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Lecture 13 Newton-Raphson Power Flow Professor Tom Overbye Department of Electrical and Computer Engineering ECE 476 POWER SYSTEM ANALYSIS

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1 Announcements Homework 6 is due on Thursday (Oct 18) Be reading Chapter 6 Homework 7 is 6.12, 6.19, 6.22, 6.45 and Due date is October 25

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2 Design Project Design Project 2 from the book is assigned today (page 345 to 348). It will be due Nov 29. For a transmission tower configuration assume a symmetrical tower configuration with individual conductor spacings as provided on the handout sheet.

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3 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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4 Newton-Raphson Method (scalar)

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5 Newton-Raphson Method, contd

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6 Newton-Raphson Example

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7 Newton-Raphson Example, contd

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8 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) ) < 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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9 Multi-Variable Newton-Raphson

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10 Multi-Variable Case, contd

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11 Multi-Variable Case, contd

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12 Jacobian Matrix

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13 Multi-Variable N-R Procedure

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14 Multi-Variable Example

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15 Multi-variable Example, contd

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16 Multi-variable Example, contd

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17 NR Application to Power Flow

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18 Real Power Balance Equations

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19 Newton-Raphson Power Flow

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20 Power Flow Variables

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21 N-R Power Flow Solution

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22 Power Flow Jacobian Matrix

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23 Power Flow Jacobian Matrix, contd

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24 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 S Base = 100 MVA.

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25 Two Bus Example, contd

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26 Two Bus Example, contd

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27 Two Bus Example, First Iteration

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28 Two Bus Example, Next Iterations

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29 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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30 Two Bus Case Low Voltage Solution

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31 Low Voltage Solution, cont'd Low voltage solution

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32 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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33 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 |V i | – V i setpoint = 0

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34 Three Bus PV Case Example

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35 Modeling Voltage Dependent Load

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36 Voltage Dependent Load Example

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37 Voltage Dependent Load, cont'd

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38 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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39 Solving Large Power Systems The most difficult computational task is inverting the Jacobian matrix – inverting a full matrix is an order n 3 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 Y bus is a sparse matrix, the Jacobian is also a sparse matrix – using sparse matrix methods results in a computational order of about n 1.5. – this is a substantial savings when solving systems with tens of thousands of buses

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40 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

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