1254 IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 29, NO. 3, MARCH 2014

Impedance Modeling and Analysis ofGrid-Connected Voltage-Source Converters

Mauricio Cespedes, Student Member, IEEE, and Jian Sun, Senior Member, IEEE

Abstract—This paper presents small-signal impedance modelingof grid-connected three-phase converters for wind and solar systemstability analysis. In the proposed approach, a converter is modeledby a positive-sequence and a negative-sequence impedance directlyin the phase domain. It is further demonstrated that the two se-quence subsystems are decoupled under most conditions and canbe studied independently from each other. The proposed modelsare verified by experimental measurements and their applicationsare demonstrated in a system testbed.

Index Terms—Converter stability, grid-connected converters,harmonic resonance, impedance modeling.

I. INTRODUCTION

THREE-PHASE voltage-source converters (VSCs) are thebasic building blocks for many applications in power sys-

tems, including grid integration of renewable energy [1] andenergy storage [2], high-voltage dc transmission [3], as wellas flexible ac transmission systems [4]. They are commonlyreferred to as grid-connected VSC in this paper. As for otherpower electronic circuits, external behavior of such VSC canbe characterized by the impedances measured at the dc and theac terminals. Depending on the direction of power flow, the acterminal impedance can be considered the input impedance (inrectification mode) or the output impedance (in inversion mode),and will be simply referred to as the impedance in this study.

One important use of the impedance of a grid-connected VSCis in the analysis of stability and resonance between the converterand the grid, including that with the filter of the converter [5]. Inparticular, it was shown in [6] that a grid-connected VSC usedfor grid integration of renewable energy can be modeled as acurrent source in parallel with an impedance, and the inverter-grid system stability can be determined by applying the Nyquiststability criterion [7] to the ratio between the grid impedanceand the VSC impedance.

Most grid-connected VSCs use current control in a rotat-ing (dq) reference frame [8], which is synchronized to thefundamental component of the grid voltages by means of aphase-locked loop (PLL) [9]. Both the dq-domain current con-

Manuscript received October 23, 2012; revised March 8, 2013; acceptedApril 10, 2013. Date of current version September 18, 2013. This work wassupported in part by GE Global Research Center and in part by the NationalScience Foundation under Award #1002265. Recommended for publication byAssociate Editor M. Liserre.

The authors are with the Department of Electrical, Computer, and SystemsEngineering, Rensselaer Polytechnic Institute, Troy, NY 12180 USA (e-mail:cespem@rpi.edu; jsun@rpi.edu).

Color versions of one or more of the figures in this paper are available onlineat http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TPEL.2013.2262473

trol and the PLL-based grid synchronization introduce nonlin-earities which cannot be removed by reduced-order modelingtechniques [10]. One method to deal with the control nonlinear-ities is to transform the converter model into the dq referenceframe [11]. This method, however, has several limitations anddisadvantages, as discussed in [12]. The harmonic linearizationmethod [13] overcomes these limitations by modeling three-phase VSC impedance directly in the phase domain.

This paper applies the harmonic linearization techniqueto develop impedance models of three-phase VSCs withPLL-based grid synchronization. A key step in the develop-ment of the impedance models is the linearization of the grid-synchronization scheme. Since there exist several synchroniza-tion schemes [14], the approach taken here is to consider a basicPLL, and show how it can be incorporated into the impedancemodels. Possible variations are reviewed to highlight their mod-eling approach. The rest of this paper is organized as follows:Section II develops impedance models assuming perfect knowl-edge of the grid voltage angle. Section III shows how to modelthe PLL, and the approach to incorporate it into the impedancemodels. Section IV includes verifications of the proposedimpedance models from both impedance measurements andtheir application in analysis of harmonic resonance. Section Vconcludes this paper.

II. IMPEDANCE MODELING WITHOUT PLL

The three-phase VSC considered in this paper is depictedin Fig. 1. Phase voltages are denoted as va , vb , and vc , whilephase currents as ia , ib , and ic . Considering the large dc buscapacitors, and the lower than fundamental frequency controlbandwidth of the dc bus voltage, Vdc is assumed constant in thisstudy. For the same reason, the active and reactive parts of thecurrent references (Idr and Iqr) are assumed constant. In thetime domain, the phase voltage with a small-signal perturbationcan be written as

va (t) = V1 cos (2πf1t) + Vp cos (2πfpt + φvp)+Vn cos (2πfnt + φvn)

(1)

where V1 corresponds to the magnitude of the fundamental volt-age at frequency f1 , Vp with φvp correspond to the magnitudeand phase of the positive-sequence perturbation at frequencyfp , and Vn with φvn correspond to the magnitude and phase ofthe negative-sequence perturbation at frequency fn . Other phasevoltages can be inferred from (1). In the frequency domain, (1)can be written as follows:

Va [f ] =

⎧⎨

⎩

V1 , f = ±f1

Vp , f = ±fp

Vn , f = ±fn

(2)

0885-8993 © 2013 IEEE

CESPEDES AND SUN: IMPEDANCE MODELING AND ANALYSIS OF GRID-CONNECTED VOLTAGE-SOURCE CONVERTERS 1255

Fig. 1. Block diagram of three-phase VSC for grid-connected applications.

where Vp = (Vp/2)e±jφvp and others follow the same notation.The current response to the voltage perturbation can be foundfrom the converter averaged model

Ld

dt

⎡

⎢⎣

ia

ib

ic

⎤

⎥⎦ =

⎡

⎢⎣

ma

mb

mc

⎤

⎥⎦Km Vdc −

⎡

⎢⎣

va

vb

vc

⎤

⎥⎦ (3)

where ma,mb , and mc are the modulating (reference) signals forthe pulse width modulation (PWM), and Km is the modulatorgain. The relationship between duty ratios and the modulatingsignal is taken as follows:

da1 = Km ma + 1/2 (4)

da2 = 1 − da1 (5)

where da1 and da2 are the duty ratios of Sa1 and Sa2 , respectively.Other phases follow the same convention.

In order to solve (3) for impedance in the frequency domain,the sequence components in the modulating signals should befound as functions of the voltage and current perturbations.Then, positive-sequence impedance is defined as the ratio ofVp to −Ip , and negative-sequence impedance is defined as theratio of Vn to −In . Coupling should also be examined. Bothphase- and dq-domain current control strategies will be consid-ered here, which use Park’s transformation defined as follows:

T (θ) =23

⎡

⎢⎣

cos θ cos (θ − 2π/3) cos (θ + 2π/3)

− sin θ − sin (θ − 2π/3) − sin (θ + 2π/3)

1/2 1/2 1/2

⎤

⎥⎦.

(6)An inductive output filter is assumed in Fig. 1. Additional

filter elements (such as in the case when an LCL is used) can behandled by simply modifying (3).

–

abc

dq–

–

ca

cb

cc

Hi(s)

Hi(s)

Hi(s)

ib ic PLL

ibr

ia

icr

iar

Iqr

Idr

Fig. 2. Block diagram of a phase-domain current controller.

A. Phase-Domain Current Control

Fig. 2 depicts a phase-domain current controller. To findthe frequency-domain response of the controller to the har-monic perturbation, first neglect the PLL dynamics, such thatθPLL(t) = θ1(t) ≡ 2πf1t. Hence the reference currents iar , ibr ,and icr are not affected by the perturbation. As a result, the se-quence components in the modulating signals may be found asfollows:

Ma [f ]=

{−Hi (s) Gi (s) Ip + Kf (s) Gv (s)Vp , f = ±fp

−Hi (s) Gi (s) In + Kf (s) Gv (s)Vn , f =±fn

(7)where Hi(s) is a current control compensator, Kf (s) is a feed-forward gain, and

Gi (s) = e−sTi1 − e−sTi

sTi

11 + s/ωi

(8)

models the current sampling delay, with Ti representing thesampling interval and ωi its ADC prefilter cutoff frequency.Similarly

Gv (s) = e−sTv1 − e−sTv

sTv

11 + s/ωv

11 + s/ωtv

(9)

models the voltage sampling delay, with Tv representing thesampling interval, ωv its ADC prefilter cutoff frequency, and ωtvits transducer delay. Since Vp does not result in any negative-sequence response in Ma , and Vn does not result in any positive-sequence response either, sequence components are decoupledfrom each other. Introducing (7) in the frequency-domain ver-sion of (3), impedance models can be found as follows:

Zp (s) = Zn (s) =Km VdcHi (s) Gi (s) + sL

1 − Km VdcKf (s) Gv (s)(10)

where Zp(s) and Zn (s) denote positive-sequence and negative-sequence impedances, respectively.

B. Dq-Domain Current Control

Fig. 3 depicts a dq-domain current controller. Recall thatcurrents id and iq are outputs of a dq-domain transformation,which in the frequency domain involves a convolution of thefrequency components in the phase currents, with the frequencycomponents in Park’s transformation. Taking θPLL(t) = θ1(t),the frequency components in Park’s transformation are easy to

1256 IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 29, NO. 3, MARCH 2014

Fig. 3. Block diagram of a dq-domain current controller.

TABLE IFREQUENCY COMPONENTS IN dq-DOMAIN CURRENT CONTROLLER OUTPUT

NEGLECTING PLL DYNAMICS

derive, and the result of the convolution is as follows:

Id [f ] =

⎧⎪⎨

⎪⎩

I1 cos φi1 , dc

Gi (s ± j2πf1) Ip , f = ± (fp − f1)

Gi (s ∓ j2πf1) In , f = ± (fn + f1)

(11)

Iq [f ] =

⎧⎪⎨

⎪⎩

I1 sin φi1 , dc

∓jGi (s ± j2πf1) Ip , f = ± (fp − f1)

±jGi (s ∓ j2πf1) In , f = ± (fn + f1)

(12)

where I1 and φi1 correspond to the amplitude and phase of thefundamental current. Sampling at the fundamental frequencyis neglected since Gi(±j2πf1) ≈ 1. From the control blockdiagram, Cd and Cq can be obtained as linear combinationsof (11) and (12) using Hi(s) and the decoupling gain Kd . Aconvolution of the frequency components in Cd and Cq withthe frequency components in the inverse Park’s transformationgives Ca , Cb , and Cc . Table I shows the possible combinationsto consider in the convolution. Note that Vp does not result inany negative-sequence response at fp , and Vn does not resultin any positive-sequence response at fn , which means thereis no impedance coupling. The nonlinear coupling at ±(fp −2f1) and ±(fn + 2f1) is neglected for impedance modelingin the phase domain. Combining the controller output with thevoltage feedforward yields the modulating signals to introducein the frequency-domain version of (3), which can be solved forsequence impedances as follows:

Zp (s) =Km Vdc [Hi (s − j2πf1) − jKd ] Gi (s) + sL

1 − Km VdcKf (s) Gv (s)(13)

Zn (s) =Km Vdc [Hi (s + j2πf1) + jKd ] Gi (s) + sL

1 − Km VdcKf (s) Gv (s). (14)

Fig. 4. Block diagram of a basic PLL.

III. IMPEDANCE MODELING WITH PLL

A. Small-Signal Modeling of the PLL

Fig. 4 depicts a basic PLL, where HPLL(s) is the loop com-pensator. The first step to develop a small-signal model for thisPLL is to model the response of vq (t) to the voltage perturbationdescribed by (1). In order to deal with the nonlinearity in Park’stransformation, we break the transformation into two parts asfollows:

T (θPLL (t))=

⎡

⎢⎣

cos (Δθ (t)) sin (Δθ (t)) 0

− sin (Δθ (t)) cos (Δθ (t)) 0

0 0 1

⎤

⎥⎦T (θ1 (t))

(15)where Δθ(t) = θPLL(t) − θ1(t). Let vdv(t) and vqv(t) be de-fined, respectively, as the d and q outputs of applying T(θ1(t))to (1), which in the frequency domain are easily found to be

Vdv [f ] =

⎧⎪⎨

⎪⎩

V1 , dc

Gv (s ± j2πf1)Vp , f = ±(fp − f1)

Gv (s ∓ j2πf1)Vn , f = ± (fn + f1)

(16)

Vqv [f ] =

{∓jGv (s ± j2πf1)Vp , f = ± (fp − f1)

±jGv (s ∓ j2πf1)Vn , f = ± (fn + f1) .(17)

For simplicity, we linearize the rotation matrix in (15) aroundthe operating point Δθ0 = 0, which is possible for balanced,nondistorted voltage conditions [9]. Then, vq (t) is given by

vq (t) ≈ −Δθ (t) vdv (t) + vqv (t) . (18)

By the harmonic linearization principle, we can remove termsproportional to second and higher-orders of the perturbation.Hence, from (16) and the fact that Δθ0 = 0, we should onlyconsider terms in Δθ(t) proportional to the first order of theperturbation. Let

Δθ [f ] ={

Gp (s) Gv (s ± j2πf1)Vp , f = ± (fp − f1)

Gn (s) Gv (s ∓ j2πf1)Vn , f = ± (fn + f1)(19)

where Gp(s) and Gn (s) are two transfer functions that need tobe determined. Then, the result of (18) is as follows:

Vq [f ] ={

[−Gp (s) V1 ∓ j] Gv (s ± j2πf1)Vp , f = ± (fp − f1)

[−Gn (s) V1 ± j] Gv (s ∓ j2πf1)Vn , f = ± (fn + f1)

(20)

where terms proportional to second or higher order of the per-turbations have been removed. Note that Δθ = HPLL(s)Vq

except for f = ±f1 ; then, from (20), we can solve for Gp(s)

CESPEDES AND SUN: IMPEDANCE MODELING AND ANALYSIS OF GRID-CONNECTED VOLTAGE-SOURCE CONVERTERS 1257

Fig. 5. Frequency response of cosine of PLL angle to perturbation in phasea voltage. Solid lines: model prediction; Dashed-dotted lines: experimentalmeasurements.

and Gn (s) as follows:

Gp (s) = [∓jHPLL (s)] / [1 + V1HPLL (s)] (21)

Gn (s) = [±jHPLL (s)] / [1 + V1HPLL (s)] . (22)

The final step is to obtain the response of cos(θPLL(t)) fromcos(Δθ(t) + θ1(t)). The Laplace transform of cos(θPLL(t)) fora positive-sequence perturbation is obtained as follows:

L {cos (θPLL (t))} =12

HPLL (s − j2πf1) Gv (s)1 + V1HPLL (s − j2πf1)

Vp (s) .

(23)For a negative-sequence perturbation

L {cos (θPLL (t))} =12

HPLL (s + j2πf1) Gv (s)1 + V1HPLL (s + j2πf1)

Vn (s) .

(24)The proposed PLL model has been verified by experi-

ments. A sweep of the response in cos(θPLL(t)) to a super-imposed perturbation in va is depicted in Fig. 5 for a 100-HzPLL bandwidth. The parameters of the experimental setup areV1 = 120

√2 V, f1 = 60 Hz, Tv = 50 μs, ωv = 2π·5 krad/s,

ωtv = 2π·5 krad/s, and HPLL(s) = (Kp+Ki /s)/s, where Kp =2.62 and Ki = 1650. For future reference, define the closed-loopgain of the PLL as

TPLL (s) = V1HPLL (s) / [1 + V1HPLL (s)] . (25)

B. Phase-Domain Current Control and PLL

Due to the PLL, the current references contain a response tothe perturbation as follows:

Iar [f ]=

⎧⎪⎨

⎪⎩

I1 , f = ±f1

[TPLL (s ∓ j2πf1) /V1 ] I1Gv (s)Vp , f = ±fp

[TPLL (s ± j2πf1) /V1 ] I∗1Gv (s)Vn , f = ±fn(26)

where I∗1 is the complex conjugate of I1 = (1/2) (Idr ± j Iqr).

Note that it is assumed that the actual converter current isequal to its reference at the fundamental frequency, such thatI1 ≡ (I1/2)e±jφ i 1 . The current regulator acts on the current ref-erence and feedback to generate Ca . Including the feedforwardpath and introducing the result in the frequency-domain versionof (3), impedance models can be found as follows:

Zp (s) = [Km VdcHi (s) Gi (s) + sL] ·{

1 − Km VdcKf (s) Gv (s)

−[

Hi (s)I1

2ejφ i 1

]

TPLL(s − j2πf1) Gv (s)Km Vdc

V1

}−1

(27)

Zn (s) = [Km VdcHi (s) Gi (s) + sL] ·{

1 − Km VdcKf (s) Gv (s)

−[

Hi(s)I1

2e−jφ i 1

]

TPLL(s + j2πf1) Gv (s)Km Vdc

V1

}−1

.

(28)

C. Dq-Domain Current Control and PLL

Due to the PLL, the current feedback after convolutionwith Park’s transformation includes frequency components pro-portional to the voltage perturbation. Neglecting second-orderterms, the convolution of phase currents with Park’s transfor-mation gives

Id[±(fp − f1)]=I1 sinφi1Gp (±j2π (fp − f1))

× Gv (±j2πfp)Vp + Gi (±j2πfp) Ip (29)

Id[±(fn + f1)]=I1 sinφi1Gn (±j2π (fn + f1))

× Gv (±j2πfn )Vn + Gi (±j2πfn ) In (30)

Iq[±(fp − f1)]=−I1 cos φi1Gp (±j2π (fp − f1))

× Gv (±j2πfp)Vp ∓ jGi (±j2πfp) Ip (31)

Iq[±(fn + f1)]=−I1 cos φi1Gn (±j2π (fn + f1))

× Gv (±j2πfn )Vn ± jGi(±j2πfn ) In . (32)

The current regulator acts on the feedback currents to gener-ate the dq-domain modulating signals. These signals are convo-luted with inverse Park’s transformation to generate their phase-domain counterparts. Table II lists the resulting frequency termsproportional to the first order of the perturbation, where non-linear coupling should be neglected. Including the feedforwardpath and introducing the result in the frequency-domain versionof (3), impedance models can be found as follows:

Zp (s) = {Km Vdc [Hi (s − j2πf1) − jKd ] Gi (s) + sL}

·{

1 − Km VdcKf (s) Gv (s) −[C1

2ejφc 1

+ Hi (s − j2πf1)I1

2ejφ i 1 −jKd

I1

2ejφ i 1

]

× TPLL (s − j2πf1) Gv (s)Km Vdc

V1

}−1

(33)

1258 IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 29, NO. 3, MARCH 2014

TABLE IIFREQUENCY COMPONENTS IN dq-DOMAIN CURRENT CONTROLLER OUTPUT

INCLUDING PLL DYNAMICS

Fig. 6. Block diagram of the SOGI-FLL.

Zn (s) = {Km Vdc [Hi (s + j2πf1) + jKd ] Gi (s) + sL}

·{

1 − Km VdcKf (s) Gv (s) −[C1

2e−jφc 1

+ Hi (s + j2πf1)I1

2e−jφ i 1 +jKd

I1

2e−jφ i 1

]

× TPLL (s + j2πf1) Gv (s)Km Vdc

V1

}−1

(34)

where C 1 = Cd/2 ± jCq/2 = (C1/2)e±jφc 1 , taking Cd as thedc component of Cd [f ], and Cq as the dc component of Cq [f ].These dc components can be computed from

C1 = {j2πf1L + V1 [1 − Km VdcKf (j2πf1)]} / (Km Vdc) .(35)

D. Other Grid Synchronization Methods

Some advanced PLL structures, such as the decoupled double-synchronous PLL [15], use the same building block of Fig. 4 inmultiple stages, such that the same modeling method is applica-ble to them. Other forms of grid-synchronization, such as thosebased on the second-order generalized integrator frequency-locked loop (SOGI-FLL) [16], need slightly different treatment,as outlined below.

Fig. 6 depicts the basic building block of the SOGI-FLL. Inthree-phase systems, two filters can be used in the αβ-referenceframe to extract sequence components. The basic functionalityof the filter is to extract a sinusoidal component in phase withvα in x1 , and a quadrature component in x2 that lags x1 by 90◦.Applying a superimposed perturbation in vα , the frequency-

Fig. 7. Magnitude plot of the frequency response of the SOGI-FLL. Solidlines: model prediction; Dots: numerical simulation results.

TABLE IIICONVERTER CIRCUIT PARAMETERS

domain response in the synchronization signals can be found asfollows:

X1 (s)Vα (s)

=kω1 +

(γV 2

1 s)/(s2 − ω2

1)

s + ω21/s + kω1 + (γV 2

1 s) / (s2 − ω21 )

(36)

X2 (s)Vα (s)

= −(γV 2

1 s)/ (2ω1)

s2 − ω21

+kω1 +

(γV 2

1 s)/(s2 − ω2

1)

s + ω21/s + kω1 + (γV 2

1 s) / (s2 − ω21 )

×[ω1

s+

γV 21 s

2ω1 (s2 − ω21 )

]

. (37)

The harmonic linearization principle has been used to removefrequency components in x1 , x2 , and x3 that are proportional tosecond or higher-order powers of the perturbation. The modelsin (36) and (37) are compared against point-by-point numeri-cal simulation of the filter frequency response in Fig. 7 (onlymagnitude plots are shown due to space limitations). The pa-rameters used in simulations are as follows: V1 = 120

√2 V,

ω1 = 2π · 60 rad/s, k = 1, and γ = 0.1. For the more com-plex three-phase application, positive- and negative-sequenceperturbations should be considered separately.

IV. IMPEDANCE MODEL VERIFICATIONS

A three-phase converter has been built and tested to verifythe proposed impedance models. The current controller wasimplemented in a DE-2 FPGA from Altera, while the cur-rent references were generated from a PLL implemented in aTMS320F28335 DSP from TI. Parameters for this experimentalsetup are provided in Table III. The converter operating point isdescribed by Vdc = 550 V, V1 = 120

√2 V, I1 = 10 A, φi 1 =

0 rad, f1 = 60 Hz, and Km = 1/2.The converter is connected to a grid as depicted in Fig. 8.

The grid impedance at the converter terminals is the same in the

CESPEDES AND SUN: IMPEDANCE MODELING AND ANALYSIS OF GRID-CONNECTED VOLTAGE-SOURCE CONVERTERS 1259

Util

ity G

rid

VSC

Vdc

LRd

Cf

vb

vc

va

ib

ic

ia

Lg

Fig. 8. Block diagram of converter connection to the grid.

Fig. 9. Impedance response with phase-domain current control. Solid lines:positive-sequence; Dashed lines: negative sequence; Dots represent frequencyresponse analyzer measurements.

positive- and the negative-sequence domain

Zg (s) ={(

sLg

)−1 +[Rd + 1/

(sCf

)]−1}−1(38)

where Lg is the grid inductance, and Rd with Cf constitute adamped filter. The grid parameters used in the experiments areLg = 3.75 mH, Rd = 1.87 Ω, and Cf = 22 μF.

A. Phase-Domain Current Control

The additional time delays due to the sampling of the currentreferences from the DSP are neglected. In this case, the currentcompensator transfer function is

Hi (s) = Kp +Kis

s2 + (2πf1)2 (39)

where Kp = 0.118 and Ki = 776. The PLL compensator gainis

HPLL (s) = (Kp + Ki/s) /s (40)

where Kp = 0.262 and Ki = 16.5. The feedforward gain Kf isset to zero.

Measurements of sequence impedances are depicted in Fig. 9.At harmonic frequencies, the impedance responses resemble se-ries CRL circuits due to the integral gain of the current controller,its proportional gain, and the converter inductance, respectively.

TABLE IVSAMPLING DELAYS WITH CURRENT CONTROL IN DSP

Fig. 10. Impedance response with dq-domain current control and feedforwarddecoupling. Solid lines: positive-sequence; Dashed lines: negative sequence;Dots represent frequency response analyzer measurements.

B. Dq-Domain Current Control

Due to the complexity of a dq-domain current controller, thecurrent controller has been moved to the DSP and only thePWM stage is left in the FPGA. Since the PWM stage in thiscase samples ma,mb , and mc , additional time delays should beincluded. In order to include them, both sampling terms Gi(s)and Gv (s) should be multiplied by

Gm (s) = e−sTm1 − e−sTm

sTm· 11 + s/ωm

(41)

where Tm is the sampling interval of the PWM and ωm isits associated ADC prefilter cutoff frequency. In this case, theparameters of the circuit stage are those listed in Table IV.

For the dq-domain current controller, the compensator is se-lected as follows:

Hi (s) = Kp +Ki

s(42)

where Kp = 0.04578 and Ki = 43.15. The decoupling gain is setto Kd = 0.00411. The PLL design is the same as in Section IV-A.The controller outputs are Cd = 0 and Cq = 0.0411 for afeedforward gain given by Kf = (Km Vdc)−1 .

Measurements of sequence impedances are depicted inFig. 10. The feedforward decoupling has been used to increasethe magnitude of the inverter impedance, but it also reducesdamping, which may lead to resonance in weak grids.

C. Effects of Unbalance

An unbalanced condition in the phase voltages causes a sec-ond harmonic oscillation of the dq-domain variables used in

1260 IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 29, NO. 3, MARCH 2014

Fig. 11. Magnitude response in q-axis voltage for a positive-sequence per-turbation in presence of unbalance: (a) response independent of unbalance; (b)response due to unbalance. Solid lines: model prediction; Dashed-dotted lines:numerical simulation.

Fig. 12. Magnitude plot of the inverter admittance swept by numerical simu-lation when grid voltage has 5% negative-sequence voltage at 60 Hz. Solid lines:balanced model prediction; Dashed-dotted lines: numerical simulation results.

the PLL for grid synchronization. The second harmonic com-ponent in Δθ(t) can lead to coupling of sequence impedances.Consider, for example, a positive-sequence perturbation of thePLL, while a small negative-sequence voltage V2 is also im-pressed on the phase voltages at the fundamental frequency.The voltage vq (t) in this case responds at two different frequen-cies ±(fp−f1) and ±(fp+f1). The former is the characteristicPLL response to the perturbation, while the latter is due to V2 .These responses can be found as follows:∣∣∣∣Vq [fp − f1 ]

Vp [fp ]

∣∣∣∣ =

∣∣∣∣

−j

1 + V1HPLL (j2π (fp − f1))

∣∣∣∣

≈ 1 ∀ (fp f1) (43)∣∣∣∣Vq [fp + f1 ]

Vp [fp ]

∣∣∣∣ =

∣∣∣∣− j [TPLL (j2πf1) − TPLL (j2π (fp − f1))]

V2 [f1 ]V1

∣∣∣∣ ≈

∣∣∣∣V2 [f1 ]

V1

∣∣∣∣∀ (fp f1) . (44)

Note that (44), responsible for any coupling of sequence com-ponents, is directly proportional to the per-unit voltage unbal-ance. Thus, at harmonic frequencies, a 10% voltage unbalanceshould result in a coupling term at least 20 dB below the char-acteristic PLL response. It is possible to verify (43) and (44) insimulations, as depicted in Fig. 11 for a 100-Hz design.

To illustrate the coupling in the sequence impedances dur-ing unbalance, a switching-circuit simulation model in Saber isused to sweep the inverter admittance, while a small grid voltageunbalance is imposed at 60 Hz. The converter power stage andcurrent control use the same parameters from the experimentalsetup with dq-domain current control, but the feedforward andtime delays are removed. The PLL bandwidth is set to 100 Hz.

Fig. 13. Harmonic resonance of the converter-grid system described in Fig. 10:(a) Phase current waveforms; (b) sequence components.

The unbalanced voltage has 5% negative-sequence. The resultsfrom sweeping the inverter admittances are depicted in Fig. 12(only magnitude plots are shown), where the following defini-tions are used:

[−Ip (s)

−In (s)

]

=[

Ypp (s) Ypn (s)

Ynp (s) Ynn (s)

] [Vp (s)

Vn (s)

]

. (45)

The off-diagonal admittances are significantly smaller above100 Hz, which verifies the fact that coupling terms can be ne-glected for conventional PLL designs not exceeding a few tensof hertz.

D. Applications of the Models

One application of the proposed impedance models is in theanalysis and mitigation of harmonic resonance problems. Be-cause of the decoupling between the two sequence subsystems,the stability criterion presented in [6] for grid-connected con-verters can be applied to each sequence impedance separatelyto determine overall converter-grid system stability. Addition-ally, the analytical impedance models also provide a basis formodification of the converter control to mitigate any harmonicresonance and other instability problems.

As an example, consider the converter-grid systems describedby Fig. 9 and Fig. 10. The time-domain waveform correspondingto Fig. 10 is shown in Fig. 13. At 440 Hz, the inverter positive-sequence impedance intersects with the grid impedance and thephase difference is almost 180◦. This explains the harmonicresonance in Fig. 13(a). Fourier analysis of the phase currentsreveals a large positive-sequence component around the seventhharmonic [see Fig. 13(b)], which correlates to the impedance in-tersection frequency. Based on the developed impedance mod-els, the harmonic resonance can be eliminated by modifying the

CESPEDES AND SUN: IMPEDANCE MODELING AND ANALYSIS OF GRID-CONNECTED VOLTAGE-SOURCE CONVERTERS 1261

Fig. 14. Phase current waveforms for the system described in Fig. 9.

inverter control to increase the phase margin at the frequencyof intersection of the two system impedances. Fig. 14 showsthe measured grid currents using the converter design corre-sponding to the impedance plots in Fig. 9, which in this caseeliminates the harmonic resonance problem. The phase-domaincurrent controller in this case presents better impedance com-patibility with the grid, because at the point of intersection ofsystem impedances, phase differences are far from 180◦.

V. CONCLUSION

Grid-connected VSC impedance models can be used to as-sess system level converter-grid compatibility and power qual-ity. Impedance modeling in the phase domain yields decoupledpositive- and negative-sequence converter impedances, whenphase- or dq-domain current control systems are implemented.As a result, the contributions in this paper enable single-inputsingle-output stability analysis of balanced three-phase con-verter systems. For the kind of nonlinearity in Park’s transfor-mation, coupling of converter sequence impedances may occurduring unbalanced phase voltage conditions. For most practicalconditions with small voltage unbalance, the coupling can beneglected.

ACKNOWLEDGMENT

The authors would like to thank Dr. Z. Jiang of GE GlobalResearch Center for his technical inputs.

REFERENCES

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[5] J. He and Y. W. Li, “Generalized closed-loop control schemes with em-bedded virtual impedances for voltage source converters with LC or LCLfilters,” IEEE Trans. Power Electron., vol. 27, no. 4, pp. 1850–1861, Apr.2012.

[6] J. Sun, “Impedance-based stability criterion for grid-connected inverters,”IEEE Trans. Power Electron., vol. 26, no. 11, pp. 3075–3078, Nov. 2011.

[7] R. D. Middlebrook, “Input filter considerations in design and applicationof switching regulators,” in Proc. Rec. IEEE Ind. Appl. Soc. Annu. Meet.,1976, pp. 366–382.

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[9] F. D. Freijedo, A. G. Yepes, O. Lopez, A. Vidal, and J. Doval-Gandoy,“Three-phase PLLs with fast postfault retracking and steady-state rejectionof voltage unbalance and harmonics by means of lead compensation,”IEEE Trans. Power Electron., vol. 26, no. 1, pp. 85–97, Jan. 2011.

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Mauricio Cespedes (S’08) received the B.S. degreefrom the University of Costa Rica in San Jose, CostaRica, and the M.S. degree from Rensselaer Polytech-nic Institute, Troy, NY, where he is currently workingtoward the Ph.D. degree, all in electrical engineering.

His research interests include modeling and con-trol of three-phase voltage-source converters, theirintegration to the electric grid, and system stabilityanalysis.

Jian Sun (M’95–SM’09) received the B.S. degreefrom the Nanjing Institute of Aeronautics, Nanjing,China, the M.S. degree from the Beijing Univer-sity of Aeronautics and Astronautics, Beijing, China,and the Dr.Eng. (Ph.D.) degree from the Universityof Paderborn, Paderborn, Germany, all in electricalengineering.

He was a Postdoctoral Fellow with the School ofElectrical and Computer Engineering, Georgia Insti-tute of Technology, from 1996 to 1997. He workedin the Advanced Technology Center of Rockwell

Collins, Inc., from 1997 to 2002, where he led research on advanced powerconversion for aerospace applications. In August 2002, he joined the RensselaerPolytechnic Institute, Troy, NY where he is currently a Professor and Directorof the New York State Center for Future Energy Systems. His research interestsare in the general area of power electronics and energy conversion, with a focuson modeling, control, as well as applications in aerospace and renewable energysystems. He has published more than 160 journal and conference papers onthese subjects, and holds nine U.S. patents.

Dr. Sun is a Senior Member of the IEEE Power Electronics Society. He cur-rently serves as the Editor-in-Chief of the IEEE POWER ELECTRONICS LETTERS

and was the Guest Editor for the IEEE TRANSACTIONS ON POWER ELECTRONICS

Special Issue on Modeling and Advanced Control published in 2009. He wasthe Chair of the IEEE Power Electronics Society’s Technical Committee onPower and Control Core Technologies until December 2012 and became theTreasurer of PELS in January 2013. He was the General Chair of IEEE COM-PEL’06 Workshop and was involved in the organization of several other PELSconferences.

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