> OAJPE-00088-2020 < 1

Distributed Mixed Voltage Angle and Frequency Droop Control ofMicrogrid Interconnections with

Loss of Distribution-PMU Measurements

S. Sivaranjani*, Etika Agarwal*, Vijay Gupta, Panos Antsaklis, and Le Xie†

Recent advances in distribution-level phasor measurement unit (D-PMU) technology have enabled the use of voltage phase anglemeasurements for direct load sharing control in distribution-level microgrid interconnections with high penetration of renewabledistributed energy resources (DERs). In particular, D-PMU enabled voltage angle droop control has the potential to enhance stabilityand transient performance in such microgrid interconnections. However, these angle droop control designs are vulnerable to D-PMU angle measurement losses that frequently occur due to the unavailability of a global positioning system (GPS) signal forsynchronization. In the event of such measurement losses, angle droop controlled microgrid interconnections may suffer from poorperformance and potentially lose stability. In this paper, we propose a novel distributed mixed voltage angle and frequency droopcontrol (D-MAFD) framework to improve the reliability of angle droop controlled microgrid interconnections. In this framework,when the D-PMU phase angle measurement is lost at a microgrid, conventional frequency droop control is temporarily used forprimary control in place of angle droop control to guarantee stability. We model the microgrid interconnection with this primarycontrol architecture as a nonlinear switched system and design distributed secondary controllers to guarantee stability of the network.Further, we incorporate performance specifications such as robustness to generation-load mismatch and network topology changesin the distributed control design. We demonstrate the performance of this control framework by simulation on a test 123-feederdistribution network.

Index Terms—Microgrids, phasor measurement units, interconnected system stability, distribution systems, droop control.

I. INTRODUCTION

PHASOR measurement units (PMUs) have been exten-sively used in real-time wide-area monitoring, protectionand control (WAMPAC) applications at the transmission level.However, in traditional distribution networks with one-waypower flows and no active loads, real-time monitoring andcontrol using phasor measurements has not been necessaryfor reliable operation [1]. Further, small angle deviationsand consequently, poor signal-to-noise ratios, make real-timeestimation of voltage phase angles at the distribution levelan extremely challenging problem. Therefore, applications ofdistribution-level PMUs (D-PMUs) have so far been con-fined to the substation-level, with angle references beingsynchronized with the transmission network [2]. However, infuture distribution systems, new architectures like microgridswith large-scale integration of renewable distributed energyresources (DERs) and flexible loads, as well as new economicparadigms like demand response, will require extensive real-time monitoring and control. In this context, D-PMUs (such asthe µPMU) that can provide accurate measurements of smallangle deviations (≈ 0.001◦) and voltage magnitude deviations

*These authors contributed equally to this work. † Corresponding author.S. Sivaranjani and Le Xie were supported in part by NSF grants ECCS-

1839616, ECCS-1611301 and CCF-1934904, and the Department of EnergyContract DE-EE0009031. S. Sivaranjani was also partially supported in thiswork by the Schlumberger Foundation Faculty for the Future fellowship.Vijay Gupta was partially supported by NSF grant CNS-1739295 and DARPAFA8750-20-2-0502. Etika Agarwal and Panos Antsaklis were partially sup-ported by the Army Research Office Grant No. ARL W911NF-17-1-0072.

S. Sivaranjani and Le Xie are with the Department of Electricaland Computer Engineering, Texas A&M University, College Station, TX;{sivaranjani,le.xie}@tamu.edu. Etika Agarwal is with GE Global Research,India; {etika.agarwal09@gmail.com}. S. Sivaranjani and Etika Agarwal werewith the Department of Electrical Engineering, University of Notre Dame,Notre Dame, IN when this work was carried out. Vijay Gupta and PanosAntsaklis are with the Department of Electrical Engineering, University ofNotre Dame, Notre Dame, IN; {vgupta2,pantsakl}@nd.edu.

(≈ 0.0002 p.u.) with high sampling rates (≈ 120s−1) haverecently been developed [3]-[5], and are expected to be criticalcomponents of future power distribution infrastructure [1][6].

In particular, consider the problem of ensuring stability andreliability in distribution networks comprised of interconnectedmicrogrids. Typically, such microgrid interconnections arecontrolled in a hierarchical manner with three layers of control[7][8] - (i) a primary control layer to ensure proper loadsharing between microgrids, (ii) a secondary control layer toensure system stability by eliminating frequency and voltagedeviations, and (iii) a tertiary control layer to provide powerreference set points for individual microgrids. The primarycontrol layer commonly comprises of frequency droop andvoltage droop controllers to regulate the real and reactivepower outputs respectively of each microgrid at the point ofcommon coupling (PCC). These droop control characteris-tics are implemented artificially using voltage-source inverter(VSI) interfaces that are designed such that individual micro-grids emulate the dynamics of synchronous generators.

However, the use of frequency droop controllers in net-works with a large penetration of low inertia VSI-interfacedmicrogrids has been demonstrated to result in numerous is-sues including chattering [9], loss of synchronization, andundesirable frequency deviations resulting from the trade-offbetween active power sharing and frequency accuracy [10].With the availability of highly accurate D-PMUs, angle droopcontrollers that directly use the voltage phase angle deviationmeasurements at the PCC for active power sharing have beenproposed to replace frequency droop controllers in the primarylayer. These angle droop designs have been demonstrated toresult in increased stability, smaller frequency deviations andfaster dynamic response [11]–[15].

A key bottleneck in the widespread adoption of D-PMUbased control designs for microgrids is the reliability of D-

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PMU voltage phase angle measurements. Phase angle mea-surements from D-PMUs require a global positioning system(GPS) signal for synchronization of the angle reference acrossthe network [1]. However, recent studies have demonstratedthat PMU GPS signals are frequently lost due to factors likeweather events and communication failure, leading to loss ofPMU measurements [16][17]. In fact, such PMU measurementlosses have been observed to occur as often as 6-10 timesa day, with each loss event ranging from an average of 6-8seconds to over 25 seconds [18]. In WAMPAC applications,loss of GPS signal for PMUs has been demonstrated toresult in severe performance degradation [19]. Furthermore,control strategies that rely on PMU measurements have beendemonstrated to be vulnerable to GPS spoofing attacks, wherea falsified GPS signal may be fed to compromise the PMUangle reference, leading to potentially catastrophic outcomeslike cascade failures [20][21]. Therefore, in the event of D-PMU measurement losses, microgrid interconnections that relysolely on angle droop control will be particularly prone to poordynamic performance and potential instability [22].

In wide-area control applications, the issue of PMU mea-surement loss is typically handled from a networked controlsystems perspective, where a popular approach is to allow thecontroller to use the last available measurement in the eventof a measurement loss, and bound the maximum allowableduration of the loss to guarantee stability [23][24]. Typically,networked control designs also assume knowledge of theprobability distributions of the loss events [25]. However,these approaches suffer from two critical issues in the contextof distribution systems. First, in D-PMUs, the durationof measurement loss may exceed the maximum allowableduration to guarantee stability. This is due to the fact thatangle droop controllers are extremely fast acting [11], and maytherefore not be able to maintain stability [22] under the longermeasurement loss durations (typically tens of seconds) seen inD-PMU GPS signal losses [18]. Second, even for measurementloss durations smaller than this threshold, large voltage angleand frequency drifts can occur due to the controller repeatedlyusing the incorrect (last available) measurement [22].

In order to address stability and performance issues resultingfrom D-PMU measurement losses in angle droop controlledmicrogrid interconnections, we introduce the idea of mixedvoltage angle and frequency droop control (MAFD). WhenD-PMU voltage angle measurements are lost at a microgriddue to loss of a GPS signal, frequency measurements maystill be available, since they can be obtained locally withouta GPS signal for synchronization. In the MAFD framework,frequency droop control is, therefore, temporarily used inplace of angle droop control for primary control of activepower sharing at particular microgrids where D-PMU mea-surements are lost [22]. In a network of microgrids underthe MAFD primary control framework, each microgrid mayoperate with either angle droop or frequency droop controlat any time instant, depending on the availability of D-PMUmeasurements at that microgrid. We therefore model the time-varying dynamics of a network of microgrids with MAFDprimary control as a nonlinear switched system. We thenshow that the MAFD framework, along with a dissipativity-

based distributed secondary controller, guarantees stability ofangle droop controlled microgrid interconnections without anyrestrictions on the duration or probability distribution of D-PMU measurement losses. We refer to this control architecturewith MAFD primary control and the proposed distributedsecondary control as the distributed MAFD (D-MAFD) frame-work. Additionally, the D-MAFD design incorporates perfor-mance specifications including robustness to disturbances andnetwork topology changes, with the view of increasing the roleof D-PMU measurements in islanding operations and plug-and-play architectures in future microgrid interconnections.Finally, we show through case studies that the D-MAFDframework significantly enhances system stability in D-PMUmeasurement loss scenarios under conditions of generation-load mismatches and is robust to network topology changesinduced by faults. The key contributions of this paper are:

• First, we develop a distributed control framework to guar-antee stability of a network of microgrids operating underangle droop control, when D-PMU angle measurements arelost. We accomplish this via an MAFD primary controlframework, where each microgrid can switch from thedefault angle droop control mode to a frequency droopcontrol mode if D-PMU angle measurements are lost atthat microgrid. In contrast to the networked control systemsapproaches are typically adopted in literature to handlemeasurement losses [23][24], the D-MAFD framework doesnot impose any bounds on the duration or probability dis-tribution of measurement loss to guarantee stability. Rather,it exploits the physics of the system (that is, the fact thatthe frequency is the derivative of the angle) to indirectlycontrol the voltage angle using the frequency when anglemeasurements are lost. Therefore, the voltage angle can becontrolled continuously without any loss of information evenunder D-PMU measurement losses.

• Second, we design a dissipativity-based distributed sec-ondary controller to guarantee stability of the network ofmicrogrids, when each microgrid can arbitrarily switchbetween angle and frequency droop control based on theavailability of D-PMU angle measurements. This designextends the passivity-based state-feedback control design fornonlinear discrete-time switched systems in [26] to a generalcontinuous-time output-feedback dissipativity framework,incorporating additional performance specifications and ro-bustness to network topology changes (such as line outages).

A preliminary version of this work was presented in [22].In this paper, we significantly expand the results in [22]by incorporating robustness and performance specifications,additional case studies, and detailed proofs of all mathematicalresults that were omitted in [22].

Notation: Let R and Rn denote the sets of real numbers andn-dimensional real vectors respectively. The (i, j)-th elementof a matrix A ∈ Rm×n is denoted by Aij and the transposeis denoted by A′ ∈ Rn×m. The identity matrix is representedby I , with dimensions clear from the context. A symmetricpositive (negative) definite matrix P ∈ Rn×n is representedby P > 0 (P < 0).

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II. MIXED VOLTAGE ANGLE AND FREQUENCY DROOPCONTROL (MAFD) MODEL

Consider a network of N microgrids where each microgridis connected to the network at the PCC as shown in Fig. 1.Define Ni to be the neighbor set of the i-th microgrid, that is,the set of all microgrids to which the i-th microgrid is directlyconnected in the network. For convenience of notation, wealso include i in this set. Considering the coupled AC powerflow model, the real and reactive power injections P jinj(t) andQjinj(t), at the j-th microgrid at time t are given by

P jinj(t) =∑k∈Nj

Vj(t)Vk(t)|Yjk| sin(δjk(t) + π/2− ∠Yjk)

Qjinj(t) =∑k∈Nj

Vj(t)Vk(t)|Yjk| sin(δjk(t)− ∠Yjk),(1)

where Vj(t) and δj(t) are the voltage magnitude and phaseangle at PCC respectively, δjk(t) = δj(t)− δk(t), and Yjk isthe complex admittance of the line between the PCC buses ofmicrogrids j and k.

As shown in Fig. 1, the primary control layer for everymicrogrid comprises of an angle droop and a voltage droopcontrol loop, which regulate the real and reactive powerinjections of the microgrid respectively to track desired ref-erence values V refi and δ

refi of the voltage magnitude and

phase angle at the PCC respectively. These angle droopcontrollers can be readily implemented by programming theVSI-interfaces at the microgrid PCC [9][27]. Alternatively,these angle droop controllers may be implemented via anglefeedback control in inertia emulating interfaces like virtualsynchronous generators [28]. As mentioned in Section I, theuse of angle droop primary control to directly regulate realpower has several advantages such as increased frequencystability and power sharing accuracy [11]. The implementationof primary angle droop control schemes requires real-timeangle measurements from D-PMUs placed at the microgridPCCs, which in turn require a GPS signal for synchronization.However, since GPS signals are frequently lost, microgridinterconnections that primarily use angle droop control willsuffer from poor performance and potential instability.

To address this issue, when D-PMU angle measurements arelost at certain microgrids due to loss of a GPS signal loss, weemploy a mixed angle and frequency droop control (MAFD)framework for primary control, as shown in Fig. 2. When D-PMU angle measurements are lost, frequency measurementscan still be obtained locally at each microgrid, without the

PCC1

PCCi

PCCN

PCCi

Programmable VSI Interface

Angle Droop ControlRenewable

energy

Load

StorageControl

Renewable

energy

Load

StorageControl

Renewable

energy

Load

StorageControl

Fig. 1. Network of interconnected microgrids with angle droop control. Someicons are licensed from Adobe Stock and modified for this illustration.

need for a GPS signal-based synchronization. In the MAFDframework, classical frequency droop control is therefore usedin place of the angle droop control to temporarily regulatereal power at those microgrids until D-PMU measurementsare restored. Thus, at any given time, some microgrids mayoperate with angle droop control while others operate withfrequency droop control depending on the availability of D-PMU measurements. At every time t, each microgrid in theMAFD framework operates in one of two modes, denoted byσi(t) ∈ {1, 2} - (i) angle droop control mode (σi(t) = 1),when real-time angle measurements are available from theD-PMU at that microgrid, and (ii) frequency droop controlmode (σi(t) = 2), when D-PMU voltage angle measurementsare lost or corrupted at that microgrid due to GPS signalloss or sensor malfunction (Fig. 2). The dynamics of the i-thmicrogrid in each of these modes is described as follows.

Angle Droop Control Mode, σi(t) = 1: In this mode, D-PMU angle measurements are available, and the microgridoperates with angle and voltage droop control laws given by

μG1

MAFD1

Primary control

μG2

MAFD2

Primary control

μG3

MAFD3

Primary control

μG4

MAFD4

Primary control

K1 K2 K3 K4

Angle measurement

available?

σi(t)

xi(t)

yi(t)

Angle,

voltage

droop control

Frequency,

voltage

droop control

wi(t)

ui(t)

MAFD primary control

Dynamic coupling

Secondary control information $ow -

Fig. 2. Schematic of representative microgrid in the D-MAFD framework with MAFD primary control and distributed secondary control.

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Jδi∆δ̇i(t) = −Dδi∆δi(t) + ∆P iext(t)−∆P iinj(t) (2)JVi∆V̇i(t) = −DVi∆Vi(t) + ∆Qiext(t)−∆Qiinj(t), (3)

where ∆Vi(t) = Vi(t) − V refi and ∆δi(t) = δi(t) −δrefi , are the deviations of the PCC voltage magnitude fromtheir reference values, ∆P iinj(t) = P

iinj(t) − P

i,refinj and

∆Qiinj(t) = Qiinj(t) − Q

i,refinj are the deviations of the real

and reactive power injections P iinj(t) and Qiinj(t) from their

nominal values P i,refinj and Qi,refinj respectively, and ∆P

iext(t)

and ∆Qiext(t) are the generation-load mismatches at the i-thmicrogrid. The droop coefficients Jδi , Dδi , JVi and DVi canbe implemented by programming the VSI interface at the PCC[13]. Additionally, the dynamics of the frequency error in theangle droop control mode is propagated as

∆ω̇i(t) =−DδiJδi

[−DδiJδi

∆δi(t) +1

Jδi∆P iext(t)

− 1Jδi

∆P iinj(t)

]− 1Jδi

∆Ṗ iinj(t),

(4)

where ∆ωi(t) = ωi(t)−ωrefi is the deviation of the frequencyof the i-th microgrid ωi(t) from its reference value ω

refi . The

derivative ∆P iinj(t) is computed from (1).

Remark 1. We note that the propagation of the dynamicsof the frequency error through (4) in the angle droop controlmode is not carried out in typical angle droop control schemes.However, in our framework, this serves the important purposeof ensuring the continuity of the state xi(t) at every instantof time, thereby allowing for seamless switching betweenthe angle droop control and frequency droop control modesas follows. The continuity of the state can be guaranteedby ensuring that (5) is satisfied at every switching instant.While this is automatically ensured for switches from thefrequency droop mode to the angle droop mode, (4) enforces∆ω̇i(t) = ∆δ̈i(t) for all time t, thereby ensuring continuityfor switches from the angle droop mode to the frequencydroop mode. Thus, through (4), the continuity of the statexi(t) is guaranteed at every time instant for arbitrary switchingbetween the angle and frequency droop control modes at thei-th microgrid.

Frequency Droop Control Mode, σi(t) = 2: When D-PMUangle measurements are not available, a frequency droopcontrol law is used to regulate the real power, and the dynamicsof the microgrid in this mode are given by

∆δ̇i(t) = ∆ωi(t) (5)

Jωi∆ω̇i(t) = −Dωi∆ωi(t) + ∆P iext(t)−∆P iinj(t) (6)JVi∆V̇i(t) = −DVi∆Vi(t) + ∆Qiext(t)−∆Qiinj(t), (7)

where Jωi and Dωi are the frequency droop coefficients.We now define the state, input and disturbance vectors of the

i-th microgrid to be xi(t) = [∆δi(t) ∆ωi(t) ∆Vi(t)]′, ui(t) =[∆P iinj(t) ∆Q

iinj(t)]

′ and wi(t) = [∆P iext(t) ∆Qiext(t)]

′

respectively. The output vector of the i-th microgrid is

yi(t) = giσi(t)

(xi(t), wi(t)), (8)

where giσi(t) = [∆δ̇i(t) ∆Vi(t)]′ when σi(t) = 1 and

giσi(t)(t) = [∆ω̇i(t) ∆Vi(t)]′ when σi(t) = 2. The dynamics

of the i-th microgrid in the MAFD primary control frameworkcan then be written as a nonlinear switched system

ẋi(t) = fiσi(t)

(xi(t), ui(t), wi(t))

ui(t) = hi(xi(t)),

(9)

where the dynamics f i1(xi(t), ui(t), wi(t)) in the angledroop control mode are defined by (2)-(4), the dynamicsf i2(xi(t), ui(t), wi(t)) in the frequency droop control modeare defined by (5)-(7), and hi(xi(t)) is given by the powerflow model (1) independent of the switching mode σi. Definethe augmented state vector for the microgrid interconnec-tion as x(t)=[x′1(t), x

′2(t), . . . , x

′N (t)]

′. Similarly, define theaugmented input, disturbance and output vectors obtained bystacking the inputs, disturbances and outputs of all micro-grids to be u(t), w(t) and y(t) respectively. Finally, definethe augmented switching vector σ(t) = [σ1(t), · · · , σN (t)]′,where every element can take values of 1 or 2, indicatingthe availability or loss of D-PMU angle measurements at thatmicrogrid. Let Σ denote the set of all possible values of thisswitching vector. We now write the dynamics of the microgridinterconnection with MAFD as the nonlinear switched system

ẋ(t) = fσ(t)(x(t), u(t), w(t))

y(t) = gσ(t)(x(t), w(t))

u(t) = h(x(t)),

(10a)

fσ(t) =

f1σ1(t)

...fNσN (t)

, gσ(t) =g

1σ1(t)

...gNσN (t)

, h =h

1

...hN

. (10b)In order to design a secondary controller that guarantees the

stability of the microgrid interconnection with MAFD, welinearize the system model (10) around the origin to obtaina linearized switched system model

ẋ(t) = Aσ(t)x(t) +B(1)σ(t)u(t) +B

(2)σ(t)w(t)

y(t) = Cσ(t)x(t) +Dσ(t)w(t)

u(t) = Hx(t),

(11a)

Aj =∂fj∂x

∣∣∣∣x=0w=0

, B(1)j =

∂fj∂u

∣∣∣∣x=0w=0

, B(2)j =

∂fj∂w

∣∣∣∣x=0w=0

Cj =∂gj∂x

∣∣∣∣x=0w=0

, Dj =∂gj∂w

∣∣∣∣x=0w=0

, (11b)

H =

∂u1∂x1

· · · ∂u1∂xN

......

...∂uN∂x1

· · · ∂uN∂xN

x=0

, (11c)

∂ui∂xk

=

∂∆P iinj∂∆δk ∂∆P iinj∂∆ωk ∂∆P iinj∂∆Vk∂∆Qiinj∂∆δk

∂∆Qiinj∂∆ωk

∂∆Qiinj∂∆Vk

, i, k ∈ {1, · · · , N}.Note that H is the power flow Jacobian pertaining to the

linearization of (1).The MAFD primary control architecture allows for indirect

control of the real power sharing in the microgrid intercon-nection in a decentralized manner even when D-PMU angle

> OAJPE-00088-2020 < 5

measurements are lost. With this architecture, a supplementarycontroller, termed as the secondary controller, must then bedesigned to eliminate the voltage and angle errors that arisedue to open-loop droop control. In the following section, wepropose a distributed secondary control design that uses onlylocal information at each microgrid to regulate angle andvoltage deviations and guarantee stability of the network ofinterconnected microgrids.

III. D-MAFD SECONDARY CONTROL DESIGNConsider the microgrid interconnection with MAFD pri-

mary control as shown in Fig. 2. Given the dynamics in(10) and its linear approximation (11), we would like todesign a secondary controller that eliminates voltage and angledeviations in the microgrid interconnection and guaranteesstability during D-PMU measurement losses. The proofs ofall the results in this section are provided in the Appendix.

A. Distributed Secondary Control Synthesis

In this section, we develop a distributed secondary con-trol design for microgrid interconnections operating in theMAFD framework, where every microgrid locally determinesits secondary control actions using information only fromits immediate neighbors. The microgrid interconnection withMAFD primary control and distributed secondary control asshown in Fig. 2 is termed as the distributed MAFD (D-MAFD) framework. For this network, we would like to designa secondary output-feedback control law ũ(t) = Kσ(t)y(t),Kj ∈ R2N×2N , j ∈ Σ, such that the microgrid interconnection(10) with u(t) 7→ u(t) + ũ(t) is L2-stable with respect todisturbances w(t) even when D-PMU angle measurements areunavailable in any number of microgrids in the network, thatis, (10) switches arbitrarily between angle droop and frequencydroop primary control modes of individual microgrids. Intu-itively, the L2 stability property guarantees that the systemoutputs (angles, frequencies and voltages) are bounded forfinite disturbances.

In order to design a distributed secondary controller thatguarantees the stability of the network of interconnected mi-crogrids with MAFD with dynamics by (10), we will enforcethe property of QSR-dissipativity, defined as follows.

Definition 1. The nonlinear switched system (10) is said to beQSR-dissipative with input w and dissipativity matrices Qj ,Sj and Rj , j ∈ Σ, if there exists a positive definite storagefunction V (x) : R3N → R+ such that,[

y(t)w(t)

]′ [Qj SjS′j Rj

] [y(t)w(t)

]≥ V̇ (x(t)) (12)

holds for all t ≥ t0 ≥ 0, where x(t) is the state at timet resulting from the initial condition x(t0) and input w(·).Further, (10) is said to be QSR-state strictly input dissipative(QSR-SSID) if, for all t ∈ R+ and j ∈ Σ,[

y(t)w(t)

]′ [Qj SjS′j Rj

] [y(t)w(t)

]≥V̇ (x(t)) + φj(w(t)) + ψj(x(t)),

(13)

where φj(·), ψj(·) are positive definite functions of w(t) andx(t) respectively. Finally, a switched system (10) is said to belocally QSR-dissipative if it is QSR-dissipative for all x ∈ Xand w ∈ W where X ×W is a neighborhood of (x,w) = 0.

QSR-dissipativity is a very useful property for nonlinearswitched systems, since it implies L2 stability as follows.

Remark 2. A QSR-dissipative switched system (10) is L2stable if Qj < 0 for every j ∈ Σ.

In addition to L2 stability, QSR-dissipativity can also beused to capture other useful dynamical properties such asrobustness and transient performance via appropriate choiceof the Qj , Sj and Rj matrices [29]. Such dissipativity-basedcontrollers have also been shown to guarantee stability andperformance in several microgrid applications [26][30]. Wenow have the following result relating the dissipativity of thenonlinear system (10) to that of its linear approximation (11).

Proposition 1. The nonlinear switched system (10) is locallyQSR-dissipative if its linear approximation (11) is QSR-SSIDwith the same dissipativity matrices and a quadratic storagefunction V (x(t)) = x(t)′Px(t), where P ∈ R3N×3N andP > 0.

Proposition 1 extends the results of [26] to continuous-timeswitched systems. Using the concept of QSR-dissipativity,we now show that a distributed secondary controller thatguarantees L2 stability of the microgrid interconnection withMAFD primary control subject to disturbances w(t) can bedesigned by solving linear matrix inequalities as follows.

Theorem 1. If there exists symmetric positive definite matrixP > 0, negative definite matrix Qj < 0, and matricesUj , Vj , Sj and Rj of appropriate dimensions such that (14)is satisfied for every switching vector j ∈ Σ, then thedistributed secondary control law u(t) 7→ u(t) + ũ(t) whereũ(t) = Kσ(t)y(t) with

Kj = V−1j Uj , ∀j ∈ Σ, (15)

obtained by solving design equations (14), where Sv is theset of all diagonal matrices and SH is the set of all matriceswith the same sparsity structure as the Jacobian matrix H in(11), is sufficient to guarantee L2 stability of the microgrid

Mj =

−P (Aj +B(1)j H)− (Aj +B

(1)j H)

′P −B(1)j UjCj − C′jU′jB

(1)′

j −PB(2)j −B

(1)j UjDj + C

′jSj −C′jQ

1/2j−

−B(2)′

j P −D′jU′jB

(1)′

j + S′jCj D

′jS + S

′jDj +Rj −D′jQ

1/2j−

−Q1/2j− Cj −Q1/2j− Dj I

> 0 (14a)PB

(1)j = B

(1)j Vj , Q

1/2j− Q

1/2j− = −Qj (14b)

Vj ∈ Sv, Uj ∈ SH (14c)

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interconnection (10) in the D-MAFD framework with respectto disturbances w(t) under arbitrary loss of D-PMU anglemeasurements.

Using Theorem 1, a distributed secondary controller canbe designed for the microgrid interconnection with MAFD toguarantee stability even when D-PMU angle measurements arelost. We note that Theorem 1 guarantees stability for arbitraryloss of D-PMU angle measurements at any microgrid, sincethe control design does not require any a priori knowledgeof the switching vector σ(t). Further, by imposing a sparsityconstraint (14c) on the structure of the matrices Vj and Uj ,j ∈ Σ, we ensure that the secondary control law for microgridi only uses measurements from its immediate neighbors Ni.Therefore, the sparsity structure of the distributed secondarycontroller Kj will be the same as that of the network Ja-cobian matrix H , that is, Kj ∈ SH . We also note thatthe D-MAFD control design can accommodate constraintson the power sharing between microgrids, which can eitherbe incorporated while solving the power flow problem (1)to provide the power reference set points, or as additionalconstraints limiting the range and/or rate of change of thecontrol input ui(t) = [∆P iinj(t) ∆Q

iinj(t)]

′ in the controldesign in Theorem 1.

B. Robustness to network topology changes

In microgrid interconnections, topology changes may fre-quently occur due to islanding of some microgrids or lineoutages. In such scenarios, it is important to ensure thatthe stability of the microgrid interconnection with the newtopology can be guaranteed without redesigning the existingcontrollers in the system, even if D-PMU angle measurementlosses occur during islanding or reconnection of microgrids.We accomplish this objective by incorporating a robustnessmargin into the D-MAFD secondary control design as follows.

Let the perturbation in the system Jacobian due to thechange in network topology be given by

∆H = H −Hnew, (16)where Hnew is the Jacobian matrix after the network topologychange. Then the stability of the microgrid interconnectionwith the new system topology with respect to disturbancesw(t) even under D-PMU measurement losses is guaranteedby the following robustness result.

Theorem 2. Given a network topology change with a newJacobian matrix Hnew, and ∆H as defined in (16), if thereexists symmetric positive definite matrix P > 0, negativedefinite matrix Qj < 0, and matrices Uj , Vj , Sj and Rj

of appropriate dimensions such that (17) is satisfied withγ = ||B(1)j ∆H||2I for every j ∈ Σ, then the distributedcontrol law u(t) 7→ u(t) + ũ(t) where ũ(t) = Kσ(t)y(t) with

Kj = V−1j Uj , ∀j ∈ Σ, (18)

guarantees that the closed loop dynamics (10) is stable in theL2 sense with respect to any disturbance w(t) for the newnetwork topology. Furthermore, the control law u(t) 7→ u(t)+ũ(t) also guarantees stability of the closed loop system (10)for any new network topology with Jacobian matrix Ĥnewsuch that ||B(1)j ∆Ĥ||2I < γ, where ∆Ĥ = H − Ĥnew.

Selection of robustness margin: Theorem 2 presents adistributed secondary control design such that the microgridinterconnection in the D-MAFD framework is not only robustto a particular topology change, but also robust to any topologychange that results in a smaller perturbation in the system Ja-cobian than the one used for the control design. Therefore, formaximal robustness, the D-MAFD secondary controller shouldbe designed for the topology change that leads to the largestperturbation in the network Jacobian, that is, by selecting therobustness margin γ to be the maximum value of ||B(1)j ∆Ĥ||2over all possible topology changes. However, in practice, sucha choice of γ will require the computation of Jacobian matriceswith respect to a very large number of network topologies, andmay also be conservative. Therefore, the secondary controldesign using Theorem 2 can be carried out to guarantee(N − 1)-robustness, that is, robustness in the scenario wherea single microgrid is islanded or an outage takes place ona single line. In this case, an (N − 1)-contingency analysisconsidering islanding or outage scenarios can be performedto select the worst-case robustness margin γ for the secondarycontrol synthesis. With this design, robustness of the microgridinterconnection to topology changes can be guaranteed even ifD-PMU measurement losses occur during islanding, outagesor restoration operations.

IV. CASE STUDIESWe demonstrate the performance of the D-MAFD control

framework by considering a test five-microgrid interconnection(Fig. 4) constructed as described in [13] from the 123-feedertest system shown in Fig. 3. For this network with MAFDprimary control, we obtain a nonlinear switched system modelof the form (10). Since every microgrid can switch betweenthe angle droop control mode and the frequency droop controlmode at any time instant, the total number of switching modesof the system of five microgrids is given by 25 = 32. Wethen linearize the system around the power flow operatingpoints in Table 1. We present two test scenarios to illustrate the

M̂j =

−P (Aj +B

(1)j H)− (Aj +B

(1)j H)

′P −B(1)j UjCj − C′jU′jB

(1)′

j − 2γP −PB(2)j −B

(1)j UjDj + C

′jSj −C′jQ

1/2j−

−B(2)′

j P −D′jU′jB

(1)′

j + S′jCj D

′jS + S

′jDj +Rj −D′jQ

1/2j−

−Q1/2j− Cj −Q1/2j− Dj I

> 0 (17a)PB

(1)j = B

(1)j Vj , Q

1/2j− Q

1/2j− = −Qj (17b)

Vj ∈ Sv , Uj ∈ SH (17c)

> OAJPE-00088-2020 < 7

External

System

Fig. 3. IEEE 123-feeder test network partitioned into five microgrids. Fig. 4. Network parameters (p.u.) for 123-feeder five-microgrid test system.

Fig. 5. A. D-PMU measurement loss pattern and disturbance pattern forScenario 1. B. D-PMU measurement loss pattern for Scenario 2.

Fig. 6. Sparsity structure of the D-MAFD secondary controller for five-microgrid test system.

performance of the D-MAFD framework - (i) under D-PMUmeasurement losses and disturbances, and (ii) during systemtopology changes due to line restoration after an outage.

A. Scenario 1: D-PMU measurement loss and load change

Here, we assess the performance of the D-MAFD frame-work under D-PMU measurement losses. We use the linearizedsystem model around the power flow solution for Condition(i) in Table 1 to design three controllers for comparison:• C1: a distributed output-feedback secondary controller

(D-MAFD controller) by solving (14)-(15) for everyj ∈ Σ,

• C2: a centralized output-feedback secondary controllerby solving (14a), (14b), (15) for every j ∈ Σ, and

• C3: a centralized secondary controller by solving (14a),(14b), (15) for the microgrid interconnection where all

TABLE IPOWER FLOW SOLUTION FOR 123-FEEDER 5-MICROGRID TEST SYSTEM

Condition P refinj Qrefinj P

refload Q

refload V

ref δref

(p.u.) (p.u.) (p.u.) (p.u.) (p.u.) (deg.)µG1 0.79 1.35 0.92 0.47 1.000 0.000

(i) SW1, µG2 0.80 0.10 0.23 0.11 1.003 0.233SW2, µG3 0.20 0.10 0.45 0.20 1.000 0.110SW3 µG4 0.80 0.20 0.27 0.12 1.003 0.158closed µG5 0.20 0.10 0.92 0.95 0.999 0.052

µG1 0.80 1.36 0.92 0.47 1.000 0.000µG2 0.80 0.10 0.23 0.11 1.008 0.493

(ii) SW1 µG3 0.20 0.10 0.45 0.20 1.002 0.227open µG4 0.80 0.20 0.27 0.12 1.016 0.808

µG5 0.20 0.10 0.92 0.95 1.000 0.071µG1 0.82 1.38 0.92 0.47 1.000 0.000µG2 0.80 0.10 0.23 0.11 0.978 0.727

(iii) SW2 µG3 0.20 0.10 0.45 0.20 0.968 0.763open µG4 0.80 0.20 0.27 0.12 0.996 0.312

µG5 0.20 0.10 0.92 0.95 0.963 0.788µG1 0.80 1.36 0.92 0.47 1.000 0.000µG2 0.80 0.10 0.23 0.11 1.013 0.699

(iv) SW3 µG3 0.20 0.10 0.45 0.20 0.997 0.012open µG4 0.80 0.20 0.27 0.12 1.006 0.292

µG5 0.20 0.10 0.92 0.95 0.998 0.038

microgrids continue to use angle droop control with thelast available measurement even when D-PMU measure-ments are lost, that is, with the dynamics correspondingto the mode j = [1 1 . . . 1] in (10).

The control design LMIs are solved using YALMIP [31]. Weconsider a test pattern of D-PMU measurement losses and adisturbance acting on all microgrids as shown in Fig. 5-A. Forthis test pattern, we simulate the nonlinear system dynamicswith each of the designed controllers and make the followingobservations about their performance:• We note that the sparsity structure of the D-MAFD

secondary controller (Fig. 6) is the same as that of thenetwork Jacobian, indicating that each microgrid onlyuses output measurements from its immediate neighbors.

• From the angle and voltage profiles in Fig. 7, we ob-serve that the D-MAFD control design is successful instabilizing the microgrid interconnection under the testD-PMU measurement loss scenario in the presence ofdisturbances. On the other hand, when all microgridscontinue to use angle droop control with the last avail-able measurement during D-PMU measurement loss, thesystem suffers from poor transient performance, and theangle and voltage droop errors continue to increase fromt = 12s until the disturbance is withdrawn at t = 14s,indicating that angle droop control alone is unable tostabilize the system in this scenario.

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Angle droop controller D-MAFD Disturbance Switching instants

A.

B.

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3 (p.u.)

Fig. 7. A. Angle errors, and B. voltage errors of the D-MAFD design (C1) compared with a traditional angle droop controller (C3) for Scenario 1.

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∆V

5 (p.u.)

A.

B.

Centralized MAFD D-MAFD Disturbance Switching instants

Fig. 8. A. Angle errors, and B. voltage errors of the D-MAFD design (C1) compared with a centralized secondary control design (C2) for Scenario 1.

• A comparison with the performance of the centralizedsecondary control design for this scenario indicates thatthe performance of the D-MAFD design is comparable(Fig. 8), despite the secondary controller in the D-MAFD framework using limited information from othermicrogrids in the network. This is an advantage, sincedistributed control designs are typically significantly out-performed by centralized designs.

• The time constant of the D-MAFD controller is slightlyhigher, that is, the D-MAFD controller has slightly slowerdynamics than the traditional angle droop controller.This is natural, since indirect control of the angle viafrequency measurements is carried out in the D-MAFDframework during angle measurement loss. Therefore,the D-MAFD controller, while being much faster thana traditional frequency droop controller, is still slightlyslower than the angle droop controller. However, wenote that the angle droop controller cannot stabilize thenetwork under loss of D-PMU angle measurements, whilethe D-MAFD controller can stabilize the network underarbitrary measurement losses, as seen in Fig. 7.

• As mentioned in Remark 1, the MAFD framework allowsfor seamless switching between the angle and frequencydroop control modes when angle measurements are lostor restored. This is confirmed by the continuity of thestate and the absence of any switching transients in ourcase study in Fig. 7.

B. Scenario 2: Line reclosing after outage

In order to demonstrate the robustness of the D-MAFDcontrol design to changes in topology, we consider three faultscenarios that result in a line outage due to the opening ofeither SW1, SW2 or SW3 in Fig. 4. The power flow solutionsfor each case are provided in Table 1. From a contingencyanalysis of the system, we determine that the opening of SW2(outage on the tie line between microgrids µG1 and µG5)results in the worst case γ in Theorem 2. We design the D-MAFD controller corresponding to this outage by solving (17).

We study the performance of this D-MAFD control designwhen the faulted line is restored by reclosing SW2 at t = 5s,under the D-PMU measurement loss scenario shown in Fig. 5-B. Prior to the reclosing operation, the system initial conditioncorresponds to Condition (iii) in Table 1. After the reclosingoperation, it is desired that the system returns to the originaloperating point corresponding to Condition (i) in Table 1.From the angle and voltage error profiles for this scenario (Fig.9-B), we observe that the D-MAFD control design maintainssystem stability even when a D-PMU measurement loss occursduring the reclosing operation.

We also evaluate the robustness of the designed controllerby evaluating its performance for two other scenarios whereSW1 (Fig. 9-A) and SW3 (Fig. 9-C) are reclosed afteroutages resulting from faults. We observe that the D-MAFDsecondary controller designed for the worst-case line outage

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(p.u.)

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B.

C.

Time (sec)

Fig. 9. Angle and voltage errors of the D-MAFD design for Scenario 2,reclosing A. SW1, B. SW2 and C. SW3.

configuration (SW2 open) is also successful in maintainingsystem stability in these two scenarios. This indicates that theD-MAFD framework is robust to network topology changesand does not require redesigning the secondary controller tomaintain stability under different interconnection topologies.

V. CONCLUSION

We presented a mixed voltage angle and frequency droopcontrol framework with a distributed secondary controller (D-MAFD framework) for distribution-level microgrid intercon-nections where loss of D-PMU angle measurements may resultin degradation of stability and performance. The proposed D-MAFD framework provides a stable and robust solution toenhance the reliability of D-PMU based control designs formicrogrid interconnections. In addition to D-PMU measure-ment loss scenarios, the D-MAFD framework can also beapplied to legacy systems in which some microgrids operatewith traditional frequency droop control and others employnewer angle droop control schemes.

APPENDIX

Here, we present the proofs of all the results in Section III.

Proof of Proposition 1: If system (11) is QSR-SSID, then(13) is true for all j ∈ Σ. Substituting (11) in (13),[

x(t)w(t)

]′Γ(j)

[x(t)w(t)

]> φj(w(t)) + ψj(x(t)), (19)

where Γ(j)11 =C′jQjCj−P (Aj +B

(1)j H)−(Aj +B

(1)j H)

′P , Γ(j)12 =

C′jQjDj + C′jSj − PB

(2)j , Γ

(j)21 = Γ

(j)′

12 , and Γ(j)22 = D

′jQjDj +

D′jSj + S′jDj +Rj . Now consider

Λj =

[y(t)w(t)

]′ [Qj SjS′j Rj

] [y(t)w(t)

]− V̇ (x(t))

− φj(w(t))− ψj(x(t)), j ∈ Σ, (20)

for the nonlinear switched system (10). Since the linearization(11) is obtained by computing a first order Taylor approxima-tion of every mode of (10), we can substitute for ẋ(t) andy(t) from (10) in (20) and write the Taylor series expansionsof fj , gj and h around x = 0 and w = 0. Using (19), wecan then show that Λj is upper bounded by a function of thehigher order terms in the Taylor series expansion. Then, alongthe lines of [32, Theorem 3.1], these higher order terms canbe upper bounded to show that Λj > 0 in a neighborhood ofx = 0, w = 0 for all j ∈ Σ, completing the proof. �

Proof of Theorem 1: The dynamics of closed loop system(11) with output feedback controller u(t) 7→ u(t) + ũ(t),ũ(t) = Kσ(t)y(t) are given byẋ(t) = Âσ(t)x(t) + B̂

(2)

σ(t)w(t), y(t) = Ĉσ(t)x(t) + D̂σ(t)w(t),

where Âσ(t) = Aσ(t) + B(1)σ(t)H + B

(1)σ(t)Kσ(t)Cσ(t), B̂

(2)σ(t) =

B(2)σ(t) + B

(1)σ(t)Kσ(t)Dσ(t), Ĉσ(t) = Cσ(t) and D̂σ(t) = Dσ(t).

Since P > 0, it is full rank. If matrices B(1)j , j ∈ Σare full column rank, then Vj satisfying (14b) are invertible.Substituting equations (15) and (14b) in (14a) gives−PÂj − Â

′jP Ĉ

′jSj − PB̂

(2)j −Ĉ

′jQ

1/2j−

S′jĈj − B̂(2)′

j P D̂′jSj + S

′jD̂j +Rj −D̂′jQ

1/2j−

−Q1/2j− Ĉj −Q1/2j− D̂j I

> 0, (21)∀j ∈ Σ, where Q1/2j− Q

1/2j− = −Qj . By taking the Schur’s

complement, it is easy to conclude that the closed loop system(11) with the controller in Theorem 1 is QSR-SSID. The resultcan now be obtained using Proposition 1 and Remark 2. �

Proof of Theorem 2: Consider the closed loop system (10)with a new Jacobian matrix Hnew and the control law u(t) 7→u(t) + ũ(t), where ũ(t) = Kσ(t)y(t), ∀j ∈ Σ is obtainedfrom (17)-(18). Now consider the matrix Mj in (14a) with itsfirst term updated to [Mj ]11 = −P (Aj +B(1)j Hnew)− (Aj +B

(1)j Hnew)

′P −B(1)j UjCj − C ′jU ′jB(1)′

j . Then,

Mj − M̂j =

−P (B(1)j ∆H)− (B

(1)j ∆H)

′P + 2γP 0 0

0 0 0

0 0 0

, (22)where γ = ||B(1)j ∆H||2I . Clearly, since γ ≥ B

(1)j ∆H , Mj −

M̂j ≥ 0. If (17) holds, Mj ≥ M̂j > 0 =⇒ Mj > 0.Thus, using Theorem 1, if (17), (18) holds, the closed loopsystem (10) is locally L2 stable with the new Jacobian matrixHnew = H+∆H . It is then fairly straightforward to show thatthis control law renders the closed loop system L2-stable forany new network topology with Jacobian matrix Ĥnew suchthat ||B(1)j ∆Ĥ||2I < γ, where ∆Ĥ = H − Ĥnew. �

ACKNOWLEDGMENT

The first author thanks Shailaja Seetharaman for the designof the illustrations in Figs. 1 and 2.

REFERENCES

[1] J. Sexauer, P. Javanbakht, and S. Mohagheghi, “Phasor measurementunits for the distribution grid: Necessity and benefits,” in IEEE PESInnovative Smart Grid Technologies (ISGT), 2013, pp. 1–6.

> OAJPE-00088-2020 < 10

[2] A. Von Meier, D. Culler, A. McEachern, and R. Arghandeh, “Micro-synchrophasors for distribution systems,” in IEEE PES Innovative SmartGrid Technologies Conference (ISGT), 2014, pp. 1–5.

[3] A. von Meier, E. Stewart, A. McEachern, M. Andersen, andL. Mehrmanesh, “Precision micro-synchrophasors for distribution sys-tems: A summary of applications,” IEEE Trans. on Smart Grid, vol. 8,no. 6, pp. 2926–2936, 2017.

[4] S. Peisert, R. Gentz, J. Boverhof, C. McParland, S. Engle, A. El-bashandy, and D. Gunter, “Lbnl open power data,” Lawrence Berke-ley National Laboratory, 2017, http://dx.doi.org/10.21990/C21599 Re-trieved from https://escholarship.org/uc/item/92f19749.

[5] M. Jamei, A. Scaglione, C. Roberts, E. Stewart, S. Peisert, C. McParland,and A. McEachern, “Anomaly detection using optimally placed µ-pmusensors in distribution grids,” IEEE Transactions on Power Systems,vol. 33, no. 4, pp. 3611–3623, 2017.

[6] G. Sanchez-Ayala, J. R. Agüerc, D. Elizondo, and M. Lelic, “Currenttrends on applications of pmus in distribution systems,” in IEEE PESInnovative Smart Grid Technologies (ISGT), 2013, pp. 1–6.

[7] J. M. Guerrero, J. C. Vasquez, J. Matas, L. G. De Vicuña, andM. Castilla, “Hierarchical control of droop-controlled ac and dc mi-crogrids—a general approach toward standardization,” IEEE Trans. onIndustrial Electronics, vol. 58, no. 1, pp. 158–172, 2011.

[8] J. C. Vasquez, J. M. Guerrero, J. Miret, M. Castilla, and L. G. De Vi-cuna, “Hierarchical control of intelligent microgrids,” IEEE IndustrialElectronics Magazine, vol. 4, no. 4, pp. 23–29, 2010.

[9] R. Majumder, B. Chaudhuri, A. Ghosh, R. Majumder, G. Ledwich, andF. Zare, “Improvement of stability and load sharing in an autonomousmicrogrid using supplementary droop control loop,” IEEE Trans. onPower Systems, vol. 25, no. 2, pp. 796–808, 2010.

[10] J. M. Guerrero, J. C. Vasquez, J. Matas, M. Castilla, and L. G. de Vicuna,“Control strategy for flexible microgrid based on parallel line-interactiveups systems,” IEEE Trans. on Industrial Electronics, vol. 56, no. 3, pp.726–736, 2009.

[11] R. Majumder, A. Ghosh, G. Ledwich, and F. Zare, “Angle droopversus frequency droop in a voltage source converter based autonomousmicrogrid,” in IEEE PES General Meeting, 2009, pp. 1–8.

[12] ——, “Power management and power flow control with back-to-backconverters in a utility connected microgrid,” IEEE Trans. on PowerSystems, vol. 25, no. 2, pp. 821–834, 2010.

[13] Y. Zhang and L. Xie, “Online dynamic security assessment of microgridinterconnections in smart distribution systems,” IEEE Trans. on PowerSystems, vol. 30, no. 6, pp. 3246–3254, 2015.

[14] ——, “A transient stability assessment framework in power electronic-interfaced distribution systems,” IEEE Trans. on Power Systems, vol. 31,no. 6, pp. 5106–5114, 2016.

[15] Y. Zhang, L. Xie, and Q. Ding, “Interactive control of coupled micro-grids for guaranteed system-wide small signal stability,” IEEE Trans. onSmart Grid, vol. 7, no. 2, pp. 1088–1096, 2016.

[16] W. Yao, Y. Liu, D. Zhou, Z. Pan, J. Zhao, M. Till, L. Zhu, L. Zhan,Q. Tang, and Y. Liu, “Impact of gps signal loss and its mitigationin power system synchronized measurement devices,” IEEE Trans. onSmart Grid, vol. 9, no. 2, pp. 1141–1149, 2018.

[17] W. Yao, D. Zhou, L. Zhan, Y. Liu, Y. Cui, S. You, and Y. Liu, “Gpssignal loss in the wide area monitoring system: Prevalence, impact, andsolution,” Electric Power Systems Research, vol. 147, pp. 254–262, 2017.

[18] C. Huang, L. Fangxing, Z. Dao, G. Jiahui, P. Zhuohong, L. Yong, andL. Yilu, “Data quality issues for synchrophasor applications part i: areview,” Journal of Modern Power Systems and Clean Energy, vol. 4,no. 3, pp. 342–352, 2016.

[19] M. S. Almas and L. Vanfretti, “Impact of time-synchronization signalloss on pmu-based wampac applications,” in IEEE PES General Meet-ing, 2016, pp. 1–5.

[20] D. P. Shepard, T. E. Humphreys, and A. A. Fansler, “Evaluation ofthe vulnerability of phasor measurement units to gps spoofing attacks,”International Journal of Critical Infrastructure Protection, vol. 5, no.3-4, pp. 146–153, 2012.

[21] “Extended loss of gps impact on reliability,” North American ElectricReliability Corporation (NERC), Tech. Rep., 2012.

[22] S. Sivaranjani, E. Agarwal, L. Xie, V. Gupta, and P. Antsaklis, “Mixedvoltage angle and frequency droop control for transient stability ofinterconnected microgrids with loss of pmu measurements,” in AmericanControl Conference (ACC), 2020, pp. 2382–2387.

[23] S. Sivaranjani and D. Thukaram, “A networked control systems perspec-tive for wide-area monitoring control of smart power grids,” in IEEEInnovative Smart Grid Technologies-Asia (ISGT Asia), 2013, pp. 1–6.

[24] ——, “Networked control of smart grids with distributed generation,”in IEEE India Conference, 2013, pp. 1–6.

[25] A. K. Singh, R. Singh, and B. C. Pal, “Stability analysis of networkedcontrol in smart grids,” IEEE Trans. on Smart Grid, vol. 6, no. 1, pp.381–390, 2015.

[26] E. Agarwal, S. Sivaranjani, and P. J. Antsaklis, “Feedback passivationof nonlinear switched systems using linear approximations,” in IndianControl Conference (ICC), 2017, pp. 12–17.

[27] R. Majumder, G. Ledwich, A. Ghosh, S. Chakrabarti, and F. Zare,“Droop control of converter-interfaced microsources in rural distributedgeneration,” IEEE Transactions on Power Delivery, vol. 25, no. 4, pp.2768–2778, 2010.

[28] Y. Zhang, “A clean slate control framework for future distributionsystems,” Ph.D. dissertation, Texas A&M University, 2016.

[29] E. Agarwal, “Compositional control of large-scale cyber-physical sys-tems using dissipativity theory,” Ph.D. dissertation, University of NotreDame, 2019.

[30] S. Sivaranjani, J. R. Forbes, P. Seiler, and V. Gupta, “Conic-sector-basedanalysis and control synthesis for linear parameter varying systems,”IEEE Control Systems Letters, vol. 2, no. 2, pp. 224–229, 2018.

[31] J. Lofberg, “Yalmip: A toolbox for modeling and optimization inmatlab,” in IEEE International Symposium on Computer Aided ControlSystems Design, 2004, pp. 284–289.

[32] H. Wang and J. Zhao, “Passivity and h∞ control of switched discrete-time nonlinear systems using linearisation,” International Journal ofSystems Science, vol. 49, no. 1, pp. 68–83, 2018.

S Sivaranjani is a postdoctoral researcher in theDepartment of Electrical and Computer Engineeringat Texas A&M University. She received her Ph.D.,M.S., and B.E. degrees, all in Electrical Engineering,from the University of Notre Dame, the Indian Insti-tute of Science, and the PES Institute of Technology,respectively. Her research interests are in data-drivenand distributed control for large-scale infrastructurenetworks, with emphasis on energy systems andtransportation networks. She is a recipient of theSchlumberger Foundation Faculty for the Future fel-

lowship (2015-18), the Zonta International Amelia Earhart fellowship (2015-16) and the Notre Dame Ethical Leaders in STEM fellowship (2016-17).

Etika Agarwal received her Ph.D. and M.S. degrees,both in Electrical Engineering, from the Universityof Notre Dame in 2019 and 2016 respectively, andher B. Tech in Avionics from the Indian Instituteof Space Science and Technology in 2012. Beforejoining the graduate school, she worked with theIndian Space Research Organization from 2012-2014. Her research interests are in computationallyefficient and scalable control of large-scale cyber-physical systems.

Vijay Gupta Vijay Gupta is a Professor in theDepartment of Electrical Engineering at the Uni-versity of Notre Dame, having joined the facultyin January 2008. He received his B. Tech degreeat Indian Institute of Technology, Delhi, and hisM.S. and Ph.D. at California Institute of Technology,all in Electrical Engineering. He received the 2018Antonio J Rubert Award from the IEEE ControlSystems Society, the 2013 Donald P. Eckman Awardfrom the American Automatic Control Council and a2009 National Science Foundation (NSF) CAREER

Award. His research and teaching interests are broadly in the interface ofcommunication, control, distributed computation, and human decision making.

> OAJPE-00088-2020 < 11

Panos Antsaklis is the H.C. & E.A. Brosey Pro-fessor of Electrical Engineering at the Universityof Notre Dame. He is graduate of the NationalTechnical University of Athens, Greece, and holdsMS and PhD degrees from Brown University. His re-search addresses problems of control and automationand examines ways to design control systems thatwill exhibit high degree of autonomy. His currentresearch focuses on Cyber-Physical Systems and theinterdisciplinary research area of control, computingand communication networks, and on hybrid and

discrete event dynamical systems. He is IEEE, IFAC and AAAS Fellow,President of the Mediterranean Control Association, the 2006 recipient ofthe Engineering Alumni Medal of Brown University and holds an HonoraryDoctorate from the University of Lorraine in France. He served as thePresident of the IEEE Control Systems Society in 1997 and was the Editor-in-Chief of the IEEE Transactions on Automatic Control for 8 years, 2010-2017.

Le Xie (S’05-M’10-SM’16) received the B.E. degreein electrical engineering from Tsinghua University,Beijing, China, in 2004, the M.S. degree in engineer-ing sciences from Harvard University, Cambridge,MA, USA, in 2005, and the Ph.D. degree from theDepartment of Electrical and Computer Engineering,Carnegie Mellon University, Pittsburgh, PA, USA,in 2009. He is currently a Professor with the De-partment of Electrical and Computer Engineering,Texas A&M University, College Station, TX, USA.His research interests include modeling and control

of large-scale complex systems, smart grids application with renewable energyresources, and electricity markets.

I INTRODUCTIONII Mixed Voltage Angle and Frequency Droop Control (MAFD) ModelIII D-MAFD Secondary Control DesignIII-A Distributed Secondary Control SynthesisIII-B Robustness to network topology changes

IV Case StudiesIV-A Scenario 1: D-PMU measurement loss and load changeIV-B Scenario 2: Line reclosing after outage

V ConclusionReferencesBiographiesS SivaranjaniEtika AgarwalVijay GuptaPanos AntsaklisLe Xie