MVDC/LVDC converter stations — feeder balancing as an optimisation
This tutorial shows how BMOPFTools models MVDC/LVDC converter stations — back-to-back soft open points (SOPs), MVDC ties, and DC feeders — by reusing the ibr object. There is no separate "converter" object: an AC/DC converter is an ibr that carries a dc_bus reference, and several converters sharing one dc_bus form a station. The DC side balances active power, so the station can route power between AC feeders that the AC topology alone could not.
Prerequisites: a Julia environment with BMOPFTools, JuMP and Ipopt installed, and familiarity with the JSON input format — see the end-to-end tutorial first. Every code block on this page runs when the docs are built, so the numbers below are real.
See the conventions for the DC terminal/pole/return and grounding rules used below.
Why this is an optimisation, not a simulation
A normally-open tie leaves two feeders radial: a power-flow simulator gives a single, determined answer. Replace the tie with a soft open point and the power split across it becomes a free decision — there is no longer one "simulate" answer. To find the best split you must solve an optimisation, and the quantity you want (the optimal transfer, and the loss/hosting-capacity it unlocks) is the solution of that problem, not an output you can read off a simulator.
The literature frames the SOP/MVDC benefit exactly this way — feeder load balancing, loss reduction, voltage support, and DG hosting-capacity enhancement, all posed as optimisation problems (1, 2, 3), with reported loss reductions and hosting-capacity gains well into the tens of percent. Below we reproduce the simplest of these — loss-minimising feeder balancing — as a single OPF.
The case: one substation, two unevenly loaded feeders
A single substation feeds two feeders of equal impedance (R = 1 Ω each). One feeder carries 50 kW, the other 10 kW. An SOP ties the two feeder ends through a DC bus — the whole MVDC subsystem is embedded in one AC system, the way a real SOP is deployed.
using BMOPFTools, JuMP, Ipopt
const OPT = optimizer_with_attributes(Ipopt.Optimizer, "print_level" => 0)
sop_case() = parse_bmopf("""
{"bus":{
"sub":{"terminal_names":["1","n"],"perfectly_grounded_terminals":["n"]},
"a":{"terminal_names":["1","n"],"perfectly_grounded_terminals":["n"],"v_min":[600.0],"v_max":[1100.0]},
"b":{"terminal_names":["1","n"],"perfectly_grounded_terminals":["n"],"v_min":[600.0],"v_max":[1100.0]}},
"linecode":{"lc":{"R_series_1_1":1.0,"X_series_1_1":0.0}},
"line":{
"La":{"bus_from":"sub","bus_to":"a","terminal_map_from":["1"],"terminal_map_to":["1"],"length":1.0,"linecode":"lc"},
"Lb":{"bus_from":"sub","bus_to":"b","terminal_map_from":["1"],"terminal_map_to":["1"],"length":1.0,"linecode":"lc"}},
"voltage_source":{
"s":{"bus":"sub","terminal_map":["1","n"],"v_magnitude":[1000.0,0.0],"v_angle":[0.0,0.0],"cost":[1.0]}},
"load":{
"La_load":{"bus":"a","terminal_map":["1","n"],"configuration":"SINGLE_PHASE","p_nom":[50000.0],"q_nom":[0.0]},
"Lb_load":{"bus":"b","terminal_map":["1","n"],"configuration":"SINGLE_PHASE","p_nom":[10000.0],"q_nom":[0.0]}},
"ibr":{
"vA":{"bus":"a","terminal_map":["1","n"],"topology":"SINGLE_PHASE","prime_mover":"GENERIC","s_max":[60000.0],"dc_bus":"dc","dc_terminal_map":["p","m"],"dc_control":"V","dc_v_set":1500.0},
"vB":{"bus":"b","terminal_map":["1","n"],"topology":"SINGLE_PHASE","prime_mover":"GENERIC","s_max":[60000.0],"dc_bus":"dc","dc_terminal_map":["p","m"]}},
"dc_bus":{
"dc":{"terminal_names":["p","m"],"pole":{"p":"POSITIVE","m":"METALLIC_RETURN"},
"v_dc_min":[1400.0,0.0],"v_dc_max":[1800.0,0.0]}},
"dc_grounding":{"g":{"dc_bus":"dc","terminal":"m"}}}
"""; from_string=true)
res = solve_opf(sop_case(); optimizer = OPT)
println("vA pg : ", round(res["ibr"]["vA"]["1"]["pg"], digits=0), " W (SOP delivers into the heavy feeder)")
println("vB pg : ", round(res["ibr"]["vB"]["1"]["pg"], digits=0), " W (SOP draws from the light feeder)")
println("v_dc : ", round(res["dc_bus"]["dc"]["p"]["v_dc"], digits=1), " V")
println("p_loss : ", round(res["losses"]["p_loss"], digits=0), " W")vA pg : 20000.0 W (SOP delivers into the heavy feeder)
vB pg : -20000.0 W (SOP draws from the light feeder)
v_dc : 1500.0 V
p_loss : 1917.0 WThe source carries a cost, so minimising generation cost minimises the power it must supply — and with the loads fixed, that is exactly the network loss. The two converters share dc_bus = "dc", which is what makes them one station; each spans the pole p and the grounded metallic return m. Per-converter results live under res["ibr"][id]["1"]["pg"] (AC-side active power, positive into the AC network) and the DC node voltage under res["dc_bus"]["dc"]["p"]["v_dc"].
Converter vA is the DC-voltage master (dc_control = "V", dc_v_set = 1500): like an AC slack bus, an MVDC zone needs one converter to set the DC voltage, or its v_dc is underdetermined (the validator flags E.INT.DC_NO_VOLTAGE_CONTROL otherwise — demonstrated below). The master holds v_dc; its AC power floats to balance the zone, while vB controls power. Real systems also use V–P droop (dc_control = "droop"), which shares DC-voltage regulation across converters and saturates at their power limits — see the conventions and the next section.
The answer the optimiser computes
The two feeders start at 50 kW and 10 kW. The OPF chooses to move 20 kW across the SOP — vA.pg = +20 kW, vB.pg = −20 kW — so both feeders end up carrying 30 kW. That is precisely the analytic loss-optimal split (P_a − P_b)/2: equalising the currents minimises Σ I²R. Removing the SOP shows what it was worth:
radial = sop_case()
for k in ("ibr", "dc_bus", "dc_grounding")
delete!(radial, k)
end
res_radial = solve_opf(radial; optimizer = OPT)
println("radial p_loss : ", round(res_radial["losses"]["p_loss"], digits=0), " W")radial p_loss : 2888.0 WThe losses fall from 2888 W (radial, no SOP) to 1917 W — a 34 % reduction — with no change to the loads served. Crucially, 20 kW is not something you could read off a load-flow run; it is the decision the optimisation makes. Drop the converters and you get the radial 2888 W; the SOP's value only appears once you let the OPF choose the transfer.
Droop control: shared DC-voltage regulation
The case above uses one V-master (dc_control = "V") — fine when a single converter can hold the DC voltage. Real MVDC zones usually prefer V–P droop, where several converters share the DC-voltage regulation, so no single converter is a single point of failure (4, 5). Droop is a control law, not free dispatch: each converter's AC-side active power follows its own P–V characteristic,
\[p_g = \texttt{dc\_p\_ref} + \frac{v_{dc} - \texttt{dc\_v\_set}}{\texttt{dc\_droop}},\]
saturated at the converter's power limits. Sign conventions matter here: pg is the AC-side injection, so the power a converter delivers into the DC link is −pg — with dc_p_ref = 0, that is (dc_v_set − v_dc)/dc_droop, positive whenever the DC voltage has drooped below the set-point.
Put two droop converters (vA, vB) on one dc_bus that jointly feed a DC load, both with dc_v_set = 1500, dc_p_ref = 0. The listing is complete — note the DC voltage window [1200, 1800] V is deliberately wide enough for the heaviest load below:
droop_case(kA, kB, P_load; smaxA = 60000.0, smaxB = 60000.0) = parse_bmopf("""
{"bus":{
"sub":{"terminal_names":["1","n"],"perfectly_grounded_terminals":["n"]},
"a":{"terminal_names":["1","n"],"perfectly_grounded_terminals":["n"],"v_min":[600.0],"v_max":[1100.0]}},
"linecode":{"lc":{"R_series_1_1":1.0,"X_series_1_1":0.0}},
"line":{
"La":{"bus_from":"sub","bus_to":"a","terminal_map_from":["1"],"terminal_map_to":["1"],"length":1.0,"linecode":"lc"}},
"voltage_source":{
"s":{"bus":"sub","terminal_map":["1","n"],"v_magnitude":[1000.0,0.0],"v_angle":[0.0,0.0],"cost":[1.0]}},
"ibr":{
"vA":{"bus":"a","terminal_map":["1","n"],"topology":"SINGLE_PHASE","prime_mover":"GENERIC","s_max":[$smaxA],
"dc_bus":"dc","dc_terminal_map":["p","m"],
"dc_control":"droop","dc_v_set":1500.0,"dc_p_ref":0.0,"dc_droop":$kA},
"vB":{"bus":"a","terminal_map":["1","n"],"topology":"SINGLE_PHASE","prime_mover":"GENERIC","s_max":[$smaxB],
"dc_bus":"dc","dc_terminal_map":["p","m"],
"dc_control":"droop","dc_v_set":1500.0,"dc_p_ref":0.0,"dc_droop":$kB}},
"dc_bus":{
"dc":{"terminal_names":["p","m"],"pole":{"p":"POSITIVE","m":"METALLIC_RETURN"},
"v_dc_min":[1200.0,0.0],"v_dc_max":[1800.0,0.0]}},
"dc_load":{"dl":{"dc_bus":"dc","terminal_map":["p","m"],"p":$P_load}},
"dc_grounding":{"g":{"dc_bus":"dc","terminal":"m"}}}
"""; from_string=true)Solving gives the textbook droop behaviour — and it is the OPF that finds the self-consistent (v_dc, P_A, P_B), not a hand-iterated controller model. We tabulate the power each converter delivers into the DC link (−pg):
println(rpad("kA", 9), rpad("kB", 9), rpad("P_load", 9),
rpad("P_A→dc", 10), rpad("P_B→dc", 10), "v_dc")
for (kA, kB, P) in ((0.005, 0.005, 10e3), (0.0025, 0.005, 10e3), (0.005, 0.005, 60e3))
r = solve_opf(droop_case(kA, kB, P); optimizer = OPT)
PA = -r["ibr"]["vA"]["1"]["pg"]
PB = -r["ibr"]["vB"]["1"]["pg"]
println(rpad(kA, 9), rpad(kB, 9), rpad("$(P/1e3) kW", 9),
rpad("$(round(PA/1e3, digits=2)) kW", 10),
rpad("$(round(PB/1e3, digits=2)) kW", 10),
round(r["dc_bus"]["dc"]["p"]["v_dc"], digits=1), " V")
endkA kB P_load P_A→dc P_B→dc v_dc
0.005 0.005 10.0 kW 5.0 kW 5.0 kW 1475.0 V
0.0025 0.005 10.0 kW 6.67 kW 3.33 kW 1483.3 V
0.005 0.005 60.0 kW 30.0 kW 30.0 kW 1350.0 VThree things to read off:
- The DC voltage droops below the 1500 V reference as load rises (1475 → 1350 V) — that voltage deviation is the shared signal that tells every converter how much to contribute: with
dc_p_ref = 0, each delivers(dc_v_set − v_dc)/dc_droopinto the link. - Power shares inversely to the droop coefficient. Halving
kA(a stiffer converter) makesvAcarry twicevB(6.67 vs 3.33 kW) at the same sharedv_dc. - Saturation is built in. Drive a converter past its headroom and its power clamps at
s_maxinstead of following the line to infinity — the curve is a saturated piecewise-linear P–V characteristic, encoded with the same smooth (softplus) machinery as the Volt-watt/Volt-var droops so it stays well-behaved for Ipopt.
The saturation point deserves a demonstration. Shrink vA to a 10 kVA nameplate and load the link with 30 kW — unsaturated droop would split it 15/15 kW at 1425 V, but vA can no longer follow its line:
r = solve_opf(droop_case(0.005, 0.005, 30e3; smaxA = 10000.0); optimizer = OPT)
println("P_A→dc : ", round(-r["ibr"]["vA"]["1"]["pg"]/1e3, digits=2), " kW (clamped at s_max)")
println("P_B→dc : ", round(-r["ibr"]["vB"]["1"]["pg"]/1e3, digits=2), " kW")
println("v_dc : ", round(r["dc_bus"]["dc"]["p"]["v_dc"], digits=1), " V")P_A→dc : 10.0 kW (clamped at s_max)
P_B→dc : 20.0 kW
v_dc : 1400.0 VvA clamps at its 10 kW limit; vB — still on its droop line — must pick up the remaining 20 kW, which requires the common DC voltage to droop all the way to 1500 − 20000 × 0.005 = 1400 V. The saturated converter no longer participates in voltage regulation; the burden shifts entirely to the ones with headroom.
Every DC zone needs at least one "V" or "droop" converter; an all-"P" zone leaves v_dc undetermined and is flagged E.INT.DC_NO_VOLTAGE_CONTROL.
MVDC feeder: a resistive DC tie
The back-to-back SOP above is a point device: both converters share one dc_bus, so the DC link is electrically a single node and its only loss is in the converters. Pull the two converters apart onto separate DC buses (dcA, dcB) and join them with a dc_branch carrying a real pole resistance, and the tie becomes a distributed MVDC feeder: the pole conductor now has its own I²R loss and a pole-voltage drop I·R, exactly like an AC line. This is the same D2/D4 physics the test-suite pins to Ohm's law.
The far feeder b now carries the whole 40 kW load; there is no AC path to it, so all of that power must cross the MVDC feeder. vscA is the DC-voltage master (pins dcA at 1500 V); vscB floats and delivers into feeder b. The pole resistance is a deliberately stiff 5 Ω so the drop is clearly visible:
const R_POLE = 5.0 # Ω, DC pole conductor
feeder_tie(; v_dc_min_far = 1200.0) = parse_bmopf("""
{"bus":{
"sub":{"terminal_names":["1","n"],"perfectly_grounded_terminals":["n"]},
"a":{"terminal_names":["1","n"],"perfectly_grounded_terminals":["n"],"v_min":[600.0],"v_max":[1100.0]},
"b":{"terminal_names":["1","n"],"perfectly_grounded_terminals":["n"],"v_min":[600.0],"v_max":[1100.0]}},
"linecode":{"lc":{"R_series_1_1":1.0,"X_series_1_1":0.0}},
"line":{
"La":{"bus_from":"sub","bus_to":"a","terminal_map_from":["1"],"terminal_map_to":["1"],"length":1.0,"linecode":"lc"}},
"voltage_source":{
"s":{"bus":"sub","terminal_map":["1","n"],"v_magnitude":[1000.0,0.0],"v_angle":[0.0,0.0],"cost":[1.0]}},
"load":{
"Lb_load":{"bus":"b","terminal_map":["1","n"],"configuration":"SINGLE_PHASE","p_nom":[40000.0],"q_nom":[0.0]}},
"ibr":{
"vscA":{"bus":"a","terminal_map":["1","n"],"topology":"SINGLE_PHASE","prime_mover":"GENERIC","s_max":[60000.0],
"dc_bus":"dcA","dc_terminal_map":["p","m"],"dc_control":"V","dc_v_set":1500.0},
"vscB":{"bus":"b","terminal_map":["1","n"],"topology":"SINGLE_PHASE","prime_mover":"GENERIC","s_max":[60000.0],
"dc_bus":"dcB","dc_terminal_map":["p","m"]}},
"dc_bus":{
"dcA":{"terminal_names":["p","m"],"pole":{"p":"POSITIVE","m":"METALLIC_RETURN"},
"v_dc_min":[1400.0,0.0],"v_dc_max":[1800.0,0.0]},
"dcB":{"terminal_names":["p","m"],"pole":{"p":"POSITIVE","m":"METALLIC_RETURN"},
"v_dc_min":[$v_dc_min_far,0.0],"v_dc_max":[1800.0,0.0]}},
"dc_branch":{
"tie":{"dc_bus_from":"dcA","dc_bus_to":"dcB","terminal_map_from":["p","m"],"terminal_map_to":["p","m"],"r":[$R_POLE,0.0]}},
"dc_grounding":{"g":{"dc_bus":"dcA","terminal":"m","r":0.0}}}
"""; from_string=true)
res = solve_opf(feeder_tie(); optimizer = OPT)
pA = res["ibr"]["vscA"]["1"]["pg"]
pB = res["ibr"]["vscB"]["1"]["pg"]
i = res["dc_branch"]["tie"]["1"]["i_dc"]
vA = res["dc_bus"]["dcA"]["p"]["v_dc"]
vB = res["dc_bus"]["dcB"]["p"]["v_dc"]
println("status : ", res["termination_status"])
println("vscA pg : ", round(pA, digits=0), " W (master, draws from feeder a into the DC tie)")
println("vscB pg : ", round(pB, digits=0), " W (delivers into feeder b's load)")
println("i_dc : ", round(i, digits=1), " A")
println("v_dc A : ", round(vA, digits=1), " V (master setpoint)")
println("v_dc B : ", round(vB, digits=1), " V (drooped by the pole drop)")status : LOCALLY_SOLVED
vscA pg : -44376.0 W (master, draws from feeder a into the DC tie)
vscB pg : 40000.0 W (delivers into feeder b's load)
i_dc : 29.6 A
v_dc A : 1500.0 V (master setpoint)
v_dc B : 1352.1 V (drooped by the pole drop)Read the physics straight off the solution — and both identities hold to the digit, because the OPF solves the DC network to the same Ohm's-law equations:
println("pole-voltage drop : ", round(vA - vB, digits=1), " V vs i·R = ", round(i * R_POLE, digits=1), " V")
println("DC feeder loss : ", round(-pA - pB, digits=1), " W vs i²·R = ", round(i^2 * R_POLE, digits=1), " W")pole-voltage drop : 147.9 V vs i·R = 147.9 V
DC feeder loss : 4376.1 W vs i²·R = 4376.1 WvscA draws 44.4 kW off feeder a to serve a 40 kW load on feeder b: the 4.4 kW gap is precisely the i²·R = 29.6² × 5 ≈ 4376 W dissipated in the pole conductor. The pole voltage sags from the master's 1500 V at dcA to 1352 V at dcB, a 148 V ≈ i·R drop — the DC analogue of an AC feeder's voltage droop. Unlike the shared-bus SOP, whose only loss was in the converters, this MVDC feeder carries a real, quantified line loss that the optimiser now trades against the AC-side losses when it chooses the transfer.
Remember to widen the far bus. The pole drop pushes dcB down to 1352 V, below the master's 1400 V floor — so if dcB inherits dcA's [1400, 1800] window, this operating point is excluded and the local solve reports infeasibility:
r_narrow = solve_opf(feeder_tie(; v_dc_min_far = 1400.0); optimizer = OPT)
println("narrow far-bus bound → ", r_narrow["termination_status"])narrow far-bus bound → LOCALLY_INFEASIBLEThe lower v_dc_min on dcB is not a solver hint; it is the physical headroom the resistive feeder needs to develop its drop. (This is also why pinning both metallic returns to exactly 0 V while the return conductor has resistance would force the tie current to zero — grounding only dcA's return leaves dcB's free to float down by i·R.)
Extending the story
- Loss vs hosting capacity. Swap a load for DER on one feeder and the same formulation finds how much extra generation the feeders can host before a voltage or thermal limit binds — the SOP redistributing the surplus (3).
- A real DC line. Putting the two converters on separate
dc_busnodes joined by a resistivedc_branchturns the tie into an MVDC feeder with its ownI²Rloss and anI·Rpole-voltage drop — worked end-to-end in MVDC feeder: a resistive DC tie above. - Bipolar / DC loads. Use 3-wire
dc_bus/dc_branchfor bipolar feeders, and adddc_load/dc_sourcefor DC-connected demand or DC-coupled PV/storage.
Validating
The static validators catch DC data problems before the solve — dangling dc_bus references, ungrounded DC islands, bad bus arity, and a converter whose AC bus is energised only through the DC link (W.INT.DC_FED_AC_ISLAND, the "MVDC not embedded in a referenced AC system" check). Here is the voltage-control check firing: demote the V-master to plain power control and no converter regulates the DC voltage.
bad = sop_case()
bad["ibr"]["vA"]["dc_control"] = "P" # no V-master or droop left in the zone
rep = analyze(bad)
for f in rep.findings
if f.code == "E.INT.DC_NO_VOLTAGE_CONTROL"
println(f.severity, " ", f.code, "\n ", f.message)
end
endERROR E.INT.DC_NO_VOLTAGE_CONTROL
DC island of 1 dc_bus(es) has no converter on DC-voltage control (dc_control = "V" or "droop") — the DC operating voltage is underdetermined. Designate a master/droop converter: dc.solution_check adds post-solve DC checks (E.SOL.DC_VOLT_VIOLATION, E.SOL.DC_THERMAL_VIOLATION). See the finding-code reference.
References
- W. Cao, J. Wu, N. Jenkins, Feeder load balancing in MV distribution networks using soft normally-open points, IEEE PES Innovative Smart Grid Technologies Europe (ISGT-Europe), 2014.
- X. Jiang, Y. Zhou, W. Ming, P. Yang, J. Wu, An Overview of Soft Open Points in Electricity Distribution Networks, IEEE Transactions on Smart Grid 13(3):1899–1910, 2022.
- R. de Oliveira, L. W. de Oliveira, E. J. de Oliveira, Optimization Approach for Planning Soft Open Points in a MV-Distribution System to Maximize the Hosting Capacity, Energies 16(3):1035, 2023, doi:10.3390/en16031035.
- F. Torres, S. Martinez, C. Roa, E. Lopez, Comparison Between Voltage Droop and Voltage Margin Controllers for MTDC Systems, IEEE ICA-ACCA, Concepción, Chile, 2018.
- K. Rouzbehi, A. Miranian, A. Luna, P. Rodriguez, A Generalized Voltage Droop Strategy for Control of Multiterminal DC Grids, IEEE Transactions on Industry Applications 51(1):607–618, 2015.