Title of Invention: Fault Location Method For Series-Compensated Double-Circuit Transmission Lines Abstract: A highly accurate fault location method for series-compensated double-circuit transmission lines having series compensation devices and metal oxide varistors (SC&MOV) at first and second terminal ends is provided. Synchronized or unsynchronized current phasors and local voltage phasors are used as input to the fault location method. The voltages and currents at the fault point are formulated as a function of the unknown fault location. Boundary conditions for the particular IED-determined fault type are used to derive the fault location formulas. Detailed Description: FIELD OF INVENTION The present application is directed to a highly precise fault location method for series-compensated double-circuit transmission lines and a derivation of synchronization angle for unsynchronized measurements used in the method. BACKGROUND Known fault location methods use the lumped parameter line model to approximate power networks for analysis. The lumped parameter line model approximates the parameters of the circuit into ideal components such as resistors, capacitors, and inductors. The lumped parameter line model does not represent a real circuit in which the parameters are distributed uniformly through the length of the line. For example, given the per kilometer (km) impedance (z), admittance (y), and line length (l) in km, the equivalent series impedance for the entire line length under the lumped parameter line model is zl. In contrast, the equivalent series impedance for the entire line length under the distributed parameter line model is √{square root over (z/y)} sin h(√{square root over (zyl)}). Further, most known fault location methods for power transmission lines ignore mutual coupling between parallel lines as well as the charging effect of the shunt capacitances of the lines. Additionally, some known fault location methods require source impedances at the two terminal substations and assume constant source impedances, so that these methods are adversely impacted by the varying source impedance. Utilizing the lumped parameter line model and/or ignoring shunt capacitance causes the accuracy of the fault location algorithm to decrease as the length of the power transmission line increases. Thus, while known fault location methods may be useful for shorter distance transmission lines, there is room for improvement in the accuracy of fault location methods for longer distance power transmission lines. SUMMARY A non-transitory computer readable storage medium has thereon a plurality of machine-readable instructions that when executed by at least one computer processor cause the at least one computer processor to perform a method for detecting a fault in series-compensated double-circuit transmission lines between first and second terminals, the plurality of machine-readable instructions having instructions to carry out the following steps: detecting a fault on a circuit of the transmission lines between the first and second terminals; determining a fault type for the detected fault; obtaining synchronized during-fault first terminal voltage measurements and first and second terminal current measurements; formulating zero, positive, and negative sequence circuits between first and second terminals by distributing as a function of the unknown fault location, m, along the transmission lines, the zero sequence parameters: equivalent mutual series impedance and mutual shunt-admittance of parallel lines between the first terminal and the fault location, equivalent mutual series-impedance and mutual shunt-admittance of parallel lines between the second terminal and the fault location, and the zero, positive and negative sequence parameters: equivalent self-series impedance and self shunt-admittance of the line between the first terminal and the fault location, and equivalent self series-impedance and self shunt-admittance of the line between the second terminal and the fault location; formulating zero-sequence voltage and negative- and positive-sequence voltage and current at the fault point as a function of unknown fault location, m, as given by: V f ( 0 ) = V p   1 ( 0 ) - [ I p   1 ( 0 ) - V p   1 ( 0 )  Y pf ( 0 ) 2 - ( V p   1 ( 0 ) - V p   2 ( 0 ) )  Y 1  m ( 0 ) 2 ]  Z pf ( 0 ) - [ I p   2 ( 0 ) - V p   2 ( 0 )  Y pf ( 0 ) 2 - ( V p   2 ( 0 ) - V p   1 ( 0 ) )  Y 1  m ( 0 ) 2 ]  Z 1  m ( 0 )   and  V f ( i ) = V p   1 ( i ) - ( I p   1 ( i ) - V p   1 ( i )  Y pf ( i ) 2 )  Z pf ( i )   and I f ( i ) = I p   1 ( i )  ( 1 + Y qf ( i )  Z pf ( i ) 2 1 + Y qf ( i )  Z qf ( i ) 2 + Y pf ( i ) + Y qf ( i ) 2  Z pf ( i ) ) + I q   1 ( i )   j   δ  ( 1 - Y qf ( i )  Z qf ( i ) 2 1 + Y qf ( i )  Z qf ( i ) 2 ) - V p   1 ( i )  [ Y pf ( i ) 2 + Y qf ( i ) 2  1 + Y pf ( i )  Z pf ( i ) 2 1 + Y qf ( i )  Z qf ( i ) 2 + Y pf ( i ) + Y qf ( i ) 2  ( 1 + Y pf ( i )  Z pf ( i ) 2 ) ] ; and determining the fault location distance from the first terminal by the boundary condition for the determined fault type and zero-sequence during-fault voltage at the fault point formulated as a function of the unknown fault location, and positive/negative-sequence during-fault voltage at the fault point and current flowing out of the fault point as a function of the unknown fault location. A non-transitory computer readable storage medium has thereon a plurality of machine-readable instructions that when executed by at least one computer processor cause the at least one computer processor to perform a method for determining a synchronization angle for use in synchronizing measurements retrieved at first and second terminals of series-compensated double-circuit transmission lines, the plurality of machine-readable instructions having instructions to: receiving pre-fault first terminal voltage measurement and first and second terminal current measurements; obtaining positive-sequence equivalent self series-impedance and positive-sequence equivalent self shunt-admittance of the line between first and second terminals; obtaining voltage and current relationships as expressed below:  j   δ = - I p   1 ( 1 )  0 - V p   1 ( 1 )  0  Y pq ( 1 ) 2 - [ V p   1 ( 1 )  0 - Z pq ( 1 )  ( I p   1 ( 1 )  0 - V p   1 ( 1 )  0  Y pq ( 1 ) 2 ) ]  Y pq ( 1 ) 2 I q   1 ( 1 )  0 using the measurements of positive-sequence pre-fault voltage phasor at the first terminal of the first circuit, positive-sequence pre-fault current phasor at the first terminal of the first circuit, positive-sequence pre-fault current phasor at the second terminal of the first circuit and the line parameters of positive-sequence equivalent self series-impedance and positive-sequence equivalent self shunt-admittance of the line between first and second terminals and synchronization angle; and calculating the synchronization angle from the expression of voltage and current relationships using the measurements of positive-sequence pre-fault voltage phasor at the first terminal of the first circuit, positive-sequence pre-fault current phasor at the first terminal of the first circuit, positive-sequence pre-fault current phasor at the second terminal of the first circuit and the line parameters of positive-sequence equivalent self series-impedance and positive-sequence equivalent self shunt-admittance of the line between first and second terminals. BRIEF DESCRIPTION OF THE DRAWINGS In the accompanying drawings, structural embodiments are illustrated that, together with the detailed description provided below, describe exemplary embodiments of a fault location method for series-compensated double-circuit power transmission lines. One of ordinary skill in the art will appreciate that a component may be designed as multiple components or that multiple components may be designed as a single component. Further, in the accompanying drawings and description that follow, like parts are indicated throughout the drawings and written description with the same reference numerals, respectively. The figures are not drawn to scale and the proportions of certain parts have been exaggerated for convenience of illustration. FIG. 1 is a schematic diagram of a series-compensated double-circuit transmission line having intelligent electronic devices (IEDs) for measuring current and voltage on the circuits between first and second terminals; FIG. 2 is a schematic of a zero-sequence model of a series-compensated double-circuit line with a fault; FIG. 3 is a schematic of a positive/negative-sequence model of a series-compensated double-circuit line with a fault; FIG. 4 is a schematic of a pre-fault positive-sequence model of a transmission line; FIG. 5 is an equivalent circuit of a series compensation device and metal oxide varistor (SC&MOV); FIG. 6 is flow chart of a fault location method for series-compensated double-circuit transmission lines; and FIG. 7 is a flow chart of a method for the derivation of the synchronization angle. DETAILED DESCRIPTION With reference to FIG. 1, a series compensation (SC) device is installed for long transmission lines to improve power transfer capability and enhance power transmission system 10 stability. The SC device is a capacitor bank or a thyristor-based power flow device that is protected by a metal oxide varistor (MOV) device. As is known, after a fault occurs, it is important to accurately locate the fault so as to quickly repair the faulted component and restore service. The fault location method disclosed herein is based on the distributed parameter line model and is developed for two-end series-compensated double-circuit lines 10 in which each of the circuits 26, 28 may be three-phase circuits. The method is able to accurately consider the mutual coupling between the parallel lines under the zero-sequence network, and fully takes into account the charging effect of the lines and thus does not lose accuracy for long lines. The fault location method utilizes synchronized or unsynchronized two-terminal current phasors and local voltage phasors P1, P2 as inputs. The voltages and currents at the fault point are formulated in terms of the unknown fault location m. Then, boundary conditions of different fault types are harnessed to derive the fault location formulas as given by Eqs. (28), (30), (32), and (34) which will be described in detail below. A new approach to calculate the synchronization angle using pre-fault quantities is disclosed for the scenario that the two-terminal measurements are not synchronized. It is assumed that the fault impedance is purely resistive and the system is transposed. The fault type may be known from the intelligent electronic devices (IEDs) such as relays or other protective devices installed at the terminals of the transmission lines in a substation. The fault type may be determined by comparing three phase and zero-sequence during-fault current phasor magnitudes with a pre-set threshold, which is a well-established method. IEDs are fed from current and voltage transformers installed on the first and second circuits 26, 28 between the first and second terminals 14, 24. A fault locator may be located at the first and/or second terminals 14, 24 to receive the information about the type of fault and timestamp of the fault occurrence. The fault location method does not require the availability of the two-terminal substation impedances and is not impacted by the fault impedance. A schematic diagram of a series-compensated double-circuit transmission line is shown in FIG. 1. In one embodiment, the series capacitors 16 are capacitor banks that are installed at first and second terminal ends P, Q of the transmission lines. The first and second terminal ends of the series-compensated transmission lines may be located at various points along the transmission lines. Alternatively, the first and second terminals may each be substations at the ends of the transmission lines. The transmission lines between first and second terminals often span hundreds of miles or more. A metal oxide varistor (MOV) 20, equipped in parallel with a series compensation device (SC) 16, will conduct when an overvoltage across the series capacitor is detected. Suppose there are voltage transformers 17 installed on the right side of the series compensators from terminal P of FIG. 1. The voltage phasors are then directly available at points P1 and P2 and the two-terminal current phasors are available from current transformers 15 installed at both the healthy and faulty circuits of two terminals. ZP and ZQ are the equivalent source impedances at first and second terminals 12, 22. The voltage and current measurements are retrieved from the IEDs installed at the respective substation upon the detection of a fault. These measurements are communicated back to the control room computer having a fault location application installed thereon. The fault location application is deployed upon the detection of a fault by the relay or other protective device. The control room computer has access to the line parameters and upon receiving the voltage and current measurements at first and second terminals 14, 24 (points P and Q in the present example), executes the fault location application and outputs the fault location estimate for crew dispatch. In some examples, non-transitory computer readable storage media 34 may have embodied thereon a plurality of machine-readable instructions 36 that when executed by at least one computer processor 32 cause the at least one computer processor 32 of the computing device 30 to perform a method for identifying a fault location in a poly-phase power network having double-circuit transmission lines 26, 28. The plurality of machine-readable instructions may include instructions to receive at least one current and/or voltage measurement measured during a fault by an intelligent electronic device (IED), current transformer 15 or voltage transformer 17 disposed within the power network, instructions to calculate a synchronization angle for unsynchronized measurements, instructions to determine voltage and current phasors from voltage and current measurements, instructions to obtain self series-impedance and self shunt-admittance of each line and mutual series-impedance and mutual shunt-admittance of the parallel lines, and instructions to determine during-fault voltage and during-fault current flowing out of the fault point. In one embodiment, the fault location application is an executable application stored on the substation computer. One example of a trigger for the fault location application is when a relay on the power transmission system being measured is tripped, signaling there is a fault. The voltage and current measurements are transmitted to the substation computer, and the fault location application uses the measurements in the equations described herein to calculate and provide the fault location on a power transmission system as output to the substation computer, mobile devices of the utility crew, IEDs installed on the transmission system of interest, fault locators, or any other medium or device that is capable of communicating or displaying the fault location. The communication of the fault location to the utility crew through an electronic device or other means allows the utility crew to respond to the fault quickly. Definitions of parameters used in the present disclosure are as follows: P,Q First and second terminals of the series-compensated transmission line; F Fault point; S The point corresponding to the fault point on the healthy circuit; i Symmetrical component index; i=0, 1, 2 for zero-, positive- and negative-sequence, respectively; m Unknown fault location in p.u. from bus P to F; l Total length of the transmission line in km; δ Synchronization angle between P and Q with P as the reference; x Parallel line indicator; x=1, 2 for each of the parallel lines; Vpx(i) ith sequence during-fault voltage phasor at P of line x; Vpx(1)0 Positive-sequence pre-fault voltage phasor at P of line x; Vqx(i) ith sequence during-fault voltage phasor at Q of line x; Vqx(1)0 Positive-sequence pre-fault voltage phasor at Q of line x; Ipx(i) ith sequence during-fault current phasor at P of line x; Ipx(1)0 Positive-sequence pre-fault current phasor at P of line x; Iqx(i) ith sequence during-fault current phasor at Q of line x; Iqx(1)0 Positive-sequence pre-fault current phasor at Q of line x; Vf(i) ith sequence during-fault voltage at fault point; If(i) ith sequence fault current flowing out of the fault point; Ipf(i) The contribution of fault current from P under ith sequence network; Iqf(i) The contribution of fault current from Q under ith sequence network; z(i),y(i) ith sequence self series-impedance and self shunt-admittance of the line per km, respectively; zm(0),ym(0) Zero-sequence mutual series-impedance and mutual shunt-admittance between the parallel lines per km, respectively; Zpf(i),Ypf(i) ith sequence equivalent self series-impedance and self shunt-admittance of the line between P and F, respectively; Zqf(i),Yqf(i) ith sequence equivalent self series-impedance and self shunt-admittance of the line between Q and F, respectively; Zpq(1),Ypq(1) Positive-sequence equivalent self series-impedance and self shunt-admittance of the line between P and Q, respectively; Z1m(0),Y1m(0) zero-sequence equivalent mutual series-impedance and mutual shunt-admittance of the parallel lines between P and F, respectively; Z2m(0),Y2m(0) Zero-sequence equivalent mutual series-impedance and mutual shunt-admittance of the parallel lines between Q and F, respectively; Zcm1 Zero-sequence mode-1 characteristic impedance of the double-circuit line; Zcm2 Zero-sequence mode-2 characteristic impedance of the double-circuit line; γm1 Zero-sequence mode-1 propagation constant of the double-circuit line; and γm2 Zero-sequence mode-2 propagation constant of the double circuit line. Zc Positive-sequence characteristic impedance of the double-circuit line; and γ Positive-sequence propagation constant of the double circuit line. Sequence circuits are used to model the double-circuit transmission lines in the present disclosure. The original double-circuit transmission lines contain two parallel three-phase transmission lines. In general, utilizing the symmetrical component transformation matrix, the original three-phase circuit is decoupled into three independent single-phase positive-, negative-, and zero-sequence circuits. For a double-circuit line, after decoupling, the original two parallel three-phase transmission lines are represented by two parallel single-phase lines in each sequence network. In a zero-sequence network, there is a mutual coupling between the two parallel lines. For positive- and negative-sequence networks, there is no mutual coupling between the parallel lines. A schematic diagram of the zero-sequence series-compensated double-circuit line with a fault is shown in FIG. 2. Since the synchronization angle is δ, then all of the unsynchronized measured phasors at Q need to be multiplied by ejδ to compensate for the angle difference, i.e. Vq1(i) becomes Vq1(i)ejδ. The parameters in FIG. 2 are calculated as follows: The zero-sequence equivalent self series-impedance of line between terminal P and the fault point F, is calculated as follows using Eq. (1): Z pf ( 0 ) = 1 2  [ Z cm   2   sinh  ( γ m   2  ml ) + Z cm   1   sinh  ( γ m   1  ml ) ] ( 1 ) The zero-sequence equivalent mutual series-impedance of the parallel lines between P and F, respectively, is calculated using Eq. (2): Z 1  m ( 0 ) = 1 2  [ Z cm   2   sinh  ( γ m   2  ml ) - Z cm   1   sinh  ( γ m   1  ml ) ] ( 2 ) The zero-sequence equivalent self shunt-admittance of the line between terminal P and fault point F is calculated using Eq. (3): Y pf ( 0 ) = 2   tanh  ( γ m   2  ml  /  2 ) Z cm   2 ( 3 ) The zero-sequence mutual shunt-admittance of the parallel lines between terminal P and fault point F is calculated using Eq. (4): Y 1  m ( 0 ) = tanh  ( γ m   1  ml  /  2 ) Z cm   1 - tanh  ( γ m   2  ml  /  2 ) Z cm   2 ( 4 ) The zero-sequence equivalent self series-impedance of the line between terminal Q and fault point F is calculated using Eq. (5): Z qf ( 0 ) = 1 2  { Z cm   2   sinh  [ γ m   2  ( l - ml ) ] + Z cm   1   sinh  [ γ m   1  ( l - ml ) ] } ( 5 ) The zero-sequence equivalent mutual series-impedance of the parallel lines between terminal Q and F, is calculated using Eq. (6): Z 2  m ( 0 ) = 1 2  { Z cm   2   sinh  [ γ m   2  ( l - ml ) ] - Z cm   1   sinh  [ γ m   1  ( l - ml ) ] } ( 6 ) The zero-sequence equivalent self shunt-admittance of the line between terminal Q and fault point F, is calculated using Eq. (7): Y qf ( 0 ) = 2   tanh  [ γ m   2  ( l - ml )  /  2 ] Z cm   2 ( 7 ) The zero-sequence equivalent mutual shunt-admittance of the parallel lines between terminal Q and fault point F is calculated using Eq. (8): Y 2  m ( 0 ) = tanh  [ γ m   1  ( l - ml )  /  2 ] Z cm   1 - tanh  [ γ m   2  ( l - ml )  /  2 ] Z cm   2 ( 8 ) where the mode-1 characteristic impedance of the double-circuit line is given by Zcm1=√{square root over ((z(0)−zm(0))/(y(0)+2ym(0)))}; the mode-2 characteristic impedance of the double-circuit line is given by Zcm2=√{square root over ((z(0)+zm(0))/y(0))}; the mode-1 propagation constant of the double-circuit line is given by γm1=√{square root over ((z(0)−zm(0))/(y(0)+2ym(0)))}; and the mode-2 propagation constant of the double-circuit line is given by γm1=√{square root over ((z(0)+zm(0))/y(0))}. For the zero-sequence network, the following equation holds for zero-sequence during-fault voltage at the fault point: V f ( 0 ) = V p   1 ( 0 ) - [ I p   1 ( 0 ) - V p   1 ( 0 )  Y pf ( 0 ) 2 - ( V p   1 ( 0 ) - V p   2 ( 0 ) )  Y 1  m ( 0 ) 2 ]  Z pf ( 0 ) - [ I p   2 ( 0 ) - V p   2 ( 0 )  Y pf ( 0 ) 2 - ( V p   2 ( 0 ) - V p   1 ( 0 ) )  Y 1  m ( 0 ) 2 ]  Z 1  m ( 0 ) ( 9 ) From Eq. (9), the expression of Vf(0) is obtained in terms of Ip1(0), Ip2(0), Vp1(0), Vp2(0) and the an unknown distance to fault location m. With reference now to FIG. 3, a schematic diagram of a positive- or negative-sequence series-compensated double-circuit line with a fault is shown. The parameters in of FIG. 3 are calculated as described below. The positive- or negative-sequence equivalent self series-impedance of the line between terminal P and fault point F is calculated using Eq. (10): Zpf(i)−Zc sin h(γml) (10) The positive- or negative-sequence equivalent self shunt-admittance of the line between terminal P and fault point F is calculated using Eq. (11): Y pf ( i ) = 2   tanh  ( γ   ml  /  2 ) Z c ( 11 ) The positive- or negative-sequence equivalent self series-impedance of the line between terminal Q and fault point F is calculated using Eq. (12): Zqf(i)=Zc sin h[γ(l−ml)] (12) The positive- or negative-sequence equivalent self shunt-admittance of the line between terminal Q and fault point F is calculated using Eq. (13): Y qf ( i ) = 2   tanh  [ γ  ( l - ml )  /  2 ] Z c ( 13 ) Where the characteristic impedance is given by Zc=√{square root over (z(i)/y(i))} the propagation constant is given by γ=√{square root over (z(i)y(i))} and i=1, 2. The following Eqs. (14), (15), (16) are derived for the positive- or negative-sequence network:  V p   1 ( i ) - V f ( i ) = ( I p   1 ( i ) - V p   1 ( i )  Y pf ( i ) 2 )  Z pf ( i ) ( 14 )  V q   1 ( i )   j   δ - V f ( i ) = ( I q   1 ( i )   j   δ - V q   1 ( i )   j   δ  Y qf ( i ) 2 )  Z qf ( i ) ( 15 ) I p   1 ( i ) - V p   1 ( i )  Y pf ( i ) 2 - V f ( i )  Y pf ( i ) 2 + I q   1 ( i )   j   δ - V q   1 ( i )   j   δ  Y qf ( i ) 2 - V f ( i )  Y qf ( i ) 2 = I f ( i ) ( 16 ) From Eq. (14), the expression of Vf(i), the during-fault voltage at the fault point for the positive- or negative-sequence network is obtained in terms of Ip1(i), Vp1(i), and m as follows: V f ( i ) = V p   1 ( i ) - ( I p   1 ( i ) - V p   1 ( i )  Y pf ( i ) 2 )  Z pf ( i ) ( 17 ) Substituting Eq. (17) into Eq. (15) leads to the expression of Vq1(i), the positive- or negative-sequence during-fault voltage phasor at Q of line 1, in terms of Ip1(i), Iq1(i), Vp1(i), and m as follows: V q   1 ( i )   j   δ = V p   1 ( i ) - ( I p   1 ( i ) - V p   1 ( i )  Y pf ( i ) 2 )  Z pf ( i ) + I q   1 ( i )  Z qf ( i ) 1 + Y qf ( i )  Z qf ( i ) 2 ( 18 ) Substituting Eqs. (17) and (18) into Eq. (16) leads to the expression of If(i), the positive- or negative-sequence fault current flowing out of the fault point, in terms of Ip1(i), Iq1(i), Vp1(i), and m as follows: I f ( i ) = I p   1 ( i ) + I q   1 ( i )   j   δ - V p   1 ( i )  Y pf ( i ) 2 - Y qf ( i ) 2  V p   1 ( i ) - ( I p   1 ( i ) - V p   1 ( i )  Y pf ( i ) 2 )  Z pf ( i ) + I q   1 ( i )   j   δ  Z qf ( i ) 1 + Y qf ( i )  Z qf ( i ) 2 - Y pf ( i ) + Y qf ( i ) 2  [ V p   1 ( i ) - ( I p   1 ( i ) - V p   1 ( i )  Y pf ( i ) 2 )  Z pf ( i ) ] ( 19 ) Further organizing Eq. (19), the positive- or negative-sequence fault current flowing out of the fault point is obtained using Eq. (20): I f ( i ) = I p   1 ( i )  ( 1 + Y qf ( i )  Z pf ( i ) 2 1 + Y qf ( i )  Z qf ( i ) 2 + Y pf ( i ) + Y qf ( i ) 2  Z pf ( i ) ) + I q   1 ( i )   j   δ  ( 1 - Y qf ( i )  Z qf ( i ) 2 1 + Y qf ( i )  Z qf ( i ) 2 ) - V p   1 ( i )  [ Y pf ( i ) 2 + Y qf ( i ) 2  1 + Y pf ( i )  Z pf ( i ) 2 1 + Y qf ( i )  Z qf ( i ) 2 + Y pf ( i ) + Y qf ( i ) 2  ( 1 + Y pf ( i )  Z pf ( i ) 2 ) ] ( 20 ) In accordance with the present disclosure, the synchronization angle is derived using pre-fault measurements, i.e. when no fault is present in the transmission system. A sample series-compensated positive-sequence transmission line without any fault is shown in FIG. 4 to facilitate the derivation of the synchronization angle δ. From the model of the pre-fault positive-sequence transmission line we can write the following two equations: V p   1 ( 1 )  0 - V q   1 ( 1 )  0   j   δ = Z pq ( 1 )  ( I p   1 ( 1 )  0 - V p   1 ( 1 )  0  Y pq ( 1 ) 2 ) ( 21 ) - I q   1 ( 1 )  0   j   δ = I p   1 ( 1 )  0 - V p   1 ( 1 )  0  Y pq ( 1 ) 2 - V q   1 ( 1 )  0   j   δ  Y pq ( 1 ) 2 ( 22 ) where the positive-sequence shunt-admittance of the line between terminals P and Q is given by: Y pq ( 1 ) = 2   tanh  ( γ   l  /  2 ) Z c ( 1 ) and the positive-sequence series-impedance of the line between terminals P and Q is given by: Zpq(1)=Zc sin h(γl). Substituting Vq1(1)0ejδ from Eq. (21) into Eq. (22) leads to the expression of ejδ as follows:  jδ = I p   1 ( 1 )  0 - V p   1 ( 1 )  0  Y pq ( 1 ) 2 - [ V p   1 ( 1 )  0 - Z pq ( 1 )  ( I p   1 ( 1 )  0 - V p   1 ( 1 )  0  Y pq ( 1 ) 2 ) ]  Y pq ( 1 ) 2 I q   1 ( 1 )  0 ( 23 ) Eq. (23) demonstrates that δ can be calculated from the pre-fault two-terminal current phasors at terminals P and Q and the local voltage phasor at terminal P. When the two-terminal measurements at terminals P and Q are synchronized, is equal 0 and there is no need to calculate this quantity. When potential transformers are not installed on the right side of SC&MOVs, there are no direct voltage measurements available at P1 and P2. However, it is possible to calculate the voltage drop on SC&MOV to arrive at Vp1 and Vp2 from the local bus voltage measurement Vp and current measurements Ip1 and Ip2. As described by Daniel L. Goldsworthy in “A Linearized Model for MOV-Protected Series Capacitors,” IEEE Transactions on Power Systems, Vol. PWRS-2, No. 4, November 1987, a function in Eqs. (24), (25), and (26) for obtaining the equivalent SC&MOV impedance using the current phasors is provided below. Using the single phase voltage Vp1 as an example, the equivalent SC&MOV resistance Req and reactance Xeq in series in FIG. 5 can be calculated as follows: R eq = { 0 if   I PU ≤ 0.98 X C  ( 0.0745 + 0.49 ·  - 0.243  I PU - 35.0 ·  - 5.0  I PU - 0.6 ·  - 1.4  I PU ) if   I PU > 0.98   X eq = { X C if   I PU ≤ 0.98 X C  ( 0.1010 - 0.005749 · I PU + 2.088 ·  - 0.8566  I PU ) if   I PU > 0.98 ( 24 ) where I PU = I p   1 I prot ( 25 ) IPU is the normalized current through SC&MOV with respect to the protection level current Iprot. Xc is the actual reactance of the installed capacitor bank. The value of Iprot in Eq. (25) is estimated using the following formula: Vpeak=√{square root over (2)}IprotXC (26) where Vpeak is the instantaneous peak value of the voltage waveform across SC&MOV and is usually 80-90% of Vref. Here, Vref is the MOV nameplate reference voltage. The single phase voltage Vp1 can be calculated from Vp, Ip1, Req and Xeq using Vp1=Vp−Ip1(Req+jXeq). The fault location formulas for different types of fault are derived below. The following boundary condition holds for a phase A to ground fault: Vf(1)+Vf(2)+Vf(0)=3RfIf(1) (27) where Vf(1), Vf(2) and Vf(0) can be obtained from Eqs. (17) and (9) and If(1) can be obtained from Eq. (20). Since Rf is a real number, it indicates that (Vf(1)+Vf(2)+Vf(0)) and If(1) are in phase in Eq. (27). The following equation then holds for a phase A to ground fault: Im  { V f ( 1 ) + V f ( 2 ) + V f ( 0 ) I f ( 1 ) } = 0 ( 28 ) where lm(•) yields the imaginary part of the argument. In Eq. (28), the required inputs are two-terminal current phasors and local voltage phasors and the only unknown is the fault location variable m. In one embodiment, the two-terminal current phasors are determined at terminals P and Q and the local voltage phasors are determined at terminal P. The iterative approach is harnessed to pinpoint the fault location. Starting with an initial value of the per unit distance to fault m, the method may utilize the Newton-Raphson technique to iteratively reach a solution for the per unit fault distance m. For a phase B to phase C fault, the following boundary condition holds: Vf(1)−Vf(2)=RfIf(1) (29) Based on Eq. (29), the fault location for a phase B to phase C fault can be calculated from the following formula: Im  { V f ( 1 ) - V f ( 2 ) I f ( 1 ) } = 0 ( 30 ) For a phase B to phase C to ground fault, the following boundary condition is satisfied at the fault point: Vf(0)−Vf(1)=3RfIf(0) (31) where the zero-sequence fault current flowing out of the fault point is given by: If(0)=−(If(1)+If(2)). The positive-sequence current flowing out of the fault point, If(1), and the negative-sequence current flowing out of the fault point, If(2), are obtained from Eq. (20). Based on Eq. (31), the fault location for a phase B to phase C to ground fault can then be calculated from the following equation: Im  { V f ( 0 ) - V f ( 1 ) I f ( 0 ) } = 0 ( 32 ) For a phase A to phase B to phase C fault, the boundary condition at the fault point is expressed by the following equation: Vf(1)=RfIf(1) (33) Utilizing Eqs. (17) and (20), the fault location formula for a phase A to phase B to phase C fault is reached as Im  { V f ( 1 ) I f ( 1 ) } = 0 ( 34 ) Fault location formulas for fault types involving other phases can be similarly deduced. Referring now to FIG. 6, a method for identifying the location of a fault in double-circuit power transmission lines extending between a first terminal and a second terminal 40, and wherein each circuit has a series compensation device and metal oxide varistor (SC&MOV) installed at first and second terminal ends has the following steps. At step 42, a fault is detected on a circuit of the transmission lines between the first and second terminals by an IED or a protective relay and a fault type is determined by the IED for the fault occurrence. At step 44, during-fault first terminal voltage measurements Vpx(i) and two-terminal current measurements Ipx(i), Iqx(i) are retrieved. At step 46, it is determined whether during-fault first terminal voltage measurements Vpx(i) are directly available from the IED. If the during-fault first terminal voltage measurements are not available the Goldsworthy Eqs. (24), (25), and (26) as mentioned above are utilized to determine equivalent SC&MOV impedance and desired during-fault first terminal voltage measurements at step 56. After obtaining the during-fault first terminal voltage measurements and first and second terminal current measurements, a check is done to determine whether the first and second terminal measurements are synchronized at step 48. If the measurements are not synchronized, at step 58 the synchronization angle is calculated as described in the flow chart of FIG. 7 using the pre-fault first and second terminal current measurements and the first terminal voltage measurement. Further at step 58, the second terminal during-fault current measurements are synchronized to the first terminal during-fault measurements using the calculated synchronization angle. When the first and second terminal measurements are obtained and synchronized, the zero-, positive-, and negative-sequence circuits between first and second terminals are represented by the distributed parameter line model. In particular, the equivalent zero-sequence parameters and positive- and negative-sequence parameters are formulated as a function of the unknown fault location, m. The zero-sequence parameters include: equivalent mutual series-impedance and mutual shunt-admittance of the parallel lines between the first terminal and the fault location, equivalent mutual series-impedance and mutual shunt-admittance of parallel lines between the second terminal and the fault location. The remaining zero-, positive- and negative-sequence parameters include: equivalent self series-impedance and self shunt-admittance of the line between the first terminal and the fault location, and equivalent self series-impedance and self shunt-admittance of the line between the second terminal and the fault location at step 50. At step 52, the zero-sequence voltage and negative- and positive-sequence voltage and current at the fault point as a function of unknown fault location, m, are obtained by the following expressions: V f ( 0 ) = V p   1 ( 0 ) - [ I p   1 ( 0 ) - V p   1 ( 0 )  Y pf ( 0 ) 2 - ( V p   1 ( 0 ) - V p   2 ( 0 ) )  Y 1  m ( 0 ) 2 ]  Z pf ( 0 ) - [ I p   2 ( 0 ) - V p   2 ( 0 )  Y pf ( 0 ) 2 - ( V p   2 ( 0 ) - V p   1 ( 0 ) )  Y 1  m ( 0 ) 2 ]  Z 1  m ( 0 )    and    V f ( i ) = V p   1 ( i ) - ( I p   1 ( i ) - V p   1 ( i )  Y pf ( i ) 2 )  Z pf ( i )    and   I f ( i ) = I p   1 ( i )  ( 1 + Y qf ( i )  Z pf ( i ) 2 1 + Y qf ( i )  Z qf ( i ) 2 + Y pf ( i ) + Y qf ( i ) 2  Z pf ( i ) ) + I q   1 ( i )   jδ  ( 1 - Y qf ( i )  Z qf ( i ) 2 1 + Y qf ( i )  Z qf ( i ) 2 ) - V p   1 ( i )  [ Y pf ( i ) 2 + Y qf ( i ) 2  1 + Y pf ( i )  Z pf ( i ) 2 1 + Y qf ( i )  Z qf ( i ) 2 + Y pf ( i ) + Y qf ( i ) 2  ( 1 + Y pf ( i )  Z pf ( i ) 2 ) ] The fault location distance from the first terminal is then determined at step 54 using the boundary condition for the determined fault type and zero-sequence during-fault voltage at the fault point formulated as a function of the unknown fault location, and positive/negative-sequence during-fault voltage at the fault point and current flowing out of the fault point as a function of the unknown fault location. With reference now to FIG. 7, when the first and second terminal current and voltage measurements are not synchronized, the method for deriving a synchronization angle 60 has the following steps. A pre-fault first terminal voltage measurement and first and second terminal current measurements are received at step 62. The positive-sequence equivalent self series-impedance and positive-sequence equivalent self shunt-admittance of the line between first and second terminals is obtained at step 64. A voltage and current relationship is derived as expressed below:  jδ = I p   1 ( 1 )  0 - V p   1 ( 1 )  0  Y pq ( 1 ) 2 - [ V p   1 ( 1 )  0 - Z pq ( 1 )  ( I p   1 ( 1 )  0 - V p   1 ( 1 )  0  Y pq ( 1 ) 2 ) ]  Y pq ( 1 ) 2 I q   1 ( 1 )  0 linking the measurements of positive-sequence pre-fault voltage phasor at the first terminal of the first circuit, positive-sequence pre-fault current phasor at the first terminal of the first circuit, positive-sequence pre-fault current phasor at the second terminal of the first circuit and the line parameters of positive-sequence equivalent self series-impedance and positive-sequence self shunt-admittance of the line between first and second terminals and synchronization angle at step 66. At step 68, the synchronization angle is calculated from the expression of voltage and current relationships using the measurements of positive-sequence pre-fault voltage phasor at the first terminal of the first circuit, positive-sequence pre-fault current phasor at the first terminal of the first circuit, positive-sequence pre-fault current phasor at the second terminal of the first circuit and the line parameters of positive-sequence equivalent self series-impedance and positive-sequence self shunt-admittance of the line between first and second terminals. Simulation results for evaluating the fault location method 40 and synchronization angle derivation method 60 of the present disclosure were obtained using Matlab SimPowerSystems. The series-compensated double-circuit line was simulated, and voltage and current phasors were generated for faults of different types, locations and resistances. These phasors were fed into the algorithm to produce the fault location estimate. The initial value for the iterative fault location estimation process is chosen as 0.5 p.u. or 0.5 per unit length in km. The accuracy of fault location estimate is evaluated by percentage error defined as: %   Error =  Actual   Location - Estimated   Location  Total   Length   of   Faulted   Line  100 ( 35 ) The sample power system studied is a 500 KV, 1000 MVA, 50 Hz double-circuit transmission line compensated at the degree of 45%. The total length of the line is 350 km, with the series compensation devices installed at two ends of the transmission lines. The synchronization angle is set to π/8 (22.5°). Other data are listed in Table I and Table II. TABLE I Voltage Source Data Quantity Source P Source Q Emf(p.u.) 1.0∠20° 1.0∠0° Pos. Seq. Imp. (Ω) 17.177 + j45.5285 15.31 + j45.9245 Zero. Seq. Imp. (Ω) 2.5904 + j14.7328 0.7229 + j15.1288 TABLE II Transmission Line Data Parameters Pos. Seq Zero. Seq Zero. Seq Mutual R(Ω/km) 0.061 0.268 0.12 L(mH/km) 1.1182 3.3012 1.1485 C(nF/km) 14.8332 8.6001 −0.6761 In one embodiment, the MOV has thirty columns of metal-oxide discs connected in parallel inside the same porcelain housing. The characteristics of each column is represented by a combination of the following three exponential functions: V V ref = k i  ( I I ref ) 1  /  α i ( 36 ) where Vref=165 kV, Iref=1 kA, i=1, 2, 3. The parameters of the three segments of (36) are: k1=0.955, α1=50 k2=1.0, α2=25 k3=0.9915, α3=16.5 The fault location results are summarized in Table III, Table IV and Table V. The first two columns display the actual fault type and fault resistance simulated. The remaining columns present the percentage fault location error under various actual fault locations. Under the assumption that the voltage measurements on the right side of SC&MOV are directly available, Table III and Table IV deliver the results when the terminals are synchronized and when the terminals have a 22.5° synchronization angle, respectively. Table V reports the results when the two terminal measurements have a 22.5° synchronization angle and the voltage measurements are not directly available on the right side of SC&MOV. TABLE III Fault Location Result - Synchronized, no voltage estimation Fault Resistance Fault Location Error (%) Fault Type (Ω) 60 km 120 km 180 km 260 km 320 km AG 1 0.0176 0.0421 0.0769 0.1816 0.2871 10 0.0178 0.0432 0.0760 0.1823 0.2899 50 0.0178 0.0413 0.0748 0.1852 0.2843 BC 1 0.0098 0.0459 0.0925 0.1575 0.6018 10 0.0098 0.0459 0.0922 0.1575 0.6021 50 0.0099 0.0467 0.0910 0.1578 0.6013 BCG 1 0.0590 0.0657 0.3203 0.4736 0.0024 10 0.0590 0.0651 0.3189 0.4724 0.0059 50 0.0584 0.0619 0.3113 0.4688 0.0199 ABC 1 0.0242 0.0437 0.0481 0.2299 0.4124 10 0.0242 0.0459 0.0482 0.2296 0.4171 50 0.0250 0.0481 0.0497 0.2282 0.4163 TABLE IV Fault Location Result - 22.5° Sync Angle, no voltage estimation Fault Resistance Fault Location Error (%) Fault Type (Ω) 60 km 120 km 180 km 260 km 320 km AG 1 0.0176 0.0421 0.0768 0.1816 0.2872 10 0.0178 0.0432 0.0760 0.1823 0.2899 50 0.0178 0.0413 0.0748 0.1850 0.2840 BC 1 0.0098 0.0459 0.0925 0.1577 0.6024 10 0.0098 0.0459 0.0922 0.1577 0.6027 50 0.0099 0.0467 0.0910 0.1579 0.6016 BCG 1 0.0590 0.0657 0.3203 0.4736 0.0024 10 0.0590 0.0651 0.3188 0.4724 0.0058 50 0.0584 0.0619 0.3113 0.4686 0.0195 ABC 1 0.0242 0.0437 0.0481 0.2301 0.4128 10 0.0242 0.0459 0.0482 0.2297 0.4174 50 0.0250 0.0481 0.0497 0.2282 0.4163 TABLE V Fault Location Result - 22.5° Sync Angle, with voltage estimation Fault Resistance Fault Location Error (%) Fault Type (Ω) 60 km 120 km 180 km 260 km 320 km AG 1 0.2360 0.4311 0.1034 0.1881 0.3141 10 0.2363 0.4321 0.1024 0.1896 0.3182 50 0.2366 0.4300 0.1008 0.1961 0.3171 BC 1 0.2000 0.0900 0.2392 0.4227 0.9959 10 0.2002 0.0897 0.2374 0.4230 0.9961 50 0.2007 0.0877 0.2296
{'app_number': None, 'application_number': '<application-reference appl-type="utility">\n<document-id>\n<country>US</country>\n<doc-number>14789042</doc-number>\n<date>20150701</date>\n</document-id>\n</application-reference>', 'doc_number': None, 'publication_number': '<publication-reference>\n<document-id>\n<country>US</country>\n<doc-number>20170003335</doc-number>\n<kind>A1</kind>\n<date>20170105</date>\n</document-id>\n</publication-reference>', 'source_file': './application/2017/ipa170105.xml', 'source_data': 'USPTO-Application'}