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J. Wang, S. Huang, W. Zuo, D. Vrabie 2021. “Occupant Preference-Aware Load
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Scheduling for Resilient Communities.” Energy and Buildings, 252, pp. 111399.
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https://doi.org/10.1016/j.enbuild.2021.111399
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Occupant Preference-Aware Load Scheduling for Resilient Communities
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Jing Wanga, Sen Huangb, Wangda Zuoa,c,
*
, Draguna Vrabieb
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a University of Colorado Boulder, Department of Civil, Environmental and Architectural
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Engineering, Boulder, CO 80309, United States
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b Pacific Northwest National Laboratory, 902 Battelle Blvd, Richland, WA 99354, United States
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c National Renewable Energy Laboratory, 15013 Denver West Parkway, Golden, CO 80401,
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United States
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Abstract
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The load scheduling of resilient communities in the islanded mode is subject to many uncertainties
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such as weather forecast errors and occupant behavior stochasticity. To date, it remains unclear
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how occupant preferences affect the effectiveness of the load scheduling of resilient communities.
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This paper proposes an occupant preference-aware load scheduler for resilient communities
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operating in the islanded mode. The load scheduling framework is formulated as a model
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predictive control problem. Based on this framework, a deterministic load scheduler is adopted as
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the baseline. Then, a chance-constrained scheduler is proposed to address the occupant-induced
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uncertainty in room temperature setpoints. Key resilience indicators are selected to quantify the
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impacts of the uncertainties on community load scheduling. Finally, the proposed preference-
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aware scheduler is compared with the deterministic scheduler on a virtual testbed based on a real-
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world net-zero energy community in Florida, USA. Results show that the proposed scheduler
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performs better in terms of serving the occupants’ thermal preference and reducing the required
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*
Corresponding author.
Email address: wangda.zuo@colorado.edu.
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battery size, given the presence of the assumed stochastic occupant behavior. This work indicates
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that it is necessary to consider the stochasticity of occupant behavior when designing optimal load
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schedulers for resilient communities.
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Keywords: Microgrid; Optimal load scheduling; Uncertainty; Occupant behavior; Resilient
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community; Model predictive control.
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Nomenclature
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Parameters
battery energy
intercept coefficient for the logistic
regression model
battery charging power
slope coefficient for the logistic regression
model
scheduled critical loads
upper limit of battery energy
curtailed PV power
mathematical constant
battery discharging power
H
MPC prediction horizon
HVAC system (heat pump) total power
N
simulation horizon
total scheduled loads
number of critical loads in each building
scheduled modulatable loads
number of modulatable loads in each
building
scheduled sheddable loads
number of sheddable loads in each building
scheduled shiftable loads
number of shiftable loads in each building
PV power
average cycle time of each shiftable load
speed ratio of the heat pump
upper limit of battery power
indoor air temperature
critical load data
starting operation time of shiftable loads
HVAC system (heat pump) nominal power
internal heat gain
predicted loads upper bound
Binary Variables
modulatable load data
binary decision variable for sheddable load
on/off status
sheddable load data
binary variable for shiftable load starting
time
3
average nominal power of each shiftable
load
Abbreviations
probability of setpoint-changing actions
BAL
building agent layer
scheduling matrix for each shiftable load
CDF
cumulative distribution function
ambient outdoor temperature
COL
community operator layer
lower room temperature bound
DER
distributed energy resource
upper room temperature bound
DR
demand response
solar irradiance
HVAC
heating, ventilation, and air-conditioning
penalty coefficients
KRI
key resilience indicator
timestep
MPC
model predictive control
,
maximum constraint violation probability
RC
resistance-capacitance
battery charging efficiency
RMSE
Root Mean Square Error
battery discharging efficiency
SOC
state of charge
mean of room temperature error distribution
PDF
probability density function
standard deviation of room temperature
error distribution
PID
proportional integral derivative
Continuous Variables
PV
photovoltaics
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1 Introduction
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Due to the increasing frequency of extreme weather events such as the 2021 Texas Power Crisis
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[1], there is an emerging need for community resilience studies. Resilient communities refer to
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those that can sustain disruptions and adapt to them quickly by continuing to operate without
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sacrificing the occupants’ essential needs [2, 3]. Enabling technologies for resilient communities
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often involve distributed energy resources (DERs) such as photovoltaics (PV) and electrical energy
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storage (EES) systems. When disconnected from the main grid, the adoption of advanced control
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techniques can help enhance community resilience.
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As an advanced control technique, optimal load scheduling determines the operation schedules of
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controllable devices in the community to achieve optimization objectives. For a resilient
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community, typical controllable assets include the EES, PV, and thermostatically controllable
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devices in buildings such as the heating, ventilation, and air-conditioning (HVAC) system.
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Building plug loads that are sheddable, shiftable, or modulatable can also be considered flexible
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loads in islanded circumstances [4]. The objectives of the load scheduling for resilient
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communities often involve maximizing the self-consumption rate of locally generated PV energy,
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minimizing PV curtailment, and minimizing the unserved ratio to critical loads.
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It is important to account for uncertainties when designing a load scheduler for resilient
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communities. Moreover, due to the limited amount of available PV generation during off-grid
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scenarios, the uncertainties need to be more carefully dealt with to ensure a satisfying control
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performance. Sources of uncertainties for a community load scheduling problem mainly lie in two
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aspects: power generation and consumption. For renewable energy generation, weather forecast
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errors play a prominent role in the cause of uncertainty. Whereas, for energy consumption,
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occupant behavior stochasticity is a major source of uncertainty.
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Much of existing load scheduling research has considered the uncertainty of weather forecasts [5–
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13]. Kou [5] proposed a comprehensive scheduling framework for residential building demand
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response (DR) considering both day-ahead and real-time electricity markets. The results
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demonstrated the effectiveness of the proposed approach for large-scale residential DR
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applications under weather and consumer uncertainties. Garifi [13] adopted stochastic
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optimization in a model predictive control (MPC)-based home energy management system. The
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indoor thermal comfort is ensured at a high probability with uncertainty in the outdoor temperature
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and solar irradiance forecasts. Faraji [6] proposed a hybrid learning-based method using an
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artificial neural network to precisely predict the weather data, which eliminated the impact of
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weather forecast uncertainties on the scheduling of microgrids. Similarly, in the authors’ previous
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publication [7], normally distributed outdoor temperature and solar irradiance forecast errors were
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introduced into the community control framework, which accounted for the uncertainties in the
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weather forecasts.
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However, the uncertainties from the power consumption perspective, especially the occupant
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behavior uncertainty, is rarely accounted for in load scheduling research [14–18]. Some efforts to
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integrate occupant behavior modeling can be found in studies of building optimal control [19–22].
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Aftab [19] used video-processing and machine-learning techniques to enable real-time building
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occupancy recognition and prediction. This further facilitated the HVAC system operation control
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to achieve building energy savings. Lim [20] solved a joint occupancy scheduling and occupancy-
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based HVAC control problem for the optimal room-booking (i.e., meeting scheduling) in
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commercial and educational buildings. Both the occupancy status of each meeting room and the
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HVAC control variables were decision variables. Mixed-integer linear programming was adopted
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to optimally solve the optimization problem.
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Notably, all of the preceding control work considered the stochasticity of building occupancy
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schedules, but the integration of other types of occupant behavior into building optimal control is
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not well studied in existing literature. Some researchers integrate the occupant thermal sensation
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feedback into the MPC for buildings [23, 24]. For instance, Chen [23] integrated a dynamic thermal
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sensation model into the MPC to help achieve energy savings using the HVAC control. For the
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occupant sensation model, the predictive performance of certainty-equivalence MPC and chance-
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constrained MPC were compared.
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To summarize, the literature review shows that current research mainly focuses on the load
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scheduling of single buildings under grid-connected scenarios. There is a lack of research on the
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optimal load scheduling of resilient communities informed by occupant behavior uncertainties in
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the islanded mode. Given this gap, this paper proposes an occupant preference-aware load
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scheduling framework for resilient communities in the islanded mode. The occupants’ thermal
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preference for indoor air temperature will be reflected in the integration of thermostat adjustment
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probabilistic models. The optimal load scheduling is formulated as an MPC problem, so the
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stochastic thermostat-changing behavior will be regarded as the uncertainty in the MPC problem.
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Different methods, such as the offset-free method and robust method, can be used to handle the
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uncertainties in MPC problems [25]. The chance-constraint method, also known as the stochastic
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MPC, was selected to deal with the uncertainty in occupant preference in our study. It allows the
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violation of certain constraints at a predetermined probability. It thus enables a systematic trade-
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off between the control performance and the constraint violations [26]. The advantage of
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addressing occupant preference uncertainty by using the chance-constraint method lies in the a
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priori handling of the uncertainty, which does not require the extra error-prediction models needed
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by other methods (i.e., offset free method), and thus simplifies the control problem [27]. Therefore,
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less computational effort is required after the control design phase. Though it requires the
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controller to know the estimated uncertainty distribution beforehand, the development of occupant
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behavior probabilistic modeling will make knowing this less challenging.
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In this work, we consider the load scheduling of a resilient community in islanded mode during
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power outages. The goal is to study the impact of occupants’ thermal preference on the operation
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of an islanded community. The load scheduling problem of the community will be solved using
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an optimization-based hierarchical control framework. Occupant thermal preference will be
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integrated through thermostat changing behavioral models to inform the development of the load
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scheduler. The major contributions of this work include (1) a proposed new preference-aware load
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scheduler for resilient communities, which assures better control performance related to satisfying
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occupants’ thermal preferences and reducing the battery size; (2) the quantification of the impact
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of occupant thermostat-changing behavior on resilient community optimal scheduling using
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selected key resilience indicators (KRIs); and (3) the testing of the proposed scheduler on a high-
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fidelity virtual testbed for resilient communities.
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The remainder of this paper is organized as follows: Section 2 details the research methodology.
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Section 3 describes the controllable device models used in this work involving the building HVAC
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models, load models, and battery models. Section 4 then discusses the deterministic versus
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stochastic scheduler formulations and proposes a chance-constrained controller for preference-
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aware load scheduling of resilient communities. Section 5 applies the theoretical work to a case
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study community and quantifies the impact of occupant preference uncertainty. Simulation results
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and discussions are presented in this section. Finally, Section 6 concludes the paper by identifying
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future work.
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2 Methodology
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In this section, we first introduce a hierarchical optimal control structure for resilient community
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load scheduling. Based on the structure, a deterministic scheduler will be implemented as the
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baseline. Further, we propose a research workflow to implement a stochastic preference-aware
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scheduler for addressing uncertainties in occupant thermostat-changing behavior. KRIs are
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proposed at the end of this section.
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2.1 Hierarchical Optimal Control for Resilient Communities
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In this study, we assume that the only energy resource accessible to the islanded community is on-
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site PV generation and the batteries for an extended period of more than 24 hours. In this problem
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setting, in order to make full use of the limited amount of PV generation and satisfy the occupants’
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essential needs, the building loads need to be shifted or modulated. The battery works as a temporal
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arbitrage for meeting the demand at night. In addition, the occupant thermal preference will affect
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the energy consumption of the HVAC system through the stochastic thermostat-changing behavior.
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To optimally control such a community, considering the above factors, we adopted a hierarchical
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control structure.
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As illustrated in Figure 1, two layers of control are formulated: a community operator layer (COL)
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and a building agent layer (BAL). The COL optimally allocates the limited amount of the on-site
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PV generation based on the load flexibility provided by each building. The calculated allowable
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load for each building is then passed down to the BAL, where each building optimally schedules
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its controllable devices (i.e., HVAC, battery, and controllable loads) to achieve its local
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optimization goals. Both layers are formulated as MPC-based optimization problems.
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Figure 1 The hierarchical optimal control structure for community operation.
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The input of the hierarchical control involves the predicted PV generation data, outdoor air dry-
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bulb temperature, and solar irradiance. The PV generation data are used by the COL to determine
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the optimal allocation among buildings. The temperature and irradiance data are used by the
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HVAC models for updating the indoor room temperature predictions. The occupant behavior
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affects the two layers differently. The COL uses building occupancy schedules to decide the
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weights of different buildings during the PV allocation (details can be found in [7]). The BAL
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considers occupant thermal preference to be the uncertainty in the indoor room temperature
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prediction.
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2.2 Proposed Workflow
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Figure 2 depicts the workflow of this paper. A deterministic optimal load scheduler without the
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occupant thermal preference uncertainty is implemented in the hierarchical control structure.
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Further, to account for the uncertainties, we propose a chance-constrained controller. It is
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developed based on the deterministic controller and involves an alteration of the room temperature
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constraints, which accounts for the uncertainties in room temperature prediction errors caused by
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the occupants’ thermostat-changing behavior. The Monte Carlo simulation method was adopted
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to cover a wide range of simulation results.
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Figure 2 Diagram of the proposed workflow.
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Further, to reflect various styles of occupant behavior, three types of occupant thermostat-changing
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models were adopted: low, medium, and high, which represent three levels of frequencies of the
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thermostat-changing activities. Here, we assume that when the occupant decides to change the
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indoor air temperature setpoint according to their preference, the predetermined optimal HVAC
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equipment control setting at the current timestep will be overridden. Instead, a new control setting
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will be calculated to achieve the occupants’ setpoint at the current timestep. At the next timestep,
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the predetermined optimal setting will still be used if the occupant is not changing the setpoint
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consecutively.
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Finally, the optimal schedules determined by the chance-constrained controller and the
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deterministic controller are tested on a high-fidelity virtual testbed [28] with respect to their
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individual performances. KRIs such as the unserved load ratio, the required battery size, and the
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unmet thermal preference hours were adopted to quantify the results.
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The unserved load ratio in this paper is defined as the relative discrepancy between the served
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load
and the originally predicted load
:
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(1)
where is the MPC simulation horizon of 48 hours. The required battery size is obtained by
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subtracting the minimum battery SOC from the maximum SOC. This gives us a sense of how much
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of the battery capacity has been used under different scenarios. Finally, we define the unmet
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thermal preference hours metric for the cumulative absolute difference between the actual and the
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preferred room temperature over the optimization horizon:
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(2)
It quantifies how well the controller performs to satisfy the occupants’ thermal preference and has
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the unit of ºC·h (degree hours).
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3 Models for Controllable Devices
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3.1 HVAC Models
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This study assumes that heating and cooling is provided by heat pumps and the heat pump energy
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consumption represents the HVAC system energy consumption. We adopted linear regression
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models for the HVAC system to predict room temperatures at each timestep. To precisely model
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the building thermal reactions, two types of parameters that contribute to the heat gain of the
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building space are considered. The first type is environmental parameters such as the outdoor air
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dry-bulb temperature and solar irradiance. The second type represents the internal heat gain due to
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the presence of the occupants and the operation of appliances. We assumed that the simulated
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buildings are well sealed and thus the interference from the infiltration can be omitted. Therefore,
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the HVAC model updates the indoor room temperature based on the room temperature at the last
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timestep, the abovementioned heat gains, and the heating/cooling provided by the heat pump
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system at every timestep. The control variable is the heat pump speed ratio, which ranges from 0
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to 1 continuously. The resulting HVAC power is equal to the speed ratio multiplied by the nominal
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heat pump power. Additionally, to better account for the effect of building thermal mass, for each
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heat gain parameter, two past terms are adopted, respectively [29]. The equations for the HVAC
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model are as follows:
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(3)
s.t.
(4)
(5)
where
, and
represent the room temperature, ambient dry-bulb temperature,
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solar irradiance, and internal heat gain at timestep , respectively. The
and are the
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heat pump speed ratio and the nominal HVAC system power. The linear regression coefficients
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are represented by . For , a negative value means cooling and positive means heating.
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In the model,
and
are related to the occupant presence and the operation of the
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building appliances. When the building is occupied, 70% of the total heat rate of a person (i.e.,
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100 W) is dissipated as sensible heat into the space and the rest 30% is latent heat [30]. The heat
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gain from appliances is calculated by the power of the appliance multiplied by its heat gain
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coefficient, which reflects how much of the consumed electric power is dissipated into the space
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as heat. Table A-1 in Appendix A lists the heat gain coefficients adopted from literature [31–33].
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Note that the controllable loads are optimization variables of the scheduling problem, which will
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be iteratively calculated at each optimization timestep. Therefore, to speed up the optimization, we
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reduced the coupling between the thermal models and the electric demand models. This was done
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by calculating the weighted average heat gain coefficients for each building based on the capacity
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of each appliance (Table A-1).
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3.2 Load and Battery Models
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The building load models in this work are categorized into four types according to their power
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flexibility characteristics: sheddable, modulatable, shiftable, and critical (Figure 3). We did the
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categorization from the perspective of the building owners during power outages. The sheddable
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loads are those that can be disconnected without affecting the occupants’ essential needs. For
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instance, the microwave in a bakery is categorized as sheddable during an outage. The modulatable
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loads are the systems that have varying power shapes such as an HVAC system with a variable
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frequency drive. The shiftable loads are the appliances that have flexible operation schedules such
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as washers and dryers. Lastly, the critical loads refer to appliances and systems related to the
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occupants’ essential needs. In this work, we consider only loads used for lighting and food
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preservation as critical loads, which aligns with the two bottom levels of Maslow’s Hierarchy of
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Needs (i.e., physiological and safety needs) [34]. The critical loads account for about 20% to 90%
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of the total building loads depending on building type and time of day.
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Figure 3 Power flexibility characteristics of the four load types [35].
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The mathematical formulation of the sheddable load is shown in Equation (6):
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(6)
where
is a binary optimization variable,
is the original sheddable load time series
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data, and is the number of sheddable loads in the building. The actual sheddable load after
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optimization
is determined by the ON/OFF status represented by the binary variable. The
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modulatable load
is formulated as a continuous optimization variable, which ranges
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between zero and its original power demand
. Equation (7) sets the lower and upper bound
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of the modulatable load.
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(7)
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The shiftable loads are scheduled through scheduling matrices [36]. First, using the power data
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[37], we extracted the average cycle time and the average power demand of each
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shiftable load. The starting operation timestep of each shiftable load is optimized over the
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MPC horizon. At the scheduled starting timestep, the binary variable
equals 1 and is 0
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otherwise:
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(8)
is the MPC prediction horizon. Once the starting time of a shiftable load is selected, the power
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demand of the load is then fixed at its average power until it finishes its cycle. The appliance must
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finish its cycle before the horizon ends (). Here, we assume that each
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shiftable load operates once and only once during each horizon, which is enforced by:
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(9)
Next, a scheduling matrix of shape is generated for each shiftable
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load. The actual power shape of the load, denoted
, is thus calculated by:
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(10)
Finally, the actual critical load
must be exactly equal to the critical power demand
,
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as enforced by:
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(11)
Summing up the four types of loads in each building, we obtain the optimization variable
as
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follows:
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(12)
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The linear battery model adopted in this work is represented by Equation (13). The battery state of
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charge (SOC)
is predicted based on the SOC of the previous timestep
, the battery
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charging
or discharging power
at each step, and the battery charging/discharging
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efficiencies and . The inequality constraints in Equations (14) and (15) enforce the
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acceptable limits for the battery charging/discharging power and SOC, where and are
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the maximum values for battery power and capacity:
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(13)
s.t.
(14)
(15)
4 Optimal Load Scheduling
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This section first presents the mathematical formulation of the deterministic load scheduler. After
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that, we will introduce the formulation of the occupant preference-aware stochastic scheduler
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containing three parts: the thermostat-changing model, the uncertainty introduction mechanism,
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and the method to address the uncertainty.
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4.1 Deterministic Scheduler
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As introduced in Section 2.1, the deterministic scheduler adopts a two-layer structure with COL
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and BAL. The objective of the COL is to minimize the community-level PV curtailment to
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facilitate better use of the limited PV power during the outage. The main constraints are the load
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flexibility of each building, building occupancy, and building priority, etc. No detailed building
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assets are simulated at the community layer. This ensures that the COL is computationally tractable,
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especially when the problem scales up and the number of controllable building assets scales up.
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The detailed mathematical formulation of the COL can be found in reference [7].
275
The objective of the BAL is to minimize the unserved load ratio of each building within the
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allowable load range allocated by the COL. This is achieved through MPC-based optimal
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scheduling of the building-owned HVAC system, controllable loads, and battery. The optimization
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14
is a mixed-integer linear programming problem, because the sheddable and shiftable load models
279
contain binary variables. Next, the mathematical formulation of the optimization problem is
280
presented. Note that the formulation applies for every individual building in the community.
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The cost function to minimize the unserved load ratio is formulated as:
282
(16)
(17)
where
is the predicted load upper bound from data. The difference between this upper bound
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and the actual operated loads
is minimized to achieve a maximum served load to the building.
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To avoid simultaneous battery charging and discharging as well as PV curtailment, the objective
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function also includes small penalizations of charging
and curtailment
[38], where
286
and are the penalization coefficients. The power balance of each building that must be satisfied
287
at each timestep is given by:
288
(18)
where PV curtailment
is limited by how much PV generation
is available:
289
(19)
The left-hand side of Equation (18) represents power generation, whereas the right-hand side
290
represents consumption. The
and
stand for the battery charging and discharging power as
291
in Equation (13). The
and
are the total building loads and the HVAC power calculated
292
in Equations (12) and (5), respectively. To assure thermal comfort of the indoor environment, a
293
temperature constraint is given by:
294
(20)
where and are the lower and upper room temperature bounds implemented as hard
295
constraints. The optimization variables in each building agent are collected in vector :
296
15
(21)
4.2 Stochastic Preference-aware Scheduler
297
To address the uncertainties of occupant thermal preference in the scheduling problem of resilient
298
communities, this section introduces the stochastic preference-aware scheduler. First, we discuss
299
the modeling of the occupant behavior uncertainties as a probability function. Then we show the
300
mechanism by which this uncertainty might affect the optimal control of the HVAC system. After
301
that, we propose using the chance-constraint method to address the uncertainty.
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4.2.1 Stochastic Thermostat-Changing Model
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The stochastic occupant thermostat-changing model adopted in this paper was proposed by Gunay
304
et al. [39]. Through continuous observation of the occupants’ thermostat keypress actions in
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private office spaces, the relationship between the thermostat-changing behavior and the
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concurrent occupancy, temperature, and relative humidity was analyzed. It was noted that the
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frequency of thermostat interactions (i.e., increasing or decreasing) can be approximated as a
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univariate logistic regression model with the indoor temperature as the independent predictor
309
variable. Though the original data set was obtained from two office buildings, Gunay et al.
310
generalized the study to understand occupants’ thermostat user behavior and temperature
311
preferences. Given the universality of their work, we have adapted their models based on our use
312
cases. Note that occupants might have varied (e.g., higher) tolerance of indoor temperature during
313
an emergency situation. The exact thresholds need further experimental study and validation,
314
which is out of the scope of this work.
315
The thermostat-changing behavior models determine whether the occupants will change the
316
setpoint temperature based on the concurrent indoor air temperature. The probability of increasing
317
and decreasing the temperature setpoint is predicted with a logistic regression model:
318
(22)
where is the probability of the changing action, is the indoor room temperature, and and
319
are coefficients. To investigate different uncertainty levels, we proposed three different active
320
16
levels by revising the coefficients of the model in Equation (22). As shown in Table 1, the low
321
active level adopts the original coefficients in [39]. Then, we proposed the medium and high active
322
levels to represent various occupant thermal preference styles. The standard errors and p-values of
323
the low active level coefficients are also provided in the table. As for the medium and high levels,
324
we do not have measurement data for the statistical analysis since we adapted the coefficients from
325
the original reference [39].
326
Table 1 Coefficients in different active levels of the occupant thermostat-changing behavior
327
model.
328
Active Level
Coefficients
Increasing
Decreasing
a
b
a
b
Low [39]
-0.179
-0.285
-17.467
0.496
Medium
7.821
-0.485
-20.667
0.696
High
15.821
-0.685
-23.867
0.896
Standard Error
1.047
0.048
0.684
0.028
p-value
0.864
0.000
0.000
0.000
329
Note that the adaptation of the original logistic regression models was made under the following
330
assumptions to ensure the adapted models remained realistic. For the setpoint increasing scenario,
331
the slope coefficient of is varied linearly to reflect a higher frequency of the changing behavior.
332
The intercept coefficient is then calculated to make sure that all active levels have the same value
333
of probability at the temperature of 40ºC. For the setpoint decreasing scenario, a similar approach
334
is taken to make sure the same value of probability at 16ºC is shared by all active levels. At each
335
thermostat interaction, we assume that 1ºC of setpoint change would take place. Figure 4 depicts
336
the probabilities of the three active levels. Note that this figure contains a wider temperature range
337
than 16ºC ~ 40ºC to show a more comprehensive performance of the behavior models.
338
339
17
340
Figure 4 Probability of different thermostat-changing behavior.
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Once the probability of the thermostat-changing behavior is determined using the above models,
342
the increasing or decreasing action is determined by comparing the probabilities with a randomly
343
generated number. At each optimization timestep, a random number between 0 and 1 is generated.
344
If the number is larger than , the action will be to increase. On the contrary, if
345
it is smaller than , the action will be to decrease. Because the sum of the increase
346
and decrease probabilities is smaller than 1 in our case, this algorithm assures at most one action
347
will be taken at each timestep.
348
4.2.2 Introducing Occupant Behavior Uncertainties in Scheduling
349
To introduce the occupant thermostat-changing uncertainties to the load scheduling problem, a
350
stochastic simulation model representing the behavior needs to be incorporated into the
351
optimization. Figure 5 shows the control signal flow for the typical indoor air temperature control,
352
which affects the HVAC system operational status and its power consumption. The occupant sets
353
the temperature setpoint according to his/her preference through the thermostat. Behind the
354
thermostat, a proportional integral derivative (PID) controller decides the next heat pump speed to
355
offset the difference between the measured room temperature and the setpoint. This heat pump
356
speed signal is then fed into the heat pump system to provide cooling for the conditioned space.
357
18
Due to the presence of the dynamic environmental and behavioral disturbances, this process will
358
need to be repeated until the measured room temperature reaches the setpoint.
359
360
Figure 5 Diagram showing the introduction of occupant thermostat-changing behavior to the
361
optimization.
362
However, in the optimal control mechanism, the optimal scheduler takes over the control of the
363
heat pump speed from the PID controller. As a result, the occupants’ preference has thus been
364
“disabled” to allow an optimal control determined by the scheduler. To mimic the overriding of
365
the room temperature setpoint by the occupants, the following algorithm was implemented in the
366
MPC problem and the pseudo code is shown below. Before each round of the optimization starts
367
(Steps 1–2), if the occupant decided to change the setpoint (Step 3), the heat pump speed for the
368
current timestep should be calculated to reach the setpoint instead of achieving the optimization
369
objective (Steps 4–7). Otherwise, the optimization runs normally because no overriding happens
370
(Step 7). After each optimization timestep, the flag variables indicating the thermostat-changing
371
actions need to be updated according to the concurrent room temperature (Step 8). It should be
372
noted that in the optimization, no PID controller has been implemented, so we assumed that
373
and the setpoint changes were directly added to the room temperature
.
374
Step 1. Start
Step 2. Initialization of flag variables: ;
Step 3. If or :
Step 4
or
;
Step 5 Calculate the corresponding
;
Step 6 Disable
from the optimization variables;
Step 7. Run MPC for timestep ;
19
Step 8. Update flag variables (i.e., and ) according to
;
Step 9. Repeat Steps 3–8 until the end of the MPC horizon of 48 hours;
Step 10. End
4.2.3 Chance-Constraint Method
375
As mentioned in Section 4.2.1, the uncertainties in the occupants’ thermostat-changing behavior
376
are a probability function. In the scheduling optimization problem, the constraint directly affected
377
by the occupants’ thermostat-changing behavior is the room temperature bounds. The uncertainties
378
related to the occupants’ adjusting the thermostat could lead to the violation of the temperature
379
bounds during the implementation of the developed control strategies. Furthermore, this could lead
380
to other control-related performances being affected, including higher building load unserved ratio
381
and larger required battery size. To address this, we adopted the chance-constraint method.
382
By definition, the chance constraint allows the violation of a certain constraint with a small
383
probability, which thus presents a systematic trade-off between control performance and
384
probability of constraint violations [40]. It can be expressed in general by the following equation:
385
(23)
where is the inequivalent constraint and is the maximum violation probability.
386
Given the uncertainties in the occupants’ thermostat-changing behavior, we assume that the
387
temperature bounds can be satisfied with a probability of . For the lower temperature
388
bounds, the chance constraint can thus be written as:
389
(24)
Then, we rewrite it as:
390
(25)
where
Let the indoor temperature be rewritten in terms of the prediction
391
error:
where
is the predicted indoor room temperature and
392
is the error caused by uncertainties. Similarly,
. For both
393
timesteps, the room temperature distribution error follows the same distribution. The hypothetical
394
20
error distributions can be in different forms and here we assume the distribution to be normal.
395
Hence, it can be represented by:
396
(26)
Therefore,
is also normally distributed with the following mean and standard deviation :
397
(27)
(28)
The chance constraint can thus be reformulated as:
398
(29)
where is the cumulative distribution function (CDF) of the standard normal distribution
399
. By taking the inverse CDF of both sides, we can get:
400
(30)
Rearrange the above equation and substitute and with Equations (27) and (28). Finally, we
401
obtain the chance constraint for ensuring the indoor temperature will not fall below the lower
402
bound of with the probability of as follows:
403
(31)
Substituting Equation (3) into (31) and rearranging, we have:
404
(32)
Similarly, we have Equation (33) for the upper bound,
405
(33)
21
Taking a similar derivation process as that in Equations (24) to (32), we can obtain the chance
406
constraint for the temperature upper bound:
407
(34)
The updated inequivalent constraints indicate that the temperature bounds for the optimization
408
should be narrower than the original temperature bounds to account for the setpoint behavioral
409
uncertainty, which is consistent with the expectations. Note that because the uncertainty-dealing
410
method is focused on the temperature constraints, one possible limitation is that the above method
411
might have limited effect on the controller design for buildings that have larger thermal masses,
412
because the building temperature is insensitive to temperature constraints. More discussion of this
413
point follows in Section 5.3.1.
414
5 Case Study
415
5.1 Studied Community
416
The case study community is a net-zero energy community located in Anna Maria Island, Florida,
417
USA, which is a cooling dominated region. The community buildings are installed with both roof-
418
top PV panels and solar carports, which harvest about 85 MWh annually for the whole community.
419
A centralized ground source heat pump system provides the HVAC needs of the whole community
420
with high efficiency. Other sustainable features include well-insulated building envelopes, solar
421
thermal water heating, and rainwater recycling. This community achieved net-zero energy in the
422
year of 2014. In the community, there are various building types such as residential, small office,
423
gift shop, etc. We would like to cover both residential and commercial buildings in the case study.
424
So, we selected one residential and two small commercial buildings based on the measurement
425
data quality. More specifically, the selected three buildings consist of a residential building (area:
426
93.8 m2), an ice cream shop (area: 160.5 m2), and a bakery (area: 410 m2). The building layout of
427
the community can be found in reference [28].
428
For the given community, a virtual testbed based on the object-oriented modeling language
429
Modelica [41] was built and validated [42]. In the testbed, the Typical Meteorological Year 3 data
430
for a nearby city, Tampa, was adopted for this case study. The building thermal models are
431
resistance-capacitance (RC) network models. For the optimal control in this work, the HVAC
432
22
models were trained using one month (i.e., August) of the simulation data exported from the
433
testbed. Table 2 lists the coefficients for the linear regression HVAC models, the Root Mean
434
Square Error (RMSE) of the models, as well as the corresponding nominal heat pump power. The
435
N/A in the table represents a coefficient that is too small and thus has been neglected in the model.
436
Three effective decimal places are provided.
437
Table 2 Coefficients and nominal power of the HVAC models.
438
Residential
Ice Cream
Shop
Bakery
Coefficients
1.429
0.502
0.977
-0.432
0.498
0.0213
0.0263
0.000295
0.00405
-0.0232
-0.000193
-0.00196
-0.210
-0.0114
-0.178
0.0151
0.0000345
0.0107
-0.00302
0.000181
-0.00621
0.00852
N/A
N/A
N/A
N/A
0.0140
RMSE []
0.160
0.0205
0.114
Nominal Power [kW]
2.140
2.830
3.770
439
Additionally, Table 3 lists the load categorization for the studied buildings following the principles
440
proposed in Section 3.2. A complete list of the building load capacities and their heat gains can be
441
found in Appendix A.
442
Table 3 Building loads categorized into four types.
443
Residential
Ice Cream Shop
Bakery
Sheddable
Computer
Coffee maker, soda
dispenser, outdoor
ice storage
Microwave
Modulatable
HVAC
HVAC
Mixer, unspecific room
plug loads, HVAC
23
Residential
Ice Cream Shop
Bakery
Shiftable
Range, washer,
dryer
None
Range, oven,
dishwasher
Critical
Lights, refrigerator
Lights, cooler,
display case
Lights, cooler, display
case
444
We designed three uncertainty levels (i.e., low, medium, high) as in Table 1 to evaluate the
445
deterministic and preference-aware schedulers in this paper. They are compared to the baseline
446
scenario, where the deterministic scheduler is applied without occupant behavior uncertainties.
447
The following results and discussion are all based on these scenarios. All scenarios were run in the
448
three buildings for 48 hours with a timestep of 1 hour in the islanded mode.
449
5.2 Settings of Chance-Constrained Controllers for Different Buildings
450
The preference-aware schedulers use chance-constrained controllers, whose settings depend on
451
individual building properties and uncertainty levels. Following the method proposed in Section
452
4.2.3, this section provides the details of the chance-constrained controller settings for three
453
individual buildings in the case study, which is based on the control outcome of the deterministic
454
schedulers under three uncertainty levels.
455
Considering the occupant-preference-driven actions as the source of “prediction errors” for the
456
room temperature, we extracted the distributions of the room temperature prediction errors. The
457
Monte Carlo simulation method [43] was adopted, where 100 repeated simulations were run using
458
the deterministic scheduler with three uncertainty levels. We used the room temperature of the
459
deterministic baseline scenario as the benchmark to calculate the errors caused by the occupant
460
setpoint-changing behavior. To describe the room temperature errors, three hypothetical
461
distributions are proposed (i.e., fit distribution in Table 4). The normal distribution is mentioned
462
in the derivation in Section 4.2.3. The half-normal distribution is a fold of a normal distribution at
463
its mean. For the residential building medium uncertainty level, a half-normal distribution was
464
adopted. This can be attributed to the fact that almost no temperature decrease action was observed
465
and thus the errors were all above zero. Constants were used for the residential building and the
466
bakery under the low uncertainty level because the frequency of the setpoint-changing is too low
467
(nearly zero) to follow any distributions.
468
24
Chi-square goodness of fit tests [44] at a rejection level of 1% were conducted to evaluate whether
469
the proposed hypothetical distributions fit well. The types of fitting distributions, p-values of the
470
tests, and the distribution parameters are reported in Table 4. In the table, µ is the mean and σ is
471
the standard deviation of the normal/half-normal distribution. The null hypothesis here is that the
472
room temperature prediction error follows the hypothetical distribution. The p-value is the
473
evidence against this null hypothesis. Since all p-values are greater than 99%, all error distributions
474
failed to reject the hypothesis at the level of 1%. This means they all follow the corresponding
475
hypothetical distribution.
476
Table 4 Chi-square goodness of fit test p-values and normal distribution parameters.
477
Building
Uncertainty
Fit Distribution
p-value
µ [ºC]
σ [ºC]
Residential
Low
Constant
1.0
-6.45E-05
N/A
Medium
Half-normal
0.999
-3.57E-01
4.35E-01
High
Normal
0.999
1.56E+00
8.17E-01
Ice Cream Shop
Low
Normal
0.999
-3.48E-03
7.86E-03
Medium
Normal
0.999
-4.45E-03
8.59E-03
High
Normal
0.999
1.60E-02
1.59E-02
Bakery
Low
Constant
1.0
-3.42E-03
N/A
Medium
Normal
0.999
3.01E-02
1.05E-01
High
Normal
0.999
5.33E-01
4.65E-01
The frequency histogram and probability density functions (PDFs) of each building under various
478
uncertainty levels are plotted in Figure 6. In the figure, it can be seen that the higher the uncertainty,
479
the wider the room temperature range. This is because in scenarios with a higher uncertainty,
480
occupants change the thermostat more frequently, which expands the possible temperature ranges.
481
We also noticed that the temperature range in the ice cream shop is relatively concentrated
482
compared to the other two buildings. This can be attributed to the large thermal mass of the
483
building.
484
25
485
Figure 6 Room temperature prediction error PDFs obtained from the Monte Carlo simulations.
486
For the scenario where the temperature prediction error follows the half-normal distribution, we
487
applied the chance constraint only to the upper bound because only increasing actions happen in
488
this scenario. For the two scenarios where the room temperature error is estimated to be a constant,
489
we adopted the original temperature bounds of [20ºC, 25ºC] because the estimated errors in both
490
scenarios are smaller than 0.01ºC. We choose the to ensure a 99% probability of
491
abidance of the temperature constraints (Equation (24)). Table 5 lists the updated room
492
temperature lower and upper bounds for each building under different scenarios.
493
Table 5 Room temperature bounds for chance-constrained optimizations.
494
Building
Uncertainty
[ºC]
[ºC]
Residential
Low
20.000
25.000
Medium
20.000
24.236
High
20.547
21.343
Ice Cream
Shop
Low
20.024
24.983
Medium
20.027
24.982
High
20.025
24.943
Bakery
Low
20.000
25.000
Medium
20.240
24.700
High
20.664
23.273
495
26
5.3 Results and Discussions
496
This section first quantifies the impact of introducing occupant behavior uncertainties to the
497
optimal scheduling problem. Then, the deterministic and chance-constrained controllers are tested
498
on the community virtual testbed. Their control performance in terms of the unserved load ratio,
499
the required battery size, and the unmet thermal preference hours are then compared.
500
5.3.1 Impact of Uncertainty
501
Figures 7 to 9 depict the occupant thermal preference and the corresponding room temperatures.
502
In the figures, the upper plots show the simulated stochastic thermostat-changing actions at
503
different uncertainty levels, where increase means a setpoint increase action, and vice versa. The
504
lower plots show the resulting room temperatures with dashed lines.
505
The results of the low uncertainty scenario overlap with that of the baseline scenario (i.e., the
506
deterministic scheduler without uncertainty) mainly due to the low probability of setpoint-
507
changing actions in this scenario. With the increase in the probability, we see more frequent
508
setpoint-changing actions in all three buildings. Further, the increase action happens more
509
frequently than the decrease action. This is because between the temperature range of 20ºC and
510
24ºC, the probability of increase is much higher than that of decrease (see Figure 4). This also
511
implies that the occupants’ temperature preference is closer to 24ºC than 20ºC. Additionally, for
512
the residential building and the bakery, the temperature difference between scenarios is more
513
noticeable than for the ice cream shop; this is attributable to the different building thermal masses
514
of the three buildings.
515
27
516
Figure 7 Residential building occupant thermostat changing actions (upper) and resulting room
517
temperatures (lower) under three levels of uncertainty.
518
519
Figure 8 Ice cream shop occupant thermostat changing actions (upper) and resulting room
520
temperatures (lower) under three levels of uncertainty.
521
28
522
Figure 9 Bakery occupant thermostat changing actions (upper) and resulting room temperatures
523
(lower) under three levels of uncertainty.
524
Table 6 lists the values of the KRIs in correspondence with Figures 7 to 9. The HVAC energy and
525
average room temperature over the optimization horizon are also provided to facilitate the analysis
526
of the results.
527
Table 6 Key resilience indicators for studied buildings under different uncertainty levels.
528
Building
Scenario
Unserved Load Ratio
Battery Size [kWh]
HVAC Energy
[kWh]
Mean Room
Temperature [ºC]
Residential
Baseline
0.0744
47.686
32.139
20.185
Low
0.0744
47.686
32.139
20.185
Medium
0.0744
47.168
32.099
20.271
High
0.0744
38.541
21.400
21.468
Ice Cream Shop
Baseline
0.0215
99.139
32.703
21.006
Low
0.0215
99.139
32.703
21.006
Medium
0.0215
99.139
32.703
21.006
High
0.0215
93.166
10.063
21.033
Bakery
Baseline
0.0247
80.007
35.144
21.579
Low
0.0247
80.007
35.144
21.579
Medium
0.0247
73.496
27.604
21.766
High
0.0247
76.801
11.310
21.973
529
29
From the table, we see that the unserved load ratio remains the same across all scenarios for each
530
building. This can be attributed to the fact that in the controller design phase, the optimization
531
objective is set to minimize the unserved load ratio. Hence, the unserved load ratios for each
532
building are already minimal and are not affected by the occupants’ thermostat-overriding
533
behavior uncertainties. Instead, the battery-charging/discharging behavior is affected, as reflected
534
by the different required battery sizes in the table. Note that the unserved load ratios are minimal,
535
but not zero, because of our assumption that each shiftable load operates once and only once per
536
day.
537
For the rest of the metrics, note that the battery size, HVAC energy, and the average room
538
temperature remain the same for the baseline and low uncertainty scenarios in all buildings. This
539
is because no setpoint-changing actions happened due to the relatively low probabilities, as shown
540
in the figures above. As for the medium uncertainty scenarios, both the residential building and
541
the bakery show higher room temperatures and lower HVAC energy while the ice cream shop still
542
has the same results as the baseline, given its large thermal mass.
543
In terms of the high uncertainty scenarios, due to the prominent increase in room temperatures, we
544
noticed more HVAC energy savings in all buildings. Note that though the average room
545
temperature increase is insignificant, the HVAC energy savings is large due to the cumulative
546
effect over the many hours of setpoint increase. Overall, we see a positive correlation between the
547
HVAC energy and the required battery size. When the PV generation and the other building loads
548
remain the same, the more HVAC energy, the larger required battery size. However, one opposite
549
case was noted in the bakery high uncertainty scenario where the required battery size is slightly
550
larger in the high uncertainty scenario than in the medium uncertainty scenario. This was caused
551
by a setpoint decrease action at hour 28, which resulted in a battery discharging during the night
552
and thus a smaller minimum SOC of the battery.
553
To summarize, occupant thermostat-changing behavior uncertainty needs to be considered when
554
designing optimal schedulers for resilient buildings because it affects the indoor room temperature,
555
the HVAC power, and thus the sizing of batteries. For the whole community, when considering
556
the highest occupant behavior uncertainty, the consumed HVAC energy can be 57.2% less and the
557
battery 8.08% smaller. Whereas the aforementioned impact depends on the uncertainty level (i.e.,
558
how frequently the occupants change the setpoint), heating or cooling season, and the occupants’
559
30
actual preference for the indoor room temperature compared to the room temperature designed by
560
the scheduler. In our case, a preferred higher indoor room temperature saves HVAC energy.
561
During the heating season, the observations could be the reversed.
562
5.3.2 Controller Performance
563
To further evaluate the performance of the chance-constrained controller in comparison with the
564
deterministic controller, tests were run on the virtual testbed [28] in a stochastic manner. In each
565
of the studied buildings, both the deterministic controller and the chance-constrained controller
566
were tested for two days (i.e., August 4 and 5) with the three levels of uncertainties. The testing
567
method is similar to the method proposed in Section 4.2.2. Additionally, the precalculated optimal
568
battery charging/discharging, as well as the optimized loads, are also implemented in the testbed.
569
One hundred repeated Monte Carlo simulations were run for each scenario to better observe the
570
controller performance. The KRIs of the unserved load ratio, the required battery size, and the
571
unmet thermal preference hours are adopted for the performance evaluation.
572
The upper plot of Figure 10 depicts the predetermined optimal schedules of the heat pump speed
573
ratio as the inputs of the test. The lower plot then shows the corresponding room temperatures
574
predicted by the linear regression models in the optimization. The data for the residential building
575
is adopted here for the analysis. The plots for the ice cream shop and the bakery can be found in
576
Appendix A. From the figure, we see that the scheduled speed ratios in the low and medium
577
uncertainty scenarios overlap with that of the deterministic scheduler. Whereas the high
578
uncertainty scenario tends to have lower speed ratios over the whole optimization horizon. This
579
can be attributed to the controller settings shown in Table 5, where the temperature bounds set in
580
the low and medium uncertainty scenarios are closer to the original bounds of [20ºC–25ºC]. Hence,
581
the temperature constraints are not binding in these two scenarios. However, in the high
582
uncertainty scenario, the temperature constraint is binding, which leads to the speed ratio
583
reductions. As a result, a higher room temperature can be seen in the high uncertainty scenario.
584
31
585
Figure 10 Optimal schedules of the heat pump speed ratio and predicted room temperatures by
586
various schedulers (residential building).
587
Figures 11 to 13 depict the room temperature boxplots as the controller testing outputs. The lower
588
and upper borders of the boxes represent the 25th and 75th percentiles of the data, respectively.
589
The longer the box, the more scattered the room temperature. The lines inside the boxes represent
590
the median values. The lines beyond the boxes represent the minimum and maximum values except
591
for outliers, which are not shown in these figures. Note in the figures that the temperatures first
592
concentrate together (shown as black lines) and then spread out (shown as boxes). This is because
593
at the beginning of the simulations, no overriding behavior of the setpoints happens and the heat
594
pump operates following the scheduled speed ratio. Once the overriding happens at a certain
595
timestep in some simulations, the room temperature trends start to deviate and become boxes. The
596
occupant-preferred temperature lines are also shown as orange lines in these figures as a reference;
597
they are average setpoints adjusted by the occupants in all the Monte Carlo tests.
598
32
599
Figure 11 Residential building room temperature boxplots for control testing results.
600
601
Figure 12 Ice cream shop room temperature boxplots for control testing results.
602
33
603
Figure 13 Bakery room temperature boxplots for control testing results.
604
In the figures, we see a general trend of narrower room temperature ranges from the low
605
uncertainty scenarios to high uncertainty scenarios. This is due to the introduction of the occupant
606
setpoint-overriding mechanism, which tends to moderate the extreme room temperatures. Also,
607
there is a plant-model mismatch, which describes the parametric uncertainty of modeling that
608
originates from neglected dynamics of the plant [25]. In our case, the mismatch exists as the
609
simulated room temperatures in the testbed are slightly higher than those predicted by the reduced-
610
order linear HVAC models. This is understandable because the physics-based testbed has a much
611
higher fidelity and simulates the non-linearity of the real mechanical systems.
612
Because the difference in the room temperature between the two controllers is not depicted in these
613
figures, Table 7 and Table 8 provide further quantitative evaluations of the room temperatures
614
along with other controller performance. Additionally, note that the optimal schedules of some
615
scenarios remain the same because of the unbinding temperature constraints, which led to the same
616
testing outputs. Here we only discuss the scenarios that have different inputs and outputs. A full
617
list of all testing results is available in Table A-2.
618
Table 7 Comparison of controller performance in the residential building high uncertainty
619
scenario.
620
34
Controller
Unmet Thermal
Preference Hours
[ºC·h]
Mean Room
Temperature [ºC]
Unserved
Load Ratio
Required
Battery Size
[kWh]
Deterministic
48.91
23.75
0.074
47.69
Chance-
constrained
46.42
23.87
0.074
44.12
621
In Table 7, we see a larger value of unmet thermal preference hours in the deterministic controller
622
than the chance-constrained one. This can be attributed to the higher room temperatures regulated
623
by the chance constraints to better satisfy the occupants’ thermal preferences. Again, the same
624
unserved load ratio is observed in both controllers because it is already minimal, which is enforced
625
by the objective function. In terms of the battery size, the chance-constrained controller shows a
626
smaller required battery size than the deterministic controller. This results from the fact that a
627
higher room temperature has led to less consumed HVAC energy in the chance-constrained
628
scenario. Thus, less discharging from the battery was happening, which led to a smaller required
629
battery size. For the bakery results shown in Table 8, the same trends for the battery size and the
630
unserved load ratio as the residential building can be observed under each uncertainty level.
631
Namely, smaller batteries and the same unserved load ratios.
632
Table 8 Comparison of controller performances in the bakery medium and high uncertainty
633
scenarios.
634
Uncertainty
Controller
Unmet Thermal
Preference
Hours [ºC·h]
Mean Room
Temperature
[ºC]
Unserved
Load Ratio
Required
Battery Size
[kWh]
Medium
Deterministic
88.80
24.27
0.025
80.01
Chance-
constrained
91.28
24.50
0.025
76.89
High
Deterministic
102.81
23.65
0.025
80.01
Chance-
constrained
101.61
23.89
0.025
76.89
635
35
As for the unmet thermal preference hours, different trends are witnessed in the medium and high
636
uncertainty levels. In the medium level, the deterministic controller shows fewer unmet preference
637
hours than the chance-constrained controller. Whereas in the high uncertainty level, an opposite
638
trend is seen. This is reasonable as we see a generally higher mean room temperature regulated by
639
the chance-constrained controller under different uncertainty levels. However, in the medium
640
scenario, a lower preference temperature line was obtained from the Monte Carlo testing, which is
641
closer to the actual room temperatures of the deterministic controller. When the preference
642
temperature rises in the high uncertainty scenario, the chance-constrained controller outperforms
643
the deterministic controller with a higher actual room temperature and thus smaller unmet thermal
644
preference hours.
645
When we compare different uncertainty levels in the bakery, we see that the mean room
646
temperature decreases with the increase in uncertainty. This is because the lower temperature
647
upper bounds shown in Table 5 have regulated the room temperature to sink when the uncertainty
648
gets higher. Additionally, as seen in Figure 4, in the temperature range of 20ºC to 24ºC, the
649
probability of increasing the temperature setpoint is much higher than that of decreasing it While
650
above 24ºC, the probability to increase and to decrease is almost the same. This has caused the
651
room temperatures to end up around 24ºC in the high uncertainty scenarios for all buildings (Table
652
A-2). This reveals that with the increase in the occupant thermostat-changing uncertainties, the
653
room temperatures tend to get closer to the occupants’ preferred room temperature.
654
Though some improvement was noticed in the chance-constrained controller compared to the
655
deterministic controller, the overall improvement was less than expected. This could be attributed
656
to the following three factors. First, the impact of the uncertainty level on the controller
657
performance improvement is prominent as we observe higher performance improvement in high
658
uncertainty scenarios. Second, the thermal property, especially thermal mass, of the building itself
659
also affects the results. Thermal mass serves as a thermal buffer to filter the impact of various
660
HVAC supply temperatures. Hence, buildings with a larger thermal mass tend to experience less
661
impact from the occupant thermal preference uncertainty. This can be demonstrated by the results
662
of the ice cream shop, where the two controllers perform the same. Third, the plant-model
663
mismatch also plays a significant role in the transition from the optimal scheduler design to its
664
implementation. In the design phase, a series of control-oriented linear regression building models
665
36
was used. However, the testing took place on a high-fidelity physics-based testbed, where the
666
complex system dynamics of the whole buildings and HVAC systems were modeled with shorter
667
simulation timesteps. This is a common source of uncertainty to be addressed for MPC design and
668
implementation.
669
In our opinion, joint effort from building scientists, modelers, and engineers is needed to facilitate
670
implementing stochasticity in the building domain and ultimately better serve the occupants. For
671
example, an open-source database focused on building performance related stochasticity such as
672
the occupant behavior and weather forecast needs to be established. Further, readily available
673
stochastic simulation tools need to be developed (e.g., Occupancy Simulator [45]). Finally,
674
stochasticity needs to be incorporated into the whole process of building modeling and design in
675
the form of boundary conditions or internal components.
676
6 Conclusion
677
In this paper, we proposed a preference-aware scheduler for resilient communities. Stochastic
678
occupant thermostat-changing behavior models were introduced into a deterministic load
679
scheduling framework as a source of uncertainty. The impact of occupant behavior uncertainty on
680
community optimal scheduling strategies was discussed. KRIs such as the unserved load ratio, the
681
required battery size, and the unmet thermal preference hours were adopted to quantify the impacts
682
of uncertainties. Generally, the proposed controller performs better in terms of the unmet thermal
683
preference hours and the battery sizes compared to the deterministic controller. Though only tested
684
on three buildings of the studied community, the methodology of introducing occupant behavior
685
uncertainty into load scheduling and testing can be generalized and applied to other building and
686
behavior types.
687
More specifically, we determined that occupant thermostat-changing behavior uncertainty should
688
be considered when designing optimal schedulers for resilient communities. For the whole
689
community, when considering the highest occupant behavior uncertainty, the consumed HVAC
690
energy can be 57.2% less and the battery 8.08% smaller. During the controller testing phase, the
691
proposed chance-constrained controller proves its advantage over the deterministic controller by
692
better serving the occupants’ thermal needs and demonstrating a savings of 6.7 kWh of battery
693
capacity for the whole community. Additionally, we noticed that with the presence of occupant
694
37
thermostat-changing uncertainties, the room temperatures tend to get closer to the occupants’
695
preferred room temperature.
696
During the simulation experiments, we noticed some limitations of the proposed work. Because
697
the proposed uncertainty method mainly deals with the uncertainty through the temperature
698
constraints, it can be less effective for buildings of larger thermal mass due to the insensitivity to
699
temperature constraints. Also, plant-model mismatch was noticed in the controller testing phase,
700
which is a common parametric uncertainty that originates from neglected dynamics of the
701
plant [25]. Finally, we used the thermostat changing models developed based on data from private
702
office spaces in different building types, which can be debatable. Future work for this research
703
includes extending the scope to heating scenarios to further generalize the findings. Additionally,
704
real-time MPC control techniques could be integrated into the framework to overcome the lack of
705
flexibility in a priori designed controllers.
706
Acknowledgements
707
This research is partially supported by the National Science Foundation under Awards No. IIS-
708
1802017. It is also partially supported by the U.S. Department of Energy, Energy Efficiency and
709
Renewable Energy, Building Technologies Office, under Contract No. DE-AC05-76RL01830.
710
This work also emerged from the IBPSA Project 1, an internationally collaborative project
711
conducted under the umbrella of the International Building Performance Simulation Association
712
(IBPSA). Project 1 aims to develop and demonstrate a BIM/GIS and Modelica Framework for
713
building and community energy system design and operation.
714
Appendix A
715
Table A-1 Complete list of building loads and heat gain coefficients [31–33].
716
Building
No.
Load
Capacity
[W]
Heat Gain
Coefficient
Heat Gain
[W]
Weighted Average
Coefficient
Residential
1
Lights
293
0.8
234.4
0.31
2
Refrigerator
494
0.4
197.6
3
Computer
18
0.15
2.7
4
Range
1775
0.34
603.5
38
Building
No.
Load
Capacity
[W]
Heat Gain
Coefficient
Heat Gain
[W]
Weighted Average
Coefficient
5
Washer
438
0.8
350.4
6
Dryer
2795
0.15
419.25
Ice Cream
Shop
1
Lights
135
0.8
108
0.35
2
Coolers
7394
0.4
2957.6
3
Display case
280
0.4
112
4
Coffee maker
2721
0.3
816.3
5
Soda dispenser
201
0.5
100.5
6
Outdoor ice
storage
1127
0
0
Bakery
1
Lights
1859
0.8
1487.2
0.38
2
Coolers
4161
0.4
1664.4
3
Display case
1011
0.4
404.4
4
Range
4065
0.15
609.75
5
Mixer
521
0.31
161.51
6
Gas oven
761
0.2
152.2
7
Room plugs
377
0.5
188.5
8
Microwave
1664
0.67
1114.88
9
Dishwasher
1552
0.15
232.8
717
718
Figure A-1 Optimal schedules of the heat pump speed ratio and predicted room temperatures by
719
various schedulers (ice cream shop).
720
39
721
Figure A-2 Optimal schedules of the heat pump speed ratio and predicted room temperatures by
722
various schedulers (bakery).
723
Table A-2 Full comparison of controller performances under different uncertainty levels in all
724
three buildings.
725
KRIs
Controller
Residential
Ice Cream Shop
Bakery
Low
Mediu
m
High
Low
Mediu
m
High
Low
Mediu
m
High
Unmet
Thermal
Preference
Hours [ºC·h]
Deterministic
33.70
47.19
48.91
70.69
85.61
86.87
89.03
88.80
102.81
Chance-
constrained
33.70
47.19
46.42
70.69
85.61
86.87
89.03
91.28
101.61
Mean Room
Temperature
[ºC]
Deterministic
24.38
23.69
23.75
21.23
22.87
23.38
25.34
24.27
23.65
Chance-
constrained
24.38
23.69
23.87
21.23
22.87
23.38
25.34
24.50
23.89
Unserved
Load Ratio
Deterministic
0.074
0.022
0.025
Chance-
constrained
0.074
0.074
0.074
0.022
0.022
0.022
0.025
0.025
0.025
Required
Battery Size
[kWh]
Deterministic
47.69
99.14
80.01
Chance-
constrained
47.69
47.69
44.12
99.14
99.14
99.14
80.01
76.89
76.89
40
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