Critical Line Research Articles

Transmission grid stability with large interregional power flows

We propose a general methodology for identifying critical lines in the long-distance transmission of power across large electric grids. When the system is pushed to its operational limit, for instance by large power imbalances such as those generated by high penetration of variable renewable energy sources, the network gets destabilized and loses synchrony. We investigate a model of the synchronous AC grid of continental Europe under tunable large interregional power flows. When those flows exceed some critical value, we find that instabilities emerge due to topological constraints. We identify two different scenarios triggering these instabilities. In the first one, specific sets of lines reach their maximal load simultaneously, causing the grid to split into two desynchronized zones. In the second one, one or a few lines become overloaded, which eventually renders one Lyapunov exponent positive. The first scenario is obviously less generic and we develop a numerical approach to force the splitting of the AC grid into disconnected areas. Remarkably, the critical lines identified in this way match those that triggered the separation of the synchronous grid of continental Europe in two instances in 2021. We further discuss how the modes of the system provide information on which areas are more susceptible to lose synchrony with each other. Published by the American Physical Society 2025

Determination of the Critical State Line in Partially Frozen Sand

A new method for measuring internal pore water pressure (PWP) is introduced to determine the critical state line (CSL) in partially frozen sand, investigating the influence of temperature and strain rate on the critical state parameters. A series of consolidated undrained and drained triaxial tests, along with internal PWP measurements, were conducted on both dense and loose specimens under different temperatures and strain rates. Similarly to unfrozen sand, a unique CSL was established for the partially frozen sand at −3 °C in both stress (q-p′) and void ratio (e-p′) space. The results show that the critical state friction angle (φcs′) is not affected by temperature (warmer than −5 °C) and strain rate, while the critical state cohesion (ccs′) varies with temperature, strain rate and failure mode. The ccs′ increases with decreasing temperature from 23 °C to −3 °C and to −10 °C, but decreases to zero when the strain rate was reduced from 1%/min to 0.1%/min. In e-p′ space, the slope of CSL could be associated with the dilation of partially frozen sand, which increases with decreasing temperature and increasing strain rate, potentially due to the increased contact area between the pore ice and sand grains.

Drained and undrained behaviour of a sandy silt gold tailings under general multiaxial conditions

A laboratory study was carried out on a sandy silt gold tailings to investigate the effect of the Lode angle (θ) on the critical state line (CSL) in the compression (e–p′) and effective stress (q–p′) planes, and to study the strength anisotropy of the tailings. Four sets of strain-controlled tests were carried out: (a) drained and undrained triaxial compression tests; (b) torsional shear hollow cylinder tests under drained and undrained conditions at constant θ and constant direction of the major principal stress relative to the vertical axis (α); (c) drained and undrained triaxial extension tests; and (d) undrained hollow cylinder simple shear tests. The tests were carried out on loose moist tamped (LMT) and dense slurry air-dried (AD) prepared specimens. The results indicated that: (a) the CSL in the compression plane does not depend on θ or on the stress path; (b) the stress-dilatancy is dependent on θ while the effect of θ on state-dilatancy is minor; (c) the undrained yield strength of LMT was found not to depend significantly on the state parameter (ψ), but it was significantly affected by the stress path followed during consolidation and subsequent shear, for example, by the pre-shearing stress ratio, stress-reversal loading, θ and α. The results demonstrate the contribution of the initial shear stress and evolving anisotropy on the undrained strength of the gold tailings in a loose state.

Random matrix ensemble for the covariance matrix of Ornstein-Uhlenbeck processes with heterogeneous temperatures

We introduce a random matrix model for the stationary covariance of multivariate Ornstein-Uhlenbeck processes with heterogeneous temperatures, where the covariance is constrained by the Sylvester-Lyapunov equation. Using the replica method, we compute the spectral density of the equal-time covariance matrix characterizing the stationary states, demonstrating that this model undergoes a transition between stable and unstable states. In the stable regime, the spectral density has finite and positive support, whereas negative eigenvalues emerge in the unstable regime. We determine the critical line separating these regimes and show that the spectral density exhibits a power-law tail at marginal stability, with an exponent independent of the temperature distribution. Additionally, we compute the spectral density of the lagged covariance matrix characterizing the stationary states of linear transformations of the original dynamical variables. Our random-matrix model is potentially interesting to understand the spectral properties of empirical correlation matrices appearing in the study of complex systems. Published by the American Physical Society 2025

Phase transitions and critical behavior in quasi-one-dimensional two-channel systems with quasiperiodic disorder

Abstract We investigate the localization properties of a quasi-one-dimensional two-channel system with symmetric and asymmetric onsite energies using the Aubry-Andr'{e} model. By analyzing the Lyapunov exponent and localization length, we characterize the phase transitions and critical behavior of the system. For the symmetric model, we obtain the phase diagram for the entire spectrum, revealing mobility edges between delocalized and localized states. In contrast, for the asymmetric model, we identify a critical line $\lambda_1^c + \lambda_2^c \approx 0.5$ marking the phase transition between delocalized and localized states. We also study the effects of the inter-channel coupling $\tilde{t}$ and observe that increasing $\tilde{t}$ reduces the delocalized phase space, shifting the transition from $\lambda_1 = \lambda_2 = 1$ at $\tilde{t} = 0$ to $\lambda_1^c + \lambda_2^c \approx 0.5$ at larger $\tilde{t}$. Additionally, we explore the influence of energy levels and the incommensurate modulation parameter, demonstrating that while the general phase transition behavior is preserved, subtle differences arise for different incommensurate modulation parameter values. Using the cost function approach, we calculate the critical potential strength $\lambda_c$ and the critical exponent $\nu$, with $\nu \approx 0.5$ for the middle of the spectrum in both symmetric and asymmetric models.

Pluralism, Ecology and Planning

In Economic Democratic Planning Robin Hahnel rearticulates and defends the model of participatory planning he developed with Michael Albert. This paper develops three lines of criticism of the model. It argues that the model’s principle of distribution of income among workers according to a metric of effort would involve pervasive surveillance of persons and potential humiliation. The use of a price metric of opportunity costs and cost-benefit analysis in the allocation of resources fails to address the implications of value-pluralism and incommensurability for their allocation. In response to Hahnel’s criticism of those who argue that environmental constraints entail limits to economic growth, it argues that we need to take those constraints more seriously than he does. The paper focuses on the second and third area of disagreement by placing those differences within the wider history of the socialist calculation debates and ecological economics.

Droplets impact on sparse microgrooved non-wetting surfaces

Droplets impinging on sparse microgrooved polydimethylsiloxane (PDMS) surfaces with different solid fractions was experimentally investigated. First, wettability and stability of droplets on these surfaces was analyzed. The advancing and receding contact angles were found to have a large difference between in the longitudinal direction and in the transverse one, which could be attributed to the anisotropy of the micropatterned surfaces. The judgement of whether a droplet on a sparse microgrooved structure is collapsed or suspended is proposed, and it was found that the droplets were in the Cassie-Baxter wetting state when the actual contact line density is greater than the critical contact line density, while they were in the Wenzel wetting state otherwise. Second, for the case of droplets impacting on sparse microgrooved PDMS surfaces, it was found that droplets can bounce off the micro-patterned surface with a solid fraction of 0.158 when the impact velocity was in a certain range. The lower limit of impact velocity for bouncing droplets can be determined by balancing the kinetic energy of the droplets with the energy barrier due to contact angle hysteresis. The upper limit of impact velocity for bouncing droplets was predicted using a theoretical model taking into account the penetration of liquid into the cavities between microstripes.

Sexual Minority Stress: Preliminary Evidence of Accelerated Pubertal Development in Early Adolescence.

Societal stressors place a tremendous burden on individuals who identify with a sexual minority identity. While minority stress experienced by racial/ethnic minority groups has been linked to accelerated aging, this link has yet to be examined among sexual minority youth. This study explores whether sexual minority youth who indicate experiencing stress at home or school (Minority Stress) due to their identity show evidence of accelerated aging (pubertal status or tempo) compared to those who do not report such experiences (No Minority Stress). Data are from the Adolescent Brain Cognitive Development (ABCD) Study. Participants were approximately 9-10 years old at baseline, 10-11 years old at Time 1, and 11-12 years old at Time 2. Measures included child-reports of sexual minority identity and stressors, and parent-reports of adolescent pubertal development. Among 432 included participants who identified with a sexual minority identity, 83.6% were assigned female at birth and 24.8% were in the Minority Stress group. There were consistent results showing that sexual minority youth in the Minority Stress group showed accelerated pubertal status over time compared to those in the No Minority Stress group. Pubertal tempo (i.e., slope) was only accelerated for those who first identified as sexual minority at Time 2. Our findings underscore that minority stress experienced by sexual minority youth may be linked to differences in pubertal development among these youth. The developmental and clinical implications of these patterns present critical lines for future research concerned with the wellbeing of sexual minority youth.

Gapless topological behaviors in a long-range quantum spin chain

Topology is a fascinating phenomena in condensed-matter physics typically associated with a bulk gap. However, recent research shifts focus to quantum critical points or phases that exhibit nontrivial topological properties. Here we explore a cluster-Ising chain with long-range antiferromagnetic interactions that decay as a power law with distance. Using large-scale density matrix renormalization group simulations, we demonstrate that the nontrivial topology at the critical point remains stable against long-range interactions, resulting in a topologically nontrivial critical line. Moreover, even within the gapped region, the interplay between topology and long-range interaction can give rise to a topological phase featuring algebraically decaying correlations and edge modes, similar to gapless topological phases. We refer to this phase as the algebraic topological phase, which exhibits nontrivial gapless topological behaviors and arises solely from long-range interactions without short-range counterparts. The findings pave the way for more studies on topological states in long-range systems.

Orbital moiré and quadrupolar triple-q physics in a triangular lattice

We numerically study orders of planer type (xy,x2−y2) quadrupoles on a triangular lattice with nearest-neighbor isotropic J and anisotropic K interactions. This type of quadrupole possesses unique single-ion anisotropy proportional to a third order of the quadrupole moments. This provides an unconventional mechanism of triple-q orders which does not exist for the degrees of freedom with odd parity under time-reversal operation such as magnetic dipoles. In addition to several single-q orders, we find various orders including incommensurate triple-q quasi-long-range orders with orbital moiré and a four-sublattice triple-q partial order. Our Monte Carlo simulations demonstrate that the phase transition to the latter triple-q state belongs to the universality class of the critical line of the Ashkin-Teller model in two dimensions close to the four-state Potts class. These results indicate a possibility of realizing unique quadrupole textures in simple triangular systems. Published by the American Physical Society 2024

Light holographic dilatons near critical points

We investigate the relation between the emergence of a dilaton in gapped (confining) field theories, and the presence of either complex fixed points or instabilities in the strongly coupled dynamics in two classes of bottom-up holographic models. We demonstrate that in one of the two classes there is a critical line of first-order phase transitions (at zero temperature) that terminates at a critical point. We calculate the mass spectrum of fluctuations of the associated regular gravity backgrounds, which we interpret as bound states in the dual field theories. In proximity to the second-order phase transition, we find a parametrically light scalar state, and its composition leads us to identify it as a dilaton. Published by the American Physical Society 2024

Critical lines identification and mitigation strategy for wind-integrated power systems under N-k contingencies

Critical lines identification and mitigation strategy for wind-integrated power systems under N-k contingencies

Lower bounds for the large deviations of Selberg's central limit theorem

AbstractLet and . We prove that, for any and as , where is large enough depending on . The result is unconditional on the Riemann hypothesis. As a consequence, we recover the sharp lower bound for the moments on the critical line proved by Heap and Soundararajan and Radziwiłł and Soundararajan. The constant is explicit and is compared to the one conjectured by Keating and Snaith for the moments.

Finite-size Nagle-Kardar model: Casimir force.

We derive exact results for the critical Casimir force (CCF) within the Nagle-Kardar model with periodic boundary conditions (PBCs). The model represents a one-dimensional Ising chain with long-range equivalent-neighbor ferromagnetic interactions of strength J_{l}/N>0 superimposed on the nearest-neighbor interactions of strength J_{s}, which could be either ferromagnetic (J_{s}>0) or antiferromagnetic (J_{s}<0). In the infinite system limit, the model exhibits in the plane (K_{s}=βJ_{s},K_{l}=βJ_{l}) a critical line 2K_{l}=exp(-2K_{s}),K_{s}>-ln3/4, which ends at a tricritical point (K_{l}=sqrt[3]/2,K_{s}=-ln3/4). The critical Casimir amplitudes are Δ_{Cas}^{(cr)}=1/4 at the critical line, and Δ_{Cas}^{(tr)}=1/3 at the tricritical point. Quite unexpectedly, with the imposed PBCs, the CCF exhibits very unusual behavior as a function of temperature and magnetic field. It is repulsive near the critical line and tricritical point, decaying rapidly with separation from those two singular regimes, and it becomes attractive when the maximum amplitude of the attraction exceeds the maximum amplitude of repulsion. This represents a violation of the widely accepted "boundary condition rule," which holds that the CCF is attractive for equivalent BCs and repulsive for conflicting BCs independently of the actual bulk universality class of the phase transition under investigation.

Elasto-plastic solution for undrained cylindrical cavity expansion in refuse soil

To address geotechnical engineering issues such as pile driving, lateral pressure tests, and static cone penetration tests on increasing number of infrastructure projects being constructed on landfills, an elasto-plastic theoretical solution for the undrained cylindrical cavity expansion in refuse soil is proposed in this paper based on an elasto-plastic constitutive model for refuse soil considering the reinforcement effect of fibers, along with a large deformation theory. The correctness of the results is validated through comparison with existing solutions based on the modified Cam-clay model. The results indicate that the response of columnar pore expansion in refuse soil is significantly different from that in ordinary soil. Near the pore, refuse soil does not enter a critical state, only the slurry-like components do. Subsequently, the reinforcement effect of the fiber material begins to manifest. Refuse soils with different fiber contents undergo both elastic and plastic stages before reaching the critical state line of the slurry-like components in the soil. They then move a certain distance along this line to reach a maximum. Given the rarity of engineering construction on similar sites in the past, the unique response of refuse soil during cylindrical cavity expansion should be given special attention.

A thermo-mechanical model for saturated and unsaturated soil–structure interfaces

Shear behaviour of soil–structure interfaces greatly affects the performance of geotechnical structures. The soil–structure interfaces in geothermal structures (e.g., energy pile and energy wall) are often subjected to varying temperature and suction conditions. However, there is no constitutive model to simulate the coupled effects of suction and temperature on the shear behaviour of soil–structure interfaces. In this study, a thermo-mechanical model was newly developed based on the bounding surface plasticity framework to predict the thermo-mechanical behaviour of saturated and unsaturated interfaces. A power function was used to calculate the degree of saturation at the interface and improve the evaluation of suction effects on interface shear strength. A linear relationship between temperature and interface critical state friction angle was proposed to incorporate thermal effects. New equations were also proposed to describe the critical state lines (CSLs) in the void ratio versus stress plane ( e-[Formula: see text]) and to model the shearing-induced deformation at various temperatures and suctions. The experimental data from different interfaces in the literature were used to evaluate the model capability. Comparisons between measured and computed results suggest that this model can well capture the coupled effects of temperature, suction, and net normal stress on the shear behaviour of interfaces.

Stability analysis and voltage improvement in DG-integrated distribution networks using VCPI-based critical buses and lines detection considering uncertain power factor

Voltage stability ensures reliable and efficient power transmission to end-users, making it essential for power system efficiency. Dynamic load changes and rising power requirements might cause voltage instability and strain performance. This paper investigates the voltage stability Indicators of distribution systems to address the voltage stability issue in power systems. The uncertain power factor and other numerous technical parameters that affect the voltage drop or voltage collapse are involved in predicting the system’s voltage stability. First, the analysis uses the voltage collapse proximity index (VCPI) technique, which considers bus voltage, impedance, uncertain power factor of load buses, and transmission line impedance at both ends. The applied technique correctly predicts and detects weak buses and transmission lines using different voltage stability indices since technical parameters are correlated with the VCP index. The index and line impedance correspond; consequently, the recommended indication compensates for voltage loss owing to impedance. Therefore, the VCPI is a better index for anticipating voltage collapse and finding network weak busses and lines. The implemented approach is simple, precise, and economical; the index technique detects weak buses and forecasts system collapse. Second, this work introduces EMFO-FPSO, a hybrid metaheuristic that improves voltage stability by combining Fractional-order particle swarm optimization (PSO) and Entropy design mouth flame optimization (MFO). The proposed approach is validated on the IEEE 33 bus system by identifying weak buses and optimizing distributed generation placement in the distribution system. The implemented solution approach based on VCPI and EMFO-FPSO optimization achieves a significant power loss reduction of 71.5% compared to the base case, whereas the reductions for ABC, ACO-ABC, and PSO are 57.3%, 60.7%, and 64.4%, respectively. Moreover, the proposed method attains a stable VCPI Index of 0.90, indicating a 10% enhancement in voltage stability relative to the base case. The study’s results, key findings, and comparisons show the relevance of the proposed method for voltage stability enhancement.

Methods for Evaluating Critical Lines and Nodes in Cyber–Physical Power Systems From Three Network Perspectives

Methods for Evaluating Critical Lines and Nodes in Cyber–Physical Power Systems From Three Network Perspectives

Results of a dilatancy round robin

The critical state approach currently finds widespread adoption in geotechnical engineering practise, particularly with respect to tailings storage facilities. The laboratory component of such works usually involves determination of the critical state line (CSL) followed by stress- and state-dilatancy behaviour in triaxial compression conditions, characterised by the volumetric coupling parameter Ntc and the state dilatancy constant χtc. While the reliability of CSL testing using current state of practise methods has been recently demonstrated, it is currently unclear how reproducibly Ntc and χtc can be obtained. To investigate this, a dilatancy round robin programme was carried out by five laboratories who regularly carry out such testing. In the first stage of the programme, each laboratory adopted a version of ‘densification in layers’, either dense moist tamping or vibration-densification. While a generally consistent Ntc was observed, significant variation in χtc was seen. To further investigate this variation, a second stage was carried out where slurry deposition and the ‘air-dried’ technique were used. These techniques showed better agreement between laboratories, while also producing lower values of χtc than densification in layers. Some potential implications of these observations on current tailings engineering laboratory testing practise are outlined.

Limits on Dark Matter Compact Objects implied by Supermagnified Stars in Lensing Clusters

Abstract Supermagnified stars are gravitationally lensed individual stars that are located close to a caustic of a lensing galaxy cluster, and have their flux magnified by a large enough factor (typically ∼ 1000) to make them detectable with present telescopes. The maximum magnification is limited by microlensing caused by intracluster stars or other compact objects, which create a network of corrugated critical lines with an angular width proportional to the surface density of microlenses. We consider a set of 9 cases of supermagnified stars reported in the literature, and derive an upper limit on the surface density of compact objects, such as primordial black holes, that might be present as a fraction of the dark matter in addition to known intracluster stars. Any such additional compact objects would widen the corrugated critical line network and therefore the width of the distribution of supermagnified stars around the modeled critical lines of the lens. We find that any compact objects, including primordial black holes, with masses above ∼10−6 M⊙ (below which the microcaustics are closer together than the typical angular size of supermagnified stars) cannot account for more than ∼ 2% of the dark matter.