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Calculation of Transmission Line Parameters: A real Case Study

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DOI: 10.1109/ISGTEurope52324.2021.9640007

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 Calculation of Transmission Line Parameters: A real
Case Study
Xenios Economides, Markos Asprou, and Andreas Stavrou

Published in:
IEEE PES Innovative Smart Grid Technologies Europe 2021 (ISGT-Europe 2021)

Publication date:
2021

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Peer reviewed version

Digital Object Identifier (DOI):

Citation for published version (IEEE format):

X. Economides, M. Asprou and A. Stavrou, "Control Scheme for Phase Balancing and Reactive Power
Support from Photovoltaic Inverters Calculation of Transmission Line Parameters: A real Case
Study," IEEE PES Innovative Smart Grid Technologies Europe 2021 (ISGT-Europe 2021), Espoo,
Finland, 2021

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 Calculation of Transmission Line Parameters: A real
Case Study

Xenios Economides and Markos Asprou Andreas Stavrou

KIOS Research and Innovation Center of Excellence and Transmission System Owner
Department of Electrical and Computer Engineering Electricity Authority of Cyprus
University of Cyprus Nicosia, Cyprus
Nicosia, Cyprus astavrou@eac.com.cy
{economides.xenios, asprou.markos}@ucy.ac.cy

Abstract— The accurate transmission line parameters are Reinforced (ACSR) according to [4] shows an increase of
important in various power system analyses, while potential around 20% for a temperature rise from 25oC to 75oC.
uncertainty in their values affects the accuracy of control center
applications and compromises the selectivity of the protection Several methods have been recently developed for the
systems. The transmission line parameters are usually calculated calculation of the transmission line parameters, which can be
based on manufacturers data, ignoring environmental factors classified into offline and online methods. In [5] and [6],
(e.g. ambient temperature) which affect the accuracy of the methods for calculating offline the line parameters are
calculated parameters. Thus, the systematic refinement of the described. These methods use the geometry of the tower, the
line parameters can be very beneficial for the situational geometry and type of the conductor, and constant ambient
awareness of the power system operators. In this paper, the conditions. In addition, according to [7], the offline methods
calculation of the positive sequence parameters of a transmission based on lumped parameter model are not accurate for long-
line of the Cyprus power system through real synchronized distance transmission lines. Since the environmental
phasor measurements is presented. The seasonal variation of the conditions vary throughout the year, the line parameters that
transmission line parameters is analyzed by calculating the are calculated through the offline methods might cause serious
parameters during different periods of the year, while the impact discrepancies between the actual and the estimated value of
of the instrument transformers static error is demonstrated in the line parameters.
this real case study.
Based on the aforementioned facts, the development of
Keywords—Instrument transformers, measurement estimation methods, that can obtain line parameters in real-
uncertainty, synchronized phasor measurements, transmission time, is an imperative need. In this attempt, the deployment of
line parameters. synchronized measurement devices, such as Phasor
Measurement Units (PMUs), contributes to the correction of
I. INTRODUCTION any erroneous parameters since PMUs can provide
Transmission line parameters, including resistance, synchronized phasor measurements from both ends of the line
reactance, and susceptance, are used in several critical control [8]. Several methods have been proposed in the literature for
center applications [1]. These parameters are usually taking advantage of online voltage and current measurements
calculated based on the ideal structure of the transmission line, [9]-[14]. The authors in [9] developed a method which uses
ignoring several factors (i.e., weather, soil resistivity, coupling only RMS voltage and power measurements for the estimation
between parallel lines, and joints at the ends of the lines) that of the parameters of a distribution line. Reference [10]
certainly affect the line parameter values. Consequently, any proposed a high accuracy estimation method which calculates
mismatches between the actual and calculated transmission the positive sequence line parameters during normal operation
line parameters directly affect the accuracy of the control by utilizing online voltage and current phasors. This method
center applications, such as the state estimator, power flow detects and removes faulty measurements, minimizes the
analysis and voltage stability analysis. At the same time, the impacts of measurement errors and improves the accuracy of
uncertainty compromises the effectiveness of the protection the estimation. A method for estimating the parameters of a
schemes; for instance, impedance relays require exact long line is proposed in [11]. The method estimates the
knowledge of the line parameters for making a correct parameters by obtaining the ABCD parameters by using
protection zone identification and tripping decision [1], [2]. voltage and current phasors from the two ends of the line.
Reference [12] presents the estimation of the line parameters
It is therefore of great importance to update the using a total least square algorithm. A moving window
transmission line parameters according to the environmental technique is proposed for using voltage, current, active and
conditions, rather than having constant parameters in the reactive power measurements in order to calculate the line
power system models. This is also valid if one considers that parameters of an equivalent pi-model. The line parameters are
the series resistance is dependent on the temperature of the estimated in [13] based on Laplace transform technique by
conductor and typically rises as the conductor temperature utilizing three sets of synchronized voltage and current
increases above a standard temperature (usually at 20oC) [3]. phasors. The authors of [14] proposed a method for estimating
Moreover, series reactance of the Aluminum Conductor Steel the line parameters using phasor measurements provided by
“This work has been supported by the European Union's Horizon 2020 only one PMU. The methodology performs well either for a
research and innovation programme under grant agreement No 739551 single or for multiple lines. However, the magnitude of the
(KIOS CoE) and from the Government of the Republic of Cyprus through measurement noise might affect the performance of the
the Directorate General for European Programmes, Coordination and methodology.
Development.” “This work was also co-funded by the European Regional
Development Fund and the Republic of Cyprus through the Research and
Innovation Foundation (Project: INTEGRATED/0916/0035)”.
 In this paper, the calculation of the positive sequence According to Fig. 1, the sending and receiving voltage
parameters of the line that connects two important substations and current phasors can be expressed using the line
(Vasilikos power station and Costas Petrou substation) of the parameters as,
Cyprus power system will be presented. The calculations were 𝑌
carried out for different seasons of the year in order to 𝐼̃𝐿 = 𝐼̃𝑅 + ( 𝑉̃𝑅 ) (1)
2
investigate the seasonality variation of the transmission line 𝑌
𝐼̃𝑅 = 𝐼̃𝐿 − ( 𝑉̃𝑅 ) (2)
parameter values. The calculation of the line parameters is 2
𝑍𝑌
done offline considering the equivalent pi-model, and 𝑉̃𝑆 = 𝑉̃𝑅 ( 1 + ) − 𝑍𝐼̃𝑅 (3)
2
synchronized voltage and current phasor measurements 𝑍𝑌 𝑍𝑌
obtained by PMUs from both ends of the line. The 𝐼̃𝑆 = 𝑉̃𝑅 [ 𝑌 (1 + )] + 𝐼̃𝑅 (1 + ) (4)
4 2
contributions of this paper are: 1) the utilization of multiple
real phasor measurements from the PMUs at Vasilikos power where, 𝑉̃𝑆 , 𝑉̃𝑅 are the sending and receiving end voltage
station and Costas Petrou substation in order to calculate the phasors and 𝐼̃𝑆 , 𝐼̃𝑅 are the sending and receiving end current
line parameters; 2) the demonstration of the impact of the phasors, respectively. 𝑍 and 𝑌 are the series impedance and
static error of the instrument transformers (ITs) and the shunt admittances of the equivalent pi-model of the medium
necessity to consider it in the process for the calculation of the line.
line parameters, and 3) the investigation of the relationship of
the transmission line parameters with environmental and The series resistance and reactance, and shunt
loading conditions. susceptance, can be obtained through the voltage and current
phasors of the sending and receiving end when captured in
The rest of the paper is organized as follows. Section II balanced three-phase conditions. Thus, using (3) and (4), one
describes the estimation method that is followed in this work, can obtain the series and shunt susceptance as,
while the calculation of the transmission line parameters is
presented in Section III along with the effect of instrument 𝐼̃𝑆 𝑉
̃𝑅 + 𝑉̃𝑆 𝐼̃𝑅 1 1
transformers static error on the calculation of the line 𝑌𝑠𝑒𝑟𝑖𝑒𝑠 = ̃𝑆2 − 𝑉
̃𝑅2
= = (5)
𝑉 𝑍𝑠𝑒𝑟𝑖𝑒𝑠 𝑅+𝑗𝑋
parameters. Section IV demonstrates two case studies 𝐼̃𝑆 −𝐼̃𝑅
regarding the line parameter calculation and their variation 𝑌𝑠ℎ𝑢𝑛𝑡 = ̃𝑆 + 𝑉 ̃𝑅 (6)
𝑉
due to environmental and loading conditions. The paper
concludes in Section V. In the case that there is a PMU at both ends of the line as
shown in Fig. 2, the sending and receiving end voltage and
II. ESTIMATION OF TRANSMISSION LINE PARAMETERS current 𝑉̃𝑆 , 𝑉̃𝑅 , 𝐼̃𝑆 , 𝐼̃𝑅 , are available. Therefore, voltage and
current phasor measurements can be used in (5) and (6) for the
The transmission lines are usually represented in a pi line parameter calculation.
model with lumped series resistance/reactance and shunt
susceptance as shown in Fig. 1. In this work, the transmission
GPS
line that connects Vasilikos and Costas Petrou substations is
represented with the equivalent pi-model of a medium length yij=g
ĨS ij+jbij ̃IL
Zseries ̃IR
line (Fig. 1). The aim of this work is to calculate the i j
Bus S Bus R
parameters of the pi model based on real time synchronized
phasor measurements provided by the two line ends. PMU PMU
+ +
Nowadays, PMUs are installed in selected substations ̃S ̃R
V ysh=gsh+jbsh
Yshunt V
(usually in the transmission level) of the power systems. The
PMU measurements are transferred to the control center of the - -
power system where they are concentrated by a Phasor Data
Concentrator. Since PMUs are GPS synchronized equipment,
measurements with the same time stamp can be used for either Fig. 2: Two-bus system with PMUs at both ends of the line
real time monitoring and control applications or for power
system model refinement such as line parameter calculation.
Regarding the latter, the measurements with the same time III. CALCULATION OF LINE PARAMETERS FOR THE VASILIKOS-
stamp provided by the two PMUs at the two ends of the line
COSTAS PETROU TRANSMISSION LINE
can be used to estimate the transmission line parameters. In
case some measurements from one of the two ends are missing The transmission line parameters were calculated with
(there is not a pair of measurements with the same time measurements of the installed PMUs in the two substations of
stamp), the time instances that correspond to the missing the Cyprus power system. The reporting rate of the PMUs is
measurements are ignored. 50 phasors per second, which corresponds to 180000
measurements per hour. Considering that the value of the
ĨS yij=g ij+jbij
Zseries ĨL ĨR parameters remains constant within one hour interval the
i j
average value of the parameters calculated in one hour (using
the 180000 measurements) was used for representing the
+ + parameters’ value of each hour.
̃S
V ysh=gsh+jbsh
̃R
V
Yshunt A. Calculation using unprocessed PMU measurements
- - Initially, for the calculation of the parameters, the
measurements were used in equation (5) as they were recorded
by the PMUs. Since it is relatively short line (approximately
Fig. 1: Equivalent pi-model of a medium length line
25 km), only the series resistance and reactance were current transformers are class PX and are characterized by
calculated. It is worth mentioning that the nominal values of high accuracy with negligible error, while the static errors of
the resistance and reactance as calculated by the geometry of the voltage transformers are shown in Table I.
the line and the manufacturers data is 1.28 Ω and 13.18 Ω
respectively. As it is shown in Fig. 3, the calculated resistance TABLE I. STATIC ERROR OF THE VOLTAGE TRANSFORMERS AT VASILIKOS
POWER STATION AND COSTAS PETROU SUBSTATION
shows a significant deviation compared to the nominal value
of the line resistance (1.28 Ω), while the calculated values of Vasilikos Power Station Costas Petrou Substation
the resistance are not correct since the calculated resistance is
Phase Magnitude (%) Angle (min) Phase Magnitude (%) Angle (min)
negative. It should be noted that the phenomenon with the
negative resistance is observed to all the dates under A +0.19 0.9 A -0.36 1.8
examination and not only for the day that is depicted in Fig. 3 B +0.19 1.3 B -0.36 0.6
(22/01/2019). In the case of the reactance, as depicted in Fig. C +0.16 1.1 C -0.30 1.7
4, the calculated values are between acceptable limits and are
close to the nominal value (13.18 Ω). Therefore, in the view of the impact of low accuracy
instrument transformer on the resistance calculation, it was
0.00
decided to add static error in the synchronized voltage phasor
Resistance (Ω)

-0.50
measurements of Vasilikos power station and Costas Petrou
-1.00
-1.50
substation. It should be noted that the static error of the voltage
-2.00
angle is negligible (as shown in Table I) and was omitted.
More specifically, the synchronized voltage phasor
Time (hours) measurements of each phase from the Vasilikos and Costas
Fig. 3: Calculated resistance with recorded data as they were measured Petrou substation were modified as,

13.10 𝑉̃𝑆𝑚 = 𝑉𝑆𝑃𝑀𝑈 (1 + 𝑒𝑆𝑉 ) ∠ (𝛿𝑆𝑃𝑀𝑈 ) ()

Reactance (Ω)

12.95
12.80 𝑉̃𝑅𝑚 = 𝑉𝑅𝑃𝑀𝑈 (1 + 𝑒𝑅𝑉 ) ∠ (𝛿𝑅𝑃𝑀𝑈 ) ()
12.65
12.50 where 𝑉̃ 𝑚 is the modified voltage phasor, 𝑉 𝑃𝑀𝑈 and 𝛿 𝑃𝑀𝑈
are the measured (from PMU) magnitude and angle of the
Time (hours)
voltage respectively, and 𝑒 𝑉 is the static error of the voltage
Fig. 4: Calculated reactance with recorded data as they were measured magnitude. The S and R subscripts refer to sending and
receiving end, respectively.
B. Calculation of line parameters considering PMU
Measurement chain uncertainties In Figs. 6 and 7, the calculated resistance and reactance
including the above static errors are shown, respectively.
Although the procedure for calculating the transmission Based on these two figures, the calculated resistance is very
line parameters through the PMU measurements is simple, the close to the reference value and the deviation from its nominal
quality of the PMU measurements should be always value is minimized considerably. Regarding the reactance of
considered in this procedure. As it is indicated in Fig. 5, the the line, it can be concluded that is not affected by the static
instrument transformers (ITs) and PMU are the two main errors of the voltage transformers as it can be concluded by
components in a PMU measurement chain that might the comparison of Figs. 4 and 7 (calculated reactance before
introduce uncertainties in the PMU measurements. Even and after the consideration of the voltage transformers static
though PMUs can be characterized as high accuracy devices, errors). In this sense, this case study demonstrates through a
the accuracy of their measurements may be low especially real-life example that the voltage transformer static error is
when ITs belong to a low accuracy class [5]. In this sense, the dominant to the calculation of the line resistance and it should
low accuracy of the instrument transformer can deteriorate be always considered to the calculation of the line parameters.
the quality of the PMU measurements and eventually lead to
1.30
the calculation of erroneous parameters.
Resistance (Ω)

1.25

Inetwork 1.20

1.15

1.10
Itransf
Instrument PMU
transformers Time (hours) Reference Value
Vtransf
Vnetwork Fig. 6: Calculated resistance including the voltage static error

13.20
Reactance (Ω)

Fig. 5: Generic PMU measurement chain [5] 12.90

In this direction, after the calculation of erroneous line 12.60

resistance, this work was focused on the errors of the voltage 12.30

and current transformers that are connected to the two PMUs.

The manufacturers of the installed voltage and current Time (hours) Reference Value
transformers at the two ends of the line provided data sheets
Fig. 7: Calculated reactance including the voltage static error
which include routine tests that indicate the static error
(magnitude and angle) of each instrument transformer. The
IV. LINE PARAMETER VARIATION DUE TO ENVIROMENTAL comparison to the winter periods. This is also verified by (14)
AND OPERATIONAL FACTORS in which the resistance varies linearly with respect to the
operating temperature. Therefore, as temperature rises, the
Based on the modified voltage phasor measurements
resistance increases linearly.
considering the voltage transformer static error, the variation
of the line parameters in different weather (Summer/Winter) 𝑅 = 𝑅𝑓 [1 + 𝑎(𝑡 + 𝜃)] ()
and loading conditions (Weekday/Weekend) is investigated in
this case study. Specifically, the line parameters were where, 𝑅𝑓 is the resistance at reference temperature, 𝑎 is the
calculated in the following days: 21st of January 2018 temperature coefficient of resistance per °C, 𝑡 is the ambient
(Sunday), 8th of July 2018 (Sunday), 22nd of January 2019 temperature and 𝜃 is the difference between the reference and
(Tuesday), and 22nd of August 2019 (Thursday) in order to the final temperature.
derive any conclusions regarding the variation of the line
Moreover, in summer the value of the resistance is very
parameters in different environmental and operational
similar either for a weekday (high loading condition) or a day
conditions.
in the weekend (low loading conditions). However, in winter
A. Variation of series resistance the resistance during the early morning hours for the weekdays
is greater than in weekends, and then the value is similar for
The variation of the series resistance (Ω) in different both cases. This is due to the relatively higher loading
weather conditions for weekdays and weekends is shown in conditions in weekday in the morning hours than in the
Figs 8 and 9, while Figs. 10 and 11 depict the variation of the weekend.
resistance (Ω) in different loading conditions in winter and
summer respectively. B. Variation of series reactance
Regarding the series reactance, Figs. 12 and 13 show the
Resistance (Ω)

1.40
variation of the series reactance (Ω) for different weather
1.30
conditions for weekdays and weekends, respectively. Further,
1.20
Figs. 14 and 15 show the variation of the series reactance (Ω)
1.10 for different loading condition in winter and summer,
respectively.
Time (hours)
22 January 19 / Tuesday 22 August 19 / Thursday 12.65
Reactance (Ω)

Fig. 8: Comparison of series resistance between different weather conditions 12.55

for weekdays 12.45

12.35
Resistance (Ω)

1.40
1.30
1.20 Time (hours)
1.10 22 January 19 / Tuesday 22 August 19 / Thursday
1.00 Fig. 12: Comparison of series reactance between different weather conditions
0.90 for weekdays
12.65
Time (hours)
Reactance (Ω)

21 January 18 / Sunday 8 July 18 / Sunday 12.55

12.45
Fig. 9: Comparison of series resistance between different weather conditions 12.35
for weekends 12.25

1.30
Resistance (Ω)

Time (hours)
1.20
21 January 18 / Sunday 8 July 18 / Sunday
1.10
Fig. 13: Comparison of series reactance between different weather conditions
1.00
for weekends
0.90

12.70
Reactance (Ω)

Time (hours) 12.60

22 January 19 / Tuesday 21 January 18 / Sunday
12.50
12.40
Fig. 10: Comparison of series resistance between different loading conditions 12.30
in winter
Time (hours)
1.45
Resistance (Ω)

22 January 19 / Tuesday 21 January 18 / Sunday

1.35
Fig. 14: Comparison of series reactance between different loading conditions
1.25
in winter
1.15
12.65
Reactance (Ω)

Time (hours)
22 August 19 / Thursday 8 July 18 / Sunday 12.55

Fig. 11: Comparison of series resistance between different loading conditions

12.45
in summer
Time (hours)
The deviation of the calculated series resistances from the 22 August 19 / Thursday 8 July 18 / Sunday

nominal during the day for these cases ranges from 9% to Fig. 15: Comparison of series reactance between different loading conditions
25%. In addition, as shown in Figs. 8 and 9, during the in summer
summer, the value of the series resistance is higher in
 The deviation of the calculated series reactance from the depends mainly on the loading conditions and more specific
nominal value during the day, for these cases, ranges between to the current flow of the transmission line.
4% to 6.5%. Based on Figs. 12 and 13, the value of the series
reactance is a bit larger in summer than in winter. Also, as Moreover, it is crucial to include the static error of the
indicated in Figs. 14 and 15, in weekdays around at 2:00-3:00 instrument transformers in the PMU measurements
am, the value of the reactance has a sharp increase for the (especially when the line resistance is calculated). It is
whole year, while this is not the case in the weekends. indicated in this paper that the calculation of the parameters
by using the measurements as they are recorded by the PMUs
However, the value of the reactance is not higher in is compromised. However, the compensation of the static
summer due to the temperature increase but it is dependent on error of the instrument transformers in the PMU
the loading condition of the line, and hence on the current of measurements results in accurate line parameter calculation.
the transmission line (that flows through the line) which is
larger in summer than in winter. Moreover, the current flow in It should be noted that as a part of the future work, the
weekdays increases sharply in the early morning hours, which PMU measurements will be used for the calculation of zero
affects the reactance’s value and as a result shows an increase sequence parameters of the line. This can be done by using
in the corresponding time. PMU measurements from time instants that an asymmetric
disturbance has occurred in the system.
This is verified in Figs. 16 and 17, in which the daily
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increases accordingly. However, the value of the reactance IEEE Transactions on Power Delivery, vol. 32, no. 6, pp. 2510-2519,
Dec. 2017

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