Grid-Forming AC/DC Converters for Meshed DC Networks via Dual-Measurement Frequency Droop

II. PROPOSED CONTROLLER

• Each terminal forms its frequency command from local DC-voltage deviation and filtered DC-port current deviation, using only these two DC measurements.At steady state, the filtered current deviation equals the unfiltered deviation.Source: §I · p. 2, §II · p. 2
• The frequency command is integrated into the converter voltage angle without a PLL, and a supplementary local loop limits sustained bidirectional active-power exchange.The steady-state and small-signal models assume the power-limit loop is unsaturated.Source: §I · p. 2, §II · p. 3

I. INTRODUCTION

• AC synchronization under DMFD closes an effective DC droop, giving a connected mesh a unique operating point without direct DC-voltage or current regulation.The paper states that this preserves the validity of conventional MTDC DC-side models.Source: §I · p. 2
• A terminal frequency bias and single-terminal islanding enter the DC nodal equations as equivalent current offsets, with island frequency set by local power balance.The resulting effects on node voltages and branch currents are computable from the linear equations.Source: §I · p. 2
• Design envelopes relate droop ratios and net DC injections to voltage bands and current limits, while small-signal damping maps guide gain selection.Voltage-only droop is marginally stable with an undamped oscillatory mode; the DMFD current channel damps it.Source: §I · p. 2

II. PROPOSED CONTROLLER

• The current-based DMFD keeps steady-state network equations linear in network variables, unlike the power-based form, which introduces operating-point values into those equations.Both forms yield identical DC-network matrices.Source: §II · pp. 2–3

III. STEADY-STATE MODEL

• The steady-state section derives and proves uniqueness of the operating point, then treats a terminal frequency bias and single-terminal islanding before constructing four-terminal-ring design envelopes.The section description identifies the analysis scope; the network equations use a standard node–branch representation.Source: §III · p. 3, A. Network Model · p. 4

A. Network Model

• The node–branch model defines oriented branches, incidence matrix, resistances, inductances, node capacitances, node voltages, branch currents, and converter and exogenous injections.Converter currents and exogenous injections are positive into the DC grid.Source: A. Network Model · pp. 3–4
• At steady state, Kirchhoff’s current law and Ohm’s law yield the network relations, with L_R denoting the resistive Laplacian of the DC mesh.The passage identifies the network laws and names the resulting resistive mesh matrix.Source: A. Network Model · p. 4

• The formulation defines scheduled DC current as positive into the grid and uses positive voltage and current droop gains for each terminal.The controller relation is expressed using terminal-specific per-unit metering and gains.Source: B. DMFD Steady-State Relation and… · p. 4
• At steady state, with converter limits inactive and zero power-limit bias, the filtered current equals the measured current.Source: B. DMFD Steady-State Relation and… · p. 4
• Solving the steady-state relation gives the physical converter current as a function of the scheduled current and DC-voltage deviation.Source: B. DMFD Steady-State Relation and… · p. 4
• The controller enters the network model through effective SI droop matrices, allowing terminals with different ratings to share one physical network model.The network matrices remain in SI units while controller gains are tuned locally in per unit.Source: B. DMFD Steady-State Relation and… · p. 4

• For an unsaturated, connected DC network with positive line resistances and positive effective droop coefficients, the steady operating-point matrix is invertible.The uniqueness argument uses nonnegative line and droop terms; connectivity and positive droop force the homogeneous voltage solution to zero.Source: C. Operating Point, Existence, and… · p. 4
• AC synchronization fixes terminal frequencies, creating effective DC droop even though the controller acts only on terminal frequency.Source: C. Operating Point, Existence, and… · p. 4
• Each specified frequency vector determines a unique DC-network solution for fixed scheduled converter currents and exogenous injections.The solution determines DC-voltage deviations and branch currents, with converter currents following from the controller relation.Source: C. Operating Point, Existence, and… · p. 4
• When scheduled converter currents do not balance exogenous injections, droop corrections absorb the mismatch and produce DC-voltage deviations.Source: C. Operating Point, Existence, and… · p. 4
• The actual operating-point values can differ from nominal voltage and scheduled current.Source: C. Operating Point, Existence, and… · pp. 4–5

1) Single-Terminal Grid-Frequency Bias:

• A single terminal's prescribed frequency deviation enters as an equivalent nodal-current offset, shifting DC voltage and converter and branch currents.The voltage shift is relative to the nominal-frequency operating point.Source: 1) Single-Terminal Grid-Frequency Bias · p. 5

2) Islanded Terminal:

• For an islanded terminal, island power balance determines its otherwise-unprescribed frequency, with the other terminals held at nominal frequency.The island demand is specified as frequency-independent, and converter losses are neglected.Source: 2) Islanded Terminal · p. 5
• The islanded-terminal equations are linearized around the nominal-frequency operating point before deriving the frequency–power map.Source: 2) Islanded Terminal · p. 5
• A unique islanded operating point additionally requires nonzero steady frequency-to-power sensitivity at the islanded terminal.For the positive gains considered, this condition is satisfied; the resulting DC voltages and currents follow from the network equations.Source: 2) Islanded Terminal · p. 5

E. Design Envelopes for a Four-Terminal Ring

• Disturbances and schedules are represented by a per-unit pattern vector D, scaled by amplitude α and mapped to amperes.Source: E. Design Envelopes for a… · p. 5
• The admissible DC-voltage band is ±0.05 pu, and the droop is uniform unless otherwise stated.Imax denotes the largest per-unit disturbance amplitude considered.Source: E. Design Envelopes for a… · p. 5
• Figure 3 plots node-wise DC-voltage deviation against disturbance amplitude for four patterns at nominal frequency and uniform droop.Dashed lines mark the ±0.05 pu band.Source: Figure 3 · p. 6
• Figure 4a gives the operating-band compliance envelope for uniform droop, while Figure 4b plots worst-case converter current against single-terminal frequency bias.Source: Figure 4 · p. 6

1) Nominal Frequency (∆ω = 0):

• At nominal frequency, node-wise voltage deviations follow near-linear slopes across the canonical disturbance patterns, and the ±Υ band identifies each pattern’s limiting node.The induced ℓ∞ bound defines a safe operating region in the (ρ, Imax) plane.Source: 1) Nominal Frequency (∆ω = 0) · p. 6

2) Single-Terminal Frequency Bias:

• Under a steady frequency bias at one terminal, the worst-case per-unit DC-current magnitude grows linearly with |∆ω|.Source: 2) Single-Terminal Frequency Bias · p. 6

3) Single-Terminal Island:

• For islanded Terminal 2, the island frequency varies with island power change, while the resulting maximum DC-voltage excursion is compared with the ±Υ band.Source: 3) Single-Terminal Island · p. 6, Figure 5 · p. 6

IV. DC NETWORK DYNAMIC MODEL WITH CURRENT-BASED DMFD CONTROL

• The dynamic model couples the DC network, DMFD, and AC active-power dynamics with references fixed and limiting loops inactive.It is intended for gain selection and network-coupled modes at the synchronized grid-connected operating point, not large-signal fault or islanding transients.Source: §IV · p. 6

A. DC-Grid Dynamic Equations

• The DC-grid dynamic equations are introduced using the network orientation defined earlier.Source: A. DC-Grid Dynamic Equations · p. 6

B. Operating Point and Small-Signal Deviations

• The operating point satisfies the steady network and controller equations, and tildes denote perturbations from that point.The voltage and current perturbations are defined relative to their operating-point values.Source: B. Operating Point and Small-Signal… · p. 6

C. Current-Based DMFD and Current Metering

• For a fixed current schedule, the filtered-error perturbation is introduced, and the DMFD relation is expressed in terms of frequency perturbations.Source: C. Current-Based DMFD and Current… · p. 6

D. AC-Side Active-Power Map and Linearization

• The AC active-power channel is represented by a lossless power-angle model and linearized about the operating point.With AC-voltage magnitudes fixed, the first two terms of the expansion are eliminated; a local grid held at operating frequency has eωg,k = 0.Source: D. AC-Side Active-Power Map and… · p. 7

E. DC-Port Power Identity and Current Deviation

• With lossless converter coupling, DC-port power satisfies Pdc,k = −Pac,k, and its linearization yields the current-deviation relation.Source: E. DC-Port Power Identity and Current… · p. 7

F. State-Space Stacking

• The state vector stacks voltage, branch-current, active-power, and filtered-current perturbations into a linear system ˙x = Ax + Buu.The third-row blocks follow from the stated DC-port and DMFD relations.Source: F. State-Space Stacking · p. 7
• Figure 6 plots dominant eigenvalue loci while sweeping mdc or mi, and Figure 7 maps the dominant oscillatory-mode damping ratio over the (mdc, mi) gain plane.Source: Figure 6 · p. 7, Figure 7 · p. 7

G. Small-Signal Dynamic Analysis Results

• At the baseline point, the linearized system is stable, with a dominant oscillatory-mode damping ratio of ζdom = 0.71.The dominant-mode damping ratio varies across the (mdc, mi) gain plane; removing current feedback leaves an undamped mode at mi = 0, while positive mi in the evaluated range moves it into the left-half plane.Source: G. Small-Signal Dynamic Analysis Results · p. 7, Power-limit controller · p. 8

V. SIMULATION RESULTS AND DISCUSSION

• An averaged EMT model of the proposed DMFD control was implemented in MATLAB/Simulink for a four-terminal meshed DC network; Table I lists key electrical parameters and controller gains.Table I is labeled “4-terminal ring: baseline values and symbols”; listed values include line resistances [0.75, 0.6, 0.9, 1.5] Ω, inductances [4, 6.36, 4.77, 7.96] mH, and 15 mF node capacitors.Source: §V · p. 8, Table I · p. 8, DC network (4-terminal ring) · p. 8
• Loading ramps keep all node voltages within the ±Υ band, and DMFD and GFL reach essentially identical steady-state operating points.Terminal-3 DC voltage settles at 0.9563 pu with DMFD and 0.9558 pu with GFL.Source: §V · p. 8, Figure 8 · p. 9
• DMFD and GFL follow the scheduled Terminal-1-to-Terminal-3 power transfer with comparable errors.The steady-state offset reflects unequal line resistances and DC voltage drops present under either controller.Source: §V · p. 8, Figure 9 · p. 9
• Islanded operating points follow the predicted frequency–power characteristic, and DC-voltage points remain within the analytical envelope.DMFD remains stable when Terminal 2 disconnects from its local grid, whereas the GFL baseline loses stability.Source: §V · p. 8, Figure 11 · p. 9, Figure 10 · p. 9
• DMFD maintains stable operation and post-fault recovery on a weak AC tie, while the GFL baseline becomes unstable under the same disturbance.The Terminal-4 tie has SCR = 2; the terminal is preloaded at 0.75 pu.Source: §V · pp. 8–9

VI. CONCLUSION

• AC synchronization under DMFD closes an effective DC droop, so the connected mesh has a unique operating point without direct DC-voltage or current regulation.An AC-frequency bias at one terminal becomes a bounded, linear DC-network quantity that can be budgeted in voltage and current envelopes.Source: §VI · p. 9
• The same mechanism permits islanding without reconfiguration, with island frequency fixed by local power balance and the DC excursion predictable from island demand.The frequency bias also redistributes power across the mesh, a coupling absent in GFL-based MTDC schemes.Source: §VI · pp. 9–10
• Removing DMFD current feedback leaves an undamped oscillatory mode and renders the system marginally stable.The conclusion identifies sensitivity to nonuniform droop and line asymmetry, and comparison with voltage-only DC droop, as future work.Source: §VI · p. 10
• Weak-tie fault ride-through and islanding stability are obtained without giving up conventional MTDC steady-state DC-droop behavior.The conclusion frames this compatibility as structural rather than dependent on tuning.Source: §VI · pp. 9–10