Non-Linear Least Squares and Maximum Likelihood Estimation of Probability Density Function of Cross-Border Transmission Losses

In the modern power system, transmission losses play an increasingly important role in determining the costs of transmission system operators, in particular in cross-border energy exchange. A variety of transmission losses calculation methods are present in scientific literature in recent years, but regularly neglecting the measurement uncertainty, which is an important contribution in calculating the final cost of exchanged energy. Due to the significant cost of transmission losses in total costs, all transmission system operators are interested in discovering the probabilistic nature of transmission losses as a fundamental requirement for finding the fair method for transmission losses allocation. In this paper, transmission losses are simulated on 110-kV cross-border transmission line using an adaptive Monte Carlo method. The probability density estimation procedure is performed by the non-linear least-squares method, using the Levenberg-Marquardt algorithm. The Gaussian, log-normal, Rayleigh, four-parameter beta, generalized trapezoidal, and the sum of uniform and normal distribution are fitted and the quality of the distribution estimates is compared according to the corresponding values of the Kolmogorov-Smirnov statistic. Furthermore, an additional example presents a distribution fitting procedure on the zero-impedance data of the same transmission line.