Dynamic Phasor Modeling of SRF-PLL based Grid-Tie Inverter

under Islanded Conditions

Venkatramanan D, Student Member, IEEE, and Vinod John, Senior Member, IEEEDepartment of Electrical Engineering, Indian Institute of Science, Bangalore

Email: venkat86ram@gmail.com

Abstract—Synchronous reference frame (SRF) phase lockedloop (PLL) is a widely employed scheme for synchronizationof three-phase grid-tied inverters (GTI). The SRF-PLL modeldiscussed in the literature for analysis assumes a voltage sourcebeing present at the point of common coupling (PCC) in the formof the grid. The case of grid-disconnection and a subsequentunintentional island formation is typically not considered. Inthis work, a dynamic-phasor based complex domain analysis ofthe SRF-PLL model is proposed that is suitable for studyingthe behavior of an unintentionally islanded GTI. The modelaccurately predicts the steady-state operating frequency of thesystem after the unintentional island formation, for UPF aswell as other operating power factors of the GTI with variedload conditions. Simulation and experimental results of islandingconducted on a 4.5kVA three-phase GTI system are presentedthat validate the proposed analytical approach and the resultingsystem model.

Keywords—SRF transformation, PLL, islanding, inverter con-trol, dynamic phasor.

I. INTRODUCTION

Recently, a notable spur has occurred toward greaterutilization of clean energy resources in order to meet thegrowing energy demand while addressing concerns on cli-mate change [1]. Pulse-width modulated (PWM) grid-tiedinverters (GTIs) form the core component that facilitate grid-integration of renewable distributed energy resources (DERs).Phase locked loops (PLL) are invariably required for grid-synchronization of GTIs [2]. The circuit schematic of a typicalsolar photo-voltaic (PV) GTI interfaced with grid, along withlocal RLC loads, is shown in Fig. 1. The interconnectionnode of the DER facility is commonly referred to as pointof common coupling (PCC). In three-phase grid-integrationapplications, synchronous reference frame (SRF)-PLL is apopularly used scheme for grid-synchronization owing to itssimple architecture and superior performance [3]. The SRF-PLL structure traditionally discussed in literature, which istypically expressed in dq-domain as indicated in Fig. 1(b),assumes a voltage source to be always present at the PCC inthe form of grid [2], [3].

The scenario of an islanded system following an uninten-tional grid-disconnection, that continues to operate with thelocal RLC loads present at PCC in current-control mode, aretypically not considered in SRF-PLL modeling and analysis.However, understanding of islanding behavior of SRF-PLLbased GTI system along with local loads is crucial for detectingunintentional island formation and for subsequently ceasingDER power injection into the grid [4]. Typically, anti-islanding

This work was supported by CPRI, Ministry of Power, Government of India,under the project Power conversion, control, and protection technologies formicrogrid.

algorithms use PCC voltage magnitude and frequency or rateof change of phase as target variables for islanding detec-tion [2].

(a)

(b)

(c)

Fig. 1: Grid-integration scheme of DER system and DPbased proposed equivalent circuit model showing (a) powercircuit schematic of a three-phase PV GTI along with digitalcontroller, and (b) DP circuit model in grid connected mode,and (c) DP circuit model in unintentional islanded mode.

Since phasors inherently carry information of magnitudeand phase of a dynamic ac quantity, phasor-domain approachshows promise for analysis of islanded GTI system. Phasordomain analysis using dynamic-phasors (DP) have been em-ployed in past literature in various applications. In [5], DPmodel is used for analyzing unbalance caused by faults inelectric machines. In [6], dynamic-phasor models are usedto enhance computational efficiency in large power systemsimulations. In [7], DP domain analysis is used to analyze

978-1-4799-7312-5/18/$31.00 ©2018 IEEE 5670

inverters, while [8], [9] employ an energy based approachto develop DP models of circuits with which stability ofac microgrids are analyzed in a multi-converter environment.In [10], instead of energy approach, DP models for passive net-works are developed using conventional KVL-KCL equationsin phasor domain, for analyzing inverters.

In this work, a dynamic-phasor (DP) based approach isadopted for modeling and analysis of SRF-PLL and local RLCloads in order to study a three-phase GTI behavior under anunintentional islanding condition, where it continues to operatein constant current-control mode without transitioning into agrid-forming mode of operation. A dynamic-phasor model isproposed that incorporates the dynamics of local RLC alongwith the SRF-PLL structure used in the GTI, where systemfrequency is treated as a state-variable. The model capturesthe effect of real and reactive current commands from the GTI,and of RLC load parameters and yields the steady-state systemfrequency accurately after island formation. It is shown that thethree-phase GTI system has multiple stable operating frequen-cies after island formation. Firstly, dynamic-phasor concept isintroduced in Section II. The proposed dynamic-phasor circuitmodel of the islanded GTI is introduced in Section III, wheretraditional dq-domain based SRF-PLL model and the proposedDP based PLL model are subsequently described. DP modelingof GTI system is performed in Section IV from which a closed-form expression for islanded system frequency is obtained.Experimental results on a 4.5kVA three-phase GTI setup arepresented in Section V that verify the proposed analyticalmodel of the system.

II. DYNAMIC PHASORS

Consider a real valued sinusoidal signal x(t) whose fre-quency is ω and period T at steady state. The phasor repre-sentation of this signal, which is a complex number, is givenby,

x(t) = Acos(ωt+Φ) (1)

X = AejΦ = A∠Φ (2)

When x(t) in (1) has not reached steady-state but is still time-evolving, amplitude and phase will not be steady. The signalcan then be represented as,

x(t) = a(t)cos[ωt+ φ(t)] ⇒ �[a(t)ej(ω+φ(t))]

The corresponding complex representation, which is a phasorand is dynamic, is given by,

x = a(t)ej(ωt+φ(t)) (3)

Alternatively, a rotating reference frame transformation usinge−jωt can be performed, that yields,

⇒ x = a(t)ejφ(t) = aejφ (4)

Equation (3)-(4) give the definition of DP in polar coordinates,where the time-varying magnitude a and phase φ form thedynamic phasor quantities, and the DP model of a systemcomprises of non-linear differential equations describing thesequantities [7]–[9]. This notation is henceforth employed todescribe electric circuit components [10]. Consider a genericimpedance Z carrying a current iz and voltage vz of frequencyω as shown in Fig. 2(a), whose respective phases are δz

Fig. 2: Dynamic phasor diagram of an impedance Z.

(a)

(b)

(c)

Fig. 3: SRF-PLL control structure in grid-connected modeshowing (a) traditional d-q domain model, (b) correspondingphasor alignments, and (c) modified phasor domain model.

and φz with respect to a common reference, as indicated inFig. 2(b). The current and voltage dynamic phasors beforerotating reference frame transformation can be represented as(5) and (6). This framework can be employed to formulatedynamic-phasor equations for the system.

vZ = vZej(ωt+φZ) (5)

iZ = iZej(ωt+δZ) (6)

III. DYNAMIC PHASOR MODEL OF GTI INTERFACE

For the grid interface circuit indicated in Fig. 1(a), a DPbased equivalent circuit in grid-connected mode is shown inFig. 1(b), where the grid or source voltage DPs are representedby [vs, θs]. The injected GTI current DPs and PCC voltage DPsare represented by [iinv, δinv] and [vg, φg] respectively. In grid-connected mode, the PCC voltage is governed predominantlyby the grid, assuming that the source impedance is negligible.In islanded mode, the PCC voltage will be governed solely bythe GTI current and load characteristics. The correspondingequivalent circuit gets modified to Fig. 1(c), where the systemdynamics are governed by that of PLL, RLC load and theinverter.

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Fig. 4: Proposed dynamic-phasor model of SRF-PLL in islanded mode.

A. SRF-PLL model

Fig. 3(a) shows the typical SRF-PLL control structure indq-domain [3], where the PCC voltage is aligned with the q-axis of GTI control, as indicated by the phasor diagram inFig. 3(b). Hence, q-axis and d-axis correspond to real andreactive power control axes of the GTI respectively. This SRF-PLL structure is modified to a phasor domain equivalent, asshown in Fig. 3(c), where the input to the PLL structure isthe PCC voltage steady-state phasor Vge

Θg . Multiplicationby e−jθe represents the SRF-transformation that provides acomplex signal with the required phase-difference information.z∗ represents a complex conjugate operation performed toobtain the corresponding real-valued signal, which is theequivalent of vd in Fig. 3(a), upon which the PI controlleracts. After an island formation, the PCC voltage and hencethe SRF-PLL input, is governed purely by dynamics of theinjected GTI current and the local load impedance, and thecontribution from grid voltage source vs is zero. Since thefrequency of the GTI current itself is variable under islandedcondition, the SRF-PLL input is represented as a dynamic-phasor. The proposed DP based SRF-PLL control structurein islanded condition is shown in Fig. 4. Gi represents thedynamics of the GTI control, which can be assumed to be unityif the GTI current-loop bandwidth is designed to be sufficientlyhigh. Thus, the dynamics of the system after islanding aredominantly governed by the dynamics of local RLC load andSRF-PLL PI control loop, that are relatively slower than thecurrent control loop. Hence, for the system DP modeling, it isadequate to consider the DP models of RLC load and SRF-PLL loop respectively.

IV. DYNAMIC PHASOR EQUATIONS

For the islanded GTI system shown in Fig. 1(c), let theGTI inject a current phasor iinv = Iinve

j(θe+δinv) into thePCC, where Iinv and δinv represent the magnitude and phaseof inverter current [10]. This is given by,

iinv = Id + jIq = Iinvej(θe+δinv) (7)

Iinv =√

I2d + I2q , δinv = tan−1

(IqId

)(8)

where, ωe be the output of the SRF-PLL’s PI controlleroutput, which is same as the system frequency after grid-disconnection, and θe = ωet. Applying KCL equation at the

PCC node, we get,

iinv = Cdvgdt

+vgR

+ iL (9)

Iinvej(θe+δinv)=C

d

dt

[vge

j(θe+φg)]

+vgRej(θe+φg) + iLe

j(θe+δL) (10)

From KVL equation for the load inductor, we get,

vg = LdiLdt

(11)

⇒ vgej(θe+φg) = L

d

dt

[iLe

j(θe+δL)]

(12)

From the SRF-PLL structure, we get,

ωe = vgcos(φg)

(kp +

kis

)(13)

⇒ dωe

dt= kp

[d

dt(vgcos(φg))

]+ ki (14)

Simplifying the complex domain differential equations (10),(12) and (14), we obtain the dynamic-phasor equations of thesystem.

dvgdt

=1

C

[Iinvcos(δinv−φg)−iLcos(δL−φg)−vg

R

](15)

dφg

dt=

1

Cvg[Iinvsin(δinv−φg)−iLsin(δL−φg)]−ωe (16)

diLdt

=1

Lvgcos(φg − δL) (17)

dδLdt

=1

LiL[vgsin(φg − δL)]− ωe (18)

dωe

dt=

kpC

[Iinvcos(δinv)− iLcos(δL)]

+ kpωevgsin(φg) +

(− kpRC

+ ki

)vgcos(φg) (19)

The simplifying steps are explained in Appendix-A. Equations(15)-(19) represent the DP model equations of the islanded GTIsystem. At steady-state, both amplitude and phase of variousquantities are constants, including the system frequency, whichmeans iL = IL, vg = Vg , δL = ΔL, φ = Φ, and ωe =We, where We represents the steady-state system frequencyin the unintentional island. In order to obtain the steady-state

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Fig. 5: Experimental setup showing (a) three-phase GTI and (b) RLC load bank.

TABLE I: GTI System ratings

Item ValueRated Power P 4.5 kW

Voltage 210 VLoad qf 1

L 112.5 mHC 90.6 μFR 35.4 Ω

ωres 312.2 rad/sLf 6 mH (6%)Vdc 800 Vfsw 10 kHz

solutions of the system, the derivatives in (15)-(19) are equatedto zero, which yields,

Φ−ΔL =π

2(20)

Vinv = WeL.IL (21)

Iinvcos(δinv − Φ) =Vinv

R(22)

Iinvsin(δinv − Φ) = WeCVinv − IL (23)

Φ =π

2(24)

Equations (20)-(24) are combined together to obtain aquadratic equation in We.

(LC)W 2e +

(L

Rcot(δinv)

)We − 1 = 0 (25)

⇒ We,1,2 =−L

Rcot(δinv)±√(

LRcot(δinv)

)2+ 4LC

2LC(26)

Equation (26) provides the closed-form expression for GTIsystem frequency after island formation, that is dependent onlocal load parameters and the ratio of q-axis and d-axis currentcommands from the GTI. Such an analytical expression forthe unintentionally islanded system for arbitrary GTI powerfactor is not available in previous literature. It can be noted thatthe DP model predicts two operating frequencies in (26) afterisland formation, and one of them is negative. This negativefrequency in DP domain indicates a second operating point ofthe three-phase islanded system with reversed phase sequence.

V. EXPERIMENTAL RESULTS

The experimental GTI system and the RLC load bank areshown in Fig. 5(a) and (b) respectively, the ratings of which areindicated in Table. I. Vector control using conventional PI con-trollers in d-q frame is employed for GTI current control [11].The controller architecture is presented in the Appendix andis digitally implemented in TI’s TMS320F28377S DSP baseddigital signal controller (DSC). The current loop bandwidth isset at 1 kHz, which is adequately high such that the injectedGTI current iinv rapidly follows the DSC current commandi∗inv = I∗d+jI∗q . Thus, when I∗d = 0, the GTI injects current intothe grid at UPF. Islanding tests are conducted in accordancewith IEEE standard 1547a-2014 [4]. The quality factor (qf )

and resonance frequency ωres of the load is defined as,

qf =

√LC

Rωres =

1√LC

(27)

A. Discussion

Fig. 6 shows the experimental behavior of GTI at theinstant of islanding for the test conditions of - GTI currentcommand I∗q = 1p.u, I∗d = 0, and qf = 1. This indicates aperfect balance between injected GTI real power, load realpower, and load reactive power, and also that the GTI isoperating at UPF. Thus the R-phase source current is<R drawnfrom grid prior to islanding is close to zero since all the loadpower is supplied by the GTI. Fig. 6(a) indicates that whenthe grid is present, the PCC voltage is slightly distorted, andafter island formation, is,R drops to zero completely, and thePCC voltage turns into a smooth sinusoid. This is becausethe injected GTI currents at PCC are almost purely sinusoidalowing to high current loop bandwidth, that lead to sinusoidalPCC voltage. It can be noted that since I∗d = 0, δinv is π

2 .

This condition in (17) yields We = 1√LC

= ωres, which is

close to 50 Hz as indicated in (18). Hence, no noticeablechange is frequency must occur after islanding. Fig. 6(b) showsthe opening of the grid-breaker Sgrid and the correspondingprofiles of R-phase source voltage vs,R, source current is,R andthe injected GTI current iinv,R. Fig. 6(c) shows the R-phasePCC voltage and its frequency trend, which indicates that thefrequency remains close to 50 Hz after island formation asexpected.

Fig. 7 shows the experimental behavior of GTI at theinstant of islanding for the test conditions of - GTI currentcommand I∗q = 1p.u, I∗d = -0.25 p.u or -3A, with qf=1 andPload = 1 p.u. This indicates a perfect balance between injectedGTI real power, load real power, and load reactive power.However, the GTI is not operating at UPF, but is injectingreactive power into the PCC with δinv = -76.5◦, whichaccording to (26), yields a frequency of 56.4Hz. Fig. 7(a)shows the measured profiles of R-phase source voltage v,Rs,source current is,R and the injected GTI current iinv,R whenSgrid opens up. As can be seen, is,R is slightly larger thanthat in Fig. 6(b) prior to islanding, due to additional reactivecurrent injection from GTI. The frequency shift of GTI currentiinv,R with respect to source voltage vs,R after islanding can be

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(a)

(b)

(c)

Fig. 6: Measured PCC voltage and frequency behavior at theinstant of islanding for GTI current command I∗q = 1 p.u, I∗d= 0 p.u and Pload = 1 p.u at qf=1, showing (a) three-phasePCC voltage, (b) R-phase inverter current, source current andsource voltage profile, and (c) corresponding frequency profile.

noticed, where the respective peaks of the two waveforms driftcontinuously after islanding. The measured frequency increaseis indicated in Fig. 7(b), which matches with that predicted by(26).

Fig. 8 shows the experimental behavior of GTI at theinstant of islanding for the test conditions of - GTI currentcommand I∗q = 1p.u, I∗d = 0, Pload = 1 p.u at qf = 0.866.This indicates that the GTI is operating at UPF with per-fect balance between injected GTI real power and load realpower. However, the load quality factor is reduced as the loadcapacitance C = 1.33 p.u, while L = 1 p.u, which causesthe source current to be non-zero and leading the sourcevoltage before islanding. Fig. 8(a) shows the measured profilesof R-phase source voltage v,Rs, source current is,R and the

(a)

(b)

Fig. 7: Measured PCC voltage and frequency behavior at theinstant of islanding for GTI current command I∗q = 1p.u, I∗d= -0.25 p.u and Pload = 1 p.u at qf=1, showing (a) R-phaseinverter current, source current and source voltage profiles, and(b) corresponding frequency profile.

injected GTI current iinv,R when Sgrid opens up. It can beseen that is,R is greater than zero prior to islanding, owing tothe mismatch between capacitive reactive power supplied andinductive reactive power consumed in the load. After islanding,since C is larger, the frequency of the system drops to 48.1 Hzafter islanding from the grid-frequency as seen in Fig. 8(b),which is in accordance with (26). This is also noticed inFig. 8(a) by observing the respective peaks of vs,R and GTIcurrent iinv,R after island formation.

Fig. 9 compares the analytical and experimentally mea-sured system frequency values after islanding for variousvalues of I∗d command from GTI. It can be noted that thereis a close match between the two, and the RMS error valuebetween them is be 2.4rad/s which is less than 1% of thenominal frequency value.

VI. CONCLUSION

A dynamic phasor model for three-phase GTI grid-interfacefunctioning in current-control mode is proposed that incor-porates the effect of SRF-PLL control and the local loaddynamics. This analytical framework is suitable for studyingthe GTI system after an unintentional island formation, theanalysis of which is presented in this paper. Using sucha framework, a set of non-linear differential equations areobtained that govern the overall system dynamics after is-landing. These equations are solved to obtain the steady-statesolutions that accurately predict the frequency at which theGTI system settles after an unintentional island formation,

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(a)

(b)

Fig. 8: Measured PCC voltage and frequency behavior at theinstant of islanding for GTI current command I∗q = 1p.u, I∗d= 0 p.u and Pload = 1 p.u at qf= 0.866, showing (a) invertercurrent, source current and source voltage profiles, and (b)corresponding frequency profile.

−4 −3 −2 −1 0 1 2 3 4250260270280290300310320330340350360370

I∗

d(A)

ωe(rad/s)

AnalyticalExperimental

Fig. 9: Comparison of analytical and measured system fre-quency values for various reactive current I∗d commands.

which has been verified experimentally for different values ofload parameters and GTI power factor. Dynamic-phasors thusprovide an alternative mathematical framework for systematicmodeling and analysis of grid-connected DER systems, whichis used used for studying scenarios such as unintentionalislanding of GTI. These results are also useful in providingsuitable settings for the GTI controller that yields rapid anti-islanding detection.

Fig. 10: Comparison of analytical and measured system fre-quency values for various reactive current I∗d commands.

APPENDIX

A. Dynamic Phasor Equation Simplification

Equation (10) reproduced below represents a complex time-domain differential equation.

Iinvej(ωet+δinv)=C

d

dt

[vge

j(ωet+φg)]

+vgRej(ωet+φg) + iLe

j(ωet+δL)

⇒ Iinvej(ωet+δinv)=C

[dvgdt

+ j

(ωe +

dφg

dt

)vg

]ej(ωet+φg)

+vgRej(ωet+φg) + iLe

j(ωet+δL) (28)

Multiplying both sides by e−j(ωet+φg), which amounts to arotating reference-frame transformation whose frequency isωe [10], we get,

Iinvej(δinv−φg) = C

[dvgdt

+ j

(ωe +

dφg

dt

)vg

]

+vgR

+ iLej(δL−φg) (29)

Comparing real and imaginary parts of (29), and rearrangingthem as two first-order differential equations yields,

dvgdt

=1

C

[Iinvcos(δinv−φg)−iLcos(δL−φg)−vg

R

](30)

dφg

dt=

1

Cvg[Iinvsin(δinv−φg)−iLsin(δL−φg)]−ωe (31)

Equations (30) and (31) are same as (15) and (16). The sameprocedure can be followed to simplify (11) as well.

B. GTI Control Architecture

The three-phase GTI switching at 10 kHz is operatedin constant-current control mode using vector control in d-qframe [11]. The power circuit is shown in Fig. 1(a), and the PIcontroller based d-q domain control architecture is shown inFig. 10. The controllers are digitally implemented in the DSCusing bilinear transformation, where the sampling frequency isset to 50μs and PWM is generated in double-update mode.

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