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ECE 476 POWER SYSTEM ANALYSIS. Lecture 13 Power Flow Professor Tom Overbye Department of Electrical and Computer Engineering. Announcements. Be reading Chapter 6, also Chapter 2.4 (Network Equations).

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Ece 476 power system analysis


Lecture 13

Power Flow

Professor Tom Overbye

Department of Electrical andComputer Engineering


  • Be reading Chapter 6, also Chapter 2.4 (Network Equations).

  • HW 5 is 2.38, 6.9, 6.18, 6.30, 6.34, 6.38; do by October 6 but does not need to be turned in.

  • First exam is October 11 during class. Closed book, closed notes, one note sheet and calculators allowed. Exam covers thru the end of lecture 13 (today)

  • An example previous exam (2008) is posted. Note this is exam was given earlier in the semester in 2008 so it did not include power flow, but the 2011 exam will (at least partially)

Possible ehv overlays for wind
Possible EHV Overlays for Wind

AEP 2007 Proposed Overlay

Two bus newton raphson example
Two Bus Newton-Raphson Example

For the two bus power system shown below, use the

Newton-Raphson power flow to determine the

voltage magnitude and angle at bus two. Assume

that bus one is the slack and SBase = 100 MVA.

Two bus solved values
Two Bus Solved Values

Once the voltage angle and magnitude at bus 2 are

known we can calculate all the other system values,

such as the line flows and the generator reactive

power output

Low voltage solution cont d
Low Voltage Solution, cont'd

Low voltage solution

Two bus region of convergence
Two Bus Region of Convergence

Slide shows the region of convergence for different initial

guesses of bus 2 angle (x-axis) and magnitude (y-axis)

Red region


to the high



while the

yellow region


to the low



Pv buses
PV Buses

  • Since the voltage magnitude at PV buses is fixed there is no need to explicitly include these voltages in x or write the reactive power balance equations

    • the reactive power output of the generator varies to maintain the fixed terminal voltage (within limits)

    • optionally these variations/equations can be included by just writing the explicit voltage constraint for the generator bus |Vi| – Vi setpoint = 0

The N-R Power Flow: 5-bus Example


800 MVA

345/15 kV






520 MVA

Line 3 345 kV

50 mi

400 MVA

15 kV

800 MVA

15 kV

400 MVA

15/345 kV

345 kV 100 mi

40 Mvar

80 MW

345 kV 200 mi

Line 2

Line 1


280 Mvar

800 MW

Single-line diagram

The N-R Power Flow: 5-bus Example

Table 1.

Bus input


Table 2.

Line input data

The N-R Power Flow: 5-bus Example

Table 3.


input data

Table 4. Input data

and unknowns

Ybus details
Ybus Details Shown)

Elements of Ybus connected to bus 2

Good power system operation
Good Power System Operation Shown)

Good power system operation requires that there be no reliability violations for either the current condition or in the event of statistically likely contingencies

Reliability requires as a minimum that there be no transmission line/transformer limit violations and that bus voltages be within acceptable limits (perhaps 0.95 to 1.08)

Example contingencies are the loss of any single device. This is known as n-1 reliability.

North American Electric Reliability Corporation now has legal authority to enforce reliability standards (and there are now lots of them). See for details (click on Standards)

Looking at the impact of line outages
Looking at the Impact of Line Outages Shown)

Opening one line (Tim69-Hannah69) causes an overload. This would not be allowed

Contingency analysis
Contingency Analysis Shown)

Contingencyanalysis providesan automaticway of lookingat all the statisticallylikely contingencies. Inthis example thecontingency set

Is all the single line/transformeroutages

Power flow and design
Power Flow And Design Shown)

One common usage of the power flow is to determine how the system should be modified to remove contingencies problems or serve new load

In an operational context this requires working with the existing electric grid

In a planning context additions to the grid can be considered

In the next example we look at how to remove the existing contingency violations while serving new load.

An unreliable solution
An Unreliable Solution Shown)

Case now has nine separate contingencies with reliability violations

A reliable solution
A Reliable Solution Shown)

Previous case was augmented with the addition of a 138 kV Transmission Line

Generation changes and the slack bus
Generation Changes and The Slack Bus Shown)

The power flow is a steady-state analysis tool, so the assumption is total load plus losses is always equal to total generation

Generation mismatch is made up at the slack bus

When doing generation change power flow studies one always needs to be cognizant of where the generation is being made up

Common options include system slack, distributed across multiple generators by participation factors or by economics

Generation change example 1
Generation Change Example 1 Shown)

Display shows “Difference Flows” between original 37 bus case, and case with a BLT138 generation outage;

note all the power change is picked up at the slack

Generation change example 2
Generation Change Example 2 Shown)

Display repeats previous case except now the change in generation is picked up by other generators using a participation factor approach

Voltage regulation example 37 buses
Voltage Regulation Example: 37 Buses Shown)

Display shows voltage contour of the power system, demo will show the impact of generator voltage set point, reactive power limits, and switched capacitors

Solving large power systems
Solving Large Power Systems Shown)

  • The most difficult computational task is inverting the Jacobian matrix

    • inverting a full matrix is an order n3 operation, meaning the amount of computation increases with the cube of the size size

    • this amount of computation can be decreased substantially by recognizing that since the Ybus is a sparse matrix, the Jacobian is also a sparse matrix

    • using sparse matrix methods results in a computational order of about n1.5.

    • this is a substantial savings when solving systems with tens of thousands of buses

Newton raphson power flow1
Newton-Raphson Power Flow Shown)

  • Advantages

    • fast convergence as long as initial guess is close to solution

    • large region of convergence

  • Disadvantages

    • each iteration takes much longer than a Gauss-Seidel iteration

    • more complicated to code, particularly when implementing sparse matrix algorithms

  • Newton-Raphson algorithm is very common in power flow analysis