D-STATCOM unbalance study

Reactive support vs. active power circulation on an unbalanced LV feeder.

This case study follows one unbalanced low-voltage (LV) feeder to show what an inverter-interfaced shunt compensator — a STATCOM, or D-STATCOM in distribution-system parlance — can and cannot do about voltage unbalance, and why the answer is an optimisation problem rather than a simulation one. Every code block runs when the docs are built, so the numbers below are real.

Prerequisites: a Julia environment with BMOPFTools, JuMP and Ipopt installed, and familiarity with the JSON input format — see the end-to-end tutorial first.

Why unbalance is hard to fix with reactive power

LV feeders are unbalanced by construction: single-phase loads (and single-phase rooftop PV) are spread unevenly across the three phases, so the phase carrying the most load sags while the lightly-loaded phases stay high (1). The instinct from transmission engineering is to throw reactive power at the problem — a STATCOM is, after all, a shunt var source, and reactive management with STATCOMs is an active research thread for four-wire LV feeders (2).

The instinct misfires in LV, and the reason is the network impedance. Voltage magnitude responds to a power injection roughly as $\Delta|V| \approx (R\,\Delta P + X\,\Delta Q)/|V|$. On the transmission grid $X \gg R$ so reactive power is the lever; on an LV feeder the cables are resistive, $X/R \approx 0.2\!-\!0.5$, so it is active power that moves the voltage (1). A var source therefore has weak authority over the very phase that needs help.

A four-wire converter has a second lever that a capacitor bank or an SVC does not: its three phase legs share one DC link. With no energy source behind that link, the converter can still circulate active power between phases — sourcing real power on the heavily-loaded phase and sinking it on the lightly-loaded ones — while the net active power stays at zero. This is exactly the DC-link power-balance constraint $\sum_k P_k = 0$ formalised for four-wire OPF in 3 and 4. It is what lets a STATCOM balance a feeder rather than merely support its average voltage.

In BMOPFTools a STATCOM is modelled as an IBR (see the OPF model); the two control philosophies are a single keyword on add_statcom!:

• reactive-only (default) — each phase's active power is clamped to zero;
• active power circulation (dc_link_coupled = true) — per-phase active power is free within the converter's apparent-power circle, coupled by $\sum_k P_k = 0$.

The rest of this page shows, on one feeder, that the first barely moves the needle and the second balances the feeder outright — and that finding the per-phase $(P,Q)$ split is intrinsically an optimisation, not a setpoint a power-flow simulation could have guessed.

1. An unbalanced four-wire LV feeder

A 230 V three-phase source feeds a short, resistive ($R/X = 5$) four-wire cable to a single load bus whose phase 1 carries six times the load of phases 2 and 3 — a deliberately stark single-phase concentration.

using BMOPFTools, JuMP, Ipopt
const OPT = optimizer_with_attributes(Ipopt.Optimizer, "print_level" => 0)

feeder() = parse_bmopf("""
{"bus":{
    "src":{"terminal_names":["1","2","3","n"],"perfectly_grounded_terminals":["n"]},
    "b1": {"terminal_names":["1","2","3","n"],"perfectly_grounded_terminals":["n"]}},
 "voltage_source":{"vs":{"bus":"src","terminal_map":["1","2","3"],
     "v_magnitude":[230.0,230.0,230.0],"v_angle":[0.0,-2.0944,2.0944]}},
 "linecode":{"lc":{"R_series_1_1":0.4,"X_series_1_1":0.08,
                   "R_series_2_2":0.4,"X_series_2_2":0.08,
                   "R_series_3_3":0.4,"X_series_3_3":0.08,
                   "R_series_4_4":0.4,"X_series_4_4":0.08}},
 "line":{"l1":{"bus_from":"src","bus_to":"b1",
     "terminal_map_from":["1","2","3","n"],"terminal_map_to":["1","2","3","n"],
     "linecode":"lc","length":1.0}},
 "load":{"ld":{"bus":"b1","terminal_map":["1","2","3","n"],"configuration":"WYE",
     "p_nom":[18000.0,3000.0,3000.0],"q_nom":[2000.0,500.0,500.0]}}}
"""; from_string=true)

The headline metric is the voltage unbalance factor (VUF) — the ratio of the negative- to positive-sequence voltage, the standard measure of unbalance — together with the per-phase voltage spread and the feeder's active losses.

solved(r) = r["termination_status"] in ("LOCALLY_SOLVED","OPTIMAL","ALMOST_LOCALLY_SOLVED")

function report(label, r, bus="b1")
    if !solved(r)
        println(rpad(label, 34), r["termination_status"]); return
    end
    b = r["bus"][bus]
    V = [b[t]["vr"] + im*b[t]["vi"] for t in ("1","2","3")]
    a  = exp(im*2pi/3)
    V1 = (V[1] + a*V[2] + a^2*V[3]) / 3
    V2 = (V[1] + a^2*V[2] + a*V[3]) / 3
    println(rpad(label, 34),
            "VUF = ", rpad(round(abs(V2)/abs(V1)*100, digits=2), 5), " %   ",
            "phase V = ", round.(abs.(V), digits=1), " V   ",
            "loss = ", round(r["losses"]["p_loss"], digits=0), " W")
end

With no compensation the feeder is badly unbalanced — phase 1 sags well below the lightly-loaded phases.

r0 = solve_opf(feeder(); optimizer = OPT)
report("no STATCOM", r0)

no STATCOM                        VUF = 5.16  %   phase V = [191.6, 224.5, 224.5] V   loss = 3723.0 W

2. The simulation reflex: a fixed reactive setpoint

What would an engineer running a power-flow simulation do? Pick a STATCOM, choose a sensible reactive setpoint — say a symmetric +5 kVAr per phase — and simulate the result with solve_pf. We fix the device (p = 0, q pinned) and let the power flow report the operating point.

sim = feeder()
add_statcom!(sim, "b1"; s_max = 15_000.0)              # reactive-only nameplate
si = sim["ibr"]["statcom_b1"]
si["p_min"] = [0.0,0.0,0.0];      si["p_max"] = [0.0,0.0,0.0]
si["q_min"] = [5000.0,5000.0,5000.0]; si["q_max"] = [5000.0,5000.0,5000.0]

report("fixed +5 kVAr/phase (solve_pf)", solve_pf(sim; optimizer = OPT))

fixed +5 kVAr/phase (solve_pf)    VUF = 5.07  %   phase V = [193.7, 226.1, 226.1] V   loss = 4009.0 W

The unbalance hardly budges. The reactive injection lifts all three phase voltages together — it cannot preferentially raise the sagging phase, because on a resistive feeder reactive power is the wrong lever (1). Worse, the guess is actively counterproductive: the extra reactive current raises the feeder losses from 3723 W to 4009 W. A different constant setpoint would land somewhere else equally arbitrary; a single fixed-setpoint power flow can only evaluate a guess — finding the right per-phase split needs a search over setpoints, i.e. the optimisation.

3. Reactive-only, optimally dispatched — still not enough

Give the device its full reactive freedom and let solve_opf choose the per-phase vars optimally, but keep it reactive-only. To make the limit concrete we now impose a lower voltage bound on the load bus and ask whether reactive support can hold it. The AS IEC 60038:2022 supply floor is $0.90 \times 230 = 207\,V$; we require 209 V — the floor plus a small operating margin.

bounded() = (n = feeder();
    n["bus"]["b1"]["v_min"] = [209.0, 209.0, 209.0];
    n["bus"]["b1"]["v_max"] = [253.0, 253.0, 253.0]; n)

for s in (15_000.0, 30_000.0, 60_000.0)
    n = bounded(); add_statcom!(n, "b1"; s_max = s); n, _ = augment_case(n)
    report("reactive-only, s_max = $(Int(s/1000)) kVA", solve_opf(n; optimizer = OPT))
end

reactive-only, s_max = 15 kVA     LOCALLY_INFEASIBLE
reactive-only, s_max = 30 kVA     LOCALLY_INFEASIBLE
reactive-only, s_max = 60 kVA     LOCALLY_INFEASIBLE

The reactive-only STATCOM cannot hold the heavy phase above its limit even at 60 kVA — twice the nameplate that will balance the feeder outright in §4, and more than twice the feeder's total load. More vars cannot substitute for the active power the phase is short of.

4. Active power circulation balances the feeder

Now flip the one keyword — dc_link_coupled = true — so the converter may circulate active power between phases under $\sum_k P_k = 0$. augment_case opens the per-phase active range to $\pm s_{\max}$ and pins the net to zero; the per-phase apparent-power circle still bounds each leg.

n = bounded()
add_statcom!(n, "b1"; s_max = 30_000.0, dc_link_coupled = true)
n, _ = augment_case(n)
inv = n["ibr"]["statcom_b1"]
println("per-phase P range : ", inv["p_min"], " … ", inv["p_max"], " W")
println("net DC-link bound : [", inv["p_dc_min"], ", ", inv["p_dc_max"], "] W")

ract = solve_opf(n; optimizer = OPT)
report("active circulation, 30 kVA", ract)

per-phase P range : [-30000.0, -30000.0, -30000.0] … [30000.0, 30000.0, 30000.0] W
net DC-link bound : [0.0, 0.0] W
active circulation, 30 kVA        VUF = 0.0   %   phase V = [215.1, 215.1, 215.1] V   loss = 1660.0 W

The same nameplate that was infeasible with reactive-only control now holds every phase inside the window. Looking at what the converter does makes the mechanism explicit: it sources real power on the heavy phase and sinks it on the light phases, summing to zero — a pure inter-phase transfer drawing no net energy.

ph = ract["ibr"]["statcom_b1"]
for t in ("1","2","3")
    println("phase $t :  P = ", rpad(round(ph[t]["pg"], digits=0), 8), " W   ",
                          "Q = ", round(ph[t]["qg"], digits=0), " var")
end
println("Σ P over phases : ", round(sum(ph[t]["pg"] for t in ("1","2","3")), digits=2), " W")

phase 1 :  P = 10000.0  W   Q = 2111.0 var
phase 2 :  P = -5000.0  W   Q = 611.0 var
phase 3 :  P = -5000.0  W   Q = 611.0 var
Σ P over phases : 0.01 W

Run on the unconstrained feeder (no voltage limits), the optimiser drives the unbalance essentially to zero and, by flattening the phase currents, cuts the feeder losses, since balanced currents minimise the resistive $\sum R|I|^2$:

n = feeder(); add_statcom!(n, "b1"; s_max = 30_000.0, dc_link_coupled = true)
n, _ = augment_case(n)
report("no STATCOM (repeat)", r0)
report("active circulation", solve_opf(n; optimizer = OPT))

no STATCOM (repeat)               VUF = 5.16  %   phase V = [191.6, 224.5, 224.5] V   loss = 3723.0 W
active circulation                VUF = 0.0   %   phase V = [215.1, 215.1, 215.1] V   loss = 1660.0 W

5. A physically faithful current limit

The apparent-power circle $P_k^2 + Q_k^2 \le s_{\max}^2$ is the constant-MVA idealisation. A real converter is limited by its current, and $|S_k| = |\Delta V_k|\,|I_k|$, so its capability shrinks as the terminal voltage sags. Supply an optional per-conductor i_max and the model captures that rolloff (3); the balancing authority is then bounded by amps, not VA. i_max is per conductor — one entry per phase plus a final entry for the neutral, which on a four-wire converter doing unbalance compensation can carry more current than any phase. On the unconstrained feeder of §4 — where active circulation alone drove the VUF to zero — adding a 40 A per-phase current cap (here with a generously-rated neutral, so the phase limit is what binds) leaves a small residual unbalance, the visible signature of the var/active rolloff under load. Printing the per-conductor current magnitudes makes "bounded by amps" concrete — the heavy-phase leg sits at its 40 A cap:

n = feeder(); add_statcom!(n, "b1"; s_max = 30_000.0, dc_link_coupled = true)
n["ibr"]["statcom_b1"]["i_max"] = [40.0, 40.0, 40.0, 120.0]   # A per conductor: a,b,c,n
n, _ = augment_case(n)
r5 = solve_opf(n; optimizer = OPT)
report("active circulation + i_max = 40 A", r5)
st = r5["ibr"]["statcom_b1"]
I  = [st[t]["cri"] + im*st[t]["cii"] for t in ("1", "2", "3")]
for (t, i) in zip(("1", "2", "3"), I)
    println("phase $t   : |I| = ", round(abs(i), digits=1), " A")
end
println("neutral   : |I| = ", round(abs(-sum(I)), digits=1), " A")

active circulation + i_max = 40 A VUF = 0.82  %   phase V = [211.5, 216.8, 216.8] V   loss = 1713.0 W
phase 1   : |I| = 40.0 A
phase 2   : |I| = 19.4 A
phase 3   : |I| = 19.4 A
neutral   : |I| = 58.7 A

Why optimisation matters

The lesson is not that a STATCOM is powerful — it is that which degree of freedom you give it, and how you dispatch it, decides everything, and both questions are optimisation questions. A power-flow simulation can score a fixed per-phase $(P, Q)$ setpoint, but the setpoint that balances the feeder lives on a constraint surface — the coupled DC-link balance $\sum_k P_k = 0$ intersected with three per-phase apparent-power circles — that no heuristic rule traces. Finding it is what the OPF does, and it is the difference between a compensator that does almost nothing and one that balances the feeder outright.

References

1. D. Pullaguram, S. Mishra, N. Senroy, Coordinated single-phase control scheme for voltage unbalance reduction in low voltage network, Philosophical Transactions of the Royal Society A 375(2100):20160308, 2017.
2. O. Rahman, D. Robinson, S. Elphick, Mitigation of Solar PV Impact in Four-Wire LV Radial Distribution Feeders Through Reactive Power Management Using STATCOMs, Electronics 14(15):3063, 2025.
3. R. Heidari, F. Geth, Improved Algebraic Inverter Modelling for Four-Wire Power Flow Optimization, arXiv:2403.07285, 2024.
4. M. Deakin, R. Heidari, X. Deng, Power Converter DC Link Ripple and Network Unbalance as Active Constraints in Distribution System Optimal Power Flow, arXiv:2512.18293, 2025.