ECE 476 POWER SYSTEM ANALYSIS Lecture 24 Transient Stability Professor Tom Overbye Department of Electrical and Computer Engineering

Announcements Be reading Chapter 10 and 13 Including CH 10 article about zone 3 relays, CH 13 DSA and blackout articles HW 11 is not turned in but should be done before final. HW 11 is 13.1, 13.7, 13.8, 13.18, and SP1 Final is Wednesday Dec 12 from 1:30 to 4:30pm in EL 269 (note room change). Final is comprehensive. One new note sheet, and your two old note sheets are allowed

In the News: Dallman Accident Cause of CWLP (Springfield, IL) Dallman generator accident is still under investigation, but after the generator tripped the main turbine steam stop valve did not close. With the generator’s electrical output at zero (i.e., it was disconnected) the turbine/rotor continued to accelerate up to over 5000 rpm (3600 normal). Then it suddenly stopped. 95 MW unit, $60 million to repair damage/replace unit On a more positive note for CWLP their Dallman 4 plant is on schedule for a January 2010 start.

After the Dallman Accident

Outside of Dallman

2007 Energy Bill Congress and President Bush are currently discussing an energy bill, with a key sticking point being whether to require investor owned utilities to provide 15% of their electricity from renewables by 2020. There is a big difference between 15% of capacity versus 15% of energy from renewables A major issue is lack of wind capacity in the southeast U.S.

US Wind Resource Map

Transient Stability Analysis For transient stability analysis we need to consider three systems Prefault - before the fault occurs the system is assumed to be at an equilibrium point Faulted - the fault changes the system equations, moving the system away from its equilibrium point Postfault - after fault is cleared the system hopefully returns to a new operating point

Transient Stability Solution Methods There are two methods for solving the transient stability problem Numerical integration this is by far the most common technique, particularly for large systems; during the fault and after the fault the power system differential equations are solved using numerical methods Direct or energy methods; for a two bus system this method is known as the equal area criteria mostly used to provide an intuitive insight into the transient stability problem

SMIB Example Assume a generator is supplying power to an infinite bus through two parallel transmission lines. Then a balanced three phase fault occurs at the terminal of one of the lines. The fault is cleared by the opening of this line’s circuit breakers.

SMIB Example, cont’d Simplified prefault system

SMIB Example, Faulted System During the fault the system changes The equivalent system during the fault is then During this fault no power can be transferred from the generator to the system

SMIB Example, Post Fault System After the fault the system again changes The equivalent system after the fault is then

SMIB Example, Dynamics

Transient Stability Solution Methods There are two methods for solving the transient stability problem Numerical integration this is by far the most common technique, particularly for large systems; during the fault and after the fault the power system differential equations are solved using numerical methods Direct or energy methods; for a two bus system this method is known as the equal area criteria mostly used to provide an intuitive insight into the transient stability problem

Numerical Integration of DEs

Examples

Euler’s Method

Euler’s Method Algorithm

Euler’s Method Example 1

Euler’s Method Example 1, cont’d xactual(t) x(t) Dt=0.1 x(t) Dt=0.05 10 0.1 9.048 9 9.02 0.2 8.187 8.10 8.15 0.3 7.408 7.29 7.35 … 1.0 3.678 3.49 3.58 2.0 1.353 1.22 1.29

Euler’s Method Example 2

Euler's Method Example 2, cont'd

Euler's Method Example 2, cont'd x1actual(t) x1(t) Dt=0.25 1 0.25 0.9689 0.50 0.8776 0.9375 0.75 0.7317 0.8125 1.00 0.5403 0.6289 … 10.0 -0.8391 -3.129 100.0 0.8623 -151,983

Euler's Method Example 2, cont'd Below is a comparison of the solution values for x1(t) at time t = 10 seconds Dt x1(10) actual -0.8391 0.25 -3.129 0.10 -1.4088 0.01 -0.8823 0.001 -0.8423

Transient Stability Example A 60 Hz generator is supplying 550 MW to an infinite bus (with 1.0 per unit voltage) through two parallel transmission lines. Determine initial angle change for a fault midway down one of the lines. H = 20 seconds, D = 0.1. Use Dt=0.01 second. Ea

Transient Stability Example, cont'd

Transient Stability Example, cont'd

Transient Stability Example, cont'd

Transient Stability Example, cont'd

Equal Area Criteria The goal of the equal area criteria is to try to determine whether a system is stable or not without having to completely integrate the system response. System will be stable after the fault if the Decel Area is greater than the Accel. Area