Symmetric AC fuzzy power flow model Eduardo M. Gouveia a, * , Manuel A. Matos b a ESTV – School of Technology of Viseu, Polytechnic Institute of Viseu, Campus Politécnico de Repeses, 3504-510 Viseu, Portugal b INESC Porto – Instituto de Engenharia de Sistemas e Computadores do Porto FEUP – Faculdade de Engenharia da Universidade do Porto, Rua Dr. Roberto Frias, n° 378, 4200-465 Porto, Portugal article info Article history: Received 30 December 2006 Accepted 11 December 2007 Available online 13 March 2008 Keywords: Linear programming Nonlinear programming Power flow Fuzzy models Optimization abstract Power flow calculations are one of the most important computational tools for planning and operating electric power systems. After the stabilization of the deterministic power flow calculation methods, the need to capture uncertainty in load definition lead first to the development of probabilistic models, and later to fuzzy approaches able to deal with qualitative declarations and other non-probabilistic infor- mation about the value of the loads. Present fuzzy power flow (FPF) calculations use typically incremental techniques, in order to obtain a good approximation of the fuzzy state variables. However, these models and procedures are not entirely satisfactory for the evaluation of the adequacy of the electric transmis- sion system, since they are not completely symmetric. In this paper, we show how to perform the detailed calculation of the state variables of the FPF problem in an exact and symmetrical way, by means of solving multiple optimization problems. The procedure is illustrated using the IEEE 118 test system. Ó 2008 Elsevier B.V. All rights reserved. 1. Introduction The recent changes in the organization of the electric sector all over the world forced the unbundling of the generation, transmis- sion, distribution and commercialization activities, previously inte- grated vertically in the traditional utilities. Among many other consequences, these changes lead to the need of evaluating the adequacy of the transmission system independently of the genera- tion system. In the European Union, for instance, Directive 2003/ 54/EC states that the transmission system operator (TSO) must en- sure the ‘‘long-term ability of the system to meet reasonable de- mands for the transmission of electricity”. Note that this is more than just performing a load forecast exer- cise, since there is also uncertainty regarding generation, due to the market influence regarding scheduled units and generation amounts. Since these phenomena are hardly described in terms of probabilistic models, fuzzy approaches seem adequate to cap- ture the possible variations in load and generation and help check- ing the adequacy of the transmission system. The ability of fuzzy models to describe mathematically qualita- tive declarations about load or generation is patent in Fig. 1, where three example situations are addressed: ‘‘load between 10 and 12 MW” (Fig. 1a), ‘‘generation around 50 MW” (Fig. 1b) and ‘‘load possibly between 50 and 60 MW, but may take values between 45 and 65” (Fig. 1c). Intervals are thus a particular case of fuzzy numbers, so we will not make a distinction between them (in fact, we will later use intervals as a basis to the detailed calculation of the fuzzy power flow). Fuzzy power flow may be described as a procedure that uses information like the one just described to obtain a mathematical description of the uncertain values of voltages and branch power flows, and check for possible violations of the branch flow limits or voltage minimum or maximum allowed values. Identification of congestion situations (characterized by a possibility value) is straightforward, conveying important information to the TSO. Sec- tion 2 of the paper reviews the main classic formulations of the FPF, which addressed this problem as an extension of the deter- ministic power flow, normally using linear models around a central operating point and fuzzy arithmetic (Miranda and Matos, 1989; Miranda et al., 1990; Saraiva et al., 1991; Miranda and Saraiva, 1992). In order to overcome some limitations of the preceding models, mainly due to the existence of a slack bus (typical of the determin- istic power flow) that may distort the results, Matos and Gouveia (2005) presented a symmetrical model for the linearized real power flow case (DC model) directly derived from the extension principle formulated by Zadeh (1965). The approach leads to a set of optimization problems (two for each variable and each a-le- vel), in this case linear programming problems. This formulation has the additional advantage of enabling the incorporation of branch limit constraints to detect situations of repression directly in the nodes. The main purpose of the present paper is to extend the methodology to the AC problem, while maintaining the sym- metry of all the buses and obtaining the exact values of the state 0377-2217/$ - see front matter Ó 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.ejor.2007.12.044 * Corresponding author. E-mail addresses: egouveia@elect.estv.ipv.pt (E.M. Gouveia), mmatos@inescpor- to.pt (M.A. Matos). European Journal of Operational Research 197 (2009) 1012–1018 Contents lists available at ScienceDirect European Journal of Operational Research journal homepage: www.elsevier.com/locate/ejor