IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 20, NO. 2, MAY2005 773 Cumulant-Based Probabilistic Optimal Power Flow (P-OPF) With Gaussian and Gamma Distributions Antony Schellenberg, William Rosehart, and José Aguado Abstract—This paper introduces the cumulant method for the probabilistic optimal power flow (P-OPF) problem. By noting that the inverse of the Hessian used in the logarithmic barrier interior point can be used as a linear mapping, cumulants can be computed for unknown system variables. Results using the proposed cumulant method are compared against results from Monte Carlo simulations (MCSs) based on a small test system. The Numerical Results section is broken into two sections: The first uses Gaussian distributions to model system loading levels, and cumulant method results are compared against four MCSs. Three of the MCSs use 1500 samples, while the fourth uses 20 000 samples. The second section models the loads with a Gamma distribution. Results from the proposed technique are compared against a 1000-point MCS. The cumulant method agrees very closely with the MCS results when the mean value for variables is considered. In addition, the proposed method has significantly reduced computational expense while maintaining accuracy. Index Terms—Cumulants, optimal power flow (OPF), proba- bilistic optimization. I. INTRODUCTION O PTIMAL POWER FLOW (OPF) is a tool that has been commonly used within the power systems industry for many years [1] and has generally been addressed as a deter- ministic optimization problem. However, it is becoming increas- ingly important that solution methods to the optimal power flow problem be developed to address probabilistic quantities and, thus, transform the optimal power flow problem into the proba- bilistic optimal power flow (P-OPF) problem [2]. Probabilistic programming, or probabilistic optimization, is concerned with the introduction of probabilistic randomness or uncertainty into conventional linear and nonlinear programs [3]. However, the randomness introduced tends to have some structure to it, and this structure is generally represented with a probability density function (PDF) [4]. The goal of the P-OPF problem is to determine the PDFs for all variables in the problem. These PDFs are the distributions of the optimal solutions. A typical example of an uncertain or probabilistic parameter in a P-OPF problem is bus loading. The cumulant method for probabilistic power flow was dis- cussed in [5] and [6]. The present paper briefly outlines the fun- damentals of the cumulant method and presents the adaptation Manuscript received June 10, 2004; revised November 17, 2004. Paper no. TPWRS-00300-2004. A. Schellenberg and W. Rosehart are with the University of Calgary, Calgary, AB T2N 1N4, Canada (e-mail: schellen@enel.ucalgary.ca; rosehart@enel.ucal- gary.ca). J. Aguado is with the University of Malaga, 29071 Malaga, Spain (e-mail: jaguado@uma.es). Digital Object Identifier 10.1109/TPWRS.2005.846184 of the method in [5] to the P-OPF problem using a logarithmic barrier interior point method (LBIPM) [7]-type solution. This paper is structured in the following manner. Section II presents information related to the Edgeworth form of the Gram–Charlier A series. It includes the A series itself, in addi- tion to some background information on Tchebycheff–Hermite polynomials and computation of A series coefficients. Next, in Section III, an overview of the pure Newton step in the LBIPM for numerical programming is provided. In Sections IV and V, the cumulant method is presented, in addition to the proposed application to the P-OPF problem. Numerical results from a system based on the Matpower 9-bus system [8] using normally (Gaussian) and Gamma distributed independent random loads with the proposed cumulant method for P-OPF are detailed in Section VI. Finally, conclusions are presented in Section VII. Two appendixes are included to provide background infor- mation in probability and statistics, focusing on moments and cumulants, as well as information on Gaussian and Gamma dis- tributions. II. GRAM–CHARLIER ASERIES The Gram–Charlier A Series allows many PDFs, including Gaussian and Gamma distributions, to be expressed as a series composed of a standard normal distribution and its derivatives. As a part of the proposed P-OPF method, distributions are re- constructed with the use of the Gram–Charlier A Series. Addi- tional information can be found in [9]. The series can be stated as follows: (1) where is the PDF for the random variable is the th series coefficient, is the th Tchebycheff–Hermite, or Hermite, polynomial, and is the standard normal distribu- tion function (see Appendix I-A). The Gram–Charlier form uses moments to compute series co- efficients, while the Edgeworth form uses cumulants. Since the work presented in this paper is based on cumulants, only Edge- worth’s form of the A series is discussed. Throughout this section, the operator is defined as the derivative with respect to to simplify notation. The remainder of this section is devoted to discussion of the Hermite polynomials and the computations of A series coeffi- cients in (1). 0885-8950/$20.00 © 2005 IEEE