In the unloaded state the compressor draws only a fraction of its full-load BHP. That fraction, \(f_{bhp,unloaded}\), represents the residual mechanical work (friction, windage, valve actuation) absorbed by the unloaded compressor. The motor efficiency in this state, \(\eta_{unloaded}\), may differ from the loaded value because motors are less efficient at light loads.

\begin{equation}\label{eq:operating-cost-unloaded} C_{unloaded} = \frac{P_{motor} \cdot k_{BHP} \cdot H_{annual} \cdot f_{bhp,unloaded} \cdot f_{unloaded} \cdot c_{elec}}{\eta_{unloaded}} \end{equation}

Symbols

\(C_{unloaded}\)	Annual electricity cost in the unloaded condition \([\unit{ \dollar\per\year}]\)
\(P_{motor}\)	Full-load motor brake horsepower \([\unit{ \bhp}]\)
\(k_{BHP}\)	BHP-to-kilowatt conversion factor (0.746) \([\unit{ \kilowatt\per\bhp}]\)
\(H_{annual}\)	Total compressor operating hours per year \([\unit{ \hour}]\)
\(f_{bhp,unloaded}\)	Percentage of full-load BHP consumed when unloaded (input ÷ 100) \([\unit{ \percent}]\)
\(f_{unloaded}\)	Run-time unloaded fraction (= (100 - f_{loaded}) ÷ 100) \([\unit{ \percent}]\)
\(c_{elec}\)	Electricity unit cost \([\unit{ \dollar\per\kilowatt\hour}]\)
\(\eta_{unloaded}\)	Motor efficiency in the unloaded condition (input ÷ 100) \([\unit{ \percent}]\)

Note: The unloaded BHP percentage, \(f_{bhp,unloaded}\), is a compressor nameplate or measured value that varies by machine type. Reciprocating compressors typically unload to 15–25 % of full-load BHP; rotary screw compressors with inlet modulation may be 25–40 %. A lower value indicates a more efficient unloaded state.