DOI 10.1007/s10749-016-0688-5 Power Technology and Engineering
Vol. 50, No. 2, July, 2016
ESTIMATION OF THE TRANSMISSION LINE PARAMETERS USING A GRID MODEL S. K. Kakovskii,1 A. A. Nebera,2 M. A. Rabinovich,1 and P. N. Kazakov2 Translated from Élektricheskie Stantsii, No. 2, February 2016, pp. 42 – 53.
The parameters of an equivalent circuit of power transmission lines in the UNPG single-line model are estimated from synchrophasor measurements of nodal voltages and currents. Systematic and random errors in PMU measurements that lead to errors in the estimates of the transmission line parameters are considered. Keywords: electric power system; measurement error; parameter estimation; UNPG model; correlation; systematic error; random error.
The transmission line parameters in models of power systems are needed for solving the following problems: state estimation, steady-state and transient analysis, state optimization, calculation of short-circuit currents, setting up of protection and control relays, etc. These parameters depend on many factors; therefore, they should be updated in real time, directly before solving a problem. As shown in [1, 2], the use of the transmission line parameters determined under normal conditions may result in a noticeable (from 5% to 200 – 300%) difference from the real values and the type of errors in measurements. The accuracy of the transmission line parameters is usually from 5 to 10% [2], which is insufficient to solve the problems mentioned above. Synchrophasor measurements of the transmission line parameters were carried out by Russian and foreign experts [2, 3] using mainly state estimation methods and taking unknown parameters of the grid into account. The detailed analysis of errors is difficult in these conditions. A stimulus for the present work was the possibility of analyzing errors and their causes in real measurements of the transmission line parameters. Problem Statement. The task is to analyze the possibility of estimating the transmission line parameters (resistance R, reactance X, shunt conductance G between conductors and ground due to corona, and shunt capacitance B between conductors and ground) in an equivalent Ð-circuit using synchrophasor measurements with additive random and systematic errors. We will consider two types of measurement errors: random errors with zero and nonzero mean. The latter errors result from systematic errors in synchrophasor measurements that are due mainly to the nonlinearities in instrument transformers (IT) and malfunctions in telemetry instruments. Systematic errors are the most difficult to separate from the measured values of parameters because their statistical
The parameters of power transmission lines (PTLs) needed for calculations of electric systems can be determined from conventional telemetry. However, since the measurements are not fully synchronized and there are other types of errors, the parameter values are not very accurate. For example, since the update time for conventional telemetry data is on the order of 10 sec, the root-mean-square error in measurements of real power flows of the order of 2000 MW is about 0.5%, and it is due to desynchronization alone. The lower the measured power flow, the greater the desynchronization error. The same is true for other measured parameters. The software used for timestamping telemetry data reduces, yet does not eliminate the desynchronization error. Using phasor measurement units (PMU), which are devices that measure the magnitude and phase angle of voltages and currents using a common time source for synchronization, allows almost complete elimination of the desynchronization error, improving (compared with conventional measurements) the accuracy of the estimates of the transmission line parameters, and analysis of the errors. Here we will theoretically estimate the transmission parameters for some types of systematic errors in PMUs. To determine the additive random errors in measurements, the transmission line parameters were determined using a model of the UPS of Russia. Unlike full-scale tests conducted at the plant, modeling allows obtaining statistical characteristics of errors in the transmission line parameters, which are usually unavailable in conventional measurements. We will present some of these characteristics below. Thus, we will deal with the errors in PMU measurements of operating-state parameters and the errors in the estimates of the transmission line parameters found with the model. 1 2
JSC “R&D Center at Federal Grid Company of United Energy System,” Moscow, Russia; e-mail:
[email protected] JSC “RTSoft,” Moscow, Russia.
224 1570-145X/16/5002-0224 © 2016 Springer Science + Business Media New York
Estimation of the Transmission Line Parameters Using a Grid Model
characteristics are very similar. The nonlinearity of ITs can be offset if their characteristics are known; however, this method is rather labor-consuming and rarely used. In addition to systematic errors, measurements can involve other types of errors: time sampling and amplitude quantization errors, channel and telemetry interference, etc. Nonsynchronous measurement of the parameters is a special type of error. All these errors can also be regarded as random. The important tasks of the present work are: — to analyze the adequacy of our method for estimating the transmission line parameters for steady-state and transient conditions; — to estimate the accuracy of instrument transformers required to obtain acceptable values of the transmission line parameters; — to determine the accuracy of the estimation of the transmission line parameters from measurements of operating-state parameters with systematic errors; — to assess the effect of the random errors in PMUs on the accuracy of measurements of the transmission line parameters; — to estimate the necessary time of averaging the estimates of the transmission line parameters. It is impossible to accomplish all these tasks within one work; therefore, some of them (such as analysis of the effect of random errors in PMUs on the accuracy of estimates) have been only formulated. Modeling Conditions. For the purpose of modeling, we used a model of the UPS of Russia consisting of 1964 nodes and 3300 PTLs and transformers. During the modeling, the transmission line parameters were estimated for the singlecircuit PTL between the Zagorskaya Pumped-Storage Power Plant (PSPP) and the Kostroma Gas-Fired Power Plant (GFPP), as an example, taking into account the systematic and random errors in synchrophasor measurements. This PTL is represented by lumped parameters in the model, as in the COSMOS Operating System. This PTL has been chosen because there are relevant experimental data. These measurements were conducted using MIP-01-00 PMUs (accuracy class 0.2) made by the RTSoft Company. The measurements of the parameters of one PTL affect in no way the measurements of the parameters of the other PTLs because the state of the grid remains the same and only synchrophasor measurements change because of the allowance made for the errors. Both steady-state and transient modeling was conducted on a standard computer with suffi-
225
R
G 2
B 2
X
G 2
B 2
Fig. 1. Equivalent circuit of power transmission line.
cient resources. The cycle of estimating the parameters of all PTLs using the model was 20 sec. The transmission line parameters were estimated in three ways: — from synchrophasor measurements of voltages and currents with errors in steady-state conditions; — from synchrophasor measurements of voltages and currents with errors during a transient in the grid; — from synchrophasor measurements of voltages and currents with random fluctuations in nodal loads using the grid model (and, certainly, with errors in measurements of nodal voltages and currents). In the latter case [4], the fluctuations of the nodal loads were independent and normally distributed and their correlation time constant was 3 min. Measurement errors were generated as independent normal variables. Systematic errors were modeled as constant biases (of the magnitude and phase of voltage or current). The transmission line parameters were estimated for power flows corresponding to low, medium, and high load on the PTL (130, 393, and 967 MW, respectively). Table 1 summarizes the parameter values at the ends of the PTL under consideration. Negative values of active or reactive power at a node mean its inflow. The power flows were specified as power imbalances at nodes at the ends of the PTL. The input parameters of the 500 kV overhead line between the Zagorskaya PSPP and the Kostroma GFPP: R = 5.82 Ù, X = 66.75 Ù, G = 9.8 ìS, B = 983.0 ìS. They were assumed constant (the real transmission line parameters depend on the environmental and operating conditions). For the purpose of modeling, we used a single-line equivalent pi circuit of the PTL (Fig. 1) represented as a two-port network (Fig. 2). Since equivalent pi circuits of PTLs are used in the majority of software systems (RASTR, COSMOS, MUSTANG), it makes no sense to use other equivalent circuits.
TABLE 1. Parameters of Three Operating States Number
P1, MW
P2, MW
Q1, MVar
Q2, MVar
U1, kV
U2, kV
Phase difference
1 2 3
–130.0 –393.5 –967.2
130.4 399.4 991.1
114.9 74.1 77.8
–145.1 –150.7 –99.4
518, 8 518.3 518.3
518.3 518.8 518.8
1.48° 5.66° 14.12°
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U1
I1
I12
I11
Y
Z
I21
I2
Y
I22
simplified Park-Gorev equations (i.e., algorithms used in the well-known MUSTANG system). In the model, additive errors were added to the exact values of magnitudes and phases of the PMU-measured parameters:
U2
u1 = Um1 + ÄU1; u2 = Um2 + ÄU2; i1 = Im1 + ÄI1; i2 = Im2 + ÄI2;
Fig. 2. Two-port network.
a1 = á1 + Äá1; a2 = á2 + Äá2; The admittance and impedance of the two-port network: Y = (I1 – I2)/(U1 + U2);
(1)
Z = (U1 – U2)/(I2 + YU2) = = (U1 – U2)/[I2 + (I1 – I2)/(U1 + U2)U2].
(2)
b1 = â1 + Äâ1; b2 = â2 + Äâ2, where ÄU, ÄI, Äá, Äâ are random measurement errors, which are independent normal random variables with given mean and variance: ÄU1 = ÄUn1 + ÄUs1; ÄU2 = ÄUn2 + ÄUs2;
The transmission line parameters were estimated with and without regard to the errors in measurements. The measured parameters are represented in the following complex form [5 – 7]:
Äá1 = Äán1 + Äás1; Äá2 = Äán2 + Äás2;
U1(t ) = Um1(t ) exp [ j (ùt + á1(t ))];
Äâ1 = Äân1 + Äâs1;Äâ2 = Äân2 + Äâs2,
U2(t ) = Um2(t ) exp [ j(ùt + á2(t ))];
where ÄUni, ÄIni, Äáni, Äâni are normal random errors with zero mean and ÄUsi, ÄIsi, Äási, Äâsi are systematic errors (i.e., average values of measurement errors), i = 1, 2. Measurement of Grid Parameters Using the Grid Model. Figure 3 shows a control panel for estimating the transmission line parameters using a grid model and setting test parameters according to formulas (1) and (2). This panel allows the user not only to get instantaneous values of the measured transmission line parameters, but also to visualize how they vary with time (Graph button). The level of random errors (root-mean-square values) in measurements of the magnitudes and phases of voltages at the nodes adjoining the branch of interest and the magnitudes and phases of nodal currents is set in the Errors section of the panel. The rootmean-square values of errors are dimensional. Not only random, but also systematic errors can be set for all parameters (magnitudes and phases of nodal voltages and currents) on the control panel (Fig. 3; syst. boxes). The user can set the error level for each type of parameters. During modeling, the nodal voltages at the ends of the PTL were maintained by nodal voltage regulators and were 518.3 kV for the Zagorskaya PSPP and 518.8 kV for the Kostroma GFPP in all measurements. Estimation of the Transmission Line Parameters from Synchrophasor Measurements without Errors. The reference measurements of the transmission line parameters (R and X in Ù, and B and G in ìS) for the three values of power flows in steady state according to formulas (1) and (2) in the absence of errors are in very good agreement with the input transmission line parameters, which validates the algorithm based on expressions (1) and (2).
I1(t ) = Im1(t ) exp [ j(ùt + â1(t ))]; I2(t ) = Im2(t ) exp [ j(ùt + â2(t ))]; where ù is the circular frequency of alternating current; Um1, Um2, Im1, Im2 are the amplitudes of voltages and currents at the ends of the PTL; á1, á2, â1, â2 are the phases of voltages and currents at the ends of the PTL about an axis synchronously rotating with circular frequency ù. The parameters can be represented in terms of envelope and phase: U1(t ) = Um1(t ) exp [ já1(t )];
(3a)
U2(t ) = Um2(t ) exp [ já2(t )];
(3b)
I1(t ) = Im1(t ) exp [ jâ1(t )];
(3c)
I2(t ) = Im2(t ) exp [ jâ2(t )];
(3d)
For narrow-band signals, the envelope and phase appear slower functions than ù. Expressions (3) are valid for arbitrary signals; however, their envelope and phase can be not only slow, but also arbitrary functions [4, 7, 8]. Formulas (3) are the basis for modeling synchrophasor measurements. In steady-state conditions at a frequency of 50 Hz, the amplitudes and phases of the parameters in the system of equations (3) can be considered constant. Modeling of Operating States. Transient conditions were modeled with the help of RETREN simulator using
ÄI1 = ÄIn1 + ÄIs1; ÄI2 = ÄIn2 + ÄIs2;
Estimation of the Transmission Line Parameters Using a Grid Model
227
Fig. 3. Panel for setting the level of errors and estimating the transmission line parameters.
The measurements conducted without errors in the magnitudes and phases of voltages in transient conditions with random fluctuations in the nodal loads gave quite accurate (1% for R and 0.02% for X ) estimates of the transmission line parameters for a power flow of 132 MW. Such measurements (without errors) at a nonrated frequency (frequency departure 0.1 Hz) in steady-state conditions did not lead to significant errors in the estimates of the transmission line parameters. Measurements of the transmission line parameters during a transient with stronger disturbances at grid nodes (such as emergency shutdown of a unit at the Surgut-2 GasFired Power Plant) led to more significant errors in the estimates (3% for R and 0.1% for X ). Thus, when synchrophasor measurements do not have errors over the whole range of real power flows and in all states (transients, abnormal states, random fluctuations of load), the transmission line parameters estimated with formulas (1) and (2) are in very good agreement with the input parameters for the grid model. Systematic Errors. The systematic errors introduced by instrument transformers and telemetry equipment are the most critical for the estimation of the transmission line parameters from measurements of the operating-state parameters. ITs have not only nonlinear, but also linear distortions determined by its frequency response. Such distortions [5],
which mainly cause a lag in measurements, can be reduced with an equalizing filter. The measured values of the transmission line parameters are known to be affected by the operating conditions, environmental temperature, humidity, etc. However, we will assume that these are not measurement errors, but rather deviations of the measured values of the transmission line parameters from the rating that should be estimated and used in operating state calculations. The systematic errors in the current and voltage magnitudes are due mainly to the nonlinearity of the frequency responses of the ITs and can be compensated for if the IT responses are known. These characteristics are not always available; therefore, we assume that the accuracy class of the ITs is known only. Modern (laser) ITs have a nearly linear frequency response, which considerably reduces the systematic error in the estimates of the transmission line parameters. Systematic errors are nonpermanent and can indirectly (through the nonlinearity of the IT) depend on the measured parameter itself. It is difficult to eliminate their effect on the estimates of the transmission line parameters by statistical treatment of single measurements. Systematic Error in the Voltage Magnitude at a Node. It is very difficult to estimate the systematic errors in measured parameters because these errors do not differ in their characteristic from the parameters Um1, Um2, Im1, Im2, á1, á2, â1, â2.
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a
b
Fig. 4. Graphs of estimates of R and X (a) and their errors (b).
Let us consider, as an example, the simple case of occurrence of a systematic error ì only in one measured parameter (say, in the voltage magnitude at one node at the end of the PTL). Let the capacitance B be known. After simple, yet tedious transformations over expressions (1), (2), and (3), we obtain the deviation of the estimate of the resistance R from R0: R – R0 = ì(I2 cos á1 cos â2 – – U2B sin á2 cos á1 + I2 sin á2 sin á1)/A,;
(4)
where A = (I2 cos â2 – U2B sin á2)2 + (I2 sin â2 + U2B cos á2)2. The error in the estimate of the reactance X can be found in a similar way: X – X0= ì(I2 sin á1 cos â2 – U2B sin á1 sin á2 – – I2 sin â2 cos á1 – U2B cos á2 cos á1)/A.
(5)
The expression for the systematic error in the voltage at the second node is similar, yet opposite in sign. When there are systematic errors in the voltage magnitudes at the ends of the PTL, the error in the estimates of the transmission line parameters appears, as a first approximation, to be the sum of these two errors. Equal systematic errors in the voltage magnitudes at the ends of the PTL almost completely cancel each other out because they have opposite signs in the estimates of the transmission line parameters. Noteworthy is the simple proportionality between the systematic error ì in the measurement of the magnitude of U at a node and the errors in the estimates of R and X. Relations (4) and (5) are plotted in Fig. 4. Figure 4a shows curves of the estimates of R and X found by formulas (4) and (5) for a wide range (–50, +50%) of systematic error in the voltage magnitude at a node, while Fig. 4b shows the deviations of R
and X from the input values for a narrower range of systematic error (–0.02, +0.02%). Figure 5 shows a parametric R-vs-X graph for systematic errors of –0.02 and +0.02% in the voltage U2 when the real power flow is 132 MW. Noteworthy is the linear behavior in these figures, though the rates of variation differs by an order of magnitude. Figures 4 and 5 lead us to the following important conclusion: for all values of systematic errors in the voltage magnitude at a node, the errors in the estimates of R and X are opposite in sign. Certainly, the real pattern of errors in the estimates of R and X because of the systematic errors in measurements is much more complex. A systematic error may also occur in the second voltage magnitude and in phase measurements. Table 2 summarizes the estimates of the transmission line parameters for several values of power flow when there is a systematic error in one of the measured voltage magnitudes. If ÄU1 = 0.1 kV (0.02%), the error in the estimate of R is 0.38 Ù for a power flow of 131.8 MW and 0.05 Ù for a power flow of 979.1 MW, i.e., the estimate of R is inversely proportional to the real power flow. The fivefold change (from 0.1 to 0.5 kV) in the systematic error at a low real power flow (131.8 MW) led (Table 2) to a considerable (fivefold) increase in the error of estimate of the resistance. The values of the other transmission line parameters remained almost the same, i.e., the error in the estimate of R is also proportional to the systematic error in the voltage magnitude at the end of the PTL. This conclusion is also true for high real power flows. If the real power flow becomes negative, the error of the estimate of R reverses sign as well. The other parameters remain almost the same, and so does the absolute value of the error. With relatively large systematic errors (0.2 – 1.0%) in the voltage magnitude, the errors in the estimates of the transmission line parameters are great as well (Table 2). For example, if the real power flow is low (131.8 MW) and the
Estimation of the Transmission Line Parameters Using a Grid Model
systematic error in the voltage magnitude is 1% of the rating, not only the error in the estimate of R is very large (approximately 400%), but also there are errors in the estimates of the other parameters: 5% for X and 1% for B. Thus, the systematic error in the voltage magnitude at a node at the end of the PTL is proportional to the errors in the estimates of the transmission line parameters. Table 2 confirms the analytical linear relationship between the errors in the estimates of R and X and the systematic error in the voltage magnitude, according to (4) and (5). Note that the reversal of the sign of the systematic error in the nodal voltage causes sign reversal in the error of the estimate of R. As mentioned above, the systematic errors have the strongest effect on the estimates (up to sign reversal in the estimate of R, which is supported by experimental data) when the real power flows in the PTL are low. In the same conditions, the changes in the estimates of X, B, and G are insignificant. Systematic Error in the Current Phase at a Node. The errors in the measurements of the parameters B and G are, as a first approximation, in linear relationship with the systematic errors in the measurements of the nodal current phase at one end of the PTL. Systematic Error in the Voltage Phase at a Node. A systematic error ä in the voltage phase at a node (ä = 0.5) leads to considerable errors in the estimates of R, X, and G when the real power flows and the error in the estimate of the capacitance of the PTL are low (Table 3). Unlike the systematic errors in the voltage magnitude, the systematic error in the voltage phase leads to errors in R and X of like sign, the error in the estimate of X being an or-
229
Fig. 5. Parametrical graph of the errors in R and X.
der of magnitude greater than this same error due to the measurement error in the voltage magnitude. When there is an error in the voltage phase, as well as in the voltage magnitude, sign reversal in the real power flow leads to sign reversal in the errors of the estimates of R and X. The errors in G and B hardly depend on the direction of the real power flow. Systematic Errors in the PMUs at Two Nodes. The systematic errors in the magnitude and phase of the nodal voltages at both ends of the PTL have, in linear approximation, an additive effect on the measured parameters. Since these errors can be of opposite signs, there may be cases where systematic errors in phasor measurements can mutually in-
TABLE 2. Estimates of the Transmission Line Parameters when there is a Systematic Error in U1 Power flow, MW
ÄU1,, kV
R, Ù
X, Ù
B, ìS
G, ìS
ÄU1,, kV
R, Ù
X, Ù
B, ìS
G, ìS
131.8 –131.8 396.4 –396.4 979.1 –979.1 131.8 396.4 979.1 —
0.1 0.1 0.1 0.1 0.1 0.1 1.0 1.0 1.0 0.0
5.44 6.21 5.69 5.95 5.77 5.87 2.00 4.55 5.32 5.82
66.80 66.73 66.77 66.75 66.76 66.75 67.26 66.94 66.86 66.75
982.9 982.9 982.9 982.9 982.9 982.9 982.0 982.0 982.0 983.0
9.82 9.83 9.82 9.83 9.82 9.84 9.80 9.77 9.70 9.83
0.5 0.5 0.5 0.5 0.5 0.5 5.0 5.0 5.0 5.0
3.90 7.75 5.19 6.46 5.57 6.08 –13.27 –0.51 3.32 5.82
67.00 66.68 66.84 66.73 66.80 66.76 69.36 67.69 67.30 66.75
982.53 982.53 982.53 982.53 982.53 982.53 978.28 978.29 978.29 983.0
9.82 9.83 9.80 9.85 9.77 9.88 9.71 9.55 9.20 9.83
Note. The last row contains the input parameter values.
TABLE 3. Estimates of Transmission Line Parameters when there is a Systematic Error in Voltage Phase ä1 Power flow, MW
ä1, deg.
R, Ù
X, Ù
B, ìS
G, ìS
ä1, deg.
R, Ù
X, Ù
B, ìS
G, ìS
131.8 396.4 979.1 —
0.1 0.1 0.1 0.0
5.36 5.65 5.72 5.82
63.25 65.61 66.30 66.75
983.0 982.9 982.9 983.0
10.69 10.69 10.69 9.83
0.5 0.5 0.5 0.5
3.58 5.01 5.33 5.82
49.27 61.06 64.49 66.75
982.89 982.95 982.89 983.0
14.12 14.12 14.11 9.83
Note. The last row contains the input parameter values.
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S. K. Kakovskii et al.
crease or decrease the errors in the estimates of the transmission line parameters. Table 4 collects the estimates of R, X, B, and G for different systematic errors in the voltages at two nodes at the ends of the PTL. Small systematic errors in measured voltages of like sign hardly affect the estimates of the transmission line parameters. However, if the systematic errors in the voltages are small (0.1 kV), yet of unlike signs and the real power flow is low, then the error in the estimate of R is large (15%) and the error in the estimate of X is small. The errors in the estimates of G and B can be neglected. If the systematic errors are still 0.1 kV, but the real power flow increases to 395 MW, the accuracy of the estimates of the transmission line parameters increases considerably (by a factor of almost 3). As the power flow further increases to 982 MW, the accuracy increases again by a factor of approximately 3. In other words, the error in the estimates of the transmission line parameters is proportional to the real power flow if the systematic errors in the voltages remain the same. Table 4 summarizes the estimates of the transmission line parameters for a power flow of 132 MW and large systematic errors in the voltage magnitudes at the ends of the PTL. If the errors of the voltage are of like sign, the estimate of R hardly changes, while the error in X is 0.2%. If the errors in the voltages are of opposite signs, the errors in the estimates of R and X are 120 and 1.4% respectively. Thus, even with relatively high accuracy (0.2%) of the instrument transformer and low real power flow, the error in the estimate of R may appear too large. The errors in the estimates of G and B
can be neglected. Note that errors in the voltage magnitude that are equal in value and sign hardly affect the estimates of the transmission line parameters. The errors in the estimates of the transmission line parameters for medium (395 MW) and high (982 MW) real power flows are much smaller, the signs of the errors in the estimates of R and X remaining opposite. Also note that in the above examples, the estimates of X are by an order of magnitude more accurate (in relative units) than the estimates of R. Table 5 summarizes the estimates of the transmission line parameters when there are systematic errors (0.2 – 0.5%) in the nodal voltage phases at the ends of the PTL. Unlike the errors in the voltage magnitudes, the systematic errors in the voltage phases lead to errors in the estimates of R and X of like signs. This may be an indication of the difference between the pairs of systematic errors in the magnitudes and phases of the voltages at the ends of the PTL. Large errors affect the measurements of R (up to sign reversal in the estimate of R ), while the errors in the estimates of the other transmission line parameters (X, B, G ) remains acceptable (less than 2%). It should be pointed out that the estimates of R are much more sensitive to systematic errors in measurements of the voltage magnitude, whereas the errors in the estimates of X are more sensitive to the systematic errors in the measured voltage phase. Estimation of the Transmission Line Parameters from Measurements with Normal Errors. Let us now con-
TABLE 4. Estimates of the Transmission Line Parameters when there are either Small or Large Systematic Errors in U1 and U2 Error, kV U1
U2
0.1 0.1 0.1 0.0 1.0 1.0 1.0 1.0
0.1 –0.1 0.0 0.1 0.1 –0.1 0.0 0.1 —
R, Ù
X, Ù
G, ìS
B, ìS
ÄR,, Ù
ÄX, Ù
ÄG, ìS
ÄB,ìS
5.82 5.02 5.42 6.22 5.83 –1.86 1.99 9.67 5.82
66.76 66.84 66.80 66.71 66.88 67.65 67.26 66.38 66.75
9.83 9.83 9.83 9.84 9.81 9.80 9.80 9.84 9.83
982.81 983.00 982.91 982.91 981.19 983.00 982.05 982.05 983.0
0.00 0.80 0.40 –0.40 0.01 7.68 3.83 –3.99 0.00
0.01 –0.09 –0.05 0.04 0.13 –0.90 –0.53 0.40 0.00
0.00 0.00 0.00 0.01 –0.02 0.03 0.03 0.04 0.00
–0.19 0.00 0.09 0.09 –1.81 0.00 0.95 0.95 0.00
Notes. 1. The last row contains the input parameter values. 2. The power flow is 132 MW.
TABLE 5. Estimates of the Transmission Line Parameters from Measurements of Voltage Phases ä1 and ä2 with Systematic Error Error, deg. ä1
ä2
0.1 0.1 0.1 0.0
0.1 –0.1 0.0 0.1 —
R, Ù
X, Ù
G, ìS
B, ìS
ÄR, Ù
ÄX, Ù
ÄG, ìS
ÄB, ìS
5.70 5.02 5.36 6.16 5.82
66.73 59.80 63.28 70.22 66.75
11.55 9.84 10.69 10.69 9.83
982.98 982.97 982.98 983.01 983.0
–0.12 –0.80 –0.46 0.34 0.00
–0.02 –6.95 –3.47 3.47 0.00
1.72 0.01 0.86 0.86 0.00
–0.02 –0.03 –0.02 0.01 0.00
Note. The last row contains the input parameter values.
Estimation of the Transmission Line Parameters Using a Grid Model
sider classical errors in synchrophasor measurements. They are uncorrelated normal random processes of given intensity. Amplitude quantization noise also occurs in digitally represented parameters of operating states. Such noise is always present in measuring devices and communication channels and usually is of low level. Let us analyzed the effect of normal errors on the accuracy of estimation of the transmission line parameters. The error levels used in the model experiment covered their real range observed in synchrophasor measurements. Statistical Characteristics of Random Errors in Synchrophasor Measurements. The errors in the magnitudes and phases of nodal voltages and currents were modeled by independent normal random variables. Table 6 gives the normalized symmetric correlation matrix (its upper triangular part) of random errors. The small values of the normalized correlation coefficients between the components of random errors are indicative of their independence for all the parameters subject to synchrophasor measurements. Note that random errors were modeled for all PMU-measured parameters. Table 7 summarized the mean values and root-meansquare deviations (RMS) of the PMU-measured parameters. Since three-fold exceedance of the RMS is highly probable, it may be assumed, as a first approximation, that the accuracy classes for the measurements of voltage magnitude, voltage phase, current magnitude, and current phase are 0.2, 1, 0.5, and 0.5%, respectively. Table 7 can be used to estimate the level of errors in individual synchrophasor measurements to reject the least accurate ones. The RMS of individual measured parameters is proportional to the level of random errors. The RMSs were measured assuming the stationarity of the processes analyzed. Low Real Power Flow in the PTL. Let the real power flow in the PTL under consideration be low (say, 132 MW). The estimation of the transmission line parameters during a short period in the presence of random errors results in poor accuracy, and the estimation of the parameters from instantaneous values for low power flows may produce an absolutely
231
n1
~ U1
U1
n2
U2
~ U2
Fig. 6. Vector diagram of synchrophasor measurements of voltages.
unreal result (including a negative estimate of R ) in some cases (Fig. 6). Figure 6 shows the input voltage vectors U1 and U2 at the ends of the PTL used for the purpose of modeling and the ~ ~ vectors U1 and U2 derived from the input vectors by adding the error vectors n1 and n2. The circles in this figure show the “wandering” of the voltage resultant because of the random errors. It can be seen that measured voltage magnitudes and phases (dashed lines) can differ considerably from the input vectors (solid lines). The effect of this difference on the estimates is especially strong for low real power flows, i.e., for small differences between the nodal voltage phases. The transmission line parameters can be estimated only from measured parameters such as the magnitudes and phase an~ ~ gles of the vectors U1 and U2 . The real difference between the voltages at the ends of the PTL (U1 – U2) that causes the ~ ~ power to flow differs from the measured difference U1 - U2 ; hence, the estimate of the power flow in the model differs from the real one. For a power flow of 132 MW, the difference of the phases at the ends of the PTL is just 1.48 deg, while the RMSs of the measured phases are 0.34 and 0.35 deg. Since three-fold exceedance of the RMS is highly probable, the sign reversal in the difference between the
TABLE 6. Normalized Correlation Matrix of Modeled Errors Parameter
U1
U2
Angle U1
Angle U2
I1
I2
Angle I1
Angle I2
U1 U2 Angle U1 Angle U2 I1 I2 Angle I1 Angle I2
1
0.02 1
0.00 0.00 1
0.02 0.02 0.02 1
0.02 –0.04 0.01 0.01 1
–0.02 0.00 0.02 0.03 0.00 1
0.02 –0.02 0.00 –0.03 –0.01 –0.01 1
–0.03 –0.01 0.01 0.05 –0.04 –0.01 –0.02 1
TABLE 7. Mean Values and RMSs of Measured Parameters Parameter
U1, kV
U2, kV
Angle U1
Angle U2
I1, A
I 2, A
Angle I1
Angle I2
Mean RMS deviation
518.8 0.32
518.3 0.32
–7.07° 0.04°
–5.59° 0.03°
173.3 0.38
197.8 0.37
–236.4° 0.37°
–131.6° 0.36°
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S. K. Kakovskii et al.
measured nodal voltage phases at the ends of the PTL is very ~ ~ likely to occur (when the vectors U1 and U2 appear in the common (intersection) region in Fig. 6). Note that the estimates of the power flow can reverse sign at such instants, and it is not the flow itself that reverses its sign, but rather its estimate. In these cases, the sign reversal in the estimate of R is highly probable. When the real power flow is high, the probability of sign reversal in its estimate is much lower (by several orders of magnitude). Normal Errors in Measurements of Voltage. Table 8 summarizes the estimates of the transmission line parameters and their RMSs ó for low real power flow and random errors of constant level (RMS = 0.1 kV) in measurements of nodal voltage magnitudes. The level of the random error corresponds to accuracy class 0.1%. The averaging time in all measurements was 10 sec. Note that the RMS of the estimates increases for all the transmission line parameters (except for G and B, which remain almost constant) with decrease in the real power flow. When there are normal errors in measurements of the voltage magnitude and phase and the real power flow is high, the errors in the estimates of the transmission line parameters tend to decrease with increase in the real power flow. Table 9 shows such dependence on the real power flow when there are random errors of constant level only in the measurements of the voltage phase. The averaging time is 10 sec. The level of the random error in the measurements of the voltage phase corresponds to accuracy class 1%. As with the random error in the voltage magnitude, the random errors in the voltage phases causes the RMS of the main (R and X ) transmission line parameters to increase with decrease in the real power flow. Recall that the data in Tables 8 and 9 were obtained for low power flows (the difference of voltage phases does not exceed 2 deg.). The mean values of the transmission line parameters in Tables 8 and 9 are random with respect to their input values.
With increase in the averaging time, their mean values approach the input values. The RMS of these parameters is a quite stable characteristic. Comparing Tables 8 and 9, we can conclude that the estimates of R and X are more sensitive to the random errors in the measurements of the voltage phases than the other parameters. Certainly, the full pattern is much more complex because the accuracy of the estimates of the transmission line parameters is also affected by the errors in the measurements of the amplitudes and phases of nodal currents. Random Errors in Measurements of Current. The errors in the estimates of R and X are similar to those in the previous case of random errors in the magnitudes and phases of nodal voltage. With decrease in the real power flow, the RMS of the errors in the estimates of R and X increases, i.e., the dependence is close to inversely proportional. The errors in the estimates of G are proportional to the power flow and hardly dependent on the random error in the measurement of the current phase, while the errors in the estimates of B are weakly dependent on the random error in the current magnitude and inversely proportional to the power flow. Improving the Accuracy of the Estimation of the Transmission Line Parameters. When there are random errors in synchrophasor measurements, the accuracy of the estimates of R, X, G, and B can be improved by averaging, i.e., by increasing the measurement time. As one would expect, increasing the averaging time for the estimates of the transmission line parameters causes their RMSs to decrease and their mean values to approach the input values. However, an increase in the averaging time leads to an increase in the dynamic error, which requires minimizing the total mean-square error of measurements. An averaging time of 10 sec may be considered the most acceptable for energy management systems. Correlation Relationships of the Transmission Line Parameters Estimated from Measurements with Random Errors. Random Errors in the Voltage Magnitude or Phase
TABLE 8. Estimates and RMS of the Transmission Line Parameters when there is a Random Error in the Magnitude of U1 Real power flow, MW
R, Ù
X, Ù
G, ìS
B, ìS
óR
óX
óG
óB
25 50 100 200 —
6.3 6.05 5.84 5.82 5.82
66.63 66.71 66.75 66.74 66.75
9.83 9.83 9.83 9.83 9.83
983.03 983.03 982.95 983.04 983.0
1.45 0.93 0.36 0.24 0.0
0.35 0.15 0.04 0.02 0.0
0.00 0.00 0.00 0.00 0.0
0.08 0.10 0.12 0.06 0.0
TABLE 9. Estimates and RMS of the Transmission Line Parameters when there is a Random Error in the Phase of U1 Real power flow, MW
R, Ù
X, Ù
G, ìS
B, ìS
óR
óX
óG
óB
25 50 100 200 —
6.42 5.08 5.65 5.77 5.82
55.39 62.07 65.52 66.45 66.75
10.13 9.62 10.14 10.16 9.83
983.00 983.00 983.00 983.00 983.0
3.95 1.39 0.42 0.16 0.0
12.3 6.83 2.46 1.52 0.0
0.57 0.52 0.63 0.53 0.0
0.01 0.01 0.01 0.01 0.0
Note. The last row contains the input parameter values.
Estimation of the Transmission Line Parameters Using a Grid Model
Fig. 7. Matrices 1 and 1a for a random error in the voltage magnitude of the first node.
at a Node. If a random error is only in one measured parameter, then the normalized correlation matrix K of the transmission line parameters consists of elements close to 1 in absolute value. This is true for the random error in either magnitude or phase of voltage. Therefore, if at least one transmission line parameter is known exactly, then the others can be determined from the regression equations. In Figs. 7 and 8, matrices 1 and 2 correspond to a positive real power flow, while matrices 1a and 2a to a negative flow. Note that the correlation coefficient KRX does not depend on the direction of the power flow, whereas the other correlation coefficients do. Low real power flows (the difference between the voltage phases is less than 2 deg in absolute value) that are the most critical for the accuracy of estimates of the transmission line parameters were used in modeling. The same measurements conducted when the nodal loads undergo random fluctuations lead to insignificant (2 – 3%) deviations of matrices 1 and 2 from unity. Measurements at a nonrated frequency (0.1 Hz) do not lead to significant deviations of the elements of these matrices from unity. Random Errors in the Voltage Magnitudes or Phases at Two Nodes. The results of measurements are the following. When there is a random error (RMS = 0.1 kV) in the measurements of the voltage magnitudes at the ends of the PTL, the normalized correlation coefficient KRX of the estimates of R and X is close to –1.0, i.e., the errors in the estimates of R and X are opposite in sign. The RMSs of R and X are insignificant: 0.37 and 0.04 Ù, respectively. The other elements of the matrix K (say, KRB ) are considerably different from 1. When there is only one source of random noise, all the elements of matrices 1, 1a, 2, 2a are positive or negative numbers close to unity, irrespective of the direction of the real power flow. If, however, random noise is present at both two nodes (random errors in the measurements of the voltage magnitudes or phases), some elements of the matrices are considerably different from unity. If the level of the random error in the measurements of one parameter is much more greater (say, by an order of magnitude) than in the other parameters, the matrix K almost fully consists of unit elements.
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Fig. 8. Matrices 2 and 2a for a random error in the voltage phase of the first node.
It is significant that the normalized correlation coefficients among the transmission line parameters are weakly dependent on the level of the random error. For example, changing the level of the random error by an order of magnitude did not affect the correlation relationships. When the real power flow is high, the correlation coefficient KRX can differ substantially from 1, and the higher the real power flow, the more the difference of KRX from 1. Recall that the accuracy of measurements of the transmission line parameters is a critical issue when the real power flow is low. Certainly, there may be situations where random errors are present in the phasor measurements of both nodal voltages and currents. In such cases, the coefficient KRX has intermediate (between –1 and +1) values from which it is difficult to ascertain the level of random errors in the phasor measurements of voltages. Random Errors in Phasor Measurements of Currents. When there is a random error in the measured magnitude of the current at one end of the PTL, the cross-correlation moments KRX of the estimates R and X are close to 1. The moment KGB between the estimates of G and B is close to 1 as well. Note that the signs of KRX and KGB are opposite. When only one measured parameter (say, the magnitude or phase of nodal voltage) has a random error, the cross-correlation coefficients of the estimates of R, X, B, and G can be used to identify the source of errors. If several parameters have random errors, and especially if these errors are of comparable level, the analysis is much more complicated. For example, if random errors are present in the measurements of the magnitude and phases of the voltages at two nodes at the ends of the PTl, the moments of the correlation matrix are considerably smaller than 1. If the level of the random error in one parameter is much (by an order of magnitude) greater than the levels of the random errors in the other parameters, the “bad” parameter can easily be identified in this way. CONCLUSIONS 1. Our algorithm for the estimation of the transmission line parameters from measurements with systematic and ran-
234
dom errors has been shown to work adequately. Accuracy and correlation characteristics of estimates of the transmission line parameters have been found for typical accuracy classes of ITs. 2. It has been shown that if there no errors in synchrophasor measurements, the accuracy of the estimates of the transmission line parameters is quite high and hardly dependent on the operating conditions (daily load profile, random fluctuations in load, etc.). 3. Systematic errors in synchrophasor measurements are the most critical. It has been shown that the errors in the estimates of the parameters R and X are proportional over a wide range to the systematic error in the phasor measurements of voltage. The other parameters (B and G ) weakly depend on the level of the systematic error in synchrophasor measurements. 4. When real power flows are low (less than 10% of throughput), the transmission line parameters can be adequately estimated if the accuracy of the instrument transformers is sufficiently high (0.1 – 0.2%). 5. The accuracy of the estimates of the transmission line parameters is almost proportional to the real power flow. For all possible values of a single systematic error in the measurements of voltage magnitude, the error in the estimates of R and X are of opposite signs. A systematic error in the measurement of the voltage phase at a node leads to errors in R and X of like signs. 6. The accuracy of the estimate of R is much lower than the accuracy of the estimate of X for the same level of (random and systematic) errors in the measured voltages at the ends of the PTL. 7. A single source of random errors in synchrophasor measurements can reliably be identified in real time even for
S. K. Kakovskii et al.
low real power flows by measuring the normalized correlation coefficients among the estimates of R, X, G, and B. 8. It has been shown that the accuracy of the transmission line parameters estimated from measurements with random errors can be improved by averaging the measured parameters. The longer the averaging time, which is limited by the dynamic error of measurements, the higher the accuracy of the estimates. 9. The results presented here suggest the necessity of further studies in this field, structurization of the results obtained, and development of algorithms for their use. REFERENCES 1. V. I. Idel’chik, A. S. Novikov, and S. I. Palamarchuk, “Errors in setting the parameters of equivalent circuits for electric-system calculations,” in: Statistical Treatment of Operational Data in Electric Power Systems [in Russian], Irkutsk (1979), pp. 145 – 152. 2. V. A. Faibisovich, Determining the Parameters of Electric Systems [in Russian], Énergoizdat, Moscow (1982). 3. Jun Zhu, “Detection and identification of network parameter errors using conventional and synchronized phasor measurements,” Electrical Engineering Dissertations, Paper 4 (2008) (http://hdl.handle.net/2047/d10018001). 4. M. A. Rabinovich, Digital Processing and Visualization of Data for Problems of Operating Control in the Electric Power Industry [in Russian], Izd. NTs ENAS, Moscow (2001). 5. M. Kraus and E.-G. Woschni, Information Measurement Systems [in German], Verlag Technik, Berlin (1975). 6. S. M. Rytov, Introduction to Statistical Radiophysics [in Russian], Nauka, Moscow (1976). 7. J. M. Wozencraft and I. M. Jacobs, Principles of Communication Engineering, Wiley, New York (1965). 8. A. A. Andronov and M. A. Rabinovich, “Distribution of abnormal frequency durations and zeros of Gaussian process,” Radiotekhninka, 33(8), 72 – 75 (1978).
