Quiet@Remove["Global`*"];
a=Exp[I*2(Pi/3)];
dosu={(Subscript[u, 1])->U,
(Subscript[u, 2])->(a^2)*U,(Subscript[u, 3])->a*U};
rceij[{i_,j_}]:=(Subscript[is, i,j])==((Subscript[u, i])-(Subscript[u, j]))*(Subscript[Y, i,j]);
rcei[{i_,j1_,j2_}]:=(Subscript[if, i])==(Subscript[is, i,j1])-(Subscript[is, j2,i])
rs=rceij/@{{1,2},{2,3},{3,1}};
rf=rcei/@{{1,2,3},{2,3,1},{3,1,2}};
rce=Union[rs,rf]/.{(Subscript[Y, 1,2])->I*Y,(Subscript[Y, 3,1])->-I*Y,(Subscript[Y, 2,3])->G};
res=Solve[rce,{(Subscript[if, 1]),(Subscript[if, 2]),(Subscript[if, 3]),(Subscript[is, 1,2]),(Subscript[is, 2,3]),(Subscript[is, 3,1])}][[1]]//Simplify;
rceSym=((Subscript[if, #])==k*G*(Subscript[u, #])&/@Range[3])/.res;
Solve[#,{k,Y}][[1]]&/@Partition[Append[rceSym,rceSym[[1]]],2,1]//Simplify;
%/.dosu//Simplify
N[Y/.%]//Chop

Quiet@Remove["Global`*"];
a=Exp[I*2*π/3];
eqn1:=U1==U;
eqn2:=U2==(a^2)*U;
eqn3:=U3==a*U;
eqn4:=I*Y*(U1-U2)-I*Y*(U1-U3)==k*U1;
eqn5:=-I*Y*(U1-U2)+G*(U2-U3)==k*U2;
eqn6:=G*(U3-U2)-I*Y*(U3-U1)==k*U3;
Solve[{eqn1,eqn2,eqn3,eqn4,eqn5,eqn6},{Y,k},{U1,U2,U3}]
FullSimplify[%]

Pz=r*(i1^2)+r*(i2^2)+r*(i3^2);
podm=u*(i1+i2+i3)-Pdod;
lagrange=Pz-λ*podm;
Solve[{D[lagrange,i1]==0,D[lagrange,i2]==0,D[lagrange,i3]==0,podm==0},{i1,i2,i3,λ}][[1]]