## 1 Introduction  
To improve the performance evaluation, fault detection and maximum energy harvesting of PV systems [1], one major focus of the research community is accurate modeling and efficient parameter extraction to closely represent the nonlinear \(I\)-\(V\) (current vs. voltage) characteristics of solar cells. Although various models have been developed to describe the electrical behavior of solar cells, only three equivalent circuit models are widely used [2]: the single diode model (SDM), the double diode model (DDM) and the triple diode model (TDM). Unfortunately, all three models are implicit nonlinear transcendental equations. This inherently implicit nature increases not only the difficulty of parameter extraction but also the complexity of PV system simulation, and thus calls for accurate, efficient and reliable methods to extract the model parameters from the measured \(I\)-\(V\) data of solar cells. For solving the parameter extraction problem of SDM, DDM and TDM, various methods have been developed. These methods can be generally classified into analytical, deterministic, population-based stochastic search and hybrid algorithms. Analytical methods take several key points to generalize all measured \(I\)-\(V\) data, but the results are inaccurate. Deterministic algorithms, such as Newton-Raphson, pattern search and Nelder-Mead simplex search, are sensitive to initial values and thus prone to being trapped in local minima. Population-based stochastic algorithms, inspired by nature [3; 4; 5; 6; 7; 8; 9], biology [10; 11; 12; 13; 14; 15; 16],swarm [17] and evolution [18; 19; 20; 21; 22; 23], usually require significant computational efforts to converge and depend strongly upon the proper tuning of control parameters, such as population size, the maximum number of function evaluations (MaxNFEs) or iterations, searching strategies and so on. Hybrid algorithms [24; 25; 26; 27; 28; 29] combine the strengths of two or more different methods in an attempt to achieve better results. For example, the ABC-TRR [29] algorithm combined the global exploration capability of the artificial bee colony (ABC) algorithm and the local exploitation ability of trust-region reflective (TRR) search to extract the parameters of SDM and DDM. However, hybrid algorithms usually mean more control parameters that need to be carefully tuned. For instance, there are seven control parameters in the ABC-TRR [29] algorithm that need to be elaborately selected, and any improper choice could result in slow convergence and premature termination of iterations. Even though these algorithms provided accurate results, there is still room for improving their computational efficiency, convergence speed and reliability. Motivated by the preceding discussion, this paper proposes an opposite-normalized trust-region reflective (ONTRR) algorithm aiming for accurate, efficient and reliable parameter extraction of SDM, DDM and TDM. Unlike the ABC-TRR [29] algorithm, the proposed ONTRR adopts random opposite initialization and min-max normalized processing to overcome the drawbacks of the standard TRR search. Initially, the proposed ONTRR generates a random point and its opposite point, and then selects the better of these two points as the initial point, which gives a higher chance of finding promising regions. In the iteration phase, the proposed ONTRR employs the unitary surrogate of model parameter values to calculate distances to the normalized boundary and truncate the unconstrained trial step to ensure a sufficient decrease of the derivative-dependent TRR search. Moreover, the proposed ONTRR omits the terminate tolerances of TRR search and adopts the only control parameter MaxNFEs to determine stop or not, which removes the burden of multi-parameter tuning and permits its use without prior knowledge about solar cell parameter extraction. In general, the main novelties/contributions of this paper are as follows:  
1.