Power Flow Solution by Holomorphic Embedding Method Part 1

In this article  the Holomorphic embedding based  power flow is explained .

Power flow problem is one of the oldest and important power system problems.It is the steady state solution of the power network. In a N bus  power system, There are 4N variables.

1.Pi-Real  Power bus number 'i'

2.Qi-Reactive   Bus  Power bus number 'i'

Si=Pi+i.Qi

The 2*N equations are formed by Kirchoffs currect law.

Though these equations are well known ,it can be observes that ,the equations are little bit manipulated. The 'i'th bus voltage Vi is moved to right hand side.This is the first step in Holomorphic embedding.

Why it is moved?

In algebra RHS(right hand side) is known values and the LHS is  unknown variables. So the Vi moved to RHS must be known. By the Halomorphic embedding it will be made known.That is the logic behind this method.The power flow problem is modified as recursive linear equations by making LHS as linear system.

The Vi is expressed as power series  polynomial in 's'.

Vi(s)=Vi[0]+s.Vi[1]+....(s^n).Vi[n]; 

Theoreticaly 'n' can be infinity.But we can choose n according to the accuracy of the solution. For the IEEE 118 bus system just 10th order polynomial is sufficient.

The coefficients are to be determined.  

The Holomorphicaly embedded equations are given below.

What is this 's' ?

s is a real variable .To solve power flow s is treated as variable. The objective is to determine the polynomial coefficients ,by recursive linear relationship evolving from the equations [H1 to H4]..  polynomial.After  determining the polynomial coefficients , Vi(s=1) will give the bus voltages for given base loading..  Vi(s=2) will give the power flow solutions 200% loading .

(To be continued)