A modified thermal Bhatnagar–Gross–Krook Lattice Boltzmann (BGK-LB) model was developed to study the convection phenomenon of non-Newtonian fluids (NNFs). This model integrates the local shear rate into the equilibrium distribution function (EDF) of the flow field and keeps the relaxation time from varying with fluid viscosity by introducing an additional parameter. In addition, a modified temperature EDF was constructed for the evolution equation of the temperature field to ensure the precise recovery of the convection–diffusion equation. To validate the accuracy and effectiveness of the proposed model, numerical simulations of benchmark problems were performed. Subsequently, we investigated the natural convection of power–law (PL) fluids and examined the impact of the PL index (n = 0.7–1.3) and Rayleigh number (Ra = 10³–5 × 10⁵) on the flow and temperature fields while holding the Prandtl number (Pr = 7) constant. The obtained results indicate that, for a given value of n, the convective intensity exhibits a positive correlation with Ra, which is illustrated by the rising trend in the average Nusselt number ($\overline{Nu}$) with increasing Ra. Additionally, shear-thinning fluid (n < 1) exhibited increased $\overline{Nu}$ values compared to the Newtonian case, indicating an enhanced convection effect. Conversely, shear-thickening fluid (n > 1) exhibits reduced $\overline{Nu}$ values, indicating weakened convective behavior.

Under the guidance of the 'dual carbon' goals and 'rural revitalization' strategy, the development of microgrids primarily based on wind, solar, and biogas energy is rapidly advancing in rural areas. A critical and challenging area of current research is how to optimally configure the capacity of these microgrids of varying sizes, taking into account the availability of resources in the system's environment and specific climatic conditions, to maximize economic benefits. Based on this, the article constructs a model of a hybrid AC/DC microgrid system powered by wind, solar, and biogas energy. It undertakes multi-objective optimization to achieve the highest utilization of renewable energy, the most economical cost, and the minimum carbon emissions while ensuring the reliability of the system's power supply. The study explores the economically and technically optimal configuration of this microgrid energy system under certain climatic conditions. The results indicate that the optimal configuration for a rural microgrid powered by wind, solar, and biogas energy should include a 2.6 kW biogas generator, 30.00 kW solar panels, 5.24 kW wind turbines, a 2.6 kW battery storage system, and a 10.00 kW bidirectional inverter. This configuration results in the lowest total net cost of the system, achieving optimal outcomes in terms of total net cost, cost per kilowatt-hour, and supply reliability.

Despite the widespread popularity of the Bhatnagar–Gross–Krook lattice Boltzmann (BGK-LB) model due to its simplicity and efficiency, its application in heat transfer involving non-Newtonian fluids (NNFs) has been limited by inherent constraints. This paper proposes a numerically stable BGK-LB model for the thermal flow of NNFs. The modified model incorporates the local shear rate into the equilibrium distribution function of the velocity field and addresses the numerical instability problems encountered in the traditional BGK-LB model under low viscosity conditions by introducing an additional parameter. In addition, a temperature evolution equation that can accurately recover the convective diffusion equation is adopted. The accuracy of the current method is validated by performing simulations of Rayleigh–Bénard convection (RBC) in a square cavity filled with Newtonian fluids and NNFs. Subsequently, simulations are conducted to investigate the behavior of RBC in power-law fluids. The analysis focuses on examining the impact of the Rayleigh number (Ra = 5 × 103 − 105) and the power-law index (n = 0.8–1.3) on the convective structure and heat transfer characteristics while maintaining a fixed Prandtl number (Pr = 7) and aspect ratio (L/H = 2). It is discovered that, for a given n value, the convection intensity and heat transfer rate increase with increasing Ra, which is supported by the increasing trend of the mean Nusselt number (Nū) with Ra. Furthermore, compared to NFs, pseudo-plastic fluids display a higher Nū value due to an augmented heat transfer rate, while dilatant fluids exhibit a lower Nū value owing to a diminished heat transfer rate.