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

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

NERC Regions

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)

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

Tree Trimming: Before

Tree Trimming: After

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.

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

Derivation of V, I Relationships

Setting up a Second Order Equation

V, I Relationships, cont’d

Equation for Voltage

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

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

Determining Line Voltage

Determining Line Voltage, cont’d

Determining Line Current

Transmission Line Example

Transmission Line Example, cont’d

Transmission Line Example, cont’d

Lossless Transmission Lines

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

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

Transmission Matrix Model, cont’d

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

Equivalent Circuit Parameters

Equivalent circuit parameters

Simplified Parameters

Simplified Parameters

Medium Length Line Approximations

Three Line Models

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

Power Transfer in Lossless Lines

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.

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.