We consider the problem of distributed secondary frequency regulation such that stability and an economically optimal allocation are attained. We study an arbitrarily connected power network with general linear generation dynamics and use a swing equation to describe frequency dynamics. We propose a distributed averaging dynamic output control (DADOC) scheme that makes use of local frequency and generation measurements and provides stability and optimality guarantees when a broad class of high order turbine governor dynamical systems with quadratic cost functions are considered. The proposed controller includes a number of design parameters that affect the stability and optimality properties of the system. We choose these parameters in a computationally efficient way by solving appropriate linear matrix inequalities (LMIs). Furthermore, we demonstrate how the proposed analysis applies to several examples of turbine governor models. Moreover, to highlight the contribution of our work, we compare and explain the main advantages of DADOC schemes over existing schemes in the literature, in terms of required measurements for their implementation and allowable generation dynamics and cost functions such that stability and optimality are guaranteed. The practicality of our analysis is demonstrated with simulations on the Northeast Power Coordinating Council (NPCC) 140-bus system that verify that our proposed controller achieves convergence to nominal frequency and an economically optimal power allocation.