# Extensions of the Dynamic Programming Framework: Battery Scheduling, Demand Charges, and Renewable Integration

## TL;DR

This paper extends the dynamic programming framework to handle non-separable objectives, introducing a tractable augmentation scheme and applying it to optimize battery scheduling with demand charges using stochastic models.

## Contribution

It develops a state-augmentation approach for non-separable DP problems, especially those with Naturally Forward Separable objectives, and applies it to renewable energy management.

## Key findings

- Proposed a tractable augmented-state DP method for NFS objectives.
- Extended the framework to stochastic DP problems with a new Principle of Optimality.
- Applied the method to optimize battery scheduling with demand charges.

## Abstract

We consider a general class of Dynamic Programming (DP) problems with non-separable objective functions. We show that for any problem in this class, there exists an augmented-state DP problem which satisfies the Principle of Optimality and the solutions to which yield solutions to the original problem. Furthermore, we identify a subclass of DP problems with Naturally Forward Separable (NFS) objective functions for which this state-augmentation scheme is tractable. We extend this framework to stochastic DP problems, proposing a suitable definition of the Principle of Optimality. We then apply the resulting algorithms to the problem of optimal battery scheduling with demand charges using a data-based stochastic model for electricity usage and solar generation by the consumer.