This paper is an extension and a generalization of the previous results, cf. \cite{DC,Benc,sal,Bench}. It is devoted to studying the finite element approximation of the non coercive system of parabolic quasi-variational inequalities related to the management of energy production problem. Specifically, we prove optimal $L^{\infty }$-asymptotic behavior of the system of evolutionary quasi-variational inequalities with nonlinear source terms using the finite element spatial approximation and the subsolutions method.