Lecture 14

Power Flows

Read: Chapter 6.1 – 6.6, 6.8 –6.11

Midterm Exam is on March 8st

Dr. Lei Wu

Department of Electrical and Computer Engineering

EE 333

POWER SYSTEMS ENGINEERING

2

IEEE Power and Energy Society—

Interest meeting--

Formation of a Clarkson Student Chapter

Wednesday March 7

5pm

CAMP 178

IEEE Power and Energy Society (PES) is the primary professional group for power engineers. Being a

member of a PES student chapter is an indication of your interest in power engineering. Members have

access to PES publications, particularly the Power and Energy Magazine, which reports on state of the art

issues in power engineering. A power engineering student chapter at Clarkson will allow us to hold power

engineering oriented meetings and projects, potential activities will be discussed at the meeting.

Outline

� Bus-branch model of power systems

� The power flow problem

� Iterative methods for solving nonlinear equations

� Approximations to the power flow problem

3

Bus-Branch Model of Power Systems

� A power system includes

� Loads

� Generators

� Transmission lines

� Transformers

4

Load Models

� Ultimate goal is to supply loads with electricity at constant

frequency and voltage

� Electrical characteristics of individual loads matter, but usually

they can only be estimated actual loads are constantly changing,

consisting of a large number of individual devices only limited

network observability of load characteristics

� Aggregate models are typically used for analysis

� Two common models

� constant power: Si = Pi + jQi

� constant impedance: Si = |V|2 / Zi

5

Generator Models

� Engineering models depend upon application

� Generators are usually synchronous machines

� For generators we will use two different models:

� a steady-state model, treating the generator as a constant power

source operating at a fixed voltage; this model will be used for

power flow and economic analysis

� a short term model treating the generator as a constant voltage

source behind a possibly time-varying reactance

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Bus-branch model of power systems

� A power system includes

� Transmission lines

� Transformers

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� Generators

� Loads

Power Flow Problem

� The most common power system analysis tool is the power flow

(also known sometimes as the load flow)

� Power flow determines how the power flows in a network under balanced

three-phase steady-state conditions.

� Also determines all bus voltages and all currents, as well as equipment

losses can be obtained

� Because of constant power models, power flow is a nonlinear problem

� Power flow is a steady-state analysis tool

� Each bus has 4 variables:

� P

� Q

� V

� δ

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� Bus types

� Reference bus (VΘ bus) : Usually select a bus with a large

generator, , calculate P and Q.

� PQ bus: load buses where P and Q are given, calculate V and δ

� PV bus: Voltage controlled buses ( typically are buses with

generators) where P and V are given, calculate Q and δ.

� Usually Q of a generator is limited by Qmin and Qmax, which are

dependent upon the generator's MW output

� In the calculation procedure

� if Q < Qmin, fix Q = Qmin

� If Q > Qmax, fix Q = Qmax

� "type-switching“: Change the bus type to PQ and continue the

calculation procedure.

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1 0o∠

� Bus types

� Reference bus (VΘ bus) : Usually select a bus with a large

generator, , calculate P and Q.

� PQ bus: load buses where P and Q are given, calculate V and δ

� PV bus: Voltage controlled buses ( typically are buses with

generators) where P and V are given, calculate Q and δ.

10

1 0o∠

Linear VS Nonlinear systems

� f(x) is a linear system as long as

� f(ax+by) = a*f(x) + b*f(y)

� The output is proportional to the input

� The principle of superposition holds

� Linear example f(x) = c*x

� Nonlinear example f(x) = cx2 , f(x) = c*sin(x)

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Linear Power System Elements

� Linear circuit elements, which may be analyzed by

superposition.

� Resistor

� Inductor

� Capacitor

� Independent voltage and current sources

V = R I

V =

1V =

j L I

Ij C

ω

ω

Nonlinear Power System Elements

� Constant power loads and generator injections are nonlinear

and hence systems with these elements can not be analyzed by

superposition

� Nonlinear problems can be very difficult to solve, and usually

require an iterative approach

� Gauss-Seidel (G-S)

� Newton-Raphson (N-R)

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Nonlinear Systems May Have Multiple Solutions or No Solution

� Example 1: x2 - 2 = 0 has real solutions x = ± 2

� Example 2: x2 + 2 = 0 has no real solution

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f(x) = x2 - 2 f(x) = x2 + 2

two solutions where f(x) = 0 no solution f(x) = 0

Gauss Iteration

� With the Gauss method, we need to rewrite the equation in an

implicit form x= h(x).

� To iterate, we first make an initial guess of x, x(0) , and then

iteratively solve x(k+1)= h(x(k)), until we find a fixed point, x, such

that x= h(x).

� Stopping criterion

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( ) ( ) ( 1) ( )

( ) 22

1

i

If x is a scalar with

If x is a vector with

OR max x

v v v v

nv

iji

i

x x x x

x x

ε

ε

+

=

∞

∆ < ∆ −

∆ < ∆ = ∆

∆ = ∆

∑x

x

≜

Gauss Iteration Example

( 1) ( )

(0)

( ) ( )

Example: Solve - 1 0

1

Let = 0 and arbitrarily guess x 1 and solve

0 1 5 2.61185

1 2 6 2.61612

2 2.41421 7 2.61744

3 2.55538 8 2.61785

4 2.59805 9 2.61798

v v

v v

x x

x x

v

v x v x

+

− =

= +

=

Power Flow Analysis - Gauss

� We know neither complex bus voltages nor complex current injections

� Rather, we know the complex power being consumed by the load, and

the power being injected by generators plus their voltage magnitudes

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bus=I Y V1

N

k kn nn

Y V=

= ∑I

* * *

1

N

k k k k k k kn nn

S P jQ V I V Y V=

= + = = ∑* * *

1

N

k k k k k k kn nn

S P jQ V I V Y V=

= − = = ∑* 1

*1 1 1

N k Nk

kn n kk k kn n kn nn n n kk

SY V Y V Y V Y V

V

−

= = = += = + +∑ ∑ ∑

* 1

*1 1

1 k Nk

k kn n kn nkk n n kk

SV Y V Y V

Y V

−

= = +

= − −

∑ ∑

Three-Bus Power Flow Example –Gauss-Seidel

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20 50 10 20 10 30

10 20 26 52 16 32

10 30 16 32 26 62bus

j j j

Y j j j

j j j

− − + − + = − + − − + − + − + −

V Θ P Q

Bus 1 1.05 0 ? ?

Bus 2 ? (1) ? (0) -4 -2.5

Bus 3 1.04 ? (0) +2 ?

Using the MVA base of 100MVA