Error analysis for differential-equation algorithms

The property identified in describing the frequency response of the differential-equation algorithms, i.e. that the correct R and L are obtained as long as v(t) and i(t) satisfy the differential equation, is a major strength of these algorithms.

The exponential offset in the current satisfies the equation if the correct values of R and L are used, so that it need not be removed. We have seen in Section 5.2 that the voltage seen by the relay just after fault inception has non-fundamental frequency components caused by the power system itself. If the faulted line can accurately be modeled as a series R-L line, then the current will respond to these non-fundamental frequency components in the voltage (according to the differential equation) and no errors in estimating R and L will result. As long as the R-L representation is correct, then the only sources of error are in the measurement of v(t) and i(t). Transducer errors, A/D errors, and errors in the anti-aliasing filters contribute portions of the measured voltage and current which do not satisfy the differential equation and which will cause errors in the estimates. The frequency

response in Figure 5.13 represents the response of the algorithm to such error signals in the measured voltage. To examine the effect of such errors in both voltage and current, let the measured current and voltage be denoted by im(t) and vm(t) where

i_m(t) = i(t) + εi(t) (5.47)

v_m(t) = v(t) + εv(t) (5.48)

and v(t) and i(t) satisfy the differential equation. The current im(t) satisfies the differential equation

Ri_m(t) + Ldim(t)

dt = v(t) + Rεi(t) + Ldεi(t)

dt (5.49)

and the current i_m(t) and voltage v_m(t) are related by:

im(t) + Ldi_m(t)

dt = v_m(t) + Rεi(t) + Ldεi(t)

dt − εv(t) (5.50) The measured voltage and current then satisfy a differential equation with an error term which is made up of the voltage error plus a processed current error term. The latter is similar to the error obtained at the output of a mimic circuit with the current error as an input (see Figure 5.16). Figure 5.13 then can finally be interpreted as the response of the algorithm to the entire error term in Equation (5.50), assuming that the sum of the last three terms on the right hand side of the equation are thought of as a signal cos (ωt); the last three terms in (5.50) also make it clear that error signals that satisfy the differential equation do not contribute a net error to Equation (5.50).

There is an additional subtlety in the three-sample differential-equation algorithm.

The denominator of Equations (5.45) and (5.46) is not a constant but rather a func-tion of time which has maxima and minima. The denominator can be simplified to

(ik+1+ i_k)(ik+2− i_k+1) − (ik+2+ i_k+1)(ik+1− i_k) = −2(i²_k+1− i_ki_k+2) If we assume that

i_k+1 = I cos(ωot) − I cos(ωot_o) e−^{− RL}(t − to)

and use 12 samples per cycle (θ = 30^o) with a line time constant of 40 ms, then 2

i²_k+1− ikik+2

= I²



0.5 − 0.5384 cos(ωot + 7.41^◦) cos(ωoto) e^{− RL}(t − to) The denominator is shown in Figure 5.14 as a function of the time of the k^th sample for the case of maximum offset (ω0t0= 0). When the denominator is small

0

t 0.5 I²

Figure 5.14 The denominator of Equations (5.45) and (5.46) for maximum offset in the current

the error terms from Equation (5.50) are amplified. In the limit as the denominator becomes zero the estimates are unacceptably sensitive to even the smallest error terms. A counting algorithm will deal with such poor estimates by indexing down because the estimate is not in the characteristic. The net effect is then only a delay in issuing the trip signal. Since the denominator is a constant if the offset is absent, it can be seen that the supposed immunity of the differential-equation algorithms to offset is a bit of myth.

The preceding has assumed that the actual voltage and current satisfy the differential equation but that there are errors made in the measurement process. As seen in Section 5.2, the largest contributions to errors in the waveform algorithms are non-fundamental frequency signals from the power system itself. If the differential-equation algorithms were immune to these signals, there would be much to recommend them. A somewhat more realistic model of the faulted line can be used to investigate the impact of these power system signals on such algorithms. The circuit shown in Figure 5.15 includes the shunt capacitance of the transmission line at the relay terminals. The voltage source v(t) represents the voltages seen in Figure 5.5 made up of a fundamental plus non-fundamental components whose frequency and phase are unpredictable. The current i(t) is the current measured by the relay. The

L C

{ i(t) - C }dv dt i(t)

v(t)

Figure 5.15 A single-phase line model with shunt capacitance

actual relationship between the measured voltage and current is given by v(t) = Ri(t) + Ldi(t)

dt − RCdv(t)

dt − LCd²v(t)

d²t (5.51)

If the algorithm of Equations (5.45) and (5.46) is used then the last two terms in Equation (5.51) must be regarded as error terms. The magnitude of these terms is a function of fault location and of the frequency of the signals in v(t). The dependence on fault location is quadratic, reaching a maximum for a fault at the end of the line. For faults at the far end of a long high-voltage line, these terms can be quite substantial, especially if high frequencies are included in v(t). On the other hand, the terms RC and LC are small for close-in faults and on lower voltage lines.

A solution to this problem, of course, is to include the capacitance in the sys-tem model. This has been proposed and central differences have been used to approximate the derivatives in Equation (5.51).¹¹ It seems desirable to integrate Equation (5.51) once to recover some similarity with Equations (5.45) and (5.46).

If this is done and 

then a more elaborate version of Equations (5.45) and (5.46) can be obtained by considering four consecutive intervals

If Equation (5.52) is thought of as a partitioned matrix in the form

M₁₁ M12

where p is the parameter vector made up of R and L, then the estimate can be formed by solving the second set of equations and substituting into the first set to obtain:¹¹ Rˆ Equation (5.54) represents a formidable amount of computation given that the matri-ces involved are formed of measured voltages and currents. It is not clear that the computation is warranted since the pi-section representation is still an approximation to the transmission line model.