power flow solution

Power Flow Analysis

In computer application in power system analysis

2

Purpose of Load Flow Calculations

Network planning tasks Determination of equipment loading Identification of weak points Impact of load increase Investigation of peak / low load and generation conditions Voltage control, reactive power compensation Security of supply (n-1 criterion) and reliability

Network operation Loss reduction Investigation of network configurations during maintenance

Initial state for Stability calculations Motor start

3

Results of load flow calculation Load currents

magnitude and angle Equipment loading, overloading

Node voltages magnitude and angle

Powers Active and reactive power balance Active and reactive power of generators Losses

4

Modeling for Load Flow

Modeling mathematically as voltage or power source

Slack bus - voltage (magnitude and angle) fixed, real and reactive power variable

PU-/PV-bus - voltage (magnitude) and real power fixed,

reactive power variable (normal operation mode of generator)

PQ-bus - real and reactive power fixed, voltage (magnitude and angle) variable

5

Importance of Slack Generator Task of slack generator (swing bus)

Fixing of voltage angle Balance of power difference between loads and

infeed

6

Fundamentals of load flow calculation

infeeds and loads, buses, branches• description of network topology, i.e. solving load flow calculation

~

Infeed

Load

Node

Branch

7

Description of infeeds

Slack feed: voltage fixed fixed

P, Q variable1 slack needed in each network to balance powers

PU-feed: voltage fixedreal power fixedQ, variable

PQ-feed: real power fixed reactive power fixedU, variable

8

Description of loads

PQ-load: real and reactive power fixed

description by P,Q

P, cos phi

S, cos phi

I, cos phi...

9

Description of branches

Impedance ZAB=RAB+ jXAB

or

AdmittanceABABAB

AB jXRZY

11

10

Description of network topology

54515451

12454343

343432313231

23232121

151312151312

00

00

0

00

0

YYYY

YYYY

YYYYYY

YYYY

YYYYYY

Y

2 3

1

5 4

~

Infeed

Load

U1 Branch

I12 Y12

I2

k node and i node between

admittance Negative

i node to connected

sadmittance all of Sum

ik

ii

Y

Y

11

Properties of admittance matrix large matrix

elements are complex numbers

sparse (for large networks only few elements non-zero)

diagonal elements positive

non-diagonal elements zero or negative

12

Load flow problem

UYI

node) at currents all of sum (signed

currents node ofmatrix I

voltages node ofmatrix U

matrix admittance Y

non-linear problem for non-impedance loads (typical)

13

Load flow problem

*3 iiii IUjQP

power at nodes

power at nodes, expressed as matrix equation

*** 33 uYuiuqp diagdiagj

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Solving technique

Guauss-Seidel method Newton-Raphson method Fast Decoupled method

15

Gauss-Seidel technique

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

x

f(x)

f1(x)=x f2(x)=exp(-x)Root

Similar to the fixed-point iteration method

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Convergence of Fixed-point iteration

0x 1x 2x

y

x

xy1 )x(gy2

0x1x 2x

y

x

xy1

)x(gy2

0x1x2x

y

x

xy1

)x(gy2

0x1x

y

x

xy1 )x(gy2

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Newton-Raphson method

xxxS1x

S1x

S2x

S2x

L1x

L1x

L2x

L2x

S3x

S3

df(xS2)dxS2

(f x)

i

ii i 1

f(x) 0f'(x)

x x

ii 1 i

i

f(x)x x

f'(x)

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Limits of load flow calculation Iteration boundary

high accuracy ( small) vs. high calculation time

Load model assumption of constant power for PQ-loads only valid near

rated voltage for low voltages load assumption too high -> voltage collapse

Possible reasons for non-convergence load too high (PQ-load instead of Z-load) reactive power problem -> voltage collapse long lines slack bus badly positioned

Steady state solution might not be reachable because of stability problems

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Principle procedure of load flow calculation by iterationStart

Start values for node voltages 0

i

ri UU

Start values for deviations 0 iiU

Adjustment of node voltages

iii

iii UUU

Calculation of node power

*** 33 uYuiuqp diagdiagj

Comparison with allowed divergence

End

Calculation of ΔUi and Δδi

nomi

nomi

QQ

PPno

yes

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Gauss-Seidel method

Calculation continues with the new values of voltage for new iteration

The process is repeated until the difference in voltage between the consecutive iterations is small enough

Converges slowly

21

Gauss-Seidel acceleration factor Correction in voltage is multiplied by the

constant

Selection of the multiplier depends on the network to be analyzed; 1.6 being a common value

22

Newton-Raphson method

f(x) = 0 Initial guess x0

Find x1 such that f(x0 + x1) = 0

Taylor series: f (x0) + f ’(x0)x1 = 0

23

Newton-Raphson method

The process is repeated with the value

x1 = x0 + x1

J is Jacobian matrix

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Newton-Raphson method Power equations for load nodes

Alternative representation of power equations

25

Newton-Raphson method

Initially guess for voltage magnitude and angle

Corresponding Pi and Qi to guessed voltage are calculated

Compare with initial data of P and Q to get mismatch Pi and Qi

Repeat until mismatches are small enough

26

Newton-Raphson method

Selection of initial values Ui0 and i0

Calculation of mismatches (actual-calculated)

Form linearization of node equations

27

Newton-Raphson method

Determine inverse Jacobian matrix and solve the corrections for angles and voltages

Substitute new values to voltages and angles and calculate the new partial derivative matrix

Calculate the new power mismatches If the mismatches > given tolerance, repeat

the process until the tolerance is small enough

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The elements of Jacobian matrix

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Newton-Raphson method – branch flow Power flow in branch is calculated by

Iij = Yij(Vi – Vj)

And Sij = ViI*ij

Loss in branch is calculated by

SL = Sij - Sji

30

Decoupled load flow (DLF)

In a power transmission network, JB and JC can be assumed zero

Therefore, construction of the Jacobian and finding its inverse become easier

31

Fast decoupled load flow

The Jacobian matrix replaced by real constant matrix has to be constructed and inverted only once

These accelerated (approximate) methods nevertheless give accurate results, because the calculated powers are always compared with the real ones

32

Possibilities to reach convergence The following tips that may help to achieve convergence. It

should remembered that changes to the network may have to be reversed again and plausibility of results must be checked.

change PQ-loads to Z-loads (impedance load conversion) change PU-generator to PQ-generator, relax operating limits of

generators set starting points change method of calculation (current iteration, Newton-

Raphson) disconnect long lines divide network in independent sub-networks try different positions of slack depending on network structure insert reactive power (capacitive

or inductive) increase number of iterations and change accuracy

requirements set tap changer to variable setting