Assessment of Nonlinear Solutions Applied to Parameter Estimation of Power Transmission Systems

Abstract

This paper proposes an analysis of the nonlinear least-squares methods applied to parameter estimation of overhead transmission lines and underground/submarine cables in offshore applications. Three optimization algorithms are applied to solve the estimation problem, varying the noise level, number of samples obtained from phasor measurement units (PMUs), and initial guess. The noise modeling is carried out based on a Gaussian distribution, i.e., white Gaussian noise. Three nonlinear solutions are applied to the estimation problem: the Gauss–Newton method, Trust-Region, and the Levenberg–Marquardt method. The accuracy and robustness of each method vary with the number of samples, noise level, load, and most importantly, initial guess. Results show that in generic and practical scenarios, in which parameters are unknown, the Levenberg–Marquardt method proved to be more robust.

Appendix A: Deductions and Expressions

1.1 Appendix A.1: Description of the Exact Matrix Equation

The elements of the exact matrix equation are described by

$$\begin{aligned}&\varvec{{\bar{Y}}}=\begin{bmatrix} {\bar{V}}_{ei}+{\bar{V}}_{si} \\ {\bar{V}}_{er}+{\bar{V}}_{sr} \\ {\bar{V}}_{er}-{\bar{V}}_{sr} \\ {\bar{V}}_{ei}-{\bar{V}}_{si} \\ {\bar{V}}_{er}-{\bar{V}}_{sr} \\ {\bar{V}}_{ei}-{\bar{V}}_{si} \end{bmatrix},\\&\varvec{{\bar{f}}}(\varvec{\theta })= \begin{bmatrix} -(2/b){\bar{I}}_{er}-(2/b){\bar{I}}_{sr} \\ (2/b) {\bar{I}}_{ei}+(2/b){\bar{I}}_{si} \\ \left( {\bar{I}}_{er}+(b/2){\bar{V}}_{ei} \right) R+\left( -{\bar{I}}_{ei}+(b/2) {\bar{V}}_{er}\right) X \\ \left( {\bar{I}}_{ei}-(b/2) {\bar{V}}_{er} \right) R+\left( {\bar{I}}_{er}+(b/2) {\bar{V}}_{ei}\right) X \\ \left( -(b/2){\bar{V}}_{si}-{\bar{I}}_{sr}\right) R+\left( -(b/2) {\bar{V}}_{sr}+{\bar{I}}_{si}\right) X \\ \left( (b/2){\bar{V}}_{sr}-{\bar{I}}_{si}\right) R+\left( -(b/2) {\bar{V}}_{si}-{\bar{I}}_{sr} \right) X \end{bmatrix}. \end{aligned}$$

1.2 Appendix A.2: Complete Expression of $r(\theta )$

The residual vector is defined as

$$\begin{aligned} \varvec{r}(\varvec{\theta })=\varvec{Y}-\varvec{f}(\varvec{\varvec{\theta }}). \end{aligned}$$

where the output vector $\varvec{Y}$ and the nonlinear vector function $\varvec{f}(\varvec{\theta })$ are composed of corrupted measures, i.e., considering noise. Using the definition of the vectors, the residual is completely described as

$$\begin{aligned} \varvec{r}(\varvec{\theta })&= \begin{bmatrix} V_{ei}+ V_{si} +(2/b) I_{er}+(2/b) I_{sr} \\ V_{er}+ V_{sr}- (2/b) I_{ei}-(2/b) I_{si} \\ V_{er}-V_{sr}-\left( I_{er}+(b/2) V_{ei} \right) R-\left( - I_{ei}+(b/2) V_{er}\right) X \\ V_{er}- V_{sr}-\left( I_{ei}-(b/2) V_{er} \right) R-\left( I_{er}+(b/2) V_{ei}\right) X \\ V_{er}- V_{sr}- \left( -(b/2) V_{si}- I_{sr}\right) -\left( -(b/2) V_{sr}+ I_{si}\right) X \\ V_{ei}- V_{si}-\left( (b/2) V_{sr}- I_{si}\right) R-\left( -(b/2) V_{si}- I_{sr} \right) X \end{bmatrix}\\&=\begin{bmatrix} r_1 \\ r_2 \\ r_3 \\ r_4 \\ r_5 \\ r_6 \end{bmatrix}. \end{aligned}$$

where

$$\begin{aligned} \varvec{\theta }=[b \quad R \quad X]^T. \end{aligned}$$

1.3 Appendix A.3: Simulated System Data

See Table 8.

Table 8 Transmission line parameters