Multi-Metric Resource Adequacy Analyses with PRAS

In practice, no single metric fully captures system adequacy. Instead, multiple complementary metrics should be considered together to understand the frequency, distribution and severity of shortfall events. (NERC (2018), EPRI, ESIG (2024), Stephen et al. 2022).

For this reason, PRAS provides multiple result specifications and derived metrics that allow different aspects of system risk to be evaluated consistently. This tutorial compares metrics that describe the temporal occurrence and magnitude of shortfalls.

Temporal Occurrence of Shortfall

Resource adequacy metrics can be understood by first defining three related concepts (Stephen et al. 2022):

• An event-period is a simulation time step in which a shortfall occurs.
• An event-day is a day containing at least one event-period.
• An adequacy event is a set of event-periods that are contiguous at the highest available temporal resolution.

These distinctions are important because different metrics count different temporal quantities. LOLE and LOLD correspond to the first two concepts:

• LOLE is the expected number of event-periods
• LOLD is the expected number of event-days

These metrics are related, but they are not interchangeable.

Shortfall Severity

LOLE and LOLD describe when shortfalls occur, but they do not describe their magnitude. EUE complements these metrics by measuring the expected total amount of unserved energy over the study horizon.

Why Multiple Metrics Matter

Another important reason to use multiple metrics, as described in (Stephen et al. 2022), is that systems with similar shortfall magnitudes or counts of event-periods can exhibit very different temporal patterns.

We can consider a simple example of two cases next, for which we assume that every shortfall hour has the same amount of unserved energy.

Case A: One day with 10 hours of shortfall

Case B: Ten days with 1 hour of shortfall each

As we can see in the table above, even though LOLE and EUE are identical in this case, LOLD reveals that shortfall events are more dispersed in Case B.

Because event-periods may be distributed across many days, a system with the same number of shortfall periods can have very different numbers of event-days. As a result, exact conversions between hourly and daily adequacy criteria are not generally possible (Stephen et al. 2022).

This behavior is reflected in PRAS results, where LOLE and LOLD provide complementary views of how shortfall events are distributed in time.

Mathematical Interpretation

In PRAS, adequacy metrics can be interpreted from Monte Carlo shortfall samples.

Using the following notation:

• $r$ indexes regions
• $t$ indexes timestamps
• $d$ indexes calendar days
• $s$ indexes Monte Carlo samples
• $e$ indexes adequacy events
• $S_{r,t,s}$ denotes the shortfall in region $r$, at timestamp $t$, in Monte Carlo sample $s$
• $T(d)$ is the set of timestamps in day $d$
• $\Delta t$ is the duration of each simulation time step

the adequacy metrics can be expressed as expectations over Monte Carlo samples:

LOLE

LOLE counts the expected number of event-periods with shortfall:

\[\mathrm{LOLE} = \mathbb{E}\left[\sum_t \mathbf{1}\left(\sum_r S_{r,t,s} > 0\right)\right]\]

LOLD

LOLD counts the expected number of days containing at least one shortfall:

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

where:

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

EUE

EUE measures expected total unserved energy across the Monte Carlo samples:

\[\mathrm{EUE} = \mathbb{E}\left[\sum_t \sum_r S_{r,t,s}\,\Delta t\right]\]

Analysis with PRAS

We revisit the RTS-GMLC system with increased load to induce shortfall, which was described in PRAS walkthrough

using PRAS
sys = PRAS.rts_gmlc()
sys.regions.load .+= 700.0

shortfall_samples, = assess(
    sys,
    SequentialMonteCarlo(samples=100, seed=1),
    ShortfallSamples(),
)

(PRASCore.Results.ShortfallSamplesResult{8784, 1, Hour, MW, MWh, ShortfallSamples}(Regions{8784, MW}(["1", "2", "3"], [1685 1686 … 1839 1781; 1803 1783 … 2004 1923; 1950 1892 … 2173 2058]), ZonedDateTime(2020, 1, 1, tz"UTC"):Hour(1):ZonedDateTime(2020, 12, 31, 23, tz"UTC"), [0 0 … 0 0; 0 0 … 0 0; 0 0 … 0 0;;; 0 0 … 0 0; 0 0 … 0 0; 0 0 … 0 0;;; 0 0 … 0 0; 0 0 … 0 0; 0 0 … 0 0;;; … ;;; 0 0 … 0 0; 0 0 … 0 0; 0 0 … 0 0;;; 0 0 … 0 0; 0 0 … 0 0; 0 0 … 0 0;;; 0 0 … 0 0; 0 0 … 0 0; 0 0 … 0 0]),)

and we calculate the metrics we discussed above:

system_lole = LOLE(shortfall_samples)
system_lold = LOLD(shortfall_samples)
system_eue = EUE(shortfall_samples)

println(system_lole)
println(system_lold)
println(system_eue)

LOLE = 85±2 event-h/8784h
LOLD = 25.8±0.5 event-day/366days
EUE = 26400±1100 MWh/8784h

We can also evaluate upper-tail severity by selecting a CVAR confidence level:

alpha = 0.95
system_cvar = CVAR(:energy, shortfall_samples, alpha)
println(system_cvar)

CVAR@0.95 = 55000±2000 MWh/8784h

In the RTS example above, the system has approximately 85 shortfall hours but only 25.8 shortfall days. This indicates that shortfall events are temporally clustered, meaning that multiple shortfall hours tend to occur within the same day rather than being evenly distributed across the year. EUE summarizes the average total unserved energy, while CVAR ($\alpha = 0.95$) summarizes unserved energy in outcomes beyond the 95th-percentile threshold.

References

• NERC (2018), Probabilistic Adequacy and Measures Technical Reference Report
• EPRI, Resource Adequacy Gap Assessment: Resource Adequacy Assessment Framework
• ESIG (2024), New Resource Adequacy Criteria for the Energy Transition: Modernizing Reliability Requirements
• Stephen et al. (2022), Clarifying the Interpretation and Use of the LOLE Resource Adequacy Metric

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