I want to ask Is Vds is equal to Vd ? and What could be the conditions for Vds= Vgs. And what is difference between Vgs and vgs?
Vds is not the same as Vd. Rather, Vds = Vd - Vs.
Vds equal to Vgs would imply the gate is connected to the drain, not an uncommon arrangement. It makes the transistor act something like a diode.
Typically, Vgs would be used to represent the total DC+AC gate-source voltage, while vgs would represent the AC component of the gate-source voltage only.