Tractable Probabilistic Models for Investment Planning

TL;DR

This work introduces a TPM-based framework using Sum-Product Networks to replace traditional finite scenario ensembles in long-horizon power system planning under uncertainty. By training SPNs on simulator-generated data and embedding their probabilistic inferences into MILP, the approach enables exact, scalable chance-constrained optimization for generation, transmission, and storage expansion. Experiments demonstrate that SPN-based methods yield more reliable and conservative investment decisions, especially under limited data, and offer favorable scalability compared to scenario enumeration. The results highlight TPMs as a practical, interpretable tool for reliability-aware, data-driven energy planning in a decarbonizing grid.

Abstract

Investment planning in power utilities, such as generation and transmission expansion, requires decade-long forecasts under profound uncertainty. Forecasting of energy mix and energy use decades ahead is nontrivial. Classical approaches focus on generating a finite number of scenarios (modeled as a mixture of Diracs in statistical theory terms), which limits insight into scenario-specific volatility and hinders robust decision-making. We propose an alternative using tractable probabilistic models (TPMs), particularly sum-product networks (SPNs). These models enable exact, scalable inference of key quantities such as scenario likelihoods, marginals, and conditional probabilities, supporting robust scenario expansion and risk assessment. This framework enables direct embedding of chance-constrained optimization into investment planning, enforcing safety or reliability with prescribed confidence levels. TPMs allow both scenario analysis and volatility quantification by compactly representing high-dimensional uncertainties. We demonstrate the effectiveness of the approach through a representative power system planning case study, illustrating its computational and reliability advantages over traditional scenario-based models.

Figures (4)

• Figure 1: Illustrative comparison of computational scaling between traditional scenario-based formulations and the proposed SPN-based chance-constrained approach. While scenario-based methods scale linearly or superlinearly with the number of scenarios, the SPN-based formulation requires only a one-time training cost and remains effectively constant afterward.
• Figure 2: A schema of the proposed framework.
• Figure 3: Comparison between empirical and SPN-estimated shortfall ratios (SPN, SPN-Max, and SPN-Piecewise): In order, the Empirical column shows the best empirical estimation possible with respect to access to more samples. The SPN column displays the learned approximation. The SPN-Max column shows the first proposed approximation (SPN-Max). The SPN-Piecewise column displays the improved version of our proposed approximation. We highlight that our best method is capable of capturing the pattern of the best empirical approximation. Although SPN-Max captures the pattern as well, the error scales with the size of the SPN, something that is minimized with SPN-Piecewise while remaining computable using MILP.
• Figure 4: Configuration space exploration across data sparsity levels (30 iterations per percentage). Point clouds show each method's configuration choices with capacity-based axes (primary) and unit counts (secondary). A gold star indicates the ground truth optimum. Red dashed boundary marks the region where the adequacy constraint ($\mathbb{P}(\text{shortfall}) = \varepsilon$, where $\varepsilon=5\%$) is respected based on ground truth. Configurations outside this boundary fail the adequacy requirement. Note that these methods may appear safe based on their own estimates, yet select configurations that exceed the true safety boundary, revealing calibration overconfidence. We added some jitter to the points to show the clusters of runs, but they concentrate around the same solution.