Extending the Velocity-dependent One-scale String Evolution Model

We provide a general overview of the velocity-dependent one-scale model for cosmic string evolution and discuss two further extensions to it. We introduce and justify a new ansatz for the momentum parameter $k$, and also incorporate the effect of radiation backreaction. We thus discuss the evolution of the basic large-scale features of cosmic string networks in all relevant cosmological scenarios, concentrating in particular on the `scaling' solutions relevant for each case. In a companion paper, we show, by comparing with numerical simulations, that this model provides an accurate description of the large-scale features of cosmic string networks.