METHOD AND SYSTEM OF REAL-TIME ESTIMATION OF TRANSMISSION LINE PARAMETERS IN ON-LINE POWER FLOW CALCULATIONS

METHODE ET SYSTEME D'ESTIMATION EN TEMPS REEL DES PARAMETRES D'UNE LIGNE DE TRANSMISSION DANS DES CALCULS DE DEBIT D'ENERGIE EN LIGNE

English Abstract

A system and method for estimating parameters of transmission lines employing phasor measurement units is provided, wherein measurements are provided from the phasor measurement units relating to a transmission line. These measurements are filtered to remove invalid measurements. Using the remaining valid measurements, resistance, reactance and grounding admittance are calculated and estimated and checked for errors.

French Abstract

Un système et une méthode d'estimation des paramètres des lignes de transmission emploient des modules de mesure de phase, dans lesquels les mesures sont fournies à partir des modules de mesure de phase relativement à une ligne de transmission. Ces mesures sont filtrées pour retirer les mesures non valides. À l'aide des mesures valides restantes, la résistance, la réactance et l'admittance terre sont calculées et estimées, puis validées.

Claims

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IN THE CLAIMS:
1. A method of filtering invalid measurements of voltage phasors, both
magnitude
and angle, and power phasors, both real and reactive powers, at the two buses
of a transmission line, comprising the steps of:
(a) receiving measurements from a first phasor measurement unit at a
sending bus i of a transmission line, and a second phasor measurement
unit at a receiving bus j of said transmission line;
(b) calculating charging reactive powers at said sending bus and said
receiving bus using the measured voltages at said buses and initially
estimated grounding admittance, a reactive power flow excluding charging
reactive power at said sending bus, and a reactive power flow excluding
charging reactive power at said receiving bus, using the said
measurements, and then further calculating a reactive power loss .DELTA.Q of
said line using the said measurements, calculated charging reactive
powers and calculated reactive power flows at the two said buses;
(c) calculating a grounding admittance Y(new) using the following equation:
<IMG>
wherein V i is a measured voltage magnitude at said sending bus, V j is a
measured voltage magnitude at said receiving bus, Q i is a measured
reactive power flow including charging reactive power at said sending bus,
Q ij is a measured reactive power flow including charging reactive power at
said receiving bus, and .DELTA.Q is a power loss of said transmission line
that is
calculated in step (b);
(d) using said grounding admittance Y(new) and measured voltage to
said charging reactive power at said receiving bus, and then further
said reactive power flow Q*ii excluding charging reactive power at said
bus using the measured reactive power and the recalculated charging
power;

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(e) calculating a resistance R ij and a reactance X ij using the following
equation:
<IMG>
wherein P ij is a measured real power flow at said receiving bus, Q* ij is a
reactive power flow that is calculated from the measured reactive flow by
excluding said charging reactive power at said receiving bus, and a and b
are calculated using the following equations:
<IMG>
wherein .theta.ji = .theta. j-.theta. i wherein .theta. i is a measured
voltage angle at said sending
bus and .theta. j is a measured voltage angle at said receiving bus.
(f) calculating an error for each of said resistance R ij, said reactance X
ij and
said grounding admittance Y, which is the difference between any newly
calculated parameter R ij, X ij or Y and any corresponding
parameter R ij, X ij or Y that was calculated at a previous time
interval, and if said error for either said resistance, or said reactance or
said grounding admittance, is larger than a predetermined threshold, the
whole set encompassing the measurements of all voltage phasors and
power phasors is identified as a set of invalid measurements and is
discarded.
2.
The method of claim 1 wherein said measurements from said first and second
measurement units include said voltage magnitude V i at said sending bus, said
magnitude V j at said receiving bus, said voltage angle .theta. i at said
sending bus, said
angle .theta. j at said receiving bus, a line power flow P ij+j Q ij including
a charging

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power at said receiving bus, and a line power flow P i +j Q i including a
charging
power at said sending bus.
3. The method of claim 1 wherein said threshold is determined using a
precision
factor associated with said measurements, an error transfer relationship
factor
between said measurements and said resistance, said reactance and said
grounding admittance, and an estimate of possible small change of said
resistance, said reactance and said grounding admittance in a short time
interval.
4. A system for filtering invalid measurements of voltage phasors, both
magnitude
and angle,and power phasors, both real and reactive powers, at the two buses
of
a transmission line, comprising:
(a) a first phasor measurement unit at a sending bus i of a transmission
line
and a second phasor measurement unit at a receiving bus j of said
transmission line, said first and second phasor measurement units
providing the measurements of voltage phasors and power phasors at the
two buses of said transmission line;
(b) a computer, said computer calculating: charging reactive powers at said
sending bus and said receiving bus using the measured voltages at said
buses and initially estimated grounding admittance, a reactive power flow
excluding charging reactive power at said sending bus, and a reactive
power flow excluding charging reactive power at said receiving bus, using
the said measurements, and then further calculating a reactive power loss
.DELTA.Q of said line using the said measurements, calculated charging
reactive
powers and calculated reactive power flows at the two said buses;
and calculating a grounding admittance Y using the following
equation:
<IMG>
wherein V i is a measured voltage magnitude at said sending bus, V j is a
voltage magnitude at said receiving bus, Q i is a measured reactive power

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including charging reactive power at said sending bus, Q ij is a measured
power flow including charging reactive power at said receiving bus, and
calculated loss of power of said transmission line;
and using said grounding admittance Y(new) and measured voltage to
recalculate the said charging reactive power at said receiving bus, and
then further calculating the said reactive power flow Q* ij excluding charging
reactive power at said receiving bus using the measured reactive power
and the recalculated charging reactive power;
and calculating a resistance R ij and a reactance X ij using the following
equation:
<IMG>
wherein P ij is a measured real power flow at said receiving bus, Q* ij is a
reactive power flow that is calculated from the measured reactive flow by
excluding said charging reactive power at said receiving bus, and a and b
are calculated using the equations:
¨<IMG> ¨ a and
<IMG> =b
wherein .theta.
ji=.theta. j-.theta. i wherein .theta. i is a measured voltage angle at said
sending
bus and .theta. j is a measured voltage angle at said receiving bus;
and further calculating an error for each of said resistance R ij, said
and said grounding admittance Y, which is the difference between any
calculated parameter R ij(new), X ij(new) or Y(new) and any corresponding

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parameter R ij(old), X ij(old) or Y(old) that was calculated at a previous
time
interval, and if said error for either said resistance, or said reactance or
grounding admittance, is larger than a predetermined threshold, the whole
encompassing the measurements of all voltage phasors and power
identified as a set of invalid measurements and is filtered out by said
5. The system of claim 4 wherein said measurements from said first and
second
phasor measurement units include said voltage magnitude V i at said sending
bus, said voltage magnitude V j at said receiving bus, said voltage angle
.theta.i at said
sending bus, said voltage angle .theta.j at said receiving bus, a line power
flow P ij+jQ ij
including a charging reactive power at said receiving bus, and a line power
flow
P i+jQ i including a charging reactive power at said sending bus.
6. The system of claim 4 wherein said computer uses the said threshold that
is
based on a precision factor associated with said measurements, an error
transfer
relationship factor between said measurements and said resistance, said
reactance and said grounding admittance, and an estimate of possible small
change of said resistance, said reactance and said grounding admittance in a
short time interval.
7. A method of estimating line parameters, which include resistance,
reactance and
grounding admittance, using first and second phasor measurement units
associated with a transmission line, comprising the steps of
(a) obtaining a plurality (M) sets of measurements of voltage phasors, both
magnitude and angle, and power phasors, both real and reactive powers,
from said phasor measurement units, in which invalid measurements have
been filtered out;
(b) for each of said sets of measurements, calculating charging reactive
sending bus i and a receiving bus j of said transmission line using the
voltages at said buses and initially estimated grounding admittance, a
power flow excluding charging reactive power at said sending bus, and a
power flow excluding charging reactive power at said receiving bus of said
using the said measurements, and then further calculating a reactive

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.DELTA.Q of said line, using the said measurements, calculated charging
and calculated reactive power flows at the two said buses;
and calculating a grounding admittance Y(new) using the following
equation:
<IMG>
wherein V i is a measured voltage magnitude at said sending bus, V j is a
measured voltage magnitude at said receiving bus, Q i is a measured
reactive power flow including charging reactive power at said sending bus,
Q ij is a measured reactive power flow including charging reactive power at
said receiving bus, and .DELTA.Q is a calculated loss of said transmission
line;
and using said grounding admittance Y(new) and measured voltage to
recalculate the said charging reactive power at said receiving bus, and the
further calculating the said reactive power flow Q*ij excluding charging
reactive power at said receiving bus using the measured reactive power
and the recalculated charging reactive power;
and calculating a resistance R ij and a reactance X ij using the following
equation:
<IMG>
wherein P ij is a measured real power flow at said receiving bus, Q*ij is a
reactive power flow that is calculated from the measured reactive flow by
excluding said charging reactive power at said receiving bus, and a and b
are calculated using the following equations:

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- V~ + V i V j Cos.theta. ij =a
V i V j Sin.theta. ji =b
wherein .theta. ji =.theta. j- .theta.i wherein .theta.i is a measured
voltage angle at said sending
bus and .theta. j is a measured voltage angle at said receiving bus;
(c) estimating a value of grounding admittance of said transmission line,
using
the equation:
<IMG>
wherein each of said Yk values is a previously calculated value of
admittance;
(d) estimating a value of resistance of said line, using the equation:
<IMG>
wherein each of said R ikj values is a previously calculated value of
resistance;
(e) estimating a value of reactance of said line, using the equation:
<IMG>
wherein each of said X ijk values is a previously calculated value of
reactance;
(f) calculating a sample standard deviation of said estimated resistance
and a
standard deviation of said estimated reactance;

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(g) if said sample standard deviation of said estimated resistance or said
sample standard deviation of said estimated reactance is greater than a
predetermined threshold, then re-estimating said resistance and said
reactance using a least squares method.
8. The method of claim 7 wherein said measurements are filtered to remove
unreliable data prior to determining said estimated reactance, said estimated
resistance and said estimated admittance.
9. The method of claim 7 wherein said M is a number of reliable sets of
measurements and should be greater than nine.
10. The method of claim 7 wherein said least squares method is used to
obtain a
least squares solution of said resistance and said reactance using said Y ijk
values, which are contained in Q*ij, said R ikj values and said X ijk
values and the following equations:
R ij + cX ij = d
R ij + eX ij = f
wherein:
<IMG>

- 34 -
<IMG>
11. A system for estimating line parameters, which include resistance,
reactance and
grounding admittance, comprising:
(a) a transmission line;
(b) first and second phasor measurement units associated with said
transmission line;
(c) a computer, said computer receiving a plurality (M) sets of
measurements
of voltage phasors, both magnitude and angle, and power phasors, both
real and reactive powers, from said phasor measurement units, in which
invalid measurements have been filtered out;
and for each of said sets of measurements, calculating charging reactive
powers at a sending bus i and a receiving bus j of said transmission line
using the measured voltages at said buses and initially estimated
grounding admittance, a reactive power flow excluding charging reactive
power at said sending bus, and a reactive power flow excluding charging
reactive power at said receiving bus, using the said measurements, and
then further calculating a reactive power loss .DELTA.Q of said line using the
said
measurements, calculated charging reactive powers and calculated
reactive power flows at the two said buses;
and calculating a grounding admittance Y(new) using the following
equation:
<IMG>
wherein V i is a measured voltage magnitude at said sending bus, V j is a
voltage magnitude at said receiving bus, Q i is a measured reactive power
including charging reactive power at said sending bus, Q ij is a measured
power flow including charging reactive power at said receiving bus, and
calculated loss of said transmission line;

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and using said grounding admittance Y and measured voltage to
recalculate the said charging reactive power at said receiving bus, and
then further calculating the said reactive power flow Q* ij excluding charging
reactive power at said receiving bus using the measured reactive power
and the recalculated charging reactive power;
and calculating a resistance R ij and a reactance X ij using the following
equation:
<IMG>
wherein P ij is a measured real power flow at said receiving bus, Q*ij is a
reactive power flow that is calculated from the measured reactive flow by
excluding said charging reactive power at said receiving bus, and a and b
are calculated using the following equations:
- V~ + V i V j Cos .theta. ij = a and
V i V j Sin.theta. ji = b
wherein .theta. ji=.theta. j .- theta. i and wherein .theta. i is a measured
voltage angle at said
sending bus and .theta.j is a measured voltage angle at said receiving bus;
and estimating a value of grounding admittance of said transmission line,
using the equation:
<IMG>

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wherein each of said Y k values is a previously calculated value of
admittance;
and estimating a value of resistance of said line, using the equation:
<IMG>
wherein each of said R ikj values is a previously calculated value of
resistance;
and estimating a value of reactance of said line, using the equation:
<IMG>
wherein each of said X ijk values is a previously calculated value of
reactance;
and calculating a sample standard deviation of said estimated resistance
and a sample standard deviation of said estimated reactance;
and if said sample standard deviation of said estimated resistance or said
sample standard deviation of said estimated reactance is greater than a
predetermined threshold, then re-estimating said resistance and said
reactance using a least squares method.
12. The system of claim 11 wherein said measurements are filtered to remove
unreliable data prior to determining said estimated reactance, said estimated
resistance and said estimated admittance.
13. The system of claim 11 wherein said M is a number of reliable sets of
measurements and should be greater than nine.

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14. The system of claim 11 wherein said least squares method is used to
obtain a
least squares solution of said resistance and said reactance by said computer
using said Y k values, which are contained in Q*ij, said R ikj values and
said X ijk values and the following equations:
R ij +c X ij = d
R ij + e X ij = f
wherein:
<IMG>

Description

CA 02602888 2007-09-18
METHOD AND SYSTEM OF REAL-TIME ESTIMATION OF TRANSMISSION
LINE PARAMETERS IN ON-LINE POWER FLOW CALCULATIONS
Field of the Invention
This invention relates to methods of estimating transmission line parameters
in power
systems and more particularly to methods of using such estimations in real
time
power flow calculations.
Background of the Invention
High voltage transmission line parameters include resistance, reactance, and
equivalent admittance which represent reactive charging power along the line.
These
parameters are a necessary and important input in power system modeling, power
flow computations, voltage stability assessment, line protection design and
other
applications. In the prior art, the transmission line parameters are
calculated using
theoretically derived formulas based on information of the line's size,
length,
structure and type, etc., which are assumed to be constant during the power
flow
modeling. However, there is a difference between the calculated and actual
parameters. Some prior art methods have been developed to measure the
resistance
and reactance of a line. However, the equivalent admittance representing
reactive
charging power is not measurable. Also, a "snapshot" off-line measurement of
resistance and reactance parameters is not sufficient for on-line and real
time
applications, because in a real live environment, resistance, reactance and
equivalent
admittance of lines vary with environment and weather (such as temperature and
wind
speed). Therefore, the assumption of constant parameters may create
unacceptable
error, particularly when the environment or weather around the line has a
relatively
large change.
US Patent No. 5631569, entitled "Impedance measurement in a high-voltage power
system" discloses a power monitoring instrument for evaluating and displaying
the
source impedance, load impedance, and distribution system impedance. The focus
of
this patent is placed on sources, loads and distribution systems but not on
transmission
line parameters. US Patent No. 5818245, entitled "Impedance measuring",
discloses a
measurement method for impedance of a power system adapted for operation at a
predetermined line frequency. This patent focuses on the effect of frequency
on the
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CA 02602888 2007-09-18
measurement and requires a testing signal of frequency. US Patent No. 6397156,
entitled "Impedance measurement system for power system transmission lines"
discloses an impedance measurement method to improve various protection
functions.
All of the above methods are not designed for an application operating in real
time or
for on-line power flow modeling and calculations, and also cannot provide an
estimation of equivalent admittance representing reactive charging power of a
transmission line.
PMU technology has developed quickly in the utility industry of both developed
and
developing countries in recent years since the basic concept was presented, as
described in A. G. Phadke, "Synchronized Phasor Measurements in Power
Systems",
IEEE Computer Applications in Power, April 1993, pp10-15; M. Zima, M. Larsson,
P.
Korba, C. Rehtanz and G. Anderson, "Design aspects for wide-area monitoring
and
control systems", Proceedings of IEEE, Vol. 93, No. 5, May, 2005, pp 980-996;
"Eastern Interconnection Phasor Project", 2006 IEEE PES Power Systems
Conference
and Exposition, 2006 (PSCE '06), Oct. 29-Nov. 1 2006, pp 336 ¨ 342; and
Xiaorong
Xie, Yaozhong Xin, Jinyu Xiao, Jingto Wu and Yingduo Han, "WAMS applications
in Chinese power systems," IEEE Power & Energy magazine, Vol. 4, No.1,
Jan/Feb,
2006 pp 54-63. The application of PMU is currently limited to phasor
monitoring,
enhancement of system state estimator and protection relays as disclosed in US
Patent
No. 684533, entitled "Protective relay with synchronized phasor measurement
capability for use in electric power systems" and US Patent No. 7069159,
entitled
"Electric power transmission network state estimation".
Summary of the Invention
The system and method according to the invention discloses a method for real
time
estimation of transmission line parameters for on-line power flow modeling or
for
similar calculations in power system applications. The basic features of the
system
and method include:
= The method uses synchronized phasor measurement units (PMU);
= Invalid PMU measurements (false or erroneous information) are filtered
and
only reliable PMU measurements are used, thereby increasing reliability and
accuracy of an estimation;
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CA 02602888 2007-09-18
= The estimation can be performed very quickly and be applied in a real
time
application environment; and
= The method is based on the fact that line parameters change with time and
can
be therefore used to enhance accuracy of state estimation and power flows in
the Energy Management System (EMS) at utility control centers, which are
performed every a few minutes. The method can be also used for other
applications that use power flow information, such as real time identification
of voltage stability.
The method according to the invention is based on the use of synchronized
phasor
measurement. The method according to the invention promotes the application of
PMUs in real time estimation of transmission line parameters and on-line power
flow
calculations.
A method of filtering invalid measurements is provided, including the steps
of: (a)
receiving measurements from a first phasor measurement unit at a sending bus i
of a
transmission line, and a second phasor measurement unit at a receiving bus j
of the
transmission line; (b) calculating charging reactive powers at the sending bus
and the
receiving bus; a reactive power flow excluding charging reactive power at the
sending
bus, a reactive power flow excluding charging reactive power at the receiving
bus,
and a reactive power loss of the line, using the measurements; (c) calculating
a
grounding admittance Y(new) using the following equation:
¨ Qi + AQ
Y(new) = ________
2 2
+ V =
wherein V, is a voltage magnitude at the sending bus, Vi is a voltage
magnitude at the
receiving bus, Q,, is a reactive power flow including charging reactive power
at the
sending bus, and Qj is a reactive power flow including charging reactive power
at the
receiving buses; and AQ is a loss of the transmission line; (d) using said
grounding
admittance Y(new) to recalculate the charging reactive power at the receiving
bus,
and the reactive power flow excluding charging reactive power at the receiving
bus;
(e) calculating a resistance Rii and a reactance X0 using the following
equation:
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al). = + bQ-*.
u
R u
,j== (new) = __________ and
2
(Qi*j )2
aQ= = ¨ bP= =
1.1 1.1
X 1.1== (new) =
2 * 2
Pij (Qij
wherein Pii is a real power flow at the receiving bus, Q*ii is a reactive
power flow
excluding the charging reactive power at the receiving bus; and a and b are
calculated
using the following equations:
+VV.Cos =a
1 I
V V Sin0 =b
wherein Oii=0J-0; wherein Oi is a voltage angle at the sending bus and Oi is a
voltage
angle at the receiving bus.
The method may further include calculating an error for each of said
resistance, said
reactance and said grounding admittance, and if said error for either said
resistance, or
said reactance or said grounding admittance, is larger than a predetermined
threshold,
discarding said measurements. Also, the measurements from the first and second
phasor measurement units may include the voltage magnitude V; at the sending
bus,
the voltage magnitude Vi at the receiving bus, the voltage angle 0, at the
sending bus
and the voltage angle Oi at the receiving bus; a line power flow Pii+j%
including a
charging reactive power at the receiving bus; and a line power flow Pi+jQ,
including a
charging reactive power at the sending bus.
The threshold may be determined using a precision factor associated with the
measurements, an error transfer relationship factor between the measurements
and the
resistance, the reactance and the grounding admittance, and an estimate of
possible
small change of the resistance, the reactance and the grounding admittance in
a short
time interval.
A system for filtering invalid measurements is provided, including a first
phasor
measurement unit at a sending bus i of a transmission line and a second phasor
measurement unit at a receiving bus j of the transmission line, the first and
second
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phasor measurement units providing measurements associated with the
transmission
line; a computer calculating: charging reactive powers at the sending bus and
the
receiving bus; a reactive power flow excluding charging reactive power at the
sending
bus, a reactive power flow excluding charging reactive power at the receiving
bus,
and a reactive power loss of the line, using the measurements; and calculating
a
grounding admittance Y(new) using the following equation:
Y(new) ¨ Q11 Qi AQ
2 2
V. + V =
wherein V, is a voltage magnitude at the sending bus, Vj is a voltage
magnitude at the
receiving bus, Q,, is a reactive power flow including charging reactive power
at the
sending bus, and Qkk is a reactive power flow including charging reactive
power at the
receiving bus; and AQ is a loss of power of the transmission line; and using
the
grounding admittance Y(new) to recalculate the charging reactive power at the
receiving bus, and the reactive power flow excluding charging reactive power
at the
receiving bus; and calculating a resistance Ro and a reactance Xo using the
following
equation:
aP= +bQ-
1,1 Y
Ry- (new) =
2'2
and
+(Q' )
Y
aQ-*- ¨bP=
X u
n==( ew) = ______
2
ij (Qi*j )2
P
wherein Po is a real power flow at the receiving bus, Q*ki is a reactive power
flow
excluding the charging reactive power at the receiving bus; and a and b are
calculated
using the equations:
¨ 17,2 + V, VI Cos 0 =a and
VV/ Sin0
wherein Offl=01-0, wherein 0, is a voltage angle at the sending bus and 01 is
a voltage
angle at the receiving bus.
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The computer may further calculate an error for each of the resistance, the
reactance
and the grounding admittance, and if the error for either the resistance, or
the
reactance or the grounding admittance is larger than a predetermined
threshold, the
computer discards the measurements. The measurements from the first and second
phasor measurement units may include the voltage magnitude V; at the sending
bus,
the voltage magnitude V., at the receiving bus, the voltage angle 0; at the
sending bus
and the voltage angle 0J at the receiving bus; a line power flow Pu+jQJ
including a
charging reactive power at the receiving bus; and a line power flow P,+jQ,,
including a
charging reactive power at the sending bus. The computer may further calculate
the
threshold using a precision factor associated with the measurements, an error
transfer
relationship factor between the measurements and the resistance, the reactance
and the
grounding admittance, and an estimate of possible small change of the
resistance, the
reactance and the grounding admittance in a short time interval.
A method of estimating line parameters using first and second phasor
measurement
units associated with a transmission line is provided, including the steps:
(a) obtaining
a plurality (M) sets of measurements from the phasor measurement units; (b)
for each
of the sets of measurements, calculating charging reactive powers at a sending
bus i
and a receiving bus j of the transmission line; a reactive power flow
excluding
charging reactive power at the sending bus, a reactive power flow excluding
charging
reactive power at the receiving bus, and a reactive power loss of the line,
using the
measurements; and calculating a grounding admittance Y(new) using the
following
equation:
Qt../ ¨ + AQ
Y(new) ¨ ________
2 2
+ V.1 -
wherein V, is a voltage magnitude at the sending bus, VJ is a voltage
magnitude at the
receiving bus, Q,, is a reactive power flow including charging reactive power
at the
sending bus, and Qu is a reactive power flow including charging reactive power
at the
receiving bus; and AQ is a loss of power of the transmission line; and using
the
grounding admittance Y(new) to recalculate the charging reactive power at the
receiving bus, and the reactive power flow excluding charging reactive power
at the
receiving bus; and calculating a resistance Ru and a reactance X,J using the
following
equation:
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aPij + bQij
R( new) = ____ 1./* and
+ (Q..)2
aQ= = ¨bPj
X 1).= (new) = ___
2 * 2
Pij (Qij
wherein Pij is a real power flow at the receiving bus, Q*ij is a reactive
power flow
excluding the charging reactive power at the receiving bus; and a and b are
calculated
using the following equations:
¨1/./2 +Vy,CosOli = a
V,V,SinO =b
wherein 0j,---A-0, wherein Oi is a voltage angle at the sending bus and Oj is
a voltage
angle at the receiving bus;
(c) estimating a value of grounding admittance of the transmission line, using
the
equation:
E Yk (new)
Y(estim) k=1
wherein each of the Yk(new) values is a previously calculated value of
admittance;
(d) estimating a value of resistance of the line, using the equation:
E Rift (new)
R . = (estim) = k=1 __
wherein each of said R1(new) values is a previously calculated value of
resistance;
(e) estimating a value of reactance of the line, using the equation:
E X (new)
Xl./..(estim)= k=1 __
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=
wherein each of the Xuk(new) values is a previously calculated value of
reactance; (0
calculating a sample standard deviation of the estimated resistance and a
sample
standard deviation of the estimated reactance; and (g) if the sample standard
deviation
of the estimated resistance or the sample standard deviation of the estimated
reactance
is greater than a predetermined threshold, then the computer re-estimating the
resistance and the reactance using a least square method.
The measurements may be filtered to remove unreliable data prior to
determining the
estimated reactance, the estimated resistance and the estimated admittance.
The
number of reliable sets of measurements may be greater than nine. The least
square
method used to obtain a least squares solution of the resistance and the
reactance
using the Yk(new) values, the Rik(new) values and the Xijk(new) values may be
the
following equations:
Rti + =d
e== +X =f
wherein:
*
lj
C
PJ
V o-V2- + .v.c se..
= ____________________________
13-
g./
A
e = and
QVVSinO
ij
f= _____________________
A system for estimating line parameters is provided, including: a transmission
line;
first and second phasor measurement units associated with the transmission
line; and a
computer receiving a plurality (M) sets of measurements from the phasor
measurement units; and calculating charging reactive powers at a sending bus i
and a
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CA 02602888 2007-09-18
receiving bus j of the transmission line; a reactive power flow excluding
charging
reactive power at the sending bus, a reactive power flow excluding charging
reactive
power at the receiving bus, and a reactive power loss of the line, using the
measurements; and calculating a grounding admittance Y(new) using the
following
equation:
Qu ¨ Qi + AQ
Y(new) = _________
2 2
V= + V =
wherein Vi is a voltage magnitude at the sending bus, Vj is a voltage
magnitude at the
receiving bus, Qi, is a reactive power flow including charging reactive power
at the
sending bus, and Qii is a reactive power flow including charging reactive
power at the
receiving bus; and AQ is a loss of power of the transmission line; and using
the
grounding admittance Y(new) to recalculate the charging reactive power at the
receiving bus, and the reactive power flow excluding charging reactive power
at the
receiving bus; and calculating a resistance Rij and a reactance X0 using the
following
equations:
aP1 -- +bQ-*-
1J
RIf== (new) =
2.1and
P,+ (Qii* )2
aQi*; ¨bPij
X lj== (new) = ___
2 * 2
Pij +(Q,,)
wherein 13,i is a real power flow at the receiving bus, Q*ii is a reactive
power flow
excluding the charging reactive power at the receiving bus; and a and b are
calculated
using the following equations:
Vj?+V,VjCosO =a
and
VV Sint9it =b
wherein 0j,=0J-0, and wherein Oi is a voltage angle at the sending bus and Eki
is a
voltage angle at the receiving bus; and estimating a value of admittance of
the
transmission line, using the equation:
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CA 02602888 2007-09-18
E Yk(new)
Y(estim)= k=1
wherein each of said Yk(new) values is a previously calculated value of
admittance;
and estimating a value of resistance of the line, using the equation:
E Rift, (new)
Ry e-( stim) = k=1
wherein each of said R,ki(new) values is a previously calculated value of
resistance;
and estimating a value of reactance of the line, using the equation:
E Xi.* (new)
eX ( stim) = k=1
wherein each of the Xuk(new) values is a previously calculated value of
reactance; and
calculating a sample standard deviation of the estimated resistance and a
sample
standard deviation of the estimated reactance; and if the sample standard
deviation of
the estimated resistance or the sample standard deviation of the estimated
reactance is
greater than a predetermined threshold, then the computer re-estimating the
resistance
and the reactance using a least square method.
The measurements may be filtered to remove unreliable data prior to
determining the
estimated reactance, the estimated resistance and the estimated admittance.
The
number of reliable sets of measurements may be greater than nine. The least
square
method used to obtain a least squares solution of said resistance and said
reactance by
said computer using the Yk(new) values, the Rikj(new) values and the Xuk(new)
values
may use the following equations:
Rjj+cXij =d
R= + eX = f
wherein:
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CA 02602888 2012-08-02
c = __
P. =
¨ oV2 V.V.0 seij
= ______________
¨Põ.=
e= ___ and
Qij
=ViV=Sno,,
f = _______
Qi;
Brief Description of the Figures
Figure 1 is a one-line diagram of a typical transmission line i-j; and
Figure 2 is a representation of a portion of the British Columbia Transmission
Corporation system.
Description of the Invention
PMU devices are installed at two sides of a transmission line of which its
parameters
are to be estimated. The estimation includes two tasks:
(1)A measurement from a PMU may be invalid data. False data that is caused by
failure or malfunction of PMUs or communication channels may or may not be
recognized using features of PMU measurements. In particular, some errors that
are only associated with accuracy of measurements cannot be identified by the
PMU itself Fortunately, the measured voltage and power flow phasors of a line
must satisfy the relationship of line flow equation. This fact enables users
to
identify and filter out invalid measurements.
(2) The line (branch) parameters (i.e. resistance, reactance of lines and
admittance
representing reactive charging power) cannot be directly measured by a PMU.
These parameters vary with environment and weather (such as temperature)
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CA 02602888 2007-09-18
(2) The line (branch) parameters (i.e. resistance, reactance of lines and
admittance
representing reactive charging power) cannot be directly measured by a PMU.
These parameters vary with environment and weather (such as temperature)
conditions. Therefore it is necessary to perform a real time continuous
estimation
of line parameters in on-line power flow calculations.
In the following discussion, the n equivalence of a line shown in Fig. 1 is
used to
explain the method. Generally, this equivalence is sufficient. It is not
difficult to
extend the concept to a multiple IT equivalence circuit in the method if it is
thought
necessary in actual applications. Note that in this document, the unit of all
quantities
is in per unit system, all quantities related to real or reactive power refer
to the total in
three phases and voltage quantities to the line voltage.
As seen in Figure 1, RifFjXii is the line impedance. Y represents half of the
grounding
admittance corresponding to charging reactive power of the line. vizei and
vi Lei are the voltage phasors at the sending and receiving buses. Pri-ja and
PrFjQ*
are the line power flows respectively before and after the charging reactive
power at
the sending bus i. Pu-FjC,/ and Pii+jaj are the line power flows before and
after the
charging reactive power at the receiving bus j. ao and Qio represent the
charging
reactive powers at the sending and receiving ends respectively. In the real
application,
only Põ a, P,1 and Q, are measurable through PMUs whereas Q*, and Q*,/ can be
calculated using Qi and Qo and the charging reactive power. Note that the
initial
measurements are voltage and current phasors but these can be easily converted
to
line power flows. The charging reactive power occurs along the line but the
total
charging reactive power can be calculated by the difference between a and Qu
minus
the reactive losses on the line.
The estimation of line parameters R,J, X,1 and Y for all lines whose
parameters need to
be estimated is performed at given time intervals (such as every 2-5 minutes
or
shorter). The PMUs can provide synchronized phasor data at a rate of about 10-
30
samples per second or faster and therefore there is considerable sampling data
available in the given interval. Note that the rate of waveform sampling can
be up to
3000 or more samples per second. While the parameters of Rv, Xo and Y may vary
with the environment and weather conditions around the line in a relatively
long
period (such as more than half an hour), however, unlike the measurements of
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CA 02602888 2007-09-18
voltages and line (branch) power flows, the parameters are sufficiently stable
(constant or a small change) in short intervals (for example a couple of
minutes).
Therefore, the parameters should be re-estimated at the given interval in a
real time
manner, whereas their stability in a very short time is used to filter invalid
measurements.
Filtering invalid measurements
A number of sets of sampling data (measurements) are taken in the given
interval.
For each set of measurements, the following data filtering process is
performed:
1. The Rd, Xd and Y from the last estimation are used as a reference. Initial
estimates of the parameters may be obtained using the prior art "one-snapshot"
method.
2. The charging reactive powers are calculated by:
Qio = vi2 Y (1)
2 v
Qio , vi (2)
3. The equivalent reactive power flows on the line within the points A and B
are
calculated by:
= Qi Qio (3)
Qi; = - Q jo (4)
4. The reactive loss on the line is estimated by:
AQI = X (Pi2 +(Q))
(5)
Vi2
AQ2 = X ij (Po + (Q ;kJ )2 )
2 (6)
V =
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CA 02602888 2007-09-18
AQ = AQI + AQ2
(7)
2
The reactive loss is estimated from the two buses respectively and Equation
(7)
provides the average estimation from the two buses.
5. The parameter Y is updated using the measured reactive power flows at the
two buses and the estimated line loss by:
Qu - Qi + AQ
Y(new) = ______ (8)
2 2
+ V =
A threshold for filtering accuracy is specified. The threshold is based on the
precision of PMU measurements, error transfer relationship between the
measurements and Y, and possible small change of Y in the given short
interval, which can be determined through testing and pre-estimation. For
example, if 5% is used as the threshold, when Y(new) is larger than 1.05x
Y(old)
or smaller than 0.95x Y(old) where Y(old) refers to the value of Y in the last
estimation, the whole set of measurements (V, 0, Vp Ofi 130 Q0 P and Qu) may
be viewed as unreliable data and abandoned.
6. The equivalent charging reactive power at the receiving bus is updated by:
Q Jo (new) =11 Y(new) (9)
7. The line reactive power on the line at the receiving end is updated by:
Qij ¨ Qio (new) (10)
8. The parameters Ru and Xij are estimated using the following line power flow
equation based method.
The line power flow equation of Pii+jQ*,) can be expressed as:
(1// + Lei -v/Lei e
+ jQi; iLO _____________ (11)
jX
wherein the symbol o denotes the conjugate operation.
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CA 02602888 2007-09-18
Separating Equation (11) into the real and imaginary parts yields:
RPij X ij = 17 2 V T7T7iVi L,OS Uji = a (12)
bR-Q== ¨ X -P= =V=V =Sin -= == (13)
ti u
wherein 0i,=0J-19,.
It can be derived from Equations (12) and (13) that:
aP1.1.= +bQlj==
n-=( ew)= 2 * (14)
2
Pij (QU
¨bPii
n== ( ew) =
2(15)
P+ (Q)2
Similarly, a threshold for filtering accuracy is specified. The threshold is
based on the precision of PMU measurements, error transfer relationship
between the measurements and Ru or X,J, and possible small change of Ru or
X,j in the given short interval, which can be determined through testing and
pre-estimation. For example, if 5% is used as the threshold, when either
R(new) is larger than 1.05xRu(old) or smaller than 0.95x Rii(old), or X(new)
is larger than 1.05xXu(old) or smaller than 0.95x Xu(old), this whole set of
measurements (Vb 0õ Vi, 9j, P, Qb Pu and Qu) may be viewed as unreliable
data and abandoned.
If the number of reliable sets of measurement are smaller than a specified
threshold
(such as 10), more sampling data should be used until the specified threshold
is met.
If in a case, all sets of sampling data for a line in the given interval are
filtered out as
invalid data, a warning message should be sent to operators. Consecutive
warning
messages indicate that the PMU devices for that particular line may be in an
abnormal
situation.
Estimating R1, Xj and Y
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CA 02602888 2007-09-18
Each of the estimated parameters in the above process is based on individual
sampling data at a time point, and is used for the purpose of filtering
invalid data.
The parameters should be re-estimated using a group of reliable sets of
sampling data
to minimize errors. It is assumed that M reliable sets of measurements are
obtained
after the filtering process.
The parameter Y is re-estimated by the average of the M estimated Y values
obtained
using the M reliable sets of measurements in the filtering process:
EYk (new)
Y(estim) = k=1 ____________________________ (16)
wherein Yk(new) is the value obtained using Equation (8) corresponding to the
kth
reliable set of measurements after filtering.
The parameters Rif and Xij are also re-estimated using the average of the M
estimated
Rif or X values, as applicable, obtained using the M reliable sets of
measurements in
the filtering process:
R( new)
Rli (estim) = k=1 (17)
I X (new)
(estim)= k=1 (18)
wherein Riik(new) and Xiik(new) are, respectively, the values obtained using
Equations
(14) and (15) corresponding to the kth reliable set of measurements after
filtering.
The standard deviations of Ry(estim) and Xigestim) are calculated using the
following
equations:
E [R new - R-y )1
estim 2
(sd) =11k=1
M-1 (19)
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CA 02602888 2007-09-18
E [X iik (new) - X u (estim)] 2
X == (sd)
1.1 k=1
M -1 (20)
If either Ri(sd)/Ry(estim) or Xe(sd)/Vestim) is larger than a threshold
(expressed as a
percentage), the estimated Rfi and xi; obtained using Equations (17) and (18)
are
abandoned and the parameters Rif and X are re-estimated using the following
method.
This threshold is generally selected as half of the threshold for filtering
accuracy (see
step 8 above).
Equations (12) and (13) are re-written as:
Rif +cXii= =d (21)
R = + eX = = = f
ij (22)
wherein:
Qi;
c = ¨ (23)
P= =
¨V2 = + ViVc s = 0 =
"
= _________________________________________ (24)
P/./= =
¨P
e _________________________________________ (25)
V=V -SinGi=
f= ________________________________________ (26)
Qi*j
Applying the least square method to Equation (21) with the M sets of reliable
measurements, results in:
(estim)= (estim) (27)
S-,
=X e= ( stim) =
(28)
Scc
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CA 02602888 2007-09-18
wherein:
E dk
k=1 (29)
Eck
= k=1 (30)
Sec/ = E(ck --E)(dk (31)
k=1
Scc = E (ck ¨E)2 (32)
k=i
Similarly, applying the least square method to Equation (22) with the M sets
of
reliable measurements results in:
R,12(estim) = jeXij2 (estim) (33)
Wef
Xii2(estim) = (34)
Wee
wherein:
/fk
= 1k=1 (35)
Eek
= k =1 (36)
Wef = /(ek e)(fk (37)
k=1
Wee = 1(ek ¨F) 2
(38)
k= 1
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CA 02602888 2007-09-18
The subscript k indicates the value corresponding to the kth reliable set of
measurements after filtering.
The Ry and Xu are estimated using:
(estim)+ Rio(estim)
Rij (estim) = _________________ 2 (39)
X (estim)+ X ii2(estim)
X ii(estim) ¨ __________________ 2 (40)
In a high voltage transmission system, Rif is much smaller than X,J, and Pu is
generally
much larger than Q*,i. It is possible that in numerical calculations, Equation
(21) is
more accurate than Equation (22) for estimation of Rif and Equation (22) is
more
accurate than Equation (21) for estimation of X1. An alternative approach in
an actual
application is to use both Equations (21) and (22) first as described above.
Then, if
the difference between Rui(estim) and Ru2(estim), or between Xu/(estim) and
Xu2(estim), exceeds a threshold (in a relative percentage), only Rui(estim)
and
Xu2(estim) are used as the final estimates.
The derivation above is based on the fact that three phases in a transmission
system
are symmetrical and therefore a single phase model is used in power flow
calculation
modeling. Similar to SCADA measurements, PMU devices provides separate
measurements of phases A, B and C, which may have slight differences among
them.
The total real and reactive power flows of the three phases can be obtained by
summing up the power flows that are calculated from measured voltage and
current
phasors of three individual phases. For voltage phasors, which are required in
the
calculations, the following two approaches can be used:
(1) The average of the measured voltage magnitudes or angles of phases A, B
and C
is used or the measured voltage magnitude and angle of one selected phase with
the best measurement precision (such as phase A) is used. This is the
traditional
method used in the existing EMS.
(2) The voltage phasors of phases A, B and C and the total three phase power
flows
are used to estimate three sets of line parameters. The final parameter
estimate is
the average of the three estimates using the voltage phasors of phases A, B
and C.
DM_VAN/260254-00106/6737484.1 19

CA 02602888 2012-08-02
Simulation results
The method according to the invention was tested using system power flow
studies.
Voltages (magnitudes and angles) and line power flows (real and reactive power
flows) obtained from a number of power flow calculations were viewed as
"measurements". The tests were conducted on IEEE test systems and the utility
system operated and planned by the British Columbia Transmission Corporation
in
Canada. In some cases, errors were intentionally introduced to the voltage
(either
magnitude or angle) or line current (either magnitude or angle). The error of
a voltage
measurement impacts both voltage itself and line power flow, and the error of
a
current measurement only impacts the line power flow. The results showed that
if no
error is introduced, the estimated parameters were the same as those specified
in the
power flow calculations. In cases where some errors are introduced, the
"measurements" with relatively large errors are filtered out and the estimated
parameters, with a few unfiltered small measurement errors, are still the same
as those
specified in the power flow calculations.
Two examples are given below to demonstrate the feasibility and effectiveness
of the
presented method.
A. IEEE 118 bus system
The IEEE 118 bus system is one of the test systems developed by IEEE PES for
various testing purposes The data and original single-line diagram of this
system is
available at the web site http://www.ee.washington.edu/researchipstca/, and
which is
hereby incorporated by reference. The system has 118 buses, 177 lines and 9
transformer branches. Bus voltage phasors and line power flows obtained via a
considerable number of power flows were used as measurements to estimate the
line
parameters. Table 1 presents 30 sets of sampling measurements (voltage phasors
and
line power flows) of the line between Bus 42 and 49. Table 2 shows the
estimates of
resistance, reactance and grounding admittance parameters of the line obtained
using
each set of the measurements without any error introduced. Table 3 shows the
first
eight (8) measurements with intentionally introduced errors, and Table 4 shows
the
estimated parameters obtained using each of the eight "polluted" measurements.
Five
percent (5%) was used as the threshold for filtering accuracy. It is seen that
four sets
of measurements with relatively large errors for parameter estimation (two for
DM VAN 240150-00071/8370710 1 20

CA 02602888 2007-09-18
resistance and two for grounding admittance) were filtered out. Table 5 shows
the
original and final estimated parameters of the line. It is observable that the
estimated
parameters, with other four acceptable measurement errors, are still the same
as those
without measurement error and the original parameters.
Table 1 Voltage phasors and power flows of line between bus 42 and 49
(Used as measurements)
V, V-0, V
P, Qi PgQij
(p.u.) (degree) (p.u.) (degree) (p.u.) (p.u.) (p.u.) (p.
u. )
1.02500 -9.16450 0.92095 -21.60700 0.68445 0.20213 0.64841 0.12095
1.02500 -9.18850 0.92048 -21.66400 0.68607 0.20361 0.64983 0.12149
1.02500 -9.21260 0.92000 -21.72100 0.68769 0.20510 0.65125 0.12202
1.02500 -9.23670 0.91952 -21.77700 0.68932 0.20659 0.65267 0.12256
1.02500 -9.26080 0.91903 -21.83400 0.69094 0.20809 0.65409 0.12309
1.02500 -9.28500 0.91855 -21.89100 0.69257 0.20959 0.65551 0.12363
1.02500 -9.30920 0.91807 -21.94800 0.69420 0.21109 0.65693 0.12416
1.02500 -9.33340 0.91758 -22.00500 0.69583 0.21260 0.65836 0.12470
1.02500 -9.35770 0.91709 -22.06300 0.69746 0.21411 0.65978 0.12524
1.02500 -9.38200 0.91661 -22.12000 0.69909 0.21563 0.66120 0.12577
1.02500 -9.40640 0.91612 -22.17800 0.70073 0.21716 0.66263 0.12631
1.02500 -9.43080 0.91563 -22.23500 0.70236 0.21868 0.66405 0.12685
1.02500 -9.45520 0.91513 -22.29300 0.70400 0.22022 0.66548 0.12738
1.02500 -9.48030 0.91462 -22.35300 0.70571 0.22183 0.66697 0.12795
1.02500 -9.50480 0.91412 -22.41100 0.70735 0.22337 0.66839 0.12849
1.02500 -9.52940 0.91363 -22.46900 0.70899 0.22492 0.66982 0.12902
1.02500 -9.55390 0.91313 -22.52700 0.71064 0.22647 0.67125 0.12956
1.02500 -9.57850 0.91263 -22.58500 0.71229 0.22803 0.67268 0.13010
1.02500 -9.60320 0.91213 -22.64300 0.71393 0.22959 0.67411 0.13063
1.02500 -9.62790 0.91163 -22.70200 0.71558 0.23116 0.67554 0.13117
1.02500 -9.65260 0.91113 -22.76000 0.71724 0.23273 0.67697 0.13171
1.02500 -9.67730 0.91063 -22.81900 0.71889 0.23431 0.67840 0.13225
1.02500 -9.70210 0.91013 -22.87800 0.72054 0.23589 0.67983 0.13278
1.02500 -9.72690 0.90962 -22.93700 0.72220 0.23747 0.68127 0.13332
1.02500 -9.75180 0.90911 -22.99600 0.72386 0.23907 0.68270 0.13386
1.02500 -9.77670 0.90861 -23.05500 0.72552 0.24066 0.68414 0.13439
1.02500 -9.80160 0.90810 -23.11400 0.72718 0.24226 0.68557 0.13493
1.02500 -9.82660 0.90759 -23.17300 0.72884 0.24387 0.68701 0.13547
1.02500 -9.85160 0.90707 -23.23300 0.73051 0.24548 0.68844 0.13601
1.02500 -9.87670 0.90656 -23.29200 0.73218 0.24710 0.68988 0.13655
Note: vi, VA, V, and V-01 are voltage magnitudes and angles at the two buses
of the line.
Põ Q,Põ and Qõ are real and reactive line power flows at the two ends of the
line.
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CA 02602888 2007-09-18
Table 2 Estimated parameters using each set of measurements
(No error introduced into measurements)
Resistance 12.8 (p.u.) Reactance X, (p.u.) Grounding admittance Y (p.u.)
7.150819053923806E-002 0.322989968322601 4.300220526445065E-002
7.149330580993267E-002 0.323002038639029 4.300189766289475E-002
7.149046946189627E-002 0.323009052958054 4.299980527569113E-002
7.149658150866126E-002 0.322991396789738 4.300055510092178E-002
7.150472406138476E-002 0.322994653692808 4.299844120684803E-002
7.150059309021398E-002 0.322996113504715 4.299835133317951E-002
7.149665765565506E-002 0.322996425440579 4.299946261069169E-002
7.150359595814752E-002 0.322989108633325 4.300074690170377E-002
7.150254278137581E-002 0.323006488974226 4.300247211567151E-002
7.149489823420041E-002 0.323001470408724 4.299968940946629E-002
7.149109097810898E-002 0.323009477800442 4.299772505322701E-002
7.149717020207620E-002 0.322998254754971 4.299993083454870E-002
7.150410198175207E-002 0.323002420925849 4.299814438170383E-002
7.149390637544904E-002 0.323005200606151 4.300047376549218E-002
7.150129364038653E-002 0.323009639489799 4.300105320744722E-002
7.149477925376078E-002 0.323007578406282 4.299851336355507E-002
7.150033822538918E-002 0.323004930334097 4.300097996821717E-002
7.150467579167154E-002 0.322999056323605 4.300044095865730E-002
7.150930537175613E-002 0.322989863806196 4.300059970852191E-002
7.150400649437677E-002 0.323001968416419 4.299837632850823E-002
7.150718318722206E-002 0.322990614500469 4.300030297576305E-002
7.149952006321696E-002 0.323000498258261 4.299752789907323E-002
7.149265451649425E-002 0.323006951425721 4.299663995904027E-002
7.149865954872979E-002 0.323005662316445 4.300231080932703E-002
7.150161838879489E-002 0.323006100812629 4.299821434436004E-002
7.149388892505171E-002 0.323002116530079 4.300135676163106E-002
7.149668413889657E-002 0.323000221611103 4.300111705480244E-002
7.149830074128569E-002 0.322990356245569 4.299994611080062E-002
7.150330688094943E-002 0.323004133163954 4.300314836950893E-002
7.150367321102724E-002 0.322989930110879 4.300335400354474E-002
Table 3 First 8 sets of measurements with introduced errors
V, V; P,= Q, P,i Q,1
(p.u.) (degree) (p.u.) (degree) (p.u.) (p.u.) (p.u.)
(p.u.) Error source
*1.03525 -9.16450 0.92095 -21.60700 *0.69129 *0.20415 0.64841 0.12095 1% error
on V;
1.02500 *-9.37230 0.92048 -21.66400 *0.68672'*0.20141 0.64983 0.12149 2% error
on V-8,
1.02500 -9.21260 *0.93840 -21.72100 0.68769 0.20510 *0.66428 *0.12446 2% error
on Vi
1.02500 -9.23670 0.91952 *-21.99500 0.68932 0.20659 *0.65313 *0.12008 1% error
on V-
1.02500 -9.26080 0.91903 -21.83400 *0.69785 *0.21017 0.65409 0.12309 1% error
on I,
1.02500 -9.28500 0.91855 -21.89100 "0.69063 *0.21590 0.65551 0.12363 2% error
on 1-ei
1.02500 -9.30920 0.91807 -21.94800 0.69420 0.21109 *0.67007 0.12665 2 /0 error
on I;
1.02500 -9.33340 0.91758 -22.00500 0.69583 0.21260 *0.65764 0.12846 1% error
on
Note: 1)1,, I-0õ Ij and I-0, are current magnitudes and angles at the two
buses of the line. V, , V-0õ V, and V-01 are
voltage magnitudes and angles at the two buses of the line. Põ Q,,Põ and Q,
are real and reactive line power
flows at the two ends of the line.
2) The measurements with errors are prefixed by *.
3) An error in the measurement of voltage magnitude or angle has impacts not
only on itself but also on real
and reactive line powers whereas an error in the measurement of current
magnitude or angle has impacts
only on real and reactive line powers.
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Table 4 Estimated parameters using each of the first 8 sets of measurements
with errors
(Errors introduced on voltage or current phasors at two buses)
Resistance R, (p.u.) Reactance X,, (p.u.) Grounding
admittance Y (p.u.) Error source Note
8.442778000797151E-002 0.327976565022061 4.149395128849082E-002 1% error on
V, filtered
7.354215645412961E-002 0.318611262005352 4.414604437753773E-002 2% error on
V-81 unfiltered
4.650541001052875E-002 0.319639262614001 4.349025118816752E-002 2% error on
V, filtered
7.030523456060858E-002 0.327814339269931 4.170960516485524E-002 1% error on
V-8; unfiltered
7.140607804964676E-002 0.323003125655167 4.276462519717335E-002 1% error on
I, unfiltered
7.012042547553828E-002 0.323108469786721 3.971573441817353E-002 2% error on
I-R filtered
7.100579512324626E-002 0.316582486741477 4.612013643859956E-002 2% error on
I, filtered
7.048479944309825E-002 0.323431732936217 4.495975726943799E-002 1% error on
I-6 unfiltered
Note: "filtered" indicating that the set of measurements is filtered by the
given threshold and "unfiltered"
indicating that the errors in the estimated parameters caused by the
measurement errors are within the
threshold and acceptable.
Table 5 Original and estimated line parameters
Original Estimated Estimated
(No measurement error) (Measurement errors filtered)
Resistance Rõ (p.u) 0.0715 0.07150 0.07149
Reactance Xõ (p.u) 0.3230 0.32300 0.32303
Grounding
Admittance Y (p.u) 0.0430 0.04300 0.04306
B. BCTC system
The system power flow case used in testing had 15,161 buses and 19,403
branches,
including the partial system model of the west USA network. Figure 2 shows a
partial
representation of the system. General information and some particulars of the
BCTC
system are available at http://www.bctc.com/the_transmission_system, which is
hereby incorporated by reference. Similarly, bus voltage phasors and line
power
flows obtained in power flow calculations were used as measurements to
estimate the
line parameters. The line whose parameters were estimated is a 500 kV overhead
line
named line 5L96. The system power flow states considered had a wide range,
including normal states to outage states that create heavy loading levels on
line 5L96
causing it come close to voltage instability. Any change in the system
operation state
DM_VAN240150-00071/8370710.1 23

CA 02602888 2007-09-18
should not have had any impact on the line parameters even if the system state
approached the voltage collapse point. Table 6 shows ten sets of sampling
measurements on line 5L96 (voltage phasors and line power flows), and Table 7
shows the estimates of resistance, reactance and grounding admittance
parameters of
the line using each set of the measurements. Table 8 shows the original and
final
estimated parameters of the line 5L96.
Table 6 Voltage phasors and power flows of Line 5L96
(Used as measurements)
Vi V-01 Vi V-f P Q Po Qq
(p.u.) (degree) (p.u.) (degree) (p.u.) (p.u.) (p.u.) (p.u.)
1.07000 8.60000 1.08400 1.50000 6.57000 -1.90700 6.51000 -0.39500
1.06000 10.40000 1.07500 2.30000 7.38000 -1.86900 7.30000 -0.62900
1.04800 13.40000 1.06300 3.70000 8.65000 -1.70500 8.55000 -0.93500
1.04200 15.50000 1.05500 4.70000 9.56000 -1.47800 9.44000 -1.08000
1.03600 17.60000 1.04600 5.60000 10.47000 -1.22400 10.31000 -1.24000
0.97400 35.40000 0.95000 10.50000 18.39000 2.82700 17.82000 -3.31000
0.94300 38.50000 0.92100 11.20000 18.79000 3.25000 18.16000 -3.97000
0.93400 39.40000 0.91300 11.50000 18.83000 3.32000 18.18000 -4.14000
0.92300 40.40000 0.90300 11.80000 18.86000 3.40000 18.19000 -4.37000
0.90400 42.00000 0.88700 12.10000 18.87000 3.63000 18.17000 -4.74000
Table 7 Estimated parameters using each set of measurements
(No error introduced into measurements)
Resistance R, (p.u.) Reactance X, (p.u.) Grounding admittance Y (p.u.)
1.548079448990775E-003 2.164686703340432E-002 1.00634041480235
1.540257804047732E-003 2.161604192015683E-002 1.00642302851147
1.507477920127257E-003 2.158783668403971E-002 1.00748042825992
1.491795456659398E-003 2.147316672004808E-002 1.00745808660354
1.573061016193664E-003 2.149563310309445E-002 1.00788947177925
1.543499388548838E-003 2.149672277048962E-002 1.00839028571578
1.507889329436661E-003 2.153452780650219E-002 1.00544332987656
1.505684381083002E-003 2.153580979630141E-002 1.01163590422382
1.537998257741009E-003 2.149474573088756E-002 1.00884387956467
1.465168579534557E-003 2.155604011536260E-002 0.94910943742322
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CA 02602888 2007-09-18
Table 8 Original and estimated parameters of 5L96
Original Estimated
Resistance R, (p.u) 0.00153 0.00152
Reactance Xõ (p.u) 0.02154 0.02154
Grounding
Admittance Y (p.u) 1.00701 1.00190
The calculations performed in the above described system and method can be
implemented as a series of instructions stored on computer readable memory
within a
computer, such as within RAM, or on computer readable storage medium. The
method and system may be expressed as a series of instructions present in a
carrier
wave embodying a computer data signal to communicate the instructions to a
networked device or server, which when executed by a processor within the
computer,
carry out the method.
Although the particular preferred embodiments of the invention have been
disclosed
in detail for illustrative purposes, it will be recognized that variations or
modifications
of the disclosed apparatus lie within the scope of the present invention.
DM_VAN/260254-00106/6737484. I 25

Representative Drawing