1. ECE 576 – Power System Dynamics and Stability Prof. Tom Overbye University of Illinois at Urbana-Champaign overbye@illinois.edu 1 Lecture 25: FFT, Power System Stabilizers

2. Mode Observability, Shape, Controllability and Participation Factors In addition to frequency and damping, there are several other mode characteristics Observability tells how much of the mode is in a signal, hence it is associated with a particular signal Mode Shape is a complex number that tells the magnitude and phase angle of the mode in the signal (hence it quantifies observability) Controllability specifies the amount by which a mode can be damped by a particular controller Participation facts is used to quantify how much damping can be provided for a mode by a PSS 2

3. Determining Modal Shape Example Example uses the four generator system shown below in which the generators are represented by a combination of GENCLS and GENROU. The contingency is a self-clearing fault at bus 1 – The generator speeds (the signals) are as shown in the right figure 3 Case is saved as HW7_B4

4. Determining Modal Shape Example Example uses the multi-signal VPM to determine the key modes in the signals – Four modes were identified, thought the key ones where at 1.22, 1.60 and 2.76 Hz 4

5. Determining Modal Shape Example Information about the mode shape is available for each signal; the mode content in each signal can also be isolated 5 Graph shows original and the reproduced signal

6. Determining Modal Shape Example Graph shows the contribution provided by each mode in the generator 1 speed signal 6 Reproduced without 1.22 and 2.76 Hz Reproduced without 2.76 Hz

7. Modes Shape by Generator (for Speed) The table shows the contributions by mode for the different generator speed signals 7 Mode (Hz)Gen 1Gen 2Gen 3Gen 4 0.2440.070.0820.0390.066 1.221.5671.4940.0970.01 1.62.3672.2032.9531.639 2.760.1740.3780.9274.913 Image on the right shows the Gen 4 2.76 Hz mode; note it is highly damped

8. Modes Depend on the Signals! The below image shows the bus voltages for the previous system, with some (poorly tuned) exciters – Response includes a significant 0.664 Hz mode from the gen 4 exciter (which can be seen in its SMIB eigenvalues) 8

9. Inter-Area Modes in the WECC The dominant inter-area modes in the WECC have been well studied A good reference paper is D. Trudnowski, “Properties of the Dominant Inter-Area Modes in the WECC Interconnect,” 2012 – Four well known modes are NS Mode A (0.25 Hz), NS Mode B (or Alberta Mode), (0.4 Hz), BC Mode (0.6 Hz), Montana Mode (0.8 Hz) 9 Below figure from paper shows NS Mode A On May 29, 2012

10. Example WECC Results Figure shows bus frequencies at several WECC buses following a large system disturbance 10

11. Example WECC Results The VPM was run simultaneously on all the signals – Frequencies of 0.20 Hz (16% damping and 0.34 Hz (11.8% damping) 11 Angle of 58.7  at 0.34 Hz and 132  at 0.20 Hz Angle of -137.1  at 0.34 Hz and 142  at 0.20 Hz

12. 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 Fast Fourier Transform (FFT) Applications: Motivational Example 12

13. 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 FFT Overview Image Source: wikipedia.org/wiki/Fourier_series 13

14. 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 Sampled Signals Image: upload.wikimedia.org/wikipedia/commons/c/ca/Fourier_transform,_Fourier_series,_DTFT,_DFT.gif 14

15. Aliasing Image: upload.wikimedia.org/wikipedia/commons/thumb/2/28/AliasingSines.svg/2000px-AliasingSines.svg.png All practical signals are time-limited (and thus not band-limited) 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 15

16. Discrete Fourier Transforms (DFTs) can be used to provide frequency information about sampled, non- periodic 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 Fast Fourier Transform (FFT) Overview 16

17. The frequency analysis display shows the original data, the FFT for each time result, and a frequency summary Frequency Analysis Display 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 17

18. 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 19318 18

19. 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) Frequency Summary Plot of Maximum and Average Values For this example the system was modified to deliberately introduce a problem at gen 19318 by artificially increasing its exciter gain 19

20. Graphs shows the bus frequency at 19318 and its generator field voltage Gen 19318 Bus Frequency (150 MW Output) 20

21. The below results show the impact of aliasing when the sampling is too slow Previous Results Showing Aliasing When Sampled at 6 Hz 21

22. While not ideally suited for damped signals, the results can still be useful FFT Characteristics Figure on right is the FFT for the signal on the left 22

23. The key FFT advantage is it is fast and robust, allowing consideration of a many signals – Possibly detecting otherwise unnoticed issues The frequency resolution of 1/(T end -T start ) is usually not high Currently results are not normalized (by say standard deviation), so it is best to compare similar values (such as speed, pu voltage, etc) Results are sensitive to the time window FFT Characteristics 23

24. FFT considers a set of time points, making the assumptions that this time set is periodic No error is introduced if the first point matches the last point; but this is seldom the case – First and last points not matching introduces error For general functions the FFT accuracy can be improved using windowing functions PowerWorld uses one of the most common windowing function, called a Hann Window or sometimes the Hanning Window FFT with a Hann (Hanning) Window 24

25. The Hann window just scales the input values, de- weighting the values at the beginning and at the end FFT with a Hann (Hanning) Window Image Source: upload.wikimedia.org/wikipedia/commons/b/b3/Window_function_and_frequency_response_-_Hann.svg The bottom left image is the original FFT (with Hann), while the right is without Hann. 25

26. Announcements Be reading Chapter 9 Homework 7 is posted; it is due on Tuesday April 26 Final exam is on Monday May 9 from 8 to 11am in regular room. Closed book, closed notes, with two 8.5 by 11 inch note sheets and regular calculators allowed Key papers for book's approach on stabilizers are – F.P. DeMello and C. Concordia, "Concepts of Synchronous Machine Stability as Affected by Excitation Control, IEEE Trans. Power Apparatus and Systems, vol. PAS-88, April 1969, pp. 316-329 – W.G. Heffron and R.A. Philips, "Effects of Modern Amplidyne Voltage Regulator in Underexcited Operation of Large Turbine Generators," AIEE, PAS-71, August 1952, pp. 692-697 26

27. Overview of a Power System Stabilizer (PSS) A PSS adds a signal to the excitation system to improve the rotor damping – A common signal is proportional to speed deviation; other inputs, such as like power, voltage or acceleration, can be used – Signal is usually generated locally (e.g. from the shaft) Both local mode and inter-area mode can be damped. When oscillation is observed on a system or a planning study reveals poorly damped oscillations, use of participation factors helps in identifying the machine(s) where PSS has to be located Tuning of PSS regularly is important 27

28. Block Diagram of System with a PSS 28 Image Source: Kundur, Power System Stability and Control

29. Power System Stabilizer Basics Stabilizers can be motivated by considering a classical model supplying an infinite bus Assume internal voltage has an additional component This will add additional damping if sin(  ) is positive In a real system there is delay, which requires compensation 29

30. Example PSS An example single input stabilizer is shown below (IEEEST) – The input is usually the generator shaft speed deviation, but it could also be the bus frequency deviation, generator electric power or voltage magnitude 30 V ST is an input into the exciter The model can be simplified by setting parameters to zero

31. Example PSS Below is an example of a dual input PSS (PSS2A) – Combining shaft speed deviation with generator electric power is common – Both inputs have washout filters to remove low frequency components of the input signals 31 IEEE Std 421.5 describes the common stabilizers

32. PSS Tuning: Basic Approach The PSS parameters need to be selected to achieve the desired damping through a process known as tuning The next several slides present a basic method using a single machine, infinite bus (SMIB) representation Start with the linearized differential, algebraic model with controls u added to the states If D is invertible then 32

33. PSS Tuning: Basic Approach Low frequency oscillations are considered the following approach A SMIB system is setup to analyze the local mode of oscillation (in the 1 to 3 Hz range) – A flux decay model is used with E fd as the input Then, a fast-acting exciter is added between the input voltage and E fd Certain constants, K 1 to K 6, are identified and used to tune a power system stabilizer 33

34. SMIB System (Flux Decay Model) SMIB with a flux decay machine model and a fast exciter 34 ~

35. Stator Equations Assume Rs=0, then the stator algebraic equations are: 35

36. Network Equations The network equation is (assuming zero angle at the infinite bus and no local load) Simplifying, ~ 36

37. Complete SMIB Model Stator equations Network equations ~ Machine equations 37

38. Linearization of SMIB Model Notice this is equivalent to a generator at the infinite bus with modified resistance and reactance values 38

39. Linearization (contd) Final Steps involve 1. Linearizing Machine Equations 2. Substitute (1) in the linearized equations of (2). 39

40. Linearization of Machine Equations Substitute (3) in (2) to get linearized model. Symbolically we have 40

41. Linearized SMIB Model Excitation system is not yet included. K 1 – K 4 constants are defined on next slide 41

42. K1 – K4 Constants K1 – K4 only involve machine and not the exciter. 42

43. Including Terminal Voltage The change in the terminal voltage magnitude also needs to be include since it is an input into the exciter 43

44. Computing While linearizing the stator algebraic equations, we had Substitute this in expression for  V t to get 44

45. Heffron–Phillips Model (from 1952 and 1969 ) Add a fast exciter with a single differential equation Linearize This is then combined with the previous three differential equations to give SMIB Model 45

46. Block Diagram K1 – K6 are affected by system loading 46

47. Numerical Example Consider an SMIB system with Z eq = j0.5, V inf = 1.05, in which in the power flow the generator has S = 54.34 – j2.85 MVA with a V t of 1  15  – Machine is modeled with a flux decay model with (pu, 100 MVA) H=3.2, T' do =9.6, X d =2.5, X q =2.1, X' d =0.39, R s =0, D=0 47 Saved as case B2_PSS_Flux

48. Initial Conditions The initial conditions are 48

49. Add an EXST1 Exciter Model Set the parameters to K A = 400, T A =0.2, all others zero with no limits and no compensation Hence this simplified exciter is represented by a single differential equation 49 V s is the input from the stabilizer, with an initial value of zero

50. Initial Conditions (contd) From the stator algebraic equation, 50

51. SMIB Results Doing the SMIB gives a matrix that closely matches the book’s matrix from Example 8.7 – The variable order is different; the entries in the w column are different because of the speed dependence in the swing equation and the Norton equivalent current injection 51 The  values are different from the book because of speed dependence included in the stator voltage and swing equations

52. Computation of K1 – K6 Constants The formulas are used. 52