Breaking the Hierarchy: Distributed Control & Economic Optimality in Microgrids

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by  Florian Dörfler

Breaking the Hierarchy: Distributed Control
& Economic Optimality in Microgrids
Florian Do¨rfler, John W. Simpson-Porco, and Francesco Bullo
Abstract—Modeled after the hierarchical control architecture economic operation [6], [7]. A variety of control and decision
ofpowertransmissionsystems,alayeringofprimary,secondary, architectures—rangingfromcentralizedtofullydecentralized
andtertiarycontrolhasbecomethestandardoperationparadigm
— have been proposed to address these challenges [5]–[8]. In
4 for islanded microgrids. Despite this superficial similarity, the
1 control objectives in microgrids across these three layers are transmissionnetworks,thedifferentcontroltasksareseparated
0 varied and ambitious, and they must be achieved while allowing in their time scales and aggregated into a hierarchy. Similar
2 forrobustplug-and-playoperationandmaximalflexibility,with- operation layers have been proposed for microgrids.
c out hierarchical decision making and time-scale separations. In ControlHierarchyinTransmissionSystems:Thefoundation
e this work, we explore control strategies for these three layers ofthishierarchy,termedprimarycontrol,mustrapidlybalance
D and illuminate some possibly-unexpected connections and de-
generationanddemand,whilesharingtheload,synchronizing
pendencies among them. Building from a first-principle analysis
8 of decentralized primary droop control, we study centralized, the AC voltage frequencies, and stabilizing their magnitudes.
decentralized, and distributed architectures for secondary fre- This is accomplished via decentralized droop control, where

] quency regulation. We find that averaging-based distributed generators are controlled such that their power injections are
C controllers using communication among the generation units
proportional to their voltage frequencies and magnitudes [9].
O offer the best combination of flexibility and performance. We
Droop controllers induce steady-state errors in frequency
further leverage these results to study constrained AC economic
h. dispatchinatertiarycontrollayer.Surprisingly,weshowthatthe and voltage magnitudes, which are corrected in a secondary
t minimizers of the economic dispatch problem are in one-to-one control layer. At the transmission level, the network is parti-
a
correspondence with the set of steady-states reachable by droop tioned into control areas, and a few selected generators then
m
control.Inotherwords,theadoptionofdroopcontrolisnecessary
balance local generation in each area with load and inter-area
[ andsufficienttoachieveeconomicoptimization.Thisequivalence
resultsinsimpleguidelinestoselectthedroopcoefficients,which power transfers. Termed automatic generation control (AGC),
2 include the known criteria for power sharing. We illustrate the this architecture is based on centralized integral control and
v performance and robustness of our designs through simulations. operates on a slower time scale than primary control [10].
7
The operating point stabilized by primary and secondary
6
controlisscheduledinatertiarycontrollayer,toestablishfair
7
1 I. INTRODUCTION load sharing among the sources, or to dispatch the generation
1. With the goal of integrating distributed renewable genera- to minimize operational costs. In conventional operation of
0 tionandenergystoragesystems,theconceptofamicrogridhas bulkpowersystems,aneconomicdispatchisoptimizedoffline,
4 recentlygainedpopularity[2]–[5].Microgridsarelow-voltage in a centralized fashion, using precise load forecasts [11]. In
1 electrical distribution networks, heterogeneously composed [12]–[19] it has been shown that the dynamics of a power
v: of distributed generation, storage, load, and managed au- transmission system with synchronous generators and AGC
i tonomously from the larger transmission network. Microgrids naturally optimize variations of the economic dispatch.
X
are able to connect to a larger power system, but are also able Adaption of Control Layers to Microgrids: With regards to
ar toislandthemselvesandoperateindependently.Suchislanding primary control in islanded microgrids, inverters are typically
could be the result of an emergency, such as an outage of the controlledtoemulatethedroopcharacteristicsofsynchronous
larger utility grid, or may be by design in an isolated grid. generators [3]–[7]. Despite forming the foundation of micro-
grid operation, networks of droop-controlled inverters have
Distributed energy sources in a microgrid generate either
onlyrecentlybeensubjecttoarigorousanalysis[20],[21].We
DCorvariablefrequencyACpower,andareinterfacedwithan
ACgridviapowerelectronicDC/ACinverters.Inislandedop- also refer to [22]–[28] for further results. To compensate for
steady-state deviations induced by droop control, secondary
eration, it is through these inverters, cooperative actions must
integral control strategies akin to AGC have been adapted
betakentoensurefrequencysynchronization,voltagestability,
tomicrogrids.Whereasfullydecentralizedintegralcontrollers
power balance, load sharing, regulation of disturbances, and
successfully regulate the frequency, they result in steady-state
ThisworkwassupportedinpartbyETHstartupfunds,theNationalScience deviationsfromthedesiredpower injectionprofile[29].Thus,
and Engineering Research Council of Canada, and the National Science distributed controllers merging primary and secondary control
FoundationNSFCNS-1135819.Apreliminaryversionofpartoftheresults
have been proposed based upon continuous-time averaging
inthisdocumenthasbeenpresentedin[1].
Florian Do¨rfler is with the Automatic Control Laboratory, Swiss with all-to-all [30], [31] or nearest-neighbor [20], [25] com-
Federal Institute of Technology (ETH) Zu¨rich, Switzerland. Email: munication. In transmission grids, the tertiary optimization
[email protected]. J. W. Simpson-Porco and F. Bullo are with the
layer can be merged with the primary and secondary layer
MechanicalEngineeringDepartment,UniversityofCaliforniaSantaBarbara.
Email:{johnwsimpsonporco,bullo}@engineering.ucsb.edu. based on continuous-time optimization approaches [12]–[19].
Similar discrete-time approaches are based on game-theoretic discovered relation between AC and DC power flows [37],
ideas [32] or discrete-time averaging algorithms [33], [34]. [38] and show that the set of minimizers of the nonlinear and
The above approaches employ varying models ranging non-convex AC economic dispatch optimization problem are
from linear to nonlinear differential-algebraic, some of which inone-to-onecorrespondencewiththeminimizersofaconvex
are not appropriate in microgrids such as lossless lines and DC dispatch problem. Our next result shows a surprising
rotationalinertia.Someoftheproposedstrategiesarevalidated symbioticrelationshipbetweenprimary/secondarycontroland
only numerically without providing further analytic insights. tertiary. We show that the minimum of the AC economic dis-
Oftentheprimaryandsecondarycontrolloopsmayinteractin patchcanbeachievedbyadecentralizeddroopcontroldesign.
an adverse way unless a time-scale separation is enforced, the Whereassimilarconditionsareknownforrelatedtransmission
gains are carefully tuned, or an estimate of the load is known. system problems [12]–[19] (in a simplified linear and convex
Transmission Level vs. Distribution & Microgrids: While setting with lossless DC power flows), we also establish a
the hierarchical architecture has been adapted from the trans- converseresult:everydroopcontrollerresultsinasteady-state
mission level to microgrids, the control challenges and archi- which is the minimizer of some AC economic dispatch. We
tecture limitations imposed by the microgrid framework are deduce, among others, that the optimal droop coefficients are
as diverse as they are daunting. The low levels of inertia in inversely proportional to the marginal generation costs, and
microgridsmeanthatprimarycontrolmustbefastandreliable the conventional power sharing objectives is a particular case.
to maintain voltages, frequencies, and power flows within In summary, we demonstrate that simple distributed and
acceptabletolerances,whilethehighlyvariableanddistributed averaging-based PI controllers are able to simultaneously ad-
nature of microgrids preclude centralized control strategiesof dressesprimary,secondary,andtertiary-levelobjectives,with-
any kind. Microgrid controllers must be able to adapt in real out time-scale separation, relying only on local measurements
timetounknownandvariableloadsandnetworkconditions.In and nearest-neighbor communication, and in a model-free
short, the three layers of the control hierarchy must allow for fashion independent of the network parameters and topology,
as close to plug-and-play operation as possible, be either dis- theloadingprofile,andthenumberofsources.Thus,ourcon-
tributedorcompletelydecentralized,withoutknowledgeofthe trol strategy is suited for true plug-and-play implementation.
system model and the load and generationprofile, and operate In Section VI, we illustrate the performance and robustness
seamlessly without a pre-imposed separation of timescales. of our controllers with a simulation study of the IEEE 37 bus
Contributions and Contents: In Section II, we present a distributionnetwork.Finally,SectionVIIconcludesthepaper.
comprehensive modeling and control framework for micro- The remainder of this section introduces some preliminaries.
grids with heterogenous components and control objectives.
Preliminaries and Notation
Our approach builds on a first-principle nonlinear differential-
algebraic model, decentralized primary droop controllers, and Vectorsandmatrices:GivenafinitesetV,let|V|denoteits
networks with constant (not necessarily zero) resistance-to- cardinality. Given a finite index set I and a real-valued one-
reactance ratios extending the conventional lossless models. dimensional array {x ,...,x }, the associated vector and
1 |I|
In Section III, we review the properties and limitations of diagonal matrix are x∈R|I| and diag({x } )∈R|I|×|I|.
i i∈I
droop control, including the conditions for the existence of Let 1 and 0 be the n-dimensional vectors of unit and zero
n n
stable synchronized solutions satisfying actuation constraints entries.WedenotethediagonalvectorspaceSpan(1 )by1
n n
and proportional load sharing. Moreover, we show the fol- anditsorthogonalcomplementby1⊥ (cid:44){x∈Rn : 1Tx=0}.
n n
lowing reachability result: the set of feasible setpoints for Algebraic graph theory: We denote by G(V,E,A) an
generation dispatch is in one-to-one correspondence with the undirectedandweightedgraph,whereV isnodeset,E ⊆V×V
set of steady-states reachable via decentralized droop control. is the edge set, and A = AT ∈ R|V|×|V| is the adjacency
InSectionIV,westudyseveraldecentralizedanddistributed matrix.Ifanumber(cid:96)∈{1,...,|E|}andanarbitrarydirection
secondary integral control strategies. We first discuss the are assigned to each edge, the incidence matrix B ∈R|V|×|E|
limitations of decentralized secondary integral control akin to is defined component-wise as B = 1 if node k is the sink
k(cid:96)
AGC. Next, we study distributed secondary control strategies node of edge (cid:96) and as B =−1 if node k is the source node
k(cid:96)
based on averaging. We provide a rigorous analysis for the ofedge(cid:96);allotherelementsarezero.TheLaplacianmatrixis
strategiesproposedin[30],[31]foraproperchoiceofcontrol L(cid:44)Bdiag({a } )BT. If the graph is connected, then
ij {i,j}∈E
gains and compare them to our earlier work [20] with regards ker(BT)=ker(L)=1 . For acyclic graphs, ker(B) = ∅,
|V|
totuninglimitationsandcommunicationcomplexity.Weshow and for every x ∈ 1⊥ there is a unique ξ ∈ R|E| satisfying
|V|
that all these distributed strategies successfully regulate the Kirchoff’s Current Law (KCL) x=Bξ. In a circuit, x are the
frequency, maintain the injections and stability properties of nodal current injections, and ξ are the associated edge flows.
theprimarydroopcontroller,anddonotrequireanyseparation Geometry on the n-torus: The set S1 denotes the circle, an
oftimescales.Finally,wedemonstratethatthesepropertiesare angle is a point θ ∈ S1, and an arc is a connected subset
maintained when only a subset of generating units participate of S1. Let |θ − θ | be the geodesic distance between two
1 2
in secondary control. The effectiveness and practical applica- angles θ ,θ ∈ S1. The n-torus is Tn = S1 ×···×S1. For
1 2
bility of the proposed distributed secondary control strategies γ ∈[0,π/2[ and a graph G(V,E,·), let ∆ (γ)={θ ∈T|V| :
G
has been validated experimentally; see [30], [35], [36]. max |θ −θ | ≤ γ} be the closed set of angle arrays
{i,j}∈E i j
In Section V, we study tertiary control policies that mini- θ =(θ ,...,θ )withneighboringanglesθ andθ ,{i,j}∈E
1 n i j
mize an economic dispatch problem. We leverage a recently no further than γ apart. Let ∆ (γ) be the interior of ∆ (γ).
G G
II. MICROGRIDSANDTHEIRCONTROLCHALLENGES After appropriate inner control loops are established, an in-
verter behaves much like a controllable voltage source behind
A. Microgrids, AC Circuits, and Modeling Assumptions
a reactance [4], which is the standard model in the literature.
We adopt the standard model of a microgrid as a syn-
chronous linear circuit. The associated connected, undirected,
and complex-weighted graph is G(V,E,A) with node set (or B. Primary Droop Control
buses) V = {1,...,n}, edge set (or branches) E ⊂ V×V, The frequency droop controller is the main technique for
impedances zij ∈ C for each {i,j} ∈ E,and symmetric primary control in islanded microgrids [3]–[7]. At inverter
weights (or admittances) aij = aji = 1/zij. The admittance i, the frequency θ˙i is controlled to be proportional to the
matrixY =Bdiag({zi−j1}{i,j}∈E)BT ∈Cn×nistheassociated measured power injection Pe,i(θ) according to
Laplacian. We restrict ourselves to acyclic(alsocalledradial)
topologies prevalent in low-voltage distribution networks. Diθ˙i =Pi∗−Pe,i(θ), i∈VI, (4)
To each node i ∈ V, we associate an electrical power
√ where P∗ ∈ [0,P ] is a nominal injection setpoint, and the
injection S = P + −1Q ∈ C and a voltage phasor i i
√e,i e,i e,i proportionality constant D ≥0 is referred to as the (inverse)
Vi = Eie −1θi ∈ C corresponding to the magnitude Ei > 0 droop coefficient. In this noitation, θ˙ is actually the frequency
and the phase angle θ ∈S1 of a harmonic voltage solution to i
i error ω −ω∗, where ω∗ is the nominal network frequency.
the AC circuit equations. The complex vector of nodal power i
The droop-controlled microgrid is then described by the
injectionsisthenS =V ◦(YV)C,whereC denotescomplex
e nonlinear, differential-algebraic equations (DAE) (2),(4).
conjugation and ◦ is the Hadamard (element-wise) product.
Remark 1: (Droop Controllers for Non-Inductive Net-
We assume that all lines in the microgrid are made from
works). The droop control equations (1)-(4) are valid for
the same material, and thus have uniform per-unit-length
√ purelyinductivelineswithoutresistivelosses.Thisassumption
resistance-to-reactance ratios. It follows that z =|z |e −1ϕ
ij ij is typically justified, as the inverter output impedances are
for some angle ϕ ∈ [−π/2,π/2] for all {i,j} ∈ E. The
√ controlled to dominate over the network impedances [39].As
associated admittance matrix is Y =e −1(π/2−ϕ)Y˜ where
discussed prior to equation (1), this assumption can be made
Y˜ =Bdiag(cid:16)(cid:8)−√−1/|z |(cid:9) (cid:17)BT , withoutlossofgeneralityinnetworkswithconstantresistance-
ij {i,j}∈E
to-reactance ratios provided that the power injections are
√
itshethpeowademrittrtaannscfeormmaattriioxnoS˜f t=heSloes√sl−es1s(ϕc−irπc/u2i)t,.tAhefteprowapeprlyflionwg transformed as S˜e =See −1(ϕ−π/2), or in components
equations are lossless and ienducetive: S˜e =V ◦(Y˜V)C. In the (cid:20)P˜e,i(cid:21)=(cid:20)sin(ϕ) −cos(ϕ)(cid:21)(cid:20)Pe,i(cid:21) . (5)
following we assume, without loss of generality, that all lines Q˜e,i cos(ϕ) sin(ϕ) Qe,i
are purely inductive ϕ=π/2; see Remark 1 for an additional
Indeed, (5) is a common transformation decoupling lossy and
discussion on this assumption and the power transformation.
losslessinjections[28],[40]thatisconsistentwiththefactthat
In summary, the active/reactive nodal power injectionsare
activeandreactivedroopcontrollawsarereversedforresistive
P =(cid:88)n Im(Y )E E sin(θ −θ ), i∈V, (1a) lines (ϕ = 0) and negated for capacitive lines (ϕ = −π/2)
e,i ij i j i j
j=1 [4, Chapter 19.4]. Finally, due to continuity and exponential
(cid:88)n
Q =− Im(Y )E E cos(θ −θ ), i∈V. (1b) stability, all of our forthcoming results also hold for networks
e,i ij i j i j
j=1 with sufficiently uniform resistance-to-reactance ratios. (cid:3)
We adopt the standard decoupling approximation [4], [9]
where all voltage magnitudes E are constant in the active
i
C. Secondary Frequency Control
power injections (1a) and P =P (θ). By continuity and
e,i e,i
exponentialstability,ourresultsarerobusttoboundedvoltage Thedroopcontroller(4)inducesastaticerrorinthesteady-
dynamics [20], [37], which we illustrate via simulations. state frequency. If the droop-controlled system (2), (4) settles
We partition the set of buses into loads and inverters, V = to a frequency-synchronized solution, θ˙ (t) = ω ∈ R for
i sync
V ∪V ,anddenotetheircardinalitiesbyn(cid:44)|V|,n (cid:44)|V |, all i ∈ V, then summing over all equations (2),(4) yields the
L I L L
and n (cid:44)|V |. Each load i∈V demands a constant amount synchronous frequency ω as the scaled power imbalance
I I L sync
of active power Pi∗ ∈R and satisfies the power flow equation (cid:80) P∗
ω (cid:44) i∈V i . (6)
0=Pi∗−Pe,i(θ), i∈VL. (2) sync (cid:80)i∈VI Di
We refer to the buses V strictly as loads, with the under- Notice that ω is zero if and only if the nominal injections
L sync
standing that they can be either loads demanding constant P∗ are balanced: (cid:80) P∗ =0. Since the loads are generally
i i∈V i
powerP∗ <0,orconstant-powersourcessuchasPVinverters unknown and variable, it is not possible to select the nominal
i
performing maximum power point tracking with P∗ >0. source injections to balance them. Likewise, to render ω
i sync
We denote the rating (maximal power injection) of inverter small, the coefficients D cannot be chosen arbitrary large,
i
i ∈ V by P ≥ 0. As a necessary feasibility condition, we sincetheprimarycontrolbecomesslowandpossiblyunstable.
I i
assumethroughoutthisarticlethatthetotalload(cid:80) P∗ is To eliminate this frequency error, the primary control (4)
i∈VL i
anetdemandserviceablebytheinverters’maximalgeneration: needs to be augmented with secondary control inputs u (t):
i
(cid:88) (cid:88)
0≤− i∈VLPi∗ ≤ i∈VI Pi. (3) Diθ˙i =Pi∗−Pe,i(θ)+ui(t). (7)
If there is a synchronized solution to the secondary-controlled controlactuation(4)canbeexplicitlymodeledbyafirst-order
equations (2),(7) with frequency ω∗ and steady-state sec- lag filter with state s ∈R and time constant T >0:
sync i i
ondary control inputs u∗=lim u (t), then we obtain itas
i t→∞ i D θ˙ =P∗−s ,
(cid:80) P∗+(cid:80) u∗ (cid:80) u∗ i i i i (11)
ω∗ = i∈V i j∈VI i =ω + j∈VI i . (8) Tis˙i =Pe,i(θ)−si.
sync (cid:80) D sync (cid:80) D
i∈VI i i∈VI i As shown in [27, Lemma 4.1], after a linear change of
Clearly, there are many choices for the inputs u∗ to achieve variables,thedynamics(11)equalthemachinedynamics(10).
i
the control objective ωs∗ync = 0. However, the inputs u∗i are Frequency-dependent loads: If the demand depends on the
typically constrained due to additional performance criteria. frequency[14]–[17],[20],[24],[29],thatis,theleft-handside
of(2)isD θ˙ withD >0,theloaddynamics(2)areformally
i i i
identicaltotheinverterdynamics(4),andthemicrogridmodel
D. Tertiary Operational Control
features no algebraic equations. This frequency-dependence
A tertiary operation and control layer has the objective to does not alter the local stability properties [37].
minimize an economic dispatch problem, that is, an appropri- In summary, all results pertaining to equilibria of the
ate quadratic cost of the accumulated generation: microgrid model (2),(4) and their local stability extend to
synchronous machines, inverters with measurement delays,
(cid:88) 1
minimize f(u)= α u2 (9a) and frequency-dependent loads. Likewise, all secondary or
θ∈Tn,u∈RnI i∈VI 2 i i tertiary control strategies can be equally applied. With these
subject to Pi∗+ui =Pe,i(θ) ∀i∈VI, (9b) extensions in mind, we focus on the microgrid model (2),(4).
P∗ =P (θ) ∀ i∈V , (9c)
i e,i L
|θ −θ |≤γ(AC) ∀{i,j}∈E, (9d)
i j ij III. DECENTRALIZEDPRIMARYCONTROLSTRATEGIES
P (θ)∈[0,P ] ∀i∈V , (9e)
e,i i I In this section, we study the fundamental properties of the
Here, α > 0 is the cost coefficient for source i ∈ V .1 The droop-controlled microgrid (2),(4). In Section IV, we design
i I
appropriate secondary controllers, which preserve the proper-
decisionvariablesaretheanglesθandsecondarycontrolinputs
ties of primary control even if the load profile is unknown.
u. The non-convex equality constraints (9b)-(9c) are the non-
linear steady-state secondary control equations, the security
constraint(9d)limitsthepowerflowoneachbranch{i,j}∈E
A. Symmetries, Synchronization, and Transformations
with γ(AC) ∈[0,π/2[, and (9e) is a generation constraint.
ij
Two typical instances of the economic dispatch (9) are as We begin our analysis by reviewing the symmetries of the
follows:ForP∗ =0,u equalsP (θ),andthetotalgeneration droop-controlled microgrid (2),(4). Observe that if the system
i i e,i
cost is penalized. If the nominal generation setpoints P∗ are (2),(4) possesses a stable and synchronized solution with fre-
i
positive (for example, scheduled according to a load and quencyωsync asgivenin(6),thenitpossessesstableequilibria
renewablegenerationforecast),thenu∗i istheoperatingreserve inarotatingcoordinateframewithfrequencyωsync.Theeffect
that must be called upon to meet the real-time net demand. of secondary control (7) regulating the frequency can be
understoodascarryingoutsuchacoordinatetransformationto
anappropriatelyrotatingframe.Inthissubsection,weformally
E. Heterogeneous Microgrids with Additional Components establish the equivalence of these three ideas so thatwecan
restrictourattentiontoashiftedsysteminrotatingcoordinates.
In the following, we briefly list additional components in a
microgrid, which can be captured by the model (1a),(2),(4). The microgrid equations (2),(4) feature an inherent rota-
tional symmetry: they are invariant under a rigid rotation of
Synchronous machines: Synchronous generators (respec-
all angles. Formally, let rot (r) ∈ S1 be the rotation of a
tively motors) are sources (respectively loads) with dynamics s
point r ∈ S1 counterclockwise by the angle s ∈ [0,2π]. For
Miθ¨i+Diθ˙i =Pi∗−Pe,i(θ), (10) (r1,...,rn)∈Tn, define the equivalence class
where M > 0 is the inertia, and D = D +D > 0 [(r ,...,r )]={(rot (r ),...,rot (r ))∈Tn :s∈[0,2π]}.
i i diss,i droop,i 1 n s 1 s n
combines dissipation D θ˙ and a droop term D θ˙ [9].
diss,i i droop,i i
The constant injection P∗ ∈R is positive for a generator and Thus, a synchronized solution θ∗(t) of (2),(4) is part of a
i one-dimensionalconnectedsynchronizationmanifold[θ∗].For
negative for a load. As shown in [41, Theorem 5.1], the syn-
ω = 0, a synchronization manifold is also an equilibrium
chronous machine model (10) is locally topologically equiva- sync
manifold of (2),(4). In the following, when we refer to a
lent to a first-order model of the form (4): both models share
synchronizedsolutionas“stable”or“unique”,theseproperties
the same equilibria and the same local stabilityproperties.
are to be understood modulo rotational symmetry.
Inverters with measurement delays: The delay between the
powermeasurementP (θ)ataninverteri∈V andthedroop Recallthat,withoutsecondarycontrol,thesynchronousfre-
e,i I
quencyω isthescaledpowerimbalance(6).Bytransform-
sync
ing to a rotating coordinate frame with frequency ω , that
1See Remark 4 for a discussion of more general objective functions and sync
theirimplicationsonthedroopcurvetradingofffrequencyandactivepower. is, θi(t) (cid:55)→ rotωsynct(θi(t)) (with slight abuse of notation, we
maintain the variable θ), a synchronized solution of (2),(4) is (i) Synchronization: thereexistsanarclengthγ ∈[0,π/2[
equivalent to an equilibrium of the shifted control system such that the shifted control system (12) possesses
a locally exponentially stable and unique equilibrium
0=P(cid:101)i−Pe,i(θ), i∈VL, (12a) manifold [θ∗]⊂∆ (γ);
G
Diθ˙i =P(cid:101)i−Pe,i(θ), i∈VI, (12b) (ii) Flow feasibility: the power flow is feasible, that is,
where the shifted power injections are P(cid:101)i = Pi∗ for i ∈ VL, Γ(cid:44)(cid:107)A−1ξ(cid:107)∞ <1. (15)
and P(cid:101)i = Pi∗ −Diωsync for i ∈ VI. We emphasize that the If the equivalent statements (i) and (ii) hold true, then the
shifted injections in (12) are balanced: P(cid:101) ∈ 1⊥. Notice that,
n quantities Γ∈[0,1[ and γ ∈[0,π/2[ are related uniquely via
equivalentlytotransformingtoarotatingframewithfrequency
ω (orreplacingP byP˜),wecanassumethatthesecondary Γ=sin(γ), and sin(BTθ∗)=A−1ξ.
sync
controlinputin(2),(7)takestheconstantvalueu =−D ω
i i sync
for all i∈V to arrive at the shifted control system (12). C. Power Flow Constraints and Proportional Power Sharing
I
We summarize these observations in the following lemma.
While Theorem 3.2 gives the exact stability condition, it
Lemma 3.1: (Synchronization Equivalences). The follow-
offers no guidance on how to select the control parameters
ing statements are equivalent: (P∗,D ) to achieve a set of desired steady-state power injec-
i i
(i) The primary droop-controlled microgrid (2),(4) pos- tions. One desired objective is that all sources meet their the
sessesalocallyexponentiallystableanduniquesynchro- actuation constraints P (θ)∈[0,P ] and share the load in a
e,i i
nization manifold t(cid:55)→[θ(t)]⊂Tn for all t≥0; fair way according to their power ratings [4]–[6].
(ii) The secondary droop-controlled microgrid (2),(7) with Definition 1: (Proportional Power Sharing). Consider an
constant secondary-control input ui = −Diωsync for equilibrium manifold [θ∗]⊂Tn of the shifted control system
aulnliqiu∈e eVqIuipliobsrsieusmsesmaanliofocaldlly[θ¯e]x⊂poTnnen;tially stable and (ti1o2n)a.lTlyheacicnovredrtienrgstVoIthsehiarrepothweertortaaltilnogasdif(cid:80)foi∈rVaLllPii,∗jp∈roVpor-
I
(iii) The shifted control system (12) possesses a locally
exponentially stable and unique equilibrium [θ˜]⊂Tn. Pe,i(θ∗)/Pi =Pe,j(θ∗)/Pj. (16)
If the equivalent statements (i)-(iii) are true, then all systems We also define a useful choice of droop coefficients.
have the same synchronization manifolds [θ(t)]=[θ¯]=[θ˜]⊂ Definition2:(ProportionalDroopCoefficientsandNomi-
Tn and the same power injections Pe(θ(t))=Pe(θ¯)=Pe(θ˜). nalInjectionSetpoints).Thedroopcoefficientsandinjections
Additionally, θ(t)=rotωsynct(ξ0) for some ξ0 ∈[θ¯]=[θ˜]. setpoints are selected proportionally if for all i,j∈VI
In light of Lemma 3.1, we restrict the forthcoming discus-
sion in this section to the shifted control system (12). Pi∗/Di =Pj∗/Dj and Pi∗/Pi =Pj∗/Pj. (17)
Observe also that equilibria of (12) are invariant under
Aproportionalchoiceofdroopcontrolcoefficientsleadstoa
constant scaling of all droop coefficients: if D is replaced by
i fairloadsharingamongtheinvertersaccordingtotheirratings
D ·β forsomeβ ∈R,thenω changestoω /β.Sincethe
i sync sync and subject to their actuation constraints – a result that also
product D ·ω remains constant, the equilibria of (12) do
i sync holds for lossy and meshed circuits [20, Theorem 7]:
notchange.Moreover,ifβ >0,thenthestabilitypropertiesof
Theorem 3.3: (Power Flow Constraints and Power Shar-
equilibriadonotchangesincetimecanberescaledast(cid:55)→t/β.
ing). Consider an equilibrium manifold [θ∗] ⊂ Tn of the
shifted control system (12). Let the droop coefficients be se-
B. Existence, Uniqueness, & Stability of Synchronization lectedproportionally.Thefollowingstatementsareequivalent:
In vector form, the equilibria of (12) satisfy (i) Injection constraints: 0≤P (θ∗)≤P , ∀i∈V ;
e,i i I
(ii) Serviceable load: 0≤−(cid:80) P∗ ≤(cid:80) P .
P(cid:101) =BAsin(BTθ), (13) Moreover,theinverterssharethetoit∈alVLloadi (cid:80) j∈PVI∗ pjropor-
where A = diag({Im(Yij)EiEj}{i,j}∈E) and B ∈ R|V|×|E| tionally according to their power ratings. i∈VL i
is the incidence matrix. For an acyclic network, ker(B) = ∅,
and the unique vector of branch flows ξ ∈ R|E| (associated
D. Power Flow Shaping
to the shifted injections P(cid:101)) is given by KCL as ξ = B†P(cid:101) =
(BTB)−1BTP(cid:101). Hence, equations (13) equivalently read as Wenowaddressthefollowing“reachability”question:given
a set of desired power injections for the inverters, can one
ξ =Asin(BTθ). (14)
select the droop coefficients to generate these injections?
Duetoboundednessofthesinusoid,anecessaryconditionfor We define a power injection setpoint as a point of power
solvability of equation (14) is (cid:107)A−1ξ(cid:107) < 1. The following balance, at fixed load demands and subject to the basic
∞
resultshowsthatthisconditionisalsosufficientandguarantees feasibility condition (15) given in Theorem 3.2.
stability of an equilibrium manifold of (12) [20, Theorem 2]. Definition3:(FeasiblePowerInjectionSetpoint).Letγ ∈
Theorem 3.2: (Existence and Stability of Synchroniza- [0,π/2[. A vector Pset ∈ Rn is a γ-feasible power injection
tion). Consider the shifted control system (12). Let ξ ∈ R|E| setpoint if it satisfies the following three properties:
betheuniquevectorofpowerflowssatisfyingtheKCL,given (i) Power balance: Pset ∈1⊥;
n
by ξ =B†P(cid:101). The following two statements are equivalent: (ii) Load invariance: Piset =Pi∗ for all loads i∈VL;
(iii) γ-feasibility: the associated branch power flows ξset = A. Decentralized Secondary Integral Control
B†Pset are feasible, that is, (cid:107)A−1ξset(cid:107) ≤sin(γ).
∞ To investigate decentralized secondary control, we partition
Thenextresultcharacterizestherelationshipbetweendroop thesetofinvertersasVI =VIP ∪VIS,wheretheactionofthe
controller designs and γ-feasible injection setpoints. For sim- VIP inverters is restricted to primary droop control, and the
plicity, we omit the singular case where ωsync = 0, since VIS invertersusethelocalfrequencyerrorforintegralcontrol:
in this case the droop coefficients offer no control over the
steady-state inverter injections Pe,i(θ∗)=Pi∗−Diωsync. ui(t)=−pi , kip˙i =θ˙i, i∈VIS, (20)
Theorem 3.4: (Power Flow Shaping). Consider the shifted ui(t)=0, i∈VIP .
control system (12). Assume ω (cid:54)= 0, let Pset ∈ 1⊥, and
sync n Consider the case |V | = 1, which mimics AGC inside a
let γ ∈[0,π/2[. The following statements are equivalent: IS
control area of a transmission network. It can be shown, as
(i) Coefficient selection: there exists a selection of droop
a direct corollary to Theorem 4.4 (in Section IV-D), that this
coefficients D , i ∈ V , such that the steady-state
i I controller achieves frequency regulation but fails to maintain
injections satisfy P (θ∗)=Pset, with [θ∗]⊂∆ (γ);
e G the power sharing. Additionally, if a steady-state exists, p
(ii) Feasibility:Pset isaγ-feasiblepowerinjectionsetpoint. must converge to the total power imbalance (cid:80) P∗, whichi
i∈V i
If the equivalent statements (i) and (ii) hold true, then the places a large and unpredictable burden on a single generator.
quantities Di and Piset are related with arbitrary β (cid:54)=0 as For|VIS|>1,itiswell-knownincontrol[42]andinpower
systems[9],thatmultipleintegratorsinaninterconnectedsys-
D =β(P∗−Pset), i∈V . (18)
i i i I temleadundesirablepropertiessuchasundesirableequilibria.
Moreover, [θ∗] is locally exponentially stable if and only if
β(P∗−Pset) is nonnegative for all i∈V . Lemma 4.1 (Steady-states of decentralized integral con-
iProofi:(i)=⇒(ii): Sinceθ∗ ∈∆ (γI)andP (θ∗)∈1⊥, trol): Consider the droop-controlled microgrid (2),(7) with
G e n decentralized secondary integral control (20) at nodes V .
Theorem3.2showsthatPset isaγ-feasibleinjectionsetpoint. IS
If the system reaches an equilibrium, the steady-state source
(ii) =⇒ (i): Let Pset be a γ-feasible injection setpoint.
injections are determined only up to a subspace of the same
Consider the droop coefficients D = β(P∗ − Pset). Since
i i i dimension as the number of decentralized integrators |V |.
ωsync (cid:54)=0,foreachi∈VI weobtainthesteady-stateinjection IS
Proof: We partition the injections according to loads V ,
L
Pe,i(θ∗)=P(cid:101)i =Pi∗−Diωsync invertersVIP =VI\VIS withonlyprimarycontrol(4),andin-
=Pi∗−β(Pi∗−Piset)β1 (cid:80) (cid:80)(iP∈V∗P−i∗Pset) =Piset, v(Pe(cid:101)rLte,rP(cid:101)sIVPI,SP(cid:101)IwSi)thanddecPene(trθa)li=zed(Pine,tLeg(θra)l,Pcoe,nItPro(θl)(,7P),e(2,I0S)(θa)s)P(cid:101)an=d
i∈VI i i obtain the closed-loop equilibria from the equations
(cid:124) (cid:123)(cid:122) (cid:125)
=1
    
where we used (cid:80) Pset =−(cid:80) P∗. Since P (θ∗)= 0 I 0 0 0 P(cid:101)L−Pe,L(θ)
P∗=Pset foreachi∈iV∈I Vi ,wehavei∈PVL(θ)i=Pset.Sien,ciePset 0=0 I 0 0P(cid:101)IP −Pe,IP(θ).
isiγ-feaisible, θ∗ is well Ldefined in ∆eG(γ). By the reasoning 00 00 00 0I 10P(cid:101)IS −−Ppe,IS(θ)
leading to Theorem 3.2 (see [20, Theorem 2]), the shifted
system (12) is stable if and only if all D are nonnegative.
i It follows that the matrix determining the inverter equilib-
Remark 2: (Generation Constraints). For a γ-feasible riuminjectionsP (θ)hasa|V |-dimensionalnullspace.
injection setpoint, the inverter generation constraint Pset ∈ e,IS IS
i Lemma 4.1 implies that the primary objectives (such as
[0,P ]isgenerallynotguaranteedtobemet.Thisconstraintis
i proportional power sharing or power flow shaping) cannot
feasible if P∗ =0 (i∈V ) and an additional conditionholds:
i I be recovered and the steady-state injections depend on initial
−(cid:88) P∗ ≤(cid:0)P /D (cid:1)(cid:88) D , i∈V . (19) values, loads, and exogenous disturbances. These steady-state
j i i j I subspacescorrespondtodifferentchoicesofu∗renderingω∗
j∈VL j∈VI sync
tozeroin(8).Onewaytoremovetheseundesirablesubspaces
The inequalities (19) limit the heterogeneity of the inverter
is to implement (20) via the low-pass filter
power injections, and are sufficient for the load serviceability
condition(3),asonecanseebyrearrangingandsummingover u (t)=−p , k p˙ =θ˙ −(cid:15)p , i∈V , (21)
all i∈V . A similar result holds for the choice P∗ =P . (cid:3) i i i i i i IS
I i i
For small (cid:15) > 0 and large k > 0 (enforcing a time-scale
separation), the controller (21) achieves practical stabilization
IV. CENTRALIZED,DECENTRALIZED,ANDDISTRIBUTED
butdoesnotexactlyregulatethefrequency[29].Inconclusion,
SECONDARYCONTROLSTRATEGIES
the decentralized control (20) and its variations generally
The primary droop controller (4) results in the static fre- fail to achieve fast frequency regulation while maintaining
quencyerrorω in(6).Thepurposeofthesecondarycontrol power sharing among generating units. Additionally, a single
sync
u (t) in (7) is to eliminate this frequency error despite un- microgridsourcemaynothavetheauthorityorthecapacityto
i
knownandvariableloads.Inthissection,weinvestigatediffer- perform secondary control. In the following, we analyze dis-
ent decentralized and distributed secondary control strategies. tributed strategies that exactly recover the primary injections.
B. Centralized Averaging PI (CAPI) Control a locally exponentially stable and unique equilibrium
Different distributed secondary control strategies have been manifold ([θ∗],p∗)⊂∆G(γ)×RnI.
proposed in [30], [31]. In [30], an integral feedback of a If the equivalent statements (i) and (ii) are true, then [θ∗] and
weighted average frequency among all inverters is proposed:3 allinjectionsareasinTheorem3.2,andp∗i =Diωsync,i∈VI.
Proof:Westartbywritingsystem(25)invectorform.Let
ui(t)=−pi , kip˙i = (cid:80)(cid:80)j∈VI DDjθ˙j , i∈VI, (22) DanIgl=esdaicacgo(r{dDinig}it∈oVlIo)a,dlsetVDtaont=di(cid:80)nvie∈rVteIrDsVi, aansdθp=art(itθion,θth)e.
j∈VI j L I L I
Withthisnotation,theclosedloop(25)readsinvectorformas
Here,p isthesecondaryvariableandk >0.ForP∗ =0,the
i i i
average frequency in (22) is the sum of the inverter injections I 0 0 0 I 0 0 P(cid:101)L−Pe,L(θ)
Pe,i(θ).Inthiscase,(22)equalsthesecondarycontrolstrategy 0 DI 0 θ˙I=0 I DI1P(cid:101)I −Pe,I(θ)
in [31], where the averaged inverter injections are integrated. 0 0 k·Dtot q˙ 0 1T Dtot −q
By counter-examples, it can be shown that the secondary (cid:124) (cid:123)(cid:122) (cid:125)
controller (22) does not have the power sharing property of (cid:44)Q1
   
the shifted control system (12) unless the values of Di and ki I 0 0 I 0 0 P(cid:101)L−Pe,L(θ)
are carefully tuned. In the following, we suggest the choice =0 DI 00 DI−1 1 P(cid:101)I −Pe,I(θ) . (26)
k =k/D , i∈V , (23) 0 0 1 0 1T Dtot −q
i i I (cid:124) (cid:123)(cid:122) (cid:125)(cid:124) (cid:123)(cid:122) (cid:125)(cid:124) (cid:123)(cid:122) (cid:125)
where k >0. The closed loop (2), (7), (22), (23) is given by (cid:44)Q2 (cid:44)Q3 (cid:44)x
ThematricesQ andQ arenonsingular,whileQ issingular
1 2 3
0=Pi∗−Pe,i(θ), i∈VL, (24a) with ker(Q3) = Span([0, DI1nI , −1]). On the other hand,
D θ˙ =P∗−P (θ)−p , i∈V , (24b) [1T 1T 0]x=0duetobalancedinjections1TP(cid:101) =0andflows
i i i e,i i I n n
kDp˙ii = (cid:80)(cid:80)j∈j∈VVIIDDjθj˙j , i∈VI. (24c) e1(1qT2uP)ileia(bnθrdi)aq=o∗f0=(2.6I0)t.afBroeyllxoTw=hse0othrnea,mtthxa3t.(cid:54)∈2i,s,kthtehere(Qeeq3qu)ua.itliTiobhnruiaxs,[θ=p∗o]s0fsnriobmlies
solvable for [θ∗]∈∆ (γ) if and only if condition (15) holds.
By changing variables q (cid:44) p /D − ω for i ∈ V and G
observing that kq˙ =(cid:80) i Diθ˙ /i(cid:80) synDc is identicIal for To establish stability, observe that the negative power flow
all i∈V , we cani rewrijte∈VthIe cjlojsed-ljo∈oVpIeqjuations (24) as Jacobian −∂Pe(θ)/∂θ equals the Laplacian matrix L(θ) =
I Bdiag({a } )BT with a (cid:44) Im(Y )E E cos(θ −
ij {i,j}∈E ij ij i j i
0=P(cid:101)i−Pe,i(θ), i∈VL, (25a) θj) as weights [37, Lemma 2]. For [θ∗]∈∆G(γ), all weights
Diθ˙i =P(cid:101)i−Pe,i(θ)−Diq, i∈VI, (25b) aitiivje>se0mairdeesfitnriictetlyLpaopslaitciivaen.foAr{liin,jea}ri∈zaEti,oanndofLt(hθe∗)DiAsEap(2o6s)-
kq˙= (cid:80)j∈VI Djθ˙j , (25c) about the regular set of fixed points ([θ∗],0) and elimination
(cid:80) D of the algebraic variables gives the reduced Jacobian
j∈VI j
(cid:20)I 0 (cid:21)(cid:20)D−1 1 (cid:21)(cid:20)−L (θ∗) 0(cid:21)
where P(cid:101)i is as in (12). In this transformed system, (25c) J(θ∗)= 0 (k·D )−1 1IT D re0d −1 ,
can be implemented as a centralized controller: it receives tot tot
(cid:124) (cid:123)(cid:122) (cid:125)(cid:124) (cid:123)(cid:122) (cid:125)(cid:124) (cid:123)(cid:122) (cid:125)
frequency measurements from all inverters and broadcasts the
(cid:44)Q(cid:101)1 (cid:44)Q(cid:101)2 (cid:44)X
secondary control variable q back to all units. Due to this
whereL (θ∗)istheSchurcomplementofL(θ∗)withrespect
insightonthecommunicationcomplexity,wereferto(22)-(23) red
totheloadentrieswithindicesV .ItisknownthatL (θ∗)is
ascentralizedaveragingproportionalintegral(CAPI)control. I red
again a positive semidefinite Laplacian [43, Lemma II.1]. The
Theorem 4.2: (Stability of CAPI-Controlled Network).
Consider the droop-controlled microgrid (2),(7) with P∗ ∈ matrixQ(cid:101)1 isdiagonalandpositivedefinite,andQ(cid:101)2 ispositive
[0,Pi] and Di >0 for i∈VI. Assume a complete commiuni- semidefinite with ker(Q(cid:101)2)=Span([(DI1nI),−1]).
We proceed via a continuity-type argument. Consider mo-
cationtopologyamongtheinvertersV ,andletu (t)begiven
bTyhethfeolCloAwPinIgcotwntorosltlaetrem(2e2n)tswaitrhe tehqeuIipvaarlaemnte:tricichoice (23). bmyenthtaeriploystihtievepedretufirnbietedmJaactoribxiaQ(cid:101)n2J,(cid:15)(cid:15)(=θ∗(cid:104))D,1wI−T1hDerto1et+Q(cid:15)(cid:101)(cid:105)2wisitrhep(cid:15)la>ce0d.
(i) Stability of primary droop control: the droop control ThespectrumofJ(cid:15)(θ∗)isobtainedfromQ(cid:101)1Q(cid:101)2,(cid:15)Xv =λv for
stability condition (15) holds; some (λ,v)∈C×CnI+1. Equivalently, let y =Q(cid:101)−1v, then
1
(ii) Stability of CAPI control: there exists an arc length
γ ∈ [0,π/2[ such that the closed loop (24) possesses −Q(cid:101)2,(cid:15)·blkdiag(Lred, 1/(k·Dtot))y =λy.
The Courant-Fischer Theorem applied to this generalized
3Asidefromtheintegralfeedback(22),thecontrollerin[30]alsocontains eigenvalue problem implies that, for (cid:15) > 0 and modulo
a proportional feedback of the average frequency. We found that such a
rotational symmetry, all eigenvalues λ are real and negative.
proportionalfeedbackdoesgenerallynotpreservetheequilibriuminjections,
and we omit it here. The controller in [30] uses an arithmetic average with Now, consider again the unperturbed case with (cid:15) = 0.
all Di = 1 in (22). Since the synchronization frequency (6) is obtained We show that the number of zero eigenvalues for (cid:15) = 0
by a weighted average and since Diθ˙i is the inverter injection Pe,i(θ) (for equals those for (cid:15) > 0 and thus stability (modulo ro-
P∗ = 0),we found the choice (22) more natural. Simulations indicate that
i
anyconvexcombinationoftheinverterfrequenciesyieldsidenticalresults. tational symmetry) is preserved as (cid:15) (cid:38) 0. Recall that
ker(Q(cid:101)2) = Span([(DI1nI),−1]), and the image of the thefrequency.Toinvestigatethisscenario,wepartitiontheset
matrixblkdiag(L , 1/(k·D ))excludesSpan([1 ,0]).It of inverters as V = V ∪V , where the action of the V
red tot nI I IP IS IP
followsthatQ(cid:101)2·blkdiag(Lred, 1/(k·Dtot))y iszeroifonlyif inverters is restricted to primary droop control, and the VIS
y ∈Span([1 ,0]) corresponding to the rotational symmetry. inverters perform the secondary DAPI or CAPI control:
nI
We conclude that the number of negative real eigenvalues of
J(cid:15)(θ∗) does not change as (cid:15) (cid:38) 0. Hence, the equilibrium Diθ˙i =Pi∗−Pe,i(θ), i∈VIP , (29a)
([θ∗],0) of the DAE (26) is locally exponentially stable. D θ˙ =P∗−P (θ)+u (t), i∈V , (29b)
i i i e,i i IS
The CAPI controller (22),(23) preserves the primary power
Observe that the V inverters are essentially frequency-
injections while restoring the frequency. However, it requires IP
dependent loads and the previous analysis applies. The fol-
all-to-allcommunicationamongtheinverters,andarestrictive
lowing result shows that partial secondary control strategies
choiceofgains(23).Toovercometheselimitations,wepresent
successfullystabilizethemicrogridandregulatethefrequency.
an alternative controller and a modification of CAPI control.
Theorem 4.4: (Partially Regulated Networks). Consider
the droop-controlled microgrid with primary and partial sec-
C. Distributed Averaging PI (DAPI) Control
ondary control (2),(29) and with parameters P∗ ∈ [0,P ],
i i
Asthirdsecondarycontrolstrategy,considerthedistributed and D > 0 for i ∈ V . For i ∈ V , let the secondary
i I IS
averaging proportional integral (DAPI) controller [20]: control inputs be given by the CAPI controller (22), (23) with
(cid:18) (cid:19) |V | ≥ 1 and a complete communication graph among the
ui =−pi , kip˙i =Diθ˙i+(cid:88)j∈VI Lij Dpii − Dpjj . (27) V|VIISS|n≥ode2sa(nrdesapeccotinvneelcy,tedbycotmhemDunAicPaItioconngtrroalplehra(m27o)ngwtihthe
IS
Here, ki > 0 and L is the Laplacian matrix of a weighted, VIS nodes). The following statements are equivalent:
connected and undirected communication graph between the
(i) Stability of primary droop control: the droop control
inverters. The resulting closed-loop system is then given by
stability condition (15) holds;
0=P∗−P (θ), i∈V , (28a) (ii) Stability of partial secondary control: there is an arc
i e,i L
D θ˙ =P∗−P (θ)−p , i∈V , (28b) length γ ∈[0,π/2[ so that the partially regulated CAPI
i i i e,i i I system(2),(22),(23),(29)(resp.DAPIsystem(2),(27),
(cid:18) (cid:19)
k p˙ =D θ˙ +(cid:88) L pi − pj , i∈V . (28c) (29))possessesalocallyexponentiallystableandunique
i i i i j∈VI ij Di Dj I equilibrium manifold ([θ∗],p∗)⊂∆G(γ)×R|VIS|.
The following result has been established in earlier work [20, If the equivalent statements (i) and (ii) hold true, then for
Theorem 8] and shows the stability of the closed loop (28). i ∈ V , the injections P (θ∗) are as in Theorem 3.2 and
Theorem 4.3: (Stability of DAPI-Controlled Network). p∗ = DISω , where ω e,i = (cid:80) P∗/((cid:80) D ). For
Consider the droop-controlled microgrid (2),(7) with parame- alil otheriinpvaretiratlers i∈V pa,rtiwale havei∈tVhatiP (θ∗i∈)V=ISP∗i.
ters Pi∗ ∈ [0,Pi], and Di > 0 for i ∈ VI. Let the secondary Proof: The proof fIoPr partial CAPI coen,itrol (respeictively,
control inputs be given by (27) with ki >0 for i∈VI and a partial DAPI control) is analogous to the proof of Theorem
connected communication graph among the inverters VI with 4.2 (respectively, [20, Theorem 8]), while accounting for the
Laplacian L. The following two statements are equivalent: partition V =V ∪V in the Jacobian matrices.
I IP IS
(i) Stability of primary droop control: the droop control We now investigate the power sharing properties of partial
stability condition (15) holds; secondary control. The steady-state injections at ([θ∗],p∗) are
(ii) Stability of secondary integral control: there exists
P (θ∗)=P∗, i∈V ∪V ,
an arc length γ ∈ [0,π/2[ such that the closed loop e,i i IP L
(24) possesses a locally exponentially stable and unique P (θ∗)=P∗−D ω , i∈V ,
e,i i i partial IS
equilibrium manifold ([θ∗],p∗)⊂∆G(γ)×RnI.
By applying Theorem 3.3, we obtain the following corollary:
If the equivalent statements (i) and (ii) are true, then [θ∗] and
Corollary4.5:(Injection Constraints and Power Sharing
allinjectionsareasinTheorem3.2,andp∗ =D ω ,i∈V .
i i sync I with Partial Regulation). Consider a locally exponentially
The DAPI control (27) regulates the network frequency,
requires only a sparse communication network, and preserves stable equilibrium ([θ∗],p∗)⊂∆G(γ)×R|VIS|, γ ∈[0,π/2[,
of the partial secondary control system (2),(29) as in Theo-
thepowerinjectionsestablishedbyprimarycontrol.Moreover,
rem 4.4. Select the droop coefficients and injection setpoints
the gains D >0 and k >0 can be chosen independently.
i i proportionally. The following statements are equivalent:
Weremarkthatahigher-ordervariationoftheDAPIcontrol
(i) Injection constraints: 0≤P (θ∗)≤P , ∀i∈V ;
(27) (additionally integrating edge flows) can also be derived (ii) Serviceable load: 0≤− (cid:80)e,i P∗ ≤i (cid:80) P I.S
from a network flow optimization perspective [14], [15]. j j
j∈VIP∪VL j∈VIS
Moreover, the inverters V performing secondary control
D. Partial Secondary Control IS
share the load proportionally according to their power ratings.
The DAPI and CAPI controller require that all invert- These results on partial CAPI/DAPI show that only a
ers participate in secondary control. To further reduce the connected subset of inverters have to participate in secondary
communication complexity and increase the adaptivity of the control, which further reduces the communication complexity
microgrid,itisdesirablethatonlyasubsetofinvertersregulate and increases the adaptivity and modularity of the microgrid.
V. DECENTRALIZEDTERTIARYCONTROLSTRATEGIES Denote the unique vector of AC branch power flows by
ξ = Asin(BTθ); see (14). For an acyclic network, we have
Inthissection,weexaminethetertiarycontrollayer.Incon-
ker(B) = ∅, and ξ ∈ Rn−1 can be equivalently rewritten as
ventional power system operation, the tertiary-level economic
ξ =ABTδ for some δ ∈Rn. Thus, we obtain
dispatch (9) is solved in a centralized way, offline, and with a
precise knowledge of the network model and the load profile. Asin(BTθ)=ABTδ. (32)
In comparison, we show that the economic dispatch (9) is
Now, we associate δ with the angles of the DC flow (30), so
minimizedasymptoticallybyproperlyscaleddroopcontrollers
that(32)isabijectivemapbetweentheACandtheDCflows.
in a fully decentralized fashion, online, and without a model.
Due to the AC security constraints (9d), the sine function
For simplicity and thanks to Lemma 3.1, we restrict our-
is invertible. If the DC security constraints (31d) satisfy
selvestotheshiftedcontrolsystem(12)withtheunderstanding
(cid:107)BTδ(cid:107) ≤max γ(DC) <1, then BTθ can be uniquely
thattheoptimalasymptoticinjectionsarealsoobtainedbyany ∞ {i,j}∈E ij
recovered from (and mapped to) BTδ via (32). Additionally,
secondarycontrolthatreachesthesteadystateu =−D ω .
i i sync
up to rotational symmetry and modulo 2π, the angle θ and be
uniquelyrecoveredfrom(andmappedto)δ.Thus,identity(32)
A. Convex Reformulation of the AC Economic Dispatch
between the AC and the DC flow serves as a bijective change
ThemaincomplicationinsolvingtheACeconomicdispatch of variables (modulo 2π and up to rotational symmetry).
optimization (9) is the nonlinearity and nonconvexity of the This change of variables maps the AC economic dispatch
ACinjectionsconstraints(1a).Inpracticalpowersystemoper- (9) to the DC economic dispatch (31) as follows. The AC
ation, the nonlinear AC injection P (θ) is often approximated injections P (θ) are replaced by the DC injections P (δ).
e e DC
by the linear DC injection P (θ) with components The AC security constraint (9d) translates uniquely to the DC
DC
(cid:88)n constraint(31d)withγi(jDC) =sin(γi(jAC))<1.TheACinjection
P (θ)= Im(Y )E E (θ −θ ), i∈V. (30)
DC,i ij i j i j constraint(9e)ismappedtotheDCinjectionconstraint(31e).
j=1
Finally,ifbothproblems(9)and(31)arefeasiblewithmini-
Accordingly, the AC economic dispatch (9) is approximated
mizersu∗ =v∗ andsin(BTθ∗)=BTδ∗,thenf(u∗)=f(v∗)
by the corresponding DC economic dispatch given by
is the unique global minimum due to convexity of (31).
minimize f(v)=(cid:88) 1α v2 (31a) Theorem 5.1 relies on the bijection (32) between AC and
δ∈Rn,v∈RnI i∈VI 2 i i DC flows in acyclic networks [37], [38]. For cyclic networks,
subject to P∗+v =P (δ) ∀i∈V , (31b) the two problems (9) and (31) are generally not equivalent,
i i DC,i I
P∗ =P (δ) ∀ i∈V , (31c) but the DC flow is a well-accepted proxy for the AC flow.
i DC,i L
We now state a rather surprising result: any minimizer
|δ −δ |≤γ(DC) ∀{i,j}∈E, (31d)
i j ij of the AC economic dispatch (9) can be achieved by an
P (δ)∈[0,P ] ∀i∈V , (31e) appropriately designed droop control (4). Conversely, any
DC,i i I
steady state of the droop-controlled microgrid (2),(4) is the
where the DC variables (δ,v) are distinguished from the AC
minimizer of an AC economic dispatch (9) with appropriately
variables (θ,u). In formulating the DC economic dispatch
chosen parameters. The proof relies on the economic dispatch
(31), we also changed the line flow parameters from γ(AC)
ij criterion [11] stating that all marginal costs α u∗ must be
to γ(DC) ∈[0,π/2[ for all {i,j}∈E. The DC dispatch (31) is i i
ij identicalfortheoptimalinjection,anditcanbeextendedtothe
a quadratic program with linear constraints and hence convex.
constrained case at the cost of a less explicit relation between
Typically, the solution (δ∗,v∗) of the DC dispatch (31)
the optimization parameters and droop control coefficients.
servesasproxyforthesolutionofthenon-convexACdispatch Theorem 5.2: (Droop Control & Economic Dispatch).
(9).Thefollowingresultshowsthatbothproblemsareequiva- ConsidertheACeconomicdispatch(9)andtheshiftedcontrol
lent for acyclic networks and appropriate security constraints. system (12). The following statements are equivalent:
Theorem 5.1: (Equivalence of AC and DC Economic
(i) Strict feasibility and optimality: there are parameters
Dispatch in Acyclic Networks). Consider the AC economic α > 0, i ∈ V , and γ(AC) < π/2, {i,j} ∈ E such
dispatch (9) and the DC economic dispatch (31) in an acyclic i I ij
that the AC economic dispatch problem (9) is strictly
network. The following statements are equivalent: feasible with global minimizer (θ∗,u∗)∈Tn×RnI.
(i) AC feasibility: the AC economic dispatch problem (9) (ii) Constrained synchronization: there exists γ ∈ [0,π/2[
with parameters γ(AC) < π/2 for all {i,j} ∈ E is anddroopcoefficientsD >0,i∈V ,sothattheshifted
ij i I
feasible with a global minimizer (θ∗,u∗)∈Tn×RnI; control system (12) possesses a unique and locally
(ii) DC feasibility: the DC economic dispatch problem (31) exponentially stable equilibrium manifold [θ]⊂∆ (γ)
G
with parameters γ(DC) < 1 for all {i,j} ∈ E is feasible meetingtheinjectionconstraintsP (θ)∈]0,P [,i∈V .
ij e,i i I
with a global minimizer (δ∗,v∗)∈Rn×RnI. Iftheequivalentstatements(i)and(ii)aretrue,then[θ∗]=[θ],
If the equivalent statements (i) and (ii) are true, then u∗ =−Dω 1 , γ =max γ(AC), and for some β >0
sync n {i,j}∈E ij
sin(γ(AC)) = γ(DC), u∗ = v∗, sin(BTθ∗) = BTδ∗, and
ij ij D =β/α , i∈V . (33)
f(u∗)=f(v∗) is a global minimum. i i I
Proof: The proof relies on the fact that branch flows are Theorem 5.2, stated for the shifted control system (12), can
uniqueinanacyclicnetwork:nodevariables(injections)P can be equivalently stated for the CAPI or DAPI control systems
be uniquely mapped to edge variables (flows) ξ via P =Bξ. (by Lemma 3.1). Before proving it, we state a key lemma.
Lemma 5.3: (Properties of strictly feasible points). If inequality constraints (9d)-(9e) are met, Popt ∈]0,P [, [θ∗]⊂
i i
(θ∗,u∗) ∈ Tn ×RnI is a strictly feasible minimizer of the ∆G(γ) with γ =max{i,j}∈Eγi(jAC), and the vector of load and
AC economic dispatch (9), then u∗ is sign-definite, that is, source injections (P∗,Popt) is a γ-feasible injection setpoint.
L I
all u∗i, i ∈ VI, have the same sign. Conversely, any strictly By Theorem 3.4 and identity (18), the droop coefficients
feasible pair (θ,u)∈Tn×RnI of the AC economic dispatch D = −β(P∗ − Popt) = βu∗, i ∈ V , guarantee that the
i i i i I
(9)withsign-definiteuisinverseoptimalwithrespecttosome shifted control system (12) possesses an equilibrium manifold
α∈Rn>I0:thereisasetofcoefficientsαi >0,i∈VI,suchthat [θ] satisfying Pe(θ)=Popt =Pe(θ∗). Forβu∗i >0 (recall u∗
(θ,u) is global minimizer of the AC economic dispatch (9). issign-definite),[θ]islocallyexponentiallystablebyTheorem
Proof: The strictly feasible pairs of (9) are given by the 3.2.Finally,theidentityP (θ)=P (θ∗)showsthat[θ∗]=[θ].
e e
setofall(θ,u)∈Tn×RnI satisfyingthepowerflowequations (ii) =⇒ (i): Any equilibrium manifold [θ] ⊂ ∆ (γ) as
G
(9b)-(9c) and the strict inequality constraints (9d)-(9e). Sum- in (ii) is a γ-feasible power injection setpoint with
ming all equations (9b)-(9c) yields the necessary solvability
condition (power balance constraint) (cid:80)i∈VI ui=−(cid:80)i∈VPi∗. P(cid:101)i =Pi∗−Diωsync =Pe,i(θ) ∀i∈VI, (36a)
To establish the necessary and sufficient optimality con- P(cid:101)i =Pi∗ =Pe,i(θ) ∀ i∈VL, (36b)
ditions for (9) in the strictly feasible case, without loss of
|θ −θ |<γ ∀i,j ∈E, (36c)
i j
generality, we drop the inequality constraints (9d)-(9e). With
λ∈Rn, the Lagrangian L:Tn×RnI ×Rn →R is given by Pe,i(θ)∈]0,Pi[ ∀i∈VI. (36d)
L(θ,u,λ)=(cid:88) 1α u2+(cid:88) λ (cid:0)u +P∗−P (θ)(cid:1) Hence, any θ ∈ [θ] is strictly feasible for the economic dis-
j∈VI 2 j j j∈VI j j j e,j patch (9) if we identify θ∗ with θ (modulo symmetry), γ with
+(cid:88)j∈VLλj(cid:0)Pj∗−Pe,j(θ)(cid:1) . mSianxce{i,uj∗i}∈iEsγsii(jAgCn)-,deafinnditeu,∗ithewictlhaim−Dfoilωloswyncs f(rmomodLuleomsmcaalin5g.3)..
The necessary KKT conditions [44] for optimality are: Inthestrictlyfeasiblecase,acomparisonofthestationarity
conditions(36a)-(36b)andtheoptimalityconditions(35)gives
∂∂θLi =0 : 0=(cid:88)j∈Vλj · ∂P∂e,θji(θ), ∀i∈V, (34a) Dtheiωdsyronco=p g−aiun∗is=areλ(cid:101)/dαefii,nwedheurpeωtosysnccaalnindgλ(cid:101), wareeocbotnasitnan(3t.3S).ince
∂L
=0 : α u =−λ , ∀i∈V , (34b) Remark 3 (Selection of droop coefficients): The equiva-
∂u i i i I
i lence revealed in Theorem 5.2 suggests the following guide-
∂L
∂λ =0 : −ui =Pi∗−Pe,i(θ), ∀i∈VI, (34c) lines to select the droop coefficients: large coefficients Di for
i desirable (e.g., economic or low emission) sources with small
∂L
∂λi =0 : 0=Pi∗−Pe,i(θ), ∀i∈VL. (34d) ccoosntnceocetefdfictioenthtseαpiro;paonrdtivoincaelvpeorwsae.rTshheasreinignsoibgjhetcsticvaen:aiflseoacbhe
SincetheACeconomicdispatch(9)isequivalenttotheconvex Pi∗ and 1/αi are selected proportional to the rating Pi, that
DC dispatch (see Theorem 5.1), the KKT conditions (34) are is, αiPi = αjPj and Pi∗/Pi = Pj∗/Pj, then the associated
also sufficient for optimality. In vector form, (34a) reads as droop coefficients (33) equal those in (17) for load sharing.(cid:3)
0 =λT∂P (θ)/∂θ,wheretheloadflowJacobianisgivenby Remark4(Beyondquadraticobjectivesandlineardroop
n e
symmetric Laplacian ∂P (θ)/∂θ = Bdiag({a } )BT slopes): As shown in Theorem 5.2, the steady states of a
e ij {i,j}∈E
withstrictly positive weights a =Im(Y )E E cos(θ −θ ) microgrid (2) with linear droop control (4) are related one-to-
ij ij i j i j
(due to strict feasibility of the security constraint (9d)). onetotheglobaloptimizersoftheeconomicdispatch(9)with
Itfollowsthatλ∈1n,thatis,λi =λ(cid:101)∈Rforalli∈V and quadratic objective (9a). If analogous proof methods are car-
forsomeλ(cid:101)∈R.Hence,condition(34b)readsastheeconomic riedoutforageneralobjectivefunctionf(u)=(cid:80)i∈VI Ci(ui)
dispatch criterion (identical marginal costs) ui = −λ(cid:101)/αi for with strictly convex and continuously differentiable functions
all i∈VI, and the conditions (34c)-(34d) reduce to Ci, the associated optimal droop controllers (4) need to
have nonlinear frequency-dependent droop slopes given by
λ(cid:101)/αi =Pi∗−Pe,i(θ), ∀i∈VI, (35a) Di = (Ci(cid:48))−1(θ˙i). Conversely, practically employed droop
0=Pi∗−Pe,i(θ), ∀i∈VL. (35b) curves with frequency deadbands and saturation [10, Chapter
9] can be related to non-smooth and barrier-type costs [15].
By summing all equations (35), we obtain the constant λ(cid:101) as
Fromsuchanoptimizationperspective,theprimarydynam-
λa(cid:101)n=dθ(cid:80)∗di∈eVterPmi∗i/n(cid:80)edif∈rVoImα(i−351.).TIthfeomlloinwismtihzaetrsu∗ariessui∗ign=-d−efiλ(cid:101)n/iαtei. ics (12) are a primal algorithm converging to a steady-state
satisfying the optimality conditions (34). Likewise, second-
By comparing the (strict) optimality conditions (35) with
orderorintegralcontroldynamicscanbeinterpretedasprimal-
the (strict) feasibility conditions (9b)-(9c), it follows that any dual algorithms, as shown for related systems [12]–[18]. (cid:3)
strictly feasible pair (θ,u) with sign-definite u is inverse
optimal for the coefficients α =−β/u with some β >0.
i i
Proof of Theorem 5.2: (i) =⇒ (ii): If the AC economic VI. SIMULATIONCASESTUDY
dispatch (9) is strictly feasible, then its minimizer (θ∗,u∗) is ThroughoutthepastsectionswedemonstratedthattheCAPI
global (Theorem 5.1), and the optimal inverter injections are control (7), (22) and DAPI control (7), (27) with properly
Popt = P (θ∗) = P∗ + u∗ with sign-definite u∗ (Lemma scaled coefficients can simultaneously address primary, sec-
i e,i i i
5.3). Since the power flow equations (9b)-(9c) and the strict ondary,andtertiary-levelobjectivesinaplug-and-playfashion.