I2R losses I²R
I²R losses are the heat generated whenever current flows through resistance: the power dissipated is current squared times resistance, so doubling the current quadruples the heat.
That quadratic is the most consequential scaling law in a battery plant's electrical design — it is why power moves at high voltage and low current wherever the architecture allows, why cable and busbar sizing is a lifetime loss budget rather than just an ampacity check, and why losses pile up in the high-current, low-voltage corner at the end of a discharge.
Every I²R watt-hour is purchased charging energy leaving the plant as heat, and it appears on several separate lines of the efficiency ledger: cell internal resistance, busbars and DC cabling, transformer windings, collection cables. In the round-trip-efficiency chain those are the load-shaped lines — they scale with the square of current, which is why an RTE figure means nothing without its C-rate.
Reviewed August 2026 by Sergey Syrvachev
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What it is (precise)
In a single conductor the dissipated power is P = I² × R; across a balanced three-phase circuit the total conductor loss is 3 × I² × R. The law is quadratic in current and only linear in resistance, and that asymmetry decides where design effort goes: halving the resistance halves the loss, but halving the current quarters it.
Current, in turn, is set by power and voltage — I = P / V on the DC side, I = P / (√3 × V(L-L) × cos φ) on the AC side — which is what turns every voltage decision into a loss decision. Datasheets and transformer test reports call the same quantity copper loss or load loss; where the resistance itself lives, from cell electrodes to cable runs to every joint between them, is the resistance entry's subject.
The cell-level arithmetic is worked in the internal-resistance entry and worth restating because the plant total surprises people. A 314 Ah cell discharging at 0.5C carries about 157 A; through an assumed 0.25 mΩ that is roughly 6 W of heat in one cell.
A 5 MWh-class container holds roughly 5,000 such cells, so the fleet inside one enclosure generates on the order of 30 kW of ohmic heat at 0.5C — and at 1C the per-cell figure quadruples to roughly 24 W, because the law is quadratic, not proportional. That heat is the load the liquid-cooling loop is sized to remove, and the pumping and chilling energy spent removing it lands in auxiliary consumption — so one I²R watt is paid for twice, once as lost throughput and again as cooling load.
Why moving a megawatt prefers high voltage
Fix the power and the current falls as 1/V, so the I²R loss in a given conductor falls as 1/V² — double the voltage, quarter the heat. Inside the fence the 1500-vdc entry works the example: a 3 MW DC feed means 3,000 A at 1000 V but 2,000 A at 1500 V, and through the same 0.5 mΩ collection path the heat drops from 4.5 kW to 2.0 kW — the factor is (2/3)², or 4/9.
The same relationship is why series connection exists at all: stacking cells raises voltage at constant current, so a rack's conductors are sized for one cell's amps rather than the rack's megawatts, and it is the argument that moved DC architectures from 1000 to 1500 VDC, with 2,000 V platforms emerging on the identical logic.
The AC side repeats the move at larger scale. A PCS delivers at low voltage — 690 V is the common grid-scale figure — where one megawatt at unity power factor is about 840 A; step up to a 34.5 kV collection bus and the same megawatt is about 17 A. Fifty times less current is 2,500 times less heat in a given conductor, which is why the step-up transformer sits at the PCS and not at the substation, and why the collection run is medium voltage rather than converter voltage.
Beyond the fence the 1/V² law is the whole story of transmission voltage — the transmission-voltage-and-distance entry owns that ladder — and the volts the same current costs along a run belong to the voltage-drop entry: same physics, read in volts there and in watts here.
The law is P = I² × R per conductor and 3 × I² × R across a balanced three-phase circuit: quadratic in current, linear in resistance. Every voltage decision in the plant is this law read backwards. At fixed power current scales as 1/V, so loss scales as 1/V² — the same 3 MW through the same conductor is 3,000 A and 4.5 kW at 1000 V-class against 2,000 A and 2.0 kW at 1500 V-class, four-ninths the heat. Stepping to MV is the same move made larger: one megawatt is about 840 A at 690 V and about 17 A at 34.5 kV, roughly 2,500 times less heat in a given conductor. At cell scale the worked figure is ~6 W per cell at 0.5C — 157 A through an assumed quarter-milliohm — and on the order of 30 kW per 5 MWh-class container, quadrupled rather than doubled at 1C. In the RTE ledger these are the load-shaped lines: cells, busbars, DC cabling, transformer load loss and collection cables, the lines that move with C-rate. A conductor at its ampacity limit runs at its maximum continuous loss; the first-pass check for enclosed copper bars is about 1.2–2 A/mm².
- The law
- P = I² × R per conductor; 3 × I² × R across a balanced three-phase circuit — quadratic in current, linear in resistance
- Scaling
- Double the current, four times the heat; at fixed power, current ∝ 1/V so loss ∝ 1/V²
- Worked cell figure
- ~6 W per cell at 0.5C (157 A through an assumed 0.25 mΩ); on the order of 30 kW per 5 MWh-class container — quadrupled, not doubled, at 1C
- 1000 V vs 1500 V
- Same 3 MW, same conductor: 3,000 A and 4.5 kW at 1000 V against 2,000 A and 2.0 kW at 1500 V — 4/9 the heat
- Stepping up to MV
- One megawatt is ~840 A at 690 V but ~17 A at 34.5 kV — 2,500 times less heat in a given conductor
- End-of-discharge corner
- Constant power at a ~1,040 V floor instead of ~1,330 V nominal takes ~28% more current — about 63% more I²R heat
- Ampacity vs loss budget
- A conductor at its ampacity limit runs at its maximum continuous loss; first-pass check for enclosed copper bars is ~1.2-2 A/mm², below bare thermal limits
- Ledger position
- The load-shaped lines of the RTE ledger — cells, busbars, DC cabling, transformer load loss, collection cables — the lines that move with C-rate
Cable sizing is a loss budget, not just an ampacity check
Ampacity answers a thermal question: the largest continuous current a conductor can carry without cooking its own insulation. It says nothing about whether the copper is earning its keep. A cable run at its ampacity limit is, by definition, running at its maximum continuous I²R loss — perfectly code-compliant and maximally wasteful at once.
The loss budget asks the economic question instead: R × I² × hours, integrated over a project life of daily cycling, is a stream of megawatt-hours bought at the charging price and never delivered, plus the cooling energy spent rejecting the heat where the run is indoors. Upsizing a cross-section buys resistance down for the life of the asset, which is why DC collection cabling on a hard-cycling plant is commonly sized above its ampacity minimum — and why the busbar entry's first-pass check for enclosed copper bars, roughly 1.2-2 A/mm², sits well below what bare thermal limits would permit.
Joints are the loss budget's failure mode. Every bolted or welded joint sits in series with the full operating current, so a joint whose resistance has crept up concentrates I²R heat at a point — a laser weld at tens of microohms dissipates about a watt at 157 A, and a degraded joint can dissipate orders of magnitude more in a spot the thermal design never allowed for.
That is why torque checks and joint-resistance measurements appear in factory and commissioning records, why thermographic scans belong in the O&M routine, and why the symptom to watch between scans is efficiency drift at the meter rather than any alarm: resistance growth in a joint shows up in the RTE trend long before it shows up as a temperature trip.
The end-of-discharge corner
Losses do not spread evenly over a cycle; they concentrate where voltage is lowest. At constant power, current rises as the string voltage falls, and a 1500 VDC-class string spans roughly 1,150-1,330 V nominal down to a low-SOC floor near 900-1,040 V. Delivering the same megawatts at a ~1,040 V floor instead of ~1,330 V nominal takes roughly 28% more current through every cell, busbar and DC cable — about 63% more I²R heat, arriving in the last fraction of the discharge, after the cooling system has already absorbed hours of dispatch.
The corner also feeds itself: the extra current deepens the I × R sag (the voltage-drop entry's territory), pulling the terminal voltage lower still and demanding yet more current to hold power, and cold makes it worse because the cold loaded sag sets the binding low-voltage corner of the design table.
The same corner is why a PCS is specified as a current-limited machine. At low DC voltage the converter must either push more current — more of its own I²R and switching heat — or shed power, and the full-power knee published in its VDC window is that trade drawn as a line.
Downstream, the corner puts a rate stamp on every energy number: at high current the deeper sag reaches the weakest rack's voltage floor sooner, so the discharge terminates earlier and a capacity test at a gentle rate and a full-power market dispatch return two different deliverable-energy figures for one plant. Resistance growth compounds the effect with age, which is why the corner cases in a supply agreement are tested at stated C-rate and temperature rather than assumed from the nominal row.
Where it sits in the efficiency ledger
The round-trip-efficiency entry writes the plant's losses as a ledger, and I²R terms appear on most of its lines: inside the DC boundary as cell ohmic heating plus busbar and DC cabling loss, at the transformer as the load loss in its windings, and between transformer and POI as collection-cable loss. The audited bands are the reference: DC RTE commonly 92-96%, AC RTE at the PCS terminals 88-93%, net AC at the POI 85-90% once transformer and auxiliaries are counted.
What distinguishes the I²R lines from the rest of the ledger is their shape. Transformer no-load loss and standby auxiliaries accrue by the hour whatever the dispatch; the I²R lines are load-shaped — quadratic in current — so they scale with the square of C-rate while throughput scales only linearly, and the per-MWh loss on those lines doubles when the dispatch rate doubles.
That shape is what contracts have to pin down. An RTE guarantee quoted without a C-rate is unfalsifiable, because the same hardware posts different numbers at 0.25C and 0.5C with nothing wrong anywhere; the test protocol's stated rate, temperature and boundary are what make the number checkable.
And the ledger drifts with age on exactly these lines: resistance growth raises I²R heat one-for-one at identical dispatch — the resistance-growth entry works 30% growth into roughly 30 kW becoming 40 kW per container — which is the mechanism behind RTE declining a point or two from beginning to end of life. When an operating plant's efficiency trend bends away from the degradation model, the I²R lines are where to look first: a rate profile that shifted, a corner visited more often, or a joint on its way to the thermography report.
I²R losses are a fixed percentage of throughput — take the plant's efficiency figure and apply it to every megawatt-hour.
In reality: The percentage moves with dispatch, because the loss is quadratic in current while throughput is linear. Double the C-rate and the resistive heat quadruples but the energy it is spread over only doubles, so the per-MWh loss on every I²R line doubles; run the same megawatts at the bottom of the voltage window and the loss rises again because the amps do. That is why a capacity test at a gentle rate and a full-power market dispatch measure different efficiencies of one healthy plant, why an RTE guarantee is only checkable at a stated C-rate, temperature and boundary, and why an efficiency trend that bends away from the degradation model points at the load-shaped lines — the rate profile, the corners, the joints — before it points at the cells.
- Round-trip efficiency Glossary
- 1500 VDC Glossary
- How a Grid-Scale BESS Works: From Cell to Grid Article
- Interactive: Cell Losses & Efficiency Interactive visual · bess.engineer
I2R losses, in context.
The Grid-Scale BESS course covers i2r losses — and the rest of the system — from the ground up, the way it actually gets deployed.