US20200348352A1  Parameter Free Identification of Fault Location in MultiTerminal Power Transmission Lines  Google Patents
Parameter Free Identification of Fault Location in MultiTerminal Power Transmission Lines Download PDFInfo
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 US20200348352A1 US20200348352A1 US16/957,408 US201816957408A US2020348352A1 US 20200348352 A1 US20200348352 A1 US 20200348352A1 US 201816957408 A US201816957408 A US 201816957408A US 2020348352 A1 US2020348352 A1 US 2020348352A1
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 G—PHYSICS
 G01—MEASURING; TESTING
 G01R—MEASURING ELECTRIC VARIABLES; MEASURING MAGNETIC VARIABLES
 G01R31/00—Arrangements for testing electric properties; Arrangements for locating electric faults; Arrangements for electrical testing characterised by what is being tested not provided for elsewhere
 G01R31/08—Locating faults in cables, transmission lines, or networks
 G01R31/081—Locating faults in cables, transmission lines, or networks according to type of conductors
 G01R31/085—Locating faults in cables, transmission lines, or networks according to type of conductors in power transmission or distribution lines, e.g. overhead

 H—ELECTRICITY
 H02—GENERATION; CONVERSION OR DISTRIBUTION OF ELECTRIC POWER
 H02J—CIRCUIT ARRANGEMENTS OR SYSTEMS FOR SUPPLYING OR DISTRIBUTING ELECTRIC POWER; SYSTEMS FOR STORING ELECTRIC ENERGY
 H02J3/00—Circuit arrangements for ac mains or ac distribution networks

 G—PHYSICS
 G01—MEASURING; TESTING
 G01R—MEASURING ELECTRIC VARIABLES; MEASURING MAGNETIC VARIABLES
 G01R31/00—Arrangements for testing electric properties; Arrangements for locating electric faults; Arrangements for electrical testing characterised by what is being tested not provided for elsewhere
 G01R31/08—Locating faults in cables, transmission lines, or networks
 G01R31/081—Locating faults in cables, transmission lines, or networks according to type of conductors
 G01R31/086—Locating faults in cables, transmission lines, or networks according to type of conductors in power transmission or distribution networks, i.e. with interconnected conductors

 H—ELECTRICITY
 H02—GENERATION; CONVERSION OR DISTRIBUTION OF ELECTRIC POWER
 H02J—CIRCUIT ARRANGEMENTS OR SYSTEMS FOR SUPPLYING OR DISTRIBUTING ELECTRIC POWER; SYSTEMS FOR STORING ELECTRIC ENERGY
 H02J13/00—Circuit arrangements for providing remote indication of network conditions, e.g. an instantaneous record of the open or closed condition of each circuitbreaker in the network; Circuit arrangements for providing remote control of switching means in a power distribution network, e.g. switching in and out of current consumers by using a pulse code signal carried by the network
 H02J13/00002—Circuit arrangements for providing remote indication of network conditions, e.g. an instantaneous record of the open or closed condition of each circuitbreaker in the network; Circuit arrangements for providing remote control of switching means in a power distribution network, e.g. switching in and out of current consumers by using a pulse code signal carried by the network characterised by monitoring

 H—ELECTRICITY
 H02—GENERATION; CONVERSION OR DISTRIBUTION OF ELECTRIC POWER
 H02J—CIRCUIT ARRANGEMENTS OR SYSTEMS FOR SUPPLYING OR DISTRIBUTING ELECTRIC POWER; SYSTEMS FOR STORING ELECTRIC ENERGY
 H02J13/00—Circuit arrangements for providing remote indication of network conditions, e.g. an instantaneous record of the open or closed condition of each circuitbreaker in the network; Circuit arrangements for providing remote control of switching means in a power distribution network, e.g. switching in and out of current consumers by using a pulse code signal carried by the network
 H02J13/00032—Systems characterised by the controlled or operated power network elements or equipment, the power network elements or equipment not otherwise provided for
 H02J13/00034—Systems characterised by the controlled or operated power network elements or equipment, the power network elements or equipment not otherwise provided for the elements or equipment being or involving an electric power substation

 H—ELECTRICITY
 H02—GENERATION; CONVERSION OR DISTRIBUTION OF ELECTRIC POWER
 H02J—CIRCUIT ARRANGEMENTS OR SYSTEMS FOR SUPPLYING OR DISTRIBUTING ELECTRIC POWER; SYSTEMS FOR STORING ELECTRIC ENERGY
 H02J2203/00—Indexing scheme relating to details of circuit arrangements for AC mains or AC distribution networks
 H02J2203/20—Simulating, e g planning, reliability check, modelling or computer assisted design [CAD]

 Y—GENERAL TAGGING OF NEW TECHNOLOGICAL DEVELOPMENTS; GENERAL TAGGING OF CROSSSECTIONAL TECHNOLOGIES SPANNING OVER SEVERAL SECTIONS OF THE IPC; TECHNICAL SUBJECTS COVERED BY FORMER USPC CROSSREFERENCE ART COLLECTIONS [XRACs] AND DIGESTS
 Y02—TECHNOLOGIES OR APPLICATIONS FOR MITIGATION OR ADAPTATION AGAINST CLIMATE CHANGE
 Y02E—REDUCTION OF GREENHOUSE GAS [GHG] EMISSIONS, RELATED TO ENERGY GENERATION, TRANSMISSION OR DISTRIBUTION
 Y02E60/00—Enabling technologies; Technologies with a potential or indirect contribution to GHG emissions mitigation

 Y—GENERAL TAGGING OF NEW TECHNOLOGICAL DEVELOPMENTS; GENERAL TAGGING OF CROSSSECTIONAL TECHNOLOGIES SPANNING OVER SEVERAL SECTIONS OF THE IPC; TECHNICAL SUBJECTS COVERED BY FORMER USPC CROSSREFERENCE ART COLLECTIONS [XRACs] AND DIGESTS
 Y04—INFORMATION OR COMMUNICATION TECHNOLOGIES HAVING AN IMPACT ON OTHER TECHNOLOGY AREAS
 Y04S—SYSTEMS INTEGRATING TECHNOLOGIES RELATED TO POWER NETWORK OPERATION, COMMUNICATION OR INFORMATION TECHNOLOGIES FOR IMPROVING THE ELECTRICAL POWER GENERATION, TRANSMISSION, DISTRIBUTION, MANAGEMENT OR USAGE, i.e. SMART GRIDS
 Y04S10/00—Systems supporting electrical power generation, transmission or distribution
 Y04S10/30—State monitoring, e.g. fault, temperature monitoring, insulator monitoring, corona discharge

 Y—GENERAL TAGGING OF NEW TECHNOLOGICAL DEVELOPMENTS; GENERAL TAGGING OF CROSSSECTIONAL TECHNOLOGIES SPANNING OVER SEVERAL SECTIONS OF THE IPC; TECHNICAL SUBJECTS COVERED BY FORMER USPC CROSSREFERENCE ART COLLECTIONS [XRACs] AND DIGESTS
 Y04—INFORMATION OR COMMUNICATION TECHNOLOGIES HAVING AN IMPACT ON OTHER TECHNOLOGY AREAS
 Y04S—SYSTEMS INTEGRATING TECHNOLOGIES RELATED TO POWER NETWORK OPERATION, COMMUNICATION OR INFORMATION TECHNOLOGIES FOR IMPROVING THE ELECTRICAL POWER GENERATION, TRANSMISSION, DISTRIBUTION, MANAGEMENT OR USAGE, i.e. SMART GRIDS
 Y04S10/00—Systems supporting electrical power generation, transmission or distribution
 Y04S10/50—Systems or methods supporting the power network operation or management, involving a certain degree of interaction with the loadside end user applications
 Y04S10/52—Outage or fault management, e.g. fault detection or location

 Y—GENERAL TAGGING OF NEW TECHNOLOGICAL DEVELOPMENTS; GENERAL TAGGING OF CROSSSECTIONAL TECHNOLOGIES SPANNING OVER SEVERAL SECTIONS OF THE IPC; TECHNICAL SUBJECTS COVERED BY FORMER USPC CROSSREFERENCE ART COLLECTIONS [XRACs] AND DIGESTS
 Y04—INFORMATION OR COMMUNICATION TECHNOLOGIES HAVING AN IMPACT ON OTHER TECHNOLOGY AREAS
 Y04S—SYSTEMS INTEGRATING TECHNOLOGIES RELATED TO POWER NETWORK OPERATION, COMMUNICATION OR INFORMATION TECHNOLOGIES FOR IMPROVING THE ELECTRICAL POWER GENERATION, TRANSMISSION, DISTRIBUTION, MANAGEMENT OR USAGE, i.e. SMART GRIDS
 Y04S40/00—Systems for electrical power generation, transmission, distribution or enduser application management characterised by the use of communication or information technologies, or communication or information technology specific aspects supporting them
 Y04S40/20—Information technology specific aspects, e.g. CAD, simulation, modelling, system security
Abstract
A method and device can be used with a power transmission line. Prefault voltage and current phasors and duringfault voltage and current phasors for each of first, second, and third terminals are determined based on disturbance records. Using an assumed faulted section, values for a propagation constant of each section, a surge impedance of each section, and a fault location parameter are computed. The computing is based on simultaneously solving prefault and duringfault objective functions for the assumed faulted section with the computed prefault and duringfault voltage and current phasors. The prefault and duringfault objective functions are formulated based on equating junction voltages determined from two of the terminals, conservation of charge at the junction, and equating fault location voltages determined from one terminal and the junction. The values determined for the propagation constant, the surge impedance, and the fault location parameter can be compared with predefined criteria.
Description
 The present subject matter relates, in general, to identification of fault location in power transmission lines and, in particular, to identification of fault location in multiterminal power transmission lines.
 Power transmission lines such as twoterminal, threeterminal power transmission lines etc., are used for bulk power transfer, i.e. for supplying power between loads and one or more power sources. With increased integration of renewable power to the grid, threeterminal/tapped lines are being more commonly used. Using threeterminal/tapped lines saves the costs associated with building substation and installing measurement transformers at tap/junction point. For example, threeterminal/tapped lines are used in cases of solar parks and offshore wind farms where the power evacuation to the main grid is done through a short line that is connected to the main transmission line.
 To supply power reliably and protect the transmission lines from thermal loading, fast restoration of outages is very important. For this, accurate identification of fault location (fault location identification) in multiterminal/tapped lines is required so that a maintenance crew may reach the fault point and undertake the repair quickly. There are numerous fault location identification techniques available for two terminal lines, but accurate fault location identification for threeterminal/tapped lines is challenging as each section of the line has a different length, per unit impedance (X/R ratio), infeed, and charging current. For fault locators with line parameters as input, the accuracy of fault location estimate is affected by the error in input parameters. In addition, accuracy of impedance based fault location identification methods depends on mutual coupling, nonhomogeneity of the line and source impedance angles, source to line impedance ratio, fault resistance and fault loop information etc., making fault location identification complicated for multiterminal/tapped lines.
 The features, aspects, and advantages of the present subject matter will be better understood with regard to the following description, and accompanying figures. The use of the same reference number in different figures indicates similar or identical features and components.

FIG. 1 illustrates an example configuration of a fault free threeterminal power transmission line. 
FIG. 2 illustrates current and voltage phasors used for computation of line parameters in a fault free threeterminal power transmission line. 
FIG. 3 illustrates a method for determining line parameters from voltage and current functions using Newton Raphson approximation, in accordance with an implementation of the present subject matter. 
FIG. 4 illustrates a device for identifying a fault location in a multiterminal power transmission line, in accordance with an implementation of the present subject matter. 
FIGS. 5A5C illustrate current and voltage phasors used for computation of line parameters and fault location in a threeterminal power transmission line with fault in different sections, in accordance with an implementation of the present subject matter. 
FIG. 6 illustrates a method for identifying a fault location in a multiterminal power transmission line, in accordance with an implementation of the present subject matter. 
FIG. 7 illustrates a method for identifying a fault location in a multiterminal power transmission line, in accordance with an implementation of the present subject matter.  The present subject matter relates to systems and methods for identifying fault location in multiterminal power transmission lines.
 Multiterminal power transmission lines, such as twoterminal, threeterminal power transmission lines etc., supply power from multiple power sources to loads. A conventional method for identifying fault location in multiterminal lines uses only selected negative sequence quantities at all terminals. Though the communication requirement between relays is reduced in such methods, the solution does not work for all fault cases. These methods also require the knowledge of source impedance magnitude and angle at all terminals, which might not always be practically available. Another conventional fault location identification method for three terminal lines uses symmetrical components of the voltages and currents measured at the terminals. One limitation of this method is that it requires the fault loop as an input. Also, since it uses zerosequence quantities to locate the fault, mutual coupling has to be taken into consideration for doublecircuit lines. This also causes the solution to fail in cases where the second circuit is open and grounded since there will be no zero sequence current measurements available from unfaulted line. Correct selection of fault location identification subroutine depends on fault resistance, which may not be reliable.
 The accuracy of conventional fault location identification methods depends on correct estimation of the line parameters, such as resistance, inductance and capacitance per unit length. These line parameters are difficult to estimate correctly and such estimation depends on many practical conditions such as loading, weather, aging, material property etc. There are a few line parameter estimation methods that require three independent observations/data sets, which may not be available in case of a fault.
 In one example, parameter estimation of a threeterminal transmission line may use synchronized measurements from all terminals. In particular, three observations of voltage and current data from all terminals of the line may be used. Such data is available using intelligent electronic devices (IED) data. This data can be recorded by the IEDs at regular intervals and can be analysed for determination of line parameters.

FIG. 1 illustrates an example configuration of a fault free threeterminal power transmission line 100 and a control system 102. The control system 102 may be implemented as any computing device which may be, but is not restricted to, a server, a workstation, a desktop computer, a laptop, and an application. In one example, the control system 102 may be implemented in a cloud network environment.  The power transmission line 100 may be used to transmit electric power. The electric power transmitted may be at high voltages, such as in the range of kilovolts, and for long distances, such as for tens or hundreds of kilometres. The power transmission line 100 includes a first terminal 104 at which the power transmission line 100 receives electric power from a first power source 106. The first power source 106 may be an electric generator in one example, however, in other examples other power sources may be used. It will also be understood that while the following description is provided with reference to power sources connected at the different terminals, in other implementations substations or loads may be connected at one or more of the terminals and all such implementations are also intended to be covered herein.
 The first terminal 104 may also be referred to as a first bus 104. The power transmission line 100 also includes a second terminal 108 at which the power transmission line 100 receives electric power from a second power source 110. The second terminal 108 may also be referred to as a second bus 108. The power transmission line 100 further includes a third terminal 112 at which the power transmission line 100 receives electric power from a third power source 114. It will be understood that there would be various other components, such as transformers, power system equipment, etc. present at each terminal, which are not shown or described for brevity.
 Each terminal may also be associated with a respective Intelligent Electronic Device (IED). For example, the first terminal 104 may be associated with a first IED 116, the second terminal 108 may be associated with a second IED 118, and the third terminal 112 may be associated with a third IED 120. As will be understood, an IED can be used to record voltage and current and control power system equipment such as a circuit breaker, disconnector, transformer, and the like, at the terminal (or location) at which it is deployed. The components and working of an IED in the context of a power transmission system is readily understood to a person skilled in the art and are hence not described in detail.
 Further, as shown in
FIG. 1 , a junction 122 may be present in the power transmission line between the three terminals 104, 108, 112. Accordingly, a section of the power transmission line between the first terminal 104 and the junction 122 may be referred to as a first section, a section of the power transmission line between the second terminal 108 and the junction 122 may be referred to as a second section, and a section of the power transmission line between the third terminal 112 and the junction 122 may be referred to as a third section.  During operation of the power transmission line 100, the line parameters, such as inductance and capacitance per unit length, may vary with variation in ambient conditions like temperature, aging of the line, etc. To obtain the accurate line parameters, online estimation of line parameters is highly desirable and especially beneficial to fault location and protection applications. In one example, parameter estimation of a threeterminal transmission line may be done using synchronized measurements of voltage and current from all terminals. For example, each of the IED devices 116, 118, and 120 may obtain synchronized voltage and current measurements and provide it to the control system 102. The synchronization of the measurements can be based on synchronization between the IED clocks, or performed at the control system. The control system 102 may then compute the line parameters as discussed below with reference to
FIGS. 2 and 3 . 
FIG. 2 illustrates current and voltage phasors used for computation of line parameters in a fault free threeterminal power transmission line. Bus M, Bus N and Bus P correspond to the three terminals of the transmission line of sectional lengths I_{MJ}, I_{NJ }and I_{PJ}. To determine the line parameters, three observations of time synchronized voltage and current data may be received from all terminals. The current and voltage phasors may be then determined and the positive sequence voltages and currents may be obtained.  The positive sequence voltages and currents of the line from bus M may be represented as V_{M} ^{1}, I_{M} ^{1}. The positive sequence voltages and currents from bus N may be represented as V_{N} ^{1}, I_{N} ^{1}. The positive sequence voltages and currents from bus P may be represented as V_{P} ^{1}, I_{P} ^{1}.
 Further, voltage and current functions may be formulated for the power transmission line considering the threeterminal power transmission line to be a distributed twoport network model.
 Considering the threeterminal line shown in
FIG. 2 , the voltage at junction J can be obtained using the positive sequence voltage and current from end M in the definition of twoport network P1 as 
V _{J,(M)} ^{1} =A _{MJ} ^{1} V _{M} ^{1} +B _{MJ} ^{1} I _{M} ^{1} (1)  Where, A_{MJ} ^{1}, B_{MJ} ^{1 }are the positive sequence ABCD parameters of the line section MJ defined as,

A _{MJ} ^{1}=cos h(γ_{M} l _{MJ}); 
B _{MJ} ^{1} =Z _{cM }sin h(γ_{M} l _{MJ})  where, l_{MJ}—length of the line section MJ
 Z_{cM}— positive sequence characteristic impedance of section MJ
 γ_{M }positive sequence propagation constant of section MJ
 V_{J,(M)} ^{1}—Voltage at junction J calculated using data from end M.
 Substituting, the definitions of A_{MJ} ^{1 }and B_{MJ} ^{1 }in (1), we have

V _{J,(M)} ^{1}=cos h(γ_{M} l _{MJ})V _{M} ^{1} −Z _{cM }sin h(γ_{M} l _{MJ})I _{M} ^{1} (2)  Similarly, junction voltage can be obtained using the twoport network definition of line section NJ using N end data as,

V _{J,(N)} ^{1}=cos h(γ_{N} l _{NJ})V _{N} ^{1} −Z _{cN }sin h(γ_{N} l _{NJ})I _{N} ^{1} (3)  l_{NJ}—length of the line section NJ
 Z_{cN}— positive sequence characteristic impedance of section NJ
 γ_{N}—positive sequence propagation constant of section NJ
 V_{J,(N)} ^{1}—Voltage at junction J calculated using data from end N.
 Similarly, junction voltage can be obtained using the twoport network definition of line section PJ using P end data as,

V _{J,(P)} ^{1}=cos h(γ_{P} l _{PJ})V _{P} ^{1} −Z _{cP }sin h(γ_{P} l _{PJ})I _{P} ^{1} (4)  l_{PJ}—length of the line section PJ
 Z_{cP}— positive sequence characteristic impedance of section PJ
 γ_{P}—positive sequence propagation constant of section PJ
 V_{J,(P)} ^{1}— Voltage at junction J calculated using data from end P.
 For a nofault condition, the voltage of junction J calculated using either end data must be equal. Therefore, for no fault in line,

Voltage of node J calculated from Bus M data (V _{J,(M)} ^{1})−Voltage of node J calculated from Bus N data(V _{J,(N)} ^{1})=0  Therefore, using (2) and (3), we define,

F _{1}=cos h(γ_{M} l _{MJ})V _{M} ^{1} −Z _{cM }sin h(γ_{M} l _{MJ})I _{M} ^{1}−cos h(γ_{N} l _{NJ})V _{N} ^{1} +Z _{cN }sin h(γ_{N} l _{NJ})I _{N} ^{1} (5) 
Similarly, 
Voltage of node J calculated from Bus M data (V _{J,(M)} ^{1})−Voltage of node J calculated from Bus P data (V _{J,(P)} ^{1})=0  Therefore, using (2) and (4), we define,

F _{2}=cos h(γ_{M} l _{MJ})V _{M} ^{1} −Z _{cM }sin h(γ_{M} l _{MJ})I _{M} ^{1}−cos h(γ_{P} l _{PJ})V _{P} ^{1} +Z _{cP }sin h(γ_{P} l _{PJ})I _{P} ^{1} (6)  The threeterminal power transmission line may be homogenous or nonhomogenous. In case of a homogenous power transmission line, the line parameters in all three sections of the power transmission line are considered to be identical. For a three terminal nonhomogenous line, two sections of the main line are considered to be identical and the Tsection (i.e. section of the other line, from the tap/junction to the terminal) parameters are considered to be different.
 The computation of line parameters is discussed below considering a nonhomogenous line and is then extended for computation of line parameters for a homogenous line.
 In the present example, it is considered that terminal P is connected to main line MN via tap at junction J. Hence, two sections of the main line corresponding to MJ and NJ are identical and the section corresponding to PJ is not identical to MJ or NJ. For such a nonhomogenous line, γ_{M}=γ_{N }and Z_{cM}=Z_{cN}. With γ_{M}=γ_{N }and Z_{cM}=Z_{cN}, using (5) and (6) for 3 observations of voltage and currents from all the terminals of the line, we have,

F _{1,1}=cos h(γ_{M} l _{MJ})V _{M,1} ^{1} −Z _{cM }sin h(γ_{M} l _{MJ})I _{M,1} ^{1}−cos h(γ_{M} l _{NJ})V _{N,1} ^{1} +Z _{cM }sin h(γ_{M} l _{MJ})I _{N,1} ^{1} (7) 
F _{2,1}=cos h(γ_{M} l _{MJ})V _{M,1} ^{1} −Z _{cM }sin h(γ_{M} l _{MJ})I _{M,1} ^{1}−cos h(γ_{P} l _{PJ})V _{P,1} ^{1} +Z _{cP }sin h(γ_{P} l _{PJ})I _{P,1} ^{1} (8) 
F _{1,2}=cos h(γ_{M} l _{MJ})V _{M,2} ^{1} −Z _{cM }sin h(γ_{M} l _{MJ})I _{M,2} ^{1}−cos h(γ_{M} l _{NJ})V _{N,2} ^{1} +Z _{cM }sin h(γ_{M} l _{NJ})I _{N,2} ^{1} (9) 
F _{2,2}=cos h(γ_{M} l _{MJ})V _{M,2} ^{1} −Z _{cM }sin h(γ_{M} l _{MJ})I _{M,2} ^{1}−cos h(γ_{P} l _{PJ})V _{P,2} ^{1} +Z _{cP }sin h(γ_{P} l _{PJ})I _{P,2} ^{1} (10) 
F _{1,3}=cos h(γ_{M} l _{MJ})V _{M,3} ^{1} −Z _{cM }sin h(γ_{M} l _{MJ})I _{M,3} ^{1}−cos h(γ_{M} l _{NJ})V _{N,3} ^{1} +Z _{cM }sin h(γ_{M} l _{NJ})I _{N,3} ^{1} (11) 
F _{2,3}=cos h(γ_{M} l _{MJ})V _{M,3} ^{1} −Z _{cM }sin h(γ_{M} l _{MJ})I _{M,3} ^{1}−cos h(γ_{P} l _{PJ})V _{P,3} ^{1} +Z _{cP }sin h(γ_{P} l _{PJ})I _{P,3} ^{1} (12)  Where, subscript M, i represents i^{th }data set at bus M and so on.
 These equations are of form,

$\begin{array}{cc}\left[\begin{array}{c}{F}_{1,1}\\ {F}_{2,1}\\ {F}_{1,2}\\ {F}_{2,2}\\ {F}_{1,3}\\ {F}_{2,3}\end{array}\right]=F\ue8a0\left({\gamma}_{M},{\gamma}_{P},{Z}_{\mathrm{cM}},{Z}_{\mathrm{cP}}\right)=F\ue8a0\left(x\right)=0& \left(13\right)\end{array}$  Where, x=(γ_{M}, γ_{P}, Z_{cM}, Z_{cP}).
The positive sequence voltage and current data can be substituted in the above equations (7) to (12) and the equations can be solved to determine the line parameters, i.e., resistance, inductance, and capacitance, in each section of the power transmission line.  As the above set of equations is of form F(x)=0, in one example, they can be solved using NewtonRaphson (NR) method. However, it will be understood that any other numerical analysis technique of solving the equations can also be used and the present subject matter is not limited to solving by NR method. For solving the equations using NR method, the Jacobian matrix of F(x) is defined as,

ΔF(x)=JΔx (14)  where Jacobian J is,

$\begin{array}{cc}J=\left[\begin{array}{cccc}\frac{\partial {F}_{1,1}}{\partial {\gamma}_{M}}& \frac{\partial {F}_{1,1}}{\partial {\gamma}_{N}}& \frac{\partial {F}_{1,1}}{\partial {Z}_{\mathrm{cM}}}& \frac{\partial {F}_{1,1}}{\partial {Z}_{\mathrm{cP}}}\\ \frac{\partial {F}_{2,1}}{\partial {\gamma}_{M}}& \frac{\partial {F}_{2,1}}{\partial {\gamma}_{N}}& \frac{\partial {F}_{2,1}}{\partial {Z}_{\mathrm{cM}}}& \frac{\partial {F}_{2,1}}{\partial {Z}_{\mathrm{cP}}}\\ \frac{\partial {F}_{1,2}}{\partial {\gamma}_{M}}& \frac{\partial {F}_{1,2}}{\partial {\gamma}_{N}}& \frac{\partial {F}_{1,2}}{\partial {Z}_{\mathrm{cM}}}& \frac{\partial {F}_{1,2}}{\partial {Z}_{\mathrm{cP}}}\\ \frac{\partial {F}_{2,2}}{\partial {\gamma}_{M}}& \frac{\partial {F}_{2,2}}{\partial {\gamma}_{N}}& \frac{\partial {F}_{2,2}}{\partial {Z}_{\mathrm{cM}}}& \frac{\partial {F}_{2,2}}{\partial {Z}_{\mathrm{cP}}}\\ \frac{\partial {F}_{1,3}}{\partial {\gamma}_{M}}& \frac{\partial {F}_{1,3}}{\partial {\gamma}_{N}}& \frac{\partial {F}_{1,3}}{\partial {Z}_{\mathrm{cM}}}& \frac{\partial {F}_{1,3}}{\partial {Z}_{\mathrm{cP}}}\\ \frac{\partial {F}_{2,3}}{\partial {\gamma}_{M}}& \frac{\partial {F}_{2,3}}{\partial {\gamma}_{N}}& \frac{\partial {F}_{2,3}}{\partial {Z}_{\mathrm{cM}}}& \frac{\partial {F}_{2,3}}{\partial {Z}_{\mathrm{cP}}}\end{array}\right]& \left(15\right)\end{array}$  For n equations and m variables, the Jacobian matrix has n rows and m columns. The first element in J is

$\begin{array}{cc}{J}_{1\ue89e1}=\frac{\partial {F}_{1,1}}{\partial {\gamma}_{M}}& \left(16\right)\end{array}$  The element J_{11 }can be obtained using finite differences. Consider a small perturbation of δ in γ_{M}, the differential term J_{11 }can be calculated as,

$\begin{array}{cc}{J}_{11}=\frac{\partial {F}_{1,1}\ue8a0\left({x}_{j}\right)}{\partial {\gamma}_{M}}=\frac{{F}_{1,1}\ue8a0\left({\gamma}_{M}+\delta ,{\gamma}_{P},{Z}_{\mathrm{cM}},{Z}_{\mathrm{cP}}\right){F}_{1,1}\ue8a0\left({\gamma}_{M}\delta ,{\gamma}_{P},{Z}_{\mathrm{cM}},{Z}_{\mathrm{cP}}\right)}{\left({\gamma}_{M}+\delta \right)\left({\gamma}_{M}\delta \right)}& \left(17\right)\end{array}$ 
$\begin{array}{cc}{J}_{1\ue89e1}=\frac{{F}_{1,1}\ue8a0\left({\gamma}_{M}+\delta ,{\gamma}_{P},{Z}_{\mathrm{cM}},{Z}_{\mathrm{cP}}\right){F}_{1,1}\ue8a0\left({\gamma}_{M}6,{\gamma}_{P},{Z}_{\mathrm{cM}},{Z}_{\mathrm{cP}}\right)}{2\ue89e\left(\delta \right)}& \left(18\right)\end{array}$  Using similar finite difference based approach, all the elements in matrix J can be calculated to form the Jacobian matrix defined in (14). The NR method uses an initial guess for variables x and calculates the Jacobian matrix using partial derivatives of the functions with respect to the variables being solved using equation (14) through (20). In every iteration, the NR solver tries to reduce ΔF(x)−the error in function value for the guessed value of x from the actual solution of F(x). The value of x is thus obtained by iteratively solving till Δx→0.

FIG. 3 illustrates a method 300 for determining line parameters from voltage and current functions using Newton Raphson approximation, in accordance with an implementation of the present subject matter.  At block 302, an initial guess x′ is taken. Typically, resistance per meter length of a transmission line is in range 2 to 10 mΩ, inductance per meter length of a transmission line is in range 8 to 15 mH and Capacitance per meter length of a transmission line is in range 8 to 30 nF. Therefore, as an approximation, x′ can calculated using R=3 mΩ, L=10 mH and C=10 nF for all line sections for starting the NR method iterations. As will be understood, the method can work for other guess value as well without any significant change in number of iterations or time requirement. However, as the variables Z_{cM }and Z_{cN }have C in the denominator, the term becomes notdeterminant for C=0 value. Therefore, with C=0 it is not possible to solve these equations and the method cannot be used for initial guess value as 0.
 At block 304, x is set at the guess x′, i.e., x=x′.
 At block 306, equations (7) to (13) are calculated based on the set value of x and elements of J are estimated using equation (18). Further, error in set value of x, Δx, is computed as,

Δx=J ^{−1} ΔF(x) (19)  At block 308, it is determined whether the error is less than a threshold to ensure that Δx→0. In one example, it may be checked if

$\uf603\frac{\Delta \ue89e\phantom{\rule{0.3em}{0.3ex}}\ue89ex}{x}\uf604<{10}^{5}$  At block 310, when the error is not less than the threshold, a new guess x′ is computed as

x′=x+Δx (20)  The method 300 then returns to block 304, where x is set to the new guess and the process is repeated iteratively till the error is less than the threshold.
 At block 312, when the error is less than the threshold, the method 300 provides the final value of x as output.
 As discussed above, with reference to equation (13),

x=(γ_{M},γ_{P} ,Z _{cM} ,Z _{cP})  Further, it is known that the characteristic impedance Z_{c }and line propagation constant γ are defined as,

$\begin{array}{cc}{Z}_{c}=\sqrt{\frac{\left(R+\mathrm{jwL}\right)}{\left(\mathrm{jwC}\right)}}& \left(21\right)\\ \gamma =\sqrt{\left(R+\mathrm{jwL}\right)\ue89e\left(\mathrm{jwC}\right)}& \left(22\right)\end{array}$  Where, w is the angular frequency of the system defined using system frequency f as w=2πf.
Rearranging (21) and (22), the line parameters (Resistance R, Inductance L and Capacitance C) can be obtained as: 
$\begin{array}{cc}C=\frac{1}{\mathrm{jw}}\ue89e\sqrt{\frac{\gamma}{{Z}_{c}}}& \left(22\right)\\ L=\frac{\mathrm{Imag}\ue8a0\left(\sqrt{{Z}_{c}\ue89e\gamma}\right)}{\mathrm{jw}}& \left(23\right)\\ R=\mathrm{Real}\ue8a0\left(\sqrt{{Z}_{c}\ue89e\gamma}\right)& \left(24\right)\end{array}$  Thus, using the value of x determined above and equations (22) to (24), the line parameters R, L, and C can be computed.
 In case of a homogenous line, the parameters of the Tsection are same as that of the main line sections, i.e. γ_{M}=γ_{N}=γ_{P }and Z_{cM}=Z_{cN}=Z_{cP}. Therefore, the parameters of the line can be obtained by solving for parameters of the main line section only. Thus, using (5) for 3 observations of voltage and currents from all the terminals of the line, we have,

F _{1,1}=cos h(γ_{M} l _{MJ})V _{M,1} ^{1} −Z _{cM }sin h(γ_{M} l _{MJ})I _{M,1} ^{1}−cos h(γ_{M} l _{NJ})V _{N,1} ^{1} +Z _{cM }sin h(γ_{N} l _{MJ})I _{N,1} ^{1} (26) 
F _{1,2}=cos h(γ_{M} l _{MJ})V _{M,2} ^{1} −Z _{cM }sin h(γ_{M} l _{MJ})I _{M,2} ^{1}−cos h(γ_{M} l _{NJ})V _{N,2} ^{1} +Z _{cM }sin h(γ_{N} l _{MJ})I _{N,2} ^{1} (27) 
F _{1,3}=cos h(γ_{M} l _{MJ})V _{M,3} ^{1} −Z _{cM }sin h(γ_{M} l _{MJ})I _{M,3} ^{1}−cos h(γ_{M} l _{NJ})V _{N,3} ^{1} +Z _{cM }sin h(γ_{N} l _{MJ})I _{N,3} ^{1} (28)  Where, subscript M, i represents i^{th }data set at bus M and so on.
 Thus, in case of homogenous lines, the equations are of form,

$\begin{array}{cc}\left[\begin{array}{c}{F}_{1,1}\\ {F}_{1,2}\\ {F}_{1,3}\end{array}\right]=F\ue8a0\left({\gamma}_{M},{Z}_{\mathrm{cM}}\right)=0& \left(29\right)\end{array}$  The above set of equations is also of form F(x)=0 and can be solved using NewtonRaphson (NR) method using equations (14) through (20) as discussed for nonhomogenous lines above. The line parameters (Resistance R, Inductance L and Capacitance C) can then be obtained from γ and Z_{c }using the relations (23), (24) and (25) as discussed above.
 In one example, the quality of the estimated parameters may be verified using a function norm(F) defined as,

norm(F)=Σ_{k=1} ^{3}(F _{1,k}^{2} +F _{2,k}^{2}) (30)  The estimated parameters can be used to calculate norm(F). Ideally, for no error in parameters and phasors, the value of norm(F) must be zero. But due to some phasor inaccuracy and inaccuracy in determined line parameters (R, L, C and lengths) function may have higher value. However, norm(F)<20 indicates the estimated parameters are close to actual with less than 5% error and hence can be considered to be accurate.
 As the above discussed equations used for line parameter estimation use three observations of voltage and current, it is essential that the recorded data sets result in three sets of equations. The above discussed parameter estimation results were found to be good (low error in output parameters) when the three observations had variation in power or current of more than 0.1% of the rated value, thereby resulting in three distinct equations. However, for smaller variations (less than 0.1%) in power/current the error in output values was found to be high as the three sets of equations obtained were not sufficiently distinct. In one example, one of the data set may also be recorded at noload.
 In case of a fault, three sets of data may not be available. Hence, a method using Disturbance recorder (DR) data can be used to formulate the equations for estimation of line parameters and fault location, as discussed below.
 The present subject matter relates to systems and methods for identifying fault location in multiterminal power transmission lines. The systems and methods of the present subject matter can be used for accurately identifying a section of a multiterminal power transmission line that is having a fault using disturbance recorder data (prefault and duringfault information). The systems and methods provide a setting free fault section and fault location identification method for multiterminal power transmission lines. Thus, line parameter information is not required as an input for faulted section identification/fault location determination. The faulted section identification is based on real and imaginary parts of surge impedance or characteristic impedance as will be discussed below.
 The above and other features, aspects, and advantages of the subject matter will be better explained with regard to the following description, appended claims, and accompanying figures. While the following description has been provided using the example of a threeterminal power transmission line, it will be understood that the present subject matter can be implemented for any power transmission system such as multiterminal/tapped power transmission lines, including for two terminal lines.

FIG. 4 illustrates a device 400 for identifying a fault location in a power transmission line 402, in accordance with an implementation of the present subject matter. The device 400 may be implemented as any computing device which may be, but is not restricted to, a server, a workstation, a desktop computer, a laptop, and an application. In one example, the device 400 may be implemented in a cloud network environment. In one example, the device 400 may be part of a control system, such as the control system 102, or may be implemented separately.  The power transmission line 402 may be used to transmit electric power. The electric power transmitted may be at high voltages, such as in the range of kilovolts, and for long distances, such as for tens or hundreds of kilometres. The power transmission line 402 includes a first terminal 404 at which the power transmission line 402 receives electric power from a first power source 406. The first terminal 404 may also be referred to as a first bus 404. The power transmission line 402 also includes a second terminal 408 at which the power transmission line 402 receives electric power from a second power source 410. The second terminal 408 may also be referred to as a second bus 408. The power transmission line 402 further includes a third terminal 412 at which the power transmission line 402 receives electric power from a third power source 414.
 The first, second, and third power sources 406, 410, 414 may be, for example, an electric generator. However, in other examples other power sources may be used. It will also be understood that while the following description is provided with reference to power sources connected at the different terminals, in other implementations substations or loads may be connected at one or more of the terminals and all such implementations are also intended to be covered herein.
 Each terminal may also be associated with a respective disturbance recorder or IED. For example, the first terminal 404 may be associated with a first disturbance recorder 416, the second terminal 408 may be associated with a second disturbance recorder 418, and the third terminal 412 may be associated with a third disturbance recorder 420.
 As will be understood, a disturbance recorder is a time sequence data recording equipment used to record instantaneous voltage and current at the terminal at which it is deployed. Generally, the disturbance data or records are generated in response to an event, such as in the event of a fault in the power transmission line. In some implementations, the disturbance recorder may be an IED. The components and working of a disturbance recorder would be understood to a person skilled in the art and are not described in detail for brevity.
 Further, as shown in
FIG. 4 , a junction 422 may be present in the power transmission line between the three terminals 404, 408, and 412. Accordingly, a section of the power transmission line between the first terminal 404 and the junction 422 may be referred to as a first section, a section of the power transmission line between the second terminal 408 and the junction 422 may be referred to as a second section, and a section of the power transmission line between the third terminal 412 and the junction 422 may be referred to as a third section.  Sometimes, an electrical fault or disturbance, commonly referred to as a fault, may occur on the power transmission line 402. For example, the fault may occur in any one of the sections in the power transmission line 402. The fault may be, for example, a phasetoground fault or a phasetophase fault. To facilitate identification of the section having the fault, the device 400 may utilize a processor 424, a phasor computation module 426, and a fault location identification module 428.
 The phasor computation module 426, and the fault location identification module 428 may be implemented in hardware, software, or combination of the two. In case the phasor computation module 426 and the fault location identification module 428 are partly or fully implemented in software, the processor 424 can fetch and execute instructions corresponding to the modules. Though not described, it will be understood that the device 400 can include other hardware and software components, such as memory, input/output interfaces, network interfaces, and various programs and applications that can aid in its functioning.
 The device 400 is communicatively linked to the disturbance recorders 416, 418, and 420, for example over a wired connection or wireless connection or combination of the two. In one example, the device 400 and the disturbance recorders 416, 418, and 420 communicate with each other over the Internet.
 In operation, the phasor computation module 426 may obtain synchronized disturbance records from the disturbance recorders 416, 418, and 420 of the terminals 404, 408, and 412 after a fault has occurred in the power transmission line and compute prefault terminal voltage and current phasors and duringfault terminal voltage and current phasors.
 The fault location identification module 428 may compute, using an assumed faulted section of the first, second, and third sections, values for propagation constant of each section, surge impedance of each section, and a fault location parameter. In one example, the computing is based on simultaneously solving a plurality of prefault objective functions and duringfault objective functions for the assumed faulted section with the computed prefault voltage and current phasors, duringfault current and voltage phasors, and line lengths of each section. For example, the prefault objective functions and duringfault objective functions may have been formulated based on equating junction voltages determined from two of the terminals, conservation of charge at the junction, and equating fault location voltages determined from one terminal and the junction. In one example, the prefault objective functions and duringfault objective functions may be stored in a memory of the device 400.
 In one example, each of the prefault objective functions and the duringfault objective functions correspond to a function of a set of variables, the function being equal to zero and the set of variables comprising the propagation constant of each section, the surge impedance of each section, and the fault location parameter. In one example, to solve the prefault functions and the duringfault functions, the fault location identification module 428 is configured to apply a numerical analysis technique, such as Newton Raphson method, to determine the propagation constant of each section, the surge impedance of each section, and the fault location parameter.
 Formulation of the prefault objective functions and duringfault objective functions and solving of the objective functions will be discussed later in detail with reference to an example shown in
FIGS. 5A5C .  The fault location identification module 428 may compare the values for propagation constant, surge impedance, and fault location parameter determined for the assumed faulted section with predefined criteria for the section, and determine a faulted section, the fault location, and the line parameters for each of the first, second, and third sections, based on the comparison.
 In one example, the predefined criteria used to determine whether a particular assumed section is a faulted section include, (a) the fault location parameter has a value between zero and one; (b) real part of square of propagation constant of each section is less than zero and real part of square of surge impedance of each section is greater than zero; and (c) imaginary part of square of propagation constant of each section is greater than zero and imaginary part of square of surge impedance of each section is less than zero.
 The method of identification of fault location is further described below with reference to
FIGS. 5A5C showing fault location identification on a threeterminal/tapped power transmission line. It will, however, be understood that similar computations can be performed for identification of fault location in any multiterminal power transmission line, albeit with modifications as will be understood to a person skilled in the art. In the discussion, the objective functions may also be interchangeably referred to as equations.  For a prefault condition, the voltage of junction J calculated using either end data must be equal. Therefore, as discussed above, with reference to
FIG. 2 , considering the threeterminal power transmission line to be a distributed twoport model, the prefault objective functions using positive or negative sequence voltage can be formulated as: 
F _{1}=cos h(γ_{M} l _{MJ})V _{M} ^{1} −Z _{cM }sin h(γ_{M} l _{MJ})I _{M} ^{1}−cos h(γ_{M} l _{NJ})V _{N} ^{1} +Z _{cM }sin h(γ_{N} l _{MJ})I _{N} ^{1} (31) 
F _{2}=cos h(γ_{M} l _{MJ})V _{M} ^{1} −Z _{cM }sin h(γ_{M} l _{MJ})I _{M} ^{1}−cos h(γ_{P} l _{PJ})V _{P} ^{1} +Z _{cP }sin h(γ_{P} l _{PJ})I _{P} ^{1} (32) 
F _{3}=cos h(γ_{N} l _{NJ})V _{N} ^{1} −Z _{cN }sin h(γ_{N} l _{NJ})I _{N} ^{1}−cos h(γ_{P} l _{PJ})V _{P} ^{1} +Z _{cP }sin h(γ_{P} l _{PJ})I _{P} ^{1} (33)  Here, F_{1 }has been formulated based on voltage calculated from M and N ends, F_{2 }has been formulated based on voltage calculated from M and P ends, and F_{3 }has been formulated based on voltage calculated from M and P ends. Further, in equations 3133,
 γ_{M}=propagation constant of section MJ
 γ_{N}=propagation constant of section NJ
 γ_{P}=propagation constant of section PJ
 Z_{cM}=characteristic impedance of section MJ
 Z_{cN}=characteristic impedance of section NJ
 Z_{cP}=characteristic impedance of section PJ
 Also by Kirchhoff's current law (KCL) for conservation of charge at junction 222, for prefault condition, the total current flowing out of the junction should be equal to the total current flowing into the junction. Hence, a prefault objective function for current can be formulated as

$\begin{array}{cc}{F}_{4}=\frac{\mathrm{sinh}\ue8a0\left({\gamma}_{M}\ue89e{l}_{\mathrm{MJ}}\right)}{{Z}_{\mathrm{cM}}}\ue89e{V}_{M}^{1}\mathrm{cosh}\ue8a0\left({\gamma}_{M}\ue89e{l}_{\mathrm{MJ}}\right)\ue89e{I}_{M}^{1}+\frac{\mathrm{sinh}\ue8a0\left({\gamma}_{N}\ue89e{l}_{\mathrm{NJ}}\right)}{{Z}_{\mathrm{cN}}}\ue89e{V}_{N}^{1}\mathrm{cosh}\ue8a0\left({\gamma}_{N}\ue89e{l}_{\mathrm{NJ}}\right)\ue89e{I}_{N}^{1}+\frac{\mathrm{sinh}\ue8a0\left({\gamma}_{P}\ue89e{l}_{\mathrm{PJ}}\right)}{{Z}_{\mathrm{cP}}}\ue89e{V}_{P}^{1}\mathrm{cosh}\ue8a0\left({\gamma}_{P}\ue89e{l}_{\mathrm{PJ}}\right)\ue89e{I}_{P}^{1}& \left(34\right)\end{array}$  The objective functions for duringfault conditions may be formulated assuming the fault location to be in any one of the sections. This will be described with reference to
FIG. 5A5C . 
FIGS. 5A5C illustrate current and voltage phasors used for computation of line parameters and fault location in a threeterminal power transmission line with fault in different sections, in accordance with an implementation of the present subject matter.  As shown in
FIG. 5A , consider a fault in section MJ at a distance dl_{MJ }from Bus M and therefore, at a distance (1−d)I_{MJ }from junction point J. Here, d corresponds to fault location parameter and is, in one example, the fraction of length of section MJ from the terminal M at which the fault point is located. It will be understood that the fault location parameter may be defined differently also such as, for e.g., distance from terminal M. The duringfault objective functions can be formulated using two synchronized sets of readings as shown below: 
${F}_{5,\mathrm{MJ}}=\mathrm{cosh}\ue8a0\left({\gamma}_{N}\ue89e{l}_{\mathrm{NJ}}\right)\ue89e{V}_{\mathrm{Nf}}^{1}{Z}_{\mathrm{cN}}\ue89e\mathrm{sinh}\ue8a0\left({\gamma}_{N}\ue89e{l}_{\mathrm{NJ}}\right)\ue89e{I}_{\mathrm{Nf}}^{1}\mathrm{cosh}\ue8a0\left({\gamma}_{P}\ue89e{l}_{\mathrm{PJ}}\right)\ue89e{V}_{\mathrm{Pf}}^{1}+{Z}_{\mathrm{cP}}\ue89e\mathrm{sinh}\ue8a0\left({\gamma}_{P}\ue89e{l}_{\mathrm{PJ}}\right)\ue89e{I}_{\mathrm{Pf}}^{1}$ ${F}_{6,\mathrm{MJ}}=\mathrm{cosh}\ue8a0\left({\gamma}_{N}\ue89e{l}_{\mathrm{NJ}}\right)\ue89e{V}_{\mathrm{Nf}}^{2}{Z}_{\mathrm{cN}}\ue89e\mathrm{sinh}\ue8a0\left({\gamma}_{N}\ue89e{l}_{\mathrm{NJ}}\right)\ue89e{I}_{\mathrm{Nf}}^{2}\mathrm{cosh}\ue8a0\left({\gamma}_{P}\ue89e{l}_{\mathrm{PJ}}\right)\ue89e{V}_{\mathrm{Pf}}^{2}+{Z}_{\mathrm{cP}}\ue89e\mathrm{sinh}\ue8a0\left({\gamma}_{P}\ue89e{l}_{\mathrm{PJ}}\right)\ue89e{I}_{\mathrm{Pf}}^{2}$ $\phantom{\rule{1.1em}{1.1ex}}\ue89e{V}_{\mathrm{Jf}}^{1}=\mathrm{cosh}\ue8a0\left({\gamma}_{N}\ue89e{l}_{\mathrm{NJ}}\right)\times {V}_{\mathrm{Nf}}^{1}{Z}_{\mathrm{cN}}\ue89e\mathrm{sinh}\ue8a0\left({\gamma}_{N}\ue89e{l}_{\mathrm{NJ}}\right)\ue89e{I}_{\mathrm{Nf}}^{1}$ $\phantom{\rule{1.1em}{1.1ex}}\ue89e{V}_{\mathrm{Jf}}^{2}=\mathrm{cosh}\ue8a0\left({\gamma}_{N}\ue89e{l}_{\mathrm{NJ}}\right)\times {V}_{\mathrm{Nf}}^{2}{Z}_{\mathrm{cN}}\ue89e\mathrm{sinh}\ue8a0\left({\gamma}_{N}\ue89e{l}_{\mathrm{NJ}}\right)\ue89e{I}_{\mathrm{Nf}}^{2}$ ${I}_{\mathrm{JMf}}^{1}=\left(\frac{\mathrm{sinh}\ue8a0\left({\gamma}_{N}\ue89e{l}_{\mathrm{NJ}}\right)}{{Z}_{\mathrm{cN}}}\ue89e{V}_{\mathrm{Nf}}^{1}\mathrm{cosh}\ue8a0\left({\gamma}_{N}\ue89e{l}_{\mathrm{NJ}}\right)\ue89e{I}_{\mathrm{Nf}}^{1}+\frac{\mathrm{sinh}\ue8a0\left({\gamma}_{P}\ue89e{l}_{\mathrm{PJ}}\right)}{{Z}_{\mathrm{cP}}}\ue89e{V}_{\mathrm{Pf}}^{1}\mathrm{cosh}\ue8a0\left({\gamma}_{P}\ue89e{l}_{\mathrm{PJ}}\right)\ue89e{I}_{\mathrm{Pf}}^{1}\right)$ ${F}_{5,\mathrm{MJ}}=\mathrm{cosh}\ue8a0\left({\gamma}_{N}\ue89e{l}_{\mathrm{NJ}}\right)\ue89e{V}_{\mathrm{Nf}}^{1}{Z}_{\mathrm{cN}}\ue89e\mathrm{sinh}\ue8a0\left({\gamma}_{N}\ue89e{l}_{\mathrm{NJ}}\right)\ue89e{I}_{\mathrm{Nf}}^{1}\mathrm{cosh}\ue8a0\left({\gamma}_{P}\ue89e{l}_{\mathrm{PJ}}\right)\ue89e{V}_{\mathrm{Pf}}^{1}+{Z}_{\mathrm{cP}}\ue89e\mathrm{sinh}\ue8a0\left({\gamma}_{P}\ue89e{l}_{\mathrm{PJ}}\right)\ue89e{I}_{\mathrm{Pf}}^{1}$ ${F}_{6,\mathrm{MJ}}=\mathrm{cosh}\ue8a0\left({\gamma}_{N}\ue89e{l}_{\mathrm{NJ}}\right)\ue89e{V}_{\mathrm{Nf}}^{2}{Z}_{\mathrm{cN}}\ue89e\mathrm{sinh}\ue8a0\left({\gamma}_{N}\ue89e{l}_{\mathrm{NJ}}\right)\ue89e{I}_{\mathrm{Nf}}^{2}\mathrm{cosh}\ue8a0\left({\gamma}_{P}\ue89e{l}_{\mathrm{PJ}}\right)\ue89e{V}_{\mathrm{Pf}}^{2}+{Z}_{\mathrm{cP}}\ue89e\mathrm{sinh}\ue8a0\left({\gamma}_{P}\ue89e{l}_{\mathrm{PJ}}\right)\ue89e{I}_{\mathrm{Pf}}^{2}$ $\phantom{\rule{1.1em}{1.1ex}}\ue89e{V}_{\mathrm{Jf}}^{1}=\mathrm{cosh}\ue8a0\left({\gamma}_{N}\ue89e{l}_{\mathrm{NJ}}\right)\times {V}_{\mathrm{Nf}}^{1}{Z}_{\mathrm{cN}}\ue89e\mathrm{sinh}\ue8a0\left({\gamma}_{N}\ue89e{l}_{\mathrm{NJ}}\right)\ue89e{I}_{\mathrm{Nf}}^{1}$ $\phantom{\rule{1.1em}{1.1ex}}\ue89e{V}_{\mathrm{Jf}}^{2}=\mathrm{cosh}\ue8a0\left({\gamma}_{N}\ue89e{l}_{\mathrm{NJ}}\right)\times {V}_{\mathrm{Nf}}^{2}{Z}_{\mathrm{cN}}\ue89e\mathrm{sinh}\ue8a0\left({\gamma}_{N}\ue89e{l}_{\mathrm{NJ}}\right)\ue89e{I}_{\mathrm{Nf}}^{2}$ ${I}_{\mathrm{JMf}}^{1}=\left(\frac{\mathrm{sinh}\ue8a0\left({\gamma}_{N}\ue89e{l}_{\mathrm{NJ}}\right)}{{Z}_{\mathrm{cN}}}\ue89e{V}_{\mathrm{Nf}}^{1}\mathrm{cosh}\ue8a0\left({\gamma}_{N}\ue89e{l}_{\mathrm{NJ}}\right)\ue89e{I}_{\mathrm{Nf}}^{1}+\frac{\mathrm{sinh}\ue8a0\left({\gamma}_{P}\ue89e{l}_{\mathrm{PJ}}\right)}{{Z}_{\mathrm{cP}}}\ue89e{V}_{\mathrm{Pf}}^{1}\mathrm{cosh}\ue8a0\left({\gamma}_{P}\ue89e{l}_{\mathrm{PJ}}\right)\ue89e{I}_{\mathrm{Pf}}^{1}\right)$ ${I}_{\mathrm{JMf}}^{2}=\left(\frac{\mathrm{sinh}\ue8a0\left({\gamma}_{N}\ue89e{l}_{\mathrm{NJ}}\right)}{{Z}_{\mathrm{cN}}}\ue89e{V}_{\mathrm{Nf}}^{2}\mathrm{cosh}\ue8a0\left({\gamma}_{N}\ue89e{l}_{\mathrm{NJ}}\right)\ue89e{I}_{\mathrm{Nf}}^{2}+\frac{\mathrm{sinh}\ue8a0\left({\gamma}_{P}\ue89e{l}_{\mathrm{PJ}}\right)}{{Z}_{\mathrm{cP}}}\ue89e{V}_{\mathrm{Pf}}^{2}\mathrm{cosh}\ue8a0\left({\gamma}_{P}\ue89e{l}_{\mathrm{PJ}}\right)\ue89e{I}_{\mathrm{Pf}}^{2}\right)$ ${F}_{7,\mathrm{MJ}}=\mathrm{cosh}\ue8a0\left({\gamma}_{M}\ue89e{\mathrm{dl}}_{\mathrm{MJ}}\ue89e{l}_{\mathrm{MJ}}\right)\ue89e{V}_{\mathrm{Jf}}^{1}{Z}_{\mathrm{cM}}\ue89e\mathrm{sinh}\ue8a0\left({\gamma}_{M}\ue89e{\mathrm{dl}}_{\mathrm{MJ}}\ue89e{l}_{\mathrm{MJ}}\right)\ue89e{I}_{\mathrm{JMf}}^{1}\mathrm{cosh}\ue8a0\left({\gamma}_{M}\ue8a0\left(1{\mathrm{dl}}_{\mathrm{MJ}}\right)\ue89e{l}_{\mathrm{MJ}}\right)\ue89e{V}_{\mathrm{Jf}}^{1}+{Z}_{\mathrm{cM}}\ue89e\mathrm{sinh}\ue8a0\left({\gamma}_{M}\ue8a0\left(1{\mathrm{dl}}_{\mathrm{MJ}}\right)\ue89e{l}_{\mathrm{MJ}}\right)\ue89e{I}_{\mathrm{JMf}}^{1}$ ${F}_{8,\mathrm{MJ}}=\mathrm{cosh}\ue8a0\left({\gamma}_{M}\ue89e{\mathrm{dl}}_{\mathrm{MJ}}\ue89e{l}_{\mathrm{MJ}}\right)\ue89e{V}_{\mathrm{Jf}}^{2}{Z}_{\mathrm{cM}}\ue89e\mathrm{sinh}\ue8a0\left({\gamma}_{M}\ue89e{\mathrm{dl}}_{\mathrm{MJ}}\ue89e{l}_{\mathrm{MJ}}\right)\ue89e{I}_{\mathrm{JMf}}^{2}\mathrm{cosh}\ue8a0\left({\gamma}_{M}\ue8a0\left(1{\mathrm{dl}}_{\mathrm{MJ}}\right)\ue89e{l}_{\mathrm{MJ}}\right)\ue89e{V}_{\mathrm{Jf}}^{2}+{Z}_{\mathrm{cM}}\ue89e\mathrm{sinh}\ue8a0\left({\gamma}_{M}\ue8a0\left(1{\mathrm{dl}}_{\mathrm{MJ}}\right)\ue89e{l}_{\mathrm{MJ}}\right)\ue89e{I}_{\mathrm{JMf}}^{2}$  Equations F_{5,MJ }and F_{6,MJ }are based on voltage at junction J computed using a twoport model from the ends where there is no fault, i.e., from N and P terminals.
 Equations for current flowing out of junction J into faulted section MJ, i.e., I_{JMf} ^{1 }and I_{JMf} ^{2 }are determined based on Kirchhoff's current law applied at Junction J.
 Equations F_{7,MJ }and F_{8,MJ }are based on voltage at fault location computed using a twoport model from the ends M and J.
where, the superscript i in the duringfault equations shown above indicates the i^{th }data set.  As shown in
FIG. 5B , consider a fault in section NJ at a distance dl_{NJ }from Bus N and therefore, at a distance (1−d)I_{NJ }from junction point J. Here, d corresponds to fault location parameter and is, in one example, the fraction of length of section NJ from the terminal N at which the fault point is located. It will be understood that the fault location parameter may be defined differently also. The duringfault objective functions can be formulated using two synchronized sets of readings in a manner similar to that discussed forFIG. 5A . The duringfault objective functions for fault in section NJ are shown below: 
${F}_{5,\mathrm{NJ}}=\mathrm{cosh}\ue8a0\left({\gamma}_{M}\ue89e{l}_{\mathrm{MJ}}\right)\ue89e{V}_{\mathrm{Mf}}^{1}{Z}_{\mathrm{cN}}\ue89e\mathrm{sinh}\ue8a0\left({\gamma}_{M}\ue89e{l}_{\mathrm{MJ}}\right)\ue89e{I}_{\mathrm{Mf}}^{1}\mathrm{cosh}\ue8a0\left({\gamma}_{P}\ue89e{l}_{\mathrm{PJ}}\right)\ue89e{V}_{\mathrm{Pf}}^{1}+{Z}_{\mathrm{cP}}\ue89e\mathrm{sinh}\ue8a0\left({\gamma}_{P}\ue89e{l}_{\mathrm{PJ}}\right)\ue89e{I}_{\mathrm{Pf}}^{1}$ ${F}_{6,\mathrm{NJ}}=\mathrm{cosh}\ue8a0\left({\gamma}_{M}\ue89e{l}_{\mathrm{MJ}}\right)\ue89e{V}_{\mathrm{Mf}}^{2}{Z}_{\mathrm{cN}}\ue89e\mathrm{sinh}\ue8a0\left({\gamma}_{M}\ue89e{l}_{\mathrm{MJ}}\right)\ue89e{I}_{\mathrm{Mf}}^{2}\mathrm{cosh}\ue8a0\left({\gamma}_{P}\ue89e{l}_{\mathrm{PJ}}\right)\ue89e{V}_{\mathrm{Pf}}^{2}+{Z}_{\mathrm{cP}}\ue89e\mathrm{sinh}\ue8a0\left({\gamma}_{P}\ue89e{l}_{\mathrm{PJ}}\right)\ue89e{I}_{\mathrm{Pf}}^{2}$ $\phantom{\rule{1.1em}{1.1ex}}\ue89e{V}_{\mathrm{Jf}}^{1}=\mathrm{cosh}\ue8a0\left({\gamma}_{M}\ue89e{l}_{\mathrm{MJ}}\right)\times {V}_{\mathrm{Mf}}^{1}{Z}_{\mathrm{cN}}\ue89e\mathrm{sinh}\ue8a0\left({\gamma}_{M}\ue89e{l}_{\mathrm{MJ}}\right)\ue89e{I}_{\mathrm{Mf}}^{1}$ $\phantom{\rule{1.1em}{1.1ex}}\ue89e{V}_{\mathrm{Jf}}^{2}=\mathrm{cosh}\ue8a0\left({\gamma}_{M}\ue89e{l}_{\mathrm{MJ}}\right)\times {V}_{\mathrm{Mf}}^{2}{Z}_{\mathrm{cN}}\ue89e\mathrm{sinh}\ue8a0\left({\gamma}_{M}\ue89e{l}_{\mathrm{MJ}}\right)\ue89e{I}_{\mathrm{Mf}}^{2}$ ${I}_{\mathrm{JNf}}^{1}=\left(\frac{\mathrm{sinh}\ue8a0\left({\gamma}_{M}\ue89e{l}_{\mathrm{MJ}}\right)}{{Z}_{\mathrm{cM}}}\ue89e{V}_{\mathrm{Mf}}^{1}\mathrm{cosh}\ue8a0\left({\gamma}_{M}\ue89e{l}_{\mathrm{MJ}}\right)\ue89e{I}_{\mathrm{Mf}}^{1}+\frac{\mathrm{sinh}\ue8a0\left({\gamma}_{P}\ue89e{l}_{\mathrm{PJ}}\right)}{{Z}_{\mathrm{cP}}}\ue89e{V}_{\mathrm{Pf}}^{1}\mathrm{cosh}\ue8a0\left({\gamma}_{P}\ue89e{l}_{\mathrm{PJ}}\right)\ue89e{I}_{\mathrm{Pf}}^{1}\right)$ ${I}_{\mathrm{JNf}}^{2}=\left(\frac{\mathrm{sinh}\ue8a0\left({\gamma}_{M}\ue89e{l}_{\mathrm{MJ}}\right)}{{Z}_{\mathrm{cM}}}\ue89e{V}_{\mathrm{Mf}}^{2}\mathrm{cosh}\ue8a0\left({\gamma}_{M}\ue89e{l}_{\mathrm{MJ}}\right)\ue89e{I}_{\mathrm{Mf}}^{2}+\frac{\mathrm{sinh}\ue8a0\left({\gamma}_{P}\ue89e{l}_{\mathrm{PJ}}\right)}{{Z}_{\mathrm{cP}}}\ue89e{V}_{\mathrm{Pf}}^{2}\mathrm{cosh}\ue8a0\left({\gamma}_{P}\ue89e{l}_{\mathrm{PJ}}\right)\ue89e{I}_{\mathrm{Pf}}^{2}\right)$ ${F}_{7,\mathrm{NJ}}=\mathrm{cosh}\ue8a0\left({\gamma}_{N}\ue89e{\mathrm{dl}}_{\mathrm{NJ}}\ue89e{l}_{\mathrm{NJ}}\right)\ue89e{V}_{\mathrm{JNf}}^{1}{Z}_{\mathrm{cN}}\ue89e\mathrm{sinh}\ue8a0\left({\gamma}_{N}\ue89e{\mathrm{dl}}_{\mathrm{NJ}}\ue89e{l}_{\mathrm{NJ}}\right)\ue89e{I}_{\mathrm{JNf}}^{1}\mathrm{cosh}\ue8a0\left({\gamma}_{N}\ue8a0\left(1{\mathrm{dl}}_{\mathrm{NJ}}\right)\ue89e{l}_{\mathrm{NJ}}\right)\ue89e{V}_{\mathrm{JNf}}^{1}+{Z}_{\mathrm{cN}}\ue89e\mathrm{sinh}\ue8a0\left({\gamma}_{N}\ue8a0\left(1{\mathrm{dl}}_{\mathrm{NJ}}\right)\ue89e{l}_{\mathrm{NJ}}\right)\ue89e{I}_{\mathrm{JNf}}^{1}$ ${F}_{8,\mathrm{NJ}}=\mathrm{cosh}\ue8a0\left({\gamma}_{N}\ue89e{\mathrm{dl}}_{\mathrm{NJ}}\ue89e{l}_{\mathrm{NJ}}\right)\ue89e{V}_{\mathrm{JNf}}^{2}{Z}_{\mathrm{cN}}\ue89e\mathrm{sinh}\ue8a0\left({\gamma}_{N}\ue89e{\mathrm{dl}}_{\mathrm{NJ}}\ue89e{l}_{\mathrm{NJ}}\right)\ue89e{I}_{\mathrm{JNf}}^{2}\mathrm{cosh}\ue8a0\left({\gamma}_{N}\ue8a0\left(1{\mathrm{dl}}_{\mathrm{NJ}}\right)\ue89e{l}_{\mathrm{NJ}}\right)\ue89e{V}_{\mathrm{JNf}}^{2}+{Z}_{\mathrm{cM}}\ue89e\mathrm{sinh}\ue8a0\left({\gamma}_{N}\ue8a0\left(1{\mathrm{dl}}_{\mathrm{NJ}}\right)\ue89e{l}_{\mathrm{NJ}}\right)\ue89e{I}_{\mathrm{JNf}}^{2}$  Equations F_{5,NJ }and F_{6,NJ }are based on voltage at junction J computed using a twoport model from the ends where there is no fault, i.e., from M and P terminals.
Equations for current flowing out of junction J into faulted section NJ, i.e., I_{JNf} ^{1}, I_{JNf} ^{2}, are determined based on Kirchhoff's current law applied at Junction J.
Equations F_{7,NJ }and F_{8,NJ }are based on voltage at fault location computed using a twoport model from the ends N and J.
where, the superscript i in the duringfault equations shown above indicates the i^{th }data set.  As shown in
FIG. 5C , consider a fault in section PJ at a distance dl_{PJ }from Bus N and therefore, at a distance (1−d)I_{PJ }from junction point J. Here, d corresponds to fault location parameter and is, in one example, the fraction of length of section PJ from the terminal P at which the fault point is located. It will be understood that the fault location parameter may be defined differently also. The duringfault objective functions can be formulated using two synchronized sets of readings in a manner similar to that discussed forFIG. 5A . The duringfault objective functions for fault in section PJ are shown below: 
${F}_{5,\mathrm{PJ}}=\mathrm{cosh}\ue8a0\left({\gamma}_{M}\ue89e{l}_{\mathrm{MJ}}\right)\ue89e{V}_{\mathrm{Mf}}^{1}{Z}_{\mathrm{cM}}\ue89e\mathrm{sinh}\ue8a0\left({\gamma}_{M}\ue89e{l}_{\mathrm{MJ}}\right)\ue89e{I}_{\mathrm{Mf}}^{1}\mathrm{cosh}\ue8a0\left({\gamma}_{N}\ue89e{l}_{\mathrm{NJ}}\right)\ue89e{V}_{\mathrm{Nf}}^{1}+{Z}_{\mathrm{cN}}\ue89e\mathrm{sinh}\ue8a0\left({\gamma}_{N}\ue89e{l}_{\mathrm{NJ}}\right)\ue89e{I}_{\mathrm{Nf}}^{1}$ ${F}_{6,\mathrm{PJ}}=\mathrm{cosh}\ue8a0\left({\gamma}_{M}\ue89e{l}_{\mathrm{MJ}}\right)\ue89e{V}_{\mathrm{Mf}}^{2}{Z}_{\mathrm{cM}}\ue89e\mathrm{sinh}\ue8a0\left({\gamma}_{M}\ue89e{l}_{\mathrm{MJ}}\right)\ue89e{I}_{\mathrm{Mf}}^{2}\mathrm{cosh}\ue8a0\left({\gamma}_{N}\ue89e{l}_{\mathrm{NJ}}\right)\ue89e{V}_{\mathrm{Nf}}^{2}+{Z}_{\mathrm{cN}}\ue89e\mathrm{sinh}\ue8a0\left({\gamma}_{N}\ue89e{l}_{\mathrm{NJ}}\right)\ue89e{I}_{\mathrm{Nf}}^{2}$ $\phantom{\rule{1.1em}{1.1ex}}\ue89e{V}_{\mathrm{Jf}}^{1}=\mathrm{cosh}\ue8a0\left({\gamma}_{M}\ue89e{l}_{\mathrm{MJ}}\right)\times {V}_{\mathrm{Mf}}^{1}{Z}_{\mathrm{cM}}\ue89e\mathrm{sinh}\ue8a0\left({\gamma}_{M}\ue89e{l}_{\mathrm{MJ}}\right)\ue89e{I}_{\mathrm{Mf}}^{1}$ $\phantom{\rule{1.1em}{1.1ex}}\ue89e{V}_{\mathrm{Jf}}^{2}=\mathrm{cosh}\ue8a0\left({\gamma}_{M}\ue89e{l}_{\mathrm{MJ}}\right)\times {V}_{\mathrm{Mf}}^{2}{Z}_{\mathrm{cN}}\ue89e\mathrm{sinh}\ue8a0\left({\gamma}_{M}\ue89e{l}_{\mathrm{MJ}}\right)\ue89e{I}_{\mathrm{Mf}}^{2}$ ${I}_{\mathrm{JPf}}^{1}=\left(\frac{\mathrm{sinh}\ue8a0\left({\gamma}_{M}\ue89e{l}_{\mathrm{MJ}}\right)}{{Z}_{\mathrm{cM}}}\ue89e{V}_{\mathrm{Mf}}^{1}\mathrm{cosh}\ue8a0\left({\gamma}_{M}\ue89e{l}_{\mathrm{MJ}}\right)\ue89e{I}_{\mathrm{Mf}}^{1}+\frac{\mathrm{sinh}\ue8a0\left({\gamma}_{N}\ue89e{l}_{\mathrm{NJ}}\right)}{{Z}_{\mathrm{cN}}}\ue89e{V}_{\mathrm{Nf}}^{1}\mathrm{cosh}\ue8a0\left({\gamma}_{N}\ue89e{l}_{\mathrm{NJ}}\right)\ue89e{I}_{\mathrm{Nf}}^{1}\right)$ ${I}_{\mathrm{JPf}}^{2}=\left(\frac{\mathrm{sinh}\ue8a0\left({\gamma}_{M}\ue89e{l}_{\mathrm{MJ}}\right)}{{Z}_{\mathrm{cM}}}\ue89e{V}_{\mathrm{Mf}}^{2}\mathrm{cosh}\ue8a0\left({\gamma}_{M}\ue89e{l}_{\mathrm{MJ}}\right)\ue89e{I}_{\mathrm{Mf}}^{2}+\frac{\mathrm{sinh}\ue8a0\left({\gamma}_{N}\ue89e{l}_{\mathrm{NJ}}\right)}{{Z}_{\mathrm{cN}}}\ue89e{V}_{\mathrm{Nf}}^{2}\mathrm{cosh}\ue8a0\left({\gamma}_{N}\ue89e{l}_{\mathrm{NJ}}\right)\ue89e{I}_{\mathrm{Nf}}^{2}\right)$ ${F}_{7,\mathrm{PJ}}=\mathrm{cosh}\ue8a0\left({\gamma}_{P}\ue89e{\mathrm{dl}}_{\mathrm{PJ}}\ue89e{l}_{\mathrm{PJ}}\right)\ue89e{V}_{\mathrm{JPf}}^{1}{Z}_{\mathrm{cP}}\ue89e\mathrm{sinh}\ue8a0\left({\gamma}_{P}\ue89e{\mathrm{dl}}_{\mathrm{PJ}}\ue89e{l}_{\mathrm{PJ}}\right)\ue89e{I}_{\mathrm{JPf}}^{1}\mathrm{cosh}\ue8a0\left({\gamma}_{P}\ue8a0\left(1{\mathrm{dl}}_{\mathrm{PJ}}\right)\ue89e{l}_{\mathrm{PJ}}\right)\ue89e{V}_{\mathrm{JPf}}^{1}+{Z}_{\mathrm{cP}}\ue89e\mathrm{sinh}\ue8a0\left({\gamma}_{P}\ue8a0\left(1{\mathrm{dl}}_{\mathrm{PJ}}\right)\ue89e{l}_{\mathrm{PJ}}\right)\ue89e{I}_{\mathrm{JPf}}^{1}$ ${F}_{8,\mathrm{PJ}}=\mathrm{cosh}\ue8a0\left({\gamma}_{P}\ue89e{\mathrm{dl}}_{\mathrm{PJ}}\ue89e{l}_{\mathrm{PJ}}\right)\ue89e{V}_{\mathrm{JPf}}^{2}{Z}_{\mathrm{cP}}\ue89e\mathrm{sinh}\ue8a0\left({\gamma}_{P}\ue89e{\mathrm{dl}}_{\mathrm{PJ}}\ue89e{l}_{\mathrm{PJ}}\right)\ue89e{I}_{\mathrm{JPf}}^{2}\mathrm{cosh}\ue8a0\left({\gamma}_{P}\ue8a0\left(1{\mathrm{dl}}_{\mathrm{PJ}}\right)\ue89e{l}_{\mathrm{PJ}}\right)\ue89e{V}_{\mathrm{JPf}}^{2}+{Z}_{\mathrm{cP}}\ue89e\mathrm{sinh}\ue8a0\left({\gamma}_{P}\ue8a0\left(1{\mathrm{dl}}_{\mathrm{PJ}}\right)\ue89e{l}_{\mathrm{PJ}}\right)\ue89e{I}_{\mathrm{JPf}}^{2}$  Equations F_{5,PJ }and F_{6,PJ }are based on voltage at junction J computed using a twoport model from the ends where there is no fault, i.e., from M and N terminals.
Equations for current flowing out of junction J into faulted section PJ, i.e., I_{JPf} ^{1 }and I_{JPf} ^{2 }are determined based on Kirchhoff's current law applied at Junction J.
Equations F_{7,PJ }and F_{8,PJ }are based on voltage at fault location computed using a twoport model from the ends P and J.
where, the superscript i in the duringfault equations shown above indicates the i^{th }data set.  Thus, the duringfault objective functions are formed considering fault in different sections of the line. Then the line parameters and fault location are simultaneously estimated for a set of equations of an assumed faulted section, where the set of equations includes prefault objective functions F1 to F4 and duringfault objective functions F5 to F8 corresponding to the assumed faulted section. As can be seen, the equations F1F8 in a particular set of equations are of the form F(x)=0 and hence can be solved simultaneously using a numerical analysis technique, for example, using the NR method as discussed above, to determine the surge impedance, propagation constants, and fault location parameter d.
 Further, the line parameters, i.e., resistance, conductance and inductance of a section can be determined from the surge impedance and propagation constant in that section as discussed earlier. In one example, the line parameters for two of the sections are equal. In another example, the line parameters for all three sections are equal.
 The correct line parameters and fault location are selected based on predefine criteria. As is known, for a transmission line with per unit series impedance z=R+jωL and per unit shunt admittance y=jωC (where, R, L and C indicate the resistance, inductance and capacitance of respective element), the characteristic impedance of the line is defined as,

${Z}_{c}=\sqrt{\frac{z}{y}}=\sqrt{\frac{R+j\ue89e\phantom{\rule{0.3em}{0.3ex}}\ue89e\omega \ue89e\phantom{\rule{0.3em}{0.3ex}}\ue89eL}{j\ue89e\phantom{\rule{0.3em}{0.3ex}}\ue89e\omega \ue89e\phantom{\rule{0.3em}{0.3ex}}\ue89eC}}=\sqrt{\frac{\left(R+j\ue89e\phantom{\rule{0.3em}{0.3ex}}\ue89e\omega \ue89e\phantom{\rule{0.3em}{0.3ex}}\ue89eL\right)\ue89e\left(j\ue89e\phantom{\rule{0.3em}{0.3ex}}\ue89e\omega \ue89e\phantom{\rule{0.3em}{0.3ex}}\ue89eC\right)}{{\omega}^{2}\ue89e{C}^{2}}}=\sqrt{\frac{{\omega}^{2}\ue89e\mathrm{LC}}{{\omega}^{2}\ue89e{C}^{2}}\frac{j\ue89e\phantom{\rule{0.3em}{0.3ex}}\ue89e\omega \ue89e\phantom{\rule{0.3em}{0.3ex}}\ue89e\mathrm{RC}}{{\omega}^{2}\ue89e{C}^{2}}}$ 
${Z}_{c}^{2}=\frac{L}{C}j\ue89e\frac{R}{\omega \ue89e\phantom{\rule{0.3em}{0.3ex}}\ue89eC}$  Also, the propagation constant of the line is defined as,

γ=√{square root over (zy)}=√{square root over ((R+jωL)(jωC))}=√{square root over (−ω^{2} LC+jωRC)} 
Therefore, 
γ^{2}=−ω^{2} LC+jωRC.  The R, L and C being positive values for a transmission line, Zc and γ should satisfy following characteristic—

 a. real(γ^{2})<0, and real(Z_{c} ^{2})>0
 b. image(γ^{2})>0, and imag(Z_{c} ^{2})<0
 Hence, when the prefault and duringfault equations for a particular section are solved simultaneously, the resulting propagation constant and surge impedance of each section, and fault location parameter should meet the following predefined criteria if that particular section is the faulted section:

 (1) The fault location parameter should have a value between 0 and 1, i.e., 0<d<1;
 (2) Real part of the square of propagation constant should be less than zero and real part of the square of surge impedance should be greater than zero; and
 (3) Imaginary part of the square of propagation constant should be greater than zero and imaginary part of the square of surge impedance should be less than zero
 Thus, the assumed faulted section for which the predefined criteria are met when the set of equations, including the prefault objective functions and the duringfault objective functions for that assumed faulted section, are solved is identified as the faulted section. Accordingly, the fault location and line parameters can be then determined from the fault location parameter, propagation constant, and surge impedance as discussed above.

FIG. 6 illustrates a method for identifying a fault location in a multiterminal power transmission line, in accordance with an implementation of the present subject matter. The order in which the method 600 is described is not intended to be construed as a limitation, and any number of the described method blocks may be combined in any order to implement the method 600, or an alternative method. Furthermore, the method 600 may be implemented by processor(s) or computing device(s) through any suitable hardware, nontransitory machinereadable instructions, or a combination thereof.  It may be understood that steps of the method 600 may be performed by programmed computing devices and may be executed based on instructions stored in a nontransitory computer readable medium. Although the method 600 may be implemented in a variety of systems, the method 600 are described in relation to the device 400 and
FIGS. 4 and 5A5C , for ease of explanation. Further, while the method 600 may be implemented for any multiterminal power transmission line, for discussion purposes, the method 600 is described for a power transmission line having three or more terminals connected at one or more junctions, with at least a first terminal, a second terminal, and a third terminal connected at a junction, wherein a first section connects the first terminal and the junction, a second section connects the second terminal and the junction, and a third section connects the third terminal and the junction.  At step 602, synchronized disturbance records are obtained from each of the first, second, and third terminals after a fault has occurred in the power transmission line.
 At step 604, prefault voltage and current phasors and duringfault current and voltage phasors are computed for each of the first, second, and third terminals based on the disturbance records
 At step 606, using an assumed faulted section from among the first, second, and third sections, values for propagation constant for each section, surge impedance for each section, and a fault location parameter are computed. The computation is based on simultaneously solving a plurality of prefault objective functions and duringfault objective functions for the assumed faulted section with the computed prefault voltage and current phasors, duringfault current and voltage phasors, and line lengths of each section.
 As discussed above, the prefault objective functions and duringfault objective functions may be formulated based on equating junction voltages determined from two of the terminals, conservation of charge at the junction, and equating fault location voltages determined from one terminal and the junction. For example, each of the prefault objective functions and the duringfault objective functions may correspond to a function of a set of variables, the function being equal to zero, wherein the set of variables comprise the propagation constant of each section, the surge impedance of each section, and the fault location parameter.
 For example, for a threeterminal power transmission line, the prefault objective functions include functions F1 to F4 as discussed above with reference to
FIG. 4 andFIGS. 5A5C and the duringfault functions may correspond to functions F5F8 as discussed above with reference toFIG. 5A5C . Thus, the prefault objective functions comprise a set of functions based on computation of voltage at the junction from each pair of terminals of the first, second, and third terminals, and a function based on conservation of charge at the junction. Further, the duringfault objective functions for a section comprise a first set of functions based on computation of voltage at the junction from each pair of terminals excluding the terminal of the section, and a second set of functions based on computation of voltage at a fault point from the junction and the terminal of the of the section.  In one example, for simultaneously solving the prefault objective functions and the duringfault objective functions, a numerical analysis technique, such as Newton Raphson method, may be applied for determining the propagation constant of each section, the surge impedance of each section, and the fault location parameter.
 At step 608, for a particular section assumed to be the faulted section, the values determined for each of the first, second, and third sections are compared with predefined criteria. In one example, the predefined criteria comprise (a) the fault location parameter has a value between zero and one; (b) real part of square of propagation constant of each section is less than zero and real part of square of surge impedance of each section is greater than zero; and (c) imaginary part of square of propagation constant of each section is greater than zero and imaginary part of square of surge impedance of each section is less than zero.
 At step 610, a faulted section, the fault location, and the line parameters for each of the first, second, and third sections, are determined based on the comparison. The line parameters may include resistance, conductance, and inductance of each section. In one example, the line parameters for two of the sections may be equal.
 In one example, the steps 606610 may be performed iteratively until the faulted section is identified, for example, as illustrated in
FIG. 7 .FIG. 7 illustrates a method 700 for identifying a fault location in a multiterminal power transmission line, in accordance with an implementation of the present subject matter. Like method 600, the method 700 may also be performed by programmed computing devices and may be executed based on instructions stored in a nontransitory computer readable medium. The method 700 is described in relation to the device 400 andFIGS. 4 and 5A5C for a power transmission line having three or more terminals connected at one or more junctions, for ease of explanation. However, the method 700 can be extended to other implementations also.  As shown, at step 702, the disturbance recorder data is obtained from the terminals, M, N, and P. At step 704, the prefault objective functions F1F4 are formulated as discussed above. Further, the steps corresponding to branches 706, 708, 710 are performed, as shown. For example, at first, the fault may be assumed to be in section MJ and branch 706 may be followed. Accordingly, duringfault objective functions F5F8 as discussed above with reference to
FIG. 5A may be used for determining the fault location. The equations F1F8 may be solved using a numerical technique to determine the fault location parameter (d) and the propagation constant and surge impedance of each section and the values may be compared to the predefined criteria. In case the predefined criteria are met, the section MJ may be determined to be the faulted section and accordingly, the fault location and line parameters may be determined.  However, if the predefined criteria are not met, a next section may be assumed to be the faulted section. For example, when the predefined criteria are not met, the steps under branch 708 may be carried out assuming section NJ to be the faulted section. Thus, the equations F5F8 as discussed above with reference to
FIG. 5B with section NJ as the assumed faulted section may be used for determining the fault location. Further, the equations F1F8 may be solved using a numerical technique to determine the fault location parameter (d) and the propagation constant and surge impedance of each section and the values may be compared to the predefined criteria. In case the predefined criteria are met, the section NJ may be determined to be the faulted section and accordingly, the fault location and line parameters may be determined.  However, if the predefined criteria are still not met, a next section may be assumed to be the faulted section. For example, when the predefined criteria are still not met, the steps under branch 710 may be carried out assuming section PJ to be the faulted section. Thus, the equations F5F8 as discussed above with reference to
FIG. 5C with section PJ as the assumed faulted section may be used for determining the fault location. Further, the equations F1F8 may be solved using a numerical technique to determine the fault location parameter (d) and the propagation constant and surge impedance of each section and the values may be compared to the predefined criteria. In case the predefined criteria are met, the section PJ may be determined to be the faulted section and accordingly, the fault location and line parameters may be determined.  Hence, the methods 600 and 700 provide for a setting free fault location identification, i.e., the fault location can be identified without prior knowledge of the line parameters.
 Although the present subject matter has been described with reference to specific implementations, this description is not meant to be construed in a limiting sense. Various modifications of the disclosed implementations, as well as alternate implementations of the subject matter, will become apparent to persons skilled in the art upon reference to the description of the subject matter. Further, although the present subject matter has been explained with reference to specific configurations of a threeterminal power transmission line, it is to be understood that the present subject matter can be used for any power transmission system such as multiterminal power transmission lines, two terminal transmission lines, mixed power transmission lines etc.
Claims (23)
110. (canceled)
11. A method for use with a power transmission line having a first terminal, a second terminal, and a third terminal connected at a junction, wherein a first section connects the first terminal and the junction, a second section connects the second terminal and the junction, and a third section connects the third terminal and the junction, the method comprising:
obtaining synchronized disturbance records from each of the first, second, and third terminals after a fault has occurred in the power transmission line;
computing prefault voltage and current phasors and duringfault voltage and current phasors for each of the first, second, and third terminals based on the disturbance records;
using an assumed faulted section from among the first, second, and third sections, computing values for a propagation constant of each section, a surge impedance of each section, and a fault location parameter, wherein the computing is based on simultaneously solving a plurality of prefault objective functions and duringfault objective functions for the assumed faulted section with the computed prefault voltage and current phasors and duringfault voltage and current phasors, and wherein the prefault objective functions and duringfault objective functions are formulated based on equating junction voltages determined from two of the terminals, conservation of charge at the junction, and equating fault location voltages determined from one terminal and the junction; and
comparing the values determined for the propagation constant, the surge impedance, and the fault location parameter with predefined criteria.
12. The method as claimed in claim 11 , further comprising determining a faulted section, a fault location, and line parameters for each of the first, second, and third sections, based on the comparing.
13. The method as claimed in claim 11 , wherein the computing is based on simultaneously solving the plurality of prefault objective functions and duringfault objective functions for the assumed faulted section with the computed prefault voltage and current phasors, the duringfault voltage and current phasors, and lengths of each section.
14. The method as claimed in claim 11 , wherein the assumed faulted section is determined as being the faulted section when the values determined from the prefault objective functions and the duringfault objective functions for the assumed faulted section meet the predefined criteria, the predefined criteria comprising:
the fault location parameter has a value between zero and one;
the real part of the square of the propagation constant of each section is less than zero and the real part of the square of the surge impedance of each section is greater than zero; and
the imaginary part of the square of the propagation constant of each section is greater than zero and the imaginary part of the square of the surge impedance of each section is less than zero.
15. The method as claimed in claim 11 , wherein two of the sections have the same line parameters.
16. The method as claimed in claim 11 , wherein
the prefault objective functions comprise a set of functions based on computation of voltage at the junction from each pair of terminals of the first, second, and third terminals, and a function based on conservation of charge at the junction; and
the duringfault objective functions for the assumed faulted section comprise a first set of functions based on computation of voltage at the junction from each pair of terminals excluding the terminal of the assumed faulted section, and a second set of functions based on computation of voltage at a fault point from the junction and the terminal of the assumed faulted section.
17. The method as claimed in claim 11 , wherein each of the prefault objective functions and the duringfault objective functions correspond to a function of a set of variables, the function being equal to zero, wherein the set of variables comprise the propagation constant of each section, the surge impedance of each section, and the fault location parameter.
18. The method as claimed in claim 11 , wherein simultaneously solving the prefault objective functions and the duringfault objective functions comprises applying a numerical analysis technique for determining the propagation constant of each section, the surge impedance of each section, and the fault location parameter.
19. The method as claimed in claim 11 , wherein the line parameters include resistance, conductance, and inductance of each section, and wherein the duringfault voltage and current phasors include at least two data sets.
20. A device for identifying a fault location in a power transmission line, the power transmission line having a plurality of terminals connected at one or more junctions, with at least a first terminal, a second terminal, and a third terminal connected at a junction, wherein a first section connects the first terminal and the junction, a second section connects the second terminal and the junction, and a third section connects the third terminal and the junction, the device comprising:
a phasor computation module configured to obtain synchronized disturbance records from each of the plurality of terminals after a fault has occurred in the power transmission line and compute prefault terminal voltage and current phasors and duringfault terminal voltage and current phasors; and
a fault location identification module configured to
compute, using an assumed faulted section from among the first, second, and third sections, values for propagation constant of each section, surge impedance of each section, and a fault location parameter, wherein the computation is based on simultaneously solving a plurality of prefault objective functions and duringfault objective functions for the assumed faulted section with the computed prefault voltage and current phasors, duringfault current and voltage phasors, and line lengths of each section, and wherein the prefault objective functions and duringfault objective functions are formulated based on equating junction voltages determined from two of the terminals, conservation of charge at the junction, and equating fault location voltages determined from one terminal and the junction;
compare the values determined for the propagation constant, the surge impedance, and the fault location parameter with predefined criteria; and
determine a faulted section, the fault location, and the parameters for each of the first, second, and third sections, based on the comparison.
21. The device as claimed in claim 20 , wherein each of the prefault objective functions and the duringfault objective functions correspond to a function of a set of variables, the function being equal to zero, wherein the set of variables comprise the propagation constant of each section, the surge impedance of each section, and the fault location parameter, and wherein to simultaneously solve the prefault functions and the duringfault functions, the fault location identification module is configured to apply a numerical analysis technique to determine the propagation constant of each section, the surge impedance of each section, and the fault location parameter.
22. The device as claimed in claim 20 , wherein the assumed faulted section is determined as being the faulted section when the values determined from the prefault objective functions and the duringfault objective functions for the assumed faulted section meet the predefined criteria, the predefined criteria comprise:
the fault location parameter has a value between zero and one;
the real part of the square of the propagation constant of each section is less than zero and the real part of the square of the surge impedance of each section is greater than zero; and
the imaginary part of the square of the propagation constant of each section is greater than zero and the imaginary part of the square of the surge impedance of each section is less than zero.
23. A device for use with a power transmission line having a first terminal, a second terminal, and a third terminal connected at a junction, wherein a first section connects the first terminal and the junction, a second section connects the second terminal and the junction, and a third section connects the third terminal and the junction, the device comprising:
a processor; and
a nontransitory memory coupled to the processor, the memory storing computer code that, when executed by the processor, causes the processor to:
compute prefault voltage and current phasors and duringfault voltage and current phasors for each of the first, second, and third terminals based on synchronized disturbance records obtained from each of the first, second, and third terminals after a fault has occurred in the power transmission line;
compute, using an assumed faulted section from among the first, second, and third sections, values for a propagation constant of each section, a surge impedance of each section, and a fault location parameter, wherein the computation is based on simultaneously solving a plurality of prefault objective functions and duringfault objective functions for the assumed faulted section with the computed prefault voltage and current phasors and duringfault voltage and current phasors, and wherein the prefault objective functions and duringfault objective functions are formulated based on equating junction voltages determined from two of the terminals, conservation of charge at the junction, and equating fault location voltages determined from one terminal and the junction; and
compare the values determined for the propagation constant, the surge impedance, and the fault location parameter with predefined criteria.
24. The device as claimed in claim 23 , wherein the computer code further causes the processor to determine a faulted section, a fault location, and line parameters for each of the first, second, and third sections, based on the comparison.
25. The device as claimed in claim 24 , wherein the line parameters include resistance, conductance, and inductance of each section, and wherein the duringfault voltage and current phasors include at least two data sets.
26. The device as claimed in claim 23 , wherein the computer code further causes the processor to compute the propagation constant of each section, the surge impedance of each section, and the fault location parameter by on simultaneously solving the plurality of prefault objective functions and duringfault objective functions for the assumed faulted section with the computed prefault voltage and current phasors, the duringfault voltage and current phasors, and lengths of each section.
27. The device as claimed in claim 23 , wherein the assumed faulted section is determined as being the faulted section when the values determined from the prefault objective functions and the duringfault objective functions for the assumed faulted section meet the predefined criteria, the predefined criteria comprising:
the fault location parameter has a value between zero and one;
the real part of the square of the propagation constant of each section is less than zero and the real part of the square of the surge impedance of each section is greater than zero; and
the imaginary part of the square of the propagation constant of each section is greater than zero and the imaginary part of the square of the surge impedance of each section is less than zero.
28. The device as claimed in claim 23 , wherein
the prefault objective functions comprise a set of functions based on computation of voltage at the junction from each pair of terminals of the first, second, and third terminals, and a function based on conservation of charge at the junction; and
the duringfault objective functions for the assumed faulted section comprise a first set of functions based on computation of voltage at the junction from each pair of terminals excluding the terminal of the assumed faulted section, and a second set of functions based on computation of voltage at a fault point from the junction and the terminal of the assumed faulted section.
29. The device as claimed in claim 23 , wherein each of the prefault objective functions and the duringfault objective functions correspond to a function of a set of variables, the function being equal to zero, wherein the set of variables comprise the propagation constant of each section, the surge impedance of each section, and the fault location parameter.
30. The device as claimed in claim 23 , wherein the computer code further causes the processor to compute the propagation constant of each section, the surge impedance of each section, and the fault location parameter by on simultaneously solving the prefault objective functions and the duringfault objective functions by applying a numerical analysis technique for determining the propagation constant of each section, the surge impedance of each section, and the fault location parameter.
31. A power transmission system comprising:
the device as claimed in claim 23 ; and
a power transmission line having the first terminal, the second terminal, and the third terminal connected at the junction, wherein the first section connects the first terminal and the junction, the second section connects the second terminal and the junction, and the third section connects the third terminal and the junction.
32. The system as claimed in claim 31 , wherein two of the sections have the same line parameters.
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CN202218000U (en) *  20100514  20120509  帕西·西姆公司  Improved surge protection device 
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US20160116522A1 (en) *  20141027  20160428  King Fahd University Of Petroleum And Minerals  Fully adaptive fault location method 
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