Three-terminal lines do pose a significant challenge to the task of validating the zero-sequence line impedance. The third terminal contributes to the total fault current and changes the impedance equations which are commonly used for two-terminal lines. Furthermore, with the introduction of a third terminal, there are now two lines whose zero-sequence line impedances have to be validated from a single fault event. Based on this aforementioned background, this section presents two approaches for calculating the zero-sequence line impedance of three-terminal 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 zero-sequence 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 three-terminal 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 line-to-ground 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 zero-sequence impedance of Line 1 and Line 2 are outlined below.
2.1.1 Step 1: synchronize terminal T with terminal G
Consider the negative-sequence network of a three-terminal line during a single or a double line-to-ground 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, ejδ1, is applied to the terminal T measurements. The value of ejδ1 can be determined by calculating VTap2 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:
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VTap2 is the negative-sequence voltage at the tap point
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VG2,VH2,VT2 are the negative-sequence fault voltages at terminals G, H and T respectively
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IG2,IH2,IT2 are the negative-sequence fault currents at terminals G,H and T respectively
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VG0,VH0,VT0 are the zero-sequence fault voltages at terminals G, H and T respectively
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IG0,IH0,IT0 are the zero-sequence fault currents at terminals G,H and T respectively
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Z2L1 is the negative-sequence impedance of Line 1
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Z2L2 is the negative-sequence impedance of Line 2
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Z0L1 is the zero-sequence impedance of Line 1
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Z0L2 is the zero-sequence impedance of Line 2
Since terminals G and T operate in parallel to feed the fault, VTap2 should be equal when calculated from either terminal. From this principle, ejδ1 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 ejδ1.
2.1.2 Step 2: synchronize terminal H with terminals T and G
After the synchronization process in Step 1, the new negative-sequence voltage and current phasors at terminal T are VT2ejδ1 and IT2ejδ1, respectively, and at terminal G are VG2 and IG2, respectively. This step synchronizes the phasors at terminal H with the phasors at terminals G and T. For this purpose, a second synchronizing operator, ejδ2, is applied to the measurements at terminal H. The value of ejδ2 can be calculated from the fact that the negative-sequence voltage at the fault point, VF2, must be the same when calculated from either terminal G or H as
$$\begin{array}{*{20}l} &Terminal G: \\ &V_{F2}=V_{G2}-\left[mZ_{2}L_{2}\times I_{G2}]-[(m-d)Z_{2}L_{2}\times I_{T2}e^{j\delta1}\right]\\ &Terminal H: \\ &V_{F2}=V_{H2}e^{j\delta2}-\left[(1-m)Z_{2}L_{2}\times I_{H2}e^{j\delta2}\right] \end{array} $$
(3)
Therefore, ejδ2 is given by
$$ {} e^{j\delta2} = \frac{V_{G2}-\left[mZ_{2}L_{2}\times I_{G2}\right]-\left[\left(m-d\right)Z_{2}L_{2}\times I_{T2}e^{j\delta1}\right]}{V_{H2}-\left[(1-m)Z_{2}L_{2}\times I_{H2}\right]} $$
(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 zero-sequence impedance of Line 2
To estimate the zero-sequence impedance of Line 2, the zero-sequence network during a single or double line-to-ground fault is shown in Fig. 3. The zero-sequence voltage at the fault point, VF0, can be calculated from terminal G and H as
$$\begin{array}{*{20}l} &{}Terminal G: \\ &{}V_{F0}\! =\! V_{G0}\! -\! (mZ_{0}L_{2}\times I_{G0})-\left[(m-d)Z_{0}L_{2}\times I_{T0}e^{j\delta1}\right] \\ &{}Terminal H: \\ &{}V_{F0}=V_{H0}e^{j\delta2}-\left[(1-m)Z_{0}L_{2}\times I_{H0}e^{j\delta2}\right] \end{array} $$
(5)
Since VF0 is equal when calculated from either line terminal, equate the two equations in (5) to solve for Z0L2 as
$$ Z_{0}L_{2} = \frac{V_{G0}-V_{H0}e^{j\delta2}}{mI_{G0}+\left(m-d\right)I_{T0}e^{j\delta1}-\left(1-m\right)I_{H0}e^{j\delta2}} $$
(6)
2.1.4 Step 4: calculate the zero-sequence impedance of Line 1
To estimate the zero-sequence impedance of Line 1, Z0L1, calculate the zero-sequence voltage at the tap point, VTap0, 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 VT0 is the zero-sequence fault voltages at terminal T. Because VTap0 from terminal G is equal to that from terminal T, we can use (7) to solve for Z0L1 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 zero-sequence 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 negative-sequence network shown in Fig. 2 is used for this purpose. When the fault is between terminal H and the tap point, VTap2 from the other two terminals are equal. Therefore, to identify the faulted section of the feeder, the approach consists of calculating VTap2 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 |VTap2| from two of the terminals will be an exact match while |VTap2| from the third terminal will be different. The fault is expected to lie between that third terminal and the tap point. Next, apply one-ended 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 zero-sequence line impedance.
2.2 Assumptions
Assumptions made by Approach 1 when estimating the zero-sequence impedances of Line 1 and Line 2 are summarized below:
1. Line 1 and Line 2 are homogeneous
2. Zero-sequence 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.
Zero-sequence mutual coupling arises when transmission lines are physically very close or parallel to each other. An example where zero-sequence mutual coupling could be present is when there are two sets of three-phase lines running on same towers. Though this configuration of lines is occasionally found in two-terminal transmission lines, it is highly unlikely and rare to have three-terminal transmission line with such a wire configuration that would have zero-sequence mutual coupling.
2.3 Approach 2 for estimating zero-sequence line impedance: data from two terminals
Although Approach 1 can successfully validate the zero-sequence impedance of three-terminal 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 zero-sequence impedance of three-terminal 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 zero-sequence impedance of Line 1 and Line 2 in such a scenario is described below:
2.3.1 Step 1: estimate the negative-sequence current from terminal T
To estimate the negative-sequence fault current from terminal T, IT2, the negative-sequence network shown in Fig. 2 is used. The approach consists of calculating VTap2 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 ZT2 is the negative-sequence source impedance of terminal T. Since VTap2 is equal when calculated from either terminal, IT2 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, ejδ, to the terminal H measurements so as to align the measurements at this terminal with those at terminals G and T. The fact that VF2 is the same when calculated from terminals G and H is used to calculate the synchronization operator as
$$ {}e^{j\delta}=\frac{V_{G2}-\left(mZ_{2}L_{2}\times I_{G2}\right)-\left[(m-d)\times Z_{2}L_{2}\times I_{T2}\right]}{V_{H2}-\left[(1-m)Z_{2}L_{2}\times I_{H2}\right]} $$
(13)
The new set of sequence voltage phasors at terminal H are VH1ejδ,VH2ejδ, and VH0ejδ. Similarly, the new set of current phasors at terminal H are IH1ejδ,IH2ejδ, and IH0ejδ.
2.3.3 Step 3: estimate the zero-sequence impedance of line 2
The zero-sequence line impedance of Line 2 is estimated from terminal H measurements using any zero-sequence line impedance estimation algorithm for two-terminal line which uses measurements from one terminal only [5, 11]. The algorithm to estimate the zero-sequence line impedance of a two-terminal 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 zero-sequence current from terminal T
The zero-sequence voltage at the fault point, VF0, is the same when calculated from terminal G or terminal H. This principle is used to calculate the zero-sequence fault current contributed by terminal T, IT0, as
$$ I_{T0}=\frac{V_{G0}-V_{H0}e^{j\delta}+Z_{0}L_{2}\left[\left(1-m\right)I_{H0}e^{j\delta}-mI_{G0}\right]}{Z_{0}L_{2}\left(m-d\right)} $$
(14)
2.3.5 Step 5: estimate the zero-sequence impedance of line 1
The fact that VTap0 is the same when calculated from terminal G or terminal T is used to estimate the zero-sequence 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 zero-sequence line impedance.
2.4 Assumptions:
Assumptions made by Approach 2 when estimating the zero-sequence line impedance are summarized below:
1. Line 1 and Line 2 are homogeneous
2. Fault location is known
3. Fault resistance is zero
4. Zero-sequence mutual coupling is absent
The assumptions that the transmission network is homogeneous and the absence of zero-sequence 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 zero-sequence line impedance of the lines.
In utility/industry practice, the process of estimating the zero-sequence 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 short-circuit 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 real-world field data in [5]. However, having to extract more information from less data requires us to make such an assumption.