In this work, the exact solution to the mean value theorem problem applied to the maximum power point (MPP) calculation is presented; besides, an improvement to this solution is developed based on an expression of the MPP that depends on the optimal load and the characteristic parameters of the photovoltaic module. Both approximations depend only on the natural logarithm, avoiding the evaluation of the Lambert-W function. The validation of the approximations is performed by comparing them with real measurements and the numerical solution of the five-parameter single-diode model using the dataset from 11 commercial modules and six different technologies, as well as by comparing them with existing approximations in the literature.

To compute the maximum power point (MPP) from physical parameters of the single-diode model (SDM), it is necessary to solve a transcendental equation using numerical methods. This is computationally expensive and can lead to divergence problems. An alternative is to develop analytical approximations which can be accurate enough for engineering problems and simpler to use. Therefore, this paper presents approximations for computing the MPP of single-junction solar cells. Two special cases are considered: (i) SDM with only series resistance, and (ii) SDM with only shunt resistance. Power series closed-form expressions for the MPP are obtained using perturbation theory and the Lagrange inversion theorem. Validation of the formulas is performed using experimental data from six different technologies obtained from the NREL database and comparing the results with the numerical solution of the SDM and three approximations from the literature. The results show an absolute percentage error (APE) of less than 0.035% with respect to the real MPP measurements. In cases with limited computational resources, this value could be further improved by using a higher- or lower-order power-series approximation.

In this work, the exact solution to the mean value theorem (MVT) problem applied to the maximum power point (MPP) calculation is presented, besides an improvement to this solution is developed based on an expression of the MPP that depends on the optimal load and the characteristic parameters of the photovoltaic (PV) module. Both approximations depend only on the natural logarithm, avoiding the evaluation of the W function. The validation of the approximations is performed by comparing them with real measurements and the numerical solution of the 5-parameter single diode model (SDM) using dataset from 11 commercial modules and 6 different technologies, as well as by comparing them with existing approximations in the literature..

Partial shading significantly impacts photovoltaic (PV) array performance, complicating design, efficiency optimization, and climate-specific simulations. Accurately predicting the current-voltage (I--V) characteristics under uneven irradiance requires solving coupled nonlinear equations—typically via computationally intensive numerical methods prone to divergence. To address this challenge, this work proposes an explicit analytical model for I--V curve calculation in Total Cross-Tie (TCT) configurations. This approach is computationally efficient and avoids numerical complexities. A parameterized model is derived using asymptotic methods, requiring only two measured points on the I--V curve. For the maximum power point (MPP), the proposed model necessitates solving an implicit equation; to address this issue, an analytical approximate solution is developed. The resulting closed-form expressions enable efficient simulation of I--V and P--V characteristics under partial shading and provide a fast analytical approximation of the global MPP in TCT arrays, making the proposed approach particularly suitable for system-level analysis, performance evaluation, and maximum power point tracking applications. The analytical formulas presented show an absolute percentage error (APE) ranging from 1.486% to 0.0014% for the power at the MPP, Pmpp, in the TCT configuration.