energies

Article Communication Characteristics of Faulted Overhead High Voltage Power Lines at Low Radio Frequencies

Nermin Suljanovi´c 1, Aljo Mujˇci´c 1 and Matej Zajc 2,* ID

1 Faculty of Electrical Engineering, University of Tuzla, Franjevaˇcka2, 75000 Tuzla, Bosnia and Herzegovina; [email protected] (N.S.); [email protected] (A.M.) 2 Faculty of Electrical Engineering, University of Ljubljana, Trzaška 25, SI-1000 Ljubljana, Slovenia * Correspondence: [email protected]; Tel.: +386-1-4768-880

Received: 19 September 2017; Accepted: 6 November 2017; Published: 8 November 2017

Abstract: This paper derives a model of high-voltage overhead power line under fault conditions at low radio frequencies. The derived model is essential for design of communication systems to reliably transfer information over high voltage power lines. In addition, the model can also benefit advanced systems for power-line fault detection and classification exploiting the phenomenon of changed conditions on faulted power line, resulting in change of low radio frequency signal propagation. The methodology used in the paper is based on the multiconductor system analysis and propagation of electromagnetic waves over the power lines. The model for the high voltage power line under normal operation is validated using actual measurements obtained on 400 kV power line. The proposed model of faulted power lines extends the validated power-line model under normal operation. Simulation results are provided for typical power line faults and typical fault locations. Results clearly indicate sensitivity of power-line frequency response on different fault types.

Keywords: high-voltage power-line; communications; fault detection; signal propagation; modelling; power line carrier (PLC)

1. Introduction Electricity grids are in transition from fossil-based towards a low-carbon scenario, primarily based on renewable generation. This concept has led to the evolution of electrical grids to smart grids, dominantly focused on incorporation of renewable energy sources (RESs) into the electrical distribution grid and the ability to enable charging of an arbitrary number of electrical vehicles. This is a significant transition from traditional electrical (power) systems in which electricity demand was met by fossil-based bulk generated energy and conveyed to loads through transmission and distribution grids. The smart grid concept requires continuous development of power grids, at the transmission and distribution levels, developing new technologies and techniques as well as repurposing the existing ones. Transmission grids remain vital also in the smart grids due to numerous reasons such as to convey energy from large renewable generation locations to demand sites, to link national markets and allow transfer of renewable electricity surpluses, and to support intermittent renewable generation to operate more efficiently. The task is to ensure the power flow by maintaining grid stability at the required power quality levels. As investments are very high, great attention is devoted to monitoring and maintaining the transmission lines. A transmission grid consists of high-voltage (HV) power lines spread over a large geographic area. In alternating current (AC) grids, electricity is transferred as three-phase voltages and currents at 50/60 Hz, which is usually denoted as the power frequency. Besides electricity transfer, HV power lines have also being utilized for communication purposes to transfer relevant operational data and

Energies 2017, 10, 1801; doi:10.3390/en10111801 www.mdpi.com/journal/energies Energies 20172017,, 1010,, 18011801 2 of 24

lines have also being utilized for communication purposes to transfer relevant operational data and protection signals (Figure1 1).). WithWith thisthis aim,aim, high-frequencyhigh-frequency (HF)(HF) signalssignals cancan bebe injectedinjected intointo powerpower lineline conductorsconductors atat oneone terminalterminal using using appropriate appropriate frequency-selective frequency-selective circuits circuits (filters) (filters) and and extracted extracted at theat the other other terminal, terminal, not not interfering interfering with with power-frequency power-frequency “signals”. “signals”. Information Information is superimposed is superimposed into HFinto signals HF signals using using analog analog or digital or digital modulation. modulation.

Figure 1. High voltage power line utilized for communications.

This telecommunication technology founded on the HF signal transmission over HV power lines This telecommunication technology founded on the HF signal transmission over HV power is known as power line carrier (PLC). Nowadays HV PLC doesn’t provide throughput that can lines is known as power line carrier (PLC). Nowadays HV PLC doesn’t provide throughput that compete with optical fiber or fourth generation (4G) cellular systems, and its application is very can compete with optical fiber or fourth generation (4G) cellular systems, and its application is very limited to specific applications in power systems. On the other hand, knowledge gained in the limited to specific applications in power systems. On the other hand, knowledge gained in the domain of HF signal transmission over HV power lines and the phenomena accompanying it can be domain of HF signal transmission over HV power lines and the phenomena accompanying it can be used for some other advanced applications such as fault detection and localization. Mathematical used for some other advanced applications such as fault detection and localization. Mathematical models primarily derived for PLC communications are fundamental for the design of such advanced models primarily derived for PLC communications are fundamental for the design of such advanced systems. Technology deployed in smart transmission grids can utilize these data as input for real- systems. Technology deployed in smart transmission grids can utilize these data as input for real-time time monitoring of overhead power lines. monitoring of overhead power lines. Electrical grids are facing the incorporation of intermittent renewable energy resources and the Electrical grids are facing the incorporation of intermittent renewable energy resources and necessity for infrastructure reinforcement [1]. Transmission system operators (TSOs) consider the necessity for infrastructure reinforcement [1]. Transmission system operators (TSOs) consider different power-line monitoring technologies to enhance operational efficiency of the existing HV different power-line monitoring technologies to enhance operational efficiency of the existing HV power lines, such as Dynamic Line Rating (DLR) [2,3]. DLR is a sensor-based system that increases power lines, such as Dynamic Line Rating (DLR) [2,3]. DLR is a sensor-based system that increases the transmission capacity by utilizing real-time data and information. These data include various the transmission capacity by utilizing real-time data and information. These data include various factors that impact temperature of overhead conductors. With such an approach, the power line factors that impact temperature of overhead conductors. With such an approach, the power line transmission capacity can reach 110–130% of the designed limits, under suitable weather conditions. transmission capacity can reach 110–130% of the designed limits, under suitable weather conditions. Besides DLR sensors, information about the temperature of overhead power line conductors can be Besides DLR sensors, information about the temperature of overhead power line conductors can be extracted from knowledge about overhead line catenary. In particular, power line sag is directly extracted from knowledge about overhead line catenary. In particular, power line sag is directly related related to the weather conditions and temperature of conductors. In [4], the authors proposed the to the weather conditions and temperature of conductors. In [4], the authors proposed the application application of PLC signals for real-time sag monitoring of HV overhead power lines. The method of PLC signals for real-time sag monitoring of HV overhead power lines. The method determines the determines the average overhead conductor height variations in real-time, correlating mathematical average overhead conductor height variations in real-time, correlating mathematical models with the models with the processed PLC signals captured at power-line terminals. processed PLC signals captured at power-line terminals. Ice load can cause severe damage on overhead power lines. The number of icing disasters on Ice load can cause severe damage on overhead power lines. The number of icing disasters overhead power lines is increasing due to prevailing macroclimate, micrometeorological, and on overhead power lines is increasing due to prevailing macroclimate, micrometeorological, microtopography conditions [5]. There are several complex methods to obtain information about the and microtopography conditions [5]. There are several complex methods to obtain information state of ice accretion based on real-time measurements with specific sensor systems [6]. Another about the state of ice accretion based on real-time measurements with specific sensor systems [6]. approach for icing thickness monitoring is based on image recognition algorithms [5]. Ice load on Another approach for icing thickness monitoring is based on image recognition algorithms [5]. Ice load overhead line conductors causes PLC signal propagation changes, which can be utilized for the on overhead line conductors causes PLC signal propagation changes, which can be utilized for the detection of ice loads [7]. In other words, ice load detection can be implemented through processing detection of ice loads [7]. In other words, ice load detection can be implemented through processing of of PLC signals propagated over the HV power line. PLC signals propagated over the HV power line. Detection and classification of faults on HV power lines is crucial for stable and reliable power system operation. Various methods for detection, classification and location of faults on HV power

Energies 2017, 10, 1801 3 of 24

Detection and classification of faults on HV power lines is crucial for stable and reliable power system operation. Various methods for detection, classification and location of faults on HV power lines have been studied, taking advantage of digital signal processing algorithms and machine learning methods [8]. The common characteristic of all available approaches is to utilize measured phase voltage and current signals or fault-generated transient phenomena [9,10]. On the other hand, faults on power lines have a significant impact on frequency response. If a signal at PLC frequency is constantly injected, faults can be classified and located by processing the received signal at the line terminal. The authors of [11] proposed a concept in which faults in multiconductor overhead distribution networks are detected using PLC devices. The method is based on detecting differences between values of metrics related to the input impedance at frequencies utilized by the PLC systems. The purpose of this work is to provide the high-frequency model of faulted HV power line, which can be utilized for fault classification and localization purposes. This information is needed for future utilization of HV power lines for transmission of HF signals for various smart grid services. Analytical model is derived for the HV system under normal operation and evaluated with computer simulations validated with real HV power line measurements. This model is further extended for a faulted HV power-line and frequency response is obtained with computer simulations for different types of faults and deployed couplings. Different couplings are considered in order to determine the coupling with H(f ) most sensitive to power-line faults. The research presented in the paper is focused on high-frequency characteristics of HV power line under faulted operation. The main contributions of the paper are:

• Telegrapher’s equations for multiconductor lines are extended with equations that provide characteristics for available couplings connecting electronic equipment to HV power-line conductors. This approach has been validated with actual measurements on 400 kV power line and outer-to-middle phase coupling. • The methodology for calculation of high-frequency characteristics of faulted HV power lines is developed for the series and shunt faults. The presented methodology is based on the validated approach used for HV power line in normal operation.

2. From HV PLC Systems to Advanced Power-Line Monitoring Applications Reliable communications are fundamental component of modern power grids. The ability to utilize power lines for communications has attracted the research community for almost a century. At the transmission level, PLC presents one of the first communication technologies inside power systems. The range and quality of HV PLC communications strongly depends on the communication characteristics of the power lines (attenuation and noise), but also interference with other systems. HV overhead power lines radiate energy at high frequencies and interfere with other systems operating at the same frequencies. In Europe, the frequency range devoted to HV PLC system is from 30 kHz to 500 kHz [12–14]. HV PLC systems are not permitted to operate in the broadband, but rather in the frequency range which is a multiple of 4 kHz bands to limit the interference. Therefore, HV PLC systems operate in a narrowband at carrier frequencies, which are determined at the deployment site to prevent interference. Available data rates are low for this reason, even at high spectral efficient digital modulation techniques (number of bits per Hz that channel transfers). On the other side, the major advantages of HV PLC communications are high reliability of these systems and the capability to transfer information over long distances, both crucial inside the smart grid concept. HV PLC channels are characterized with very low attenuation, therefore communication signals can be transferred over very long distances. The distance is limited by the noise, generated by corona phenomena (discharges generated by rated high voltage) and interfering signals [12–14]. Even though corona noise intensity varies with power frequency voltage, it doesn’t make huge impact on ultra-narrow band systems. It means that ultra-narrow band communication signals can be reliably used for control and command signals, such as trip signal for distant protection of HV power lines. Consequently, for decades HV PLC systems have provided a communication link to reach different Energies 2017, 10, 1801 4 of 24 power system components for control reasons and applications that require high reliability, low delays, and small throughput.

EnergiesHV 2017 power, 10, 1801 lines employ three phase conductors (usually denoted as A, B and C) for electricity4 of 24 transfer and two shield wires for lightning protection. Schemes that describe how two terminals of communicationcommunication equipmentequipment areare connectedconnected toto phasephase conductorsconductors areare knownknown asas couplings.couplings. ShieldShield wireswires areare groundedgrounded atat eacheach power-linepower-line polepole andand cannotcannot bebe usedused for for signal signal transmission. transmission. CouplingCoupling equipmentequipment blocks blocks power-frequency power-frequency voltages voltages and forwardsand forwards HF communication HF communication signals. Communicationsignals. Communication signals aresignals blocked are blocked from propagating from propagating towards towards transformers transformers at both at both power-line power- terminalsline terminals by frequency-tuned by frequency-tuned circuits circuits named named line trapline trap units units (LTUs). (LTUs). PLC PLC channel channel characteristics characteristics are differentare different for individual for individual couplings couplings schemes. schemes. High-frequency High-frequency signal signal transmission transmission over the over three the phase three conductorsphase conductors is coupled is coupled therefore therefore the three phasethe three conductors phase conductors cannot be analyzed cannot be as threeanalyzed independent as three channels.independent For channels. this reason, For modal this reason, analysis modal is used analys to decoupleis is used PLC to decouple channel into PLC three channel virtual into modal three channels.virtual modal While channels. modal channelsWhile modal are characterizedchannels are characterized with different with attenuations, different attenuations, their excitation their is directlyexcitation correlated is directly with correlated the selected with coupling the selected scheme coupling [12–14 scheme]. [12–14]. ForFor advancedadvanced HVHV power-linepower-line monitoringmonitoring applications,applications, aa similarsimilar topologytopology isis usedused forfor HFHF signalsignal injectioninjection and and detection detection (Figure (Figure2). Each2). Each individual individual application application defines defines the shape the andshape frequency and frequency content ofcontent injected of HFinjected signals HF at signals the transmitting at the transmitting site as well site as as processing well as processing algorithms algorithms of captured ofsignals captured at thesignals receiving at the site. receiving The condition site. The of condition HV power of line HV is power superimposed line is superimposed into its frequency into response its frequencyH(f ). Completeresponse understandingH(f). Complete of understandingH(f ) is crucial for of design H(f) is of advancedcrucial for power-line design of monitoring advanced applications power-line utilizingmonitoring HF applications signal propagation. utilizing HF signal propagation.

E()x dx I + rdx Ldx I + dI

Cdx J ()x dx V + dV V gdx

dx x l − x

FigureFigure 2.2. EquivalentEquivalent circuitcircuit representingrepresenting one-conductorone-conductor transmissiontransmission lineline lump.lump.

3. High-Voltage Overhead Power Line High Frequency Characteristics in Normal Operation 3. High-Voltage Overhead Power Line High Frequency Characteristics in Normal Operation Voltage and current propagation along a transmission line can be described with telegrapher’s Voltage and current propagation along a transmission line can be described with telegrapher’s equations. Since HV power line is a multiconductor line, telegrapher’s equations take a matrix form. equations. Since HV power line is a multiconductor line, telegrapher’s equations take a matrix form. Telegrapher’s equations in the matrix form are derived from Maxwell equations and can be simplified Telegrapher’s equations in the matrix form are derived from Maxwell equations and can be simplified for the case of sinusoidal voltages and currents [15,16]. for the case of sinusoidal voltages and currents [15,16]. In order to obtain frequency response of a HV power line, we consider propagation of time- In order to obtain frequency response of a HV power line, we consider propagation of harmonic voltages and currents along the power line. Our analysis uses two assumptions: time-harmonic voltages and currents along the power line. Our analysis uses two assumptions: 1. HV power line is a linear system. 1.2. HVOnly power stationary line is states a linear are system.considered. 2.3. OnlyTransmission stationary line states is assumed are considered. to be infinitely long. 3. Transmission line is assumed to be infinitely long. Telegrapher’s equations for the case of sinusoidal voltages and currents are derived starting with the equivalentTelegrapher’s electrical equations scheme for theof infinitely case of sinusoidal small line voltages lumps [17]. and currents are derived starting with the equivalent electrical scheme of infinitely small line lumps [17]. 3.1. Wave Propagation on Infinetely Long Homogoneous Multiconductor Transmission Line 3.1. Wave Propagation on Infinetely Long Homogoneous Multiconductor Transmission Line In order to describe wave propagation over a multiconductor transmission line, we use the equivalentIn order electrical to describe circuit wave of a propagation line lump [17]. over In a multiconductorthe first step, the transmission transmission line, line we with use one the equivalentcylindrical electricalconductor circuit parallel of with a line the lump earth [17 plane]. In and the harmonic first step, sources the transmission at the line line terminals with one are considered. Voltage and current at any arbitrary point on the line are determined with concentrated harmonic sources at the line terminal as well as distributed sources along the line, together with the transmission line characteristics. This is modelled with an equivalent electrical circuit of line lump with concentrated parameters as is shown in Figure 2 [17].

Energies 2017, 10, 1801 5 of 24 cylindrical conductor parallel with the earth plane and harmonic sources at the line terminals are considered. Voltage and current at any arbitrary point on the line are determined with concentrated harmonic sources at the line terminal as well as distributed sources along the line, together with the transmission line characteristics. This is modelled with an equivalent electrical circuit of line lump with concentrated parameters as is shown in Figure2[17]. Distributed voltage and current sources exist due to different types of noise (e.g., Corona noise) appearing on the transmission line. The line is characterized with passive elements per unit length: r, L, g and C. Element r corresponds to the line resistance per unit length while L is the inductance per unit length. The shunt passive elements g and C represent a conductance and a capacitance of transmission line per unit length. The analysis is reduced to harmonic voltage and current propagation, presented as phasors rotating with angular frequency f. According to Figure2, the increment of the voltage and current across the line lump can be written as [17]:

−dV = zI + E(x)dx (1) −dI = yV + J(x)dx where z = r + jωL is the line impedance per unit length and y = g + jωC is the shunt admittance per unit length. Separating variables in the coupled Equation (1) and neglecting the term dV in comparison with V, equations describing voltage and current along the transmission line are obtained:

d2V dE(x) − zyV = − + zJ(x) dx2 dx

d2 I dJ(x) − yzI = − + yE(x) (2) dx2 dx These equations are also known as telegrapher’s equations for the transmission line with a single conductor above the ground. Equation (2) can be extended to the multiconductor case. A major difference with the previous case is the existence of magnetic and electrostatic couplings between the conductors. The magnetic coupling is modeled with the distributed voltage sources per unit length Eij and Eji, and mutual impedances zij = zji. Electrostatic coupling is represented with admittance yij between the conductors. A number of equations increases with a number of conductors and therefore it is appropriate to write Equation (2) in a matrix form [16,17]:

dV − = ZI + E(x) dx dI − = YV + J(x) (3) dx where Z and Y are square matrices:

Z(i, k) = zik, i, k = 1, . . . n

n Y(i, i) = ∑ yij, Y(i, j) = −yij, i, k = 1, . . . n, i 6= k (4) j=1 and n is the number of conductors. Vectors V, I, E and J have all dimension n × 1. Variable separation in Equation (3) derives matrix telegrapher’s equations:

d2V dE(x) − ZYV = − + ZJ(x) dx2 dx Energies 2017, 10, 1801 6 of 24

d2I dJ(x) − YZI = − + YE(x) (5) dx2 dx In the case of a passive network, distributed sources are equal to zero and the telegrapher’s equations become: d2V − ZYV = 0 dx2 d2I − YZI = 0 (6) dx2 Matrices Z and Y in Equation (3), also found in the literature as primary parameters of a transmission line, define how the harmonic voltage and current propagate over a multiconductor transmission line. The detailed calculation of the HV overhead power line primary parameters is given in [12–14].

3.2. Solution of the Telegrapher's Equations The non-diagonal elements of matrices Z and Y, corresponding to couplings between the phases, make solution of telegrapher’s equations difficult. Therefore, let’s replace the product of matrices Z and Y with the matrix Γ taking into account that matrices Z and Y are symmetrical [17]:

Γ2 = ZY (7)

furthermore:  T Γ2 = (ZY)T = YTZT = YZ (8)

In analogy with the propagation constant, we will denote this matrix as the propagation matrix. Equation (6) can be rewritten as: d2V − Γ2V = 0 dx2 d2I  T − Γ2 I = 0 (9) dx2 In case that Γ2 is a diagonal matrix, the previous equation is equal to the set of n independent differential equations. Therefore, we need to find an appropriate linear transformation that results in the diagonal matrix Γ2. This linear transformation is founded on matrices S and Q, correlating voltages and currents in natural (phase) coordinates V and I with voltages and currents in modal coordinates V(m) and I(m) [16–18]: V = SV(m)

I = QI(m) (10)

Substituting the expression for the voltage and current vectors defined with Equation (10), Equation (9) becomes: d2V(m) = S−1Γ2SV(m) dx2 d2I(m)  T = Q−1 Γ2 QI(m) (11) dx2 The matrix S is determined to diagonalize the product [18,19]:

−1 2 2 n 2 2 2 o S Γ S = γ = diag γ1, γ2 ... γn (12)

We conclude that elements of the diagonal matrix γ2 are eigenvalues of the matrix Γ2, where columns of the matrix S are their corresponding eigenvectors [18,20,21]. Energies 2017, 10, 1801 7 of 24

Matrices ZY and YZ are similar since:

YZ = Z−1(ZY)Z (13) and therefore the product YZ has the same eigenvalues [18–20]. The columns of the matrix Q are eigenvectors of the matrix ΓT = YZ, resulting in:

−1 2 n 2 2 2 o Q YZQ = γ = diag γ1, γ2 ... γn (14)

The presented analysis that decouples telegrapher’s equations into 2 × n ordinary differential equations is known as modal analysis. The modal analysis introduces decoupled modal channels. Propagation of a modal voltage and current through the modal channel k is described by a modal propagation constant γk. The modal voltage and current in the modal channel k at the arbitrary point x can be expressed as:

( )+ ( )− (k) k −γk x k γk x (k)+ (k)− V (x) = V1 e + V1 e = V (x) + V (x)

( )+ ( )− (k) k −γk x k γx (k)+ (k)− I (x) = I1 e − I1 e = I (x) − I (x) (15) (k)+ (k)− Constants I1 and I1 are calculated using boundary conditions at the power-line terminal. A column vector of the matrix S defines distribution of the modal voltage between the phase conductors. Distribution between the power-line conductors i and j is defined with the ratio:

V(i) S(i, k) = (16) V(j) S(j, k)

Matrix S is usually normalized with respect to the first row that is multiplied with a transformation matrix in the manner that elements in the first row are all equal to one. We denote such normalized matrix S with λ. Similar expressions can be derived for currents:

I(i) Q(i, k) = (17) I(j) Q(j, k) where matrix Q is normalized in respect to the first row, denoted with δ.

3.3. Characteristic Impedance Matrix Voltage V(k) and current I(k) propagate through the modal channel k independently from other modal channels. At an arbitrary point x on a power line, the modal voltage and current are equal to the sum of incident and reflected waves, as it is written in Equation (15). The modal characteristic impedance establishes a relation between the modal incident voltage and current with modal reflected voltage and current: ( ) ( ) ( )+ (k)+ k (k)+ k k −γk x V = Zc I = Zc I e 1 (18) ( ) ( ) ( )− (k)− k (k)− k k γk x V = Zc I = Zc I1 e for k = 1, . . . n, or rewritten in a matrix form:

(m)+ (m) (m)+ (m) (m)+ −γx V (x) = Zc I (x) = Zc I1 e

(m)− (m) (m)− (m) (m)− γx V (x) = Zc I (x) = Zc I1 e (19) (m) where Zc is a diagonal matrix containing characteristic impedances of modal channels:

(m) n (1) (n)o Zc = diag Zc ... Zc (20) Energies 2017, 10, 1801 8 of 24

Matrix eγx is a diagonal matrix defined as:

eγx = diag{eγ1x ... eγn x} (21)

Modal voltages at an arbitrary point x equal to the sum of incident and reflected waves:

(m) (m)+ (m)− (m)+ −γx (m)− γx V (x) = V (x) + V (x) = V1 e + V1 e (22) while corresponding modal currents are:

(m) (m)+ −γx (m)− γx (m) (m)+ −γx (m)− γx I (x) = I e − I e = Yc V1 e − V1 e (23)

(m)+ (m)− The boundary condition at the first line terminal determines constants V1 and V1 . Substituting the modal voltage in modal telegrapher’s Equation (11) with equations (22) and (23), modal characteristic impedance [17] is derived:

(m)  −1  −1 Zc = S ZQ γ (24)

The modal characteristic admittance matrix is defined by:

(m) (m)−1 n (1) (1)o Yc = Zc = diag Yc ... Yc (25)

Voltage and current vectors in natural (phase) coordinates are computed from modal voltage and current vectors obtained as a solution of the telegrapher’s equation. Equations (10) and (19) also correlate incident and reflected waves in natural coordinates with incident and reflected waves in the modal coordinates: + (m)+ (m) (m)+ (m) −1 + + V = SV = SZc I = SZc Q I = ZcI − (m)− (m) (m)− (m) −1 − − V = SV = SZc I = SZc Q I = ZcI (26) where V+ and I+ are the incident voltage and current vectors in natural coordinates. Characteristic impedance matrix Zc is computed as: (m) −1 Zc = SZc Q (27) while the characteristic admittance matrix is:

−1 (m) −1 Yc = Zc = QYc S (28)

Substitution of the modal voltage and current in Equation (10) with Equations (24) and (25) yields [19]: (m)  −γx −1 (m)+ γx −1 (m)+ V(x) = SV (x) = S e S V1 + e S V1

(m)  −γx −1 (m)+ γx −1 (m)− I(x) = QI (x) = Yc Se S SV1 − Se S SV1 (29) We introduce the exponential function:

EXP(±Γx) = Se±γxS−1 (30) and denote: + (m)+ − (m)− V1 = SV1 , V1 = SV1 (31) then Equation (29) can be rewritten in the form:

+ − V(x) = EXP(−Γx)V1 + EXP(Γx)V1 Energies 2017, 10, 1801 9 of 24

+ −
I(x) = Yc EXP(−Γx)V1 − EXP(Γx)V1 (32) + − Vectors V1 and V1 are computed from boundary conditions defined at the power-line terminal. + − When boundary conditions are V(0) = V1 and I(0) = I1, vectors V1 and V1 are found to be:

1 1 V+ = (V + Z I ), V− = (V − Z I ) (33) 1 2 1 c 1 1 2 1 c 1 Substitution of previous expressions into Equation (32) yields [17]:

1 1 V(x) = (EXP(Γx) + EXP(−Γx))V − (EXP(Γx) − EXP(−Γx))Z I 2 1 2 c 1 1 1 I(x) = − Y (EXP(Γx) − EXP(−Γx))V + Y (EXP(Γx) + EXP(−Γx))Z I (34) 2 c 1 2 c c 1

3.4. Characteristics of HV Power Line Terminated with Impedance Communication equipment can be connected to a HV power line in different couplings (to one, two or three phase conductors), employing different power-line channels for data transmission. Each power-line channel is characterized with its own frequency-dependent transfer function since different couplings excite different modes. There are two major approaches to HV power-line modeling based on the modal analysis [16]. The first approach utilizes the power-line multiport representation and reduction of ports in respect to the used coupling, without computation of voltages and currents. The second approach computes received voltages and currents with known transmitted voltages and currents for the selected coupling. This approach incorporates the reflection coefficient and it is usually denoted as a method of matrix reflection coefficients [7,17,21]. If V1 and I1 are known harmonic voltage and current of an arbitrary frequency f at the power-line transmitting end and V2 and I2 are received voltages at the other line end, the frequency-dependent power-line transfer function is defined with the amplitude α and phase characteristic β:

H(jω) = eα(ω)+jβ(ω) = Aejβ(ω) (35)

where:

1 V2 I2 V2 I2 α = ln [Np] or α = 10 log [dB] 2 V1 I1 V1 I1  V I  β = Im ln 2 2 [rad] (36) V1 I1

3.4.1. Voltage Transfer Matrix The voltage and current at the transmitting line terminal are interrelated through the input admittance (or input impedance) matrix and at the receiving end with the load admittance matrix:

−1 I1 = YinV1 = Zin V1

−1 I2 = YLV2 = ZL V2 (37) The voltage and current at the line terminal is a sum of incident and reflected components:

+ − V2 = V2 + V2

+ − + −
I2 = I2 − I2 = Yc V2 − V2 (38) Energies 2017, 10, 1801 10 of 24

Solving the set of Equation (38) per incident and reflected voltages, and taking into account Equation (37), yields: 1 1 V+ = (I + Z Y )V , V− = (I − Z Y )V (39) 2 2 c L 2 2 2 c L 2 A reflection coefficient represents a ratio between the reflected and incident waves. Since a power line is a multiconductor system, the reflection coefficient is a square matrix correlating the reflected and incident waves at different ports:

− + − + V1 = K1V1 , V2 = K2V2 (40)

Utilizing Equations (39) and (40), the matrix reflection coefficient at the receiving terminal is:

−1 K2 = (I + ZcYL) (I − ZcYL) (41)

Similarly, the voltage at the transmitting line terminal is expressed in terms of the incident wave and matrix reflection coefficient K1 [7]:

+ − + V1 = V1 + V1 = (I + K1)V1 (42)

Since the incident voltage at the receiving line end is determined by the propagation function and the power-line length: + + V2 = EXP(−Γl)V1 (43) the voltages at the receiving and sending line ends are interrelated with:

−1 V2 = (I + K2)EXP(−Γl)(I + K1) V1 = TV1 (44) where matrix T is the voltage transfer matrix. The matrix reflection coefficient is obtained by interrelating reflected and incident waves at the transmitting line terminal [16,22]:

K1 = EXP(−Γl)K2 EXP(−Γl) (45) while the input admittance is: −1 Yin = Yc(1 − K1)(1 + K1) (46) If a reflection is considered in the modal coordinates, we deal with the modal incident and reflected waves are interrelated with the modal reflection coefficient. The relation between the incident and reflected waves in phase (original) and modal coordinates is established with:

(m)+ + (m)− − Vx = SVx , Vx = SVx (47) where index x is used to denote both line ends. The modal reflection coefficient is then derived from Equations (40) and (47): (m) −1 Kx = S KxS (48) The transformation defined by Equation (48) results in a non-diagonal matrix, what causes power exchange between modes at the termination points. A conclusion is that modal waves propagate along the modal channels without interacting between each other. However, interaction occurs at the termination points. On the other hand, any inhomogeneity on a power line produces reflected waves and power exchange between modes. Energies 2017, 10, 1801 11 of 24 Energies 2017, 10, 1801 11 of 24

3.4.2. Computation of High-Frequency Power-Line Transfer FunctionFunction forfor DifferentDifferent CouplingsCouplings In thisthis section section we we focus focus on twoon two types types of coupling of coupling utilizing utilizing one or twoone phaseor two conductors, phase conductors, presented inpresented Figure3 .in Figure 3.

z z z z g g eg g g + z e z z z eg eg g + + 2 2 +

a) b) FigureFigure 3. 3.PrincipalPrincipal schemes schemes of of(a) ( aa) phase a phase to toground ground coupling coupling and, and, (b ()b a) phase a phase to to phase phase coupling. coupling.

Voltages at the transmitting and receiving ends of a HV power line are correlated with voltage Voltages at the transmitting and receiving ends of a HV power line are correlated with voltage transfer matrix T. Corresponding currents are calculated using termination admittances at both transfer matrix T. Corresponding currents are calculated using termination admittances at both power-line ends, denoted with matrices Yg and Y . power-line ends, denoted with matrices Yg and YL. L When thethe voltage voltage transfer transfer matrix matrix is determined, is determined, high-frequency high-frequency power-line channelpower-line characteristics channel forcharacteristics different couplings for different can becouplings obtained can with be twoobtained approaches: with two approaches: 1. The first is to assume a value of the source (voltage or current) and to compute voltages and 1. The first is to assume a value of the source (voltage or current) and to compute voltages and currents at both line terminals. High-frequency power-line channel characteristics are then currents at both line terminals. High-frequency power-line channel characteristics are then computed from the Equation (36). In case of very long lines, such approach can cause a computed from the Equation (36). In case of very long lines, such approach can cause a significant significant numerical inaccuracy that increases with the frequency. numerical inaccuracy that increases with the frequency. 2. In the second approach, the high-frequency power-line characteristics are computed directly 2. In the second approach, the high-frequency power-line characteristics are computed directly from from the voltage transfer matrix, utilizing information of relations between the voltages at the the voltage transfer matrix, utilizing information of relations between the voltages at the both both line ends contained in the matrix T. line ends contained in the matrix T. A problem that arises in such formulae derivations is related to a multiconductor system, where voltagesA problem of all output that arises ports in suchare interrelated formulae derivations with all voltages is related of to all a multiconductorinput ports. In system,the first where step, voltagesfrequency of response all output is calculated ports are interrelatedfor the phase with to ground all voltages coupling. of all We input suppose ports. that In the the transmitter first step, frequency response is calculated for the phase to ground coupling. We suppose that the transmitter is modeled as the voltage generator Eg and impedance Zg. Since the voltage source impedance is is modeled as the voltage generator Eg and impedance Zg. Since the voltage source impedance is zg, zg, the expression that defines voltages of the three phases at the transmitting line terminal is: the expression that defines voltages of the three phases at the transmitting line terminal is: −1 1 −1 V = YΣ I = ()Y + Y E = LE 1 −1 g z1 g in
−1g g (49) V1 = YΣ Ig = g Yg + Yin Eg = LEg (49) zg = I11 = YYininVV11 (50)(50) Let’s considerconsider a a transmitter transmitter connected connected between between the phasethe phasek and k the and ground the ground and a receiver and a betweenreceiver thebetween phase thet and phase the ground,t and the as presented ground, inas Figurepresented3a. The in amplitudeFigure 3a. and The phase amplitude characteristics and phase are computedcharacteristics as: are computed as: V (t)I (t) = 2 ()2 () [ ] α 20 log V2 t I 2 t dB α = 20 log V1()(k)I1 ()(k) [dB] V1 k I1 k  V (t)I (t)  β = Im ln 2 2 [rad] (51) V1(()k)I1 ()(k)  V2 t I 2 t  β = Imln () () [rad] (51) The k-th element in the vector Eg equals V1egk,I while1 k  other elements are zero. Equation (49) simplifies to: The k-th element in the vector E equals e , while other elements are zero. Equation (49) g V1 = gLkeg (52) simplifies to:

Energies 2017, 10, 1801 12 of 24

where Lk is k-th column of the matrix L. In the next step all phase voltages should be expressed in term of one parameter. For instance, voltage of the first port V1(1) at the sending end can be chosen as such parameter. The ratio between the voltage of the phase i and the voltage of phase 1 is:

V (i) L(i, k) 1 = , i = 1, . . . n (53) V1(1) L(1, k) vector LkR contains elements defined by Equation (53). Now, vector V1 is expressed in the term of V1(1) V1 = LkRV1(1) (54)

Further we rewrite the voltage at the receiving end in the term of parameter V1(1):

V2 = TLkRV1(1) (55) and the voltage of the phase t at the receiving end is found to be:

V2(t) = TtLkRV1(1) (56) where Tt is a row t of the matrix T. The current at the sending end equals to:

I2 = YLTLkRV1(1) = FLkRV1(1) (57)

k where F is the auxiliary square matrix. If with Yin we denote row k in the input admittance matrix Yin and with Ft row t in the matrix F, then the voltage of phase k at the sending end and the voltage of the phase t at the receiving end equal to:

k I1(k) = YinLkRV1(1)

I2(t) = BtLRV1(1) (58) Substituting the expressions for the both line ends in Equation (51) we derive:

Yk L TtLR in kR α = 20 log [dB] LkR(k) BtLkR ( ) T L Yk L β = Im ln t kR in kR [rad] (59) LkR(k) BtLkR Equation (59) defines the high-frequency power-line channel characteristics for the phase to ground coupling. A similar analysis can be conducted also for phase to phase coupling (Figure3b). Let’s assume the coupling phase j to phase k at the transmitting power-line terminal and phase s to phase t at the receiving terminal. The elements of the voltage vector Eg equal zero, except s-th and k-th elements that are: Eg(j) = eg/2, Eg(k) = −eg/2 (60)

If Lj and Lk denote j-th and k-th columns of the matrix L respectively, the voltage at the sending terminal is: eg eg V = L − L
= L (61) 1 j k 2 jk 2

Likewise, the phase to ground coupling, elements of the voltage vector V1 are expressed in the term of voltage V1(1) utilizing the vector column LjkR whose elements are:

V1(i) Ljk(i) L(i, j) − L(i, k) LjkR = = = (62) V1(1) Ljk(1) L(1, j) − L(1, k) Energies 2017, 10, 1801 13 of 24

The voltage at the transmitting end is then:

V1 = LjkRV1(1) (63) and for the selected coupling:   V1(j) − V1(k) = LjkR(j) − LjkR(k) V1(1) (64)

Voltage at the receiving end is correlated to the voltage at the sending end with the voltage transfer matrix: V2 = TLjkRV1(1) (65) and the voltage at the receiver is:

V2(s) − V2(t) = (Ts − Tt)LjkRV1(1) (66)

The currents at the power line terminals are:

I1 = YinLjkRV1(1)

I2 = YLTLjkRV1(1) = FLjkRV1(1) (67)

j When Yin represents the j-th row of matrix Yin and Fs corresponds to the s-th row of the matrix B, currents at the receiver and at the transmitter respectively are:

j I1(j) = YinLjkRV1(1)

I2(s) = BsLjkRV1(1) (68) The high-frequency power-line channel characteristics are then derived from Equations (64), (66) and (68) as:

V (s) − V (t) I (s) (Ts − Tt)L BsL = 2 2 2 = jkR jkR [ ] α 20 log 20 log j dB V1(j) − V1(k) I1(j) LjkR(j) − LjkR(k) YinLjkR

   ( !) V (s) − V (t) (Ts − Tt)LjkR β = Im ln 2 2 = Im ln [rad] (69) V1(j) − V1(k) LjkR(j) − LjkR(k)

4. High-Frequency Characteristics of Faulted HV Overhead Power Line In this section we analyze high-frequency response of HV power line under different fault conditions. For this purpose we start with previously derived model of a HV power line in normal operation, validated with actual measurements. Faulted HV power line model is derived for all known HV power line faults, which are divided into two categories [23,24]:

• shunt faults, • series faults.

The shunt faults refer to all types of short circuits that are further categorized as [25]:

• short circuit between one phase and the ground (phase to ground fault), • short circuit between two phases, with possibility of simultaneous grounding (phase to phase fault and grounded phase to phase fault), • short circuit of all three phases (three phase fault). Energies 2017, 10, 1801 14 of 24

• short circuit between one phase and the ground (phase to ground fault), Energies• short2017 ,circuit10, 1801 between two phases, with possibility of simultaneous grounding (phase to 14phase of 24 fault and grounded phase to phase fault), • short circuit of all three phases (three phase fault). Short circuits are also denoted as permanent or temporary, depending on whether they still existsShort when circuits power are line also is not denoted rated by as a permanent high voltage. or Thetemporary, permanent depending faults are on caused whether by breakingthey still phaseexists when or shielding power conductorsline is not rated and shortby a high circuiting voltage. with The the permanent other phase faults conductors are caused or by the breaking ground. Thephase temporary or shielding short conductors circuits exist and dueshort to circuiting arcs and with disappear the other after phase the power conductors lineis or switched the ground. off. TemporaryThe temporary faults short can circuits be initiated exist bydue the to temporaryarcs and disappear insulation after breakdown the power due line to is short-duration switched off. atmosphericTemporary faults discharges. can be Sinceinitiated a large by the amount tempor of faultsary insulation are “self-healing”, breakdown the due automatic to short-duration reclosure processatmospheric is utilized discharges. instead Since of permanent a large amount power-line of switchfaults are off. “self-healing”, Automatic reclosure the automatic attempts toreclosure switch onprocess the power is utilized line until instead the short-circuit of permanent path power-lin is deionized.e switch After theoff. predefinedAutomatic number reclosure of unsuccessfulattempts to attemptsswitch on (usually the power up toline three until times) the short-circuit the power line path remains is deionized. switched After off [26 the]. predefined number of unsuccessfulThe series attempts faults correspond (usually up to to a phasethree ti conductorsmes) the power breaking line without remains connection switched off with [26]. other phase conductorsThe series or the faults ground. correspond Together, to the a seriesphase and conduc shunttors faults breaking on the twowithout sides connection of the fault locationwith other on thephase power conductors line form or the the hybrid ground. type Together, fault. The the most series common and shunt faults faults are phaseon the to two ground sides faults. of the Theyfault contributelocation on three the power quarters line of form all faults the hybrid while thetype rarest fault. are The the most three common phase faults faults [27 are]. phase to ground faults.Next They we contribute derive the three method quarters to computeof all faults characteristics while the rarest of theare faultedthe three power phase line.faults It [27]. should be noticedNext we that derive short the circuits method are to alwayscompute followed characteri bystics the of power the faulted line trip power in orderline. It to should prevent be thenoticed severe that damage short circuits of the are equipment always followed due to by high the power short-circuit line trip currents. in order to In prevent the analysis the severe we assumedamage thatof the tripped equipment power due line to high is terminated short-circuit with currents. modified In the impedances analysis we what assume causes that changed tripped high-frequencypower line is characteristics.terminated with modified impedances what causes changed high-frequency characteristics. 4.1. Fault Modeling with Shunt Impedance 4.1. Fault Modeling with Shunt Impedance The shunt impedance on the HV power line is defined as an electrical system that consist of passiveThe elements shunt impedance connected on between the HV power-linepower line (phaseis defined or shield) as an electrical conductors system and thethat ground consist orof betweenpassive elements power-line connected conductors between only (Figurepower-line4). Inhomogeneity (phase or shield) (fault conductors location) isand modelled the ground with or a shuntbetween admittance power-line matrix conductorsYsh. only (Figure 4). Inhomogeneity (fault location) is modelled with a shuntThere admittance are two matrix approaches Ysh. to incorporate the shunt impedance in the derived high-frequency power-lineThere are model. two approaches The first approach to incorporate utilized the the sh methodunt impedance of the matrix in the reflectionderived high-frequency coefficients in compliancepower-line model. with the The previous first approach section. Itutilized is obvious the thatmethod a reflected of the wavematrix appears reflection at the coefficients shunt point in wherecompliance the impedance with the previous changes. section. This phenomenon It is obvious that is incorporated a reflected wave in the appears analysis at withthe shunt the matrix point reflectionwhere the coefficient impedance computed changes. as This [27]: phenomenon is incorporated in the analysis with the matrix reflection coefficient computed as [27]: −1 K = I + Z Y
I − Z Y
(70) sh = ()()+ c eq −1 − c eq K sh I Z c Yeq I Z c Yeq (70) where Yeq is the admittance matrix terminating the line section before inhomogeneity and equals: where Yeq is the admittance matrix terminating the line section before inhomogeneity and equals:

Yeq ==Ysh ++ Yin2 (71) Yeq Ysh Yin2 (71)

a b c

y13

y12 y23

y10 y20 y30

Figure 4. HV overhead power line faultfault modeledmodeled withwith shuntshunt impedance.impedance.

Energies 2017, 10, 1801 15 of 24

Energies 10 The2017 admittance, , 1801 matrix Yin2 is the input admittance of the second line section. 15 of 24

4.2. Short Circuits The admittance matrix Yin2 is the input admittance of the second line section. A short circuit represents a fault when a power line conductor forms an electrical contact with 4.2.another Short conductor Circuits or the ground through some impedance. Such fault is therefore represented by the shuntA impedance short circuit at representsthe fault location. a fault whenThe faulted a power power-line line conductor high-frequency forms an characteristics electrical contact are withthen anotherobtained conductor as described or the in Section ground 4.1. through some impedance. Such fault is therefore represented by the shuntThe impedance fault analysis at the is fault often location. conducted The with faulted an power-lineassumption high-frequency that a fault impedance characteristics equals are zero. then It obtainedimplies that asdescribed the power-line in Section channel 4.1. attenuation with the phase to phase coupling tends to infinity whenThe grounded fault analysis phase isto oftenphase conducted fault or three with phas an assumptione fault occurs. that However, a fault impedance it can be found equals in zero. the Itliterature implies that theattenuation power-line introduced channel attenuationby a short ci withrcuit the doesn’t phase overreach to phase coupling 40 dB to tends 50 dB to and infinity that whenreduces grounded with frequency phase to[7]. phase A magnetic fault or coupling three phase between fault two occurs. line sections However, separated it can be byfound the fault in can the literaturebe represented that attenuationas inductive introduced reactance connected by a short at circuit the fault doesn’t location. overreach The expressions 40 dB to 50 that dB define and that the reducesfault impedances with frequency for different [7]. A magnetic couplings coupling are available between in [7,17,27]. two line sections separated by the fault can be representedThe fault impedance as inductive between reactance the connected phase conductor at the fault and location.the ground The (for expressions phase to ground that define fault) the is faultcomputed impedances as [7,27]: for different couplings are available in [7,17,27]. The fault impedance between the phase conductor= + ω and the ground (for phase to ground fault) is Z f rf j L fg (72) computed as [7,27]:

where rf is the sum of a grounding resistanceZf = r(betweenf + jωL f g10 Ω and 20 Ω) and a conductor caused(72) the L whereshort circuitr f is the (or sum arc). of According a grounding to [7,27], resistance fg for (between phase to 10 groundΩ and fault 20 Ω )is and computed a conductor as: caused the short circuit (or arc). According to [7,27], L f g for phase to ground fault is computed as:   = ⋅ − 7 2h − L fg 2 10 h ln 1.06  (73)  r  −7  20h  L f g = 2 · 10 h ln − 1.06 (73) r0 where h and r0 are the length and radius of the grounding conductor. The previous expression > whereholds whenh and ther0 are distance the length to the and fault radius from of the the coupling grounding device conductor. is lf 2 Theh [7]. previous expression holds > whenIn the the distance case of toa phase the fault to phase from the fault, coupling without device grounding, is l f 2theh [ 7fault]. impedance is computed as a shortIn circuit the case between of a phase the two to phase parallel fault, lines. without The resi grounding,stance is neglected, the fault impedance while inductance is computed equals as to a short[27]: circuit between the two parallel lines. The resistance is neglected, while inductance equals to [27]:   −  S  L = =· ⋅ −77S  −  f L f 2 21010 S lnln 11.06.06  (74)(74)  r0 

The grounded phasephase toto phase fault is a combinationcombination of the above defineddefined impedances. The short circuit isis schematicallyschematically representedrepresented inin FigureFigure5 5..

rf rf L f L f

L L f L f fg Lfg

Phase to Phase to phase Grounded phase Three phase ground fault fault to phase fault fault

Figure 5. Short-circuit types.

4.3. Series Faults A series fault occursoccurs whenwhen oneone ofof phasephase conductorsconductors breaksbreaks withoutwithout anyany connectionconnection betweenbetween phases or between a phase and ground [[23].23]. However, it isis commoncommon that one of phasesphases or bothboth after

Energies 2017, 10, 1801 16 of 24 Energies 2017, 10, 1801 16 of 24 the breaking fallfall downdown onon thethe groundground whatwhat causescauses aa phasephase toto groundground fault.fault. Such case is denoted as a hybrid fault. We consider one phase break, with possibilitypossibility that a phase conductor at both sides can also be grounded (falling(falling downdown onon thethe ground),ground), presentedpresented inin FigureFigure6 6..

a a I = 0 1 y10 z fg z fg y z fg 12 y32 b y20 b

I c 2 y23

Yin2 I3 Fault y30 c Yin2 location (a) (b)

Figure 6.6. Schematically represented series fault. (a) Principle scheme; (b) Termination impedanceimpedance atat thethe faultfault location.location.

A fault divides the line into two sections. In order to determine the admittance terminating the A fault divides the line into two sections. In order to determine the admittance terminating the first line section, let’s assume that the fault occurs on the first phase conductor. We consider two first line section, let’s assume that the fault occurs on the first phase conductor. We consider two options: phase conductor can be open-circuited or short-circuited via impedance zfg. The impedance options: phase conductor can be open-circuited or short-circuited via impedance zfg. The impedance zfg corresponds to the phase to ground impedance, computed in the previous sub-section. The input zfg corresponds to the phase to ground impedance, computed in the previous sub-section. The input impedance of the second section Yin2 is schematically represented in Figure 6b and can be determined impedance of the second section Yin2 is schematically represented in Figure6b and can be determined for the line termination admittance YL . The interrelation between voltages and node currents on the for the line termination admittance YL. The interrelation between voltages and node currents on the right side of the fault location (Figure 6) is established as: right side of the fault location (Figure6) is established as: (y + y )V + y V + y V = I  11 fg 1 12 1 13 1 1 y11 + y f g V1 + y12V1 + y13V1 = I1 + + = y 21 V 1 y 22 V 1 y 23 V 1 I 2 y21V1 + y22V1 + y23V1 = I2 + + = y 31 V 1 y 32 V 1 y 33 V 1 I 3 (75) y31V1 + y32V1 + y33V1 = I3 (75) where I , I and I are node currents. Admittance y is an element (i, j) of the matrix Y where I1,1 I2 and2 I3 are3 node currents. Admittance yij is anij element (i, j) of the matrix Yin2 whilein 2 y = z = y = = I whilef g 1y/fgf g 1or/ z fgf g or 0y.fg When0. When the first the phase first conductorphase conductor is faulted, is faulted, the node the current node current1 equals I zero1 equals and:

zero and: 1 V1 = − (y12V2 + y13V3) (76) y +1y V = − 11 f g ()y V + y V 1 + 12 2 13 3 (76) y11 y fg Substituting previous equation into Equation (75), a relation between the two other voltages and nodeSubstituting currents can previous be written equation in the matrix into Equation form as: (75), a relation between the two other voltages and node currents can be written in the matrix form as:   y − y y / y + y y − y y / y + y " # " # " # 22 21 12 11− f g ( +23 ) 21 13− 11 ( f g+ ) V2 V2 I2   y22 y21 y12 /y11 y fg y23 y21 y13 / y11 yfg V  = YVp  I  = (77) 2 = Y 2 = 2 y32 − y31y12/ y11− + y f g ()y+33 − y31y13−/ y11 + ()y f g+ V3 p   V3  I3 (77)  y32 y31 y12 / y11 y fg y33 y31 y13 / y11 y fg V3  V3  I3 

It isis obviousobvious thatthat thethe secondsecond and third phasesphases of thethe firstfirst lineline sectionsection areare terminatedterminated withwith thethe second lineline section section while while the the first first phase phase is either is eith short-circuiteder short-circuited via fault via impedance fault impedance or open-circuited. or open- Therefore,circuited. Therefore, the admittance the admittance terminating terminating the first line the section first line is: section is:

" 0 or y 0  # Y = 0 or y fg 0 =f  f g  (78) Y f 0Yp (78)  0 Yp If the termination admittance at the fault location is known then the first-section voltage transfer matrix T1 is determined, and voltages at the sending line terminal and the left-side fault locations can be calculated:

Energies 2017, 10, 1801 17 of 24

If the termination admittance at the fault location is known then the first-section voltage transfer matrix T1 is determined, and voltages at the sending line terminal and the left-side fault locations can be calculated: V f L = T1V1 (79)

When the line is terminated with the load admittance YL, the second-section voltage transfer matrix is found in compliance with the analysis conducted in the previous sub-section. Voltage at the line end equals to: V2 = T2V f R (80) where V f R is the voltage vector at the beginning of the second line section. Furthermore, the voltages at the both sides of the fault location are interrelated with the fault voltage transfer matrix:

V f R = T f V f L (81) where T f is calculated taking into account Equation (77) as:

      0 −y12/ y11 + y f g −y13/ y11 + y f g   T f =  0 1 0  (82) 0 0 1

The voltage transfer matrix is: T = T2T f T1 (83) Power-line high-frequency characteristic are computed from the equations derived in sub-Section 3.4. Due to simplicity, the expressions are derived for the first phase while analogy is valid for the faults of the two other phases.

5. Simulation Results with Discussion The methodology for high-frequency characteristics computation proposed in this paper is simulated and validated for a 52 km long 400 kV overhead power line with horizontal disposition of conductors (AppendixA). Measurement methodology and results, with the detailed description of the power line is available in [13]. Its characteristics are measured for the middle phase to the outer phase coupling when the power line is in normal operation. The proposed computation methodology is validated for the normal power line operation. Characteristics of the faulted power line are only simulated, since there is no opportunity to measure characteristics of faulted power lines. The methodology for power lines under normal operation is extended also for faulted power lines with an assumption that simulation results are valid.

5.1. Validation of Computation Methodology for 400 kV Overhead Powerline in Operation Due to high-frequency characteristics of the installed LTU, the computation of the HV power-line amplitude characteristic and group delay is validated in the frequency range from 150 kHz to 350 kHz. The validation in this frequency range also means possible extrapolation in the wider high-frequency range. The comparison of simulated high-frequency PLC channel characteristics with the measured is carried out for the case of warm weather (measurements conducted in March with fair weather). Detailed measurement methodology and impact analysis of coupling equipment, LTU, impedance mismatch and optical cable used as return path for the network analyzer, on measurements are given in [13]. In this section we use measurements under fair weather conditions to validated model presented in Section3. Validation of the models presented in the paper for amplitude characteristics are given in Figure7. The red line represents the measurement while the black line corresponds to the simulated characteristic. We assumed the 6 m sag and the ground resistance per unit length 50 Ωm (swampy ground). Since the Energies 2017, 10, 1801 18 of 24 phase at both sides is connected to the HV transformer with large inductivity, non-operating phase is consideredEnergies 2017 to, 10 be, 1801 open circuited. 18 of 24

-3 -6

-4 -7

-5 -8 -6 -9 -7 -10 -8

Amplitude [dB] characteristic -11 Amplitude characteristic [dB] characteristic Amplitude -9

-10 -12 150 160 170 180 190 200 200 210 220 230 240 250 Frequency [kHz] Frequency [kHz] (a) (b)

-6 -6

-7 -7

-8 -8

-9 -9

-10 -10

-11 -11 Amplitude [dB] characteristic Amplitude characteristic [dB] Amplitude characteristic -12 -12

-13 -13 250 260 270 280 290 300 300 310 320 330 340 350 Frequency [kHz] Frequency [kHz] (c) (d)

-8 -10

-10 -12

-12 -14

-14 -16 Amplitude characteristic [dB] characteristic Amplitude Amplitude characteristic [dB] characteristic Amplitude -16 -18

350 360 370 380 390 400 400 410 420 430 440 450 Frequency [kHz] Frequency [kHz] (e) (f)

FigureFigure 7. 7.Comparison Comparison of of the the simulated simulated amplitude amplitude characteristic characteristic of of the the 400 400 kV kV PLC PLC channel channel with with the the measuredmeasured for for different different frequency frequency intervals intervals (a (–af–):f): black-simulated, black-simulated, red-measured. red-measured.

Oscillations in measured and simulated amplitude characteristics are caused by reflection at Oscillations in measured and simulated amplitude characteristics are caused by reflection at power-line terminals. Echoes (reflected waves) on power line make ripples on the amplitude power-line terminals. Echoes (reflected waves) on power line make ripples on the amplitude characteristic. Significant deviations between measured and simulated characteristics present in characteristic. Significant deviations between measured and simulated characteristics present in Figure 7a,d,e,f are caused by operating frequency range of LTU. The capacity of the coupling Figure7a,d–f are caused by operating frequency range of LTU. The capacity of the coupling capacitor capacitor is 4400 pF, while LTU has the minimal resistance 600 Ω in the frequency range from 168 to is 4400 pF, while LTU has the minimal resistance 600 Ω in the frequency range from 168 to 304 kHz. 304 kHz. Measurement equipment is connected with the coupling device via a coaxial cable. The Measurement equipment is connected with the coupling device via a coaxial cable. The estimated estimated length of the coaxial cable is 500 m and its attenuation varies linearly from 0.2 to 0.35 dB/100 length of the coaxial cable is 500 m and its attenuation varies linearly from 0.2 to 0.35 dB/100 m in the m in the frequency range from 150 to 350 kHz. frequency range from 150 to 350 kHz. The measurements proved existence of oscillations in the amplitude characteristic caused by The measurements proved existence of oscillations in the amplitude characteristic caused by reflection while the period of oscillations in the measured and simulated characteristics match. The reflection while the period of oscillations in the measured and simulated characteristics match. difference between measured and simulated characteristics in the operational range is within 1 dB. The difference between measured and simulated characteristics in the operational range is within This is acceptable because we deal with the estimated high-frequency characteristics of the coupling devices.

Energies 2017, 10, 1801 19 of 24

1 dB. This is acceptable because we deal with the estimated high-frequency characteristics of the coupling devices.

5.2. SimulationEnergies 2017 of High-Frequency, 10, 1801 Characterstics of Faulted HV Power Line 19 of 24 Energies 2017, 10, 1801 19 of 24 High-frequency5.2. Simulation of characteristics High-Frequency Char areacterstics calculated of Faulted in MATLAB HV Power Line (R2015a, MathWorks, Natick, MA, Energies 2017, 10, 1801 19 of 24 USA) for5.2. 400 SimulationHigh-frequency kV overhead of High-Frequency powercharacteristics line Char for areacterstics different calculated of Faulted type in MATLAB ofHV faults Power (R2015a, andLine locations. MathWorks, Results Natick, are MA, presented in Figures5.2.USA)8– Simulation11High-frequency for. The400 kV selectedof overheadHigh-Frequency characteristics frequency power Char line areacterstics rangefor calculated different isof Faulted from typein MATLAB 200 ofHV faults Power to 250 and(R2015a, Line kHzlocations. whileMathWorks, Results we assumedare Natick, presented MA, that the in Figures 8–11. The selected frequency range is from 200 to 250 kHz while we assumed that the fault fault occursUSA) afterHigh-frequency for 400 25 kV km overhead from characteristics the power sending line are for point.calculated different The typein non-operating MATLAB of faults and(R2015a, locations. phases MathWorks, Results are terminated are Natick, presented MA, with 400 occurs after 25 km from the sending point. The non-operating phases are terminated with 400 Ω Ω resistanceUSA)in Figures atfor both 400 8–11. kV line The overhead terminalsselected power frequency for line a for phase range different is to from ground type 200 of to faults coupling250 kHz and whilelocations. and we with assumed Results 600 are thatΩ presentedfor the afault phase to resistance at both line terminals for a phase to ground coupling and with 600 Ω for a phase to phase inoccurs Figures after 8–11. 25 kmThe fromselected the frequencysending point. range Theis fr omnon-operating 200 to 250 kHz phases while are we terminated assumed that with the 400 fault Ω phase coupling.coupling. occursresistance after at 25both km line from terminals the sending for a phase point. to The ground non-operating coupling and phases with are 600 terminated Ω for a phase with to 400phase Ω coupling. resistance at both line terminals for0 a phase to ground coupling and with 600 Ω for a phase to phase -2 Faulted coupling. 0 Normal -4 -2 Faulted -60 Normal -4 Faulted -8-2 -6 Normal -10-4 -8 -12-6 -10 -14-8 -12 -16-10 -14 Amplitude characteristic[dB] -18-12 -16 -14 Amplitude characteristic[dB] -20 -18200 205 210 215 220 225 230 235 240 245 250 -16 Frequency [kHz] Amplitude characteristic[dB] -20 -18200 205 210 215 220 225 230 235 240 245 250 Frequency [kHz] Figure 8. Amplitude characteristics-20 of the faulted power line. The middle phase to ground coupling. Figure 8. Amplitude characteristics200 of the205 210 faulted215 220 power225 230 235 line.240 The245 250 middle phase to ground coupling. TheFigure broken 8. Amplitude middle phase characteristics 25 km away of thefrom faulted theFreq senduenc powey ing[kHz]r end, line. with The themiddle middle phase phase to ground conductor coupling. open- The broken middle phase 25 km away from the sending end, with the middle phase conductor circuitedThe broken at themiddle both phase sides 25at thekm faultaway location. from the sending end, with the middle phase conductor open- open-circuitedFigure at8. Amplitude the bothsides characteristics at the fault of the location. faulted power line. The middle phase to ground coupling. circuited at the both sides at the fault location. The broken middle phase 25 km0 away from the sending end, with the middle phase conductor open- circuited at the both sides at the-2 fault location. Faulted 0 Normal -4 -2 Faulted 0 Normal -6-4 -2 Faulted -8 -6 Normal -4 -10-8 -6 -12 -10 -8 -14-12 -10 Amplitude characteristic [dB] -16 -14 -12

Amplitude characteristic [dB] -18-16 -14200 205 210 215 220 225 230 235 240 245 250 Frequency [kHz] -18 Amplitude characteristic [dB] -16200 205 210 215 220 225 230 235 240 245 250 Frequency [kHz] Figure 9. Amplitude characteristics-18 of the faulted power line. The middle phase to ground coupling. The broken middle phase 25 km200 away205 from210 215 the220 sendin225 230g end,235 240 with245 the250 middle phase conductor short- Figure 9. Amplitude characteristics of the faultedFrequenc powey [kHz]r line. The middle phase to ground coupling. Figure 9. circuitedTheAmplitude broken at themiddle characteristics left sidephase and 25 open-circuitedkm ofaway the from faulted atthe the sendin power rightg side end, line. at with the The faultthe middle middle location. phasephase conductor to ground short- coupling. Figure 9. Amplitude characteristics of the faulted power line. The middle phase to ground coupling. The brokencircuited middle at the phase left side 25 and km open-circuited away from atthe the right sending side at end, the fault with location. the middle phase conductor The broken middle phase 25 km0 away from the sending end, with the middle phase conductor short- short-circuited at the left side and open-circuited at the rightFaulted side at the fault location. circuited at the left side and open-circuited-2 at the right side at the fault location. 0 Normal -4 -2 Faulted 0 Normal -6-4 -2 Faulted -8 -6 Normal -4 -10-8 -6 -12 -10 -8 -14-12 -10 Amplitude characteristic [dB] -16 -14 -12 -18 Amplitude characteristic [dB] -16 -14200 205 210 215 220 225 230 235 240 245 250 -18 Frequency [kHz] Amplitude characteristic [dB] -16200 205 210 215 220 225 230 235 240 245 250 Frequency [kHz] Figure 10. Amplitude characteristics-18 of the faulted power line. The middle phase to ground coupling. The broken middle phase 25 km200 away205 from210 215 the220 send225ing230 end,235 240 with245 the250 middle phase conductor open- Figure 10. Amplitude characteristics of the faultedFrequenc powery [kHz] line. The middle phase to ground coupling. circuitedThe broken at themiddle left sidephase and 25 short-circuitekm away fromd atthe the send righting side end, at with the faultthe middle location. phase conductor open- Figure 10. Amplitude characteristics of the faulted power line. The middle phase to ground coupling. Figure 10.circuitedAmplitude at the left characteristics side and short-circuite of the faultedd at the powerright side line. at the The fault middle location. phase to ground coupling. The broken middle phase 25 km away from the sending end, with the middle phase conductor open- The brokencircuited middle at the phase left side 25 and km short-circuite away fromd at the the right sending side at end, the fault with location. the middle phase conductor open-circuited at the left side and short-circuited at the right side at the fault location.

Energies 2017, 10, 1801 20 of 24 Energies 2017, 10, 1801 20 of 24 Energies 2017, 10, 1801 20 of 24 0 0 Faulted -2 NormalFaulted -2 Normal

-4 -4

-6 -6 -8 -8

-10 Amplitude characteristic [dB] -10 Amplitude characteristic [dB] -12 -12200 205 210 215 220 225 230 235 240 245 250 200 205 210 215Fre220quenc225y [230kHz]235 240 245 250 Frequency [kHz] Figure 11. Amplitude characteristics of the faulted power line. The middle phase to ground coupling. FigureFigure 11. 11. Amplitude characteristics characteristics of of the the faulted faulted power power line.line. The middle The middle phase to phase ground to coupling. ground The broken outer phase 25 km away from the sending end, with the outer phase conductor short- coupling.The broken The outer broken phase outer 25 phase km away 25 km from away the from send theing sending end, with end, the with outer the outerphase phase conductor conductor short- circuited at the left side and open-circuited at the right side at the fault location. short-circuitedcircuited at the at left the side left sideand andopen-circuited open-circuited at the at right the right side side at the at fault the fault location. location. HV power-line faults are classified into two groups, short circuits and series faults. The short HVHV power-line power-line faults faults are are classified classified into into two two groups, groups, short short circuits circuits and and series series faults. faults. The The short short circuit schemes are given in Figure 5. Short-circuit impedances are calculated using Equations (73)– circuitcircuit schemes schemes are are given given in Figurein Figure5. Short-circuit 5. Short-circui impedancest impedances are calculated are calculat usinged using Equations Equations (73)–(75). (73)– (75). For the phase to ground fault, we assumed the grounding conductor’s length equals the For(75). the For phase the to phase ground to fault,ground we assumedfault, we theassume groundingd the conductor’sgrounding conductor’s length equals length the equivalent equals the equivalent phase conductor’s height 12.5 m and the radius 1 mm. In our analysis, grounding phaseequivalent conductor’s phase height conductor’s 12.5 m and height the radius12.5 m 1 mm.and Inthe our radius analysis, 1 mm. grounding In our resistance analysis, is grounding assumed resistance is assumed to be equal 10 Ω. The computed fault inductance is Lfg = 2.2667 × 10−5 H. In the −5 fg −5 toresistance be equal 10is assumedΩ. The computed to be equal fault 10 inductanceΩ. The computed is Lfg = fault 2.2667 inductance× 10 H. is In L the = 2.2667 analysis × 10 of phase H. In tothe analysis of phase to phase fault, the calculated fault inductance between the phases is Lf = 1.63 × 10−5 −5 f −5 phaseanalysis fault, of thephase calculated to phase fault fault, inductance the calcul betweenated fault the inductance phases is Lbetweenf = 1.63 ×the10 phasesH, for is L distance = 1.63 × S 10 = H, for distance S = 10 m. In the case of series fault analysis, impedance zfg corresponds to the phase to fg 10H, m. for In distance the case ofS = series 10 m. faultIn the analysis, case of impedanceseries fault zanalysis,fg corresponds impedance to the z phase corresponds to ground to inductancethe phase to ground inductance Lfg = 2.2667 × 10−5 H. L ground= 2.2667 inductance× 10−5 H. Lfg = 2.2667 × 10−5 H. fg The results presented in Figures 8–11 show distortion of power-line amplitude characteristic TheThe results results presented presented in in Figures Figures8– 118–11 show show distortion distortion of of power-line power-line amplitude amplitude characteristic characteristic caused by series faults. We considered breaking of the operating phase conductor as well as non- causedcaused by by series series faults.faults. We We cons consideredidered breaking breaking of of the the operating operating phase phase conductor conductor as well as well as non- as operating phase for the two couplings: middle phase to ground and middle phase to outer phase. As non-operatingoperating phase phase for the for two the couplings: two couplings: middle middle phase phaseto ground to ground and middle and middlephase to phase outer tophase. outer As it was expected, the broken non-operating phase does not have a significant influence on high- phase.it was As expected, it was expected, the broken the brokennon-operating non-operating phase does phase not does have not a have significant a significant influence influence on high- on frequency characteristics. In other words, a small additional attenuation appears as reflected waves high-frequencyfrequency characteristics. characteristics. In other In other words, words, a small a small additional additional attenuation attenuation appears appears as reflected as reflected waves from the fault location distort high-frequency characteristics. wavesfrom fromthe fault the faultlocation location distort distort high-frequency high-frequency characteristics. characteristics. Influence of fault location on amplitude characteristics is presented in Figure 12. We observe InfluenceInfluence of of fault fault location location on on amplitude amplitude characteristics characteristics is presented is presented in Figure in Figure 12. We 12. observe We observe that that on such short power line distance of the fault location from the power line terminal does not onthat such on short such powershort power line distance line distance of the faultof the location fault location from the from power the power line terminal line terminal does not does have not have a crucial influence. When the middle phase to ground coupling is used, the middle phase ahave crucial a influence.crucial influence. When the When middle the phasemiddle to phase ground to coupling ground iscoupling used, the is middleused, the phase middle breaking phase breaking increases attenuation about 10 dB while oscillations have higher amplitude levels. The series increasesbreaking attenuation increases attenuation about 10 dB about while 10 oscillationsdB while oscillations have higher have amplitude higher amplitude levels. The levels. series The faults series faults that are closer to the line terminal will increase attenuation more than remote faults. thatfaults are that closer are to closer the line to the terminal line terminal will increase will increase attenuation attenuation more than more remote than faults.remote faults. 0 0 Fault 5 km -2 Fault 5 km -2 Fault 30 km -4 NormalFault 30 km [dB] -4 Normal c [dB] i -6 c st i i -6 st i -8 -8

aracter -10 h aracter -10 h e c -12 d e c -12 tu d li

tu -14

li -14 mp A

mp -16

A -16 -18 -18 200 205 210 215 220 225 230 235 240 245 250 200 205 210 215Fre220quenc225y [kHz]230 235 240 245 250 Frequency [kHz] Figure 12. Amplitude characteristics of the faulted power line for fault distance 5 and 30 km from the Figure 12. Amplitude characteristics of the faulted power line for fault distance 5 and 30 km from the Figuresending 12. terminal.Amplitude The characteristics middle phase ofto theground faulted coupling. power The line broken for fault middle distance phase 5 and conductor 30 km fromshort- sending terminal. The middle phase to ground coupling. The broken middle phase conductor short- thecircuited sending at terminal.the left side The and middle open-circuited phase to groundat the right coupling. side at Thethe fault broken location. middle phase conductor short-circuitedcircuited at the at left the side left sideand andopen-circuited open-circuited at the at right the right side side at the at fault the fault location. location. For the case of middle phase to outer phase coupling, a series or hybrid fault occurrence also For the case of middle phase to outer phase coupling, a series or hybrid fault occurrence also degrades amplitude characteristic (Figures 13 and 14). The worst-case degradation is the middle degrades amplitude characteristic (Figures 13 and 14). The worst-case degradation is the middle phase break resulting in increased attenuation for more than 10 dB. The fault influence is smaller in phase break resulting in increased attenuation for more than 10 dB. The fault influence is smaller in

Energies 2017, 10, 1801 21 of 24

For the case of middle phase to outer phase coupling, a series or hybrid fault occurrence also degradesEnergies 2017 amplitude, 10, 1801 characteristic (Figures 13 and 14). The worst-case degradation is the middle phase21 of 24 Energies 2017, 10, 1801 21 of 24 break resulting in increased attenuation for more than 10 dB. The fault influence is smaller in the scenariothe scenario of outer of outer phase phase break. break. This can This be can explained be explai byned the by fact the that fact the that middle the middle phase participatesphase participates in the the scenario of outer phase break. This can be explained by the fact that the middle phase participates phasein the mode phase with mode the with smallest the attenuation.smallest attenuation. The non-operating The non-operating phase break phase introduces break introduces a high-frequency a high- in the phase mode with the smallest attenuation. The non-operating phase break introduces a high- characteristicsfrequency characteristics distortion caused distortion by reflection caused by at areflec faulttion location. at a fault This location. distortion This can distortion be neglected canin be frequency characteristics distortion caused by reflection at a fault location. This distortion can be comparisonneglected in with comparison the faulted with operating the faulted phase operating effects. phase effects. neglected in comparison with the faulted operating phase effects.

0 0 -2 -2 -4 -4 -6 -6 -8 -8 -10 -10 -12 -12 -14 -14 -16 -16 Amplitude characteristic [dB]

Amplitude characteristic [dB]-18 -18 -20 -20200 205 210 215 220 225 230 235 240 245 250 200 205 210 215Fre220quenc225y [kHz]230 235 240 245 250 Frequency [kHz] Figure 13. Amplitude characteristics of the faulted power line. The middle phase to the outer phase FigureFigure 13. 13.Amplitude Amplitude characteristics characteristics of of the the faulted faulted power powe line.r line. The The middle middle phase phase to to the the outer outer phase phase coupling. The fault distance is 25 km from the sending terminal. Legend: black-normal, blue-phase C coupling.coupling. The The fault fault distance distance is is 25 25 km km from from the the sending sending terminal. terminal. Legend: Legend: black-normal, black-normal, blue-phase blue-phaseC C broken, red-phase B broken, magenta-phase A broken. broken,broken, red-phase red-phaseB broken,B broken, magenta-phase magenta-phaseA Abroken. broken. 0 0 -2 -2 -4 -4 -6 -6 -8 -8 -10 -10 -12 -12 -14 -14 -16 -16 -18 -18 Amplitude [dB] characteristic

Amplitude [dB] characteristic -20 -20 -22 -22200 205 210 215 220 225 230 235 240 245 250 200 205 210 215Fre220quenc225y [kHz]230 235 240 245 250 Frequency [kHz] Figure 14. Amplitude characteristics of the faulted power line for broken middle phase conductor. Figure 14. Amplitude characteristics of the faulted power line for broken middle phase conductor. FigureThe middle 14. Amplitude phase to characteristicsthe outer phase of coupling. the faulted Le powergend: blue-5 line for km broken fault middledistance, phase red-30 conductor. km fault The middle phase to the outer phase coupling. Legend: blue-5 km fault distance, red-30 km fault Thedistance, middle black-normal. phase to the outer phase coupling. Legend: blue-5 km fault distance, red-30 km fault distance,distance, black-normal. black-normal. The final conclusion based on the presented simulations is that faulted operating phase causes The final conclusion based on the presented simulations is that faulted operating phase causes a significantThe final conclusionattenuation based increase on the and presented group delay simulations distortion. is that The faulted influence operating of the phase faulted causes non- a a significant attenuation increase and group delay distortion. The influence of the faulted non- significantoperating attenuation phase can be increase neglected and while group the delay middle distortion. phase is The denoted influence as the of themost faulted sensitive non-operating to the series operating phase can be neglected while the middle phase is denoted as the most sensitive to the series phaseand hybrid can be neglectedfaults. The while presented the middle analysis phase is isessent denotedial for as implementation the most sensitive of to different the series algorithms and hybrid to and hybrid faults. The presented analysis is essential for implementation of different algorithms to faults.detect, The classify presented and analysislocate faults is essential on HV for power implementation lines using of HF different signals. algorithms Furthermore, to detect, the obtained classify detect, classify and locate faults on HV power lines using HF signals. Furthermore, the obtained andresults locate show faults that on HV power lineslines usingcan provide HF signals. communications Furthermore, using the obtained PLC technology results show even that under HV results show that HV power lines can provide communications using PLC technology even under powerfault conditions. lines can provide communications using PLC technology even under fault conditions. fault conditions.

6.5. Conclusions Conclusions 5. Conclusions InIn this this paper paper we we have have presented presented the the model model of of a a faulted faulted HV HV power-line power-line in in the the low low radio radio frequency frequency In this paper we have presented the model of a faulted HV power-line in the low radio frequency range.range. TheThe modelmodel is is derived derived from from the the multiconductor multiconductor system system analysis analysis and and electromagnetic electromagnetic wave wave range. The model is derived from the multiconductor system analysis and electromagnetic wave propagation described by telegrapher’s equations in a matrix formulation. Analysis is done for propagation described by telegrapher’s equations in a matrix formulation. Analysis is done for typical HV power line faults, short circuits and series faults. The developed model is validated for typical HV power line faults, short circuits and series faults. The developed model is validated for the HV power line in normal operation using 400 kV HV power line measurements. The presented the HV power line in normal operation using 400 kV HV power line measurements. The presented model is further extended for HV power line under faulted conditions. model is further extended for HV power line under faulted conditions.

Energies 2017, 10, 1801 22 of 24 propagation described by telegrapher’s equations in a matrix formulation. Analysis is done for typical HV power line faults, short circuits and series faults. The developed model is validated for the HV Energiespower 2017 line, 10 in, normal1801 operation using 400 kV HV power line measurements. The presented model22 of 24 is further extended for HV power line under faulted conditions. The main focus is placed on determination of HV power-line characteristics when one phase The main focus is placed on determination of HV power-line characteristics when one phase conductor is faulted for which an adequate model is derived. The phase conductor break-down conductor is faulted for which an adequate model is derived. The phase conductor break-down (series (series or hybrid fault) is interesting for analysis since handling such faults is demanding and time or hybrid fault) is interesting for analysis since handling such faults is demanding and time consuming consuming while it is desired to maintain reliable communications over the HV power line. The while it is desired to maintain reliable communications over the HV power line. The conclusion is made conclusion is made that faulted operating phase causes serious degradation of power-line high- that faulted operating phase causes serious degradation of power-line high-frequency characteristics frequency characteristics while fault on a non-operating phase can be neglected. Presented simulation while fault on a non-operating phase can be neglected. Presented simulation results describe the results describe the impact of different faults on frequency response of the high voltage overhead impact of different faults on frequency response of the high voltage overhead power lines. Beside the power lines. Beside the design of PLC communication systems resilient to power line faults, analysis design of PLC communication systems resilient to power line faults, analysis presented in this paper presented in this paper is useful for the design of smart systems for fault classification and is useful for the design of smart systems for fault classification and localization, utilizing low radio localization, utilizing low radio frequency signal propagation over the power lines. frequency signal propagation over the power lines. Acknowledgments: This work was supported in part by ARRS under the Scientific Program P2-0246. Acknowledgments: This work was supported in part by ARRS under the Scientific Program P2-0246. Author Contributions: N.S., A.M. and M.Z. conceived and designed the experiments; N.S. performed the experiments; N.S. N.S. and and M.Z. analyzed the data; N.S. and A.M. conducted measurements; N.S. and M.Z. wrote the paper.

ConflictsConflicts of Interest: The authors declare no conflictconflict ofof interest.interest.

Appendix Appendix A

dx15 dx25 dx24 dx34

5 4 = h4 h5

1 400 mm 23 dx12 dx23

= = h1 h2 h3

Figure A1. 400400 kV kV power-line pole.

Energies 2017, 10, 1801 23 of 24

Table A1. Geometrical 400 kV power-line data.

Disposition of Conductors Horizontal

Height of phase conductors [m] h1 = h2 = h3 = 20 (maximal)

Height of shield wires [m] h4 = h5 = 23.7 (average) dx(12) = 10 dx(13) = 20 dx(14) = 16 dx(15) = 4 dx(23) = 10 Horizontal distance between conductors dx [m] dx(24) = 6 dx(25) = 6 dx(34) = 4 dx(35) = 16 dx(36) = 12 [1 or 2 or 3] and [4 or 5] Vertical distance between conductors [m] 3.7

Table A2. Electrical 400 kV power-line data.

Rated voltage [kV] 380 Length [km] 50 Number of phase conductors 3 Number of conductors in the bundle 2 Distance between conductors in bundle [mm] 400 Phase conductor material Al Fe 490/65 mm2 Phase conductor diameter [mm] 30.6 Number of shield wires 2 Shield wire diameter [mm] 18 Shield wire material Al Mg FE 170/70 mm2 Ground resistance per unit length [Ωm] 50 (approximately)

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