Inverter Efficiency Research Articles

On the Construction of Energy Efficiency-based Degradation Indicator for Photovoltaic Solar Inverters

Photovoltaic systems are essential in the renewable energy sector, addressing global energy needs. PV inverters, which convert DC from solar panels to AC for grid use, are the most failure-prone components in these systems. This study aims to develop a degradation indicator based on energy efficiency for PV inverters to enhance their reliability and lifetime management. Through analysis of data from 35 PV plants in central Europe, involving eight inverter brands and fourteen models, this research identifies trends in inverter efficiency degradation. Despite literature suggesting minimal efficiency impact over time, our findings demonstrate measurable efficiency degradation, providing a new key performance indicator for proactive maintenance and replacement strategies.

Performance analysis of a photovoltaic component integrated into a hybrid power plant in Southeast Mauritania

This study investigated the performance of photovoltaic components of the 1.3MW KIFFA hybrid power plant in Mauritania. Data from the plant's monitoring system (January-December 2021) was used to assess various performance metrics. The analysis revealed a high daily reference yield (5.60 h/d), indicating good solar resource availability. However, final and array yields (4.78 h/d and 4.86 h/d, respectively) suggested potential for improvement. System component efficiencies were within acceptable ranges, with particularly high inverter efficiency (98.31%). Array capture losses were moderate (0.74 h/d), and system losses were minimal (0.08 h/d). The annual performance ratio (86.33%) and capacity factor (19.91%) indicated good overall plant performance. These findings were then compared with data from similar installations in various climate zones to understand the impact of climatic variations on photovoltaic performance. Compared to installations in temperate zones with lower irradiation levels, the KIFFA plant's reference yield was significantly higher. However, the final and array yields were closer due to potentially higher operating temperatures in Mauritania affecting module efficiency. Interestingly, comparisons with installations in other desert regions with similarly high irradiation levels revealed lower performance, particularly in terms of final yield, (4.71 h/d) in Algéria (Adrar) and (4.10 h/d) in Oman (Muscat). This suggests that climatic factors beyond just sunlight availability, such as dust accumulation, may have played a significant role in their performance compared to the KIFFA plant.

High-Efficiency e-Powertrain Topology by Integrating Open-End Winding and Winding Changeover for Improving Fuel Economy of Electric Vehicles

The fuel economy of electric vehicles (EVs) is an important factor in determining the competitiveness of EVs. Since the fuel economy is affected by the efficiency of an e-powertrain composed of a motor and inverter, it is necessary to select a high-efficiency topology for the e-powertrain. In this paper, a novel topology of e-powertrains to improve the fuel economy of EVs is proposed. The proposed topology aims to improve the system efficiency by integrating open-end winding (OEW) and winding changeover (WC). The proposed OEW-PMSM with WC enables to drive a permanent magnet synchronous motor (PMSM) in four different modes. Each mode can increase inverter efficiency and motor efficiency by changing motor parameters and maximum modulation index. In this paper, the system efficiency of the proposed topology was evaluated using electromagnetic finite element analysis and a loss model of power semiconductors. In addition, the vehicle simulations were performed to evaluate the fuel economy of the proposed topology, thereby proving the superiority of the proposed topology compared with the conventional PMSM.

Hybrid control for capacitor-assisted Z-source inverter in grid-connected photovoltaic system

Renewable energy integration into the power grid is essential for sustainable energy systems, but maintaining efficiency and reliability in such systems remains a key challenge. This study introduces a novel hybrid control topology for a Capacitor-Assisted Extended Boost Z-Source Multilevel Inverter in a grid associated solar photovoltaic (PV) structure. The suggested hybrid technique implies a united implementation of a Gannet Optimization Algorithm and Spiking Deep Residual Network and usually referred as GOA-SDRN technique. Initially, the modelling of the inverter is collected to get the best signal by the proposed controller. The proposed inverter configuration consists of a minimal quantity of diodes, switches, and sources. This configuration also offers advantages like less total harmonic distortion (THD) and reduced electromagnetic interference, making it a favourable choice for PV integration. The GOA is employed to determine the most favourable gain limiting factor based on a variation of power from their regular values. This control method ideally satisfies the load demand while reducing changes in the system limiting factor and external disturbances. The proposed control topology is executed in MATLAB and the concept is contrasted to different techniques. The THD values of existing methods such as Particle Swarm Optimization, Genetic Algorithm, and Grasshopper Optimization Algorithm are 1.36 %,0.89 %, and 1.99 % respectively, while the THD value of the proposedmethod is 0.63 %, demonstrating its optimal performance over existing methods. The proposed GOA-SDRN technique provides an effective solution for enhancing inverter efficiency and reducing distortion in grid-associated PV systems.

Analysis of Inverter Efficiency Using Photovoltaic Power Generation Element Parameters

Photovoltaic power generation is influenced not only by variable environmental factors, such as solar radiation, temperature, and humidity, but also by the condition of equipment, including solar modules and inverters. In order to preserve energy production, it is essential to maintain and operate the equipment in optimal condition, which makes it crucial to determine the condition of the equipment in advance. This paper proposes a method of determining a degradation of efficiency by focusing on photovoltaic equipment, especially inverters, using LSTM (Long Short-Term Memory) for maintenance. The deterioration in the efficiency of the inverter is set based on the power generation predicted through the LSTM model. To this end, a correlation analysis and a linear analysis were performed between the power generation data collected at the power plant to learn the power generation prediction model and the data collected by the environmental sensor. With this analysis, a model was trained using solar radiation data and power data that are highly correlated with power generation. The results of the evaluation of the model’s performance show that it achieves a MAPE of 7.36, an RMSE of 27.91, a MAE of 18.43, and an R2 of 0.97. The verified model is applied to the power generation data of the selected inverters for the years 2020, 2021, and 2022. Through statistical analysis, it was determined that the error rate in 2022, the third year of its operation, increased by 159.55W on average from the error rate of the power generation forecast in 2020, the first year of operation. This indicates a 0.75% decrease in the inverter’s efficiency compared to the inverter’s power generation capacity. Therefore, it is judged that it can be applied effectively to analyses of inverter efficiency in the operation of photovoltaic plants.

A Noninvasive Sliding Mode Observer Based Approach for Detecting Open-Circuit Faults in HERIC Inverters

This article presents a new technique for detecting and localizing the open-circuit (OC) faults in the semiconductor switches of the Highly Efficient and Reliable Inverter Concept (HERIC) converter. It is a transformerless Photovoltaic (PV) inverter based on the full-bridge topology, with two extra switches on the AC side called ac bypass switches. The proposed method uses a sliding mode observer (SMO) to detect faults in the AC bypass switch and main switches of the full bridge inverter. To diagnose OC faults in the AC bypass leg, two parallel SMOs are created based on the converter’s state space model. Similarly, two SMOs are designed for the main switches. A residual is generated by combining the observed and measured grid currents to accurately locate the OC fault within the inverter system. To localize the faulty main switch, once the fault is detected the inverter is reconfigured and based on the grid current the faulty switch is identified. A major benefit of this method is that it does not rely on additional sensors for fault diagnosis, making the system more robust. The effectiveness and reliability of this approach are demonstrated through simulation studies. In addition, a laboratory prototype is developed to validate the practical applicability of the method.

Size, Efficiency, Reliability, and Cost Analysis of a Multiport Traction Inverter With Downsized DC–DC Converter for a Catenary/Battery Tram

Size, Efficiency, Reliability, and Cost Analysis of a Multiport Traction Inverter With Downsized DC–DC Converter for a Catenary/Battery Tram

Assessing PV inverter efficiency degradation under semi-arid conditions: A case study in Morocco

The worldwide shift towards renewable energy sources, including solar and wind, is progressing rapidly. This change is driven by decreasing prices and technical advancements that are becoming more centralized. The objective is to develop a cutting-edge approach and technology that seamlessly incorporates photovoltaic (PV) energy sources into a power network while ensuring grid stability. This research evaluates the lifetime and degradation of PV inverters under real operating conditions, focusing on semi-arid climate scenarios. Current papers demonstrate a yearly failure rate of 1–15% for PV inverters, highlighting the need for a thorough reliability evaluation. This investigation research applied a unique technique that included continuous monitoring of PV inverter performance parameters by comparing the measured with the predicted normal operating output. The results show that Sandia's inverter performance model is highly suitable for accurately modeling normal PV inverter behavior. Furthermore, weighted efficiency dropped by 3.96%, 0.63% and 1.29% respectively for 7kW, 15kW and 20kW photovoltaic systems over the five years. These findings have implications for comprehensive maintenance strategies, including regular inspections, safety protocols, and remote monitoring solutions. Ultimately, this research paper sheds light on the causes of declining solar inverter performance and provides suggestions for enhancing PV plant maintenance and reliability. It also represents an outstanding reference given the literature shortage.

Long-term power performance evaluation of vertical building integrated photovoltaic system

As the mandatory scope of zero-energy building certification has expanded, the integration of renewable energy systems into building designs and energy operations has significantly increased. The long-term maintenance of these systems is crucial for sustaining zero-energy goals. The primary key to maintaining renewable energy systems is to assess quantitative performance factors, such as degradation and malfunction. This study presents a comprehensive evaluation of the long-term power performance of a large-scale vertical building-integrated photovoltaic system, which has been monitored over 9 years (2012–2020). The system, comprising 22 arrays with a total installed capacity of 142.654 kWp, was analyzed to derive quantitative performance indicators that highlight the importance of continuous maintenance. The photovoltaic system's arrays were investigated to separate them into analysis groups, and simulation-based analysis was carried out for the detailed assessments. The findings revealed that 19 of 22 arrays were operating normally, while three experienced malfunctions. In 2020, a 19.12 % performance degradation was observed in normally operating arrays compared to 2012. Post-malfunction, the affected arrays performed at approximately half the efficiency of the normal arrays. Despite theses issues, inverter efficiency remained stable across all arrays. However, a noticeable decline in array efficiency indicated performance degradation due to specific array issues. When compared to simulation-based theoretical performance, in 2020, the malfunctioned arrays exhibited nearly triple the performance degradation of the normally operating arrays. These results underscore the critical role of quantitative performance indicators in maintenance strategies and demonstrate how inadequate maintenance can lead to substantial power generation losses.

Maximizing solar energy efficiency with efficient interleaved boost converter with Three-Level neutral point clamped inverter for Grid-Connected photovoltaic using hybrid approach

The efficient interleaved boost converter (IBC) combined with the 3-level neutral point clamped (NPC) inverter for grid-connected photovoltaic systems (GCPVS) maximizes solar energy efficiency are presented. The proposed hybrid technique implies the combination of Double Attention Convolutional Neural Network (DACNN) and Starling Murmuration Optimization (SMO) algorithm and is usually referred as DACNN-SMO technique. Enhancing power quality at the Point of Common Coupling (PCC) while utilizing the rated capacity of the inverter into consideration is the main goal of the proposed approach. The grid-connected photovoltaic system additionally includes an efficient three-level NPC inverter and interleaved boost converter to decrease DC link voltage oscillation. In addition to preventing overheating, the inverter current controls reactive power adjustment, active power injection, and current harmonic filtering. Utilizing the SMO technique, By employing the SMO technique, the system effectively regulates output voltage, ensuring that the inverter output voltage. The DACNN enhances the system’s ability to monitor and diagnose faults. Using MATLAB, the proposed topology is implemented, and the results are contrasted with those of other existing techniques. The existing methods such as Heap Based Optimizer (HBO), Circle Search Algorithm (CSA) and Sparrow Search Algorithm (SSA) attains a higher THD of 3.7%, 4.7% and 5.3% respectively. The proposed method DACNN-SMO attains a THD of 2.5%. The proposed technique displays the efficiency is 99.86%. When compared to existing strategies, the proposed technique displays lower THD and high efficiency.

Performance Improvement of Wireless Power Transfer System for Sustainable EV Charging Using Dead-Time Integrated Pulse Density Modulation Approach

The recent developments in electric vehicle (EV) necessities the requirement of a human intervention free charging system for safe and reliable operation. Wireless power transfer (WPT) technology shows promising options to automate the charging process with user convenience. However, the operation of the WPT system is designed to operate at a high-frequency (HF) range, which requires proper control and modulation technique to improve the performance of power electronic modules. This paper proposes a dead-time (DT) integrated Pulse Density Modulation (PDM) technique to provide better control with minimal voltage and current ripples at the switches. The proposed technique is investigated using a LCC-LCL compensated WPT system, which predominantly affects the high-frequency voltage and current ripples. The performance analysis is studied at different density conditions to explore the impact of the integrated PDM approach. Moreover, the PDM technique gives better control over the power transfer at different levels of load requirement. The simulation and experimental analysis was performed for a 3.7 kW WPT prototype test system under different modes of operation of the high-frequency power converters. Both the simulated and experimental results demonstrate that the proposed PDM technique effectively enhances the efficiency of the HF inverter while significantly reducing output current ripples, power dissipation and improving the overall WPT system efficiency to 92%, and leading to a reduction in the power loss in the range of 10% to 20%. This leads to improved overall system control and performance.

Profile of Round-Trip Efficiency of Lithium-Ion Battery Energy Storage System

Lithium-ion battery energy storage systems (Li-BESSs) are expected to adjust variable renewable energies such as photovoltaic energy. In the viewpoint of the energy management cost, profiles of round-trip efficiency (RTE) are important characteristics of Li-BESSs. The profiles of RTE are related to the characteristics of both LIBs and inverters installed on Li-BESSs. The characteristics of Li-BESSs would be various because of reusing and degradation of lithium-ion batteries (LIBs). And inverters made by a lot of manufacturers would be so on. Accordingly, it could be said that the understanding of the profile of each Li-BESS’s RTE is complex and difficult. Therefore, we studied the time series analysis of operation data and tried to estimate the characteristics of both LIBs and inverters. The important ones of LIB are full-charge capacity (FCC), open-circuit voltage (OCV), and internal resistance (R). In addition, characteristics of OCV and R can be expressed as function profiles with state of charge (SOC) as elements. Similarly, the important one of inverter is the typical efficiency curve. In this study, we proposed the estimation method of the profiles of OCV and R and inverter efficiency curve. On the other hand, FCC was given as assuming that general diagnosis results would be provided. Then, a RTE profile of Li-BESS (including 9.94kWh LIB and 10kW inverter) was tried to establish from the results of it. The profiles of OCV and R were estimated by the method called MGFFD (Modified Gaussian Free-Form Deformation) that we have been established [1]. The novelty of this study could be described as: Development of the estimation method of an efficiency curve of inverter installed on BESS during operation (Fig.1). Quantitatively visualized profile of RTE of Li-BESS by combining results of MGFFD estimation applied for LIBs and inverter (Fig.2). The profile described on Fig.2 is the relationship between SOC, input-output power rate of inverter, and RTE of Li-BESS. To understand Li-BESS’s charge-discharge efficiency of from state at that time to the end of an operation plan, it would be necessary information. This proposed method could clarify various RTE of Li-BESSs, consequently, be useful for its efficient operation by information communication technology.Reference[1] M. Arima, L. Lin, and M. Fukui, "Kalman-Filter-Based Learning of Characteristic Profiles of Lithium-Ion Batteries", Sensors, vol. 22, no. 14, 5156. Figure 1

Investigation of Dead Time Losses in Inverter Switching Leg Operation: GaN FET vs. MOSFET Comparison

This paper investigates the commutation transients of MOSFET and GaN FET devices in motor drive applications during hard-switching and soft-switching commutations at dead time operation. This study compares the switching behaviors of MOSFETs and GaN FETs, focusing on their performance during dead time in inverter legs for voltage source inverters. Experimental tests at various phase current levels reveal distinct switching characteristics and energy dissipation patterns. A validated simulation model estimates the experimental energy exchanged and dissipated during switching transients. The results demonstrate that GaN FETs exhibit lower overall losses at shorter dead times compared to MOSFETs, despite higher reverse conduction voltage drops. The study provides a quantitative framework for selecting optimal dead times to minimize energy losses, enhancing the efficiency of GaN FET-based inverters in low-voltage motor drive applications. Finally, a dead time optimization strategy is proposed and described.

Hybrid SVM for Enhanced Efficiency of 3L ANPC Inverter With Full ZVS Range

Hybrid SVM for Enhanced Efficiency of 3L ANPC Inverter With Full ZVS Range

Decentralized Retrofit Model Predictive Control of Inverter-Interfaced Small-Scale Microgrids

In recent years, small-scale microgrids have become popular in the power system industry because they provide an efficient electrical power generation platform to guarantee autonomy and independence from the power grid, which is a critical feature in cases of catastrophic events or remote areas. On the other hand, due to the short distances among multiple distribution generation systems in small-scale microgrids, the interconnection couplings among them increase significantly, which jeopardizes the stability of the entire system. Therefore, this work proposes a novel method to design decentralized robust controllers based on a retrofit model predictive control scheme to tackle the issue of instability due to the short distances among generation systems. In this approach, the retrofit model predictive controller receives the measured feedback signal from the interconnection current and generates a control command signal to limit the interconnection current to prevent instability. To design a retrofit controller, only the model of a robust closed-loop system, as well as an interconnection line, is required. The model predictive control signal is added in parallel to the control signal from the existing robust voltage source inverter controller. Simulation results demonstrate the superior performance of the proposed technique as compared with the virtual impedance and retrofit linear quadratic regulator techniques (benchmarks) with respect to peak-load demand, plug-and-play capability, nonlinear load, and inverter efficiency.

Techno-economic optimization of photovoltaic (PV)-inverter power sizing ratio for grid-connected PV systems

- The accurate sizing of the inverter, specifically the power sizing ratio (PSR) plays a vital role in maximizing energy production and economic benefits. Existing studies often overlook the complex interplay between maximizing energy capture and minimizing inverter-related economic costs when selecting the optimal PSR. This research presents a techno-economic approach to optimizing the PSR for grid-connected photovoltaic (PV) systems. A simulation model is developed, incorporating real weather data and inverter efficiency curves. The model is calibrated using a Pattern Search Algorithm (PSA) to ensure an accurate representation of real-world inverter behavior by achieving a minimum relative difference of 13.78 % (Norm-RMS) in AC power output. The analysis explores the trade-off between PSR, annual energy yield, and inverter clipping. An optimal PSR of 1.19 is identified, balancing energy capture (up to 2000W inverter capacity) and economic efficiency. This approach promotes cost-effective inverter selection and wider PV adoption.

Development and assessment of a floating photovoltaic-based hydrogen production system integrated with storage options

The integrated system approach utilized in the current study represents an innovative approach to harnessing solar energy through a floating photovoltaic-based integrated plant featuring a diverse array of energy storage options. As part of this research, a solar energy system integrating both floating and concentrating solar technologies was designed at the specified site. Despite the constraints presented by the cold temperature at the chosen location, particularly during the winter months when solar irradiation is restricted, strategies are employed, such as exploiting the high albedo effect caused by the freezing of the water body, optimizing the application of solar radiation, and assuring the system's resilience and energy efficiency. Floating photovoltaic and concentrated solar panels are integral components of this advanced system, which is supplemented by underground hot and cold storage units, underground hydro-pumped storage, a multi-effect desalination system with freshwater storage, a lithium bromide absorption cooling system, and an alkaline electrolysis system for hydrogen and oxygen storage. This comprehensive energy strategy includes diverse storage and production technologies. The thermodynamic analyses and assessments employing the Engineering Equation Solver and the System Advisor Model provide some quantitative performance evaluations. The study considers time-dependent operational characteristics, illustrates the water withdrawal rates, energy and exergy efficiencies of the heating space temperature, inverter efficiency, GHI irradiance, mass, and volume values of the heat transfer fluid in both hot and cold tanks within thermal energy storage, annual cooling generation over a year, power production, hydropower storage, and hydrogen storage over a year. The system's concentrated solar panels are projected to generate an annual AC output of 180.79 GWh, with a yearly water usage of 14,055 m3, while floating panels are predicted to provide 7000 kWh with a 0.85 performance ratio. The overall energy efficiency of the current system is determined to be 51.76 %, and the exergy efficiency is found to be 55.41 %, respectively. Optimizing energy efficiency, utilizing diverse storage solutions, and utilizing aquatic environments minimizes environmental impact, and time-dependent analysis enhances perceptions of performance throughout different times and circumstances.

Stabilized voltage source inverter for sensitive loads in nuclear installations

The present paper proposes a novel design of a stabilized single-phase voltage-source inverter with pure sinusoidal output voltage for photovoltaic systems employed for feeding sensitive loads in nuclear installations. Such types of loads require a voltage source with quite stabilized magnitude and frequency and pure sinusoidal shape that is free from higher-order harmonics. The inverter is based on an H-bridge four insulated gate bipolar transistors (IGBTs) with gate drivers and optocoupler isolators. A control system based on a fast DSP unit and a microcontroller is designed for cancelation of the higher-order harmonics to produce a pure sinusoidal output with almost zero total harmonic distortion (THD). The sinusoidal purity, frequency and magnitude stabilization are achieved through pulse width modulation (PWM) controlled by a fast-response feedback system that measures the time wave form of the output voltage applied to the load and then calculates the THD, frequency and amplitude. The sinusoidal shaping is performed with aid of a sinusoidal shaping filter (SSF) to fasten the operation of higher-order harmonic cancellation. The frequency measurement is achieved through a frequency counter whereas the amplitude measurement is carried out through a differential amplifier that amplifies the difference between the output voltage of the inverter (while being applied to the load) and reference voltage that determines the desired amplitude. A prototype of the entire system of the proposed voltage-source inverter is fabricated for experimental evaluation of its performance. It is shown through simulation and experimental work that the proposed control system of the inverter output voltage is able to stabilize both the amplitude and frequency of the voltage applied to the load irrespective of the load impedance. Also, it is shown that the THD of the output voltage is -57dB\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$-57 \ ext{ dB}$$\\end{document} (0.00025%\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$0.00025 \\%$$\\end{document}), which is almost zero. Moreover, the switching losses are considerably reduced and can be negligible owing to the use of the appropriate IGBTs. The efficiency of the proposed inverter is obtained by simulation taking into account all types of losses. Also, the dependence of the inverter efficiency on the load current is investigated. The average efficiency of the proposed voltage-source inverter is shown to be higher than 97%\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$97\\%$$\\end{document}.

Digital Functional Blocks Implementation of PWM and Control for a High-Frequency Interleaved Y-Inverter Motor Drive

This paper is about the development and demonstration of a motor drive for e-transport applications based on an innovative hybrid Si-SiC dual switching frequency interleaved buck–boost Y-inverter and a single-rotor Halbach machine. In particular, the focus is the implementation of the required discontinuous inverter modulation scheme, input voltage feed-forward and motor currents feedback control loops, as well as over-voltage, over-current, and over-temperature protections, using an off-the-shelf commercial hardware platform enabling straightforward Simulink-environment programming of all parts, while ensuring the high switching frequency capability required by wide-band-gap semiconductors. Experimental results showed good inverter efficiency, transient dynamic response to load changes, input voltage regulation, and fully functional protection capability, validating the proposed approach as a powerful option for reducing system development time and cost.

Efficiency Optimization in Parallel LLC Resonant Inverters with Current-Controlled Variable-Inductor and Phase Shift for Induction Heating

This paper presents a comprehensive analysis of a novel control approach to improve the efficiency of parallel LLC resonant inverters using a combination of a current controlled variable inductor (VI) and phase shift (PS). The proposed control aims to reduce the Root Mean Square (RMS) current, thereby reducing conduction and switching losses, and improving power transfer efficiency, while considering zero-voltage-switching (ZVS) operation, output power variations and load changes. Furthermore, a design methodology for the variable inductor is proposed, and design considerations relevant to this application are discussed. The effectiveness of the proposed approach is evaluated through mathematical modeling and experimental results. The mathematical modeling results show that the proposed approach, utilizing SiC MOSFETs, maintains a maximum efficiency of 98.5% for a 20 kW inverter over a wider range of output power, which is a significant improvement over existing approaches. The experimental results also confirm the effectiveness of the proposed control in improving the efficiency of parallel LLC resonant inverters.