Lecture 12 Power Flow

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ECE 476POWER SYSTEM ANALYSIS

• Lecture 12Power Flow
• Professor Tom Overbye
• Department of Electrical andComputer Engineering

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Announcements

• Homework 5 is due now
• Homework 6 is 2.38, 6.8, 6.23, 6.28 you should
  do it before the exam but need not turn it in.
• First exam is 10/9 in class closed book, closed
  notes, one note sheet and calculators allowed
• Abbott power plant and substation field trip,
  Tuesday 10/14 starting at 1230pm. Well meet at
  corner of Gregory and Oak streets.
• Be reading Chapter 6 exam covers up through
  Section 6.4 we do not explicitly cover 6.1.

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Newton-Raphson Algorithm

• The second major power flow solution method is
  the Newton-Raphson algorithm
• Key idea behind Newton-Raphson is to use
  sequential linearization

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Newton-Raphson Method (scalar)
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Newton-Raphson Method, contd
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Newton-Raphson Example
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Newton-Raphson Example, contd
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Sequential Linear Approximations
At each iteration the N-R method uses a
linear approximation to determine the next
value for x
Function is f(x) x2 - 2 0. Solutions are
points where f(x) intersects f(x) 0 axis
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Newton-Raphson Comments

• When close to the solution the error decreases
  quite quickly -- method has quadratic convergence
• f(x(v)) is known as the mismatch, which we would
  like to drive to zero
• Stopping criteria is when ?f(x(v)) ? lt ?
• Results are dependent upon the initial guess.
  What if we had guessed x(0) 0, or x (0) -1?
• A solutions region of attraction (ROA) is the
  set of initial guesses that converge to the
  particular solution. The ROA is often hard to
  determine

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Multi-Variable Newton-Raphson
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Multi-Variable Case, contd
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Multi-Variable Case, contd
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Jacobian Matrix
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Multi-Variable N-R Procedure
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Multi-Variable Example
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Multi-variable Example, contd
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Multi-variable Example, contd
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Power System Planning MISO
Source Midwest ISO MTEP08 Report
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MISO Generation Queue
Source Midwest ISO MTEP08 Report
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MISO Conceptual EHV Overlay
Black lines are DC, blue lines are 765kV, red are
500 kV
Source Midwest ISO MTEP08 Report
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Illinois EHV Overlay
Source Midwest ISO MTEP08 Report
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Transmission Siting Example
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Madison Transmission Siting Example Lots of
Support and Opposition
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NR Application to Power Flow
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Real Power Balance Equations
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Newton-Raphson Power Flow
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Power Flow Variables
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N-R Power Flow Solution
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Power Flow Jacobian Matrix
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Power Flow Jacobian Matrix, contd
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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.
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Two Bus Example, contd
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Two Bus Example, contd
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Two Bus Example, First Iteration
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Two Bus Example, Next Iterations
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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
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Two Bus Case Low Voltage Solution
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Low Voltage Solution, cont'd
Low voltage solution
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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 converges to the high voltage
solution, while the yellow region converges to
the low voltage solution
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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

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Three Bus PV Case Example
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Modeling Voltage Dependent Load
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Voltage Dependent Load Example
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Voltage Dependent Load, cont'd
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Voltage Dependent Load, cont'd
With constant impedance load the MW/Mvar load
at bus 2 varies with the square of the bus 2
voltage magnitude. This if the voltage level is
less than 1.0, the load is lower than 200/100
MW/Mvar
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Solving Large Power Systems

• 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

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Newton-Raphson Power Flow

• 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