# An Incentive-Based Optimization Approach for Load Scheduling Problem in Smart Building Communities

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## Abstract

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## 1. Introduction

#### 1.1. Literature Review

#### 1.2. Study Contributions

- The impact of thermal assets on the building energy consumption and human comfort are considered in the formulation.
- The user inconvenience levels for time-shiftable and thermal assets are considered in the formulation.
- The DR program participants’ incentives are included in the optimization model that helps them get incentivized based on the amount of energy they are willing to shift.
- For solving the nonlinear optimization problem, the method of feasible direction is discussed and employed.
- A complete data set for residential and office buildings’ appliances is prepared that can be used by other researchers in the field.

## 2. Problem Statement and Preliminaries

## 3. Problem Formulation

#### 3.1. Time-Shiftable Assets

#### 3.2. Thermal Assets

#### 3.2.1. HVAC Systems

#### 3.2.2. Electric Water Heater Systems

#### 3.2.3. The Inconvenience Level for Thermal Assets

#### 3.3. Objective Functions

#### 3.4. The Optimization Model

#### 3.5. The Solution Approach

## 4. Case Study Design

#### 4.1. Case Scenarios

#### 4.2. Building Functionalities

## 5. Results and Discussion

#### 5.1. The Small-Scale Community

#### 5.1.1. The Baseline Case

#### 5.1.2. The Case without Building Collaboration

#### 5.1.3. The Case with Building Collaboration

#### 5.1.4. Comparison between Case Studies

#### 5.2. The Large-Scale Building Community

#### 5.3. Discussion

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Nomenclature

${\mathrm{x}}_{\mathrm{i},\mathrm{j}}\left(\mathrm{t}\right)$ | a binary number to represent the optimal on or off status of a flexible time-shiftable asset i of building j at time t, which equals 1 if asset i is to be turned on at time t and zero otherwise |

${\mathrm{y}}_{\mathrm{i},\mathrm{j}}\left(\mathrm{t}\right)$ | an auxiliary binary decision variable to state the status operation of asset i of building j at time t. If $\text{}\mathrm{it}\text{}\mathrm{is}\text{}\mathrm{zero}$, the operation of this asset is just completed during time slot t and the corresponding ${x}_{i,j}\left(t\right)$ must be zero |

${\mathrm{z}}_{\mathrm{i},\mathrm{j}}\left(\mathrm{t}\right)$ | a binary number for the inconvenience, which equals 1 if there is a miss-match between the preferred schedule and the optimal schedule for deferrable asset i of building j at time t |

${\mathrm{w}}_{\mathrm{i},\mathrm{j}}\left(\mathrm{t}\right)$ | a binary number for incentives, which equals 1 if consumers earn incentives since they switched off asset i of building j at time t, against their preference, and zero otherwise |

${\mathrm{u}}_{\mathrm{i},\mathrm{j}}\left(\mathrm{t}\right)$ | the consumer’s preferred on-off status of a flexible time-shiftable asset i of building j at time t |

${\mathrm{v}}_{\mathrm{i},\mathrm{j}}\left(\mathrm{t}\right)$ | the inconvenience level for thermal asset i of building j at time t |

${\mathrm{P}}_{\mathrm{HVAC},\mathrm{j}}\left(\mathrm{t}\right)$ | power consumption of the HVAC system in building j at time t (kW) |

${\mathrm{P}}_{\mathrm{EWH},\mathrm{j}}\left(\mathrm{t}\right)$ | power consumption of the EWH system in building j at time t (kW) |

${\mathrm{T}}_{\mathrm{in}}\left(\mathrm{t}\right)$ | building indoor temperature at time t ($\mathbb{C}$) |

${\mathrm{T}}_{\mathrm{EWH}}\left(\mathrm{t}\right)$ | hot water temperature at time t ($\mathbb{C}$) |

${\mathsf{\alpha}}_{\mathrm{i},\mathrm{j}}$ | the start time of operation range for asset i of building j |

${\mathsf{\beta}}_{\mathrm{i},\mathrm{j}}$ | the end time of operation range for asset i of building j |

${\mathsf{\omega}}_{\mathrm{i},\mathrm{j}}$ | the required number of time slots to operate the time-shiftable asset i of building j |

$\mathrm{d}\left(\mathrm{t}\right)$ | the incentive offered at time t |

$\mathsf{\Delta}\mathrm{t}$ | the time step length (h) |

${\mathrm{S}}_{\mathrm{i},\mathrm{j}}$ | rated power of time-shiftable asset i of building j |

${\mathrm{t}}_{\mathrm{s}}$ | start time |

$\mathsf{\tau}$ | planning horizon (h) |

$\mathsf{\epsilon}$ | factor of inertia |

$\Delta {\mathrm{T}}_{\mathrm{i},\mathrm{j}}\left(\mathrm{t}\right)$ | difference between the actual temperature and the desired one ($\mathbb{C}$) |

$\mathrm{K}$ | thermal conductivity (kW/$\mathbb{C}$) |

$\mathsf{\phi}$ | coefficient of performance |

mc | total thermal mass (kWh/$\mathbb{C}$) |

${\mathrm{T}}_{\mathrm{iw}}$ | the temperature of incoming water to the electric water heater system ($\mathbb{C}$) |

${\mathrm{T}}_{\mathrm{amb}}$ | ambient temperature ($\mathbb{C}$) |

${\mathrm{V}}_{\mathrm{tank}}$ | the capacity of the tank ($\mathrm{l}$) |

${\mathrm{A}}_{\mathrm{tank}}$ | the surface area of the tank (${\mathrm{m}}^{2}$) |

${\mathsf{\rho}}_{\mathrm{water}}$ | the density of water ($\mathrm{kg}/{\mathrm{m}}^{3}$) |

${\mathrm{Cp}}_{\mathrm{water}}$ | the specific heat of water ($\mathrm{J}/\left(\mathrm{kg}\xb7\mathbb{C}\right)$) |

${\dot{\mathrm{m}}}_{\mathrm{water}}$ | the flow rate of water ($\mathrm{l}/\mathrm{h}$) |

${\mathrm{L}}_{\mathrm{j}}\left(\mathrm{t}\right)$ | total hourly load for building j |

R | the thermal resistance of the water storage tank (${\mathrm{m}}^{2}\xb7\mathbb{C}/\mathrm{W}$) |

${\mathsf{\psi}}_{\mathrm{i},\mathrm{j}}$ | maximum allowable difference between the desired temperature and actual temperature (${(\mathbb{C})}^{2}$) |

${\mathsf{\mu}}_{1}$, ${\mathsf{\mu}}_{2}$, ${\mathsf{\mu}}_{3}$ | non-negative constants for the quadratic cost function |

Indices | |

i | index of building assets |

j | index of buildings |

t | index of time |

Sets | |

${\mathrm{A}}_{\mathrm{j}}$ | set of assets in building j |

$\mathrm{N}$ | set of buildings in the community |

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**Figure 1.**The interaction between a building and the other end-users as well as the utility company in a smart connected community.

**Figure 4.**The operation of some assets in one of the residential buildings in the case where they did not collaborate.

**Figure 5.**The operation of some assets in one of the office buildings in the case where they did not collaborate.

**Figure 6.**The operation of some assets in one of the residential buildings in the case they collaborate.

No. | Asset | Rated Power (kW) | User Preferred Time | The Operation Range | Duration (Minutes) |
---|---|---|---|---|---|

1 | Clothes washing machine | 3.5 | 17:20–18:00 | 15–21 | 40 |

2 | Clothes dryer | 3.2 | 18:10–19:20 | 15–22:30 | 70 |

3 | Dishwasher | 2.8 | 20:10–22:10 | 18–23:30 | 120 |

4 | Microwave | 0.9 | 18:30–18:40 | 18:20–19:10 | 10 |

5 | Electric kettle | 1.8 | 7:30–7:40 19–19:10 | 7:10–7:50 18:50–19:30 | 10 |

6 | Electric stove | 5.2 | 7–7:40 18–18:40 | 6:30–7:50 17:30–19:20 | 40 |

7 | Blender | 0.8 | 17:40–17:50 | 17:20–18:40 | 10 |

8 | Hair dryer | 1.5 | 6:40–6:50 | 6:30–8 | 10 |

9 | Steam iron | 1.4 | 20–20:20 | 19:30–23:30 | 20 |

10 | Vacuum cleaner | 1.35 | 19:30–20 | 14–21 | 30 |

11 | Coffee maker | 1.1 | 7:10–7:20 | 6:40–8 | 10 |

12 | Phone charger | 0.01 | 21:30–23:30 | 18–1 (next day) | 120 |

No. | Asset | Rated Power (kW) | User Preferred Time | The Operation Range | Duration (Minutes) |
---|---|---|---|---|---|

1 | Microwave | 0.9 | 12:10–12:20 12:20–12:30 12:30–12:40 | 11:40–13:00 | 10 |

2 | Electric kettle | 1.8 | 8:30–8:40 14–14:10 | 8:00–9:20 13:30–14:30 | 10 |

3 | Bottleless water cooler and heater | 5.1 | 9–9:30 14–14:30 | 8:30–10 12:30–15 | 30 |

4 | Paper shredder | 0.15 | 15–15:20 | 14–17 | 20 |

5 | Coffee maker | 1.1 | 8:20–8:30 11–11:10 14–14:10 | 8–8:40 10:30–11:30 13:30–15 | 10 |

Building Type | Number of PEVs | User Preferred Time | The Operation Range | Charging Duration for One Car (mins) |
---|---|---|---|---|

Residential unit, Type 1 | 1 | 17–22 | 17–6 (next day) | 300 |

Residential unit, Type 2 | 2 | 19–24 | 18:30–6 (next day) | 300 |

Residential unit, Type 3 | 1 | 17–18 21–2 (next day) | 17–18 21–6 (next day) | 360 |

Office building | 10 | 9–12 | 8:30–16:30 | 180 |

Parameters | Value |
---|---|

Thermal conductivity | $K=0.45\left(\mathrm{kW}/\mathbb{C}\right)$ |

The total thermal mass of the fluid of the cooling system | $mc=6.3\text{}\left(\mathrm{kWh}/\mathbb{C}\right)$ |

The coefficient of performance of the cooling system | $\phi =3.2$ |

The temperature of incoming water to the EWH system | ${T}_{iw}=25(\mathbb{C})$ |

The volume of the storage tank of the EWH system | ${V}_{tank}=150(l)$ |

The surface area of the storage tank of the EWH system | ${A}_{tank}=0.04$(m^{2}) |

The density of water | ${\rho}_{water}=998\left(\mathrm{kg}/{\mathrm{m}}^{3}\right)$ |

The specific heat of water | $C{p}_{water}=4186.8\left(\mathrm{J}/\left(\mathrm{kg}\xb7\mathbb{C}\right)\right)$ |

The thermal resistance of the water storage tank | R = 1.309(m^{2}°C/W) |

The flow rate of the hot water | ${\dot{m}}_{water}=40\left(\mathrm{l}/\mathrm{h}\right)$ |

Power range of the EWH system | ${P}_{hvac,j}^{t}\text{}\in \left[0,\text{}4.5\right]\mathrm{kW}$ |

Power range of the cooling system | ${P}_{ewh,j}^{t}\text{}\in \left[0,\text{}2.8\right]\mathrm{kW}$ |

Case | Energy Consumption Reduction | Peak Demand Reduction | Energy-Cost Saving | |
---|---|---|---|---|

Small-scale building community | Without collaboration | 3.98% | 17.35% | 8.76% |

With collaboration | 5.31% | 44.15% | 10.47% | |

Large-scale building community | Without collaboration | 4.05% | 26.13% | 9.91% |

With collaboration | 5.43% | 53.15% | 13.02% |

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## Share and Cite

**MDPI and ACS Style**

Nazemi, S.D.; Jafari, M.A.; Zaidan, E. An Incentive-Based Optimization Approach for Load Scheduling Problem in Smart Building Communities. *Buildings* **2021**, *11*, 237.
https://doi.org/10.3390/buildings11060237

**AMA Style**

Nazemi SD, Jafari MA, Zaidan E. An Incentive-Based Optimization Approach for Load Scheduling Problem in Smart Building Communities. *Buildings*. 2021; 11(6):237.
https://doi.org/10.3390/buildings11060237

**Chicago/Turabian Style**

Nazemi, Seyyed Danial, Mohsen A. Jafari, and Esmat Zaidan. 2021. "An Incentive-Based Optimization Approach for Load Scheduling Problem in Smart Building Communities" *Buildings* 11, no. 6: 237.
https://doi.org/10.3390/buildings11060237