Optimality Conditions and Cost Recovery in Systems with High Shares of Variable Renewable Energy and Energy Storage

We formulate generation capacity portfolio planning in the power grid as a least-cost optimization problem and derive analytic expressions for the optimality conditions for dispatchable generation, variable renewable energy (VRE), and energy storage systems (EES) using a generalized duration curve approach. This is done for a range of different operating conditions for EES with and without VRE in the system. For all studied combinations of technologies and operation conditions, it is shown that all units, including VRE and EES, recover their costs and maximizes their profits in the system optimum, for an ideal short-term market based marginal cost pricing. Furthermore, a simplified, linear duration curve representation is introduced, where we derive explicit equations for the optimal capacities of all technologies present in the system. We verify the theoretical findings through a numerical example, where it is shown that the general duration curve approach gives identical results as a standard capacity expansion model based on sequential simulation of the optimal operation of units. Finally, we report the results of the linear duration curve model, which are in line with the general model. The results highlight that the duration curve models presented in this paper can be a useful supplement to more detailed simulation studies of markets with high penetration of VRE and EES, to better understand the underlying factors that determine the optimal capacity mix and profitability of each technology in electricity markets based on marginal cost pricing.