Bregman storage functions for microgrid control

Transcription

1 1 Bregman storage functions for microgri control C. De ersis an N. Monshizaeh arxiv: v5 [math.oc] 4 Aug 2016 Abstract In this paper we contribute a theoretical framework that shes a new light on the problem of microgri analysis an control. he starting point is an energy function comprising the kinetic energy associate with the elements that emulate the rotating machinery an terms taking into account the reactive power store in the lines an issipate on shunt elements. We then shape this energy function with the aition of an ajustable voltage-epenent term, an construct socalle Bregman storage functions satisfying suitable issipation inequalities. Our choice of the voltage-epenent term epens on the voltage ynamics uner investigation. Several microgris ynamics that have similarities or coincie with ynamics alreay consiere in the literature are capture in our incremental energy analysis framework. he twist with respect to existing results is that our incremental storage functions allow for a large signal analysis of the couple microgri obviating the nee for simplifying linearization techniques an for the restrictive ecoupling assumption in which the frequency ynamics is fully separate from the voltage one. A complete Lyapunov stability analysis of the various systems is carrie out along with a iscussion on their active an reactive power sharing properties. I. INRODUCION Microgris have been envisione as one of the leaing technologies to increase the penetration of renewable energies in the power market. A thorough iscussion of the technological, physical an control-theoretic aspects of microgris is provie in many interesting comprehensive works, incluing [60], [59], [25], [4], [40]. ower electronics allows inverter in the microgris to emulate esire ynamic behavior. his is an essential feature since when the microgri is in gri forming moe, inverters have to inject active an reactive power in orer to supply the loas in a share manner an maintain the esire frequency an voltage values at the noes. Hence, much work has focuse on the esign of ynamics for the inverters that achieve these esire properties an this effort has involve both practitioners an theorists, all proviing a myria of solutions, whose performance has been teste mainly numerically an experimentally. he main obstacle however remains a systematic esign of the microgri controllers that achieve the esire properties in terms of frequency an voltage regulation with power sharing. he ifficulty lies in the complex structure of these systems, comprising ynamical moels of inverters an loas that are physically interconnecte via exchange of active an reactive power. In quasi steay state working conitions, these quantities are sinusoial terms epening on the voltage phasor relative phases. As a result, mathematical moels of microgris C. De ersis an N. Monshizaeh are with ENEG an the J.C. Willems Center for Systems an Control, University of Groningen, the Netherlans. {c.e.persis, n.monshizaeh}@rug.nl. reuce to high-orer oscillators interconnecte via sinusoial coupling, where the coupling weights epen on the voltage magnitues obeying aitional ynamics. he challenges with these moels lie in the presence of highly nonlinear terms an the strict coupling between active an reactive power flow equations. o eal with the aforementione complexity of these ynamical moels common remeies are to ecouple frequency an voltage ynamics, an to linearize the power flow equations. While the former enables a separate analysis of the two ynamics ([42]), the latter permits the use of a small signal argument to infer stability results; see e.g. [46], [47]. Results that eal with the fully couple system are also available [41], [54], [35]. In this case, the results mainly concern network-reuce moels with primary control, namely stability rather than stabilization of the equilibrium solution. Furthermore, lossy transmission lines can also be stuie [20], [54], [6], [54], [35], an also [15]. Main contribution. In spite of these many avances, what is still missing is a comprehensive approach to eal with the analysis an control esign for microgris. In this paper we provie a contribution in this irection. he starting point is the energy function associate with the system, a combination of kinetic an potential energy. Relying on an extene notion of incremental issipativity, a number of so-calle Bregman storage functions whose critical points have esire features are constructe. he construction is inspire by works in the control of networks in the presence of isturbances, which makes use of internal moel controllers ([9], [34]) an incremental passivity ([49]). he storage functions that we esign encompass several network-reuce versions of microgri ynamics that have appeare in the literature, incluing the conventional roop controller [60], [41], the quaratic roop controller [46], an the reactive power consensus ynamics [42]. Our analysis, however, suggests suitable moifications such as an exponential scaling of the averaging reactive power ynamics of [42], an inspires new controllers, such as the socalle reactive current controller (we refer to [7] for a relate controller). he approach we propose has two aitional istinguishing features: we o not nee to assume ecouple ynamics an we perform a large signal analysis. Our contribution also expans the knowlege on the use of energy functions in the context of microgris. Although historically energy functions have playe a crucial role to eal with accurate moels of power systems ([52], [16], [14]), our approach base on the incremental issipativity notion shes a new light into the construction of these energy functions, allows us to cover a wier range of microgri ynamics, an paves the way for the esign of ynamic controllers, following the combination of passivity techniques an internal moel principles as in [9]. We refer the reaer to e.g. [36], [19] for

2 2 seminal work on passivity-base control of power networks. In this paper we focus on network reuce moels of microgris ([41], [54], [35], [47]). hese moels are typically criticize for not proviing an explicit characterization of the loas ([46]). Focusing on network reuce moels allows us to reuce the technical complexity of the arguments an to provie an elegant analysis. However, one of the avantages of the use of the energy functions is that they remain effective also with network preserve moels ([52]). In fact, a preliminary investigation not reporte in this manuscript for the sake of brevity shows that the presente results exten to the case of network preserve moels. A full investigation of this case will be reporte elsewhere. he outline of the paper is as follows. In Section II, etails on the moel uner consieration are provie. In Section III the esign of Bregman storage functions is carrie out an incremental issipativity of various moels of microgris associate with ifferent voltage ynamics is shown. A few technical conitions on these energy functions are iscusse in Section IV, an a istribute test to check them is also provie. Base on the results of these sections, attractivity of the prescribe synchronous solution an voltage stability is presente in Section V, along with a iscussion on power sharing properties of the propose controllers (Subsection V-A). ower sharing in the presence of homogeneous lossy transmission lines is stuie in Subsection V-B. he paper ens with concluing remarks in Section VI. II. MICROGRID MODEL AND A SYNCHRONOUS SOLUION We consier a network-reuce moel of a microgri operating in islane moe, that is isconnecte from the main gri. his moel is given by θ = ω ω = (ω ω )K ( )+u V = f(v,,u ) where θ n is the vector of voltage angles, ω R n is the frequency, R n is the active power vector, R n is the reactive power vector, an V R n >0 is the vector of voltage magnitues. he integer n equals the number of noes in the microgri an I := {1,2,...,n} is the set of inices associate with the noes. he matrices, V, an K are iagonal an positive efinite. he vectors ω an enote the frequency an active power setpoints, respectively. he vector may also moels active power loas at the buses (see Remark 2). he vector u is an aitional input. he function f accounts for the voltage ynamics/controller an is ecie later. he moel (1) with an appropriate selection of f escribes various moels of network-reuce microgris in the literature, incluing conventional roop controllers, quaratic roop controllers, an consensus base reactive power control schemes ([60], [45], [41], [46], [42]). However, while [45], [46], [43] consier network-preserve moels of microgris, in this paper network-reuce moels are consiere. We refer the reaer to [43] for a compelling erivation of microgri moels from first principles. (1) Our goal here is to provie a unifying framework for analysis of the microgri moel (1) for ifferent types of voltage controllers, an stuy frequency regulation, voltage stability, an active as well as reactive power sharing. A key point of our approach is that it oes not rely on simplifying an often restrictive premises such as the ecoupling assumption an linear approximations. Active an reactive power. he active power i is given by i = j N i B ij V i V j sinθ ij, θ ij := θ i θ j (2) an the reactive power by i = B ii Vi 2 j N i B ij V i V j cosθ ij, θ ij := θ i θ j. (3) Note that here B ii = ˆB ii + j N i B ij, where B ij = B ji > 0 is the negative of the susceptance at ege {i,j} an ˆB ii 0 is the negative of the shunt susceptance at noe i. 1 Hence, B ii j N i B ij for all i. It is useful to have compact representations of both active an reactive power. Setting Γ(V) = iag(γ 1 (V),...,γ m (V)), γ k (V) = V i V j B ij, with k E := {1,2,...,m} being the inex corresponing to the ege {i,j} (in short, k {i,j}), the vector of the active power at all the noes writes as = DΓ(V)sin(D θ). where D = [ ik ] is the incience matrix of the graph escribing the interconnection structure of the network, an the vector sin( ) is efine element-wise. Let us now introuce the vector A 0 = col(b 11,...,B nn ). Since ik cos( ik θ i + jk θ j ) = cos(θ i θ j ), for k {i,j}, the vector of reactive power at the noes takes the form = [V][A 0 ]V D Γ(V)cos(D θ), where D is obtaine by replacing each element ij of D with ij. 2 Moreover, here an throughout the paper, the notation [v] represents the iagonal matrix associate with vector v. Another compact representation is useful as well. o this en, introuce the symmetric matrix A(cos(D θ)) = B 11 B 12 cosθ B 1n cosθ 1n B 21 cosθ 21 B B 2n cosθ 2n B n1 cosθ n1 B n2 cosθ n2... B nn 1 See Remark 2 for a iscussion on the physical meaning of these shunt susceptances. 2 In fact, enote by η the vector D θ, the entry ij of the matrix D Γ(V)cos(D θ) writes as m [ D Γ(V)cos(D θ)] ij = ik γ k (V)cos(η k ) k=1 = ik V i V j B ij cos( ik θ i + jk θ j ) = k {i,j} j N i V i V j B ij cos(θ i θ j )

3 3 he vector becomes = [V]A(cos(D θ))v, (4) where again we are exploiting the ientity cos( ik θ i + jk θ j ) = cosθ ij. As a consequence of the conition B ii j N i B ij for all i, provie that at least one ˆB ii is non-zero (which is the staning assumption throughout the paper), the symmetric matrix A(cos(D θ)) has all strictly positive eigenvalues an hence is a positive efinite matrix. Note that the matrix A can be interprete as a loopy Laplacian matrix of the graph. Before proceeing further, we remark on the aopte moel. Remark 1. (Lossless an lossy network) he power lines are assume to be lossless in (1). his is vali if the lines are ominantly inuctive, a conition which can be fulfille by tuning output impeances of the inverters; see e.g. [31]. As will be observe in Subsection V-B, the lossless assumption can be relaxe by consiering lossy, yet homogenous, power lines. Remark 2. (Loas) here are a few loa scenarios that can be incorporate in the microgri moel (1). he first scenario accounts for purely inuctive loas, see [42, Remark 1]. Whether these loas are collocate with inverters or appear as iniviual noes, they will lea to nonzero shunt amittances at the noes of the reuce network, where the latter follows from Kron reuction. he resulting shunt amittances constitute the nonzero shunt susceptance ˆB ii introuce after (3), see also [47, Section V.A] an [42]. As for the active power loas, following [41, Remark 3.2], one can consier negative active power setpointsi for the inverteri, which correspons to the inverter i connecting a storage evice to the gri, in which case the evice is acting as a frequency an voltage epenent loa (see also [35, Section 2.4]). Another possibility is to consier constant active power loas collocate with the inverters by embeing the constant active power consumption in the term. We remark that the controllers stuie in the paper o not rely on the knowlege of i, an are therefore fully compatible with the case in which i are not completely known ue to uncertainties in the loas. Finally, the extension of our analysis to the lossy lines in Subsection V-B allows us to accommoate loas as homogenous RL circuits. As an interesting special case of this, the forthcoming issipativity/stability analysis carries over to the case of microgris with (purely) resistive lines an loas. More etails on this case are provie in Subsection V-B. i o pursue our analysis, we emonstrate an incremental cyclo-issipativity property of the various microgri moels, with respect to a synchronous solution. he notion of issipativity aopte in this paper is introuce next, an synchronous solutions will be ientifie afterwars. Definition 1. System ẋ = f(x,u),y = h(x), x X, X the state space, y,u R m, is incrementally cyclo-issipative with state-epenent supply rate s(x,u,y) an with respect to a given input-state-output triple (u, x, y), if there exist a continuously ifferentiable function S : X R, an state-epenent positive semi-efinite 3 matrices W,R : X R m m, such that for all x X, u R m an y = h(x), y = h(x) 4 with f(x,u)+ f(x,u) s(x,uu,y y) x x s(x,u,y) = y W(x)y +y R(x)u. (5) We remark that at this point the functions is not require to be non-negative nor boune from below an that the weight matrices W,R are allowe to be state epenent. he use of the qualifier cyclo in the efinition above stresses the former feature [53, Def. 2]. Remark 3. In case the matrices W an R are state inepenent, some notable special cases of Definition 1 are obtaine as follows: i) W 0, R = I, S 0 (incremental passivity) ii) W > 0, R = I, S 0 (output-strict incremental passivity) iii) W 0, R = I (cyclo-incremental passivity) iv) W > 0, R = I (output-strict cyclo-incremental passivity). Synchronous solution. Given the constant vectors u an u, a synchronous solution to (1) is efine as the triple (θ(t),ω(t),v(t)) = (θ,ω,v), where θ = ωt+θ 0, ω = 1ω 0, the scalar ω 0 an the vectors θ 0 an V R n >0 are constant. In aition, where 0 = (ω ω )K ( )+u 0 = f(v,,u ), = DΓ(V)sin(D θ) = DΓ(V)sin(D θ 0 ), = [V]A(cos(D θ))v = [V]A(cos(D θ 0 ))V. (7) Notice that the key feature of a synchronous solution is that the voltage phase angles are rotating with the same frequency, namely ω 0, an the ifferences of these angles are thus constant. Another feature is that the voltage amplitues are constant. III. DESIGN OF BREGMAN SORAGE FUNCIONS A crucial step for the Lyapunov base analysis of the couple nonlinear moel (1) is constructing a storage function. o this en, we start off with the following classical energybase function, e.g. [38] U(θ,ω,V) = 1 2 ω K 1 ω (6) = 1 2 ω K 1 ω V A(cos(D θ))v, (8) 3 A state-epenent matrix M : X R m m is positive semi-efinite if y M(x)y 0 for all x X an for all y R m. If M is positive semiefinite an y M(x)y = 0 y = 0 then M is calle positive efinite. 4 We are slightly abusing the classical notion of incremental issipativity [18], for we o not consier pairs of arbitrary input-state-output triples, but pairs in which one of the two triples is fixe. For aitional work on incremental issipativity we refer the reaer to [48], [49].

4 4 where we have exploite (4) to write the secon equality. Notice that the first term represents the kinetic energy (in quotes because the term has the units of power an it oes not correspon to the physical inertia), an the secon one the sum of the reactive power store in the links an the power partly associate with the shunt components. Next, we compute the graient of the storage function as follows: U = = DΓ(V)sin(D θ), θ U = [V] 1 = [A 0 ]V [V] 1 D Γ(V)cos(D θ). In the equality above, we are implicitly assuming that each component of the voltage vector never crosses zero. In fact, we shall assume the following: Assumption 1. here exists a subset X of the state space n R n R n >0 that is forwar invariant along the solutions to (1). Conitions uner which this assumption is fulfille will be provie later in the paper. Notice that the voltage ynamics ientifie by f has not yet been taken into account in the function U. herefore, to cope with ifferent voltage ynamics (or controllers) we a another component, namely H(V), an efine S(θ,ω,V) = U(θ,ω,V)+H(V). (9) We rest our analysis on the following founational incremental storage function S(θ,ω,V) = S(θ,ω,V)S(θ,ω,V) θ (θ θ) ω (ω ω) (V V) (10) where we use the conventional notation F x = F x (x), F x = ( F x (x)) for a function F : X R. he storage function S, in fact, efines a istance between the value of S at point (θ,ω,v) an the value of a first-orer aylor expansion of S aroun (θ,ω,v). his construction is referre to as Bregman istance or Bregman ivergence following [8], an has foun its applications in convex programming, clustering, proximal minimization, online learning, an proportional-integral stabilisation of nonlinear circuits; see e.g. [8], [3], [12], [55], [28]. In thermoynamics, the Bregman istance has its anteceents in the notion of availability function [30], [1], [56]. he function S can be ecompose as S = U +H (11) where U(θ,ω,V) = U(θ,ω,V)U(θ,ω,V) U θ (θ θ) U ω (ω ω) U (V V) an H(V) = H(V)H(V) H (V V). he above ientities show that the critical points of S occur for ω = ω an = which is a esire property. he critical point of S with respect to the V coorinate is etermine by the choice of H which epens on the voltage ynamics. o establish a suitable incremental issipativity property of the system with respect to a synchronous solution, we introuce the output variables with y = 1 an input variables y = col(y, y ) (12) ω = K1 ω, y = 1, u = col(u, u ). (13) In what follows, we ifferentiate among ifferent voltage controllers an ajust the analysis accoringly by tuning H. A. Conventional roop controller he conventional roop controllers are obtaine by setting f in (1) as f(v,,u ) = V K +u (14) where K = [k ] is a iagonal matrix with positive roop coefficients on its iagonal. Note that u is ae for the sake of generality an one can set u = u = K +V for nominal constant vectors V an to obtain the well known expression of conventional roop controllers, see e.g. [13], [60]. For this choice of f, we pick the function H in (9) as ([41], [50]) H(V) = 1 K V (+K 1 V) ln(v), (15) with +K 1 V = K1 u R n >0 an ln(v) efine element-wise. his term has two interesting features. First, it makes the incremental storage function S raially unboune with respect to V on the positive orthant. Moreover, it shifts the critical points of S as esire. Noting that by (6) 0 = V K +u, straightforwar calculations yiels V = K [V] +u u. (16) In the following subsections we will erive analogous ientities an then use those for concluing incremental cycloissipativity of the system.

5 5 B. uaratic roop controller Another voltage ynamics propose in the literature is associate with the quaratic roop controllers of [46], which can be expresse as (1) with f(v,,u ) = K [V](V u ), (17) where again K = [k ] collects the roop coefficients. he quaratic roop controllers in [46] is obtaine by setting u = V for some constant vector V. Notice however the ifference: while [46] focuses on a network preserve microgri moel in which the equation above moels the inverter ynamics an are ecouple from the frequency ynamics, here a fully couple network reuce moel is consiere. Moreover, note that the scaling matrix [V] istinguishes this case from the conventional roop controller. For this case, we aapt the storage function S by setting H(V) = 1 2 V K 1 V. (18) Recall that S = U +H. Note that S is efine on the whole n R n R n an not on n R n R n >0. he resulting function S can be interprete as a performance criterion in a similar vein as the cost function in [46]. Noting that it is easy to verify that 0 = K [V](V u ), V = K [V] +[V](u u ). (19) C. Reactive current controller he frequency ynamics of the inverters in microgris typically mimics that of the synchronous generators known as the swing equation. his facilitates the interface of inverters an generators in the gri. o enhance such interface, an iea is to mimic the voltage ynamics of the synchronous generators as well. Motivate by this, we consier the voltage ynamics ientifie by f(v,,u ) = [V] 1 +u. (20) his controller aims at regulating the ratio of reactive power over voltage amplitues, which can be interprete as reactive current ([32]). For this controller, we set H = 0 (21) meaning that S = U an no aaption of the storage function is neee. It is easy to observe that V = +u ū, (22) where ū = [V] 1 is again the feeforwar input guaranteeing the preservation of the steay state. D. Exponentially-scale averaging reactive power controller In this subsection, we consier another controller which aims at achieving proportional reactive power sharing f(v,,u ) = [V]K L K +[V]u (23) wherek = [k ] is a iagonal matrix anl is the Laplacian matrix of a communication graph which is assume to be unirecte an connecte. Compare with the controller in [42], here the the voltage ynamics is scale by the voltages at the inverters, namely [V], the reactive power is not assume to be inepenent of the phase variables θ, an an aitional input u is introuce. It is easy to see that the voltage ynamics in this case can be equivalently rewritten as χ = K L K +u V = exp(χ) (24) where can be expresse in terms of χ as [exp(χ)]a(cos(d θ))exp(χ) with exp(χ) = col(e χi ). Hence, we refer to this controller as an exponentially-scale averaging reactive power controller (E-AR). Now, we choose H as H(V) = lnv, (25) with as in (7), an obtain = [V]1 (). (26) Note that, in fact, our treatment here together with the above equality hints at the inclusion of the matrix [V] into the controller, or equivalently at an exponential scaling of the reactive power averaging ynamics (see (23), (24)). his, as will be observe, results in reactive power sharing for the fully couple nonlinear moel (1). By efining the voltage ynamics can be rewritten as u = K L K, (27) V = [V]K L K [V] +[V](u u ). (28) where we have set = I. Having unitary time constants is assume for the sake of simplicity an coul be relaxe. On the other han, requiring them to be the same is a purely technical assumption, motivate by the ifficulty of analysing the system without such conition. E. Incremental issipativity of microgri moels In this subsection, we show how the caniate Bregman storage functions introuce before allow us to infer incremental issipativity of the microgris uner the various controllers. heorem 1. Assume that the feasibility conition (6) amits a solution an let Assumption 1 hol. hen system (1) with output (12), input (13), an, respectively, 1) f(v,,u ) given by (14); 2) f(v,,u ) given by (17); 3) f(v,,u ) given by (20);

6 6 4) f(v,,u ) given by (23); is incrementally cyclo-issipative with respect to a synchronous solution (θ,ω,v), with 1) incremental storage function S efine by (8),(9),(10),(15) an supply rate (5) with weight matrices W(V) = ï ò K 0, R = 0 K [V] ï ò I 0 ; 0 I 2) incremental storage function S efine by (8),(9),(10),(18) an supply rate (5) with weight matrices ï ò K 0 W(V) =, 0 K [V] ï ò I 0 R(V) = ; 0 [V] 3) incremental storage function S efine by (8),(9),(10),(21) an supply rate (5) with weight matrices W = ï ò K 0, R = 0 ï ò I 0 ; 0 I 4) incremental storage function S efine by (8),(9),(10),(25) an supply rate (5) with weight matrices ï ò K 0 W(V) =, 0 [V]K L K [V] ï ò I 0 R(V) =. 0 [V] roof: 1) Recall that ω = K1 (ω ω), θ = DΓ(V)sin(D θ)dγ(v)sin(d θ 0 ) = ( ). (29) hen t S = (ω ω) K 1 ω +(DΓ(V)sin(D (θ)) DΓ(V)sin(D θ 0 )) θ +( ) V = (ω ω) K 1 ((ω ω)k ( )+(u u )) +(DΓ(V)sin(D θ)dγ(v)sin(d θ 0 )) (ω ω) +( ) 1 (K [V] +u ū ) = (ω ω) K 1 ((ω ω)k ( )+(u u )) +( ) (ω ω)( ) 1 K [V] +( ) 1 (u ū ) where the chain of equalities hol because of the feasibility conition an (16). Hence t S = (ω ω) K 1 (ω ω)+(ω ω) K 1 (u u ) ( ) 1 K [V] +( ) 1 (u ū ). (30) Observe now that by efinition = an that represents the output component at a synchronous solution. Hence equality (31) at the top of the next page can be establishe. We conclue incremental cyclo-issipativity of system (1), (12), (13), (14) as claime. 2) If in the chain of equalities efining S above, we use t (19) instea of (16), we obtain that t S = (ω ω) K 1 (ω ω)+(ω ω) K 1 (u u ) ( ) 1 K [V] +( ) 1 [V](u u ) (32) which shows incremental cyclo-issipativity of system (1), (12), (13), (17). 3) For this case, aopting the equality (22) results in the equality t S = (ω ω) K 1 (ω ω)+(ω ω) K 1 (u u ) ( ) (u u ), (33) from which incremental cyclo-issipativity of (1), (12), (13), (20) hols. 4) Finally, in view of (28), t S = (ω ω) K 1 (ω ω)+(ω ω) K 1 (u u ) ( ) [V]K L K [V] +( ) [V](u u ) (34) which implies incremental cyclo-issipativity of (1), (12), (13), (23). IV. FROM CYCLO-DISSIAIVIY O DISSIAIVIY he issipation inequalities proven before can be exploite to stuy the stability of a synchronous solution. Recall that heorem 1 has been establishe in terms of cyclo-issipativity rather than issipativity, i.e. without imposing lower bouneness of the storage function S. However, in orer to conclue the attractivity of a synchronous solution we ask for incremental issipativity of the system, an require the storage function to possess a strict minimum at the point of interest. o this en, we investigate conitions uner which the Hessian of the storage function S is positive efinite at the point of interest ientifie by a synchronous solution. It is not ifficult to observe that ue to the rotational invariance of θ variable, the existence of a strict minimum for S cannot be anticipate. o clear this obstacle, we notice

7 t S = î (ω ω) K 1 + î (ω ω) K 1 ( ( ) 1 ) 1 ó ï òñ K 0 0 K [V] ó ï òï ò I 0 u ū. 0 I u ū K 1 1 ( ô (ω ω) ) 7 (31) that the phase angles θ appear as relative terms, i.e. D θ, in (8) an thus in S as well as S. Motivate by this observation, we introuce the new variables [57] ϕ i = θ i θ n, i = 1,2,...,n1. (35) hese can be also written as ϕ 1 θ 1. ϕ n1 =. θ n1 1θ n. 0 θ n Let us partition D accoringly as D = col(d 1,D 2 ), with D 1 an (n1) m matrix an D 2 a 1 m matrix. Notice that D 1 is the reuce incience matrix corresponing to the noe of inex n taken as reference. hen D an therefore ϕ 1. ϕ n1 0 = D 1 ϕ, with ϕ := D θ = D 1 ϕ. ϕ 1. ϕ n1 More explicitly, given θ R n, we can efine ϕ R n1 from θ as in (35), an the equality D θ = D 1 ϕ hols. Hence, U(θ,ω,V) = 1 2 ω K 1 ω V A(cos(D θ))v = 1 2 ω K 1 ω V A(cos(D 1 ϕ))v an we set, by an abuse of notation, U(ϕ,ω,V) := 1 2 ω K 1 ω V A(cos(D 1 ϕ))v. Furthermore, we can efine U(ϕ,ω,V) = U(ϕ,ω,V)U(ϕ,ω,V) U ϕ (ϕϕ) U ω (ω ω) U (V V) (36) where, ϕ i := θ i θ n, i = 1,2,...,n 1, (hence D θ = D1 ϕ), an to have S(ϕ,ω,V) = U(ϕ,ω,V)+H(V) (37) S(θ,ω,V) = S(ϕ,ω,V). (38) A. Strict convexity of Bregman storage functions Observe that (ϕ,ω,v) is a critical point of S. Next, we compute the Hessian as 2 S (ϕ,ω,v) 2 = D 1 Γ(V)[cos(D1 ϕ)]d K 1 0. D 1 [V] 1 D Γ(V)[sin(D1 ϕ)] 0 A(cos(D 1 ϕ))+ 2 H 2 (39) Notice that in all the previously stuie cases, the matrix 2 H 2 is iagonal. In particular, 2 H 2 = K +[V] 2 [+K 1 V], 2 H 2 = K1 2 H 2 = 0,, 2 H 2 = [V]2 [], (40) for conventional roop, quaratic roop, reactive current controller, an exponentially-scale averaging reactive power controller, respectively. Now, let [h(v)] := 2 H 2, (41) an h(v) = col(h i (V i )). hen, the following result, which establishes istribute conitions for checking the positive efiniteness of the Hessian, an hence strict convexity of the Bregman storage function, can be proven: roposition 1. Let η := D θ 0 = D1 ϕ (π 2, π 2 )m, V R n >0, an m ii := ˆB ii + Ç B il 1 V å l sin 2 (η k ) +h i (V i ). V i cos(η k ) hen the inequality hols if k {i,l} E m ii > for all i = 1,2,...,n. 2 S (ϕ,ω,v) 2 > 0 (42) k {i,l} E B il sec(η k ) (43) roof: he proof is given in the appenix. Remark 4. (Isolate minima) he result shows that the conition (43) for positive efiniteness are met provie that at the point (ϕ,ω,v) the relative voltage phase angles are small enough an the voltages magnitues are approximately the same. his is a remarkable property, stating that if the equilibria of interest are characterize by small relative voltage

8 8 phases an similar voltage magnitues, then they are minima of the incremental storage function S(θ,ω,V), an equivalently isolate minima of S(ϕ,ω,V). Remark 5. (Hessian) he Hessian of energy functions has always playe an important role in stability stuies of power networks (see e.g. [52], an [41] for a microgri stability investigation). Conitions for assessing the positive efiniteness of the Hessian of an energy function associate to power networks have been reporte in the literature since [52], an use even recently to stuy e.g. the convexity of the energy function ([24]). Our conitions however are ifferent an hol for more general energy functions. B. An instability conition Conversely, one can characterize an instability conition that shows how, for a given vector of voltage values, equilibria with large relative phase angles are unstable. o this en, first observe that a negative eigenvalue of the Hessian matrix implies instability of the equilibrium (ϕ,ω,v) of system (1), with f(v,,u ) given by (14), (17), (20), expresse in the ϕ coorinates an with u = u, u = u : Lemma 1. Suppose that the Hessian 2 S (ϕ,ω,v) 2 has a negative eigenvalue. hen the equilibrium (ϕ,ω,v) is unstable. roof: he proof is given in the appenix. Before proviing sufficient conitions uner which the Hessian in Lemma 1 has a negative eigenvalue, we first provie conitions uner which the matrix at the center of the prouct in (44), enote as M when evaluate at (ϕ,ω,v), has a negative eigenvalue. M is symmetric an as such iagonalizable. Using the iagonal form, it is immeiate to notice that if there exists a vector v = (v (1),v (2) ) 0 such that v Mv < 0, then the matrix M has a negative eigenvalue. A characterization of the conition v Mv < 0, or equivalently the existence of a negative eigenvalue of the matrix M, is now stuie. o this en, it is instrumental to introuce a class of cut-sets of the graph, as in the following efinition: Definition 2. A cut-set K E is sai to have non-incient eges if for each k {i,j} K an k {i,j } K, with k k, all the inices i,j,i,j are ifferent from each other. he class of cut-sets with non-incient eges is enote by K. In wors, the property amounts to the following: given any two eges in the cut-set, the two pairs of en-points associate with the two eges are ifferent from each other. he set of graphs for which these cuts exists is not empty an inclues trees, rings an lattices. Complete graphs o not amit this class of cuts. he following hols: Lemma 2. Let V R n >0. If there exists a cut-set K in the class K such that, for all k {i,j} K, sin(η k ) 2 > β k (V i,v j )cos(η k ), (45) where η = D 1 ϕ an β k (V i,v j ) = 2max{ (Bii+hi(V i))v i (Bjj+hj(V j))v j, }, B ijv j B ijv i an h i is efine in (40), (41), then the matrix M at the center of the prouct in (44) evaluate at ϕ,v has a negative eigenvalue. roof: he proof is given in the appenix. he two lemma above lea to the following conclusion: roposition 2. An equilibrium (ϕ,ω,v), with V R n >0, is unstable if there exists a cut-set K in the class K such that the inequality (45) hols for all k {i,j} K. roof: he proof is given in the appenix. From the relation above, we see that for equilibria for which the components of V have comparable values, inequality (45) is likely to be fulfille as η k iverges from 0, thus showing that equilibria with large relative phase angles are likely to be unstable. Remark 6. (lastic coupling strength) It is interesting to establish a connection with existing stuies on oscillator synchronization arising in ifferent contexts. Once again, this connection leverages the use of the energy function. If the coupling between any pair of noes i,j is represente by a single variable v ij, moeling e.g. a ynamic coupling, instea of the prouct of the voltage variables V i V j, then a ifferent moel arises. o obtain this, we focus for the sake of simplicity on oscillators without inertia, an replace the previous energy function (8) with U(θ,v) = 1 n v ij B ij cos(θ j θ i )+ 1 vij i=1 j N i hen U v ij = B ij cos(θ j θ i )+v ij, an the resulting (graient) system becomes {i,j} E θ i = j N i v ij B ij sin(θ j θ i ), i = 1,2,...,n v ij = B ij cos(θ j θ i )v ij, {i,j} E, which arises in oscillator networks with so-calle plastic coupling strength ([39], [27], [33]) an in the context of flocking with state epenent sensing ([39], [23], [44]). Although stability analysis of equilibria have been carrie out for these systems, the investigation of the methos propose in this paper in those contexts is still unexplore an eserves attention. V. FREUENCY CONROL WIH OWER SHARING In this section, we establish the attractivity of a synchronous solution, which amounts to the frequency regulation (ω = ω ) with optimal properties. Moreover, we investigate voltage stability an reactive power sharing in the aforementione voltage controllers. Recall from (6) that for a synchronous solution we have 0 = K (DΓ(V)sin(D θ 0 ) )+u. (46)

9 = ï ò D1 0 0 I D 1 Γ(V)[cos(D1 ϕ)]d 1 [sin(d1 ϕ)]γ(v) D [V] 1 D1 D 1 [V] 1 D Γ(V)[sin(D1 ϕ)] A(cos(D1 ϕ))+ 2 H 2 Γ(V)[cos(D1 ϕ)] [sin(d1 ϕ)]γ(v) D [V] 1 ï ò D [V] 1 D Γ(V)[sin(D1 ϕ)] A(cos(D1 ϕ))+ 2 H 1 0 > 0. 0 I 2 9 (44) Among all possible vectors u satisfying the above, we look for the one that minimizes the quaratic cost function C(u ) = 1 2 u K 1 u. (47) his choice is explicitly compute as [2], [22], [51] 1 u = 1 1 K 1 1. (48) Note that in (47) any positive iagonal matrix, say Σ, coul be use instea of K 1. However, the choice Σ = K1 yiels more compact expressions, an results in proportional sharing of the active power accoring the roop coefficients k,i, see Subsection V-A. Replacing u in (6) with its expression in (48), an replacing with its explicit efinition via the loopy Laplacian, the feasibility conition (6) can be restate as follows: Assumption 2. here exist constant vectors V R n an θ 0 n such that an DΓ(V)sin(D θ 0 ) = (I K K 1 (49) 1) 0 = f(v,[v]a(cos(d θ 0 ))V,u ). (50) Remark 7. Similar to [51, Remark 5] it can be shown that if the assumption above is satisfie then necessarily V R n >0. Furthermore, in case the network is a tree, it is easy to observe that (49) is satisfie if an only if there exists V R n >0 such that Γ(V) 1 D (I K K 1 < 1, 1) with D enoting the left inverse of D. In the case of the quaratic voltage roop an reactive current controllers, explicit expressions of the voltage vector V can be given (see Subsection V-A), in which case the conition above becomes epenent on the voltage phase vector θ 0 only. o achieve the optimal input (48), we consier the following istribute active power controller ([45], [22], [10]) u ξ = L ξ +K 1 (ω ω) = ξ (51) where the matrix L is the Laplacian matrix of an unirecte an connecte communication graph. Here, the term ω ω attempts to regulate the frequency to the nominal one whereas the consensus base algorithm L ξ steers the input to the optimal one given by (48) at steay-state. For the choice of the voltage/reactive power control u, we set u = u where u is a constant vector enforcing the setpoint for the voltage ynamics. he role of this setpoint will be mae clear in Subsection V-A. hen, the main result of this section is as follows: heorem 2. Suppose that the vectors θ 0 n an V R n are such that Assumption 2 an conition (42), with ω = ω, hol. Let u be given by (51), u = u R n, an u the optimal input (48). hen, the following statements hol: (i) he vector (D θ,ω,v,ξ) with (θ,ω,v,ξ) being a solution to (1), (51), with the conventional roop controller (14), quaratic roop controller (17), or reactive current controller (20), locally 5 converges to the point (D θ 0,ω,V,ξ). (ii) he vector (D θ,ω,v,ξ) with (θ,ω,v,ξ) being a solution to (1), (51), with the E-AR controller (23), locally converges to a point in the set {(D θ,ω,v,ξ) ω = ω,ξ = u, Moreover, for all t 0, =,L K = K 1 u } 1 K 1 ln(v(t)) =1 K 1 ln(v(0)). roof: First recall that ϕ = E θ, ϕ = E θ, an D 1 ϕ = D θ = D θ 0 with E = [I n1 1 n1 ] an noting that ED 1 = D. By the compatibility property of the inuce matrix norms, we have ϕ(0) ϕ E θ(0) θ(0), thus showing that a choice of θ(0) sufficiently close to θ 0 = θ(0), returns an initial conition ϕ(0) sufficiently close to ϕ. We then consier a solution (θ,ω,v,ξ) to the closeloop system an express the solution into the new coorinates as (ϕ, ω, V, ξ). Define the incremental storage function Notice that ξ im 1. hen C(ξ) = 1 2 (ξ ξ) (ξ ξ). (52) t C = (ξ ξ) L (ξ ξ)(ξ ξ) K 1 (ω ω) = (ξ ξ) L (ξ ξ)(u u ) K 1 (ω ω). 5 locally refers to the fact that the solutions are initlialize in a suitable neighborhoo of (θ,ω,v,ξ).

10 10 By (38), the time erivative of S(θ,ω,V) is equal to that of S(ϕ,ω,V), with ϕ obtaine from (35), namely (with (38) in min) t S(θ,ω,V) = S(ϕ,ω,V). (53) t Hence, from the proof of heorem 1 we infer that t S(ϕ,ω,V) = (ω ω) K 1 (ω ω)+(ω ω) K 1 (u u ) ( ) X(V) +( ) Y(V)(u ū ), (54) or [V]K L K [V] an,1 [V],[V] epening on the voltage controller aopte. Observe that, by setting u = u an bearing in min (54), the equalities (30), (32), (33), an (34) can be written in a unifie manner as where X(V) = 1 K [V], 1 Y(V) = 1 t S(ϕ,ω,V) = (ω ω) K 1 (ω ω) Å ã X(V) +(ω ω) K 1 (u u ) where X is a positive (semi)-efinite matrix efine above. Now taking S +C as the Lyapunov function, we have t S + t C = (ω ω) K 1 (ω ω) Å ã X(V) (55) (ξ ξ) L (ξ ξ). By local strict convexity of S + C (thanks to (42)), we can construct a forwar invariant compact level set Ω aroun (ϕ, ω, V, ξ) an apply LaSalle s invariance principle. Notice in particular that on this forwar invariant set V(t) R n >0 for all t 0. hen the solutions are guarantee to converge to the largest invariant set where ω = ω 0 = L (ξ ξ) (56) Å ã 0 = X(V) he first equality yiels = 0 on the invariant set. Recall ω that ξ im 1. Hence, on the invariant set, L ξ = 0 an thus ξ = γ1 for some γ R. Note that, by (51), γ has to be constant given the fact that ω = ω an L ξ = 0. Also note that u = K (DΓ(V)sin(D θ 0 ) ) on the invariant set. Multiplying both sies of the above equality by 1 K 1 yiels γ1 K 1 1 = 1. herefore, ξ = 11 1 K 1 1, an on the invariant set, u is equal to the optimal input u given by (48). his also means that C ξ = 0. Notice that any solution (ϕ, ω, V, ξ) on the invariant set satisfies 0 = 1 K E ϕ 2 K C ω +1 ξ. Hence, evaluating the ynamics above on the invariant set yiels ϕ = 0 noting that the matrix E has full column rank. Furthermore, by (30), (32), an (33), the matrix X(V) is positive efinite for the roop controller, quaratic roop controller, an the reactive current controller. Hence, the thir equality in (56) yiels = 0 on the invariant set. herefore, the partial erivatives of S + C vanish on the invariant set. Now, as the solution is evolving in a neighborhoo where there is only one isolate minimum(ϕ, ω, V, ξ) of S+C, then the invariant set only comprises such a minimum, an therefore convergence to the latter is guarantee. his verifies the first statement of the theorem noting that the convergence of ϕ to ϕ implies that of D θ to D θ 0 by continuity an the equality ED 1 = D. For the E-AR controller, we have X(V) = [V]K L K [V] as evient from (34). Hence, by (26) an the thir equality in (56), on the invariant set we obtain that L K = L K. (57) By (27) an (57), the vector satisfies on the invariant set L K = K 1 u. (58) Notice that, for the E-AR controller, we have so far shown that the solutions (ϕ,ω,v,ξ) converge to the set := {(ϕ,ω,v,ξ) Ω ω = ω,ξ = u, =,L K = K 1 u }. Next, we establish the convergence of trajectories to a point in. o this en, we take the forwar invariant set Ω sufficiently small such that 2 (S +C) (ϕ,ω,v,ξ) 2 > 0 (59) for every (ϕ,ω,v,ξ). Note that this is always possible by (42) an continuity. Observe that any solution (ϕ,ω,v,ξ) satisfies ϕ = E K 1 ω ω = 1 K E ϕ 2 K C ω +1 ξ V = 1 X(V) C ξ = L C ξ K 1 ω. It is easy to see that every point of is an equilibrium of the system above, an by (59) is Lyapunov stable. In fact, by (59), the incremental storage function S+C can be analogously efine with respect to any point in to establish Lyapunov stability by the inequality S + C 0. herefore, the positive limit set associate with any solution issuing from a point in Ω contains a Lyapunov stable equilibrium. It then follows by [29, roposition 4.7] 6 that this positive limit set is a singleton which proves the convergence to a point in. his proves the claim in the secon statement of the theorem given the relationship between θ an ϕ variables exploite before. 6 For the correcte version, see the errata an aena in

11 11 Finally, by (27), the E-AR controller can be written as Hence, we have V = [V]K L K (). t (1 K 1 lnv) = 1 K 1 [V]1 [V]K L K () = 0, as 1 L = 0, which proves that 1 K 1 ln(v)is a conserve quantity. Remark 8. (Stability uner feeforwar control) When the inputu is set to the optimal feeforwar inputu, rather than being generate by the feeback controller (51), the closeloop system takes the form ϕ = E K 1 ω ω = 1 K E ϕ 2 K ω V = 1 X(V). he same arguments as in the proof above then show that solutions to this close-loop system locally converge to the equilibrium point (ϕ,ω,v). Hence, the stability of this equilibrium is an intrinsic property of the close-loop system obtaine setting u = u, u = u. he aoption of the istribute integral controller (51) is require to overcome the lack of knowlege ofu, which epens on global parameters. A. ower sharing heorem 2 portrays the asymptotic behavior of the microgri moels iscusse in this paper, namely frequency regulation an voltage stability. In aition, optimal active power sharing for the couple nonlinear microgri moel (1) is achieve if the roop coefficients K are suitably chosen. In fact, substituting (48) into (46) yiels or, component-wise, = K K 1 1, i = i (k ) 1 1 i 1 K 1 1, where K = [k ]. In case roop coefficients are selecte proportionally ([45], [22], [2], [10], [51]), i.e. for all i,j, we conclue that (k ) i i = (k ) j j, (k ) i i = (k ) j j, which accounts for the esire proportional active power sharing base on the iagonal elements of K as expecte. Next, we take a closer look at other consequences an implications of heorem 2 for ifferent voltage ynamics. 1) Conventional roop controller: he vectors of voltages an reactive powers converge to V an satisfying K +V = u (60) which yiels (k ) i i +V i (k ) j j +V j = (u ) i (u ) j. (61) his results in partial voltage regulation an reactive power sharing for the roop controlle inverters. In fact, for small values of k, u regulates the voltages following (60). On the other han, if the elements of k are sufficiently large, reactive power is share accoring to the elements of u as given by (61). his tunable traeoff between voltage regulation an reactive power sharing is consistent with the finings of [47]. 2) uaratic roop controller: he vector of voltages an reactive power converge to V an with his implies that K [V] 1 +V = u. (k ) i i +V 2 i (k ) j j +V 2 j = (u ) i (u ) j which again results in a partial voltage regulation an reactive power sharing by an appropriate choice of k an u. Moreover, in this case, the voltage variables at steay-state are explicitly given by V = (I +K A(cos(D θ 0 ))) 1 u. 3) Reactive current controller: In this case, we have which results in i V i = j V j [V] 1 = u (u ) i (u ) j = ( V j V i ) ( i j ). he first equality provies the exact reactive current sharing, whereas the secon equality can be interprete as a mixe voltage an reactive power sharing conition. Moreover, the voltage variables at steay-state are given by V = A 1 (cos(d θ 0 ))u. 4) Exponentially-scale averaging reactive power controller: In this case, the exact reactive power sharing can be achieve as evient from the secon statement of heorem 2, withu = 0. In particular, by equality (58) with u = 0 we obtain that = αk 1 1 for some α R. Multiplying both sies of the above equality by 1 yiels α = 1 1 K 1 1. Clearly, α > 0, by efinition of an as the matrix A is positive efinite. herefore, as a consequence of heorem 2, the vector of reactive power converges to a constant vector R n >0 where (k ) i i = (k ) j j, (62)

12 12 which guarantees proportional reactive power sharing accoring to the elements of k as esire. Notice that the quantity 1 K 1 lnv is a conserve quantity in this case. Hence, the point of convergence for the voltage variables is primarily etermine by the initialization V(0). B. Lossy lines Uner appropriate conitions, the stability of the system ynamics uner the various controllers are preserve in the presence of lossy transmission lines that are homogeneous, namely whose impeences Z ij equal Z ij e 1φ, with φ [0, π 2 ]. Consistently, shunt components at the buses that are a series interconnection of a resistor an an inuctor whose impeance is ˆr ii + 1ˆx ii are consiere. Assuming homogeneity of the shunt elements, i.e. ˆr ii + 1ˆx ii = ˆr 2 ii + ˆx 2 ii e ˆx 1arctan ii ˆr ii = Z ii e 1arctanφ, where φ = arctan ˆxii ˆr ii for all i, routine erivations (see e.g. [60], [35]) show that the total active an reactive power i l,l i exchange by the inverter i in the lossy network is equal to ï ò ï ò l i = Φ(φ) (63) i where i l i ï ò sinφ cosφ Φ(φ) =, cosφ sinφ an i, i, have the same expressions as in (2) an (3). Hence, the matrix Φ(φ) will moify the expressions of the active an reactive power exploite previously, an thus the frequency an voltage ynamics of the inverters will be change accoringly, isrupting the convergence of the solutions. A natural way to counteract this moification is to exploit the inverse of Φ(φ) an use l sinφ l cosφ an l cosφ+ l sinφ, with l = col( l i ) an l = col( l i ), in (1) instea of an, respectively. In this way, the lossless expressions of i an i as in (2) an (3) will be recovere. Notice that, however, the implementation of these controllers requires the knowlege of the parameter φ which is assume to be available. An interesting special case is obtaine for φ = 0 meaning that the network is purely resistive. In that case, in (1) shoul be replace by l, an by l, which is consistent with the use of roop controllers in resistive networks (see e.g. [7, Sec. II.A]). As a result of the aaptation above, the same conclusions 7 as in heorem 2 hols for the lossy network with moifie inverter ynamics. Notice, however, that the actual active power l will no longer be optimally share in a lossy network with the conventional roop controller (14), quaratic roop controller (17), or the reactive current controller (20). Remarkably, in the case of the E-AR controller, one can aitionally prove that both active as well as reactive power sharing continues to hol. Because of its importance, the result is formalize below. roposition 3. Suppose that Assumption 2 with f(v,,u ) = [V]K L K an conition (42), 7 In these conitions, whenever relevant, the negative of the susceptances ˆB ii,b ij shoul be replace by Ẑii 1, Z ij 1. with ω = ω an ˆB ii,b ij replace by Ẑii 1, Z ij 1, respectively, hol. hen the vector (D θ,ω,v,ξ) with (θ,ω,v,ξ) a solution of θ = ω ω = (ω ω )K ( l sinφ l cosφ )+u V = [V]K L K ( l cosφ+ l sinφ) (64) an u given by (51), locally converges to a point in the set {(D θ,ω,v,ξ) ω = ω,ξ = u, =,L K = 0}. Moreover, 1 K 1 ln(v(t)) =1 K 1 ln(v(0)), for all t 0. Finally, l, l converge to constant vectors l, l that satisfy provie that (k ) i l i = (k ) j l j (k ) i l i = (k ) j l j, (65) (k ) i (k ) j = (k ) i (k ) j, i,j. (66) roof: As remarke above, the convergence of the solutions is an immeiate consequence of heorem 2. hus, we only focus on the power sharing property. By conition (66) an relation (63) at steay state, l i = i sinφ+ i cosφ = (k ) j (k ) i j sinφ+ (k ) j (k ) i j cosφ = (k ) j (k ) i l j. Similarly, for the reactive power l i = i cosφ+ i sinφ = (k ) j (k ) i j cosφ+ (k ) j (k ) i j sinφ = (k ) j (k ) i ( j cosφ+ j sinφ) = (k ) j (k ) i l j. C. Dynamic extension Another interesting feature is that thanks to the incremental passivity property the static controller u = u can be extene to a ynamic controller. By heorem 1 an keeping in min Definition 1 together with (12) an (13), the incremental input-output pair of the voltage ynamics appears in the time erivative of the storage function S as ( ) 1 R 2(u u ) (67) where R 2 is the lower iagonal block of R in heorem 1. Clearly this cross term is vanishe by applying the feeforwar

13 13 input u = u. But an alternative way to compensate for this term is to introuce the ynamic controller λ = R2 u = K λ λ (68) for some positive efinite matrix K λ. Notice that that the controller above is ecentralize for a iagonal matrix K λ. hen, enoting the steay state value of λ by λ, the incremental storage function C (λ) = 1 2 (λλ) K λ (λλ) satisfies Note that t C = (u u ) 1 = R 2. (69) herefore, (69) coincies with the negative of (67), an thus the same convergence analysis as before can be constructe base on the storage function S +C +C. Consequently, the result of heorem 2 extens to the case of ynamic voltage/reactive power controller (68). For illustration purposes, below we provie the exact expression of the controller above in case of the conventional roop controller: λ = [V] 1 K 1 (K ()+V V) u = K λ λ which by setting K λ = K reuces to u = [V] 1 (K ()+V V) Note that here the constant vectors V an are interprete as the setpoints of the ynamic controller. It is easy to see that this controller rejects any unknown constant isturbance entering the voltage ynamics (14). Other possible avantages of these ynamic controllers require further investigation, which is postpone to a future research. VI. CONCLUSIONS We have presente a systematic esign of incremental Lyapunov functions for the analysis an the esign of networkreuce moels of microgris. Our results encompass existing ones an lift restrictive conitions, thus proviing a powerful framework where microgri control problems can be naturally cast. he metho eals with the fully nonlinear moel of microgris an no linearization is carrie out. wo major extensions can be envisione. he first one is the investigation of similar techniques for network-preserve moels of microgris. Early results show that this is feasible an will be further expane in a follow-up publication. he secon one is how to use the obtaine incremental passivity property to interconnect the microgri with ynamic controllers an obtain a better unerstaning of voltage control. Examples of these controllers are iscusse in [47] but many others can be propose an investigate. A more general question is how the set-up we have propose can be extene to eal with other control problems that are formulate in the microgri literature. Furthermore, the propose controllers exchange information over a communication network an woul be interesting to assess the impact of the communication layer on the results. In that regar, the use of Lyapunov functions is instrumental in avancing such research, since powerful Lyapunov-base techniques for the esign of complex networke cyber-physical systems are alreay available (see e.g. [17]). AENDIX roof of roposition 1. For the sake of notational simplicity, in this proof we omit the bar from all V, ϕ. Clearly, the Hessian (39) is positive efinite if an only if (44) hols. he latter is true if an only the matrix M below Γ(V)[cos(D1 ϕ)] [sin(d 1 ϕ)]γ(v) D [V] 1 [V] 1 D Γ(V)[sin(D1 ϕ)] A(cos(D1 ϕ))+ 2 H 2 is positive efinite. In fact recall that the matrix in (44) can be written as the prouct ï ò ï ò D1 0 D M 1 0, 0 I 0 I an our claim escens from D 1 D1 being nonsingular, the latter holing for D 1 D1 is the principal submatrix of the Laplacian of a connecte graph. Furthermore, note that by assumptionγ(v)[cos(d1 ϕ)] is nonsingular. hen the Hessian is positive efinite, or equivalently (44) hols, if an only if Γ(V)[cos(D1 ϕ)] an Ψ(D 1 ϕ,v) := A(cos(D 1 ϕ))+[h(v)][v]1 D Γ(V) [sin(d 1 ϕ)]2 [cos(d 1 ϕ)]1 D [V] 1 > 0. Introuce the iagonal weight matrix, where η = D 1 ϕ, W(V,η) := Γ(V)[sin(η)] 2 [cos(η)] 1. For each k {i,j} E, its kth iagonal element is W k (V i,v j,η k ) := B ij V i V j sin 2 (η k ) cos(η k ). Furthermore, it can be verifie that [ D Γ(V)[sin(η)] 2 [cos(η) ] 1 D ] ij sin 2 (η k ) B il V i V l cos(η k ) = k {i,l} E B ij V i V j sin 2 (η k ) cos(η k ) if i = j if i j, from which [ [V] 1 D Γ(V)[sin(η)] 2 [cos(η) ] 1 D [V] 1 ] ij = On the other han, k {i,l} E B ij sin 2 (η k ) cos(η k ) V l sin 2 (η k ) B il V i cos(η k ) if i = j if i j. [A(cos(η))+[h(V)]] ij ˆB ii + B il +h i (V i ), if i = j = k {i,l} E B ij cos(η m ), if i j