1 On the Properties of the Compound Nodal Admittance Matrix of Polyphase Power Systems Andreas Martin Kettner, Member, IEEE, and Mario Paolone, Senior Member, IEEE Abstract—Most techniques for power system analysis model a subblock of the nodal admittance matrix, which is obtained the grid by exact electrical circuits. For instance, in power flow by removing the rows and columns associated with one single study, state estimation, and voltage stability assessment, the use node (i.e., the slack node), has full rank in practice [21], [22]. of admittance parameters (i.e., the nodal admittance matrix) and hybrid parameters is common. Moreover, network reduction However, this finding cannot be generalized straightforwardly techniques (e.g., Kron reduction) are often applied to decrease the (i.e., for generic polyphase grids, or removal of several nodes). size of large grid models (i.e., with hundreds or thousands of state Recently, the authors of this paper proved stronger properties variables), thereby alleviating the computational burden. How- for Kron reduction and hybrid parameters for monophase grids ever, researchers normally disregard the fact that the applicability (see [23]). Two conditions were used in order to prove these of these methods is not generally guaranteed. In reality, the nodal admittance must satisfy certain properties in order for hybrid properties: i) the connectivity of the network graph, and ii) parameters to exist and Kron reduction to be feasible. Recently, the lossiness of the branch impedances. If these conditions are this problem was solved for the particular cases of monophase and satisfied, then Kron reduction can be performed for any set balanced triphase grids. This paper investigates the general case of zero-injection nodes, and a hybrid parameters matrix can of unbalanced polyphase grids. Firstly, conditions determining be constructed for any partition of the nodes. The theorems the rank of the so-called compound nodal admittance matrix and its diagonal subblocks are deduced from the characteristics proven in [23] only apply to monophase grids, and polyphase of the electrical components and the network graph. Secondly, the grids that can be decomposed into decoupled monophase grids implications of these findings concerning the feasibility of Kron using the method of symmetrical components [24]. As known reduction and the existence of hybrid parameters are discussed. In from power system analysis, the sequence decomposition only this regard, this paper provides a rigorous theoretical foundation works for balanced triphase grids (i.e., grids composed of for various applications in power system analysis. elements whose impedance/admittance matrices are symmetric Index Terms—Admittance parameters, hybrid parameters, circulant), because they can be reduced to equivalent positive- Kron reduction, multiport networks, nodal admittance matrix, sequence networks. Therefore, the generic case of unbalanced polyphase power systems, unbalanced power grids polyphase grids cannot be treated. This paper develops the theory for the generic case, namely I. INTRODUCTION unbalanced polyphase grids. Since the method of symmetrical NHERENTLY, techniques for power system analysis need components cannot be applied, the grid has to be represented I an exact analytical description of the grid. This description by a polyphase circuit, so that the electromagnetic coupling in is normally deduced from an equivalent electrical circuit. For between the phases can be accounted for properly. Therefore, instance, in Power Flow Study (PFS) [1]–[3], State Estimation the generalization to the polyphase case is actually non-trivial. (SE) [4]–[6], and Voltage Stability Assessment (VSA) [7]–[9], More precisely, the electromagnetic coupling is modeled using the use of admittance parameters (i.e., the nodal admittance compound electrical parameters [25]. It is argued that physical matrix) or hybrid parameters (i.e., hybrid parameters matrices) electrical components are represented by polyphase two-port is a common practice. As the solution methods employed for equivalent circuits, whose compound electrical parameters are these applications are computationally heavy (e.g., [10]–[14]), symmetric, invertible, and passive. Via mathematical derivation network reduction techniques, such as Kron reduction [15], are and physical reasoning, it is proven that the diagonal subblocks often applied in order to reduce the problem size. Thereby, the of the compound nodal admittance matrix have full rank if the computational burden is decreased, and the execution speed is network graph is weakly connected. Using this property, it is increased without the use of high-performance computers (e.g., shown that the feasibility of Kron reduction and the existence [16]). However, neither the reducibility of the nodal admittance of hybrid parameters are guaranteed under practical conditions. arXiv:1712.08764v4 [cs.SY] 5 Oct 2018 matrix nor the existence of hybrid parameters are guaranteed In that sense, this paper provides – for the first time in the a priori. In order for this to be the case, the nodal admittance literature – a rigorous theoretical foundation for the analysis matrix has to satisfy certain properties (i.e., the corresponding of polyphase power systems, in particular for applications like diagonal subblocks have to be invertible). PFS, SE, and VSA (cf. [26]–[28]). Interestingly, most researchers and practitioners apparently The remainder of this paper is structured as follows: First, ignore this fact. There exist a handful of publications which the basic theoretical foundations are laid in Sec. II. Thereupon, investigate the feasibility of Kron reduction (e.g., [17]) and the properties of the compound nodal admittance matrix are the existence of hybrid parameters matrices (e.g., [18]–[20]), developed in Sec. III. Afterwards, the implications with respect but their validity is limited. The former base their reasoning to Kron reduction and hybrid parameters matrices are deduced upon trivial cases (i.e., purely resistive/inductive monophase in Sec. IV. Finally, the conclusions are drawn in Sec. V. grids), and the latter establish feeble guarantees (i.e., hybrid multip parameters may exist for solely one partition of the nodes). Other works, which deal with triphase power flow, show that II. FOUNDATIONS This work was supported by the Swiss National Science Foundation through A. Numbers the National Research Programme NRP-70 “Energy Turnaround”. The authors are with the Swiss Federal Institute of Technology of Lausanne Scalars are denoted by ordinary letters. The real and imagi- (EPFL), CH-1015 Lausanne, Switzerland. E-mail: forename.surname@epfl.ch. nary part of a complex scalar z 2 C are denoted by <fzg and 2 =fzg, respectively. Thus, z can be expressed in rectangular Let fRi j i 2 Ig and Cj j j 2 J be partitions of R and C, coordinates as z = <fzg + j=fzg. The complex conjugate of where I := f1; ··· ; jIjg and J := f1; ··· ; jJjg. The block z is denoted by z∗. The absolute value and the argument of formed by the intersection of the rows R (i 2 I) with the i z are denoted by jzj and arg (z), respectively. Thus, z can be columns Cj (j 2 J) is denoted by Mij. That is, M = Mij . 0 0 expressed in polar coordinates as z = jzj \ arg (z). Let MI0×J0 (I ⊆ I, J ⊆ J) be the submatrix consisting of 0 0 the blocks Mij (i 2 I , j 2 J ). Now, consider the particular 0 0 B. Set Theory case I := I n fjIjg and J := J n fjJjg, and define Sets are denoted by calligraphic letters. The cardinality of AB MI0×J0 MI0×{|Jjg a set A is denoted by jAj. The (set-theoretic) difference A n B M = := (9) CD MfjI|}×J0 MjIjjJj of two sets A and B is defined as If D is invertible, the Schur complement M=D of D in M is A n B := fx j x 2 A; x2 = Bg (1) M=D := A − BD−1C (10) The Cartesian product A × B of A and B is defined as The following properties hold (see [32]). A × B := f(a; b) j a 2 A; b 2 Bg (2) Lemma 6. det(M) = det(D) det(M=D). A partition of a set A is a family of sets fAk j k 2 Kg, where K := f1; ··· ; jKjg is an integer interval, for which Lemma 7. Let i 2 I0 and j 2 J0. Then Ak ⊆ A 8k 2 K (3) −1 Aij Bi (M=D)ij = Aij − BiD Cj = = D (11) Cj D Ak 6= ; 8k 2 K (4) Ak \ Al = ; 8k; l 2 K; k 6= l (5) The Kronecker product A ⊗ B of two generic matrices A [ A = A (6) and B is a block matrix, whose blocks (A ⊗ B)ij are defined k as the product of the corresponding element A of A and B. k2K ij That is, the parts Ak are non-empty and disjoint subsets of A, A ⊗ B :(A ⊗ B)ij = AijB (12) whose union is exhaustive. If Ak ( A (8k 2 K), which means The following property holds (see [33]). that fAk j k 2 Kg 6= fAg, the partition is non-trivial. Lemma 8. rank(A ⊗ B) = rank(A) · rank(B). C. Linear Algebra Matrices and vectors are denoted by bold letters. Consider a D. Graph Theory matrix M = (M ) (r 2 R, c 2 C), where R := f1; ··· ; jRjg rc A directed graph G = (V; E) consists of a set of vertices and C := f1; ··· ; jCjg are the sets of row and column indices. T T V( G) := V and a set of directed edges E( G) := E, where The transpose of M is denoted by M . If M = M , then M is called symmetric. The rank of a matrix is the dimensionality E( G) ⊆ f(v; w) 2 V( G) × V( G) j v 6= wg (13) of the vector spaces spanned by its row or column vectors. The following properties hold The sets of outgoing and incoming edges of a vertex v 2 V( G) T jR|×|Cj are denoted by Eout ( G; v) and Ein ( G; v).