 Original research
 Open Access
Estimating zerosequence impedance of threeterminal transmission line and Thevenin impedance using relay measurement data
 Swagata Das^{1},
 Sundaravaradan Navalpakkam Ananthan^{2}Email authorView ORCID ID profile and
 Surya Santoso^{2}
https://doi.org/10.1186/s416010180108y
© The Author(s) 2018
 Received: 4 March 2018
 Accepted: 26 September 2018
 Published: 6 November 2018
Abstract
Current and voltage waveforms recorded by intelligent electronic devices (IEDs) are more useful than just performing postfault analysis. The objective of this paper is to present techniques to estimate the zerosequence line impedance of all sections of a threeterminal line and the Thevenin equivalent impedance of the transmission network upstream from the monitoring location using protective relay data collected during shortcircuit ground fault events. Protective relaying data from all three terminals may not be always available. Furthermore, the data from each terminal may be unsynchronized and collected at different sampling rates with dissimilar fault time instants. Hence, this paper presents approaches which use unsynchronized measurement data from all the terminals as well as data from only two terminals to estimate the zerosequence line impedance of all the sections of a threeterminal line. An algorithm to calculate positive, negative and zerosequence Thevenin impedance of the upstream transmission network has also been presented in this paper. The efficacy of the proposed algorithms are demonstrated using a test case. The magnitude error percentage in determining the zerosequence impedance was less than 1% in the test case presented.
Keywords
 Zerosequence
 Threeterminal line
 Thevenin Impedance
1 Introduction
One of the key features in modern intelligent electronic devices (IEDs) such as digital relays and digital fault recorders is the generation of event reports during faults. By analyzing fault event reports, system operators can understand what happened during the event and the cause of the event but event reports contains much more valuable information. The authors in [1–3] have used event reports to glean information about relay misoperations and estimate a variety of system parameters.
Transmission line parameter estimation using different methods and a variety of data has been a topic of interest for researchers [4, 5]. The zerosequence impedance of an overhead line must be specified by protection engineers in relay settings and plays a key role in distance and directional protection [6, 7], and fault location calculations. The zerosequence impedance of a line depends on earth resistivity. A commonly used practice to determine the zerosequence impedance is by using Carson’s equation and using a typical value of 100 Ωm as earth resistivity [6, 8]. As the earth resistivity depends on various factors such as soil type, temperature and moisture content in the soil [9] and is difficult to measure accurately, the zerosequence line impedance is subject to much uncertainty. As a result, to avoid relay misoperations and incorrect fault location, efforts must be made to validate the zerosequence line impedance using fault event data.
Threeterminal transmission lines are frequently used by utilities to transfer bulk electrical power and support loads from three generating sources [10]. Often, utilities upgrade an existing twoterminal line to a threeterminal line by simply connecting a line with a generating source to it. Building threeterminal lines have several advantages. There are low or no costs associated with constructing a new substation and procuring power system equipment. No rightofway and regulatory approvals are required. As a result, threeterminal lines are expeditious in increasing the operational support necessary to meet system demands.
Several authors in the past have researched on validating zerosequence line impedance but only for twoterminal lines [6, 11]. Very little work, if any, has been conducted on validating the zerosequence line impedance of threeterminal transmission lines. The event reports from IEDs at different ends of the line may detect the fault at slightly different time instants and have different sampling rates. Furthermore, they can be unsynchronized. Therefore, it is necessary to devise a methodology that can use unsynchronized measurements to confirm the zerosequence impedance of threeterminal transmission lines.
Using the same set of waveform data, the Thevenin impedance of the transmission network upstream from the monitoring location can be estimated. The Thevenin impedance, often referred to as the shortcircuit impedance, plays an important role when calculating the currents during a balanced or an unbalanced fault [12]. System operators obtain the Thevenin impedance by performing a shortcircuit analysis on the circuit model. However, to avoid any erroneous fault current calculations due to an inaccurate circuit model, it is a good practice to validate the shortcircuit impedance from the circuit model with that calculated from the fault data. The Thevenin impedance is also required by the Eriksson, Novosel et al., and other faultlocating algorithms to track down the location of a fault [8, 13, 14]. Furthermore, the Thevenin impedance calculated at regular intervals during a long duration fault can provide insight into the state of the transmission network upstream from the fault. Authors in [15] have attempted to estimate the Thevenin impedance but have not provided the equations for the same. Authors in [15, 16] have also highlighted the importance and uses of estimating the Thevenin impedance. Hence, it is essential to estimate the Thevenin impedance of the transmission network upstream from the monitoring location using any available fault data.
Based on the aforementioned background, the primary objective of this paper is to estimate the zerosequence impedance of threeterminal line from waveform data captured by intelligent electronic devices during a ground fault. The contribution lies in developing algorithms which either use unsynchronized data from all the three terminals or use unsynchronized data from only two terminals. The second objective of this paper is to estimate the Thevenin impedance of the transmission network upstream from the monitoring location using the same set of fault waveform data. The authors had developed algorithms for estimating zerosequence line impedance for twoterminal lines [5] but this paper explores algorithms for estimating zerosequence impedance of threeterminal lines which has barely been studied previously. This paper serves as a further extension and as a sequel to the aforementioned paper.
The proposed methods for both estimating the zerosequence line impedance as well as estimating Thevenin impedance from fault waveforms are validated using a test case. The approach which estimates zerosequence line impedance using data from three terminals requires lesser assumptions than the method which uses data from only two terminals. Both the methods produced accurate zerosequence impedance estimates and the magnitude error percentage was less than 1%. Similarly, results indicate the success of Thevenin impedance estimation algorithm as the magnitude error was less than 1% in the test case.
2 Estimating the zerosequence impedance of threeterminal lines
Threeterminal lines do pose a significant challenge to the task of validating the zerosequence line impedance. The third terminal contributes to the total fault current and changes the impedance equations which are commonly used for twoterminal lines. Furthermore, with the introduction of a third terminal, there are now two lines whose zerosequence line impedances have to be validated from a single fault event. Based on this aforementioned background, this section presents two approaches for calculating the zerosequence line impedance of threeterminal lines. The proposed algorithm uses fault current and voltage phasors from the line terminals for its analysis. The proposed technique should therefore be applied on the steady state portion of fault waveforms to obtain accurate results. Hence, it is suitable for application in ground fault scenarios which contain steady state fault waveforms. Approach 1 requires the availability of voltage and current waveforms from all the three terminals while Approach 2 uses waveforms captured at two terminals only.
2.1 Approach 1 for estimating zerosequence line impedance: data from three terminals
The steps to validate the zerosequence impedance of Line 1 and Line 2 are outlined below.
2.1.1 Step 1: synchronize terminal T with terminal G

V_{Tap2} is the negativesequence voltage at the tap point

V_{G2},V_{H2},V_{T2} are the negativesequence fault voltages at terminals G, H and T respectively

I_{G2},I_{H2},I_{T2} are the negativesequence fault currents at terminals G,H and T respectively

V_{G0},V_{H0},V_{T0} are the zerosequence fault voltages at terminals G, H and T respectively

I_{G0},I_{H0},I_{T0} are the zerosequence fault currents at terminals G,H and T respectively

Z_{2}L_{1} is the negativesequence impedance of Line 1

Z_{2}L_{2} is the negativesequence impedance of Line 2

Z_{0}L_{1} is the zerosequence impedance of Line 1

Z_{0}L_{2} is the zerosequence impedance of Line 2
In effect, this step calculates the phase angle mismatch between terminal G and terminal T measurements, and accordingly adjusts the phasors at terminal T by e^{jδ1}.
2.1.2 Step 2: synchronize terminal H with terminals T and G
At the end of this step, the voltage and current measurements at terminals T and H are synchronized with respect to those at terminal G.
2.1.3 Step 3: calculate the zerosequence impedance of Line 2
2.1.4 Step 4: calculate the zerosequence impedance of Line 1
In this way, Approach 1 is successful in validating the zerosequence impedance of Line 1 and Line 2. If the actual fault location is not available, then an additional step (Step 0) must be performed to track down the exact fault location before applying the steps described above.
2.1.5 Step 0: identify the fault location
The estimated V_{Tap2} from two of the terminals will be an exact match while V_{Tap2} from the third terminal will be different. The fault is expected to lie between that third terminal and the tap point. Next, apply oneended fault location algorithm to the voltage and current waveforms at the third terminal and estimate the distance to the fault.
2.2 Assumptions
Assumptions made by Approach 1 when estimating the zerosequence impedances of Line 1 and Line 2 are summarized below:
1. Line 1 and Line 2 are homogeneous
2. Zerosequence mutual coupling is absent
A transmission network is homogeneous when the local and remote source impedance have the same impedance angle as the transmission line. Assumption 1 is a reasonable assumption to make as local and remote source impedance angles are often close to that of the transmission line.
Zerosequence mutual coupling arises when transmission lines are physically very close or parallel to each other. An example where zerosequence mutual coupling could be present is when there are two sets of threephase lines running on same towers. Though this configuration of lines is occasionally found in twoterminal transmission lines, it is highly unlikely and rare to have threeterminal transmission line with such a wire configuration that would have zerosequence mutual coupling.
2.3 Approach 2 for estimating zerosequence line impedance: data from two terminals
Although Approach 1 can successfully validate the zerosequence impedance of threeterminal transmission lines, voltage and current waveforms from all the three terminals may not always be available. For this reason, this subsection develops a methodology that can use data from only two terminals to estimate the zerosequence impedance of threeterminal lines. To illustrate the approach, consider the scenario shown in Fig. 1. Suppose that the measurements captured by digital relays at terminals G and H are available while measurements from terminal T are missing. The procedure to estimate the zerosequence impedance of Line 1 and Line 2 in such a scenario is described below:
2.3.1 Step 1: estimate the negativesequence current from terminal T
Note that since terminal G measurements are being used to estimate the fault current from terminal T, measurements of these two terminals are automatically synchronized.
2.3.2 Step 2: synchronize terminal H with terminals G and T
The new set of sequence voltage phasors at terminal H are V_{H1}e^{jδ},V_{H2}e^{jδ}, and V_{H0}e^{jδ}. Similarly, the new set of current phasors at terminal H are I_{H1}e^{jδ},I_{H2}e^{jδ}, and I_{H0}e^{jδ}.
2.3.3 Step 3: estimate the zerosequence impedance of line 2
The zerosequence line impedance of Line 2 is estimated from terminal H measurements using any zerosequence line impedance estimation algorithm for twoterminal line which uses measurements from one terminal only [5, 11]. The algorithm to estimate the zerosequence line impedance of a twoterminal line using data from only one terminal assumes that the fault resistance is zero and the fault location is known.
2.3.4 Step 4: estimate the zerosequence current from terminal T
2.3.5 Step 5: estimate the zerosequence impedance of line 1
2.4 Assumptions:
Assumptions made by Approach 2 when estimating the zerosequence line impedance are summarized below:
1. Line 1 and Line 2 are homogeneous
2. Fault location is known
3. Fault resistance is zero
4. Zerosequence mutual coupling is absent
The assumptions that the transmission network is homogeneous and the absence of zerosequence mutual coupling are same as that of Approach 1. Since Approach 2 has lesser data availability than Approach 1, further assumptions must be made to obtain the zerosequence line impedance of the lines.
In utility/industry practice, the process of estimating the zerosequence line impedance is an offline post fault analysis process using event reports recorded by the relay. The electric power utility performs a thorough investigation of the fault event as soon as the fault has occurred. As a result, it is expected that the utility identifies the fault location and hence, it is a reasonable assumption that the fault location is known while implementing the proposed algorithm.
It is highly unlikely to have a completely bolted fault where the fault resistance is exactly zero. However, it is a reasonable assumption to make when the fault resistance value is small and close to zero. Though not every shortcircuit fault has a fault impedance value close to zero, the occurrence of this type of fault where the fault impedance is close to zero is relatively common in transmission lines. Such a scenario has also been demonstrated and analyzed using realworld field data in [5]. However, having to extract more information from less data requires us to make such an assumption.
3 Demonstration of estimation of zerosequence impedance using a test case
This section demonstrates the efficacy of Approach 1 and Approach 2 in estimating the zerosequence line impedance of a threeterminal line during a single linetoground fault. The system parameters used for the study are described below.
The threeterminal network shown in Fig. 1 was modeled in PSCAD simulation software [17]. The model was used to replicate actual shortcircuit faults that occur on a transmission line and generate the corresponding voltage and current waveforms.
The positive and zerosequence source impedances at terminal G are Z_{G1} = 3.75 ∠86° Ω and Z_{G0} = 11.25 ∠86° Ω, respectively. The positive and zerosequence source impedances at terminal H are Z_{H1} = 12 ∠80° Ω and Z_{H0} = 36 ∠80° Ω, respectively. The positive and zerosequence source impedances at terminal T are Z_{T1} = 5 ∠83° Ω and Z_{T0} = 12 ∠83° Ω, respectively.
When a bolted phase Atoground fault occurs at a distance of 13 miles from terminal G, monitors at terminals G, H, and T capture the voltage and current waveforms at 32, 64, and 128 samples per cycle, respectively. The waveforms are, therefore, not synchronized with each other.
For the test scenarios, the fundamental frequency magnitude and phase angle of the fault current and voltages were extracted using Fast Fourier Transform. To avoid inaccuracies due to transients and DC offset, a cycle with stable fault current is used. For this purpose, the third cycle after fault inception was chosen in the simulation results to estimate the zerosequence line impedance. Fast Fourier Transform is a commonly used method to extract fundamental frequency components and filter out decaying DC offset [8, 19]. If the DC offset remains significant after the third cycle, more advanced filters [20, 21] can be used or appropriate fault cycle can be chosen to obtain accurate current and voltage measurements from fault records. This ensures that the error in determining the current and voltage phasors is minimized and hence the zerosequence line impedance estimation will not be affected by it.
Estimated vs. Actual Zerosequence Line Impedance of Line 1
Approach  Zerosequence line impedance (Ω)  Magnitude error (%)  Phase angle error (degrees)  

Actual  Estimated  
Approach 1  4.98+j10.42  4.99+j10.39  0.19  0.11 
Approach 2  4.95+j10.46  0.20  0.22 
Estimated vs. Actual Zerosequence Line Impedance of Line 2
Approach  Zerosequence line impedance (Ω)  Magnitude error (%)  Phase angle error (degrees)  

Actual  Estimated  
Approach 1  15.08+j31.44  15.13+j31.33  0.22  0.15 
Approach 2  15.12+j31.38  0.10  0.10 
Although Approach 2 uses data from only two terminals to estimate the zerosequence line impedance of all the sections of the threeterminal line, it requires the knowledge of source impedance parameters of the third terminal. For this purpose, the next section provides approaches to calculate the source impedance or the Thevenin impedance of the transmission network upstream from the monitoring location.
In [5], the transmission line, whose zerosequence estimation was performed, was a twoterminal line whereas threeterminal transmission lines are analyzed in this paper. In a threeterminal transmission line, there is shortcircuit current contribution from all three terminals unlike twoterminal transmission line where there is current contribution from only two terminals. Hence, the circuit explored, system equations and method of analysis are different compared to the algorithms presented in [5]. The data availability for a threeterminal line is also different from that of a twoterminal line. Moreover, there is a need to synchronize measurements from different terminals of the line based on data availability before estimating the zerosequence line impedance. Furthermore, careful system analysis is required to calculate the zerosequence impedance of both Line 1 and Line 2 as seen in the two approaches discussed above unlike having to estimate the zerosequence impedance of only one line as in [5]. In addition to estimating the zerosequence impedance of the lines, this paper also presents a method for estimating the Thevenin impedance of the transmission network upstream from the monition location using the same set of waveform data.
4 Estimating the Thevenin impedance of upstream network
It should be noted that Z_{G1} is not assumed to be equal to Z_{G2} and is calculated separately using (19). This is because the positivesequence impedance equals the negativesequence impedance in static electrical components such as lines and transformers. However, the sequence impedances are not equal to each other in rotating machines.
5 Demonstration of estimation of Thevenin impedance of upstream network using a test case
Actual vs. Estimated Thevenin impedances at Terminal G for the threeterminal line test scenario
Source impedance (Ω)  Magnitude error (%)  Phase angle error (degrees)  

Actual  Estimated  
Z _{ G1}  3.75 ∠86°  3.78 ∠89.2°  0.80  3.2 
Z _{ G2}  3.75 ∠86°  3.75 ∠86°  0.00  0.00 
Z _{ G0}  11.25 ∠86°  11.25 ∠86°  0.00  0.00 
As seen from Table 3, the estimated source impedances matched well with the actual impedances used in the simulation model. The magnitude error percentages of the estimated Thevenin impedances are less than 1%. The positivesequence impedance shows a small deviation from the actual value. Most likely, the error stems from the constant current load model assumption in (19). Similarly, the Thevenin impedance can be estimated at Terminals H and T as well using sequence currents measured at their respective terminals.
The process of estimating the Thevenin impedance of the upstream network is done for each terminal of a transmission line. It requires data from only one terminal to estimate the Thevenin impedance of the upstream network of that terminal as shown in (17)–(19). It does not require data or information from other terminals of the line.
The calculation of Thevenin impedance or source impedance is done for the upstream network. Hence, the upstream Thevenin impedance is not affected by components downstream of the monitoring location. Therefore, the transmission line that is present after the monitor does not affect the Thevenin impedance of the upstream network.
6 Conclusion
This paper presents methods to estimate the zerosequence line impedance of all the sections in a threeterminal transmission line using fault data from all three terminals as well as using data from only two terminals. When data from only two terminals are available, the algorithm requires source impedance data from the third terminal, assumes that the fault location is known and the fault resistance is zero. These assumptions are not required when data from all three terminals are available. The accuracy of the proposed methods are demonstrated using a test case. Both approaches produced accurate zerosequence line impedance estimates when the assumptions were satisfied and the magnitude error percentage was less than 1%. Additionally, the paper also presented a technique for estimating Thevenin equivalent impedance of the transmission network upstream from the monitoring location. The efficacy of this method was demonstrated using a test case. The magnitude error percentage in estimating the Thevenin impedance was less than 1% in the test case. Hence, it can be concluded that the presented methods provide a reliable estimation of the zerosequence impedance of the line and Thevenin impedance of the upstream network which can be used to verify relay settings.
Declarations
Authors’ contributions
SD and SNA had developed the algorithms, performed the analysis and drafted the manuscript. SS was the technical advisor and supervised the analysis and submission of the manuscript. All authors read and approved the manuscript.
Competing interests
The authors declare that they have no competing interests.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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