True RMS Power

This calculation is for a constant speed or two-speed, constant-volume system. Measured input data is average hourly power draw (in kW) as measured by a data logging power meter for the supply and return fans in the AHU.

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

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

\begin{equation} \bar{P}_{d,h,f} = \frac{\sum_{n_f = 1}^{N_f(d,h)} P_f(t_f)}{N_f(d,h)} \end{equation}

Where,

$\bar{P}_{d,h,f} =$ average motor power for given day of week, d, and hour of day, h, for either suply or return fan, f, kW

$P_f(t_f) =$ measured motor power for either supply or return fan, f, kW

$t_f =$ index for measured data points for either supply or return fan, f, datasets

$N_f(d,h) =$ total number of measured data points that fall on day of week, d, and hour of day, h, for either supply or return fan, f

$n_f \in t_f(d,h) =$ index for subset of measured data points that fall on day of week, d, and hour of day, h, for either supply or return fan, f

2. Calculate energy used for the simulation period that users input for each fan (Worksheet: “Step 4. Results”).

\begin{equation} E_f = \sum_{d = 1}^7 \sum_{h = 1}^24 \bar{P}_{d,h,f} * [\textit{operating weeks per year}] \end{equation}

Where,

$E_f =$ annual energy for either supply or return fan, f, kWh

$\bar{P}_{d,h,f} =$ average motor power for given day of week, d, and hour of day, h, for either supply or return fan, f, kW

3. Total energy used by the AHU (Worksheet: “Step 4. Results”).

\begin{equation} E = E_{supply} + E_{return} \end{equation}

Where,

$E =$ total energy usage for AHU, kWh

This calculation is for a cooling tower with constant-speed fans. Measured input data is average hourly power draw (in kW) as measured for each fan in the cooling tower.

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

1. Total energy for each hour (Worksheet: “Step 2 Energy Calcs”). Since the measured power value (kW) is the average for an hour, it is equivalent to the energy for the hour (kWh).

\begin{equation} \bar{E}(t) = \sum_{f = 1}^F \bar{E}_{f}(t) \end{equation}

Where,

$\bar{E}_{f}(t) =$ average hourly fan motor energy on for each fan, f, kWh

$\bar{E} =$ total average energy of all fan motors, kWh

$F =$ total number of fans in the cooling tower

$t =$ time index

2. Wet-bulb temperature is calculated from the measured dry-bulb temperature and measured relative humidity with the following empirical equation (Worksheet: “Step 3 Wetbulb Calcs”)[1].

\begin{equation} T_{wb} = T_{db}tan^{-1}[0.151977(RH+8.313659)^{1/2}] \end{equation}

\begin{equation*} +tan^{-1}(T_{db}+RH) \end{equation*}

\begin{equation*} -tan^{-1}(RH-1.676331)+0.00391838(RH)^{3/2} \end{equation*}

\begin{equation*} *tan^{-1}(0.023101*RH) - 4.686035 \end{equation*}

Where,

$T_{wb} =$ Wet-bulb temperature, C

$T_{db} =$ Dry-bulb temperature, C

$RH =$ Relative Humidity, %

3. To model the total fan energy at different wet-bulb temperatures, a second-order polynomial regression model is fitted, with total fan energy as a function of wet-bulb temperature (Worksheet: “Step 4 Regression”).

\begin{equation} E(T_{wb}) = a*T_{wb}^{2} + b*T_{wb} + c \end{equation}

Where,

$E(T_{wb}) =$ Average hourly fan energy, kWh

$a,b,c =$ Regression Coefficients

4. Calculate operating hours (Worksheet: “Step 5 Daily Op Hrs”).

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

\begin{equation} \overline{On}_{d,h} = \begin{cases} 1 & \text{if } \overline{E}(n) > 0 \text{ for any } n \in N(d,h) \\ 0 & 0 \end{cases} \end{equation}

Where,

$On(t) =$ Cooling tower fans are on, binary

$\overline{On}_{d,h} =$ Cooling tower is on for a given day of the week, d, and hour of the day, h, binary

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

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

5. Calculate energy used for full year for each fan (Worksheet: “Step 6 Results”).

\begin{equation} E = \sum_{t=1}^{8760} \overline{On}_{d,h}(t) * CoolingSeason(t) * E(T_{wb}(t)) \end{equation}

Where,

$E =$ Annual energy, kWh

$CoolingSeason(t) =$ Does the given hour, t, fall within the cooling season, binary

This calculation is for VFD-controlled fans that are operated at different speeds proportional to the heating/cooling load. Because the heating/cooling load of a facility is mainly affected by outdoor air temperature (OAT), this data must be collected as well. Measured input data include average hourly power draw (kW) and average hourly OAT.

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

1. Perform second-order polynomial regression analysis of true RMS power as a function of OAT. (Worksheet: “Step 3. Regression”) for supply and return fans.

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

Where,

$P_{f} =$ Average hourly true RMS Power for either supply or return fan, f, kW

$a,b,c =$ Regression coefficients

$OAT_{f} =$ Outdoor air temperature for either supply or return fan, f

2. Determine if the supply and return motors are on or off at each measurement interval based on the input true RMS power values (Worksheets: “Step 4. Schedule Calcs”).

\begin{equation} On_{f}(t_{f}) = \begin{cases} 1 & \text{if } P_{f}(t_{f}) > 0\\ 0 & otherwise \end{cases} \end{equation}

Where,

$P_{f}(t_{f}) =$ Measured power

$On_{f}(t_{f}) =$ Motor is on at time, $t_{f}$ for either supply or return fan, f, binary

3. Average hourly % motor is on for each hour of each day of the week, for supply and return fan motors (Worksheet: “Step 5. Daily Avg Schedule Calc”).

\begin{equation} \overline{\%FanOn}_{d,h,f} = \frac{\sum_{n_{f}=1}^{N_{f}(d,h)} On_{f}(t_{f})}{N_{f}(d,h)} \end{equation}

Where,

$\overline{\%FanOn}_{d,h,f} =$ Average % time motor is on per hour for given day of week, d, and hour of day, h, for either supply or return fan, f, %

$N_{f}(d,h) =$ Total number of measured data points that fall on day of week, d, and hour od day, h, for either supply or return fan, f

$n_{f} \in t_{f}(d,h) =$ Index for subset of measured data points that fall on day of week, d, and hour of day, h, for either supply or return fan, f

4. Calculated energy (kWh) every hour based on normalized annual OAT for the supply and return fan motors (Worksheet: “Step 6. Energy Calcs,” column F & G).

\begin{equation} E_{f} = \sum_{t=1}^{8760} P_{f}(OAT'(t))*\overline{\%FanOn}_{d,h,f}(t) \end{equation}

Where,

$E_{f} =$ Annual energy usage for either supply or return fan, f, kWh

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

5. Total energy used by Fan Motor (Worksheet: “Step 7. Results”).

\begin{equation} E = E_{supply} + E_{return} \end{equation}

Where,

$E =$ Total energy usage for the Fan Motor, kWh

This calculation is for a cooling tower with variable-speed fans. Measured input data is average hourly power draw (in kW) as measured by a power logger for one fan in the cooling tower. It is assumed that all the remaining fans are run at the same speed.

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

1. Total energy for each hour (Worksheet: “Step 2 Energy Calcs”). Since the measured power value (kW) is the average for an hour, it is equivalent to the energy for the hour (kWh).

\begin{equation} \overline{E}(t) = \overline{E}_{1}(t) * F \end{equation}

Where,

$\overline{E}_{1}(t) =$ Average hourly fan motor energy for the measured fan, kWh

$\overline{E} =$ Total average energy of all fan motors, kWh

$F =$ Total number of fans in the cooling tower

$t =$ Time index

2. Wet-bulb temperature is calculated from the measured dry-bulb temperature and measured relative humidity with the following empirical equation (Worksheet: “Step 3 Wetbulb Calcs”)[1].

\begin{equation} T_{wb} = T_{db}tan^{-1}[0.151977(RH+8.313659)^{1/2}] \end{equation}

\begin{equation*} +tan^{-1}(T_{db}+RH) \end{equation*}

\begin{equation*} -tan^{-1}(RH-1.676331)+0.00391838(RH)^{3/2} \end{equation*}

\begin{equation*} *tan^{-1}(0.023101*RH) - 4.686035 \end{equation*}

Where,

$T_{wb} =$ Wet-bulb temperature, C

$T_{db} =$ Dry-bulb temperature, C

$RH =$ Relative Humidity, %

3. To model the total fan energy at different wet-bulb temperatures, a second-order polynomial regression model is fitted, with total fan energy as a function of wet-bulb temperature (Worksheet: “Step 4 Regression”).

\begin{equation} E(T_{wb}) = a*T_{wb}^{2} + b*T_{wb} + c \end{equation}

Where,

$E(T_{wb}) =$ Average hourly fan energy, kWh

$a,b,c =$ Regression Coefficients

4. Calculate operating hours (Worksheet: “Step 5 Daily Op Hrs”).

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

\begin{equation} \overline{On}_{d,h} = \begin{cases} 1 & \text{if } \overline{E}(n) > 0 \text{ for any } n \in N(d,h) \\ 0 & 0 \end{cases} \end{equation}

Where,

$On(t) =$ Cooling tower fans are on, binary

$\overline{On}_{d,h} =$ Cooling tower is on for a given day of the week, d, and hour of the day, h, binary

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

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

5. Calculate energy used for full year for each fan (Worksheet: “Step 6 Results”).

\begin{equation} E = \sum_{t=1}^{8760} \overline{On}_{d,h}(t) * CoolingSeason(t) * E(T_{wb}(t)) \end{equation}

Where,

$E =$ Annual energy, kWh

$CoolingSeason(t) =$ Does the given hour, t, fall within the cooling season, binary

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

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

The following equations are used to calculate the annual energy consumption of a lighting electrical distribution system where energy in kilowatt hours (kWh) is measured at the output end of an electrical panelboard or switchgear. It is assumed that data is collected for six (6) weeks at one-hour intervals with a data-logging power meter that has the capacity to measure three-phase power.

The calculator uses the following methodology to compute annual energy consumption of a lighting fixture system:

1. Find hourly average energy use for each hour of each day of the week (Worksheet: “Step 2. Avg Energy Calcs,” column J.)

\begin{equation} \overline{kWh}_{d,h} = \frac{\sum_{1}^{N_{f}(d,h)} kWh_{n}}{N_{f}(d,h)} \end{equation}

Where,

$\overline{kWh}_{d,h} =$ Average energy of each hour for each day of the week (in kWh)

$kWh_{n} =$ Measured hourly energy kWh from the data-logging power logger (in kWh)

$N_{f}(d,h) =$ Total number of measuremed data points that fall on day of week, d, and hour the day, h

2. Find total daily energy consumption (Worksheet: “Step 3. Results,” column C, “Step 3. Hourly Results.”)

\begin{equation} \overline{kWh}_{d} = \sum_{h = 0}^{23} \overline{kWh}_{d,h,n} \end{equation}

Where,

$\overline{kWh}_{d} =$ Sum of energy consumption for each day of the week (in kWh)

$\overline{kWh}_{d,h} =$ Average energy of each h hour for each day of the week (in kWh)

$h =$ Hour of the day where 0 is 12:00 a.m. and 23 is 11:00 p.m.

3. Calculate the average energy consumption for weekdays (Worksheet: “Step 3. Results,” cell D3.)

\begin{equation} \overline{kWh}_{Wd} = \frac{\sum_{d = 2}^{6} \overline{kWh}_{w,d}}{5} \end{equation}

Where,

$\overline{kWh}_{Wd} =$ Average energy consumption for weekdays (in kWH)

$\overline{kWh}_{w,d} =$ Average energy of each n weekday (in kWh)

$d =$ day of week: (2 = Monday, 3 = Tuesday, ... 6 = Friday)

4. Calculate the average energy consumption for weekend days (Worksheet: “Step 3. Results,” cell D6.)

\begin{equation} \overline{kWh}_{WEd} = \frac{\overline{kWh}_{we,1} + \overline{kWh}_{we,7}}{2} \end{equation}

Where,

$\overline{kWh}_{WEd} =$ Average energy consumption for each day of the week (in kWh)

$\overline{kWh}_{we,d} =$ Average energy of each n weekend day (in kWh)

$d =$ Day of week: (7 = Saturday, 1 = Sunday)

5. Calculate the annual weekday energy consumption (Worksheet: “Step 3. Results,” cell E3.)

\begin{equation} \overline{kWh}_{WdYr} = \overline{kWh}_{Wd} * (261 \hspace{2mm} weekdays \hspace{2mm} per \hspace{2mm} year - X) \end{equation}

Where,

$\overline{kWh}_{WdYr} =$ Annual weekday energy consumption (in kWh)

$\overline{kWh}_{Wd} =$ Average weekday energy consumption (in kWh)

$X =$ Number of weekdays that are adjusted to use weekend day average energy consumption

6. Calculate the total weekend day energy consumption (Worksheet: “Step 3. Results,” cell E6.)

\begin{equation} \overline{kWh}_{WEdYr} = \overline{kWh}_{WEd} * (104 \hspace{2mm} weekend \hspace{2mm} days \hspace{2mm} per \hspace{2mm} year + X) \end{equation}

Where,

$\overline{kWh}_{WEdYr} =$ Annual weekend day energy consumption (in kWh)

$\overline{kWh}_{Wd} =$ Average weekday day energy consumption (in kWh)

$X =$ Number of weekdays that are adjusted to use weekend day average energy consumption

7. Calculate total annual estimate energy consumption (Worksheet: “Step 3. Results,” cell F3.)

\begin{equation} \overline{kWh}_{anm} = \overline{kWh}_{WdYr} + \overline{kWh}_{WEdYr} \end{equation}

Where,

$\overline{kWh}_{ann} =$ Annual energy consumption (in kWh)

$\overline{kWh}_{WdYr} =$ Annual weekday energy consumption (in kWh)

$\overline{kWh}_{WEdYr} =$ Annual weekend day energy consumption (in kWh)

8. If more than one electric panel or switchgear was measured, sum the annual energy consumption of all panels to find the total energy consumption of the electrical distribution system.

\begin{equation} \overline{kWh} = \overline{kWh}_{{n}_{1}} + \overline{kWh}_{{n}_{2}} + ... \end{equation}

Where,

$\overline{kWh} =$ Total energy consumption for the electrical distribution system (in kWh)

$\overline{kWh}_{n} =$ Total energy consumption for each panel or switchgear(in kWh)

$n =$ Number of panels or switchgear measured