Fig. 1. Cascaded control scheme for three-phase grid-connected inverter.
This work presents a method for adaptive control of grid-connected three-phase inverters. The method adds an additional algorithm to converter control, which measures the grid impedance during the start up, and chooses the most suitable control parameters from preset table. The control parameters are chosen based on generalized Nyquist criterion assessment (GNC) of the grid interface, and the use of sen- sitivity function for ensuring sufficient stability margins. In this work, the adaptively adjusted parameters are bandwidth of the phase-locked loop (PLL) and gain of proportional grid- voltage feedforward, as these have been shown to have great impact on the converter output impedance [13]–[15].
The method is verified with power hardware-in-the-loop (PHIL) experiments consisting of three-phase inverter, DC source, linear amplifier acting as grid emulator, and real- time grid simulator. The method is tested in varying grid conditions, including strong, weak, and resonant grids. The adaptive scheme shows improved performance in strong grids and ensures stability in challenging grid conditions.
II. THEORY A. Three-Phase Grid-Connected Inverter
Sinusoidal three-phase waveforms can be presented in a synchronous reference frame as two DC-valued signals by applying Park’s transformation. The attained DC values are direct (d) and quadrature (q) components of the three-phase signals. Converting the sinusoidal values into dq-domain allows linearization of the system around its operating point.
The control of the inverter can be then designed by using a small-signal model derived from the linearized equations [16].
The linearized model of the inverter can be transformed into frequency domain and used to form open-loop transfer functions between input and outputs of the system. The transfer function can then be formed into matrix form
Adaptive Method for Control Tuning of Grid-Connected Inverter Based on Grid Measurements During Start-Up
RoniLuhtala1 and Tommi Reinikka1 and HenrikAlenius2 andTomi Roinila1and TuomasMesso2
Abstract— As the share of grid-connected converters in- creases, potential stability issues in the grid interface of the converters are emphasized. Impedance-based stability criterion can be used to assess the stability of the interface accurately, and the grid conditions should be considered in control design of the converter. However, the grid impedance is often an unknown parameter, and thus, adaptive control is a favorable method for addressing the uncertainties in the control design. This work presents an algorithm, in which the full-order grid impedance is measured during the start-up of the device, and control bandwidth of phase-locked loop and grid-voltage feedforward gains are adjusted to optimize sensitivity function in the grid interface, based on generalized Nyquist criterion. The method can ensure robust control in weak grid conditions, while still achieving good control performance in strong grids. The method is verified w ith p ower h ardware-in-the-loop e xperiments with a kW-scale grid-connected three-phase inverter.
I. INTRODUCTION
The electricity production is shifting towards renewable energy sources with a growing rate [1], resulting in increas- ing share of grid-connected power electronics. The dynamics of energy production through inverters differ drastically from conventional rotating machines, mainly due to very fast dynamics of switched-mode devices [2]. Stability and power qualityissueshavealreadybeenreportedingridswithhigh penetrationof converters [3]–[5].
The grid-interface of the converter is prone to adverse interactions, especially in weak grids with high equivalent impedance.Impedance-basedstabilityassessmenthasproven to predict these interactions accurately, by comparing the out- put impedance of converter and grid impedance [6]. The con- verter impedance can be modeled reasonably accurately, but the grid impedance is often unknown and possibly resonant or time-variant [7], [8]. Consequently, accurate consideration of grid conditions is difficult in control design of converters, which often results in unnecessarily high stability margins and robustness of converter controls. However, unnecessarily robust control causes decreased performance in strong grid conditions.
Inordertoaddresstheseissues,adaptivecontrolbasedon measurements of the grid has beenproposed [9], [10].The grid impedance can be measured directly with the inverter [11], [12], and the obtained results can be used in tuning of control parameters to better match the grid conditions.
By adapting the converter to grid, the performance can be optimized while still preserving sufficient s tability margins.
1Luhtala,Reinikka,andRoinilaarewithLaboratoryofAutomationand Hydraulics,TampereUniversityofTechnology,Finland
2AleniusandMessoarewithLaboratoryofElectricalEnergyEngineer- ing,TampereUniversityofTechnology,Finland
Fig. 2. Effect of different PLL bandwidths and feedforward gains to inverter output impedance qq component.
representing the system in dq-domain including the cross- couplings between the two components [17].
vdc idq
=
Zin-o Toi-o Gci-o Gio-o −Yo-o Gco-o
⎡⎣ idc
vdq ddq
⎤
⎦ (1)
The open-loop transfer functions are used to design the cascaded control scheme of the inverter, which is shown in Fig. 1. The inner loop of the control is the output current con- trol. The d-component current control reference is provided by the DC-voltage control and the q component is regulated for unity power factor. The control is synchronized to power grid with the use of the PLL. Grid-voltage feedforward is used for mitigating effect of grid-voltage harmonics on output currents.
B. Stability assessment
The small-signal stability of the grid connection of the inverter can be assessed by the ratio of the inverter output impedanceZo and the power grid impedanceZgat the point of common coupling (PCC). In dq-domain, the impedances are 2x2 matrices with direct components (Zdd and Zqq) and cross-couplings (Zdq andZqd). In the impedance-based stability analysis the inverter is simplified into a Norton equivalent circuit and the power grid to a Thevenin equivalent circuit. The simplification can be done when the inverter is assumed stable when connected to an ideal power grid and the grid is assumed stable without the inverter. This allows the stability to assessed by applying the GNC to the return ratio [18]
Gs(s) = [I+Z−1o (s)Zg(s)]−1 (2) The PCC is stable if a GNC loci of the return-ratio matrix Z−1o (s)Zg(s) satisfies the GNC and the grid connection is robust under distortion if it has sufficient stability margins.
The sufficient stability margins are defined by maximum peak criterion (MPC) which defines a forbidden zone in the Nyquist diagram [19], [20].
C. Impact of Control Design to Inverter Output Impedance The inverter output impedance qq component (and slightly qd component) is affected by the PLL which introduces
negative-resistance-like behavior below its control band- width. Due to PLL, the inverter output impedance loses its passive characteristics (phase decreases below -90 degrees) in low frequencies [21]. The stability issues occur when inverter with the high-bandwidth PLL is connected to a grid with high impedance and a phase close to 90 degrees in relatively low frequencies. To avoid the stability issues, the PLL bandwidth is usually limited, which weakens its phase- tracking performance.
The grid-voltage feedforward is used to mitigate low- order harmonics from output currents, originating from grid-voltage disturbances through an inverter open-loop ad- mittance Yo-o. So, the feedforward introduces a virtual impedance which compensates the open-loop admittance, and thus increases the inverter output impedance. The ideal gain for the feedforward path, that fully compensates the open-loop admittance, is based on the DC-side voltage as KFF = 1/VDC. [14] However, the average control delay of 1.5 times the switching period produces significant phase lag which causes inverter to lose its passivity. Thus, the feedforward gain has to be limited with certain types of grid impedance to avoid impedance-based stability issues. [22]
Fig. 2 shows the impact of different PLL bandwidths and feedforward gains on the inverter output impedance qq component. The figure illustrates the PLL bandwidths from 1 to 100 Hz and feedforward gains from 0 % to 100 % of its ideal gain, so that the lowest values are represented in blue and the highest in red. Impact of the PLL is limited to low frequencies where the magnitude is decreased and the phase stays close to -180 degrees, starting to increase around frequency of the PLL control bandwidth. Impact of the feedforward occurs in higher frequencies than PLL, where the phase of the inverter impedance is decreased but the magnitude is increased. Due to high feedforward gain, the inverter output impedance can be non-passive up to 300 Hz. While PLL has a clear impact only to qq component and slight impact on the crosscouplings, the feedforward affects both the dd and qq components in a similar manner.
III. METHODS
A. Test environment
The experiments are performed with a PHIL test bench, which allows connecting the tested inverter to an emulated model of a power grid [23]. This allows realistic and sophisti- cated test scenarios in comparison with simplified hardware implementations of the grid impedance. The status of the power grid can be modified while the system is online, for example by simulating grid faults or other changing conditions.
The interface between grid emulation and hardware can face stability issues, as the delay in the system can destabilize the PHIL setup. The delay is caused by conversion between analog and digital signals, calculation time of the simulation, and response time of the amplifier. Due to potential issues at the interface, the stability of the system should be assessed before turning the system online.
Fig. 3. Grid impedance models for three different cases.
The stability of the system can be evaluated by examining the interface between the emulated power grid and the hardware under test, by analyzing the loop gain of the emulation [24], [25]. This requires identifying the interface algorithm used for the setup, transfer functions of the used equipment and the total delay in the system. The loop gain can be represented as
Ggrid=e−sΔTd Zs
ZHW
TampTfilt (3) where e−sΔTd is the delay of the emulation control loop, Zs is the simulated impedance, ZHW is the hardware impedance,Tampis the voltage amplifier transfer function, and Tfilt is the low-pass filter transfer function. Nyquist stability criterion can be used to evaluate the stability of the power grid emulation interface.
B. Grid impedances for case studies
Fig. 3 shows the three different grid conditions (strong, weak, resonant), which are used for testing the adaptive control method. In all cases, the inverter is connected to PCC through transmission line and isolation transformer, which produce a series resistance and inductance. The middle section of the grid is either
• direct connection to strong grid (a)
• weak grid connection through significant line induc- tance (b)
• resonant grid connection caused by e.g. capacitive VAR compensator (c)
The high voltage grid is assumed to have very high short- circuit ratio, and thus, negligible impedance. Table I presents the equivalent grid parameters.
TABLE I
PARAMETERS FOR AGGREGATED GRID IMPEDANCE MODEL. Parameter Symbol Value
PCC inductance Ltr 0.9 mH
PCC resistance Rtr 0.4Ω
Case B inductance Lb 8.0 mH Case C inductance Lc 4.0 mH Case C resistance Rc 15Ω Case C capacitance Cc 400μF
C. Grid impedance measurements
The grid impedance is measured with two broadband excitations. The dq-domain impedance is 2 x 2 matrix which can be simultaneously measured by using maximum-length
binary sequence (MLBS) and its inverse-repeated sequence (IRS). The length of IRS is double to MLBS but has even order harmonics suppressed, resulting in similar frequency resolution to MLBS. The sequences are orthogonal, and thus, their responses can be separated from each other in (nearly) linear systems [26]. The MLBS is injected to the d-component reference of AC-current control and IRS to q- component reference. The inverter produces the sequences to its output currents, and the current and voltage responses from the grid side are measured and transformed to frequency domain with fast Fourier transformation (FFT). From the measured responses, the complete dq-domain grid impedance can be calculated as
Zg-dd(s) Zg-qd(s) Zg-dq(s) Zg-qq(s)
=
Vd(s)/Id(s) Vd(s)/Iq(s) Vq(s)/Id(s) Vq(s)/Iq(s)
(4) The measurement time is
tmeas=NMLBS∗PMLBS/fgen (5) whereNMLBS is length of the MLBS, PMLBS is the number of MLBS response averaging, and fgen is the generation frequency.
D. Control Design
The inverter is controlled using cascaded control scheme.
This paper considers adaptive design rules for the PLL and the grid-voltage feedforward to ensure robustness and to improve control performance.
The stability issues related to the PLL in high-impedance grids are widely studied, and it is known that the PLL bandwidth has to be limited when the inverter is connected to a weak grid [21], [27], [28]. However, overly conservative PLL tunings weaken the phase-tracking performance and affect, for example, the accuracy of the angle of synchronous- reference frame.
The grid-voltage feedforward is an optional addition to control scheme to improve power quality. The feedforward has a dual impact on the impedance-based stability as higher gains produce more phase lag (decreasing stability margins) to the inverter impedance, while the feedforward also in- crease the magnitude (increasing stability margins). Thus, design of the feedforward gains requires detailed information of the grid, because resonances and other uncertainties in the grid impedance may cause stability issues.
E. Adaptive Control Algorithm
Optimal control parameters for the grid-connected inverter depend on the interfaced grid impedance, and therefore, the proposed adaptive control algorithm takes the grid impedance into account. Fig. 4 shows a flowchart of the proposed method which starts by measuring the grid-impedance before powering up the inverter. Based on the grid-impedance measurements, the GNC loci in the interface is calculated applying the analytical inverter output impedances with different control parameters, and the optimal parameters are chosen. Then, normal inverter operations are continued. The algorithm can also be executed during normal operation of the inverter with normal power production [27], [28].
Fig. 4. Flowchart of the adaptive control algorithm.
Nine different control-parameter sets are shown in Table II, with three different PLL bandwidths and feedforward gains. Lower PLL bandwidths are suitable for weak grids to increase robustness while higher bandwidths can be used in strong grids for faster phase-tracking performance. The optimal feedforward gain depends on the shape of the grid impedance, due to dualistic impact of the feedforward gains.
Thus, the possible gains are chosen as 0, 50, and 100 % of the ideal gain for achieving wide tuning range. A simple set of control parameters is applied as a proof-of-concept, and for clear demonstration. The method can be extended to include AC-current and DC-voltage control parameters, or to consist of drastically larger set of parameters.
TABLE II
PARAMETERS OF THE INVERTER. Parameter Set
1 2 3 4 5 6 7 8 9
PLL (Hz) 10 10 10 50 50 50 100 100 100
FF (p.u.) 0 0.5 1.0 0 0.5 1.0 0 0.5 1.0
Design rule for choosing the control parameter set is based on MPC with sensitivity peak as Speak = 2, which is equivalent to 1/2 as the minimum distance of the GNC loci to the critical point (-1 + j0) in complex plane. If the distance is less, the desired sensitivity peak is exceeded and the system robustness is compromised [20]. The distance to critical point is
d(s) =
(Re(s)−(−1))2+ (Im(s))2 (6) where Re(s)is a real part and Im(s)an imaginary part of the GNC loci, and dmin is the minimum distance to the critical point. If the closest point of the locus has negative imaginary part, the locus can be unstable without breaching MPC, and thus, corresponding parameter sets are neglected. The control parameter sets are compared using a penalty factor
P = dmin−0.5, if dmin≥0.5.
10(0.5−dmin)−1, if dmin<0.5. (7) which estimates how accurately they satisfy MPC (Speak= 2 i.e. dmin = 0.5). The design rule allows minor breaches
Fig. 5. Experimental setup.
of MPC, but weights penalty factor with higher coefficient inside MPC. Lowest penalty factor implies optimal behavior, and consequently, the set with lowest penalty factor is chosen.
IV. EXPERIMENTS
A. Power Hardware-in-the-Loop Setup
The experiments are carried out with a laboratory setup consisting of DC source, 10 kW three-phase inverter, LCL- filter, isolation transformer, and linear voltage amplifier. The linear voltage amplifier emulates the grid voltages and is controlled in real-time simulation, where the remainder of the power grid is simulated. Fig. 5 presents a block diagram of the setup.
B. Grid impedance measurements
The complete dq-domain grid impedance is measured using 2047-bit-long MLBS and corresponding IRS with generating frequency of 4 kHz. MLBS is injected to d component and IRS to q component of the current references.
The grid-side voltage and current responses are averaged over 100 periods for MLBS and 50 periods for IRS. Thus, frequency resolution is 1.95 Hz and measurement time 51.175 s. However, the binary sequence parameters can be changed to adjust the measurement resolution, accuracy, and duration.
Fig. 6 presents the measurements for the three grid config- urations presented in Fig. 3; weak (green), resonant (blue), and strong (red) grids show very different behaviour. The results are as presumed, as the inductance increase is clearly seen in impedance magnitudes. In resonant grid, the parallel capacitance produces an additional resonant characteristic in frequencies between 300 and 500 Hz, which can affect adversely on the stability of the interface. In addition, all the measurements show mirror-frequency decoupled characteris- tics typical for passive circuits, given as
Zg-dd=Zg-qq and Zg-qd=−Zg-dq (8) C. Stability impact of PLL and feedforward
The PLL bandwidth and feedforward gain both have significant impact on stability of the inverter-grid interface.
Fig. 7 illustrates the effect of increasing PLL bandwidth (from 1 to 140 Hz) and increasing feedforward gain (from 0 % to 100% of 1/VDC) on the Nyquist contour of the interface. The critical point is marked with an X-marker. In resonant grid, increasing feedforward gain clearly decreases
Fig. 6. Grid impedance models for three different cases.
Fig. 7. GNC loci of return ratio with increasing feedforward gain (left) and increasing PLL bandwidth (right).
stability margins and eventually destabilizes the system, as the contour encircles the critical point. The same applies to PLL bandwidth in weak grid; the increasing bandwidth diminishes the stability margins as the countour shifts left.
D. Adaptive Start-Up Routine
Table III shows the penalty factors for every control parameter set under strong, resonant, and weak grids. The lowest penalty factors are bolded in the table and applied in that grid case. Infinite penalty factor value implies insta- bility. The trends in table can be noticed and they follow predictions; high PLL bandwidths can be applied in strong grid, feedforward decreases stability when low-frequency resonances occur in the grid impedance, and low PLL bandwidth have to be applied in weak grids.
TABLE III
WEIGHTED DISTANCES TO0.5CICRCLE.
Strong Resonant Weak
PLL\FF 0 0.5 1.0 0 0.5 1.0 0 0.5 1.0
10 Hz 0.304 0.329 0.361 0.061 1.362 1.531 0.126 0.018 Inf 50 Hz 0.215 0.261 0.232 0.031 0.647 Inf 0.381 0.446 Inf 100 Hz 0.073 0.154 0.284 0.305 0.289 Inf 1.671 0.932 Inf
Fig. 8 shows the optimized GNC loci with adaptively ad- justed control parameters under all test grids, where the circle represents the forbidden zone of the MPC. As the design rule indicates, the closest distance from the loci to the critical point is very close to 0.5. Thus, the control performance is maximized, while still ensuring sufficient stability margins based on MPC. For the strong grid the minimum distance is dmin = 0.573 > 0.5, while in weak and resonant grids the MPC is marginally violated, as the distances are 0.487 and0.492, respectively. However, this is in accordance with
Fig. 8. GNC loci with MPC under different grid conditions using adaptive control parameters.
the design rule as the optimal control is a trade-off between performance and robustness. In resonant grid, the resonance peak is very abrupt, and due to limited frequency resolution, the corresponding GNC locus is indicative. However, the resolution is sufficient to set the feedforward gain to a very low value. A larger set of available control parameters could be used for fine-tuning of the control. As predicted, the PLL bandwidth was reduced in weaker grid conditions, and the optimal feedforward gains require more precise information of the grid conditions (e.g. whether the grid is resonant or not).
The inverter was tested with two different control param- eters for verification of presented methods. Fig. 9 shows the critical GNC loci in weak grid when the feedforward gain was set to 0 and PLL bandwidth either to 10 or 100 Hz. The corresponding penalty factors given from design rule are 0.126 and 1.671, respectively. Thus, the control is assumed to be very robust with low PLL bandwidth, and to significantly breach MPC with higher bandwidth. Fig. 10 presents the inverter output current (A phase) when the PLL bandwidth is changed from 10 Hz to 100 Hz att= 0.05s.
As predicted from the GNC locus, the harmonic distortion in current increases significantly after transient (resulting from low stability margins and exceeded sensitivity peak).
The excessive harmonic current causes the inverter to trip at t= 0.08s, as the over-current limit is exceeded.
V. CONCLUSION
The stability and disturbance sensitivity of the grid- interface of an inverter can be assessed based on ratio of grid impedance and inverter output impedance. The chosen inverter control parameters drastically affect the shape of output impedance, and consequently, the grid impedance should be taken into account in control design to satisfy
Fig. 9. Critical GNC loci in weak grid with different PLL bandwidths.
Fig. 10. Phase current when PLL bandwidth is increased in a weak grid.
maximum-peak criterion at the interface. This work presents an adaptive start-up routine for optimization of phase-locked loop bandwidth and grid voltage feedforward gains, to ensure system stability while maximizing the performance. In the method, the grid impedance is measured with a broadband excitation during the start up, and the control parameters chosen based on design rule, which optimized the trade- off between robustness and performance. The method is verified by PHIL experiments, where the control parameters are accurately chosen for different grid types. The GNC loci was used to accurately predict stability and power quality when the PLL bandwidth was changed. This paper presented simple proof-of-concept for start-up adaptivity of a grid-connected converter, and the adaptive tuning can be extended to other control parameters (for example, AC- current controller). Further optimization of design rule and parameter sets remains a future topic.
REFERENCES
[1] B. Bose, “Global Energy Scenario and Impact of Power Electronics in 21St. Century,”IEEE Transactions on Industrial Electronics, vol. 60, no. 7, pp. 2638–2651, 2013.
[2] Y. Cho, K. Hur, Y. C. Kang, and E. Muljadi, “Impedance-based stabil- ity analysis in grid interconnection impact study owing to the increased adoption of converter-interfaced generators,”Energies, vol. 10, no. 9, 2017.
[3] C. Li, “Unstable Operation of Photovoltaic Inverter from Field Experiences,” IEEE Transactions on Power Delivery, vol. 8977, no. c, pp. 1–1, 2017. [Online]. Available:
http://ieeexplore.ieee.org/document/7827164/
[4] C. Buchhagen, C. Rauscher, A. Menze, and J. Jung, “BorWin1 - First Experiences with harmonic interactions in converter dominated grids,”
International ETG Congress 2015; Die Energiewende - Blueprints for the new energy age, pp. 27–33, 2015.
[5] P. Belkin, “Event of 10/22/09,”CREZ TEchnical Conference, Electri- cal Reliability Council of Texas, 2010.
[6] J. Sun, “Impedance-based stability criterion for grid-connected invert- ers,” IEEE Transactions on Power Electronics, vol. 26, no. 11, pp.
3075–3078, 2011.
[7] L. Jessen, S. Gunter, F. W. Fuchs, M. Gottschalk, and H. J. Hinrichs,
“Measurement results and performance analysis of the grid impedance in different low voltage grids for a wide frequency band to support grid integration of renewables,”2015 IEEE Energy Conversion Congress and Exposition, ECCE 2015, pp. 1960–1967, 2015.
[8] H. Alenius, T. Messo, T. Reinikka, and T. Roinila, “Aggregated Model- ing and Power Hardware-in-the-Loop Emulation of Grid Impedance,”
inIEEE Energy Conversion Congress and Exposition 2018, 2018, pp.
4179–4186.
[9] R. Luhtala, T. Roinila, and T. Messo, “Implementation of Real-Time Impedance-Based Stability Assessment of Grid-Connected Systems Using MIMO-Identification Techniques,” IEEE Transactions on In- dustry Applications, vol. 9994, no. c, 2018.
[10] M. Cespedes and J. Sun, “Impedance Modeling and Analysis of Grid- Connected Three-Phase Converters,” IEEE Transactions on Power Electronics, vol. 29, no. 3, pp. 1254–1261, 2014.
[11] T. Roinila, M. Vilkko, and J. Sun, “Broadband methods for online grid impedance measurement,”2013 IEEE Energy Conversion Congress and Exposition, ECCE 2013, pp. 3003–3010, 2013.
[12] M. Cespedes and J. Sun, “Online grid impedance identification for adaptive control of grid-connected inverters,” 2012 IEEE Energy Conversion Congress and Exposition, ECCE 2012, pp. 914–921, 2012.
[13] L. Harnefors, “Modeling of Three-Phase Dynamic Systems Using Complex Transfer Functions and Transfer Matrices,”IEEE Transac- tions on Industrial Electronics, vol. 54, no. 4, pp. 2239–2248, 2007.
[14] T. Messo, A. Aapro, T. Suntio, and T. Roinila, “Design of grid- voltage feedforward to increase impedance of grid-connected three- phase inverters with LCL-filter,”2016 IEEE 8th International Power Electronics and Motion Control Conference, IPEMC-ECCE Asia 2016, pp. 2675–2682, 2016.
[15] J. Z. Zhou, H. Ding, S. Fan, Y. Zhang, and A. M. Gole, “Impact of short circuit ratio and phase locked loop parameters on the small signal behavior of a VSC HVDC converter,”IEEE Transactions on Power Delivery, vol. 29, no. 5, pp. 2287–2296, 2014.
[16] T. Messo, A. Aapro, and T. Suntio, “ Generalized multivariable smallsignal model of three-phase grid-connected inverter in dq-domain ,” in IEEE 16th Workshop on Control and Modeling for Power Electronics (COMPEL), 2015, pp. 1–8.
[17] R. Dorf and R. Bishop,Modern Control Systems, Prentice-Hall, 2000.
[18] J. Sun, “ Impedance-Based Stability Criterion for Grid-Connected Inverters ,”IEEE Transactions on Power Electronics, pp. 3075–3078, 2011.
[19] S. Vesti, T. Suntio, J. Oliver, R. Prieto, and J. Cobos, “ Impedance- Based Stability and Transient-Performance Assessment Applying Maximum Peak Criteria ,”IEEE Transactions on Power Electronics, pp. 2099–2104, 2013.
[20] T. Reinikka, R. Luhtala, T. Messo, and T. Roinila, “Impedance-based sensitivity criterion for photovoltaic inverter,” in 10th Annual IEEE Energy Conversion Congress and Exposition (ECCE 2018), September 2018.
[21] L. Harnefors, X. Wang, A. G. Yepes, and F. Blaabjerg, “Passivity- based stability assessment of grid-connected vscs - an overview,”IEEE Journal of Emerging and Selected Topics in Power Electronics, vol. 4, no. 1, pp. 116–125, March 2016.
[22] T. Messo, R. Luhtala, A. Aapro, and T. Roinila, “Accurate impedance model of grid-connected inverter for small-signal stability assessment in high-impedance grids,” inIPEC-Niigata 2018 -ECCE ASIA, May 2018, pp. 1–8.
[23] T. Reinikka, R. Luhtala, H. Alenius, T. Roinila, and T. Messo, “ PHIL Test Bench for Online-Identification Methods of Complex Power Grid ,” in18th IFAC Symposium on System Identification, 2018.
[24] F. Lehfuss, G. Lauss, P. Kotsampopoulos, N. Hatziargyriou, P. Crolla, and A. Roscoe, “Comparison of multiple power amplification types for power Hardware-in-the-Loop applications ,” in2012 Complexity in Engineering (COMPENG), 2012, pp. 1–6.
[25] T. Reinikka, H. Alenius, T. Roinila, and T. Messo, “ Power Hardware- in-the-Loop Setup for Stability Studies of Grid-Connected Power Converters ,” in 2018 International Power Electronics Conference (IPEC 2018), 2018, pp. 1704–1710.
[26] K. R. Godfrey,Perturbation Signals for System Identification. Pren- tice Hall, UK, 1993.
[27] M. Cespedes and J. Sun, “Adaptive control of grid-connected inverters based on online grid impedance measurements,” IEEE Trans. on Sustainable Energy, vol. 5, pp. 516–523, 2014.
[28] R. Luhtala, T. Messo, T. Roinila, T. Reinikka, J. Sihvo, and M. Vilkko,
“Adaptive control of grid-connected inverters based on real-time measurements of grid impedance: Dq-domain approach,” in2017 IEEE Energy Conversion Congress and Exposition (ECCE), Oct 2017, pp.
69–75.
