P e X I E E X E1E X 1 F I ( Post fault) sin ( Faulted ) sin ( Post fault) 3

33 TEF for SMIB System d max M P sin (1) m Pe dt The right hand side of (1) can be written as V ( ) P P cos () d dt d dt d dt PE Multiplying (1) by i.e i.e M 1 [ V (, )] V (, ) d dt M 1 m V d dt V PE 0 M max e, re-write PE ( ) ( ) Hence, the energy function is V PE 0 ( ) 0 since V PE,where d dt 33

34 TEF for SMIB System (contd) The equilibrium point is given by 0 P P sin ( 1) m max e P sin ( ) s 1 m max Pe This is the stable e.p. Can be verified by linearizing. Eigenvalues on j axis. (Marginally Stable) With slight damping eigenvalues are in L.H.P. TEF is still constructed for undamped system. 34

35 TEF for SMIB System The energy function is V (, ) V V ( ) KE PE M max e There are two UEP: u1 = p- s and u = -p- s A change in coordinates sets V PE =0 for = s s s max s VPE (, ) Pm ( ) Pe (cos cos ) With this, the energy function is V M P P 1 P P 1 s max s (, ) m( ) e (cos cos ) V KE V PE s (, ) m cos The kinetic energy term is V KE M 1 35

36 Equal-Area Criterion During the fault A 1 is the gain in the kinetic energy and A 3 the gain in potential energy Figure from course textbook 36

37 Energy Function for SMIB System V(,) is equal to a constant E, which is the sum of the kinetic and potential energies. It remains constant once the fault is cleared since the system is conservative (with no damping) V(,) evaluated at t=t cl from the fault trajectory represents the total energy E present in the system at t=t cl This energy must be absorbed by the system once the fault is cleared if the system is to be stable. The kinetic energy is always positive, and is the difference between E and V PE (, s ) 37

38 Potential Energy Well for SMIB System Potential energy well or P.E. curve How is E computed? 38

39 Structure Preserving Energy Function If we retain the power flow equations 39

40 Structure Preserving Energy Function Then we can get the following energy function 40

41 Energy Functions for a Large System Need an energy function that at least approximates the actual system dynamics This can be quite challenging! In general there are many UEPs; need to determine the UEPs for closely associated with the faulted system trajectory (known as the controlling UEP) Energy of the controlling UEP can then be used to determine the critical clearly time (i.e., when the faulton energy is equal to that of the controlling UEP) For on-line transient stability, technique can be used for fast screening 41