System Element Modeling Overview

Purpose

This spec describes the physical components of a hydrothermal power system as modeled by Cobre: what each element represents, its decision variables, how it connects to other elements, and its role in the optimization objective. It serves as the conceptual foundation between the SDDP algorithm description and the full LP formulation — the reader should understand what is being optimized before seeing how the constraints are assembled.

Reading order: SDDP algorithm → this spec → LP formulation → equipment formulations

For variable naming conventions and index sets, see notation conventions.

Variable Units Convention

All decision variables in Cobre use rate units: electrical quantities in MW, hydraulic flows in m³/s. Storage (hm³) is inherently an absolute quantity. The block duration $\tau_k$ [hours] enters the LP as an external multiplier — it appears in the objective function coefficients and in the water balance conversion factor, but not in the variable bounds or constraint matrix.

This is a deliberate design decision with consequences across the formulation:

Aspect	Consequence
Objective function	All cost terms are scaled by $\tau_k$: the coefficient for a $100/MWh thermal is $\tau_k \times 100$
Variable bounds	Block-invariant — a 500 MW capacity bound is the same regardless of block duration
Constraint matrix	Clean coefficients: load balance has ±1, production function uses $\rho$ [MW/(m³/s)] directly
Duals	Load balance dual $\pi_{b,k}$ has units $/MW. To obtain the marginal cost (CMO) in $/MWh, divide by $\tau_k$
Cut coefficients	Unaffected — coupling state variables (storage in hm³, AR lags in m³/s) are independent of this choice

Why not absolute units (MWh, hm³ per block)? The alternative — internalizing $\tau_k$ into the variables — was evaluated and rejected for three reasons:

1. FPHA incompatibility: The FPHA hyperplane $g \leq \gamma_0 + \gamma_v \cdot v^{avg} + \gamma_q \cdot q + \gamma_s \cdot s$ relates instantaneous rates to storage level. Converting to absolute units does not eliminate $\tau_k$ from the FPHA constant term, and introduces a $10^6/3600 \approx 278$ scaling factor on flow coefficients.

2. Constraint matrix quality: With rate units, the constraint matrix (excluding the objective row) spans ~3 orders of magnitude ($\rho \approx 0.05$–$5$, $\eta \approx 0.95$–$1.0$, conversion factors ~$0.01$–$3$). Absolute units inflate this to ~5 orders via the 278 factor in FPHA rows. A tighter constraint matrix improves simplex pivot stability.

3. Block-varying bounds: Absolute units make every variable bound depend on $\tau_k$ ($\bar{G} \cdot \tau_k$ instead of $\bar{G}$), complicating LP construction and making the LP structure vary with block configuration.

The objective row does carry large coefficients — up to $\tau_k \times c^{def} \approx 730 \times 10{,}000 = 7.3 \times 10^6$ for deep deficit in monthly problems. Modern LP solvers handle objective scaling effectively through internal prescaling, and this is where the scaling challenge is most tractable.

Output convention

All output reports and user-facing marginal costs must apply the $\div \tau_k$ conversion.

1. System Architecture Overview

A hydrothermal power system in Cobre consists of interconnected physical elements that work together to meet electricity demand at minimum cost under inflow uncertainty:

System element overview — buses, a hydro plant with its reservoir (inflow in), a thermal plant, an NCS (wind/solar) source, a transmission line between buses, a demand draw (d) off each bus, and a deficit slack (δ, dashed) backstopping unserved load. Key flow variables are labelled (f line flow, q hydro turbined, g thermal, gⁿᶜ NCS, δ deficit, d demand).

The optimizer determines generation and flow decisions at each stage to minimize total expected cost (thermal generation + deficit penalties + regularization costs) while respecting physical constraints and preparing for uncertain future inflows.

Entity Commissioning Windows

Most entity types may enter service or be decommissioned partway through the horizon, so that planning studies can represent new plants coming online and aging units retiring. An entity carries an optional commissioning window defined by two stage indices, entry_stage_id and exit_stage_id. The window is half-open: the entity is active at stage $t$ when

$$\text{entry\_stage\_id} \leq t < \text{exit\_stage\_id}$$

so the entry stage is inclusive and the exit stage is exclusive — the entity is gone from its exit stage onward. A null entry_stage_id means “active from the first stage”; a null exit_stage_id means “never decommissioned”. The default for an entity that sets neither is to be active at every stage.

This window applies uniformly to transmission lines, thermal plants, non-controllable sources, pumping stations, and contracts (and, for generation, to hydro plants). Outside its window an entity contributes nothing to the dispatch: its decision columns are present in the LP but pinned to zero, so it injects no power, withdraws no power, and consumes no resource. Decommissioning is symmetric to commissioning — both are expressed by the same two fields.

Two element-specific lifecycle mechanisms layer on top of this generic window:

• Hydro dead-volume filling (§5) — a hydro plant may exist but be unable to generate while its reservoir is still filling toward the dead volume; this is a distinct commissioning state with its own per-stage storage floors, not just a presence gate.
• Anticipated thermals (§4) — a commitment column is opened only when its delivery stage falls inside the study horizon, an additional horizon predicate beyond the entry/exit window.

2. Buses (Regional Subsystems)

Physical Meaning

A bus represents a node in the power network where electrical energy balance must be maintained. The granularity is user-defined: a bus may represent a large regional subsystem, a single substation, or any aggregation level in between. The model scales from a handful of buses to hundreds or thousands without structural changes.

Example

In studies of the Brazilian interconnected system (SIN), buses typically correspond to the four major regions (Southeast/Midwest, South, Northeast, North), but finer decompositions are equally valid.

Decision Variables

Variable	Units	Description
$\delta_{b,k,s}$	MW	Load deficit (unserved energy) at bus $b$, block $k$, segment $s$
$\epsilon_{b,k}$	MW	Excess generation at bus $b$, block $k$

Connections to Other Elements

Each bus serves as the energy balance node where:

• Inflows: Generation from hydro plants, thermal plants, and import contracts connected to the bus
• Outflows: Demand, export contracts, pumping station consumption, and transmission to other buses

Key Parameters

Parameter	Units	Description
$D_{b,k}$	MW	Load demand at bus $b$, block $k$
$c^{def}_{b,s}$	$/MWh	Deficit cost (value of unserved energy), segment $s$
$c^{exc}_b$	$/MWh	Excess generation penalty (regularization)
$\bar{d}_{b,s}$	MW	Deficit segment depth

Role in Objective Function

$$\sum_{k \in \mathcal{K}} \tau_k \left[ \sum_{b \in \mathcal{B}} \sum_{s \in \mathcal{S}_b} c^{def}_{b,s} \cdot \delta_{b,k,s} + \sum_{b \in \mathcal{B}} c^{exc}_b \cdot \epsilon_{b,k} \right]$$

• Deficit cost: Very high penalty ($1,000–10,000/MWh) representing value of lost load
• Excess cost: Small regularization term to eliminate spurious slack generation

LP Constraint Preview

For each bus $b$ and block $k$, the load balance constraint enforces:

$$\text{(generation at } b\text{)} + \text{(imports)} - \text{(exports)} - \text{(pumping)} + \sum_{s} \delta_{b,k,s} - \epsilon_{b,k} = D_{b,k}$$

For the assembled constraint, see LP formulation.

3. Transmission Lines

Physical Meaning

A transmission line represents the interconnection between two buses, allowing power transfer subject to capacity limits and transmission losses. Lines are bidirectional.

Decision Variables

Variable	Units	Description
$f^+_{l,k}$	MW	Direct flow on line $l$ (source → target), block $k$
$f^-_{l,k}$	MW	Reverse flow on line $l$ (target → source), block $k$

Modeling note

Cobre uses two non-negative variables ($f^+$, $f^-$) rather than a single signed variable. This simplifies bound handling and naturally prevents simultaneous bidirectional flow through regularization costs.

Connections to Other Elements

Each line connects exactly two buses:

• Source bus: Exports $f^+_{l,k}$, receives $\eta_l \cdot f^-_{l,k}$
• Target bus: Receives $\eta_l \cdot f^+_{l,k}$, exports $f^-_{l,k}$

Key Parameters

Parameter	Units	Description
$\bar{F}^+_l$	MW	Capacity limit (direct direction); may vary by stage via exchange factors
$\bar{F}^-_l$	MW	Capacity limit (reverse direction); may vary by stage via exchange factors
$\eta_l$	—	Transmission efficiency: $\eta_l = 1 - \text{losses}/100$
$c^{exch}_l$	$/MWh	Exchange cost (regularization)

Role in Objective Function

$$\sum_{k \in \mathcal{K}} \tau_k \sum_{l \in \mathcal{L}} c^{exch}_l \cdot (f^+_{l,k} + f^-_{l,k})$$

The exchange cost is a regularization term (typically $0.01–1.00/MWh) that prevents degenerate solutions with unnecessary power circulation and guides the solver toward physically meaningful flow patterns.

LP Constraint Preview

Capacity bounds:

$$0 \leq f^+_{l,k} \leq \bar{F}^+_l, \quad 0 \leq f^-_{l,k} \leq \bar{F}^-_l$$

Load balance contribution at source bus: $-f^+_{l,k} + \eta_l \cdot f^-_{l,k}$

For detailed constraints, see equipment formulations.

4. Thermal Plants

Physical Meaning

A thermal plant represents dispatchable generation using fuel (natural gas, coal, oil, biomass, nuclear). Thermal plants have fuel costs modeled with piecewise-linear cost curves.

Decision Variables

Variable	Units	Description
$g_{j,k,s}$	MW	Generation at thermal plant $j$, block $k$, cost segment $s$

The total generation is $g_{j,k} = \sum_s g_{j,k,s}$.

Connections to Other Elements

• Bus connection: Each thermal plant connects to exactly one bus, contributing to its energy balance
• No cascade coupling: Unlike hydro plants, thermals are independent of each other

Key Parameters

Parameter	Units	Description
$\bar{G}_j$, $\underline{G}_j$	MW	Generation bounds (capacity, minimum stable load)
$c^{th}_{j,s}$	$/MWh	Marginal cost for segment $s$ (fuel + O&M)
$\bar{g}_{j,s}$	MW	Segment $s$ capacity

Role in Objective Function

$$\sum_{k \in \mathcal{K}} \tau_k \sum_{j \in \mathcal{T}} \sum_{s} c^{th}_{j,s} \cdot g_{j,k,s}$$

Thermal costs represent actual operating expenses ($50–500/MWh depending on fuel type) and constitute the primary controllable cost in the objective function.

LP Constraint Preview

Segment bounds (cost curve linearization): $0 \leq g_{j,k,s} \leq \bar{g}_{j,s}$ for all $s$.

Total generation bounds: $\underline{G}_j \leq \sum_s g_{j,k,s} \leq \bar{G}_j$

Note on continuous LP relaxation

Cobre uses a continuous LP relaxation — there are no binary commitment variables. The minimum generation bound $\underline{G}_j$ represents either “fully off” (0) or “minimum stable” operation. This is a deliberate modeling choice: the LP may dispatch a thermal at an intermediate level between zero and $\underline{G}_j$, which is physically unrealistic but acceptable for the long-term planning horizon targeted by SDDP.

For detailed constraints, see equipment formulations.

Anticipated Thermal Plants

A thermal plant may be flagged as anticipated by attaching a per-plant lead $K \geq 1$. The physical motivation is fuel-ordering lead time: LNG (originally Gás Natural Liquefeito) terminals, long-haul coal contracts and similar arrangements require the committed dispatch quantity to be locked $K$ stages before the energy is physically delivered. Anticipated thermals therefore split the per-stage thermal decision into two coupled variables — the commitment, made at the decision stage $t$, and the delivered generation, which is forced to match the matured commitment at the delivery stage $t + K$.

The lead is encoded as an integer plant-specific parameter $K_i \geq 1$ (lead_stages on the plant entity). A plant with $K_i = 1$ commits one stage ahead; a plant with $K_i = 3$ commits three stages ahead. Plants without the anticipation flag use the standard thermal model from the preceding subsections.

State variables and ring-buffer layout

Each anticipated plant $i$ carries a ring-buffer pipeline of $K_i$ pending commitments through the Bellman recursion alongside hydro storage and inflow lags. Anticipated thermals are the only non-hydro elements with state variables in the SDDP formulation.

State variable	Slot	Meaning at the start of stage $t$
$x^{\mathrm{a}}_{0,i,t}$	0	Commitment maturing this stage — sets the delivered generation at $t$
$x^{\mathrm{a}}_{1,i,t}$	1	Commitment placed at $t - K_i + 1$, scheduled to deliver at $t + 1$
$\vdots$	$\vdots$	…
$x^{\mathrm{a}}_{K_i - 1,i,t}$	$K_i - 1$	Most-recent commitment, written at $t - 1$, scheduled to deliver at $t + K_i - 1$

The state vector reserves $K_{\max} = \max_i K_i$ slots per plant for layout uniformity across stages; slots beyond $K_i$ are padded to zero by an LP equality and carry no information.

Decision variable and fishing constraint

At each stage $t$ the plant exposes a commitment column $d^i_t$ on the LP. Its bounds replicate the generation bounds of the delivery stage $t + K_i$:

$$d^i_t \in \begin{cases} [\,\underline{G}_i(t + K_i),\ \bar{G}_i(t + K_i)\,] & t + K_i < T \\ \{\,0\,\} & t + K_i \geq T \quad \text{(presolved out)} \end{cases}$$

The horizon predicate $t + K_i < T$ guarantees the commitment is only opened when there is a delivery stage inside the study horizon; commitments whose delivery would fall outside $T$ are pinned to zero.

The fishing constraint binds the per-block generation of plant $i$ to the matured commitment in slot 0 at every stage $t \in [0, T - 1]$:

$$\sum_{b} h_b \cdot g_{i,b,t} \;=\; H_t \cdot x^{\mathrm{a}}_{0, i, t}$$

where $h_b$ is the duration of block $b$ and $H_t = \sum_b h_b$ is the total stage hours. The fishing row is active at every stage, including the pre-horizon stages where slot 0 is fed by the seeded past_anticipated_commitments (see below) rather than by an LP-decided $d^i$.

Column bounds pin each slot to its incoming value, and a separate decoupled state-out column writes the LP commitment $d^i_t$ into the post-shift slot $K_i - 1$ for the next stage. The decoupling prevents a write into slot 0 from overlapping a read in the same stage, which is what makes the fishing-at-every-stage predicate safe even at $K_i = 1$.

Pre-horizon seed

Because anticipated dispatch carries state across stages, plant $i$ must be primed with $K_i$ pre-horizon commitments — the values that were committed at the negative stages $-K_i, \ldots, -1$ and are scheduled to mature at stages $0, \ldots, K_i - 1$. These live in initial_conditions.past_anticipated_commitments[i].values_mw, an array of exactly $K_i$ entries with values_mw[k] carrying the commitment that delivers at study stage $k$. The values must satisfy the plant’s generation bounds; both the bounds-check and the array-length check are enforced by the semantic validator. An empty list is the default for studies with no anticipated plants.

The seed enters the LP via the slot-0 fishing constraint at stages $0, \ldots, K_i - 1$: at $t = 0$ the LP reads values_mw[0] out of slot 0; the ring-buffer shift then moves values_mw[1] into slot 0 for stage $1$; and so on. From stage $K_i$ onward, slot 0 carries the LP-decided commitment $d^i_{t - K_i}$.

Cost discounted to the delivery stage

Anticipation costs are charged at the decision stage but discounted to the delivery stage in the same way a real-world option is priced. The objective coefficient on $d^i_t$ is

$$\mathrm{obj}\bigl[\,d^i_t\,\bigr] \;=\; c_i(t + K_i) \cdot H_{t + K_i} \cdot d^{\mathrm{NPV}}_{t + K_i}$$

where $c_i(\cdot)$ is the unit cost in $/MWh, $H_{t + K_i}$ is the total hours at the delivery stage, and $d^{\mathrm{NPV}}_{t + K_i}$ is the cumulative NPV discount factor at the delivery stage. The per-block thermal column cost is zeroed at delivery stages for anticipated plants so the same energy is not double-priced once via the commitment and once via the per-block dispatch.

Cut machinery

Benders cuts propagate the marginal value of the matured commitment back through the ring buffer. The reduced cost of the pinned slot-0 column at stage $t + 1$ is the partial $\partial Q_{t+1} / \partial x^{\mathrm{a}}_{0, i, t+1}$ and is non-zero only when the fishing constraint has bound the LP there. The cut-row builder remaps this coefficient to the predecessor’s commitment column (for $K_i = 1$) or to the predecessor’s outgoing-state slot one step deeper into the ring buffer (for $K_i \geq 2$), so the subgradient ultimately lands on the original $d^i$ decision regardless of how many stages have elapsed between commitment and delivery.

5. Hydro Plants

Hydro plants are the central elements of the SDDP formulation because:

1. Reservoir storage creates temporal coupling (water saved today is available tomorrow)
2. Inflows are stochastic (uncertain future rainfall/snowmelt)
3. The water value (opportunity cost of using water now vs. saving it) emerges from the optimization

Physical Meaning

A hydro plant converts the potential energy of stored water into electricity. Each plant has a reservoir (storage), turbines (conversion), and spillways (excess water release). Hydro plants are typically arranged in cascades where upstream releases become downstream inflows.

Operating Status

Cobre distinguishes two hydro plant subsets based on their operational state:

Subset	Symbol	Description
Operating	$\mathcal{H}^{op}$	Plants that can generate electricity; subject to generation constraints
Filling	$\mathcal{H}^{fill}$	New plants under commissioning, filling dead volume; no generation

Most plants are in $\mathcal{H}^{op}$. Filling hydros have per-stage target-storage floors instead of generation constraints. Some plants have negligible storage capacity (run-of-river) and must pass all inflows through turbines and spillways within the same stage.

Decision Variables

Variable	Units	Description
$v_h$	hm³	End-of-stage reservoir storage (state variable)
$a_{h,\ell}$	m³/s	AR lag $\ell$ for inflow model (state variable, see note below)
$q_{h,k}$	m³/s	Turbined flow (through generators), block $k$
$s_{h,k}$	m³/s	Spillage (released without generation), block $k$
$u_{h,k}$	m³/s	Diversion flow (bypassed to separate channel), block $k$
$e_{h,k}$	m³/s	Evaporation (see note below), block $k$
$r_{h,k}$	m³/s	Water withdrawal (see note below), block $k$
$g_{h,k}$	MW	Hydro generation, block $k$
$o_{h,k}$	m³/s	Total outflow: $o_{h,k} = q_{h,k} + s_{h,k}$ (downstream channel flow)

State variables ($v_h$ and $a_{h,\ell}$) link stages through the Bellman recursion. The storage $v_h$ tracks reservoir volume and is a true decision variable within each stage (the optimizer chooses its end-of-stage value). The AR lags $a_{h,\ell}$ carry inflow history for the PAR(p) model: they are state variables in the SDDP sense (passed between stages and subject to Benders cuts), but are fixed at the beginning of each stage to the realized inflow values — they are not free for the optimizer to choose. All other variables are control variables determined within each stage.

Evaporation and withdrawal

$e_{h,k}$ and $r_{h,k}$ are parameter-driven quantities computed from input data (reservoir surface area, withdrawal schedules). They appear in the LP as fixed contributions to the water balance, with slack variables to ensure feasibility when the computed values cannot be physically sustained (e.g., drought reducing available water below the withdrawal commitment).

Connections to Other Elements

• Bus connection: Each hydro plant connects to one bus for energy delivery
• Cascade topology: Upstream plants’ outflows ($q + s + u$) become downstream plants’ inflows, with optional water travel time delay
• Diversion targets: Some plants can divert water to a separate downstream plant (not the immediate cascade successor), bounded by $\bar{U}_h$
• Pumping stations: May receive pumped water (increasing storage) or supply water to pumps (decreasing storage)

Key Parameters

Parameter	Units	Description
$\bar{V}_h$, $\underline{V}_h$	hm³	Storage bounds (useful volume)
$\bar{Q}_h$, $\underline{Q}_h$	m³/s	Turbined flow bounds (machine limits)
$\bar{O}_h$, $\underline{O}_h$	m³/s	Outflow bounds (environmental flow, flood control)
$\bar{G}_h$, $\underline{G}_h$	MW	Generation bounds (user-defined, not derived from flow)
$\bar{U}_h$	m³/s	Maximum diversion flow
$\rho_h$	MW/(m³/s)	Productivity (constant model)
$\gamma^m_0, \gamma^m_v, \gamma^m_q, \gamma^m_s$	—	FPHA hyperplane coefficients for plane $m$
$a_h$	m³/s	Incremental inflow (stochastic, from PAR model)
$\hat{v}_h$, $\hat{a}_{h,\ell}$	hm³, m³/s	Incoming state (from previous stage)

Water Balance Phenomena

The reservoir dynamics account for all water flows in and out of the plant:

Term	Direction	Description
$\hat{v}_h$	Initial	Incoming storage from previous stage
$a_h$	Inflow	Incremental inflow (lateral catchment, stochastic)
$\sum_{i \in \mathcal{U}_h}(q_i + s_i + u_i)$	Inflow	Upstream cascade outflows (with travel time delay)
$\sum_{i:\text{div}=h} u_i$	Inflow	Diverted water received from other plants
$\sum_{j:\text{dest}=h} p_j$	Inflow	Pumped water received from pumping stations
$q_h + s_h + u_h$	Outflow	Turbined + spillage + diversion (released downstream)
$e_h$	Outflow	Evaporation (reservoir surface loss; can be negative for net precipitation)
$r_h$	Outflow	Water withdrawal (consumptive use, removed from system)
$\sum_{j:\text{src}=h} p_j$	Outflow	Pumped water extracted by pumping stations

Note on outflow

Outflow $o_h = q_h + s_h$ is the water released to the downstream channel (affects tailrace level). Withdrawal $r_h$ and diversion $u_h$ exit through different paths and do not affect the main tailrace.

Production Function (Water to Power)

Cobre supports three models for converting turbined flow to electrical generation, in increasing order of complexity:

1. Constant Productivity: $g_{h,k} = \rho_h \cdot q_{h,k}$ — simple linear relationship with fixed $\rho_h$ [MW/(m³/s)], suitable for plants with stable head.

2. Linearized Head (simulation-only): Adjusts productivity based on head variation with storage level. Requires hydro geometry data (Volume-Height-Area curve). This model is excluded from training because the bilinear term ($q \times v^{avg}$) requires re-fixing $v^{avg}$ between iterations, changing the LP and breaking SDDP convergence guarantees. See hydro production models §3.

3. FPHA (Função de Produção Hidrelétrica Aproximada): Piecewise-linear approximation via hyperplanes that captures head variation with storage level and accounts for tailrace effects from spillage. Each plane $m$:

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

Training models

Only constant_productivity and fpha are valid during training (policy construction). The linearized head model is available during simulation (policy evaluation) only.

The production model can vary by stage or season per hydro.

For the complete FPHA formulation and model comparison, see hydro production models.

Operational Constraints (Soft)

Several hydro constraints are enforced as soft constraints with slack variables and penalties:

Constraint	Meaning	Slack Variable
$v_h \geq \underline{V}_h$	Minimum storage (dead volume)	$\sigma^{v-}_{h}$
$q_{h,k} \geq \underline{Q}_h$	Minimum turbined flow (equipment limits)	$\sigma^{q-}_{h,k}$
$o_{h,k} \geq \underline{O}_h$	Minimum outflow (environmental flow)	$\sigma^{o-}_{h,k}$
$o_{h,k} \leq \bar{O}_h$	Maximum outflow (flood control)	$\sigma^{o+}_{h,k}$
$g_{h,k} \geq \underline{G}_h$	Minimum generation (grid services)	$\sigma^{g-}_{h,k}$
$e_{h,k}$ feasible	Evaporation within physical limits	$\sigma^{e\pm}_{h,k}$
$r_{h,k}$ met	Water withdrawal commitment	$\sigma^{r}_{h,k}$
$v_h \geq V^{\text{target}}_t$ (filling floor)	Per-stage filling target during commissioning	$\sigma^{fill}_{h}$

Soft constraints allow the optimizer to violate bounds when physically necessary (e.g., drought conditions preventing minimum outflow), with high penalty costs signaling undesirable operation. Maximum storage ($\bar{V}_h$) is a hard physical limit — excess water is handled by emergency spillage, not a slack variable. For the penalty priority ordering and cost magnitudes, see Penalty System. For the complete constraint formulations, see equipment formulations.

Dead-Volume Filling

Cobre models the commissioning of new hydro plants with a filling period ([start_stage_id, entry_stage_id)) during which the reservoir accumulates water to reach the dead volume ($\underline{V}_h$). During this period:

• The hydro has no generation: $q_{h,k} = 0$, $g_{h,k} = 0$ (hard constraint)
• Natural inflow flows freely through the ordinary water balance — there is no retention or impound cap diverting inflow to storage
• Storage is allowed below $\underline{V}_h$ (the operating min-storage slack is inactive)
• Outflow is limited to spillage (turbines not operational), with the outflow_violation_below slack if environmental flow cannot be met
• A per-stage filling floor $v_h \geq V^{\text{target}}_t$ requires the reservoir to stay on a minimum-accumulation schedule set by filling_min_rate_m3s. The floor ramps up to $\underline{V}_h$ at the last filling stage; its slack $\sigma^{fill}_h$ is priced below deficit (a fill schedule is not defended as hard as load serving)

When the plant enters service at entry_stage_id, the ordinary soft minimum-storage floor takes over. For the per-stage floor trajectory and the penalty ordering, see Penalty System.

Role in Objective Function

$$\sum_{k \in \mathcal{K}} \tau_k \sum_{h \in \mathcal{H}} \left[ c^{spill}_h \cdot s_{h,k} + c^{fpha}_h \cdot q_{h,k} + c^{div}_h \cdot u_{h,k} \right] + \text{(slack penalties)}$$

• Spillage cost: Small regularization ($0.001–0.01 per m³/s·h) to prefer turbining over spilling
• FPHA turbined cost: Regularization applied only to hydros using the FPHA production model. Must be > spillage_cost for the same plant to prevent interior FPHA solutions. See Penalty System.
• Diversion cost: Small regularization, typically higher than spillage (water leaves main cascade)
• Slack penalties: High costs for constraint violations — storage below dead volume, outflow violations, generation violations, evaporation violations, water withdrawal shortfall. See Penalty System for the full penalty taxonomy and priority ordering.
• No generation cost: Hydro generation has zero marginal fuel cost — its “cost” is the opportunity cost of depleting storage, captured through the value function $V_{t+1}(v_h)$

LP Constraint Preview

Water balance (reservoir dynamics):

$$v_h = \hat{v}_h + \zeta \cdot \left[ a_h + \sum_{k} w_k \cdot \text{net\_flows}_{h,k} \right]$$

where $\text{net\_flows}_{h,k}$ includes all inflow and outflow terms listed above.

AR lag pinning (inflow history): $a_{h,\ell} = \hat{a}_{h,\ell}$ for all $\ell \in \{1, \ldots, P_h\}$, enforced by equal column bounds (not a constraint row)

Outflow definition: $o_{h,k} = q_{h,k} + s_{h,k}$

Generation constraint (depends on production model):

• Constant productivity: $g_{h,k} = \rho_h \cdot q_{h,k}$
• FPHA: $g_{h,k} \leq \gamma_0^m + \gamma_v^m \cdot v_h^{avg} + \gamma_q^m \cdot q_{h,k} + \gamma_s^m \cdot s_{h,k}$ for each plane $m$

For the fully assembled constraints, see LP formulation.

6. Non-Controllable Generation Sources

Physical Meaning

A non-controllable source represents intermittent generation (wind farms, solar plants, small run-of-river hydros, etc.) whose available output depends on external conditions (weather, river flow) rather than dispatch decisions. The solver receives a stochastic availability value per scenario from the scenario pipeline, and can only curtail generation below that availability — it cannot dispatch upward beyond what nature provides.

Non-controllable sources have near-zero marginal cost. The cost of curtailing available generation is a regularization penalty (Category 3 in the Penalty System), analogous to spillage_cost for hydros — curtailment discards available “free” energy.

Curtailable vs. Must-Run

A per-source flag $\chi_r \in \{\text{curtailable}, \text{must-run}\}$ selects between two LP behaviours for the realized availability $A_r$:

• Curtailable (default): the generation column has bounds $[0, A_r]$, the LP is free to curtail any amount below $A_r$, and curtailment is regularised by curtailment_cost. This is the standard model for stand-alone wind and solar plants where ramping down is physically feasible and economically justified by the regularisation cost.
• Must-run: the generation column is pinned to the realized availability, $g^{nc}_{r,k} = A_r$, by setting both lower and upper bounds to $A_r$ on every scenario. The curtailment penalty is unused because the curtailment slack is zero by construction. This model is required when the scenario pipeline feeds the LP with an aggregate that has already been pre-netted from demand by the upstream model — for example, NEWAVE-derived geracao_usinas_nao_simuladas totals over PCH, PCT, EOL, UFV, and MMGD: those have already been subtracted from MERC upstream, so allowing the LP to curtail them double-discounts the must-run contribution and produces a cheaper hydrothermal dispatch than the reference model. On the bundled deterministic 1983 case, leaving the aggregates curtailable gives ≈ 18 % of total NCS supply curtailed, a ≈ +15 % hydro-dispatch swing, and ≈ −23 % spillage versus NEWAVE; pinning them as must-run restores parity while keeping per-source observability in the simulation outputs.

The pin is applied per scenario by overwriting the upper and lower bounds on the NCS column with $A_r$ before each stage solve. The availability ratio $\alpha = \mathrm{clamp}(\mu + \sigma \cdot \eta,\,0,\,1)$ and the per-(stage, block) shape factor multiply the installed capacity exactly as in the curtailable case — only the lower bound differs.

Operative States

Non-controllable sources follow the generic commissioning window (§1, Entity Commissioning Windows), controlled by entry_stage_id and exit_stage_id:

State	Condition	LP Variables
non_existing	Before entry_stage_id	Pinned to zero
operating	$\text{entry\_stage\_id} \leq t < \text{exit\_stage\_id}$	generation, curtailment (derived)
decommissioned	At or after exit_stage_id	Pinned to zero

Decision Variables

Variable	Units	Description
$g^{nc}_{r,k}$	MW	Generation at non-controllable source $r$, block $k$

Curtailment is not a separate LP decision variable — it is derived as $\kappa_{r,k} = A_{r} - g^{nc}_{r,k}$, where $A_r$ is the stochastic available generation for the current (stage, scenario). The implementation may equivalently formulate curtailment as a reward for dispatching (negative cost on $g^{nc}_{r,k}$) rather than a penalty on $(A_r - g^{nc}_{r,k})$; either is valid. For must-run sources, $\kappa_{r,k} \equiv 0$ by construction (the lower bound on the generation column equals $A_r$), so the regularisation term contributes zero to the objective regardless of how it is signed.

Connections to Other Elements

• Bus connection: Each source connects to exactly one bus, contributing generation to its energy balance

Key Parameters

Parameter	Units	Description
$\bar{G}_r$	MW	Installed capacity (hard physical upper bound)
$A_r$	MW	Available generation for current (stage, scenario), from scenario pipeline. Bounded by $[0, \bar{G}_r]$
$c^{curt}_r$	$/MWh	Curtailment cost (regularization penalty)

Role in Objective Function

$$\sum_{k \in \mathcal{K}} \tau_k \sum_{r \in \mathcal{R}} c^{curt}_r \cdot (A_r - g^{nc}_{r,k})$$

The curtailment cost is a small regularization term ($0.001–0.01/MWh) that incentivizes the solver to use all available generation before curtailing.

LP Constraint Preview

Generation bounds (hard constraints, no slack variables):

$$g^{nc}_{r,k} \in \begin{cases} [\,0,\ A_r\,] & \text{if } \chi_r = \text{curtailable} \\ [\,A_r,\ A_r\,] & \text{if } \chi_r = \text{must-run} \end{cases}$$

Load balance contribution at connected bus: $+g^{nc}_{r,k}$ (generation injected).

Note on NCS stage-varying bounds

Unlike hydro and thermal plants, non-controllable sources have no stage-varying operational bounds beyond the stochastic availability. Their output is entirely determined by the scenario model.

For detailed constraints, see equipment formulations.

7. Pumping Stations

Physical Meaning

A pumping station transfers water from one reservoir (source) to another (destination), consuming electrical power in the process. Pumping enables elevation transfer, basin transfer, and storage arbitrage (pumping during low-demand periods, generating during high-demand).

Decision Variables

Variable	Units	Description
$p_{j,k}$	m³/s	Pumped water flow at station $j$, block $k$

Connections to Other Elements

• Source hydro: Water is withdrawn from this reservoir
• Destination hydro: Water is added to this reservoir
• Bus connection: Pumping consumes power at the connected bus

Key Parameters

Parameter	Units	Description
$\underline{P}_j$	m³/s	Minimum pumped flow
$\bar{P}_j$	m³/s	Maximum pumped flow
$\gamma_j$	MW/(m³/s)	Power consumption rate

Role in Objective Function

Important

Pumping stations do not have a direct cost term in the objective function. The cost of pumping is implicitly captured through the energy consumed, which appears as load in the power balance.

LP Constraint Preview

Flow bounds: $\underline{P}_j \leq p_{j,k} \leq \bar{P}_j$

Water balance impact: Source hydro: $-p_{j,k}$ (water removed); Destination hydro: $+p_{j,k}$ (water added).

Load balance impact at connected bus: $-\gamma_j \cdot p_{j,k}$ (power consumed).

For detailed constraints, see equipment formulations.

8. Import/Export Contracts

Physical Meaning

Contracts represent agreements to buy (import) or sell (export) electricity with external systems outside the modeled region, providing flexibility during shortages and revenue opportunity for surplus.

Each contract is unidirectional: it is either an import contract or an export contract, identified by a type field.

Decision Variables

Variable	Units	Description
$\chi_{c,k}$	MW	Dispatched power for contract $c$, block $k$

Note on Exchange Cost

$\chi$ (Greek chi) is used for contracts to avoid confusion with cost parameter $c$ and flow variable $q$.

Connections to Other Elements

Each contract connects to exactly one bus, contributing to its energy balance:

• Import contracts ($c \in \mathcal{C}^{imp}$): Add $\chi_{c,k}$ to the bus (power entering the system)
• Export contracts ($c \in \mathcal{C}^{exp}$): Remove $\chi_{c,k}$ from the bus (power leaving the system)

Key Parameters

Parameter	Units	Description
$\underline{C}_c$, $\bar{C}_c$	MW	Minimum and maximum contract dispatch limits
$c^{ctr}_c$	$/MWh	Contract price: positive for imports (cost), negative for exports (revenue)

Role in Objective Function

$$\sum_{k \in \mathcal{K}} \tau_k \sum_{c \in \mathcal{C}} c^{ctr}_c \cdot \chi_{c,k}$$

Because import prices are positive and export prices are negative, this single summation naturally adds import costs and subtracts export revenue.

LP Constraint Preview

Capacity bounds: $\underline{C}_c \leq \chi_{c,k} \leq \bar{C}_c$. A non-zero lower bound $\underline{C}_c$ acts as a hard take-or-pay floor — the LP must dispatch at least that quantity at the contract price. Contracts are stateless and honor the generic commissioning window (§1).

For detailed constraints, see equipment formulations.

9. Summary: Physical Elements to LP Components

The following table maps each physical system element to its LP representation:

Physical Element	State Variables	Control Variables	Key Constraints	Objective Role
Bus	—	$\delta_{b,k,s}$, $\epsilon_{b,k}$	Load balance	Deficit penalty (high), Excess penalty (low)
Transmission Line	—	$f^+_{l,k}$, $f^-_{l,k}$	Capacity bounds	Exchange cost (regularization)
Thermal Plant	—	$g_{j,k,s}$	Generation bounds, Segment limits	Fuel cost
Thermal (anticipated)	$x^{\mathrm{a}}_{s,i,t}$	$d^i_t$, $g_{j,k,s}$	Ring-buffer slot pinning + fishing equality	Commitment cost discounted to delivery stage
Hydro Plant	$v_h$, $a_{h,\ell}$	$q_{h,k}$, $s_{h,k}$, $u_{h,k}$, $g_{h,k}$	Water balance, Generation function	Spillage/diversion cost (regularization)
Non-Controllable	—	$g^{nc}_{r,k}$	Availability bound	Curtailment penalty (regularization)
Pumping Station	—	$p_{j,k}$	Flow bounds (min/max)	None (cost via energy consumption)
Contract	—	$\chi_{c,k}$	Dispatch bounds (min/max)	Import cost or Export revenue

Key insight: The hydro storage variables $v_h$, the AR lag variables $a_{h,\ell}$, and the anticipated-thermal ring-buffer slots $x^{\mathrm{a}}_{s,i,t}$ are the state variables that link stages through the Bellman recursion. All other elements contribute control variables that are determined within each stage. This structure enables SDDP’s decomposition: the stage subproblem optimizes all control variables given the incoming state, and Benders cuts approximate the future cost as a function of the outgoing state.

Cross-References

• Notation conventions — variable naming conventions and index sets used throughout
• LP formulation — fully assembled LP constraints combining all elements
• Equipment formulations — detailed per-equipment constraint derivations
• Hydro production models — FPHA and linearized head alternatives for the hydro production function
• Penalty System — penalty costs, priority ordering, and the full penalty taxonomy