ECE 476 Power System Analysis Lecture 8: Transmission Line Parameters, Transformers Prof. Tom Overbye Dept. of Electrical and Computer Engineering University of Illinois at Urbana-Champaign Special Guest Lecturer: TA Won Jang

Announcements Please read Chapters 5 and then 3 Quiz today on HW 3 H4 is 4.34, 4.41, 5.2, 5.7, 5.16 It should be turned in on Sept 24 (hence no quiz next week) 1

Leidos Engineering

Transmission Line Equivalent Circuit Our current model of a transmission line is shown below Units on z and y are per unit length! 3

Derivation of V, I Relationships 4

Setting up a Second Order Equation 5

V, I Relationships, cont’d 6

Equation for Voltage 7

Real Hyperbolic Functions For real x the cosh and sinh functions have the following form: 8

Complex Hyperbolic Functions For x =  + j  the cosh and sinh functions have the following form 9

Determining Line Voltage 10

Determining Line Voltage, cont’d 11

Determining Line Current 12

Transmission Line Example 13

Transmission Line Example, cont’d 14

Transmission Line Example, cont’d 15

Lossless Transmission Lines 16

Lossless Transmission Lines If P > SIL then line consumes vars; otherwise line generates vars. 17

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”. VSVS VRVR ISIS IRIR Transmission Line 18

Transmission Matrix Model, cont’d 19

Equivalent Circuit Model Next we’ll use the T matrix values to derive the parameters Z' and Y'. 20

Equivalent Circuit Parameters 21

Equivalent circuit parameters 22

Simplified Parameters 23

Medium Length Line Approximations 24

Three Line Models 25

Power Transfer in Short Lines Often we'd like to know the maximum power that could be transferred through a short transmission line V1V1 V2V I1I1 I1I1 Transmission Line with Impedance Z S 12 S 21 26

Power Transfer in Lossless Lines 27

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. 28

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 I 2 X. As reactive power increases the voltage falls, resulting in a potentially cascading voltage collapse. 29

Transformers Overview Power systems are characterized by many different voltage levels, ranging from 765 kV down to 240/120 volts. Transformers are used to transfer power between different voltage levels. The ability to inexpensively change voltage levels is a key advantage of ac systems over dc systems. In this section we’ll development models for the transformer and discuss various ways of connecting three phase transformers. 30

Transmission to Distribution Transfomer 31

Transmission Level Transformer 32