Two-Terminal Algorithm Analysis for Unsymmetrical Fault Location on 110 kV Lines

energies Article Two-Terminal Algorithm Analysis for Unsymmetrical Fault Location on 110 kV Lines Zuzana Bukvisova *, Jaroslava Orsagova , David Topolanek and Petr Toman Department of Electrical Power Engineering, Brno University of Technology, Technicka 12, 61600 Brno, Czech Republic; [email protected].cz (J.O.); [email protected] (D.T.); [email protected].cz (P.T.) *Correspondence: [email protected].cz; Tel.: +420-606-202-347 Received: 5 March 2019; Accepted: 25 March 2019; Published: 27 March 2019   Abstract: This work analyses a two-terminal algorithm designed to locate unsymmetrical faults on 110 kV power transmission lines. The algorithm processes synchronized voltage and current data obtained from both ends of the protected transmission line and calculates the distance of the fault. It is based on decomposing the equivalent circuit into the positive-, negativeand zero-sequence components and finding the point where the output voltages of the right and the left side of the transmission line are equal. Compared to the conventional distance relay locator, the accuracy of this method is higher and less influenced by the fault resistance, the parallel-operated line effect and line asymmetry, as discussed in this work. It is, however, very sensitive to the synchronization accuracy. The mathematical model of the power system was created in the PSCAD (Power Systems Computer Aided Design) environment and the computational algorithm was implemented in Mathematica software. Keywords: unsymmetrical fault location; two-terminal algorithm; sequence components 1. Introduction To locate a fault during single-phase faults in a 110 kV line, a fault locator, which is one of the functions of distance protection, is currently used. The basic parameters of a protected line are set in the relay and used to compute the fault distance and usually to initiate the locator function as well. Basically, after the pick up or the trip of the distance protection, a short-circuit loop is determined and the currents and voltages measured in this loop are used for the short-circuit loop reactance and resistance calculation. After that, a fault loop and its impedance ZLf through the measured phase currents and voltages are determined. If a ground fault location is calculated, it is necessary to take account of values of the residual compensation factors R E /R 1 ,X E /X 1 as well. The computational algorithm used for the fault impedance ZLf determination is based on the solution of the short-circuit loop using the value of the voltage measured at the protection connection point, the current in the forward direction (to the fault) and the current in the backward direction (back to the point of measurement). In the case of a single-phase fault in phase i(i= 1, 2, 3) it will be: ZLf =ULi ILi−kEIE . (1) The fault distance is evaluated from the reactance to the fault XLf , which is the imaginary component of the fault impedance ZLf . Thus, the correct evaluation of the single-phase fault distance is always strongly dependent on the reactance per kilometer XL and residual compensation factor kE of the protected line set in the distance relay. Since most of the 110 kV lines are not fully transposed, fault locator errors are often caused by unequal impedance values of the phase-ground loops and mutual impedances of these loops. Another Energies 2019,12, 1193; doi:10.3390/en12071193 www.mdpi.com/journal/energies Energies 2019,12, 1193 2 of 14 error can be caused by non-homogeneity of the transmission line. Non-homogeneity means that the line parameters are not constant for the whole line length but vary in individual sections. Typical examples are changes in cross-section or different ground wire conductivity. Negative effect of the fault resistance on fault location is described in [ 1 , 2 ]. In systems with parallel-operated lines, the accuracy of the fault location is also greatly influenced by mutual impedances. In general, the ground wire conductivity and the short-circuit contribution of the parallel line power supply will have the major influence on the locator error rate. Many researchers have focused their attention on possible error elimination. The proposed algorithms differ depending on the input data they have available and the method of calculation. The methods can be divided into three main groups: •Travelling wave-based methods, •Artificial intelligence-based methods, •Impedance-based methods. 1.1. Travelling Waveand Artificial Intelligence-Based Methods Travelling wave-based computational methods are usually based on the correlation between the forward and backward waves that travel alongside the transmission line. When a fault occurs, the waves travel from the point of the fault to both ends of the transmission line. The fault is then identified using the transient-state analysis. Some studies [ 3 , 4 ] suggest creating a database of various fault scenarios to make the transient analysis easier. These methods can work with oneor multiple-terminal data. The accuracy of these algorithms is mainly dependent on the data sampling rate. For the fault detection, the wavelet transform proves to be very reliable. Intelligent techniques help to improve efficiency of the fault detection and classification. The advantage of using neural network is its ability to recognize a pattern and categorize the data. According to [5], the most used techniques based on the artificial intelligence are: •Expert System Techniques, •Artificial Neural Networks, •Fuzzy Logic Systems. Recently, a use of time-time (TT) transform in signal processing has been discussed [ 6 ]. Reference [ 7 ] proposes to apply TT-transform to series-compensated lines, which proved to be efficient even when processing a signal influenced by high noise. In [ 8 ], the TT-transform and a fault classification based on support vector machine (SVM) is used to locate fault on a hybrid line. Although this technique processes transient voltage signal obtained from only one end, the fault was identified and located with a high accuracy. 1.2. Impedance-Based Methods The basic principle of the impedance-based algorithms for the fault location calculation is simple. The protection relay uses positive and zero sequence impedances and measured voltages and currents to determine the distance to the fault by calculating the impedance, as described by Equation (1). These methods are simple and commonly used and their accuracy can be significantly improved if data from multiple terminals is acquired. Algorithms that use only local measurement data are called single-ended (or one-terminal) algorithms. These methods are often implemented, e.g., in microprocessor-based protective relays [9,10] . Their advantages are simplicity due to their lack of communication requirements. However, the accuracy of these methods is greatly influenced by fault resistance, load flow and source impedances. In [ 11 ], a single-ended technique, which is based on using the current and voltage measurements at local terminal and an estimated short-circuit capacity of the remote system, is proposed. Although this method is not unaffected by the problems mentioned above, the errors were within acceptable limits. Energies 2019,12, 1193 3 of 14 Two-terminal algorithms work with the measurements from both ends of the transmission line. These algorithms can be further divided into methods using synchronized or unsynchronized measurements. It is also possible to process data from multiple terminals to improve the accuracy of the calculation. Using the synchronized two-terminal voltage and current phasors, the calculation of the fault location is significantly improved [ 1 ]. The data collected from digital recorders is evaluated in a central computer using specialized software. With the telecommunications development, the synchronization and fast and reliable measured phasor data exchange are becoming easier and therefore it is possible to implement these algorithms directly into the protection. The advantages of these methods are: • Elimination of fault location errors caused by inaccuracies in residual compensation factors determination, •Suppression of an error caused by the fault resistance, •Reduction of the effects of mutual coupling and line asymmetry. The impedance-based methods are currently the most widely used methods of fault location and with the development and installation of the phasor measurement units (PMUs) or digital relays with global positioning systems, the techniques based on the fundamental power frequency components can be improved. The biggest advantage of these techniques, proposed e.g., in [ 12 – 14 ], is that the error caused by variations in the source and fault impedances is eliminated. Extensive placing of the PMUs is, however, very limited by the high installation costs and therefore, some works focus on developing an optimal PMU placement strategy [ 12 ]. Moreover, the developed techniques are often influenced by other factors, such as need of high data sampling rate [14]. Reference [ 13 ] proposes a method that utilizes synchronized voltage and current data derived at both ends of the transmission line and expresses the voltage across the fault in terms of the measured data. This work is based on similar principle and it will be discussed later in this text. Compared to [ 13 ], however, the exact evaluation of the fault distance is not performed. Instead, the accuracy of the analyzed algorithm is improved by using not only positive-, but negative-sequence components as well. Then, the least square method is applied to find the point of the fault. Some papers [ 15 – 17 ] try to remove the need of obtaining synchronized data. Reference [ 15 ] suggests modifying the technique introduced in [ 13 ] by considering only the magnitudes of the voltage at the fault point. This assumption allows to use unsynchronized phasors of measured voltages and currents and remove the error caused by the unsynchronized data. Methods described in [ 16 , 17 ] are based on a simple assumption that a fault impedance is purely resistive. Finding a solution to this condition is then used to find the synchronization angle. Although these algorithms enable to use the unsynchronized measurements, which is a big improvement, the main aim of this work is to find a usable method for the fault location on 110 kV lines in the Czech Republic, which are short and often parallel-operated. Methods proposed in [ 15 – 17 ] do not deal with the effect of the parallel line. In this work, the effect of mutual coupling is fully considered. The rest of the paper is organized as follows: Section 2describes the basic principle of the algorithm. Section 3discusses the model used for testing the method and results for various scenarios. Section 4gives the conclusions. 2. Description of the Analyzed Algorithm Firstly, a method using the one-terminal approach will be described to outline a basic principle of the proposed impedance-based algorithm and the errors that can occur. The parallel-operated line effect is described here as well. Later in this section, an analyzed two-terminal algorithm will be discussed. Energies 2019,12, 1193 4 of 14 2.1. Basic Description of the One-Terminal Algorithm Figure 1displays a situation when a single-phase fault with a fault resistance R f occurs in a transmission line. The fault is located in the distance dmeasured from the locator at the x-point, the total line length is l. Using this example, an idea of one-terminal algorithm errors can be given. Energies 2018, 11, x 4 of 15 2.1. Basic Description of the One-Terminal Algorithm Figure 1 displays a situation when a single-phase fault with a fault resistance Rf occurs in a transmission line. The fault is located in the distance d measured from the locator at the x-point, the total line length is l. Using this example, an idea of one-terminal algorithm errors can be given. Figure 1. A simplified faulty transmission line scheme. An equivalent circuit is formed by the interconnection of impedance sequence component circuits and the fault resistance—see Error! Reference source not found.. The indexes for the positive, negative and zero-sequence components are 1, 2 and 0, respectively. If the Kirchhoff’s circuit laws are applied in individual loops of the equivalent circuit, an evaluation of the fault distance d using the current x Iand voltage x U ratio used by the locator placed at the x-point can be done: () 1f 2f 0f 1x 1L 2x 1L 0x 0L 0x 1L 0x 1L 1x 2x 0x x x 1x2x0x 1x2x0x U U U dIZ IZ IZ IZ IZ UU U U IIII III +++ + + + − ++ == ++ ++ , (2) ()() 1f 2f 0f 1L 1x 2x 0x 0x 0L 1L x x1x2x0x ..UU U dZI I I IdZ Z U IIII +++ ++ + − =++ . (3) If 0L 1L E 1L 3 Z ZkZ−= (where 0L Z and 1L Z are line impedances per unit length of a zeroand positive-sequence component, respectively, and E k is a residual compensation factor) is assumed, then: 0x E 1L 1f 2f 0f 1f 2f 0fxEx 1L 1L E xx1x2x0x x1x2x0x 31 IkZ U U U U U U UI dZ dZ k I I II I I II I   ++ ++ =+ + = + +   ++ ++   , (4) where the voltage at the fault point can be written as: ()()()() 1f 2f 0f f 1x 1y f 2x 2y f 0x 0y f Ex Ey 33 3U U U RI I RI I RI I RI I++= += + = + = + . (5) Applying these equations, the impedance measured by the locator at the x-point is: Ex Ey xEx 1L E f xxx .1 II UI dZ k R III +  =++   (6) and similarly, for the locator placed at the y-point: () yEyExEy 1L E f yyy .1 UIII ldZ k R III  + =− + +    . (7) If the line parameters 1L Z and E k, fault resistance Rf and the earth currents Ey I and Ex I acquired from both ends are known, the Equations (6) and (7) can be used to precisely calculate the fault distance. If these values are not obtained, the locator algorithm correction method will always be just an estimation and therefore potentially able to actually increase the locator error. Even more complicated situation occurs for the parallel-operated lines, when the equivalent circuit contains also the parallel-line sequence impedances—in the Error! Reference source not found. drawn in red. Figure 1. A simplified faulty transmission line scheme. An equivalent circuit is formed by the interconnection of impedance sequence component circuits and the fault resistance—see Figure 2. The indexes for the positive, negative and zero-sequence components are 1, 2 and 0, respectively. If the Kirchhoff’s circuit laws are applied in individual loops of the equivalent circuit, an evaluation of the fault distance dusing the current Ix and voltage Ux ratio used by the locator placed at the x-point can be done: Ux Ix =U1x +U2x +U0x I1x +I2x +I0x =U1f +U2f +U0f +dI1xZ1L +I2xZ1L +I0xZ0L +I0xZ1L −I0xZ1L I1x +I2x +I0x , (2) Ux Ix =U1f +U2f +U0f +d.Z1L.I1x +I2x +I0x+I0xdZ0L −Z1L I1x +I2x +I0x . (3) If Z0L −Z1L = 3 kEZ1L (where Z0L and Z1L are line impedances per unit length of a zeroand positive-sequence component, respectively, and kE is a residual compensation factor) is assumed, then: Ux Ix =d Z1L +I0x3kEZ1L Ix!+U1f +U2f +U0f I1x +I2x +I0x =dZ1L1+kE IEx Ix+U1f +U2f +U0f I1x +I2x +I0x , (4) where the voltage at the fault point can be written as: U1f +U2f +U0f =3RfI1x +I1y=3RfI2x +I2y=3RfI0x +I0y=RfIEx +IEy. (5) Applying these equations, the impedance measured by the locator at the x-point is: Ux Ix =d.Z1L1+kE IEx Ix+Rf IEx +IEy Ix (6) and similarly, for the locator placed at the y-point: Uy Iy =(l−d).Z1L 1+kE IEy Iy!+Rf IEx +IEy Iy . (7) If the line parameters Z1L and kE , fault resistance R f and the earth currents IEy and IEx acquired from both ends are known, the Equations (6) and (7) can be used to precisely calculate the fault distance. If these values are not obtained, the locator algorithm correction method will always be just an estimation and therefore potentially able to actually increase the locator error. Even more Energies 2019,12, 1193 5 of 14 complicated situation occurs for the parallel-operated lines, when the equivalent circuit contains also the parallel-line sequence impedances—in the Figure 2drawn in red. Energies 2018, 11, x 5 of 15 Figure 2. The equivalent circuit for parallel-operated transmission lines. From the circuit diagram it is clear that the positive and the negative sequence components of the impedance measured by the locator remain the same, but the zero sequence does not. This is caused by the common path of the earth current for both transmission lines formed by the ground and the ground wire. The mutual coupling is symbolized in the zero-sequence component by four additional induced voltage sources. Two of them are part of the zero-sequence component of the faulted section. These voltages are induced by the zero-sequence current 0x par I of the parallel line: 0A 0M 0x par ..UdZI=, (8) () 0B 0M 0x par ..UldZI=− . (9) The remaining two voltage sources are located in the zero-sequence component of the healthy line and are induced by the zero-sequence current 0x I of the faulty line: 0C 0M 0x ..UdZI=, (10) () () 0D 0M 0x f ..UldZII=− − . (11) Voltages can be substituted for product of currents that induce these voltages and mutual zero sequence impedance 0M Z according to Equations (8)–(11). Then, the equivalent circuit of the zerosequence impedance can be modified, as shown in Figure 3. The locator placed at the x-point calculates the impedance as () 1x 2x 0xx x1x2x0x 1f 2f 0f 1x 1L 2x 1L 0x 0L 0x 1L 0x 1L 0x par 0M 0y 0L 1x 2x 0x UU U U IIII U U U dIZ dIZ dIZ dIZ dIZ lI Z l dIZ II I ++ =++ +++ + + + − + −− =++ , (12) () xE yE 0x par 0M 0y 0L xEx 1L E f xxxxx .1 () II I ZIZ UI dZ k R l l d I III I +  =++ + −−   . (13) Figure 2. The equivalent circuit for parallel-operated transmission lines. From the circuit diagram it is clear that the positive and the negative sequence components of the impedance measured by the locator remain the same, but the zero sequence does not. This is caused by the common path of the earth current for both transmission lines formed by the ground and the ground wire. The mutual coupling is symbolized in the zero-sequence component by four additional induced voltage sources. Two of them are part of the zero-sequence component of the faulted section. These voltages are induced by the zero-sequence current I0x par of the parallel line: U0A =d.Z0M.I0x par, (8) U0B =(l−d).Z0M.I0x par. (9) The remaining two voltage sources are located in the zero-sequence component of the healthy line and are induced by the zero-sequence current I0x of the faulty line: U0C =d.Z0M.I0x, (10) U0D =(l−d).Z0M.I0x −If. (11) Voltages can be substituted for product of currents that induce these voltages and mutual zero sequence impedance Z0M according to Equations (8)–(11). Then, the equivalent circuit of the zero-sequence impedance can be modified, as shown in Figure 3. Energies 2019,12, 1193 6 of 14 The locator placed at the x-point calculates the impedance as Ux Ix=U1x+U2x+U0x I1x+I2x+I0x =U1f+U2f+U0f+dI1xZ1L+dI2xZ1L+dI0xZ0L+dI0xZ1L−dI0xZ1L+lI0x parZ0M−(l−d)I0yZ0L I1x+I2x+I0x , (12) Ux Ix =d.Z1L1+kE IEx Ix+RfIxE +IyE Ix +lI0x parZ0M Ix −(l−d)I0y Z0L Ix . (13) For the locator placed at the y-point can be written as: Uy Iy =(l−d).Z1L 1+kE IEy Iy!+RfIEx +IEy Iy +lI0y parZ0M Iy −(l−d)I0x Z0L Iy . (14) It is obvious that due to the number of quantities that influence the impedance calculation, it is much more effective to use two-terminal algorithms. Energies 2018, 11, x 6 of 15 For the locator placed at the y-point can be written as: () () Ex Ey yEy0y par0M 0x 0L 1L E f yyyyy .1 () II UI IZ IZ ldZ k R l ld IIIII +  =− + + + −−    . (14) It is obvious that due to the number of quantities that influence the impedance calculation, it is much more effective to use two-terminal algorithms. . Figure 3. A zero-sequence component equivalent circuit for parallel-operated transmission lines. 2.2. Two-Terminal Algorithm The equivalent circuit used for two-terminal algorithms is displayed in Error! Reference source not found.. The transmission line between the source and the fault point is substituted by the distributed parameter line model with series impedances and shunt admittances. The complete equivalent circuit consists of individual sequence component circuits connected to the fault resistance Rf. Figure 4. An equivalent circuit for a single-phase fault. To calculate the fault location, individual sequence component voltages at the fault point (where i = 1, 2, 0) are determined. These voltages represent the output voltages of the respective transmission line model and can be calculated using the input values of currents in xi I and voltages in xi U at the x-point: out in in fx x xv cosh( . ) sinh( . ) ii i i ii i UU U dlIZ dl γγ == − , (15) or using the input values of currents in yi I and voltages in yi U at the y-point: Figure 3. A zero-sequence component equivalent circuit for parallel-operated transmission lines. 2.2. Two-Terminal Algorithm The equivalent circuit used for two-terminal algorithms is displayed in Figure 4. The transmission line between the source and the fault point is substituted by the distributed parameter line model with series impedances and shunt admittances. The complete equivalent circuit consists of individual sequence component circuits connected to the fault resistance Rf. To calculate the fault location, individual sequence component voltages at the fault point (where i= 1, 2, 0 ) are determined. These voltages represent the output voltages of the respective transmission line model and can be calculated using the input values of currents Iin ix and voltages Uin ix at the x-point: Uif=Uout ix=Uin ixcosh(γid.l)−Iin ixZivsinh(γid.l), (15) or using the input values of currents Iin iyand voltages Uin iyat the y-point: Uif=Uout iy=Uin iycosh(γi(1−d).l)−Iin iyZivsinh(γi(1−d).l), (16) where γi=pZiL.YiL is the propagation constant and Ziv =rZiL YiL is the characteristic impedance of the protected transmission line (i= 1, 2, 0 for the positive-, negativeand zero-sequence component of the line impedance per unit length ZiLand the line admittance per unit length YiL). If Uif=Uout ix=Uout iy , then the fault location dcan be determined. For this purpose, the positive-sequence component can be used because its parameters γ1=pZ1L.Y1L and Z1v =rZ1L Y1L are known and their calculation is not influenced by the soil resistivity. Energies 2019,12, 1193 7 of 14 Energies 2018, 11, x 6 of 15 For the locator placed at the y-point can be written as: () () Ex Ey yEy0y par0M 0x 0L 1L E f yyyyy .1 () II UI IZ IZ ldZ k R l ld IIIII +  =− + + + −−    . (14) It is obvious that due to the number of quantities that influence the impedance calculation, it is much more effective to use two-terminal algorithms. . Figure 3. A zero-sequence component equivalent circuit for parallel-operated transmission lines. 2.2. Two-Terminal Algorithm The equivalent circuit used for two-terminal algorithms is displayed in Error! Reference source not found.. The transmission line between the source and the fault point is substituted by the distributed parameter line model with series impedances and shunt admittances. The complete equivalent circuit consists of individual sequence component circuits connected to the fault resistance Rf. Figure 4. An equivalent circuit for a single-phase fault. To calculate the fault location, individual sequence component voltages at the fault point (where i = 1, 2, 0) are determined. These voltages represent the output voltages of the respective transmission line model and can be calculated using the input values of currents in xi I and voltages in xi U at the x-point: out in in fx x xv cosh( . ) sinh( . ) ii i i ii i UU U dlIZ dl γγ == − , (15) or using the input values of currents in yi I and voltages in yi U at the y-point: Figure 4. An equivalent circuit for a single-phase fault. If the right sides of the Equations (15) and (16) are equal, then it is possible to use only the equations for the positive-sequence voltage at the fault point Uin 1x cosh(γ1d.l)−Iin 1xZ1vsinh(γ1d.l)−Uin 1y cosh(γ1(1−d).l) + Iin 1yZ1vsinh(γ1(1−d).l) = 0. (17) Assuming γ1=γ2=pZ1L.Y1L and Z1v =Z2v =rZ1L Y1L , to increase the calculation reliability, the negative-sequence voltage is used as well Uin 2x cosh(γ1d.l)−Iin 2xZ1vsinh(γ1d.l)−Uin 2y cosh(γ1(1−d).l) + Iin 2Z1vsinh(γ1(1−d).l) = 0. (18) For each equation, a deviation from zero for various values of dcan be determined ε1(d)=Uout 1x −Uout 1y , (19) ε2(d)=Uout 2x −Uout 2y . (20) After obtaining the deviations for positiveand negative-sequence component and applying the least square method, a simple function F is created. This function reaches its minimum in the distance d, which corresponds to the fault point: F(d) = |ε1(d)|2+|ε2(d)|2. (21) The output of this algorithm is the value of distance dwith the minimal F(d) value. 2.3. Synchronization To correctly calculate the distance of the fault dusing this algorithm, it is necessary to measure the input currents and voltages at xand y-points synchronously. This, however, is mostly impossible in current 110 kV distribution networks. The distance protection locators do not use mutual time synchronization; therefore, the data records must be additionally synchronized before performing the calculations. This can be done with use of the transient data records in the time domain, e.g., Energies 2019,12, 1193 8 of 14 capturing the moment of the transient inception (a sufficient data sampling rate required). Another method is based on the phasor correction by estimating the input quantities mutual phase shift. Using this estimated phase shift value, the corresponding quantities are shifted by angle δ -synchronization operator. If Iin xis the synchronizing quantity, then: •Phasor correction at the x-point: Iin x=Iin x∠(α−α),Uin x=Uin x∠(β−α), •Phasor correction at the y-point: Iin y=Iin y∠(ϕ−ϕ+δ),Uin y=Uin y∠(γ−ϕ+δ), where α , β,γ , ϕ are the angles of the unsynchronously recorded phasors and δ is the synchronization operator determined by the phase shift of the currents Iin xand Iin y. Considering the 110 kV distribution networks operation, the phase shift between the input currents can be determined using the short-circuit impedances Zxs and Zys representing the protected line load, as shown in a simplified diagram in Figure 5. According to [ 1 ], to estimate the synchronization operator δ , the input current and voltage data captured during the normal operation or during the fault (or their combination) can be used. The same principle of the synchronization in [13] is used. Energies 2018, 11, x 8 of 15 Figure 5. A simplified equivalent circuit for a single-phase fault. Another method is, as mentioned above, based on transient observation. To analyze the frequency components of the signal, the Fourier transform is commonly used. The problem with using Fourier transform is that it is not capable to determine when the particular frequency changes. To acquire information about both time and frequency, a short-time Fourier transform, which uses a sliding window, can be applied. This technique, however, limits the frequency resolution. Better solution is to use the wavelet transform [18]. The wavelet transform decomposes the signal into functions located both in Fourier and the real-time space. It is basically an infinite set of various transforms. The problem with records synchronization using the transient study is the unequal time that the transient needs to get from the fault point to the point of measurement. The transient arrives sooner at the closer terminal, so capturing the moment of the transient inception at both terminals and subsequent comparison can be quite inaccurate. Future research will be focused on this topic. The synchronization method should be chosen based on the data availability. However, the preferable one is the transient analysis method. If the required data are not available, the synchronization operator needs to be calculated. 3. Analysis of the Two-Terminal Algorithm Testing 3.1. Mathematical Model of a 110 kV Line The presented two-terminal algorithm was implemented in the Mathematica software (version 11.1.1.0, Wolfram Research, Champaign, IL, USA) to process the voltages and currents measured at both ends of the 110 kV line modelled in PSCAD software (version 4.6.0.0, Manitoba HVDC Research Centre, Winnipeg, MB, Canada). The voltage and current data are obtained synchronously, which means that quantities mutual time shift is zero. The created model consists of two non-transposed parallel-operated 110 kV lines with six conductors and one ground wire, as in Figure 6. One side of the tower carries the conductors of the first line, the other side carries the second line. The mathematical model of this line is built according to the real distribution parallel-operated 110 kV line. The transmission line is modelled using the frequency dependent line model. The soil resistivity is 50 Ωm, the length of the line is 27.88 km. There are six 240 AlFe4 conductors and a combined ground wire with 48 fibers (0.2 Ω DC resistance and 18 mm diameter). Line parameters were derived from series impedance and shunt admittance matrixes in PSCAD line constant program output file. The results are listed in Table 1. Figure 5. A simplified equivalent circuit for a single-phase fault. Another method is, as mentioned above, based on transient observation. To analyze the frequency components of the signal, the Fourier transform is commonly used. The problem with using Fourier transform is that it is not capable to determine when the particular frequency changes. To acquire information about both time and frequency, a short-time Fourier transform, which uses a sliding window, can be applied. This technique, however, limits the frequency resolution. Better solution is to use the wavelet transform [ 18 ]. The wavelet transform decomposes the signal into functions located both in Fourier and the real-time space. It is basically an infinite set of various transforms. The problem with records synchronization using the transient study is the unequal time that the transient needs to get from the fault point to the point of measurement. The transient arrives sooner at the closer terminal, so capturing the moment of the transient inception at both terminals and subsequent comparison can be quite inaccurate. Future research will be focused on this topic. The synchronization method should be chosen based on the data availability. However, the preferable one is the transient analysis method. If the required data are not available, the synchronization operator needs to be calculated. 3. Analysis of the Two-Terminal Algorithm Testing 3.1. Mathematical Model of a 110 kV Line The presented two-terminal algorithm was implemented in the Mathematica software (version 11.1.1.0, Wolfram Research, Champaign, IL, USA) to process the voltages and currents measured Energies 2019,12, 1193 9 of 14 at both ends of the 110 kV line modelled in PSCAD software (version 4.6.0.0, Manitoba HVDC Research Centre, Winnipeg, MB, Canada). The voltage and current data are obtained synchronously, which means that quantities mutual time shift is zero. The created model consists of two non-transposed parallel-operated 110 kV lines with six conductors and one ground wire, as in Figure 6. One side of the tower carries the conductors of the first line, the other side carries the second line. The mathematical model of this line is built according to the real distribution parallel-operated 110 kV line. Energies 2018, 11, x 9 of 15 Figure 6. Transmission line conductor arrangement. Table 1. Line parameters. Positiveand negative-sequence impedance () 121 1 j 3.513 j 11.096ZZ R X==+ = + Ω Per kilometer: () 1L 2L 1L 1L j 0.126 j 0.398 /kmZZ R X==+ = + Ω Zero-sequence impedance () 00 0 j 7.165 j 32.369ZR X=+ = + Ω Per kilometer: () 0L 0L 0L j 0.257 j 1.161 /kmZR X=+ = + Ω Positiveand negative-sequence admittance () 75 12 1 1 jB 2.788.10 j 8.263.10 SYY G −− ==− = − Per kilometer: () 86 1L 2L 1L 1L j 10 j 2.964.10 S/kmYY G B −− ==− = − Zero-sequence admittance () 75 00 0 j 2.788.10 j 4.851.10YG B −− =− = − Ω Per kilometer: () 86 0L 0L 0L j 10 j 1.740.10 S/kmYG B −− =− = − Residual compensation factor 0L 1L EREXE 1L j 0.346 j 0.639 3. ZZ kkk Z − ==+=+ Using the created PSCAD model, it is also possible to calculate the fault distance. To do this, the Equation (1) is used similarly to the common fault locator. The computational algorithm is built with logical blocks and functions implemented in the PSCAD library. It utilizes obtained current and voltage data processed using the Fast Fourier Transform. The output of this calculation can be compared to the output of the proposed algorithm. The simulation data are stored in a Comtrade file format. The PSCAD software contains a component called RTP/COMTRADE Recorder, which is able to record the simulated data a save them as Comtrade. Therefore, the data can be further viewed in a software that uses this format. 3.2. Impact of the Fault Resistance Several tests were performed to assess the impact of the fault resistance value on the proposed algorithm. The fault occurred in the L1 phase in the distance of 4.49 km from the terminal at the xpoint. The fault resistance was changed up to 100 Ω, which appears very rarely in the 110 kV network and even a fault with such high resistance was successfully localized, as seen in Error! Reference source not found., where the minimum of the function [ ] ln F( )d denotes the calculated fault location. Figure 6. Transmission line conductor arrangement. The transmission line is modelled using the frequency dependent line model. The soil resistivity is 50 Ω m, the length of the line is 27.88 km. There are six 240 AlFe4 conductors and a combined ground wire with 48 fibers (0.2 Ω DC resistance and 18 mm diameter). Line parameters were derived from series impedance and shunt admittance matrixes in PSCAD line constant program output file. The results are listed in Table 1. Table 1. Line parameters. Positiveand negative-sequence impedance Z1=Z2=R1+jX1=(3.513 +j 11.096)Ω Per kilometer : Z1L =Z2L =R1L +jX1L =(0.126 +j 0.398)Ω/km Zero-sequence impedance Z0=R0+jX0=(7.165 +j 32.369)Ω Per kilometer : Z0L =R0L +jX0L =(0.257 +j 1.161)Ω/km Positiveand negative-sequence admittance Y1=Y2=G1−jB1=2.788.10−7−j 8.263.10−5S Per kilometer : Y1L =Y2L =G1L −jB1L =10−8−j 2.964.10−6S/km Zero-sequence admittance Y0=G0−jB0=2.788.10−7−j 4.851.10−5Ω Per kilometer : Y0L =G0L −jB0L =10−8−j 1.740.10−6S/km Residual compensation factor kE=Z0L−Z1L 3.Z1L =kRE +jkXE =0.346 +j 0.639 Using the created PSCAD model, it is also possible to calculate the fault distance. To do this, the Equation (1) is used similarly to the common fault locator. The computational algorithm is built with logical blocks and functions implemented in the PSCAD library. It utilizes obtained current and voltage data processed using the Fast Fourier Transform. The output of this calculation can be compared to the output of the proposed algorithm.