Stability, Power Sharing, & Distributed SecondaryControl in Droop-Controlled Microgrids

John W. Simpson-Porco, Florian Dorfler, Francesco BulloCenter for Control, Dynamical Systems and Computation

University of California at Santa BarbaraSanta Barbara, California 93106

Email: johnwsimpsonporco@engineering.ucsb.edu

Qobad Shafiee, Josep M. GuerreroInstitute of Energy Technology

Aalborg UniversityAalborg East DK-9220, Denmark

Email: qsh@et.aau.dk

Abstract—Motivated by the recent and growing interest inmicrogrids, we study the operation of droop-controlled DC/ACinverters in an islanded microgrid. We present a necessary andsufficient condition for the existence of a synchronized steadystate that is unique and locally exponentially stable. We discussa selection of controller gains which leads to a sharing of poweramong the generators, show that this proportional selectionenforces actuation constraints for the inverters. Moreover, wepropose a distributed integral controller based on averagingalgorithms which dynamically regulates the system frequency inthe presence of a time-varying load. Remarkably, this distributed-averaging integral controller has the additional property that itmaintains the power sharing properties of the primary droopcontroller. Finally, we present experimental results validating ourcontroller design, along with simulations of extended scenarios.Our results hold without assumptions on uniform line admit-tances or voltage magnitudes.

I. INTRODUCTION

Microgrids are low-voltage electrical distribution networks,heterogeneously composed of distributed generation, storage,load, and managed autonomously from the larger primarynetwork. Microgrids are able to connect to the wide areaelectric power system through a Point of Common Coupling(PCC), but are also able to “island” themselves and operate in-dependently [1]. Energy generation within a microgrid can behighly heterogeneous, including photovoltaic, wind, geother-mal, micro-turbines, etc. Many of these sources generate eithervariable frequency AC power or DC power, and are interfacedwith a synchronous AC grid via power electronic DC/ACinverters. In islanded operation, it is through these invertersthat actions must be taken to ensure synchronization, security,power balance and load sharing in the network [2].

Inspired by control architectures from transmission levelpower systems, control in microgrids is generally approachedin a hierarchical manner [1]. The first and most basic level isprimary control, which is concerned with the stability of —and load sharing within — the electrical network. Althoughcentralized architectures have been used for primary control,in order to enhance redundancy and enable “plug-and-play”functionality the standard is to employ decentralized propor-tional control loops locally at each inverter [2]–[7]. Whilegenerally successful, these decentralized “droop” controllerstypically force both the local voltages and the steady-statenetwork frequency to deviate from their nominal values.

Fig. 1. Schematic illustration of a microgrid, with four inverters (nodesVI ) supplying six loads (nodes VL) through an acyclic interconnection. Thedotted lines between inverters represent communication links, which will beused exclusively in Section VI for secondary control.

This leads naturally to the next level in the control hierarchy,termed secondary control. Generally speaking, the goal ofsecondary control is to remove the aforementioned devia-tions in both global frequency and local voltage. Centralizedtechniques for secondary control have been well studied inclassical wide-area transmission and distribution networks, see[8]. These centralized strategies have also been applied inthe context of microgrids, and the term “secondary” has beenbroadened to include additional control goals such as harmoniccompensation and voltage unbalance, see [1], [2], [9], [10] forvarious works.

In this work we present recent theoretical and experimentalresults on primary and secondary control in microgrids [11].After a review of the droop control method and secondarycontrol (Section II), we provide necessary and sufficient condi-tions for the existence of a stable operating point for a networkof droop-controlled inverters and loads (Section III–IV), andrigorously establish control parameter selections and boundson loads which result in the inverters meeting given actuationconstraints (Section V). In Section VI we propose a noveldistributed secondary controller, based on Laplacian averagingalgorithms, which quickly regulates the network frequency to anominal value. Remarkably, this controller accomplishes thistask while maintaining the power sharing properties of theprimary droop controller. In Sections VII and VIII we provide

experimental and simulation results validating our controllerdesigns, before offering conclusions and future directions inSection IX. Detailed proofs of all results can be found in [11].

II. REVIEW OF DISTRIBUTED CONTROL IN MICROGRIDS

A. Problem Setup and Review of Circuit TheoryIn this work we model an AC microgrid by a connected,

undirected, and complex-weighted graph with nodes (or buses)V = 1, . . . , n, edges (or branches) E ⊂ V × V , andsymmetric edge weights (or admittances) −Yij = −Yji ∈ Cfor every branch i, j ∈ E . We partition the set of busesas V = VL ∪ VI , corresponding to the loads and inverters.To each bus i ∈ V , we associate a complex power injectionSi = Pi + jQi and the phasor voltage variable Eie

jθi

corresponding to the magnitude and the phase shift of asolution to the AC power flow equations. For inductive lines,the power flow equations are

Pi =∑n

j=1Im(Yij)EiEj sin(θi − θj) , i ∈ V , (1a)

Qi = −∑n

j=1Im(Yij)EiEj cos(θi − θj) , i ∈ V. (1b)

If a number ` and an arbitrary direction is assigned toeach branch i, j ∈ E , the incidence matrix A is definedcomponent-wise as Ak` = 1 if bus k is the sink bus of branch` and as Ak` = −1 if bus k is the source bus of branch `,with all other elements being zero. In the inductive networkcase, for every set of balanced power injections Pi there existsa branch vector ξ satisfying Kirchoff’s Current Law (KCL)P = Aξ [12]. For tree networks such as distribution networks,ξ is unique and is given by ξ = (ATA)−1ATP [13]. Thevector P is interpreted as bus injections, with ξ being theassociated branch flows. We denote by ξij the component ofthe branch-vector ξ corresponding to the branch i, j ∈ E .

As is standard in the microgrid literature, we model aninverter as a controllable voltage source behind a reactance.This model is widely adopted among experimentalists in themicrogrid field. Further modeling explanation can be found in[14]–[16] and the references therein.

B. Review of Primary Droop ControlThe conventional droop controller is the foundational tech-

nique for primary control (synchronization and power bal-ancing) in islanded microgrids, and is a heuristic based onthe classic active/reactive decoupling assumption for smallpower angles and non-mixed line conditions, see [1]–[3], [5],[6], [9], [17]–[21]. For the case of inductive lines, the droopmethod specifies both the inverter frequencies ωi and voltagemagnitudes Ei by

ωi = ω∗ −mi(Pi − P ∗i ) , i ∈ VI , (2a)

Ei = E∗i − ni(Qi −Q∗i ) , i ∈ VI , (2b)

where ω∗ is a nominal network frequency, E∗i (resp. P ∗i , Q∗i )is the nominal voltage set point (resp. nominal active/reactivepower injection) for the ith inverter, and Pi (resp. Qi) isthe measured active (resp. reactive) power injection. The con-troller gains mi, ni > 0 are referred to as droop coefficients.

Fig. 2. Schematic of inverters operating in parallel.

In islanded operation, one typically sets P ∗i = Q∗i = 0. From(2a), it is clear that if an inverter injects an amount of powerPi which differs from its pre-determined nominal injectionP ∗i , its frequency will deviate from ω∗. While small signalstability analysis of (2a) is standard, in this work we presentlarge-signal results specifying when a steady-state is feasiblefor the network.

C. Review of Classical Secondary Control

The removal of the steady state frequency deviation gen-erated by the droop controller is accomplished by so-called“secondary” integral controllers. If the primary controllerstabilizes the network, then the frequency of each inverter hasconverged to a steady state value ωss, and a slower additionalcontrol loop can then be used locally at each inverter [9].Each local secondary controller slowly modifies the controllergain P ∗i until the frequency deviation is zero. This procedureimplicitly assumes that the measured local frequency is agood approximation of the steady state network frequency,and relies on a separation of time-scales between the fast,synchronization-enforcing primary droop controller and theslower, secondary integral controller [7], [9], [10]. For smalldroop coefficients mi, this approach can be particularly slow,leading to an inability of the method to dynamically regulatethe network frequency in the presence of a time-varyingload. Moreover, these decentralized (i.e., local) secondary con-trollers may destroy the power sharing properties establishedby the primary droop controller.

Alternatively, a centralized microgrid controller may beused to perform secondary actions. This has the obviousdisadvantage of introducing a single point of failure in thesystem, and may induce communications over prohibitivelylarge distance.

III. STABILITY RESULTS FOR PRIMARY CONTROLLER

While the inverters are controlled according to the P − ωdroop method (2a) with Pi given by (1a), the constant powerloads P ∗i at load buses i ∈ VL must satisfy the power balancerelations

0 = P ∗i −∑n

j=1|Yij |EiEj sin(θi − θj) , i ∈ VL. (3)

Together, equations (2a) and (3) constitute the model for thenetwork. A natural question now arises: under what conditions

on the loads, nominal power injections, network topology,admittances, and droop coefficients is the network stable? Thefollowing result provides the definitive answer for general treenetworks. This class of networks includes the standard parallelmicrogrid topology (Figure 2), and most distribution networksafter islanding events.

For simplicity in stating our results, we define the vectorof loads and nominal power injections P ∗ := (P ∗1 , . . . , P

∗n),

the diagonal matrix of inverse droop coefficients M† :=diag(0, . . . , 0,m−11 , . . . ,m−1|VI |), and we let 1n be the n-dimensional vector where each entry is equal to one.

Theorem 1. (Existence and Stability of Sync’d Solution).Consider the frequency-droop controlled system (2a) withloads (3). Define the steady state network frequency ωss by

ωss := ω∗ +

∑ni=1 P

∗i∑

i∈VI m−1i

,

and let ξ be the unique vector of branch active power flows,satisfying KCL, namely P ∗ − (ωss − ω∗)M†1n = Aξ. Thefollowing two statements are equivalent:

(i) Synchronization: There is a number 0 ≤ γ < π/2 suchthat the closed-loop system (2a)–(3) possess a locallyexponentially stable and unique synchronized solutionθ∗(t) with |θ∗i − θ∗j | ≤ γ for all branches i, j ∈ E;

(ii) Flow Feasibility: The active power flow is feasible, i.e.,

Γ := maxi,j∈E

∣∣∣∣ ξijEiEj |Yij |

∣∣∣∣ < 1 (4)

The parameter Γ and the angular spread γ are then relateduniquely via Γ = sin(γ), the network synchronizes to steadystate frequency ωss, and the power angles satisfy sin(θ∗i −θ∗j ) = ξij/EiEj |Yij | for each branch i, j ∈ E .

Proof: See [11] for all proofs.

As outlined in [11], physically the parametric condition (4)is the very mild stipulation that the active power flow alongeach branch be feasible, i.e., less than the physical maximumξij,max := EiEj |Yij |. The quantity Γ can therefore be usedfor monitoring as an indicator of network stress. If Γ ' 1, thenetwork is close to collapsing, while if Γ ' 0, safe operationis assured. In practice, thermal constraints may place an upperbound of Γ ' 0.25 or roughly max(ij)∈E |θi − θj | ' 15.

IV. ROBUSTNESS TO VOLTAGE DYNAMICS

The analysis so far has been based on the assumption thatthe product EiEj |Yij | is a constant and known parameterfor all branches i, j ∈ E . In a realistic power system,both effective line susceptances and voltage magnitudes areonly approximately known, and are dynamically adjusted byadditional controllers, such as the Q−E droop controller (2b).The following result states that as long as these additionalcontrollers can regulate the effective susceptances and nodalvoltages above prespecified lower bounds, the stability resultsof Theorem 1 go through with little modification.

Corollary 2. (Robustified Stability Condition). Considerthe frequency-droop controlled system (2a) with loads (3).

Assume that the bus voltage magnitudes and branch suscep-tance magnitudes are above a lower bound. That is, theysatisfy Ei > Ei > 0 for all buses i ∈ 1, . . . , n, and|Yij | ≥ |Y ij | > 0 for all i, j ∈ E . The following twostatements are equivalent:

(i) Robust Synchronization: For all possible voltage mag-nitudes Ei > Ei and line susceptances |Yij | ≥ |Y ij |,there exists a number 0 ≤ γ < π/2 such that theclosed-loop system (2a) with loads (3) possess a locallyexponentially stable and unique synchronized solutionθ∗(t) with |θ∗i − θ∗j | ≤ γ for all branches i, j ∈ E;

(ii) Worst Case Flow Feasibility: The active power flow isfeasible for the worst case voltage magnitudes and linesusceptances, that is,

maxi,j∈E

∣∣∣∣ ξijEiEj |Y ij |

∣∣∣∣ < 1 . (5)

V. POWER SHARING AND POWER INJECTION LIMITS

While Theorem 1 addresses the concern of stability forthe frequency-droop controlled system (2a) with loads (3), itdoes not take into account that the inverter power injectionsmust satisfy actuation constraints. That is, we must have0 ≤ Pi ≤ P i for some P i > 0 called the rating of inverter i.The following definition gives the proper criteria for selection.

Definition 1. (Proportional Droop Coefficients). The droopcoefficients are selected proportionally if miP

∗i = mjP

∗j and

P ∗i /P i = P ∗j /P j for all i, j ∈ VI .

While the first condition in Definition 1 is standard inthe microgrid literature, the second is a generalization of thechoice P ∗i = P i. The next result shows that this proportionalselection leads to desirable steady state power injections.

Theorem 3. (Power Flow Constraints and Power Shar-ing). Consider a synchronized solution of the frequency-droop controlled system (2a) with loads (3), and let thedroop coefficients be selected proportionally. Define the totalload PL :=

∑i∈VL P

∗i . The following two statements are

equivalent:

(i) Injection Constraints: 0 ≤ Pi ≤ P i, ∀i ∈ VI ;(ii) Load Constraint: −

∑j∈VI P j ≤ PL ≤ 0.

Moreover, the inverters share the total load PL proportionallyaccording to their power ratings, that is, Pi/P i = Pj/P j foreach i, j ∈ VI .

In particular, note that the implication (ii)⇒(i) of Theorem3 shows that if the total load is feasible for the inverters toservice, then Theorem 3 guarantees that every inverter satisfiesits actuation constraint. Also, for any load PL satisfyingTheorem 3 (ii), we can combine the expression for ωss fromTheorem 1 with the bounds in Theorem 3 (ii) to calculate that0 ≤ ωss − ω∗ ≤ ∆ωmax := miP

∗i . W We therefore obtain

the classic selection of droop coefficients mi = P ∗i /∆ωmax,where ∆ωmax is selected to be, say, (2π · 0.1)Hz [21].

Fig. 3. Schematic of DAPI control architecture.

VI. A NOVEL DISTRIBUTED SECONDARY CONTROLLERUSING AVERAGING

As is evident from the expression for the steady statefrequency ωss in Theorem 1, the frequency-droop methodalmost always leads to a deviation of the steady state operatingfrequency from the nominal value ω∗.

In what follows, we pursue a scheme for frequency restora-tion which does not implicitly rely on a separation of time-scales as in [7], [9], [10]. Assuming the existence of a sparsecommunication network among the inverters, we expand onthe conventional frequency-droop design (2a) and propose thedistributed-averaging proportional-integral (DAPI) controller

ωi = ω∗ −mi(Pi − P ∗i + pi) , (6a)

kidpidt

= ωi − ω∗ +∑j∈VI

Lij (mipi −mjpj) , (6b)

where pi is an auxiliary power variable and ki > 0 is a gain,for each i ∈ VI . The matrix L is the Laplacian (or Kirchoff)matrix corresponding to a weighted, undirected and connectedcommunication graph between the inverters, see Figure 1.The control architecture is depicted in Figure 3. The DAPIcontroller has the following three key properties:

(i) the controller is able to quickly regulate the networkfrequency under large and rapid variations in load,

(ii) the controller accomplishes this regulation while pre-serving the power sharing properties of the primarydroop controller (2a) (see Section V), and

(iii) the communication network between the inverters needonly be connected – we do not require all-to-all or all-to-one communication among the inverters (Figures 1and 3).

While we omit a formal statement due to space, one can showthat the secondary controller succeeds in restoring the networkfrequency and sharing power among the inverters if and onlyif the primary droop control stability condition (4). That is,the secondary control works if the primary control works, andvice versa [11].

A. Communication Complexity vs. Redundancy

The key feature of our control architecture is that it isdistributed. That is, the inverters do not need to communicateinformation to a central microgrid controller. All that is

PC-Simulink

RTW & dSPACE

Control Desk

Inverter 1

DC

Power

Supply

650 V

Inverter 2

DC

Power

Supply

650 V

io1v1

io2v2

iL1

iL2

LCL Filter

LCL Filter

NLLNLC

( )LR t

Nonlinear Load

Fig. 4. Schematic illustration of experimental setup.

required is that every inverter is able to at least indirectlycommunicate with every other inverter, see Figure 1. Thisdistributed architecture for the cyber-physical layer of thecontroller allows for flexibility in the design. Specifically, oneis able to trade off communication complexity against commu-nication redundancy without impacting the functionality of thesecondary control. While for mathematical simplicity, we havepresented our theoretical results for the case of continuoustime communication, all results extend to discrete time andasynchronous communication, see [22]. Moreover, it has beenshown that secondary control strategies are generally robustagainst packet losses and delays, see [23]

VII. EXPERIMENTAL RESULTS

Experiments were performed at Aalborg University in orderto evaluate the performance of the DAPI controller (6a)–(6b).A schematic of the experimental setup is shown in Figure 4,in which two inverters operating in parallel supply power toa nonlinear load.

Figure 5 shows the experimental setup, which consisted oftwo Danfoss R© 2.2 kW inverters operating at 10kHz with LCLoutput filters, a dSPACE R© 1103 control board, and LEM R©

voltage and current sensors. A diode rectifier was used as anonlinear load, loaded by a capacitor, and 200 Ω resistor. Theelectrical setup and control system parameters are detailed inTable I. Control parameters for both inverters were identical,and the inverter voltages were controlled according to the E−Q droop method (2b) with Q∗i = 0 for i ∈ 1, 2.

Experimental results are shown in Figure 6. Figure 6 (a)shows the microgrid frequency being regulated to its nominalvalue by the DAPI controller. During the first two secondsof operation under only the droop controller (2a), a steadystate frequency deviation exists. When the full DAPI controlleris implemented at t=2s, the system frequency is successfullyregulated. In the latter half of the experiment, the load wasquickly varied twice: at t=9s (from 200Ω to 400Ω) andat t=14s (from 400Ω to 200Ω) respectively. As seen, thecontroller is able to quickly regulate the network frequencydespite rapid load variation. For clarity, the secondary gains

Fig. 5. Labeled diagram of experimental setup.

0 2 4 6 8 10 12 14 16 18 2049.4

49.6

49.8

50

50.2

50.4

Inverter Frequencies

Time (s)

Frequen

cy(H

z)

0 2 4 6 8 10 12 14 16 18 20300

400

500

600

700

Inverter Active Power Injections

Time (s)

ActivePower

(W)

(b)

(a)

DAPI Controller

DroopOnly DAPI Controller + Load Switching

Fig. 6. Performance of DAPI Controller (6a)-(6b) for (a) frequencyrestoration, and (b) active power sharing.

ki have been tuned so that the frequency restoration visible.The transient deviations in frequency caused by the load stepscan be further suppressed, and their duration decreased, bydecreasing ki. Figure 6 (b) shows the corresponding activepower injections at the inverters. This illustrates that the

primary P − ω droop method is sufficient to share the activepower accurately between the two inverters, and that the DAPIcontroller preserves the power sharing properties establishedby the primary controller. The small increase in active power att=2s is due to the fact that the load is not a true constant powerload, and indeed contains a frequency dependent component.

VIII. SIMULATION EXTENSIONS

To demonstrate the robustness of our approach in a largelossy network, in this section we present a simulation of ouralgorithm applied to the IEEE 37-bus distribution network(Figure 7). After an islanding event, the distribution networkis disconnected from the larger transmission grid, and thedistributed generators must taken action to ensure stabilityin the network while regulating the frequency and sharingboth active and reactive loads. The cyber layer describingthe communication capabilities of the distributed generatorsis shown in dotted blue.

Fig. 7. Islanded IEEE37 bus distribution network. Red notes are strictlyloads, while blue nodes represent distributed generation and load. The dottedblue lines represent communication links.

The DAPI controller (6a)–(6b) is used to regulate the P −ω dynamics in the network, while the Q − E dynamics aregoverned by the quadratic droop controller [24]

d

dtEi = −CiEi(Ei − E∗i )−Qi

along with an additional secondary control loop to ensurereactive power sharing. Figure 7 shows the performance ofthe algorithm, when one of the loads suddenly doubles att = 5s. The controllers are seen to quickly restore thenetwork frequency while maintaining exact sharing of activeand reactive power demands in proportion to the respectivecapabilities of the generation.

2 4 6 8 10−20

−15

−10

−5

0

5

10

Active Power Injections

Pi(p

.u.)

Time (s)

Loads

DGs

2 4 6 8 10−10

−5

0

5

Reactive Power Injections

Qi(p

.u.)

Time (s)

2 4 6 8 1059

59.2

59.4

59.6

59.8

60

Inverter Frequencies

fi(H

z)

Time (s)2 4 6 8 10

0.975

0.98

0.985

0.99

0.995

1

1.005

1.01

1.015

Voltage Magni tudes

Ei(p

.u.)

Time (s)

Fig. 8. Simulation results for IEEE37 bus distribution network.

IX. CONCLUSIONS

In this work we have surveyed our recent theoretical resultson synchronization, power sharing, and secondary control inmicrogrids. In addition, we have presented experimental andsimulation results validating our calculations. While through-out we have assumed inductive branches, in the case heavilymixed resistive/inductive lines, the primary control law (2a)–(2b) is inappropriate. A provably functional control strategyfor general interconnections and line conditions is an openand exciting problem. Further work will examine provablyrobust controllers for mixed line networks, and how distributedcontrol can be used to achieve additional goals such asharmonic compensation and fault tolerance.

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TABLE IPARAMETER VALUES FOR EXPERIMENTS.

Parameter Symbol Value

Electrical Setup

Nom. Frequency ω∗/2π 50 HzDC Voltage Vdc 650 VNom. Voltages E∗

i 311 VFilter Capacitance C 25µFFilter Inductance Lf 1.8 mHOutput Impedance L0 1.8 mHResistive Load RL(t) RL ∈ 200, 400Ω

Inner Voltage/Current Control Loops

Current Propor. Gain kpi 0.35 V/ACurrent Integral Gain kii 200 Vs/AVoltage Propor. Gain kpv 0.35 A/VVoltage Integral Gain kiv 400 As/V

DAPI Control

Inv. Ratings (P ) Q∗i 0 kvar

Inv. Ratings (Q) P ∗i = P i 2.2 kW

P − ω Droop Coeff. mi 8× 10−4 W · s/rad

Q− E Droop Coef. ni 0.16 Vvar

Sec. Droop Coeff. ki 1 sComm. Graph Gc Two nodes, one edgeComm. Weight ` 1 Ws

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