3.1.2. Chilled Water Production

In many sites, chilled water is produced at a central location and used for conditioning the air in local buildings. There is a cost associated with the production of this chilled water, and it is important to capture in an economic control method. In the presented case study, chilled water provided by the central plant is passed through the UBO's AHU chilled water coil to lower the temperature of the moving air as well as reduce the air's humidity. The amount of chilled water passing through the

coil is controlled by a valve which is actuated by a PID control loop. This control loop is driven by a difference in a discharge air temperature setpoint and the discharge air temperature. The discharge air temperature setpoint is the output of a supervisory PID control loop (see Figure 3 and Table 1) that is trying to maintain a cooling demand setpoint. A cooling demand calculation supplies the feedback signal for this supervisory controller that is a weighted combination of the average and maximum cooling loopout values from the individual zone control loops.

Figure 5 details the chilled water loop. A chilled water pump works to maintain a specific supply pressure to provide the required chilled water flow for the AHU. The chilled water supply and return temperatures are available to measure within the energy managemen<sup>t</sup> system.

**Figure 5.** Chilled water loop for the Utilities Business Office (UBO) at Texas A&M University.

## 3.1.3. Conditioned Zones in Buildings

Zones within buildings can consist of one or multiple rooms, and can be serviced by one or multiple pieces of equipment. Frequently, zones in commercial buildings are assigned a VAV terminal box or damper that regulates the amount of conditioned air that is introduced to that space. There is typically a thermostat placed in the zone to provide feedback for the zone controller to regulate temperature.

As described above, the UBO consists of 11 zones, 10 of which are actively controlled. Each controllable zone is serviced by a VAV terminal box equipped with hot water reheat capabilities, an example of which is shown in Figure 6. The flow of conditioned air into the room is regulated by a damper in the terminal box whose position is determined by a PID control loop. The error signal for the control loop is the difference between the respective room temperature setpoint and the measured room temperature while the output is the damper position (see Table 1). The room temperature setpoint is determined by whether the room is occupied or unoccupied and if the zone is in heating or cooling mode. A supervisory deadband control method (see Figure 3) is employed such that if the room is occupied, the zone VAV will heat the room to 70 °F or cool the room to 74 °F. If the room is unoccupied, the VAV will heat the room to 60 °F or cool the room to 85 °F. If the occupancy sensors in all the rooms read unoccupied, then the main AHU will turn off and the room temperatures will freely fluctuate.

**Figure 6.** Variable Air Volume (VAV) box in the Utilities Business Office (UBO) at Texas A&M University.

#### *3.2. Economic Cost of Operating Equipment*

Having described the existing equipment and controls, the following sections will outline the economic cost functions for said equipment, particularly an AHU and individual zones. Examining a typical AHU, two sources of economic cost become apparent: (1) the cost of electricity used by the AHU fan to move the air; and (2) the cost associated with the production of chilled water for cooling of the air. At the zone level, there is usually no building equipment that consumes a significant amount of power or resources (when considering cooling only applications; VAVs with local reheat would be a different situation). For example, the damper in a VAV terminal box is the only actuated component and the power required to move it is negligible.

For the presented case study, an economic objective function that minimizes the cost of occupant discomfort as a measure of the loss of productivity at the zone level was developed. As mentioned in the Background Section, select previous building energy optimizations have included some form of occupant comfort, whether simply as constraints on the optimization or as a measure to be minimized; however, to the knowledge of the authors, no previously simulated or implemented building energy optimization has considered the economic cost of occupant comfort. The following sections detail the development of these objective functions.

#### 3.2.1. AHU Fan Economic Objective Function Development

As described above, the fan in an AHU works to maintain an end static pressure in the duct to move the required amount of air to condition the individual zones. To accomplish this, the fan motor requires electricity and there is a cost associated with the power used. A simple way to measure the power consumed by a fan motor would be to use a power meter on the electrical lines to the fan and measure the power consumed; however, power meters are not often included on individual fans within existing building energy systems. Therefore, for the presented case study of the UBO, the power consumed by the fan was instead determined by other available data. Specifically, the change in pressure across the fan and the volume flow rate of air were used. With these two values, the work being performed by the fan on the air can be estimated and converted into a measure of power. The equation for work performed by a fan is given in Equation (3):

$$P\_{fan} = 0.1175 \cdot \frac{q\_{AHL} \cdot \Delta P}{\mu\_f \cdot \mu\_b \cdot \mu\_m} \tag{3}$$

where *Pf an* is the power consumed by the fan [W], 0.1175 is a conversion factor for imperial units, *qAHU* is the air flow rate through the AHU [cfm], Δ*P* is the change in pressure across the fan [in. H2O], and *μf* , *μb*, and *μm* are efficiencies for the fan blade, the fan belt, and the fan motor, respectively. The efficiencies were all assumed to be 0.9 based on comparable values to reflect some degradation from the ideal efficiencies of new/perfectly tuned equipment. The change in pressure across the fan was taken from the end static pressure sensor in the AHU. As for the volume flow rate, the flow rate

meters included in each of the VAVs were used in summation to calculate the total air flow rate through the AHU, assuming minimal duct losses and leaks.

The most appropriate control action *u* for the AHU fan objective function would be the end static pressure setpoint. Assuming the local PID controllers have zero steady state error and sample significantly faster than the new supervisory controller, a reasonable assumption can be made that the end static pressure will equal the end static pressure setpoint. As such, assuming an electric utility rate of \$0.12 per kWh, the fan power cost can be calculated as:

$$J\_{fan} = \frac{0.1175}{1000} \cdot \frac{\mathbb{C}\_{elcc} \cdot q\_{AHUI} \cdot P\_{EDS}^\* \cdot t\_s}{\mu\_f \cdot \mu\_b \cdot \mu\_m} \tag{4}$$

where *Celec* is the rate of electricity cost [\$/kWh], *<sup>P</sup>*<sup>∗</sup>*EDS* [in. H2O] is the end static pressure setpoint, and *ts* [hr] is the sampling time of the supervisory controller. Reformulating Equation (4) to fit Equation (1) gives:

$$\begin{aligned} I\_{fan} &= \mathbf{e}^T Q \mathbf{e} + \mathbf{e}^T R + \mathbf{u}^T S \mathbf{u} + \mathbf{u}^T T \\ Q &= 0 \quad R = 0 \quad S = 0 \\ \mathbf{u} &= P\_{EDS}^\* \\ T &= \frac{0.1175 \cdot \mathbf{C}\_{dlec} \cdot q\_{AHL} \cdot \mathbf{t}\_s}{1000 \cdot \mu\_f \cdot \mu\_b \cdot \mu\_m} \end{aligned} \tag{5}$$

thus establishing an objective function that can minimize the cost of electricity used by the AHU's VAV fan by optimizing the end static pressure setpoint.

#### 3.2.2. AHU Chilled Water Economic Objective Function Development

As described in the building operation details, chilled water is used in the AHU to condition the zone supply air. If specific details about the chiller are known, the consumption of power can be calculated and then used to determine the cost of producing said chilled water. In the case study, utility usage data were leveraged to determine this cost. The UEM Office at Texas A&M University maintains utility usage data, specifically the cost per unit of energy of the respective utility. For chilled water, this is the dollar cost associated with producing one mmBtu of chilled water at the campus wide chilled water temperature of 45 °F. In the AHU, the discharge air temperature setpoint is tracked by a PID loop that actuates the valve metering how much chilled water flows through the chilled water coil. The energy associated with chilled water usage can be determined by Equation (6):

$$
\hat{Q} = \dot{m} \cdot \mathcal{c} \cdot \Delta T \tag{6}
$$

where *Q* ˙ is the rate of change of heat, or energy, *m*˙ is the mass flow rate, *c* is the specific heat of the respective fluid, and Δ*T* is the change in temperature of the fluid. If a mass flow rate sensor is available on the AHU for the chilled water, then this measurement can be used to calculate the rate of change of energy of the chilled water and thus the overall cost; however, mass flow rate sensors are not usually installed on chilled water lines at the AHU level. If this is the case, then a volume flow rate can be used such that:

$$
\dot{m} = \rho \cdot q \tag{7}
$$

where *ρ* is the density of water and *q* is the volume flow rate. With the UBO, the maximum flow rate through the chilled water valve was determined to be 3.41*e*<sup>−</sup><sup>3</sup> m3/s (54.05 gal/min). Assuming a linear valve/flow relationship, the chilled water valve position multiplied by the maximum possible flow will give the current volume flow rate of the chilled water. Using data values for the discharge air temperature setpoint and the chilled water valve position, a fit was generated to transform the setpoint to a valve position [31]. Combining the relationships gives Equation (8)

$$\begin{aligned} J\_{\rm CHW} &= \mathbf{a} \cdot q\_{\rm max} \cdot \boldsymbol{\rho} \cdot \mathbf{c} \cdot \Delta T\_{h\_2o} \cdot t\_s \cdot 0.00341214 \cdot \frac{3682}{241} \\ &\quad a = \left( 3.021 T\_{DA}^{\*2} - 109 A T\_{DA}^{\*} + 1002 \right) \end{aligned} \tag{8}$$

where *α* is the fitted relationship between the discharge air temperature setpoint ( *<sup>T</sup>*<sup>∗</sup>*DA* [ ◦C]) and the chilled water valve position [%], *qmax* [m3/s] is the maximum flow through the valve (at 100% opening), *ρ* [kg/m3] is the density of water, *c* [kJ/kg ◦C] is the specific heat of water, Δ *Th*2*<sup>o</sup>* [ ◦C] is the change in temperature of the supply and return chilled water, *ts* [h] is the sample time of the supervisory controller, 0.00341214 [mmBtu/kWh] is a conversion factor from kWh to mmBtu, and the last term is the economic cost for the UBO of the consumed chilled water. Reformulating Equation (8) to fit Equation (1), where *u* equals the discharge air temperature setpoint, gives:

$$\begin{aligned} I\_{CHW} &= e^T Q e + e^T R + u^T S u + u^T T \\ Q &= 0 \quad R = 0 \quad S = 0 \end{aligned}$$

$$\mu = \mu = \left( 3.021 \cdot T\_{DA}^{\*2} - 109.4 \cdot T\_{DA}^\* + 1002 \right) \tag{9}$$

$$T = q\_{\text{max}} \cdot \rho \cdot c \cdot \Delta T\_{h\_{20}} \cdot t\_s \cdot 0.00341214 \cdot \frac{3682}{241}$$

where the linear cost term *uTT* is equal to Equation (8). This result actually lends itself well to traditional convex optimization of the setpoint *<sup>T</sup>*<sup>∗</sup>*DA*, due to the quadratic nature of the fit.

#### 3.2.3. Zone Occupant Comfort Economic Objective Function Development

To measure an occupants level of discomfort, many have relied on the use of predicted mean vote (PMV), developed by Fanger in the 1970s [29]. PMV is a measure on the American Society of Heating, Refrigeration, and Air-Conditioning Engineers (ASHRAE) thermal sensation scale of −3 to 3, where negative numbers represent being too cold, positive numbers represents being too warm, and a value of zero represents being comfortable. Performing a large study of people, Fanger collected data regarding occupants votes and created an equation to determine the PMV across a variety of environmental factors. These factors include air temperature, air relative humidity, relative air speed, mean radiant temperature, an occupant's insulation level due to clothing, an occupant's metabolic rate, and an occupant's work output, among several other variables. The details of the equation can be found in [29]. From PMV, Fanger determined the Predicted Percentage of Dissatisfied (PPD) of people. The relationship of PPD to PMV can be seen in Figure 7a.

 **Figure 7.** Curves used in the zone occupant comfort economic objective function development. These equations/relationships and their values are defined in [29,32]. (**a**) Relationship between PPD and PMV as calculated in Fanger [29]. (**b**) Shape of Loss Of Productivity function as calculated in [32].

One can see that at −3 and 3 PMV approximately 100% of the population would be dissatisfied. Of note is that, even at 0 PMV, 5% of the population, on average, will still be dissatisfied, attesting to the fact that each individual has specific preferences. While PMV determines how many people will be dissatisfied, a relationship to tie this to an economic cost is still needed.

Fortunately, researchers have investigated the effect of occupant comfort on worker productivity, as mentioned in the literature review. One effort in particular tied PMV to a measure of Loss of Productivity (LOP) [%]. By using regression analysis, a direct relation can be calculated between a worker's loss of performance and the PMV of an indoor climate by including the calculations of equivalent thermal situations from Gagge's two-layer human model [33] and Fanger's comfort equation [29]. For a detailed explanation of the relationship, see [32]. The results of Roelofsen's work [32] are two sets of coefficients for a regression fit for the cold side of the PMV comfort zone and for the warm side of the PMV comfort zone. The regression is the sixth-order fit shown in Equation (10):

$$LOP = c\eta + c\_1 PMV + c\_2 PMV^2 + c\_3 PMV^3 + c\_4 PMV^4 + c\_5 PMV^5 + c\_6 PMV^6 \tag{10}$$

where *LOP* is the loss of productivity and *c*0, ... , *c*6 are the regression coefficients. The value of the Roelofsen's coefficients are repeated in Table 2, for reference. Roelofsen constructed the regression on the cold side to be zero at −0.5 PMV and for the warm side to be zero at 0 PMV, leaving a region between −0.5 and 0 PMV where LOP is zero. This bias is because several studies found a region of conditions near and below 0 PMV that had a negligible effect on productivity. Figure 7b shows the change in LOP as PMV varies.


**Table 2.** Regressions for loss of productivity fit of PMV; values presented in [32].

To connect LOP to an economic cost, a multiplication of the lost productivity percentage by the amount of salary an employee earns over the sample period can be used, as shown in Equation (11):

$$
\beta = LOP \cdot \left(\frac{p\_{ycar} \cdot t\_s}{52 \cdot 40}\right) \tag{11}
$$

where *β* [\$] is the lost productivity in wages, *pyear* [\$] is the annual salary for the zone, and *ts* [h] is the sampling time. A 40-h work week for the entire year was used as a conservative assumption. In reality, there will be some variation due to holidays, vacation, and overtime.

Considering the individual zones in the presented case study with respect to Equation (1), it is noted that, while there is no control action *u* at the zone level that consumes energy, there is an error signal present. That is the error *e* in Equation (1), which is given by:

$$
\sigma = T\_{zone} - T\_{zone}^\* \tag{12}
$$

where *Tzone* and *T*<sup>∗</sup>*zone* are the zone temperature and zone temperature setpoint, respectively. The error is defined in this manner as this publication focuses on the systems when in cooling mode; thus, the error will be mostly positive during cooling mode. If the zone were in heating mode, the sign of the error term would need to reversed. Recognizing that the zone temperature setpoint can be optimized by the supervisory controller to minimize the LOP by optimizing the zone's PMV, and that the error signal is a difference of temperatures, the sensitivities of the above relationships can be determined and combined to give an economic objective function.

In determining PMV, only the zone air temperature and zone air relative humidity are changing. As such, a fit of the PMV equation was created and used to reduce computation time and complexity. A metabolic rate of 70 [W/m2], a clothing insulation factor of 0.75 [m2K/W], and a relative air velocity of 0.2 [m/s] were assumed. Additionally, the mean radiant temperature was assumed to be equal to the zone air temperature. The fit is defined as:

$$PMV = 0.5542 \cdot Rh\_{zonte} + 0.23 \cdot T\_{zonte} - 5.44 \tag{13}$$

where *Rhzone* [%] is the relative humidity of the zone and *Tzone* [ ◦C] is the zone air temperature. Equation (13) is used to determine the sensitivity of PMV to changes in air temperature. The resulting combination of sensitivities is shown in Equation (14):

$$\begin{aligned} f\_{\text{room}} &= e^T Q \varepsilon + e^T R + \mu^T S \mu + \mu^T T\\ Q &= 0 \quad S = 0 \quad T = 0\\ R &= \left[\frac{\partial P MV}{\partial T\_d}\right] \cdot \left[\frac{\partial LOP}{\partial PMV}\right] \cdot \left[\frac{\partial \beta}{\partial LOP}\right] \end{aligned} \tag{14}$$

where the sensitivities are determined to be:

$$
\left[\frac{\partial PMV}{\partial T\_a}\right] = 0.23\tag{15}
$$

$$
\left[\frac{\partial LOP}{\partial PMV}\right] = \mathfrak{c}\_1 + 2\mathfrak{c}\_2 PMV + 3\mathfrak{c}\_3 PMV^2 + 4\mathfrak{c}\_4 PMV^3 + 5\mathfrak{c}\_5 PMV^4 + 6\mathfrak{c}\_6 PMV^5 \tag{16}
$$

$$
\left[\frac{\partial\beta}{\partial LOP}\right] = \left(\frac{p\_{year} \cdot t\_s}{52 \text{wks} \cdot 40 \text{hrs}}\right).
\tag{17}
$$

The cost calculation assumes that, if more than one occupant is in a zone, the sum of the occupant's salaries is used for *pyear*. For Equation (16), the coefficients are as described in Table 2. If the zone's PMV falls within the range of −0.5 to 0, then Equation (16) is equal to zero. The overall trend of the objective function as the zone air temperature varies is shown in Figure 8. The abrupt changes occur at the PMV values of −0.5 and 0, due to the fit of the LOP equation.

**Figure 8.** The change in objective cost of the loss of productivity due to discomfort due to changes in zone air temperature during a 15 min period.

This curve will shift depending on other variables in the objective function that are varied, such as annual salary and relative humidity of the zone. In addition, depending on the value of the user-defined setpoint, the objective cost can become negative. This does not mean that the zone is earning money, but occurs because of the structure of the objective function. An optimization competition can occur between the error term and the coefficient *R*. As the error is defined as the difference in the zone temperature and zone temperature setpoint, the objective cost will be zero when the zone reaches this defined setpoint; however, if this setpoint is not equal to the optimal comfort temperature, as determined by the PMV function, then the objective cost will also be zero if the zone air temperature is equal to the optimal comfort temperature.

This behavior presents an interesting control question. To operate at the most cost effective point, as defined by the objective function, would require that the zone temperature setpoint be set to the PMV optimal temperature, but, in doing so, the ability for occupants or building managers to provide the system feedback about their comfort is removed. Thus, which is more important: the ability for occupants to choose their zone temperatures or allowing the system to determine what is best for the occupants? This question deserves additional investigation but is beyond the scope of this publication. Fortunately, with this objective function, the system will propose a compromise: a temperature between the user-defined setpoint and the PMV optimal comfort temperature.

For a centralized implementation, the objective function then becomes:

$$J\_{\text{total}} = J\_{\text{fan}} + J\_{\text{CHW}} + \sum\_{i=1}^{n} J\_{\text{zone},i} \tag{18}$$

where *n* is the number of zones in the system. Equation (18) is the system objective function that will be minimized in the optimization.
