8 th International Conference on Probabilistic Methods Applied to Power Systems, Iowa State University, Ames, Iowa, September 12-16, 2004 Copyright Iowa State University, 2004 Abstract—The aim of this paper is to present a methodology based on interval mathematics to deal with load uncertainty in power electric systems. The approach is suitable to model two types of uncertainty in power flow analysis, namely: (i) the influence of the load measurement errors at all system buses on the voltage profile; (ii) the voltage profile behavior under a load variation during a specific period of time. This methodology entails an interval power flow model and the resulting interval non-linear system is solved using Krawczyk’s method. We implemented the algorithm in the Matlab environment using the IntLab toolbox. Results for IEEE test-systems and a comparison with related work are presented. Index Terms—interval mathematics, interval non-linear systems, IntLab library, Krawczyk’s algorithm, load uncertainty, load variation, power flow. I. INTRODUCTION URRENTLY a reliable and economical analysis of power systems is essential for their operation. The loads at buses are usually uncertain since they can vary in a fast and disordered way. It is desirable to study and develop a reliable methodology in order to deal with load uncertainty in power flows. We attempt the solution of the following problems: (a) What is the influence of the load measurement errors on the voltage profile in a power electric system? (b) How can we estimate the voltage profile behavior under a load variation in a given period of time? The solution of problem (a) is a classic task of interval mathematics [1]-[3]. Regarding problem (b) and considering that both minimum and maximum load values in a specific period of time can be estimated, it is possible to evaluate the respective minimum and maximum voltage levels using a traditional power flow analysis [4]-[8]. However, if an interval approach is used to deal with the whole interval of load variation, then all possible combinations of load values at all system buses are taken into account. In this sense, the former conventional analysis (i.e., just considering either minimal or maximal values) is a particular case. In addition, interval methods are conservative and provide automatically This work was supported in part by CTINFO/CNPq and FAPERGS. Luciano V. Barboza is with the Electrical Engineering Department, Catholic University of Pelotas, 96010-000, Pelotas, RS, Brazil (phone: +55 53 284-8288; fax: +55 53 225-3105; e-mail: luciano@ucpel.tche.br). Graçaliz P. Dimuro and Renata H. S. Reiser are with the Computer Science Department, Catholic University of Pelotas, 96010-000, Pelotas, RS, Brazil, e- mail: {liz,reiser}@ucpel.tche.br. verified results, that is, interval voltages that include all possible punctual results and computational errors. In order to assess the reliability of the proposed methodology, we have analyzed the results for IEEE test- systems. In this paper, we consider just the systems with 6 and 118 buses. The implementation is carried out in the Matlab environment using the IntLab toolbox [9], [10]. This paper is organized as follows. Section 2 presents a summary of Interval Arithmetic and IntLab toolbox. The interval methods for solution of non-linear systems are discussed in Section 3, emphasizing the Krawczyk’s operator [11], [12]. Section 4 states the interval load flow problem. The numerical results are discussed in Section 5. The conclusions and comparison to related work are presented in Section 6. II. INTERVAL ARITHMETIC AND INTLAB TOOLBOX Interval Mathematics [1]-[3] considers a set of methods for handling intervals that approximate uncertain data. These methods are based on the definition of both interval arithmetic and optimal scalar product [13]. The maximal accuracy principle guarantees (by means of the outward rounding) the automatic control of errors in numerical computation. A real interval X is a nonempty subset of real numbers R, [ ] { } 1 2 1 2 , R| X x x x x x x = = ∈ ≤ ≤ , where x 1 is the infimum and x 2 is supremum. The set of real intervals is denoted by IR. An interval [ ] 1 2 , IR X x x = ∈ may not be representable on a machine if 1 2 and x x are not machine numbers. In order to obtain a rounded interval X % such that X X ⊆ % (i.e., X % is an approximation of X), 1 2 and x x must be rounded downward and upward, respectively, which is called outward rounding. The midpoint, the diameter and the radius of an interval X are given, respectively, by ( ) 1 2 1 ( ) 2 mid X X x x = = + ( , 2 1 ( ) diam X x x = - and ( ) ( ) 2 diam X rad X = . X can be also denoted by ( ) , ( ) X mid X rad X = . Interval arithmetic operations are defined such that the interval result encloses all possible real results [1], which guarantees the reliability of interval methods. The elementary operations are defined by { } { } | , , for ,,, X Y x y x X y Y ∗ = ∗ ∈ ∈ ∗∈ +-× ÷ , and, for Towards Interval Analysis of the Load Uncertainty in Power Electric Systems Luciano V. Barboza, Graçaliz P. Dimuro, and Renata H. S. Reiser C