Presentation is loading. Please wait.

Presentation is loading. Please wait.

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

Similar presentations


Presentation on theme: "Lecture 13 Newton-Raphson Power Flow Professor Tom Overbye Department of Electrical and Computer Engineering ECE 476 POWER SYSTEM ANALYSIS."— Presentation transcript:

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

2 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

3 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.

4 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

5 4 Newton-Raphson Method (scalar)

6 5 Newton-Raphson Method, contd

7 6 Newton-Raphson Example

8 7 Newton-Raphson Example, contd

9 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

10 9 Multi-Variable Newton-Raphson

11 10 Multi-Variable Case, contd

12 11 Multi-Variable Case, contd

13 12 Jacobian Matrix

14 13 Multi-Variable N-R Procedure

15 14 Multi-Variable Example

16 15 Multi-variable Example, contd

17 16 Multi-variable Example, contd

18 17 NR Application to Power Flow

19 18 Real Power Balance Equations

20 19 Newton-Raphson Power Flow

21 20 Power Flow Variables

22 21 N-R Power Flow Solution

23 22 Power Flow Jacobian Matrix

24 23 Power Flow Jacobian Matrix, contd

25 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.

26 25 Two Bus Example, contd

27 26 Two Bus Example, contd

28 27 Two Bus Example, First Iteration

29 28 Two Bus Example, Next Iterations

30 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

31 30 Two Bus Case Low Voltage Solution

32 31 Low Voltage Solution, cont'd Low voltage solution

33 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

34 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

35 34 Three Bus PV Case Example

36 35 Modeling Voltage Dependent Load

37 36 Voltage Dependent Load Example

38 37 Voltage Dependent Load, cont'd

39 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

40 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

41 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


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

Similar presentations


Ads by Google