Cost Estimation in Unit Commitment Problems Using Simulation-Based Inference

TL;DR

The paper tackles estimating unknown UC cost parameters from public historical data by casting parameter inference as a simulation-based Bayesian inverse problem. It employs Neural Posterior Estimation to learn a neural density $q_{\phi}(\boldsymbol{\theta}|\boldsymbol{G},\boldsymbol{\delta})$ from simulations, enabling amortized, fast inference of $p(\boldsymbol{\theta}|\boldsymbol{G},\boldsymbol{\delta})$ without MCMC. Through experiments on a 9-unit UC with a DSM unit and a 24-hour horizon, the study shows the learned posteriors are well-centered with slight overconfidence, validated by posterior predictive checks and coverage analyses. The results demonstrate the potential for operators to forecast a range of costs from past generation schedules, improving short-term forecasts and robustness, while highlighting avenues for improving calibration, scalability, and applicability to longer horizons and renewables.

Abstract

The Unit Commitment (UC) problem is a key optimization task in power systems to forecast the generation schedules of power units over a finite time period by minimizing costs while meeting demand and technical constraints. However, many parameters required by the UC problem are unknown, such as the costs. In this work, we estimate these unknown costs using simulation-based inference on an illustrative UC problem, which provides an approximated posterior distribution of the parameters given observed generation schedules and demands. Our results highlight that the learned posterior distribution effectively captures the underlying distribution of the data, providing a range of possible values for the unknown parameters given a past observation. This posterior allows for the estimation of past costs using observed past generation schedules, enabling operators to better forecast future costs and make more robust generation scheduling forecasts. We present avenues for future research to address overconfidence in posterior estimation, enhance the scalability of the methodology and apply it to more complex UC problems modeling the network constraints and renewable energy sources.

Figures (4)

• Figure 1: Corner plot shows marginal posterior distributions on the diagonal and joint posterior distributions for unit pairs elsewhere. These are evaluated using the observed generation schedule $\boldsymbol{G}^*$, with a Masked Autoregressive Flow (blue) and a Neural Spline Flow (orange). Black dots represent the true parameter values $\boldsymbol{\theta}^*$ that generated this schedule.
• Figure 2: Coverage plot assessing the computational faithfulness of $q_{\phi}(\boldsymbol{\theta}|\boldsymbol{G}, \boldsymbol{\delta})$ in terms of expected coverage. The coverage probability is under the credibility level $1 - \alpha$, which indicates that the posterior approximations produced by NPE are slightly overconfident.
• Figure 3: Learning curves of the 2 trained flows
• Figure 4: Sanity checks of the MAF flow on all power plants. To obtain the posterior predictive distribution $p(\boldsymbol{G}|\boldsymbol{G}^*)$, we sample several $\boldsymbol{\theta}$'s from $q_{\phi}(\boldsymbol{\theta}| \boldsymbol{G}^*, \boldsymbol{\delta}^*)$ and then pass them through the UC problem defined before. This results in a list of generation schedules, from which we computed the $68.7\%$, $95.5\%$ and $99.7\%$ quartiles, as well as the mean and median. This diagnostic is an indication of the good quality of the inference results obtained with the trained flow. In particular, they demonstrate that the generation schedules produced by the parameters sampled from $q_{\phi}(\boldsymbol{\theta}| \boldsymbol{G}^*, \boldsymbol{\delta}^*)$ are close to the real generation schedule $\boldsymbol{G}^*$.