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Comparison of Virtual Oscillator and Droop Control Preprint Brian Johnson and Miguel Rodriguez National Renewable Energy Laboratory

Mohit Sinha and Sairaj Dhople University of Minnesota

Presented at the IEEE COMPEL 2017 Stanford, California June 9—12, 2017

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Comparison of Virtual Oscillator and Droop ControlBrian Johnson, Miguel RodriguezPower Systems Engineering Center

National Renewable Energy LaboratoryGolden, CO 80401

Email: brian.johnson@nrel.gov, miguelrg@gmail.com

Mohit Sinha, Sairaj DhopleDepartment of Electrical & Computer Engineering

University of MinnesotaMinneapolis, MN 55455

Email: sinha052,sdhople@umn.edu

Abstract—Virtual oscillator control (VOC) and droop controlare distinct methods to ensure synchronization and power sharingof parallel inverters in islanded systems. VOC is a controlstrategy where the dynamics of a nonlinear oscillator are usedto derive control states to modulate the switch terminals ofan inverter. Since VOC is a time-domain controller that reactsto instantaneous measurements with no additional filters orcomputations, it provides a rapid response during transients andstabilizes volatile dynamics. In contrast, droop control regulatesthe inverter voltage in response to the measured average real andreactive power output. Given that real and reactive power arephasor quantities that are not well-defined in real time, droopcontrollers typically use multiplicative operations in conjunctionwith low-pass filters on the current and voltage measurementsto calculate such quantities. Since these filters must suppresslow frequency ac harmonics, they typically have low cutofffrequencies that ultimately impede droop controller bandwidth.Although VOC and droop control can be engineered to producesimilar steady-state characteristics, their dynamic performancecan differ markedly. This paper presents an analytical frameworkto characterize and compare the dynamic response of VOC anddroop control. The analysis is experimentally validated with three120 V inverters rated at 1 kW, demonstrating that for the samedesign specifications VOC is roughly 8 times faster and presentsalmost no overshoot after a transient.

I. INTRODUCTION

Ac microgrids based on electronics generally contain mul-tiple inverter interfaced energy sources connected across an acbus where the system of inverters must collectively maintainsystem voltage and frequency. To enhance modularity andresilience to failures, decentralized controllers are generallypreferred such that inverters can be added and removed asnecessary and load sharing without communication is guar-anteed. In this work, we examine the performance of twodecentralized controllers for inverter-based microgrids: droopcontrol and virtual oscillator control (VOC). Droop controlis a classic method that draws inspiration from synchronousmachine systems [1]–[4] and VOC is a recently developedapproach that leverages contemporary advances in nonlinearcontrol [5]. The main objective of this work is to compare thedynamic response of these two control strategies and uncoverperformance gains that one strategy may have over the other.Towards that end, we establish an analytical framework toassess the eigenvalues of both VOC and droop. After showing

Funding support from the the U.S. Department of Energy (DOE) SolarEnergy Technologies Office under Contract No. DE-EE0000-1583, and theNational Science Foundation under the CAREER award, ECCS-1453921, andgrant ECCS-1509277 is gratefully acknowledged.

that the linearized response of VOC indicates a more rapidand damped response in comparison to droop control, wefurther substantiate this result with an experimental setupconsisting of three parallel inverters serving load. In particular,measurements show that VOC can synchronize roughly 8 timesfaster than droop for the particular hardware setup and designconsidered here.

At its core, the operation principle behind droop controlis the enforcement of a linear relationship between measuredreal and reactive power with respect to frequency and voltage,respectively, at the inverter terminals. Since real and reactivepower are not well-defined in real time and are generallyassumed to be averaged quantities, we refer to droop controlas being a phasor-domain controller to emphasize the fact thatthe control variables are essentially averaged phasors. Froman implementation standpoint, this implies that computationsfrom sinusoidal measurements must undergo processing andfiltering to obtain the phasor quantities that form the basisof the droop control law. For instance, algebraic relationshipsare typically used to first obtain a preliminary set of realand reactive power values from the raw voltage and currentmeasurements. However, because these algebraically computedvalues are susceptible to line frequency harmonics and dis-tortion, they must be filtered before obtaining frequency andvoltage commands via the linear droop laws. It should be notedthat if the aforementioned filters were bypassed, any harmonicsfrom the sinusoidal measurements would propagate to thecontrolled inverter voltage. So although the utilization of filtersblocks undesirable harmonics from reaching the voltage andfrequency commands, it should be clear that these filters forma key component of the control feedback path and may act asa performance bottleneck and lead to a sluggish response. Toprovide some context, it is not uncommon to see droop controlfilter cutoff frequencies below 10 Hz. Taken together, thisindicates that droop controllers exhibit a fundamental trade-off between immunity to harmonics and control bandwidth.

VOC is a method that is based on digitally emulating anonlinear oscillator circuit such that measurements act as aninputs to the oscillator and its states are used to modulatethe inverter terminals directly. In contrast to droop control,VOC is a time-domain controller that acts on the real-timesinusoidal current delivered by the inverter such that it reactsnearly instantaneously to system disturbances without theneed for additional filters or averaging. While VOC mayappear to be a highly unconventional approach, recent workhas shown that its steady-state behavior exhibit a droop-like

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characteristic between frequency and voltage with respect toreal and reactive power. [6] In essence, VOC subsumes thefunctionality of droop in steady-state while providing superiorspeed during transients. Furthermore, fast synchronization forparallel-connected inverters by using oscillator based nonlinearcontrol was also reported in [7] where the authors also pointedout the limitations of the phasor-based methods for controlduring transients.

While the enhanced speed of VOC over droop has been pre-viously reported via simulation [8], here we take the first stepat analytically characterizing such differences and providing anexperimental validation. In Section II we outline foundations ofthe inverter model. Section III outlines the dynamical modelsof droop control and VOC as well as their correspondinglinearized representations. Simulation and experimental resultsare given in Section IV and concluding statements are inSection V.

II. PRELIMINARIES

We base our analysis on the inverter models in Fig. 1. Ineach case, an inverter is connected to bus voltage, vb, withRMS amplitude Vb, instantaneous angle θb, and frequencyωb = θb (i.e., vb =

√2Vb cos θb). Ultimately, each con-

troller produces a sinusoidal voltage command of the form,v =

√2V cos θi, where V and θi are the terminal RMS

voltage amplitude and angle, respectively. The modulationsignal is derived from the voltage command via m = v/vdcand pulse width modulation is applied to obtain the switchstates where v denotes the voltage across the switch terminals.From here forward, we utilize an averaged value model whereit is assumed that the switch terminal voltage perfectly tracksthe commanded value such that v = v. This can be justifiedunder the setting where m varies on a much slower timescalein comparison to the switch period, Tsw, [9] and the followingexpression holds

1

Tsw

∫ t

s=t−Tsw

v(s)ds = m(t)vdc = v(t). (1)

Subsequent analysis is carried out in the inverter referencedq frame which rotates with angle θi. To facilitate analy-sis, we will use the inverter-bus angle difference, denotedas δ = θi − θb, rather than the inverter and bus anglesthemselves. This choice is motivated by the fact that whilethe instantaneous angles grow continuously, their differencereaches an equilibrium if the system reaches steady-state. Sincethe model is in the inverter reference frame, we note that[vd, vq]

> = [V, 0]>, where vd and vq are the dq componentsof the inverter terminal voltage. The RL output filter has aninductance and resistance of Lf and Rf and the dq componentsof the current flowing through this branch are denoted as idand iq . The state equations for the current and inverter-busangle difference are [10]

d

dt

[idiq

]=

−Rf

Lfωi

−ωi −Rf

Lf

[idiq

]+

[V − vbd−vbq

], (2)

dδ

dt=ωi − ωb. (3)

PWM

m×÷

+ +−+

i

ip

V q

dcvv bv

bv

v

2/fL 2/fR

2/fR2/fL

fω

fω

powercalculation

drooplaws filters

fp

fq

iωfp

fqV

∫θ

θcosV2√

(a)

PWM

m×÷

+ +−+

i

dcvv

−

+

vκ iκ

L Ciiκ

virtual oscillator

CvLi

vεκ

bv

σ1−

Cv

v

C3αv

2/fL

2/fL

]sinΦ−,cosΦ

[

2/fR

2/fR

(b)Fig. 1: Typical implementations of droop controlled and VO-controlledinverters are given in (a) and (b), respectively. In the subsequent analysis,we adopt an averaged value model where we assume that the voltage acrossthe switch terminals, v, follows the control voltage v(t).

The state equations in (2)–(3) are of particular importancesince they will be used for both models in Fig. 1. Note thatalthough the analysis here can be easily extended to morecomplex filter structures, we restrict our focus to the RL filter.The bus voltages, vbd and vbq in (2), which are specified inthe inverter reference frame, can be obtained via the followingtransformation[

vbdvbq

]=

[cos(θi − θb) sin(θi − θb)− sin(θi − θb) cos(θi − θb)

] [vbDvbQ

], (4)

where vbD and vbD are the D and Q components of the busvoltage in its own frame rotating at angle θb. Then, withoutloss of generality, we can specify [vbD, vbQ]> = [Vb, 0]>.

III. MODELING OF CONTROLLED INVERTERS

A. Droop control

A single inverter with droop control is illustrated inFig. 1(a). The first stage of the controller is a power com-putation which uses the filter current and bus voltage, i andvb, to compute the instantaneous real and reactive power, pand q, respectively. To remove ac harmonics, these signals areprocessed by a first-order low-pass filter to obtain the filteredsignals pf and qf . Lastly, the droop laws yield the voltageamplitude, V , and frequency, ωi, that are used to constructthe switch terminal voltage waveform, v =

√2V cos θi, where

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θi is obtained by integrating ωi. The instantaneous real andreactive power computations and droop laws can be specifiedin the inverter frame dq coordinates as[

pq

]=

[vbd vbqvbq −vbd

] [idiq

], (5)

V =Vnom −mq(qf − q∗), (6)ωi =ωnom −mp(pf − p∗), (7)

where Vnom and ωnom are the nominal voltage and frequencywhile p∗ and q∗ are the associated setpoints. The voltage andfrequency droop slopes are mq and mp. The state equationsthat are unique to the droop controlled inverter are

d

dt

[pfqf

]=− ωf

[pfqf

]+

[pq

], (8)

where ωf is the cutoff frequency for the active and reactivepower filter. Considering the inverter states in Section II, theoverall droop system in Fig. 1(a) has the following states:pf , qf , id, iq, and δ.

The expressions in (2), (3), and (8) specify the non-lineardynamics of a single droop-controlled inverter connected to avoltage bus. To gain insights on the dynamic response of thesystem, we linearize these expressions to obtain the followingsmall-signal model:

d

dt

∆pf∆qf∆δ∆id∆iq

=

−ωf 0 0 Vbωf 00 −ωf 0 0 −Vbωf

−mp 0 0 0 0

0 −mq

Lf0 −Rf

Lfωi

0 0VbLf

−ωi −Rf

Lf

∆pf∆qf∆δ∆id∆iq

+[0 0 −1 0 0

]>∆ωb, (9)

where ωi is the inverter frequency at equilibrium, the equi-librium currents id and iq are set to zero, the equilibriumangle difference is assumed to be negligible, and the busfrequency deviation, ∆ωb, is interpreted as a model input.The negligible angle difference approximation is reasonableto study the response to a system transient since steady-stateangle differences just before a transient are typically small.1

In the context of this model, system disturbances, such asthe addition and removal of inverters, will be interpreted asa change in bus frequency. This framework can be extendedto accommodate bus voltage disturbances. Finally, note thatthe setpoints p∗ and q∗ have been set to 0. From (9), we canderive the characteristic equation, as given below, whose rootsgive us the system eigenvalues:

D(s) = L2f s

5 + (2ωfL2f + 2Rf)s

4

+ (L2f ω

2f + L2

f ω2i + 4LfωfRf +R2

f )s3

+ (2L2f ωfω

2i + 2Lfω

2f Rf + VbmqLfωfωi + 2ωfR

2f )s2

+ (L2f ω

2f ω

2i +mpLfV

2b ωfωi +mqLfVbω

2f ωi + ω2

f R2f )s

+mqmpV3b ω

2f + LfmpωiV

2b ω

2f . (10)

Here we note that the linearized formulation in (9) is similarto that used in [11]–[14], but with a reduced number of states

1Note that the linearized model can be straightforwardly linearized aroundnon-zero currents. Here we set them to zero as it was observed that their valuemade a trivial impact on the numerical results.

to facilitate the calculation of (10). Furthermore, our resultin (10) is obtained from first principles as opposed to the morecomplex methods used in [15].

B. Dynamic Model of Virtual Oscillator Control

The dynamics of the states within the virtual oscillator aregiven by

LdiLdt

= vC ,

CdvCdt

= −αv3C + σvC − iL − κii,(11)

where vC and iL are the virtual capacitor voltage and inductorcurrent, respectively. The virtual capacitance, inductance, andconductance are given by C,L, and σ, and the nonlinearvoltage dependent current source is parameterized by α. Thecontroller is interfaced via voltage and current scaling values,κv and κi. As illustrated in Fig. 1(b), the output voltagecommand is given by

v = κv (vC cos Φ− εiL sin Φ) , (12)

where ε =√L/C and Φ is a fixed user-defined angle,

typically chosen as either 0 or π/2, that specifies the type ofdroop-like characteristic the VO-controlled inverter exhibits insteady-state.

Since VOC is a time-domain controller and its implementa-tion in Fig. 1 offers no clear notion of phasor quantities suchas amplitude, angle, frequency, and power, we must utilizean averaged model that is written in these terms in order toenable a comparison with the droop control. As delineated in[16], such a model can be obtained by averaging the trajectoryof the oscillator in Fig. 1(b) over one ac cycle such that weobtain the following amplitude and frequency dynamics

dV

dt=

σ

2C(V − β

2V 3)− κiκv

2CVq, (13)

dθidt

= ωi = ωnom −κiκv

2CV 2p, (14)

where β = 3α(κ2vσ

)−1and the following model is obtained

with Φ = π/2 such that the VOC system exhibits V -versus-qand ωi-versus-p droop-like relationships [8] in alignment withthe form of the conventional droop controller considered inSection III-A. Although we use Φ = π/2 to ensure a similarsteady-state relationship between the two controllers underconsideration, it should be pointed out that p and q in (13)–(14)are computed using unfiltered signals and are given by[

pq

]=

[V 00 −V

] [idiq

]. (15)

This follows from the fact that VOC is a time-domain con-troller and its averaged formulation, as we consider here,naturally incorporates a one-cycle moving average without anyadditional filters or computations. Note that (5) and (15) takeon slightly different form. This discrepancy is due to the factthat whereas droop control typically uses the measured realand reactive power at the bus terminals, the physical structureof the VOC feedback loop in Fig. 1(b) naturally implies apower computation at the switch terminals.

After substituting (14) into (3), the expressions in (2), (13)and the inverter-bus angle difference can be linearized around

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Table I: Parameters of the inverter hardware, droop controller in Section III-A,and oscillator-based controller in Section III-B.

Lf = 1 mH Rf = 0.7 Ω Lo = 0.2 mHRo = 0.12 Ω Cf = 24µF Vdc = 220 V

Snom = 750 VA Vnom = 120 V ωnom = 2π60 rad/s

mp = 2π0.5rad/sSnom

mq = 0.05Vnom

Snomωf = 2π5 rad/s

L = 39.9µH C = 176.3 mF σ = 11.4 Ω−1

α = 7.58 A/V3 κv = 120 V/V κi = 0.16 A/A

the steady-state values, V , p, and q. This yields the followingrepresentation

d

dt

∆δ∆id∆iq∆V

=

0 − κiκv2CV

0 0

0 −Rf

Lfωnom

1

LfVbLf

−ωnom −Rf

Lf0

0 0κiκv2C

χ

∆δ∆id∆iq∆V

(16)

+[−1 0 0 0

]>∆ωb,

where χ = σ(1 − 3βV2/2)/(2C) and again we assume the

current and angle differences at equilibrium are negligible.Note that the VOC system has one less state in comparisonto droop since it does not contain low-pass filters and thoseassociated states but has one additional voltage dynamic stategiven by (13). The characteristic equation for the VOC systemtakes the following form:

D(s) =(8C2L2

f V)s4

+(

16RfC2LfV + 6βσCL2

f V3 − 4σCL2

f V)s3

+(

8C2L2f V ωnom + 8C2V R2

f

+12βσCLfV3Rf − 8σCLfV Rf

)s2 (17)

+(

6CβσV 3L2f ω

2nom − 4CσV L2

f ω2nom

+4CκiκvLfV ωnom + 4CVbκiκvLfωnom

+6CβσV3R2

f − 4CσV R2f

)s

+ 3LfVbβσωnomV2κiκv + 2Vbκ

2i κ

2v

− 2LfVbσωnomκiκv.

IV. SIMULATION AND EXPERIMENTAL RESULTS

In this section, we first present numerical results basedon the analysis in Sections III-A and III-B. Thereafter, wecompare the measured response of a hardware system under atransient for both droop control and VOC. In all the ensuinganalysis, the numerical paramaters are based on the experi-mental hardware for a hardware prototype rated at 1 kVA and120 V RMS.

A. Eigenvalue analysis

Here we assess the roots of (10) and (17) as we sweepcertain parameters from the nominal values specified in TableI. Figures 2(a) and 2(b) show the roots of (10) and (17) as afunction of the inverter output inductance and resistance, Lf

and Rf . For the linearized operating conditions, we assume

-100 -80 -60 -40 -20 0-100

-50

0

50

100

voc

droop

fRfR

(a)

-100 -80 -60 -40 -20 0-50

0

50

voc

droop

fL fL

(b)

0 0.2 0.4 0.6 0.8 10

0.5

1

1.5

droop

voc

, [s]t

(c)Fig. 2: (a) Root locus of (10) and (17) for Rf = 0.1Ω → 1Ω. (b) Root locusof (10) and (17) for Lf = 0.1mH → 1mH. (c) Unit step response of thetransfer function 1/D(s).

Vb = 120 V and ωi = 2π60 rad/s. Furthermore, mq and mp

are chosen to achieve a maximum 5% frequency and voltagevariation over the full-rated operating regime for both for droopand VOC (see [8], [16] for how mp and mq can be translatedto VOC parameters for equivalent steady-state performance).In Figs. 2(a)–(b), it is clear that the smallest negative real partsof the VOC eigenvalues are consistently farther away from theimaginary axis in comparison to those of the linearized droopsystem. This implies a faster and more damped response forVOC. The normalized step response of 1/D(s) response inFig. 2(c) corroborates this intuition.

B. Hardware validation

A setup consisting of three 1 kVA parallel inverters operatedat 120Vrms in an islanded configuration was used to validate

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o1i

o2i

o3i

control

on/off ov

b1v

1i

2i

3i

b2v

b3v

control

control

fL oL

Cdcv

dcv

dcv

oR

fL oL

C

oR

fL

fR

fR

fR oL

C

oR

Fig. 3: Diagram of experimental setup consisting of three parallel connectedinverters serving a load. To compare the performance of the oscillator- anddroop-based controllers, we subject the system to identical transient conditionsunder each control strategy and assess their dynamic performance. For bothtests, the system is first energized with two inverters and once they reachsteady state the third inverter is turned on abruptly.

the analytical findings. Each unit exhibited the physical char-acteristics in Table I, was controlled with a TI F28335 DSP,and utilized an LCL output filter where Cf denotes the filtercapacitance, and the grid-side branch inductance and resistanceare denoted as Lo and ro, respectively (see Fig. 3 to clarify thenotation of the filter parameters). Note that it can be shown viafrequency-domain analysis that the component values underconsideration allow us to approximate the LCL filter by asimple LR filter with relatively high accuracy up to 500 Hz.Accordingly, the analysis in prior sections retains sufficient ac-curacy for the actual experimental setup and for the timescalesof interest. To reiterate, the controllers were both designed forequivalent performance such that ωnom = 2π60 rad/s, mp andmq were identical for both droop control and VOC. Figures4(a)–(b) show the oscilloscope captures of the transient afterenergizing the power semiconductors of the third unit in thesystem for VO and droop controlled inverters, respectively. Toelaborate further, the controller for inverter #3 was configuredto synchronize its internal states to the measured terminalvoltage and wait for a zero crossing before engaging thegate signals. This procedure was utilized for both controllersand was motivated by the need to avoid destructive transientcurrents during turn-on.

Visually inspecting the results in Fig. 4, it is readily apparentthat VOC is substantially faster than VOC. To further elucidatethe performance gains of VOC and obtain a concrete metricbetween both systems, we employ the projector matrix [5]to quantify the synchronization error, which is denoted anddefined as Π := IN − 1N1>N/N , where N is the numberof operational inverters, IN is an N ×N identity matrix and1N is a N × 1 column-vector of ones. As shown in [17], theEuclidean norm ‖Πx‖2 is non-negative and is proportional tothe proportional differences between each and every elementin the N × 1 vector x. Hence, because ‖Πx‖2 approacheszero only in the case when each element is synchronized,

ov

o2, io1i o3i

(a)ov

o2, io1i o3i

(b)Fig. 4: Addition of inverter 3 when inverters 1 and 2 are supplying a 1 kWload. In each plot, the timescale is 100 ms/div, 2A/div (for io1, io2, io3 ), and100 V/div for vg . (a) and (b) show the response of the inverter system withVOC and droop control, respectively. Here, the superior speed and transientresponse of VOC is evident.

we will use it to quantify the synchronization error betweenmultiple inverters on a system. Referring to Fig. 5, we plotthe synchronization error in the vector of measured outputcurrents io = [io1, io2, io3]>. As illustrated, the VOC anddroop-controlled systems fall below and remain under the errorthreshold in 45 ms and 346 ms, respectively, where the thresh-old is defined as the measured maximum steady-state erroramong both controllers and is equal to approximately 1.45 Afor our system. Hence, VOC synchronizes approximately 7.7(or roughly 8) times faster than droop for this particular design.

0 0.2 0.4 0.6 0.8

0

5

10

0 0.2 0.4 0.6 0.8

0

5

10droopt346ms≈

45ms≈voct

,[A

]2‖

oiΠ‖

,[A

]2‖

oiΠ‖

Fig. 5: Measured current synchronization error where the quantity ‖Πio‖2is proportional to the differences between the currents in the vector io =[io1, io2, io3]>.

5This report is available at no cost from the National Renewable Energy Laboratory at www.nrel.gov/publications.

V. CONCLUSIONS

Droop and VOC are two methods to ensure power sharingand synchronization of parallel inverters in islanded micro-grids. In this work, their dynamic performance was studiedusing a unified analytical framework. A dynamic model ofeach inverter system was developed and linearized to obtaintheir eigenvalues. Analytical and experimental results showthat the droop method suffers from inherent limitations in itsdynamic performance due to the need to measure and filter realand reactive power. In contrast, VOC acts on instantaneousmeasurements and inherently provides a faster and better-damped response. Simulation and experimental results cor-roborate these findings and hardware measurements indicatea speed enhancement factor of approximately 7.7 with VOCfor the particular design considered here.

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6This report is available at no cost from the National Renewable Energy Laboratory at www.nrel.gov/publications.