Notation

This page defines the complete mathematical notation used across the Cobre documentation: index sets, parameters, decision variables, and dual variables. It is the canonical reference for symbol meanings, ensuring consistency across all mathematical and data model content.

1. General Notation Conventions

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

2. Index Sets

3. Parameters

3.1 Time and Conversion

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:

$$\zeta =0.0036\times \sum _{k\in \mathcal{K}}{\tau }_k\text{[hm³/(m³/s)]}$$

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):

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

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

3.3 Hydro Parameters

3.4 Transmission and Contract Parameters

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 $m(t)=((t-1)\mathrm{mod}M)+1$ converts stage index $t$ to season index $m\in \{1,\dots ,M\}$.

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 $\chi$ (chi) with direction superscript: ${\chi }^{in}$, ${\chi }^{out}$
• Slack variables use $\sigma$ with constraint-type superscript

Symbol Selection Rationale:

• $q$ (turbined): from “vazão turbinada” (Portuguese) or “discharge through turbines”
• $s$ (spillage): standard hydrology notation
• $u$ (diversion): “bypass” or “desvio” — avoids confusion with demand $D$ or deficit $\delta$
• $o$ (outflow): total downstream flow affecting tailrace level
• $r$ (withdrawal): “retirada” — consumptive removal from the system
• $\chi$ (contract): Greek chi, visually distinct from cost symbol $c$

4.1 Per-Block Variables

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

4.2 Stage-Level State Variables

4.3 Slack Variables

Slack variables for soft constraints:

5. Dual Variables

Dual variables are essential for cut coefficient computation in the SDDP backward pass. This section describes how state-linking constraints are formulated in the LP for efficient solver updates, and how the resulting dual variables 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, we formulate constraints so that:

Design Principle: All incoming state variables are isolated on the right-hand side (RHS) of their respective constraints.

This allows the hot path (forward/backward passes) to update subproblems by simply modifying row bounds via changeRowBounds() without rebuilding constraint matrices. The LP matrix coefficients remain constant across all subproblem solves within a stage.

5.2 Water Balance: LP Form

The water balance is mathematically:

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

For LP implementation, we rearrange to isolate ${\hat{v}}_h$ on the RHS:

$$v_h-\zeta \cdot a_h-\zeta \sum _{k\in \mathcal{K}}{w}_k\cdot {\text{net\_flows}}_{h,k}={\hat{v}}_h$$

Expanding the net flows and collecting all LP variables on the LHS:

$$\boxed{v_h-\zeta \cdot a_h-\zeta \sum _k{w}_k[\sum _{i\in {\mathcal{U}}_h}(q_{i,k}+s_{i,k}+u_{i,k})+\sum _{i:\text{div}=h}{u}_{i,k}+\sum _{j:\text{dest}=h}{p}_{j,k}-q_{h,k}-s_{h,k}-u_{h,k}-e_{h,k}-r_{h,k}-\sum _{j:\text{src}=h}{p}_{j,k}]={\hat{v}}_h}$$

LP Structure:

• LHS: Linear combination of LP variables (storage $v_h$, flows $q$, $s$, $u$, etc.)
• RHS: Incoming state ${\hat{v}}_h$ only (set via row bounds in hot path)
• Constraint type: Equality ($=$)

5.3 AR Lag Constraints: LP Form

For autoregressive inflow state variables, the constraint fixes the current-stage lag variable to the incoming value:

$$\boxed{a_{h,\ell }={\hat{a}}_{h,\ell }}\forall h\in \mathcal{H},\ell \in \{1,\dots ,P_h\}$$

LP Structure:

• LHS: Single LP variable $a_{h,\ell }$ (coefficient = 1)
• RHS: Incoming lag value ${\hat{a}}_{h,\ell }$ (set via row bounds in hot path)
• Constraint type: Equality ($=$)

5.4 Cut Coefficient Derivation from Duals

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

$${\theta }_{t-1}\ge \alpha +\sum _h{\pi }_h^v\cdot v_h+\sum _{h,\ell }{\pi }_{h,\ell }^{lag}\cdot a_{h,\ell }$$

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

Key principle: For a constraint written as $\text{LHS}=\text{RHS}$, the dual variable $\pi$ represents:

$$\pi =\frac{\partial Q^{\ast }}{\partial \text{RHS}}$$

where $Q^{\ast }$ is the optimal objective value. This is the marginal cost with respect to increasing the RHS.

Storage Dual (${\pi }_h^{fix}$)

For the storage fixing constraint (see LP Formulation §4a):

$$\underset{\text{LHS}}{\underbrace{v_h^{in}}}=\underset{\text{RHS}}{\underbrace{{\hat{v}}_h}}$$

The dual ${\pi }_h^{fix}$ 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: $\frac{\partial Q^{\ast }}{\partial {\hat{v}}_h}<0$
• By LP convention (minimization), this gives ${\pi }_h^{fix}<0$

Cut coefficient:

$$\boxed{{\pi }_h^v={\pi }_h^{fix}}$$

No sign change is needed because the incoming state ${\hat{v}}_h$ appears directly on the RHS with coefficient $+1$. By the LP envelope theorem, this dual automatically captures all downstream effects — water balance, FPHA hyperplanes, and generic constraints — without manual combination of duals from multiple constraint types. See Cut Management.

AR Lag Dual (${\pi }_{h,\ell }^{lag}$)

For the lag fixing constraint:

$$\underset{\text{LHS}}{\underbrace{a_{h,\ell }}}=\underset{\text{RHS}}{\underbrace{{\hat{a}}_{h,\ell }}}$$

The dual ${\pi }_{h,\ell }^{lag}$ 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, ${\pi }_{h,\ell }^{lag}<0$ (increasing the lag decreases cost)

Cut coefficient:

The cut coefficient for lag $\ell$ is the dual variable ${\pi }_{h,\ell }^{lag}$ directly, since ${\hat{a}}_{h,\ell }$ appears on the RHS with coefficient $+1$. No sign change or scaling is needed.

5.5 Summary Table

5.6 Implementation Notes

The hot-path solver update pattern (modifying RHS via changeRowBounds for incoming state, extracting duals via getRowDual for cut coefficients) is documented in the Solver HiGHS Implementation and Solver Abstraction specs. The key property: since incoming state variables appear on the RHS of fixing constraints with coefficient $+1$, no sign change is needed when mapping duals to cut coefficients (${\pi }_h^v={\pi }_h^{fix}$; for AR lags, ${\pi }_{h,\ell }^{lag}$ is used directly). Cut coefficient extraction is a single contiguous slice read: dual[0..n_state].

Verification check: In a typical hydrothermal system:

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

Cross-References

• Theory: SDDP Theory — Algorithm overview and cut generation process
• Theory: Cut Management — Cut coefficient computation and aggregation details
• Theory: PAR(p) Autoregressive Models — Detailed PAR(p) model using inflow parameters defined here
• Specs: LP Formulation — Complete LP subproblem using this notation
• Specs: Cut Management — Formal cut management specification
• Specs: PAR Inflow Model — PAR(p) inflow model specification
• Specs: Hydro Production Models — FPHA plane coefficients and productivity
• Specs: Equipment Formulations — Thermal, contract, pumping variable notation