DC networks

BMOPFTools models an MVDC/LVDC network — DC buses, cables, groundings, loads, and sources — that AC/DC converters share. This is how converter stations, back-to-back soft open points (SOPs), and MVDC ties are formed. DC quantities have no angle: each DC terminal holds a single real, signed voltage to earth (positive pole $>0$, negative pole $<0$, metallic return $\approx 0$). This page groups the DC objects and the converter coupling; each follows the foundational → implementation split. Symbols are defined in Notation.

The DC subsystem is a BMOPFTools extension with no counterpart in the Task Force PDF (see the reconciliation note).

DC buses

Data model (dc_bus)

Variables

Each DC terminal $p$ holds a real signed voltage $v^{\text{dc}}_{b,p}\in\mathbb{R}$ (perfectly grounded terminals fixed to $0$). A free earth current $i^{\text{gnd}}_{b,p}$ is added at each perfect ground.

Constraints

DC KCL at every terminal (currents sum to zero; grounded terminals keep the equation, balanced by the earth current) — the DC analogue of the AC bus:

\[\sum \text{(branch, converter, load, source, ground currents)} = 0.\]

Signed line-to-ground bounds are variable bounds on $v^{\text{dc}}_{b,p}$. Line-to-neutral / line-to-line magnitude bounds stay linear because the pole roles fix the sign of each difference: for an oriented difference $\Delta\ge 0$, $\textcolor{red}{v^{\min}}\le\Delta\le\textcolor{red}{v^{\max}}$ (a POSITIVE/NEGATIVE role is required, else a hard error — never a non-convex squared form).

DC branches

Data model (dc_branch)

Variables and constraints

Per conductor $k$, a real current $i^{\text{dc}}_{\ell,k}$ from from to to. Ohm's law (or an ideal conductor when $\textcolor{red}{r_k}=0$):

\[i^{\text{dc}}_{\ell,k} = \frac{v^{\text{dc}}_{b^{\text{fr}},k} - v^{\text{dc}}_{b^{\text{to}},k}}{\textcolor{red}{r_k}}.\]

It enters DC KCL with opposite sign at each end. Thermal and power limits: $(i^{\text{dc}}_{\ell,k})^2 \le (\textcolor{red}{i^{\max}_{\ell,k}})^2$ and, on the pole conductor, $(v^{\text{dc}}\,i^{\text{dc}})^2 \le (\textcolor{red}{p^{\max}_\ell})^2$.

DC groundings, loads, and sources

dc_grounding

Sets the signed-voltage reference of a DC island. r = 0 (or omitted) is perfect grounding ($v^{\text{dc}}=0$ with a free earth current); r > 0 is grounding through an impedance, drawing $i^{\text{earth}} = v^{\text{dc}}/\textcolor{red}{r}$ from the node. At least one grounding per connected DC island is required.

dc_load and dc_source

A DC load draws constant power across a terminal pair; a DC source injects a dispatched power. With port voltage $\Delta v^{\text{dc}}$ (pole − return, or pole − earth) and port current $I$:

\[\Delta v^{\text{dc}}\, I = \textcolor{red}{p} \quad(\text{load, drawn}), \qquad \Delta v^{\text{dc}}\, I = P \quad(\text{source, injected}),\]

with the source power $P$ either a fixed setpoint (p) or dispatchable within $[\textcolor{red}{p^{\min}},\textcolor{red}{p^{\max}}]$. The port current enters DC KCL at the two terminals.

AC/DC converters

An IBR that references a dc_bus (via dc_bus + dc_terminal_map) becomes an AC/DC converter: its AC side is the IBR model; its DC port injects into the shared DC node. Converters are lossless here — the DC-port power equals the AC active power.

Coupling equality

With DC-port voltage $\Delta v^{\text{dc}}_r$ (pole − return) and port current $I_r$:

\[\Delta v^{\text{dc}}_r\, I_r = \sum_k P_{r,k},\]

and $I_r$ enters DC KCL at the port terminals. A converter station / back-to-back SOP / MVDC tie emerges automatically when several converters share one dc_bus and balance through DC KCL.

DC-side control mode (dc_control)

• P (default): the OPF dispatches the converter power (no extra constraint).
• V (DC-voltage master): pins $\Delta v^{\text{dc}}_r = \textcolor{red}{v^{\text{set}}_r}$; the AC power floats to balance the zone.
• droop (saturated V–P): $\sum_k P_{r,k} = f(\Delta v^{\text{dc}}_r)$, a piecewise-linear characteristic rising with DC voltage, flat within an optional dead-band around $\textcolor{red}{v^{\text{set}}_r}$, and clamped at the converter power limits.

Each connected DC island needs at least one V or droop converter, else the DC voltage is underdetermined.

Implementation in BMOPFTools

Realisation

• DC variables — signed node voltages v_dc (grounded fixed to 0), branch currents idc_br, converter port currents idc_conv, load/source currents and source power (dcnetwork.jl:_add_dc_variables!).
• DC KCL — a per-terminal accumulator mirroring the AC pattern, enforced $=0$ (_init_dc_kcl, _add_dc_kcl_constraints!).
• Branches, groundings, loads, sources — stamped in _add_dc_network_constraints!; degenerate voltage bands (e.g. a return pinned to 0) are fixed rather than bounded, and ideal-conductor equalities to a fixed node are skipped.
• Converter coupling — the lossless bilinear balance and the P/V/droop control law are added by _couple_converter_to_dc! (called from the IBR builder, which supplies each converter's AC active power). The droop reuses the same smoothed piecewise-linear operator as the AC Volt-Watt curves.
• Warm start — DC node voltages are seeded at v_dc_nom (or the mid-band) so the bilinear converter balance starts away from the degenerate $v=0$ point (_set_dc_start_values!).

Source map

Reconciliation note