Electrical overhead lines parameters estimation using ...

Electrical overhead lines parameters estimation using synchrophasor measurements Ph.D, Eng. Bogdan VICOL1, Prof. Mihai GAVRILAS1, Ph.D, Eng. Alexandru KRIUKOV1 Rezumat—Lucrarea de faţă propune o metodă de estimare a parametrilor longitudinali ai liniilor electrice aeriene de transport cum sunt rezistenţa şi reactanţa, transversali cum sunt conductaţa şi susceptanţa, precum şi lungimea pe baza măsurărilor fazoriale sincronizate prelevate de dispozitivele de măsurări fazoriale amplasate la ambele capete ale liniei. Aceşti parametric constituie datele de intrare pentru numeroase aplicaţii din domeniul energeticii cum ar fi estimarea stării statice a sistemelor sau aplicaţii privind reglarea adaptivă a protecţiilor digitale. Mai mult, lungimea şi temperature conductorului pot fi folosite în aplicaţii de încărcare dinamică a liniilor electrice aeriene pentru creşterea capacităţii de transport. Cuvinte cheie-dispozitive de măsurări fazoriale, linii electrice aeriene, aproximarea funcţiilor neliniare Abstract—This paper presents a method for estimating transmission line parameters such as series resistance, reactance and shunt conductance, susceptance and line length, based on the synchronous voltage and current phasors measured by phasor measurement units (PMU) installed at both ends of the line. Transmission line parameters are essential inputs for various applications such as state estimation, protective relaying applications. Furthermore, the line length and conductor temperature can be used in dynamic thermal line rating algorithms for increased power transfer capacity. Numerical examples and preliminary results are presented. Keywords: overhead transmission lines, electrical parameters, phasor measurement unit, non-linear estimation theory.

Notations OHL – Overhead Line PMU – Phasor Measurement Unit GPS – Global Positioning System PLL - Phase Locked Loop EKF – Extended Kalman Filter Contributions The authors’ original contribution is a new method that proposes the integration of the lumped parameter model with the distributed parameter model of electrical lines for estimating electrical line parameters, per length unit and global values, using the Levenberg-Marquardt estimation model. 1.

General considerations

A large number of power systems applications used in the industry require the knowledge of precise values for the electrical lines’ parameters such as resistance, reactance, conductance and susceptance, and also of the exact line length. It is well known that some of these parameters, such as the resistance, vary with the line length and with the change of conductor temperature, and others, such as the reactance and the susceptance, are dependent only on the line length. Also, the overhead lines’ length is directly influenced by the weather conditions (environmental temperature, wind speed and strength, frost precipitations) and current flows, through the Joule-Lenz effect. Furthermore, the knowledge of OHL parameters’ exact values adds performance to other power systems operation fields such as adaptive transmission system relaying, dynamic line rating or state estimation. Fig. 1 describes the main applications that benefit from an accurate estimation of electrical line parameters.

1

Department of Power Systems Engineering, University “Gheorghe Asachi” of Iasi, Romania

Proof of the relevance of this problem is the various mathematical models and algorithms for real time line parameter estimation, many of which use precise synchrophasor measurements provided by phasor measurement units installed at both ends of the line. For instance, [1] proposes an algorithm in which the line resistance, reactance and susceptance are recursively estimated with an EKF, using PMU current and voltage measurements from both ends of the line, and then, taking into account the actual weather conditions, the conductors’ average temperature along the line is computed using the estimated resistance and thermal equations. In [2-4], the OHL parameters are estimated using PMU measurements and the line sag is determined using the weighted least squares method.

Fig.1. Power system applications that require the knowledge of exact line parameters

In the above methods, the assumption is that the PMU measurements are completely error-free ant that the line length is known. However, the PMU measurements are always prone to various errors, such as measurement transformers’ saturation, conversion and communication errors. Also, the line length changes with its load and the environmental temperature variation. At the same time, the line parameters’ estimation algorithms proposed in the literature compute the line length using the average conductors’ temperature, the conductors’ reference temperature and Young modulus of elasticity. Frequently, these values are not available or become outdated over the years, as the OHL structure can change because of various reasons. This paper proposes a electrical line parameters’ and length estimation method which uses the weighted least square algorithm. It is well known that the line resistance has an almost linear variation with the temperature, and also that the line length varies widely with the temperature, mainly because of the changes in conductor sag and tension. Thus, if the line length is known, the average sag can be determined based on the average span length. 2.

Synchrophasor measurements

Standard data acquisition systems such as EMS/SCADA are capable of measuring sampling rates down to a few seconds, which, for phasor measurements, leads to significant synchronization errors for system wide spread measurements. This shortcoming has made impossible until recently to use angle measurements in state estimation studies.

After the development of the GPS system and modern communication systems, high-performance digital measurement equipment has been developed, which uses time synchronization and provide high sampling rates, up to microseconds for accurate voltage and current measurements. The measured values are then converted with numeric algorithms into phasors. Such measuring devices are known as phasor measurement units or PMUs. A PMU is an electronic device that uses state of the art signal processing to measure 50-60 Hz voltage and current waves at very high sampling rates. The analog AC waveforms are converted into numeric signals by an analog/numeric convertor, one for each wire of the three phase circuit. A PLL oscillator and the GPS time reference signal allow a sampling rate of up to one microsecond. Digital signal processing algorithms are used to compute the current and voltage amplitude and angle phasors (Fig. 2). The synchrophasor technology used in power systems is governed by the C37.118.1-2011 standard [5], which establishes the mandatory procedures for measurement acquisition, a method of measurement precision assessment and an acquisition guideline for normal operation conditions.

Fig. 2. The block-diagram of a phasor measurement unit

3.

The Levenberg – Marquardt method

The Levenberg-Marquardt method is widely used for solving weighted least squares algorithms [6], which seek the minimization of the square error between values measured and computed with a given mathematical model. These methods are suited when the dependence between the state variables, which need to be computed, and the measurements, is non-linear. For this type of problem, the solution is found via an iterative process. The Levenberg-Marquardt method is a combination of two optimization algorithms: the gradient method, which is a descent method, and the Gauss-Newton method, which is a least squares method. In the gradient descent method, the solutions’ initial approximation is updated by applying successive corrections scaled and oriented according to the optimized function gradient’s direction. In the Gauss-Newton method, the sum-squared error is minimized considering the assumption that the minimized function is locally quadratic, and its minimum is determined by linearization. The Levenberg-Marquardt method behaves like the gradient descent method when the solution approximation is far from the optimal value, and like the Gauss-Newton method near the optimal solution. 4.

Overhead line parameters estimation

By using PMUs placed at both ends of a OHL, voltage and current magnitude and angle synchrophasors can be obtained, which can be used to compute the real values of the line’s real electrical parameters, which, can differ at one given moment from the rated parameters. The reasons for this difference can be various. The line’s power rating and length can vary because of electrical or non-electrical factors such as environment temperature, wind speed and strength, sag due to self-weight of the conductor, frost precipitations. The standard model for electrical lines’ representation is the lumped parameter PI model, represented in Fig. 3. The following equations can be written:

Fig. 3. The lumped parameter PI model for overhead lines

Y Y  I 1  U 1  2  I 2  U 1  2  0   U  Z   I  U  Y    U  0   1 2 2  1 2  

(1)

where: U1, U2, I1, I2 – voltages and currents at both ends of the line; Zπ, Yπ - global impedance and admittance of the line. According to [2], in order to estimate the electric parameters of overhead lines, resistance, reactance, conductance and susceptance, using measurements from PMUs installed at both ends of the line, it is enough to rewrite (1) as four equations, for the real and imaginary parts of the state variables, and to solve the resulting equations system with a least squares approach. But for finding the real line length, additional equations are required. For this, the paper proposes the use of the distributed parameters model, illustrated in Fig. 4 and described by the mathematical model (2).

Fig. 4. The distributed parameters line model

   

   

 U 2  ch   l  Z c  I 2  sh   l  U 1  0   Y c  U 2  sh   l  I 2  ch   l  I 1  0 where: γ, Zc – line propagation constant and characteristic impedance;

Yc 

1 - line characteristic admittance; Zc

l – line length. The relation between the lumped parameter and the distributed parameter representation is given by:

(2)

 

Z   Z c  sh   l

Y 

(3)

    Z c  sh  l 

2 ch   l  1

(4)

 Z Y  ch 1 1     2    l

Zc 

(5)

Z  Z Y  1  4   Y   

(6)

z0  Z c 

y0 

(7)

 Zc

(8)

where z0 and y0 – are the line impedance and admittance per unit of length. Thus, for each voltage and current measurement set provided by PMUs from both ends of the line, the four complex equations that describe the lumped and distributed parameters model for electrical lines can be rewritten as 8 equations with real numbers, for the real and imaginary parts of the state variables, which are complex numbers. To solve this equations system for the 9 unknown state variables (the real and imaginary part of Zπ, Yπ, Zc, γ and l, the line length), the assumption that the admittance of the line per unit of length, y0, remains constant. 5.

Case study

The proposed line parameters and length estimation algorithm, which uses PMU measurements, has three main steps: a)

Give the initial approximation of the parameters to be estimated, (R, X, B, l) as [0, 0, 0, 1];

b) Build the equation system to be solved, by grouping (1) and (2) and consider the limitations described in the above paragraph; c)

Using the PMU measurements from both ends of the studied overhead line, given as [U1, U2, I1, I2], the equations system built at step 2 is solved with the Levenberg-Marquardt method, obtaining the real R, X, B and l values.

For testing the method, the PMU measurements were simulated using a load flow algorithm. Voltage and current measurements were taken from both ends of an overhead transmission line. The used parameters are given in Table 1. Table 1 Simulated OHL parameters Rated voltage Nominal frequency Line length Electrical parameters per unit of length Conductor type Simulated measurements

R Ω/km 0.034 PMU 1 U1 = 241.0869 + 21.8931i kV I1 = 0.3388 - 0.0354i kA

400 kV 50 Hz 200 km X Ω/km 0.330 ACSR

B µS/km 3.473 PMU 2 U2 = 230.9401 + 7.1054e-15i kV I2 = 0.3464 - 0.2000i kA

The algorithm’s sensitivity to measurement error was tested by altering the reference PMU 1 and PMU 2 voltage and current values. The results in Table 2 show minor changes in parameters’ value when erroneous measurements are used to estimate the line parameters. Table 2 Simulation results Measurement error % 0 0.1 0.5 1 5 10

Estimated R 0 Ω/km 0.034002 0.033858 0.03331 0.032354 0.035794 0.041193

Estimated R Ω 6.697042 6.667842 6.558927 6.605356 7.095839 8.008731

Estimated X Ω 65.50401 65.49711 65.48676 65.41487 65.9853 64.82743

Estimated B S 0.000697 0.000697 0.000698 0.000698 0.000717 0.000796

length km 200.0057 199.9843 199.9538 199.7340 201.5269 198.1170

In Table 3, values for the line parameters obtained with the Levenberg-Marquardt algorithm are presented. The parameters from Table 1 are the rated values, for a temperature of 20 °C on the conductors. When this temperature rises, the voltages and currents measured by the PMU will change, and the algorithm will estimate new values for R, X, G, B and l. Table 3 Simulation results Estimated R Ω

Estimated X Ω

Estimated B S

length km

00:00:00

Estimated R 0 Ω/km 0.038538

7.590245

63.50492

0.000697

199.9073

01:00:00

0.038155

7.523744

63.5816

0.000698

200.1527

02:00:00

0.035559

7.011433

63.57667

0.000698

200.1398

03:00:00

0.037899

7.46967

63.54998

0.000697

200.052

04:00:00

0.037899

7.472553

63.57476

0.000698

200.1312

05:30:00

0.037643

7.422103

63.57467

0.000698

200.1311

06:30:00

0.038027

7.497448

63.57209

0.000698

200.1225

07:30:00

0.03841

7.573476

63.57629

0.000698

200.1355

08:30:00

0.038921

7.674577

63.58053

0.000698

200.1484

09:30:00

0.036393

7.176793

63.58505

0.000698

200.1657

10:30:00

0.037048

7.302907

63.55852

0.000697

200.0802

11:30:00

0.037183

7.330385

63.5655

0.000698

200.1023

Time of the day

Conclusions The paper proposes a method for estimating the real values of electrical lines’ parameters in real time, to take into account the variations that can occur due to the operating conditions. The key requirement for estimating these parameters is the use of precise PMU measurements from both ends of the line. The results obtained with the algorithm show variations in line parameters that should be taken into account when estimating the state of the system. An advantage of the method is the possibility to extend it for other types of network elements, such as transformers.

References [1] Janecek E., Hering P., Janecek P., Popelka A., Transmission line identification using PMUs, Environment and Electrical Engineering (EEEIC), 2011 10th International Conference, , May, 2011, pp. 1 - 4. [2] Tianshu B., Jinmeng Chen, Jingtao Wu, Qixun Yang, Synchronized phasor based on-line parameter identification of overhead transmission line, „Electric Utility Deregulation and Restructuring and Power Technologies”, (Third International Conference), April, 2008, pp. 1657 - 1662. [3] Di Shi, Tylavsky D., Logic N., Koellner K., Identification of Short Transmission-Line Parameters from Syncrophasor Measurements, (Power Symposium NAPS 40th North American), 2008, pp. 1 - 8. [4] Di Shi, Tylavsky D., Logic N., Koellner K., Wheeler D., Transmission-Line parameter identification using PMU measurements, (European transactions on electrical power), May, 2011, pp. 1574–1588. [5] Madsen K,. Nielsen H.B., Tingleff O., Methods for Non-Linear Least squares problems, 2nd Edition, Informatics and Mathematical Modelling, Technical University of Denmark, April 2004. [6] IEEE Std. C.37-118.1.2011 IEEE Standard for Syncrophasors for Power Systems, IEEE Power Engineering Society 2011.