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This lecture explores transient stability analysis in power systems, including the generator electrical and mechanical models, single machine infinite bus (SMIB) model, equilibrium points, and post-fault dynamics. It also covers the equal area criteria for stability assessment and the basics of power system harmonics.
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ECE 476POWER SYSTEM ANALYSIS Lecture 25 Transient Stability, Harmonics Professor Tom Overbye Department of Electrical andComputer Engineering
Announcements • Be reading Chapter 13. • HW 11 is not turned in but should be done before final. HW 11 is 13.1, 13.7, 13.8, 13.18, and the special problem (see website for complete assignment) • Final is Tuesday Dec 16 from 7 to 10pm in EL 165 (web it now correct). Final is comprehensive. One new note sheet, and your two old note sheets are allowed. • Special Office Hours Dec 15 from 1 to 3pm
Generator Electrical Model • The simplest generator model, known as the classical model, treats the generator as a voltage source behind the direct-axis transient reactance; the voltage magnitude is fixed, but its angle changes according to the mechanical dynamics
Generator Mechanical Model Generator Mechanical Block Diagram
Single Machine Infinite Bus (SMIB) • To understand the transient stability problem we’ll first consider the case of a single machine (generator) connected to a power system bus with a fixed voltage magnitude and angle (known as an infinite bus) through a transmission line with impedance jXL
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
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
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
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
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 DecelArea is greater than the Accel. Area
Power System Harmonics • So far class has talked about fundamental frequency analysis. Many traditional loads only consume power at the fundamental frequency. However, some loads, mostly electronic-based, tend to draw current in non-linear pulses, which gives rise to harmonics. • If current has half-wave-symmetry (values are equal and opposite when separated by T/2) then there are no even harmonics
Switched-Mode Power Supply Current Source: www.utterpower.com/commercial_grid.htm
Harmonic Current Specturm • The below figure shows the harmonic current components for an 18-W, electronic-ballast compact fluorescent lamp. Source: Fig 2.34 of “Renewable and Efficient Electric Power Systems” by Masters
Key Problems with Harmonics • A key problem with the third harmonic is neutral current since the fundamental 120 degree phase shift becomes 360 degrees for the third harmonic so the third harmonic values do not cancel (also true for other triplen harmonics) • Delta-grounded wye transformers prevent triplen harmonic currents from flowing into the power grid • Harmonics cause transformer overheating since core losses are proportional to frequency • Harmonic resonance, particularly with shunt capacitors (can be around 5th or 7th harmonic values)
