The Power Flow Problem

Scott NorrFor

EE 4501April, 2015

Electric Concepts

• Ohm’s Law: V = IR (V = IZ)• Kirchoff: KCL: ∑ i = 0 (at any node)

∑ V = 0 (on closed path)• Power: P = VI (S = VI*)

= V2/R = I2R• NOTE: i means electric current, j2 = -1

Previously: DC Circuits (RI=V)

Resistance Matrix

AC Circuits: Phasor AnalysisZI = V (Thanks to Euler, Steinmetz)

Impedance Matrix

DC Power:All electrical systems naturally seek an equilibrium point of lowest entropy

Important to recognize that P ᴕ V2

(try to find the proportional

symbol in powerpoint

sometime…..)

AC Power

• The complex power, S = VI* = P + jQ• P is the average power (“real” power) in watts,

attributable to resistive loads • Q is the reactive power (“imaginary” power) in

VAr, attributable to capacitive and inductive loads

The Power Problem:

• On AC power systems, we don’t pre-determine the phase angles on the sources, they are determined by the system (additional unknowns to solve for!)

• Power is injected into nodes in the system via sources and is removed at nodes via loads (consumption points)

• Additionally, power is lost in the network

Consider an Example: 3 Node System

Unknowns:

• At each bus (node) there are 4 parameters: P, Q, V and Ө

• There are three types of buses:– Load Buses: P, Q are known, V, Ө are unknown– Generator Buses: P, V are known, Q, Ө are unknown– Slack Bus: (unique) V, Ө are known, P, Q unknown(this special generator node is allowed to accumulate errors in the iterative solution of the system of equations)

So, for N nodes, 2N unknown node parameters

Balancing Power at Each Node:• ∑Si = o• SG-SL = Vi∑Ip*

• SG-SL = Vi∑Vp*Yp*

• Can separate the real (P) from the

imaginary (Q)to form two equations at each

Bus

• A system of 2N equations

• Sparse, largely diagonalized matrices

Solve for Node Voltages and Angles:

• Vi new = (1/Yii)(Si/Viold - ∑Vp*Yp*)

• An iterative process, involving an initial starting estimate and convergence to a pre-determined tolerance.

• This is called the Gauss-Seidel Solution Method

A better Method:

• For analytic, complex differentiable systems, can compute the low order terms of the Taylor series and solve using Netwon’s method.

• In two variables, an iterative approach:f1(x,y) = K = f(xo + Δxo, yo + Δyo)g1(x,y) = L = g(xo + Δxo, yo + Δyo)

Computing the Taylor Series, and truncating ityields an equation exploiting a Jacobian matrix

Newton – Raphson Solution:

Conclusions:• Powerflow Software is used by every electric

utility in the world. Many models contain 10,000 nodes or more.

• There are quite a few solution techniques that are more efficient than the G-S and N-R methods outlined here:

Fast-Decoupled N-R – decouples P,O from Q,V and solves the two, smaller systemsInterior Point Newton - calculates a Hessian Mtx!

PowerWorld Simulator

References:

• Stevenson, William D., Elements of Power System Analysis, McGraw-Hill, 1982

• Tylavsky, Daniel, Lecture Notes #19, EEE 574, Arizona State University, 1999

• PowerWorld Simulator, www.powerworld.com,2014