Towards time-variant scenario reduction for energy system optimization modeling under uncertainty

Technical Summary: Towards Time-Variant Scenario Reduction for Energy System Optimization Modeling Under Uncertainty

Problem Statement
Long-term two-stage stochastic energy system optimization models (ESOMs) are critical for decision-making under uncertainty, particularly regarding intermittent renewable power production. However, these models face a scalability challenge: spanning time horizons of 20–60 years while representing uncertainty with multiple scenarios (e.g., 10–40 historical weather years) leads to rapid growth in model size. Existing scenario reduction (SR) methods, termed "time-invariant," are often inefficient in this context. They assume scenario probabilities remain constant over time and typically focus on short-time horizons (e.g., single time steps or 24-hour windows). When applied to long-term planning with limited available scenarios (e.g., ~40 historical years), time-invariant methods cannot adjust marginal distributions over time, potentially distorting the optimal solution and failing to accurately capture the realizations and probabilities of critical scenarios that drive first-stage investment decisions.

Methodology
The authors propose a novel time-variant scenario reduction framework that explicitly allows scenario aggregations to change over time. The methodology involves the following steps:

1. Temporal Decomposition: Unlike conventional methods that cluster scenarios across the entire time horizon simultaneously, the proposed approach breaks the temporal chronology of the stochastic process.
2. Time-Step Clustering: For each time step $t \in T$, the $k$-means clustering algorithm is applied individually to the scenario vectors $w_{t,i}$ (representing joint realizations of variables like PV and wind power).
3. Intertemporal Assignment: To restore chronological consistency for the reduced set of scenarios $J$, a linear intertemporal assignment problem is solved for each time step transition ($t$ to $t+1$). This minimizes the cost of transitioning between reduced scenarios, defined as the squared Euclidean distance between their values at consecutive time steps.
4. Time-Variant Probabilities: The procedure yields subsets of original scenarios $I_{j,t}$ that are aggregated into reduced scenario $j$ at specific time steps $t$. Consequently, scenario probabilities $\pi_{t,j}$ become time-variant, calculated as the sum of the probabilities of the aggregated original scenarios at that specific time step:
   $$\pi_{t,j} = \sum_{i \in I_{j,t}} \pi_i$$
   This contrasts with time-invariant methods where $\pi_{t,j} = \pi_j$ for all $t$.

Key Contributions

• Novel Framework: Introduction of a time-variant SR method that decouples scenario aggregation from a fixed temporal structure, allowing probabilities to evolve over time.
• Enhanced Flexibility: The method enables the accurate capture of scenario realizations and their probabilities specifically at time steps that drive first-stage investment decisions, a capability lacking in traditional time-invariant approaches.
• Proof of Concept: The application of this framework to a two-stage stochastic generation expansion planning (SGEP) problem involving uncertain renewable production and battery storage investment.

Results
The proposed method was evaluated using a joint probabilistic forecast of 20 equiprobable scenarios for PV and wind power over a 96-hour horizon, reduced to solve an SGEP problem. The results, measured by Relative Approximation Error (RAE) against the full 20-scenario solution, demonstrate:

• Superior Accuracy with Fewer Scenarios: The time-variant SR method achieved an RAE of 3.06% using only 4 reduced scenarios. In contrast, the time-invariant method yielded an RAE of 18.71% with the same number of scenarios.
• Investment Decision Quality: The time-invariant method with 4 scenarios significantly underestimated thermal generation capacity investment ($x^*$) and overestimated storage investment ($s^*$), leading to a massive increase in Expected Energy Not Served (EENS) from 113.9 MWh to 2815.8 MWh. The time-variant method accurately reproduced optimal investments, resulting in an EENS of 712.9 MWh.
• Efficiency: The time-variant approach achieved performance comparable to a time-invariant method using at least 11 scenarios, representing an approximate 80% reduction in model size (number of variables) for equivalent accuracy.
• Convergence: The time-variant method converged to the full scenario set solution much faster and more robustly as the number of reduced scenarios increased.

Significance and Claims
The paper claims that the proposed time-variant framework substantially increases flexibility compared to traditional time-invariant methods. By allowing scenario aggregations and probabilities to vary over time, the method overcomes the inefficiencies of existing SR techniques when applied to large-scale, long-term planning ESOMs with limited historical data. The authors assert that this approach unlocks the ability to reduce model complexity while maintaining the accuracy of critical investment-driving scenarios.

The paper concludes modestly, noting that future research is required to derive appropriate time-variant scenario distance metrics and algorithms to avoid the current two-step procedure (clustering followed by assignment). Additionally, extending mathematical stability results to the time-variant case to provide formal performance guarantees is identified as a necessary direction for future work.