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
This approach requires the availability of voltage and current phasors at all the three terminals during a fault and is illustrated using the threeterminal transmission line shown in Fig. 1. Line 2 is homogeneous and connects terminal G with terminal H. Line 2 is of length LL per unit. Line 1 is also homogeneous and connects terminal T with Line 2 at a distance of d per unit from terminal G. When a single or double linetoground fault occurs on Line 2 at m per unit distance from terminal G, all three sources contribute to the fault. Digital relays at each terminal capture the voltage and current phasors during the fault. The phasors need not be synchronized with each other.
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
Consider the negativesequence network of a threeterminal line during a single or a double linetoground fault as shown in Fig. 2. Let δ1 represent the synchronization error between the measurements at terminal T and terminal G. Therefore, to align the voltage and current phasors at terminal T with respect to terminal G, a synchronization operator, e^{jδ1}, is applied to the terminal T measurements. The value of e^{jδ1} can be determined by calculating V_{Tap2} from both terminals as
$$\begin{array}{*{20}l} Terminal \thinspace G: & ~ V_{Tap2} ~ = \thinspace V_{G2}  \left(d\times Z_{2}L_{2}\times I_{G2}\right) \\ Terminal \thinspace T: & ~ V_{Tap2} ~ = \thinspace V_{T2}e^{j\delta1}  \left(Z_{2}L_{1}\times I_{T2}e^{j\delta1}\right) \end{array} $$
(1)
Notations in the Figs. 2 and 3 are defined as follows:

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
Since terminals G and T operate in parallel to feed the fault, V_{Tap2} should be equal when calculated from either terminal. From this principle, e^{jδ1} can be solved as
$$ e^{j\delta1} = \frac{V_{G2}\left(d\times Z_{2}L_{2}\times I_{G2}\right)}{V_{T2}\left(Z_{2}L_{1}\times I_{T2}\right)} $$
(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
After the synchronization process in Step 1, the new negativesequence voltage and current phasors at terminal T are V_{T2}e^{jδ1} and I_{T2}e^{jδ1}, respectively, and at terminal G are V_{G2} and I_{G2}, respectively. This step synchronizes the phasors at terminal H with the phasors at terminals G and T. For this purpose, a second synchronizing operator, e^{jδ2}, is applied to the measurements at terminal H. The value of e^{jδ2} can be calculated from the fact that the negativesequence voltage at the fault point, V_{F2}, must be the same when calculated from either terminal G or H as
$$\begin{array}{*{20}l} &Terminal G: \\ &V_{F2}=V_{G2}\left[mZ_{2}L_{2}\times I_{G2}][(md)Z_{2}L_{2}\times I_{T2}e^{j\delta1}\right]\\ &Terminal H: \\ &V_{F2}=V_{H2}e^{j\delta2}\left[(1m)Z_{2}L_{2}\times I_{H2}e^{j\delta2}\right] \end{array} $$
(3)
Therefore, e^{jδ2} is given by
$$ {} e^{j\delta2} = \frac{V_{G2}\left[mZ_{2}L_{2}\times I_{G2}\right]\left[\left(md\right)Z_{2}L_{2}\times I_{T2}e^{j\delta1}\right]}{V_{H2}\left[(1m)Z_{2}L_{2}\times I_{H2}\right]} $$
(4)
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
To estimate the zerosequence impedance of Line 2, the zerosequence network during a single or double linetoground fault is shown in Fig. 3. The zerosequence voltage at the fault point, V_{F0}, can be calculated from terminal G and H as
$$\begin{array}{*{20}l} &{}Terminal G: \\ &{}V_{F0}\! =\! V_{G0}\! \! (mZ_{0}L_{2}\times I_{G0})\left[(md)Z_{0}L_{2}\times I_{T0}e^{j\delta1}\right] \\ &{}Terminal H: \\ &{}V_{F0}=V_{H0}e^{j\delta2}\left[(1m)Z_{0}L_{2}\times I_{H0}e^{j\delta2}\right] \end{array} $$
(5)
Since V_{F0} is equal when calculated from either line terminal, equate the two equations in (5) to solve for Z_{0}L_{2} as
$$ Z_{0}L_{2} = \frac{V_{G0}V_{H0}e^{j\delta2}}{mI_{G0}+\left(md\right)I_{T0}e^{j\delta1}\left(1m\right)I_{H0}e^{j\delta2}} $$
(6)
2.1.4 Step 4: calculate the zerosequence impedance of Line 1
To estimate the zerosequence impedance of Line 1, Z_{0}L_{1}, calculate the zerosequence voltage at the tap point, V_{Tap0}, from terminals G and T as
$$\begin{array}{*{20}l} &Terminal ~ G: V_{Tap0}=V_{G0}(dZ_{0}L_{2}\times I_{G0})\\ &Terminal ~ T: V_{Tap0}=V_{T0}e^{j\delta1}\left(Z_{0}L_{1}\times I_{T0}e^{j\delta1}\right) \end{array} $$
(7)
where V_{T0} is the zerosequence fault voltages at terminal T. Because V_{Tap0} from terminal G is equal to that from terminal T, we can use (7) to solve for Z_{0}L_{1} as
$$ Z_{0}L_{1} = \frac{V_{T0}e^{j\delta1}V_{G0}+\left(dZ_{0}L_{2}\times I_{G0}\right)}{I_{T0}e^{j\delta1}} $$
(8)
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
Before computing the distance to the fault, it is necessary to identify whether the fault is on Line 1 or on Line 2. The negativesequence network shown in Fig. 2 is used for this purpose. When the fault is between terminal H and the tap point, V_{Tap2} from the other two terminals are equal. Therefore, to identify the faulted section of the feeder, the approach consists of calculating V_{Tap2} from each terminal as
$$\begin{array}{*{20}l} Terminal ~ G: ~ \lvert V_{Tap2}\rvert=&\lvert V_{G2}\left(d\times Z_{2}L_{2}\times I_{G2}\right)\rvert\\ Terminal ~ H: ~ \lvert V_{Tap2}\rvert=&\lvert V_{H2}\! \! \left((LL\! \! d)\! \times\! Z_{2}L_{2}\! \times\! I_{H2}\right)\rvert\\ Terminal ~ T: ~ \lvert V_{Tap2}\rvert=&\lvert V_{T2}\left(Z_{2}L_{1}\times I_{T2}\right)\rvert \end{array} $$
(9)
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.
Figure 4 shows a flow diagram which summarizes Approach 1 for calculating the zerosequence line impedance.
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
To estimate the negativesequence fault current from terminal T, I_{T2}, the negativesequence network shown in Fig. 2 is used. The approach consists of calculating V_{Tap2} from terminal G and terminal T as
$$\begin{array}{*{20}l} Terminal ~ G: V_{Tap2}=& \thinspace V_{G2}(d\times Z_{2}L_{2}\times I_{G2}) \end{array} $$
(10)
$$\begin{array}{*{20}l} Terminal ~ T: V_{Tap2}=& \thinspace (Z_{T2}+Z_{2}L_{1})I_{T2} \end{array} $$
(11)
where Z_{T2} is the negativesequence source impedance of terminal T. Since V_{Tap2} is equal when calculated from either terminal, I_{T2} can be estimated as
$$ I_{T2}=\frac{\left(d\times Z_{2}L_{2}\times I_{G2}\right)V_{G2}}{Z_{T2}+Z_{2}L_{1}} $$
(12)
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
This step applies a synchronization operator, e^{jδ}, to the terminal H measurements so as to align the measurements at this terminal with those at terminals G and T. The fact that V_{F2} is the same when calculated from terminals G and H is used to calculate the synchronization operator as
$$ {}e^{j\delta}=\frac{V_{G2}\left(mZ_{2}L_{2}\times I_{G2}\right)\left[(md)\times Z_{2}L_{2}\times I_{T2}\right]}{V_{H2}\left[(1m)Z_{2}L_{2}\times I_{H2}\right]} $$
(13)
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
The zerosequence voltage at the fault point, V_{F0}, is the same when calculated from terminal G or terminal H. This principle is used to calculate the zerosequence fault current contributed by terminal T, I_{T0}, as
$$ I_{T0}=\frac{V_{G0}V_{H0}e^{j\delta}+Z_{0}L_{2}\left[\left(1m\right)I_{H0}e^{j\delta}mI_{G0}\right]}{Z_{0}L_{2}\left(md\right)} $$
(14)
2.3.5 Step 5: estimate the zerosequence impedance of line 1
The fact that V_{Tap0} is the same when calculated from terminal G or terminal T is used to estimate the zerosequence impedance of Line 1 as
$$ Z_{0}L_{1}=\frac{\left(d\times Z_{0}L_{2}\times I_{G0}\right)V_{G0}}{I_{T0}}Z_{T0} $$
(15)
Figure 5 shows a flow diagram which summarizes Approach 2 for calculating the zerosequence line impedance.
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.