RU2801438C1 - Method for determining location of a short circuit on an overhead power line with unsynchronized measurements from its two ends - Google Patents

Abstract

FIELD: electric power industry.

SUBSTANCE: invention can be used to determine the location of a short circuit on power lines during unsynchronized measurements from its two ends. Essence: measured from both ends of the line (' - one end of the line,'' - the second end of the line) instantaneous values of phase currents (i'_A, i'_B, i'_C), (i''_A, i''_B, i''_C) and stresses (u'_A, u'_B, u'_C), (u''_A, u''_B, u''_C) during a short circuit, receive their oscillograms. The oscillograms from the two ends of the line are combined along the cut of the beginning of the short circuit. Arrays are formed from successive instantaneous values for each of the oscillograms of the current and voltage of the damaged phase i' (m), i'' (m), u' (m), u'' (m), determine the first time derivatives of the currents for the selected arrays of instantaneous values di'(m)/ dt _m , di''(m)/ dt _m , as well as the first and second derivatives of the voltages du'(m)/ dt _m , du''(m)/ dt _m , d ² u'(m)/ dt ² _m , d ² u''(m)/ dt ² _m . The voltage matrices U 1 and U 2 are formed using the active resistance R, inductance L and transverse capacitance C of the power line, instantaneous values of arrays of currents, voltages i'(m), i''(m), u'(m), u''(m), as well as the instantaneous values of the derivatives of currents and voltages. The relative value of the distance to the short circuit n is determined using the voltage matrices U 1 and U 2 and the least squares method. Based on the obtained relative value of the distance to the place of the short circuit n, the distance to the place of the short circuit is determined from the side of the end with the index '.

EFFECT: improving the accuracy of determining the location of a short circuit under conditions of deviations of currents and voltages from a sinusoidal shape, taking into account the distributed capacitance of power transmission lines.

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Description

The present invention relates to the electric power industry and can be used to determine the location of a short circuit on power lines with unsynchronized measurements from its two ends.

A known method of determining the location of damage on overhead power lines [RF Patent No. 2526095, IPC G01R 31/08, publ. 20.08.2014, Bull. No. 23], in which phase voltages and currents are measured from both ends of the line, they are converted into calculated complex values according to the proposed expressions and, using the imaginary parts of the calculated values, the relative and physical distances of the damage site from the ends of the power line are calculated by calculation. In this method, the equivalent parameters of the supply systems are not used; the influence of the transient resistance is eliminated.

The disadvantage of this method is the need to use only the imaginary components of the calculated values. This shortcoming can lead to an error in determining the location of the damage due to the insufficient amount of parameters taken into account, in particular, due to the lack of consideration of the distributed capacitance of the power line.

A known method of determining the location of damage on overhead power lines [Technology of vector registration of parameters and its application to control the modes of the UES of Russia, Electro. Electrical engineering, Electric power industry, Electrotechnical industry. 2011. №2. S. 2-5], in which the need to use only the imaginary components of the calculated values is eliminated. In this method, the angle between the currents at the ends of the line is preliminarily measured, which is done by using digital communication channels between the ends of the power line or by using satellite time synchronization. When using digital communication channels, the angles are determined by performing time-synchronized samples or by constantly calculating the signal transit time between the line differential protection half-sets. When using satellite time synchronization, time synchronization pulses are received from GLONASS (GPS) signal receivers. Next, phase voltages and currents are measured from both ends of the line, converted into calculated complex values according to the proposed expressions and, using the full parts of the calculated values, the relative and physical distances of the damage site from the ends of the power line are calculated by calculation. In this method, the equivalent parameters of the supply systems are not used; the influence of the transient resistance is eliminated.

The disadvantage of this method is the need to use complex equipment and algorithms, for example, digital communication channels between the ends of the line, when the angles are determined by performing synchronized samples in time or by constantly calculating the signal transit time between the differential protection half sets of the line, or satellite time synchronization equipment, when the synchronization pulses time is received from GLONASS (GPS) signal receivers.

The closest technical solution is a method for determining the location of a short circuit on an overhead power line with unsynchronized measurements from its two ends [RF Patent No. 2508556, IPC G01R 31/08, publ. 27.02.2014, Bull. No. 6], having a length l , active R and inductive resistance X _L , connecting two supply systems, in which phase currents and voltages not synchronized in time during a short circuit are measured from both ends of the line, damaged phases are determined, the relative value of the distance to places of short circuit n and physical distance to the place of short circuit from the end of the line with index ' according to the expression l ' = n ⋅ l . According to the proposal, measure from both ends of the line (' -

one end of the line, "- the second end of the line) instantaneous values of phase currents ( i ' _A , i ' _B , i ' _C ), ( i " _A , i " _B , i " _C ) and voltages ( u ' _A , u ' _B , u ' _C ), ( u " _A , u " _B , u " _C ) during a short circuit, receive oscillograms of currents and voltages, combine the oscillograms from both ends of the line along the cut of the beginning of the short circuit, select in the interval of two to ten periods from the beginning of the short circuit, the section on the oscillograms of the current and voltage of the damaged phase, take the instantaneous values of the currents i ', i ”and voltages u ', u " in the section and at neighboring points, calculate the derivatives of the currents with respect to time di '/ dt , di "/ dt , determine the relative value of the distance to the place of the short circuit according to the expression (1):

where n is the relative value of the distance to the short circuit; u ', u "- instantaneous voltage values obtained in the section of the oscillograms of the damaged phase voltages from one and the second ends of the line (B); i ', i "- instantaneous current values obtained in the section of the oscillograms of the damaged phase currents from one and the second ends of the line (A); di '/ dt , di "/ dt - time derivatives of currents (A / s); R , X _L - active and inductive phase resistances of the power line (Ohm).

The disadvantage of the prototype method for determining the location of a short circuit (OMKZ) on an overhead power line (TL) with unsynchronized measurements from its two ends are errors in determining the distance to the place of a short circuit when the quality of electrical energy (PCEE) deviates even within the limits of standard values, and also due to the lack of consideration of the distributed capacitance of the overhead power transmission line (OL).

For example, the standard of the organization PJSC "FGC UES" STO 56947007-29.240.55.224.2016 "Guidelines for determining the location of damage to overhead lines with a voltage of 110 kV and above", introduced on August 17, 2016, it is assumed that there is no need to take into account the reactive (capacitive) conductivity must be pre-assessed. Neglecting this conductivity (especially in manual calculations) is permissible if the resulting distance calculation error does not exceed 2%.

The error from neglecting reactive conductivity depends on the length of the overhead line and the resistance of the networks adjacent to it.

The specified standard of the organization introduces the following recommendations. When using zero sequence parameters, it is permissible to neglect this conductivity for overhead lines up to 100 km long, when using negative sequence parameters - up to 120-150 km. For long-range overhead lines, where the ratio of the resistances of the adjacent network and overhead lines becomes of primary importance, it is allowed to ignore the reactive conductivity for power lines up to 200-250 km long, if this ratio is in the range of 0.1-0.5.

When taking into account the transverse capacitive conductivity, the equivalent circuit for the OMKZ of an overhead power line (Fig. 1) changes and takes the form of Fig. 2. Accordingly, the differential equations that describe the ratio of currents and voltages of a damaged power line change. And the voltage drop to the short circuit point from the two ends of the line (Fig. 2) corresponds to the following expression:

where L and C are the longitudinal inductance and transverse capacitance of the overhead transmission line, corresponding to the equivalent circuit of Fig. 2.

After the transformation of expression (2), we obtain an expression for calculating the distance to the place of the OMKZ of an overhead power line in relation to the equivalent circuit of Fig. 2:

It should be noted that, in contrast to relation (1) in expression (3), along with the parameters of the overhead transmission line, voltages u ' and u '', currents i ' and i '', as well as their derivatives di '/ dt , di ''/ dt , the first du '/ dt , du ''/ dt are additionally involved, as well as the second d ² u '/ dt ² , d ² u ''/ dt ² derivatives of voltages u ' and u ''. It is the use of the first and second derivatives of the voltage that makes it possible to take into account the transverse capacitive conductance of the equivalent circuit of Fig. 2.

It is known [for example, Kulikov A.L., Kolobanov P.A., Falshina V.A. Algorithm for digital measuring devices of distance protection based on differential equations of the line. Elektricheskie stantsii. 2014. No. 8. S. 44-49] that the use of differential equations corresponding to the power transmission line model (Fig. 2) makes it possible to take into account, when processing oscillograms of emergency events, both the aperiodic current component and the transient high-frequency components of voltage and current signals without any or additional filtering. Thus, the method for determining the location of a short circuit on an overhead power line with unsynchronized measurements from its two ends, using the model of an overhead power line (Fig. 2), involves the inclusion of sources of transient voltage and current signals in the algorithm for calculating the distance to the fault site (expression (3 )) and does not require additional filtering of the components of transients in the oscillograms of emergency events.

The reasons for the occurrence of errors OMKZ on power lines in case of deviations of the SCEE from the standard values can, for example, be:

- deviations of the industrial frequency from the value f = 50 Hz, which determine the errors caused by the inappropriate sampling rate in the implementation of the calculation algorithms;

- saturation of current transformers and the corresponding appearance of an aperiodic (constant) component in sinusoidal emergency current oscillograms;

- the presence of multiple harmonics and interharmonics in emergency oscillograms of currents and voltages, caused, for example, by the influence of electrical consumers of industrial consumers;

- distorting effect of various interference components in the form of flicker and high-frequency industrial interference;

- other factors.

Since in the prototype method, its calculated ratios, it is provided that emergency currents and voltages are represented by ideal sinusoidal signals of frequency f = 50 Hz without interference, and calculations are carried out using only individual instantaneous values, then due to deviations of private SCEE (or their total impact), OMCB errors can be significant. Moreover, the deviations of the SCEE may be within the permissible values, and their cumulative effect on the accuracy of the OMKZ of power lines may be significant.

To assess the errors of OMCB caused by deviations of the SCEE, we will use the calculation example given in the prototype method, as well as the calculation expression (1), including the instantaneous values of current and voltage on both sides of the power line. We consider that i '( m ) and di '( m )/ dt _m , as well as i ''( m ) and di ''( m )/ dt _m are quadrature (orthogonal) components obtained at some point in time ( m ). Then the amplitudes of the currents at the ends of the transmission line can be determined from expressions (4):

where the function sqrt { ⋅ } means the square root operation.

The voltage drop to the short circuit point on both sides of the power transmission line (Fig. 1) will correspond to the equation:

The formation of voltage oscillograms at the ends of the power transmission line can be implemented using expressions (6) - (7):

as well as expressions for the quadrature components of the currents at the ends of the transmission line:

Then the instantaneous values of the voltage waveforms will correspond to expressions (10)-(11):

Let's calculate the value of U by substituting the data of a numerical example of the prototype method: u ' = 100051.9 (B); u "= 79091 (B); i '= 11274.6 (A); di' / dt = 8143.71 (A / s); i '' = 7336.6 (A); di ''/ dt = 5264 ,74 (A / s), n = 0.5. In this case, we select the specific parameters of the 110 kV power transmission line, using the data [Panova E.A., Albrecht A.Ya. Refined specific electrical parameters of double-circuit power lines 110 kV for remote determination of the fault location // Electrotechnical Systems and Complexes, 2016, No. 4 (33), pp. 35-40, doi: 10.18503/2311-8318-2016-4(33)-35-40] and accepting Rsp values for the AC-120 _wire = 0.25 (Om/km) X _{L beats} = 0.404 (Om/km) Taking into account the length of the power line l = 50 km, we have:

Let us calculate the amplitudes of the currents I ' and I '', using equalities (4):

For example, based on the values of variables at one of the ends of the power transmission line and the rounding error, we have:

Thus, taking into account (10)-(11), the expressions for the formation of voltage oscillograms at the ends of the power transmission line take the form:

We introduce the sampling rate, for example, corresponding to the variable ( m ) and the component N = 8 samples per period of the industrial frequency (sampling interval t _d = 0.0025 s). With respect to the beginning of the analysis time of the oscillograms, we choose a delay, for example, nine signal counts (current, voltage) t _z = 9⋅ t _d = 0.0225 s. It should be noted that the numerical choice of the value ( m ) does not affect the amplitude-phase ratios of currents and voltages and the accuracy of calculations according to expression (1), since the validity of equality (1) must be observed for each discrete time moment ( m ).

To verify and determine the accuracy of setting model analytical expressions (4) - (16), we implement the OMKZ procedure for power lines under the assumption that the industrial frequency takes the values f = 50 Hz, and the value of the discrete variable m = 1. In this case, the total discrete time will be t = t _h + t _d = 10⋅ t _d and the following relations hold:

Thus, the distance to the damage site is x ' = n l = 0.5000099·50 = 25.000495 km and corresponds to the calculated example of the prototype method.

Let us assume that, for example, from the side of system-1 (Fig. 1), the discrete instantaneous values of the current i '( m ) are distorted by flicker [for example, GOST 32144-2013]. The distorted current signal i '( m ) is illustrated in FIG. 3(a), and the mathematical expression for instantaneous current values, taking into account the delay t _z, corresponds to the relationship:

where k is a number (constant coefficient) characterizing the "depth of distortion" by flicker; rnd ( m ) - a random number (for example, uniformly distributed in the interval [0; 1], formed at each discrete time value m .

In particular, setting the numerical values of the example of the prototype method leads to the dependence:

and the implementation of the current waveform at N = 20 ( t _d = 10 ^-3 s), k = 0.15 (15% of the amplitude I '), t _c = 9⋅ t _d = 9⋅10 ^-3 s is shown in Fig. 3(a).

Note that the expression for the instantaneous voltage values u '( m ) takes the form:

Let us assume that from the side of the system -2 (Fig. 1) there is a non-linear load that outputs, for example, interharmonics to the electrical network. For simplicity of the example, we assume that the instantaneous values of the current i ''( m ) are distorted by the interharmonic frequency f _and = 135 Hz with the amplitude I _and = 0.15⋅ I '' and zero initial phase:

The oscillogram of the current i ''( m ) for a given variant of interharmonic distortion is illustrated in Fig. 3(b). Then the expression for the instantaneous voltage values from the side of system-2 takes the following form:

Thus, the calculated expression for the damage site in the presence of flicker and interharmonics of frequency f _and = 135 Hz corresponds to the equality:

Setting numerical values in accordance with Table 1 and expression (27) at k = 0.15 leads to the following results.

Table 1. Values of calculated parameters

Parameter I '
(A) I ''
(A) f
(Hz) t _d
(With) L
(H) R
(ohm) f _and
(Hz) U
(IN) n Meaning 13908.15 9030.13 50 0.0025 0.0643 12.5 135 29323.83 0.5

- at m = 20; n _and (20) = 0.486; Δ x = l ⋅( n - n _and ) = 50⋅(0.5 - 0.486) = 0.7 (km);

- at m = 60; n _and (60) = 0.526; Δ x \u003d l ⋅ ( n - n _and ) \u003d 50 ⋅ (0.5 - 0.526) \u003d -1.30 (km).

The analysis of the obtained results of calculations shows that the errors of the OMKZ of power transmission lines can have both positive and negative signs, they are distributed unevenly with respect to the use of different points in time.

The accuracy of calculating the distance to damage is set based on the requirements of operational practice. The operational and technological personnel of electrical networks consider the accuracy of calculating the distance to the fault site to be high if the calculation error does not exceed the length of one span of the line [for example, Kulikov A.L., Obalin M.D. Adaptive determination of the location of damage to power lines by the parameters of the emergency mode. Part 2. - M.: NTF "Energoprogress", 2019. - 80 p.]. The length of one span of an overhead power transmission line with a voltage of 110 kV is 0.15-0.2 km, so the required accuracy of calculating the distance to the damage site is Δ x = 0.2 km.

Thus, in most calculated cases, subject to deviations of currents and voltages from the sinusoidal form, the prototype method does not allow calculating the distance to the fault with the required accuracy.

The objective of the invention is to improve the accuracy of determining the location of a short circuit on an overhead power line with unsynchronized measurements from its two ends under conditions of deviations of currents and voltages from a sinusoidal shape, taking into account the distributed capacitance of the overhead power line.

The task is achieved by a method for determining the location of a short circuit on an overhead power line with unsynchronized measurements from its two ends, having a length l , active resistance R and inductance L , connecting two supply systems, in which phase currents not synchronized in time are measured from both ends of the line and voltage during a short circuit, determine the damaged phases, determine the relative value of the distance to the place of the short circuit n and the physical distance to the place of the short circuit from the end of the line with the index ' by the expression l ' = n l , measure from both ends of the line (' - one end of the line, "- the second end of the line) instantaneous values of phase currents ( i ' _A , i ' _B , i ' _C ), ( i " _A , i " _B , i " _C ) and voltages ( u ' _A , u ' _B , u ' _C ), ( u " _A , u " _B , u " _C ) during a short circuit, receive oscillograms of currents and voltages, combine the oscillograms from both ends of the line along the cut of the beginning of the short circuit, select in the interval of two to ten periods from the beginning of a short circuit on the oscillograms of the current and voltage of the damaged phase, instantaneous values, calculate the derivatives of the currents with respect to time di '/ dt , di "/ dt .

According to the proposal, when determining the location of a short circuit, the values of the transverse capacitance C of the overhead power line are used, they are selected at an interval of two to ten periods from the beginning of the short circuit and arrays are formed from M ( m = 1, ..., M ) consecutive instantaneous values for each of the current oscillograms and faulty phase voltages i '( m ), i ''( m ), u '( m ), u ''( m ), determine the first derivatives of the currents with respect to time for the selected arrays of instantaneous values di '( m )/ dt _m , di ''( m )/ dt _m , as well as the first and second derivatives of voltages du '( m )/ dt _m , du ''( m )/ dt _m , d ² u '( m )/ dt ² _m , d ² u ''( m )/ dt ² _m , voltage matrices U 1 and U 2 are formed using active resistance R , inductance L and transverse capacitance C of the power line, instantaneous values of arrays of currents, voltages i '( m ), i ''( m ), u '( m ), u ''( m ), as well as the calculated instantaneous values of derivative currents and voltages di '( m )/ dt _m , di ''( m )/ dt _m , du '( m )/ dt _m , du ''( m )/ dt _m , d ² u '( m )/ dt ² _m , d ² u ''( m )/ dt ² _m according to the expressions:

U1 = ,

U2 = ,

Where

U1( m ) = R ⋅( i '( m ) + i ''( m )) + L ⋅( di '( m )/ dt _m + di ''( m )/ dt _m ) - R ⋅ C ( du '( m )/ dt _m + du ''( m )/ dt _m ) -

- L ⋅ C ( d ² u '( m )/ dt ² _m + d ² u ''( m )/ dt ² _m ),

U2( m ) = [ u '( m ) - u ''( m )] + i ''( m )⋅ R + L ⋅ di ''( m )/ dt _m - R ⋅ C du ''( m ) / dt _m - L ⋅ C d ² u ''( m )/ dt ² _m ,

determine the relative value of the distance to the short circuit n using the voltage matrices U 1 and U 2 and the least squares method by the expression:

n = ( U1 ^T U1 )- ¹ ⋅ U1 ^T ⋅ U2 ,

according to the obtained relative value of the distance to the place of the short circuit n, the distance to the place of the short circuit from the side of the end with the index ' is determined.

In FIG. 1 shows a single-line equivalent circuit of an overhead transmission line in relation to the calculation example, having a length ( l = 50 km) 1, phase active resistance ( R ) 2 and inductance ( L ) 3, connecting buses 4 and 5 of two systems 6 and 7. The line shows short circuit 8 behind the contact resistance ( Z _P ) 9 at a distance ( x ) 10 from one end of the line. If a short circuit occurs on the line, current ( i ') flows through it from bus 4 and current ( i ") from bus 5. At the same time, on buses 4 and 5, instantaneous values of phase currents (not synchronized in time) are measured from both ends of the line ( i ' _A , i ' _B , i ' _C ), ( i " _A , i " _B , i " _C ) and voltages ( u ' _A , u ' _B , u ' _C ), ( u " _A , u " _B , u " _C ) at the time of the short circuit.

In FIG. 2 shows a single-line equivalent circuit of an overhead transmission line, similar to FIG. 1, but taking into account its distributed capacitance.

In FIG. 3 shows current waveforms distorted by: a) flicker; b) interharmonic frequency f = 135 Hz.

Fig. 4 illustrates voltage waveforms corresponding to discrete sample values of the matrices U 1 and U 2.

The method for determining the location of a short circuit on an overhead power line with unsynchronized measurements from its two ends is implemented as follows.

To implement the method, both under conditions of deviations of currents and voltages from a sinusoidal form, and in the absence of such deviations, instantaneous values of phase currents ( i ' _A , i ' _B , i ' _C ), not synchronized in time, are measured from both ends of the line, ( i '' _A , i '' _B , i '' _C ) and voltages ( u ' _A , u ' _B , u ' _C ), ( u '' _A , u '' _B , u '' _C ) at the moment short circuit, determine the damaged phases, obtain oscillograms of currents and voltages, combine the oscillograms from the two ends of the line along the cut of the beginning of the short circuit, select at the interval of two to ten periods from the beginning of the short circuit and form arrays from M ( m = 1, ..., M ) successive instantaneous values for each of the waveforms of the current and voltage of the damaged phase i '( m ), i ''( m ), u '( m ), u ''( m ), determine the first derivatives of the currents with respect to time for the selected arrays of instantaneous values di '( m )/ dt _m , di ''( m )/ dt _m , as well as the first and second derivatives of voltages du '( m )/ dt _m , du ''( m )/ dt _m , d ² u '( m )/ dt ² _m , d ² u ''( m )/ dt ² _m , voltage matrices U 1 and U 2 are formed using active resistance R , inductance L and transverse capacitance C of the transmission line, instantaneous values of arrays of currents, voltages i '( m ), i ''( m ), u '( m ), u ''( m ), as well as the calculated instantaneous values of derivatives of currents and voltages di '( m )/ dt _m , di ''( m )/ dt _m , du '( m )/ dt _m , du ''( m )/ dt _m , d ² u '( m )/ dt ² _m , d ² u ''( m )/ dt ² _m according to the expressions:

U1 = ,

U2 = ,

Where

U1( m ) = R ⋅( i '( m ) + i ''( m )) + L ⋅( di '( m )/ dt _m + di ''( m )/ dt _m ) - R ⋅ C ( du '( m )/ dt _m + du ''( m )/ dt _m ) -

- L ⋅ C ( d ² u '( m )/ dt ² _m + d ² u ''( m )/ dt ² _m ),

U2( m ) = [ u '( m ) - u ''( m )] + i ''( m )⋅ R + L ⋅ di ''( m )/ dt _m - R ⋅ C du ''( m ) / dt _m - L ⋅ C d ² u ''( m )/ dt ² _m ,

determine the relative value of the distance to the short circuit n using the voltage matrices U 1 and U 2 and the least squares method by the expression:

n = ( U1 ^T U1 )- ¹ ⋅ U1 ^T ⋅ U2 ,

according to the obtained relative value of the distance to the place of the short circuit n, the distance to the place of the short circuit from the side of the end with the index ' is determined.

Let us explain the above signal processing operations necessary to implement the proposed method for determining the location of a short circuit on an overhead power line with unsynchronized measurements from its two ends.

We introduce a discrete time variable and group the components of the expression, writing it (2) as:

where m is a discrete variable corresponding to the time points of current and voltage oscillogram readings; m = 1, 2, ..., M , where M is the number of readings of the current (voltage) components selected for the implementation of digital signal processing operations of the OMKZ transmission line method.

Let us denote by column matrices U1 and U2 the expressions for different moments of discrete time m , which are part of equality (28):

Thus, for the implementations of expression (28) at different times m , the following matrix expression is valid

To find a more accurate n that characterizes the location of the OMKZ of an overhead power transmission line with unsynchronized measurements from its two ends under conditions of deviation of currents and voltages from a sinusoidal shape, we will use the least squares method. In this case, a complete array of M discrete realizations of currents and voltages (expression (31)) is used, which ensures minimization of the OMCF error under the influence of random factors, in contrast to the prototype method, which uses only individual "time slices" of the oscillograms.

We write equality (31) as

where e is a column vector of measurement errors associated with distorted instantaneous values of current and voltage oscillograms.

The problem of minimizing errors in calculating the value of n is reduced to minimizing the square of the norm of the error vector e , that is:

We transform equality (33) using substitution (32):

Then:

To find the minimum, it is necessary to calculate the partial derivative with respect to n of equation (35) and equate it to zero:

From the last relation, we find the desired value of n according to the expression:

Thus, the estimation of the relative distance to the place of the OMKZ of an overhead power transmission line with unsynchronized measurements from its two ends under the conditions of deviation of currents and voltages from a sinusoidal form using the least squares method is reduced to finding the results of the matrix product using expression (37).

Let us illustrate with an example the calculation of the distance to the OMCF site in relation to the above conditions of current distortion i '( m ) by the flicker (Fig. 3(a)), as well as i ''( m ) by the interharmonic frequency f _u = 135 Hz with amplitude I _u = 0 ,15⋅ I '' (Fig. 3(b)).

According to the previously obtained expressions (22)-(27), the introduced delay t _from the matrices U1 and U2 take the form:

Where:

U1( m ) = ( I '⋅(1 - k ⋅ rnd ( m )) + I '')⋅[ R ⋅ sin (2π⋅ f ⋅( t _s + m ⋅ t _d ))+ L ⋅ cos (2π ⋅ f ⋅( t _h + m ⋅ t _d ))] +

+ 0.15⋅ I ''⋅[ R ⋅ sin (2π⋅ f _and ⋅( t _s + m ⋅ t _d ))+ L ⋅ cos (2π⋅ f _and ⋅( t _s + m ⋅ t _d ))] - R ⋅ C ( du '( m )/ dt _m + du _and ''( m )/ dt _m ) -

- L ⋅ C ( d ² u '( m )/ dt ² _m + d ² u _and ''( m )/ dt ² _m );

U2( m ) = [( u '( m ) - u _and ''( m ))+ I ''⋅[ R ⋅ sin (2π⋅ f ⋅( t _s + m ⋅ t _d ))+ L ⋅ cos ( 2π⋅ f ⋅( t _h + m ⋅ t _d ))] +

+ 0.15⋅ I ''⋅[ R ⋅ sin (2π⋅ f _and ⋅( t _s +m ⋅ t _d ))+ L ⋅ cos (2π⋅ f _and ⋅( t _s +m ⋅ t _d ))] ] - R ⋅ C du _and "( m )/ dt _m - L ⋅ C ⋅ d ² u _and "( m )/ dt ² _m ;

u '( m ) = U + I '⋅(1 - k ⋅ rnd ( m ))⋅[ n ⋅ R ⋅ sin (2π⋅ f ⋅( t _h + m ⋅ t _d )) + n ⋅ L ⋅ cos ( 2π⋅ f ⋅( t _c + m ⋅ t _d ))];

u _and ''( m ) = U + (1 - n )⋅ R ⋅[ I ''⋅ sin (2π⋅ f ⋅( t _h + m ⋅ t _d )) +0.15⋅ I ''⋅ sin ( 2π⋅ f _and ⋅( t _h + m ⋅ t _d ))] +

+(1 - n )⋅ L ⋅[ I ''⋅ sin (2π⋅ f ⋅( t _s + m ⋅ t _d )) + 0.15⋅ I ''⋅ sin (2π⋅ f _and ⋅( t _s + m ⋅ t _d ))],

du '( m )/ dt _m = ( u '( m +1) - u '( m -1))/(2⋅ t _d ), du _and ''( m )/ dt _m = ( u _and '' ( m +1) - u _and ''( m -1)) /(2⋅ t _d ),

d ² u '( m )/ dt ² _m =( u '( m +1)-2⋅ u '( m )+ u '( m -1))/( t _d ) ² , d ² u _and '' ( m )/ dt ² _m =( u _and ''( m +1)-2⋅ u _and ''( m )+ u _and ''( m -1))/( t _d ) ² .

The numerical values of the parameters for the calculations are presented in Table 1,

The specific transverse capacitive component Csp of a 110 kV transmission _line (Fig. 2) was selected according to [Panova E.A., Albrecht A.Ya. Refined specific electrical parameters of double-circuit power lines 110 kV for remote determination of the location of damage // Elektrotekhnicheskie sistemy i kompleksy. 2016. No. 4 (33). pp.35-40. doi: 10.18503/2311-8318-2016-4(33)-35-40]. The capacity of the transmission line C for calculations was

In FIG. 4 shows the voltage waveforms corresponding to the discrete sample values of the matrices U 1 and U 2.

To implement the proposed method, it is necessary to set the value of the variable M , which determines the number of discrete readings of currents and voltages involved in the calculations by the least squares method. It is advisable to choose the value of M a multiple of N - the number of readings per period of industrial frequency (for example, for the example under consideration N = 20), and the full sample of consecutive instantaneous values of currents and voltages, as in the prototype method, should be carried out within the interval of two to ten periods from the beginning of a short circuit for the current and voltage waveforms of the damaged phase. We choose the value M = 80, and set the delay relative to the beginning of the short circuit, as before, equal to t _s = 9⋅ t _d = 0.0225 s, k = 0.15.

Calculations were made of the distance to the place of a short circuit under conditions of deviation of currents and voltages from a sinusoidal form with the substitution of the values of M , t _s , as well as the values of the parameters in expressions (38), (39) in the MATCAD software package. The final results of calculations using expression (37) made it possible to obtain an updated distance to the short circuit point:

n _and = ( U1 ^T U1 ) - ¹ ⋅ U1 ^T ⋅ U2 = 0.499.

Taking into account the previously performed calculations, the error of the OMKZ PTL will be:

Δ x = l ⋅( n - n _and ) = 50⋅(0.5 - 0.499) = -0.05 (km)

or (with the length of the transmission line l = 50 km) the value -0.05/50 = - 0.1% of the length of the transmission line.

Thus, the proposed method for determining the location of a short circuit on an overhead power line with non-synchronized measurements from its two ends provides a highly accurate calculation of the distance to the fault location of an overhead power line under conditions of current and voltage deviations from a sinusoidal shape and meets the accuracy requirements of operational practice.

It should be noted that the proposed OMKZ method for power transmission lines can be implemented based on the use of modern digital relay protection terminals interconnected by a communication channel for exchanging current and voltage oscillograms [for example, https://ekra.ru/]. The requirements for modern digital relay protection terminals [Federal Grid Company of the Unified Energy System PJSC STO 56947007-29.120.70.241-2017 "Technical requirements for relay protection microprocessor devices"] determine the mandatory implementation of the function of determining the fault location on power lines. It is also possible to implement the proposed method in the network control centers of electrical network enterprises, since the calculation of the distance to the place of damage is assigned to the operational and technological personnel who manage overhead power lines. At the same time, oscillograms of emergency currents and voltages are collected remotely from the emergency event recorders (relay protection terminals) and the subsequent calculation of the distance to the fault site is performed using computer technology.

Claims (10)

A method for determining the location of a short circuit on an overhead power line during unsynchronized measurements from its two ends, having a length l , active resistance R and inductance L , connecting two supply systems, in which phase currents and voltages not synchronized in time are measured from both ends of the line during short circuit, determine the damaged phases, determine the relative value of the distance to the short circuit n and the physical distance to the short circuit from the end of the line with the index ' by the expression l ' = n l , measure from two ends of the line (' - one end of the line , " - the second end of the line) instantaneous values of phase currents ( i ' _A , i ' _B , i ' _C ), ( i " _A , i " _B , i " _C ) and voltages ( u ' _A , u ' _B , u ' _C ), ( u " _A , u " _B , u " _C ) during a short circuit, receive oscillograms of currents and voltages, combine the oscillograms from both ends of the line along the cut of the beginning of the short circuit, select at an interval of two to ten periods from the beginning of the short short circuits on the oscillograms of the current and voltage of the damaged phase instantaneous values, calculate the derivatives of the currents with respect to time di '/ dt , di "/ dt , characterized in that when determining the location of the short circuit, the values of the transverse capacitance C of the overhead power line are used, are selected in the interval of two - ten periods from the beginning of the short circuit and form arrays of М ( m = 1, …, М ) consecutive instantaneous values for each of the current and voltage waveforms of the damaged phase i '( m ), i ''( m ), u '( m ) , u ''( m ), determine the first derivatives of currents with respect to time for the selected arrays of instantaneous values di '( m )/ dt _m , di ''( m )/ dt _m , as well as the first and second derivatives of voltages du '( m ) / dt _m , du ''( m )/ dt _m , d ² u '( m )/ dt ² _m , d ² u ''( m )/ dt ² _m , form the stress matrices U 1 and U 2 using active resistance R , inductance L and transverse capacitance C of the power line, instantaneous values of arrays of currents, voltages i '( m ), i ''( m ), u '( m ), u ''( m ), as well as the calculated instantaneous values of derivatives currents and voltages di '( m )/ dt _m , di ''( m )/ dt _m , du '( m )/ dt _m , du ''( m )/ dt _m , d ² u '( m )/ dt ² _m , d ² u ''( m )/ dt ² _m according to the expressions: U1 = , U2 = , Where U1( m ) = R ∙( i '( m ) + i ''( m )) + L ∙( di '( m )/ dt _m + di ''( m )/ dt _m ) − R ∙ C ( du '( m )/ dt _m + du ''( m )/ dt _m ) – − L ∙ C ( d ² u '( m )/ dt ² _m + d ² u ''( m )/ dt ² _m ), U2( m ) = [ u '( m ) – u ''( m )] + i ''( m )∙ R + L ∙ di ''( m )/ dt _m − R ∙ C du ''( m ) / dt _m − L ∙ C d ² u ''( m )/ dt ² _m , determine the relative value of the distance to the short circuit n using the voltage matrices U 1 and U 2 and the least squares method by the expression: n = ( U1 ^T U1 ) ⁻¹ ∙U1 ^T ∙U2 , according to the obtained relative value of the distance to the place of the short circuit n, the distance to the place of the short circuit from the side of the end with the index ' is determined.