Refrigeration Product Test Procedure 

Theoretical Analysis of Potential Measurement Error for Incomplete Cycling Products 

2/5/13 

Incomplete cycling products have compressor cycles that last longer than 12 hours. For these 

products, the DOE test procedure calls for a test period that is 24 hours long (see, for example, 

10 CFR 430, subpart B, appendix A, section 4.1). 

Assuming that the duration of compressor on-cycles1

 and off-cycles2 remains consistent and does 

not vary, the long-term average energy use of the product should be equal to the average energy 

use associated with a full compressor cycle, consisting of an on-cycle and an off-cycle. This is 

because the energy use is dominated by the energy consumed during the on-cycles and the total 

time duration is equal to the number of cycles multiplied by the duration of a single cycle. For 

example, after 1000 cycles the remaining energy use associated with compressor operation 

beyond the 1000th cycle, referred to here as dE, is much smaller than the energy use associated 

with the 1000 on-cycles. Likewise, the remaining time associated with operation beyond the 

1000th cycle, referred to here as dT, is much smaller than the duration of 1000 on-cycles. Hence, 

the average energy use of a single cycle is approximated as 

݁ݏܷ	ݕ݃ݎ݊݁ܧ

ܶ݅݉݁ ൌ 	

ܧ݀ ൅ ܧ1000

	 ൎ ݐ݀ ൅ 1000ܶ

ܧ1000

1000ܶ ൌ 	

ܧ

ܶ

where E is the energy used during a compressor on-cycle and T is the duration of a full 

compressor cycle. 

Because long-term average energy use is equal to the average energy use for a full compressor 

cycle, stopping the energy test at 24 hours can introduce error into the measurement. This 

analysis illustrates the potential magnitude of this error. 

The analysis assumes that the duration of compressor on-cycles and off-cycles remains constant 

(i.e. that the unit has reached steady state, as required by the DOE test procedure prior to 

measuring energy use, see Appendix A, sections 2.9 and 4), power input is constant at a level P 

(with units of Watts) throughout the on-cycle, and power input is zero during the off-cycle. The 

duration of a full compressor cycle is designated T (with units of hours). The duty cycle  is 

defined as the ratio of the duration of a compressor on-cycle to the duration of the complete 

compressor cycle. Hence, the duration of the on-cycle is T and the duration of the off-cycle is 

(1-)T. 

As discussed above, the long-term average energy use of the product is equal to E/T, where E is 

the energy use associated with the full compressor. In terms of the parameter P defined above, 

this average energy use is equal to: 

݁ݏܷ	ݕ݃ݎ݊݁ܧ ൌ௚ ௩௔ܧ

ܶ݅݉݁ ൌ 	

ܧ

ܶ ൌ 	

݅݉݁ݐ	݈ܿ݁ݕܿ ∙ ܱ݊ ൈ ܲ

ܶ ൌ ܲ߶ܶܶ ൌ ܲ߶

                                                            

1

 Time periods when the compressor is operating. 

2

 Time periods when the compressor is not operating.
2

The error associated with use of a 24-hour test period depends on the time at which the test 

period starts. This analysis compares test periods that begin at the start of a compressor on-cycle 

and test periods that begin at the end of a compressor on-cycle. Measurement error is generally 

positive for the former case and negative for the latter case. Figure 1 and Figure 2 below 

illustrate the refrigeration product power input as a function of time for each of these cases. The 

figures also illustrate four regions of time for each of these cases when a 24-hour test period 

could end. The equation for measurement error is different for each of these eight cases. 

Figure 1: Power Input Trace for Test Period Start at the Start of an On-Cycle 

Figure 2: Power Input Trace for Test Period Start at the End of an On-Cycle 

T

0 T T T(1+) 2T

P

12 34 On‐cycle Off‐cycle

Power Input

Time

Four possible time ranges for the

end of the test period (24 hours).

T

0 T(1‐) T T(2‐) 2T

P

12 34 On‐cycle Off‐cycle

Power Input

Time

Four possible time ranges for the

end of the test period (24 hours).
3

The measured energy use for the 24-hour test period is equal to the power input P times the 

duration of compressor operation during the 24-hour period. To determine average energy use 

per unit of time, this value is divided by the test period duration, 24 hours. Table 1 below shows 

calculations for the duration of compressor operation and the average energy use per unit of time 

for the 24-hour test period for the eight cases identified by Figures 1 and 2. The table also shows 

the fraction of error in the measurement—this value can be converted to percent error by 

multiplying it by 100%. 

Table 1: Assessment of Measured Energy Use and Measurement Error 

Test 

Period 

Start Time 

Time Range 

of Test 

Period End 

Duration of 

Compressor 

Operation 

Ton (hours) 

Average Energy Use 

Em = P×Ton/24 

(Watts) 

Fraction of Error in 

Measurement 

Err = ( ா௠

ா௔௩௚

െ 1) 

Start of 

On-Cycle 1 24 ܲ ൬

1െ߶

߶ ൰

2 ܶ߶ ܲܶ߶

24 ൬

ܶ

24 െ 1൰

3 24 െ ܶሺ1െ߶ሻ ܲ

24 ሾ24 െ ܶሺ1െ߶ሻሿ ൬

1െ߶

߶ ൰ ൬1 െ

ܶ

24൰

4 2ܶ߶ ܲܶ߶

12 ൬

ܶ

12 െ 1൰

End of OnCycle 1 0 0 -1 

2 24 െ ܶሺ1െ߶ሻ ܲ

24 ሾ24 െ ܶሺ1െ߶ሻሿ െ ൬

1െ߶

߶ ൰ ൬ ܶ24 െ 1൰

3 ܶ߶ ܲܶ߶

24 െ ൬1 െ

ܶ

24൰

4 24 െ 2ܶሺ1െ߶ሻ ܲ

24 ሾ24 െ 2ܶሺ1െ߶ሻሿ െ ൬

1െ߶

߶ ൰ ൬ ܶ12 െ 1൰

In order to illustrate the potential magnitude of the error in the 24-hour test period, the percent 

error is plotted in Figure 3 for compressor cycle durations from 12 to 36 hours and duty cycles 

equal to 25%, 50%, and 75%. The plot shows that the error can be quite large depending on the 

compressor cycle duration and the duty cycle. 
4

Figure 3: Percent Error Using a 24-Hour Test Period as a Function of Compressor Cycle 

Duration and Duty Cycle 

‐100%

‐80%

‐60%

‐40%

‐20%

0%

20%

40%

60%

12 16 20 24 28 32 36

Percent Error Using 24‐hour Test Period

Compressor Cycle Duration (hours)

0.25

0.5

0.75

0.25

0.5

0.75

Duty Cycle

Test Period

Start with

On‐Cycle Start

Test Period

Start with

On‐Cycle End