Solar photovoltaic (PV) cell modeling is crucial to understanding and optimizing solar energy systems. While the single-diode model (PVSDM) is commonly used, the double-diode model (PVDDM) offers improved accuracy at a reasonable level of complexity. However, finding analytical closed-form solutions for the current-voltage (I-U) dependency in PVDDM circuits has remained a challenge. This work proposes two novel configurations of PVDDM equivalent circuits and derives their analytical closed-form solutions. The solutions are expressed in terms of the Lambert W function and solved using a special transcendental function approach called Special Trans Function Theory (STFT). The accuracy of the proposed equivalent circuits is demonstrated on two solar cells/modules, RTC-F and MSX-60, showing equal or better performance than the standard PVDDM equivalent circuit. Further testing on a commercial solar panel under different irradiance and temperature conditions confirms the applicability of the proposed models. To address the parameter estimation problem, a novel metaheuristic algorithm, the chaotic honey-badger algorithm, is developed and evaluated. The results obtained validate the accuracy and practicality of the proposed PVDDM equivalent circuit configurations.