Generalized Envelope-Based Modeling of Single-Phase Grid-Connected Power Converters

In-depth models of single-phase grid-tied power converters facilitate the examination of low-frequency (LF) interactions among loads, distributed energy resources (DERs), and synchronous generators by operators and designers. These interactions are becoming increasingly significant with the growing integration of power electronics into electrical grids. This article extends the envelope modeling (EM) technique to develop LF linear time-invariant (LTI) circuit models for single-phase grid-tied power converters. The models utilize an independent phase signal that aligns with the most appropriate reference frame. This methodology preserves the LF dynamics inherent to the power converter and control system. The practicality of this method is evidenced by constructing a model for a bridgeless totem-pole power factor corrector (PFC), which includes a zero-crossing detector (ZCD) and operates without closed-loop regulation. The outcomes from this model are juxtaposed with those from a switched model and other well-stablished modeling techniques for comparison. Furthermore, a commercially available circuit design featuring current and voltage control loops is simulated, and the results are corroborated with experimental data. These experiments are conducted under disturbances influencing the converter's performance within its linear operational range.


I. INTRODUCTION
T HE increasing penetration of power electronics within electrical grids intensifies low-frequency (LF) interactions among loads, distributed energy resources (DER), and synchronous generators.It exacerbates the risk of instability in power grids [1], [2].Threat assessment necessitates the evaluation of the time domain transient responses for diverse scenarios.Moreover, at the power converter side, LF corrective actions due to outer/slow control loops, such as DC voltage regulation in AC/DC converters or synchronization and activereactive droop control in DC/AC converters, induce interactions at the point of connection (PC) that warrant in-depth analysis [3], [4].
Most research efforts are focused on grid-connected power converters rated over tens of kilowatts.However, 1φ configurations, around 3.6 kW and below, dominate low-voltage (LV) residential grids.These configurations are used for home appliances, plug-and-play photovoltaics, domestic power storage, electric vehicles, and light mobility.Extracting LF characteristics in 1φ power converters is challenging since only one set of phase voltage and current measurements are available, and effects at twice the grid frequency have to be rejected for LF studies.
Analytical-and circuit-oriented approaches have been proposed for studying LF interactions in LV 1φ power systems under diverse scenarios.The target is to transform averaged models using the switching period of single-phase power converters into models retaining the most relevant characteristics of the controlled power converter to identify the interaction with the grid.
To address this problem, in [5], [6], and [7], the Park (dq) transformation is used to represent the grid voltage and line current in the rotating reference frame (RRF), which requires synchronization with the AC side of the power converter.An intermediate step is shifting the AC electrical quantities by 90 • , i.e., one auxiliary imaginary orthogonal circuit (IOC) [8], [9], [10], [11], [12], or −120 • and 120 • , i.e., two auxiliary circuits [13], [14], to transform these quantities into the complex stationary reference frame (SRF).The resulting linear timeinvariant (LTI) model in the RRF, under low harmonic distortion conditions or poor synchronization, is a mildly periodic system retaining most of the power converter characteristics [13].The modeling approach can be extended to cope with the harmonic distortion by using multiple dq transformations at integer frequency multiples of the fundamental.A phase-locked loop (PLL) is used or assists in transforming the electrical quantities into an RRF aligned with the grid voltage.So, the effects of the PLL dynamics cannot be studied independently.
Recently, dynamic phasors (DPs) have been used for modeling 1φ power converters for a certain set of frequencies of interest.In [15], DPs allow LF interactions of the electrical grid and 1φ PV inverters with active and reactive power control to be evaluated.Increasing the number of phasors permits the match of the dominant harmonics at both the AC and DC sides of the 1φ power converter, as in [16], [17].DPs do not require grid synchronization since they correspond to time-varying coefficients for Fourier series expansions approximating quasi-Tperiodic electrical quantities with a sliding time window length equal to T = (2π/ω DP ), where each k ∈ Z multiple of ω DP results in a k-order phasor [18], i.e., The resulting phasors avoid frequency overlapping and perform as stationary phasors in a steady state but retain slow amplitude and frequency variations of electrical quantities under dynamic conditions [19].Windowing mismatches due to grid frequency variations might deteriorate the model performance, but under slow ones, the results are considered accurate enough [20], [21], [22].
Both the dq transformation and DPs, when used for modeling 1φ power converters, require a proper representation of the electrical quantities used for modeling purposes, which involves a certain previous knowledge about the grid angle or frequency and limits the performance of these methods for analyzing large frequency variations.The generalized envelope modeling (GEM) for 1φ grid-connected power converters presented in this article, whose basic equations were introduced in [23], avoids this issue through the key contribution of using a reference angle signal as an independent variable for AC quantities representation.Other contributions in Section II are the detailed derivation of the switching cell model and the evaluation of consistency with accepted power theories.Moreover, the GEM is compared with dqand DP-modeling approaches in simulations, using a precalculated duty cycle to avoid biases due to the controller used, in Section III.This article also validates GEM experimentally with laboratory tests using a commercial evaluation board for a 1φ bridgeless totem-pole power factor corrector (PFC) in Section IV.Finally, conclusions are provided.

II. GENERALIZED ENVELOPE-BASED MODELING (GEM)
OF SINGLE-PHASE GRID-CONNECTED POWER CONVERTERS

A. Electrical AC Quantities Representation
The GEM approach represents electrical AC quantities as time-varying phasors, e.g., v(t) and ī(t).The notation in [24] and [25] is adopted here.The amplitude of each phasor retains the whole spectrum of the represented electrical quantity, avoiding the decomposition and addition of the components at different frequencies.Phasors are initially considered uncorrelated and are represented using the same reference frame, and then, the relative position of v(t) and ī(t) changes instantaneously depending on the frequency spectra, the actual grid frequency, and the converter operation.Having an infinite set of complex reference frames for the representation of v(t) and ī(t), a certain SRF and RRF set, related through a certain angle θ, is selected, as depicted in Fig. 1.As a novelty, in this modeling approach, the positioning of these reference frames only depends on θ, i.e., which can be related or not to the electrical grid angle or frequency.Despite the simplicity of (2), selecting the most appropriate θ is crucial to achieving a complete modeling of the interaction of the converters with the grid.While ω represents the switching frequency for resonant converters in [26] and θ would result from integrating ω, in this case, the model incorporates θ(t) from different origins depending on the study target, e.g., propagation of the power converter effects through the electrical grid.
From Fig. 1, the instantaneous AC quantities in that particular set of RRF and SRF are defined as v(t) = Re v(t)e jγ(t) e jθ(t) i(t) = Re ī(t)e jδ(t) e jθ(t) ( where x represents the amplitude and time-varying angles γ and δ depend on the relative position of the RRF and the voltage and current phasors, respectively.Then, where d and q subscripts correspond to in-phase and inquadrature quantities in the RRF, respectively.With (3), ( 4) is rewritten as 2 . ( Given these AC quantities representations, large-and smallsignal analysis can be carried out depending on the study target.Moreover, diverse values for ω(t) = θ(t) can be selected with this signal representation without deteriorating the dynamic information.8), (b) AC inductors in (9), and (c) AC/DC switching cells in (11) and (12).

B. AC Passives
Phasors in (4) enable the converter passives modeling without artificially influencing the obtained equivalent, as in dqor DP-based methods, since the derivatives are obtained from the instantaneous AC quantities in (3).As in EM for resonant converters, if inductive and capacitive elements are used with the AC signal quantities in (5), the following relationships are obtained: From ( 6) and ( 7), two RRFs, at ω(t) and −ω(t), relate inphase and in-quadrature components for capacitors and inductors at the AC side.Since the RRF at −ω(t) is a replica, the obtained relationships are given in the RRF at ω(t), i.e., Fig. 2(a) and 2(b) shows the equivalent inductor and capacitor models obtained from GEM for the RRF defined by the selected reference angle.

C. Switching Cells
The relationships between input and output electrical quantities depend on the switching cell topology and the gate signals.
The instantaneous power at the AC side, p(t) = v(t)i(t), is represented using the phasors in (5), resulting in sin 2θ(t) The last term in (10) inherently depends on the value θ(t) used to represent electrical quantities.It approximates the active power transferred P (t), when θ(t) = ω(t) is close to the actual grid frequency.However, when θ(t) deviates from the actual grid frequency, this term varies periodically depending on the frequency deviation and the frequency spectra of the phasors.The amplitude of the terms oscillating at 2θ(t), i.e., p c and p s , depends on the AC phasors amplitudes and their relative angle, i.e., |γ(t) − δ(t)| in Fig. 1.For the proposed GEM, θ(t) ≈ 0, and then, θ(t) = ω varies slowly enough.Both the AC and the DC sides are linked through the power transferred, with , where i o (t) and v o (t) are the DC current and voltage, respectively.The proposed GEM neglects the terms with amplitude p c (t) and p s (t) in (10) since those terms, depending explicitly on the reference frame set used, i.e., 2θ(t), do not contribute to power transfer.
In the case of 1φ grid-connected power converters with capacitive DC bus, the AC voltage depends on v o and the switching function used, i.e., m(t) = m d (t) + jm q (t), through where the frequencies of interest for the subsequent analysis are well below the switching frequency f sw = 1/T sw .By replacing (11) into (10) and using GEM assumptions, the DC side current of the switching cell is obtained as From ( 12), i o (t) depends on AC side quantities m and i, exclusively.
For further clarification, the large-signal GEM is consistent with well-established electrical power theories [27], [28] and phasorial representations [29].The Fryze-Buchholz-Depenbrock (FBD) theory [30] uses the averaged conductance, G, for the decomposition of the AC current into the component contributing to the active power i p and the powerless current i z as ) By replacing (5) into ( 13), the large-signal GEM equivalent of the FBD definitions is obtained as follows: where ), see (10) and the associated explanation, and the squared rms voltage V 2 is obtained from the squared voltage represented through the GEM, given (5), as Under the same assumptions as in (10), the first term in (15) inherently depends on the selected reference frame set while the two terms explicitly dependent on 2θ(t) are neglected for G(t) evaluation, i.e., Additionally, the large-signal GEM equivalent for the nonactive power [28], N , is defined as

III. COMPARISON OF PARK, DP, AND GEM APPROACHES
The dq and DP approaches, following the assumptions and procedures in [13] and [31], respectively, have been used for modeling the 1φ bridgeless totem-pole PFC depicted in Fig. 3.The switched model is simulated along with two dq, two DP, and two GEM models for validation and comparison purposes, using PLECS, from Plexim, and Simulink/MatLab, from Mathworks.The comparison uses the same conditions without a closed-loop controller.

A. Models Used for Comparison
Two dq-based models, DQ nom and DQ PLL , are tested using a SOGI PLL for AC quantities transformation to the RRF.However, DQ nom uses the nominal grid frequency for AC variables decoupling, i.e., ω DQ = ω 0 in Fig. 4(a) [13], while DQ PLL uses the frequency estimation provided by the PLL, i.e., ω DQ = ω PLL .
Two DP-based models, DP simple and DP full , are also tested.DP simple [Fig.4(b)] uses first-and zeroth-order phasors with v g , i g , and v o quantities, as in [31], in the averaged model of the 1φ bridgeless totem-pole PFC by assuming that ω DP = ω 0 and, then, the fundamental or the DC component of these quantities.The model DP full considers the DC side ripple at twice the grid frequency by including second-order phasor at the DC side, as shown in Fig. 4(c).
Two GEM, i.e., GEM 1 and GEM 2 , with different complex reference frame sets are tested to validate the robustness of the proposal independently of the reference angle used, i.e., Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.the envelopes obtained are identical independently of the reference angle using the proposed GEM approach.Both GEMs follow the scheme depicted in Fig. 5, obtained by replacing the electrical quantities, passives and the switching cell at the AC side in Fig. 3 with GEM equivalents in Fig. 2. Passive components at the DC side are retained and connected to the active power port of the switching cell GEM equivalent.As a result, two AC-coupled circuits are obtained to represent the AC side while the DC side remains unchanged.The GEM presented demonstrates that perturbing the load results in model changes while the grid voltage, specifically v gq , results in deterioration of the zero-crossing detection, producing model perturbations independently of the complex reference frame set selected.On the DC side, nonactive power interactions are also modeled.

B. Simulation Conditions
The grid frequency used, ω g , for testing all the models is shown in Fig. 6(a).The reference angle for each GEM is obtained using (2), changing the initial conditions and using slightly different frequency profiles, as shown in Fig. 6(b) and 6(c) through Δω i = ω GEM,i − ω g and Δθ i = θ GEM,i − θ g , respectively.The frequency profiles are accomplished with EN-50 160 limits [32] and are obtained by adding a white noise sequence (0.005 pu power) filtered out through a 2.5 Hz second-order low-pass filter, which forces deviations of the complex reference frame set used for each GEM.
The models are compared using the precalculated duty cycle for power factor correction operation presented in [33].
LF switches, S 3 and S 4 , are operated as diodes, and the highfrequency branch, S 1 and S 2 , provides three-level PWM voltage at the AC side.High-frequency components in the line current are filtered out through an LCL filter, i.e., the boost inductor, L, and the differential LC due to the EMI filter, L d and C d .Precise zero-crossing grid voltage detection is required to achieve a low THD line current [34], [35].Gate signals are routed to the power devices depending on the grid voltage polarity, i.e., d = t on /T sw  for S 2 during the positive grid semiperiod and S 1 during the negative semiperiod.Therefore, in this converter, See that (1 − d) and m have the same envelope, so it is interesting to detect the effect of a phase perturbation linked to the rectification and polarity detection methods used.
Following standard design criteria for grid filters [36], [37], v(jω) ≈ v g (jω) at approximately the grid frequency.Therefore, when the reference angle θ(t) is selected to match the electrical angle at the PC, the following relationship must be accomplished: where the effects of C d and grid-side harmonic distortion are neglected; unity power factor operation and small-ripple approximation for C (i.e., v o ≈ V o ) are assumed.Equation (19) includes the effects of parasitic inductor resistances, R Ld and R L , and the AC measurement resistor, R meas .Parameters for comparison tests are given in Table I.

C. DC Load Step
A 340 Ω to 220 Ω load step is applied at 0.2 s.Fig. 7(a), 7(b), and 7(c) shows the simulation results for v g , i g , and v o ,   Yes, including more RRFs [38] Yes, phasors for harmonics [17] No Frequency separation?
Yes, through ω RRF for different RRFs

Yes, set of phasors
No, envelope of all freq.< ω GEM /2 Note: N : Max.phasor order in DP.
respectively.The GEMs result in envelopes following v g and i g amplitudes in the switching model.At the DC side [Fig.7(c)], the GEM 1 and GEM 2 obtain the same output voltage, rejecting the oscillation at double the grid frequency and retaining the LF dynamics.DP simple exhibits small amplitude oscillations due to the grid frequency variations.DP full tracks the 2ω g ripple at the DC side, but the effect of the grid frequency variations is greater than DP simple .Both DQ nom and DQ pll result in the highest ripple and error, DP pll the worst, due to the large cutoff frequency of the PLL, i.e., T settling = 0.8 s.
Fig. 8(a), 8(b), and 8(c) compares P , S, and G quantities with the same simulation conditions.The switching model evaluates them by averaging the instantaneous quantities with a sliding and adjustable window, minimizing errors due to the frequency profile used.A ripple-less and matching estimation of P , with differences below 0.3% in steady state, is obtained with both GEMs.The instantaneous power p due to the switching model is plotted in dashed line for the accuracy of GEM evaluating P .The computation of P from the switched model deviates from the actual value due to windowing mismatches while averaging the instantaneous quantities.Due to the grid frequency variations, DQ simple and DQ pll result in P with ripple.DP simple and DP full approximate the GEM performance but with ripple due to window length mismatches with the grid frequency.Differences among S evaluations due to the GEMs and the switched model increase to 1.4% [Fig.8(b)], caused by high-frequency harmonics in the AC current, which the GEMs neglect.This effect worsens for DP simple and DP full , showing a ripple exacerbated by the grid frequency excursion from the nominal conditions.Again, DP simple and DP full approximate the GEM response but with ripple due to the window mismatch.From the average load conductance [Fig.8(c)], the switched model and the values obtained with the GEM, i.e., (16), diverge less than 0.15% in steady state.The adjustable length of the averaging window used is the origin of larger response times of the evaluated electrical quantities due to the switching model.DP simple and DP full translate the ripple in P and S to the load conductance, while DP simple provides a practical ripple-less conductance.On average, DP full approximates the switched model conductance the best but exhibits the highest ripple of the DP and GEM models.

D. Low-Frequency Grid Voltage Amplitude Variations
The results for amplitude variations of the grid voltage, from 185 V to 230 V, at 0.96 Hz (a maximum 127.3 V s −1 slope) are shown in Fig. 9.The GEMs track both the AC and the DC variables retaining their LF characteristics up to ω GEM without ripple, since envelope quantities are obtained for both semiperiods, i.e., twice the grid period.DP simple and DP full result in electrical quantities with ripple, depending on the grid frequency excursion.DP simple approximates the most to the GEM responses, while DP full exhibits a greater ripple.Moreover, from Fig. 9(c), the amplitude variations of v g translate to the models, deviating the output voltage ripple from the switched model.
Table II compares key aspects due to dq, DP, and GEM approaches.Since the proposal is circuit-oriented, most analyses, such as component tolerance with Monte Carlo (MC), are enabled in a circuit-oriented simulation tool.

IV. EXPERIMENTAL VALIDATION OF THE GEM
The 1φ bridgeless totem-pole PFC in Fig. 3 is controlled using the multiloop scheme in Fig. 10 for experimental validation of the GEM approach.The envelopes predicted by the GEM are compared with the electrical quantities obtained experimentally.The experimental setup is shown in Fig. 11.The current controller uses the rectified grid voltage and line current.Rectification is achieved with the help of a zero-crossing detector (ZCD), which identifies the grid voltage polarity and initializes the calculation of the rms value V as represented in Fig. 10 [39].V is low-pass filtered with a first-order f cutoff ≈ 4.5 Hz filter.This value and the active power required by the power converter, evaluated through the outer voltage loop, i.e., G v (s), are used to obtain the average load conductance.The inner current controller, i.e., G i (s), is assisted by feedfowarding the AC voltage.The controller also implements antiwindup for the controller integrators, soft starting after zero voltage crossings for the active high-frequency switching power device.Power devices switching at the grid frequency are stopped when |v g | reaches a threshold voltage, i.e., V g,th .Table III summarizes the most relevant operation parameters for the experimental tests.
The reference angle for GEM is obtained by integrating the grid frequency estimation given by the ZCD, i.e., θ(t) = θ ZCD (t) = t 0 (2π/T ZCD (τ ))dτ , where T ZCD is evaluated by counting the sampling periods between edges of the polarity signal.

A. DC Load Step
Fig. 12 shows the experimental results of the described setup with a 340 Ω to 220 Ω DC load step and the corresponding AC   The deviation occurs because the step load transient contains high-frequency components that fall outside the modeling scope of the GEM.However, the controller employed by the evaluation board mitigates their impact, e.g., it prevents integrators wind-up.

B. Low-Frequency Grid Voltage Amplitude Variations
Fig. 13 shows the experimental results for amplitude variations of the grid voltage, from 185 V to 230 V at 0.96 Hz, with a maximum 127.3 V s −1 slope.On the AC side, the GEM retains LF variations of the grid voltage amplitude and LF effects on the line current.On the DC side, the effects of these variations, through the outer control loop, are also accurately captured.

V. CONCLUSION
An innovative approach to modeling single-phase gridconnected power converters, characterized by an envelopebased technique with an autonomous grid angle reference, has been introduced.This method facilitates the derivation of straightforward and precise LTI models for single-phase converters, applicable across both grid-following and grid-forming modes of operation.The technique is uniquely tailored for single-phase systems and aims to elucidate the dynamic interaction between the converter and the electrical grid.This concept was validated through comparative analysis of the envelope and switched models' steady-state and transient behaviors, focusing on voltage, current, and power metrics within a single-phase totem-pole PFC.This analysis was conducted independently of any synchronization mechanisms.Although the resulting circuit model resembles those derived via DPs or Park transformation, incorporating a grid angle reference and a distinct input parameter significantly mitigates transient and steady-state ripple discrepancies.Furthermore, empirical evidence from a prototype design corroborates the model's fidelity.The singlephase converter model notably encapsulates the LF phenomena attributable to the DC-side controller and the employed phase reference generation subsystem.

Fig. 1 .
Fig. 1.Time-varying phasors of the grid voltage, the line current, and their representation in a certain set of SRF and RRF defined by the reference angle θ.

Fig. 5 .
Fig. 5. Generalized envelope-based model of 1φ totem-pole PFC.Controlled voltage sources v gd and vgq depend on vg and the selected ω GEM .
Francisco J.Azcondo (Senior Member, IEEE) received the B.Eng. and M.Eng.degrees in electrical engineering from the Universidad Politécnica de Madrid, Madrid, Spain, in 1989, and the Ph.D. degree from the University of Cantabria, Santander, Spain, in 1993.He was a Visiting Researcher with the Department of Electrical, Computer, and Energy Engineering, University of Colorado, Boulder, CO, USA, from 2004 to 2010; the Department of Electronics and Communication Engineering, University of Toronto, Toronto, ON, Canada, in 2006; and Utah State University Power Electronics Laboratory, Department of Electronics and Communication Engineering, Utah State University, Logan, UT, USA, in 2013.He is currently a Professor with the Department of Electronics Technology, Systems and Automation Engineering and the Dean of the Doctorate School, University of Cantabria.His current research interests include modeling and control of switch-mode power converters and resonant converters, digital control capabilities for switched-mode power supplies, and current sensorless control for grid-connected converters and applications, such as outdoor lighting, electrical discharge machining, and welding arc.Dr. Azcondo was the Chair of the IEEE Industrial Electronics Society-Power Electronics Society Spanish Joint Chapter from 2008 to 2011.He has been an Associate Editor of IEEE JOURNAL OF EMERGING AND SELECTED TOPICS IN POWER ELECTRONICS and IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS and the Vice Chair of IEEE PELS TC 1: Power & Control Core Technologies.He is an Associate Editor of IEEE TRANSACTIONS ON POWER ELECTRONICS.

TABLE I SIMULATION
PARAMETERS FOR OPEN-LOOP COMPARISON OF dq, DP-MODELING AND GEM APPROACHES