Discipline: Technology and Engineering
Subcategory: Electrical Engineering
Sheriff Sadiqbatcha - California State University, Bakersfield
Co-Author(s): Saeed Jafarzadeh, California State University, Bakersfield
Uncertainty stems from lack of complete information and comes in many guises and is independent of the kind of methodology used to handle it. At the empirical level, uncertainty is an inevitable companion of any measurement, resulting from errors and limitations. Regardless of their sources, uncertainties are classified into two major classes: vagueness and randomness. Vagueness often results in linguistic uncertainties, and randomness results in stochastic uncertainties. Most research studies on renewable energies rely only on stochastic methodologies to represent and address these uncertainties in renewable resources. However, renewable energy generation tightly depends on meteorological data, and meteorological quantities contain linguistic uncertainties. Therefore, the available methodologies lack a way of representing linguistic uncertainties. Fuzzy logic is a powerful tool for analysis of linguistic uncertainties. In this research study, we consider systems of linear equations with fuzzy coefficient.
It has been shown in a recent study that fuzzy linear equations are direct extensions of interval linear systems of equations. In our study we adhere to this concept where we use alpha cuts to break down the given fully fuzzy system of equations into a number of interval equations. We then systematically solve the resulting system of interval equations at each alpha value to obtain the solution of the fuzzy linear equations. Solving a system of fuzzy equations using interval theory has been proposed in a number of other studies, but the key point that differentiates this study is that we investigate this problem to its deepest roots. We first start with the simplest form of the interval equation, which is the first order equation in the form Ax=b. Some studies in the literature on this topic fail to properly solve the interval equation by using certain algebraic properties that do not hold for interval equations. For instance, division in interval arithmetic is not the inverse of multiplication. We propose a new method of solving this first order interval equation, which we utilize extensively in the process of solving the fuzzy equations.
At each alpha cut, the interval linear system obtained from the fuzzy system is solved using LU decomposition on the coefficient matrix, then forward and backward substitutions are used to solve for the unknown values, whilst the whole time heeding to the proposed method of solving first order interval equations. The resulting fuzzy membership functions have been found to precisely satisfy the original fully fuzzy system.
Applications for our proposed method can be found in, but not limited to, the fields of Economics and Finance, Electric Power Systems, Circuits and Systems, Biological System, and Meteorological Sciences. Essentially, our method can be utilized at any scenario where a linear set of equations with fuzzy coefficients needs to be solved to obtain the unknown values. However, our main focus in this study is on applications of the theoretical results on addressing the intermittency of renewable energies on electric power systems. In particular, we use the theoretical results for power flow analysis in a power system with high levels of renewable energy penetration. The outcome of this research study can significantly advance the state-of-the-art in the fields of renewable energy and power systems by better quantifying natural uncertainties that exists in these systems.
Funder Acknowledgement(s): United States Department of Defense; National Science Foundation.
Faculty Advisor: Saeed Jafarzadeh, firstname.lastname@example.org