The Voltage-Squared CVR Saving

Ada re-derives why a small voltage cut yields a squared energy cut

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ADA · CALCULATION AUDIT

Foundations · optional physics

The Voltage-Squared CVR Saving

A utility serving 500,000 customers and 15 TWh a year trims distribution voltage from 120 V to 116 V — a mere 3.33% cut. Because a resistive load’s power scales with voltage squared, the chapter claims that small trim becomes a 6.56% energy cut worth $39.3M a year against a $0.5–2M investment. This audit squares the ratio line by line and asks whether that exponent really turns a modest tap-change into a 20–80× first-year payback — provided the loads truly are resistive.

Companion to the chapter Smart Grid and Energy IoT — every number here comes from that chapter.

Ada's Calculation Audit: the voltage-squared CVR saving — why a 3.3% voltage cut yields a 6.56% energy cut, ~4 minutes

Conservation Voltage Reduction leans on one physics fact: a resistive load's power scales with voltage squared. Squaring the ratio is where the saving actually comes from.

See the relationship before changing it

The figure reads from left to right. The blue card is delivered voltage. The middle card applies this page's rule. The green card is resistive energy cut. Walk the arrows once: set the input, apply the rule, then read the result with its unit.

The retained audit below checks several chapter fixtures. This model keeps those stated values fixed and changes only delivered voltage, so the numeric fixture does not switch without explanation.

Walk the arrows. Voltage-squared savings apply to the resistive share, not every customer load.

Derive the baseline in four named moves

1
Name the input. The chapter baseline is 116 V.
2
Name the relationship. energy cut = (1 - (voltage / 120 V)^2) x 100
3
Substitute with units. (1 - (116 / 120)^2) x 100 = 6.56%
4
Read the result. Keep the unit beside the value. Use it only inside the technical boundary on this page.

Predict, then change delivered voltage

Try Predict the direction of energy cut = (1 - (voltage / 120 V)^2) x 100. Test another delivered voltage, then compare resistive energy cut.

Observe Voltage-squared savings apply to the resistive share, not every customer load. Reset delivered voltage to 116 and compare resistive energy cut.

Explain Voltage-squared savings apply to the resistive share, not every customer load.

Check yourself

What should you do before trusting a moved-control result?

Answer: Predict its direction, apply the shown relationship, keep the units, and reset to the worked baseline.

What does this small model leave out?

Answer: Only delivered voltage moves here. Field effects named in the technical boundary stay fixed.

Try

Select Calculate for 15 TWh/year with voltage reduced from 120 V to 116 V and a 40% resistive share.

Observe

Check squares the 0.9667 ratio to show 6.56% energy reduction, 0.393 TWh/year, and about $39.3M value.

Explain

Resistive power follows voltage squared, so the 3.33% voltage trim becomes a 6.56% cut only for the modeled resistive load fraction.

Technical boundaries

For the CVR ledger, excluded from this fixed arithmetic are load voltage dependence outside the stated squared-power model, feeder losses, regulator limits, customer behaviour, or rebound effects.

The working

1. Square the voltage ratio. 116 V ÷ 120 V = 0.9667, a 3.33% trim. For a resistive load P = V²/R, so the power ratio is the square:

power ratio = (116 ÷ 120)² = (0.9667)² = 0.9344 → energy cut = 1 − 0.9344 = 6.56%

Squaring nearly doubles the 3.33% voltage trim into a 6.56% energy cut on resistive load.

2. Apply it to the resistive share. 15 TWh/yr × 0.40 resistive = 6.0 TWh, so savings = 6.0 × 0.0656 = 0.393 TWh/yr.

3. Turn energy into money and ROI. 0.393 TWh = 393 million kWh × $0.10 = $39.3M/yr. Against a $0.5–2M VVO investment that is $39.3M ÷ $0.5M = 79× down to $39.3M ÷ $2M = 20× first-year ROI.

Step	Arithmetic	Result
Voltage ratio	116 ÷ 120	0.9667 (−3.33%)
Power ratio (V²)	0.9667²	0.9344
Energy cut	1 − 0.9344	6.56%
Resistive energy	15 × 0.40	6.0 TWh/yr
Savings	6.0 × 0.0656	0.393 TWh/yr
Value	0.393e9 kWh × $0.10	$39.3M/yr
ROI	$39.3M ÷ $0.5–2M	20–80×

What the audit buys you: the whole business case rides on the exponent — a 3.3% voltage trim would be trivial if power were linear in voltage, but the V² law doubles it to 6.56%, turning a modest tap-change into a $39M-a-year, 20–80× payback, provided the loads really are resistive.

Every number above is taken from the chapter's own CVR voltage-saving example and re-derived step by step.