Innovative cooling systems and solar panel modeling approaches have become critical for effective photovoltaic module performance and higher energy efficiency. In the present study, a novel cooling system for thermoelectric generator integrated photovoltaic module is proposed. Three dimensional coupled photovoltaic module/thermoelectric generator/cooling channel system is numerically analyzed by using Galerkin weighted residual finite element method. The cooling channels include rotating circular cylinders, with ternary nanofluid serving as the cooling medium. Three dimensional coupled numerical simulations are carried out for various values of cylinder’s rotational speed (rotational Reynolds number, Rew between 0 and 5000), number of channels with rotating cylinders (N between 2 and 8), size of the cylinders in the cooling channel (RC between 0.006H and 0.433H) and nanoparticle amount in water (ϕ between 0 and 3%). Higher rotational speed of the cylinders, higher nanoparticle loading and higher channel number with cylinder contribute positively to the performance enhancement of the photovoltaic-thermoelectric generator coupled system. In the case of motionless cylinder in cooling channels, larger cylinder sizes have negative impact on the performance. By using cylinders that rotate at the fastest possible rate together with the usage of ternary nanofluid and only water, photovoltaic-cell temperature drops of 61.6 °C and 61 °C in comparison to un-cooled photovoltaic are achieved. When compared to motionless cylinder arrangements, cell temperature reductions of 4.3 °C and 3 °C are achieved by using the fastest rotation speeds for the base fluid and nanofluid. The temperature of the photovoltaic cells decreases and the thermoelectric power increases as the number of rotating cylinders in the cooling channels increases while temperature of the cell drops by 2 °C from N=2 to N=8 and becomes 4.3 °C as compared to motionless cylinder configuration. An efficient computational method by using proper orthogonal decomposition is proposed. While the computational time is reduced by a factor of 1/50, the approach is effective in properly predicting the variation in photovoltaic-cell temperature. The proposed cooling system and practical computational method are useful for further development and optimization of efficient thermal management of photovoltaic panels and integrated systems.