Power Flow Analysis
Assist. Prof. MOHAMMAD AL SAMMAN
Faculty of Engineering and Natural Science, Electrical and
Electronics Engineering, International University of Sarajevo
Power Markets & Economics

Outline
2
◎ Introduction
◎ Bus Admittance matrix
◎ Solution of Non-linear Algebraic Equations
◎ Power Flow Solution
◎ Gauss-Seidel Power Flow Solution
◎ Line Flows and Losses
◎ Newton-Raphson Power Flow Solution
For further reading, please check R1/Ch 6

Introduction
3
IEEE 39-Bus System

Introduction
4
• The system is operating under balanced conditions
• The system is represented by a single phase network
• All the nodes and branches’ impedances are specified in per unit on a common MVA
base.
• The network is formulated using node-voltage method.
• Power flow (load flow) studies are the backbone of the power system analysis

Bus Admittance matrix
5

Bus Admittance matrix
6

Bus Admittance matrix
7

Bus Admittance matrix
8
𝑰𝒃𝒖𝒔 is the vector of the injected bus
currents (external current sources)
𝑽𝒃𝒖𝒔 is the vector of bus voltages
measured from the reference node.
𝒀𝒃𝒖𝒔 is the bus admittance matrix.

Solution of Nonlinear Algebraic Equations
9
Gauss-Seidel Method
𝑓 𝑥 = 0
𝑥 = 𝑔 𝑥
𝑥(𝑘) is an initial estimate of the variable x, hence
𝑥(𝑘+1) − 𝑥(𝑘) ≤ 𝜀
𝜀 is the desired accuracy

Solution of Nonlinear Algebraic Equations
10
In general; for a system with n equations in n variables
Gauss-Seidel Method

Solution of Nonlinear Algebraic Equations
11
𝑓 𝑥 = 𝑐
𝑥(0)
is the initial estimate of the solution and ∆𝑥(0)
is a small deviation from the correct
solution,
𝑓 𝑥 0 + ∆𝑥 0 = 𝑐
Expanding the left-hand side of the equation using Taylor’s series about 𝑥(0) yields
Assuming the error ∆𝑥 0 is very small, hence
where
Newton-Raphson Method

Solution of Nonlinear Algebraic Equations
12
Newton-Raphson Method
where

Solution of Nonlinear Algebraic Equations
13
Newton-Raphson Method
In general; for a system with n equations in n variables

Solution of Nonlinear Algebraic Equations
14
Newton-Raphson Method
We can rewrite it in a matrix form
or

Power Flow solution
15
To solve Power flow (Load flow), we need to have four parameters: the voltage
magnitude, the phase angle; the real power; and the reactive power
The system buses are divided into:
Slack bus: (swing bus). It is the reference bus, where the voltage magnitude and the phase
angle are known
Load bus: here the active and reactive power are known
Regulated buses: (generator buses) or voltage controlled buses. Here the real power and
the voltage magnitude are specified. Also known as P-V buses.

Power Flow solution
16
IEEE 39-Bus System

Power Flow Equation
17

Gauss-Seidel Power Flow Solution
18

Line Flows and Losses
19

20

Newton-Raphson Power Flow Solution
21
For large power systems, Newton-Raphson is found to be more practical and efficient as
compared to Gauss-Seidel

Newton-Raphson Power Flow Solution
22

Newton-Raphson Power Flow Solution
23
Expanding using Tylor’s series

Newton-Raphson Power Flow Solution
24

Newton-Raphson Power Flow Solution
25
For 𝐽1 For 𝐽2
For 𝐽3 For 𝐽4

Newton-Raphson Power Flow Solution
26
The power residuals are the differences between the scheduled and calculated values

Newton-Raphson Power Flow Solution
27
Ex. 6.10: obtain the NR power flow solution for the system given in Ex 6.8
The base is 100 MVA. Find the line flows and
line losses

Next week
28
Optimization in Power Systems