Due to the intermittent nature of solar energy, accurate photovoltaic power predictions are very important for energy integration into existing energy systems. The evolution of deep learning has also opened the possibility to apply neural network models to predict time series, achieving excellent results. In this paper, a five layer CNN-LSTM model is proposed for photovoltaic power predictions using real data from a location in Temixco, Morelos in Mexico. In the proposed hybrid model, the convolutional layer acts like a filter, extracting local features of the data; then the temporal features are extracted by the long short-term memory network. Finally, the performance of the hybrid model with five layers is compared with a single model (a single LSTM), a CNN-LSTM hybrid model with two layers and two well known popular benchmarks. The results also shows that the hybrid neural network model has better prediction effect than the two layer hybrid model, the single prediction model, the Lasso regression or the Ridge regression.

Following the work of Leutheusser [Physica A 127, 667 (1984)], the solution to the Percus-Yevick equation for a seven-dimensional hard-sphere fluid is explicitly found. This allows the derivation of the equation of state for the fluid taking both the virial and the compressibility routes. An analysis of the virial coefficients and the determination of the radius of convergence of the virial series are carried out. Molecular dynamics simulations of the same system are also performed and a comparison between the simulation results for the compressibility factor and theoretical expressions for the same quantity is presented.

A recently derived method [R. D. Rohrmann and A. Santos, Phys. Rev. E 76, 051202 (2007)] to obtain the exact solution of the Percus–Yevick equation for a fluid of hard spheres in (odd) d dimensions is used to investigate the convergence properties of the resulting virial series. This is done both for the virial and compressibility routes, in which the virial coefficients Bj are expressed in terms of the solution of a set of (d−1)/2 coupled algebraic equations which become nonlinear for d≥5. Results have been derived up to d=13. A confirmation of the alternating character of the series for d≥5, due to the existence of a branch point on the negative real axis, is found and the radius of convergence is explicitly determined for each dimension. The resulting scaled density per dimension 2η1/d, where η is the packing fraction, is wholly consistent with the limiting value of 1 for d→∞. Finally, the values for Bj predicted by the virial and compressibility routes in the Percus–Yevick approximation are compared with the known exact values [N. Clisby and B. M. McCoy, J. Stat. Phys. 122, 15 (2006)].