Power System Dynamics and Stability Lecture Details

ECE 576 Power System Dynamics and Stability Lecture 25: Distributed PV, Small Signal Stability Prof. Tom Overbye University of Illinois at Urbana-Champaign overbye@illinois.edu 1

Announcements Read Chapters 8 and 9 Homework 8 should be completed before final but need not be turned in Final Exam is Wednesday May 14 at 7 to 10pm in classroom. Closed book, closed notes, your two previous note sheets and one new note sheet allowed, simple calculators allowed 2

Status of Nuclear Power Worldwide In the USA, the five reactors under construction (about 1200 MW each) are 1) two units at the Vogtle plant in Georgia (2017) 2) two units in South Carolina (2017/9) 3) TVA's Watts Bar Unit 2 (2015) 3 Source: Fortune Magazine, April 2014

Distributed PV System Modeling PV in the distribution system is usually operated at unity power factor There is research investigating the benefits of changing this IEEE Std 1547 prevents voltage regulation, but would allow non-unity power factor A simple model is just as negative constant power load An issue is tripping on abnormal frequency or voltage conditions IEEE Std 1547 says, "The DR unit shall cease to energize the Area EPS for faults on the Area EPS circuit to which it is connected. (note EPS is electric power system) 4

Distributed PV System Modeling An issue is tripping on abnormal frequency or voltage conditions IEEE Std 1547 says, "The DR unit shall cease to energize the Area EPS for faults on the Area EPS circuit to which it is connected. (note EPS is electric power system) This is a key safety requirement! Small units (less than 30kW) need to disconnect if the voltage is < 0.5 pu in 0.16 seconds, and in 2 seconds if between 0.5 and 0.88 pu; also in 1 second if between 1.1 and 1.2, and in 0.16 seconds if higher Small units need to disconnect in 0.16 seconds if the frequency is > 60.5 Hz, or less than 59.3 Hz Reconnection is after minutes 5

Distributed PV System Modeling Below is a prototype model for distributed solar PV 6

Oscillations An oscillation is just a repetitive motion that can be either undamped, positively damped (decaying with time) or negatively damped (growing with time) If the oscillation can be written as a sinusoid then it has the form ( ) ( cos e a t b + ) ( ) ( ) = + t t sin cos t e C t b = + = 2 2 where and tan C A B a And the damping ratio is defined as (see Kundur 12.46) + damping ratio multiplied by 100 = The percent damping is just the 2 2 7

Power System Oscillations Power systems can experience a wide range of oscillations, ranging from highly damped and high frequency switching transients to sustained low frequency (< 2 Hz) inter-area oscillations affecting an entire interconnect Types of oscillations include Transients: Usually high frequency and highly damped Local plant: Usually from 1 to 5 Hz Inter-area oscillations: From 0.15 to 1 Hz Slower dynamics: Such as AGC, less than 0.15 Hz Subsynchronous resonance: 10 to 50 Hz (less than synchronous) 8

Example Oscillations The below graph shows an oscillation that was observed during a 1996 WECC Blackout 9

Example Oscillations The below graph shows oscillations on the Michigan/Ontario Interface on 8/14/03 10

Small Signal Stability Analysis Small signal stability is the ability of the power system to maintain synchronism following a small disturbance System is continually subject to small disturbances, such as changes in the load The operating equilibrium point (EP) obviously must be stable Small system stability analysis (SSA) is studied to get a feel for how close the system is to losing stability and to get additional insight into the system response There must be positive damping 11

Small Signal Stability Analysis Model based SSA is performed by linearizing about an EP, and then calculating the associated eigenvalues (and other properties) of the linearized system With the advent of PMUs, measurement based techniques are becoming increasingly common; this approach is typically broken into two types Ringdown analysis is performed after the power system has experienced a significant disturbance that has moved it away from its EP Ambient analysis is performed when the power system is operating in quasi-steady state 12

An On-line Oscillation Detection Tool Image source: WECC Joint Synchronized Information Subcommittee Report, October 2013 13

Model Based SSA Assume the power system is modeled as in our standard form as ( ) , = x f x y 0 = g(x,y) The system can be linearized about an equilibrium point + x = A x B y 0 = C x+D y models then D is the power flow Jacobian;otherwise it also includes the stator algebraic equations If there are just classical generator Eliminating y gives ( ) = -1 x = A BD C x A x sys 14

Model Based SSA The matrix Asys can be calculated doing a partial factorization, just like what was done with Kron reduction SSA is done by looking at the eigenvalues (and other properties) of Asys 15

SSA Two Generator Example Consider the two bus, two classical generator system from lectures 18 and 20 with Xd1'=0.3, H1=3.0, Xd2'=0.2, H2=6.0 Bus 2 Bus 1 GENCLS GENCLS X=0.22 slack 11.59 Deg 1.095 pu 0.00 Deg 1.000 pu Essentially everything needed to calculate the A, B, C and D matrices was covered in lecture 20 16

SSA Two Generator Example The A matrix is calculated differentiating f(x,y) with respect to x (where x is 1, 1, 2, 2) d dt d 1 P P D dt 2H d dt d 1 P P D dt 2H ( P Ei Di Di Di i Qi Qi Qi E E V G E E V ( cos sin Di Qi i i E jE E j + = + = 1 . 1 pu s ( ) , 1 pu = . M1 E1 1 1 pu 1 = 2 . 2 pu s ( ) , 2 pu = . M 2 ) E2 2 1 pu ) ) 2 ( ( ) = + + 2 2 G Di Qi E V E V B i Qi Di i i 17

SSA Two Generator Example Giving . 0 376 99 0 0 0 0 0 0 0 0 0 761 . 0 0 = A . 376 99 0 0 389 . B, C and D are as calculated previously for the implicit integration, except the elements in B are not multiplied by t/2 0 0 0 2889 0 6505 0 0 0 0 0 0 0 0 0 0 . . = B . . 0 0833 0 3893 18

SSA Two Generator Example The C and D matrices are = . . 0 0 . . 3 903 0 1 733 0 0 0 0 0 0 7 88 0 4 54 0 0 4 54 0 9 54 . . 0 9 54 7 88 0 4 54 4 54 = C D , . . . 0 0 0 4 671 0 1 0 . . . 0 Giving . 0 376 99 0 0 0 0 0 0 0 229 . 0 . 0 229 0 0 114 = = -1 A A-BD C sys . 376 99 0 . . 0 114 19

SSA Two Generator Calculating the eigenvalues gives a complex pair and two zero eigenvalues The complex pair, with values of +/- j11.39 corresponds to the generators oscillating against each other at 1.81 Hz One of the zero eigenvalues corresponds to the lack of an angle reference Could be rectified by redefining angles to be with respect to a reference angle (see book 226) or we just live with the zero Other zero is associated with lack of speed dependence in the generator torques 20

SSA Two Generator Speeds The two generator system response is shown below for a small disturbance 60.5 60.45 60.4 60.35 60.3 60.25 60.2 60.15 60.1 60.05 Notice the actual response closely matches the calculated frequency 60 59.95 59.9 59.85 59.8 59.75 59.7 59.65 59.6 59.55 59.5 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 Speed, Gen Bus 1 #1 Speed, Gen Bus 2 #1 g f e d c b g f e d c b 21

SSA Three Generator Example The two generator system is extended to three generators with the third generator having H3 of 8 and Xd3'=0.3 Bus 1 Bus 2 GENCLS X=0.2 GENCLS slack X=0.2 X=0.2 3.53 Deg 1.0500 pu 0.00 Deg 1.0000 pu -3.53 Deg 1.050 pu Bus 3 GENCLS 200 MW 0 Mvar 22

SSA Three Generator Example Using SSA, two frequencies are identified: one at 2.02 Hz and one at 1.51 Hz We next discuss modal analysis to determine the contribution of each mode to each signal 60.0100 60.0090 60.0080 60.0070 60.0060 60.0050 60.0040 60.0030 60.0020 60.0010 60.0000 59.9990 59.9980 59.9970 59.9960 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 Speed_Gen Bus 1 #1 Speed_Gen Bus 2 #1 Speed_Gen 3 #1 g f e d c b g f e d c b g f e d c b 23

Large System Studies The challenge with large systems, which could have more than 100,000 states, is the shear size Most eigenvalues are associated with the local plants Computing all the eigenvalues is computationally challenging, order n3 Specialized approaches can be used to calculate particular eigenvalues of large matrices See Kundur, Section 12.8 and associated references 24

Single Machine Infinite Bus A quite useful analysis technique is to consider the small signal stability associated with a single generator connected to the rest of the system through an equivalent transmission line Driving point impedance looking into the system is used to calculate the equivalent line's impedance The Zii value can be calculated quite quickly using sparse vector methods Rest of the system is assumed to be an infinite bus with its voltage set to match the generator's real and reactive power injection and voltage 25

Small SMIB Example As a small example, consider the 4 bus system shown below, in which bus 2 really is an infinite bus Bus 1 Bus 2 X=0.2 GENCLS Bus 4 Infinite Bus X=0.1 slack Bus 3 11.59 Deg 1.0946 pu X=0.1 X=0.2 6.59 Deg 1.046 pu 4.46 Deg 1.029 pu 0.00 Deg 1.000 pu To get the SMIB for bus 4, first calculate Z44 25 0 10 10 0 1 0 0 Y j 10 0 15 0 10 0 0 13 33 Z44 is Zth in parallel with jX'd,4 (which is j0.3) so Zth is j0.22 = = . Z j0 1269 bus 44 . 26

Small SMIB Example The infinite bus voltage is then calculated so as to match the bus i terminal voltage and current = V V Z I While this was demonstrated on an extremely small system for clarity, the approach works the same for any size system inf i i i * + P jQ = where i i I i V i In the example we have * * + + . j0 220 P jQ 1 j0 572 + = = . 4 4 1 j0 328 . . V = = 1 072 4 ( 1 0 ) ( ) + . . ( . ) . V 1 072 j0 220 j0 22 1 j0 328 inf . V inf 27

Calculating the A Matrix The SMIB model A matrix can then be calculated either analytically or numerically The equivalent line's impedance can be embedded in the generator model so the infinite bus looks like the "terminal" This matrix is calculated in PowerWorld by selecting Transient Stability, SMIB Eigenvalues Select Run SMIB to perform an SMIB analysis for all the generators in a case Right click on a generator on the SMIB form and select Show SMIB to see the Generator SMIB Eigenvalue Dialog These two bus equivalent networks can also be saved, which can be quite useful for understanding the behavior of individual generators 28

Example: Bus 4 SMIB Dialog On the SMIB dialog, the General Information tab shows information about the two bus equivalent 29

Example: Bus 4 SMIB Dialog On the SMIB dialog, the A Matrix tab shows the Asys matrix for the SMIB generator In this example A21 is showing P 1 1 6 1 ( ) ( ) , 4 pu = = , E 4 . cos . 1 2812 23 94 0 3 0 22 + . . 2H 0 3753 = 4 4 4 . 30

Example: Bus 4 SMIB Dialog On the SMIB dialog, the Eigenvalues tab shows the Asys matrix eigenvalues and participation factors (which we'll cover shortly) Saving the two bus SMIB equivalent, and putting a short, self-cleared fault at the terminal shows the 1.89 Hz, undamped response 31

Example: Bus 4 with GENROU Model The eigenvalues can be calculated for any set of generator models This example replaces the bus 4 generator classical machine with a GENROU model There are now six eigenvalues, with the dominate response coming from the electro-mechanical mode with a frequency of 1.83 Hz, and damping of 6.92% 32