A hybrid solar$-$wind energy system is designed to supply power to a remote area. The system includes both solar panels and wind turbines, along with energy storage technologies to ensure continuous power supply. Given the following information:

Solar panel capacity: $50$ kW

Solar panel efficiency: $18\%$

Average solar irradiance: $800$ W$/$m Wind turbine capacity: $100$ kW

Wind turbine efficiency: $30\%$

Average wind speed: $5$ m$/$s

Battery capacity: $200$ kWh

Inverter efficiency: $90\%$

Load demand: $150$ kW

a$).$ Calculate the total power output of the solar panels. 

b$).$ Determine the total power output of the wind turbines.

c$).$ Calculate the total energy storage capacity required for the system.

d$).$ Determine the overall efficiency of the system in supplying power to the load

e$).$ Calculate the state of charge $($SOC$)$ of the batteries after $24$ hours of operation, assuming e constant load demand and renewable energy generation patterns.

f$).$ Discuss the importance of battery sizing and control strategies in ensuring reliable power supply in hybrid energy systems.