Probabilistic Techniques for Three-phase Load Flow Analysis P. Caramia, IEEE Member G. Carpinelli, IEEE Member V. Di Vito P. Varilone, IEEE Member Abstract-- Some techniques are presented to obtain the probability density functions of phase-voltages in an unbalanced three-phase power system. The techniques are based on the Monte Carlo simulation applied to the non linear and linear three-phase load flow equations, on convolution process and on the Pearson distributions. These methods are compared in terms of time execution and accuracy in the evaluation of phase-voltage and unbalance factor probability density functions and in particular regarding their 95 th percentiles, being these quantities the statistical measures of greatest interest in many International Standards for Power Quality. Several numerical applications are presented and discussed with reference to the three phase unbalanced IEEE 13-bus distribution test system. Index Terms: Three-phase load flow - Unbalances – Probabilistic approaches I. INTRODUCTION As known, the unbalances in power systems are not always negligible, as in the case of single-phase AC traction plants, electrical furnaces and long untransposed lines [1]; moreover, several distribution systems are known to have unbalanced lines and line sections carrying a mixture of single, double or three-phase loads [2]. In such systems, unavoidable uncertainties affect the input data of the modeling for the evaluation of the unbalances; these uncertainties are mainly due to time variations of phase- load demands and of network configurations. The variations have a random character and the only way can describe the behaviour of such characteristics is in statistical terms. Probabilistic modeling for the evaluation of the unbalances proposed to date provides for the evaluation of the voltage unbalances with Monte Carlo simulation procedures [3] or with simplified methods [4]; these last methods allow to strongly reduce the computational efforts in evaluating mean value and covariance matrix, but they do not allow to determine the true analytical form of the phase-voltage or unbalance factor probability density functions. Nowadays, however, the interest of Standards is devoted to the percentile evaluation [5, 6]; in the European Standard EN 50160 [6], for example, the 95 th probability weekly value of the unbalance factor should not exceed the specified limit. Therefore, the knowledge of the whole probability density functions is mandatory for Standard application. Pierluigi Caramia, Vittorio Di Vito and Pietro Varilone are with the Department of Industrial Engineering of the University of Cassino, Cassino (FR), Italy – Guido Carpinelli is with the Department of Electrical Engineering of the University of Napoli Federico II, Napoli (Italy) , (e-mail: caramia@unicas.it, v.divito@unicas.it, varilone@unicas.it, carpinelli@unicas.it) In this paper, several techniques are compared for calculating the phase- voltage and unbalance factor probability density functions and, therefore, their percentiles. The probabilistic techniques are: • non linear Monte Carlo simulation; • linear Monte Carlo simulation; • convolution process; • Pearson’s distributions. The first technique applies Monte Carlo simulation to the non linear Three-phase Load Flow equations. The second technique applies Monte Carlo simulation to the non linear equation system linearized around the expected value region. The third technique is based on a convolution process applied after the previous linearization. The last technique is based on the use of Pearson distribution functions [7], that are used to approximate the phase-voltage or unbalance factor probability density functions of interest. In the last part of the paper the implementation and practical application of all the techniques are discussed, comparing them on the unbalanced IEEE 13-bus distribution test system [2]. II. PROBABILISTIC THREE-PHASE LOAD FLOW The Three-phase Load Flow equations here considered are expressed as [1]: ( ) ( ) Φ , ) ( Φ , ) ( sp sp U Q Q U P P = = (1) ( ) ( ) ( ) Φ , ) ( Φ , sp sp U U U U P P gen gen gen gen = = (2) where in the case of probabilistic three phase load flow: ( ) ( ) sp sp , Q P input random vectors of active and reactive powers specified at each of the three phases of load and generator terminal busbars, ( ) sp gen P input random vector of three phase active power specified at each internal generator busbar without the slack, ( ) sp gen U input random vector of voltage regulator law specified at each terminal generator busbar, U, Φ state random vectors of phase-voltage magnitudes and arguments. The equations (1) represent the phase power balance equations (active and reactive) at load and generator terminal busbars while the equations (2) represent the active power and voltage regulation balance equations at generator busbars. Voltage regulators and Q-limits of generators can be included. Equations (1) and (2) constitute a system of non linear algebraic equations; it can be expressed in compact form as: 0-7803-7967-5/03/$17.00 ©2003 IEEE Paper accepted for presentation at 2003 IEEE Bologna Power Tech Conference, June 23th-26th, Bologna, Italy