A technique for simultaneous parameter identification and measurement calibration for overhead transmission lines

Transcription

1 A technique for simultaneous parameter identification and measurement calibration for overhead transmission lines PAVEL HERING University of West Bohemia Department of cybernetics Univerzitni 8, 36 4 Pilsen CZECH REPUBLIC phering@y.zcu.cz EDUARD JANEČEK University of West Bohemia Department of cybernetics Univerzitni 8, 36 4 Pilsen CZECH REPUBLIC janece@y.zcu.cz Abstract: he paper deals with the on-line parameter identification of transmission lines from synchronized measurements of current and voltage phasors provided by the phasor measurement units. Particularly because the current measurements contain a systematic error, which strongly influences the identified parameters, a technique for simultaneous parameter identification and measurement calibration will be proposed. he series and shunt line parameters are recursively estimated from the phasor measurements and weather conditions using the extended Kalman filter. Actual line temperature and reference resistance are estimated instead of actual series resistance, which is then computed from these estimates. wo case studies based on data measured from two V overhead transmission lines are presented to demonstrate the effectiveness of the proposed approach. Key Words: On-line parameter identification, measurement calibration, overhead transmission line, phasor measurement unit, possitive sequence component. Introduction As the demand for electricity transmission capacity increases, the nowledge of transmission line parameters involving series impedance and shunt admitance is crucial to determine the line ampacity. While construction parameters are commonly valid when the line is designed, actual values differ due to line aging, ambient conditions and the load of line. It is therefore necessary to compute the actual parameters to ensure effective and safe operation of the transmission line. Development of the phasor measurement units (PMUs, which guarantee the high precision of mutual synchronization of phasor measurements from both ends of the transmission line, maes possible on-line parameter identification. A number of papers deal with the utilization of synchronized phasors in wide area measurement systems, e.g. [] and other references mentioned therein. Several approaches for identification of the line parameters from the measured phasors were discussed e.g. in []. he weighted least squares (WLS method is widely used for power system state and parameter estimation, where the weights are commonly set to be the inverse of the measurement error variances [3]. At each, the line parameters are estimated based on a redundant number of current and voltage measurements taen at different instants. he number of measurements used is limited by numerical stability and must be sufficiently large to yield reliable instantaneous estimates of varying parameters [4]. he accuracy of the identified parameters is significantly affected by random and systematic measurement errors. While the existing methods wor with random errors, the suppression of the influence of systematic errors have not been adequately addressed yet. herefore, a new technique using extended Kalman filter (EKF [7] for parameter identification and simultaneous measurement calibration is proposed in this paper. he paper is organized such that Section describes the proposed model of the transmission line. Section 3 introduces the EKF based parameter identification and measurement calibration method. he proposed approach is illustrated in two case studies for V overhead transmission lines in Section 4. he results are summarized in Section 5. Model Description Let us assume a three-phase transmission line, where PMUs are installed at both ends of each phase for synchronous measurement of voltage phasors, U = U Re + ju Im, and current phasors, I = I Re + ji Im. he objective is to estimate the line parameters Z = R + jx and Y = G + jb, where R is series resis- ISBN:

2 can be described by an average temperature of the line, which is given in the IEEE standard [5] by the differential equation τ d dt = l R( I(t (t q c ( q r ( +q s, (5 R( = R ( + α (, (6 Figure : Model of the transmission line. tance, X is reactance, G is shunt conductance and B is susceptance, from the measurements. Instead of independent estimatinon of the line parameters from the measurements on each of the three phases, the possitive sequence component for the voltage and current phasors will be computed [6]. he transmission line model represented by equivalent π circuit is shown in Figure. Phasors of possitive sequence components measured at one end will be mared with superscript ( and at the other end with superscript (. As the current is usually measured in the lower part of the measuring range of the meter, it is appropriate to consider the influence of systematic errors in the current measurements. So, let the relation between measurements of the currents and calibrated current be considered in the following equation ˆ I = I I + c I, ( where ˆ I is calibrated (true value, I is measured value, I, c I C are calibration constants, which are considered for measuring devices at both ends of the line to be equal. Using relation ( in the line model, which is displayed in Figure, the following relations hold U ( U ( = Z I, ( ( I ( I ( = I ( Y ( U ( + U (, (3 where ( I = I ( I ( + I ( Y ( U ( U ( +c I. (4 he line parameters change due to line currents and ambient weather conditions. o obtain accurate parameter estimates, it is desirable to include their effects to the model. Let us assume that we have measurements of ambient temperature and speed and direction of wind. hen, the dependence of resistance where τ total heat capacity of conductor, l length of the transmission line, I(t absolute value of current through the series impedance at t, see equation (4, q c ( convected heat loss rate per unit length, q r ( radiated heat loss rate per unit length, q s solar heat gain rate per unit length, R( AC resistance of conductor at average temperature, R AC resistance of conductor at reference temperature C, α temperature coefficient of resistance. Averaged conductor temperature is understood as the average between conductor core and surface temperatures. Heat losses and gain are determined by the following expressions: he solar heat gain: q s = aq se sin θa, (7 where θ = arccos [cos(h c cos(z c Z ], a is solar absorptivity, H c is altitude of the sun, Z c is azimuth of the sun, Z is azimuth of the line, A is projected area of conductor, Q s is total solar and sy radiated heat flux. he radiated heat loss: [ ( ( ] a q r ( =.78Dε (8 where D is conductor diameter, ε is emissivity, a is ambient temperature. he convected heat loss is determined according to the IEEE standard [5] as the maximum from convected heat losses computed for natural convection q cn, low wind speed q c and high wind speed q c : q c ( = max(q c, q c, q cn, (9 ISBN:

3 where q c = q c = [ ( ] Dρf V.5 w.+.37 f K a ( a, µ f [ ( ] Dρf V.6 w.9 f K a ( a, µ f q cn =.5ρ.5 f D.75 ( a.5. ( Computation of coefficients K a, µ f, ρ f, f is adressed in the IEEE standard [5]. Identification of the line resistance through R and average temperature of the conductor is advantageous because it allows to use entire measurement history. Similarly to relation (6, the impact of the conductor temperature on the reactance X could be modeled, which is reflected through thermal expansion of the conductor. However, this impact is negligible and the reactance X will be assumed to be constant. Shunt parameters G and B obviously also depend on weather conditions. However, decription of the dependence based on physical laws is not possible. So, their variability will be modeled by a random wal. he next section focuses on description of a technique of identification of the line parameters and calibration coefficients using EKF. 3 Parameter identification by EKF Let us have a measured possitive sequence component of voltage and current phasors at discrete instants with sampling period t. When equation (5 is discretized with period t, the following difference equation will be obtained: + = + t c p l R,( + α( I + + t c p q s t c p (q c + q r. ( We define a vector s of the line parameters and calibration coefficients, which has to be identified, as s = R, X G B I,Re I,Im c I,Re c I,Im ( hen, a system of equations describing the dynamics of changes of identified parameters is given as f (s s + =. + f 9 (s w,. w 9, = f(s + w, (3 where f (s is the right hand side of equation ( and f i (s, i =,..., 9, is the i th element of the vector s. he noise w maes it possible to represent uncertainty in the development of parameter values. As the parameters, G and B are assumed to be a varying, the elements w i,, i = {, 4, 5}, are considered as random variables with zero mean and variances σ i and the remaining elements are set to zero for all. he measurement equations are given in the following form U Re, z = U Im, I Re, = h (s + e. (4 I Im, Substituting from ( and (3 to (4 we obtain R I,Re X I,Im {( R I,Im + X I,Re z = U ( + U ( G +jb {( U ( + U ( I }Re G +jb I }Im + e, e, e 3, e 4,, (5 where I,Re and I,Im are real and imaginary part of the current I, which is computed according to (4 from the measurements obtained at. he symbols { } Re and { } Im denote real and imaginary part of the expression in bracets. e denotes random errors of measurement devices with E{e } =, cov(e = R = diag ( σ, σ, σ 3, σ 4, (6 where variances are given according to the degrees of precision of the measurement instruments. As the values of z are assumed to be the differences between measurements at line ends and, the resulting variance is the sum of the variances of measuring instrument errors. Uncertainty in the difference of phasors is a crucial issue in parameter estimation due to the low signal to noise ratio. herefore it is necessary to use as much of data as possible. he EKF [7] taes into account the entire measurement history up to over the traditional WLS approach and is defined by the following ISBN:

4 relations: s = ( s + K F, z h (s,(7 K F, = P H [ H P H + R] (8, P = P K F, H P, (9 s + = f(s, ( P + = F P F, ( where s is the parameter estimate at based on measurement history up to, s + is the one-step ahead prediction of the parameter values at + based on measurement history up to, H = h s = s =s F = f s = s =s ( h, s ( h, s ( h3, s ( h4, s, [ ( ] f s, 8 I is an 8-by- vector of zeros and I 8 8 is identity matrix of dimension 8. 4 Case studies his section is focused on two case studies demonstrating the effectiveness of the proposed approach. Both are performed on real data from V transmission lines. Identification of the lines is performed recursively, in order to simulate the activity in real. he proposed estimation algorithm was implemented in MALAB. 4. Identification of the transmission line A Let us consider the V overhead transmission line with the maximum permissible current 483A. he aim is to identify the transmission line parameters based on measurements of possitive sequence currents and voltages, respectively, from period of April 4 to September 9 with sampling period t = min. Development of parameter estimates is displayed in Figure and Figure 3. In able, the final parameter estimates obtained with and without calibration of measurements of possitive sequence current are compared to the designed values. Improved quality of the estimate of the line resistance can be seen if the simultaneous calibration is used in comparison with the case of the identification without the calibration. R [Ω] X[Ω] G[µS] B[µS] Identified value Designed value Figure : Development of identified parameters according to the increasing number of measurements from the transmission line A. [ C] 6 4 R[Ω] Figure 3: Development of the identified temperature of conductor and actual resistance of the transmission line A. he bias of the computed voltage drop on the series impedance, which is caused by systematic error of the current measurement, is clearly seen in Figure 4. When the current measurements are calibrated, the computed voltage drop corresponds to the measured values. he significant systematic error of the current measurements is caused due to measuring at the bottom of the measuring range of the meter, which is 58A, see Figure 5. ISBN:

5 U Re [V] /9 4/ a able : Designed and identified parameters of the V overhead transmission line A. Parameter Designed Identified value Identified value value with calibration without calibration R [Ω] X[Ω] G[µS].4 B[µS] U Im [V] b /9 4/ R [Ω] X[Ω] Figure 4: Comparison of real and imaginary parts of measured voltage drop on the series impedance of the transmission line A (dotted line with the voltage drop computed from measured current and designed parameters (dot-and-dash line and identified parameters with measured current calibration (solid line, respectively. G[µS] B[µS] Identified value Designed value I ( [A] (I ( [rad] Figure 5: Measurements of possitive sequence current on the transmission line A. Figure 6: Development of identified parameters according to the increasing number of measurements from the transmission line B. able : Designed and identified parameters of the V overhead transmission line B. Parameter Designed Identified value Identified value value with calibration without calibration R [Ω] X[Ω] G[µS].55. B[µS] Identification of the transmission line B Let us consider another V overhead transmission line with the maximum permissible current 7A, where the measurements of possitive sequence currents and voltages, respectively, are obtained from period of June 5 to September 9 with sampling period t = min. As in the previous example, the aim is to recursively identify the line parameters. Development of identification of the line parameters is shown in Figure 6 and Figure 7. Comparison of final parameter estimates with designed values is presented in able. Again, the calibration of current measurement brings an improvement of quality of the parameter estimates. ISBN:

6 [ C] R[Ω] [] Y. Liao, Some Algorithms for ransmission Line Parameter Estimation. 4st Southeastern Symposium on System heory, University of ennessee Space Institute ullahoma,n, USA, 9. [] A. Phade and B. Kasztenny, Synchronize phasor and frequency measurement under transient conditions, IEEE ransactions on Power Delivery, vol. 4, no., 9. [3] A. Abur and A. Expósito, Power System State Estimation, heory and Implementation. Marcel Deer INC., New Yor, 4. [4] Y. Liao and M. Kezunovic, Online optimal transmission line parameter estimation for relying applications, IEEE ransactions on Power Delivery, vol. 4, no., 9. [5] IEEE standard for calculating the current temperature of bare overhead conductors, IEEE std 738 M 6, 7. [6] J. Blacburn, Symmetrical components for power systems engineering. Marcel Deer INC., New Yor, 993. [7] A. P. Grewal, M. nad Andrews. Kalman filtering. J. Wiley and Sons,..5.4 Figure 7: Development of the identified temperature of conductor and actual resistance of the transmission line B. 5 Conclusion he paper was devoted to the online transmission line parameter identification. A new algorithm was proposed for online parameter identification and simultaneous calibration of the current measurement. It was shown, that measurement calibration significantly improves the quality of the parameter estimates. he transmission line model incorporating average temperature of conductor is used. he estimation of reference resistance and actual conductor temperature instead of actual resistance allows to identify parameters of the entire data history and thus improves resultant estimates of the overhead transmission line parameters. Acnowledgements: he wor was supported by echnology Agency of the Czech Republic under projects A865 and E97. References: ISBN: