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EE 369 POWER SYSTEM ANALYSIS. Lecture 13 Newton-Raphson Power Flow Tom Overbye and Ross Baldick. Announcements. Read Chapter 12, concentrating on sections 12.4 and 12.5. Homework 11 is 6.43, 6.48, 6.59, 6.61, 12.19, 12.22, 12.20, 12.24, 12.26, 12.28, 12.29; due November 21.

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Ee 369 power system analysis l.jpg


Lecture 13Newton-Raphson Power Flow

Tom Overbye and Ross Baldick

Announcements l.jpg

  • Read Chapter 12, concentrating on sections 12.4 and 12.5.

  • Homework 11 is 6.43, 6.48, 6.59, 6.61, 12.19, 12.22, 12.20, 12.24, 12.26, 12.28, 12.29; due November 21.

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Using the Power Flow: Example 1

Usingcasefrom Example6.13

Dishonest newton raphson l.jpg
Dishonest Newton-Raphson

  • Since most of the time in the Newton-Raphson iteration is spent dealing with the Jacobian, one way to speed up the iterations is to only calculate (and factorize) the Jacobian occasionally:

    • known as the “Dishonest” Newton-Raphson or Shamanskii method,

    • an extreme example is to only calculate the Jacobian for the first iteration, which is called the chord method.

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Dishonest N-R Example, cont’d

We pay a price

in increased

iterations, but

with decreased


per iteration

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Two Bus Dishonest ROC

Slide shows the region of convergence for different initial

guesses for the 2 bus case using the Dishonest N-R

Red region


to the high



while the

yellow region


to the low



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Honest N-R Region of Convergence

Maximum of 15


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Decoupled Power Flow

  • The “completely” Dishonest Newton-Raphson (chord) is not usually used for power flow analysis. However several approximations of the Jacobian matrix are used that result in a similar approximation.

  • One common method is the decoupled power flow. In this approach approximations are used to decouple the real and reactive power equations.

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Fast Decoupled Power Flow

  • By further approximating the Jacobian we obtain a typically reasonable approximation that is independent of the voltage magnitudes/angles.

  • This means the Jacobian need only be built and factorized once.

  • This approach is known as the fast decoupled power flow (FDPF)

  • FDPF uses the same mismatch equations as standard power flow so it should have same solution if it converges

  • The FDPF is widely used, particularly when we only need an approximate solution.

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FDPF Three Bus Example

Use the FDPF to solve the following three bus system

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“DC” Power Flow

  • The “DC” power flow makes the most severe approximations:

    • completely ignore reactive power, assume all the voltages are always 1.0 per unit, ignore line conductance

  • This makes the power flow a linear set of equations, which can be solved directly:

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DC Power Flow 5 Bus Example

Notice with the dc power flow all of the voltage magnitudes are 1 per unit.

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Power System Control

  • A major problem with power system operation is the limited capacity of the transmission system

    • lines/transformers have limits (usually thermal)

    • no direct way of controlling flow down a transmission line (e.g., there are no low cost valves to close to limit flow, except “on” and “off”)

    • open transmission system access associated with industry restructuring is stressing the system in new ways

  • We need to indirectly control transmission line flow by changing the generator outputs.

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Indirect Transmission Line Control

What we would like to determine is how a change in

generation at bus k affects the power flow on a line

from bus i to bus j.

The assumption is

that the change

in generation is

absorbed by the

slack bus

Power flow simulation before l.jpg
Power Flow Simulation - Before

  • One way to determine the impact of a generator change is to compare a before/after power flow.

  • For example below is a three bus case with an overload.

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Power Flow Simulation - After

  • Increasing the generation at bus 3 by 95 MW (and hence decreasing generation at the slack bus 1 by a corresponding amount), results in a 31.3 MW drop in the MW flow on the line from bus 1 to 2.

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Analytic Calculation of Sensitivities

  • Calculating control sensitivities by repeated power flow solutions is tedious and would require many power flow solutions.

  • An alternative approach is to analytically calculate these values