A Distributed Frequency Regulation Architecture for Islanded … · 2017. 4. 25. · 1 A Distributed Frequency Regulation Architecture for Islanded Inertia-Less AC Microgrids Stanton

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A Distributed Frequency Regulation Architecture forIslanded Inertia-Less AC Microgrids

Stanton T. Cady, Student Member, IEEE, Madi Zholbaryssov, Student Member, IEEE,Alejandro D. Domı́nguez-Garcı́a, Member, IEEE, and Christoforos N. Hadjicostis, Senior Member, IEEE

Abstract—We address the problem of frequency regulationin islanded ac microgrids with no inertia, i.e., those consistingentirely of generators interfaced through power electronics. Thecontrol architecture we propose to achieve this is designed todrive the average frequency error to zero while ensuring thatthe frequency at every bus is equal and that the operatingpoint that results is stable. We also introduce a distributedimplementation of the proposed control architecture that relieson a combination of three distributed algorithms. Two of thealgorithms, which are well-established consensus-type algorithms,allow the generators and loads to acquire global informationneeded for making control decisions; the third algorithm, whichwe propose herein, enables the generators to obtain output valuesthat balance the total demand for load without violating line flowconstraints. Collectively, these algorithms eliminate the need fora centralized entity with complete knowledge of the network,its topology, or the capabilities or properties of the generatorsand loads therein. Moreover, the distributed implementation wepropose relies on minimal measurements, requiring only thatthe power injection at each bus be measured. To verify ourproposed control architecture and the algorithms on which itsdistributed implementation relies, we analytically show that theresulting closed-loop system is stable and establish convergenceof our proposed algorithm. We also illustrate the features ofthe architecture using numerical simulations of three test casesapplied to six- and 37-bus networks.

I. INTRODUCTION

Although the nascency of microgrids precludes a strict def-inition, any collection of interconnected generators and loadsthat is capable of islanded operation is generally considered tobe one. It follows, then, that there are no formal restrictionson the types of generators that may be present in a microgrid;however, like the prototypical example of a neighborhoodcomprising homes with rooftop-mounted photovoltaic (PV) ar-rays [1], microgrids are typically envisaged to consist entirelyof generators that are interfaced through power electronics.Consequently, without the spinning mass inherent in traditionalsynchronous generators, this class of microgrid has little tono effective inertia. Moreover, irrespective of the presence ofinertia, the power demands of loads in a microgrid can belarge relative to the output capabilities of each generator.

Even though the properties mentioned above may notcharacterize all types of microgrids, e.g., dc microgrids or

S. T. Cady, M. Zholbaryssov, and A. D. Domı́nguez-Garcı́a are with theECE Department at the University of Illinois at Urbana-Champaign, Urbana,IL 61801, USA. E-mail: {scady2, zholbar1, aledan}@ILLINOIS.EDU.

C. N. Hadjicostis is with the ECE Department at the University ofCyprus, Nicosia, Cyprus, and also with the ECE Department at the Uni-versity of Illinois at Urbana-Champaign, Urbana, IL 61801, USA. E-mail:[email protected].

microgrids with traditional generation (see, e.g., [2] and thereferences therein), in this paper, we restrict considerationto islanded ac microgrids with purely power electronics-interfaced generators, i.e., those with no inertia, which wehenceforth refer to simply as microgrids. Furthermore, whilethese properties, among others, complicate the problem offrequency regulation, microgrids are unencumbered by therequirements and well-established concepts of their largercounterparts (see, e.g., [3]), making them amenable to newcontrol paradigms. In particular, whereas generation controlarchitectures for large power systems typically depend on acentrally located decision-maker, numerous distributed archi-tectures have been proposed for microgrids (see, e.g., [4], [5]).Through a combination of computations performed by proces-sors located at each bus and information exchanged betweenneighboring processors, these distributed architectures, whichcan meet the same objectives as their centralized counterparts,obviate the need for an entity with complete information aboutthe number, type, or capabilities of the generation units in thesystem. Additionally, by eliminating the need for a centralizedprocessor and a communication network connecting it to eachgenerator, these distributed approaches can achieve highersystem-level efficiency, reliability, and adaptability.

Regardless of the control paradigm employed, a frequencyregulation architecture for microgrids must be designed toensure that stable operation results and that the frequency atevery bus in the system is equal to the desired reference value.While numerous control schemes for microgrids have beenproposed in the literature, designing a frequency regulationarchitecture that achieves the aforementioned properties hasreceived limited attention thus far. In fact, to the authors’knowledge, no previous work has presented a scheme forfrequency control in inertia-less ac microgrids for whichstability and system-wide operation at the same frequency areguaranteed. Next, we discuss some of the work that has beenpresented in the literature on control for microgrids includingfrequency regulation.

The authors in [6]–[10] propose centralized approaches tomicrogrid control, with overall control schemes presented in[9] and strategies for optimizing operation provided in [7].While [8]–[10] propose control architectures for frequencyregulation of microgrids with power electronics-interfacedgenerators, none of these references provides analytical resultsguaranteeing both stability and system-wide operation at thedesired frequency. Distributed approaches to frequency controlhave been proposed in [5], [11]–[15], as well as our ownprior work in [16]. The authors in [11], [12], [14] proposed

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control schemes for frequency regulation that they analyticallyshowed to be stable; however, [11] relies on a linearizednetwork model, [12] can only guarantee stability for certaininitial conditions, and [14] does not guarantee that every busoscillates at the same frequency. In our previous work in [16],we proposed a distributed approach to frequency regulationin microgrids, including those with generation resources in-terfaced through power electronics. Although we were ableto experimentally validate the proposed architecture, we onlydid so using a system with synchronous generators, andcould not analytically guarantee that our controller resultedin closed-loop stability or that the frequency at every buswould be equal. Finally, the authors in [5], [15] proposed acontrol architecture that they analytically showed to result insystem-wide operation at the desired frequency, but could onlyguarantee local stability.

Despite the lack of existing control architectures designedto ensure stable operation at a common reference frequency,the authors in [17] provide a condition for a broad classof coupled oscillators, which can model the generators andloads in an inertia-less microgrid, that is sufficient for meetingthe so-called phase-cohesiveness requirement. In the contextof power flows in a microgrid, satisfying this condition isequivalent to ensuring that the angle difference between everypair of connected buses in steady state will be strictly less thanπ2 . Moreover, if the condition for phase cohesiveness is met,the natural frequencies of the oscillators—equivalent to thepower injected at each bus in a microgrid—and the couplingbetween them—equivalent to the admittances of the branchesinterconnecting buses in a microgrid—are such that the systemexhibits coherent behavior, i.e., the angle of every oscillatorevolves at the same rate. Thus, when combined with a schemefor regulating the average frequency error, ensuring phasecohesiveness is sufficient for ensuring stability and system-wide operation at a common frequency. While we providedsome details for a control architecture that makes use of thephase-cohesiveness condition in our earlier work in [18], wedid not specifically address the problem of choosing feedbackgains that ensure stability of the closed-loop system, nor didwe have a proof for one of the main algorithms on which theoperation of the architecture relies; we address both of theseand other issues in this paper.

In the discussion that follows, we provide an overview of thedistributed architecture we propose for frequency regulation inislanded inertia-less ac microgrid and outline the contributionspresented herein. As in [18], our proposed control architectureis designed to take advantage of the results in [17] by trackinggenerator set-points that, for some initial load demand, areknown to result in phase-cohesive operation. Following one ormore small perturbations to the power demanded by the loads,the architecture iteratively adjusts the generator set-points todrive the average frequency to some reference value while alsoensuring that the operating point that results is phase cohesive.By regulating the average frequency around an operating pointthat is known to be phase cohesive, our controller ensuressmall-signal stability of the closed-loop system while alsoguaranteeing that the frequency at every bus is the same. Tohandle larger load perturbations, we also provide a method

for triggering the recomputation of the generator set-points tobe tracked based upon an estimate for the amount by whichthe system has deviated from the original phase-cohesiveoperating point. Additionally, we propose and provide a proofof convergence for a distributed algorithm that enables thecomputation of generator output values that collectively meetthe total load power demand without violating generationor line flow limits. Using this distributed algorithm andtwo consensus-type algorithms, we also outline a distributedimplementation of our proposed control architecture. Thesethree algorithms enable the acquisition of global informationwith which processors located at each bus can make decisionsto collectively achieve the system-level objectives of ourproposed control architecture. Finally, we provide analyticalcriteria for choosing gains that result in closed-loop stabilityand demonstrate the operation of our proposed architecture andits distributed implementation using numerical simulations.

II. PRELIMINARIES

We begin this section by outlining a model to represent thephysical network of a microgrid. The model uses notions fromgraph theory to represent the interconnections between busesin the system as well as differential equations to represent thedynamic behavior of the generators and loads that compriseit. Next, we introduce another graph-theoretic model to rep-resent the communication network on which the distributedimplementation of our proposed control architecture relies.Finally, we formally define the notion of phase cohesivenessand outline a sufficient condition for achieving it; we thenpose the criterion for achieving phase cohesiveness and othercontrol objectives as a feasibility problem.

A. Physical Layer Model

We consider ac microgrids comprising loads and generatorsthat can be represented by a first-order dynamical model; wefurther restrict consideration to systems in which all gener-ation units are interfaced through power electronics, i.e., weconsider systems comprising generators with no inertia; non-dispatchable generators are considered as loads injecting nega-tive power. Although electrical lines in typical microgrids havenon-negligible losses, we make the reasonable assumption thatthe resistance-to-reactance ratio (commonly referred to as theR/X ratio) is homogeneous across all lines, which allows usto use a linear transformation in [15] to recover a losslessmodel. To facilitate the analytical discussion presented herein,we assume that the generators have no internal impedance,the power network is lossless, and the voltage magnitude atevery bus is constant and unity. Although these assumptionsmay lead to modeling inaccuracies under extreme loadingscenarios, the control architecture we propose in Section III isdesigned for operation under nominal conditions where suchmodels justifiably approximate the behavior of generators andloads (see, e.g., [17]).

For an n-bus microgrid, let Gp = (Vp, Ep) be an undi-rected simple graph representing the interconnections betweenbuses. The vertex—or bus—set, Vp, is defined to be Vp :=

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{1, 2, . . . , n} = V(g)p ∪V(`)p , where V(g)p and V(`)p denote gener-ator and load bus sets, respectively. Without loss of generality,we partition the bus set such that V(g)p := {1, 2, . . . ,m},V(`)p := {m + 1,m + 2, . . . , n}, and V(g)p ∩ V(`)p = ∅,i.e., each bus has only a generator or load attached, butnot both. The edge—or branch—set, Ep, is defined to beEp ⊆ {{i, j} : i 6= j, i, j ∈ Vp}, where the edge {i, j} ∈ Epif buses i and j, i 6= j, are connected electrically. Wedenote the set of buses to which each bus i is connected byNp(i) := {j ∈ Vp : {i, j} ∈ Ep}, and denote the number ofsuch buses by δp(i) = |Np(i)|. Finally, we assume that noislands exist in the microgrid such that the graph Gp consistsof a single connected component.

If we arbitrarily assign a direction to each edge e = {i, j} ∈Ep, the oriented incidence matrix, denoted by M ∈ Rn×|Ep|,is defined as M := [mie] where

mie =

−1, if i is the sink node of edge e,1, if i is the source node of edge e,0, otherwise.

(1)

Furthermore, we define the weighted Laplacian matrix of Gpas L = Mdiag({Bij : {i, j} ∈ Ep})MT, where diag({Bij :{i, j} ∈ Ep}) is an |Ep| × |Ep| diagonal matrix consisting ofsusceptances, Bij = Bji, {i, j} ∈ Ep, between electricallyconnected buses i and j. Given that the Laplacian is singular(due to its zero eigenvalue), we will utilize its Moore-Penrosepseudo inverse, denoted by L† (see, e.g., [19] for details onhow to compute it).

At time t > 0, let the voltage angle at bus i ∈ Vp be denotedby θi(t); similarly, let the set-point of generator i ∈ V(g)p bedenoted by ui(t), and let ui and ui denote lower and upperbounds such that 0 ≤ ui ≤ ui(t) ≤ ui. Additionally, let`i(t) := `

0i + ∆`i(t) ≥ 0 denote the real power demand at

load bus i ∈ V(`)p where `0i ≥ 0 is the demand at t = 0, and∆`i(t) is a load perturbation that occurs at some t > 0. Foreach inverter we adopt a controllable voltage source behinda reactance model (see, e.g., [15]), and write the microgriddynamics as follows:

Didθi(t)

dt= Diω0 + ui(t)−

∑j∈Np(i)

Bij sin(θi(t)− θj(t)),

(2)

for i ∈ V(g)p , and

Didθi(t)

dt=Diω0 − (`0i + ∆`i(t))

−∑

j∈Np(i)

Bij sin(θi(t)− θj(t)), (3)

for i ∈ V(`)p , where Di > 0 is a time constant associatedwith the dynamics of the generator or load at bus i ∈ Vp, andω0 denotes the reference frequency. The model in (2) – (3)is based on the standard decoupling approximation where thefrequency and voltage control loops are decoupled so that thefrequency is regulated to adjust the active power assumingconstant voltage magnitudes, and the voltage magnitude isregulated to adjust the reactive power [15].

B. Cyber Layer Model

In Section V, we will introduce a distributed implementationof our proposed frequency regulation architecture that relieson processors located at each bus. To realize this distributedimplementation, we assume that each processor is capable ofperforming low-complexity computations, e.g., addition andmultiplication, based upon information obtained locally andfrom the processors of neighboring buses. In this section,we introduce a second graph-theoretic model to representthe communication network over which the local processorscan exchange information with other neighboring processors.While the connectedness of the graph used to model thephysical layer is sufficient for the algorithms on which ourdistributed implementation relies, we abstract the cyber layerfrom the underlying physical layer to enable a more generalcommunication modality.

To allow for the possibility of unidirectional transfers ofinformation between neighboring processors, we represent thecyber layer by a directed graph, which we denote by Gc =(Vc, Ec). The vertex—or node—set, Vc, consists of processors,one for each bus in the physical layer, i.e., Vc := {1, 2, . . . , n}.Although Vc and Vp are defined identically, the physical andcyber layer vertex sets need not be equal; however, since eachload and generator is outfitted with a processor, there must bea one-to-one correspondence between the vertex sets of thetwo layers, i.e., there must exist a bijection hc : Vp 7→ Vc.The communication graph edge set, Ec, is defined to be Ec ⊆Vc × Vc, where the ordered pair (i, j) ∈ Ec if node i canreceive information from node j.

We refer to the set of nodes from which node i can receiveinformation as its in-neighborhood and denote it by N−c (i) :={j ∈ Vc : (i, j) ∈ Ec}. Similarly, we refer to the set of nodesto which node i can send information as its out-neighborhoodand denote it by N+c (i) := {j ∈ Vc : (j, i) ∈ Ec}. Thenumber of nodes that can receive information from node i,which we refer to as its out-degree, is the cardinality of theout-neighborhood and we denote it by δ+c (i) := |N+c (i)|.

For a pair of nodes i, j ∈ Vc, and for ν0, ν1, . . . , ντ ∈ Vcand e0, e1, . . . , eτ−1 ∈ Ec, we refer to the alternating sequenceof nodes and edges i ≡ ν0, e0, ν1, e1, ν2, . . . , ντ−1, eτ−1, ντ ≡j such that ei = (νl+1, νi) for l = 0, 1, . . . , τ−1 as a directedpath of length τ between nodes i and j. Furthermore, werefer to the minimum distance from i to j, i 6= j, as theshortest-length directed path between the nodes, and denoteits value by dc(i, j), with dc(i, j) = ∞ if no path exists.The diameter of Gc, which we denote by ∆c, is defined tobe the longest shortest path between any two nodes, i.e.,∆c := maxi,j∈Vc, i 6=j dc(i, j). Finally, for the algorithmsintroduced in Section IV that rely on Gc, we require thatthe graph modeling the communication network be stronglyconnected, i.e., we require that the diameter of Gc be finite.

C. Phase Cohesiveness Criterion

The control architecture we will propose in Section III isdesigned to adjust the output of the generators in a microgridto track pre-determined set-points while simultaneously elimi-nating the frequency error that results from small perturbations

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in load. These pre-determined set-points are chosen such that,when applied, the system is stable around the resultant operat-ing point, the average frequency equals some reference value,and the frequency at every bus is the same. In the discussionthat follows, we formalize the properties that characterize suchan operating point, and introduce a feasibility problem forchoosing generator set-points, the solution of which meetsthese properties.

Given constant generator set-points, which we denote by u∗i ,i ∈ V(g)p , and the initial load power demands `0i , i ∈ V

(`)p , as

described in Section II-A, let θ∗ =[θ∗1 , θ

∗2 , . . . , θ

∗n

]Tdenote

a steady-state operating point of the system in (2) – (3).Then, for some reference frequency, ω0, we would like todetermine the u∗i ’s such that, when applied to the generators,the operating point θ∗ exists and is characterized by thefollowing properties:

P1. the average frequency in the system equals the reference,i.e.,

∑i∈Vp θ̇i(t)

n = ω0;P2. the frequency at every bus is equal, i.e., |θ̇i(t)−θ̇j(t)| = 0

for i, j ∈ Vp; andP3. the system is stable around the operating point θ∗, i.e.,

−∇θ(t)h(θ(t))∣∣∣∣θ=θ∗

� 0, where θ :=[θ1(t), . . . , θn(t)

]Tand h(θ(t)) = [he(θ(t))] with he(θ(t)) := Bij sin(θi −θj) for e = {i, j} ∈ Ep.

While we will show in Section III that meeting PropertyP1 can be achieved by balancing generation and demand,i.e., choosing the u∗i ’s such that

∑i∈V(g)p

u∗i =∑i∈V(`)p

`0i ,ensuring that Properties P2 and P3 are met is more difficult,especially when trying to do so in a distributed fashion.However, if generation and demand are balanced and the so-called phase-cohesiveness condition is met, i.e., |θ∗i − θ∗j | ≤ φfor {i, j} ∈ Ep and φ ∈ [0, π/2) [17], it can be shownthat the resultant operating point, θ∗, exists and is char-acterized by Properties P1 – P3. To that end, if we letu∗ =

[u∗1, u

∗2, . . . , u

∗m

]Tand `0 =

[`0m+1, `

0m+2, . . . , `

0n

]T,

then it follows from the results in [17] that, for a broad classof network topologies, if∥∥∥∥MTL† [ u∗−`0

]∥∥∥∥∞≤ sin(φ), (4)

for some angle φ ∈ [0, π/2), the resulting operating point, θ∗,exists and is phase cohesive. In subsequent developments, werefer to such u∗i ’s as phase-cohesive set-points; although thisterm is a slight abuse of nomenclature, we use it as short-handto refer to set-points that result in phase-cohesive operation,subject to the imposed load demands. Despite being provenfor several topologies, including acyclic networks and thosecomprising low-dimensional cycles, the condition in (4) isnot sufficient for general networks. As a result, we restrictconsideration to network topologies for which (4) holds (werefer the reader to [17] and supporting information for details).

Beyond finding phase-cohesive set-points that are balancedwith demand, we must further restrict the problem to findingthose set-points that lie within the individual bounds of thegenerators, i.e., ui ≤ u∗i ≤ ui, i ∈ V

(g)p . Thus, the task

of finding generator set-points that meet all of the above-

described requirements can be summarized by the followingfeasibility problem:

find u

subject to∑i∈V(g)p

ui =∑i∈V(`)p

`0i ,∥∥∥∥MTL† [ u−l0]∥∥∥∥∞≤ sin(φ),

ui ≤ ui ≤ ui, ∀i ∈ V(g)p .

(5)

III. FREQUENCY REGULATION ARCHITECTUREIn this section, we propose a control architecture that is

designed to regulate the frequency in an inertia-less ac micro-grid. We begin by providing an overview of the architectureand its operation. Then, we formalize the control scheme andoutline criteria for choosing gains that yield closed-loop sta-bility. Finally, we describe a method for determining when torecompute generator set-points to ensure phase cohesiveness.

A. Overview

From a system-level perspective, the scheme we propose forfrequency regulation is similar to a discrete-time proportionalintegral controller. More specifically, after determining set-points that satisfy (5), which we denote by u∗i , i ∈ V

(g)p , the

set-point of each generator i is incrementally adjusted awayfrom u∗i over several discrete time intervals. Following oneor more small perturbations in the power demanded by theloads, these incremental changes serve to drive the resultingfrequency error in the system to zero. However, in general,these incremental adjustments move the system away from anoperating point that is known to be phase cohesive. To counterthis adverse effect, our control architecture includes a methodfor determining when to recompute the u∗i ’s based upon anestimate for the amount by which the system has deviatedfrom the phase-cohesive operating point.

Without loss of generality, if we transform the microgridmodel in (2) – (3) to a reference frame rotating at the referencefrequency, ω0, we see that an operating point at which thefrequency of every bus equals ω0 is equivalent to one at whichthe derivative of the voltage angle at each bus is zero, i.e.,dθi(t)dt = 0, i ∈ Vp. [To keep notation simple, we use the same

θ in the original and the rotating reference frames.] Let ∆ω(t)denote the average frequency error in this reference frame,weighted by the time constants associated with the dynamicsof the generators and loads, i.e.,

∆ω(t) :=

∑ni=1Di

dθi(t)dt∑n

i=1Di, (6)

for t > 0. Replacing the numerator in (6) with a summationof all of the equations in (2) and (3), it follows that the valueof the average frequency error at time t > 0 is given by

∆ω(t) =1∑n

i=1Di

∑i∈V(g)p

ui(t)−∑i∈V(`)p

`i(t)

. (7)As mentioned above, our frequency regulation architecture

incrementally adjusts the generator set-points over several

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discrete time intervals. We refer to the intervals during whichthe controller is executed as rounds and index them byr = 0, 1, 2, . . . . To simplify notation, we reset the round indexevery time the set-points to be tracked are computed, i.e., theu∗i ’s are always determined immediately prior to round r = 0.We denote the duration of the rounds by T0 and define thetime at the beginning of each round r to be tr := rT0. Forour controller to eliminate the frequency error that results fromchanges in load, we assume that a sufficient number of roundshave elapsed between load perturbations. As a result, for theanalysis of the control scheme we present next, we assumethat the power demanded by the loads is constant, i.e., if thepower demanded by load i ∈ V(`)p is perturbed by ∆`i(t) att = t0, then `i(t) = `0i + ∆`i(t0) for t0 < t < t0 + r0T0 andr0 sufficiently large.

Let ui[r] := ui(t), tr ≤ t < tr+1, be the set-point ofgenerator i ∈ V(g)p during round r, which is adjusted at thebeginning of the round and held constant for the remainingduration. Additionally, define D := 1∑n

i=1Di; then, given the

assumption that the power demanded by the loads is constant,it follows from (7) that, for tr ≤ t < tr+1,

∆ω[r] := D

∑i∈V(g)p

ui[r]−∑i∈V(`)p

(`0i + ∆`i)

. (8)

B. Control Scheme

From (8), if we assume that ui[r] = u∗i , for i ∈ V(g)p and

r = 0, 1, 2, . . . i.e., the set-points of the generators are constantand satisfy (5), it is clear that, following one or more changesto the power demanded by the loads, the average frequencyerror at round r is given by

∆ω[r] = D

∑i∈V(g)p

u∗i −∑i∈V(`)p

`0i −∑i∈V(`)p

∆`i

. (9)From (9), we see that, for constant generator set-points, thevalue of the average frequency error is equal to the additiveinverse of the sum of the load perturbations, weighted by thesum of the generator and load time constants, i.e., ∆ω[r] =−D

∑i∈V(`)p

∆`i. Thus, in order to drive the frequency errorto zero, it is sufficient to adjust the ui[r]’s such that

limr→∞

∑i∈V(g)p

ui[r] =∑i∈V(`)p

`0i + ∆`i. (10)

Given previously determined phase-cohesive set-points, u∗i ,i ∈ V(g)p , and in order to satisfy (10), our control architectureadjusts the set-point of generator i at round r according to

ui[r] = u∗i + ∆ui[r], (11)

where ∆ui[r] denotes the incremental amount by which ui[r]is adjusted away from u∗i . We define the incremental set-pointto be ∆ui[r] := αiei[r], where the value of ei[r] is updatedrecursively as

ei[r + 1] = ei[r] + κi∆ω[r], (12)

for appropriately chosen gains αi and κi for i ∈ V(g)p , andwith ei[0] = 0, i ∈ V(g)p . [Note that, in addition to resettingthe round index, the value of ei, i ∈ V(g)p must also be resetto zero when the u∗i ’s are recomputed.] Next, we establishcriteria for choosing gains that ensure the closed-loop systemis stable and that ∆ω[r]→ 0 as r →∞.

C. Stability Analysis and Criteria for Choosing Gains

If we define α :=[α1, . . . , αm

]T, κ :=

[κ1, . . . , κm

]T, and

e[r] :=[e1[r], . . . , em[r]

]T, then, by stacking (8) with (12) for

i ∈ V(g)p , we can represent the closed-loop system by[∆ω[r + 1]e[r + 1]

]=

[β DαT

κ Im

]︸ ︷︷ ︸

:=Φ

[∆ω[r]e[r]

]+

[−10m

]D∑i∈V(`)p

∆`i,

(13)

where Φ ∈ R(m+1)×(m+1), β := D∑i∈V(g)p

αiκi, and Im and0m represent the m-dimensional identity matrix and all-zerosvector, respectively.

To ensure that the closed-loop system in (13) is stable,the αi’s and κi’s specified above must be chosen such thatthe spectral radius of Φ lies on the boundary of or withinthe unit circle. More specifically, for marginal stability, wemust choose the αi’s and κi’s such that ρ(Φ) ≤ 1, i.e.,|λj | < 1 for j = 1, 2, . . . ,m + 1, where λj is the jtheigenvalue of Φ. Given such gains, we can compute theasymptotic value that

[∆ω[r], e[r]

]Ttakes, which we denote

by[∆ωss, ess

]T. As r →∞, we have that

[∆ω[r], e[r]

]T=[

∆ω[r + 1], e[r + 1]]T

=[∆ωss, ess

]T, and (13) becomes[

1− β −DαT−κ 0m×m

] [∆ωss

ess

]=

[−10m

]D∑i∈V(`)p

∆`i, (14)

where 0m×m is an m×m matrix of all zeros. The followingproposition, a proof of which is provided in Appendix A,establishes the criteria for choosing gains such that the closed-loop system is marginally stable and ∆ωss = 0.

Proposition 1: Consider the system in (13). If αi and κi fori ∈ V(g)p are chosen such that −2 < β = D

∑i∈V(g)p

αiκi < 0,then the system is marginally stable and the average frequencyerror asymptotically approaches zero, i.e., ρ(Φ) ≤ 1 and∆ω[r]→ 0 as r →∞. Additionally, the value of the averagefrequency error at any round r is given by

∆ω[r] = (1 + β)r−1(β − 1)D∑i∈V(`)p

∆`i. (15)

D. Estimating Deviation From Phase-Cohesive Operation

Recall the criterion for phase cohesiveness in (4), whichwas defined for generator set-points and load demands u∗

and `0, respectively. More generally, for any set-points andload demands, u =

[u1, . . . , um

]Tand ` =

[`m+1, . . . , `n

]T,

respectively, we can define the function

hp(u, `) :=

∥∥∥∥MTL† [ u−`]∥∥∥∥∞, (16)

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where the operating point that results from the injections atthe generators and loads is phase cohesive if the value that thefunction hp(·, ·) takes is smaller than sin(φ) for φ ∈ [0, π/2).

Given set-points found to result in phase-cohesive operationsubject to the initial power demanded by the loads, u∗ and `0,respectively, we define ∆u :=

[∆u1, . . . ,∆um

]Tand ∆` :=[

∆`m+1, . . . ,∆`n]T

to be vectors representing the amountby which the generators and loads deviate from the initial set-points and demands, respectively. Then, to ensure the systemremains phase cohesive as it evolves away from the operatingpoint for which the u∗i ’s were found, we must have h

p(u∗ +∆u, `0 + ∆`) ≤ sin(φ), which implies that hp(u∗ + ∆u, `0 +∆`) ≤ [sin(φ)−hp(u∗, `0)] for φ ∈ [0, π/2). In the discussionthat follows, we describe a method for estimating the value thathp(·, ·) takes as our control architecture adjusts the generatorset-points in response to changes in load.

By inspecting the update rule in (11) and the values onwhich the incremental set-point adjustment for each generatordepends, we see that ∆ui[r] ∝

∑j∈V(`)p

∆`j . More specifi-cally, if we substitute the value of ∆ω[r] from (15) into (12),we see that the value of ei[r] at any round r for i ∈ V(g)p is

ei[r] = ei[0] + κi

r−1∑s=0

∆ω[s], (17)

= ei[0] + κiD

∑i∈V(`)p

∆`i

(r−1∑s=0

(1 + β)s

), (18)

and, given that ei[0] = 0, i ∈ V(g)p , we have that theincremental set-point of generator i at round r is given by

∆ui[r] = αiκiD

∑j∈V(`)p

∆`j

(r−1∑s=0

(1 + β)s

). (19)

Thus, since the ∆ui[r]’s depend on the load perturbations, wecan define a new function,

hl(`0 + ∆`) := hp(u∗ + ∆u(∆`), `0 + ∆`),

=

∥∥∥∥MTL† [u∗ + ∆u(∆`)−(`0 + ∆`)]∥∥∥∥∞, (20)

that takes the value of the load demands as input, and returnsthe same value hp(·, ·) takes when evaluated at u = u∗ + ∆uand ` = `0 + ∆` for the closed-loop system. If we define thegradient of hl(·) as

∇hl(`) =[∂hl(`)∂`m+1

, · · · , ∂hl(`)∂`n

], (21)

it follows that, for small perturbations in load around `0, wecan estimate (20) as

hl(`0 + ∆`) ≈ hl(`0) +∇hl(`)∣∣∣∣`=`0

∆`,

= hl(`0) +∑i∈V(`)p

∂hl(`)

∂`i

∣∣∣∣`=`0

∆`i. (22)

By maintaining an estimate for the value of ∇hl(`)∣∣∣∣`=`0

∆`

as the amount of power demanded by the loads changes, the

point at which phase-cohesive set-points should be recomputedcan be determined. For example, one strategy could be toinitiate a recomputation of the u∗i ’s when

ξ(c) :=

∑i∈V(`)p

∂hl(`)

∂`i

∣∣∣∣`=`0

∆`i

c(23)

exceeds unity, where 0 < c <[sin(φ)− hl(`0)

]is a constant

that controls the frequency with which set-points are computedand the margin of phase cohesiveness.

IV. DISTRIBUTED ALGORITHMSIn this section, we introduce three distributed algorithms

that will be used as primitives for distributively implementingthe frequency regulation architecture proposed in Section III.We begin by providing an overview of two consensus-typealgorithms, both of which rely on a communication networkthat is described by the strongly connected directed graphmodel Gc = (Vc, Ec) introduced in Section II-B. The firstenables the computation of the maximum of locally maintainednodal values in finite time; the second enables the nodes toasymptotically determine a ratio of sums of values knownlocally by each of the individual nodes; in order to convergeto desired values, both algorithms require Gc to be stronglyconnected (see, e.g., [20], [21]). The third algorithm, whichwe refer to as the feasible flow algorithm, enables the com-putation of generator outputs and the resultant branch powerflows that collectively meet the total power demanded by theloads without violating any branch flow limits; the feasibleflow algorithm requires a communication network topology tomatch that of the physical layer which also necessitates havingbidirectional communication links. We index the iterationsover which all three algorithms update locally maintainedvalues by k = 0, 1, 2, . . . .

A. Max ConsensusConsider the vector η :=

[η1, η2, . . . , ηn

]T, where the value

of ηi, i ∈ Vc, is known only by node i, and suppose the nodesare interested in finding the maximum value among the ηi’s,which we define to be µ := maxi∈Vc ηi = ‖η‖∞. Let µi[k]be an estimate for µ maintained by node i ∈ Vc at iteration k.Then, as shown in [20], if the nodes initialize their estimatesas µi[0] = ηi and update them according to

µi[k + 1] = maxj∈N−c (i)∪{i}

µj [k] (24)

for each iteration k, it follows that, after a finite number ofiterations bounded from above by the diameter of the graph,every node can obtain the value of µ, i.e., for km ≥ ∆c,µi[km] = µ for i ∈ Vc.

If, in addition to iteratively updating the value of µi[k], thenodes also maintain a second value, denoted by νi[k], andupdate it at each k as

νi[k + 1] = argmax{µj : j∈N−c (i)∪{i}}

µj [k], (25)

it follows that νi[km] = argmax{ηj : j∈Vc}

ηj for i ∈ Vc and km ≥ ∆c,

i.e., every node can obtain the index of the node that has the

7

maximum ηi at the same iteration that µ is acquired. [Notethat in the case that two or more nodes have the maximizingvalue, e.g., ηj1 = ηj2 = µ, the value of νl[km + 1] can betaken to be the largest index between j1 and j2.]

B. Ratio Consensus

Similar to the max-consensus algorithm described above,nodes participating in the so-called ratio-consensus algorithmiteratively update two state variables, which we denote by yi[k]and zi[k] for i ∈ Vc. As shown in [21], by updating the statevariables according to

yi[k + 1] =∑

j∈N−c (i)∪{i}

1

δ+c (j) + 1yj [k], (26)

zi[k + 1] =∑

j∈N−c (i)∪{i}

1

δ+c (j) + 1zj [k], (27)

the ratio of the state variables, which we define to be γi[k] :=yi[k]zi[k]

, converges to the same value ∀i ∈ Vc; more specifically,each of the γi[k]’s asymptotically approaches

γ = limk→∞

γi[k] =

∑nj=1 yj [0]∑nj=1 zj [0]

, i ∈ Vc. (28)

By properly initializing the state variables, as we will showin Section V, the ratio-consensus algorithm can enable eachgenerator controller to obtain system-level information withwhich it can make local decisions.

Remark 1: Although the discussion above implies that theprocess used to obtain the value of γ must extend over aninfinite number of iterations, it is possible to distributivelydetermine the iteration at which γ is computed to within apre-specified bound of the asymptotic value. As shown inour previous work in [22], by maintaining and periodicallyreinitializing min- and max-consensus algorithms as describedin Section IV-A (the formulation of an algorithm for min-consensus follows analogously), the nodes can compute anapproximation to the asymptotic value of γ after a finitenumber of iterations, kr. To obtain the simulation resultspresented in Section VI, we make use of this variant ofthe ratio-consensus algorithm. However, when using suchapproximations for frequency regulation, large approximationerrors should be avoided because they might lead to largeerrors in the computation of the set-points, which mightcause instability problems if the resulting equilibrium pointis no longer phase-cohesive. However, by choosing φ in (4)strictly smaller than π2 , we allow our controller to have anadditional stability margin to provide resiliency against smallapproximation errors. �

C. Feasible Flow Algorithm

We begin the following discussion by introducing the prob-lem of determining individual generator outputs that collec-tively balance the total power demanded by the loads in amicrogrid without violating any generation or branch powerflow limits. After formalizing the objectives of this problem,which we refer to as the feasible flow problem, we propose analgorithm for distributively solving it.

1) Feasible Flow Problem Statement: For each edge{i, j} ∈ Ep, let fij denote the power flow between nodes iand j, where fij > 0 if the actual direction of the flow is frombus i to bus j. We restrict each flow to be within lower andupper limits, f

ijand f ij , respectively, i.e., f ij ≤ fij ≤ f ij .

For each generator node i ∈ V(g)p , let gi denote its output,with 0 ≤ g

i≤ gi ≤ gi, where gi and gi denote lower

and upper limits, respectively. [Note that, while related, theset-point of generator i ∈ V(g)p and its respective limits, i.e.,ui, ui, ui, defined in Section II-A are not necessarily equivalentto gi, gi, gi.] Similarly, for each load node i ∈ V

(`)p , let `i ≥ 0

denote its power demand. Finally, let

bi :=

{−∑j∈Np(i) fij + gi, i ∈ V

(g)p ,

−∑j∈Np(i) fij − `i, i ∈ V

(`)p ,

(29)

be the flow balance at each node i ∈ Vp.Based upon the definitions above, the feasible flow problem

is summarized as follows. Suppose that the power demandsat the load buses, i.e., the `i’s, are known; then, the objectiveis to assign values to each of the flows fij , {i, j} ∈ Ep andgenerator outputs gi, i ∈ V(g)p , such that the flow balance ateach node is zero, the total generator outputs are balancedwith the load power demands, and all flow assignments andgenerator outputs are within limits. More specifically, theobjective is to obtain a set of flows {fij : {i, j} ∈ Ep} andgenerator outputs {gi : i ∈ V(g)p } such that:

F1. fij≤ fij ≤ f ij for {i, j} ∈ Ep;

F2. gi≤ gi ≤ gi for i ∈ V

(g)p ; and

F3. bi = 0 for i ∈ Vp.

If we define f := [fe], f := [fe], f := [fe] ∈ R|Ep| for e ∈ Ep

and g := [gi], g := [gi], g := [gi] ∈ R|V(g)p |, for i ∈ V(g)p , the

feasible flow problem can be written in matrix form as: findf and g such that Mf −

[gT,−`T

]T= 0|Vp|, f ≤ f ≤ f ,

and g ≤ g ≤ g, where 0|Vp| is the |Vp|-dimensional all zerosvector.

2) Feasible Flow Algorithm: In the discussion that follows,we introduce an algorithm that iteratively adjusts locallymaintained estimates for flows and generator outputs such thatthe estimates asymptotically approach values that satisfy theconstraints of the feasible flow problem. Unlike the consensus-type algorithms presented in Section IV-A and Section IV-B,which were defined in terms of a directed graph, the commu-nication network on which this algorithm relies must conformto the undirected graph modeling the physical layer, Gp.

To support the distributed nature of our proposed algorithm,each local processor maintains an estimate for the value of theflows along edges connecting it to all of its neighbors. Morespecifically, for each node i ∈ Vp, the estimate maintained byi for the flow to each j ∈ Np(i) at iteration k = 0, 1, 2, . . .is denoted by f (i)ji [k]. Additionally, for each generator nodei ∈ V(g)p , we denote an estimate for its output at iteration kby gi[k]. Finally, based upon the flow and generator outputestimates, the value of the flow balance at each node is

8

computed at each iteration as

bi[k] :=

{−∑j∈Np(i) f

(i)ij [k] + gi[k], i ∈ V

(g)p ,

−∑j∈Np(i) f

(i)ij [k]− `i, i ∈ V

(`)p .

(30)

The algorithm we propose for distributively solving thefeasible flow problem is given by the following procedure inwhich the flow and generator output estimates are initializedand iteratively updated using a three-step process.

[Initialization] Each node initializes its flow estimates tobe the average of its respective lower and upper limit, i.e.,f

(i)ij [0] =

12 (f ij + f ij), i ∈ Vp, j ∈ Np(i). Analogously, the

estimate for the output of each generator node is initialized asgi[0] =

12 (gi + gi), i ∈ V

(g)p .

[Step 1] Node i adjusts its estimate for each flow in such away that drives the flow balance estimate to zero, i.e., bi[k]→0 as k →∞. More specifically, node i adjusts its estimate foreach flow as

f̃(i)ij [k + 1] = f

(i)ij [k] +

bi[k]

wi, j ∈ Np(i), (31)

where wi := δp(i) + 1. If node i ∈ V(g)p , its output estimateis adjusted in a similar fashion to the flows, i.e.,

ĝi[k + 1] = gi[k]−bi[k]

wi. (32)

[Step 2] Since each node updates its flow estimates duringStep 1 independently, two neighboring nodes may have differ-ent estimates for the flow between them. Thus, by exchangingflow estimates with neighboring nodes, each node updates itslocally maintained estimates as

f̂(i)ij [k + 1] =

1

2

(f̃

(i)ij [k + 1]− f̃

(j)ji [k + 1]

). (33)

for j ∈ Np(i). Note that while the generator outputs areanalogous to positive injections, only each respective localprocessor maintains an estimate for its output, eliminating theneed for an agreement step.

[Step 3] Finally, during the first two steps, each flowestimate may have been adjusted in such a way that violateslimits. To ensure the flow assignment to which the nodesconverge is feasible, any flow estimate exceeding its upper orlower bound is clamped to be within limits, i.e., for j ∈ Np(i),

f(i)ij [k + 1] =

f ij , if f̂

(i)ij [k + 1] > f ij ,

fij, if f̂ (i)ij [k + 1] < f ij ,

f̂(i)ij [k + 1], otherwise.

(34)

Similarly, the estimates for each generator output must beclamped to be within limits:

gi[k + 1] =

gi, if ĝi[k + 1] > gi,gi, if ĝi[k + 1] < gi,

ĝij [k + 1], otherwise.(35)

Given that the nodes incident to any given edge clamp theirflow estimates to the same limits during Step 3, it follows thatat the beginning of each iteration, the estimates maintained byboth nodes are additive inverses, i.e., f (i)ij [k] = −f

(j)ji [k] =:

fij [k], {i, j} ∈ Ep; thus, the progress of the algorithm in Steps

Algorithm 1: Distributed feasible flow algorithm

Input: fe, fe, e ∈ Ep; gl, gl, l ∈ V

(g)p ; `j , j ∈ V(`)p

Output: f (i)∗ij {i, j} ∈ Ep; g∗l , l ∈ V(g)p

Each node i ∈ Vp separately does the following:begin

initializef

(i)ji [0] =

12 (f ji + f ji), j ∈ Np(i)

gi [0] =12 (gi + gi), i ∈ V

(g)p

foreach iteration, k = 0, 1, . . . , kf docompute

bi[k] ={−∑j∈Np(i) f

(i)ij [k] + b̃i[k] + gi[k], i ∈ V

(g)p

−∑j∈Np(i) f

(i)ij [k] + b̃i[k]− `i, i ∈ V

(`)p

transmitbi[k]/wi to j ∈ Np(i)

receivebj [k]/wj from j ∈ Np(i)

computef̂

(i)ij [k+ 1] = f

(i)ij [k]−

bj [k]2wj

+ bi[k]2wi , j ∈ Np(i)ĝi[k + 1] = gi[k]− 12

bi[k]wi

, if i ∈ V(g)pset

forj ∈ Np(i), f (i)ij [k + 1] =f ij , if f̂

(i)ij [k + 1] > f ij ,

fij, if f̂ (i)ij [k + 1] < f ij ,

f̂(i)ij [k + 1], otherwise,

gi[k + 1] =gi, if ĝi[k + 1] > gi,gi, if ĝi[k + 1] < gi,

ĝi[k + 1], otherwise,if i ∈ V(g)p

For kf sufficiently large, set:f

(i)∗ij = f

(i)ij [kf ], j ∈ Np(i); g∗l = gl[kf ], l ∈ V

(g)p

1-3 can be summarized by the following iterations:

fij [k + 1] =

[fij [k] +

1

2

bi[k]

wi− 1

2

bj [k]

wj

]fijfij

, ∀{i, j} ∈ Ep,

(36)

gi[k + 1] =

[gi[k]−

1

2

bi[k]

wi

]gigi

, ∀i ∈ V(g)p ,

(37)

where [·]xx denotes the projection onto [x, x].A summary of the feasible flow algorithm is given in Algo-

rithm 1. The following proposition establishes the convergenceof the algorithm to a solution that satisfies the feasible flowproblem (see Appendix A for a proof).

Proposition 2: Suppose, for given load power demands,`, a solution to the feasible flow problem specified by the

9

r = 0 r = 1 r = 2 r = r0 r = 0

FeasibleFlow

MaxConsensus

RatioConsensus

kf

u∗, hl(`0)

kf

{f (i)∗ij (�vl), f(i)∗ij (−�vl)}

σkm

kr

km

∆hl(`)∆`i

σkm

kr

σkm

kr

σkm

kr

γg → α, κ ξ(c),∆ω[0]

ξ(c),∆ω[1]

ξ(c),∆ω[r0]

Compute phase-cohesive criterion-dependent values Execute frequency regulation scheme

Fig. 1: Timeline of distributed implementation of frequency regulation architecture.

objectives in F1 – F3 exists and let f∗e for e ∈ Ep and g∗l forl ∈ V(g)p denote the flows and generator outputs that satisfyit. Then, Algorithm 1 is guaranteed to yield f∗ and g∗ such

that Mf∗ −[g∗

−`

]= 0|V|, and f ≤ f∗ ≤ f and g ≤ g∗ ≤ g,

where f∗ := [f∗e ] for e ∈ Ep and g∗ := [g∗i ] for i ∈ V(g)p , i.e.,

as k →∞, fe[k]→ f∗e , e ∈ Ep; gl[k]→ g∗l , l ∈ V(g)p .

From the summary in Algorithm 1, we can think of thefeasible flow algorithm as a function that takes flow andgenerator output limits and load demands as inputs and yieldsflows and generator outputs that satisfy the feasible flowproblem, i.e., the algorithm can be represented by

hf : (f, f , g, g, `) 7→ (f∗, g∗). (38)

Remark 2: As with ratio consensus, the algorithm wepropose to solve the feasible flow problem must extend overan infinite number of iterations. In reality, by stopping theiterative process after a finite number of iterations, whichwe denote by kf , the worst-case error between the flow andgenerator output estimates and their true asymptotic valuescan be made arbitrarily small. Additionally, similar to thefinite-time approach proposed in [22], the nodes can use maxconsensus, re-initialized every ∆d iterations, to periodicallycompute the value of �b[∆dk] := maxi∈Vp bi[∆dk]. Given thatthe values of the flow and generator output estimates dependon the flow balances, it may be possible to use this instanceof max consensus to distributively determine the iteration atwhich all values of f (i)ij [k], {i, j} ∈ Ep and gi[k], i ∈ V

(g)p are

within some bound of their true asymptotic values. �

V. DISTRIBUTED IMPLEMENTATION

As described in Section III, our proposed frequency reg-ulation architecture requires global information to be im-plemented. In particular, the following values on which thearchitecture relies require system-level information to becomputed: the average frequency error, ∆ω[r]; the value ofβ = D

∑i∈V(g)p

αiκi, for ensuring closed loop stability; the

phase-cohesive set-points, u∗i , i ∈ V(g)p ; and the value of

∇hl(`)∣∣∣∣`=`0

∆` to determine ξ(c). In this section, we describe

how the three algorithms outlined in Section IV—the max- andratio-consensus and feasible flow algorithms—can collectivelyenable the distributed computation of all these values. Beyondeliminating the need for global information, the distributed im-plementation we propose does not rely on time-synchronized

measurements of the frequency or phase angle; instead, onlythe injection at each bus is required.

We begin the following discussion by outlining how ratioconsensus can be used to compute the value of ∆ω[r] at eachround and to compute gains that ensure the closed loop systemis stable. Then, we describe how the feasible flow algorithmcan be used to compute the set-points that satisfy (5) and, whencombined with the max- and ratio-consensus algorithms, canenable the distributed computation of an estimate for the valueof ξ(c) as defined in (23). Finally, we discuss the timeline overwhich each of the necessary values is distributively computed.

A. Computing the Average Frequency Error

By inspection of (8), we see that the definition of the averagefrequency error at each round r is a ratio of sums of valuesknown by each local processor. In particular, if we define

xi[r] =

{ui[r], if i ∈ V(g)p ,−`i, if i ∈ V(`)p ,

(39)

it is clear that ∆ω[r] =∑

i∈Vp xi[r]∑i∈Vp Di

. Thus, if we use an

instance of ratio consensus at each round r, where y(r)i [k]and z(r)i [k] denote the states maintained by node i, which areinitialized as y(r)i [0] = xi[r] and z

(r)i [0] = Di, it follows that

the average frequency error can be asymptotically computed,i.e., limk→∞

y(r)i [k]

z(r)i [k]

= ∆ω[r], i ∈ Vc.

B. Computing Stable Gains

From the analysis in Section III-C, it can be shown that, inthe limit as r → ∞, if the generator set-points are updatedaccording to the control scheme in (11) – (12), the steady-statevalue of generator i’s set-point is given by

ussi = u∗i +

αiκi∑l∈V(g)p

αlκl

∑j∈V(`)p

∆`j , (40)

which implies that the product of gains αiκi affects the amountby which generator i will adjust its set-point away fromu∗i in order to meet the total incremental demand for load.Furthermore, from (15), we see that if we choose the αi’sand κi’s such that β := D

∑i∈V(g)p

αiκi = −1, it followsthat the frequency error that results from one or more changesin load can be eliminated after one round of our proposedcontrol architecture. Thus, to ensure that β = −1, we seethat the summation −

∑i∈Vp Di must be divided among the

10

generators, and that the specific choice of gains will dictatethe proportion of the total incremental demand attributed toeach generator.

Let ∆ui = ui − u∗i and ∆ui = ui − u∗i be the lower andupper bounds on the amount by which generator i can adjustits set-point away from u∗i without violating its output limits

and define γg :=−

∑i∈Vp Di−

∑i∈V(g)p

∆ui∑i∈V(g)p

∆ui−∆ui. Then, if we choose

αiκi = hgi (γg), where

hgi (γg) :=∆ui + γg(∆ui −∆ui),=ui − u∗i + γg(ui − ui),

i ∈ V(g)p , (41)

it is easy to see that∑i∈V(g)p

αiκi = −∑i∈Vp Di, and that

the summation −∑i∈Vp Di is divided among the generators

proportionally to their incremental set-point limits.Similar to the so-called fair splitting allocation scheme in

[16], we can use ratio consensus to compute the value of γg .More specifically, given the dependence of the value of γgon the u∗i ’s, we can use an instance of ratio consensus eachtime the phase-cohesive set-points are computed. We denotethe states maintained by node i for each of these instancesby y(g)i [k] and z

(g)i [k], and, if we initialize them as y

(g)i [0] =

−Di − ∆ui if i ∈ V(g)p , y

(g)i [0] = −Di if i ∈ V

(`)p , and

z(g)i [0] = ui − ui if i ∈ V

(g)p , z

(g)i [0] = 0 if i ∈ V

(`)p , it

follows from (28) that limk→∞y(g)i [k]

z(g)i [k]

= γg . By using this

value as the argument of the hgi (·) function defined in (41),each node can compute the product of its gains as αiκi =hgi (y

(g)i [k0]/z

(g)i [k0]), for sufficiently large k0. Since there are

no constraints on the individual αi’s and κi’s, we choose gainssuch that αi = h

gi (y

gi [k0]/z

gi [k0]) and κi = 1. [Note that if

we use the approximate ratio consensus algorithm described inRemark 1 to compute γg , the approximation error that resultswill prevent the value of β from exactly equalling −1; thus, inreality, it may take more than one round to drive the frequencyerror to zero.]

Remark 3: Although using the function in (41) to assigngains ensures that β = −1, it only guarantees that theincremental set-point of every generator is within its respectiveincremental limits for small enough collective load perturba-tions,

∑i∈V(`)p

∆`i. However, given that the control schemein (11) – (12) is designed to regulate the frequency for smallperturbations away from a pre-determined phase-cohesive op-erating point, we can assume that the load perturbations aresufficiently small such that this choice of gain ensures allincremental set-points are within limits for the operating rangein which they are used.

An alternative function for choosing the gains that ensuresβ = −1 and guarantees all incremental set-points are withinlimits is:

h̃gi (γ̃g, γ̃d) := −γ̃d (∆ui + γ̃g(∆ui −∆ui)) , (42)

for i ∈ V(g)p , where γ̃g :=∑

i∈V(`)p∆`i−

∑i∈V(g)p

∆ui∑i∈V(g)p

∆ui−∆uiand

γ̃d :=

∑i∈Vp Di∑

j∈V(`)p∆`j

. If we assume that the total amount by

which the load deviates is within the collective incremental ca-pacity of the generators, i.e.,

∑i∈V(g)p

∆ui ≤∑i∈V(`)p

∆`i ≤

Load Bus

Generator Bus1

4

5

2

3

6

Fig. 2: Physical layer model for six-bus microgrid.∑i∈V(g)p

∆ui, this choice of gains will divide the incrementaldemand among the generators while also ensuring β = −1.Although this alternative function has advantages comparedto hg(·), it requires that each load knows the value of ∆`i inorder to compute γ̃d. Additionally, while ratio consensus canbe used to compute γ̃d, doing so requires another operationbefore the incremental set-points can be determined. �

C. Computing Phase-Cohesive Set-Points

By inspection, we see that ensuring the phase cohesivenesscriterion in (4) is satisfied is equivalent to ensuring that allnetwork flows are within limits. More specifically, when thevector of injections at the generator and load buses is pre-multiplied by the matrix product MTL†, the elements of theresultant vector are the network power flows, normalized bythe susceptances of the branches connecting each pair of buses,i.e., the value of each element is equal to sin(θi(t) − θj(t))for {i, j} ∈ Ep. To ensure phase-cohesive operation results,every normalized flow in the network must be less than sin(φ),i.e., MTL†

[uT,−`T

]T ≤ sin(φ)1|Ep|, where 1|Ep| is the |Ep|-dimensional all-ones vector and φ ∈ [0, π/2). Thus, we seethat the problem of choosing phase-cohesive set-points subjectto the power injections at the load buses is analogous tochoosing generator outputs such that no branch power flowexceeds its limit.

From the discussion above, it follows that, if a solution tothe feasible flow problem exists for given load power demands,we can use Algorithm 1 to find the generator set-pointsthat result in an operating point that satisfies the feasibilityproblem in (5), i.e., the operating point is phase-cohesive, thetotal demand for load is balanced by the collective generatorset-points, and all the generator set-points are within limits.In order to enforce appropriate limits for the feasible flowalgorithm that ensure phase cohesiveness, we see that themaximum branch power flow on any given line is upperbounded by its susceptance. Thus, if we enforce lower andupper limits on the flow along each edge in the feasible flowproblem equal to f

ij= −Bij sin(φ) and f ij = Bij sin(φ),

{i, j} ∈ Ep, respectively, we can find set-points that satisfy(5) using Algorithm 1 as follows. Let u :=

[u1, . . . , um

]Tand u :=

[u1, . . . , um

]Tbe vectors representing the lower

and upper set-point limits of the generators and define B′ :=[Bij sin(φ) : {i, j} ∈ Ep] ∈ R|Ep|; then, using the functionalrepresentation of Algorithm 1, i.e., hf (·), with parameters(−B′, B′, u, u, `0), the generator outputs that result satisfy (5),i.e., u∗ = g∗.

D. Computing Phase Cohesiveness Margin

In Section III-D, we showed that by monitoring the value ofξ(c) as the amount of power demanded by the loads changes,

11

0 2 4 6 8−0.2

−0.15

−0.1

−0.05

0

0.05

time, t [s]

θ i(t

)[r

ad]

θ1(t) θ2(t) θ3(t)

θ4(t) θ5(t) θ6(t)

(a) Bus voltage angles.

0 2 4 6 8

0

−0.01

−0.02

−0.03

−0.04

−0.05

−0.06

time, t [s]

∆ω

(t)

[rad

/s]

(b) Weighted average frequency error.

0 2 4 6 80.75

1

1.25

1.5

time, t [s]

ui(t

)[p

u]

u1(t) u2(t) u3(t)

u∗1 u∗2 u

∗3

(c) Generator set-points.

Fig. 3: Case I simulation results with loads `4, `5, and `6 perturbed at t = 2 s, t = 4 s, and t = 6 s, respectively.

the nodes can estimate when to recompute phase-cohesivegenerator set-points. Recall from (23) that the sensitivity ofthe function hl(·) to small perturbations in load, evaluated at`0, ∇hl(`)

∣∣∣∣`=`0

, is required to compute ξ(c). By noting that

the output of the function hl(·) is the maximum normalizedbranch flow subject to injections at the generators and loads,i.e., given set-points u∗ + ∆u(∆`) and demands `0 + ∆`,∥∥∥∥MTL† [u∗ + ∆u(∆`)−(`0 + ∆`)

]∥∥∥∥∞

= max(i,j)∈Ep

sin(θi(t)− θj(t)),

(43)

we can use the following procedure, which combines thefeasible flow algorithm and max consensus, to distributively

compute ∇hl(`)∣∣∣∣`=`0

.

From the functional representation of Algorithm 1 definedin (38), we see that, in addition to the generator outputs g∗i , i ∈V(g)p , the set of flows {f (i)∗ij : {i, j} ∈ Ep} is also computed.Thus, if we use the max-consensus algorithm, with the value

initially known by node i given by ηi = maxj∈Np(i)|f(i)∗ij |

Bij sin(φ),

it follows that, without measurements of θi(t), i ∈ Vp, asimplied by (43), we can compute the value of∥∥∥∥MTL† [g∗−`

]∥∥∥∥∞

= max(i,j)∈Ep

|f (i)∗ij |Bij sin(φ)

= maxl∈Vc

ηl,

subject to the parameters passed to the hf (·) function.

If we approximate the gradient of hl(·) by

∇hl(`) ≈[

∆hl(`)∆`m+1

, · · · , ∆hl(`)

∆`n

], (44)

we can compute it using 2|V(`)p |-instances of the feasible flowand max-consensus algorithms. More specifically, given thatwe are interested in determining the value of ξ(c) for theclosed-loop system as it evolves away from the operatingpoint for which the u∗i ’s were computed, we can approximatethe sensitivity of hl(·) to small changes in demand at loadi ∈ V(`)p , i.e., the value of ∆h

l(`)∆`i

∣∣∣∣`=`0

, subject to our proposed

control architecture, as follows. Define vi ∈ R|Vp| to be avector with 1 in the ith coordinate and 0’s elsewhere; then,for each i ∈ V(`)p , let `(�vi) := `0 + �vi be a vector of load

demands with the ith load perturbed by �. From (40), let

uj(�) := u∗j + �αjκj/

∑l∈V(g)p

αlκl (45)

be the set-point of generator j ∈ V(g)p given one such loadperturbation, subject to the control scheme in (11) – (12).Furthermore, let f∗e (�vi), e ∈ Ep be the flows that result fromAlgorithm 1 with parameters (−B′, B′, u(�), u(�), `(�vi)),where u(�) :=

[u1(�), . . . , um(�)

]T. Then, we can use an

instance of the max-consensus algorithm, where the valueknown by node j is

ηj(�vi) = maxl∈Np(j)

|f (j)∗jl (�vi)|Bjl sin(φ)

, (46)

to compute the value of hl(`(�vi)) = max(j,l)∈Ep|f(j)∗jl (�vi)|Bjl sin(φ)

=

maxj∈Vc ηj(�vi). If we use the central difference approxima-

tion to estimate the value of ∆hl(`)

∆`i

∣∣∣∣`=`0

, then it follows that

∆hl(`)

∆`i

∣∣∣∣`=`0

≈ hl(`(�vi))− hl(`(−�vi))

2�, (47)

where hl(`(−�vi)) is computed analogously to hl(`(�vi)) with`(−�vi) = `0 − �vi and uj(−�) = u∗j −

�αjκj∑l∈V(g)p

αlκl.

Given that the value of ξ(c) depends on an appropriatechoice of c, which depends on the value of hl(`) for ` = `0,we can use the process described above for computing theindividual hl(`(�vi))’s to determine a value of c that is lessthan sin(φ) − hl(`0). By using max consensus to computehl(`0), which can be combined with a pre-determined localrule for choosing c, the node with the maximizing flow can

also be determined, i.e., if ηi = maxj∈Np(i)|f(i)∗ij |

Bij sin(φ), the

node l = argmax{ηj : j∈Vc}

ηj is the one with the maximizing flow.

Then, by combining the appropriately chosen value of c withthe approximation to the sensitivity of the function hl(·) foundusing the process described above, the nodes can compute thevalue of ξ(c) as follows. Let y(p)i [k] and z

(p)i [k] denote the

states maintained by node i, where y(p)i [0] =∆hl(`)

∆`i

∣∣∣∣`=`0

∆`i

if i ∈ V(`)p , and y(p)i [0] = 0 otherwise; and z(p)i [0] = c

if i = argmax{ηj : j∈Vc}

ηj and z(p)i [0] = 0 otherwise. Then, by

12

0 2 4 6 80.18

0.19

0.2

0.21

0.22

time, t [s]

Max

imum

Net

wor

kFl

ows Estimated Actual

Fig. 4: Estimated vs. actual maximum flowfor Case I.

0 5 10 15 20 25 30 35 40 45 50

−1−0.5

0

0.5

1

iterations, k

b̂ i[k

]

b1[k] b2[k] b3[k] b4[k] b5[k] b6[k]

Fig. 5: Nodal balance estimates as feasible flow algorithm evolves for initialcomputation of u∗ for Case I.

updating the states according to (26) – (27), the nodes canasymptotically obtain

limk→∞

y(p)i [k]

z(p)i [k]

=

∑i∈V(`)p

∆hl(`)

∆`i

∣∣∣∣`=`0

∆`i

c.

E. Timeline

In the preceding discussion, we showed how each of thevalues necessary to implement our proposed frequency reg-ulation architecture can be acquired using a combination ofthe distributed algorithms introduced in Section IV. Next, weprovide an overview of the timeline over which our proposeddistributed implementation operates, specifically discussingthe order in which the algorithms must be executed to properlycompute the necessary values.

As discussed in Section III-B, at each round of our proposedfrequency regulation architecture, in addition to the weightedaverage frequency error, ∆ω[r], the output of each generatori ∈ V(g)p is adjusted according to its phase-cohesive set-point,u∗i , and gains, αi and κi, all of which depend on globalinformation. Additionally, as discussed in Section III-D, toensure the system remains phase cohesive for large changesin load, the nodes monitor the value of ξ(c), the definition

of which depends on ∆hl(`)

∆`i

∣∣∣∣`=`0

, i ∈ V(`)p , as the system

evolves. Given that these quantities—u∗i , αi, κi, i ∈ V(g)p

and ∆hl(`)

∆`i

∣∣∣∣`=`0

, i ∈ V(`)p , which we collectively refer toas phase-cohesive criterion-dependent values—must be knownto compute the generator set-points and ξ(c), it is clear thatthey need to be determined before our proposed frequencyregulation scheme can be executed. Moreover, from (41), we

see that, in order to compute the ∆hl(`)

∆`i

∣∣∣∣`=`0

’s, the controller

gains, αi and κi, i ∈ V(g)p , must be known beforehand.Similarly, from (45), we see that the u∗i ’s must be known inorder to compute the αi’s and κi’s. Thus, it is clear that thephase cohesive criterion-dependent values must be computedin a specific order; we provide a detailed description of thisorder next.

Prior to operation and immediately following the roundfor which the value of ξ(c) is found to exceed unity, i.e.,before round r = 0, the nodes use the feasible flow algorithmto compute the phase-cohesive set-points, u∗. The generator

nodes then use the u∗i ’s to determine their incremental limits,∆ui and ∆ui, i ∈ V

(g)p and, together with the processors

located at the load buses, use ratio consensus to determinethe value of γg . With the value of γg , or an approximationthereof, each generator processor then determines its controllergains, αi and κi, according to the function h

gi (·). Using

2|V(`)p |-instances of the feasible flow algorithm, the nodes thendetermine the sets of flows {f (i)∗ij (�vl) : l ∈ V

(`)p , {i, j} ∈ Ep}

and {f (i)∗ij (−�vl) : l ∈ V(`)p , {i, j} ∈ Ep}, where f (i)∗ij (�vl)

and f (i)∗ij (−�vl) are the flow assignments that result when loadl ∈ V(`)p is perturbed by � and −�, respectively, and the set-point of generator j ∈ V(g)p is lower- and upper-bounded byu(�) and u(−�), respectively. From the sets of flows that result,the nodes can use max consensus to determine the values ofhl(`(�vl)) and hl(`(−�vl)) for l ∈ V(`)p , with which the loadprocessors can compute an estimate of ∆h

l(`)∆`l

∣∣∣∣`=`0

. After the

phase-cohesive set-points, generator gains, and load sensitivityestimates are computed, the frequency regulation scheme canbegin operation, with the nodes using separate instances ofratio consensus to compute the values of ξ(c) and ∆ω[r] ateach round. An overview of the order in which each value iscomputed is illustrated by the timeline in Fig. 1.

VI. SIMULATION RESULTS

In this section, we present numerical simulation results forthree test cases in which our proposed control architectureand its distributed implementation are used. For each case,we demonstrate closed-loop operation given a series of smallperturbations to the loads and discuss several metrics thatillustrate its effectiveness. We consider a six-bus ring networkfor the first two test cases; the network consists of threegenerators and three loads. For the third case, we utilize a treenetwork consisting of 37-buses, 15 of which are generators,and 22 of which are loads. Generator, load, network, andsimulation parameters for each of the cases can be foundin [23, Appendix B]; furthermore, in all cases, we makeuse of the finite-time ratio-consensus algorithm mentioned inRemark 1.

A. Cases I and II: Six-Bus Ring Network

We consider a six-bus microgrid, the physical layer model ofwhich is illustrated by the graph in Fig. 2. In Cases I and II,

13

0 2 4 6 8

−0.15

−0.1

−0.05

0

0.05

time, t [s]

θ i(t

)[r

ad]

θ1(t) θ2(t) θ3(t)

θ4(t) θ5(t) θ6(t)

(a) Bus voltage angles.

0 2 4 6 8

0.05

0.04

0.03

0.02

0.01

0

−0.01−0.02−0.03−0.04

time, t [s]

∆ω

(t)

[rad

/s]

(b) Weighted average frequency error.

0 2 4 6 80.80.9

11.11.21.31.41.5

time, t [s]

ui(t

)[p

u]

u1(t) u2(t) u3(t)

u∗1 u∗2 u

∗3

(c) Generator set-points.

Fig. 6: Case II simulation results with loads `4, `6, and `5 perturbed at t = 2 s, t = 4 s, and t = 6 s, respectively.

the total power initially demanded by the loads is taken tobe∑i∈V(`)p

`0i = 3.3 pu and at time t = 0−, prior to the

computation of u∗i ’s, the generators equally share the load suchthat ui[0−] = 1.1 pu, i = 1, 2, 3. Using the method describedin Section V-D, the sensitivity of the function hl(·) to changesin load `6 is found to be negative, i.e., a decrease in the powerdemanded by `6 will lead to an increase in the value of hl(·).

1) Case I: For this case, we consider the closed-loopresponse of the six-bus microgrid subject to an increase inpower demand by loads `4, `5, and `6 at time t = 2, t = 4,and t = 6, respectively. Each load demand is perturbed byan increase of at most 7.5% of its original value, with `4perturbed by 5% such that ∆`4 = 0.0575 pu, `5 perturbed by7.5% such that ∆`5 = 0.09375 pu, and `6 perturbed by 5%such that ∆`6 = 0.045 pu. Given these perturbations and thechoice of c = 0.01, our proposed control architecture triggersa recomputation of the u∗i ’s at time t ≈ 4 s, immediatelyafter load `5 is perturbed. The response of the closed-loopsystem to the three perturbations and the recomputation of u∗

is illustrated by the plots in Fig. 3. From Fig. 3a and Fig. 3b, itcan be seen that, following each load perturbation, the voltageangles stabilize and the weighted average frequency errorthat results from the changes in load is quickly eliminated.Additionally, from Fig. 3c, in which the generator set-pointsand their respective values of u∗i are illustrated, it can beseen that the generators increase their output according to thecontrol scheme in (11) – (12) at times t = 2 s and t = 6 s andthat new values of u∗ are computed and applied following theperturbation to load `5 at t = 4 s.

To demonstrate the effectiveness of our proposed methodfor estimating the value of the function hl(·) as the loads areperturbed, the actual maximum normalized flow as computedat each time step, i.e., max(i,j)∈Ep sin(θi(t) − θj(t)), and its

estimate, i.e., hl(`0)+∇hl(`)∣∣∣∣`=`0

, are shown in Fig. 4. From

the figure, we see that the estimated maximum flow closelytracks the true value, with an error on the order of 0.001throughout the simulation period. Additionally, we see that theestimate properly predicts that an increase in `6 will lead to adecrease in the maximum network flow. Finally, we illustratethe evolution of the nodal flow balances maintained by eachnode for the first 50 iterations of the feasible flow algorithmas it is used to compute the initial u∗i ’s in Fig. 8. From this

figure, it can be seen that the flow balances maintained by allnodes quickly approach zero, with every node’s estimate beingwithin 0.001 of zero within the first 50 iterations.

2) Case II: For this case, we again consider the six-busmicrogrid, but, in order to demonstrate the negative sensitivityof the function hl(·) to changes in demand at load `6, weevaluate its closed-loop response to increases in demand atloads `4 and `5, and a decrease in demand at load `6.Specifically, the loads are perturbed as follows: `4 is increasedby 5% at t = 2 s such that ∆`4 = 0.0575 pu, `6 is decreasedby 7.5% at t = 4 s such that ∆`6 = −0.0675 pu, and `5increased by 4.8% at t = 6 s such that ∆`5 = 0.06 pu. Giventhese perturbations and the choice of c = 0.01, our proposedcontrol architecture triggers a recomputation of the u∗i ’s (andthe load sensitivities) at time t ≈ 6 s, immediately after load`5 is perturbed.

The response of the closed-loop system to the three pertur-bations and the recomputation of u∗ is illustrated by the plotsin Fig. 6. As in Case I, Fig. 6a and Fig. 6b, illustrate that,following each load perturbation, the voltage angles stabilizeand the weighted average frequency error that results from thechanges in load is quickly eliminated. Additionally, Fig. 6cillustrates the generator set-points and their respective u∗i ’sas the control architecture responds to the load perturbations.The figure illustrates that the third load perturbation, at timet = 6 s, increases the estimate for hl(·) such that, given thevalue of c, a recomputation of the u∗i is performed.

B. Case III: 37-Bus Tree Network

For this case, we consider the closed-loop response of the37-bus tree network, the topology of which is illustrated inFig. 7. Additionally, the total power initially demanded by theloads is taken to be

∑i∈V(`)p

`0i = 28.92 pu and at time t = 0−,

prior to the computation of u∗i ’s, the generators equally sharethe load such that ui[0−] = 1.928 pu, i = 1, . . . , 15.

To demonstrate the closed-loop response of our proposedfrequency regulation architecture using the 37-bus network,we increase the amount of power demanded by loads `20,`25, and `33 by 15% at t = 3 s, t = 5 s, and t = 7 s,respectively, such that ∆`20 = 0.2143 pu, ∆`25 = 0.2624 pu,and ∆`33 = 0.2195 pu. Given these load perturbations andthe choice of c = 0.01, a recomputation of the u∗i ’s and the

14

Load Bus

Generator Bus

17

16

18

19

20

21 23

22

24

25

26

27

28

29 30

31

32

34

33

35 3637

1 2

3 4

5 6

7

8

9

10

11 12 13 14

15

Fig. 7: Graph-theoretic model of physical layer of 37-bus microgrid.

0 1 2 3 4 5 6 7 8 90.34

0.36

0.38

0.4

time, t [s]

Max

imum

Net

wor

kFl

ows Estimated Actual

Fig. 8: Estimated vs. actual max flow for Case III.

sensitivities of hl(·) to changes in load is performed shortlyafter load `33 is perturbed at t ≈ 7 s. The response ofthe closed-loop system is illustrated by the plots in Fig. 9.Specifically, the evolution of the weighted average frequencyerror is shown in Fig. 9a, from which it can be seen thatthe frequency error that results from the load perturbations isquickly eliminated. Additionally, the generator set-points andrespective u∗i ’s (illustrated by the dotted traces) are shownin Fig. 9b. From the figure, it can be seen that the amountby which the generators deviate away from the originallycomputed u∗i ’s is relatively small, and that the new u

∗i ’s are

computed at t ≈ 7 s after load `33 is perturbed.As in the six-bus cases, we illustrate the evolution of

the estimated and computed values for the maximum flowthroughout the simulation in Fig. 8. Similar to the smallernetwork cases, the estimated value of hl(·) closely tracks thetrue computed value, with errors on the order of 0.0001 forthe entire 9-second duration.

VII. CONCLUDING REMARKS

In this paper, we introduced a control architecture suitablefor regulating the frequency in an islanded ac microgrid withno inertia. The approach we proposed is designed to regulatethe average frequency subject to small load perturbations byadjusting the output of the generators in the system around set-points that are known to result in phase-cohesive operation. Tohandle larger perturbations in the loads, we also proposed amethod that enables the computation of an estimate for theamount by which the system has deviated from the phase-cohesive operating point; the controller monitors this estimateas the system evolves to determine when the set-points to betracked should be recomputed. We also proposed a distributedimplementation of our proposed architecture, including theintroduction of an algorithm that enables processors locatedat each bus to determine phase-cohesive set-points and thenetwork flows that result. Finally, we demonstrated our pro-posed control architecture and its distributed implementationusing three case studies applied to two test systems.

APPENDIX AANALYTICAL RESULTS

A. Proof for Proposition 1

In order to choose αi’s and κi’s such that ρ(Φ) ≤ 1, whereΦ is defined in (13), we consider the characteristic equation of

Φ given by p(λ) = det(Φ− λIm+1). By taking advantage ofthe structure of Φ − λIm+1 and using its cofactor expansionalong row 1, p(λ) becomes

p(λ) = (β − λ)(1− λ)m +Dm∑j=1

(−1)jαj det(M1,j+1),

(48)

where M1,j+1 is the matrix Φ−λIm+1 with the first row and(j + 1)th column deleted. Using cofactor expansion along thefirst column of M1,j+1, it can be shown that

det(M1,j+1) = (−1)j+1κj(1− λ)m−1. (49)

Thus, the characteristic equation becomes

p(λ) =(β − λ)(1− λ)m

+D(1− λ)m−1m∑j=1

(−1)j(−1)j+1αjκj

=(1− λ)m−1λ[λ− (β + 1)], (50)

from which it follows that λ1 = β + 1, λm = 0, and λj = 1,j = 3, 4, . . . ,m+ 1.

From the analysis above, it is clear that by appropriatelychoosing gains such that −2 ≤ β ≤ 0, all of the eigenvaluesof Φ will lie on the boundary of or within the unit circle.However, for marginal stability, given that λ = 1 has algebraicmultiplicity m− 1, we must show that Φ is nondefective, i.e.,we must show that the geometric multiplicity of λ = 1 ism − 1. If we consider the characteristic equation in (50) forλ = 1, we see that since the rightmost m columns of thematrix Φ−Im+1 are linearly dependent, the rank of Φ−Im+1is 2. Given the relationship between the rank and nullity of amatrix, it is clear that the m− 1 eigenvectors associated withλ = 1 are linearly independent, which implies that λ = 1 hasgeometric multiplicity m− 1.

Given that Φ is nondefective, we can write its Jordanform as Φ = UJU−1 where J := diag({λj}) and U is anonsingular matrix composed of columns corresponding to theeigenvectors of Φ given by

U =

[β −1 0 . . . 0κ κ w1 . . . wm−1

], (51)

where wj ∈ Rm, j = 1, 2, . . . ,m−1, are linearly independentand orthogonal to α. It then follows that we can write the value

15

0 1 2 3 4 5 6 7 8 9

0.01

0

−0.01−0.02−0.03−0.04−0.05−0.06−0.07

time, t [s]

∆ω

(t)

[rad

/s]

(a) Weighted average frequency error.

0 1 2 3 4 5 6 7 8 9

1.4

1.6

1.8

2

2.2

2.4

2.6

time, t [s]

ui(t

),i

=1,...,1

5[p

u]

(b) Generator set-points.

Fig. 9: Case III simulation results with loads `20, `25, and `33 perturbed at t = 3 s, t = 5 s, and t = 7 s, respectively.

of[∆ω[r], e[r]

]Tin matrix form as[

∆ω[r]e[r]

]=UJrU−1

[∆ω[0]e[0]

]+

[r−1∑s=0

UJsU−1

] [−10m

]D∑i∈V(`)p

∆`i, (52)

where ∆ω[0] = D∑i∈V(`)p

∆`i and e[0] = 0m. Note that

(U(Jr −r−1∑s=1

Js))1 = [β(1 + β)r − β

r−1∑s=1

(1 + β)s, 0Tm],

(53)

where (U(Jr −∑r−1s=1 J

s))1 is the first row of U(Jr −∑r−1s=1 J

s). Then, by using (52) – (53), we obtain that

∆ω[r] =

(((1 + β)r −

r−1∑s=1

(1 + β)s)β(U−1)11 − 1

)∆ω[0],

(54)

where (U−1)ij is the (i, j)-th entry of the matrix U−1. Bydirectly computing ∆ω[1] and solving the following equation:

∆ω[1] = (β − 1)∆ω[0] = (β(1 + β)(U−1)11 − 1)∆ω[0],(55)

we find that (U−1)11 = 11+β . Then, by using (54), weobtain that the value of the average frequency error given thecontroller in (11) – (12) at any round r is given by

∆ω[r] = (1 + β)r−1(β − 1)D∑i∈V(`)p

∆`i. (56)

From (56), we see that choosing gains such that β = −2 willresult in oscillatory behavior from the closed loop system;similarly, choosing gains such that β = 0 will lead toconstant frequency error, i.e., ∆ω[r] ≡ −D

∑i∈V(`)p

∆`i,r = 0, 1, 2, . . . . Thus, to ensure that ∆ω[r] → 0 as r → ∞,we must enforce strict inequalities on the value of β, i.e., werestrict to choosing gains such that −2 < β < 0.

B. Proof for Proposition 2

Consider the following quadratic optimization problem:

ming,f

12b

TQb

s.t. fij≤ fij ≤ f ij , {i, j} ∈ Ep

gi≤ gi ≤ gi, i ∈ V

(g)p ,

(57)

which is a quadratic programming problem, where Q is a diag-onal weight matrix with entries Qii = 1wi =

1δp(i)+1

, ∀i ∈ Vp.To solve this problem, we define Dg = {g : gj ≤ gj ≤gj , j ∈ V

(g)p } and Df = {f : f ij ≤ fij ≤ f ij , {i, j} ∈ Ep},

and apply the gradient projection method, the iterations ofwhich are given by [24, Section 2.3]:

f [k + 1] =

[f [k]− s

2

(∂bTQb

∂f[k]

)T]+Df

, (58)

g[k + 1] =

[g[k]− s

2

(∂bTQb

∂u[k]

)T]+Dg

, (59)

where [·]+D denotes the projection onto the set D. The iterationsin (58) – (59) can be simplified as follows:

f [k + 1] =

[f [k]− s

(∂b

∂f

)TQb[k]

]+Df

=[f [k] + sMTQb[k]

]+Df, (60)

g[k + 1] =

[g[k]− s

(∂b

∂g

)TQb[k]

]+Dg

= [u[k]− sQb[k]]+Dg . (61)

The iterations in (60) – (61) can be written as:

fij [k + 1] =

[fij [k] + s

bi[k]

wi− sbj [k]

wj

]fijfij

, ∀{i, j} ∈ Ep

gi[k + 1] =

[gi[k]− s

bi[k]

wi

]gigi

, ∀i ∈ V(g)p ,

which are identical to iterations (36) – (37) when s = 1/2.Now, we show that, for 0 < s ≤ 0.8, iterations (60) – (61)converge globally to a phase-cohesive solution. Define x[k] =

16

[gT[k], fT[k]

]Tand ∆x[k + 1] = x[k + 1]− x[k]. Then,

‖∆x[k + 1]‖ =∥∥[Ax[k]]+ − [Ax[k − 1]]+∥∥ , (62)

where [·]+ is the projection onto the positive orthant, A =[Im − sQ sQMsMTQ I|Ep| − sMTQM

]. Since [·]+ is a non-expansive

mapping, we have that ‖[Ax[k]]+ − [Ax[k − 1]]+‖ ≤‖A∆x[k]‖ [24, Proposition 2.1.3]; therefore,

‖∆x[k + 1]‖ ≤ ‖A∆x[k]‖ . (63)

Define F =[

Q QMMTQ MTQM

], which is a symmetric

and positive-semidefinite matrix, and let λmax(F ) denote thelargest eigenvalue of F . Clearly, if 0 < s < 2λmax(F ) , then,−1 < λ(A) ≤ 1, where λ(A) is any eigenvalue of A. Next,we find γ such that γ ≥ λmax(F ). By using the determinantformula for the Schur complement of Q−λIm, where λ is aneigenvalue of F , we have that

0 = det(F − λIm+|Ep|),= det(Q− λIm) det(MTQM − λI|Ep| −H), (64)

where H = MTQ(Q − λIm)−1QM . Let λmax denote thelargest eigenvalue of MTQM , and choose γ ≥ λmax + 0.5.Since Q ≤ 0.5Im, we have that γIm − Q ≥ λmaxIm ≥2λmaxQ; thus,

MTQM − γI|Ep| −H ≤MTQM(1 +

0.5

λmax)− γI|Ep| ≤ 0,

which implies that λmax(F ) ≤ γ. The absolute sum of theentries in each row corresponding to an edge {i, j} ∈ Ep ofMTQM can be easily shown to be equal to δp(i)δp(i)+1 +

δp(j)δp(j)+1

.Therefore, by Gershgorin disc theorem (see [25, Theo-rem 6.1.1]), λmax ≤ max

{i,j}∈Ep

δp(i)δp(i)+1

+δp(j)δp(j)+1

. Define χ =

max{i,j}∈Ep

δp(i)δp(i)+1

+δp(j)δp(j)+1

. Then, we can choose γ = χ+ 0.5

and any 0 < s < 2γ . Note that χ < 2 and γ < 2.5,which implies that we can choose any 0 < s ≤ 0.8 tohave 1 ≥ λ(A) > −1. Since A is symmetric, we candecompose x[k] as follows: x[k] = v[k] + c[k], whereAv[k] 6= v[k], Ac[k] = c[k] and v[k] ⊥ c[k]. Define∆v[k+ 1] = v[k+ 1]− v[k] and ∆c[k+ 1] = c[k+ 1]− c[k].Then, ‖A∆x[k]‖ ≤ σ2(A) ‖∆v[k]‖+‖∆c[k]‖, where σ2(A) isthe second largest singular value of A. By using (63), we havethat ‖∆v[k + 1]‖+‖∆c[k + 1]‖ ≤ σ2(A) ‖∆v[k]‖+‖∆c[k]‖.Then, clearly, ‖∆v[k]‖ → 0 as k → ∞ since σ2(A) < 1.Suppose that v[k]→ v and v[k] = v+ �[k] for some sequence�[k]. To show that ‖∆x[k]‖ → 0, it suffices to show that‖∆c[k]‖ → 0. The iterations in (60) – (61) can be written as:

v[k + 1] + c[k + 1] = [Ax[k]]+ = v[k] + c[k]− sX[k]Fv[k],(65)

where X[k] is an (m+ |Ep|)×(m+ |Ep|) diagonal matrix with0 ≤ Xii[k] ≤ 1. Then, from (65), we have that

v + �[k + 1] + c[k + 1] =v + �[k] + c[k]− sX[k]F (v + �[k]),

or c[k+1] = �[k]+c[k]−sX[k]F (v+�[k])−�[k+1]. Finally,from the last equation we obtain that

limk→∞

(∆c[k + 1] + sX[k]Fv) = limk→∞

(�[k]− sX[k]F�[k]

− �[k + 1]) = 0.

If limk→∞

sX[k]Fv = 0, then, limk→∞

∆c[k + 1] = 0, whichwould establish the convergence of the algorithm. Suppose thatsX[k]Fv does not converge to 0. Since Xi[k](Fv)i ≥ 0 orXi[k](Fv)i ≤ 0 for all k, then, without loss of generality, wecan assume that there exists some i for which Xi[k](Fv)i ≥ δ,∀k > N , for some δ > 0 and N . From (65), we have that

xi[k + 1] = xi[k]− sXi[k](Fv)i − sXi[k](F�[k])i. (66)

Since �[k] → 0, (66) implies that xi[k] converges to itsupper limit and Xi[k](Fv)i → 0, which is a contradiction.Therefore, lim

k→∞sX[k]Fv = 0, lim

k→∞∆c[k] = 0, and x[k]

converges to a stationary point [24, Proposition 2.3.2], whichis also a global minimum since the cost function 12b

TQb isconvex in f and g [24, Proposition 2.1.2].

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