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 Drives

Christian-Albrechts-University of Kiel Kiel, Germany

nho@tf.uni-kiel.de / fwf@uni-kiel.de

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 impedance steps 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. INTRODUCTION 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 power generated by distributed energy sources in addition to the slow process in the expansion of the public grid network structures leads to a high utilization of the existing grid structures. This high penetration of distributed energy sources has a strong impact 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 connected to the grid.

Distribution networks are characterized by a non- negligible grid impedance. The value of the grid impedance and the amount of connected apparent power determine the network distortions generated by electrical loads and distributed energy sources. To guarantee a defined and standardized electromagnetic compatibility (EMC) level for devices 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 grid impedance).

Besides the network distortions a non-negligible grid impedance is an important parameter that determines the interaction between distributed energy sources (and loads) among each other [5,6] and that determines the stability limits of the whole electrical power system [3,7]. Beyond that, the grid impedance influences the control performance of grid connected power converters [4,8]. Especially, when LCL type line filters are used that are not passively damped, a variation of the grid impedance can lead to drastic control performance degradation or even to unstable converter operation [8]. Furthermore, an active-filter-functionality to mitigate voltage unbalances and low order harmonic voltage contents can be implemented to the control of grid connected power converters used for distributed power sources dependent on the actual grid impedance conditions [9,10]. These aforementioned aspects reveal that the time and frequency dependent grid impedance is one key parameter to optimize the operation and performance of distributed energy sources in the public electrical distribution networks.

To optimize the performance of individual converter-based energy sources in relation to their control, their interaction with other converters and their performance in the overall distributed power system it is essential to determine the grid impedance during the converter operation. Basically, the grid impedance can be determined in two different ways: Based on several measurements in the distributed power network (often in addition to the injection of forced network disturbances) the grid impedance characteristic can be calculated offline by means of a detailed frequency-response analysis [11-15]. Usually, these grid impedance measurement techniques achieve a high precision over a wide frequency band but are not able to provide the information about the determined grid impedance in an online manner. Further, these methods often use additional power electronic hardware that has to be connected to the power network under investigation.

To determine the grid impedance characteristic online by utilizing the power converter setups that are already used to feed-in the generated electrical power several methods are documented in literature [16-21]. These methods can be classified in passive-methods [19] (or non-invasive methods),

978-1-4673-0803-8/12/$31.00 ©2012 IEEE 922

in active-methods [16,21] (or invasive methods) and in quasi-passive methods [17,18,20]. Passive methods are referred to as methods that utilize the existing disturbances that are already present in the power networks. These methods often detect the grid impedance at individual characteristic disturbances, e.g. the fundamental- or low-order harmonic frequencies. Active methods use a forced disturbance that is injected to the grid in parallel to the regular operation of the power converter. Based on the signal used for injection these methods are able to determine the grid impedance at individual frequencies or in characteristic frequency bands. Quasi passive methods are referred to as impedance determination methods that use hybrid identification methods. This is often achieved by forced operation point changes of the power converter.

This paper aims to contribute to the online grid impedance identification by presenting a new parameter estimation method that is based on an Extended Kalman-Filter (EKF). To achieve a practical estimation algorithm that can be used in addition to the control of grid connected converters, the proposed EKF is designed to utilize the disturbances that are already presented in typical power networks. One of the main goals is to use as less as possible forced operation point changes to estimate the grid impedance in an acceptable precision range. The study is focused on typical low voltage distributed power generation networks where the grid impedance is (in most cases) inductive-resistive. Special attention is paid to practical implementation aspects and the performance of the proposed EKF during transient grid impedance changes. Measurements are carried out utilizing a 22 kW laboratory test-bench.

This paper is structured as follows: In the second chapter a system description is given. The general idea of an EKF used for parameter estimation is presented in the third chapter. The system modeling and the state prediction that is needed for the EKF algorithm is explained in the fourth chapter. To validate the theoretical design a measurement study the fifth chapter presents a measurement analysis. The paper is closed by a critical discussion of the presented EKF algorithm and a conclusion.

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

II. SYSTEM DESCRIPTION In Fig. 1 the system under investigation is illustrated. The

system consists of converter that is connected to a time variant and frequency dependent power network.

A. Grid connected converter A two-level voltage source converter is considered as a

typical and widespread converter topology that is used for grid connection of distributed (renewable) energy sources in low voltage applications. A LCL-type line filter is used to reduce the voltage distortions that are generated by the switching transitions of the power semiconductors, usually insulated gate bipolar transistors (IGBTs). A pulse width modulation (PWM) based on space-vector modulation (SVM) and the well known voltage-oriented control (VOC) scheme are applied to the converter to control the DC link voltage and the currents at the point of common coupling (PCC) [22]. Even though no passive damping 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 control design [23]. To synchronize a converter with the fundamental grid voltage under highly distorted grid conditions a double second-order generalized integrator (DSOGI) phase-locked loop (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 many

cases a complex structure that consists of power transformers, transmissions lines, linear and non-linear loads and distributed energy sources. Due to transient load conditions, the connection or disconnection of individual energy sources (and sinks) or even the connection or disconnection of whole sub-networks, the network structure (and thus the resultant grid impedance) is non-stationary and time-variant. Low voltage distribution networks reveal (in most cases, but not in general) a time and frequency dependent resistive-inductive grid impedance characteristic which is confirmed by several measurement studies performed in representative power distribution networks [25-27]. Bearing in mind that the grid impedance (which affects a power converter) dependents on the localization of the converter in the whole network, the Thévenin equivalent is used to model the time variant and frequency dependent grid impedance. In Fig. 2 the used one-line block diagram of the (Thévenin) equivalent grid impedance model is presented. The model consists of an equivalent grid inductance Lgrid, an equivalent grid resistance Rgrid and an equivalent grid voltage Ugrid. It is assumed that the power network is composed of a three-phase three-wire system and that the equivalent grid impedance is (almost) symmetrical.

923

III. EXTENDED KALMAN-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 well established 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 theoretical understanding of the EKF applied for parameter estimation some 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 state vector, u as the input and y as the output vector. The system is superimposed 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 Cd are time invariant. 1 (1)

To use an observer based parameter estimation approach it is convenient to summarize all relevant time dependent system parameters in a parameter vector θ. This parameter vector is added to the system state vector which results in a non linear system description of the linear system with time variant parameters (2). Now, the process noise model has to be extended to the process noise in relation to the primary state-vector εx and the process noise in relation to the parameter-vector εθ. The physical measurement model remains unaffected. 11 , , 00 0 0

(2)

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

divided into a prediction and an update step. First, the a priori system 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 noise characteristic and is independent from the measurement noise the a priori error-covariance Pk| a priori is estimated. Here, Qk is referred to as the process noise covariance matrix which includes the process noise model of the system. In the next step of the EKF algorithm, the update step, the actual system measurements y are used to calculate the Kalman gain Kk that is 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 Pk| a posteriori is calculated that is used for the prediction step in the next sampling instant. This recursive nature of the EKF algorithm ensures a fast and accurate state estimation even in highly distorted and noisy measurement environments 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. The following paragraphs present the development of the model used for the prediction step and update step of the EKF algorithm.

A. Basic system model A general system description is presented in the second

chapter whereas the model of the equivalent grid impedance is illustrated in Fig. 2. The three-phase PCC voltages and currents of the three-phase three-wire system are measured. A convenient state-space presentation is achieved by selecting the α- and β-components of the PCC currents for the state-vector formulation, 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 Ugrid are used to expand the state-vector for a convenient state-space formulation (4).

,, ,, 000000 000

000

,, ,,0 000 00 ,,

(4)

Fig. 3: Cycle-diagram of Extended Kalman-Filter algorithm for

parameter estimation [31]

924

To study the observability of the presented disturbance observer formulation (when the PCC currents and voltages are measured) the Kalman criterion is used. The analysis reveals that the system is observable for positive values (unequal zero) of Lgrid and Rgrid.

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 reformulate the disturbance observer model presented in (4) with the goal to include the measured PCC voltages into the measurement model of the observer formulation. This leads to the observer formulation presented in (5). In respect to the underlying EKF algorithm this mathematical reformulation provides the advantage to include the measurement noise models of both, the voltage- and current-measurements into the EKF algorithm.

,,,,,,00000

00000

0000000000

0000000000:

,,,,,,

(5)

,,,,1000

01000010

00010000

0000,,,,,,

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

D. Observer formulation in discrete time-domain The EKF algorithm is implemented in a sampled system

with zero-order hold characteristic. The zero-order hold equivalent in discrete time-domain of the state-space model formulation in continuous time-domain (5) is derived by applying the transformation law presented in (6). |

(6)

The resultant observer formulation in discrete time-domain is presented in (7) whereas the measurement matrix Cd remains unaffected by the transformation. , 1, 1, 1, 1, 1, 1

0000000000

0100000100

0001000001:

,,,,,,

(7)

exp

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

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

discrete time-domain (7). This model is insufficient when the EKF algorithm is applied to sinusoidal voltage and current waveforms that are distorted with unbalances and harmonics. A more precise formulation of the expected disturbances of the equivalent grid voltages is necessary. A practicable and adequate choice of the extended disturbance model is presented in (8).

, 1, 1, 1, 1

exp Δ0000exp Δ00

00exp 5Δ0000exp 7Δ

,,,, (8)

The equivalent grid voltage is modeled as a superposition of a positive Ugrid,1+ and negative fundamental Ugrid,1- voltage sequence. Further, the 5th harmonic (negative sequence, Ugrid,5-) and 7th harmonic (positive sequence, Ugrid,7+) voltages are included into the disturbance model. The prediction model is calculated by considering the rotation angle resolution Δθ to the sinusoidal disturbance model in the stationary αβ-reference frame. The rotation angle resolution depends on the applied sampling time Ts and the fundamental grid frequency fgrid, see (9). Δ 2 (9)

F. Introduction of time variant parameter vector The final step of the mathematical problem formulation is

to extend the state-vector with the time-variant system parameters that have to be estimated (here: Lgrid, Rgrid). Two simplifications are used to achieve a convenient mathematical formulation of the observer problem. First, the exponential dependency (see (7)) of the system parameters that results from the discrete observer formulation is simplified using Taylor series expansion. exp 1 (10)

Second, to greatly simplify the calculation of the Jacobean matrix Fk during the prediction step of the EKF algorithm (see Fig. 3) the substitution presented in (11) is used. 1

(11)

The proposed EKF algorithm estimates the reciprocal inductance lgrid to reduce the calculation burden of the prediction step. To achieve the actual inductance Lgrid the estimated value is then inverted.

G. Final observer formulation, prediction and measurement model The aforementioned simplifications and the proposed

observer problem formulation in the discrete time-domain lead to the final observer formulation that is presented in (12). The resultant state-vector consists of the measured PCC currents and voltages, the equivalent grid voltages including the extended disturbance model formulation and the equivalent grid parameters. The additional terms qi and ri are introduced

925

to highlight that the proposed model includes process noise and measurement noise. Further, the measurement noise covariance matrix R has a 4x4 form, is assumed to be time independent and assumed to have a diagonal-from. The process noise covariance matrix Q has a 14x14 form, is also assumed to be time-independent and assumed to have a diagonal form to simplify the EKF tuning. , … , , , … , ,

(13)

V. MEASUREMENT ANALYSIS Measurements are carried out to validate the theoretical

analysis under laboratory conditions.

A. Test-bench description and measurement methodology A 22 kW test setup is used to verify the proposed EKF

algorithm for equivalent grid parameter estimation. The setup consists 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 impedance emulator. The grid impedance emulator is used to emulate (step wise) grid impedance changes under laboratory conditions. The emulator is connected between the grid connected converter and the actual grid of the institute’s laboratory. Three multi tap inductors are used to emulate different relative short circuit powers (Ssc/SN) 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, 111

1 , , , , , ,1 , , , , , ,,,, cos Δ , sin Δ, sin Δ , cos Δ, cos Δ , sin Δ, sin Δ , cos Δ, cos 5Δ , sin 5Δ, sin 5Δ , cos 5Δ, cos 7Δ , sin 7Δ, sin 7Δ , cos 7Δ:

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

(12)

, 1, 1, 1, 1

10000100

00100001

00000000

00000000

00000000

00000000

00000000

,,,,

‐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 (Lgrid 0.65 mH to 1.15 mH at t = 0 s): (a) estimated equivalent grid

inductance Lgrid, (b) estimated equivalent grid resistance Rgrid, (c) estimated relative grid short-circuit power Ssc,pu, (d) estimated and measured PCC-currents Ipcc and (e) estimated and measured PCC-voltages (line-to-neutral) Upcc

926

performance of the grid impedance estimation algorithm additional IGBTs (1200V, 200A) are used to switch between the taps of the inductors [32]. Further, the actual equivalent grid impedance parameters (i.e. when the grid impedance emulator and converter are disconnected) of the institutes laboratory are measured for each experiment with an additional grid impedance analyzer [25] that is connected to the PCC of the test setup. The control and estimation algorithms are implemented on a dSPACE DS1006 system.

TABLE I MEASUREMENT SYSTEM PARAMETERS

Symbol Quantity Value (per unit) ULL Line-to-Line Voltage (rms) 400 V (1.0) UDC DC-link voltage 700 V (1.75) ω Angular line frequency 2π 50 Hz (1.0) IL Rated converter current (rms) 32 A (1.0) Lfg Line-side filter inductance 1.5 mH (0.06) Rfg Line-side filter resistance 90 mΩ (0.01) Lfc Converter-side filter inductance 2.5 mH (0.11) Rfc Converter-side filter resistance 150 mΩ (0.021) Cf Filter capacitance 16.2 μF (0.04)

CDC DC-link capacitance 2200 μF (5.0) fcon/fcarr Control-/Carrier-frequency 5 / 2.5 kHz (100/50)

B. Extended Kalman-filter tuning The tuning of the EKF is a complex problem. The

measurement noise covariance matrix R consists of 4 unknowns and the process noise covariance matrix Q consists of 14 unknowns that influence the estimation performance of the EKF algorithm. Even though some tuning guidelines can

be found in literature (e.g. in [30,33]) the tuning of the EKF algorithm remains difficult. Here, the EKF parameter tuning is based on a trial-and-error procedure since the main focus of this work is set on proofing the effectiveness of the EKF applied for grid parameter estimation. Several experiments under different grid and operation point conditions are performed to achieve an EKF covariance matrix parameter setting that leads to an acceptable grid parameter estimation performance. The selected EKF tuning parameters are summarized 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 of 13.97 (grid condition II, additional 1.0 mH). The emulated grid conditions (including the actual measured laboratory grid conditions) are summarized in Table III.

A grid impedance step between the grid conditions I and II is performed with the grid impedance emulator whereas a data acquisition system (a Dewetron DEWE-2010 system) is triggered on the impedance step. The results are summarized in Fig. 4 and Fig. 5. In Fig. 4 (a) and (b) the estimated and actual equivalent grid inductance and -resistance are presented. It can be seen that the grid inductance is estimated with a bias of approximately 50 μH 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 measurement results into a per-unit perspective Fig. 4 (c) presents the comparison of the estimated and measured relative short-circuit power of the emulated grid conditions. This comparison

TABLE II KALMAN-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 (Lgrid 0.65 mH to 1.15 mH at t = 0 s): (a) estimated positive sequence Upcc,1+, (b) estimated negative sequence Upcc,1-, (c) estimated 5th-harmonic Upcc,5-

and (d) estimated 7th-harmonic Upcc,7+

927

reveals that the equivalent grid parameter estimation accuracy is 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). A settling-time of approximately two fundamental grid voltage periods is measured whereas the rise-time is approximately half of the fundamental grid voltage period.

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

Fig. 5 presents the estimated equivalent grid voltage waveforms. The measurements reveal small unbalanced voltage contents as well as high 5th and 7th harmonic voltage contents of the equivalent grid voltage. The converter’s operation 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 A is used to manipulate the converter control’s d and q component reference currents.

VI. DISCUSSION The effectiveness of the presented EKF algorithm for

equivalent grid parameter estimation is presented by theoretical and practical analysis. However, there are some limitations and open issues of the proposed EKF algorithm that are discussed in this section.

First, it is assumed that the equivalent grid impedance is inductive-resistive. Once the power network reveals capacitive components (e.g. due to capacitor-banks used for reactive power compensation) this impedance model is insufficient. Basically, this problem can be solved by including the capacitive components in the grid model which will be the scope 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 for the Q and R matrices. This will be the scope of future research.

Third, the analysis of the EKF algorithm is performed by using a single grid connected converter. Once multiple converters are connected to one PCC bus, these converters will influence the PCC voltages dependent on their operation conditions. These grid distortions generated by other power converters might affect the detection performance of the proposed EKF algorithm. Further analysis has to be carried out to address this issue.

VII. CONCLUSION A real-time estimation of the equivalent grid impedance

and the equivalent grid voltage seen from a power converter connected to an inductive-resistive distributed power network by means of Extended Kalman-Filter is presented. The Extended Kalman-Filter approach used for equivalent grid impedance 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 grid voltage conditions, the tuning of the noise covariance matrices and implementation issues on a real-time system are discussed.

The theoretical analysis is verified on a 22 kW laboratory system where a grid impedance emulator is used to simulate grid impedance steps. The proposed Kalman Filter approach utilizes the noise that is already present at the point of power converter coupling to overcome the need of active disturbance injection to estimate the equivalent grid impedance. Thus, electrical equipment connected close to the grid connected converter is only affected marginally by the equivalent grid impedance estimation technique. The measurement results reveal that the proposed extended Kalman Filter for grid impedance identification is able to detect the actual grid conditions with a adequate accuracy and in real-time during regular converter operation with excellent detection dynamics in 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 under Cewind 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 contributions during the theoretical and practical analysis.

I q/

A

TABLE III GRID IMPEDANCE CONDITIONS Grid condition in Laboratory

Lgrid = 150 μH, Rgrid = 325 mΩ, Ssc = 487 kVA, Ssc,pu = 22.15

Emulated grid condition I

Ltotal = 650 μH, Rtotal = 350 mΩ, Ssc = 395 kVA, Ssc,pu = 17.95 Emulated grid condition II

Ltotal = 1150 μH, Rtotal = 375mΩ, Ssc = 307 kVA, Ssc,pu = 13.97

(a) (b) Fig. 6: System-excitation during impedance-step (Lgrid 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

928

REFERENCES [1] F. Blaabjerg, F. Iov, T. Terekes, R. Teodorescu, and K. Ma, "Power

electronics - key technology for renewable energy systems," in Proc. Power Electronics, Drive Systems and Technologies Conference (PEDSTC), 2011 2nd, 2011, pp. 445-466.

[2] A. M. Azmy and I. Erlich, "Impact of distributed generation on the stability of electrical power system," in Proc. Power Engineering Society General Meeting, 2005. IEEE, 2005, pp. 1056-1063.

[3] S. Jian, "Small-Signal Methods for AC Distributed Power Systems - A Review," Power Electronics, IEEE Transactions on, vol. 24, no. 11, pp. 2545-2554, Nov.2009.

[4] M. Liserre, R. Teodorescu, and F. Blaabjerg, "Stability of photovoltaic and wind turbine grid-connected inverters for a large set of grid impedance values," Power Electronics, IEEE Transactions on, vol. 21, no. 1, pp. 263-272, Jan.2006.

[5] J. L. Agorreta, M. Borrega, J. Lopez, and L. Marroyo, "Modeling and Control of N-Paralleled Grid-Connected Inverters With LCL Filter Coupled Due to Grid Impedance in PV Plants," Power Electronics, IEEE Transactions on, vol. 26, no. 3, pp. 770-785, Mar.2011.

[6] J. H. R. Enslin and P. J. M. Heskes, "Harmonic interaction between a large number of distributed power inverters and the distribution network," Power Electronics, IEEE Transactions on, vol. 19, no. 6, pp. 1586-1593, Nov.2004.

[7] S. Jian, "Impedance-Based Stability Criterion for Grid-Connected Inverters," Power Electronics, IEEE Transactions on, vol. 26, no. 11, pp. 3075-3078, Nov.2011.

[8] S. Cobreces, E. Bueno, F. J. Rodriguez, F. Huerta, and P. Rodriguez, "Influence analysis of the effects of an inductive-resistive weak grid over L and LCL filter current hysteresis controllers," in Proc. Power Electronics and Applications, 2007 European Conference on, 2007, pp. 1-10.

[9] N. Hoffmann, F. W. Fuchs, and L. Asiminoaei, "Online grid-adaptive control and active-filter functionality of PWM-converters to mitigate voltage-unbalances and voltage-harmonics - a control concept based on grid-impedance measurement," in Proc. Energy Conversion Congress and Exposition (ECCE), 2011 IEEE, 2011, pp. 3067-3074.

[10] A. Tarkiainen, R. Pollanen, M. Niemela, and J. Pyrhonen, "Identification of grid impedance for purposes of voltage feedback active filtering," Power Electronics Letters, IEEE, vol. 2, no. 1, pp. 6-10, Mar.2004.

[11] A. de Oliveira, J. C. de Oliveira, J. W. Resende, and M. S. Miskulin, "Practical approaches for AC system harmonic impedance measurements," Power Delivery, IEEE Transactions on, vol. 6, no. 4, pp. 1721-1726, Oct.1991.

[12] J. P. Rhode, A. W. Kelley, and M. E. Baran, "Complete characterization of utilization-voltage power system impedance using wideband measurement," Industry Applications, IEEE Transactions on, vol. 33, no. 6, pp. 1472-1479, Nov.1997.

[13] Z. Staroszczyk, "A method for real-time, wide-band identification of the source impedance in power systems," Instrumentation and Measurement, IEEE Transactions on, vol. 54, no. 1, pp. 377-385, Feb.2005.

[14] M. Sumner, B. Palethorpe, D. W. P. Thomas, P. Zanchetta, and M. C. Di Piazza, "A technique for power supply harmonic impedance estimation using a controlled voltage disturbance," Power Electronics, IEEE Transactions on, vol. 17, no. 2, pp. 207-215, Mar.2002.

[15] M. Sumner, B. Palethorpe, and D. W. P. Thomas, "Impedance measurement for improved power quality-Part 1: the measurement technique," Power Delivery, IEEE Transactions on, vol. 19, no. 3, pp. 1442-1448, July2004.

[16] L. Asiminoaei, R. Teodorescu, F. Blaabjerg, and U. Borup, "Implementation and Test of an Online Embedded Grid Impedance Estimation Technique for PV Inverters," Industrial Electronics, IEEE Transactions on, vol. 52, no. 4, pp. 1136-1144, Aug.2005.

[17] M. Ciobotaru, R. Teodorescu, P. Rodriguez, A. Timbus, and F. Blaabjerg, "Online grid impedance estimation for single-phase grid-

connected systems using PQ variations," in Proc. Power Electronics Specialists Conference, 2007. PESC 2007. IEEE, 2007, pp. 2306-2312.

[18] M. Ciobotaru, V. Agelidis, and R. Teodorescu, "Line impedance estimation using model based identification technique," in Proc. Power Electronics and Applications (EPE 2011), Proceedings of the 2011-14th European Conference on, 2011, pp. 1-9.

[19] S. Cobreces, E. J. Bueno, D. Pizarro, F. J. Rodriguez, and F. Huerta, "Grid Impedance Monitoring System for Distributed Power Generation Electronic Interfaces," Instrumentation and Measurement, IEEE Transactions on, vol. 58, no. 9, pp. 3112-3121, Sept.2009.

[20] M. Liserre, F. Blaabjerg, and R. Teodorescu, "Grid Impedance Estimation via Excitation of LCL-Filter Resonance," Industry Applications, IEEE Transactions on, vol. 43, no. 5, pp. 1401-1407, Sept.2007.

[21] A. V. Timbus, R. Teodorescu, F. Blaabjerg, and U. Borup, "Online grid measurement and ENS detection for PV inverter running on highly inductive grid," Power Electronics Letters, IEEE, vol. 2, no. 3, pp. 77-82, Sept.2004.

[22] N. Hoffmann, F. W. Fuchs, and J. Dannehl, "Models and effects of different updating and sampling concepts to the control of grid-connected PWM converters - A study based on discrete time domain analysis," in Proc. Power Electronics and Applications (EPE 2011), Proceedings of the 2011-14th European Conference on, 2011, pp. 1-10.

[23] J. Dannehl, C. Wessels, and F. W. Fuchs, "Limitations of Voltage-Oriented PI Current Control of Grid-Connected PWM Rectifiers With LCL Filters," Industrial Electronics, IEEE Transactions on, vol. 56, no. 2, pp. 380-388, Feb.2009.

[24] N. Hoffmann, R. Lohde, M. Fischer, F. W. Fuchs, L. Asiminoaei, and P. B. Thogersen, "A review on fundamental grid-voltage detection methods under highly distorted conditions in distributed power-generation networks," in Proc. Energy Conversion Congress and Exposition (ECCE), 2011 IEEE, 2011, pp. 3045-3052.

[25] A. Knop and F. W. Fuchs, "High frequency grid impedance analysis by current injection," in Proc. Industrial Electronics, 2009. IECON '09. 35th Annual Conference of IEEE, 2009, pp. 536-541.

[26] M. Sigle, W. Liu, and K. D. Karlsruhe, "On the impedance of the low-voltage distribution grid at frequencies up to 500 kHz," in Proc. Power Line Communications and Its Applications (ISPLC), 2012 16th IEEE International Symposium on, 2012, pp. 30-34.

[27] Z. Staroszczyk, "Impedance in voltage-current relations description of the power system PCC - experimental investigations of the accuracy of the LTI system model," in Proc. Electrical Power Quality and Utilisation, 2009. EPQU 2009. 10th International Conference on, 2009, pp. 1-6.

[28] R. Isermann and M. Münchhof, Identification of Dynamic Systems - An Introduction with Applications, 1 ed Springer Verlag, 2011.

[29] S. Bolognani, R. Oboe, and M. Zigliotto, "Sensorless full-digital PMSM drive with EKF estimation of speed and rotor position," Industrial Electronics, IEEE Transactions on, vol. 46, no. 1, pp. 184-191, Feb.1999.

[30] S. Bolognani, L. Tubiana, and M. Zigliotto, "Extended Kalman filter tuning in sensorless PMSM drives," Industry Applications, IEEE Transactions on, vol. 39, no. 6, pp. 1741-1747, Nov.2003.

[31] G. Welch and G. Bishop, "An Introduction to the Kalman Filter," Tech. Rep. SIGGRAPH 2001 Course at University of North Carolina at Chapel Hill, 2004.

[32] F.W.Fuchs, F.Gebhardt, N.Hoffmann, A.Knop, R.Lohde, J.Resse, and C.Wessels, "Research Laboratory for Grid-Integration of Distributed Renewable Energy Resources - Design and Realization," in Proc. Energy Conversion Congress and Exposition (ECCE), 2012 IEEE, accepted for publication.

[33] R. Mehra, "On the identification of variances and adaptive Kalman filtering," Automatic Control, IEEE Transactions on, vol. 15, no. 2, pp. 175-184, Apr.1970.

929