ECE 530 – Analysis Techniques for Large-Scale Electrical Systems Prof. Hao Zhu Dept. of Electrical and Computer Engineering University of Illinois at Urbana-Champaign 8/31/ Lecture 3: Power Flow Problem

Ybus Matrix 2

Power Flow Analysis When analyzing power systems we know neither the complex bus voltages nor the complex current injections Rather, we know the complex power being consumed by the load, and the power being injected by the generators plus their voltage magnitudes Therefore, we can not directly use the Y bus equations, but rather must use the power balance equations 3

Power Flow Problem Classic paper for this topic is W.F. Tinney and C.E. Hart, “Power Flow Solution by Newton’s Method,” IEEE Power App System, Nov 1967 Basic power flow is also covered in essentially power system analysis textbooks. At Illinois we use the term “power flow” not “load flow” since power flows not load. Also, the power flow usage is not new (see title of Tinney’s 1967 paper, and note Tinney references Ward’s 1956 paper) 4

Power Balance Equations 5

Power Balance Equations, cont’d 6 = = = = = = (conductance and susceptance) (polar coordinates)

Real-Valued Balance Equations 7 Less popular alternative formulation based on rectangular coordinates using V i = e i + j f i

Problem Formulation 8

Slack Bus We can not arbitrarily specify S at all buses because total generation must equal total load + total losses We also need an angle reference bus. To solve these problems we define one bus as the "slack" bus. This bus has a fixed voltage magnitude and angle, and a varying real/reactive power injection. In an actual power system the slack bus does not really exist; frequency changes locally when the power supplied does not match the power consumed 9

Three Types of Buses 10

Solving Nonlinear Equations 11 Difficult problem Easy problem

Linear Equations 12

Nonlinear Equations 13

Nonlinear Equations The notation f(x) is short-hand for the vector function so the problem is to solve n equations for n unknowns We assume f(x) be a continuous real-valued function with at least the first-order derivative 14

Nonlinear Example of Multiple Solutions and No Solution 15 f (x) = x f (x) = x two solutions where f(x) = 0 no solution f(x) = 0 Example 1:x = 0 has solutions x =  1.414… Example 2: x = 0 has no real solution A key challenge with nonlinear equations is there may be one, none or multiple solutions!

Newton-Raphson Method Newton developed his method for solving for the roots of nonlinear equations in 1671, but it wasn’t published until 1736 Raphson developed a similar method in 1690; Raphson’s approach was actually simpler than Newton’s, and is what is used today It’s an iterative method – divergence or convergence speed could be an issue 16

Newton-Raphson Method (Scalar) 17

Newton-Raphson Method, cont’d 18

Newton-Raphson Example 19

Newton-Raphson Example, cont’d 20

Sequential Linear Approximations 21 Function is f(x) = x = 0. Solutions are points where f(x) intersects f(x) = 0 axis

Geometric interpretation x (3) x (2) x (0) root x * x (4) x (1) x f (x) 22

Example 2 Find the positive root of using Newton’s method starting Computation must be done using radians! 23

Example 2 Graphical View 24

Example 2 Iterations We continue the iterations to obtain the following set of results iteration number v

Example 2, Changed Initial Guess If we start at , the solutions is also iteration number v

Remarks on Convergence 27

Normal Convergence desired root f (x) 28

Oscillatory Convergence x (3) x (1) x (2) x (0) x (4) f (x) 29

Convergence to an Unwanted Root x desired root undesired root x (1) x (0) f (x) 30

Divergence x (1) x (0) x (2) x f (x) 31

Multi-Variable Newton-Raphson 32

Multi-Variable Case, cont’d 33

Multi-Variable Case, cont’d 34

Jacobian Matrix 35

Multi-Variable N-R Procedure 36

Multi-Variable Example 37

Multi-variable Example, cont’d 38

Multi-variable Example, cont’d 39

Stopping Criteria: Vector Norms 40 = =