ECE 476 POWER SYSTEM ANALYSIS Lecture 12 Power

ECE 476 POWER SYSTEM ANALYSIS Lecture 12 Power Flow Professor Tom Overbye Department of Electrical and Computer Engineering

Announcements l l l Homework 5 is due now Homework 6 is 2. 38, 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 12: 30 pm. We’ll 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. 1

Newton-Raphson Algorithm l l The second major power flow solution method is the Newton-Raphson algorithm Key idea behind Newton-Raphson is to use sequential linearization 2

Newton-Raphson Method (scalar) 3

Newton-Raphson Method, cont’d 4

Newton-Raphson Example 5

Newton-Raphson Example, cont’d 6

Sequential Linear Approximations Function is f(x) = x 2 - 2 = 0. Solutions are points where f(x) intersects f(x) = 0 axis At each iteration the N-R method uses a linear approximation to determine the next value for x 7

Newton-Raphson Comments l l l 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 8

Multi-Variable Newton-Raphson 9

Multi-Variable Case, cont’d 10

Multi-Variable Case, cont’d 11

Jacobian Matrix 12

Multi-Variable N-R Procedure 13

Multi-Variable Example 14

Multi-variable Example, cont’d 15

Multi-variable Example, cont’d 16

Power System Planning: MISO Source: Midwest ISO MTEP 08 Report 17

MISO Generation Queue Source: Midwest ISO MTEP 08 Report 18

MISO Conceptual EHV Overlay Black lines are DC, blue lines are 765 k. V, red are 500 k. V Source: Midwest ISO MTEP 08 Report 19

Illinois EHV Overlay Source: Midwest ISO MTEP 08 Report 20

Transmission Siting Example 21

Madison Transmission Siting Example: Lots of Support and Opposition 22

NR Application to Power Flow 23

Real Power Balance Equations 24

Newton-Raphson Power Flow 25

Power Flow Variables 26

N-R Power Flow Solution 27

Power Flow Jacobian Matrix 28

Power Flow Jacobian Matrix, cont’d 29

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. 30

Two Bus Example, cont’d 31

Two Bus Example, cont’d 32

Two Bus Example, First Iteration 33

Two Bus Example, Next Iterations 34

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 35

Two Bus Case Low Voltage Solution 36

Low Voltage Solution, cont'd Low voltage solution 37

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 38

PV Buses l 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 39

Three Bus PV Case Example 40

Modeling Voltage Dependent Load 41

Voltage Dependent Load Example 42

Voltage Dependent Load, cont'd 43

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 44

Solving Large Power Systems l 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 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 n 1. 5. this is a substantial savings when solving systems with tens of thousands of buses 45

Newton-Raphson Power Flow l Advantages – – l Disadvantages – – l fast convergence as long as initial guess is close to solution large region of convergence 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 46