0. ECE 476 POWER SYSTEM ANALYSIS
Lecture 11
Ybus and Power Flow
Professor Tom Overbye
Department of Electrical and Computer Engineering

1. Announcements Be reading Chapter 6.
HW 4 is 3.4, 3.10, 3.14, 3.19, 3.23, 3.60; due September 29 in class.
First exam is October 11 during class. Closed book, closed notes, one note sheet and calculators allowed

2. Multiple Solution Example 3
The dc system shown below has two solutions:
where the 18 watt
load is a resistive
load
What is the
maximum
PLoad?

3. Bus Admittance Matrix or Ybus
First step in solving the power flow is to create what is known as the bus admittance matrix, often call the Ybus.
The Ybus gives the relationships between all the bus current injections, I, and all the bus voltages, V, I = Ybus V
The Ybus is developed by applying KCL at each bus in the system to relate the bus current injections, the bus voltages, and the branch impedances and admittances

4. Ybus Example Determine the bus admittance matrix for the network
shown below, assuming the current injection at each
bus i is Ii = IGi - IDi where IGi is the current injection into the
bus from the generator and IDi is the current flowing into the
load

5. Ybus Example, cont’d

6. Ybus Example, cont’d symmetric matrix (i.e., one where Aij = Aji)
For a system with n buses, Ybus is an n by n
symmetric matrix (i.e., one where Aij = Aji)

7. Ybus General Form
The diagonal terms, Yii, are the self admittance terms, equal to the sum of the admittances of all devices incident to bus i.
The off-diagonal terms, Yij, are equal to the negative of the sum of the admittances joining the two buses.
With large systems Ybus is a sparse matrix (that is, most entries are zero)
Shunt terms, such as with the p line model, only
affect the diagonal terms.

8. Modeling Shunts in the Ybus

9. Two Bus System Example

10. Using the Ybus

11. Solving for Bus Currents

12. Solving for Bus Voltages

13. Power Flow Analysis
When analyzing power systems we know neither the complex bus voltages nor the complex current injections
Rather, we know the complex power being consumed by the load, and the power being injected by the generators plus their voltage magnitudes
Therefore we can not directly use the Ybus equations, but rather must use the power balance equations

14. Power Balance Equations

15. Power Balance Equations, cont’d

16. Real Power Balance Equations

17. In the News: Renewable Energy Finance
Last week IEEE Spectrum had an opinion piece on how wind, water and solar could power the world.
The piece was very sparse on financial details
Yesterday the News—Gazette had an article on the difficulties the first US offshore wind farm, Cape Wind, has in attracting customers
National Grid is buying one half of their output at an average cost of 24 cents per kWh ($240/MWh!!)
Without committed buyers Cape Wind is unlikely to find financing for the full $2.6 billion project (454 MWs of capacity)

18. Power Flow Requires Iterative Solution

19. Gauss Iteration