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 ∈ C are denoted by <{z} and 2

={z}, respectively. Thus, z can be expressed in rectangular Let {Ri | i ∈ I} and Cj | j ∈ J be partitions of R and C, coordinates as z = <{z} + j={z}. The complex conjugate of where I := {1, ··· , |I|} and J := {1, ··· , |J|}. The block z is denoted by z∗. The absolute value and the argument of formed by the intersection of the rows R (i ∈ I) with the i z are denoted by |z| and arg (z), respectively. Thus, z can be columns Cj (j ∈ J) is denoted by Mij. That is, M = Mij . 0 0 expressed in polar coordinates as z = |z| ∠ arg (z). Let MI0×J0 (I ⊆ I, J ⊆ J) be the submatrix consisting of 0 0 the blocks Mij (i ∈ I , j ∈ J ). Now, consider the particular 0 0 B. Set Theory case I := I \ {|I|} and J := J \ {|J|}, and define Sets are denoted by calligraphic letters. The cardinality of AB MI0×J0 MI0×{|J|} a set A is denoted by |A|. The (set-theoretic) difference A \ B M = := (9) CD M{|I|}×J0 M|I||J| of two sets A and B is defined as If D is invertible, the Schur complement M/D of D in M is A \ B := {x | x ∈ A, x∈ / B} (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 := {(a, b) | a ∈ A, b ∈ B} (2) Lemma 6. det(M) = det(D) det(M/D). A partition of a set A is a family of sets {Ak | k ∈ K}, where K := {1, ··· , |K|} is an integer interval, for which Lemma 7. Let i ∈ I0 and j ∈ J0. Then Ak ⊆ A ∀k ∈ K (3) −1 Aij Bi (M/D)ij = Aij − BiD Cj = / D (11) Cj D Ak 6= ∅ ∀k ∈ K (4)

Ak ∩ Al = ∅ ∀k, l ∈ 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. k∈K 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 (∀k ∈ K), which means The following property holds (see [33]). that {Ak | k ∈ K}= 6 {A}, 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 ∈ R, c ∈ C), where R := {1, ··· , |R|} rc A directed graph G = (V, E) consists of a set of vertices and C := {1, ··· , |C|} 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) ⊆ {(v, w) ∈ V( G) × V( G) | v 6= w} (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 ∈ V( G) T |R|×|C| are denoted by Eout ( G, v) and Ein ( G, v). Formally Lemma 1. rank(M M) = rank(M) ∀M ∈ C . |R|×|R| |C|×|C| Eout ( G, v) := {e ∈ E( G) | e = (v, w), w ∈ V( G)} (14) Lemma 2. Let the matrices A ∈ C and B ∈ C |R|×|C| E ( , v) := {e ∈ E( ) | e = (w, v), w ∈ V( )} (15) be non-singular. Then, for any matrix M ∈ C , it holds in G G G that rank(AM) = rank(M) = rank(MB). A set of internal edges Eint ( G, W) with respect to W ⊆ V( G) A non-singular complex matrix is called unitary if its inverse contains all directed edges that start and end in W. So equals its conjugate transpose, that is M−1 = (M∗)T . Eint ( G, W) := {(u, v) ∈ E( G) | u, v ∈ W ⊆ V( G)} (16) |R|×|R| T Lemma 3. Let M ∈ C and M = M . Then, M can be T |R|×|R| A cut-set E ( , W) with respect to W V( ) contains all factorized as M = U DU, where U ∈ C is unitary, cut G ( G |R|×|R| directed edges that start in W and end in V( ) \ W. That is and D ∈ R is non-negative diagonal (Autonne-Takagi G factorization, see [29]). If M is non-singular, then D is positive ( ) u ∈ W V( G) , diagonal. ( Ecut ( G, W) := (u, v) ∈ V( G) (17) v ∈ V( G) \ W A real symmetric matrix is called positive definite (M 0) or negative definite (M ≺ 0), respectively, when The connectivity of the graph G is given by its (edge-to-vertex) incidence matrix A , whose elements are given by (see [34]) M 0 : xT Mx > 0 ∀x 6= 0 (7) G  T +1 if e ∈ E ( , v) M ≺ 0 : x Mx < 0 ∀x 6= 0 (8)  i out G A G,iv := −1 if ei ∈ Ein ( G, v) (18) If the inequality is not strict, M is called positive semi-definite  0 otherwise (M 0) or negative semi-definite (M 0). A directed graph G is said to be weakly connected if there |R|×|R| Lemma 4. Let M ∈ C and <{M} 0. Then, M is exists a connecting path (which need not respect the directivity non-singular and <{M−1} 0 (for proof, see [30]). of the edges) between any pair of vertices. |R|×|R| Lemma 5. If M ∈ C and ={M} 0. Then, M is Lemma 9. If the directed graph G is weakly connected, then −1 non-singular and ={M } ≺ 0 (for proof, see [31]). rank(A G) = | V( G) | − 1 (for proof, see [35]). 3

m ∈ NE (m, n) ∈ LΠ n ∈ NE polyphase nodes (i.e., physical and virtual). The vertices are V := N ∪ G (19)

ZΠ,(m,n) The edges fall into two categories, namely polyphase branches and polyphase shunts. The former connect a pair of polyphase nodes, the latter a polyphase node and ground. Let LΠ ⊆ N×N and LT ⊆ N × NT be the polyphase branches associated with

YΠ,m|(m,n) YΠ,n|(m,n) the Π-section and T-section equivalent circuits, respectively. The set of all polyphase branches is L := LΠ ∪ LT. Similarly, let TE := NE × G and TT := NT × G be the polyphase shunts g ∈ G associated with NE and NT, respectively. Thus, the set of all polyphase shunts is T := TE ∪ TT. The edges are obtained as (a) Π-section equivalent circuit. E := L ∪ T (20) m ∈ NE (m, x) ∈ LT x ∈ NT (n, x) ∈ LT n ∈ NE The polyphase branches are related to the longitudinal electrical parameters of the polyphase two-port equivalents. ZT,(m,x) ZT,(n,x) More precisely, every polyphase branch ` ∈ L is associated with a compound branch impedance Z`, which is given by ZΠ,(m,n) if ` = (m, n) ∈ LΠ Z` := (21) ZT,(n,x) if ` = (n, x) ∈ LT YT,x Similarly, the polyphase shunts are related to the transversal electrical parameters of the polyphase two-port equivalents. The aggregated shunt admittance YΠ,n resulting from the Π- g ∈ G section equivalent circuits connected to the polyphase node n ∈ N is given by (b) T-section equivalent circuit. E X X Fig. 1. Polyphase two-port equivalent circuits of the components of the grid. YΠ,n := YΠ,n|(n,m) + YΠ,n|(m,n) (22)

(n,m)∈LΠ (m,n)∈LΠ Accordingly, the compound shunt admittance Y associated E. Power System Analysis t with a polyphase shunt t ∈ T is given by Consider the case of an unbalanced polyphase power system, YΠ,n if t = (n, g) ∈ TE which is equipped with a neutral conductor. With regard to the Yt := (23) YT,x if t = (x, g) ∈ TT wiring, the following is assumed: With respect to the compound electrical parameters of the grid Hypothesis 1. The reference point of every voltage or current model, the following assumption is made: source is connected with the neutral conductor. Furthermore, the neutral conductor is grounded using an effective earthing Hypothesis 3. The compound branch impedances Z` defined system, which is capable of establishing a null voltage between by (21) are symmetric, invertible, and passive. That is  T the neutral conductor and the physical ground (see [36], [37]). Z` = Z` ∀` ∈ L :  −1 (24) Therefore, the phase-to-neutral voltages effectively correspond  ∃Y` = Z` to phase-to-ground voltages. Let G := {0} be the ground node, <{Z`} 0 and P := {1,..., |P|} the phases. An array of terminals which In particular, this implies Z 6= 0. Conversely, the compound belong together (one for every phase p ∈ P) form a polyphase ` shunt admittances Y defined by (23) may be zero. If not, then node. Define N := {1,..., |N |} as the set of the physically t E E they are also symmetric, invertible, and passive. That is existent polyphase nodes of the grid (i.e., where actual voltages  T and currents could be measured). The grid consists of electrical Yt = Yt components that link the polyphase nodes with each other and −1 t ∈ T with Yt 6= 0 :  ∃Z = Y (25) the ground. As to the grid, the following is presumed:  t t <{Yt} 0 Hypothesis 2. The grid consists of electrical components that In practice, the above-stated assumptions are valid for a broad are linear and passive. Further, electromagnetic coupling only variety of power system components, like transmission lines, matters inside of electrical components, but not between them. transformers, and various FACTS devices. Further information Hence, in a per-unit grid model, they are represented either by about this subject is given in App. App:Equipment. Π-section or T-section polyphase two-port equivalent circuits Let V and I be the phasors of the nodal voltage and without mutual coupling (see Fig. 1 and [25]). n,p n,p the injected current in phase p ∈ P of the polyphase node Every T-section equivalent circuit comes with an additional n ∈ N. By definition, the nodal voltage is referenced to the polyphase node (see Fig. 1b). These nodes are purely virtual. ground node, and the injected current flows from the ground That is, they are part of the model, but do not exist in reality. node into the corresponding terminal (see Fig. 2). Analogous Let NT encompass all virtual polyphase nodes originating from quantities are defined for a polyphase node as a whole the T-section equivalent circuits. The topology of the grid V := col (V ) (26) model is described by the directed graph G = (V, E), which is n p∈P n,p In := colp∈P(In,p) (27) constructed as follows. Define N := NE ∪ NT as the set of all 4

u ∈ N `j = (n, u) ∈ L I1 I|N| I1 I|N| Z 0 `j V

...... G0 B B m ∈ N ` = (m, n) ∈ L 0 i B n ∈ N V1 V|N| V0 Z g `i G G

(a) Yt = 0 ∀t ∈ T. (b) ∃t ∈ T s.t. Yt 6= 0. Fig. 3. Proof of Theorem 1 (surfaces indicate weakly connected graphs). I V Y n,p n,p tn = (n, g) ∈ T tn A. Rank Theorem 1. Suppose that Hypotheses 1–3 hold. If the branch g ∈ G graph B = (N, L) is weakly connected, it follows that (|N| − 1)|P| if Y = 0 ∀t ∈ T Fig. 2. Definition of the injected currents, nodal voltages, compound branch rank(Y) = t (35) impedances, and compound shunt admittances. |N||P| otherwise In other words, Y is non-singular if there is at least one non- and for the grid as a whole zero shunt admittance in the circuit, and singular otherwise. The proof is conducted separately for the two cases. V := coln∈N(Vn) (28) Proof (Case I: Y = 0 ∀t ∈ T, see Fig. 3a.). From (31), it I := coln∈N(In) (29) t follows that YT = 0. Therefore, (34) simplifies to where the operator col constructs a (block) column vector. P T P Y = (AB) YLAB (36) The primitive compound branch admittance matrix YL and the primitive compound shunt admittance matrix YT are According to (30), YL is block diagonal. By Hypothesis 3, its blocks Y` (` ∈ L) are symmetric and invertible. Therefore, YL := diag`∈L(Y`) (30) YL is also symmetric and invertible. According to Lemma 3, YT := diagt∈T(Yt) (31) there exists a factorization of the form diag T where the operator constructs a (block) diagonal matrix. YL = ULDLUL (37) Let B := (N, L) represent the subgraph of G comprising the P branches only. Define its polyphase incidence matrix A as where UL is unitary, and DL is positive diagonal. Therefore, B T there exists a matrix EL so that DL = ELEL, which is itself P AB := AB ⊗ diag(1|P|) (32) positive diagonal. Hence, Y can be written as T P where 1|P| is a vector of ones with length |P|. The compound Y = MLML, ML = ELULAB (38) nodal admittance matrix Y, which relates I with V via Note that the term ELUL is non-singular. Therefore, it follows I = YV (33) from Lemmata 1 & 2 that P (i.e., Ohm’s law) is given by (see [25], [35]) rank(Y) = rank(ML) = rank(AB) (39) P P T P Recall the definition of AB (32). According to Lemma 8 Y = (AB) YLAB + YT (34) rank(AP ) = rank(A ) · rank diag(1 ) (40) By definition (28)–(29), the vectors V and I are composed of B B |P| blocks which correspond to the polyphase nodes of the grid. = rank(AB) · |P| (41) Therefore, Y can be written in a block form as Y = (Y ), mn Since = (N, L) is weakly connected, rank(A ) = |N| − 1 where Y relates I with V (m, n ∈ N). As known from B B mn m n according to Lemma 9. This proves the claim. circuit theory, it holds that (see [35]) P P Proof (Case II: ∃t ∈ T s.t. Yt 6= 0, see Fig. 3b.). Introduce a Lemma 10. t = (n, g) ∈ T: Ynm = Ymn = Yt. 0 m∈N m∈N virtual ground node G , and build a modified grid by turning the original ground node into another polyphase node. That is

III.PROPERTIES N0 := N ∪ G (42)

The properties of the compound nodal admittance matrix are Further, let V0 be the counterpart of V referenced w.r.t. G0, and developed in two steps. First, it is proven that the compound 0 0 Vg the voltage of G w.r.t. G . Obviously, polyphase shunts with shunt admittances determine the rank of the whole matrix non-zero admittance in the original grid become polyphase in Sec. III-A. Afterwards, it is shown that the compound branches in the modiﬁed grid. That is branch impedances determine the rank of its diagonal blocks 0 in Sec. III-B). L = L ∪ {t ∈ T | Yt 6= 0} (43) 5

Hypothesis 3 implicates that the compound branch impedances B 0 0 Z`0 (` ∈ L ) are symmetric and invertible. By construction 0 Z` if ` = ` ∈ L Z 0 = (44) ` Y−1 if `0 = t ∈ L0 \ L t Ecut (B, Mk)

The claimed properties follow from (24) and (25), respectively. C Ck 0 0 0 Let B := (N , L ) be the analogon of B for the modified grid. 0 E (B, N \ Mk) Obviously, if B is weakly connected, B is weakly connected, cut too. Clearly, the modified grid satisfies the conditions required for the application of Theorem 1. Observe that the compound shunt admittances from the polyphase nodes N0 to the virtual ground node G0 are zero by construction. 0 0 0 0 Fig. 4. Branch graphs and cut-sets in the fictional grid used for the proof of Y 0 = 0 ∀t ∈ T := N × G (45) t Theorem 2 (surfaces indicate weakly connected graphs). Therefore, the first part of (35), which has already been proven, ˜ can be applied. Accordingly, the compound nodal admittance ` ∈ Lk m ∈ Mk ` ∈ Xk n ∈ N \ Mk matrix Y0 of the modified grid has rank

0 0 Z rank(Y ) = (|N | − 1)|P| = |N||P| (46) Ze` = Z` `˜

Let yT be the column vector composed of the compound shunt admittances Yt (t ∈ T). Namely

yT := colt∈T(Yt) (47) ˜ Y t = (m, g) ∈ Tk t˜ Ye t˜ Given that the voltages of the modiﬁed grid are referenced to the virtual ground node, Ohm’s law (33) reads as follows

     0  I Y −yT V g ∈ G   =     (48) I −yT P Y V0 g T t∈T t g Fig. 5. Compound branch impedances and shunt admittances in the fictional grid used for the proof of Theorem 2. That is, Y0 is given in block form as   0 Y −yT Y =   (49) Proof. As known from circuit theory, YM×M relates IM and T P −yT t∈T Yt VM for VN\M = 0 (i.e., the polyphase nodes N\M are short- circuited). This is due to the assumption that the state of each It is known that elementary operations on the rows and column electrical component is composed solely of ground-referenced of a matrix do not change its rank. Hence, one can add the first nodal voltages. In that sense, Y can be regarded as the |N| block rows/columns of Y0 to the last block row/column M×M compound nodal admittance matrix of a fictional grid which is without affecting its rank. Using Lemma 10, one finds obtained by grounding the polyphase nodes N \ M. Therefore,   Y −y the polyphase branches internal to M (i.e., Eint (B, M)) persist, T X   row|N|+1 += rown (50) whereas those interconnecting M and N \M (i.e., Ecut (B, M) −yT P Y T t∈T t n∈N and Ecut (B, N \ M)) become polyphase shunts in the modified grid. The remaining branches are described by the graph Y −y X T col += col (51) 0 0 |N|+1 n n∈N C := (M, Eint (B, M)) (56) Y 0 (52) In general, C is disconnected. However, since B is weakly 0 0 connected as a whole, there exists a partition {Mk | k ∈ K} It follows straightforward that of M so that the subgraphs Ck associated with Mk 0 rank(Y) = rank(Y ) = |N||P| (53) Ck := (Mk, Eint (B, Mk)), k ∈ K (57) which proves the claim. are weakly connected, and mutually disconnected (see Fig. 4). Therefore, YM×M is block diagonal. Namely B. Block Rank Y = diag (Y ) (58) Theorem 2. Suppose that Hypotheses 1–3 hold. If the branch M×M k∈K Mk×Mk graph B = (N, L) is weakly connected, and all the compound Y In turn, Mk×Mk can be interpreted as the compound nodal branch impedances Z` (` ∈ L) are strictly passive admittance matrix of a fictional grid, which is constructed by grounding the polyphase nodes N \ M . Define <{Z`} 0 ∀` ∈ L (54) k then it follows that Lk := E ( Ck) = Eint (B, Mk) ⊂ L (59)

Tk := Mk × G ⊂ T (60) rank(YM×M) = |M||P| ∀M ( N (55)

That is, every proper diagonal subblock of the compound nodal The compound branch impedances Ze` and compound shunt admittance matrix Y has full rank. admittances Ye t of this ﬁctional grid satisfy Hypothesis 3. 6

The grounding process does not affect the compound branch to experience, Kron reduction is feasible in practice, but there impedances. Therefore (see Fig. 5) was no theoretical proof for this empirical observation until recently. Lately, some researchers (i.e., [17], [23]) examined ` ∈ L : Z = Z (61) k e` ` this problem for monophase grids. The results of these works do apply to balanced triphase grids (respectively, the equivalent Therefore, (24) of Hypothesis 3 obviously applies to Ze`, too. In contrast, the compound shunt admittances are modified, as positive-sequence networks), but not to generic unbalanced some of the polyphase branches become polyphase shunts. The polyphase grids. This gap in the existing literature is closed by Corollary 1, which provides a guarantee for the feasibility set Xk containing these polyphase branches is given by of Kron reduction in generic polyphase grids. Xk := Ecut (B, Mk) ∪ Ecut (B, N \ Mk) (62) Corollary 1. Suppose that Theorem 2 applies. Let Z ( N be Since B is weakly connected, it holds that Xk 6= ∅ ∀k ∈ K. a non-empty subset of N with zero current injection. ˜ Therefore, ∃` ∈ Xk ∀k ∈ K, which is a polyphase branch in the original grid, and a polyphase shunt in the modified grid. IZ = 0 (67) The compound branch impedances of the grounded polyphase Then, the voltages at the zero-injection buses Z linearly depend branches contribute to the compound branch admittances of on those at the other buses Z := N \ Z, and (33) reduces to the modified grid. Namely (see Fig. 5) { " IZ = YVb Z , Yb := Y/YZ×Z (68) t ∈ Tk, X X { { : Ye t = Yt + Y` + Y` (63) t = (m, g) The reduced compound nodal admittance matrix Yb satisfies `∈Xk `∈Xk `=(m,n) `=(n,m) rank(Y ) = |M||P| ∀M Z (69) b M×M ( { If the sums are empty, (25) of Hypothesis 3 obviously applies. Otherwise, it follows from Hypothesis 3, (54), and Lemma 4 That is, Yb has the same properties as Y w.r.t. the rank of its proper diagonal subblocks. that Ye t is symmetric and invertible, and has positive definite real part. Hence, (25) of Hypothesis 3 applies in this case, too. Observe that, according to (67), a zero-injection node has As Hypothesis 3 holds, Theorem 1 can be applied. Due to zero current injection in every phase. Moreover, according to the fact that X 6= ∅ ∀k ∈ K, it follows that ∃t˜∈ T ∀k ∈ K, k k (68), Yb is the Schur complement of Y w.r.t. YZ×Z. In order for which Y˜ 6= 0, even if Y = 0 ∀t ∈ T (see Fig. 5). Thus e t t k for this Schur complement to exist, YZ×Z has to be invertible.

rank(YM ×M ) = |Mk||P| ∀k ∈ K (64) Proof. Observe that Z and Z form a partition of N. Therefore, k k { Ohm’s law (33) can be written in block form as Since YM×M is block diagonal with blocks YM ×M k k X IZ YZ ×Z YZ ×Z VZ rank(YM×M) = rank(YM ×M ) (65) { = { { { { (70) k k IZ YZ×Z YZ×Z VZ k∈K { X According to Theorem 2, YZ×Z has full rank. Therefore = |P| |Mk| = |P||M| (66) k∈K −1 ∃ZZ×Z := YZ×Z (71) This proves the claim. From IZ = 0, it follows straightforward that −1 IV. IMPLICATIONS VZ = −YZ×ZYZ×Z VZ (72) { { Using the properties developed in the pevious section, the So, VZ is a linear function of VZ as claimed. Substitute this findings of [23] can be extended to polyphase power systems. formula back into Ohm’s law, and{ obtain In Sec. IV-A, it is is proven that Kron reduction is feasible for any subset of the polyphase nodes with zero current injections. IZ = YZ ×Z VZ + YZ ×ZVZ (73) { { { { { In Sec. IV-B, it is shown that hybrid parameters matrices exist −1 = YZ ×Z VZ − YZ ×ZYZ×ZYZ×Z VZ (74) for arbitrary partitions of the polyphase nodes. { { { { { { = (Y/YZ×Z)VZ (75) A. Kron Reduction { This proves the first claim (68). According to Lemma 7, the Ohm’s law (33) establishes the link between injected current diagonal block of Yb associated with M ( Z is given by phasors and nodal voltage phasors through the grid. Obviously, { the current injected into a polyphase node also depends on the Yb M×M = (Y/YZ×Z)M×M (76) devices which are connected there. In particular, a polyphase YM×M YM×Z z ∈ N = / YZ×Z (77) node , in which no devices are present, has zero injected YZ×M YZ×Z current (i.e., Iz = 0). Let Z ( N (Z 6= ∅) be a set of such zero- = Y(M∪Z)×(M∪Z)/YZ×Z (78) injection nodes (i.e., IZ = 0). As known from power system analysis, (33) can be reduced by eliminating the zero-injection From M Z = N\Z and Z N, it follows that M∪Z N. ( { ( ( nodes Z through Kron reduction (see [15]). This yields a model That is, both M∪Z and Z are proper subsets of N. According of the grid with fewer unknowns. Therefore, computationally to Theorem 2, the diagonal subblocks of Y associated with heavy tasks like PFS, SE, and VSA can be accelerated without M ∪ Z and Z therefore have full rank. By consequence using high-performance computers (see [10]–[14], [16]). D := det(Y ) 6= 0 (79) In order for Kron reduction to be applicable, the diagonal M∪Z (M∪Z)×(M∪Z) : subblock YZ×Z of Y has to be invertible (this will be shown DZ = det(YZ×Z) 6= 0 (80) shortly). Interestingly, researchers and practitioners hardly ever From Lemma 6, it follows that check whether this condition is actually satisfied. In fact, even the inventor (i.e., [15]) did not consider this issue. According det(Yb M×M) = DM∪Z · DZ 6= 0 (81) 7

Therefore, Yb M×M has full rank. Namely By Theorem 2, YM×M has full rank, and is hence invertible. −1 Define ZM×M := YM×M, and solve for VM. rank(Yb M×M) = |M||P| (82) −1 VM = YM×M(IM − YM×M VM ) (89) As M Z is arbitrary, this proves the second claim (69). { { ( { = HM×MIM + HM×M VM (90) Moreover, Corollary 1 itself has a fundamental implication. { { as claimed in (84)–(85). Write Ohm’s law (33) for IM , and Observation 1. Kron reduction preserves the property which substitute the above formula for V . This gives { enables its applicability in the first place (i.e., that every proper M diagonal subblock of a compound nodal admittance matrix has IM = YM ×M VM − YM ×MVM (91) { { { { { full rank). Therefore, if Z is partitioned as {Zk | k ∈ K}, the  −1 YM ×MYM×MIM { parts Zk can be reduced one after another, and the (partially or =  +YM ×M VM (92) fully) reduced compound nodal admittance matrices obtained  { { { −1 after every step of the reduction also exhibit the rank property. −YM ×MYM×MYM×M VM { { { = HM ×MIM + HM ×M VM (93) Performing the reduction sequentially rather than “en bloc” is { { { { beneficial in terms of computational burden, because the Schur as claimed in (86)–(87). complement (10) requires a matrix inversion. This operation is computationally expensive, and scales poorly with problem It is worth noting that compound hybrid parameters matrices size (even if the inverse is not computed explicitly). also exist for Kron-reduced grids. B. Hybrid Parameters Observation 2. The existence of compound hybrid parameters matrices is based on the same rank property as the feasibility Evidently, the circuit equations (33) are in admittance form. of Kron reduction. Since Kron reduction preserves this property Namely, the injected current and nodal voltage phasors appear (recall Observation 1), compound hybrid parameters matrices in separate vectors, which are linked by the nodal admittance can be obtained from unreduced, partially reduced, and fully matrix Y. In power system analysis, it is often more convenient reduced compound nodal admittance matrices. In this regard, to write the circuit equations in hybrid form (if this is feasible). Y can be replaced by Y in Corollary 2. The corresponding system of linear equations features vectors b composed of both voltage and current phasors, and is described by a so-called hybrid parameters matrix H. In this context, V. CONCLUSIONS observe that there is no guarantee for the existence of a hybrid This paper examined the properties of the compound nodal representation. Whether a suitable matrix H exists, depends admittance matrix of polyphase power systems, and illustrated both on the nodal admittance matrix Y of the grid and on the their implications for power system analysis. Using the concept partition of the nodes underlying the hybrid representation. of compound electrical parameters, and exploiting the physical Various researchers have treated the subject of hybrid pa- characteristics of electrical components, rank properties for the rameters matrices for monophase grids. Some authors plainly compound nodal admittance matrix and its diagonal subblocks describe how a hybrid parameters matrix can be built, provided were deduced. Notably, it was proven that the diagonal blocks that it exists at all (e.g., [38]–[40]). Others do provide criteria have full rank if the grid is connected and lossy. Based on these for the existence of hybrid parameters matrices, but only for findings, it was shown that the feasibility of Kron reduction and some (i.e., at least one) partition of the nodes (e.g., [18]–[20]). the existence of hybrid parameters are guaranteed in practice. One recent work establishes a criterion for arbitrary partitions Thus, this paper provided a rigorous theoretical foundation for of the nodes [23]. All of the aforementioned works only study the analysis of generic polyphase power systems. the monophase case. Accordingly, those results may apply to balanced triphase grids (respectively, their equivalent positive- APPENDIX A sequence networks), but not to generic unbalanced polyphase POWER SYSTEM COMPONENTS grids. In contrast, Corollary 2 ensures the existence of hybrid A. Transmission Lines parameters matrices for arbitrary partitions of the polyphase nodes of a generic polyphase grid. Consider a transmission line with |P| phase conductors and one neutral conductor. Let v(z, t) and i(z, t) be the vectors of Corollary 2. Suppose that Theorem 2 applies. Let M ( N be phase-to-neutral voltages and phase conductor currents, i.e. non-empty, that is M and M := N \M form a partition of N. { Then, there exists a compound hybrid parameters matrix H, v(z, t) = colp∈P vp(z, t) (94) and the grid is described by the hybrid multiport equations i(z, t) = colp∈P ip(z, t) (95) " # " # IM VM where z is the position along the line. If (i) the electromagnetic { = H { (83) VM IM parameters of the line are state-independent, (ii) the conductors are parallel, and the perpendicular distance between any two of The blocks of H are given as follows them is much shorter than the wavelength, (iii) the conductors −1 have finite conductance, (iv) the electromagnetic field outside HM×M = YM×M (84) −1 of the conductors produced by the charges and currents inside HM×M = −YM×MYM×M (85) { { of them is purely transversal, and (v) the sum of the conductor −1 HM ×M = YM ×MYM×M (86) currents is zero, Maxwell’s equations simplify to the so-called { { HM ×M = Y/YM×M (87) telegrapher’s equations (see [41]) { { 0 0 Proof. Write Ohm’s law (33) for IM ∂zv(z, t) = −(R + L ∂t)i(z, t) (96) 0 0 ∂zi(z, t) = −(G + C ∂t)v(z, t) (97) IM = YM×M VM + YM×MVM (88) { { 8

R0 and L0 are the resistance and inductance per unit length of m n the conductors, G0 and C0 are the conductance and capacitance per unit length of the dielectric. These matrices are symmetric. ZΛ,(m,n) ZΓ,(m,n) Consider a segment of inﬁnitesimal length ∆z at position z. Let Ee(z, t) and Em(z, t) be the energy stored in the electric and magnetic ﬁeld, respectively. They are given by (see [41]) series compensator

∆z YΛ,m|(m,n) YΛ,n|(m,n) E (z, t) = (v(z, t))T C0v(z, t) (98) e 2 ∆z T 0 Em(z, t) = (i(z, t)) L i(z, t) (99) 2 g

As Ee(z, t) > 0 ∀v(z, t) 6= 0 and Em(z, t) > 0 ∀i(z, t) 6= 0, (a) Series compensation. it follows straightforward that C0 0 and L0 0. Similarly, m n let Pc(z, t) and Pd(z, t) be the power dissipated in the conductors and the ambient dielectric. They are given by (see [41]) ZΛ,(m,n) T 0 Pd(z, t) = ∆z · (v(z, t)) G v(z, t) (100) T 0 Pc(z, t) = ∆z · (i(z, t)) R i(z, t) (101)

Since real systems are lossy, Pd(z, t) > 0 ∀v(z, t) 6= 0 and Y shunt Y Y 0 0 Λ,m|(m,n) compensator Γ,n Λ,n|(m,n) Pc(z, t) > 0 ∀i(z, t) 6= 0. Therefore, G 0 and R 0. If a transmission line is electrically short (i.e., its total length |zm − zn| is significantly shorter than the wavelength), it can g be represented by a Π-section equivalent circuit with (b) Shunt compensation (for simplicity, only one compensator is shown). 1 0 0 Y = (G + jωC )|z − z | (102) Fig. 6. Incorporation of FACTS devices into the transmission line model. Π,m|(m,n) 2 m n YΠ,n|(m,n) = YΠ,m|(m,n) (103) 0 0 C. FACTS Devices ZΠ,(m,n) = (R + jωL )|zm − zn| (104) There exist three families of FACTS devices, namely series 0 0 0 0 Since G , C , R , and L are real positive definite, YΠ,m|(m,n), compensators, shunt compensators, and combined series-and- YΠ,n|(m,n), and ZΠ,(m,n) are symmetric with positive definite shunt compensators (see [43]). Here, the first two cases are real part. So, according to Lemmata 4 & 5, they are invertible. discussed. Let ZΛ,(m,n), YΛ,m|(m,n), and YΛ,n|(m,n) be the This is in accordance with Hypothesis 3. compound electrical parameters describing a transmission line without compensation (see App. A-A). If a series compensator is installed, the compound branch impedance of the respective B. Transformers transmission line is altered. In the Π-section equivalent circuit, When analyzing power systems operating at rated frequency, this is reflected by adding the compound impedance ZΓ,(m,n) transformers are represented by a T-section equivalent circuits, of the compensator to the compound branch impedance of the which correspond one-to-one to the composition of the devices transmission line (see Fig. 6a). Namely (see [42]). Consider a polyphase transformer connecting two ZΠ,(m,n) = ZΛ,(m,n) + ZΓ,(m,n) (108) polyphase nodes m, n ∈ N. Let m be its primary side, and n its secondary side. In this case, the parameters of the T-section If shunt compensators are installed, the compound admittances equivalent circuit are given as YΓ,m and YΓ,n of the compensators add to the compound shunt admittance of the transmission line (see Fig. 6b). Namely ZT,(m,x) = Rw,1 + jωL`,1 (105) YΠ,m|(m,n) = YΛ,m|(m,n) + YΓ,m (109) ZT,(x,n) = Rw,2 + jωL`,2 (106) YΠ,n|(m,n) = YΛ,n|(m,n) + YΓ,n (110) YT,x = Gh + jωBm (107) Usually, such compensators are built from banks of capacitors

The resistance matrices Rw,1 and Rw,2 represent the winding or inductors, which can be stepwise (dis)connected. This kind resistances, and the inductance matrices L`,1 and L`,2 the of devices are symmetrical w.r.t. the phases, and lossy. Hence leakage inductances of the coils on the primary and secondary " Z = ZT side, respectively. The former are positive diagonal, the latter series compensation : Γ,(m,n) Γ,(m,n) (111) <{Z } 0 are positive definite. The conductance matrix Gh represents Γ,(m,n) the hysteresis losses, and the susceptance matrix B the " T m YΓ,m/n = YΓ,m/n magnetization of the transformer’s core. The former is positive shunt compensation : (112) <{Y } 0 diagonal, the latter is positive definite. Therefore, according to Γ,m/n Lemmata 4 & 5, ZT,(m,x), ZT,(x,n), and YT,x are invertible. Since the compound electrical parameters of transmission lines This holds irrespective of whether the polyphase transformer is satisfy Hypothesis 3 (see App. A-A), the compound electri- built with one single multi-leg core or several separate cores. In cal parameters (108) and (109)–(110) are symmetric, have the latter case, L`,1, L`,2, and Bm are diagonal, since separate positive definite real part, and are invertible (by Lemma 4). cores are magnetically decoupled. In either case, the compound Accordingly, transmission lines equipped with series or shunt electrical parameters satisfy Hypothesis 3. compensators satisfy Hypothesis 3. 9

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