Estimation of transmission line parameters by using two least‐squares methods

Funding information CAPES; CNPq, Grant/Award Number: 408681/2016-0; FAPESP, Grant/Award Number: 2019/05381-9 Abstract This work proposes two new approaches based on the ordinary least-squares method and the total least-squares method to estimate the parameters of a balanced three-phase transmission line using voltage and current measurements from phasor measurement unit. First, a new model for the steady-state phasorial equations of a medium-length transmission line is proposed. Then, the noises acting upon each measurement on the ordinary least-squares setup are considered, and for the total least-squares setup, noise acting upon the observation matrix in order to account for model uncertainties and non-linearities is also considered. The methods are tested in simulation data of a real medium-size transmission line.s The main goal is to compare both methods and show their complexities. The results show good performances for both methods and, indeed, the total least-squares setup had better performance than other reported total least-squares estimators, which use a different phasorial set of equations and oversimplified noise modelling. It is concluded that for the ordinary least-squares, the solution is well known and its behavior is predictable. While for the total least-squares, the solution requires more sophisticated methods of matrix decomposition and its behavior is not as predictable. Therefore the implications of these new approaches, where new considerations about the modelling of the noises are made and where a new phasorial set of equations is used are significant, given that the many works in the literature make use of these common-place tools.


INTRODUCTION
The knowledge of the characteristics of a power system is crucial for analysing the reliability and operation of the network as a whole. The accuracy in which such parameters are identified is directly related to the system performance in several aspects, such as the tuning of the protection system against overvoltages; detection and location of faults in overhead and underground networks; isolation coordination; surge propagation characteristics in transmission lines; prediction of possible electromagnetic transients and overvoltages; analysis of the system stability in terms of voltage levels and power demand. The parameters of overhead transmission lines are conventionally calculated from the physical and geometric characteristics of towers and conductors, such as the height of the conductors, geometry of the towers, soil conductivity, effect of the return current through the soil, and skin effect in the conductors [1,2].
The conventional calculation methods are based on several approximations, which generally lead to inaccuracies in the estimation of the transmission line parameters. For example, the electric permissivity varies along the length of the transmission line according to humidity and variations in the electrical insulation and soil characteristics. These variations are not taken into account when parameters are calculated using the Carson formulation with constant soil conductivity. Furthermore, several other approximations are considered in the tower geometry, bundle conductors, and environmental characteristics [3].
These inaccuracies in the calculation are directly related to the performance of the power system in terms of power and voltage stability, overvoltage protection, insulation coordination, and electromagnetic wave propagation through the line. In addition, the estimation of the electrical parameters of transmission lines is a mathematical dynamical system, which depends on technical and environmental aspects. In this context, the new concept of smart grids and modern transmission systems requires a more efficient methodology for estimating the state of power systems, taking into account its dynamic behaviour and timevarying parameters.
A few estimation methods have taken place in the last decade as an alternative procedure to determine the series impedance and shunt admittance of power transmission lines. The parameter estimation is carried out in the time-domain employing current and voltage measurements at both the input and output terminals, for example, fault records obtained by digital relays in the time-domain [4,5]. Other estimation methods are developed in the frequency domain, by using phasor measurements Units (PMUs) records at both terminals of the line [6][7][8][9].
An interesting method is proposed from the resolution of transmission line equations, which can be represented as an equivalent -circuit, and phase measurements at both line terminals. This estimation method is developed from the set of equations that represents the Kirchhoff 's laws by phase and the systematic errors in the synchrophasor measurements. In this sense, an algorithm was proposed based on first-order approximations of the Taylor series and cost function minimisation for errors and system parameters [10]. This method has proven to be effective for real-time estimation of transmission line parameters with short and medium lengths. Another relevant contribution is presented in [6], where the authors present a method based on multiple measurements, selected in an adaptative form. These measurements compose a system of non-linear exponential equations, where each parameter has a standard deviation, and the solution is found using a numerical method based on Newton's method. Moreover, the authors in [11] used a moving window technique and a variation of the total least-squares (TLS) method to obtain voltage, current and complex power of PMUs and estimate the parameters of the transmission line using a pi-equivalent circuit. Finally, in [12], a TLS approach for line parameter estimation is proposed using only phasor measurements. The work focuses on noise modelling that considers the differences between current transformer (CT) measures and capacitive voltage transformer (CVT) measures. Besides, the different samples are concatenated in one unique matrix to solve the TLS problem.
In this work, we propose and compare two approaches for estimating transmission line parameters based on the least square minimisation. Both methods use measurements from the input and output terminals of a -modelled network. In the first method, noise modelling is represented by an additive term acting upon the network measurements, and thus, the problem has an ordinary least-square (OLS) formulation. In the second approach, noise is also incorporated in the observation matrix, resulting in an estimation problem that can be solved using the TLSmethod. Differently from other studies where the TLS is applied, we have simplified the noise modelling since -model for a medium-length transmission line the equations incorporate many products of Gaussian random noises and we have settled upon a solution that is a compromise between the most general formulation in which we have obtained the convergence for the TLS and oversimplified modelling that is commonly found in other studies. In addition, we model each entry of the matrices as unique random variables, so that when two entries are equal, then the noise acting upon them is also the same. The methods are tested in simulation data of a real medium-size transmission line and the influence of the noise modelling on the estimation of line parameters is analysed by comparing the results obtained from both the OLS and TLS methods. Therefore, we can highlight the main contributions of this research as: first, we propose a new set of steady-state phasorial equations for a medium-length transmission line. Then, we consider the noises acting upon each measurement on the OLS setup, and for the TLS setup, we have also considered noises acting upon the observation matrix in order to account for model uncertainties and non-linearities. Finally, we setup two estimation methods using the classical solution of the OLS and the TLS. We finish the paper by comparing the performances of the estimators and reasoning about their solutions according to their complexities.
The work is structured as follows: Section 2 describes the mathematical modelling of the transmission line and the problem statement; Section 3 introduces the OLS solution for the problem whereas Section 4 describes the TLS solution; Section 5 presents the phasor measurement simulations and estimation results; finally, Section 6 presents the main conclusion obtained from results and complementary discussions.

TRANSMISSION LINE MODEL
For the modelling proposed in this work, a symmetric and balanced three-phase transmission line is used, classified as of medium-length according to [13]. The equivalent circuit (positive sequence) for this line is presented in Figure 1, wherė V s ,V r ,̇I s ,̇I r are, respectively, the voltage and the current at the sending and receiving ends. From the equivalent circuit in Figure 1, it is possible to describe the phasor equations of the proposed estimation method for the transmission line parameters R, b, X . With b and where is the angular frequency of the system. By applying such laws to the circuit, the following equations are obtained: where V sr , V rr , I sr , I rr are the real parts of the phasor measurements and V si , V ri , I si , I ri are the imaginary parts of the phasor measurements. Note that the parameter b is calculated by using just measurements of the real and imaginary parts of the current and voltage phasors. Thus, it is possible to replace b in (2)(3)(4)(5). In this sense, these equations do not depend on b. Therefore, the calculation can be separated into two equations, one for b and other matrix equation for parameters R and X : where,Ȳ is a 4 × 1 vector, is a 2 × 1 vector, andH is a 4 × 2 matrix. The elements of the matrix equation are described as where

MODELLING AND SOLUTION USING OLS METHOD
Each phasor measurement is obtained from measurement devices and by using a particular estimation method, for example, fast Fourier transform (FFT). Although PMUs are extremely accurate, when estimating parameters using measurements, one must consider the presence of uncertainties in the measurements and incorporate noise modelling into the model [10,14].
Note that Equation (6) is scalar and composed of phasor measurements. Then, it is possible to model the error incorporating a term of additive noise w 1 where n = 1, 2, … , N is the index of each observation, N is the total number of observations, w 1 [n] is a zero mean Gaussian noise with variance 2 represents the expected value of a random variable, therefore, In the matrix equation represented in (7), an initial approach to modelling the noise is to consider it acting upon the output vector Y[n] where n = 1, 2, … , N . It is important to state thatỸ [n] ∼  (0; C). The vectorȲ has just two different elements, then the covariance matrix C has the following format where 2 1 is the variance associated with measure V sr − V rr and 2 2 is the variance associated with measure V si − V ri . In addition, it is assumed that the different measures are uncorrelated.
In the OLS approach, it is necessary to find the smallest vector correction Y such that Thus, the OLS solution̂[n] is obtained as an optimisation problem described by where || Y [n]|| 2 a 2-norm of the correction vector.
Under the condition that H[n] is full column rank, the problem has a unique solution̂[n], given bŷ Note that, if the system operates at the same conditions during the observations (n samples), then the matrix H[n] is fixed and thus the covariance matrix is also fixed for all n. Besides, since the estimator is unbiased, it is possible to state that

MODELLING AND SOLUTION USING TLS METHOD
The problem of estimating the parameters of a transmission line is basically a problem of error in variable, called EIV by [15]. Section 3 presented the approach that considered possible errors and inaccuracies acting upon the output vector (Y) by means of an additive Gaussian white noise. In this case, it is possible to obtain the solution using OLS method, that is, in terms of implementation, a easier method compared to others estimation methods [16].
By observing (7), matrix H is also composed of phasor measurements. Then, one should also consider a term that incorporates possible inaccuracies into this matrix. A important point that must be stressed out is that given that matrix H[n] is composed of multiplications and sums of measurements, the correct modelling should consider the presence of noise in all terms of the matrix, making the model more accurate. Indeed, if one would correctly describe these noises, more complex distributions than the Gaussian distribution should be used in the matrix terms since there are random variables multiplications. In this sense, it would result in great computational burden and sensitivity to noise due to the characteristic of least-squares estimators. Therefore, a simplification is made by considering a Gaussian term that incorporates noise into the whole matrix. This optimisation problem presented above is the best physical description to the problem of errors in variables, but in some cases, called in the literature as non-generic TLS problems, this formulation does not have a solution [18]. There is an alternative way to write the problem of classical TLS, using the following reasoning: If there is a vector [n] ∈ ℝ 2×1 such that

E = [H[n] + H[n] Y [n] + Y [n]],
it is possible to affirm that the rank of matrixÊ is less than or equal to 2, that is: Therefore, the optimisation problem that describes the TLS method can be written as a low rank approximation problem For this problem, it is possible to find a unique solution, under the conditions described by the following theorem: Theorem 1. (Matrix approximation)( [19]). Let E = U V T be the singular value decomposition of E ∈ ℝ m×(k+d ) and partition the matrices U ∈ ℝ m×(k+d ) , =: diag( 1 , … , k+d ) ∈ ℝ (k+d )×(k+d ) , and V ∈ ℝ m×(k+d ) as follows where U 1 ∈ ℝ m×k , 1 ∈ ℝ k×k and V 1 ∈ ℝ (k+d )×k . Then the rankk matrix E * = U 1 1 V T 1 is such that The solution E * is unique if only if k+1 ≠ k .
Now, consider the singular value decomposition of the expanded matrix E ] T .
Using (1) and some algebraic manipulation [20], it is possible to write that, if V YY [n] is non-singular for all n, a TLS solution exists and, if k+1 ≠ k , this solution is unique. In the case when the TLS solution exists and is unique, it is given bŷ Note that, for each observation, that is, for each value of n, it is necessary that V YY [n] is non-singular and k+1 ≠ k in the singular value decomposition. Furthermore, this method can be classified as a matrix approximation method, in particular, this means that it does not have a relation with the minimisation of the standard deviation of the noise. Thus, it is expected that the estimator will be sensitive to noise modelling.

RESULTS
In this section, the application and efficacy of the estimation methods will be discussed through computational simulations. The transmission line analysed in this work has the parameters presented in the Appendix, according to what was developing in [21]. Each lumped parameter R, L, and X is calculated by multiplying the distributed parameter R ′ , L ′ , and X ′ to the line length l , that is: In Figure 2, R shunt is necessary to ensure numerical convergence of the simulation, and its numerical value should be small compared to the other lumped parameters of the line. Therefore, the shunt resistance used was R shunt = 0.05 Ω. The load impedance (Z load ) is chosen so that the line transmits 90% of its characteristic power. In this case, this impedance is equal to 653.2 Ω. With those values, the matrixH has the following inputs:

Ordinary least-squares results
In order to complete the problem description for OLS method, it is necessary to describe the noise data used, that is, describe w 1 [n] andỸ using the data from the chosen transmission line. The standard deviation of the noise w 1 [n] was chosen as This value was chosen as 12% of real susceptance value. Considering the systematic error present in PMU measurements and the noise modelling for the system [10,14,22], such value is adequate.
Using this same methodology, that is, considering 12% of deviation for the measurements, the covariance matrix C is Fixing N = 100000 in (10) and (12), it is possible to generate three histograms, one for each parameters b, R and X , as depicted in Figure 3.
To determine the estimated value of each parameter, the average value over each set of N points is calculated using the following:x The dispersion of the data obtained is measured by the standard deviation (16) Then, it is possible to determine the absolute value of the relative error where x ′ is the exact value for each parameter. The statistical indicators of each parameter, using 100,000 realisations, are summarized in Table 1 and the absolute value of the relative error is presented in Table 2.
Using the OLS solution, the percentage deviation obtained, shown in Table 2, is small compared to others solutions presented by the references [6,7,11]. Another relevant aspect that must be pointed out is that, by using this modelling, the OLS method is an unbiased estimator. Thus, if the number of samples goes to infinity the vector parameter estimated converge to the true vector parameter.

TLS solution
In order to establish a comparison between the two methods, the noise modelling for b andȲ , represented by w 1 [n] andỸ [n], was maintained the same as represented in Section 5.1.
To complete the problem description in the case of the TLS solution, it is necessary to describe the matrixH[n]. First, the matrix, or formally the vectorisation of the matrix, has zero mean. The covariance matrix C H depends on the elements that composeH. If two entries ofH are equal or differ only by the signal, the noise model employed for them will be identical.Thus, observing the matrixH, there are four different noise modelling, that can be interpreted as four different standard deviations. Besides, each different entry of the matrixH[n] is uncorrelated with the others. Therefore, the covariance matrix C H has the following format Where, each standard deviation i j is equal to 12% of the true value of each matrix entry (h i j ) 9.
Taking N = 100, 000 and using (14), the histograms for susceptance, resistance, and reactance were generated and presented in Figure 4.
For the TLS solution, the statistical performance indicators for the estimated parameters are presented in Table 3 and the relative deviation to the true value of each parameter is shown in Table 4.
One can note that the estimated value for parameter b had no change in the different methods, as the modelling of this parameter is the same in both methods. For the parameters R and X , the standard deviation and the relative error are bigger in the TLS estimation. Nevertheless, the relative error of the estimated parameters is still small [6,7,11], less than 2%. This shows that the estimation method applied is adequate for the proposed noise modelling. For the first noise modelling, the OLS estimator is unbiased [23], thus with an increase in the number of realisations (N ) estimation vector (̂) is expected to approximate the true parameter vector ( ).
In the TLS estimation, this is not true. There is a systematic error in the estimated parameters that depend on noise modelling and it does not converge to zero when N goes to infinity. In the proposed model this systematic error is less than 2%, but it can be increased if the noise modelling becomes more complex. Besides, as the TLS estimation does not have an intrinsic relationship with variance minimisation, the dispersion obtained on the estimation becomes unpredictable as further noise terms are inserted into the model.
In [12] an estimation method based in the TLS is also proposed; however, the modelling equations are different and do not predict that the susceptance can be estimated directly from the measures. The matrix used in the estimation of the parameters has the same dimension as that presented in this work, therefore the computational complexity is the same. Furthermore, the matrix (H ), in [12], not very sensitive to variations and it is necessary to carry out a treatment in the phasors, which is not necessary for the method proposed in this work. Also, the values used for the standard deviation of noises were less than 0.05 in [12], and even so, for the constant load condition, the relative percentage deviations obtained for the parameters were significant.

CONCLUSION
Both OLS-and TLS-based estimation methods have shown acceptable relative deviations from the real line parameters when compared with well-established recent estimation methods in the literature. In the first estimation method, solved by the OLS method, noise modelling is more straightforward, which makes the dispersion of the estimation bounded.
In the TLS method, noise modelling could be more complex and the estimation results show a strong dependency on the way noise is modelled. In the proposed method, we have simplified the noise modelling by considering a Gaussian white noise acting upon the observation matrix. The results showed that there was no significant compromise in the estimated parameters, that is, no significant increase in the standard deviation and the relative percentage deviation in comparison with the first method. Nevertheless, unlike the first method, it is not possible to predict the performance of the TLS method when there is a variation in the system parameters or even when noise modelling is made different.