Post-Study Boundary

1. The Right Boundary

In finite (acyclic) horizon mode, the terminal future-cost function is zero by default: no state carried past the last stage $T$ has any value in the model (see Horizon Modes §1). A study may instead import a terminal future-cost function trained by an upstream run — a right boundary, in the sense that it prices the horizon from its far edge rather than replacing a value at its near edge. The upstream run’s terminal cuts are injected as a fixed boundary condition at the study’s own terminal stage, replacing the zero terminal value with an already-informed continuation value. This is the same terminal-boundary-cut mechanism SDDP Algorithm §7 and Weekly+Monthly Coupled Studies §2 describe for the storage boundary; this chapter is about what a right boundary additionally makes possible for state that would otherwise leave the modelled system at the horizon edge instead of being priced.

A right boundary is anchored by a post-study calendar segment — a run of stages that begins exactly where the study horizon ends and exists purely to give a delivery or maturity deadline falling after $T$ a definite place on the calendar. 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 sole role is to let a date past the horizon still resolve to a calendar position, so the reconciliation in section 4 has somewhere to land. Whatever the post-study segment “contains” is priced back onto the study’s own terminal stage through the imported cut — never solved in its own right.

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:

• Post-horizon anticipated-commitment lanes. An anticipated thermal may commit generation in-study whose delivery stage falls after $T$ (see System Elements §4 and LP Formulation §5c). Absent a right boundary, a commitment whose delivery stage lies outside the horizon is pinned to zero — the plant is not permitted to make it, because there is nothing on the far side of the horizon to price it against.
• Terminal deep-lag in-transit buckets. Water released late enough in the horizon that its travel-time delay would carry it past $T$ is, absent a right boundary, dropped rather than credited to terminal storage (see System Elements — Cascade Travel Time and LP Formulation §5d). The deepest maturity lag reachable at each stage is capped so that no bucket ever points beyond the horizon.

A right boundary changes both defaults. Instead of being pinned to zero or capped away, the post-horizon commitment lanes and the deep-lag buckets are held live as genuine terminal state, carried into the terminal stage’s incoming-state vector exactly like storage or AR lags. Holding a family live is what lets the boundary price it: a coordinate that has already been pinned to zero or dropped before reaching the terminal stage leaves nothing for a cut to act on.

The commitment lane and the in-transit bucket are carried by the same ring-buffer and column-pinning machinery already used for their in-study counterparts — a right boundary does not introduce a new state-carrying mechanism. It only changes which lags and slots are permitted to survive to the terminal stage rather than being capped or zeroed away before they get there.

3. Boundary Pricing (β·x)

An imported terminal cut carries an intercept $\alpha$ and a coefficient $\beta$ for every coordinate of the terminal incoming-state vector — one entry per hydro storage, per AR lag, per in-transit bucket, and, once held live by section 2, per post-horizon commitment lane. Each cut is the familiar affine floor on the terminal future-cost variable,

$$\theta \;\geq\; \alpha \;+\; \beta^{\top} x,$$

evaluated through the same cut row every other Benders cut uses (see SDDP Algorithm §6 for the single-cut form). Because the carried state is pinned by column bounds like any other incoming state, the coordinate of $\beta$ paired with a held-live commitment lane or in-transit bucket is read back the same way any cut coefficient is read back — as the reduced cost of that pinned column — and the backward pass propagates it to the deciding stage through the same ring-buffer remapping LP Formulation §5c and §5d already use for in-study lanes and buckets. A right boundary does not add a second pricing mechanism; it supplies the $\beta$ that the existing remapping had nothing to carry before the state was held live.

Pricing the carried state through $\beta^{\top} x$ is deliberately kept separate from pricing the fuel an anticipated commitment consumes. The commitment’s delivery-anchored fuel cost is booked on its own decision column at the stage it is decided — the same commitment column priced in-study (see the objective contributions in LP Formulation §5c) — while $\beta \cdot x$ prices the state the commitment leaves behind in the carried lane. State valuation and fuel booking are disjoint columns: one is a term in $\beta^{\top} x$ on the pinned state 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 the plant decided before the study whose delivery also falls past the horizon is priced differently again. It is not a decision the study makes — the quantity is fixed exogenously, already committed — so it carries neither a decision column nor a coordinate of $\beta$. Its cost is a sunk cost, folded once, at load, as a constant into the intercept $\alpha$ of every boundary cut whose delivery date it covers. Moving only the intercept and never a coefficient, it shifts the terminal future-cost floor by a fixed amount without changing how any live state is priced — the accounting a commitment already paid for before the study demands. Such a pre-decided post-horizon commitment is reported at its real delivery date rather than left implicit in the shifted intercept.

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, a dated, hour-weighted fan-out. A window that falls entirely inside a single priced source month is a straight copy of that month’s share; a window straddling more than one source month, or straddling into a stretch the source never priced, is renormalized over only the span the source actually covers, so the fanned-out coefficient never overstates or understates the value the source expresses.

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. A differing state dimension is thus no longer a rejection; only a coordinate the current study models but the source cannot identify is defaulted rather than sourced.

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, how many were fanned out across more than one source month, how many had no corresponding source information and were defaulted rather than guessed, and how many source coordinates the current study does not model and dropped. 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 reconciliation source is required to resolve to a single leaf pool: the upstream run’s terminal state must trace back to exactly one unambiguous cut pool. A source whose matching terminal state is shared by more than one scenario branch at the source’s own terminal stage is rejected rather than guessed at, because there is no principled way to prefer one sibling branch’s state over another’s as the boundary.

Within the fan-out, a delivery or maturity that falls past the study horizon is decided by exactly one in-study stage — its decider. The decider’s position relative to the study’s own last operative stage matters conceptually, not just calendrically. A decider strictly before the last operative stage is classified Trunk: one decision shared across every branch of the terminal fan, because every scenario has already passed through that stage by the time the decision is made. A decider that lands exactly at the last operative stage is classified TerminalFan instead: decided independently by each scenario, because at that point the scenarios have not yet been forced back together. The distinction governs how a post-horizon commitment is attributed for pricing — a Trunk-decided commitment is one shared value the boundary prices once; a TerminalFan-decided commitment is priced per scenario, matching the branch that actually made it.

Cross-References

• LP Formulation — §5c anticipated-thermal state pinning, fishing constraint, and cut subgradient remapping; §5d in-transit bucket state, pinning, and the horizon-limitation cap that a right boundary lifts.
• System Elements — §4 anticipated-thermal ring-buffer state and cascade travel time, the element-level source of the two carried families.
• Weekly+Monthly Coupled Studies — the storage-only terminal boundary cut import this chapter extends to delivery-side state.
• Horizon Modes — the zero terminal value a right boundary replaces, and the finite-horizon context the post-study segment attaches to.