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
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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
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Importance of Slack Generator Task of slack generator (swing bus)
Fixing of voltage angle Balance of power difference between loads and
infeed
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Fundamentals of load flow calculation
infeeds and loads, buses, branches• description of network topology, i.e. solving load flow calculation
~
Infeed
Load
Node
Branch
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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
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Description of loads
PQ-load: real and reactive power fixed
description by P,Q
P, cos phi
S, cos phi
I, cos phi...
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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
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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
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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)
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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
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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
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
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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
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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
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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
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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
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Newton-Raphson method
Selection of initial values Ui0 and i0
Calculation of mismatches (actual-calculated)
Form linearization of node equations
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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
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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
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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
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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