In this paper we develop a continuous time counterpart of the balancing approach based on the extended Gramians for discrete time LTI systems. The extended Gramians are solutions to LMI's and can be seen as an extension/generalization of generalized Gramians which are solutions of the standard Lyapunov inequalities. These LMI's are rather different from the discrete time case, and are coupled to dissipativity arguments related to storage functions that are instrumental for obtaining an a priori error bound if balanced truncation is applied based on these extended Gramians. The approach provides flexibility in choosing the Gramians based on which the model is truncated, in order to get a low error bound, and possibly preserve a desired structure such as a network structure.