The main emphasis of the current study is to improve the performance of hydrogen (H2) production under various working conditions via electrolysis of water. A mathematical model is designed and analyzed considering the influence of several parameters such as electrolyte type, its concentration, electrode material with its surface area, number of electrolysis cells, generated thermal energy, applied pressure, temperature and current density. The system has been designed using Engineering Equation Solver (EES) Software. Results showed that the optimum cell voltage was achieved when the concentration of KOH solution as an electrolyte was 30% wt. On the other hand, the most suitable material for coating electrodes is iridium-dioxide (IrO2) but its cost represented a challenge, stainless-steel can be used, alternatively, it is recommended to use substituted materials such as mixed metal oxides or non-precious metal catalysts. It was also found that the rate of the H2 production and generated thermal energy increased as the surface area of the electrode plate, the number of cells, and current density were increased. the influence of increasing temperature on the cell voltage is studied, which causes the cell voltage to rise because of the ohmic resistance reduction, resulting in lower operating costs. It is found that the cell voltage recorded 1.851 V at 40 °C, and 1.677 V at 80 °C. It is found that the operating pressure affects the reversible voltage, where a higher operating pressure leads to a non-linearly higher reversible voltage, resulting in a higher overall cell voltage.