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This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/TEC.2018.2881710, IEEETransactions on Energy Conversion

1

Reduced-order Aggregate Model forParallel-connected Single-phase Inverters

Victor Purba, Student Member, IEEE, Brian B. Johnson, Member, IEEE, Miguel Rodriguez,Saber Jafarpour, Member, IEEE, Francesco Bullo, Fellow, IEEE, and Sairaj V. Dhople, Member, IEEE

Abstract—This paper outlines a reduced-order aggregate dy-namical model for parallel-connected single-phase grid-connectedinverters. For each inverter, we place no restrictions on theconverter topology and merely assume that the ac-side switch-averaged voltage can be controlled via pulse width modulation.The ac output of each inverter interfaces through an LCLfilter to the grid. The closed-loop system contains a phaselocked loop for grid synchronization, and real- and reactive-power control are realized with inner and outer PI current-and power-control loops. We derive a necessary and sufficientset of parametric relationships to ensure that a reduced-orderaggregated state-space model for an arbitrary number of suchparalleled inverters has the same model order and structure asany single inverter. We also present reduced-order models forthe settings where the real- and reactive-power setpoints aredifferent and where the inverters have different power ratings.We anticipate the proposed model being useful in analyzingthe dynamics of large collections of parallel-connected inverterswith minimal computational complexity. The aggregate model isvalidated against measurements obtained from a multi-inverterexperimental setup consisting of three 750VA paralleled grid-connected inverters, hence establishing robustness of the analyt-ical result to parametric variations seen in practice.

Index Terms—Model reduction, phase-locked loop, single-phase inverter, voltage-source inverter.

I. INTRODUCTION

Rapid adoption of renewable sources of generation (e.g.,photovoltaic (PV) energy conversion systems) and flexibleloads (e.g., electric vehicles) has increased the number ofpower electronics inverters installed on the ac power grid.Scalable models that present limited computational burdenwill be critical to model and analyze the collective dynam-ics of large numbers of inverters in next-generation powernetworks [1]. To motivate the need for modeling strategiesthat can be applied to complex inverter systems, considerthe relatively small island of Oahu which already has over800, 000 microinverters [2]. This number is expected to growsignificantly as Hawaii aims to meet the goal of obtaining

V. Purba and S. V. Dhople are with the Department of Electrical andComputer Engineering at the University of Minnesota, Minneapolis, MN(email: purba002,[email protected]); B. B. Johnson is with the Depart-ment of Electrical and Computer Engineering at the University of Wash-ington, Seattle, WA (e-mail: [email protected]); M. Rodriguez was withthe Power Systems Engineering Center at the National Renewable En-ergy Laboratory, Golden, CO (now with Advanced Micro Devices, Inc.,Fort Collins, CO, e-mail: [email protected]); S. Jafarpour and F.Bullo are with the Department of Mechanical Engineering at the Universityof California, Santa Barbara, Santa Barbara, CA (email: saber.jafarpour,[email protected]). This work was supported in part by the U.S.Department of Energy under Contract No. DE-EE0000-1583 and the NationalScience Foundation under grant 1453921.

100% of its energy from renewable sources [3]. Although theHawaiian system is at the forefront of renewable adoption, itpresents a glimpse at anticipated worldwide trends [4].

The disparity in ratings between individual inverters (nolarger than a few MVA) and synchronous generators (severalhundred MVA) implies that if the same net load were servedwith power electronics instead of generators, there would bean orders-of-magnitude increase in the number of energy-conversion interfaces (from a few thousand generators to po-tentially millions of inverters across a large-scale synchronousgrid). Evaluating the stability and resilience of future powernetworks will therefore require accurate dynamical models forlarge collections of inverters that present limited computationalburden. However, development of such models is challengedby the complexity of inverter dynamics (for instance, theparticular model we examine in this paper is nonlinear, andcomposed of 16 states) and the sheer number of invertersthat will eventually be commonplace on the ac power grid.To address the challenge of model complexity in multi-inverter systems, we propose an aggregate reduced-order state-space model for an arbitrary number of single-phase grid-tiedinverters connected in parallel. While our analytical result ispresented for identical inverters, we experimentally validateour findings which immediately establishes robustness to para-metric variations that are likely to be seen in practice. We alsopresent extensions of the main result on model reduction tocover cases when the power setpoints of the inverters are alldifferent and the power ratings of the inverters are all different.

We examine the ac-timescale dynamics of a single-phasevoltage source inverter (VSI) with an output LCL filter. Toensure broad applicability across VSI topologies, we only as-sume that the switch-averaged voltage across the ac terminalsis controllable via pulse width modulation and we neglectswitch-level dynamics. The control architecture is composedof an inner current-control loop, an outer power-control loop,and a phase locked loop (PLL) for grid synchronization. Thisfilter and control architecture are prototypical and it ensuresbroad applicability of the results. The state-space model thatcaptures the dynamics of the inverter is composed of 16 states.The contributions of the paper are threefold:

1) For a parallel collection of N inverters, we derive anecessary and sufficient set of parametric relationshipsfor an aggregate reduced-order inverter model to havea state-space model with the same structure and modelorder (i.e., composed of the same 16 states referencedabove) as any single inverter in the collection. (Figure 1



2

illustrates the idea.)2) We derive parameters for an aggregate reduced-order

model (with the same order and stucture as any individualinverter) for the case where the real and reactive-powersetpoints for the inverters are all different.

3) We derive parameters for an aggregate reduced-ordermodel (with the same order and structure as any indi-vidual inverter) for the case where the power ratings ofthe inverters are all different.

In general, the identical state-space model structure impliesthat from a topological vantage point, the aggregated equiv-alent model also maps to an inverter with an LCL filter, aninner current-control loop, an outer power-control loop, anda PLL for grid synchronization, except with different filterparameters and control gains. With reference to contribution1) above, while some aspects of the aggregate model appearintuitive in hindsight (e.g., given the parallel arrangement, theoutput inductances in the aggregate model are 1/N times thosein any individual inverter), the parametric dependencies inpertinent control gains are not. What is more, the fact thatthe parametric relationships we derive in 1) are necessaryand sufficient implies that we exhaustively enumerate allalternative possibilities to obtain a structurally similar reduced-order model. Contributions in 2) and 3) follow as corollaries tothe main result in 1) and enable extending the result to obtainaggregate models under heterogeneous settings that are likelyto be noticed in practice. Finally, we believe that the generalstate-space modeling formulation and the proof strategy canbe extended to other control strategies to establish similaraggregate models.

Figure 1: (left) System of N parallel grid-connected single-phaseinverters. Controller for the `-th inverter regulates the injected gridcurrent, ig,`(t) to deliver (the commanded) real power p∗ and reactivepower q∗ into the grid terminals. We examine a prototypical 16-th order inverter dynamical model, which implies that the parallelsystem is described by a 16N -order dynamical model. (right) Ag-gregated equivalent has state-space model with the same dimensionand structure as any individual inverter, i.e., it is described by a 16-th order dynamical model, and Ar, Br and gr(·, ·) have the sameform as A, B and g(·, ·). We derive the model parameters for theaggregated-inverter filter and controllers such that with power inputsNp∗ and Nq∗, the instantaneous injected grid current is the sum ofall individual currents

∑N`=1 ig,`(t) (other states scale systematically

as well).

A majority of prior literature pertaining to model reductionfor energy conversion interfaces has understandably focusedon the dynamics of fossil-fuel-driven synchronous genera-tors [5]–[9]. More recently, there has been increased attentiondevoted to aggregate models for wind-turbines in utility-scalecollector systems [10], electric vehicles [11], and demandresponse systems [12]. However, these are focused at a macrolevel that disregards the dynamics at faster time scales whicharise from inverter filters and time-domain controllers. Withregard to inverter dynamics, most of the related literature haspredominantly focused on reduced-order models for individualgrid-connected [13], islanded [14]–[16], and resonant invert-ers [17]. (Tangentially related is literature on model reductionof dc-dc converters [18]–[20] and induction machines [21],[22].) Model-reduction methods focused on collections ofinverters have been limited to islanded settings [23], [24]and inverters with virtual-inertia emulation [25]. Both areapplication domains where inverters are controlled to emulatethe dynamics of synchronous generators, and therefore, thereis a natural translation of classical model-reduction methodsfor synchronous generators mentioned previously.

Given the landscape of related work discussed above, thiswork addresses a key gap in the literature pertaining to thedynamics of grid-connected multi-inverter systems. This papersignificantly builds upon and extends our preliminary workin [26] where we developed similar aggregated models forparallel-connected three-phase inverters. Here, we examine the(admittedly different) filter and controller dynamics for single-phase inverters which will conceivably be more dominantin number in future distribution networks. As a further andimportant contribution, we provide experimental validation ofour approach with a multi-inverter setup composed of three750 VA grid-tied single-phase inverters. We note that theparallel aggregation approach proposed here provides the basefor a broader set of aggregation techniques that can account forthe network connection of the inverters. For instance, in [27]three-phase inverters are transferred to an auxiliary bus withthe aid of auxiliary transformers, and subsequently aggregatedusing the parallel aggregation approach discussed here.

The remainder of this manuscript is organized as follows:In Section II, we establish mathematical notation and describethe grid-tied single-phase inverter model. The reduced-ordermodel for a collection of these inverters connected in parallel isderived in Section III. We validate the model-reduction methodby comparing numerical simulation results with results fromthe experimental prototype in Section IV. Finally, concludingremarks and directions for future work are in Section V.

II. PRELIMINARIES AND INVERTER DYNAMICAL MODEL

In this section, we first introduce mathematical notationused in the manuscript. Then we describe the single-phaseinverter model, and develop a standard state-space modelrepresentation.

A. NotationFor a vector x ∈ RN , diag(x) ∈ RN×N returns a diagonal

matrix with diagonal entries composed of entries of x. All-ones and all-zeros vectors of length N are denoted by 1N and



3

0N , respectively. Finally, 0M×N denotes the all-zeros matrixof size M -by-N .

B. Dynamical Model of Single-phase Inverter

A block diagram of the grid-connected single-phase inverteris illustrated in Fig. 2. The PLL is designed to track theinstantaneous angle of the grid voltage at the point of commoncoupling, the power controller regulates the real and reactivepower delivered into the ac grid, and the current controllergoverns the current delivered by the switch terminals. Thismodel represents a prototypical implementation in a single-phase grid-connected setting and captures all relevant ac-sidesystem dynamics. Since the focus of the paper is at the ac-sidepoint of common coupling, dynamics of the dc-link and anyother converter stages that precede the dc-link are neglected.We briefly overview the reference-frame transformations andthe dynamics of the filter and controllers next.

Reference-frame Transformations: The controllers illus-trated in Fig. 2 are implemented in the dq domain. To enablethis, the Hilbert transform [28] (denoted by Gπ/2) is firstutilized to generate orthogonal signals with quarter-cycle phaselag for each sinusoidal measurement, i.e., it yields signals inthe αβ domain [29]. Each signal and its corresponding phase-shifted counterpart is subsequently processed by an αβ to dqtransformation. The dq signals are then used in the PLL andcurrent controller. Note that although the above formulationutilizes the Hilbert transform as a means of generating quarter-cycle phase-shifted waveforms, these signals can also be real-ized with a quarter-cycle delay buffer or an all-pass filter withappropriate phase response. The remainder of the manuscriptfocuses exclusively on the Hilbert transform without loss ofgenerality.

The transfer function of the Hilbert transform is given by

Gπ/2(s) =ωPLL − sωPLL + s

, (1)

where ωPLL is the frequency returned by the PLL. As shownin Fig. 2, we will consider the measured signal to be theα-component, and the corresponding output of the Hilberttransform as the β component. Next, signals in the αβreference frame (xα, xβ) are transformed to the dq reference-frame (xd, xq) with the following rotation matrix [29]:[

xd

xq

]=

[cos δ sin δ− sin δ cos δ

] [xα

xβ

], (2)

where δ is the instantaneous angle generated by the PLL.As seen in Fig. 2, the PLL is in feedback with the dqtransformation. The role of the PLL is to modulate the valueof the PLL angle, δ, such that the d-axis component of thegrid voltage, vdg , is driven asymptotically to zero. From thedefinition of the α- and β-components of vg and the dqtransformation in (2), it can be shown that if vdg = 0, thenδ is the instantaneous phase angle of vg. (See Appendix A fora short derivation.)

Controller and Filter Dynamics: The internal controllersin the PLL comprise a low-pass filter with cut off frequencyωc,PLL and a PI controller with proportional and integral gains

given by kpPLL and kiPLL, respectively. The PLL dynamics aregiven by

d

dtvPLL = ωc,PLL(vdg − vPLL), (3a)

d

dtφPLL = −vPLL, (3b)

d

dtδ = ωnom − kpPLLvPLL + kiPLLφPLL =: ωPLL, (3c)

d

dtvβg = ωPLL(vg − vβg )− d

dtvg, (3d)

where ωnom is the nominal grid frequency (e.g., 2π × 60 or2π× 50 rad/s). We apply (1) to vg to obtain the dynamics ofvβg in (3d), and we apply (2) to vg and vβg to obtain vdg whichfeeds into (3a). From above, we can see that vdg = vPLL = 0in steady-state. Furthermore, when the grid frequency is ωnom,it follows that δ = ωPLL = ωnom. Note that we assume thefirst derivative of vg, i.e., d

dtvg, to be well defined.The LCL filter is composed of inverter-side inductance

Li, grid-side inductance, Lg, and filter capacitance, Cf . Thedynamics introduced by the LCL filter in the αβ frame aregiven by

d

dtiαi =

1

Li(−Rii

αi + vαi − vαf ), (4a)

d

dtiβi = ωPLL(iαi − i

βi )− d

dtiαi , (4b)

d

dtiαg =

1

Lg(−Rgi

αg + vαf − vg), (4c)

d

dtiβg = ωPLL(iαg − iβg )− d

dtiαg , (4d)

d

dtvαf = Rf

(d

dtiαi −

d

dtiαg

)+

1

Cf(iαi − iαg ), (4e)

d

dtvβf = ωPLL(vαf − v

βf )− d

dtvαf , (4f)

where the α-component dynamics are derived from funda-mental circuit laws, and the β-component expressions resultfrom the application of (1) to the corresponding α-componentdynamics.

The power controller (PC) consists of two PI controllerswith gains kpPC and kiPC and two low-pass filters with cut-off frequency ωc,PC (for the d and q components). The real-and reactive-power setpoints, p∗ and q∗, act as inputs to thepower controller and its outputs are current references for thedownstream current controller. These are generated as follows:

id∗i = kpPC (q∗ − qavg) + kiPC

∫(q∗ − qavg) , (5a)

iq∗i = kpPC (p∗ − pavg) + kiPC

∫(p∗ − pavg) , (5b)

where pavg and qavg are the outputs of the low-pass filters,with inputs to be the inverter real- and reactive-power outputsmeasured at the grid terminals, p and q, respectively. In par-ticular, with reference to Fig. 2 and with the aid of elementarytrigonometric operations we have

p =1

2(vgi

αg + vβg i

βg ), q =

1

2(vβg i

αg − vgiβg ), (6)



4

Figure 2: Block diagram of the single-phase inverter and adopted shorthand.

and as discussed above,d

dtpavg = ωc,PC(p− pavg),

d

dtqavg = ωc,PC(q− qavg). (7)

The real-power setpoint, p∗, reflects the real power that isultimately generated by an upstream input-stage controllerand dc-link voltage controller acting in concert. For instance,for a PV application, p∗ could be approximated from scaledirradiance data assuming accurate and fast maximum powerpoint tracking. Similarly, the reactive power setpoint, q∗, couldeither be fixed at zero to reflect unity power factor operationor may alternatively be generated by a Volt/VAR controller.For the sake of generality, we will simply consider p∗ and q∗

as generic model inputs for the remainder of the paper.The current controller (CC) is composed of two PI con-

trollers with gains kpCC and kiCC, and as outputs, it generatesthe voltage references for the PWM modulation block:

vd∗i = vdf + kpCC

(id∗i − idi

)+ kiCC

∫ (id∗i − idi

), (8a)

vq∗i = vqf + kpCC

(iq∗i − i

qi

)+ kiCC

∫ (iq∗i − i

qi

). (8b)

The addition of the feedforward terms vdf and vqf (obtained byapplying (2) to vαf and vβf ) is standard practice, and intendedto improve dynamic performance [30]. Suppose the VSI isideal (see Fig. 2), then the terminal inverter voltage is givenby:

vi ≈ vα∗i = vd∗i cos δ − vq∗i sin δ, (9)

where δ is the instantaneous PLL angle. This approximationimplies that the inverter terminal voltage follows the com-manded reference perfectly and without delay.

C. State-space Representation of Inverter Dynamics

The dynamics of the LCL filter, PLL, power controller,and current controller for an individual inverter are nowexpressed in state-space form to facilitate analysis. To thisend, corresponding to the power and current controllers, wewill find it useful to introduce the auxiliary dynamics

d

dtφp = p∗ − pavg,

d

dtφq = q∗ − qavg, (10)

d

dtγd = id∗i − idi ,

d

dtγq = iq∗i − i

qi . (11)

With these definitions in place, the dynamics (3a)–(11) can berepresented in a compact state-space form

x = Ax+B1u1 +B2u2 + g(x, u1, u2), (12)

where the state vector, x, and inputs u1, u2 are given by

x = [iαi , iβi , i

αg , i

βg , v

αf , v

βf , γ

d, γq, pavg, qavg, φp, φq,

vβg , vPLL, φPPL, δ]T, (13)

u1 = [p∗, q∗]T, u2 = [vg,d

dtvg]T. (14)

In order to show the entries of matrices A ∈ R16×16,B1 ∈ R16×2, and B2 ∈ R16×2, let us partition the statevector as x = [xTLCL, x

TCC, x

TPC, x

TPLL]T, where xLCL =

[iαi , iβi , i

αg , i

βg , v

αf , v

βf ]T, xCC = [γd, γq]T, xPC = [pavg, qavg,

φp, φq]T, and xPLL = [vβg , vPLL, φPLL, δ]T. Then, we can

write (12) asxLCLxCC

xPC

xPLL

=

ALCL 06×2 06×4 06×4

02×6 02×2 ACC 02×4

04×6 04×2 APC 04×4

04×6 04×2 04×4 APLL

xLCLxCC

xPC

xPLL

+

06×2

BCC

BPC

04×2

u1 +

BLCL02×2

04×2

BPLL

u2 + g(x, u1, u2), (15)

where entries of the nonzero sub-matrices ALCL, ACC,APC, APLL, BCC, BPC, BLCL, BPLL, and the functiong(x, u1, u2) : R16 × R2 × R2 → R16 are spelled out inAppendix B.

III. AGGREGATION OF PARALLEL-CONNECTED INVERTERS

In this section, we first introduce parametric scalingsrequired to realize the aggregate model for the parallel-connected inverters. Next, we prove that the aggregate modelindeed captures all the scalings in pertinent states (currents,voltages, internal-control states) for the uniform setting as wellas in cases with heterogeneous power setpoints and ratings.

A. Parametric Scalings and Structure of Aggregate Model

We consider N identical single-phase inverters (with modeldescribed in Section II) with the same setpoints, p∗ and q∗,connected in parallel to a grid bus. We are interested in anaggregated reduced-order model with the same structure anddimension as the model in (12):

xr = Arxr +Br1u

r1 +Br

2ur2 + gr(xr, ur1, u

r2). (16)

In particular: we desire matrices Ar ∈ R16×16, Br1 ∈ R16×2,

Br2 ∈ R16×2, and function gr : R16 × R2 × R2 → R16



5

(a) (b)

Figure 3: (a) Reduced-order aggregate single-phase inverter model and adopted shorthand representation. The inverter-side inductance andresistance are related to those in the individual model as Rr

iLr

i= Ri

Li. (b) Dynamics of the parallel connection of N single-phase inverters can

be captured by the aggregate inverter model in Fig. 3a.

have the same structure and dimension as A, B1, B2, andg in (12) (implying that the control architecture and output-filter arrangement of the aggregated model are the same asthat in an individual inverter); entries of state vector xr andinputs ur1, ur2 to have the same connotation as states in x andinputs u1, u2. In our main result, we demonstrate that if andonly if the following relationships hold between parameters ofthe reduced-order and original models:

Crf = NCf , R

rf =

Rf

N, Lr

g =Lg

N, Rr

g =Rg

N, (17a)

Rri

Lri

=Ri

Li,kp,rCC

Lri

=kpCC

Li,ki,rCC

Lri

=kiCC

Li, (17b)

and the reference power settings of the lumped-parameteraggregate inverter model in Fig. 3a are N times thoseof the inverter model in Fig. 2, the current- and power-related states iα,ri , iβ,ri , iα,rf , iβ,rf , γd,r, γq,r, pravg, q

ravg, φ

p,r, φq,r

in the reduced-order model are N times the corre-sponding ones in an individual inverter. Furthermore, thevoltage- and PLL-related states in the reduced-order modelvα,rf , vβ,rf , vβ,rg , vrPLL, φ

rPPL, δ

r are the same as those in anyinverter in the parallel combination. This is consistent withthe electrical behavior of a parallel connection of current (orpower) sources.

In establishing the above result, we will have establishedthat the reduced-order aggregate model in Fig. 3a capturesthe dynamics of the parallel collection. Put differently, wewill mathematically establish the equivalence illustrated inFig. 3b. It is worth emphasizing that the dynamical modelof an individual inverter has 16 states, and so modeling thedynamics of every inverter in an N -inverter parallel collectionwould require a 16N -order state-space model. By contrast, thereduced-order model has the same structure as any individualinverter, and is hence described only by 16 states.

B. Main Result: Validating the Aggregate Model

We now state and prove the main result of this paper.Theorem 1. (Aggregation of parallel-connected identicalsingle-phase inverters) Consider the dynamical model for thesingle-phase inverter specified in (12). Permute x in (13) as

x = [iαi , iβi , i

αg , i

βg , γ

d, γq, pavg, qavg, φp, φq, vαf , v

βf ,

vβg , vPLL, φPPL, δ]T, (18)

and also permute xr (corresponding to the reduced-ordermodel (16)) the same way, denoting the permuted vector by xr.Denote x(t) to be the solution to the permuted version of (12)with initial condition x(t0) and inputs u1, u2; and xr(t) tobe the solution to the permuted version of (16) with initialcondition xr(t0) and inputs ur1, u

r2. Suppose initial conditions

are such that xr(t0) = diag(Ψ)x(t0), where the scaling vector,Ψ := [N1T10, 1

T6 ]T, and the inputs are: ur1 = Nu1, u

r2 = u2

(see (14)). The states of the reduced-order model and theindividual inverter are related as:

xr(t) = diag(Ψ)x(t), ∀t ≥ t0, (19)

if and only if their parameters are related as (17a)–(17b).In particular, given the definition of the scaling vector,

Ψ, (19) establishes the following relationships between statesof the reduced-order model and those in the individual invertermodel ∀t ≥ t0:

[iα,ri , iβ,ri , iα,rg , iβ,rg , γd,r, γq,r, pravg, qravg, φ

p,r, φq,r]T

= N [iαi , iβi , i

αg , i

βg , γ

d, γq, pavg, qavg, φp, φq]T, (20a)

[vα,rf , vβ,rf , vβ,rg , vrPLL, φrPPL, δ

r]T

= [vαf , vβf , v

βg , vPLL, φPPL, δ]

T. (20b)

Proof. Let us define z := xr − diag(Ψ)x. The dynamics of zare given by

z = ˙xr− diag(Ψ) ˙x = Arxr + Br

1ur1 + Br

2ur2 + g(xr, ur1, u

r2)

− diag(Ψ)Ax− diag(Ψ)B1u1 − diag(Ψ)B2u2

− diag(Ψ)g(x, u1, u2), (21)

where matrices A, B1, B2, Ar, Br1, Br

2 and functionsg(x, u1, u2), g(xr, ur1, u

r2) are appropriately permuted versions

of corresponding matrices and functions in (12) and (16).We will now show that z = 0, ∀t ≥ t0 when z(t0) =xr(t0) − diag(Ψ)x(t0) = 016. This would further imply thatz(t) = xr(t)− diag(Ψ)x(t) = 016 ∀t ≥ t0, as claimed in thestatement of the Theorem.

Partition x = [xT1 , xT2 ]T, where x1 = [iαi , i

βi , i

αg , i

βg , γ

d, γq,

pavg, qavg, φp, φq]T and x2 = [vαf , v

βf , v

βg , vPLL, φPPL, δ]

T,and we also partition xr the same way. Then, we partitionthe appropriately permuted versions of (12) and (16) as[

˙x1˙x2

]=

[A11 A12

A21 A22

] [x1x2

]+

[B11

B12

]u1 +

[B21

B22

]u2



6

+ g(x, u1, u2), (22)[˙xr

1˙xr

2

]=

[Ar

11 Ar12

Ar21 Ar

22

] [xr1xr2

]+

[Br

11

Br12

]ur1 +

[Br

21

Br22

]ur2

+ gr(xr, ur1, ur2). (23)

From the definition of matrices ALCL, ACC, APC, APLL,BCC, BPC, BLCL, BPLL in Appendix B, and the parametricscalings established in (17a)–(17b), we note the following:

Ar11 = A11, A

r12 = NA12, A

r21 =

1

NA21, A

r22 = A22,

Br11 = B11, B

r12 =

1

NB12, B

r21 = NB21, B

r22 = B22.

(24)

Then, we have

diag(Ψ)A =

[NA11 NA12

A21 A22

]=

[NAr

11 Ar12

NAr21 Ar

22

]= Ardiag(Ψ), (25)

diag(Ψ)B1 =

[NB11

B12

]=

[NBr

11

NBr12

]= NBr

1, (26)

diag(Ψ)B2 =

[NB21

B22

]=

[Br

21

Br22

]= Br

2. (27)

The next step is to show that gr(diag(Ψ)x, ur1, ur2) =

diag(Ψ)g(x, u1, u2). Let g`(x) and gr`(diag(Ψ)x) denote the`-th entry of g(x, u1, u2) and gr(diag(Ψ)x, ur1, u

r2), respec-

tively. The nonzero entries of gr(diag(Ψ)x, ur1, ur2) are related

to the corresponding entries of g(x, u1, u2) through:

gr1(diag(Ψ)x) =N

Li

[(kpCC

N

(kpPC(Nq∗ −Nqavg) + kiPCNφ

q

−Nidi)

+kiCC

NNγd

)cos δ −

(kpCC

N(kpPC(Np∗ −Npavg)

+kiPCNφp −Niqi

)+kiCC

NNγq

)sin δ

]= Ng1(x),

gr2(diag(Ψ)x) = η(Niαi −Niβi )− gr1(diag(Ψ)x) = Ng2(x),

gr5(diag(Ψ)x) = −Niαi cos δ −Niβi sin δ = Ng5(x),

gr6(diag(Ψ)x) = Niαi sin δ −Niβi cos δ = Ng6(x),

gr7(diag(Ψ)x) =ωc,PC

2(vgNi

αg + vβgNi

βg ) = Ng7(x),

gr8(diag(Ψ)x) =ωc,PC

2(vβgNi

αg − vgNiβg ) = Ng8(x),

gr11(diag(Ψ)x) =Rf

Ngr1(diag(Ψ)x) = Rf g1(x) = g11(x),

gr12(diag(Ψ)x) = η(vαf − vβf )− gr11(diag(Ψ)x) = g12(x),

gr13(diag(Ψ)x) = η(vg − vβg ) = g13(x),

gr14(diag(Ψ)x) = ωc,PLL(vg cos δ + vβg sin δ) = g14(x),

gr16(diag(Ψ)x) = ωnom = g16(x),

with η := −kpPLLvPLL + kiPLLφPLL. Therefore we have

diag(Ψ)g(x, u1, u2) = gr(diag(Ψ)x, ur1, ur2). (28)

Notice that the PLL dynamics ((3a)–(3d)) are decoupled fromthe remainder of the states in the state-space model. Therefore,

the parameters of the PLL in the individual and reduced-ordermodels are the same, and we can conclude that:

vβ,rg = vβg , vrPLL = vPLL, φrPLL = φPLL, δr = δ. (29)

Now, consider the function h(xr, ur1, ur2) : R16 × R2 ×

R2 → R16, which is defined to have the same structure asgr(xr, ur1, u

r2) except that its 13th, 14th, and 16th entries are

0. (See Appendix B for details.) Then, the following holds

gr(xr, ur1, ur2)− gr(diag(Ψ)x, ur1, u

r2)

= h((xr − diag (Ψ)x), 02, ur2) . (30)

Using identities (25)–(30) in (21), we have

z = Ar(xr − diag(Ψ)x) + h((xr − diag (Ψ)x), 02, ur2)

= Arz + h(z, 02, ur2). (31)

If we initialize z(t0) = 016, we have z(t) = 016,∀t ≥ t0, dueto the fact that h(016, 02, u

r2) = 016. By the definition of z(t),

we have xr(t) = diag(Ψ)x(t),∀t ≥ t0.For the other direction, given that z(t) = xr −

diag(Ψ)x, ∀t ≥ t0, ur1 = Nu1, and ur2 = u2, (21) can bewritten as

015 = (Ardiag(Ψ)− diag(Ψ)A)x+ (NBr1 − diag(Ψ)B1)u1

+ (Br2 − diag(Ψ)B2)u2 + gr(diag(Ψ)x, Nu1, u2)

− diag(Ψ)g(x, u1, u2). (32)

The equality above is satisfied when the following identitieshold:

Ardiag(Ψ) = diag(Ψ)A, NBr1 = diag(Ψ)B1,

Br2 = diag(Ψ)B2,

(33)

gr(diag(Ψ)x, Nu1, u2) = diag(Ψ)g(x, u1, u2). (34)

It emerges that Ri, Li, kpCC, and kiCC always appear in the

identities above as fractions: Ri

Li, kpCC

Li, and kiCC

Li. Therefore,

these parameters relate to those in the reduced-order modelthrough (17b). The remainder of the parameters can be de-termined straightforwardly from (33) and (34); they are givenuniquely by (17a), including the unchanged parameters (i.e.,those which are not mentioned in (17a) and (17b)). Thisconcludes the proof.

C. Corollaries for Heterogeneous Settings

We now present two corollaries. In the first, we examine in-verters with different reference-power setpoints for both activeand reactive power. In this particular case, the relationshipsbetween the states of the reduced-order model and those ofthe individual inverters ` = 1, . . . , N are as follows ∀t ≥ t0:


p,r, φq,r]T

=

N∑`=1

[iαi,`, iβi,`, i

αg,`, i

βg,`, γ

d` , γ

q` , pavg,`, qavg,`, φ

p` , φ

q` ]

T,

[vα,rf , vβ,rf ]T =1

N

N∑`=1

[vαf,`, vβf,`]

T,

[vβ,rg , vrPLL, φrPPL, δ

r]T = [vβg,`, vPLL,`, φPPL,`, δ`]T,∀`.



7

This model is useful in, e.g., PV systems where the incidentirradiation might be different for different inverters (henceresulting in different values for p∗) and where local-voltagecontrol may be implemented by modulating reactive-powerinjections (hence resulting in different values of q∗).

In the second corollary, we examine inverters with differentpower ratings, and derive an aggregate model with currentsthat scale systematically. To formalize this, we define a power-scaling parameter κ` for the `-th inverter as [26]:

κ` =prated,`pbase

, (36)

where prated,` and pbase denote the rated power of the `-thinverter in the parallel system and system-wide base value,respectively. Without loss of generality, we assume that theinverter model in Fig. 2 has a power rating equal to the basevalue. We also introduce the notion of an equivalent power-scaling parameter:

κ :=N∑`=1

κ`. (37)

The states of the reduced-order model relate to those in theindividual inverters ` = 1, . . . , N and the unscaled inverter(i.e., inverter model with rating equal to the base value) asfollows ∀t ≥ t0:


p,r, φq,r]T

=N∑`=1

[iαi,`, iβi,`, i

αg,`, i

βg,`, γ

d` , γ


p` , φ

q` ]

T

= κ[iαi , iβi , i

αg , i

βg , γ

d, γq, pavg, qavg, φp, φq]T,

[vα,rf , vβ,rf , vβ,rg , vrPLL, φrPPL, δ

r]T

= [vαf,`, vβf,`, v

βg,`, vPLL,`, φPPL,`, δ`]

T,∀`= [vαf , v

βf , v

βg , vPLLφPPL, δ]

T.

Formal results establishing these two aspects follow next.

Corollary 1. (Aggregation of parallel-connected identicalsingle-phase inverters with different reference-power set-points) Let us denote x`, p∗` , and q∗` as the state vector, real-,and reactive-power setpoints of the `-th inverter in the parallelsystem. Permute x` the same way as in (18), denoting thepermuted vector as x`. Partition x` = [λT` , v

αβTf,` , xTPLL,`]

T,where λ` = [iαi,`, i

βi,`, i

αg,`, i

βg,`, γ

d` , γ


p` , φ

q` ]

T,vαβf,` = [vαf,`, v

βf,`]

T, and xPLL = [vβg,`, vPLL,`, φPPL,`, δ`]T.

We also permute and partition xr the same way, denotingthe permuted vector as xr = [λrT, vαβ,rTf , xrTPLL]T. Supposethe initial conditions are such that λr(t0) =

∑N`=1 λ`(t0),

vαβ,rf (t0) = 1N

∑N`=1 v

αβf,` (t0), and xrPLL(t0) = xPLL,`(t0),∀`,

and the inputs are:

ur1 =N∑`=1

u1,`, ur2 = u2. (39)

It follows that, ∀t ≥ t0:

λr(t) =N∑`=1

λ`(t), vαβ,rf (t) =1

N

N∑`=1

vαβf,` (t),

xrPLL(t) = xPLL,`(t),∀`,(40)

if and only if the parameters of the reduced-order are relatedto the individual inverters through (17a)–(17b).

Proof. The proof is provided in Appendix C.

Corollary 2. (Aggregation of parallel-connected single-phaseinverters with heterogeneous power ratings) The parametersof each inverter are related to the unscaled inverter through

Cf,` = κ`Cf , Rf,` =Rf

κ`, Lg,` =

Lg

κ`, Rg,` =

Rg

κ`, (41a)

Ri,`

Li,`=Ri

Li,kpCC,`

Li,`=kpCC

Li,kiCC,`

Li,`=kiCC

Li, (41b)

and parameters not mentioned are unchanged. Suppose thereference-power setpoints for each inverter are p∗` = κ`p

∗ andq∗` = κ`q

∗. The parameters of the reduced-order model arerelated to the unscaled inverter through

Crf = κCf , R

rf =

Rf

κ, Lr

g =Lg

κ, Rr

g =Rg

κ, (42a)

Rri

Lri

=Ri

Li,kp,rCC

Lri

=kpCC

Li,ki,rCC

Lri

=kiCC

Li. (42b)

Parameters not mentioned are unchanged. Let x, x`, xr

denote the state vectors of the unscaled inverter model, `-th inverter of the parallel system, and the reduced-ordermodel, respectively. Permute the state vectors the same wayas (18), denoting the permuted vectors as x, x`, xr. Par-tition the permuted state vector x = [λT, ψT]T, whereλ = [iαi , i

βi , i

αg , i

βg , γ

d, γq, pavg, qavg, φp, φq]T and ψ =

[vαf , vβf , v

βg , vPLL, φPLL, δ]

T. We also partition x` and xr thesame way: x` = [λT` , ψ

T` ]T, xr = [λrT, ψrT]T. Suppose

the initial conditions are such that λr(t0) =∑N`=1 λ`(t0) =

κλ(t0), ψr(t0) = ψ`(t0) = ψ(t0),∀`, and the inputs are:

ur1 =

N∑`=1

u1,` = κu1, ur2 = u2,` = u2,∀`. (43)

It follows that for t ≥ t0:

λr(t) =

N∑`=1

λ`(t) = κλ(t), ψr(t) = ψ`(t) = ψ(t),∀`, (44)

if and only if the parameters of the reduced-order model arerelated to the unscaled inverter through (42a)–(42b).

Proof. Each of the inverters in the parallel system can beviewed as the aggregate of κ` inverters, while keeping in mindthat κ` is not necessarily an integer. The rest of this proof isstraightforward from Theorem 1.

IV. EXPERIMENTAL VALIDATION & SIMULATION RESULTS

In this section, we outline results from an experimentalprototype and an exhaustive simulation study to demonstratevarious aspects of the reduced-order model. The purpose andscope of the experiments is to demonstrate the validity andestablish the accuracy of the reduced-order model (underuniform and symmetric settings) and this is done by comparingthe net current injected by the parallel system of invertersin hardware to the output current of the aggregated reduced-order model. The experiments also establish robustness of the



8

Table I: Inverter LCL-filter and controller parameters.

LCL filter Current Controller PLLLi = 1.0 mH kpCC = 6 V/A kpPLL = 1.25 rad/(V · s)Ri = 0.7 Ω kiCC = 350 V/(A · s) kiPLL = 10 rad/(V · s2)Cf = 24µF ωc,PLL = 2π × 200 rad/sRf = 0.02 Ω Power ControllerLg = 0.2 mH kpPC = 0.01 A/VARg = 0.12 Ω kiPC = 0.1 A/(VA · s)

ωc,PC = 50.26 rad/s

reduced-order model to parametric variations that are indeedinescapable in any hardware setup. Following the experimentalresults, we also include an exhaustive simulation study that:validates the reduced-order model derived for hetereogeneoussettings (Corollaries 1 and 2), investigates robustness of thereduced-order model to variations in filter parameters, anddemonstrates the computational benefits of the reduced-ordermodel.

A. Hardware Setup

To validate the reduced-order model, we built an experimen-tal system comprised of three identical 750 VA single-phaseinverters connected in parallel across a stiff voltage source.The hardware setup is illustrated in Fig. 4(a). It consists ofthree distinct inverters each with a dedicated power stage anda TI F28335 DSP controller. Each inverter utilizes the controlstructure shown in Fig. 2. Controllers are discretized with astep size of 1/(15× 103) s and unipolar sine-triangle PWM isutilized with a switching frequency of 30 kHz. The single-phase 60 Hz, 120 V RMS ac system voltage (i.e., the gridpoint of interconnection) is realized with an Ametek MX-45grid simulator. Subsequently, measurements obtained from themulti-inverter system are compared to a software simulation ofa single aggregated inverter model (see Figs. 3a and 4(b)). Thesimulation of the aggregated inverter, as given in Fig. 4(b), wascarried in MATLAB with the ODE45 solver, performed on acomputer with Intel Core i7-7700HQ processor @ 2.80GHzCPU and 8GB RAM. The parameters of the experimental setupare summarized in Table I. Simulation parameters used in theaggregated inverter model are obtained from these, and thescalings reported in (17a)–(17b).

B. Validation of Reduced-order Model

To validate the proposed reduced-order model, we com-pared the measured and simulated dynamic responses undera comprehensive set of step changes in both real and reactivepower. The real-power steps are representative of, e.g., suddenirradiance transients that a microinverter system might contendwith. The reactive-power steps are representative of, e.g.,ancillary services that grid-connected inverters may provide.Results are plotted in Fig. 5. The plot pair in each subfigureillustrates:

1) The measured net sinusoidal current injected into thegrid by the parallel inverters overlaid with the simulatedcurrent from the aggregated-inverter model. The measure-ment point and corresponding point in the reduced-ordermodel are marked prominently in Fig. 4.

Figure 4: (a) Experimental setup consisting of three parallel-connected single-phase inverters rated at 750VA. The system ofthree inverters are given real- and reactive-power step commands togenerate the results in Fig. 5 (Currents plotted in Fig. 5 are shown indashed boxes, marked with the same color scheme above.) (b) Thereduced-order aggregated model where the multi-inverter system isrepresented as one equivalent inverter.

2) Pertinent d- or q-axis current waveforms measured ateach inverter output in addition to measured and sim-ulated net current injection.

It is worth emphasizing that we focus just on the net currentat the point of grid interconnection and compare that with thecurrent suggested by the aggregate model. The match betweenthese through a variety of large-signal changes—as suggestedin Fig. 5—validates the accuracy of the aggregate model.Furthermore, note that in this case, the parallel collection ofinverters are collectively described by a 48-state model, whilethe simulations are performed with the reduced-order 16-statemodel.

C. Simulation Study

Next, we establish the accuracy and computational benefitsof the proposed reduced-order model (for a system of 100parallel-connected inverters) in heterogeneous settings withnumerical simulation results. The parameters of the inverterwith nominal power rating are listed in Table I. We considerthe following cases: #1) Inverters have heterogeneous powerratings with power-scaling parameters κ selected to be uni-formly distributed between 0.5 and 5. #2) All inverters haveratings that match the nominal power ratings, but their LCL-filter parameters vary between ±10% of their nominal values.#3) Same setup as #2, but the LCL-filter parameters of theinverters vary between ±80% of their nominal values. For allcases, the real- and reactive-power setpoints of the invertersare assumed to be uniformly distributed between 0 − 200 Wand 0−100 VAR, respectively, and we perform a step change



9

(a)

(b)

(c)

(d)

Figure 5: Comparison of experimentally measured and simulatedwaveforms: (a) Real-power step up p∗ : 30W → 600W with fixedq∗ = 0VAR, (b) Real-power step down p∗ : 700W → 50W withfixed q∗ = 0VAR, (c) Reactive-power step up q∗ : 0VAR →500W with fixed p∗ = 200W, (d) Reactive-power step downq∗ : 500VAR→ 0W with fixed p∗ = 250W.

to both setpoints, with the values again selected to be uni-formly distributed between 400−600 W and 300−500 VAR,respectively. The step change is introduced at t = 2 s, and we

stop the simulations at t = 4 s. We note that case #2 and #3have the same reduced-order model. The parameter scalings ofthe reduced-order models for case #1 and #2 (#3) are givenby (42a)–(42b) and (17a)–(17b), respectively. The net currentinjection of the multi-inverter system and the reduced-ordermodels for case #1, #2, and #3 are shown in Fig. 6. We canclearly see in Fig. 6a that for case#1, the output current ofthe reduced-order model is exactly the same as the net currentinjection of the parallel system—this validates Corollaries 1and 2. Furthermore, Fig. 6b shows that the reduced-ordermodel is quite robust with respect to the parametric variationsin the LCL filter parameters with discrepancies obvious inhigh-frequency content. For larger variation (±80%), Fig. 6cshows that the reduced-order model captures the dynamicsof the multi-inverter system, albeit with degraded accuracyduring the transient. Finally, the computation time for the1600-th order multi-inverter system simulation for cases #1and #2 are 58.23 s, 66.97 s, and 145.08 s, respectively, andof the reduced-order 16-th order aggregate model are 1.89 s,1.62 s, and 1.62 s, respectively. This clearly establishes thecomputational benefits of the proposed model.

V. CONCLUDING REMARKS AND DIRECTIONS FORFUTURE WORK

In this paper, we derived a reduced-order aggregated modelfor identical parallel-connected grid-tied single-phase invert-ers and extensions covering cases when the inverter power-setpoints are different and the inverter power ratings aredifferent. The reduced-order model preserves the structure andhas the same order as any individual inverter in the parallelcollection. Experimental validation was provided to establishthe accuracy of the reduced-order model in capturing ac-side dynamics of inverters during large-signal transients, andsimulation results were provided to demonstrate computationalbenefits and robustness to parametric variations. Directions forfuture work include analytically establishing error bounds onthe trajectories returned by reduced-order models in the faceof parametric variations in the filter and control parameters.

APPENDIX

A. Steady-state Operation of PLL

Express the grid voltage as vg = Vg sin(δg), where Vgand δg are the voltage amplitude and angle, respectively. Thecorresponding αβ components of vg are given by

vαg = vg = Vg sin(δg),

vβg = Vg sin(δg −

π

2

)= −Vg cos(δg).

The d-axis component of vg is obtained from (2) as

vdg = Vg cos(δ) sin(δg)− Vg sin(δ) cos(δg) = Vg sin(δg − δ).

From above, it follows that when δ = δg, vdg = 0.



10

(a) Case #1

(b) Case #2

(c) Case #3

Figure 6: Simulation results comparing the injected current for asystem of 100 parallel-connected inverters with all inverter dynamicssimulated superimposed to results from the reduced-order model forthe following cases: (a) heterogeneous power ratings with power-scaling parameters (κ) vary between 0.5 and 5, (b) identical powerratings with κ = 1 and LCL-filter parameters vary between ±10%of their nominal values,(c) same setup as (b), but with variation of±80%.

B. State-space Model Particulars

ALCL =

−Ri

Li0 0 0

ωnom + Ri

Li−ωnom 0 0

0 0 −Rg

Lg0

0 0 ωnom +Rg

Lg−ωnom

−RfRi

Li+ 1

Cf0 Rf

Rg

Lg− 1

Cf0

RfRi

Li− 1

Cf0 −Rf

Rg

Lg+ 1

Cf0

0 00 01Lg

0

− 1Lg

0

−Rf

Lg0

ωnom + Rf

Lg−ωnom

, ACC =

0 −kpPC

−kpPC 00 kiPC

kiPC 0

T

,

APC =

−ωc,PC 0 0 0

0 −ωc,PC 0 0−1 0 0 00 −1 0 0

, BCC =

[0 kpPC

kpPC 0

],

APLL =

−ωnom 0 0 0

0 −ωc,PLL 0 00 −1 0 00 −kpPLL kiPLL 0

, BPLL =

0 −10 00 00 0

,

BLCL =

[0 0 0 0 0 0

0 0 − 1Lg

1Lg

Rf

Lg−Rf

Lg

]T, BPC =

0 00 01 00 1

.Lastly, the entries of g(x, u1, u2), with g` denotes the `-thentry of g(x, u1, u2), are

g1 =1

Li

(kpCC(id∗i − iαi cos δ − iβi sin δ) + kiCCγ

d)

cos δ

− 1

Li

(kpCC(iq∗i + iαi sin δ − iβi cos δ) + kiCCγ

q)

sin δ,

g2 = η(iαi − iβi )− g1, g3 = 0, g4 = 0,

g5 = Rfg1, g6 = η(vαf − vβf )− g5,

g7 = −iαi cos δ − iβi sin δ, g8 = iαi sin δ − iβi cos δ,

g9 =ωc,PC

2(vgi

αg + vβg i

βg ), g10 =

ωc,PC

2(vβg i

αg − vgiβg ),

g11 = 0, g12 = 0, g13 = η(vg − vβg ),

g14 = ωc,PLL(vg cos δ + vβg sin δ), g15 = 0, g16 = ωnom,

where η := −kpPLLvPLL+kiPLLφPLL, id∗i = kpPC (q∗ − qavg)+kiPCφ

q, and iq∗i = kpPC (p∗ − pavg) + kiPCφp.

C. Proof of Corollary 1

We begin by noting that the PLL dynamics are decou-pled, and the its parameters in the individual and reduced-order models are the same, therefore ∀t ≥ t0, xrPLL(t) =xPLL,`(t) ∀` if we initialize xrPLL(t0) = xPLL,`(t0) ∀`. Next,partition the permuted versions of (12) and (16), excluding thePLL dynamics, as[

λ`vαβf,`

]=

[A11 A12

A21 A22

] [λ`vαβf,`

]+

[B11

B12

]u1,` +

[B21

B22

]u2

+

[g1(x`, u1,`, u2)g2(x`, u1,`, u2)

], (45)[

λr

vαβ,rf

]=

[Ar

11 Ar12

Ar21 Ar

22

] [λr

vαβ,rf

]+

[Br

11

Br12

]ur1 +

[Br

21

Br22

]ur2

+

[gr1(xr, ur1, u

r2)

gr2(xr, ur1, ur2)

], (46)



11

where g1 : R16×R2×R2 → R10 and g2 : R16×R2×R2 → R2

are the nonlinear parts of the dynamics of λ` and vαβf,` ,respectively (similarly for gr1 and gr2). We bring to note a slightabuse of notation in terms of the submatrices in (45) and (46)and those in (22) and (23). Furthermore, the submatricesin (45) and (46) also follow the relationships in (24). Definez1 := λr −

∑N`=1 λ` and z2 := Nvαβ,rf −

∑N`=1 v

αβf,` . The

dynamics of z1 and z2 are:

z1 = λr −N∑`=1

λ` = Ar11λ

r + Ar12v

αβ,rf + Br

11ur1 + Br

21ur2

+ gr1(xr, ur1, ur2)−

N∑`=1

(A11λ` + A12v

αβ,rf,` + B11u1,`

+B21u2 + g1(x`, u1,`, ur2)), (47)

z2 = Nvαβ,rf −N∑`=1

vαβf,` = N(Ar

21λr + Ar

21vαβ,rf + Br

12ur1

+Br22u

r2 + gr2(xr, ur1, u

r2))−

N∑`=1

(A21λ` + A22v

αβ,rf,`

+B12u1,` + B22u2 + g2(x`, u1,`, ur2)). (48)

Next, we will show that

gr1(xr, ur1, ur2)−

N∑`=1

g1(x`, u1,`, u2) = g1(χ, 02, u2), (49)

Ngr2(xr, ur1, ur2)−

N∑`=1

g2(x`, u1,`, u2) = g2(χ, 02, u2), (50)

where χ := [zT1 , zT2 , x

rTPLL]T. Let g1,k(x`), g2,k(x`), gr1,k(xr),

and gr2,k(xr) denote the k-th entries of g1(x`, u1,`, u2),g2(x`, u1,`, u2), gr1(xr, ur1, u

r2), and gr2(xr, ur1, u

r2), respec-

tively. Then, we have

gr1,1(xr)−N∑`=1

g1,1(x`) =N

Li

[(kpCC

N

(kpPC(

N∑`=1

q∗` − qravg)

+kiPCφq,r − id,ri

)+kiCC

Nγd,r

)cos δr −

(kpCC

N

(kpPC(

N∑`=1

p∗`

−pravg) + kiPCφp − iq,ri

)− kiCC

Nγq,r

)sin δr

]−

N∑`=1

((kpCC

Li(kpPC(q∗` − qavg,`) + kiPCφ

q` − i

di,`

)+kiCC

Liγd`

)cos δ` +

kpCC

Li((kpPC(p∗` − pavg,`) + kiPCφ

p − iq,ri

)− kiCC

Nγq,r

)sin δ`

)=

(kpCC

Li

(kpPC(0− (qravg −

N∑`=1

qavg,`)) + kiPC(φq,r

−N∑`=1

φq` )− (id,ri −N∑`=1

idi,`)

)+kiCC

Li(γd,r −

N∑`=1

γd` )

)cos δr

+

(kpCC

Li

(kpPC(0− (pravg −

N∑`=1

pavg,`)) + kiPC(φp,r

−N∑`=1

φp` )− (iq,ri −N∑`=1

iqi,`)

)+kiCC

Li(γq,r −

N∑`=1

γq` )

)sin δr

= g1,1(χ, 02, u2), (51)

gr1,2(xr)−N∑`=1

g1,2(x`) = (−kpPLLvrPLL + kiPLLφ

rPLL)(iα,ri

− iβ,ri )− gr1,1(xr)−N∑`=1

((−kpPLLvPLL,` + kiPLLφPLL,`)(i

αi,`

− iβi,`)− g1,1(x`))

= (−kpPLLvrPLL + kiPLLφ

rPLL)

((iα,ri

−N∑`=1

iαi,`) + (iβ,ri −N∑`=1

iβi,`))−(gr1,1(xr)−

N∑`=1

g1,1(x`))

= g1,2(χ, 02, u2), (52)

gr1,3(xr)−N∑`=1

g1,3(x`) = 0 = g1,3(χ, 02, u2), (53)

gr1,4(xr)−N∑`=1

g1,4(x`) = 0 = g1,4(χ, 02, u2), (54)

gr1,5(xr)−N∑`=1

g1,5(x`) = −iα,ri cos δr − iβ,ri sin δr

−N∑`=1

(−iαi,` cos δ` − iβi,` sin δ`

)= −(iα,ri −

N∑`=1

iαi,`) cos δr

− (iβ,ri −N∑`=1

iβi,`) sin δr = g1,5(χ, 02, u2), (55)

gr1,6(xr)−N∑`=1

g1,6(x`) = iα,ri sin δr − iβ,ri cos δr

−N∑`=1

(iαi,` sin δ` − iβi,` cos δ`

)= (iα,ri −

N∑`=1

iαi,`) sin δr

− (iβ,ri −N∑`=1

iβi,`) cos δr = g1,6(χ, 02, u2), (56)

gr1,7(xr)−N∑`=1

g1,7(x`) =ωc,PC

2(vgi

α,rg + vβ,rg iβ,rg )

−N∑`=1

ωc,PC

2

(vgi

αg,` + vβg,`i

βg,`

)=ωc,PC

2

(vg(iα,rg −

N∑`=1

iαg,`)

+ vβ,rg (iβ,rg −N∑`=1

iβg,`))

= g1,7(χ, 02, u2), (57)

gr1,8(xr)−N∑`=1

g1,8(x`) =ωc,PC

2(vβ,rg iα,rg − vgiβ,rg )

−N∑`=1

ωc,PC

2

(vβg,`i

αg,` − vgi

βg,`

)=ωc,PC

2

(vβ,rg (iα,rg

−N∑`=1

iαg,`)− vg(iβ,rg −N∑`=1

iβg,`))

= g1,8(χ, 02, u2), (58)

gr1,9(xr)−N∑`=1

g1,9(x`) = 0 = g1,9(χ, 02, u2), (59)



12

gr1,10(xr)−N∑`=1

g1,10(x`) = 0 = g1,10(χ, 02, u2), (60)

Ngr2,1(xr)−N∑`=1

g2,1(x`) = NRf

Ngr1,1(xr)−Rf

N∑`=1

g1,1(x`)

= Rf(gr1,1(xr)−

N∑`=1

g1,1(x`)) = g2,1(χ, 02, u2), (61)

Ngr2,2(xr)−N∑`=1

g2,2(x`) = N(−kpPLLvrPLL + kiPLLφ

rPLL)

(vα,rf − vβ,rf )−Ngr2,1(xr)−N∑`=1

((−kpPLLvPLL,`

+ kiPLLφPLL,`)(vαf,` − v

βf,`)− g2,1(x`)

)= (−kpPLLv

rPLL

+ kiPLLφrPLL)

((Nvα,rf −

N∑`=1

vαf,`)− (Nvβ,rf −N∑`=1

vβf,`))

− (Ngr2,1(xr)−N∑`=1

g2,1(x`)) = g2,2(χ, 02, u2). (62)

Therefore, (49) and (50) hold. Using identities (24), (49)and (50), we can write the dynamics of z1 and z2 as

z1 = A11z1 +A12z2 + g1(χ, 02, u2), (63)z2 = A21z1 +A22z2 + g2(χ, 02, u2). (64)

If we initialize z1(t0) = 010 and z2(t0) = 02, we havez1(t) = 010, z2(t) = 02,∀t ≥ t0 since g1(χ, 02, u2) = 010 andg2(χ, 02, u2) = 02 when z1 = 010 and z2 = 02. By the defi-nition of z1 and z2, we have λr(t) =

∑N`=1 λ`(t), v

αβ,rf (t) =

1N

∑N`=1 v

αβf,` (t),∀t ≥ t0.

For the other direction, given that ∀t ≥ t0: z1(t) = λT(t)−∑N`=1 λ`(t) = 010, z2(t) = Nvαβ,rf (t) −

∑N`=1 v

αβf,` (t) =

02, xrPLL(t) = xPLL,`(t),∀`, (47) and (48) can be written as

010 = (Ar11 − A11)λr + (Ar

12 −NA12)vαβ,rf + (Br11 − B11)ur1

+ (Br21 −NB21)ur2 + gr1(xr, ur1, u

r2)−

N∑`=1

g1(x`, u1,`, ur2),

02 = (NAr21 − A21)λr + (NAr

22 −NA22)vαβ,rf + (NBr21

− B21)ur1 + (NBr22 −NB22)ur2 +Ngr2(xr, ur1, u

r2)

−N∑`=1

g2(x`, u1,`, ur2).

These equalities are satisfied when the following identitieshold:

Ar11 = A11, A

r12 = NA12, A

r21 =

1

NA21, A

r22 = A22,

Br11 = B11, B

r21 = NB21, B

r12 =

1

NB12, B

r22 = B22,

(65)

gr1(xr, ur1, ur2) =

N∑`=1

g1(x`, u1,`, u2), (66)

Ngr2(xr, ur1, ur2) =

N∑`=1

g2(x`, u1,`, u2). (67)

It is straightforward to see that (17a) and the unscaled pa-rameters are the only set of parameters that satisfy (65). Forthe rest of parameters, i.e., Ri, Li, k

pCC, and kiCC, it can be

derived that they always appear in (65)–(67) as fractions ofRi

Li, k

pCC

Li, and kiCC

Li. Therefore, they are related to those in the

reduced-order model through (17b). This concludes the proof.

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Victor Purba (S’15) received the Bachelor’s andM.S. degrees in electrical engineering in 2014 and2016, respectively, from University of Minnesota,Minneapolis, MN, USA, where he is currently pur-suing the Ph.D. degree in electrical engineering. Hisresearch interests include modeling, analysis, andcontrol of power systems with the integration ofrenewable energy sources.

Brian Johnson (S’08, M’13) obtained the M.S.and Ph.D. degrees in Electrical and Computer En-gineering from the University of Illinois at Urbana-Champaign, Urbana, in 2010 and 2013, respectively.He is the Washington Research Foundation Innova-tion Assistant Professor within the Department ofElectrical and Computer Engineering at the Univer-sity of Washington. Prior to joining the Universityof Washington in 2018, he was an engineer with theNational Renewable Energy Laboratory in Golden,CO. He was awarded a National Science Foundation

Graduate Research Fellowship in 2010, and currently serves as an AssociateEditor for the IEEE Transactions on Energy Conversion. His research interestsare in renewable energy systems, power electronics, and control systems.

Miguel Rodriguez (S’06, M’11) was born in Gijon,Spain, in 1982. He received the M.S. and Ph.D.degrees in telecommunication engineering from theUniversity of Oviedo, Oviedo, Spain, in 2006 and2011, respectively. From 2011 to 2013, he was aPost-Doctoral Research Associate with the ColoradoPower Electronics Center, University of Coloradoat Boulder, Boulder, CO, USA. In 2013 he joinedAdvanced Micro Devices, Inc., in Fort Collins, CO,USA, as a member of the Low Power AdvancedDevelopment Group. His current research interests

include dc/dc conversion and digital control, analysis of power deliverynetworks, and linear and switched integrated voltage regulators for micro-processors and SoCs.

Saber Jafarpour (M’16) is a Postdoctoral re-searcher with the Center for Control, DynamicalSystems and Computation at the University of Cali-fornia, Santa Barbara. He received his Ph.D. in 2016from the Department of Mathematics and Statisticsat Queen’s University. His research interests focuson network systems and synchronization of coupledoscillators with application to power grids.

Francesco Bullo (S’95, M’99, SM’03, F’10) is aProfessor with the Mechanical Engineering Depart-ment and the Center for Control, Dynamical Systemsand Computation at the University of California,Santa Barbara. He was previously associated withthe University of Padova (Laurea degree in Elec-trical Engineering, 1994), the California Institute ofTechnology (Ph.D. degree in Control and DynamicalSystems, 1999), and the University of Illinois. Hisresearch interests focus on network systems anddistributed control with application to robotic coor-

dination, power grids and social networks. He is the coauthor of “GeometricControl of Mechanical Systems” (Springer, 2004) and “Distributed Controlof Robotic Networks” (Princeton, 2009); his “Lectures on Network Systems”(CreateSpace, 2018) is available on his website. He is a Fellow of IEEE andIFAC. He has served on the editorial boards of IEEE, SIAM, and ESAIMjournals, and serves as 2018 IEEE CSS President.

Sairaj V. Dhople (M’13) received the B.S., M.S.,and Ph.D. degrees in electrical engineering from theUniversity of Illinois at Urbana-Champaign, Urbana,IL, USA, in 2007, 2009, and 2012, respectively.He is currently an Associate Professor with theDepartment of Electrical and Computer Engineering,University of Minnesota, Minneapolis, MN, USA,where he is with the Power and Energy Systemsresearch group. His research interests include mod-eling, analysis, and control of power electronics andpower systems with a focus on renewable integra-

tion. Dr. Dhople received the National Science Foundation CAREER Awardin 2015. He is an Associate Editor for the IEEE Transactions on EnergyConversion and the IEEE Transactions on Power Systems.

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