Dynamic Phasor Research Articles

Harmonic Sequence Component Model-Based Small-Signal Stability Analysis in Synchronous Machines during Asymmetrical Faults

Power systems are complex and often subject to faults, requiring accurate mathematical models for a thorough analysis. Traditional time-domain models are employed to evaluate the dynamic response of power system elements during transmission system faults. However, only the positive sequence components are considered for unbalanced faults, so the small-signal stability analysis is no longer accurate when assuming balanced conditions for asymmetrical faults. The dynamic phasor approach extends traditional models by representing synchronous machines with harmonic sequence components, making it suitable for an unbalanced condition analysis and revealing dynamic couplings not evident in conventional methods. By modeling electrical and mechanical equations with harmonic sequence components, the study implements an eigenvalue sensitivity analysis and participation factor analysis to identify the variable with significant participation in the critical modes and consequently in the dynamic response of synchronous machines during asymmetric faults, thereby control strategies can be proposed to improve system stability. The article validates the dynamic phasor model through simulations of a single-phase short circuit, demonstrating its accuracy and effectiveness in representing the transient and dynamic behavior of synchronous machines, and correctly identifies the harmonic sequence component with significant participation in the critical modes identified by the eigenvalue sensitivity to the rotor angular velocity and rotor angle.

Dynamic phasor model of HVDC links for linear analysis of dynamic performance

This paper presents a linearized HVDC link model, which was implemented in PacDyn software and is applicable to dynamic performance studies of power electronic equipment considering electrical network electromagnetic transients, differently from electromechanical studies where the network dynamics is usually neglected. The proposed HVDC model was developed employing equivalent switching functions concepts and dynamic phasors. Simulations and results for small and large-scale test systems are presented. Model validation is performed through comparisons between results obtained by PacDyn and PSCAD. In sequence, there are practical examples of controller tunning considering adverse dynamic interaction mitigation between HVDC Link and electrical grid. The final results on a large-scale model of the Brazilian power system demonstrate that the proposed HVDC model can be used in actual engineering studies.

Wireless Power Transfer System Model Reduction with Split Frequency Matching

Reduced-order dynamic models of wireless power transfer (WPT) systems are desired to simplify the analysis and design of power control, phase synchronization, and maximum efficiency tracking. The reduced-order dynamic phasor model is a good choice because of its straightforward physical meaning and concise mathematical formula. However, the model relies on the assumption of loose coupling and loses accuracy when the coupling becomes stronger. In this paper, a model reduction method with split frequency matching is proposed to improve model accuracy under relatively strong coupling conditions, which is suitable for most short-distance WPT applications, such as wireless electrical vehicle charging. Split frequency matching is achieved through a pair of conjugate equivalent mutual inductances, which are derived from the asymmetry characteristics of the full-order dynamic phasor model in the positive and negative frequency domains. The proposed model retains the advantages of the existing model while significantly improving the accuracy under strong coupling conditions. Its characteristics are verified by comparing the experimental results and model predictions under both large step changes and small-signal perturbations.

Modular Dynamic Phasor Modeling and Simulation of Renewable Integrated Power Systems

Modular Dynamic Phasor Modeling and Simulation of Renewable Integrated Power Systems

Disturbance Rejection Control of Grid-Forming Inverter for Line Impedance Parameter Perturbation in Weak Power Grid

When a grid-forming (GFM) inverter is connected to a low- or medium-voltage weak power grid, the line impedance with resistive and inductive characteristics will cause power coupling. Typical GFM decoupling control strategies are designed under nominal line impedance parameters. However, there are deviations between the nominal line impedance and actual parameters, resulting in poor decoupling effects. Aiming at this problem, this paper proposes a power decoupling strategy based on a reduced-order extended state observer (RESO). Firstly, the power dynamic model of the GFM is established based on the dynamic phasor method. Then, the model deviation and power coupling due to line impedance parameter perturbation are estimated as internal disturbances of the system, and the disturbances are compensated on the basis of typical power control strategy and virtual impedance decoupling. Good decoupling performance is obtained under different impedance parameters, improving the control strategy’s robustness. Finally, the effectiveness of the proposed method is verified by the results of RT Box hardware-in-the-loop experiments.

ClosedLoop Control of Pulse-Density-Modulated Wireless Power Transfer with Fast MEPT

Efficient wireless power transfer (WPT) requires closedloop control to perform maximum efficiency point tracking (MEPT) against the dynamic changes of operating conditions. In this paper, we took the latest deltasigma pulse-density-modulated (PDM) WPT system as the plant and presented a systematic control design procedure. The WPT system was modeled by the reducedorder dynamic phasor method and further decoupled under the tuned condition. It was identified that the coupled resonators behave like a secondorder system, and its natural frequency is much lower than the resonant frequency. This was considered in the closedloop design to derive a suitable loop gain and ensure stability. The nonlinearity caused by the dualside PDM control is limited to a small inner loop so that the entire loop is linear and simple. Experimental results showed that the closedloop voltage regulation and MEPT took only about 10 ms after load stepping. The system DC output voltage was always tightly regulated, and the steadystate efficiency was very close to the theoretical maximum value under different coupling conditions.

Dynamic phasor measurement algorithm based on high-precision time synchronization

Ensuring the swift and precise tracking of power system signal parameters, especially the frequency, is imperative for the secure and stable operation of power grids. In instances of faults within the distribution network, abrupt changes in frequency may occur, presenting a challenge for existing algorithms that struggle to effectively track such signal variations. Addressing the need for enhanced performance in the face of frequency mutations, this paper introduces an innovative approach—the Covariance Reconstruction Extended Kalman Filter (CREKF) algorithm. Initially, the dynamic signal model of electric power is meticulously analyzed, establishing a dynamic signal relationship based on high-precision time source sampling tailored to the signal model’s characteristics. Subsequently, the filter gain, covariance matrix, and variance iteration equation are determined based on the signal relationship among three sampling points. In a final step, recognizing the impact of the covariance matrix on algorithmic tracking ability, the paper proposes a covariance matrix reset mechanism utilizing hysteresis induced by output errors. Through extensive verification with simulated signals, the results conclusively demonstrate that the CREKF algorithm exhibits superior measurement accuracy and accelerated tracking speed when confronted with mutating signals.

Measurement accuracy verification of phasor measurement unit with dynamic phasor estimation

Measurement accuracy verification of phasor measurement unit with dynamic phasor estimation

Dynamic Phasor Simulation of Systems with High Shares of Inverter-Based Resources

Integrating inverter-based resources into power systems increases their complexity, and makes their operation more challenging. The conventional approach for simulating power systems neglects transients of the power grid, using steady-state phasors and considering only fundamental frequency values. However, to accurately predict the dynamic behavior of systems with high shares of inverter based resources, electromagnetic transient simulations (EMTSs) are necessary. The primary limitation of classical time-domain EMTSs is their frequent time-consuming nature, making them impractical for larger systems. In this work, we explore the use of dynamic phasors (DPs) as a technique to accelerate EMTSs. For this purpose, we first adapt the power grid and inverter model to DP simulation. Subsequently, we assess the effectiveness of this approach through three numerical examples, demonstrating a reduction in the simulation times by a factor of 4 to 40.

Novel VSC-Based STATCOM Model Based on Dynamic Phasors for Unbalanced Distribution Networks

Novel VSC-Based STATCOM Model Based on Dynamic Phasors for Unbalanced Distribution Networks

Fast Dynamic Phasor Estimation Algorithm Considering DC Offset for PMU Applications

Because of the volatility and intermittence of the renewable energy units in the novel power system, the electrical variables usually exhibit dynamic characteristics. Under fault conditions, the electrical variables usually contain a decaying DC (DDC) component. Currently, when considering the DDC component and the dynamic characteristics of electrical variables, it is difficult to obtain highly precise estimation results in most references. In this paper, a fast dynamic phasor estimation algorithm considering the DC offset is proposed to obviate errors caused by dynamic characteristics and the DDC component. The algorithm consists of two parts. The first part establishes the mathematical model of DDC using exponential functions and obtains the exact DDC component by iteration. The second part establishes the compressive sensing Taylor Fourier Multi-Frequency (CSTFM) model of the dynamic phasor. The improved sparsity adaptive matching pursuit (ISAMP) algorithm is used to quickly and accurately estimate the dynamic phasor, frequency and rate of change of frequency (ROCOF). Simulation and field data test results show that the proposed algorithm can accurately estimate the fault current phasor, frequency, and ROCOF compared with existing phasor estimation algorithms.

Using Dynamic Phasors To Model and Analyze Selective Harmonic Compensated Single-Phase Grid-Forming Inverter Connected to Nonlinear and Resistive Loads

In an islanded microgrid, a grid-forming inverter (GFMI) is required to maintain a stable sinusoidal voltage at nominal frequency and amplitude with minimal distortion regardless of loading conditions. As a result, the accurate design of GFMI control schemes in an islanded microgrid requires accurate understanding of the converter dynamics and system harmonics. Electromagnetic transient (EMT) simulations based on detailed switching models give an accurate picture of the GFMI's behaviour. However, EMT simulations take a relatively long time to produce accurate results in large-scale systems. On the other hand, conventional phasor models are unsuitable for harmonic studies. To address this limitation, this article leverages the dynamic phasor (DP) method to develop a new model of a single-phase GFMI connected to single-phase nonlinear (diode-bridge rectifier) and resistive loads. The benefit of the DP method lies in its inherent feature of transforming time-domain signals into slow-varying or DC signals, thereby enabling faster simulations. In this article, the relationship between the control gains of the proposed DP model and the conventional detailed model is established for the first time through small-signal and frequency-domain-based comparative analyses. User-written codes of the proposed DP model are scripted in MATLAB, and simulations on MATLAB/Simulink show that the proposed DP model results have a high degree of correlation with the detailed switching model results. A significant savings in computation time is exhibited by the proposed DP models. The simulation results are validated against experimental results to confirm the accuracy of the proposed DP model.

Co-simulation of real-time electromagnetic transient and transient stability simulations using dynamic phasor T-Line model

This paper presents techniques to interface a Transient Stability Analysis (TSA) based model to an ElectroMagnetic Transient (EMT) based model running on a Real Time Digital Simulator. Practical challenges of implementing such an interface are discussed. The proposed TSA model can be simulated using a substantially larger time-step compared to the EMT model. The interface between the EMT model and the TSA model is implemented using a transmission line modeled using Dynamic Phasors (DP). Time-step delay is the primary cause of numerical instability in co-simulations. In this work, a traveling wave model of a transmission line is used to decouple the EMT and TSA networks. The interface requires that the propagation delay of the interfaced transmission lines be greater than the EMT time-step even though the TSA is simulated using a much larger time-step than the EMT model. The proposed technique is validated using the IEEE 39 bus system and a power system with 500 buses.

A multi-solver framework for co-simulation of transients in modern power systems

This paper develops a novel multi-rate, multi-solver co-simulation framework combining dynamic phasors, transient stability, base-frequency dynamic phasors for frequency-adaptive simulation of transients, and electromagnetic transient (EMT) models. This framework subdivides a given power network into several types of subsystems based on the connected devices, required accuracy in representing dynamic details, electrical distance from perturbations, and the intended purpose of the study; as such, the paper describes methods and guidelines to simulate each subsystem using the most appropriate solver and time-step size to maximize simulation efficiency and accuracy. It also addresses the tasks of multiple interfacing and solver interactions that are essential in coupling different solvers. The proposed framework is built around an industrial-grade EMT simulator, to which other solvers are interfaced, enabling access to a variety of power system models and distinct features. The accuracy and efficiency of the framework are demonstrated through co-simulations carried out on a modified version of the 118-bus network, which includes an MMC-HVDC system.

High-Frequency Oscillation Mechanism Analysis and Suppression Strategy of Grid-Forming Control MMC-HVDC

Recently, several high-frequency oscillation (HFO) events have occurred in the modular multilevel converter (MMC) based HVDC (MMC-HVDC) projects, threatening the safety and reliability of power systems. Existing studies only focus on the HFO of MMC-HVDC with grid-following control (GFWC) but less on the HFO of MMC-HVDC with grid-forming control (GFMC). Therefore, this paper establishes the small-signal model of MMC-HVDC with GFMC through the dynamic phasor method. Then, the impact of different parameters on the HFO is discussed. Unlike MMC-HVDC with GFWC, the outer loop control of MMC-HVDC with GFMC has a great influence on HFO. Meanwhile, the existing HFO suppression strategies by adding a low-pass filter (LPF) to feedforward control of terminal voltage for MMC-HVDC with GFWC will increase the HFO risk of MMC-HVDC with GFMC. Further, the HFO suppression strategies suitable for MMC-HVDC with GFMC are proposed based on the mechanism analysis results. Finally, the obtained results are verified by the time-domain simulation of the studied system in RT-LAB.

A modeling method for the DC side simplified impedance of line commutated converter based on the simplified switching function

With a large amount of new energy access and the wide application of high voltage direct current (HVDC) transmission technology based on line commutated converter (LCC), it is of great significance to study the impedance frequency characteristics of the converter to analyze the stability of the LCC-HVDC system. Due to the complexity and inefficiency of the existing modeling techniques, in this paper, a simplified impedance modeling method for the DC side of LCC based on the simplified switching function is proposed. Firstly, the commutation process of LCC is deduced according to the converter valve opening sequence and circuit principle, and the simplified switching function of LCC is proposed. Next, based on the dynamic phasor method, through the analysis and calculation of harmonic transmission between the AC and DC sides of LCC, the simplified impedance of the DC side of LCC is obtained. Then, a simulation model is established in PSCAD, and the impedance scanning results of the simulation model show that the error is small between the method proposed in this paper and the method without simplification, which verifies the validity of the proposed model. The proposed method can reduce modeling complexity and provide a fast calculation model of the DC side impedance of LCC for analyzing the stability of large complex transmission systems.

Dynamic phasor-based hybrid simulation for multi-inverter grid-connected system

To realize the efficient transient simulation of a grid-connected power generation system based on multiple inverters, this paper proposes a hybrid simulation method integrating the models of electromagnetic transient and dynamic phasors. Based on a demonstration of the concepts and properties of dynamic phasors, the models of single-phase and three-phase inverters described by dynamic phasors are established first. Considering the numerical compatibility problem between dynamic phasors and instantaneous values, an interface scheme between dynamic phasors and instantaneous values is designed, and the efficiency and precision differences of various transformation methods are compared in detail. Finally, by utilizing MATLAB/Simulink, a hybrid simulation platform of a multi-inverter grid-connected system is built, and the efficiency and accuracy of the hybrid simulation are validated via comparison with the full electromagnetic transient simulation.

Dynamic Phasor-Based Modeling and Simulation of a Single-Phase Diode-Bridge Rectifier

This paper presents dynamic phasor (DP) models of a single-phase diode-bridge rectifier (DBR). Single-phase DBRs serve as a primary building block of consumer electronics, whether with a simple filter or coupled with additional circuitry to produce unity power factor rectification. Four DP models are proposed, two models (DP-SF and DP-AW) suitable for systems with negligible line inductance and two models (DP-Full and DP-Fund) which can be used when the line inductance causes significant commutation overlap. The proposed models are implemented in MATLAB and simulation results are compared against a detailed switching model simulation. A significant reduction in simulation time is demonstrated by the proposed DP models. The simulation results are validated against experimental data, confirming the accuracy of the proposed modeling approaches.

Noisy-Intermediate-Scale Quantum Electromagnetic Transients Program

Quantum-empowered electromagnetic transients program (QEMTP) is a promising paradigm for tackling EMTP’s computational burdens. Nevertheless, no existing studies truly achieve a practical and scalable QEMTP operable on today’s noisy-intermediate-scale quantum (NISQ) computers. The strong reliance on noise-free and fault-tolerant quantum devices–which appears to be decades away– hinder practical applications of current QEMTP methods. This paper devises a NISQ-QEMTP methodology which for the first time transitions the QEMTP operations from ideal, noise-free quantum simulators to real, noisy quantum computers. The main contributions lie in: (1) design of shallow-depth QEMTP quantum circuits for mitigating noises on NISQ quantum devices; (2) practical QEMTP linear solvers incorporating executable quantum state preparation and measurements for nodal voltage computations; (3) a noise-resilient QEMTP algorithm leveraging quantum resources logarithmically scaled with power system dimension; (4) a quantum shifted frequency analysis (QSFA) for accelerating QEMTP by exploiting dynamic phasor simulations with larger time steps; (5) a systematical analysis on QEMTP’s performance under various noisy quantum environments. Extensive experiments systematically verify the accuracy, efficacy, universality and noise-resilience of QEMTP on both noise-free simulators and IBM real quantum computers.

Review of Discrete Fourier Transform During Dynamic Phasor Estimation and the Design of Synchrophasor Units

In a large and complex interconnected power system, the measurement of synchronized bus voltage and line current plays a vital role in the monitoring and precise control of various sophisticated electrical equipment for secure and reliable operation. Phasor measurement units (PMUs) are incorporated into a wide area of the power system to extract the different signals of synchronized phasors. In this paper, the capacity of the PMU phasor estimation algorithm is explored based on discrete Fourier transform (DFT) under different sampling frequency rates during various dynamic scenarios in accordance with the IEEE C37.118.1a-2014 standard. Furthermore, the performance of the DFT algorithm varies according to the phase angle and dynamic parameters such as frequency, frequency ramp rates, modulation frequency, harmonic levels, step change, decaying dc, and noise levels. The simulation results reveal that accuracy of the phasor estimation algorithm based on DFT can be achieved at high sampling frequency rates. Furthermore, the results of DFT-based phasor estimation are compared with Shank’s estimation method (SEM) and the least-squares estimation method (LEM). The presented method is best suited to PMU algorithms development based on DFT for better visualization of the smart electric grid.