A Tutorial Review of the Solar Power Curve: Regressions, Model Chains, and Their Hybridization and Probabilistic Extensions

Owing to the persisting hype in pushing toward global carbon neutrality,the study scope of atmospheric science is rapidly expanding.Among numerous trending topics,energy meteorology has been attracting the most attention hitherto.One essential skill of solar energy meteorologists is solar power curve modeling,which seeks to map irradiance and auxiliary weather variables to solar power,by statistical and/or physical means.In this regard,this tutorial review aims to deliver a complete overview of those fundamental scientific and engineering principles pertaining to the solar power curve.Solar power curves can be modeled in two primary ways,one of regression and the other of model chain.Both classes of modeling approaches,alongside their hybridization and probabilistic extensions,which allow accuracy improvement and uncertainty quantification,are scrutinized and contrasted thoroughly in this review.

Discussion on relationship between minimal energy and curve shapes

Energy minimization has been widely used for constructing curve and surface in the fields such as computer-aided geometric design, computer graphics. However, our testing examples show that energy minimization does not optimize the shape of the curve sometimes. This paper studies the relationship between minimizing strain energy and curve shapes, the study is carried out by constructing a cubic Hermite curve with satisfactory shape. The cubic Hermite curve interpolates the positions and tangent vectors of two given endpoints. Computer simulation technique has become one of the methods of scientific discovery, the study process is carried out by numerical computation and computer simulation technique. Our result shows that： （1） cubic Hermite curves cannot be constructed by solely minimizing the strain energy; （2） by adoption of a local minimum value of the strain energy, the shapes of cubic Hermite curves could be determined for about 60 percent of all cases, some of which have unsatisfactory shapes, however. Based on strain energy model and analysis, a new model is presented for constructing cubic Hermite curves with satisfactory shapes, which is a modification of strain energy model. The new model uses an explicit formula to compute the magnitudes of the two tangent vectors, and has the properties： （1） it is easy to compute; （2） it makes the cubic Hermite curves have satisfactory shapes while holding the good property of minimizing strain energy for some cases in curve construction. The comparison of the new model with the minimum strain energy model is included.

Efficient Construction of B-Spline Curves with Minimal Internal Energy

In this paper,we propose an efficient method to construct energy-minimizing B-spline curves by using discrete mask method.The linear relations between control points are firstly derived for different energy-minimization problems,then the construction of B-spline curve with minimal internal energy can be addressed by solving a sparse linear system.The existence and uniqueness of the solution for the linear system are also proved.Experimental results show the efficiency of the proposed approach,and its application in 1 G blending curve construction is also presented.

Ab initio calculation of accurate dissociation energy, potential energy curve and dipole moment function for the A^1∑＋ state ^7LiH molecule

The reasonable dissociation limit of the A^1∑＋ state ^7LiH molecule is obtained. The accurate dissociation energy and the equilibrium geometry of this state are calculated using a symmetry-adapted-cluster configuration-interaction method in complete active space for the first time, The whole potential energy curve and the dipole moment function for the A^1∑＋ state are calculated over a wide internuclear separation range from about 0.1 to 1.4 nm. The calculated equilibrium geometry and dissociation energy of this potential energy curve are of Re=0.2487 nm and De=1.064eV, respectively. The unusual negative values of the anharmonicity constant and the vibration-rotational coupling constant are of ωeXe=-4.7158cm^-1 and αe=0.08649cm^-1, respectively. The vertical excitation energy from the ground to the A^1∑＋ state is calculated and the value is of 3.613eV at 0.15875nm （the equilibrium position of the ground state）. The highly anomalous shape of this potential energy curve, which is exceptionally flat over a wide radial range around the equilibrium position, is discussed in detail. The harmonic frequency value of 502.47cm^-1 about this state is approximately estimated. Careful comparison of the theoretical determinations with those obtained by previous theories about the A^1∑＋ state dissociation energy clearly shows that the present calculations are much closer to the experiments than previous theories, thus represents an improvement.

Potential energy curves and analytical potential energy functions of the metastable states of B2^＋＋

The multi-reference configuration interaction method and aug-cc-pvqz （AVQZ） have been used to calculate potential energy curves （PECs） of the singlet and triplet states of the riu and rig symmetry of B2＋＋. All of the four states （^l∏u, ^1∏g, ^3∏u and ^3∏g） are found to be metastable states, though the potential well of ^3∏u symmetry is very shallow. Based on the PECs, the analytical potential energy functions （APEFs） of these states have been fitted using the least square fitting method and two models of function. The spectroscopic parameters of each state are also calculated, and are compared with other investigations in the literature. The credibility and veracity of the two functions are evaluated. Some ideas to improve the fitting accuracy are presented. Also the vibrational levels for each state are predicted by solving the SchrSdinger equation of nuclear motion.

Multireference configuration interaction potential curve and analytical potential energy function of the ground and low-lying excited states of CdSe

The potential energy curves （PECs） of the ground state （^3∏） and three low-lying excited states （^1∑, ^3∑,^1∏） of CdSe dimer have been studied by employing quasirelativistic effective core potentials on the basis of the complete active space self-consistent field method followed by multireference configuration interaction calculation. The four PECs are fitted to analytical potential energy functions using the Murrel-Sorbie potential function. Based on the PECs, the vibrational levels of the four states are determined by solving the Schrodinger equation of nuclear motion, and corresponding spectroscopic constants are accurately calculated. The equilibrium positions as well as the spectroscopic constants and the vibrational levels are reported. By our analysis, the ^3∏ state, of which the dissociation asymptote is Cd（^1S） ＋ Se（^3p）, is identified as a ground state of CdSe dimer, and the corresponding dissociation energy is estimated to be 0.39eV. However, the first excited state is only 1132.49cm^-1 above the ground state and the ^3∑ state is the highest in the four calculated states.

Adaptive Error Curve Learning Ensemble Model for Improving Energy Consumption Forecasting

Despite the advancement within the last decades in the field of smart grids,energy consumption forecasting utilizing the metrological features is still challenging.This paper proposes a genetic algorithm-based adaptive error curve learning ensemble(GA-ECLE)model.The proposed technique copes with the stochastic variations of improving energy consumption forecasting using a machine learning-based ensembled approach.A modified ensemble model based on a utilizing error of model as a feature is used to improve the forecast accuracy.This approach combines three models,namely CatBoost(CB),Gradient Boost(GB),and Multilayer Perceptron(MLP).The ensembled CB-GB-MLP model’s inner mechanism consists of generating a meta-data from Gradient Boosting and CatBoost models to compute the final predictions using the Multilayer Perceptron network.A genetic algorithm is used to obtain the optimal features to be used for the model.To prove the proposed model’s effectiveness,we have used a four-phase technique using Jeju island’s real energy consumption data.In the first phase,we have obtained the results by applying the CB-GB-MLP model.In the second phase,we have utilized a GA-ensembled model with optimal features.The third phase is for the comparison of the energy forecasting result with the proposed ECL-based model.The fourth stage is the final stage,where we have applied the GA-ECLE model.We obtained a mean absolute error of 3.05,and a root mean square error of 5.05.Extensive experimental results are provided,demonstrating the superiority of the proposed GA-ECLE model over traditional ensemble models.

Ab initio MRCI+Q study on potential energy curves and spectroscopic parameters of low-lying electronic states of CS^+

Carbon monosulfide molecular ion （CS＋）, which plays an important role in various research fields, has long been attracting much interest. Because of the unstable and transient nature of CS＋, its electronic states have not been well investigated. In this paper, the electronic states of CS＋ are studied by employing the internally contracted multireference configuration interaction method, and taking into account relativistic effects （scalar plus spin–orbit coupling）. The spin–orbit coupling effects are considered via the state-interacting method with the full Breit–Pauli Hamiltonian. The potential energy curves of 18 Λ–S states correlated with the two lowest dissociation limits of CS＋ molecular ion are calculated, and those of 10 lowest Ω states generated from the 6 lowest Λ–S states are also worked out. The spectroscopic constants of the bound states are evaluated, and they are in good agreement with available experimental results and theoretical values. With the aid of analysis of Λ–S composition of Ω states at different bond lengths, the avoided crossing phenomena in the electronic states of CS＋ are illuminated. Finally, the single ionization spectra of CS （X1Σ＋） populating the CS＋（X2Σ1/2＋, A2Π3/2, A2Π1/2, and B2Σ1/2＋） states are simulated. The vertical ionization potentials for X2Σ1/2＋, A2Π3/2, A2Π1/2, and B2Σ1/2＋ states are calculated to be 11.257, 12.787, 12.827, and 15.860 eV, respectively, which are accurate compared with previous experimental results, within an error margin of 0.08 eV^0.2 eV.

Potential Energy Curves &Material Properties

Potential energy curves govern the properties of materials. A critical analysis of the potential energy curve helps better understand the properties of the material. Potential energy curve and in turn the properties of any material depend on the composition, bonding, crystal structure, their mechanical processing and microstructure. The type, strength, and directionality of atomic bonding controls the structure and material properties viz., melting temperature, thermal expansion, elastic stiffness, electrical properties, ductility and toughness etc. This paper attempts to bring out the correlation between the potential energy curves with the properties of materials.

Nonlinear density wave and energy consumption investigation of traffic flow on a curved road

A new car-following model is proposed based on the full velocity difference model（FVDM） taking the influence of the friction coefficient and the road curvature into account. Through the control theory, the stability conditions are obtained,and by using nonlinear analysis, the time-dependent Ginzburg-Landau（TDGL） equation and the modified Korteweg-de Vries（mKdV） equation are derived. Furthermore, the connection between TDGL and mKdV equations is also given. The numerical simulation is consistent with the theoretical analysis. The evolution of a traffic jam and the corresponding energy consumption are explored. The numerical results show that the control scheme is effective not only to suppress the traffic jam but also to reduce the energy consumption.

Energy-Minimizing Curve Fitting for High-Order Surface Mesh Generation

We investigate different techniques for fitting Bézier curves to surfaces in context of high-order curvilinear mesh generation. Starting from distance-based least-squares fitting we develop an incremental algorithm, which incorporates approximations of stretch and bending energy. In the process, the algorithm reduces the energy weight in favor of accuracy, leading to an optimized set of sampling points. This energy-minimizing fitting strategy is applied to analytically defined as well as triangulated surfaces. The results confirm that the proposed method straightens and shortens the curves efficiently. Moreover the method preserves the accuracy and convergence behavior of distance-based fitting. Preliminary application to surface mesh generation shows a remarkable improvement of patch quality in high curvature regions.

Accurate calculation of the potential energy curve and spectroscopic parameters of X^2Σ^+ state of ^(12)Mg^1H

High level calculations on the ground state of 12Mg1H molecule have been performed using multi-reference configuration interaction （MRCI） method with the Davidson modification. The core-valence correlation and scalar relativistic corrections are included into the present calculations at the same time. The potential energy curve （PEC） of the ground state, all of the vibrational levels and spectroscopic parameters are fitted. The results show that the levels and spectroscopic parameters are in good agreement with the available experimental data. The analytical potential energy function （APEF） is also deduced from the calculated PEC using the Murrell-Sorbie （M-S） potential function. The present results can provide a helpful reference for the future spectroscopic experiments or dynamical calculations of the molecule.

Self-optimizing Diffusion Quantum Monte Carlo Calculation: the Potential Energy Curve of C_2

In this paper we propose a novel diffusion quantum Monte Carlo algo-rithm, it is a self-optimizing and self-improving procedure. The method has been em-ployed to calculate the potential energy curve of C2. The total energies for the X 1Σg+state of C2 were calculated at seven values of the bond length: 0. 106, 0. 111, 0. 124,0. 132, 0. 143, 0. 159 and 0. 185 nm; and a smooth potential energy curve was ob-tained, because when the self-optimizing technique is used, the statistical error decreas-es tremendously. The calculation results on the potential energy curve of C2 show thatthe self-optimizing diffusion quantum Monte Carlo method proposed in the present pa-per is successful.

Theoretical study of potential energy curves,spectroscopic constants,and radiative lifetimes of low-lying states in an SeO molecule

The low-lying potential energy curves of the SeO molecule are computed by means of an ab initio multireference configuration interaction technique, taking into account relativistic （scalar plus spin~）rbit coupling） effects. The spectroscopic constants of Ω states for X3∑-, a1△, b2∑＋, A3П , A′3△, and A″ 3∑＋ states are obtained, and they are in good accordance with available experimental values. The Franck-Condon factors and transition dipole moments to the ground state are computed, and the natural radiative lifetimes of low-lying Ω states are theoretically obtained. Comparisons of the natural lifetimes of Ω states with previous experimental results and those of isovalent TeO molecule are made.

Potential energy curves, transition dipole moments, and radiative lifetimes of KBe molecule

An ab initio calculations on the ground and low-lying excited statesX2∑＋,2^2∑＋,3^2∑＋,1^4∏,2^4∏,1^4∑＋,2^4∑＋,and 3^4∑＋of KBe molecule have been performed using multireference configuration interaction （MRCI） plus Davidsoncorrections （MRCI＋Q） approach with all electron basis set aug-cc-pCV5Z-DK for Be and def2-AQZVPP-JKFI for K.3^2∑＋,1^4∏,2^4∏,1^4∑＋,2^4∑＋and3^4∑＋states are investigated for the first time. Inner shell electron correlations are computed on the potential energy curves （PECs） calculations. The spectroscopic and molecular parameters are also predicted. In addition, The transition properties including transition dipole moment, Franck-Condon factors qv′v″, Einstein coefficients Av′v″, and the radiative lifetimesτ′for the2^2∑＋-X2∑＋,3^2∑＋-X2∑＋,and 2^4∏-1^4∏ transitions are predictedat the same time.

Potential energy curves and spectroscopic properties of X^2Σ^+ and A^2Π states of ^(13)C^(14)N

The potential energy curves （PECs） of X2∑ and A2П states of the CN molecule have been calculated with the multi- reference configuration interaction method and the aug-cc-pwCVSZ basis set. Based on the PECs, all of the vibrational and rotational levels of the 13C14N molecule are obtained by solving the Schrrdinger equation of the molecular nuclear motion. The spectroscopic parameters are determined by fitting the Dunham coefficients with the levels. Both the levels and the spectroscopic parameters are in good qualitative agreement with the experimental data available. The analytical potential energy functions are also deduced from the calculated PECs. The present results can provide a helpful reference for future spectroscopy experiments or dynamical calculations of the molecule.

Theoretical Study on the Potential Energy Curve and Vibration-rotation Spectra of BeF Radical

The potential energy curve （PEC） of BeF（X2Σ＋） radical is investigated by using the complete active space self-consistent field （CASSCF） method followed by the highly accurate valence internally contracted multireference configuration interaction （MRCI） approach over the internuclear separation range from 0.0522 to 2.0472 nm. The PEC is fitted to the analytic Murrell-Sorbie function, which is employed to accurately determine the spectroscopic parameters. The present D0, De, Re, ωe, ωeχe, αe and Be are 6.14 eV, 6.22 eV, 0.1372 nm, 1236.12 cm-1, 9.11 cm-1, 0.0175 cm-1 and 1.4651 cm-1, respectively. These parameters have been compared with those of previous investigations reported in the literature. With PEC determined at the present level of theory, a total of 75 vibrational states have been predicted for the first time by numerically solving the radial Schrdinger equation of nuclear motion using the Numerov method. For each vibrational state, the complete vibrational levels, classical turning points, inertial rotation and centrifugal distortion constants are determined for the first time. Comparing with the available experiments and other theories, we find that the present spectroscopic parameter and molecular constant results are more accurate and complete than the previous theoretical investigations.

Planar Cardinal Spline Curves with Minimum Strain Energy

In this paper, we focus on how to use strain energy minimization to obtain the optimal value of the fl＇ee parameter of the planar Cardinal spline curves. The unique solution can be easily obtained by minimizing an appropriate approximation of the strain energy. An example is presented to illustrate the effectiveness of our method.

A new method for testing pile by sin- gle-impact energy and P-S curve

By studying the pile-formula and stress-wave methods (e.g., CASE method), the authors propose a new method for testing piles using the single-impact energy and P-S curves. The vibration and wave figures are recorded, and the dynamic and static displacements are measured by different transducers near the top of piles when the pile is im- pacted by a heavy hammer or micro-rocket. By observing the transformation coefficient of driving energy (total energy), the consumed energy of wave motion and vibration and so on, the vertical bearing capacity for single pile is measured and calculated. Then, using the vibration wave diagram, the dynamic relation curves between the force (P) and the displacement (S) is calculated and the yield points are determined. Using the static-loading test, the dynamic results are checked and the relative constants of dynamic-static P-S curves are determined. Then the sub- sidence quantity corresponding to the bearing capacity is determined. Moreover, the shaped quality of the pile body can be judged from the formation of P-S curves.

The Origin, Properties and Detection of Dark Matter and Dark Energy

The pictures from the James Webb Space Telescope (JWST) suggest that massive galaxies were already at the beginning of the expansion of the Universe because there was too short time to create them. It is consistent with the new cosmology presented within the Scale-Symmetric Theory (SST). The phase transitions of the initial inflation field described in SST lead to the Protoworld—its core was built of dark matter (DM). We show that the DAMA/LIBRA annual-modulation amplitude forced by the change of the Earth’s velocity (i.e. baryonic-matter (BM) velocity) in relation to the spinning DM field in our Galaxy’s halo should be very low. We calculated that in the DM-BM weak interactions are created single and entangled spacetime condensates with a lowest mass/energy of 0.807 keV—as the Higgs boson they can decay to two photons, so we can indirectly detect DM. Our results are consistent with the averaged DAMA/LIBRA/COSINE-100 curve describing the dependence of the event rate on the photon energy in single-hit events. We calculated the mean dark-matter-halo (DMH) mass around quasars, we also described the origin of the plateaux in the rotation curves for the massive spiral galaxies, the role of DM-loops in magnetars, the origin of CMB, the AGN-jet and galactic-halo production, and properties of dark energy (DE).