ECE 476 POWER SYSTEM ANALYSIS

1. ECE 476POWER SYSTEM ANALYSIS Lecture 7 Transmission Line Models Professor Tom Overbye Department of Electrical andComputer Engineering

2. 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

3. NERC Regions

4. 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. 565-574)

5. Ike Electrical System Damage Conroe, Tx Beaumont, Tx Source: Entergy Website, www.entergy.com

6. Tree Trimming: Before

7. Tree Trimming: After

8. 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.

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

10. Derivation of V, I Relationships

11. Setting up a Second Order Equation

12. V, I Relationships, cont’d

13. Equation for Voltage

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

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

16. Determining Line Voltage

17. Determining Line Voltage, cont’d

18. Determining Line Current

19. Transmission Line Example

20. Transmission Line Example, cont’d

21. Transmission Line Example, cont’d

22. Lossless Transmission Lines

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

24. IS IR + + Transmission Line VS VR - - 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”.

25. Transmission Matrix Model, cont’d

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

27. Equivalent Circuit Parameters

28. Equivalent circuit parameters

29. Simplified Parameters

30. Simplified Parameters

31. Medium Length Line Approximations

32. Three Line Models

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

34. Power Transfer in Lossless Lines

35. 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.

36. 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.