Notation Conventions

Purpose

This chapter defines the complete mathematical notation used across the Novomodelo methodology chapters: index sets, parameters, decision variables, and dual variables. It serves as the canonical reference for symbol meanings, ensuring consistency across the methodology chapters.

1. General Notation Conventions

This document follows the SDDP.jl notation conventions of Dowson & Kapelevich (2021) for consistency with the broader SDDP literature:

Convention	Meaning
$t \in \{1, \ldots, T\}$	Stage index; $T$ is the number of study stages
$\omega \in \Omega_t$	Opening (a realization of the stage noise) at stage $t$
$j \in \{1, \ldots, N_t\}$	Opening index within the opening tree of stage $t$
$N_t$	Branching factor: the number of openings of stage $t$
$p(\omega)$	Probability of opening $\omega$, uniform: $p(\omega) = 1/N_t$; on a policy graph, $p_{n'}(\omega)$, uniform within node $n'$
$n$	Node of a policy graph
$P(n \to n')$	Transition probability of the edge from node $n$ to its child $n'$ (the policy-graph diagrams label the edges p₁, p₂, …); the probabilities of a node’s edges sum to $1$
$x_t$	State vector at the end of stage $t$; $x_0$ is the initial state
$\hat{x}_{t-1}$	Incoming state of stage $t$ (the trial point)
$u_t$	Control vector of stage $t$
$\mathcal{X}_t(x_{t-1}, \omega_t)$	Feasible set of the stage-$t$ state and control, given the incoming state $x_{t-1}$ and the opening $\omega_t$; written $\mathcal{X}_t(\omega_t)$ when the incoming state is fixed
$c_t(x_t, u_t)$	Immediate (stage) cost of stage $t$, undiscounted
$x_j$	Variable of column $j$ in a generic statement about a stage LP
$Q_t$	Optimal value of the stage-$t$ LP as a function of its incoming state and opening, $Q_t(\hat{x}_{t-1}, \omega)$
$V_t(x)$	Value function (cost-to-go) at stage $t$
$\mathbb{E}_{\omega_t}$	Expectation over the opening of stage $t$
$\theta_t$	Epigraph variable approximating $V_{t+1}(x_t)$
$\pi$	Dual variables (row Lagrange multipliers)
$(\beta_0, \beta)$	Cut intercept and coefficients: the cut $\theta_t \geq \beta_0 + \beta^\top x_t$
$\hat{x}_j$, $\beta_j$	Component $j$ of the incoming state and of the cut slope; $j$ indexes the state coordinates
$i$	Cut index ($\beta_{0,i}$, $\beta_i$)
$\underline{V}_\tau(x)$	Outer (cut) approximation of the value function, $\max_i \{\beta_{0,i} + \beta_i^\top x\}$, per stage, or per season $\tau$ of a cyclic policy graph
$k$	Iteration counter
$\underline{z}^k$	Lower bound at iteration $k$: the first stage’s risk-adjusted value over its openings with the current cuts
$\bar{z}^k$	Upper-bound estimate at iteration $k$: the mean discounted cost of the iteration’s $M$ forward-pass trajectories, a statistical estimate; exact, $\bar{z}_{\text{exact}}$, under an enumerated forward pass
$M$, $m$	Number of trajectories averaged in the upper-bound estimate, and the trajectory index $m \in \{1, \ldots, M\}$
$N_{\text{forward\_passes}}$	Number of forward-pass trajectories per iteration
$n_{\text{sample}}$	Number of openings a sampled backward pass would draw per stage in place of all $N_t$
$\text{gap}^k$	Optimality gap at iteration $k$: the upper bound minus the lower bound, $\bar{z}^k - \underline{z}^k$; its percent form normalises it by the lower bound; see Stopping Rules
$P(\ell)$, $C(\ell)$	Probability and total discounted cost of leaf path $\ell$ of an enumerated scenario tree
$\bar{z}_{\text{exact}}$	Exact upper bound of an enumerated forward pass: $\sum_\ell P(\ell)\, C(\ell)$ under the expectation, the nested root value under a uniform CVaR
$w_m$	Census weight of simulation scenario $m$: its leaf-path probability, $\sum_m w_m = 1$
$\Delta_{95}$	Half-width of the 95 % normal-approximation confidence interval of a sampled cost estimate
$V^\star$	Optimal value of the multistage objective
$V'(v)$	Slope of the one-reservoir value function at a trial storage $v$: the slope of the tangent cut in the value-function figures
$\text{iterations}$	Number of training iterations, in growth orders such as $\mathcal{O}(\text{iterations} \times N_{\text{forward\_passes}})$
$\mathcal{O}(\cdot)$	Asymptotic order of a count or cost
$\rho^{\lambda, \alpha}$	Convex-combination risk measure, $\rho^{\lambda, \alpha}[Z] = (1 - \lambda)\, \mathbb{E}[Z] + \lambda\, \text{CVaR}_\alpha[Z]$; per stage, $\rho_t$, the measure of the stage that owns a cut, which aggregates the next stage’s openings into it
$\lambda$	Risk-aversion weight, $\lambda \in [0, 1]$: $0$ is risk-neutral, $1$ is pure CVaR
$\mu^*$	Risk-adjusted probability weights of a cut: $\mu^* = (1-\lambda)\, p + \lambda\, q^*$, between the floor $(1-\lambda)\, p_\omega$ and the cap $(1-\lambda)\, p_\omega + \lambda\, p_\omega / \alpha$
$\alpha$	CVaR tail fraction, $\alpha \in (0, 1]$; $\alpha = 1$ gives the expectation
$\text{CVaR}_\alpha$	Conditional value-at-risk: the expected cost over the worst $\alpha$-fraction of outcomes
$\mathrm{VaR}_\alpha$	Value-at-risk: the $(1 - \alpha)$-quantile of the cost, the optimal $\eta$
$\eta$	Threshold variable of $\text{CVaR}_\alpha(Z) = \min_{\eta \in \mathbb{R}} \{\eta + \tfrac{1}{\alpha} \mathbb{E}[(Z - \eta)^+]\}$
$Z$, $f(Z)$	Random cost a risk measure applies to, and its probability density
$\rho_{\text{nested}}$	Nested (time-consistent) risk functional: the stage measure applied stage by stage, $\rho_1[\rho_2[\cdots \rho_{T-1}[\cdot]]]$
$\rho_{\text{end-of-horizon}}$	End-of-horizon risk functional: one measure applied to the whole-path total cost

Declared Scoped Reuse

A glyph takes a second meaning only on pages that never carry its first; each such reuse is declared here:

• $k$ indexes the blocks of a stage, $k \in \mathcal{K}$, in the System Modelling chapters and Scenario Generation; it counts training iterations in SDDP Algorithm, Cut Management, LP Warm-Start, Stopping Rules, Upper Bound Evaluation, Horizon Modes, Discount Rate Formulation, Risk Measures and the worked examples; in PAR(p) Inflow Model it is the candidate order of the partial-autocorrelation test.
• $k_{max}$ is the number of slots in every anticipated thermal’s commitment ring in State Augmentation and Policy Checkpoint; in Stopping Rules it is the iteration limit.
• $\ell$ is the lag index of the autoregressive inflow model; in Stopping Rules and Upper Bound Evaluation it indexes the leaf paths of an enumerated scenario tree.
• $\lambda$ is the risk-aversion weight of the convex-combination risk measure; in PAR(p) Inflow Model $\lambda_i$ are the eigenvalues of the correlation matrix, and in LP Formulation, Block Formulation Variants and Hydro Production Function Models $\lambda_{h,b}$ is the share of cell $(h, b)$ in the turbine capacity of plant $h$.
• $L$ is the last filling stage of a filling hydro in LP Formulation and Penalty System; in PAR(p) Inflow Model it is the lower-triangular Cholesky factor of the correlation matrix, in State Augmentation $L_h$ is the bucket depth of receiving plant $h$, and in Upper Bound Evaluation $L_t$ is the vector of per-state-component Lipschitz constants $L_{t,j}$ of the stage-$t$ value function.
• $\psi_{m,\ell}$ is the autoregressive coefficient of the inflow model ($\psi^*_{m,\ell}$ standardized; $\psi^{A*}_m$ and $\psi^A_m$ the annual coefficient of PAR(p)-A); in Risk Measures $\psi(p, \mu)$ is the penalty function of the dual representation of a convex risk measure.
• A hat on a state quantity marks its incoming (trial) value ($\hat{x}_{t-1}$, $\hat{v}_h$, $\hat{a}_{h,\ell}$); a hat on a model parameter marks its sample estimate from the historical record ($\hat{\mu}_m$, $\hat{s}_m$, $\hat{\rho}_m(\ell)$); the two never decorate the same base symbol.
• $\tau_k$ is the duration of block $k$; in Stopping Rules $\tau$ is the bound-stalling window, and in Horizon Modes and Upper Bound Evaluation it indexes the seasons of a cyclic policy graph.
• $\phi(v, q, s)$ is the exact hydro production function in Hydro Production Function Models; in LP Formulation and State Augmentation $\phi_{h,k}$ is the arrival density that spreads a maturing in-transit volume over the blocks of its arrival stage; in PAR(p) Inflow Model $\phi$ names the AR-coefficient notation of other implementations, which that chapter writes $\psi$.
• $A_{ij}$ is an entry of the constraint matrix $A$ of the stage LP in LP Layout and Scaling; in System Element Modeling Overview and Equipment-Specific Formulations $A_{r,k}$ is the available generation of non-controllable source $r$ in block $k$, in PAR(p) Inflow Model $A_{h,t-1}$ is the annual regressor of PAR(p)-A, and in State Augmentation the bare $A$ is the number of anticipated thermals.
• $D_{b,k}$ is the load at bus $b$, block $k$, in LP Formulation and System Element Modeling Overview, written $D$ and $D_b$ in the worked examples; in LP Layout and Scaling $D_r$ and $D_c$ are the diagonal row and column scaling matrices.
• $i$ is the Benders cut index; in PAR(p) Inflow Model $(i, j)$ index the rows and columns of the periodic Yule-Walker system, in Scenario Generation $i$ indexes the entities of a correlation group, in Upper Bound Evaluation it indexes the vertices of the inner approximation, in State Augmentation, System Element Modeling Overview, SDDP Algorithm, Post-Study Boundary & Chained Studies and the Glossary it indexes the anticipated thermal plants, in LP Layout and Scaling it is the LP row index, and in Hydro Production Function Models it indexes the storage points of the volume-height curve, $v^{(i)}$ and $h^{(i)}$, and of the FPHA fitting grid, $V_i$.
• $j$ is the thermal-plant index; in Scenario Generation, SDDP Algorithm and the worked examples it indexes the openings of a stage, in Cut Management and Upper Bound Evaluation it indexes the components of the state vector, in LP Layout and Scaling it is the LP column index, and in Hydro Production Function Models it indexes the turbined-flow points $Q_j$ of the FPHA fitting grid.
• $K_i$ is the ring depth of anticipated thermal $i$ in State Augmentation, System Element Modeling Overview and the Glossary; in LP Layout and Scaling and Cut Management the bare $K$ is the cost-scale factor.
• $M$ is the number of seasons in the cycle of the inflow model and of a cyclic policy graph; in Hydro Production Function Models it is the number of FPHA hyperplanes of a plant, and in SDDP Algorithm, Discount Rate Formulation and Upper Bound Evaluation the number of forward-pass trajectories of an iteration.
• $m$ indexes the seasons of the inflow model, with $m(t)$ the season of stage $t$, and, in $\gamma^m$ and $\pi^{fpha}_m$, the planes $m \in \mathcal{M}_h$ of an FPHA model; in SDDP Algorithm, Discount Rate Formulation and Upper Bound Evaluation it indexes trajectories, forward-pass or simulated, and in State Augmentation, System Element Modeling Overview and Post-Study Boundary & Chained Studies it is the delivery stage of an anticipated commitment.
• $N$ is the number of hydro plants; in Scenario Generation it is the uniform branching factor of the scenario tree ($N_t = N$), and in Upper Bound Evaluation the number of out-of-sample simulation scenarios.
• $n$ is a node of a policy graph or of an enumerated scenario tree; in Horizon Modes it counts cycle repetitions, in LP Formulation, System Element Modeling Overview, Equipment-Specific Formulations and Penalty System it is the transmission-line index, in PAR(p) Inflow Model it is the dimension of the correlation matrix with eigenvalues $\lambda_1, \ldots, \lambda_n$, and in Hydro Production Function Models the superscript of $h_{tail}^{(n)}$ indexes the segments of a piecewise-quartic tailrace curve.
• $p(\omega)$ is the probability of opening $\omega$; in Hydro Production Function Models $p$ is the security-curve fraction of the maximum stored energy, and in LP Formulation, System Element Modeling Overview and Equipment-Specific Formulations $p_{y,k}$ is the pumped flow of station $y$.
• $Q_t$ is the optimal value of the stage-$t$ LP as a function of its incoming state; in Hydro Production Function Models $Q$ is the turbined-flow coordinate of the FPHA fitting grid.
• $q_{h,k}$ is the turbined flow of hydro $h$; in Upper Bound Evaluation $q_{n \to n'}$ is the conditional probability of reaching child node $n'$ from node $n$ of an enumerated scenario tree; in Risk Measures $q^*$ is the vector of CVaR tail weights.
• $r_h$ is the water-withdrawal target of hydro $h$ in LP Formulation, System Element Modeling Overview, Block Formulation Variants and Penalty System; in PAR(p) Inflow Model $r_m$ is the standardized innovation scale of season $m$, in Discount Rate Formulation $r_t$ is the annual discount rate that applies to stage $t$, and in State Augmentation $r_i(m)$ is the ring position of anticipated thermal $i$‘s delivery at stage $m$.
• $s$ indexes the deficit segments of a bus in LP Formulation, System Element Modeling Overview and Penalty System; in State Augmentation, SDDP Algorithm and Policy Checkpoint it indexes the slots of an anticipated thermal’s commitment ring. The spillage $s_{h,k}$ and the standard deviations $s_m$, $s^{\text{load}}_{b,t}$ and $s^{nc}_r$ are distinct forms.
• $u_{h,k}$ is the diversion flow of hydro $h$, and $u$ indexes the unit groups of a (hydro, bus) cell, in the System Modelling chapters; in SDDP Algorithm, Risk Measures, Upper Bound Evaluation, Horizon Modes, Policy Graphs, Discount Rate Formulation, The SDDP Framework in One Page and the Glossary $u_t$ is the control vector of the stage problem.
• $w_k$ is the weight of block $k$ in the System Modelling chapters; in Upper Bound Evaluation and the Glossary $w_m$ is the census weight of simulation scenario $m$, in PAR(p) Inflow Model $w$ indexes the rolling windows of a season bucket, and in Multi-Resolution Studies $w_{t,\mathcal{W}}$ is the share of lag period $\mathcal{W}$ covered by stage $t$.
• $y$ indexes the pumping stations in the System Modelling chapters; in Scenario Generation $y \in W$ is a window year of historical replay.
• $Z$ is the random cost a risk measure applies to; in PAR(p) Inflow Model it is the standardised series on which the PAR(p)-A conditional partial autocorrelation conditions.
• $z$ is the vector of independent standard normal draws that the correlation factor maps to correlated noise; in LP Formulation, State Augmentation and LP Layout and Scaling $z_h$ is the realized-inflow column of hydro $h$.
• $\delta_{b,k,s}$ is the load deficit at bus $b$, block $k$, segment $s$; in Hydro Production Function Models $\delta$ is the normalised mean-squared generation difference of two FPHA planes, and in Scenario Generation $\delta_t$ is the offset from the window year to the year of stage $t$‘s season occurrence.
• $\varepsilon$ with an entity, stage or source index is a noise innovation ($\varepsilon_t$, $\varepsilon^{\text{load}}_{b,t}$, $\varepsilon^{nc}_r$) and with a text subscript a tolerance ($\varepsilon_{\text{viol}}$, $\varepsilon_{\text{stall}}$, $\varepsilon_{\text{abs}}$, $\varepsilon_{\text{rel}}$); in Hydro Production Function Models the bare $\varepsilon$ is the merge tolerance of FPHA plane reduction.
• $\eta$ with an entity index is an efficiency ($\eta_h$ of a turbine, $\eta_n$ of a transmission line); in Risk Measures and The SDDP Framework in One Page $\eta$ is the threshold variable of the CVaR minimization formula.
• $\theta_t$ is the future-cost epigraph variable, approximating $V_{t+1}(x_t)$; in Hydro Production Function Models $\theta$ is the angle between the normals of two FPHA planes.
• $\kappa_{r,k}$ is the curtailment of non-controllable source $r$ in System Element Modeling Overview and Equipment-Specific Formulations; in Hydro Production Function Models $\kappa$ is the intercept-only correction factor of precomputed FPHA planes, and in Penalty System $\kappa = 10^6/3600$ is the number of (m³/s)·h in one hm³.
• $\mu$ with a season, bus or source index is a mean ($\mu_m$, $\mu^A_m$, $\mu^{\text{load}}_{b,t}$, $\mu^{nc}_r$); in Risk Measures $\mu$ is a risk-adjusted probability vector.
• $\epsilon_{b,k}$ is the excess generation at bus $b$, block $k$, in LP Formulation, System Element Modeling Overview and Penalty System; in Cut Management $\epsilon$ is the cut-activity tolerance of periodic pruning.
• $\mathcal{C}$ is the contract set, with $\mathcal{C}^{imp}$, $\mathcal{C}^{exp}$ and their per-bus subsets, in the System Modelling chapters; in Horizon Modes $\mathcal{C}_\tau$ is the set of stages that occupy season $\tau$ of a cyclic policy graph.
• $\mathcal{P}$ is the pumping-station set, with per-bus subsets $\mathcal{P}_b$, in the System Modelling chapters; in Risk Measures it is the probability simplex of the scenario probabilities.
• $\mathcal{M}_h$ is the FPHA hyperplane set of hydro $h$; in Risk Measures $\mathcal{M}(p)$, $\mathcal{M}_\alpha(p)$ and $\mathcal{M}^{EAVaR}(p)$ are risk sets of the dual representation of a convex risk measure.
• $\mathcal{X}_t(x_{t-1}, \omega_t)$ is the feasible set of the stage-$t$ state and control; in Cut Management $\mathcal{X}_t$ is the feasible state set on which a cut is valid.
• $I_t$ is the number of vertices stored at stage $t$ in Upper Bound Evaluation; in LP Formulation $I_{h,k}$ is the inflow of hydro $h$ in block $k$ that a hydro-inflow term reads.
• $\xi_r$ is the availability ratio of non-controllable source $r$ in System Element Modeling Overview, Equipment-Specific Formulations and Scenario Generation; in Inflow Non-Negativity Solution Methods $\xi_h$ is the noise-adjustment slack of the reference design.

Stage Indexing

Math formulas throughout this corpus index stages starting at $1$: $t \in \{1, \ldots, T\}$ (the convention already fixed above). Novomodelo’s configuration files and Parquet outputs instead identify a stage by its declared stage_id — the integer id the stage carries in stages.json. Declared ids need not start at $0$ or be contiguous (a pre-study stage may carry a negative id); stages are ordered by id ascending, and $t$ is a study stage’s position in that order.

Context	Convention
Math (this corpus)	$t \in \{1, \ldots, T\}$ — position in ascending-id order
Config stage_id and fields built on it	The declared stage id from stages.json
Output stage_id column (simulation, training)	The same declared stage id, unchanged
Mapping, when ids are declared densely from $0$	$\text{stage\_id} = t - 1$

Every math-layer chapter uses the 1-based $t$. Every JSON config field, Parquet output column, and CLI reference named stage_id, and every stage-window field built on it, carries the declared id, not a position, and a stage window is compared against declared ids. The offset $\text{stage\_id} = t - 1$ holds only for a case whose study stages are declared $0, 1, \ldots, T-1$. No other chapter restates this mapping; it defers here.

Terminology

Two term choices are pinned corpus-wide:

• Opening is the canonical term for a single realization drawn from a stage’s set of pre-generated noise vectors (e.g. “every opening $\omega \in \Omega_t$”). “Branch”/“branching” is reserved for the abstract scenario-tree structure itself — the branching factor $N_t$ (how many children a node has) — used only where a chapter discusses the tree’s topology, such as Scenario Generation, never for a specific drawn realization.
• Cost-to-go is the canonical term for the value function $V_t(x)$ (§1 above) in math-layer prose. “FCF” (Função de Custo Futuro) is reserved for the bilingual Glossary, which maps terms from other planning tools.

2. Index Sets

Symbol	Description
$t \in \{1, \ldots, T\}$	Stages
$k \in \mathcal{K}$	Blocks within stage
$\mathcal{B}$	Buses, indexed by $b$
$\mathcal{H}$	Hydro plants, indexed by $h$
$\mathcal{H}^{op} \subseteq \mathcal{H}$	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, indexed by $j$
$\mathcal{R}$	Non-controllable generation sources, indexed by $r$
$\mathcal{L}$	Transmission lines, indexed by $n$
$\mathcal{C}$	All contracts ($\mathcal{C}^{imp} \cup \mathcal{C}^{exp}$), indexed by $c$
$\mathcal{C}^{imp}$, $\mathcal{C}^{exp}$	Import/export contracts
$\mathcal{P}$	Pumping stations, indexed by $y$
$\mathcal{G}$	Generic constraints, indexed by $g$
$\mathcal{S}_b$	Deficit segments for bus $b$, indexed by $s$
$\mathcal{B}_h \subseteq \mathcal{B}$	Buses hosting one of hydro $h$‘s (hydro, bus) cells
$(h, b)$	(hydro, bus) cell: the unit groups of plant $h$ that share bus $b \in \mathcal{B}_h$, indexed by $u$; turbined flow and generation are tracked per cell
$\mathcal{H}_b$, $\mathcal{T}_b$, $\mathcal{R}_b$, $\mathcal{P}_b$	Hydros with a cell at bus $b$; thermals, non-controllable sources and pumping stations connected to bus $b$
$\mathcal{C}^{imp}_b$, $\mathcal{C}^{exp}_b$	Import and export contracts connected to bus $b$
$\mathcal{M}_h$	FPHA planes for hydro $h$, indexed by $m$
$\mathcal{U}_h$	Upstream hydros of $h$, indexed by $h'$
$\Omega_t$	Openings of stage $t$

Entity Indexing

Every operational entity index set above ($\mathcal{B}$, $\mathcal{H}$, $\mathcal{T}$, $\mathcal{R}$, $\mathcal{L}$, $\mathcal{C}$, $\mathcal{P}$) is enumerated in the canonical $(\text{operational\_start\_date}, \text{id})$ order: primarily by each entity’s operational start date, then — among entities sharing a date — by its unique id. This is the order every state block, the dense LP column layout, and the output column order follow. See Determinism & Provenance and LP Layout and Scaling.

3. Parameters

3.1 Time and Conversion

Symbol	Units	Description
$\tau_k$	hours	Duration of block $k$
$H_t = \sum_{k \in \mathcal{K}} \tau_k$	hours	Total duration of stage $t$
$w_k = \tau_k / H_t$	-	Block weight (fraction of stage)
$\zeta$	hm³/(m³/s)	Time conversion: m³/s over stage → hm³
$\zeta_k = 0.0036 \times \tau_k$	hm³/(m³/s)	Block conversion: m³/s over block $k$ → hm³; $\zeta_k = w_k\,\zeta$ and $\sum_{k \in \mathcal{K}} \zeta_k = \zeta$
$d_{t \to t+1}$	-	Discount factor of the transition from stage $t$ to $t+1$, applied to $\theta_t$ in the stage-$t$ objective
$d_{1 \to t}$	-	Cumulative discount factor of stage $t$: the product of the one-step factors of stages $1$ to $t - 1$, with $d_{1 \to 1} = 1$
$d_{t_1 \to t_2}$	-	Discount from stage $t_2$ back to stage $t_1 \le t_2$, $d_{1 \to t_2} / d_{1 \to t_1}$
$d_{\text{cycle}}$	-	Cumulative discount around one cycle of a cyclic policy graph, $d_{\text{cycle}} = \prod_{t \in \text{cycle}} d_{t \to t+1} < 1$

Time Conversion Factor Derivation

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

$$\zeta = 0.0036 \times \sum_{k \in \mathcal{K}} \tau_k \quad [\mathrm{hm}^3/(\mathrm{m}^3/\mathrm{s})]$$

Dimensional Analysis:

$$[\zeta] = \frac{\text{s}}{\text{h}} \times \frac{\mathrm{hm}^3}{\mathrm{m}^3} \times \text{h} = \frac{\mathrm{hm}^3}{\mathrm{m}^3/\mathrm{s}}$$

3.2 Load and Costs

Cost coefficients use $c$ with a superscript naming the cost type.

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
$c^{th}_j$	$/MWh	Marginal cost of thermal plant $j$
$c^{spill}_h$	$/(m³/s·h)	Spillage cost
$c^{div}_h$	$/(m³/s·h)	Diversion cost
$c^{tc}_h$	$/(m³/s·h)	Turbined-flow regularization cost of hydro $h$, charged on the turbined flow of every cell
$c^{sv-}_h$	$/hm³	Storage-below-minimum penalty, pricing the soft dead-volume floor of a filling hydro once it operates
$c^{fill}_h$	$/hm³	Filling-target shortfall penalty
$c^{tv-}_h$	$/(m³/s·h)	Turbined-flow-minimum violation penalty, charged on every cell of hydro $h$
$c^{ov-}_h$, $c^{ov+}_h$	$/(m³/s·h)	Outflow below-minimum and above-maximum violation penalties
$c^{gv-}_h$	$/MWh	Generation-minimum violation penalty, charged on every cell of hydro $h$
$c^{ev+}_h$, $c^{ev-}_h$	$/(m³/s·h)	Evaporation above-target and below-target violation penalties
$c^{wv+}_h$, $c^{wv-}_h$	$/(m³/s·h)	Water-withdrawal over-delivery and under-delivery penalties
$c^{inf}_h$	$/(m³/s·h)	Inflow non-negativity penalty (penalty-based inflow methods)
$c^{exch}_n$	$/MWh	Exchange (transmission) cost
$c^{curt}_r$	$/MWh	Curtailment regularization cost of non-controllable source $r$
$c^{ctr}_c$	$/MWh	Contract price (signed: + import cost, − export revenue)
$c_i(t)$	$/MWh	Unit cost of anticipated thermal $i$ at stage $t$; a commitment for delivery stage $m$ is priced at $c_i(m)$ on its decision column
$K$	-	Cost-scale factor: every objective coefficient except that of $\theta$ is divided by $K$, so cuts are held in scaled cost units (LP Layout and Scaling §2.1)

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	Plant turbined-flow bounds; the plant maximum caps every cell
$\bar{Q}_{h,b}$, $\underline{Q}_{h,b}$	m³/s	Cell turbined-flow bounds: the maximum sums the cell’s unit-group maxima (under constant productivity each also limited by its generation maximum) and is capped by the plant maximum; the minimum sums their minima and is a soft floor
$\bar{Q}_u$	m³/s	Turbined-flow maximum of unit group $u$
$\bar{G}_h$, $\underline{G}_h$	MW	Plant generation bounds; the plant maximum caps every cell
$\bar{G}_{h,b}$, $\underline{G}_{h,b}$	MW	Cell generation bounds: the maximum sums the cell’s unit-group maxima and is capped by the plant maximum; the minimum sums their minima and is a soft floor
$\lambda_{h,b}$	-	Share of cell $(h, b)$ in plant $h$‘s declared turbine capacity; it apportions the flow-independent part of each FPHA plane among the cells
$\bar{O}_h$, $\underline{O}_h$	m³/s	Outflow bounds
$\bar{U}_h$	m³/s	Maximum diversion flow
$r_h$	m³/s	Water withdrawal target — stage-level, signed fixed RHS parameter (not a per-block LP decision variable); negative = inter-basin return/addition. See LP Formulation.
$\rho_h$	MW/(m³/s)	Productivity (constant model)
$V^{min}_h$, $V^{max}_h$	hm³	Physical storage range — stage-invariant plant property (dead-volume floor, full-reservoir ceiling); distinct from the operative storage-variable bounds $\underline{V}_h$, $\bar{V}_h$. See Hydro Production Function Models.
$\rho_{eq,h,t}$	MW/(m³/s)	Equivalent productivity at the reference operating point. See Hydro Production Function Models.
$\rho_{acum,h,t}$	MW/(m³/s)	Accumulated cascade productivity (plant plus downstream), reference-point evaluator. See Hydro Production Function Models.
$\bar\rho_{eq,h,t}$	MW/(m³/s)	Useful-range mean equivalent productivity — forebay level averaged over $[V^{min}_h, V^{max}_h]$. See Hydro Production Function Models.
$\bar\rho_{acum,h,t}$	MW/(m³/s)	Useful-range mean accumulated cascade productivity. See Hydro Production Function Models.
$E^{max}_{h,t}$	MW/(m³/s)·hm³	Maximum stored energy $\bar\rho_{acum,h,t}\,(V^{max}_h - V^{min}_h)$ (raw unit, not MWh). See Hydro Production Function Models.
$\gamma_0^m, \gamma_v^m, \gamma_q^m, \gamma_s^m$	-	FPHA plane $m$ coefficients — intercept ($\gamma_0^m$), storage/volume ($\gamma_v^m$), turbined flow ($\gamma_q^m$), spillage ($\gamma_s^m$); already $k_{FPHA}$-scaled. Lowercase by convention — never $\Gamma$.
$k_{FPHA}$	-	FPHA least-squares fit-correction factor; scales the fitted plane set. See Hydro Production Function Models.
$\phi(v, q, s)$	MW	Exact hydro production function: generation at storage $v$, turbined flow $q$ and spillage $s$, proportional to $q$ and to the net head $h_{net}$. See Hydro Production Function Models.
$h_{net}(v, q, s) = h_{fore}(v) - h_{tail}(q + s) - h_{loss}$	m	Net head, clamped at zero: the forebay level $h_{fore}$, a function of storage, minus the tailrace level $h_{tail}$, a function of total outflow, minus the hydraulic head losses $h_{loss}$
$V_i$, $Q_j$	hm³, m³/s	Storage point $i$ and turbined-flow point $j$ of the FPHA fitting grid, on the storage and turbined-flow coordinates $V$ and $Q$
$L$	-	Last filling stage of a filling hydro, the stage before its entry stage
$\text{rate}_t$	m³/s	Minimum accumulation rate of a filling hydro at stage $t$
$V^{\text{target}}_t$	hm³	Minimum end-of-stage storage of a filling hydro at stage $t$ (the filling floor), reaching $\underline{V}_h$ at stage $L$
$L_h$	stages	Bucket depth of receiving plant $h$: the deepest maturity lag any travel-time arc into $h$ reaches on the stage calendar
$\phi_{h,k}$	-	Arrival density of plant $h$‘s maturing in-transit volume over the blocks $k$ of a chronological stage, $\phi_{h,k} \geq 0$, $\sum_k \phi_{h,k} = 1$; on a parallel stage $\phi_{h,k} = w_k = \tau_k / H_t$ in a hydro-inflow term (LP Formulation §10)
$\Delta^{tt}_{h'}$	hours	Travel time of the main cascade arc of upstream hydro $h'$, $0$ when none is declared
$\nu_{h',t,0}$	-	Same-stage share of $h'$‘s release on the downstream water balance at stage $t$, $(H_t - \Delta^{tt}_{h'})^+ / H_t$
$\nu^{k' \to k}_{h',t}$	-	Within-stage routing share on a chronological stage, from $h'$‘s block $k'$ to the downstream block $k \ge k'$
$\gamma^{ev}_{0,h}$, $\gamma^{ev}_{v,h}$	m³/s, (m³/s)/hm³	Linearized net-evaporation intercept and storage slope of hydro $h$ at the current stage

3.4 Thermal, Network and Equipment Parameters

Symbol	Units	Description
$\bar{G}_j$, $\underline{G}_j$	MW	Thermal generation bounds: capacity and minimum stable load
$K_i$	-	Ring depth of anticipated thermal $i$; the lead for a stage-count lead
$k_{max} = \max_i K_i$	-	Number of slots in every anticipated thermal’s commitment ring
$t_i(m)$	-	Decision stage of anticipated thermal $i$‘s delivery at stage $m$: $m - K_i$ under a stage-count lead, and under a physical lead the stage containing the instant one lead time before the end of stage $m$ (an instant on a stage boundary belongs to the earlier stage); a delivery decided before the study has none
$r_i(m)$	-	Ring position of anticipated thermal $i$‘s delivery at stage $m$; the delivery holds slot $r_i(m) \bmod k_{max}$
$\bar{F}^+_n$, $\bar{F}^-_n$	MW	Line capacity (direct/reverse)
$\eta_n = 1 - \text{losses}/100$	-	Reported line efficiency: scales the post-solve reported transmission losses, $(1-\eta_n)(f^+ + f^-)$; it does not enter the dispatch LP, whose line flows carry coefficient ±1. Distinct from the PAR innovation $\varepsilon_t$ (§3.5).
$\text{loss}_{n,k}$	MW	Reported transmission loss of line $n$ in block $k$, $(1-\eta_n)(f^+_{n,k} + f^-_{n,k})$
$\bar{C}_c$, $\underline{C}_c$	MW	Contract capacity bounds
$\rho^{pump}_y$	MW/(m³/s)	Power consumption rate of pumping station $y$
$\bar{P}_y$, $\underline{P}_y$	m³/s	Pumped-flow bounds of station $y$
$\bar{G}_r$	MW	Installed capacity of non-controllable source $r$
$A_{r,k}$	MW	Available generation of non-controllable source $r$ in block $k$ for the current stage and scenario, $\bar{G}_r \, \xi_r \, f_{r,k}$
$\xi_r$	-	Availability ratio of non-controllable source $r$ for the current stage and scenario, in $[0, 1]$
$\mu^{nc}_r$, $s^{nc}_r$	-	Mean and standard deviation of the unclamped availability factor of non-controllable source $r$ at the current stage
$\varepsilon^{nc}_r$	-	Standard-normal noise of non-controllable source $r$: $\xi_r = \mathrm{clamp}(\mu^{nc}_r + s^{nc}_r \, \varepsilon^{nc}_r, 0, 1)$
$f_{r,k}$	-	Block factor of non-controllable source $r$ in block $k$

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. Seasons are numbered $m \in \{1, \ldots, M\}$. The season $m(t)$ of stage $t$ is the stage’s resolved season: the season the stage declares or, when it declares none, the season its start date falls in on the calendar. The season $\ell$ lags before season $m$ is its $\ell$-th calendar predecessor, so season arithmetic such as $m - \ell$ is taken modulo $M$ on $\{1, \ldots, M\}$.

Symbol	Units	Description
$\mu_m$	m³/s	Seasonal mean inflow for season $m$
$s_m$	m³/s	Seasonal sample standard deviation of season $m$ (population divisor)
$\psi_{m,\ell}$	-	AR coefficient for season $m$, lag $\ell$ (original units)
$\psi^*_{m,\ell}$	-	Standardized AR coefficient, $\psi^*_{m,\ell} = \psi_{m,\ell}\, s_{m-\ell} / s_m$
$P_h$	-	AR order of hydro $h$; its lags are $\ell \in \{1, \ldots, P_h\}$
$P^{\max}$	-	Lag depth of the inflow-lag state, the same for every hydro: the largest AR order, $\max_h P_h$, raised to at least twelve when any hydro carries the annual component, and to the deepest lag a terminal boundary references (State Augmentation §4)
$r_m$	-	Standardized innovation scale, $r_m = \sigma_m / s_m \in (0, 1]$
$\sigma_m$	m³/s	Innovation (residual) standard deviation for season $m$, $\sigma_m = s_m\, r_m$
$b_{h,m(t)}$	m³/s	Deterministic base of the PAR(p) inflow equation at season $m = m(t)$: $\mu_m - \sum_{\ell=1}^{P_h} \psi_{m,\ell}\, \mu_{m-\ell}$
$\mu^A_m$, $\sigma^A_m$	m³/s	Seasonal mean and standard deviation (population divisor) of season $m$‘s annual regressor in PAR(p)-A
$\rho_m(\ell)$	-	Periodic autocorrelation at lag $\ell$ for season $m$
$N_m$	-	Number of historical observations of season $m$
$z_{0.975}$	-	Standard-normal 0.975 quantile: the critical value of the PACF significance test at the 95 % level
$\varepsilon_t$	-	PAR innovation: standardized noise term, $\varepsilon_t \sim \mathcal{N}(0,1)$ (distinct from the line efficiency $\eta_n$, §3.4, and the excess-generation variable $\epsilon_{b,k}$, §4.1). See PAR(p) Inflow Model.
$z$	-	Vector of independent standard normal draws, $z \sim \mathcal{N}(0, I)$, mapped to correlated noise
$C$	-	Spatial correlation matrix of a correlation group
$U$, $\Lambda$	-	Eigendecomposition $C = U \Lambda U^\top$: orthogonal eigenvectors $U$ and the diagonal $\Lambda = \mathrm{diag}(\lambda_1, \ldots, \lambda_n)$ of the eigenvalues $\lambda_i$
$\tilde{\Lambda}^{1/2}$	-	Square roots of the eigenvalues clipped at zero, $\mathrm{diag}(\sqrt{\max(\lambda_1, 0)}, \ldots, \sqrt{\max(\lambda_n, 0)})$
$C^{1/2}$	-	Spectral correlation factor $U \Lambda^{1/2} U^\top$, negative eigenvalues clipped to zero; the correlated noise is $C^{1/2} z$
$a_h$	m³/s	Incremental inflow of hydro $h$ at the current stage
$\hat{a}_{h,\ell}$	m³/s	Incoming AR lag $\ell$ (state)

4. Decision Variables

Notation Convention

• Generation variables use $g$ with entity subscript: $g_{h,b,k}$ (hydro, per (hydro, bus) cell), $g_{j,k}$ (thermal plant $j$), $g^{nc}_{r,k}$ (non-controllable source $r$)
• 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 $b$
$f^+_{n,k}$	$[0, \bar{F}^+_n]$	MW	Direct flow on line $n$
$f^-_{n,k}$	$[0, \bar{F}^-_n]$	MW	Reverse flow on line $n$
$g_{j,k}$	$[\underline{G}_j, \bar{G}_j]$	MW	Generation of thermal plant $j$
$q_{h,b,k}$	$[0, \bar{Q}_{h,b}]$	m³/s	Turbined flow of cell $(h, b)$; its minimum $\underline{Q}_{h,b}$ is a soft floor (slack $\sigma^{q-}_{h,b,k}$)
$q_{h,k}$	-	m³/s	Plant turbined flow, $q_{h,k} = \sum_{b \in \mathcal{B}_h} q_{h,b,k}$
$s_{h,k}$	$\geq 0$	m³/s	Spillage at hydro $h$
$g_{h,b,k}$	$[0, \bar{G}_{h,b}]$	MW	Hydro generation of cell $(h, b)$, injected at bus $b$; its minimum $\underline{G}_{h,b}$ is a soft floor (slack $\sigma^{g-}_{h,b,k}$)
$g_{h,k}$	-	MW	Plant hydro generation, $g_{h,k} = \sum_{b \in \mathcal{B}_h} g_{h,b,k}$
$v_{h,k}$	$[\underline{V}_h, \bar{V}_h]$	hm³	Storage at the end of block $k$ on a chronological stage, $v_{h,0} = \hat{v}_h$; only $v_{h,\lvert\mathcal{K}\rvert}$ is state; the column lower bound is $0$, not $\underline{V}_h$, for a filling hydro and for a hydro not in service (see LP Formulation §8)
$u_{h,k}$	$[0, \bar{U}_h]$	m³/s	Diversion/bypass flow (to separate channel)
$o_{h,k}$	-	m³/s	Total downstream outflow: $o_{h,k} = q_{h,k} + s_{h,k}$
$e_{h,k}$	bounded	m³/s	Net evaporation flow (negative for net rainfall input): one stage-level value $e_h$ on a parallel stage, one per block $k$ on a chronological stage; its magnitude has a per-stage bound
$\text{net\_flows}_{h,k}$	-	m³/s	Net per-block flow terms of hydro $h$‘s water balance in block $k$: the turbined and spilled release credited from upstream and the flows diverted and pumped in, minus the plant’s own turbined, spilled and diverted flow and its pumped-out flow
$p_{y,k}$	$[\underline{P}_y, \bar{P}_y]$	m³/s	Pumped flow at station $y$
$\chi_{c,k}$	$[\underline{C}_c, \bar{C}_c]$	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
$g^{nc}_{r,k}$	$[0, A_{r,k}]$	MW	Generation of non-controllable source $r$
$\kappa_{r,k}$	-	MW	Curtailment of non-controllable source $r$, $\kappa_{r,k} = A_{r,k} - g^{nc}_{r,k}$ (derived)

4.2 Stage-Level State Variables

Variable	Domain	Units	Description
$v_h$	$[\underline{V}_h, \bar{V}_h]$	hm³	End-of-stage storage; the column lower bound is $0$, not $\underline{V}_h$, for a filling hydro and for a hydro not in service (see LP Formulation §8)
$v^{in}_h$	fixed	hm³	Incoming-storage variable of the stage LP, fixed at the incoming storage $\hat{v}_h$
$v^{avg}_h$	-	hm³	Average storage during stage: $(\hat{v}_h + v_h)/2$
$a_{h,\ell}$	fixed	m³/s	AR lag $\ell$ (fixed by state transition)
$x^{\mathrm{a}}_{s,i}$	free; $0$ when no row holds it	MW	Outgoing slot $s$ of anticipated thermal $i$‘s commitment ring, $s \in \{0, \ldots, k_{max} - 1\}$
$x^{\mathrm{a,in}}_{s,i}$	fixed	MW	Incoming slot $s$, fixed at its trial value $\hat{x}^{\mathrm{a}}_{s,i}$
$b^{\mathrm{out}}_{h,d}$	$\geq 0$	hm³	In-transit water destined for downstream plant $h$ at maturity lag $d$, carried to the next stage; it enters $h$‘s water balance $d$ stages after the current stage
$g^{\mathrm{a}}_{i,t}$	$[\underline{G}_i(m), \bar{G}_i(m)]$	MW	Anticipated-thermal commitment decided at stage $t$ for its delivery stage $m$, $t_i(m) = t$ ($m = t + K_i$ under a stage-count lead)
$x^{in}$	fixed	-	Incoming-state variable of the stage LP, fixed at the trial point: $x^{in} = \hat{x}_{t-1}$
$\theta$	$\geq 0$	$	Future-cost epigraph variable $\theta_t$ of the stage-$t$ LP, approximating $V_{t+1}(x_t)$

The state dimension is $n_{\text{state}} = N(1 + P^{\max}) + B + A \, k_{max}$, with $N = \lvert\mathcal{H}\rvert$ hydros, $B = \sum_h L_h$ in-transit buckets and $A$ anticipated thermals.

4.3 Slack Variables

Slack variables for soft constraints:

Variable	Domain	Units	Constraint
$\sigma^{v-}_h$	$\geq 0$	hm³	Storage below the dead volume of a filling hydro, from its entry stage on (every other operating hydro’s dead volume is a hard bound)
$\sigma^{fill}_h$	$\geq 0$	hm³	Per-stage filling-floor shortfall
$\sigma^{q-}_{h,b,k}$	$\geq 0$	m³/s	Turbined flow below minimum — one per (hydro, bus) cell $b \in \mathcal{B}_h$ of a split plant
$\sigma^{o-}_{h,k}$	$\geq 0$	m³/s	Outflow below minimum (per plant — no per-cell outflow column to attribute a floor to)
$\sigma^{o+}_{h,k}$	$\geq 0$	m³/s	Outflow above maximum (per plant)
$\sigma^{g-}_{h,b,k}$	$\geq 0$	MW	Generation below minimum — one per (hydro, bus) cell $b \in \mathcal{B}_h$ of a split plant
$\sigma^{e+}_{h,k}$, $\sigma^{e-}_{h,k}$	$\geq 0$	m³/s	Evaporation above-target and below-target violation (per plant): one stage-level pair $\sigma^{e\pm}_h$ on a parallel stage, priced over the stage hours $H_t$; one pair per block on a chronological stage, priced over $\tau_k$
$\sigma^{w-}_h$, $\sigma^{w+}_h$	$\geq 0$	m³/s	Water withdrawal under-/over-delivery relative to the target $r_h$ (stage-level, not per-block); priced by $c^{wv-}_h$ / $c^{wv+}_h$ respectively
$\sigma^{inf}_h$	$\geq 0$	m³/s	Inflow non-negativity (if enabled)

5. Duals and Cut Slopes

A row dual $\pi$ is the Lagrange multiplier of one LP row: the rate at which the optimal stage cost changes per unit increase of the row’s right-hand side. The cut slope $\beta$ is a subgradient of the value function $V_t$ with respect to the incoming state at the trial point $\hat{x}_{t-1}$; each component equals the dual of the bound that pins its state coordinate at the trial value. Its components include the storage slope $\beta^v_h$, the AR-lag slope $\beta^{lag}_{h,\ell}$ and the slopes on the in-transit buckets and the anticipated ring slots; with the stored intercept $\beta_0$ they form the cut $(\beta_0, \beta)$ of §1. Sign convention: more incoming storage lowers the cost-to-go wherever the extra water displaces thermal generation or deficit, so the storage slope $\beta^v_h$ is non-positive there, and it can turn positive where the extra water can only leave the reservoir at a cost; see Cut Management for the cut coefficients, LP Formulation for the rows and State Augmentation for the state pinning.

Symbol	Row	Meaning
$\pi^{lb}_{b,k}$	Load balance, bus $b$, block $k$	Marginal cost of energy
$\pi^{wb}_h$	Water balance, hydro $h$ (block $k$ on a chronological stage, $\pi^{wb}_{h,k}$)	Water value
$\pi_m^{fpha}$	FPHA hyperplane $m$	Marginal value of the generation limit set by plane $m$

Cross-References

• LP Formulation — Complete LP subproblem using this notation
• State Augmentation — the state vector, its pinning, the outgoing state and the state families’ mechanics
• LP Layout and Scaling — column and row layout, prescaling and cost scale
• SDDP Algorithm — Algorithm overview and cut generation process
• Cut Management — Cut coefficient computation and aggregation details
• PAR(p) Inflow Model — Detailed PAR(p) model using inflow parameters defined here
• Hydro Production Function Models — FPHA plane coefficients ($\gamma$) and productivity ($\rho$)
• Equipment-Specific Formulations — Thermal, contract, pumping variable notation
• What Novomodelo Solves — the methodology principles: reproducibility, determinism, declaration order invariance and agent-readability