Transient Stability

1
ECE 476POWER SYSTEM ANALYSIS

• Lecture 24
• Transient Stability
• Professor Tom Overbye
• Department of Electrical andComputer Engineering

2
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
  (note this is NOT what the web says). Final is
  comprehensive. One new note sheet, and your two
  old note sheets are allowed

3
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

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Generator Mechanical Model
Generator Mechanical Block Diagram
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Generator Mechanical Model, contd
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Generator Mechanical Model, contd
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Generator Mechanical Model, contd
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Generator Swing Equation
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Single Machine Infinite Bus (SMIB)

• To understand the transient stability problem
  well 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

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SMIB, contd
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SMIB Equilibrium Points
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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

13
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

14
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 lines circuit
  breakers.

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SMIB Example, contd
Simplified prefault system
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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
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SMIB Example, Post Fault System
After the fault the system again changes
The equivalent system after the fault is then
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SMIB Example, Dynamics
19
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

20
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

21
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

22
Numerical Integration of DEs
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Examples
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Eulers Method
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Eulers Method Algorithm
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Eulers Method Example 1
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Eulers Method Example 1, contd
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Eulers Method Example 2
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Euler's Method Example 2, cont'd
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Euler's Method Example 2, cont'd
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Euler's Method Example 2, cont'd
Below is a comparison of the solution values for
x1(t) at time t 10 seconds
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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
  Dt0.01 second.

Ea
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Transient Stability Example, cont'd
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Transient Stability Example, cont'd
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Transient Stability Example, cont'd
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Transient Stability Example, cont'd
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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