Multi-Resolution Studies

Purpose

A multi-resolution study is a single study whose stages span more than one temporal resolution — most commonly a monthly head followed by a quarterly tail, or a weekly head followed by a monthly tail. All stages belong to one SDDP horizon and share one set of value-function cuts; no coupling boundary exists. The modelling challenge is that the PAR(p) inflow model is indexed by season, yet stages of different length follow one another: each stage must take the parameters of a season at its own resolution, and its lags must be values of whole season periods at that resolution. This chapter describes how Novomodelo handles that challenge.

Boundary with chained studies. A multi-resolution study is one study with one cut set. A chained study is instead two studies: the second imports a fixed terminal function, the cuts of one pool of the first study’s policy, chosen by date and not necessarily the pool of the first study’s last stage (see Post-Study Boundary & Chained Studies). This chapter owns the sub-period lag accumulation (§2).

1. Season Map

The PAR(p) model associates each stage $t$ with a season $m(t)$, and the stages of one season share its parameters $\mu_m$, $s_m$ and $\psi_{m,\ell}$. A multi-resolution study declares its coarse seasons — for example quarters — as seasons of their own, with identifiers distinct from those of the fine seasons whose calendar span they overlap. When the season map layers two resolutions over the same calendar days, a stage’s dates alone do not determine its season, so every stage names its season; a coarse stage therefore takes its coarse season and that season’s PAR parameters. When no two seasons overlap, a stage’s season follows from its start date, and a stage shorter than its season period may name the season it subdivides.

Fine stages that subdivide one season — for example weekly stages that name their month’s season — share that season’s parameters. The season map thus fixes the parameters of every stage; §2 and §3 describe how the lags those parameters act on are formed when stages differ in length.

2. Lag Accumulation

The lags of the PAR model are values of completed season periods, not of single stages. A lag period $\mathcal{W}$ is one occurrence of a stage’s season period: the calendar window that the season covers in one cycle, such as one particular month or one particular quarter. Writing $t$ also for the calendar window of stage $t$, the share of the period that stage $t$ covers is

$$w_{t,\mathcal{W}} = \frac{\lvert t \cap \mathcal{W} \rvert}{\lvert \mathcal{W} \rvert}$$

with $\lvert t \cap \mathcal{W} \rvert$ the hours of stage $t$ inside $\mathcal{W}$ and $\lvert \mathcal{W} \rvert$ the hours of the period. The value a completed period contributes to the lag state of hydro $h \in \mathcal{H}$ is the duration-weighted mean of the realized inflows $a_{h,t}$ of the stages $t$ that overlap it:

$$\frac{\sum_{t} w_{t,\mathcal{W}}\, a_{h,t}}{\sum_{t} w_{t,\mathcal{W}}}$$

A stage that straddles a period boundary contributes to both periods, each with its own share (spillover). The period completes at the last stage of its season occurrence: there the completed value becomes lag 1 and every older lag moves back by one, so the lag $\ell$ that a stage reads is the value of the $\ell$-th most recent period completed before it. At every earlier stage of the period the lag state is unchanged, so the stages inside one period see the same lags. The partial accumulation, the two running sums of the mean, travels with each trajectory and is not a coordinate of the cut state.

In a uniform single-resolution study, where every stage spans exactly one season period, every stage is one period with share 1 and the lag is the stage’s inflow. The one exception is a weekly cycle in a year of 53 ISO weeks: the 53rd week takes the season of the 52nd, so those two stages form one period and the lag they complete is the mean of their two inflows.

The figure below draws this accumulation for weekly stages inside one month and for a monthly head before a quarterly tail. In the upper panel each week feeds the month’s lag period with its share of the month’s hours; the last week straddles the month’s end, feeds the next month’s period with its spillover and, as the month’s last stage, completes the period, whose duration-weighted mean becomes lag 1. The bracket around the weeks is their noise group (§4): in the opening tree they share one innovation per opening. In the lower panel, drawn for a coarse order of at least 2, the months of the window that the coarse order reaches feed both their monthly lag periods and the quarterly periods (§3), and at the first quarterly stage the lag state is rebuilt from the completed quarterly periods, newest first.

3. Resolution Change

A coarse season’s AR coefficients act on lags of coarse periods, so the lag state changes resolution with the stages. When fine stages precede coarse stages that carry coarse seasons of their own (§1) — a monthly head before a quarterly tail — the fine stages of the window before the first coarse stage also accumulate into the coarse periods, as many of them as the coarse model’s order reaches back, with the shares and spillover of §2. At the first coarse stage the lag state is rebuilt from those completed coarse periods, newest first: the most recent becomes lag 1, the one before it lag 2, and so on. From there the accumulation runs at the coarse resolution.

4. Noise Groups

In the opening tree, consecutive stages that share a season and the calendar year of their start form one noise group and share one innovation per opening, as when several weekly stages fall inside one monthly season. Within the opening tree, a season’s innovation $\varepsilon_t$ is therefore not resampled for every stage the season is subdivided into. A stage without a season forms its own group, and in a uniform single-resolution study every group is a single stage, so every stage draws independently, except that on a weekly cycle the 53rd week of a 53-week year takes the season of the 52nd and joins its group. The sampling rule is stated in Scenario Generation; the group assignment depends on the stage calendar alone, so every MPI rank computes the same groups (see Determinism & Provenance).

Along a trajectory, the stages inside one lag period see the same lags (§2).

5. Duration-Weighted Aggregation for Fitting

The seasonal statistics of a season must refer to the resolution at which its PAR model is evaluated. When a season is coarser than the observation record — a quarterly season over a monthly record — the observations of each fully observed occurrence of that season are collapsed into their duration-weighted mean before estimation, so the statistics of the season refer to its own resolution. The aggregate is a mean, not a sum: it keeps the units of a flow rate, and observations of unequal duration, such as months of unequal length, weigh in proportion to their duration. The PAR estimation procedure of PAR(p) Inflow Model §3 then applies per season without modification, and each stage takes its season’s parameters when the runtime PAR quantities are built (PAR(p) Inflow Model §4.2). A coarse season’s fit sees only coarse-resolution variation: the variation among the fine observations inside one occurrence does not reach its parameters.

Cross-References

• PAR(p) Inflow Model — The fitting procedure (§3) that applies to aggregated statistics; the parameter set that the aggregated statistics populate; the LP-ready form that quarterly stages use at runtime.
• Scenario Generation — Opening-tree generation that produces per-stage noise vectors; the noise-group sampling rule.
• Post-Study Boundary & Chained Studies — Two studies coupled by a terminal function imported from the upstream policy, rather than one study with mixed stages; the lag seeding at a study’s start.