Bilinear Theory Research Articles

Bilinear Quadratic Feedback Control of Modular Multilevel Converters

The present paper introduces the formulation and development of a bilinear quadratic control algorithm for Modular Multilevel Converter (MMCs), with a specific emphasis on achieving internal energy stabilization and balance within the converter. A bilinear average model of the MMC is employed, enabling the separation between the DC voltage and the voltage generated by submodules. The algorithm proposed in this study is formulated using bilinear theory and is founded on quadratic feedback control principles. The stability of the suggested controller is scrutinized using a meticulous mathematical approach based on Lyapunov theory. Subsequently, the theoretical findings are assessed using a comprehensive MMC switching model implemented in Matlab Simscape Electrical. The utilization of a phase-shift PWM technique, accompanied by a sorting algorithm, is considered in the study. Additionally, a comparison between the proposed bilinear controller and a standard PI controller is conducted. The outcomes demonstrate that the proposed controller effectively facilitates the regulation of circulating and AC currents, along with managing the internal energy of MMCs. Consequently, this achievement makes a noteworthy contribution to the field, as it introduces an innovative bilinear control approach capable of stabilizing all the state variables of MMCs converters using a single control law.

Novel distributed state estimation method for the AC‐DC hybrid microgrid based on the Lagrangian relaxation method

The AC-DC hybrid microgrid is a credible evolution path for the microgrid. State estimation in complex distribution network is a significant foundation for the safe operation. In AC-DC hybrid microgrid, centralised state estimation algorithms have quite a few deficiencies such as large communication capacity and privacy protection. In response to these obstacles, this paper proposed a novel distributed state estimation algorithm. First, a three-stage state estimation model was established for AC-DC hybrid microgrid, in which the second stage is a nonlinear transform for the linearisation of the nonlinear state estimation progress so that the first and third stage can solve the linear state estimation problem. Furthermore, the Lagrangian relaxation method was ultilised for the distributed problem solution. In the proposed algorithm, AC subsystems and DC subsystems respectively execute state estimation through local information, and the overall consistency of the system state estimation can be eventually achieved according to the transfer iteration of the boundary information. The computational efficiency and convergence of the proposed method was ensured owing to the combination of bilinear theory and Lagrangian relaxation. Simulation results indicate that the proposed method can achieve its goal and be superior to existing method in efficiency.

The state of stress in SW Iran and implications for hydraulic fracturing of a naturally fractured carbonate reservoir

The first part of this study provides an interpretation of the present-day stress state and mechanical parameters for the giant Bangestan reservoir in SW Iran. A total of 28 drilling induced tensile fractures (DITF) and 172 breakouts were mapped in seven wells in the Bangestan reservoir using image logs. The average maximum principal horizontal stress (SH) direction for each well is given a quality ranking according to the World Stress Map (WSM) criteria. The mean orientation of SH finds to be N44°(±9.95°). A correlation is seen between this data and stress orientations from earthquake focal mechanism solution. This suggests that stresses are related to tectonic forces generated by Arabia–Eurasia collision. Estimates of vertical principal stress (Sv) gradients vary between 23.3 and 24.7 kPa/m (1.03–1.09 psi/ft), based on empirical methods and density log data, respectively. Minimum horizontal stress (Sh) gradient estimates vary from 15.2 to 17.4 kPa/m (0.67–0.77 psi/ft) using extended leak-off test (XLOT) and step-rate test results, respectively. Uncertainties remain in estimating SH from acid fracturing in the deep naturally fractured carbonate reservoirs, even though numerous papers have been written on the subject. Herein, an approach called Deformation/Diffusion/Thermal (DDT) is suggested to estimate SH in the Bangestan Reservoir. The method combines poroelastic theories for the circular underground cavities, bilinear theory for fluid flow through the preexisting induced fracture and thermal stress. Using this method, SH gradient is calculated to be 18.9 kPa/m (0.83 psi/ft). For comparison, horizontal stress magnitudes and mechanical parameters were computed from sonic wave velocity using poroelastic-tectonic stress equations over the full reservoir layer thickness. All data have been calibrated with field tests and lab results. The measured SH was in good agreement with the previous analytical solution. The study suggests that the contemporary state of stress in the reservoir is characterized by a normal faulting (NF) stress regime. Next, current and prospective hydraulic fracturing operations in the field are discussed. Based on the NF stress regime, a vertical plane fracture is expected and the wells drilled in the direction of SH are likely to intersect the most natural fractures.

Three-Stage Distributed State Estimation for AC-DC Hybrid Distribution Network Under Mixed Measurement Environment

The ac–dc hybrid distribution network is a credible path for the future evolution of distribution network. State estimation is a paramount foundation for the safe and stable operation of the complex distribution network. The application of the centralized state estimation method in an ac–dc distribution network has some obstacles, such as low computational efficiency, large communication capacity, and privacy protection problem. Based on the three-stage state estimation theory, this paper established a three-stage state estimation model for the ac–dc hybrid distribution network integrating supervisory control and data acquisition system and phasor measurement unit. The proposed three-stage estimation model achieves linearization of the nonlinear state estimation for the ac–dc hybrid distribution network. That is, the first and third stages simply solve the linear state estimation problem, and the second stage is a one-step nonlinear transform. Moreover, the alternating direction method of multipliers (ADMM) is applied to solving the distributed problem. Thus, an accurate and efficient three-stage distributed state estimation method for the ac–dc hybrid distribution network is proposed. In this method, ac subsystems and dc subsystems execute state estimation tasks based on local information and eventually achieve the overall consistency of the system state estimation through the transfer iteration of the boundary information, respectively. The combination of bilinear theory and ADMM ensures the convergence of distributed state estimation while improving computational efficiency. The simulation results verified the advantage of the proposal over the existing methods in many aspects.

Abstract Hardy spaces with variable exponents

In this paper, we introduce the definition of abstract Hardy spaces with variable exponents via the atomic decomposition and molecular decomposition. We prove the continuity from our variable Hardy space Hatop(⋅) into Lp(⋅), where p(⋅):Rn→(0,1],0<p−⩽p+⩽1. Moreover, we investigate the bilinear theory. Finally, we give an application of the abstract Hardy spaces with variable exponents.

Solitary waves, homoclinic breather waves and rogue waves of the [formula omitted]-dimensional Hirota bilinear equation

In this paper, the (3+1)-dimensional Hirota bilinear equation is investigated, which can be used to describe the nonlinear dynamic behavior in physics. By using the Bell polynomials, the bilinear form of the equation is derived in a very natural way. Based on the resulting bilinear form, its N-solitary waves are further obtained by using the Hirota’s bilinear theory. Finally, by using the Homoclinic test method, we obtain its rational breather wave and rogue wave solutions, respectively. In order to better understand the dynamical behaviors of the equation, some graphical analyses are discussed for these exact solutions.

Swimming of a linear chain with a cargo in an incompressible viscous fluid with inertia

An approximation to the added mass matrix of an assembly of spheres is constructed on the basis of potential flow theory for situations where one sphere is much larger than the others. In the approximation, the flow potential near a small sphere is assumed to be dipolar, but near the large sphere it involves all higher order multipoles. The analysis is based on an exact result for the potential of a magnetic dipole in the presence of a superconducting sphere. Subsequently, the approximate added mass hydrodynamic interactions are used in a calculation of the swimming velocity and rate of dissipation of linear chain structures consisting of a number of small spheres and a single large one, with account also of frictional hydrodynamic interactions. The results derived for periodic swimming on the basis of a kinematic approach are compared with the bilinear theory, valid for small amplitude of stroke, and with the numerical solution of the approximate equations of motion. The calculations interpolate over the whole range of scale number between the friction-dominated Stokes limit and the inertia-dominated regime.

Stokesian spherical swimmers and active particles.

The net steady state flow pattern of a distorting sphere is studied in the framework of the bilinear theory of swimming at low Reynolds number. It is argued that the starting point of a theory of interacting active particles should be based on such a calculation, since any arbitrarily chosen steady state flow pattern is not necessarily the result of a swimming motion. Furthermore, it is stressed that as a rule the phase of stroke is relevant in hydrodynamic interactions, so that the net flow pattern must be used with caution.

Fatigue Life Prediction Based on Strengthening and Damaging

Based on moving S-N curve, a new method of fatigue life prediction for vehicle component in random loading spectra was put forward. In the new method, both strengthening and damaging of loads are fully considered in fatigue life prediction of vehicle component. The results of fatigue life prediction showed that the new method was more exacter than that of other life prediction models, such as Miner's rule, modified Miner's rule, Manson's bilinear theory and Corten-Dolan's theory. Only this new method could exactly predict the fatigue life of vehicle component in original or slightly strengthened loading spectra.

The swimming of animalcules

Animalcules can swim in a viscous fluid at low Reynolds number and low Stokes number by moving their body parts in a periodic coherent fashion. The swimming motion is analyzed in a simple model of beads subject to periodic one-body forces. In the model the animalcule is held together by reactive two-body forces. The nonlinear equations of Stokesian dynamics are formulated on the basis of the Oseen tensor. Under suitable conditions the solution of the equations of motion has a limit cycle character. The limit cycle is analyzed for small amplitude motion in the framework of a bilinear theory. The linearized equations of motion are solved analytically for longitudinal and transverse modes of motion for a linear trimer, and expressions are derived for the swimming velocity and the mean dissipation to second order in the force amplitude. The results of the bilinear theory are compared to numerical solution of the nonlinear equations of motion. A similar comparison is made for chains of twelve beads.

A generalized algorithm for the study of bilinear vibrations of cracked structures

Structural cracks may cause variations in structural stiffness and thus produce bilinear vibrations to structures. This study examines the dynamic behavior of structures with breathing cracks. A generalized algorithm based on the finite element method and bilinear theory was developed to study the influence of a breathing crack on the vibration characteristic. All the formulae derived in the time domain were applied to estimate the period of the overall bilinear motion cycle, and the contact effect was considered in the calculations by introducing the penetration of the crack surface. Changes in the dynamic characteristics of cracked structures are investigated by assessing the variation of natural frequencies under different crack status in either the open or closed modes. Results in estimation with vibrational behavior variation are significant compared with the experimental results available in the literature as well as other numerical calculations.

Effective bending modulus of carbon nanotubes with rippling deformation

This paper concerned with the relation of the bending moment to the bending curvature during bending of carbon nanotubes, and the relation between the rippling formation and the bending modulus. Based on the three-dimensional orthotropic theory of finite elasticity deformation, a non-linear bending moment–curvature relationship of carbon nanotubes which is the appearance of wavelike distortion on the inner arc of the bent nanotubes is simulated by using an advanced finite element analysis package, ABAQUS. Utilizing the non-linear bending moment–curvature relationship, the effective bending modulus of carbon nanotubes with different cross-sections are obtained by means of a bi-linear theory and a simplified vibration analysis method. The effective bending modulus of carbon nanotubes simulated in the paper is close to the measuring result presented in reference [Science 283 (1999) 1513].

Nonlinear waves in thermo-magnetoelasticity I. Basic equations

A complete set of field equations, constitutive relations and boundary conditions is derived in material form, within the framework of rigorous thermodynamics, to study the nonlinear thermo-magnetoelastic interactions in anisotropic continuous media for which the specific heat capacity depends on strain and magnetic field. The proposed model includes several other couplings as well and may be considered as a refinement of other existing models. When the medium has finite electric conductivity, it is appropriate to represent the electromagnetic field in terms of the electromagnetic potentials. This has the advantage of reducing the number of field equations. The special case of the perfect electric conductor is briefly mentioned and the linear theory for the isotropic medium of finite electric conductivity is investigated. Comparison is carried out with previous results. The bilinear theory and the propagation of surface regimes and surface magnetoacoustic waves will be dealt with separately.

Bilinear Theories in Plasticity and an Application to Two-Dimensional Wave Propagation

The Koehler and Seitz bilinear theory is generalized and related to a similar theory given by Swainger. It is shown that, in contradistinction to the corresponding Hencky theory, the generalized theory depends partially on the strain path, and further leads to linearization of the governing equations. An alternative form analogous to the generalized Hooke’s law is given. Displacement equations of motion for the bilinear model are derived and explicit expressions for plastic wave velocities obtained. Dynamic equations for cases previously considered in the literature are compared. As an application, the initial stress discontinuities for the problem of scattering of a plane compressional step wave by a rigid, perfectly dense cylinder are obtained.

Functionals of Bounded Frechet Variation

In a series of papers which will follow this paper the authors will present a theory of functionals which are bilinear over a product A × B of two normed vector spaces A and B. This theory will include a representation theory, a variational theory, and a spectral theory. The associated characteristic equations will include as special cases the Jacobi equations of the classical variational theory when n = 1, and self-adjoint integrodifferential equations of very general type. The bilinear theory is oriented by the needs of non-linear and non-bilinear analysis in the large.