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Online Grid Impedance Estimation for the Control of Grid

Connected Converters in Inductive-Resistive Distributed Power- Networks Using Extended Kalman-Filter

Nils Hoffmann & Friedrich W. Fuchs

Institute for Power Electronics and Electrical DrivesChristian-Albrechts-University of Kiel

Kiel, [email protected] / [email protected]

Abstract — Real-time estimation of the equivalent grid impedance

and the equivalent grid voltage seen from a power converter

connected to the public electric distribution network by means of

Extended Kalman-Filter is addressed. The theoretical

background of the Extended Kalman-Filter used for equivalent

grid impedance estimation is introduced. Practical aspects like

the use of the filter in an environment with highly distorted

voltage waveforms, the tuning of the noise covariance matrices

and the implementation on a laboratory system are discussed.

The theoretical analysis is verified on a 22 kW test-bench where a

grid impedance emulator is used to simulate grid impedancesteps in the laboratory environment. The proposed Extended

Kalman-Filter is designed to utilize the noise that is already

present at the connection point of the power converter to

overcome the need of active disturbance injection to estimate the

equivalent grid impedance.

I. I NTRODUCTION

In the last years the number of converter-based distributed(or decentralized) energy production units connected to the

public energy distribution networks has increased significantly[1]. This trend is mainly driven by the efforts to push electrical

power generation towards green and sustainable energy production. This enormously amount of electrical powergenerated by distributed energy sources in addition to the slow

process in the expansion of the public grid network structuresleads to a high utilization of the existing grid structures. Thishigh penetration of distributed energy sources has a strongimpact to the stability of the overall electrical system [2-4].This is especially of high interest in low-voltage networks,where the majority of renewable energy sources are connectedto the grid.

Distribution networks are characterized by a non-negligible grid impedance. The value of the grid impedanceand the amount of connected apparent power determine thenetwork distortions generated by electrical loads anddistributed energy sources. To guarantee a defined andstandardized electromagnetic compatibility (EMC) level fordevices connected to public distribution networks international

standards, like the IEC 61000 series or the IEEE 519-1992,define harmonic current emission limits in relation to the short-

circuit power of the grid (which is anti proportional to the gridimpedance).

Besides the network distortions a non-negligible gridimpedance is an important parameter that determines theinteraction between distributed energy sources (and loads)among each other [5,6] and that determines the stability limitsof the whole electrical power system [3,7]. Beyond that, thegrid impedance influences the control performance of gridconnected power converters [4,8]. Especially, when LCL typeline filters are used that are not passively damped, a variationof the grid impedance can lead to drastic control performancedegradation or even to unstable converter operation [8].Furthermore, an active-filter-functionality to mitigate voltageunbalances and low order harmonic voltage contents can beimplemented to the control of grid connected power convertersused for distributed power sources dependent on the actual gridimpedance conditions [9,10]. These aforementioned aspectsreveal that the time and frequency dependent grid impedance isone key parameter to optimize the operation and performanceof distributed energy sources in the public electricaldistribution networks.

To optimize the performance of individual converter-basedenergy sources in relation to their control, their interaction withother converters and their performance in the overall

distributed power system it is essential to determine the gridimpedance during the converter operation. Basically, the gridimpedance can be determined in two different ways: Based onseveral measurements in the distributed power network (oftenin addition to the injection of forced network disturbances) thegrid impedance characteristic can be calculated offline bymeans of a detailed frequency-response analysis [11-15].Usually, these grid impedance measurement techniquesachieve a high precision over a wide frequency band but arenot able to provide the information about the determined gridimpedance in an online manner. Further, these methods oftenuse additional power electronic hardware that has to beconnected to the power network under investigation.

To determine the grid impedance characteristic online byutilizing the power converter setups that are already used tofeed-in the generated electrical power several methods aredocumented in literature [16-21]. These methods can beclassified in passive-methods [19] (or non-invasive methods),

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in active-methods [16,21] (or invasive methods) and in quasi- passive methods [17,18,20]. Passive methods are referred to asmethods that utilize the existing disturbances that are already

present in the power networks. These methods often detect thegrid impedance at individual characteristic disturbances, e.g.the fundamental- or low-order harmonic frequencies. Activemethods use a forced disturbance that is injected to the grid in

parallel to the regular operation of the power converter. Basedon the signal used for injection these methods are able todetermine the grid impedance at individual frequencies or incharacteristic frequency bands. Quasi passive methods arereferred to as impedance determination methods that usehybrid identification methods. This is often achieved by forcedoperation point changes of the power converter.

This paper aims to contribute to the online grid impedanceidentification by presenting a new parameter estimationmethod that is based on an Extended Kalman-Filter (EKF). Toachieve a practical estimation algorithm that can be used inaddition to the control of grid connected converters, the

proposed EKF is designed to utilize the disturbances that arealready presented in typical power networks. One of the maingoals is to use as less as possible forced operation pointchanges to estimate the grid impedance in an acceptable

precision range. The study is focused on typical low voltagedistributed power generation networks where the gridimpedance is (in most cases) inductive-resistive. Specialattention is paid to practical implementation aspects and the

performance of the proposed EKF during transient gridimpedance changes. Measurements are carried out utilizing a22 kW laboratory test-bench.

This paper is structured as follows: In the second chapter asystem description is given. The general idea of an EKF usedfor parameter estimation is presented in the third chapter. Thesystem modeling and the state prediction that is needed for theEKF algorithm is explained in the fourth chapter. To validatethe theoretical design a measurement study the fifth chapter

presents a measurement analysis. The paper is closed by acritical discussion of the presented EKF algorithm and a

conclusion.

Fig. 1: Grid connected converter with LCL filter and time-variant andfrequency-dependent power network

II. SYSTEM DESCRIPTION

In Fig. 1 the system under investigation is illustrated. Thesystem consists of converter that is connected to a time variantand frequency dependent power network.

A. Grid connected converter

A two-level voltage source converter is considered as atypical and widespread converter topology that is used for gridconnection of distributed (renewable) energy sources in lowvoltage applications. A LCL-type line filter is used to reducethe voltage distortions that are generated by the switchingtransitions of the power semiconductors, usually insulated gate

bipolar transistors (IGBTs). A pulse width modulation (PWM)

based on space-vector modulation (SVM) and the well knownvoltage-oriented control (VOC) scheme are applied to theconverter to control the DC link voltage and the currents at the

point of common coupling (PCC) [22]. Even though no passivedamping resistors are used to damp the LCL filter resonance,no active resonance damping is added to the proportional-integral-based (PI) current control to simplify the controldesign [23]. To synchronize a converter with the fundamentalgrid voltage under highly distorted grid conditions a doublesecond-order generalized integrator (DSOGI) phase-lockedloop (PLL) with additional second-order lead compensators(SOLC) tuned to the 5th

and 7th voltage harmonics is used [24].

Fig. 2: One line diagram of equivalent grid impedance model

B. Equivalent grid parameter

A low voltage public distributed power network is in manycases a complex structure that consists of power transformers,transmissions lines, linear and non-linear loads and distributedenergy sources. Due to transient load conditions, theconnection or disconnection of individual energy sources (andsinks) or even the connection or disconnection of whole sub-networks, the network structure (and thus the resultant gridimpedance) is non-stationary and time-variant. Low voltagedistribution networks reveal (in most cases, but not in general)a time and frequency dependent resistive-inductive gridimpedance characteristic which is confirmed by severalmeasurement studies performed in representative powerdistribution networks [25-27]. Bearing in mind that the gridimpedance (which affects a power converter) dependents onthe localization of the converter in the whole network, theThévenin equivalent is used to model the time variant andfrequency dependent grid impedance. In Fig. 2 the used one-line block diagram of the (Thévenin) equivalent gridimpedance model is presented. The model consists of anequivalent grid inductance L grid , an equivalent grid resistance

R grid and an equivalent grid voltage U grid . It is assumed that the

power network is composed of a three-phase three-wire systemand that the equivalent grid impedance is (almost)symmetrical.

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III. EXTENDED K ALMAN-FILTER ALGORITHM FOR

PARAMETER ESTIMATION

The principles of using an Extended Kalman-Filter for parameter estimation are well documented in literature, e.g. in[28]. In power electronic applications the EKF is wellestablished for the parameter estimation of electrical machines.Some outstanding works to indentify the parameters of

permanent synchronous machines (PMSM) are presented in[29,30]. However, to ensure a suitable theoreticalunderstanding of the EKF applied for parameter estimationsome basic aspects are briefly discussed in the following

paragraphs.In (1) the generic state-space model for a linear system

(without direct feed-through capability) in the discrete time-domain is presented. Here, x is referred to as the system’s statevector, u as the input and y as the output vector. The system issuperimposed by process noise ε and measurement noise ν .When the system parameters are constant the state matrix Ad ,the input matrix Bd and the output matrix C d are time invariant. 1

(1)

To use an observer based parameter estimation approach itis convenient to summarize all relevant time dependent system

parameters in a parameter vector θ . This parameter vector isadded to the system state vector which results in a non linearsystem description of the linear system with time variant

parameters (2). Now, the process noise model has to beextended to the process noise in relation to the primary state-

vector ε x and the process noise in relation to the parameter-

vector ε θ . The physical measurement model remainsunaffected.

1 1 , ,

00 0

0

(2)

In Fig. 3 a cycle-diagram of the applied EKF algorithm for parameter estimation is presented. The illustration and theunderlying algorithm are based on [31]. The EKF algorithm is

divided into a prediction and an update step. First, the a priorisystem states (including the system parameters) are predicted

by the non linear system model f(x,u,θ ) that is included in (2).Based on the assumption that the process noise has white noisecharacteristic and is independent from the measurement noisethe a priori error-covariance P k| a priori is estimated. Here, Qk isreferred to as the process noise covariance matrix whichincludes the process noise model of the system. In the next stepof the EKF algorithm, the update step, the actual systemmeasurements y are used to calculate the Kalman gain K k thatis used to update the a posteriori estimated system states. Here,

Rk is referred to as the measurement noise covariance matrix.In the last step of the EKF algorithm the a posteriori error-covariance matrix P k| a posteriori is calculated that is used for the

prediction step in the next sampling instant. This recursivenature of the EKF algorithm ensures a fast and accurate stateestimation even in highly distorted and noisy measurementenvironments with a reduced calculation burden.

IV. SYSTEM MODELING AND STATE PREDICTION

The EKF algorithm is used to estimate the equivalent grid parameters online during the converters operation. Thefollowing paragraphs present the development of the model

used for the prediction step and update step of the EKFalgorithm.

A. Basic system model

A general system description is presented in the secondchapter whereas the model of the equivalent grid impedance isillustrated in Fig. 2. The three-phase PCC voltages and currentsof the three-phase three-wire system are measured. Aconvenient state-space presentation is achieved by selecting the

α - and β -components of the PCC currents for the state-vectorformulation, see (3). ,, 00

,, 00 ,,

00 ,,

(3)

B. Disturbance observer formulation

The state-space model presented in (3) is used to achieve a

disturbance observer formulation. Here, the α - and β -components of the equivalent grid voltage U grid are used toexpand the state-vector for a convenient state-spaceformulation (4).

,,,,

00

0

000

000

000

,,,,

0 0 00 00 ,,

(4)

Fig. 3: Cycle-diagram of Extended Kalman-Filter algorithm for parameter estimation [31]

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To study the observability of the presented disturbanceobserver formulation (when the PCC currents and voltages aremeasured) the Kalman criterion is used. The analysis revealsthat the system is observable for positive values (unequal zero)of L grid and R grid .

C. Observer forumlation for available measurement signals

To determine the equivalent grid impedance with the proposed EKF algorithm the PCC voltages (line-to-neutral)

and PCC currents are measured. It is beneficial to reformulatethe disturbance observer model presented in (4) with the goalto include the measured PCC voltages into the measurementmodel of the observer formulation. This leads to the observerformulation presented in (5). In respect to the underlying EKFalgorithm this mathematical reformulation provides theadvantage to include the measurement noise models of both,the voltage- and current-measurements into the EKF algorithm.

,,,,,,

00000

00000

00000

0 0000

00000

00000

:

,,,,,,

(5)

,,,, 1000

01000010

00010000

0000 ,,,,,,

An analysis of the observability of this reformulationreveals that the system (5) is observable.

D. Observer formulation in discrete time-domain

The EKF algorithm is implemented in a sampled systemwith zero-order hold characteristic. The zero-order holdequivalent in discrete time-domain of the state-space modelformulation in continuous time-domain (5) is derived byapplying the transformation law presented in (6).

| (6)

The resultant observer formulation in discrete time-domainis presented in (7) whereas the measurement matrix C d remainsunaffected by the transformation.

, 1, 1, 1, 1, 1, 1

00000

00000

01000

0 0100

00010

0 0001

: ,,,,,,

(7)

exp

E.

Extension of disturbance observer formulationIn (4) the disturbance observer formulation is introduced.

This disturbance formulation identifies the PCC voltage as aconstant disturbance. This can also be seen from the model in

discrete time-domain (7). This model is insufficient when theEKF algorithm is applied to sinusoidal voltage and currentwaveforms that are distorted with unbalances and harmonics.A more precise formulation of the expected disturbances of theequivalent grid voltages is necessary. A practicable andadequate choice of the extended disturbance model is presentedin (8).

,

1, 1, 1, 1 exp Δ000 0expΔ00 00exp5Δ0 000exp 7Δ ,

, , , (8)

The equivalent grid voltage is modeled as a superpositionof a positive U grid,1+ and negative fundamental U grid,1- voltagesequence. Further, the 5

th harmonic (negative sequence, U grid,5-)

and 7th

harmonic (positive sequence, U grid,7+) voltages areincluded into the disturbance model. The prediction model is

calculated by considering the rotation angle resolution ∆θ to

the sinusoidal disturbance model in the stationary αβ -referenceframe. The rotation angle resolution depends on the appliedsampling time T s and the fundamental grid frequency f grid , see(9).

Δ 2 (9)

F. Introduction of time variant parameter vector

The final step of the mathematical problem formulation isto extend the state-vector with the time-variant system

parameters that have to be estimated (here: L grid , R grid ). Twosimplifications are used to achieve a convenient mathematicalformulation of the observer problem. First, the exponentialdependency (see (7)) of the system parameters that results fromthe discrete observer formulation is simplified using Taylorseries expansion.

exp 1 (10)

Second, to greatly simplify the calculation of the Jacobean

matrix F k during the prediction step of the EKF algorithm (seeFig. 3) the substitution presented in (11) is used. 1 (11)

The proposed EKF algorithm estimates the reciprocalinductance l grid to reduce the calculation burden of the

prediction step. To achieve the actual inductance L grid theestimated value is then inverted.

G. Final observer formulation, prediction and measurement

model

The aforementioned simplifications and the proposedobserver problem formulation in the discrete time-domain leadto the final observer formulation that is presented in (12). Theresultant state-vector consists of the measured PCC currentsand voltages, the equivalent grid voltages including theextended disturbance model formulation and the equivalentgrid parameters. The additional terms qi and r i are introduced

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to highlight that the proposed model includes process noiseand measurement noise. Further, the measurement noisecovariance matrix R has a 4x4 form, is assumed to be timeindependent and assumed to have a diagonal-from. The processnoise covariance matrix Q has a 14x14 form, is also assumed to

be time-independent and assumed to have a diagonal form tosimplify the EKF tuning.

, … , ,, … , , (13)

V. MEASUREMENT A NALYSIS

Measurements are carried out to validate the theoreticalanalysis under laboratory conditions.

A. Test-bench description and measurement methodology

A 22 kW test setup is used to verify the proposed EKFalgorithm for equivalent grid parameter estimation. The setupconsists of a two-level grid connected voltage source converter

with a LCL filter (setting see Fig. 1, parameters see Table I)and a custom designed and self build 30 kVA grid impedanceemulator. The grid impedance emulator is used to emulate(step wise) grid impedance changes under laboratoryconditions. The emulator is connected between the gridconnected converter and the actual grid of the institute’slaboratory. Three multi tap inductors are used to emulatedifferent relative short circuit powers (S sc /S N ) between 22(actual laboratory grid conditions) and 1 (relative short circuit

power with maximum inductance). To study the transient

, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 1 1

1 , , , , , ,1 , , , , , ,,,, cosΔ , sinΔ, sinΔ , cosΔ, cosΔ , sinΔ, sinΔ , cosΔ, cos5Δ , sin5Δ, sin5Δ , cos5Δ, cos7Δ , sin7Δ, sin 7Δ , cos7Δ

:

,,,,,,,,,,,,

(12)

, 1, 1, 1, 1 1000

01000010

00010000

00000000

00000000

00000000

00000000

0000

,,,,

‐2 0 2 4 60.5

0.6

0.7

0.8

0.9

1

1.1

1.2

1.3

1.4

estimated equivalent grid‐inductance Lgrid

t / s

L meas

L EKF

‐2 0 2 4 6

340

345

350

355

360

365

370

375

380

estimated equivalent grid‐resistance Rgrid

t / s

R meas

R EKF

(a) (b) (c)

(d) (e)

Fig. 4: Measurement results of equivalent grid impedance detection during impedance-step ( L grid 0.65 mH to 1.15 mH at t = 0 s): (a) estimated equivalent gridinductance L grid , (b) estimated equivalent grid resistance R grid , (c) estimated relative grid short-circuit power S sc,pu, (d) estimated and measured PCC-currents I pcc and

(e) estimated and measured PCC-voltages (line-to-neutral) U pcc

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performance of the grid impedance estimation algorithmadditional IGBTs (1200V, 200A) are used to switch betweenthe taps of the inductors [32]. Further, the actual equivalentgrid impedance parameters (i.e. when the grid impedanceemulator and converter are disconnected) of the instituteslaboratory are measured for each experiment with an additionalgrid impedance analyzer [25] that is connected to the PCC ofthe test setup. The control and estimation algorithms areimplemented on a dSPACE DS1006 system.

TABLE I

MEASUREMENT SYSTEM PARAMETERS

Symbol Quantity Value (per unit)

U LL Line-to-Line Voltage (rms) 400 V (1.0)

U DC DC-link voltage 700 V (1.75)ω Angular line frequency 2π 50 Hz (1.0)

I L Rated converter current (rms) 32 A (1.0)

L fg Line-side filter inductance 1.5 mH (0.06)

R fg Line-side filter resistance 90 mΩ (0.01)

L fc Converter-side filter inductance 2.5 mH (0.11)

R fc Converter-side filter resistance 150 mΩ (0.021)

C f Filter capacitance 16.2 µ F (0.04)

C DC DC-link capacitance 2200 µ F (5.0)

f con /f carr Control-/Carrier-frequency 5 / 2.5 kHz (100/50)

B. Extended Kalman-filter tuning

The tuning of the EKF is a complex problem. Themeasurement noise covariance matrix R consists of 4unknowns and the process noise covariance matrix Q consistsof 14 unknowns that influence the estimation performance ofthe EKF algorithm. Even though some tuning guidelines can

be found in literature (e.g. in [30,33]) the tuning of the EKFalgorithm remains difficult. Here, the EKF parameter tuning is

based on a trial-and-error procedure since the main focus ofthis work is set on proofing the effectiveness of the EKFapplied for grid parameter estimation. Several experimentsunder different grid and operation point conditions are

performed to achieve an EKF covariance matrix parametersetting that leads to an acceptable grid parameter estimation

performance. The selected EKF tuning parameters aresummarized in Table II.

C. Extended Kalman-filter performance during transient grid

impedance conditions

Two grid conditions are considered for the experiment:

First, a relative short-circuit power of 17.95 (grid condition I,additional 0.5 mH ) and second, a relative short-circuit power of13.97 (grid condition II, additional 1.0 mH ). The emulated gridconditions (including the actual measured laboratory gridconditions) are summarized in Table III.

A grid impedance step between the grid conditions I and IIis performed with the grid impedance emulator whereas a dataacquisition system (a Dewetron DEWE-2010 system) istriggered on the impedance step. The results are summarized inFig. 4 and Fig. 5. In Fig. 4 (a) and (b) the estimated and actualequivalent grid inductance and -resistance are presented. It can

be seen that the grid inductance is estimated with a bias of

approximately 50 µ H whereas the estimated grid resistance is

estimated with a bias of 10 mΩ at grid condition I and with a

bias of 5 mΩ at grid condition II. To put these measurementresults into a per-unit perspective Fig. 4 (c) presents thecomparison of the estimated and measured relative short-circuit power of the emulated grid conditions. This comparison

TABLE II

K ALMAN-FILTER TUNING

Measurement noise covariance R 5 · 10

30 · 10

Process noise covariance Q

10 · 10 55 · 10 ,

100 · 10 , 15 · 10 , , 10 · 10 , , , 10 · 10 , 10· 10

Ω , 50 · 10

(a) (b)

(c) (d)

Fig. 5: Measurement results of equivalent grid voltage detection during impedance-step ( L grid 0.65 mH to 1.15 mH at t= 0 s): (a) estimated positive sequence U pcc,1+, (b) estimated negative sequence U pcc,1-, (c) estimated 5th-harmonic U pcc,5-

and (d) estimated 7th-harmonic U pcc,7+

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reveals that the equivalent grid parameter estimation accuracyis satisfactory under real laboratory conditions even if the

parameters are estimated with a small steady-state bias.To evaluate the settling time of the EKF used for parameter

estimation a zoomed time-axis plot is added to Fig. 4 (c). Asettling-time of approximately two fundamental grid voltage

periods is measured whereas the rise-time is approximatelyhalf of the fundamental grid voltage period.

In Fig. 4 (d) and (e) a comparison of the estimated andmeasured PCC voltages and currents is presented. From thesefigures the basic operation principle of the EKF can be seen.Once the grid impedance change occurs (here at t = 0 s) theestimation error of the PCC currents is increased drastically.This increased estimation error is decreased by adjusting theequivalent grid parameters to minimize the estimation error.

Fig. 5 presents the estimated equivalent grid voltagewaveforms. The measurements reveal small unbalancedvoltage contents as well as high 5

th and 7

th harmonic voltage

contents of the equivalent grid voltage. The converter’soperation point during the emulated impedance step is

presented in Fig. 6. A 7 bit pseudo-random bit-sequence(PRBS) with a signal length of 2.54 s and an excitation of 8 Ais used to manipulate the converter control’s d and qcomponent reference currents.

VI.

DISCUSSION The effectiveness of the presented EKF algorithm for

equivalent grid parameter estimation is presented by theoreticaland practical analysis. However, there are some limitations andopen issues of the proposed EKF algorithm that are discussedin this section.

First, it is assumed that the equivalent grid impedance isinductive-resistive. Once the power network reveals capacitivecomponents (e.g. due to capacitor-banks used for reactive

power compensation) this impedance model is insufficient.Basically, this problem can be solved by including thecapacitive components in the grid model which will be thescope of future work.

Second, the tuning of the covariance matrices of the EKF is based on trial-and-error procedure. Additional analysis has to be performed to present a straight forward tuning guideline forthe Q and R matrices. This will be the scope of future research.

Third, the analysis of the EKF algorithm is performed byusing a single grid connected converter. Once multipleconverters are connected to one PCC bus, these converters willinfluence the PCC voltages dependent on their operationconditions. These grid distortions generated by other powerconverters might affect the detection performance of the

proposed EKF algorithm. Further analysis has to be carried outto address this issue.

VII. CONCLUSION

A real-time estimation of the equivalent grid impedanceand the equivalent grid voltage seen from a power converterconnected to an inductive-resistive distributed power network

by means of Extended Kalman-Filter is presented. TheExtended Kalman-Filter approach used for equivalent gridimpedance estimation is introduced by providing a detailed

physical, an observer and a mathematical problem formulation.Practical aspects like the use of the filter in distorted gridvoltage conditions, the tuning of the noise covariance matricesand implementation issues on a real-time system are discussed.

The theoretical analysis is verified on a 22 kW laboratorysystem where a grid impedance emulator is used to simulategrid impedance steps. The proposed Kalman Filter approachutilizes the noise that is already present at the point of powerconverter coupling to overcome the need of active disturbance

injection to estimate the equivalent grid impedance. Thus,electrical equipment connected close to the grid connectedconverter is only affected marginally by the equivalent gridimpedance estimation technique. The measurement resultsreveal that the proposed extended Kalman Filter for gridimpedance identification is able to detect the actual gridconditions with a adequate accuracy and in real-time duringregular converter operation with excellent detection dynamicsin the range of two fundamental voltage periods.

ACKNOWLEDGMENT

This work is financed by the Ministry of Schleswig-Holstein and the European Union and is operated underCewind e.G. Center of Excellence for Wind energy in

Schleswig-Holstein.The authors would also like to thank P.B. Thøgersen, L.Asiminoaei and A. Osmanbasic for their valuable contributionsduring the theoretical and practical analysis.

I q

/ A

TABLE III

GRID IMPEDANCE CONDITIONS

Grid condition in Laboratory

L grid = 150 µ H, R grid = 325 mΩ ,

S sc = 487 kVA, S sc,pu = 22.15

Emulated grid condition I

Ltotal = 650 µ H, Rtotal = 350 mΩ ,

S sc = 395 kVA, S sc,pu = 17.95

Emulated grid condition II

Ltotal = 1150 µ H, Rtotal = 375mΩ ,

S sc = 307 kVA, S sc,pu = 13.97

(a) (b)Fig. 6: System-excitation during impedance-step ( L grid 0.65 mH to 1.15 mH at t = 0 s): (a) reference- and measured

q-component PCC-currents and (b) reference- and measured d-component PCC-currents

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