IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 20, NO. 4, NOVEMBER 2005 1843 Probabilistic Load-Flow Computation Using Point Estimate Method Chun-Lien Su, Member, IEEE Abstract—A new probabilistic load-flow solution algorithm based on an efficient point estimate method is proposed in this paper. It is assumed that the uncertainties of bus injections and line parameters can be estimated or measured. This paper shows how to estimate the corresponding uncertainty in the load-flow solution. The proposed method can be used directly with any existing deterministic load-flow program. For a system with uncertain parameters, it uses calculations of load flow to cal- culate the statistical moments of load-flow solution distributions by weighting the value of the solution evaluated at locations. The moments are then used in the probability distribution fitting. Performance of the proposed method is verified and compared with those obtained from Monte Carlo simulation technique and combined simulation and analytical method using several IEEE test systems. Index Terms—Point estimate method, power system planning, probabilistic load flow. I. INTRODUCTION L OAD flow study is required in power system expansion planning, operational planning, real-time operations and control. It provides the analyst with the steady state of the system for a specified set of power generation, loads, and network conditions. Traditional load-flow approach is the deterministic load flow where the system condition represents a snapshot in time or a set of deterministic values chosen by the analyst for each input variable. So, its accuracy depends on the knowledge of the input data. In an open access environment, the generation patterns are not certain, the paths of supply are more diverse, and future load characteristics become more unpredictable. In the case of statistical uncertainty associated with the input data, a point estimate does not exactly indicate the whole result. In system planning, it is desirable to assess bus voltages and line flows for a range of loads, generations, and network conditions. Performing conventional load-flow computations for every possible or probable combination of bus loads, generating patterns, and network topologies is impractical because of the large computation efforts required. From a system planning point of view, it has been shown worthwhile to approach the problem as a probabilistic one. The probabilistic load-flow (PLF) study could take into account uncertainty in Manuscript received December 27, 2004; revised June 23, 2005. This work was supported by the National Science Council of Taiwan under Grant NSC 93-2213-E-022-004. Paper no. TPWRS-00683-2004. The author is with the Department of Marine Engineering, National Kaoh- siung Marine University, Kaohsiung 805, Taiwan, R.O.C. (e-mail: cls@mail. nkmu.edu.tw). Digital Object Identifier 10.1109/TPWRS.2005.857921 the load-flow computations and calculate the system states and branch flows based not only on expected average or peak values of input data. Instead of obtaining a point estimate result by the deterministic load flow, the PLF algorithm evaluates probability density functions and/or statistical moments of all state variables and output network quantities to indicate the possible ranges of the load-flow result. Many PLF methods have been proposed to study load-flow uncertainty problem [1]–[9]. These methods can be classified as simulation method, analytical method, or by using a combi- nation of both. The simplest evaluation of the PLF problem is through Monte Carlo simulation (MCS). This method requires that the data involved to be assigned a probability distribution that characterizes the possible variation in the parameters. The random values from these distributions are selected and used to arrive at an estimate of load-flow solution. A large computation effort is required for the MCS method. To reduce the computational effort in solving PLF problem, several analytical approaches were proposed to estimate the load-flow solution distributions [1]–[6]. In [1], a probabilistic dc load-flow model was proposed to consider nodal data uncertainty and to find the distributions of branch flows. [2] used a direct and efficient approach based on the principle of statistical least square estimation to analyze the effects of nodal data uncertainty on all network output quantities and to obtain the expected value and variance of the load-flow solution. A discrete frequency-domain convolution technique based on fast Fourier transformation and linearized power flow equation was used in [3] to enhance the computation accuracy. [4] used a dc load-flow model combining the concept of Cumulants and Gram-Charlier expansion theory to consider the bus injection uncertainties and to compute cumulative distributions of net- work branch flows with less computation effort. [5] proposed a new PLF algorithm based on linearized models to compute the load-flow solution distributions through nonlinear power flow equations. A PLF method was proposed in [6] to consider the bus power injection uncertainty and system operating strategy. The main advantage of the analytical approaches mentioned previously is to avoid a large number of simulations required in MCS method, but more assumptions and complicated math- ematical computations are required for these methods. A MCS based on linear power flow equations combined with analytical convolution technique is used in [7] to simplify computation process and maintain sufficient computation ac- curacy. [8] proposed a new PLF algorithm combining MCS and multilinearized load-flow equations to sufficiently and efficiently evaluate all load-flow result quantities. To consider 0885-8950/$20.00 © 2005 IEEE