Contents The Role of Affine Arithmetic in Robust Optimal Power Flow Analysis

In many real-life situations, we do not know the actual dependence y = f(x1, . . . , xn) between the physical quantities xi and y, we only know expert rules describing this dependence. These rules are often described by using imprecise (“fuzzy”) words from natural language. Fuzzy techniques have been invented with the purpose to translate these rules into a precise dependence y = f̃(x1, . . . , xn). For deterministic dependencies y = f(x1, . . . , xn), there are universal approximation results according to which for each continuous function on a bounded domain and for every ε > 0, there exist fuzzy rules for which the resulting approximate dependence f̃(x1, . . . , xn) is ε-close to the original function f(x1, . . . , xn). In practice, many dependencies are random, in the sense that for each combination of the values x1, . . . , xn, we may get different values y with different probabilities. It has been proven that fuzzy systems are universal approximators for such random dependencies as well. However, the existing proofs are very complicated and not intuitive. In this paper, we provide a simplified proof of this universal approximation property. 1 Formulation of the Problem It is important to determine dependencies. One of the main objectives of science is to find the state of the world and to predict the future state of the world – both in situations when we do not interfere and when we perform a certain action. The state of the world is usually characterized by the values of appropriate physical quantities. For example: – we would like to know the distance y to a distant star, – we would like to predict tomorrow’s temperature y at a given location, etc. In some cases, we can directly measure the current value of the quantity y of interest. However, in many practical cases, such a direct measurement is not possible – e.g.: – while it is possible to measure a distance to a nearby town by just driving there,