Complete matrix formulation of fault-position method for voltage-dip characterisation G. Carpinelli, P. Caramia, C. Di Perna, P. Varilone and P. Verde Abstract: The method of fault position is useful for characterising power-system performance in the presence of voltage dips due to faults. It is based on short-circuit simulations repeated for all the system nodes and for many points along the system lines: fault voltages that are below a preset threshold are the required voltage dips. These dips are stored in so-called dip matrices which contain only the dips in all the system nodes when faults occur at points along the lines. The paper proposes a new compact analytical formulation of dip matrices for balanced and unbalanced faults in terms of bidimensional vector relations and for site- and system-voltage dip indexes. Compact formulations are very useful tools when several sensitivity analyses have to be conducted to estimate variation of site- and system-voltage dip indexes in relation to possible reinforcement and/ or compensation devices. Graphical presentation of dip matrices is also proposed as a valuable tool to ascertain the critical area for system performance. Numerical applications to an actual transmission system are presented to demonstrate the easy applicability of the model. List of principal symbols # Kronecker tensor product which allows construction of an (MN QP) matrix C starting from an (N P) matrix A and an (M Q) matrix B; each element c ij is an (M Q) submatrix given by a ij B. Hadamard product which allows con- struction of an (N M) matrix D starting from an (N M) matrix E and an (N M) matrix F; each element d ij is given by e ij f ij . diag(A) Diagonal matrix whose off-diagonal ele- ments are zero, and whose (i , i)th diagonal element is the (i ,i )th element of matrix A. j Aj Matrix that contains the amplitude of each element of matrix A. T ¼ 1 1 1 1 _ a 2 _ a 1 _ a _ a 2 2 4 3 5 transformation matrix. I N (N N) identity matrix. u i N (N 1) extraction vector whose i th ele- ment is equal to 1, all other elements being zero. _ a e j2=3p intðAÞ Function that defines a matrix whose each element is the nearest integer of each element of matrix A. k (1 N) failure-rate vector. V df (3N N) fault phase-voltages matrix. _ A (3N 3N) transformation matrix from fault sequence voltages to fault phase voltages. V opn (3N N) fault-sequence-voltage matrix. V p , V n , V o (N N) fault-voltage matrix of positive, negative and zero sequence, respectively. D p , D n , D o (3N N) extraction matrix of positive, negative, and zero sequence, respectively. _ Z p , _ Z n , _ Z o (N N) impedances matrix at positive, negative, and zero sequence, respectively. E pf (N N) positive-sequence prefault vol- tage matrix. E (N 1) positive-sequence prefault voltage vector. V dipx (3N N) dip matrix for assigned thresh- old x. d x (3N N) ‘dip-markers’ matrix for assigned threshold x; each (i, j) element of d x is equal to 1 if the (i, j) element of the matrix j V df j is below the threshold X, otherwise is equal to zero. N dip (N 1) vector of the total number of dips in each node. RFI-X (N 1) RFI-X index vector. K (3N N) failure-rate matrix. VDA (N 1) voltage-dip-amplitude index vec- tor. RECRFI-90 (1 N) vector of the reciprocal values of the RFI-90 indices in each node. W (N 1) weighting vector. 1 Introduction The voltage-dip phenomenon is recognised as one of the most severe disturbances in power quality [1] . It is defined by EN 50160 [2] as ‘sudden reduction of the supply voltage E-mail: verde@unicas.it G. Carpinelli is with Dipartimento di Ingegneria Elettrica, Universit " a degli studi di Napoli Federico II, Napoli, Italy P. Caramia, C. Di Perna, P. Varilone and P. Verde are with Dipartimento di Ingegneria Industriale, Universit " a degli studi di Cassino, Cassino, Italy r The Institution of Engineering and Technology 2007 doi:10.1049/iet-gtd:20050544 Paper first received 7th December 2005 and in final revised form 4th April 2006 56 IET Gener. Transm. Distrib., Vol. 1, No. 1, January 2007