Voltage Harmonics Research Articles

Linearity analysis of FE-based graded channel junctionless FET obtaining negative capacitance for low power applications

This paper reports ferroelectric (FE) oxide based graded-channel junctionless FET featuring the negative capacitance effects in nano-scale regime. The linearity nature of its response is analysed on the basis of third order interception point (PIP3), harmonic interception voltages of 2nd and 3rd orders (VIP2, VIP3) and intermodulation distortion (IMD3). Influence of fundamental device’s parameters such as, gate and underlap length, ferroelectric oxide thickness, graded channel doping and operating temperature on its linear behavior is observed and analysed in detail. The subthreshold slope is also found to go below 60mV/dec for optimum features, obtaining the NC characteristics. In its circuit application part, a cascode amplifier is designed using the proposed device showing variations due to the change in the proposed device dimensions. The proposed device is designed and simulated using Silvaco ATLAS device simulator, which is calibrated with the available experimental results. Therefore, the present study is quite relevant in recent days for low power analog applications.

Comprehensive assessment of factors affecting the operational reliability of low-voltage asynchronous electric motors of a coal mining complex

THE RELEVANCE of this study lies in solving the problem of reliable functioning of elements of energy systems and ensuring their effective operation during open-pit mining. The paper presents a methodological approach that allows us to assess the level of reliability of low-voltage asynchronous electric motors in a coal mine with complex mining, geological and climatic conditions of production. THE PURPOSE of the work is to quantify the operational factors affecting the service life of an asynchronous electric motor with a short-circuited rotor. Attention is focused on how influencing factors, different in physical basis, among which electrical parameters are highlighted – the coefficient of voltage asymmetry in the reverse sequence, the coefficients of harmonic voltage components and environmental parameters – temperature, air humidity and dustiness, affect the dynamics of changes in the service life of an asynchronous electric motor. Asynchronous electric motors used at one of the enterprises of JSC SUEK in the Trans-Baikal Territory are considered as an object of research. METHODS. The main tool for the implementation of research tasks is computer modeling based on the Matlab software package. RESULTS. The studies were performed on an asynchronous electric motor with a short-circuited rotor with Ph = 3 kW, rated speed 1500 rpm. The simulation of the operating mode of the electric motor under study was performed by changing the experimental parameters: the values of the total coefficient of harmonic voltage components (KU) – from 4 to 12% with a range of 2%; the values of the voltage asymmetry coefficient in the reverse sequence (K2U) – from 0 to 4% with a range of 1%; the ambient temperature (∆Tокr) – from -30°C to +30°C with a range of 10°C; air humidity values (Vc) – 20% and 60%; dust thermal conductivity coefficient values (Kт) – 0,12 and 0,28. Based on the obtained research results, diagrams are constructed that clearly illustrate the dynamics of changes in the service life of an asynchronous electric motor depending on the combination of influencing factors. CONCLUSION. The trends in solving the issues of reliable functioning of a coal mining enterprise with the presence of asynchronous electric motors and ensuring their efficient operation are the most priority, mainly focused on maintaining the stability of technological processes and maintaining the production volumes of a coal mining enterprise. The stability of the electric motor to certain influencing factors is estimated and their critical values are established. Recommendations on the use of asynchronous electric motors are formulated, taking into account their operating conditions.

Point of Common Connection Voltage Modulated Direct Power Control with Disturbance Observer to Increase in Renewable Energy Acceptance in Power System

In this paper, we present a disturbance observer-based point of common connection voltage-modulated direct power control (PCCVM-DPC) system, which increases the robustness of the PCCVM-DPC system. First, the mathematical analysis of the disturbances for the step-up transformer’s nonlinearity, the grid voltage harmonics, and the parameter uncertainties is presented. By analyzing the disturbance terms of the PCCVM-DPC system, we present the disturbance observer (DOB) for the PCCVM-DPC system. To assess the efficacy of our approach, we perform comparative studies of the PCCVM-DPC without DOB and PCCVM-DPC with DOB by constructing the simulation environment based on the commercial step-up transformer and ESS inverter datasheet. We have validated that the active and reactive power control performance of the PCCVM-DPC with DOB outperforms the PCCVM-DPC without DOB from the observation that the current total harmonic distortion reduced by more than 40% compared to the PCCVM-DPC without the DOB.

An Enhanced Edge Computing Technique for Detection of Voltage Fluctuation in Grid-tied Renewable Energy

Renewable energy sources (RES) such as solar photovoltaic and wind are becoming the most attractive power generation options in many nations. Even while high penetration seems likely, power quality anomalies such as voltage fluctuation, harmonics, and frequency fluctuation associated with RES hinder seamless integration. The variability and unpredictability of these sources create the most oddities. In grid-tied renewable energy, monitoring power quality efficiently is crucial. Power grid monitoring solutions in related literature use sensor-based cloud and edge computing techniques. The existing systems struggle with excessive latency when delivering large amounts of generated data to the cloud. To fill this gap, a new approach for the detection and localization of voltage fluctuation is proposed in this study. The approach integrated three techniques namely; feed-forward neural network (FFNN), Stockwell transform, and anomaly-aware edge computing to detect and locate voltage fluctuation in a GtRE. Using MATLAB/Simulink, virtual emulation of a modified IEEE 33 Bus and a GtRE representing a section of Ado Ekiti (in Nigeria) low-voltage distribution grid are carried out for data generation and system evaluation. Feature extraction was carried out in a Python IDE using Stockwell transform. The voltage fluctuation events are detected and localized based on the extracted features using the trained FFNN model deployed and evaluated within three microcontroller-based computing devices. The proposed approach integrated anomaly-aware with edge computing to send only voltage data that are considered abnormal to a dedicated data center for visualization and storage. Performance evaluation of the proposed technique on the simulated GtRE demonstrates a significant decrease of 98% and 90% in latency when compared to cloud computing and conventional edge computing respectively. Comparison of the proposed approach to two closely related solutions in literature also demonstrates a 50% and 92.5 % reduction in latency. The contribution of the study is the reduced latency and minimal bandwidth utilization achieved by the implementation of the developed technique.

PV Based Single Stage Cascaded Seven Level Modular Multilevel Converter with ZSV Control for Grid Interfaced System

Objectives: Climatic perturbation and environmental factors affect solar power generation. When these solar panels are connected to the grid it causes power quality issues such as voltage harmonics, voltage distortion, etc. These problems are addressed in this paper using a distributed MPPT algorithm. Methods: Multilevel inverters have made remarkable contributions to renewable energy, high voltage, and other high-power applications. Hence, a single-stage Cascaded Seven-Level Modular Multilevel Converter (CSLMMC) is used to address the issues. Zero Sequence Voltage (ZSV) control is proposed for producing balanced power during varying irradiance conditions. The proposed method is simulated using MATLAB SIMULINK software. Partial shading is conquered by using distributed MPPT. Findings: The simulation of the proposed system is performed for a three-phase system. The three-phase Cascaded Modular Multilevel Converter is connected to the grid through two loads which are controlled by a Zero Sequence Voltage Controller. Power quality parameters such as current, harmonics, and power are analyzed for the proposed system. Novelty: THD analysis for a five-level cascaded H bridge converter and the proposed systems are compared and presented. Keywords: Photo voltaic, Cascaded seven-level modular multilevel converter, Zero sequence control, Maximum power point tracking, Voltage Source Converter

Real‐Time Implementation of Unknown Input Observer‐Based State and Unknown Disturbance Estimation for Series Active Filters

ABSTRACTSeries active filter (SAF) is generally utilized in compensating harmonic voltage source‐type loads. In this paper, a combined action of controller and observer is proposed for SAF. Full state feedback (SFB) and linear quadratic regulator (LQR) controllers are posited in order to estimate states, and fault. SFB utilizes the pole placement method, whereas LQR utilizes the tuning of matrices by Bryson's rule. Luenberger observer (LO), proportional–integral observer (PIO), and unknown input observer (UIO) were also designed. The efficacy of this scheme is illustrated through the simulations, in the presence of faults. The state dynamics of SFB‐ and LQR‐based control of SAF with LO, PIO, and UIO in different faulty cases are compared. It can be clearly seen through simulations that the state dynamic response of LQR‐based control of SAF is faster when compared to SFB‐based control of SAF in both cases. Further, LO and PIO established a steady‐state error between the actual state and its estimate, with reference to desired input, in the presence of a fault. UIO perfectly tracked the desired input, even in the presence of various unknown fault, when compared to LO and PIO. Furthermore, UIO also estimates the unknown faults better when compared to PIO. Finally, in a dynamic operating scenario, the effectiveness of the suggested control strategy has been assessed and verified using MATLAB/SIMULINK and the output of a real‐time simulator (OPAL‐RT OP4510).

Power quality enhancement of grid-integrated solar photovoltaic system with unified power quality conditioner

Introduction. To enhance the quality of power and ensure a consistent electricity supply, this study proposes the utilization of a unified power quality conditioner (UPQC) system integrated with solar photovoltaic (PV) technology. The innovation involves single DC-link connecting back-to-back voltage-compensating components arranged in series and shunt, forming the PV-UPQC. The shunt compensator utilizes energy from a PV array to address harmonics in the load current. The objective is to mitigate voltage dips and spikes by injecting voltage that is either in phase with or out of phase with the common coupling point through a series compensator. The method combines the benefits of generating renewable energy to enhance electrical quality. The goal of the paper is the power quality enhancement of grid-integrated solar PV system. The novelty of the proposed work consists of enhancement of grid-integrated solar PV system with UPQC. The purpose of integrating a UPQC into a grid-connected solar PV system is to enhance power quality by mitigating issues such as voltage fluctuations, harmonics and reactive power imbalance. Methods. The proposed topology is implemented in MATLAB/Simulink with grid-integrated solar PV system with UPQC. Results. Integrating UPQC with a grid-connected solar PV system yields substantial improvements in power quality. This includes effectively mitigating voltage fluctuations and harmonics, resulting in smoother operation and reduced disturbances on the grid. Practical value. The proposed topology has proven to be extremely useful for grid-integrated solar PV system with UPQC applications. References 15, table 2, figures 9.

Hilbert-Huang Transform and machine learning based electromechanical analysis of induction machine under power quality disturbances

Monitoring and predicting Power Quality (PQ) is crucial for quickly minimizing risk, protecting induction machines (IMs), and increasing productivity. This paper proposes an electromechanical analytical framework for IM working with various PQ disturbances based on machine learning (ML). This work collects the motor characteristics and PQ data, motor vibration signals, and other sensor data to identify the failures. The complicated electromechanical data is broken down into intrinsic mode functions (IMFs) using the Hilbert-Huang Transform (HHT), which reveals the modes present in the signals. This frequency-domain data is retrieved, paired with time-domain characteristics, and used as input features for ML algorithms. Support Vector Machines (SVMs) trained on labeled datasets in which features extracted from PQ disturbances are used to categorize the disturbances into several groups. The PQ disturbances caused by voltage sags, swells, harmonics, and transients can all be efficiently classified using SVM classifiers, allowing for real-time determination of the kind of disturbance influencing the IM. The accuracy for the SVM in the proposed scheme is 97.2 %. The SVM method is trained with PQ data and classification results using MATLAB classifier and Python software.

Decentralized Goal-Function-Based Microgrid Primary Control with Voltage Harmonics Compensation

Decentralized Goal-Function-Based Microgrid Primary Control with Voltage Harmonics Compensation

Resonance Suppression Method Based on Hybrid Damping Linear Active Disturbance Rejection Control for Multi-Parallel Converters

The parallel operation of multiple LCL-type converters will result in a deviation of the resonant frequency and resonance phenomena. The occurrence of harmonic resonance can cause problems such as an increase in harmonic voltage and current. This can lead to the malfunction of relay protection and automatic devices, causing damage to system equipment. In severe cases, it can cause accidents and threaten the safe operation of the power system. A hybrid damping active disturbance rejection control (HD-ADRC) method is proposed in this paper to suppress the harmonic resonance of parallel LCL-type converters. First, a third-order linear disturbance rejection controller (LADRC) including the linear extended-state observer and the error-feedback control rate is designed based on LCL-type converter model analysis. The proposed method considers the resonance couplings caused by both internal and external disturbances as the total disturbance, thus improving the anti-disturbance capabilities as well as the operational stability of converters in parallel. Then, a hybrid damping control is proposed to reconstruct the damping characteristics of converters to suppress the parallel resonance spike and reduce the resonance frequency offset. And the parameter selection of the control system is optimized through a stability analysis of the tracking performance and anti-disturbance performance of the HD-ADRC controller. Finally, all the theoretical considerations are verified by simulation and experimental results based on the Matlab/Simulink 2018B and dSpace platform. The simulation and experimental results show that the PI controller gives a THD of 5.33%, which is reduced to 4.66% by employing the HD-LADRC, indicating an improved decoupling between the converters working in parallel with the proposed control scheme.

Harmonic voltage compensation and harmonic current sharing strategy of grid-forming inverter

In a multi-inverter parallel system, the harmonic governance control strategy may worsen the system's current circulation problem due to the line impedance difference, which could further affect the stable operation of the system. To solve the problem, the strategy considering both power sharing and harmonic control is proposed in this paper. First, the microgrid system with background harmonics is modeled and analyzed. Based on the modeled system, a comprehensive control strategy for the grid inverter with harmonic voltage compensation and harmonic current sharing is established. In this paper, the output signal is controlled by frequency division. This paper adopts grid-forming control in the fundamental wave domain and employs harmonic voltage compensation control in the harmonic domain. The harmonic voltage is synthesized by collecting the specific harmonic current through the band-pass filter, compensating for the harmonic voltage drop on the line to improve the harmonic voltage quality of the point of common coupling(PCC). Aiming at the sharing problem caused by line impedance difference, virtual impedance control is introduced, harmonic impedance is redesigned according to the adjusted output impedance, and the harmonic output impedance before and after adjustment is measured and compared. The simulation results show that the proposed control strategy can not only reduce the harmonic content of the PCC voltage but also achieve the effect of equal harmonic current division.

Reactive Power Compensation for Standalone Hybrid Power System Using Facts Devices

Reactive power compensation is essential in hybrid grid-connected systems because power electronic inverters utilized for supplying DC energy into the grid causes a reduction in the overall power factor of the power systems. Due to the ability to provide a reliable and efficient power supply to remote and off-grid areas, hybrid power systems are growing in popularity. It can provide greater opportunities for operational growth by combining economic and technology advancement. To meet the demand for electrical energy in both regular and critical situations, operators must be in charge of the power systems reliable and secure operation. The primary objective of the work is to control the reactive power flow in a grid-connected hybrid renewable energy system (PV-wind-battery). The power quality problems that these systems frequently experience include voltage sags, harmonics, and flicker. A FACTS device Unified Power Quality controller (UPQC) with a Genetic Algorithm based PI controller is suggested to handle these power quality issues. In order to improve the performance of the power system, the proposed optimization approaches are used to tune the UPQC in a multiline transmission system. The model was developed with the help of the MATLAB/Simulink work framework.

Revealing the flashover mechanism of EP/GF composite insulation under DC combined harmonic voltage

Abstract In ultra-high voltage converter stations, the phenomenon of flashover along the outer insulation of dry hollow reactors made of epoxy/glass fiber composites poses a potential threat to the stability of the power system. To enhance its flashover performance under varying complex conditions, it is necessary to conduct in-depth research on the flashover mechanism of this material under different voltage forms. This study conducted tests on the flashover voltage, surface space charge distribution, and partial discharge parameters under varying AC-DC ratios and AC frequencies, and exploring their relationships. The results show that the DC content in the AC-DC ratio decreases, the flashover voltage decreases, the apparent total discharge of partial discharges increases, and the maximum surface space charge density decreases. The main influencing factor is the increase in the number of seed charges involved in gas ionization. Increasing the AC frequency, the flashover voltage decreases, the apparent total discharge increases significantly, and the surface charge density remains basically unchanged. The main reason is that the change in the number of alternating cycles further increases the number of seed charges. This study reveals the flashover mechanism of epoxy/glass fiber composites under different voltage forms, which provides theoretical support for subsequent material modification.

Experimental study and techno-economic evaluation of an active fault detection kit in the prospect of future zero energy building installations

This article presents an active fault detection kit, applicable to low voltage single-phase electrical installations, in the prospect of the Zero Energy Building concept, integrating on-site renewable energy generation and energy storage. This simple, flexible, self-powered and compact kit is capable of detecting faults, such as power theft (meter tampering), unintentional islanding and neutral conductor loss. Its operation is based on harmonic voltage injection, in series with the electrical installation, through a low-power H-bridge inverter and a current transformer, along with the corresponding harmonic current measurement, to estimate the impedance and effectively detect faulty conditions; the fast and robust Goertzel algorithm is utilized. Moreover, it features IoT communication capabilities, employing the ESP32 microcontroller, to exchange data and information with the installation meters. The functionality and effective fault detection of the proposed device are validated, through experimental tests on a custom-developed hardware prototype; IoT connectivity and data uploading are experimentally tested and verified, too. Finally, a sustainability assessment study is performed, using the Life Cycle Cost Analysis tool.

Impact of Harmonic Voltage on the Partial Discharge Properties of Eco-Friendly Nanofluid

The Insulation is crucial in transformers to maintain electrical isolation between components and windings, ensuring safe and efficient operation. Mineral oil acts as both insulation and coolant in transformers. It is a substance that does not naturally degrade and can persist in the environment for an extended period if accidentally released. Therefore, Sunflower oil is used as a substitute for mineral oil. However, harmonic distortion is a common issue in power systems, leading to increased dielectric loss and partial discharge activity, ultimately causing insulation failure and equipment damage. Hence, it is vital to analyze the partial discharge performance of pure Sunflower oil under varying harmonic AC voltages to ensure the reliable operation of electrical devices. The study examines the partial discharge characteristics of a sunflower oil-based nanofluid under harmonic AC voltages, which is a blend of pure sunflower oil and SiO2 nanoparticles. Two nanofluid mixtures were created, one with a 0.01% mass ratio and the other with 0.05% sunflower oil. Various PD characteristics were evaluated, including Partial Discharge significance, starting voltage, duration of increase, maximum amplitude, Phase-Resolved PD sample, and frequency and time graphs of the Partial Discharge signal. All tests were carried out under IEC regulations. A comparison was made between the outcomes of Nano blended sunflower oil and natural sunflower edible oil. Experiments were conducted on partial discharges in Sunflower oil using specific harmonic superimposed AC Voltages, utilizing the third and fifth harmonics. The findings reveal that: a) PDIV value decreases with increasing harmonic order when using AC voltage with superimposed harmonics, b) PD characteristics are closely related to the voltage waveform, and the addition of Silica to sunflower oil results in reduced PD amplitude, pulse duration, and time duration compared to pure sunflower oil.

Reliability Enhancement of a Double-Switch Single-Ended Primary Inductance–Buck Regulator in a Wind-Driven Permanent Magnet Synchronous Generator Using a Double-Band Hysteresis Current Controller

The wind power exchange system (WECS) covered in this paper consists of a voltage source inverter (VSI), a DSSB regulator, and an uncontrolled rectifier. An AC grid or a heavy inductive or resistive load (RL) can be supplied by this system. The DSSB is a recently developed DC-DC regulator consisting of an improved single-ended primary inductance regulator (SEPIC) followed by a buck regulator. It has a peak efficiency of 95–98% and a voltage gain of (D (1+D)/(1−D). where D is the regulator transistor’s on-to-off switching ratio. The proposed regulator improves the voltage stability and MPPT strategy (optimal or maximum power-point tracking). The combination of the DSSB and the proposed regulator improves the efficiency of the system and increases the power output of the wind turbine by reducing the harmonics of the system voltages and current. This method also reduces the influence of air density as well as wind speed variations on the MPPT strategy. Classical proportional–integral (PI) controllers are used in conjunction with a vector-controlled voltage source inverter, which adheres to the suggested DSSB regulator, to control the PMSM speed and d-q axis currents and to correct for current error. In addition to the vector-controlled voltage source inverter (which follows the recommended DSSB regulator), classical proportional–integral controllers are used to regulate the PMSM speed and d-q axis currents, and to correct current errors. In addition, a model Predictive Controller (PPC) is used with the pitch angle control (PAC) of WECS. This is done to show how well the proposed WECS (WECS with DSSB regulator) enhances voltage stability. A software-based simulation (MATLAB/Simulink) evaluates the results for ideal and unoptimized parameters of the WT and WECS under a variety of conditions. The results of the simulation show an increase in MPPT precision and output power performance.

Observer-based robust servomechanism control of a three-phase induction motor drive

Induction motors (IMs) are widely used in industrial applications, but harmonic distortion in IM drives can lead to inefficiencies and reduced lifespan. Traditional control strategies and filtering methods have limitations in mitigating lower-order harmonics, particularly the 5th harmonic, which can generate reverse torque and increase overheating risk. Existing solutions often struggle to fully address these issues, especially in high-power applications with dynamic loading conditions. This paper proposes a control strategy to reduce voltage and current harmonics in a Z-source inverter (ZSI)-fed IM drive. The system incorporates an observer-based output feedback robust servomechanism controller, designed using Linear Quadratic Regulator (LQR) and Linear Matrix Inequality (LMI) techniques. It ensures closed-loop stability, precise tracking regulation, and effective disturbance rejection, while also featuring a speed adaptation mechanism for sensorless operation. Simulation results demonstrate significant harmonic mitigation, with the controller achieving 1.01% lower voltage THD and 1.03% lower current THD compared to conventional approaches. This improvement aligns with industry standards and outperforms several existing techniques. The proposed method contributes a comprehensive solution for harmonic mitigation in high-power IM drives, applicable to various industrial applications requiring high-quality power and precise speed control.

Robust Control Scheme for Optimal Power Sharing and Selective Harmonic Compensation in Islanded Microgrids

In power systems, nonlinear loads cause harmonic distortion, adversely affecting sensitive equipment such as induction motors, power electronics, and variable-speed drives. This paper presents a novel control strategy that integrates with existing hierarchical control systems to mitigate voltage imbalances and harmonic disturbances in AC-islanded microgrids. The proposed method utilizes selective harmonic order filtering through multiple second-order generalized integrators (MSOGI) to extract negative, positive, and harmonic order components. The distributed generation (DG) unit control mechanism is designed to immediately correct voltage imbalances and harmonic disruptions, distributing the compensatory load evenly to rectify real and reactive power imbalances and harmonic disturbances. The microgrid’s control architecture primarily includes droop controllers for real and reactive power of positive sequences, voltage and current regulation inner control loops, an additional loop for correcting imbalances and harmonics, and secondary controllers to maintain voltage magnitude and frequency at nominal levels, ensuring high-quality voltage at the point of common coupling (PCC). The effectiveness of this approach is demonstrated through simulation results on the MATLAB/Simulink platform, proving its ability to effectively mitigate voltage imbalances and harmonic issues with the total harmonic of voltage reduced to approximately THDv = 0.5% and voltage unbalance factor (VUF) within approximately 0.1%.

A simple 60‐pulse voltage source inverter using voltage reinjection strategy to improve voltage harmonics for high‐power local loads

AbstractThe voltage reinjection strategy is an effective solution to improve the voltage quality of the voltage source inverters for high‐power applications. In this article, the fundamentals of voltage reinjection strategy are analyzed to reduce voltage harmonics of the output voltage. Then, a novel voltage reinjection circuit is proposed which can generate a 5‐level voltage to be injected on the voltages of the DC sources. The reinjected voltage can create a 60‐pulse voltage waveform for a three‐phase load. Turn ratio optimization of the reinjection transformer has a significant effect on the improvement of the harmonic situation of the output voltage. Furthermore, the proposed reinjection circuit provides ZVS conditions for the main switches to reduce switching losses considerably. The simulation results indicate that the proposed reinjection circuit can reduce the output voltage THD to 3.16% without any type of filter or modulation, and the load variation does not have a considerable effect on the voltage quality. The experimental results approve the analytical trend and simulation process and also confirm the reinjection circuit performance. According to the results, the proposed method has generated a semi‐sinusoidal voltage waveform with in the laboratory.

The HVDC commutation failure mechanism impacted by harmonics and its suppression method

Harmonics are an important cause of voltage distortion, and voltage reduction and voltage distortion are the main causes of commutation failure. But most of the current research studies on commutation failure are based on the drop in fundamental voltage after a fault, and there are few reports on the impact of voltage distortion on commutation failure. Aiming to solve the above problem, this article first analyzes the generation and transmission characteristics of harmonics in AC and DC systems, as well as the coupling degree of harmonics in DC systems. Then, starting from the mechanism of the commutation process, a commutation failure analysis method based on the harmonic voltage time area is proposed. This method can quantitatively analyze the impact of different harmonics on commutation failure. On this basis, the harmonic criteria for causing commutation failure are established, and the harmonic characteristics of AC voltage when harmonics cause subsequent commutation failure are quantitatively analyzed. Finally, a preventive controller is proposed to suppress commutation failure caused by different harmonics, where the above analysis is verified by the simulations.