Post-Study Boundary & Chained Studies

1. The Right Boundary

In finite (acyclic) horizon mode the terminal condition is $V_{T+1} = 0$ : no state carried past the last stage $T$ has any value in the model (see Horizon Modes §1). A study may instead import a right boundary, in the sense that it prices the horizon from its far edge. The cuts of one pool of an upstream policy (§5.1) become the terminal stage’s cut pool, which holds exactly those cuts and to which training never adds or removes one, so the imported terminal function is fixed and replaces $V_{T+1} = 0$ . It prices every coordinate of the terminal state the source models: storage, AR lags and the carried state of §2. Chaining two studies this way is the subject of §5.

The post-study calendar — a run of stages that begins exactly where the study horizon ends — is independent of the boundary: a study may declare post-study stages with or without a loaded boundary. A post-study stage is never dispatched, never joins the study’s own stage chain, and never accumulates or carries a Benders cut of its own: it contributes no LP subproblem and no backward pass. Its role is to let a date past the horizon resolve to a calendar position — so the reconciliation in section 4 has somewhere to land — and, for an anticipated thermal, to declare the capability and cost of a delivery there. Without a boundary the state carried to the terminal stage has zero terminal value (§2); with one, the imported cuts price it (§3).

2. Held-to-Terminal State

Two families of state would otherwise leave the modelled system at the horizon edge instead of being carried forward and priced:

Commitments delivered past the horizon. A commitment decided in the study for a delivery stage after $T$ exists only when that delivery stage lies on the declared post-study calendar and the plant is in service there; it is then a genuine decision, bounded and costed by the post-study stage’s declared capability and cost, and carried in its ring slot to the terminal stage (see System Element Modeling Overview §4 and State Augmentation — Ring Rows). A delivery past $T$ outside that calendar has no decision. Without a loaded boundary the carried commitment has zero terminal value while its fuel is still charged on its decision column, and the study setup warns about it.
Terminal deep-lag in-transit buckets. Water released late in the horizon may still be in transit at $T$ (see System Element Modeling Overview — Cascade Travel Time). With a boundary loaded, every lag a stage’s releases reach is held live to the terminal stage; without one, a lag that would mature past the horizon is capped away and its water is dropped (see State Augmentation — Horizon limitation).

The two families are gated differently: the post-study calendar decides whether a post-horizon commitment is made, and the boundary decides whether a deep-lag bucket survives to the terminal stage. A coordinate that reaches the terminal stage is held live in the terminal stage’s outgoing state, like storage or AR lags, and the boundary prices it; a coordinate fixed at zero or dropped before the terminal stage leaves nothing for a cut to act on.

The figure follows a commitment decided at stage $t$ for a post-study delivery stage $m$ . It is deposited into its ring slot at $t$ , carried to the terminal stage $T$ and valued there by the boundary cut through the slot’s coefficient, while its fuel is charged at $t$ ; the delivery stage $m$ itself lies on the post-study calendar past the boundary. A commitment decided before the study for a post-study stage holds no slot (the dashed path), and its state contribution enters the boundary cut’s intercept.

Both families are carried by the in-study state machinery — ring slots and bucket columns, their incoming copies pinned by column bounds — and a right boundary adds no state-carrying mechanism: for these two families it only changes which bucket lags reach the terminal stage.

3. Boundary Pricing (`β·x`)

An imported terminal cut carries an intercept $\beta_0$ and a coefficient $\beta$ for every coordinate of the terminal stage’s outgoing state — one entry per hydro storage, per AR lag, per in-transit bucket and per commitment-ring slot. Each cut is the familiar affine floor on the terminal future-cost variable,

\theta \;\geq\; \beta_0 \;+\; \beta^{\top} x_T,

evaluated through the same cut row every other Benders cut uses (see SDDP Algorithm §6 for the single-cut form). The coefficient of a ring slot or a bucket multiplies the terminal stage’s outgoing column for that coordinate, which the commitment’s deposit or carry row, or the bucket’s definition row, ties to the decision or release that produced it (State Augmentation — Ring-Slot Cut Coefficient; State Augmentation — Bucket definition rows). A right boundary adds no second pricing mechanism; it supplies the coefficients of the state held live at the terminal stage.

Pricing the carried state through $\beta^{\top} x_T$ is deliberately kept separate from pricing the fuel an anticipated commitment consumes. The fuel of a post-horizon commitment is booked on its decision column at its decision stage, at the post-study stage’s declared cost and discounted from the delivery stage (State Augmentation — Objective contributions; Discount Rate Formulation — Post-Study Extension), while $\beta^{\top} x_T$ prices the state the commitment leaves behind in its ring slot. State valuation and fuel booking are disjoint columns: one is a term in $\beta^{\top} x_T$ on the outgoing slot column, the other is the commitment’s own objective coefficient on its decision column. Because no single column carries both roles, the two compose without double-counting the same delivered energy — the same discipline the in-study fishing and objective machinery already applies to delivery inside the horizon.

A commitment decided before the study for a delivery past the horizon is fixed: it has no decision column, no ring slot and no coordinate of $\beta$ . Its fuel is sunk and enters no objective. When a boundary is loaded (Boundary Cuts), its state contribution — the value the boundary cut assigns to the committed rate over the source slots its delivery window overlaps, hour-weighted — is added once, at load, to the intercept $\beta_0$ of every boundary cut; no coefficient changes. With no boundary loaded it enters no term, and the study setup warns about it when its committed rate is non-zero. Either way it is reported at its real delivery date (Simulation Output).

4. Calendar Reconciliation (Fan-Out)

An upstream run’s terminal state is expressed on its own calendar, which need not share the current study’s stage boundaries — a monthly source informing a weekly or monthly study is the typical case. Loading the boundary therefore reconciles the source’s dated state onto the current study’s own calendar before any coefficient is used: every source month is distributed across the study’s own delivery windows in proportion to the hours each shares with it (each weight is the shared hours divided by the source month’s hours), a dated, hour-weighted fan-out. A window that falls entirely inside a single priced source month takes that month’s share alone; a window drawing on more than one source month takes the hour-weighted share of each; only a window straddling into a stretch the source never priced is renormalized over the span the source actually covers: the source’s value is taken as spread uniformly over the window, so the covered part’s density is extended over the uncovered part.

The source and the current study need not even model the same set of state coordinates. A source trained without in-transit buckets, or with monthly anticipated slots where the current study carries weekly ones, presents a terminal state of a different shape. Reconciliation therefore matches each target coordinate to the source by entity identity and delivery date, never by position in the state vector: a storage coordinate binds to the same reservoir, an anticipated slot to the same plant-and-delivery-date, an in-transit bucket to the same arc-and-maturity. A source coordinate the current study does not model has nowhere to land and is dropped — counted, per family, in the reconciliation summary below rather than silently discarded — and the load still succeeds by default. In the other direction, a storage or AR-lag coordinate of the current study that the source does not price refuses the load (§5.2), and only a commitment-ring slot or in-transit bucket with no overlapping source interval takes a zero coefficient. A stricter admission is available that instead rejects a superset source — one pricing state the current study does not model — rather than dropping and reporting it; see Compatibility requirements for how the software layer requires it.

The fan-out is produced once, at load, and its result is summarized rather than left implicit: a per-family reconciliation summary reports, for storage, for inflow lags, for in-transit buckets, and for anticipated commitments, how many target coordinates were copied identically from a source coordinate matched by entity identity, how many were resolved by the dated fan-out (a window inside a single source month counts here, as a one-term blend, and those renormalized for reaching into a stretch the source never priced are also counted separately as straddling), how many had no corresponding source information and took a zero coefficient rather than a guessed value, and how many source coordinates the current study does not model and dropped. For storage and inflow lags the fanned-out and zero counts are always zero on a load that succeeds, because an unpriced coordinate of either family refuses the load (§5.2). The summary is a load-time diagnostic, not a state variable — it exists so an inconsistency between the source and current state or calendars is visible rather than silently absorbed.

The source state comes from the one upstream pool that §5.1 selects by date.

A delivery past the horizon that the study decides has exactly one decision stage $t_i(m)$ (State Augmentation §5). At that stage each scenario decides it like any other stage decision and carries it in its ring slot to the terminal stage, and the boundary prices the carried value through the boundary cuts of that scenario’s terminal-stage LP.

The figure draws the fan-out past the horizon edge, the end of stage $T$ , in the post-study segment that begins there. A priced source month is distributed across the study’s delivery windows it overlaps, drawn here for a commitment, each with an hour weight: the hours they share divided by the source month’s hours. The window inside the month takes that month’s share alone. The straddling window reaches into a stretch the source never priced and is renormalized over the span the source covers, so its uncovered part takes the covered part’s density.

5. Chained Studies

A chained study couples two studies through a right boundary. An upstream study trains a policy over a longer or coarser horizon, and a downstream study imports one of its pools as its right boundary (§1). The dependency runs one way: the upstream study never reads the downstream one, and re-running the downstream study needs only the archived upstream policy. Each study draws its own scenario tree, and the downstream stages may be finer than the upstream’s — weekly stages under a monthly policy, for example. One study whose own stages mix resolutions is a different case, covered in Multi-Resolution Studies.

5.1 Source Pool

Each pool of the upstream policy prices the state that leaves its stage, at that stage’s end date. The source is the pool priced at the downstream horizon end, the end date of the downstream study’s last stage; it need not be the pool of the upstream study’s terminal stage. The load is refused when no pool, or more than one, is priced at that date, and also when the selected pool is shared by more than one node of the upstream policy graph.

The figure draws the upstream and downstream studies as two timelines. Each upstream pool is priced at its stage’s end date, and the source is the pool priced at the downstream horizon end, drawn here as an interior pool with later upstream stages after it. Its cuts become the fixed terminal pool of the downstream study’s stage $T$ , and the single edge between the studies is one-way.

5.2 Compatibility Conditions

A boundary load imposes these conditions on the source pool and the downstream study:

(a) Source pool. The source is the pool of §5.1; otherwise the load is refused.
(b) Seasons. When the study declares a season map, the source carries the same season cycle and the same number of seasons and, for every hydro whose inflow the study models, an entry with the same PAR order in every season where the study models it; otherwise the load is refused.
(c) Storage and lag correspondence. Every storage coordinate of the study’s terminal state has a source counterpart for the same reservoir, and every AR-lag coordinate one for the same hydro and lag depth that, where both sides date it, references the same past period; otherwise the load is refused. A pool prices a storage or AR-lag dimension only when the cut projection that sets its dimension, its successor stage’s selection, carries it (State Augmentation — Cut-state projection); the terminal pool always does, so an interior source pool chosen under a storage-only projection has no AR-lag counterpart.
(d) Lag depth. The study’s lag state is extended to the deepest lag that the cuts of any pool of the boundary policy reference: its depth is the larger of the study’s maximum AR order and that lag, so every lag coefficient of the imported cuts has a coordinate.
(e) Forward-dated state. Commitment-ring slot and in-transit bucket coordinates are reconciled by date (§4) and take a zero coefficient where the source has none.
(f) Surplus source state. A source coordinate the study does not model is dropped and reported, or, under the stricter admission, refuses the load.

A source that prices only storage and AR-lag coordinates needs no post-study calendar, because those coordinates are matched by identity; only the forward-dated coordinates of §4 are dated onto it.

The check order and the refusal messages are in Policy Management — Compatibility requirements.

5.3 Consistency

The imported function is a valid outer approximation of the upstream model’s cost-to-go, and of no other; the downstream policy and its lower bound are valid relative to it. A downstream model that differs from the upstream one — in its inflow-model coefficients, in the reservoirs only the upstream study models, or in its penalties — receives a terminal value priced for a different system, and the difference biases the downstream policy near the horizon; training does not correct it, because the terminal pool never changes (§1). Running with the zero terminal value instead decouples the two studies, at the price of the end-of-world effect (Horizon Modes §1).

6. Lag Seeding

Every study seeds the AR lags of its first stage from its realized inflow record: the historical record together with the recent observations of its initial conditions, the latter replacing the former on every day both cover, never averaged with it. Each lag is seeded from one of the season periods that precede the first stage’s own — the period immediately before it for the first lag, the one before that for the second, and so on, walking back through the season cycle — and its seed is the record’s duration-weighted mean over the covered days of that period.

When the study starts inside a season period, the elapsed part of that period seeds the period’s partial accumulation, and the first stages complete the period by the lag accumulation of Multi-Resolution Studies. An in-progress period the record covers only in part is accepted, with a warning.

The autoregressive order — supplied, or chosen by the order selection of PAR(p) Inflow Model §3.6 — is the number of lags each stage’s autoregressive coefficients read. When the study supplies its autoregressive coefficients, the seed of every lag up to the largest of those orders must be fully covered by the record, or the case is refused at load, unless the first stage’s season cannot be resolved: those lags then seed to zero, with a warning. A lag is never set to its season’s mean.

The seeds condition the first stages’ inflows through the autoregressive terms without changing the inflow model’s parameters, so a study seeded from recent observations answers for the conditions it starts from. In a chained study the downstream study seeds its own lags this way, while the boundary supplies only the terminal function (§1). The fitting-time estimation of the statistics of lag seasons outside the study window is a different mechanism; see PAR(p) Inflow Model §3.8.

Cross-References

State Augmentation — §5 the commitment ring (hold ring, ring rows, ring-slot cut coefficient, objective contributions); §6 in-transit bucket state, pinning, and the horizon-limitation cap that a right boundary lifts.
System Element Modeling Overview — §4 the anticipated-thermal commitment ring and §5 cascade travel time, the element-level source of the two carried families.
Horizon Modes — the zero terminal value a right boundary replaces, and the finite-horizon context the post-study segment attaches to.
Discount Rate Formulation — the cumulative factor extended over the post-study stages.
SDDP Algorithm — §7 the terminal-boundary summary.
Multi-Resolution Studies — one study with mixed resolutions; the lag accumulation, and the partial accumulation of the period a study’s first stages complete, which §6 of this chapter seeds.
PAR(p) Inflow Model — order selection; the pre-study lag window of the fit.
Running Novomodelo: Policy Management — boundary configuration and compatibility requirements.