Abstract:
Faults on power lines are detected by measuring the voltage and current at one end of a power line and using these measurements, together with values of series and shunt parameters of the line, to calculate values of voltage and current at an intermediate point on the line—the so-called “reach point”. At least the calculated voltage is used to determine whether a fault lies within a particular zone on the line. The calculation utilizes a value of a derivative of the measured current with respect to time and is able to take into account the sectioning of a line into discrete lengths corresponding to, for example, overhead and cable sections. To increase accuracy, the line or each discrete section of the line is conceptually divided into subsections each having its own parameter values. In a second aspect, detection of whether a fault lies within a particular zone involves the measurement of the voltage and current at one end of the line both before and after the fault, the calculation of a complex impedance between an intermediate point of the line and a reference point, normally earth, the use of these measured values and values relating to line parameters and to a source impedance at the second end of the line, and the determination of the sign of the complex impedance.

Description:
FIELD OF THE INVENTION 
     The invention relates to a fault-detection apparatus for detecting the presence of a fault in an electrical power line, and in particular, but not exclusively, a multi-phase underground-cable power line and a composite line including an underground-cable power line. 
     BACKGROUND OF THE INVENTION 
     Fault-detection arrangements are known in which the line being monitored for faults is modelled simply as series inductance and resistance elements, any shunt capacitance that may exist between the line and any other line or earth being ignored. Such a line will normally be an overhead power line. FIG. 1 shows this scenario, in which a line  10  is considered to have a lumped impedance Z, consisting of an inductive reactance X L  associated with an inductive component L, and a resistance R L  and is monitored in terms of voltage V r  and current I r  at one end of the line. The other end of the line may be connected to a load which may be another AC power connection. 
     The voltage V r  (known as the “relaying voltage”) is related to the current Ir (known as the “relaying current”) by the equation: 
     
       
         V r =I r Z  (1) 
       
     
     Since the relaying voltage and current are known by measurement, a value for Z can be calculated and, if the line impedance parameters per unit-length are known, an indication can be derived as to whether the fault lies within the particular section of line being monitored by the relaying equipment. 
     Discrete signal processing techniques allow calculation of Z in terms of R and L using the equation:                v   r     =       Ri   r     +     L               i   r            t                   (   2   )                                
     The use of the differential equation (2) rather than the phase equation (1) removes the need for the relay input signals v r  and i r  to be at a specified frequency. Thus equation (2) could be equally valid when applied to a 50 Hz or 60 Hz system, so that R and L could in principle be solved over a range of discrete time-signal frequencies of the relay inputs v r  and i r . 
     The value of shunt capacitance present in overhead lines is sufficiently low to be neglected at power-system frequency so that equation (2) can be used to determine the values of R and L. However, the presence of capacitance causes high-frequency travelling waves to be generated when a fault occurs and these can cause errors in the calculation of R and L. It is therefore necessary to remove the travelling-wave frequencies from the signals v r  and i r  before calculating R and L using equation (2). Since these frequencies are relatively high with respect to the power-system frequency, they can be removed by short-window digital filters without causing the relay operating time to be extended beyond one power-frequency cycle. 
     On cable circuits, however, the values of shunt capacitance are much greater and the travelling-wave noise is far lower in frequency, so that its effects are less easily removed by filtering. On this basis the values of R and L calculated from equation (2) no longer accurately represent the location of the fault. 
     SUMMARY OF THE INVENTION 
     In accordance with the invention, there is provided relay equipment for an electrical power line, characterized by fault-detection means comprising: 
     a signal-deriving means for deriving first and second signals representative of a voltage and current, respectively, at a first end of said line; 
     a calculating means for calculating from said first and second signals and from defined values of series and shunt parameters of said line a value of voltage and current associated with a point along said line intermediate said first end and a second end of said line; and 
     a fault-location determining means for determining from said calculated voltage and current values whether a fault lies between said first end and said intermediate point. 
     The calculating means may calculate said intermediate-point voltage and current values using a value of a derivative of said second signal with respect to time, and preferably employing the matrix relationship:          [           v   1               i   1           ]     =       [         1         -     (       R   0     +       L   0                  t           )                   -     C   0                    t             1         ]                [           v   0               i   0           ]                            
     where v 1 , i 1  are said voltage and current at said intermediate point, v 0 , i 0  are said voltage and current at said first end of the line, R 0  and L 0  are said series parameters of the line and C 0  is a shunt parameter of the line. 
     The calculating means may be arranged to model said line as a plurality p of line-portions in series having respective series and shunt parameters, said matrix relationship being:          [           v   1               i   1           ]     =       ∏     m   =   1     p                       [                    1         -     (       R     0      m       +       L     0      m                    t           )                   -     C     0      m                      t             1                    ]                [           v   0               i   0           ]                              
     Corresponding parameters of said line-portions may have substantially the same value. 
     The line may be composed of a plurality q of sections in series having respective series and shunt parameters, each of said plurality q of sections being composed of a plurality p of line-portions in series having respective series and shunt parameters, said matrix relationship being:          [           v   q               i   q           ]     =       ∏     n   =   0       q   -   1                         ∏     m   =   1     p                       [                    1         -     (       R     n                 m       +       L     n                 m                    t           )                   -     C     n                 m                      t             1                    ]                [           v   0               i   0           ]                                
     At least the shunt parameters of said sections may not all be of substantially the same value. The sections may correspond to respective distinct forms of line including at least one overhead line section and at least one cable section. 
     Each of said line or line-portions is preferably modelled by said calculating means as a Γ-network, though a π-network may also be employed. 
     The fault-location determining means may comprise means for detecting whether said voltage at said intermediate point is zero, positive or negative relative to a reference quantity, a zero result indicating that there is a fault at said intermediate point, a positive result indicating that there is a fault between said point and one end of said line and a negative result indicating that there is a fault between said point and the other end of said line. The reference quantity may be the current associated with said intermediate point. 
     The calculating means may comprise means for calculating from said calculated voltage and current values a value of a complex impedance between said first end and said intermediate point, and said fault-location determining means may comprise means for evaluating a sign of said complex impedance. The calculating means may calculate said value of a complex impedance by means of the equation:        v   =     Ri   +     L             i          t                                  
     where v and i are said intermediate-point voltage and current values, respectively, and R and L are components of said complex impedance. 
     A negative result may indicate that there is a fault between said intermediate point and said first end of said line and a positive result indicate that there is a fault between said intermediate point and said second end of said line. 
     The intermediate point will usually lie approximately 80% of the way along said line from said first end of the line. 
     The signal-deriving means may be adapted to derive said first and second signals both before and after the occurrence of a fault on said line, said calculating means may be adapted to calculate from said pre-fault and post-fault first and second signals and from said parameters and from a value of a source impedance of said second end of said line a change in a value of voltage and current associated with said intermediate point and to calculate from said change a value of a complex impedance existing between said intermediate point and a reference point, and said fault-location determining means may be adapted to determine a sign of an imaginary part of said complex impedance and to infer from said sign a position of said fault relative to said intermediate point. 
     In accordance with a second aspect of the present invention, a method of determining if a fault on a power line lies within a particular zone of said line comprises the steps of: 
     measuring a voltage and current at a first end of said line before and after the occurrence of a fault; 
     calculating the complex impedance between a point on said line intermediate said first end and a second end of the line and a reference point, said complex fault impedance calculation utilizing the measured voltage and current values and values relating to parameters of the line and to a source impedance at said second end of the line; 
     determining a sign of an imaginary part of said complex impedance; and 
     inferring from said sign a position of said fault relative to said intermediate point on said line. 
     The inference of said fault position may follow the rule that a negative sign indicates a fault between one of said ends and said intermediate point of said line, a positive sign indicates a fault between the other of said ends and said intermediate point of said line and a zero imaginary part indicates a fault at said intermediate point itself. 
     The step of calculating the complex impedance may comprises a calculation of the voltage and current at said intermediate point using the equation:          [           v   1               i   1           ]     =       [         1         -     (       R   0     +       L   0                  t           )                   -     C   0                    t             1         ]                [           v   0               i   0           ]                            
     where v 0 , i 0  are the voltage and current at said first end of the line, v 1 , i 1  are the voltage and current at said intermediate point and R 0 , L 0  are series parameters and C 0  is a shunt parameter of said line. 
     The line may be divided into a plurality q of sections and said step of calculating the complex impedance may comprise a calculation of a voltage and current at said intermediate point using the equation:          [           v   q               i   q           ]     =       ∏     n   =   0       q   -   1                         [                    1         -     (       R   n     +       L   n                  t           )                   -     C   n                    t             1                    ]                [           v   0               i   0           ]                              
     where v 0 , i 0  are the voltage and current at said first end of the line, v q , i q  are the voltage and current at said intermediate point and R n , L n  are series parameters and C n  is a shunt parameter of respective said sections of the line. 
     The line or each of said sections may be conceptually subdivided into p line-portions having respective line parameters, and said step of calculating the complex impedance may comprise a calculation of a voltage and current at said intermediate point using the equation:          [           v   q               i   q           ]     =       ∏     n   =   0       q   -   1                         ∏     m   =   1     p                       [                    1         -     (       R     n                 m       +       L     n                 m                    t           )                   -     C     n                 m                      t             1                    ]                [           v   0               i   0           ]                                
     The intermediate point may lie approximately 80% of the way along the line from said first end thereof. 
    
    
     BRIEF DESCRIPTION OF THE DRAWINGS 
     Embodiments of the invention will now be described, by way of example only, with reference to the drawings, of which: 
     FIG. 1 is an equivalent circuit of a section of power line monitored for faults; 
     FIG. 2 is an equivalent circuit as in FIG. 1 but with the inclusion of shunt capacitance; 
     FIG. 3 is an equivalent circuit of a composite power line; 
     FIG. 4 is an equivalent circuit of one section of the composite power line of FIG. 3, the section being subdivided into subsections; 
     FIG. 5 is a diagram of a voltage characteristic against line length for various positions of fault along the line; 
     FIG. 6 is a graph showing a first type of relaying characteristic which can be applied to the present invention; 
     FIG. 7 is a graph showing a second type of relaying characteristic which can be applied to the present invention; 
     FIG. 8 shows a polarizing characteristic as applicable to the present invention; 
     FIG. 9 is a diagram showing the effect of severe remote-end infeed on relaying performance; 
     FIGS.  10 ( a ), ( b ) and ( c ) are equivalent circuits illustrating the three circuit states associated with a superposition direction-finding technique according to a second aspect of the present invention; 
     FIG. 11 is an equivalent circuit of a three-phase power line in its superimposed state as applied to a fault-detection arrangement according to the present invention; and 
     FIG. 12 is a diagram showing a characteristic achieved by the superposition directional technique in the second aspect of the present invention. 
    
    
     DETAILED DESCRIPTION OF PREFERRED EMBODIMENTS 
     The present invention takes into account the significant level of shunt capacitance which exists in cable circuits and employs the following relationship between the relaying voltage and current signals:                [           V   r               I   r           ]     =       [         A       B           C       D         ]                [           V   f               I   f           ]             (   3   )                                
     where A, B, C and D represent the constants of the circuit up to the fault point and V f  and I f  represent the voltage between the faulted point on the line and a reference point, which is normally earth but may alternatively be another phase, and the current through the fault branch, respectively. Constants A, B, C and D would normally be hyperbolically derived for maximum accuracy. 
     The fundamental difficulty with solving equation (3) is the extra number of variables involved. Although a direct solution is not impossible, it involves very complicated real-time processing. By contrast, if it is assumed that the fault is at a known position—the so-called “reach point” (see later)—calculation of the fault-point voltage and current (i.e. the “reach-point” voltage and current) is considerably facilitated. Thus equation (3) can be recast as:                [           V   f               I   f           ]     =         [         A       B           C       D         ]       -   1            [           V   r               I   r           ]               (   4   )                                
     Referring to FIG. 2, this shows a “gamma” (Γ) circuit model of a section of line or cable to be monitored, but a “pi” (π) model could equally well be used. The voltage and current values, v 1 , i 1  at the right hand side of the section are calculated using a knowledge of the voltage and current values, v 0 , i 0  at the left hand side of the section, by the equation:                [           v   1               i   1           ]     =                  [                    1         -     (       R   0     +       L   0                  t           )                   -     C   0                    t             1                    ]                [           v   0               i   0           ]             (   5   )                                
     where R 0 , C 0  and L 0  are the resistance, capacitance and inductance values of the circuit section. Again, the differential form of this equation allows the calculation to be made over a range of frequencies and not be restricted to solely the power-system frequency. 
     The invention caters also for composite forms of line involving different parameters; an example is where a line takes the form of an overhead stretch of line in one location, continues as a cable section in another location and then becomes overhead again in a third location, and so on. This is shown in FIG.  3 . Under these conditions the calculation of the right hand-side voltage and current signals is achieved using:                [           v   3               i   3           ]     =                    [                    1         -     (       R   2     +       L   2                  t           )                   -     C   2                    t             1                    ]                [                    1         -     (       R   1     +       L   1                  t           )                   -     C   1                    t             1                    ]                 [                    1         -     (       R   0     +       L   0                  t           )                   -     C   0                    t             1                    ]                [           v   0               i   0           ]                 (   6   )                                
     In order to achieve the correct bandwidth of operation and to retain accuracy by more closely approximating a hyperbolic circuit form, a preferred embodiment of the invention divides each circuit section into several cascaded matrices of the form used in equations (5) and (6). Thus, for example, the first matrix on the right hand side of equation (6) can be expressed as (assuming the first overhead line section to be notionally divided into three subsections):                               [                    1         -     (       R     2        -        1       +       L     2        -        1                    t           )                   -     C     2        -        1                      t             1                    ]                [                    1         -     (       R     2        -        2       +       L     2        -        2                    t           )                   -     C     2        -        2                      t             1                    ]               [                    1         -     (       R     2        -        3       +       L     2        -        3                    t           )                   -     C     2        -        3                      t             1                    ]                 (   7   )                                
     This situation is illustrated in FIG. 4, in which the second overhead line portion (line 2) is represented as being composed of three subsections in series, each having the parameter designations indicated. 
     An equation of the form of equation (6) in conjunction with the matrix terms (7) allows an accurate discrete-time calculation of the voltage and current at the “reach point”. This calculation is valid over a range of frequencies dependent on the number of subsections incorporated. In practice, four or five subsections are sufficient for most ehv (extra-high voltage) composite-circuit applications, and an operating bandwidth of several hundred Hertz is obtainable. Since this calculation obviates the need for filtering over its operating bandwidth, less overall filtering is needed. 
     A general expression for the case where q line sections (e.g. overhead, cable) are employed, each divided, for the sake of modelling, into p portions or subsections, is:                [           v   q               i   q           ]     =       ∏     n   =   0       q   -   1                         ∏     m   =   1     p                       [         1         -     (       R     n                 m       +       L     n                 m                    t           )                   -     C     n                 m                      t             1         ]                [           v   0               i   0           ]                 (   8   )                                
     The R, L, C parameters will normally be the same within each section, i.e., over the p portions for a particular section. Subdivision into portions applies not only to the case where the line is composed of a number of discrete physical sections, but also to the case where the line is an integral whole, i.e., where q=1. 
     The “reach point” mentioned earlier is now described with reference to FIG.  3 . In FIG. 3 the voltage and current V 3 , i 3  correspond to a point on the line which is somewhat short of the right hand end of the line. In practice the line is usually a section of line under the supervision of relaying equipment  100  at the end  22  of the line section. Relay  100  incorporates the fault-detection means of the present invention, implemented as microprocessor-based computing equipment programmed with the algorithms described herein. Further sections of line may be connected to the end  20  and these may be monitored by other relays. Thus, while the end of the relevant line section is shown as  20 , the calculations performed in the relay  100  relate to a point  21  on the line. This is the “reach point” and is typically around 80% of the total line-section length. 
     It is possible, in the present invention, to differentiate between a fault which occurs at the reach point itself, a fault which occurs between the reach point  21  and the end  20  of the section and a fault which occurs between the relaying end  22  of the section and the reach point  21 . Referring to FIG. 5, it is assumed that there is a solid fault (i.e., a fault of substantially zero resistance) either at point A, point B or the reach point  21 . Where the fault is at the reach point itself, the calculated voltage is equal to zero. For a fault beyond the reach point, e.g., at point B, the reach-point calculations are correct and show the voltage to have a small positive value. For an in-zone fault, e.g., at point A, the reach-point calculation will be incorrect (inasmuch as the result will not necessarily reflect the true reach-point conditions); however, the voltage will have a small negative value. Under no-fault conditions the reach-point values will be correctly calculated. 
     The reach-point voltage calculated by this method contains a considerably smaller proportion of travelling-wave frequencies than would be achieved by using a series R-L model. A small degree of filtering then allows phasor techniques to be used to determined the positive/negative nature of the reach-point voltage. A convenient reference phasor for this purpose is the reach-point current. There will be a phase reversal of the reach-point voltage as the location of the fault moves from one side of the reach point to the other. This phase behavior can be determined in practice by calculating the impedance apparent to the reach point using, for example, a solution to equation (2). Thus, if it is imagined that v 1 , i 1  in FIG. 2 represent the reach-point voltage v r  and current i r , respectively, then, taking these quantities to be the values derived by means of, for example, equation (8), a value of R and L can be derived from equation (2) to yield the complex impedance Z. A negative imaginary value of Z will indicate an in-zone fault, whereas a positive imaginary value will signify an out-of-zone fault. 
     It has so far been assumed that the impedance of the fault will be negligible. This is not always the case, however, and where fault impedance is significant an infeed of current from the remote end  20  of the line section can give rise to misleading indications of fault voltage and current. Because of this it may be necessary to employ some form of relay “characteristic”, which will now be described. 
     The concept of the relay characteristic is illustrated by reference to FIG.  6 . In this diagram the impedance Z (cf. FIG. 1) to the fault is represented by a line  24  situated in the complex R-X plane and having its origin at a point  25  corresponding to the relaying point  22 . The impedance equals zero at the relaying point, which is to be expected, and increases to a maximum at a point  26  corresponding to the end  20  of the line section, passing on its way through a value  27  corresponding to the reach-point value. For fault impedances close to zero, Z will be somewhere along the line  24 . When the fault has finite resistance, however, the total impedance value will be displaced from the line  24 , taking a value along the line  28 , for example, for a fault midway between the relaying point and the reach point and having pure resistance and negligible infeed from the remote end of the line section. Other lines  29 ,  30  relate to similar faults at locations nearer the reach point. 
     Where fault resistance and remote-end infeed are not negligible, a phase shift is introduced between the current in the fault path and that measured by the relay; hence the line  28 , for instance, possesses a reactive component, assuming the orientation shown as dotted line  28 ′ or  28 ″. The former is for power flow from relaying end  22  to remote end  20 , the latter for power flow in the reverse direction. At all events, steps are taken to ensure that the relay does not trip the associated circuit breakers when the calculated value of impedance lies outside a locus formed by the circle  31 . The X-R values forming this characteristic are simply supplied to the relay processor which then uses a suitable algorithm to effect a comparison between these characteristic values and the calculated values. 
     Since the use of a circular characteristic  31  restricts the value of fault resistance that can be tolerated while still being registered as a fault and hence used to trip the circuit breakers (cf. the shortness of line  30  compared with line  28 ), it is possible to employ a non-circular characteristic allowing for a more uniform fault resistance value. A suitable characteristic is a quadrilateral locus (see FIG. 7) having as its essential elements a straight upper side  40  passing through the reach-point impedance value  27  and a right hand side  41  approximately parallel to the line  24 . Thus now lines  28  and  30  can be of equal length and still serve to signal a genuine fault. 
     For faults occurring close to the relay, the voltage input will be close to zero. Under these circumstances, the relay is unable to differentiate between faults in the protected zone, for which the relay must operate, and faults behind the relay location, where the relay is required to block. To overcome these difficulties the relay is provided with an estimate of the phase of the pre-fault voltage signal so that the direction of the fault can be ascertained; this is referred to as “polarization”. 
     Polarization can be achieved by using the voltages from phases not involved in the fault—so-called “sound (i.e., good) phase polarization”—which are vectorially manipulated to give an approximation to the faulted-phase pre-fault voltage. Alternatively, memory polarization, where samples of the pre-fault voltage signal are stored in memory, can be used. The combination of measured and polarized signals is formed into a directional element as shown in the lower line of the quadrilateral of FIG.  8 . For either of these approaches, the signals employed for the directional element will be based on the signals at the relay point close to the busbar (i.e., measured signals) and not the reach-point values calculated by means of, for example, equation (8). 
     In a practical power-line system there may be reactive elements involved apart from the line&#39;s own intrinsic series and shunt parameters. Such additional reactive elements include capacitance, which is deliberately included in series with the line in order to compensate the series inductance L of the line, and inductance which is shunted between the line and ground/associated lines in order to compensate for the shunt capacitance C of the line when the line is long. These elements, where they lie within the protected zone of the relay, will tend to affect the accuracy of the reach-point calculations unless they are included in the modelling described earlier. Ideally, therefore, these elements are taken into account in this manner. 
     As previously stated, when a fault has a significant resistive component any infeed at the remote end of the line section introduces phase shifts which can adversely affect the accuracy of the reach-point voltage and current calculations. In particular, the effect of a change in load can be to trip the relay (i.e., signal a fault) when a fault that has occurred lies in another zone, not the zone protected by the relay in question. This is illustrated by reference to FIG. 9, in which a resistive fault lying outside the protected zone and involving no infeed is shown as line  50 , whereas the same fault involving a transfer of power in one direction is shown as line  51  and in the other direction as line  52 . Although the fault lies outside the relevant zone, since line  52  passes through the quadrilateral characteristic  53  the relay in question will signal a fault in its zone. 
     In order to compensate for the effects of infeed on the reliability of the fault detection function, it has been proposed in the past to approximate the current in the fault impedance by employing a model based on symmetrical component sequence networks, in particular a negative sequence network. This model, however, has the drawback of being limited to the fundamental system-frequency only. 
     The present invention approaches this problem by incorporating in a preferred embodiment an improved reach-point directional element which will now be described. 
     The basic principle of the “superimposition” technique is discussed in the UK Patent GB 2286088B, granted on Sep. 24, 1997. This technique, which in this patent is employed in a context different from that of the present application, namely the determining of the exact location of a fault on a power line, is described here with reference to FIG.  10 . FIG. 10 a  represents a three-phase line under conditions of a fault between phase a and earth, FIG. 10 b  represents the same three-phase line as it appears just before the fault and FIG. 10 c  corresponds to the same line in its “superimposed” state, i.e. highlighting those quantities that have changed during the occurrence of the fault. The voltages and currents measured at the relaying end (the lefthand side of the figures) are as designated for all three states, the following relationship holding: 
     
       
         Δ V   rx   =V   rx   −V   srx   (9) 
       
     
     where x=a, b or c according to phase. 
     The impedance in the fault path is shown as impedance Z f  and the superposition circuit (FIG.  10 ( c )) is energized by an emf−V sfa  which is the pre-fault voltage at the fault point. 
     The invention applies this superposition principle as shown in FIG.  11 . In FIG. 11 the fault point is assumed to be the reach point which is situated, as mentioned earlier, so that approximately 80% of the line impedance lies between the reach point and the relaying end and 20% lies between the reach point and the remote end of the line. Included with the 20% line impedance is the source impedance of the remote source which appears in FIG. 10 as impedances Z sa , Z sb , Z sc . FIG. 11 shows the superimposed state, the various superimposed voltage and current parameters having “Δ” designations. 
     The superimposed voltages are given by equation (9) above; similarly, the superimposed currents are given by: 
     
       
         Δ I   rx   =I   rx   −I   srx   (10) 
       
     
     The reach point superimposed voltage and current values can be similarly calculated using the faulted and pre-fault values from the calculation of equation (6), (7) or (8), as appropriate. Algebraically, the superimposed voltage at the reach point for phase a is: 
     
       
         Δ V   fa   =V   fa   −V   sfa   (11) 
       
     
     and likewise, mutatis mutandis for the other phases. As far as the superimposed currents are concerned, we have: 
     
       
         Δ I   fa   =I   ra   −I   sra   
       
     
     
       
         Δ I   fb   =I   rb   −I   srb   
       
     
     
       
         Δ I   fc   =I   rc   −I   src   (12) 
       
     
     The unknown quantities in FIG. 11 are Z f , Z sq  and ΔI aa . Z f  can be calculated from:                Z   f     =           Δ                   V   fa       +     V   sfa         Δ                   I     Z   f           =         Δ                   V   fa       +     V   sfa           Δ                   I   fa       -     Δ                   I   aa                     (   13   )                                
     Assuming that the remote source impedance Z sq  is known, ΔI aa  is calculated from:                [           Δ                   I   aa                 Δ                   I   fb                 Δ                   I   fc             ]     =         [           Z   s         Zsubm         Z   m               Z   m           Z   s           Z   m               Z   m           Z   m           Z   s           ]       -   1                  [           Δ                   V   fa                 Δ                   V   fb                 Δ                   V   fc             ]             (   14   )                                
     where Z s  and Z m  are the self and mutual impedances of the circuit section between the reach point and the remote end and including the assumed value of the remote source impedance. Since high-resistance earth faults are only likely where the reach point is on an overhead line rather than cable, the discrete time calculation of ΔI aa  can be made by representing Z s , Z m  as series resistance and inductance elements only, i.e. by using models of the form of equation (2) rather than equation (6) or (8). Z f  is calculated by substitution of the value derived for ΔI aa  from equation (14) into equation (13). 
     The behavior of Z f  can be summarized as follows: 
     
       
         
               
               
               
               
             
           
               
                   
               
               
                 Fault Position 
                 Re{Z ƒ } 
                 Im{Z ƒ } 
                 Remarks 
               
               
                   
               
             
             
               
                 In-zone 
                 Pos. or Neg. 
                 Negative 
                 — 
               
               
                 Reach point 
                 Pos. or zero 
                 Zero 
                 Re{Z ƒ } = true fault resistance 
               
               
                 Out-of-zone 
                 Positive 
                 Positive 
                 — 
               
               
                   
               
             
          
         
       
     
     Thus, it is possible to use the change in sign of the imaginary component of Z f  as the basis for a directional decision, i.e., whether a fault is in- or out-of-zone. Furthermore, since the superimposed network is not influenced by prefault loading, discrimination of the reach point will not be affected by fault resistance, as it is affected in the conventional methods of direction evaluation described earlier. Hence there is no need to perform any check on the superimposed reach-point resistance, or to evaluate it at all. Investigations have shown that for ehv applications the assumed value of the remote source impedance can vary by an order of magnitude without significantly affecting the accuracy of the relay. Further to this, in equation (14) the quantities ΔI fb , ΔI fc , ΔV fa , ΔV fb  and ΔV fc  are all known from the reach-point calculation. Hence equation (14) could be solved for Z s  and Z m  (the impedance to the right of the fault in FIG.  11 ). In principle, the accuracy of the relaying process can be improved by using calculated values of Z s  and Z m  rather than the assumed values described earlier. 
     The effect of the superimposition technique just described is to provide a characteristic such as that shown in FIG. 12, in which the righthand line, which normally limits the allowed value of fault resistance, is missing. 
     Although the superimposed reach-point technique has been described in connection with the specific line protection arrangement according to the invention, it is envisaged that it could also be employed with more conventional relaying techniques such as that described in the opening part of the present specification. 
     Also, whereas the faults covered in the description of the superimposition direction-finding technique according to the invention were assumed to be faults from line to earth, in principle the method could also cover faults between phases.