Small-Signal Stability Analysis for Droop-Controlled Inverter-based Microgrids with Losses and Filtering
Abdullah Al Maruf, Mohammad Ostadijafari, Anamika Dubey, Sandip Roy
SSmall-Signal Stability Analysis for Droop-ControlledInverter-based Microgrids with Losses and Filtering
Abdullah Al Maruf
Washington State [email protected]
Mohammad Ostadijafari
Washington State [email protected]
Anamika Dubey
Washington State [email protected]
Sandip Roy
Washington State [email protected]
ABSTRACT
An islanded microgrid supplied by multiple distributed energy re-sources (DERs) often employs droop-control mechanisms for powersharing. Because microgrids do not include inertial elements, andlow pass filtering of noisy measurements introduces lags in control,droop-like controllers may pose significant stability concerns. Thispaper aims to understand the effects of droop-control on the small-signal stability and transient response of the microgrid. Towardsthis goal, we present a compendium of results on the small-signalstability of droop-controlled inverter-based microgrids with hetero-geneous loads, which distinguishes: (1) lossless vs. lossy networks;(2) droop mechanisms with and without filters, and (3) mesh vs.radial network topologies. Small-signal and transient characteris-tics are also studied using multiple simulation studies on IEEE testsystems.
CCS CONCEPTS • Networks → Network dynamics . KEYWORDS microgrid islanded operation, droop control with first-order lag,structure-preserving model, angle stability.
ACM Reference Format:
Abdullah Al Maruf, Mohammad Ostadijafari, Anamika Dubey, and SandipRoy. 2019. Small-Signal Stability Analysis for Droop-Controlled Inverter-based Microgrids with Losses and Filtering. In
Proceedings of the TenthACM International Conference on Future Energy Systems (e-Energy ’19), June25–28, 2019, Phoenix, AZ, USA.
ACM, New York, NY, USA, 12 pages. https://doi.org/10.1145/3307772.3328310
A microgrid is an interconnected low/medium-voltage power dis-tribution network primarily supplied by inverter-based distributedenergy resources (DERs) and can operate in both grid connected andislanded modes [2, 4]. To enable a stable and economical operation,
Permission to make digital or hard copies of all or part of this work for personal orclassroom use is granted without fee provided that copies are not made or distributedfor profit or commercial advantage and that copies bear this notice and the full citationon the first page. Copyrights for components of this work owned by others than ACMmust be honored. Abstracting with credit is permitted. To copy otherwise, or republish,to post on servers or to redistribute to lists, requires prior specific permission and/or afee. Request permissions from [email protected]. e-Energy ’19, June 25–28, 2019, Phoenix, AZ, USA © 2019 Association for Computing Machinery.ACM ISBN 978-1-4503-6671-7/19/06...$15.00https://doi.org/10.1145/3307772.3328310 a hierarchical control framework comprised of primary, secondaryand tertiary controllers operating at different time-scales is oftenemployed [11, 12]. In islanded-mode, the primary controller is re-sponsible for maintaining a stable grid operation and often utilizes adroop controller that mimics the inertial response of a synchronousgenerator for a highly inductive grid [6, 29]. As it is mentionedin [23], increasing the droop gains improves the power sharingwhile it adversely affects the overall system stability. Despite thisinherent trade-off between power sharing and stability, the abilityto distribute the total demand among DERs using local feedback sig-nals, without the need for communications, makes the droop-basedcontrollers an appropriate choice for providing a fast response [9].However, because microgrids do not contain inertial elements andcontrols involve fast, noisy measurements, droop-like controllersimplemented in DERs may pose significant stability concerns, andhence require a comprehensive stability analysis [28].Small-signal stability analyses of islanded microgrid dynamicsusing linearized models have been recently considered in the litera-ture [1, 21, 24–27]. The analyses that evaluate network-level prop-erties often draw on algebraic-graph-theory formalisms, whichprovide useful constructs for the dynamic analysis of intercon-nected systems. In particular, algebraic graph theory has been usedto understand the relationship between a droop-controlled micro-grid network’s structure and its dynamical behavior [1, 25, 26]. Forexample, [25], presents a model for lossless inductive microgridwith frequency-droop control similar to Kuramoto model of phase-coupled oscillators, and derives conditions for obtaining a stableand synchronous solution for the network. Building on [25], [1]proposes structure-preserving models for lossless frequency-droopcontrolled inverter-based microgrid and derives the necessary con-dition for frequency synchronization. The concept of the activepower flow graph is introduced in [26] where the authors prove thatsmall-signal stability is equivalent to the positive semi-definitenessof the resulting Laplacian matrix.The algebraic graph theory formalism developed in [1, 25, 26]provides a foundation for the modeling and stability analyses ofdroop-controlled microgrids. The models in the aforementionedliterature, however, do not represent the lag/sluggishness in droopcontrollers that is either inherently present, or deliberately intro-duced due to low-pass filters employed to decrease the measurementnoise in controller’s input variables. In [20], the authors recognizethat the delay introduces an inertial response that makes the fre-quency change at a slower rate when compared to the changes inactive power flow, however, a formal network-level analysis is not a r X i v : . [ ee ss . S Y ] J u l -Energy ’19, June 25–28, 2019, Phoenix, AZ, USA Maruf, Ostadijafari, Dubey and Roy undertaken. A model for internal inverter control together withoutput filter is developed in [17]. However, these studies do notdetermine the dependence of the dynamics on the network topol-ogy and the filter parameters introducing first-order lag. Also, theyassume a lossless highly inductive microgrid, and do not investi-gate the effects of time-lagged droop-control on the stability ofmicrogrid with lossy lines.Although several theoretical frameworks have been proposed tounderstand the angle stability of the droop-controller microgrids,there remain a critical gap with regard to theoretical stability guar-antees in the presence of filtering in droop control and/or powersystem losses due to resistive lines. Both filtering and resistive/lossylines are practical realities for a microgrid and cannot be neglectedin any stability argument. To address this critical gap in the existingliterature, in this paper, we present a compendium of results statinggraph-theoretic conditions to ensure angle stability of inverter-based microgrids in the small-signal sense with first-order laggeddroop controls and network losses. Specifically, we present theo-retical results on the effects of filtering and the network topologyon angle stability of droop-controlled microgrids. Towards thisgoal, we develop appropriate mathematical models for microgridwith low-pass-filtered droop control and thoroughly investigatethe effects of the filtering on the dynamic stability of both lossyand lossless microgrids when subjected to small disturbances. Weemphasize that the filter model used in this paper may representan actual deployed low-pass filter used to reduce the measurementnoise or may be a first-order approximation for a delay.A microgrid employs droop-control to modulate both the gener-ated real power and reactive power in terms of measured frequencyand voltages respectively. Because our focus here is on small-signalstability, it is reasonable for us to focus solely on the modulation ofthe real power (i.e. P − droop ). Specifically, just as in [1, 25–27], weassume the bus voltage magnitude to be constant but not necessar-ily identical, and in consequence we can ignore the Q − V dynamics.In analogy with the bulk grid, this assumption is often reasonablebecause there is a time-scale separation between the grid’s voltagedynamics and angle dynamics: in particular, the typical time-framefor the angle-stability analysis ranges from 0.1 to 10 seconds, whilefor voltage stability it is in the range of 10-20 seconds. Under theseassumptions, we can only evaluate the impacts of P − droop onthe small-signal stability of the microgrid. We acknowledge thatthere are circumstances where the voltage and angle dynamicsfor a distribution system may be entangled [19, 31]. Such caseswill require a full model that simultaneously considers the angleand voltage dynamics. However, the analysis becomes considerablymore complicated and the simpler case considered here is importantas a baseline even if there is some entanglement.The major contributions are detailed below:
1) Microgrid Dynamic Models.
A nonlinear differential-algebraicequation (DAE) model is developed for lossless and lossy microgrids,with an accurate representation of droop control with filteringand heterogeneous load types. A structure-preserving linearizeddifferential-equation model is also obtained to facilitate the small-signal stability analysis of the model.
2) Stability Analysis.
We provide several graph-theoretic con-ditions for lossless and lossy microgrids that guarantees stabilityregardless of lags in droop control. For lossless network, we present sufficient and necessary conditions of stability and discuss graph-theoretic implications. For lossy networks without filtering, wepresent sufficient conditions for small-signal stability using theextension of the Sylvester’s inertia theorem for non-symmetricmatrices [5] and derive equivalent concepts for critical lines forlossy case. When filtering are included in lossy case, we present acondition on filtering time constant when otherwise stable homoge-neous micgrogrid becomes unstable. Finally, we discuss the specialcase of a radial lossy microgrid (as opposed to a mesh network),and verify that stability is maintained regardless of the filtering.
3) Transient Response.
The effects of filtering on the transientresponse of both lossless and lossy microgrid is shown using simula-tions carried on modified IEEE 9-bus and IEEE 57-bus test systems. Itis observed that the stable lossless microgrid with droop-controllershaving higher lags (due to higher filtering time constant) showshigher overshoot and settling time. An example simulation casestudy for lossy microgrid is also detailed that demonstrates thedestabilization of otherwise stable equilibrium point on increasingthe lag or filtering time constant. As a special case, for a radial lossymicrogrid, we demonstrate that the small-signal stable equilibriumpoint remains stable regardless of the filtering in droop controlsimilar to the lossless case.It should be noted that the motivation for this work is closelyrelated to communications aspects of the grid infrastructure. Specif-ically, a droop control in a microgrid requires filtering, and is subjectto time-lag, because the controller is implemented using measure-ment data which must be transmitted to the inverter-based controls;this contrasts with droop controls in the bulk grid, which are im-plemented directly in the prime mover of synchronous generators.This characteristic in the microgrid yields a model with intrinsiclags or filtering, and hence motivates the analysis considered inthe paper. From this perspective, our study is also aligned witha body of work at the interface of communications and controlsengineering, which is concerned with feedback controls that areimplemented over a communication channel (which may be subjectto lags, communication errors, etc).
A mathematical model for the transient and small-signal dynam-ics of a droop-controlled microgrid operating in islanded modeis developed in this section. Models for the dynamics of a droop-controlled microgrid have been developed in several recent studies[1, 24, 25]. The model presented here incorporates: 1) time-lagsand/or low-pass filtering that is present in real-power droop con-trols implemented at the inverters; and 2) heterogeneous busesincluding inverter based generator buses and frequency depen-dent/independent load buses and 3) losses in network lines. First,a nonlinear DAE model is developed. Then, two simplifications– approximation of the algebraic equations as dynamic ones viasingular perturbation, and linearization – are undertaken to enablesmall-signal stability analysis.
Islanded operation of inverter-based microgrids requires an ap-propriate mechanism for power-sharing among the DERs. Onepromising approach for power sharing is to regulate the power mall-Signal Stability Analysis: Droop-Controlled Microgrids e-Energy ’19, June 25–28, 2019, Phoenix, AZ, USA injection by each DER based on the local frequency measured atits connecting inverter, in a way that mimics droop control forsynchronous generators in the bulk grid. In an important recentwork, Song and co-workers have modeled the transient dynamicsof a microgrid network when such droop controls are used, andthen examined the small-signal stability of the network model via alinearization of the model [27]. The study [27] assumes an instanta-neous feedback of the frequency signal by the droop controller, inanalogy with the bulk grid. However, in the microgrid setting, incor-porating droop controls requires direct measurement of electricalfrequency signals using power electronics, which thus necessitatesuse of low-pass filters for noise reduction and/or incurs time-lag. Inthis study, following [26], we incorporate models of filtering/time-lag in droop controls into the microgrid-network model. We alsoexplicitly model frequency dependent heterogeneous load busesand line losses in the network.We present the microgrid network model in three steps: 1) themodel for the droop control (Section 2.1.1); 2) the network power-flow and load model (Section 2.1.2); and 3) the full DAE model whichcombines the network power-flow models with dynamic models ofthe DERs and loads (Section 2.1.3). It is important to note that themicrogrid network’s dynamics involve both active and reactive pro-cesses; active and reactive droop controls affect angle and voltagemagnitude, respectively [12]. However, as mentioned earlier, basedon the assumption that the angle dynamics are significantly fasterthan the voltage dynamics, we model only the real-power droopcontrol ( P − droop ) and dynamics. Hence, only angle and frequencydynamics are considered. These assumptions are standard in thesmall-signal analysis of the distribution grid [1, 25, 26]. The P − droop control mech-anism has an inherent delay, although it is typically small. In addi-tion, a low pass filter is incorporated to suppress the high frequencyvariations in the measured power (see Fig. 1). A few recent studieson small-signal stability of droop-controlled microgrids have repre-sented such delay and/or filtering [24]; however, these studies havenot fully incorporated the complexities of microgrids (e.g., losses,heterogeneity of loads). Here, we adopt a linear model for the droopcontrol, which allows evaluation of the microgrid dynamics in thepresence of these factors. Specifically, in our framework, the timedelay and any filter in the P − droop are abstracted as a first-orderlow pass filter. For an inverter at bus i , the power and frequencyrelationship is given in the Laplace domain as follows: P G i − P G i ( s ) D R i × + T D i s + ω = ω i ( s ) (1)where, ω is the nominal angular frequency; ω i ( s ) is the Laplacetransform of frequency ω i ; P G i is the reference power which isthe generated active power at the nominal frequency; P G i ( s ) is theLaplace transform of generated power P G i ; D R i ≥ T D i is the time constantof the low pass filter. Converting (1) to the time domain and setting ω = P G i = P G i − D R i (cid:219) θ i − D R i T D i (cid:220) θ i . (2)We notice that the droop control with filter involves a feedback ofthe angular acceleration, i.e. it mimics an inertial response. Figure 1: Block Diagram of P-Droop Control with Filtering.
A network model (power flow model) isdeveloped here which incorporates lossy lines. While stability anal-yses of the bulk grid often assume lossless transmission lines, mi-crogrids operating at medium or low voltage level require modelingof line losses.We consider a connected microgrid network with n buses, labeled i = , . . . , n . The admittance of the line between buses i and k obtained from the bus-admittance matrix is denoted by Y ik ∠ ϕ ik = G ik + jB ik , where G ik ≤ B ik ≥ i is given as the following [1]: P G i = P L i + n (cid:213) k = V i V k Y ij cos ( θ i − θ k − ϕ ik ) (3) = P L i + V i G ii + (cid:213) k ∈ adj ( i ) V i V k Y ik cos ( θ i − θ k − ϕ ik ) where, P G i and P L i are the active power generation and demand atbus i , respectively. adj ( i ) refers to the set of buses that are adjacentto bus i (or equivalently, the set of buses connected to bus i viaa distribution line). Note that the bus voltages are assumed to beconstant but not necessarily identical at all the buses.We use Bergen and Hill’s model for frequency-dependent systemloads [3]. The load at bus i can then be written as: P L i = P L i + D L i (cid:219) θ i (4)where, P L i and D L i ≥ i . In developing the DAE model for thedynamics, we distinguish frequency independent loads ( D L i = The DAE model forthe microgrid is detailed in this section. Let us first denote theset of buses that have inverters as V A . The governing differentialequation for the bus i ∈V A is derived using (2-4) and is given by: D R i T D i (cid:220) θ i = ( P G i − P L i − V i G ii )− (cid:0) D L i + D R i (cid:1) (cid:219) θ i − (cid:213) k ∈ adj ( i ) V i V k Y ik cos ( θ i − θ k − ϕ ik ) , i ∈V A (5)We next present the governing equations for the buses that do nothave inverter-based generators ( P G i = ) . It is convenient to subdi-vide these buses into sets with and without frequency-dependentloads. We denote the set of buses that do not have inverters buthave frequency dependent loads as V B . The governing equationfor each i ∈V B is given by:0 = (− P L i − V i G ii )− D L i (cid:219) θ i − (cid:213) k ∈ adj ( i ) V i V k Y ik cos ( θ i − θ k − ϕ ik ) , i ∈V B (6)Next, we denote the set of buses that have neither inverter-basedgeneration nor frequency dependent loads as V C . The governing -Energy ’19, June 25–28, 2019, Phoenix, AZ, USA Maruf, Ostadijafari, Dubey and Roy equation for each i ∈V C is given by:0 = (− P L i − V i G ii ) − (cid:213) k ∈ adj ( i ) V i V k Y ik cos ( θ i − θ k − ϕ ik ) , i ∈V C (7) We develop a linear differential-equation approximation of the non-linear DAE to enable formal analysis of small-signal stability. First,a singular perturbation argument is applied to obtain a differential-equation approximation of the DAE in a way that maintains thetopological structure of the network (hence, we refer to this asa structure-preserving model). Then, the nonlinear differential-equation model is linearized to obtain the small-signal model of themicrogrid dynamics.
Stability analysis of DAE modelsis usually undertaken in one of the two ways: 1) the algebraic equa-tions are solved (the so-called "Kron" reduction) and the stability ofthe resulting purely differential equation is determined; or 2) a sin-gular perturbation argument is applied to approximate the algebraicequations as fast dynamics yielding a differential-equation model,whereupon the stability can be assessed. The singular-perturbationapproach has the advantage of preserving the topological structureof the microgrid, which is useful for developing topological char-acterizations of stability and other properties. Here, we develop astructure-preserving model approximation for the microgrid DAEmodel using the singular-perturbation approach.Specifically, per a singular-perturbation approach, we approxi-mate (6) and (7) as: ϵ (cid:220) θ i = ( − P L i − V i G ii )− (cid:0) + D L i (cid:1) (cid:219) θ i − (cid:213) k ∈ adj ( i ) V i V k Y ik cos ( θ i − θ k − ϕ ik ) , i ∈V B (8) ϵ (cid:220) θ i = ( − P L i − V i G ii )− ϵ (cid:219) θ i − (cid:213) k ∈ adj ( i ) V i V k Y ik cos ( θ i − θ k − ϕ ik ) , i ∈V C (9)where, ϵ and ϵ are sufficiently small positive numbers.According to the singular perturbation theory, the system de-scribed by (5), (8), (9) and the system described by (5), (6), (7) havethe same equilibrium solutions. Additionally, provided that ϵ and ϵ are chosen appropriately (specifically, ϵ is sufficiently small and ϵ is on the order of ϵ ), the local stability of the equilibrium is alsomaintained. Hence, we use differential equation set (5), (8) and (9)for our analysis. Here we clarify that because the non-linear modelis a differential-algebraic equation, the constants ϵ and ϵ are ex-actly equal to zero in the correct linearization; the approximatedequations are purely algebraic in the context of the swing-dynamicsmodels. The use of the factors ϵ and ϵ is a construct which allowsdevelopment of the formal results in the paper: the proofs givecharacterizations in the limit that these factors are zero. For thepurpose of simulation, it is important that sufficiently small valuesfor ϵ and ϵ are used (as example see Section 4.1). The appropriatebounds can be derived from the singular-perturbation literature[16]. Another approach and the typical way of selecting these val-ues is by validating simulations against numerical solutions of thedifferential-algebraic equations. In close analogy with Song et al [26], we find it convenientto present a compact form of these equations based on a graph-theoretic description of the microgrid. To do so, let us consider adirected graph G defined for the microgrid, where each bus is repre-sented by a vertex, and the line between buses i and k correspondsto two directed edges ( i , k ) and ( k , i ) . Hence, the vertex set V hascardinality n and edge set E ⊆ V × V has cardinality 2 l , where l isthe number of lines in the microgrid. It is convenient to arbitrarilylabel and order the edges, with edge e m ( m = , . . . , l ) representingedge ( i , k ) of the graph. Then, the incidence matrix E ∈ R n × l of G is defined as E im = E km = − e m ∈ E , with all otherentries being zero. Furthermore, the orientation matrix is definedfor the directed graph (see [10]) as a matrix C ∈ R n × l with entries C im = E im = x = [ x , x , ..., x n ] T ∈ R n as x = [ x i ] ,and a diagonal matrix D ∈ R n × n as diaд ( D , D , · · · , D nn ) where D , D , · · · , D nn are the diagonal entries. Using these notations,we can write (5), (8), and (9) as a single vector equation, as follows: M (cid:220) θ = P − D (cid:219) θ − CB l cos ( E T θ − ϕ ) (10)where, θ = [ θ i ] ∈ R n is the vector of bus phase angles; M ∈ R n × n isa diagonal matrix imitating inertia with M ii = D R i T D i if i ∈V A , oth-erwise, M ii = ϵ ; P = [ P i ] ∈ R n is a vector representing effectiveinjected power in each bus with P i = P G i − P L i − V i G ii ; D ∈ R n × n is a diagonal matrix of total damping coefficients with D ii = ϵ if i ∈V C , otherwise, D ii = D L i + D R i ; ϕ = [ ϕ ik ] ∈ R l is a vec-tor of phase angles of the admittance of each directed lines and B l ∈ R l × l = diaд ( V i V k Y ik ) represents the magnitude of activepower flow through the edges. Note that in (10) the cosine function cos ( . ) is meant in the hadamard sense (i.e. component-wise).Equation (10) is in suitable form to develop the small-signalmodel that is used to characterize the stability from a structural orgraph-theoretic perspective. Linearization is used to obtain a small-signal differential-equation model for the the microgrid network,which allows analysis of stability. For this purpose, let us define thestate vector as x = (cid:16) θ (cid:219) θ (cid:17) . By linearizing (10) around the equilibriumpoint x = (cid:18) θ (cid:219) θ (cid:19) , we get ∆ (cid:219) x = (cid:18) ∆ (cid:219) θ ∆ (cid:220) θ (cid:19) = J ( x ) (cid:18) ∆ θ ∆ (cid:219) θ (cid:19) = J ( x ) ∆ x (11)where J (cid:0) x (cid:1) is the Jacobian matrix of the nonlinear model evalu-ated at the equilibrium. The Jacobian is found to be: J ( x ) = (cid:20) n × n I n × n − M − C W ( θ ) E T − M − D (cid:21) (12)where, W ( θ ) = ∂ B l cos ( E T θ − ϕ ) ∂ ( E T θ − ϕ ) = diaд (cid:0) − B l sin ( E T θ − ϕ ) (cid:1) ∈ R l × l . Here, n × n and I n × n denote n × n zero matrix and n × n unit matrix respectively. Note that as (10) depends on the relativeangles of the buses, therefore, x and x + k v refer to physicallysame equilibrium points of (10) where, v = (cid:20) n n (cid:21) , and n and n are vectors in R n with all entries equal to 1 and 0, respectively.Therefore, the linearization undertaken in (10) can be understood mall-Signal Stability Analysis: Droop-Controlled Microgrids e-Energy ’19, June 25–28, 2019, Phoenix, AZ, USA as a linearization over the manifold x + k v and thus, there ex-ists an invariant manifold where all the angles are synchronizedand all the frequency deviations are zero in the linearized system(11). Again in analogy with [26], the form of (12) motivates usto define W ( θ ) as the edge weights of the graph G . This defini-tion allows us to rewrite the dynamics in terms of the (directed)Laplacian matrix of the graph, which is helpful for the analysisof the stability. Like [26], we refer to the weighted directed graph G( x ) = (V , E , W ( θ )) as active power flow graph . We also notethat the edge weights in the graph G( x ) depend on the operatingpoint x . The Laplacian matrix for this directed weighted graphcan be written as L (G( x )) = C W ( θ ) E T . Therefore, from (11) and(12) we can write: ∆ (cid:219) x = J ( x ) ∆ x = (cid:20) n × n I n × n − M − L (G( x )) − M − D (cid:21) ∆ x (13)The developed small-signal model can be further simplified if weassume the microgrid network is lossless, or there is no filtering inthe droop control. For a lossless microgrid, the admittance of the linebetween buses i and k is given by Y ik ∠ ◦ = jB ik and therefore, theedge weights of both ( i , k ) and ( k , i ) edges become same and equalto V i V k B ik cos ( θ i − θ k ) in active power flow graph G . Thus, G canbe considered as an undirected graph where each line between bus i and k corresponds to an undirected edge ( i , k ) . For undirectedgraph, the Laplacian matrix is symmetric and can be written as L (G( x )) = E u W u ( θ ) E Tu where, E u ∈ R n × l is the incidence ma-trix and W u ( θ ) = diaд ( V i V k Y ik ) cos ( E Tu θ ) = B l , u cos ( E Tu θ ) ∈ R l × l is the weights of the edges in the undirected graph G . In thecase when there is no filtering in any of the droop control, the struc-ture preserving model (10) reduces to D (cid:219) θ = P − CB l cos ( E T θ − ϕ ) and small-signal model (13) reduces to the following: ∆ (cid:219) x = J ( x ) ∆ x = − D − L (G( x )) ∆ x (14)where, the state vector x = θ . Note that (14) is similar to the modeldeveloped in [26] but here L (G( x )) can be asymmetric also. Here, we analyze the small-signal stability of the microgrid, asdefined by the small-signal model (13). Our main aim is to under-stand the impact of the lag/ filtering time constant and the networktopology on stability. In our analysis, we distinguish stability char-acteristics for lossless vs. lossy networks and also for general vs.radial topologies.Before proceeding, first we review the notion of small-signalstability in our context, see the standard literature on power-systemanalysis [18]. To begin, we note that the vector v = (cid:20) n n (cid:21) is in thenull space of the Jacobian matrix J ( x ) . Therefore, as mentionedearlier, the manifold where all the angles are synchronized and allthe frequency deviations are zero is an invariant manifold. Theexistence of an invariant manifold, rather than a single invariantpoint, is apparent as the non-linear dynamics (10) depends on theangle differences and frequencies only. Therefore, the small-signalmodel (13) is said to be stable when the invariant manifold whereall the angles are synchronized and all the frequency deviationsare zero is asymptotically stable in the sense of Lyapunov [14].With this context, next we present our results on the conditions for stability of lossless microgrid, and then we expand the results forthe general lossy microgrid cases. There has been a considerable effort to characterize microgrid sta-bility for a lossless network model [1, 25, 26]. While we focus onthe lossy case here, in this subsection we review and develop thesmall-signal stability analysis of microgrids when lossless assump-tion is made, for the sake of completeness. Recall that for losslesscase the active power flow graph is undirected and the Laplacianmatrix is symmetric.We begin by reviewing the Laplacian-based necessary and suffi-cient condition for stability for the lossless microgrid when thereis no filtering/lag in the droop control. The result is similar to thatof Lemma 1 and the Theorem 1 of [26] and therefore presentedwithout proof.
Theorem 1:
For a lossless microgrid without lag (filtering) in theP-droop control, the small-signal model (14) is asymptotically stableif and only if the Laplacian L (G( x )) of the active power flow graphis positive semidefinite with a non-repeated eigenvalue at zero. Next, the stability of the microgrid is examined when the droopcontrol is subject to filtering. Note that the small-signal modeldescribed by (13) is similar to the structure-preserving classicalsmall-signal model developed for the bulk power grid with synchro-nous generators. The droop-controlled network with filtering hasbeen considered in [24] with the assumption of uniform damping,however the analysis does not give necessary and sufficient condi-tions, and also does not consider the singular perturbation modelwhich captures heterogeneous load. To develop a necessary andsufficient condition, we instead exploit a result that was developedin [27] for stability analysis of the bulk grid. Specifically, our resultuses Theorem 1 of [27], which is based on relating the number ofeigenvalues of the Jacobian having positive real part with the num-ber of eigenvalues of the Laplacian having negative real part usingthe Sylvester’s law of inertia [13] and the Proposition 4-2 of [7].Here, we express the small-signal model for the droop-controlledlossless microgrid network with filtering in an appropriate form,and then directly apply Theorem 1 of [27]. Here is the criterion:
Theorem 2:
Assume that the zero eigenvalue of the Laplacian matrix L (G( x )) of the active power-flow graph has algebraic multiplicity of (i.e., is non-repeated). The small-signal model of a lossless microgridas described by (13) is asymptotically stable for any filtering timeconstant if and only if the Laplacian matrix is positive semi-definite.If the condition is not satisfied, the small-signal model is not stablefor any filtering time constant. Proof:
We show that the state matrix of the linearized model (13) hasthe same form as that of the Jacobian matrix in [27] and, therefore,Theorem 1 of [27] establishes our result. Since the nonlinear modeldescribed by (10) and its small-signal model (13) depends on therelative angle of the bus voltages, without loss of generality, we takethe voltage angle at bus n as the reference and apply the followingtransformation, α = Tθ , where, α ∈ R n − is the vector of relativebus voltage angles with respect to bus n and T = [ I ( n − )×( n − ) -Energy ’19, June 25–28, 2019, Phoenix, AZ, USA Maruf, Ostadijafari, Dubey and Roy − n − ] ∈ R ( n − )× n . Let us define the new state vector as x n = (cid:16) α (cid:219) θ (cid:17) .The linearization around the equilibrium point x n = (cid:18) α (cid:219) θ (cid:19) yields ∆ (cid:219) x n = J ( x n ) ∆ x n . Using (13), where L (G( x )) = E u W u ( θ ) E Tu forthe undirected graph, we obtain J ( x n ) as: J ( x n ) = diaд ( T , I n × n ) J ( x ) diaд ( T † , I n × n ) = (cid:20) ( n − )×( n − ) T − M − T T E s W u ( θ ) E Ts − M − D (cid:21) (15)Here, we have used E u = T T E s , where E s ∈ R ( n − )× l consists ofthe first ( n − ) rows of E u and T † = T T ( TT T ) − , where T † is theright pseudo-inverse of T . Note that (15) is identical to the equation(7) of [27] with T L = , T G = T , D G = D . Therefore, the necessaryand sufficient condition of asymptotic stability given in Theorem 1of [27] implies that the manifold where all of the angles are synchro-nized and all the frequency deviations are zero is asymptoticallystable, if and only if, the Laplacian matrix L (G( x )) is positivesemi-definite. Note that from (13) of [27], it is clear that the as-sumption made on the nonsingularity of F ( α ) = E s W u ( θ ) E Ts in[27] is equivalent to the Laplacian L (G( x )) having a non-repeatedeigenvalue at zero. □ Theorem 2 demonstrates that filtering in the real-power droopcontrol does not alter the microgrid’s small-signal stability, pro-vided that the model is lossless. Algebraic and graph-theoreticconditions under which L (G( x )) satisfies the conditions – i.e., thematrix is positive semidefinite, and has a non-repeated eigenvalueat zero – can readily be determined. The Laplacian is guaranteedto be positive semi-definite and has a non-repeated eigenvalue atzero, if its off-diagonal entries are nonnegative and the matrix isirreducible (equivalently, its associated graph is connected and haspositive edge weights). For the Laplacian matrix in our formula-tion, the weight of the edge between vertices i and k is given by V i V k B ik cos ( θ i − θ k ) . This weight is positive if the difference be-tween the power-flow bus angle across the corresponding line isless than π /
2, or more generally | θ i − θ k | mod π < π /
2. If this con-dition is not satisfied by any line, the line is called critical [26, 27].As a special case, if the distribution grid is radial (i.e. the graph ofthe Laplacian is a tree) , there exists a solution to the power-flowequation for which no line is critical (see also the discussion for thelossy case in the next section). Hence this equilibrium (manifold) isasymptotically stable for any lag. These observations are summa-rized in the following corollaries to Theorem 2 (presented withoutproof, since they are automatic consequences of the graph-theoreticanalysis of the Laplacian).
Corollary 1:
Consider the active power flow graph G( x ) at the equi-librium of interest. If all the edge weights are positive or equivalently,there are no critical lines, then the small-signal model (13) of a losslessmicrogrid is asymptotically stable for any filtering time constant. Corollary 2:
Consider a microgrid whose active power flow graph isradial. The power flow has an equilibrium solution for which no line iscritical. For this equilibrium, the small-signal model is asymptoticallystable for any filtering time constant.Discussions:
Note that the stability criterion given in Theorem 2is independent of the droop control’s lag (time constant of the low-pass filter), hence it implies Theorem 1 as a special case. Thecondition given in Corollary 1 is a sufficient condition: stability maybe achieved even when some lines are critical. We refer the readerto [26, 27] for characterization of such cases. Even though any lagsdo not influence stability under broad conditions in the losslesscase, simulations suggest that lags may have significant impact onthe damping and settling time of the microgrid, see Section 4.
Unlike the bulk grid, the line resistances for MV/LV power dis-tribution systems are relatively high. Thus, the assumption thatmicrogrid is lossless is often inaccurate. In this subsection, we an-alyze small-signal stability for the general microgrid model. Forconvenience of presentation, we refer to the model as the lossymicrogrid model, although it encompasses the lossless case. Westress that the challenge in analyzing the lossy case comes from thefact that the the Laplacian matrix associated with the power-flowgraph is non-symmetric.To develop results for the general lossy case, we use an extensionof the Sylvester’s inertia theorem for non-symmetric matrices. First,we define mathematical notions on inertia [13]. For a square matrix A , we denote the number of eigenvalues of A with positive, negativeand zero real parts as i + ( A ) , i − ( A ) and i ( A ) , respectively. We callthe ordered triple, (cid:0) i + ( A ) , i − ( A ) , i ( A ) (cid:1) , the inertia of A . We usethe notation sym ( A ) to indicate the symmetric part of A which isgiven by ( A + A T ) . Next, we state an extension of Sylvester’s inertiatheorem, which is contained in Theorem 3 of [5]. Lemma 1:
Let A be a square matrix and H be a symmetric matrixof same dimension. If sym ( AH ) is positive semidefinite with rank r ,then (cid:0) r − i + ( A ) (cid:1) ≤ i − ( H ) [5]. This lemma allows us to develop Laplacian-based sufficient con-dition for stability for a lossy microgrid without any filtering in thedroop controls.
Theorem 3:
Consider a lossy microgrid without lag (filtering) in theP-droop control, and assume that the symmetric part of the Lapla-cian matrix of the active power flow graph sym ( L (G( x ))) is positivesemidefinite and has a non-repeated eigenvalue at zero. Then thesmall-signal model described by (14) is asymptotically stable. Proof:
From (14), it follows that: L (G( x )) = (− D J ( x ) D − ) D . When sym ( L (G( x ))) is positive semidefinite and has a non-repeatedeigenvalue at 0, the rank r of sym ( L (G( x ))) is ( n − ) . Now we useLemma 1 considering A ≡ (− D J ( x ) D − ) and H ≡ D . According toLemma 1 we can say: i − ( J ( x )) = i + (− D J ( x ) D − ) ≥ ( n − ) since i − ( D ) =
0. Since, n is a vector in the null space of ( J ( x )) , theinertia of ( J ( x )) is given by ( , n − , ) . Hence, the small-signalmodel described by (14) is asymptotically stable in our context. □ Unlike the lossless microgrid, the Laplacian based conditiongiven in Theorem 3 is not a necessary condition. We can developanother sufficient condition for the small-signal stability of lossymicorgrid without filtering in the droop control, in terms of theedge weights of the active power flow graph. We note that this term “inertia" is unrelated to the notion of generator inertia; weuse this language in keeping with the terminology used in [13]. mall-Signal Stability Analysis: Droop-Controlled Microgrids e-Energy ’19, June 25–28, 2019, Phoenix, AZ, USA
Lemma 2:
Consider a lossy microgrid without lag (filtering) in theP-droop control. If all the edge weights in W ( θ ) are positive, thesmall-signal model described by (14) is asymptotically stable. Proof:
If the edge weights are all positive, then the Laplacian L (G( x )) is the negative of a Metzler matrix, which further has the propertythat each row sums to zero. Since, D is diagonal and positive definite, D − L (G( x )) is also the negative of a Metzler matrix, for whicheach row sums to zero. From Gershgorin’s circle theorem [13] andthe connectivity assumption of the microgrid, it is straightforwardto verify that that the small-signal model (14) is asymptoticallystable. □ Lemma 2 motivates us to extend the notion of critical lines forthe lossy network. For the lossy case, the edges ( i , k ) and ( k , i ) willboth have positive weights if 0 < ( ϕ ik ± | θ i − θ k |) mod 2 π < π is satisfied. We define a line to be critical when the above an-gle condition is not satisfied. For a directed graph, positive edgeweights do not guarantee that the symmetric part of the Laplacianis positive semidefinite. Therefore, the sufficient condition men-tioned in Theorem 3 may not be necessary. As a counterexample,if L (G( x )) = [ . , − . , − . − . , . , − . − . , − . , . ] thenall the edge weights are positive and therefore, by Lemma 2, small-signal model (14) is stable. However, sym ( L (G( x ))) is not positivesemidefinite, as one of its eigenvalue is -0.1339.We now focus our attention to the case when the droop-controlsin the lossy microgrid have filtering. It can be shown that a Laplacian-based sufficient condition similar to Theorem 3 does not hold any-more, when there is filtering in the active power droop control. Toshow this, we first study a homogeneous microgrid where each bushas inverter with the same frequency droop gain and filtering timeconstant. For this special model, we derive a condition on filteringtime constant for which (13) becomes unstable. Lemma 3:
Consider a homogeneous lossy microgrid where each bushas inverter with same droop control. Let the reciprocal of frequencydroop gain and the time constant of low pass filter in the droop controlof each bus are d and r respectively. If r > Re ( µ )( Im ( µ )) d , where µ isany of the eigenvalues of the Laplacian L (G( x )) of the active powerflow graph and Re ( µ ) and Im ( µ ) are its real and imaginary partrespectively, the small-signal model (13) is unstable. Proof:
Here, M = rd I n × n and D = d I n × n . First we will sim-plify the characteristic equation of the Jacobian J ( x ) utilizingthe properties of the determinant of block matrix. According to(13) the characteristic equation of J ( x ) can be written as fol-lows: det (cid:0) λ I n × n − J ( x ) (cid:1) = det (cid:16) λ (cid:16) λ + r (cid:17) I n × n + rd L (G( x )) (cid:17) .If L (G( x )) has an eigenvalue µ , then by comparing between thecharacteristic equations of L (G( x )) and J ( x ) we get: λ (cid:16) λ + r (cid:17) = − µrd . Equivalently, rd λ + d λ + µ =
0. Now if r > Re ( µ )( Im ( µ )) d for anyeigenvalue µ of L (G( x )) , then the system has eigenvalue in righthalf plane, making the lossy network unstable. □ Lemma 3 shows that increasing the filtering time constant (low-ering the bandwidth of the filters) in droop controls may destabilizethe small-signal model. In particular, for the lossy case, L (G( x )) isnon-symmetric and can potentially admit complex eigenvalues, andhence stability may be lost for sufficiently small filter bandwidth. For instance, the non-symmetric Laplacian matrix L (G( x )) = [ . , − . , − . − . , . , − . − . , − . , . ] has positive edgeweights and also has positive semidefinite sym ( L (G( x ))) with anon-repeated eigenvalue at zero, hence stability of the microgridmodel is guaranteed in the non-lagged case. The eigenvalues ofthe Laplacian for this example are µ = , . ± j . L (G( x )) hasright-half-plane but complex eigenvalues, small filtering time con-stant will maintain stability while large ones will cause instabilityper Lemma 3. In Section 4, we construct a small 3-bus network thatexhibits instability for larger time constant. From the expressionof λ , it is also evident that within the stability region, the increaseof time constant decreases the damping and increases the settlingtime. Our simulation results shown in Section 4 also illustrate thistransient behavior. However, if the Laplacian matrix L (G( x )) satis-fies certain special properties, it can be shown that the small-signalmodel (13) is asymptotically stable for any lag. For example, stabil-ity can be guaranteed if the Laplacian matrix is similar to a positivesemidefinite matrix, via a diagonal similarity transform. We presentthe result next: Theorem 4:
Consider a lossy microgrid for which the Laplacian L (G( x )) of the active power flow graph has a non-repeated eigen-value at zero, and further there exists a diagonal matrix K such that L s (G( x )) = KL (G( x )) K − is positive semidefinite. Then the small-signal model (13) of the lossy microgrid is asymptotically stable forany filtering time constant . Proof:
Consider the following similarity transformation of the linearizedmodel’s state matrix: J s ( x ) = diaд ( K , K ) J ( x ) diaд ( K − , K − ) = (cid:20) n × n I n × n − M − L s (G( x )) − M − D (cid:21) (16)Note, in (16) we have used the fact that the matrix multiplicationof diagonal matrices of same order is commutative. Now sincethe similarity transformation does not change the eigenvalues and L s (G( x )) is positive semidefinite with a non-repeated eigenvalueat zero, it is immediate from Theorem 2 that the small-signal modelsgoverned by J s ( x ) and hence J ( x ) are asymptotically stable. □ Next we identify graph theoretic conditions for which the Lapla-cian L (G( x )) has a non-repeated eigenvalue at zero and thereexists a diagonal matrix K such that L s (G( x )) = KL (G( x )) K − is positive semidefinite. When the network is radial and the ac-tive power flow graph has positive weights, it is known that thereexists a diagonal matrix K such that L s (G( x )) = KL (G( x )) K − is symmetric. This follows from the standard results on diagonalsymmetrizability of Metzler matrices whose associated graphs areradial [22]. Therefore, if the microgrid’s active power flow graphis radial with positive edge weights, L s (G( x )) is guaranteed tobe positive semidefinite with a non-repeated eigenvalue at zero.As mentioned before, for the lossy case, all edge weights will bepositive if the inequality 0 < ( ϕ ik ± | θ i − θ k |) mod 2 π < π is satis-fied for all edges ( i , k ) in the active power flow graph. For a radialnetwork, the power flow equations can be shown to always have a -Energy ’19, June 25–28, 2019, Phoenix, AZ, USA Maruf, Ostadijafari, Dubey and Roy solution where this condition is met. This can be verified througha recursive argument, where the the angle for each leaf node (bus)is calculated first, and then the calculation is repeated recursivelyfor remaining buses for which only one angle is unknown in thepower flow equations. This analysis yields the following corollary:
Corollary 3:
Consider The small-signal model (13) for the lossymicrogrid. The small-signal model is asymptotically stable for anyfiltering time constant, when the mircogrid is radial, and the activepower flow graph does not have any critical lines. For a radial network,an equilibrium always exists such that the active power flow graphdoes not have any critical lines.
Discussion : Illustrated results in Sections 3.1 and 3.2 are based onthe known and fixed loads/generations. Microgrid planning underuncertainty in loads/generations is discussed widely from differentaspects in current literature [15, 30]. Uncertainty in loads/generationalter the operating solution, and hence the linearized model. Thecombined effect of this variability and lag/ filtering depends on thenetwork structure. Specifically, in the lossless case, the impact oftime-lag on stability is orthogonal to the impact of uncertainty, asthe stability property is maintained regardless of the lag. Thus, ifthe no-lag model is stable over the range of operating uncertainties,the lagged model will be stable as well. In the lossy case, however,lags may alter the stability property, and hence there is a complexrelationship between uncertainty and lag tolerance. We note thatcharacterizing load/generation profiles which yield stability is animportant (and difficult) question in its own right, even withoutconsidering lag/filtering, and it is out of the scope of this paper.
In this section we aim to show simulation results which supportour presented theoretical results. First we demonstrate simulationresults showing the effect of lag/filtering on the stability and tran-sient response of lossless microgrid. Next we present further resultsfor lossy microgrids with both mesh and radial topologies. Thesimulations of transients are undertaken using the nonlinear DAEmodel developed in Section 2.1.3 if not mentioned otherwise.
Here we demonstrate the effects of filtering on the transient re-sponse of a droop-controlled lossless microgrid. Specifically, weevaluate the effects of increasing lag on damping and settling timefor the bus angles of microgrid. We first present the simulationresults using modified IEEE 9-bus test system with the same modelparameters as those used in [26]. In the modified test system (seeFig. 2), buses 1, 2 and 3 are interfaced with inverter-based DERs(belong to V A ), buses 5, 7 and 9 are non-inverter buses supplyingfor frequency dependent loads (belong to V B ), and buses 4, 6 and8 are supplying constant (i.e. zero) power loads (belong to V C ).The bus and line parameters are detailed in Table 1 and Table 2,respectively in the appendix section.The operation of the network is analyzed at two different equi-librium points: A and B. The bus angles obtained from power flowsolution corresponding to both equilibrium points are shown inTable 2. They admit the following characteristics: Deletion of edge for which edge weight is zero from active power flow graph doesnot change the Laplacian and thus we assume there are no such lines.
Figure 2: Diagram of the IEEE 9-Bus Test System.
Bus 3 T D =0.1T D =1T D =10 B u s A ng l e ( deg r ee s ) → Bus 4
Time (second) → -3.8-3.75-3.7 Bus 5
Bus 3 T D =0.1T D =1T D =10 B u s A ng l e ( deg r ee s ) → Bus 4
Time (second) → -1000100 Bus 5
Figure 3: Small-Disturbance Angle Response of ModifiedIEEE-9 Bus Lossless System for T D =
1) Point A is the normal operating point for the system. Noneof the lines satisfy the definition of a critical line. Therefore, forequilibrium point A, L (G( x )) is positive semidefinite.
2) Point B represents an unstable operating point for the testsystem. Corresponding to this point, lines ( , ) and ( , ) satisfythe definition for critical lines. It is found that L (G( x )) is not pos-itive semidefinite. The angle response obtained from simulatingtransient behavior of test system due to small disturbances aroundboth equilibrium points, A and B, are shown in Figs. 3(a) and 3(b),respectively. The angle response are shown using three represen-tative buses one from each category: inverter interfaced (bus 3),supplying frequency-dependent loads (bus 5), and supplying con-stant power loads (bus 4). The bus angles are plotted with respect tothe angle measurement of bus 1. Note that the same time constant, T D , is used for all the droop-controlled inverters. Three cases with mall-Signal Stability Analysis: Droop-Controlled Microgrids e-Energy ’19, June 25–28, 2019, Phoenix, AZ, USA Figure 4: Diagram of the IEEE 57-Bus Test System. different time constant ( T D = 0.1, 1, and 10 sec) are simulated forboth equilibrium points.As it can be seen from Fig. 3(a) the system is small-signal stableat point A regardless of the time constant of low pass filter of thedroop controller. Similar conclusion is drawn from Theorem 2 asthe Laplacian L (G( x )) at point A is positive semidefinite with anonrepeated eigenvalue at zero. Fig. 3(a) also depicts the effectof lag on the settling time for microgrid. It is observed that thehigher the time constant T D , the longer the system takes to settleto the equilibrium point. Furthermore, buses having inverters, forexample bus 3, shows a comparatively low damping and highersettling time compared to the load buses.It can be observed from Fig. 3(b) that the other equilibrium, pointB, is not small-signal stable. Since the corresponding L (G( x )) isnot positive semidefinite, the instability of point B is immediatefrom Theorem 2. The unstable behavior of point B can be furtherverified from the eigen analysis of the corresponding Jacobian ma-trix. When evaluated at point B, J ( x ) admits one eigenvalue inthe right half plane (i.e. λ unstable = .
42 for T D = . D R i = D L i = ϵ = . ϵ = . T D . The angleresponse for buses 2, 16 and 7, that belong to V A , V B and V C ,respectively are shown in in Fig. 5. The angle response are shownfor three different values of time constant, T D , where angles aremeasured with respect to angle at bus 1. Similar to the results ofmodified IEEE 9-bus system, we see that the system remains stable Bus 2 T D =0.01T D =1T D =10 B u s A ng l e ( deg r ee s ) Bus 16
Time (second) -10-5
Bus 7
Figure 5: Small-Disturbance Angle Response of ModifiedIEEE-57 bus system for T D = irrespective of time constant while damping decreases and settlingtime increases with the increase in the magnitude of the time con-stant. Also, as before, the inverter based buses, for example bus 2,show low damping and higher settling time compared to the loadbuses. In the subsequent subsections we show simulation results for lossymesh and lossy radial microgrids.
We present a simulation case study toverify the effects of filtering/lag for a lossy microgrid with meshtopology. A simple three-bus lossy microgrid shown in Fig. 6 is usedfor simulations. The bus and line parameters are detailed in Table 3and 4 in the appendix section. Table 4 also includes an equilibriumpoint for the lossy three-bus system. We consider each filter hassame time constant T D . We show using simulations that the givensystem has a stable equilibrium for smaller time constant. However,for very large time constant, the system becomes unstable.For the three-bus system, the Laplacian is given by: L (G( x )) = [ . , − . , − . − . , . , − . − . , − . , . ] which has all of its eigenvalue in the right half plane. Note that W ( θ ) is positive definite here and thus the system has no criticallines. Fig. 7 shows angle response of bus 2 for two different lags. Itcan be observed that for smaller lag i.e. T D = s , the angle reachesits equilibrium value. Whereas, for the larger lag, i.e. T D = s ,the system is unstable as angle diverges from the equilibrium point.This unstable behaviour can further be verified from the eigen-analysis of the corresponding Jacobian matrix. The J ( x ) admitsunstable eigenvalue λ unstable = . ± j . T D = -Energy ’19, June 25–28, 2019, Phoenix, AZ, USA Maruf, Ostadijafari, Dubey and Roy Time (second) → B u s A ng l e ( deg r ee s ) → Bus 2 T D =10 T D =1000 Figure 7: Small-Disturbance Angle Response of Bus 2 of Con-sidered Mesh Network for T D =
10 and 1000 sec. time constant that is greater than 522 . Here, we investigate the effects offiltering on small-signal stability of a lossy radial mircrogrid. Weobtain a radial microgrid by removing the line from bus 5 to bus 6 inFig. 2. Here we consider each line to be lossy with R / X ratio as 0.5.All the load and generation parameters for buses in the obtainedradial system are same as the ones in Table 1 except the bus 1 whoseactive power generation changed to 1 . T D as before. It can be seen in Fig. 8that the equilibrium is small-signal stable regardless of the timeconstant of the low pass filter in the droop control. It is immediatefrom Corollary 3 as the network is radial and its corresponding ac-tive power flow graph has positive edge weights (equivalently, doesnot have any critical lines). Note that, there exists a diagonal ma-trix K = diaд ( . , . , . , . , . , . , . . , . ) for which L s (G( x )) = KL (G( x )) K − is positivesemidefinite with a non-repeated eigenvalue at zero. Therefore,stability of the equilibrium is also guaranteed by Theorem 4.The simulation results demonstrated here verify the theoreticalresults obtained in Section 3. From simulations, the following arethe key observations for a droop controller with filtering : (1) Forlossless connected microgrid, lag/filtering can not destabilize thesystem irrespective of its topology; (2) An increased lag (due toincrease of filter time constant) results in less damping and increasesthe settling time for the transient response of the bus angles; (3)The effects of lag damping and settling time for bus angles are morepronounced for generator buses; (4) For a lossy mesh microgrid,a large lag can destabilize the otherwise stable equilibrium pointof the system, (5) For a lossy radial microgrid, larger lag cannotdestabilize the system but affects the damping and settling time. In this paper, we thoroughly investigated the effects of lag/ filteringin frequency-droop control on the angle stability of inverter-basedlossless and lossy microgrids. We developed a nonlinear DAE model
Bus 3 B u s A ng l e ( deg r ee s ) → Bus 4
Time (second) → -11-10-9 Bus 5 T D =0.1T D =1T D =10 Figure 8: Small-Disturbance Angle Response of ModifiedIEEE-9 Bus Lossy System with Radial Topology for T D = and a structure preserving linear differential-equation approximatemodel for a microgrid with low-pass-filtered droop control. Usingthe proposed models, we derived necessary and sufficient conditionfor lossless microgrid that guarantees a stable operation regardlessof lag in droop control and also discussed graph-theoretic implica-tions. Simulation case studies are presented using IEEE 9-bus and57-bus test systems to demonstrate the transient response of thelossless microgrid with lagged droop-control. It is observed that onincreasing lag the system takes longer time to settle to the equilib-rium point. For the microgrid with lossy lines, sufficient conditionshave been derived when there are no lag/filtering in the droopcontrol and later by providing a condition we prove that the previ-ously derived sufficient conditions do not apply in general whenlag/filtering is present and instability may result on increasing thelag. A simulation case study is also included for a 3-bus lossy micro-grid that demonstrates the destabilization of a stable equilibriumsystem on increasing the lag. Further, for a tree/radial lossy micro-grid, sufficient condition for stability is derived. It is demonstratedusing modified IEEE 9-bus test system that for a radial microgrid,the droop-control with lag/filtering does not affect the otherwisestable equilibrium point, however, damping and settling times areaffected. These crucial observations caution us against generalizingthe stability conditions obtained using structure-preserving modelsfor a practical lossy microgrid that usually admits a time-laggeddroop response.Future research directions include the analysis of the voltagedynamics and consequently the effects of Q − droop on microgrid’ssmall signal stability. Another important future direction is to studythe impacts of uncertainty in loads/generation on the small-signalstability of the microgrid. Finally, it would be interesting to improvethe stability of the microgrid by designing the droop controllers’gain. This could be formulated as an optimization problem wherethe objective could be to optimize a stability metric (e.g. an optimaldamping ratio) given bounds on the lag introduced due to filtering.Finally, although our focus here is on a primary control, overlaidsecondary and tertiary controllers for microgrids depend on longer-range communications, and hence are also critically impacted bytime-lags; co-design of primary and secondary controls in a waythat accounts for the lags is an important direction of future work. mall-Signal Stability Analysis: Droop-Controlled Microgrids e-Energy ’19, June 25–28, 2019, Phoenix, AZ, USA APPENDIXA TEST SYSTEMS
Here we detail the parameter values of the test systems that wehave used in our simulation in the following tables.
Table 1: Bus Generation and Load Parameters of ModifiedIEEE-9 Bus Lossless Test System
Bus P G i (p.u.) P L i (p.u.) V i (p.u.) D R i (s) D L i (s)1 0.67 0 1 5 02 1.63 0 1 5 03 0.85 0 1 5 04 0 0 1 0 05 0 0.90 1 0 26 0 0 1 0 07 0 1.00 1 0 28 0 0 1 0 09 0 1.25 1 0 2 Table 2: Line Parameters and Angle Differences across Linesat Equilibrium Point A and Point B for Modified IEEE-9 BusLossless Test System
Line X (p.u.) θ i − θ k at A θ i − θ k at B(1, 4) 0.0576 2.21 ◦ ◦ (4, 5) 0.0920 1.53 ◦ -22.04 ◦ (5, 6) 0.1700 -5.96 ◦ -122.17 ◦ (3, 6) 0.0586 2.86 ◦ ◦ (6, 7) 0.1008 1.38 ◦ -24.60 ◦ (7, 8) 0.0720 -3.14 ◦ ◦ (8, 2) 0.0625 -5.85 ◦ -5.85 ◦ (8, 9) 0.1610 8.05 ◦ -145.71 ◦ (9, 4) 0.0850 -1.85 ◦ -23.81 ◦ Table 3: Bus Generation and Load Parameters of 3-Bus LossySystem
Bus P G i (p.u.) P L i (p.u.) V i (p.u.) D R i (s) D L i (s)1 0.5295 2 1 0.1 02 3.0860 2 1 0.1 03 3.0650 2 1 0.1 0 REFERENCES [1] Nathan Ainsworth and Santiago Grijalva. 2013. A structure-preserving modeland sufficient condition for frequency synchronization of lossless droop inverter-based AC networks.
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Table 4: Line Parameters and Angle Differences across Linesat Equilibrium Point of 3-Bus Lossy System
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