Equivalent Linear Model Research Articles

Proposal of an equivalent linearization method to predict seismic hysteretic energy demand considering stiffness degradation effects

SummaryTo overcome complexities and shortcomings of previous studies, a new method is proposed to derive an equivalent linear model for predicting seismic hysteretic energy demand of bilinear single degree of freedom (SDOF) models. A new displacement spectrum is defined, which represents hysteretic energy. It is found that by increasing initial period and damping of a nonlinear system in the correct proportion and defining a linear model with these characteristics, the new developed displacement can be achieved. Error minimization is applied through an algorithm to find the optimum equivalent period corresponding to an equivalent damping utilizing two sets of far‐field and near‐field earthquakes. To analyze the effects of stiffness degradation, the proposed algorithm has been implemented on modified Clough hysteretic model as well. Comparing the results, effects of stiffness degradation on the ratio of equivalent to initial period is evident in the short period range, while with increasing initial period, the effect can almost be neglected at higher values of ductility. Nonlinear regression analysis is carried out to provide the equations for predicting equivalent linear parameters as a function of ductility. Despite the previous predictive equations, the proposed model is independent of earthquake characteristics and response‐related parameters, which has increased efficiency as well as simplicity.

Numerical simulation of damage evolution of Daikai station during the 1995 Kobe earthquake

Abstract Three-dimensional (3D) finite element (FE) analyses are performed to identify the collapse mechanism of the Daikai station, which suffered major structural damage during the 1995 Kobe earthquake. To investigate the damage evolution in the reinforced concrete structure, concrete and steel rebars are modeled separately and assembled in a soil–tunnel model. Bilinear models are used for both the concrete and steel rebars, whereas a nonlinear hysteretic model is used for the soil. Simulation results illustrate that the 3D FE model is capable of accurately reproducing the structural collapse and surface settlement. The cracks are observed to form initially at the outer walls of the tunnels and subsequently in the center column. As the ground motion intensity increases, longitudinal and shear cracks form in the center column, inducing a significant loss of axial capacity and eventually leading to a structural collapse. Irrecoverable damage initiates in the center column under the combined shear force (V) and bending moment (M). Therefore, it is recommended to utilize the V-M failure surface to estimate the onset of structural damage. The damage evolution of the Daikai station is also correlated with the drift ratio of the center column. Additional analyses show that when the equivalent linear soil model is used, the calculated peak drift ratio is similar to that obtained using the nonlinear model. However, it results in unrealistic gapping at the tunnel–soil interface and lower surface settlement. The no-slip condition for the tunnel–soil contact interface produces a pronounced lower response of the tunnel and therefore should not be used in a damage analysis.

Jitter Minimization in Digital PLLs with Mid-Rise TDCs

This paper analyzes the absolute jitter performance of digital phase-locked loops and compares the case when either a multi-bit time-to-digital converter with mid-rise characteristic or a bang-bang phase detector is adopted. The linear equivalent model of the PLL and expressions for random-noise and limit-cycle jitter are first derived for the case of a 2-bit time-to-digital converter with a mid-rise characteristic, and the optimal TDC resolution is determined. The analysis, which account for TDC mismatches, shows that, compared to the 1-bit one, the 2-bit time-to-digital converter can substantially reduce the quantization noise in the case of dominant random-walk noise at the TDC input. Moving to the $N_{b}$ -bit midrise TDC case, the quantization noise can be further reduced at the cost of higher complexity and finer time resolution. The choice of $N_{b}=2$ seems to be the best compromise between jitter reduction and complexity increase. Time-domain simulations assess the theoretical framework and demonstrate the validity of the assumptions made throughout the paper.

Data driven NARMAX modeling for PEMFC air compressor

Air compressor of proton exchange membrane fuel cell (PEMFC) system is usually nonlinear and strong coupled. It is difficult to establish a online optimization oriented model. In order to solve this problem, this paper proposed a nonlinear autoregressive moving average with exogenous inputs (NARMAX) model for air compressor of PEMFC system. The NARMAX model is an equivalent time-varying linear model, and the time-varying parameters are identified by recurrent neural network (RNN). Simulation results show that the proposed method has small fitting error, the error of air flow and pressure ratio approximate zero, while the mean square error (MSE) of air flow and pressure ratio are 1.5171e-07 and 6.3767e-05, respectively. Therefore, the established air compressor model is accurate and effective.

Two Variable-Weather-Parameter Models and Linear Equivalent Models Expressed by Them for Photovoltaic Cell

For the conventional four-parameter model of photovoltaic (PV) cell, to simplify the number of its mathematical equations, in this paper, a variable-weather-parameter cell model (VWP model) is proposed. Furthermore, to decrease the number of its model parameters, a simplified variable-weather-parameter cell model (SVWP model) is also proposed. Based on them, the model parameters of linear equivalent models can be replaced by the conventional four-parameter model parameters, which finds the direct relationships between linear equivalent cell model and four-parameter cell model. Finally, some simulation experiments are done by comparing the proposed VWP and SVWP models with conventional four-parameter model of PV cell. The results show that, on the one hand, their output characteristics almost correspond with each other. On the other hand, these proposed cell models are very accurate under varying irradiance or temperature conditions. Meanwhile, some simulation experiments also verify that the linear equivalent models expressed by the VWP model and SVWP model are feasible, available and reasonable. By this work, both the conventional four-parameter model and linear equivalent models are greatly evolved and improved. On the one hand, the conventional four-parameter model can be greatly simplified, which makes its modeling and application easier. On the other hand, the bridge between it and linear equivalent models can be built successfully, which makes these linear models more standard and practical.

Site-specific ground response analysis at a site in the affected area of the 2016 Pidie Jaya earthquake

Analytical models have demonstrated that they are able to simulate reasonably well the seismic motions at the ground level. The most widely used model is the equivalent linear approach. This equivalent linear model was used to compute the free-field response of Meureudu-Pidie Jaya, Aceh Indonesia’s soft soils during the 2016 Pidie Jaya earthquake. The model computes the ground response of horizontally layered soil deposits subjected to transient and vertically propagating shear waves through the one-dimensional soil column. Each soil layer is assumed to be homogeneous, visco-elastic and infinite in the horizontal extent. The equivalent linear estimation of soil properties is taken to express the nonlinearity of the soil’s shear modulus and damping values. These values are assumed to be a function of shear strain amplitude and determined by an iterative process that must be consistent with the level of the effective strain induced in each sub-layer. Starting with the highest shear modulus and a low damping value, the shear modulus and the damping ratio of each sub-layer are modified. The modification is based on the applicable relationship between both properties and the shear strain. The calculation is repeated until strain-compatible modulus and damping values converge within a tolerance of 1%. This research reveals the ground motions of Pidie Jaya’s soils. The results of the analysis are presented.

Computing invariant sets of discrete-time nonlinear systems via state immersion

In this paper, we propose a method for computing invariant sets of discrete-time nonlinear systems by lifting the nonlinear dynamics into a higher dimensional linear model. In particular, we will focus on the maximal invariant set. Some special types of nonlinear systems can be considered as the projection of a higher dimensional linear system with a state immersion transformation. For such systems, the equivalence between invariant sets of the nonlinear system and its linear equivalent can be also established, which allows to characterize the maximal invariant set of the nonlinear system using a lifted linear model. For general nonlinear systems, we will use linear approximations because equivalent linear models cannot be achieved exactly. To handle mismatch errors, we tighten the constraint set of the lifted linear model, which will lead to an invariant inner approximation of the maximal invariant set.

Optimal Control for Discrete-Time, Linear Fractional-Order Systems with Markovian Jumps

This paper considers a finite-horizon linear quadratic (LQ) optimal control problem for a class of stochastic discrete-time, linear systems of fractional order which are generated by the operator involved in the definition of the fractional-order derivative of Grunwald-Letnikov type. This subject is new for discrete-time, linear, fractional-order systems (DTLFSs) with infinite Markovian jumps. We use an equivalent linear expanded-state model of the DTLFS with jumps and an equivalent quadratic cost functional to reduce the original optimal control problem to a similar one for discrete-time, linear, integer-order systems with Markovian jumps. The obtained optimal control problem is then solved by applying a dynamic programming technique.

Robust Optimization of a Distribution Network Location-Routing Problem Under Carbon Trading Policies

Taking carbon emissions into account in decision-making on distribution network operations contributes to achieving the goal of promoting energy conservation and emissions reduction. The focus of this paper is to research multicapacity hierarchical location-routing robust optimization in distribution network design under carbon trading policies. First, this problem is described as a mixed integer nonlinear programming model. Then, based on strong duality theory, the nonlinear model is transformed into a linear robust equivalent model. Finally, GUROBI software is used for numerical calculation and analysis. The results suggest the following: carbon trading policies have a carbon abatement effect; with a decrease in the carbon emissions cap and an increase in carbon trading prices, carbon emissions undergo a ladder-like downward trend; uncertain fluctuations in freight units will influence the optimal decision-making patterns of enterprises; and making more vehicles available will reduce carbon emissions. The government should set a reasonable carbon emissions cap according to market conditions. Enterprises could adopt robust control parameters on the basis of their decision-making preferences and consider the impact of carbon trading policy in formulating and adjusting an optimal decision-making scheme.

Oil volatility pass through and real exchange misalignment in leading commodity exporting countries

Past research has shown how real Exchange rates follow a univariate nonlinear process that approximates their behavior in terms of transaction costs. However, little or nothing has been said about alternative sources of nonlinearity in commodity exporting countries. Our paper investigates the missing link between the Real Exchange Rate Commodity Prices equilibrium by employing an oil price volatility measure as an external source of short-term fluctuations. Our estimates show that the Real Exchange Rate Commodity price relationship appears to be nonlinear with respect to oil price variation, and that the goodness of fit of the nonlinear specifications appears to outperform that of the equivalent linear models. The equilibrium speed of adjustment appears to be different in the two branches of the relationship: in the majority of the threshold models, the negative volatility regime presents a faster speed of adjustment and in some cases a most significant one.

Seismic Response Validation of Simulated Soil Models to Vertical Array Record During A Strong Earthquake

Several soil models, such as linear elastic, equivalent linear, and non-linear models, are employed in seismic ground response analysis. The aim of this study was to validate the seismic responses at ground surface of several soil models with the vertical array record of the Kobe earthquake. One-dimensional seismic response analyses were performed at Port Island using several soil models. The responses at ground surface from the simulated soil models were validated with the vertical array record of the Kobe earthquake. The results showed that the extended hyperbolic model yielded the most appropriate response according to the Kobe earthquake's recorded motion. This means that this model can be considered a suitable soil model to predict the response of strong earthquakes. In general, the results support the recommendation to select the most appropriate soil model for seismic ground response analysis.

Channel Magnitude-Based MIMO With Energy Detection for Internet of Things Applications

In this article, three low complexity multiple-input multiple-output (MIMO) receivers using energy detection and multilevel amplitude-shift keying (ASK) for Internet of Things (IoT) applications are described. To circumvent the inherent nonlinearity of the noncoherent MIMO system, an equivalent linear system model for the channel is introduced. This allows the proposed receivers to be configured analogously to coherent-based approaches. The tradeoff is that increased numbers of receive antennas need to be used but the overall complexity is decreased compared to existing noncoherent approaches. The receivers can also easily be integrated with energy harvesting systems. The three receiver structures considered are zero forcing (ZF), ZF with ordered successive interference cancellation (ZF-OSIC), and weighted envelope detection (WENVD), and their performance in flat Rayleigh fading channels with multilevel ASK is investigated. For the WENVD receiver, the error performance under the condition of high signal-to-noise ratio is also analyzed, and closed form approximations of the conditional (conditioned on the channel matrix) symbol vector error probability (SEP) and the average of the conditional SEP are derived. It is found that the ZF receiver has the lowest complexity and the WENVD receiver the highest; however, error performance comparison through numerical results reveals that the WENVD receiver has the best performance and the ZF-OSIC receiver has the next best. When combined with multiport energy harvesting technology the receivers have significant potential for use in IoT.

On Linear Analysis of the Power Flow Equations for DC and AC Grids With CPLs

This express brief presents an approximation of the power flow problem for alternating-current (ac) and directcurrent (dc) distribution networks by using a linear representation of the hyperbolic constraints i = p/v ↔ II* = ℤ/V related to the power balance at each constant power load node. Taylor's or Laurent's series expansion methods are not required to obtain an equivalent linear power flow model. The proposed linear method allows us to achieve a high quality approximation of the power flow modeling without iterative procedures. Our simulation results show the accurate estimation of the voltage profile in distribution networks by the proposed linear approach in comparison to existing methods in specialized literature for ac and dc networks, including linear estimators or classical numerical methods, such as Gauss-Seidel and Newton-Raphson approaches. Numerical implementation of those approaches is carried out in the MATLAB 2017a programming environment.

Proposed SMA Tension Sling Damper for Passive Seismic Control of Building

Passive Shape Memory Alloy (SMA) based Tension Sling Damper (TSD) is proposed to control seismic response of building represented as dynamic SDOF system. Non-linear hysteretic behavior of SMA is characterized by Tanaka model with seismic excitations as input. It is converted to equivalent piece-wise linear elastic viscous damping model described by stiffness constant ‘keff’ and damping constant ‘ceff’ to implement it with linear dynamic system. Damping constant ‘ceff’ is evaluated for TSD for given seismic excitation at each instance of time defined as instantaneous damping, unlike constant damping used in other studies, to derive accurate seismic response of the dynamic system. Seismic response control parameters – peak and rms perfor-mance indices (PI) for displacement and acceleration, peak damper force, maximum damping ratio are evalu-ated for flexible, moderately stiff and stiff dynamic systems fitted with TSD under pulse and strong ground motion type seismic excitations with varied PGA. Substantial reduction in PI is observed for controlled systems through varying design parameters of TSD – cross-sectional area and length of SMA sling. TSD yields better seismic response control for moderately stiff and flexible system due to super-elastic properties of SMA. Fur-ther, it works efficiently for seismic excitations with low to moderate PGA.

Analysis and Modeling of Conducted EMI from an AC–DC Power Supply in LED TV up to 1 MHz

Critical conduction mode (CRM) boost power factor correction (PFC) converter is widely used in ac–dc power supplies to achieve high power factor. The switching frequency varies in a half-line cycle. In this article, both the differential-mode (DM) and common-mode (CM) electromagnetic interference (EMI) noises below 1 MHz from the ac–dc power supply in a LED TV are analyzed and modeled. The power supply consists of two parts: CRM boost PFC converter and LLC resonant converter. The conducted EMI noise and noise source voltages are measured in the time domain and then converted to the frequency domain via short-time fast Fourier transform (STFFT). Through the joint time-frequency analysis using STFFT, the drain-to-source voltage of the power MOSFET in the PFC converter is identified as the dominant noise source of both CM and DM EMI noises below 1 MHz from the power supply. The EMI current path is different during different periods of a cycle of the line voltage. During most time of a cycle, two diodes of the bridge rectifier are forward biased, and the bridge rectifier can be treated as short circuit. The current paths of DM and CM EMI noises are explained and modeled by a linear equivalent circuit model. Three parasitic capacitances need to be considered to model the CM EMI noise in this device under test. From the circuit model, transfer functions relating DM and CM EMI spectra to noise source voltage spectrum are obtained. The conducted EMI spectra are predicted by multiplying the spectrum of noise source voltage by the transfer functions. The prediction matches with measurement. The effects of parasitic capacitances on CM EMI noise are analyzed by simulation and then validated by measurement. The analysis results can help with EMI design to reduce the CM EMI.

A New Prospect in Road Traffic Energy Harvesting Using Lead-Free Piezoceramics.

In this paper, a new prospect using lead-free piezoelectric ceramics is presented in order to determine their behavior in piezoelectric-based road traffic energy harvesting applications. This paper will describe the low-cost and fully programmable novel test bench developed. The test bench includes a traffic simulator and acquires the electrical signals of the piezoelectric materials and the energy harvested when stress is produced by analogous mechanical stimuli to road traffic effects. This new computer-controlled laboratory instrument is able to obtain the active electrical model of the piezoelectric materials and the generalized linear equivalent electrical model of the energy storage and harvesting circuits in an accurate and automatized empirical process. The models are originals and predict the extracted maximum power. The methodology presented allows the use of only two load resistor values to empirically verify the value of the output impedance of the harvester previously determined by simulations. This parameter is unknown a priori and is very relevant for optimizing the energy harvesting process based on maximum power point algorithms. The relative error achieved between the theoretical analysis by applying the models and the practical tests with real harvesting systems is under 3%. The environmental concerns are explored, highlighting the main differences between lead-containing (lead zirconate titanate, PZT) and lead-free commercial piezoelectric ceramics in road traffic energy harvesting applications.

The Role of Shear Wave Velocity and Non-Linearity of Soil in the Seismic Response of a Coupled Tunnel-Soil-Above Ground Building System

The presence of tunnels close to aboveground structures may modify the response of these structures, while the contrary is also true, the presence of aboveground structures may modify the dynamic response of tunnels. In this context, the dynamic properties of the soil through which the aboveground and underground structures are “connected” could play an important role. The paper reports dynamic FEM (Finite Element Method) analyses of a coupled tunnel-soil-above ground structure system (TSS system), which differ in regards to the soil shear wave velocity and in turns for the damping ratio, in order to investigate the role of these parameters in the full-coupled TSS system response. The analyses were performed using three different seismic inputs. Moreover, the soil non-linearity was taken into account adopting two different constitutive models: i) an equivalent linear visco-elastic model, characterized by degraded soil shear moduli and damping ratios, according to suggestions given by EC8 in 2003; and ii) a visco-elasto-plastic constitutive model, characterized by isotropic and kinematic hardening and a non-associated flow rule. The seismic response of the system was investigated in the time and frequency domains, in terms of: acceleration ratios; amplification ratios and response spectra; and bending moments in the tunnel.

Temporal Decomposition for Security-Constrained Unit Commitment

This paper proposes a temporal decomposition strategy to reduce the computation time of security-constrained unit commitment (SCUC). The novelty of this paper is twofold. The scheduling horizon is decomposed into multiple subhorizons. The concept of coupling intervals is introduced, and a set of auxiliary counting variables and logical expressions along with their equivalent linear models are formulated to handle intertemporal ramp constraints and minimum on/off times between consecutive subhorizons. An accelerated analytical target cascading (A-ATC) algorithm is developed to coordinate SCUC subproblems and find the optimal solution for the whole operation horizon in a distributed manner. An initialization strategy is presented to enhance the convergence performance of A-ATC. The proposed algorithm is tested on four test systems.

Dynamic characteristics of liquid sloshing in cylindrical tanks filled with porous media

The dynamic characteristics of liquid sloshing in cylindrical tanks filled with porous media are analyzed in this paper. As the tank is subjected to the small external excitations, the flow is assumed to be in Darcy’s flow regime, and then the porous medium can provide damping effect linearly proportional to the fluid velocity. Based on the pore flow theory, the linear equivalent mechanical model with the damping effect due to porous medium is established. This equivalent model consists of a rigidly fixed mass and infinite sets of mass-damping-spring systems, which can exert the same total dynamic force as the linearized actual fluid field. The equivalent system can explain the dynamic characteristics of each modal response of sloshing in a cylindrical porous-medium tank. Therefore it is more convenient to analyze the sloshing behavior from the structural dynamic analysis point of view. By this model, the natural frequency and damping ratio of the cylindrical porous-medium tank are revealed. Both parameters are related to the tank dimensions, porosity and permeability of the porous medium.

Reference Modulation for Performance Enhancement of Motion Control Systems With Nonlinear Parameter Variations

In general, it is not intuitive to quantify the performance of nonlinear systems; thus, designing controllers and/or estimators is not easy. Moreover, there are situations in which a system designed with a fixed feedback controller is not customizable, and yet its closed-loop performance (e.g., disturbance attenuation) is not satisfactory. In this study, we first present an equivalent linear model for motion control systems with nonlinear parameter variations. The linear model facilitates the quantitative analysis of the motion control system. Furthermore, we introduce a reference modulation technique that enhances the performance by boosting loop-gain over a desired range of frequencies. The proposed approach, by introducing an additional modulated function and a direct feedforward channel, facilitates the separation of the closed-loop sensitivity function from other issues of the closed-loop system design so that both disturbance attenuation and tracking performance can be improved. We also provide robust stability analysis of the motion control systems by reference modulation. The simulations and experiments are performed on a permanent magnet synchronous motor system with a predesigned feedback controller. The simulation and experimental results show that both the tracking and disturbance attenuation performance of the closed-loop system are enhanced via reference modulation.