Loads

A load draws a specified power at a bus. Its power may be constant, or vary with voltage (ZIP / exponential). Three connection configurations are supported: single-phase, wye (with neutral return), and delta. Parts 1–5 state the foundational model; part 6 records how BMOPFTools realises it. Symbols are defined in Notation. For choosing between the load models — and identifying one from measurements — see the load-model tutorial.

1. Data model

A load is an entry of the top-level load object, keyed by its string ID $d$.

2. Input symbols

A sub-load $k$ is one two-terminal power element of the load: a phase-to-neutral branch (WYE), the single terminal pair (SINGLE_PHASE), or one line-to-line branch (DELTA).

3. Variables

Each sub-load $k$ draws a complex current $\textcolor{blue}{I_{d,k}}$, stacked into $\textcolor{blue}{\mathbf{I}_{d}}$. These are the currents the load contributes to KCL; the neutral return current is not an independent variable.

4. Equality constraints

Sub-load voltage drop and power

Let $\Delta\textcolor{blue}{U_{d,k}}$ be the voltage across sub-load $k$: phase-to-neutral for WYE, the terminal-pair difference for SINGLE_PHASE, line-to-line for DELTA. The complex power the sub-load absorbs is

\[\textcolor{blue}{S_{d,k}} = \Delta\textcolor{blue}{U_{d,k}}\,(\textcolor{blue}{I_{d,k}})^{*} = P_{d,k} + \textcolor{brown}{j}\,Q_{d,k}.\]

Constant power (default) pins it to the nominal setpoint:

\[P_{d,k} = \textcolor{red}{P^{\text{nom}}_{d,k}}, \qquad Q_{d,k} = \textcolor{red}{Q^{\text{nom}}_{d,k}}.\]

Voltage-dependent (ZIP) scales the setpoint by impedance/current/power fractions of the voltage-magnitude ratio $|\Delta\textcolor{blue}{U_{d,k}}|/\textcolor{red}{V^{\text{nom}}_{d,k}}$:

\[\begin{aligned} P_{d,k} &= \textcolor{red}{P^{\text{nom}}_{d,k}}\left(\textcolor{red}{\alpha_Z}\,\frac{|\Delta\textcolor{blue}{U_{d,k}}|^2}{(\textcolor{red}{V^{\text{nom}}_{d,k}})^2} + \textcolor{red}{\alpha_I}\,\frac{|\Delta\textcolor{blue}{U_{d,k}}|}{\textcolor{red}{V^{\text{nom}}_{d,k}}} + \textcolor{red}{\alpha_P}\right),\\ Q_{d,k} &= \textcolor{red}{Q^{\text{nom}}_{d,k}}\left(\textcolor{red}{\beta_Z}\,\frac{|\Delta\textcolor{blue}{U_{d,k}}|^2}{(\textcolor{red}{V^{\text{nom}}_{d,k}})^2} + \textcolor{red}{\beta_I}\,\frac{|\Delta\textcolor{blue}{U_{d,k}}|}{\textcolor{red}{V^{\text{nom}}_{d,k}}} + \textcolor{red}{\beta_P}\right). \end{aligned}\]

Exponential uses a power law $P_{d,k} = \textcolor{red}{P^{\text{nom}}_{d,k}}\,(|\Delta\textcolor{blue}{U_{d,k}}|/\textcolor{red}{V^{\text{nom}}_{d,k}})^{\textcolor{red}{\gamma_P}}$ (and likewise $Q$ with $\textcolor{red}{\gamma_Q}$). Constant-impedance, constant-current and constant-power are the special cases $\alpha=(1,0,0),(0,1,0),(0,0,1)$ and integer exponents $\gamma\in\{2,1,0\}$.

Current conservation

The sub-load currents sum to zero over the load's terminals (the neutral / return carries $-\sum_k\textcolor{blue}{I_{d,k}}$), giving the KCL contribution at each host terminal.

5. Inequality constraints

None (foundational). A load has no rating bounds in this model — it is a fixed or voltage-dependent power sink. (See part 6 for the auxiliary variable box bounds the implementation adds purely for conditioning.)

6. Implementation in BMOPFTools

Realisation

• Rectangular bilinear power. With $\Delta v^r,\Delta v^i$ the drop and crd/cid the sub-load current, the code stamps $P = \Delta v^r\,\text{crd} + \Delta v^i\,\text{cid}$ and $Q = \Delta v^i\,\text{crd} - \Delta v^r\,\text{cid}$ (load.jl:_add_load_constraints!), then pins them per model (_add_subload_power!).
• Quadratic ZIP/exponential. A squared-voltage-drop variable $W=\Delta v^{r2}+\Delta v^{i2}$ (and, when a constant-current term is present, $s=\sqrt{W}$) is introduced so ZIP terms stay quadratic. Integer exponents are routed to the constant-Z/I/P terms; only a genuinely non-integer exponent emits the nonlinear term $\textcolor{red}{P^{\text{nom}}}(W/\textcolor{red}{V^{\text{nom}}}^2)^{\gamma/2}$.
• SINGLE_PHASE two-terminal handling. A length-2 map is treated as one element between the two listed terminals — phase-to-neutral or phase-to-phase — not as a phase-to-ground load, so 240 V split-phase loads are modelled correctly.
• KCL contributions are added at the host terminals (_kcl_add!); the neutral return is implicit as the sum of phase currents.

Cartesian bounds (conditioning only)

The auxiliary $W$ (and $s$) carry box bounds at $[0.5\,\textcolor{red}{V^{\text{nom}}},1.5\,\textcolor{red}{V^{\text{nom}}}]^2$ — deliberately wider than any supply standard. These bound the auxiliary variables for solver conditioning; the operational voltage limits live on the bus, not here.

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