Transformer tap optimisation

Choosing on-load tap-changer (OLTC) and regulator settings as a continuous decision variable — and why it is an optimisation, not a guess.

Most distribution transformers and every step-voltage regulator carry a tap: a small, switchable change to the turns ratio that trades primary against secondary voltage. Picking the tap is a control decision — too low and the feeder end under-voltages, too high and it over-voltages or wastes capacity. Classically a regulator picks its tap from a local voltage measurement; but the best tap for the whole feeder (lowest losses, all buses in band) depends on the network and the load, so it is naturally a decision the OPF should make.

This page makes the tap a free continuous variable and lets solve_opf choose it. Every code block runs when the docs are built, so the numbers are real.

Prerequisites: a Julia environment with BMOPFTools, JuMP and Ipopt (Pkg.add(["JuMP", "Ipopt"])), and basic familiarity with the JSON input format from the end-to-end tutorial. The result-dictionary layout used below is documented in OPF result dictionary. The VVWO tutorial covers the complementary voltage-control lever — IBR droop — on a real LV feeder.

The model

A transformer tap is exposed with the same implicit free-variable pattern as generators and IBRs: bounds make it optimisable.

• Ordinary transformers (single_phase, center_tap, delta_wye, wye_delta) take a dimensionless multiplier tap on the nominal from-side ratio $N_0 = v_{\text{nom,from}}/v_{\text{nom,to}}$ (the v_nom_from/v_nom_to fields), with tap_min/tap_max. For center_tap this taps the HV winding, so both LV legs scale together.
• Regulator subtypes (single_phase_autotransformer, open_delta_regulator) make their native tap_ratio free with tap_ratio_min/tap_ratio_max.

n_winding is the one subtype without tap support: its ratio is always held at the nominal turns ratio. Supplying a tap field on an n_winding transformer or its windings raises ArgumentError at native OPF-build time. Use a supported two-bus subtype when tap regulation is required.

If tap_min < tap_max the tap becomes a decision variable; otherwise it is fixed (so existing data is unchanged). Internally the solved tap enters the winding constraints as the effective ratio $N = N_0\cdot t$. The from-referred leakage of an OLTC scales with the winding turns, i.e. with $t^2$; written on the to side the leakage is constant and — using the ideal-core coupling $N\,I_{\text{series}} = -I_{\text{to}}$ — the voltage drop stays quadratic in the tap variable, so no solver change is needed (the OPF is already an Ipopt NLP). At $t = 1$ the model is identical to the fixed-tap transformer, so the tap variable adds no error.

using BMOPFTools, JuMP, Ipopt
const OPT = optimizer_with_attributes(Ipopt.Optimizer, "print_level" => 0)
solved(r) = r["termination_status"] in ("LOCALLY_SOLVED","OPTIMAL","ALMOST_LOCALLY_SOLVED")

1. A single-phase feeder that needs its tap

An 11 kV / 240 V, 50 kVA single-phase transformer feeds a heavy 40 kW + 13 kVAr load. The source has a unit cost, so minimising it minimises imported power — i.e. losses.

zbf = 11_000.0^2 / 50_000.0
zbt =    240.0^2 / 50_000.0
feeder(tapfields, vbounds) = parse_bmopf("""
{"bus":{
   "hv":{"terminal_names":["1","n"],"perfectly_grounded_terminals":["n"]},
   "lv":{"terminal_names":["1","n"],"perfectly_grounded_terminals":["n"]$vbounds}},
 "voltage_source":{"src":{"bus":"hv","terminal_map":["1"],
     "v_magnitude":[11000.0],"v_angle":[0.0],"cost":[1.0]}},
 "load":{"ld":{"bus":"lv","terminal_map":["1","n"],
     "configuration":"SINGLE_PHASE","p_nom":[40000.0],"q_nom":[13000.0]}},
 "transformer":{"single_phase":{"t1":{
     "bus_from":"hv","bus_to":"lv",
     "terminal_map_from":["1","n"],"terminal_map_to":["1","n"],
     "v_nom_from":11000.0,"v_nom_to":240.0,"s_rating":50000.0,
     "r_series_from":$(0.01*zbf),"x_series_from":$(0.04*zbf),
     "r_series_to":$(0.01*zbt),"x_series_to":0.0$tapfields}}}}
"""; from_string=true)

vlv(r) = round(r["bus"]["lv"]["1"]["vm"], digits=2)
ploss(r) = round(r["losses"]["p_loss"], digits=1)

At the fixed nominal tap the secondary sags well below a 236 V floor:

r0 = solve_opf(feeder("", ""); optimizer = OPT)
println("fixed tap = 1.0 :  |V_lv| = ", vlv(r0), " V,  losses = ", ploss(r0), " W")

fixed tap = 1.0 :  |V_lv| = 233.39 V,  losses = 748.2 W

A 236 V lower limit is therefore infeasible at the nominal tap — there is no dispatch lever, the load and source are fixed:

r_lim = solve_opf(feeder("", ""","v_min":[236.0],"v_max":[246.0]"""); optimizer = OPT)
println("fixed tap, [236,246] V band  → ", r_lim["termination_status"])
@assert !solved(r_lim)

fixed tap, [236,246] V band  → LOCALLY_INFEASIBLE

Now free the tap over $[0.9, 1.1]$ and re-solve. The optimiser lowers the tap (a lower from-side ratio raises the secondary), pulling the feeder back into band and cutting losses at the same time:

free = ""","tap":1.0,"tap_min":0.9,"tap_max":1.1"""
rf = solve_opf(feeder(free, ""","v_min":[236.0],"v_max":[246.0]"""); optimizer = OPT)
t★ = rf["transformer"]["t1"]["tap"]
println("free tap  →  status = ", rf["termination_status"])
println("  optimal tap   = ", round(t★, digits=4), "   (binding bound: ",
        rf["transformer"]["t1"]["tap_binding"], ")")
println("  |V_lv|        = ", vlv(rf), " V")
println("  losses        = ", ploss(rf), " W   (was ", ploss(r0), " W at tap 1.0)")

free tap  →  status = LOCALLY_SOLVED
  optimal tap   = 0.9514   (binding bound: false)
  |V_lv|        = 246.0 V
  losses        = 673.5 W   (was 748.2 W at tap 1.0)

The single number tap is the whole story: an infeasible feeder becomes feasible, losses fall, and the result reports whether the tap bound is binding — a binding tap means the regulation range is exhausted and another lever (a second regulator, DER, reconductoring) is needed.

2. The tap is load-dependent: a tap schedule

Because the right tap depends on loading, sweeping the load traces the tap an OLTC should follow — an optimised tap schedule:

for mult in (0.4, 0.7, 1.0, 1.3)
    net = feeder(free, ""","v_min":[232.0],"v_max":[250.0]""")
    net["load"]["ld"]["p_nom"] = [40000.0 * mult]
    net["load"]["ld"]["q_nom"] = [13000.0 * mult]
    r = solve_opf(net; optimizer = OPT)
    println("load ×", mult, "  →  tap = ", round(r["transformer"]["t1"]["tap"], digits=4),
            ",  |V_lv| = ", vlv(r), " V")
end

load ×0.4  →  tap = 0.9507,  |V_lv| = 250.0 V
load ×0.7  →  tap = 0.9438,  |V_lv| = 250.0 V
load ×1.0  →  tap = 0.9369,  |V_lv| = 250.0 V
load ×1.3  →  tap = 0.9323,  |V_lv| = 250.0 V

Loss minimisation holds the LV bus at the top of its band; the tap is lowered further as load grows (a deeper boost — recall a lower multiplier raises the secondary) to keep it there. This tap-vs-load curve is the optimised schedule a line-drop compensator approximates, here derived directly from the network optimum.

3. Continuous vs. a real discrete regulator

A physical regulator moves in discrete steps (a 32-step ±10 % regulator has 0.625 %/step). The continuous optimum is the relaxation; rounding to the nearest step recovers an implementable setting and shows how little is lost:

step = 0.10 / 16                      # ±10 % over 32 steps
t_disc = 1 + round((t★ - 1) / step) * step
rd = solve_opf(feeder(""","tap":$t_disc""", ""); optimizer = OPT)   # tap fixed at the step
println("continuous tap = ", round(t★, digits=4), " (|V_lv| = ", vlv(rf), " V)")
println("nearest step   = ", round(t_disc, digits=4), " (|V_lv| = ", vlv(rd), " V)")

continuous tap = 0.9514 (|V_lv| = 246.0 V)
nearest step   = 0.95 (|V_lv| = 246.38 V)

Rounding to the nearest physical step lands a few tenths of a volt off the continuous optimum — the small, quantifiable cost of discretisation. The continuous relaxation is both the achievable bound and an excellent guide to the step to dial in.

4. Three-phase Dy

The same one-keyword change works for a three-phase delta-wye transformer. Here a 500 kVA, 11 kV(Δ) / 415 V(Y) unit feeds a balanced load; freeing the tap holds the LV bus in a tight band. (The delta source is set per phase-to-ground, i.e. $11\,\text{kV}/\sqrt3 = 6350.85$ V; the wye secondary is reported phase-to-neutral.)

zf = 11_000.0^2 / 500_000.0
zt =    415.0^2 / 500_000.0
dy(tapfields, vb) = parse_bmopf("""
{"bus":{
   "hv":{"terminal_names":["1","2","3"]},
   "lv":{"terminal_names":["1","2","3","n"],"perfectly_grounded_terminals":["n"]$vb}},
 "voltage_source":{"src":{"bus":"hv","terminal_map":["1","2","3"],
     "v_magnitude":[6350.85,6350.85,6350.85],
     "v_angle":[0.0,-2.0944,2.0944],"cost":[1.0,1.0,1.0]}},
 "load":{"ld":{"bus":"lv","terminal_map":["1","2","3","n"],"configuration":"WYE",
     "p_nom":[110000.0,110000.0,110000.0],"q_nom":[35000.0,35000.0,35000.0]}},
 "transformer":{"delta_wye":{"t1":{
     "bus_from":"hv","bus_to":"lv",
     "terminal_map_from":["1","2","3"],"terminal_map_to":["1","2","3","n"],
     "v_nom_from":11000.0,"v_nom_to":415.0,"s_rating":500000.0,
     "r_series_from":$(0.01*zf),"x_series_from":$(0.02*zf),
     "r_series_to":$(0.01*zt),"x_series_to":$(0.02*zt)$tapfields}}}}
"""; from_string=true)

rdy = solve_opf(dy(""","tap":1.0,"tap_min":0.95,"tap_max":1.05""",
                   ""","v_min":[236.0,236.0,236.0],"v_max":[240.0,240.0,240.0]""");
                optimizer = OPT)
println("Dy free tap  →  status = ", rdy["termination_status"],
        ",  tap = ", round(rdy["transformer"]["t1"]["tap"], digits=4))
for t in ("1","2","3")
    println("  |V_lv.", t, "| = ", round(rdy["bus"]["lv"][t]["vm"], digits=2), " V")
end

Dy free tap  →  status = LOCALLY_SOLVED,  tap = 0.9771
  |V_lv.1| = 240.0 V
  |V_lv.2| = 240.0 V
  |V_lv.3| = 240.0 V

The load is balanced, so the optimiser boosts all three LV phases together into the tight 236–240 V band with a single tap — the three-phase analogue of Section 2's rescue, with no per-phase code changes.

What single-phase and Dy tell us about the general picture

(A developer-oriented aside — skip to Validation if you only want to use the feature.)

Working the two cases out side by side surfaces the design choice for extending taps to every transformer:

• The single-phase (YY) tap admits an exact, degree-2 OLTC model. Scaling the from-winding leakage by $t^2$ and referring it to the to side gives a constant leakage and a voltage drop that is linear-in-current and quadratic-in-tap — and it matches the OpenDSS turns-scaled Yprim at the optimised tap to the usual ~0.3 V validation floor.
• The split-phase (center_tap) tap also admits an exact, degree-2 model. Its free tap reuses the same coupled-coil Yprim as the fixed model, but with the HV ratio promoted to a variable and the voltage drop written in the degree-2 T-model form ($N\cdot v$ and $N\cdot Z_2 I$ products only). It is algebraically identical to the fixed Yprim — re-solving with the tap fixed to $t^\star$ reproduces the free solution to 0.0 V — so the leakage referral is exact, not approximate, even under heavy drop and leg unbalance.
• The Dy/Yd coupled delta-arm model also carries the exact $t^2$ referral. OpenDSS scales winding 1's self-impedance by $t^2$, so the short-circuit impedance referred to the tapped (from) side scales as $t^2$ and the non-tapped side is held at nominal (verified directly against OpenDSS's short-circuit Yprim). The coefficients are degree-1 in the tap variables ($n_\text{eff}$ and its reciprocal), so the coupled-arm drop stays degree-2 for a free tap, and the same referral is used in the exported Yprim — so the two paths agree at every tap.

The optimiser, the JSON schema and the result reporting are uniform across all the covered subtypes, and the leakage referral is now exact for every two-bus subtype. What remains to generalise is free taps for the n_winding subtype, to-side taps, and discrete steps.

Validation

Each case is checked against OpenDSS: the optimised continuous tap $t^\star$ is set on the corresponding OpenDSS transformer winding, the power flow is solved, and the bus voltages are compared to the OPF solution (see test/powerflow_comparison_tests.jl, optimised tap vs OpenDSS). Because the variable-tap model is the fixed-tap model with the ratio promoted to a variable, agreement at $t^\star$ is the validation of the feature.

For center_tap the cross-check is deliberately stressed: a 5 km feeder with heavy, unequal 120 V leg loads (high voltage drop + unbalance) — node voltages agree with the OpenDSS turns-scaled Yprim at $t^\star$ to ≈ 0.25 V and total losses to ≈ 2.5 %, and the free T-model reproduces the fixed-Yprim re-solve at $t^\star$ exactly (0.0 V).