Estimating zero-sequence impedance of three-terminal transmission line and Thevenin impedance using relay measurement data

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, 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 negative-sequence voltage at the tap point

  

• V_G2,V_H2,V_T2 are the negative-sequence fault voltages at terminals G, H and T respectively

• I_G2,I_H2,I_T2 are the negative-sequence fault currents at terminals G,H and T respectively

• V_G0,V_H0,V_T0 are the zero-sequence fault voltages at terminals G, H and T respectively

• I_G0,I_H0,I_T0 are the zero-sequence fault currents at terminals G,H and T respectively

• Z₂L₁ is the negative-sequence impedance of Line 1

• Z₂L₂ is the negative-sequence impedance of Line 2

• Z₀L₁ is the zero-sequence impedance of Line 1

• Z₀L₂ is the zero-sequence 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 negative-sequence voltage and current phasors at terminal T are V_T2e^jδ1 and I_T2e^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 negative-sequence voltage at the fault point, V_F2, must be the same when calculated from either terminal G or H as

$$\begin{array}{*{20}l} &Terminal G: \\ &V_{F2}=V_{G2}-\left[mZ_{2}L_{2}\times I_{G2}]-[(m-d)Z_{2}L_{2}\times I_{T2}e^{j\delta1}\right]\\ &Terminal H: \\ &V_{F2}=V_{H2}e^{j\delta2}-\left[(1-m)Z_{2}L_{2}\times I_{H2}e^{j\delta2}\right] \end{array} $$

(3)

Therefore, e^jδ2 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, V_F0, can be calculated from terminal G and H as

$$\begin{array}{*{20}l} &{}Terminal G: \\ &{}V_{F0}\! =\! V_{G0}\! -\! (mZ_{0}L_{2}\times I_{G0})-\left[(m-d)Z_{0}L_{2}\times I_{T0}e^{j\delta1}\right] \\ &{}Terminal H: \\ &{}V_{F0}=V_{H0}e^{j\delta2}-\left[(1-m)Z_{0}L_{2}\times I_{H0}e^{j\delta2}\right] \end{array} $$

(5)

Since V_F0 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, 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 zero-sequence 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 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, 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 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.

2.3 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, I_T2, the negative-sequence 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 negative-sequence 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

$$ {}e^{j\delta}=\frac{V_{G2}-\left(mZ_{2}L_{2}\times I_{G2}\right)-\left[(m-d)\times Z_{2}L_{2}\times I_{T2}\right]}{V_{H2}-\left[(1-m)Z_{2}L_{2}\times I_{H2}\right]} $$

(13)

The new set of sequence voltage phasors at terminal H are V_H1e^jδ,V_H2e^jδ, and V_H0e^jδ. Similarly, the new set of current phasors at terminal H are I_H1e^jδ,I_H2e^jδ, and I_H0e^jδ.

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, V_F0, 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, I_T0, 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 V_Tap0 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.

This section demonstrates the efficacy of Approach 1 and Approach 2 in estimating the zero-sequence line impedance of a three-terminal line during a single line-to-ground fault. The system parameters used for the study are described below.

The three-terminal network shown in Fig. 1 was modeled in PSCAD simulation software [17]. The model was used to replicate actual short-circuit faults that occur on a transmission line and generate the corresponding voltage and current waveforms.

The positive- and zero-sequence source impedances at terminal G are Z_G1 = 3.75 ∠86° Ω and Z_G0 = 11.25 ∠86° Ω, respectively. The positive- and zero-sequence source impedances at terminal H are Z_H1 = 12 ∠80° Ω and Z_H0 = 36 ∠80° Ω, respectively. The positive- and zero-sequence source impedances at terminal T are Z_T1 = 5 ∠83° Ω and Z_T0 = 12 ∠83° Ω, respectively.

The distance between terminal G and the tap point is 6.21 miles. Line 2 is 18.64 miles long and has the same configuration as shown in Fig. 6. Shield wires S₁ and S₂ protect phase conductors A, B, and C from direct lightning strikes. The positive- and zero-sequence line impedances calculated using modified Carson’s model [18] at an earth resistivity value of 100 Ω-m are Z₁L₂ = 16.15 ∠70° Ω and Z₀L₂ = 34.87 ∠64° Ω, respectively. Line 1 is 6.21 miles long and also has the same configuration as Fig. 6. However, the earth resistivity was changed to 80 Ω-m. The positive- and zero-sequence line impedances are Z₁L₁ = 5.38 ∠70° Ω and Z₀L₁ = 11.55 ∠65° Ω, respectively.

When a bolted phase A-to-ground 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 zero-sequence 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 zero-sequence line impedance estimation will not be affected by it.

Although Approach 2 uses data from only two terminals to estimate the zero-sequence line impedance of all the sections of the three-terminal 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 zero-sequence estimation was performed, was a two-terminal line whereas three-terminal transmission lines are analyzed in this paper. In a three-terminal transmission line, there is short-circuit current contribution from all three terminals unlike two-terminal 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 three-terminal line is also different from that of a two-terminal line. Moreover, there is a need to synchronize measurements from different terminals of the line based on data availability before estimating the zero-sequence line impedance. Furthermore, careful system analysis is required to calculate the zero-sequence impedance of both Line 1 and Line 2 as seen in the two approaches discussed above unlike having to estimate the zero-sequence impedance of only one line as in [5]. In addition to estimating the zero-sequence 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.

Voltage and current waveforms captured by IEDs during a fault can be used to estimate the Thevenin impedance of the transmission network upstream from the monitoring location as illustrated in Fig. 7.

Figures 8, 9 and 10 represent the negative-, zero- and positive-sequence network respectively. The rest of the down stream network is represented as RDSN in Figs. 8, 9 and 10.

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 positive-sequence impedance equals the negative-sequence impedance in static electrical components such as lines and transformers. However, the sequence impedances are not equal to each other in rotating machines.

To demonstrate the approach described in Section 4 for estimating the Thevenin impedance of the upstream network, the three-terminal line test case analyzed in Section 3 is used. The same fault scenario, a single line-to-ground (SLG) fault on phase A at a distance of 13 miles from terminal G with monitors at terminal G, H and T is used. The current and voltage waveforms recorded at terminal G are shown in Fig. 11.

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 positive-sequence 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.