Pump Motor Energy Consumption

This calculation tool is for a constant speed, constant volume system. Measured input data include spot measurements for true power and motor operational time per hour, as measured by motor on/off loggers.

The calculator uses the following methodology to compute annual energy consumption of a constant speed pump:

1. Convert seconds on per hour to percent on per hour (Worksheet: “Step 2. Percent Runtime Calcs”).

\begin{equation} \%On(t) = \frac{On(t)}{3600} \end{equation}

Where,

$\%On(t) =$ Percent of an hour that the pump motor is on, %

$On(t) =$ Measured time that pump motor is on, seconds

$t =$ Index for each measured data point

2. Average % hour motor is on for each hour of each day of the week (Worksheet: “Step 3. Daily Avg Runtime Calcs”).

\begin{equation} \overline{\%On}_{d,h} = \frac{\sum_{n=1}^{N(d,h)} \%On(n)}{N(d,h)} \end{equation}

Where,

$\overline{\%On}_{d,h} =$ Average % time motor is on per hour for given day of week, d, and hour of the day, h, %

$N(d,h) =$ Total number of measured data points that fall on day of week, d, and hour of the day, h

$n \in t(d,h) =$ Index for the subset of measured data points that fall on day of week, d, and hour of the day, h

3. Find the average hours per day pump is on (Worksheet: “Step 4. Results”).

\begin{equation} \overline{HrsOn}_{d} = \sum_{h=1}^{24} \overline{\%On}_{d,h} \end{equation}

Where,

$\overline{HrsOn}_{d} =$ Average hours per day for given day of week, d, hours

4. Calculate energy used for full year (Worksheet: “Step 4. Results”).

\begin{equation} WeeksPumpOn = (DayOfYear(PumpOperationEndDate) \end{equation}

\begin{equation*} - DayOfYear(PumpOperationStartDate)) * \frac{WeeksPerDay}{DaysPer Year}\end{equation*}

\begin{equation} E = WeeksPumpOn * P\sum_{d=1}^{7} \overline{HrsOn}_{d} \end{equation}

Where,

$E =$ Annual pump energy, kWh

$DayOfYear =$ Function to convert a date to the n^{th} day of the year

$PumpOperationEndDate =$ Cooling season end date, mm/dd/yyyy

$PumpOperationStartDate =$ Cooling season start date, mm/dd/yyyy

$WeeksPumpOn =$ Pump operating time, weeks

$WeeksPerYear =$ 52

$DaysPerYear =$ 365.24

$P =$ Measured pump power, kW

This calculation is for a constant speed, constant volume pump. Measured input data is average hourly power draw (in kW) as measured by a power logger.

The calculator uses the following methodology to compute annual energy consumption of a constant one- or two-speed pump:

1. Average % hour motor is on for each hour of each day of the week (Worksheet: “Step 3. Avg Day of Week Calcs”).

\begin{equation} \bar{P}_{d,h} = \frac{\sum_{n=1}^{N(d,h)} P(t)}{N(d,h)} \end{equation}

Where,

$\bar{P}_{d,h} =$ average motor power for given day of week, d, and hour of day, h, kW

$P(t) =$ measured motor power, kW

$t =$ index for measured data points

$N(h,d) =$ total number of measured data points that fall on day of week, d, and hour of the day, h

$n \in t(d,h) =$ index for subset of measured data points that fall on day of week, d, and hour of the day, h

2. Calculate energy used for full year (Worksheet: “Step 4. Results”). WeeksPumpOn is from Equation 5.

\begin{equation} E = WeeksPumpOn * \sum_{d = 1}^7 \sum_{h = 1}^{24} \bar{P}_{d,h} \end{equation}

Where,

$E =$ annual pump energy, kWh

$WeeksPumpOn =$ Pump operating time, weeks

1. Convert average hourly current to percent on per hour (Worksheet: “Step 2. Percent Runtime Calcs”).

\begin{equation} \%On(t) = \begin{cases} \frac{I(t)}{I_{max}} & \text{if } I(t) < I_{max} \\ 100\% & \text{o.w.} \end{cases} \end{equation}

Where,

$\%On(t) =$ percent of an hour that the pump motor is on, %

$I(t) =$ measured time that pump motor is on, seconds

$t =$ index for each measured data point

$I_{max} =$ current measured at the maximum constnat speed, Amps

3. Find the average hours per day pump is on (Worksheet: “Step 4. Results”).

\begin{equation} \overline{HrsOn}_{d} = \sum_{h=1}^{24} \overline{\%On}_{d,h} \end{equation}

Where,

$\overline{HrsOn}_{d} =$ Average hours per day for given day of week, d, hours

4. Calculate energy used for full year (Worksheet: “Step 4. Results”).

\begin{equation} WeeksPumpOn = (DayOfYear(PumpOperationEndDate) \end{equation}

\begin{equation*} - DayOfYear(PumpOperationStartDate)) * \frac{WeeksPerDay}{DaysPer Year}\end{equation*}

\begin{equation} E = WeeksPumpOn * P\sum_{d=1}^{7} \overline{HrsOn}_{d} \end{equation}

Where,

$E =$ Annual pump energy, kWh

$DayOfYear =$ Function to convert a date to the n^{th} day of the year

$PumpOperationEndDate =$ Cooling season end date, mm/dd/yyyy

$PumpOperationStartDate =$ Cooling season start date, mm/dd/yyyy

$WeeksPumpOn =$ Pump operating time, weeks

$WeeksPerYear =$ 52

$DaysPerYear =$ 365.24

$P =$ Measured pump power, kW

This calculation is for VFD-controlled pumps that are operated at speeds proportional to the heating/cooling load as represented by proxy with OAT. Measured input data include average hourly power draw (kW) as measured by a DENT data logging power logger, and the average hourly OAT as measured by a temperature/RH logger.

The calculator uses the following methodology to compute annual energy consumption of a variable speed pump:

1. Fit a second-order polynomial regression model of true RMS power as a function OAT. (Worksheet: “Step 3. Regression”).

\begin{equation} P(OAT) = a * OAT^{2} + b * OAT + c \end{equation}

Where,

$P =$ Average hourly true RMS power

$a,b,c =$ Regression coefficients

$OAT =$ Outdoor air temperature, F

2. Average % hour motor is on for each hour of each day of the week (Worksheets: “Step 4. Schedule Calcs” and “Step 5. Daily Ave Schedule Calc”).

\begin{equation} On(t) = \begin{cases} 1 & \text{if } P(t) > 0\\ 0 & otherwise \end{cases} \end{equation}

\begin{equation} \overline{\%On}_{d,h} = \frac{\sum_{n=1}^{N(d,h)} On(t)}{N(d,h)} \end{equation}

Where,

$P(t) =$ Measures power, kW

$On(t)=$ Motor is on at time, t, binary

$\overline{\%On}_{d,h} =$ Average % time motor is on per hour for given day of week, d, and hour of the day, h, %

$N(d,h) =$ Total number of measured data points that fall on day of week, d, and hour of the day, h

$n \in t(d,h) =$ Index for subset of measured data points that fall on day of week, d, and hour of the day, h

3. Total annual pump energy (Worksheets: “Step 6. Energy Calcs” and “Step 7. Results”).

\begin{equation} E = WeeksPumpOn * \sum_{t=1}^{8760} P(OAT'(t)) * \overline{\%On}_{d,h}(t) \end{equation}

Where,

$E =$ Annual energy usage, kWh

$OAT'(t) =$ Hourly climate normal outside air temperature from National Weather Service at station closest to site, F

$WeeksPumpOn =$ Pump operating times, weeks

This calculation is for VFD-controlled pumps using hourly average motor speed (in RPM) as downloaded from a BAS or VFD, and spot measurements for true power.

1. Convert the speed data from the BAS or VFD to power (Worksheet: “Step 2. Aggregate Data”)

\begin{equation} P(t) = P_{max}(\frac{\omega(t)}{\omega_{max}})^{2.5} \end{equation}

Where,

$P(t) =$ average hourly true RMS power, kW

$P_{max} =$ full speed true RMS power, kW

$\omega_{max} =$ maximum motor speed, rpm

$\omega(t) =$ motor speed, rpm

This equation represents the pump affinity law, where theoretically the exponent is 3. To represent motor losses, the Consortium for Energy Efficiency recommends using a value of 2.5.[1]

2. Fit a second-order polynomial regression model of true RMS power as a function OAT. (Worksheet: “Step 3. Regression”).

\begin{equation} P(OAT) = a * OAT^{2} + b * OAT + c \end{equation}

Where,

$P =$ Average hourly true RMS power

$a,b,c =$ Regression coefficients

$OAT =$ Outdoor air temperature, F

3. Average % hour motor is on for each hour of each day of the week (Worksheets: “Step 4. Schedule Calcs” and “Step 5. Daily Ave Schedule Calc”).

\begin{equation} On(t) = \begin{cases} 1 & \text{if } P(t) > 0\\ 0 & otherwise \end{cases} \end{equation}

\begin{equation} \overline{\%On}_{d,h} = \frac{\sum_{n=1}^{N(d,h)} On(t)}{N(d,h)} \end{equation}

Where,

$P(t) =$ Measures power, kW

$On(t)=$ Motor is on at time, t, binary

$\overline{\%On}_{d,h} =$ Average % time motor is on per hour for given day of week, d, and hour of the day, h, %

$N(d,h) =$ Total number of measured data points that fall on day of week, d, and hour of the day, h

$n \in t(d,h) =$ Index for subset of measured data points that fall on day of week, d, and hour of the day, h

4. Total annual pump energy (Worksheets: “Step 6. Energy Calcs” and “Step 7. Results”).

\begin{equation} E = WeeksPumpOn * \sum_{t=1}^{8760} P(OAT'(t)) * \overline{\%On}_{d,h}(t) \end{equation}

Where,

$E =$ Annual energy usage, kWh

$OAT'(t) =$ Hourly climate normal outside air temperature from National Weather Service at station closest to site, F

$WeeksPumpOn =$ Pump operating times, weeks