Presentation on theme: "Announcements Homework 6 is due on Thursday (Oct 18)"— Presentation transcript:
0 ECE 476 POWER SYSTEM ANALYSIS Lecture 13 Newton-Raphson Power FlowProfessor Tom OverbyeDepartment of Electrical and Computer Engineering
1 Announcements Homework 6 is due on Thursday (Oct 18) Be reading Chapter 6Homework 7 is 6.12, 6.19, 6.22, 6.45 and Due date is October 25
2 Design ProjectDesign 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.
3 Newton-Raphson Algorithm The second major power flow solution method is the Newton-Raphson algorithmKey idea behind Newton-Raphson is to use sequential linearization
8 Newton-Raphson Comments When close to the solution the error decreases quite quickly -- method has quadratic convergencef(x(v)) is known as the mismatch, which we would like to drive to zeroStopping 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
24 Two Bus Newton-Raphson Example For the two bus power system shown below, use theNewton-Raphson power flow to determine thevoltage magnitude and angle at bus two. Assumethat bus one is the slack and SBase = 100 MVA.
32 Two Bus Region of Convergence Slide shows the region of convergence for different initialguesses of bus 2 angle (x-axis) and magnitude (y-axis)Red regionconvergesto the highvoltagesolution,while theyellow regionto the lowsolution
33 PV BusesSince 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 equationsthe 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
38 Voltage Dependent Load, cont'd With constant impedance load the MW/Mvar load atbus 2 varies with the square of the bus 2 voltagemagnitude. This if the voltage level is less than 1.0,the load is lower than 200/100 MW/Mvar
39 Solving Large Power Systems The most difficult computational task is inverting the Jacobian matrixinverting a full matrix is an order n3 operation, meaning the amount of computation increases with the cube of the size sizethis amount of computation can be decreased substantially by recognizing that since the Ybus is a sparse matrix, the Jacobian is also a sparse matrixusing 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
40 Newton-Raphson Power Flow Advantagesfast convergence as long as initial guess is close to solutionlarge region of convergenceDisadvantageseach iteration takes much longer than a Gauss-Seidel iterationmore complicated to code, particularly when implementing sparse matrix algorithmsNewton-Raphson algorithm is very common in power flow analysis