Accurate and effective optimization of photovoltaic (PV) systems is still a challenging task because solar cell current–voltage (I–V) characteristics are strongly sensitive to environmental conditions, which behave nonlinearly. This paper presents an analytical, non-iterative method to achieve the global maximum power point (MPPT) of PV cells and panels based on the Lagrange Multiplier Method (LMM). Compared with traditional iterative and empirical methods, the proposed method can obtain a mathematical optimization solution directly for output power maximization of PV systems in terms of single-diode and double-diode equivalent circuit models with series and parallel resistances.
An objective function and a constraint are set up for the optimization problem, consisting of the desired output power and nonlinear current–voltage solar cell equation. The problem becomes a matter of a reduced analytical approach by normalizing the variables with respect to current and voltage, which leads to the application of constrained optimization theory. Then, the Lagrange Multiplier Method is used under standard regularity and optimality conditions such as the KKT, Fritz John, Mangasarian–Fromovitz (MFCQ), or Linear Independence Constraint Qualification. These statements ensure that there is an optimal solution, which further means that the condition of optimality is satisfied.
Regarding the single-diode model, an analytical solution results in a second-order polynomial equation that describes the efficiency gap against the current density [12], which is of third-order type for the double-diode model. The optimal current, voltage, and load resistance corresponding to the maximum power point are analytically obtained in closed forms. The physical meaning of the Lagrange multiplier is also presented, emphasizing its connection with the characteristic resistance of the solar cell at the maximum power point.
To assess the efficiency and applicability of the proposed method, a series of simulations are performed based on experimental data recorded from R.T.C France (silicon solar cell) and Photowatt-PWP 201 (photovoltaic module). Further validation is carried out on the different PV technologies, such as monocrystalline silicon, polycrystalline silicon, amorphous silicon, and copper indium diselenide (CIS) modules. The results indicate that the analytically calculated maximum power point corresponds well with experimental current–voltage and power–voltage characteristics for various operating temperatures and technologies.
The Lagrange-based optimization is an effective, fast, and low-cost approach compared to the conventional heuristic search techniques of AI for MPPT. Its non-iterative nature, analytical transparency, and versatility make it especially well-suited for real-time PV analysis, system sizing, and performance estimation tasks, as well serving as a bridge rom optimization environments to larger PV plant and system hybrids.
