Notation Conventions

Purpose

This spec defines the complete mathematical notation used across all Cobre specification documents: index sets, parameters, decision variables, and dual variables. It serves as the canonical reference for symbol meanings, ensuring consistency across all math and data model specs.

1. General Notation Conventions

This document follows SDDP.jl notation conventions for consistency with the broader SDDP literature:

Convention	Meaning
$t \in \{1, \ldots, T\}$	Stage index
$\omega \in \Omega_t$	Scenario realization at stage $t$
$x_t$	State variables at end of stage $t$
$\hat{x}_{t-1}$	Incoming state (from previous stage)
$V_t(x)$	Value function (cost-to-go) at stage $t$
$\theta_t$	Epigraph variable approximating $V_{t}$
$\pi$	Dual variables (row Lagrange multipliers)
$(\alpha, \pi)$	Cut intercept and coefficients
$k$	Iteration counter

2. Index Sets

Symbol	Description
$t \in \{1, \ldots, T\}$	Stages
$k \in \mathcal{K}$	Blocks within stage
$\mathcal{B}$	Buses
$\mathcal{H}$	Hydro plants
$\mathcal{H}^{op} \subseteq \mathcal{H}$	Operating hydros (can generate)
$\mathcal{H}^{fill} \subseteq \mathcal{H}$	Filling hydros (no generation)
$\mathcal{H}^{fpha} \subseteq \mathcal{H}$	Hydros using FPHA production model
$\mathcal{H}^{const} \subseteq \mathcal{H}$	Hydros using constant productivity (complement of $\mathcal{H}^{fpha}$ within $\mathcal{H}^{op}$)
$\mathcal{T}$	Thermal plants
$\mathcal{R}$	Non-controllable generation sources
$\mathcal{L}$	Transmission lines
$\mathcal{C}$	All contracts ($\mathcal{C}^{imp} \cup \mathcal{C}^{exp}$)
$\mathcal{C}^{imp}$, $\mathcal{C}^{exp}$	Import/export contracts
$\mathcal{P}$	Pumping stations
$\mathcal{G}$	Generic constraints
$\mathcal{S}_b$	Deficit segments for bus $b$
$\mathcal{M}_h$	FPHA planes for hydro $h$
$\mathcal{U}_h$	Upstream hydros of $h$
$\Omega_t$	Scenario realizations at stage $t$

3. Parameters

3.1 Time and Conversion

Symbol	Units	Description
$\tau_k$	hours	Duration of block $k$
$w_k = \tau_k / \sum_j \tau_j$	-	Block weight (fraction of stage)
$\zeta$	hm³/(m³/s)	Time conversion: m³/s over stage → hm³

Time Conversion Factor Derivation

The factor $\zeta$ converts a flow rate in m³/s to a volume in hm³ accumulated over the stage duration.

Fundamental Relationship:

$$\text{Volume} = \text{Flow Rate} \times \text{Time}$$

Unit Conversion Chain:

1. Flow rate: $Q$ [m³/s]
2. Time period: $\tau$ [hours]
3. Target volume: $V$ [hm³] = $10^6$ m³

$$V \text{ [hm³]} = Q \text{ [m³/s]} \times \tau \text{ [hours]} \times \frac{3600 \text{ s}}{1 \text{ hour}} \times \frac{1 \text{ hm³}}{10^6 \text{ m³}}$$ $$V = Q \times \tau \times \frac{3600}{10^6} = Q \times \tau \times 0.0036$$

For a stage with multiple blocks: If the stage has blocks $k \in \mathcal{K}$ with durations $\tau_k$ hours, and the flow is assumed constant across the stage (parallel blocks), the total time is $\sum_k \tau_k$ hours:

$$\zeta = 0.0036 \times \sum_{k \in \mathcal{K}} \tau_k \quad \text{[hm³/(m³/s)]}$$

Dimensional Analysis:

$$[\zeta] = \frac{\text{s}}{\text{h}} \times \frac{\text{m³}}{\text{hm³}} \times \text{h} = \frac{\text{hm³}}{\text{m³/s}}$$

Worked Example (Monthly Stage):

Block	Name	Duration $\tau_k$ (h)
1	LEVE	200
2	MÉDIA	300
3	PESADA	228
Total		728

$$\zeta = 0.0036 \times 728 = 2.6208 \text{ hm³/(m³/s)}$$

Verification: A constant inflow of $Q = 100$ m³/s over the month yields:

$$V = Q \times \zeta = 100 \times 2.6208 = 262.08 \text{ hm³}$$

Direct calculation: $100 \text{ m³/s} \times 728 \text{ h} \times 3600 \text{ s/h} / 10^6 = 262.08 \text{ hm³}$ ✓

3.2 Load and Costs

Symbol	Units	Description
$D_{b,k}$	MW	Load at bus $b$, block $k$
$c^{def}_{b,s}$	$/MWh	Deficit cost at bus $b$, segment $s$
$\bar{d}_{b,s}$	MW	Deficit segment depth
$c^{exc}_b$	$/MWh	Excess generation cost
$c^{th}_{j,s}$	$/MWh	Thermal cost at plant $j$, segment $s$
$c^{spill}_h$	$/(m³/s·h)	Spillage cost
$c^{div}_h$	$/(m³/s·h)	Diversion cost
$c^{exch}_l$	$/MWh	Exchange (transmission) cost
$c^{ctr}_c$	$/MWh	Contract price (signed: + import cost, − export revenue)

3.3 Hydro Parameters

Symbol	Units	Description
$\hat{v}_h$	hm³	Incoming storage (state from previous stage)
$\bar{V}_h$, $\underline{V}_h$	hm³	Storage bounds
$\bar{Q}_h$, $\underline{Q}_h$	m³/s	Turbined flow bounds
$\bar{G}_h$, $\underline{G}_h$	MW	Generation bounds
$\bar{O}_h$, $\underline{O}_h$	m³/s	Outflow bounds
$\rho_h$	MW/(m³/s)	Productivity (constant model)
$\gamma^m_0, \gamma^m_v, \gamma^m_q, \gamma^m_s$	-	FPHA plane coefficients (already $\alpha_{FPHA}$-scaled)
$\alpha_{FPHA}$	-	FPHA least-squares fit-correction factor; scales the fitted plane set, distinct from the Benders cut intercept $\alpha$. See Hydro Production Models.

3.4 Transmission and Contract Parameters

Symbol	Units	Description
$\bar{F}^+_l$, $\bar{F}^-_l$	MW	Line capacity (direct/reverse)
$\eta_l = 1 - \text{losses}/100$	-	Line efficiency
$\bar{C}_c$, $\underline{C}_c$	MW	Contract capacity bounds

3.5 Inflow Model Parameters

Note on Periodicity

The PAR(p) model uses periodic parameters that repeat with a cycle length $M$. Common configurations:

• Monthly stages: $M=12$ (seasons = months)
• Weekly stages: $M=52$ (seasons = weeks)
• Custom resolution: $M$ = number of distinct periods in the cycle

We use “season $m$” as a generic term for the position within the cycle, avoiding the term “month” which is resolution-specific. The mapping $m(t) = ((t-1) \mod M) + 1$ converts stage index $t$ to season index $m \in \{1, \ldots, M\}$.

Symbol	Units	Description
$\mu_m$	m³/s	Seasonal mean inflow for season $m$
$\psi_{m,\ell}$	-	AR coefficient for season $m$, lag $\ell$
$\sigma_m$	m³/s	Residual standard deviation for season $m$
$\hat{a}_{h,\ell}$	m³/s	Incoming AR lag $\ell$ (state)

4. Decision Variables

Notation Convention

• Generation variables use $g$ with entity subscript: $g_h$ (hydro at plant $h$), $g_j$ (thermal at plant $j$)
• Flow variables use intuitive single letters: $q$ (turbined), $s$ (spillage), $u$ (diversion/bypass)
• Total outflow is explicitly defined: $o_h = q_h + s_h$ (downstream channel flow)
• Contract variables use a single $\chi_c$ (chi) per contract; direction is carried by set membership ($c \in \mathcal{C}^{imp}$ or $c \in \mathcal{C}^{exp}$), not a superscript — each contract is unidirectional
• Slack variables use $\sigma$ with constraint-type superscript

4.1 Per-Block Variables

Per-block variables are indexed by $k \in \mathcal{K}$:

Variable	Domain	Units	Description
$\delta_{b,k,s}$	$[0, \bar{d}_{b,s}]$	MW	Deficit at bus $b$, segment $s$
$\epsilon_{b,k}$	$\geq 0$	MW	Excess generation at bus $b$
$f^+_{l,k}$	$[0, \bar{F}^+_l]$	MW	Direct flow on line $l$
$f^-_{l,k}$	$[0, \bar{F}^-_l]$	MW	Reverse flow on line $l$
$g_{j,k,s}$	$[0, \bar{g}_{j,s}]$	MW	Thermal generation at plant $j$, segment $s$
$q_{h,k}$	$[\underline{Q}_h, \bar{Q}_h]$	m³/s	Turbined flow at hydro $h$
$s_{h,k}$	$\geq 0$	m³/s	Spillage at hydro $h$
$g_{h,k}$	$[\underline{G}_h, \bar{G}_h]$	MW	Hydro generation at plant $h$
$u_{h,k}$	$[0, \bar{U}_h]$	m³/s	Diversion/bypass flow (to separate channel)
$o_{h,k}$	-	m³/s	Total downstream outflow: $o_{h,k} = q_{h,k} + s_{h,k}$
$e_{h,k}$	free	m³/s	Evaporation (can be negative for condensation)
$r_{h,k}$	signed	m³/s	Water withdrawal; pinned to a signed target (negative = inter-basin return/addition); the realized value cannot cross zero past the target
$p_{j,k}$	$[\underline{P}_j, \bar{P}_j]$	m³/s	Pumped flow at station $j$
$\chi_{c,k}$	$[\underline{C}_c, \bar{C}_c]$	MW	Contract dispatch (import if $c \in \mathcal{C}^{imp}$, export if $c \in \mathcal{C}^{exp}$); $\underline{C}_c > 0$ is a take-or-pay floor

4.2 Stage-Level State Variables

Variable	Domain	Units	Description
$v_h$	$[\underline{V}_h, \bar{V}_h]$	hm³	End-of-stage storage
$v^{avg}_h$	-	hm³	Average storage during stage: $(\hat{v}_h + v_h)/2$
$a_{h,\ell}$	fixed	m³/s	AR lag $\ell$ (fixed by state transition)
$x^{\mathrm{a}}_{s,i,t}$	fixed	MW	Slot $s$ of plant $i$‘s anticipated-thermal ring buffer at stage $t$ (fixed by state transition); slot 0 matures here.
$d^i_t$	$[\underline{G}_i, \bar{G}_i]$	MW	Anticipated-thermal commitment placed at stage $t$ for delivery at stage $t + K_i$
$\theta$	$\geq 0$	$	Future cost (cost-to-go approximation)

4.3 Slack Variables

Slack variables for soft constraints:

Variable	Domain	Units	Constraint
$\sigma^{v-}_h$	$\geq 0$	hm³	Storage below minimum
$\sigma^{fill}_h$	$\geq 0$	hm³	Per-stage filling-floor shortfall
$\sigma^{q-}_{h,k}$	$\geq 0$	m³/s	Turbined flow below minimum
$\sigma^{o-}_{h,k}$	$\geq 0$	m³/s	Outflow below minimum
$\sigma^{o+}_{h,k}$	$\geq 0$	m³/s	Outflow above maximum
$\sigma^{g-}_{h,k}$	$\geq 0$	MW	Generation below minimum
$\sigma^{e+}_{h,k}$, $\sigma^{e-}_{h,k}$	$\geq 0$	m³/s	Evaporation violation
$\sigma^{r}_{h,k}$	$\geq 0$	m³/s	Water withdrawal violation
$\sigma^{inf}_h$	$\geq 0$	m³/s	Inflow non-negativity (if enabled)

5. Dual Variables and Reduced Costs

Cut coefficients in SDDP are state sensitivities. This section describes how the incoming state is carried in the LP for efficient solver updates, and how the resulting sensitivities — row duals for true constraints, reduced costs for the pinned state columns — map to cut coefficients.

5.1 LP Formulation Strategy for Efficient Hot-Path Updates

In the SDDP algorithm, each subproblem solve requires setting the incoming state values (storage volumes and AR lags from the previous stage). For computational efficiency with solvers like HiGHS, Cobre gives each incoming-state coordinate its own LP column and pins it by setting equal lower and upper column bounds:

Design Principle

Each incoming state coordinate is carried on a dedicated incoming-state column and pinned by equal column bounds; constraints that use incoming state reference that column rather than a constant RHS.

This design allows the hot path to update incoming state values by patching column bounds (and, for the realized-inflow noise, a single row RHS) without rebuilding the constraint matrix — only scalar values change between solves, not LP structure. The cut coefficient for each state coordinate is then the reduced cost of its pinned column (§5.4).

5.2 Water Balance: LP Form

The water balance is mathematically:

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

For LP implementation, the incoming storage is carried as a dedicated LP variable $v^{in}_h$ (the storage_in column) rather than a constant, and all LP variables are collected on the LHS with a zero RHS:

$$\begin{aligned} & v_h - v^{in}_h - \zeta \cdot a_h - \zeta \sum_{k} w_k \Big[ \sum_{i \in \mathcal{U}_h} (q_{i,k} + s_{i,k} + u_{i,k}) + \sum_{i:\text{div}=h} u_{i,k} \\ & \qquad + \sum_{j:\text{dest}=h} p_{j,k} - q_{h,k} - s_{h,k} - u_{h,k} - e_{h,k} - r_{h,k} - \sum_{j:\text{src}=h} p_{j,k} \Big] = 0 \end{aligned}$$

LP Structure:

• LHS: Linear combination of LP variables (storage $v_h$, incoming storage $v^{in}_h$, flows $q$, $s$, $u$, etc.)
• RHS: $0$ — the incoming state is not a constraint RHS; it is pinned separately on the $v^{in}_h$ column (§5.1)
• Constraint type: Equality ($=$)

5.3 AR Lag Pinning: LP Form

For autoregressive inflow state variables, the lag column is pinned to the incoming value by equal column bounds:

$$\underline{a}_{h,\ell} = \bar{a}_{h,\ell} = \hat{a}_{h,\ell} \quad \forall h \in \mathcal{H}, \; \ell \in \{1, \ldots, P_h\}$$

LP Structure:

• Column: $a_{h,\ell}$ (the inflow_lags column), bounds set equal to the incoming lag value
• Pin value: Incoming lag $\hat{a}_{h,\ell}$ (set via column bounds in the hot path)
• No constraint row: pinning is a bound, not an equality row

5.4 Cut Coefficient Derivation from Duals

The SDDP cut at stage $t-1$ has the form:

$$\theta_{t-1} \geq \alpha + \sum_{h} \pi^v_h \cdot v_h + \sum_{h,\ell} \pi^{lag}_{h,\ell} \cdot a_{h,\ell}$$

where $\alpha$ is the intercept and $\pi$ are the coefficients with respect to state variables.

Key principle: For an incoming-state column pinned at $\underline{x} = \bar{x} = \hat{x}$, the column’s reduced cost $\bar{c}$ represents:

$$\bar{c} = \frac{\partial Q^*}{\partial \hat{x}}$$

where $Q^*$ is the optimal objective value — the marginal cost of increasing the pinned bound. By KKT parity this equals the multiplier the equivalent equality row $x^{in} = \hat{x}$ would have carried.

Storage Reduced Cost ($\bar{c}^{in}_h$)

The incoming storage column $v^{in}_h$ is pinned at $\underline{v}^{in}_h = \bar{v}^{in}_h = \hat{v}_h$ (see LP Formulation §4a). Its reduced cost $\bar{c}^{in}_h$ measures: “How does optimal cost change if incoming storage $\hat{v}_h$ increases by 1 hm³?”

Economic interpretation:

• More incoming storage means more water available for generation
• Water has value (can displace thermal generation or avoid deficit)
• Therefore, increasing $\hat{v}_h$ decreases cost: $\frac{\partial Q^*}{\partial \hat{v}_h} < 0$
• By LP convention (minimization), this gives $\bar{c}^{in}_h < 0$, hence $\pi^v_h < 0$

Cut coefficient:

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

The reduced cost is divided by the column’s prescaler factor $d^{col}_h$ to recover the original-unit sensitivity (§12 of LP Formulation); no sign change is needed. By the LP envelope theorem, the reduced cost automatically captures all downstream effects — water balance, FPHA hyperplanes, and generic constraints — without manual combination of duals from multiple constraint types. See Cut Management §2.

AR Lag Reduced Cost ($\bar{c}^{lag}_{h,\ell}$)

The lag column $a_{h,\ell}$ is pinned at $\underline{a}_{h,\ell} = \bar{a}_{h,\ell} = \hat{a}_{h,\ell}$. Its reduced cost measures: _“How does optimal cost change if incoming lag $\hat{a}_{h,\ell}$ increases by 1 m³/s?”_

Economic interpretation:

• Higher historical inflow (in the PAR model) correlates with higher expected current inflow
• Higher inflows reduce cost (more hydro generation possible)
• Therefore, $\bar{c}^{lag}_{h,\ell} < 0$ (increasing the lag decreases cost)

Cut coefficient:

The cut coefficient for lag $\ell$ is the reduced cost of the pinned lag column, divided by its prescaler factor: $\pi^{lag}_{h,\ell} = \bar{c}^{lag}_{h,\ell} / d^{col}_{h,\ell}$. No sign change is needed.

5.5 Summary Table

Symbol	Source	Pinned value / RHS	Cut Coefficient
$\bar{c}^{in}_h$	Reduced cost of pinned $v^{in}_h$ column	$\hat{v}_h$	$\pi^v_h = \bar{c}^{in}_h / d^{col}$ (captures all downstream effects)
$\bar{c}^{lag}_{h,\ell}$	Reduced cost of pinned $a_{h,\ell}$ column	$\hat{a}_{h,\ell}$	$\pi^{lag}_{h,\ell} = \bar{c}^{lag}_{h,\ell} / d^{col}$
$\pi^{lb}_{b,k}$	Load balance (row dual)	$D_{b,k}$	Marginal cost of energy
$\pi^{wb}_h$	Water balance (row dual)	-	Not used directly for cut coefficients
$\pi_m^{fpha}$	FPHA hyperplane $m$ (row dual)	-	Captured via $\bar{c}^{in}_h$ automatically
$\pi^{gen}_c$	Generic constraint $c$ (row dual)	-	Captured via $\bar{c}^{in}_h$ automatically
$\lambda_i$	Benders cut $i$ (row dual)	$\alpha_i$	Cut activity indicator

5.6 Implementation Notes

The key property: since incoming state coordinates are pinned at equal column bounds, each column’s reduced cost is the bound sensitivity directly, so no sign change is needed when mapping to cut coefficients — only the per-column prescaler unscaling ($\pi^v_h = \bar{c}^{in}_h / d^{col}_h$; likewise for AR lags). Cut coefficient extraction reads the reduced costs of the contiguous incoming-state column regions (storage_in, inflow_lags, and, when present, anticipated_state).

Verification check: In a typical hydrothermal system:

• $\bar{c}^{in}_h < 0$ (water has value, more storage reduces cost)
• $\pi^v_h < 0$ (cut value increases as storage decreases — future is more expensive with less water)
• The cut $\theta \geq \alpha + \pi^v \cdot v$ correctly penalizes low storage

Cross-References

• LP Formulation — Complete LP subproblem using this notation
• SDDP Algorithm — Algorithm overview and cut generation process
• Cut Management — Cut coefficient computation and aggregation details
• PAR Inflow Model — Detailed PAR(p) model using inflow parameters defined here
• Hydro Production Models — FPHA plane coefficients ($\gamma$) and productivity ($\rho$)
• Equipment Formulations — Thermal, contract, pumping variable notation
• What Cobre Solves — methodology principles (reproducibility, determinism, declaration order invariance, code as ground truth, agent-readability) that frame this book