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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,

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, 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].

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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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is a mixed-integer linear programming problem, because the sheddable and shiftable load models

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contain binary variables. Next, the mathematical formulation of the optimization problem is

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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:

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(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

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and are the penalization coefficients. The power balance of each building that must be satisfied

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at each timestep is given by:

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(18)

where PV curtailment

is limited by how much PV generation

is available:

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(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

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in Equations (12) and (5), respectively. To assure thermal comfort of the indoor environment, a

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temperature constraint is given by:

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(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

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To address the uncertainties of occupant thermal preference in the scheduling problem of resilient

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communities, this section introduces the stochastic preference-aware scheduler. First, we discuss

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the modeling of the occupant behavior uncertainties as a probability function. Then we show the

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mechanism by which this uncertainty might affect the optimal control of the HVAC system. After

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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

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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

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variable. Though the original data set was obtained from two office buildings, Gunay et al.

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generalized the study to understand occupants’ thermostat user behavior and temperature

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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

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an emergency situation. The exact thresholds need further experimental study and validation,

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which is out of the scope of this work.

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The thermostat-changing behavior models determine whether the occupants will change the

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setpoint temperature based on the concurrent indoor air temperature. The probability of increasing

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and decreasing the temperature setpoint is predicted with a logistic regression model:

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(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

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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,

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we do not have measurement data for the statistical analysis since we adapted the coefficients from

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the original reference [39].

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Table 1 Coefficients in different active levels of the occupant thermostat-changing behavior

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model.

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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,

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the slope coefficient of is varied linearly to reflect a higher frequency of the changing behavior.

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The intercept coefficient is then calculated to make sure that all active levels have the same value

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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

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thermostat interaction, we assume that 1ºC of setpoint change would take place. Figure 4 depicts

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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.

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339

17

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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,

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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.

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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.

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4.2.2 Introducing Occupant Behavior Uncertainties in Scheduling

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To introduce the occupant thermostat-changing uncertainties to the load scheduling problem, a

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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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