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System Modeling and Power Flow Analysis ECG 740
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Page 1: Power Flow Studies - University of Nevada, Las Vegaseebag/740 Power Flow Analysis.pdf · Introduction to Power Flow •A power flow study (load-flow study) is a steady-state analysis

System Modeling and Power Flow Analysis

ECG 740

Page 2: Power Flow Studies - University of Nevada, Las Vegaseebag/740 Power Flow Analysis.pdf · Introduction to Power Flow •A power flow study (load-flow study) is a steady-state analysis

Synchronous Generators

Page 3: Power Flow Studies - University of Nevada, Las Vegaseebag/740 Power Flow Analysis.pdf · Introduction to Power Flow •A power flow study (load-flow study) is a steady-state analysis

Synchronous Generator Rating

• The purpose of ratings is to protect the machine from damage. Typical ratings of synchronous machines are voltage, speed, apparent power (kVA), power factor, field current.

– The rated frequency of a synchronous machine depends on the power system to which it is connected. Once the operation frequency is determined, only one rotational speed in possible for the given number of poles.

– For a given design, the rated voltage is limited by the flux that is capped by the field current. The rated voltage is also limited by the windings insulation breakdown limit.

– The maximum acceptable armature current sets the apparent power rating for a generator. The power factor of the armature current is irrelevant for heating the armature windings.

Page 4: Power Flow Studies - University of Nevada, Las Vegaseebag/740 Power Flow Analysis.pdf · Introduction to Power Flow •A power flow study (load-flow study) is a steady-state analysis

Generator Loading Capability Curve

Page 5: Power Flow Studies - University of Nevada, Las Vegaseebag/740 Power Flow Analysis.pdf · Introduction to Power Flow •A power flow study (load-flow study) is a steady-state analysis

Power Transformers

Page 6: Power Flow Studies - University of Nevada, Las Vegaseebag/740 Power Flow Analysis.pdf · Introduction to Power Flow •A power flow study (load-flow study) is a steady-state analysis

Transformer ratings: Voltage and Frequency

The voltage rating on each side is a) used to protect the winding insulation

from breakdown; b) related to the magnetization current of the transformer

(more important)

If a steady-state voltage

( ) sinMv t V t

is applied to the transformer’s

primary winding, the transformer’s

flux will be

( ) c) o1

s( M

p p

v t dV

t tN

tN

An increase in voltage will lead to a

proportional increase in flux.

However, after some point (in a

saturation region), such increase in

flux would require an unacceptable

increase in magnetization current!

flux

Magnetization

current

Page 7: Power Flow Studies - University of Nevada, Las Vegaseebag/740 Power Flow Analysis.pdf · Introduction to Power Flow •A power flow study (load-flow study) is a steady-state analysis

Transformer ratings: Voltage and Frequency

Therefore, the maximum applied voltage (and thus the rated voltage) is set by

the maximum acceptable magnetization current in the core.

We notice that the maximum flux is also related to the frequency:

maxmax

p

V

N

Therefore, to maintain the same maximum flux, a change in frequency (say, 50

Hz instead of 60 Hz) must be accompanied by the corresponding correction in

the maximum allowed voltage. This reduction in applied voltage with frequency

is called derating. As a result, a 50 Hz transformer may be operated at a 20%

higher voltage on 60 Hz if this would not cause insulation damage.

Page 8: Power Flow Studies - University of Nevada, Las Vegaseebag/740 Power Flow Analysis.pdf · Introduction to Power Flow •A power flow study (load-flow study) is a steady-state analysis

Transformer ratings: Apparent Power

The apparent power rating sets (together with the voltage rating) the

current through the windings. The current determines the i2R losses and,

therefore, the heating of the coils.

Overheating shortens the life of transformer’s insulation. Under any

circumstances, the temperature of the windings must be limited.

A typical power transformer rating would be OA/FA/FOA where • The OA (open-air) rating is the base power rating. • The FA (forced-air) rating is the power carrying capability when forced air

cooling (fans) are used. • The FOA (forced-oil-air) rating is the power carrying capability when forced

air cooling (fans) are used in addition to oil circulating pumps. Each cooling method provides nearly an additional 1/3 capability. For

example, a common substation transformer is rated at 21/28/35 MVA.

Page 9: Power Flow Studies - University of Nevada, Las Vegaseebag/740 Power Flow Analysis.pdf · Introduction to Power Flow •A power flow study (load-flow study) is a steady-state analysis

Transformer equivalent circuit model

Page 10: Power Flow Studies - University of Nevada, Las Vegaseebag/740 Power Flow Analysis.pdf · Introduction to Power Flow •A power flow study (load-flow study) is a steady-state analysis

Transmission Lines

10

Page 11: Power Flow Studies - University of Nevada, Las Vegaseebag/740 Power Flow Analysis.pdf · Introduction to Power Flow •A power flow study (load-flow study) is a steady-state analysis

Equivalent Circuit of Transmission Line

• Total shunt admittance:

• Total series impedance:

𝑍 = 𝑧𝑑 = 𝑟 + 𝑗𝑥 𝑑 = 𝑅 + 𝑗 𝑋

• where y, r, x are the shunt admittance, series resistance, and series reactance per unit length, and d is the length of the transmission line.

• The line resistance depends on the conductor type and size.

• The line reactance and admittance depend on the conductor radius, GMR, and line geometric mean distance between conductors

11

Page 12: Power Flow Studies - University of Nevada, Las Vegaseebag/740 Power Flow Analysis.pdf · Introduction to Power Flow •A power flow study (load-flow study) is a steady-state analysis

Load Modeling in Power Flow Analysis

• ZIP model:

• For simplicity, we often represent the load by the constant power component (P,Q).

Page 13: Power Flow Studies - University of Nevada, Las Vegaseebag/740 Power Flow Analysis.pdf · Introduction to Power Flow •A power flow study (load-flow study) is a steady-state analysis

Single line diagram

Page 14: Power Flow Studies - University of Nevada, Las Vegaseebag/740 Power Flow Analysis.pdf · Introduction to Power Flow •A power flow study (load-flow study) is a steady-state analysis

Introduction to Power Flow

• A power flow study (load-flow study) is a steady-state analysis whose target is to determine the voltages, currents, and real and reactive power flows in a system under a given load conditions.

• The purpose of power flow studies is to plan ahead and account for various hypothetical situations. For example, if a transmission line is be taken off line for maintenance, can the remaining lines in the system handle the required loads without exceeding their rated values.

Page 15: Power Flow Studies - University of Nevada, Las Vegaseebag/740 Power Flow Analysis.pdf · Introduction to Power Flow •A power flow study (load-flow study) is a steady-state analysis

Power-flow analysis equations

11 12 13 14 1 1

21 22 23 24 2 2

31 32 33 34 3 3

41 42 43 44 4 4

Y Y Y Y V I

Y Y Y Y V I

Y Y Y Y V I

Y Y Y Y V I

The basic equation for power-flow analysis is derived from

the nodal analysis equations for the power system: For

example, for a 4-bus system,

where Yij are the elements of the bus admittance matrix, Vi are

the bus voltages, and Ii are the currents injected at each node.

The node equation at bus i can be written as

n

j

jiji VYI1

Page 16: Power Flow Studies - University of Nevada, Las Vegaseebag/740 Power Flow Analysis.pdf · Introduction to Power Flow •A power flow study (load-flow study) is a steady-state analysis

Power-flow analysis equations

Relationship between per-unit real and reactive power

supplied to the system at bus i and the per-unit current

injected into the system at that bus:

where Vi is the per-unit voltage at the bus; Ii* - complex

conjugate of the per-unit current injected at the bus; Pi and Qi

are per-unit real and reactive powers. Therefore,

iiiii jQPIVS *

** /)(/)( iiiiiiii VjQPIVjQPI

n

j

ijij

n

j

jijiii VVYVYVjQP1

*

1

*

Page 17: Power Flow Studies - University of Nevada, Las Vegaseebag/740 Power Flow Analysis.pdf · Introduction to Power Flow •A power flow study (load-flow study) is a steady-state analysis

Power flow equations

Let

Then

Hence,

and

iiiijijij VVandYY ||||

)(||||||1

ijij

n

j

ijijii VVYjQP

)cos(||||||1

ijij

n

j

ijiji VVYP

)sin(||||||1

ijij

n

j

ijiji VVYQ

Page 18: Power Flow Studies - University of Nevada, Las Vegaseebag/740 Power Flow Analysis.pdf · Introduction to Power Flow •A power flow study (load-flow study) is a steady-state analysis

• There are 4 variables that are associated with each bus:

o P,

o Q,

o V,

o δ.

• Meanwhile, there are two power flow equations associated

with each bus.

• In a power flow study, two of the four variables are defined

an the other two are unknown. That way, we have the

same number of equations as the number of unknown.

• The known and unknown variables depend on the type of

bus.

Formulation of power-flow study

Page 19: Power Flow Studies - University of Nevada, Las Vegaseebag/740 Power Flow Analysis.pdf · Introduction to Power Flow •A power flow study (load-flow study) is a steady-state analysis

Each bus in a power system can be classified as one of three types:

1. Load bus (P-Q bus) – a buss at which the real and reactive

power are specified, and for which the bus voltage will be

calculated. All busses having no generators are load busses. In

here, V and δ are unknown.

2. Generator bus (P-V bus) – a bus at which the magnitude of the

voltage is defined and is kept constant by adjusting the field

current of a synchronous generator. We also assign real power

generation for each generator according to the economic

dispatch. In here, Q and δ are unknown

3. Slack bus (swing bus) – a special generator bus serving as the

reference bus. Its voltage is assumed to be fixed in both

magnitude and phase (for instance, 10˚ pu). In here, P and Q

are unknown.

Formulation of power-flow study

Page 20: Power Flow Studies - University of Nevada, Las Vegaseebag/740 Power Flow Analysis.pdf · Introduction to Power Flow •A power flow study (load-flow study) is a steady-state analysis

Formulation of power-flow study

• Note that the power flow equations are non-linear, thus cannot

be solved analytically. A numerical iterative algorithm is

required to solve such equations. A standard procedure

follows:

1. Create a bus admittance matrix Ybus for the power system;

2. Make an initial estimate for the voltages (both magnitude

and phase angle) at each bus in the system;

3. Substitute in the power flow equations and determine the

deviations from the solution.

4. Update the estimated voltages based on some commonly

known numerical algorithms (e.g., Newton-Raphson or

Gauss-Seidel).

5. Repeat the above process until the deviations from the

solution are minimal.

Page 21: Power Flow Studies - University of Nevada, Las Vegaseebag/740 Power Flow Analysis.pdf · Introduction to Power Flow •A power flow study (load-flow study) is a steady-state analysis

Example

Consider a 4-bus power system below. Assume that

– bus 1 is the slack bus and that it has a voltage V1 = 1.0∠0° pu.

– The generator at bus 3 is supplying a real power P3 = 0.3 pu to the system with a voltage magnitude 1 pu.

– The per-unit real and reactive power loads at busses 2 and 4 are P2 = 0.3 pu, Q2 = 0.2 pu, P4 = 0.2 pu, Q4 = 0.15 pu.

Line impedances (in pu): Line 1-2: Z= 0.1+j0.4 Line 2-3: Z = 0.1+j0.5 Line2-4: Z = 0.1+j-.4 Line3-4: Z= 0.05+j0.2 Line 4-1: Z = 0.05+j0.2

Page 22: Power Flow Studies - University of Nevada, Las Vegaseebag/740 Power Flow Analysis.pdf · Introduction to Power Flow •A power flow study (load-flow study) is a steady-state analysis

Example (cont.)

• Y-bus matrix (refer to example in book)

• Power flow solution:

• By knowing the node voltages, the power flow (both active and reactive) in each branch of the circuit can easily be calculated.

Page 23: Power Flow Studies - University of Nevada, Las Vegaseebag/740 Power Flow Analysis.pdf · Introduction to Power Flow •A power flow study (load-flow study) is a steady-state analysis

Newton-Raphson Algorithm

• The key idea behind Newton-Raphson is to use sequential linearization

• Consider the case of one variable:

General form of problem: Find an such that

( ) 0

x

f x

Page 24: Power Flow Studies - University of Nevada, Las Vegaseebag/740 Power Flow Analysis.pdf · Introduction to Power Flow •A power flow study (load-flow study) is a steady-state analysis

Newton-Raphson Method (scalar)

( )

( )

( ) ( ) ( ) ( ) ( )

2 2( ) ( )

2

1. Represent by a Taylor series about the

current guess . Write for the deviation

from :

( ) ( ) ( )

1( )

2

higher order terms.

v v v v v

v v

f

x x

x

dff x x f x x x

dx

d fx x

dx

Page 25: Power Flow Studies - University of Nevada, Las Vegaseebag/740 Power Flow Analysis.pdf · Introduction to Power Flow •A power flow study (load-flow study) is a steady-state analysis

Newton-Raphson Method, cont’d

( ) ( ) ( ) ( ) ( )

( )

1( ) ( ) ( )

2. Approximate by neglecting all terms

except the first two

( ) ( ) ( )

3. Set linear approximation equal to zero

and solve for

( ) ( )

4. Sol

v v v

v

v v v

f

dff x x f x x x

dx

x

dfx x f x

dx

( 1) ( ) ( )

ve for a new estimate of solution:

v v vx x x

Page 26: Power Flow Studies - University of Nevada, Las Vegaseebag/740 Power Flow Analysis.pdf · Introduction to Power Flow •A power flow study (load-flow study) is a steady-state analysis

Newton-Raphson - Example

2

1( ) ( ) ( )

( ) ( ) 2

( )

( 1) ( ) ( )

( 1) ( ) ( ) 2

( )

Use Newton-Raphson to solve ( ) 0,

where: ( )= 2.

The iterative update is:

( ) ( )

1(( ) 2)

2

1(( ) 2).

2

v v v

v v

v

v v v

v v v

v

f x

f x x

dfx x f x

dx

x xx

x x x

x x xx

Page 27: Power Flow Studies - University of Nevada, Las Vegaseebag/740 Power Flow Analysis.pdf · Introduction to Power Flow •A power flow study (load-flow study) is a steady-state analysis

Newton-Raphson - Example (cont’d)

( 1) ( ) ( ) 2

( )

(0)

( ) ( ) ( )

3 3

6

1(( ) 2)

2

Matlab code: x=x0; x = x-(1/(2*x))*(x^2-2).

Guess 1. Iteratiting, we get:

( )

0 1 1 0.5

1 1.5 0.25 0.08333

2 1.41667 6.953 10 2.454 10

3 1.41422 6.024 10

v v v

v

v v v

x x xx

x

x f x x

Page 28: Power Flow Studies - University of Nevada, Las Vegaseebag/740 Power Flow Analysis.pdf · Introduction to Power Flow •A power flow study (load-flow study) is a steady-state analysis

Sequential Linear Approximations

Function is f(x) = x2 - 2. Solutions to f(x) = 0 are points where f(x) intersects x axis.

At each iteration the N-R method uses a linear approximation to determine the next value for x

Page 29: Power Flow Studies - University of Nevada, Las Vegaseebag/740 Power Flow Analysis.pdf · Introduction to Power Flow •A power flow study (load-flow study) is a steady-state analysis

Newton-Raphson Comments

• When close to the solution the error decreases quite quickly -- method has what is known as “quadratic” convergence:

– number of correct significant figures roughly doubles at each iteration.

• f(x(v)) is known as the “mismatch,” which we would like to drive to zero.

• Stopping criteria is when f(x(v)) <

Page 30: Power Flow Studies - University of Nevada, Las Vegaseebag/740 Power Flow Analysis.pdf · Introduction to Power Flow •A power flow study (load-flow study) is a steady-state analysis

Newton-Raphson Comments

• Results are dependent upon the initial guess. What if we had guessed x(0) = 0, or x(0) = -1?

• A solution’s region of attraction (ROA) is the set of initial guesses that converge to the particular solution.

• The ROA is often hard to determine.

Page 31: Power Flow Studies - University of Nevada, Las Vegaseebag/740 Power Flow Analysis.pdf · Introduction to Power Flow •A power flow study (load-flow study) is a steady-state analysis

Multi-Variable Newton-Raphson

1 1

2 2

Next we generalize to the case where is an -

dimension vector, and ( ) is an -dimensional

vector function:

( )

( )( )

( )

Again we seek a solution of ( ) 0.

n n

n

n

x f

x f

x f

x

f x

x

xx f x

x

f x

Page 32: Power Flow Studies - University of Nevada, Las Vegaseebag/740 Power Flow Analysis.pdf · Introduction to Power Flow •A power flow study (load-flow study) is a steady-state analysis

Multi-Variable Case, cont’d

i

1 11 1 1 2

1 2

1

1 21 2

The Taylor series expansion is written for each f ( )

( ) ( ) ( ) ( )

( ) higher order terms

( ) ( ) ( ) ( )

( ) higher order terms

nn

n nn n

nn

n

f ff x x f x x x x x

x x

fx x

x

f ff x x f x x x x x

x x

fx x

x

x

Page 33: Power Flow Studies - University of Nevada, Las Vegaseebag/740 Power Flow Analysis.pdf · Introduction to Power Flow •A power flow study (load-flow study) is a steady-state analysis

Multi-Variable Case, cont’d

1 1 1

1 21 1

2 2 22 2

1 2

1 2

This can be written more compactly in matrix form

( ) ( ) ( )

( )

( ) ( ) ( )( )( )

( )

( ) ( ) ( )

n

n

nn n n

n

f f f

x x xf x

f f ff x

x x x

ff f f

x x x

x x x

x

x x xxf x +Δx

x

x x x

higher order terms

nx

Page 34: Power Flow Studies - University of Nevada, Las Vegaseebag/740 Power Flow Analysis.pdf · Introduction to Power Flow •A power flow study (load-flow study) is a steady-state analysis

Jacobian Matrix

1 1 1

1 2

2 2 2

1 2

1 2

The by matrix of partial derivatives is known

as the Jacobian matrix, ( )

( ) ( ) ( )

( ) ( ) ( )( )

( ) ( ) ( )

n

n

n n n

n

n n

f f f

x x x

f f f

x x x

f f f

x x x

J x

x x x

x x xJ x

x x x

Page 35: Power Flow Studies - University of Nevada, Las Vegaseebag/740 Power Flow Analysis.pdf · Introduction to Power Flow •A power flow study (load-flow study) is a steady-state analysis

Multi-Variable N-R Procedure

Derivation of N-R method is similar to the scalar case

( ) ( ) ( ) higher order terms

( ) ( ) ( )

To seek solution to ( ) 0, set linear

approximation equal to zero: 0 ( ) ( ) .

f x x f x J x x

f x x f x J x x

f x x

f x J x x

x1

( 1) ( ) ( )

( 1) ( ) ( ) 1 ( )

( )

( ) ( )

( ) ( )

Iterate until ( )

v v v

v v v v

v

J x f x

x x x

x x J x f x

f x

Page 36: Power Flow Studies - University of Nevada, Las Vegaseebag/740 Power Flow Analysis.pdf · Introduction to Power Flow •A power flow study (load-flow study) is a steady-state analysis

Multi-Variable - Example

1

2

2 21 1 2

2 22 1 2 1 2

1 1

1 2

2 2

1 2

Solve for = such that ( ) 0 where

( ) 2 8

( ) 4

First symbolically determine the Jacobian

( ) ( )

( ) =

( ) ( )

x

x

f x x x

f x x x x x

f fx x

x x

f fx x

x x

x f x

J x

Page 37: Power Flow Studies - University of Nevada, Las Vegaseebag/740 Power Flow Analysis.pdf · Introduction to Power Flow •A power flow study (load-flow study) is a steady-state analysis

Multi-variable Example, cont’d

1 2

1 2 1 2

11 1 2 1

2 1 2 1 2 2

4 2( ) =

2 2

4 2 ( )Then

2 2 ( )

Matlab code: x1=x10; x2=x20;

f1=2*x1^2+x2^2-8;

f2=x1^2-x2^2+x1*x2-4;

J = [4*x1 2*x2; 2*x1+x2 x1-2*x2];

[x1;x2] =

x x

x x x x

x x x f

x x x x x f

J x

x

x

[x1;x2]-inv(J)*[f1;f2].

Page 38: Power Flow Studies - University of Nevada, Las Vegaseebag/740 Power Flow Analysis.pdf · Introduction to Power Flow •A power flow study (load-flow study) is a steady-state analysis

Multi-variable Example, cont’d

(0)

1(1)

1(2)

( )

1Initial guess

1

1 4 2 5 2.1

1 3 1 3 1.3

2.1 8.40 2.60 2.51 1.8284

1.3 5.50 0.50 1.45 1.2122

At each iteration we check ( ) to see if it

x

x

x

f x

(2)

is

0.1556below our specified tolerance : ( )

0.0900

If = 0.2 then done. Otherwise continue iterating.

f x

Page 39: Power Flow Studies - University of Nevada, Las Vegaseebag/740 Power Flow Analysis.pdf · Introduction to Power Flow •A power flow study (load-flow study) is a steady-state analysis

Power Balance Equations

1

1

For convenience, write:

( ) ( cos sin )

( ) ( sin cos )

The power balance equations are then:

( ) 0

( ) 0

n

i i k ik ik ik ikk

n

i i k ik ik ik ikk

i Gi Di

i Gi Di

P V V G B

Q V V G B

P P P

Q Q Q

x

x

x

x

Page 40: Power Flow Studies - University of Nevada, Las Vegaseebag/740 Power Flow Analysis.pdf · Introduction to Power Flow •A power flow study (load-flow study) is a steady-state analysis

Power Flow Variables

2

n

2

Assume the slack bus is the first bus (with a fixed

voltage angle/magnitude). We then need to determine

the voltage angle/magnitude at the other buses.

We must solve ( ) , where:

n

V

V

f x 0

x

2 2 2

2 2 2

( )

( )( )

( )

( )

G D

n Gn Dn

G D

n Gn Dn

P P P

P P P

Q Q Q

Q Q Q

x

xf x

x

x

Page 41: Power Flow Studies - University of Nevada, Las Vegaseebag/740 Power Flow Analysis.pdf · Introduction to Power Flow •A power flow study (load-flow study) is a steady-state analysis

N-R Power Flow Solution

(0)

( )

( 1) ( ) ( ) 1 ( )

The power flow is solved using the same procedure

discussed previously for general equations:

For 0; make an initial guess of ,

While ( ) Do

[ ( )] ( )

1

End

v

v v v v

v

v v

x x

f x

x x J x f x

Page 42: Power Flow Studies - University of Nevada, Las Vegaseebag/740 Power Flow Analysis.pdf · Introduction to Power Flow •A power flow study (load-flow study) is a steady-state analysis

Power Flow Jacobian Matrix

1 1 1

1 2 2 2

2 2 2

1 2 2 2

2 2 2 2 2 2

1 2

The most difficult part of the algorithm is determining

and factorizing the Jacobian matrix, ( )

( ) ( ) ( )

( ) ( ) ( )( )

( ) ( )

n

n

n n n

f f f

x x x

f f f

x x x

f f f

x x x

J x

x x x

x x xJ x

x x2 2

( )n

x

Page 43: Power Flow Studies - University of Nevada, Las Vegaseebag/740 Power Flow Analysis.pdf · Introduction to Power Flow •A power flow study (load-flow study) is a steady-state analysis

Power Flow Jacobian Matrix, cont’d

1

Jacobian elements are calculated by differentiating

each function, ( ), with respect to each variable.

For example, if ( ) is the bus real power equation

( ) ( cos sin )

i

i

n

i i k ik ik ik ik Gik

f x

f x i

f x V V G B P P

1

( ) ( sin cos )

( ) ( sin cos ) ( )

Di

ni

i k ik ik ik iki k

k i

ii j ij ij ij ij

j

fx V V G B

fx V V G B j i

Page 44: Power Flow Studies - University of Nevada, Las Vegaseebag/740 Power Flow Analysis.pdf · Introduction to Power Flow •A power flow study (load-flow study) is a steady-state analysis

Two Bus Newton-Raphson - Example

Line Z = 0.1j

One Two 1.000 pu 1.000 pu

200 MW

100 MVR

0 MW

0 MVR

For the two bus power system shown below, use the Newton-Raphson power flow to determine the voltage magnitude and angle at bus two. Assume that bus one is the slack and SBase = 100 MVA.

2

2

10 10Unkown: , Also,

10 10bus

j j

V j j

x Y

Page 45: Power Flow Studies - University of Nevada, Las Vegaseebag/740 Power Flow Analysis.pdf · Introduction to Power Flow •A power flow study (load-flow study) is a steady-state analysis

Two Bus Example, cont’d

1

1

General power balance equations:

( cos sin ) 0

( sin cos ) 0

For bus two, the power balance equations are

(load real power is 2.0 per unit,

while react

n

i k ik ik ik ik Gi Dik

n

i k ik ik ik ik Gi Dik

V V G B P P

V V G B Q Q

2 1 2

22 1 2 2

ive power is 1.0 per unit):

(10sin ) 2.0 0

( 10cos ) (10) 1.0 0

V V

V V V

Page 46: Power Flow Studies - University of Nevada, Las Vegaseebag/740 Power Flow Analysis.pdf · Introduction to Power Flow •A power flow study (load-flow study) is a steady-state analysis

Two Bus Example, cont’d

2 2 2

22 2 2 2

2 2

2 2

2 2

2 2

2 2 2

2 2 2 2

( ) 2.0 (10sin ) 2.0

( ) 1.0 ( 10cos ) (10) 1.0

Now calculate the power flow Jacobian

( ) ( )

( )

( ) ( )

10 cos 10sin

10 sin 10cos 20

P V

Q V V

P Px x

V

Q Qx x

V

V

V V

x

x

J x

Page 47: Power Flow Studies - University of Nevada, Las Vegaseebag/740 Power Flow Analysis.pdf · Introduction to Power Flow •A power flow study (load-flow study) is a steady-state analysis

Two Bus Example, First Iteration

(0)2(0)

(0)2

(0) (0)2 2

(0)

2(0) (0) (0)2 2 2

(0) (0) (0)2 2 2(0)

(0) (0)2 2

0For 0, guess . Calculate:

1

(10sin ) 2.0 2.0( )

1.0( 10cos ) (10) 1.0

10 cos 10sin( )

10 sin 10cos

vV

V

V V

V

V

x

f x

J x(0) (0)2 2

1(1)

10 0

0 1020

0 10 0 2.0 0.2Solve

1 0 10 1.0 0.9

V

x

Page 48: Power Flow Studies - University of Nevada, Las Vegaseebag/740 Power Flow Analysis.pdf · Introduction to Power Flow •A power flow study (load-flow study) is a steady-state analysis

Two Bus Example, Next Iterations

(1)

2

(1)

1(2)

0.9(10sin( 0.2)) 2.0 0.212( )

0.2790.9( 10cos( 0.2)) 0.9 10 1.0

8.82 1.986( )

1.788 8.199

0.2 8.82 1.986 0.212 0.233

0.9 1.788 8.199 0.279 0.8586

(

f x

J x

x

f(2) (3)

(3)2

0.0145 0.236)

0.0190 0.8554

0.0000906( ) Close enough! 0.8554 13.52

0.0001175V

x x

f x

Page 49: Power Flow Studies - University of Nevada, Las Vegaseebag/740 Power Flow Analysis.pdf · Introduction to Power Flow •A power flow study (load-flow study) is a steady-state analysis

Two Bus Solved Values

Line Z = 0.1j

One Two 1.000 pu 0.855 pu

200 MW

100 MVR

200.0 MW

168.3 MVR

-13.522 Deg

200.0 MW 168.3 MVR

-200.0 MW-100.0 MVR

Once the voltage angle and magnitude at bus 2 are known we can calculate all the other system values, such as the line flows and the generator real and reactive power output

Page 50: Power Flow Studies - University of Nevada, Las Vegaseebag/740 Power Flow Analysis.pdf · Introduction to Power Flow •A power flow study (load-flow study) is a steady-state analysis

Two Bus Case Low Voltage Solution

(0)

(0) (0)2 2

(0)

(0) (0) (02 2 2

This case actually has two solutions! The second

"low voltage" is found by using a low initial guess.

0Set 0, guess . Calculate:

0.25

(10sin ) 2.0

( )

( 10cos )

v

V

V V

x

f x2)

(0) (0) (0)2 2 2(0)

(0) (0) (0) (0)2 2 2 2

2

0.875(10) 1.0

10 cos 10sin 2.5 0( )

0 510 sin 10cos 20

V

V V

J x

Page 51: Power Flow Studies - University of Nevada, Las Vegaseebag/740 Power Flow Analysis.pdf · Introduction to Power Flow •A power flow study (load-flow study) is a steady-state analysis

Low Voltage Solution, cont'd

1(1)

(2) (2) (3)

0 2.5 0 2 0.8Solve

0.25 0 5 0.875 0.075

1.462 1.42 0.921( )

0.534 0.2336 0.220

x

f x x x

Line Z = 0.1j

One Two 1.000 pu 0.261 pu

200 MW

100 MVR

200.0 MW

831.7 MVR

-49.914 Deg

200.0 MW

831.7 MVR

-200.0 MW

-100.0 MVR

Low voltage solution

Page 52: Power Flow Studies - University of Nevada, Las Vegaseebag/740 Power Flow Analysis.pdf · Introduction to Power Flow •A power flow study (load-flow study) is a steady-state analysis

Two Bus Region of Convergence

Graph shows the region of convergence for different initial guesses of bus 2 angle (hor. axis) and magnitude (vertical axis).

Red region converges to the high voltage solution, while the yellow region converges to the low voltage solution

Maximum of 15

iterations

Page 53: Power Flow Studies - University of Nevada, Las Vegaseebag/740 Power Flow Analysis.pdf · Introduction to Power Flow •A power flow study (load-flow study) is a steady-state analysis

Solving Large Power Systems

Most difficult computational task is inverting the Jacobian matrix (or solving the update equation): – factorizing a full matrix is an order n3 operation,

meaning the amount of computation increases with the cube of the size of the problem.

– this amount of computation can be decreased substantially by recognizing that since Ybus is a sparse matrix, the Jacobian is also a sparse matrix.

– using sparse matrix methods results in a computational order of about n1.5.

– this is a substantial savings when solving systems with tens of thousands of buses.

Page 54: Power Flow Studies - University of Nevada, Las Vegaseebag/740 Power Flow Analysis.pdf · Introduction to Power Flow •A power flow study (load-flow study) is a steady-state analysis

Decoupled Power Flow

( ) ( )

( ) ( )( )

( )( ) ( ) ( )

( )2 2 2

( )

( )

Standard form of the Newton-Raphson update:

( )( )

( )

( )

where ( ) .

( )

Note

v v

v vv

vv v v

vD G

v

vn Dn Gn

P P P

P P P

P P

θθ V P xf x

Q xVQ Q

θ V

x

P x

x

that changes in angle and voltage magnitude

both affect (couple to) real and reactive power.

Page 55: Power Flow Studies - University of Nevada, Las Vegaseebag/740 Power Flow Analysis.pdf · Introduction to Power Flow •A power flow study (load-flow study) is a steady-state analysis

Decoupling Approximation

( ) ( )

( )

( ) ( )( )

( ) ( ) ( )

Usually the off-diagonal matrices, and

are small. Therefore we approximate them as zero:

( )( )

( )

Then the update c

v v

v

v vv

v v v

P Q

V θ

P0

θ P xθf x

Q Q xV0

V

1 1( ) ( )( )( ) ( ) ( )

an be decoupled into two separate updates:

( ), ( ).v v

vv v v

P Qθ P x V Q x

θ V

Page 56: Power Flow Studies - University of Nevada, Las Vegaseebag/740 Power Flow Analysis.pdf · Introduction to Power Flow •A power flow study (load-flow study) is a steady-state analysis

Off-diagonal Jacobian Terms

So, angle and real power are coupled closely, and

voltage magnitude and reactive power are coupled cl

Justification for Jacobian approximations:

1. Usually , therefore

2. U

os

su

ely.

ally is s

ij ij

ij

r x G B

mall so sin 0

Therefore

cos sin 0

cos sin 0

ij

ii ij ij ij ij

j

ii j ij ij ij ij

j

V G B

V V G B

P

V

Q

θ 56

<< <<

Page 57: Power Flow Studies - University of Nevada, Las Vegaseebag/740 Power Flow Analysis.pdf · Introduction to Power Flow •A power flow study (load-flow study) is a steady-state analysis

Fast Decoupled Power Flow

• By further approximating the Jacobian we obtain a typically reasonable approximation that is independent of the voltage magnitudes/angles.

• This means the Jacobian need only be built and factorized once.

• This approach is known as the fast decoupled power flow (FDPF)

• FDPF uses the same mismatch equations as standard power flow so it should have same solution if it converges

• The FDPF is widely used, particularly when we only need an approximate solution.

Page 58: Power Flow Studies - University of Nevada, Las Vegaseebag/740 Power Flow Analysis.pdf · Introduction to Power Flow •A power flow study (load-flow study) is a steady-state analysis

FDPF Approximations

( ) 1 ( ) 1 ( )

( ) 1 ( ) 1 ( )

The FDPF makes the following approximations:

1. 0

2. 1 (for some occurrences),

3. sin 0 cos 1

Then: {| | } ( ),

{| | } ( )

Where is just the imaginary pa

ij

i

ij ij

v v v

v v v

G

V

diag

diag

θ B V P x

V B V Q x

B bus bus bus

bus

( )

rt of the ,

except the slack bus row/column are omitted. That is,

is , but with the slack bus row and column deleted.

Sometimes approximate {| | } by identity. v

j

diag

Y G B

B B

V

Page 59: Power Flow Studies - University of Nevada, Las Vegaseebag/740 Power Flow Analysis.pdf · Introduction to Power Flow •A power flow study (load-flow study) is a steady-state analysis

FDPF Three Bus Example

Line Z = j0.07

Line Z = j0.05 Line Z = j0.1

One Two

200 MW

100 MVR

Three 1.000 pu

200 MW

100 MVR

Use the FDPF to solve the following three bus system

34.3 14.3 20

14.3 24.3 10

20 10 30

bus j

Y

Page 60: Power Flow Studies - University of Nevada, Las Vegaseebag/740 Power Flow Analysis.pdf · Introduction to Power Flow •A power flow study (load-flow study) is a steady-state analysis

FDPF Three Bus Example, cont’d

1

(0)(0)2 2

3 3

34.3 14.3 2024.3 10

14.3 24.3 1010 30

20 10 30

0.0477 0.0159

0.0159 0.0389

Iteratively solve, starting with an initial voltage guess

0 1

0 1

bus j

V

V

Y B

B

(1)2

3

0 0.0477 0.0159 2 0.1272

0 0.0159 0.0389 2 0.1091

Page 61: Power Flow Studies - University of Nevada, Las Vegaseebag/740 Power Flow Analysis.pdf · Introduction to Power Flow •A power flow study (load-flow study) is a steady-state analysis

FDPF Three Bus Example, cont’d

(1)

2

3

1

(2)2

3

1 0.0477 0.0159 1 0.9364

1 0.0159 0.0389 1 0.9455

( )( cos sin )

0.1272 0.0477 0.0159

0.1091 0.0159 0.0389

ni Di Gi

k ik ik ik iki ik

V

V

P x P PV G B

V V

(2)

2

3

0.151 0.1361

0.107 0.1156

0.924

0.936

0.1384 0.9224Actual solution:

0.1171 0.9338

V

V

θ V

Page 62: Power Flow Studies - University of Nevada, Las Vegaseebag/740 Power Flow Analysis.pdf · Introduction to Power Flow •A power flow study (load-flow study) is a steady-state analysis

Assignment:

• Verify the solution on Example 6A in your book using PowerWorld software.


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