Harmonic State Space Model Research Articles

Harmonic State-Space Modeling and System Characteristic Analysis of Grid-Connected Inverter Parallel-Operation System Considering Asynchronous Carriers

Harmonic State-Space Modeling and System Characteristic Analysis of Grid-Connected Inverter Parallel-Operation System Considering Asynchronous Carriers

UHV AC/DC power grid geomagnetically induced currents harmonic characteristics analysis

The power grid’s geomagnetically induced currents (GIC) during geomagnetic storms, are associated with a broad and abrupt harmonic impact, with the potential for grid paralysis and extensive power outages in severe scenarios. Therefore, the harmonic characteristics of geomagnetically induced currents in China’s ultra-high voltage (UHV) AC/DC power grid are analyzed in this paper. Firstly, a harmonic equivalent model of transformer GIC is established, and the injection of harmonics into the UHV AC/DC power grid during GIC events is quantified.Secondly, a harmonic state-space model is established by combining the 12-pulse converter switching function theory with the exponential Fourier series. The harmonic transmission and dynamic coupling characteristics of GIC in the UHV AC/DC power grid are analyzed using the harmonic equivalent model of the main transformer GIC and the harmonic equivalent model of the converter transformer GIC. Finally, a harmonic simulation model of the UHV AC/DC power grid under GIC is constructed using PSCAD, and the distribution of harmonics in the UHV AC/DC power grid under GIC is calculated. The calculations indicate that with 30 A injected GIC in the sending end transformers, the 2nd harmonic current on the sending end network side alone is 61.45 A, and the 3rd harmonic voltage on the valve side is 8.57 kV. The characteristics of harmonic transmission in the power grid result in a 2nd harmonic current of 34.87 A on the receiving end network side. Simultaneously, the total harmonic distortion of the AC power grid at the sending end is 2.35 %, surpassing the permissible limit established by the national standard. Consequently, this issue should be prioritized by the relevant power authorities.

Harmonic State-Space Modeling and Closed-Loop Control of Single-Stage High-Frequency Isolated DC–AC Converter

The port harmonic features of the single-phase single-stage high-frequency isolated dc-ac converter is not analyzed so far and the corresponding theoretical basis for improving the current quality is still a research gap. In this paper, a linear model in frequency domain of the single-stage high-frequency isolated dc-ac converter is originally derived based on the combined harmonic state space and extended description function modeling technique. The systematic modeling process is analyzed and the accuracy of the modeling is verified by the simulation results. Combined with the model analysis, the harmonic coupling characteristics between the ac side and the dc side are clearly presented. On this basis, this paper proposes a closed-loop control strategy to reduce the decrease of the dc-side and ac-side current quality caused by the harmonic coupling at the control level. Furthermore, this paper constructs the compound controllers based on the P-type iterative learning to suppress the harmonic currents of the dc side and the ac side, respectively. The composite controller does not distinguish the harmonic frequency. Hence, the proposed controller achieves the effective suppression of the harmonic currents in real time. The effectiveness and feasibility of the proposed control strategy are verified by the experiments.

Harmonic Interaction Analysis of Inverters Based on Harmonic State-Space Model Considering Dead-Zone Effect

In the context of increasing device switching frequencies year by year, non-ideal factors such as dead-zone effects in inverters not only cause output waveform distortion, but also generate harmonics that make the interaction between inverters and power grids more complex. In order to accurately describe and analyze this phenomenon and establish a more practical grid-connected inverter model, this paper proposes and establishes a three-phase LCL grid-connected inverter harmonic state space model that considers dead-zone effects. By incorporating dead-zone effects into the modeling process, the system transfer function matrix is derived to quantify the harmonic interaction coefficient between the inverter and the power grid, and the influence mechanism of dead-zone effects on the harmonic interaction process at high switching frequencies is analyzed in depth. The necessity of considering dead-zone effects is illustrated through simulation, and the effectiveness and accuracy of the inverter HSS model considering the effects of dead-zone effects are verified through experiments.

Vienna Rectifier Modeling and Harmonic Coupling Analysis Based on Harmonic State-Space

Due to the high permeability characteristics of power electronic devices connected to the distribution grid, the potential harmonic coupling problem cannot be ignored. The Vienna rectifier is widely utilized in electric vehicle charging stations and uninterruptible power supply (UPS) systems due to its high power factor, adaptable control strategies, and low voltage stress on power switches. In this paper, the three-level Vienna rectifier is studied, and the harmonic state-space (HSS) method is used to model the rectifier. The proposed model can reflect the harmonic transfer characteristics between the AC current and the DC output voltage at various frequencies. Finally, the model’s accuracy and the corresponding harmonic characteristics analysis are further verified by simulation and experimental test results. The results show that the harmonic state-space modeling used for Vienna rectifiers can reflect the harmonic dynamics of the AC and DC sides, which can be used in stability analysis, control parameter design, and other related fields.

Active disturbance rejection controller design for harmonic suppression in MPC optimal control based on harmonic state space modeling

SummaryThis paper proposes a harmonic suppression strategy for photovoltaic‐voltage source converters (PV‐VSC). The proposed method utilizes a harmonic state space (HSS) model‐combine model predictive control (MPC) algorithm, which provides a comprehensive representation of the global characteristics of AC/DC harmonic coupling impedance in photovoltaic systems. The HSS‐based converter model operates as a linear time‐varying periodic system, which allows for the implementation of MPC to optimize the HSS equation. This optimization yields voltage increments across various harmonic orders, providing feedforward compensation for the current controller. By incorporating feedback signals into the current control, harmonic compensation is achieved, resulting in improved power quality. Furthermore, a decoupled proportional resonance‐linear adaptive disturbance rejection control (PR‐LADRC) is employed to effectively mitigate steady‐state errors and decrease total harmonic distortion in the presence of DC‐side disturbances. This is accomplished by decoupling the disturbance observer from the controller and integrating the increment feedback signal derived from the MPC‐HSS. Finally, the efficacy of the proposed method and the analysis results are corroborated through numerical simulations.

Integrated Design of Filter and Controller Parameters for Low-Switching-Frequency Grid-Connected Inverter Based on Harmonic State-Space Model

In high-power grid-connected inverter for new energy power generation system, low switching frequency makes the control loop, <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">LCL</i> -filter resonant peak and sideband harmonics generated by modulation coupled, so that the parameter design of filter and controller affects each other. This increases the difficulty of design of filter parameters and controller parameters. To solve this problem, an accurate current harmonic calculation method based on harmonic state-space model and considering switching frequency and control delay is proposed in this article firstly. Secondly, based on the above model, the parameter optimization process considering current harmonic distortion rate, filter cost and so on is constructed. The integrated optimization design of filter and controller parameters of low-switching-frequency <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">LCL</i> grid-connected inverter is completed through intelligent algorithm. Thus the filter cost is minimized on the premise that inverter performances meet the specific requirements. In addition, considering the non-ideal characteristic of grid, a scheme of integrated parameter design which can adapt to the large-scale change of grid strength is proposed. Finally, the effectiveness of the proposed method and scheme is verified by simulation and experiment.

A Multifrequency Small-Signal Model for the MLCL-Filtered Grid-Connected Inverter Considering the FCE of Nonlinear Inductors

To study the frequency coupling effect (FCE) of nonlinear inductors in frequency domain, a multifrequency small signal model is developed in this paper for the modified LCL (MLCL) filtered grid-connected inverter. This model is realized by extracting side band coupling (SBC) units out of the harmonic state-space (HSS) model and superposing them to the perturbation frequency loop. Compared with the complicated reductant HSS model, the proposed multifrequency model provides an intuitive efficient accurate way to describe the system. Besides, notions in classical control theory can be directly utilized with the multifrequency model. With this model, the current control loop of the inverter is studied. Besides, by introducing a simplified linear time invariant (LTI) small signal model (the FCE is ignored), it is found that the FCE effect is non-negligible especially in the frequency range around resonance points, which affects the stability of the differential mode resonance (DMR) and common mode resonance (CMR). The proposed multifrequency model accurately predicts the DMR and CMR stability, while the simplified LTI model fails. Furthermore, damping strategies are developed to suppress the DMR and CMR. Finally, simulations and experiments based on a 228 kW PV inverter are conducted to verify the analytical result.

Truncation Number Selection of Harmonic State-Space Model Based on the Floquet Characteristic Exponent

Harmonic state-space (HSS) theory has been an effective method for studying system stability with time-periodic characteristics, such as modular multilevel converters (MMCs) and renewable power generation under unbalanced grid conditions. However, the original state-space formulation of the HSS model comprises infinite-dimensional matrices. Truncation is necessary before practical computing application. Therefore, this letter proposes a truncation number selection method for the HSS model. The theoretical basis of the proposed method is based on the Floquet characteristic exponent of the linear time-periodic model, which has a straightforward physical meaning. Finally, a case study of an MMC grid-tied system with tests in RT-LAB illustrates the validity of the proposed method.

Research on Harmonic State-Space Modeling and Calculation Analysis of Low-Switching-Frequency Grid-Connected Inverter Considering the Impact of Digitization

Under low-switching-frequency condition, there is harmonic coupling in grid-connected inverter. The harmonic state-space (HSS) modeling method is an effective means to analyze the output harmonic characteristics of inverter. When the switching frequency is low, the sampling/holding and control delay make the update of pulsewidth modulation pulses more untimely, which will have a non-negligible impact on system output. However, the current relevant research fails to take these factors into account in the HSS model. In addition, the existing work fails to thoroughly study the calculation and analysis methods of HSS model, and the relevant work is insufficient. Based on this, this article studies the closed-loop HSS model and its calculation and analysis methods of three-level grid-connected inverter considering the digitization impact. First, according to the different calculation and analysis methods of the HSS model of the grid-connected inverter, the methods considering the digitization impact are proposed, respectively. Second, according to the proposed method, the harmonic characteristics of low-switching-frequency grid-connected inverter are analyzed, and the necessity of considering the digitization impact is explained. Finally, the effectiveness and accuracy of the proposed HSS-model calculation and analysis methods considering the digitization impact are verified by simulation and experiment.

Modeling and Analysis on AC-DC Harmonic Coupling of the Three-Phase Voltage Source Converter under Asymmetric Condition

A three-phase voltage source converter (VSC) is the most common power electronic device in power systems, but its inherent nonlinearity leads to complex AC-DC harmonic coupling phenomena. Existing studies focus on the harmonic coupling characteristics of three-phase VSCs under three-phase symmetrical conditions, but the problem under asymmetrical conditions is rarely investigated. This paper proposes a practical modeling method for harmonic state space (HSS) modeling of the VSC topology, together with a classic T/4-delay-based decoupled double synchronous reference frame (DDSRF) control system. The steady-state phase shift characteristics of the T/4 delay link for each frequency component are first analyzed, and the expansion method of the DDSRF controller in the HSS frame is derived. Compared with the conventional method, the proposed technique can provide an accurate description of the steady-state effect of the delay link without introducing additional state variables and, therefore, can reduce model complexity and computation burden. Then, the small-signal close-loop HSS model of the VSC is established, and a harmonic transfer matrix between the AC positive sequence/negative sequence and DC current of different frequencies is developed, which reveals the global harmonic coupling relationship between AC and DC grids across the VSC. Analysis shows that the T/4-delay-link significantly weakens the AC-DC inter-harmonic coupling near the fundamental frequency. Finally, the electromagnetic transient simulations are carried out based on MATLAB/Simulink, and the validity of the proposed modeling method and the model and the correctness for the analysis of harmonic coupling characteristics are verified.

Stability analysis of Three-phase Grid-Connected inverter under the weak grids with asymmetrical grid impedance by LTP theory in time domain

The Grid-connected inverter (GCI) often operates in the weak grid with asymmetrical grid impedance due to the unbalanced and single-phase loads. However, the time-periodic dynamic behavior effect of the Phase-Locked Loop (PLL) on the GCI operating in an asymmetrical system is not investigated in the time domain. This paper explores the influence of the asymmetrical grid impedance on the stability of the weak grid with GCI. Firstly, GCI's complete harmonic state-space (HSS) model considering PLL under the asymmetrical network is established. Secondly, in view of the developed HSS model, the predominant parameters are determined through the eigenvalues of the system state matrix, along with the participation factors. Afterward, with eigenvalue locus analysis, the stability boundaries of the system's critical parameters involving the current reference, current controller, and PLL controller are determined under different grid impedance asymmetrical indexes. It is found that the stability region of system parameters is narrowed with increasing impedance asymmetrical index. Lastly, simulations and experiments are carried out to verify the effectiveness of the proposed HSS model and the theoretical analysis results.Index terms— Grid-connected inverter, asymmetrical grid impedance, Phase Lock Loop, harmonic state-space model, stability boundaries, participation factors.

Small-Signal Modeling and Interaction Analysis of LCC-HVDC Systems Based on Harmonic State Space Theory

High-voltage direct current (HVDC) technology based on Line Commutated Converter (LCC) is widely used in modern power systems due to its numerous benefits. However, as a typical power electronic device, LCC-HVDC has caused significant changes in the dynamic behavior of modern power systems due to largely replaced conventional power supplies in power systems. The interactions among sending-terminal and receiving-terminal system become stronger and have caused some stability issues, but the mechanism behind this has yet to be fully revealed. In order to analyze the interactions among sending-terminal and receiving-terminal system of LCC-HVDC, a small-signal model of modular HSS modeling is proposed based on Harmonic State Space (HSS) theory. Comparison of time-domain response with detailed switching model shows that the proposed can well reflect the dynamic behavior of the detailed switching model. Based on the model, the influence of controller’s parameters on system stability is analyzed, as well as the dynamic interactions of LCC-HVDC among sending-terminal and receiving-terminal system are investigated. Analytical results demonstrate that there exists interactions among sending-terminal and receiving-terminal system which have an influence on the stability of the system.

H∞ Optimal Control Design of Static Var Compensator Coupling Hybrid Active Power Filter Based on Harmonic State-Space Modeling

Compared with the conventional active power filter (APF), the static var compensator coupling hybrid active power filter (SVC-HAPF) has distinct characteristics of low DC-link operating voltage, which can lower the system and operational costs. It also has a wider operational range than the conventional hybrid active power filter (HAPF). In this paper, a harmonic state-space (HSS) modeling is firstly proposed to address the nonlinearity issue of the SVC-HAPF. The HSS based SVC-HAPF model is linear time-invariant (LTI), which enables the optimal control techniques. Then, the linear matrix inequality (LMI) based H ∞ optimal control design is proposed, which finds the optimal controller parameters via convex optimization algorithm. The proposed controller for the SVC-HAPF achieves high steady-state accuracy, fast transient response, and fixed switching frequency simultaneously. Finally, simulation and experimental results are also provided to verify the effectiveness and performance of the optimal controller for the SVC-HAPF in comparison with the recent developed state-of-the-art multi-quasi-proportional-resonant (MQPR) controller, which shows superior control performances.

Small-Signal Modeling of AC Power Electronic Systems: Critical Review and Unified Modeling

The harmonic state-space (HSS), the dynamic phasor (DP), and the generalized dq (GDQ) modeling are three widely used methods for small-signal analysis of ac power electronic systems. By reviewing their principles and deriving their mathematical relationships, this paper proposes a unified framework for all the three approaches. The unified modeling reveals that the linearization and transformation can be exchanged flexibly in the modeling process, and the initial phase takes a role in transforming the GDQ model into the HSS or DP model. Case studies on a three-phase voltage-source converter in unbalanced power grids are provided for validation. The relationships of three modeling methods are verified by mathematical proofs and time-domain simulations. The unified frequency-domain model is further validated through the frequency scan in experiments. Insights of the unified modeling framework and recommendations from engineering perspectives are finally discussed.

Comparative Study of Dynamic Phasor and Harmonic State-Space Modeling for Small-Signal Stability Analysis

This paper conducts a comparison of two promising frequency-lifted representations used in the state-space modeling of power-electronic converters: dynamic phasors and harmonic state-space. These methods originate from similar hypotheses and aim to tackle the time-periodicity problem observed in systems with multiple harmonic components in steady state. The paper derives the theoretical foundations of the methods with an emphasis on their similarities and differences, and applies them to a two-level voltage-source converter. The application demonstrates the effects of different truncation orders of the models infinite formulations. Stability analysis tools such as eigenvalues and transfer functions are examined and benchmarked against a classical dq-frame small-signal model.

Disturbance-State Modeling and Oscillation Analysis of Modular Multilevel Converters under P/Q Control Mode

To investigate the oscillation problems in power systems with modular multilevel converters (MMCs), a disturbance-state model of the MMC under P/Q control mode is established based on the harmonic state-space (HSS) method. Firstly, the basic structure of the MMC with P/Q control is presented, including the circuit structure and the controller structure. Next, the dynamic characteristics of each module in the MMC are described with the time-domain state-space model or the s-domain transfer function. The complete controller structure is considered, including the P/Q outer-loop controller, the inner-loop control, the phase-locked loop (PLL), the circulating current suppression control (CCSC) and the control delay. On the basis, the disturbance-state HSS model of the MMC and its corresponding steady-state condition are established by the HSS method. Lastly, the case studies are performed to verify the accuracy of the proposed model, based on an MMC based test system in PSCAD/EMTDC. Moreover, one oscillation phenomenon in test system is illustrated by analyzing the AC-side impedance characteristics based on the proposed model. The results indicate that the MMC system under P/Q control mode has certain oscillation risk due to its negative resistance effect and the proposed model could be applied to its oscillation analysis.

Small Signal Modeling and Stability Analysis of Modular Multilevel Converter Based on Harmonic State-Space Model

The stability of Modular multilevel converter (MMC) itself is the premise of analyzing the stability of MMC cascading with other cells, so this paper addressed the small-signal stability of MMC with dc voltage control mode which is the common operation mode of grid-side MMC in high-voltage direct current (HVDC) systems. First, the small signal model of MMC with open-loop control mode and dc voltage control mode while considering the dc voltage controller and current controller are established with the harmonic state-space (HSS) modeling method. Subsequently, eigenvalues and participation factors are analyzed to estimate the stability of MMC system and the influence of controller parameters, especially the proportional coefficients of dual-loop controllers, which can improve the design efficiency of control system. Finally, time-domain simulations in MATLAB/Simulink are provided to verify the accuracy of HSS small signal model and the effectiveness of the theoretical stability analysis.

Stability analysis of modular multilevel converter based on harmonic state‐space theory

Modular multilevel converter (MMC) is regarded as one of the most potential topologies for high-voltage direct current transmission system. However, the multi-frequency behaviour and complex control structure of MMC bright great challenges to the stability analysis, identification of factors affecting stability, and controller design. To address this problem, the harmonic state-space (HSS) theory is applied to MMC in this study. First, the small-signal HSS model is established to describe the multi-frequency behaviour of MMC. Capacitor voltage control, circulating current control, and output current control are also taken into account in the modelling. On this basis, the stability analysis is performed by evaluating the eigenvalues of the HSS model, and participation factor is introduced to identify the factors affecting stability. In addition, the proposed approach reveals the influence of different controllers on the system stability, providing guidance for the tuning of multiple controller parameters. Finally, the effectiveness and accuracy of the proposed approach are verified by simulation and experiment results.

Generalized Single-Phase Harmonic State Space Modeling of the Modular Multilevel Converter With Zero-Sequence Voltage Compensation

The modular multilevel converter (MMC) has attracted extensive research in recent years. An appropriate model is necessary to analyze stability or to design MMC controllers. Several published MMC models have been derived in single-phase form to simplify the modeling mathematics. However, little attention is given to the zero-sequence voltage, which introduces coupling in the single-phase model and leads to significant error. In this paper, after revealing the mechanism behind the zero-sequence voltage, a more accurate generalized single-phase MMC model is proposed, which eliminates the zero-sequence voltage coupling effect and is linearized based on harmonic state space (HSS) theory to precisely characterize the internal harmonic features of MMC. A systematic HSS modeling process is presented for both open-loop and closed-loop conditions. And the proposed model is generalized as it can incorporate different control strategy by controller transfer function substitution. Hence it is valuable to analyze MMC stability and dynamics. Model effectiveness is verified by simulation results.