Abstract:

Within the BSc in Electrical Engineering – Energy Systems at the School of Engineering (ISEP), Polytechnic of Porto, the Mathematical Laboratories I (LabMat1) course has been redesigned to foster the integration of mathematical modelling, computation, and experimental learning in photovoltaic (PV) systems. The course adopts an inquiry-based, student-centred approach in which learners analyse current–voltage (I–V) and power–voltage (P–V) characteristics using physically grounded models, including the single-diode model and its parameters: photocurrent Iₚₕ, diode saturation current I₀, ideality factor n, series resistance Rₛ, and shunt resistance Rₛₕ.

Instead of relying on closed-form solutions, students engage in parameter estimation as an open-ended, data-driven task. Experimental I–V and P–V curves are obtained under controlled irradiance conditions by varying an external load, and subsequently used to calibrate mathematical models. This process promotes active learning and encourages students to confront discrepancies between theoretical predictions and experimental observations.

For future editions of the course, to support this pedagogical framework, an optimisation-based methodology is introduced for parameter identification in both single- and two-diode models. The approach combines nonlinear least-squares fitting (using SciPy), the Newton iterative method for solving implicit equations, and a genetic algorithm for robust initialisation. Importantly, these tools are presented in a transparent manner, enabling students to understand modelling assumptions, numerical limitations, and the implications of algorithmic choices. This transition from basic experimentation to advanced computational modeling ensures that students develop high-level technical competences in scientific programming and numerical analysis.

From an educational perspective, this activity promotes the development of both technical competences and soft skills, including analytical problem-solving, critical thinking, autonomy, and the ability to interpret non-intuitive numerical results. Students are also encouraged to communicate findings and justify modelling decisions, reinforcing transversal skills relevant to engineering practice. This methodology shifts the focus from merely obtaining a result to understanding the process of engineering inquiry, where the ability to argue the validity of a model is as important as the model itself.

Results from multiple optimisation runs indicate that the single-diode model may yield ideality parameters inside physically meaningful ranges, reflecting its approximation to real PV devices. Nevertheless, these outcomes are deliberately used as reflective learning opportunities, highlighting that a good numerical fit does not necessarily imply physical validity.

By interpreting estimated parameters as “effective” quantities that compensate for model limitations, students develop a more nuanced understanding of the relationship between mathematical abstraction and real-world systems. Overall, this approach enhances student engagement and supports the development of critical and transferable competences aligned with contemporary STEM education.

Keywords:

PV modelling, single-diode model, parameter estimation, genetic algorithm, I–V characteristics, soft skills.