# Analytical Modeling of the Maximum Power Point with Series Resistance
## 1 Introduction  
[PERSON]'s diode equation describes how a solar cell responds to bias and illumination [1]. Analytical expressions for photovoltaic parameters, such as the open-circuit voltage, \(V_{\mathrm{oc}}\), or the short-circuit current, \(i_{\mathrm{sc}}\), can easily be derived from it. When series resistance is accounted for, however, the diode equation becomes an implicit expression of the current, which is not as straightforward to work with. Some work has been performed aiming to quantify analytically the effect of series resistance on various solar cell parameters. [PERSON] et al. showed that [PERSON]'s W function allows for a closed-form expression of the current when the effect of series resistance is considered [2]. [PERSON] et al. derived, in [3], an analogue to [PERSON]'s expression that also accounted for the effect of shunt resistance. The latter authors later made use of this expression to derive analytical expressions for \(V_{\mathrm{oc}}\) and \(i_{\mathrm{sc}}\). The maximum power point (MPP) has been little explored from an analytical perspective, although some exceptions exist [4, 5, 6, 7, 8]. The reason for this lies in the fact that, when series resistance is considered, deriving an expression for the maximum power point voltage, \(V_{\mathrm{mpp}}\), involves solving a transcendental equation. Such equations often do not have closed-form solutions and need to be solved numerically. It is worth mentioning the work presented in [4], where [PERSON] was able to obtain an approximate closed-form expression of \(V_{\mathrm{mpp}}\), in terms of \(V_{\mathrm{oc}}\). In contrast with a numerical model, an analytical model would allow identifying the physical parameters affecting the MPP. Accounting for the series resistance in the analytical model for the MPP would also allow for better characterization of real solar cells. In this sense, such a model would describe the MPP with higher accuracy than other analytical models that do not account for the effect of series resistance. The lack of an accurate analytical model for the MPP, which includes series resistance, in the scientific literature represents a research gap that this work aims to fill. To this end, a closed-form expression for the maximum power point voltage that accounts for the effect of the series resistance is derived. The starting pointof the derivation is the analytical expression for the current, derived by [PERSON] in [2], in terms of Lambert's W function. It is then argued that, at the maximum power point, the argument of the W function is small enough to accurately approximate the function value. This makes the transcendental problem analytically solvable. The accuracy of the derived model was found to increase with increasing bandgap energy and to decrease with increasing series resistance. From the new expression for \(V_{\text{mpp}}\), an analytical expression for the series resistance was then derived. Additionally, approximate analytical expressions for the maximum power point current, \(i_{\text{mpp}}\), and power, \(P_{\text{mpp}}\), were derived. Numerical results were calculated using parameters typical for seven different solar cell technologies. The accuracy of the model was tested through a comparison with reference values that were obtained from a numerical single-diode model. The results showed that the analytic model can predict the maximum power with relative errors below 0.1% when compared to the numerical model, when typical values of the series resistance are used. Still, it must be kept in mind that the analytical approach derived in the present work corresponds to a one-diode model. This implies that it should not be used with solar cells that follow, e.g., the double-diode equation. For such cells, different techniques, such as those proposed in [9], should be employed. Finally, the practical limitations of the model are discussed based on its experimental validation, provided in [10].