State-space Averaging Research Articles

A Building-Block Approach to State-Space Modeling of DC-DC Converter Systems

Small-signal models of DC-DC converters are often based on a state-space averaging approach, from which both control-oriented and other frequency-domain characteristics, such as input or output impedance, can be derived. Updating these models when extending the converter by filters or non-trivial loads, or adding control loops, can become a tedious task, however. To simplify this potentially error-prone process, a modular modeling approach is being proposed in this article. It consists of small state-space models for certain building blocks of a converter system on the one hand, and standardized operations for connecting these subsystem models to an overall converter system model on the other hand. The resulting state-space system model builds upon a two-port converter description and allows the extraction of control-oriented and impedance characteristics at any modeling stage, be it open loop or closed loop, single converter or series connections of converters. The ease of creating more complex models enabled by the proposed approach is also demonstrated with examples comprising multiple control loops or cascaded converters.

Stability Analysis of Grid-Connected Photovoltaic Systems with Dynamic Phasor Model

The typical layout of power systems is experiencing significant change, due to the high penetration of renewable energy sources (RESs). The ongoing evaluation of power systems is expecting more detailed and accurate mathematical modeling approaches for RESs which are dominated by power electronics. Although modeling techniques based on state–space averaging (SSA) have traditionally been used to mathematically represent the dynamics of power systems, the performance of such a model-based system degrades under high switching frequency. The multi-frequency averaging (MFA)-based higher-index dynamic phasor modeling tool is proposed in this paper, which is entirely new and can provide better estimations of dynamics. Dynamic stability analysis is presented in this paper for the MFA-based higher-index dynamical model of single-stage single-phase (SSSP) grid-connected photovoltaic (PV) systems under different switching frequencies.

The Analysis of Energy Transition Processes in Boost Converter

Introduction. This article presents the analysis of energy transition processes in boost converter under the conditions of load changing. The boost converter laboratory layout is investigated with forms of the transient process caused by load changing. The transient currents and voltages are measured in harmonic and anharmonic modes. The parameters of boost converter transient modes are estimated. The analysis of energy transition processes is performed using two boost converter models. Problem statement. The boost converter circuits are used in the areas of electronics where the minimization of energy losses is strictly required. The boost converter energy losses depend on the form of transient processes caused by load changing. The purpose of energy losses minimization requires exact analysis of energy transition processes which appear under load changing. The presented work describes the responses of boost converter circuit to changing the load twice with four different transient modes. Results. The analysis of energy transition processes is performed with two models. The first model is based on the iterative mapping technique, where the currents and voltages of each period of gate driving signal are determined from the circuit parameters and initial conditions given by the previous period of driving signal. Such model allows to perform an exact analysis of current and voltage ripple transient processes. But the iterative mapping model is characterized by numerical approximation errors and requires more computation time. The second model describes the envelopes of boost converter current and voltage transient processes analytically. This model is based on the state-space averaging method which is widely used for modelling of switching circuits. Such model does not take into account the waveforms of current and voltage ripples, but it provides a more simple description of transient process envelopes which are useful for the circuit design purposes. The modeling results obtained from iterative mapping and state space averaging, match with the experimental data. Conclusions. The performed analysis shows that the energy losses depend on the transient process mode significantly. During the load transient process time, the smallest energy losses can be obtained under the critical load transient process form which is situated between periodic and aperiodic modes. Such result is obtained experimentally and confirmed by the both iterative mapping and state-space averaging models. The analytical results are confirmed by optimization procedure in MATLAB environment.

Large-signal modeling of three-phase dual active bridge converters for electromagnetic transient analysis in DC grids

The three-phase dual active bridge (3p-DAB) converter is widely considered in next-generation DC grid applications. As for traditional AC grids, the successful integration of power electronic converters in DC grids requires accurate time-domain system-level studies. As demonstrated in the existing literature, the development and efficient implementation of large-signal models of 3p-DAB converters are not trivial. In this paper, a generalized average model is developed, which enables system-level simulation of DC grids with 3p-DAB converters in electromagnetic transient type (EMT-type) programs. The proposed model is rigorously compared with alternative modeling techniques: ideal-model, switching-function and state-space averaging. It is concluded that the generalized average model provides an optimal solution when accuracy of transient response, reduction in computation time, and wideband response factors are considered.

State-space averaged modeling and transfer function derivation of DC-DC boost converter for high-brightness led lighting applications

This paper presents dynamic analysis of a boost type DC-DC converter for high-brightness LED (HBLED) driving applications. The steady state operation in presence of all system parasitics has been discussed for continuous conduction mode (CCM). The state-space averaging, energy conservation principle and standard linearization are used to derive ac small signal control to inductor current open-loop transfer function of the converter. The derived transfer function can be further used in designing a robust feed-back control network for the system. In the end frequency and transient responses of the derived transfer function are obtained for a given set of component values, hence to provide a useful guide for control design engineers.

The Analysis of Power supply Characteristics of Aircraft Power supply System

With the rapid development of multi-electric aircraft technology, the aircraft electrical power system is becoming more and more complex and significant. This paper analyzes power supply characterisitics of aircraft power supply system. First, the Generalized State Space Averaging (GSSA) technique is adopted to realize the digital simulation of complex aircraft electrical system. Next, AC vector coordinate transformation, digital zero-crossing detection and least squares polynomial fitting are used to analyze power supply characteristics of aircraft power supply system. Finally, the power quality of the power supply system is analyzed in the time domain and the trend analysis diagram of these power quality parameters with the variation of time is given.

Nonlinear Voltage Regulation Algorithm for DC-DC Boost Converter with Finite-Time Convergence

For the DC-DC Boost converter system, this paper employs the finite-time control technique to design a new nonlinear fast voltage regulation control algorithm. Compared with the existing algorithm, the main advantage of the proposed algorithm lies in the fact that it can offer a fast convergent rate, i.e., finite-time convergence. Based on the average state space model of Boost converter system and finite-time control theory, rigorous stability analysis showed that the output voltage converges to the reference voltage in a finite time. Simulation results demonstrate the efficiency of the proposed method. Compared with PI control algorithm, it is shown that the proposed algorithm has a faster regulation performance and stronger robust performance on load-variation.

Mathematical modelling and analysis of voltage super-lift power dc-dc converter for enhanced dynamic characteristics in CCM

PurposeThe purpose of this paper is to propose a mathematical model for voltage super-lift dc-dc power converter in continuous conduction mode (CCM). Using the presented mathematical model, the analysis of dynamics of power stage for voltage super-lift dc-dc power converter can be performed.Design/methodology/approachThe proposed method is based on the average state space model using the state equations of the dc-dc power converter. In the proposed method, the converter is represented as a set of differential equations derived for each switching state of the power switch in terms of inductor current and capacitor voltage. The proposed method describes the dynamic behaviour of the system. The controller is designed to meet performance requirement of the system such as to maintain the dynamics such as stability, steady-state accuracy and the speed of response of the system. Using the obtained model, the analysis of dynamic response of the voltage super-lift dc-dc power converter can be performed.FindingsThe converter is modelled and verified using conventional circuit analysis method employing state-space averaging technique, and their corresponding transfer function is also derived. The dynamics of the converter is investigated using frequency response characteristics obtained using MATLAB programming environment. In addition, to improve the stability of the converter, proportional-integral controller is designed using Ziegler–Nichols tuning rules, and the effect of the compensator in the plant is also investigated.Originality/valueThe proposed method can be used for analysing the dynamics of power stage for voltage super-lift DC-DC power converter.

DC transformer modeling and control of DC-DC buck converter

Switching Power Converters convert one form of power to another with high ef-ciency and accurate control.One of the most widely used DC-DC Converter is Buck Converter. Control is invariably required to maintain the output voltage/current in spite of variations in source/load. In order to design the controller and gain insight about the system a dynamic model needs to be developed. Modeling techniques widely used are state space averaging and PWM switch model. In this paper DC transformer modeling technique is used to develop the averaged model of the converter. One of the advantages of this model is that it can be implemented in Spice simulator using basic circuit elements. The same model can be used for time domain as well as frequency domain analysis. Analog type-II compensator is designed to compensate the system. Simulation and experimental results for start-up transient and load transient are shown to validate the model.

Advanced small‐signal‐based analytical approach to modelling high‐order power converters

This study devises an advanced small-signal based approach (SSA) to model high-order power converter system (HOPCS); in detail, a 3-Z-network converter which employs a single switch to achieve high voltage gain is utilised as a typical example of HOPCS. Therein, four operating modes are detailed in terms of the states of electric current flowing through the inductors. The proposed SSA is performed by means of the state-space averaging technique and small-signal equivalent circuits in frequency domain. Based on the proposed approach, the derived transfer functions for input voltage, output voltage and control input yield deep insight into the dynamic responses of the converter in each working mode. In order to well validate the effectiveness of the proposed approach, numerical results and experimental prototype are conducted and measured, the results agree well with the theoretical analyses.

Electromagnetic Transient Modeling and Simulation of Power Converters Based on a Piecewise Generalized State Space Averaging Method

Large-scale photovoltaic (PV) power plants connected to the grid leads to the complex calculations in modeling and simulation of the power system, especially in the electromagnetic transient progress. Common averaging methods are studied for modeling a grid-connected converter to simplify the calculation and balance accuracy and efficiency in the simulation. The piecewise technique is applied to the generalized state space averaging model of converters, which combine time segment with similar operating characteristic. In addition, a new piecewise generalized state space averaging (P-GSSA) model is derived and a multi time scale modeling is achieved for the grid-connected converters in PV systems. The model error and accuracy of the P-GSSA model are analyzed, the physical significance and the influence factors in the model error of P-GSSA model are discussed. The proposed P-GSSA model is flexible to be used in a PVs system simulation, and it can accurately reflect the steady-state and transient characteristics of a converter. The research shows that P-GSSA model improved the simulation efficiency over 20% by the index of consumed time, meanwhile, the model error analysis is under 5%. The analysis of the P-GSSA model is verified by simulating a DC/AC converter and a photovoltaic system.

A Non-Isolated Hybrid-Modular DC-DC Converter for DC Grids: Small-Signal Modeling and Control

This paper presents small-signal modeling, stability analysis, and controller design of a non-isolated bidirectional hybrid-modular DC-DC Converter for DC grid applications. The DC-DC converter can be used to interconnect two different DC voltage levels in a medium-/high-voltage DC grid. Half-bridge Sub-Modules (SMs) and a high-voltage valve are the main components of the converter. The high-voltage valve can be implemented via employing series-connected Insulated-Gate Bipolar Transistors (IGBTs). Operation with zero voltage switching of the involved high-voltage valve is feasible, i.e., there is no concern pertinent to dynamic voltage sharing among the series-connected IGBTs. The power is transferred from one side to another through the involved SMs, where their capacitors are connected in series across the high-voltage side, while they are connected sequentially across the low-voltage side. In this paper, the state-space averaging technique is employed to derive the small-signal model of the presented converter for controller design. Closed-form expression of the duty cycle-to-inductor current transfer function is extracted. Comparison between simulation results of the small-signal model and the detailed circuit model is presented to authenticate the accuracy of the derived small-signal model. Finally, a scaled-down prototype is used to verify the accuracy of the small-signal model.

Robust $\mu$ -Synthesis Control of Dual LCL Type IPT System Considering Load and Mutual Inductance Uncertainty

The output voltage of a dual LCL inductive power transfer (IPT) system varies with the load and mutual inductance, which can affect the system performance, e.g., the charging speed of electrical equipment. This paper proposes a robust controller to maintain constant output voltage against the load and mutual inductance variations. A μ-synthesis method is proposed based on structured singular value to design the dual LCL IPT closed-loop system. The frequency-domain nominal model of the system is established by using the generalized state-space averaging (GSSA) method. A standard M-Δ configuration is generalized through separating the nominal and uncertain block by upper linear fractional transformation (LFT). The μ-controller is finally obtained by D-K iteration. The robust performance analysis and time-domain performance analysis are also carried out. The simulation and experimental results verified the effectiveness of the proposed modeling and controller design, and demonstrated the robust stability and robust performance of the dual LCL IPT system.

Electromagnetic transient modelling and simulation of power converters based on a piecewise generalised state–space averaging method

Common averaging methods are studied for modelling a grid-connected converter in renewable energy domain to balance accuracy and efficiency of electromagnetic transient simulation. A piecewise generalised state–space averaging (P-GSSA) method is proposed for converters with large-scale renewable energy connected to the grid. The piecewise technique is applied to the GSSA model of converters, which combine time segment with similar operating characteristics. In addition, multi-time scale modelling is successfully achieved for the grid-connected converters in the renewable energy domain. An example is simulated according to the P-GSSA model proposed in this paper, and simulation results show that the proposed model can provide an efficient simulation of power converters connected to the grid in the domain of large-scale renewable energy.

Review and Selection Strategy for High-Accuracy Modeling of PWM Converters in DCM

Among various modeling methods for DC-DC converters introduced in the past two decades, the state-space averaging (SSA) and the circuit averaging (CA) are the most general and popular exhibiting high accuracy. However, their deduction approaches are not entirely equivalent since they incorporate different averaging processes, thus yielding different small signal transfer functions even under identical operating conditions. Some research studies claimed that the improved SSA can obtain the highest accuracy among all the modeling methods, but this paper discovers and clearly verifies that this is not the case. In this paper, we first review and study these two modeling methods for various DC-DC converters operating in the discontinuous conduction mode (DCM). We also streamline the general model-deriving processes for DC-DC converters, and test and compare the accuracy of these two methods under various conditions. Finally, we provide a selection strategy for a high-accuracy modeling method for different DC-DC converters operating in DCM and verified by simulations, which revealed necessary and beneficial for designing a more accurate DCM closed-loop controller for DC-DC converters, thus achieving better stability and transient response.

Day-to-Day Reliability of Nonlinear Methods to Assess Walking Dynamics

The present study investigated the day-to-day reliability (quantified by the absolute and relative reliability) of nonlinear methods used to assess human locomotion dynamics. Twenty-four participants of whom twelve were diagnosed with knee osteoarthritis completed 5 min of treadmill walking at self-selected preferred speed on two separate days. Lower limb kinematics were recorded at 100 Hz and hip, knee, and ankle joint angles, three-dimensional (3D) sacrum marker displacement and stride time intervals were extracted for 170 consecutive strides. The largest Lyapunov exponent and correlation dimension were calculated for the joint angle and sacrum displacement data using three different state space reconstruction methods (group average, test-retest average, individual time delay and embedding dimension). Sample entropy and detrended fluctuation analysis (DFA) were applied to the stride time interval time series. Relative reliability was assessed using intraclass correlation coefficients and absolute reliability was determined using measurement error (ME). For both joint angles and sacrum displacement, there was a general pattern that the group average state space reconstruction method provided the highest relative reliability and lowest ME compared to the individual and test-retest average methods. The DFA exhibited good reliability, while the sample entropy showed poor reliability. The results comprise a reference material that can inspire and guide future studies of nonlinear gait dynamics.

Influence factors analysis of rail potential in urban rail transit

Rail potential is affected by a variety of influence factors in the urban rail transit. This paper proposes a state space average (SSA) model of the traction power system to explore the effects caused by the traction network resistance, inductance, capacitance and rail-to-ground resistance. The SSA model solutions prove that the traction network inductance and capacitance have significant influences on rail potentials. Moreover, based on the SSA model, the rail potential distribution of a metro line during the train timetable period is obtained for rail protections, which is conductive to improve the reliability of the traction power system.

Robust Control for Inductively Coupled Power Transfer Systems With Coil Misalignment

Coil misalignment in inductively coupled power transfer (ICPT) systems will lead to fluctuation of transferred power, and even instability of the ICPT systems. To obtain robust control for ICPT system with coil misalignment, a μ-controller based on structured singular value (SSV) is used to set up a closed-loop system in this paper. A generalized state-space averaging (GSSA) model is established using GSSA method first. Then a standard M-Δ configuration is obtained by a linear fractional transformation of the nominal and uncertainty parts of the GSSA model. Finally, a μ-controller is designed by D-K iteration. Besides, the theoretical results are validated by computer simulations and practical experiments.

Hybrid Average Modeling of Three-Phase Dual Active Bridge Converters for Stability Analysis

The three-phase dual active bridge (3p-DAB) converter is widely addressed in emerging power systems applications such as solid-state transformer and dc microgrids. Its successful integration requires accurate modeling of its small-signal characteristics. Due to its dc–ac–dc structure, the DAB converter brings many challenges to small-signal modeling. The state-space averaging (SSA) has been the first proposed methodology to approximate the control-to-output and line-to-output transfer functions of the 3p-DAB. However, as shown in this paper, SSA is not precise for the stability analysis of 3p-DAB converters. A generalized state-space averaging (GSSA) model based on the dynamic phasor concept is developed in this paper for the Y-Δ 3p-DAB. A hybrid SSA and GSSA model representation is then proposed for the evaluation of all the converter transfer functions. The developed models are validated with detailed time-domain switch-level simulations in an electromagnetic transient-type (EMT-type) program. They are also used for the accelerated stability prediction in an EMT-type program.

Stability Analysis of Digital-Controlled Single-Phase Inverter With Synchronous Reference Frame Voltage Control

Stability analysis of single-phase power converters controlled in stationary reference frame is now mature and well developed, by using either linear or nonlinear methods. However, for the single-phase converters with synchronous reference frame (SRF) control loops, little work has been done on the evaluation of the nonlinear approaches for stability analysis. In this paper, the stability of a digital-controlled single-phase voltage-source inverter (VSI) with SRF voltage control loop is investigated from the perspective of nonlinear system. The analysis is based on the discrete-time model defined by the stroboscopic map, which is derived using the state-space averaging (SSA) technique. Furthermore, two different nonlinear analysis methods, the Jacobian matrix method and the Lyapunov exponent method, are adopted to analyze the fast-scale stability and the slow-scale stability of the pulsewidth-modulated (PWM) inverter under variations of control parameters; hence, the stability regions can be obtained. The theoretical results indicate that, for the established stroboscopic models, the Jacobian matrix method and the Lyapunov exponent method are mathematically equivalent, which means that the fast-scale stability and the slow-scale stability of the studied single-phase VSI are consistent, especially under linear load conditions. Experimental results under resistive load, inductive–resistive load, and diode rectifier load conditions are presented to support the theoretical results, which also proves that the discrete-time model plus the Jacobian matrix method or the Lyapunov exponent method is capable to investigate the stability of a converter with SRF control loops accurately.