Hydro Production Function Models

Purpose

This chapter defines the hydro generation constraint models supported by Novomodelo, which relate turbined flow and reservoir storage to electrical output. Two production functions shape dispatch: constant productivity and FPHA. A third model name — linearized head — is a reserved alias for constant productivity (see §3). The choice between constant productivity and FPHA trades off accuracy vs. computational cost, and can vary by stage per hydro.

All decision variables use rate units (MW, m³/s) — see the Variable Units Convention in System Element Modeling Overview. For variable definitions see Notation Conventions; for LP integration see LP Formulation; for hydro element descriptions see System Element Modeling Overview.

1. Constant Productivity Model

The simplest model assumes a linear relationship:

$$g_{h,k} = \rho_{h,t} \cdot q_{h,k}$$

where $\rho_{h,t}$ (MW per m³/s) is the hydro productivity for stage $t$:

$$\rho_{h,t} = \frac{9.81 \times \eta_h \times H^{ref}_{h,t}}{1000}$$

with:

• $\eta_h$ = turbine efficiency (a dimensionless factor in $(0, 1]$)
• $H^{ref}_{h,t}$ = reference net head (meters), evaluated at the reference storage level and varying by stage

Per-stage productivity: the productivity coefficient is authored per (hydro, stage) rather than per plant. A plant can therefore carry a stage-varying constant productivity — useful when the reference head differs between near-term and far-future stages of the same study, or when the constant model is being used as a coarse approximation that needs different operating points across the horizon. Section 5.1 describes how that per-(hydro, stage) value is supplied.

LP treatment: constant productivity adds no generation column and no production row: the turbined-flow column of each (hydro, bus) cell (§2.9) enters its bus’s load balance with coefficient $\rho_{h,t}$ in every block, so $g_{h,k} = \rho_{h,t} \cdot q_{h,k}$ holds by construction. Simple and fast, but ignores head variation with storage within a stage.

Data requirements: a per-stage productivity scalar per hydro plant. No geometry or hyperplane data needed.

2. FPHA (Approximate Hydroelectric Production Function)

For accurate modeling of hydroelectric generation, FPHA (Approximate Hydroelectric Production Function) captures the nonlinear relationship between storage, flow, spillage, and generation through a piecewise-linear approximation. The approach follows the piecewise-linear hydro production model of Diniz & Maceira (2008); that model is four-dimensional (forebay/head, turbined flow, and tailrace/spillage), whereas Novomodelo fits a reduced variant over storage and turbined flow at spillage = 0, capturing the spillage effect through the per-plane lateral-flow secant of §2.7 rather than an explicit spillage axis.

2.1 Notation

This section uses the notation of the LP Formulation; for equivalent terms in other planning tools, see the Glossary.

Note on lateral flow

The tailrace level depends on the outflow below the plant, which this model takes as $q_{out} = q + s$ (the plant’s own turbined and spilled flow). The lateral-flow secant (§2.7) captures the spillage effect, and upstream releases or other inflows below the dam do not enter $q_{out}$.

2.2 Exact Production Function

The exact hydroelectric production function relates generation to the operating state:

$$\phi(v, q, s) = (9.81\,\eta_h/1000) \cdot q \cdot h_{net}$$

where:

• $v$ = reservoir storage volume (hm³)
• $q$ = turbined flow (m³/s)
• $s$ = spillage flow (m³/s)
• $h_{net}$ = net head (m), clamped to $\ge 0$
• $\eta_h$ = turbine efficiency (section 1), constant per plant; the factor $9.81\,\eta_h/1000$ (MW·s/m⁴) converts hydraulic power to electrical power

The net head is computed as:

$$h_{net}(v, q, s) = h_{fore}(v) - h_{tail}(q + s) - h_{loss}$$

clamped to $\max(h_{net}, 0)$, where:

• $h_{fore}(v)$ = forebay (upstream reservoir) level as function of storage
• $h_{tail}(q + s)$ = tailrace (downstream channel) level as function of total outflow
• $h_{loss}$ = hydraulic head losses (from the factor or constant model)

Why linearization is needed: $\phi$ is nonlinear in $(v, q)$ due to the bilinear product $q \times h_{net}$, nonlinear topology functions $h_{fore}(v)$ and $h_{tail}(q_{out})$, and flow-dependent hydraulic losses. For LP formulation, Novomodelo approximates $\phi$ with a set of linear hyperplanes.

2.3 Topology Functions

Novomodelo uses tabular data with linear interpolation for the forebay curve — more transparent and easier to validate against surveyed data than polynomial fits.

Forebay Level $h_{fore}(v)$

The upstream water level is read from the plant’s volume-height curve. For storage $v$ where $v^{(i)} \leq v < v^{(i+1)}$:

$$h_{fore}(v) = h^{(i)} + \frac{h^{(i+1)} - h^{(i)}}{v^{(i+1)} - v^{(i)}} \times (v - v^{(i)})$$

Tailrace Level $h_{tail}(q_{out})$

The downstream water level depends on total outflow. Three representations are supported:

Polynomial model:

$$h_{tail}(q_{out}) = c_0 + c_1 q_{out} + c_2 q_{out}^2 + c_3 q_{out}^3 + c_4 q_{out}^4$$

Piecewise-linear model: tabular breakpoints with linear interpolation between points.

Piecewise-quartic families (exact tailrace): an optional per-plant tailrace table provides piecewise degree-4 polynomial segments (evaluated via Horner’s method), grouped into backwater families keyed by the downstream reservoir’s reference forebay level — see §2.3.1.

Total downstream flow in LP: $q_{out} = q + s$ (turbined flow + spillage). For FPHA fitting, spillage is fixed at $s = 0$ when building the generation cloud; the lateral-flow secant (§2.7) captures the spillage correction per plane.

2.3.1 Piecewise-Quartic Tailrace Families (Backwater Coupling)

When plants are hydraulically close, plant $h$‘s tailrace level depends on the downstream reservoir’s forebay. An optional per-plant tailrace table provides piecewise-quartic tailrace segments: the flow domain $[Q_{jus,lo}, Q_{jus,hi}]$ is split into contiguous segments, each a degree-4 polynomial

$$h_{tail}^{(n)}(q_{jus}) = c_0^{(n)} + c_1^{(n)} q_{jus} + c_2^{(n)} q_{jus}^2 + c_3^{(n)} q_{jus}^3 + c_4^{(n)} q_{jus}^4$$

evaluated via Horner’s method for numerical stability. Consecutive segments are contiguous in outflow and agree in level at their shared boundary (C0 continuity) to calibration precision (Implementation notes, “Computed-FPHA fitting behavior”). These segments are grouped into backwater families keyed by the downstream reservoir’s reference forebay level ($HrefJus$, in metres). At fitting time, the active family is linearly interpolated by the downstream plant’s resolved stage reference level — clamped to the calibrated level range, never extrapolated. Plants with a single keyless family (no backwater coupling) evaluate that family directly regardless of the downstream level. Plants without a tailrace table use the entity-level polynomial or piecewise-linear tailrace.

Hydraulic Losses $h_{loss}$

Two models are supported:

Factor model — proportional to gross head:

$$h_{loss} = k_{loss} \times (h_{fore} - h_{tail})$$

where $k_{loss}$ is a small dimensionless fraction of the gross head.

Constant model — fixed head loss:

$$h_{loss} = \Delta h_{const} \quad \text{(metres)}$$

2.4 Productivity

The exact production function converts hydraulic power to electrical power with the factor $9.81\,\eta_h/1000$ (MW per (m³/s · m)), so $\phi = (9.81\,\eta_h/1000) \cdot q \cdot h_{net}$ in MW. Novomodelo uses constant efficiency $\eta_h$ per plant. The specific productivity $\rho_{esp}$, in the same unit, is a separate plant input that the exact production function and the FPHA fit do not use.

An FPHA plant has no single scalar productivity $\rho$. The fitted planes follow from $\eta_h$ and the topology functions alone, while the equivalent productivity $\rho_{eq}$ at the reference operating point is derived from a specific productivity $\rho_{esp}$ supplied for the plant (overridable per stage) and the VHA geometry; the derivation is documented in section 5.1.

2.5 FPHA Hyperplanes

The FPHA approximation replaces the nonlinear production function $\phi(v, q, s)$ with a set of $M$ linear hyperplanes that form a concave approximation of the exact surface (an outer approximation of the capped cloud before the $k_{FPHA}$ correction of §2.6.3). Each hyperplane $m$ defines an upper bound on generation:

$$g_{h,k} \leq \gamma_0^m + \gamma_v^m \cdot v_h^{avg} + \gamma_q^m \cdot q_{h,k} + \gamma_s^m \cdot s_{h,k}$$

The plane is fitted once per plant (§2.1–§2.8); a plant split across buses evaluates it per (hydro, bus) cell at LP-integration time, apportioning its flow-independent terms by each cell’s share of the plant’s turbine capacity — see §2.9 for the exact per-cell form.

Physical interpretation of coefficients:

Coefficient	Sign	Meaning
$\gamma_0^m$	—	Intercept (MW at zero storage, flow, spillage); its sign is not validated
$\gamma_v^m$	≥ 0	Higher storage → higher forebay → more generation
$\gamma_q^m$	≥ 0 (> 0 on at least one plane per stage for precomputed planes)	More turbined flow → more generation
$\gamma_s^m$	≤ 0	More spillage → higher tailrace → less net head

Source of hyperplanes: planes are either pre-computed (supplied with the case) or computed from topology data during preprocessing. The computed fit is described in §2.6.

2.6 Computed FPHA Fitting Pipeline

The computed-FPHA path produces hyperplanes from topology data in four stages: convex-hull fit → $k_{FPHA}$ correction → lateral-flow secant → optional plane reduction. The fit is resolved per production-model entry (per season or stage range), so planes can differ across the horizon for the same plant. Results are expanded to per-stage hyperplane rows.

2.6.1 Grid and Cloud

The fitter evaluates the exact production function $\phi = (9.81\,\eta_h/1000) \cdot q \cdot h_{net}$ on a uniform two-dimensional grid over the fitting window, with spillage $s = 0$ and lateral flow $= 0$. The grid has:

• Volume axis: a configurable number of uniformly spaced values spanning the fitting window $[v_{min}, v_{max}]$. The window lies within the plant’s forebay storage range (the full range unless narrowed) and is distinct from the reference volume (§5.1), which fixes an operating point, not this window. The point counts and the window are set on the Configure tab.
• Flow axis: a configurable number of uniformly spaced values spanning $[0, q_{max}]$. The axis starts at $q = 0$, where generation is zero; this zero-flow column anchors the lower closure of the cloud and eliminates the need for any synthetic closing point.

Each cloud point is capped at the plant’s installed capacity $\bar{G}_h$. Spillage is not a cloud dimension — it is fixed at zero throughout.

Plants without turbine capacity: when $q_{max} = 0$ the flow axis collapses onto $q = 0$, where generation is zero, so the cloud carries no production and no plane survives the hull. Such a plant is modelled with zero productivity at every stage instead of a plane set; the implementation notes give the capacity threshold and how the plant is reported.

Run-of-river plants: when the plant has a single fitting volume ($v_{min} \approx v_{max}$, or a single volume sample requested), two nearby volume samples are synthesized to keep the 3-D hull non-degenerate (Implementation notes, “Run-of-river support”). The resulting $\gamma_v^m$ is snapped to exactly 0 when its magnitude is negligible, enforcing the run-of-river semantics ($\gamma_v^m = 0$).

Determinism: the cloud points and the hull output are canonically sorted, so the fitted hyperplanes are bit-identical regardless of input ordering and MPI rank count.

2.6.2 Convex Hull

The 3-D convex hull of the $(V, Q, \text{generation})$ cloud is computed via the qhull library. The upper-envelope facets — those whose outward normal has a positive generation component — are selected. Each is read as $\text{generation} = \gamma_0^m + \gamma_v^m V + \gamma_q^m Q$. Before the $k_{FPHA}$ correction of §2.6.3, the result is a concave outer approximation (the smallest concave function lying above $\min(\phi, \bar{G}_h)$ at the capped cloud points); non-concave regions of $\phi$ fall inside the hull and their facets drop out. Near-parallel coplanar facets arising from hull triangulation are deduplicated by exact coefficient comparison.

2.6.3 Least-Squares $k_{FPHA}$ Correction

The raw hull envelope $FPHA_0$ is optimistic where $\phi$ is non-concave and pessimistic where it is concave. A single scalar $k_{FPHA}$ corrects the bias by minimising the mean-squared error between $k_{FPHA} \cdot FPHA_0$ and the capped production $\min(\phi, \bar{G}_h)$ over the spill$=0$ grid:

$$k_{FPHA} = \frac{\sum_{i,j} FPHA_0(V_i,Q_j)\,\min\big(\phi(V_i,Q_j), \bar{G}_h\big)}{\sum_{i,j} FPHA_0(V_i,Q_j)^2}, \qquad FPHA = k_{FPHA}\cdot FPHA_0$$

Key properties:

• The regression uses the pointwise minimum over the raw hull planes as $FPHA_0(V_i, Q_j)$, because the LP applies planes as $g \le \gamma_0^m + \gamma_v^m V + \gamma_q^m Q$ for every $m$, so the binding cap is the minimum, not the maximum.
• The regression is over the spill$=0$ grid only — adding a spillage axis would pull $k_{FPHA}$ toward the larger-deviation spill region and degrade the no-spill operating region.
• $k_{FPHA}$ scales the whole affine function ($\gamma_0^m, \gamma_v^m, \gamma_q^m$ alike), not just the intercept. It therefore differs from the factor $\kappa$ of precomputed planes, which scales the intercept only.
• $k_{FPHA}$ may be greater or less than 1 (an MSE balance, not a one-sided shrink). Validation requires $k_{FPHA} > 0$. A degenerate denominator (all-zero production) yields the neutral $k_{FPHA} = 1$.
• Validation also requires $\gamma_v^m \ge 0$, $\gamma_q^m \ge 0$, $\gamma_s^m \le 0$.

Fit-quality diagnostic: after the full pipeline, the relative mean-absolute-deviation of the emitted min-envelope vs the capped production $\min(\phi, \bar{G}_h)$ over the spill$=0$ grid is computed. A warning is emitted (in canonical plant/stage order) when it exceeds a threshold (Implementation notes, “Fit-quality warning”) — typically indicating a strongly non-concave surface that no single $k_{FPHA}$ can track well.

Precomputed input: precomputed planes may carry an intercept factor $\kappa \in (0, 1]$ (Configure tab) that scales each plane’s intercept only ($\gamma_0^m \leftarrow \gamma_0^m \cdot \kappa$), unlike the whole-affine $k_{FPHA}$. The computed path does not use $\kappa$; its $k_{FPHA}$ correction plays that role.

The figure below shows a synthetic plant with normalized axes: storage, flow and generation are fractions of $v_{max}$, $q_{max}$ and $\bar{G}_h$. It slices the fit at one grid volume, which the example takes as the reference volume $V^{ref}$ of §5.1, and plots five series against turbined flow: the cloud points at that volume; the exact production $\phi$ capped at $\bar{G}_h$; the raw hull $FPHA_0$, the minimum over the hull planes, which lies on or above every cloud point; the corrected envelope $k_{FPHA} \cdot FPHA_0$, which rescales the hull by the single scalar of this section; and the dashed line $\rho_{eq}\,q$ of §5.1, which, as the example sets $\rho_{esp}$ to $9.81\,\eta_h/1000$, meets the uncapped $\phi$ at $q_{max}$, above the capped curve where the capacity binds.

2.7 Lateral-Flow Secant

Spillage raises the tailrace level, lowering net head and generation. The $\gamma_s^m$ coefficient for each plane is fit by a per-plane 1-D ordinary-least-squares secant of generation vs lateral flow (the plant’s own spillage $s$) over evenly spaced samples of $q_{lat} \in [0, S_{max}]$, evaluated at the plane’s representative (active-maximum) operating point:

$$S_{max} = \begin{cases}2 \cdot \text{MLT} & \text{if MLT} > 0 \\ 2 \cdot q_{max} & \text{if MLT} = 0\end{cases}$$

where MLT is the long-term mean inflow (m³/s). The representative operating point for each plane is the spill$=0$ grid point where that plane is the active (tightest) upper bound and attains the largest generation — the operating region the plane actually governs. The secant samples the uncapped production function $\phi$ (not clipped at installed capacity $\bar{G}_h$), so the spillage sensitivity is read from the raw head curve and is not flattened wherever the capacity ceiling binds — unlike the cloud and the $k_{FPHA}$ regression (§2.6), which both use the capacity-capped output.

Sign: $\gamma_s^m \le 0$ (more lateral flow raises the tailrace, reduces generation). A slope that is non-negative or negligibly negative is snapped to exactly 0 (Implementation notes, “Computed-FPHA fitting behavior”) to protect LP column scaling from near-zero structural coefficients.

Lateral axis: the plant’s own spillage ($q_{lat} = s$); upstream releases and post-incremental inflows do not enter it.

2.8 Similar-Hyperplane Reduction

An optional post-fit step merges consecutive near-parallel or near-coincident planes into their mean hyperplane to shrink the LP. Two mutually-exclusive methods are supported:

• Angle: merge consecutive pairs whose normal-vector angle $\theta = \arccos(\mathbf{n}_1 \cdot \mathbf{n}_2 / \|\mathbf{n}_1\|\|\mathbf{n}_2\|)$ satisfies $\theta < \varepsilon$ (strict), with $\theta$ and the tolerance $\varepsilon$ in degrees. Fully deterministic from coefficients.
• Distance: merge consecutive pairs whose normalised mean-squared generation difference $\delta = \text{EQM}/\phi_{\max}^2 < \varepsilon/100$, where $\phi_{\max}$ is the largest value of the exact production $\phi$ (uncapped) over the fitting grid, with the tolerance $\varepsilon$ in percent and EQM estimated over sampled operating points. Uses a deterministically-seeded PRNG (seeded from stable plant/stage/plane-pair identity, never from wall clock or MPI rank), so results are bit-identical across input ordering and rank count.

Origin-plane invariant: the plane through the origin ($\gamma_0^m \approx 0 \wedge \gamma_v^m \approx 0$, generating zero power at zero turbining) is never merged. This guarantees the zero-generation floor.

Reduction is optional and applies only when selected (Configure tab).

2.9 LP Integration

A hydro plant is partitioned into one or more (hydro, bus) cells — one per distinct bus among its declared unit groups (see System Element Modeling Overview §5) — and FPHA is evaluated per cell, not per plant: a plant’s fitted hyperplane set (§2.1–§2.8) is fitted once per plant and consumed by every one of its cells, apportioned as described below. For a single-cell plant the per-cell constraint is the plant-level constraint.

Final FPHA Constraint

For each cell $(h, b)$ of a hydro $h$ using FPHA, block $k$, and plane $m \in \mathcal{M}_h$:

$$g_{h,b,k} - \lambda_{h,b} \, \gamma_v^m \cdot v_h^{avg} - \gamma_q^m \cdot q_{h,b,k} - \lambda_{h,b} \, \gamma_s^m \cdot s_{h,k} \leq \lambda_{h,b} \, \gamma_0^m$$

Every variable sits on the left with its plane coefficient negated, and the apportioned intercept is the row’s upper bound; the row is the cap $g_{h,b,k} \leq \lambda_{h,b} \big( \gamma_0^m + \gamma_v^m \cdot v_h^{avg} + \gamma_s^m \cdot s_{h,k} \big) + \gamma_q^m \cdot q_{h,b,k}$.

The coefficients are already $k_{FPHA}$-scaled — there is no separate pre-scaling step. These are hard constraints — no slack variables.

$\lambda_{h,b}$ apportions the plane’s flow-independent part — the intercept $\gamma_0^m$, the storage term, and the spillage term — by cell $(h,b)$‘s share of the plant’s declared turbine capacity:

$$\lambda_{h,b} = \frac{\sum_{u \,\in\, (h,b)} \bar{Q}_u}{\sum_{u \,\in\, h} \bar{Q}_u}$$

($0$ when the plant’s declared total is $0$), a study-time constant derived from each unit group’s declared maximum turbined flow — never recomputed per stage or block, and never re-derived from a resolved per-stage override, which would double-count an availability derate already enforced on the flow path by each cell’s own column bounds. It satisfies $\sum_{b \in \mathcal{B}_h} \lambda_{h,b} = 1$, and a single-cell plant has $\lambda_{h,b} = 1$ exactly. The flow coefficient $\gamma_q^m$ stays on the cell’s own $q_{h,b,k}$ unscaled — it is the only term homogeneous in the cell partition, which is what makes $\sum_{b} g_{h,b,k}$ at fixed $v^{avg}_h$, $s_{h,k}$, and $\sum_b q_{h,b,k}$ recover the plant-level bound this per-cell form replaces.

Average Storage Computation

The average storage $v^{avg}_h$ over the stage:

$$v^{avg}_h = \frac{v^{in}_h + v_h}{2}$$

where $v^{in}_h$ is the incoming storage LP variable (pinned to $\hat{v}_h$ via column bounds — see State Augmentation §2) and $v_h$ is end-of-stage storage — a single plant-level quantity shared by every cell, since storage is not partitioned. Both are LP variables, so $v^{in}_h$ appears in every cell’s FPHA constraint with coefficient $-\lambda_{h,b} \, \gamma_v^m / 2$ on the left-hand side, as does $v_h$, on a parallel stage; on a chronological stage, in block 1’s rows, and $v_h$ in block $|\mathcal{K}|$‘s. The LP solver automatically accounts for this in the reduced cost of the pinned $v^{in}_h$ column.

On a chronological stage the row of block $k$ uses that block’s own average storage $(v_{h,k-1} + v_{h,k})/2$ in place of $v^{avg}_h$, with $v_{h,0} = v^{in}_h$ (see Block Formulation Variants).

Generation as Independent Variable

When using FPHA, the generation variable $g_{h,b,k}$ is not directly computed from turbined flow. Instead:

1. Generation is a free LP variable per cell, bounded by $[0, \bar{G}_{h,b}]$, where $\bar{G}_{h,b}$ sums the cell’s own unit groups’ declared maximum generation and closes against the plant’s own resolved maximum (see LP Formulation §6)
2. FPHA constraints (one per plane $m$, per cell) provide upper bounds relating generation to storage, flow, and spillage
3. Where generation has value, the optimizer raises it until an FPHA plane or a tighter bound such as $\bar{G}_{h,b}$ binds
4. At an optimum where generation has value, each cell’s generation lies on one of its FPHA hyperplane facets or on a tighter bound, such as $\bar{G}_{h,b}$

Key insight: Because minimizing cost includes maximizing hydro generation (which has zero fuel cost), the optimizer naturally pushes generation to the FPHA surface boundary or to a tighter bound such as $\bar{G}_{h,b}$. Where generation has no value, turbined flow beyond what the planes convert produces nothing, and the turbined cost of section 2.10 decides between turbining that water and spilling it; on a binding plane with $\gamma_q^m = 0$, spillage still lowers the plane through $\gamma_s^m$ (section 2.7), which the LP weighs against that cost.

2.10 Turbined Cost

A small regularization cost $c^{tc}_h$ is charged on the turbined flow of every cell of every hydro, whatever its production model:

$$\sum_{k} \tau_k \sum_{b \in \mathcal{B}_h} c^{tc}_h \, q_{h,b,k}$$

Where generation has no value, the LP is indifferent between turbining water without generating from it and spilling it; a cost above the spillage cost, $c^{tc}_h > c^{spill}_h$, tips it toward spilling. On a binding plane with $\gamma_q^m = 0$, more turbined flow adds no generation, but each unit of spillage lowers the plane by $\lambda_{h,b} \lvert \gamma_s^m \rvert$ (sections 2.7 and 2.9), so the LP spills that water only where $c^{tc}_h - c^{spill}_h$ exceeds the value of the generation lost. A hydro using FPHA requires $c^{tc}_h \geq 0$ (Penalty System — Category 3).

For the full penalty taxonomy and priority ordering, see Penalty System.

2.11 Impact on Benders Cuts

The FPHA formulation affects water value computation. Because the incoming storage variable $v^{in}_h$ appears in every one of the plant’s cells’ FPHA constraints (via $v^{avg}_h = (v^{in}_h + v_h)/2$, each apportioned by that cell’s $\lambda_{h,b}$, section 2.9) — on a parallel stage; on a chronological stage, in block 1’s rows — the FPHA hyperplane duals of every cell contribute to the marginal value of incoming storage. However, the implementation does not require manually combining duals from the water balance and FPHA constraints. Instead, pinning $v^{in}_h$ to $\hat{v}_h$ by column bounds (see State Augmentation §2) makes its reduced cost capture the total sensitivity $\beta^v_h$ automatically — the LP solver propagates every cell’s FPHA contribution through the single shared $v^{in}_h$.

The cut coefficient for storage is simply the reduced cost of the pinned $v^{in}_h$ column:

$$\beta^v_h = \bar{c}^{in}_h / d^{col}_h$$

This reduced cost implicitly includes the water balance contribution (that of the row reading $v^{in}_h$: $\pi^{wb}_h$ on a parallel stage, $\pi^{wb}_{h,1}$ on a chronological stage), the FPHA contribution summed over every cell of the plant ($\frac{1}{2} \sum_{b \in \mathcal{B}_h} \lambda_{h,b} \sum_m \pi_{h,b,m}^{fpha} \cdot \gamma_v^m$, with $\pi_{h,b,m}^{fpha}$ summed over the blocks whose rows read $v^{in}_h$: every block on a parallel stage, block 1 on a chronological stage), the dual of the evaporation row (LP Formulation — Evaporation Row) weighted by $\gamma^{ev}_{v,h}/2$ for a hydro with an evaporation model (block 1’s row only on a chronological stage), and any generic constraint contributions — all resolved by the LP solver without explicit dual combination.

For the complete cut coefficient computation, see Cut Management.

Production Model and Cut Validity

A production model that varies by stage or season is part of the model the policy is trained on: its cuts lower-approximate that model’s cost-to-go whichever production model each stage uses (Cut Management — when bounds and certificates hold). A policy simulated under a different production model than the one it was trained on is a heuristic, with no bound guarantee: the trained lower bound need not bound the simulated model’s optimal value, and the gap between the two certifies nothing.

3. Linearized Head Model (Reserved — Resolves to Constant Productivity)

Linearized head is a reserved production-model name: it is equivalent to constant productivity (§1) in every phase — training (backward and forward passes) and simulation alike — and selecting it applies no head correction and imposes no phase restriction.

Generation is therefore $g_{h,k} = \rho_{h,t} \cdot q_{h,k}$ with the plant’s per-(hydro, stage) productivity (§5.1). It still requires the plant’s productivity input.

4. Model Selection Guidelines

Training (Policy Construction)

The substantive choice during training is between constant productivity and FPHA (linearized head is an alias for the former, §3). That choice trades accuracy against computational cost:

Scenario	Recommended Model	Rationale
High-head storage reservoirs	FPHA	Significant head variation
Large storage variation plants	FPHA	Operating across wide volume range
Run-of-river plants	FPHA or Constant	The hull fit handles run-of-river plants (§2.6.1)
Initial algorithm testing	Constant productivity	Fast iteration, debug focus
Near-term stages	FPHA	Accuracy for operational decisions
Far-future stages	Constant productivity	Computational efficiency

Simulation (Policy Evaluation)

During simulation, constant productivity and FPHA behave as they do in training, and linearized head resolves to constant productivity (§3):

Scenario	Recommended Model	Rationale
Plants trained with FPHA	FPHA	Consistency with training model
Plants trained with constant, low head variation	Constant productivity	No benefit from head correction
Plants selecting linearized head	Constant productivity (equivalent)	Linearized head resolves to the same constant-productivity source (§3); no head correction is applied
Post-optimization validation	Compare all models	Verify approximation quality

Simulating a plant under another production model than the one it was trained with evaluates the policy heuristically (§2.11).

The production model may vary by stage or by season per hydro (Configure tab).

5. Energy-Conversion Quantities

The three production models of sections 1–3 describe how generation depends on the operating state. For accounting purposes — natural-inflow energy (ENA), stored reservoir energy (EARM), and per-stage MW/MWh reporting — Novomodelo reduces each plant’s production model to a small set of per-(hydro, stage) scalars produced by two evaluators: one evaluated at a representative reference operating point (§5.1–§5.2), and one averaged over the plant’s physical storage range (§5.3). These scalars are computed once at study setup and reused on every stage of every scenario.

5.1 Equivalent Productivity

The equivalent productivity $\rho_{eq,h,t}$ (MW per m³/s) is the single-scalar productivity that the plant would carry at the reference operating point $(V^{ref}_{h,t},\, Q^{ref}_{h,t})$. The derivation depends on the plant’s declared generation model, not on the model a stage selects:

Generation model	$\rho_{eq,h,t}$ derivation
Constant productivity	A per-(hydro, stage) numeric value authored by the case (see below).
Linearized head	A per-(hydro, stage) numeric value authored by the case, supplied as for constant productivity.
FPHA	$\rho_{eq,h,t} = \rho_{esp,h,t} \cdot h_{eq}(V^{ref}_{h,t},\, Q^{ref}_{h,t})$, where $h_{eq}$ is the net head computed from the VHA geometry (section 2.3) at the reference point and $\rho_{esp,h,t}$ is resolved per stage: a per-stage override, else a per-plant default override, else the plant’s own value. FPHA hydros author no separate $\rho_{eq}$ scalar — it is derived, and an override is accepted (see below).

Reference volume: the reference operating volume $V^{ref}_{h,t}$ is declared once per production-model entry (a stage range or a season), either as an absolute volume or as a fraction of the useful volume between $V^{min}_h$ and $V^{max}_h$; when absent a default fraction of the useful volume applies (Configure tab). This single declaration is the source of truth for the $\rho_{eq}$ derivation above and for the reservoir reference level at which the computed-FPHA piecewise-quartic tailrace families are interpolated (§2.3.1 — a plant’s reference volume sets the downstream forebay level seen by the plant immediately upstream). It does not set the computed-FPHA volume fitting window $[V_{min}, V_{max}]$: that window is configured separately on the Configure tab, defaulting to the plant’s full forebay storage range (§2.6.1).

The reference flow $Q^{ref}_{h,t}$ at which both evaluators compute the net head is the plant’s installed turbined-flow capacity.

Authoring: a non-FPHA plant’s equivalent productivity $\rho_{eq,h,t}$ is authored per hydro and stage. An FPHA plant’s equivalent productivity is derived as in the table above and may be overridden; the override replaces the derived value in both evaluators, the reference-point value here and the useful-range mean value of §5.3. A zero equivalent productivity at a stage where the plant does not use the FPHA model is a planned outage: the plant generates nothing at that stage. At a stage that uses the FPHA model, generation follows the fitted planes, not this scalar. The authoring sources, their resolution order and the load-time checks are on the Configure tab.

5.2 Accumulated Cascade Productivity

The accumulated productivity $\rho_{acum,h,t}$ (MW per m³/s) is the energy that one m³/s of incremental inflow into plant $h$ contributes once it is routed through plant $h$ and every plant downstream of $h$ along the cascade:

$$\rho_{acum,h,t} \;=\; \rho_{eq,h,t} \;+\; \sum_{h' \,\in\, \text{downstream}(h)} \rho_{eq,h',t}$$

The sum is taken in topological order over the cascade (see System Element Modeling Overview for the cascade topology). Plants with no downstream successors have $\rho_{acum} = \rho_{eq}$. The accumulation is per-stage because each summand can vary by stage.

5.3 Useful-Range Mean Productivity

The reference-point scalars of §5.1–§5.2 value water at a single operating volume. A quantity that values the whole stored volume — stored energy, or a floor on it — needs a productivity representative of every volume the reservoir can occupy. The second evaluator supplies it by averaging the forebay level over the plant’s physical storage range.

Physical storage range. Each plant carries a physical storage range $[V^{min}_h, V^{max}_h]$: a stage-invariant property of the plant (its dead-volume floor and its full-reservoir ceiling). It is distinct from the per-stage operative bounds $[\underline V_h, \bar V_h]$ on the storage variable, which may be tighter at a given stage (for example under a flood-control ceiling). Every quantity in this subsection and in §5.4 reads the physical range, never the operative bounds, so an active operative restriction never truncates a stored-energy value.

Mean forebay level. The forebay level is averaged over the physical range:

$$\bar h_{fore,h} \;=\; \frac{1}{V^{max}_h - V^{min}_h} \int_{V^{min}_h}^{V^{max}_h} h_{fore}(v)\, dv$$

Because $h_{fore}$ is the piecewise-linear volume–height curve of §2.3, this integral is evaluated exactly (a composite trapezoid over the curve’s breakpoints), not approximated by quadrature.

Mean equivalent productivity. With $h_{eq}(h_f, Q) = h_f - h_{tail}(Q) - h_{loss}$ the net-head function of §5.1, evaluated from a forebay level $h_f$ and the reference flow $Q$:

$$\bar\rho_{eq,h,t} \;=\; \rho_{esp,h,t} \cdot h_{eq}\big(\bar h_{fore,h},\; Q^{ref}_{h,t}\big)$$

It is the forebay level that is averaged, not the net head: the tailrace level and the hydraulic losses are then evaluated at $Q^{ref}_{h,t}$ exactly as the reference-point evaluator evaluates them. The two evaluators therefore differ only in the forebay level they feed to the same net-head function.

Resolution order. For each (hydro $h$, stage $t$):

1. An explicit equivalent-productivity override replaces both evaluators with the same value, so $\bar\rho_{eq,h,t} = \rho_{eq,h,t}$ equals the override.
2. Otherwise, a collapsed physical range ($V^{min}_h = V^{max}_h$), or a plant without volume–height geometry or without a specific productivity $\rho_{esp,h,t}$, gives $\bar\rho_{eq,h,t} = \rho_{eq,h,t}$: a zero-width range reduces the average to a point evaluation.
3. Otherwise, $\bar\rho_{eq,h,t}$ is the integrated value above.

The integrating evaluator is gated by the plant’s geometry, not by its generation model: it applies to any plant with volume–height geometry and a specific productivity, whether its LP production model is FPHA or constant productivity. A non-positive mean net head is rejected at study setup. The collapsed-range rule is how a run-of-river plant falls out of the data: with a collapsed range it behaves identically under either evaluator.

Mean accumulated productivity. $\bar\rho_{acum,h,t}$ follows from the mean own terms by the same downstream recurrence as §5.2:

$$\bar\rho_{acum,h,t} \;=\; \bar\rho_{eq,h,t} \;+\; \sum_{h' \,\in\, \text{downstream}(h)} \bar\rho_{eq,h',t}$$

Maximum stored energy. The energy content of the plant’s full useful volume is

$$E^{max}_{h,t} \;=\; \bar\rho_{acum,h,t}\,\big(V^{max}_h - V^{min}_h\big)$$

Its unit is the raw product MW/(m³/s) × hm³ — not MWh (no hm³-to-hours factor is applied).

Security-curve identity. A security curve is a per-stage floor on stored energy expressed as a fraction $p$ of the maximum stored energy. Written on the outgoing storage $v_h$ with both sides on the mean evaluator,

$$\bar\rho_{acum,h,t}\,\big(v_h - V^{min}_h\big) \;\ge\; p\,E^{max}_{h,t} \;\;\Longleftrightarrow\;\; v_h - V^{min}_h \;\ge\; p\,\big(V^{max}_h - V^{min}_h\big)$$

states “at least a fraction $p$ of the useful volume, in energy terms”. The per-plant coefficient cancels only because both sides use the same (mean) evaluator and the same raw unit; mixing the reference-point coefficient on one side with $E^{max}_{h,t}$ on the other, or converting one side to MWh alone, would not reduce to the useful-volume fraction. How such a row is authored is described in LP Formulation §10.

5.4 Inflow and Storage in Energy Units

$\rho_{acum}$ and $\bar\rho_{acum}$ convert hydraulic quantities to energy units that downstream reporting expects:

Incremental inflow energy (MW):

$$\text{ENA}_{h,k} \;=\; \rho_{acum,h,t} \cdot a_{h,k}$$

This is the rate-form natural energy inflow in MW. Stagewise energy (MWh) is recovered by multiplying by block duration $\tau_k$ in hours.

Stored reservoir energy (MWh):

$$\text{EARM}^{\,\text{init}}_h \;=\; (V^{init}_h - V^{min}_h) \cdot \bar\rho_{acum,h,t} \cdot \frac{10^6}{3600} \qquad \text{EARM}^{\,\text{final}}_h \;=\; (V^{final}_h - V^{min}_h) \cdot \bar\rho_{acum,h,t} \cdot \frac{10^6}{3600}$$

where $V^{min}_h$ is the physical minimum storage of §5.3 (not the stage’s operative lower bound) and $\bar\rho_{acum,h,t}$ is the useful-range mean accumulated productivity. The conversion factor $10^6 / 3600$ converts hm³ to m³ and seconds to hours so that storage in hm³ multiplied by productivity in MW/(m³/s) yields MWh; no stage duration enters, so EARM is independent of stage length.

Stored reservoir energy, power form (MW):

$$\text{EARM}^{\,\text{MW}}_h \;=\; \frac{\text{EARM}_h}{\sum_{k \in \mathcal{K}} \tau_k}$$

for either the initial or the final value, where the divisor is the stage’s total block hours — the same divisor for every block of the stage.

EARM and ENA use different evaluators on purpose: ENA values an inflow at the plant’s representative operating point, while EARM values the stored volume over the whole range it can occupy.

These quantities do not enter the LP — they are accounting outputs derived from the LP solution. Their methodology relevance is that they make the production model auditable in the same energy units used by the load forecast and the cost objective.

5.5 Why a Scalar Reduction Exists at All

The full FPHA production function (section 2) is multi-dimensional and concave; constant productivity is a scalar but per-plant. Neither can be summed across a cascade or scaled by inflow without a reference operating point. The energy-conversion scalars resolve this: each model is reduced to two numbers per (hydro, stage) — one from the reference-point evaluator, one from the useful-range mean evaluator — and those numbers are what the cascade-summation, ENA, EARM, and maximum-stored-energy formulas above can consume uniformly. The LP’s production model does not use these scalars; it continues to enforce the full production model. They are accounting quantities that may also be used as coefficients of user-authored generic constraints, such as the security curve of §5.3 (see LP Formulation §10).

Implementation in Novomodelo

The methodology above defines the hydro production models; the tabs below cover how Novomodelo’s software surface configures, feeds, and reports on them.

Configure

Novomodelo’s system/hydro_production_models.json file authors the production-model selection the methodology above describes, and the generation block of system/hydros.json supplies each plant’s default model and its turbine and generation bounds. The equations these fields feed are in the sections above (Hydro Production Function Models §1–§5). The rest of the plant registry is on System Element Modeling Overview — Configure.

system/hydros.json — generation block

The generation block configures the turbine model for dispatch purposes and provides the default production function used when no hydro_production_models.json entry lists this plant. All variants share turbine bounds (min_turbined_m3s, max_turbined_m3s) and generation bounds (min_generation_mw, max_generation_mw); the model key selects the production function.

"generation": {
  "model": "constant_productivity",
  "min_turbined_m3s": 0.0,
  "max_turbined_m3s": 50.0,
  "min_generation_mw": 0.0,
  "max_generation_mw": 50.0
}

Field	Type	Description
model	string	Production function variant. See the model table below.
min_turbined_m3s	number	Minimum turbined flow [m³/s]. Non-zero values model a minimum stable turbine operation.
max_turbined_m3s	number	Maximum turbined flow (installed turbine capacity) [m³/s].
min_generation_mw	number	Minimum electrical generation [MW].
max_generation_mw	number	Maximum electrical generation (installed capacity) [MW].

Available production function models:

Model	model value	Description
Constant productivity	"constant_productivity"	power = productivity * turbined_flow. Independent of reservoir head. Productivity is supplied per stage range or season in system/hydro_production_models.json.
FPHA	"fpha"	Piecewise-linear envelope of the nonlinear production function. Head-dependent. Configured via hydro_production_models.json — see below.
Linearized head	"linearized_head"	Reserved model name; it resolves to the same constant-productivity source as constant_productivity (see methodology §3). Requires the plant’s productivity input.

Every model applies in training and in simulation.

For most initial studies, constant_productivity is the natural choice. The productivity coefficient encodes the plant’s average efficiency and net head: for a plant with 80 m net head and 90% efficiency, the theoretical productivity is approximately 9.81 × 80 × 0.90 / 1000 ≈ 0.706 MW/(m³/s).

system/hydro_production_models.json — Production Model Selection

This optional file maps each hydro plant to a production-model selection strategy, overriding generation.model from hydros.json for the stages or seasons it lists. Two selection strategies are supported.

stage_ranges — assigns a model to each contiguous stage interval:

{
  "production_models": [
    {
      "hydro_id": 1,
      "selection_mode": "stage_ranges",
      "stage_ranges": [
        {
          "start_stage_id": 0,
          "end_stage_id": null,
          "model": "fpha",
          "fpha_config": { "source": "precomputed" }
        }
      ]
    }
  ]
}

seasonal — assigns a model by season index, with a fallback for seasons not listed:

{
  "production_models": [
    {
      "hydro_id": 1,
      "selection_mode": "seasonal",
      "default_model": "constant_productivity",
      "seasons": [
        {
          "season_id": 0,
          "model": "fpha",
          "fpha_config": {
            "source": "computed",
            "volume_discretization_points": 7,
            "turbine_discretization_points": 7
          }
        }
      ]
    }
  ]
}

Season indices are 0-based and match the season map in stages.json.

reference_volume

Each stage range or season entry may carry an optional reference_volume sibling of fpha_config (not nested inside it), declaring the reference operating volume the equivalent-productivity derivation and the computed-FPHA tailrace backwater interpolation consume. Set exactly one of:

• volume_hm3 — an absolute value in hm³ (finite, > 0.0).
• percentile — a fraction in [0.0, 1.0] of the plant’s operating range.

When reference_volume is omitted entirely, the reference volume defaults to 0.65 of the operating band (equivalent to { "percentile": 0.65 }).

"reference_volume": { "percentile": 0.65 }

Hyperplane Sources

When a plant is configured with model: "fpha", fpha_config.source selects where the hyperplane coefficients come from.

source: "precomputed" — hyperplanes are loaded directly from system/fpha_hyperplanes.parquet:

"fpha_config": { "source": "precomputed" }

No additional fpha_config fields are needed; discretization and fitting options are ignored. The parquet schema:

Column	Type	Required	Description
hydro_id	Int32	Yes	Hydro plant identifier
stage_id	Int32	No	Stage the plane applies to (null = all stages; an absent column means the same)
plane_id	Int32	Yes	Plane index within this hydro
gamma_0	Float64	Yes	Intercept coefficient (MW)
gamma_v	Float64	Yes	Volume coefficient (MW/hm³). Must be non-negative (>= 0); run-of-river plants have gamma_v = 0.
gamma_q	Float64	Yes	Turbined flow coefficient (MW per m³/s). Must be non-negative (>= 0); each stage a plant runs on precomputed planes needs at least one plane with gamma_q > 0, checked when the study is set up.
gamma_s	Float64	Yes	Spillage coefficient (MW per m³/s). Must be <= 0.
kappa	Float64	No	Intercept-only correction factor, defaulting to 1.0; validated in (0, 1]. Applies only to precomputed planes — the computed path does not use kappa (see Implementation notes).
valid_v_min_hm3	Float64	No	Minimum volume where this plane is valid (hm³)
valid_v_max_hm3	Float64	No	Maximum volume where this plane is valid (hm³)
valid_q_max_m3s	Float64	No	Maximum turbined flow where this plane is valid (m³/s)

source: "computed" — hyperplanes are fitted at runtime from the plant’s physical geometry. Novomodelo evaluates the exact production function on a (volume, turbined-flow) grid at spillage = 0 — the flow axis starts at q = 0, whose zero-flow column anchors the lower closure, so there is no synthetic closing point and no spillage axis in the cloud — takes the 3-D convex hull of the resulting cloud with vendored qhull, applies a least-squares $k_{FPHA}$ correction that scales the whole affine plane (intercept and slopes together, not just the intercept), and fits a per-plane lateral-flow secant. Fits are resolved independently per stage range or season. Run-of-river plants (a single fitting volume) are supported: the fitter synthesizes two nearby volume samples to keep the hull non-degenerate, then snaps the resulting gamma_v to exactly 0. See Implementation notes for the full fitting-pipeline summary and methodology §2.6–§2.7 for the derivation.

This source requires tailrace, hydraulic_losses, and efficiency models in hydros.json, plus geometry rows in system/hydro_geometry.parquet for the plant. An FPHA-configured plant needs at least 1 row — a single row is valid for run-of-river plants (a constant forebay, so gamma_v = 0). A plant configured with linearized_head needs at least 2 rows when the file holds any row; that higher minimum is a load-time validation rule keyed on the configured model name, not a runtime head fit — linearized_head resolves to constant productivity (methodology §3).

"fpha_config": {
  "source": "computed",
  "volume_discretization_points": 5,
  "turbine_discretization_points": 5,
  "spillage_discretization_points": 5,
  "max_planes_per_hydro": 10,
  "fitting_window": null
}

All fields except source are optional:

Field	Default	Description
volume_discretization_points	5	Number of volume grid points for fitting. 1 selects the single-volume fit of a run-of-river plant; otherwise it must be >= 2. With a single fitting volume the fitter synthesizes two nearby volume samples and snaps gamma_v to exactly 0 (see Implementation notes).
turbine_discretization_points	5	Number of turbined-flow grid points. Must be >= 2.
spillage_discretization_points	5	Validated (must be >= 2) but does not add a spillage axis to the fitting grid — the cloud and the $k_{FPHA}$ regression fix spillage at 0, and the lateral-flow secant sweeps its own sample range independently (§2.7). The field is retained for input round-tripping.
max_planes_per_hydro	10	Accepted and validated (>= 1) but does not trim the plane set — the hull fitter emits exactly the upper-envelope facets.
fitting_window	null	Optional volume range for fitting; absent uses the full forebay range of the plant’s system/hydro_geometry.parquet volume-height curve.

fitting_window restricts which portion of the operating range builds the grid — useful when the plant rarely operates near one extreme:

"fitting_window": { "volume_min_hm3": 1000.0, "volume_max_hm3": 40000.0 }

"fitting_window": { "volume_min_percentile": 0.05, "volume_max_percentile": 0.95 }

Absolute (_hm3) and percentile (_percentile) bounds are mutually exclusive per bound (min and max may use different modes). For the 5% fit-quality warning emitted after fitting, and for the round-trip parquet export of computed planes, see Implementation notes.

Plane Reduction (fpha_plane_reduction)

The optional file-level fpha_plane_reduction block merges near-parallel or near-coincident FPHA planes after fitting. Off by default (absent = no reduction); applied uniformly to every plant in the file.

Angle method — merges planes whose normal vectors are within tolerance_deg degrees:

"fpha_plane_reduction": { "method": "angle", "tolerance_deg": 2.0 }

Field	Required	Description
method	Yes	Must be "angle".
tolerance_deg	Yes	Maximum angle between plane normals to merge them. Finite, in [0.0, 90.0].

Distance method — merges planes whose sampled mean-squared distance stays within tolerance_pct, using n_samples sample points:

"fpha_plane_reduction": { "method": "distance", "tolerance_pct": 0.01, "n_samples": 200 }

Field	Required	Description
method	Yes	Must be "distance".
tolerance_pct	Yes	Tolerance, in percent, on the normalised mean-squared generation difference: two planes merge when that difference, as a fraction, is below tolerance_pct / 100. Finite, >= 0.0.
n_samples	Yes	Number of sample points used to estimate the distance. Must be >= 1.

Supplying a field belonging to the other method is a load-time error. The origin plane (zero generation at zero turbining) is never merged. The distance method’s sample draws are deterministically seeded — bit-identical across input ordering and rank count.

Per-Range and Per-Season Productivity

For constant_productivity and linearized_head hydros, the equivalent productivity for each (hydro, stage) pair is supplied by exactly one of two sources:

• system/hydro_production_models.json — an inline productivity_mw_per_m3s field on the matching stage_range or seasonal entry. Natural for “this productivity is constant for the next five stages.”
• system/hydro_energy_productivity.parquet — a row in the equivalent_productivity_mw_per_m3s column; a row with stage_id refines a single stage, a row with stage_id = null is a per-hydro default. Natural for per-stage numerical refinement.

As with the generic-constraint inputs, the JSON file owns model selection and range-level values, and the parquet file owns per-stage numerical refinement.

Resolution order at load time:

1. Parquet stage-specific row (exact stage_id match).
2. Parquet per-hydro default row (stage_id = null).
3. JSON productivity_mw_per_m3s on the matching stage range or season.

Supplying a value from both sources for the same (hydro, stage) is a load-time error; supplying neither is also a load-time error. The conflict error names both files and the offending (hydro_id, stage_id) and asks for the value from exactly one source; the coverage error names the pair and both files.

A hydro whose generation.model is fpha is outside both checks: its equivalent productivity is derived from the VHA geometry and the specific productivity (methodology §5.1). An equivalent_productivity_mw_per_m3s row for such a hydro (stage-specific, else the per-hydro default) is an override that replaces the derived value for both evaluators — the reference-point value (§5.1) and the useful-range mean value (§5.3).

In system/hydro_energy_productivity.parquet, equivalent_productivity_mw_per_m3s and specific_productivity_mw_per_m3s_per_m must each be finite and non-negative when set: 0 is accepted and a negative value is rejected at load. A 0 equivalent productivity at a stage that does not use the "fpha" model marks a planned-outage stage, as 0.0 does in the JSON field below. A 0 specific productivity zeroes the head-derived equivalent productivity and does not stop generation.

{
  "start_stage_id": 12,
  "end_stage_id": 24,
  "model": "constant_productivity",
  "productivity_mw_per_m3s": 0.72
}

Field	Type	Required	Description
productivity_mw_per_m3s	number	Optional (non-FPHA)	Finite and non-negative (>= 0.0) when present; 0.0 marks a planned-outage stage. Rejected on FPHA — FPHA derives productivity from VHA geometry, not a scalar coefficient.

Load-Time Validation Rules

Rule	Error Class	Description
FPHA geometry coverage	Dimensional error	When system/hydro_geometry.parquet holds any row, every fpha plant must have at least 1 row of its own (a single row is valid for run-of-river plants) and every linearized_head plant at least 2 rows; with no row in the file this rule does not run.
FPHA plane coverage	Dimensional error	Every (hydro_id, stage_id) group in system/fpha_hyperplanes.parquet must have at least 1 plane.
FPHA coefficient signs	Semantic error	gamma_v must be non-negative (>= 0; run-of-river plants have gamma_v = 0); gamma_s must be non-positive.
Geometry monotonicity	Semantic error	volume_hm3 must be strictly increasing; height_m and area_km2 must be non-decreasing.

Inputs & Outputs

This is a topic-scoped index of the files hydro production touches and of the fields the methodology reads from them — it names each file and its role and maps fields to the quantities they feed; it does not repeat their field-by-field schemas. The exhaustive, field-by-field case-directory and output reference is owned by the Reference corpus (Case Format and Output Format pages).

Inputs

File	Role
system/hydros.json	Hydro plant registry — core fields, reservoir, outflow, generation model, the required unit_groups bus partition, and the advanced blocks (see the System Element Modeling Overview Configure tab).
system/hydro_production_models.json	Optional per-hydro production-model selection (stage_ranges / seasonal), FPHA fitting configuration, and plane reduction.
system/hydro_geometry.parquet	Volume-Height-Area (VHA) curves — required for any hydro using computed FPHA and for the evaporation area-volume conversion; when the file holds any row, every FPHA plant needs at least 1 row of its own and every linearized_head plant at least 2; also read, for any plant with a specific productivity, by the useful-range mean evaluator (Hydro Production Function Models §5.3).
system/hydro_energy_productivity.parquet	Per-(hydro, stage) overrides: equivalent productivity (both evaluators), specific productivity (both evaluators’ head term and the specific_productivity tag), reference turbined flow (the reference_turbine tag only) — the tabular authoring source of the Configure tab’s resolution order, and the FPHA override.
system/tailrace_curves.parquet	Optional piecewise-quartic tailrace-level curves — tailrace_level(outflow) per (hydro_id, family_id, segment_id) — feeding the computed-FPHA fit. A hydro with no curve uses its entity-level tailrace model.
system/fpha_hyperplanes.parquet	Pre-fitted FPHA hyperplane coefficients, read when a plant’s fpha_config.source is "precomputed".
constraints/hydro_unit_group_bounds.parquet	Optional stage-varying, optionally per-block overrides of a unit group’s four declared bounds (see the System Element Modeling Overview Configure tab). Rows are keyed by (hydro_id, hydro_unit_group_id, stage_id, block_id?), where hydro_unit_group_id matches a group’s own id in hydros.json; a group with no override row reads its declared value.
constraints/hydro_bounds.parquet	Optional stage-varying (and optionally per-block) overrides of a hydro plant’s own bounds (maximum turbined flow, storage, outflow, maximum generation, diversion, spillage, filling rate, withdrawal). Its min_turbined_m3s and min_generation_mw columns reach no LP row: the minimum turbined-flow and minimum-generation rows read the unit groups’ resolved minima. Keyed by (hydro_id, stage_id) with an optional block_id; an absent/null cell falls back to the declared base value. Distinct from the per-unit-group hydro_unit_group_bounds.parquet above — a separate, plant-level file (see the System Element Modeling Overview Configure tab).
initial_conditions.json (hydro rows)	Per-hydro initial state: storage or filling_storage (exactly one), and optional past_defluences seeding the pre-study releases in transit on a declared travel-time arc. AR-lag seeding for the PAR inflow model derives from scenarios/inflow_history.parquet and recent_observations instead — see the PAR(p) Inflow Model Inputs & Outputs tab.

For the complete field-by-field schema of each file above, see the Case Format reference page in the Reference corpus.

Data requirements by quantity

Each row names a data source, the fields the methodology above reads from it, and the quantity those fields feed:

Data source	Fields	Used for
Hydro plant entity	tailrace (polynomial or piecewise)	$h_{tail}(q_{out})$ computation
Hydro plant entity	hydraulic_losses (factor or constant)	$h_{loss}$ computation
Hydro plant entity	efficiency (constant)	Turbine efficiency $\eta$
Hydro plant entity	specific_productivity_mw_per_m3s_per_m (FPHA, and the useful-range mean evaluator for any model)	$\rho_{esp}$ for $\rho_{eq}$ derivation (§5.1) and $\bar\rho_{eq}$ (§5.3)
Hydro plant entity	Cascade topology (downstream pointer)	$\rho_{acum}$ topological sum (§5.2)
Hydro production models input	Range-level productivity per stage_range / seasonal entry (non-FPHA, optional)	$\rho_{h,t}$ for §1 and §3 (§5.1)
Hydro production models input	reference_volume per stage_range / seasonal entry (absolute or percentile)	$V^{ref}$ for the energy-conversion reduction (§5.1) and the tailrace backwater reference level (§2.3.1)
Hydro production models input	FPHA fitting_window per stage_range / seasonal entry (absolute or percentile bounds, optional)	Computed-FPHA volume grid $[v_{min}, v_{max}]$ (§2.6.1)
Hydro production models input	FPHA fitting configuration and optional fpha_plane_reduction block	Grid sizes, plane reduction method (§2.6–§2.8)
Hydro energy productivity input	Per-(hydro, stage) equivalent_productivity_mw_per_m3s and specific productivity	$\rho_{eq,h,t}$ for both evaluators: the authoring source of a non-FPHA plant, the override of an FPHA plant (§5.1); $\rho_{esp,h,t}$ for both evaluators’ head-derived value (§5.1, §5.3)
Hydro geometry	volume_hm3, height_m	$h_{fore}(v)$ interpolation (§2.3); mean forebay level $\bar h_{fore,h}$ (§5.3)
Pre-fitted FPHA planes	gamma_0, gamma_v, gamma_q, gamma_s (plus the optional kappa intercept factor, default 1.0)	Optional alternative to in-process fitting

Outputs

File	Role
hydro_models/fpha_hyperplanes.parquet	Fitted hyperplane coefficients when fpha_config.source is "computed", written only when at least one plant has fitted planes (for example, a computed-FPHA plant with no turbine capacity has none) — the same schema as the system/fpha_hyperplanes.parquet input, so it round-trips: copy it into system/ and switch source to "precomputed" to reuse a prior fit without refitting. See the Implementation notes tab, “Round-trip parquet export”, for the workflow.
hydro_models/evaporation_models.parquet	Fitted per-hydro evaporation-model coefficients — written whenever any evaporation model was built (not gated on an exports flag; absent when no plant declares evaporation).
hydro_models/fpha_deviation_points.parquet	Per-plane FPHA fit-deviation sample points — a fitting diagnostic. Written only when the exports.fpha_deviation_points flag is enabled (default off) and at least one deviation point exists; absent otherwise.
Hydro generation and storage output columns	Per-(stage, block, hydro) dispatch results (turbined flow, spillage, generation, storage) written alongside every other entity’s output rows — the plant total, unchanged by bus-partitioning.
Energy-conversion columns of simulation/hydros/	The reference-point (equivalent_productivity_mw_per_m3s, accumulated_productivity_mw_per_m3s) and useful-range mean (integrated_*) productivities, incremental_inflow_energy_mw, and stored energy in MWh and MW (stored_energy_{initial,final}_{mwh,mw}) — see §5 above and the Output Format reference.
simulation/hydro_bus_generation/	Per-(stage, block, hydro, bus) cell dispatch results — turbined_m3s, generation_mw, generation_mwh; bus_id always non-null. See Implementation notes for the exact-sum caveat on the generation_mw column.
simulation/in_transit/	Per-(stage, downstream hydro, maturity lag) in-transit water volumes on declared travel-time arcs, including the delayed arrival that enters the receiving plant’s water balance; written only when a travel-time arc is declared (schema: Simulation Output).
simulation/transit_seed/	Scenario-level rolling release-window seed for a declared travel-time arc — five columns (scenario_id, hydro_id, start_date, end_date, value_m3s), carrying no stage/node prefix. Written only when a travel-time arc is declared; absent otherwise. Feeds a continuing run’s own initial_conditions.past_defluences (study chaining).

For the complete output schema (columns, types, file layout), see the Output Format reference page in the Reference corpus.

Implementation notes

Non-normative software behavior for the hydro production models — what Novomodelo does at runtime, beyond the equations above. This tab references the methodology body for the derivations rather than restating them.

Computed-FPHA fitting behavior

The computed path (§2.6–§2.7 above) evaluates the exact production function on a (volume, turbined-flow) grid at spillage fixed to 0. The flow axis starts at q = 0; that zero-flow column is generation-zero by construction and anchors the lower closure of the cloud, so the fitter needs neither a synthetic closing point nor a spillage axis in the cloud itself — spillage sensitivity is captured separately by the per-plane lateral-flow secant (§2.7), not by adding a third grid dimension. That secant uses nine evenly spaced samples of the lateral flow.

The 3-D convex hull’s upper-envelope facets are corrected by a single least-squares $k_{FPHA}$ scalar. That correction scales the whole affine plane — intercept and both slopes together — not just the intercept, unlike the kappa factor of precomputed planes, which scales the intercept only (validated in (0, 1]).

Lateral-flow slopes above -1e-10 — a near-zero negative residual or any positive one — are set to exactly 0, so no near-zero coefficient reaches the spillage column.

When any plant uses computed FPHA, the families of system/tailrace_curves.parquet are checked as the production models are built: consecutive segments of a family must meet (the previous outflow_max_m3s and the next outflow_min_m3s differ by at most 1e-6 m³/s) and give the same level at the shared boundary, to within the larger of 0.001 m and 1e-4 of the larger of the two levels. A gap, an overlap or a level mismatch beyond these bounds is an error that names the plant by id and the boundary.

Run-of-river support

A plant whose fitting volume range collapses to a single point (constant forebay) is fit by synthesizing two nearby volume samples so the 3-D hull stays non-degenerate, then snapping the resulting gamma_v residual to exactly 0 when its magnitude is at most 1e-6 MW per hm³. This enforces the correct run-of-river semantics — generation on these planes does not vary with storage — and is why run-of-river plants are supported on the computed path rather than rejected.

The two samples sit either side of that single volume, each offset by the larger of half a percent of the plant’s useful storage and 1 hm³, so a plant with no useful storage still gets two distinct samples; when the volume-height curve spans a range of volumes, both samples are clamped into it.

FPHA without turbine capacity

A plant that requests FPHA, computed (fpha_config.source: "computed" in any of its production-model entries) or precomputed with no row in system/fpha_hyperplanes.parquet, and declares max_turbined_m3s at or below 1e-9 is not fitted; it is modelled with zero productivity at every stage (§2.6.1 above). The computed-FPHA prerequisites apply to a computed plant: tailrace, hydraulic_losses and efficiency in hydros.json and geometry rows in system/hydro_geometry.parquet; a missing one is a load-time error.

A warning names the plant; novomodelo validate and novomodelo run both print it:

 WARN hydro UHE1 (id=0) requests FPHA but has no turbine capacity (max_turbined_m3s = 0); modeling it with zero productivity

Every MPI rank prepares the production models, so a multi-rank run prints the warning once per rank.

The plant counts as constant productivity: in n_constant of training/hydro_models.json, in its no_turbine_capacity list, in n_constant of the Python hydro_models summary, and in the constant count of the CLI Production: line. It is absent from n_fpha, total_planes and fpha_details, from n_fpha_computed_from_geometry and n_fpha_precomputed_hyperplanes in training/model_provenance.json, and from hydro_models/fpha_hyperplanes.parquet.

Fit-quality warning

After the full pipeline, Novomodelo compares the fitted min-envelope against the exact production function on the spillage-0 grid and computes their relative mean-absolute deviation. When it exceeds 5%, a warning is logged naming the plant and stage:

FPHA fit for hydro UHE Example (stage 3) deviates 6.2% from the exact production function (mean |Δ| 4.1 MW, max 9.7 MW); the convex-hull approximation is poor here — typically a strongly non-concave production surface that no single α correction can track

The warning is informational — the run continues with the fitted planes. A high deviation typically indicates a strongly non-concave production surface that no single $k_{FPHA}$ scalar can track well; narrowing fitting_window or raising the discretization point counts are the usual remedies.

Determinism guarantees

The fitting cloud points and the hull output are canonically sorted, so fitted hyperplanes are bit-identical regardless of input ordering and MPI rank count. The fpha_plane_reduction distance method’s sampled draws are seeded from stable plant/stage/plane-pair identity — never wall clock or MPI rank — so its merges are equally reproducible.

Evaporation: disabled on zero area, rejected on missing geometry

A hydro that declares evaporation.coefficients_mm needs an area-volume curve — the rows in system/hydro_geometry.parquet — to linearize its evaporative flux, and the two ways that curve can be absent are handled differently:

• No geometry rows at all is a hard load-time error. A hydro that declares evaporation.coefficients_mm but contributes no rows to system/hydro_geometry.parquet is rejected at load, naming the plant: the evaporation linearization requires area-volume curve data and none is present.
• Rows present but every area_km2 is zero is disabled with a warning. The curve exists but describes zero surface area throughout, so evaporative flux is identically zero regardless of the authored coefficients. Novomodelo disables evaporation (zero flux) for that hydro and logs a warning naming the plant rather than failing the run.

Two conditions raised while building the per-stage linearization remain hard load-time errors: a stage whose total block duration is not positive, and a computed evaporation coefficient — volume slope or intercept — that is not finite (the signature of a degenerate area-volume curve). The calendar month that selects each stage’s monthly evaporation coefficient is taken from that stage’s start date. See Penalty System — Signed Net Evaporation for the water-balance treatment of the resulting flux.

Commissioning: the PreFilling state

A hydro’s entry_stage_id/exit_stage_id commissioning window is honored even for a non-filling hydro (one with no filling block). At any stage outside its window — before entry_stage_id, or from exit_stage_id onward — the plant is PreFilling: the site is not in service, so its turbine, spillage, and diversion columns are pinned to zero, its storage is frozen by the identity $v_h = v^{in}_h$, and its local inflow, the upstream releases, the flows diverted in and the in-transit water maturing into it are routed past the site to the first downstream plant that is not PreFilling (or leaves the modelled system if there is none). Unlike the generic outside-window zero-pin used for the other entity types, the river is never trapped or duplicated at a site that is not in service. See LP Formulation — Lifecycle Phases for every phase and Penalty System for the spillage-frozen-versus-costed distinction.

Water travel time: in-transit seed and validation

When a plant declares a travel-time arc, the releases already in transit at the study start must be supplied so the first stages’ delayed-arrival buckets are seeded rather than starting empty. This history lives in initial_conditions.past_defluences: windowed pre-study release records (a hydro_id, a release window, and an average rate in m³/s) for the plants whose outflow feeds a declared arc.

Load-time validation of a declared arc:

• History depth. The past_defluences windows must cover the arc’s in-transit span at study start — the interval [start_0 − travel_time, start_0) ending at the study start. A gap, or no windows at all, is a load-time error, and no window may be future-dated past the study start — there is no proxy or fallback seed from any other source.
• Downstream must be operating. A release at a stage before the downstream plant’s entry_stage_id — while it is PreFilling or Filling — is rejected; the check covers each in-study release stage, the first included, so the downstream plant must have entered service by the first study stage.
• Heterogeneous travel time under chronological blocks. When two or more declared arcs feed the same downstream plant with differing travel_time_hours and at least one study stage runs in chronological block mode, the case is rejected at load with a NotImplemented error, whose message reads “unsupported in v1”, naming the downstream plant and the conflicting arcs; align the arcs’ travel_time_hours, or keep every study stage in parallel block mode. Equal travel times into a confluence, or a study with no chronological stage, are unaffected. See Error Codes — NotImplemented.
• Zero, absent, or invalid travel time. A zero or absent travel_time_hours is treated as undeclared — an instantaneous transfer that creates no cross-stage arc, adds no in-transit bucket, and needs no past_defluences seed. An explicit 0.0 additionally logs a model-quality warning that it is being read as undeclared, so a deliberate zero is flagged while an omitted field stays silent. A negative or non-finite travel_time_hours is a hard load-time error. See Error Codes.
• Model-quality warnings (non-blocking). Two further conditions on a declared arc are reported as warning: lines after Valid case: …; the case still loads and runs. See Error Codes — ModelQuality.
  • Negligible travel time. When travel_time_hours is below 1 % of every study stage’s length (travel_time_hours / stage hours < 0.01 on every stage), the arc carries a negligible cross-stage fraction; the warning suggests not declaring it.
  • Longer than the remaining study horizon. When travel_time_hours exceeds the remaining study horizon from some stage, releases from that stage onward never arrive within the study, and the warning names the first such stage as the point from which the arc is economically inert. The in-transit sizing stays safe.

In-transit water that matures into a plant at a stage where the plant is PreFilling enters the row of the first non-PreFilling plant downstream, and leaves the modelled system only when no such plant exists.

Horizon limitation. Without a loaded terminal boundary, in-transit water that would mature after the last stage is dropped; with one, it is carried to the terminal stage and priced there (State Augmentation — Horizon limitation, Post-Study Boundary & Chained Studies).

Bus-partitioned hydro: cell collapse and the generation-sum caveat

A plant’s unit_groups partition it into one cell per distinct bus_id: the LP carries one turbine column and one FPHA-generation column per cell. When every group shares one bus, the plant collapses to a single cell — the same LP shape, byte-identical, as a plant declaring no groups beyond the required one. Same-bus groups sharing a cell’s turbine and generation columns is exact for the box constraints (each group’s own bounds fold independently before summing into the cell), because every group on a plant shares the same production model and productivity coefficients — there is no per-group productivity field.

The simulation/hydro_bus_generation/ output sums bit-exactly to the plant’s own turbined_m3s on every model, since both are a plain sum of the same per-cell turbine columns. generation_mw sums exactly only for FPHA plants and single-cell plants: for ConstantProductivity/LinearizedHead, the plant-level row computes (Σ_c turbined_c) × ρ — sum the cells’ turbined flow first, then multiply once — while each bus-cell row computes turbined_c × ρ independently. The two expressions are equal in exact arithmetic, but floating-point multiplication does not distribute over addition bit-for-bit, so the plant total need not equal the sum of the cell rows to the last bit once a plant spans more than one cell. FPHA plants are exempt because both the plant row and the cell rows sum the same underlying per-cell FPHA generation columns the same way; a single-cell plant is exempt because there is only one term either way.

Round-trip parquet export

When hyperplanes are fitted at runtime (source: "computed"), the fitted coefficients are automatically written to output/hydro_models/fpha_hyperplanes.parquet using the same schema as the system/fpha_hyperplanes.parquet input. To reuse a prior fit without refitting on a subsequent run, copy this output file into system/ and change source to "precomputed" in hydro_production_models.json.

A plant with no turbine capacity has no rows in the exported file and needs none: copied to system/, the file loads unedited, and the plant stays at zero productivity.

Cross-References

• Notation Conventions — variable and set definitions ($g_h$, $q_h$, $v_h$, $s_h$, $\rho_h$)
• System Element Modeling Overview — hydro plant element description, decision variables, Variable Units Convention
• LP Formulation — how production constraints integrate into the assembled LP
• Penalty System — turbined-cost regularization, penalty priority ordering
• Cut Management — Benders cut generation affected by FPHA dual variables
• Equipment-Specific Formulations — thermal and hydro equipment constraint patterns