Zonal Math Formulation#

This page documents the zonal SDOM formulation used when:

  • Network = AreaTransportationModelNetwork

It extends the base formulation in SDOM Formulation.

Sets#

  • \(\mathcal{A}\): areas (model.A)

  • \(\mathcal{L}\): interconnections (model.L)

  • \(\mathcal{H}\): hours (model.h)

For each area \(a \in \mathcal{A}\):

  • \(\mathcal{L}_{in}(a)\): lines ending in \(a\) (model.L_in[a])

  • \(\mathcal{L}_{out}(a)\): lines starting in \(a\) (model.L_out[a])

Parameters#

  • \(\overline{F}^{FT}_{l,h}\): capacity in from -> to direction (model.LineCap_FT[l,h])

  • \(\overline{F}^{TF}_{l,h}\): capacity in to -> from direction (model.LineCap_TF[l,h])

Variables#

  • \(f_{l,h} \in \mathbb{R}\): signed line flow (model.f[l,h])

Reporting expressions:

  • \(f^{FT}_{l,h} = f_{l,h}\) (model.f_FT[l,h])

  • \(f^{TF}_{l,h} = -f_{l,h}\) (model.f_TF[l,h])

In reports, SDOM clips directional values to non-negative values.

Network Constraints#

Directional capacity limits are modeled as two explicit constraint blocks:

\[ f_{l,h} \leq \overline{F}^{FT}_{l,h} \]
\[ -f_{l,h} \leq \overline{F}^{TF}_{l,h} \]

These correspond to model.f_upper[l,h] and model.f_lower[l,h].

Per-Area Supply Balance#

For each area and hour, SDOM enforces:

\[ \text{Demand}_{a,h} + \text{Charge}_{a,h} - \text{Discharge}_{a,h} - \text{Gen}_{a,h}^{all} - \sum_{l \in \mathcal{L}_{in}(a)} f_{l,h} + \sum_{l \in \mathcal{L}_{out}(a)} f_{l,h} = 0 \]

Implemented as model.area[a].SupplyBalance[h].

Sign convention:

  • Positive \(f_{l,h}\) means flow in from -> to direction.

  • Net inflow to area \(a\) is subtracted from the left-hand side in the implementation.

System-Wide Clean Share Constraint#

The carbon-free target remains system-wide (not per area). It is built as top-level model.GenMix_Share.

Objective#

Zonal objective is the sum of area costs plus a transmission placeholder:

\[ \min Z = \sum_{a \in \mathcal{A}} \left(Z^{fixed}_a + Z^{variable}_a\right) + Z^{trans} \]

Current implementation uses:

\[ Z^{trans} = 0 \]

Transmission investment is intentionally deferred.