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ECE 476 POWER SYSTEM ANALYSIS

ECE 476 POWER SYSTEM ANALYSIS. Lecture 13 Newton-Raphson Power Flow Professor Tom Overbye Department of Electrical and Computer Engineering. 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 6.50. Due date is October 25.

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ECE 476 POWER SYSTEM ANALYSIS

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  1. ECE 476POWER SYSTEM ANALYSIS Lecture 13Newton-Raphson Power Flow Professor Tom Overbye Department of Electrical andComputer Engineering

  2. 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 6.50. Due date is October 25

  3. 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.

  4. 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

  5. Newton-Raphson Method (scalar)

  6. Newton-Raphson Method, cont’d

  7. Newton-Raphson Example

  8. Newton-Raphson Example, cont’d

  9. 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 solution’s region of attraction (ROA) is the set of initial guesses that converge to the particular solution. The ROA is often hard to determine

  10. Multi-Variable Newton-Raphson

  11. Multi-Variable Case, cont’d

  12. Multi-Variable Case, cont’d

  13. Jacobian Matrix

  14. Multi-Variable N-R Procedure

  15. Multi-Variable Example

  16. Multi-variable Example, cont’d

  17. Multi-variable Example, cont’d

  18. NR Application to Power Flow

  19. Real Power Balance Equations

  20. Newton-Raphson Power Flow

  21. Power Flow Variables

  22. N-R Power Flow Solution

  23. Power Flow Jacobian Matrix

  24. Power Flow Jacobian Matrix, cont’d

  25. 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.

  26. Two Bus Example, cont’d

  27. Two Bus Example, cont’d

  28. Two Bus Example, First Iteration

  29. Two Bus Example, Next Iterations

  30. 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

  31. Two Bus Case Low Voltage Solution

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

  33. 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

  34. 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

  35. Three Bus PV Case Example

  36. Modeling Voltage Dependent Load

  37. Voltage Dependent Load Example

  38. Voltage Dependent Load, cont'd

  39. 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

  40. 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

  41. 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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