Limit Of Inertia Research Articles

Inertia and Governor Ramp Rate Constrained Economic Dispatch to Assess Primary Frequency Response Adequacy

The contribution of non-dispatchable, renewable generation to both total grid inertia and grid frequency response is significantly lesser than that of dispatchable, thermal generation. In large non-dispatchable generation penetration scenarios, the effect of inertia and frequency response diminution on primary frequency control reserves adequacy will be important. This paper develops a constraint on the economic dispatch problem that assures that frequency nadirs after a given sudden loss of generations will not be lower than a given limit. This constraint considers system inertia and governor ramp rate limits. A simulation using the IEEE Reliability Test System is presented.

The influence of an outer bath on the dewetting of an ultrathin liquid film

We report a theoretical and numerical investigation of the linear and nonlinear dynamics of a thin liquid film of viscosity μ sandwiched between a solid substrate and an unbounded liquid bath of viscosity λμ. In the limit of negligible inertia, the flow depends on two non-dimensional parameters, namely, λ and a dimensionless measure of the relative strengths of the stabilizing surface tension force and the destabilizing van der Waals force between the substrate and the film. We first analyze the linear stability of the film, providing an analytical dispersion relation. When the viscosity of the outer bath is much larger than that of the film, λ≫1, the most amplified wavenumber decreases as km∼λ−1/3, indicating that very slender dewetting structures are expected when λ becomes large. We then perform fully nonlinear simulations of the complete Stokes equations to investigate the spatial structure of the flow close to rupture revealing that the flow becomes self-similar with the minimum film thickness scaling as hmin=K(λ)τ1/3 when τ→0, where τ is the time remaining before the singularity. It is demonstrated that the presence of an outer liquid bath affects the self-similar structure obtained by Moreno-Boza et al. [“Stokes theory of thin-film rupture,” Phys. Rev. Fluids 5, 014002 (2020)] through the prefactor of the film thinning law, K(λ), and the opening angle of the self-similar film shape, which is shown to decrease with λ.

Collective behavior of soft self-propelled disks with rotational inertia

We investigate collective properties of a large system of soft self-propelled inertial disks with active Langevin dynamics simulation in two dimensions. Rotational inertia of the disks is found to favor motility induced phase separation (MIPS), due to increased effective persistence of the disks. The MIPS phase diagram in the parameter space of rotational inertia and disk softness is reported over a range of values of translation inertia and self-propulsion strength of the disks. Our analytical prediction of the phase boundary between the homogeneous (no-MIPS) and MIPS state in the limit of small and large rotational inertia is found to agree with the numerical data over a large range of translational inertia. Shape of the high density MIPS phase is found to change from circular to rectangular one as the system moves away from the phase boundary. Structural and dynamical properties of the system, measured by several physical quantities, are found to be invariant in the central region of the high density MIPS phase, whereas they are found to vary gradually near the peripheral region of the high density phase. Importantly, the width of the peripheral region near the phase boundary is much larger compared to the narrow peripheral region far away from the phase boundary. Rich dynamics of the disks inside the high density MIPS phase is addressed. Spatial correlation of velocity of the disks is found to increase with rotational inertia and disk hardness. However, temporal correlation of the disks’ velocity is found to be a function of rotational inertia, while it is independent of disk softness.

Reentrant phase separation of a sparse collection of nonreciprocally aligning self-propelled disks.

We study a model of aligning self-propelled disks that nonreciprocally reorient the self-propulsion directions along the interparticle separation and towards the other disks. In the limit of small inertia and large softness, where conventional motility-induced phase separation is absent, we demonstrate that the homogeneous system at a small area fraction phase-separates into clusters and a low-density phase that, eventually, reenters the homogeneous phase with a monotonic increase in alignment strength. The disks inside the clusters move with a finite space-dependent speed, constantly shuttling between clusters through the surrounding low-density homogeneous phase while maintaining the hexatic structure properties within the clusters. The area fraction gradually increases from the periphery towards the center of the clusters with a negligible correlation of the velocity and propulsion direction inside the clusters. The novel collective behavior of reentrant phase separation is found to follow from both the limits of hard disks and extremely small inertia, tending towards the overdamped limit. However, important differences in the structural and dynamical properties are shown in the limit of hard disks and extremely small inertia, as compared to that for soft disks at finite inertia. We show that the cluster phase is associated with an effective temperature for a wide range of values of alignment strength, whereas an effective temperature is associated with the specific range of alignment in the low-density phase. We believe that the reentrant phase behavior in the limit of small area fraction and the remarkable properties of the clusters should be useful in understanding a wide range of physics issues, ranging from clogging and unclogging to information exchange and transport, in biological and synthetic self-propelled systems.

Inertio-thermal vapour bubble growth

Our understanding of homogeneous vapour bubble growth is currently restricted to asymptotic descriptions of their limiting behaviour. While attempts have been made to incorporate both the inertial and thermal limits into bubble growth models, the early stages of bubble growth have not been captured. By accounting for both the changing inertial driving force and the thermal restriction to growth, we present an inertio-thermal model of homogeneous vapour bubble growth, capable of accurately capturing the evolution of a bubble from the nano- to the macro-scale. We compare our model predictions with: (a) published experimental and numerical data, and (b) our own molecular simulations, showing significant improvement over previous models. This has potential application in improving the performance of engineering devices, such as ultrasonic cleaning and microprocessor cooling, as well as in understanding of natural phenomena involving vapour bubble growth.

Effective Moment of Inertia and Slenderness Limits of Reinforced Concrete and Fiber-Reinforced Concrete Slabs

Effective Moment of Inertia and Slenderness Limits of Reinforced Concrete and Fiber-Reinforced Concrete Slabs

Zero inertia limit of incompressible Qian–Sheng model

The Qian–Sheng model is a system describing the hydrodynamics of nematic liquid crystals in the Q-tensor framework. When the inertial effect is included, it is a hyperbolic-type system involving a second-order material derivative coupling with forced incompressible Navier–Stokes equations. If formally letting the inertial constant [Formula: see text] go to zero, the resulting system is the corresponding parabolic model. We provide the result on the rigorous justification of this limit in [Formula: see text] with small initial data, which validates mathematically the parabolic Qian–Sheng model. To achieve this, an initial layer is introduced to not only overcome the disparity of the initial conditions between the hyperbolic and parabolic models, but also make the convergence rate optimal. Moreover, a novel [Formula: see text]-dependent energy norm is carefully designed, which is non-negative only when [Formula: see text] is small enough, and handles the difficulty brought by the second-order material derivative.

The zero inertia limit from hyperbolic to parabolic Ericksen-Leslie system of liquid crystal flow

The original Ericksen-Leslie consists the conservation laws of the nematic liquid crystals, which includes the inertial effect. Thus the full Ericksen-Leslie system has hyperbolic feature. In this paper we study the zero inertia limit that is from the hyperbolic to parabolic Ericksen-Leslie's liquid crystal flow. By introducing an initial layer and constructing an energy norm and energy dissipation functional depending on the solutions of the limiting system, we derive a global in time uniform energy bound to the remainder system under the small size of the initial data. This work justifies the validity of the well-known parabolic Ericksen-Leslie system without inertial term.

From particle swarm optimization to consensus based optimization: Stochastic modeling and mean-field limit

In this paper, we consider a continuous description based on stochastic differential equations of the popular particle swarm optimization (PSO) process for solving global optimization problems and derive in the large particle limit the corresponding mean-field approximation based on Vlasov–Fokker–Planck-type equations. The disadvantage of memory effects induced by the need to store the local best position is overcome by the introduction of an additional differential equation describing the evolution of the local best. A regularization process for the global best permits to formally derive the respective mean-field description. Subsequently, in the small inertia limit, we compute the related macroscopic hydrodynamic equations that clarify the link with the recently introduced consensus based optimization (CBO) methods. Several numerical examples illustrate the mean field process, the small inertia limit and the potential of this general class of global optimization methods.

A filament-level analysis on failure mechanism and ballistic limit of real-size multi-layer 2D woven fabrics under FSP impact

This paper investigates the failure mechanism and fabric ballistic performance of real-size multi-layer 2D woven fabrics impacted by sharp-edge fragment simulating projectile (FSP). First, the relations between digital fiber shear force and bending rigidity are established under the modified digital element approach (DEA) framework. Then, a systematic parametric study was carried out on the ballistic impact of a 4-inch-long single yarn and 4-inch by 4-inch 2D woven fabric at near fiber level to solve for the relations of digital fiber moment of inertia and ballistic limit. The results show that for the same number of digital fibers per yarn model, the simulated ballistic limits are in direct proportion to digital fiber moment of inertia. The increase of the number of digital fibers per yarn, however, decreases the digital fiber moment of inertia effect on ballistic limits. Second, the 1- to 28-layer real-size 2D woven Kevlar KM2 fabrics are simulated at filament level against FSP on the cluster to estimate the V50 zone. The perforation process and failure mechanism of 4-layer fabric is investigated and analyzed in detail. The simulation results demonstrate the deformed fabric shape with respect to time and the damage modes at the impact area. Numerical results are compared with standard ballistic test results.

Paths to caustic formation in turbulent aerosols

The dynamics of small, yet heavy, identical particles in turbulence exhibits singularities, called caustics, that lead to large fluctuations in the spatial particle-number density, and in collision velocities. For large particle, inertia the fluid velocity at the particle position is essentially a white-noise signal and caustic formation is analogous to Kramers escape. Here we show that caustic formation at small particle inertia is different. Caustics tend to form in the vicinity of particle trajectories that experience a specific history of fluid-velocity gradients, characterised by low vorticity and a violent strain exceeding a large threshold. We develop a theory that explains our findings in terms of an optimal path to caustic formation that is approached in the small inertia limit.

Far-field flow and drift due to particles and organisms in density-stratified fluids.

In the limit of small inertia, stratification, and advection of density, Ardekani and Stocker [Phys. Rev. Lett. 105, 084502 (2010)PRLTAO0031-900710.1103/PhysRevLett.105.084502] derived the flow due to a point-force and force-dipole placed in a linearly density-stratified fluid. In this limit, these flows also represent the far-field flow due to a towed particle and a neutrally buoyant swimming organism in a stratified fluid. Here, we derive these two far-field flows in the limit of small inertia, stratification but at large advection of density. In both these limits, the flow in a stratified fluid decays rapidly and has closed streamlines but certain symmetries present at small advection are lost at large advection. To illustrate the application of these flows, we use them to calculate the drift induced by a towed drop and a swimming organism, as a means to quantify the mixing caused by them. The drift induced in a stratified fluid is less than that in the homogeneous fluid. A towed drop induces a large drift relative to its own volume at small advection while it induces at least an order of magnitude smaller drift at large advection. On the other hand, a swimming organism induces a large partial drift as compared with its own volume irrespective of the magnitude of advection, unless the stresslet exerted by the swimmer is small. These results are useful in understanding the stratification effects on the drift-based contributions to mixing.

The effect of wall slip on the dewetting of ultrathin films on solid substrates: Linear instability and second-order lubrication theory

The influence of wall slip on the instability of a non-wetting liquid film placed on a solid substrate is analyzed in the limit of negligible inertia. In particular, we focus on the stability properties of the film, comparing the performance of the three lubrication models available in the literature, namely, the weak, intermediate, and strong slip models, with the Stokes equations. Since none of the aforementioned leading-order lubrication models is shown to be able to predict the growth rate of perturbations for the whole range of slipping lengths, we develop a parabolic model able to accurately predict the linear dynamics of the film for arbitrary slip lengths.

Self-Similar Liquid Lens Coalescence.

A basic feature of liquid drops is that they can merge upon contact to form a larger drop. In spite of its importance to various applications, drop coalescence on prewetted substrates has received little attention. Here, we experimentally and theoretically reveal the dynamics of drop coalescence on a thick layer of a low viscosity liquid. It is shown that these so-called "liquid lenses" merge by the self-similar vertical growth of a bridge connecting the two lenses. Using a slender analysis, we derive similarity solutions corresponding to the viscous and inertial limits. Excellent agreement is found with the experiments without any adjustable parameters, capturing both the spatial and temporal structures of the flow during coalescence. Finally, we consider the crossover between the two regimes and show that all data of different lens viscosities collapse on a single curve capturing the full range of the coalescence dynamics.

Alfvén waves in temperature anisotropic Cairns distributed plasma

The dispersion relations and Landau damping of Alfvén waves in kinetic and inertial limits are studied in temperature anisotropic Cairns distributed plasma. In the case of kinetic Alfvén waves (KAWs), it is found that the real frequency is enhanced when either the electron perpendicular temperature or the non-thermal parameter Λ increases. For inertial Alfvén waves (IAWs), the real frequency is slightly affected by the electron temperature anisotropy and Λ. Besides the real frequency, the damping rate of KAWs is reduced when the electron perpendicular temperature or Λ increases. In the case of IAWs, the temperature anisotropy and Λ either enhance or reduce the damping rate depending upon the perpendicular wavelength. These results may be helpful to understand the dynamics of KAWs and IAWs in space plasmas where the non-Maxwellian distribution of particles are routinely observed.

Curtailment analysis for the Nordic power system considering transmission capacity, inertia limits and generation flexibility

Although regular curtailment of wind power has not been necessary in the Nordic power system so far, rapidly increasing wind power capacity means that it may be needed in the future. To estimate the amount of curtailment in the future Nordic power system we develop an hourly dispatch model based on open data. The model is validated against historical data and used to perform a case study for the Nordic power system in 2025 to estimate the amount of wind power curtailment under different assumptions. Curtailment is found to be below 0.3% of available generation for a 26 GW wind scenario and below 1.7% for a 33 GW wind scenario, when considering trade with neighbouring systems. The most important measures for decreasing curtailment are found to be increased transmission capacity, particularly between the areas in Sweden and those in Norway and Denmark, as well as flexibility of nuclear generation. Inertia requirements are found to have a limited impact on curtailments.

Understanding the impact of non-synchronous wind and solar generation on grid stability and identifying mitigation pathways

High penetrations of non-synchronous renewable energy generation can decrease overall grid stability because these units do not provide rotational inertia in the same way as traditional synchronously-connected generators. Many recent studies have investigated 100% renewable energy generation scenarios, but few have explored the trade-offs associated with an electricity grid dominated by non-synchronous generation (i.e. wind and solar). Fast frequency response from grid-forming inverters—along with other technology changes—could help mitigate low system inertia levels, but the impact of this response is unknown. An inertia-constrained unit commitment and dispatch model was used to study the stability of future grid scenarios with high penetrations of non-synchronous renewable energy generation under a variety of technology scenarios. The Texas grid (the Electric Reliability Council of Texas – ERCOT) was used as a test case and instances when the system inertia fell below 100 GW·s (the grid’s current minimum level) were recorded. When the modeled critical inertia limit was reduced to 80 GW·s to represent changes in grid operation, no critical inertia hours occurred for renewable energy penetrations up to 93% of annual energy. The critical inertia limit could drop to 60 GW·s if the largest generators in ERCOT (two co-located nuclear plants) were retired, but emissions increased by ~25% in these scenarios. If the critical inertia limit was kept the same (100 GW·s), adding 525 MW of fast frequency response from wind, solar, and energy storage could reduce the number of critical inertia hours by up to 95%. These results show that changes to grid operating practices and generator retirements reduced critical inertia hours more than fast frequency response from inverter-connected resources. Each of these mitigation pathways has associated trade-offs, so the transition to a grid dominated by non-synchronous energy generation should be handled with care, but high renewable energy penetrations (i.e. >80%) might be feasible in Texas.

Leidenfrost suppression and contact time reduction of a drop impacting on silicon nanowire array-coated surfaces

Drop impacting on a superheated surface is of practical concern in several thermal systems. When the surface temperature is higher than the Leidenfrost point (LFP), a stable vapor cushion forms on the solid surface and the drop hovers over the thin vapor layer. This results in a drastic reduction in heat and mass transfer due to the thermally insulating vapor layer. The contact time of an impacting drop is the amount of time during which the drop comes in contact with the solid surface. A short contact time enhances heat transfer. Thus, a high LFP and a short drop contact time are of interest to researchers. The present study revealed substantial Leidenfrost suppression and contact time reduction on a superheated superhydrophilic silicon nanowire (SiNW) array-coated surface. For an impacting drop of 14 μL with a Weber number (We) of 2.0, a LFP of 655 ± 14 °C was obtained on the SiNW array-coated surface with a nanowire height of 90 μm. This LFP was the highest reported value in the literature. A theoretical modeling revealed that the elevated LFP was caused by a superior capillary force and a strong vapor dispersion into the nanowire array. In addition, contact time was substantially reduced on the superheated SiNW surface in the jet-bouncing regime. The contact time of 1.3 ± 0.1 ms was 91% lower than the capillary–inertia limit. It was also the lowest reported value in the literature. The contact time reduction on the superheated SiNW surface resulted from the application of an upward thrust on the impacting drop generated by the vigorous bubbling on the SiNW surface.

Surface modification of polyetheretherketone by grafting amino groups to improve its hydrophilicity and cytocompatibility

Although special engineering plastics polyether ether ketone (PEEK) have many excellent properties, its hydrophobicity and biological inertia limit its application in the field of implant materials. In this study, amino modification of PEEK was conducted to endow it with good hydrophilicity and bioactivity. PEEK was sulfonated with concentrated sulfuric acid, and reacted with sulfoxide chloride to obtain aromatic sulfonyl chloride. Finally, amino (–NH2) was introduced onto the surface of PEEK by acylation reaction between acyl chloride group and ethylenediamine (EDA) to improve its hydrophilicity and cell compatibility. The PEEKs before and after modification were characterized by FTIR, XPS and WCA. The cell compatibility was evaluated by human umbilical vein endothelial cells (HUVEC). The results showed that –NH had been successfully introduced onto the surface of PEEK, and the hydrophilicity of the aminated PEEK was significantly improved. Compared with the PEEK before amino modification, the aminated PEEK significantly promoted adhesion and spread of cells, indicating good cell compatibility. This study breaks through the limitations of traditional wet chemical methods and provides a new way for further modification of PEEK.

A variational approach to the quasistatic limit of viscous dynamic evolutions in finite dimension

In this paper we study the vanishing inertia and viscosity limit of a second order system set in an Euclidean space, driven by a possibly nonconvex time-dependent potential satisfying very general assumptions. By means of a variational approach, we show that the solutions of the singularly perturbed problem converge to a curve of stationary points of the energy and characterize the behavior of the limit evolution at jump times. At those times, the left and right limits of the evolution are connected by a finite number of heteroclinic solutions to the unscaled equation.