Line Charge Research Articles

Blackout Recovery Scenario in a Combined-Cycle Power Plant via Line Charging and Internal Cross-Supply: A Techno-Economic Comparative Analysis

The readiness of fast response power plants, such as Combined-Cycle Power Plant (CCCP), following a blackout in the power system shall be maintained to preserve the availability of the supply. Hence, blackout recovery scenario is usually prepared and considered as one of the measures to achieve the system readiness after blackout. This study presents a techno-economic comparative analysis between two blackout recovery methods, namely via line charging and internal cross-supply, in CCCP Priok, Indonesia. It analyzes the historical data of the relationship of the active power contribution to the frequency, and then obtains the appropriate settings for the power plant parameters. From the technical perspective, the gain value or participation factor of this plant is 49 MW/Hz with 6% droop setting and 0.029 Hz of deadband frequency. It is found that a load set point lower than 2.49 MW can lead to grid synchronization failure since there are self-consumption loads on each gas turbine. Moreover, to prevent the risk of reverse power and to achieve a successful internal cross-supply scenario, the minimum load setting shall be adjusted to 3 MW. Meanwhile, from an economic perspective, the results show that a successful internal cross-supply method may save up to IDR 2.7 billion compared with line charging method.

Development of a MATLAB‐based graphical user interface to illustrate Coulomb's law and Gauss's law

AbstractCoulomb's law and Gauss's law are two fundamental laws in electrostatics that involve electric charges and electric fields. These concepts that are not visible physically in real life can be challenging for students to understand, especially when it involves complex charge distributions. This paper presents a MATLAB‐based graphical user interface (GUI) to visualise the concept of Coulomb's law and Gauss's law as well as a serious snake game to reinforce users' knowledge. This MATLAB GUI features a diverse range of 3‐dimensional (3D) visualisations illustrating electric field attributes such as electric force, charge distribution and electric flux. These visualisations dynamically adapt to user‐defined parameters, offering a rich and varied exploration of the electric field's characteristics without confining the scope to a specific count of models. In Coulomb's law section, the GUI plots the electric forces in 3‐D for dipole, three‐point charges, and unlimited charge. The Gauss's law section consists of windows for illustration of the fundamentals of Gauss's law for the electric field, and the illustration for the application of Gauss's law in finding the electric field for a point charge, infinite line charge and infinite plane charge. The serious snake game developed in this work allows students to engage in quizzes related to Coulomb's law and Gauss's law to serve as a tool for reinforcing their learning outcomes. The MATLAB‐based GUI was chosen as the platform due to its excellent visualisation capabilities, ease of programming and capability to act as a powerful tool to enhance the learning of Coulomb's law and Gauss's law for students.

ВИЗНАЧЕННЯ ЗАРЯДНОЇ ЄМНОСТІ ПОВІТРЯНИХ ЛІНІЙ ЕЛЕКТРОПЕРЕДАЧІ З УРАХУВАННЯМ ПРОВИСАННЯ ПРОВОДУ

The paper presents the results of studying the effect of wire sagging on the charging capacity of an overhead power line. A refined mathematical model of the overhead line charging capacity is obtained. The results of the study of the error of neglect of wire sagging for typical structures of overhead power lines with voltage from 35 to 750 kV with the arrangement of wires in 2-3 tiers and horizontal arrangement are presented. The influence of changes in the wire sagging boom on the value of the power line charging capacity due to changes in the operating temperature of the wire and due to ice deposits is investigated. It is shown that for 750 kV power lines, such a change can reach 1-1.5%, which, in combination with the long length of the main power lines, can cause a change in the balance of reactive power in the line itself and in adjacent power systems. The possibility of monitoring the wire sagging boom in a run is investigated and the requirements for the accuracy of such measurements are determined. Bibl. 14, fig. 3, table.

Long-term irradiation study of sMDT drift tubes with an integrated accumulated charge of 60 C per wire using beta-electrons from a 90Sr source

Two ATLAS sMDT drift tubes have been irradiated for almost 1 year using a 90Sr beta-decay source. An integrated charge of 62C has been accumulated on each of both anode wires over an anode-wire region of about 7.5 cm. Taking into account the intensity distribution of the irradiation corresponds to a maximum accumulated line charge density of about 14C/cm. At the innermost position of the ATLAS forward muon spectrometer 10C/cm are expected for 10 years of high-luminosity LHC operation and for this detector type at gas gain 20000. To investigate potential outgassing, the endplug region of the drift-tubes, where no gas amplification occurs, was irradiated additionally using about half the beta-electrons emitted from the source. The other beta-electrons were irradiating an active part of the gas volume for monitoring purpose. During four months the endplugs were irradiated by 5C/cm equivalent. All observed anode currents were very stable over the whole period of irradiation and thus no sign of deterioration in the performance of both drift tubes was observed. This indicates that no ageing effects occurred and that no performance loss due to outgassing of any plastic surfaces has been observed. All components that have potential contact to the detector gas Ar:CO2 with a mixture of 93:7 (percent volume) have been carefully and properly chosen. The required cleanliness of all tube- and gas components has been achieved during construction and operation of these drift tubes.

Engineering 2D Material Exciton Line Shape with Graphene/h-BN Encapsulation.

Control over the optical properties of atomically thin two-dimensional (2D) layers, including those of transition metal dichalcogenides (TMDs), is needed for future optoelectronic applications. Here, the near-field coupling between TMDs and graphene/graphite is used to engineer the exciton line shape and charge state. Fano-like asymmetric spectral features are produced in WS2, MoSe2, and WSe2 van der Waals heterostructures combined with graphene, graphite, or jointly with hexagonal boron nitride (h-BN) as supporting or encapsulating layers. Furthermore, trion emission is suppressed in h-BN encapsulated WSe2/graphene with a neutral exciton red shift (44 meV) and binding energy reduction (30 meV). The response of these systems to electron beam and light probes is well-described in terms of 2D optical conductivities of the involved materials. Beyond fundamental insights into the interaction of TMD excitons with structured environments, this study opens an unexplored avenue toward shaping the spectral profile of narrow optical modes for application in nanophotonic devices.

Experimental Study and 3-D Modeling of Positive Streamer Discharges Under the Effect of Humidity

The research on streamer discharges not only provides support for insulation design of high-voltage power transmission systems, but also lays a theoretical foundation for lightning discharges. The macroscopic parameters of streamer discharges have been investigated, but there is a lack of analysis on the development and branching characteristics of streamers under the effect of humidity. Besides, different types of models have been built to simulate streamer propagation. However, there are few 3D models which simulate branching process of multiple streamers at different humidity. In this work, a discharge experimental platform with controllable environmental parameters has been built. The effect of the humidity of 4.5~19.1 g/m <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">3</sup> on streamer propagation and branching has been experimentally studied. Then the 3D “discharge tree” model of collective streamers is improved, in which the streamer branching incorporates the photoionization in humid air. Furthermore, we compare the results of the 3D simulation model with the experimental data. Through the simulation model, the streamer parameters such as line charge density and electric field distribution are obtained. The results show that the electric field of streamer channels are larger when the humidity is higher, but the propagation velocities are reduced. The humidity has limited effect on the streamer branching ratio.

Parallelization of Garfield++ and neBEM to simulate space-charge effects in RPCs

Numerical simulation of avalanches, saturated avalanches, and streamers can help us understand the detector physics of Resistive Plate Chambers (RPC). 3D Monte Carlo simulation of an avalanche inside an RPC, the transition from avalanche to saturated avalanche to streamer may help the search for the optimum voltage and alternate gas mixtures. This task is dauntingly resource hungry, especially when space-charge effects become important, which often coincides with important regimes of operation of these devices. By modifying the electric field inside the RPC dynamically, the space charge plays a crucial role in determining the response of the detector. In this work, a numerical model has been proposed to calculate the dynamic space-charge field inside an RPC and the same has been implemented in the Garfield++ framework. By modeling space charge as large number of line charges and using the multithreading technique OpenMP to calculate electric field, drift line, electron gain, and space-charge field, it has been possible to maintain time consumption within reasonable limits. For this purpose, a new class, pAvalancheMC has been introduced in Garfield++. The calculations have been successfully verified with those from existing solvers and an example is provided to show the performance of pAvalancheMC. Moreover, the details of the transition of an avalanche into a saturated avalanche have been discussed. The induced-charge distribution is calculated for a timing RPC and results are verified with the experiment.

Numerical simulation of electric field distribution inside streamer zones of positive and negative lightning leaders

It is widely known that there is an asymmetry in development modes of positive and negative leaders. Negative leaders in air always propagate in a step-wise manner, while positive ones can grow either continuously (at least optically) or via steps (restrikes). The step-formation mechanism in negative and especially positive leaders is poorly understood. In particular, it is still unknown how space stems, precursors of space leaders, form inside and/or at the boundary of the streamer corona, where an expected electric field is sufficiently smaller than the dielectric strength of air. The study presents numerical simulation of electrostatic conditions inside streamer zones of positive and negative lightning leaders. It is shown that, regardless of polarity, streamer coronas are characterized by highly nonuniform electric field distribution. Moreover, it appears that they always contain the areas of locally increased electric fields sometimes exceeding the breakdown level that can be the potential places of space stems origin. Due to the well-known asymmetry between propagation threshold fields of positive and negative streamers, the number of such zones and the magnitudes of electric fields inside them are sufficiently bigger for negative leaders. A significant factor influencing the level of electric field amplification is found to be the share of streamer zone channels that deviate from an expected direction of growth. The role of the streamer corona fractal dimension and the leader channel sheath line charge density is also analyzed. The model predicts that the space stem formation conditions worsen with increasing altitude above the mean sea level and get better with rising humidity which is in line with observations. The study is helpful for further development of the theory of propagation mechanisms of positive and negative leaders.

On the Child–Langmuir law in one, two, and three dimensions

We consider the limiting current from an emitting patch whose size is much smaller than the anode–cathode spacing. The limiting current is formulated in terms of an integral equation. It is solved iteratively, first to numerically recover the classical one-dimensional Child–Langmuir law, including Jaffe's extension to a constant, nonzero electron emission velocity. We extend to two-dimensions in which electron emission is restricted to an infinitely long stripe with infinitesimally narrow stripe width so that the emitted electrons form an electron sheet. We next extend to three-dimensions in which electron emission is restricted to a square tile (or a circular patch) with an infinitesimally small tile size (or patch radius) so that the emitted electrons form a needlelike line charge. Surprisingly, for the electron needle problem, we only find the null solution for the total line charge current, regardless of the assumed initial electron velocity. For the electron sheet problem, we also find only the null solution for the total sheet current if the electron emission velocity is assumed to be zero, and the total maximum sheet current becomes a finite, nonzero value if the electron emission velocity is assumed to be nonzero. These seemingly paradoxical results are shown to be consistent with the earlier works of the Child–Langmuir law of higher dimensions. They are also consistent with, or perhaps even anticipated by, the more recent theories and simulations on thermionic cathodes that used realistic work function distributions to account for patchy, non-uniform electron emission. The mathematical subtleties are discussed.

Monovalent cations have different effects on the assembly kinetics and morphology of α-synuclein amyloid fibrils

Formation of α-synuclein amyloid fibrils is a pathological hallmark of Parkinson's disease and a phenomenon that is strongly modulated by environmental factors. Here, we compared effects of different monovalent cations (Li+, Na+, K+) on the formation and properties of α-synuclein amyloid fibrils. Na+ > Li+ were found to have concentration-dependent catalytic effects on primary nucleation whereas K+ ions acted inhibitory. We discuss this discrepancy in terms of a superior affinity of Na+ and Li+ to carboxylic protein groups, resulting in reduced Columbic repulsion and by considering K+ as an ion with poor protein binding and slight chaotropic character, which could promote random coil protein structure. K+ ions, furthermore, appeared to lower the β-sheet content of the fibrils and increase their persistence lengths, the latter we interpret as a consequence of lesser ion binding and hence higher line charge of the fibrils. The finding that Na+ and K+ have opposite effects on α-synuclein aggregation is intriguing in relation to the significant transient gradients of these ions across axonal membranes, but also important for the design and interpretation of biophysical assays where buffers containing these monovalent cations have been intermixedly used.

Mono-valent salt corrections for RNA secondary structures in the ViennaRNA package

BackgroundRNA features a highly negatively charged phosphate backbone that attracts a cloud of counter-ions that reduce the electrostatic repulsion in a concentration dependent manner. Ion concentrations thus have a large influence on folding and stability of RNA structures. Despite their well-documented effects, salt effects are not handled consistently by currently available secondary structure prediction algorithms. Combining Debye-Hückel potentials for line charges and Manning’s counter-ion condensation theory, Einert et al. (Biophys J 100: 2745-2753, 2011) modeled the energetic contributions of monovalent cations on loops and helices.ResultsThe model of Einert et al. is adapted to match the structure of the dynamic programming recursion of RNA secondary structure prediction algorithms. An empirical term describing the salt dependence of the duplex initiation energy is added to improve co-folding predictions for two or more RNA strands. The slightly modified model is implemented in the ViennaRNA package in such way that only the energy parameters but not the algorithmic structure is affected. A comparison with data from the literature show that predicted free energies and melting temperatures are in reasonable agreement with experiments.ConclusionThe new feature in the ViennaRNA package makes it possible to study effects of salt concentrations on RNA folding in a systematic manner. Strictly speaking, the model pertains only to mono-valent cations, and thus covers the most important parameter, i.e., the NaCl concentration. It remains a question for future research to what extent unspecific effects of bi- and tri-valent cations can be approximated in a similar manner.AvailabilityCorrections for the concentration of monovalent cations are available in the ViennaRNA package starting from version 2.6.0.

Quantitative electric field mapping between electrically biased needles by scanning transmission electron microscopy and electron holography

Stray electric fields in free space generated by two biased gold needles have been quantified in comprehensive finite-element (FE) simulations, accompanied by first moment (FM) scanning TEM (STEM) and electron holography (EH) experiments. The projected electrostatic potential and electric field have been derived numerically under geometrical variations of the needle setup. In contrast to the FE simulation, application of an analytical model based on line charges yields a qualitative understanding. By experimentally probing the electric field employing FM STEM and EH under alike conditions, a discrepancy of about 60% became apparent initially. However, the EH setup suggests the reconstructed phase to be significantly affected by the perturbed reference wave effect, opposite to STEM where the field-free reference was recorded subsequently with unbiased needles in which possibly remaining electrostatic influences are regarded as being minor. In that respect, the observed discrepancy between FM imaging and EH is resolved after including the long-range potential landscape from FE simulations into the phase of the reference wave in EH.

SOLPS-ITER predictive simulations of the impact of ion-molecule elastic collisions on strongly detached MAST-U Super-X divertor conditions

The role of ion-molecule () elastic collisions in strongly detached divertor conditions has been studied in the MAST-U Super-X configuration using SOLPS-ITER. Two strongly detached steady state solutions were compared, one obtained through a main-ion fuelling scan and the other through a nitrogen seeding scan at fixed fuelling rate. A significant difference in the electron–ion recombination (EIR) levels was observed; significant EIR in strongly detached conditions in the fuelling scan and negligible EIR throughout the seeding scan. This is partly because the fuelling scan achieves electron temperatures () as low as 0.2 eV near the divertor target, compared to 0.8 eV in the seeding scan (EIR increases strongly below eV), and partly due to higher divertor plasma densities achieved in fuelling scan. Features of the strongly detached seeded cases, i.e. higher temperatures and negligible EIR, are recovered in the fuelling scan by turning off elastic collisions. Analysis suggests that dissipation mechanisms like line radiation and charge exchange (important for detachment initiation) become weak when falls below 1 eV, and that elastic collisions are necessary for further heat dissipation and access to strongly recombining conditions in the fuelling scan. In the seeding scan, heat dissipation through elastic collisions is weak. This could be because our nitrogen seeding simulations do not include interactions between nitrogen ions and neutrals, and the strongly detached cases contain high levels of in the divertor. As a result, the acts like a reservoir of energy and momentum which appears to weaken the impact of elastic collisions on the divertor plasma energy and momentum balance, making it more difficult to access recombining conditions. This suggests that some of the differences between seeding and fuelling scans could be because energy and momentum exchange between impurities and neutrals is not sufficiently captured in our simulations.

Application of the Wave Channels Method to Analyze the Switching Overvoltages in Energizing a High-voltage Cable Line

A study of cable line failures shows that a large number of faults occur in the connecting and terminal couplings of 110--500 kV power lines. Overvoltages caused by operational switching in the substation switchgear circuits contribute to the total number of failures. The switching overvoltage amplitude and frequency depend on the parameters of the source, switched circuits, and switching moment. The aim of the study is to identify the network and switched circuit parameters affecting the maximum overvoltage excited by energizing a 400 kV cable line and to determine the degree of their influence. The following factors are considered: switching phase, cable line length, source inductance, and line residual charge. The analysis was carried out by the wave channels method. To achieve this goal, an analysis model was developed in the EMTP-RV software environment, which made it possible to calculate the electromagnetic transients excited by cable line switching. The model has been verified with respect to the surge propagation velocity along a coaxial wave channel. Overvoltage calculations have shown that the switching phase and the line charge availability in the pre-switching operation mode have the most significant effect on the overvoltage value. The obtained study results apply to high-voltage electrical networks containing cable lines. The study results made it possible to quantify the degree to which the considered factors influence the overvoltages excited by switching 400 kV cable lines and to assess possible exceeding of the values permitted by the relevant standards, which will make it possible to have a preventive influence on the number of failures.

Modelling needle domains with dislocations: A dislocation pile-up model and comparison with synchrotron experiment

The shape and strain field of a needle domain in a barium titanate single crystal are modelled using a distribution of dislocations and line charges. The arrangement of these dislocations and charges is a result of the balance of modified Peach–Koehler forces acting among the dislocations and a lattice friction assumed to act at each dislocation site. Based on measurements of needle shape by synchrotron X-ray diffraction, dislocation pile-up theory is used to compute the distribution of discrete dislocations along the needle and hence estimate the lattice friction. It is found that the lattice friction in this model is proportional to the opening angle of a wedge-shape needle domain and consistent with the observed magnitude of stress required to mobilize needle domains. The microstrain distribution around an a-a needle domain tip, obtained from X-ray diffraction measurement, is further used to test the dislocation model, with a similar pattern and magnitude of strains identified in the model and the experiment.

Emergency Collision Avoidance and Mitigation Using Model Predictive Control and Artificial Potential Function

Although extensive research in emergency collision avoidance has been carried out for straight or curved roads in a highway scenario, a general method that could be implemented for all road environments has not been thoroughly explored. Moreover, most current algorithms don't consider collision mitigation in an emergency. This functionality is essential since the problem may have no feasible solution. We propose a safe controller using model predictive control and artificial potential function to address these problems. A new artificial potential function inspired by line charge is proposed as the cost function for our model predictive controller. The vehicle dynamics and actuator limitations are set as constraints. The new artificial potential function considers the shape of all objects. In particular, the artificial potential function we proposed has the flexibility to fit the shape of the road structures, such as the intersection. We could also realize collision mitigation for a specific part of the vehicle by increasing the charge quantity at the corresponding place. We have tested our methods in 192 cases from 8 different scenarios in simulation with two different models. The simulation results show that the success rate of the proposed safe controller is 20 <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\%$</tex-math></inline-formula> higher than using HJ-reachability with system decomposition by using a unicycle model. It could also decrease 43 <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\%$</tex-math></inline-formula> of collision that happens at the pre-assigned part. The method is further validated in a dynamic bicycle model.

Varvec{J}-, M- and varvec{L}-integrals of line charges and line forces

In this work, the varvec{J}-, M- and varvec{L}-integrals of two line charges and two line forces in generalized plane strain are derived in the framework of three-dimensional (3D) electrostatics and three-dimensional compatible linear elasticity, respectively, in order to study the interaction between them. The key point in this derivation is achieved by expressing the varvec{J}-, M- and varvec{L}-integrals of point charges and point forces in three dimensions in terms of the corresponding three-dimensional Green functions, that is, the three-dimensional Green function of the Laplace operator and the three-dimensional Green tensor of the Navier operator, respectively. The major mathematical tool used in deriving the varvec{J}-, M- and L_3-integrals of line charges and line forces from the corresponding varvec{J}-, M- and varvec{L}-integrals of point charges and point forces is the method of embedding or method of descent of Green functions to two dimensions (2D) from the corresponding Green functions in 3D. The analytical expressions of varvec{J}-, M- and L_3-integrals of line charges and line forces in antiplane and plane strain are derived and discussed. The varvec{J}-integral is the electrostatic part of the Lorentz force (electrostatic interaction force) between two line charges in electrostatics and the Cherepanov force between two line forces in elasticity. The M-integral of two line sources (charges or forces) equals half the electrostatic interaction energy in electrostatics and half the elastic interaction energy in elasticity of these two line sources, respectively. The L_3-integral of two line sources (charges or forces) is the z-component of the configurational vector moment or the rotational moment representing the total torque about the z-axis caused by the interaction of the two line sources. An important outcome is that the varvec{J}-integral is twice the negative gradient of the M-integral, leading to the result that the {varvec{J}}-integral for line charges and line forces is a conservative force (with the corresponding interaction energy playing the role of the potential energy) and consequently an irrotational vector field. Finally, the obtained varvec{J}-, M- and L_3-integrals being functions of the distance and of the angle, are able to describe the physical behaviour of the interaction of two line sources (charges and forces), since they represent the fundamental and necessary quantities, that is, the interaction force, interaction energy and total torque produced by the interaction of these two line sources.

A New Model to Calculate Contributions of the Distributed Power

Besides metering, a more transparent load with known power distribution is valuable to draft energy strategies, especially in a deregulated power market. Thus, the usage-of-transmission can be considered properly. There are many studies published on related subjects and every research tried to solve a particular part of the problem. There are three basic categories for discussing power distributions: (i) the real power distribution, (ii) the reactive power distribution, and (iii) the loss allocation. These categories were very often treated separately, with the mutual coupling terms, counter flows, and line charging largely neglected. However, we know that these entities are non-separable. These are inter-related entities; the change of one entity will cause the change to every other entity. A good method should consider these entities altogether, while satisfying all electrical theories. This study developed a method to solve the above problem, with all electrical entities solved, satisfying all electrical circuit theories. With several matrix formulations, this method is capable of solving and tracing all electrical entities, including the current flow, the real and reactive power, the counter flow, and the couplings between the active and reactive power. The algorithm can also allocate power distributions and loss among participants effectively. Besides, a line usage idea is formed to allocate the loss to each generator, where counter flows are not necessarily penalized. It can be awarded sometimes. The idea can integrate with the existent tariffs in a deregulated market.

Enhanced two-terminal impedance-based fault location using sequence values

&lt;span lang="EN-US"&gt;Fault at transmission line system may lead to major impacts such as power quality problems and cascading failure in the grid system. Thus, it is very important to locate it fast so that suitable solution can be taken to ensure power system stability can be retained. The complexity of the transmission line however makes the fault point identification a challenging task. This paper proposes an enhanced fault detection and location method using positive and negative-sequence values of current and voltage, taken at both local and remote terminals. The fault detection is based on comparison between the total fault current with currents combination during the pre-fault time. While the fault location algorithm was developed using an impedance-based method and the estimated fault location was taken at two cycles after fault detection. Various fault types, fault resistances and fault locations have been tested in order to verify the performance of the proposed method. The developed algorithms have successfully detected all faults within high accuracy. Based on the obtained results, the estimated fault locations are not affected by fault resistance and line charging current. Furthermore, the proposed method able to detect fault location without the needs to know the fault type.&lt;/span&gt;

Multidimensional Optimization of CSM With Different Algorithms for Investigating the Electric Field in the Vicinity of a Point Electrode

In this article, the combined charge simulation method (CSM) with the optimization methods for calculating the electric field produced by a point electrode (hyperbolic needle) is introduced. The simulation has been carried out using the combination between ring charges and finite line charges with variable charge density for the onset field simulation. In CSM, the combination of different types of these fictitious charges increases the degree of consistency between the results and avoids disparity. To determine the location of these charges, three different optimization algorithms are used [particle swarm optimization (PSO), gray wolf optimization (GWO), and bald eagle search (BES)] as a comparative case study. The presented model for investigating the electric field due to a point electrode is a multidimensional field problem. Optimization results and discussion confirm that the BES results, compared with other used algorithms, are the best results and offer the most stable performance with repeated operations. The best value of maximum percent error of electric field is 0.068 using the BES algorithm. Finally, the simulation results have been compared with the available analytical results, at axial line of needle-to-plane configuration, and previous works.