0. ECE 476 POWER SYSTEM ANALYSIS
Lecture 7
Transmission Line Models
Professor Tom Overbye
Department of Electrical and Computer Engineering

1. Announcements For lectures 7 to 9 please be reading Chapter 5.
HW 3 is 4.8, 4.9, 4.23, 4.25 (assume Cardinal conductors; temperature is just used for the current rating) is due Thursday

2. NERC Regions

3. In the News: Restoration
Hurricane Ike left millions without electric power across its path from southeast Texas and then extending to the north and east.
While most of the restoration issues will focus on the distribution system, Ike also knocked out hundreds of transmission lines, including six 345 kV transmission lines in ERCOT.
Book has article on power system restoration at the beginning of Chapter 11 (pp )

4. Ike Electrical System Damage
Conroe, Tx
Beaumont, Tx
Source: Entergy Website,

5. Tree Trimming: Before

6. Tree Trimming: After

7. Transmission Line Models
Previous lectures have covered how to calculate the distributed inductance, capacitance and resistance of transmission lines.
In this section we will use these distributed parameters to develop the transmission line models used in power system analysis.

8. Transmission Line Equivalent Circuit
Our current model of a transmission line is shown below
Units on
z and y are
per unit
length!

9. Derivation of V, I Relationships

10. Setting up a Second Order Equation

11. V, I Relationships, cont’d

12. Equation for Voltage

13. Real Hyperbolic Functions
For real x the cosh and sinh functions have the following form:

14. Complex Hyperbolic Functions
For x =  + j the cosh and sinh functions have the following form

15. Determining Line Voltage

16. Determining Line Voltage, cont’d

17. Determining Line Current

18. Transmission Line Example

19. Transmission Line Example, cont’d

20. Transmission Line Example, cont’d

21. Lossless Transmission Lines

22. Lossless Transmission Lines
If P > SIL then line consumes
vars; otherwise line generates vars.

23. Transmission Matrix Model
Oftentimes we’re only interested in the terminal characteristics of the transmission line. Therefore we can model it as a “black box”. VS VR + - IS IR
Transmission
Line

24. Transmission Matrix Model, cont’d

25. Equivalent Circuit Model
Next we’ll use the T matrix values to derive the
parameters Z' and Y'.

26. Equivalent Circuit Parameters

27. Equivalent circuit parameters

28. Simplified Parameters

29. Simplified Parameters

30. Medium Length Line Approximations

31. Three Line Models

32. Power Transfer in Short Lines
Often we'd like to know the maximum power that could be transferred through a short transmission line V1 V2 + - I1
Transmission
Line with Impedance Z
S12
S21

33. Power Transfer in Lossless Lines

34. Limits Affecting Max. Power Transfer
Thermal limits
limit is due to heating of conductor and hence depends heavily on ambient conditions.
For many lines, sagging is the limiting constraint.
Newer conductors limit can limit sag. For example, in 2004 ORNL working with 3M announced lines with a core consisting of ceramic Nextel fibers. These lines can operate at 200 degrees C.
Trees grow, and will eventually hit lines if they are planted under the line.

35. Other Limits Affecting Power Transfer
Angle limits
while the maximum power transfer occurs when line angle difference is 90 degrees, actual limit is substantially less due to multiple lines in the system
Voltage stability limits
as power transfers increases, reactive losses increase as I2X. As reactive power increases the voltage falls, resulting in a potentially cascading voltage collapse.