Steady state characterization and frequency synchronization of a multi-converter power system on high-order manifold
Taouba Jouini1, Zhiyong Sun2
Abstract—We investigate the stability properties of a multi- AC-sides and the intermediate switching block, which can converter power system model, defined on a high-order manifold. structurally match that of synchronous machines [4]. Recently, For this, we identify its symmetry (i.e., rotational invariance) the matching control has been proposed in [4] as a promising generated by a static angle shift and rotation of AC signals. We characterize the steady state set, primarily determined by the control strategy, which achievesa structural equivalence of steady state angles and DC power input. Based on eigenvalue the two models, and endows the closed-loop system with conditions of its Jacobian matrix, we show asymptotic stability of advantageous features (droop properties, power sharing, etc.). the multi-converter system in a neighborhood of the synchronous By augmenting the system dynamics with a virtual angle, steady state set by applying the center manifold theory. We the frequency is set to be proportional to DC-side voltage guarantee the eigenvalue conditions via an explicit approach. Finally, we demonstrate our results based on a numerical example deviations, constituting a measure of power imbalance in the involving a network of identical DC/AC converter systems. grid. This leads to the derivation of higher-order models that describe a network of coupled DC/AC converters on nonlinear manifolds with higher order than the circle. I.INTRODUCTION Similar to the physical world, where the laws governing in- Electricity production is one of the largest sources of teractions in a set of particles are invariant with respect to static greenhouse gas emissions in the world. Carbon-free electricity translations and rotations of the whole rigid body [5], power will be critical for keeping the average global temperature system trajectories are invariant under a static shift in their within the United Nation’s target and avoiding the worst angles, or said to possess a rotational invariance. The symme- effects of climate change [1]. Prompted by these environmental try of the vector field describing the power system dynamics, concerns, the electrical grid has witnessed a major shift in indicates the existence of a continuum of steady states for the power generation from conventional (coal, oil) into renewable multi-converter (with suitable control that induces/preserves (wind, solar) sources. The massive deployment of distributed, angle symmetry) or multi-machine dynamics. In particular, renewable generation had an elementary effect on its operation the rotational invariance is the topological consequence of via power electronics converters interfacing the grid, deemed the absence of a reference frame or absolute angle in power as game changers of the conventional analysis methods of systems and regarded thus far as a fundamental obstacle for power system stability and control. defining suitable error coordinates. To alleviate this, a common Literature review: Modeling and stability analysis in power approach in the literature is to perform transformations either system networks is conducted as a matter of perspective from resulting from projecting into the orthogonal complement, if two different angles. First, network perspective suggests an the steady state set is a linear subspace [6], or grounding up to bottom approach, where DC/AC converter dynamics are a node [7], where classical stability tools such as Lyapunov regarded as controllable voltage sources and voltage control is direct method can be deployed. directly accessible. The most prominent example is droop con- To analyze power system stability, different conditions have trol that leads to the study of second-order pendulum dynam- been proposed. In [4] and [8], sufficient stability conditions are ics, emulating the Swing equation of synchronous machines obtained for a single-machine/converter connected to a load. [2], which resembles the celebrated Kuramoto-oscillator [3]. In [3], a sufficient algebraic stability condition connects the arXiv:2007.14064v3 [math.OC] 5 Aug 2021 The analogy drawn between the two models has motivated synchronization of power systems with network connectivity a vast body of literature that harness the results available and power system parameters. Although these conditions give for synchronization on the circle via Kuramoto oscillators qualitative insights into the sensitivities influencing stability, to analyze the synchronization in power systems. Second, they usually require strong and often unrealistic assumptions. a bottom to up approach derives DC/AC converter models For example, the underlying models are of reduced order from first-order principles, where the dynamics governing (mostly first or second order) [3], [9]. Reduced-order systems, DC/AC converters are derived from the circuitry of DC-and where one infers stability of the whole system from looking at only a subset of variables, are not a truthful representation *This work has received funding from the European Research Council of the full-order system if important assumptions are not (ERC) under the European Union’s Horizon 2020 research and innovation program (grant agreement No: 834142) and from the European Union’s met [10]. Some stability conditions are valid only in radial Horizon 2020 research and innovation program under grant agreement No: networks [6]. Moreover, explicit stability conditions require 691800 and ETH Zurich¨ funds. assumptions like strong mechanical or DC-side damping [3], 1 Taouba Jouini is with the Department of Automatic Control, LTH, Lund whereas implicit conditions are based on semi-definite pro- University, Ole Romers¨ vag¨ 1, 22363 Lund, Sweden. 2Zhiyong Sun is with Department of Electrical Engineering, Eindhoven University of Technology, gramming and thus not very insightful [11]. the Netherlands. E-mails: [email protected], Contributions: In this work, we ask in essence two funda- [email protected]. mental questions: i) Under mild assumptions on input feasi- bility, how can we describe the behavior of the steady state presents the model setup based on a high-fidelity nonlinear trajectories of the nonlinear power system, in closed-loop with power system model. Section III studies the symmetry of a suitable control, that induces/preserves the symmetry, e.g. its vector field, characterizes the steady state set of interest. the matching control [4], [12]? ii) Based on the properties of Section IV studies local asymptotic stability of the nonlinear the steady state manifold, can we ensure local stability, i.e., power system model. Section V shows asymptotic stability of synchronization? the linearized power system dynamics and provides interpreta- To answer the first question, we study the behavior of the tions of our results. Finally, Section VI exemplifies our theory steady state manifold. For this, we derive a steady state map, via simulations in two test cases. which embeds known steady state angles into the DC power Notation: Define an undirected graph G = ( , ), where input as a function of the network topology and converter is the set of nodes with = n and V E is the parameters. We show that the steady state angles fully describe setV of interconnected edges with|V| = m.E We ⊆ V assume × V that the steady state behavior and determine all the other states. the topology specified by is arbitrary|E| and define the map E The steady state map depends on network topology, which is , which associates each oriented edge ei j = (i, j) to E → V V ∈ E known to play a crucial role in the synchronization of power an element from the subset = 1,0,1 | |, resulting in the n mI {− } systems [5], [6]. Since the vector field exhibits symmetry with incidence matrix R × . We denote the identity matrix respect to translation and rotation actions, i.e., under a shift 1 0 B ∈ I = 0 1 , and I the identity matrix of suitable dimension in all angles and a rotation in all AC signals, the steady state 0 1 p N, and J = I J2 with J2 = 1− 0 . We define the rotation manifold inherits the same property and every steady state ∈ h cos⊗(γ) sin(γ) i matrix R(γ) = − and R(γ) = I R(γ). Let diag(v) trajectory is invariant under the same actions. This allows us to sin(γ) cos(γ) ⊗ denote a diagonal matrix, whose diagonals are elements of the define set of equilibria that are generated under these actions. vector v and Rot(γ) = diag(r(γk)), k = 1...n, with r(γk) = In this manner, we gain an overall perspective of the behavior sin(γk) cos(γk) >. Let 1n be the n-dimensional vector characterizing the steady state set of the power system model. − n 1 1 We address the second question by showing asymptotic sta- with all entries being one and T = S S the n- dimensional torus. We denote by d( , ) be× a distance··· × metric. bility of the nonlinear trajectories confined to a neighborhood n · · Given a set R , then d(z, ) = inf d(z,x) and Tz is the to the steady state set of interest. For this, we study the stability x A A ⊆ A n A of the nonlinear dynamics as a direct application of the center tangent space of at z. Given a vector∈ v R , we denote A ∈ manifold theory to the multi-converter power system. We by v⊥ its orthogonal complement, vk its k-th entry and ⊕ assume that the eigenvalues of the Jacobian evaluated at a is the direct sum. For a matrix A, let A 2 = σ(A) denote k k point on the synchronous steady state set can be split into its 2-norm and σ(A) denote its maximum singular value. For ∂ f (x) one zero mode and the remainder with real part confined convenience, we denote by Jf (x∗) = the Jacobian ∂x x=x∗ to the left half-plane. Accordingly, we then decompose the of f . nonlinear dynamics into two subsystems, whose dynamics are zero and Hurwitz, respectively. This allows to define a center manifold upon modal transformation, where we use the II.POWER SYSTEMMODELINCLOSED-LOOPWITH reduction principle [13, p.195] to deduce the stability of the MATCHING CONTROL trajectories of the multi-converter system from the dynamics evolving on the center manifold. The point-wise application of A. Multi-source power system dynamics the center manifold theory allows to construct a neighborhood of the steady state set of interest and thereby showing its local R L i asymptotic stability. ik net,k i + x,k + + To satisfy the Jacobian eigenvalue condition in an explicit idc∗ ,k way, we study the linearized system trajectories and pursue v v v a parametric linear stability analysis approach at a frequency dc,k Gdc Cdc x,k C k G synchronous steady state. Towards this, we develop a novel stability analysis for a class of partitioned linear systems − − − characterized by a stable subsystem and a one-dimensional invariant subspace. We propose a new class of Lyapunov functions characterized by an oblique projection onto the Fig. 1: Circuit diagram of a balanced and averaged three-phase µ µ complement of the invariant subspace, where the inner product DC/AC converter with ix,k = 2 r>(γk)ik and vx,k = 2 r(γk)vdc,k, is taken with respect to a matrix to be chosen as solution to see e.g., [14]. Lyapunov and ∞ Riccati equations. Our approach has natural cross-links withH analysis concepts for interconnected systems, We start from the following general model describing the e.g., small-gain theorem systems. For the multi-source power evolution of the dynamics of n identical three-phase balanced system model, we arrive at explicit stability conditions that and averaged DC/AC converters− given in Figure 1 in closed- depend only on the converter’s parameters and steady-state loop with the matching control [4], a control strategy that values. In accordance with other works, our conditions require renders the closed-loop DC/AC converter structurally similar sufficient DC-side and AC-side damping. to a synchronous machine, based on the concept of matching Paper organization The paper unfurls as follows: Section II their dynamics; see Section V-C. The converter input uk is controlled as a sinusoid with constant magnitude µ ]0,1[ and describes the line dynamics and in particular, the evolution ∈ frequency γ˙ R given by the DC voltage deviation. h i> 2m of the line current i` := i> ,...,i> R , where R` > 0 ∈ `1 `m ∈ γ˙k = η(vdc k v∗ ) (1a) is the line resistance, L` > 0 is the line inductance, B = , − dc I and K = diag(I,Cdc I,L I,C I,L I). It is noteworthy sin(γk) B ⊗ ` uk = µ − , k = 1,...,n, (1b) that, inet = Bi . The multi-converter input is represented by cos(γk) ` n u = idc∗ 1,...,idc∗ n > R . 1 , , ∈ where γk S is the virtual angle after a transformation into Let N be the dimension of the state vector z = ∈ a dq frame, rotating at the nominal steady state frequency v x >, where we define the relative DC voltage R t γ> ˜dc> > ω∗ > 0, with angle θdq(t) = ω∗ dτ (by the so-called Clark 0 v˜ = v v 1 , AC signals x = i v i > and the input transformation, see [2]) and η > 0 is a control gain. The dc dc dc∗ n > > `> − N u = 0>,u>,...,0> > R given by the vector (3). converters are interconnected with m identical resistive and ∈ inductive lines. The closed-loop converter− dynamics are given By putting it all together, we arrive at the nonlinear power by the following set of first-order differential equations in system dynamics compactly described by, dq frame. For the simplicity of notation, we will drop the z˙ = f (z,u), (4) subscript dq from all the AC signals. N ˙ 0 for all z R , where is a smooth manifold and f (z,u) γk η(vdc,k vdc∗ ) ∈ X ⊆ X − µ denotes the vector field (3). Cdcv˙dc,k Kp(vdc,k vdc∗ ) 2 r(γk)>ik idc∗ ,k = − − µ− + , (2) Li˙ (RI+Lω∗ J)ik+ r(γk)vdc,k vk 0 k − 2 − Remark 1. Without loss of generality, we assume that the (GI+C ω J)v +i i Cv˙k − ∗ k k− net,k 0 DC/AC converters are identical and interconnected via iden- tical RL lines which is a common assumption in the analysis Let the nominal frequency be given by ω∗ R and ∈ of power system stability, see e.g., [9], [15]. Nonetheless, our vdc,k R denote the DC voltage across the DC capacitor ∈ analysis carries on to the more general heterogeneous setting, with nominal value v . The parameter Cdc > 0 represents dc∗ where the converters and the lines can be parameterized the DC capacitance and the conductance Gdc > 0, together differently. See also Section V-C. with the proportional control gain Kˆp > 0, are represented by Kp = Gdc +Kˆp > 0. This results from designing a controllable current source idc k = Kˆp(vdc k v ) + i , where we denote , , − dc∗ dc∗ ,k III.CHARACTERIZATION OF THE STEADY STATE SET by idc∗ ,k R a constant current source representing DC side ∈ 2 A. Steady state map input to the converter. Let ik R be the inductance current 2 ∈ and vk R the output voltage. The modulation amplitude µ, Lemma III.1 (Steady state map). Consider the nonlinear ∈ feed-forward current idc∗ and the control gain Kbp are regarded power system model (4). Given the steady state angles γ∗ as constants usually determined offline or in outer control satisfying γ˙∗ = 0. Then, a feasible input u is given by, loops. See [4] for more details. On the AC side, the filter 1 resistance and inductance are represented by R > 0 and L > 0 u = ξ Rot(γ∗)>Y Rot(γ∗) n, (5) respectively. The capacitor C > 0 is set in parallel with the load v where ξ = µ2 dc∗ > 0 and Y = (Z + (Z + B Z 1 B ) 1) 1. conductance G > 0 to ground and connected to the network 4 R C `− > − − 2 via the output current inet,k R . Proof. To begin with, we solve for the steady state z∗ by ∈ 1 Observe that the closed-loop DC/AC converter dynamics (2) setting (4) to zero. Note that ZC + BZ`− B> and ZR + (ZC + match one-to-one those of a synchronous machine with single- 1 1 BZ`− B>)− are non-singular matrices due to the presence of pole pair, non-salient rotor under constant excitation [4]. Thus, the resistance R > 0 and the load conductance G > 0, where all the results derived ahead can conceptually also be applied 1 BZ`− B> is a weighted Laplacian matrix. The steady state of to synchronous machines (see also our comments in Section 1 the lines is described by i`∗ = Z`− B>v∗, from which follows V-C). 1 1 that v∗ = (ZC +BZ`− B>)− i∗ for the output capacitor voltage By lumping the states of n identical converters and at steady state. The steady state inductance current is given by − 1 1 m identical lines and defining the impedance matrices ZR = i = µY Rot(γ )v 1 and finally from µRot (γ )i = u, − ∗ 2 ∗ dc∗ n 2 > ∗ ∗ R I + Lω∗ J, ZC = G I +C ω∗ J, Z` = R` I + L`ω∗ J, we obtain we deduce (5). the following power system model, 2n 2n Notice that the matrix Y R × in (5) has an admittance- γ˙ 1 0 ∈ η(vdc v∗ n) like − dc structure which is customary in the analysis of power v˙dc 1 1 u Kp(vdc vdc∗ n) 2 µRot(γ)> i system models and encodes in particular the admittance of the ˙ 1 − − 1 − 1 i = K− ZR i+ µRot(γ)v v + K− 0 , (3) − 2 dc− transmission lines according to the network topology given v˙ ZC v+i Bi` 0 1 − − by the weighted network Laplacian B Z− B>, as well as ˙ Z` i`+B> v ` i` − 0 the converter output filter parameters given by the impedance matrices Z and Z . Once we solve for the steady state angles > n R C where we define the angle vector γ = γ1,...,γn T , n n∈ γ∗ T , we recover the full steady state vector z∗ . It with DC voltage vector v = v ,...,v > , the dc dc,1 dc,n R is noteworthy∈ that, each angle vector γ determines a∈ unique M n ∈ ∗ inductance current i i i > 2 and output capacitor = 1>,..., n> R steady state z∗ , which induces a steady state manifold 2n ∈ ∈ M voltage v = v>,...,vn> > R . The last equation in (3) (z∗) as described in (9). 1 ∈ S Equation (5) can be understood as a steady state map (in Consider the steady state manifold described by (6). the sense of [16]), Observe that a steady state z pertainsM to a continuum ∗ ∈ M n n of equilibria, as a consequence of the rotational symmetry (7) : T R , γ∗ ξ Rot(γ∗)> Rot(γ∗)1n, P → 7→ Y and given by, taking as argument the desired steady state angle γ∗ and n 1o (z∗)= (γ∗ + θ1n)> 0> (R(θ)x∗)> > , θ S , (9) mapping into the feasible input u pertaining to the set . The S ∈ steady state angles in (5) are obtained from solvingU an AC that is, for all z , it holds that (z ) . optimal power flow problem. Equation (5) is a power balance ∗ ∈ M S ∗ ⊂ M equation between electrical power Pe∗ = vdc∗ ξRot>(γ∗)i∗ = 1 IV. LOCAL SYNCHRONIZATION OF MULTI-CONVERTER vdc∗ ξ Rot(γ∗)> Rot(γ∗) n and DC power given by Pm∗ = vdc∗ u. In the sequel,Y we denote by the set of all feasible inputs u POWERSYSTEM U given by (5). In this section, we study local asymptotic stability of the steady state set (z ) in (9), as an application of the center S ∗ B. Steady state set manifold theory [13], [17, p.195]. Take u and let be a non-empty steady-state manifold resulting∈ U fromM setting ⊂ X (4) to zero and given by, A. Preliminaries We provide some background theory on center manifold = z∗ f (z∗,u) = 0 . (6) M { ∈ X | } theory [17] which is our main tool for proving asymptotic We are particularly interested in a synchronous steady-state stability. with the following properties: We review some key concepts from the center manifold The frequencies are synchronized at the nominal value theory. For this, consider a dynamical system in normal form, • ω mapped into a nominal DC voltage v 1. ∗ dc∗ ≥ n [ω] = ω R ω = ω∗1n , y˙ = A y + f (y,ρ), (10a) { ∈ 0| } y 1 R≥n 1 [vdc] = vdc 0 vdc = vdc∗ n . ρ˙ = Bρ ρ + f2(y,ρ), (10b) { ∈ ≥ | } The angles are stationary c c • where Ay R × has eigenvalues with zero real part and (n ∈c) (n c) Tn Bρ R − × − has eigenvalues with negative real parts [γ] = γ γ˙∗ = 0 . ∈ { ∈ | } (or Hurwitz), and f1 and f2 are nonlinear functions with the The inductor currents, capacitor voltage and line current following properties, • are constant at steady state in a rotating frame f1(0,0) = 0,Jf (0,0) = 0, (11) 2n 2n 1 [i] = i R i˙∗ = 0 ,[v] = v R v˙∗ = 0 , ∈ | ∈ | f2(0,0) = 0,Jf2 (0,0) = 0. (12) 2m [i`] = i` R i˙∗ = 0 . ∈ | ` An invariant manifold c is a center manifold of (10), if it can be locally representedW as, C. Symmetry of the vector field c Consider the nonlinear power system model in (4). For all = (y,ρ) ρ = h(y) , (13) 1 W { ∈ O| } θ S , it holds that, where is a sufficiently small neighbourhood of the origin, ∈ O f s S z u f z u S f z u h(0) = 0 and (θ 0 + (θ) , ) = ( ( ), ) = (θ) ( , ), (7) S dh Jh(0) = = 0. s 1 > where we define the translation vector 0 = n> 0> 0> , dy y=0 I 0 0 the matrix S(θ) = 0 I 0 , and the set 0 0 R(θ) It has been shown in [10, Thm. 8.1] that a center manifold always exists and the dynamics of (10) restricted to the center n 1o (z)= (γ + θ1n)> v˜> (R(θ)x)> > , θ S . (8) manifold are described by, S dc ∈ ˙ The symmetry (7) follows from observing that the rotation ma- ξ = Ayξ + f1(ξ,h(ξ)), (14) trix R(θ), commutes with the impedance matrices Z , Z , Z , R C ` for a sufficiently small ξ c. Note that ξ is a parametric the skew-symmetric matrix J and the incidence matrix B. R representation of the dynamics∈ along points on the center Notice that for θ = 0, it holds that S(z) = z and hence manifold c in (13). z S(z). In fact, the symmetry (7) arises from the{ } fact that the The stabilityW of the system dynamics (10) is analyzed from nonlinear∈ power system model (4) has no absolute angle: A the dynamics on the center manifold using the reduction shift in all angles γ Tn, corresponding to a translation by s , 0 principle described in the following theorem. induces a rotation in∈ the angles of AC signals by R(θ). Up to re-defining the dq transformation angle to θdq0 (t) = θdq(t)+θ, Theorem IV.1 ( [13], p.195). If the origin is stable under the vector field (4) remains invariant under the translation s0 (14), then the origin of (10) is also stable. Moreover there and rotation action S(θ) in (7). exists a neighborhood of the origin, such that for every O (y(0),ρ(0)) , there exists a solution ξ(t) of (14) and where z is near the origin 0 (z ). Next, by defining (y,ρ) = ∈ O ∈ S ∗ constants c1,c2 > 0 and γ1,γ2 > 0 such that, T z, we arrive at the following system in normal form
y(t) = ξ(t) + r1(t), y˙ = f1(y,ρ) (17a)
ρ(t) = h(ξ(t)) + r2(t), ρ˙ = Bρ + f2(y,ρ), (17b)
γi t where ri(t) < ci e− , i = 1,2. where f1(0,0) = 0, f2(0,0) = 0 and Jf1 (0,0) = Jf2 (0,0) = 0. k k Now, we show that, Next, we provide background on set stability in the follow- ing definition. c := (y,ρ) ( z (0)) (y,ρ) = T z , W { | ∃ ∈ S × } Definition IV.2 (Set stability [18]). A set is called stable is a center manifold for the system dynamics (17). K with respect to the dynamical system (4), if for all ε > 0, there First, c is invariant because it consists of equilibria of exists δ > 0, so that, (17). Second,W c is tangent to the y-axis at y = 0. To see this, define W d(z , ) δ = d(z(t,z ), ) < ε, t 0 (15) 0 0 1 y K ≤ ⇒ K ∀ ≥ f˜(y,ρ) := f T − = f (z). A set as in Definition IV.2 is called asymptotically stable ρ K with respect to a dynamical system (4), if (15) holds and Then c = (y,ρ) f˜(y,ρ) = 0 . The row vectors of the JacobianW given{ by | } lim d(z(t,z0), ) = 0. t ∞ K → ∂ f˜1(0,0) ∂ f˜1(0,0) ∂y ∂ρ B. Local asymptotic stability . . d f 1 1 Jf˜(0,0) = . . = T − = Jf (0)T − dz z=0 Next, we present our main result on local asymptotic ∂ f˜N (0,0) ∂ f˜N (0,0) ∂y ∂ρ stability of the set (z∗) with respect to the multi-converter dynamics (4). S span the normal space of c at 0. Since the columns of 1 W The following assumption on the eigenvalues of the Jaco- T − = (v(0),...) consist of the right eigenvectors of Jf (0), by 1 bian of the multi-converter system (4) is crucial to derive our means of Jf (0)v(0) = 0, Jf (0)T − has a zero first column. main result. This shows that Jf˜(0,0) has its first entry (corresponding to y component) equal to zero. As a consequence, there Assumption 1. Consider the linearized system described by − dh exists a function h(y) such that h(0) = 0 and dy y=0 = 0 in a the following equations, c | neighborhood 0 of 0, where 0 = (y,ρ) ρ = h(y) . W W ∩ W { | } δz˙ = Jf (z∗)δz. (16) It follows that the dynamics restricted to 0 are given by ξ˙ = 0 because c is an equilibrium manifoldW to (17) d f W Assume that Jf (z∗) = z z in (16) representing the Jaco- and thus f (ξ,h(ξ)) = 0. This shows that ξ(t) = ξ(0). By dz | = ∗ 1 bian of multi-converter system (4) linearized at z∗ , has applying Theorem IV.1, the solutions for y ρ starting in ∈ M ( , ) 0 only one eigenvalue at zero and the real-parts of all other are described by, W eigenvalues are in the open-left half plane. y(t) = ξ(t) + r1(t), Remark 2. In Section V, we provide an approach on how ρ(t) = h(ξ(t)) + r (t), to satisfy the eigenvalue condition in Assumption 1 for the 2 γ t multi-converter system in an explicit way. where ri(t) < cie i , i = 1,2 for some constants ci,γi > 0. k k − We now present our main result in the following theorem. This implies that, lim(y(t),ρ(t)) = (ξ(0),h(ξ(0))), Theorem IV.3 (Local asymptotic stability). Consider the t ∞ power system dynamics in (4) under Assumption 1 with a → and thus, feasible input u . Then, (z∗) is locally asymptotically ∈ U S 1 stable. Moreover, there exists a neighborhood of (z∗) such lim z(t) = T − (ξ(0),h(ξ(0))) (0). D S t ∞ ∈ S that for every z(0) , there exists a point s (z∗), where → ∈ D ∈ S This argument can be repeated for each point on (0) to lim z(t) = s. S t ∞ obtain a cover k of (0). Since (0) is compact, we → can construct a finite{W } subcoverS to form aS neighbourhood = Proof. To prove that (z∗) is stable, we consider the system S D S k k of (0). Local asymptotic stability of (0) follows dynamics (4) under Assumption 1. directly.W S S Without loss of generality, assume z∗ = 0. From Assumption N N 1, we know there exists a transformation T R × , such that Note that our results conceptually apply to prove local 1 ∈ TJf (0)T − is block diagonal, where Jf (0) is given in (22), asymptotic stability of a synchronous steady state set with with a zero for the first component and a block B that is respect to trajectories of high-order dynamics of synchronous Hurwitz. We rewrite the dynamics of (4) as machines and find an estimate of their region of attraction, based on the structural similarities between synchronous ma- z˙ = Jf (0)z + ( f (z,u) Jf (0)z) − chines and DC/AC converter in closed-loop with the matching control [4]. Our local analysis also paves the way for a global In standard Lyapunov analysis, one seeks a pair of matrices analysis of the stability of high-order multi-machine or multi- (P, ) with suitable positive (semi-) definiteness properties so Q converter system with non-trivial conductance, which is an that the Lyapunov equation PA + A>P = is met. In the open problem in the power system community [19], [20]. following, we apply a helpful twist and parameterize−Q the - matrix as a quadratic function (P) of P, which renders theQ Q V.S UFFICIENT CONDITIONS FOR THE STABILITY OF THE Lyapunov equation to an ∞ algebraic Ricatti equation. We choose the following structureH for the matrix (P), LINEARIZEDSYSTEM Q This section derives sufficient conditions to satisfy the H (P) P 1 > (20) eigenvalue decomposition described in Assumption 1 in an ( ) = Q 1 , Q H(P) H(P) 1− H(P)> + 2 explicit way for a class of linear systems that applies to the Q Q where 1 is a positive definite matrix, 2 is a positive semi- stability of the linearized multi-converter system. Q Q definite matrix with respect to span p2 , P is block-diagonal, { } P 0 A. Lyapunov stability of vector fields with symmetries P = 1 , (21) 0 P In this section, we develop a stability theory for a gen- 2 eral class of linear systems enjoying some of the structural with P1 = P1> > 0 and P2 = P2> > 0, i.e., the Lyapunov function properties featured by the Jacobian matrix in (4). For this, we is separable, and finally H(P) = A12> P1 +P2 A21 is a shorthand. consider a class of partitioned linear systems of the form We need to introduce a third and final assumption. 1 A11 A12 Assumption 4. Consider the matrix F = A22 + A21 − P1A12 x˙ = x, (18) Q1 A21 A22 and the transfer function,
1 where x = [x1> x2>]> denotes the partitioned state vector, and the = (sI F)− B, G C − block matrices A11,A12,A21,A22 are of appropriate dimensions. 1/2 1 1/2 In the following, we assume stability of the subsystem with B = A21 − , = (A> P1 − P1A12 + 2) . Assume Q1 C 12 Q1 Q characterized by A and the existence of a symmetry, i.e., that F is Hurwitz and that ∞ < 1. 11 kGk an invariant zero eigenspace. Assumption 4 will guarantee suitable definiteness and decay Assumption 2. In (18), the block diagonal matrix A11 is properties of the Lyapunov function (21) under comparatively Hurwitz. mild conditions discussed in Section V-C. Assumptions 2, 3, and 4 recover our requirement for positive Assumption 3. There exists a vector p = [p1> p2>]>, so that definiteness of the matrix P in (21) and semi-definitness (with A span p = 0. respect to span p ) of (P) in (20) as shown in the following. · { } { } Q Proposition V.1. Under Assumptions 2, 3 and 4, the matrix We are interested in asymptotic stability of the subspace P in (21) exists, is unique and positive definite. span p : all eigenvalues of A have their real part in the open{ left-half} plane except for one at zero, whose eigenspace Proof. By calculating PA + A P = (P), where A is as in > −Q is span p . Recall that the standard stability definitions and (18), P is as in (21), and (P) is as in (20), we obtain { } Q Lyapunov methods extend from stability of the origin to P1 A11+A11> P1 H(P)> Q1 H(P)> stability of closed and invariant sets when using the point- = 1 , H(P) P A +A P H(P) H(P)Q− H(P) +Q2 to-set-distance rather than merely the norm in the comparison 2 22 22> 2 − 1 > functions; see e.g., [21, Theorem 2.8]. In our case, we seek a the block-diagonal terms of which are quadratic Lyapunov function that vanishes on span p , is pos- 1 P1 A11 + A11> P1 = 1, itive elsewhere and whose derivative is decreasing{ everywhere} −Q 1 2 P2 A22 + A22> P2 = H(P) 1− H(P)> 2, outside span v . − Q − Q where H(P) = A> P1 +P2 A21. Since A11 is Hurwitz, there is a We start by{ } defining a Lyapunov function candidate, 12 unique and positive definite matrix P1 solving 1 . Moreover,
Ppp P specification 2 is equivalent to solving for P2 in the following V x x P > x (19) ( ) = > , ∞ algebraic Riccati equation: − p>Pp H 1 1 where P is a positive definite matrix. Our Lyapunov candidate P2 A21 − A> P2 + P2F + F>P2 + A> P1 − P1A12 + 2 = 0, Q1 21 12 Q1 Q construction is based on two key observations: 1 where F = A22 +A21 1− P1A12. Under Assumption 4, the pair First, the function V(x) is defined via an oblique pro- Q 1/2 • F B is stabilizable with B A − and for 1, jection of the vector x n parallel to span p onto ( , ) = 21 1 ∞ < R Theorem 7.4 in [22] implies thatQ no eigenvalueskGk of the x n p Px = 0 . If ∈P = I, then V is the{orthogo-} R > h FBB i nal{ ∈ projection| onto} p V x Hamiltonian matrix = > are on the imaginary span ⊥. Hence, ( ) vanishes on H C> C F> { } 1 − − 1/2 span p and is strictly positive definite elsewhere. axis with = (A P1 − P1A12 + 2) . By Theorem 7.2 { } C 12> Q1 Q Second, the positive definite matrix P is a degree of in [22], there exists a unique stabilizing solution P2 to • 1 1 freedom that can be specified later to provide sufficient 2 . Define E = A P1 − P1A12 + 2 + P2 A21 − A P2 0. 12> Q1 Q Q1 21> ≥ stability conditions. From Ap = 0 follows that A12 p2 = A11 p1 = 0 and since − 6 d f 2 p2 = 0, ker 2 kerA12 = 0 . This shows that E is non- The system matrix is the Jacobian Jf (z∗) = dz z=z , δz = Q Q ∩ { } | ∗ singular and thus E is positive definite. Since F is Hurwitz, z z > T z δ 1> δ 2> z∗ , corresponding to the partition δ 1 = by standard Lyapunov theory [10], the Lyapunov equation ∈ Mn n δγ δv > 2 , δz 6 . The matrices in (22) are P F +F P +E = 0 admits a positive definite solution P . > dc> R 2 R 2 > 2 2 given by, ∈ ∈ Lemma V.2. Under Assumptions 2, 3 and 4, the matrix v 2 dc∗ 2 1 (P) in (20) is positive semi-definite. Additionally, ker(A) = ∇ U(γ∗) = µ diag(Rot>(γ∗) J> Rot(γ∗) n), Q 4 Y ker( (P)) = span p . 1 Q { } = µdiag((JRot(γ∗))>i∗), Proof. First, note that by Proposition V.1, the matrix P = P > 2 > 1 0 and observe that the matrix (P) in (20) is symmetric and Ξ(γ∗) = µJRot(γ∗), Q 2 the upper left block 1 > 0 is positive definite. By using the 1 Schur complement andQ positive semi-definiteness of , we Λ(γ ) = µv Rot(γ ), 2 ∗ 2 dc∗ ∗ obtain that (P) is positive semi-definite. Second, byQ virtue of p (P)pQ= p (PA + A P)p = 0 due to Assumption 3, it where we consider the smooth potential function, >Q > > follows that span p ker( (P)). Third, consider a general n 1 1 { }⊆ Q U : T R, γ ξ n> Rot>(γ)J> Rot(γ∗) n. vector s = [ s>s> ]>, so that (P)s = 0. Given H(P) = A P1 + → 7→ − Y 1 2 Q 12> P2 A21, we obtain the algebraic equations 1s1 + H(P)>s2 = Note that the Jacobian Jf (z∗) has one-dimensional zero 1 Q 0, H(P)s1 + H(P) 1− H(P)> + 2 s2 = 0. One deduces that eigenspace denoted by, s = 0 and thusQ s span pQ . The latter implies s 2 2 2 2 1 Q 1 ∈ { } ∈ v z 1 x > T − H(P) span p2 span p1 because (P)span p = 0. span ( ∗) = span n> 0> (J ∗)> z∗ , −Q1 > { } ∈ { } Q { } { } { } ⊂ M s s Thus, it follows that s span [ 1> 2> ]> = span p and we ∈ { } { } with Jx Ji Jv Ji >. In particular, we deduce that ker( (P)) = span p . Fourth and finally, for the ∗ = ( ∗)> ( ∗)> ( `∗)> sake of contradiction,Q take a{ vector} v ˜ / span p , so that can establish a formal link between the linear subspace ∈ { } span v(z∗) and the steady state set (z∗) in (9) as follows. v˜ ker(A) v˜> A>P + PA v˜ = 0 v˜> (P)v˜ = 0 v˜ { } 1 S ker∈( (P)).⇒ This is a contradiction to⇒ ker(Q(P)) = span⇒ p∈. For all θ S , Q Q { } ∈ Hence, we conclude that ker(A) = ker( (P)) = span p . Z θ 1 Q { } Z θ n (z∗) = z∗ + v(z∗) ds,= z∗ + 0 ds, Next, we provide the main result of this section. S 0 0 JR(s)x∗ Lemma V.3. Consider the linear system (18). Under Assump- which follows from (9). In fact, v(z ) is the tangent vector of tions 2, 3 and 4, span p is an asymptotically stable subspace ∗ (z ) in the θ- direction and lies on the tangent space T . of A. { } ∗ z∗ Hence,S (z ) is the angle integral curve of span v(z ) . M S ∗ { ∗ } Proof. Consider the function V(x) in (19). The matrix P in It can be deduced from (7), that by expanding Taylor series 1/2 1 (21) is positive definite by Proposition V.1. By taking y = P x around (θ 0,z∗), θ 0 S , z∗ of left and right terms in (7) ∈ ∈ M and w = P1/2 p, the function V(x) can be rewritten as V(y) = and comparing the terms of their first derivatives with respect y I ww> y y y I ww> to θ, we recover Jf (z∗)v(z∗) = 0, as follows, > w w = >Πw . The matrix Πw = w w is a pro- − > − > jection matrix into the orthogonal complement of span(w), and d f (z) dS(θ) is hence positive semi-definite with one-dimensional nullspace z∗ + s0 (θ θ 0) = 0, dz z z dθ − corresponding to P1/2span p . It follows that the function = ∗ θ=θ 0 { } d f (z) dS V(x) is positive definite for all x span p ⊥. By means of where z=z = Jf (z∗), z∗ + s0 = v(z∗) (by the ∈ { } dz | ∗ dθ |θ=θ 0 Ap = (P)p = 0 in p>PA = p>( (P) A>P) = 0, we obtain I 0 0 Q Q − definition of the set (9)), S(θ) = 0 I 0 , and s0 = V˙ (x) = x> (P)x. By Lemma V.2, it holds that V˙ (x) is 0 0 R(θ) negative− definiteQ for all x span p . We apply Lyapunov’s ⊥ 1 0 0 >. method and Theorem 2.8∈ in [21]{ to} conclude that span p is n> > > { } Next, we consider the linearized model (22) and identify asymptotically stable. the matrices h 0 ηI i h 0 0 0 i A = 1 2 1 , A = 1 , 11 C ∇ U(γ ) C KpI 12 C− Λ(γ∗)> 0 0 B. Stability of the linearized multi-DC/AC converter − dc− ∗ − dc− − dc " 1 1 # 1 1 L− ZR L− I 0 L− Ξ(γ∗) L− Λ(γ∗) − 1 − 1 1 Our next analysis takes under the loop the behavior of A = , A = C− I C− ZV C− B . 21 0 0 22 − − linearized trajectories described by the Jacobian of (4) at 0 0 0 L 1B L 1Z `− > − `− ` z = z . For this, we consider the linearized system described ∗ Define the Lyapunov function V z as in (19) with p by the following equations, ( ) = v(z∗) = [v∗>,v∗>]>. Hence, V(z) is positive semi-definite with 1 2 0 ηI 0 0 0 respect to span v(z∗) . Next, we fix the matrix (P) given by 2 ∇ U(γ∗) KpI Λ(γ∗)> 0 0 A A { } Q 1 − − − 11 12 (20), where we set 1 = I, 2 = I v∗v∗>/v∗>v∗ and search δz˙ = K− Ξ(γ∗) Λ(γ∗) ZR I 0 δz = δz. 2 2 2 2 − − Q Q − 0 0 I ZC B A21 A22 for the corresponding matrix P so that, 0 0 0 −B −Z > − ` (22) PJf (z∗) + Jf (z∗)>P = (P). −Q Analogous to (21), we choose the block diagonal matrix The feasibility of specification 2 with the positive semi- v v definite matrix = I 2∗ 2∗> is given by 2 v v P11 P12 0 Q − 2∗> 2∗ P1 0 P = P12 P22 0 = , (23) 0 P2 P2 A21A21>P2 + P2F + F>P2 + NN> + Q2 = 0, (27) 0 0 P33 where F = A22 + A21P1A12 and N = A12> P1. where P11,P12 and P22 are matrices of appropriate dimensions. If Assumption 3 is satisfied, then there exists a positive Notice that the chosen structure of P1 and the zeros in the definite matrix P2 that satisfies the ∞ ARE in (27). off-diagonals in P originate from the physical intuition of Next, we find sufficient conditions,H − for which F the tight coupling between the angle of the converter and its satisfies the Lyapunov equation PF F + F PF = F . > −Q corresponding DC voltage (proportional to the AC frequency), We choose PF and F to be block-diagonal as enabled by the matching control (1). The same type of h L 0 0 i Q h Γ 0 0 i matrices PF = 0 C 0 , F = 0 2GI 0 , with coupling comes into play in synchronous machines between 0 0 L` Q 0 0 2R`I 1 the rotor angle and its frequency, due to the presence of the Γ = 2RI + Cdc− Ξ(γ∗)P12Λ(γ∗)> + Λ(γ∗)P12Ξ(γ∗)> + 1 electrical power in the swing equation [2]. The 4 4 matrix 2Cdc− Λ(γ∗)P22Λ(γ∗)> being itself block-diagonal. Aside × P Q P2 is dense with off-diagonals coupling at each phase, the from Γ, all diagonal blocks of F and F are positive inductance current of one converter with the others. definite. We evaluate the block-diagonal matrix Γ for positive In the sequel, we show that this structure allows for suffi- definiteness by exploring its two-by-two block diagonals, cient stability conditions. where trace and determinant of each block are positive under µ2v 2 Q = 1 v µ(r(γ )) J i > dc∗ , Assumption 5 (Parametric synchronization conditions). Con- x∗,k 2 dc∗ k∗ > > k∗ 16R 1 1 Furthermore, we impose the condition ∞ < 1, by sider Px,k = v∗ µr>(γ∗)i∗ > 0,Qx,k = v∗ µr>(γ∗)J>i∗ > 0 1 kGk 2 dc k k 2 dc k k equivalently setting sup ( jζI F)− B 2 < 1, where = and the matrix F as given in Assumption 4. Assume the ζ RkC − k C following condition is satisfied, ∈ 1/2 (Jx∗)(Jx∗)> A12> P1>P1A12 + I , B = A21. It is sufficient − (Jx∗)>(Jx∗) s 2 2 1 2 α to consider < (sup ( jζI F)− 2 B 2)− . Using the cos 1 (24) kCk2 k − k k k (φk) < 2 2 , ζ R − Px,k + α ∈ 2 triangle inequality for the 2-norm, it holds that 2 A P P A + . Since = 1, we considerkCk in-≤ Px,k 12> 1> 1 12 2 2 2 2 2 where cos(φk) = q [0,1[ is the power fac- k k kQ k kQ k Q2 +P2 ∈ stead, x,k x,k µ2v 2 µ v 2 1 2 tor of k th converter, α = max dc∗ , dc∗ , Y = A12> P1>P1A12 2 (sup ( jζI F)− 2 B 2)− 1. 16R 2 k k ≤ k − k k k − − 4√Y − 1 ζ 1 1 1 − 2 µvdc∗ L− sup ( jζI F)− 2 with the condition Y < 1. ζ k − k 2 2 2 Ξ>ΞΞ>Λ From B 2 = A21 2 = L− 2 = Additionally, assume that, k k k k k Λ>ΞΛ>Λ k 2 Ξ>Ξ 0 2 1 2 µ 1 L− 2 = L− ( 2 µvdc∗ ) because vdc∗ 1. 2 (1 + η Cdcvdc∗ Qx−,k ) k 0 Λ>Λ k ≥ < Kp. (25) 1 1 1 q 4 1 2 Y L v j F Y 2 1 µ2v 2 Q Define = 2 − µ dc∗ sup ( ζI )− 2. For v 2 ( − ) 4 dc∗ x−,k k − k dc∗ − − ζ Y < 1, straightforward calculations show that Next, we provide the main result of this section. 2 2 2 A12> P1>P1A12 2 = σ(Cdc− Λ(γ∗)(P12 + P22)Λ(γ∗)>) = kmax d σ r kγ r γ 4 Y 2 1 , with k ( ( k∗) >( k∗))) < v 2 ( − ) Lemma V.4. Consider the linearized closed-loop multi- k=1,...,n dc∗ − converter model (22). Under Assumption 5, the subspace 2 1 1 12 dk = (r>(γ∗)J>i∗)− + 2 1 + ηCdc( µr>(γ∗)J>i∗)− . k k k 4Kp 2 k k span v(z∗) is asymptotically stable. 2 2 2 { } Under 4/vdc∗ (Y − 1) max (rk>(γk∗)J>ik∗)− > 0, we − − k=1,...,n Proof. Since v(z∗) ker(Jf (z∗)), Assumption 3 is satisfied. If ∈ solve for the gain Kp with σk = σ(r(γk∗)r>(γk∗)) = 1, to find Condition (24) is true, then r>(γk∗)J>ik∗ > 0, for all k = 1,...n, the sub-matrix A is Hurwitz and hence Assumption 2 is also 11 v valid. u µ2 1 1 2 u max (1 + ηCdc( µ r>(γk∗)J>ik∗)− ) u k=1,...,n 4 2 Next, we verify Assumption 4. First, the matrix u < K . P P t 4 2 2 p 11 12 2 (Y − 1) max (r>(γk∗)J>ik∗)− P1 = can be identified from specification 1 with vdc∗ k=1,...,n P21 P22 − − 1 = I by the following expressions, 2 2 Q This can be simplified into (25). The condition 4/vdc∗ (Y − 2 2 2 − 1 1 ∇ U(γ ) 1) max (rk>(γk∗)J>ik∗)− > 0 can be written as Q x,k > P K 2U 1 ∗ I C 2U 1 − k=1,...,n 11 = p(∇ (γ∗))− + ( + η dc(∇ (γ∗))− ) , 2 4 η 2 2Kp µ vdc∗ 2 , under the condition that Y < 1 and we deduce that, 1 16(Y − 1) P = P = (∇2U(γ )) 1C , − 12 12> 2 ∗ − dc µ2v 2 µ v 2 Cdc dc∗ dc∗ P = I + ηC (∇2U(γ )) 1. max , < Qx,k. 22 2K dc ∗ − 16R 4√Y 2 1 p − − Px,k From the definition of the power factor cos(φk) = q , Q2 P2 x,k+ x,k we arrive at (24). In summary, we arrive at the sufficient con- ditions (25) and (24). By applying Theorem IV.3, we deduce that span v(x∗) is asymptotically stable for the linearized system (22).{ }
C. Results contextualization In what follows, we discuss Assumption 5. Generally speak- ing, Condition (24) can be regarded as a condition on the AC side, whereas (25) is a condition on the DC side control. Both conditions are sufficient for stability and can be evaluated in a centralized fashion. Condition (24) connects the efficiency of the converter given by the power factor that defines the amount of current producing useful work to the lower bound α > 0. From (24), the power factor may approaches 1, as α 0. If µ2v 2 µv 2 µ2v 2 → max dc∗ , dc∗ = dc∗ , then condition (24) depends 16R 2 16R Fig. 2: Three converter setup with system dynamics described 4√Y − 1 on the converter’s− resistance R, modulation amplitude µ, by (3), composed of identical three-phase converters C1,C2 and C3 in closed-loop with the matching control and interconnected nominal DC voltage vdc∗ , and the steady state current i∗. This is a known practical stability condition [23]. In fact from via identical RL lines. The internal dynamics of the converters (24), sufficient resistive damping is often enforced by virtual are modeled according to Figure 1. impedance control which makes α 0. µ2v 2 µv 2 µ→v 2 If max dc∗ , dc∗ = dc∗ , then we can again 16R 4√Y 2 1 4√Y 2 1 Let us consider three identical DC/AC converter model in − − − − deploy ∞ control to make Gac ∞ arbitrarily small and thus closed-loop with the matching control depicted in Figure 2 and α 0.H We note that, the ACk sidek feedback control is crucial connected via three identical RL lines, as in (4) and connected to→ achieve desired steady states for our power system model to an inductive and resistive load. Table I shows the converter (4). This can be implemented e.g., via outer loops that take parameters and their controls (in S.I.). measurements from the AC side and using the classical vector First, we start by verifying the parametric conditions estab- control architecture for the regulation of the inductance current lished in Assumption 5 via (24) and (25). We tune the filter and the output capacitor voltage [24]. resistance R > 0 (e.g. using virtual impedance control) so that The condition Y < 1 translates into the requirement that, (24) is satisfied. Next we choose the DC side gain Kp > 0 so that (25) is satisfied. 2L G < β, β = , Second, we numerically estimate the region of attraction ac ∞ v k k µ dc∗ of (z ) in the angle or γ space by initializing sample D S ∗ − 1 trajectories of the angles depicted in Figure 2 at various where Gac( jζ) = ( jζI F)− asks for 2 gain from the disturbances on the AC− side to AC signalsL to be less than locations and illustrate the evolution of the nonlinear angle trajectories of (4) to estimate the region of attraction . As β. This can be achieved via ∞ control [25]. D H predicted by Theorem IV.3, we observe that the set (z∗) Condition (25) depends on the steady state angles γ∗ and S the converter and network parameters and asks for damping as restricted to the angles (relative to their steady state) space, and represented by span 13 is asymptotically stable for the the case for other stability conditions obtained in the literature { } on the study of synchronous machines [3], [4]. The smaller sampled angle trajectories of (4). is the synchronization gain η > 0, the larger is the operating Figure 3 depicts a projection onto the relative (γ1,γ2,γ3) space of the estimate of (in rad). The convergence of angle− range of the DC damping gain Kbp. D1 For a more general setting with heterogeneous converters solutions to the subspace 3 is guaranteed for initial conditions and transmission lines parameters, our stability analysis can at distance of 3.1 resulting from varying the initial angles, be applied and analogous sufficient conditions to (24) and (25) while keeping the remaining initial states fixed. In particular, can be derived. DC voltages and AC currents are also initialized close to their steady state values, as shown in Figure 4. Our simulations show that the DC capacitor voltage vdc in Figures 4 and the VI.SIMULATIONS AC output capacitor voltage in abc frame, namely v converge − The goal of this section is to assess the asymptotic sta- to a corresponding steady state. This validates our theoretical bility of the trajectories of the nonlinear power system (4) results from Section IV. in Theorem IV.3 locally, i.e., by numerically estimating the For completeness, we have also illustrated a projection region of attraction in the neighborhood of (z ) for of the level sets of the Lyapunov function given in D S ∗ z∗ = [γ∗>,0>,vc∗>,i`∗>]>. (19) of an example network consisting of two DC/AC for a corresponding matrix P > 0 as defined in (23). The function V(x) takes positive values everywhere and is zero on the subspace spanned by v(x∗).
Fig. 3: A plot of the region of attraction and the steady D state set (z∗) restricted to (γ1 γ1∗,γ2 γ2∗,γ3 γ3∗) space of the threeS DC/AC converter angles− and− convergence− − of the sample angle trajectories of (4) to the subspace 13 within a distance of d = 3.1. This results from varying the initial angles, while keeping the remaining initial states fixed. A sample of angles deviations initialized within the green area and denoted by different stars converge towards the stable set, while some angle trajectories initialized outside the estimated region are divergent. All the angles are represented in rad.
1,000 ) V ( 3 , 2 , 600 Fig. 5: 3D representation of the Lyapunov function V x
1 ( ) dc,
v in (19) for two DC/AC converters in closed-loop with the 200 matching control and connected via an RL line in (3) after
0 1 1 1 1 1 a projection into (γ1 γ∗,γ2 γ∗) space for P > 0 as in (23) 0 1 10− 2 10− 3 10− 4 10− 5 10− 1 2 · · · · · − − Time(s) and the subspace spanning v(x∗). The parameter values can be found in Table I.
200
) Ci, i = 1,2,3 RL Lines V
( { } 3 ,
2 i 16.5 – , 0 dc∗ 1
c, v∗ 1000 – v dc 3 Cdc 10− – 200 5 − Gdc 10− –
1 1 1 1 1 KP 0.099 – 0 1 10− 2 10− 3 10− 4 10− 5 10− · · · · · η 0.0003142 – Time(s) µ 0.33 – 4 L 5 10− – Fig. 4: Synchronization of DC capacitor voltages corresponds · 5 C 10− – to frequency synchronization at the desired value. Hereby the G 0.1 – angles are initialized at ( 6, 2, 13.15) (in rad) and belong R 0.2 – − − − to the projected region of attraction shown in green in Figure R` – 0.2 5 L` – 5 10− 3. The output capacitor voltage vc in (abc) frame converges · to a sinusoidal steady state v as shown in Figure 3. c∗ TABLE I: Parameter values of DC/AC converters and the RL lines (in S.I). converters interconnected via RL line that have the same dynamics as in (4) into (γ1 γ ,γ2 γ ) VII.CONCLUSIONS − 1∗ − 2∗ − space in Figure 5. The parameter values can be taken We investigated the characteristics of a high-order steady from Table I. We set v(z∗) = v>(z∗) v>(z∗) > state manifold of a multi-converter power system, by exploit- 1 2 ∈ ker(A(z∗)), v1(z∗) = [ 0.043, 0.043, 0, 0 ]>, and v2(z∗) = ing the symmetry of the vector field. We studied local asymp- [ 0.0033 0.0023 0.0033 0.0023 0.7034 0.0108 0.7034 0.0108 0 0 ]> totic stability of the steady state set as a direct application − − − − − − − − of the center manifold theory and provided an operating range [20] H.-D. Chiang, “Study of the existence of energy functions for power for the control gains and parameters. Future directions include systems with losses,” IEEE Transactions on Circuits and Systems, vol. 36, no. 11, pp. 1423–1429, 1989. finding better estimates of the region of attraction using [21] Y. Lin, E. D. Sontag, and Y. Wang, “A smooth converse advanced numerical methods and more detailed simulations Lyapunov theorem for robust stability,” SIAM Journal on Control of high-order power system models. and Optimization, vol. 34, no. 1, pp. 124–160, Jan. 1996. 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