Solving a complicated time-dependent function

I want to solve the following time-dependent system of equations:

Clear["Global`*"]; q = (((1602176634)/10^9))*10^(-19); k = (((1380649)/10^6))*10^(-23); \[Eta] = 3/2; Td = 20 + (5463/20); R1 = 10; Vi = Piecewise[{{u*Sin[\[Omega]*t],       0 <= t <= ((2*Pi)/\[Omega])/2}, {0, t > ((2*Pi)/\[Omega])/2}}]; u = 230*Sqrt[2]; \[Omega] = 2*Pi*50; Is = 5*10^(-9); FullSimplify[  Solve[{I1 == Is*(Exp[(q*(Vi - V1))/(\[Eta]*k*Td)] - 1),     I1 == Is*(Exp[(q*(V1 - V2))/(\[Eta]*k*Td)] - 1), I1 == V2/R1}, {I1,     V1, V2}]] 

But it spits out nothing, just the simplified version of the input. Is there a way to solve for the unknowns $ I_1$ , $ V_1$ and $ V_2$ ?