LOLD Implementation

[akrivi]

This PR:

• Implements the Loss of Load Days (LOLD) metric as defined in [1]
• Adds a new documentation page: "Interpreting Resource Adequacy Metrics"

Mathematical formulation

We define the Loss of Load Days (LOLD) metric as:

$$ \mathrm{LOLD} = \mathbb{E}\left[\sum_{d} I_{d,s}\right] $$

where:

$$ I_{d,s} = \begin{cases} 1 & \text{if } \exists t \in {T}(d) \text{ such that } \sum_r S_{r,t,s} > 0 \\ 0 & \text{otherwise} \end{cases} $$

and

• $r$ indexes regions in the system
• $t$ indexes timestamps
• $d$ indexes calendar days
• $s$ indexes Monte Carlo samples
• $S_{r,t,s}$ is the shortfall in region $r$, time $t$, sample $s$
• ${T}(d)$ is the set of all time periods belonging to day $d$

We note that LOLD is not naturally defined at a single timestamp, since it aggregates over all time periods within a day. As a result, additional computation is needed compared to LOLE, namely:

1. grouping time periods by calendar day
2. aggregating within each day

Benchmarks

Complexity wise, the system wide and full-horizon LOLD scales with total simulation size, while the day-specifc queries scale only with the size of the selected day.

We also provide below a comparison of the benchmarks between LOLD and LOLE:

References:
[1] G. Stephen et al., "Clarifying the Interpretation and Use of the LOLE Resource Adequacy Metric," 2022 17th International Conference on Probabilistic Methods Applied to Power Systems (PMAPS), Manchester, United Kingdom, 2022, pp. 1-4, doi: 10.1109/PMAPS53380.2022.9810615.