# Demand Side Management for Smart Grid Based on Smart Home Appliances with Renewable Energy Sources and an Energy Storage System

Authors: Zhihong Xu; Yan Gao; Muhammad Hussain; Panhong Cheng
Journal: Mathematical Problems in Engineering
 (2020)

---

## Abstract

A novel energy management system to improve the efficiency of renewable energy and storage system by scheduling various types of household appliances is developed. The end users schedule appliances optimally taking into account maximum utility (a measure of the satisfaction level of user’s electricity consumption) as well as minimum user’s cost of energy as competitive objectives. A random disturbance (reflecting an uncertainty) is introduced to describe the indeterminate amount of the electricity produced from the renewable energy sources at the adjacent slots. By applying the probability theory, the uncertain optimization model is transformed into a convex optimization problem. Then, the optimal solution is obtained using a quasi-Newton method. The rationality of our proposed model is verified through numerical simulations. According to the results of simulation studies, it is demonstrated that our proposed model not only enhances users’ utility but also reduces energy consumption cost.


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## Body

## 1. Introduction

### 1.1. Literature Review

The energy shortage and aggravation of environmental pollution are serious problems in a traditional power grid system. Power management is currently undergoing through a dramatic transformation from a traditional power system (TPS) into a smart grid [1]. The smart grid is an intellectualized power network system, which is based on highly integrated two-way communication networks and information systems. It is reliable, economical, and environmentally friendly and can be applied in sensing technologies and control infrastructures. Compared with traditional power grids, the smart grid has the advantages of flexible structure, excellent system performance, and high quality, which can improve the social development. Therefore, smart grid has become a beacon of development in the direction toward a modern power system [2–4].Demand side management (DSM) in smart grid is a kind of management activity which can effectively promote and mobilize the enthusiasm of users to change or transfer the power consumption according to the dynamic prices (e.g., real-time pricing, time of use pricing, and inclining block rates) [5]. It is a hot topic for how to mobilize the enthusiasm of users to participate in their energy management. With improving the technology of renewable energy sources (RES) in recent years, many users are equipped with rooftop solar panels or small wind turbines. The RES is reducing the reliance on the TPS and relieving the supply pressure of power grids [6–9]. To alleviate the randomness and intermittent characteristics of the RES, an energy storage system (ESS) is being applied in smart grid, for example, popular electric vehicles that require both energy availability and storage power [10–13].Scheduling the RES and ESS, power grids are becoming more secure and efficient in the electricity market [10, 14–22].


### 1.2. The Motivation of the Work and Our Contributions

Research on the management of RES and ESS are at their infancy [23–26]. Moreover, in current studies, RES and ESS have mainly focused on the foundation level of load prediction [10, 16, 18, 19, 27–33]. An optimization method [10] based on teaching-learning and adaptive neuro-fuzzy inference is used to estimate the maximum actual capacity of a charging state. A charging and discharging strategy of a future battery energy storage system was studied using an open distribution system simulator software to predict the daily load and photovoltaic (PV) generation curves [31]. The authors in [32], presented a simple method for determining the optimal capacity of PV power generation with a battery system based on the total annual solar radiation. The grid dependency can be obtained based on the total annual solar radiation without considering the time-series waveform of solar radiation. The studies need to know the production power distribution of the RES. However, in our model, a random disturbance (reflecting an uncertainty) is introduced to describe the indeterminate amount of the electricity produced from the RES at the adjacent slots. It is unnecessary to know about the distribution of electricity produced from the RES which is our first novelty.Considering the RES or ESS, the classification of appliances is given in some studies [16, 34–42]. A cost minimization problem focusing on the uninterruptible operation with an uncertainty of renewable energy generation at different time intervals was raised in [34]. In this study, uninterruptible and immovable electrical appliances were involved, whereas elastic appliances (such as air conditioners and water heaters) and semielastic ones (such as washing machines, dishwashers, and electric cookers) were not. A user convenience model employing a heuristic algorithm was proposed to minimize a user’s dissatisfaction and the supplier’s cost [36]. The model only involved appliances with beginning operation time and ending time. A robust optimization technique for scheduling smart appliances and electrical energy storage was applied to minimize the electricity bill and the CO2 emissions [37]. Chen et al. utilized a Lyapunov optimization approach to minimize the total payment from the grid and also have focused on semielastic appliances [38]. Similarly, authors in [39–42] elaborated on multilevel management with various RES and controllable loads, taking into account the energy saving and comfortable lifestyle as objectives of a realistic smart home energy management system.Nevertheless, these works failed to consider the impacts of RES and ESS by scheduling different types of smart appliances, which is our second novelty.To address these issues, an efficient model for demand side management is proposed, wherein, not only it is unnecessary to know about the distribution of electricity produced from the RES but also the impacts of RES and ESS are taken into account through managing various types of household appliances. In this way, users can optimally schedule operating set times of their smart home appliances and reasonably equip with the ESS and RES to achieve their energy management.In our model, an expectation (caused by the uncertainty of RES) users’ welfare maximization model is established. The users’ utility maximization and bill minimization constitute the objective function. The users can design their electricity consumption mode according to the objective function. The constraint reflects the fluctuation (uncertainty) of electricity caused by the intermittency of the RES. The objective function is transformed into a deterministic problem and the constraint into the deterministic one. The model is then solved by applying the dual method. Finally, the rationality of the model and the validity of the algorithm are verified through numerical simulations. Our contributions are as follows:(1)
A random disturbance is introduced to describe the intermittence and instability of the RES at the adjacent slots. In this model, it is unnecessary to know about the distribution of electricity produced from the RES.(2)
An efficient model for demand side management is proposed so that users can reasonably arrange different smart appliances and consume the electricity supplied from the TPS, produced from the RES, or stored in the ESS concerning the users’ utility maximization and their bill minimization.The remainder of this paper is organized as follows. InSection 2, the classification of different appliances and the problem statement are formulated. We present the proposed model in Section 3. The results of numerical simulations are provided in Section 4, and some concluding remarks are given in Section 5.The list of abbreviations is shown in Appendix. The architecture of the proposed system model in time slotk is illustrated in Figure 1.Figure 1
Architecture of the proposed system model in time slotk.


## 1.1. Literature Review

The energy shortage and aggravation of environmental pollution are serious problems in a traditional power grid system. Power management is currently undergoing through a dramatic transformation from a traditional power system (TPS) into a smart grid [1]. The smart grid is an intellectualized power network system, which is based on highly integrated two-way communication networks and information systems. It is reliable, economical, and environmentally friendly and can be applied in sensing technologies and control infrastructures. Compared with traditional power grids, the smart grid has the advantages of flexible structure, excellent system performance, and high quality, which can improve the social development. Therefore, smart grid has become a beacon of development in the direction toward a modern power system [2–4].Demand side management (DSM) in smart grid is a kind of management activity which can effectively promote and mobilize the enthusiasm of users to change or transfer the power consumption according to the dynamic prices (e.g., real-time pricing, time of use pricing, and inclining block rates) [5]. It is a hot topic for how to mobilize the enthusiasm of users to participate in their energy management. With improving the technology of renewable energy sources (RES) in recent years, many users are equipped with rooftop solar panels or small wind turbines. The RES is reducing the reliance on the TPS and relieving the supply pressure of power grids [6–9]. To alleviate the randomness and intermittent characteristics of the RES, an energy storage system (ESS) is being applied in smart grid, for example, popular electric vehicles that require both energy availability and storage power [10–13].Scheduling the RES and ESS, power grids are becoming more secure and efficient in the electricity market [10, 14–22].


## 1.2. The Motivation of the Work and Our Contributions

Research on the management of RES and ESS are at their infancy [23–26]. Moreover, in current studies, RES and ESS have mainly focused on the foundation level of load prediction [10, 16, 18, 19, 27–33]. An optimization method [10] based on teaching-learning and adaptive neuro-fuzzy inference is used to estimate the maximum actual capacity of a charging state. A charging and discharging strategy of a future battery energy storage system was studied using an open distribution system simulator software to predict the daily load and photovoltaic (PV) generation curves [31]. The authors in [32], presented a simple method for determining the optimal capacity of PV power generation with a battery system based on the total annual solar radiation. The grid dependency can be obtained based on the total annual solar radiation without considering the time-series waveform of solar radiation. The studies need to know the production power distribution of the RES. However, in our model, a random disturbance (reflecting an uncertainty) is introduced to describe the indeterminate amount of the electricity produced from the RES at the adjacent slots. It is unnecessary to know about the distribution of electricity produced from the RES which is our first novelty.Considering the RES or ESS, the classification of appliances is given in some studies [16, 34–42]. A cost minimization problem focusing on the uninterruptible operation with an uncertainty of renewable energy generation at different time intervals was raised in [34]. In this study, uninterruptible and immovable electrical appliances were involved, whereas elastic appliances (such as air conditioners and water heaters) and semielastic ones (such as washing machines, dishwashers, and electric cookers) were not. A user convenience model employing a heuristic algorithm was proposed to minimize a user’s dissatisfaction and the supplier’s cost [36]. The model only involved appliances with beginning operation time and ending time. A robust optimization technique for scheduling smart appliances and electrical energy storage was applied to minimize the electricity bill and the CO2 emissions [37]. Chen et al. utilized a Lyapunov optimization approach to minimize the total payment from the grid and also have focused on semielastic appliances [38]. Similarly, authors in [39–42] elaborated on multilevel management with various RES and controllable loads, taking into account the energy saving and comfortable lifestyle as objectives of a realistic smart home energy management system.Nevertheless, these works failed to consider the impacts of RES and ESS by scheduling different types of smart appliances, which is our second novelty.To address these issues, an efficient model for demand side management is proposed, wherein, not only it is unnecessary to know about the distribution of electricity produced from the RES but also the impacts of RES and ESS are taken into account through managing various types of household appliances. In this way, users can optimally schedule operating set times of their smart home appliances and reasonably equip with the ESS and RES to achieve their energy management.In our model, an expectation (caused by the uncertainty of RES) users’ welfare maximization model is established. The users’ utility maximization and bill minimization constitute the objective function. The users can design their electricity consumption mode according to the objective function. The constraint reflects the fluctuation (uncertainty) of electricity caused by the intermittency of the RES. The objective function is transformed into a deterministic problem and the constraint into the deterministic one. The model is then solved by applying the dual method. Finally, the rationality of the model and the validity of the algorithm are verified through numerical simulations. Our contributions are as follows:(1)
A random disturbance is introduced to describe the intermittence and instability of the RES at the adjacent slots. In this model, it is unnecessary to know about the distribution of electricity produced from the RES.(2)
An efficient model for demand side management is proposed so that users can reasonably arrange different smart appliances and consume the electricity supplied from the TPS, produced from the RES, or stored in the ESS concerning the users’ utility maximization and their bill minimization.The remainder of this paper is organized as follows. InSection 2, the classification of different appliances and the problem statement are formulated. We present the proposed model in Section 3. The results of numerical simulations are provided in Section 4, and some concluding remarks are given in Section 5.The list of abbreviations is shown in Appendix. The architecture of the proposed system model in time slotk is illustrated in Figure 1.Figure 1
Architecture of the proposed system model in time slotk.


## 2. Classification of Appliances and Problem Statement

A system that includes a power supplier and multiple residential users is considered. The supplier and all users are linked using a communication infrastructure to realize information exchange. An energy hub (a detailed description given in [43]) of each user contains a rooftop solar panel, a small wind turbine, several batteries and electrical vehicles, some smart home appliances, and an electric quantity controller (EQC) with a smart meter. The smart meter not only bears a function of information transmission but also has the function of regulating and controlling household electricity information. It transmits the information of household electrical equipment consumption to users in order to better control the consumption and cost of electricity and sends relevant information to the power supplier to achieve load control and electricity price. The electricity price is given by the power supplier. Consumers can adjust the mode of electricity consumption by smart meters to achieve their maximum utility and minimum bill. In this model, it is supposed that users cannot sell redundant energy to the grid or other users, and thus the excess power from the RES is discarded [36].Within the scope of our model, the devices’ cost and the reactive power are not considered, and frequency and voltage stability are guaranteed [44]. The appliances are linked with the smart meter, where t is the initial time, and not is necessarily 1, K is the number of time slots, and K=t,t+1,…,K is the entire cycle. The indexes, functions and variables, and parameters are listed in Tables 1–3, respectively. The modeling of the energy system components is detailed as follows.Table 1
List of indexes.
Indexes
Description
Unit
t
Initial time
h
K
The entire cycle
h
k
Index of time slot
h
K
Number of time slots
h
i
Index of users
—
n
Number of users
—
A
i
Set of must-run appliances
—
B
i
Set of elastic appliances
—
C
i
Set of semielastic appliances
—
a
u
,
i
Electrical appliance
—
r
a
u
,
i
k
Electricity consumed by must-run applianceau,i
kWh
K
a
u
,
i
Running time of must-run applianceau,i∈Ai
h
M
A
i
Number of must-run appliancesAi
—
M
B
i
Number of elastic appliancesBi
—
M
C
i
Number of semielastic appliancesCi
—
E
a
u
,
i
Total electricity consumed by must-run appliance
kWh
r
a
u
,
i
k
max
Maximum electricity consumed by elastic appliance
kWh
R
a
u
,
i
k
max
Maximum total electricity consumed by elastic appliances
kWh
E
a
u
,
i
k
′
Total electricity of semielastic appliances
kWh
α
a
u
,
i
Beginning time of semielastic appliances
h
β
a
u
,
i
End time of semielastic appliances
h
K
a
u
,
i
′
Working time of semielastic appliances
h
C
b
Capacity of the ESS
kWh
g
b
Maximum value of the ESS
kWh
−
g
b
Minimum value of the ESS
kWh
s
i
0
Initial charging state of the ESS
kWh
s
i
k
Charging state of the ESS
kWh
X
Set of electricity supplied from the TPS
kWh
Y
Set of electricity produced from the RES
kWh
Z
Set of electricity consumed in the ESS
kWh
G
t
Total electricity quantity of the power system
kWhTable 2
List of functions and variables.
Functions and variables
Description
Unit
x
i
k
Electricity supplied from the TPS
kWh
y
i
k
Electricity produced from the RES
kWh
z
i
k
Charging and discharging rates of the ESS
kWh
U
l
i
k
,
ω
i
k
Utility function of useri
$
p
k
Price
Cents/kWh
l
i
k
Total power consumed by useri in time slot k
kWh
l
a
u
,
i
k
Electricity consumed by applianceau,i∈Ai∪Bi∪Ci
kWh
P
l
i
k
Payment function of users
$
Φ
x
Distribution function
—Table 3
List of parameters.
Parameters
Description
Unit
α
i
Preset parameter
—
ω
i
k
Elastic coefficient
—
m
k
,
n
k
Differentiated prices
Cents/kWh
b
k
Energy consumption threshold in time slotk
kWh
ρ
i
k
Random disturbance to describe the indeterminate load produced by the RES
—
ε
,
η
Positive numbers
—
λ
,
θ
,
γ
Lagrangian multipliers
—


### 2.1. Classification of Appliances

Every user has three types of appliances, namely,Ai, Bi, and Ci. The first type of household appliances, Ai, which are must-run appliances, are those household electrical appliances that must be operated. Owing to their inelasticity, this type of electrical appliances does not depend on the change of electricity price. For instance, lighting appliances, televisions, and refrigerators are not controlled by an EQC unit. The second category of household appliances, Bi, includes elastic equipment, such as air conditioners, water heaters, and other adjustable equipment. The use of these appliances is directly affected by the electricity price. When the electricity price rises, the electricity consumption decreases, and vice versa. The last type of household appliances, Ci, includes semielastic equipment, such as washing machines, dishwashers, electric cookers, and dryers. These appliances all require a certain amount of working time to complete an established task. Similar to elastic devices, semielastic devices are affected by the electricity price, whereas the working time of semielastic devices is fixed.


#### 2.1.1. Must-Run Appliances

Letlau,ik denote the power consumed by appliance au,i∈Ai∪Bi∪Ci of user i in time slot k; Kau,i denotes the running time of a must-run appliance au,i∈Ai, which consumes rau,ik in time slot k; and Kau,i⊆K. Denote MAi as the number of must-run appliances, and let Eau,i=∑u=1MAilau,ik denote the total power consumption of must-run appliances during an operation cycle.


#### 2.1.2. Elastic Appliances

Denoterau,ikmax and MBi as the maximum electricity consumed by elastic appliance au,i and the number of elastic appliances, respectively, and Rau,ikmax=∑u=1MBirau,ikmax denotes the maximum electricity consumed by all elastic appliances. The working models of these appliances are varied. The total electricity consumption ∑u=1MBilau,ik is given. Referring to this value, users can make their own unique electricity consumption mode. This means that, for the same total electricity consumption, the user has the opportunity to keep the water heater at a higher temperature during the winter or turn an electric fan on.


#### 2.1.3. Semielastic Appliances

Semielastic appliances complete a task within a fixed time interval. Every household hasMCi semielastic appliances. Denote Eau,i′ as the total electricity of semielastic appliances. The working hours of these appliances are continuous. Once semielastic appliances are operated, they need to complete their tasks without stop. Here, Kau,i′ is the working time of semielastic appliances. The beginning time of an electrical appliance is αau,i, and the end time is βau,i. The working time can be expressed as αau,i,αau,i+1,…,βau,i∈Kau,i′. The satisfaction of the user will be seriously hindered if semielastic appliances are interrupted during an operation.The three types of appliances satisfy the available electricity consumption, as indicated in Table4.Table 4
Available electricity consumption of appliances.
Appliances
Parameters
Available electricity consumption
Must-run appliancesAi
K
a
u
,
i, rau,ik
l
a
u
,
i
k
=
r
a
u
,
i
k
,
0
,
k
∈
K
a
u
,
i
otherwise
Elastic appliancesBi
r
a
u
,
i
k
max
0
≤
l
a
u
,
i
k
≤
r
a
u
,
i
k
max, ∀k∈K
Semielastic appliancesCi
r
a
u
,
i
k
max, Kau,i′, Eau,i′
0
≤
l
a
u
,
i
k
≤
r
a
u
,
i
k
max, ∀k∈Kau,i′
l
a
u
,
i
k
=
0, ∀k∉Kau,i′
∑
k
∈
K
a
u
,
i
′
l
a
u
,
i
k
=
E
a
u
,
i
′


### 2.2. Power Sources

#### 2.2.1. Traditional Power System

In this paper, TPS refers to the electricity supplied from the power supplier. Letxik denote the electricity supplied from a TPS in time slot k.


#### 2.2.2. Renewable Energy Sources

The RES are deemed as significant generation alternatives in smart grid owing to their nonexhausted nature and benign environmental effects. Both a stochastic approach and a robust one are optimization methods for uncertain situations. The RES power generation is stochastic and intermittent and will be disturbed by many independent random factors. In the stochastic approach, the probability distribution of random parameters is known; in the robust one, the probability distribution of random parameters is unknown, but its fluctuation range needs to be known. In our model, unnecessary to know about the distribution of electricity produced from the RES, a random disturbanceρik is introduced to describe the uncertainty of the RES with expectation value Eρik=μik and variance Dρik=σik2. Then,(1)yik+1=yik+ρik+1,∀k∈K,where yik denotes the electricity produced from the RES in time slot k.By applying the Central Limit Theorem, when the number of users is large enough or segmentation of time slots is small enough, the sum of all random variables∑i=1n∑j=t+1kρij is subject to a normal distribution:(2)∑i=1n∑j=t+1kρij∼Nμ,σ2,where N⋅ denotes a normal distribution function, μ=∑i=1n∑j=t+1kμij, and σ2=∑i=1n∑j=t+1kσij2.Letw=−∑i=1n∑j=t+1kρij+μ/σ, the standardization of the normal distribution can be obtained as follows: w∼N0,1; therefore, the distribution of all random variables is known why a stochastic approach is chosen instead of a robust one in our study.


#### 2.2.3. Energy Storage System

Electric vehicles and batteries can be classified as the ESS, which can be charged or discharged.Cb is denoted as the capacity of the ESS. In time slot k, the charge rate zik is positive and the discharge rate zik is negative. In addition, gb and −gb denote the maximum and minimum capacity values of the ESS, −gb≤zik≤gb, sik is denoted as the capacity of the ESS at the beginning time slot k, and si0 is the initial charging state of the ESS. At every time slot, the stored energy cannot exceed the storage capacity:(3)sik=si0+∑j=1k−1zij,0≤sik≤Cb.


### 2.3. Real-Time Pricing

The existing price forms mainly include fixed pricing, step pricing, time of use pricing (TOU), real-time pricing (RTP), and inclining block rates (IBR). Among them, RTP and IBR models have been extensively studied. It refers to the cost occurring at a certain moment during electricity sales, which reflects the characteristics of electricity prices with respect to time. Practically, power suppliers design the real-time price considering the time of demand and the power availability. Contrastingly in the IBR pricing model, as time goes on, prices remain unchanged, while incurring an increase when the energy consumption of a user reaches a predetermined threshold [19].In our model, combining RTP and IBR price tariffs, the pricepk is denoted as follows (see [45]):(4)pklik=mk,0≤lik≤bk,nk,lik>bk,where mk and nk are differentiated prices, in which mk≤nk; lik denotes the total electricity consumption of user i in time slot k; and bk is the energy consumption threshold in time slot k.


### 2.4. Utility Function of a User

The concept of a utility function in microeconomics is widely used to characterize a user’s satisfaction with a certain amount of electricity consumption. The existing studies have shown that the user’s response can be expressed by a utility function. The electricity demand varies with each individual and depends upon the time of a day, climate, and price. The value of utility function denotes the satisfaction level of each user after using the electricity purchased.Ulik,ωik is the utility function of user i, lik denotes the amount of consumed energy, and ωik is an elastic coefficient and is a known parameter, and the utility value is expressed in a monetary form. Different utility functions represent different types of users and are distinguished by different elastic coefficients. Under the same conditions, the larger the ωik value is, the higher the user’ satisfaction. Usually, the general utility functions satisfy the following basic assumptions:(1)
The utility function is a nondecreasing function, i.e.,∂Ul/∂l≥0.(2)
Marginal utility is nonincreasing, i.e., the utility is a concave function. Thus,∂2Ul/∂l2≤0.(3)
When the user does not consume electricity, the utility is zero, that is,U0=0.In this paper, utility functions are considered as follows:(5)Ulik,ωik=ωiklik−αi2lik2,0≤lik≤ωikαi,ωik22αi,lik>ωikαi,where αi is a preset parameter, and ωik and lik are the elastic coefficient and power consumed by user i in time slot k, respectively.


### 2.5. Payment Function of Users

The termPlik denotes a user’s payment function. In our model, the devices’ cost is not considered. Based on the price shown in (4), for a total load lik in time slot k, the user’s cost Plik is determined as the maximum of two intersecting lines [45]:(6)Plik=pklik=mklik,Plik=pklik=nklik+mk−nkbk.The user’s payment function is obtained from the following equation:(7)Plik=pklik=maxmklik,nklik+mk−nkbk.


## 2.1. Classification of Appliances

Every user has three types of appliances, namely,Ai, Bi, and Ci. The first type of household appliances, Ai, which are must-run appliances, are those household electrical appliances that must be operated. Owing to their inelasticity, this type of electrical appliances does not depend on the change of electricity price. For instance, lighting appliances, televisions, and refrigerators are not controlled by an EQC unit. The second category of household appliances, Bi, includes elastic equipment, such as air conditioners, water heaters, and other adjustable equipment. The use of these appliances is directly affected by the electricity price. When the electricity price rises, the electricity consumption decreases, and vice versa. The last type of household appliances, Ci, includes semielastic equipment, such as washing machines, dishwashers, electric cookers, and dryers. These appliances all require a certain amount of working time to complete an established task. Similar to elastic devices, semielastic devices are affected by the electricity price, whereas the working time of semielastic devices is fixed.


### 2.1.1. Must-Run Appliances

Letlau,ik denote the power consumed by appliance au,i∈Ai∪Bi∪Ci of user i in time slot k; Kau,i denotes the running time of a must-run appliance au,i∈Ai, which consumes rau,ik in time slot k; and Kau,i⊆K. Denote MAi as the number of must-run appliances, and let Eau,i=∑u=1MAilau,ik denote the total power consumption of must-run appliances during an operation cycle.


### 2.1.2. Elastic Appliances

Denoterau,ikmax and MBi as the maximum electricity consumed by elastic appliance au,i and the number of elastic appliances, respectively, and Rau,ikmax=∑u=1MBirau,ikmax denotes the maximum electricity consumed by all elastic appliances. The working models of these appliances are varied. The total electricity consumption ∑u=1MBilau,ik is given. Referring to this value, users can make their own unique electricity consumption mode. This means that, for the same total electricity consumption, the user has the opportunity to keep the water heater at a higher temperature during the winter or turn an electric fan on.


### 2.1.3. Semielastic Appliances

Semielastic appliances complete a task within a fixed time interval. Every household hasMCi semielastic appliances. Denote Eau,i′ as the total electricity of semielastic appliances. The working hours of these appliances are continuous. Once semielastic appliances are operated, they need to complete their tasks without stop. Here, Kau,i′ is the working time of semielastic appliances. The beginning time of an electrical appliance is αau,i, and the end time is βau,i. The working time can be expressed as αau,i,αau,i+1,…,βau,i∈Kau,i′. The satisfaction of the user will be seriously hindered if semielastic appliances are interrupted during an operation.The three types of appliances satisfy the available electricity consumption, as indicated in Table4.Table 4
Available electricity consumption of appliances.
Appliances
Parameters
Available electricity consumption
Must-run appliancesAi
K
a
u
,
i, rau,ik
l
a
u
,
i
k
=
r
a
u
,
i
k
,
0
,
k
∈
K
a
u
,
i
otherwise
Elastic appliancesBi
r
a
u
,
i
k
max
0
≤
l
a
u
,
i
k
≤
r
a
u
,
i
k
max, ∀k∈K
Semielastic appliancesCi
r
a
u
,
i
k
max, Kau,i′, Eau,i′
0
≤
l
a
u
,
i
k
≤
r
a
u
,
i
k
max, ∀k∈Kau,i′
l
a
u
,
i
k
=
0, ∀k∉Kau,i′
∑
k
∈
K
a
u
,
i
′
l
a
u
,
i
k
=
E
a
u
,
i
′


## 2.1.1. Must-Run Appliances

Letlau,ik denote the power consumed by appliance au,i∈Ai∪Bi∪Ci of user i in time slot k; Kau,i denotes the running time of a must-run appliance au,i∈Ai, which consumes rau,ik in time slot k; and Kau,i⊆K. Denote MAi as the number of must-run appliances, and let Eau,i=∑u=1MAilau,ik denote the total power consumption of must-run appliances during an operation cycle.


## 2.1.2. Elastic Appliances

Denoterau,ikmax and MBi as the maximum electricity consumed by elastic appliance au,i and the number of elastic appliances, respectively, and Rau,ikmax=∑u=1MBirau,ikmax denotes the maximum electricity consumed by all elastic appliances. The working models of these appliances are varied. The total electricity consumption ∑u=1MBilau,ik is given. Referring to this value, users can make their own unique electricity consumption mode. This means that, for the same total electricity consumption, the user has the opportunity to keep the water heater at a higher temperature during the winter or turn an electric fan on.


## 2.1.3. Semielastic Appliances

Semielastic appliances complete a task within a fixed time interval. Every household hasMCi semielastic appliances. Denote Eau,i′ as the total electricity of semielastic appliances. The working hours of these appliances are continuous. Once semielastic appliances are operated, they need to complete their tasks without stop. Here, Kau,i′ is the working time of semielastic appliances. The beginning time of an electrical appliance is αau,i, and the end time is βau,i. The working time can be expressed as αau,i,αau,i+1,…,βau,i∈Kau,i′. The satisfaction of the user will be seriously hindered if semielastic appliances are interrupted during an operation.The three types of appliances satisfy the available electricity consumption, as indicated in Table4.Table 4
Available electricity consumption of appliances.
Appliances
Parameters
Available electricity consumption
Must-run appliancesAi
K
a
u
,
i, rau,ik
l
a
u
,
i
k
=
r
a
u
,
i
k
,
0
,
k
∈
K
a
u
,
i
otherwise
Elastic appliancesBi
r
a
u
,
i
k
max
0
≤
l
a
u
,
i
k
≤
r
a
u
,
i
k
max, ∀k∈K
Semielastic appliancesCi
r
a
u
,
i
k
max, Kau,i′, Eau,i′
0
≤
l
a
u
,
i
k
≤
r
a
u
,
i
k
max, ∀k∈Kau,i′
l
a
u
,
i
k
=
0, ∀k∉Kau,i′
∑
k
∈
K
a
u
,
i
′
l
a
u
,
i
k
=
E
a
u
,
i
′


## 2.2. Power Sources

### 2.2.1. Traditional Power System

In this paper, TPS refers to the electricity supplied from the power supplier. Letxik denote the electricity supplied from a TPS in time slot k.


### 2.2.2. Renewable Energy Sources

The RES are deemed as significant generation alternatives in smart grid owing to their nonexhausted nature and benign environmental effects. Both a stochastic approach and a robust one are optimization methods for uncertain situations. The RES power generation is stochastic and intermittent and will be disturbed by many independent random factors. In the stochastic approach, the probability distribution of random parameters is known; in the robust one, the probability distribution of random parameters is unknown, but its fluctuation range needs to be known. In our model, unnecessary to know about the distribution of electricity produced from the RES, a random disturbanceρik is introduced to describe the uncertainty of the RES with expectation value Eρik=μik and variance Dρik=σik2. Then,(1)yik+1=yik+ρik+1,∀k∈K,where yik denotes the electricity produced from the RES in time slot k.By applying the Central Limit Theorem, when the number of users is large enough or segmentation of time slots is small enough, the sum of all random variables∑i=1n∑j=t+1kρij is subject to a normal distribution:(2)∑i=1n∑j=t+1kρij∼Nμ,σ2,where N⋅ denotes a normal distribution function, μ=∑i=1n∑j=t+1kμij, and σ2=∑i=1n∑j=t+1kσij2.Letw=−∑i=1n∑j=t+1kρij+μ/σ, the standardization of the normal distribution can be obtained as follows: w∼N0,1; therefore, the distribution of all random variables is known why a stochastic approach is chosen instead of a robust one in our study.


### 2.2.3. Energy Storage System

Electric vehicles and batteries can be classified as the ESS, which can be charged or discharged.Cb is denoted as the capacity of the ESS. In time slot k, the charge rate zik is positive and the discharge rate zik is negative. In addition, gb and −gb denote the maximum and minimum capacity values of the ESS, −gb≤zik≤gb, sik is denoted as the capacity of the ESS at the beginning time slot k, and si0 is the initial charging state of the ESS. At every time slot, the stored energy cannot exceed the storage capacity:(3)sik=si0+∑j=1k−1zij,0≤sik≤Cb.


## 2.2.1. Traditional Power System

In this paper, TPS refers to the electricity supplied from the power supplier. Letxik denote the electricity supplied from a TPS in time slot k.


## 2.2.2. Renewable Energy Sources

The RES are deemed as significant generation alternatives in smart grid owing to their nonexhausted nature and benign environmental effects. Both a stochastic approach and a robust one are optimization methods for uncertain situations. The RES power generation is stochastic and intermittent and will be disturbed by many independent random factors. In the stochastic approach, the probability distribution of random parameters is known; in the robust one, the probability distribution of random parameters is unknown, but its fluctuation range needs to be known. In our model, unnecessary to know about the distribution of electricity produced from the RES, a random disturbanceρik is introduced to describe the uncertainty of the RES with expectation value Eρik=μik and variance Dρik=σik2. Then,(1)yik+1=yik+ρik+1,∀k∈K,where yik denotes the electricity produced from the RES in time slot k.By applying the Central Limit Theorem, when the number of users is large enough or segmentation of time slots is small enough, the sum of all random variables∑i=1n∑j=t+1kρij is subject to a normal distribution:(2)∑i=1n∑j=t+1kρij∼Nμ,σ2,where N⋅ denotes a normal distribution function, μ=∑i=1n∑j=t+1kμij, and σ2=∑i=1n∑j=t+1kσij2.Letw=−∑i=1n∑j=t+1kρij+μ/σ, the standardization of the normal distribution can be obtained as follows: w∼N0,1; therefore, the distribution of all random variables is known why a stochastic approach is chosen instead of a robust one in our study.


## 2.2.3. Energy Storage System

Electric vehicles and batteries can be classified as the ESS, which can be charged or discharged.Cb is denoted as the capacity of the ESS. In time slot k, the charge rate zik is positive and the discharge rate zik is negative. In addition, gb and −gb denote the maximum and minimum capacity values of the ESS, −gb≤zik≤gb, sik is denoted as the capacity of the ESS at the beginning time slot k, and si0 is the initial charging state of the ESS. At every time slot, the stored energy cannot exceed the storage capacity:(3)sik=si0+∑j=1k−1zij,0≤sik≤Cb.


## 2.3. Real-Time Pricing

The existing price forms mainly include fixed pricing, step pricing, time of use pricing (TOU), real-time pricing (RTP), and inclining block rates (IBR). Among them, RTP and IBR models have been extensively studied. It refers to the cost occurring at a certain moment during electricity sales, which reflects the characteristics of electricity prices with respect to time. Practically, power suppliers design the real-time price considering the time of demand and the power availability. Contrastingly in the IBR pricing model, as time goes on, prices remain unchanged, while incurring an increase when the energy consumption of a user reaches a predetermined threshold [19].In our model, combining RTP and IBR price tariffs, the pricepk is denoted as follows (see [45]):(4)pklik=mk,0≤lik≤bk,nk,lik>bk,where mk and nk are differentiated prices, in which mk≤nk; lik denotes the total electricity consumption of user i in time slot k; and bk is the energy consumption threshold in time slot k.


## 2.4. Utility Function of a User

The concept of a utility function in microeconomics is widely used to characterize a user’s satisfaction with a certain amount of electricity consumption. The existing studies have shown that the user’s response can be expressed by a utility function. The electricity demand varies with each individual and depends upon the time of a day, climate, and price. The value of utility function denotes the satisfaction level of each user after using the electricity purchased.Ulik,ωik is the utility function of user i, lik denotes the amount of consumed energy, and ωik is an elastic coefficient and is a known parameter, and the utility value is expressed in a monetary form. Different utility functions represent different types of users and are distinguished by different elastic coefficients. Under the same conditions, the larger the ωik value is, the higher the user’ satisfaction. Usually, the general utility functions satisfy the following basic assumptions:(1)
The utility function is a nondecreasing function, i.e.,∂Ul/∂l≥0.(2)
Marginal utility is nonincreasing, i.e., the utility is a concave function. Thus,∂2Ul/∂l2≤0.(3)
When the user does not consume electricity, the utility is zero, that is,U0=0.In this paper, utility functions are considered as follows:(5)Ulik,ωik=ωiklik−αi2lik2,0≤lik≤ωikαi,ωik22αi,lik>ωikαi,where αi is a preset parameter, and ωik and lik are the elastic coefficient and power consumed by user i in time slot k, respectively.


## 2.5. Payment Function of Users

The termPlik denotes a user’s payment function. In our model, the devices’ cost is not considered. Based on the price shown in (4), for a total load lik in time slot k, the user’s cost Plik is determined as the maximum of two intersecting lines [45]:(6)Plik=pklik=mklik,Plik=pklik=nklik+mk−nkbk.The user’s payment function is obtained from the following equation:(7)Plik=pklik=maxmklik,nklik+mk−nkbk.


## 3. Expectation Welfare Maximization Model

### 3.1. Problem Formulation

In our model, based on scheduling household appliances and considering the fluctuation of electricity produced from the RES, an expectation model of users’ welfare maximization is taken as follows:(P1)(8)maxE∑k=tK∑i=1nUlik,ωik−Plik,subject to(9a)0≤lau,ik≤rau,ikmax,∀k∈K,∀au,i∈Bi,(9b)0≤lau,ik≤rau,ikmax,∀k∈Kau,i′,∀au,i∈Ci,(9c)lau,ik=0,∀k∉Kau,i′,∀au,i∈Ci,(9d)∑k∈Kau,i′lau,ik=Eau,i′,∀au,i∈Ci,(9e)lik=Eau,ik+∑au,i∈Bilau,ik+∑au,i∈Cilau,ik,∀k∈K,(9f)yik+1=yik+ρik+1,∀k∈K,(9g)zik∈−gb,gb,sik=si0+∑j=1k−1zij,0≤sik≤Cb,∀k∈K,(9h)E∑i=1nlik≤Gk,∀k∈K,(9i)lik=xik−yik+zik+,where Ulik,ωik and Plik are the utility function and payment function of user i in time slot k, respectively, ωik(reflecting the satisfaction level of each user after using the electricity purchased) is an elastic coefficient and is a known parameter, xik is the amount of electricity supplied from the TPS, yik is the amount of electricity produced from the RES, and zik is the electricity charged or discharged in the ESS. In addition, xik, yik, and zik are variables which the amount is eight hundred and forty and Gk is the total electricity quantity of a power system.The constraints (9a) and (9b)–(9d) characterize the electric performance of elastic and semielastic appliances, respectively. The compositions and sources of a user’s electricity consumption are expressed as constraints (9e) and (9i). Constraint (9f) is the power fluctuation generated from the RES between two time slots. This constraint on the RES is not considered in [46]. Besides, determining the relationship between yik and yit, yik can be expressed as follows:(10a)yik=yit+∑j=t+1kρij.Constraint (9g) describes the electric performance of the ESS. Constraint (9h) shows that the expected total electricity consumption of users does not exceed the electricity production, which reflects the correlation and coupling between the electricity consumption of users. Samadi et al. do not take the constraint into account [35].


### 3.2. Convex Optimization Problem Formulation

The transformation of the objective function and constraint (9h) are shown as Appendixes A and B.According to AppendixesA and B, a convex optimization problem from problem (P1) is obtained as(P2)
(11)
max
∑
k
=
t
K
∑
i
=
1
n
U
˜
l
i
t
,
ω
i
k
−
P
˜
l
i
k
,subject to (9a)–(9e), (10a), (9g), and (9i)(12a)∑i=1nxit−yit+zit+≤Gt,k=t,(12b)∑i=1nxit−yit+zik+≤Gk+φk,k=t+1,…,K,in which,(13)U˜ilit,ωik=ωiklit−αi2lit2−ωik∑j=t+1kμij+αilit∑j=t+1kμij−αi2∑j=t+1kμij2−αi2∑j=t+1kσij2,0≤lit≤ωikαi,ωik22αi−ωik∑j=t+1kμij+αilit∑j=t+1kμij−αi2∑j=t+1kμij2−αi2∑j=t+1kσij2,lit≥ωikαi.P˜lik=maxmklit−∑j=t+1kμij+,nklit−∑j=t+1kμij++mk−nkbk,Eρik=μik and Dρik=σik2 are the expectation and variance of the random disturbance ρik, respectively.Among these constraints, constraints (9g), (12a), and (12b) are our inequality constraints, and constraints (9d) are our equality ones. The type of decision variables is real, and there are eight hundred and forty variables, one hundred and thirty-nine inequality constraints, twenty equality constraints, and eight hundred and forty boundaries.Theorem 1.
Problem (P2) is a convex optimization and has an optimal solution.Proof of Theorem1 is shown in Appendix C.


### 3.3. Dual Problem

From Theorem1, (P2) is a convex optimization and has an optimal solution. It is equivalent to its dual problem. Therefore, it can be solved using a dual method [20].The Lagrangian function of (P2) is(14)LX,Y,Z,λ,θ,γ=∑k=tK∑i=1nU˜lit,ωik−P˜lik−λ∑i=1nxit−yit+zit+−Gt+∑k=t+1K∑i=1nxit−yit+zik+−Gk−φk−∑i=1n∑au,i∈Ciθau,iEau,i′−∑k∈Kau,ilau,ik−γ∑k=tK∑i=1nsik−Cb=∑i=1n∑k=tKU˜lit,ωik−P˜lik−∑i=1n∑k=tKλxit−yit+zik++λ∑k=tKGk+λ∑k=t+1Kφk−∑i=1n∑au,i∈Ciθau,iEau,i′−∑k∈Kau,i′lau,ik−γ∑k=tK∑i=1nsik−Cb,where X=x1t,x2t,…,xnt, Y=y1t,y2t,…,ynt, Z=z1t,z2t,…,znt, λ≥0, θ=θau,iMCi×n∈RMCi×n, and γ∈R+. The objective function of the dual optimization problem is as follows:(15)Dλ,θ,γ=maxxik,yik,zikLX,Y,Z,λ,θ,γ0≤lau,ik≤rau,ikmax,∀au,i∈Bi,Ci,zik∈−gb,gb,∀k∈K,lik=xik−yik+zik+.The dual problem is the following:(16)minλ,θ,γ∈R+Dλ,θ,γ.


### 3.4. Smoothing Method

#### 3.4.1. Introduction of Smoothing Methods

The payment functionP˜lik of (P2) contains of the plus and maximum functions, which are nonsmooth. It is impossible to solve this problem directly using classical optimization methods. Smooth approximation functions can be found to replace the abovementioned functions. The basic idea of a smoothing method is to replace the nonsmooth function with a smooth approximation function and solving the original problem is equivalent to solving the traditional optimization problem. There are many smoothing methods, such as quadratic function method, aggregate function method, and density method (see [47]).Absolute value functiony=x is convex on R1 and smooth everywhere except x=0. Given a small constant ε>0, the following smooth function approximates it:(17)fεx=−x,x≤−ε,12εx2+ε2,−ε<x<ε,x,x≥ε,which is called quadratic function method.The aggregate function method is considered. The maximum function is as follows:(18)fx=max1≤i≤mfix,where fixi=1,…,m is a second-order continuous differentiable function on Rn. The following aggregate function is a smooth approximation function of formula (18) [47]: (19)fx,π=πln∑i=1mexpfixπ,where π is a small positive number.Next, the density method is discussed.Letρx be a probability density function from R1 to R+ and satisfy(20)∫−∞+∞sρsds=κ<+∞.The smooth approximation function ofx+=maxx,0 is as follows:(21)Pμ,x=∫−∞+∞x−μs+ρsds,where μ>0 is a smaller number.Because the plus function is inside and the maximum function is outside of the payment functionP˜lik, the plus function is considered first smoothing and then smoothing the maximum function.


#### 3.4.2. First Smoothing

The quadratic function and aggregate function method are simple, practical, and effective, and these smoothing methods are adopted to deal with the nonsmoothness of payment functionP˜lik. The quadratic method is first adopted.Whenx=0, ψx=maxx,0 is nonsmooth, and let(22)ψtx=x,x>t,x24t+x2+t4,−t≤x≤t,0,x<−t.Theorem 2.
For∀t>0, ψtx, in formula (22) is a smooth approximation function of ψx=maxx,0.Proof of Theorem2 is shown as Appendix D.According to Theorem2, x of formula (22) is directly replaced by lit−∑j=t+1kμij+ and the smoothing function ψtlit−∑j=t+1kμij can be obtained. Then,(23)P˜lik=maxmklit−∑j=t+1kμij+,nklit−∑j=t+1kμij++mk−nkbk=maxmkψtlit−∑j=t+1kμij,nkψtlit−∑j=t+1kμij+mk−nkbk.


#### 3.4.3. Second Smoothing

During the second smoothing process, the aggregate function method is used to smooth the maximum function of the payment functionP˜lik.LetP˜lik=maxf1x,f2x, where f1x=mkψtlit−∑j=t+1kμij and f2x=nkψtlit−∑j=t+1kμij+mk−nkbk.Thus,P˜liku=πlnexpf1/π+expf2/π. LX,Y,Z,λ,θ,γ is smoothed to (24)LtuX,Y,Z,λ,θ,γ=∑k=tK∑i=1nU˜lit,ωik−P˜liku−λ∑i=1nxit−yit+zit+−Gt+∑k=t+1K∑i=1nxit−yit+zik+−Gk−φk−∑i=1n∑au,i∈Ciθau,iEau,ik′−∑k∈Kau,i′lau,ik−γ∑k=tK∑i=1nsik−Cb,and the model in (15) can be transformed to find the maximum value of the smoothing approximation function:(25)Dλ,θ,γ=maxxik,yik,zikLtuX,Y,Z,λ,θ,γ0≤lau,ik≤rau,ikmax,∀au,i∈Bi,Ci,zik∈−gb,gb,∀k∈K,lik=xik−yik+zik+.The optimal solution of (25) can be obtained using a quasi-Newton method [34]:(1)
TakingX0, H0, and parameter ε≥0, let k=0.(2)
Computingdk=−Hkgk and Xk+1=Xk+αkdk, where Xk=xik,yik,zik, gk=∂Ltu/∂XX=Xk, αk is a step generated by a line search.Ifgk+1≤ε, the algorithm stops; otherwise, moves to the next step.(3)
LetHk+1=Hk+sk−Hkyksk−HkykT/sk−HkykTyk, where yk=gk+1−gk, and k=k+1, go to (2).Thus,X∗λ,θ,γ is obtained and is substituted into (25). We can then have Dλ,θ,γ. Computing λ, θ, and γ, the solution of the model is obtained.A flowchart of the algorithm is shown in Figure2.Figure 2
A flowchart of the algorithm.


## 3.1. Problem Formulation

In our model, based on scheduling household appliances and considering the fluctuation of electricity produced from the RES, an expectation model of users’ welfare maximization is taken as follows:(P1)(8)maxE∑k=tK∑i=1nUlik,ωik−Plik,subject to(9a)0≤lau,ik≤rau,ikmax,∀k∈K,∀au,i∈Bi,(9b)0≤lau,ik≤rau,ikmax,∀k∈Kau,i′,∀au,i∈Ci,(9c)lau,ik=0,∀k∉Kau,i′,∀au,i∈Ci,(9d)∑k∈Kau,i′lau,ik=Eau,i′,∀au,i∈Ci,(9e)lik=Eau,ik+∑au,i∈Bilau,ik+∑au,i∈Cilau,ik,∀k∈K,(9f)yik+1=yik+ρik+1,∀k∈K,(9g)zik∈−gb,gb,sik=si0+∑j=1k−1zij,0≤sik≤Cb,∀k∈K,(9h)E∑i=1nlik≤Gk,∀k∈K,(9i)lik=xik−yik+zik+,where Ulik,ωik and Plik are the utility function and payment function of user i in time slot k, respectively, ωik(reflecting the satisfaction level of each user after using the electricity purchased) is an elastic coefficient and is a known parameter, xik is the amount of electricity supplied from the TPS, yik is the amount of electricity produced from the RES, and zik is the electricity charged or discharged in the ESS. In addition, xik, yik, and zik are variables which the amount is eight hundred and forty and Gk is the total electricity quantity of a power system.The constraints (9a) and (9b)–(9d) characterize the electric performance of elastic and semielastic appliances, respectively. The compositions and sources of a user’s electricity consumption are expressed as constraints (9e) and (9i). Constraint (9f) is the power fluctuation generated from the RES between two time slots. This constraint on the RES is not considered in [46]. Besides, determining the relationship between yik and yit, yik can be expressed as follows:(10a)yik=yit+∑j=t+1kρij.Constraint (9g) describes the electric performance of the ESS. Constraint (9h) shows that the expected total electricity consumption of users does not exceed the electricity production, which reflects the correlation and coupling between the electricity consumption of users. Samadi et al. do not take the constraint into account [35].


## 3.2. Convex Optimization Problem Formulation

The transformation of the objective function and constraint (9h) are shown as Appendixes A and B.According to AppendixesA and B, a convex optimization problem from problem (P1) is obtained as(P2)
(11)
max
∑
k
=
t
K
∑
i
=
1
n
U
˜
l
i
t
,
ω
i
k
−
P
˜
l
i
k
,subject to (9a)–(9e), (10a), (9g), and (9i)(12a)∑i=1nxit−yit+zit+≤Gt,k=t,(12b)∑i=1nxit−yit+zik+≤Gk+φk,k=t+1,…,K,in which,(13)U˜ilit,ωik=ωiklit−αi2lit2−ωik∑j=t+1kμij+αilit∑j=t+1kμij−αi2∑j=t+1kμij2−αi2∑j=t+1kσij2,0≤lit≤ωikαi,ωik22αi−ωik∑j=t+1kμij+αilit∑j=t+1kμij−αi2∑j=t+1kμij2−αi2∑j=t+1kσij2,lit≥ωikαi.P˜lik=maxmklit−∑j=t+1kμij+,nklit−∑j=t+1kμij++mk−nkbk,Eρik=μik and Dρik=σik2 are the expectation and variance of the random disturbance ρik, respectively.Among these constraints, constraints (9g), (12a), and (12b) are our inequality constraints, and constraints (9d) are our equality ones. The type of decision variables is real, and there are eight hundred and forty variables, one hundred and thirty-nine inequality constraints, twenty equality constraints, and eight hundred and forty boundaries.Theorem 1.
Problem (P2) is a convex optimization and has an optimal solution.Proof of Theorem1 is shown in Appendix C.


## 3.3. Dual Problem

From Theorem1, (P2) is a convex optimization and has an optimal solution. It is equivalent to its dual problem. Therefore, it can be solved using a dual method [20].The Lagrangian function of (P2) is(14)LX,Y,Z,λ,θ,γ=∑k=tK∑i=1nU˜lit,ωik−P˜lik−λ∑i=1nxit−yit+zit+−Gt+∑k=t+1K∑i=1nxit−yit+zik+−Gk−φk−∑i=1n∑au,i∈Ciθau,iEau,i′−∑k∈Kau,ilau,ik−γ∑k=tK∑i=1nsik−Cb=∑i=1n∑k=tKU˜lit,ωik−P˜lik−∑i=1n∑k=tKλxit−yit+zik++λ∑k=tKGk+λ∑k=t+1Kφk−∑i=1n∑au,i∈Ciθau,iEau,i′−∑k∈Kau,i′lau,ik−γ∑k=tK∑i=1nsik−Cb,where X=x1t,x2t,…,xnt, Y=y1t,y2t,…,ynt, Z=z1t,z2t,…,znt, λ≥0, θ=θau,iMCi×n∈RMCi×n, and γ∈R+. The objective function of the dual optimization problem is as follows:(15)Dλ,θ,γ=maxxik,yik,zikLX,Y,Z,λ,θ,γ0≤lau,ik≤rau,ikmax,∀au,i∈Bi,Ci,zik∈−gb,gb,∀k∈K,lik=xik−yik+zik+.The dual problem is the following:(16)minλ,θ,γ∈R+Dλ,θ,γ.


## 3.4. Smoothing Method

### 3.4.1. Introduction of Smoothing Methods

The payment functionP˜lik of (P2) contains of the plus and maximum functions, which are nonsmooth. It is impossible to solve this problem directly using classical optimization methods. Smooth approximation functions can be found to replace the abovementioned functions. The basic idea of a smoothing method is to replace the nonsmooth function with a smooth approximation function and solving the original problem is equivalent to solving the traditional optimization problem. There are many smoothing methods, such as quadratic function method, aggregate function method, and density method (see [47]).Absolute value functiony=x is convex on R1 and smooth everywhere except x=0. Given a small constant ε>0, the following smooth function approximates it:(17)fεx=−x,x≤−ε,12εx2+ε2,−ε<x<ε,x,x≥ε,which is called quadratic function method.The aggregate function method is considered. The maximum function is as follows:(18)fx=max1≤i≤mfix,where fixi=1,…,m is a second-order continuous differentiable function on Rn. The following aggregate function is a smooth approximation function of formula (18) [47]: (19)fx,π=πln∑i=1mexpfixπ,where π is a small positive number.Next, the density method is discussed.Letρx be a probability density function from R1 to R+ and satisfy(20)∫−∞+∞sρsds=κ<+∞.The smooth approximation function ofx+=maxx,0 is as follows:(21)Pμ,x=∫−∞+∞x−μs+ρsds,where μ>0 is a smaller number.Because the plus function is inside and the maximum function is outside of the payment functionP˜lik, the plus function is considered first smoothing and then smoothing the maximum function.


### 3.4.2. First Smoothing

The quadratic function and aggregate function method are simple, practical, and effective, and these smoothing methods are adopted to deal with the nonsmoothness of payment functionP˜lik. The quadratic method is first adopted.Whenx=0, ψx=maxx,0 is nonsmooth, and let(22)ψtx=x,x>t,x24t+x2+t4,−t≤x≤t,0,x<−t.Theorem 2.
For∀t>0, ψtx, in formula (22) is a smooth approximation function of ψx=maxx,0.Proof of Theorem2 is shown as Appendix D.According to Theorem2, x of formula (22) is directly replaced by lit−∑j=t+1kμij+ and the smoothing function ψtlit−∑j=t+1kμij can be obtained. Then,(23)P˜lik=maxmklit−∑j=t+1kμij+,nklit−∑j=t+1kμij++mk−nkbk=maxmkψtlit−∑j=t+1kμij,nkψtlit−∑j=t+1kμij+mk−nkbk.


### 3.4.3. Second Smoothing

During the second smoothing process, the aggregate function method is used to smooth the maximum function of the payment functionP˜lik.LetP˜lik=maxf1x,f2x, where f1x=mkψtlit−∑j=t+1kμij and f2x=nkψtlit−∑j=t+1kμij+mk−nkbk.Thus,P˜liku=πlnexpf1/π+expf2/π. LX,Y,Z,λ,θ,γ is smoothed to (24)LtuX,Y,Z,λ,θ,γ=∑k=tK∑i=1nU˜lit,ωik−P˜liku−λ∑i=1nxit−yit+zit+−Gt+∑k=t+1K∑i=1nxit−yit+zik+−Gk−φk−∑i=1n∑au,i∈Ciθau,iEau,ik′−∑k∈Kau,i′lau,ik−γ∑k=tK∑i=1nsik−Cb,and the model in (15) can be transformed to find the maximum value of the smoothing approximation function:(25)Dλ,θ,γ=maxxik,yik,zikLtuX,Y,Z,λ,θ,γ0≤lau,ik≤rau,ikmax,∀au,i∈Bi,Ci,zik∈−gb,gb,∀k∈K,lik=xik−yik+zik+.The optimal solution of (25) can be obtained using a quasi-Newton method [34]:(1)
TakingX0, H0, and parameter ε≥0, let k=0.(2)
Computingdk=−Hkgk and Xk+1=Xk+αkdk, where Xk=xik,yik,zik, gk=∂Ltu/∂XX=Xk, αk is a step generated by a line search.Ifgk+1≤ε, the algorithm stops; otherwise, moves to the next step.(3)
LetHk+1=Hk+sk−Hkyksk−HkykT/sk−HkykTyk, where yk=gk+1−gk, and k=k+1, go to (2).Thus,X∗λ,θ,γ is obtained and is substituted into (25). We can then have Dλ,θ,γ. Computing λ, θ, and γ, the solution of the model is obtained.A flowchart of the algorithm is shown in Figure2.Figure 2
A flowchart of the algorithm.


## 3.4.1. Introduction of Smoothing Methods

The payment functionP˜lik of (P2) contains of the plus and maximum functions, which are nonsmooth. It is impossible to solve this problem directly using classical optimization methods. Smooth approximation functions can be found to replace the abovementioned functions. The basic idea of a smoothing method is to replace the nonsmooth function with a smooth approximation function and solving the original problem is equivalent to solving the traditional optimization problem. There are many smoothing methods, such as quadratic function method, aggregate function method, and density method (see [47]).Absolute value functiony=x is convex on R1 and smooth everywhere except x=0. Given a small constant ε>0, the following smooth function approximates it:(17)fεx=−x,x≤−ε,12εx2+ε2,−ε<x<ε,x,x≥ε,which is called quadratic function method.The aggregate function method is considered. The maximum function is as follows:(18)fx=max1≤i≤mfix,where fixi=1,…,m is a second-order continuous differentiable function on Rn. The following aggregate function is a smooth approximation function of formula (18) [47]: (19)fx,π=πln∑i=1mexpfixπ,where π is a small positive number.Next, the density method is discussed.Letρx be a probability density function from R1 to R+ and satisfy(20)∫−∞+∞sρsds=κ<+∞.The smooth approximation function ofx+=maxx,0 is as follows:(21)Pμ,x=∫−∞+∞x−μs+ρsds,where μ>0 is a smaller number.Because the plus function is inside and the maximum function is outside of the payment functionP˜lik, the plus function is considered first smoothing and then smoothing the maximum function.


## 3.4.2. First Smoothing

The quadratic function and aggregate function method are simple, practical, and effective, and these smoothing methods are adopted to deal with the nonsmoothness of payment functionP˜lik. The quadratic method is first adopted.Whenx=0, ψx=maxx,0 is nonsmooth, and let(22)ψtx=x,x>t,x24t+x2+t4,−t≤x≤t,0,x<−t.Theorem 2.
For∀t>0, ψtx, in formula (22) is a smooth approximation function of ψx=maxx,0.Proof of Theorem2 is shown as Appendix D.According to Theorem2, x of formula (22) is directly replaced by lit−∑j=t+1kμij+ and the smoothing function ψtlit−∑j=t+1kμij can be obtained. Then,(23)P˜lik=maxmklit−∑j=t+1kμij+,nklit−∑j=t+1kμij++mk−nkbk=maxmkψtlit−∑j=t+1kμij,nkψtlit−∑j=t+1kμij+mk−nkbk.


## 3.4.3. Second Smoothing

During the second smoothing process, the aggregate function method is used to smooth the maximum function of the payment functionP˜lik.LetP˜lik=maxf1x,f2x, where f1x=mkψtlit−∑j=t+1kμij and f2x=nkψtlit−∑j=t+1kμij+mk−nkbk.Thus,P˜liku=πlnexpf1/π+expf2/π. LX,Y,Z,λ,θ,γ is smoothed to (24)LtuX,Y,Z,λ,θ,γ=∑k=tK∑i=1nU˜lit,ωik−P˜liku−λ∑i=1nxit−yit+zit+−Gt+∑k=t+1K∑i=1nxit−yit+zik+−Gk−φk−∑i=1n∑au,i∈Ciθau,iEau,ik′−∑k∈Kau,i′lau,ik−γ∑k=tK∑i=1nsik−Cb,and the model in (15) can be transformed to find the maximum value of the smoothing approximation function:(25)Dλ,θ,γ=maxxik,yik,zikLtuX,Y,Z,λ,θ,γ0≤lau,ik≤rau,ikmax,∀au,i∈Bi,Ci,zik∈−gb,gb,∀k∈K,lik=xik−yik+zik+.The optimal solution of (25) can be obtained using a quasi-Newton method [34]:(1)
TakingX0, H0, and parameter ε≥0, let k=0.(2)
Computingdk=−Hkgk and Xk+1=Xk+αkdk, where Xk=xik,yik,zik, gk=∂Ltu/∂XX=Xk, αk is a step generated by a line search.Ifgk+1≤ε, the algorithm stops; otherwise, moves to the next step.(3)
LetHk+1=Hk+sk−Hkyksk−HkykT/sk−HkykTyk, where yk=gk+1−gk, and k=k+1, go to (2).Thus,X∗λ,θ,γ is obtained and is substituted into (25). We can then have Dλ,θ,γ. Computing λ, θ, and γ, the solution of the model is obtained.A flowchart of the algorithm is shown in Figure2.Figure 2
A flowchart of the algorithm.


## 4. Numerical Simulations

A power system that includes a power supplier with five users is considered. One day is divided into twenty-four periods. Each family is equipped with a rooftop solar panel, a small wind turbine, several batteries and electrical vehicles, and ten electric appliances. The must-run appliancesa1, a2∈Ai consume electricity at any time during the day. The electricity consumption of these appliances is set as 1 kWh and 3 kWh. The elastic appliances Bi, include four appliances a3, a4, a5, and a6. Their electricity consumption is considered as a whole, so long as the total electricity consumption will not exceed Rau,ikmax=∑au,i∈Birau,ikmax=10kWh. The semielastic appliances, include four appliances a7, a8, a9, and a10. The total amount of electricity required for their tasks is 5 kWh, 3 kWh, 4 kWh, and 2 kWh, respectively, and the expected working periods are {3, 4, …, 9}, {5, 6, …, 11}, {4, 5, …, 14}, and {9, 10, …, 18}. The capacities of the RES and ESS are set as 20 kWh and 5 kWh, respectively. All the computations are performed by Matlab R2016a. The running time for all the algorithm simulations is less than 1 ms. The abovementioned data are shown in Table 5.Table 5
The parameters of appliances.
The types of Appliances
Must-run appliancesAi
Elastic appliancesBi
Semielastic appliancesCi
Appliance
a
1
a
2
a
3
a
4
a
5
a
6
a
7
a
8
a
9
a
10
Electricity (kWh)
[1, 3]
<10
5
3
4
2
Schedule time frame (h)
—
—
—
—
—
—
3 am–9 am
5 am–11 am
4 am–2 pm
9 am–6 pmThe differentiated pricesmk and nk are set as follows:(26)mk=3cents/kWh,0:00,10:00,7cents/kWh,10:00,16:00,10cents/kWh,16:00,20:00,8cents/kWh,20:00,24:00,nk=9cents/kWh,0:00,8:00,7cents/kWh,8:00,13:00,15cents/kWh,13:00,24:00.Letbk = 4 kW during the entire period.The utility function of useri is set as follows:(27)Ulik,ωik=ωiklik−αi2lik2,0≤lik≤ωikαi,ωik22αi,lik>ωikαi,where ωik is an elastic utility coefficient and is a known parameter. Under the same conditions, the larger is the value of ωik, the higher is the power consumption while increasing the pressure on the TPS (further information regarding the utility function can be found in [48]). Therefore, ωik needs to be taken into account as a proper range. In the first scenario, different ranges of the elastic coefficient ωik are analyzed. In the other scenario, the impacts of the RES and ESS by manipulating semielastic and elastic appliances are studied under the appropriate range of ωik in various models.


### 4.1. Discussion under Different Elastic Coefficients

ω
i
k is selected randomly from the intervals [1, 3, 5, 7, 9, 11] and [18, 20], respectively. “w=2” means ωik∈1,3 for the lowest satisfaction level; “w=6” and “w=10” mean ωik∈5,7 and ωik∈9,11 for the normal satisfaction levels, respectively; and “w=20” means ωik∈19,21 for the highest satisfaction level. The specifications of these units are summarized in Table 6.Table 6
The indexes used in the models.
Indexes
w
=
2
w
=
6
w
=
10
w
=
20
The intervals ofωik
[1, 3]
[5, 7]
[9, 11]
[19, 21]
The satisfaction levels of users
Lowest
Normal
HighestThrough the simulations, the electricity consumption from the TPS by the semielastic appliancea7 at each hour is computed and their average values are determined. The cycle of appliance a7 is denoted as {3, 4, …, 9} (see in Table 5) and the assignment must be completed during this period. As shown from Figure 3, the appliance a7 is operated at time 4:00 am and is turned off at 8:00 am. Its assignment is completed within the designated time {3, 4, …, 9}. The appropriate ranges of ωik in semielastic appliances are discussed, as shown from Figures 3–6. From Figure 3, when w=2,6,10, the electricity consumption from the TPS by scheduling the semielastic appliance a7 is relatively smaller and stabler (relieving the supply pressure of the TPS) than that when w=20. The similar conclusions of other semielastic appliances are obtained, as shown from Figures 4–6. Therefore, the value of ωik under [1, 11] is more appropriate.Figure 3
The average of electrici1ty consumed from the TPS by semielastic appliancesa7.Figure 4
The average of electricity consumed from the TPS by semielastic appliancesa8.Figure 5
The average of electricity consumed from the TPS by semielastic appliancesa9.Figure 6
The average of electricity consumed from the TPS by semielastic appliancesa10.Multiple simulations are conducted; the electricity consumption from the TPS by elastic appliancesBi for five users is computed at each hour, and the average value is determined. The smaller and stabler electricity consumption from the TPS by scheduling elastic appliances is seen from, Figure 7, when w=2 and w=6. Consequently, ωik is suitable under [5, 11] considering the influence on the TPS.Figure 7
The average of electricity consumed from the TPS by elastic appliancesBi.Users are enthusiastic for making full use of electricity consumption from the RES whenωik is suitable under [11, 20], as shown from Figure 8. While w=10, the availability of the ESS is relatively higher, especially in peak periods (10:00–20:00) discharged and off-peak periods charged, as presented in Figure 9.Figure 8
The average of electricity consumed from the RES.Figure 9
The average of electricity consumed from the ESS.As for the objective of this model, the higher the user’s welfare value is whenw=20, the worser the stability is. While not only a higher welfare value but also an optimal solution is obtained when w=10, as shown in Figure 10. The users’ welfare values are relatively stabler as w=2 and w=6; however, it is abnormal to get the negative values (e.g., w=2). In summary, w=10 (i.e., ωik∈9,11) is the suitable range considering both the interest of users and the influence on the TPS.Figure 10
The welfare of users in four models.


### 4.2. Discussion under the Same Elastic Coefficient

In this scenario, four models to illustrate the impacts of the RES and ESS by manipulating semielastic and elastic appliances are studied underωik∈9,11. The indexes used in the four models are listed in Table 7.Table 7
The indexes used in the four models.
Model
ESS
RES
Expectation
Allno-model
×
×
×
ESS-model
○
×
×
RES-model
×
○
○
All-model (our proposed)
○
○
○As can be observed from Figure11, the average of electricity consumed from the TPS by semielastic appliances a7 in the all-model is the least during the working periods. It is shown that the RES and ESS play positive roles as auxiliary energies making users using less power consumption from the TPS. It should be also noted, owing to the continuous working cycle of the semielastic appliances, users’ aggregate power consumption (green line) from the TPS in the ESS-model is much less than that, in the RES-model (blue line). Therefore, users have priority to consume the electricity stored in the ESS and then produced from the RES and from the TPS last. By giving aggregate electricity consumption shown from Figure 11, users can reasonably arrange batteries or electrical vehicles according to their semielastic home appliances, to a large extent, alleviating the pressure of the traditional power grid. Likewise, the same conclusions are obtained from Figures 12–14.Figure 11
The average of electricity consumed from the TPS by semielastic appliancesa7.Figure 12
The average of electricity consumed from the TPS by semielastic appliancesa8.Figure 13
The average of electricity consumed from the TPS by semielastic appliancesa9.Figure 14
The average of electricity consumed from the TPS by semielastic appliancesa10.From Figure15, it can be seen that the electricity consumption from the TPS by elastic appliances Bi in the all-model is the most stable at each moment, and the range of fluctuation is the smallest of the four models. At the same time, it is observed that the ESS-model and RES-model have a similar impact in terms of electricity consumption from the TPS by elastic appliances.Figure 15
The average of electricity consumed from the TPS by elastic appliancesBi.From Figure16, it can be demonstrated that the electricity consumption from the RES in the all-model is relatively higher during peak periods than that in the RES-model, which indicates that users make full use of the RES in our proposed model. It can be seen from Figure 17 that the ESS is in the discharge states (positive values) during peak periods while in the charge states (negative values) during off-peak periods in the all-model. To some extent, the ESS relieves the power pressure on the TPS.Figure 16
The average of electricity consumed from the RES.Figure 17
The average of electricity consumed from the ESS.Without loss of generality, we consider the difference in electricity consumed from the TPS by three types of appliances for user 1 at each hour in four models. Figure18 shows the electricity consumed from the TPS by all appliances for user 1 at each hour. The electricity consumption from the TPS in our proposed model is much less than those other models overall.Figure 18
The electricity consumed from the TPS by all appliances for user 1.Figure19 shows that the users’ welfare value in the all-model (i.e., our proposed model) is significantly higher than that of other models. Users’ welfare values in the all-model are more than double those of other models under ωik∈9,11. In particular, the welfare value of user 4 reaches 448 dollars in the all-model, while it reaches 175 dollars, 180 dollars, and 200 dollars in other models, respectively. This verifies the rationality of our proposed model and feasibility of the method.Figure 19
The welfare of users in four models.It can be demonstrated from Figure20 that the proposed algorithm has remarkable convergence performance, and it converges at 16 times keeping the total welfare of users around 2250 dollars.Figure 20
The algorithm convergence and iteration times of the all-model (our proposed model).


## 4.1. Discussion under Different Elastic Coefficients

ω
i
k is selected randomly from the intervals [1, 3, 5, 7, 9, 11] and [18, 20], respectively. “w=2” means ωik∈1,3 for the lowest satisfaction level; “w=6” and “w=10” mean ωik∈5,7 and ωik∈9,11 for the normal satisfaction levels, respectively; and “w=20” means ωik∈19,21 for the highest satisfaction level. The specifications of these units are summarized in Table 6.Table 6
The indexes used in the models.
Indexes
w
=
2
w
=
6
w
=
10
w
=
20
The intervals ofωik
[1, 3]
[5, 7]
[9, 11]
[19, 21]
The satisfaction levels of users
Lowest
Normal
HighestThrough the simulations, the electricity consumption from the TPS by the semielastic appliancea7 at each hour is computed and their average values are determined. The cycle of appliance a7 is denoted as {3, 4, …, 9} (see in Table 5) and the assignment must be completed during this period. As shown from Figure 3, the appliance a7 is operated at time 4:00 am and is turned off at 8:00 am. Its assignment is completed within the designated time {3, 4, …, 9}. The appropriate ranges of ωik in semielastic appliances are discussed, as shown from Figures 3–6. From Figure 3, when w=2,6,10, the electricity consumption from the TPS by scheduling the semielastic appliance a7 is relatively smaller and stabler (relieving the supply pressure of the TPS) than that when w=20. The similar conclusions of other semielastic appliances are obtained, as shown from Figures 4–6. Therefore, the value of ωik under [1, 11] is more appropriate.Figure 3
The average of electrici1ty consumed from the TPS by semielastic appliancesa7.Figure 4
The average of electricity consumed from the TPS by semielastic appliancesa8.Figure 5
The average of electricity consumed from the TPS by semielastic appliancesa9.Figure 6
The average of electricity consumed from the TPS by semielastic appliancesa10.Multiple simulations are conducted; the electricity consumption from the TPS by elastic appliancesBi for five users is computed at each hour, and the average value is determined. The smaller and stabler electricity consumption from the TPS by scheduling elastic appliances is seen from, Figure 7, when w=2 and w=6. Consequently, ωik is suitable under [5, 11] considering the influence on the TPS.Figure 7
The average of electricity consumed from the TPS by elastic appliancesBi.Users are enthusiastic for making full use of electricity consumption from the RES whenωik is suitable under [11, 20], as shown from Figure 8. While w=10, the availability of the ESS is relatively higher, especially in peak periods (10:00–20:00) discharged and off-peak periods charged, as presented in Figure 9.Figure 8
The average of electricity consumed from the RES.Figure 9
The average of electricity consumed from the ESS.As for the objective of this model, the higher the user’s welfare value is whenw=20, the worser the stability is. While not only a higher welfare value but also an optimal solution is obtained when w=10, as shown in Figure 10. The users’ welfare values are relatively stabler as w=2 and w=6; however, it is abnormal to get the negative values (e.g., w=2). In summary, w=10 (i.e., ωik∈9,11) is the suitable range considering both the interest of users and the influence on the TPS.Figure 10
The welfare of users in four models.


## 4.2. Discussion under the Same Elastic Coefficient

In this scenario, four models to illustrate the impacts of the RES and ESS by manipulating semielastic and elastic appliances are studied underωik∈9,11. The indexes used in the four models are listed in Table 7.Table 7
The indexes used in the four models.
Model
ESS
RES
Expectation
Allno-model
×
×
×
ESS-model
○
×
×
RES-model
×
○
○
All-model (our proposed)
○
○
○As can be observed from Figure11, the average of electricity consumed from the TPS by semielastic appliances a7 in the all-model is the least during the working periods. It is shown that the RES and ESS play positive roles as auxiliary energies making users using less power consumption from the TPS. It should be also noted, owing to the continuous working cycle of the semielastic appliances, users’ aggregate power consumption (green line) from the TPS in the ESS-model is much less than that, in the RES-model (blue line). Therefore, users have priority to consume the electricity stored in the ESS and then produced from the RES and from the TPS last. By giving aggregate electricity consumption shown from Figure 11, users can reasonably arrange batteries or electrical vehicles according to their semielastic home appliances, to a large extent, alleviating the pressure of the traditional power grid. Likewise, the same conclusions are obtained from Figures 12–14.Figure 11
The average of electricity consumed from the TPS by semielastic appliancesa7.Figure 12
The average of electricity consumed from the TPS by semielastic appliancesa8.Figure 13
The average of electricity consumed from the TPS by semielastic appliancesa9.Figure 14
The average of electricity consumed from the TPS by semielastic appliancesa10.From Figure15, it can be seen that the electricity consumption from the TPS by elastic appliances Bi in the all-model is the most stable at each moment, and the range of fluctuation is the smallest of the four models. At the same time, it is observed that the ESS-model and RES-model have a similar impact in terms of electricity consumption from the TPS by elastic appliances.Figure 15
The average of electricity consumed from the TPS by elastic appliancesBi.From Figure16, it can be demonstrated that the electricity consumption from the RES in the all-model is relatively higher during peak periods than that in the RES-model, which indicates that users make full use of the RES in our proposed model. It can be seen from Figure 17 that the ESS is in the discharge states (positive values) during peak periods while in the charge states (negative values) during off-peak periods in the all-model. To some extent, the ESS relieves the power pressure on the TPS.Figure 16
The average of electricity consumed from the RES.Figure 17
The average of electricity consumed from the ESS.Without loss of generality, we consider the difference in electricity consumed from the TPS by three types of appliances for user 1 at each hour in four models. Figure18 shows the electricity consumed from the TPS by all appliances for user 1 at each hour. The electricity consumption from the TPS in our proposed model is much less than those other models overall.Figure 18
The electricity consumed from the TPS by all appliances for user 1.Figure19 shows that the users’ welfare value in the all-model (i.e., our proposed model) is significantly higher than that of other models. Users’ welfare values in the all-model are more than double those of other models under ωik∈9,11. In particular, the welfare value of user 4 reaches 448 dollars in the all-model, while it reaches 175 dollars, 180 dollars, and 200 dollars in other models, respectively. This verifies the rationality of our proposed model and feasibility of the method.Figure 19
The welfare of users in four models.It can be demonstrated from Figure20 that the proposed algorithm has remarkable convergence performance, and it converges at 16 times keeping the total welfare of users around 2250 dollars.Figure 20
The algorithm convergence and iteration times of the all-model (our proposed model).


## 5. Conclusions

In this paper, an optimal residential energy consumption model is proposed, which aims to achieve a trade-off between maximizing users’ utility and minimizing their payment based on scheduling household appliances. The optimal solution of the model is obtained using a smoothing quasi-Newton method. Several cases are discussed under different elastic coefficientsωik which is demonstrated that ωik∈9,11 is the suitable range considering not only the interest of users but also the influence on the TPS. Afterwards, the impacts of the ESS and the RES are discussed in several models. The discussion results show that users’ aggregate power consumption from the TPS by semielastic appliances in the ESS-model is much less than that in the RES-model. However, in terms of electric power consumed from the TPS by elastic appliances, the ESS-model and the RES-model have little difference. Finally, the simulation results show that our proposed model is superior to other models in terms of not only users’ welfare but also electricity consumption from the TPS.Future studies will be conducted to extend as follows:(1)
An optimal residential energy consumption model is proposed considering thermoelectrical load modeling of a house(2)
Similar models are solved adopting a robust approach or artificial intelligence algorithms such as machine-learning algorithm(3)
We should consider extending the proposed model to take care of other uncertainties (such as in the customer behavior, for scheduling appliances, or in real-time electricity price)



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*Source: 9545439-2020-04-27.xml*