Single, internally adjustable modular battery systems are provided, for handling power delivery from and to various power systems such as electric vehicles, photovoltaic systems, solar systems, grid-scale battery energy storage systems, home energy storage systems and power walls. Batteries comprise a main fast-charging lithium ion battery (FC), configured to deliver power to the electric vehicle, a supercapacitor-emulating fast-charging lithium ion battery (SCeFC), configured to receive power and deliver power to the FC and/or to the EV and to operate at high rates within a limited operation range of state of charge (SoC), respective module management systems, and a control unit. Both the FC and the SCeFC have anodes based on the same anode active material and the control unit is configured to manage the FC and the SCeFC and manage power delivery to and from the power system(s), to optimize the operation of the FC.