Transformer Hot Spot, Ageing and Harmonic Derating

The thermal screen uses the fixed-time-constant exponential method restated by Dong et al. (arXiv 1805.00630, PDF pp. 2–3), naming IEEE C57.91-2011. Let k be total RMS current / rated current. Rated rises are Δθor above ambient and Δθhr above top oil; R is rated total load loss / no-load loss.

For harmonic currents aₕ = Iₕ/I₁, including a₁ = 1, FHL = Σ(aₕ²h²)/Σaₕ² and FSTR = Σ(aₕ²h⁰·⁸)/Σaₕ². On a rated I²R-loss base, e and s are average winding-eddy and other-stray loss ratios. The total load-loss multiplier is Mₗ = (1 + e FHL + s FSTR)/(1 + e + s). For the local winding hot spot, Mᵥ = (1 + eₕ FHL)/(1 + eₕ). These loss equations follow the open IET 2020 paper, section 2, and Rex Power Magnetics' worked C57.110 example.

With harmonic heating enabled, ultimate rises are Δθo,u = Δθor[(1 + R Mₗ k²)/(1 + R)]ⁿ and Δθh,u = Δθhr(Mᵥ k²)ᵐ. With it disabled, both multipliers are one. Coupling harmonic loss multipliers into the rise equations is an engineering screening extension derived here, not the complete IEC differential model or IEEE Annex G model.

Each period follows x(t) = xᵤ + (x₀ − xᵤ) exp(−t/τ). Oil and winding gradient are independent states; θh = θair + Δθo + Δθh. The periodic option solves the affine one-day map exactly for x₀ = xend, eliminating warm-up error. The preload option starts from steady prior load.

Ageing acceleration is exp(15000/383 − 15000/(θh + 273)), retaining the published historical 273 K offset so 110 °C gives exactly one (Dong p. 3; WPRC 2022 pp. 1–3). Temperature plotting samples are spaced by at most one minute and one eighth of either thermal time constant, including every period boundary. Ageing is evaluated from the exact exponential temperatures at additional midpoint samples. Composite Simpson panels are bisected until the sum of absolute panel error estimates, |Sfine − Scoarse|/15, divided by total equivalent ageing is at most 10⁻⁶ (0.0001%). Summing absolute panel differences prevents cancellation of local errors. The cumulative curve uses the refined integrals, not trapezoids through the plotted ageing rates. If the refinement limit is reached before this estimated tolerance, an error condition holds back results. This is a quadrature error target, not a physical-model accuracy claim. Divide equivalent hours by 24 h for the daily factor and by the assumed normal life for life fraction. Temperature peaks remain maxima of the plotting samples; the animation cursor interpolates the computed curves.

Equal-loss current capabilities are 1/√Mₗ and 1/√Mᵥ; the smaller is the reported screening capability. It is not applied automatically to the entered cycle. At rated voltage, multiply by nameplate kVA. FHL is also the spectrum K-factor when normalized to total RMS current. The separately reported peak rated-current-base K-factor is kpeak² FHL; it is not a certified K-rating.

• Mineral-oil immersed transformer with thermally upgraded cellulose insulation and a 110 °C ageing reference. Dry-type temperatures and non-upgraded paper ageing are outside this model.
• Six piecewise-constant load intervals; constant ambient, cooling mode and thermal time constants. The winding exponent is m, so the sinusoidal current power is 2m.
• ONAN defaults n = m = 0.8 are stated by Dong et al., PDF pp. 2–3. Default rises (55 K and 25 K), time constants (180 min and 7 min), R = 5, loss split and daily loads are illustrative editable engineering assumptions.
• Spectrum shape stays fixed over the day and scales with total RMS current. Include every significant harmonic in the six available rows; omit phases, DC and interharmonics. Magnitudes are RMS, not peak.
• Rated winding and stray losses are specified on a consistent rated I²R base; the local hot-spot eddy ratio is separate from the average. FSTR uses the conservative h⁰·⁸ weighting. No skin-depth correction is applied to the h² eddy weighting.
• The 180,000 h normal-life default is cited in WPRC 2022 p. 1. Life consumption is thermal insulation ageing only. Scene temperatures are explicitly labelled in °C; physical input and result cards support the viewer's unit conversion.

Screening and education only. A qualified engineer must check all loading decisions. Methods are identified with IEEE C57.91-2011, IEC 60076-7:2017 and IEEE C57.110-2018; check against the standard in force and the manufacturer's loading guide.

This is not a complete implementation or compliance assessment of those standards. In particular, fixed oil time constants, independent first-order winding response and a constant ambient omit cooling-stage changes, load-dependent time constants, oil-flow overshoot, tap changes, resistance-temperature feedback, voltage harmonics, neutral/triplen circulation, unbalance, enclosure ventilation and accessory limits. The equal-loss capability does not establish allowable cyclic or emergency loading.

Use validated heat-run and local loss data; the illustrative defaults cannot establish a loading rating. Loads above twice rated current are outside this screen. Very high computed temperatures are outside useful model accuracy, and a temperature under any particular threshold is not a safety approval. Temperature plotting is one minute or finer, resolving each time constant with at least eight intervals and including every period boundary. Ageing integration refines independently to an estimated relative quadrature error of 0.0001%; inspect the reported quadrature error under How it’s calculated. Sampled temperature maxima can still miss extrema between samples, and cursor values interpolate the curves; compare finer measurements for rapid duty.

Moisture, oxygen, paper condition and prior ageing are not modelled. Percentage life used is based on an assumed normal insulation life, not an estimate of remaining transformer life or failure probability. Installation work belongs to a qualified person.