Stochastic power flow using cumulants and Von Mises functions L A Sanabria* and T S Dillon** * Dept. of Electrical Engineering, Monash University, Clayton, Victoria 3168, Australia ** Dept. of Computer Science, Latrobe University, Bundoora, Victoria 3083, Australia A method of solution of the stochastic load flow problem based on the method of cumulants and Von Mises function is developed. The method has significant computational advantages when compared to presently existing methods. In addition, it can be used for the solution of problems, where arbitrary distribution functions are used to specify special loads existing in the system. Keywords: distribution systems, load flow, mathematical programming I. Introduction In recent years there has been an increased interest in find- ing ways to solve the probabilistic load flow problem when the random variables involved are not normally distributed. Researchers 1'9 have shown that, generally, the output random variables of the probabilistic load flow problem are not normally distributed, and that the Central Limit Theorem cannot be used as a basis for asserting that they are. Even if the input variables are normally distributed, the output variables will not be normals due to the non- linearity of the power flow equations. On the other hand, for a more realistic representation of the problem, input variables other than normals must be used. The starting point for the probabilistic load flow problem is to find approximations for the power equations. Various first-order and second-order transformations have been proposed. The second-order ones are based on the expan- sion of the equations around a solution point, using the Taylor series2'3. The first-order ones use a linearization of the equations around the deterministic solution 1. The next step is the mathematical manipulation of the input random variables to obtain the output random variables. In the second-order transformation, Sauer-Heydt 2 and Brucoli et al. 3 have represented the random variables by their statistical moments. For the first-order transformations, Received: 21 November 1984, Revised: 20 February 1985 the classical convolutions of random variables using Laplace Transform is normally used. In this paper we solve the probabilistic load flow problem by using cumulants. We will show that the method of cumulants gives more accurate results, and works faster than other recent methods to convolve random variables, such as the Fast Fourier Transform, and the convolution by Laplace Transform. An additional advantage is that our method uses the same subroutines employed in the deter- ministic solution of the problem, so the computer imple- mentation of the method is very simple. It is useful to discuss the method for solving the general probabilistic load flow problem, using a second-order transformation and convolution by using statistical moments 2. From now on we will write RVs for random variables. They will be represented by capital letters. In this method a power equation of the form Y = f(V) (1) is expanded using Taylor series around an operating point (V o, yo). A second-order approximation for Y can be obtained by consitlering V as an RV. The random nature of the variable V is dne to random changes around V°, i.e. V=V°+AV (2) hence, .l.r AVT"I ~V ] iv] = [Wl + [av] +: i_X.~)~.ii~.S~j (3) where: J'J= .=¢o H~k- ~2fi I ~V~ v=¢' Vol 8 No 1 January 1986 0142-0615/86/010047-14 © 1986 Bultterworth & Co (Publishers) Ltd 47