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Frequency Multiplier Algorithm Based Fundamental Active Current Extraction and Phase Locked Loop for the Control of 3-Phase Shunt Active Power Filter
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Amit V. SANT1, Arpitkumar J. PATEL1, Josep M. GUERRERO2
CPSS Transactions on Power Electronics and Applications | 2024, 9(4) : 384 - 394
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CPSS Transactions on Power Electronics and Applications | 2024, 9(4): 384-394
• Original articles •
Frequency Multiplier Algorithm Based Fundamental Active Current Extraction and Phase Locked Loop for the Control of 3-Phase Shunt Active Power Filter
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Amit V. SANT1, Arpitkumar J. PATEL1, Josep M. GUERRERO2
Affiliations
  • 1 Pandit Deendayal Energy University Department of Electrical Engineering, School of Energy Technology Gandhinagar India
  • 2 Universitat Politècnica de Catalunya (UPC) Barcelona Spain

    • Amit V. Sant received Ph.D. degree in Power Electronics from the Indian Institute of Technology Delhi, New Delhi, India, in 2013. Presently, he is an Associate Professor in the Department of Electrical Engineering, School of Energy Technology, Pandit Deendayal Energy University (PDEU), Gandhinagar, Gujarat, India. Before joining PDEU, he was a Postdoctoral Research Researcher at the Masdar Institute of Science and Technology, Abu Dhabi, United Arab Emirates. His present research focuses on multilevel inverters, z-source inverters, high gain DC-DC converters, grid integration of renewables, charging infrastructure for electric vehicles, power quality enhancement, smart metering, and applications of artificial intelligence in power electronic systems.

    Arpitkumar J. Patel is currently pursuing a Ph.D. in Electrical Engineering at Pandit Deendayal Energy University, focusing on the multifunctional control of custom power devices for EV charging stations incorporating Solar PV systems. He holds an M.Tech in Electrical Engineering (Power Systems) from Pandit Deendayal Energy University, where he worked on the control of grid-tied inverters. He completed his B.E. in Electrical Engineering from Gujarat Technological University. His research interests include electric vehicle charging systems, power electronic converters, power quality, and control algorithms.

    Josep M. Guerrero received the B.S. degree in telecommunications engineering, the M.S. degree in electronics engineering, and the Ph.D. degree from the Technical University of Catalonia, Barcelona, Spain, in 1997, 2000, and 2003, respectively. In 2019, he was a Villum Investigator with the Villum Fonden, which supports the Center for Research on Microgrids (CROM), Aalborg University, Aalborg, Denmark, where since 2011, he has been a Full Professor with the Department of Energy Technology. His research interests include different microgrid aspects, including applications as remote communities, energy prosumers, and maritime and space microgrids.

Published: 2024-12-10 doi: 10.24295/CPSSTPEA.2024.00022
Outline
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Control of shunt power active filter (SAPF) necessitates estimation of the fundamental active component (FAC) of load current and unit voltage templates (UVTs). In this paper, a frequency multiplier based FAC and UVT extractor is proposed, wherein the αβ quantities of load current and supply voltage undergo frequency multiplier action to obtain the respective components with fundamental frequency four times the power frequency. With the band pass filtering of these signals, the components corresponding to four times the power frequency are determined. Thus obtained current components are further processed to extract the FAC of load current. Similarly obtained voltage signals are used by the synchronous reference frame theory based phase locked loop for accurate UVT extraction with the help of the designed synchronizing logic. The comparative analysis performed using an experimental setup demonstrates faster dynamic response and accurate estimation with a developed extractor compared to earlier reported schemes. The performance of SAPF controlled with the proposed extraction algorithm is investigated in PSIM software. Further, experimental validation is also presented. The SAPF operation with the proposed control scheme ensures unity powerfactor operation and adherence to total harmonic distortion (THD) limits by drawing sinusoidal currents from the grid.

Active filters  /  frequency multiplier  /  harmonics  /  power quality  /  shunt active power filter
Amit V. SANT, Arpitkumar J. PATEL, Josep M. GUERRERO. Frequency Multiplier Algorithm Based Fundamental Active Current Extraction and Phase Locked Loop for the Control of 3-Phase Shunt Active Power Filter[J]. CPSS Transactions on Power Electronics and Applications, 2024 , 9 (4) : 384 -394 . DOI: 10.24295/CPSSTPEA.2024.00022
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I. Introduction
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WTTH the advances in signal and power electronics, power converters have emerged as the most suitable option for the modulation of power from supply to load in an efficient and automated manner. Moreover, with the increasing focus on energy conservation, power converters are increasingly being employed in lighting, electronic appliance, transportation, industrial and utility sectors [1]. However, the power converters usually necessitate the drawing of non-sinusoidal currents from the utility grid and thus, are seen as a nonlinear load from the utility end.
The increasing penetration of nonlinear loads is liable for power quality (PQ) degradation through current distortions and consequent voltage distortions [2]-[4]. These PQ issues have a detrimental effect on distribution system equipment and connected loads. These distortions are also accountable for increased line losses, overheating of cables, failure of protective systems and critical loads, etc.[2],[4]-[8]. As the PQ problems are directly or indirectly responsible for economic losses and an impediment to energy savings, it is necessary to mitigate them [9]. Hence, installation of a shunt active power filter (SAPF) for employing shunt compensation, at end user premise is advisable to prevent the deterioration of PQ due to the proliferation of current harmonics into the supply [10]. With SAPF supplying the harmonic and reactive components of load current, the grid needs to provide only the fundamental active component (FAC).
To control SAPF, extraction of FAC of load current and unit voltage template (UVT) extraction is critical. For estimating the FAC of load current, use of self-tuning filters [11], II-order filters [5],[7],[12], generalized integrators [13], neural networks [2],[14],[15], variants of least mean square (LMS) algorithm [16]-[19], and $I\cos \left(\phi \right)$ algorithm [20] are reported. The performance of generalized integrators and self-tuning filters is highly affected by the values of the constants used [5]. It necessitates proper selection of the constants to obtain optimal performance in terms of dynamic response and accuracy [5]. Training neural networks require a large data set corresponding to each possible case for accurate estimation when implemented in real-time [5],[21]. LMS algorithms are computationally intensive, which is further compounded when a variable convergence factor is employed for the performance enhancement of the LMS algorithm [5],[22]. With a fixed convergence factor, tuning is required and the LMS algorithm cannot provide optimal FAC extraction with respect to dynamic and steady-state response. In $I\cos \left(\phi \right)$ algorithm, the load current is filtered to obtain the fundamental component with a phase shift of ${90}^{\circ }\left\lbrack {20}\right\rbrack$. The FAC is the value of filtered current at the phase angle of $\pi .I\cos \left(\phi \right)$ algorithm is computationally simple and determines the peak value of FAC once every cycle [20]. A self-constructing fuzzy neural fractional order sliding mode control of active power filter is proposed in [23]. This control algorithm processes the reference harmonic current and the actual harmonic current for the generation of reference signal for the sinusoidal pulse width modulation technique. The control scheme is free from performance deterioration under variations in load or passive elements. In [24], fuzzy neural super twisting sliding mode control of SAPF using nonlinear extended state observer is proposed. This control scheme utilizes fast harmonic detection algorithm to compute the reference current.
UVT extraction is another significant part of the control of SAPF. For UVT extraction, synchronous reference frame (SRF) theory-based phase-locked loop (PLL) and generalized integrators are largely employed [25]. For FAC and UVT extraction, the challenge is maintaining the operation undisturbed without compromising the accuracy under non-ideal grid conditions and fast dynamic response. Also, a minimum requirement for the determinations of involved constants is preferable.
In this paper, the frequency multiplier based FAC and UVT extractor is proposed for the control of SAPF. This extraction method can also be extended for controlling distributed static compensator and unified power quality conditioner. In the proposed method, $\alpha -\beta$ axis quantities of load current and supply voltage with fundamental frequency as four times the power frequency are computed with the designed frequency multiplier. Thus, obtained current and voltage signals are filtered using band pass filter (BPF) to extract the components corresponding to four times the power frequency. Synchronizing logic is developed to assist the SRF-PLL in processing the determined voltage signals for the extraction of UVTs corresponding to fundamental positive sequence component (FPSC) of supply voltage. The obtained voltage and current signals are processed by the FAC extractor for computing peak value and power-factor (PF) of the FPSC of load current. Based on these computations, the FAC of load current is extracted. The proposed methodology results in accurate extraction of FAC and UVTs even under non-ideal operating conditions. With frequency multiplier action, faster extraction can be achieved and consequently dynamic response of SAPF can be improved. A comparative analysis of the proposed FAC extraction method with state-of-the-art techniques is presented in Table I, wherein, the proposed FAC extraction technique is compared with ADALINE-LMS and SRF theory-based FAC extraction methods [5],[19],[26]-[27].
Highlights of the proposed work are:(a) frequency multiplier algorithm based on fundamental active current and unit voltage template extractor for the control of shunt active power filter is proposed,(b) proposed extraction algorithm offers a fast dynamic and accurate steady-state response,(c) control of SAPF with the proposed frequency multiplier based FAC and UVT extraction method is evaluated through simulation and experimental studies, and (d) SAPF operation using the proposed technique ensures power quality compliance with sinusoidal and balanced supply currents and unity PF.
II. Generalized Control of Shunt Active Power FILTER
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Fig. 1 demonstrates the power structure of SAPF consisting of a voltage source inverter (VSI) integrated with the grid at the point of common coupling (PCC) via coupling inductors, ${L}_{\mathrm{{Fa}}}- {L}_{\mathrm{{Fb}}}- {L}_{\mathrm{{Fc}}}$. In the power structure, ${C}_{\mathrm{{dc}}}$ and ${v}_{\mathrm{{dc}}}$ represent the ${DC}$ -link capacitor and the voltage measured across it. The three-phase voltages measured at PCC and the grid voltages are denoted as ${v}_{\mathrm{{Sa}}}- {v}_{\mathrm{{Sb}}}- {v}_{\mathrm{{Sc}}}$ and ${v}_{\mathrm{{Ga}}}- {v}_{\mathrm{{Gb}}}- {v}_{\mathrm{{Gc}}}$, respectively. ${R}_{\mathrm{s}}$ and ${L}_{\mathrm{s}}$ represents the line resistance and inductance, respectively. ${i}_{\mathrm{{La}}}- {i}_{\mathrm{{Lb}}}- {i}_{\mathrm{{Lc}}}$ and ${i}_{\mathrm{{Sa}}}- {i}_{\mathrm{{Sb}}}- {i}_{\mathrm{{Sc}}}$ are the load currents and currents drawn from the grid, respectively. ${i}_{\mathrm{{Fa}}}- {i}_{\mathrm{{Fb}}}- {i}_{\mathrm{{Fc}}}$ are the currents supplied by SAPF for the requisite shunt compensation.
In case of non-linear loads, ${i}_{\mathrm{{La}}}- {i}_{\mathrm{{Lb}}}- {i}_{\mathrm{{Lc}}}$ are distorted and for the ${k}^{\text{th }}$ phase can be represented as
${i}_{\mathrm{L}k}\left(t\right)= {i}_{\mathrm{L}k\mathrm{{FAC}}}\left(t\right)+ {i}_{\mathrm{L}k\mathrm{{FRC}}}\left(t\right)+ {i}_{\mathrm{L}k\mathrm{{HC}}}\left(t\right)$
where, ${i}_{\mathrm{L}k\mathrm{{FAC}}}$ is the FAC of ${i}_{\mathrm{L}k},,{i}_{\mathrm{L}k\mathrm{{HC}}}$ is the harmonic component (HC) of ${i}_{\mathrm{L}k},{i}_{\mathrm{L}k\mathrm{{FRC}}}$ is the fundamental reactive component of (FRC) of ${i}_{\mathrm{L}k}, t$ is time, and $k$ can be a, b or c.
The three components of ${i}_{\mathrm{L}k}$ are ${i}_{\mathrm{L}k\mathrm{{FAC}}}$ (fundamental active part responsible for supplying active power to the load), ${i}_{\mathrm{L}k\mathrm{{FRC}}}$ (fundamental reactive part responsible for load magnetization) and ${i}_{\mathrm{L}k\mathrm{{HC}}}$ (harmonics part responsible for the PQ degradation). In absence of SAPF, ${i}_{\mathrm{L}k\mathrm{{FAC}}},{i}_{\mathrm{L}k\mathrm{{FRC}}}$, and ${i}_{\mathrm{L}k\mathrm{{HC}}}$ are supplied by the grid. ${i}_{\mathrm{L}k\mathrm{{FRC}}}$ and ${i}_{\mathrm{L}k\mathrm{{HC}}}$ are responsible for grid congestion and reduced power transfer capability. Also, the flow of ${i}_{\mathrm{L}k\mathrm{{HC}}}$ through the line is responsible for the non-sinusoidal voltage drop across the line impedance, resulting in distorted grid voltages and further deterioration of PQ. To collectively solve these issues, SAPF can be installed at the consumer premises to ensure that ${i}_{\mathrm{{Sa}}}- {i}_{\mathrm{{Sb}}}- {i}_{\mathrm{{Sc}}}$ are sinusoidal with unity PF. For this, ${i}_{\mathrm{{Fa}}}- {i}_{\mathrm{{Fb}}}- {i}_{\mathrm{{Fc}}}$ need to be controlled as per (2). This results in ${i}_{\mathrm{S}k}$ being equal to ${i}_{\mathrm{L}k\mathrm{{FAC}}}$ part of ${i}_{\mathrm{L}k}$. Thus, the proliferation of FRC and HC into the grid and occurrence of consequent issues, will be prevented, resulting in energy conservation and reduced energy bills.
${i}_{\mathrm{F}k}\left(t\right)= {i}_{\mathrm{L}k\mathrm{{FRC}}}\left(t\right)+ {i}_{\mathrm{L}k\mathrm{{HC}}}\left(t\right)$
The generalized control of SAPF, presented in Fig. 2， involves computation of reference source currents, ${i}_{\mathrm{{SaR}}}- {i}_{\mathrm{{SbR}}}- {i}_{\mathrm{{ScR}}}$, for the indirect control of ${i}_{\mathrm{{Fa}}}- {i}_{\mathrm{{Fb}}}- {i}_{\mathrm{{Fc}}}$. As per (3), computation of ${i}_{\mathrm{{SaR}}}- {i}_{\mathrm{{SbR}}}- {i}_{\mathrm{{ScR}}}$ requires extraction of peak value of FAC, ${I}_{\mathrm{{Fl}}}$, and UVT, ${u}_{k}$, corresponding to ${v}_{\mathrm{S}k}\cdot {u}_{k}$ is stated in (4), where ${\omega }_{1}$ is the fundamental power frequency and ${p}_{k}$ is the phase difference between ${u}_{k}$ and UVT for the a-phase.
${i}_{\mathrm{S}k\mathrm{R}}\left(t\right)= {I}_{\mathrm{F}1}\left(t\right){u}_{k}\left(t\right)$
${u}_{k}\left(t\right)= \sin \left({{\omega }_{1}t -{p}_{k}}\right)$
The DC-link control of SAPF involves processing ${e}_{\mathrm{{dc}}}$, the difference between ${v}_{\mathrm{{dc}}}$ and its reference value, ${v}_{\mathrm{{dcR}}}$, through a proportional-integral (PI) controller for determining the active current required to be drawn from the grid, ${i}_{\mathrm{{dcR}}}$, for the necessary regulation. Subsequently, to incorporate regulation of ${v}_{\mathrm{{dc}}},{i}_{\mathrm{{SkR}}}$ is modified as per (5). Gate pulses for the SAPF are generated upon comparing ${i}_{\mathrm{S}k\mathrm{R}}$ with the respective ${i}_{\mathrm{S}k}$ using hysteresis current controller.
${i}_{\mathrm{S}}\left(t\right)= \left\lbrack {{I}_{\mathrm{F}1}\left(t\right)+ {i}_{\mathrm{{dcR}}}\left(t\right)}\right\rbrack {u}_{k}\left(t\right)$
III. Frequency Multiplier Based UVT and FAC EXTRACTION
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A. Proposed Frequency Multiplier Action
3-phase voltages or currents can be represented in $\alpha -\beta$ reference frame as given in (6)-(7), where $x$ represents voltage or current, ${X}_{\mathrm{m}}$ and ${\theta }_{1x}$ are the peak value and phase angle of FPSC, and ${H}_{\alpha }- {H}_{\beta }$ are the respective harmonic components. The frequency multiplier action is employed in two steps. First, ${x}_{\alpha }- {x}_{\beta }$ are individually processed by the frequency multiplier to determine ${x}_{\alpha 2}- {x}_{\beta 2}$, whose fundamental frequency is twice that of ${x}_{\alpha }- {x}_{\beta }$. Thereafter, ${x}_{\alpha 2}- {x}_{\beta 2}$ undergo frequency multiplier action for obtaining ${x}_{\alpha 4}- {x}_{\beta 4}$, which are having the fundamental frequency as four times that of ${x}_{\alpha }- {x}_{\beta }$. Fig. 3 displays the block diagram depiction of the computational process for frequency multiplier action as per (8)-(19). The steps involved in frequency multiplier action are elaborated in the Appendix.
${x}_{\alpha }= {X}_{\mathrm{m}}\sin {\theta }_{1x}+ {H}_{\alpha }$
${x}_{\beta }= {X}_{\mathrm{m}}\cos {\theta }_{1x}+ {H}_{\beta }$
${x}_{\alpha 2}= 2{x}_{\alpha }{x}_{\beta }$
${x}_{\alpha 2}= {X}_{\mathrm{m}}^{2}\sin 2{\theta }_{1x}+ {H}_{\alpha 2}$
${H}_{\alpha 2}= 2\left({{H}_{\alpha }{H}_{\beta }+ {H}_{\alpha }{X}_{\mathrm{m}}\cos {\theta }_{1x}+ {H}_{\beta }{X}_{\mathrm{m}}\sin {\theta }_{1x}}\right)$
${x}_{\beta 2}= {x}_{\beta }^{2}- {x}_{\alpha }^{2}= {X}_{\mathrm{m}}^{2}\cos 2{\theta }_{1x}+ {H}_{\beta 2}$
${x}_{\beta 2}= {X}_{\mathrm{m}}^{2}\cos 2{\theta }_{1x}+ {H}_{\beta 2}$
${H}_{\beta 2}= {H}_{\beta }^{2}- {H}_{\alpha }^{2}+ 2{X}_{\mathrm{m}}\left({{H}_{\beta }\cos {\theta }_{1x}- {H}_{\alpha }\sin {\theta }_{1x}}\right)$
${x}_{\alpha 4}= 2{x}_{\alpha 2}{x}_{\beta 2}$
${x}_{\alpha 4}= {X}_{\mathrm{m}}^{4}\sin 4{\theta }_{1x}+ {H}_{\alpha 4}$
${H}_{\alpha 4}= {H}_{\alpha 2}{H}_{\beta 2}+ {H}_{\alpha 2}{X}_{\mathrm{m}}^{2}\cos 2{\theta }_{1x}$
${x}_{\beta 4}= {x}_{\beta 2}- {x}_{\alpha 2}$
${x}_{\beta 4}= {X}_{\mathrm{m}}^{4}\sin 4{\theta }_{1x}+ {H}_{\beta 4}$
${H}_{\beta 4}= {H}_{\beta 2}{}^{2}- {H}_{\alpha 2}{}^{2}+ 2{X}_{\mathrm{m}}{}^{2}\left({\sin 2{\theta }_{1x}\cos 2{\theta }_{1x}}\right)$
The frequency spectrum for the distorted ${x}_{\alpha 4}$, having the fundamental frequency of ${200}\mathrm{\;{Hz}}$, is tabulated in Table II. It is clear that both harmonics and sub-harmonics are present. Using second-order BPF with center frequency as the nominal frequency, the fundamental component of ${x}_{\alpha 4}- {x}_{\beta 4}$ are obtained as shown in (20)-(21). Now, the fundamental orthogonal components ${v}_{\mathrm{S}{\alpha 4}\mathrm{\;F}}- {v}_{\mathrm{S}{\beta 4}\mathrm{\;F}}$ and ${i}_{\mathrm{L}{\alpha 4}\mathrm{\;F}}- {i}_{\mathrm{L}{\beta 4}\mathrm{\;F}}$ can be computed as shown by (22)-(25), where ${V}_{1}$ and ${I}_{1}$ are the peak amplitudes of PSFC of ${v}_{\mathrm{S}\alpha }- {v}_{\mathrm{S}\beta }$ and ${i}_{\mathrm{L}\alpha }- {i}_{\mathrm{L}\beta }$. The phase angle of PSFC of ${v}_{\mathrm{S}a}$ and displacement angle are denoted by ${\theta }_{1}$ and ${\phi }_{1}$, respectively.
${x}_{{\alpha 4}\mathrm{\;F}}= {X}_{\mathrm{m}}^{4}\sin 4{\theta }_{1x}$
${x}_{{\beta 4}\mathrm{\;F}}= {X}_{\mathrm{m}}^{4}\cos 4{\theta }_{1x}$
${v}_{\mathrm{S}{\alpha 4}\mathrm{\;F}}= {V}_{1}^{4}\sin \left({4{\theta }_{1}}\right)$
${v}_{\mathrm{S}{\beta 4}\mathrm{\;F}}= {V}_{1}^{4}\cos \left({4{\theta }_{1}}\right)$
${i}_{\mathrm{L}{\alpha 4}\mathrm{\;F}}= {I}_{1}^{4}\sin \left({4{\theta }_{1}- 4{\phi }_{1}}\right)$
${i}_{\mathrm{L}{\beta 4}\mathrm{\;F}}= {I}_{1}^{4}\cos \left({4{\theta }_{1}- 4{\phi }_{1}}\right)$
B. Proposed FAC Extraction Scheme
The block diagram representation of the proposed algorithm for extracting ${I}_{\mathrm{F}1}$ and generating of ${i}_{\mathrm{{SaR}}}- {i}_{\mathrm{{SbR}}}- {i}_{\mathrm{{ScR}}}$ is shown in Fig. 4. As reported in [20], ${I}_{\mathrm{F}1}$ can be computed by sampling the fundamental component of ${i}_{\mathrm{L}}$ phase shifted by ${90}^{\circ }$ at the zero-crossing instant of the fundamental supply voltage. However, ${I}_{\mathrm{F}1}$ can also be computed by sampling the $\left|{i}_{\mathrm{L}}\right|$ at the peak of $\left|{v}_{\mathrm{s}}\right|$. Both the quantities should comprise of respective fundamental component only. Based on this alternate concept, the developed algorithm employs sample and hold logic for sampling $\left|{i}_{\mathrm{L}{\alpha 4}\mathrm{\;F}}\right|$ at the maximum value of $\left|{v}_{\mathrm{S}{\alpha 4}\mathrm{\;F}}\right|$. The sampled current, ${I}_{\mathrm{F}{\alpha 4}}$, is mathematically expressed in (26). The determination of ${V}_{1}$ and ${I}_{1}$ are also included in this proposed algorithm. The peak value is computed by observing three consecutive samples. If the middle sample is greater than both the adjacent samples than that value represents the peak value.
Using (27), ${\phi }_{1}$ is calculated. ${\phi }_{1}$ can vary from ${0}^{\circ }$ to $-\pi /2$ for inductive loads and $4{\phi }_{1}$ can range from 0 to $-{2\pi }$. Generally, in software the inverse cosine operation computes angles in the range of $-\pi$ to $\pi$. When ${\phi }_{1}$ ranges from 0 to $-\pi /4$, the computation of $4{\phi }_{1}$ will be correct since it ranges from 0 to $-\pi$. However, when ${\phi }_{1}$ ranges from $-\pi /4$ to $-\pi /2,4{\phi }_{1}$ varies between $-\pi$ to $-{2\pi }$, which leads to erroneous result. Hence, to overcome this error the corrected displacement angle, ${\phi }_{2}$, is calculated based on the polarity of ${v}_{\mathrm{S}{\alpha 4}\mathrm{\;F}}$ and $\mathrm{d}{i}_{\mathrm{L}{\alpha 4}\mathrm{\;F}}/\mathrm{d}t$ as per the look-up table given in Table III. Next, the fundamental power-factor, $P{F}_{1}$, and ${I}_{\mathrm{F}1}$, are computed as per (28)-(29), respectively. During steady state, ${I}_{\mathrm{F}1}$ is filtered through a moving average filter for removing the high frequency components and noise.
${I}_{\mathrm{F}{\alpha 4}}= {\left.\left|{\dot{i}}_{\mathrm{L}{\alpha 4}\mathrm{\;F}}\right|\right|}_{\theta ={22.5}\&\theta ={67.5}}= {I}_{1}^{4}\cos \left({4{\phi }_{1}}\right)$
${\phi }_{1}= {\cos }^{-1}\left(\frac{{I}_{\mathrm{F}{\alpha 4}}}{{I}_{1}^{4}}\right)/4 $
$ P{F}_{1}= \cos \left({\phi }_{2}\right)$
${I}_{\mathrm{F}1}= {I}_{1}\times \cos {\phi }_{2}$
C. Proposed UVT Extraction Scheme
Fig. 5 displays the block diagram representation of the proposed frequency multiplier based SRF-PLL with synchronizing logic for UVT extraction. ${v}_{\mathrm{S}{\alpha 4}\mathrm{F}}- {v}_{\mathrm{S}{\beta 4}\mathrm{\;F}}$ are utilized by the SRF-PLL structure for determining the d-q axis components, ${v}_{\mathrm{S}{d4}\mathrm{\;F}}- {v}_{\mathrm{S}{q4}\mathrm{\;F}}$. ${v}_{\mathrm{{Sq4F}}}$ is processed by a PI controller for computing ${\omega }_{4}\left({ = 4{\omega }_{1}}\right)$. Thereafter, for computing ${\theta }_{4}\left({ = 4{\theta }_{1}}\right)$ and ${\theta }_{1},{\omega }_{4}$ is individually processed by integrators ${\mathrm{Y}}_{2}$ and ${\mathrm{Y}}_{1}$, which are correspondingly reset at ${2\pi }$ and ${8\pi }.{\theta }_{1}\left({ ={\omega }_{1}\mathrm{t}}\right)$ is obtained by dividing the output of ${\mathrm{Y}}_{1}$ by four.
Referring to Fig. 6， the exact instant to reset ${\theta }_{4}$, at which ${\theta }_{1}$ is also reset, needs to be determined for ensuring proper synchronization. By the time ${\theta }_{1}$ reaches from 0 to ${2\pi },{\theta }_{4}$ has been reset four times. Hence, the initial starting point for ${\theta }_{1}$ is to be determined for proper synchronization. This is ensured by the developed synchronizing logic shown in Fig. 5. In this logic, SR flip-flop SR-2 is initially reset, which results in the input and output of ${\mathrm{Y}}_{1}$ being held at zero. At start, the first negative half cycle of ${v}_{\mathrm{S}\alpha \mathrm{F}}$, the fundamental component of ${v}_{\mathrm{S}\alpha }$, results in output of comparator ${\mathrm{C}}_{1}$ being set to logic one. This in turn sets SR flip-flop ${SR}$ -1 . But, the output of AND gate is logic zero as the output of comparator ${\mathrm{C}}_{2}$ is low. Upon the occurrence of subsequent positive half cycle, output of ${\mathrm{C}}_{2}$ and consequently that of AND gate goes high. This sets ${SR}- 2$ and enables ${\mathrm{Y}}_{1}- {\mathrm{Y}}_{2}$. Thus, the synchronizing logic ensures that ${\mathrm{Y}}_{1}-$ ${\mathrm{Y}}_{2}$ are enabled at the beginning of positive half cycle of ${v}_{\mathrm{{S\alpha F}}}$ and thereby results in synchronization of ${\theta }_{1}$ with ${v}_{\mathrm{S}\alpha \mathrm{F}}$. Based on ${\theta }_{1}$, the UVTs are computed and subsequently, based on ${u}_{k},{I}_{1}$, ${i}_{\mathrm{{dcR}}}$ and ${\phi }_{2},{i}_{\mathrm{S}k\mathrm{R}}$ is computed.
IV. Control of Shunt Active Power Filter With FREQUENCY MULTIPLIER BASED UVT AND FAC EXTRACTION
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The block diagram illustration of the control of SAPF with the proposed frequency multiplier algorithm based FAC and UVT extraction is shown in Fig. 7. Based on ${v}_{\mathrm{{Sa}}}- {v}_{\mathrm{{Sb}}}- {v}_{\mathrm{{Sc}}}$, ${v}_{\mathrm{S}{\alpha 4}}- {v}_{\mathrm{S}{\beta 4}}$, having fundamental frequency as ${\omega }_{4}$ (i.e. $4{\omega }_{1}$) are obtained using (8)-(19). The fundamental components, ${v}_{\mathrm{{S\alpha 4F}}}$ ${v}_{\mathrm{S}{\beta 4}\mathrm{\;F}}$, are further determined by processing ${v}_{\mathrm{S}{\alpha 4}\mathrm{\;F}}- {v}_{\mathrm{S}{\beta 4}\mathrm{\;F}}$ through BPF. As shown in Fig. 5， SRF-PLL assisted with synchronizing logic is implemented for the computation of ${\theta }_{1}$ and ${u}_{\mathrm{a}}- {u}_{\mathrm{b}}- {u}_{\mathrm{c}}$. Similarly, using (8)-(19), the frequency multiplier algorithm computes ${i}_{\mathrm{L}{\alpha 4}}- {i}_{\mathrm{L}{\beta 4}}$, having fundamental frequency as ${\omega }_{4}.{i}_{\mathrm{L}{\alpha 4}\mathrm{F}}-$ ${i}_{\mathrm{L}{\beta 4}\mathrm{\;F}}$ are then obtained by filtering ${i}_{\mathrm{L}{\alpha 4}}- {i}_{\mathrm{L}{\beta 4}}$. As shown in Fig. 4， using alternate implementation of $I\cos \left(\phi \right)$ algorithm, $\left|{i}_{\mathrm{L}{\alpha 4}\mathrm{\;F}}\right|$ is sampled at the peak of $\left|{v}_{\mathrm{S}{\alpha 4}\mathrm{\;F}}\right|$. The sampled value is used to obtain ${I}_{\mathrm{F}1}$ as per (26)-(29) and Table III. Further, the regulation of ${v}_{\mathrm{{dc}}}$ is implemented as per the discussion presented in Section II and as shown in Fig. 7. The peak value of active current to be drawn from the grid is summation of ${I}_{\mathrm{F}1}$ and ${i}_{\mathrm{{dcR}}}$, which is multiplied with ${u}_{\mathrm{a}}- {u}_{\mathrm{b}}- {u}_{\mathrm{c}}$ to generate the ${i}_{\mathrm{{SaR}}}- {i}_{\mathrm{{SbR}}}- {i}_{\mathrm{{ScR}}}$ as per (5). Hysteresis current controller generates the appropriate gate pulses for SAPF by comparing ${i}_{\mathrm{{Sa}}}- {i}_{\mathrm{{Sb}}}- {i}_{\mathrm{{Sc}}}$ with ${i}_{\mathrm{{SaR}}}- {i}_{\mathrm{{SbR}}}- {i}_{\mathrm{{ScR}}}$. The produced gate signals ensure the VSI operation that results in requisite harmonic current mitigation and reactive power compensation as mandated by the established standards.
V. Results and Discussions
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For validating the performance of SAPF with the proposed frequency multiplier based FAC and UVT extractor, simulation as well as experimental studies are carried out. For simulation study, the system is modeled in PSIM software and its performance is analyzed for non-ideal supply and non-linear loading. The developed laboratory prototype model utilized to validate the performance of SAPF controlled with proposed technique is shown in Fig. 8 wherein major components of experimental set-up are labelled. The system parameters and part numbers of the components used in this study are given in Table IV. The SAPF comprise of IGBT’s with part number SKM100GB12T4 is interface with the grid through ${L}_{\mathrm{{Fa}}}- {L}_{\mathrm{{Fb}}}- {L}_{\mathrm{{Fc}}}$ of ${10}\mathrm{{mH}}$. The value of ${C}_{\mathrm{{dc}}}$ is ${1800\mu }\mathrm{F}$. 3-phase,122.47 V AC supply having nominal frequency of ${50}\mathrm{\;{Hz}}$ is utilized for the study. Same parameters for grid, ${L}_{\mathrm{{Fa}}}- {L}_{\mathrm{{Fb}}}- {L}_{\mathrm{{Fc}}}$ and ${C}_{\mathrm{{dc}}}$ are selected for the simulation studies. In experimental set-up, current sensors (LA55-P/SP1) and voltage sensors (LV 25-P/SP2) are utilized. These sensors will pass sensed signals to dSPACE MicroLab Box 1202, which implements the developed algorithm in real-time. The waveforms are observed with digital storage oscilloscope DSO-X2002A (Agilent Technologies) and DS1074Z (RIGOL). To validate the performance of SAPF controlled with the proposed technique under various operating conditions, different load configurations are utilized. Load-I is a 3-phase diode bridge rectifier (DBR) with ${50\Omega }$ load, Load-II is parallel combination of Load-I and 3-phase load of $\left({{50}+ \mathrm{j}{\omega }_{1}{0.2}}\right),\left({{50}+ \mathrm{j}{\omega }_{1}{0.2}}\right)$ and $\left({{100}+ \mathrm{j}{\omega }_{1}{0.2}}\right)\Omega$, Load-III is a 3-phase DBR with ${25\Omega }$ load, Load-IV is parallel combination of Load-I and 3-phase load of $\left({{50}+ \mathrm{j}{\omega }_{1}{0.2}}\right)$, $\left({{50}+ \mathrm{j}{\omega }_{1}{0.2}}\right)$ and $\left({{50}+ \mathrm{j}{\omega }_{1}{0.2}}\right)\Omega$, and Load-V is parallel combination of Load-III and 3-phase load of $\left({{50}+ \mathrm{j}{\omega }_{1}{0.2}}\right),({50}$ $\left.{+\mathrm{j}{\omega }_{1}{0.2}}\right)$ and $\left({{50}+ \mathrm{j}{\omega }_{1}{0.2}}\right)\Omega$.
Based on simulation studies, the performance of 3-phase SAPF controlled using the developed frequency multiplier based FAC and UVT extractor under steady state and load change is illustrated in Figs. 9 and 10, respectively. ${v}_{\mathrm{{Sa}}}- {v}_{\mathrm{{Sb}}}- {v}_{\mathrm{{Sc}}}$ are unbalanced and distorted with the corresponding peak values of 105-105-82 V and total harmonic distortion (THD) of 18-18- 23.5%. For steady state analysis with Load-I, ${i}_{\mathrm{{La}}}- {i}_{\mathrm{{Lb}}}- {i}_{\mathrm{{Lc}}}$ have the %THD and peak amplitudes of 17.5-23-28 and 3-3-2.8 A, respectively. ${I}_{\mathrm{F}1}$ and $P{F}_{1}$ determined with the proposed method are ${2.5}\mathrm{\;A}$ and 0.89, respectively. The control of SAPF with frequency multiplier based FAC and UVT extractor results in ${i}_{\mathrm{{Sa}}}- {i}_{\mathrm{{Sb}}}- {i}_{\mathrm{{Sc}}}$ being balanced with %THD and peak amplitude of 3.5 and ${2.31}\mathrm{\;A}$, respectively. With the change in load from Load-I to Load-II at 0.4 s, the peak values of ${i}_{\mathrm{{La}}}- {i}_{\mathrm{{Lb}}}- {i}_{\mathrm{{Lc}}}$ increase to 4.1-4.0-3.4 A. At the new steady state, they have %THD value of 12.8-17.7-24.3. The highest value of load current in the dynamic state is ${4.2}\mathrm{\;A}$. The new steady state is attained in ${0.011}\mathrm{\;s}$ with the ${I}_{\mathrm{F}1}$ and $P{F}_{1}$ recorded as ${3.3}\mathrm{\;A}$ and0.84. SAPF operation results in ${i}_{\mathrm{{Sa}}}- {i}_{\mathrm{{Sb}}}- {i}_{\mathrm{{Sc}}}$ having %THD of 2.7 and the peak value of 3.08 A. In both cases, the SAPF operation with the proposed extraction method results in supply current being sinusoidal, balanced and in phase with the respective phase voltage irrespective of non-linear and unbalanced loading or unbalanced supply. Also, ${v}_{\mathrm{{dc}}}$ is regulated at the reference value of ${250}\mathrm{\;V}$.
Fig. 11 shows the comparison of FAC extraction with the proposed algorithm and SRF theory employing the cut-off frequency, ${f}_{\mathrm{c}}$, of ${10}\mathrm{\;{Hz}}$ and ${50}\mathrm{\;{Hz}}$ for the low pass filter. Upon switching of load, the extracted value of ${I}_{\mathrm{F}1}$ increases from 2.997 A to 5.284 A. The time required to attain the new steady state after the load change is 14 ms. In case of ${f}_{\mathrm{c}}= {10}\mathrm{\;{Hz}}$, extracted FAC, ${I}_{\mathrm{{FLPF}}}$, demonstrates sluggish response, whereas with ${f}_{\mathrm{c}}= {50}\mathrm{\;{Hz}}$ the dynamic response of ${I}_{\mathrm{{FLPF}}}$ is comparable to that of ${I}_{\mathrm{F}1}$. With ${f}_{\mathrm{c}}= {50}\mathrm{\;{Hz}}$, there is presence of steady state ripple in ${I}_{\mathrm{{FLPF}}}$ when the load is unbalanced, which is unacceptable. The average value of ${I}_{\mathrm{F}1}$ matches that of the ${I}_{\text{FLPF }}$ with ${f}_{\mathrm{c}}$ as 10 and ${50}\mathrm{\;{Hz}}$. The proposed frequency multiplier based FAC extraction provides faster dynamic response without any steady state ripple even under non-ideal operating conditions.
The comparison of the proposed algorithm with the adaptive linear neuron (ADALINE) based LMS, reported in [19], is shown in Fig. 12. In [19], convergence factor, $\mu$, is considered as 0.01 . Similar implementation of ADALINE based LMS with $\mu ={0.01}$ results in sluggish dynamic response for FAC
extraction in the considered system. On the contrary, when $\mu$ is increased to 0.05 for improving the dynamic response, FAC extracted with ADALINE based LMS algorithm exhibits faster dynamic response with persistent oscillations at steady state. For load-I, FAC calculated using ADALINE based LMS, ${I}_{\mathrm{{FLMS}}}$, are ${3.13}\mathrm{\;A}$ and ${3.11}\mathrm{\;A}$ for $\mu$ as 0.01 and 0.05, respectively. Similarly, for Load-III, ${I}_{\mathrm{{FLMS}}}$ is recorded as 5.31 and 5.26 for $\mu$ selected as 0.01 and 0.05, respectively. However, the developed scheme demonstrates faster dynamic response and no oscillations at the steady state.
For the experimental analysis of the proposed frequency multiplier based SAPF operation under steady state, three cases are considered:(I) unbalanced ${v}_{\mathrm{{Sa}}}- {v}_{\mathrm{{Sb}}}- {v}_{\mathrm{{Sc}}}$ and Load-III,(II) unbalanced ${v}_{\mathrm{{Sa}}}- {v}_{\mathrm{{Sb}}}- {v}_{\mathrm{{Sc}}}$ and Load-IV, and (III) unbalanced ${v}_{\mathrm{{Sa}}}- {v}_{\mathrm{{Sb}}}- {v}_{\mathrm{{Sc}}}$ and Load-V. For the three cases, the measured currents and voltages are tabulated in Table V. Experimental results for case-I are shown in Fig. 13. The peak values and $\%\mathrm{{THD}}$ of ${v}_{\mathrm{{Sa}}}- {v}_{\mathrm{{Sb}}}- {v}_{\mathrm{{Sc}}}$ are 104-101-85 V and 2.51-2.61-3.87, respectively. Also, ${i}_{\mathrm{{La}}}- {i}_{\mathrm{{Lb}}}- {i}_{\mathrm{{Lc}}}$ are observed to be non-sinusoidal with the corresponding %THD as 17.77-16.34-21.08. Moreover, with the unbalanced supply voltages, the peak values of ${i}_{\mathrm{{La}}}- {i}_{\mathrm{{Lb}}}-$ ${i}_{\mathrm{{Lc}}}$ are measured as ${5.8}- {5.8}- {5.3}\mathrm{\;A}$, respectively. The frequency multiplier algorithm based control of SAPF injects ${i}_{\mathrm{{Fa}}}- {i}_{\mathrm{{Fb}}}-$ ${i}_{\mathrm{{Fc}}}$, having peak values ${3.3}- {4.3}- {3.9}\mathrm{\;A}$, to ensure ${i}_{\mathrm{{Sa}}}- {i}_{\mathrm{{Sb}}}- {i}_{\mathrm{{Sc}}}$ are maintained sinusoidal with $\%$ THD of 3.98-4.44-4.01. ${i}_{\mathrm{{Sa}}}$ -${i}_{\mathrm{{Sb}}}- {i}_{\mathrm{{Sc}}}$ are observed to have the peak values of 5.9-5.7-5.7 A, respectively. Sensor gain for ${v}_{\mathrm{{sa}}}- {v}_{\mathrm{{sb}}}- {v}_{\mathrm{{sc}}}$ in 0.01 .
Fig. 14. Experimental results for steady-state operation of SAPF with the proposed frequency multiplier based FAC and UVT extractor with Load-III for (a) phase-a,(b) phase-b, and (c) phase-c.
The experimental results for the steady state operation of SAPF using the developed algorithm for case-II are shown in Fig. 14， where ${v}_{\mathrm{{Sa}}}- {v}_{\mathrm{{Sb}}}- {v}_{\mathrm{{Sc}}}$ are unbalanced with peak values 105-104-94 V. Peak amplitudes and %THD of ${i}_{\mathrm{{La}}}- {i}_{\mathrm{{Lb}}}- {i}_{\mathrm{{Lc}}}$ are noted as 4.1-4.4-4.1A and 16.38-15.72-17.87, respectively. SAPF injects ${i}_{\mathrm{{Fa}}}- {i}_{\mathrm{{Fb}}}- {i}_{\mathrm{{Fc}}}$, having the peak values 2.8-3.1-3.2 A, to implement shunt compensation so that ${i}_{\mathrm{{Sa}}}- {i}_{\mathrm{{Sb}}}- {i}_{\mathrm{{Sc}}}$ are observed to be sinusoidal with %THD of 4.45-4.20-3.92. The maximum values of ${i}_{\mathrm{{Sa}}}- {i}_{\mathrm{{Sb}}}- {i}_{\mathrm{{Sc}}}$ are noted as 4.6-4.4-4.6 A.
The experimental results for the steady state performance of SAPF controlled with the proposed algorithm for case-III are shown in Fig. 15. ${v}_{\mathrm{{Sa}}}- {v}_{\mathrm{{Sb}}}- {v}_{\mathrm{{Sc}}}$ are unbalanced with the peak values measured as 103-104-84 V. With the increased loading due to Load-V, the peak values of ${i}_{\mathrm{{La}}}- {i}_{\mathrm{{Lb}}}- {i}_{\mathrm{{Lc}}}$ are increased to 6.6-7.0-6.3 A. Despite high %THD of ${i}_{\mathrm{{La}}}- {i}_{\mathrm{{Lb}}}- {i}_{\mathrm{{Lc}}}$, recorded as ${14.86}- {13.40}- {18.05},{i}_{\mathrm{{Sa}}}- {i}_{\mathrm{{Sb}}}- {i}_{\mathrm{{Sc}}}$ are controlled to be sinusoidal having %THD values as 4.38-4.63-3.64. The maximum values f ${i}_{\mathrm{{Sa}}}- {i}_{\mathrm{{Sb}}}- {i}_{\mathrm{{Sc}}}$ and ${i}_{\mathrm{{Fa}}}- {i}_{\mathrm{{Fb}}}- {i}_{\mathrm{{Fc}}}$ are correspondingly observed as 6.7-6.4-6.6 A and 4.0-5.0-4.5 A, respectively.
Figs. 11-12 validate the faster dynamic response of the proposed algorithm for computing the FAC of load current. Further, the experimental results for the dynamic performance evaluation of SAPF controlled using developed algorithm for switching of load from Load-IV to Load-V are presented in Fig. 16(a). After the load change, the peak magnitudes of load, supply and injected current increases from 4.1-4.6-2.8 A to 6.6-6.7-4.0 A, respectively. $\%$ THD for ${i}_{\mathrm{{Sa}}}$ is 4.45 before the load change and decreases to 4.38 after the change in load. The new steady state is attained before one cycle without any overshoots, oscillations or any other abnormality.
Fig. 16(b) demonstrates the effectiveness of the proposed SRF-PLL assisted with synchronizing logic for phase angle extraction and subsequent computation of UVTs. Even for the unbalanced grid voltages, ${\theta }_{1}$ is accurately estimated without any presence of oscillations or any other abnormalities. This results in accurate UVT generation which are essentially required to ensure that the generated ${i}_{\mathrm{{SRa}}}- {i}_{\mathrm{{SRb}}}- {i}_{\mathrm{{SRc}}}$ and consequently ${i}_{\mathrm{{Sa}}}- {i}_{\mathrm{{Sb}}}- {i}_{\mathrm{{Sc}}}$ are in phase with the respective FPSC of ${v}_{\mathrm{{Sa}}}- {v}_{\mathrm{{Sb}}}- {v}_{\mathrm{{Sc}}}$.
The presented simulation and experimental study reveal that the SAPF operation using the developed frequency multiplier algorithm based control results in sinusoidal grid currents and unity PF operation. The faster dynamic and accurate steady state response of the proposed scheme for computing the FAC of load current is also evident from the presented results. The simulation and experimental results presented in this section validate the feasibility, applicability, fast dynamic response and compliance with established power quality standards under ideal and non-ideal operating conditions.
VI. Conclusion
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The control of SAPF with frequency multiplier based FAC and UVT extraction scheme is presented in this paper. The proposed algorithm individually processes the $\alpha -\beta$ components of load current and supply voltage through frequency multiplier and BPF to obtain the respective fundamental quantities with frequency four times the power frequency. Thus computed $\alpha$ -axis load current is sampled at the peak of the obtained $\alpha$ -axis supply voltage. PF is estimated using this sampled value and lookup table. FAC of load current is computed by multiplying PF and peak value of $\alpha$ -axis fundamental current raise to the power of 0.25 . The obtained $\alpha -\beta$ voltages are used by SRF-PLL assisted with the designed synchronizing logic to correctly estimate the phase angle of the FPSC of supply voltage and UVTs corresponding to the power frequency. The simulation and experimental results validate that the proposed frequency multiplier based extraction scheme provides faster and accurate extraction of FAC and UVT regardless of the operating condition. The experimental studies for SAPF operation with the proposed extractor demonstrate drawing of sinusoidal and balanced source currents with THD less than 5% and in-phase with the supply voltages even under abnormal operating conditions.
Appendix
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As mentioned in Section III, the frequency multiplier based algorithm is elaborated in this Appendix. ${x}_{\alpha 2}$ is defined in (8) as $2{x}_{\alpha }{x}_{\beta }$. Using (6) and (7), this can be expanded as
${x}_{\alpha 2}= 2\left\lbrack {{X}_{\mathrm{m}}\sin \left({\theta }_{1x}\right)+ {H}_{\alpha }}\right\rbrack \left\lbrack {{X}_{\mathrm{m}}\cos \left({\theta }_{1x}\right)+ {H}_{\beta }}\right\rbrack $
For separating the fundamental and harmonic components, ${x}_{a2}$ can be rewritten as
${x}_{\alpha 2}= 2{\left({X}_{\mathrm{m}}\right)}^{2}\sin \left({\theta }_{1x}\right)\cos \left({\theta }_{1x}\right)+ \\ 2{H}_{\alpha }{X}_{\mathrm{m}}\cos \left({\theta }_{1x}\right)+ \\ 2{H}_{\beta }{X}_{\mathrm{m}}\sin \left({\theta }_{1x}\right)+ 2{H}_{\alpha }{H}_{\beta }$
The first term of (31) can be written in form of $2{\theta }_{1x}$ component to obtain component having frequency twice that of the fundamental component as
${x}_{\alpha 2}= {\left({X}_{\mathrm{m}}\right)}^{2}\sin \left({2{\theta }_{1x}}\right)+ 2{H}_{\alpha }{X}_{\mathrm{m}}\cos \left({\theta }_{1x}\right)+ $
$ 2{H}_{\beta }{X}_{\mathrm{m}}\sin \left({\theta }_{1x}\right)+ 2{H}_{\alpha }{H}_{\beta }\\{x}_{\alpha 2}= {\left({X}_{\mathrm{m}}\right)}^{2}\sin \left({2{\theta }_{1x}}\right)+ {H}_{\alpha 2}$
where, ${H}_{\alpha 2}$ is as defined in (10).
Similarly, frequency multiplier operation is performed to obtain ${x}_{\beta 2}$. For this,(11) can be written in expanded form using (6) and (7) as
${x}_{\beta 2}= {\left({X}_{\mathrm{m}}\cos {\theta }_{1x}+ {H}_{\beta }\right)}^{2}- {\left({X}_{\mathrm{m}}\sin {\theta }_{1x}+ {H}_{\alpha }\right)}^{2}$
$\begin{aligned}x_{\beta 2}= & \left(X_{\mathrm{m}}\right)^{2} \cos ^{2}\left(\theta_{1 x}\right)+2 X_{\mathrm{m}} \cos \left(\theta_{1 x}\right) H_{\beta}+H_{\beta}^{2}- \\& \left(X_{\mathrm{m}}\right)^{2} \sin ^{2}\left(\theta_{1 x}\right)-2 X_{\mathrm{m}} \sin \left(\theta_{1 x}\right) H_{\alpha}-H_{a}^{2}\end{aligned}$
$\begin{array}{c}x_{\beta 2}=\left(X_{\mathrm{m}}\right)^{2} \cos ^{2}\left(\theta_{1 x}\right)-\left(X_{\mathrm{m}}\right)^{2} \sin ^{2}\left(\theta_{1 x}\right)+ \\2 X_{\mathrm{m}}\left[H_{\beta} \cos \left(\theta_{1 x}\right)-\right. \\\left.H_{a} \sin \left(\theta_{1 x}\right)\right]+H_{\beta}^{2}-H_{a}^{2}\end{array}$
The first two terms of (36) have frequency twice that of the fundamental frequency and the remaining terms can be represented as ${H}_{\beta 2}$, which is defined in (13). Based on this, ${x}_{\beta 2}$ can now be expressed as
${x}_{\beta 2}= {\left({X}_{\mathrm{m}}\right)}^{2}\cos \left({2{\theta }_{1x}}\right)+ {H}_{\beta 2}$
Similar steps as mentioned in (30)-(37) can be used to process ${x}_{\alpha 2}$ and ${x}_{\beta 2}$ to further multiply the frequency and obtain components having fundamental frequency four times that of the grid frequency, ${x}_{\alpha 4}$ and ${x}_{\beta 4}$. The final expression for ${x}_{\alpha 4}- {x}_{\beta 4}$ are expressed in (15)-(18), respectively.
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Appendix
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Year 2024 volume 9 Issue 4
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Cite this Article
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Article Info
doi: 10.24295/CPSSTPEA.2024.00022
  • Receive Date：2024-01-16
  • Online Date：2025-07-05
  • Published：2024-12-10
Article Data
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History
  • Received：2024-01-16
  • Revised：2024-08-28
  • Accepted：2024-09-26
Affiliations
    1 Pandit Deendayal Energy University Department of Electrical Engineering, School of Energy Technology Gandhinagar India
    2 Universitat Politècnica de Catalunya (UPC) Barcelona Spain

Corresponding:

Amit V. Sant.
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表12种不同金属材料的力学参数
科
Family		 属数
Number of
genus		 种数
Number of
species		 占总种数比例
Percentage of
total species (%)		 属
Genus		 种数
Number of
species		 占总种数比例
Percentage of total
species (%)
鹅膏菌科Amanitaceae 2 11 5.26 鹅膏菌属 Amanita 10 4.78
小菇科 Mycenaceae 2 12 5.74 丝盖伞属 Inocybe 5 2.39
多孔菌科 Polyporaceae 8 14 6.70 蜡蘑属 Laccaria 5 2.39
红菇科 Russulaceae 3 23 11.00 小皮伞属 Marasmius 6 2.87
小菇属 Mycena 11 5.26
光柄菇属 Pluteus 5 2.39
红菇属 Russula 17 8.13
栓菌属 Trametes 5 2.39
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