Emission assessment of single-phase switch-mode PFC loads up to 150 kHz: Experimental analysis and modelling

Electric Power Systems Research 220 (2023) 109236 Available online 15 March 2023 0378-7796/© 2023 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Contents lists available at ScienceDirect Electric Power Systems Research journal homepage: www.elsevier.com/locate/epsr Emission assessment of single-phase switch-mode PFC loads up to 150 kHz: Experimental analysis and modelling Adam J. Collin a, Jiri Drapela b, Roberto Langella c,∗, Alfredo Testa c aDepartment of Engineering, University of Sannio, Italy bDepartment of Electrical Power Engineering, Brno University of Technology, Czech Republic cDepartment of Engineering, University of Campania Luigi Vanvitelli, Italy ARTICLE INFO Keywords: Ac–dc power converters Load modelling Low and high frequency emissions Power quality Power system harmonics ABSTRACT The ongoing integration of new power electronic interfaces is increasing emissions in the frequency range between 2 and 150 kHz. This paper considers the emission assessment from 0 to 150 kHz of the switch-mode power factor correction converter topologies widely utilised in modern single-phase loads. First, experimental analysis of a load constituted by a variable switching frequency converter is performed to introduce modelling considerations and proposals for the extended frequency range. Then, a full circuit time domain model of the considered load is developed. Following this, a fast and accurate hybrid modelling technique which simultaneously models the low frequency (𝑓 < 2 kHz) and high frequency (2 ≤𝑓 < 150 kHz) emissions is presented. The proposed hybrid technique can be readily applied to a range of topologies and control algorithms and easily integrated in Iterative Harmonic Analysis for system level analysis of distortion up to 150 kHz. 1. Introduction Recent IEC Standards [1] for limiting harmonic current emissions have driven the integration of new power electronic interfaces for low voltage loads. Such converters, commonly referred to as active power factor correction (a-PFC), include flexible dc–dc converters at the output of the diode bridge rectifier (DBR) and are widely utilised in single-phase loads, e.g. in heating, ventilation and air conditioning (HVAC) systems, electric vehicle (EV) battery chargers, switch-mode power supplies (SMPSs) and light-emitting diode (LED) lamps. As the emissions of these loads are manifest as distortion at the system level, their proliferation has contributed to the changing characteristics of distortion in power systems and extended the range of interest from 2 kHz up to 150 kHz [2]. There is particular interest in the 2–150 kHz frequency range as it is not well covered by existing standards; it is also the range in which several mains communication systems (MCSs) operate. Although emissions in this range are generally small, e.g., from μA to hundreds of mA for household appliances characterised as current sources [3–5], disturbance levels up to 3 V have been observed for voltage source converters such as photovoltaic inverters (PVIs) and EV battery chargers [6–8]. There are several reports of interference caused by the conducted disturbances of such emissions, e.g., [9,10]. As the interaction and ∗Corresponding author. E-mail address: [email protected] (R. Langella). propagation of these components will tend to be localised, they are particularly important to consider in the design of residential, commercial and industrial installations. However, there are several cases of wider area propagation, and further work is required to fully characterise and understand their impact. Particularly, resonance phenomena [11,12] and interaction with MCS [13] must be analysed and managed to maintain the integrity of the network. A key part of this is developing accurate models of emission sources, i.e., power converters, in the 2–150 kHz interval and their interaction with network components [2]. 1.1. Modelling emissions up to 150 kHz The vast majority of previous research in this area has focussed on observing the characteristics of emissions, e.g., [3–7]. Literature on representing emission sources for harmonic penetration studies is still relatively scarce. Time domain models (TDMs) are the most accurate representation of power converters. They are used in the design process, and, as such, there is considerable knowledge regarding the representation of converter circuits and control algorithms, e.g., [14,15]. However, they generally require to model the switching components in detail with very small integration time steps (resulting in long simulation times), https://doi.org/10.1016/j.epsr.2023.109236 Received 2 December 2022; Received in revised form 13 February 2023; Accepted 20 February 2023 Electric Power Systems Research 220 (2023) 109236 2 A.J. Collin et al. which limits their use to detailed studies of specific equipment and precludes their use in system level studies. Frequency domain models (FDMs), based on Thevenin/Norton equivalent circuit models, are computationally efficient, and some models for power system studies have been proposed. In such approaches, the emissions are aggregated into a single frequency band around the switching frequency, neglecting the remaining parts of the spectra. This approach has been applied to model the emissions of homogeneous aggregate loads, e.g. [16], and individual fixed switching frequency pulse width modulation (PWM) converters, i.e., PVIs [7,8,17] and an EV battery charger [8]. In [7,8], the emission source is parameterised with respect to operating conditions, e.g., PVI power output and dc link voltage in [7], with the supply voltage magnitude also included in [8]. Clearly, as the number of parameters increases, the model form and parameter estimation become more complex. Alternatively, hybrid model (HM) approaches, which combine time domain and frequency domain methods, may be applied. One hybrid modelling approach is proposed in [18,19], for single and three phase rectifiers, respectively, for the design of electromagnetic interference (EMI) filters. This approach starts from the analytical representation of fixed switching frequency PWM converters which is transferred across the device input impedance using a frequency domain representation of the line commutated rectifier switching function. 1.2. Contribution From the review of existing literature, it is clear that models of emission sources are still not widely available. The focus has been on characterising and modelling specific fixed switching frequency converters, with contributions intended for system level analysis utilising frequency domain approaches with aggregated emission bands. Thus, further work is required to characterise and model variable switching converters and intra-cycle variations (which are present in both fixed and variable switching frequency converters). This is addressed in this paper, which contributes to the research area by proposing a two-step HM approach to develop models of the fixed and variable switching frequency switch-mode a-PFC converter topologies widely utilised in single-phase loads. The approach can reproduce the periodic steady-state intra-cycle variations which are inherent in the operation of the power converter. The modelling approach is introduced starting from experimental analysis and the development of a detailed TDM, with a variable switching frequency buck-boost converter LED lamp driver used as an example. This clearly demonstrates the advantages of the proposed approach versus TDMs (with respect to simulation time) other modelling approaches (which are limited to fixed switching frequency PWM converters). However, it is more generally applicable and can be applied to model the switch-mode a-PFC converter topologies that are widely utilised in power electronic loads which: (i) contain ac–dc rectifiers with small or zero sized output capacitors in order to satisfy the requirements for harmonic current limits in [1], and (ii) operate with current mode control. 1.3. Structure The rest of the paper is structured as follows: Section 2provides details of the preliminary experimental analysis, which are used to introduce the modelling considerations of the extended frequency range up to 150 kHz in Section 3; Section 4discusses the TDM considerations and the proposed two-step HM approach is described in Section 5; Section 6presents numerical results comparing the performance of the TDM and HM approaches. Brief conclusions and areas of further work are included in Section 7. 2. Experimental analysis The device used to illustrate the emission characteristics and modelling considerations is an LED lamp equipped with a buck-boost singlestage LED driver. The single-stage driver sets the output to the LED chain and shapes the ac line current iac, with variable switching frequency peak current control [20]. The choice of an LED lamp was made due to the following factors: (i) the power rating was compatible with the available laboratory set-up, (ii) due to the relative simplicity of the circuit, the authors could observe all stages of power conversion, and, importantly (iii) as a variable switching frequency converter, it presents an important new modelling case not yet discussed in literature. The aim of the experimental activity is to measure iac and identify the characteristics of the load to be modelled. The experimental analysis can also identify sensitivity to changes in the supply conditions which serve during the model evaluation process. 2.1. Measurement setup and supply voltage conditions The measurement chain consisted of a Data AcQuisition (DAQ) system, a Current Transducer (CT) and a Voltage Transducer (VT). A programmable power source was used to supply the equipment under test. The National Instruments (NI) PCI eXtension for Instrumentation (PXI) 5922 board (24 bit, 5 V, 500 kHz), housed in a PXI chassis running the measurement software (developed in LabVIEW), was used as the DAQ. Due to the resolution afforded by the 24 bit DAQ, it was not necessary to utilise a high-pass filter in the measurement chain (as demonstrated by the characterisation of 16 bit and 24 bit DAQs in [21]). The CT utilised was the Pearson Current Monitor Model 411 (50 A/5 V, 1 %, 1 Hz–20 MHz) and the VT used was the LEM CV3-1000 (700 V/7 V, 0.2%, 0–500 kHz). The programmable Power Source was the Pacific Power 3120 AMX (3-ph, 12 kVA, 20 Hz–5 kHz). Further details of the measurement chain and its characterisation are available in [3]. Nine different operating supply voltage conditions have been implemented: specifically, sinusoidal, flat top and peak top voltage waveforms with 0.9, 1.0 and 1.1 pu supply voltage magnitude. Nominal supply conditions are defined as a sinusoidal waveform with V1,rms = 230 V and f1= 50 Hz. The flat top and peak top voltage waveform are typical of the voltages present in residential/commercial and industrial networks, respectively, with definitions in [22]. 2.2. Results Fig. 1 shows the experimental results corresponding to nominal supply conditions, while the impact of changes in the supply voltage waveform (the nine different conditions previously mentioned) are reported later in Figs. 9 and 10 of Section 6. Fig. 1(a) shows one cycle (at fundamental frequency) of the measured input voltage and current waveform. It is possible to observe the long conduction period of the DBR; this is characteristic of DBRs with small (or zero) sized output capacitor and is reflected in the low magnitude of the low order harmonic components, shown in Fig. 1(b) and Fig. 1(c). The spectrogram of the input current and the corresponding magnitude according to the IEC metrics, i.e., the method in [23] extended to 150 kHz [24], are also shown in Fig. 1(d) and Fig. 1(e), respectively. The spectrogram has been calculated using a Hanning window of 0.5 ms width, with a time shift of 1 μs. The main HF emissions, not visible in Fig. 1(a), are observed around 50 kHz, and its multiples. Fig. 1(d) evinces the use of a variable switching frequency control strategy which produces intra-cycle (temporal) variations of the emissions. From 20 to 40 kHz some noise is observed from the interaction between the device and the Power Source, as the lamp exhibits a passive impedance response in the presence of background distortion [3]. This interaction can be removed by introducing a line impedance stabilisation network into the measurement chain [3]. Electric Power Systems Research 220 (2023) 109236 3 A.J. Collin et al. Fig. 1. Measured data at nominal supply conditions: (a) time domain voltage and current waveforms; low frequency current components: (b) magnitudes and (c) phase angles; high frequency current components: (d) spectrogram and (e) frequencies versus magnitudes. The results of the other supply conditions (explicitly referred to later in Figs. 9 and 10 of Section 6) demonstrate the impact of supply conditions on the emissions observed in the extended frequency range up to 150 kHz. The experimental analysis demonstrates the need for accurate models to account for the great complexity of the emissions and their sensitivity to supply conditions. 3. Modelling considerations As shown in the previous section, it is necessary to separate the emissions of the considered device into low frequency (LF), f<2 kHz, i.e., the traditional harmonic frequency range, and high frequency (HF), 2 ≤f<150 kHz, due to their different orders of magnitude and characteristics. Accordingly, the ac current absorbed by singlephase switch-mode a-PFC converter loads can be described by the superposition of LF and HF components: 𝑖𝑎𝑐 (𝑡)=𝑖𝑙𝑓 𝑎𝑐 (𝑡)+𝑖ℎ𝑓 𝑎𝑐 (𝑡),(1) where the HF emissions are produced exclusively by the operation of the dc–dc converter when pre-existing HF distortion in the supply voltage is not present. In steady-state conditions, the power electronic device can be assumed to be a time periodic system. The LF harmonic components are considered time invariant during the fundamental period T1of the supply voltage. However, in general, the amplitude I, angular frequency 𝜔and phase angle 𝜙of the HF components vary during T1due to the operation of the dc–dc switching converter: 𝑖ℎ𝑓 𝑎𝑐 (𝑡)=𝐼ℎ𝑓 𝑎𝑐 (𝑡)sin (𝜔ℎ𝑓 𝑎𝑐 (𝑡)𝑡+𝜙ℎ𝑓 𝑎𝑐 (𝑡)).(2) For the dc–dc converter considered in this paper, intra-cycle (temporal) variations in magnitude and frequency within the fundamental period can be clearly observed in the measured HF emissions in Fig. 1(d) and Fig. 1(e). 3.1. Requirements and assumptions Similar to the techniques used in the traditional LF frequency range, e.g., [25–27], the load models used for analysing HF distortion in ac networks are required to: (i) reproduce the magnitude and phase response for representation of attenuation and cancellation effects and (ii) have fast computation times for use in system level analysis. The extended frequency range up to 150 kHz introduces new considerations which could form additional modelling requirements; specifically: (iii) representation of the steady-state periodic intra-cycle variations of HF distortion shown in Fig. 1 and reported in, e.g., [28–31] and (iv) dependency of HF emissions on the LF characteristics of the supply network, e.g. [3,32], which points towards a model capable of emulating both LF and HF emissions. Modelling the power system at the load’s terminals as a pure voltage source (sinusoidal or distorted), i.e. assuming zero impedance, is a common assumption when developing a model of a single load. The interactions with the power supply system (impedance) can then be modelled inside a wider simulation framework, i.e., iterative harmonic analysis (IHA), for system level studies [25,33]. IHA has several advantages in the extended frequency range up to 150 kHz due to the fact that it represents the supply network in the frequency domain, while the loads can be modelled in either the time domain or the frequency domain. To model the supply system components, e.g., lines, cables, transformers etc., in the time domain requires the use of distributed parameters, e.g., hyperbolic functions [34,35], including parasitic components, which can significantly increase the modelling requirements and computation time. Therefore, the representation of the series and parallel resonances which are always present is extremely difficult to accurately realise with time domain equivalent circuits based on discrete components. Conversely, the complexity of the network impedance has no impact on the IHA process, which represents the network impedance using a FDM. This will be fully addressed in another paper. For the type of modelling approaches considered in this paper, it is necessary to know the converter topology, the control technique adopted, including its parameters, and the values of the main circuit components. This can be obtained following two different methods. The first is ‘‘design based’’, which starts from a generic functional description, allowing for the selection of a converter topology and the specification of circuit components and control strategy to then be theoretically developed. Examples of this approach include [18,19,36, 37]. The second method, better suited to existing equipment, is based on reverse engineering. This starts from a preliminary experimental observation of the device under study to identify characteristics relating to, e.g., the switching frequency, by observing LF and HF emissions. Examples include [38,39]. 4. Time domain modelling A TDM has been developed with respect to the requirements and assumptions discussed in Section 3.1. The resulting model is a trade-off between the accuracy and the computation time. For example, generic switching components and ideal control are utilized, while, on the other hand, all the main components are included in the model. Electric Power Systems Research 220 (2023) 109236 4 A.J. Collin et al. Fig. 2. Detailed schematic of the buck-boost converter supplying an LED chain. 4.1. Development The schematic of the TDM developed by the authors is shown in Fig. 2. The key components of the circuit topology and functionality have been identified by reverse engineering. Then, the values of the key components have been obtained by direct measurement or by a parameter estimation technique when direct measurement was not possible (e.g. the LED chain in Fig. 2). The control strategy of the switch sw has been identified directly from the datasheet [40]. 4.2. Implementation The model was implemented in MathWorks®Simulink®and consists of impedances Z1and Z2, representing the input and output filter impedances (i.e. the EMI filter) of the DBR, a DBR and a buck-boost dc–dc converter supplying an LED chain (represented by the third order approximation model). The component values of Fig. 2 are included in Table A.1 in Appendix A. The variable switching frequency peak current control algorithm is defined by Imax, the current limit, and Ton,min/max and Toff,min/max, which are the on and off time constraints of the electronic switch of the buck-boost converter, and was implemented using the parameter values specified by the datasheet [40]. The values are reproduced in Table A.2 of Appendix A. Due to the high level of detail, the developed TDM is able to accurately reproduce the LF and HF characteristics of the measured data. The results at nominal conditions are so close to those of Fig. 1 that they are not reported here for the sake of brevity. However, in Section 6the results of the TDM model corresponding to the nine supply conditions considered are reported alongside those of the HM approach. 4.3. Discussion In general, TDMs inherently satisfy criteria (i), (iii) and (iv) described in Section 3. However, they will struggle to satisfy criterion ii as they generally require long simulation times that limits their use in system level studies. One of the main limitations of using time domain approaches is manifest in the modelling of the converters themselves; this requires extremely small integration time steps to model the power electronic switching components and their control. When there is the need to model many devices and, in addition, embed these in the probabilistic frameworks typically utilised for network studies, e.g., to develop generic aggregates of low voltage power electronic equipment, simulation times can become prohibitive. This is discussed further in Section 6. 5. Hybrid modelling The HM approach, which is the main original contribution of this paper, is introduced and illustrated referring to the circuit in Fig. 2. It is a two step procedure. In Step 1, the emission source, i.e., the dc current at the input terminals of the dc–dc switching converter idc, is modelled in the time domain. In Step 2, the ac line current iac, which inherently includes LF and HF emissions, is obtained by combining frequency and time domain methods. Fig. 3. Hybrid model Step 1: Circuit to attain idc. Table 1 Input and output data overview of the proposed hybrid modelling approach. Name Description Step 1 Step 2 Topology dc–dc converter topology Input – L dc–dc converter inductor value Input – vodc–dc converter output voltage Input – Control Control algorithm type and settings Input – Z1, Z2Input and output impedances (EMI filter) – Input vac Supply voltage Input Input idc dc–dc converter input current Output Input iac ac line current – Output 5.1. Overview In Step 1, the dc-dc converter of the schematic in Fig. 2 is emulated using the simplified representation in Fig. 3. The input voltage viis modelled as a pure voltage source (sinusoidal or distorted), the DBR is assumed to be an ideal, lossless rectifier, with a zero-sized output capacitor, and the dc-dc converter output voltage vois assumed to be constant, i.e. assuming an infinite valued output filter capacitor and no voltage ripple or the presence of an outer control loop regulating vo. In Step 2, the ac line current iac is obtained using the circuits shown in Fig. 4, where the component labels refer to Fig. 2. This defines iac as consisting of 𝑖′ 𝑎𝑐 and 𝑖′′ 𝑎𝑐 obtained from the two time domain sources: idc in Fig. 4(a) and vac in Fig. 4(b). The required input and output information of the proposed technique is summarised in Table 1. 5.2. Step 1 With reference to the circuit of Fig. 3, the input voltage viis defined as: 𝑣𝑖(𝑡)=𝑎𝑏𝑠 (𝑣𝑎𝑐 (𝑡))=𝑎𝑏𝑠 (∑ ℎ 𝑉𝑎𝑐,ℎ sin (𝜔𝑎𝑐,ℎ𝑡+𝜙𝑎𝑐,ℎ))(3) where Vac,h,𝜔𝑎𝑐,ℎ and 𝜙𝑎𝑐,ℎ are the magnitude, angular frequency and phase of vac harmonic orders h = 1 ... H. While vois emulated as a controlled constant voltage source. To solve the circuit in Fig. 3, the proposed approach considers two separate sub-systems: the switching converter and the control circuit. The state of the switch in the switching converter is set by the control circuit via a feedback loop. Electric Power Systems Research 220 (2023) 109236 5 A.J. Collin et al. Fig. 4. Hybrid model Step 2: The circuits utilised to calculate the ac line current iac:(a) transfer of idc from dc to ac for 𝑖′ 𝑎𝑐 (b) excitation of the device input impedance by the power supply voltage vac for 𝑖′′ 𝑎𝑐 . 5.2.1. Switching converter model Emulating viand voas controlled voltage sources in Fig. 3 allows the switching converter to be modelled using only the inductor current iL. The 𝑣−𝑖relationship of the ideal inductor 𝐿can be solved at each simulation time step tsafter discretisation using any suitable integration technique [41]. For the results presented in this paper, the backward Euler method was used: 𝑖𝐿(𝑡𝑠)=𝑖𝐿(𝑡𝑠−𝛥𝑡)+1 𝐿𝑣𝐿(𝑡𝑠)𝛥𝑡, (4) where: 𝛥tis the simulation time step. For the buck-boost converter in Fig. 3, when the switch is closed, the source viis directly connected across inductor 𝐿and idc is equal to iL, which will charge proportionally to the voltage magnitude (4). When the switch is open, the energy stored in inductor 𝐿is transferred to the load; in this state idc is equal to zero and the voltage across 𝐿is equal to the negative value of voand inductor current iLwill decrease (4). 5.2.2. Control circuit model The control circuit determines the binary state sw of the dc–dc switching converter in order to regulate the measurand iLwith respect to a reference value Imax and the set of switch time limits Tlims: 𝑇𝑙𝑖𝑚𝑠 = {𝑇𝑜𝑛,𝑚𝑖𝑛, 𝑇𝑜𝑛,𝑚𝑎𝑥, 𝑇𝑜𝑓𝑓,𝑚𝑖𝑛, 𝑇𝑜𝑓 𝑓 ,𝑚𝑎𝑥},(5) where: Ton,min,Ton,max,Toff,min and Toff,max are the minimum and maximum on and off intervals of the switch (see Table A.2). The functionality of the control circuit can be emulated using a counter Tsw to track the time spent in the current switch position and simple combinational logic to compare the values of Tsw and iLagainst their respective limits. At each time step ts, the counter is compared to Tlims (5) and iL(4) against Imax to determine the switch state sw. Values of idc and vLare set as previously discussed in Section 5.2.1 depending on the position of the switch. This can be illustrated using the two different operating regimes presented in Fig. 5(a) and Fig. 5(b). Starting from 0.99 ms in Fig. 5(a), the sw is closed and the inductor will charge with respect to (4). Eventually, the time in the closed position will exceed limit Ton,max and the combinational logic will change the switch position to open and Tsw will reset to zero. In this switch position,idc is set to zero, while the inductor will discharge with respect to (4) until it is fully discharged, at which time the combinational logic will change the switch position to closed. In Fig. 5(a), it is clear that the magnitude of the inductor current iLis far from the current limit Imax. Therefore, the binding constraint is Ton,max. However, due to simple nature of the logical comparison all constraints are handled in the same way. Fig. 5(b) shows an exemplar section when the binding constraint is Imax. The difference in switching frequency can also be observed: in Fig. 5(a) the period is 0.015 ms (≃67 kHz), while, in Fig. 5(b), the period is 0.017 ms (≃58 kHz). By repeating this procedure, it is possible to obtain idc for one half period of the fundamental waveform. In the case of half cycle symmetry, the full period waveform of idc is obtained by appending a replica of idc to itself. Fig. 5(c) shows one half period of the fundamental waveform, demonstrating the intra-cycle variation in the peak magnitude of idc, reflecting the change in rate of charge of Lin response to changes in vi. Further detail of the extent of the intra-cycle variations of the magnitude and frequency of idc is presented in Fig. 6, where the results of TDM and HM are reported. The switching periods, of which examples are shown in Fig. 5(a) and Fig. 5(b), have been used to calculate both the frequency, as the reciprocal of each switching period, and, subsequently, the rms value of each switching period. These results clearly quantify the variation in Fig. 1(d) and Fig. 1(e). As the HF emissions in idc propagate towards the supply network, reproducing this is crucial for accurate network analysis. The full implementation algorithm is included in Appendix B. 5.3. Step 2 With reference to the circuits of Fig. 4, the non-linear operation of the DBR is taken into account by multiplying, ideally in the first instance, i.e., neglecting the conduction/commutation process, idc and vac by the signum function of vac. Assuming that the impedances Z1and Z2consist of only linear passive components, both circuits of Fig. 4 can be considered linear time invariant systems and modelled as a two-port network comprised of the cascade connection of Z1and Z2. Specifically, the circuit in Fig. 4(a) is represented by transmission parameters, with current gain parameter Dbetween the input 𝑖′ 𝑎𝑐 and output idc current, and the circuit in Fig. 4(b) is represented by admittance parameters, with the admittance Y11 connecting the input current 𝑖′′ 𝑎𝑐 and the input voltage vac. Thus, the currents 𝑖′ 𝑎𝑐 and 𝑖′′ 𝑎𝑐 are obtained in the frequency domain as: 𝑰′ 𝑎𝑐 (𝜔)={𝑖𝑑𝑐 (𝑡𝑠)⋅𝑠𝑔𝑛 (𝑣𝑎𝑐 (𝑡𝑠))}⋅𝐷(𝜔)(6) and: 𝑰′′ 𝑎𝑐 (𝜔)={𝑣𝑎𝑐 (𝑡𝑠)⋅𝑠𝑔𝑛 (𝑣𝑎𝑐 (𝑡𝑠))}⋅𝒀11 (𝜔)(7) where: idc is the current obtained during Step 1 of the modelling process, vac is the known ac line voltage, sgn and are the signum and Fourier transform operators, respectively, and Dand Y11 are the frequency domain representations of Dand Y11, respectively. It is then possible to model the conduction/commutation process of the DBR observed in Fig. 1 as a simplified switching function SFDBR: 𝑆𝐹DBR (𝑡𝑠)={1if: 𝑖′ 𝑎𝑐 (𝑡𝑠)+𝑖′′ 𝑎𝑐 (𝑡𝑠)>0, 0if: 𝑖′ 𝑎𝑐 (𝑡𝑠)+𝑖′′ 𝑎𝑐 (𝑡𝑠)≤0,(8) where 𝑖′ 𝑎𝑐 (𝑡𝑠)=−1{𝑰′ 𝑎𝑐 }and 𝑖′′ 𝑎𝑐 (𝑡𝑠)=−1{𝑰′′ 𝑎𝑐 }, respectively, and −1 is the inverse Fourier transform operator. This is visualised in Fig. 7 for a half cycle of the fundamental period, where the two components 𝑖′ 𝑎𝑐 ,𝑖′′ 𝑎𝑐 and their summation 𝑖′ 𝑎𝑐 +𝑖′′ 𝑎𝑐 are shown for nominal supply conditions together with the DBR switching function SFDBR. The final model output iac is then calculated in the time domain as: 𝑖𝑎𝑐 (𝑡𝑠)= (𝑖′ 𝑎𝑐 (𝑡𝑠)+𝑖′′ 𝑎𝑐 (𝑡𝑠))⋅𝑆𝐹 𝐷𝐵𝑅 (𝑡𝑠).(9) Electric Power Systems Research 220 (2023) 109236 6 A.J. Collin et al. Fig. 5. Hybrid model Step 1: Switching converter input current idc of the proposed model for nominal supply conditions: (a) operating regime 1 (b) operating regime 2 (c) half cycle of the line voltage fundamental period. Fig. 6. Intra-cycle variations of the switching converter input current idc for nominal supply conditions from the time domain model (TDM) and the hybrid model (HM): (a) frequency (b) current rms. Fig. 7. Hybrid model Step 2: Visualisation of the ac current calculation for nominal supply conditions showing 𝑖′ 𝑎𝑐 ,𝑖′′ 𝑎𝑐 and SFDBR. 5.4. Results The final output iac of the hybrid model, obtained by applying (9) to the results in Fig. 7, is shown in Fig. 8, again for a half cycle of the fundamental period at nominal supply conditions. The output of the TDM is also included and it is clear that the HM is able to accurately reproduce the time domain waveform of iac. The results at nominal conditions in the same terms of Fig. 1 are again so close that they are not reported here for the sake of brevity. It is worthwhile noting that the DBR’s switching function can be calculated with more accurate models accounting for the voltage drop Fig. 8. Comparison of final output iac of the hybrid model (HM) with that of the time domain model (TDM) at nominal supply conditions. across Z1, e.g., [42], but the proposed approach has been shown to give satisfactory results. 5.5. Generalisability The proposed methodology can be readily applied to model a wide variety of loads that contain ac–dc rectifiers with small or zero sized output capacitors with current mode control. Different converter topologies (i.e., buck, boost and buck-boost) with different fixed and variable switching frequency control strategies (i.e., peak, average, hysteresis and borderline control) can be represented. Such loads are commonly utilised in single-phase applications, with the average Electric Power Systems Research 220 (2023) 109236 7 A.J. Collin et al. Fig. 9. Distribution across all nine supply conditions of the low frequency harmonics of ac line current iac of the measured data, the time domain model (TDM) and the proposed hybrid model (HM): (a) magnitude (b) phase angle. current control boost a-PFC topology being the most diffuse in higher power applications, e.g., HVAC and EV battery chargers. The authors have successfully applied the modelling approach to a number of devices. Although the results are omitted here due to space limitations, they are available on request. 6. Model comparison versus experiments The evaluation of the HM versus the TDM and experimental results was performed considering the nine supply conditions described in Section 2.1. 6.1. Accuracy Results for the LF range are reported in Fig. 9 using boxplots to convey the distribution across the nine supply conditions of the magnitude and phase angles of the main input current harmonics. Both the TDM and HM accurately reproduce the measured data; the small differences are due to the assumptions made for both modelling approaches, e.g., when measuring/estimating circuit parameters. Concerning the deviations of the HM with respect to the TDM, in absolute terms, the differences are very small, reaching a largest magnitude deviation of less than 1.5×10−1 mA. The differences in phase angles of the HM with respect to the TDM, which are more evident for higher order harmonics with lower magnitude, are an artefact of the simplified model of the DBR system but are acceptable for the intended application, i.e., the reproduction of harmonic cancellation and attenuation effects. Results for the HF range are presented in Fig. 10 in terms of the mean and min–max range of the IEC groups magnitude (again according to the IEC metrics, i.e., the method in [23] extended to 150 kHz [24]). A visual inspection reveals that the variable frequency switching characteristic is well represented using both the TDM and the HM. The major deviations between the TDM and HM results and the measurements are present in the range from about 10 kHz to 35 kHz; this is of minor interest as it is far from the LF components and from the switching frequency. To provide a quantitative and compact comparison, the IEC groups were aggregated to provide a total harmonic current value around the switching frequency of the two variable components identified between 45–60 kHz and 90–120 kHz. The mean error is 6.0% for iac. The results in Figs. 9 and 10 also demonstrate the impact of changes in the supply conditions on the LF and HF characteristics. The impact on LF characteristics is as expected, with small variations in magnitude and phase angle due to the changes in the conduction period of the Table 2 Mean solution times. Implementation Solution time (s) TDM 12.33 HM 0.16 DBR. However, it is of great importance to observe that, in the HF interval (in particular, in the ranges 45–60 kHz, in which the switching converter operates, and 90–120 kHz), the current magnitude is influenced by the supply condition, with variations reaching one or more order of quantities. Both the TDM and the HM are able to accurately reproduce the LF and HF variations present in the measured data. 6.2. Computation time To compare the solution time of the TDM and the proposed HM, each supply condition was solved 100 times. All simulations were performed on a laptop computer with an Intel®Core™i7-85508 CPU (1.8 GHz) and 16 GB RAM. The Simulink accelerator mode – with a discrete time step of 0.01 μs – has been used to have comparable performance with other simulation tools (e.g. PSCAD). The mean solution times are presented in Table 2, demonstrating the acceleration provided by the proposed HM. The computation time is particularly relevant when considering system level analysis of distortion up to 150 kHz, as the interactions with the power supply system (impedance) must be modelled inside an IHA framework [25,33] which requires solving the load model several times until system convergence is achieved. 7. Conclusions This paper has considered the emission assessment, in the frequency range from 0 up to 150 kHz, of the switch-mode power factor correction converter topologies widely utilised in modern single-phase loads. Experimental analysis has been performed using an LED lamp to introduce modelling considerations and proposals. Then, a full circuit time domain model, which can accurately reproduce the emission characteristics of the measured load, has been developed. Following this, the main contribution of the paper was presented: a fast and accurate hybrid technique which simultaneously models the low frequency (f <2 kHz) and high frequency (2 ≤f<150 kHz) emissions of switch mode a-PFC converters. It was shown that the proposed model can accurately reproduce intra-cycle variations at solution times compatible Electric Power Systems Research 220 (2023) 109236 8 A.J. Collin et al. Fig. 10. Mean and min–max range of the high frequency emissions for all nine supply conditions: (a) measured data (b) time domain model (c) hybrid model. with system level analysis. The case study of a variable switching frequency converter clearly demonstrates the advantages of the proposed approach versus time domain models (with respect to simulation time) and other modelling approaches (which are limited to fixed switching frequency PWM converters). Although the approach has been demonstrated using an LED lamp, it is more widely applicable and the authors have already successfully modelled a number of devices, with results omitted due to space limitations. The significant computational advantages compared to the time domain model, achieved with only a small reduction in accuracy, are of particular importance in system levels studies of multiple devices. Work is ongoing to solve the multiple converter systems that appear of paramount importance to support standardisation activities related to power system distortion in the frequency range up to 150 kHz. CRediT authorship contribution statement Adam J. Collin: Conceptualization, Methodology, Data curation, Software, Validation, Writing – original draft, Visualization, Investigation, Writing – review & editing. Jiri Drapela: Conceptualization, Methodology, Resources, Software, Validation, Visualization, Investigation, Writing – review & editing. Roberto Langella: Conceptualization, Methodology, Resources, Software, Validation, Visualization, Investigation, Writing – review & editing. Alfredo Testa: Conceptualization, Methodology, Supervision, Writing – review & editing. Declaration of competing interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Data availability Data will be made available on request. Acknowledgments This work was supported in part by the Centre for Research and Utilization of Renewable Energy and in part by the Ministry of Education, Youth and Sports of the Czech Republic under Brno University of Technology Specific Research Program, Project FEKT-S-23-8403. Appendix A. Considered load data The component values of the schematic shown in Fig. 2 are reported in Table A.1, while the control algorithm parameters are shown in Table A.2. Table A.1 Considered load data: component values. R1 R2 R3 Rdc Rled L1 L2 Ldc C1 C2 C3 Co Vled (Ω) (Ω) (Ω) (Ω) (Ω) (mH) (mH) (mH) (nF) (nF) (nF) (nF) (V) 100 5k 1.5k 3 30 2 15 2.6 22 110 240 22 × 10466.1 Table A.2 Considered load: control algorithm parameters. Imax (A) Toff,min (μs) Toff,max (μs) Ton,min (μs) Ton,max (μs) 0.385 1.5 40 (0) 6.2 Fig. B.1. Algorithm for calculating the input current idc of the dc–dc switching converter in Step 1. Appendix B. HM step 1 implementation The switching converter and the control circuit models are connected by the algorithm shown in Fig. B.1. Starting from the switching converter model, iLis calculated for time step ts(4) and idc is set according to the switch state. The control algorithm procedure is then called: after checking the switch status sw, the operating limits are checked. This compares the measurand iLagainst the threshold value, either Imax or zero, depending on the switch position, and checks the Electric Power Systems Research 220 (2023) 109236 9 A.J. Collin et al. time spent in the current switch position against the time limits Tlims. The comparison with Tlims is achieved using the inloop counter Tsw, which tracks the time spent in the current state and resets when the switch changes state. The combinational logic comparisons on Lines 15 and 21 set sw for the next iteration at ts+𝛥t. 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