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**ECE 476 POWER SYSTEM ANALYSIS**

Lecture 11 Ybus and Power Flow Professor Tom Overbye Department of Electrical and Computer Engineering

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**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

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**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?

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**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

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**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

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Ybus Example, cont’d

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**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)

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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.

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**Modeling Shunts in the Ybus**

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Two Bus System Example

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Using the Ybus

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**Solving for Bus Currents**

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**Solving for Bus Voltages**

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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

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**Power Balance Equations**

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**Power Balance Equations, cont’d**

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**Real Power Balance Equations**

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**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)

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**Power Flow Requires Iterative Solution**

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Gauss Iteration

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ECE 476 Power System Analysis Lecture 11: Ybus, Power Flow Prof. Tom Overbye Dept. of Electrical and Computer Engineering University of Illinois at Urbana-Champaign.

ECE 476 Power System Analysis Lecture 11: Ybus, Power Flow Prof. Tom Overbye Dept. of Electrical and Computer Engineering University of Illinois at Urbana-Champaign.

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