Inflow Non-Negativity Solution Methods

Purpose

This spec defines the four methods available for handling negative inflow realizations produced by the PAR(p) model, including their LP formulations, objective function modifications, and trade-offs.

1. Problem Statement

The PAR(p) model can generate negative inflow realizations:

$$a_h = \underbrace{\mu_m - \sum_{\ell=1}^{p} \psi_\ell \mu_{m-\ell}}_{\text{deterministic base}} + \underbrace{\sum_{\ell=1}^{p} \psi_\ell \cdot \hat{a}_{h,\ell}}_{\text{lag contribution}} + \underbrace{\sigma_m \cdot \eta}_{\text{noise term}}$$

When $\eta$ is sufficiently negative (e.g., $\eta < -2$), the total can become negative, which is physically impossible.

2. Penalty Classification

The inflow non-negativity penalty $c^{inf}$ is a Category 2 constraint violation penalty — it provides slack for a physical constraint (non-negative inflow) that may be impossible to satisfy under extreme noise realizations. Its position in the penalty hierarchy (see LP Formulation):

$$c^{tv-}, c^{ov\pm}, c^{gv-}, c^{ev}, c^{wv}, c^{inf} > c^{th}, c^{ctr}$$

Since inflow $a_h$ is defined per stage (not per block), the inflow non-negativity penalty appears outside the block summation in the objective, alongside storage violation penalties:

$$+ \sum_{h \in \mathcal{H}} c^{inf} \cdot \sigma^{inf}_h \cdot T$$

where $T = \sum_k \tau_k$ is the total stage duration in hours. The product $\sigma^{inf}_h \cdot T$ converts the slack rate (m³/s) to an energy-equivalent dimension over the full stage.

3. Method: none

Configuration:

{ "modeling": { "inflow_non_negativity": { "method": "none" } } }

LP Formulation: Standard AR constraint (unchanged):

$$a_h = \text{deterministic\_base} + \sum_{\ell=1}^{p} \psi_\ell \cdot a_{h,\ell} + \sigma_m \cdot \eta$$

Implications:

• LP may become infeasible when $a_h < 0$ causes water balance violation
• Useful only for debugging or when the AR model guarantees positive outputs
• Not recommended for production

4. Method: penalty

Configuration:

{
  "modeling": {
    "inflow_non_negativity": {
      "method": "penalty"
    }
  }
}

The penalty cost $c^{inf}$ is authored separately in the penalties file under the hydro section, not inside this block.

Additional Variables:

Variable	Domain	Units	Description
$\sigma^{inf}_h$	$\geq 0$	m³/s	Inflow non-negativity slack

Modified AR Constraint:

$$a_h + \sigma^{inf}_h = \text{deterministic\_base} + \sum_{\ell=1}^{p} \psi_\ell \cdot a_{h,\ell} + \sigma_m \cdot \eta$$

Interpretation: When the AR model produces negative $a_h$, the slack $\sigma^{inf}_h$ absorbs the violation, making the effective inflow:

$$a_h^{effective} = a_h + \sigma^{inf}_h \geq 0$$

Objective Function Addition (outside block summation):

$$+ \sum_{h \in \mathcal{H}} c^{inf} \cdot \sigma^{inf}_h \cdot T$$

where $c^{inf}$ is the penalty cost (default: 1000 $/(m³/s·h)) and $T = \sum_k \tau_k$ is the total stage duration in hours.

Advantages:

• LP always feasible
• Clear cost signal for negative inflow events
• Preserves AR dynamics for positive realizations

Disadvantages:

• Adds variables and constraints
• Slightly affects marginal water values

5. Method: truncation

Configuration:

{ "modeling": { "inflow_non_negativity": { "method": "truncation" } } }

Scenario Generation:

$$a_h = \max\left(0, \text{deterministic\_base} + \sum_{\ell=1}^{p} \psi_\ell \cdot \hat{a}_{h,\ell} + \sigma_m \cdot \eta\right)$$

LP Formulation: Standard AR constraint with the already-truncated $a_h$ value:

$$a_h = \text{(truncated value from scenario)}$$

Advantages:

• No additional LP variables or constraints
• Straightforward formulation

Disadvantages:

• Biases the distribution: Shifts mean upward
• Breaks AR dynamics: When truncation occurs, temporal correlation is disrupted
• May affect long-term storage dynamics

6. Method: truncation_with_penalty — Production Design

Configuration:

{
  "modeling": {
    "inflow_non_negativity": {
      "method": "truncation_with_penalty"
    }
  }
}

The slack-column penalty cost $c^{inf}$ is sourced from the penalties file under the hydro section (same key as for the pure-penalty method).

This is the production method Cobre uses for inflow non-negativity. The two preceding methods each handle one side of the problem well but leave the other unaddressed: pure truncation keeps the LP lean but biases the inflow distribution upward; pure penalty preserves distribution fidelity but relies entirely on LP slack to absorb every negative excursion. The hybrid combines both mechanisms to cover the full range of noise excursions efficiently.

6.1 Production formulation: clamp outside the LP, slack inside the LP

The PAR(p) noise is clamped outside the LP before the scenario is patched in, exactly as in the truncation method:

$$a_h = \max\left(0, \text{deterministic\_base} + \sum_{\ell=1}^{p} \psi_\ell \cdot \hat{a}_{h,\ell} + \sigma_m \cdot \eta\right)$$

Inside the LP, penalty slack columns $\sigma^{inf}_h$ are added to the water-balance constraint, exactly as in the penalty method:

Additional Variables:

Variable	Domain	Units	Description
$\sigma^{inf}_h$	$\geq 0$	m³/s	Inflow non-negativity slack

Modified AR Constraint (inside LP, using the clamped $a_h$):

$$a_h + \sigma^{inf}_h = \text{deterministic\_base} + \sum_{\ell=1}^{p} \psi_\ell \cdot a_{h,\ell} + \sigma_m \cdot \eta$$

Objective Function Addition (outside block summation):

$$+ \sum_{h \in \mathcal{H}} c^{inf} \cdot \sigma^{inf}_h \cdot T$$

6.2 Why the hybrid

Clamping and slack columns serve complementary roles that together preserve relatively complete recourse:

• Clamping handles the common case cheaply. Most negative excursions are small — the noise term dips slightly below zero for a handful of stages in a scenario tree. Clamping those excursions to zero outside the LP adds no LP variables and no solver work. The inflow handed to the LP is always non-negative, so the water-balance constraint is never violated by the noise term alone.
• Slack columns absorb rare large excursions without rejecting the scenario. When the PAR(p) model produces an extreme realisation, the deterministic base and lag contribution combined with the noise term can still yield a zero inflow after clamping, and the LP’s water-balance may still be infeasible without relief. The $\sigma^{inf}_h$ slack column lets the solver relax the non-negativity at a known cost rather than declaring infeasibility. The stage is kept in the training set; the penalty signal propagates into future-cost cuts.
• Together they guarantee LP feasibility (Category 1 recourse) across the full noise distribution, without biasing the distribution upward beyond what truncation already does for small excursions, and without adding LP slack columns for every stage regardless of whether they are needed.

6.3 Reference design and equivalence

The literature formulates the same problem using a dimensionless noise-adjustment slack $\xi_h$. This reference design is presented here so readers familiar with the Brazilian stochastic-dispatch literature can map between the two formulations.

Additional Variables:

Variable	Domain	Units	Description
$\xi_h$	$\geq 0$	-	Noise adjustment slack (dimensionless)

Modified AR Constraint (two parts):

Part A — Modified noise term:

$$\eta_h^{adj} = \eta_h + \xi_h$$

where $\eta_h$ is the original (possibly very negative) noise realization.

Part B — Inflow with adjusted noise:

$$a_h = \text{deterministic\_base} + \sum_{\ell=1}^{p} \psi_\ell \cdot a_{h,\ell} + \sigma_m \cdot \eta_h^{adj}$$

Non-negativity constraint:

$$a_h \geq 0$$

Interpretation: The optimizer chooses $\xi_h$ to be the minimum adjustment needed to make $a_h \geq 0$:

$$\xi_h = \max\left(0, -\eta_h - \frac{\text{deterministic\_base} + \sum_\ell \psi_\ell \cdot \hat{a}_{h,\ell}}{\sigma_m}\right)$$

Objective Function Addition (outside block summation):

$$+ \sum_{h \in \mathcal{H}} c^{inf} \cdot \sigma_m \cdot \xi_h \cdot T$$

The penalty is proportional to $\sigma_m \cdot \xi_h$, which is the actual inflow adjustment in m³/s. Note that $\sigma_m$ varies by season, so the effective penalty for a given noise adjustment $\xi_h$ is larger in high-variability seasons and smaller in low-variability seasons. This is by design — a given noise adjustment represents a larger physical inflow correction when $\sigma_m$ is large.

Equivalence with the production formulation: The $\xi_h$ reference design and the production clamp-plus-slack formulation are economically equivalent when both use the same penalty cost $c^{inf}$. In both cases the objective penalty equals $c^{inf}$ multiplied by the physical inflow correction in m³/s·h. The production formulation reuses the existing truncation and penalty mechanisms without introducing a separate noise-adjustment constraint inside the LP, which simplifies the solver’s constraint matrix.

7. Comparison Summary

Method	LP Size	Bias	AR Preservation	Feasibility	Recommendation
none	Base	None	Full	May fail	Debugging only
penalty	+vars/cons	Minimal	Full	Guaranteed	Alternative
truncation	Base	Upward	Partial	Guaranteed	Quick studies
truncation_with_penalty	+vars/cons	Minimal	Full	Guaranteed	Production

8. Reference

Larroyd, P.V., Pedrini, R., Beltran, F., Teixeira, G., Finardi, E.C., & Picarelli, L.B. (2022). “Dealing with Negative Inflows in the Long-Term Hydrothermal Scheduling Problem.” Energies, 15(3), 1115. https://doi.org/10.3390/en15031115

Cross-References

• LP Formulation — Objective function structure and penalty taxonomy where $c^{inf}$ is a Category 2 constraint violation penalty
• PAR Inflow Model — Defines the PAR(p) model that produces the inflow realizations handled here
• Penalty System — Penalty hierarchy and cascade resolution
• Scenario Generation — The noise term $\eta$ in the inflow equation comes from the fixed opening tree (pre-generated noise vectors), not from per-iteration random sampling