# Mixed-Integer Nonlinear Differentiable Predictive Control for Underground Pumped Hydro Energy Storage Systems

arXiv: https://arxiv.org/abs/2609.17964 · HTML: https://mlsanity.com/paper/2609.17964/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 15 of its 34 references that are in the map.
- First posted on arXiv 16 Sep 2026.

## Builds on (most cited first, 15 of 15)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [Categorical Reparameterization with Gumbel-Softmax](https://mlsanity.com/paper/1611.01144/) (2016)
- [Estimating or Propagating Gradients Through Stochastic Neurons for Conditional Computation](https://mlsanity.com/paper/1308.3432/) (2013)
- [The Concrete Distribution: A Continuous Relaxation of Discrete Random Variables](https://mlsanity.com/paper/1611.00712/) (2016)
- [Accelerated Policy Learning with Parallel Differentiable Simulation](https://mlsanity.com/paper/2204.07137/) (2022)
- [Efficient representation and approximation of model predictive control laws via deep learning](https://mlsanity.com/paper/1806.10644/) (2018)
- [Learning an Approximate Model Predictive Controller with Guarantees](https://mlsanity.com/paper/1806.04167/) (2018)
- [Online Mixed-Integer Optimization in Milliseconds](https://mlsanity.com/paper/1907.02206/) (2019)
- [Differentiability in Unrolled Training of Neural Physics Simulators on Transient Dynamics](https://mlsanity.com/paper/2402.12971/) (2024)
- [CoCo: Online Mixed-Integer Control via Supervised Learning](https://mlsanity.com/paper/2107.08143/) (2021)
- [Controlling Continuous Relaxation for Combinatorial Optimization](https://mlsanity.com/paper/2309.16965/) (2023)
- [Learning to Optimize for Mixed-Integer Non-linear Programming with Feasibility Guarantees](https://mlsanity.com/paper/2410.11061/) (2024)
- [Data Center Chiller Plant Optimization via Mixed-Integer Nonlinear Differentiable Predictive Control](https://mlsanity.com/paper/2603.28323/) (2026)
- [Accelerating Underground Pumped Hydro Energy Storage Scheduling with Decision-Focused Learning](https://mlsanity.com/paper/2512.20880/) (2025)
- [End-to-End Battery Dispatch with Exact Rainflow Degradation via Mixed-Integer Differentiable Predictive Control](https://mlsanity.com/paper/2609.12968/) (2026)

## Abstract (arXiv)

This paper extends Mixed-Integer Differentiable Predictive Control (MI-DPC) to multi-modal discrete decisions and nonconvex polynomial dynamics arising in Underground Pumped Hydro Energy Storage Systems (UPHES). A neural policy mapping problem parameters to continuous setpoints and integer mode selections via a Gumbel-Softmax layer is trained in a self-supervised manner by differentiating the expectation of the finite horizon control objective through the nonlinear dynamics model. Three methodological contributions enable this extension: a parallel differentiable simulator that preserves gradient magnitude, a Transformer encoder that captures long-range temporal dependencies, and a Gumbel-Softmax temperature annealing schedule that regularizes the combinatorial search. We demonstrate the framework on day-ahead scheduling of a UPHES, a large-scale mixed-integer optimal control problem with nonlinear unit performance curves and volume-head coupling. MI-DPC achieves only 1.6% suboptimality relative to a piecewise mixed-integer quadratic programming baseline, while providing five orders of magnitude speedup in online scheduling time.