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Identifying DQ-Domain Admittance Models of a2.3-MVA Commercial Grid-Following Inverter Via

Frequency-Domain and Time-Domain DataLingling Fan, Senior Member, IEEE, Zhixin Miao, Senior Member, IEEE, Przemyslaw Koralewicz, Member,

IEEE, Shahil Shah, Member, IEEE, and Vahan Gevorgian, Senior Member, IEEE

Abstract—In this paper, we present two methods to identifythe admittance of a 2.3-MVA commercial grid-following inverterthrough frequency-domain data and time-domain data. In ad-dition to the well-known harmonic injection method to obtainthe admittance at frequency points followed by vector fitting, weadopt a step response-based method where the step response dataof the converter are collected and their s-domain expressionsare obtained. In turn, the s-domain admittance model of theconverter is found. The two methods can be used for cross vali-dation. Step responses and frequency-domain responses obtainedfrom the two methods are compared. Results show that the time-domain data-based method leads to an admittance comparablewith that from the frequency-domain measurement in the rangeless than 60 Hz. In addition, two insights are obtained from thefrequency-domain admittance measurements. First, when the dq-frame is aligned to the point of common coupling (PCC) voltage,the per unit DQ-domain admittance directly reflects operatingconditions. Second, existence of negative-sequence control can bedetected via sequence-domain admittances.

Index Terms—Admittance; grid-following inverter; frequencyscan; system identification

I. INTRODUCTION

MORE and more wind and solar are being integrated intopower grids. Unlike conventional generators, which

rely on a rotating magnetic field to convert mechanical energyto electric energy and which can be directly connected to an acgrid, solar and wind rely on power electronic converters as theinterface to the grid. They are called inverter-based resources(IBRs).

High penetrations of IBRs significantly change grid dy-namic characteristics. Unprecedented dynamics have occurred

This work was authored in part by the National Renewable EnergyLaboratory, operated by Alliance for Sustainable Energy, LLC, for the U.S.Department of Energy (DOE) under Contract No. DE-AC36-08GO28308.

This material is based upon work supported by the U.S. Department ofEnergy’s Office of Energy Efficiency and Renewable Energy (EERE) underthe Solar Energy Technologies Office Award Number DE-EE0008771.

The views expressed herein do not necessarily represent the views ofthe U.S.Department of Energy or the United States Government. The U.S.Government retains and the publisher, by accepting the article for publication,acknowledges that the U.S. Government retains a nonexclusive, paid-up,irrevocable, worldwide license to publish or reproduce the published formof this work, or allow others to do so, for U.S. Government purposes.

L. Fan and Z. Miao are with Dept. of Electrical Engineering, Universityof South Florida, Tampa FL 33620 (email: linglingfan@usf.edu;zmiao@usf.edu).

P. Koralewicz, S. Shah, and V. Gevorgian are with the NationalRenewable Energy Laboratory, Golden, CO 80401 USA (e-mails: przemyslaw.koralewicz@nrel.gov; Shahil.Shah@nrel.gov; vahan.gevorgian@nrel.gov).

in power grids. Examples include 20–30-Hz oscillations thatoccurred in 2009 and 2017 in Texas when wind farms ra-dially connected to series capacitors [1], 9-Hz oscillationsthat occurred in August 2019 in an offshore wind farm inthe United Kingdom before the power disruption [2], andsubcyle overvoltage dynamics that triggered large-scale solarphotovoltaic tripping in California in 2017 and 2018 [3].

A hurdle to the dynamic study of IBR grid integrationsystems is that IBRs are black boxes because of proprietaryinformation; thus, impedance or admittance-based stabilityanalysis has become attractive for the grid industry becauseimpedance/admittance can be characterized via measurement[1]. Impedance-based stability analysis has proven effectivefor the evaluation of control interactions and stability problemsinvolving IBRs [4], [5].

This paper presents the admittance measurement of autility-scale 2.3-MVA inverter using a 13.8-kV/7-MW gridsimulator called the controllable grid interface (CGI) at theFlatirons Campus of the National Renewable Energy Labora-tory (NREL). The admittance of an inverter can be measuredby injecting voltage at different frequencies from its terminalssuperimposed on the voltages at the fundamental frequency.Such perturbations excite the inverter dynamics over a broadfrequency range. In the literature, this method is referred to asthe well-known harmonic injection method or the frequency-scanning method [6], [7].

The sine-signal injection frequency-domain measurementis known for accuracy. However, it is also known astime consuming. Thus, we also implement a time-domainmeasurement-based method, which relies on step response datagenerated by two experiments at minimum. The concept of themethod has been presented in [8], along with demonstrationsin computer simulation environment. This paper reports thefirst-time implementation of the proposed time-domain methodin a MW-level hardware test bed.

With two measurement methods, cross validation is con-ducted via two approaches. In the first approach, vector fitting[9] is applied to fit the frequency-domain measurement to alinear model. This model is then subject to step changes. Itsstep responses are compared with the time-domain measure-ment data.

In the second approach, system identification is carried outusing the time-domain data to lead to an admittance model.The frequency-domain responses of the admittance model arecompared with the frequency-domain measurements.

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In the second approach, time-domain measurement data-based model identification is used to find the DQ-domainimpedance. This method is more efficient because it requiresonly two experiments at minimum, as shown in [10]. In[10], two sets of event data are recorded following post-processing to filter out noise. The data are then fed to a systemidentification algorithm: the output error method. This leadsto an input and output model. However, an excellent match isstill lacking, as shown in the comparison results in Fig. 8 ofin [10].

Most recently, a time-domain data-based method relyingon subspace identification was proposed in [8], [11]. It wasshown to arrive at an excellent match for the model outputversus the measurement data. Tests have been conducted fordata generated from electromagnetic transient simulation testbeds. In this research, this method will also be adopted foradmittance measurement for the 2.3-MVA inverter.

Thus, the main work of this paper is to identify theDQ-domain admittance of a 2.3-MVA inverter using bothfrequency-domain data and time-domain data.

Because admittance is essentially a linearized model basedon an operating condition, different operating conditions havebeen considered in the measuring process, and their influenceon admittance is demonstrated and analyzed. It is foundthat for DQ-domain admittance, low-frequency admittanceresponse indeed reflects the operating condition. In addition,whether unbalanced control has been included can be predictedfrom the admittance models.

The contribution of this paper is twofold.• We not only obtain frequency-domain admittance mea-

surement data of a 2.3-MVA commercial converter atvarious operating conditions using advanced hardware,but also draw two important insights from the per unitizedfrequency-domain admittance measurements. The firstinsight is that the low-frequency admittance measurementin a dq-frame aligned to the PCC voltage directly reflectsP , Q operating conditions. The second insight is that theexistence of the unbalanced control can be detected afterconverting the DQ-domain admittance to the sequence-domain admittance.

• We implement a transient response data-based admittancemodel identification method in a MW-level hardware testbed. Compared to the frequency scanning method, theproposed method requires few experiments. This researchleads to valuable experience in utilizing transient responsedata for admittance identification in real world. Further,feasibility and limitations of the time-domain method areidentified.

To the best of the authors’ knowledge, the two insightshave not been stated in the state-of-the-art. In addition, thetransient response data-based method is not a conventionalwide-band signal injection method seen in the literature. Theexisting methods relying on wide-band signal injections, e.g.,impulse signal [12], [13], chirp signals [14], pseudo-randombinary sequence signals [15], and ternary-sequence signals[16], can be classified as frequency-domain methods sincediscrete Fourier transform (DFT) is used to obtain frequency-domain data.

A significant difference between the proposed time-domainmethod versus those frequency-domain methods is that Fouriertransform of measurement data is not used. The prposedmethod views the transient response data as the output ofa linear time invariant (LTI) dynamic system expressed asx = Ax. It relies on data analytics methods, e.g., dynamicmode decomposition (DMD), to extract the LTI system in-formation from a large-size data Hankel matrix. This Hankelmatrix is based on time-domain measurements. The informa-tion obtained from DMD includes the system matrix A, itseigenvalues, and residues. With the information of the LTImodel, s-domain expressions of measurement data are found.In the experiment design, with the voltage perturbation’s s-domain expressions known and the current response s-domainexpressions found, admittance models are identified.

The remainder of the paper is organized as follows. SectionII describes the measurement test bed. Section III presentsthe frequency-domain and time-domain measurement data.Section IV presents the cross-validation results. Section Vpresents insights obtained from the admittance responses.Section VI concludes this paper.

II. THE MEASUREMENT TEST BED

A 2.3-MVA inverter driven by a 1-MW battery will bemeasured for its admittance. The control structure of theinverter is not known. The inverter can follow real powerorder and reactive power order. Thus, this converter is a grid-following voltage source converter operating in real power andreactive power regulation mode.

This converter is connected to a 13.8-kV/7-MW grid sim-ulator, or the CGI [17], via a 400-V/13.2-kV 1-MVA step-uptransformer. The CGI is operating at 13.2 kV. Fig. 1 presentsthe measurement test bed constructed at NREL. The circuitparameters are given in the caption. The measurement pointis at the grid bus or at the CGI and the dq-frame is chosen tobe aligned with the measurement point voltage exactly or thePCC voltage approximately; thus, the admittance includes theRC filter at 13.2 kV, the transformer, and the shunt cap at 400V.

T1

1 MVA

400 V: 13.2 kV

VPCCI1

L1 R1

GSC

PGSC

Control

i1,abc vPCC,abc

vabc

Q

13.2 kV

Grid

VgV

ConverterPCC Meaurement point

P

Q

C1

RC

filter

Fig. 1: The admittance measurement test bed. The dq-frame is aligned withthe grid voltage. The RC filter parameters are: R = 15 Ω, and C = 6.39µF.The transformer’s per-unit impedance is 0.02+j0.0721. The 400-V capacitorparameter is 444 µF. The RL filter parameters are R1 = 0.001 Ω and L1 =46 × 10−6 H.

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Per-unit analysis is conducted for the circuit for a powerbase of 1-MVA. For the 13.2-kV system, the RC filter’sadmittance per-unit value is:

YRC =Zb

R− jXc=

13.22

1

15− j 1377×6.39×10−6

(1)

= 0.0151 + j0.4192.

Thus, at the nominal voltage, the reactive power sent by theRC filter is approximately 0.4 p.u. or 400 kVAr.

1) Grid Simulator: The CGI at NREL is shown in Fig.2. It is rated for 13.8-kV, 7-MVA continuous power, and ithas a 39-MVA short-term rating for 2 seconds. It is custom-designed based on ABB’s ACS6000 industrial-grade medium-voltage drive technology. It features a 9-MVA active rectifierunit at the utility side that regulates the intermediate dc busvoltage. The controllable grid is established by four front-endneutral-point-clamped inverters, each with a rated output of 3.3kV, connected in parallel. A specialized multi-winding outputtransformer, also shown in Fig. 2, is used to synthesize 17-level low-distortion voltage waveforms by combining three-level phase voltages from the inverters. The transformer alsosteps up the voltage to 13.8 kV. The CGI is designed to si-multaneously meet the following requirements: multimegawattpower rating, sub-1% total harmonic distortion in the voltagewaveforms, and an extremely fast response time of less than 1ms. The fast dynamics of the CGI enable it to behave like anideal voltage source that can be used for transient experimentsand to inject perturbations over a broad frequency range forimpedance measurements. The control system of the CGI iscoupled with a Real-Time Digital Simulator (RTDS), whichcan provide dynamic voltage references for all three phasesand the neutral wire for different grid integration tests; thedesired perturbation for admittance measurement of the 2.3-MVA inverter is configured inside the RTDS.

Grid-sidetransformer

Outputtransformer

4 NP-VSCin parallel

ARU +

Fig. 2: T13.8-kV/7-MVA grid simulator, called a controllable grid interface(CGI), custom designed using ABB’s medium-voltage ACS6000 drive tech-nology. Source: Mark McDade, NREL.

III. MEASUREMENT DATA: FREQUENCY-DOMAIN ANDTIME-DOMAIN

A. Frequency-Domain Measurements

Fig. 3 presents the Bode plots of the dq-admittance mea-sured by the CGI at four different operating conditions. Thisadmittance is viewed at the CGI, and thus it includes the effectof the RC filter. The CGI sets the supplying voltage in steady-state as 1 0 p.u.. The four operating conditions are:

• Case 1: P = 0 kW, Q = 0 kVAr• Case 2: P = 500 kW, Q = 0 kVAr• Case 3: P = 0 kW, Q = 500 kVAr• Case 4: P = 1000 kW, Q = 0 kVAr

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Fig. 3: The measured DQ-domain admittance after being per unitized. Thefour operating conditions are represented by different colors —Case 1: black.Case 2: green. Case 3: blue. Case 4: red.

The effect of the RC filter can be easily removed bysubtracting the RC filter’s admittance from the measuredadmittance. Fig. 4 presents the DQ-domain admittance viewedat the measurement point.

B. Time-Domain Measurement Data

The experiments are designed for three operating condi-tions: Case 1, Case 2, and Case 3. For each operating condi-tion, two experiments are conducted for the same step changesize: vd step-down and vq step-up. For each experiment,currents flowing out of the converter, id and iq , are recorded.A series of step change sizes are used: 1%, 2%, 5%, 10%,15%, 20%, 30%, 40%, and 50%.

The CGI outputs three-phase voltages with the magnitudeand angle programmable. In the initial steady state, the CGI’svoltage is operating in nominal condition with the phase angleat 0. To have a 10% vd step-down, the magnitude of the voltagewill be changed from 1 to 0.9. In order to have 10% vq step-up, both the magnitude and the angle need to be changed. As

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Fig. 4: The DQ-domain admittance viewed at the measurement point.

shown in the following, the magnitude is increased from 1 to1.005, and the angle increases from 0 to 5.7:

V = 1 + j0.1 = 1.005 5.7. (2)

Fig. 5 presents the input and output data for the experimentsin the Case 1 operating condition when the inverter delivers 0real and reactive power. The presented data are the filtered dataafter a third-order Butterworth filter with 500-Hz bandwidth.

Results show that in the initial condition, the CGI voltageis at vd = 1 and vq = 0. Fig. 5(a) presents the vd step changedata, and Fig. 5 presents the vq step change data. Three stepsizes are used: 5%, 10%, and 15%.

IV. CROSS VALIDATION

A. First Approach: Step Response Comparison

1) Vector-Fitting Results: For each operating condition,approximately 50 measurement frequency points are selectedto conduct harmonic injection, time-domain data recording,and Fourier transformation. The resulting admittance is a setof measurement points instead of a linear model. To arrive ata linear state-space model or a transfer function matrix, thevector-fitting method [9] is applied to the measurement data.The MATLAB toolbox is available in the public domain forresearch. A system can be represented as a transfer functionmatrix with each transfer function represented by a numer-ator and a denominator. Both numerator and denominatorare polynomials of the Laplace operator, s. Vector fitting isessentially a least-squares estimation method to estimate thenumerator and denominator’s coefficients by minimizing thesum of the squared errors between the model output and themeasurements.

To start the method, the model’s order or the system’s rankis first chosen along with the poles of the system. Throughiterations, the model and the measurement data achieve ahigh-degree match. In this example, the four linear systemscorresponding to the four operating conditions are found using

1.9 1.95 2 2.05 2.1 2.15 2.2 2.25 2.30.8

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0.1Vq[pu] pu=7620V

1.9 1.95 2 2.05 2.1 2.15 2.2 2.25 2.3-0.5

0

0.5Id[pu] pu=61.86A

1.9 1.95 2 2.05 2.1 2.15 2.2 2.25 2.3

Time (s)

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0.5Iq[pu] pu=61.86A

(a)

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0

Iq[pu] pu=61.86A

(b)

Fig. 5: (a) Event 1: step change in vd. (b) Event 2: step change in vq .

the Ydd, Ydq , Yqd, and Yqq measurement data. The orders ofthe systems are all assumed to be 16.

Fig. 6 presents the comparison between the admittancemodel obtained from vector fitting and the measurement data.As shown, high-degree matching is obtained.

2) Comparison with the Time-Domain Data: Fig. 7 presentsthe comparison of the measured step response data for the10% and 20% step changes and the step response generatedby the models obtained after vector fitting. Results show thatthe models generate comparable step responses.

B. Second Approach: Frequency-Domain Response Compari-son

In this approach, the admittance models will be identifiedbased on the measured step response data. Our approach is to

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Fig. 6: Comparison of the admittance model from vector fitting and the admittance measurement points. Solid lines: vector fitting. Crosses: discrete frequency-domain measurements.

find the s-domain expression of the currents using the systemidentification algorithm. Because the inputs are voltage stepchanges with their transfer functions known, the s-domainadmittance can be found by using the following procedure.

The inverter’s current flowing out and its terminal voltageare related to its admittance as follows:[

−id(s)−iq(s)

]=

[Ydd(s) Ydq(s)Yqd(s) Yqq(s)

]︸ ︷︷ ︸

Y

[vd(s)vq(s)

], (3)

where Y is the admittance of the inverter.For Event 1, we have vq not perturbed but vd perturbed with

a step-down change −ps where p is the perturbation size. Then

the resulting responses, i(1)d (s) and i(1)q (s), should reflect thefirst column of the Y matrix: Ydd(s) and Yqd(s).

[−i(1)d (s)

−i(1)q (s)

]=

[Ydd(s) Ydq(s)Yqd(s) Yqq(s)

] [(−p/s)

0

]=

[Ydd(s)Yqd(s)

]−ps(4)

Similarly, for Event 2 where vd is not perturbed while vqis perturbed with a step-up change p

s , the relation becomes:[−i(2)d (s)

−i(2)q (s)

]=

[Ydd(s) Ydq(s)Yqd(s) Yqq(s)

] [0

(−p/s)

]=

[Ydq(s)Yqq(s)

]−ps.

(5)Thus,

Y =s

p

[i(1)d (s) −i(2)d (s)

i(1)q (s) −i(2)q (s)

], (6)

where the superscript (i) notates events.To find the s-domain expressions for the four currents,

system identification algorithms will be applied. Ho andKalman, in 1965 [18], have discovered the seminal state-spacerealization theory: from measurement data, an LTI system’sstate-space model can be recovered.

We view the linearized models for the same operatingcondition subject to Event 1 and Event 2 are the same. Thus,we may use the four measurements i(1)d (t), i(1)q (t), i(2)d (s) andi(2)q (s)generated from vd perturbation and vq perturbation to

identify an LTI model. In addition, data from both 5% and 10%perturbation levels are used. Thus, the data Hankel matrix has

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Fig. 7: Comparison of step responses: measurement (left panel) and theoutput of the fitted model based on the frequency-domain data (right panel).Operating condition: Case 3, P = 0, Q = 0.5 p.u..

a row dimension of 8. Its column dimension is same as thenumber of samples.

The sampling frequency 5,000-Hz and 0.8-second datastarting from t = 2 are used. The resulting column dimensionis 4,000. In this research, the DMD algorithm is applied todeal with the data matrix and leads to a system matrix A.Eigenvalues of A are associated with the dynamic modes ofthe step response data.

This algorithm was proposed by P. Schmid in 2010 for fluiddynamics analysis [19]. DMD has been applied to real-worldphasor measurement unit data collected by ISO New Englandto identify dynamic modes and reconstruct the measurement[20]. The tool has shown efficacy in identifying modes andreconstructing signals with excellent matching based on theidentified modes and their residues.

The main idea of DMD is that the dynamic system understudy (in this case, the inverter) is in fact of low order. Fromthe measurement data, a Hankel matrix with a large dimensionis formed. This Hankel matrix is closely associated with thesystem dynamic matrix with a low order; thus, through matrixdecomposition using singular value decomposition (SVD) andby keeping only the first n singular values and the associatedSVD unitary matrices, a low-rank data matrix is constructed.Further, the system dynamic matrix A can be recovered byfactorizing the low-rank matrix. Readers may refer to [8] and[20] for details of data analytics.

The time-domain expression of a time-domain response x(t)can also be constructed using the eigenvalues and eigenvectorsof A:

x(t) =

n∑j=1

φjeωjtbj = ΦeΩtb. (7)

where Ω is a diagonal matrix that contains the continuouseigenvalues ωj ., φ is the right eigenvector matrix of A, andb is the initial state projected to the eigenvector basis ( b =Φ−1x0, where x0 is the initial state).

The eigenvector matrix, Φ, the eigenvalue matrix, Ω, andthe initial state projected to the eigenvector basis, b, will allbe identified from the measurement data by DMD.

With the DMD’s outputs, Φ, b, and Ω, it is easy toreconstruct the input measurement data using (7) as well as tofind the s-domain expression of kth measurement signal as

xk(s) =

n∑j=1

φkjbjs− ωj

. (8)

where φkjbj is the residue of the jth eigenvalue ωj , n is theorder of the system.

Fig. 8 presents both the measurement data (solid lines) andthe reconstructed signals (dotted lines). An excellent matchhas been achieved. With the s-domain expressions for currentsfound, the admittance model can also be found using (6).

The resulting frequency-domain responses of the admittancemodels for three operating conditions are compared with thefrequency-domain measurements, as shown in Fig. 9.

Remarks:• Results show that in the frequency range less than 60 Hz,

the time-domain data-based admittance models have com-parable frequency-domain responses with the frequency-domain measurements.

• Mismatch is obvious for frequency ranges greater than60 Hz.

• If admittance at a frequency range is small, the time-domain based method leads to more error. This is reason-able because smaller admittance leads to indistinguish-able current responses from noise.

V. INSIGHTS OBTAINED FROM THE DQ-DOMAINADMITTANCE MEASUREMENTS

A. Insight 1: Admittance at Low-Frequency Range DirectlyReflecting Operating Conditions

In the dq-frame, the active power, P , and reactive power,Q, exported by the converter and measured at the PCC buscan be expressed as follows:

P = vdid + vqiq

Q = −vdiq + vqid(9)

where v and i are the PCC bus voltage and the currentmeasured at the PCC bus.

By linearizing P and Q, the small-signal model is:[∆P∆Q

]=

[id iq−iq id

]︸ ︷︷ ︸

G1

[∆vd∆vq

]+

[vd vqvq −vd

]︸ ︷︷ ︸

G2

[∆id∆iq

](10)

If we assume that the dq-frame’s d-axis is aligned with thePCC voltage’s space vector, then vd = V , vq = 0, where V isthe magnitude. Also, according to (9), id = P

V , iq = −QV . If

the converter is in PQ control mode, in steady state, ∆P = 0and ∆Q =0.

Eq. (10) becomes the following.[00

]=

1

V

[P −QQ P

]︸ ︷︷ ︸

G1

[∆vd∆vq

]+

[V 00 −V

]︸ ︷︷ ︸

G2

[∆id∆iq

](11)

7

0 0.02 0.04 0.06 0.08 0.1-0.2

0

0.2

0.4i d(1

)Case 1

0 0.02 0.04 0.06 0.08 0.1-0.4

-0.2

0

0.2

i q(1)

0 0.02 0.04 0.06 0.08 0.1-0.2

-0.1

0

0.1

i d(2)

0 0.02 0.04 0.06 0.08 0.1

Time (s)

-0.4

-0.2

0

0.2

i q(2)

10%

5%

(a)

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10.4

0.6

0.8

1

i d(1)

Case 2

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1-1

-0.5

0

0.5

i q(1)

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10

0.5

1

i d(2)

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1

Time (s)

-0.4

-0.2

0

0.2

i q(2)

10%

5%

(b)

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1-0.5

0

0.5

i d(1)

Case 3

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1-1.5

-1

-0.5

0

i q(1)

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1-1

-0.5

0

0.5

i d(2)

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1

Time (s)

-1

-0.5

0

i q(2)

10%

5%

(c)

Fig. 8: Comparison of measurement data (solid lines) and the reconstructed signals (dotted lines). Blue: 10% perturbation. Red: 5% perturbation. (a) Case 1.(b) Case 2. (c) Case 3. Excellent matching is achieved.

Thus, the admittance can be found from (11):

0 =1

V 2

[P −Q−Q −P

]︸ ︷︷ ︸

YVSC

[∆vd∆vq

]+

[∆id∆iq

](12)

YVSC =1

V 2

[P −Q−Q −P

], ZVSC =

V 2

|S|2

[P −Q−Q −P

](13)

Remarks: Equation (13) indicates that in steady state, i.e.,60 Hz in the static frame and 0 Hz in the dq-frame, the DQ-domain admittance directly reflects the operating conditions.

The admittance expression based on steady-state analysishas been investigated in M. Belkhayat’s thesis [21]. Althoughthe analysis in [21] is in some ways similar to how we haverelated the low-frequency response of DQ-domain admittancewith the steady-state P and Q values, the admittance expres-sion in [21] is more complicated. The difference between theanalysis of [21] and ours lies in two aspects: (i) the assumptionof a general dq-frame versus a dq-frame aligned at the PCCbus; (ii) the adoption of real-value versus per unit system.

In this research, we adopt per unit system and a PCC voltagealigned dq-frame. Thus vq = 0 and the expression of admit-tance is greatly simplified. To the authors’ best knowledge, wehave not seen anybody explicitly showing this relationship ofper-unit values of the admittance matrix at low frequenciesrelated to the per-unit values of P and Q. This insight isvaluable as it can directly show the effect of operation pointand also serve as a quick sanity-check tool for admittancemeasurements.

As shown in Fig. 4 and Fig. 6, for Case 4 (P = 1 andQ = 0), Ydd and Yqq at a very low frequency range (< 1 Hz)have a magnitude of 0 dB or 1 p.u. For Case 2 (P = 0.5 andQ = 0), Ydd and Yqq at a very low frequency range (< 1 Hz)have a magnitude of −6 dB or 0.5 p.u. In addition, Yqq hasa phase angle of 180, wheras Ydd has a phase angle of 0.Obviously, Ydd = P and Yqq = −P when the PCC voltage iskept at nominal.

For both Case 4 and Case 2, the nondiagonal componentshave very small magnitudes at the low-frequency range. This

fact corroborates Ydq = Yqd = −Q when the PCC voltage iskept at nominal.

For Case 3 (P = 0 and Q = 0.5), Ydq has a magnitudeof −6.8 dB or approximately 0.45 p.u.; and Yqd has amagnitude of −8 dB or 0.40 p.u. According to (13), these twonondiagonal components should have a value of 0.5 p.u. at alow frequency. Note that the admittance refers to the inverteradmittance viewed at the PCC point. On the other hand, themeasured admittance includes the effect of the transformer.This causes the discrepancy.

For Case 1 (P = 0 and Q = 0), all four components at alow-frequency range have very small magnitudes. The DQ-domain admittance measurement corroborates the analysis.

B. Insight 2: Detection of Unbalanced Control via Sequence-Domain Admittance

We further proceed to plot sequence-domain admittanceto develop more insights. The relationship between the DQ-domain and sequence-domain admittance is as follows [22]:[

Ypp(s) Ypn(s)Ynp(s) Ynn(s)

]︸ ︷︷ ︸

Y mpn

=1

2

[1 j1 −j

]︸ ︷︷ ︸

H

Y mdq (s)

[1 1−j j

]︸ ︷︷ ︸

H∗

. (14)

The sequence-domain admittance evaluated at ωp associatesthe current phasors at positive sequence of ωp + ω1 andnegative sequence ωp − ω1 (where ω1 is the nominal angularfrequency and ω1 = 377 rad/s) with the respective voltagephasors.

In the low-frequency range, the sequence-domain admit-tance and impedance are expressed as follows:

Ympn =

1

V 2

[0 P − jQ

P + jQ 0

](15)

Fig. 10 presents the sequence-domain admittance measure-ments. In the low-frequency range, the diagonal componentshave very small magnitudes, whereas the nondiagonal compo-nents have magnitudes and angles corresponding to P − jQand P + jQ, respectively.

Finally, we examine whether unbalanced current control canbe reflected in the sequence-domain admittance.

8

100

102

-40

-20

0

20

40

Ma

g (

dB

)

100

102

-200

0

200

Ph

ase

(d

eg

ree

)

100

102

-40

-20

0

20

40

100

102

-200

0

200

100

102

-40

-20

0

20

40

Ma

g (

dB

)

100

102

Frequency (Hz)

-200

0

200

Ph

ase

(d

eg

ree

)

100

102

-40

-20

0

20

40

100

102

Frequency (Hz)

-200

0

200

P=0, Q=0

Ydd dqY

qdY qqY

(a)

100

102

-40

-20

0

20

40

Ma

g (

dB

)

100

102

-200

0

200

Ph

ase

(d

eg

ree

)

100

102

-40

-20

0

20

40

100

102

-200

0

200

100

102

-40

-20

0

20

40

Ma

g (

dB

)

100

102

Frequency (Hz)

-200

0

200

Ph

ase

(d

eg

ree

)

100

102

-40

-20

0

20

40

100

102

Frequency (Hz)

-200

0

200

P=0.5, Q=0

Ydd Ydq

Yqd Yqq

(b)

100

102

-40

-20

0

20

40

Mag (

dB

)

100

102

-200

0

200

Phase (

degre

e)

100

102

-40

-20

0

20

40

100

102

-200

0

200

100

102

-40

-20

0

20

40

Mag (

dB

)

100

102

Frequency (Hz)

-200

0

200

Phase (

degre

e)

100

102

-40

-20

0

20

40

100

102

Frequency (Hz)

-200

0

200

P=0, Q=0.5

Ydd

Yqq

Ydq

Yqd

Fig. 9: Comparison of the admittance measurement and the admittance learnedfrom the time-domain data. The system order is assumed to be 16. Solidlines: learned model from step responses. Crosses: discrete frequency-domainmeasurements.

10-1 100 101 102 103-40

-20

0

20

40

Ma

g (

dB

)

10-1 100 101 102 103-200

-100

0

100

200

Ph

ase

(d

eg

ree

)

10-1 100 101 102 103-40

-20

0

20

40

10-1 100 101 102 103-200

-100

0

100

200

10-1 100 101 102 103-40

-20

0

20

40

Ma

g (

dB

)

10-1 100 101 102 103

Frequency (Hz)

-200

-100

0

100

200

Ph

ase

(d

eg

ree

)

10-1 100 101 102 103-40

-20

0

20

40

10-1 100 101 102 103

Frequency (Hz)

-200

-100

0

100

200

Ypp pnY

nnYnpY

Fig. 10: The sequence-domain admittance viewed at the measurement point.

In particular, we examine sequence-domain admittance at120 Hz (ωp = 2π × 120 rad/s). This admittance associatesthe two current phasors and two voltage phasors. The twovoltage (current) phasors are referred to the phasors at positive-sequence 180 Hz and negative-sequence 60 Hz:[

Ip(j(ωp + ω1))In(j(ωp − ω1))

]=

[Ypp(jωp) Ypn(jωp)Ynp(jωp) Ynn(jωp)

] [V p(j(ωp + ω1))V n(j(ωp − ω1))

](16)

Unbalanced current converter control aims to reduce thenegative-sequence current component at 60 Hz when the volt-age contains the negative-sequence component. This require-ment indicates that the admittance associating the negative-sequence 60-Hz current phasor with the negative-sequence 60-Hz voltage phasor should be small. i.e., Ynn(s) evaluated at120 Hz should be small.

In Fig. 10, for both Ypn and Ynn, the components of thesecond column of the admittance matrix, a dip is shown at120 Hz in the magnitudes. This indicates that the resultingcomponents in the current will be suppressed because of thedip. The dip at 120 Hz observed in Ypn and Ynn also indicatesthat unbalanced current control is in place.

VI. CONCLUSION

In this paper, a 2.3-MVA grid-following inverter operatingin real and reactive power regulation mode was measuredto arrive at its admittance models, under various operatingconditions. Two methods —frequency-domain based and time-domain based —are used for cross validation. Both methodscan lead to linear models through either vector fitting offrequency-domain data or the DMD-based method to fit thetime-domain data. From the frequency-domain admittancemeasurements, insights of the inverter can be revealed, includ-ing its operating condition and whether unbalanced control isincluded.

9

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Lingling Fan (SM’08) received the B.S. and M.S. degrees in electricalengineering from Southeast University, Nanjing, China, in 1994 and 1997,respectively, and the Ph.D. degree in electrical engineering from West Virginia

University, Morgantown, in 2001. Currently, she is a full professor at theUniversity of South Florida, Tampa, where she has been since 2009. Shewas a Senior Engineer in the Transmission Asset Management Department,Midwest ISO, St. Paul, MN, form 2001 to 2007, and an Assistant Professorwith North Dakota State University, Fargo, from 2007 to 2009. Her researchinterests include power systems and power electronics. Dr. Fan serves as theeditor-in-chief for IEEE Electrification Magazine and associate editor for IEEETrans. Energy Conversion.

Zhixin Miao (SM’09) received the B.S.E.E. degree from the HuazhongUniversity of Science and Technology,Wuhan, China, in 1992, the M.S.E.E.degree from the Graduate School, Nanjing Automation Research Institute(Nanjing, China) in 1997, and the Ph.D. degree in electrical engineering fromWest Virginia University, Morgantown, in 2002.

Currently, he is with the University of South Florida (USF), Tampa. Priorto joining USF in 2009, he was with the Transmission Asset ManagementDepartment with Midwest ISO, St. Paul, MN, from 2002 to 2009. His researchinterests include power system stability, microgrid, and renewable energy.

Przemyslaw Koralewicz (M’17) received the M.S.E.E. degree from SilesianTechnical University, Gliwice, Poland, in 2010. He specializes in modeling,detailed analysis, and testing of smart inverters and complex power systemsincluding microgrids. He is utilizing the National Renewable Energy Labo-ratory Controllable Grid Interface, a new, groundbreaking testing apparatusand methodology to test and demonstrate many existing and future advancedcontrols for various renewable generation technologies on the multimegawattscale and medium-voltage levels.

Shahil Shah (S’13-M’18) received the B.E. degree from Government En-gineering College, Gandhinagar, India, in 2006, the M.Tech. degree fromthe Indian Institute of Technology Kanpur, Kanpur, India, in 2008, and thePh.D. degree from Rensselaer Polytechnic Institute (RPI), Troy, NY, USA, in2018, all in electrical engineering.,Before joining the Ph.D. program withRPI in 2012, he worked with Bhabha Atomic Research Center, SiemensCorporate Technology, and GE Global Research Center, India. He is currentlya Research Engineer with the Power Systems Engineering Center, the NationalRenewable Energy Laboratory, Golden, CO, USA. His research interestsinclude applications of modeling and control for dynamic and transientstability of power systems with high penetration of renewable generation.,Dr.Shah was the recipient of the 2018 RPI Allen B. Dumont prize, given toa Doctoral Candidate “who demonstrates high scholastic ability and makessubstantial contribution to his/her field.”

Vahan Gevorgian (M’97–SM’17) received the Ph.D. degree in electrical engi-neering from the State Engineering University of Armenia, Yerevan, Armenia,in 1993. He joined National Renewable Energy Laboratory (NREL) in October1994 and has served many roles over the years. He is currently working withthe Power Systems Engineering Center focused on renewable energy impactson transmission and interconnection issues and dynamic modeling of variablegeneration systems. He research interests include dynamometer and fieldtesting of large and small wind turbines, dynamometer testing of wind turbinedrivetrain components, development of advanced data acquisition systems, andwind turbine power quality. He provides technical support to NREL industrypartners and major U.S. wind turbine manufacturers. He is member of the IECTeam for wind turbine power quality standards. His contributions to NRELresearch have been recognized through multiple Outstanding Individual andTeam Staff awards.