**2. Background**

Control of buildings presents several unique challenges, many of which arise form the interconnected nature of building systems. For example, in a large building there may be multiple chillers that are used to chill a secondary fluid, such as water. This chilled water is then pumped to various systems and areas of buildings where heat exchangers in air handling units (AHUs) use the chilled water to cool and dehumidify air streams. A network of fans and ducts then deliver the cooled air to the desired locations. The flow of this cooled air into the zones can be controlled by variable air volume (VAV) units, in which there may be another heat exchanger that utilizes heated water to warm the air, if necessary. The heated

water for this process is provided by a different set of centralized pumps and heat exchangers. The zones themselves are connected to one another, either by conduction through barriers or convection through shared doorways/open spaces. All these interconnections and couplings result in coordinated control problems.

Another difficulty comes from nonlinear dynamics evolving over multiple time scales across these various building systems. While changes in damper position of a VAV or fan speed in an AHU are relatively fast (on the order of seconds), changes in desired chilled water temperatures can take longer (minutes), or changes in room air or wall temperature can be even slower (hours). There are also slow, overarching changes to take into consideration such as the shift in solar loads as the sun moves throughout the day, gradual changes in outdoor air temperature due to change in weather or seasons, and the slow deterioration of equipment through use over time. These all contribute to shifting system behaviors and disturbances that can cause undesired performance. There are also discrete changes to take into account, such as whether an area/room is occupied, how many people are in the room, opening and closing of windows or doors, or changes in real-time pricing of utilities. Furthermore, the system sensors are distributed (not always equally), monitoring is performed centrally, and devices are driven by localized controls. All of this occurs within constraints, either due to hardware limitations, limited resources, or issues of health and comfort.

While the challenges of building control are numerous, one control method that has emerged as a capable solution is model predictive control (MPC), also known as receding horizon control. MPC has been chosen as the most appropriate control method in several building thermal control research projects [9–11]. MPC predicts the change in dependent variables of a modeled system by optimizing independent variables. Using the current state information, dynamic models of the system, and an objective function, MPC will determine the changes in the independent variables that will minimize the user-defined objective function while honoring given constraints on both dependent and independent variables. Once this series of changes is determined, the controller will apply the first determined control action, and then repeat the calculations for the next time step. Figure 1 displays how a typical reference tracking MPC implementation would behave. It can be seen at time *k* that the controller determines what the predicted output would be along with its optimal control trajectory. After completing the computations, the control would be applied and the system would move on to time *k* + 1, repeating the predictions and optimization with the new measurements. More details about MPC can be found in [12].

MPC applications for buildings have been studied mostly in simulation [13–22], with a few experimental efforts [9–11,23–25]. In simulation, MPC has been adapted for controlling building systems such as floor heating [26], water heating [27], cooling [11], and ventilation [28], among others. There is a large body of work on MPC in buildings, but what is not as clear is if full MPC is needed to solve the challenges presented by controlling buildings. Thus, the research presented here proposes a steady-state method that is inspired by previous MPC efforts.

For most reported applications of MPC for buildings, the cost takes the form of economic MPC (E-MPC), where the objective function is a linear combination of the monetary cost of building energy consumption [17]. In these applications when E-MPC is used, the amount of energy consumed is being minimized, while authors occasionally account for occupant comfort limits through constraints on the variables [13,15,17,18,20,23,24]. These proposed control strategies are able to respond to real-time changes in utility pricing and optimize energy usage; however, occupant comfort becomes a second priority with limits on the ranges of parameters such as temperature and lighting levels. These limits ensure the controller does not drift too far from the user-defined comfort zone, but does not actually optimize occupant comfort or the associated cost of discomfort. In fact, the energy optimal solution generally maximizes discomfort within prescribed limits.

**Figure 1.** How MPC, or receding horizon control, typically functions.

Some researchers have included comfort in the optimization beyond limits on parameters, e.g., Corbin et al. [19] minimized the sum of electricity used by all HVAC equipment and a comfort penalty. This comfort penalty was defined as an area-weighted sum of the number of zone occupied hours outside of a predicted mean vote (PMV) threshold of ±0.5. PMV is an index that determines the thermal comfort of an average individual dependent upon a variety of factors, including air temperature, relative humidity, relative air velocity, metabolic rate, clothing insulation level, work output, and several other variables [29]. While the researchers of [19] worked to optimize occupant comfort, they did so with the focus of reducing the cost of energy usage, neglecting the economic cost associated with occupant comfort and productivity.

The authors of [21] also included a discomfort cost with their monetary energy cost that was based on different lower and upper thermal limits; however, the physical meaning of this discomfort cost is arbitrary as the cost increases to unity until the temperature limits are exceeded and then becomes significantly large, not following any physical or measured relationship. The cost function in [22] included regulation of occupant comfort based on PMV, though it took the quadratic cost form, with the first term being the difference between the predicted PMV of the zone and the PMV setpoint for the zone, quantity squared, multiplied by a weighting factor. The second term consisted of the square of the change in control action, or increment, multiplied by a weighting factor. With this form, the MPC will balance maintaining the desired zone PMV while limiting large control action rates, specifically changes in water flow and air velocity. This will help keep occupants comfortable but the economic cost of the control actions is not accounted for. Morosan et al. [16] used a linear objective function that penalized the error between the predicted room temperature and the future room temperature reference as well as the energy usage to condition the room. In their efforts, comfort is accounted for as a comfort index that acts as a penalty when the room temperature does not meet its setpoint. However, in the presented simulations, the temperature setpoints are arbitrarily chosen and not dependent on any comfort information. Additionally, this method, similar to the previous ones mentioned, does not account for the economic aspect of occupant comfort.

One objective function from the literature that appears more unique than others appeared in [25]. This objective function consisted of three different linear terms: (1) a weighting coefficient multiplied by the predicted percentage of dissatisfied (PPD) people, which PPD can be calculated from PMV; (2) a weighting coefficient multiplied by the summation of cost of energy consumed by the heating and cooling devices; and (3) a weighting coefficient multiplied by the summation of the green house gas intensities of the various energy sources (electricity and natural gas). This objective function displays the power of MPC to determine optimal control actions with respect to a user's desired metrics, in this case occupant comfort, monetary cost of energy, and environmental impact of energy sources. While providing grea<sup>t</sup> flexibility in allowing the building operator to prioritize the three metrics with the weighting factors, the respective economic impact of the three areas is not represented in the objective function due to differing units and arbitrary weights.

Overall, previous methods have accounted for the economic cost of energy usage and/or attempted to maintain occupant comfort through optimization constraints or optimizing comfort itself, but none have accounted for the economic aspect of comfort on occupant productivity alongside utility costs. Additionally, it is unknown whether comparable results can be achieved with less computational burden by using a steady-state predictive model as opposed to a full dynamic MPC solution. This research effort aimed to develop a novel supervisory control method that optimizes a building's economic cost, due to both the consumption of utilities and the economic cost of loss of productivity due to occupant discomfort. This control method emulates MPC in that it uses models to predict the system's behavior at steady-state, such as was done by Elliott and Rasmussen [30]. The focus of this paper is on the development of the economic costs of the system, while a full MPC implementation will be completed in future work. Through analysis of the performance of this control method, the authors intend to identify areas having the most potential for savings and guide the priorities of building managers and researchers for future work.

#### **3. Development of Economic Objective Function for Advanced Building Systems Control**

Considering the literature and previous works, a general component-level objective function of the form shown in Equation (1) was chosen. The quadratic terms (first and third) were included to allow for standard convex optimization when desired, where *e* represents the error for the system, *Q* the weighting placed upon the error, *u* the control action, and *S* the weighting placed on control action. The linear terms (second and fourth) were included to facilitate calculating cost purely in economic terms (dollars), as opposed to nonsensical units such as dollars squared. *R* and *T* can be formulated in such a manner to transform *e* and *u* into economic cost. This is demonstrated in subsequent sections using a specific building at Texas A&M's campus as a case study.

$$J\_{component} = e^T Q \varepsilon + e^T R + u^T S u + u^T T \tag{1}$$
