ECE 476 POWER SYSTEM ANALYSIS Lecture 24 Transient

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ECE 476 POWER SYSTEM ANALYSIS Lecture 24 Transient Stability Professor Tom Overbye Department of

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

Announcements l l l Be reading Chapter 13. HW 11 is not turned in

Announcements l l l 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 10 pm 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 1

Generator Electrical Model l The simplest generator model, known as the classical model, treats

Generator Electrical Model l 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 2

Generator Mechanical Model Generator Mechanical Block Diagram 3

Generator Mechanical Model Generator Mechanical Block Diagram 3

Generator Mechanical Model, cont’d 4

Generator Mechanical Model, cont’d 4

Generator Mechanical Model, cont’d 5

Generator Mechanical Model, cont’d 5

Generator Mechanical Model, cont’d 6

Generator Mechanical Model, cont’d 6

Generator Swing Equation 7

Generator Swing Equation 7

Single Machine Infinite Bus (SMIB) l To understand the transient stability problem we’ll first

Single Machine Infinite Bus (SMIB) l 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 j. XL 8

SMIB, cont’d 9

SMIB, cont’d 9

SMIB Equilibrium Points 10

SMIB Equilibrium Points 10

Transient Stability Analysis l 1. 2. 3. For transient stability analysis we need to

Transient Stability Analysis l 1. 2. 3. 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 11

Transient Stability Solution Methods l 1. There are two methods for solving the transient

Transient Stability Solution Methods l 1. There are two methods for solving the transient stability problem Numerical integration l 2. 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 l mostly used to provide an intuitive insight into the transient stability problem 12

SMIB Example l Assume a generator is supplying power to an infinite bus through

SMIB Example l 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. 13

SMIB Example, cont’d Simplified prefault system 14

SMIB Example, cont’d Simplified prefault system 14

SMIB Example, Faulted System During the fault the system changes The equivalent system during

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 15

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

SMIB Example, Post Fault System After the fault the system again changes The equivalent system after the fault is then 16

SMIB Example, Dynamics 17

SMIB Example, Dynamics 17

Transient Stability Solution Methods l 1. There are two methods for solving the transient

Transient Stability Solution Methods l 1. There are two methods for solving the transient stability problem Numerical integration l 2. 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 l mostly used to provide an intuitive insight into the transient stability problem 18

Transient Stability Analysis • 1. 2. 3. For transient stability analysis we need to

Transient Stability Analysis • 1. 2. 3. 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 19

Transient Stability Solution Methods • 1. There are two methods for solving the transient

Transient Stability Solution Methods • 1. There are two methods for solving the transient stability problem Numerical integration l 2. 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 l mostly used to provide an intuitive insight into the transient stability problem 20

Numerical Integration of DEs 21

Numerical Integration of DEs 21

Examples 22

Examples 22

Euler’s Method 23

Euler’s Method 23

Euler’s Method Algorithm 24

Euler’s Method Algorithm 24

Euler’s Method Example 1 25

Euler’s Method Example 1 25

Euler’s Method Example 1, cont’d t xactual(t) x(t) Dt=0. 1 x(t) Dt=0. 05 0

Euler’s Method Example 1, cont’d t xactual(t) x(t) Dt=0. 1 x(t) Dt=0. 05 0 10 10 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 26

Euler’s Method Example 2 27

Euler’s Method Example 2 27

Euler's Method Example 2, cont'd 28

Euler's Method Example 2, cont'd 28

Euler's Method Example 2, cont'd x 1 actual(t) x 1(t) Dt=0. 25 0 1

Euler's Method Example 2, cont'd x 1 actual(t) x 1(t) Dt=0. 25 0 1 1 0. 25 0. 9689 1 0. 50 0. 8776 0. 9375 0. 7317 0. 8125 1. 00 0. 5403 0. 6289 … … … 10. 0 -0. 8391 -3. 129 100. 0 0. 8623 -151, 983 t 29

Euler's Method Example 2, cont'd Below is a comparison of the solution values for

Euler's Method Example 2, cont'd Below is a comparison of the solution values for x 1(t) at time t = 10 seconds Dt x 1(10) actual -0. 8391 0. 25 -3. 129 0. 10 -1. 4088 0. 01 -0. 8823 0. 001 -0. 8423 30

Transient Stability Example • A 60 Hz generator is supplying 550 MW to an

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 31

Transient Stability Example, cont'd 32

Transient Stability Example, cont'd 32

Transient Stability Example, cont'd 33

Transient Stability Example, cont'd 33

Transient Stability Example, cont'd 34

Transient Stability Example, cont'd 34

Transient Stability Example, cont'd 35

Transient Stability Example, cont'd 35

Equal Area Criteria • The goal of the equal area criteria is to try

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 36