A fast power calculation algorithm for three-phase droop-controlled-inverters using combined SOGI filters and considering nonlinear loads

Citation: Li, M.; Matas, J.; Mariachet, J.E.; Branco, C.G.C.; Guerrero, J.M. A Fast Power Calculation Algorithm for Three-Phase Droop-ControlledInverters Using Combined SOGI Filters and Considering Nonlinear Loads. Energies 2022,15, 7360. https://doi.org/10.3390/en15197360 Academic Editor: Nicu Bizon Received: 13 September 2022 Accepted: 6 October 2022 Published: 7 October 2022 Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. Copyright: © 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/). energies Article A Fast Power Calculation Algorithm for Three-Phase Droop-Controlled-Inverters Using Combined SOGI Filters and Considering Nonlinear Loads Mingshen Li 1, Jose Matas 1,* , Jorge El Mariachet 1, Carlos Gustavo C. Branco 1and Josep M. Guerrero 2 1Electric Engineering Department, Politechnic University of Catalonia (EEBE-UPC), 08019 Barcelona, Spain 2Energy Teknik Department, Aalborg University (ET-AAU), 9220 Aalborg, Denmark *Correspondence: [email protected] Abstract: The power calculation is an indispensable element in droop-controlled inverters because the bandwidth of the measured power has a direct impact on the controller performance. This paper proposes a fast and accurate power calculation algorithm based on the combined Second Order Generalized Integrator (SOGI) filters in stationary coordinates for a three-phase system, which takes into consideration the use of nonlinear loads. The power calculation scheme is formed by the two-stage SOGI filters that are employed for obtaining the active and reactive powers required to perform a droop-based inverter operation, respectively. From the two-stage structure, the first SOGI is used as a band-pass filter (BPF) for filtering harmonics and obtaining the fundamental current of the nonlinear load; The second SOGI is used as a low-pass filter (LPF) for extracting the DC-component, which corresponds with the average power. A small-signal model of a two droopcontrolled inverters system is built to obtain the dynamical response and stability margin of the system. And compared it with the dynamical behaviour of a standard droop-control method. Next, the proposed power calculation system is designed in order to achieve the same ripple amplitude voltage as that obtained with the standard droop-control method by adjusting the bandwidth gains. Through simulation and hardware in the loop (HIL) validation, the proposed approach presents a faster and more accurate performance when sharing nonlinear loads, and also drives the inverters’ output voltage with lower distortion. Keywords: three-phase paralleled inverters; averaged power calculation; droop-control; SOGI filter; small-signal model; nonlinear loads 1. Introduction Microgrid technology and application could be an effective solution for the nextgeneration power system due to its flexibility and capability to integrate distributed generation sources [ 1 , 2 ]. As interfaces between the distributed resources and a microgrid, the power converters had been investigated in aspects of topology and control [ 3 , 4 ]. color The control strategies for power converters such as pulse width modulation (PWM), synchronization technology, and optimal control algorithm. are one of the most important issues of the system. For microgrid typical operating modes, islanded or grid-tied, the converters should have the ability to deliver power to the loads or to the grid. Based on this, several effective control strategies have been proposed, like the droop-control, direct power control, model predictive control, etc. In general, the power calculation is an indispensable part of many controllers, either in rotating or stationary coordinate frames. In addition, due to the low capacity and scattered distribution of DG, the connection of power electronic devices and nonlinear loads not only leads to voltage and current distortions, but also to a poor power harmonic sharing inside a microgrid [5,6]. The droop control is a well-established technique for a parallel inverter system, which is able of sharing the power between the inverters. The control strategy is implemented by Energies 2022,15, 7360. https://doi.org/10.3390/en15197360 https://www.mdpi.com/journal/energies Energies 2022,15, 7360 2 of 16 simulating the drooping characteristics of the traditional generators, in which it presents the proportional droop of the active power-frequency and reactive power-voltage curves, respectively. Therefore, the system voltage and frequency are modulated to avoid the circulating current between the paralleled inverters [ 7 – 9 ]. In the standard droop method, a LPF is commonly employed to filter the harmonics and achieve power control accuracy of the active and reactive calculated powers [ 10 – 12 ], because loads in general can induce distortions in the voltage and current. In [ 13 , 14 ], these studies dealt with the virtual impedance and droop improvement by using the first-order LPF. And, the cut-off frequency of this LPF should be designed much smaller than the pass-band of the inverter’s voltage control in order to guarantee a proper operation and lower the total harmonic distortion (THD) of the inverter output voltage [ 15 ]. For supplying nonlinear loads, ref. [ 16 ] proposed a multi-functional controller which employed a second-order LPF to filter the harmonics. In [ 17 ], the quantitative analysis of the harmonic powers were studied. In the power calculation stage, a LPF corresponding to the harmonic power calculation was added to the power calculation scheme. However, the lower cut-off frequency of the LPF slows down the system dynamics, and even lead to inverter instability. In [ 18 ], the first-order LPF of droop controller was investigated by using the small signal model, and the paper reported that the cut-off frequency has an impact on the dynamic performance, steady-state accuracy and system stability. Ref. [ 19 ] presented an nonlinear neural network to improve the power sharing accuracy, particularly for the voltage and frequency regulation in a hierarchical control scheme, in which the LPF is still employed for primary control. The SOGI filter has been studied to estimate the harmonic components [ 20 , 21 ], in this case, the droop frequency calculation determines and provides the frequency to the SOGI frequency input. Besides, due to the resonance behaviour of the SOGI, the input signal is filtered without delay at the center frequency as a BPF, and the quadrature output is filtered as a LPF. In [ 22 ], a virtual impedance loop was designed using a SOGI, which could achieve better output-voltage THD because of the SOGI-BPF capability. In [ 23 ], a SOGI filter was used to improve the inherent voltage regulation. However, the power calculation method only filters the double-frequency components and only considers linear loads. In order to improve the transient response and accuracy of the averaged power calculation for nonlinear loads, refs. [ 24 , 25 ] proposed a double structure SOGI filter applied to the current and the power calculation, respectively. However, the method was only applied to single-phase systems. The presented work seeks to contribute to the improvement of the dynamic behavior and stability for the droop-controlled three-phase voltage source inverter (VSI) when sharing a nonlinear load. The combined SOGI filters are designed to obtain the averaged powers for the droop-control. In this two-stage structure, based on αβ frame, the fundamental current signals are obtained from the the measured load current through the first-stage SOGI-BPF. Then after the instantaneous power calculation, the second-stage SOGI-LPF is employed to each instantaneous power to acquire averaged power with the less harmonics. The Small-signal model of a two parallel inverters system is formulated to investigate the dynamics performance and stability margin compared with the standard droop-LPF method. This paper is organized as follows: the control scheme with the combined SOGI filter and comparison with LPF based on droop control are indicated in Section 2. Section 3 presents the small signal analysis and corresponding root locus. The simulation and HIL validation of the proposed method are given in Section 4, including the performance comparison with the LPF-droop method. 2. Combined SOGI Power Calculation Method The scheme of a VSI controlled by the droop method when sharing a linear or nonlinear load is depicted in Figure 1. In this figure, it can be seen that the control scheme includes the power calculation block, the droop algorithm block, the voltage reference generation, Energies 2022,15, 7360 3 of 16 and PWM block. According to the P−ω and Q−V characteristics, the load can be shared among inverters. a V gb i gc i PWM ga i abc  a L b L c L abc  dc v a R b R c R b V c V i  * V  Droop controller f C AC Bus Load abc  VSI Power calculation V     0.5 abc U U  Q P Breaker Voltage reference generation   V P  − QV− Figure 1. The scheme of the Droop controlled Three-phase inverter in islanded mode. In order to make the bandwidth of the power controller smaller than the voltage controller and avoid the signal distortion [ 15 ], the average power should be calculated by filters. Figure 2illustrates the conventional average powers calculation and the proposed one for a three-phase system. As shown in Figure 2a, the average powers are obtained by using first-order LPF, whose bandwidth is designed to be much smaller than the bandwidth of the voltage loop. Therefore, motivated by the faster transient response characteristics of the double SOGI, the proposed method is able to be designed as the two stages: the first SOGI is used as a BPF for obtaining fundamental current, and the second SOGI is used as a LPF for extracting DC-component as shown in Figure 2b. abc  abc I abc  abc U SOGI BPF I  I  0  1  U  Power Calcul ation i P i Q SOGI LPF SOGI LPF 1c  2c  2  2  P Q abc  abc I abc U U  I  Power Calcul ation P Q P Q abc  c c s   + c c s   + (a) Conventional power calculation method for droop control (b) The SOGI power calculation method for droop control Figure 2. Block diagram of the power calculation schemes: ( a ) conventional LPF filter ( b ) combined SOGI method. Energies 2022,15, 7360 4 of 16 In Figure 3, the nonlinear load is chosen as a three-phase diode-bridge rectifier with a RC load with currents THD that are higher than 25% [ 26 ]. In the circuit, the inductor L=60 µH . The capacitance of the RC load is 650 µ F, and the resistances of the RC load are RL1=RL2=60 Ω. Diobe Bridge 1L R 2L R 1 S 1 C abc U RC Load L Figure 3. Three-phase Diode-bridge rectifier nonlinear load with a RC load. According to the Clarke transformation, the inverter voltages and currents can be obtained in the stationary coordinate frame as: vαβ(t) = Tαβ  va(t) vb(t) vc(t) ,iαβ(t) = Tαβ  ia(t) ib(t) ic(t) ,Tαβ =2 3"1 0−1 2 −√3 2 −1 2 −√3 2#(1) Assuming that the AC bus voltage is balanced and sinusoid. Obviously, the distorted load currents are analysed by using the Fourier series decomposition. Taking the phase A as an example, the load current can be expressed as: ia=2√3 πIdsin ωt+∑ n=6k±1 (−1)k√2Insin nωt(2) where Id and n are the rectifier DC-side current and harmonic order, respectively. Therefore, the load current can be decomposed into fundamental and n -order harmonics by using the Clarke transformation. A SOGI is a linear filter, which provides both the filtered output as well as a quadratureshifted output as shown in Figure 4. The SOGI in-phase and quadrature-phase outputs behave as BPF and LPF, respectively, with the following transfer functions: GBPF(s) = 2ζωcs s2+2ζωcs+ω2 c (3) GLPF(s) = 2ζω2 c s2+2ζωcs+ω2 c (4) where ζ is the damping factor and ωc= 2 πfc is the tuning center angle frequency. Figure 5 shows the Bode plot of the transfer functions (3) and (4) when the damping factor varies from 0 to 1. It can be observed that the center angle frequency of SOGI is equal to ωc . In addition, ζ affects the system gain and bandwidth. Considering the phase plot, the phase lag of SOGI-LPF is 90 ◦ , and there is almost no phase shift for the lower frequency values for both filters, i.e., <8 Hz. Energies 2022,15, 7360 5 of 16 Resonator q V d V 2    input V + − c  + − Figure 4. Block diagram of the SOGI filter. 20.707  = 10.35  = 30.9  = SOGI-LPF SOGI-BPF SOGI-BPF SOGI-LPF 30.9  = 20.707  = 10.35  = −20 −40 −60 −80 −100 −45 −90 −135 −180 Figure 5. Bode plot of SOGI-LPF transfer function with different damping factors. In the combined SOGI approach, the first SOGI that acts as a BPF is employed for extracting the fundamental frequency components ω0= 2 πf0 of the inverter current, so its output signal is synchronized with the input, since there is no phase shift at the center frequency as shown in Figure 5. Considering that there is not a current dc-offset, the filtered current can be expressed as: iα0(t) = Iα0sin(ωt+ϕ0) iβ0(t) = Iβ0cos(ωt+ϕ0)(5) where Iαβ0 are the filtered current amplitudes, which correspond with the amplitude of the fundamental components in stationary coordinate frame, and ϕ0 is the phase of the fundamental component. Therefore, according to (1) and (5), the instantaneous active and reactive power can be formulated as: P=vαiα0+vβiβ0=3VI cos(ωt)−3VI cos(2ωt−ϕ0) + 3VI cos(nωt−ϕn) Q=vβiα0−vαiβ0=3VI sin(ωt)−3VI sin(2ωt−ϕ0) + 3VI cos(nωt−ϕn)(6) where V and I are the voltage and current amplitudes in αβ frame, respectively, n is the harmonic components order and ϕnrefers to the harmonics phase. Therefore, instantaneous powers can be expressed as the sum of three components: the averaged powers, the double frequency ripples, and the high order frequency components: Energies 2022,15, 7360 6 of 16 P(t) = P+˜ P+Pn Q(t) = Q+˜ Q+Qn (7) where the superscripts “ − ” and “ ∼ ” correspond to the average and the second order frequency oscillating powers, PQn are the high order components. In order to remove the double frequency and high order components, an additional SOGI used as a LPF can be employed in next stage where the damping coefficient is ζ2. SOGI-BPF will provide the fundamental component of the current with a frequency that corresponds to the operating frequency delivered by the droop-method. Therefore, the signal is going to be in phase with the input, but its envelope is going to track the following first-order transfer function (the cut-off frequency is ζ1ωn ). Therefore, the transfer function of the combined SOGI proposal could be expressed as the combination of the previous transfer function cascaded with the transfer function corresponding to the SOGI that acts as a LPF, i.e.,: ¯ P(s) = 2ζ1ζ2ωnω2 c1 (s+ζ1ωn)(s2+2ζ2ωc1s+ω2 c1)P(s) ¯ Q(s) = 2ζ1ζ2ωnω2 c2 (s+ζ1ωn)(s2+2ζ2ωc2s+ω2 c2)Q(s) (8) In order to compare the transient response between double SOGI proposal and the single-LPF corresponding to the standard droop method, the bode plot of two power calculation methods is shown in Figure 6. −3dB 1 c f= 1.5 c f= 3 c f= 13 c f= 18 c f= 32 c f= −20 −40 −60 −80 −100 −120 −45 −90 −135 −180 −225 −270 Figure 6. Bode plot of transfer function of the LPF and proposed method with different cut-off requency. It can be observed that the combined SOGI method behaves a LPF as well, and its cut-off frequency (intersect with − 3 dB) is slightly larger than LPF-droop based when their output averaged power are same. Meanwhile, the steady-state phase shit of the combined SOGI filter is − 270 ◦ at high frequencies due to the fact that (8) is a third-order system. At − 3 dB, There are delays for the combined SOGI method, which are similar with the Energies 2022,15, 7360 7 of 16 phase delays of LPF-droop method. Note that the single-LPF is nearly flat when the cut-off frequency varies from 1 Hz to 3 Hz. By contrast, the stop band of the combined SOGI is much narrower than that of the single-LPF even though SOGI’s fc varies from 13 Hz to 32 Hz, which indicates the frequency response of proposed method is faster. It can be concluded that the combined SOGI method is third order LPF-filter, which has the ability to suppress the second and high-frequency components with a fast frequency response. 3. Small Signal Analysis The objective of this section is to analyse the effects of the conventional and combined SOGI filters’ cut-off frequency on the stability and performance of system. The simplified interface circuit for two paralleled VSIs connect to the common AC bus through line impedance is presented in Figure 7. The voltage at the inverter nodes is E∠θ referring to an AC bus voltage U∠0. AC Bus VSI-1 VSI-2 𝑅1 𝑅2 𝑗𝑋1 𝑗𝑋2 𝑅𝐿𝑜𝑎𝑑 𝐼1 𝐼2 11 E   22 E   0U Inverters side Load side Figure 7. The equivalent circuit of two paralleled VSIs in islanding operation. Considering an inductive impedance line [ 15 ], the active and reactive powers from inverters to the AC bus can be derived as: P=1 R2+X2(RU2−REU cos θ+XEU sin θ) Q=1 R2+X2(XU2−XEU cos θ−REU sin θ)(9) With regard to the small disturbances, (9) can be linearized at an equilibrium point at steady-state, the disturbance errors of the powers can be expressed as: ∆P=kPE∆E+kPθ∆θ ∆Q=kQE∆E+kQθ∆θ(10) where ∆ denotes the small deviation of the respective variable from the equilibrium point, and the expansion form of the parameters kare: kPE =1 R2+X2(2RE −RU cos θ+XU sin θ) kPθ=1 R2+X2(REU sin θ+XEU cos θ) kQE =1 R2+X2(2XE −XU cos θ−RU sin θ) kQθ=1 R2+X2(XEU sin θ−REU cos θ) (11) Next, in frequency domain, the small disturbance of frequency and voltage can be given as: ∆ω(s) = −m∆P=Gf(s)(kPE∆E(s) + kPθ∆θ(s)) ∆E(s) = −n∆Q=Gf(s)(kQE∆E(s) + kQθ∆θ(s)) (12) where m and n are the droop coefficient of the active and reactive powers, respectively, and Gf(s) is the transfer function of the LPF filter used in the power calculation. Substituting ∆ω(s) = s∆θ(s) into (10) and (12), the characteristic equation of the system considering the active power loop can be derived as: Energies 2022,15, 7360 8 of 16 (nGf(s)kQE +1)s∆θ(s) + mnGf2(s)∆θ(s)(kPθkQE −kPEkQθ) + mkPθGf(s)∆θ(s) = 0(13) In order to analyse the effects of the cut-off frequency of the LPF-droop method and of the proposed combined SOGI method, substitute the transfer function of LPF-droop and the proposed method (8) into (13), then the dynamic response of system can be obtained based on the different power calculation methods. As shown in Figure 8, the root locus for both methods is indicated when the cut-off frequency fc varies from 0.1 Hz to 1 Hz. For the conventional droop method, the pair of dominant poles move along the negative real axis and towards infinite negative when fc increases, which implies that the system’s dynamic response becomes faster and the system presents overdamped characteristics. In comparison, the dominant poles of the combined SOGI method also are located further from the imaginary axis, and the system indicates an underdamped feature when fc= 0.1 ∼ 1 Hz. Thus, the combined SOGI method can be concluded to be faster than the LPF-droop-method considering the same cut-off frequency situation. Moreover, the stability margin of the combined SOGI method is much wider. LPF-droop ：𝑓 𝑐𝑖𝑛𝑐𝑟𝑒𝑎𝑠𝑒𝑠 𝑓𝑟𝑜𝑚 0.1𝐻𝑧 𝑡𝑜 1𝐻𝑧 Combined SOGI：𝑓 𝑐𝑖𝑛𝑐𝑟𝑒𝑎𝑠𝑒𝑠 𝑓𝑟𝑜𝑚 0.1𝐻𝑧 𝑡𝑜 1𝐻𝑧 Figure 8. The root locus of the LPF-droop method and the proposed combined SOGI method when fcvaries from 0.1 Hz to 1 Hz. Figure 9illustrates the phase response of the two methods. It can be seen that the dynamic response of the proposed method is faster and without oscillation, because the eigenvalues locates near the real axis. To sum up, from the power calculation perspective, the cut-off frequency will affect the stability and dynamic response of the system, and the larger fc makes the system faster. To compare with the LPF-droop controller, the proposed method indicates the faster dynamic response and larger stability margin under the same conditions. Energies 2022,15, 7360 9 of 16 Time(s) 𝜃(rad) Combined SOGI：𝑓 𝑐𝑖𝑛𝑐𝑟𝑒𝑎𝑠𝑒𝑠 𝑓𝑟𝑜𝑚 10𝐻𝑧 𝑡𝑜 15𝐻𝑧 LPF-droop：𝑓 𝑐𝑖𝑛𝑐𝑟𝑒𝑎𝑠𝑒𝑠 𝑓𝑟𝑜𝑚 1𝐻𝑧 𝑡𝑜 1.5𝐻𝑧 10−2 Figure 9. The phase step response of the LPF-droop method and the proposed combined SOGI method when fcvaries from 0.1 Hz to 1 Hz. 4. Simulation and HIL Validation 4.1. Simulation Results In order to test the performance of proposed combined SOGI method and compare it with the conventional LPF-droop control, the simulation model of two parallel inverters with different types of loads was built under Matlab/PLECS environment. In the simulation model, the electrical system includes the DC source, two three-phase H-bridge inverters, the L filter , transmission line and the shared linear/nonlinear load. The controller block contains two parts: power calculation blocks and droop controller according to Figure 1. Each inverter implements the P - ω and Q - V droop controller with the same slope coefficients, and the parameters of the system are listed in Table 1. Table 1. System parameters used in simulation. Paremeters Value Nominal output voltage 220 V Nominal frequency 50 Hz L-type filter 0.1 Ω+ 2 mH Transmission line 0.01 Ω+ 0.32 mH SOGI-LPF damping coefficient ζ20.707 Active power droop coefficient m0.001 Reactive power droop coefficient n0.0001 4.1.1. Case Study I: A Linear Load Step Change The dynamical performance of two methods under a resistive load step change are illustrated in Figure 10. The cut-off frequency of the LPF-droop controller is designed as 1Hz. For the proposed method, the damping factors of the SOGI-BPF and the SOGI-LPF are all set as 0.707. The center frequency of SOGI-BPF is equal to the instantaneous frequency in P - ω loop, and the cut-off frequency of SOGI-LPF is set to fc1= 15 Hz to obtain the same average power as with the LPF-droop method. Figure 10b shows the transient response to an output current step change from 15 A to 30 A supplying a linear load. In this response, from the active power variation of average power, it can be observed that the proposed combined SOGI method is much faster than the LPF-droop method with transient time that Energies 2022,15, 7360 16 of 16 11. Li, Z.; Chan, K.W.; Hu, J.; Guerrero, J.M. Adaptive droop control using adaptive virtual impedance for microgrids with variable PV outputs and load demands. IEEE Trans. Ind. Electron. 2020,68, 9630–9640. [CrossRef] 12. Zhang, W.; Zheng, Z.; Liu, H. A novel droop control method to achieve maximum power output of photovoltaic for parallel inverter system. CSEE J. Power Energy Syst. 2021, early access. 13. Lu, F.; Liu, H. An Accurate Power Flow Method for Microgrids with Conventional Droop Control. Energies 2022 ,15, 5841. [CrossRef] 14. Buraimoh, E.; Aluko, A.O.; Oni, O.E.; Davidson, I.E. Decentralized Virtual Impedance-Conventional Droop Control for Power Sharing for Inverter-Based Distributed Energy Resources of a Microgrid. Energies 2022,15, 4439. [CrossRef] 15. Coelho, E.A.A.; Cortizo, P.C.; Garcia, P.F.D. Small-signal stability for parallel-connected inverters in stand-alone AC supply systems. IEEE Trans. Ind. Appl. 2002,38, 533–542. [CrossRef] 16. Vasquez, J.C.; Mastromauro, R.A.; Guerrero, J.M.; Liserre, M. Voltage support provided by a droop-controlled multifunctional inverter. IEEE Trans. Ind. Electron. 2009,56, 4510–4519. [CrossRef] 17. Wang, W.; Zeng, X.; Tang, X.; Tang, C. Analysis of microgrid inverter droop controller with virtual output impedance under non-linear load condition. IET Power Electron. 2014,7, 1547–1556. [CrossRef] 18. An, R.; Liu, J.; Wu, T.; Wang, S.; Liu, B. Analysis and design of cutoff frequency for power calculation low-pass filters in droop control. In Proceedings of the 2017 IEEE 3rd International Future Energy Electronics Conference and ECCE Asia (IFEEC 2017-ECCE Asia), Kaohsiung, Taiwan, 3–7 June 2017; pp. 1596–1600. 19. Baghaee, H.R.; Mirsalim, M.; Gharehpetian, G.B. Power calculation using RBF neural networks to improve power sharing of hierarchical control scheme in multi-DER microgrids. IEEE J. Emerg. Sel. Top. Power Electron. 2016,4, 1217–1225. [CrossRef] 20. Matas, J.; Martin, H.; de la Hoz, J.; Abusorrah, A.; Al-Turki, Y.A.; Al-Hindawi, M. A family of gradient descent grid frequency estimators for the SOGI filter. IEEE Trans. Power Electron. 2017,33, 5796–5810. [CrossRef] 21. Matas, J.; Martín, H.; de la Hoz, J.; Abusorrah, A.; Al-Turki, Y.; Alshaeikh, H. A new THD measurement method with small computational burden using a SOGI-FLL grid monitoring system. IEEE Trans. Power Electron. 2019,35, 5797–5811. [CrossRef] 22. Matas, J., J.M.; Castilla, M.; De Vicuña, L.G.; Miret, J.; Vasquez, J.C. Virtual impedance loop for droop-controlled single-phase parallel inverters using a second-order general-integrator scheme. IEEE Trans. Power Electron. 2010,25, 2993–3002. 23. Tolani, S.; Sensarma, P. An improved droop controller for parallel operation of single-phase inverters using RC output impedance. In Proceedings of the 2012 IEEE International Conference on Power Electronics, Drives and Energy Systems (PEDES), Bengaluru, India, 16–19 December 2012; pp. 1–6. 24. El Mariachet, J.; Guan, Y.; Matas, J.; Martín, H.; Li, M.; Guerrero, J.M. HIL-assessed fast and accurate single-phase power calculation algorithm for voltage source inverters supplying to high total demand distortion nonlinear loads. Electronics 2020 , 9, 1643. [CrossRef] 25. El Mariachet, J.; Matas, J.; Martín, H.; Li, M.; Guan, Y.; Guerrero, J.M. A power calculation algorithm for single-phase droopoperated-inverters considering linear and nonlinear loads HIL-assessed. Electronics 2019,8, 1366. [CrossRef] 26. Wallace, I. Key Changes and Differences between the New IEEE 519-2014 Standard and IEEE 519-1992. Alcatel Telecommun. Rev. 2014,11. 27. Rodriguez, F.; Bueno, E.; Aredes, M.; Rolim, L.; Neves, F.A.; Cavalcanti, M.C. Discrete-time implementation of second order generalized integrators for grid converters. In Proceedings of the 2008 34th Annual Conference of IEEE Industrial Electronics, Orlando, FL, USA, 10–13 November 2008; pp. 176–181.