Small Grid Research Articles

Data analysis and management for optimal application of an advanced ML-based fault location algorithm for low voltage grids

As the need for automatization of the electricity grid’s fault diagnosis schemes is rising, the application of technologies such as the artificial intelligence (AI) can provide practical solutions to the problem. AI can overcome the challenges that complex topologies like those of the low voltage (LV) smart grids pose and prove to be a powerful tool in the development of advanced fault diagnosis methods. An important parameter for the success of any AI-based method is the quality of data. Therefore, in this paper a data analysis is performed in order to evaluate the type of data produced by a small LV grid and an representative AI algorithm’s response to those. In the context of this analysis, the most important features and meters were identified. Furthermore, as a response to the large volume of available data, a data management strategy is proposed. The strategy combines original and reshaped features. For this purpose, five dimensionality reduction methods are tested and compared. Truncated-SVD is deemed the most appropriate and is subsequently utilized for the reshaping of the dataset that is introduced to the XGBoost fault location model. The integration of the dimensionality reduction technique in the algorithm results in the decrease of the computational time and the dataset’s size and in a higher generalizability of the algorithm. Thus, the application of the proposed method is not limited by the grid’s topology. The method’s robustness was verified against various influencing parameters such as the fault resistance, the size of the dataset, the loss of data and the photovoltaics’ penetration level. The overall algorithm achieved a mean squared error of 13.26 and a training and test accuracy of more than 99% when tested on the CIGRE LV benchmark grid.

A Survey on the Investigation and Analysis for a Power System (Micro-Grid) with Stochastic Harmonic Distortion of Multiple Converters

The current study is based on a survey on the exploration and analysis of stochastic harmonic distortion in power systems using multiple converters (Micro-Grid). When Renewable Energy Systems are integrated into EPS, they may create clean energy, fulfill consumer energy needs, and help to protect fossil fuel supplies, which are rapidly dwindling. These renewable energy sources (RES) are frequently connected to the grid via power converters (Voltage Source Converters) to provide the necessary energy regulation and conversion. However, Voltage Source Converter (VSCs) generates both current and voltage harmonics, has a negative impact on the Power Quality (PQ) of a small grid and has the potential to cause damage or equipment failure. In the face of uncertainties, such as those resulting from design parameter selection or system parameter changes, the amount of harmonic distortion of many VSCs may be greatly influenced and difficult to forecast. When dealing with VSC harmonic distortion levels in the face of uncertainty, it is necessary to use statistical methodologies.

Predicting basin stability of power grids using graph neural networks

The prediction of dynamical stability of power grids becomes more important and challenging with increasing shares of renewable energy sources due to their decentralized structure, reduced inertia and volatility. We investigate the feasibility of applying graph neural networks (GNN) to predict dynamic stability of synchronisation in complex power grids using the single-node basin stability (SNBS) as a measure. To do so, we generate two synthetic datasets for grids with 20 and 100 nodes respectively and estimate SNBS using Monte-Carlo sampling. Those datasets are used to train and evaluate the performance of eight different GNN-models. All models use the full graph without simplifications as input and predict SNBS in a nodal-regression-setup. We show that SNBS can be predicted in general and the performance significantly changes using different GNN-models. Furthermore, we observe interesting transfer capabilities of our approach: GNN-models trained on smaller grids can directly be applied on larger grids without the need of retraining.

A Probabilistic Forecast-Driven Strategy for a Risk-Aware Participation in the Capacity Firming Market

This paper addresses the energy management of a grid-connected renewable generation plant coupled with a battery energy storage device in the capacity firming market, designed to promote renewable power generation facilities in small non-interconnected grids. The core contribution is to propose a probabilistic forecast-driven strategy, modeled as a min-max-min robust optimization problem with recourse. It is solved using a Benders-dual cutting plane algorithm and a column and constraints generation algorithm in a tractable manner. A dynamic risk-averse parameters selection strategy based on the quantile forecasts distribution is proposed to improve the results. A secondary contribution is to use a recently developed deep learning model known as normalizing flows to generate quantile forecasts of renewable generation for the robust optimization problem. This technique provides a general mechanism for defining expressive probability distributions, only requiring the specification of a base distribution and a series of bijective transformations. Overall, the robust approach improves the results over a deterministic approach with nominal point forecasts by finding a trade-off between conservative and risk-seeking policies. The case study uses the photovoltaic generation monitored on-site at the University of Li\`ege (ULi\`ege), Belgium.

DEH Scheme DGTD-Based Transient Modeling Approach for the Cole–Cole Dispersive Media Using Tustin’s Method

The Cole–Cole model can simulate the electromagnetic properties of biological tissues more accurately in a wide frequency band. The current simulation of wave propagation in the Cole–Cole dispersive media usually uses the finite-difference time-domain (FDTD) method, but its requirements in small grid size and time step interval for curved objects limit its application in calculating multiscale complex biological tissues. The discontinuous Galerkin time-domain (DGTD) method is more suitable for simulating multiscale complex targets because it can use unstructured grids and high-order basis functions. Therefore, this article proposes a transient modeling approach for the Cole–Cole dispersive media based on DGTD method, in which the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$z$ </tex-math></inline-formula> transform is performed on the Cole–Cole model, and the fractional power terms are approximated to integer power polynomials according to the continued fraction theory. Moreover, in order to avoid the frequency-dependent permittivity in Maxwell’s equations, the DEH scheme is constructed and a weak form equation for the constitutive relationship between D and E is created. Maxwell’s equations and the constitutive relationship are updated by the leap-frog time-stepping method. The accuracy and ability of the new method are verified by simulating the plane wave propagation in various Cole–Cole media.

Analysis of Control-System Strategy and Design of a Small Modular Reactor with Different Working Fluids for Electricity and Hydrogen Production as Part of a Decentralised Mini Grid

Hydrogen is increasingly being viewed as a significant fuel for future industrial processes as it offers pathways to zero emission. The UK sees hydrogen as one of a handful of low-carbon solutions for transition to net zero. Currently, most hydrogen production is from steam reforming of natural gas or coal gasification, both of which involve the release of carbon dioxide. Hydrogen production from mini decentralised grids via a thermochemical process, coupled with electricity production, could offer favourable economics for small modular reactors (SMRs), whereby demand or grid management as a solution would include redirecting the power for hydrogen production when electricity demand is low. It also offers a clean-energy alternative to the aforementioned means. SMRs could offer favourable economics due to their flexible power system as part of the dual-output function. This study objective is to investigate the critical performance parameters associated with the nuclear power plant (NPP), the cycle working fluids, and control-system design for switching between electricity and hydrogen demand to support delivery as part of a mini grid system for a reactor power delivering up to approximately 600 MWth power. The novelty of the work is in the holistic parametric analysis undertaken using a novel in-house tool, which analyses the NPP using different working fluids, with a control function bolt-on at the offtake for hydrogen production. The results indicate that the flow conditions at the offtake can be maintained. The choice of working fluids affects the pressure component. However, the recuperator and heat-exchanger effectiveness are considered as efficiency-limiting factors for hydrogen production and electricity generation. As such, the benefit of high-technology heat exchangers cannot be underestimated. This is also true when deciding on the thermochemical process to bolt onto the plant. The temperature of the gas at the end of the pipeline should also be considered to ensure that the minimum temperature-requirement status for hydrogen production is met.

Microphysics of Water Clouds in the Atmospheres of Y Dwarfs and Temperate Giant Planets

Abstract Water clouds are expected to form on Y dwarfs and giant planets with equilibrium temperatures near or below that of Earth, drastically altering their atmospheric compositions and their albedos and thermal emission spectra. Here we use the 1D Community Aerosol and Radiation Model for Atmospheres (CARMA) to investigate the microphysics of water clouds on cool substellar worlds to constrain their typical particle sizes and vertical extent, taking into consideration nucleation and condensation, which have not been considered in detail for water clouds in H/He atmospheres. We compute a small grid of Y-dwarf and temperate giant-exoplanet atmosphere models with water clouds forming through homogeneous nucleation and heterogeneous nucleation on cloud condensation nuclei composed of meteoritic dust, organic photochemical hazes, and upwelled potassium chloride cloud particles. We present comparisons with the Ackerman &amp; Marley parameterization of cloud physics to extract the optimal sedimentation efficiency parameter (f sed) using Virga. We find that no Virga model replicates the CARMA water clouds exactly and that a transition in f sed occurs from the base of the cloud to the cloud top. Furthermore, we generate simulated thermal emission and geometric albedo spectra and find large, wavelength-dependent differences between the CARMA and Virga models, with different gas absorption bands reacting differently to the different cloud distributions and particularly large differences in the M band. Therefore, constraining the vertically dependent properties of water clouds will be essential to estimate the gas abundances in these atmospheres.

Community‐Driven Code Comparisons for Three‐Dimensional Dynamic Modeling of Sequences of Earthquakes and Aseismic Slip

AbstractDynamic modeling of sequences of earthquakes and aseismic slip (SEAS) provides a self‐consistent, physics‐based framework to connect, interpret, and predict diverse geophysical observations across spatial and temporal scales. Amid growing applications of SEAS models, numerical code verification is essential to ensure reliable simulation results but is often infeasible due to the lack of analytical solutions. Here, we develop two benchmarks for three‐dimensional (3D) SEAS problems to compare and verify numerical codes based on boundary‐element, finite‐element, and finite‐difference methods, in a community initiative. Our benchmarks consider a planar vertical strike‐slip fault obeying a rate‐ and state‐dependent friction law, in a 3D homogeneous, linear elastic whole‐space or half‐space, where spontaneous earthquakes and slow slip arise due to tectonic‐like loading. We use a suite of quasi‐dynamic simulations from 10 modeling groups to assess the agreement during all phases of multiple seismic cycles. We find excellent quantitative agreement among simulated outputs for sufficiently large model domains and small grid spacings. However, discrepancies in rupture fronts of the initial event are influenced by the free surface and various computational factors. The recurrence intervals and nucleation phase of later earthquakes are particularly sensitive to numerical resolution and domain‐size‐dependent loading. Despite such variability, key properties of individual earthquakes, including rupture style, duration, total slip, peak slip rate, and stress drop, are comparable among even marginally resolved simulations. Our benchmark efforts offer a community‐based example to improve numerical simulations and reveal sensitivities of model observables, which are important for advancing SEAS models to better understand earthquake system dynamics.

Removing the stability limit of the time-space domain explicit finite-difference schemes for acoustic modeling with stability condition-based spatial operators

The time step and grid spacing in explicit finite-difference (FD) modeling are constrained by the Courant-Friedrichs-Lewy (CFL) condition. Recently, it has been found that the spatial FD coefficients may be designed through simultaneously minimizing the spatial dispersion error and maximizing the CFL number. This allows one to stably use a larger time step or a smaller grid spacing than usually possible. However, when using such a method, only second-order temporal accuracy is achieved. To address this issue, I propose a new method to determine the spatial FD coefficients, which simultaneously satisfy the stability condition of the whole wavenumber range and the time-space domain dispersion relation of a given wavenumber range. Therefore, stable modeling can be performed with high-order spatial and temporal accuracy. The coefficients can adapt to the variation of velocity in heterogeneous models. In addition, based on the hybrid absorbing boundary condition, I develop a strategy to stably and effectively suppress artificial reflections from the model boundaries for large CFL numbers. Stability analysis, accuracy analysis, and numerical modeling demonstrate the accuracy and effectiveness of my proposed method.

How to Morph a Tree on a Small Grid

How to Morph a Tree on a Small Grid

Power output control method of small power photovoltaic grid connected power generation under complex illumination

Due to the irregular change of lighting conditions, there is a large error in the control of photovoltaic grid connected power output. Therefore, this paper proposes proposes a power output control method for low-power photovoltaic grid-connected power generation considering complex illumination. Based on the analysis of the basic characteristics of photovoltaic grid connected power generation, the output characteristics of photovoltaic array under complex lighting conditions are studied. Combined with its variation characteristics, based on fuzzy control strategy, the output of photovoltaic grid connected energy storage system is adjusted by filter control with variable time constant, and economic operation and stable operation are taken as constraints. The test results show that the design method has a high degree of fit between the control of photovoltaic grid connected power output and the actual results, and the error is within a reasonable range, which is obviously due to the two groups of comparison methods.

Dynamic state power system fault monitoring and protection with phasor measurements and fuzzy based expert system

The main aim of this paper is building of a small micro grid test bed system with a few sources, and few loads with various types of lumped resistive (R), capacitive (C) and inductive (L) elements. Before the advent of electric smart grid every load is facilitated with its own electromagnetic. Slowly with an advancement of protection system numerical relays came into existence. Because in the numerical relays the communication between the relays and communication with central computer is possible, with this a coordinative protection system was evolved. For the monitoring and protection of system in dynamic state many of the numerical relays were now replaced by intelligent electronic devices. The system is to be centrally controlled with an improvement in data communications and the artificial intelligence into power system made it is possible to control the power system centrally. In this paper, it is presented that the fault is monitoring and protection against different types of faults, such as frequency variations, voltage variations and different fault conditions using fuzzy based expert system (FBES). The proposed protection scheme is adaptive for frequency variations, voltage variations and fault levels. The FBES scheme is implemented in the LabVIEW software with MAMDANI structure.

Reactive grid-assisted co-sputtering of titanium and chromium in a pure nitrogen atmosphere: Uniformity, optics, and structure of the Ti–Cr–N films

This study provides the results of the first attempt to deposit thin films by reactive grid-assisted co-sputtering with the deployment of small grounded grids having two varying apertures for enabling the transfer of particles. The diameter of all the grids utilized was 50 mm, and their apertures' diameters were changed from 5 mm to 15 mm. Furthermore, the ratios of the grid area to the chamber wall area were lower than 0.0161, preventing the formation of ion-rich sheaths and their adverse impact upon discharge and plasma stability. Accordingly, Ti 1 Cr 1-x N films (0.88 < x < 0.97) with low chromium contents were deposited on soda-lime glass substrates by using pure nitrogen as the sputtering gas. The grazing incidence X-ray diffraction patterns of the films, the average thicknesses of which were lower than 50 nm, showed no sign of crystallinity in the films. Ranging from 3.26 to 3.79 eV, the optical bandgap of the films changed by altering the apertures’ size; chromium content, internal stress, and quantum confinement effect appeared to be the main contributing factors. The photoluminescence intensities appertaining to trap state emissions reflected a decreasing trend by increasing the chromium content, which can be ascribed to the capability of the chromium particles included in the surface, structure, and grain boundaries of the films to prevent photoinduced electron-hole pairs from recombination. It was found that not only is the grid-assisted co-sputtering method an effective tool whereby the doping range of conventional co-sputtering methods can be significantly extended but also it can cover a spectrum of surface morphologies. However, in comparison with the conventional co-sputtering method, the utilization of the grids with fairly small apertures can limit the thickness uniformity of the films; this effect may be attenuated by changing the geometry and design of grids.

A hybrid fault-tolerant routing based on Gaussian network for wireless sensor network

In this paper, we have proposed a hybrid fault-tolerant routing to solve fault-tolerant issue in wireless sensor networks (WSNs) based on hierarchical topology. The hierarchical topology is a combination of clustering and the labeling of sensor nodes as Gaussian integers. Accordingly, the network area is divided into small square grids, the cluster head of each grid is represented by a Gaussian integer. These cluster heads are connected together to create a Gaussian network. Through node symmetry, and the shortest distance in the Gaussian network, as well as the advantages of multi-path routing, this paper proposes a hybrid fault-tolerant clustering routing protocol based on Gaussian network for wireless sensor network (FCGW). The purpose of FCGW is to improve fault tolerance, increase data reliability and reduce energy consumption for wireless sensor networks. The experimental results of the proposed scheme show that FCGW protocol has high data reliability. In addition, the FCGW protocol consumes about 48% of the energy in the network, while other protocols consume 70% more energy.

Influence of the Gravitational Darkening Effect on the Spectrum of a Hot, Rapidly Rotating Neutron Star

Abstract In this paper, we discuss the influence of the gravitational darkening effect on the emergent spectrum of a fast-rotating, flattened neutron star. Model atmosphere codes always calculate spectra of emergent intensities and fluxes emitted from the unit surface on the star in plane-parallel geometry. Here we took a step beyond that and calculated a small sample grid of theoretical spectra integrated over the distorted surface of a sample rotating neutron star seen by a distant observer at various inclination angles. We assumed parameters like two dimensionless angular velocities Ω ¯ 2 = 0.30 and 0.60, the effective temperature of a nonrotating star T eff = 2.20 × 107 K, the logarithm of the surface gravity of a spherical star log ( g ) = 14.40 (cgs), and inclination angles from i = 0° to i = 90° with step Δi = 10°. We assumed that the atmosphere consists of a mixture of hydrogen and helium with M H = 0.70 and M He = 0.30. At each point on the neutron star surface, we calculated true intensities for local values of parameters (T eff and log ( g ) ), and these monochromatic intensities are next integrated over the whole surface to obtain the emergent spectrum. In this paper, we compute for the first time theoretical spectra of the fast-rotating neutron star. Our work clearly shows that the gravitational darkening effect strongly influences the spectrum and should be included in realistic models of the atmospheres of rotating neutron stars.

Adaptive variational preparation of the Fermi-Hubbard eigenstates

Approximating the ground states of strongly interacting electron systems in quantum chemistry and condensed matter physics is expected to be one of the earliest applications of quantum computers. In this paper, we prepare highly accurate ground states of the Fermi-Hubbard model for small grids up to 6 sites (12 qubits) by using an interpretable, adaptive variational quantum eigensolver(VQE) called ADAPT-VQE. In contrast with non-adaptive VQE, this algorithm builds a system-specific ansatz by adding an optimal gate built from one-body or two-body fermionic operators at each step. We show this adaptive method outperforms the non-adaptive counterpart in terms of fewer variational parameters, short gate depth, and scaling with the system size. The fidelity and energy of the prepared state appear to improve asymptotically with ansatz depth. We also demonstrate the application of adaptive variational methods by preparing excited states and Green functions using a proposed ADAPT-SSVQE algorithm. Lower depth, asymptotic convergence, noise tolerance of a variational approach, and a highly controllable, system-specific ansatz make the adaptive variational methods particularly well-suited for NISQ devices.

An energy- and charge-conserving electrostatic implicit particle-in-cell algorithm for simulations of collisional bounded plasmas

The present work investigates the applicability of the implicit energy- and charge-conserving particle-in-cell method to collisional bounded plasmas. The possible modifications might require modeling of the charge dynamics associated with a particle impinging on a bounding surface of different materials and coupling to an external RLC circuit. The former requires no change if the lowest order shape function is used for the current density assignment, whereas the method needs to be altered when the shape function is linear and particle recombination at the surface of a perfect conductor is assumed. The proposed fix consists of adding a virtual oppositely charged particle symmetrically with respect to the conductor boundary. A newly derived numerical form of the power balance equation contains only physical terms and demonstrates that the numerical heating is controlled solely by the accuracy of the energy-conserving iteration procedure. The external circuit's contribution is incorporated into the field iterations through Kirchhoff's laws for a system consisting of the plasma reactor and the external circuit. Due to the implicit nature of the proposed algorithm modification, it does not lead to numerical instabilites, and thus is more robust than explicit analogs. The energy-conserving PIC method can be employed for numerical modeling of a wide range of technological bounded plasmas, including those with high densities, which is demonstrated with 1D simulations of a capacitively coupled plasma discharge in Cartesian geometry. It is found that the algorithm remains stable even if the iterative procedure needed for the consistent update of the particle orbits and the electric field in the framework of an implicit method is stopped after the first iteration and the lowest order shape function is used for the electric field interpolation along with the conduction current assignment. The resulting solutions prove reasonably close to the more precise counterparts obtained with the energy conservation ensured to much better accuracy and with higher order shape functions. Since this makes the computational complexity of the new algorithm comparable to that of the explicit PIC code whereas utilizing much smaller grids is possible, the new algorithm should be preferred over the latter when simulating high-density plasmas.

Assessing the Geothermal Resource Potential of an Active Oil Field by Integrating a 3D Geological Model With the Hydro-Thermal Coupled Simulation

Assessment of available geothermal resources in the deep oil field is important to the synergy exploitation of oil and geothermal resources. A revised volumetric approach is proposed in this work for evaluating deep geothermal potential in an active oil field by integrating a 3D geological model into a hydrothermal (HT)-coupled numerical model. Based on the analysis of the geological data and geothermal conditions, a 3D geological model is established with respect to the study area, which is discretized into grids or elements represented in the geological model. An HT-coupled numerical model was applied based on the static geological model to approximate the natural-state model of the geothermal reservoir, where the thermal distribution information can be extracted. Then the geothermal resource in each small grid element is calculated using a volumetric method, and the overall geothermal resource of the reservoirs can be obtained by making an integration over each element of the geological model. A further parametric study is carried out to investigate the influence of oil and gas saturations on the overall heat resources. The 3D geological model can provide detailed information on the reservoir volume, while the HT natural-state numerical model addressed the temperature distribution in the reservoir by taking into account complex geological structures and contrast heterogeneity. Therefore, integrating the 3D geological modeling and HT numerical model into the geothermal resource assessment improved its accuracy and helped to identify the distribution map of the available geothermal resources, which indicate optimal locations for further development and utilization of the geothermal resources. The Caofeidian new town Jidong oil field serves as an example to depict the calculation workflow. The simulation results demonstrate in the Caofeidian new town geothermal reservoir that the total amount of geothermal resources, using the proposed calculation method, is found to be 1.23e+18 J, and the total geothermal fluid volume is 8.97e+8 m3. Moreover, this approach clearly identifies the regions with the highest potential for geothermal resources. We believe this approach provides an alternative method for geothermal potential assessment in oil fields, which can be also applied globally.

Blockchain-Based Distributed Renewable Energy Management Framework

Most of the world&#x2019;s countries are concerned with reducing harmful gas emissions. Some governments have made considerable attempts to address this problem. Saudi Arabia, for example, has taken significant steps to utilize renewable energy (RE) sources in addition to oil and gas. Consumers are encouraged to build small RE systems. These small grids will help people meet their daily electrical energy requirements. They can sell the excess to other customers. One of the major issues is managing the distribution and sale of RE. Blockchain-based peer-to-peer (P2P) networks can help overcome several obstacles to the implementation of a distributed RE management system (DREMS). However, several impediments may still stand in the way of its execution. Scalability and productivity are two of the most important considerations. The number of transactions made to the Blockchain network will increase in lockstep with the number of energy consumers. This will result in a significant lag in response. Understanding the renewable distributed energy system will aid in minimizing the effects of these roadblocks. Therefore, this research identifies the RE systems installation approaches, and how Blockchain technology can be utilized. It provides the solution requirements of any DREMS. Moreover, it proposes a new Blockchain-based framework for DREMS. And designs selective protocols of the proposed framework. The designed protocols are evaluated through a comparative analysis with the state of the art identified requirements.

Close binary evolution based on Gaia DR2

Context. The observed late-type WC Wolf-Rayet stars (WC7-9) with low luminosity below log L/L⊙ &lt; 5.4 in the HR diagram cannot be reproduced satisfactorily by the evolutionary track of single stars. The mass transfer due to Roche lobe overflow drastically modifies the internal structure and surface compositions of two components. Therefore, binaries provide a very promising evolutionary channel to produce these WC stars. Aims. The Gaia satellite provides accurate distances to WC stars and confirms the luminosities of WC stars. Based on a small grid containing single stars and binaries, we aim to investigate the extent to which the evolution of a single or a close binary can reproduce the properties of these stars. Methods. We considered single-star models with masses between 20 and 40 M⊙. We calculated the evolution for three binaries with a 30 M⊙ primary star with a 27 M⊙ companion star with initial orbital periods of 6.0, 20.0, 500.0, and 1000.0 days. Results. The rotating single star can evolve into a late-type WC star but with high luminosity (i.e., log L/L⊙ &gt; 5.4). Enhanced wind mass loss rates during RSG and WR stages, as proposed in the literature, can cause the star to approach the observational range of low-luminosity WC stars and favor the formation of low-luminosity WO stars. In a wide binary system with initial Porb = 1000 days, the primary star can evolve into a late-type WC star and be compatible with the observed properties of the low-luminosity WC stars. The result is almost insensitive to the adopted accretion efficiency 1 − β. Conclusions. Compared with single stars, the low brightness is due to a smaller temperature gradient inside the star after the Case C Roche lobe overflow, while the low effective temperature is due to envelope expansion. There are four physical reasons for the formation of the expanding envelope. Firstly, less helium envelope can be transferred to the companion star in this system. Heavy helium envelopes can be heated by the helium burning shell and this creates the necessary conditions for the envelope expansion. Secondly, the expansion of the helium envelope can also be boosted by the sharp shrinkage of the larger carbon-oxygen core through the mirror effect. Thirdly, a more massive WC star can attain a higher Eddington factor because of its higher L/M ratio. The increase in L/M with mass is the primary cause for the extended envelopes in WC stars. Finally, the iron opacity bump at T ∼ 105.25 K may also trigger envelope inflation because it can lead to a larger Eddington factor.