We study equilibria in traffic networks and in power system networks with storage in the presence of logic constraints. These constraints consist of binary variables that are added to complementarity-based equilibrium models. Although these models have been thoroughly studied, the addition of logic constraints can provide additional benefits for practical applications. The main contribution of this work is to demonstrate, using two specific examples of applications, that logic constraints can render classical equilibrium models more realistic by allowing the inclusion of useful features such as equity in network flows or threshold events. Specifically, for the traffic equilibrium problem, we show how logic constraints can introduce some equity in the assignment of traffic when more than one equilibrium exists. For power system networks, we show that the presence of a storage operator acting as a service provider will not only support the operation of a power grid, but will also help stabilise the price of electricity and avoid the well-documented price-shifting effect. Unlike previous works, our model considers the storage operator as a service provider rather than a competitor to the producers. We also consider the minimum power output of production. We present results illustrating the expanded capabilities and insights provided by these new paradigms.