Electrical Current

1. Convert seconds fan is on per hour to percent fan is on per hour (Worksheet: “Step 2. Percent Runtime Calcs”) for supply and return fans.

\begin{equation} \%FanOn_{high,f}(t_{f}) = \begin{cases} 100\% & \text{if } i_{high,f} * (1+\varepsilon) < i_{f}(t_{f}) \\ \frac{i_{f}(t_{f}) - i_{low,f}}{i_{high,f} - i_{low,f}} & \text{if } i_{low,f} * (1+\varepsilon) < i_{f}(t_{f}) < i_{high,f}* (1-\varepsilon) \\ 0\% & \text{if } i_{f}(t_{f}) < i_{low,f} * (1+\varepsilon) \end{cases} \end{equation}

\begin{equation} \%FanOn_{low,f}(t_{f}) = \begin{cases} 100\% & \text{if } i_{f}(t_{f}) < i_{low,f} * (1+\varepsilon) \\ 1 - \frac{i_{f}(t_{f}) - i_{low,f}}{i_{high,f} - i_{low,f}} & \text{if } i_{low,f} * (1+\varepsilon) < i_{f}(t_{f}) < i_{high,f}* (1-\varepsilon) \\ 0\% & \text{if } i_{high,f} * (1-\varepsilon) < i_{f}(i) \end{cases} \end{equation}

Where,

$\%FanOn_{high,f}(i) =$ Percent of an hour that motor is running at high speed for either supply or return fan, f, %

$\%FanOn_{low,f}(i) =$ Percent of an hour that motor is running at low speed for either supply or return fan, f, %

$i_{f}(t_{f}) =$ Measured current for either supply or return fans, f, Amps

$t_{f} =$ Index for each measured data point for either supply or return fans, f, datasets

$i_{high,f} =$ One-time measured current at high speed setpoint for either supply or return fan, f, Amps

$i_{low,f} =$ One-time measured current at the low speed setpoint for either supply or return fan, f, Amps

$\varepsilon =$ Error tolerance to classify current as representing high speed or low speed operation

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

\begin{equation} \overline{\%FanOn}_{s,d,h,f} = \frac{\sum_{n_f=1}^{N_{f}(d,h)} \%FanOn_{s,f}(n)}{N_{f}(d,h)} \end{equation}

Where,

$\overline{\%FanOn}_{s,d,h,f} =$ Average % time motor is at either low speed or high speed setpoint, s, per hour for given day of wekk, d, and hour of the day, h, for either supple of return fan, %

$N_{f}(d,h) =$ 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

$n_{f} \in i_{f}(d,h) =$ 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

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

\begin{equation} \overline{HrsOn}_{s,d,f} = \sum_{h=0}^{23} \overline{\%FanOn}_{s,d,h,f} \end{equation}

Where,

$\overline{HrsOn}_{s,d,f} =$ Average hours per day for given day of week, d, for either supply or return fan, f, hours

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

\begin{equation} E_{f} = \sum_{d=1}^{7} (\overline{HrsOn}_{low,d,f} * P_{low,f} + \overline{HrsOn}_{high,d,f} * P_{high,f}) * 52 [weeks \hspace{2mm} per \hspace{2mm} year] \end{equation}

Where,

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

$P_{s,f} =$ Measured power for either the low speed or high speed setpoint, s, and for either supply or return fan, f, kW

5. Total energy used by Fan Motor (Worksheet: “Step 4. Results”).

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

Where,

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

1. Total energy for each hour (Worksheet: “Step 2 Energy Calcs”). Since the measured current (Amps) is the average for an hour, Equation 10 yields the average energy per hour (kWh) assuming voltage is constant.

\begin{equation} \overline{E}_{f}(t) = \frac{i_{f}(t) * P_{f}}{I_{f}} \end{equation}

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

Where,

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

$i_{f}(t) =$ Average hourly current for each fan, f, Amps

$t =$ Time index

$P_{f} =$ One-time power measurement for each fan, f, kW

$I_{f} =$ One-time current measurement for each fan, f, Amps

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

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

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

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

Lighting fixtures generally require single-phase power to operate, but electrical distribution systems are commonly three-phase. When measuring individual circuits of a panelboard, it is essential to know which phase the circuit is connected to. A proper line-to-neutral connection is required to accurately measure the circuit’s voltage.

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

This calculation tool requires the phase of the circuit to be specified in order to calculate annual energy consumption. If multiple circuits are measured and are connected to the same phase, then only the voltage of that phase is necessary Additionally, the user should ensure that the panelboard is balanced (i.e., all three electrical lines, or phases, must have the same current and line-to-line voltage). If the panelboard is not balanced, then it must be specified in the calculation tool (see worksheet ‘Step 1.1 Circuit Raw Data’).

The following equations are used to calculate the annual energy consumption of a lighting electrical distribution system where AC current is measured at the output of circuit breakers in an electrical panelboard. AC current should be measured using a data logger with current transformers (e.g., Onset HOBO 4-channel analog logger and Onset CTV-x current transformer sensors) for one or more circuit breakers; a group of circuits can be measured with a single transformer. Current data should be at one-hour intervals and data should be averaged with a sample of measurements[1] for each one-hour interval. Obtain voltage from spot measurements (i.e., measured one time) with a power meter at the circuit breakers.

It is assumed that multiple spot measurements are taken and averaged (e.g., measure line-to-neutral voltage for a circuit breaker three times at five-minute intervals and calculate the average) see Equation (9). The average value should be used with these equations to reduce measurement uncertainty. In this scenario, the line-to-neutral voltage is measured for a single circuit. If measuring multiple circuits, be sure to take multiple spot measurements of line-to-neutral voltage for all circuits.

\begin{equation} V_{LN,avg} = \frac{V_{t1,n} + V_{t2,n} + V_{t3,n} + V_{tx,n}}{x} \end{equation}

Where,

$n =$ The electrical line that was measured of the three-phase system

$V_{avg,n} =$ Average line-to-neutral voltage of a circuit in electrical line n

$V_{t1,n} =$ First measurement of voltage for electrical line n

$V_{t2,n} =$ Second measurement of voltage for electrical line n, at least five minutes after the first measurement

$V_{t2,n} =$ Third measurement of voltage for electrial line n, at least five minutes after the second measurement

$x =$ Number of spot measurements taken, at least five minutes apart

1. For each circuit or group of circuits measured, calculate average power for each hour interval (Worksheet: “Step 2. Power Calcs.”). Power is calculated using line-to-neutral voltage and current of the circuit.

\begin{equation} \overline{kW}_{h,n} = \frac{I_{n} * V_{LN,avg}}{1000} \end{equation}

Where,

$\overline{kW}_{h,n} =$ Average hourly power for a circuit n (in kW)

$I_{n} =$ Measured average houry current for a circuit (in Amps)

$V_{I,N,avg} =$ Measured average line-to-neutral voltage for a circuit (in V)

2. Calculate average energy use for each hour of each day of the week (Worksheet: “Step 3. Avg Energy Calcs, column C, D, E, F.”). In this step, the hourly power draw (kW) gets converted to hourly energy consumption (kWh) because data is in one-hour intervals. kWh = kw * h, where h = 1.

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

Where,

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

$kW_{h} =$ Total power at each hour of each day of each week (in kW)

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

3. Calculate the sum of average hourly energy of all circuits (Worksheet: “Step 3. Avg Energy Calc” column G.)

\begin{equation} \overline{kWh}_{d,h,n} = \sum_{c=1}^{4} \overline{kWh}_{d,h,c} \end{equation}

Where,

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

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

$c =$ Circuit that was measured

4. Calculate total daily energy consumption for a given day of the wekk (Worksheet: “Step 4. Results,” column C.)

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

Where,

$\overline{kWh}_{d,w} =$ Total average energy for each day of the week (in kWh)

$\overline{kWh}_{d,h} =$ Total 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.

5. Calculate the average energy consumption for weekdays in a week for all circuits (Worksheet: “Step 4. Results,” cell D3.)

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

Where,

$\overline{kWh}_{WdCt} =$ Average energy on a weekday (in kWh)

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

$n =$ Day of week (2 = Monday, 3 = Tuesday, ..., 6 = Friday)

$24 =$ Hours per day

6. Calculate the average energy consumption for weekend days in a week for all circuits (Worksheet: “Step 4. Results,” cell D6.)

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

Where,

$\overline{kWh}_{WEdCT} =$ Average energy on a weekend day (in kWh)

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

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

$24 =$ Hours per day

7. Calculate the total annual weekday energy consumption (Worksheet: “Step 4. 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 weekly weekday energy consumption (in kWh)

$X =$ Number of weekdays that are considered holidays and adjusted to weekend day average energy consumption

8. Calculate the total annual weekend day energy consumption (Worksheet: “Step 4. Results,” cell E6.)

\begin{equation} \overline{kWh}_{WEdYr} = \overline{hWh}_{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)

$\overline{kWh}_{WEd} =$ Average weekly weekend day energy consumption

$X =$ Number of weekdays that are considered holidays and adjusted to weekend day average energy consumption

9. Calculate total annual estimated energy consumption.

\begin{equation} \overline{kWh}_{ann} = \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)

Equation (15) is the annual energy consumption of the measured circuits for a panelboard. If additional circuits were measured for the same panelboard with a different data logger, be sure to consolidate the results (annual energy consumption) to obtain the total panelboard annual energy consumption. Additionally, if multiple panelboards were measured, be sure to sum up the annual consumption of all panelboards to obtain the total system energy consumption.