Generators
A generator injects a dispatchable power at a bus — active and reactive power lie within bounds rather than being fixed (a fixed injection is modelled as a negative load). It shares the load's bilinear power form and connection configurations. Parts 1–5 state the foundational model; part 6 records how BMOPFTools realises it. Symbols are defined in Notation.
1. Data model
A generator is an entry of the top-level generator object, keyed by its string ID $g$.
| Field | Type | Unit | Req. | Description |
|---|---|---|---|---|
bus | string | – | ✔ | Host bus ID $i$ |
terminal_map | string[] | – | ✔ | Conductor→terminal map $\textcolor{purple}{\mathbf{N}_{g}}$ |
configuration | string | – | ✔ | WYE, SINGLE_PHASE, or DELTA |
p_min, p_max | number[] | W | Per-phase active-power bounds | |
q_min, q_max | number[] | var | Per-phase reactive-power bounds | |
s_max | number[] | VA | Per-phase apparent-power rating | |
i_max | number[] | A | Per-conductor current-magnitude limit (incl. optional neutral entry) | |
cost | number[] | $/kWh | Per-phase linear generation cost |
2. Input symbols
| Field | Symbol | Notes |
|---|---|---|
p_min, p_max | $\textcolor{red}{P^{\min}_{g}},\ \textcolor{red}{P^{\max}_{g}}$ | per phase |
q_min, q_max | $\textcolor{red}{Q^{\min}_{g}},\ \textcolor{red}{Q^{\max}_{g}}$ | per phase |
s_max | $\textcolor{red}{\mathbf{S}^{\max}_{g}}$ | per phase |
i_max | $\textcolor{red}{\mathbf{I}^{\max}_{g}}$ | per conductor (last entry may bound the neutral return) |
cost | $\textcolor{red}{\mathbf{c}_{g}}$ | per phase |
3. Variables
Each phase conductor $k$ injects a complex current $\textcolor{blue}{I_{g,k}}$ (the current sent to the bus, opposite sign to a load), stacked into $\textcolor{blue}{\mathbf{I}_{g}}$. The neutral return is implicit in KCL.
4. Equality constraints
With $\Delta\textcolor{blue}{U_{g,k}}$ the sub-generator voltage (phase-to-neutral for WYE/SINGLE_PHASE, line-to-line for DELTA), the injected complex power is
\[\textcolor{blue}{S_{g,k}} = \Delta\textcolor{blue}{U_{g,k}}\,(\textcolor{blue}{I_{g,k}})^{*} = P_{g,k} + \textcolor{brown}{j}\,Q_{g,k}.\]
Current conservation over the generator's terminals gives its KCL contribution (injection positive at the phase terminal, return at the neutral).
5. Inequality constraints
Cartesian variable bounds
When i_max is present, a box on the current components $|\mathfrak{R}(\textcolor{blue}{I_{g,k}})|,\,|\mathfrak{I}(\textcolor{blue}{I_{g,k}})|\le\textcolor{red}{I^{\max}_{g,k}}$ bounds the search; implied by the current circle below.
Engineering bounds
Active/reactive power box (the dispatch range):
\[\textcolor{red}{P^{\min}_{g,k}} \le P_{g,k} \le \textcolor{red}{P^{\max}_{g,k}}, \qquad \textcolor{red}{Q^{\min}_{g,k}} \le Q_{g,k} \le \textcolor{red}{Q^{\max}_{g,k}}.\]
Apparent-power circle (optional):
\[P_{g,k}^2 + Q_{g,k}^2 \le (\textcolor{red}{S^{\max}_{g,k}})^2.\]
Current-magnitude circle (optional), per conductor:
\[\textcolor{blue}{I_{g,k}}(\textcolor{blue}{I_{g,k}})^{*} \le (\textcolor{red}{I^{\max}_{g,k}})^2.\]
For a star-connected generator whose i_max carries a trailing neutral entry, the neutral return current $-\mathbf{1}^{\text{T}}\textcolor{blue}{\mathbf{I}_{g}}[\mathcal{P}]$ is additionally bounded by that entry.
6. Implementation in BMOPFTools
Realisation
- Rectangular bilinear power. The code stamps $P = \Delta v^r\,\text{crg} + \Delta v^i\,\text{cig}$, $Q = \Delta v^i\,\text{crg} - \Delta v^r\,\text{cig}$ and applies the P/Q box bounds directly (
generator.jl:_add_generator_constraints!). - Apparent-power circle via auxiliaries. To keep the constraint quadratic, real variables
pg,qgare pinned to the bilinear $P,Q$ and bounded by $\text{pg}^2+\text{qg}^2\le\textcolor{red}{S^{\max}}^2$. - Current circle + box.
crg^2+cig^2 ≤ i_max^2plus_limit_current_box!. - Neutral-conductor limit. With a trailing
i_maxentry (≥2 phases), the return current $-\sum_k\textcolor{blue}{I_{g,k}}$ is bounded via_neutral_current_limit!. For a single-phase device (phase and return are the same current) the entries collapse to one circle at the tighter limit. - KCL:
+crgat the phase terminal,-crgat the neutral (_kcl_add!).
Source map
| Constraint | Code location |
|---|---|
| Currents, KCL, P/Q box, s/i limits | generator.jl:_add_generator_constraints! |
| Neutral-return current limit | _neutral_current_limit! |
Objective (per-phase cost) | objective.jl |
The generator cost is a per-phase array ($/kWh), and s_max/i_max are the apparent-power and current ratings above. In the Task Force PDF these appear in its schema-update addendum, including the neutral-return i_max entry; the code implements all of them. Fold into the primary generator data model when the PDF is superseded.