Electrical Networks and Algebraic Graph Theory: Models, Properties, and Applications Florian Dorﬂer,¨ Member, IEEE, John W

Electrical Networks and Algebraic Graph Theory: Models, Properties, and Applications Florian Dorﬂer,¨ Member, IEEE, John W. Simpson-Porco, Member, IEEE, and Francesco Bullo, Fellow, IEEE

Abstract—Algebraic graph theory is a cornerstone in the We then study general models of electrical networks, start- study of electrical networks ranging from miniature integrated ing from elementary models and building up to a prototypical circuits to continental-scale power systems. Conversely, many circuit, with several instructive special cases. Our proposed fundamental results of algebraic graph theory were laid out by early electrical circuit analysts. In this paper we survey prototypical circuit is a Π-line-coupled RC circuit with non- some fundamental and historic as well as recent results on linear sources and loads. This prototypical nonlinear RLC how algebraic graph theory informs electrical network analysis, circuit has numerous interesting features. First, our prototypi- dynamics, and design. In particular, we review the algebraic and cal circuit generalizes the widely-studied resistive circuit and spectral properties of graph adjacency, Laplacian, incidence, and features rich dynamical behaviors, including synchronization resistance matrices and how they relate to the analysis, network- reduction, and dynamics of certain classes of electrical networks. and consensus behaviors. Second, power system network mod- We study these relations for models of increasing complexity elling is essentially based on this circuit (Π-line transmission ranging from static resistive DC circuits, over dynamic RLC models, charging capacitors at the buses, and ZIP loads, circuits, to nonlinear AC power flow. We conclude this paper by including modern constant-power devices). Third and final, it presenting a set of fundamental open questions at the intersection showcases popular energy-based, power-based, and compart- of algebraic graph theory and electrical networks. mental modeling approaches, and it is sufficiently general to admit a variety of graph-theoretic analysis approaches. Based on algebraic graph theory methods, we then study the I.INTRODUCTION analysis, network-reduction, and dynamics of our prototypical The study of electrical networks, the theory of graphs, circuit and its variations, in linear and nonlinear as well as and their associated matrices share a long and rich history static and dynamic settings. Thereby we consider models of in- of synergy and joint development. Starting from the founda- creasing complexity ranging from static resistive circuits, over tional classical work by Gustav Kirchhoff [87], modeling and dynamic RLC networks, to nonlinear AC power flow models. analysis of electric circuits has motivated the birth and the We motivate our treatment with a few interesting examples, development of a broad range of graph-theoretical concepts review a few fundamental and historic results in a tutorial and certain classes of matrices. Vice-versa, algebraic graph exposition, and also showcase related recent developments. theory concepts and constructions have enabled fundamental Our focus is on static and dynamic analysis of DC circuits, advances in the theory of electrical networks. As is well except for Section VI-B where we explicitly focus on steady- known, it is in graph-theoretical language that Kirchhoff’s laws state analysis of AC circuits through the lens of graph theory. are most succinctly and powerfully expressed, and it is via It is important to clarify that this article does not aim to be matrix theory that the discrete nature of graphs is most pow- comprehensive in its scope, nor does it present multiple view- erfully analyzed. To this day, graph theory, matrix analysis, points on the given material, as both algebraic graph theory and electrical networks inspire and enrich one another. and electrical circuits are mature and broadly developed fields. In this paper we survey some fundamental and historic In the context of algebraic graph theory, we refer interested as well as recent results on how algebraic graph theory readers to the textbooks [16], [19], [72] and, for example, the informs electrical network analysis, dynamics, and design. In surveys [102], [97], [17]. There are numerous complementary particular, we review the algebraic and spectral properties of viewpoints on electrical network modeling and analysis. We graph adjacency, Laplacian, Metzler, incidence, and effective mention the well-established linear network theory [6], [144], resistance matrices; we review the basic notions from algebraic [101], [145]; classical network analysis in the nonlinear setting potential theory, including cycle and cutset spaces. [37], [36], [124]; the signals, systems, and control viewpoint [4]; the behavioral approach [148] and its application F. Dorfler¨ is with the Automatic Control Laboratory, ETH Zurich,¨ 8092 to circuits [149]; energy-based Port-Hamiltonian approaches Zurich,¨ Switzerland. Email: dorfl[email protected]. J. W. Simpson-Porco is with the Department of Electrical and Computer [135], [104], [133], [95]; and power-based Brayton-Moser Engineering, University of Waterloo, Waterloo, ON N2L 3G1, Canada. Email: approaches [25], [26], [83], [85], [84] among others. Our [email protected]. exposition and treatment highlights the algebraic graph theory F. Bullo is with the Center for Control, Dynamical Systems and Com- putation, University of California, Santa Barbara, CA 93106, USA. Email: perspective on electrical networks, with examples colored by [email protected]. our own research interests and experiences. This work was supported in part by the SNF AP Energy Grant #160573, the The remainder of the paper is organized as follows. We U.S. Department of Energy (DOE) Solar Energy Technologies Office under Contract No. DE-EE0000-1583, by the National Science and Engineering begin with a set of motivating examples in Section II that Council of Canada Discovery RGPIN-2017-04008, and by ETH Zurich¨ funds. outline the themes of the paper. Section III briefly reviews 2 relevant results of algebraic theory. In Section IV we present may appear either surprising or obvious to electrical engineers the general modeling of electrical networks based on the depending on their physical intuition. In Section V, we will language of graph theory and also introduce a prototypical show through the lens of algebraic graph theory that this effect network model that we will frequently revisit in the course is to be expected and can be analyzed at a similar complexity of the paper. Section V showcases the tools of algebraic as the analysis of a single tank circuit. graph theory to analyze the structure and dynamics of linear electrical networks, and Section VI addresses the nonlinear B. RC circuits as compartmental systems case. Finally, Section VII concludes the paper and outlines a few open and worthwhile research directions at the intersection Consider the electrical network in Figure 3 consisting of of electrical networks and algebraic graph theory. resistive branches connecting capacitors with a current source and at least one so-called shunt resistor connected to ground.

II.MOTIVATING EXAMPLES We begin by laying out a set of motivating examples with current apparently complex behavior, whose analysis becomes crisp source and clear by using the tools of algebraic graph theory. We will revisit each of these examples in the course of the paper.

A. Synchronization of resonant LC tanks Consider the electrical network in Figure 1 consisting of identical resonant tank circuits interconnected through resistive branches. Each tank circuit consists of a parallel connection of Fig. 3. Resistive network of capacitors with a current source and a shunt an inductor and a capacitor with identical values of inductance resistor. ` > 0 and capacitance c > 0. It is possible and natural to model this dynamical system as a compartmental system, i.e., a collection of compartments r in which electrical charge ﬂows into the system through the r ` c current source, through the network, and out through the shunt r r resistor. In other words, in every compartment the stored, supplied, and dissipated charge is balanced. The study of compartmental systems is rooted in algebraic ` c ` c ` c graph theory; see Appendix A and [140], [79], [27]. A key result states that, if each node has a directed path to the shunt resistance, (i.e., the compartmental graph is out-ﬂow connected), then the system has a unique, positive, globally Fig. 1. Network of resistively interconnected `c-tanks; image courtesy of [27]. asymptotically stable equilibrium point.

As known from undergraduate engineering education, each C. Steady-state feasibility of direct-current networks tank circuit in isolation exhibits harmonic oscillations with Consider the DC circuit shown in Figure 4(a), which con- natural frequency ω0 = 1/√`c and phase and amplitude depending on its initial conditions for current and voltages. When sists of an ideal DC voltage supply providing power to a load coupled through a connected resistive circuit, the voltages through a resistance r. vi(t) across the capacitors of all tank circuits synchronize to a common harmonic oscillating voltage of frequency ω ; see 0 f V Figure 2 for a simulation. This spontaneous synchronization r V0

v(t) 4 + V0 = V 2

Pcrit 0 Pload 2 − (a) (b) 4 − t Fig. 4. (a) An ideal voltage supply connected to a nonlinear load. (b) Locus of solutions for constant-power load model. 10 20 30

Fig. 2. Synchronization of capacitor voltages of the resistively interconnected The load consumes a power P (V ) as a function of the identical tank circuits in Figure 1; image courtesy of [27]. voltage V across its terminals. Since the current f which ﬂows 3

from the supply is f = (V0 V )/r, the load dissipates a and the formulas for the three-node mesh in Figure 6 are power V f. This must in turn equal− its consumed power P (V ), red r14r34 + r34r24 + r24r14 yielding the power balance r23 = , r14 V (V0 V )/r = P (V ) . (1) red r14r34 + r34r24 + r24r14 − r12 = , (2) r34 This is a nonlinear equation in the load voltage V , the solutions red r14r34 + r34r24 + r24r14 of which will determine the feasible values for the voltage V . r13 = . r24 Let us ﬁrst consider a resistive load of resistance rload > 0, 2 with power consumption P (V ) = V /rload. In this case, the At ﬁrst glance, the circuit reduction formulae (2) appear power balance (1) always has two solutions, given by convoluted and provide little immediate insight. In Section V r however, we will show how these formulae can be insightfully V = 0 and V = V . derived by means of linear algebra and intuitively interpreted r + r 0 load in terms of graph theory. Indeed, the series-circuit contraction Now, instead, consider a load consuming a constant power and Y-∆ transformation are special cases of the more general 2 P (V ) = Pload 0, and let Pcrit = V /4r. If Pload/Pcrit 1, Kron reduction [89] that permits an elegant analysis via ≥ 0 ≤ then a simple calculation shows that (1) has solutions algebraic graph theory.

V P V = 0 1 1 load , 2 P III.RELEVANT RESULTS IN ALGEBRAIC GRAPH THEORY ± r − crit !

Figure 4(b) plots these solutions as a function of Pload; depend- This section provides a concise self-contained review of ing on the ratio Pload/Pcrit, the circuit can have two, one, or algebraic graph theory, Perron-Frobenius theory, and their zero real-valued solutions. This example illustrates that even applications to row-stochastic and Laplacian matrices. We the existence of solutions depends heavily on the chosen load refer interested readers to the textbooks [16], [19], [72], [99] model. In Section VI-A we will revisit this feasibility problem and, for example, the surveys [102], [97], [17]; this section for networks, and we will see that the maximum transfer follows the treatment in [27]. limit Pcrit generalizes as a Laplacian-like matrix encoding the 1) Notation: We brieﬂy introduce the notation used in the topology and weights of the circuit graph. remainder of the paper. For a vector x Rn, the notation ∈n n diag(x) denotes a diagonal matrix in R × with the ith diagonal element being xi, the average of its entries is denoted D. Series circuit contraction and star-triangle transformation n by average(x) = i=1 xi/n, and the extremum entries are Classic methods in the study of electric circuits are the xmax = maxi 1,...,n xi and xmin = mini 1,...,n xi . ∈{ }{ } ∈{ }{ } contraction of a series of resistive circuit elements and the We denote the realP part (respectively, imaginary part) of a Y-∆ transformation; these methods date back to the work by complex number z C by (z) (respectively, by (z)). Arthur E. Kennelly [86] and are depicted in Figures 5 and 6. ∈ < = The vector ei denotes the ith canonical basis vector (with a non-zero and unit-entry at position i) in appropriate dimension. red r12 r23 r The symbols 0n m and 1n m denote the (n m)-matrices 13 × ×

8 8 × 8 30 8 of all zero and unit entries. We avoid the subscript m in the 1 2 3 1 3 vector-valued case m = 1 and entirely avoid subscripts when the dimension is clear from the content. The matrix Πn = Fig. 5. Contraction of a series of resistive circuit elements to a single resistor. 1 In n 1n n denotes the orthogonal projection operator onto − × n T the subspace 1⊥ = x R 1 x = 0 . n { ∈ | n } Element-wise (Hadamard) multiplication and division of

2 2 8 8 matrices are denoted by and .

2) Nonnegative matrices and digraphs: Given n 2, r24 ≥ rred rred an n n matrix A is nonnegative (resp. positive) if each 30 12 23 × r14 r34 entry is nonnegative (resp. positive); we write A 0 and 4 red ≥

r13 A > 0, respectively. A weighted digraph is a triplet 8 1 8 8 3 1 8 3 ( 1, . . . , n , ,A), where 1, . . . , n is a set ofG nodes, is a{ set of directed} E edges (i.e.,{ ordered} pairs of nodes), and AE is Fig. 6. Y-∆ transformation of a resistive radial circuit to a meshed circuit. a weighted adjacency matrix (i.e., a nonnegative matrix) with the property that a > 0 if and only if (i, j) . If (i, j) , The reduced circuits are equivalent in their electrical behav- ij ∈ E ∈ E ior as seen from the terminals 1, 3 (respectively, 1, 2, 3 ) we say i is the source and j is the sink of the directed edge. of the remaining nodes in the reduced{ } single-resistor{ (respec-} Given a nonnegative A, the weighted digraph associated to A has node set 1, . . . , n and edges deﬁned by the patterns tively, three-node mesh) circuit. The well-known formula for { } the remaining single resistor in Figure 5 is of non-zero entries of A. An undirected graph has undirected edges (i.e., the set consists of unordered pairs of the form rred = r + r , i, j ) and a symmetricE adjacency matrix A = AT. 13 12 23 { } 4

T 1 4 (resp. A 1n = 1n). The spectral bound implies that ρ(A) = 1 0 1 1 0 for any row-stochastic A. Clearly, the right dominant eigen- 1 0 1 0 A = . vector of a row-stochastic matrix A is 1n. A square matrix   k 1 1 0 1 A is semi-convergent if limk A is finite. If A is row- →∞ 2 3 0 0 1 0 stochastic and primitive, then it is also semi-convergent and   k T   limk A = 1nvleft. More generally, a row-stochastic matrix →∞ Fig. 7. An unweighted undirected graph and its adjacency matrix. A is semi-convergent if and only if each strongly connected component of without out-going edges is aperiodic. Proofs A directed path (resp. path) in a digraph (resp. graph) is an of these statementsG are given for example in [27, Chapters 4 ordered sequence of nodes such that any pair of consecutive and 5]. nodes in the sequence is a directed edge (resp. an undirected 6) Laplacian matrices and their algebraic connectivity: n n edge). A simple directed path (resp. path) is one with no A matrix R × is Laplacian if 1n = 0n and its repeated nodes, except possibly the first and last nodes. A off-diagonalL entries ∈ are nonpositive. TheL Laplacian matrix cycle in an undirected graph is a simple path that starts and of a weighted digraph is defined by = diag(A1n) A; ends at the same vertex and has at least three distinct vertices. vice versa, the weighted digraph associatedL to a Laplacian− G A directed cycle in a directed graph is a directed path that starts matrix is defined by setting aij = ìj for i = j. is and ends at the same vertex. For the example in Figure 7, the said toL be irreducible if is strongly− connected.6 LaplacianL sequence (4, 3, 1, 3) is a path, (1, 2, 3, 4) is a simple path, and matrices have remarkableG properties. The matrix is singular (1, 2, 3, 1) is a cycle. and all its eigenvalues different from zero haveL positive real 3) Matrix irreducibility and digraph connectivity: A di- part. The matrix is irreducible if and only if the rank of graph (resp. graph) is strongly connected (resp. connected) if L is n 1; in this case Im( ) = 1n⊥. If is symmetric there exists a directed path (resp. path) from any node to any (orL equivalently,− is undirected),L then theL eigenvalues of G other node. A subgraph is a strongly connected component of a can be ordered as 0 = λ1 λ2 λn and digraph if it is strongly connected and no other node can be Lis connected if and only if λ >≤ 0. The≤· smallest · · ≤ non-zeroG G 2 added to it while maintaing the subgraph strongly connected. eigenvalue λ2 is called the algebraic connectivity of . Finally, n 1 k G A nonnegative matrix A is irreducible if k=0− A > 0 let exp denote the matrix exponential operation; if contains and primitive if there exists a number such that k . G T k A > 0 a globally reachable node, then limt + exp( t) = 1nvleft, P → ∞ The matrix A 0 is irreducible if and only if its associated where v denotes the nonnegative left eigenvector−L of with ≥ left digraph is strongly connected. The matrix A 0 is primitive eigenvalue 0, normalized to satisfy 1Tv = 1. TheL rate ≥ n left if and only if its associated digraph is strongly connected and of convergence of exp( t) is determined by the algebraic −L aperiodic, i.e., there exists no natural number (except one) connectivity λ2. Proofs of these statements are given for dividing the length of all directed cycles in . Clearly, if A is example in [27, Chapters 6 and 7]. G primitive, then it is irreducible; the converse is not true. For the 7) Metzler matrices and their properties: A matrix M is example in Figure 7, one can see that A is irreducible because Metzler if its off-diagonal entries are nonnegative. If is A + A2 > 0, that A is primitive because A4 > 0, and that Laplacian, is Metzler. The weighted digraph associatedL −L G the digraph associated to A (whereby each undirected edge to M is defined by aij = mij for i = j. The Perron-Frobenius is regarded as two directed edges) is strongly connected and Theorem for Metzler matrices states6 that, for any Metzler aperiodic. matrix M, 4) Perron-Frobenius theory for nonnegative matrices: The (i) there exist a real eigenvalue λ (µ) for all other Perron-Frobenius Theorem states that, for any A 0, eigenvalues , and right and left nonnegative≥ < eigenvectors ≥ µ (i) there exist a real eigenvalue λ µ for all other v and v , ≥ | | right left eigenvalues µ, and right and left nonnegative eigenvectors (ii) if M is irreducible, λ is simple and vright and vleft are vright and vleft, unique and positive. (ii) if A is irreducible, λ is positive and simple and vright and A Metzler matrix is Hurwitz if its dominant eigenvalue (and vleft are unique and positive, therefore all its eigenvalues) has negative real part. A Metzler (iii) if A is primitive, λ > µ for all other eigenvalues µ. matrix M is Hurwitz if and only if M is invertible and | | 1 Proofs of these statements are given for example in [99, M − is nonnegative. Moreover, if M is Metzler, Hurwitz, − 1 Chapter 8]. The eigenvalue λ is called the dominant eigenvalue and irreducible, then M − is a strictly positive matrix. One of A and, for irreducible matrices, its eigenvector (unique up can show that (i) any− Metzler matrix M can be written as to scaling) is called the dominant eigenvector. M0 + diag(v), where M0 has zero row-sums (or alternatively The spectral radius of a square matrix A, denoted by ρ(A), column-sums) and v Rn, and that (ii) if v has non-positive is the largest magnitude of its eigenvalues. The dominant entries and at least one∈ entry strictly negative, then the Metzler eigenvalue of a nonnegative matrix is also its spectral radius. A matrix M0 + diag(v) is Hurwitz. Proofs of these statements known spectral bound is mini(A1n)i ρ(A) maxi(A1n)i. are given for example in [27, Chapter 9]. For the example in Figure 7, one can≤ see that≤ the dominant Metzler matrices and their algebraic and graph-theoretical eigenvaue λ 2.17 lies in the interval [1, 3]. properties are the central objects in the study of linear com- 5) Row-stochastic≈ matrices: A matrix A 0 is partmental systems, an instance of which are circuits. We refer ≥ row-stochastic (resp. column-stochastic) if A1n = 1n to Appendix A for a self-contained treatment. 5

8) Incidence matrices and their properties: Given an undi- 1 1 rected graph with n nodes and m edges, assign to each edge of a uniqueG index e 1, . . . , m and an arbitrary direction. 1 2 1 2 G ∈ { } n m The (oriented) incidence matrix B R × of is deﬁned by ∈ G 2 3 3 2 3 3 +1, if the edge e is (i, j) for some j, Bie =  1, if the edge e is (j, i) for some j, Fig. 9. A simple cycle γ and a cut χ with corresponding signed path vector − vγ = [+1 − 1 + 1]T and cutset orientation vector vχ = [−1 0 + 1]T. 0, otherwise.

T T Clearly, 1nB = 0m. Moreover, if diag( ae e 1,...,m ) is { } ∈{ } The cutset space of is subspace of Rm spanned by the cutset the diagonal matrix of edge weights, then the Laplacian of G T orientation vectors corresponding to all cuts of the nodes of satisﬁes = B diag( ae e 1,...,m )B . The undirected G L { } ∈{ } , that is, span vχ Rm χ is a cut of . The following graph is connected if and only if the rank of B is n 1. G { ∈ | G} WeG next consider an illustrative example in the next ﬁgure.− linear algebraic statements follow [27, Chapter 8] easily: (S1) the cycle space is Ker(B), T 1 4 1 4 (S2) the cutset space is Im(B ), and (S3) Ker(B) Im(BT) and Ker(B) Im(BT) = Rm. 1 2 ⊥ ⊕ 4 Statement (S3) is also known as a statement in the fundamental theorem of linear algebra. 2 3 2 3 3

IV. GENERALMODELSOFELECTRICALNETWORKS Fig. 8. Numbering and orienting the edges of an undirected graph. In the following, we develop a rather general electrical network model and connect it to the algebraic graph theory For the undirected unweighted graph on the left (and its concepts introduced in Section III. We will use the language oriented version on the right), the incidence matrix and the B of graph theory, but we remark that many alternative termi- Laplacian matrix are, respectively, L nologies arose in different disciplines of electrical engineer- +1 +1 0 0 2 1 1 0 ing (circuits, electronics, power, etc.). For example, graph- 1 0 +1 0 1− 2 −1 0 theoretic concepts such as nodes and edges are often referred B = −  , = − −  . 0 1 1 +1 L 1 1 3 1 to as buses, terminals, branches, lines, and so on. Likewise, − − − − −  0 0 0 1  0 0 1 1  all of the aforementioned graph matrices can be found under  −   −      multiple different names and sign-conventions, but we will As one can verify, 1n>B = 0m> and is positive semidefinite. consistently use the terminology from Section III. 9) Cycle and cutset spaces: LetL be an undirected unweighted graph with node set 1, . . . ,G n and m edges. Num- ber the edges of with a unique{ identifier} e 1, . . . , m A. Electrical network modeling and assign an arbitraryG direction to each edge. ∈ { } 1) Topology and variables: An electrical network is an A cut χ of is a strict non-empty subset of nodes. A cut undirected graph = 1, . . . , n , composed of n nodes G and its complement χc define a partition 1, . . . , n = χ χc. and m undirectedG edges{{ . Additionally,} E} we introduce a sepa- { } ∪ Given a cut χ, the set of edges that have one endpoint in each rate ground node 0 denotingE the common electrical ground { } subset of the partition is called the cutset of χ. and a separate edge set 0 = i 1,...,n i, 0 connecting each Given a simple undirected path γ in , the signed path vec- node i 1, . . . , n toE the ground.∪ ∈{ Without}{ } loss of generality, tor vγ 1, 0, +1 m of γ is defined by,G for e 1, . . . , m , we select∈ { an arbitrary} orientation for each undirected edge ∈ {− } ∈ { } i, j 0 and, specifically, orient all edges of the form +1, if e is traversed positively by γ, {i, 0} ∈as E(i, ∪0) E. For each directed edge (i, j) we define γ { } ve = 1, if e is traversed negatively by γ, − an oriented current flow fij R, and  • ∈ 0, otherwise. an oriented voltage drop uij R. • ∈ χ As in Section III-8, the topology of the oriented edges among Given a cutχ of , the cutset orientation vector v 1, 0, +1 m of theG cut has components ∈ the nodes 1, . . . , n (excluding the ground node) is encoded {− } by the oriented{ incidence} matrix B of . When including the +1, if e has its source in χ and sink in χc, ground node 0 and its associated edges,G the overall incidence { } vχ = 1, if e has its sink in χ and source in χc, matrix takes the form e −  T T 0, otherwise. 1n 0m (1+n) (n+m) Bground = − R × . In B ∈ Here the source (resp. sink) of a directed edge (i, j) is the node i (resp j). Figure 9 illustrates these notions. We remark that the choice of edge numbering and orientation The cycle space of is the subspace of Rm spanned by the is arbitrary; the latter simply reflects the reference directions G signed path vectors corresponding to all simple undirected cy- for the variables uij and fij. We will later also use the vectors γ m cles in , that is, span v R γ is a simple cycle in . u and f that collect the variables uij and fji, for i, j , G { ∈ | G} { } ∈ E 6 with elements ordered in correspondence with the numbering spaces in Section III-9, is known as Tellegen’s Theorem [127] of the edges. Next we introduce Kirchhoff’s laws and reveal in network analysis. Tellegen’s Theorem is extremely general the role of the ground node 0 . and holds independently of the (linear or possibly nonlinear) 2) Kirchhoff’s laws: In our{ graph-theoretic} setting we take dynamic and static characteristics of an electrical network. Kirchhoff laws as two fundamental axioms that define the 3) The ground node: It is convenient to specify for each fundamental physics of electrical networks [87]: node i 1, . . . , n two separate variables denoting its current ∈ { } Kirchhoff’s current law (KCL): For each node in the flow and voltage with respect to the ground. We define∗ • network, the algebraic sum of all current flows incident the current flow I = f from the ground to each • i i0 R to the node must be zero. Specifically, for all nodes node i 1, . . . , n in− KCL∈ (4) as the external current i 0 1, . . . , n , the current flow balance is: injection∈; { and } ∈ { } ∪ { } a potential Vi R as an auxiliary variable for each node 0 = f f , • ∈ ij − ji i 1, . . . , n 0 to specify KVL (6). j s.t. (i,j) 0 j s.t. (j,i) 0 ∈ { } ∪ { } X∈E∪E X∈E∪E Observe that KVL (6) defines the potentials Vi only up to an T where, slightly abusing notation, (i, j) 0 denotes arbitrary reference since 1 Ker(B ). It is convenient ∈ E ∪ E n+1 ground the oriented edge i, j taken from the original set of to define the potential of the electrical∈ ground as zero: { } non-oriented edges 0. More generally, for any graph ground potential: the ground has zero potential: V0 = 0. cut χ with cutset orientationE ∪ E vector vχ, • We can now write Kirchhoff’s laws in a way that is χ 0 = ve fe, (3) more familiar to scholars of graph theory [17] and dynamical e 0 systems [135]. KCL (4) reads for nodes 1, . . . , n as ∈E∪EX { } These equations can be rewritten compactly as follows. I = Bf . (7) Equation (3) implies that the vector of current flows m For the ground 0 , KCL gives the current injection balance f R is perpendicular to the cutset orientation vector { } χ∈ T n v of every cut and, thus, to the subspace Im(Bground). Ii = 0 . (8) Statements (S1) and (S3) in Section III-9 imply that i=1 f Ker(Bground) so that Observe that this balanceX equation is redundant as it can also ∈ T f be found by multiplying KCL (7) from the left by 1n. 0 = B 0 , (4) n+1 ground f Given that V0 = 0, KVL (6) gives the voltage ui0 between node i and the ground 0 simply as the potential Vi: n where f0 R is the vector of components fi0. { } ∈ Kirchhoff’s voltage law (KVL): For all simple undirected ui0 = Vi , for all i 1, . . . , n . • ∈ { } cycles in the network, the algebraic sum of all directed m Let u R be the vector that collects all other voltages uij voltage drops along the (oriented) edges of the cycle must for i,∈ j with appropriate numbering. Then KVL (6) can be zero. Specifically, for each simple undirected cycle γ be written{ } ∈ as E with signed path vector vγ , the voltage drop sum is: u = BTV, (9) γ 0 = v uij , (5) (i,j) that is, the voltage drops equal potential differences uij = (i,j) γ X∈ Vi Vj for each i, j with the sign convention as These equations can be rewritten as follows. Equality (5) specified− by the oriented{ } incidence ∈ E matrix B. implies that the vector of voltage drops u Rm is Kirchhoff’s laws (7), (9) define n+m linear equations relat- ∈ perpendicular to the signed path vector vγ of every cycle ing the 2n+2m variables (V, I, f, u). We further complement and, thus, to the subspace Ker(Bground). Statements (S2) these equations through constitutive relations relating fij and and (S3) in Section III-9 imply that u Im(BT ) and, u for any pair i, j of connected nodes. ∈ ground ij { } thus, that there exist so-called potential variables V0 R 4) Constitutive relations: Here we consider three basic ∈ and V Rn such that linear circuit elements: resistors, inductors, and capacitors. ∈ u V These three elements are illustrated with their circuit symbols 0 = BT 0 , (6) u ground V in Figure 10. The voltage uij over a circuit element and current flow f through it satisfy the following well-known n ij where u0 R is the vector of components ui0. ∈ constitutive relations: Note that the fundamental theorem of linear algebra, as resistor: u = r f , where r > 0 is a resistance; • ij ij ij ij presented in statement (S3) in Section III-9, together with inductor: ` d f = u , where ` > 0 is an inductance; • ij dt ij ij ij equations (4) and (6), imply that capacitor: c d u = f , where c > 0 is a capacitance. • ij dt ij ij ij T f u The constitutive relation for a resistor, uij = rijfij known as 0 0 = 0 , f u Ohm’s law gives together with KVL (9) the current flow over the resistor as fij = uij/rij = (Vi Vj)/rij. It is instructive that is, the sum of all instantaneous power flows fij uij − · along all edges i, j in the network equals zero. This general ∗ { } Our notation follows the convention that nodal (respectively, edge) vari- fact, direct consequence of the properties of cutset and cycle ables are denoted by capital (respectively, lower-case) letters. 7

rik, respectively. We show these resistances for the following ìj rij i j i j reason: When we set ri = rki and Vk∗/rki = Ii∗, then by Ohm’s law these two models are delivering the same current (a) resistive branch (b) inductive branch I = I∗ V /r = (V ∗ V )/r . i i − i i k − i ki c ij Thus, an ideal voltage source can always be converted to an i j ideal current source and vice versa. In the following, we focus without loss of generality on constant current sources. (c) capacitive branch I I Fig. 10. Circuit symbols for resistors, inductors, and capacitors i i k i Vk⇤ rki i + to remark that this flow function f = (V V )/r can also Ii⇤ ri = ij i − j ij be derived as the unique flow-characteristic that minimizes ri = rki the network losses subject to KCL (7) and assuming anti- symmetry fij = fji of the flow. This result, known as Fig. 12. Equivalent constant current and constant voltage sources Thomson’s Principle−, is nowadays an integral part of textbooks on algebraic graph theory and Markov chains [77], [55], [61]. We will collectively refer to resistors, inductors, and capac- B. Different branch models itors as impedances, a term which is also often used when Kirchhoff’s laws, the constitutive relations, and the models multiple basic circuit elements are lumped into a single one. for loads and sources provide the required ingredients for our 5) Load models: For the ground 0 we omit the double- network model. We connect the loads and sources through indexing of adjacent circuit elements,{ } and use c , l , and r i i i a network whose branches are modeled by lumped circuit instead of c , l , and r . Circuit elements (or a collection i0 i0 i0 elements taking into account losses, charging, waves, and thereof) connected to the ground are referred to as shunt other effects. A widely used branch model is the -model impedances, and they are often used to model loads. In Π depicted in Figure 13. The Π-model consists of a series particular, a shunt resistor r injects a load current I = i load,i resistive-inductive impedance modeling the branch inductance f = u /r = V /r and models so-called active i0 i0 i i i and losses as well as a shunt capacitor to ground at each end −power loads− which dissipate− energy. On the contrary shunt of the branch modeling the cable charging. Typically, the two capacitors and inductors model so-called reactive power loads shunt capacitors take identical values. that merely transform energy; see Section VI-B. Aside from The Π-model can be used to model various branch charac- such impedance loads, which draw a current I linearly load,i teristics, including long high-voltage transmission lines (domi- depending on the potential V , another popular load model i nantly inductive), underground cables (with additional resistive is a constant current demand Iload,i = Ii∗ R 0 or more ≤ and capacitive components), and short wires (dominantly general nonlinear relations between load current∈ I and load,i resistive) [90], [106]. Note that if there are multiple branches the potential V , e.g., a load injecting a constant instantaneous i connected to a node, each modeled by the Π-model, we can power P = I V 0. A load model aggregating constant i∗ load,i i merge the multiple parallel shunt capacitors into a single one. impedance, constant≤ current, and constant power loads is normally called a ZIP load [90]. We refer to Figure 11 for r `

an illustration of such load models. i ij ij j + i ci cj

ri `i ci Ii⇤ Pi⇤ -

Fig. 13. Π-model of a branch in electrical network between nodes i and j.

Fig. 11. A load model aggregating a shunt impedance (ri, li, ci), a constant ∗ ∗ C. A prototypical electrical network current load Ii , and a constant power load with constant Pi = Iload,iVi. In what follows, we consider a prototypical electrical net- 6) Source models: A device that provides a constant current work model to illustrate applications of algebraic graph theory. injection Ii = Ii∗ R 0 or a constant potential Vi∗ R 0 For each branch we consider a Π-model as in Figure 13. When (relative to the ground)∈ ≥ at a node i is termed an ideal current∈ ≥ multiple Π-models are connected to the same node, we lump source or an ideal voltage source, respectively. Figure 12 all parallel capacitors into a single equivalent capacitance. At depicts an ideal current source and voltage source in com- each node i 1, . . . , n we thus consider an equivalent bination with a shunt resistor r and with a series resistance capacitance c ∈> { 0, a shunt} resistance r 0, a constant i i i ≥ 8

current injection I∗ R, and a constant power injection 1) Resistive-capacitive interconnection: If all branches are i ∈ Pi∗ R modeling sources and loads as in Figures 11 and purely resistive (R positive definite and L = 0m m), the 12. In∈ this case, the network equations are model (11) reduces to × ˙ KCL: (10a) CV = ( R + G) V + I∗ + P ∗ V, (12) I = Bf , − L T 1 T KVL: u = B V, (10b) where R = BR− B is the so-called conductance matrix, a LaplacianL matrix associated with the (undirected) graph of ground: I = Iload CV,˙ (10c) − the electrical network with weights given by inverse resistance branch: Lf˙ = u Rf , (10d) − 1/rij for edge i, j ; see Section III-8. We will later reveal { } load: Iload = I∗ + P ∗ V GV , (10e) various properties of the nonlinear dynamic equations (12) by − studying the properties of the matrix BR 1BT + G. where R, L, C, G are diagonal matrices of r , ` , c , and the − ij ij i In the absence of constant-power loads (P = 0 ), equi- symbol conventionally denotes the shunt conduc- ∗ n gi = 1/ri librium points of (12) are determined by a static and well- tance (reciprocal of resistance). Finally, and I∗ = (I1∗,...,In∗) studied model characterized by the Laplacian matrix and P are the vectors of constant current and power injections. R ∗ the conductance loads G: L It is convenient to reduce the network equations (10) to a state-space model defined in terms of the variables V and f I∗ = ( R + G)V. (13) associated with the capacitive and inductive storage elements. L 2) Lossless inductive-capacitive case: In the absence of By inserting (10a), (10e) in (10c), respectively, (10b) in (10d), loads (G = 0n n and I∗ = P ∗ = 0n) and dissipative elements we obtain × (R = 0m m) — that is, in an entirely lossless circuit — the × C V˙ G B V I∗ + P ∗ V electrical network model (11) can be re-written as = − T − + . (11) L f˙ B R f 0m − C V˙ 0 B V = − , (14) L f˙ BT 0 f A block-diagram of the electrical network model (10), respectively, (11), is shown in Figure 14. Observe the separation of or by taking another derivative of V , as the dynamics associated to n nodes (ground and loads) and ¨ m edges (branches), the exogenous current injections I and CV = LV, (15) ∗ −L nonlinear power injections P ∗, as well as the interconnection 1 T where L = BL− B is the weighted Laplacian matrix with through Kirchhoff’s current and voltage laws via B and B>. L 1 weights L− . 3) Homogeneous case: Consider a slightly more complex .. . Ii⇤ ,Pi⇤ scenario, where all network branches are made of the same I⇤ + P ⇤/Vi giVi material, and thus the ratio of l /r = τ is constant for all i i . ij ij .. edges i, j . Assume also that there are no constant- { } ∈ E loads current or constant-power loads, so that I∗ = P ∗ = 0n. By taking a second derivative of V in (11), substituting for f˙, and Iload . + .. finally eliminating f, the electrical network model (11) can be dVi re-written as ci = Iload Ii _ dt . . τCV¨ + (τG + C)V˙ + ( R + G)V = 0n . (16) I . V L ground dynamics For τ = 0, we recover the model (12) and for τ = and G = ∞ KCL B BT KVL 0n n, we recover the model (15). The equations (16) are also reminiscent× of the resistively coupled LC tanks introduced in . Section II-A. The LC tanks can be modeled by the equations f .. u ¨ ˙ 1 df ij CV + RV + L− V = 0n , (17) ìj = uij rijfij L dt . .. where V is the vector of voltages across every LC tank circuit [49]. In the homogeneous case in Section II-A, i.e., for branch dynamics R = rIm, C = cIn, and L = Ìn, the model (17) reduces to Fig. 14. Block-diagram of the electrical network model (10). 2 V¨ + τ 0 V˙ + ω V = 0 , L 0 n where we recall that ω0 = 1/√`c is the natural frequency of oscillations, = BBT is the unweighted Laplacian of D. Noteworthy special cases L the network, and τ 0 = 1/rc is a uniform time-constant that The electrical network model described in equations (10) determines the relaxation time to the synchronous solution. and (11), respectively, is fairly general and admits a few In the following sections, we analyze the static and dynamic special cases that we will repeatedly revisit as pedagogical properties of the electrical network model (10) and its special examples. cases from the viewpoint of algebraic graph theory. We remark 9 that most of the following approaches extend (either directly (i) the Moore-Penrose inverse of given by L or at least conceptually) to richer classes of electrical networks 0 1 with switching behavior as in power electronics [60], [162], λ2 T † = V . V (20) multi-physical dynamics as in synchronous generators [63], L  ..  [74], or nonlinear oscillators [152], [47], [39], among others. 1  λn  is symmetric positive semidefinite, with zero row and V. STRUCTUREAND DYNAMICSOF LINEAR ELECTRICAL column sums, and satisfies † = † = Πn; NETWORKS L L LLβ T (ii) the regularized Laplacian reg = + n 1n1n with β > 0 In what follows, we explain how the structure of an electri- is non-singular, positive definite,L L and satisfies cal network (in terms of its topology and impedances) reveals 1 various insights about the associated electrical dynamics. The 1 β T − 1 T reg− = + 1n1n = † + 1n1n interplay of structure and dynamics is revealed through the L L n L β n 1 algebraic graph theory methods introduced in Section III. β 1 (21) This section focuses on the case of linear electrical networks, λ2 T =V  .  V ; described by special cases of the general model (11). The study .. of nonlinear networks is deferred to Section VI.  1   λn    (iii) the shunted Laplacian shunt = + εIn with ε > 0 is A. Static resistive networks non-singular, positive definite,L andL satisfies We begin our analysis with the case of a static resistive 1 ε network with no constant power loads, as described by (13). 1 1 λ2+ε T For simplicity of notation, let us drop the super- and subscripts shunt− = V  .  V ; (22) L .. in this section and simply rewrite (13) as 1  λ +ε   n  I = ( + G)V, (18)  (n1) (n 1) (iv) the grounded Laplacian ground R − × − is the L L ∈ T n n leading principal (n 1) (n 1) submatrix of the where = R × is a symmetric and irreducible − × − L L ∈ n n Laplacian matrix (after removing the nth row and Laplacian matrix, G R × is a diagonal matrix with L ∈ n nth column) and has the following properties: ground is nonnegative diagonal entries, and I,V R are constant L ∈ non-singular and positive definite, ground is a Metzler vectors. We remark that equation (18) is also of interest −L independently of circuits, as linear diffusive equations with matrix, and its inverse is a nonnegative matrix satisfying Laplacian matrices arise all throughout the sciences [138]. 1 T ground− = (ei en) †(ej en) . (23) 1) Characteristics of solutions and Laplacian inverses: We L ij − L − explore the solution space of the resistive circuit equation (18). The regularized Laplacian reg, the shunted Laplacian We consider the singular and non-singular case separately. L shunt, and the grounded Laplacian ground are all non-singular Singular circuit equations: When G = 0n n, we know L L × and positive definite matrices. Due to these favorable proper- from Section III-6 that is singular with Ker( ) = span(1 ) L L n ties they are often preferred for numerical computations or for and with Im( ) = 1⊥. Hence, equation (18) admits a solution L n analytic studies, as compared to the singular Laplacian that if and only if I 1n⊥, that is, the current injections are L T ∈ requires careful treatment of its nullspace. Note that the regu- balanced: 1nI = 0. In this case, the solution is given by larized and shunted Laplacians reg and shunt have the same L L V = Vhom + Vpart = α 1n + †I, (19) eigenvectors as the Laplacian , but the zero (respectively, · L all) eigenvalues are shifted to theL right. The inverse Laplacian 1 where the homogeneous solution Vhom = α n with α R is matrices (20)-(22) are sometimes referred to as Green matrices flat-voltage profile · ∈ the without current flows, and the particular and studied in different applications [21], [92], [61], [31]. On solution is Vpart = †I 1n⊥, where † is the Moore- L ∈ L the other hand, the grounded Laplacian ground and its inverse Penrose inverse of the Laplacian matrix [99]. The following do not preserve the eigenstructure of L; see [109], [153] for proposition collects some properties ofL the various possible L relations between and ground. generalized inverses of a Laplacian matrix. Aside from mathematicalL L convenience, the grounded Lapla- Proposition 5.1 (Inverse Laplacian matrices [27], [52], cian ground is also a natural concept in circuit analysis when [75], [38], [69]): Consider a symmetric and irreducible Lapla- the nLth node is literally the ground of the circuit [37]. Vice n n cian matrix R × with singular value decomposition versa, the matrix + G in (13) can be thought of as the L ∈ LR 0 grounded Laplacian of a network with n + 1 nodes. Along the λ2 1 1 T √ 1 v ... v √ 1 v ... v same lines, the shunted Laplacian shunt corresponds to adding = [ n n 2 n ] . [ n n 2 n ] , L  ..  a shunt resistor to ground of value εLat every node or grounding λ T =V n =V a network with n + 1 nodes. Aside from circuits, natural   where λ2|, . . . , λ{zn > 0 }and v2, . . . , vn | 1n are{z the nonzero} applications of grounded Laplacian matrices are networks with eigenvalues of and associated eigenvectors⊥ collected in the reference nodes as in opinion dynamics with stubborn agents L matrix V. The following statements hold: [109], [153], distributed estimation with partial measurements 10

[10], or platooning of vehicles [76]. The grounded Laplacian Proposition 5.2 (Effective resistance): The effective re- eff ground is also an interesting algebraic graph and matrix theory sistance rij between two nodes i, j 1, . . . , n of an conceptL in its own right and studied in [100], [69]. undirected, connected, and weighted graph∈ { with Laplacian} Non-singular circuit equations: In case that G has at matrix is given by L least one positive diagonal entry, then ( + G) is a Hurwitz eff T − L r = (e e ) †(e e ) = † + † 2 † . (25) Metzler matrix, as discussed in Section III-7 and Appendix A. ij i − j L i − j Lii Ljj − Lij The solution to (18) is therefore unique, and is given by We remark that the effective resistance and all of its proper- 1 V = ( + G)− I. (24) ties derived below can be obtained analogously if G = 0n n L or with the regularized or grounded Laplacian matrices6 [52].× The matrix + G is sometimes called a loopy Laplacian The effective resistance is also referred to as the resistance L matrix since the non-zero diagonal entry represents a self-loop distance [88] since it defines a distance metric on a graph (it is in the graph [52]. This matrix can also be thought of as the symmetric, nonnegative, and satisfies the triangle inequality). grounded Laplacian matrix of an appropriate - (n+1) (n+1) Proposition 5.3 (Effective resistance is a distance [88]): dimensional Laplacian matrix [52]. Since ( +G) is× Metzler, − L Consider an undirected, connected, and weighted graph with Hurwitz, and irreducible, then we know from Section III-7 that eff 1 n nodes. The associated effective resistances rij satisfy ( + G)− is a positive matrix. An important consequence is L (i) nonnegativity: reff 0 for all i, j 1, . . . , n and reff = that if the current injections I in (24) are nonnegative with at ij ≥ ∈ { } ij least one strictly positive injection, then the unique voltage 0 if and only if i = j; (ii) symmetry: reff = reff for all i, j 1, . . . , n ; and solution V is a strictly positive vector; this is in contrast ij ji ∈ { } to the singular case (19). For example, this occurs if the (iii) triangle inequality: reff reff + reff for all i, j, k ij ≤ ik kj ∈ current injections I arise from converting voltage sources into 1, . . . , n . { } current sources (Section IV-A6). The matrices reg, ground, L L shunt, and ( + G) are all positive definite, and their inverses r/3 r/3 Lcan be furtherL characterized in terms of their so-called decay r/3 r/3 r/3 r/3 properties [46], [15], [100]. These decay properties reveal that 1 2 1 2 the effect a current injection Ii at node i on the potential Vj of r another node j diminishes according to the distance between r/3 r/3 r/3 r/3 nodes i and j; we now explore this distance concept further. r/3 r/3 2) The effective resistance and its properties: Consider - + - + an undirected, connected, and weighted graph and an as- e↵ e↵ sociated connected resistive electrical network governed by r12 = r/2 r12 = r/3 the equations (18) with G = 0n n. The effective resistance reff between any pair of (not necessarily× neighboring) nodes Fig. 16. Adding an edge to the circuit in the panel lowers the effective ij resistance between nodes {1, 2} in the circuit in the right panel. In other i, j 1, . . . , n is defined as the potential difference V V ∈ { } i − j words, the effective resistance takes parallel paths into account and is a between these nodes when a unit current is injected into node monotonically non-increasing function of topology and weights. i and extracted from node j; see Figure 15 for an illustration. For the example in Figure 5 the effective resistance between Compared to other distance metrics on graphs, e.g., the eff red topological distance given by the length of the shortest (possi- nodes 1 and 3 takes the well-known value r13 = r13 = r12 + r23 sometimes referred to as reduced or equivalent resistance. bly weighted) path between nodes [72], the effective resistance takes into account all parallel paths. For example, in the left panel of Figure 16, nodes 1 and 2 are connected by two i parallel paths each of resistance r. They have a resistance distance reff = r/2 whereas the weighted shortest path takes 1A ij the value r. Hence, the effective resistance is the preferred distance metric in electrical networks, e.g., see [40], and also in non-technological applications where parallel paths need to be taken into account such as chemistry [81], ecology [96], e↵ 1A rij and disease spreading [2], amongst others. j - + Related to these parallel paths is the fact that the effective resistance characterizes an average performance measure for Fig. 15. The effective resistance between nodes i and j is the potential random walks in Markov chains [55], distributed estimation difference when a unit current of 1 A is injected in i and extracted in j. [10], average consensus [156], [92], and other diffusive graph algorithms [61]; see also Proposition 5.11. Indeed, com- Note that the potential difference between nodes i and j is pared to worst-case performance measures related to dominant (e e )TV , the current injection takes the form I = e e = eigenvalues of adjacency and Laplacian matrices, the sum of i − j i − j V , and accordingly V = †(ei ej) from (19). From these all effective resistances is related to the harmonic mean of all Lsimple facts we obtain theL following− result. non-zero Laplacian eigenvalues. 11

Proposition 5.4 (Effective resistance and Laplacian eigen- algebraic elimination procedure is termed Kron reduction after values [154]): Consider an undirected, connected, and Gabriel Kron [89]. The reader is invited to verify that the n n weighted graph with n nodes, its Laplacian matrix R × well-known transformations in Subsection II-D are special L ∈ with spectrum spec( ) = 0, λ2, . . . , λn , its effective resis- cases of Kron reduction, and so are many related circuit eff L { } tances rij in (25) for all i, j 1, . . . , n , and deﬁne the total transformations [142], [117], [115]. An application to a power ∈ { n } eff effective resistance as Rtot = i,j=1,i

8 8 8 15 37 10 9 9 10 10 Another important property of the effective resistance is 37 02 10 29 38 10 30 25 03 30 25 26 28 26

d 04 2 38 2 29

18 27 ra 5 05 9 1 24 / 6

i 1 3 27 35 6 1 Rayleigh’s monotonicity law δ 1 28 stating that the effective resis- 17 39 0 18 22 F 16 4 6 9 17 21 3

15 -5 8 1 35 0 2 45 6 8 10 tances are monotonically increasing functions of the branch 21 24 39 22 7 4 14 12 14 15 23

31 6 36

5 15 16 2 7 6 12 19 23 13 7 2 06 11 resistances, e.g., compare the two networks and effective 13 20 36 07 10 10 7 31 11 19 08 10 34 33 7 32 d 20 09 8 2 33

5 4 ra 32 5 3

/ 3 34 4 9 i 4 resistances in Figure 16. We state Rayleigh’s monotonicity 3 δ 5 0 5 Fig. 9. The New England test system [10], [11]. The system includes -5 10 synchronous generators and 39 buses. Most of the buses haveconstant 0 2 4 6 8 10 law in the language of algebraic graph theory below. activeand reactive power loads. Coupled swing dynamics of 10 generators TIME / s are studied in the case that a line-to-ground fault occurs at point F near bus 16. Fig. 10. Coupled swing of phaseangle δi in New England test system. Fig. 17. Illustration of theTh IEEEe fault duration i 39s 20 cycle News of a 60-Hz si Englandne wave. The result is obt powerained system [105] with Proposition 5.5 (Rayleigh’s monotonicity law [55]): Con- by numericalintegration of eqs. (11). tegeneratorst system canbe repres nodesented by {1,..., 10} depicted as circles on the left panel; graph- δ˙i = ωi, sider two symmetric and irreducible adjacency matrices 10 theoreticHi abstraction2 in thea middlere provided tod paneliscuss whethe withr the instab nodesility in Fig. 10{1,..., 10} outside the ω˙ i = Diωi + Pmi GiiEi EiEj ⎫ (11) occurs in thecorresponding real power system. First, the πfs − − − · n n j=1,j=i ⎪ classical model with constant voltage behind impedance is ˜ !̸ ⎬ { } A, A corresponding to two undirected, connected, and dashedGij co circle;s(δi δj) + B andij sin(δi theδj) , Kron-reducedused for first swing crit networkerion of transient sta reducedbility [1]. This is to nodes 1,..., 10 . R × · { − − } where i = 2,...,10. δ is the rotor angle of generato⎪r i with because second and multi swings maybeaffected by voltage ∈ i ⎭ fluctuations, damping effects, controllerssuch as AVR, PSS, respecttobus 1, and ωi the rotor speeddeviation of generator weighted graphs with identical node sets but possibly different i relative to system angular frequency (2πf = 2π 60 Hz). and governor. Second, the fault durations, which we fixed at s × 20 cycles, are normally less than 10 cycles. Last, the load δ1 is constant for theaboveassumption. The parameters f , H , P , D , E , G , G , and B are inper unit condition used above is different from the original one in s i mi i i ii ij ij [11]. We cannot hence argue that globalinstabilityoccurs in edge sets and edge weights. Consider the associated effective system except for Hi and Di in second, and for fs in Helz. The mecPropositionhanicalinput power P togener 5.6ator i and t (Kronhe the real system reduction. Analysis, however, doessho [132],w a possibility [52]): Consider mi of globalinstability in real power systems. eff eff ˜ magnitude Ei of internal voltage ingenerator i areassumed resistances r and r˜ for i, j 1, . . . , n . If Aij Aij for tobeconstant for transient stability studies [1], [2]. Hi is IV. TOWARDS A CONTROL FOR GLOBAL SWING ij ij ththee inertiacons resistivetant of generator i, Di its d networkamping coefficient, equationsINSTABILIT (27)Y parameterized by the and they areconstant. G is the internal conductance, and eff ∈ {eff } ≥ ii Globalinstability is related to the undesirable phenomenon Gij + jBij the transfer impedance betweengenerators i n n all i, j 1, . . . , n , then r˜ij rij for all i, j 1, . . . , n . and j; They are the parameters which change withnetwork that shouldbeavoided by control. We introduce a key topoirreduciblelogy changes. Note that electrica conductancelloads in the test system mechani matrixsm for thecontrol problem andR discuss×control satisfying 1n = ∈ { } ≤ ∈ { } are modeled as passive impedance [11]. strategies for preventing or aLvoiding ∈the instability. n L Aside from this important monotonicity property exploited B.0Numericaandl Experiment the balancedA. currentInternal Resonance as injectionsAnother Mechanism I satisfying Coupnled swing dynamics of 10 generators in the Inspired by [12], we here describe the globalinstability R test system are simulated. Ei and the initial condition with dynamical systems theory close to internal resonance ∈ T [23], [24]. Consider collective dynamics in the system (5). in many algorithmic applications, the effective resistance is (δ1i(0), ωiI(0) ==0) for 0ge.nera Considertor i are fixed through pow alsoer the associated Kron-reduced network flow calculation. Hi is fixed atthe original values in [11]. For the system (5) with small parameters pm and b, the set n 1 Pmi and constant power loads areassumed tobe 50% attheir (δ, ω) S R ω = 0 of states in the phase plane is { ∈ × | } ratings [22]. The damping Di is 0.005 s for all generators. called resonant surface [23], and its neighborhood resonant red also known to be a strictly convex function of the graph Gequations, G , and B arealsobas (28)ed on the orig parameterizedinal line data band. The phase plane i bys decompo thesed into th reducede twoparts: injections ii ij ij I in [11] and the power flow calculation. Itis assumed that resonant band and high-energy zone outside of it. Here the the test system is in a steady operating condition at t = 0 s, initial conditiredons of local and mode disturbances in Sec. II weights [70]. The latter fact makes the effective resistance thandat a line-to-g conductanceround fault occurs at point F near bu matrixs 16 at indeed existinside .the r Theesonant band following. Thecollective motion statements hold: t = 1 s 20/(60 Hz), and that line 16–17 trips at t = 1 s. The before the onset of coherent growing is trappednear the fault du−ration is 20 cycles of a 60-Hz sine wave. The fault resonantLband. On the other hand, after thecoherent growing, 7 attractive for circuit design as well as the synthesis and tuning issimulated by adding a small impedance (10− j) between it escapes from 1the resonant band asshown in Figs. 3(b), bus 16(i)and groundThe. Fig. 10 sho matrixws coupled swings ofrotor 4(12b), 5, and− 8(b) and (cis). The tr nonnegativeapped motion is almost and column- angle δi in the test system. The figure indicates that all rotor integrableand22is regarded as a captured state in resonance anglesstarttogrow coherently at about 8 s. Thecoh−Lerent [23]. AtLa moment, the integrable motion maybe interrupted of diffusive algorithms leveraging the analogy to electrical growing is globalinstability. by small kicks that happen during the resonant band. Thatis, red stochastic, and thusthe so-ca thelled release reducedfrom resonance [23] h currentappens, and the injections I = C. Remarks collective motion crosses the homoclinic orbit in Figs. 3(b), networks [55], [10], [156], [70], [92]. It was confirmed thatthe system (11) in the N1ew Eng- 4(b), 5, and 8(b) and (c), and hence itTgoes redaway from land test systemIsho1ws globalinst12ability. A f22−ew commI2ents arethe reson balanced:ant band. Itis therefore said1that globIalinstabilit=y 0. In conclusion, the effective resistance is motivated from − L L (')$ red (ii) The reduced conductance matrix = 11 electrical networks, but it has now a firm place in graph theory Authorized licensed use limited to: Univ of Calif Santa1 Barbara.T Downloaded on June 10, 2009 at 14:48 from IEEE Xplore. Restrictions apply. L L − 12 22− 12 is a nonnegative, symmetric, and irreducible and its applications. We refer to [55], [154], [88], [52], [10], LaplacianL L L matrix satisfying red1 = 0. [70], [69], [157], [75], [61] for further exploration of the rich L literature. Now that we have established that Kron reduction is well- 3) Network Kron reduction: We revisit the series circuit posed and defines an electrical network, we are interested in its contraction and the star-triangle transformation from Subsec- graph-theoretic properties. We begin by studying the topology tion II-D and analyze them through algebraic graph theory. and weights of the Kron-reduced network. Observe from the Consider again the connected and resistive electrical net- examples in Figures 5, 6, and 17 that the graph associated to work model (18), and assume for simplicity here that G = the Kron-reduced network is always denser than the original graph both in terms of topology and weights. This statement 0n n. We partition the nodes into two sets 1, . . . , n = × { } can be made precise as follows. 1 2 that we term boundary nodes 1 and interior nodes U ,∪ e.g., U = 1, 3 and = 2 forU the series circuit in Proposition 5.7 (Graph-theoretical properties of Kron re- U2 U1 { } U2 { } Figure 5 and 1 = 1, 2, 3 and 2 = 4 for the star in duction [52]): Consider the network equations (27) and the Figure 6. The associatedU { partitioned} U current-balance{ } equations Kron-reduced equations (28) with conductance matrices and (18) are red, respectively. The graph associated to the conductanceL L red I1 11 12 V1 matrix has an edge between boundary nodes i, j if = L L . (27) 1 I T V and onlyL if either ∈ U 2 L12 L22 2 Observe that the lower-right block is a loopy Laplacian i, j is an edge in the original graph associated to the 22 • { } matrix and is thus (as we observedL in Section V-A1) non- conductance matrix , or L there is a path i, k , . . . , k , j in the original graph singular. By eliminating the voltages V2 of the interior nodes • 1 m 1 1 T { } as V = − I − V , we obtain the reduced model between nodes i and j passing through only interior nodes 2 L22 2 − L22 L12 1 1 1 T k1, . . . , km 2. I − I = − V , (28) { } ⊂ U 1 − L12L22 2 L11 − L12L22 L12 1 Moreover, for all distinct i, j , it holds that red , ∈ U1 Lij ≤ Lij Ired red that is, the weights are non-decreasing. L where Ired| and{z red can} be| interpreted{z as reduced} current A consequence of Proposition 5.7 is the following char- injections and reducedL conductance matrix, respectively. This acterization that is intuitive from the Y-∆ transformation in 12

Figure 6: if a set of interior nodes κ 2 forms a connected when G has at least one strictly positive diagonal element. subgraph in the original network, then⊆ U the boundary nodes In this case, the associated compartmental system is outflow- adjacent to κ form a clique in the Kron-reduced network. connected (see Appendix A and Section II-B), and the matrix 1 The careful reader may have observed that the series con- C− ( + G) is Metzler, Hurwitz, and irreducible. The traction in Figure 5 is an instance of Kron reduction network −followingL proposition summarizes this discussion. where the reduced resistance takes the same value as the Proposition 5.9 (Stability of dissipative RC network): Con- red effective resistance between nodes 1 and 3: r13 = r12 +r23 = sider the dissipative RC network dynamics (29) and assume eff r13. In general, the Kron-reduced matrix and the effective that G has at least one strictly positive diagonal element. Then resistance admit such a direct relationship only in very uniform from every initial voltage profile V (t = 0), the dynamics networks; see [52] for details. However, it is always true that (29) converge exponentially to the unique equilibrium voltage the effective resistance is invariant under Kron reduction. profile 1 Proposition 5.8 (Invariance of effective resistance [52]): limt V (t) = ( + G)− I∗ . Consider the network model (27) and the Kron-reduced model →∞ L (28) with conductance matrices and red, respectively. Then Next, consider the case without shunt conductances when L L eff for any boundary nodes i, j 1, the effective resistances rij G = 0n n: ∈ U red × can be equivalently computed from or : CV˙ (t) = V (t) + I∗ . (30) L L −L eff T T red † Recall from Section V-A that the network dynamics (30) admit rij = (ei ej) †(ei ej) = (ei ej) (ei ej) . − L − − L − T an equilibrium as in (19) if and only if 1nI∗ = 0. To further We remark that all of the above results on Kron reduction characterize the degree of freedom α R of the equilibrium ∈ can be adapted to the case when the network features shunt (19), note that the total charge is conserved: resistors (G = 0n n), and related topological, spectral, and 6 × d T T algebraic properties can be derived; see [52] for further details. 1 CV = 1 I∗ = 0 , (31) dt n n The graph-theoretic perspective on Kron reduction finds T T direct application in the synthesis and analysis of circuits where we have used the fact that 1n = 0n. Accordingly, 1TCV (t) = 1TCV for all t 0, whereL V = V (t = 0). It [132], [136], [150] particularly in the context of large-scale n n 0 ≥ 0 integration chips [113], [3], power system and power elec- follows by substituting (19) into this conservation law that tronics model reduction [143], [94], [28], [47], [129], smart 1TC(V I ) n 0 † ∗ (32) grid monitoring [48], [123], electrical impedance tomography α = T − L . 1 C1n [23], [41], and many other domains of electrical networks. In n a general context, algebraic equations governed by Laplacian To show stability of the equilibrium profile (19) with α as in matrices such as (18) are encountered in many scientific (32), we define the voltage error coordinate disciplines. Thus, Kron reduction can be found under different V˜ = V α1 †I∗ , (33) names and with a graph-theoretic perspective in Gaussian − n − L elimination of sparse matrices [65], [71], [114], sparse grid and and a quick calculation (making use of Proposition 5.1) finite-element solvers [139], [43], [138], statistical mechanics shows that V˜ satisfies the differential equation CV˜˙ = V˜ . [107], data mining [155], [69], reduction of Markov chains ˜ 1 ˜−LT ˜ Consider now the energy-like function W (V ) = 2 V CV . [98], [18], signal processing on graphs [108], [159], and pure It can be verified that this energy is non-increasing along algebraic graph theory [66], [62], [125] among others. trajectories: We conclude by remarking that this rich literature dating d T 2 λ2 back to the early days of electrical circuits is still active today. W (V˜ ) = V˜ V˜ λ2 V˜ W (V˜ ) , (34) Many applications and graph-theoretic properties are still to be dt − L ≤ − k k ≤ −cmax explored. Even apparently simple extensions to directed and where λ2 is the second-smallest eigenvalue of the Laplacian complex-valued graphs (as occurring later in Section VI-B) are matrix known as the algebraic connectivity; see Section III-6. mostly open to the best of our knowledge; see the concluding From this so-called dissipation inequality [147], we obtain the λ2 Section VII or, e.g., the recent article [134] discussing classical exponential decay estimate W (V˜ (t)) W (V˜0) exp( t), ≤ − cmax and open problems in linear resistive networks. which again implies that V˜ (t) converges exponentially:

˜ ˜ cmax λ2 B. Dynamic resistive-capacitive (RC) networks V (t) V0 exp t . (35) ≤ cmin −2cmax Now that we have thoroughly examined static resistive This discussion is summarized in the following proposition. networks, we move on towards dynamic RC networks with the Proposition 5.10 (Stability of RC network without shunt con- first-order dynamics (12) in the linear setting when P ∗ = 0n: ductances): Consider the RC network dynamics (30) without ˙ T CV (t) = ( + G) V (t) + I∗ . (29) shunt conductances and assume that 1nI∗ = 0. Then for every − L n initial voltage profile V0 R , the dynamics converge to the We will assume that C is a diagonal and positive definite unique and exponentially∈ stable equilibrium voltage profile matrix of capacitances, and the conductance matrix is an L irreducible Laplacian matrix. Consider first the dissipative case limt V (t) = α1n †I∗, →∞ − L 13 where α is given by (32). The convergence is exponential with circuit after being subjected to impulsive current inputs arising, a decay rate proportional to λ2 as in (35). Moreover, the total e.g., from line faults [38], [126]. While admittedly, the average charge is conserved along solutions as in (31). performance index (36) is of minor importance to circuits, The following remarks are in order. In the absence of it plays a key role in the design of distributed algorithms with diffusive dynamics [29], [7], [156], [55], [69], [92], external injections, I∗ = 0n, the voltages equalize to the flat T 1nCV0 [61], where the analogy to the effective resistance provides profile T 1n; this constant uniform voltage is a weighted 1nC1n average of the initial voltage values, with weights depending important intuition, and well-known concepts on the electrical on the capacitances. The features of the particular solution side (such as Rayleigh’s monotonicity law) inspire and inform the design and analysis of algorithms. Finally, we remark that †I∗ have been discussed in detail in Section V-A. The voltage Lerror coordinate (33) is related to the disagreement vector the above result can be extended to more general cost functions studied in consensus problems [103], [33]. The exponential than (36), higher-order dynamics, and discrete-time settings [29], [7], [156], [110], [38], [130], [38], [130]. Extensions to decay estimate λ /c in (35) depends on the maximum 2 max nonlinear circuits remain an open problem. capacitance as well as on the algebraic connectivity λ2 of the network. The algebraic connectivity is a well-studied quantity in algebraic graph theory dating back to the seminal C. Dynamic resistive-inductive-capacitive (RLC) networks work by Fiedler [64]. For example, λ2 is a popular metric In this section we analyze the full RLC network model (11) in graph partitioning and community detection [111], [67] from Section IV-C, repeated here for convenience: as it quantifies the smallest bottleneck in the graph, where C V˙ G B V I∗ + P ∗ V “smallest” is understood in our context as the minimal current = − T − + . L f˙ B R f 0m flow over any cut separating the nodes of an electrical network. − The exponential decay estimate (35) is achieved for a worst- Our approach will be to leverage the algebraic graph theory case initial condition V˜0 aligned with the eigenvector v2 asso- methods introduced in Section III. We begin by highlighting ciated to the eigenvalue λ2. However, often one is interested the energy conservation and dissipation properties of the in an average integral-quadratic performance criterion network system (11). Consider the electric and the magnetic energy associated to the network storage elements: ∞ ˜ T ˜ E V (t) ΠnV (t) dt , (36) 1 1 0 (V, f) = V TCV + f TLf . (38) Z H 2 2 where the expectation is with respect to a random initial The time derivative of the energy (38) along trajectories of the error voltage profile V˜ with zero mean and V˜ V˜ T = Π 0 E 0 0 n power balance (possibly due to a random realization of the current demands network dynamics (11) is given by the I or initial voltages V ). The projector matrix Π induces T T 0 n ˙ V B V V G V the global voltage error V˜ TΠ V˜ = V˜ average(V˜ )1 2, (V, f) = T − n k − nk H f B f − f R f and discards values of V˜ (t) and V˜0 aligned with 1n that =0 (lossless power circulations) 0 (power losses) do not affect the transient dynamics. The integral quadratic ≤ performance metric criterion (36) is well-known in control and T T |+V I∗ +{z1 P ∗ , } | {z (39)} signal processing under the name of an -norm of a system n H2 [161], and it is well-studied for the system (30) in the context (external power supplied) of consensus systems [29], [7], [156], power systems [126], T T [110], and random walks [55], [69], among others. In our case where we used| the identity{z V} (P ∗ V ) = 1nP ∗. The last and for identical capacitors C = In, the average performance term in the power balance equation (39) corresponds to the criterion (36) evaluates to an average of the inverse non-zero external power supplied to the network through the external Laplacian eigenvalues as in the total effective resistance (26). current and power injections, the central term corresponds to Proposition 5.11 (Average performance of RC network dissipation induced by shunt and branch resistances, and the first term evaluates to zero due to skew-symmetry of matrix in [27]): Consider the RC network dynamics (30) with identical the quadratic form. To further understand the role of the first capacitors C = I , the voltage error coordinate (33), and n term, consider the network (14) without branch dissipation the average integral quadratic performance criterion (36) for ˜ n R = 0 and without loads (G = 0,I∗ = P ∗ = 0n). In this a random initial condition V0 R with zero mean and ˙ ˜ ˜ T ∈ case, the total energy is preserved (V, f) = 0, and thus E V0V0 = Πn. The performance criterion (36) evaluates to H (V (t), f(t)) = (V0, f0) for all t 0, which says the n Hellipsoidal level setsH of the energy function≥ (38) are invariant. ∞ ˜ T ˜ 1 E V (t) ΠnV (t) dt = = Rtot/n , (37) λ In particular, since the system (14) is linear, these level sets are 0 i=2 i Z X the images of oscillating harmonic trajectories (V (t), f(t)). where Rtot is the total effective resistance (26). Thus, the dynamic behavior of the lossless network (14) and There are several equivalent interpretations of the 2-norm the first term in the general power balance (39) correspond to (36) aside from characterizing the average convergenceH rate lossless energy exchange (i.e., power circulations) between the (37) [161]. For example, an equivalent interpretation is the inductive and capacitive storage elements. For identical time steady-state voltage variance when subjecting each node to constants C = In, the dynamics (14) reduce to the Laplacian noisy current inputs or the transient energy dissipated by the oscillator (15) and the solutions can be further characterized. 14

Proposition 5.12 (Circulations in lossless circuit [49]): inductances and capacitances L and C do not change the sta- Consider the lossless network model (14). The solution is a bility of (which can be seen, e.g., by changing coordinates 1 A 1 superposition of n undamped harmonic signals. Moreover, if to C 2 VL 2 f ). Property 1) guarantees that the dynamics C = In, then the frequencies† of these harmonic signals are (40) are always marginally stable, possibly with sustained 1 √λi, i 1, . . . , n , where λi are the eigenvalues of the L− - oscillations or constant non-decaying modes. Whenever G ∈ { } 1 T weighted Laplacian matrix = BL− B . (respectively, R) is positive definite, then property 4) (respec- LL The power balance equation (39) is a special case of a so- tively, property 5)) guarantees that no sustained oscillations called dissipation (in)equality (more specifically a passivity can occur. However, in this case it is not true that all signals inequality) [147], [131], and the insights gained from it lay the will necessarily settle to zero. For example, assume that that foundations for further analysis of general nonlinear electrical G is positive definite and R = 0; this satisfies the assumptions networks [95], [128] and other interconnected systems [133], of property 4). Then a possible equilibrium for (40) is [135], [104]. For example, another key insight is that, for V ∗ = 0n , f ∗ Ker(B) , positive definite matrices G and R, the right-hand side of (39) ∈ is strictly negative for sufficiently large values of voltages V that is, the equilibrium current flows f ∗ live in the cycle space and currents f. It follows that in this case, the trajectories Ker(B) of the graph; see Section III-9. Hence, any initial of the nonlinear network dynamics (11) are always bounded. current circulation f0 Ker(B) is persistent and does not Notice also that the analysis of linear RC circuits, treated dissipate. Of course, this∈ is ruled out if either B has full rank previously in Subsection V-B, can be equivalently performed (i.e., the graph is acyclic) or R is positive definite so that based on matrix theory or dissipation inequalities such as (34). dissipation forces all circulating flows vanish. Similarly, in We will defer a further nonlinear analysis to Section VI-A the scenario of property 5) with G = 0n n and R positive and focus now on the linear and homogeneous case when definite, we observe that a possible equilibrium× is P = I = 0 to showcase the tools of algebraic graph theory ∗ ∗ n T V ∗ Ker(B ) = span(1 ) , f ∗ = 0 , from Section III. Consider the network dynamics ∈ n m C V˙ G B V which allows for any uniform potential vector V ∗ analogous = − − . (40) L f˙ BT R f to the conservation of charge in (33). If at least one shunt − resistance Gii > 0 is present, then dissipation forces V ∗ = 0n. = A Several of the special cases we have considered thus far The network matrix is related to so-called saddle or KKT allow for a more detailed analysis of the linear network A | {z } matrices [14] in quadratic optimization programs with linear dynamics (40). In particular, recall the Laplacian oscillator equality constraints (where R = 0). We collect some proper- dynamics (15) analyzed in Proposition 5.12, the homogeneous ties in the following proposition that is proved in Appendix B. network dynamics (16), and the coupled `c-tanks (17) in Proposition 5.13 (Spectrum of saddle matrices): Consider Figure 2. These dynamics are all instances of the more general the network matrix in (40), where G and R are positive second-order Laplacian flow semidefinite, and theA graph associated to the incidence matrix V¨ + (kdI + γd )V˙ + (kpI + γp )V = 0 , (41) B is connected. The matrix has the following properties: n L n L n A 1) all eigenvalues are in the closed left half-plane: which can be written in state-space form as spec( ) λ C (λ) 0 . Moreover, all eigenvalues A ⊂{ ∈ | < ≤ } d V 0n n In V on the imaginary axis have equal algebraic and geometric = × , (42) dt V˙ kpIn γp kdIn γd V˙ multiplicities; − − L − − L 2) if G and R are zero matrices, then all eigenvalues = Q of are on the imaginary axis and spec( ) = where n| n is an irreducible{z and symmetric} Laplacian 0, 0A, iλ ,..., iλ , where λ , . . . , λ are theA non- R × 2 n 2 n matrix,L and ∈ k , k , γ , and γ are scalar and nonnegative gains zero{ eigenvalues± ± of the} unweighted{ Laplacian} BBT; d p d p that (if positive) induce a diffusive coupling or a resistive 3) if G and R are positive definite, then is Hurwitz; A dissipation on the voltages V and their drifts V˙ , respectively. 4) if Ker(G) Im(B) = 0 , then has no eigenvalues n The dynamics (42) can be elegantly analyzed via a change on the imaginary∩ axis{ except} for A0. Moreover, if G is √1 1 of coordinates V V , where = [ n n v2 ... vn ] positive definite and B has full rank (i.e., the graph is V V collects the eigenvectors→ of the Laplacian matrix as in acyclic), then is Hurwitz; and L A Proposition 5.1. In these coordinates and after permuting the 5) if Ker(R) Im(BT) = 0 , then has no eigenvalues n entries appropriately, the matrix governing the second-order on the imaginary∩ axis except{ } for 0A. Moreover, if R is Laplacian flow (42) is similar toQ a block-diagonal matrix with positive definite and G > 0 for at least one element ii n blocks, each of the form i 1, . . . , n , then is Hurwitz. ∈ { } A 0 1 The above matrix spectrum results for translate quickly , i 1, . . . , n , A kp γpλi kd γdλi ∈ { } into dynamic stability results for the system (40), since the − − − − where 0 = λ1 < λ2 λn are the eigenvalues of the † Note that the 1st mode λ1 = 0 corresponding to average(V (t)) results Laplacian matrix . We≤· can · · ≤ draw the following conclusions. constant d in (0-frequency) average voltage since dt average(V (t)) = 0. L 15

Proposition 5.14 (Second-order Laplacian flows [27]): Con- (i) nonlinear constitutive relations / circuit elements (e.g., sider the second-order Laplacian flow (42), where voltage-controlled resistances, diodes, transistors) [37]; n n L ∈ R × is an irreducible and symmetric Laplacian matrix and (ii) nonlinear load or source models (e.g., constant power kd, kp, γd, γp 0 are scalar and nonnegative gains. The loads or converter-interfaced sources) [9]; and following statements≥ hold. (iii) power-oriented modeling of AC circuits [83], [84], [85]. 1) given the eigenvalues λ , i 1, . . . , n , of , the 2n We do not examine nonlinearities arising from (i) in this i ∈ { } L eigenvalues ηi, , i 1, . . . , n , of are solutions to article, and refer the interested reader to classic literature ± ∈ { } Q such as [151], [37] for various results. Instead, this section 2 η + (kd + γdλ )η + (kp + γpλ ) = 0, i 1, . . . , n ; i i ∈ { } examines some specific instances of nonlinearity where the methods of algebraic graph theory continue to provide insights 2) the second-order Laplacian flow (42) achieves a consen- into both circuit solutions and dynamic stability. Section VI-A sus on a voltage profile, that is, V V 0 and i j studies the existence of equilibrium points for the nonlinear V˙ V˙ 0 as t for all i, j| −1, . .| . , → n if and | i − j| → → ∞ ∈ { } RLC model (11) with constant power loads. Section VI-B only if the 2(n 1) eigenvalues ηi, , i 2, . . . , n , ± continues this examination for the case of AC circuits, where of the second-order− Laplacian matrix ∈have { strictly} all steady-state signals are required to be sinusoidal, leading negative real part; and Q to a discussion of the AC power flow equations. 3) the average voltage dynamics satisfy d average(V (t)) 0 1 average(V (t)) A. Dynamic resistive-inductive-capacitive (RLC) networks = . dt average(V˙ (t)) kp kd average(V˙ (t)) with constant-power loads − − 1) Equilibrium analysis: Moreover, if the voltages achieve a consensus on a voltage A first challenge when studying profile, then the steady-state profile equals the average initial nonlinear circuits is that even the existence of constant steady- voltage. state operating points is no longer guaranteed. Indeed, the example of Section II-C shows that nonlinear circuits can We can now interpret the previous dynamics as special cases possess multiple steady-state solutions, or possess no solutions of Proposition 5.14. For example, the coupled `c-tanks (17) in at all. We now extend the arguments of Section II-C to the case 2 Figure 2 define a second-order Laplacian flow with kp = ω0, of networks. In particular, we return to the nonlinear dynamic γd = τ 0, and kd = γp = 0. Accordingly, all eigenvalues ηi, , model (11), and seek to determine sufficient conditions for i 2, . . . , n of are in the open left-half plane, and the± ∈ { } Q the existence and uniqueness of an equilibrium point. The asymptotic dynamics are governed by the average voltage: equilibria of (11) are solutions of the nonlinear algebraic d average(V (t)) 0 1 average(V (t)) equations = . dt average(V˙ (t)) ω2 0 average(V˙ (t)) G B V I + P V − 0 0 = + ∗ ∗ . (43) n+m −BT −R f 0 Observe that the asymptotic dynamics equal those of a single − m `c-tank in isolation. The claimed synchronization of the volt- If we assume that R is positive definite, the second equation 1 T ages to the average can be observed in Figure 2. Analogous in (43) may be uniquely solved for f = R− B V . By comments apply to the Laplacian oscillator (15) and the substituting this expression into the first equation, we obtain homogeneous network dynamics (16). the set of n nonlinear equations We make two closing remarks concerning the analyses of 0n = ( R + G)V + I∗ + P ∗ V, this section. First, similar analyses apply in the presence of − L 1 T exogenous constant current inputs I by defining appropriate where R = BR− B . Note that this is exactly the equi- ∗ L error coordinates. Second, in the case that all signals are librium equation for the dynamic RC model (12); when R sinusoidal of constant and identical frequency and the under- is positive definite, the two models therefore share the same lying circuit is asymptotically stable, the above results can be equilibria. If we assume further that (i) at least one element extended to the alternating current (AC) domain by resorting of G is strictly positive, and that (ii) I∗ is nonnegative with to complex-valued variables (so-called phasors), depicting the at least one strictly positive entry (i.e., a current source), then amplitude and phase of all signals [4], [37]. Though generally the equilibria are equivalently determined by more complex dynamic phenomena (e.g., harmonic resonance) 1 1 V = ( R + G)− I∗ +( R + G)− (P ∗ V ) , (44) can occur; see Section VI-B and (54) for a steady-state model. L L VI∗

where V ∗ is the solution of the linear resistive network VI.STRUCTUREAND DYNAMICSOF NONLINEAR I | {z } studied in Section V-A, and has strictly positive components. ELECTRICAL NETWORKS We continue by introducing the change of variables In Section V we restricted our attention to a class of δ = (V V ∗ ) V ∗ , linear circuits, leveraging algebraic graph theory to study static − I I properties of solutions and dynamic stability. While Kirch- which is simply the percentage deviation of V from VI∗ . The hoff’s current and voltage laws are always linear, nonlinearity equilibrium equation (44) can then be further rewritten as arises frequently in circuit analysis for a number of reasons, 1 1 including but not limited to δ = F (δ) = P − (P ∗ (1 + δ)), (45) 4 crit n 16

where we defined the critical power flow matrix V = VI∗ . Many of the insights developed in Section V-A concerning inverse Laplacian matrices and effective resistance 1 n n Pcrit = diag(VI∗ )( R + G) diag(VI∗ ) R × . (46) 4 L ∈ can be leveraged to further characterize the solution. For example, just like the matrix ( + G) studied previously, One may verify easily that P is a Hurwitz Metzler matrix, R crit the matrix P is also a loopyL Laplacian matrix, but for a with the same sparsity pattern− as . The equation δ = F (δ) crit LR graph whose edges have been reweighted using the potentials in (45) is a fixed-point equation for the potential deviation 1 V ∗ . In turn, the inverse matrix P − satisfies a decay property, δ, and can be studied using the contraction mapping principle I crit quantified in terms of the effective resistance in this reweighted [116, Theorem 9.32]. The following result provides an intuitive graph. It follows from (49) then that to first order, these condition under which the nonlinear circuit (11) possesses an effective resistances quantify the sensitivity of the potential equilibrium. Vi at node i with respect to the power injection Pj∗ at node Proposition 6.1 (Existence and uniqueness of an equilibrium j. A complete discussion of many of these conclusions in the for nonlinear RLC network): Consider the equilibrium equa- context of an AC circuit model can be found in [121]. As tion (43) associated with the nonlinear circuit network model we will see shortly, the condition (47) is also instrumental in (11). Assume that R is positive definite, that G has at least one assessing local stability of the associated equilibrium point for strictly positive entry, and that I∗ is element-wise nonnegative the RLC and RC network models. with at least one strictly positive entry. If Finally, we note that convex optimization approaches have 1 recently been devised for assessing the existence of equilib- Pcrit− diag(P ∗) < 1 , (47) k k∞ rium points for static and dynamic nonlinear circuits. An op- 1 T then (43) possesses a solution (V ∗, f ∗) with f ∗ = R− B V ∗. timization formulation exploiting the Metzler structure of the Moreover, the potentials V ∗ = diag(VI∗ )(1n + δ∗) are close graph matrices is presented in [91]. Linear matrix inequality to VI∗ in the sense that (LMI) conditions for equilibrium infeasibility (i.e., necessary

(V ∗ VI∗ ) VI∗ = δ∗ δmax , (48) conditions for equilibrium existence) are presented in [9], [93], k − k∞ k k∞ ≤ and a convex programming approach for certifying feasibility where of AC power flow may be found in [56]. To our knowledge 1 1 however, the presented conditions do not have straightforward δmax = 1 1 Pcrit− diag(P ∗) [0, 1/2) , 2 − − k k∞ ∈ graph-theoretic interpretations. q and (V ∗, f ∗) is the only solution satisfying the inequality (48). 2) Dynamic analysis: Aside from the energy and power The proof of this result is nearly identical to the proof of analysis in Section V-C, a few further fundamental insights [121, Theorem 1]. The intuition behind Proposition 6.1 is that into dynamic stability can be obtained by studying the lin- for the existence of an equilibrium, the maximum size of the earization of the nonlinear circuit model (11) around the solution derived in Proposition 6.1: constant power demands P ∗ must be limited. (V ∗, f ∗) Existence/uniqueness conditions similar to (47) have been C δV˙ G (P ∗ (V ∗ V ∗)) B δV derived by several authors in the context of power distribution = − − − . L δf˙ BT R δf systems [22], [158], [141], [11], [5] and microgrids [120], − [122], [13] with constant power loads; [22] gives an inter- = (V ∗) A (50) pretation of their condition in terms of the maximum path | {z } length in the associated graph. All of these conditions have These linearized dynamics (50) are exactly of the form (40) in common that some measure of the size of the constant studied previously in Section V-C, and results developed for the network matrix in Proposition 5.13 can now be power load should be small compared to a measure of the A applied to the matrix (V ∗). Based on the properties of coupling strength in the network, quantified in terms of system A impedance and nominal voltage level. For the condition (47) the equilibrium in Proposition 6.1, the following parametric 1 stability condition can be established. above, the quantity Pcrit− diag(P ∗) provides a dimension- k k∞ less measure of how large P ∗ is, normalized by the critical Proposition 6.2 (Local stability of dynamic RLC network): power flow matrix Pcrit defined in (46). When the inequality Consider the dynamic RLC network model (11). Assume the (47) holds, we are guaranteed the existence of an equilibrium conditions of Proposition 6.1 hold, and let VI∗ and δmax be which is close to the solution VI∗ studied in Section V-A; in as in Proposition 6.1. Then the equilibrium point (V ∗, f ∗) is general, there can be many equilibria, there is exactly one equi- locally exponentially stable if n librium in the box V R (V VI∗ ) VI∗ δmax . ∞ P Another useful perspective{ ∈ | k on− the condition k (47)≤ comes} i∗ g˜i := gi + 2 2 0 , i 1, . . . , n , (V ∗ ) (1 δ ) ≥ ∈ { } from studying the linearization of the right-hand side of I i − max equilibrium equation (44) around V = VI∗ and P ∗ = 0n. with strict inequality for at least one value of i. Performing this computation, one finds that The proof of this result may be found in Appendix C. The 1 1 quantity g˜i can be interpreted as an effective shunt conductance V VI∗ 1n + Pcrit− P ∗ . (49) ≈ 4 at node i 1, . . . , n , in which the constant power load Pi∗ The condition (47) can therefore be interpreted as restricting has been∈ converted { into} a shunt conductance. The stability the first-order term of the linearized solution at the point result may then be seen as a case of item 5) in Proposition 17

5.13. The effective shunt conductances g˜i depend on δmax, Here Ii∗ , Vi , fi 0 are the constant steady-state root- 1 | | | | | | ≥ which in turn depends on the key quantity Pcrit− diag(P ∗) mean-square amplitudes of the waveforms, and φi, θi, ϕi are from Proposition 6.1. Stability therefore dependsk directlyk on∞ the respective phase shifts. The RLC network (11) exhibits the spectral properties of the graph matrix Pcrit. such steady-state solutions (V (t), f(t)) whenever the dynam- As noted earlier, (V ∗, f ∗) is an equilibrium point of the ics are internally stable and the injections I∗ are as in (51a). RLC model (11) (with R positive definite) if and only if Unlike DC power, AC power is typically transmitted in a V ∗ is an equilibrium point of the RC model (12). We can three-phase configuration in which three wires (plus a neutral therefore also assess the local stability of the equilibrium point wire) are used. Therefore, for each node (resp. edge) there in Proposition 6.1 for the RC model. are three potential and current injection (resp. voltage drop Proposition 6.3 (Local stability of dynamic RC network): and current flow) waveforms to consider. When such a three- Consider the dynamic RC network model (12). Assume the phase circuit is balanced, the harmonic waveforms on the conditions of Proposition 6.1 hold, and let be the specified three wires are separated by 120◦, and a standard equivalent V ∗ ± equilibrium point. Then V ∗ is locally exponentially stable. circuit technique allows the three-phase circuit to be studied in terms of a single-phase equivalent circuit [73, Chapter 1]. We The proof of this result may be found in Appendix D. therefore proceed with a single-phase analysis in this section, To conclude this section, we now examine some large-signal with the understanding that the results can quickly be applied stability properties for the dynamic RC network model (12). to balanced three-phase AC circuits. We perform a nonlinear analysis in the spirit of Brayton-Moser 1) Phasor analysis of AC circuits: We may equivalently [25], [26], [124], [83], [85] and adapt the energy function (38) express the harmonic signals in (51) as the real parts of towards a potential function‡ accounting for resistive power complex-valued signals losses and power dissipation by the (constant impedance, jφi jωt jθi jωt constant current, and constant power) loads as I∗(t) = R √2 I∗ e e ,V (t) = R √2 V e e , i | i | i | i| 1 ∗ T T T i C i C (V ) = V ( R + G)V ln(V ) P ∗ V I∗ , I ∈ V ∈ W 2 L − − jϕi jωt | {zfi(t)} = R √2 fi e e | . {z } T | | where lnV = (ln V1,..., ln Vn) . A straightforward calcula- i C tion then shows that the RC network model (12) reads as F ∈ (52) The complex quantities in brackets| {z are referred} to as a phasors, CV˙ = (V ) , −∇W with i∗, i and i being the complex “amplitudes” of the meaning that the potentials V evolve according to the gradi- phasors.I TheV classicF approach to study AC electrical networks ent of the power-like function (V ). Critical points of the is to represent all potentials and current flows using phasors, W function (V ) are therefore equilibrium points of (12), and and subsequently derive algebraic equations relating the vec- W vice-versa. It follows by standard results for gradient systems tors of complex amplitudes , , and ∗. With this phasor that all bounded trajectories of (12) converge to equilibrium substitution, the RLC circuitV equationsF (11)I reduce to the set points [146, Chapter 15]. By requiring the Hessian of (V ) of complex-valued algebraic equations§ to be positive definite, one can show strict convexityW of jωC = G B + ∗, (53a) (V ) and that the equilibrium specified by Proposition 6.1 V − V − F I Wis locally asymptotically stable, which recovers the result of jωL = BT R , (53b) F V − F Proposition 6.3. An estimate of the region of attraction of V ∗ where , , are the vectors of phasor amplitudes for can be obtained by finding a compact sublevel set of (V ) ∗ potentials,V F edgeI currents, and constant-current injections, re- containing V . We refer to [112], [32], [42] for related stabilityW ∗ spectively. Solving (53b) for 1 T and analyses of DC networks, to [35], [105] for a comprehensive = (R + jωL)− B eliminating from (53a), the electricalF network is compactlyV survey concerning energy functions in AC power systems, and described byF the algebraic equation to [84], [104], [135] for further reading on power and energy- based approaches to nonlinear networks. 1 T ∗ = Y = B(R + jωL)− B + (G + jωC) . (54) I V V B. AC circuits and power networks complex-weighted network complex shunts n n This section examines the important case where, in steady- The so-called admittance| {z matrix Y} | C{z× is} a sum of ∈ state, all current sources and all internal voltages and currents two terms, the first being a Laplacian matrix = LR+jωL in the RLC network (11) are harmonic with synchronous B(R + jωL) 1BT with complex weights 1 termed − rij +jωìj alternating current (AC) angular frequency ω: admittances. The second term in (54) is a complex diagonal matrix modeling the shunt admittance elements connected I∗(t) = √2 I∗ cos(ωt + φ ) , (51a) i | i | i to ground at each node. The linear equation (54) is exactly Vi(t) = √2 Vi cos(ωt + θi) , (51b) analogous to the linear resistive circuit equation (13) studied | | in Section V-A, with analogous solution 1 , and the fi(t) = √2 fi cos(ωt + ϕi) . (51c) = Y − ∗ | | corresponding real harmonic signals areV recoveredI via (52). ‡We remark that the considered RC circuit (12) is a simple yet illustrative subcase within the general Brayton-Moser modeling and analysis framework §Constant power loads are omitted, and will be treated independently in that can also account for inductive dynamics and more general nonlinearities. our subsequent discussion of AC power flow. 18

While the static resistive circuit solution of Section V-A where we have inserted (54) to eliminate ∗. Expanding the generalizes to the present AC case (54), many questions con- right-hand side and separating real and imaginaryI parts, we cerning effective resistance and Kron reduction for complex- arrive at the AC power flow equations [90] weighted graphs remain unresolved; see [52]. n 2) Instantaneous power, complex power, and power flow P = V V (R(Y ) cos(θ θ ) + I(Y ) sin(θ θ )) , i | i|| j| ij i − j ij i − j equations: Often in AC circuit analysis — and in particular, j=1 X in the study of power systems — one is interested in specifying n the sources and loads of a circuit in terms of power and Qi = Vi Vj (R(Yij) sin(θi θj) I(Yij) cos(θi θj)) , | || | − − − j=1 voltage, rather than current and voltage. For example, the X energy production of a synchronous generator is scheduled in for i 1, . . . , n . These nonlinear equations relate the active terms of power and not current, since power is more directly ∈ { } and reactive power injections Pi and Qi at each node to the related to engineering specifications and monetary cost. To jθi complex voltage variables Vi e at the neighboring nodes. move towards such a power-focused formulation of the circuit The power flow equations| are| highly nonlinear; in Section equations, define the instantaneous power pi(t) injected into VI-B3 we examine this nonlinearity in some detail. Quick node i 1, . . . , n as p (t) = V (t)I∗(t). Using the harmonic ∈ { } i i i insight however can be gained by studying their linearization representations (51), some simple trigonometry shows that around θ = 0 and V = 1 , which is given by [20] n | | n pi(t) = Vi Ii∗ cos(θi φi) 1 + cos(2(ωt + θi)) P I(Y ) R(Y ) θ | || | − − . (55) + V I∗ sin(θ φ ) sin(2(ωt + θ )). Q ≈ R(Y ) I(Y ) V | i|| i | i − i i − − | | The instantaneous power consists of two oscillatory terms, Each subblock of the block matrix in (55) is a Laplacian (or only the first of which has non-zero mean. The average (also negative Laplacian) matrix, possibly with additional diagonal called active or real) power injected into node i over one cycle elements in each subblock. Note that R(Y ) arises from the is defined as this mean component of the first term: resistive component of the interconnection between nodes, while I(Y ) arises from inductances. In other words, when 1 T P = p (t) dt = V I∗ cos(θ φ ) , written in this form, a multigraph structure naturally appears. i T i | i|| i | i − i Z0 To our knowledge, the algebraic and spectral properties of the where T = 2π/ω. Note that the active power Pi is zero if matrix in (55) are not well understood, and an analysis of (55) potential and current waveforms are 90◦ out of phase: θi in the spirit of that from Section V-A remains open. φ = π/2. The second oscillatory term in p (t) has zero mean,| − i| i 3) AC power flow problems: The AC power flow equations but its amplitude are the heart of almost all power system analysis, operations, optimization, and control. The analytic study of these equa- Qi = Vi I∗ sin(θi φi) | || i | − tions dates back at least fifty years; we do not attempt to is commonly referred to as reactive (or imaginary) power present a comprehensive overview of the area, but will present [1]. Reactive power reflects the zero-average energy exchange a selection of results based on the authors’ experiences. We between inductive or capacitive elements, with power flowing refer the reader to [118], [119] for a recent literature review. into the element for half an AC cycle and out of the element In an AC power flow problem, a subset of the variables during the remaining half (cf., the circulating power in (39)). ¶ Vi , θi,Pi,Qi are specified at a subset of nodes, and the Again, it is most convenient to study power in AC circuits problem{| | is to solve} the nonlinear AC power flow equations to using phasors. To develop a phasor representation, define the determine the remaining unspecified variables. This is directly complex power Si = Pi + jQi C injected at node i ∈ ∈ analogous to the problem of solving a static resistive circuit as 1, . . . , n using the voltage and current phasors and ∗ as { } Vi Ii in Section V-A, or the problem of determining an equilibrium 1 point for a nonlinear circuit as in Section VI-A. As a specific S = ∗ . i 2ViIi instance of an AC power flow problem, we examine the case of a purely inductive interconnection with L positive definite where ∗ is the complex conjugate of . We have through i i∗ and R = 0, with no shunt conductances G = 0 and no simpleI calculations then that I shunt capacitances C = 0. The important feature here is the

Si = Vi Ii∗ cos(θi φi) +j Vi Ii∗ sin(θi φi) . lack of resistive elements; the network transfers power without | || | − | || | − resistive power losses. This instance is complex enough to =P =Q i i highlight the important role of algebraic graph theory concepts, The real part| of Si{zis exactly} the| real power{z Pi, while} the but simple enough to actually permit some insightful analysis. imaginary part Qi is the reactive power. In vector notation, The formal modeling for a power flow problem proceeds as the complex power is given by follows. We begin with an undirected, connected and weighted 1 1 graph . The nodes 1, . . . , n of the graph are partitioned S = P + jQ = ∗ = (Y ), G { } 2V I 2V V into two subsets: a non-empty subset PV 1, . . . , n and a disjoint strict subset PQ 1, . . . , n such⊆ that { 1, . .} . , n = ¶We remark that when a circuit does not a reach a sinusoidal steady state, PQ PV. At node i ⊂PV {, the average} power{ injection}P e.g., in presence of nonlinear elements, then the definition and interpretation ∪ ∈ i of reactive power is still a controversial and contested subject [82]. and the RMS potential V are specified; in power systems, | i| 19 such a node corresponds to a generator with a scheduled which is a linear Laplacian equation analogous to (13). To power production Pi and a voltage level Vi regulated by further emphasize the role of the topology, we can rewrite a local controller. At node i PQ, the real| | power P and (58) as the two coupled equations ∈ i reactive power Qi are specified; this corresponds to a con- P = B ψ , (60a) stant power injection/demand, modeling a load or converter- K interfaced renewable energy source. In networks with non- ψ = sin(BTθ) , (60b) zero edge resistances or non-zero shunt conductances, one where ψ m is an auxiliary vector of variables. Some generally requires the presence of a third node type called R immediate∈ insights can be gained by comparing (60) to the a slack node, where θ and V are specified with P and i i i KCL and KVL equations (7) and (9) from Section IV-A. The Q undetermined. The primary| | purpose of this node is to i first equation (60a) corresponds to the KCL equation (7), with compensate for the resistive power dissipation in the network current injections I = P and current flows f = ψ.A by supplying additional power. We will restrict our attention comparison of (60b) and (9) shows that (60b) is a typeK of to lossless systems and will therefore omit the slack bus. nonlinear KVL equation with potentials θ and voltage drops Given the above specified quantities, the AC power flow ψ. The combined equation (60) is directly analogous to the problem is to determine the phase angles θ for i PQ PV i resistive network nodal current balance equation (18). and the RMS potentials V for i PQ. ∈ ∪ i In practice we are interested in solutions of (60) for which Under these modelling| assumptions,| ∈ the admittance ma- the differences between phase angles of neighboring buses are trix becomes purely imaginary and reduces to Y = π 1 T relatively small. To quantify this, for γ [0, 2 ), define jB(ωL)− B = j ωL. The power flow equations simplify ∈ − − L n to ∆(γ) = θ T θi θj γ for i, j n { ∈ | | − | ≤ { } ∈ E} n P = V V I(Y ) sin(θ θ ) , i PQ PV, (56a) as the subset of T (the n-torus) where phase angle differences i | i|| j| ij i − j ∈ ∪ j=1 along edges i, j are less than γ. X { } ∈ E n Proposition 6.4 (PV node power flow problem [54], [80]): Qi = Vi Vj I(Yij) cos(θi θj) , i PQ . (56b) Consider the PV node power flow problem (57) with 1 , − | || | − ∈ P n⊥ j=1 and let γ [0, π/2). The following statements hold: ∈ X ∈ If the equations (56) can be solved for the PV/PQ node phase (i) Every solution of (60a) is of the form angles θi and the PQ bus potentials Vi , the remaining un- T | | ψgeneral = B † P + ψhom , (61) specified reactive power injections Qi for i PV are uniquely LK ∈ T determined by back substitution. An immediate insight is where = B B and ψhom Ker(B); LK K K ∈ obtained from (56a) by summing over all i PQ PV and (ii) there exists a solution θ∗ ∆(γ) if and only if ∈ ∪ ∈ using the fact that sin( ) is odd, from which we find that there exists ψhom such that ψgeneral sin(γ) and · T k k∞ ≤ n arcsin(ψgeneral) Im(B ) (modulo 2π); Pi = 0 P 1n⊥ . ∈ i=1 ⇐⇒ ∈ (iii) for graphs containing no cycles, there exists a unique G X solution θ∗ ∆(γ) if and only if This simply says that in such a lossless network, the real power ∈ generation must always be balanced by the real power demand. T B † P sin(γ); (62) An analysis of the lossless power flow problem (56) is k LK k∞ ≤ (iv) there exists a unique solution θ∗ ∆(γ) if possible, but becomes notationally quite involved and uses ∈ some additional graph-theoretic constructions that go beyond T B † P 2 sin(γ) . (63) the scope of this survey; we refer the reader to [118], [119] k LK k ≤ for a detailed analysis. To study a comparatively tractable Equation (61) shows that every solution of the KCL-like scenario, consider the case where PQ = ; this specific equation (60a) can be decomposed into two terms: a par- model arises in the study of dynamic stability∅ in a network ticular solution belonging to the cutset space Im(BT), and of synchronous generators [51]. In this case, the products a homogeneous component belonging to the weighted cycle 1 T kij = Vi Vj I(Yij) are constants, and (56) simplifies to space − Ker(B). In fact, the particular solution B † P is | || | K LK n exactly the edge-wise phase differences one would calculate Pi = kij sin(θi θj) , i 1, . . . , n , (57) by solving the linearized equation (59). Item (ii) of Proposition j=1 − ∈ { } 6.4 connects this linear solution to the full nonlinear equation or in vectorizedX notation (58) by requiring that ψgeneral satisfies a boundedness condi- P = B sin(BTθ) , (58) tion ψgeneral sin(γ) along with the cycle constraints K arcsink (ψ k∞) ≤ Im(BT) (modulo 2π); the latter ensures general ∈ where = diag( ke e 1,...,m ) and sin(x) = that phase angle differences add up to a multiple of 2π K T { } ∈{ } (sin(x1),..., sin(xn)) . The network topology clearly around any undirected cycle of the graph (cf. KVL (5)). In affects the solvability of (58) through the incidence matrix graphs without cycles (item (iii)) the cycle constraints can B. In fact, linearizing the equation (58) around θ = 0n, we be discarded, and the existence of a solution is equivalent find that to satisfaction of the boundedness condition. The analysis of P B BTθ = θ, (59) (60) becomes considerably more challenging in networks with ≈ K LK 20 cycles, and only conservative sufficient conditions ensuring lossy setting, the basic existence and uniqueness questions are existence are known; we refer to [54], [53], [?], [?] for details still unresolved, and the study of dynamics in such networks and recent results. Specifically, the sufficient condition (iv) and is a wide open field that is currently seeing much activity. a generalized version of fact (iii) are proved in [80]. With regards to our prototypical model (11), we performed At this point in our discussion of AC power flow we a thorough linear analysis of the dynamics, though a full have reached the research frontier concerning graph-theoretic nonlinear large-signal stability analysis is still open. insights and analytic approaches to the solution space of the Finally, a topic of historic interest was to reverse the AC power flow equations. The development of graph-theoretic reduction of electrical networks [12], [68], [137], [8], e.g., conditions for the existence and uniqueness of AC power flow by embedding a cyclic network into a higher-dimensional solutions (and stability conditions for associated power system equivalent acyclic network as in Figure 6. Since many circuit dynamics, which we have not discussed here) remains an area and power flow problems are analytically and computationally of active research. Among many recent works, we refer to tractable only in acyclic networks, it would be of interest to [121], [58], [57], [22], [118], [119], [141], [59]. find more general high-dimensional network embeddings and equivalence transformations. Likewise, the passive network synthesis problem [24] has recently received a revived interest VII.CONCLUSIONSANDAVENUESFORFUTURE and triggered many open questions [78]. RESEARCH

The field of algebraic graph theory has been initially devel- APPENDIX oped, amongst others, by electrical engineers to abstract and facilitate the study of electrical networks. Conversely, several A. Compartmental systems fundamental concepts in algebraic graph theory were born out In this appendix we present a self-contained concise treat- of concrete circuit problems. In this article, we highlighted the ment of compartmental systems. For more complete treatments rich interplay between these two disciplines in the static and and proofs of the following statements we refer to [140], [79] dynamic, linear and nonlinear, and real-valued and complex- and [27, Chapter 9]. valued cases. We reviewed classical results from the early days A compartmental system is a dynamical system in which of network analysis as well as recent results. Our developments material is stored at storage nodes, called compartments, and were centered around a single yet rich prototypical model of is transferred along the edges of directed graph, called the an electrical network. compartmental digraph; see Figure 18(b). Each compartment The literature on the topic of this article is vast and mature. We hope to have delivered a survey and a tutorial exposition, u3 q3 F4 3 q4 F4 0 ! ! as seen from the perspective of algebraic graph theory, that ui qi Fi 0 ! brought the reader from the basics all the way to the research F3 2 ! F frontier. We want to conclude by listing a few fundamental Fi j 2 4 ! Fj i ! ! F2 3 open problems at the intersection of electrical networks and ! u1 q1 F1 2 q2 F2 0 algebraic graph theory. The list is by no means complete and ! ! is colored by our own interests and experiences. (a) (b) In this paper we have studied networks in which each constitutive element is a one-port, described by a single voltage- Fig. 18. (a) A compartment with inflow ui, outflow Fi→0, and inter- current relationship between its terminals. More generally, compartmental flows Fi→j . (b) A compartmental system with two inflows and two outflows; images courtesy of [27]. electric elements such as transformers and gyrators require the adoption of 2-port representations. A future research direction contains a time-varying quantity qi(t) and each directed arc is to adopt algebraic graph theory to model and study two-port (i, j) represents a mass flow, denoted Fi j, from compart- networks. Along the same lines, much of the presented theory ment i to compartment j. The inflow from→ the environment still has to be extended to circuit elements with nonlinear into compartment i is denoted by ui and the outflow from constitutive relations such as diodes and transistors. compartment i into the environment is denoted by Fi 0. We have shown that AC networks give rise to complex- In a linear compartmental system, we assume → valued graph matrices. In general, many of the results we presented for real-valued Laplacian (and more general Met- Fi j(q, t) = fijqi, for j 1, . . . , n , → ∈ { } zler) matrices have few (or no) complex-valued counterparts. Fi 0(q, t) = f0iqi, and ui(q, t) = ui. Yet the study of complex-valued matrices and their associated → graphs is of the utmost importance for large-scale AC power for appropriate flow rate coefficients. More precisely, a linear system applications. Another frontier in this regard are hybrid compartmental system consists of DC/AC networks and unbalanced multi-phase networks. (i) a nonnegative n n matrix F = (fij)i,j 1,...,n with × ∈{ } At the intersection of nonlinear and AC networks lie the zero diagonal, called the flow rate matrix, celebrated power flow equations. The characterization of so- (ii) a vector f0 0n, called the outflow rates vector, and ≥ lutions to these equations, as a function of the network param- (iii) a vector u 0n, called the inflow vector. ≥ eters and topology, has a long history and has witnessed some The flow rate matrix F is the adjacency matrix of the compart- exciting recent developments. Yet in the full nonlinear and mental digraph (a weighted digraph without self-loops). GF 21

The instantaneous flow balance provides the affine dynamics Regarding the fourth statement 4): To prove the first part, of the system: we follow the proof method of [34, Lemma 5.3]. Recall n n from statement 1) that the spectrum of is restricted to the A q˙ (t) = f + f q (t) + f q (t) + u . closed left half-plane. We aim to prove that, under the stated i − 0i ij i ji j i j=1,j=i j=1,j=i assumptions, no imaginary eigenvalues other than zero can X6 X6 occur. We reason by contradiction. Let jσ, with σ = 0, be an The compartmental matrix C = (cij)i,j 1,...,n of a linear 6 ∈{ } imaginary eigenvalue of with corresponding non-zero eigen- compartmental system is a Metzler matrix defined by A n+m vector x + jy, where x = [x1 x2]>, y = [y1 y2]> R . ∈ fji, if i = j, Then the real and imaginary parts of the eigenvalue equation cij = n 6 ( f0i h=1,h=i fih, if i = j. G B x1 + jy1 − − 6 jσ(x + jy) = (x + jy) = − − A B> R x2 + jy2 Equivalently, C = F > Pdiag(F 1n+f0) and q˙(t) = Cq(t)+u. − In the compartmental− digraph, a set of compartments S is yield the set of equations outflow-connected if there exists a directed path from every Gx1 Bx2 = σy1, (65a) compartment in S to the environment, that is, to a compart- − − − Gy1 By2 = σx1, (65b) ment j with a positive flow rate constant f0j > 0. Moreover − − inflow-connected B>x1 Rx2 = σy2, (65c) a set of compartments S is if there exists a − − directed path from the environment to every compartment in B>y1 Ry2 = σx2. (65d) S, that is, from a compartment i with a positive inflow u > 0. − i We pre-multiply equation (65a), respectively (65c), by , re- Recall that, given a digraph , its condensation is a digraph x1> spectively by , and arrive at , whose nodes are the strongly connectedG components of and x2> x1>Gx1 x1>Bx2 = σx1>y1 respectively − − . By adding− these whose edges are defined by corresponding edges in . G x2>B>x1 x2>Rx2 = σx2>y2 equations we get − − . With these definitions, the following graph-theoreticalG and x1>Gx1 x2>Rx2 = σ(x1>y1 + x2>y2) Via analogous manipulations− − of equations− (65b) and (65d), algebraic statements are known to be equivalent: we obtain y1>Gy1 y2>Ry2 = σ(x1>y1 + x2>y2). These two (i) the system is outflow-connected, conditions− imply that− (ii) each sink (node without outgoing edges) of the condensation of is outflow-connected, and x1 > G x1 y1 > G y1 GF = . (iii) the compartmental matrix C is Hurwitz. x2 R x2 − y2 R y2 Theorem A.1 (Asymptotic behavior of compartmental sys- Since G and R are positive semi-definite (possibly zero tems): Consider a linear compartmental system (F, f0, u) with matrices) by assumption, we obtain x ker(G), y ker(G) 1 ∈ 1 ∈ compartmental matrix C and compartmental digraph F . If the and x ker(R), y ker(R). By further using x G 2 ∈ 2 ∈ 1 ∈ system is outflow-connected, then ker(G), y1 ker(G) in equations (65a) and (65b) we obtain (i) the compartmental matrix C is invertible, Bx = ∈σy and By = σx , that is, x Im(B), − 2 − 1 − 2 1 1 ∈ (ii) every solution tends exponentially to the unique equilib- y1 Im(B). We have now established that both x1 and y1 1 ∈ rium q∗ = C− u 0 , and belong to ker(G) Im(B) and therefore x = y = 0 by − ≥ n ∩ 1 1 n (iii) in the ith compartment qi∗ > 0 if and only if the ith assumption. Similarly, we obtain x2 = y2 = 0m. But it is compartment is inflow-connected to a positive inflow. a contraddiction to have x + jy equal to serve since it is an eigenvector. B. Proof of Lemma 5.13: Spectrum of the saddle matrix Now we prove the second part of statement 4). If G is positive definite, the Schur determinant formula [160] yields The following proof adopts elements from [34], [50], [49]. Regarding the first statement 1), we resort to a Lyapunov- G B 1 det − − = det(G) det(R + B>G− B) . based proof. Consider the auxiliary linear dynamical system B> R · − x˙ 1 G B x1 x1 Note that det(G) = 0. If B has full rank m, i.e., the graph is = − − = , (64) acyclic (see statement6 (S1) in Section III), then 1 has x˙ 2 B> R x2 A x2 B>G− B 1 − full rank, and det(R + B>G− B) = 0. Due to these facts, 1 2 1 2 6 and the Lyapunov function (x1, x2) = 2 x1 + 2 x2 . the matrix has no eigenvalue at zero and is thus Hurwitz. The derivative of (x, y) is givenV by k k k k A V The first part of the proof of statement 5) is analogous to ˙ that of statement 4). To prove the second part, we assume that (x1, x2) = x1>Gx1 x2>Rx2 0 . V − − ≤ R is positive definite and apply the Schur determinant formula: It follows that the state (x1, x2) is bounded. Thus, the matrix G B 1 admits only eigenvalues with strictly negative real part or det − − = det(R) det(G + BR− B>) . A B> R · eigenvalues on the imaginary axis with equal algebraic and − 1 geometric multiplicity. Note that det(R) = 0 and BR− B> = R is a Laplacian Statement 2) is a corollary of Proposition 5.12. matrix associated to6 a connected graph. If LG has at least one If G and R are positive definite, then ˙ (x1, x2) is negative diagonal element, then R + G is a nonsingular matrix, as definite. It follows that the dynamics (64)V are asymptotically discussed in Section III-7.L Due to these facts, the matrix stable and is Hurwitz. This proves the third statement 3). has no eigenvalue at zero and is thus Hurwitz. A A 22

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Florian Dorﬂer¨ (S’09–M’13) is an Assistant Pro- fessor at the Automatic Control Laboratory at ETH Zurich.¨ He received his Ph.D. degree in Mechan- ical Engineering from the University of California at Santa Barbara in 2013, and a Diplom degree in Engineering Cybernetics from the University of Stuttgart in 2008. From 2013 to 2014 he was an As- sistant Professor at the University of California Los Angeles. His primary research interests are centered around distributed control, complex networks, and cyber-physical systems currently with applications in energy systems and smart grids. His students were winners or ﬁnalists for Best Student Paper awards at the European Control Conference (2013), and the American Control Conference (2016), and the PES PowerTech Conference 2017. His articles received the 2010 ACC Student Best Paper Award, the 2011 O. Hugo Schuck Best Paper Award, and the 2012-2014 Automatica Best Paper Award, and the 2016 IEEE Circuits and Systems Guillemin-Cauer Best Paper Award. He is a recipient of the 2009 Regents Special International Fellowship, the 2011 Peter J. Frenkel Foundation Fellowship, and the 2015 UCSB ME Best PhD award.

John W. Simpson-Porco (S’11–M’16) received the B.Sc. degree in Engineering Physics from Queen’s University, Kingston, ON, Canada in 2010, and the Ph.D. degree in Mechanical Engineering from the University of California at Santa Barbara, Santa Bar- bara, CA, USA in 2015. He is currently an Assistant Professor of Electrical and Computer Engineering at the University of Waterloo, Waterloo, ON, Canada. He was previously a visiting scientist with the Au- tomatic Control Laboratory at ETH Zurich,¨ Zurich,¨ Switzerland. His research focuses on the control and optimization of multi-agent systems and networks, with applications in modernized power grids. Prof. Simpson-Porco is a recipient of the 2012–2014 IFAC Automatica Prize and the Center for Control, Dynamical Systems and Computation Best Thesis Award and Outstanding Scholar Fellowship.

Francesco Bullo (S’95-M’99-SM’03-F’10) is a Pro- fessor with the Mechanical Engineering Department and the Center for Control, Dynamical Systems and Computation at the University of California, Santa Barbara. He was previously associated with the University of Padova, the California Institute of Technology, and the University of Illinois. His research interests focus on network systems and distributed control with application to robotic co- ordination, power grids and social networks. He is the coauthor of Geometric Control of Mechanical Systems (Springer, 2004) and Distributed Control of Robotic Networks (Princeton, 2009); his forthcoming ”Lectures on Network Systems” is available on his website. He received best paper awards for his work in IEEE Control Systems, Automatica, SIAM Journal on Control and Optimization, IEEE Transactions on Circuits and Systems, and IEEE Transactions on Control of Network Systems. He is a Fellow of IEEE and IFAC. He has served on the editorial boards of IEEE, SIAM, and ESAIM journals, and will serve as IEEE CSS President in 2018.