Voltage Source Converter (VSC) Model Implementation

Implementations in PSSE

PSS/E represents a point-to-point VSC HVDC link with the steady-state VSC DC Transmission Line record: two converter records, each carrying a DC-control TYPE (1 = DC-voltage control, 2 = MW/power control, 0 = out of service) and an AC-control MODE (1 = AC-voltage control, 2 = fixed power factor), plus a single fixed setpoint DCSET (scheduled DC voltage in kV for TYPE 1, MW demand for TYPE 2). When both converters are in service exactly one must be TYPE 1.

The steady-state record has no representation of DC-voltage droop control — there is no droop slope, gain, or deadband field, and the field set is stable across PSS/E v33/v34/v35. Droop is a dynamics-level concern in PSS/E (.dyr models such as VSCDCT/CDC4T) or an external/multi-terminal add-on. Consequently, when Sienna exports to PSS/E .raw, a droop-controlled converter is written as MW control (TYPE 2) at its scheduled active power, and the droop characteristic is dropped (see PSSEExportPowerFlow); only a strict DC-voltage terminal is exported as TYPE 1.

Implementation in Sienna

Every PSY DC component lowers into a single internal DCNetwork that the AC power flow solves jointly with the AC buses. A point-to-point TwoTerminalVSCLine becomes two converters, two DC nodes, and one DC branch; a multi-terminal grid (InterconnectingConverter on DCBus nodes joined by TModelHVDCLine branches) lowers into the same structure, so all residual/Jacobian kernels are written once. The escape hatch solver_settings = Dict(:model_dc_network => false) restores the historical behavior (DC components ignored in the AC solve).

The state vector is extended with a VSC tail: per converter the AC injections $(P_c, Q_c)$, and per DC node the DC voltage $V_{dc}$. Each converter contributes two control rows ($r_1$, $r_2$) and each DC node one DC-KCL row, matching the added states. VSC extensions are implemented for all three AC formulations — ACPolarPowerFlow, ACRectangularPowerFlow, and ACMixedPowerFlow — sharing the same converter-physics kernels; only the bus-coupling rows differ (polar uses the bus power balance; rectangular/mixed use the bus current-injection balance).

Control modes

A converter's per-terminal PSY DC-side and AC-side control enums lower into one VSCControlMode, which selects the two control-row residuals:

where $k$ is the DC-voltage droop gain. With $P_c$ the converter's AC-side injection, the droop row $r_1 = (V_{dc} - V_{set}) - k\,P_c$ is equivalently $V_{dc} = V_{set} - k\,P_{dc,inj}$ where $P_{dc,inj}$ is the power injected into the DC grid — the standard Beerten droop convention: a converter injecting into the DC grid operates below its voltage reference, and a high DC voltage drives converters to withdraw more. A mode pins the DC-node voltage (makes its node a DC slack) iff it is a $V_{dc}$-control mode; a wholly droop-controlled DC subnet is anchored by the droop relation instead. The AC-voltage row uses the raw $\lvert V_{ac}\rvert^2$ (floor-free, like the rectangular PV pin) for a derivative consistent with the residual.

DC network balance and converter losses

For each DC node the DC-KCL row enforces the nodal current balance through the dense DC nodal conductance matrix $G_{dc}$ (built from the branch conductances) plus the converter current injections:

\[\sum_{j} G_{dc}[k, j]\, V_{dc}[j] + \sum_{c \,\in\, k} \frac{P_{dc,c}}{V_{dc}} = 0\]

The converter draws $P_{dc} = P_c + P_{loss}$ from its DC node, with a quadratic converter-loss curve in the converter current magnitude $I_c$:

\[P_{loss} = a + b\,I_c + c\,I_c^2, \qquad I_c = \frac{S}{\lvert V_{ac}\rvert}, \qquad S = \sqrt{P_c^2 + Q_c^2}\]

Both $S^2$ and $\lvert V_{ac}\rvert^2$ are floored at V_FLOOR2 to keep $I_c$ and its derivatives finite at the origin (the $P_c/S$, $Q_c/S$ terms are otherwise singular at $S = 0$).

Jacobian

The control-row partials depend only on the control mode (and on $\lvert V_{ac}\rvert$ for the AC-voltage and loss terms), so each is a small branch on VSCControlMode rather than method dispatch:

\[\begin{aligned} \frac{\partial r_1}{\partial P_c} &= \begin{cases} 0 & V_{dc}\text{-control} \\ -k & \text{droop} \\ 1 & P\text{-control} \end{cases} &\qquad \frac{\partial r_1}{\partial V_{dc}} &= \begin{cases} 1 & V_{dc}\text{-control or droop} \\ 0 & \text{otherwise} \end{cases} \\[1em] \frac{\partial r_2}{\partial Q_c} &= \begin{cases} 0 & \text{AC-voltage control} \\ 1 & \text{otherwise} \end{cases} &\qquad \frac{\partial r_2}{\partial \lvert V_{ac}\rvert} &= \begin{cases} 2\lvert V_{ac}\rvert & \text{AC-voltage control} \\ 0 & \text{otherwise} \end{cases} \end{aligned}\]

The converter's $(P_c, Q_c)$ inject into the bus balance rows (with a $-1$ coupling), and the DC-KCL row couples to $(P_c, Q_c, \lvert V_{ac}\rvert, V_{dc})$ through $P_{dc}/V_{dc}$. The $\lvert V_{ac}\rvert$-coupling enters the Jacobian only where $\lvert V_{ac}\rvert$ is a state — i.e. at PQ buses in the polar formulation (at PV/REF buses the AC magnitude is fixed, so those derivatives vanish and no Jacobian entry is written). The loss-term DC-side derivatives are

\[\frac{\partial P_{dc}}{\partial P_c} = 1 + \frac{dP_{loss}}{dI_c}\frac{\partial I_c}{\partial P_c}, \qquad \frac{\partial P_{dc}}{\partial Q_c} = \frac{dP_{loss}}{dI_c}\frac{\partial I_c}{\partial Q_c}, \qquad \frac{\partial P_{dc}}{\partial \lvert V_{ac}\rvert} = -\frac{dP_{loss}}{dI_c}\frac{I_c}{\lvert V_{ac}\rvert}\]

with $\dfrac{dP_{loss}}{dI_c} = b + 2 c\,I_c$, $\dfrac{\partial I_c}{\partial P_c} = \dfrac{P_c/S}{\lvert V_{ac}\rvert}$, and $\dfrac{\partial I_c}{\partial Q_c} = \dfrac{Q_c/S}{\lvert V_{ac}\rvert}$.

Assembling those into the DC-KCL row for node $k$ (the $P_{dc}/V_{dc}$ term differentiated, on top of the constant conductance terms):

\[\frac{\partial \text{KCL}_k}{\partial V_{dc,k}} = G_{dc}[k,k] - \frac{P_{dc}}{V_{dc,k}^2}, \qquad \frac{\partial \text{KCL}_k}{\partial V_{dc,j}} = G_{dc}[k,j]\ \ (j \neq k), \qquad \frac{\partial \text{KCL}_k}{\partial x} = \frac{1}{V_{dc,k}}\frac{\partial P_{dc}}{\partial x}\ \ (x \in \{P_c,\, Q_c,\, \lvert V_{ac}\rvert\})\]

In rectangular/MCPB the $\lvert V_{ac}\rvert$ terms are chain-ruled onto the bus $(e, f)$ states via $\partial \lvert V_{ac}\rvert/\partial e = e/\lvert V_{ac}\rvert$ (and likewise for $f$).

Code map

The residual and Jacobian kernels are shared across formulations (only the bus-coupling rows differ), so each equation above maps to one internal method in vsc_utils.jl (polar Jacobian bus-coupling lives in ac_power_flow_jacobian.jl):